All solids can be classified in to two categories; porous and non-porous solids. Porous solids are those that have high surface area and high pore volume where as non-porous solids are those that have low surface area and low pore volume. In general, all solids to some extend are porous except ceramics fired at high temperatures.
The three measure of a porous solids are surface area, pore size and its distribution and pore volume.
Pores can be open or closed. Open pores are accessible where as closed pores are inaccessible. Open pores
Pores can be open or closed. Open pores are accessible where as closed pores are inaccessible. Open pores
Transcript of "Mate 280 characterization of powders and porous materials"
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Characterization of Powders, Porous Solids and Suspensions Lecture 8
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Main Characteristics of Powders and Porous Solids <ul><li>Particle size </li></ul><ul><li>Surface area </li></ul><ul><li>Porosity </li></ul>
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Why We Care About Particle Size and Surface Area <ul><li>These characteristics control many properties of materials: </li></ul><ul><ul><li>Flowability; </li></ul></ul><ul><ul><li>“ Filter-ability” </li></ul></ul><ul><ul><li>Viscosity-Reology; </li></ul></ul><ul><ul><li>Agglomeration; </li></ul></ul><ul><ul><li>Dusting tendency; </li></ul></ul><ul><ul><li>Settling rate; </li></ul></ul><ul><ul><li>Activity/Reactivity rate (e.g. of catalyst); </li></ul></ul><ul><ul><li>Dissolution rate (of pharmaceutical); </li></ul></ul><ul><ul><li>Gas absorption; </li></ul></ul><ul><ul><li>Hydration rate (of cement); </li></ul></ul><ul><ul><li>Moisture absorption; </li></ul></ul><ul><ul><li>Entry into lungs (shape dependency too); </li></ul></ul><ul><ul><li>Combustion rate (of fuel) </li></ul></ul><ul><ul><li>Etc… </li></ul></ul>
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What is Particle Size? SEM of real ibuprofen particles
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A Concept of Equivalent Sphere <ul><li>Due to symmetry, size of sphere is completely determined by only one parameter – it’s diameter (radius) </li></ul><ul><li>Other properties of sphere are easily computed from its size: </li></ul><ul><li>Sphere is just a convenient model! This is why it is found throughout the particle size analysis </li></ul>
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Particle Size Measurement Techniques <ul><li>Direct observation (image analysis) </li></ul><ul><li>Sieving; </li></ul><ul><li>Sedimentation – settling rate; </li></ul><ul><li>Coulter counter – electrozone sensing; </li></ul><ul><li>Gas adsorption – BET (SSA back extrapolation to size); </li></ul><ul><li>Permeability (gas or liquid) e.g. Blaine, FSSS </li></ul><ul><li>Light scattering – laser diffraction and Photon Correlation Spectroscopy / Dynamic Light Scattering </li></ul>
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And What Do They Measure <ul><li>Direct observation (image analysis) – usually some 2-D representation of a particle. Which dimension is viable?; </li></ul><ul><li>Sieving – combination of particle size and shape; </li></ul><ul><li>Sedimentation – settling rate. Stokes Law (spheres, straight line settling); </li></ul><ul><li>Coulter counter – electrozone sensing; </li></ul><ul><li>Gas absorption / Permeability – surface area. Extrapolate to average particle size only. – BET (SSA back extrapolation to size); </li></ul><ul><li>Light scattering – equivalent scatterers; </li></ul>
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Particle Size by Direct Observation Google for ImageJ
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Dynamic Light Scattering (DLS) <ul><li>DLS measures Brownian motion and relates this to the size of the particles. </li></ul><ul><li>The larger the particle the slower the Brownian motion will be. Smaller particles are “kicked” further by the solvent molecules and move more rapidly. </li></ul><ul><li>The velocity of Brownian motion is defined by a property known as the translational diffusion coefficient (D). </li></ul><ul><li>The size of a particle is calculated from the translational diffusion coefficient by using the Stokes-Einstein equation: </li></ul><ul><li>d(H) – hydrodynamic diameter, D – translational diffusion coefficient, k – Boltzmann’s constant, T – temperature, η - viscosity </li></ul>
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What We Measure in DLS? <ul><li>The diameter that is measured in DLS is a value that refers to how a particle diffuses within a fluid so it is referred to as a hydrodynamic diameter </li></ul><ul><li>The diameter that is obtained by this technique is the diameter of a sphere that has the same translational diffusion coefficient as the particle </li></ul><ul><li>The translational diffusion coefficient will depend not only on the size of the particle “core”, but also on any surface structure, as well as the concentration and type of ions in the medium </li></ul>Particle core Shell formed by solvent particles, ions etc. Low conductivity medium will produce an extended double layer of ions around the particle, reducing the diffusion speed and resulting in a larger, apparent hydrodynamic diameter. Thus, the measurements are usually done in 10mM NaCl (ISO13321 Part 8 1996)
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How DLS Works <ul><li>The dark spaces in the speckle pattern produced by light scattering are where the phase additions of the scattered light are mutually destructive. The bright spots of light in the speckle pattern are where the light scattered from the particles arrives with the same phase and interfere constructively. </li></ul><ul><li>The observed signal depends on the phase addition of the scattered light falling on the detector. In example A, two beams interfere and “cancel each other out” resulting in a decreased intensity detected. In example B, two beams interfere and “enhance each other” resulting in an increased intensity detected. </li></ul>
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How DLS Works <ul><li>For a system of particles undergoing Brownian motion, a speckle pattern is observed where the position of each speckle is seen to be in constant motion. This is because the phase addition from the moving particles is constantly evolving and forming new patterns. </li></ul><ul><li>The rate at which these intensity fluctuations occur will depend on the size of the particles. Figure above schematically illustrates typical intensity fluctuations arising from a dispersion of large particles and a dispersion of small particles. </li></ul><ul><li>The small particles cause the intensity to fluctuate more rapidly than the large ones. </li></ul><ul><li>It is possible to directly measure the spectrum of frequencies contained in the intensity fluctuations arising from the Brownian motion of particles, but it is inefficient to do so. The best way is to use a device called a digital auto correlator. </li></ul>
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How an Auto Correlator Works <ul><li>If the intensity of a signal is compared with itself at a particular point in time and a time much later, then for a randomly fluctuating signal it is obvious that the intensities are not going to be related in any way, i.e. there will be no correlation between the two signals. </li></ul><ul><li>However, if the intensity of signal at time t is compared to the intensity a very small time later (t+δt), there will be a strong relationship or correlation between the intensities of two signals. </li></ul><ul><li>Perfect correlation is indicated by unity (1.00) and no correlation is indicated by zero (0.00). </li></ul><ul><li>If the signals at t+2δt, t+3δt, t+4δt etc. are compared with the signal at t, the correlation of a signal arriving from a random source will decrease with time until at some time, effectively t = ∞, there will be no correlation. </li></ul><ul><li>If the particles are large the signal will be changing slowly and the correlation will persist for a long time. If the particles are small and moving rapidly then correlation will reduce more quickly. </li></ul>
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Different Forms of Particle Size Distribution <ul><li>Consider 2 populations of spherical particles of diameter 5nm and 50nm present in equal numbers. </li></ul><ul><li>If a number distribution of these 2 particle populations is plotted, a plot consisting of 2 peaks (positioned at 5 and 50nm) of a 1 to 1 ratio would be obtained. </li></ul><ul><li>If this number distribution was converted into volume, then the 2 peaks would change to a 1:1000 ratio (because the volume of a sphere is proportional to d 3 ). </li></ul><ul><li>If this was further converted into an intensity distribution, a 1:1000000 ratio between the 2 peaks would be obtained (because the intensity of scattering is proportional to d 6 from Rayleigh’s approximation). </li></ul><ul><li>In DLS, the distribution obtained from a measurement is based on intensity. </li></ul>
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Measurement of Porosity and Specific Surface Area by Gas Adsorption
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What are Porous Materials? F. Rouquerol, J. Rouquerol, K. S. W. Sing, Adsorption by Powders and Porous Solids, Academic Press, 1-25, 1999 <ul><li>Non-porous solid </li></ul><ul><li>Low specific surface area </li></ul><ul><li>Low specific pore volume </li></ul><ul><li>Porous solid </li></ul><ul><li>High specific surface area </li></ul><ul><li>High specific pore volume </li></ul>Porous materials have highly developed internal surface area that can be used to perform specific function. Almost all solids are porous except for ceramics fired at extremely high temperatures
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Measure of Porosity Pore size and its distribution Specific Surface Area, m 2 /g = Porosity There are three parameters used as a measure of porosity; specific surface area, specific pore volume or porosity, and pore size and its distribution. Mass of the solid, g Total surface area, m 2 Specific Pore volume, cm 3 /g Mass of the solid, g Total pore volume, cm 3 = Porosity, % = Volume of solid (including pores) Volume of pores X 100
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Concept of Porosity: Open vs. Closed Pores Dead end (open) Closed Inter-connected (open) Passing (open) F. Rouquerol, J. Rouquerol, K. S. W. Sing, Adsorption by Powders and Porous Solids, Academic Press, 1-25, 1999 Open pores are accessible whereas closed pores are inaccessible pores. Open pores can be inter-connected, passing or dead end.
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Size of Pores (IUPAC Standard) 2 nm 50 nm Micropores Mesopores Macropores Zeolite, Activated carbon, Metal organic framework Mesoporous silica, Activated carbon Sintered metals and ceramics Porous material are classified according to the size of pores: material with pores less than 2 nm are called micropores, materials with pores between 2 and 50 nm are called mesopores, and material with pores greater than 50 nm are macrospores Sing, K. S. W. et al. Reporting Physisorption Data for Gas/Solid Systems. Pure & Appl. Chem. 57, 603-619 (1985).
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Shapes of Pores Conical Interstices F. Rouquerol, J. Rouquerol, K. S. W. Sing, Adsorption by Powders and Porous Solids, Academic Press, 1-25, 1999 Slits Cylindrical Spherical or Ink Bottle Pore Shapes
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Techniques for Porosity Analysis Mercury porosimetry TEM SEM Small angle X-ray scattering Small Angle Neutron scattering Gas adsorption Techniques
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Techniques for Porosity Analysis <ul><li>Can measure only open pores </li></ul><ul><li>Pore size : 0.4 nm – 50 nm </li></ul><ul><li>Easy </li></ul><ul><li>Established technique </li></ul>Mercury porosimetry TEM SEM Small angle X-ray scattering Small Angle Neutron scattering Gas adsorption Techniques
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Techniques for Porosity Analysis <ul><li>Similar to gas adsorption </li></ul><ul><li>Can measure only open pores </li></ul><ul><li>Pore size >1.5 nm </li></ul><ul><li>Easy </li></ul><ul><li>Established technique </li></ul>Mercury porosimetry TEM SEM Small angle X-ray scattering Small Angle Neutron scattering Gas adsorption Techniques
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<ul><li>Provide information regarding pore connectivity </li></ul><ul><li>Pore size can be measured if the materials contains ordered pores </li></ul><ul><li>Rarely used for pore analysis </li></ul>Techniques for Porosity Analysis Mercury porosimetry TEM SEM Small angle X-ray scattering Small Angle Neutron scattering Gas adsorption Techniques
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Techniques for Porosity Analysis <ul><li>Pore size > 5nm </li></ul><ul><li>Rarely used for pore analysis </li></ul>Mercury porosimetry TEM SEM Small angle X-ray scattering Small Angle Neutron scattering Gas adsorption Techniques
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Techniques for Porosity Analysis <ul><li>Any pore size </li></ul><ul><li>Open + Close porosity </li></ul>Mercury porosimetry TEM SEM Small angle X-ray scattering Small Angle Neutron scattering Gas adsorption Techniques
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Techniques for Porosity Analysis <ul><li>Any pore size </li></ul><ul><li>Open & Close porosity </li></ul><ul><li>Costly </li></ul>Mercury porosimetry TEM SEM Small angle X-ray scattering Small Angle Neutron scattering Gas adsorption Techniques
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Adsorption Process Adsorption is brought by the forces acting between the solid and the molecules of the gas. These forces are of two kinds: physical (physiosorption) and chemical (chemisorption) Adsor bent - the solid where adsorption takes place Adsor bate - the gas adsorbed on the surface of solids Adsor ptive - adsorbate before being adsorbed on the surface
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Physisorption vs Chemisorption http://www.soton.ac.uk PHYSISORPTION CHEMISORPTION WEAK, LONG RANGE BONDING Van der Waals interactions STRONG, SHORT RANGE BONDING Chemical bonding involved. NOT SURFACE SPECIFIC Physisorption takes place between all molecules on any surface providing the temperature is low enough. SURFACE SPECIFIC E.g. Chemisorption of hydrogen takes place on transition metals but not on gold or mercury. Δ H ads = 5 ….. 50 kJ mol-1 Δ H ads = 50 ….. 500 kJ mol -1 Non activated with equilibrium achieved relatively quickly. Increasing temperature always reduces surface coverage . Can be activated, in which case equilibrium can be slow and increasing temperature can favour adsorption. No surface reactions. Surface reactions may take place:- Dissociation, reconstruction, catalysis. MULTILAYER ADSORPTION BET Isotherm used to model adsorption equilibrium. MONOLAYER ADSORPTION Langmuir Isotherm is used to model adsorption equilibrium.
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Adsorption Process 1 . Diffusion to adsorbent surface 2 . Migration into pores of adsorbent 3 . Monolayer builds up of adsorbate 1 <ul><li>Gas molecules admitted under increasing pressure to a clean, cold surface. </li></ul><ul><li>Data treatment techniques find the quantity of gas that forms the first layer. </li></ul>S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 2 3 1 2 3
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Adsorption Process Adsor bent Adsor bate S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 Equation of adsorption isotherm
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Gas Sorption: Isotherm Adsorption isotherm <ul><li>Isotherm is a measure of the volume of gas adsorbed at a constant temperature as a function of gas pressure. </li></ul><ul><li>Isotherms can be grouped into six classes. </li></ul>V a Desorption isotherm p p o
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Gas Sorption: Isotherm V a Type I or Langmuir S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 <ul><li>Concave to the P/P o axis </li></ul><ul><li>Exhibited by microporous solids ( < 2nm ) </li></ul>Type II <ul><li>Exhibited by nonporous or macroporous solids ( > 50nm ) </li></ul><ul><li>Unrestricted monolayer-multilayer adsorption </li></ul><ul><li>Point B indicates the relative pressure at which monolayer coverage is complete </li></ul>B V a 1 P/P o 1 P/P o
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Gas Sorption: Isotherm V a <ul><li>Convex to the P/P o axis </li></ul><ul><li>Exhibited by nonporous solids </li></ul>V a P/P o Type IV <ul><li>Exhibited by mesoporous solids </li></ul><ul><li>Initial part of the type IV follows the same path as the type II </li></ul>S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 1 P/P o Type III 1
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Gas Sorption: Isotherm V a Type V Type VI <ul><li>Highly uncommon </li></ul><ul><li>Exhibited by mesoporous solids </li></ul><ul><li>Exhibited by nonporous solids with an almost completely uniform surface </li></ul>S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 V a 1 P/P o 1 P/P o
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Gas Sorption: Hysteresis <ul><li>Hysteresis indicates the presence of mesopores. </li></ul><ul><li>Hysteresis gives information regarding pore shapes . </li></ul><ul><li>Types I, II and III isotherms are generally reversible but type I can have a hysteresis. Types IV and V exhibit hysteresis. </li></ul>Hysteresis V a S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 1 P/P o
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Gas Sorption: Hysteresis V a Type A Cylindrical Slits Type B Type C Type E Conical Bottle neck 1 P/P o 1 P/P o 1 P/P o Type D 1 P/P o 1 P/P o
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Adsorption Theories: Langmuir Adsorbate Adsorbent <ul><li>Assumptions: </li></ul><ul><li>homogeneous surface (all adsorption sites energetically identical) </li></ul><ul><li>monolayer adsorption (no multilayer adsorption) </li></ul><ul><li>no interaction between adsorbed molecules </li></ul>I. Langmuir The Constitution and Fundamental Properties of Solids and Liquids. Part I. Solids. J. Am. Chem. Soc. , 1916, 38 (11), 2221-2295
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Adsorption Theories: BET <ul><li>Modification of Langmuir isotherm </li></ul><ul><li>Both monolayer and multilayer adsorption </li></ul><ul><li>Assumptions: </li></ul><ul><li>gas molecules physically adsorb on a solid in layers infinitely; </li></ul><ul><li>there is no interaction between each adsorption layer; </li></ul><ul><li>the Langmuir theory can be applied to each layer. </li></ul>S . Brunauer, P.Emmett, E . Teller Adsorption of Gases in Multimolecular Layers, J. Am. Chem. Soc., 1938, 60 (2), pp 309–319 Adsorbate Adsorbent
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Specific Surface Area Calculation At least three data points in the relative pressure range 0.05 to 0.30 P/P o 1 V[(P o /P)-1] 0-1 0-2 0-3
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Single Point BET <ul><li>Single-point method offers the advantage of simplicity and speed, often with little loss in accuracy. </li></ul>i.e. V m = 1/ slope <ul><li>A relative pressure of 0.3 gives good general agreement with the multi-point method. </li></ul><ul><li>Correction of single point “error” at P/P 0 = 0.3 by multiplying the single point BET value by C/C-2 decreases the difference. </li></ul>Sample No. Multi-point BET (m 2 /g) Uncorrected single-point (m 2 /g) Uncorrected difference (%) Corrected single –point (m 2 /g) Corrected difference (%) 1 4.923 4.241 -13.9 4.948 0.51 2 4.286 3.664 -14.5 4.275 -0.26 3 8.056 6.867 -14.8 8.011 -0.56 4 5.957 5.194 -12.8 6.060 +1.73
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Pore Size Distribution V a Pore diameter, d Narrow pore size distribution Pore diameter, d Multimodal pore size distribution The distribution of pore volume with respect to pore size is called a pore size distribution. V a Broad pore size distribution Unimodal pore size distribution
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Pore Size Distribution G ads = RT(lnP ads - lnP 0 ) G des = RT(lnP des - lnP 0 ) G des < G ads 1 P/P o ( P/P o ) des ( P/P o ) ads <ul><li>Adsorption or desorption isotherm. </li></ul><ul><li>The desorption isotherm is preferred over adsorption isotherm. </li></ul>V a
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Pore Size: Kelvin Equation <ul><li>Multilayer formation occurs in parallel to capillary condensation. </li></ul><ul><li>Capillary condensation is described by the Kelvin equation. </li></ul>
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Pore Size: Kelvin Equation Actual radius of the pore Kelvin radius of the pore Thickness of the adsorbed layer Prior to condensation, some adsorption has taken place on the walls of the pore, r k does not represent the actual pore radius. Adsorbed layer
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Methods for Calculation of Pore Size Distribution <ul><li>BJH (Barrett, Joyner and Halenda) method </li></ul><ul><li>DH (Dollimore Heal) method </li></ul><ul><li>Dubinin-Astakhov method </li></ul><ul><li>HK (Horvath-Kawazoe) method </li></ul><ul><li>Saito-Foley method </li></ul>Mesoporous solids Microporous solids <ul><li>NLDFT (Non Local Density Functional Theory) and Monte Carlo simulation method </li></ul>Microporous and Mesoporous solids
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Porosity Analyzer Outgassing station Analysis station Liquid nitrogen bath
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Sample Preparation (Outgassing) <ul><li>Surface contamination is removed by application of: </li></ul><ul><ul><li>Temperature </li></ul></ul><ul><ul><li>Flowing gas (helium or nitrogen) or vacuum </li></ul></ul><ul><li>Backfill can be done using helium or adsorbate gas. </li></ul><ul><li>According to IUPAC standards, materials should be outgassed for at least 16 hours . </li></ul>Adsorbate Helium Vacuum P o Outgassing station Analysis station Sample Cell
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Adsorption Analysis <ul><li>Adsorbate (nitrogen, argon, carbon dioxide, krypton) </li></ul><ul><li>Analysis temperature (liquid nitrogen, liquid argon, 0 o C) </li></ul><ul><li>Quantity of sample (1 mg sample is sufficient) </li></ul><ul><li>Number of points (single point, five points, seven points, eleven points, full analysis) </li></ul>Adsorbate Helium Vacuum P o Outgassing station Analysis station Sample Cell
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Interpretation OUTPUT Points P/P o Volume adsorbed 1 2 3 Weight of sample Pore shape Specific surface area Pore volume Pore size & distribution Results
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Common Adsorbates Gas Temperature Cross sectional area (nm 2 ) N 2 <ul><li>-195.8 o C (liquid nitrogen) </li></ul><ul><li>-183 o C (liquid argon). </li></ul>0.162 Ar <ul><li>-183 o C (liquid argon). </li></ul><ul><li>-195.8 o C (liquid nitrogen) </li></ul>0.142 CO 2 <ul><li>-78 o C, -25 o C, 0 o C </li></ul>0.195 CO <ul><li>-183 o C (liquid argon) </li></ul>0.163 Kr <ul><li>-195.8 o C (liquid nitrogen) </li></ul>0.205 O 2 <ul><li>-183 o C (liquid argon) </li></ul>0.141 C 4 H 10 <ul><li>0 o C, 25 o C </li></ul>0.469
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Choice of Adsorptive <ul><li>N 2(g) in N 2(l) is the most commonly used adsorbate. </li></ul><ul><li>Not completely inert. </li></ul><ul><li>Dipole movement and thus can have localized adsorption. </li></ul><ul><li>Cross-sectional area of 0.162 nm 2 is questionable. </li></ul><ul><li>S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 </li></ul><ul><li>Quantachrome Autosorb-I Operational Manual </li></ul>
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Choice of Adsorptive <ul><li>S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 </li></ul><ul><li>Quantachrome Autosorb-I Operational Manual </li></ul><ul><li>Ar (g) in Ar (l) is preferable but because of unavailability of Ar (l) (87K), N 2(l) (77 K) is used. </li></ul><ul><li>Ar can reach to somewhat smaller pores than N 2 . </li></ul><ul><li>Accurate measurement of micropores is possible using Ar. </li></ul>
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Choice of Adsorptive <ul><li>S. Lowell & J. E. Shields, Powder Surface Area and Porosity, 3rd Ed. Chapman & Hall, New York, 1991 </li></ul><ul><li>Quantachrome Autosorb-I Operational Manual </li></ul><ul><li>In case of activated carbon, CO 2 is often the most preferred adsorbate. </li></ul><ul><li>Adsorption analysis of CO 2 takes less time. </li></ul><ul><li>Limited to micropore analysis. </li></ul>
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Validity of BET - Method <ul><li>The BET method depends on the cross-sectional area of adsorbate. </li></ul><ul><li>Monolayer structure is same on all the surface. </li></ul><ul><li>Localized monolayer coverage. </li></ul>K. S. W. Sing, The Use of Nitrogen Adsorption for the Characterisation of Porous Materials, Colloids and Surfaces, 187 – 188, 2001, 3 - 9 Adsorbate Adsorbent
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Validity of Kelvin Equation <ul><li>Is relation between the meniscus curvature and the pore size and shape valid? </li></ul><ul><li>Is it applicable for micropores and narrow mesopores ? </li></ul><ul><li>Does surface tension varies with pore width ? </li></ul>F. Rouquerol, J. Rouquerol, K. S. W. Sing, Adsorption by Powders and Porous Solids, Academic Press, 203, 1999
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Shape of Microporous Materials V a Type I or Langmuir <ul><li>Type I isotherms don’t have hysteresis. </li></ul><ul><li>Pore shape cannot be determined by isotherm . </li></ul><ul><li>As various methods for pore size calculation are based on shape of pores, reliability of pore size calculation is questionable . </li></ul>F. Rouquerol, J. Rouquerol, K. S. W. Sing, Adsorption by Powders and Porous Solids, Academic Press, 439-446, 1999 1 P/Po
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2 nm 50 nm Micropores Mesopores Macropores Choice of Method <ul><li>P. A. Webb, C. Orr, Analytical Methods in Fine Particle Technology, Micromeritics, 53 – 152, 1997 </li></ul><ul><li>Quantachrome Autosorb-I Operational Manual </li></ul>Methods Assumption Pore Shape Based on .. Brunauer MP method Cylindrical or Slit shaped de Boer’s t-method Dubinin-Astakhov method - <ul><li>Polanyi potential theory </li></ul><ul><li>Independent of Kelvin equation </li></ul>HK (Horvath-Kawazoe) method Slit <ul><li>Everett and Powl method </li></ul><ul><li>Independent of Kelvin equation </li></ul>Saito-Foley method Cylindrical HK method
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2 nm 50 nm Micropores Mesopores Macropores Choice of Method <ul><li>P. A. Webb, C. Orr, Analytical Methods in Fine Particle Technology, Micromeritics, 53 – 152, 1997 </li></ul><ul><li>Quantachrome Autosorb-I Operational Manual </li></ul>Methods Assumption Pore Shape Based on .. BJH (Barrett, Joyner and Halenda) method Cylindrical, Slit-shaped Kelvin equation DH (Dollimore Heal) method Cylindrical t-method
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2 nm 50 nm Micropores Mesopores Macropores Choice of Method <ul><li>P. A. Webb, C. Orr, Analytical Methods in Fine Particle Technology, Micromeritics, 53 – 152, 1997 </li></ul><ul><li>Quantachrome Autosorb-I Operational Manual </li></ul>Methods Assumption Pore Shape Based on .. NLDFT (Non Local Density Functional Theory) and Monte Carlo simulation method Cylindrical and slit Statistical thermodynamics
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