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2 the kinetic of emulsion polymerisation

  1. 1. The Kinetic of Emulsion polymerisation Dipl.ing. Anton Adžaga
  2. 2. The Kinetic of Emulsion Polymerisation Table of ContentsIntraduction ........................................................................................................................................................ 11 General Features ............................................................................................................................................ 2 1.1 Polymerization techniques ....................................................................................................................... 2 1.2 Basic principles of Emulsion polymerisation .......................................................................................... 22 Emulsion Polymerization Kinetics .................................................................................................................. 5 2.1 Generally Accepted Kinetics Scheme (particle formation and particle growth) .................................... 5 2.2 Summary of the Smith-Ewart Theory...................................................................................................... 83 Kinetics and Mechanisms of Emulsion Polymerization.................................................................................. 9 3.1 Radical Entry ........................................................................................................................................... 9 3.2.1 Diffusion-Controlled Entry................................................................................................................ 9 3.2.2 Propagation-Controlled Entry.......................................................................................................... 11 3.3 Radical Desorption (Exit) ...................................................................................................................... 12 3.3.1 Desorption in Homopolymer Systems ............................................................................................. 12 3.3.2 Desorption in Copolymer Systems .................................................................................................. 14 3.4 Particle Formation and Growth ............................................................................................................. 15 3.4.1 Particle Formation ........................................................................................................................... 15 3.4.2 Particle Growth in Homopolymer Systems ..................................................................................... 17 3.4.3 Particle Growth in Copolymer Systems........................................................................................... 19 3.4.4 Monomer Concentration in Polymer Particles................................................................................. 214 Concluding Remarks ..................................................................................................................................... 23References: ....................................................................................................................................................... 24 1
  3. 3. The Kinetic of Emulsion PolymerisationIntroductionIn recent years, these emulsion polymerisation reactions have become more and more importanttechnologically and commercially, not only as methods for producing high performance polymeric materialsin a low cost, but also from an environmental friendly process. The drive to develop environmentally friendlyproduction methods for polymers has resulted in widespread development and implementation of thepolymerization technique. When combined with novel polymerization mechanisms and kinetics the processcan give rise to a range of polymer products with particularly useful properties.Emulsion polymerisation is a complex process, governed by the interplay of chemical and physical propertiesincluding polymerisation kinetics and dispersion stability.In this report, recent developments in emulsion polymerization are reviewed from kinetic and mechanisticperspectives.1 General Features [1] [2]1.1 Polymerization techniquesPolymerizations may be categorized by (1) the polymerization mechanism (e.g. radical polymerization, anionic polymerizationetc.) and by (2) polymerisation technique. We distinguish four different polymerization systems (techniques): bulk, solution,suspension and emulsion polymerization. Suspension and emulsion polymerization are heterogeneous systems, bulk andsolution are homogeneous, although in later stages the polymer formed may become insoluble in the reaction medium(precipitation polymerization). A third factor is how the reactor is operated: in batch mode, or semicontinuous (addingmonomers during the process) or by continuous operation.1.2 Basic principles of Emulsion polymerisationAt emulsion polymerization use emulsifier (soap) much more than at suspension polymerization. Final particles are more than1000 times smaller than in the suspension polymerization (the size order of 0.1 µm).Final particles form stable latex in water.For this reason emulsion polymerisation technique is used at the production of synthetic rubbers, like as SBR (styren-butadien-rubber), and as coating or lacquersThe most important components (ingredients) in an emulsion polymerisation are: monomer, the dispersing medium (usuallywater), soap or detergent and a water soluble initiator.By the high soap (emulsifying surfactant) concentration, above the critical micelle concentration CMC, the soap moleculesform in the water spherical aggregates so-called micelles. In these micelles the hydrophobic tails are oriented inwards, while thepolar heads are located at the interface to the water-phase. These micelles include approx. 50-150 soap molecules and have asize of approx. 2-10 µm. To form enough micelle, the soap concentration must be high enough to come above the criticalmicelle concentration. This is usually the case when added 0,5-5% soap with respect to monomer , what result in the presenceof 1018 − 10 21 micelle per liter.When is added a water-soluble or a little soluble monomer, solves in water a small part of monomers from the monomerdroplets. A monomer is usually only slightly soluble in water. Furthermore a small part of monomers solve in the micelles (theysolubilized in the interior of the micelles). The largest part of monomers (> 95%) becomes dispersed in the water in the form ofmonomerdroplets which has surrounded with stabilizing monomolecular layer of soap molecules. These drops have now a sizeof 1 to 100 µm and a concentration of 1012 − 1014 drops per liter. This means that the quantity micelles are larger a lot oftimes than the number of monomerdroplets. Moreover their total surface area with the water medium is a lot of times largerthan that of the monomerdroplets. The monomer consumption will be compensated by diffusion of monomer molecules fromthe large monomer droplets .The polymer containing former micelles are called latex particles. The monomer must have a lowsolubility, less than 0.004 moles of monomer per liter of water. With too high of a solubility, there would be solutionpolymerization in the water phase, which is undesirable.The initiator are chosen that not solved in the monomerdroplets or soap micelles like at suspensionpolymerisation , butcorrectly in the surrounding water-phase, for example: potassium persulphate.Smith and Ewart presented the next mechanism. The monomermolecules make diffusion from the monomerdroplets by meansof the water phase to the micelles, in which them become soluble and become surround with the apolar tails of the soapmolecules. The initiator is only present at the water-phase and produce there approx. 1013 radicals per seconds per liter. Theseradicals initiate polymerisation in the water phase with monomer, and probably thereby oligomere radicals with an averagechain length of 2-5 and those diffusion to a micelle.The oligomere radical enter in to a micelle and react with in this the solubilized monomermolecules and form a long chain.Because the total surface of micelles which containing monomers are much larger that total surface of the will be micellesmainly hit with oligomere radical. Per a micelle grows one chain at the same time and this propagation as long there come anext radical from the water phase and direct occur termination by combination. Polymerisation again occurs in such a micellewhen there came a new radical, etc. The periods between polymerisation and no polymerisation in a micelle have the order of10 s. A polymer micelle is also called latex droplets and these particles will grow as long as the monomer in the micelle will becompleted with new monomer, which diffuse throughout the water phase, from the monomerdroplets. 2
  4. 4. The Kinetic of Emulsion PolymerisationA typical emulsion polymerisation can be subdivided at three stages, based on the particles number per volume, on present ofseparate monomerphase and the rate of polymerisation.Separate monomerphase can be found in the stage I and II but she has disappeared in the stage III. The num-ber of latexparticles increases in the time during phase I and remains constant in the phase II and stage III.In the stage I the system undergoes large changes. The number of particle, in which is present polymer, increase and afterwardsstabilizes.The particles (micelles) also grow themselves and thereby them need more and more soap molecules for the itself stabilize.These are taked from the solution. As a result, the concentration soap molecules in the solution also decreases and comes at agiven moment below CMC, which manifests in an increase of the surface tension of the water.The polymerisation rate in the stage II is constant because the monomerconcentration in the micelles are constant and inbalance with the monomerconcentration in the water and with the pure monomerdroplets. During this phase micelles becometherefore larger and the monomerdroplets smaller. Phase II finishes if all monomerdroplets are disappeared and there arepresent only still monomers in water and in micelles. Then polymerisation rate decrease as a result of decreasingmonomerconcentration in both.Some emulsion polymerizations show a gel-effect at the end of phase II and the end of phase III.The overall rate of monomer consumption in the stage II with a constant concentration of latex particles N and a constantmonomer concentration [M.] in latex particles can be expressed by: = k p [M ]∑ [M i ⋅] d[M]- dtThe concentration growing chain ends ∑ [M i ⋅] are defined by the number of micelles. If we adopt that all micelles containsonly one radical, then it is clear that this radical terminate when one another radical comes in micelle. For this reasonpolymerisation in the micelle occur in half cases. In a latex particle polymerisation goes further when a next radical comes inthe particle. From that follows that the concentration growing chain ends:∑[M ⋅] = 2 N And for the overall rate of monomer consumption: - d[M] = k p [ M ] N i dt 2So, during the phase II the overall rate of polymerisation is constant and independent of the rate initiation and monomerconversion. However the rate increases with the soap concentration because N will be higherThe rate of propagation in an active latex particle is equal to k p [M ] , and the lifetime of a growing chain is the reciprocal ofthe frequency with which radicals enter in a latex particle. If the rate of radical production per unit of time second and peremulsion volume is given by ρ, and all latex particles have equal probability to be hit by a radical, then the average lifetime ofchain growth will be equal to N/ρ and thus the number average degree polymerisation is: NP n = k p [M] ρSo, both the overall rate and the number average degree of polymerisation increase with N and thus with the soap concentration.A highly idealized schematic representation of an emulsion polymerisation is shown in next figure s(a,b,c,d,e,f): M icelle M icelle M M M M R PPPP M M M M M M M M M M M M M M M M M M R● M M M M M M M M M M M M M M M M M M R● M M M M M o n o m eM r M o n o m eM r d r o p le t M M d r o p le t M Ma) prior to initiation contains: b) polymerisation phase I contains:1)external water phase -1-6 as in ( a)2)low conc. of molecularly dissolved monomer molecules(M) -latex particles containing polymer(P) swollen with monomer3)molecularly dissolved emulsifier -initiator free radicals(R●)4)emulsifier micelles (lasts to about 10-20%conversion)5)monomer droplets dispersed6)emulsifier micelles containing solubilized monomer 3
  5. 5. The Kinetic of Emulsion Polymerisation M PM M P P PPPPPM P P P M PM P P P M P P P P P P M P MMP P P P MP P M P P P MMP M PPPM M PM PM P M PPPMR● M PM R● R● M M M R● M M M M omom PP d r o p le t P PP PM P P Mc) polymerisation phase II contains: d) polymerisation phase III-external water phase -external water phase-low conc. of molecularly dissolved monomer -no dissolved monomer-no dissolved emulsifier -no dissolved emulsifier-no emulsifier micelles -no emulsifier micelles-fewer monomer droplets, smaller surface area -no monomer droplets-no micelles swollen with monomer -no monomer-swollen micelles-growing latex particles -latex particles with depleting monomer concentration-initiator free radicals -initiator free radicals(lasts to about 20-60% conversion) (lasts to about 60-70% conversion) I II III PPP X PPPPP There are monomer PPPPPP droplets PPPPP There are no monomer PPP droplets N PPP PPPPP PPPPPP PPPPP PPP dX/dt Timee)end of polymerisation contains: f)Clasical qualitative desorption ofphases (intervals) I, II and III as-only by emulsifier(latex particles) a function of the evolution of monomer conversion (X), rate of dispersed through aqueous phase polymerisation (dX/dt) and numer of particles (N) [ 3] (100%conversion)It is essential that the simple qualitative representation in figures above of the mechanism of an “ideal”emulsion polymerisation be understood before the kinetics can be discussed or many deviations understood.These deviations can be caused by the following:-monomer solubility,-organic solubility of initiator andinitiator types,-emulsifier types,-inhibitors and schortstops.,retarders,-modificers,-various secundaryingredients, eg.stabilizers,electrolites,-phase ratio,-incremental addition of emulsifier,initiator or modifier,-particle size control-tapering,-seeding. 4
  6. 6. The Kinetic of Emulsion Polymerisation2 Emulsion Polymerization KineticsThere are four main types of liquid-phase heterogeneous free-radical polymerization:1.microemulsion polymerization,2.emulsion polymerization, 3.miniemulsion polymerization and 4.dispersionpolymerization. Emulsion polymerization is sometimes called macroemulsion polymerization.It is well known that microemulsion, miniemulsion and dispersion polymerizations bare many similarities toemulsion polymerization in the kinetics of particle nucleation and growth and in polymer structuredevelopment. For optimal design and operation of these heterophase free radical polymerizations, it isimportant to have detailed knowledge of the kinetics and mechanisms of emulsion polymerization.2.1 Generally Accepted Kinetics Scheme (particle formation and particle growth) [3a] [3b] [3c]Particle FormationThree major mechanisms for particle formation have been proposed to date.Figure 1a shows the proposed scheme for particle formation in emulsion polymerization initiated by water-soluble initiators.Particle formation occur by the following mechanisms:(1) a free radical in the aqueousphase enters a monomer-swollen emulsifier micelle and propagation proceeds therein( micellar nucleation),(2) the chain length of a free radical growing in the aqueous phase exceeds its solubility limit and precipitatesto form a particle nucleus (homogeneous nucleation),(3) a combination of these: homogeneous-coagulativenucleation and(4)coagulative nucleation ((5) a free radical growing in the aqueous phase enters a monomerdroplet and propagation proceeds therein (droplet nucleation)).A scheme of these mechanisms is shown in Fig. 1.The micellar nucleation mechanism consists of the diffusion of radicals from the aqueous phase to the innerpart of monomer-swollen micelles and subsequent polymerization to form a polymer-latex particle (see Fig.1(a)). The homogeneous nucleation mechanism occurs by precipitation of propagating radicals in the aqueousphase, when a critical degree of polymerization is reached (see Fig. 1(b)).The homogeneous-coagulative nucleation mechanism (see Fig.1(c)) consists of the coagulation of the so-called primary or precursor particles formed by homogeneous nucleation. Coagulative nucleation (see Fig.1(d)], includes coagulation of primary particles formed by both micellar and homogeneous nucleation. Therole of the micelles in the homogeneous and homogeneous-coagulative mechanisms is only to provide therequired surfactant to stabilize the polymer-latex particle, whereas in the micellar and coagulativemechanisms, besides this function, micelles act as nucleation loci. In the micellar and homogeneousmechanisms, the particles grow only by propagation of radicals and by the swelling of the polymer formed.In the homogeneous-coagulative and coagulative mechanisms, the particles grow mainly by “limitedcoagulation.” The argument of the coagulative theories is that in the primary particles, the lipophilic phase isincompletely formed, compared with the mature latex particles. There is no clear segregation between thehydrophilic and lipophilic phases so that there are insufficient hydrophilic groups to form a polar “shell”enclosing a non-polar phase. Accordingly, primary particles are unstable and the concentration of monomerin the polymer particles (and therefore their rate of growth) is low. Coagulation of primary particles leads tomature particles with higher surface charge density. The interfacial area of a mature particle is smaller thanthe total interfacial area of the primary particles from which the former was formed, thus, surface chargedensity (and therefore stability) increases. Coagulation occurs, or is limited to, the moment at which polymerparticles are stable and their growth is mainly propagative. For this reason, the process described is known as“limited coagulation.”However, if the resultant polymer particles are not stable enough, the final number of polymer particlesproduced, regardless of the mechanism of particle formation, is determined by coagulation between theexisting particles (coagulative nucleation).In the process of particle growth, various chemical and physical events occur in both the aqueous and particlephases, as illustrated in Fig. 1b [1][ 4].We now know that the polymerization takes place exclusively in theresultant polymer particle phase, wherever the free radicals are generated. Smith and Ewart [5] were the firstto establish a quantitative description of the processes of parti-cle formation and growth in emulsionpolymerization on the basis of the achievements made by Harkins et al. [6]. This is now called the Smith-Ewart theory. It is not an exaggeration to say that almost all of the theoretical developments in emulsionpolymerization that have been made so far are based on the Smith-Ewart theory. 5
  7. 7. The Kinetic of Emulsion Polymerisation Aqueous phase termination Radical Formation +M + mM ¯SO4 M ¯SO4 M* M ¯SO4 M* .......... R* (radicals) Particle Formation a) Micellar nucleation Propagation Primary(precursor) polymer particle R* (radical) (colloidally stable) Monomer swollen micelle b) Homogeneous nucleation M(monomer) Primary(precursor) polymer particle R* (radical) (colloidally stable above CMC) (Propagation in the aqueous phase and adsorption of surfactant) c) Homogeneous-coagulative nucleation M(monomer) R* “Slow”propagation (not significant) Radicals M(monomer) coagulation R* (Propagation in the aqueous phase and adsorption of surfactant; “Mature” polymer particle deficient adsorption of Primary (precursor) (colloidaly stabile) surfactant)) polymer particles (colloidally unstables) d) Coagulative nucleation R* “Slow”propagation Swollen micelle (not significant) Radicals coagulation R* (Propagation in the aqueous phase and adsorption of surfactant; “Mature” polymer particle deficient adsorption of Primary (precursor) (colloidaly stabile) surfactant)) polymer particles (colloidally unstables) e) Droplet nucleation R* (radical) Monomer dropletFig.1. Several nucleation mechanism reported in the literature:(a) micellar, (b) homogeneous, (c) homogeneous- coagulative, (d) coagulative and (e) droplet 6
  8. 8. The Kinetic of Emulsion Polymerisation Particle Growth initiator I* aqueous-phase propagation aqueous-phase termination R* entry propagation termination transfer propagation + M* oligomer exit aqueous-phase terminaion M* re-entry propagation re-escape M* terminationFig.2 Various chemical and physical events that occur during the process of particle growth in a emulsionpolymerisation. 7
  9. 9. The Kinetic of Emulsion Polymerisation2.2 Summary of the Smith-Ewart TheoryThe Smith and Ewart theory (the S-E theory) describes the basic concept of emulsion polymerization. Itsmain points are briefly reviewed here. Smith and Ewart showed that the rate of emulsion polymerization,which proceeds exclusively in the polymer particles, is given by RP - rate of emulsion polymerisation [Μ P ] - monomer concentration in the monomer-swollenR P = k P [Μ P ] n N T NT - number of monomerswollen polymer particles per unit volume of water kP - propagation rate constant n - average number of radicals per particlen defined as: n - free radicals ∝ ∝ ∝n = ∑ nN n ∑N =∑ n nN n NT Nn - number of polymer particles containing n free radicals n =1 0 n −1N n is described by the following balance equation that takes into account three rate processes: (1) radicalentry into, (2) radical desorption (exit) from, and (3) bimolecular radical termination inside the polymerparticle: [ ] [ ]dN n / dt = (ρ e / N T )N n −1 + k f (n + 1)N n +1 + k tp (n + 2)(n + 1) / v p − N n {ρ e / N T + k f n + k tp (n (n − 1) / v p ) } = 0 kf - rate coefficient for radical desorption per particle vp - volume of a polymer particle k tp - rate coefficient for bimolecular radical termination inside the polymer particles ρe - the overall rate of radical entry into polymer particlesρ e defined by: ρw - rate of emulsion polymerisation k tw -rate coefficient for bimolecular radical termination in the aqueous phaseρ e = ρ w + k f n N T − 2k tw R ∗ w [ ] [R ] 2 ∗ w - radical concentration in the aqueous phaseOn the other hand, they derived an expression that predicts the number of polymer particles produced, N T ,assuming that (1) a monomer-swollen emulsifier micelle is transformed into a polymer particle by capturing afree radical from the aqueous phase, (2) the volumetric growth rate per particle m is constant, at least duringparticle formation, and (3) free radical activity does not transfer out of a growing particle k - constant between 0.37 and 0.53 a S - surface area occupeied by a unit amount of emulsifierN T = k (ρ w / µ ) (a SS 0 )0.6 0.4 N T - number of polymer particles produced S 0 - initial emulsifier concentration(conc. of emulsifier forming micelles) ρ w - rate of radical generation per unit volume of waterρ w is given by: k d - rate constant for initiator deconposition f - initiator efficiencyρ w = 2k d f [I 0 ] [I 0 ] - initial initiator concentrationSince the appearance of the S-E theory, much effort has been directed into investigating the physicalmeanings of various parameters such as ρ w , k f and k tp , and the effects of these parameters on the three keyfactors of emulsion polymerization, [Μ P ], n and N T . 8
  10. 10. The Kinetic of Emulsion Polymerisation3 Kinetics and Mechanisms of Emulsion Polymerization3.1 Radical EntryOne of the most important parameters in the S-E theory is the rate coefficient for radical entry. When a water-soluble initiator such as potassium persulfate (KPS) is used in emulsion polymerization, the initiating freeradicals are generated entirely in the aqueous phase. Since the polymerization proceeds exclusively inside thepolymer particles, the free radical activity must be transferred from the aqueous phase into the interiors of thepolymer particles, which are the major loci of polymerization. Radical entry is defined as the transfer of freeradical activity from the aqueous phase into the interiors of the polymer particles, whatever the mechanism is.It is believed that the radical entry event consists of several chemical and physical steps. In order for aninitiator-derived radical to enter a particle, it must first become hydrophobic by the addition of severalmonomer units in the aqueous phase. The hydrophobic oligomer radical produced in this way arrives at thesurface of a polymer particle by molecular diffusion. It can then diffuse (enter) into the polymer particle, orits radical activity can be transferred into the polymer particle via a propagation reaction at its penetratedactive site with monomer in the particle surface layer, while it stays adsorbed on the particle surface. Anumber of entry models have been proposed: (1) the surfactant displacement model; (2) the collisionalmodel; (3) the diffusion-controlled model; (4) the colloidal entry model, and; (5) the propagation-controlledmodel. The dependence of each entry model on particle diameter is shown in Table1. [7].Table1. Dependence of entry rate coefficient on particle diameter, as predicted by different models Entry model Dependence on d p 8 Surfactant displacement model [ ] none Collisional model [ 9] d2 p Diffusional-controlled model [ 10] dp 11 Colloidal entry model [ ] dp 12 13 Propagational-controlled model [ , ] no dependenceHowever, some of these models have been refuted, and two major entry models are currently widelyaccepted,however, Liotta et al. [14] claim that the collision entry is more probable. One is the diffusion-controlled model, which assumes that the diffusion of radicals from the bulk phase to the surface of apolymer particle is the rate-controlling step.The other is the propagationcontrolled model, which assumes thatsince only z-mer radicals can enter the polymer particles very rapidly, the generation of z-mer radicals from(z–1)-mer radicals by a propagation reaction in the aqueous phase is the rate-controlling step.3.2.1 Diffusion-Controlled EntrySmith and Ewart [5] first proposed: that the transfer of free radical activity into the interior of a polymerparticle takes place by the direct entry of a free radical into a polymer particle. They pointed out that the rateof radical entry into a polymer particle is given by the rate of diffusion of free radicals from an infinite ∗medium of concentration [R w ] into a particle of diameter d p with zero radical concentration. [R ∗ ] w - free radicals concentration from an infinite medium ρe / N T - rate of radical entry into a polymer particle ∗ (1) 2 π D w d p [R ] - rate of diffusion of free radicals w dp - diameter of a particle with zero radical concentration ∗ ∗ρ e / N T = 2 π D w d p [R ] = k ep [R ] w w NT - number of polymer particles produced D w - diffusion coefficient for the radicals in the water phase k ep - mass transfer coefficient for radical entry into a particleHowever, for simplicity, they actually used a rate coefficient that is proportional to the square of the diameter(the surface area). Since then, most researchers have treated the problem of particle formation by assuming 9
  11. 11. The Kinetic of Emulsion Polymerisationthat the rate of radical entry into a micelle and a polymer particle is proportional to the surface area (thecollisional entry model) [9, 15].On the other hand, Nomura and Harada [16] proposed a kinetic model for the emulsion polymerization ofstyrene (St), where they used Eq. 1 to predict the rate of radical entry into both polymer particles andmonomer-swollen micelles. In their kinetic model, the ratio of the mass-transfer coefficient for radical entryinto a polymer particle k ep to that into a micelle k em , k ep / k em , was the only one unknown parameter (Eq. 5).They determined the value of k ep / k em to be about 10 3 by comparing the model’s predictions withexperimental results. However, the observed value of kep/kem was at least two orders of magnitud greaterthan that predicted by Eq. 1, because kep/kem=dp/dm (dm is the diameter of a micelle) according to Eq. 1 andthe value of dp/dm would be 10 at the most during particle formation. This was considered to indicate that theradical capture efficiency of a micelle is a factor of about 100 less than that of a particle. Taking this intoconsideration, they implicitly introduced a concept called the “radical capture efficiency of a micelle relativeto a polymer particle” to adjust for this disagreement and pointed out two possible reasons for the lowerradical capture efficiency of a micelle. One is that the energy barrier against the entry of charged radicals intomicelles may be higher than that into polymer particles. The other is that an oligomeric radical, havingentered a micelle, may pass through the micelle without adding at least one extra monomer unit because thevolume of the micelle is so small that the mean residence time of the radical in the micelle is too short for theradical to add another unit.The concept of “radical capture efficiency” was further elaborated on by Hansen et al.[17,18,19]. Byapplying the theory of mass transfer with simultaneous chemical reactions, they proposed the followingexpression to represent the net rate of radical absorption by a particle, introducing an “absorption efficiencyfactor” F.ρ e / N T = 2 π D w d p [R ∗ ] F = k ep [R ∗ ] w wF represents a factor that describes the degree to which absorption is lowered compared to irreversiblediffusion, and is given by: λ -equilibrium partition coefficient between particles and water for radicals1  Dw  = (X coth X − 1)−1 + W r X = (d p / 2){(k p [M] p + n k tp / v p ) / D p } 1/ 2 λD F  p  W r - potential energy barrier analogous to Fuchs’ stability factor D p - diffusion coefficient for the radicals inside a particle k p - propagation rate constant n - number of radicals in a particle [M] p - monomer concentration in particles v p - particle volumeIt should be noted that radicals are captured inside the particles only if they react therein; otherwise they willeventually diffuse out and back to the water phase.Value of F for a particle containing radicals is higher than that for a particle containing no radicals.A much simpler model for probability F (the radical capture (absorption) efficiency), can be derived byintroducing the concept of radical desorption from a polymer particle, developed in Section 2.3.1. Theprobability F for a radical to be captured inside a particle containing n radicals by any chemical reaction(propagation or termination) is given by: F - probability for a radical to be captured inside a particle (2) n - radicals inside a particle K 0 - overall radical desorption rate constant for a particle k p [M] p + k tp (n / v p ) F= K 0 + k p [M] p + k tp (n / v p ) d p - diameter of a particle with zero radical concentration N T - number of polymer particles produced 10
  12. 12. The Kinetic of Emulsion Polymerisation D w - diffusion coefficient for the radicals in the water phase k ep - mass transfer coefficient for radical entry into a particleIn the case where K 0 〉〉 k p [M ] p , k tp (n / v p ) , substitution of Eq. 2*(secion2.3.1) into Eq. 2 leads to:F = k p [M] p / K 0 ∝ d 2 pUnzueta et al. [143] derived a kinetic model for the emulsion copolymerization of methyl methacrylate(MMA) and butyl acrylate (BA) employing both the micellar and homogeneous nucleation mechanisms andintroducing the radical absorption efficiency factor for micelles, Fm , and that for particles, Fp . Theycompared experimental results with model predictions, where they employed the values of Fp = 10 −4 andFm = 10 −5 , respectively, as adjustable parameters.3.2.2 Propagation-Controlled EntryMaxwell et al. [13] proposed a radical entry model for the initiator-derived radicals on the basis of thefollowing scheme and assumptions. The major assumptions made in this model are as follows: An aqueous-phase free radical will irreversibly enter a polymer particle only when it adds a critical number z of monomerunits. The entrance rate is so rapid that the z-mer radicals can survive the termination reaction with any otherfree radicals in the aqueous phase, and so the generation of z-mer radicals from (z–1)-mer radicals by thepropagation reaction is the rate-controlling step for radical entry. Therefore, based on the generation rate of zmer radicals from (z–1)-mer radicals by propagation reaction in the aqueous phase, they considered that theradical entry rate per polymer particle, ρ(ρ = ρ e / N T ) is given by ρ(ρ = ρ e / N T ) - radical entry rate per polymer particle ∗ρ = k pw [I M z −1 ][M] w N T z - critical number z of monomer units(critical chain length of entry radicals) k pw - propagation rate constant in the aqueous phase [M] w - monomer concentration in the aqueous phase ∗By substituting the steadystate concentration of (z–1)-mer radicals [IM z −1 ] into eqation above, theapproximate expressions for ρ and the initiator efficiency, f entry are derived, respectively, as: 1− z 1− z 2 k [I]  k d [I]k t , w    2 k [ I]  k d [I]k t , w    ρ= d  + 1 = d f entry f entry = + 1 N p  k p, w [M ]w ;    Np  k p , w [M] w   Maxwell et al.[13] proposed a semi-empirical thermodynamic model to predict the value of z forpersulfatederived oligomeric radicals, which is given by: (z ≅ 1 + int − 23 kJmol −1 /{RT ln[M sat ] w } )where the integer function (int) rounds down the quantity in parentheses to the nearest integer valueand [M sat ] w is the saturation solubility of the monomer in mol dm–3.On the other hand, Sundberg et al. [20]proposed a thermodynamic method for estimating the critical chain length z of entry radicals with ahydrophilic end group (such as SO4–) using a simple two-layer lattice model.Several research articles have been published that deal with the methodology for determining the radicalentry rate ρ , the initiator efficiency f entry and the actual values of z.Hawkett et al. [21] developed a method for determining the value of ρ along with the desorption ratecoefficient k f , termed the slope-andintercept method.This method is experimentally simple,but has severaldrawbacks [22]. For example, it is only applicable to the so-called zero-one system (n–≤0.5) with negligibleradical termination in the aqueous phase. It is usually very difficult to judge whether or not the radicaltermination in the aqueous phase is negligible. Moreover, it gives a large error if an induction period causedby any trace of impurity exists. 11
  13. 13. The Kinetic of Emulsion PolymerisationMarestin et al. [23] proposed an experimental method for directly determining the entry rate of a critical sizeMMA oligomer into the polymer particle using the seeded emulsion polymerization of MMA initiated byKPS. The initial seed latex used was synthesized so as to have radical traps (TEMPO) covalently bound ontothe particle surface. When an aqueous phase-propagating radical entered a seed particle, the nitroxide moietyled to the formation of a stable alkoxyamine. Therefore, the kinetics of radical entry into the seed particleswas followed by monitoring the decay of the ESR signal from the nitroxide in the samples withdrawn fromthe reactor. They obtained f entry =0.36 for KPS at 70 °C and f entry =0.33 for V-50 at 70 °C, respectively.Maxwell et al. [13] obtained the value of z≈2 by comparing the model predictions with the experimentalresults in the emulsion polymerization ,ofSt. Schoonbrood et al. [24] reported z≈18 for a 80:20/St:MA emulsion copolymerization system.3.3 Radical Desorption (Exit)3.3.1 Desorption in Homopolymer SystemsThe desorption (exit) of free radicals from polymer particles into the aqueous phase is an important kineticprocess in emulsion polymerization. Smith and Ewart [5] included the desorption rate terms into the balanceequation for N n particles, defining the rate of radical desorption from the polymer particles containing n freeradicals in equation: [ ] { [( )]}dN n / dt = (ρ e / N T )N n −1 + k f (n + 1)N n +1 + k tp (n + 2)(n + 1) / v p − N n ρ e / N T + k f n + k tp n (n − 1) / v p = 0 ask f nN n . However, they did not give any detailed discussion on radical desorption. Ugelstad et al. [25]pointed out that the rate coefficient for radical desorption (the desorption rate coefficient) could be a functionof particle size, the rate of chain transfer to monomer, the rate of polymerization and the diffusioncoefficients involved in the transport processes leading to desorption of radicals.On the other hand, Nomura and Harada [14,26,27,28,29] pointed out that radical desorption from thepolymer particles and micelles plays a decisive role in particle formation and growth, and further that thereare many examples of kinetic deviations from the S-E theory that are attributable to radical desorption. First,they theoretically derived the desorption rate coefficient from both stochastic [26] and deterministicapproaches, [14,26,27] based on a scheme consisting of the following three consecutive steps: (1) chain-transfer of a polymeric radical to a monomer molecule or a species like CTA (chain transfer agent) in apolymer particle, followed by (2) diffusional transportation of the resulting low molecular weight radical tothe particle-water interface, and (3) successive diffusion into the bulk water phase through a stagnant filmadjacent to the surface of the particle. In modeling the rate coefficient for radical desorption, the followingassumptions were made:1. Polymer particles contain at most one radical (zero-one system).2. An oligomeric radical with no more than s monomer units can desorb from and re-enter into a polymer particle with the same rate, irrespective of its chain length.3. Instantaneous termination takes place when another radical enters a particle already containing a radical. 4. No distinction is made between radicals with or without an initiator fragment on its end.5. Water-phase reactions such as propagation, termination, and chain-transfer to a monomer are negligible for the desorbed radicals. This means that all of the desorbed radicals would re-enter particles and the loss of these radicals occurs only through the event given by the assumption A3).6. The physical and chemical properties of chain transfer agent (CTA) radicals are approximately equal to those of monomer radicals. Based on these assumptions, they derived the desorption rate coefficient k f as: ρ w (1 − n ) ρ w (1 − n )k i j    k mf k Tf [T] p  S  k p [M] p k f = k oI   + Ko  + +  ⋅ ∑    K oI n + k i [M] p nN p     k p k p [M] p K oI n + k i [M] p k p nN p  j=1  K o n + k p [M] p    When it is assumed that initiator-derived radicals do not exit and only monomeric and CTA radicals producedby chain transfer to a monomer and/or a CTA can desorb (s=1), equation above can be simplified as:  k p [M ]p   k mf k Tf [T ]p kf = Ko  +   K n + k [M ]p   k k p [M ]p  (1*)  o p  p  12
  14. 14. The Kinetic of Emulsion Polymerisationwhere k mf and k Tf are the chain transfer rate constants to monomer and to CTA, respectively, and K o is theoverall desorption rate constant per particle for monomeric (or CTA) radicals,which is approximately givenby [26,27,28 ]: −1  2 π D w d p   ΨD w   Ko =   1 +  =  12 D w δ  (2*)  m v  m D   m d2   d p  d p   d p  δ = (1 + ΨD w / m d D p ) - denotes the ratio of the aqueous- side to overall diffusion resistance −1 Ψ - a numerical constant between 1 and 6 that depends on the mass-transfer coefficient employed m d - partition coefficient for monomer radicals between the polymer particle and aqueous phases, defined by: m d = [M ] p /[M ] w ΨD w / m d D p - is the ratio of the particle-side to water-side diffusion resistanceWhen the diffusion resistance inside the particle is far greater than that in the aqueous-side effective diffusionfilm, that is, D w / m d D p 〉〉 1 , equation above gets: −1  2 π D w d p   ΨD w   Ko =   1 +  =  12 D w   m v  m D   d2   d p  d p   p Equation (1*) was derived under the assumption that the physical and chemical properties of a CTA radicalare approximately equal to those of a monomeric radical. However, if it is necessary to take into account thedifferences in the physical and chemical properties between monomeric and CTA radicals, equation 1* canbe modified approximately as:  k mf [M ]p   k Tf [T ]p k f = K om   + K ot    K n + k [M ]p   K n + k [M ]   om p   oT iT p where K om and K oT are the overall desorption rate constants per particle for monomeric and CTA radicals,respectively.Asua et al. [30,31,32] modified equation (1*) in the absence of a CTA to include more general cases, takinginto account the fate of the desorbed radicals (both chemical reactions in the water phase and re-entry) as:  Ko k f = k mf [M] p    K β + k [M ]p   o p where β stands for the fraction of desorbed radicals that cannot re-enter because of the aqueous phasetermination or propagation, and is given by: [T ∗ ] w - total radical concentration in the water phase k p [M] w + k tw [T] wβ= ka - mass-transfer coefficient for radical entry k p [M] w + k tw [T ∗ ] w + k a N T [M] w- monomer concentration in the water phaseCasey and Morrison et al. [33,34]derived the desorption rate coefficient for several limiting cases incombination with their radical entry model, which assumes that the aqueous phase propagation is theratecontrolling step for entry of initiator-derived free radicals. Kim et al. [35] also discussed the desorptionand re-entry processes after Asua et al. [30] and Maxwell et al. [13] and proposed some modifications. Fanget al. [36] discussed the behavior of free-radical transfer between the aqueous and particle phases (entry anddesorption) in the seeded emulsion polymerization of St using KPS as initiator. 13
  15. 15. The Kinetic of Emulsion Polymerisation3.3.2 Desorption in Copolymer SystemsNomura et al. [29,37] first derived the rate coefficient for radical desorption in an emulsion copolymerizationsystem by extending the approach developed for emulsion homopolymerization under the same assumptionsas 1–6 given in Section 2.3.1. This methodology is now termed the “pseudo-homopolymerizationapproach”.According to this approach, the average rate coefficient for radical desorption, defined, forexample, in a binary emulsion copolymerization system with monomers A and B, k f , is given by: k fA - desorption rate coefficient for A-monomeric radicalsk f =k fA (n A / n t ) + k fB (n B / n t ) = k fA f A + k fB f B n t - average number of total radicals per particle n A - average number of A-radicals per particle( n = n A + n B ) f A - fraction of A-radicals in the article phase and is ,expressed, at steady-state, as a function of the propagation rate constant k p , the monomer reactivity ratio r ,and the monomer concentration in the polymer particles [M ] p , in the following form: n = n A + n BIn the case where all the desorbed A-monomeric radicals reenter the polymer articles, the desorption ratecoefficient for A-monomeric radicals k fA is given by: C mBA -chain transfer constant for a B-radical to A-monomer  C mAA rA [M A ]P + C mBA [M B ]P k fA = K oA    rB {(K oA n t / k pAA ) + [M B ]P } + [M A ]P  K oA -overall desorption rate constant per particle for A-   monomeric radicals given by equation 2 in Section 3.2.1Lopez et al.[38] investigated the kinetics of the seeded emulsion copolymerization of St and BA inexperiments where the diameter and number of seed particles, and the concentration of initiator were widelyvaried. The desorption rate coefficient for the A-monomeric radical that they used is given by:   k fA f A = (k mf ,AA f A + k mf ,BA f B )[M A ]P  K oA   β A K oA + k ,pAA [M A ]p + k ,pAB [M B ]   p ;where k mf , AB denotes the chain transfer constant of the A-radical to the B-monomer and β is the fraction ofthe desorbed A-monomeric radicals that cannot reenter the polymer particles because of the aqueous phasetermination or propagation.Barudio et al. [39] developed a simulation model for emulsion copolymerization based on the pseudo-homopolymerization approach, where they used the average rate coefficient for radical desorption given by : ( )(k de = 12D w z / m d d 2 k mf / k p p )Saldivar et al. have presented a survey of emulsion copolymerization models that have been published in theliterature, and a comprehensive mathematical model for emulsion copolymerization [40], along with itsexperimental verification [41,42].They present a detailed discussion on the average rate coefficient forradical desorption,which is applicable to a multimonomer system.Vega et al.[43] modeled the batch emulsion copolymerization of AN and Bu in order to simulate an industrialprocess and improve the final polymer quality. The mathematical model they used was an extended versionof that developed by Guliotta et al. [44] for the continuous emulsion polymerization of St and Bu.Barandiaran et al. [45] proposed a method to estimate the rate coefficient for radical desorption in emulsioncopolymerization and gave the values of this parameter for the MA-VAc and MMA-BA emulsioncopolymerization systems. 14
  16. 16. The Kinetic of Emulsion Polymerisation3.4 Particle Formation and Growth3.4.1 Particle FormationAs we mentioned in Section 2.1 (Fig.1), there are three major models for particle formation in emulsionpolymerization.According to these models, polymer particles are formed:1. When a free radical in the aqueous phase enters a monomer-swollen emulsifier micelle and polymerization proceeds therein (micellar nucleation).2. When the chain length of a free radical growing in the aqueous phase exceeds its solubility limit and precipitates to form a particle nucleus (homogeneous nucleation).3. When a free radical growing in the aqueous phase enters a monomer droplet and polymerization proceeds therein (droplet nucleation).However,when the resultant polymer particles become unstable and coagulate,then whatever the mechanism of particle formation is, the final number of polymer particles produced isdetermined by a limited coagulation between existing polymer particles (coagulative nucleation).Smith and Ewart [5 ] derived an expression that can predict the number of polymer particles produced, byassuming that:1. A monomer-swollen emulsifier micelle is transformed into a polymer particle by capturing a free radical from the aqueous phase [5,6].2. The volumetric growth rate per particle µ is constant, at least during particle formation (µ = dv p / dt = const.) .3. Free radical activity does not transfer out of a growing particle (k i ≅ 0) .4. The amount of emulsifier that dissolves in the water phase without forming micelles and adsorbs on the surface of emulsified monomer droplets may be neglected.Based on these assumptions, two limiting cases were discussed:Case A: The rate of radical entry into micelles that results in the formation of new particles is approximatelyequal to the rate of radical generation in the water phase (ρ w ) , as long as emulsifier micelles are present:dN T = (ρ w ) dtParticle formation stops at the time t c , when the emulsifier micelles have just disappeared because all of theemulsifier molecules comprising the emulsifier micelles have been transferred to the surfaces of growingpolymer particles for adsorption. The volume v p, c at time t c of a particle formed at time τ is v p , c = µ (t c − τ) , [and so the surface area a P , C of this particle at time t c is given by a P , C = σ (t c − τ) where σ = (4π) 3µ 1/ 2 ] 2/3 .Thetotal surface area A P , C of all the polymer particles present at time t c is given by: tcA p , c = ∫ σ (t c − τ ) ρ w dt = 3 / 5σ ρ w t 5 / 3 2/3 0No micelles exist ( A m =0) at time t c , and so all of the charged emulsifier molecules are adsorbed onto thesurfaces of polymer particles present. Therefore, it holds that A p , c = 3 / 5σ ρ w t 5 / 3 = a sSo . In this case, thenumber of polymer particles produced ( N T ) can be obtained by substituting t c into N T = ρ w t c as:N T = 0.53(ρ w / μ )0.4 (a s S o )0.6where A m and A p are the total surface area of the micelles and the total surface area of the polymer particlesper unit volume of water, respectively, a s is the surface area occupied by a unit amount of emulsifier, andS o is the amount of initially charged emulsifier per unit volume of water (the initial emulsifierconcentration).Case B: Radicals enter both micelles and polymer particles at rates that are proportional to their surface areas(collision theory), so that the rate of new particle formation is given by: 15
  17. 17. The Kinetic of Emulsion Polymerisation A ρ m w = 0.37(υ w / µ ) (a s S o )0.6 0.4dN / dt = ρ = ;It follows that N T w A +A 1 + A /A T m p p mOn the other hand, Nomura et al. [16]proposed a different approach for predicting the number of polymerparticles produced, where the new concept of “radical capture efficiency” of a micelle relative to a polymerparticle was proposed. The assumptions employed were almost the same as those of Smith and Ewart, exceptthat the volumetric growth rate m of a polymer particle was not considered to be constant. It was alsoassumed that all of the radicals formed in the aqueous phase enter either micelles or polymer particles withnegligible termination in the aqueous phase. The following equations, describing the balance of radicals inthe aqueous phase and the rate of particle formation, were obtained: N T -the total number of polymer particles produced ( N T = N ∗ + N 0 )d[R ∗ ] ∗ w = ρ w − k em m s [R ∗ ] − k ep N T [R ∗ ] (3) w w ( N -number of active particles containing a radical; N 0 - number of dt dead particles containing no radical)dN T = k em m s [R ∗ ] w (4) [R ∗ ] - concentration of free radicals in the aqueous phase w dt k em - rate constant for radical entry into micelles k ep - rate constant for radical entry into particles ∗Introducing the aqueous phase concentration [R w ] , obtained by applying the steady state assumption toequation 3 into equation. 4, and rearranging leads to:dN T ρw ρw = k em m s [R ∗ ] = = 1 + (k ep N T / k em m s ) 1 + (εN T / S m ) w (5) dtwhere k ep N T / k em m s denotes the ratio of the rate of radical entry into polymer particles to that intomicelles and is rewritten as εN T / S m ,wher ε = (k ep / k em )M m and ε is the one unknown parameter, whichaffects the number of polymer particles produced. S m is the total number of emulsifier molecules formingmicelles, and M m is the aggregation number of emulsifier -molecules per micelle( M m = S m / m s ). Bysolving a set of simultaneous differential equations can be predicted with respect to the initial emulsifier ( S o )and initiator concentrations ( I o ) (or ρ w = 2k d f [I o ] ) as shown by N T ∝ ρ 0.3 S 0.7 . w 0Particle formation below the critical micellar concentration (CMC) in emulsion polymerization is nowaccepted to take place according to the homogeneous nucleation mechanism. Among several quantitativetreatments of homogeneous particle formation in emulsion polymerization, the best-known model was thatproposed by Fitch and co-workers [46]. Their model is basedon the assumption that when the chain length ofa free radical growing in the aqueous phase reaches its solubility limit (critical chain length), it precipitates toform a primary particle, and that particle formation will be hindered if these growing oligomers are absorbedin polymer particles formed earlier. Hansen [47] made significant improvements on the Fitch model [theHUFT (Hansen-Ugelstad-Fitch-Tsai) model]. According to Hansen et al, the rate of particle formation isgiven by: = k pw M w (R Ijcr + R Mjcr )dN T k pw - propagation rate constant in the aqueous phase dt M w - the monomer concentration in the aqueous phaseR Ijcr - concentrations of oligomer radicals(aqueous phase) with critical chain length derived from initiatorR Mjcr - concentrations of oligomer radicals(aqueous phase) with critical chain length derived from monomer radicalsThey, along with several assumptions, obtained: 16
  18. 18. The Kinetic of Emulsion PolymerisationdN T = ρ w (1 + k tw R w / k pw M w + k c N T / k pw M w ) − jIcr k c -average rate coefficient for radical entry into polymer particles dt R w -total radical concentration in the aqueous phase,and gets: j cr - critical degree of polymerization {[N T (t ) = k 1ρ w jcr t + (k 2 + 1) cr j ] 1 / jcr } − k 2 − 1 / k1Tauer et al. [48] developed a framework for modeling particle formation in emulsion polymerization on thebasis of a combination of classical nucleation theory with radical polymerization kinetics and the Flory-Huggins theory of polymer solutions.Usually particle formation by initiation in the monomer droplets (droplet nucleation) is not consideredimportant in conventional emulsion polymerization. This is because of the low absorption rate of radicals intothe monomer droplets, relative to the other particle formation rates.Although it is now accepted that particle formation below the CMC in emulsion polymerization takes placeaccording to the homogeneous nucleation mechanism, there has been debate as to whether homogeneousnucleation is still operative even above the CMC, especially when relatively water-soluble monomers arepolymerized in emulsion in the presence of emulsifier micelles.Therefore, the mechanism of particle formation is still anything but a settled question, even in the emulsionpolymerization of St.Only a few papers [49,50,51,52] have been published so far that discuss methodologies that could be used todiscriminate experimentally between micellar and homogeneous nucleations.Semibatch seeded emulsion polymerizations are quite common in industrial operations. One of the mostimportant problems in semibatch seeded emulsion polymerization is how to control secondary particleformation. It is well known that the amount of emulsifier must be carefully fed during starved-fed semibatchseeded emulsion polymerization. Too little emulsifier leads to emulsion instability and hencecoagulation,while too much emulsifier leads to secondary particle formation by the micellar mechanism.It has been reported that both the surface charge density and the degree of surface coverage by emulsifier onthe seed particles affect the behavior of secondary particle nucleation in seeded emulsion polymerizationbecause these factors control the rate of radical entry into seed particles.There are an enormous variety of commercial emulsifiers that are employed in emulsion polymerization.Emulsifiers are generally categorized into four major classes: anionic, cationic,nonionic and zwitterionic(amphoteric). The anionic and nonionic emulsifiers are the most widely used. In addition,mixtures ofemulsifiers are also often used. Since the effects of the molecular structure and chemical and physicalproperties of an emulsifier on particle formation are still far from being well understood,numerousexperimental investigations on particle formation have been carried out to date with various nonionicemulsifiers [53,54,55,56],mixed emulsifiers (ionic and nonionic emulsifiers) [143,57,58,59,60] and reactivesurfactants [61,62,63,64,65]. Recently, polymeric surfactants have become widely used and studied inemulsion polymerizations [66,67,68,69,70,71]. A general review of polymeric surfactants was published in1992 by Piirma [72].Recently, emulsion polymerization stabilized by nonionic and mixed (ionic andnonionic) emulsifiers was reviewed by Capek [73].Reactive surfactants have also been used in emulsion polymerization [61,62.63,64,65,69]. This is because thedisadvantages of the surfactants that are typically used in emulsion polymerization, such as instability of thelatex and surfactant migration during film formation, can be overcome in theory by using a reactivesurfactant.Recently, polymeric surfactants have received considerable attention in industry. They provide the stericrepulsion between interacting particles, which gives the latex excellent stability against high electrolyteconcentration, freeze-thaw cycling and high shear rates.3.4.2 Particle Growth in Homopolymer SystemsAs is clear from equation R P = k P [Μ P ] n N T ., the rate of particle growth ( R P / N T ) is proportional to themonomer concentration( M p ), and the average number of radicals per particle( n ). n is one of the basicparameters that characterize the kinetic behavior of particle growth in an emulsion polymerization system. 17
  19. 19. The Kinetic of Emulsion PolymerisationSmith and Ewart [5] solved it for three limiting cases at steady-state conditions, that is, dN n / dt = 0 .Case 1.The number of radicals per particle is smaller than unity.In this case, it holds that, ρ e / N T 〈〈 k f , (ρ e / N T ) N o ≅ k f N 1 , N o ≅ N T .a. When radical termination in the water phase is dominant; in other words: ρ e ≅ 2k tw [R ∗ ] 2 , w k tw - termination rate constant in the water phase then: n = (ρ w / 2k tw ) m d v p 〈〈 0.5 ρe 1/ 2 - rate of radical generation per unit volume of water [R ∗ ] - concentration of radicals in the water phase w m d - partition coefficient of radicals between the water and the polymer particle phasesb. When termination in the polymer particles is dominant ; n = (ρ w / 2k tw N T ) 〈〈 0.5 1/ 2 ( 2 2 2 ) The requirement for this condition is obtained as 4π D w d p N T / k f 〉〉 k tw from additional assumptions that ρ e ≅ 2πD w d p [R w ]N T and 2k tw [R w ]〈〈 2(ρ e / N T )N 1 ,where D w is the diffusion coefficient for the ∗ ∗ radicals in the water phase and d p the diameter of the particles.Case 2.The number of radicals per particle is approximately equal to 0.5. The requirements for this case are givenas:. k f 〈〈 ρ e / N T 〈 k tp / v p , then: n = 0.5Case 3.The number of radicals per particle is larger than unity. This situation will prevail when the average timeinterval between successive entries of radicals into a polymer particle is much smaller than the average timefor two radicals in the same particle to coexist without mutual termination; in other words:ρ e / N T 〉〉 k tp / v p ; n = (ρ e v p / 2k tp ) 〉〉 0.5 1/ 2Moreover, when both radical termination in the water phase and radical desorption from the particles are (negligible, n = ρ e v p / 2k tp )1/ 2 〉〉 0.5 is reduced to n = (ρ e v p / 2k tp N T ) 〉〉 0.5 1/ 2In this case, the kinetic behavior is quite similar to that of suspension polymerization, except that the polymerparticles are supplied with free radicals from the external water phase. When the polymerization proceeds (according to n = ρ e v p / 2k tp N T ) 1/ 2 〉〉 0.5 , the system is sometimes referred to as obeying “pseudo-bulk”kinetics.A general solution to equation [ ] [ ]dN n / dt = (ρ e / N T )N n −1 + k f (n + 1)N n +1 + k tp (n + 2)(n + 1) / v p − N n {ρ e / N T + k f n + k tp (n (n − 1) / v p ) } = 0 wasprovided by Stockmayer [74] with minor corrections by O’Toole [75]. On the other hand, Ugelstad et al.[76] proposed the most useful and widely applicable expression for n – given by: I m (a ) 2αn = (a / 4) = (1 / 2) I m −1 (a ) (5) 2α m+ 2α m +1+ m + 2 + ...I m (a ) - modified Bessel function of the first kind; m = k i v p / k tp ; α = a 2 / 8 = ρ e v p / k tp N T . [ ] ∗ 2The radical balance in the water phase ( ρ e = ρ w + k f n N T − 2k tw R w ) leads to the following relationshipusing the non-dimensional parameters: α , α w , m and Y. α = α w + mn − Yα 2 , where ; α w = ρ w v p / k tp N T and Y = 2k tw k tp / k a N T v p 2 18
  20. 20. The Kinetic of Emulsion PolymerisationNomura et al. [77] provided a semi-theoretical expression for n – corresponding to Y=0, and compared itwith the experimental data [78]: 1  αw  α w   αw    1 αw  2 1/ 2 1/ 2    1n =  α w +  + 2 α w +  −  α w +  + +  − 2  m   m    m   4 2  2  The values predicted by equation above agree well with those predicted by equation 5 within less than 4%.This type of plot is called a “Ugelstad plot”(comparison between predict and observed values of n ) and hasbeen applied as a criterion to determine whether a system under consideration obeys either zero-one kinetics( n ≤0.5) or pseudo-bulk kinetics ( n >0.5).Nomura et al. [26,27,79] showed that when the value of the term k tp / v p is very large (the rate ofbimolecular termination in the polymer particles is very rapid), n is expressed by: ( )n = − C + C + 2C / 2 ;where C = ρ w / k f N T . 2Hawkett et al. [21] developed a method for determining ρ and k f termed the slope-and-intercept method.Asua et al. [7,80] proposed a new approach for the estimation of kinetic parameters such as the entry anddesorption rate coefficients, the termination rate constant in the aqueous phase, the rate coefficient forinitiator decomposition and the propagation rate constant in emulsion homopolymerization systems underzero-one conditions.Recently, several modeling papers have been published which are useful for the design and operation ofemulsion homopolymerization processes [81,82,83,84,85,86]Mendoza et al. [84] developed a mathematical model that could predict the monomer conversion, particlediameter, number of polymer particles produced, and the number-average and weight-average molecularweights in the unseeded emulsion polymerization of St using n-dodecyl mercaptan as CTA.Kiparissides et al. [85] proposed a comprehensive mathematical model to quantify the effect of the oxygenconcentration on the polymerization rate and PSD in the unseeded emulsion polymerization of VCl.Asua et al. [86] developed a mathematical model for seeded emulsion polymerization stabilized withpolymerizable surfactants (surfmers).Herrera-Ordonez et al. [22,80,81] proposed a detailed mathematical model of the kinetics of St emulsionpolymerization. They applied the model to the emulsion polymerization of MMA above the CMC of theemulsifier to discuss the mechanism of particle formation and growth in this system.The emulsion polymerization of VAc is already one of the most studied systems, research articles on thistopic are still being published [87,88,89,90,91].Gilmore et al. [87,88] presented a mathematical model for particle formation and growth in the isothermalsemibatch emulsion polymerization of VAc stabilized with poly(vinyl alcohol) (PVA).Budhiall et al. [89] investigated the role of grafting in particle formation and growth during the emulsionpolymerization of VAc with partially hydrolyzed PVA as the emulsifier and KPS as the initiator. 40M.Nomura et al.Shaffie et al. [90] studied the kinetics of the emulsion polymerization of VAc initiated by redox initiationsystems of different persulfate cations such as KPS, sodium persulfate (NaPS), and ammonium persulfate(APS); each of them was coupled with a developed acetone sodium bisulfate adduct as the reducing agent.Chern et al. [92] also developed a model that includes particles containing at most two free radicals.Bruyn et al. [91] proposed a kinetic model that considers a zero-one system with instantaneous radicaltermination inside the particles. They [93] also studied the kinetics and mechanisms of the emulsionpolymerization of vinyl neo-decanete(VnD),a practically water-insoluble monomer at 50 °C using sodiumpersulfate (NaPS) as the initiator and SDS as the emulsifier. They explained the kinetic behavior of thissystem with the same mechanisms and model applied to the VAc system [91].3.4.3 Particle Growth in Copolymer SystemsBallard et al. [94]presented an extended S-E theory that provides a descriptionof the emulsioncopolymerization system during Interval II and III and suggested the possibility of using an “average” ratecoefficient to treat the copolymerization system. Nomura et al. [45,47,95] first developedan approach to generalize the S-E theory for emulsion homopolymerization to emulsion copolymerization byintroducing “average (or mean) rate coefficients” for propagation, termination and radical desorption. This 19
  21. 21. The Kinetic of Emulsion Polymerisationmethodology was termed the “pseudo-homopolymerization approach”[96]or the “pseudo-kinetic rateconstant method” [97] . Nomura et al. [47,122a,98] demonstrated that the equations derived so far foremulsion homopolymerization can also be applied without any modification to a binary emulsioncopolymerization system with monomers A and B by substituting the following “average rate coefficients”for the corresponding rate constants for emulsion homopolymerization.The polymerization rate for the A-monomer is expressed as:rpA = k pa [M A ]p n t N T [M A ]p - concentration of A-monomer in the polymer particles n t - average number of total radicals per particle n t = n A + n BThe overall rate of copolymerization is defined by: (R p = k p [M ]p n t N T = rpA + rpB = k pA [M A ]p + k pB [M B ]p n t N T ;)k p -is the overall propagation rate coefficient and is a function of the propagation rate constant, monomerreactivity ratio and mole fraction of each monomer in the polymer particles.In the case of a binary emulsion copolymerization system, for example, the average rate coefficients k p aredefined as:1. The average rate coefficient for the propagation of the A-monomer, k pA , is given: k pBB rA [M A ]p k pA = k pAA f A + (k pBB / rB )f B ; nA fA = = n t k pAA rB [M B ]p + k pBB rA [M A ]p k pBA -rate constant for the propagation of a B-radical to an A-monomer fA - fraction of A-radicals in the particle phase (f A + f B = 1) rB - B-monomer reactivity ratio.2. The average rate coefficient for radical desorption, k f , is defined using the equations:  C mAA rA [M A ]p + C mBA [M A ]p  k f = k fA (n A / n t ) + k fB (n B / n t ) = k fA f A + k fB f B ; k fA = K oA   { }  rB (K oA n t / k pAA ) + [M B ]p + [M ]A    k fA - desorption rate coefficient for the A-monomeric radicals C mBA - chain transfer constant of a B-radical to an A-monomer K oA - desorption rate constant for A-monomeric radicals3. The average rate coefficient for radical termination in the particle phase, k tp , is defined by: k tp = k tpAA f A + 2k tpAB f A f B + k tpBB f B 2 2k tpAB - bimolecular radical termination rate constant between A- and B-radicals.Giannetti [99]concluded that the pseudo-homopolymerization approach represents a suitable approximationfor most copolymerization systems of practical interest.Since the appearance of the pseudo-homopolymerization approach, a wide variety of mathematical modelshave been developed for emulsion copolymerization systems using this approach, in order to thoroughlyunderstand the mechanisms involved in particle formation, growth processes, and to predict thecopolymerization rate, the properties of the copolymer obtained (molecular weight and copolymercomposition), and colloidal characteristics (the particle number and PDS)[143,24,38,40,41,42,43,44,45,77,98,100,101,102,103,104].Nomura et al. [77] first proposed a kinetic model that introduced the pseudo-homopolymerization approach.Barandiaran et al. [63] also developed a mathematical model based on the pseudo-homopolymerizationapproach.Schoonbrood et al. [24]carried out a kinetic study of the seeded emulsion copolymerization of St with therelatively water-soluble monomer MA to investigate the mechanisms of radical entry into particles, radicaldesorption from particles, and the fate of radical species in the aqueous phase. 20
  22. 22. The Kinetic of Emulsion PolymerisationLopez et al. [38] used calorimetric measurements to study the kinetics of the seeded emulsioncopolymerization of St and BA.Martinet et al. [102] carried out the emulsion copolymerization of a-methyl styrene (aMSt) and MMA atvarious temperatures (60, 70,85 °C) in order to study the kinetic behavior, investigating theconversion,particle size, and the average number of radicals per particle, as well as the copolymercomposition, microstructure, molecular weight distributions (MWDs), and the glass transition temperature. (Unzueta et al. [18] proposed a mathematical model for emulsion copolymerization with mixed emulsifiersystems, and carried out the seeded and unseeded emulsion copolymerizations of MMA and BA.Vega et al. [61,104] and Dube et al. [101] both developed mathematical models for the emulsioncopolymerization of AN and Bu initiated by a redox initiator system.Saldívar et al. [58-60,100] carried out extensive investigations on emulsion copolymerization.In industrial emulsion polymerization processes, a small amount of watersoluble carboxylic monomer (suchas AA(acrylic acid) is often added to improve the colloidal stability and surface properties of the resultinglatex particles. Numerous studies have been carried out to date to clarify the influence of the AA monomer onthe kinetic behavior of the emulsion copolymerization of St and AA [105,106,107,108,109,110] and ofemulsion terpolymerizations including AA [111,112,113].Xu et al. [114] studied the emulsifier-free emulsion terpolymerization of St, BA and the cationic monomer N-dimethyl, N-butyl, N-ethyl metacrylate ammonium bromide (DBMA) using oil-soluble azobis (isobutyl-amidine hydrochloride) (AIBA) as the initiator.Fang et al. [115] investigated the kinetics and the colloidal properties of the resulting polymer latexes in theemulsifier-free emulsion copolymerization of St and the nonionic water-soluble comonomer AAm, using anamphoteric water-soluble initiator, 2,2¢-azobis[N-(2-carboxyethyl)- 2–2-methylpropionamidine]-hydrate(VA057).Kostov et al. [116,117] carried out a kinetic and mechanistic investigation of tetrafluoroethylene andpropylene with a redox system containing tert-butylperbenzoate (TBPB).Noel et al. [118] studied the effect of water solubility of the monomers on the copolymer composition drift inthe emulsion copolymerization of MA and vinyl ester combinations.Urretabizkaia et al. [119] investigated the kinetics of the high solids content semicontinuous emulsionterpolymerization of VAc, MMA and BA.Ge et al. [120] studied the inverse emulsion copolymerization of (2-methacryloyloxyethyl) trimethylammonium chloride and AAm initiated with KPS.3.4.4 Monomer Concentration in Polymer ParticlesFrom R P = k P [Μ P ] n N T ,the monomer concentration in a polymer particle is one of the three key factorsthat control the particle growth rate, and accordingly, the rate of polymerization. In emulsion polymerization,the course of emulsion polymerization is usually divided into three stages, namely, Intervals I, II and III. InIntervals I and II of emulsion homopolymerization, the monomer concentration in the polymer particles isassumed to be approximately constant. In Interval III, it decreases with reaction time.Two methods are now used to predict the monomer concentration in the polymer particles in emulsionhomopolymerization: empirical and thermodynamic methods.According to the empirical method [14,121,122], the monomer concentration in Intervals I and II can beexpressed as:[Μ P ] = [M]pc ; ( [M ] pc -constant monomer concentration in polymer particles at saturation swelling)Interval III begins when the monomer droplets disappear from the system at the monomerconversion X Mc (critical monomer conversion where monomer droplets disappear from the aqueous phase) .The monomer concentration in this interval ( X M 〉 X Mc ) is approximately given by:  [M]p = [M]pc  1 − X M     1 − X Mc Several researchers [123,124,125,126]have tried to thermodynamically describe the swelling behavior ofpolymer particles by one monomer. The thermodynamic method now used is based on the so-called Mortonequation given by: 21