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  • 1. ric AAI, Inc. ~ i l m i n ~ ~~ o r t h ~ r o l i n ~ on, C Larry L. Augsburger David E. Nichols University of ~ a r y l a n d ~ u r d u e University Baltimore, ~ a r y l a n d West Lafayette, Indiana Douwe D. Breimer S t e ~ ~ e nS c h u l m ~ n G. Gorlaeus Laboratories University of Florida Leiden, The l ether lands Gainesvilie, Florida Trevor M. Jones Jerome P. Skelly The Association of the Alexandria, Virginia ~ritish Pharmaceutic~lIndustry London, United Kingdom Hans E. Junginger Felix Theeuwes Leiden/Amsterdam Center Aka Cor~oration for Drug Research Palo Alto, C~lifornia Leiden, The Netherlands Vincent H. L. Lee~niversity of Southern California University of Sheffield Los Angeles, California Royal Hallamshire H o s ~ i t ~ i Sheffield, United K i n ~ d o m Peter G. ~ e ~ ~ i n lnstitut de Recherche Jouvein~l Fresnes, France
  • 2. tical Statistics: Practical a R. evised and Expanded, edjted by J o s e ~ h Robjnson and ~ n c e nH. Lee t rd A. Guarin~31 . Transdermal Controlled Systemic ~edications,edjted by Yje W. C ~ j e n32. elivery Devices: Fundamentals and Ap~lications, edjted by ravee en33. Pharmacokinetics; Regulatory lndustria~ erspect~ves, ed~ted by ~ e t e G. Welljn~ Francis L. S. Tse r and ~ l ~ n i cDrug Trials and Tribulations, edited by ~ l l e n Cat0 ai E. . ~ransdermalDrugDelivery: Deveiop~entalIssuesandResearchInitiatives, edjted by Jonathan a d ~ r a f t Ri~hard Guy ~ and H. DosageForms, edjtedby Coatingsfor P ~ a r ~ a c e u t i c a l ~ j l b S. ~ a n k eand Chrjsto~her7: ~hodes e r nufacturing and Pro~uctionTechnolo- W. ions, edjted by ~ a v j d Osborne and ~ n t o n H. A ~ a n n Stability: Principles and Practices, Jens 7. Carstensen istics: Practical and Clinical Applications, Second Edition, ed, Sanford ~ o l t o n radablePolymers as Drug De~ivery Systems, edjtedby ~ a asin in r ~46. ~ r e c l i n i c a ~ Disposition: A LaboratoryHandbook, ~rancjsf. S. 7seand Drug J a ~ e J. Jaffe s LC in the Pharmaceutical Industry, e ~ j t e d ~ o d ~W. n by j ~~n~ and Stanley r m a c e ~ t i ~ a l ~ i o e q u i v a l e n c e , by ~ e t e E. Well;n~, ~ranc;s edjted r f. and S ~ ~ j k a V.t ~ j n ~ h e n49. Pharmaceutica~ Dissolution Testing, ~ ~ e V. ~ a n a k a ~ s h50, Novel Drug ~elivery Systems:SecondEdition,Revisedand Expan~ed, Yie W. Chjen the Clinical Drug ~ e v e l o p m e n t Process, avid M. Cocchetto and ~ardj nufacturing Practices for ~harmaceuticals: A Plan dition, edited by Sjdney H. W;llj~ and J a ~ e s53. Prodrugs: Topical and Ocular Drug Delivery, edjted by ~ e n n54. P h ~ r m a c e u t i c a l i n h a ~ ~Aerosol Technolo~y, tion ed;ted by A~thony ~ ; c ~ e y J.
  • 3. 55, Radiopharmaceuticals: ~ h ~ m i s t rand y Pharmacology, ~ ~ n n nd ~hrjster y s t r ~ m ~ Delivery, e ~ j t by~ j c h a eJ. ~athbone e ~ i in Pharmaceutical Development, e~ited fopment Increasing Process: Efficiency and Cost- Effectiveness, ~ d i t e dby Peter G, w1~lijng,Louis Las a, an^ U ~ e s hV. n~kar c r o ~ a ~ i c u l a tSystems for the Delivery of Proteins and Vaccines, e ~ d j ~ e y Srnadar ohe en and ~ o ~ a %ernstejn r d78. Good ~ a n u f a c t u r i n gPr ces for Pharmaceuticals: A Plan Control,FourthEdition,visedand Expanded, ~ j ~ n e y N, R. ~ t o k e r79. ~ q u e o ~Polymeric ~ o a t i n g s for s Pharmaceutical Dos Edition, Revised and Expanded, e ~ i t e d James w1 ~ c ~ j ~ j t y by Statistics:Practicaland ~ l i n ~ c Applicatjons,ThirdEdit~on, al81 . andb book ofPhar~aceutica~Granu~ationTechnoio~y, ~ ~ j t e d by 2. ~ n o l o ~ ~ntibiotics: Second ofy Edition, Revised and Ex~ande~, ~ j ~ e e by ~ j i i R. ~ t r~ h i j ~ o
  • 4. rocess E n g ~ n e e r i n ~ ~ t ~ o nJ. n y ~epyrogenation, Second
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  • 6. MARCELDEKKER
  • 7. This book is printed on acid-free paper. ~~rters L Dekker, Inc. adison Avenue, New York, N Y 10016tel: 212-696-9000; fax: 212-685-4540Marcel Dekker ACHutgasse 4, Postfach 8 12, CH-4001 Basel, Switzerlandtel: 41-51-261-8482; fax: 41-51-261-8896http://www.dekker.comThe publisher offers discounts on this book when ordered in bulk quantities. For morei n f o ~ a t i o nwrite to Special Sales/Professional Marketing at the headquarters address above. ,Neither this book nor any part may be reproduced or transmitted in any form or by anymeans, electronic or mechanical, including photocopying, micro~lming, recording, or by andany infor~ationstorage and retrieval system, without permission in writing from thepublisher.Current printing (last digit):10987654321 CA
  • 8. o my wife with gratitude for her~~nderstandi~g, support, and love
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  • 10. This book is an outgrowth of my notes for a graduate course given at the ~niversityof Wisconsin for several decades. It focuses on the ~ r i ~ c i ~ Z e s science of phar- of themaceutical sciences, not necessarily on details or particular examples, except whenthey are s~pportive material for the text. The solids area of the pharmaceutical sciences has been explored more often inthe last decade than in prior times. This, in particular, is due to the advent ofsophisticated instrumentation and computer access. However, such advantages canlead to a certain mental laziness, and much of what is written in today’s literature isdisregardful, in fact at times ignorant, of the principles on which the instruments andprograms are based, and much misinterpretation occurs. Parts of this book addressthis aspect. In so doing, the ref~rences often not new, but rather give credit to the arescientists of yore who really were the innovators. The book also presents some entirely new aspects, not pr viously published,concerning the proper basic consideration in the approach to certain areas of phar-maceutical solid science. The book is written for those who are interested in the actual pro~esses theonmicroscopic level, with particular emphasis on elucidating models for systems so thatthey can be of general use. The book should appeal to pharmaceutical scientists inindustry, as well as the more sophisticated segment ofpharmaceutical manufacturingpersonnel. It should appeal to scientists in government agenciproblem areas that might have bearing on, for example, New (NDAs). It should have appeal to attorneys in patent law as well as patent exam-iners, because it elucidates whether a given. type of solution to a problem is reallypatentable. Also, it should be appealing to graduate students and to advanced under- graduate students who desire a place in the pharmaceutical solid sciences area.
  • 11. This Page Intentionally Left Blank
  • 12. Preface v ne ~omponent Systems 1 operties of Solids 13 3. Solubility 27 51 61 6. ~rystallization 89 7. Amorphates 107 117 erms with Crystalline Solids 133 e~o~si~era~ions 159 iagrarns and Eutectics 169 Particles and Surfaces 191 209 14.Solid State Sta~ility 223 olid-State Stability 267 lumes and Densities 28 1
  • 13. viii17. Cohesion 299 30919. ~ o m m i n ~ t i o n 323 33521. et ~ r a ~ ~ ~ a t i o n 353 ard Shell Capsules 37523. Tablet Physics 387 rinciples of Tablets 407 sintegration and Dissolution 427 439 27. ~ o a t i n g Tablets of 455 28. Single Unit S ~ s t a i n e ~ Release Dosage 469 29. Sustained Release by ~icroenca~sulation 493 Index 51 1
  • 14. 2 namic Functions 2 3 4 1.5. Methods 6 6 1.7. Introduction to Polymorphism 7 attice Energy, for Ionic Compounds 8 Symbols 11 eferences I1The purpose of pharmaceutical research is to explore the causes of properties ofdosage forms, in this case, solid dosage forms, The properties of the dosage foand a host of its qualities are a function of the neat drug. Characteri~ation the ofdosage form, therefore, requires characterization of the drug substance and what itsproperties are, so that the sources of derivative properties in the dosage form canbeadequately assigned. It is granted that such sourcing is never complete. Is the dis-solution rate of a drug in a dosage form, for instance, a function of the dissolutionrate of the drug substance, or is it influenced more by the excipients? Suchquestionscannot be answered a priori, but before an answer is attempted, the dissolution rateof the drug substance must first be known. Hence, this property (and many other)properties of the drug substance must be explored. Tools exist, nowadays, that allow sharp definition of a solid. Such character-ization of solid-state forms encompass microscopy, infrared (IR) spectroscopy, dif-ferential scanning calorimetry (DSC), t h e r ~ o g r a v i ~ e t r ianalysis (TGA), Karl cFischer titration, X-ray powder diffraction analysis, single-crystal X-raydiffraction,
  • 15. and (at times) solution calorimetry (Ledwige, 1996). eference will be made to thesemethods in appropriate places in this book. n solid pharmaceutical-marketed products, both the drug substance and exci- are present. (The word “drug,” at times, also refers to the drug product, ction made in the present text now seems to be the accepted h research dealing with pharmaceutical products is directed ke them possible and also addresses the failures that mightor do occur. though many such failures stem from manufacturing and excipients, manyalso stem from the drug substance itself. It is, therefore, of importance to discuss the ropert ties and testing approaches of the neat drug (the “drug,” the “drug sub-stance”) to assess the properties and difficulties asso~iatedwith the final product(the “dosage form” or the “drug product”).There are three states of matter: (a) gases, (b) liquids, and (c) solids. Their definitionsare intuitive, but if defined in words, a gas needs a three-dimensional, closed con-tainer to contain it, a Z i q ~ needs simply an open three-dimensional container, and a i~s ~ Z simply needs a two-~imensiona~ i~ planar support. The definition, however, isnot specific in the terms of solids. As shall be seeninlater chapters, solids are either crystalline or amor~hous, amorphoussolids may and(above their glass temperature, ir’) be rubbery, and below this temperature, they areglassy. In the rubbery state they are to be likened to (or actually are) supercooledmelts or li~uids and, as such, are liquids. In the glassy state, however, a substancewill mimic many of the qualities of a crystalline solid; hence it may be considered ve a viscosity, it will,in this text, be conventional to even if it is amorphous, if its viscosity is higher thanwhat it is at the glass transition temperature. A viscosity at 2’of 10l2“often used (Lu and Zografi, 1997) and this willbe employedhere as the cutoffpoint for a solid. In this book, the followi~g terminology will be used for the four t h e ~ o d y n a ~ i c functions: E is free energy, F is Helmholz free energy, G is Cibbs’ energy, and N is enthalpy, and in differe~tialform they are related as follows, whereiris temperature, S is entropy, V is volume, P is pressure: (1.1)
  • 16. The chemical energy terms are not included in the foregoing, but with these, it is inparticular Eq. (1.3) that is affected. dG ==: -SdT + VdP + (13where I;L is chemical potential and n is number of i-species transferred. It is particu-larly noted that dG = 0 during equilibrium, and that, for a voluntary process,dG < 0. G is a convenient function in that d T and d P are zero at constant tempera-ture and pressure, and that, under these conditions, (1*6) lpy change at constant pressure. An outcome of this is that (1*7)Often, in a chemical situation, at constant T? it is possible to indepen~entlydeter-mine AG and AH, and it is then possible to calculate A S fromAnother frequently employed relation isFor instance, for a chemical reaction with equilibrium constant (1.10)Inserting Eq. (1.9) into Eq. (1. 10)then gives d{AG/T}/d~ -~dln[K]/dT = -AH/T2 = (1.11)If ln[K] is known at several temperatures, A H for the reaction may be found, andA S may now be found from Eq. (1.8). ost often, in chemistry, systems are constant-~ressuresystems.h o ~ e v ~situations arise that call for constant-volume considerations a r,case, the chemical equilibrium criterion is that A F , not AG, be zero. It should finally bementioned that the entropy S, of a system isa measure of its oltzmanns law states that: (1.12) mber of ways in which a system can be made up. It is of interest to estimate the number of phases that can be present under one particular energetic condition. Suppose an ensemble in equilibrium consists of rz c o ~ p ~ n e n tand p phases. ecause there is equilibrium between phase 1 and 2, s, between phase 2 and 3, and so on the following holds. Note that Eq. (1.13) constitute p - 1 equations. There are - 1) equations for each of the n compounds, so the total number of equations is n(p - 1). temperature are variables and there are (n - 1) independent concentrations per
  • 17. +phase, so that the number of variables is p(n - 1 ) 2. The number of degrees offreedom is the n u m ~ e r variables minus the number of equations, i.e., of d =p(n-1)+2-n(p--l)=n-p+2 f (1.14)This means that there are d variables that may be changed without the system f“losing” a phase. As an example, a beaker of water has one component, there are two phases(liquid and gas); hence, n = 1 and p = 2, so that by Eq. (1.14) there is 1 degree offreedom (i.e., one variable [either T or P may be changed). If the temperature is Iincreased a bit, no phase will be lost. However, it is not possible to change both Tand P at will, because a given T dictates a certain P and vice versa. The situation is different at the freezing point. Here, there are three phases, ice(solid), water (liquid), and vapor. Hence d = 0, and neither T nor P may be cham- fged without losing a phase. Increasing the t~mperature loses the solid phase (the icemelts) and lowerin it loses the liquid phase (the water freezes). Such a point is calleda t r ~ l point. e The use of Eq. (1.14) is often difficult and it is st~essedthat it applies only to anequili~riu~ sit~ation. When in doubt, it is prudent to actually do the derivation leading to Eq. (1.14) for the particular system and obtain [df - 21 as the difference between the number of equations and the number of unknowms. The term degree o ffree do^ in this contextis exactly the oppositef its statistical~ e a n i n ~ s o (where itis the n ~ ~ o ~po~nts i ~ the s f e r~ u number o equations~. f LLA lattice is a periodic array. Points in the (ideal) lattice are occupied by molecules orions, and these may arrange themselves in different fashions (Fig. 1.1). There are seven different crystal systems, as shown in Table 1.1 ositioning of atoms, molecules, or ions in the lattice may be visualized as aseries of layers. Depending on which direction the lattice is viewed,there are different“layers” in different directions. The distance between these layers is denoted d below,and the manner in which d is d e t e ~ i n e d as follows: is To get an idea, first of all, ofthe magnitude of d consider a solid compound ofmolecular weight 180 and a true density 1.5 The molar voll.”lle of such a corn- *pound would be 18011.5 = 120 cm3jmol. cause there are 6 x molecules in amole, each of these occupies 120/(6 x = 200 x cm3. If, for order of mag- b Example of crystal forms. The angles, u, v, and w are shown in the left figures,as are the possible distances, a, b, and e. These are referred to in Table 1.1.
  • 18. Angle between Length of side Alternateaxes Examples name distances System a=b=c Regular Cubic NaCI a=b$c Tetragonal Pyramidal Rutil aZb#c 0rthorhombic AgN03 Rhombic a # W c Monoclinic ~-~minobenzoic acidu#v#w#9O0 a#b+ Triclinic K2Cr07u=t1=~#90" a=b=c Trigonal ~hombohedral NaN03u = w = 90" a =b =c # d Hexagonal Graphitew = 120"nitude calculations~the arrangement is assumed to be cubic, the side length of thecube encasing the molecule would be given by d3 = 200 x cm3 (1.15)or d = 5.85 x cm3 = 5.85 A (1.16) Awhere 1 is defined as 10" cm3.X-rays are of this order of magnitude and are usedfor measurement of atomic, molecular, and ionic distances within a lattice. This is performed according to rags law, which relies on the fact, that whentwo X-rays are in-phase, they will then reinforce one another, and the principle onwhich it is carried out is shown in Fig. 1.2.Two X-rays, 1 and 2, strike a surface at an angle of II . Ray 2 traverses a distanceABC (in bold in the figure) longer than ray 1; hence, for them to be in-phase, thisdistance must be a multiple of the wavelength h of the ray. The distance A.shown, by simple trigonometry, to be equal to 2d sin[q Le., for attenuation to bemonitored at the collector Q , this distance must be equal to nh; that is, 2d sin[U ] = nh (1.17) Schematic for Braggs law. The incident angle, U, equaling thereflected angle, U,isusually referred to as 8.
  • 19. Ledwidge et al. (1996), for instance, reports an X-ray diffraction pattern. (using -X rays with h = 1.5418tf)of diclofenac ~-(2-hydroxyethyl)pyrrolidin.e andthe smallest 20-value where a peak occurs is 7.6". The d i s ~ ~ (i.e.,e the d-value) ~ccorresponding to this would be d = 1.54/(2 sin 3.8") = 11.6 A (1.18)The most common method is powder X-ray d ~ ~ ~ ~ In t this method, powder is c i u ~ .packed into a cell, and this is subjected to the type of detection shown in Fig. 1.2. lanes present themselves in sufficient abundance to allow determination of thecrystal lattice constants without determination of the position and direction ofatoms, molecules, or ions in the lattice ~ingle-crystalX-ray crystallography allows determination of the position anddirection of the ions, atoms, and molecules in the lattice. For instance, Turel et al.(1997) usedX-ray crystallography to determine the crystal structure of ciprofloxacinhexahydrate, and showed that it exists in zwitterionic form in the solid state. Thecarboxylic proton is present by the piperaz terminal nitrogen. Adjunctly, however, employed they , Raman spectroscopy, and thermalmethods to determine that the water in the hexahydrate was present in complicated anetwork governed by hydrogen bonding.Inorganic ionic compounds consist of fairly spherical entities, and their packing isrelated to the relative radii of the two components of the systems. Consider, for instance, the situation in Fig. 1.3, in which, a c o ~ p o u n d consistsof two ions, one smaller, with radius r, and one larger, with radius R. It is obviousfrom the figure at the right t~iangle, Ahypotenuse, CA = 2fz + 2r. Hence, (2Ry + (2R)2= (211 + 2r)2 (1.19)or (1.20) Schematic for derivation of the radius ratio rule.
  • 20. which has the positive root: r = R(1 - V2) = 0.141R (I 21)Similar relations can be obtained for other arrangements (crystal systems), and therules in Table 1.2 emerge.Whereas inorganic compounds often (if not most often) crystallize in one particularcrystal system, organic compounds have the capability of c~stallizing several indifferent ~ Q Z forms ( ~ o r ~ ~ s )this p~enomenon denoted ~ o Z ~ ~ o r ~ ~ ~ ) and , is ecause there are seven crystal systems, it might be tempting to think that therecould, at most, beseven different po~ymorphsof one compound; however, thenumber is not limited to that. The molecules may be in different lattices, because their orientation is differentin thetwo different polymorphs (of the same crystal system). The lattice constants, a,6, and e, then might or would be different. olymorphs will be su et to a special chapter (see Chapter 8) but at this pointthe following will be noted. two polymorphs, one (form I) will be (con~guration-ally) more stable than the other (e.g., form 11) for the following reasons. 1. There areno rules for the nomenclature I, 11, and so on. numbers simply signify the chronological order in which they were pro- duced. 2. The less stable form, at a given temperature, will have a higher vapor pressure. 3. The less stable form, at a given temperature, will have a higher ap~arent solubility. This concentration of drug in the solvent is reproduci~le, but the solution is not thermodynamically stable. Eventually precipitation of a more (the more) stable form will occur, and the concentration will level off at the the~odynamic e~uilibrium solubility. 4. It is not possible, in a practical sense, to talk about the “most stable” polymorph, for a more stable polymorph may be discovered at a later time. From a fictional point of view, this is the subject of the book ~ ~ t Examples of the Radius Ratio Rule ApplicationsRatio = r/R coordinatio~ number Lattice Example 0-0.155 2 Carbon dioxide0.155-0.225 3 Hexagonal Boron nitride0.225-0.414 4 Tetrahedral blende Zinc0.414-4.733 6 Octahedral NaCX0.733-1 8 Body-centered cubic csc121 12 Face-centered cubic and also hexagonal
  • 21. r l ~ r ~ ~ l Kurt ~ o n n e g u t . ere a more stable, higher-melting form of by e , water (Ice Nine) eventually causes the world’s oceans to freeze over. 5. The molecules in solutions created by either a less or more stable poly- morph are the same.The section to follow has been developed quite rigidly for inorganic ions.extensions to organic crystals are possible. In the development, the term ~ ~ r t j c Z e willoften be used to signify “ion” or in some cases “molecule,” hen bonding occurs between two molecules, a minimum will occur in thepo~ential energy curve that exists betweenthem. This distance is known as the latticeconstant Ro (Fig. 1.4). nergy curves, as a function of atomic or molecular distance, are rationalized aron and Prutton, 1965) by the existence of two opposing forces between the ms or molecules: an attractive force and a repulsive force. The attractive force is, theoretically, inversely proportionalto the seventhpower of the interatomic or intermolecular distance. The repulsive term dependson distance by some (the nth) power of the separation, The potential energy u‘ ofthe interaction between two neighboring ions, therefore, may be written as -~ ~ d =( ~ 1 (By) ) (I .22) The value o f n is, ordinarily, from 10 to 13. ach interparticular distance (rY)is expressedas a number (pii) multiplied with ration (R)between two particles. Examplewise, in the situation shown in Fig.1.5, thejth and the ith molecule wouldseparated by 2 “units” so that rij would equal In general this may be written as rij = pijR (123) ) isnowsummedover all interaction possibilities, which then gives theenergy, ( , one ~ ~ r t i c ~ e . pfor ”I Potential energy curve.
  • 22. <"_"""" "" """"> r10,12 Schematic of interacting atoms or molecules. (1.24)where summationis over all i # j . The following termsare introduced for thesake ofconve~ience: (1.25)and (1.26)For a given crystal a and b are constants, so that Eq. (1.24) becomes: 4 = ( A u / R 7 )- ( ~ b / R n ) (1.27) The distance at equilibrium, Ro, is obtained by obtaining the first ~erivativeand equating it to zero (and at the same time ensuring that thesecond derivative will ~ 4 / = (-7Au/R8) ~ R +( ~ ~ b / R n " ) (12 8 ) occur this must be zero, soFor equilibriu~ to + (-7AaIR~) ( ~ ~ b / R ~ " ) (1.29)or 7AaIR; = n ~ ~ / R ~ ~ ~ (1.30)or (7/n)(Aa)/R: = ~ b / R ~ (1.31)This is now inserted into Eq. ( 1 2 7 ) to give 40 [Aa/u:l - [ ( 7 / ~ ) ( A a ) / RAla / R ; J I { l - ( 7 1 ~ ) ) [: = (1.32) of The energy per mole U can now be obtained by multi~lication 4 0 , ~ i t h theAvogadro number N,so that
  • 23. u =N ~ A ~ / - (7/n)}~ l R ~ ~ (1.33) onv vent ion ally? U is equated with the enthalpy of sublimation, because solidsare considered constant-volume (rather than constant-pressure) situations. Theactual value of a is obtained geometrically. For ions, the terms alternate in sign(each secondion being negative,thus exerting attraction, each second beingoppositein sign and giving rise to attraction). For molecules (many organic molecules) theforce terms are all positive.The large negative value ofn, the exponent in the secondterm in Eq. (1.27), is most often acceptable to use only one or two terms m a ~ i n g itthe s~mmation fairly simple. Ro values may be obtained by X-ray analysis, leaving but two unknowns, Aand y1 in Eq. (1.33). The value of n may be obtained by lo~-temperature compres-sibility measurements. The definition for compressibility k, is k -(l/V)dV/dP (1.34) t low temperature, the 7“ term in the definition for U vanishes so that dU =: -PdV, ‘35) (1so that 1/k = Vd2U/dV2 (1.36) olar volume with the nomenclature used here is v = NR3 (1.37)so that dU/dV = ( d U / d ~ ) ( d R / d ~ ) (1.38) he second derivative, hence, is + d2 U / d V 2 = (dU/d~)(d2R/dV2) (d2U/dR2)(dR/d~)2 (1.39) t equilibrium^ dU/dR = 0, so that the first term vanishes, reducing Eq. (1.39) toU/d d2 V 2 = (d2U/dR2)(dR/d V)2 (1 .40) rom Eq. (1.36) we have (dR/dV)2 = (3NR2)2== 9N4R4 (1.41) q. (1.40) becomes d2U/d V2 = (d2U/dR2)(9N4R4) (1.42)This is combined with Eq. (1.36) to give ~)(9N4R~)(d2 U/dR2) = 9N3Ri(d2 / d R 2 ) U (1.43)Equation (1.43) when differentiated twice gives d2U / d R 2 = 56(NAa)[1 - (7/n)](R~) (1.44)which inserted in I l k = 504N4(~a)[1 (7/n)](R~2) - (1.45)which allows calculation o f n.
  • 24. A constant in the energy versus distance equation = adelung constant adelung constant B = constant in the energy versus distance equation d = distance between molecular layers E = energy H = enthalpy anns constant; (b) compressibility N = Avogadros number n = (a) number of particles, ions, molecules; (b) integer in (c) exponent in potential energy versus distance equation. P = pressure pij = factor forexpressing the distance between the ith andjth ion in units of = distance between the ith and jth ion = (a) distance between particles, ions, molecules; (b) ionic radius of larger ion r = ionic radius of a smaller ion Ro = ~quilibrium distance between particles, ions, molecules S = entropy 1 = absolute temperature ( " U = crystal energy u = potential energy between two ions Y = volume W = number of ways of building up a system , = chemical potential u 8 = incident angle of an X-ray h = wavelength enettlnetti 6, Giordano F, Fronza 6 , Italia A, Pellegata R, Villa M, Ventura P (1990). J Pharm Sci 79:470.Carstensen JT (198 1). Solid Phar~aceutics,MechanicalProcesses andRate "Phenomena. Academic Press, New York, pp 6-7.Kittel (1962). Introduction to Solid State Physics. pp 70"79. T, Draper SM, Wilcock DJ, Corrlgan 0 1 (1996). J Pharm Sci. 85:16.Lu Q, Zografi G (1997). J Pharm Sci 86: 1374.Maron SM, Prutton CF (1965). ""Principles o f PhysicalChemistry,4th ed. London, pp 728-729. (1997). Int J Pharm 15259.
  • 25. This Page Intentionally Left Blank
  • 26. 2.1. 13 2.2. 16 2.3. Classic Heat Capacity 18 2.4. The Einstein Equation 19 2.5. 22 2.6. 2.7. References 25The text, as mentioned earlier, will deal first with properties of solids that are not,primarily, a function of their subdivision. In essence they may be considered theproperties of an infinitely large slab of the solid, Later chapters will deal with proper-ties that are a function of the subdivision of the solid (e.g., particle size).Crystals are never perfect. As they grow (a point that will be discussed later) planesmay grow over one another on the surface (Fig. 2.1), shunting out areas of voids.They may also grow as a screw (a so-called screwdislocation), and in this case thereis a sort of pore that penetrates the crystal as the axis of the screw. Several types ofdefects are depicted in Figs. 2.2 and 2.3. From a statistical-mechanical point of view, defects are to be expected. Thedevelopment of this concept in the following is based on the Schottky defect, but itwould also apply to vacancies of other natures. Suppose (Fig. 2.3), that a crystalcontains nine molecules. There is but one wayof arranging them. If one of theinternal molecules is moved to the surface, there will be { lo1}= 10 ways of doing
  • 27. A E 3 c D A nucleus or crystal (A) grows on the surface, and two sites are shown. Furthergrowth and a site in a second layer are shown in (C) and in (D) the growth in “higher” sitesgrow over the lower sites creating a “hole.” Schottky Frenkel Vacancy Frenkel, Schottky, and screw defects. 0 000 0 0 000 9 Posit’ions 10 Positions 11 Posltlons One Way 10 Ways 55 ways Situation where one and two Schottky defects are created in a crystal with (origin- ally) nine lattice sites.
  • 28. MW 2 x 74.5Cl K Cl K C1 KK 6 K C1 K C1 1C K C1 K C1 K lK C1 K C1 K C1K C1 C1 K C1 KC1 K 6C1 Ca 1K C1 K C1 K C1C1 K C1 K Cl K MW 75.5 + 35.5 Calcium replacing IC. as a means of creating a vacancy.this. Itf two molecules were moved the surface, then the number of ways would be to{112}= 11 x 10/2 = 55 ways. olt~mann states that the entropy of a system S is proportional to the lawlogarithm of the number of ways in which it can be made up: S = kln(ways the system can be made up) (2.1) oltzmann constant. For a system of N + n positions with n vacancies?the entropy would be S = k ln[(N + n)!/{N!n!} (.) 22Use is now made of Sterlings formula lnN! = N l n N - NLe., as applied to this system ln[(N + n)!/{N!n!} N l n N - ( N - n ) l ~ ( N n) - nln(n) = -so that S = k{NlnN - ( N - n)ln(N - n) - nln(n)} (2.5) olid systems are usual1 onsidered constant volume systems, so that in equilibriumconsiderations?it is the lmholtz free energy (rather than the Gibbs energy) that isapplied. IC = nEs - TSwhere Es is the energy associated with one vacancy. This isnow differentiatedrelative to n to give the equilibrium condition: dF/dn = EL* kT ln([N - n]/n)= 0 - (2.3where the argum~nt Eq. (2.5) differentiated relative to n. This rearranges to: is E:s= -kT ln(n/[N - n]) (2.8)or n == NemEsk 1 / "Normal range of vacancies is ofthe order of 0.001%.
  • 29. efects are often creating by doping (i.e., introducing a foreign molecule into ce of the compound in question). For instance, with KCl, the potassium ion ) may be replaced with relative ease bycalcium ion (MW 40), becausetheir asizes are appro~imatelyequal. If one considers a crystal with N ions of KCl, theneach time a calcium ion (MW 40) is introduced, a hole with one missing K 39) ~~~is created. The loss in weight,therefore, is 38 per calcium ion. If there are n calciumions in a crystal with N positive ion sites, then the density is calculated as follows(Figs. 2.4 and 2.5) without vacancies, should be (~~ of C1 being 35.5) N“ = 2(N + n)74.5 (2.10)The weight W+ with vacancies would be W+ = 2N74.5 + n75.5 + n35.5 (2.11)The difference between these two numbers is A W = -38n (2.12)The volume of the crystal is N x V , where V is the molecular volume, so that thedifference in density would be(2.13)A ~ / =N-38n/NV ~ttenrauch (1983), H~ttenrauch and einer(1979 a,b), Longuemard et al. ersey and Krycer (1981), Moriata etal. (1984), Grant and York (1986), and nd Grant (1987) have called attention to the fact that processing of solidscauses lattice defects,givingrise to an increase in disorder. ancock and Zografi(1997) claim that this would give the particle a certain viscoelasticity. Hiestand (1997) states that “the ever present, plastic deformation profiles anexplanation why lot-to-lot problems are c o ~ m o n . ” yield value ofthe particles is Thedependent on defects in the crystals, and changes may occur in nearly all processing.Usual production sp~cifications not include criteria for mechanical properties. doThere are several different definitions of ~ e n ~ i tThe ideal density ofa crystal can be y.calculated from knowledge of its lattice parameters and the molecular weight. n/N x 10000 Change in density of KC1 doped with calcium ions. The lower line is the theoreticalline, the upper line the experimental line. (Data from Pick and Weber, 1950.)
  • 30. If, for instance, the lattice parameters of a orthorhombic crystal are 5, 7 , and8A, and its molecular weight is 240, then the mass of one molecule is180/(6 x =3 x g. The volume it occupies is 5x 7x 8x= 2.8 x em3, so that the crystallographic density wouldbe 3/28 = 3/1.92 =1.07 g/cm3.Nowadays, crystallographic densities are reported routinely in studies ofthe crystallographic details of a particular form of the compound. As an example,Ceolin (1997) has reported the volume of the triclinic unit cell of carba~azepine tobe 2389 A3. ecause of lattice defects and vacancies, the actual density would be less, Theactual particle density is determined by either wet pycnometry or by helium pycno-metry (Fig. 2.6). In wet pycnometry, a liquid in which the solid is insoluble, is selected (e.g.,water for a poorly water-soluble compound). The pycnometer has a given volume V em3, and the weight of the contents W is determined. The pycnometer is filledto amark giving the density p l , of the solvent: Now M grams of solidare added, having the (unknown) density o f p2. Thesegrams occupy M/p2 em3, so that the liquid now occupies{ V - ( M / p 2 ) } em3. The netweight ( M 2 )of the ensemble is obtained ex~erimentally(Fig. 2.7), and is given byThe only unknown is p2, which is the quantity sought. isadvantages are (a) that the solid may be somewhat soluble in the pycn-ometer liquid, and (b) air entrapment. Theformer is marginal at best if the solvent isselectedwith care. At high dilution, ideal solutions are approached, so that thevolume contraction or expansion considerations are negligible. None of theseproblems exist inthe use of the helium pycnometer, which workson the same principle, except the “liquid” is helium. Therefore, it is not to be expectedthat theparticle density isthe “true” density.This could be derived only by knowing the lattice para~eters, ~ozwiakowski ai. (1996) reported on the solubility behavior of lamivudine etand in this process report on the lattice constants of the compound.for the compound is C8N I 1016S. The following program in BAmolecular weight em~loying precise atomic weights the Ground Glass Stopper With Capillary Inserting Glass Stopper Allows Excess Liquld t o Escape, Yielding A n Exact Volume Liquid pycnometer,
  • 31. y - 0.70668 + - 5.0131e-3x RA2 0.591 l.G r 120 130 140 150 160 170 Molecular Weight .7 Densities as afunction.ofsubstituent for aseriesofmonoclinic 4“ubstitutedbenzoic acids. (Data from Musa, 1972.) 1 = (8 * 12.01115) X3 = 11 * 1.00797 X4 = 3 * 15.9994 2 + x 3 + X4 + x 5 ”; X6 U1 = 10.427 U4 = Ul*~2*Y3/20 ecVol in AA3 ”; U4 = INT “Mol Vol = ”;21 INT “Vol/grarn = ”; 2 2 st. Density = ”; 2 3 eat capacity plays a part in several pharrnaceutical considerations on a theoreticalplane. If a process goes from stage A to stage A+ (2.16)is accompanied by an enthalpy. The process could be, for instance, solubility, andthe heat associated with this would be the heat of solution. This is often considered aconstant. The heat capacity Cp of the solution is given by Cp = dAH/dT (2.17)
  • 32. and assuming that it is constant then implies that AH is temperature-indepen~ent.There is but little difference between Cp and Cvfor solids, and they may be inter-changed freely. he considerations to be outlined in the following are mostly based on work etals, but they translate to organic molecules as well. The heat capacity is assumed to be associated with the energy E of the mole-cules in the lattice, and these are assumed to be harmonic oscillators. In classictheory, the average energy of a system is kT per degree of freedom, where k is the oltzmann constant and 7 is absolute temperature. For an ensemble of N harmonic oscillators, with three degrees of freedom (themolecule may oscillate in three directions), the average energy is: Eavg =3 N ~ ~ (2.18)or, for a mole Eavg 3RT = (2.29) (~.20)So that, for a solid, the heat capacity should be Cv = 6 cal/deg-mol (2.22)Table 2.1 shows examples of this. It will be shown later that indium is used as a calibrator for diff~rential scan-ning calorimetry ( equation. The equation is called the ~ e t i t - ~ ~ Z o ~ ~The foregoing holds in a classic sense, but the problem with it is that it predictsconstancy. The data in the table fairly well substantiates the Petit-but at lower temperatures, the heat capacities begin to drop. with systems for which one assumes constant enthalpies in atemperature range, it is possible to ~ o m m i terrors, and it becomes important toobtain an idea of at which te~perature(a,, the so-called Einstein te~perature),deviations may start to occur. Heat Capacities at 25°C Heat capacity Molecular Heat capacityCompound cal/"-g weight cal/"rnolCa 0.156 40.08 6.25CU 0.092 63.54 5.85In 0.056 114.82 6.43Mg 0.243 24.31 5.90CO 0.109 58.93 6.42
  • 33. In this type of development, the quantum mechanical concept that the energytakes on values only as integers of one another, is used. The energy, for a harmonicoscillator is given by E = nhu = n(h/211.)(~211.) n h u = (2.22) cks constant (6.624 erg-s); v is frequency; h (i.e., h/211.) the is ks constant (1.054 and u is the angular frequency; y1 is ntum number, and is an integer. In an ensemble of N molecules, there will be various energy levels, El (withy1 = l), E2 (with n = 2), and so on. The fraction Vn) the molecules in energy of state n iven by the ~ o l t ~ m a n n distribution, Le., total number of molecules is givenby (2.24)The energy of all the molecules is given by (2.25) y introducing Eq. (2.20), the average energy may now be calculated as (2.26)~ntroducingEq. (2,19) - ~ ~ / R= -hu/RT = X T (2.27)we may write Eq. (2.24) as (2.28)If we use the notation Y = (1 + ex + e2dV,+ - a (2.29)then dY/dx = (e" + 2e" + - .) (2.30)so that, in Eq. (2.28) Eavg nhud In Y/dx = (2.3 1) .29)] is a geometric series with factor ex, so that the sum is Y = 1/(1 - ex) (2.32) ence, - Eavg hu/{(exp(hu/~T) 1) = (2.33)This should be applicable at all temperatures, but at high temperatures k + ( e x ~ ( ~ ~-/ 1 x~ 1) (ho/rcT) + . . - 1 = (hu/kT) (2.34)
  • 34. so that Eavg ho/(ho/kT) == ==I kT (2.35)that is, at temperatures higher than a given temperature c3pE (the so-called Einsteintemperature), the energy equals the classic energy. The Einstein model gives profiles in reasonable agreement with experimentaldata, provide^ a suitable choice is made of the fundamental oscillator frequency.Both the terns h and kT are energy terms, and it is more convenient to talk about otemperatures than about f~equencies, so it is conventional to tie this in with theEinstein tem~erature by: .ho z== k@E (2.3~)~ i t this terminology, Eq. (2.31) becomes h Eavg = kc3p,/{eXP(@E/T) - 1) (2.37)so that the heat capacity becomes - LdE/dT = Lk(c3p,lT)2{exp(c3p,/T))/{exp(c3p~/T) (2.38)where L is Avogadro’s number. ith experimental data, it is possible nowto find (byiteration) a value of c3pE that makes the data fit the best. Figure 2.8 is an example ofthis. The severe assumption in the Einstein model is that there is only one funda-mental frequency. (There should at least be three, one for each degree of freedom.)Debye later refined the model to include many frequencies and obtained an evenbetter fit. The important lesson to draw from this is that heat ca~acities ( ~ e ~ eon ~ i are nthe c o m ~ o u ~ d ) times s u f ~ c i e n t ltewlperature-dependent and thatthis ~ h o ~ be at ~ ldtaken into account. The most common appro~imation that is hH=Q+gT (2.39)For instance, ~ n t e ~ n a t i o n a l~ r i t i cTables uses this, and higher polynomial ~lap~roximations,when tabulating heat capacities and enthalpies as a function ofte~perature. 0.0 0.2 0.4 0.6 0.8 1.0 TI0 iarnond heat capacities compared with the Einstein. model with a 4of 1320 >,
  • 35. 0th liquids and solids have vapor pressures. Vapor pressures ofsolids may be quite ow, although some pharmaceutical substances (nitroglycerin7ibuprofen) have sig-nificant vapor pressures at room temperature. Vapor pressure of a solid is measured by means of a so-calledwhich measures the escaping tendency of the gas through a capillary. For less precise, but more easily attainable vapor pressures, thermal gravi-metric analysis (TGA) is employed. A covered pan with a pinhole is a l l o ~ e d tostay at a given temperature in the TGA, and the loss rate (dWx/d~) measured. isThis loss rate is proportional to the vapor pressure in the pan. A substance withknown vapor pressure PC ( e g , benzoic acid) is run in parallel, and the vapor pres-sure of the solid (Py) is The development to follow holds for any condensed phase of a one-componentsystem. It will be assumed that the equilibrium is between a solid and its vapor. ibbs, phase rule states that cf = C - - P + 2 i (2.41)where G is number of components, P is number of phases, and df is the degrees of hases. For a one-component system this becomes df= --E"+ (2.42) n e~uilibrium two phases have the same chemical potential, that is, the Pg = Ps (2.43)where the subscript g denotes gas and s denotes solid. The phase rule states that inthe described situation there will be two degrees of freedom (e.g., temperature andpressure) that may be changed, so that at equilibrium, the following must hold:It is recalled that dG -SdT + Vdp (2.45)and that lu, is the G function per mole, SO that { d P ~ / d T= ~ } "8s (2.46)andwhere s and v are molar entropy and volume. A similar set of equations for the solid ence, Eq. (3.2) may be written (2.48) (2.49)or
  • 36. (2.50) (~.51)where A H is the heat absorbed at constant temperature and pressure when X mol ofsubstance passes from the solid to the gaseous state (Le.? it is the molar heat of egarding the volumes, the molar volume in the s d stateis negligi~le that in the gas phase, and if this is considered (VI - v,) 25 VI =RT/P (2,52)~ntroductionof Eqs. (2.51) and (2.52) into Eq. (2.50) gives: (2.53) (2.54)This integrates to lnb] = - ~ ~ / /( ~ ~ ) 3 + (2.55)whereis an integra~ion constant. An example of this is the vapor pressure of benzoic acid, The direct data are plotted in Fig. 2.9 and the logarithmic transformation isplotted in Fig. 2.10. It is noted that the heat of vaporization is AH == 1.99 x ’7.685 = 15.4 ~cal/mol It may also be noted that it is assumed that the enthalpy of vaporization is nottemperature-depend~nt, (from the source) it is not so in the temperature interval andshown. If a substance is at a temperature suf~cientlyhigh for it to be in a melted ~ondition, its vapor pressure curve will follow the Clausius Clapeyron equation, except that now the slope is AHvap (i.e., the heat of vaporization). Vapor Pressure of Benzoic Acid as a Function o f Tempe~ature P = vapor pressure Temp (“C) lOOO/T K_-l 1nVl 60 0.1065 3.002 -2.244 70 0.2085 2.914 -1.568 80 0.3928 2.832 -0.934 90 0.7147 2.754 -0.336 100 1.2592 2.680 0.230 110 2.1539 2.610 0.767 Source: West and Selby (1967).
  • 37. 50 60 70 80 9 0 100 110 120 Temp ("C) Vapor pressure of benzoic acid as a function of temperature. y - 20.826 - 7 . 6 8 5 1 ~R"2 - 1.000 1 0 -12.8 2.7 "2.6 2.9 3.0 3.1 1000/T Data in Table 2.2 (see Fig. 2.9) treated according to Eq. (2.55). The heat of fusion AHmelt is the difference between the two, that is, Vapor pressure curves (Fig. 2.11) and melting points will assume a special signifi- cance when further discussion on polymorphism is presented. olymor~hism the phenomenon of a chemical entity being able to exist in two is different crystal forms. It will be discussed ingreater detail elsewhere in this text, but a few points and examples are appropriate to mention at this point. Ceolin et ai. (1997), have reported on p , T diagrams of carbamazepine. Car~amazepine (USP) is monoclinic, but other polymorphic forms exist. ~ u b l i ~ ~ t gives a triclinic polymorph, but single crystals are ~ i f ~ c u l tproduce ion to in this manner. The authors produced a crystal of dimensions 10 x 70 x 430 p m that they used for single-crystal characterization of the polymorph. They show the following topological p , T diagram (Fig. 2.12).
  • 38. 51 Melting Point v40 GO 80 100 120 140 160 Temp ("C) Vapor pressure diagram of benzoic acid ( ~ e l t i n g point 122°C). E Vapor Phase !32 190 Temperature ( " C ) The literature caption (the reference Fig. 4) should read: B is the triple pointbetween triclinic, monoclinic and vapor; D is the triple point between triclinic, liquid, andvapor. (Data from Ceolin et al., 1977.) They found the transition point by using a tube heated at the position of thesolid, and by monitor in^ the deposit and the temperature alongthe tube; they foundthat 132 was the triple point. , Toscanini S , Gardette M.-F, Agafonov VN, Dzyabchen~oAV, Bachet B (1997). J Pharm Sci 86:1062.Einstein A (1907). Ann Physik 22: 180.Grant DJW, York P (1986). Int J P h a m 30:161.Hersey JA, Krycer I (1981). Int J Pharm Techno1 Prod Manuf 2(2):55.Hiestand E (1997). J P h a m Sci 86:987.Huttenrauch R (1983). P h a m Ind 45(4):435.Huttenrauch R, Keiner I (1979a). Int J P h a m 259.Hutten~auchR, Keiner I (1979b). Powder Techno1 22289.
  • 39. Jozwiakowsk~ MJ,Nguyen NT, Sisco JJ, Spankcak CW (1996). J Pharrn Sci 87:193.Longuemard P, Jbilou My Guyot-Herrnann A- , Guyot J-C (1998). Int J P h a m 17051.Moriata M, Nakai Y, Kukuoka E, Nakajima SI (1984). Chern Pharrn (19’72). PhD dissertation, University of Wisconsin, Madiso try and Physics, 48th ed. The Ch~micai ubber Co., Cleveland, OH, p. D 141. 143.
  • 40. 3.1. E ~ u i l i b r i u ~ Solubility 28 3.2. eat of Solution 28 3.3. etermination: Effect of Temperature 32 3.4. trolytes on Solubility 37 3.5. ixed Solvent Systems 37 3.6. lectric Constant on Solubility Parameters 38 3.7. ultiple Solubility Peaks 39 3.8. ~ o ~ ~ ~ e ~ a t i o ~ 41 3.9. ~ yclodextri~s 423.10. Solub~lityand p 423.11. rediction Equations for Solubility in 443.12. 443.13. 453.14. ffect of Surfactants 463.15. 46 46 47Solubility of c o ~ ~ o u nis of great importance in pharmaceutics, and the subject has dsbeen subdivide^ into the foregoing subtopics.
  • 41. henever the tern solubility is employed, it is tacitly assumed that it is equilibriumsolubility. In other words, it assumes that a (stable) solid (the solute) is laced incontact with a li~uid (the o l ~ eand~the system is all owe^ to be agitate^ for a long s ~ ~ , while, or by other ~ e a n allowed to reach a state o e ~ u i l i b r i ucharacteri~ed the s f ~, byfact t ~ a the co~cent~ation solute has reached a co~stant t of level. This de~nition by no means easy to establish in practice. Such things as small is temperature ~uctuations,and that solubility maybe a function of particle size,makes the experimental establishment of solubili~yof acompound dif~cult to achieve. Add to that the fact that solids of higher energetics(metastable poly~orphs or amorphates) have higher apparent solubilities also confounds the issue. At times (e.g., in the case of benzodiazepam), the drugsubstance, as first produced (in clinical trials), turned out to be a metastable polymorph. Apparent equilibrium solubilities esta~lishedand were t h o u ~ h t be true equilibrium solubilities because the to figures were reproducible, until one day the more stable form happened to be pro- duced, and this had a lower solubility. Because it is never reallycertain that any drug substance produced is a c ~ ~ a lthe stable polymorph, the term e ~ u i l i b r i u ~ ly solubility is clouded to some degree with uncertainty. Inthis chapter to follow it is going to be assumed that solubility is exactly what the foregoing italicized de~nition purports it to be. For the purposes of this book, there are four types of equilibri~msituations ill, 1933) that may be considered: e solid phase is a pure compound, and there is one liquid phase. e solid phase is a pure compound, and there is more than one liquid phase. 3. The two components form a solid solution in such a way that there is unlimited solubility in the solid phase. 4. There aretwo solid solutions forming (Le., there is limitedsolubility in the solid phase). f these, case 1 is ove~helmingly most cosituation. the Case 2 is at portant in differential scanning calorimetry work. If the melts ofcompoundAand compound Bare immiscible, then the DSC thermogram willshowtwo peaks, one at each compound’s melting point, otherwiseone broadpeak will occur. This will be discussed in a later chapter. When a substance (the solute) dissolves in a solvent there are certain changes that ome solutions are ideal solutions, and in such solutions the volumes, for property thatis of importance in the following isthe heat associated with the solution of a solid drug substance in a solvent (most often water), and it will become a ~ ~ a r e that the effect of temperature on solubility is associated with an aspect of nt this thermal phenomenon. There is a fair amount of misinterpretation of the “heats of solution” in lit- erature, and in this aspectitis fruitful to quote a very old, but comprehensive
  • 42. reference (Taylor, 193 1). If solubility ofa compound in a solvent isplotted versus thetemperature then, in its simplest form, the curve will either rise or fall. If heat isevolved when the solid is dissolved in an ( a l ~ o s st ~ t u r a t e d s o l ~ tthen, the solubi- ~ io~lity of the compound will decrease with increasingtemperature, and the opposite, inthe simple case, is also true. owever, to quote Taylor (1931): nowledge that when water is poured upon solid p o ~ ~ s s i uhydroxide, m much heat is evolved; if one deduced therefrom that the solubility of the compound decreased with the temperature, the error would be flagrant. The initial heat of solution is positive; it may be that the total heat of solution is positive, but the final heat of solution, representing the dissolving of the last increment entering the solution at the saturation point, is negative,and hence a rise of temperature will result in the dissolving of another increment. The relations between partial molarand inte~ralheats of solutions are ransted (1943a),in the followingwords,directly translated (theword t ~ ~ ~ ~ o d uy ~ t ia n~ i ~ the tern A in the translatio~): f n n o used for To visualize the connection one may utilize a graphic presentation in which one most advantageously utilizes the x-concentration scale and in place of A which applies to n1 + n2 molecules of mixture [utilizes] Ai, the integral mixing [ ~ h e ~ o d y ~ a m i c ] func~ion for one mole of mixture. The equation corresponding to this may be derived in direct analogy with the [previously cited equations] containing nl + n2 moles but may also be obtained by introducing: A = (nl + n2)Al and One, hence, obtains the following equation, valid at constant temperature and pressure: as well as the relation between the differential [thermodynamic functions] The connection between AI, Az, and Ai is shown in Fig. 2 [re~onstructed as 3.1 in Fig. this text]. At a further point in the text ansted (1943b) states that A the t h e ~ o d y -namic function “can be the S, V, E, 6, or H functions.” The foregoing text talks to the difference between differential heats of solutionand integral heats of solution; examplewise the heat evolved per mole of sulfuric acidadded to 1 mol of water. The heat of solution (IT)of a mixture of n1 moles of acompound A in n2 moles of a solvent
  • 43. Enthalpy N M 0 x=o x= 1 ti20 Solute “The abscissa is x (Br~nsted,1933~): The thermodynamic function in the figure is alpy H . The distance DJ is equal to the slope at L, mu~tipliedbyJD = (1 - x ) a ~ / a ~which, according to the Brransted Eq. (20) equals At - A,. ),value at D corresponds to H , = aHr/ax. The distance QP equals x times the slope at L, (Le.,x a H / ~ ~which according to the Brransted Eq. (20) equals A, - A 2 . Hence, the ordinate value ),at P equals A2 = a ~ ~ / a ~ .whereand ted the partial molar quantities of compone (subscri~t and compo- 1) ubscript 2). Table 3.1 lists the heats of sol of the sulfuric acid watersystem. that the first column is the third column divide the second column. se data are plotted, then a graph, such as shown ig. 3.2 ensues. If thetangent is drawn at any point of the curve, then the interce- th the left axis givesHI = 3H/dnl and the intercept with the right axis gives p 2 = 3H/dn2 where n1 andn2 are the molesof water and acidin the particular amount of solution i.e., +x = ~ / ( n l na). his is shownfor a composition of 0.55 mol ofsulfuric acid added ater (i.e., a mole fraction of 0. ). It is seen that the partial molar lution of water (A) and sulfuric acid ( are 6200 and 510 cal, respectively. is dissolved in water, there is a limiting value for x; namely, thatcorrespond in^ to the solubility. Hence onlypart of graph would apply, as shown in
  • 44. Water and Sulfuric Acid. Heat of Solution as a Function of Compos~tion ole fraction 1000 x heat evolved mole per A H per mole ofacid ( X ) acid of solution0 0 00.1 15.6 -1.560.2 12.94 -2.590.3 10.71 -3.210.35 9.65 -3.380.4 8.63 -3.4520.45 7.68 -3.4560.5 6.73 -3.370.55 5.81 -3.200.6 4.87 -2.930.65 4-06 -2.640.7 3.2 -2.300.75 2.6 - 1.950.8 1.97 -1.580.85 1.42 -1.210.9 0.93 -0.840.95 0.45 -0.431.oo 0 0Source: Data from Brmsted, 1909; Marshall, 1933. 0.0 0.2 0.4 0.G 0.8 1.0 x eats of solution of the sulfuric acid system: The abscissa is the mole fraction ofsulfuric acid.
  • 45. Fig. 3.3. It is noted that in Fig. 3.3 the heats of solution are terminated by thesolubility X . statements may (incorrectly) imply that there is either anincrease or a decrease in solubility of a compound with temperature. There aremany exceptions. ~ m m o n i u mnitrate solubility in water, for instance, exhibitsbreaks at 32"C, 83"C, and 126°C Maxima and minimain solubility/temperaturecurves also occur, and some of the situations of this will be discussed later in thischapter.The subject of eutectic diagrams willbe taken up in a later chapter, but a shortoutline will be given at this point. The simple solution situationreferred to in Sec. 3.1 exhibits a eutectic diagramsuch as shown in ig.3.4a. The so-called liquidous line in the right part of theeutectic, QU, is a mpositional line where, at a given temperature, T, there is anequilibrium between solid solute I and a solution of in water of composition x. 3This, in essence, is a solubility curve, and if the axes are ~ i p p e as ,shown in Fig. ~3Ab, then a conventional repr~sentation solubility versus temperature results. of Solubility of solids are determined by placing an excess of solid in contact withthe solvent in a hermetic containers (ampoule or closed testtube) and agitating it a inconstant te~perature bath. It is conventional to use 7 2 h for e~~ilibration. If less time is used, then the solubility may be obtained by iterative extrapola-tion, as demonstrated Table 3.2. Samples are taken after certain in times (here multi-ples of 12 h), and the supernatant is assayed. The concentrations are then plotted asa function of time, as shown in Fig. 3.5. It is seen that the data"seem" to level off at59, so the solubilities are subtracted from 59 (see column 3 inlogarithm takenof these numbers. These are plotted in Fig. 3.6. 5: X Q 0 Q c( - I V 0.0 0.2 0.4 0.G 0.8 1.0 x Heats of s o l u t i ~ ~ s depicted in Fig. 3.1 but t e r ~ i n a t e ~ the solubility X , repre- bysenting the highest concentration.
  • 46. ilitMeltingPomt,T Mole Fraction x 1 L 1 L L U 1 $0 Solid B+ Water Q a Ice + Solid 8 4 1 Mole Fraction, X Temperature, T H20 Solute, B (a) Eutectic diagram of water and a solute, B. (b) The right side of the eutecticdiagram from Fig. 3.4a plotted with reversed axes (i.e., solubility as a functi~n oftemperature). 60M00.-.L M 40 30 20 10 0 0 20 40 GO 80 Time (hours) Data from Table 3.2. Example of ~olubilityDetermination by Iterative ~ x t r a p o l a ~ i o ~Time (h) Solub~lity(g/lOOO 59 g) -s h[59 - 21 0 4.078 0 5912 3.367 30 2924 2.639 45 1436 52.5 6.5 1.87248 56 3 1.09960 0.18 57.8 1.2
  • 47. y = . 4 . 1 3 8 2 - 6.4403e-2x R*2 0.998 5 4 3 2 1 0 0 20 40 60 T m e (Hours) Data from Fig. 3.4 treated by iteration.repeated with a figure different from 59. The value of the iterant thatgives the best fit(the least sum of residual squares) is then assigned as the solubility. lubility is best expressed as molality or as weightof solute per gram of .e., not per cubic centimeter of solution). The conventional t r e a t ~ e n t ofsolubility as a function of temperature is to note that the chemical potential of acompound in solution, at a concentration level correspo~ding to an activity of a, isgiven by p1 = po + RTlna (3.4) ere, po is a reference state, and obviouslyis the chemical potential when the is unity (i.e., when a = 1 molal). hen there is eq~ilibrium (of between a solid and a saturated solution activity the chemical potential of the solid p,, equals that of the compound insolution, given by Eq. (3.4), that is, (3.5) ividing through by T and di~erentiatingrelative to T now gives = T + { a ( ~ . ~ / T ) / a T } ~ d{a(po/aT}~dT Rd(ln a,) (3.6)It is recalled thatThis when inserted in Eq. (3.6) then gives (after rearrangement) --{(h, - ho~/T2}dT ~ d ( l n ~ a , ~ ) =h" - h = -(hs - ho)is the enthalpyassociated with tran~ferri~g , 1 mol of solid into a quantity of saturated solution and h" - h, is commonly simply denoted h nes h, as "the partial molar enthalpy of the component in the erefore, at a given tem~erature,be the partial (3.9) (3.10)
  • 48. where , is an integration constant; a, is the activity of the solute at saturation andis &given by a, = Y,S (3. 11)where S is the saturation concentration (in molality) and y, is the activity coat saturation. If this is assumed to be unity, then Eq. (3.10) becomesthe welland often used equation ln[q = AH/(^^)} +BThis is referred to as a Van’ ff plot (although this latter, properly, iswith equilibrium constants9 solubilities). More correctly Eq. (3.12)written:If y, is temperature-independent9then the logarithm of the saturation on cent rationis linear in reciprocal absolute temperatur a plotting mode that is often useexample of this is shown in Linearity of the Van’t ependent, and (b) activity coefficient (y,) is temperature-independent. If they are no off plot will not be linear. An example of this is shown in Fig. 3 rant et al. (1984) hypothesizedthat if, rather than “the partial molof solution of the solute, is inde~endent temperature, we assume that it is a oflinear function oftempe , as follows: AH; = a + bT.’9They interpret that “ amaybeconsidered to be the ~ y ~ ~ t ~valueof c AH; at the absolute zero of e t ~ a ~tem~erature b is the change in the apparent partial molar heat capacity of t,he andsolute at constant pressure, AC;2, whichisitselfassumed to be independent oftemperature. There isevidence that the introduction of terns containing hipowers of T, e.g., cT2’ etc., is unnecessary.” If the curvature in Fig. 3.by the heat of solutio^ not being temperature-independent, i.e., (3.15) d ln S/dT = A H / ( ~ ~ 2 ) (3.16)~ombining with Eq. (3.15) then gives this Td In S/dT = { A / ( (3.1’7) §olubility of ~ r t h ~ r h o m ~u~fanilamide Ethanol bi~ inTemp (“6) Solubility (g/ 1000 g) 1n[sl47.4 28.22 3.34 31 1240.3 23.34 3.15 3.1929.6 16.78 2.82 3.3024.1 14.15 2.65 3.36Source: Data from ~ i l o s o ~ i c 1964. h,
  • 49. y = 12.380 - 2 . 8 9 6 1 ~R*2 - 1.000 1000/T able 3.3 plotted according to Eq. (3.12).Equation (3.17) integrates to Ins= (3.18) This may be fittedby nonlinear programs, butfor these to work, one must havea good estimate of A and B. To obtain good estimates, most graphing programswillcalculate (d In Cs)which may then be multiplied by I“ and plotted by way of Eq.(3.17) versus 1/T. This should produce a straight line with intercept B / R and slope A and 13 may now be estimated from the slope and intercept of this line, andmay be used a s . ~ r s t a p p r o x i ~ a ~ iin n s~ o n l i program, This approach has been oa ~ e ~ ~employed byseveralrecent investigators (~udipeddi,1998; ~ozwiakows~i ai., et1~96). It should, again, be emphasized that the enthalpy term in both(3.18) corresponds to “the partial molar enthalpy o f the c o ~ p o n e n tin the . . .solution . , . ii.e.1 the heat absorbed, at constant temperature and pressure, when 1mole of the component dissolves in the . . . solution.” ( ~ e n b i g h1961). T h i s ~ a c tin , ,itself, ~ a ~ itequite unde~standablewhy the Van’t ~o~ can not be expected to be slinear. Consider the diagram in Fig. 3.1. Suppos the depicted compound at a tern-perature of TI had a solubility corresponding to and at a higher tem~erature a had 1.9 1.8 1.7 1.6 1.5 1.4 3.1 3.4 3.2 3.3 3.5 1000/T Solubility of dl -p[pseudoephedrine]. (Data from ~ u ~ i p e d d1996.) i,
  • 50. y 331.78 - 9 3 . 3 6 2 ~R-2 = 0.992 4 0 F"25 10 3.15 I I I 1000/T Derivative curve (d I S ) of data in Fig. 3.8 versus n T". solubility corresponding to L, then the ~ ~ f e r e n t i a l e n t h ~ ~ i e s ~ a of solution^ w o u be ~ f ~ ~ c t i of nt e ~ ~ e r a t ~ r e ; it is not unexpected that the Vant Hoff plot is not hence, linear, but it is rather to be expected. To assign the change in heat capacity as an explanation to the nonlinearity is rational only in the sense that the composition changes with temperature; hence, the change in heat capacity also changes. There are many examples of this; for instance, Longuemard et al. (1998) have reported on the solubility of aspirin in 38% alcohol; they failed to obtain linearity according to the Vant Hoff, although in this case the curvature may be because the ordinate is in grams per liter (g/L), rather than in grams per 1000 g (g/lOOO g) of solvent * The solvent has a great i ~ ~ u e n on solubility and should always be speci~ed. ce aqueous solutions, the concentration of electrolytes may greatly affect the sol~bility of a compound. (It will be seen later, that this is particularly true for a compoun~ that is, itself, an electrolyte). Figure 3.10 shows the effect of sodium chloride con- centration on the solubility of a bisnaphthalimide derivative. The use of mixedsolvent systems isoften necessary in pharmaceuti~s when a drug is poorly soluble. Cosolvents used are Ethanol Propylene glycol Glycerin olyoxyethylene glycols Ternary diagrams are used to visualize wheremaximum solubility occurs when more than one solvent is used (Fig. 3.1 1). The length of PA is the percentage of water, the length WB is the amount of ethanol and, here, EC is the percentage of glycerol. The lines in this presentation mode are parallel to the sides in the triangle. In a different presentation mode they
  • 51. 0 -1 -2 -3 -4 -5 0.00 0.05 0.10 INaCl]. M Effect of salt concentration on the solubility of a bisnaphthalimide. (Data from et al., 1996.)are cast perpendicularly to the axes. A point inside the triangle, such asone given composition. If solubilities are determined for many solvent compositions, then the solubilitywould be the same (10 mglg, 20 mglg, etc.) for given compositions of the solvent, andsuch points can be connected to form isotherms and diagrams, such aswould result. The figure to the left in Fig. 3.12implies a maximum solubility,whereas in the other diagram, the more of one cosolvent that is added, the largerthe solubility is. requently the solubility is a fun~tionof the dielectric constant of the medium. ften, the relation is that of the Jaffe equation: + In[SJ = ( A / & ) l3 . (3.19)where A and B are constants and E is the dielectric constant of the solvent. Anexample of this is shown in Table 3.4, in which the solubility of a c o m ~ o u n ~ istabulated as a function of the dielectric constant of the medium (glycerin/water Water Glycerol Ternary diagram.
  • 52. H 20 Hz0Glyerol Ethanol Glycerol Ethanol Ternary diagrams of the two types of solubility. ost often, with hydrophobic drugsthe solubility decreases withincreas- constant. The opposite happens at times, and an example of this isshown in Table 3.4 and Fig. 3.13. It is particularly useful, from a practical point of view, to carry out solubilitiesin solvent pairs of different ratios tovary the dielectric constant. Graphswill often belinear when plotted asin Fig. 3.13, but they will often show maximumsolubility at agiven dielectricconstant (Fig. 3.14), and the practical part of this is that once this isestablished, almost any other solvent pair willshow maximum stability atthat her than using dielectric constant as a measure, the Hildebrand solubility~arameter is often employed. Shino~a 6 (1978) defines this as (3 20) s the heat of vaporization of the solvent, V its molar volume, and 3.15 gives an example both of plotting the solubility of a com~ound(caffeine) in solvent rnistures with different solubility parameters, the plotting as afunction of their dielectric constant. olubility profiles vis-a-vis the solubility parameter of the solvent at times showsmult~plepeaks. This is the so-called cha~eleonic effect (Sunwoo and Eisem, 1971; Effect of Dielectric Constant on Solubility of ~isnaphthalimideDielectric constant Solubility ( S , mg/mL) 1nES3 1O O / ~ ~78.5 2.49 0.912 1.27474.9 2.30 0.833 1.33565.9 2.00 0.693 1.51752.6 1.42 0.35 1 1.90045.45 1.09 0.086 2.20042.5 0.9% -0.02 4.333
  • 53. , y = 2.0006 - 8 6 . 4 7 6 ~ R^2 = 0.999 -0.2 0.0 12 1 0.0 17 I 0.022 l/(l)ielectric Constant) ffect of dielectric constant on the solubility of bisnaphtha.limide. (Data from aghavan et al., 1996.) 0 10 20 30 40 50 Dielectric Constant Effect of dielectric constant onthe solubility of phenobarbital in four systems: A, ropyleneglyco1:ethanol; € ,g1ycerin:ethano~; watecethanol; D, propylene glyco1:water. 3 C, ata from Lordi et al., 1964.) Dielectric Constant 20 40 60 80 100 " 5 10 155 2 20 Solubility Parameter The solubility of caffeine in a solvent cansisti~gof dioxane and water at 25°C:Top curve (with top abscissa) is solubility versus dielectricconstant, and the lower curve (withthe lower abscissa) is the solubility versus the solubility parameter (ai). (Data from Martinetal., 1961.)
  • 54. ustamante et al., 1994; Romero et al., 1996); it also exists for and the molecules appear to adjust their solubility to fit the tin et ai., 1985). ~ystems this type are often characterized by ofnonspecific van der ~ a a l forces as well as strong specific interactions? so that the sHildebrand solubility parameters no longer can explain theinstance, polar solutes in semipolar (or polar) solvents (JouybaAcree, 1998).Drug substances maycomplexwithcomplexing agents. An exampleis ascorbicacid/niacinamide (niacinamide ascorbate). In general, one of the two componentsof the system (e.g., drug A)? is called the substrate and the otherligand. ~omplexation often applicable to solubility problems in pha~aceutics. is Adrug A (the substrate) will react with another compound I (the ligand) and form a 3weak e ~ u i l i b r i u ~ . (3.2~)The equilibri~mconstant of this (the stability constant) is K =[ ~ ~ l / [ ~ ~ [ ~ l (3.21) The concentration of u n c o ~ p l e ~ substrate is the solubility when no ligand is edpresent. [A] = SThe ~uantities brackets are actual concentrations in the complexed system. If, by in , we denote the concentration of ligand calculated based on theamount added, then (B) will be this a ~ o u nless the amount complexed. tThe measured solubility Sobs is the solubility S plus the amount complexed. (324)so (3.25)and (3.26) Inserting the expressions for [ ] in the equilibrium equation [see Eq.(3,21)] gives (32’7)which rearranges to (3.28) ence by measuring the solubility as a function of the added ligandconcentration), a straight line should ensue with a slope of b, given by
  • 55. lope = B = K S / ( 1 + KS) (3.29) hen S is small, then the equation becomes, approximately S=S,+K (3.30)as demonstrated in Fig. 3.16. The complexation equations have been e x a ~ i n e d extensively for 1:1 complexes iguchi and Connors, 1965; L6pez et al., 198’7; Ahmed et al., 1991). ~omplexationis useful in severalapplications of solids; for instance, ascorbic acid and niacinamidecomplexes . If the two compounds are placed, for example, in a soft-shell capsule,they will i n t e ~d ~ t (and the complex is yellow). So in general they are pre- i time n ~reacted by placin~them in a mixer, adding alcohol, which allows the complex toform, and then drying the powder mass.A special note on cyclodextrins is in p, order. These compounds (a, and y dependingon the size of the ring), form inclusion compounds with many drugs. They take upeither the entire molecule or some hydrophobic portion of it into their “cavity.” Thisaffectsmany of the physicochemical properties of the complexed drug, without ng pharmacological properties ( ouessidjewe, 1996; Loftson rewster, 1996; Irie and and Uekama, Stella, 1996). The pro-blem may be considered in opposite sense, and Loftson and the ~ridriksdottir (1998)have shown the effect of water-soluble polymers and of a series of drugs on thesolubility of p-cyclodextrin in water (Figs. 3.17 and 3.18). he following ons side rations deal with the solubility of an acid as a functibut the inverse problem of an amine and its solubility as a function offollow the same lines. Mostlyan acid is less solublethan its salts, and at 1 asslebach equation will predict that y 1.3778 + - 4 3 . 7 3 3 ~ RA2 1.000 100 80 60‘r: 40 d 3 20 0 0 1 2 3 ~ ~ a c i n ~ mConcentration, M ide Effect of a ligand ( n i a c i n a ~ i d on the solubility of a b i s n ~ p ~ t h a l i ~(Data. ~) id~from Raghavan et al., 1996.)
  • 56. 70 -1 E HPMC Percent W/V Effect of HPMC on p-cyclodextrin solubility in solutions saturated in carbama-zepine. (Data from Loftson and Fridriksdottir, 1998.)orwhere CHA is the solubility of the acid form, CA is the concentration of the acidanion, andply: is the pK of the acid. If the solubility of HA, denoted SHA, is lessthanwhat is calculated from the solubility product of the metal counterion (3.33)then the solubility S is given by curve would have the appearance of a titration ould be given by Eq. (3.33). A is increased by addition of an alkali , then the amount in solution will increase by Eq. (3.32) until such t this point, let us assume that p moles ofbeen added, and th is ~uf~ciently that high the solubility is simthe concentration of A", so that both [ 1 and [A"] is equal to p , 52 60 70 I65 mg Drug/g Cyclodextrin NoPol CMC PVP HPMC The solubility of a hydrocor /y-cyclodextrin complex in aqueous solutions of , PVP, or 0.1% w/v HPMC. from Loftson and Fridriksdottir, 1997.)
  • 57. K = p2 (3.35)If a bit more MOH is added, the concentration of each ion would bep + A where Ais the small amount. But Cp+A}2>P2=K (3.36)so that MA will precipitate out: a plateau would be reached. product of zZA. is smaller than If the acid is dibasic, then often, the solubilit~that of HA and in this case the solubility versus pH curve would decrease abovethe second pH of the compound.There have been a series of attempts in literature (Martin et al., 1980; ~alkowskyand Roseman, 1981; Williams and Amidon, 1984; Ochsner et al., 1985; Acree et al,1991; Acree, 1992,1996;Barzegar-Jalali and Hanaee, 1994; arzegar-~alaliandJouyban-~haramaleki, 1996) to establish a reliable predictor for an unexperiencedsolute solubility in binary mixtures of solvents with know neat properties. One ofthese is the Redlich-Kister (or, the CNIElS/.R-K) equation, Here Xm, the mole fraction of the solubility is related to fa and ji, the volumefractions of the two solvents A and El when no solute is present, where ‘ and X X bdenote the molefraction solubility in the neat solvents A and I of the solute, and /lo, 3/11, and ,B2 are least-squares-~tted constants in the equation:~ o u y b a n ~ a r a m a l eand Hanaee (1997) have investigated this equation for a series kiof hydroxybenzoic acid esters.~lthough perhaps not strictly applicable to “solids,” solubility in micellar systems isof importance. Often the important aspect of solubility is indissolution testing, and ashort note on the effect of micellar systems is of importance. Micellar systems, in a certain way, are similar to two-phase systems with onebig exception. For instance, whereas in oil in water systems the oil droplets arecontinents unto themselves, and the oil molecules in a droplet are fairly much thesame independently oftime, in micellarsystems there is an e ~ ~ i Z between j ~ i ~ ~monomers in solution, and molecules of the amphiphile in the micelle. As far as solubility is concerned, there is a p a r t i t ~ o ~ solute molecules ofbetween solute in the aqueous phase and solutes in the micellar phase, so that inthat aspect, the phenomenon is one of partition, ~ j the e ~ c e ~ t that~the linear t ioincrease in solubility of, for example, the drug, does not start taking place before thecritical micelle concentration (CMC) is achieved. Solubilityplots, therefore, have theappearance shown in Fig. 3.19. Hammad and Muller (1998), for instance, has reported on the solubility ofclonazepam in mixed micelles.
  • 58. -1&a“I0v) 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.G SPC Mol Fraction Solubility of chlonazeparn in soya phosphatidyl-choline. (Data from H a ~ r n a dand Muller, 1998.) Niacinamide is often used to increase solubility of drugs (Fawzi et al., 1980; ruelove et al., 1984; ~ a l a v i o l l e al., 1987;Chen et al., 1994). For instance, et ogdanova et al. (1998) have shown that melts made with niacinamide and indo-methacin give rise to a maximum solubility at an indo~ethacinconcentration ofabout 7.5%.When a compound is poorly stable in aqueous solution, then the Nogami methodmay be particularly useful, because the longer the solubility experiment goes on, themore drug substance will degrade. Then, it is worthwhile selecting smaller value for athe sampling interval, q, for instance, 20 min. The method of approaching this is as follows: the rate with which the com-pound goes into solution in V milliliters of liquid is given by: d ~ / d = --VdC/dt = -kAS t (3.38)or (3.39) is mass not dissolved at time t, A is the surface area, S is the solubility, andk is the intrinsic dissolution rate constant. The rate with which the drug decomposes is given by: dC/dt = -klC (3.40)where first-order kinetics are assumed, and where k l is the decomposition rate con-stant. The con cent ratio^ profile, therefore, is governed by the following equation: dC/dt =I= ( k A S / V ) - kl C = kl(,8 - C) (3.41)where , Z= ( k A S / k ~ ) 8 V (3.42)This may be rewritten dC/(B - C ) = -kldt (3.43)
  • 59. which integrates to 1np - C] =: ”k,t + ln[P - Co] (3.44)The value of may be found by iteration, and this then gives both the value of S andkl (provided the surface area is known). urfactants, such as the polysorbates, willsolubilize drug compounds when thesurfactant is present in excess of its critical micelle concentration ( solubility will increase linearly with surfactant concentration. rfactants also aid in the wetting down of a solvent, and they are often used in on tests for poorly soluble drugs. here will be more description of this in the next chapter. stwald (1900) and ~reundlich (1922) postulated that the size of a particle affectedIts ‘‘solubility,’’ The equation, known as the stwald-Freundlich equation, or as the stwald ~ p e n i n g effect, relates the solubilities S1and S2 of particles of size rl and r2by the following equation: (3.45)where CT is the inte cia1 energy between solid and liquid, M is molecular weight,the gas constant, s absolute te~perature, p is density. The derivation of andwill be shown in Chapter 6, on crystal~ization. T a ~ e n its fullest, the equation predicts that real equilibrium existsonly tobetween an infinitely large amount of liquid with a single, in~nitely large crystal. is difficult to prove e~perimenta~ly with prevalent values of and, T p ) , real differences would be difficult to pts at this have been made ( ~ ~ o l and en 1972; Jeannin et al., 1975), but the equation has also been. refuted ne reality of the equation is that, in polydisperse suspensions, smaller parti-cles often disap~ear at expense of larger ones (ripenin. ), but other circu~stances the(temperature ~uctuations) could also account for this. = constant in the extended Van’t off solubility equation 3 =concentration of A, ligand or substrate 1 = ther~odynamic function of c o ~ p o n e n t 2 = Ther~odynamic function of co~pollen = t h e r ~ o d y n a ~function of ~ i x t u ~ e ic l/dnl = partial molal thermo~ynamicfunction of comp ~ A 2 / d ~partial t h e ~ o d y n a m i c = ~ function of component Jdx = slope of thermodyna~ic function versus c o ~ p o s i t i ox curve (~) a, = activity of component A in saturated solution
  • 60. = constant in the e off solubility equation 1 = concentration o = concentration h e = heat capaci differencebetween component A in solution and solid ions two solvents A and €3 when no solute is present /dnl = slope of integral heat versus composition (n) curve olal enthalpy of component A olal enthalpy of component I3 enthalpy of solution, enthalpy associated with transfer- ring I mol of solid into a (large quantity of ) saturated solution = heat of vaporization enthalpy of solid h, = enthalpy of component A in saturated solution R = gas constant K = complexation constant = (a) molecular weight or (b) counter ion 1 = concentration of metal counter ion, M rl = size of a large particle r2 = size of a smaller particle S = solubility of component A SI = solubility of large particle S2 = solubility of smaller particle T = absolute temperature V = molar volume nl = number of moles of component y 2 = number of moles of component 1 x = mole fraction Xa,X b = ~olubility mol action of a compound in neat solvents in /3 = constant in the Van? ff equation for solubility BO,P I , /32 = least-squares-fitted constants in Eq. (3.37) ivity coefficient of component A in saturated solution ebrand solubility parameter /A,chemical potential of component A in saturated solution po = standard chemical potential of Component A in solution p = density CT = interfacial tension between solid and liquidAcree WE Jr (1991). Thermochi~ Acta 178:152.Acree WE Jr (1992). Thermochirn Acta 198:71.Acree WE Jr (1996). Int J Pharrn 127:27.Aguiar AJ, Krc J Jr, Kinkel AW, Samyn JC (1967). J Pharrn Sei 56:847.Ahmed SM, Naggi A, Guerrmi M, Focher B (1991). Int J Pharrn 77:247. , Hanaee J (1994). Int J Pharm 198:281. , Jouyban-Gharamal~ki A (1996). Int J P h a m 140:237.Bikeman JJ (1970). Physical Surfaces. Academic Press, New York, 216.
  • 61. ~ogdanovaS, Sidzhakova D, Karaivanova V, Georgieva S (1998). Int J Pharm 163:1. ransted JN (1909). 2, Phys Chem 68:700. r0nsted JN (1933). Fysisk Kemi. Munksgaard, Copenhagen, p 139.Brsnsted JN (1933b). Fysisk Kemi. Munksgaard, Copenhagen, p 140. nsted JN (1933~). Fysisk Kemi. ~ u n k s g a a r d ,openh hag en, p 141. tamante C, Qchoa R, Reillo A, Escalera JB (1994). Chem tamante C, Qchoa R, Rei110 A, Escalera JB (1994). Chem arstensen JT (1977). FormulationandPreparation of Dos Elsevier/North-Holland ~iomedical, Amsterdam, pp 197-21 5 ,Chen A, Zito S, Nash R (1994). Pharm Res 11:398.Diaz D, Bernard MJB, Mora JG, Lianos CME (1998). Pharm Dev Techno1 3:395.Denbigh I (1961). "he PrinciplesofChemical Equilibrium. Cambridge UniversityPress, C London, p 257.DucheneD, Wouessidjewe D (1996). In: Durnitriu S, ed. Pharmaceuticaland Medical dextrins. Marcel Dekker, New York, pp 575-602. , Martin A (1994). J Pharm Pharmacol 46172. M (1980). J P h a m Sci 69: 104.Freundlich H (1922). Colloid and Capillary Chemistry. Dutton, New York, p 155.Grant DJW, Medhizadeh M, Chow AHL, Fairbrother JE (1984). Int J Pharm 18:27.Hammad MA, Muller BW (1998). Int J Pharm 169:55.Higuchi T, Connors K A (1965). Adv Anal Chem Instr 4: 116.Hi~uchi (1958). J Am P h a m Assoc Sci Ed 47:657. THill AE (1931). In: aylor HS, ed. A Treatise on Physical Chemistry. Van Nostrand, New York, pp 336-338; 536-539. oy K (1970). J Paint ~ e c h n o 42:76. l rie T, Uekama K (1997). J Pharm Sci 86:147. , Carst€nsen JT (1975). Ann Pharm Fr 33:433. i A, Hanaee J (1997). Int J Pharm 154:245. i A, Acree WE (1998). Int J P h a m l67:177.Joz~iakowski MJ, Nguyen NT, Sisco JJ, ~ p a ~ k c a k (1996). J P h a m Sci 87:193. CW (1972). J P h a m Sci 61:281. M, ~uyot-Hermann A-M, Guyot J-C (1998). Int J P h a m 17051. Galan YCR (1987). Cienc Ind Farm 6:325. ir H (1998). Int J P h a m 163:115.Loftsson T, Brewster ME (1996). J Pharm Sci 85: 1017.Lordi N, Sciarrone , Ambrosio T, Parta AN (1964). J Pharrn Sci 53:463. alaviolle I, ~ e M a u r y Chauvet A, Terol A, Masse J (1987). ~hermochim 6, Acta 121:283. arshall AL (1931). In: Taylor, HS, ed. A Treatise on Physical Chemistry. Van Nostrand, New York, pp 336-338; 336-339. artin A, Newburger J, Adjei A (1980). J Pharm Sci 69:487.Martin A, Paruta A , Adjei A (1981). J Pharm Sci 70: 1115.Martin A, Wu PL, Liron Z, Cohen S (1985). J Pharm Sci 74:638. Pharm Sci 74: 132. School of Pharmacy, University of Wisconsin, A, Gray DB, Hussain MA (1996). Pharrn Dev Techno1 1231. 996). J P h a m Sci 85:1142. 6). Chern Pharm Bull 44: 1061.Squillante E, Needham T, Zia W (1997). Int J P h a 159: 171. ~
  • 62. Sunwoo C, Eisen H (1971). J Pharm Sci 60238. , Chen N, ~ u s s a i n (1984). Int J F h a 19:17. A ~ keda M, ~ishimuraK, Yamamoto K (1997). Int J~ i l l i a m NA, Amidon CL (1984). J Pharrn Sei 79:9. sYalk SH, Roseman TJ (1981). In: Yalkowsky SH, ed. Techniques of Solu~ilizationof . Dekker, New York, pp 91-134.
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  • 64. 4.1. 52 4.2. 52 4.3. 53 4.4. 53 4.5. 5 4.6. nd Cas Adsorption 4.7. 55 4.8. lectronic Counters and Laser Counters 56 4.9. 574.10. eflectance Infrared (F 584.11. 58 59 59In the previous chapters the subdivisions of the solid has been of no importance stwald-~reundlich equation.In general, however, solids are multi- a solid sample is usually more than one particle). ome of the methods to be d e s c ~ i b e ~ old and tested, but today, there are are vailable for particle size and particle distri~utionassessme~t ith a multitude of methods, it becomes important to be nguishbetween the many definitions of particle size that 990). The most im~ortant be discussed in the following. will
  • 65. The three most prominent subdivisions of multiparticulates are illustrated in Fig. 4.1. A. ~ o ~ ~ d i s ~powder is one for which all the particles are the same size. A. erse erse powder is one, the particles of which are not the same size. article is one of whichthe surface s may be expressed as the ratioof the surface, s to the two-thirds power of its volume v. s = r(v2l3) (4-1)where I“ is denoted the general shape factor. This will be discussed further whenshapes are discussed; here, it will suffice to say that there are three, common iso-metric shapes; that is, a cube (with shape factor 6), a sphere, and a right cylinder (onefor which its diameter equals its height).E ~ ~ 4.1 ~ ~ l eCalculate the shape factor for a sphere.A ~ 4.1~ ~ ~ r lume of a sphere is d3n/6 and the 2/3-power of the volume is d2{n/6}2/3; the is nd2, so the ratio is ~ / { n / 6 = ~ / ~ = 4.84, } 62/3n1/3 he “size” ofa particle would be easyto define, if the particle were either a sphere ora cube, but once the shape of the particle deviates from that, more than one defini-tion becomes possible. igure 4.2 shows two situations for which the type “diameter” must be defined.In the parallelepiped, it could be either the small dimension (the height, h) AB, thelong di~ension(the length, 1 ) AC, the width or breadth, b ( ~ ~or) one of the, hich dimension is chosen is often a question of which measuring method is 0th b and E may be recorded, but h is usually “hidden” becausethe particle lies on its short side. The same is true for the a ~ o r p h o u particle, but in sthis case one often records the diagonal. ouu Monodisperse Isometric Polydisperse 0 Isometric i Polydisperse Non-lsometrlc States of subdivision.
  • 66. crystal (a para~lelepiped)and an oddly shaped particle (e.g., an An orthorho~bica ~o r p h ate) . Isometric particles are frequently approximated by spheres. In this writing,schematic, isometric particles will always be a ~ ~ r o x i m a t e d cubes, because this byshape is closer to a that of a real particle. The scientific field concerned with these matters, as well as with distributionsand surface areas, is usually referred to as ~ i c ~ o ~ e ~ i t i c ~ .The foregoing definitions refer to single particles, but in general, particles exist in a~ o ~ ~(i.e., multiparticulate). The easiest method of differentiating between the l a ~ ~ o ~various types of particle sizes isto describe, briefly, how theyare measured. The mostcommon methods are ~icrosco~y Screen (sieve) analysis Electronic counting (Coulter, ~edimentation methods (And In the following, the measured, or treated, particle dimension will be denotedby a,In optical microscopy, a very dilute suspension of a sample (e.g., in mineral oil) ismade, and placed on a hemocyto~eter slide. The number of particles of arange in the field (e.g., between zero and 10 pm) are counted and noted, then thenumber between 10 and 25, and so on. The results may present themselves (in asimpli~ed fashion) as shown in Table 4.1. An average particle diameter would then, logically, be given by(2 X 1) + (3 X 5.5) + (4 X 12.5) + (1 X + + 4 + 1) = 106/10 = 10.6 pm 3 3~.5)/(~ This type of diameter is called the ~ ~ i t ~ ~ e~ t i i a ~ ~ and e denoted by ~e~~ ~ is ~ e ~
  • 67. Example of Microscopic Particle Count~ u ~ b e ~ 2 3 4 1Particle size (a) p m 1 5.5 37.5 12.5The problem with microscopicdiameters is that the sample size is verysmall; hence,the measured diameteris not necessarily representative of the larger lot from whichit t is collventional to measure in such a fashion that the total number of articles is about 300. Withscanning optical microscopy (S ), it ispossible toincrease the measured number considerably, but the sample size is still small. uch larger magni~cations than achievable b ical microscopy are achieved by . Ceolin et al. (1997), for instance, used to distinguish between trigonaland triclinic phases of carbamazepine.In permeametr~ is actually the surface area that is measured and this method will itbe treated in more detail at a later point. The type diameter obtained by pemea-metry is called th surface ~ o l ~ ~ a ~ iea ~ e t eIf . e ~n r one considers the volume V andthe surface area of a sphere, with. diameter a, then the ratio of V to A has thedimension of a diameter V / A = {(n/6)a3)/(na2) a/6 = (4.3)so that for a nonspherical particle one may generalize that as,,= ~ V / S (4.4)asv is denoted the surface volume mean diameter. For isometric shapes this is inde- iven a particle population o f rzl particles of diameter a l , rz2 particles ofdiameter a2, and so on (Table 4.2), it is seen that Example of Microscopic Particle CountNumber 2 3 4 1~ a r ~ i csize (4 pm le 1 37.5 5.5 12.5 Ea3 2 499.125 78 12.75 52734.38 3 90.75 625 1406.25 yla 2
  • 68. so thatwhere V and S are the volume and surface area of the ~ o ~ ~ Z ~ tthe o ~ (i.e., i sample). From the sums it can be calculated that as, = 28.74211 = 28.7 pm r such conversions if they are often needed. A The surface-volume mean diameter is fre~uently called the ~ ~ ~dia- rmeter, and the arithmetic mean diameter is, in similar fashion called t h e ~ ~ ~ ~~ i ~ ~ ethe e ~ ? t moment denoting the power of the n u ~ e r a t o r . ng the specific surface area, A, per gram of solid, for instance, by T), surface area measurements (to be discussed later), the volum V = l/pwhere p is the density. ultiplying this by 6 and dividing by A, would then give thesurface-volume diameter a,, = ~ ~ ( p A s )Scanning electron microsco~y (SI3 ) may be used for small particle sizes and theprocedures used are quite the same.A. very common manner of measuring particle size in industry ~n-process sieve isanalysis (Fig. 4.3). The openin~s the screens are described by a U.S. esh Number, which in ofindicates the n u m b ~ r strands per inch. As the wire has a width of itscannot deduce the size of the opening by dividing 1 in. by the number.shows common mesh sizes. Program for Converting a, to a,,100 I ~ ‘“Number of DataTSets = ”;Ql ~ ~110 READ N1,DI120 Q2 = Q2 + 1140 VV2 = Nl*(D1*2)150W3 = W3 + W1160W4 = W4 + W2170 D2 = W 3 ~ ~ ~180 IF Q2 = Q1 GOTO 500190 GOTO 400400 DATA 2,1,3,5.5,4,12.5,1,37.5410 GOTO 110500 ~~~~~ “”Diameter = ”;I22
  • 69. A B Example of sieve analysis. If a sieve analysis is conducted on a W gram sample, then the masses (weights)of the fractions collected on the various sieves are denoted w l , w2, - - *. Each sieve is,conventionally? assumed to collect particles of diamters of d l , d2 . ., of which thediameters are the average valuesof the diameters of the confiningscreens. Forexample, a sample that went through a 60-mesh screen and was retained by a 70- +mesh screen, is assumed to have an average diameter of d = (0.25 0.21)/2 = 0.23mm or 230 pm. The “average” diameter of the entire sample may be expressed as (4.11) w. This is a fourth-~oment diameter and is denoted as the weightmean diameter.In the ~ o ~ l t c o ~ ~ t ean?electrode with an aperture is employed. The electrode is er rplaced in a dilute suspension of the drug substance, which is pumped and circulatedthrough the aperture. The electric conductivity over the aperture is then measured.Every time a particle (with essentially negligible conductivity) passes through theaperture, the conductivity is reduced by an amount corresponding to the volume of U.S. Mesh OpeningsMesh Opening (mm) Mesh Opening (mm)10 2 80 0.17720 0.84 100 0.14925 0.69 120 0.12530 0.59 200 0.07440 0.42 230 0.06350 0.297 270 0.05360 0.250 325 0.04470 0.210 400 0.037
  • 70. liquid it replaces. The instrument is adjusted to a threshold value T l ,so that only thenumber of particles ofa given volume V I , is counted. This threshold is then changedto a different threshold T2,counting the number of particles of volume larger thanV2,and so on, so that, in the end, results may appear as a cumulative ~istributionfunction. The possibility of two particles passing at the same time is taken care of by acoincidence factor. It is possible to convert the cumulative distribution to a fre-quency function, so that one knows the number of particles rzi, that are in a certaininterval of volumes, u and vb, If the average of these is denoted ui, then an aver~ge ,diameter can be calculated from this. This introduces the concept of a v o Z u ~ e mearz~i~~eter. 113 (4.12)In a similar fashion, the alvern counter employs a laser beam that is interfered withby particles flowing in its path. This leads to the concept of a cross-sectional defini-tion of a diameter, denoted the s u ~ f a c ~ d i a ~ e t e r . mean (4.13) Andrks et al. (1998) have shown that comparing a set of data of particle sizedistributions of fenofibrate, obtained by microscopy, led to a monomo~al distribu-tion, whereas laser lightscattering detected a trimodal distribution, one (weak) modeabout 1 pm, a size simply not detected in optical (projected) light microscopy.The Andreasen apparatus (Fig. 4.4) depends on Stokes law. Particles from a popula-tion are sampled and added to water to a concentration no larger than 2%. IJsuaily,sodium metaphosphate is added as a deglomeration agent. If the particles have a hydrodynamic radius of a, then the steady-state velocityv of a particle, with a density that is A p larger than the dispersion ~ e d i u mwithviscosity r] will be given bywhere g is gravitational acceleration. The suspension is dispersed, and attime zero, asample is taken through the stopcock. An assay of this then gives the analyticalconcentration C ofsolids in the dispersion. At a giventime t, the procedure isrepeated. Particles, larger than a, = [9~20/(2Ap~)~ havepassed the 20-cm will . The new concentration C1 may now be used to calculate the fraction, (C - f particles with a particle size larger than a,. y modern standards, the method is slow, but it has several advantages: 1. The sample size is large compared with other methods. 2. It gives, directly, the oversize distribution. 3. It gives a defined radius (the hydrodynamic radius).
  • 71. S topcock - Liquid Level o f Suspension L Andreasen apparatus: At a given time t, particles possessing a radius that is largerthan a, =: [9q20/(2Apt)] willbebelow the mark B. These particles, therefore, will not bepipetted out.larger than in the other “precision” methods. The obtained results give good correlation between diameters observed andthose observed from some (but not all) of the other methods mentioned. It was mentioned earlier that it is important to disting~ish between the variouskinds of dia~eters, and the quoted article does not do that, leading to some spec-u~ation about why the diameter values from the FTZ diffuse method differed froma value obtained by SE e will mention more about particle diameters related to their shapes as this text to say at this point, that there are several ways of expressing iameters are often determined microscopically. ne presentation method fora ~ o u n t i n g the shape is the ~ r o J e c tse~~r f a for a ~ i a ~ e ~ e r , is the diameter whichof a circlewith the same area as the particle. ctual geometric s~rface area (thearea ~ a l ~ u l a from the geometry of the te~ , assuming to be it completelysmooth) is often calculated. The e Z o ~ g ~ tfactor is the ratio of largest to smallest io~diam~ter (diagonal) is used as a measure of particle shape. Another method is the ~ e ~ factor, o o attempts to describe the circu- ~ which ~larity of the particle projection (microscopically). It is the ratio of thmeter to the ~ e r i m e t ofrthe circle with the same area as the particle. ~close to circular, then the eywood factor will be close to unity.
  • 72. Chebli and Cartilier (1998) have determined that samples of microcrystalline 101) and cross-linked cellulose (CLC) of particle dia-meters of about 50 pm and found their eywood factors to be 125 and 179, respec-tively. a = size, diameter a, = X ~ Z ~ / Xarithmetic mean diameter ~Z, asu = ~ r ~End2, surface volume mean diameter d ~ / a," = { X r ~ d ~ / X n } volume mean diameter "~, a, = (Xna2/Xn}I 2 ,surface mean diameter a, = hydrodynamic radius A, = specific surface area, area per gram h = height (small dimension) of a particle AB E = the long dimension of a particle b = breadth of a particle dum = ~ ~ j d Xi n i d ~ / ~ ~ i d ~ mean diameter in sieve analysis = / ~ X weight dj = average diameter insieveanalysis,which equals average ofconfining screen o~enings g = gravitational acceleratio~ i = ru~ning index t = time s = surface of a single particle S = surface area of a popluation (sample) ZI = (a) volume of a single particle; (b) Stokes velocity V = volume of a population (sample) W = X w i = weight of a sample for sieve analysis 17 = overall shape factor Ap = difference in densities of solid and liquid in a settling sus~ension q = viscosityAndrks C, Bracconi P, RCginault P, louquin P, Rochat MH, Pourcelot Y (1 968).Int J P h a m 147: 129. Lafaye A, Lafaye JM (1990). Powder Techno1 60:205. Toscanini S, Cardette M-E;, Agafonov VN, Dzyabchenko AV, BachetChebli C, Cartilier L (1998). Int J Pharm 171:lOl. , Matsuda Y (1996). J Pharm Sci 85:112.
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  • 74. 5.1. 62 5.2. Distribution Types 62 5.3. istributions: The atch-Choate Relations 63 5.4. 65 5.5. Adsorption Isotherms 66 5.6. FreundSich Isotherms 67 5.7. dsorption Isotherms 67 5.8. er, Emmett, and Teller) Isotherms 69 5.9. 745.10. 75 5.11. Porosity 75 5.12. Permeametry (Carman- 78 5.13. Surface Areas from article Size Distributions 79 5.14. 79 5.15. Shape Factors by Way of Fractal Dimensions 81 5.16. olydisperse Particle Populations 85 Symbols 85 References 87~ulti~articulates often, as seen in the last chapter, are notmonodisperse; they mostcontain a spectrum of particle sizes. The expected distributions will be overvie~ed in
  • 75. efore discussing this it is worthwhile to have an overview of theconcept and the definitions associated with “diameters” or “sizes.”There are three commonly used dimensions employed in microscopy.For crystallinemate~als, uses height, length, and breadth. For amorphous materials, there are oneseveral ways, some of which also apply to crystalline samples. The diameter of acircle with the same area as the (microscopically d e t e ~ i n e d ) cross-section of theparticle is one, the Martin’s and the Feret’s diameters are two others. These areshown in Fig. 5.1. in’s diameter is along a line that dissects the rticle in two equal areas. erdan (1961) “Feret’s diameter is the mean leof the distance between opposite sidesof the apparent out1 f the particle parallel to direction and irrespective of the orientation of each particle There are different definitions for diameters, some of which havealready been~ i s c u s s eThey are repeated in Table 5.1 for convenience, and include some defini- ~.tions not yet touched on.The most common distribution^ of numbers are normal, lognormal, oisson, and binary. The latter two are not too significant in d i s t r i ~ ~ t i o ofs multi- nparticulates, but the former, as well as bimodal distributions are. The normal frequency function, f ( a ) is given by Martin Area 1 = Area 2 ~chematic showing the definitions of a artin and a Feret diameter.
  • 76. Diameter DefinitionsaCna/Cn = a,: arithmetic mean diameter~ n a ~ / = asu:a ~ ~ n surface volume mean diameterCna4/Cna3 = awnz sum: weight mean diameter’ = = volume mean diameter’ = surface mean diameter“The symbol a denotes size or “diameter.”bThis notation has been used in this text to distinguish between two diameters. The “weight meandiameter” is, at times, referred to as “the volume mean diameter,” but to avoid confusion that conventionwill not be followed in this text.where~denotes “function of,” a is “diameter” (or size) , aavgis ~verage “diameter”(or size), and cr is the standard deviation of the population. The normal distrib~tionfunction is the integral of this function. The probability of a particle having adiameter smaller than a* is then given by he lognormal distribution is defined in ec. 5.3. The Weibull istribution has thefollowing form: ln{- Ink]} = - ln[a] +C ( 5* 3)where J is cumulative frequency. The mast common distribution encountered in multiparticulate solids is thelognormal distribution.Often distributions are lognormal, in that, instead of plotting sizes (diameters) on thex-axis of a frequency plot or a distribution plot, one plots the logarit~ms the sizes. ofThe mean diameter is denoted dg (the g e o ~ e t ~~ c e ~ iaa ~~ ~ e r ) . i e ence, this is defined by Eq. (5.4).where d represents some of the aforementioned diameters. The resent at ion in dis-tributional form is then given by dlf(a)] is the number fraction of particles with diameters a, the logarithms of +which are between ln[a] and ln[a] aln[a], where the symbol a is the differentia^notation (used to distinguish it from the diameter notation of a). The number average of such a population is given by
  • 77. The following substitution is now made: u = (ln[a] - ln[ag1)/21/2 a] In (5.7)SO that when a = 0, u = -DO. This may be rearranged to ln[a/agl= u[2I2~ l a)] n (5 -8)or (5.9)It is noted that (5.10)Inserting these equations into Eq. (5.6) now gives a, = [y/2 ln[al/ ln[a]y/2n] agexp( 42~n[cr]u - u2]}du (5.11)The term under the exponent sign may be rewritten + y/2 ln[a]u- u2 = -{u - (2/2/2) l n [ ~ ] } ~ ln2CY 0.5 (5.12)The substitutio~is introduced, noting that dm = du (5.14)and this inserted in Eq. (5.11) gives 00 exp(0.5 ln2a)e x p ( - ~ 2 ) d ~ J-00 (5.15) = (ag/2/7r)(0.5 a)2(2/~/2) exp(0.5 ln2a + In a,) ln2 =where usehas been made of the gamma function in the evaluation of the integral. Eq.(5.15) may be r e ~ r i t t e ~ ln[a,] = 0.5 ln2a a, + In (5.16)and is the first of the Hatch-Choate equations. The re~aining equations are shown in Table 5.2 [Eq. (5.16) has been repeatedfor convenience]. The relations correlate the ea^ ~ i a m e t e ofs number d~~tributions those ~ withof weight distributions; however, one of the distributions may be truly lognormal,a~cordin~ly, other will not, but the mean diameter calculated on the assumption theof l o ~ n o ~ a l iwould have the value stated in the table. ty
  • 78. The Hatch-Choate Relations"Relation Equation no.ln[aJ = ln[a,] + 0.5 ln2 a (5.16) +~n[a,l=1n[a,] In2a (5.17) +ln[a,]= 1n[ag1 1.5 In2a (5.18) +ln[a,,] = ln[a,] 2.5 In2a (5.19) +ln[ag] = ln[aF] 2.5 ln2 a" (5.20)ln[a,] = ln[aF] - 2.5 In2a" (5.21)~n[a,l== lnia;] - 2 In2a" (5.22)~n[a,]= 1n[41- I In2cro .5 (5.23)ln[as,] = ln[ar] - 0.5 ln2a" (5.24)aSuperscript w implies distribution by weight, and lack of superscript implies distribution by number. Fordiameter definitions consult Table 5.1 or list of symbols at the end of the chapter.For narrow particle size, distributions are often normal. The equation for this typeof distributio~ Eq. (5.1). is In microscopy, a very small sample is taken from the population and a slide isprepared (usually a very dilute suspension in oil). A measuring device (e.g., a hemo-cytometer) allows the viewer to count the number of particles in certain particleranges. An example of this from microscopy is shown in Table 5.3. The Z-values are found from normal error curve tables. The frequencies may be presented in histogram form, but it is more advanta-geous to plot the Z-value (obtained from the ~ u m u l a t i v ~ frequency) as a function ofparticle size to see if the distribution is normal. In Fig. 5.2, a set of similar data (40/50 mesh) are plotted in this fashion. The data seem to be normally distributed. The least-squares equation is Z = -6.8862 + 0.031355b Data Generated for 1 0 0 ~ 2 Mesh Salicylic Acid ~0Range bavg Frequency, Cumulative Normal(Pm) (Pm) Count f f 2-value"202 215 14 0.056 0.056 -I .590228 24 1 32 0.127 0,183 -0.905254 267 43 0.171 0.354 -0.360280 293 44 0.175 0.529 0.075306 3 19 45 0.179 0.708 0.550332 345 40 0.159 0.867 1.115358 37 1 18 0.072 0.939 1.550384 397 9 0.036 0.976 1.980410 423 6 0.024aObtained from a normal error table.
  • 79. 300 Normalized presentation of size d i st ri b~t i o~s a 40/50 mesh cut of oxalic acid in ata from Dali and Carstensen, 1999.)where b is the breadth of the particle. The mean (2= 0) is at = ~.88~2/0.031355220 pm (5.24) the standard deviation is 1/0.031355 = 32 pm (5.25)It wiltbe seen in the following that gas adsorption is employed extensi~elyin theme~surem~nt surface areas. Three of the conventionally accepted ty of (type I, TI, and 111) are shown in Fig. 5.3. n the isotherms, the adsorbed volume ofgas v is plotted as a function of~ r e s s ~P e the gas. In type I isotherms v ap oaches an e ~ u i l i b r i u ~ increas- r of withing pressure, whereas this is nottruefor either 11, norfor the isotherm tobe dis~ussed next.A~ount Type II Type I I I Type IVAbsorbec Pressure Four of the five conventional types of gas adsorption isotherms.
  • 80. If a solid, of volume Vs mL9is suspended ina liquid (of volume V mL) in which thesolid is virtually insoluble, and if this liquid before the addition contains Cmaterial per milliliter,then part of this will adsorb ontothe surface of the solid.equilibration, the supernatant is separated by centrifugation (not filtration, befilter material may also be adsorbed), and is assayed and now contains C mThe adsorbed amount m is obtained as m = V(C - C) (5.26)And the relation between C and m is often givenby the so-called Freun~lichequation. yM = qc/" (5.27)where q and n are constants. In logarithmic terms this becomes:The equation is empirical, and the value of m does not approach an asymptoticvalue.As ~entioned, surface areas (but not particle size distributions) may be o ~ t a i n efrom gas adsorption or, oc asionally, by adsorption of solutes from liquids in whichthe solid isinsoluble. Type isotherms are usually explained by means of Langmuirsequation (Langmuir9 1916, 1918). It is assumed that 1. The surface of the solid is smooth. 2. There is no interaction betweensites. 3. All the sites are identi~al. This equation willbededucedin the following for gas sorption, but thearguments hold equally well for solute adsorption ( e g , adso m a solution). This is shown schematically in Fig. 5.4. as molecules willadsorb onto the surface of the solid with a rate /3+, which isproportional to the activity a of the gas, and proportionally to the fraction (1 - 8),which is not already covered with gas. # += B k+(1 - 8)P (5.29)where k is the adsorption rate constant and P is the pressure tion ofgaswill occur with a rate /3-, whichis pro~ortional to theam0 red by the gas, that is, e+ = k-B (5.3~~where k- is the d e s o ~ t i o n constant. At equilibrium the two rates will equal one rateanother, so that k-8 = k+(l - 8)P hen using the terminology for the equilibrium constant K
  • 81. AdsorptionFraction Adsorbed Proporttonal to Fractton Not Covered / and t o Vapor Pressure Desorptfon Proportional t o Fraction Covered Schematic o f Langmuir adsorption. =kJk, (5.32) q. (5.32) b e c o ~ e s (5.33) he amount of gasor solute M , which isadsorbed for each square centimeter or eachgram of adsorbate would be ~ r o ~ ~ r t i oto a l surface fraction covered, so that n the (5.34)where 4 is a proportionality constant. Taking inverses gives (5.35)where a is the gas activity, given by a = PIP0 (5.36) In Figure 5.5 the asymptote is estimated at 0.151 and the surface area of thesolid can be estimated from this, if the cross-sectional area of the gas or solute is e~nements of the asymptote calculation can be made statistically , 1996b). The data in Fig. 5.5 are plotted according to Eq. (5.28) to 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 Activity fP/Po) Curve characteristic of a Langmuir isotherm.
  • 82. y - ~ . O O O O 0 . 7 6 9 2 3 ~R*2 - 1.000 + 14 12 8 “ 0 2 4 6 10 12 1 /a Data from Fig. 5.4 plotted according to Eq. (5.35). The Langmuir adsorption equationis based on theassumption that all surfacesare smooth, and that all the particle sites are equally energetic. t also assumes nonearest-neighbor interactions between sorbed molecules. Inverse plots, in general, ~ayznotbe obtaine~ p~ottiyz~ inve~ses by st- by thes ~ ~ a$ , and use of nonlinear programs are advocated (Carstensen, 1998). 3 ~ e sType I1 isotherms are usually explainedby the derivationof Br~nauer, Emmett, and 0). The assumption made in the derivation of the Lan~muir iso-therm is that only one (mo -) layer is allowed. Aside from the other assumptio ’the Langrnuir model, the T modelassumes that multiple layers may for5.7). It is assumed that the first layer may not be complete before the secondand the third layer may form before both of the underlayers are completesituation depicted isat agiven pressure p . The surface area of the solid is Aone part (so) is not covered. One part (sl) has one layer, one part (s2) has two layers,and so on. It is assumed in the following that yz layers may form. The moleculesadsorbed take up a volume of vo/m2of layer. The volume adsorbed in Fig. 5.7 is,therefore, so + SlVO + 2s2vo + ?s3vo + * * - ~ ~ ~ The rate with which adsor~tion occurs on the uncovered surface (to create thefirst layer of adsorption) is proportional to the uncovered area so. It is also propor-tional to the gas pressure p , and the rate consta~t denoted a l . The s u b s c ~ ‘‘I” is ~there denotes the first layer that is sorbed. The rate, hence, isalso. The d e s o ~ t i o n rateis ~roportional to sl, the area of the first-sorbed layer, and the rate constant iskl = bl e x ~ ( E / ~Here, E is the energy required for the adsorption of the first- ~). he following equation, therefore, holds at equilib~umfor the first alsop = blsl exp(--E/ (5.38)
  • 83. Schematic for the assumptional mechanism I n BET isotherms. The second layer is formed by molecules sorbing on an area of sl(m2) of thefirst layer, so that the sorption rate is a2s1ptand the desorption rate is now propor-tional to the surface area of the second layer (s2), and the rate constant isbz exp(-E2/RT). he equation for this situation is, in analogy with Eq. (5.39), a2slp = b2s2exp(-EL/RT) (5.39) T treatment now makes the following assumption: e value of EL is the enthalpy of liquefa~tion (condensation) of the gas on the surface. 5. This value applies to all s u b s e ~ layers. ~ ~e~An equation similar to Eq. (5.39) may now be written for the third layer. a 3 s s = b3s3 exp(--EL/RT) (5.40)and so on. The next approximation made is: 6. The ratio a j / b jis constant and simply denoted a/b for the first and higher layers (i.e., i # 1). The term Y = P(al/W exP(E/RT) (5.41) is now introduced into Eq. (5.40), which becomes s1 = YSO (5.42) (5.43)The expression for the ~lausius-~lapeyron ~ u a t i o ~this notation, is po = (a/b) e inexp(-EL/RT) so that x = PI~O (5.44)where po is the equilibrium pressure for a bulk mass of the adsorbent at the tem-perature T of the experiment.
  • 84. s 2 = xs1In general si = sp"xso thatOne now denotes c = y/xso that c = { [ ~ 1 / ~ 1 1 / ( exp{(E - ~ ~/~)} ~ ) / ~ ~(5.49) ~We may hence combine qs. (5.47) and (5. s1 = csox (5.50)The total area (see Fig. 5.7) is given by A = s o + s i + ~ ~ + ~ ~ ~ ~ +,~ = ~~ o c x ~ ~ sO ( ~+ ~ E+ x~ f ~ ~ ~ 2 ~ s o + = o l c i (5.51) i ig. 5.7 the total volume v, of adsorbant is v = vo{sl + 2s2 + 3s3 + . - ns,} = vo{cs,x -I-&sox 2 + 3CSoX 3 + - - ncsoxn} (5.52) = csovoXix, The aim is to find the value of the volume v,, of an adsorbed m o ~ o ~ a y e r ,because it may be converted to the number of molecules that are adsorbed and,knowing their effective cross section, the area of the solid A may be calculated.From the beginning and Eq. (5.50) it follows that qs. (5.51) and (5.53), Eq. (5.54) is obtained. (5.54)The geometric series Ex = x/( 1 - x). It is also apparent that Eix = x( aXxz/laJc]=x{a[x/(l - x)]ax} = x/(l - x)2 so that introducin~ these terms into the summatio~sin Eq, (5.54) will give v/v, = cx/[( 1 - x)( 1 - x + cx)] (5.55)Introducing Eq. (5.44) into Eq. (5.55) now gives (5.57)
  • 85. terso that plotting the parameter @/po)/{v(l - @/po)) versus @/po) should yield a ht line. If a nitrogen isotherm is carried out at liquid nitrogen tempe~ature, 0 = 1 atm, so that it is simply a matter of plottingp, rather t h a n p / p ~ . Equations (5.56) and (5.57) are the BET equations, and they account for type ption isotherms when c is not too large, and for large values of c, account for If plotting is carried out according to Eq. (5.57), then the slope/intercept ratiowill be ~lope/intercept= [c - l]/c (5.58) v is the amount of gas adsorbed converted to standard temperature and pres- rom the isotherm it is possible to calculate the value of v (and c) and from v , , sible to calculate the number of molecules N in a monolayer, For surface areamea~urements, nitrogen is the most frequently used gas (krypton is also employed). trogen has a projected adsorbed area of 16 A2 = 16 x ern2, so that the area the solid measured would be 16 x N. runauer et al. (1959, 1961) laternoted that nitrogen adsorption often gives low results, and quotes that for tober-rnorite the surface areas ranged from 20 to 90% of those obtained by water adsorp-tion, and that this latter was c o n f i ~ e d means of low-angle X-ray scattering. by wever, moisture isotherms are often associated with ater bonding to “internal” s (e.g., in the case of microcrystalline cellulose) ( arshall et al., 1974, 1975; ollenbeck et al., 1978); thus, for dry solids the value obtained with nitrogen is areliable measure of the actual surface area. The value of surface area measurements in pharmaceutics is its relation directly ilability (because often this increases with surface area of the drug), and to dissolution rates. Because this latter is associated with wetted surfaces,the dry surface area may not be all that meaningful at times.~ ~ a 5.1 p ~ ~ e iven the data in Table 5.4 for a 10-g sample of a solid, calculate the surface areaand the specific surface area. The gas used is nitrogen at liquid nitrogen temperature. hen the second column is plotted versus the first a typical type I1 isotherm results(Fig. 5.8). Nitrogen Adsorption onto a Solid Sample0 0 00.2 23.81 0.0110.4 32.52 0.0210.6 49.18 0.0310.8 98.77 0.041
  • 86. 1 P (atm) Data from Table 5.4. The data in Table 5.4 are now treated by way of Eq. (5.57) and shown in Fig.5.9. It is seen that the intercept is 5 xand the slope is 0.05, so that accord in^ toEq. (5.58) (C - l)/c = 5 X 10-4/0.05 = 0.01 (5.59)that is, c = 110.99 = 1.01 (5.60)The slope is (C - ~)/(v,c) O.Ol/(l.Olv,) = 0.05 (5.61)so that v, = 20 mL (STP) (5.62) Y . 5.0083e-4 + 4.9998e-2x - R 2 1,000 0.00 0.2 I 04 I 0.6 1 0.8 1 1.o p(atm) Data in. Table 5.4 treated by Eq. (5.47).
  • 87. his corresponds to 20 x lO""22.4 = 0.893 x lW3 mol = 5.36 x lo2 molecules at16 x 1@" cm2; hence, the surface area of the solid sample is A == f5.36 x 102]f16 x cm2 = 85 x IO4 cm2 = 85 m2 (5.63) s adso~tion measures both external (real surface) area and internal (pore If the surface area A of a sample is divided by its mass ee ~ rA,, ~results. At times the y o l u ~ e t ~spec$ic s u ~ ~ ar c ~ ~ , ic a ing the surface area of a sample by its real volume. . dsorption is associated with an enthalpy of adsorption, as stated in the foregoing. he isosteric heat of adsorption is obtained in the manner shown in The isosteric diffe~entialheat of adsorption q (Jacobs and q == ~ ~ 2 { ( ~ l n ~ ) / ~ ~ } ~ (5.64)where @ is the fraction covered. The equation, strictly speaking, was derived for a muir isotherm only, but may also be applied to the low-pressure region of a curve. If Eq. (5.64) is integrated it becomes (5.65)where /? is a constant. udipeddi et al. (1995)havemodified a thermal activity monitor to allowmeasurements of eat of adsorption directly. In so doing, they can also constructthe entire adsorption isotherm. P2 Isotherms at three different t e ~ p e r a t ~ r eTI < s, T2 < T3.
  • 88. There are several, rather severe ass~mptions the two models presented so far. inis that of active sites. It has tacitly been assumed that all sites are equally ener e manner in which this maybe investigated is through the isosteric heat of orption. If this is plotted versus temperature (i-e., if the data in Fig. 5.8 aretreated at different levels of coverage), it becomes apparent at times, that there isa very distinct difference between the heat at low coverage and that at higher cover-age)* In some isotherms this is direct1 demonstrable. ~ d s o r p t i o nisotherms of aron cadmium been bromide reported Olivier by (1960) and are typi-cally of type T (see V al. et (1961) explain that“the experimentalisothermshown in [Fig. 11 scribed quantitatively by a dual distributionof the adsorptive energie~.’~ The differencein heat of adsorption at different degrees of coverage, might alsobe attributable to anotherassumption; namely, the notion thatthe adsorbed layer isassociated with one energy of adsorption, and all the others are unaffected by thesolid, only by the heat of condensation, Guggenheim (1966), Ander oor (1968),haveproposed and f o r ~ u l a t e da model, the Gaccounts for an intermediate state between the first, and tightly bound layer andthe bulk layer, which is associated with higher pressures. This model will be discussed in further detail in Chapter 8, dealing with moist- ure is di (1996), has demonstrated this directly and states: It should be noted that in real systems the heat ofinteraction of the adsorbatewith the solid surface isnot constant as assumed by the BET or its analogous model. The heat of adsorption decreases as a function ofcoverage to a constant value (close to the heat of condensation of the adsorbate). A fair amount of adsorption work is performedby the heat of immersio~, first oyd (1942) and Jura and Harkins (1943).If a solid is “all solid,” then the considerations alluded to in the foregoing hold true,but most parti~ulate solids exhibit some degree of p~rticZe ~orosity. (distinguished from bed porosity, which will be covered in later chapters.) A liquidcondensed in a pore of radius r will have a lower vapor pressure P” than that of thebulk liquid, Po, and the relation is given by the Kelvin equation: ln[P”/Po]= e x p [ ( - 2 ~ ~ c o s ~ / ~ ~ r ) ] (5.66)where IV is the molar volume of the liquid, y is the interfacial tension between liquidand solid and 0 is the contact angle. It is noted that anexternal vapor pressure has toexceed P” before condensation of the “adsorbent” can begin, and this pressure isoften referred to as the b r ~ ~ ~ t h pressure. ~ rou~ If all the pores have the same size and are evacuated fully, and the solid thenexposed to vapor of a gas, below its critical temperature, then, as the pressure isincreased in the low-pressure range, a conventional isotherm (type I or 11, forinstance) will result as shown by 0 * in Fig. 5.1 la. Capillary condensation will
  • 89. Am t. Adsorbed Amt, Adsorbed C L 0 P* Pressure (4 Isotherm for vapor adsorption an ideal (A) and a real (B) porous body, (Data byfrom Defay and Prigogine, 1966.)commence at point A, and this will continue (at the same pressure-the break-through pressure) until the pores are filled, and after this the adsorption (AB) willonce again be conventional surface adsorption. hen a distribution of pore sizes occur, there will be a different breakthrough for each size pore, and the situation will be as shown in y hysteresis looping (C‘C) during the desorption, it is possible to obtain thepore slze distribution. For actual pore size distribution, however, mercury intrusionpermeametry is the method of choice, If there are large “pockets” in a solid of volumeV* (so-called inkwell porosity),and these are connected to the surface by smaller pores of radius r, then, the dis-tribution will fallaciously appear as V* larger at radius r, than it really is. It is seen, however, that gas permeametry will account for complete surfaces(i.e., the surfaces of the pores as well). It is often the e ~ t e surface area that is of ~ ~ ~ Zmost importance, and insuch cases thearea as given by gas adsorption is irrelevant. ore size distributions are usually elucidated byway of mercury intrusion etry, For this, the contact angle is about 135” (Le., above 90”) so that anexternal pressure is required to intrude the merc~ry into the pore. The Kelvin equa-tion [see Eq. (5,66)] still applies, so that the smaller the pore radius r, the higher thepressure needed to obtain intrusion (Fig. 5.12) The placing of a powder sample in a cuvette, of known volume, and filling thisto a given mark, allows calculation of the apparent volume of the sample, so that theapparent density p’ can be calculated. By now increasin~ pressure of the mercury thesystematically and measuring the volume 660utside” the cuvette, the difference inbetween two readings will give the volume u intruded at a given pressure P, (Fig. 5.13). ~ s u a l l yporosimeters have a maximum of 30,000 psi, but can go as high as , ence, the very smallest of pores (e.g,, less than 0.01 pm) will not beaccounted for. The total‘ porosity, I , may be obtained from knowled~e the true of~ensity and the apparent density, as E is given by p*
  • 90. r(3) r(3) Lower Pressure, Hg Intrusion o f Larger Pores Schematic representing pores and defects and the principle of mercury intrusion:r(4) < r(3) > r(2) > r(1). (5.67)The porosity, E,, measured at the highest pressure, will be smaller than E , and thedifference E - E, can be converted to a radius that will represent the "average" ofsmaller pores. This unmeasured porosity represents small pores as well as defects(which are obviously not pores). The problem of inkwell pores has already beentouched on. It is apparent that the data will represent themselves as volumes ai, ~ p r ~ s e n t - ring a certain pore radius rl, and one may, therefore, define the surface area A,, of theparticle using volume fractions wi, as Ap = cwinr; (~.68)U (Volume) U (Volume) Schematic of mercury porosimetry trace.
  • 91. is is obviously based on all the pores being cylinders (the so-called bundle of inders model) and more directly, and more correctly, A, maybe obtained bythe consideration that the work W exerted by intruding a volume of du at a pressureof P is = Pdu (5.69)It is,however, also equal to the surface dA,, times the contact angle, times the tension, so thatinterfa~ial (5.70)Equating Eqs. (5.69) and (5.70) gives ~~~ = Pdu/y[cos8] (5.71) rated this becomes Ap = ~ l / ~ ~ C O S 0 ~ } (5.72) PJSllflwhere the inte~ral represents the (cross-hatched) area under the curve in he total volume V of the sample is known from its weight and true density; there-fore, a surface-volume mean diameter, dzl, of the pores, may formally be calculatedas V/6A, = d:v (5.73)External surface areas may be obtained bygas permeametry. ~ccordingto oisseuille’s law for a liquid flowing under a pressure head of AP, a volume of Ywill pass through a capillary of radius r and length h in a time element t, if theviscosity of the liquid is q. his may be rearranged to ~ ~= 8qvf/{xu4} / h (5.75)where v is the velocity of flow through the capillary. In a powder bed with speci~csurface area A,/mL of solid, and porosity E , the so-called hydraulic radius r* is givenby r* = ( 1 / 2 A , ) [ ~ / (- E ) ] 1 (5.76)If liquid approaches the bed with a velocity v*, then the velocity in the pores is largerby a factor of E , that is, V* = V’/E (5.77) oducin~this e~pression and the expression for the hydraulic radius into the seuille e~uation now gives
  • 92. h P / h = 32v’q(l - E ) ~ (~.78) This is the Car ny equation (Carman, 1937, 193whichis employed in ry (e.g., the Fisher subsieve sizer).of powder that corresponds to 1 cm3 o solid (obtained by taki fmaterial weighing l / p where p is the particle density of the solicross-section of 1 cm2 is used, so that the bed volume is the heiallows calculation of E . ir (or another gasof known viscosthrough the bed at measured vel and the pressure diffeand efflux streams are measured. e, all quantities excepthis latter can be calculated. ecause gas adsorption measures both externaZ and * s of a solid, and because the latter are not necessarilyavailable for dissolution, permeametry is often a better meansareas that have meaning in dissolution testing.If a particle size distrib~tion~(a) a function of size (a) is known, then it is possible asto calculate the surface area, ass~ming that the particles are spherical and smooth.Such a surface area is denoted the geo~etric s~rface (not to be confused with area thegeometric mean). For a weight mean diameter-type calculation the area will be A, = ~ ~ w ~ a 2 (5.79)and the weight mean diameter will be Such calculations are often carried out forsieve analysis. If the external surfacearea A* is known, this will larger be be may attribute^on an overall scale/surface gosi sit^, = A*/A, It is, as will be discussedlater, possible to assign v o Z ~ ~ e t r i c ~ ~ h a ~a,,~to c t o r s e ~particulate solids, and this converts a “size” a (however, that isdiameter or projection diameter) to the particle volume v.A com~only used shape factor converts the volume sf a ~articZe,v, to its size,volumetric mean, a,”(see Table 5.1). 3 v = a,av (5.82)Similarly a shape factor may be defined that converts the surface area s o a ~ a ~ t i c Z e fto its surface mean a,. s = a,a 2 (5.83)
  • 93. If a particle p~pulation fairly monodisperse, such as for a narrow mesh cut, isthen 1 g of the cut will contain N particles, and if the particle density is p, then forthe 1 g sample Nu = N ~ , = 1 a ~ (5 34)The surface area of the particle is s = asa2, so the specific surface area A,, is this ure divided by the mass of the particle a,,& A, = a,a 2 /a,a,, 3 (5.85)It is a s s u ~ e d that the narrow mesh cut is lognormally distrib~ted, that by intro- so the ap~ropr~ate Hatch-~hoate relations [see Eqs. (5.17) and (5.18)] fromTable 1 into Eq. (5.77) gives (5.86) ecause N , p, and ag are measurable, a, can be calculated. A similar approach willshow that (5.87)where now all quantities except a, are known so that this latter can be calculated.A. prerequisite for this is that the distribution be narrow, so thatbe relied on, Another approach is tedious accounting for N by countingmethods. y formally be calculated from microscopy, ifthe total number may be obtained experimentally, The total number of parti- represents a weight, which may be converted to a volume E . ecause the count consists of, for each interval, listing the appropriate ni particles of zes l e ~ ~ai h breadth bj.The sum ~ a j a may be calculated, and assuming the t and ~ b ~ (5.88)from which a, may be calculated. However, the ass~mption made that the dimen- ission that is hidden in microscopy, the height h is the same as the breadth. a, may beo b t a i ~ indire~tly} more l o ~ i ~ a l l y } f r o m e~ but d~ssolution data, this will be covered and A shape factor (simply denoted the s ~ ~ ~ e ~ a c . o roften calculated by the It)is ,surface of the sample A and its true volume V by the formula: (5.89) or a particle the relation between a,, a,,and I is obviously given by: (5.90)
  • 94. This, however, is not correct for values of the shape factor obtained by dividing the #area of a s u ~ ~ with the two-thirds power of its volume, because [Xniu:]2/3 ZeThe introduction of fractal geometry as a mathematical tool is attributableto~ a n d e l b r o t(1983). There are many applications of the concept, and the intenthere is,first to describe what it is, and then to showhow it can beapplied topha~aceutical problems. ~ i t h o udelving into the intricacies of this approach, the general philosophy of t f, for instance, the length of a contour, such as depicted in ured, then the length obtained would depend on the “lengmeasuring stick,” the scaling length, h; if, for instance, this were reduced to q thenthe periphery measured would be longer than if one had used a larger-scaling length.This is the origin of the practical application of fractal dimensions, because ~nterestin it started with the work of Richardson (1961), who had the task of measurin~ thelength of the coastline of In general it can be shown that the length I, of a perimeter depends on. thescaling length, q, by the relation: log[L] = ”(1 - I))logl[q] + (5.91) s a constant and 1) is referred to as the fractal dimension [and as demon- Eq. (5.91) emergesas thenegative of the slope of a logarithmic L versus g- plots are known as Richardson plots, coast line plots, or walking yard-stick plots, because of their origin in geographic and topological science. An intuitive understanding of I) is de~onstrated Fig. 5.15. For the straight- inline on top of the figure, the dimension is 1. For the wiggly lineon the bottom of thefigure, the space is to a great extent filled up by the line [it is not possible to entirelyfill up space witha line], and one could visualize this as having a dimension of 2. Thetopologic dimension is still 1, but the Euclidean dimension is 2. For the line in themiddle the fractal dimension could be visualized as being between 1 and 2. It is now possible to define the surface irregularity of a particle by the fractaldimension D [definedin Eq. (5.91)]. To do so it wouldbenecessary, by imageanalysis, to obtain a cross-sectional representation of the particle and, from this, ~ c ~ e ~ a t i c principle of ~ e a s u r e ~ e oft fractal di~ensions~ showing n
  • 95. D=1D=ca.l;Z Fractal dimensions of a contour. The curves and concepts are approximate andare shown for definition purposes only. The point where a curve becomes a plane-filling curve(e.g., a Peano curve) is complex and beyond this writing.obtain the fractal dimension. This indeed has been done, in the pharmaceuticalliterature, and Fig. 5.16 is taken from the work by Ramadan and The slope is H=S-D ( 5 -92)the fractal i ~ c r e ~ ~ ~ is ,a measure of the surface roughness. The curve is not which tlinear (although the authors have treated it as such). 0th the photomicrographs h spheres with “pimples”) published by the authors and the upper curves in ig. 6 imply that there are two surface pop~lations (perimet~r lengths La and d the same group of in~estigators(Thibert et al., S988), indeed, reportedlater such a behavior in the fractal analysis of lactose granules; as exemplified in Again, there are two distinct line segments, indicating two types of surfacem o ~ h o l o g yWith lactose, assumedly, fine structure of surface pores has a fractal .~imension different from that of the nonporous part of the surface. 0 1 2 3 4 log steplength L as a function 1 (step length) for natural microspheres. The lower curve shows allthe data plotted in simple linear regression. The points in the upper graph are those in thelower graph + 0.5. This is done for graphic clarity. In the upper graph, the points are shownas bimodal, indicating that there are two self-similarity populations. (Data from Ramadanand Tawashi, 1990.)
  • 96. 4.88 0 1 2 3 4 log [Steplength] The fractal character of lactose granulations. (Data from Thibert et al., 1988.) rojection of cross-sectional images can be misleading in that the observationcan be a function of the orientation of the solid particle, This method is better, themore sp~erical particle is. It is time-consuming, and sampling (as in any other thetype of microscopy) represents a problem. ible to probe surfaces in a more convenient manner^ that is, by gas re the property values are averages over the entire surface; hence, of a problem (although sample sizesare still smallin such work, theyare not simply one particle). ractal approaches to surface sampling by gas analysis are based on the prin- tlined inFig. 5.18. Ifa small adsorbent molecule is employed (seeFig. 5.18a),then more of the roughness will manifest itself than if a larger molecule (see ere, again, the measured surface should be the larger the smaller t or adsorption, the cross-sectional area of the molecule is a function its dia-meter h, squared. If the molecule has a circular cross-section ofarea b andby square arrange~ent, then the sorbed area 6 is simp~y per mo1ecule. h2 /3= h2 (~,~3)or h = (8)"2 (594) mploying an approach similar to Eq. (5.91) gives Coverage of an irregular surface by different-sized adsorbant molecules.
  • 97. + ~n[n,] = -D l n [ ( ~ > ” ~ Q = - ( ~ / 2 )ln(P) + ] (5.95) where r is the number of molecules in a monolayer. The more general case, where z, the molecular packing is other than square, packing can be treated similarly, now a =Jh 2 (5.96) Figure 5.19 is constructed from data published by Avnir et al. (1983). It is seen that the slope is --L)/2 = -1.0581 so that L) = 2.16 n the foregoing example, the molecules are fairly spherical, and if an adsor- flat on a surface, then the fractal equation becomes(5.99) n= log[v] ( 4 + 1) where v is the molar volume of the sorbed molecule. ~ p e r i ~ e n tsuch as those described, are still rather cumbersome, and it is s, more convenient (although still not practical from a quality control point of view),to do nitrogen adsorption on various mesh fractions of the solid. Figure 5.20 shows the BET (nitrogen) surface area of different-sized fractions of Aerosil (colloidal silica). When this approach is used, the ap~licable equation is: log[A] == (D- 3) log[d] + constant (5.100) where A is the surface area obtained by gas (nitrogen) adsorption anddis the particle diameter. Fini et al. (1996 a,b,c) reported that the fractal dimensions often depend on the e of crystalli~ation.These authors (Fini et al., 199’7 a,b) studied the physical properties of salts of ursodeosycholic acid, and reported on the fractal dimensions of the surface (D,) the dissolution reactive dimension (D,) reported by Farin and and as Avnir (1987). . Fractal plot of carbo? black: Amount o f adsorba (mmolis) in monolayer as a tion of cross-sectional area (A2) o f adsorbing molecule. ata from et ala, 1983.)
  • 98. 2.6 1 - y = 3.3996 1.0722X R”2 0.995 2.4 2.0 1.8 1.6 BET surface area, A (m2/g) as a function of particle diameter, d (nm) of variousAerosils. (Data from Avnir et al., 1983.)~lectronic counters and laser counters are the methods of choice in many present-day situations. For instance, Zhang and Johnson (1997)used a Coulter Counter (Coulter Electronics, Hialeah, FL) to measure the particle size distribu-tion of an experimental drug (CP 118 954, Pfizer). As electrolyte they use a 2% w ~ vsolution of maleic acid, adjusted to pH of 5.4, containing 0.005% of Tween 80, afterwhich they saturate it with drug. Their investigation dealt with a lognormal particle size distribution, and theyemployed jet-milled and bantam-milled material and “spiked” it with larger particlesto make the distribution log-normal by weight. A = surface area of a sample A” = external surface area of porous solid (in a bed) A, = X Z W , ~ = geometric surface area ~ A, = surface area of a sample divided by its real volume A, = specific surface area Ap = surface area of the particle as A, = volumetric specific surface area (per cm3 of solid) a = (a) activity (= PIPo); (b) length of a particle; (c)“diameter” (or size); (d) rate constant of adsorption ( a, = X:na/Cn, arithmetic mean diameter aavg= average“diameter” b = (a) rate constant of desorption (BET equation); (b) br bi = pre~xponential factor for adsorption of the ith layer C” = concentration before adsorption C = concentration after adsorption D = fractal dimension d = general size term for diameters
  • 99. dg = geometric mean diameter = exp( xrzi ln[di]/Gni]d: = ~ / = surface volume mean pore size 6 ~ ~dg = geometric mean diameter = exp f Eni ln[di]/dwm = ~ ~ ~ = weight mean diameter a / x ~ ~dh[f(a)] = number fraction of particles with diameters a, the logarithms of which are between ln[a] and ln[a]E = enthalpy of adsorption of the first layer ( quation)E L = enthalpy of condensation (BET equationf = “function of”f(a> = (a) { T / [ O ( ~ X ) ” ~ } exp{(a - a , , , ~ ) ~ / 2 a ~ } normal frequency function; = (b) generally particle size distribution as a function of size (a) = slope of fractal dimension plot = (a) “length” of adsorbate molecule; (b) length measuring stick (in fractal dimensions); (c) length of capillary; (d) height of particle i = running index * = cumulative frequency in Weibull distribution = equilibrium constant in Langmuir equation k , = rate constant of adsorption k- = rate constant of desorption I; = length of perimeter (in fractal dimensions) c) = standard deviation for lognormal distribution r = amount of gas adsorbed on a solid w1 = (a) u - (y/2/2)ln[rr] = integral substituent dummy variable; (b) adsorbed amount (in Freundlich adsorption) N = number of particles in a population n = Freundlich exponent = gas pressure * - (a) pressure that is lower than the equilibrium pressure of a gas at a - given temperature; (b) breakthrough pressure in intrusion porosimetry. Po = equilibrium vapor pressure of a gas at a given temperature Pr(a < a”) = probability of a size a being smaller than a* + = (a) Weibull distribution = In{- ln[j]} = - ln[a] C; (b) constant (fractal dimension equation) q = (a) ~ ~ 2 { ( ~ l n ~ ) / ~ heat; ~ scaling factor; (c) factor in = isosteric ~ } (b) Langmuir equation; (d) Freundlich prefactor R = gas constant r = (a) radius of capillary; (b) pore radius r* = hydraulic radius rl = pore radius of the ith pore s = surface area of a par ’ so = surface not covered s1 = surface covered with one adsorbent layer ( s, = surface covered with i layers of adsorbent T = absolute temperature t = time f, = (a) molecular volume (in elvin equation); (b) liquid volume Y” = inkwell pore volume ‘c/T/ = work (in mercury intrusion)
  • 100. u = (a) substituent in integral = (ln[a] - l n [ ~ z ~ ] ) / lna]; (b) volume intruded 2~/~ at a mercury pressure of P v = (a) molar volume of the sorbed molecule (in fractal dimension); (b) liquid velocity; (c) volume of a particle; (d) volume of adsorbent gas at a given v = velocity of flow v* = approach velocity of a liquid to a bed vo = volume of molecules per square meter of layer v = volume of a monomolecular layer of adsorbent gas ( , W = mass of a sample w, = weightof fr on of particles withsize a x = p(a/b)exp( Y =p(a/~) exP(-~~/~~) Z = standard normal deviate a = j h 2 = scaling factor when adsorbate is not of square con~guration a, = a2/s = surface shape factor of a particle a, = volumetric shape factor of a particle = a:/v #3 = square of scaling factor in three dimensions (h2 per molecule) B+ = rate with which gas molecules will adsorb onto a surface 6- = rate with which gas molecules will desorb from a surface r = . A / ( v ~ /(general) shape factor = ~) = ~x,/[a:/~](general) shape factor for a particle = y = interfacial tension between adsorbate and solid 8 = the differential notation E = particle por~sity E, = total porosity measured at the highest intrusion pressure S2 = A*/A, = rugocity 8 = (a) contact angle; (b) fractional coverage of a surface with adsorbed gas p = particle apparent density p* true particle density q = viscosity a = standard deviation of a population ln(a) = standard deviation for lognormal distributionAnderson RB (1946). J Am Chem Soc 68:686. , Pfeifer P (1983). J Phys Chem 97:3566. ett PH, Teller E (1938). J Am Ghem SOC 60:309.Brunauer S, Emmett PH, Teller E (1940). J Am Chem Soc 62: 1723.Carman PC (1937). Trans Inst Chem Eng Lond 15: 150.Carman PC (1938). J Soc Chern Ind 57:225. Modeling andData Treatmentinthe Pha~aceutic~l Sciences. ing, Lancaster, PA, pp 63-73.Carstensen JT (1996bj. Modeling andData Treatment in the Pharmaceutical Sciences. Technomic publish in^, Lancaster, PA, p 39.Dali MV, Carstensen JT (1999). Drug Dev Ind Pharm 25:347.DallaValle JM (1943). Microrneritics. Pitman Publishing, New York, p 28.
  • 101. deBoor JH (1968). TheDynamicalCharacterofAdsorption, 2nd ed. Clarendon Press, Oxford. 1987). J Phys Chem 91:5517. olgado MA, Fernandez-Hervas MJ, Rabasco AM (1996a). Eur J PharmFini A, Fazio 6 , Holgado MA, Fer~andez-Hervas MJ, Rabasco AM (1996b). J Pharm Sci 8597 1.Fini A, Fazio G, Holgado MA, Fernandez-Hervas MJ, Rabasco AM (1996~).Eur J Pharm Sci 4:231.Fini A, Fernandez-Hervas MJ, Holgado MA (1997a). J Pharm Sci 86:1303.Fini A, Fazio 6 , Fernandez-Hervas MJ, Holgado MA (1997b). Int J Pharm 171:45.~ ~ g g e n h e iEA (1966). Application of Statistical ~ a c h a n i c s Clarendon Press, Oxford. m .Harkins WD, Boyd C E (1942). J Am Chem SOC 64:1195,Herdan C (1960). Small Particle Statistics. Butterworths, London, p 45.Hollenbeck RC, Peck GE, Kildsig DO (1978). J Pharm Sci 67:1599.Jura G Harkins WD (1943). J Chem Phys 11:561. J (1927). Royal Acad Sci Vienna Proc Class I 136:271. ir I(1916). J Am Chem SOC 38:2221.Langmuir I (1918). J Am Chem SOC 38:2221. (1983). The Fractal Geometry of Nature. WH Freeman & Co, New York.Marshall I(, Sixsmith D, Stanley-Wood N C (1972). J Pharm Pharmacol 24:138.Marshall I , Sixsmith D, Stanley-Wood N C (1974/1975). Drug Dev Ind Pharm 1:51. (Olivier JP (1960). PhD dissertation, Rensselaer Polytechnic Institute.Pudipedi M (1996). PhD dissertation, University of Wisconsin, p 184.Pudipeddi M, Sokoloski TD, Duddu SP, Carstensen JT (1995). J Pharm Sci 85:381.Richardson LF (1961). General Systems Yearbook 6:139.Ramadan MA, Tawashi R (1990). J Pharm Sci 79:929.Ross S, Olivier JP, Hinchen JJ (1961). In: Copeland LE,Beebe RA, Graham DP,Zettlemoyer AC, Zisman WA, eds. Solid Surfaces. American Chemical Society, Washington, DC, p 319.Thibert R, Akbarieh M, Tawashi R (1988). J Pharm Sci 77:724.Zhang Y, Johnson KC (1997). Int J Pharm 154: 179.~~~b~~~~ h ~ 5 ~ t ~ ~ C lot the data in Table 5.3 and ascertain whether they are normal or lognormal.Calculate the appropriate mean and the standard deviation.
  • 102. 6.1. Crystallization 90 6.2. etastable Zones and Nucleation 90 6.3. Nucleation and Critical Nucleus Size 91 6.4. The ~ q u i l i ~ r i u m 92 6.5. omogeneous Nucleation 93 6.6. 94 6.7. Yield and Metasta~le Zones 94 6.8. Crystal Growth 95 6.9. istributions After ~omogenous Nucleation 966.10. Nucleation 1006.11. Temperature and Solubility rofiles During Thermal Recrystallization 101 6.12. Particle Size ~istribution After Thermal ~ecrystallization 102 6.13. et~rogeneous Nucleation 104 6.14. 104 eferences 105The last step in drug substance manufacture is puri~cation,and this, most often,consists of r~crystallization. The conditions which this i s carried out is of underimportance in pha~aceutics, because the shape of the particle may affect machin-ability (e.g., needle-shaped particles may logjam when they flow through a hopper),and themorphology and the shape may affectdissolution. Therefore, a discussion ofsome of the fundamental factors affecting crystallization is presented.
  • 103. rug manufacture (synthesis), the drug isusually not pure when the overall is complete. For instance, succinimide has a solubility of 1 g/20 g of ethanol and 1 g/4 g at 60°C. the material is produced, it may contain, for example, 5% of an impurity solubility of 2 g/20 g of ethanol at 25°C. If 10 g of crude material (contain- f pure chemical and 0.5 g of impurity) are dissolved in 40 nd then cooled to 25"C, 8.5 g of pure material will then precipitate out (1 g in solution) and the 0.5 g of impurity will stay in solution. ecrystallizations may also be carried out by dissolving the substance in onesolvent, and adding another in which it is insoluble; thereby, precipitatin~out thepure chemical and (providing the solubilities allows it) keeping the impurities insolution. In this case one speaks of r e ~ r e ~ ~ i t ~ ~ ~ o ~ . stwald (1899) formulated arule of stages: when a system firststarts c~stallizing, itwill initially create the crystal structure that forms the smallest loss of free energy,and these crystals will later transform, stagewise, to the most (or a more) stablecrystal structure. This will be dealt with futher in Chapter 8. It is a common misunderstan~ingthat precipitationandrecrystallizatio~occurs from saturated solutions, whereas they actually occur from s ~ ~ e r s ~ ~ 1s is illustrated in Fig. 6.1. If 300 mg of material is added to 1 g of solventand heated to 70°C (point B), then it will go into solution.will not precipitate until point A (57°C). The width of this zmetastable zone. yvlt (1971)has shown that thewidth of the zone maybe a functionof to howhigh a temperaturethe solution has been heated and for how long. That is, if, in theforegoing example, the heating was carried out at 90"C, then the zone would be u10 20 30 40 50 60 70 80 Te~peratur~O C If, for instance, a solution at 300 rng/g is heated to 70°C (point past B), and thencooled, precip~tationwill not occur {in a reasonable length of time) until 57°C (A) is reached.
  • 104. wider, and if it were kept there for 1 h, rather than for 5 min, it would then also bewider. It is speculated that complete randomness of the solution is not ascertaineduntil it has been kept at a temperature well above the solution temperature angiven length of time, and that if complete rando~nessis not at hand, then thenucleation will occur more readily.In this chapter, the symbol r denotes dimension (radius, diagonal). For a cubicalnucleus of size r AG = -pr3 + 6ar2 (6.1)where p is chemical potential and cr interfacial tension. AG is maximum when d ~ G / d = -p?r; r + 12ar, =0 (6.2)that is, when r, = 4a/pThis then is the critical nucleus size, because beyond size, the growth of a n ~ c l e ~ s this ied by a negative AG (Fig 6.2). argument that follows is,strictly speaking, incorrect ( lassic form in ~arstensen (1980). alk about solubility, then AG from solid to solution must be zero. p is potential per cubic centimerer (cm?), so to determine it per mole, it mustbedivided by the density p (to o in the chemical potential per gram) and bemultiplied by the molecular weight ) to obtain the chemical potential per mole.or 10 0(3 -10 -20 0 2 4 6 8 1 0 1 2 Size (d) Chemical potential and nucleus size.
  • 105. ut the left-hand sideof the equation is AGO+ RTln[C,], where AGO refers tostandard state, so If a system is taken from point A (with dimension r l ) to pointsion rz), then (6.8) ut this means that his is known as the Ostwald-Freundlich equation (Ostwald, 1898). The equation, seemingly, predicts that solubility is inversely proportional tothe size of a particle, but there are problems withthe argument, in that AG cannot be quation (6.7) pre icts that the equilibrium state of a solid is an infinitely largecrystal, or more correctly, it predicts that if a multiparticulate system is placed ina liquid, then the crystals will grow (Ostwald ripening),until there is only one crystalleft. The size of that crystal will be such that the concentration of the supernatantwill be given by the point on curve ABC which will give mass balance. This is not reasonable, but it is difficult to disprove, because the time it wouldtake, were it true, would be so long that it could not be carried out. In solubility work, it is conventional to require 72 h for equilibrium to beattained. One might invoke criteria such as that if one tested the con cent ratio^every 24 h, solubility had been reached when there was no “detectable” increase ut this is no guarantee that the concentrations over a 6-month periodwould not increase; or would decrease, if the crystals grew and the equation werecorrect. It is possible, indeed very possible, that the basic equation [Eq. (6.2)] is incom-plete. For instance there is no accounting for defect f o ~ a t i o nand the interfacial ,energy is simply assumed to be proportional to some “size.” It is more logical tothink that the real e uation would be one leading to a curve as shown in Fig. 6.3. 3 may be ap~roximated a by (6.10)where (b is a constant and where (6.1 1)If, when r is large, (B/r) > ((b/r2), then the equation reduces to > WC,1 = B(l/r) (6.12)so (6.13)
  • 106. -&V20 0 10 Length ig. odified modelchemical of potential as a function of size. I The work required to create a nucleus of size d, is given by W = 6d2g (6.14) It is seen from this that if a solution is supersaturated to a degree of S == C/C2(> l), then (6.15) or (6.16) Inserting Eq. (6.14) into (6.15) gives (6.17) ere, S is the supersaturation ratio, and it shows when this is unity, 1n[q is 0 (i.e., finite work is required to form a nucleus). The higher the supersaturation ratio is, the more easily a nucleus will form. ullin (1961) reports the following times (Table 6.1) for a nucleus to sponta- neously form in supercooled water vapor. Time Required for Nucleation to Take Place Supersaturation ratio Time Infinite yr lo3 yr 0.1 s s 10-1~ Source: Mullin (1961).
  • 107. mpurities in the intermediate drug substance are usually removed by recrystalliza- ion. It is assumedin calculations that if the impurity is hi her than its solubility limitat the conditions of precipitation of the drug substance?then it is “removed,” in that owever inco~oration (doping) of solids by introducing guest ~olecules into ssible, and this often happens. igure 6.4 is an exam e an asymptotic limit to the uptake (in this case, he inclusion also affects the ability of the crystal to contain water (up to a se (as in the pr~viously cited case) the inclusion leads to latticevaca~cies? “space” created presumably allows “room” for the water molecules. theChow et al. ( 1 ~ 8 5 )st ed the effect of additives in the mother liquor on theoutcomes of crystalliz n. One effect is on the yield, and this maybe an impor-tant consideration? becauseonly relative small amounts of additive ( i ~ ~ u r i t y ) y affect a great number of properties of the crystallization and the crystals t firstsight it might be speculated that~cetoxyacetanilide increases thesolu~ility acetaminophen, but even though this is true, the extent of s~lubility ofincrease (about 6% at the highest co~centration additive) does not explain the ofdramatic decrease in yield, and the explanation lies in an expansion of the meta- homogeneous nucleation there is often a la time before c~stallization e foregoing example, Chow reported that thoat seeding and the pre- toxyace~anilide system did not start crystallizing in 2 h. the example of homogeneous nucleation is suspensions of amor~hous fruse- mide). The amorphous stateis much more energetic (more soluble) than states, and in a suspensio~ amorpho of ort the following c~stallization profile 5 4
  • 108. The effect of ~-acetoxyacetanili~e on yield of a c e t a ~ i n o p ~ e n 30°C. the at contained 9 g of a c e t a ~ i n o ~ h e n 390 cm3 of water.solution ori~inally per he decrease in concentration ( C ) follows a curve of the type c - cS (eo CJe-4 = - (6.18) ut it should be noted that q is neither a growth nor a nucleation rate ons st ant. llization, also used in early dissolution work, was first tney (1897a,b~. They also assumed that dissolution wasthe reverse of crystallization, an at the crystallization and dissolution rates werethe same. Reference is m a film at the crystal surface exists. ordin ding to Noyes and / d t = k&4(C1 - 6 2 )where 6 2 is solubility. erthoud and Valeton have shown that a reaction term mustbe i~cluded: v)2 0$ 0 50 100 150 Time (hours) t a C ~ ~ s t a l ~ i ~ aofi o n 16% arnorphous furosemide (frusernide) suspension. (from Doherty and York, 1987.)
  • 109. where C3 > Cs. In ~ ~= KA(C1 -~ / C3) t (6.21)where, as shall be discussed shortly (1Im = (1lkd) + (1lkP) (6.22)where kpis a "reaction on st ant^ and kd is the diffusion constant. The reason for theexpression in Eq. (6.22) is that if the film were stagnant, then ( kd = I)/h (6.23) it stands to reason that the film thickness would have to depend on the speed, and Marc (1908), indeed, found h to be zero at high velocities. Thiswould implyan infinite growth rate at high liquid velocity. A model overcoming thisproblem was proposed by Berthoud (1912) and Valeton (1924), who suggested thatthere were two processes, one a dislodging of molecules from the surface (the so-called reaction rate k,, referred to in the foregoing), and the second being the diffu-sion as discussed in the foregoing. igbie (1935) and Dankwerts (1951) suggesteda surface renewal theory where,simply kd = (Df)/2 (6.24)f is, here, the fraction rate of surface renewal. orph is allowed to dissolve, as shown in a previous higher solubilitynumber than a more stable form. Inthe metastable form would approach a concentration of C = 200 mg/g if no con- 1 wever, at a given point in time (10 time units), precipitation ens that in a time period (20-40 time units in the figure), thecon~entration stays constant. At a point ,all the metastable material in steady-statedis~ontinues and concentration decreases toward the saturation con cent ratio^ C2 theof the more stable poly~orph.CrystalSurface "". 1" h u- chernatic of concentration profile at a crystal-su~ersaturate~ solution interface.
  • 110. The steady-state phenomenon happens because the rate at which a suspendedmetastable drug substance dissolves (-kl(Cl - C)) equals the rate with which thestable form precipitates (-kZ(C - C2). Csteady is given by c"kI(c1 - csteady)] = c-kZ(Csteady - CZ)] (6.2~)that is, Csteady = (klC1 - k2C2>/(kl - k 2 ) (6.26) e is actually never quite accomplished, but the fact remains that where the concentration is fairly constant. Under such conditions, the nucleation and growth rates of the crystals areconstant (Fig. 6.8) This situat~on an o~ersimplifie~ for general crystallizations, but serves is modelwell as an introduction into how distributions are arrived at The question that posesitself first is, what is the rate of nucleation? We will see shortly that, most often, it isdependent on the degree ofsupersaturation, but in the foregoing situation the degreeof supersaturation (between A and wouldbe constant. Wewill assume, for them o ~ e n tthat a crystallization event the range AB takes T time units, and that pz ,nuclei are formed each time unit, that is, dpz/dt = 4 (6.~7)Growth rates are also a function of supersatura~ion, in AB this is constant and it butwill also be assumed that the growth of the size, a, follows Mac = da/d~ k (6.28) These are obviously severe assumptions for a general case, but they introduceus to the manner in which crystallization events are translated into particle sizedistributi~ns the ensuing product. of Assume, first, that the T time units are divided into intervals, each of 1 timeunit. Then there will be pz nuclei that will have grown for T time units, n that havegrown for T - 1 time units and so on. The crystal that was born in the first time interval will have the size: a. = k? (~.2~) 0 20 40 60 80 100 120 Time Conce~tration-precipitationtime curve. (Data from Dofierty and York, 1987.)
  • 111. The one born in the second time interval will have the size a1 = k(T - 1)3 (6.30)so that the total weight of the crystals will be + = pkfl(T3 ( T - 1)3, +- .(Z3 + 13)} (6.3 1) ecause the time unit is one, the sum of the series equals the integral: T T3dT = pknT4/4 (6.32)If one considers the amount of material that has sources from times 0 to t, then this + = pkn(T3 ( T - I ) ~ + - . * ( T- t13) , (6.33)This material will have sizes at or above a* = k(T - t ) (6.34) q. (6.33) is equal to T-t T3dT = pkn(T - t)4/4 (6.35)so that the weight fractionf(> a") of material with a particle size larger than a* is f ( > a") = ( T - t ) 4 / T 4 (6.36)or by taking logarithms ln[f( > a*)] = 4 ln[(T - t)4/T4] (6.37) ut (T- t)4/T4 = a*/aw,, (6.38)so that ln[f(> a*>]= 4 ln[a*/amax] (6.39) weight frequency of particles larger than a* versus the straight"1ine with slope 4. tsuda (1998), however, unlike what their article og. If the data points are taken off their Fig. 1 carried out), the distribution looks as shown in Table 6.2 and ated in Table 6.2 to allow plotting according to either a normal Asor a lognormal distr~bution. shown in 6.9 the distribution, if simply judgedvisually, could be either, but Fig. 6.10 (plo from data in Table 6.2) shows it to be he geometric mean is seen to be given by 373/ 1.SO5 = 4.97 (6.40)that is,
  • 112. Example of Lognormal CalculationSize Undersized(4 ("/I 2-vaf ue Mil 20 0.2 -2.880 2.996 30 1.1 -2.190 3.401 40 3 - 1.880 3.689 50 5.5 - 1.600 3.912 60 9 - 1.340 4.094 70 15 - 1.045 4.248 80 20 -0.840 4.382 90 25 -0.670 4.500125 45 -0.01 3 4.828150 55 0.013 5.01 1200 70 0.525 5.298220 80 0.845 5.394Soztrce: Otsuka and Natsuda (1998). 100 8044s;r3 20$ 0 - 0 100 200 300 Particle Size ~ ~ i ~ r o n ) Graph of data from Otsuka and Natsuda (1996). Y .. - 7.4373 + - 1 . 5 1 0 5 ~R"2 0.994 yi 1 0N -1 -2 -3 2 4 5 6 lnldl The data in Table 6.2 treated as a lognormal d i s t ~ b ~ t i o n .
  • 113. agM, 144 pm = (6.41) This corresponds fairly well with the "mean" arrived at by Otsuka and atsuda (1996). The standard deviation is the inverse of the slope, ln[cr,] = 1/1.51 = 0.6625 (6.42) nowledge ofthe geometric meandiameter allows calculation with theChoate relation: ln[d,,] = ln[ag,] - 0.5 ln[a,] (6.43)from which It is obvious that this figure is considerably higher than the value that theauthors found from BET surface measurements; therefore, it is legitimate to statethat the two methods give different results. It is, however, not legitimate simply tocompare agw with a,,, because (as demonstrated) the higher-moment diameter is, bydefinition, larger for a multiparticulate. he chemical energy AG associated with 1 mol transferring from a supersaturatedsolution of concentration C to a saturated solution (i.e., a type of situation that willoccur in thermal recrystallization) is given by AG = -RTln{C/S) = -RT ln[q] (6.45)where R is the gas constant and T is absolute temperature. C/S is denoted thesupersaturation ratio and is, in the following, symbolized with the letter q: 4 = C/S (6.46) ecause S represents the concentration in a solution in equilibrium with the solidstate, Eq. (6.45) represents the energy of transfer of one mole from solution to thecrystalline state. For crystallization to occur, there must be a nucleus from which thec ~ s t a l ~grows. The rate of cr~stallization therefore, in some form or manner, s) is,associated with the rate of nucleation. Denoting time with the symbol (9 and thenumber of nuclei at any given time by N , the rate may be expressed as dN/~(9. q. (6.2) represents the energy of activation for nucleation. Thenucleation rate, J , is given by ost often this is associated with an exponent l/n, so that the expression, if C > S, >becomes J = a(C - S)/" (6.48)
  • 114. Frequently the value of y2 is 2 (i.e., 1/ n = 1/2). In the writing to follow the aim is todeduce what type of particle size distribution would result from thermalrecrystallization.In thermal recrystallization, excess drug is dissolved in solvent at a higher tem~era-ture at which its solubility is more than its ambient solubility, and the temp~rature isthen allowed to drop by cooling, either natural or induced. The question is:would the solubilities of the compound be as a function of cooling time? Heat transfer usually results in temperatures following a s i ~ a - m i n ufunction, sthat is, T = To[l - exp(-k@)] (6.49)An example of this when the ambient temperature is 23"C, the starting temperatureis 40"C, and the harvesting temperature is 25"C, is shown in Fig. 6.1 1. If the solubility of a compound is assumed to follow a vant Hoff equation,then ln[q = ( - - A H / R T ) p + (6.50)where p is a constant and AH is the heat of solution. Combining Eqs. (6.49) and (6.50) would then give the temperature as a func-tion of time. Rather than arriving at the complex relations that would arise fromthis, an approximation approach has been taken as shown in the following example:~ ~ 6.1 a ~ ~ ~ eSuppose a recrystallization takes place and the temperature is at 40°C at time 0 andat 25" at time 10. Assume the ambient (or cooling) temperature is 23°C. Assume thesolubilities at 40°C is50 and at 25°C is30. Rraw the temperature versus timeand thesolubility versus time curves.A n ~ 6.1 e ~ ~In the stated case, Eq. (6.49) would take the f o m : 40 20 0 2 4 6 8 1 0 1 2 Time 1 Cooling curve starting at 40°C toward ambient temperature of 23°C. Recrystalli~atio~ stopped at 25°C. is
  • 115. T = 23 + (40 - 23)(1 - exp(-k@) (6.51)or, since at 4 = 10, 7 = 25: ‘ 25 = 23 + 17(1 - exp(-k@) (6.52)from which k = 0.2 +The curve T = 23 17(1 - exp(-0.2@) is shown in Fig. 6.11. gain assuming a van’t Hoff equation to hold then, given the solubility of thecompound to be recrystallized is at 40°C and 30 at 25°C then it is easilycalculated 50that the solubility would be a function of temperature by way of ln[q = (-3163.5/T) (6.54) + 14.014 This is now converted to solubility versus time, most easily by programming(e.g., as the program in BASIC; Tables 6.3 and 6.4). The tem~erature versus time curve is shown in Fig. 6.1 1. The solubility versustime curve is shown logarithmic fashion in Fig. 6.12, and the curve is a logarithmic indecay by way of Eq. (6.55): ln[C - S(23)I = 3.05 - 0.23# (6.55)C - S(23) is the supersaturation and is given by C - S(23) = 21.6 exp(--0.23@) (6.56) rom the example it is seenthat it is reasonable to assume that the s ~ ~ e ~ s a t ~ r a t i asa function of time may be given by h = (Co - S ) exp(-k@) (6.57)where Co is the concentration at the beginning temperature and S is the solublity atthe ambient temperature. ates of growth may frequently be expressed as (6.58)where A is the surface area, and A is a growth rate constant and g is an exponent, 4usually of value close to 2. Program for ~ ~ ~ e rValues~ nWay of Eq. (4.51) a t by ~For T1 = 0 to 10T2 I= 23 + l~*EX~(-O.2*T) 1 = -3163.5/(T2 + 273.15)S2 = SI + 14.014NEXT TI
  • 116. Printout of Data Generated in Table 6.3Time Temperature Solubility 0 40 49.4 1 36.9 44.7 2 34.4 41.1 3 32.3 38.3 4 30.6 36.2 5 29.3 34.5 6 28. 1 33.2 7 27.2 32.1 8 26.4 31.3 9 25.8 30.610 25 30.0 The mass of one single cubic particle, with side length r, is m = r3p (6.59)so that dm/dQ, = 3r2p(dr/dQ,) (6.~Q)The area A of the surface of the cube is A = 6r2 (6.61) Eqs. (6.6Q)and (6.61) now give dm/dQ, = 3r2p(dr/d#) = ~ 6 r 2 ~ g e x p ( - g ~ # ) (6.62)or (6.63) y ..3.0476 - 0 . 2 3 3 8 6 ~ R*2 = 1.000 3.5 Time ig. Concentr~tion ess saturation concentration at 23°C plotted in a s e m i l o ~ a r i t ~ i ~ Lfashion versus time.
  • 117. The length of time given the crystallization is denoted t. The size ofa particle that tois born at time # is given by: exp(-gk#)d4 = ( 2 M / p ) A ~ [ e - e-gkz] -~~~ (6.64)The size of the largest particle ro is obtained by inserting # = 0 in this equation: ro = (2M/p)Ag[l - e-gk"] (6.65) he number of particles that are born between time Q1, and dq5 is given by: J = dN/d@= a[(C(#)- SI1/" = aACIn x p ( - ~ # / ~ ) e (6.66)The total number of particles is obtained by integrating this from 0 to t z exp(-~#/~)d# ah/"{1 - e"kt1n1 = (6.67) y the same argument, the number of particles with particle size larger than r4 isdenoted N , and is given by integration of the integral in Eq. (6.67) from # to t: N > r ~ @aA/"{e-k@/" ~ __ (6.68)Equation (6.64) may be written = + [e-gk@] [pr4/(2MAg)] e-gkz (6.69)Inserting this in Eq. (6.68) now gives (6.70)which, when the symbol r is substituted for r(#) is the cumulative distribution func-tion, (Eq. (6.70) divided by Eq. (6.67)). Carstensen (1980) and Rodriquez (1985) have shown that these functions, forn = 1/2 and g = 2, resemble lognormal distribution functions. s mentioned in the foregoing, there is often a lag time before nucleation starts.This, in some ways, is tied in with the metastable zone. It is customary to seed a crystallization with seeds of the drug substance. Thismay eliminate the lag time and, often, reduces the energy ofactivation for the criticalnucleus formation (i.e., AG in Fig. 6.2). nce a nucleus isformed at or beyond the critical size it will continue to grow. It caneither grow equally rapidly in all directions (situation i in Fig. 6.13), or the growthmay be i ~ p a iin one direction (see ii in Fig. 6.13), in which casea plate results. If r ~ ~the growth is impaired in two directions, then a needle results (see situation iii in Fig.6.13). The drug substance, per se, may be such that one of the three situations ispreferred. There are some compounds that always crystallize out as needles.
  • 118. iii Creation of different crystal habits from a nucleus. (i) all directional growth ratesare equal; (ii) one directional growth rate is lower than the other two; (iii) two directionalgrowth rates are lower than the third. 0 1 2 Additive g/L ig. ~ o d i ~ c a t i o nacetaminophen crystals: of influence of growth in aqueous solutioncontaining ~-aceto~ytacetanilide crystal properties. (Data from Chow et al., 1985.) on owever, additives affect the dimensions in the crystallization of certain sub-stances. Figure 6.14 shows the results from the presence of ~-aceto~ytacetanilide onthe dimensions (len~th~breadth ratio) of acetaminophen (Chow et al., 1985).Berthold A (1912). J Chim Phys 10:624.Bikeman JJ (1970). Physical Surfaces. Academic Press, New York, p 215.Carstensen JT (1980).Solid Pharmaceutics: Mechanical Properties and Rate Phenomena. Academic Press, New York, pp 30-32.Carstensen JT, Rodrugue~-~ornedo (1985). J Pharm Sci 74: 1322. NChow A H-L, Chow PKK, Zbognshan W, Grant DJW (1985). Int J Pbarm 24239-258.Dankwerts PV (1951). Ind Eng Chem 43: 1460.Doherty C, York P (1987). Int J Pharm 34:197-205.
  • 119. Higbie R (1935). Trans Am fnst Chem Eng 31:365.Marc R (1908). Z Phys Chem 61:385.Mullin JW (1961). Crystallization. Butterworths, London, p 106.Noyes AA, Whitney WR (1897). J Am Chem SOC19:930.Noyes AA, Whitney WR (1 897).Z Phys Chem 47:689.Nyvlt J (1971). Industrial Crystallisatio~from Solutions. Chemical Rubber, Cleveland, OH.Ostwald W (1897). 2, Phys Chem 22:289.Otsuka N, Matsuda Y (1996). J P h a m Sci 85:112.Valeton JJP (1923). Z Krist~llogr59:135; 335.
  • 120. 7.1. ethods of Preparation 108 7.2. Amorphates S9 O 7.3. Class Transition Temperatures of Mixtures 109 7.4. Use of Modulated ~ifferential Scanning Calorimetry 110 7.5. Water Absorption “Isotherms” into Amorphates 110 7.6, etermination of Amorphates 112 7.7. Crystallization of Amorphates 112 7.8. Polymers I14 Symbols 115 eferences 115 t was previously mentioned that ~ ~ o r ~ ~ simplyedefinedas materials that are are ~ t snot crystalline. In general, they are more energetic (less stable) than any cr stallineform, although there have been some exceptions reported in the literof this they have higher dissolution rates and apparent solubilities eezer, Ahmed 1992; et al., 1998), stability (Carstensen andCarstensen et al., 1993; Pikal, 1978). At times materials are produced in amorphous form by methods usually usedfor producing crystalline modi~cations. ecrystallization from different solvents isnot always successful. Chow and Grant (1988, 1989) have described that recrystalli-zation of acetaminophen from a series of solvents gave rise to amorphous materialand different crystal forms. The theory of ‘Crolmer and Web ker and Doring (the ~~states that crystals form as a function ter formation (of volumesat~ration as an end result, the nucleation rate J is a function of interfacial tension S;0 between solid and liquid by the following formula:
  • 121. er J = exp( - 1 6 ~ ~ 3 u 3 ~ / { 3 R 3 T 3 ( l n ~ ) 2 } ]where N is Avogadro’s number, R is the gas constant, and Tis absolute temperature.The equation holds well for vapors and solutions (Mullins and Leci, 1969), but doesnot apply well to supersaturation situations or melts, particularly for more complexmolecules. Tamann (1926) showed that for melts there is maximum in J at a particulartemperature. Turnbull and Fisher (1949) modified the equation to read: ~~~ + f = e x ~ [ ~ - 1 6 ~ ~ ~ u ~ ~ / { 3 R AG,/RT] l n ~ ) ~ ~ ] ~T~(where AG, is activation energy for motion of molecules across the matrix~lusterinterface. AG, is highly dependent on the viscosity of the melt. It is obvious, therefore, that certain substances that possess high viscosity attheir melting point may be prone to become amorphous on melting and recoolingAt times it is actually difficult to prepare an organic compound in crystalline form,and in such a case, the problem in lies producing the crystalline substance. echanical interaction is often a means; it is remembered from organic laboratorycourses that students will produce adispersion ina test tube, and then scrape the sideof the test tube with a spatula. In general terms, some means of nucleationmust becreated. Hildebrand and M~ller-~oymann (1967) report on the produketoprofen by neutralization of ketoprofen with sodium hydroxide.and water as solvents produce a hygroscopic glass. However, if this glass is sus-pended in 95% ethanol and stirred at length, a crystalline sodiumsalt will eventuallyoccur. The opposite, to create a substance in amorphous form when it is easily crystal-lized may be achieved, in general, in the following manners: Sublimation Supercooling of melts ~eutralization an acid with a base (if the drug is an acid) or vice versa of Recrystallization from a variety of solvents ehydration of hydrates ophilization (e.g., by “kugeln”) ray-drying To name some examples, hard candy, produced by ~eZting sucrose, is amor-phous. If it crystallizes during storage, then it becomes cloudy and is considereddefective candy. Amo~icillin trihydrate becomes amorphous on ~ e ~ ~ ~ ~ ~ t i o net al. (1997) prepared three crystalline forms of ciprofloxacin HCl; furthermore, they re pared the amorphous form by Z ~ o ~ ~ i ~ i ~yophilization sucrose produces z~ti~n. ofamor~hous sugar and will be touched on later in the chapter. As an example of~ ~ ~ u ~ - ~ ~ lactose ~ g a high amorphate content. spray-dried ~ i has ,
  • 122. Solids that are not crystalline are denoted amorphous. If one melts a (stable) solidand recools it, then it should crystallize when the melting point is reached. This requires nucleation, and nucleation propensity is a function of the visc-osity of the liquid in which it occurs. Materials that are viscous about their meltingpoint are, therefore, prone to form supercooled solutions. At a given high viscosity(attained at or lower than the melting point), the meltwill have the appearance of a solid, and this is the type of material referred to asamorphous. Just below the melting point, the molecules will have no specific orientation,and molecular movements will be random in direction and magnitude (within thelimits of the system), as opposed to a crystalline material, in which the molecules arearranged in lattices (ordered arrays) and the orientation of each molecule is set. At a temperature T,, lower than the melting point, there will be a physical in the amorphate. An example of this is shown in Fig. 7.1. etween points A and 313 the properties of the amorphate is often similar to thatof the melt, and is referredto asthe “rubbery” state, and below C it is referredto as aglass. At the glass transition temperature T,, the viscosity of the melt is often of themagnitude of l0l2 Pa s (Lu and Zografi, 1997), and this is the “cutoff point’’between a “liquid” and a “solid.” For lyophilized materials that produce amorphous cakes, the “collapse tem-perature” is essentially the temperature at which the viscosity drops below a criticalviscosity (e.g., 1 0 ’ ~a s) that will allow the cake to deteriorate.It is often of importa~ce toestimate the glass transition of an amorphate that has acertain water (or solvent) content. If values of T (Tsl and Ts2)are known at two ’different water contents (ml and m2), then Tg at other water content may be esti-mated by using the Cordon-Taylor equation (Cordon and Taylor, 1952). Tg = TgI 4- ~ m 2 T , 2 ) ~ ( m4- Km2) l (7.3)where ’1 ‘O Rubbery -50 50 T(m) 150 250 Temperature O C Molecular volume as a function of temperature of a solid prone to forming anamorphate.
  • 123. This is referred to in the following as MDSC. Hill et al. (1998) have described thistechni~ue, which rather than using a linear cooling or heating ramp, a sinusoidal intemperature profileisused (Reading et al., 1993).Hill et al. (1998) investigatedamorphous a-lactose and were able to measure the heat capacity at Tg separatelyfrom the endotherm.A~orphates solids that are not cryst~lline. Itis assumed at this point that the areterm solid is self-evident, although amorphates in the rubbery state (just below themelting point of the crystalline form of the compound) are actually highly viscousli~uids. When exposed to h u ~ i atmospheres, they will pick moisture in a fashion d upthat is not like that of a BET isotherm (to be covered shortly). The moisture actuallypenetrates into the solid, and it may be considered a s o Z ~ t i o ~ . In an ideal situation, the water activity a, will decrease linearly with (1 - x)where x is the molefraction of solute. At a given point (x = 0.24 in Fig. 7.2) thesolution becomes saturated. (This concentration differs from compound to com- yond this concentration, the solution itselfwillbe saturated, and the ure will not change with further addition of compou~d, rather the corn-position will change, but the vapor pressure’willstay constant. In this type of graph rdinates are in a direction opposite that of a usual isotherm. an amorphous form of the compound is produced and exposed to differentrelative humidities, then the isotherm is often quite linear if the amount of waterabsorbed is expressed as olef fraction (line DE in Fig. 7.2). As shown by Carstensen and ~ a n ~ c o(1989) for amorphous sugar, this line is an extension of the solution ikvapor pressure line (see AB in Fig. 7.2), and one may consider the moist amorphate as a highly concentrated, supersaturated “solution.” Dilute Solutlon f .- - .- 0 I Solutlon + Preclpitate 6u, te 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (1-4 Moisture isothermfor an amorphous solid. (Data from Carstensen and VanScoik, 1988.)
  • 124. ecause of the random arrangement and the mobility of the molecules in anamorphate, as opposed to a crystalline modification, amorphates are usually chemi-cally less stable than crystalline modi~cations (Carstensen et al., 1993). Carstensen and VanScoik (1988) were the first to point out that for an amor-phous substance, it is illogicalto use the traditional moisture isotherms, because hereit is probably not an adsorption, but rather, an absorption that is at play. y exposing a m o ~ h o u sucrose to various relative humidities, various moist- sure leveIs were reached. If these moisture levels were expressed as molefraction ofsucrose, then the vapor pressures fell in line with vapor pressure curve of sucrose thesolutions itself. The fraction to the right of point is the principle used for salt solutions toobtain constant relative humidity in de cators. With electrolytes, the vapor pres-sure depression is larger (owing to the two or threefold number of ionic particles,over that of the molarity of the salt), and the solubilities often high, so that these arepreferred for creating constant relative humidity in desiccators. Lehto and Laine (1998),haveshown moisture isotherms of cefadroxil, foramorphate, crystalline anhydrate, and hydrate. The amorphate yields an isothermthat is constantly increasing (i.e.? not of the BET-type) up to a relative humidity of82%, at which point the isotherm suddenly drops, owing to crystalli~ation andbecause the crystalline phase only can handle surface adsorbed moisture (Le,,much less than the amorphous rubbery phase). u et al. (1996) found that [6-fluoro-2-(2’-fluoro-l,l-bipheny1-4-y1)-3-met~yl]4-quinolinec~rboxylic acid sodium salt (brequinar sodium) ean amorphous form or as a hemihydrate. When it is exposed to 75%water rapidly and changes into the hemihydrate. Both forms are quite water-soluble. ancock and Zografi (1993) later used this principle in their investigation ofwhether solution theory could be applied to macromolecules. To quote “If one considers the absorption process to be completelyanalogous to the solution p~ocess, then it should be possible to use basic solution theories to model the data..,” Their data for polyvinylpyrrolidone (PVP) K30 are shown in Fig. 7.3. 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Water Activity Fit of vapor pressure data of aqueous solutions of PVP K30 at 30°C to the Flory-Huggins equation. (Data from Hancock and Zografi, 1993.) The points are taken frorn theirFig. 6 and Fig. 7 as accurately as possible, as is the trace of the Flory-Huggins equation.
  • 125. 1 ter The importantfeature in Fig. 7.3 is that the data fit neither the Flornor the Vrentas equation. The Vrentas equation probably provides a better fit, butthe adherence at activities above 0.9 are not shown; in any event, data becomeslightly uncertain at such high h~midities.Traditionally, the fraction of a solid that is amorphous has been d e t e ~ i n e d bymeans of X-ray diffraction. Black and Lovering, (1977) d e t e ~ i n e ~ fraction thecrystallinity in samplesofdigoxin powder, and Junginger (1977) ~ e t e r m i n e ~ thedegree of phase t r a n s f o ~ a t i o n this manner, Bernabei et 81. (1983) have investi- ingated the effect that crystallinity has on the enzymatic hydrolysisof the palmitate ofchloramphenicol, and Ryan (1986) optimized crystallinity of lyophilizates this.way.Amorphous materials, as shown by Carstensen and Morris (1989), are less chemi-cally stable than their crystalline counte~arts, also shown by Imaizini etal, (1980) asand by ~ ~ b s k a et aal. (1995). y icrocalorimetry has been a useful tool in the detection of minor contents of ous m a t e ~ acaused by, for instance, milling (Briggner et al., 1994; Sebhatu let al,, 1994; B u c ~ t et ~ 1995; Ahmedet al., 1996), and contents of as little as 1% o al.,can be d e t e ~ i n e dwhich is better than d e t e ~ i n a t i o n X-ray diffraction. , by Phillips (1997) described a means of estimation the content of amorphate in haspharm~ceutical powders by means of calorimetry. It is based on comparing the sizeof enthalpic changes in fusion and crystallization. However, because melting andcrystallization occur at different points (Hancock, 1998), enthalpies are subjected to correction to bring them to “the same temperature’’ by a method forwarded by offmann (1958). Hancock (1998) has cautioned that there are several shortcomingsof this method; for one, it is difficult to obtain a sufficiently, crystallographicallypure, sample to compare the test sample against. Also, events suchas transitions anddesolvations, may occur in the same temperature range as melting (Ford andTimmins, 1989). §tubberud and Forbes (1998) used microgravimetric method (CISORPAutomated Sorption oni it or), to study the crystallization of amorphous lactose.They f o u ~ dPVP to act as aninternal desiccant and delay the onset of crystallization,but many nonhygroscopic tablet excipients accelerate it. Cases often exist in which a drug substance or excipient is partially crystallineand partially amorphous. In such cases, the ~uantitativecontent of amor~housc o ~ ~ o n emay be obtained by microcalorimetry. Density measurements may also ntbe used. Densities ofamorphous materials (pa) are most often less than those (pa) ofcrystalline solids, so that the content f of amorphous component may be assessedfromThe most common method of measuring the transformation of the etas stable amor-phate to more stable crystalline forms is bywayof X-ray if fraction.
  • 126. crystalline content as high as 10% may go undetected by this method (Ahmed et al.,1998; Saleki-~erhardt et al., 1994). The kinetics of transformation has been discussed by several authors. Ahmedet al., (1998)employ first-order kineticsin the transformation of amorphous tocrystalline griseof~lvin. Carstensen and VanScoik (1989, 1990) employed weight gain (Fig. 7.4) as ameans of studying the conversion of amorphous sucrose into crystalline sucrose. They produced amorphous sugar spheres by pipetting sucrose solutions intoliquid nitrogen (so-called kugeln), and lyophilizing them on petri dishes in a fashionsuch that no sphere touched another sphere. After freeze-drying, the petri disheswereexposed to different relative humidities and temperatures, and the weightchecked as a function of exposure time. The first event that occurs is a contractionof the spheres in size, presumably owing to a change from rubbery to glassy state.The glass transition temperature is a function of moisture content, and as thisincreases, apparently Tg decreases, so that the transition is facilitated. This ispoint A in Fig. 7.4. A plateau is then reached, and at a certain given time, corre- , the sucrose will begin crystallizing. The crystals cannot “hold” waterin the same fashion that the amorphous phase can, so that the weight drops, and theweight drops until all the sucrose has crystallized. The weight gain at a certain relative humidity, traditionally, would be part ofan isotherm, but these isotherms are not of the conventional type, but rather, suchthat the amorphous, moist state behaves similar to a “solution” (Le,, a very con-centrated, supersaturated solution of sucrose in water). As seen in Fig. 7.5, the vapor pressure curve is in line with the vapor pressurecurve of sucrose at less than saturation. Hence, it is logical to view this amor~hous state as a supersaturated solution. Carstensen and VanScoik found the points after the drop in weight (see phase CD Fig, 7.4) to follow a probit function. The levels, themselves (AB) may, as men- tioned, be considered solubility, and as such should follow a van? Hoff plot, as they indeed do (Fig. 7.6). The plot is plotted as the plateau level, which is the inverse of the solubility and, hence, the plot has a positive slope. ~icrocalorim~tric methods have also been used to study amorphous tocrystal- line t r a ~ s ~ o ~ a t i o n s (Hansen et al,, 1996a,b; Angberg et al., 1991a,b,1992a,b). 0 0 IO 20 30 40 Time (days) Weight gain at 23°C and 33% RH.
  • 127. 1.2 I .oe,0.8rc 0.6 0.4 0.2 00 0.I 0.2 0.3 0.4 0.5 0.6 Mole Fraction Sucrose y= - 6.4827+ 2.4566~RA2= 0.965 1.97 3.1 3.2 3.3 3.4 1ooorr Vant Hoff plot of plateau levels in moisture uptake by amorphates.Angberg et al. (1991a,b) employedthe method to study the transformatio~ amor- ofphous lactose into the crystalline hydrate. Larsen et al. (1997), also employing thismethod, showed thatamorphous acadesinecrystallizes bywayof a metastablehydrate. This decomposes very rapidly into the anhydrate. Transfo~ationsmay also betestedbyway of dissolution. As is true withmetastable polymorphs, concentrations will first rise to a high level (the apparentsolubility of the amorphate), or approach it, but on nucleation, precipitation willoccur, and the concentration will decreaseto the level ofthe solubility of a crystallineform. olymers will be the subject of Chapter 26 and constitute a special case of pharma-ceutical solids. The aspect of amorphicity is, here, of great importance. The rubberystate confers elasticity to the polymer film, so it is important that Tgbe as low aspossible. Plasticizers are added to polymers to achieve this, and one means of asses-sing the effectiveness ofa plasticizer is to record the glass transition temperature as afunction of plasticizer content.
  • 128. For water-soluble polymers, water is most often a good plasticizer.soft gelatin capsules is controlled to within close limits. Above a criticalcapsule will become too soft and deform in the bottle. Below a critical limit thecapsules will become brittle. The same holds true for wet granulations. Compressibility of tablets madefrom granulations is a function of the moisture content, and often, this is due tothe elasticity of the bonding bridge of the binder that keeps the particles together. Ifit deforms easily, then a compressed mass is easily formed. If the granulation isoverdried, then it becomes brittle and shatters during comminution, giving rise to nts of fines which, in turn, impedes the compression process. vinyl pyrollidone (PVP) is a frequently used binder, and 90) have shown that the glass transition temperature ofdependent on moisture content. If it rises above room temperature, then the polymerwillbein the glassy state at the time of grinding, willbe brittle, and highfines~roduction result. will f = fraction of amorphate in a batch of drug or excipient G, = Gibbs’ energy for transport of a mole from cluster to solution J = nucleation rate m1 = mass of amorphous component 1 in a mixture of amorphates m2 = mass of amorphous component 2 in a mixture of amorphates IV = Avogadro’s number PVP = polyvinylpyrollidone S = supersaturation T = absolute temperature Tg = glass transition temperature Tgl = glass transition temperature of component 1 in a mixture Tg2 = glass transition temperature of component 2 in a mixture K = weighted ratio constant in a mixture of amorphates pa = density of amorphous phase in a mixture of amorphous and crystalline phase pc = density of crystalline phase in a mixture of amorphous and crystalline phase p1 = density of component 1 in a mixture of amorphates p2 = density of component 2 in a mixture of amorphates v = cluster volume cr == interfacial tension , Buckton 6 , Rawlins DA (1996). Int J Pharm 130:195.Ahmed H, Bucktor! C , Rawlins D A (1998). Int J Pharm 167: 139.Angberg M, Nystrorn C, Castensson S (1991a). Int J Pharrn 73:209.Angberg M, Nystrorn C, Castensson S (1991b). Int J Pharm 77269.Angberg M, Nystrorn C, Castensson S (1992a). Int J Pharrn 8 1:153.Angberg M, Nystrom C, Castensson S (1992b). Int J Pharm 83:11,
  • 129. Becker R, Doring W (1935). Ann Physik 24:719.Bernabei MT, Forni F, Coppi G, Iannucelli V, Cameroni R (1983). Farm Ed Prat 38:391,Black DB, Lovering EG (1988). J Pharm Phamacol 29:634.Briggner L-E, Bucton G, Bystrom K, Darcy P (1994). Int J Pharm 105:125.Buckton G, Beezer AE (1992). Int J Pharm 82:R7-10.Buckton G, Darcy P, Greenleaf D, Holbrook P (1995). Int J Pharm 116:113.Carstensen JT, Morris T (1993). S Pharm Sci 82:657.Carstensen JT, Franchini M, Pudipeddi M, Morris T (1993). Drug Dev Ind Pharm 19:1811.Chow AH, Grant DJW (1988). Int J Pharm 51:115.Chow AH, Grant DJW (1989). Int J Pharm 52:123.Ford JJ, Timmins P (1989). Pharmaceutical Thermal Analysis: Techniques and Applications. John Wiley & Sons, New York.Gordon M, Taylor JS (1952). J Appl Chem 2428.~ u b s ~ a AV, Lisnyak YV, Blagoy YP (1995). Drug Dev Ind Pharm 21:1953. yaHi~debrand GE, Muller-Goymann CC (1997). J P h a m Sci 86:854.Hancock BC (1998). Int J Pharm 160: 131.Hansen LD, Cravvford JW, Keiser DR, Wood RW (1996a). Int J P h a m 135:3 1.Hansen LD, Pyne MT, Wood RW (1996b). Int J Pharm 137: 1.Hill VL, Craig DQM, Feely LC (1998). Int J Pharm 161:93. offmann JD (1958). J Chem Phys 29:1192.Imaizini H, Nambu N, Nagai T (1980). Chem P h a m Bull 28:2565.Junginger H (1977). Dtsch Apoth Ztg 117:456.Makkhar AP, Singh M, Mendiratta A (1997). Drug Dev Ind Pharm 23:1063.Larsen MJ, Hemming DJB, Bergstrom RG, Wood RW, Hansen LD (1997). Int J P h a m 154: 103.Lehto E, Laine E (1998). Int J Pharm 163:198.Lu Q, Zografi G (1997). 3 Pharm Sci 86: 1374.~ullins JW, Leci CL (1969). J Cryst Growth 5:75.Oksanen, CA, Zografi G (1990). Pharm Res 79:1374.Phillips EM (1997). Int J Pharm 149:267.Reading M, Elliot D and Hill VL (1993). J Therm Anal 40:949.Ryan JA (1986). J Pharm Sci 7:654. Sebhatu T, Angberg M, Ahlneck C (1994). Int J Pharm 104: 135. Stubberud L, Forbes RT (1998). Int J Pharm 163:145. ama an^ G (1926). The States of Aggregation. Mehl RF, trans. Van Norstrand, New York, p 105. Turnbull D, Fisher JC (1949). J Chem Phys 17:71. Vesa-Pekka L, Laine E (1998). Int J P h a m 163:49. Volmer M, Weber A (1925). Z Physk Chem 119277. Wu L-S, Pang J, Hussain MA (1996). Pharrn Dev Techno1 1:43.
  • 130. 8.1. Polymorphs, Methods and Detection 118 8.2. ~nantiotropes and Monotropes 119 8.3. Stability of Metastable Polymorphs~The “Disappearing” 121 8.4. s of Conversion in Moist Storage 123 8.5. 125 8.6. Pseudopolymorphism 125 8.7. ~olubility Thermodynamic Functions and 126 8.8. Mixtures of Polymorphs 127 8.9. Dissol~tion Rates of Polymorphs and ~seudopo~ymorphs 128 8.10. Rates of Conversion in 129 Symbols 129 References 130The pharmaceutical interest in polymorphs is attributable to the work by Aguiar etal. (1967), who demonstrated that different poly~orphic forms of chloramphenicolgave not only different dissolution rates, but also distinctly different degrees ofbiological absorption, Inorganic (particularly ionic) solids usually are associated with one and onlyone crystal system. ell-~nown to all is that sodium chloride is cubic. Organic solids, however, depending on how they are recrystallized, may occurin several different crystal modi~cations (polymorp~s). There are two types of poly-morphism, enantiotropes and monotropes. They are distinguished by their vaporpressure diagrams and differential scanning calorimetry (DSC) traces (see Figs. 8.1through 8.4).
  • 131. t may almost always be assumed that more than one polymorph exists in anew drug substance. One of the tasks of the p h a ~ a c e u t i c a l scientists is then toproduce as many polymorphs as possible at the earliest time possible during theproduct development stages. ods for (attempts at) producing different crystal forms most often take the of recrystallization from different solvents. If the compound is heat-stable,thensub1imation (e.g., Schnitzer et al., 199’7) maybe attempted.(1998) studied crystal forms o f piroxicam pivalate by recrystallization fromtoluene, ethyl acetate, ethyl ether,andethanol (Table 8.1). was This done atroom temperature (RT) or ice cooling (I) with (S) or without ( ~ O magnetic)stirring, Two polymorphs (i and ii) were obtained, under the mentioned ne way of distinguishing between different polymorphs is by difference in X-ray patterns. Often, however, infrared spectra (IR) show distinct difference8.2 shows the wave numbers (reciprocal centimeters) of certain bands in thetransform infrared (FTIR) spectra of the two poly~orphs. ecrystallization from different solvents is not always successful. Chow andGrant (1988, 1989) have described that recrystallization of acetaminophen from aseries of solvents gave rise to amorphous material and different crystal forms, butthis could not be duplicated~in the same solvents, by deyilleiers et al. (1998). Pyroxicam Pivalate Polymorphs ~~Solvent RT-S RT-WO I-s I-woToluene 1 i 1 iEthyl ether i i 1 1Ethyl acetate 1 1 11 i -tiiEthanol 1 I I 1Source: Giordano et al. (1998). FTIR Spectra of the TWOPolymorphsFunctional group Polymorph i Polymorph iiNH 3253 3291, 3350C=O, ester 1760 1750,1767 1682 1887 Source: Giordano et al. (1998).
  • 132. 0 0 90 100 150 200 Temp "C Vapor pressure diagram of an enantiotropic pair.
  • 133. 0 20 40 6 0 80 100 120 1 4 0 Temperature "C Possible DSC traces resulting from heating of the room temperature-stable form ofan enantiotropic pair. The other case is monotropism (ie., the situation where one form (form 11) ismetastable throughout the melting range). This is exemplified in Fig. 8.3. The DSC trace of such a pair may take one of several forms. The stable formwill simply show up as a trace with one endothem (the melting point of the stableform). Traces of the metastable form may either show up this way, or they may showup as the middle trace in Fig. 8.4. If the compound is stable to melting, it is advisable to recool the mass andrecord the melting point on the down trace. Most often, however, decomposition ofthe solid and melt preclude conclusions from cooling curves. It follows from thermodynamics that the change in Cibbs energy by a pathfrom metastable to stable form AG, is given by In the top trace, it is the stable polymorph in Fig. 8.4 that is heated, and thetwo lower traces are the heating of the metastable polymorph, which may eithersim~lymelt (lower trace) or,as shown in the middle trace: melt, precipitate(exotherm) as the stable form I, and then (second en~otherm) remelt. It is negative, so the form with the highest vapor pressure at a given tempera-ture is the least stable (metastable) compound. 0 50 100 150 200 Temp "C Graph of vapor pressures for a monotropic pair.
  • 134. his 10 r - f O * r " l " " * 0 2 0 40 GO 80 100 120 140 Temperature O C Some possible DSC traces of the heating of polymorphs that are monotropic.Ostwald (1 899) formulated a rule of stages: when system first starts crystallizing, it ainitially will create the crystal structure that forms the smallest loss of free energy,and these crystals will later transform, stagewise, to the most (or a more) stablecrystal structure. There are several types of ~ e c h a n i s mthat can occur when a metasta~le,dry,polymorph transforms to a less energetic crystal form. If a denotes the amounttransformed at time t, then some of the possibilities are elaborated in the following. If the nucleation event is such that it can occur throughout the solid then thereare three cases: 1. There are no complicating factors, the roba ability is time-independent, andthe rate of transformation is given by da/dt = k (8*2)where a is fraction converted, so that a simply increases linearly with time. is the This~olany-~in~er equation: a = kt (8* 3) 2. The rate of transformation is directly proportional to the amount of solidnot yet nucleated (frequently denoted "random nucleation"): da/dt = k( 1 - a) (84which integrates to - lnf1 - a]= kt 3. The nucleation rate may also be proportional with time (i-e., the longer theelapsed time, the more likely is it that a site will transform. da/dt = k*(l - a)t (8.6)If, as an input function, the term k* = 2k2is introduced, then
  • 135. -dln[1 - a ] = 2k2t (8.8)which integrates to - In[(1 -a)] = k2t2 (8-9)This is a form of the Avrami-Erofeev equation. An example ofthis is shown in Fig.8.5. 4. Similarly, if the nucleation rate is proportional to t, but the nucleation cantake place in three directions, then --In[(l - a)]= k3t3 (8.10)This, also, is a form of Avrami-Erofeev equation. f, in two dimensions (exemplified by a cylinder transforming in a radial direc- ly; Fig. 8.6), the nucleation starts at the surface and works its way in, then thefraction not decomposed is given by (1 - a)= 7t(R- q)2/7tR2 7t(R- kt)2/7tR2= [I - (k/R)tI2 = (8.1 1)where q is the thickness of the transformed layer, and R is the radius of the cylinder.This may now be written: E - (1 - a)’/2] ( k / R 2 ) t 1 = (8.12)Note that the rate constant ( k / R 2 )is larger, the smaller the particle ( R ) . It is easily shown that in three-dimensional diffusion, this becomes [I - (1 - 4 l i 3 1 = ( k / ~ 3 ) t (8.13) the same type dependence of particle size as in the cylinder example. mples of work in the pharmaceutical area are the publications of Umeda et dealing with transf~rmati~n aceta~olamidepoly~orphsand the of eniwa et al. (1985). Transformations are followed by disappearance(or appearance) of X-ray peaks, and the data are then plotted by the various equa-tions, and the best-fit isfound. In general, this is not a particularly good method (thedata may plot wellby many different equations), but in transformation rates, itworks quite well. I I * ’ I ” I ” ( 20 40 60 a0 too Time (hrs) Graph dealing with the conversion of (a) pure a-form and (b) crystals containing 1% of the y-form.Adherence to Eq. (8.9) is better than that to Eq. (8.10). (Data from ~ a n e n i w a al., 1985.) et
  • 136. h Schematic of cylindrical model for linear decomposition. lagden et al. (1998a) reported on four different polymorphs of sulfathiazole.They (1998b)reported that ani ~ p u r i t y sulfathiazole synthesis, etha~idosulfathia- inzole, in concentrations as small as 1 mol%, stabilizes two of the metastable forms,form IT and 111, of the drug. nd ways of stabilizing metastable polymorphs is of great indus- ym et al., 1996), because the metastable polymorph maygive d bioavailability. Often impu~ties induce twinning, which willmay inhibit the t r a n s f o ~ a t i o n ,as with terephthalic acid(Davey et al., 1994).Sometimes additives may be used (e.g., polymers prevent the change of the centric toform of 3-~-ethamido-4-~-pyrollidino nitrobenzene into the noncentric form)(Davey et al., 1997). If the “initial’? form of a new drug is a ~ o n o t r o p e ,and the stable form isunknown, then at one point in the development, seeds of the stable form may occur,and after this point it may be impossible to produce the metastable monotrope ernstein, 1995). I have had personal experience with such “diisappear-ing” polymorphs in that, in the early X960s, a batch of benzodiazepam was madethat had a slightly (l°C) higher melting point than usual. The conventional wisdomwas that thenew batch was purer (for it hada lower meltingpoint), but the truth was that it was a different polymorph, and after that batch had been made, it wasimpossible to recreate the “old” form. This, in turn, led to a large amount of duplication of clinical work, because the clinical results, up until that time, had been based on a metastable (more soluble), now unavailable polymorph. It can beshown by ’s law that solubilities are (ap~roximately) linearly related to vapor pressures ly activities such as solubility are linearly related to fuga- cities). The graphs in Figs. 8.1 and 8.3 then become as shown in Figs. 8.7 and 8.8. enry’s law argument is applied to Eq. (8.l), then
  • 137. r 0 0 20 40 60 80 Temp, O C .7 §olubilities (in mass of solute per mass of solvent) of an ena~tiotropi~ pairwhere S denotes solubility, R the gas constant, and 7 absolute temperature. There are examples for which the solubilities are close overthe entire tempera-ture range ( ~ a r s t e ~ s e n Fr~nchini, and 1984a,b) and, in such cases, may be difficult itto separate the two polymorphs in the final puri~cation (recrystallization or repre-cipitation), and there are cases where companies have been forced to suggest speci-fications that stipulate a minimum and a maximum of one polymorph in relation toanother. raphs, such as those in Figs. 8.7 and 8.8 are often presented in log-inversefom: ln[S] = (-hH~/~)(lOOO/T)B + (8.15)where S is solubility, AHs is the heat of solution at saturation, R is the gas constant,1 is absolute te~perature, /3 is a constant. It is recalled, however, AHs is not and thtnecessarily te~perature-independent (see Chapter Z), and if this is not true, then the rant equation (hdipeddi, 1998; J o z w i a ~ o ~et ~ i 1996) applies. s al., In S = - A / T + Bln[iT] + (8.16) c Form II Temp, "C §olubilities (in mass of solute per mass of solvent) of a ~ o n o ~ r o pair. pi~
  • 138. eference is made to Figs. 8.7 and 8.8. The tendency of a metastable polymorph toconvert to a more stable polymorph is a ~ n c t i o nof the difference in chemicalenergy. This, in turn [see Eq. (8.2)) is a function of their solubilities. If e curvesin Figs. 8.7 and 8.8 are veryclose to one another (Carstensen and1994a,b), inparticular, if the compound is very soluble, then the rate of t r a n s f o ~ a -tion can be exceedingly slow, and the possibility of compounds crystallizing out inthe two different crystal forms exists. The regulatory authorities, presumably, are interested in the morphicpurity ofcompounds because of the effect of polymorphism on bioavailability, and this, inturn, is tied to the solubility of polym~rphs.The metastable forms have higherap~arent solubilities than the stable forms; hence, they are likely to have higherbioavailabilities. However, in a situation as just described, the solubilities can besufficiently close that one form is as bioavailable as another, undersimilar conditions(particle size, moisture content).One aspect of polymorphism is that the metastable form will have a higher “solu-bility” than the stable form. The word s o Z ~ ~ i Zhasybeen placed inquotation marks, ~tbecause theoretically a compound can only have one solubility. It has been seen, inChapter 6, crystallization, that real equilibrium solubility happens only at infinitesize of the particles, or at a secondary energy minimum. The point is that when thesolubility is determined, an excess of solid is placed in contact with liquid that isstirred until“e~uilibrium”occurs. The facet of this solubility is that it is repeatable,so that for a metastable compound a reproducible number is arrived at, and thisnumber is higher than the solubility of the more stable polymorph. The molecules insolution, however, are thesame, and the saturated solutionofthe metastable polymerissimply asupersaturatedsolution of the compound.Seeding it or waiting for a sufficiently long time will result in precipitation of themore stable polymorph. One facet of polymorphism is, therefore, that solutions made from differentpolymo~hs contain the same compound. If a hydrateis dissolvedin water, then thesolution will contain the same molecules as a solution made from the anhydrousmaterial. Forthis reason, hydrates arecalled p s ~ ~ d o p o Z y ~ o ~ p prefix is derived The ~ s .from the fact that the solid composition differs (by water of crystallization^. Thesame argument holds for a solvate and solutions in the solvent in question. zuel(1991) and Golic et al. (1992) showed that norfloxacin forms different Sustar et al. (1993) showed that it formsat least two different crystal- azuel (1991) and Golic et al. (1992) elucidated the crystal structure ofnorfloxacin, and Turel(l997) that cipro~oxacin of hydrates. The water is present in acomplicated structure of hydrogen bonding. The manner in which the hydrmade is as follows: The ciprofloxacin was dissolved in 1 1 molar ratio of Cs a :water, and an addition a few drops of 2 M sodium hydroxide would then ofthe solution. The crystals would grow in a couple of days ~hexahydratammonia isused to dissolve the ciprofloxacin, then, depending on the a m ~ o n i aco~centration, either a tetra- or a hexahydra~eis formed.
  • 139. There are often several (e.g., three different) polymorphic forms of an anhydrate, aswell as solvates (Schnitzer et al., 1997) (e.g., there are three anhydrous, crystallinemodi~cations prema~oxacin[forms I to 1 1 and two solvates [a hydrate and a of 11met ha no late^). The DSC trace of form I is shown in Fig. 8.9. Note, that the upwardpeaks in this presentation are exotherms and that the events at appro~imately145"and 170°C are endothermic conversions, with a subsequent exotherm (indi~atingachange in morphism to a more stable form), and that the lone endotherm at about205°C implies a single endothermic change (e.g,, melting) to a (physically) stablestate (e.g., melt). They determined the enthalpy of solution of each form and found values of-33.2 kJ/mol for form 1 and -24.4 kJ/mol for f o m 111. The difference is 8.8 kJ/mol, form I11 having the lower enthalphy solid phase. They take this difference to be ifference in molar entropy of the two forms: AHI-+III= -8.8 kJ/mol (8.17) The solubilities in ethyl acetate were sI = 3.23 mg/mL for formsIII 0.14 mg/mL for form 111. They then employed the (approximate) Cibbs rela- =tion.to calculate that A G I - . + =~-7.6 kJ/mol ~ ~ (8.19) y employing the relation AGI-bIII = AffI+III - TASI-411 (820)and noting that both AH and AG were determined at T = 298 I it follows that ( A S = E--7.6 - (8.8)]/298 = 0.004 IkJfmol = 4 J/mol (8.21)It follows, therefore, that the entropy term is rather insigni~cant comparison with inthe enthalpy difference, and the authors conclude that forms I and I11 ~ o n s t i t a t ~ ~ -6 -8 100 120 1-40 160 180 200 220 Temperature, "C DSC trace of form I of premafloxacin. Cooling and reheating produces only the endotherm at about 200°C. (Data from Schinzer et al., 1997.)
  • 140. ~ o ~ o t ~pair (i.e., the free energy of form 111 is lower than that of form I at all o ~ i ctemperatures below the melting point). There are two dangers in the approach taken bythese authors (and manyothers, for that matter). The first is that Eq. (8.17) is based on bulk calor~metry;therefore, H is the integral heat of solution. To apply to saturated solutions, thee~thaipy term s ~ o u l d the ~ifferential be heat of solution at saturation (or near-satura-tion) conditions (Pudipeddi, 1996). The other is that Eq. (8.18) is correct ~ ~ c t i v i t i e s ,not c o ~ c e n t ~ a t i oare ,e ~ ~ ~ is true that in.dilute solutions activity coefficients ~s It o ~ e ~are close to unity, but they need not be so in concentrated solutions, and the impor-tant question is the~ a g n i t u d of the ratio of activity coefficientsat Slrl and S I . If one ewere, for example, 0.8and the other 1.2, it would not seem to be all that serious, butthe ratio would be off by 33% (or 50% whichever way one looks at it). ecause S is calculate^ from the difference of twonumbers that aresuspect, itis always dangerous to ~r~~ co~cZ~sions from its magnitude, in particular, if it issmall. There is an a~undanceof reports on poly~orphs today’s phar~aceutical inliterature. For instance, Giordano et al, (1998) have reported on two polymers ofpiroxicam pivalate (PIP: . ~olymorph had the higher-melting point ( Ipolymorph 1 the lower 36.5”). Their fusion enthalpies were found by 178.8 and 81.4 J/g. The “heat of fusion rule” (Ciordano etal., 1998; Yu,that if AHII - AHI is positive, then enantiotro~y what is expected. is Kakkar et al. (1997) have prepared three crystalline forms of cipro~oxacin e l , form I by cool evaporation from water, form 11 from coo1 eva~oration from 1:2 ~ a t e r / m e t h aboth~with a crystallization time of 50 h, and form 1 1 by cool ~~ , 1 evapo~ation from dimet~ylformamide. They furthermore prepared the amor~hous form by lyophilizati~n andby spray-d~ying. Some of the properties are shown in Table 8.3. Some trends are noteworthy, and often apply. The physically least stable is the amorphous and the most stable is form Ijudging from the solubilities. The least stable crystalline form has the lowest density. . ixtures of ~olymorphsmay occur, but the extent to which this may occur be of cangreat importance and cause great difficulty in product development. As an example, Cipro~o~acin-HCl Forms No. of solvent Solubility at Denslty ~elting point moles per mole of 37°CSample (g/cm3) (“C) drug (mgicm3)Amorp~ous 316.7a 70313.5 IForm 0.796 3 54Form I1 0.980 3 12 1.5 45Form316.3 I11 1.042 0.5 34 aSoftenmg point.
  • 141. 1 terBergren et al. (1996) have described two forms of delaviridine mesylate (forms VI11and form XI) both of which are anhydrous and nonhygroscopic. Sarver et al. (1998), however, crystallized delaviridine mesylate from acetoni-trile at room temperature asforms I and 11. Both forms were very hygroscopic, andthey subse~u~ntly recrystallized the compound from methanol under reflux. Acetonewas added as acosolvent. This produced either form XI or form VIII, depen~ing onthe amount of cosolvent used. In distinguishing between the many polymorphs, Sarver et al. employed,factor alinowski and Howery, 1981), in which large sets of data canbe segmen- aller sets of orthogonal components. Of these, the first describes the e data, and followi~gcomponents deal with variance of less this means it is possible to distinguish between polymorphs inmixtures.It has been shown in the previous sections that the less physically stable the poly-morph or solvate, the higher is the solubility. Because dissolution rate, that is, thecharacteristic that is of importance biopharma~utically, directly related to solu- isbility, there must be a connection between the two, This has, indeed, beenshown for phenylbutazone ( ~ a t s u n g aet al., 1976;Ibrahim et al., 1977; Muller, 1980; Tuladhar et al., 1983; ~ a t s u m o t e al., 1988; etKaneniwa, 1988); for mefenamicacid (Aguiar and Zelmeer, 1969); for diflunisal(Martinez-Ohariz et al., 1994); for indomethacin (Kaneniwa et al., 1985); for tenox-icam (Nabulsi et al., 1992); and for oxyphe~butazone (Stoltz et al., 1988). Tros de Ilarduya et al.(1997) have described the dissolution behavior of pseu-dopolymorphs of sulindac. Table 8.4 shows the effect of solvates on dissolutionrates. It would appear from the data in the table that the solvates are metastable inrelation to the non-solvate forms. ~ ~ y~ a ~ i ta affect dissolution rates as well (~arstensen,1973; ~ t may ~Mitchel, 1980, 1981; Chow and Grant, 1989). The dissolution pattern of a metastable polymorph can be one of two types asshown in Fig. 8.10. The metastable solution may be fairly stable, so that the con- Dissolution Rates of Polymorphic Forms of SulindacI 0.036I1 (tabular habit) 0.03 1I1 (hexagonal habit) 0.036Acetonate 0.076~h ~o r o f or mat e 0.076Source: Tros de Ilarduya et al. (1997).
  • 142. centrations with time will approach the metastable solubility (the middle curve); atone point in time, the stable modification may start precipitating out, and the con-centrations will drop and eventually approac~ that of the stable modification. The latter phase is one manner in which r e c r ~ ~ ~ a Z Z i zcani be carried out a~ o~strictly isothermally. The latter points of the precipitation in Fig. 8.10 (the micurve) are frequently plotted semilogarit~mically versus time.Good stability of a metastable compound can be achieved by (a) low temperature,(b) coarse crystals, and (c) dry storage. The moisture is the most serious contributorto conversion. oisture will condense onto the surface of the metastable form (11), and willthen saturate the moisture layer to form a solution that supersaturated in (I). This iswill eventually nucleate, and all of the I1 will convert to I. The conversion rate, therefore, is a function of the nucleation rate in “s0lu-tion?” and it is a well known fact ( ~ u l l i n ,196l), that the n~cleationrate J isinversely proporti~nalto the viscosity of the solution, and also to the supersatura-tion ratio AS, by the following relation: J = A exp(q / [ {T 3In AS) + (AGJRT)]) (8.22)an equation discussed in Chapter 7 . For very soluble compounds, A S will be a verysmall number, and the tendency for one polymorph to change into another will bevery small. F ~ r t ~ e r m o rif ,the solubility is high, then the AGm terms will not differ emuch. An example of this is ranitidine (Carstensen and Franchini, 1994a,b). AHaT= heat of solution at saturation, lsCal/mole R = gas constant S = solubility, weight per weight of solvent Metastable Pre~ipitationof 0I 0 5 10 Tine iss solution of (squares) a metastabl~~ o l y m ~ r p h ; (circles) a metastable poly-morph that on dissolution c o m m e n ~ s convert to a stable m o d i ~ c a t i ~ and (triangles) a to n,stable polymorph.
  • 143. T = absolute t ~ ~ ~ e ~ a t ~ ~ e = constant in. the Van’t
  • 144. Sustar I3, Bukovec N, Bukovec P (1993). J H e m Anal 40:475.Tros de Ilarduya MC, Martin C, Goni MM, Martinez-~h~rriz MC (1997). Pharm 23: 1095.Tuladar MD, Carless JE, Summers MP (1983). J P h a m Pharmacol 35:208.devilliers MM, Wurster DE, van der Watt JG, Ketkar A (1998). Int J Pharm 163:219.Yu L (1995). J Pharm Sei 84:966.
  • 145. This Page Intentionally Left Blank
  • 146. 9.1. Substances that 134 isture Adsorption or Absorption on or into Large, Crystalline 138 stalline, Non-~ydrate-Forming, Water-Soluble Substances 1389.4. Condensation 1409.5, Critical Moisture Content 1429.6. Equilibr~umMoisture Curves €or Salt Hydrates 144 Mode of ~ a t e Vapor ~ressure9.7. Presentatio~ r Versus ~ o ~ p o s i ~ i o n Diagrams 1459.8. E ~ u i l ~ ~ r iC o ~ p o u n d Forming a Crystalline Anhydrate and of a s 1469.9. Critical T ~ m ~ e r a t uand Pressure re 1489.10. Equilibria of Compounds Forming a Crystalline Anhydrate and Two Hydrate Forms 1509.11. bria of Co Forming a Crystalline Anhydrate and than Two Forms 1519.12. Moisture Equilibrium Curves of a Smooth Nature 1539.13. Solvates 156 Symbols 156 eferences 157
  • 147. hen a solid is placed in atmosphere, it will adsorb (or absorb, depending on the ansubstance) moisture from the atmosphere, The rate and extent to which this occurs is Asusually referred to as ~ygrosco~icity. well as a kinetic property it also contains athermodynamic one, and definitions, such as those used for solubility (e.g.,veryslightly and slightlysoluble) are not possible. At best one may talk about veryhy~roscopicsubstance (choline salts, for instance) and very nonhygroscopic sub-stances (sand, for instance), but the large gamut of substances call for more detail ribe their hygroscopic classification. y its nature, the concept involves pickup of moisture, and that such pickupmay be moderate at certain relative humidities and extensive at others indicates thatclassifications such as those used for solubility are not possible. There are seven distinct categories of solids, which will be treated in separatesection in the following: 1. ubstances that are “completely” water-insoluble (e.g., silica) 2. ubstances that do not form stoichiometric hydrates, but can ~ ~ s o r moisture by penetration (e.g., montmorillinite or cornstarch) 3. Crystalline substances that are(moderately or very) water-soluble, but do not form hydrates 4. Amorphous substances 5. Crystalline (anhydro~s) substances that f o m one hydrate 6. Amorphous anhydrates that form one crystalline hydrate 7. Crystalline anhydrates that form several crystalline hydratesThere is always somewater s~lubility associated with a compound, however poorlysoluble, and the characterization “completely water-insoluble” should be taken inthis vein. An example of this is silica gel.This substance, owing to its large surface area, as a desiccant in packaging of moisture-sensitive drugs and drug pro- silica gel is exposed an atmosphere of a given relativehumidity (RH), to n in Fig. 9.1 and Table 9.1, the weight of the sample will first rise fairly inetic phase) and the rate of this is referred to as the ~ o i s t ~~ e ~ t a ~ r In Fig. 9.1 it is noted that there is an (approximately) linear rate at low timepoin these are shown in Fig. 9.2, and are seen to befaidy linear inlowlues. In Fig. 9.1, it is also noted that the curves eventually 1e~uilibrium level is a function of the relative humidity at which the e ~ ~ e r i m eis t ncarried out, Table 9.2shows an exampleofthese equilibrium values at variousrelative humidities. The levels are tabulated in the second column of Table 9.2.The e~uilibrium values are plotted as a function of relative humidity in Fig. 9.3. It is customary in isotherm work to convert these adsorbed amounts to thevolume that would have been occupied 0°C and 1 atm, andthis can easily bedone; atfor example, for the first in row Table 9.2, the number of is molesn = 35 x 10-3/18 = 19.5 x mol. The volumeof this at 0°C and 1 atm wouldbe IV = ~~~/~= 19.5 x x 8.2 x 273/1 = 43.7 mL. These figures are shown in
  • 148. s 60%RH 4011 RH 20% RH I f 10 20 Time (days) Moisture uptake curves for a sample of silica gel at 20, 40, 60, and 80% R y - 0.12100 + - 3.2550e-2x R*2 0.966 0 RH Moisture uptake rates as a function of relative humidity for a “~ater-i~soluble”compound. Typical Adsorption Data as a Function of TimeTime (days) 20% RH 40% RH 60% RH 0 0 0 0 0 113.3 10 41.5 20.623.9 2 18 74.7 37.134.5 5 26 107.9 53.610 31 41.2 64.0 128.61543.8 33 136.9 64.12045.1 34 70.1 141* 1
  • 149. .2 Equilibrium Values and BET Parameters from the Data in Table 9.1 H Adsorbed (mg) V (mL, STP) RW/[ 100 - RH)] V(0 0 020 35.0 43.6 0.00640 47.8 59.5 0.01 160 72.2 89.9 0.01780 145.2 180.9 0.022the third column and are denoted Y . The R-value used is in units of cubic centimetersatmosphere (cm3atm). As mentioned in Chapter 5, curves of this type are called BET isotherms [seeEqs. (5.57) and (5.58)]. The data in the third column are shown in Fig. 9.4. It can beshown that such data follow the BET equation: - + RH/[~{lOO RH}] = (1/{Ymc~((c - l ) / ~ c Y ~ ) } ~ ~ H / l O O ~ (9.1) (STP) adsorbed; Vmis the volume of a monolayer; c is the c- equation, and the value of c is often large, so that the value of - RH/~V{lOO RH)] =: I (l/~~)[RH/lOO] (9.2)Treatment of the data in Table 9.2 by Eq. (9.1) is shown in Fig. 9.4. Yjyt, as mentioned, is the adsorbed volume (OOC, 1 atm) of water that consti-tutes just one layer on the entire surface of the solid sample. RH/[ Y(100 -been calculated in Table 9.2 (last column), and is plotted in Fig. 9.4 versus t is seen (as mentioned in Chapter 5) that 20 40 GO 80 100 RW Data from Table 9.2 plotted as adsorbed amount as a function of relativehumidity.
  • 150. y - 2.7475e-4 + 2.?3 12e-2x R*2 = 1.000 0.0 0.2 0.4 0.6 0.8 1.0 RH/ 100 Data in Table 9.2 treated by Eq. (9.3). ~lope/intercept c - 1 =so that c - 1 = 0.0273/2.75 x lov4= 99 (9.3)so c = 100Hence, we may use Eq. (9.2), such that I / Vm= slope = 0.027that is, vrn 1/0.027 = 37 cm3 =This can be converted to moles (n) and then to molecules (N): n, = PVm/RT = 1 x 37/[82 x 2731 = 16.5 x lom4mol (9.7) =6 X X 16.5 x = 1021moleculesA cross-sectional area of 12.5 x m2/molecule water is of usually employed( ~ ~ d i p e d d1996),so that in this example the entire surface area wouldbe the i,number of molecules times the area of each molecule: lo2 X (12.5 X = 125 m2 (9.8) If the weight of the sample were 4 g, then the specific surface area would be31.25 rn2/g. If a bag of silica is placed in a bottle with a dosage form, then, if there is acritical moisture content beyondwhich the dosage form becomes unstable, it ispossible to calculate from the isotherm of the dosage form, at which relative humid-ity this occurs, From the silica isotherm, one may then calculate how much moistureis taken up by the silica bag at this point, and dividing this figure by the moist~repenetration of the package, it is possibleto calculate the length of time the product isgood. This will be covered further in the section under pharmaceutical packaging.
  • 151. oisture isotherms are of great significance in ph aceutics. Cases in points moisture isotherms of polyvinylpyrrolidone (P and misoprostol-hydro-xypropyl methylcellulose complex.For an organic compound, such as starch, a smooth equilibrium moisture curve will again, there is the sharp upswing at very high relative humidities. riments, such as exemplified in the forego in^ are carried out on corn-starch, for example, then results of the type shown in Tables 9.1 and 9.2 and Fig. 9.3result. When RM/[V(l - P)]is plotted versus RH, then a Other examples include, for example, microcrystallingivesrise to BET isotherms. The surface area obtainedm any fold larger than the area obtained by nitrogen ad(1972) and Zografi and Kontny (1986) have shown that water penetrates the solid,and that each OH group in the MCC co~stitutes adsorption (absorption) site. anCompounds of this nature are, for instance, organic, nonprotic c ~ m p ~ u n (e.g., dssucrose), organic electrolytes(e.g., choline salts), and several electrolytes (e.g.,sodium chloride). This phenomenon has been dealt with by Van c amp en et al. (1980), Thepurpose here is to derive a rational equation for the rate with which moisture isadsorbed onto a water-soluble solid. As me~tioned, a solid is evacuated (Fig. 9.5a) and then placed in an atmo- ifsphere that has a vapor pressure Pa, which is lower than the vapor pressure P, of thesaturated solution of the compound, then (see Fig.9.5a,b), moisture will adsorb ontoits surface by the same process as nitrogen in a Modes of moisture adsorptlon: (a) evacuated solid; (b) active sites (BET modelapplies, below the critical relative humidity; (c) bulk, saturated solution at exactly the criticalrelative h u ~ i ~ i t y ; bulk, unsaturated solution. at RH values above the criticalrelative (d)humidity.
  • 152. owever (seeFig. 9.5c,d), once the vapor pressure in the atmosphere Pa equalsthat of the saturated solution of the compound Ps, then the condition exists in whicha bulk layer o~moisture thermodynamically feasible. is If P, > Ps and the volume of the atmosphere is infinite, then water will con-dense onto the solid, and the s ution formed willbe saturated (because it is inequilibrium with a solid phase). his will continue until all the solid has dissolved,a phenomenon known as deliquescence. It will, however, continue until the concen-tration of the now unsaturated solution is such that its vapor pressure ma~ches that ofthe atmosphere (Le., until it is P,), because nature will attempt to establish equili-brium. A consequence ofthis is that if a solid is placedin an atmosp~ere ~ fRH, lthen con~ensationwill continue ad i n ~ n i t u m ~ the volumeof the vapor phaseis ifinfinite, because in this case nature will attempt to establish the equilibrium thatexists at a concentration of solute of zero (pure water). ecause vapor-phase volumes of infinity are not within the realm of thesible, it is worthwhileto consider a more realistic situation forwhich these principlesapply * The following nomenclature will be used in the following: 6 = volume of vapor phase . P, water vapor pressure of vapor phase before condensation = P, water vapor pressure of vapor phase after condensation = Po = waters vapor pressure at a temperature of T R = ideal gas constant 77 = absolute temperature w = moles of water condensed s = solubility of compound in moles/mole of water n = number of moles of solid dissolved e now distinguish between two situations: (a) one where there is insufficientwater present in the vapor phase to dissolveall the drug (and a so-called b ~ lmoisture layer is formed), and (b) where there is sufficient moisture in the vapor to dissolve all the drug and form an unsaturated solution (deli~~escence). ith the cited nomenclature and the situation depicted in Fig. 9.6 it followsthat, with insufficient moisture for deliquescence, the amount of moisture condensedis such that the amount of solid dissolved is given by ws = n. w = n / s (9.9) After condensation the vapor pressure will be Ps. This quantity is very signifi-cant because, when the subject is the stability of drugs in exposure to moist atmo-spheres, this is the a ~ o u n t the sorbed bulk moisture layerand this dictate^ the ofexpected stability rate of the drug s~bstance. In the case of deliquescence, the solution formed is not saturated, but will havea con cent ratio^ of n/w (i.e., the mole fraction in solution will be x = {n/w)/{1 + (n/w)>= n/(n + w) (9.10) If the solution is considered ideal, then the vapor pressure P,, after equilibriu~has set in, will be + P, = Po(1 - x ) = P(){w/(n w)} (9.11)
  • 153. V ~ V a~ u ~ a l e V ~ Vapor V o ~ V u ~ ~ Vapor Pressure Vapor Pressure Pa A f t e r Cond. = PX n Moles of Mater C~nd~ns~ Schematic for moisture adsorption on a solid in a closed container.Here, n is known, as it is the number of moles of solid (all of which have dissolved).Px is related to w’by the ideal gas relation (Pa P x ) ~ w f ~ ~ - =/ (9.12)Eqs. (9.11) and (9.12) constitute two equations with two unknowns (Pxand wf), andcan be solved.In the following, it will be assumed that the foregoing situation (a) exists (i.e,, thatthe condensed water will dissolve solid), and it willbe assumed that the sorbedsolution is saturated at all times. The question is, what sort of curve might beexpected for the extent of moisture uptake withtime (the moisture uptake ratecurve; the MUR curve). A further assumption is that the amount of moisture adsorbed does not, to agreat extent, change the vapor pressure in the atmosphere surrounding the solidparticles. Assume that, at time t (Carstensen, 1986), a certain amount of moisture, w(grams) has been adsorbed by a particular solid particle weighing w1 grams and ofdiameter do, at which the subscript denotes the condition before moisture adsorp-tion. At time t, moisture will have adsorbed, some solid will have dissolved, and thediameter d of the solid itself will have decreased its original value. The diameter fromof the ensemble D is the sum of the diameter of the remaining solid, and the thicknessh of the moisture layer. It is assumed in the following that 1 g of solid is studied and that the sample is~onodisperse. Such a sample would consist of N particles where: NTE= Npndo3/6 == 1 (9.13) The amount of solid present at time t is given by the original amount less theamount dissolved. If there are W grams of water adsorbed by 1 g of solid (i.e., wgrams dissolved per particle), then,
  • 154. ist ~ ( ~ 7 t / 6 )= N(m - wS) = 1 - WS d3 (9.14)where S is solubility in gram/gram. Therefore: d3 = (1 - W S ) / ( N ( p ~ / 6 ) (9.15)The volume of liquid adsorbed by one solid particle has a volume ofthe total particleminus the solid particle; that is, (9.16)where p* is the density of the adsorbed liquid. Because it is assumed that it is alwayssaturated, it is time-independent, and under ideal conditions it would be P* = (1 - X,)Po + X,P (9.17)where (1- x) and x, are the volume fractions of liquidand solid, respectively, inthe ,ensemble particle, and po and p are the respective densities.It follows from Eq. (9.17)that the amount of moisture adsorbed per gram can be expressed in terms of dia-meters as follows: w = p*N(7t/6)D3- [p*(l - WS)/p]= QD3 - I ; + FSW (9.18)where I; = P * l P (9.19) e =p * ~ ( ~ l 6 ) (9.20) quat ti on (9.18) may be written:or D = {[F+ (1 - FAY) w]/Q}1/3 (9.22)The area a of the particle (solid plus liquid) is, hence, + a = 7 t { [ ~ (1 - F A Y ) W J / Q }= / ~ i w ~ B[E - ]~/~ (9.23)where E = {I;Q/(l - FAY)}2/3 (9.25) The rateof condensation ( d w / d ~ is proportional tothe pressure gradient (Le, )the difference between the water vapor pressure P in the atmosphere and the vaporpressure P, over a saturated solution). At a given atmospheric milieu, this gradient isa constant. It is also proportional to the surface area a, by a mass transfer coefficient k , sothat we may write: dW/dt = ka(P, - P,) = k(P, - P.,)B[E + v2/3 (9.26)where Eq. (9.23) has been used for the last step. This may be written:
  • 155. dW/{[E + = 3Gdt (9.27)where 3 6 = k(P - P,)B (9.28) quation (9.27) integrates towhere the initial conditions that W = 0 at t = 0 has been imposed. ~ ~ u a t i o n (9.29)can be solved by iteration. As an example of this,~ a n C a ~ p et nal.(1980) studied the moisture pickup in ea vacuum system by usinga Cahn balance, and exposing the evacuated head space torelative humidities created by salt baths. They also reported moisture uptake rates ofcholine chloride at room temperature and different relative humidities using a desic-cator method, An example of their results obtained by the latter method is is shownin Fig. 9.7. here are humidities below which a solid will not adsorb (considerable amounts) ofmoisture (Le.,will not form a “bulk-sorbed” layer). As already covered to someextent, these are dictated by the solubility of the compound. ~akobsen al. (1997) et scussed this situation for three drug substances. uppose a solid with a high solubility is placed in a room of a givenshown in Fig. 9.8. If the RH were 30%, then it ght pick up moisture at a givenrate, at 50% RH at a higher rate, and at 80% at an evenhigher rate. The rate with which it picks up moisture is determined by weighing the sampleat given intervals, as demonstrated in Table 9.3. It is noted that there is a linearsection of the curve (up to 6 days) as shown in Fig. 9.9. The slope of this linearsegment is the moisture uptake rate ( ~ ~ RThe . actual uptake rates (determined )from the linear portions) are shown in Table 9.4. The uptake rates can be simply obtained by weighing the sample after a giventime (6 days), but in such a case it is assumed that the moisture uptake is still in the 100 80 60 40 20 0 0 300100 200 Hours Data for choline chloride rnolsture adsorption at 100%V a n ~ a ~ et al,, 1980.) pe~
  • 156. of Mecha~ism moisture uptake. 0 100 200 300 400 Days at 50%RH uptake data from Table 9.3. ~oisture Moisture Uptake of a Highly ~ a t e r - ~ o l u bCompound at 50% R lestored Days at 50% RH Moisture pickup (mg/g) 2 0.5 6 1.5 38 2.25 36 3.4 €00 3.0 144 4.2 288 4.3
  • 157. ter Moisture Uptake Rate of ~ a t e r - ~ o l uCompound ~le25 0.150 0.2580 0.45 It is known as the cri~ical o i s t u r e o n t e ~ and the critical relati~e ~ c t humidity, fora non-hydrate-forming compound. Curves, such as the one shown in Fig. 9.10, formost salts intersect at much higher relative humidities. Because the value is related tothe solubil~ty the compoun~, this, on a mole fraction scale, is a rather small of andnumber, the saturated solution is often (nearly) ideal, and the reverse procedure maybe used; namely, from plots such as that in Fig. 9.10 and the deduced value of thecritical relative humidity RH*, the mole fraction at saturation y may be calculatedfrom *=l-y (9.30)The previous section dealt with the rate with which moisture is taken up. As shown ig. 9.9, at longer time periods, the moisture level (the weight of the sample) willtaper off and plateau at an equilibriu~value. This equilibrium value is also afunction of RH, and tbere are two types of curves that occur when equili~riumvalues are plotted against RH: salt pairs and continuous adsorption. The formerwill be discussed first. Many co~pounds, especially ionic c o ~ ~ o ~ nform, hydrates. A hydrate is dsdefined as a ~ h e ~ i cc ol~ p o u n d a with a rational ratio of water to anhy~rous o ~ p o ~ n d cat d ~ f e r e n t tem~eratures. It is visualized that the water molecules occupy definitive positions in thecrystal lattice. In some cases (e.g., ~ontmorillonite), different amounts of water RW oisture uptake rate as a function of RH. Least-squares fit is y = -0.06264 + O.O06374x, with R2 = 0.999.
  • 158. may be adsorbed or absorbed, and the crystal spacings between the layers of alumi-num magnesium silicates increase in proportion, but the curve is continuous (Le.,shows no inclination to be stepwise in nature). This is an i n t e ~ e d i a t e case, andhydrates usually have stepwise profiles when equilibrium vapor pressure is plottedversus composition. The question is then whether this water sits “in a channel” (asappa~ently it does in montmo~llonite)or is bound in a different manner.Occasionally, the water molecules are part of the coordination shell of an ion, asfor instance in magnesiumchloride, which existsas, among others, a dihydrate and atetrahydrate. The anhydrate can be produced by interaction between metallic mag-nesium and hydrochloric acid gas. Heating magnesium chloride tetrahydrate to 80“100°C will remove two of the molecules of water. ut further heating results in theremoval of 2 mol of hydrochloric acid, leaving magnesium hydroxide behind. The cases that will be discussed in the following are of the type for which ananhydrate can be produced by heating or by vacuum.The vapor pressure of a salt hydrate as a function of “composition’’ will be referredto in the following simply as the vapor pressure pro~Ze the h y ~ r ~ tSome of the of e.concepts to bediscussed are quite ancient, but since t h e ~ o d y n a ~ i c s not a arefunction of the calendar time at which they were formulated, they are presentedwith reference to original works. There is a great deal of rejnve~tjon in the field,primarily because a u t o ~ a t e d literature searches usually do not go farther back than1970. There are different conventions for presentation and the one proposed by rransted (1943) is as follows (Fig. 9.1 1).PH20 1 P U E H H20 cuso4 Mole Fra~tion CuSO4 --> Vapor pressure profile for CuS04 hydrates. (Data from Brmnsted, 1943.)
  • 159. rmnsted (1943), as opposed to convention nowadays, considers water the leftordinate axis, and starting at point P, pure water’s vapor pressure, salt is added toform unsaturated solution (u). At point A the aqueous phase is saturated withcopper sulfate, and the solid phase is the pentah~drate. suf~cient If copper sulfateis added to point , then the interchange inIn the situation depicted in the previous sections, the compound (copper sulfate) wascapable of existing in a crystalline anhydrate and several hydrate forms. Frequently,only one hydrate form exists, such as lactose ~ o n o ~ y d r aand p o t a s s i u ~ te tartrate ract with moisture to give a hydrate, say an x-hydrate, (9.32)The equilibrium constant is given by K = P&,* (9.33) The general situation is depicted in Fig.9.12. This diagram applies to oneparticular temperature and when the water activity (a = Hiloo) is low, only theanhydrate exists. At a given value ofa , a lhowever, the x-hydrate will start formi~g, ,
  • 160. 2 Constant Temperature (e.g. 25°C) AX+Y X X - P diagram for a compound that forms (only) a monohydrate.and withinfinitevolume vapor phase the reaction according toEq, (9.32)will . Increasing the vapor pressure to between al and a2 will not (but will cause some surface adsorption). At a2, however, thevapor pressure is equal to the vapor pressure of a saturated solution, so water willstart ing, and (with an infinite vapor phase volume) adsorption will continue(line il all the salt has just dissolved to create one phase. e vapor pressure is further increased, then dilute sol~tions form and willthe profile, if the solutions are ideal, will be as shown in the C E part of the curve. From Eq. (9.33) it follows thatwhich, i A H is te~perature-in~~pendent be integrated to f can + ln[P] = - ~ A H / ~ ~ ~ } (q l / ~ ) (9.35)F is here water vapor pressure, A H is the enthalpy involved with one ~ o l e c u l e ofwater, x is the number of moles of hydrate water per mole of compo~nd, q is a andconstant. It is seen in Fig. 9.13 (potassium tartrate dihydrate) that when A H is fairlytemperature-inde~endent, then log-inverse t e ~ ~ p e r aplots are linear. t~~e The slopes of the vapor pressure over the salt hydrate and the saturated solu-tion differ because the enthalpy from the slope o f the hydrate is for the reactionshown in Eq. (9.33), wher~asthe e~thalpyfor the saturated solution is for thereaction: (9.36) ~ a p o pressures over salt hydrate pairs also increase with te~perature r (9.5). The situation, hence, is that the vapor pressure of the x-hydrate will increasewith temper~ture will the vapor pressure of a saturatedsolution of the c ~ m p o u n ~ . asIn Fi . 9.13 neither the salt pair nor saturated solutions have a vapor pressure of 760 (1 atm) until a temperature of 115°C is reached.
  • 161. y - 6.9742 - 0 , 5 6 5 2 7 ~ R-2 - 0.975 0 h[sI y . 19.199 - 4 . 8 7 3 0 ~ R-2 - 1.000 0 1nlPI 2.4 2.6 2.8 3.0 3.2 3.4 3.6 1000/T 3 Vapor pressure data and solubility data of KzC4H406. (Data from Krack, 1998.) ATIn the situation shown in Fig. 9.13, the hydrate stays as such in the entire tempera-ture range of 10-1 15", and the general shape of the diagram in Fig. 9.12 wouldapplyat any temperature in this region. The vapor pressure of both salt hydrate and saturated solution increases withtemperature, and (as seen in Fig. 9.13) whenthe solubility increases with temperatureits vapor pressure most often increases more rapidly for the hydrate than for thesaturated solution. Therefore, there will often be the case(see Fig. 9.14)wherethe x-hydrateachieves the same vapor pressure as the saturated solution vapor pressure and tern-perature. This is denoted the critical t e ~ ~ e ~ and at ~ e a t ~ temperatures above thistemperature only the anhydrate is thermodynamically stable. For a monohydrate as depicted in Fig. 9.14,the increase in vapor pressure overthe salt hydrate will increase drastically once more than 1 mol of water is present per Influence of Temperature on Vapor Pressure( m ~ H of Salt Pairs ~ ) NaHP04.12Hz0 -+ SrC12.6H20 -+~ e ~ p e r a t u ("C) re NaHP04.7HzO + 5HzOa SrC12.62120 + 4H20b 0 2.66 1.2315 8.95 4.020 12.93 8.425 19.1830 27.05Source: aPartington and Winterton (1930); bBaxter and Lansing (1920).
  • 162. FEE 0 1 2 Mole W ~ t ~ r / ~ Solid ole Single salt pair (monohydrate)vapor pressures as a function of temperatur~. Theline at point A has been drawn slightly to the left for graphic clarity. It occurs at 1 mol ofwater per mole of solid. (Data from Carstensen, 1986.)mole of solid. Moisture will then keep on condensing and converting the monohy-drate to saturatedsolution, and this will continue until all is dissolved. After that thevapor pressure will increase so that it is always inequilibrium with the concentrationin the (now) unsaturated solution. The diagram in Fig. 9.14 is at a given tem~erature and shows a compoundcapable of forming a monohydrate at different temperatures. At the temperature T3the line for the salt pair has “caught up” with that o f the saturated solution. AboveT3 the salt would have a higher vapor pressure than the saturated solution, but this isthermodynamically untenable, and T3is simply the highest temperature (and a triplepoint) at which the monohydrate exists. It is the critica2 temperature for the hydrate. If a saturated solution consisted of 1 mol of salt and y mol of water, thenvapor pressure of the saturated solution, if it were ideal, would be (9.37)At higher relative humiditi~s,water will condense on the solid to form an unsatu-rated solution. If this is ideal, then adding z mol of water to 1 mol of solid wouldgive a mole fraction of a = P/Po = 1 - [l/(z + 1) + l)] = z / ( z (9.38)so that z = a/(l - a) (9.39)as a ”+ 1; z + 00. The shape of this curve is as shown in the curve in the right handsection (CDE) of Fig. 9.12. The data presentation in mole fraction (see Fig. 9.11) is simpler, because thesection PA is simply a straight line, if the solution is ideal. It follows from Eq. (9.39) H the system in equilibrium is infinite dilution (pure water), and if adiagram as this ised out to 100°/~RH, then a sharply increasing curve shouldresult at very high
  • 163. If there are two salt pairs, an m-n, an n-0 pair (m n), there are two possible (ICsituations. One is that the m-forms critical temper~ture in caseis belowthat of the n-forms critical temperature ( L in case B in Fig. 9.1s). The other(hypothetical) case is where the m-forms critical temperature ( N in case A in Fig.9.19, is above the critical tem~erature , of the n-form (case A. in Fig. 9. IS). In the Mlatter, hypothetical case the m-n vapor pressure curve crosses that of the saturatedsolution (denoted s) at T*,and above this te~perature would (if it existed) give rise itto a situation in which the vapor pressure of the m-salt would be higher than that ofthe saturated solution. This is not feasible thermodyna~ically,so that the criticaltemperature for the m-hydrate in this case would have to be the point at which itreaches the vapor pressure of saturated solution. Such a case has been reported byChen and Grant (1998) for nedocrornil the sodium trihydrate/monohydrate system. 1 of that publication fails to show three plateaus, and the nick in tion vapor pressure curve(s) is absent. he correct profile for this situation is depicted ~raphicallyas case9.15, and schematically in Fig. 9.16. Noteworthy are the nicks in the solubility curve at tem~eratures and N and at X and L. T" ften, when there is a transition between two salt pairs there will be an inter- S . between m-n and n-0, and above the transitio the stable specieswillbe ( ed, 1943). A diagram similar to that of Fig. 9.1isshown for cases A and Fig. 9.16. Case A -- 7 SatdSoln. easeB n 0 10 20 30 5 4 0 0 GO 70 Temperature, * C Water vapor pressure diagrams of a compound forming two hydrates. In the first here is a conversion below the boiling point of water, in the second the mn-hydratevapor pressure reaches the vapor pressure of the saturated solution above the boiling point ofwater. Note the nicks in the saturation curve at points E and L. Often (but not shown in the ;figure), if there is a tra~sitionbetween two salt pairs there will be an intersection between, for ample, m-n and n-0, and above the transition the stable specieswillbe m/O. (From
  • 164. T = 25°C T = T* P X X (a) Case A o/ 1 /2 012 Thereare two distinct situations hypothetically possible: (i) as temperatureincreases, the dihydrate vapor pressurewould reach its critical point T", when its vaporpressure would equal that of the saturated solution (Fig. 9.16(b)). At T > T* the diagramwould sunply bea P - x diagram of a single hydrate (the monohydrate, Fig. 9.16(c)). The solidphase in equilibrium with saturated solution would be the monohydrate. There would be anick in the vaporpressure curve at T". (b) As temper~ture increases, the vapor pressure theofmonohydrate will reach that of the dihydrate at a critical temperature, T", for the monohy-drate (Fig. 9.16(d)). Above T * , the ~onohydrate, it existed, would have a vapor pressure ifabove that of the dihydrat~, which is not possible, so at T > T" the P - x diagram will be thattypical of a single hydrate (i.e., the dihydrate,Fig. 9.26(e)). It is noted that there is a nick in thevapor pressure curve.An example of this situation has already been described, albeit it not indebil, in Fig.9.11. The numerical data ( ~ a ~ H ~ shown in Table 9.6. The compound can are ~ ~ ) ,form three hydrates (2, 7, and 12) aside from a crystalline anhydrate. In the usualpresentation mode (i.e., not using molar concentration and content units), the per-centage of moisture in the dihydrate, for example, is calculated as follows: disodiumhydrogen phosphate has a molecular weightof 142; hence, the dihydrate has amolecular weight of 142 + 36 = 178; accordingly, the moisture percentage is100 x (36/178) = 20%. The moisture contents for the remaining hydrates areshown in Table 9.6. It is seen in the table (and from Fig. 9.17) that the relativeh~midityof the at~osphereabove a mixture of anhydrous disodium hydrogenphosphate and the dihydrate is g 100 x (9/24) = 38% RH. Any orof the anhydrous salt and the rate will give this relative h u ~ i d i t y .disodium hydrogen phosphate containing between 0 and 20% moisture will have
  • 165. ~haracteristicsof Disodium Hydrogen Phosphate Moisture in Water activity solid (YO) pHz 0 (RH/lOO)Anhydrous 0 Pair 9 0.38Dihydrate 20 Pair 14 0.58Heptahydrate 47 Pair 18 0.75Dodecahydrat~ 60 0.92 Pair 22Saturated solution(100 g ~ater/4.5 salt) gSource: Maron and Prutton (1965).above it an atmosphere of 38% RH. Similarly, as shown in the table, the heptahy-drate contains 47% moisture, and mixtures of di-and heptahydrate give riseto watervapor pressures of 14 mmHg (58% RH). Similar plateaus exist for heptahydrate anddodecahydrate. Two further points need to be mentioned: (a) If disodium hydrogen phosphateis stored at an RHbetween 38 and 58%, it will not pick up moisture (or will pick uponly surface moisture). Once the relative humidity is raisedto (slightly above) 58%,then it will start picking up substantial quantities of moisture until it has completelyconverted into the heptahydrate. (b) If the relative humidity is raised to (slightlyabove) 92% RH, then the dodecahydrate isconverted to saturated solution. Athigher RH values the equilibrium willbe dictated by the water vapor pressureover the now unsaturated solution. In the solu~ilit~ there is a l w a ~ a nick in the curve at t h e ~ o i nwhere there is plots s ta c~iticaZt e ~ p e r a t u ~It .is obvious that the heats of solution of two hydrate forms ewould be different, and this causes a different slope of the solubility curve. This is M Dodecahydrate Dodecahydrateb Heptahydrate Dihydrate Anhydrate x .r( U rrl Temp "C 7 Solubilty of Na2HP04 hydrates as a function temperature. of (Data fromBrrirnsted,1928.)
  • 166. Decahydrate b-Heptahydrate PPDecahydrateI 1 PPHeptahydrate/ 0 0 10 20 30 Solubility and vapor pressure data of sodium carbonate hepta- and decahydrates.PP denotes vapor pressurecurves in centimeters mercury. Solubility data (grams of solid per100 g of water). (Data from Kracek, 1928.)exemplified for the disodium phosphate system in Fig. 9.18. It is seen here (for thesolubility curve of the anhydrate at higher temperatures), that solubility does notalways increase with temperature. In this case the heat of solution is of a differentsign so the solubility decreases with temperature. In general, however, there is acorrelation between water vapor pressure and solubility of hydrates. There are situations for which A H is not constant with temperature, but insolubility plots it often is. It is necessary to express the solubility in moles of soluteper mole of solvent, The situation is shown for sodium carbonatedecahydrate and ~-heptahydratewhere both vapor pressure and solubilities are listed. It is seen that the transiti~nte~perature (the critica~ te~perature) is apparent (32°C) fro^ both types o curve. fIt was mentioned earlier that compounds, such as gelatin, exhibit watervapor inter-actions that give rise to smooth (not stepwise) isotherms, and that these may be ofa ET nature. If such a substance is evacuated and allowed to adsorb moistsure up toa water activity closeto unity, then a curve such as (a) in Fig. 9.19 will result. If thepressure is reduced again, then a curve will result that is different from the adsorp-tion curve (a): desorption curve (b). This pheno~enon known as hysteresis. The isordinates will be denoted simply as x and y in the following. It is noted that yd is not anequilibrium condition. Obviously, AG is negative ingoing from the down-curve to the up-curve, because
  • 167. 4 Down Curve (b) 0.2 0.0 0.4 0.6 0.8 1.0 1.2 Water Activity (RHll 00) ET adsorption and desorption moisture isotherm. everal common tablet excipients give riseto Langmuir isotherms. In excipientstudy by ~angvekar (1974), when all the data are lumped together, they follow anequation of the type: + 1/y = ( A l p ) B (9.41) Usually, in pharma~utical engineering literature, the moisture equilibrium andcurves are shown in a sense opposite that shown in Fig. 9.19, that is, (9.42) The high RH tail of the curve is usually above 85% R and, therefore, is notapplicable to most realistic pharmaceutical conditions, but it is applicable to one cted test (46C, 75%RW). r routine isotherms, the high relative humidity tail is difficult to obtain with precision, and one approach (Carstensen, 1980) is to approximate them h isotherms (Le., not use the high end portion at all). dosage form (e.g., a tablet) is usuallymade to a given moisture content(e.g.,1.8g/100 g ofsolid;Fig.9.20).Because the drug and the exci ients have~ifferentmoisture isotherms, they havewill different equilibriumThere can, however, be only one RH condition in the pore space of theform, so the result is that compound b will pick up moisture (move from 0)0 ablet 0) 0 10 20 30 Water Vapor Pressure (Torr) Fr~undlichmoisture isotherm presentationof initial part of a BET isotherm.
  • 168. compound a will lose moisture (moving from A to C). The question is to estimate,quantitatively, where (at what R ) the line DC will be. Two moisture e~uilibrium rvesmay(in an abbreviated fashion) be repre- Freundlich isotherms. This can be verified byinspection of Fig. 9.19, where would both, fairly well, adhere to a Freundlich isotherm. [(9.43)]:where q is a constant (Carstensen, 1980). This may be used to estimate the moisturemovement ina solid dosage form after it is manufactured. If we consultassume that the up curve isthat of drug (A) and the down curve that of excipientthere are m A grams of A on an ~nhydrous basis, and A contains a fraction (on abasis) ofq A moisture (i.e., a total of m A q A grams of water). There are mB grams oon an anhydrous basis, and contains a fraction (on a dry basis) of moisture (i.e.,a total of mBqB grams of water). + The dry weight of the dosage form, therefore, is m A mB, and as the dosageform (e.g., tablet) is made, it is made at a particular moisture content of a fraction y basis) of q moisture (i.e., a total of mq = [mB+ mA]q grams of water). cause, as seen from the figure, the relative humidity (the vapor pressure P) inthe pore space must be one particular ure (P), it follows that A must giv (from point A to point C) and must take up moisture (from point The moisture isotherms are of the type (9.44)and (9.45)The values of yz usually do not differ much (and the two isotherms, therefore, can berepresented as differing only in the values ofthe es). The areas have not been takeninto account, and the isotherms apply to two samples of material (to account for thearea, plotting by ET would have to be done), In the situation for which a known amount of A m~ is mixed with a knownamount of B mB, mass balance (assuming no loss of moisture) gives: YC[~A + + ~ B =I Y A ~ A Y B ~ B (9.46)or: YC + CYA~A YB~B)/[~A +m ~ l (9.47)and the amount of moisture lost can then be gauged from oisture loss in A = mAbA - yc) (9.48)and for (9.49)As y C is known, then P is also known.
  • 169. ter If for instance the two compounds aremixed together, moisture added (as in agranulation), and this is dried, then xc is known. Mass balance about AC9.20 then gives that the moisture loss experienced by A (9.50)must equal the moisture gained by B: (9.51)All quantities are known, so that P[= P C = PD]can be calculated(Le., both moisturelosses and gains, and the final relative humidity may be calculated). In this lattercase, the isotherms should be determined on samples that have been wettedand driedthe same way the final mix has been wetted and dried (because the surface areachanges).What has been said inthe foregoing also applies the situation in which solvates to areformed. In these, solvent(methanol, ethanol, or other) occupy sites in much same thefashion as water occupies sites in hydrates, and what has been said about vaporpressures also applies in this case, except is now the vapor pressure of the solvent, itnot of water, that is of importance. A = general symbol for a hydrate-forming compound a = area of solid plus condensed liquid B = x[(l - 1”s}/Q]2/3 BET = Brunauer, Emmett, and Teller c = BET constant d = diameter of particle after condensation do = diameter of particle before condensation I) = diameter of particle plus condensed water E = [FQ/{l- Fs}]2/3 h = thickness of adsorbed moisture layer 1“ = P*/P G = k(P - Ps)B/3 EST = enthalpy 15: = equilibrium constant k = mass transfer rate constant m = mass of one solid particle N = number of particles n = number of molecules (a) adsorbed, (b) dissolved n, = number of molecules in a monolayer Po = water’s vapor pressure at a temperature of I ‘ Pa = water vapor pressure of vapor phase before condensation Ps = vapor pressure over a saturated solution PH,O = water vapor pressure
  • 170. Px == water vapor pressure of vapor phase after condensation Q =p * ~ ~ ~ 6 R = gas constant RH = relative humidity s or S = solubility of a compound in rnoleslmole of water STP = standard pressure and temperature T = absolute t ~ ~ e r a t u rK , e T* = critical temperature for a hydrate t = time q = general symbol for a constant V = (a) volume adsorbed; (b) volume of vapor phase Vm = volume (STP) of adsorbate in a monolayer w r = moles of water condensed w = weight of water adsorbed per particle W = weight of water dissolved per gram of solid x = (a) mole fraction; (b) symbol for number of moles of water in a hydrate x, = volume fraction of solid in solution z = moles of water per mole of solid for an unsaturated solution y = (a) amount adsorbed; (b) mole fraction at saturation; (c) moles of water per mole of salt at saturation p = density of liquid p = density of solid o p* = density of solid plus adsorbateBaxter AB, Lansing CD (1920). J Am Chem Soc 45:419.Bray MI, (1999). P h a m Dev Techno1 4:81.Brransted JN (1928). In: Washburn EW, ed. International Critical Tables, vol4. McGraw-Hill, New York, p 237.Bransted JN (1943). Fysisk Kemi. Munksgaard, Copenhagen, pp 181-185.Carstensen JT (1986). Pharm Technol 9 (Sept):4l.Carstensen JT, Danjo IS, Yoshioka S, Uchiyama M (1987). J P h a m Sci 76548.Chen LR, Grant DJW (1998). Pharrn Dev Technol 4:487.Grant DJW, Medhizadeh N,Chow AHL, Fairbrother JE (1984). Int J Pharm 1825.Jakobsen DF, Frokjaer S, Larsen C, Nieman H, Buur A (1997). Int J Pharm 156:67.Kracek FC (1928). In: Washburn EW, ed. International Critical Tables, vol 3. Mc~raw-Hill, New York, p 373,Maron SH, Prutton CF (1965). Principles of Physical Chemistry, 4th ed. M a c ~ i l l a n New , York, p 253.Partington AB, Winterton CD (1930). J Chem Soc 132:635.Vancampen L, Zografi 6, Carstensen JT (1980). Int J Pharm 5:l.Zografi CyHancock P (1993). Int J P h a m 10:1263.Zografi G, Kontny M (1986). P h a m Res 3: 187.
  • 171. This Page Intentionally Left Blank
  • 172. 10.1 Salt Selection 160 10.2 -Change Precipitation 160 10.3 cation by Use of Mixed Solvent Technique 161 10.4 162 10.5 163 10.6 164 10.7 165 10.8 ehydration Kinetics 167 10.9 Solvates 167 ~ymbols 168 ferences 168It goes without saying that the drug substance is the most important part of apharmaceutical solid-dosage form (except for placebos, and they are only importantbefore the mar~eting a drug product). of The syntheses of the drug is, therefore, the first step in development and dis-covery, and once a company decides to proceed with the development of a drug,there are a series of problems that are encountered and decisions that have to bemade. he actual synthesis of drugs is not the subject of this text, but there are aspects at have a direct bearing on further development. a~ticularly, is the purification of the raw chemical that is of importan~e, itFirst of all, what chemical form ~ ~ ~ Zfor instance) is the one that should be t ~ o r ~pursued? What r e c r y s t ~ l Z i z ~ t~ o ~ e should be used? Will these decisions have an i ~ i ~ ~in~uence p o l ~ ~ ~ ror ~~y g r o~~ c o ~ i c ~ ~ y ? on p i s
  • 173. 1 8-1 These are problems thatboth the innovatorand the generics encounter,because it is of importance to place specifications on the physical state of the drugsubstance. Other aspects, such as its m ~ c ~ i ~ ~ ~ i Z whetherit is easilymade into (i.e., i t ~tablets or capsules) should not be impaired.Drugs with ionizable functional groups are produced, most often, as specific salts(sodium salts, amine hydrochlorides, for example) and the reasons for using certainsalts rather than the corresponding free bases or acids, include the following: 1. The base or acid may be an oil. 2. Most salts of acids have higher solubility than the free acids, alts most often crystallize more easily.Clavulanic acid, which is a B-lactamase inhibitor used in ~ u ~ e n t(SKB) is an oil. inbut its potassium salt is well defined, (The salt is also highly soluble; hence, it has alow critical humidity, a point that was discussed in the previous chapter). igh solubility is usually desired, but excessive solubility may be a drawback. lubility usually results in bioavailability that is better than (or at least equalto) that of a less-soluble form, but excessive solubility causes higher hygroscopicity.It rnay also give rise to highly viscous, saturated solutions, and in this manner mayimpair the rates of solubility. In general, drugs that have ionizable groups are prepared as either sodium orpotassium salts; for drugs that contain carboxylic acids or those with an aminegroup, an addition salt,such as a hydrochloride rnay be used. For ~ ~ ~ ~ o t e ~ i eom-~ othere ~ the possibility of having either an addition salt, a free base or, for is ~ ~ ~ ,example, a sodium salt. The sodium salt of amphoteric compounds are quite soluble, and hygroscopic.In such drugs a method of approaching formulation may be to employ the acidaddition compound of the drug (the claimed substance) and neutralize it duringwet granulation with sodium carbonateor sodium bicarbonate.The reaction isthen brought to completion, and the tablet made. Examples of this are the sodium il, described by Sherman (1996a,b) in U. S. patent 5,573,780 and in et al,, 1990) where this type of approach is described. urification is the final step of drug substance synthesis. Thermal recrystallizationand p~ecipitation are most common methods of achieving purity. ~ublimation the isresorted to at times, but is not commonly used. The precipitation may be accomplished in several ways. If the compound is aprotolytic substance in solution, then a change in pH may be used for purificatio~.An example of a compound that might be purified in this fashion is naproxen,because it has a solubility of 0.0159 mg/cm3 and sodium naproxen has a solubilityat 25°C of196.7 mg/cm3 in water at a pH of about 8 (Gu et al., 1990). If, forinstance, 200 g of naproxen were added to approximately 1000 g of water at 25°Cand made alkaline to dissolve it, and the pH then lowered to below 4, where freenaproxen is the primary species, then 200 - 0.0159 g would precipitate out. This
  • 174. ruwould be freed ofany impurity that had a solubility higher than the final conditionswould dictate. If the naproxen used was not quite pure, but contained, for example, 1% ofimpurity, and if this impurity had a solubility in excess of 2 g/lOOO g of water at p4, then 198 g of naproxen would precipitate and, theoretically, this would be free othe impurity. If the impurity had a solubility of less than 2 g, for instance, 1 g/lOOO g ofwater at pH 4, then 199 g of solid would precipitate, 198 g being naproxen and 1 gbeing impurity, so that the drug substance had been made purer (Le., from contain-ing 1% of impurity, it now contained only 0.5% impurity). The purification process is, therefore, a function not only of the solubilityprofileof the drug substance, but also of the impurities. Adequate conditions(amount of water or other solvent, final pH) maybe arrived at to optimize thepurification. The attainment of zero percent impurity by any form of precipitation methodis ideal, rather than realistic. Adsorption will always occur, as well as occlusion. Adsorption may be investigated research-wise,and most often Freundlich iso-therms may be used to estimate gravity of impurity retention by adsorption.Freundlich equation, the amount adsorbed C", from solution is given by ln[C*] = In q + n ln[hf] (10.1)where q is a constant andhf is the amount in solution. In this manner it is possible toassess the severity of adsorption of different solvents and impurities. IXIf two solvents are miscible and the drug substance to be purified is soluble in oneand poorly soluble in the other, then a precipitation by solvent change can beaccomplished. In the example in Fig. 10.1 the drug is soluble in water only to the extent of < 0.02 mg/g of water, but is soluble in isopropanol to the extent of 3.9 mg/g ofisopropanol. If 3.9 g of drug substance is dissolved in 1000 of isopropanol, and 4000 gwater then added, the solubility of the drug substance then drops to 0.02 mg/g of(mixed) solvent (i.e., total of 5000 g of solvent iscapable of dissolvingonly 0.1 g of adrug substance), so that this water addition would allow 3.8 g of purer drug sub-stance. The purity obtained willbe a function of the levelof impurities in thecompound before reprecipitation and of their solubilities. An example ofthe potential use ofsolubility in mixedsolvents for precipitationpurification was published by Joszwiakowski et al. (1996). Residual solvent is a problem in precipitation purification. Residual solvent isremoved by drying by heat or by vacuum (or by both). Microwave drying wouldwork only if the energy frequency was adjusted to the particular solvent. One situation that may arise is that the drug substance forms a solvate, and inthis case, the ease of removal of solvent would depend on the equilibrium vaporpressure (at the drying temperature) of the solvent over the solid. Vacuum drying
  • 175. -0 20 40 60 80 100 Weight Percent Isopropanol Example of ~re~ipitation puri~ca~iont e r / i ~ o p r o ~ a n o ~ . in ~ amight be used to bring the pressure below that of the equilibrium vapor pressure ofthe solvent. sidual solvent may also be the result of solvent ~ntrapped crystals as they in te (i.e., the solvent may occupy defect sites in crystal), The best method theof ~‘freeing’’solvent of this type is by way of comminution, because the milling may(a) expose the defect sites, or (b) make them sufficiently mobile allow escape of to the The third situation is that surface removal of solvent forms an impenetrablecrust, trapping solvent on the inside,Thismay happen when hard vacuumise ~ p l o y e and in such a case, it may be corrected by using a lower vacuum and a ~,longer drying time.This has, to a great extent, been covered in Chap. 6, but some c ~ m ~ e n t this point at s Asare of import~nce. an example,‘ assumethat a compoundis soluble to the extentshown in Table 10.1, and an impurity has the solubility ~haracteristi~s shown. ~ s s u m that a particular batch of the drug s u b s ~ contains 2% of impurity e n~(i.e., 98% of drug substance). Taking 10 g of the batch (i.e., 9.8 g of drug substance),adding toit 1000 g of solvent and heating it to 60°C will dissolveit all. By cooling itto 25”C, 9.5 g will precipitate out (under ideal, equilibriu~ conditions). The 0.2 g ofimpurity will present a concentration less than sat~ration (0.5 g~l000 of solvent), gand the precipitated drug substance will, theoretically, be “pure.” ecause of adsorp-tion and possible inclu~ion, is never quite true, and limits on impurities, there- thisfore, are always finite, not zero. Solubilities of a Drug Substance and an Impurity Solubility of drug Solubility of i ~ p u r i t y (g/lOOO g of solvent) (g/lOOO g of solvent)25 0.5 0.560 10 2
  • 176. A special case ofpurification is that of optical isomers. omp pounds with one chiralcenter may occur as a d-form, an E-form, (denoted enantiomers), or (in a racemiccompounds) as a dE-form. Equimolar mixtures of chiral compounds (denoted dl)may, depending on the compound in question, exist as racemic compounds or con-glomerates. The expression ~ a c e ~ is often used, generically,to simply describean a~eequimolar composition of the two enantiomers without signifying whether it is aconglomerate or a racemic com~ound. racemic compound is, as the word implies, a compound, and may be con- as a strong complex between the two components. If the latter did not exist,then the mixture is a conglomerate (i,e., there is no chemical interaction between d-and E-forms) and, in that case, the solubilities of (excess amounts of) a mixture of acertain amount of the d-form and a certain (not necessarily the same) amount of theE-form would simply (approximately) be the sum of the solubilities of the two. presenceof a d,Z-form the situation is, however, different. dipeddi (1995)have reported on the isomersof pseudoe~he hase diagram of this system is shown in Fig. 10.2. It is noted in Fig. 10.2, whencompared with a melting point phase diagram of amolecular compound to be treated shortly, that the solubility plot is exactly theupside-down inverse of the melting point plot. A situation similar to that of pseu-doephedrine exists for daxclamol hydrochloride (Liu and Hurwitz, 1978). from Fig. 10.2 that the dl-form is less soluble than either of the hiral purity” appears to be mandatory for new drugs, and situati~ns makes separation by recrystallization impossible, A means of , however, is to derivatize the dZ-compound with an optically a , the d-form derivative would have a different solubility thatform derivative, and fractional recrystallization can now be carried out. The resolved 4 0.0 0.4 0.6 0.2 0.8 1.0 Mole Fraction ‘I-form ~olubility-phase diagram of the p s e u ~ o e ~ h e d ~system. (Rata from Pudipeddi, ine 1995.)
  • 177. enantiomer is then re-formed by dederivitization. This is tedious, costly, and yieldsare meager at times, and it adds to the cost of the drug. It is, however, often anecessity because of the toxicity of one enantiomer (the one not wanted). In other cases, it is actually unnecessary (e.g., ~Z-tochopherol).requirement for chiral purity seems to be only possible for “grandfather” drugs.After purification of a drug substance by rec~stallization orreprecipitation, it isfiltered or centrifuged to a certain degree of dryness, but a drying step is usuallynecessary. For crystalline compounds that are not hydrates, drying is simply removal ofsurface moisture. Some pore space drying may occur (e.g.,if agglomerates areformed). Micropores may also be dried out down to a certain pore size, but theirmoisture is often part of the residual moisture allowed by specifications. The mannerin which speci~cations set for moisture content is discussed in Chaps 14 and 15 aredealing with stability. If it is the drying of surface moisture, then the drying rate, dm/dt would bedictated by q’ = dm/dt = kA(P0 - P ) (10.2)where m i s mass ofwater or othersolvent, t is time,k is a mass-transfer coefficient, Ais the surface area of the solid, Po is the vapor pressure of water or the solvent at thedew point of the airstream drying the solid, and P is the partial vapor pressure ofwater or the solvent in the drying airstream. A high k-value requires good heat andmass transfer such as in a fluid bed dryer or in spray drying, Equation (10.2) inte-grates to m = mo - q’t (10.3)where q is given by Eq. (10.2). The integration requires that A is (fairly) constant,which can be expected in most drying conditions. It also applies only to the phasewhere drying occurs, because once drying is complete, P is no longer the pressure atthe dew point, but rather, the pressure at the temperature of the airstream. Thisallows drying to be monitored for the end point, for a rise in temperature of the exitairstream indicates that water or solvent has been completely removed (i.e., more noevaporation is taking place). Some dry solids, such as zeolites or bentonites, contain internal water that driesby diffusion. The same holds true for amorphous solids. Drying of such drugs ingeneral follows usual diffusion kinetics (Jost, 1960). For simplicity, the solid particle will be considered spherical with a radius of ro.The initial, uniform concentration of water or solvent is denoted co, and the final,uniform concentration is denoted c,, and at time t the average concentration isdenoted c. The expression for the average concentration c, at time t (Jost, 1960) isgiven by (10.4)
  • 178. where u is a running index and where summation is withu going from 1 to infinity.is a constant that is dependent on geometry (e.g., for a cylinder it is 6), but may beexpressed by considering that at t = 0, (10.5)so that (10.6) 1 The term (1/u2)exp[-u2n2Dt/ri] decreases drastically with increasing value of1/u2 (e.g., it becomes four times smaller as u increases from 1 to 2). Add to that theeffect that increasing U-values have on the term exp[-v2n2Dt/ri] and it is seen thatquite an adequate approximation would beor + ln{(co- c)/(co - e,)} = - { - t / ~ } ln[6/n2] (10.8)where (10.9)and where, depending on the geometry of the solid, Q may vary from 6 to 8. It isnoted that the a~proximation gives some zero time deviation from intercept withzero for Eq. (10.3) (Pitlcin and Carstensen, 1973). To convert Eq. (10.2) to mass ofwater or solvent, m, in a particle, rather than c~ncentration within a particle, it isnecessary to multiply by the volume V , and if it is assumed that the final amount ofwater or solvent, m, = 0, Eq. (10.9) becomes 1 - ( m (10. oexp(--t/z) / ~= ~ lo)The left~hand side isthe fraction (or if multiplied by 100, percentage) of moisture theleft at time t. This shows that the drying process in a diffusional phase is loglinear in time,and shows that it is the more rapid, the smaller the particle. The effect of ~eometryissuch that the factor (A/n)romost often is close to unity. Usually, drying curves are separated into the three sections shown in Fig. 10.3:(a) drying of surface moisture (the linear drying phase), (b) drying by diffusion (thefalling-rate phase), and (c) overdrying. Overdrying is particularly directed towarddrying of hydrates, as shall be discussed in the next section. .7.As mentioned in Chap. 9, the water in hydrates is partly held in the coordinationshell about the ions in the lattice, and partly it occurs as ~ t ~ ~ water. The e t ~ ~ ~ Z
  • 179. Moisture Content A onstant Rate PeriodR@nlOV-ableMoisture D dMolsture 0 The different drying phases.structural water is held much less tightly than the coordination water, For instance, 0 fails to give up the last two molecules of water when heated above rather gives off HCl. In the following, the “drying” of the hydrate isconsidered to be twofold, depending on purpose. Either it is desirable to remove thesurface moisture and not the water of hydration, or (b) it is desired to remove the(structural) water of hydration. It is apparent from the previous chapter that at any given temperature belowthe critical temperature T” of a salt hydrate, there is the possibility of removal ofwater (or solvent) of hydration. Often (e.g., ampicillin, amo~icillin, and cephalos-porins), it is the salt hydrate that is the desired form of the drug substance, but onthe0 nd, adsorbed moisture may be deleterious to the compound as well asd ion of the hydrate. rying with airstreams with a relative humidity equaling that of the equili-brium relative humidity of the salt hydrate will remedy that, but on the other hand,drying is more rapid if lower humidities or higher temperatures are used. The finesse,then, is to stop the drying at the right point. ying, at times, causes undesired effects, depending on drying conditions. for instance (Zoglio et al., 1975) forms an, at times water-impenetrable,crust during fluid bed drying under some conditions. ~acuum drying causes similar crusts to form (Garner, 1953)in the case ofcopper su~fate pentahydrate and magnesium tartrate dihydrate. In vait is often advantageous not to employ a hard vacuum, as shown inthe cited cases (Fig. 10.5), X-ray examination of the dried compound has demon- strated that vacuum drying forms a skin possessing no crystalline structure,whereas drying in moister atmospheres does not iverise to this phenomenon. In the hard vacuum, moisture evaporates off of the surface, creating an anhydrous ionic network, whichis unstable (except for zeolites). It is unstable rearranges to a phase that has no well-defined crystallinity (seeFig. 1 This further nucleates (see Fig. 10.5, phase C), a process that is accelerated by presenceof water. This nucleation and crystallization gives cracks at right angles to the interface. The drying then takes place through a continuous layer ) and a reduced surface (the cracks in C).
  • 180. 0 1 2 3 4 5 6 Water Vapor Pressure (mm Hg) Drying rates of CuS04-5H20as a function of water vapor pressure.Taylor and York (1998) studied the dehydration of trehalose dihydrate, and foundthat none of the conventional equations would fit the dehyration data well.It is not only water that may become part of the lattice of a compound. Solvents(ethanol, methanol, and such) may also occupy lattice sites, and in that case, onetalks about solvates. Pohlrnan et al. (1975), for instance, have shown that at least three polymorphsexist of carbamazepine, the first being monoclinic (Reboul et al., 198 1,1981), the second one being trigonal (Lowes et ai., 1981). The strucure of a dihydrate A B c Hard vacuum drying of CuSO4 (A) causes an amorphous, anhydrous subphase(B) that then crystallizes to a phase (C) containing cracks. Drying is impaired by the moisturefirst having to penetrate layer € , then being able to escape only through narrow cracks. 3and
  • 181. eck and Dietz, 1986) and that of an acetonate (Terrence et al., 1983) have alsobeen reported. A = surface area of the solid C* = amount in solution in equilibrium with a solid of concentration M c = average concentration of water (or solvent) in a particle being dried co = value of c at time zero ,c = value of c after drying is complete 1 = diffusion coefficient of water (or solvent) through a solid ) k = mass transfer coefficient M = (a) amount adsorbed (Freundlich e~uation), m~lecular (b) weight y z = mass of water or other solvent in a solid being dried ) mo = orginal mass of water in a solid being dried n = exponent in the Freundlich equation Po = water (or solvent) vapor pressure at dew point P = water (or solvent) vapor pressure Q = coefficient in drying equation, depending on shape q = constant in the Freundlich equation q = drying rate constant in the constant rate drying period ro = radius of a spherical particle being dried T* = critical temperature t = time u = running index z = ~ ~ / ( ~ 1unit that reduces drying time to reduced, no~~imensional = )) timeDuddu SP (1993). PhD dissertation, University of Minnesota, Minneapolis, MN.Gu L, Strickley RG, Chi L-H, Chowhan ZT (1990). Pharm Res 7:379.Himes VL, Mighell AD, Decamp WH (1981). Acta Crystallogr B37:2242.Jost W (1960). Diffusion in Solids, Liquids, Gases, 3rd printing. Academic Press, New York, p 46.J~zwiakowski MJ, Nguyen N-T, Sisco JJ, Spankcak CW (1996). J Pharm Sci 85193.Liu S, Hurwitz A (1978). J Pharm Sci 67:636.Lowes MMJ, Cairfa MR, Lotter AP, Van Der Watt JG (1987). J Pharm Sci 76:744.Pitkin C, Carstensen JT (1973). J Pharm Sci 62: 1215.Pudipeddi M (1995). PhD dissertation, University of Wisconsin, Madison, WI, p 77.Pohlman € ,3 Culde C, Jahn R, Pfeiffer S (1975). P h a ~ a z i e 30H.11:709.Reboul JP, Cristau B, Soyfer J-C, Astier J-P (1981). Acta Crystallogr B37: 1844.Reck G, Dietz C (1986). Cryst Res Techno1 21:1463.Sherman BC (1960a). U. S. Patent 5,573,780.Sherman BC (1960b). U. S. Patent 5,573,962.Taylor LS, York P (1998). Int J Pharm 167:215.Terrence CF, Sax M, Fromm GH, Chang C-H, Yo0 C (1983). Pharm a~ol ~gy 27:85.Zoglio MA, Steng WH, Carstensen JT (1975). J Pharm Sci 64:1869.
  • 182. 11.1. Freezing of Ideal Solutions and Ideal Solubility 17011.2. Melting Point Depressions and Purity Assessment by the Van Laar Equation 17111.3. Eutectic Diagrams 17211.4. olecular Compounds 17411.5. Solid Solutions 17511.6. Hydrous Amorphates 176 11.7. Lyophilization: Amorphous Cakes 176 11-8. Immiscible Melts 178 11.9. Miscible Melts 17911.10. Solid Solutions of the First Kind 18211.11. Partially Miscible Melts 18311.12, The Separated Phase: Solid Solutions of the Second Kind 18411.13. Melts 18411.14. Coprecipitates 18511.15. Cogrinds 18611.16. issolution of Solid Dispersions 186 Symbols 187 References 3 87
  • 183. efore discussing melting point diagrams, a note on ideal solubility relations is inorder. Assume that a crystalline substance A is dissolved in water, and assume thatthe two do notform solid solutions. If a plot is made of the mole fraction x, at whichone or the other solid phase (ice or drug) is in equilibrium with a solution, then adiagram such as shown in Fig. 11.1 results. If cooling is carried out fromcomposition S, then there will be a separation ofice at a temperature of R, and in similar fashion a solution of composition V willprecipitate drug at point Q.It is assumed that the separated phases are crystalline.The freezing point trace of water is the section NU and at the other side of point 77 .the curve U Q is denoted the solubility curve of A. It is noted (as opposed to con-ventional eutectic diagrams) the curve is not continued all the way to the right y-axis,because an upper temperature (e.g. the boiling point of water) is usually indicated(composition W ) . The condition of equilibrium is that the chemical potentials of the solute insolid and dissolved form, p, and p b are the same. We may write k =k b s (11.1)but kb = + RT ln[a] = + RT In[.] (1 1.2)since for an ideal liquid, a may be substituted by x, the mole fraction. J I L ~is then thes t a ~ ~ achemical energy of a solution o f a mole fraction of unity. rd This may be rearranged to read In[xl = -bg/[RT11 $- {kb/[RTI) (1 1.3) 00°C I .o v w Mole Fraction Freezing point diagram of a solid and water
  • 184. he temperature dependence of x may be written (and expanded by ( 1nfx113T)p= - { ( l / R ) ~ [ ~ ( ~ ~ / I " ) / 3 ~ } { ( l / ~ ) { f 3 ( ~ b / ~ ) / 3 I " l } 3 = -{(H - ~ ) / R I " 2 r partial molar heat of the solid compound in ideal solution and solid per mole, both at a temperature of at absorbed, I;, at constant pressure and te { ln[x1/3 I " } ~= L / R T ~ 3 (11.5)For the left side ofthe freezing point diagram, this equation is known as the freezingpoint equation, and atthe right-hand side ofpoint U it is the solubility equation ofin the liquid A (e.g.,,water).Integration of Eq. (1 1.5) gives rise to the Van Laar equation, which allows assess-ment of the purity of a drug substance by obtaining its "melting point T . For the ram in Fig. 11.1 to the left of U , the separating phase is water, and n of water, with the terminology used, would be (1 - x If the) tes a mole fraction x and the melting point of the pure drug sub- en the melting point of the contaminated drug substance 2"given by: lnfl - x] * --x= -AH/R{(l/I") - (1/T0)} (11.6)or x =4 N / R { 4 ~ / ~ ~ } (1 1.7)R is the gas constant and 4 N the heat of fusion. It is common to employ differential scanning calorimetry (DSC) for this type ofdetermination, and it is possible, in so doing to construct the "eutectic curve,, in itsentirety. For instance if AN equaled 8000 cal/mol and T0were 2OO"C, it is possibleto develop the entire melting point curve as follows: I" is calculated for several of x-values by using Eq. (1 1.6).achieve this is shown in Table 11.1. E~uation 1.6) may be (1 written: (l/I") = (1/T0) - ( ~ / ~ ~ ) l-nx) l ( (11.8) ram gives x up to 0.6, can beextended by changing the upper limit in -TO statement to 1.0. t eutecticshave intersections in "the middle." Inserting, for instance A H = 8200 and To = 190 givesthe results in Table 11A similar program may be written for the right-hand side of the diagram (by sstituting (1 - Y l ) for Y l and rewriting the appropriate lines. It is noted that althoug~ may be determined in this fashion, it gives no xinformation about what the conta~inant and, hence, not knowing its molecular isweight, the value of x cannot be translated into weight percent.
  • 185. r l Program for Eq. (1 1 .8) INPUT “HEAT O F FUSION = ”;Q1 INPUT “ELT.TEMP.”C = ”;F1 FOR Y1 = 0 TO .6 STEP .1 F2 = F1 + 273.15 F3 = lfF2 Y2 = 1.991Q1 Y3 = LOC(1-U1) Y4 = U2*Y3 Y5 = F3-Y4 Y6 = l/Y5 Y7 = U6-273.15 PRINT Yl,Y7 NEXT Y 1 The eutectic point is rarelya rational fraction and is, in essence, intersection thebetween two solubility curves. The most common way to assess impurities in p r ~ f o r ~ u l a t i o nby the use of isDSC. A schematic of a DSC trace is shown in Fig. 11.2. TIA literature example (Giordano et al., 1998) of a eutectic diagram is shown in Fig.11.3. In this figure the melting points of mixtures of piroxicam pivalate (PIRP)(polymor~h with piroxicam (PIR) areshown. A melt at 170°C of 0.12 mol I) fraction will start “freezing” (showing separation out of a solid phase) at 150°C. Thesolid phase is PIRP and,depending on the rate of cooling, the crystals formed are orcan be fairly large (in a relative sense). As the temperature decreases, there will bemore PIRP precipitati~g out, andthe liquid (melt) with which is in equilibriu~ isgiven by the corresponding composition of the liquidus line. At point C the follow-ing occurs: if one considers the lines AC and DC solubility curves, it follows thatlowering the t ~ m ~ e r a t u r e 140°Cwillcause below supersaturation of both com- Results from Table 111 Using A H = 8200cal/mol and To= 190°CMole fraction x Melting point0 1900.1 184.60.2 178.70.3 172.10.4 164.90.5 156.50.6 146.8
  • 186. 3 2 -1 140 120 160 180 200 220 Temperature ("C) Schematic of DSC trace of compound containing an impurity.pounds; therefore, precipitation of both will occur. Because supersaturation willcause precipitation of small particles, the eutectic ~ i x t u r e precipitate out (soli- willdify) as a finely divided mixture of PIRV and PIR, in addition to the coarser PIRalready precipitated. Similar considerations apply for cooling along the lineexcept hereit is PIR which constitutes the coarser part, which precipitates out bethe eutectic precipitation. The eutectic composition is not (necessarily) a rational ratio between the twocompounds, and is not to be considered a "compound." elow line BHE (the eutectic temperature) only solid phase exists, and aboveline ACD only liquid (melt) exists. area ACB consists of PIRP plus melt, and the Thearea CDE is an area where PIR plus melt exists, C is denoted the eutectic composi-tion. Heating solid along composition xF or xH will cause an onset of melting attemperatures TF (or TH),and at temperatures between 7;; and TG(or between THand TJ),there will be two phases present. In the former, solid PIRP and a liquidconsisting of a mixture o f PIR and PIRP. The liquidus line is essentially a lineindicating the solubility of PIRP in molten PIR at the given temperature. 220 D E 120 0.0 *i 0.2 0.4 i 0.6 * 0.8 1 .o Mole Fraction PIR Binary phase diagrams piroxicampivalate (polymorph I) with piroxicam.(Data offrom Giordano et al., 1998.)
  • 187. Similarly, the line CJD is the solubility-temperature line of PIthe latter, the solubility (in mole fraction) is (1 - x). The diagram is established by DSC, and the eutectic temperature is the onset ofthe endotherm, and the complete “melting” temperature, TGor TJ as the end of theendotherm. knowing the value of AHI and AHII of the two polymorphs I and I1 of their at their melting points allows calculation of one leg of the eutectic , and knowing A H of PIR at its melting point then allows calculation of theother leg. Ciordano (1998) from this (according to Yu, 1995; 1997) calculated x E l = 0.26, TE1 = 140.7IS, xE2 = 0.18, TE1 AG to be proportional to T allowed them to calculate the transition tempe~atureto be 32°C and to establish the Cibbs energy diagram.In certain instances the binary melting point between two c o ~ p o u n d will have an sappearance as shown in Fig. 11.4. One may think of the diagram consisting of t~djacent9 simple eutectic diagrams. The compounds, in that case are A and [A,for one and B and [AXBY] the other, i.e., [A,B,,] where x and y are simple for is a chemical compound. ebrand and Muller-~o~mann (1997) have reported on mixed crystals of sodium salt. ~etoprofen (DSC) shows a melting endotherm with d a mixed crystal onset of 400 IS. If th e first peak should not have occurr oymann studied mixtures of the acid and sodium salt in various ratios and then und three peaks, with a local maximum in enthalpy of fusion at about 33%ketoprofic acid. This would imply a molecular compound of two sodium salts toone acid. To demonstrate this experimentally the authors prepared sodium sa€t/acid ratio of 2: 1 and the crystals thusly formed exhibited only one melti~g peak at . X-ray and scanning electron microscopy (§EM) showed these crystals to be - Compd. P Compd. Q Mole Fraction Q inary melting diagram with molecular compoundformation. (Data from ~ilde~rand ~uller-~oy~ann, and 1997.)
  • 188. quite different from either sodium salt or acid. It is noted that the point C shouldhappen at a rational compositio~ 1, 1 2 , 2 3 , 1:3). (1: The foregoing is distinguishedfrom saZid ~ i s ~ e r s i Q ~ s , willbe discussed in whichSec. 11.14. It is also noted that Fig. 11.4 isthe (horizontally flipped) mirror image of ig. 10.1, showing the close relation between solubility and melting point diagrams.not,Temperature Liquid Liquld Sol Id So!id "D a b c M o l e Fraction B Case ( i ) Case (ii) Case (iii) Three situations in which solid solutions occur.
  • 189. Temperature Te I a c b Fraction B Binary melting point diagram for two c o ~ p o ~ n dA , and B, which form solid ssolutions and an eutectic. hese systems have not been fre~uently reported in the pharmaceutical litera- t have been reported in the metall~rgicalliterature). Some reports in thepha~aceutical literature have dealt with systems of the type shown in Fig. 11.6. Inthis i~stance point c is, indeed, an eutectic point, but the (finely subdivid~d) the solid parate out are compositions a and b, not of pure compoun~s and of A ms have been reported for p~armaceutical systems by (~ekiguchi and i~uchi al., 1963, 1964a; Goldberg et al., 1965; Guillory et al., 1969; et zi, 1971). ~arstensen Anik (1976) have reported on the propor- andtional re~uirements must be met for a solid solution composition to occur. the olids that are notcrystalline are denoted ~ ~An important category of this ~ ~ ~is lyophilized cakes (for intravenous reconstitution). These are formed by freezingaqueous solutions. On such freezing (when part of the solid comes out as an amor-phate), ice will first freeze out, and then the remaining solution (which under otherconditions might crystallize as a eut ctic), will supercool and will become 66solid.” t here the ‘‘solid9’ is simplyvery viscoussolution. An example ofthis is Fig. 11.7 a er and Nail, 1994). In practice this usually refers to lyophilize^ cakes. The glass transition tem-perature can usually be arrived at from thermal analysis, as shown in FiThe S O Z ~in freeze-drying is (when dried) referred to as a lyophilized cake. It is ~mostly amor~hous, and the lass transit~ontemperature can be arrived at fromthermal analysis (see Fig. 11.S). The collapse temperature in Fig. 11.8 is a tempera-ture dictated by mechanical properties. Just above the glass transition te~perature,
  • 190. 40 20 i tsovtscosity Solution Curves 0 - 20 - 40 - 60 0 20 40 60 80 100 Solute Concentr~tion, YO .7 An example of a supercooled viscous solution. (Data from Her and Nail, 1994.)sucrose solutions, for instance, have viscosities of about lo6 Pa/s, but below Tgthisfigureis 1Ol2 Pals. The generalsequence of eventsinfreeze-dryingisshown inFig. 11.9. The primary drying (see Fig. 11.9) consists of the evaporation of the crystallineice, so that the cake is left with “holes” in it, and a glass of a water content in therange of 12-15% results. If the tem~erature belowthe glass transition temperature, isthen this glass has a high viscosity and will dry slowly, because the diffusion coeffi-cient 23 for evaporation of water, will be high. If, after the primary drying, the initial freezing te~perature 240 K ( is 11.10) and the solids conten is so%, then the composition would be at point C ,between the Tcand Tgcurve. ut if sublimation were continuously carried out at this 300 200 100 -50 -40 -30 -20 - 10 0 IO 20 T e ~ ~ e r a t u*C re Thermogram of aqueous solution of 10% PVP. The relative magnitudes of theendotherms for glass transition vxs-a-vis melting is shown. (Data from Her and Nail, 1994.)
  • 191. Schematic of freeze-drying events.temperature, then, at point B, the glass transition would be passed, and the viscocitywould become very high, and sublimation would be very slow. The temperature,therefore, is continuously increased, such that the lyophilization temperature canstay within the bounds of the two curves. Some proteins have stabilities that depend on cooling rate, but this is primaril~due to electrolytes (e.g., sodium chloride) and stabilizers (e.g., glycine) in the com-position. These will crystallizeout andgive the cake structural strength, such that Tcincreases, but their presence, as well as the initial freezing rate, willmodify thepositions of the two curves, so that a slow-cooling rate may provide a different(and sometimes worse) curve than when a fast-cooling rate is employed. These aspects have been discussed in detail (Franks, 1990; Levine and Slade, acKenzie, 1977; Suzuki and Franks, 1993). Turel et al. (1997) have shown that the water in c i ~ r o ~ o ~ ais ipresent in a c ncomplicated hydrogen-bonded network. utectics have beentreated, in an initial sense, in t foregoing, and this is ofinterestwhenever a binary or multinary system is melted, s the heading implies, there is a 300 250 200 0 20 40 60 80 100 Solute Conc Limiting phases in a lyophilization event.
  • 192. cticseries of different systems that may arise. The systems are considered binary in thefollowing chapter, and the components are denoted A and l3. It is assumed in thefollowing that A has the lower melting point. If two substances, A an , are mixed, and if their melts are totally immiscible,then heating a solidmixture the two will first result in A melting, producing a in molten A. Then, on further heating, B will melt, and the twoliquids will be immiscible (Le., form two phases). A DSC thermogram of such a mixture would simply show a sharp-meltingpoint for A, followed by a sharp-melting point for l3. Systems of this kind are rareandarenot of muc 1 interest, otherthan serving asanintroductionto the *concepts to follow. , mixtures of inorganic electrolytes (sodium chloride)and organic materials would be of that ilk, but the experiment described would be atheoretical exercise,becausemost organic materials decompose at or before thetemperatures at which inorganic electrolytes melt.The commonly referred to situation of eutectic diagrams is the one shown in Fig.11.11 (which is repeated for convenience). The melting point of a mixtfrom the pure compound; for example, if a mixture of A and a little(point V),and then cooled along the line CQ, then solid phase willseparate out whenthe temperature at C is reached. This “precipitate” may be fairly coarse. As the cooling progresses (e.g., to thetemperature corresponding to point W ) ,more and more solid w(separated) out, and the liquid will become richer and richer inliquid compos~tion be X,and the amount of liquid, mL, v willsolid m,, is given by the so-called weight arm rule: {VW}m,= {WX}rnL 1.9) (1If the composition at Cr is denoted x and the composition at point X is denoted -xL, q. (1 1.9)translates to xm, = ( X L - X ) m L (11.10)If this is divided by the total mass (m, + mL),then Eq. (1 1. 10) becomes: 220 T2 0.0 0.2 0.4 0.G 0.8 1.0 A Mole Fraction B Eutectic diagram.
  • 193. 1 (11.11)wheref, is the mass fraction of solid and f L is the mass fraction of liquid. When the point E (the eutectic point, corresponding to the eutectic tempera-ture and the eutectic composition) is reached, then the following dilemma occurs:Line TE, the so-called Z i ~ ~ line represents the solubility curve of i d ~ ~ET2 is the other liquidus line and represents the solubility of A in B). If the tem-perature were to drop below the eutectic temperature, then the solubility of A. in l 3and the solubility of B in A would be superseded. In an e ~ ~ i lsituation, this~ ~ i ~ ~ icannot occur, so that the situation is resolved innature by both A and €3 precipitat-ing. It follows from the type of situation that large crystals of either would not bepossible ( ~ o u l d result in too large an increase above solubility of either compound),so that what will happen at further cooling (i.e., removal ofheat) is that a very finelysubdivided mixture of A and 1-3, the eutectic ~ i x t ~will occur. As this precipitation reoccurs, r e ~ o v a o heat will not result in a reduction in t e ~ ~ e r a t ~Not. until the lf reentire mass has frozen will the temperature drop again. Along the line UCWQ Y in Fig. 11.I 1 the t e ~ p e r a t ~profile, assuming con- re stant heat removal, would be as shown in Fig. 11.12(a) and at the eutectic composi-tion, x(E) it would have the appearance in Fig. 11.12(b). The latter profile is exactly the same as for a pure compound, but for a eutectic, x(E) wouldbe not be a convenient ratio (1:1, 1:2, 1:3, for instance). The conventional eutectic diagram, withsolubleliquid phases, divides the space intofour areas, as shown in Fig. 11.11. The area above theline TET2, where the system is liquid, the area below the line QE where the system is solid, consisting of coarse crystals of one of the components and a "eutectic mix" of finely subdivide^ crystals of A and B, and the two triangular area consists of melt plusA or melt plus B. If it were melt plusA plu of phasesp would be four (including vapor), and by Cibbs phase rule the degrees of freedom n, would be given by 1 L L U T"CTOG U Melt Melt olid T out 1fx=O Time Time Temperature profile during cooling along line UCWQY in Fig. 11.11. (a) repre- sents cooling of a noneutectic composition, whereas (b) is either one of the pure co~pounds (x = 0 or x = 1) or the eutectic composition [i.e., x = x(E) in Fig. 11.1 11.
  • 194. n=f-p+2=2-4+2=0 (11.12)where f,the number of components, is 2. This means that temperature cannot bechanged if both A, and melt are present, and this is exactly the situation depictedby Fig. 11.12. The lines TE and T2-E are solubility curves, where the solubility is expressedin mole fraction. The solubility equation for section TE would be TE : ln(1 - xB) = --{AHA/RT] + /31 1.13) (1where AHA is the heat of fusion of A, XB is the mole fraction of B, R is the gasconstant, T is absolute temperature, and is a constant applying to €3 in the system. For the section E-Ti the same type equation applies: E - T2: ln(l - xA)= - ( h H b / R T } + P2 (11.14) If xA and "xBare known at two different temperatures, then the curve may beco~structed (and the eutectic point may be calculated as the intercept between thetwo curves, or the root of the two equations). A H may not be temperature-independent, in which case, as shown in Chap. 3,a logarithmic term has to be added. Eutectic phase diagrams may be obtained by DSC, and one method for arriv-ing at the diagram is the following: A finely ground mixture of A andand mixed well, and heated in the calorimeter. Reference is made to Fig. 3 1.13,where it is assumed that the heating causes the first thermal response (the eutectictemperature) at 40°C and the last at 120"C, the liquidus line. -xB is known from thecomposition, and this gives one point where the points C (liquidus te~perature) andQ (eutectic temperature) can be plotted. It is possible to carry out the trace with just one DSC determination, if it isassumed that AH is temperature-inde~endent,AHtotal for the entire melting is theobtained (by comparison with i n d i u ~ traces) from the area, Atotalunder the entire 15 10 - t." n 20 40 GO 80 100 1 2 0 1 4 0 Temperature, "C Schematic of a DSC trace, for which the eutectic temperature is about 40°C and the liquidus line is at 120°C, at the cornposition in question.
  • 195. trace. AHw is obtained by the cross-hatched area, Aw, by comparing it with AHtotal.The fraction meltedfL, is now given by: (1 1.16)fs is 1 -fw, so that by use of Eq. (1 1.16)it is possible to calculate xL. he trace may, therefore, be divided into, for example, ten portions and the reas computed for each T-value, and the value of f L for each plotted versus tem- erature to give the entire l i ~ u i ~ line. us There have been occasional reports in literature pre~entingeutectic data assoZu~ilit~ data. An example is the work by Bogdanova et al. (1998). These authors studied melts of indomethacin and nicotinamide. Figure 11.14.shows the solubility of indomethacin as a function of its concentration in an nico-tinam~de-indomethaCinmelt. The inter~retationof the data issimply that of aeutectic diagram. Note that the eutectic c o ~ ~ o s i tis o ~ (or only by accident) a rational frac- i nottion of moles of A and €3.For a molecular Compound,as m~ntioned Sec. 11.4, the insituation is different, and a diagram such as shown in Fig. 11.12a wouldlack the linesegment CQ when a composition of the molar ratio is heated or cooled [i.e., wouldappear similar to Fig. 11.12b if only the right (or entirely the left)of the molarcon~position were considered].In some systems, the solid phase crystallizing out in the areas depicting solid plusmelt are not the pure compound (e.g., A on the left of the eutectic), but rather it is asolid that is a solid solutio^ of I in A (or A in on the right-hand side of the 3 lgure 11.15 serves demonstrate the definition a solid solution in the strictest to thermodynamic sense. If a composition at H is allowed to cool, then at a tem era- ture corresponding to H , solid willprecipitate. This, however, willnot be pure rather, will contain an amount of A corresponding to the point M . If a comp were cooled from the melt, the solid would be containing an a m o u ~ of A t corresponding to N . 0 0 00 0 0 20 40 60 80 100 % Indomethacin in Melt ~olubilityof indomethacin as a function of its concentration in the nicotina- ~ide-indomet~acin melt. (Data from Bogdanova et al., 1998.)
  • 196. ctie Mole Fraction Schematic of a situation leading to strictly solid solutions. The situation would require equilibrium, and would take long times to estab- would have to be chemically quite similar and, forinstance, KC SCN form solid solutions, There are inorganics that form solid solutions over theentire composition scale (e.g., Au and Ag) and, in that event, there is no eutectic atall. In pharmaceutics9 there are no solidly documented cases of solid solutions.There cases were reported in the 1960s and 1970s (Sekiguchi andSekiguchi et al., 1963,1964; Goldberg et al., 1965; Guillory et al., 1and Niazi, 1971), but the strict criteria for solid solutions as described in the fore-going may be missing in most of these (Carstensen andAnik,1976; Carstensen91981). efore continuing, a couple of words on miscibility of ~ i is in ~ order. ~ i ~ ~miscible liquids may exhibit different temperature behaviors. The most c o ~ m o n isthe situation depic in Fig. 1l.l6(i). The two liquids are partially miscible betweenroom temperature T) and the boiling point (Bp). If a composition ofat temperature F , then the liquid mass will separate intotwo phases, a f ~ a c t i of ~ ~ ~ oB B A B A B A B x(Q) (1) (ii) (lii) (1v) Schematic of partial miscibility diagrams.
  • 197. composition x F ,and a fractionfG of composition xG.The weight arm rule applies, sothat (11.17)At times (situation in Fig. 1l.l6(ii)) there is complete solubility above a given tem-perature below the boiling point. At times (situation (iii)) there is a temperaturerange overwhich there isonly partial miscibility and at times (situation (iv)),there is full miscibility at a certain temperature below the boiling point. elts are liquids, and miscibility of A and I3 may be limited. In case (i) it isnecessary to heat the mixture to a temperature above the melting point ofhigher melting constituent), and even so, there will be two phases. In cases (iii) or(iv), depending on the melting point of l3, there will be a temperature range overwhich there is a single phase, and such systems, although they may exhibit phaseseparation at certain temperature, lowering the temperature will bring them into thecase discussed in Secs. 11.8 and 11.9. In cases (iii)and (iv) meltingat a temperature above the melting point of B willcause a single phase, but on further heating there will be a phase separation.cases(i) and (ii)maylead to "separate,, portions of a phase diagram, and theappearance of this mayresemble that of truly solid solution discussed in Secs.11.5 and 11.10, ATThere are several possibilities for what will separate out from a molten mix as it iscooled and point C (see Fig. 11,ll) is reached. There are also several possibilitiesforthe makeup of point Q (the eutectic). For materials that are neatly crystalline, the situation is as described in Sec.11.2. If the composition to start with is to the left of the eutectic, then all o f B is in avery jinely s~bdivided state. A situation a bit more complicated is that A crystallizes out but that, at theeutectic temperature, I3 remains amorphous. If this occurs, the attainment of a solidstate is the point at which the rubbery amorphate phase is sufficiently viscous. Thefinal product, then, is crystalline A dispersed in amorphous , a solid d i s ~ e ~ s i o ~ . y, it is obviously a solid dispersion of in a m ~ ~ h oA su that results, third case is one in which both comp nds remain amorphous. Here, ashas been discussed under Sec. 11.6, a ~ o r p h a ~ ethe situation may be one of two s,cases: 1. The amorphates are mutually soluble; conse~uently, is a solid sol~tion it o the second kind that results. f 2. The amorphates are only partially miscible, accordingly, it is a second case of a solid dispersion that occurs.Solid dispersions were originally suggested by Chiou and Riegelrnan (197 1) whodealt with the dispersion of drugs in a base of pol~ethyleneglycol (PEG). This
  • 198. was shown to give enhanced blood levels. It is a principle that has been successfulcommercially (GRISPEG). The process of solid dispersions is carried out by (a) either comelting the drugand a meltable polymer, such as PEG, or (b) by coprecipitating the drug with thepolymer from water or a solvent. Lipman and Summers (1980) and Hornand(1982), likewise have discussed subject. The comelt process will be discussed the first. Lloyd et al. (1997) and Craig and Newton (1991)have made paracetamol(aceta~inophen) and PEG 4000, solid dispersions containing 20% paracetamol bycomelting in a DSC pan at 1-h storage at 70°C. On melting a single endothermoccurs at 55”C, and by cooling, recrystal~zationthe exotherm occurs at 40°C and,contains a doublet. After reheating, a doublet occurs at 52” and 55°C. The authorssuggest that the dispersions contain recrystallized polymerin both an extended stablechain form and in a metastable form, folded once. If water is the %ft” component, then the right half of the diagram constitutesthe solubility~temperaturecurve of the “right” compound in water ( ~ a r s t e ~ s e n ,1977; Denbigh, 1961). Eutectic diagrams differ if different polymorphs are used as the second com-ponent; for example, Giordano et al. (1998) employing DSC, determined eutecticdiagrams of piroxicam pivalate and prioxicam and found them to be different for thetwo different polymorphs of piroxicam pivalate. This point is of interest because, interns of labeling, if a label states piroxicam pivalate, x mg, then the contents of thecontainer (e.g., tablets) should contain x mg us ~ i v ~ ~ u t e . In certain pharmaceusituations, a salt dissociates in the solid state (e.g., if the microenvironmental pincreased), and strictly speaking the drug substance decreases (because there is lesssolid present as the salt; e.g,, pivalate). DSC traces may detect this type of dissocia-tion. Strict interpretation in terms of law, would dictate that if the pivalate is the‘4drug,”then the free base is a “derivative drug.”The solid dispersion process is carried out (a) by either comelting the drug and ameltable polymer, such as PEG; or (b) by coprecipitating the drug with the polymerfrom water or a solvent. Lipman and Summers (1980)and Horn and Dittert (1982),have discussed these aspects of the subject. The question is, whether these solid dispersions are (a) solid solutions in thestrictest sense, (b) finely subdivided, crystalline drug substance in a polymer matrix,or (c) amorphous drug in a polymer matrix. devilliers et al. (1998) have describedcoprecipitates of acetaminophen with polyvinylpyrrolidone (PVP) formed by (a)coprecipitation, (b) recrystalli~ation,(c) mechanical mixing, and (d) freeze-drying,and have assessed the products by X-ray diffraction and by solubility. Decrease incr~stallinity was observed only in cases for which both the PVP and the acetamino-phen were soluble or partly soluble (ethanol and water), and the formed amorphousphase was a glass-like, solid solution. ases other than PEG have been suggested over the years [e.g., Brachais et al.(1998)havesuggestedpoly(methylglyxy1ate) as a base for oral, controlled drugdelivery systems]. ~oprecipitation yielded a better dispersion of drug in the basethan comelts. The coprecipitates reported by Brachais et al, (1998) lend themselveswell to compression, without slowing down the release to any extent.
  • 199. he question is whether these solid dispersions are (a) solid solutions in thestrictest sense, (b) finely subdivided,crystalline drug substance in a polymer matrix,or (c) amorphous drug in a polymer matrix,It is a common practice to cogrind drugs with polymers, such as ~-cyclodextrin( ~ i t r e v e et al., 1996; Arias et al.,1997), chitin and j h et al., 1986a,b), microcrystalline cellulose ( ~ a m a m o t o al., 1974, et t al., 1978), and gelatin (Kigasawa et ai., 1981). Chitosan (Portero et al., 1998)is ,&(1-4~-2-amino-2-deoxy-~-glucose is and y ~ " d e a c e t ~ l a t the~polysaccharide chitin. This is a substanc~ in that is nature, being the princi~al component of crustaceans, insects, and shells 1977). Chitosan is a good direct compression ingredient (Nagai et al.,1984; Upadrashta etal., 1992) whichenhances dissolution of many compounds (e.g.,nifedipine; Portero et al., 1998). hin et al. (1998) studied cogrinds of furose~ide with crosspovidone (poly-plas~one).The general purpose of solid dispersions is to improve dissolution rates. The litera-ture is replete with examples, most of them usingPEG as a dispersion vehicle. Figure11-17demonstrates this with. a solid dispersion of ofloxacin (Okonogi et al., 1997). Usui et al. (1998) have reported on the improve~entin dissolution of()k -~-(4-cyanoanilino) -5,6-dihydro-~-hydroxy-solvent-method-produced solid dispersions usi he dissolution equation is given by Idt = -kA(S - C) 1.18) (1 60 40 20 0 0 20 40 GO 80 Time ( m i d 7 Dissolution profilesof pure ofloxacin(circles) and in 1:4 solid dispersion in method. Dissolution of medium: distilled water,mannitol/urea made by the c o p r e ~ i p i ~ t i o nusing USP paddle method at 37°C at 100 rpm from a 10 m m ~ i a m e t e r tablet consisting 100mg of ofloxacin, compressed at 2 t/cm2. (Data from Okonogi et al., 1997.)
  • 200. where M is the amount notdissolved.,k is the intrinsic dissolution rate constant,C isconcentration at time t, S is solubility, and A is area. For a m o ~ h a t e s and (with the ~stwald Freu~dlich equation) also for veryfinely subdivided crystalline solids, the solubility S is increased over coarse crystal-line material. It is also seen from the foregoing text, that the surface area of eithermay be greatly enhanced by forming a solid dispersion. If the drug is compothen it is noted that forthe increased surface area and, perhaps, solubility to happen,the composition must be to the left of the eutectic. This is an advantage, because it that the d~ug-loading can be considerable. owever, most successful solid dispersions have a substantial amount ofassociated with them, and a disadvantage of the systems is actually that high-leveldosage forms do not 1 themselves to the approach. This is because a melt has tobe produced in which soluble. There is often a limit on the temperature to whichthe melt may be heated, so that the limitation lies in producing a homogeneo~s meltat the higher temperature. Increased dissolution rates, however, are also aided in that many me~t-produ-cing substances, such as P G, complex with the drug substance, and the complexes en have increased solubility. achais et al. (1998) have suggested poly(methyl~lyxy1ate) a base for solid asdispersion. If the dissolution occurred through erosion, then the dissolution profileshould be a cube-root plot, but their data do not lend themselves to this type ofplotting or to square-root in time plotting. The best plotting mode as shown in Fig. 11.18 is by a loglinear decay with a slight initial burst. /= I ( ) component A, (b) area a Aw = amount melted deduced from area in a DSC peak 1 = notation for a molecular, solid compound. at time t in a dissolution experiment bs9 degrees of freedom y = - 0.37 105 - 0.22 1 1 4 ~ = 0.999 R"2 09 0 2 4 6 8 10 Time (hr) from rachais et al., 1998,) Solid dispersion produced by copre~ipitation. (Data
  • 201. fL = A ~ / = fraction melted ~ A ~ ~ ~ ~ fs = 1 -fw = fraction not melted H = partial molar heat of the solid compound in ideal solution h = enthalpy of the pure solid per mole A H = heat of fusion AHA = heat of fusion of A AHb = the heat of fusion of B ~ H = the cross-hatched area (fraction melted) w AH^^^^* = heat of transition k = intrinsic dissolution rate constant I; = ( H A ) M is amount not dissolved m, = fs = mass fraction of solid phase mL = f L = mass fraction of liquid phase n = number of compounds p = mumber of phases R = the gas constant S = solubility [r = absolute temperature To = melting point of pure compound t = time XB = mole fraction of B in solid solution XA = mole fraction of A in solid solution is the mole fraction of A in a mixture x = mole fraction xL = mole fraction in liquidus phase x, = mole fraction in solidus phase B1 = a constant applying to B in the system B2 = a constant applying to A in the systemBogdanova S, Sidzhakova D, Karaivanova V, Georgieva S (1998). Int J Pharm 163:l.Brachiais C-€3, Duclos R, Vaugelade C, Huguet J, Capelle-Hue M-L, Bunel C (1998). Int J P h a m 169:23.Burger A, Ramberger R (1979). Microchim Acta 1979:259.Carstensen JT (1977). Pharmaceutics of Solids and Solid Dosage Forms. John Wiley & Sons, New York.Carstensen JT (1980).Solid Pharmaceutics, Mechanical Properties and Rate Phenomena:. Academic Press, New York, pp 105-110.Carstensen JT, Anik S (1976). J Pharm Sci 65:158.Chiou WL, Niazi S (1971). J P h a m Sci 60: 1333.Chiou WL, Riegelman S (1971). J Pharm Sci 60:1333 1281, 1376, 1569.Craig DQM (1990). Rrug Dev Ind Pharm 16:250.Craig DQM, Newton JM (1991). Int J Pharm 76: 17.Denbigh K (1961). The Principles of Chemical Equilibrium. ~ a ~ b r i ~niversityPress, d~e pp 256-264.De~illiers MM, Wurster DE, Van der Watt JG, Ketkar A (1998). Int J P h a m 163:219.Ford JL (1986). P h a m Acta Helv 61:69.
  • 202. Franks F (1990). Cry0 Lett 11:93.Frazer JCW (1931). In: Taylor HS ed, Treatise on Physical Chemistry. Van Norstrand, New York, p. 356, 556.Garner WE (1955). Chemistry of the Solid State. Butterworths Scientific, London, pp 213- 216.Giordano F, Gazzaniga A, Moyano JR, Ventura P, Zanol M, Pever T, Carima L (1998). J P h a m Sci 87:333.Goldberg AH, Gibaldi M, Kanig JL (1965). J Pharm Sci 54:1145.6uillory JK, Huang S, Lach J (1969). J Pharm Sci 58:301.Hildebrand GE, Muller-Goymann CC (1997). J P h a m Sci 86:854.Himes VL, ~ i g h e l AD, Decamp WH (1981). Acta Crystallogr B37:2242. lHorn D, Dittert W (1982). J Pharm Sci 71:1021.Levine H, Slade L (1988). Cry0 Lett 9:21.Lippman EC, Summers IvIP (1980). J Pharm Pharmacol 32:21P.Lloyd GR, Graig DQM, and Smith A (1997). J Pharm Sci 86:991.Lowes MMJ, Cairfa MR, Lotter AP, Van Der Watt JG (1987). J Pharm Sci 76:744.MacKenzie AP (1977). Dev Biol Stand 3651.Muzarelli RAA (1977). Chitin, Pergamon, Oxford.Nagai T, Sa~ayanagi Nambu N (1984). Chitin, Chitosan and Related Enzymes, Academic Y, Press, Orlando, FL, pp 21-39.Okonogi S, Oguchi T, Yonemochi E, Puttipipatkhachorn S, Yamamoto K (1997). Int J Pharm 156:175.Pohlman H, Gulde C, Jahn R, Pfeiffer S (1975). Pharmazie, 30, H11:709.Porter0 A, Remu~an-Lopez C, Vila-Jato JL (1998). lnt J P h a m 175:75.Reboul JP, Cristau B, Soyfer J-C, Astier J-P (1981). Acta Crystallogr B37: 1844. eck 6,Dietz G (1986). Cryst Res Techno1 21:1463.Sekiguchi IC, Obi N (1961). Chem P h a m Bull, 9366.Sekiguchi K, Obi N, Ueda Y, Nakamori Y (1963). Chem Pharm Bull, 11:1 108, 1123.Sekiguchi IC, Obi N, Ueda Y (1964). Chem Pharm Bull, 12: 134, 164.Shin S-C, Oh I-J, Lee Y-B, Choi H-IC, Choi J-S (1998). Int J P h a m 175: 17.Suzuki T, Franks, E; (1993). J Chem SOCFaraday Trans, 89:3283.Terrence, CF, Sax M, Fromm GH, Chang C-H, Yo0 C (1983). Pharmacology, 27235.Turel I, Bukovec P, Quiros M (1997). Int J P h a m 15259.Upadrashta SM, Katikaneni PR, Nuessle NO (1992). Drug Dev Ind P h a m 18: 1701.Usui F, Maeda K, Kusai A, Ikeda M, Nishimura K, Yamamoto K (1998). Int J Pharm 170:247.Yu L (1995). J Pharm Sci 84:966.
  • 203. This Page Intentionally Left Blank
  • 204. 12.1. The Noyes-Whitney Equation 19112.2. The Wood’s Apparatus: Sink Plate iss solution 19212.3. issolution by ~alorimetry 19412.4, No~sink, o n s ~ aSurface Area ~ nt 194 12.5. ffect of Variables 194 12.6. ilm Theory and the Levich Equation 195 12.7. The Nelson and Shah Equation 197 12.8. ixson-Crowell (Cube 198 12.9. Constancy of the Shape Factor 20212.10. ependence of the Shape Factor During 203 Symbols 206 References 208The rate with which a drug substance dissolves either from neat drug or from adosage form is of great i ~ ~ o r t a because it often governs the biophar~aceutical n~eprofiles of the drug.The equation developed by Noyes and Whitney in 1887, states that (Fig. 12.1), whena substance (with solubility S ) dissolves from a planar surface of surface A, then its~issolu~ion -dm/dt (where w1 is mass and t is time), is given by rate, -dm/dt = kA(S - C ) (12.1)
  • 205. ter 1 Punch Shaft Powder, Tablet Platen .1 Wood’s apparatus.where k is the intrinsic dissolution rate constant (cm/s), and C is the on cent ration attime t. If the dissolution takes place into a volume of dissolution liquid V, then theconcentration will change with time by a modification of Eq, (12.1) dm/dt = VdC/dt = -kA(S - C) (12.2) s: any criticisms have been voiced against Eq. (12.2), but in general it is correct, andit will be assumedto be so in the following, ~xperimentationcan be carried out withconstant surface as when using a Wood’s apparatus as shown in Fig. 12.1. In this, adie (as shown in the upper left of the figure), is placed on a platen and filled withpowder. The powder is then compressed. The die is removed,and a shaft screwed onto it (as shown in the upper right of the figure). This is then lowered into a dissolu-tion container, the shaft is attached to a motor, and the die is rotated, most often at50 rpm. With smaller amounts of drug available it suffices to make a small pellet andencasing it in wax and exposing only one face to a dissolution medium. Alternativelya fairly constant surface area can be assured by simply employing a large excess ofpowder so that only a small amount of the solid, in the long run, dissolves. In all ofthese cases Eq. (12.2) may be integrated to give ln[l - ( C / S ) ]= -(kA/V)t (12.3)or C = S[l - exp(--(kA/V)t)’J (12.4)A typical curve following Eq. (12.4) is shown in Fig. 12.2.
  • 206. 40 r v - 04 0 20 60 80 Time (min) Dissolution of pure ciprofloxacin in distilled water, using USP paddle methodat 37°C at 100 rpm from a 10-mm-diameter tablet containing 100 mg of OFX compressedat 2 t/cm2. (Data from Okonogi et al., 1997.) The use ofa cornpressed disk is referred to asa Woods apparatus ( ~ o o et al., d1963) whenthe setup is as shown in the second drawing in Fig. 12.1. this manner it Inis felt that the area will stay constant (see Fig. 12.2). Plate dissolution is at best an estimate, and probably a fairly inaccurate esti-mate, of dissolution. The general idea is that the surface that is exposedto the liquid(Fig. 12.3A) will stay constant during the dissolution and simply recede (see owever, it is more likely that it will become uneven during dissolution(Carstensen, 1974, 1977) (see Fig. 12.3C). The other problem is that often too littlecompression pressure is applied to prevent the compact from being porous. This,essentially, should cause curvature in the dissolution plotted according to Eq. (12.4).Often as little as 1-ton/cm2pressure is applied, and only for substances with very lowelastic yield value, will this suffice to make a nonporous compact. c evert he less the method is useful, because it provides some measure of com-parison (e.g., between the k-values for different salts of a drug substance). But evenunder such condition direct comparison is not exacting because the substances mayhave different yield values; hence, they give different porosities. The best method is to repeat an experiment several times using different com-pression pressures until a consistent value of k is obtained, t o w pressures lead to wrong k-values and erroneous conclusions (Chen and Grant, 1998; Grant et al., 1984). A G .3 Types of compact formations in plate dissolution.
  • 207. ne reason that lowpressures are preferred by many investigators is that pressures may lead to pol~morphictransformations. It is a l ~ a a ~sv i s a b ~ e ~to r e ~ o v e s ~ a l l a ~ o u oft solid from the back of the c o ~ ~ a and test it for a n ct~ o r ~ h o (Cuillory, 1992). l o ~ ~ In the critical time path for product development, solid-dosage forms (tabletsor capsules) must eventually manufactured for the clinic (e.g., in clinical be phase 11). drug substance per se is subjected to a dissolution test in a Wood’s odet al., 1963). Thistestisuseful, although quite dependent on onditions, as shall be discussed shortly. It is possible from data ofthe fore oing type to calculate k (cmls). concern with iss solution is that of bioavailability. It has be lman, 1979) that if k is obtained under sink conditions over a p 37°C in a USP vessel by way of Eq. (12.3) at 50 rpm, then if the onstant ( k A / V ) is greater than 1 mg min-’ cm , then the drug isnot prone to give dissolution-rate-limited absorption problems. n the other hand,if the value is less than 0.1, such problems can definitely be anticipated, and com-pounds with values of kA/ V of from 0.1 to 1 mg min-’ cm-2 are in a gray area. For ound selectivity it is frequently useful to express dissolution findings in terms ofk (Le., in cm/s).For a small amount of powder, dissolution of the particulate material can often beassessed (and compar d with that of other compounds), by placing the powder in aca~orimeter (Iba et al., 1991) and measuring the heat evolved as a function of time.The surface area must be assessed microscopically (or by image a n a l y ~ e rand the ~,data must be plotted by a cube-root equation. (Hixson and Crowell, 1931),a point tobe discussed presently. 1 - [ M / ~ 0 ] 1 = -(2kS/pr)t ’3 (12.5) is mass not dissolved, Mo the initial amount subjected to dissolution, p istrue density, S is solubility, and r is the mean “radius” of the particle. The method issimply comparative, not absolute, owing to the hydrodyna~ics being different inthecalorimeter than it would be in a dissolution apparatus. / ~ 0 7 here proportional isto the area under the (differential) calorimetric curve at time t, divided by the totalarea under the calorimetric curve.If constant surface area dissolution is carried beyond the sink level, then curvatureresults. Figure 12.4 demonstrates this (Usui et al., 1998). It is noted that i~itially (upto 10 min) the curve is fairly straight, but then begins to curve. If plotted logarith-mically, it linearizes (Fig. 12.5).The variables in Eq. (12.1) are the solubility, the surface area, and the dissolutionrate constant. Although k is thought of as a constant, it is only a constant at a given
  • 208. Time (min) Dissolution o f (f)4-(4-cyanoanilino-5,6-dihydro-7-hydroxy-7~-cyc~opental~~pyrimidine) hydrochloride. (Data from Usui et al., 1998.)temperature and under given hydrodynamic conditions. Its hydrodynamic variabilitywill be discussed in a subsequent section. Relative to surface area, Carstensen (1977) has pointed out that asthe surface(see Fig. 12.1) recedes, the area may not be “smooth,” and the cross section of thedie, ~ssumed be the surface area, may not be so. There is also the problem with toadhered “dust,” which may give initial burst. The powder used,should be fine,for anotherwise, particles can. “fall out.” e hidden variable, rarely discussed, is the pressure at which the compact is carry out the experiment in a duplicable fashion, several curves should begenerated using different compression pressures. The pressurewhich the curvesbecome duplicable is then the pressure that should be indicated. uthors most oftenindicate the pressure used, but do not justify the choice.Equation (12.1) was, for a while, explained in the following fashion: reference i smade to Fig. 12.6. A plane surface allows dissolution of the solid into solution. y = 2.3454 - 9.4882e-2x RA2 = 0.997 “ 0 10 20 30 Time (min) Data in Fig. 12.4 treated according to Eq. (12.3) assuming a s~lubility 10 pg/ ofmL.
  • 209. Dissolution Bulk Solution Schematic of film model.is assumed that there is a film, of thickness h, which is attached to this, and that thelayer adjacent to the surface is saturated, whereas at distance h, the concentration isthat of the bulk solution (i.e., C). Fick’s law now gives: J = --D(dC/dy)where D is the diffusion coefficient, y is distance perpendicular to the plate, and J isthe flux. This latter is (1/A) dC/dt, so that Eq. (12.14) becomes: (1/ ~ ) d ~ = --D(dC/dy) / d ~ (12.7)It is noted that the amount dissolved A4 equals the volume V of the dissolutionmediumtimes the concentration. Therefore, if it isassumed that D is distance-independent, then dC/dy ==: (C - S)/h (12.8)so that (Eq. 12.7) becomes dA4/d~ -VdC/dt = ( ~ / h ) A (- C) = S (12.9)hence, dC/dt = ( ~ / h ) ( A / ~ ) ( ~ - C) (12.10)that is, the intrinsic dissolution rate constant from Eq. (12.2) becomes k =D/h (12.11) quat ti on (12.2) is often written in the fashion of Eq. (12.10). The ~tokes-Einstein equation states that (12.12)where IC is the Boltzmann constant, T is absolute temperature, q is viscosity of thedissolution medium, and a is the molecular “radius.” This inserted in Eq. (12.19)gives k =~ ~ ~ ( 6 h n ~ a ) (12.13)or + ln(k/[T) = - 1n[q] ln[~/(6na)] (12.14)
  • 210. ~arstensen Pate1 (1975) studied the dissolution of oxalic acid at different tem- andperatures and reasoned that In[k/Tl should have the temperature dependence of theviscosity: that is,where B is a constant. It is noted that viscosity decreases withincreasing temperatureso that for viscosity the activation energy, Ea, is negative. Theyfound that the slopeof the modified Arrhenius plot [see Hq. (12.14)] was very close to the activationenergy for water’s viscosity. In spite of this evidence, there is good reason to believe that it is the Levichequation, rather than the film theory as explained here, that applies to plate dissolu-tion. The Levich equation states that (12.16)where Q is a constant, v is kinematic viscosity, and w is rotational speed of the plate. It is noted that by conducting the experiment at different rotational speeds it ispossible, by plotting the dissolution rate versus the square root of w to obtain thediffusion coefficient. .In a stream passing over a plate, the dissolution is dictated by the Ficksian equation D32C/3y2- Vx(y)3C/3x= 0 (12.17)Here (as denoted in Fig. 12.79, D is diffusion coefficient, C is concentration, y isdistance from and pe~endicular tothe plate, x is distance along the x-axis, Vx isvelocity in the x direction (the dependence on y-value is denoted vx(y) ), and C isconcentration. The first term is diffusional and the second one is convectional.Ass~ming,as is shown in the leftprofile in Fig. 12.7 that the velocityincreaseslinearly with distance from the plate, then V x @ > = BY (12.18)V,(y) is a linear velocity that may be converted to mass flow rate (Q cm3/s) by~ u l t i p l y i n ~by the cross section of the channel. itThe initial and boundary conditions are C,=:o;x=o’o(y<~ (12.19) C , = S ; y = O , O < x= O ; y = O o C,<L x=o .7 Schematic of dissolution from a plate by a stream.
  • 211. ao et al. (1997) concluded that the dissolution from a Shah-Nelson plate(Shah and Nelson, 1975) of griseofulvin into a surfactant solution should exhibitthe following flux, J J = 0.8080~3S,,["/N2B]3bhL23 (12.20)where D, is a composite diffusion coefficient, Sm is the solubility of griseofulvin inthe micellar solution, v denotes the rate of flow,B and H are the width and height ofthe channel and I; and b are the length and width of the compress. A similare~uation, without the m-subscript, will hold with the foregoing boundary and initialconditions, for diffusion into a simple solution. According to this the log of _R should be linear in the log of Q with a slope of1/3, and indeed they find such a relation. It is seen from Fig. 12.8 that the relation is linear, but that the position of the identical with the theoretical line. ne problem with the experiments is the assumption of linearity of Vx(y), and relation is probably more parabolic as shown in the right-hand profile inThe foregoing text has concentrated on the basic mechanisms for dissolution fromt~eparti~le s~~face into moving body ofwater.For dosage forms, the drug substanceis usually present as a multiparticulate population. If the particles are all the samesize then the population is m o n o ~ i s ~ e r sIf .not, then it is denoted ~ o ~ y ~ i s p e r s e . e Thefirst topic at this point will be the dissolution of a monodisperse powder under sinkconditions. The dissolution of a particle and a ~ o n o d i s p e ~particle pop~lations se were first ixson and Crowell (1931). They assumed the particles to be spherical,and their derivation, in modified form, will be presented here, with exception that the y = 0.18130 + 0.33721~ RA2 = 0,999 n Theory y = 0.27599 + 0,30659~ RA2= 0.998 0 Experimental 0.9 x 0.8 0 d) 0.7 0.61.7 1.5 1 .3 1.6 1.4 1.8 1.9 2.0 Dissolution of griseofulvin into solutions of sodium dodecyl sulfate (20 mM) solution. (Data from Rao et al., 1997.)
  • 212. id aca cubical particle shape will be assumed, and the equation will be derived for a singleparticle of sidelength b. If we assume the particle to be isotropic and dissolving under sink conditions, oyes-~hitney equation becomes: drnldt = -ksS (12.21)lowercase s denotes the surface of the particle. From the geometry it follows that rn = pb3and s = 6b2 (12.23) q. (12.21) becomes (1 3pb2dbldt = --6kb2S 2.24)or (12.25)This, after initial conditions, integrates to b = bo - (2kSlp)t (12.26)or bo - b = Kt (12.27)where the cube-root dissolution rate constant K is given by = (~kS1p)t (12.28)It is noted that (12.29)so that Eq. (12.26) may be rewritten: (12.3 1)The rate constants K and K are referred to as the parti~Ze cube-root rate c~nstants. Shape factors have been dealt with before, but to repeat, the surface shapefactor, a,,is defined by the equation s = a,d2 (12.32)where s is the surface of the particle, and d is its "size" (i.e., a defined d~mension),such as the width of the particle, b. Similarly the volume v of the particle is given. by( ~ a l l a ~ a l l1948) as e, 21 = a,d3 (12.33) The ove~aZZshape factor I is given by the equation
  • 213. r l ??-Values for Some Isometric Shapes _____Shape Shape factor Numerical valueSphere [6]2/3~- 1/3 4.834Cube 6 6Isometric cylindera f .5 [4]2/3n-/3 5.54"A cylinder for which the diameter equals the height. r =s / ( v ~ / ~ ) (12.34)so that, by combining these three equations it follows that s =* ~ / ( * ~ / 3 ) (12.35)The overall shape factor will be referred to as the ~articZes ~ actor in the text ~ ~ eimmediately following. Particle shapes for which the shape factor is independent ofsize are denoted isometric, and examples of this are listed in Table 12.1 with thea~propriate I"-values. The shape factor for a sphere is smaller than that for any otherparticle shape. In the following there will be two situatio~s that will be considered: (a) thedissolution of a monodisperse powder (Fig. 12.9), and it is noted here that all theparticles disappear at the same time. (b) The other situation i s for oae particle. The~ o y e s - ~ h i t n eequation dictates that for N particles y dm/dt =L= -kNsS (12.36)If N ~ a r ~ i c l eeach of mass m, volume v, and density p are allowed to dissolve, then s,the mass m of a particle is the density p times the volume v and it follows that Eq.(12.36) for N particles is (Np)dv/dt = NksS (12.37)or €l ! Zero €l ! ZeroTime Zero tu 1 Oissolu~ofl or Critical Time 2.9 The situation during dissolution of a monodisperse powder. Note that all theparticles disappear at the same time.
  • 214. dv/dt = kFv2I3S/p (12.38) ecause s = h213,therefore dv/[v2’3 ] = (krS/p)dt (12.39)This integrates (after initial conditions are imposed) to v:I3 - v‘’~= (kFS/3p)t = Kvt ( 12.40)where Kvis the cube-root dissolution rate constant based on volume for one particle. 1 - [ v / v ~ ] ” ~ {(krS)/(p3v:I3)}t = (12.41)It is noted that vivo = N ~ / N p v o M / M o = (12.42)where A is the total mass of the undissolved powderat time t, and Mo is the amount 4before dissolution. Hence, the more familiar form of Eq. (12.40) emerges. 1 - [M/Mo]’/3 {(~rs)/(p3v:13)}t = (12.43)Since v;l3 = (~01p)113 ~ = ~ ~ l ( ~ ~ ) ~ ~ 1 ~ (1 2.44)Equation (12.43) may be rewritten 1 - [M/MoI1l3= {(krs)/(p3(Mo/Np)’13)Jt (1 2.45) = {(krS)p-213M~”3N113/3)tK ~ n t =where f& ( (krs)p-213A4-~13N‘13 0 13 (12.46)is the relative rate constant based on a population o N p~rticles. fInserting Eq. (12.44) in Eq. (12.40) gives - [ M I ( N ~ =I( ~ w 3 p ) t ) k~~ (12.47)or M;13 __ A4113 = [ ( ~ p ) l l / ~ ( k r s / 3 p=tKmnt ) 2.48) (1where ICmn = N1I3krSpV2l3 13 2.49) (1Kmnis the cube-~oot constant of N particles based on mass. The various rate constantsare summarized inthe symbol listat the end of the chapter. It is importa~t keep an toaccount of which dissolution rate constant isbeingdiscussed,because incorrectconclusions may be drawn from employing an incorrect definition. It i s worth repeating that cube-root dissolution in the strictest sense is derivedfrom the following assumptions: 1. The particles are isometric.
  • 215. rl 2. The population is monodisperse. 3. The dissolution is isotropic. 4. Sink conditions prevail, owever, the first assumption may be studied separately. It is important to investi-gate whether the cube-root equation also holds for situations in which the particle isnot isometric. xson-Crowell equation, it isassumed that the shape-factor is constant dissolution event(Le., an isometric particle will dissolve according tothis equation). It is, however,not only the effect of the shape factor thatis of importance. Theintrinsic dissolution rate constant k is assumed to be a constant for the substance. Much dissolution work has been done with this assumption, in spite of thethrough workofPedersen and Brown(1976),who studied the dissolution fromvarious faces of crystals of different morphology, and derived equations thatdescribe their dissolution behavior. The interest in this lies in (a) whet he^ there aresituations for which the shape factor stays constant, and (b) if it does not, to studythe shape factor as a function of dissolution time. The former will be addressed firstand after that a derivation will be presented to study the extent to which changes inshape factor may affect the cube-root law. The initial steps o f the derivations followfairly closely those of Pedersen and Brown (1975, 1976), and that the shape factorcalculations are the ones unique to our theme developed thereafter. There is one (probably very common) situation that allows theequation to be used without adjustment for change in shape factor during the dis- solution, It is often postulated that dissolution rate constants (k; crn/s) are numeri-cally identical, but opposite in sign to crystallization rate constants. If a crystal grows from a nucleus in the 6, h, and L-directions (Fig. 12.10) with constants kb, kh, and kl, given by kh = (h1b)k~ or khlh = kb/b (12.50) kt = (L/6)kb or k L / L = kblb (12.5 1)then, if these constants are also dissolution rate constants, then a parallelepiped withsides b, h and I will dissolve by the equations b Schematic of ideal crystal.
  • 216. (12.52) (12.53) (12.54)so v2I3 = {L0b0h0}2f3{ (kb/bo)t]2 1 (12.57) -That is, the shape factor s/u2I3 8(Lo + ho + b o ) / { [ L ~ b o h ~ ] ~ ’ ~ } (12.58)This is independentof time; hence, the shape factor, under these conditions, will stayconstant during the dissolution event.It is for the reasons stated in Sec. 12.9 that most often the Hixson-Crowell equationworks well. At times, the shape factor changes suf~ciently during a dissolution runthat there are deviations from the cube root law. The extent to which this happenswill be discussed in the following. It has been shownthat fora single isometric particle [see Eq. (12.3 1) multipliedt h r o ~ g h p] by pdvldt = kirv2I3S (12.60) ai and Carstensen (1978) studied the effect for isotropically dissolving oxalicacid cylinders to establish the effect of the shape factor on the dissolution pattern. oni isotropic shapes were simulated by cylinders, and the effect of dissolution on thetheoretical shape factor determined. This was then compared with the actual dis-solution of a cylinder and good agreement wasobserved. It is only when a particle isisometric (i.e., when l? is independent of dimension) that this may be expected. Lu et al. (1993) used the concept proposed by Lai and Carstensen (1978) ofimitating the shape of a crystal by a cylinder, and found that this model yields abetter fit of dissolution data for hydrocortisone particles than a similar model usingspheres as a model.
  • 217. Lu et al. (1993) assume the shape factor to be constant during the dissolution event, a fact that is not correct and the rami~cations this will be discussed in the of following. Consider a parallelipiped of length L breadth b, and height h as shown in Fig. 12.10. The area s and volume v of a particle with such a geometry are given by the follow in^ equations. s = 2(Lb + hL + bh) 1) (12.6 v = Lbh The Noyes-Whitney equation under sink conditions states thatdt dm/dt = -ksS (12.63) ~ombining Eqs. (12.61), (12.62), and (12.63) now gives pdvldt = 2kS(Lb (12.64) bh) hL -I- + Since v is a composite function of L, b, and h, the rate of change of volume relative to time can be written as + dvfdt = {dv/dL}{dL/dt} {dv/dt}{db/dt} {dv/dh}{dh/dt} + (12.65) + = bh(dL/dt) Lh(db/dt) Lb(dh/dt) + This is introduced into Eq. (12.63) to give bh{dL/dt] + Lh{db/dt} + lb(dh/dt} =T= -{2kS/p}(Lb + hL + bh)(12.66) or (12.67) This differential equation is in the separate variable form. Hence, it follows that dL/dt I- db/dt I=I dh/dt = -K (12.68) where 1°C = 2kS/p (12.69) This implies isotropic dissolution. These expressions can be integrated to obtain relations between the crystal dimensions and time of dissolution. If Zo, bo, and ho are the initial length, breadth, and height of the crystal, then integration of Eq. 12.68 gives L = Lo - Kt = Lo(1 - (K/L*)t] (12.70) h = ho - Kt = ho(1 - (K/ho)t} (12.71) b = bo - Kt = bo{1 - (K/bo}t} (12.72) enoting: 1 - ( h / ~ o= 1 - (b/bo) = (1 - L/Lo) = u ) (12.73)
  • 218. where u is dimensionless, it follows that u = Kt/ho = Kt/bo = Kt/Lo (12.74)Employing these i ~ e ~ t i t i Eqs. (12.78 to 12.80) become: es L =I: Lo - uho (12.75) h = ho - uho (12.76) b = bo - uho (12.77)It should be noted that just before dissolution starts h = ho and, at the point whenthe crystal completely dissolves, h = 0. Hence, the domain of u is [O,l]. The followingequations ensue by inserting Eq. (12.69) into Eq. (12.74) u ={2~~1(~hO)}t (12.78)To determine u,we consider the fraction of amount undissolved (I;= ~ / ~ oI;)can.be expressed as the ratio of instantaneous volume of the dissolving particle to itsinitial volume, density being constant, that is, I; = ~ b h / ( ~ o b o= (Lo - uho)(bo - uho)(ho - ~ h o ) / ( L o b o h ~ ) ~o) (12.79) = { 1 - (ulLo)ho}{ - (ulbo)ho}{ - (ulho)} 1 1 Two shape ratios are now defined inthe following fashion which are indicativeof the shape of the crystal, namely, P = Lolho (12.80) 4 = bolho (12.8 1)Rearranging the foregoing equation using these two ratios gives the following result = (1 - uH1 - (ulq)l{l - U l P } (12.82)This is a third-degree equation in u.The criteria for choosing the “correct” root ofthe possible three is that it should be a real number between 0 and 1. F can beobtained from dissolution data, which enables one to solve Eq. (12.82) for u. FromEq. (12.78), it is clear that a linear relation between u and t exists. Adequate linearityfor a plot of u versus t has been demonstrated by Lai and Carstensen (1978) forcylindrical tablets of oxalic acid. The slope of sucha plot gives the value of the intrinsic dissolution rateconstant if the solubility, density, and initial dimensions of the dissolving particleare known. Tsotropicity and isonletricity are some of the basic assumptions in the deriva-tion of the Hixson-Crowell cube-root law. A cube, a sphere, and a right circularcylinder are examples of isometric geomtries because their shape factors are inde-pendent of their dimensions. Real particles are far from being isometric. The shapefactor for one particle is defined as r =: sv-2f3 (12.83)so for the parallelipiped model described in the foregoing r = 2(Lb + bh + Lh)(lbh)-2f3 (12.84)
  • 219. Inserting Eqs. (12.83), (12.841, and (12.85) into Eq. (12.92) yields (12.85) earranging this equation to write it in terms of the shape ratios p and q, gives (12.86) of this expression it is possible to calculate the shape factor for thepara1 geometry considered here, as a function of reduced time. Lai andCarstensen (1978) followed a similar approach for cylindrical tablets with differentradius~hei~htratios. They found that, expectedly, for an isometric tablet (there was no change in the shape factor as the dissolution proceeded.signi~cantchangesin the shape factor wereencountered as this ratio increasedabove unity. For a ratio of 2.75, they observed that r changed si~nificantly after50% dissolution. This is exemplified in Figs. 12.11 and 12.12. A = surface area of a plane b = (a) breadth or (b) size of a dissolving particle bo = original (a) breadth or (b) size of a dissolving particle I) = diffusion coefficient d = general term for the size of a particle y = - 1.0795e-2 -t 4.4607e-2x RA2 = 0.999 Time (min) Fraction retained [see Eq. 12.861 and reduced time [see Eq. 12.741 as a function oftimeofdissolution of a ~ o t a s s i u ~ dichromate crystal of d i m e ~ s i o ~ s= 1.120 cm, lo bo = 0.518 cm, and h, cm. (Data from Dali, 1997.)
  • 220. ict 30 20 n 0 10 20 Time ( ~ i n u t ~ s ) Shape factor as a function of dissolution time of a potassium dichromate crystalof dimensions lo = 3.120 cm, bo == 0.518 cm, and ho cm as a function of dissolution time. (from Dali, 1997.) h = (a) thickness of film layer (b) height of a crystal J = flux K = ~ k S / p cube-root dissolution rate constant for a cube = Kv = k r S / 3 p = cube-root dissolution rate constant for a particle with a shape factor of r KmB = N1/3krSp-2/3/3= cube-root dissolution rate constant for apo~ulation of N particles = (kI"S)N1/3p "2/3"1/3 /3 = relative cube-root dissolution rate constant for a population of N particles k = intrinsic dissolution rate constant kb = intrinsic dissolution rate constant in b-direction kh = intrinsic dissolution rate constant in h-direction kL = intrinsic dissolution rate constant in L-direction L = length of a crystal = mass of a po~ulation monodisperse particles of = original mass of a population of monodisperse particles m = mass not dissolved at time t for a single particle mo = original mass of a single particle = ~ o / ~ o = mass flow rate (cm3/s) 4 = bolho s = surface area of a single particle S = solu~ility t = time u = reduced time V = volume of liquid v ~ c ~ ) velocity Q (cm3/s) = linear y = direction ~erpendicular a dissolving surface to
  • 221. r l a, = surface shape factor of a particle av = volume shape factor of a particle /3 = constant in the flow conversion equation lr = shape factor p = density sCarstensen JT (1974). In: Leeson L, Carstensen JT, eds. Dissolution Technology. The Academy of Pharmaceutical Sciences. American Pharmaceutical Association, X, p 5. ~as h in gt on,Chen LR, Grant DJW (1998). P h a m Dev Techno1 4:487.Dali M (1997). PhD dissertation, University of isc cons in, Madison, WI.Dali MV, Carstensen JT (1996). P h a m Res 13: 155.~allavalle JM (1948). In: Micromeritics, 2nd ed. Pitman Publishing, New York, p 142. M,Grant DJW, Medhi~adeh Chow AHL Fairbrother JE (1984). Int J Pharm 18:25.Cuillory K (1992). Personal Cornmunication.Iba I(, Arakawa E Morris T, Carstensen JT (1991). Drug Dev Ind Pharm 17:77.Lai TY-F, Carstensen JT (1978). Int J Pharm 1:33.Levich VC (1962). Physiochemical Hydrodynamics. Printice Hall, Englewood Cliffs, NJ, pp 87-1 16.Lu ATK, Frisella ME, Johnson KC (1993). Pharm Res 10:308.Noyes A, Whitney W (1897). J Am Chem SOC 23:689.Olconogi S , Oguchi T, ~onemochi Puttipipatkhachorn S, Y a m a ~ o t o (1997). Int J P h a m E, K 156: 175.Pedersen PV, Brown KF, (1976). J P h a m Sci 64: 198 1.Rao MR, Lin M, Larive CK, Southard MZ (1997). J Pharm Sci 86:1132.Riegelrnan S (1979). Dissolution testing in drug development and quality control. The Academy of Phamaceutlcal Sciences, Task Force Committee, American Pharmaceutical Association, p 3 1.Shah AC, Nelson KC, (1975). J Pharm Sci 64: 151824.
  • 222. 13.1. Dissolution of Polydisperse Powders 209 rticle Size ~istributions l is solution and 210 ssolution After the Critical Time, t* 217 219 Symbols 220 Appendix 22 1 eferences 22 1If powders are polydisperse, then if the population is “infinite,” that is starting at sizezero (with an infinitely small probability or fraction) and on the high side ending atan infinite large size ( a g a i ~with an infinitely small probability or fraction), then it ispossible to solve dissolution patterns in closed form. As we will see in the following, a more realistic powder population is one inwhich there is a finite, smallest particle and a finite largest particle (Fig. 13.l), and forsuch systems the type of dissolution pattern of the type in Eq. (12.48) (i.e., a cube-root law adherence) will prevail. ~ a r s t e n ~ e n ~ u s (1972) applied si~ulation a truncated l o ~ n o r ~dis- and a to altribution of spherical particles to the principles of dissolution, and this was followedup by a solution in closed form by Brooke (1973, 1974). In the latter case, threeadditional assumptions were made: namely, that (a) the smallest particle was zerosize; (b) the expression for the distribution remained correct during the dissolutionprocess; and (c) a number-based lognormal distribution would also be l o g n o ~ a lafter t r a n s f o ~ a t i o n the Hatch-Choate equation. by Brooke (1973,1974) arrived at the conclusion that for a polydisperse, lognor-mal powder, the plot of amount retained versus time would follow a cube-rootequation, but the slope would be inversely proportional to ln[a]. This was confirmedexperimentally by Carstensen and Patel (1975) who found that, approximately
  • 223. Size Zeroa(rnax) cl Zeroa(m1nf Zero Time Zero C r i Ti ic a e t ml t(3) Dissolution Time The situation in the dissolution of a polydisperse powder. (13.1)where h,, is the geometric mean of the particle population and ln[o-] is its standarddeviation. articulate solids resulting from unit processes such as crystallization, precipitation, milling, exhibit skewed particle size distributions (Carstensen and Rodriguez- nedo, 1985; Steiner et. al., 1974; Carstensen and Patel, 1975). The importance of particle size distribution of powder substances for dissolu- n well documented inthe pharmaceutical literature (~arstensen and ke, 1973, 1974; Higuchi and Hiestand, 1963; Higuchi et al., 1963; intz and Johnson, 1989 Pedersen and rown, 1975). Simply said, smaller particlesdissolve at a faster rate than larger particles. ~ l t ~ o u g h treatable from a theoretical point of view, there are practical veryproblems thatare associated with particle size distribution and dissolution. Intoday’s climate, virtually all solid-dosage form products are routinely subjected todissolution testing, and the most common cause for product recalls is failure of a roduct to meet dissolution specifications (Cabana and article size limits are included in drug substance specifications, because particle sizeaffects both dissolution characteristics and machinability (flow or compression) ofthe substance. However, most particle size distributions are derived from volume-based measurements (e.g., Coulter counter) of the particles from a random sampleand suffer from the fact that (a) the volume is converted to an equivalent sphericalradius, and (b) that the sample size is always very small,havecautioned researchers about the misuseof the spherical approximation bystating that, “...direct comparisons between microscope-derived mean particle sizeparameters and Coulter~ounter~derived data lead to erroneous conclusions if canno consideration isgiven to the shapes of particles...,” ( oughton and Amidon, 1991).Size determination of needle-shap~dparticles b techniques such as a oulter counter can lead to experimental difficulties such as 66coincidence’7 twoofor more particles at the orifice leading to faulty results. It further has the dis~dvan-
  • 224. tage that it does not directly address the main reason for its execution: namely,dissolution in a USP dissolution apparatus. croscopy will yield information about the c ~ ~ a Z a dimensions of nonspherical . In this, a particular dimension (length, breadth,Ferret diameter) isselected, and a certain number of particles examined, and classified in size ranges.The numbers in the ranges may be converted to fraction of particles, and the curvemay be normalized, as shown in Fig. 13.2, so that the area under the curve ( ~is ~unity. The major advantage of this method is that it can furnish size as well as shapeinformation about nonisometric solids. On the other hand, the inherent tediousnessand time-consuming nature of this procedure limits its use, Also, the user is restrictedto a relatively small sample s , based on which the representativeness of the powderpopulation has to be relied. ughton and Amidon (1991) have suggested a micro-scopic-based image analysi utine procedure to have a check on the lot-to-lot e particle size and shape characteristics of three lots of an investiga- ughton and Amidon, 1991). Invariably, such methods rely on the re-defined size and shape parameters. Powder dissolution of poly-disperse samples can be used, with distinct advantage, t o o ~ t a i meaningful infor- nmation on particle size distribution of crystalline substances of nonspherical nature (Dali, 1997). If a particle has an initial breadth bo, and volumetric shape factor avothen the original mass, m, of that particle is given by m = pv = p(avo)b~ (13.2)where v is volume, and p is particle density It is possible, by microscopy, to deter-minetwoof the three dimensions and to plot these by a normalized fre~uencyfunction f(bo), so the number fraction iVb of particles between the infinitesimallysmall interval (bo,bo + dbo) would be given by 0.008 0.006 & 0,004 n 0.002 180 230 280 330 430 380 Breadth (microns) Frequency distribution function o f a -401 + 50 sieve fraction of oxalic aciddifiy- drate.
  • 225. Denoting the initial mass of the whole powder population by Mo, it would given by (1 3.4)b,, and bm,,, here denote the largest and smallest dimension of the particles in thepowder population and bo has reference to the fact that the particles will be placed ina iss solution medium at a given time of zero. Equation (13.4) applies to a polydis-perse system ofparticles (for instance, a sieve fraction), and anaverage valuefor theinitial volume shape factor has been ascribed to that particular sieve fraction. If the powder is dissolved under sink conditions, then the dimensions of theparticle willdecreaselinearly with time (Carstensen and Musa, 1972;Carstensen, 1996; Edmundson and Lees, 1965; Schoonen et al. 1979). b = bo - Kt (13.5)Note that the initial distribution function can be used to calculate the mass undis-solved until the critical time t*. It is at the critical time that the smallest particledisappears from the dissolution medium, and up to this point in time (t*),the totalnumber of particles in the system remains the same (Carstensen and Musa, 1972;Carstensen and Patel, 1975; Dali, 1997). If the powder is allowed to dissolve then, attimes t < t", the mass undissolved will be (13.6)If the cubed term is expanded, then M == A1 - B,t + C,t2 - n , t 3 (13.7)where the coefficients A I , B1, and Dl are elaborated on in the following. An C,,example is shown in Fig. 13.3. (1 3.8)This term is obviously the original mass of the powder sample and p3 is the thirdmoment of the probability ~ i s t r i ~ u t i o n f u ~ c (Bennett and Fran~lin, tion 1961). Thethird coefficient in Eq. (13.8) is (13.9)
  • 226. 0 0 0 0.0 0 100 200 300 400 500 600 Time (s) Amount undissolved for the dissolution of a -40/ + 50 sieve fraction of oxalicacid dihydrate in 0.1 N HCI at 25°C and 50 rprn.Where p l is the first moment of the probability density function and also the mean ofthe distribution ( ennett and Franklin, 1961). The second coefficient in the expan-sion is (13.10)where p2 is the second moment of the probability density function (Bennett andFranklin, 1961). The variance of the powder population is given by 2 s2 = lLL2 - Pl (13.11)The coefficient to the last term is given by (13.12)where use has been made of the fact that the probability density function as used isnormalized? Le., (13.13)Equation (13.7) may be divided through by A = Mo, after which it takes the form: ~ 1 = 1 - B,t ~ 0 c.t2 - D,t3 (13.14)and the coefficients with subscript “2” are then the coefficients with subscript “1”divided by Mo.
  • 227. The coefficients of the terms in t, t2, and t3 in Eq. (1 3.14) are given by thefollowing equations: 1 2 = 3KP2/P3 3 c = 3K2P*/P3 2 (13.16) D2 = K 3 / P 3 (13.17)In a typical research and development setting, in the event that a new drug candidateis recognized by the drug discovery group, then the dissolution rate constant K , forthat compound under specified h y ~ r o d y n a ~conditions can be determined from icpowder dissolution data and particle size analysis by microscopy (can be done with Eqs. (13.14) through (13.17). From thedissolutio 2 is obtained and through the results from microscopy the moments ,x2 and p3can be evaluated. Similarly, from Eqs. (13.2) through ( 1 3 4 , by knowing N , theinitial number of particles, and the density of the solid, the average initial volumeshape factor for a polydisperse powder be can estimated ( issolution studies and particle size analysis on three sieve fractions of oxalic acid lhydrate: -301 40; + -401 + 50; and -501 + 60, yielded a K value of(1.42 x 0.19) x lom4cm/s when dissolution was carried out in 0. I NP paddle apparatus at 25°C and 50 rpm. The K value should be independent ofparticle size. The results for volume shape factor obtainedby two methods are quitecomparable. problem can be considered in the opposite direction (~arstensen and th the knowledge of K value for a compound (for instance oxalic acid dihydrate) under specified hydrodynamic conditions, the fraction undissolved as a function of time, the moments of the distribution function of a “dimensio~ sig- of nificance” can be obtained. Only the dissolution data up to the critical time are utilized in this manner (Fig. 13.4). At the critical time, there is a change in slope 0.55 0.45r^cx?4 - 0 0.35 I?I - 0.15 0 100 200 300 400 500 Time (s) Cube-root law plot for the dissolution of a -40/ + 50 mesh fraction of oxalicacid dihydrate showing the crltical time.
  • 228. in the cube-root law plot (Hixson and Crowell, 1931; Carstensen and Patel, 1975).The fraction undissolved data until the critical time can be least-square-~tted to athird-degree polynomial in time as dictated by Eq. (13.14). The moments of distribu-tion, p l , and p3 can be evaluated from Eqs. (13.15) through (13.17), with three p2,equations used to solve for three unknowns. To obtain an estimate of a K value for for a compound (e.g., oxalic aciddihydrate), the moments of the distribution function had to be known. Thus therestriction. of breadth being the de~ningdimension was imposed on the integralsbefore they could be evaluated numerically. In the process of working backward, toobtain the distribut~on parameters from powder is solution data, the integrals thatdefine the moments of distribution function are allowed to “float.” In other wordsno restriction on the kind of dimension is imposed at this point. So it is of interest todetermine which of the three dimensions of the particle is perceived by this approach.The discussion pertaining to this aspect will be resumed subsequently. ali (1998), to exemplify these ideas, carried out dissolution ofthree sieve fractions of oxalic acid dihydrate. For these sieve fractions the distribu-tion parameters for the lengths and breadths of the particles were known. This wasnecessary to have assurance about the validity of the approach. + Figure 13.3 shows the dissolution curve for a -401 50 mesh fraction o acid dihydrate. The cube-root law plot for the same event is shown in Fig. 13which an estimate of the critical time was obtained. From the least-square-fit to the data the coefficients of terms in t as per Eqs. (13.4) through (13.7) can be obtained. The mean and §tandard deviation for a particular sieve fraction can be calcu- lated using the following equations: ean = p I (13.18) The results for the three sieve fractions are shown in Table 13.1 and Fig. 13.5.The mean and standard deviation obtained are compared with those for the breadthof the particles from respective sieve fractions obtained from microscopy. In all thethree sieve fractions, the mean obtained directly from dissolution data is less than themean from microscopy. In the wake of this observation, the following questionarises: Is it possible, by not imposin~ any restrictions about the dimension of the Distribution Parameters for Sieve Fractions of Oxalic Acid DihydrateDetermined from Dissolution Data and Comparison with Those Obtained from Microscopy Standard Mean deviation p(bo) from oi, from from from n = PredictedSieve microscopy microscopy dissolution dissolution mean ~ ( ~ o )fraction (pm) (Pi4 (Pm> ( P 4 (holbo) (w-4-30/+40 410 85 84 0.56-40/+50 299 55 70 0.47--5O/-t- 60 240 32 33 0.47 _ _ _Source: Carstensen and Dale, 1998.
  • 229. ter 1 Third Derivatives of Dissolution Curve in Fig. 13.6Time d3M/dt3 0 30 -0.30 60 0.00 90 0.10120 -0.30I50 -0.10180 0.00210240x y = 0,78888 + - 1 . 4 6 7 2 ~ R-2 0.998 14 12 10 8 G 4 2 3 4 5 6 ’ 7 8 9 Shape Factor From Dissolution Correlation between microscopically and dissolution-dete~inedshape factors. 5.0046 - 3.5459e-2x + - 1 .2648e-4xA2- 1.8768e-7x^3 R*2 1.000 Mass undissolved for the dissolution of a 5-g 30/40-mesh fraction of oxalic aciddihydrate.
  • 230. particles on the integrals comprising the moments of distribution, that the smallestdimension (height) of the particles is recognized? To answer this query, the authorsresorted to the volume shape factor data for these sieve fractions that were obtainedmicroscopically(see Table 13.1and Fig. 13.5). The ratio of mean height to the meanbreadth can be calculated from the volume shape factors obtained frommicroscopyand dissolution. Thus the mean of height ( ~ = Mho) for particles belonging to a ~ oparticular sieve fraction can be predicted. These values can be compared with themeans obtained directly from dissolution data (Fig. 13.6). As shown in Table 13.1,these two set of values in excellentagreement. Also the standard deviations of the arebreadth of particles are comparable with those obtained from dissolution. It is seen from Eq. (13.14.) that the third derivative should be independent oftime. That this is (approximately) so is shown in Fig. 13.7. It is obvious that the longer the precritical time is, the better the assess~entofthe coefficients. It is natural to carry out the dissolution in water, but just for thepurpose of determination of distribution parameters, other solvents and apparatusesmay be used. If a solvent exerting less solubilizingpower on the substance is used, oran apparatus allowing slower dissolution is employed, then longer time intervalspriorto t* maybe used, thereby improving precision. Ifhowever, the value of(aqueous) K is sought under US.? type dissolution apparatus conditions, then thisapparatus should be used, and water, N/10 hydrochloric acid or sim~lated gastricfluids could be used as the dissolution media.After the smallest particle has dissolved, the model for dissolution must, by necessity,change. Y- - 1.5556e-3 + - 4.7619e-6x Rn2 0.025 0.002 0.00 1 0.000;? -0.001Pzr;3 -0.002 -0.003 -0.004 0 100 200 Time ig. 13.7 Third derivative from data from a dissolution run of a 30/40 mesh cut of oxalicacid dihydrate.
  • 231. Frequency a(mm) Slze a* a(max) kt Schematic of dissolution of a multipa~t~culate. Consider a (normalized) distribution shown as an example in Fig. 13.8. The~istributionshown is that of the powder before dissolution, and, €or instance aparticle of size a* has a frequency denoted fi (the length of the chord AB). Thenumber of particles of this size is N times.fi where N is the total number of particlesin the population. After a given time t, all the particles will have become smaller byan amount of [kt](Le., the particles originally of sizea* would have a size of a*-[kt],but the ~ u ~ b ofthe ~ a ~ t i c lwould still bef2N).The smallest particle after a time of ey est would be kt (point 6 )a particle just about todisappear (or just disappeared). Thenumber of particles at this point would be [ k t ] f i .wherefi is the number denoted bythe chord FG. Assuming a cubical particle, the mass remaining M , after dissolutionhas taken place for a time of t, is, therefore, y ,. - 0.29233 + 2.3933s-3x - 4.0376e-6~*2 R-2 0.866 Parabolic approximation of particle population.
  • 232. (13.19)where t > t" (i.eS7 larger than the critical time). A procedure similar to the one in theprevious sections is not possible in this instance because the lower limit isnot a givensize. f ( a ) is usually, for populations, taken aseither normal orlognormal, but ithasbeen seen in Fig. 13.2 that, for a sieve cut, it is at best normal. In fact the data shown ig. 13.2 are more likely to fit a second-degree polynomial (Fig. 13.9). Ifis inserted in Eq. (1 3.19)then a ~ i ~ t ~ - ~~e Z y~ o ~ei a Z t results. (This is shown in o g n e in issolution curves beyond t" should, therefore, be plottable in this ecause there is alwaysa nick in the curve at short times, it is possible to assignvalues to both amln and am,,,, In particular the latter can be obtained easily by thepoint where the curve intersects with the x-axis.Water or aqueous solvents as a dissolution medium have tacitly been assumed to bethe case in the preceding writing. There are examples for which the solubility of a compound is sufficiently lowthat normal USP volumes (900 mL) are insufficient to dissolve all the solid, andthere are three principles thatare used to compensate for this: (a)the useofsurfactants may sufficiently increase the solubility so that meaningful dissolutioncan be carried out, or (b) a mixed solvent maybe used, or (c) the useofcom-plexation may be employed. In work by Diaz et al. (Fig. 13.10), dealing with the 40 1 0 20 40 60 80 100 ( 10^4)xCD (mol/l) Dissolution rate constants (in M/s) of albendazole as a function of cyclodextrin concentration (CD). (Data from Diaz et al., 1998.) * The authorsemploy 1 mgt2.5 mL of water, which is far above the solubility of the compound,so that the amount of mass dissolved in this experiment 1s also insuf~cientto change the surface area.
  • 233. complexation of albendazole with cyclodextrins, it was found that the purely aqu-eous solubility So, increased to a total solubility of S, at a given ligand concentra-tion of Ll. The authors tested the dissolution under sink conditions (the initialparts of their curves) and constant surface area* and found dissolution to be fairlylinear in time. C =(kAS~t/Y)t (13.21)where A is area, St is the solubility at the ligand concentration in question Ll, and Yis the dissolution volume. If k were independent of the ligand (i.e., of the medium),then the slopesoftheselines should be proportional to St, since A and Y areconstant. Since S is linear in Lt, the slopes should be linear in ELt], but as seen inthe figure they are not. A = surface area a, = amax= size(length, breadth, or width) of largest particle a. = amln size(length, breadth, or width) of smallest particle = A = surface area of the dissolving solid at time t b = width of a particle b,, =geometric mean of a lognormal particle population 13 = width of channel for dissolution study C = concentration f ( b ) = normalized frequency function for the width of a particle h = height of a particle m = mass of an undissolved particle M = ~ultipart~culate undissolved mass Mo = initial multiparticulate mass before dissolution k = intrinsic dissolution rate constant (cmls) IC = linear (cube-root) dissolution rate constant N = number or particles in a multiparticulate sample Q = a constant r = radius of particle R = the gas constant s = standard deviation of sizes in a particle population ln[s] = standard deviation of a lognormal particle population S = solubility SL = ligand solubility in the presence of substrate t = time (of dissolution) t* = critical time T = absolute temperature, Y = volume of dissolution medium avo= volume shape factor K = ~olt~mann constant p l = mean (first-moment) of a particle population p2 = second-moment of a particle population ,u3 = third-moment of a particle population
  • 234. p = particle densityIn this section, it is assumed that particle size distribution data exist and may beapproximated by a parabola, as shown in Fig. 13.9. The parabola is expressed inequation form as:If the distribution is known, then the values of jo,jl and j2 are known from thefollowing facts. For convenience, the maximum and minimum diameters are denoteda, and ao. The maximum frequency occurs at f3 and is zero at the extremes ( 1 3A.2) ( 1 3A.3)With knowledge of f3, ao(a,,~) and al(a,,,) the values of jo,j,, and j 2 maybe he amount remaining at time t > t* is the value of the integral: M/{pa,} = f " f ( a ) ( a - ktgda kt J,": (13A.5) = {(io +j l a +j2a2}(a3- 3a2(kt)+ 3a~kt)2 ( l ~ t ) ~ } d a -The lower integration limit is (kt), rather than zero or a,,* for reasons stated in thetext. The integral has a solution of the following type: ( 1 3A.8) (13A.9) (13A.10) (13A.11) ( 1 3A. 12) (13A.13)Barnett MI, Nystrorn C (1982). Pharrn Techno1 6:49.
  • 235. ennett CA, FranklinNL (1961). Statistical Analysis in Chemistry and the Chemical Industry. John Wiley & Sons, New York.Brooke D (1973). J Pharm Sci 62:795.Brooke D (1974). J Pharm Sci 63:344.Cabana BE, O’Neil R (1980). Pharm Forum 6:71.Carstensen JT (1966). Modeling and Data Treatment in the Pharmaceutical Sciences. Technomic, Lancaster, PA.Carstensen JT, Dali MV (1998). Drug Dev Ind Pharm 24:637.Carstensen JT, Musa MN (1972). J Pharm Sci 61:223.Carstensen JT, Patel M (1975). J Pharm Sci 64: 1770.Carstensen JT, Rod~guez-Horned0 N (1986). J Pharm Sci 74:1322.Dali MV, (1997). PhD dissertation, University of VIisconsin-Madison.Dali MV Carstensen JT (1996). Pharm Res. 13:l 55. Micromeritics, 2nd ed. Pitman Publishing, New York, p 142. ora JG, Lianos CME (1998). Pharm Dev Techno1 3(3):395. (1965). J Pharm Pharmacol 17:193.Higuchi WI, Hiestand EN (1963). J Pharm Sci 5257.Higuchi WI, Rowe EL, Hiestand EN (1963). J Pharm Sci 52:163. intz RJ, Johnson KS (1989). Int J Pharm 51:9. ixson A, Crowell J (1931). lnd Eng Chem 23:923. oughton ME, Amidon GE (1991). Pharm Res 95356.Pedersen PV, Brown KF (1976). J Pharm Sci 64: 1981. VI, de Vries-Nijboer T, ~uizinga (1979). J Pharrn Sci 68:163. Steiner G, Patel M, Carstensen JT (1974). J Pharm Sci 63:1395.
  • 236. 14.1. ando om ~ e c o m ~ o s i t i o n ~ Amorphates-Spontaneous ~eactions in the Crystalline 22414.2. Topochemical Reactions 22814.3. The Avrami-Erofeyev Equations 23014.4. Nucleation F o l l o ~ e d Fast inetics (Poly~orphic by Transfor~ations) 23414.5. Surface Nucleation (Prout-Tomp~ins Model) 234 14.5.1. The solid to solid-~lus-gas reaction 235 14.5.2. ~ e ~ p e r a t udependence of the solid to solid-plus- re gas reaction 238 14.6. The Ng Equation 239 14.7. The Solid to L i ~ u i d - p l ~ s - ~ a s 240 14.8. iffusion Controlled Interactions 245 14.9. General Interactions in Dosage Forms 249 14.9.1 Tartaric acid and sodium bicarbonate 25014.10. 25414.11. Pseud~polymorphicTransformations 25514.12. Equilibria and Effects of Applied Pressure 25614.13. Photolysis in the Solid State 25614.14. Choice of ~ o d e l 25614.15. ~nteract~ons Involving a Liquid Phase 25714.16. Cases of Interaction of a Liquid with a Poorly Soluble Drug 26 114.17. eactions via the Gas Phase 26 1 262 263
  • 237. The subject of solid-state stability is of great importance in pharmaceutics. Stabilitypatterns of solid dosage forms are partly a function of the stability of the drugsubstance in the dosage form, but also, as shall be seen in the Chap. 15, a functionof moisture.It can be shown, as is to be described (Carstensen and Morris, 1993), that reactionsin the rubbery amorphous state are akin to solution kinetics. Amorphous materi-als, as shown by Carstensen and Morris (1989), are less chemically stable than theircrystalline counterparts. This has also been demonstrated by I ~ a i ~ i et ial. (1980) nand by Gubskaya et al. (1995). Reactions in the crystalline state can be attributedto the presence of moisture or light, but solids may also undergo deco~position orsolid-state reactions in the “dry” state (i.e., without the interference of water orlight) (Carstensen, 1980; Byrn et al., 1996). For instance Shalaev et al. (1997) haveshown parallel reactions occurring in the solid-state methyl transfer of tetracyclinemethyl ester because fitting of the data gives good biexponential fits. They attributethis to (a) presence of an amorphous phase (if material has been milled or freeze-dried), or (b) that processing“increases the extent of disorder in the remainingcrystal lattice,” and associate this with different types of lattice defects. The par-allel reactions (i.e., the amorphous versus the defect pathway) give rise to the samereaction products. The most interest and the largest body of work of amorphates is in the field ofmacromolecules. These usually possess a glass transition temperature Tg, and thestates are referred to as “glassy” below (the highest Tg in multiple glass transitionte~peratures) and “rubbery’, above Tg. Only a few articles have appeared in the literature on the subject of chemicalstability of amorphates. In general, a compound is more stable in the crystalline statethan in an amorphousstate, but exceptions exist (Sukenik al., 1975, O’Donnel and etWhittaker, 1992; Stacey et al., 1959). There are examples that have been reported (Lemmon et al,, 1958) for which the crystalline state is less solublethan the moleculein solution, but they are rare. in general, in a crystalline state, molecules are, to a great extent, fixedin position. If the situation exists wherea group from one molecule reacts with another group in a neighbor, the situation, as shown in Fig. 14.1, arises. Pothisiri and Carstensen (1975) have shown that, in a situation such as with substituted benzoic acids, the decomposition is between two groups in the same molecule. Suppose parts A and B of the molecule depicted in Fig. 14.1 react. If this occurs, arrangement C would give better stability than arrangement would be farther away from I3 in the former arrangement. Arrangement I> can also be more adverse than a random orientation, and if that is true, then the amorphous form would be more stable than the crystalline (arrangement D). This is the excep- tion, rather than the rule.
  • 238. oli ilit Afrangement C Arrangement D Different possiblearrangements of a molecule in solid state, implying different thedistance between possibly interacting groups (A and B). In the presence of moisture, conversions from amorphous to crystalline mod-ifications are promoted (Carstensen and Van Scoik, 1990; Van Scoik Carstensen, and1990) and the material developed in the following all refers to anhydrous conditions. In the work by Carstensen and Morris (1993), amorphous indomethacin wasproduced by melting a crystalline form of it to above melting (162°C) and recool-ing it to below 162°C. Amorphates made in this manner are morphologically stabledown to 120°C so that their chemical stability can be monitored (If the tempera-tures are lowered rapidly, then stable amorphates can be formed at room tempera-ture, but kinetics cannot befollowedeasilybecauseof the slow reaction rate atroom temperature.) At a range of temperatures below this temperature crystal-lization occurs too rapidly to permit assessment of amorphous stability.Amorphous samples wereplaced at several constant temperature stations (145, 150,155,165,175, and 185°C) and assayed from time to time. The content ofintact indomethacin was assessed by using the U.S. Pharmacopeia ( U P ) methodof analysis. The decomposition curves of amorphous indomethacin and a melt of indo-methacin at different temperatures is shown in Figs. 14.2 and 14.3. The pattern isstrictly a first-order one. Of the few reports in literature dealing with the chemicalstability of compounds in the amorphous state, a m o r ~ h cephalosporins (~feiffer o~et al., 1976; Oberholtzer and Brenner, 1979; Pika1 et al., 1977) also adhere to a first-order pattern. One purpose of the following writing is to seek an explanation for thispseudo-first-order (or indeed, truly first-order) pattern. The explanation must lie, in some manner, with the fact that in the rubbery state, the molecules can arrange themselves in a random fashion, in a somewhat frozen (or much slowed) manner of that of the melt above the traditional melting point. The results obtained from the melt are shown in Fig. 7.1, and as seen a first- order plot results. If an Arrhenius plot is drawn of the data from 14.2, then Fig. 14.3 results. It is seenthat the Arrhenius plot of the amorphate continues into the Arrhenius plot of the melt. An attempt to explain this is made in the foll~wing. If the substance in Fig. 14.1 was a crystalline solid, then the potential energy between molecules would be inversely proportional to a power function of their
  • 239. 0 20 40 60 80 Time (hours) Decomposition of amorphous indomethacin: Symbols; 0,145°C (k = 0,015 h-);A, 155°C (k = 0.036 h-). (Data from Carstensen and Morris, 1993.)distance (the lattice constant) (~arstensen and orris, 1993~; thatis, it wouldbeakin to a Lennard-Jones potential (Lennard-Jones, 193 1).phous state, if the decomposition is an intermolecular (rathan intramolecu-lar) reaction, then a group A in molecule a interacts with grmolecule b. The energy of the molecular pair willbetween the group A in one of the pair, and groupwould be assumed to be randomly distributed, and a certamolecular pairs would be at or above a critical energy Ei, necessary for reactionbetween A and l . The fraction of molecules that have this given energy El, is given 3 olt~mann distribution (Mo~lwyn-Hu~hes, 1961): (14.1) -1 -2E-c -3 -4 -5 2.1 2.2 2.3 2.4 1000/T Arrhenius plot of indomethacin decomposition: Squares are amorphous and solidcircles are melt. Circles, 165°C (rate constant 0.05 h-); squares, 175°C (rate constant 0.13 h";triangles, 185°C (rate constant 0.19 h"). (Data from Carstensen and Morris, 1993.)
  • 240. where N is the total number of moleculesand where the summation is overall energylevels. The fraction of molecules having energies in excess of E>then N , j / N , is givenby k=o k=o k==l k=OThere are several ways of approaching these summations (e.g., by considering theenergy differences small and integrating). Another, discrete approach is to assumethat the energy differenceAE between adjoining energy states is constant.(14.2) may be written: IV> I / N = Ie-Ei/RT+ e4Ei+AE)IRT + ...]/[e-EolRT+ e-(EotAE)lRT+ ***I - I~-E;/RT[~ __. +e-AE/RT+e-2AE/RT +...~}/(e-~o/~~[~ ... e-2AEIRT+ 11 (14.3)that is, >z / N = e-Ei/RT/e-EolRT = e4Ei-Eo>/RT (14.4) Alternatively, if the difference between energy levels large compared with the is round-state energy, one may simply approximate the series in the numerator anddenominator of these equations with their leading terns. This leads to the sameresult: [exp(-Eo/RT)] = exp[-(Ej - EO)/RT] (14.5)If, in a time element dt, a fraction of the molecules (dXN/N) reaching Ei (or higher)react, then, denoting this fraction q (l/XN)dN/dt = 4IIV>,lW= ~ e ~ P ~ ( E j(14.6)) / ~ ~ l ~o = klwhere kl (by definition in differential form) is a first-order rate constant; that is, byintegrating Eq. (14.6) and imposing IV = No at time t = 0 ln[N/No] = -kl tthat is, first order, where the rate constant is given by kl = 4 ~ x ~ [ - ( E ~ E o ) / ~ T I (14.8)or its logarith~ic equivalent: ln[kl] = ln[q] - E ~ / ~ ~ (14.9)that is, an Arrhenius equation where the activation energy is given by Ea = (Ei - Eo) (14.10)The data in Fig. 14.2 demonstrate the correctness of Eq. (14.7) (Le., the expectaof a first-order decomposition), and Fig. 14.3 demonstrates the correctness of(14.8). ere have been proposals ( elwyn-Hughes,1961 ; Franks, 1989) that the is best described in terns of the ~ i ~ l i a ~ s - equation (Williams et al., 1955):
  • 241. (14.11)where C2 and C1 are constants. It is far from certain that this equation would applyto chemical reactions, but Fig. 14.4 shows its application to the data in Fig. 14.3.Several different values of C and Tg will give reasonable fits, as seen. It would seemintuitive that if the Arrhenius equation fits, then there would be values of C2 thatwould make the VVLF equation fit as well. ~chmitt al. (1999) described the crystallization of amorphous lactose above etthe temperature of glass transition to follow the Johnson-~ehl-Avrami (Johnsonand Mehl, 1939; Avrami, 1939) equations: (14.12)where x is amount decomposed, n is an integer between 1 and 4, k is a rate constantand t, is a lag time. Pika1 et al. (1977) employed solution calorimetry to determine the amorphouscontent of cephalothin sodium, cefazolin sodium, cefamandole nafate, and cefaman-dole sodium. Because the amorphous forms are more energetic, they have a higherheat of solution, and the percentage of amorphate may be obtained, if the heat ofsolution of amorphate and crystalline forms separately is known. Lo (1976) showed that ampicillin trihydrate dehydrated to amorphous ampi-cillin that had much poorer stability than the trihydrate. On storage the decomposi-tion appears biphasic. L TIThere are theories, akin to the foregoing, that simply, empiricallystate thata decom-position starts at the surface of the solid and works inward. This may be visualizedastwo-dimensional (the cylinder in Fig. 14.5) or as three-dimensional (as demonstratedin the sphere in Fig. 14.5). 0.80 0.85 0.90 (T-Tg)/(C+T-Tg) Possible dependencies of ln[k] in Figs. 14.2 and 14.3 as a function of assumed ofglass transition temperature, plotted by the inverse function of the WLF equation. Triangles: +Tg= 80", C2 = 10: ln[k] = 25.40 33.1 17{T T g } { / C (2" Tg)} R = 0.977; Circles: Tg = loo", +C2 = 6: ln[k] = 45.48 - 54.47 { T - Tg} / ( C (2"- Tg)} R = 0.97; Squares: TS = 120", C2 = +5 : { T - TS}/{ (1"- T,)}] = 0.771 In[k] + 0.0289 R = 0.982. (Data from Carstensen and CMorris, 1993.)
  • 242. Examples of topochemical reactions, For a cylinder ofradius R, the decomposition will work inward in a zero-orderfashion (i.e., a layer h will have decomposed at time t ) and h =I= k2tThe amount remaining undecomposed at time t, therefore, would be HZ= H z ( R - h)2 = H z ( R - k2t)2where H is the height of the cylinder. Because the original volume is H ~ z ( R the )~retained fraction (1 - x is ) (1 - x = H2n(R - k2t)2/{H2Z(R)2] (1 - [k2t/R])2 ) = (14.15)For three-dimensional, directional diffusion, the solid can be visualized as a cubeoriginally with side a. em, so that after a given time the side length a, would be a = a. - kt (14.16)That is, it is assumed that the decomposition "front" progresses in a linear fashion.This is akin to physical phenomena such as crystal growth (the so-called McCabelaw). At time t, therefore, there will be an amount undecomposed given by Npa3 = Np[ao - ktI3 (14.17)where N is the number of particles in the sample and p is the density of the solid. Theoriginal volume of the solid was Nai so that the fraction not decomposed (1 - x, )would be given by 1 - x = ~ p a / [ ~ p a= ] ~ o [a/ao13= [I - (k/ao)t13 (14.18)It is noted from Eq. (14.18) that the rate constant (k/ao)is particle-size dependent. An example of this type of decomposition pattern is aspirin in an alkalineenvironment (Nelson et al., 1975). This is shown in Fig. 14.6. In general it is not possible to distinguish between a reaction of the typedescribed by Eq. (14.18) and a first-order reaction. It is d ~ ~ c to dis~inguish e t ~ e e n u l ~ breaction orders in the solid state on purely stat~stical ~ o u n dand other infor~ation ~ s,must be available before a mech~nisticmodel can be assigne~.Only with excellentprecision, and with a fairly large number of assays and a sufficiently large decom-position, will it be possible to distinguish between the two.
  • 243. 1 2 3 Days Aspirin decomposition in a solid, alkaline environment. (Data from Nelson et al.,1975.) l- ecomposition is most often associated with active sites that start as nuclei. Jacobsand Tompkins (1955) have summarized the Avrami-Erofeyev equations as follows: When the nucleation is according to an exponential law; that is,when thenumber of nuclei follow dN/dt = Noe-klt (14.19)where No is the original number of nuclei at the temperature to which the solid hasbeen brought from a low temperature, and if this is followed by rapid two-dimen-sional growth, thenI;,k,, and k here are constants. Such a model would require a first-order decay. Thisrarely occurs, although some instances have been reported in literature. Shefter andKmaclc(1967) studied the dehydration of theop~yllinehydrate and found it tofollow a first-order pattern. Shefter et al. (1974) have shown first-order decomposi-tion to occur for the dehydration of ampicillin trihydrate. In Bawn kinetics, to becovered later, the decomposition in the solid phase of the decompositio~,is firstorder, and Pothisiri and Carstensen (1974) have shown this to be true also for p-aminosalicylic acid. In many situations the nuclei will grow and then overlap, and when there isingestion of nucleation sites and the growth nuclei can overlap, then, the Avrami-Erofeyev equation takes the form - ln[l - x] = Q{e-kt - 1 + kl t - [(klt)2/2!]+](kl t)3/3!J) (1421) f the lag time is denoted t,, then in the decay period (t > ti) this reduces to > - 111 - X] kt3 111 (14.22)which is one common form of the Avrami equation.
  • 244. The program in Table 14.1 and the printout in Table 14.2 demonstrates thedanger in simply applying Eq. (14.22) to decomposition data. The program calcu-lates a series of data according to Eq (14.21) and the tabulation (see Table 14.2) givesthe possibility of g~aphing according to q. (14.21) (Fig. 14.7), and Eq. (14.22) (Fig.14.8). The previous sections have dealt with decompositions that occur randomlyin a space or on a surface. The section to follow will deal with the situation inwhich decom~ositi~n tied to particular sites that are created as a function of istime. This type of reaction has been assigned quite frequently in recent literature,in particular, to pseudopolymorphic transformations and dehydration kinetics ofhydrates. In the hydrate water molecules form part of the matrix. Dehydration kineticsof hydrates has had the attention of the pharmaceutical scientist for some time.(1982) has developed a generalized kinetic theory for isothermal reaction in solids,and theophylline has been used as a model for several studies of this kind (Lin and yrn, 1979; Suzuki et al., 1989; Agbada and York, 1994). The Avrami-Erofeyev model used for this type of kinetics (Avrami, 1939) willbe dealt with in the following in a somewhat simplified manner. The model assumesthat volumes within the solid at a given time t are activated, and that decompositionmay occur in these areas and not in the areas that at time t still remain 6 6 n o n ~ u - Program for Eqs. (14.21) and (14.22)100 FOR T = 0 TO 1.5 STEP .1110 X1 = EXP(-T)120 X2 = (TA2)/2130 X3 = (TA3)/3140 V1 = X1-1 +T+X2-X3150Y2 = 1-Y1160 Y3 = -LOG(Y2)1702, = T A 3150 PRINT T,Y 1,Y3,2:160 NEXT T Decomposition Data According to Eqs. (14. 21) and (14.22) __Time, kt X - ln(-x) (k03 --0 0 0 00.1 0.00950 0.0955 0.0010.2 0.0362 0.036 0.0800.3 0.0’77 0.080 0.0270.4 0,129 0.211 0.0640.5 0.327 0.396 0.2160.6 0.399 0.509 0.512
  • 245. ter 1 Time, kt .7 Decomposition. data in Table 14.2 plotted according to Eq. (14.21).cleated.” This may occur in strings (one-dimensional diffusion), areas (two-dimen-sional diffusion), and volumes (three-dimensionaldiffusion). The a p p r o ~ i ~ amath- teematical development follows the same path in the different cases, and only the latterwill be derived. For simplicity it is assumed, in Fig. 14.9, that only the volume cornered by A isnucleated and the rest of the solid isnot. This could equally well have been scatteredvolumes ofa total volume equal to the condensed volume shown in Fig. 14.9, and theresult, therefore, will be the same, except that, in the scattered case, the volumes may“grow together.” This is not considered in the model (but will be considered in the~rout-To~p~ins model). If the nucleation occurs zero order ineach. direction, then the sideof thenucleated cube, at time t, is kt, so that the number of nuclei, N 3 , at time t is 1.5 0.5 0.0 0 1 2 3 (kt)*3 Deco~position data Table 14.2 plotted according to Eq. (14.22). in
  • 246. Schematic for approximate Avrami-Ereyefov model. N3 = (k*t)3 (14.23)If the nucleation occurs in a plane, then (two-dimensional case) N2 = (kt)2 (14.24)and if it occurs along a line (a string), then (one-di~ensional case) N , = (kt) (14.25)Figure 14.9 appliesto the three-di~ensionalcase, and the decomposition is assumedto be (a) inlinewith first-order kinetics proportionalto the concentration ofunreacted solid in the nucleated volume; (b) proportional to the number of nucleiand, hence, inview of Eq. (14.23) proportional to(kt)3; (c) not occurring at all in andthe nonnucleated volume. This reasoning leads to d( 1 - x)/dt = -q(l - x)(kt)3 (14.26a)in the three-dimensional case, and in general to d(l - x)/dt = -4(1 - X)(&)" (14.260)n being unity, two or three, depending on the dimension. Equation (14.26b) may berewritten: d ln[l - x] = -qkntn (14.27)which integrates to ln[l - x] = -[qk"/(n + l)]{(t""f)) = - exp(&~)t(*")) (14.28)in line with the expected linearity in Fig. 14.8. Qr = + [gkn/(yl l)], here is a constant.Taking logarithms of Eq. (14.28) now gives In( In{- 1 - x = &I )} + 1) + (nln[t] (14.29)which is the conventional plotting mode, as employed by Dudu et al. (1995). Theseauthors used micr~calori~etric methods and showed the dehydration of theop~yl-line hydrate to be a two-step process obeying the equation [-- ln(1 - = kt (14.30)which is a variant of Eq. (14.28), with y1 = 3. Hence, in their case, the process is athr~~-dimensio~al, diffusional process.
  • 247. r l olymorphic transformation rates have lately become of importance; an example isa recent article by York et al. (1994), dealing with the dehydration kinetics oftheophylline. The article by Ng (1972) is similarly instructive in the sense that itreviews all the equations that have been developed for polymorphic transformationkinetics. Usually the transformationkinetics are S-shaped curves, and before any modelis imposed on the data, the following model should be considered. (This is compar-able with the model proposed by Carstensen and Van Scoik, 1990): If the phenom-enon that governs the t r a n s f o ~ a t i o n essentially the nucleation lag time, then the iscurves may be considered as representing either a noma1 ora lognormal error curveand the mean would be the mean (or geometric mean) nucleation time. What thisstates is that each particle, in a sense, acts as itsown entity, that there is a nucleationtime (with an error or a variance attached to it), and the particle will endure thenucleation time, and then decompose, individually, very rapidly. The reason for the lognormal relation is not difficult to rationalize. Solids areusually lognormally distributed. If the nucleation time is inversely p~oportional tosize, then it, too, would be lognormally distributed. TOjudge whether such a relation pertains, the fraction decomposed is, there-fore, converted to a cumulative Z-value (by means of normal error table), and this ais plotted versus either t or ln[t], to yield a straight line: (14.31) = 0 corresponds to the average nucleation time, tavg, that is tavg = exp(Q1/kl or tavg = Q2lk (14.32) Q-values would correspond to the standard deviation of the nucleation time. ehydration, at times, results in a morphic transformation. For instance, Lo(1976) showed that the transformation of crystalline ampicillin trihydrate to amor-phous penicillin was primarily first-order and either was first-order or followed acontracting cylinder model [(l - x)1/2being proportional to time]. f a solid is placedin a vacuum and exposed to temperatures at which it decomposesat a measurable rate, one of the following situations may arise: I Solid + solid + solid II Solid + solid + liquid III Solid + liquid + liquid IV Solid "+ solid + gas V Solid "+ liquid + gas VI Solid -+ gas + gas ther schemes are theoretically possible, but notlikely.Of the foregoing, it isschemesIV and V that willbe treated insome detail in the following, becausethey are the ones most investigated in the pharmaceutical sciences. It will later be
  • 248. shown that most pharmaceutical systems will not be of such a “purist” nature, butthe experiences gathered from examining themwill throw light on several important,real-life situations.Not all S-shaped curves will neatly fit topochemical or Avrami equations. The datain Table14.3 represent an S-shaped curve and were obtained by a reaction thatproduced a solid and a gas, and if plotted by Eq. (14.29) then Fig. 14.10 results. The plot may, at first glance, seem fairly linear, but the point is tt ~ ein~ ~ the deviations from the line are (+ ) (part AB), (-) in part that ,again (+) in part CD. Itis visually obvious, as well, that the curve is still S-shaped.Such curves also fail to give an integer (2, 3, or 4) as dictated by the mo The solid -+ solid + gas type of reaction has been investigated byTompkins (1944), used who potassium permanganate as a model substance. tical solids have been tested later [e.g., ~-aminosalicy~ rnblum and Sciarrone (1964) and by Carstensen andtypical example of such reaction is shown in Table 14.3and the readers may satisfy athemselves by plotting x versus t, that the plot is, indeed S-shaped. No solid has a smooth surface (Le., there are always surface imperfections).These could be“steps” in the surface or they could be crystal defects. These sitesaremore energetic than the remaining sites. They are most likely to occur at surfaces,which, in any event, are populated with molecules that are unlike the molecules inthe bulk of the crystal. For instance they have at least one less neighbor than bulkmolecules. It is assumed that decomposition is more likely to occur at such “”acti-vated” sites (Fig. 14.11). Once a molecule decomposesat anactivated site it changes its geometry; hence, the neighboring molecules are more likely to decompose. There will then be a chain or plane of activated molecules forming, with a probability of a (see secondfigure in Fig. 14.1 1). The a, of formation of activated molecules, N in number at time 1, is rate dlV/dt, and this is proportional to N , Initially this is then given by [dN/dt]o = a [ N + No] (14.33) Decomposition Data of 4.6 mol of a Solid Following the ~ r o u t - T o m ~ ~ i n sModel Gas (mrnol)/ Mole fraction xTime ( t ) 4.6 mmol solid decomposed ln{xl(l - 4 10 01 0.08 0.017 -4.0342 0.46 0.1 -2.1973 1.16 0.252 1.0874 2.37 0.515 0.0614.5 3.20 0.696 0.82’75 3.76 0.817 1.4996 4.15 0.902 2.222
  • 249. y = - 4.1317 -t 2.7981~ R"2 =Z 0.995 Data in Table 14.3 treated according to Eq. (14.29).It is obvious that after even a short period of time N becomes much larger than No,so that this latter can be dropped at times even remotely larger than zero. After a certain while (seelast inset in Fig. 14.1 planes will start tomerge, and l),hence there will be a termination probability / , so that at measurable times, Eq. 3(1 4.33) becomes d N / d t = {a- /3}N (14.33) 0th a and /3 are functions o f t (or what is equivalent, to the fraction decomposed x).It is reasonable to assume that a =b at t = tl/2 (or x(14.34) = 0.5)that is, at the time point at which one-half of the substance has decomposed. Also, /?=O at t=O (orx=O) (14.35)for there can be no termination probability at time zero. One (not necessarily thecorrect) function which satisfies this condition is /? = 2xa (14.36) hen this is inserted in Eq. (14.33) one obtains d N / d t = a[1 - 2x]N (14.37) tion Schematic of model leading to Prout-Tompkms kinetics: A and B are activesurface sites. Propagation of A proceeds AC (third inset), as propagation at I starts. 3Branc~ingthen occurs at C, and finally there is termination of one (or the other) of thebranches.
  • 250. 7The decomposition rate dx/dt is proportional to N; Le., dx/dt = kN or N = (l/k){dx/dt} (1438)Equation (14.37) can now be written dN/dt = (a/k)[ - 2x1 dx/dt 1 (14.39)Chain differentiation of dN/dt gives dN/dt = [dN/dx] [dx/dt] (14.40)Introducing Eq. (14.39) into Eq. (14.40) gives dN/dt = [ d N / d ~[dxldt] (a/k)[l ] - 2x]dx/dt (14.41)dx/dt is canceled out of the last part of this equation to give dN/dx = {a[l - 2x]/k} (14.42)which integrates to N == (a/k)(x- x2 ) (14.43)Since, by (Eq. 14.38), N = (Ilk) {dxldt}, it follows from Eq. (14.43) that (I / k ) dx/dt = ( ~ / k ) x (- X ) 1 (14.44)which integrates to ln[x/(l - x)] = a(t - t1,2) (14.45) The equations have a zero time problem, because the equation is not definedfor x = 0. This is a consequence of neglecting No. Similar paradoxes exist in thescientific literature. The Gibbs adsorption isotherm, for instance, is not defined forconcentration, C = 0 (i.e., for a liquid without surfactant). In solid-state stability, itmight be thought of in the vein, that as the material is being produced (Le., at timezero; e.g., through recrystallization)?it is already decomposing (however little). Data are plotted according to Eq. (14.35) in Fig. 14.12, and the linearity isgood. There are several other aspects that may convince a scientist that this is thetype of reaction at hand. First of all, Arrhenius plotting is good, and the activationenergy is us~ally three to four times as high as in that o other reactions in the solid f - 6 1 " " " " * " " f 0 1 2 3 4 5 6 7 Time Data from Table 14.3.
  • 251. (and ~ i state.~The reason ifor this is that the rate-determining parameter in Eq. ~ ~ ~(14.45) is a (i.e., it is actually a propagation probability that is measured, not a rateconstant in the usual sense). Whenever a compound “melts with decomposition,”then there is a good possibilitythat the melting range” depicts the interval in whichthe reaction occurs with a measurable rate, that it is too slow below this range, andtoo fast above the range, and in such a circumstance the activation energy is high, ompkins reaction may most likely be applicable. n the half-life is in order. There is frequently a substantial lag time (and other solid) type reactions. ecause many are carried out undervacuum (e.g., when break-seal tubes are used, or when manometers are glass-blowndirectly unto the reaction vessel), and heat transmission, therefore, is poor, so that itwill be a while before the solid itself actually attains the elevated temperature. Anexperimental remedy is to test the heat transmission by checking the length of time ittakes for a stable solid substance with knownmelting point and heat capacity to meltat that temperature, and to do this with three substances (benzoic acid being one), a calibration curve. If it is then calculated that at a given test tem- , it takes t minutes for the solid to attain the given ternpe e may be obtained by the integral mean value theorem. e subtracted from the time points used. s plotting, this does not apply, but it may be a source of Thesolid ”+ solid-plus-gas reaction embodies the dehydration of solid tes. Leung et al. (1998a,b)haveshown that aspartame 2.5 hydrate cyclizes out-Tompkins kinetics andthattherateconstants follow an ~ r r h e n i u sequation.I t s h o u l ~ pointed out, thatthe solid to solid-plus-gasreaction may be so only over bea certain temperature range, or to a certain degree of decomposition. Figure 14.13shows the eutectic diagram of a compound A with its solid decompositionproductIf the study is carried out at temperatures below the eutectic temperature T*, thenthe reaction will be solidto solid-plus-gas.If above the eutectic t~mperature, then thereaction will be solidto solid plus liquid plus gas. (If abovethe highest melting point,then it will be liquid kinetics.) The compounds reported in literature to be of the solid solid-plus-gas type are most often inorganic salts (e.g., p~tassium permanganate rout and Tompkins, 1944); silver permanganate (Coldstein and Flanagand some organic compounds, such as oxalic acid, ~-aminosalicylic acid ( arrone, 1964; Pothisiri, 1975a,b), or indomethacin (~arstensen a Isen et al. (1997) showed cefaclor monohydrate to decompose (as judged byrelated substances) by first-order kinetics. The rate constants could be plotted by~rr~enius plotting and were consistent with ambient rate constants. The reactionscheme, whenamorphous material was present, was such that the rates were faster at time points and then becoming equal to those of the cry stall in^ m~dification. onclusion was that the initial phase was decom~osition a ~ o r ~ h ocontent of usparallel to conversion of amorphous to crystalline drug.
  • 252. Liquid E Tompkins Kinetics Compound A omp pound B Decomposition Product Mole ----r Fraction of B Eutectic diagram of a compound and its decomposition product: At ternpera-tures higher than the melting point of B only liquid kinetics would be expected. At tempera-tures lower than the eutectic point only solid state kinetics (e.g., rout-Tompkins kinetics)would be expected. In i ~ t e r ~ e d i atemperatures, so-called Bawn kinetics apply. te At times the solid-state reaction cannot be completely specified, yet may bedescribed in analytical terms. Tzannis and Prestrels~i (1999) described the effect ofsucrose on the stability of trypsinogen, during spray-drying, by plotting denaturationt e ~ p e r a t u r e ~ a function ofmelting temperature and found a linear increase asbetween residual activity after spray-drying, and melting temperat~re.Adler andLee (1999) have reported on the stability of lactate dehydrogenase in spray-driedtrehalose.There are a m~ltitude “types’’ of S-shaped curves, and one, somewhat distorted, ofshape is as shown in Fig. 14.14. Ng (1975) suggestedthe f o l l o ~ i global, em~irical n~equation for this and other types of solid-state decomposition: dx/dt = kx”(l - X)’ (14.46)If both y1 and p are unity, then the equation becomes the rout-Tompkins equation.A set ofdata illustrating this is shownin Table 14.4. Theseare the data onwhich Fig.14.14 is based. n the first two columns of the table the time required for decompositions of 0,0.1,0.2, *, have been d e t e r m i ~ ~(Data treatment is actually easier if random times 0 d.are used, with the associated fractions decomposed.) The average decompositi~nsat interval midpoints are then determined (col-umns 3 and 4), and the value of dx/dt is then calculated (as shown in the table footnote^. The Ng equation may be expressed in logarithmic form. + ln[dx/dt] = y1 ln[x] p ln[l - x] + ln[k]
  • 253. 10 r Time S-shaped curve following the Ng equation: data in Table 14.4.If the data in Table 14.4 are transformed and l n [ d ~ / d is Imultiply regressed against ~ln[x] and ln[l - XI, then values of yz = 2 and p = 3 are obtained. any more compounds seem to decompose by this reaction scheme than by the solidto solid-plus-gas one. As mentioned in the caption to Fig. 14.13, this type of reaction Example of Data Amenable to Treatment by the Ng Equation Average Fraction x Average fraction, x dxldtTime decomposed time, t decomposed (from curve)0 0 0.686 0.05 0.07291.371 0.1 1.615 0.15 O.204ga1.859 0.2 2.021 0.25 0.30872.183 0.3 2.328 0.35 0.34502,472 0.4 3.632 0.45 0.31252.792 0.5 3.009 0.55 0.23043.226 0.6 3.604 0.65 0.13233.982 0.7 4.959 0.75 0.05125.935 0.8 12.107 0.85 0.08 1018.280 0.9"Obtained by: 0.1/(1.859--1.371) = 0.1/0.488 = 0.2048
  • 254. kinetics is usually referred to as awn kinetics (Bawn, 1955). The sit~ation at time fis as shown in Fig. 14.15 and, as seen, there willbe a certain amount of liquiddecomposition product. This amount corresponds to the amount of drug decom-posed. However, the liquid decomposition product will dissolveparent compound tothe extent, S (mole drug per mole decomposition product), to which it is soluble, sothat the amount present in the solid state at time t is the original number of molesAo,minus the amount decomposed Aox, minus the amount dissolved, AoSx. The rate of decomposition would be the sum of the rates of decomposition inthe solid state (assumed firstorder with rate constant k,, time" 1)and in the dissolvedstate (assumed first order with rate constant kl time"). The rate equation is hence dA/dt = -k,[Ao(l - X ) - AoxS] - kl[Ao~S] (14.48)Noting that A/Ao = (1 - x) (14.49)it follows, by division through by A. that d( 1 - x)/dt --k,[l - X - XS] - k l ~ S (14.50)or, noting that d(l - x = -dx ) dx/dt =I k,[l - X - xS] + k l S ~ k,[l + Bx] = (14.51)where B = [(kJk,) - l]S - 1 (14.52) quation (14.52) may be integrated, and yields ln[l + {Bx}] Bk,t = (14.53)Using B as an adjustable parameter, it is possible to find the value that makes thedata profile through the origin, as dictated by Eq. (14.53), and also gives the best fit. Figure 14.16 and Table 14.5 show an example of data from decomposition of~-methylaminobenzoic acid. To plot this according to Eq. (14.53) it is necessaryto assume values ofB, plotthe data, and assess the goodness offitbysome criterion. A different valueofB is then chosen, and this process repeated until a "best" value of B is arrived at.It is possible to show that in general the sumsof the squares of the deviations Situation leading to Bawn kinetics.
  • 255. 40 20 0 200 Hours Data from Table14.5: ~ e c o m ~ o s i t i oof ~-m~thylaminobenzoic n acid. (Datafrom Garstensen and Musa, 1972.),:s( = x(y- ~ ) ~ /-( 2)) of the points from the ensuing line is used a criterion. A n asdifferent criterion is the correlation coefficient. Frequently, this is also not a goodcriterion, and criteria for linearity (e.g., ~urbin- ats son statistics) are the best. Fordata fitting to Eq. (14.53) the line must pass through the origin. Fitting the data inthis fashion is shown in Table 14.6for three values of B (0.1,0.85, and2.0). It is bestto do this by computer, and a simple program in BASIC is shown in Table 14.7. The number ofdata points are inserted, the assumed value ofthe program run. One can then in three or four tries arrive at a “best” value forB(= 0.85). InEq. (14.53),using the correlation coefficientis nota good parameter,because it simply increases with increasin~ values of B up to avery high (unrealistic)value, also resulting in a very highintercept. All the correlation coefficients are good.The best criterion would be a criterion that dealt with curvature, but a simpler one,as stated, is simply to note the intercept, which should be zero. Studies of this type are usually performed on a vacuum rack. In this, thepressure is monitored as a function of time, and the sample can be observed. At agiven time point (which is quite reproducible), the last trace of solid will disappear(Fig. 14.18). At this time point, t*, the amount not decomposed, Ao(l - x), is justsufficient to dissolve the amount of liquid Aox, present at time t*. s = (1 - x*)/.* (14.54)where x* is the mole fraction decomposed at time t*. Therefore, Eq. (14.53) is validfrom time zero to time t*. If t* = 350 (as in the example used here), and x* = 0.45 atthis point, it follows that S = 0.55/0.45 = 1.22 (14.55) m0l/m01 Decol~~osition for ~ - ~ e t h y l a m i ~ o b e n zAcid Data oicTime (h) 0 50 290 110 210 150 310 350 0 4 1 2 27.9 8 20.5 45
  • 256. Data in Table 14.5 Treated by Eq. (14.53) ln[ 1 + Bx]Time (h) B == 0.1 B = 0.85 B=:!50 0.095 0.615 1.099100 0.182 0.993 1.610150 0.334 1.481 2.200210 0.588 2.054 2.830290 1.099 2.890 3.7103 10 1.335 3.210 4.040350 1.705 3.677 4.510 .7 Program for Obtaining BestValuesby Manual Iteration100 PRINT “Type in data as x,y, in 400 block”110 INPUT “Number of Data Points = ”;N1120 UT “Iteration Parameter, B = ”;B130 NT “T”;SPC(6);“X7’;SPC (6);“LN(1+ BX)140 PRINT ‘‘ ¶>200 READ A,C210 X = A220 Y = LOG(1 + B*C)230 X1 = X1 + x240 x 2 = x 2 + (XA2)250 Y1 = Y1 + Y260 Y2 = Y2 + (YA2)270 z 1 = 21 + (X*Y)280 N2 = N2+1300 PRINT ~;SPC(6);C;SPC(6);Y310 IF N2 = N 1 goto 700400 DATA 50,l410 DATA 100,2420 DATA 150,4430 DATA 210,8440 DATA 290,20450 DATA 310,28460 DATA 350,45700 Z2 = X2 - ((X1 “2)/N2)710 z 3 = Y2 - ((YlA~)/N2)720 z 4 = z 1 - (Xl*Yl/N2)730 Z5 = 24/22740 Z6 =: I(Y l - ( Z 5 * x l ) ) / ~ 2750 PRINT760 RINT “Slope = ”;Z5770 PRINT “Intercept = ”; Z6780 27 = (Z4^2)/(Z3*22)790 Z8 = (27)^(0.5)800 PRI~T “Correlation Coefficient = ”;Z8810 Z9 = (Z3 - ((Z5*2)*Z2))/(N2-2~820 PRINT “syxA2 ”;Z9=
  • 257. 1 -4 0 2 4 6 Storage (Years) .17 Data from Table 14.6 treated by Eq. (14.53).The slope in this case is 0.01 h". Since the slope is [L3k,] it follows that k, = slope/B = 0.01/0.85 = 0.012 h" (14.56)kl is now calculated from Eq. (14.52). 0.85 = [(k1/0.012) - 11 1.22 - 1 (14.57) that is kl = 0.03 h" (14.58) eyond t* the systemis a solution system, and should decompose by first-orderkinetics. The density of the liquid will actually change with time, but it is assumedthat both parent drug and decomposition product have appro~imatelythe samedensity. The Moles/cm3 density is denoted p and since there is a total number ofA. mol, the volume of liquid is Ao/p. The initial molar concentration (at time t*)is,therefore, Ao(l - x*)/[Ao/p] (I - x*) p. The time is counted from t = t*, and the =concentration at time ( t - t*) is (1 - x p, so that ) * In[( 1 - x - p] = k l t ) + In[( 1 - x*) p] a (14.59)or In[(l - x/ I - x) = -kl ( t - t*) )( *] (14.60)or ata of this type, for ~-methylaminobenzoic acid, are presented in Figs. 14.18 and14.19. It isseen that the data are quite first order. The first order rate constantobtained from this plot is kl = 0.040 h- in quite good agreement with the valueof 0.03 found from the first part of the curve. It is noted that when the total curve is plotted (Le., when Figs. 14.18 and 14.19are combined), then an S-shaped curve results. Unlike the Prout-Tomp~ins curve, awn curve is a two-phase curve, one part relating to the phase where there issolid present, the other to the part where all solid has dissolved.
  • 258. ilit 1.o 0.8 06 . of ~ecomposition ~-methylaminobenzoicacid after t* (350 h), at which pointx = 0.45 Le,, 1 - x = 0.55. (Data from Carstensen and Musa, 1972.) The values of x* obtained at t* will differ from temperature to temperaturebecause the solubility is a function of temperature. This is actually the value of theliquidus line on a eutectic diagram. The melting point depression curve (Maron andPrutton, 1958) is given by ln(1 -x*) = (~ff/R)[(l/Tf) (1/T)] - (14.62)Such plots are quite linear, as shown in Fig. 14.20.Figure 14.21 shows a situation where an ideally shaped solid A, is in contact withanother such solid . The contact area is assumed to be 1 cm2. It is assumed that A in this situation; that is, A+B+C 00 . -0.5 -1.o -1.5 -2.0 350 375 400 425 450 Hours Data in Fig. 14.18 treated according to Eq. (14.61). (Data from Carstensen andMusa, 1972.)
  • 259. .73 2.68 1ooonr ln[l - x*] as a function of lOOO/T: least-squares equation; y = 16.19 - 6 . 3 7 ~ (R= 1.00). (Data from Carstensen and Kothari, 1981.) As the reaction proceedsdecomposition product C will accumulatebetween A t a giventime t , compound A must diffuse to the surface of hrough a layerof compound C, h-cm thick, for the reaction to density of I3 is denoted p. A layer of B, h-cm thick would contain (14.64) icks first law, dB/dt is inversely proportional to h, so that we may write pdh/dt = q/h (14.65) (14.66) Interaction betweentwo solids withdecompositionlayerseparating the two reacting species, necessitating diffusion of one of the reactants through the decomposition layer. (Data from Carstensen and Musa, 1972.)
  • 260. ilitThis may be integrated to h2 = [24/p]t = k’t (14.67)or: h = [kI t]1/2 (14.68) k’ = 2g/p. If, as indicated in the lower line of Fig. 14.21 are cubical, of side length a. initially, and a at time t, and ifA h=ao-a (14.69) he amount retained at time t is (14.70)or { 1 - (1 - x ) ’ / ~= kt/ao }~ 2 (14.71)where x is fraction decomposed. It is seen that the rate constant is related to theparticle size (i.e., the finer the particles the larger the rate constant). A system of thistype is, again, the aspirin- odium bicarbonate system, but at lower t e ~ ~ e r a t u r e s .higher temperatures, the autodecomposition of aspirin is higher than the diffusicoefficient (r~lated q), and the reaction at higher temperatures then follows [see to .IS)] (Nelson et al., 1974). ecently, it has become customary to compare polymorphic and p s e u d o ~ ~ l y - c transformation data with prevailing solid-state equations (e.g., forms of theNg equation). Several such equations, some of them already alluded to, are listed inTable 14.8. There has been a tendency in recent literature to simply fit data to several (orall) o f these equations, and the equation that gives the “best fit” is then assumed tobe the mecha~ism, Figure 14.23, for instance, shows a literature example of suchdata. It is claimed that these data best fit a Jander equation (and such treatment is shown in Fig. 14.24), but first of all the fit is not good, and second, it is obvious thatthe phase C in the Jander model (see Fig. 14.21) cannot possibly apply to a poly-morphic transformation where the reaction is simply A -+ e ~ p h ~ s i hered t ~ u sorting out ~ e c h a ~by s~@tistical ~e t i s ~ ~ analysis can be fal~acious, ~q u atio ns Relating to Decomposition in the Solid State In@/( 1 - x)) = kt Surface nucleation, Prout-Tompkins equation (-1n(1 - x))”/”== kt n-Dimensional nuclear growth (Avrami and Erofeyev) ~ - ~ i ~ e n s i onucleus growth nal 1 - (1 - x)l’n = kt n-Dimensional boundary reaction x2 = kt Diffusion in one dimension (1 -x)ln(l - - x ) + x = k t Diffusion. in two dimensions (1 - (1 - x)1/3)2= kt ~ i f f ~ s i o nthree dimensions (Jander equation) in
  • 261. y = 52.296 - 24.723~ R"2 = 0.994 l/(Tmax) Kissinger plot of polymorph T of glybuzole. (Data from Otsuka et al., 1999.) I Several modelistic investigations in this field have appeared in recent years.Fini et al. (1999) have studied the dehydration and rehydration of diclofenac salthydrate at ambient temperature. Otsuka et al, (1999) investigated three forms ofglybuzole (I, 11, and amorphate), (Figs. 14.23 and 14.24) and found all to have fairlymuch the same solubility. Neither form I nor 1 changed after storage at 40°C at 75% 1 for 2 months. DSC for form I showed no peak other than a sharp them at 167.4"C, form I1 showed a slight endotherm at 116.8"C and asharp endotherm at 166.6"C. The amorphate showed a (slight) exotherm peaking at81.5"C, presumabl~owing to crystallization, and a sharp endotherm at 167.3"C.From this it would be reasonable to conclude that form I1 is stable at room tem-perature and t r a n s f o ~ s I at 116.842, this latter form being stable at the higher tote~peratu~es. The authors estimated the polymorphic stability of form I bywayof the T issinger equation (Kissinger, 1956). (14.72) 0.0 0.5 1.o 1.5 Time (hours) Literature data dealing with two polymorphic transformations allegedly diffu-sional because it ahderes (somewhat) to a Jander model.
  • 262. 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2-0.00.2 0.4 0.6 0.8 1.0 1.2 Data from form B in. Fig. 14.22 treated according to a Jandermodel. The curvefollows the least-squares fit equation: J ( t ) = -0.194 + 0 . 6 5 2 4 ,where 43 is the rate of heating, Tmax the temperature at the peak maximum in the is SC, Ea is the activation energy, and R is the gas constant. If the experiment isconducted at different heating rates, different Tmaxvalues result, and in the case ofglybuzole there were four such values. It can be seen from their graph that the activation energy is 24.723 x 1.99 =49.2 kCal/mol. Otsuka et ai. (1991, 1993, 1999) employed the Jander equation toexplain crystallization rates of compounds (e.g., amorphous glybuzole). However,the Jander equation is based on an assumption of a layer of “reaction product,” andsuch a layer (i.e., such a model) cannot be conceived of in a polymorphic transfor-mation, because what would be the “reaction product”?It is tempting to think of a tablet as an agglomeration of individual particles, inde-pendent of one another, but this cannot be true. By their mere nature, particles arefused together (by either brittle fracture or by plastic deformation in tablets ortamping in capsules), and if the created contact area is between two different com-ponents of the tablet (one being the drug), then there is the possibility ofinteraction.It is highly likely that moisture plays a part in all of these. In fact, in one of the casesto be discussed later (tartaric acid + sodium bicarbonate) this is true (in spite of thefact that the tablet can, for all practical purposes, be anhydrous at the onset). The most common type ofinteraction in solid dosage forms is actually betweenwater and drug.This is a large topic in itself, and Chap. 15 is devoted to it. The topicdiscussed here will be of special cases in which water is not the interactant (or themain interactant). The following illustrative examples will be discussed: 1. Tartaric acid and sodium bicarbonate 2. Aspirin and phenylephrine 3. Aspirin and lubricants In addition to the points made, it is noted in the curve in Fig. 14.24 that a lagtime sometimes has to be invoked for the data to linearize.
  • 263. The formation of molecular compounds as discussed in Chap. 11, under theheading of ~ ~ t e c t i ca~type of solid-state interaction. It, attimes, is of importance is ,in solid dosageform formulation. For instance, the author was in charge of scale-theup of a soft-shell capsule product, Filibon, once marketed by American CyanamidCompany (Lederle). It contained among other vitamins, niacinamide and ascorbic n small scale, in which time lapses are short, the product was quite “stable,”but in large-scale production, duringwhich the capsule contents were exposed to themoisture in the soft shell for longer times, the capsule “hardened up,” in fact becamea “bullet.” Theproduct wasa (molecular compound type) interaction betweenniacinamide and ascorbate, and the problem was rectified by carrying out the reac-tion before blending the powders. The niacinamide and ascorbic acid were simplymixed in a blender and “granulated” with ethanol. The resulting powder was brightyellow. When dissolved in water the individual components will regenerate. There have been occasional reports of solid-state interactions in the pharma-ceutical literature. ogdanova et al. (1998) shownhave a solid-state interactionbetween indomethacin and nicotinamide. The solubility of the complex varies in afashion, such that the solubility is maximum at a given indomethacin concentration.This is a common combination in effervescent tablets. When the tablet is added towater, the acid and the base will react, forming carbon dioxide, which produces thedesired bubble effect. 2(COOH)2+ 2NaHC03 -+ R2(COO-), + 2Naf + 2To be strictly correct, the left-hand side should be written in ionic form as well. t is necessary that this reaction does not take place before the time it reachesthe consumer, because if the reaction does occur in the solid state, then (a) carbondioxide will form in the container, (b) the tablet willbecome softer, and (c) on“reconstitution” the bubbleeffectwillbereduced to the extent carbon dioxidewas lost in storage. The evolution of carbon dioxide would normally build up pressure in a glassbottle, but the tubes in which effervescent products used to be sold were not tight,and the carbon dioxide could escape. The same is true to a great extent in plasticbottle and in plastic blister packs, but the problem that the reaction (as shall bedemonstrated later) is catalyzed by moisture, in. other words, that thecontainer is nothermetic in this aspect, is a disadvantage. This is so sensitive that during manufac-ture extra precautions are taken to keep the relative humidity of the processing areaslow. Hence, one must also pack the products in hermetic containers, and the alumi-num foil has become apopular means of doing this. If, however, the initial moistureis not low enough, then the reaction will proceed, and the internal pressure will causethe aluminum foil to “balloon.” The solid-state reaction has been investigated by Usui and Carstensen (1986)and Wright and Carstensen (1987). Whenthe reaction occurs in the solid state, thereare two questions that present themselves: 1. Is moisture important, and if so in what sense? 2. What is the stoichiometry? Is it that of Eq. (14.73) or is it
  • 264. 2(COOH)CO~~a HZ0 + + CO2 (14.74) Usui checked the weight loss of heated samples in hermetic containers?utili~ingdifferent ratios of acid and base and established that the stoichiometry is that of Eq.(14.74); that is, the mole-to-mole interaction of tartaric acid and sodium bicarbo-nate. e next studied the weight loss in open containers and demonstrated that the acid did not lose weight,and that the sodium bicarbonate and the mixture ofsodium bicarbonate and tartaric acid, lost weight at a low rate, corresponding to thatof the sodium bicarbonate itself. In other words in an open container, there was nointeraction, simply decarboxylation of the bicarbonate itself. e next studied the effect of compression on the decomposition of sodiumbicarbonate. Characteristic curves are showninFig.14.25. It is noted that thedecomposition rates are a function of applied pressure. In the following it is assumedthat the particles are isometric and that the reaction rate is proportional to thesurface area of unreacted sodium bicarbonate. The following nomenclature isused: there are M g of unreacted sodium bicarbonate at time t, and M0 initially.There are N particles each of area a, volume v and density p. The surface area isproportional to the two-thirds power of the volume by the isometry factor r, that is, a = rv2I3 = ~p 213 m2J3 A = Nrp-2~3m2~3 =I N1~3rrp-2~3M2~3 4.76) (1It follows that (14.77) 15 k kP 20 40 60 80 100 Time (Hours) Effect oftableting pressure on sodium bicarbonate decomposition at 70°C,from Usui and Garstensen, 1985.)
  • 265. earrangement of Eq. (14.78) gives d ~ / = kt ~ ~ ~ 3 (14.80)This maybe integrated, and when initial conditions are imposed the followingespression results: ( ~ / ~ ~ ) = (13 1 / - X ) l l 3 = 1 - kt 1)(14.8where x is mole fraction decomposed, and whereEliminating N by inserting Eq. (14.77) into Eq. (14.82) gives (14.83)The data should, therefore, plot by a cube-root equation and Fig. 14.26, indeed,shows this to be so. The rate constants according to Eq. (14.83) should be proportional to thespecific surface area at time zero ( ~ ~ / M ~ )this is true is shown in Fig. 14.27. That .The rate constants follow an Arrhenius plot, and are in line with data reported by theSchefter et al. (1974). In a closed system there is a rapid interaction between the sodium bicarbonateand tartaric acid in compressed tablets. Even though the system is supposedly dry, itis assumed that there is a very slight amount ( z mol) of water present in the tabletinitially and that the reaction starts in a dissolved stage. If this is true, then, as wateris produced in the reaction, there will be an acceleration. The data can be modeled inthe fashion shown in the following. The nomenclature used is: A is the number of 4moles ofsodium bicarbonate left at time t , and M6 is the initial number of moles, S isits sol~bility water and C is the concentration in the water present at time t. S is in Ithe solubility of the tartaric acid in water. 0 100 200 300 Hours at 55°C Cube-root plot of sodium bicarbonate decomposition at 55°C: least-squares fitequations: 0 kP; y == 1 - 0.0015~ R = loo), and 15 kP; y = 1 - 0 . 0 0 3 ~ R = 1.00). (Data ( (from Usui and Carstensen, 1985.)
  • 266. 10CYCYrc 5 0 0.2 0.4 0.6 S p ~ c ~ fSurface Area (sq. mlg) ic 7 Cube-root constants from Fig. 14.25 versus specific surface areas: least-squares +fits; 70°C, y = -1.534 + 19.447 ( R = 0.99); and 55"C, y = 0.788 3. 188~ = 1-00). (Data (Rfrom Usui and Carstensen, 1985.) According to the reaction scheme the number of moles of water present at timet then is + z)mol = (Mh - M + z)O.O18The isa appearance rate of sodium bicarbonate in solution is given by -dC/dt = k2SjS (14.85)where k2 is the second-order rate constant. To express this as number of molesdecomposed, this figure is multiplied by volume of water present [i.e., the expres- thesion in Eq. (14.84)J: dM/dt ==: -k * ( ~ -hM + Z ) (14.86)where -0.50 -0.75 -1 0 0 -1.25 -1.50 0 20 40 Hours Decomposition of tartaric acid plus sodium bicarbonate tablets at 55°C (5 kP +force): least-squares fit;ln{X} = -1.3225 0.0291 * t (R= 0.98). (Data from Usui andCarstensen, 1985.)
  • 267. k* O.OI8k2S1S (14.87) quation (14.86) can now be recast in the following form: In(MA - 1 ” + z = k*t + h[z] ) (14.88)or, employing x, the mole fraction decomposed is (14.89) ecalling that z is a small number, the term z / M o is small, and Eq. (14.89) thensim~lifies to ln[x] = k*t + ln(z/kfL] (1 4.90) ata areplotted in this fashion in Fig. 14.28. It is seenthat the linearity is quite ood. The valueof z maybe estimated from the intercept and comes to about0.1 mgltablet, which is a reasonable figure. This, in essence, shows that the theoriessuggested by Wright (1983) are correct. It is obviously of pharmaceutical importance in most situations to slow downthe reaction in the solid state, and yet maintain the reactivity in the solid state. (Anexception to this is when a reaction is purposely carried out during a granulation, forinstance). One way of retarding the reaction rate i s to preheat the bicarbonate to95°C for a certain length of time CJVhite, 1963; Mohrle, 1980). This will react by thescheme -+ Na2C03+ H 2 0 1) (14.9The water formed granulates the mixture, and makes it easier to compress.importantly the sodium carbonate formed can form double salts with the bicarbo-nate. These are dodecahydrates, and act as moisture scavengers. They hence stabilizethe acid/base mixture in the solid state (if reasonable moisture barriers are provided):any ~ ~ amountZ of moisture created by a beginning reaction of the type of Eqs. ~ Z(14.73) or (14.74,will react with a mixture of the carbonate and bicarbonate to forma double salt hydrate.In the strictest sense, pH is not a term that is defined ina solid system.For it to havemeaning, there must be some water mediation, but both tochopheryl acetcalcium pantothenate are cases in point. The former is sensitive to highformer to low pH, Calcium pantothenate is frequently admixed with magnesiumoxide and granulated separately from the remaining i~gredients.In this manner analkaline microenvironment is created, which ascertains the stability of the vitamin. n the case oftocopheryl acetate, the hydrolysis is accelerated hydroxyl ions. byAgain it is noted that the reaction must be associated with some dissolution step insmall amounts of water. The produced tocopherol is much less stable; hence, thehydrolysis and the presence ofwater are contraindicated. This is a particular instancewhere the useof alkaline excipients(e.g., hydroxyapatite) can be deleterious athigher temperatures. In the absence of (or at low levels of) moisture the reactionmay not proceed. It is also characteristic that often, higher temperatures are notindicative of what will happen at room temperature.
  • 268. ilit If it is desired to control the of the microenvironment then citric, tartaric,and fumaric acids are the acids of ice. They are, however, all corrosive, and theirpharmaceutical andl ling is far from ideal. With an alkali, sodium bicarbonate,sodium carbonate, and mag~esiumand calcium oxides are common, and are notas corrosive as the acids mentioned, but they are abrasive, and they, too, are not the a1 substances to handle in a tablet or capsule. r certain compounds it is necessary to control the “micr~environm~nt~~ ineven more drastic fashion. Gu et al. (1990) report on drug excipient incompatibilitystudies of moexipril hydr loride, and demonstrate that (even “wet”) adjustment ofthe microenvironmental (i.e., adding small amounts of water to a mixture of thedrug with sodium bicarb e or sodium carbonate), did not sufficiently stabilize the ut when the mixture was wet granulated, and when s t o i e h i o ~ e ~ r i c a ~ o uf n t s oal~aZi were used, then stabili~ation resulted. This essentially means that, in the solidstate, the s o d i u ~ is sta~Ze opposed to the acid. It might be argued that in such salt asa situation the sodium salt should be manufactured and used as such. It might beargued that it should be claimed as the active ingredient (equivalent to a certainamount of free acid, or in amphoteric substances, the acid addition salt), but oftenthe salt is very soluble and hygroscopic (e.g., potassium clavulanate); hence, they aredifficult to produce. The situation is referred to in the Federal ~ e ~ i s tasra ~ e r i v ~ t i v e edrug.Dehydration, as mentioned before, may result in amorphous anhydrates, but mayalso result in another crystalline phase (e.g., a lower hydrate or a crystalline anhy-drate). These are, properly speaking, p s e ~ d o p o l ~ ~ o rt~ h incs f o ~ ~ a t i oThere are ~a ns.several steps in dehydration of a hydrate, and theymaybe su~~marized the in denotes solid, D denotes drug, V denotes vapor, and I , §uryanarayanan, 1997). (enthalpy of dehydration = A ~ ~ ) (enthalpy of vaporization = AHv) *x (enthalpy of transition = AHT)that is, (14.92)so that different results may be obtained in DSC experiments depending on whethera crimped or open pan is used. ray et al. (1999)have shown such a diagram for ~2(~)-~-toluenesulfonylamino~-3-[[[5,6,7,8-tetrahydro-4-oxo-5-(-pyrazolo~l,5-a][l,4~diazepin-2-yl]carbonyl]amino]-propio Suihko et al., (1997) employed have show that dehydration of theo-phylline mono~ydrate a two-step process. is
  • 269. There are times in which equilibrium sets up in the solid state. Vitamin A beadletsequilibrate at about 75% of the original vitamin A potency, and tocopherol acetate,likewise, can achieve an equilibrium state in solid-dosage forms. These equilibriamay, or may not, be pressure induced. More convincingly, Wurster and Ternik (1995) havereported data that imply apressure-induced activity loss in solid-state catalase (Figs. 14.29 and 14.30). Theremay not be a total loss, even at high pressures, because the figures seem to taper offwith increasing pressure, and by iteration it maybe found that, expressing thenumbers as percent, 67% of activity left, even at high pressure, gives the best biasfit, and this is shown in Fig. 14.30.Not much systematic work has been reported on photolysis of solids. Lachman et al.(1961) pointed out that, most often, a solid tablet will decompose by photolyticdecomposition only in the surface area, so that if one broke a “discolored,” exposedtablet, then the color would be unaffected in the interior. owever, Kaminski et al. (1979) reported on a case where a combination ofmoisture and light caused an interaction between a dye and a drug (ethinyl estradiol)that permeated the entire tablet. Tonnesen et al. (1997) have reported on the photo-reactivity of mefloquine hydrochloride in the solid state.Carstensen (1980) noted that topical reaction profiles were literally indistinguishablefrom first-order decomposition profiles. It is possible, at times, to invoke Arrheniusfitting to distinguish between reaction mechanisms as pointed out by Nelson et al.(19’?4),and at times, valuable information may be gleanedin this fashion.(Carstensen, 1977) Arrhenius plotting of a first-order reaction, and the same datatreated by zero-order kinetics give fairly much the same goodness of fit and activa-tion energy. Often, data are fitted to a series of equations, and the model chosen is the onethat fits the data “the best” (Sharp et ai., 1996). Carstensen (1995), Sharp et ala 0.4 d) 0 9 0.3 e 2 9 0.2 0 10 20 30 40 Time (sec) Activity loss of catalase in the solid state inducedbypressure. (Data from Wurster and Ternic, 1995.)
  • 270. n y = 3.5279 - 4.4139e-3x RA2 = 0.958 3.6b 3.4rr($ 3.2 3.0c 2.8$ 2.63 0 300 100 200I.-l Pressure (MPa) Data from Fig. 14.29 treated by subtracting 67% from the percentage of zeropressure content, and plotting loglinearly against applied pressure. Figure is not part of thereference publication. (Data from Wurster and Ternic, 1995.)(1996), Ledwige and Corrigan (19969, and Taylor and York (1998) have cautionedagainst that “lack of discrimination of the different best fitting models.” The original suggestions by Nelson et al. (1974) and Carstensen (1980) wereinvestigated by Taylor and York (1998), who fitted dehydration data to a series ofoft-used kinetic equation and applied the rate constants to the Arrhenius equation.They, as did Carstensen (1980), found that fits and activation energies from thedifferent models remained fairly invariant. At times, models can be ruZe~ The data in Fig. 14.31 is the data in Table out.14.6 treated by the Prout-Tompkins equation. It is seen that there is d~finite curva-ture in the plot, sufficient to rule out the model as representing the decompositionmechanism. At times an active ingredient or a decomposition product in a solid dosage form is a liquid, and this may interact with other ingredients in the dosage form. A typical example is the work by Troup and Mitchner (1964) dealing with aspirin and pheny- he authors showed that the decomposition of phenylephrine was linearly related to the formation of salicylic acid. They showed that the decomposition of ~henylephrine an acetylation. This can be thought of in many ways. There has to was be some moisture present to permit the hydrolysis of aspirin. If the salicylic acid is 0 620 400 80 100 120 Time (minut~s) ig. Data from Table 14.4 treated by rout-Tompkins kinetics.
  • 271. formed by interaction of aspirin with traces of water, then the acetic acid formed may react with the phenylephrine [R(OH)3], again liberating water, so that the moisture does not play a part, quantitatively in the overall reaction, in other words 3) + HZ0 "+ C6H4(0 (14.93) 3COOH + 1/3R(OH), "+ 1/3R(OC (14.94) 3) + 1/3R(OH)3 -+ 1/3 (14.95) An alternate explanation would be that phenylephrine interacted directly with aspirin in an anhydrous solid state to transacetylate, which is not probable. The question is whether the acetic acid (which a sizable vapor pressure) interacts with has the phenylephrine as a gas with a solid reaction (to be covered shortly) or as a liquid with a solid reaction. There are other examples of the interaction of acetic acid with active ingredi- ents (e.g., the work by Jacobs et al., 1966, in which acetylation of codei~e aspirin1 in codeine combinations was demonstrated). Again, whether the acetylation is achieved by acetic acid in vapor phase or in the liquid state or (more unli~ely) the whether it is a direct solid-to-solid interaction, is not yet resolved. Ifit were the latter, then Jander kinetics should actually apply. But it is difficult to distinguish this and pseudo-first- order reactions. If it i s an interaction in the liquid state, then it probably occurs by the ~henylephrine dissolving in the acetic acid formed. In more general terms, it is assumed that there are two drugs, A and decomposes (e.g., by hydrolysis) to form a liquid decomposition product C . The reactions then are: (rate constant k ) (14.96) +C "+ (decomposition) (rate constant k) (14.97) C is the species that is liquid. In this case a saturated solution (S mol/mol) of is formed, and it is assumed that dissolution is fast. Let A be the number of moles of drug present at time t, C the number of moles of acetic acid, and let M denote the molarity of the liquid decomposition product (e.g., for acetic acidat 25OC the density is1.05 g/mL, so that, because its molecular weight is 60, A would be 1005160 == 4 16.75). The rate at which I> disappears is the question to be solved.It is assumedthat the disappearance rate of A is pseudo-~rst-order, thatis(14.98) A = A. exp(--kt) The disappearance rate of D depends on how much C is present, so the equation for st first beestablished and solved. C is created at a rate of kA, but it is consumed . The rate of the latter step isgiven by a second-order reaction term. The concentration of D is S , and the molecular weight of C is M . The amount of C at time t is C, so that (in terms of moles) dC/dt = kA - kSCA4 (14.99) nserting Eq. (14.98), using and denoting ~SMa = (14.100)
  • 272. where a is constant, we arrive at the following equation: dC/dt = kAo exp(-kt) - aC (14.101)Laplace transformation, using L-notation, gives: SL- 0 = [kA,/(s + k)]a (14.102)or + = EkAo/(a - k ) ~ [ { l / ( ~ - U/(S k)) + 411so by taking anti- C = [kAo/(krSM- k)]{exp(-kt) - exp(-k’SMt)) (14.104.)It follows that the decomposition rate of I is given by ) /dt = krSCM = aC (14.105)by integrating Eq. (14.105) and multiplying by a, we obtain - - = [ k u A o / ( k r S ~k)][{[e~p(-k’SM211/krSM) ( ~ e ~ ~ ( - k t ) } / ~ ) (1~.106) ]An example of this is shown in Fig. 14.32 usingA = 50, k = 0.2, and k r S = 0.1. A ~different situation ariseswhen an insoluble component interacts with a drug in sa). An example of this is the intera~tion be OHR”) and substituted furoic acids (RCOO furoic acids decompose when heated by thedecomposition product and carbon dioxide. In the presence of microcrystalline cel-lulose, however, the mixture f o m s carbon monoxide: pi” + Q + Q’ + CO (14.107)Q is a liquid, which will dissolve furoic acid to the extent of its solubility, and willspread over the microcrystalline cellulose. There will a number of contact points N ,at which interaction can take place (essentially the “wetted” f the microcrystal-linecellulose). There willbe a reaction probability a, asso witheach contactpoint. The reaction accelerates because the larger the extent it has reacted, the moreliquid there will be to dissolve the furoic acid; hence, the more contact points. At a 0 10 20 Time Stability profile using A = 50, k = 0.2, and a = 1.
  • 273. r l given time point there will be overcrowding, because dissolved molecules be next will to contact points that have already reacted. Hence, there is also a t e ~ i n a t i o n probability 6. But unlike the Prout-Tompkins model, this is finite at time zero. It might be argued that the external surface of the microcrystalline cellulose would be insufficient to account for the total decomposition. There are, however, two types of sudace present in microcrystalline cellulose: nitrogen adsorption gives low surface areas (the external area); whereas, for instance, water isotherms give surface areas 100 times as large (Hollenbeck, 1978;Marshall et al,, 1972; Zografiand Kontny, 1986). y the decompositionat a contact point, it is assumed that the decomposition, creating one liquid decomposition molecule, will dislodge (dissolve) S molecules of furoic acid at the contact point. If the initial number of contact points is No, then dN/dt = 1-6 + a(S - 1)]N = qiV (14.108) + where q = -6 a(S - 1). The factor arises from the fact that when a molecules react, then OS new contact points are created and one(the one at which the reaction took place) is lost. It follows then from integrating Eq. (14.108) (whichcan be done, since a and b are assumed constant), that(14.109) N = No exp(qt) Since, at a given time t, the rate of decomposition is proportional to the number of contact points, then, L being the number of intact alkoxyfuroic acid molecules dL/dt = gN 10) (14.1 where g is a constant. From the definition of L it follows that the mole fraction x ~ecomposed given by is x = (Lo - L)/Lo (14.1 11) or dx/dt I= -( l/Lo)dL/dt (14.112) E~uation (14.1 10) inserted inthis gives dx/dt = (1/Lo)giV 13) (14.1 Su~stituting (14.109) into this gives Eq. This integrates to where the term A = (Loq/gNo)has been introduced for convenience. Equation (14.1 15) isequivalent to Zn[l + Ax] = qt (14.116) Figure 14.33 shows data treated in this fashion.
  • 274. 0 10 20 30 40 50 Hours Furoic acid data treated according to Eq. (14.1 16). (Data from Carstensen andKothari, 1983.)There are cases for which there are liquids in a solid dosage form. An example ispanthenol in a multivitamin tablet. Here it is customary to adsorb the liquid onto asolid carrier and for panthenol, magnesium trisilicate is used. At elevated ternpera-tures (and at room temperature under compression as well) the panthenol will oozeout of the carrier, and corne in intimate contact with other solids. If interactionpotentials exist, then separation tech ues, such as triple-layer tablets (or compres-sion-coated tablets) are resorted to re, the liquid will stillooze into the layercontaining its interactant, but the process will be diffusion controlled. It can beshown (Jost, 1962) that the average concentration C of the liquid in the neighboringlayer with which it is in contact, is given by: (14.117)where Cf is the on cent ration at infinite time. The tern on the right-hand side isactually the leading term of an infinite series.Sometimes the vapor pressure of a drug is sufficiently highthat it may interact withother substances via the vapor phase. An example is ibuprofen (B). This is a Lewisacid, and may interact with Lewis bases.Usual measures, such as triple-layer tablets,do not work in this case, for the interactant will be present in the gas phase. If the reaction with another drug ( ) is I) D + B "+ deco~position (14.1 18)then the initial reaction rate is given by d{D}/dt kPB[D]A (14.119)
  • 275. where {Dlis the surface density of D-molecules (number of molecules/cm2)at time tand A is the surface area. As long as there is no penetration into the crystals, thereaction will, therefore, be a first-order reaction, since Eq. (14.119) integrates to ln[D] = -kAP,t (14.120) + ln[Do]where Do is the initial concentration. This will be true if only the surface of the solidinteractant is affected. The extent of decomposition will be slight, because (unlessthedrug is extremely finely subdivided) only a small fraction of the molecules are on thesurface. If, however, the ibuprofen penetrates the crystal, then Jander kinetics shouldprevail. A similar situation may be at work inthe aspirin incom~atibilities mentionedearlier. A = number of moles of unreacted solid at time t Ao = original number of moles of unreacted solid a = side of a cube at a time t after reaction has started a = side of a cube before decomposition 0 13 = (kl/k,) - 1 - S = iterant in the C = (a) generalsymbol for concentration, (b) concentration in the water present in an effervescent tablet at time t CI = constant in the WLF equation Cz = constant in the WLF equation Cf = the concentration at infinite time D = diffusion coefficient DSC = differential scanning calorimetry E = energy Ea = activation energy E,, = energy levels above Ei F = (a) a constant, (b) pree~ponential constant in first-order decay H = height of a cylinder A H = heat of fusion AHd = enthalpy of dehydration AHf = enthalpy of transition AH,”= enthalpy of vaporization h = thickness of a reacted layer k = general term for rate constant k l = first-order rate constant k2 = rate constant for two-dimensional diffusional decomposition k* = rate constant in effervescent interaction I, = (a) symbol for Laplace operator, (b) number of intact alkoxyfuroic acid molecules at time t Lo = number of intact a l ~ o x y f ~ r oacid molecules, initially ic & = mass of intact sample at time t ! M0 = initial mass of intact sample M‘ = number of moles of sodium bicarbonate left at time t ik?; = initial number of moles of sodium bicarbonate w1 = mass not reacted
  • 276. N = (a) number of nuclei, (b) number of particles in a sample No = initial number of intact molecules N1,N 2 , N3 = number of nuclei in one, two or three dimensions inapproximate Avrami model N,, = number of molecules with energy levels above El rz = (a) exponent in the Ng equation, (b) an integer between 1 and 4 (Avrami- Erofeyef equation) p = exponent in the Ng equation Q = constant in the expanded Avrami model + Ql = (a) constant in the slow-nucleation,fast-reaction model, (b) [qkn/(n l)], a constant in the Avrami treatment q = constant in the (a) Avrami treatment, (b) the Arrhenius equation, (c) Jander equation R = (a) ideal gas constant, (b) property (e.g., heat capacity or rate constant)of an amorphate at a temperature below or above its glass transition tem- perature, (c) radius of a cylinder R, = the property R of an amorphate at the glass transition temperature S = (a) solubility in water (for components of effervescent tablet), (b) S = solubility (mol/mol)of a solid compound in its liquiddecomposition product SI = solubility of tartaric acid in water 1 = absolute temperature, I " ( if* = eutectic temperature Tf = melting point T, = glass transition temperature Tmax temperature at the peak maximum in a DSC = t = time t, = lag time u = particle volume illiams- ande el-Ferry x = (a) fraction, (b) mole fraction, (c) fraction decomposed, (d) number of moles of water in a hydrate x* = the mole fraction inBawn kinetics where just enough material has decomposed to just dissolve the remainder of the parent compound Z = normal standard deviate z = original, very small amount of water present in an effervescent tablet a = propagation probability or rate / = termination probability or rate I I" = shape factor Cp = the rate of heating p = density of a solid , Lee G (1999). J Pham Sci 88:199.Agbada CO, York P (1994). Int J Pharrn 106:33.Anderson NR, Banker GS, Peck GE (1982). J Pharrn Sci 71:7.Avrami M (1939). J Chern Phys 7:1103.
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  • 279. This Page Intentionally Left Blank
  • 280. 15.1. Amorphates 26815.2. Nonhydrate-Forming Drug Substances 26815.3. 26815.4. 26915.5. Moisture Amounts at the Critical Moisture Content 27 115.6. ound Water 273 15.7. Excess Water 274 15.8. roenvironmental pH 275 15.9. drate-Forming Drugs 27615.10. 277 Symbols 278 References 278Stability of drug substances in dosage forms is affected not only by their chemistry.but also by their environ~ent. Compatibility studies are generally carried out withnew drug substances in combination with common tablet or capsule i n ~ r e ~ i e nto tsascertain that the excipients chosen are not detri~ental the integrity of the drug to(or of as little damage as possible). When such programs are carried out, it is con-ventional ( ~ a r ~ t e net e ~ 1964) to study combinations both in the absence cirict tlw s al.,presence o ~ ~ a t eThis is because, of a11 the types of substances one encounters in r.tablet and capsule formulations, in general, the most ~ e t r i ~ e ~ t water. is a l he chapter to follow will deal with the nature of the interaction between wateror water vapor with drug substances.
  • 281. ter 1 sAmorphous substances in the presence of water degrade according to first-orderkinetics (Pikal, 1977; Morris, 1990). This is not surprising in light of the previousfindings thatamorphates are somewhatlikeliquids. Carstensen and VanScoik(1990) showed that water vapor pressure over amorphous sucrose that containedwater corresponded to a value that could be extrapolated from the vapor pressurecurve of unsaturated solutions ofsucrose at the other endof the concentration ence, such systems may be considered solutions and, as such, shouldbehave, kinetically, as solution systems.If a substance does not form a hydrate, then moisture present on or in it will be ofthe types shown in Fig. 15.1. It can be moisture that is adsorbed in an amount lessthan thatcorresponding to a monolayer (see Fig. 15.la), orstarting to forma bilayer(see Fig. 15.lb), ora multilayer (not shown). Oncethe critical vapor pressure for thecompound (the water vapor pressure over a saturated solution) is reached (see Fig.15.lc), mois