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**Physisorption Methods and Techniques**

Quantachrome I N S T R U M E N T S Physisorption Methods and Techniques

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**Pore Size by Gas Sorption**

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**Micro and Mesopore Size Determination by Gas Sorption**

First: Quantitative estimation of micropore volume and area… T-plot and DR methods.

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**Multilayer adsorption**

Type II, IV Low slope region in middle of isotherm indicates first few multilayers, on external surface including meso and macropores… before the onset of capillary condensation Volume adsorbed After the knee, micropores cease to contribute to the adsorption process. Relative Pressure (P/Po)

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**Estimation of Micropores... the t-plot method**

This method uses a mathematical representation of multi-layer adsorption. The thickness, t, of an adsorbate layer increases with increasing pressure. The t-curve so produced is very similar in appearance to a type II isotherm. For every value of P/Po, the volume adsorbed is plotted against the corresponding value of “t”. If the model describes the experimental data a straight line is produced on the t-plot...

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**The t-plot Resembles a type II Statistical thickness**

A statistical multilayer A statistical monolayer Relative Pressure (P/Po)

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**t-plot Method (mesoporous only)**

Slope = V/t = A Zero intercept

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**t-plot Method showing a “knee”**

C B Slope A - slope B = area contribution by micropores size C

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? Quiz What is an s plot? s (for Ken Sing) is a comparison plot like the t-plot but its slope does not give area directly. A

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**Estimation of Micropores Dubinin-Radushkevich (DR) Theory**

W = volume of the liquid adsorbate W0 = total volume of the micropores B = adsorbent constant = adsorbate constant A linear relationship should be found between log(W) and log2(Po/P)...

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**Estimation of Micropores Dubinin-Radushkevich (DR) Plot**

Log (W) Extrapolation yields Wo Log2(Po/P)

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**Pore Size Determination**

Requires a recognition and understanding of different basic isotherm types.

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**t-plot Method (in the presence of micropores)**

Intercept = micropore volume

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**Types of Isotherms Type II Type I Volume adsorbed Type V Type IV**

Type III Type IV Relative Pressure (P/Po) Volume adsorbed Type V

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**Types of Isotherms Volume adsorbed Type I or pseudo-“Langmuir”**

Relative Pressure (P/Po) Volume adsorbed Steep initial region due to very strong adsorption, for example in micropores. Limiting value (plateau) due to filled pores and essentially zero external area.

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**Quiz Why pseudo Langmuir? A**

Langmuir applies to monolayer limit, not volume filling limit. A

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**Types of Isotherms Type II Volume adsorbed Relative Pressure (P/Po)**

Rounded knee indicates approximate location of monolayer formation. Absence of hysteresis indicates adsorption on and desorption from a non-porous surface.. Low slope region in middle of isotherm indicates first few multilayers

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**Types of Isotherms Type III Volume adsorbed Relative Pressure (P/Po)**

Lack of knee represents extremely weak adsorbate-adsorbent interaction BET is not applicable Example: krypton on polymethylmethacrylate

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**Types of Isotherms Type IV Volume adsorbed Relative Pressure (P/Po)**

Rounded knee indicates approximate location of monolayer formation. Low slope region in middle of isotherm indicates first few multilayers Hysteresis indicates capillary condensation in meso and macropores. Closure at P/Po~0.4 indicates presence of small mesopores (hysteresis would stay open longer but for the tensile-strength-failure of the nitrogen meniscus.

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**Types of Isotherms Type V Volume adsorbed Relative Pressure (P/Po)**

Example: water on carbon black Type V Volume adsorbed Lack of knee represents extremely weak adsorbate-adsorbent interaction BET is not applicable Relative Pressure (P/Po)

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**Types of Hysteresis Large pores/voids Gel Volume adsorbed Mesopores**

MCM Relative Pressure (P/Po)

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**MesoPore Size by Gas Sorption (BJH)**

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**Analyzer measures volume of pores: Yes or No?**

Quiz Analyzer measures volume of pores: Yes or No? NO! It measures what leaves supernatent gas phase A

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**Pore Size Distribution**

Hysteresis is indicative of the presence of mesopores and the pore size distribution can be calculated from the sorption isotherm. Whilst it is possible to do so from the adsorption branch, it is more normal to do so from the desorption branch... Mesopore (Greek meso = middle): 2nm - 50 nm diameter Macropore (Greek macro = large): >50 nm diameter Micropore (Greek micro = small): 0 nm - 2 nm diameter

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**Adsorption / Desorption**

multilayer formation Desorption = meniscus development

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Kelvin* Equation * Lord Kelvin a.k.a. W.T. Thomson

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**Pore Size rp = actual radius of the pore**

rk = Kelvin radius of the pore t = thickness of the adsorbed film

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**Statistical Thickness, t**

Halsey equation Generalized Halsey deBoer equation Carbon Black STSA

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**BJH Method (Barrett-Joyner-Halenda)**

Pore volume requires assumption of liquid density!

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**Pore Size Distribution**

Artifact dV/dlogD 40 Pore Diameter (angstrom)

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~ 0.42 Amount adsorbed Relative Pressure (P/Po)

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Pore Size Data Volume and size of pores can be expressed from either adsorption and/or desorption data. The total pore volume, V, is taken from the maximum amount of gas adsorbed at the “top” of the isotherm and conversion of gas volume into liquid volume. The mean pore diameter is calculated from simple cylindrical geometry: where A is the BET surface area.

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**Pore size analysis of MCM 41 (Templated silica) by N2 sorption at 77 K**

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**Pore size analysis of MCM 41: Calculations compared**

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Calculation Models

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**Gas Sorption Calculation Methods**

Comparisons Gas Sorption Calculation Methods P/Po range Mechanism Calculation model 1x10-7 to 0.02 micropore filling DFT, GCMC, HK, SF, DA, DR 0.01 to 0.1 sub-monolayer formation DR 0.05 to 0.3 monolayer complete BET, Langmuir > multilayer formation t-plot (de-Boer,FHH), > capillary condensation BJH, DH 0.1 to capillary filling DFT, BJH in M41S-type materials

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**Different Theories of Physisorption**

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**HK & SF Horvath-Kawazoe & Saito-Foley**

Direct mathematical relationship between relative pressure (P/Po) and pore size. Relationship calculated from modified Young-Laplace equation, and takes into account parameters such as magnetic susceptibility. Based on slit-shape pore geometry (e.g. activated carbons). Calculation restricted to micropore region ( 2nm width). SF Similar mathematics to HK method, but based on cylindrical pore geometry (e.g. zeolites). Calculation restricted to micropore region ( 2 nm diameter).

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**DA & DR Dubinin-Astakov and Dubinin-Radushkevic**

Closely related to DR calculation based on pore filling mechanism. Equation fits calculated data to experimental isotherm by varying two parameters, E and n. E is average adsorption energy that is directly related to average pore diameter, and n is an exponent that controls the width of the resulting pore size distribution. The calculated pore size distribution always has a skewed, monomodal appearance (Weibull distribution). DR Simple log(V) vs log2(Po/P) relationship which linearizes the isotherm based on micropore filling principles. “Best fit” is extrapolated to log2(Po/P) (i.e. where P/Po = 1) to find micropore volume.

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BET The most famous gas sorption model. Extends Langmuir model of gas sorption to multi-layer. BET equation linearizes that part of the isotherm that contains the “knee” , i.e. that which brackets the monolayer value. Normally solved by graphical means, by plotting 1/(V[(Po/P)]-1) versus P/Po. Monolayer volume (Vm) is equal to 1/(s+i) where s is the slope and i is the y-intercept. Usually BET theory is also applied to obtain the specific surface area of microporous materials, although from a scientific point of view the assumptions made in the BET theory do not take into account micropore filling. Please note, that for such samples the linear “BET” range is found usually at relative pressures< 0.1, in contrast to the classical BET range, which extends over relative pressures between 0.05 – 0.3.

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Langmuir Adsorption model limited to the formation of a monolayer that does not describe most real cases. Sometimes can be successfully applied to type I isotherms (pure micropore material) but the reason for limiting value (plateau) is not monolayer limit, but due to micropore filling. Therefore type I physisorption isotherm would be better called “pseudo-Langmuir” isotherm.

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**t-plot Statistical Thickness**

Multi-layer formation is modeled mathematically to calculate a layer “thickness, t” as a function of increasing relative pressure (P/Po). The resulting t-curve is compared with the experimental isotherm in the form of a t-plot. That is, experimental volume adsorbed is plotted versus statistical thickness for each experimental P/Po value. The linear range lies between monolayer and capillary condensation. The slope of the t-plot (V/t) is equal to the “external area”, i.e. the area of those pores which are NOT micropores. Mesopores, macropores and the outside surface is able to form a multiplayer, whereas micropores which have already been filled cannot contribute further to the adsorption process. It is recommended to initially select P/Po range 0.2 – 0.5, and subsequently adjust it to find the best linear plot.

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**BJH & DH Barrett, Joyner, Halenda and Dollimore-Heal**

Modified Kelvin equation. Kelvin equation predicts pressure at which adsorptive will spontaneously condense (and evaporate) in a cylindrical pore of a given size. Condensation occurs in pores that already have some multilayers on the walls. Therefore, the pore size is calculated from the Kelvin equation and the selected statistical thickness (t-curve) equation. DH Extremely similar calculation to BJH, which gives very similar results. Essentially differs only in minor mathematical details.

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**Other Methods FRACTAL DIMENSION**

The geometric topography of the surface structure of many solids can be characterized by the fractal dimension D, which is a kind of roughness exponent. A “flat” surface is considered D is 2, however for an irregular (real) surface D may vary between 2 and 3 and expresses so the degree of roughness of the surface and/or porous structure. The determination of the surface roughness can be investigated by means of the modified Frenkel-Halsey Hill method, which is applied in the range of multilayer adsorption.

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**Example Data : Microporous Carbon**

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**BET : Not strictly applicable**

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**Example Data : Microporous Carbon**

Tag all adsorption points Analyze behavior Note knee – transition from micropore filling to limited multilayering (plateau).

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**Example Data : Microporous Carbon**

Use Langmuir (Monolayer model) / DR for Surface Area, Micropore Volume Usue Langmuir in range of > 0.2 (monolayer)

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**Example Data : Microporous Carbon**

Langmuir Surface Area

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**Example Data : Microporous Carbon**

DR Method for surface area, micropore volume Choose low relative pressure points (up to P/P0 = 0.2)

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**Example Data : Microporous Carbon**

Reports micropore surface area, and micropore volume. Note Langmuir, DR surface areas very close (1430 m2/g vs m2/g)

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**Example Data : Macroporous Sample**

Little or no “knee”, isotherm closes at 0.95

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**Example Data : Macroporous Sample**

BET Plot = OK Surface area ca. 8m2/g (low) Note hysteresis above P/P0 = 0.95 Pores > 35 nm

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**Example Data : Macroporous Sample**

Intercept = (-), no micropore volume.

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**Example Data : Macroporous Sample**

BJH Shows pores > 20nm, to over 200 nm

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**Example Data : Mesoporous Silica**

Hysteresis => mesopores Also micropores ?? Test using t-method

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**Example Data : Mesoporous Silica**

BET Surface area = 112m2/g Classic mesoporous silica !

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**Example Data : Mesoporous Silica**

Intercept ~ 0 Look at tabular data MP SA = 8m2/g (total SA = 112) Statistical Thickness => Use de Boer for oxidic surfaces = silicas

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**Example Data : Mesoporous Silica**

Use BJH – shows narrow pore size distribution in 14-17nm range (mesopores)

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**MicroPore Size by Gas Sorption**

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**Available Calculation Models**

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**Pore filling pressures for nitrogen in cylindrical pores at 77 K, (Gubbins et al. 1997)**

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**Pore filling pressures for nitrogen in cylindrical silica pores at 77 K (Neimark et al., 1998)**

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**Pore size analysis of MCM 41 by silica by N2 sorption at 77 K**

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**Gas- and liquid density profiles in a slit pore by GCMC (Walton and Quirke,1989)**

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**NLDFT / GCMC (Monte Carlo) Kernel File Applicable Pore Diameter Range **

Examples NLDFT– N2 - carbon kernel at 77 K based on a slit-pore model 0.35nm-30 nm Carbons with slit-like pores, such as activated carbons and others. NLDFT– N2 – silica equilibrium transition kernel at 77 K, based on a cylindrical pore model 0.35nm- 100nm Siliceous materials such as some silica gels, porous glasses, MCM-41, SBA-15, MCM-48 and other adsorbents which show type H1 sorption hysteresis. NLDFT– N2 - silica adsorption branch kernel at 77 K, based on a cylindrical pore model 0.35nm-100nm Siliceous materials such as some controlled pore glasses, MCM-41, SBA-15, MCM-48, and others. Allows to obtain an accurate pore size distribution even in case of type H2 sorption hysteresis NLDFT– Ar zeolite/silica equilibrium transition kernel at 87 K based on a cylindrical pore model 0.35nm -100nm Zeolites with cylindrical pore channels such as ZSM5, Mordenite, and mesoporous siliceous materials (e.g., MCM-41, SBA-15, MCM-48, some porous glasses and silica gels which show type H1 sorption hysteresis).

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**NLDFT / GCMC (Monte Carlo) Kernel File Applicable Pore Diameter Range **

Examples NLDFT – Ar-zeolite/silica adsorption branch kernel at 87 K based on a cylindrical pore model 0.35nm-100nm Zeolites with cylindrical pore channels such as ZSM5, Mordenite etc., and mesoporous siliceous materials such as MCM-41, SBA-15, MCM-48, porous glasses some silica gels etc). Allows to obtain an accurate pore size distribution even in case of H2 sorption hysteresis. NLDFT – Ar-zeolite / silica equilibrium transition kernel based on a spherical pore model (pore diameter < 2 nm) and cylindrical pore model (pore diameter > 2 nm) Zeolites with cage-like structures such as Faujasite, 13X etc. , and mesoporous silica materials (e.g., MCM-41, SBA-15, porous glasses, some silica gels which show H1 sorption hysteresis). NLDFT – Ar-zeolite / silica adsorption branch kernel at 87 K based on a spherical pore model (pore diameter < 2 nm) and cylindrical pore model (pore diameter > 2 nm) Zeolites with cage-like structures such as Faujasite, 13X, and mesoporous silica materials (e.g., MCM-41, SBA-15, controlled-pore glasses and others). Allows to obtain an accurate pore size distribution even in case of H2 sorption hysteresis.

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**NLDFT / GCMC (Monte Carlo) Kernel File Applicable Pore Diameter Range **

Examples NLDFT – Ar - carbon kernel at 77 K based on a slit-pore model 0.35 nm - 7 nm Carbons with slit-like pores, such as activated carbons etc. NLDFT - CO2 - carbon kernel at 273 K based on a slit-pore model 0.35nm-1.5 nm GCMC – CO2 - carbon kernel at 273 K based on a slit-pore model

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RECENT ADVANCES IN THE PORE SIZE ANALYSIS OF MICRO- AND MESOPOROUS MOLECULAR SIEVES BY ARGON GAS ADSORPTION

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**Micropore Size Characterization**

Physical adsorption in micropores, e.g. zeolites occurs at relative pressures substantially lower than in case of adsorption in mesopores. Adsorption measurements using nitrogen at 77.4 K is difficult, because the filling of nm pores occurs at P/Po of 10-7 to 10-5, where the rate of diffusion and adsorption equilibration is very slow.

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**Advantages of Using Argon**

Advantage to analyze such narrow micropores by using argon at liquid argon temperature (87.3 K). Argon fills these micropores (0.5 – 1nm) at much higher relative pressures (i.e., at relative pressures 10-5 to 10-3) compared to nitrogen.

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**Advantages of Higher Temperature & Pressure**

Accelerated diffusion. Accelerated equilibration processes. Reduction in analysis time.

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**Argon Adsorption at 87.3 K versus Nitrogen Adsorption at 77.4 K**

The different pore filling ranges for argon adsorption at 87.3K and nitrogen adsorption at 77.4K in faujasite-type zeolite are illustrated above.

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**Micropore Size Calculation**

Difficulties are associated with regard to the analysis of micropore adsorption data. Classical, macroscopic, theories [1] like DR and semiempirical treatments such those of HK and SF do not give a realistic description of micropore filling This leads to an underestimation of pore sizes for micropores and even smaller mesopores [2]. [1] F. Rouquerol, J. Rouquerol & K. Sing, Adsorption by Powders & Porous Solids, Academic Press, 1999 [ 2 ] P. I Ravikovitch, G.L. Haller, A.V. Neimark, Advcances in Colloid and Interface Science , 203 (1998)

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New Calculation To overcome the above mentioned problems we introduce a new method for micropore analysis based on a Non-local Density Functional Theory (NLDFT) model by Neimark and Co-workers [3-5]. The new DFT-method is designed for micro-mesopore size characterization of zeolitic materials ranging in size from 0.44 to 20 nm using high-resolution low-pressure argon adsorption isotherms at 87.3 K. [3] P.I. Ravikovitch, G.L. Haller, A.V. Neimark, Advances in Colloid and Interface Science, 76 – 77 (1998), [4] A.V. Neimark, P.I Ravikovitch, M. Gruen, F. Schueth, and K.K. Unger, J. Coll. Interface Sci., 207, (1998) 159 [5] A.V. Neimark, P.I. Ravikovitch, Microporous and Mesoporous Materials (2001) 44-45, 697

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**Systematic, Experimental Study**

To evaluate the application of argon sorption for micro- and mesopore size analysis of zeolites and mesoporous silica materials including novel mesoporous molecular sieves of type MCM-41 and MCM-48. The sorption isotherms were determined using a static volumetric technique Samples were outgassed for 12 h under vacuum (turbomolecular pump) at elevated temperatures (573 K for the zeolites and 393 K for MCM-41/MCM-48).

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Results Argon adsorption isotherms at 87 K on MCM-41, ZSM-5 and their mixture.

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**Results Adsorption, [mmol/g] 5 10 15 20 25 0.000001 0.00001 0.0001**

5 10 15 20 25 0.0001 0.001 0.01 0.1 1 P/Po Adsorption, [mmol/g] MCM-41 ZSM-5 50-50

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**ZSM D, [Å] dV/dD [cm3/g Vcum, [cm3/g] histogram integral 0.02 0.04**

0.02 0.04 0.06 0.08 0.1 0.12 1 10 100 1000 D, [Å] dV/dD [cm3/g -0.05 0.05 0.15 0.2 0.25 0.3 0.35 0.4 Vcum, [cm3/g] histogram integral

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**MCM histogram integral D, [Å] 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16**

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 1 10 100 1000 D, [Å] dV/dD [cm3/g] /g] 0.3 0.4 0.5 0.6 0.7 Vcum, [cm3/g] histogram integral

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**Evaluation of DFT Algorithm**

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**Pore Size Distribution**

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1 10 100 1000 D, [Å] dV/dD [cm3/g] 0.2 0.3 0.4 0.5 0.6 Vcum [cm3/g] histogram integral

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Discussion Argon sorption at 77 K is limited to pore diameters smaller than 12 nm. i.e. no pore filling/pore condensation can be observed at this temperature for silica materials containing larger pores. This lack of argon condensation for pores larger than ca. 12 nm is associated with the fact, that 77 K is ca. 6.8 K below the bulk triple point [4,5] . [4] M. Thommes, R. Koehn and M. Froeba, J. Phys. Chem. B (2000), 104, 7932 [5] M. Thommes, R. Koehn and M. Froeba, Stud. Surf. Sci. Catal., (2001),

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Discussion These limitation do not exist for argon sorption at its’ boiling temperature, i.e. ca. 87 K. Pore filling and pore condensation can be observed over the complete micro- and mesopore size range .

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Discussion Results of classical, and semi-empirical methods (e.g., BJH, SF etc) indicate that these methods underestimate the pore size considerably. Deviations from the DFT-results are often in a range of ca. 20 % for pore diameters < 10 nm.

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Summary Our results indicate that argon sorption data at 87 K combined with the new NLDFT-methods provides a convenient way to achieve an accurate and comprehensive pore size analysis over the complete micro-and mesopore size range for zeolites, catalysts, and mesoporous silica materials.

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Acknowledgements Special thanks go to Alex Neimark and Peter Ravikovitch at TRI Princeton, New Jersey, USA.

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**References to research work of nitrogen, argon and krypton in MCM-48/MCM-41 materials**

(1) M. Thommes, R. Koehn and M. Froeba, “ Systematic Sorption studies on surface and pore size characteristics of different MCM-48 silica materials”, Studies in Surface Science and Catalysis 128, 259 (2000) (2) M. Thommes, R. Koehn and M. Froeba, “Sorption and pore condensation behavior of nitrogen, argon and krypton in mesoporous MCM-48 silica materials” J. Phys. Chem. B 104, 7932 (2000) (3)M. Thommes, R. Koehn and M. Froeba, “Sorption and pore condensation behavior of pure fluids in mesoporous MCM‑48 silica, MCM-41 silica and controlled pore glass, Studies in Surface Science and Catalysis, 135, 17 (2001) (4)M. Thommes, R. Koehn and M. Froeba, “Characterization of porous solids: Sorption and pore condensation behavior of nitrogen, argon and krypton in ordered and disordered mesoporous silica materials (MCM-41, MCM-48, SBA-15, controlled pore glass, silica gel) at temperatures above and below the bulk triple point”, Proceedings of the first topical conference on nanometer scale science and engineering” (G.U. Lee, Ed) AIChE Annual Meeting, Reno, Nevada, November 4-9, 2001 (5)M. Thommes, R. Koehn and M. Froeba, “Sorption and pore condensation behavior of pure fluids in mesoporous MCM-48 silica, MCM-41 silica and controlled pore glass at temperatures above and below the bulk triple point”, submitted to Applied Surface Science, (2001)

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**Rapid Micropore Size Analysis by CO2 Adsorption**

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**CO2 Adsorption at 0oC on Carbon**

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**RAPID MICROPORE ANALYSIS**

The advantages of micropore analysis with Quantachrome’s Density Functional Theory (DFT) and CO2 include: Speed of analysis; with the higher diffusion rate at K, analysis times are reduced as much as 90%. Carbon dioxide at K permits probing pores from about 2 angstroms (0.2 nm).

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DFT ADVANTAGE DFT has recently been applied to describe the behavior of fluids that are confined in small pores. The current popular gas sorption models, e.g. BJH, HK, SF, DA, etc., assume that the density of the adsorbed phase remains constant, regardless of the size of the pores that are being filled. Packing considerations suggest that these models are less than satisfactory for analyses of pores less than 2 nm.

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DFT “Fitting” For a given adsorbate-adsorbent system, DFT calculates the most likely summation of "ideal isotherms“ calculated from "ideal pores" of fixed sizes needed to match the experimental results.

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CO2 for Speed! Typically, micropore analyses with nitrogen as adsorbate will require 24 hours or more to run. Using carbon dioxide as adsorbate provides several advantages. Carbon dioxide molecules are slightly thinner than nitrogen molecules (2.8 angstroms radius vs angstroms) and will fill smaller pores than nitrogen. The use of carbon dioxide allows the measurements to be made at K, typically with an ice/water bath. There is no longer any need to provide and maintain or replenish a level of liquid nitrogen during the analysis.

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CO2 Benefits At this temperature, the diffusion rate of molecules moving through small and tortuous micropores is much higher than at 77.35K. This so-called "activated adsorption" effect led to the popularization of the use of carbon dioxide to characterize carbonaceous material since the early 1960s.

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CO2 Benefits This higher diffusion rate is responsible for reducing the analysis time to a few hours for a complete adsorption experiment. The faster rate also provides for the possibility of using larger samples than with nitrogen adsorption, thus reducing sample weighing errors. Pore size distributions thus obtained are comparable to those from a 24-hour nitrogen/77.35K analysis.

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N2 77K: 40 hours

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**CO2 adsorption at 273K: 2.75 hours**

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CO2 Adsorption at 0oC Density Functional Theory Micropore Distribution

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CO2 Adsorption at 0oC Monte Carlo Simulation Micropore Distribution

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**How to do it? Hardware requirements for this new method are minimal:**

a wide- mouth dewar and a water-level sensor. The proprietary Quantachrome Autosorb® software provides the DFT data reduction capabilities to do the rest. Pore size distributions from about 2 angstroms can be determined from the data taken at K. Currently, calculation parameters are optimized for studies on carbon surfaces.

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**BIBLIOGRAPHY for Rapid Micropore Size Analysis by CO2 Adsorption**

1. J. Garrido, A. Linares-Solano, J.M. Martin-Martinez, M. Molina-Sabio, F. Rodriguez-Reinoso, R. Torregosa Langmuir, 3, 76, (1987) 2. F. Carrasco-Martin, M.V. López-Ramón, C. Moreno-Castilla. Langmuir, 9, 2758 (1993) 3. P. Tarazona. Phys.Rev.A 31, 2672 (1985) 4. N.A. Seaton, J.P.R.B. Walton, N. Quirke. Carbon, 27, 853 (1989) 5. C. Lastoskie, K.E. Gubbins, N. Quirke. J.Phys.Chem., 97, 4786 (1993) 6. J.J. Olivier. Porous Materials 2, 9 (1995) 7. P.I. Ravikovitch, S.C. Ó Domhnaill, A.V. Neimark, F. Schüth, K.K. Unger. Langmuir, 11, 4765 (1995) 8. A.V. Neimark, P.I. Ravikovitch, M. Grün, F. Schüth, K.K. Unger. COPS-IV, 1997 (in press) 9. P.I. Ravikovitch P.I., D. Wei, W.T. Chuen, G.L. Haller,A.V. Neimark. J.Phys.Chem., May 1997 10. E.J. Bottani, V. Bakaev, W.A. Steele. Chem.Eng.Sci. 49, 293 (1994) 11. M.M. Dubinin. Carbon 27, 457 (1989)

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Micropore Size Calculations

Micropore Size Calculations

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