Determination of Micropore Size Distribution from Grand Canonical

Langmuir 1997, 13, 2795-2802

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Determination of Micropore Size Distribution from Grand Canonical Monte Carlo Simulations and Experimental CO2 Isotherm Data S. Samios, A. K. Stubos,* and N. K. Kanellopoulos NCSR Demokritos, 15310 Ag. Paraskevi Attikis, Greece

R. F. Cracknell,† G. K. Papadopoulos, and D. Nicholson Department of Chemistry, Imperial College of Science, Technology & Medicine, London SW7 2AY, U.K. Received December 17, 1996. In Final Form: March 3, 1997X A method for the determination of micropore size distribution is developed based on grand canonical Monte Carlo (GCMC) simulations and measured isotherm data. The technique is applied in the case of a microporous carbon membrane for which the CO2 isotherm at 195.5 K has been experimentally obtained. Simulations in slit-shaped graphitic pores of various sizes have been run offering insight to the structure of the CO2 molecules packing in the individual pores at different pressure levels. On the basis of these results and a postulated pore size distribution (PSD) function, the isotherm can be reconstructed and compared with its experimental counterpart. The PSD that gives the best match between measured and computed isotherm data is thus determined. The most probable pore size agrees with that found by conventional nitrogen porosimetry and the Dubinin-Radushkevich (DR) approach. The present method exhibits high sensitivity to changes in the range or mean value of the resulting optimal PSD. The GCMC simulations have also provided useful evidence about the effect of including the quadrupole moment in the model and the densification process in the micropores.

1. Introduction Pores, and especially micropores (of size less than 2nm according to IUPAC classification), play an essential role in determining the physical and chemical properties of industrially important materials like adsorbents, catalysts, soils, biomaterials, etc. Their characterization (in terms of pore size distribution and other structural information) is indispensable for the utilization and design of improved porous systems in several applications. While for mesopores and macropores there exist a host of more or less established characterization methods, the assessment of microporosity is much less advanced despite the recent interest on microporous systems like zeolites, activated carbons, and clay minerals.1 A commonly employed method is based on the thermodynamic approach of Dubinin using nitrogen adsorption at 77 K. He assumed that the micropore filling process is governed by a socalled adsorption potential characterizing the adsorbed molecules and that the micropore size distribution is Gaussian. The Dubinin-Radushkevich2 (DR) equation relates the adsorbed amount per unit micropore volume to the temperature, relative pressure, and the characteristic energy and affinity coefficients (which are in turn related to the isosteric heat of adsorption). The DR method has been subject to criticism mainly because the mechanism of molecular adsorption in micropores is still under active debate. Conventional macroscopic descriptions of states of matter are generally invalid in micropores and this adds to the accumulating evidence that none of the usually employed adsorption methods of pore characterization is entirely satisfactory.3 Improved approaches to * To whom correspondence should be addressed. † Present address: Shell Research & Technology Centre, Thornton, P.O. Box 1, Chester CH1 3SH, U.K. X Abstract published in Advance ACS Abstracts, April 15, 1997. (1) Kaneko, K. J. Membr. Sci. 1994, 96, 59. (2) Dubinin, M. M. Chem. Rev. 1960, 60, 235. (3) Nicholson, D. J. Chem. Soc., Faraday Trans. 1994, 90 (1), 181.

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the problem, based on molecular level theories, need to be developed.4 In recent years, it has become apparent that density functional theory5,6 in a sufficiently elaborate form can provide an accurate description of simple (in practice atomic) fluids in geometrically simple confined spaces. Spherical nitrogen models have been used in this context to investigate phase diagrams7 and develop practical methods for the evaluation of the pore structure over a wide range of pore sizes.8-10 Molecular simulation, and in particular the Monte Carlo technique, is an alternative approach which has been found to be a promising tool in the study of adsorption of pure or multicomponent gases in zeolites and other microporous solids.11-16 In the present work, the method is used in its grand ensemble variant in combination with experimental isotherm data to characterize microporous media and obtain the corresponding pore size distribution (PSD). Specifically, the mean CO2 density inside a single slit-shaped graphitic pore of given width is found on the basis of grand canonical Monte Carlo (GCMC) simulations for a predefined temperature and different relative pressures. Starting from (4) Nicholson, D. J. Chem. Soc., Faraday Trans. 1996, 92 (1), 1. (5) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1992, 96, 845. (6) Kierlik, E.; Rosinberg, M. L. Phys. Rev. A 1991, 44, 5025. (7) Balbuena, P.; Gubbins, K. E. Langmuir 1993, 9, 1801. (8) Seaton, N. A.; Walton J. P. R. B.; Quirke, N. Carbon 1989, 17, 853. (9) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (10) Aukett, P. N.; Quirke, N.; Riddiford, S.; Tennison, S. R. Carbon 1992, 30, 913. (11) Woods, G. B.; Panagiotopoulos, A. Z.; Rowlinson, J. S. Mol. Phys. 1988, 49, 49. (12) Karavias, F.; Myers, A. L. Mol. Simul. 1991, 8, 51. (13) Razmus, D. M.; Hall, C. K. AIChE J. 1991, 37, 5. (14) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Phys. 1993, 80 (4), 885. (15) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Simul. 1994, 13, 161. (16) Cracknell, R. F.; Nicholson, D. J. Chem. Soc., Faraday Trans. 1994, 90 (11), 1487.

© 1997 American Chemical Society

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Table 1. Potential Parameters Used in the Simulation pair

σ (nm)

CO2-CO2 (3CLJ)

σCC ) 0.2824 σOO ) 0.3026 σCO ) 0.2925 C(graphite)-C(graphite) 0.340 0.3112 C(graphite)-C(CO2) 0.3213 C(graphite)-O(CO2) a

/K (K)

l (nm)

ref

CC/K ) 26.3 OO/K ) 75.2 0.2324 17, 18 CO/K ) 44.5 28.0 19 27.1 a 45.9

By means of the Lorenz-Berthelot rules.

an initial PSD guess, it is then possible to produce a computed CO2 sorption isotherm and compare it to the measured one. The PSD of a microporous activated carbon membrane, obtained by nitrogen porosimetry via the conventional (DR) approach, is employed for the sake of comparison. 2. Potential Energy Model The realistic character of simulations and the accuracy of the results depend largely upon the potential energy model used. Here, the carbon dioxide molecule is modeled as Lennard-Jones interaction sites on the atoms plus point charges to account for the quadrupole (three-center LJ model17,18 ). The Lennard-Jones (12-6) potential between sites i and j on two molecules is given by

uij ) -4ij[(σij/rij)6 - (σij/rij)12]

(1)

The interactions are cut (but not shifted) at 2.0 nm. Because of this relatively large cutoff and the confinement of the molecules in the micropores, no long range corrections have been employed. The graphitic surface is treated as stacked planes of Lennard-Jones atoms. The interaction energy between a fluid particle and a single graphitic surface is given by the 10-4-3 potential of Steele19 as

usf(z) ) 2πdssfσsf2∆{2/5(σsf/z)10 3

where ∆ is the separation between graphite layers and ds is the number of carbon atoms per unit volume in the graphite layer. The values used for ∆ and ds are 0.335 nm and 114 nm-3, respectively. The solid-fluid LennardJones parameters σsf and sf are calculated by combining the graphite parameters of Table 1 with the appropriate fluid parameters according to the Lorentz-Berthelot rules. The “10-4-3” potential is obtained by the summation of the Lennard-Jones potential between an atom site in the adsorbate molecule and each carbon atom of the individual graphite planes. The external field, u(1) for a single Lennard-Jones site in a slit pore of width H is the sum of the interactions with both graphitic surfaces and can be expressed as

u(1) ) usf(z) + usf(H - z)

temperatures.3 The effective pore width H′ (which is determined by the experiments) is in general given by

H′ ) H - ∆

(4)

Alternatively, it has been proposed20,21 to use an estimation of the effective pore width based on the reduction in accessible by the adsorbate pore volume, caused by the physical size of the carbon atoms. Taking as the distance of closest possible approach of adsorbate to adsorbent the position z0, where the potential function for a plane wall passes through zero, results in

H′ ) H - 2z0 + σg

(5)

where σg is the hard sphere diameter of an adsorbate atom. 3. Simulation Model

(σsf/z) - σsf/3∆(0.61∆ + z) } (2) 4

Figure 1. Comparison of experimental data for CO2 adsorption on Vulcan 3G at 313 K with GCMC simulation using the potential parameters of Table 1.

(3)

where H is the C center-C center separation across the pore. This equation ignores the surface corrugation, which is unlikely to significantly affect the results at high enough (17) Murthy, C. S.; O’Shea, S. F.; McDonald, I. R. Mol. Phys. 1983, 50, 531. (18) Hammonds, K. D.; McDonald, I. R.; Tildesley, D. J. Mol. Phys. 1990, 70, 175. (19) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974.

The grand canonical Monte Carlo (GCMC) method is ideally suited to adsorption problems because the chemical potential of each adsorbed species is specified in advance.22 At equilibrium, this chemical potential can then be related to the external (bulk) pressure making use of an equation of state. Consequently, the independent variables in the GCMC simulations are the temperature, the pressure, and the micropore volume, i.e., a convenient set since temperature and pressure are the adsorption isotherm independent variables. Therefore, the adsorption isotherm for a given pore can be obtained directly from the simulation by evaluating the ensemble average of the number of adsorbate molecules whose chemical potential equals that of a bulk gas at a given temperature and pressure. Specifically, the GCMC technique establishes an algorithm generating a Markov chain of grand ensemble configurations. In this procedure, three types of trial are used,22,23 i.e., attempts to move (translate or reorient) particles, attempts to delete particles, and attempts to create particles in the simulation box. A decision is made on whether to accept each trial or return to the old configuration based on a probability which in (20) Kaneko, K.; Cracknell, R. F.; Nicholson, D. Langmuir 1994, 10 (12), 4606. (21) Cracknell, R. F.; Nicholson, D.; Tennison, S. R.; Bromhead, J. Adsorption 1996, 2, 193. (22) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (23) Norman, G. E.; Filinov, V. S. High Temp. (Transl. Teplofiz. Vys. Temp.) 1969, 7, 216.

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Figure 2. Computed CO2 isotherms for different pore widths at 195.5 K.

the case of an attempted move takes the form

pmove ) min[exp(-∆U/kT); 1]

(6)

where ∆U ) Unew - Uold is the difference in the potential energies of the new and old configurations. A detailed presentation of the method is given elsewhere.13,22,24 Periodic boundary conditions have been applied in the directions other than the width of the slit. For a given simulation, the size of the box (i.e., the two dimensions other than H) was varied in order to ensure that sufficient particles (ca. 200-450) remained in the simulation at each pressure and was always greater than twice the cutoff distance. Statistics were not collected over the first 2 × 106 configurations to assure adequate convergence of the simulation. The uncertainty on the final results (ensemble averages of the number of adsorbate molecules in the box (24) Allen, M.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, 1987.

and the total potential energy) is estimated to be less than 4%. Typical calculations for a single point require between 1.7 and 4.2 h of CPU time on a Convex 3820. 4. GCMC Simulation Results An initial validation of the adsorbate-adsorbent potential functions can be made by comparing isosteric heats of adsorption at zero coverage with experimental data. The theoretical heat for CO2 adsorption on graphite at 313 K is found as 11.4 kJ/mol using Monte Carlo integration with 1 × 107 trial insertions and a large slit width. The experimental value reported in the literature25 is 12.4 kJ/mol. A more stringent test is to compare experimental and simulated isotherms on nonporous surfaces. A good level of predictive agreement has been found between measured CO2 isotherms (at 313 K) for a well-defined nonporous carbon21,25 (Vulcan 3G) and GCMC results of the code used in this study (Figure 1). For these (25) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121.

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Figure 3. Density versus pore width for different pressures at 195.5 K (from GCMC simulations).

comparisons (and the others to follow in this work) with experimental isotherms, the simulation results have been corrected so as to give an adsorption excess using H′ from eq 4 or 5. Although it is not possible to calculate excess quantities in a completely unambiguous way, the number of molecules which would occupy the pore space in the absence of adsorption forces has been subtracted from the average number of molecules obtained in a simulation.20 This correction becomes insignificant as the temperature decreases. The GCMC method has been extensively used to simulate CO2 sorption isotherms at 195.5 K in single graphitic pores of various sizes in the micropore range. The selection of the fluid and the temperature is based on practical considerations regarding the relative ease of obtaining experimental isotherms at dry ice temperature with a molecule that is known for its ability to enter into the narrow microporosity and the realistic equilibration times required. In addition, the selected temperature is conveniently low enough to avoid large effects from excess quantities calculations and, at the same time, high enough to ignore corrugation of the solid surface. The detailed CO2 density profiles across the pore have been computed for the whole range of micropore widths (from 0.65 to 1.95 nm, in steps of 0.1 nm). From this information, the isotherms and the average fluid density in the micropores as a function of the pore width at various pressures are plotted in Figures 2 and 3, respectively. It

is observed in Figure 2 that at low chemical potential (or equivalently pressure), the adsorbate density is highest in the smaller pores, while at high chemical potential the larger pores show higher adsorptive capability. This reversal of preference can be explained with reference to the adsorbate-adsorbate (aa) and adsorbate-pore (ap) interaction energies.26 At low loadings, the adsorbates tend to occupy the energetically most favorable positions in the pore and the aa interaction is much smaller than the ap interaction. The attractive potentials due to each wall overlap most in the smallest pore, resulting in deep energy wells. In the wider pores at high loadings, molecules can occupy the central region, as well as the wall regions of the pore. This increased packing efficiency means that higher maximum densities can be attained in the pore.26 Figure 2 also indicates that in all cases a sudden jump in density occurs at a pressure (fugacity) value that varies from less than 0.001 bar to almost 0.5 bar as the pore size increases. A remarkable rise in density is identified in Figure 3 as H increases at constant pressure. This takes place at about 1.0 nm for pressures above 0.01 bar and reflects the formation of a new layer of fluid molecules in the pore (see below). As the pores become larger at constant pressure, the densities fall to the bulk value corresponding to the prevailing pressure (Figure 3). (26) Keffer, D.; Davis, H. T.; McCormick, A. V. Adsorption 1996, 2, 9.


a

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a

b b

Figure 4. Singlet density of carbon atom on CO2 and angular distribution function at 195.5 K as a function of distance from pore center: (a) with quadrupole; (b) with no quadrupole.

The above features are exemplified by considering the singlet density of the position of the carbon atom in CO2 under various conditions. The singlet density is the local number density at a distance z from the pore center. Computationally it is obtained by dividing the pore width into several parts (presently 100) and keeping record of the number of times a molecule enters each part during the simulation (see Figures 4-9, with z ) 0 at the pore centre). The corresponding local orientation of the molecules is also indicated in these figures as the ensemble average of the cosine of the angle of the molecule axis relative to the normal to the pore walls. It can be seen that for H ) 0.95 nm the CO2 molecule tends to lie flat against the wall and this energetically favorable configuration (all three atoms of the molecule in the potential minimum) leads to high average densities in the pore even at relatively low loadings (Figures 4a and 5a). At both pressures represented in these figures (0.01 and 0.1 bar) only a small proportion of the molecules span the pore with the oxygen atoms lying in the potential well of each wall and the carbon atom near the pore center (high cosφ values close to z ) 0 associated with low local densities). At low loadings where ap interactions prevail, a larger pore (H ) 1.35 nm in Figure 6a) exhibits similar profiles (although the adsorbed phase structure is more disordered away from the wall) but the absolute density values are now considerably lower. The situation is reversed at higher pressures (0.1 bar in Figure 7a) where the proportion of CO2 molecules lying almost normal to the pore walls has increased substantially (at z around 0.25 nm) while a new layer of molecules is formed at the pore center (their orientation being almost parallel to the walls). This configuration is characterized by enhanced average density. The observed abrupt fluid densification in the isotherms of Figure 2 is illustrated in Figure 8 for H ) 1.75 nm. As the pressure increases from 0.3 to 0.4 bar, the formation of two new molecular layers in the pore (at a distance of about 0.18 nm from the center) results in a significant density rise.


a

b


The singlet density profiles offer interesting evidence relating to the trends shown in Figure 3. As the pore size varies under constant pressure (e.g., 0.1 bar), the initial (energetically favorable) configuration with two layers of molecules lying flat on the walls (Figure 5a, H ) 0.95 nm) changes gradually to that of Figure 7a (H ) 1.35 nm)

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a

a

b

b


where efficient packing with a new layer at the pore center gives rise to increased average density. At even higher pore widths the density starts falling toward its bulk value as the pressure is insufficient to sustain the additional (internal) layers of molecules (Figure 9, H ) 1.45 nm). The effect of suppressing the quadrupole has also been investigated. In previous work on nitrogen adsorption in slit pores,20 the quadrupole interactions were found to have no particular significance at high (ambient) temperatures. In Figure 10 the average fluid densities in the micropores without quadrupole may be compared with those in Figure 3. It can be noted that densities tend to become generally lower. Differences are more pronounced at sizes larger than 1.25 nm where the isotherms become smoother while the decline of density toward the bulk value starts at lower H at a given pressure. In Figures 4b-7b the corresponding singlet density and orientation profiles are included for the nonpolar CO2. The effect of quadrupoles ranges from minimal (Figures 4-6) to very important (Figure 7) depending on the conditions. In the latter case, the quadrupole moment contributes essentially to the formation of the central layer (by lowering the potential barrier there) and its suppression leads to the elimination of this density enhancing feature. 5. Micropore Size Distributions A microporous carbon membrane has been tested using the Quantachrome Autosorb-1 Nitrogen porosimeter equipped with Krypton upgrade. The total pore volume has thus been found at 77 K (0.54 cm3/g). It compares favorably to the value estimated from the measured CO2 isotherm (at 195.5 K). To start with, a skewed triangular pore size distribution has been postulated making sure that the total pore volume equals the experimentally determined value. The micropore range (from 0.5 to 2.0 nm) has been subdivided in equidistant spaces with 0.1 nm width. The most probable pore size (slit width) and

Figure 8. Singlet density of carbon atom on CO2 and angular distribution function at 195.5 K as a function of distance from pore center, H ) 1.75 nm: (a) P ) 0.3 bar; (b) P ) 0.4 bar.

Figure 9. Singlet density of carbon atom on CO2 and angular distribution function at 195.5 K as a function of distance from pore center.

the standard deviation of the assumed distribution have been varied systematically. For each case, the fraction of the total pore volume associated with each class of pores (the aforementioned subdivisions of the overall pore range) has been calculated (keeping the total pore volume of the assumed distribution equal to the measured one) and the amount of gas (CO2) adsorbed in every class at a certain pressure has been computed using the GCMC code. In this way, a computed isotherm has been reconstructed up to pressure values of 0.9 bar. By comparison of these results to the corresponding experimental isotherm counterpart, the most suitable micropore size distribution has been selected (triangular shape in Figure 11 where the PSD obtained from nitrogen porosimetry at 77 K using the DR method27 is also shown). It must be noted that such a comparison between simulated and measured isotherms requires a relation between H and H′ (eq 4 was used here). The most probable pore size (0.65-0.75 nm) found is in good agreement with the independent estima(27) Dubinin, M. M.; Astakhov, V. A. Adv. Chem. Ser. 1971, No. 102, 69.


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Figure 10. Density versus pore width for different pressures (with no quadrupole).

for the optimal triangular distribution and for two other cases in which the width of the distribution and the most probable pore size have been varied (again respecting the total pore volume found experimentally). The above described trial and error procedure can be freed of the need to specify a certain type of pore size distribution function. The numerical solution of a minimization problem under certain constraints has been carried out in order to provide the optimal distribution that fits best the selected segment of the measured isotherm data (Figure 12). This involves the minimization of the function k

Figure 11. Pore size distributions obtained by nitrogen porosimetry at 77 K (DR approach) and by the present method (assuming triangular shape of PSD).

tion of 0.7-0.8 nm coming from tests performed by BP staff28 at Sunbury, U.K.. The sensitivity of the method is demonstrated (Table 2) by obtaining significant deviations from the measured isotherm at certain pressures (28) Tennison, S. R. Personal communication.

Qi -

dijVj ∑ j)1

at different pressure values Pi. In the above, Qi is the experimentally found sorbed amount at pressure Pi, dij is the calculated fluid density in a pore of width Hj at the same pressure Pi, and Vj represents the volume of the pores with size Hj (as j changes from 1 to k, the whole micropore range from 0.65 to 1.95 nm is spanned with a step of 0.1 nm). The resulting elements of the vector Vj

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Table 2. Sensitivity of the Method (%)a P ) 0.01 bar optimal triangular distribution narrower range of distribution from 0.6-1.80 nm to 0.6-1.50 nm change most probable H value from 0.85 to 1.05 nm

1.5 30.8 -28

P ) 0.1 bar 0.4 8.4 -1.6

a Percentage difference of computed and measured adsorbed quantity.

Figure 14. Optimal pore size distributions using eqs 4 and 5 to relate physical and chemical pore widths.

Figure 14, where this time (for comparison with experiments) 0.24 nm should be subtracted from H to define H′. 6. Concluding Remarks

Figure 12. Experimental and computed isotherm data from optimal PSD of Figure 13.

Figure 13. Optimal pore size distributions with and without quadrupole.

are subject to two constraints. They should be nonnegative and their sum should be equal to the measured total pore volume. A software routine solving linearly constrained linear least-squares problems based on a twophase (primal) quadratic programming method (NAG library) has been implemented, and the resulting distribution is included in the form of a histogram in Figure 13. It is interesting to note that the PSD exhibits a broad band of prevalent pore sizes between 0.75 and 1.15 nm. The effect of neglecting the quadrupole on the estimated pore size distribution is also presented in the same figure. It can be seen that the inclusion of the quadrupole results in a wider distribution containing relatively more pores at the high end of the micropore range. This is due to the aforementioned general decrease of the density values (especially in large pores) when the quadrupole moment is suppressed in the simulations. Note again that in order to convert the physical width H to the experimentally meaningful chemical width H′, relation 4 has been used. If we adopt eq 5 instead, the resulting PSD is shown in

A method for the determination of micropore size distribution has been presented based on GCMC simulations and measured isotherm data. The technique is applied in the case of a microporous carbon membrane for which the CO2 isotherm at 195.5 K has been experimentally obtained. Simulations in slit-shaped graphitic pores of various sizes have been run, offering insight to the structure of the CO2 molecules packing in the individual pores at different pressure levels. Based on these results and a postulated PSD, it is possible to reconstruct the isotherm and compare it with its experimental counterpart. The PSD that gives the best fit between measured and computed isotherm data is determined after a few iterations. The most probable pore size agrees with that found by conventional nitrogen porosimetry and the DR approach. The present method exhibits high sensitivity in changes of the range or mean value of the resulting PSD. The GCMC simulations have also provided useful evidence about the densification process in the micropores and the effect of omitting the quadrupole moment. The main uncertainties are the reliability of the potential parameters used in the simulations, the possible effect of corrugated potentials, the estimation of the excess adsorption from the computed densities, the relationship between H and H′ and the assumption concerning pore geometry. For the application described in this work, the intermolecular potentials employed have been validated against measured isosteric heats of adsorption and sorption data from nonporous carbons. The temperature chosen (195.5 K) is high enough to safely neglect surface corrugation and at the same time low enough to render the effect of excess adsorption corrections almost negligible. However, the relation of H to H′ needs further investigation while the influence of using other pore geometries (e.g., cylindrical) is currently under study by the authors. In addition, we are attempting to validate the method by estimating the PSD of the same material based on calculations and measurements of CO2 isotherms at higher (near the ambient) temperatures or on the use of nitrogen instead of carbon dioxide as the adsorbate. Acknowledgment. This work has been partly supported by the JOULE Program of the European Commission (Contract JOF3-CT95-0018). LA962111A

Determination of Micropore Size Distribution from Grand Canonical

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