Determination of the Pore Size Distribution and Network Connectivity

M. Pérez-Mendoza , C. Schumacher , F. Suárez-García , M.C. Almazán-Almazán , M. Domingo-García , F.J. López-Garzón , N.A. Seaton. Carbon 2006 44 (4), ...
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Langmuir 1997, 13, 4435-4445

4435

Determination of the Pore Size Distribution and Network Connectivity in Microporous Solids by Adsorption Measurements and Monte Carlo Simulation M. V. Lo´pez-Ramo´n,†,‡ J. Jagiełło,§ T. J. Bandosz,| and N. A. Seaton*,† Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, U.K., Department of Chemical Engineering and Materials Science, Syracuse University, 320 Hinds Hall, Syracuse, New York 13244-1190, and Department of Chemistry, The City College of The City University of New York, Convent Avenue at 138th Street, New York, New York 10031 Received February 24, 1997. In Final Form: May 5, 1997X

We carried out Monte Carlo simulations of the adsorption of three speciessCH4, CF4, and SF6sin model pores of various sizes. By comparing the simulated isotherms, integrated over a pore size distribution, with experimental isotherms for the adsorption of these species on a microporous carbon, estimates of the pore size distribution (PSD) of the carbon were obtainedsone for each adsorptive. Because the adsorptives have different molecular sizes and strengths of interaction with the adsorbent, they probe different ranges of pore size; each adsorptive thus provided a partial PSD. By combining the partial PSDs, we were able to obtain a much more complete PSD than could be obtained with a single adsorptive. Comparison of the PSDs obtained with CF4 and SF6, which substantially overlap, shows molecular sieving. By analyzing these two PSDs, using percolation theory, we extracted an estimate of the connectivity of the pore network, an important determinant of the transport properties of the solid. Our approach is applicable to microporous solids in general.

Introduction The designer, manufacturer, or user of a porous adsorbent or catalyst support often wishes to obtain information about the pore structure of the solid. Adsorption is widely used as a probe of pore structure. The measured adsorption behavior of a sample reflects, in an aggregate way, the adsorption behavior of individual pores. If adsorption in individual pores can be quantified by a sufficiently realistic mathematical model, this model can be used to extract information about the pore structure of the solid. The pore size distribution (PSD) is closely related to both the equilibrium and kinetic properties of the solid (pure species and multicomponent adsorption equilibrium, diffusion and other transport properties, and, if the solid is reactive (as a catalyst or a reactant in a noncatalytic reaction), the reaction rate). Because of this, the PSD has been used since the early 1950s as a predictor or correlator of the performance of porous solids in industrial applications. For many applications the connectivity of the pore network (i.e. the way in which the pores are connected together) is also important. A convenient way to quantify the connectivity is in terms of the mean coordination number of the pore network (i.e. the mean number of pores meeting at an intersection), Z. The equilibrium adsorption of a mixture in which all the species are smaller than all the pores present in the solid is an important problem in which connectivity is irrelevant (provided it is truly an equilibrium process under the conditions of interest). In many other processes of industrial and scientific interest, however, connectivity * Author to whom correspondence should be addressed. Telephone: (+44) 1223 334786. Fax: (+44) 1223 334796. E-mail: [email protected]. † University of Cambridge. ‡ Current address: Departamento de Quı´mica Inorga ´ nica, Grupo de Investigacio´n en Carbones, Universidad de Granada, Granada18071, Spain. § Syracuse University. | The City College of The City University of New York. X Abstract published in Advance ACS Abstracts, July 15, 1997.

S0743-7463(97)00197-2 CCC: $14.00

plays an important role. In general, connectivity is important in any process in which the presence of constrictions in the pore network inhibits the passage of one or more adsorbed species. Examples include: (i) Diffusion. The higher the value of Z, the more readily the molecules can bypass small pores that act as “bottlenecks” in the pore network, and the higher the diffusion coefficient. (These pores need not be of close to molecular dimensions; in general, pores having a relatively small cross section (compared with the mean pore size) act as bottlenecks.) (ii) Reaction. The connectivity affects the degree of diffusional limitation of the reaction, via its effect on the effective diffusivity. For reactions in which fouling occurs, the resistance to fouling depends on the connectivity;1,2 more highly connected networks provide more alternative paths to avoid blocked pores. (iii) Molecular sieving. When the adsorptive molecules are similar in size to the pores, molecules of one or more species are excluded from some of the pores. These pores include both pores that are too small to accommodate the molecules and pores that are large enough for a molecule to enter but are rendered inaccessible by adjacent smaller pores. The relative number of pores in this second category depends on the connectivity. Although molecular sieving is in the limit an equilibrium phenomenon, when the molecular sizes of the species are similar, the connectivity has a strong influence on kinetic separation, for example in carbon molecular sieves.3,4 In this paper we concern ourselves with obtaining both the PSD and the mean coordination number of the pore network from the analysis of adsorption measurements. The pore size distribution is related to the adsorption isotherm by the Adsorption Integral Equation: (1) Sahimi, M.; Tsotsis, T. T. J. Catal. 1985, 96, 552. (2) Zhang, L.; Seaton, N. A. Chem. Eng. Sci. 1996, 51, 3257. (3) MacElroy, J. M. D.; Seaton, N. A.; Friedman, S. P. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W., Eds.; Elsevier: Amsterdam, 1996. (4) Seaton, N. A.; Friedman, S. P.; MacElroy, J. M. D.; Murphy, B. J. Langmuir 1997, 13, 1199.

© 1997 American Chemical Society

4436 Langmuir, Vol. 13, No. 16, 1997

N(P) )

∫0∞F(P,w) f(w) dw

Lo´ pez-Ramo´ n et al.

(1)

In eq 1, N(P) is the experimental number of moles adsorbed, per unit mass of adsorbent, at pressure P. F(P,w) is the “single-pore isotherm function” (the density of adsorbate in a pore of width w at pressure P) generated by the model for adsorption in individual pores. f(w) is the PSD, strictly dV/dw, where V is the total pore volume. The PSD is obtained by varying f(w), using some suitable numerical method, until eq 1 is satisfied as closely as possible. The accuracy of the PSD depends on the realism of the model for adsorption in individual pores. For sufficiently large pores, classical thermodynamics provides an adequate description of adsorption at the single-pore level. The PSD of mesoporous solids (2 nm < w < 50 nm) can be obtained by studying the condensation of a subcritical adsorptive (usually nitrogen). The condensation pressure depends strongly on the pore size, and this relationship is given by a variant of the Kelvin equation, modified to account for precondensation adsorption on the pore walls. This is the basis of the widely-used method of Barrett et al.5 (When this and similar methods were originally presented, F(P,w) was not defined explicitly and the formalism of eq 1 was not used; however, very recently Olivier6 recast the Kelvin-equation analysis in this form.) For small mesopores, the molecular texture of the adsorbate becomes important and classical thermodynamics can no longer, by itself, relate pore structure to single-pore adsorption behavior. For nitrogen adsorption at 77 K, Seaton et al.7 and Lastoskie et al.8 have shown that Kelvin-equation-based methods are already significantly in error for pore widths smaller than 5 nm or so. At some point near the upper limit of the micropore size range (w < 2 nm), the condensation transition disappears (as the pore-fluid critical point decreases with pore size) and a Kelvin-equation-based analysis is not applicable. Thus, for solids with small mesopores an alternative approach is desirable, and such an approach is necessary for microporous solids. Statistical mechanics provides the means of relating a physical model for adsorption in individual pores, expressed in terms of solid-fluid and fluid-fluid interactions, and the single-pore isotherm of eq 1. [Classical approaches also exist to obtaining the PSDs of microporous solids.9 Although these methods are grounded in the physics of adsorption, they do not provide a rigorous link between the molecular interactions and the single-pore isotherm.] Statistical-mechanical methods for obtaining single-pore isotherms may be usefully divided into two classes: theory and Monte Carlo simulation. Statisticalmechanical theories simplify the interactions present in the adsorbed phase and provide a computationally fast route to approximate solutions. Monte Carlo simulation provides a stochastic solution to the single-pore model that is, in principle, exact. That is, the accuracy of the simulation can be made arbitrarily high by carrying out a sufficiently long run, in a simulation cell that is large enough to make the effect of the finite system size negligible. In practice, with modern computers, Monte Carlo simulations of the adsorption of small molecules can be carried out in a sufficiently large simulation cell, (5) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc. 1951, 73, 373. (6) Olivier, J. P. Paper presented at the 4th International Symposium on the Characterisation of Porous Solids, September 15-18, 1996. (7) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (8) Lastoskie, C.; Gubbin, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4586. (9) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982.

and for enough steps, to produce results that are accurate enough for practical purposes. (Of course, to say that a solution is “essentially exact” does not imply precise correspondence with reality, which depends also on the realism of the single-pore model.) The drawback of Monte Carlo simulation, compared with statistical-mechanical theory, is that it is computationally much slower. The first statistical-mechanical methods for PSD determination were based on theory, rather than Monte Carlo simulation. Seaton et al.7,10 used the density functional theory (DFT) of Evans and Tarazona11 to calculate single-pore isotherms for nitrogen adsorption at 77 K and used these results to obtain PSDs for microporous and mesoporous solids. Lastoskie et al.8 and Olivier et al.12 developed improved PSD methods, using a more accurate variant of density functional theory due to Tarazona.13 (Olivier et al. analyzed the adsorption of argon as well as nitrogen.) More recent applications of DFT to the determination of PSDs include the work of Neimark et al.,14 Koch et al.,15 and Sosin and Quinn16 (who used the DFT results of Tan and Gubbins17 ). In the last few years, the availability of faster computers (typically Unix workstations) has allowed Monte Carlo (MC) simulation to be used to generate the single-pore isotherms used in PSD determinations. Gusev et al.18 were the first to use MC simulation to extract PSDs from adsorption data, which they did by analyzing the adsorption of methane on an activated carbon. It is of interest to assess the realism of the PSDs obtained from this type of analysis, i.e. the extent to which the PSD obtained from the analysis reflects the structure of the real solid. A partial test of realism is to use a PSD obtained by analyzing the adsorption of a particular species at a particular temperature to predict either the adsorption of that species at a different temperature or the adsorption of another species. (This is strictly a test of consistency, rather than realism. Consistency supports, but does not prove, realism; lack of consistency indicates lack of realism.) Gusev et al.18 obtained the PSD of a microporous carbon by using MC simulation to analyze the adsorption of methane at 308 K. They then predicted methane adsorption at two higher temperatures (333 and 373 K) by integrating simulated isotherms at the higher temperatures over the PSD obtained at 308 K. In a subsequent publication, Gusev and O’Brien19 adopted a similar approach to predict the adsorption of ethane on the basis of the PSD obtained by the analysis of methane adsorption. In both these comparisons, they obtained good agreement with experiment, suggesting that there is at least a broad measure of self-consistency in the use of MC simulation to obtain PSDs. Less good agreement was found by Aukett et al.20 and Quirke and Tennison.21 They obtained the (10) Jessop, C. A.; Riddiford, S. M.; Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. In Characterization of Porous Solids (COPS-II). Proceedings of the IUPAC Symposium; Rodrı´guez Reinoso, F., Rouquerol, J., Sing, K. S. W., Eds.; Elsevier: Amsterdam, 1991; p 123. (11) Evans, R.; Tarazona, P. Phys. Rev. Lett. 1984, 52, 557. (12) Olivier, J. P.; Conklin, W. B.; v. Szombathely, M. In Characterization of Porous Solids (COPS-III). Proceedings of the IUPAC Symposium; Rodrı´guez Reinoso, F., Rouquerol, J., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; p 81. (13) Tarazona, P. Phys. Rev. A 1985, 31, 2672. (14) Neimark, A. V.; Ravikovitch, P. I.; Gru¨n, M.; Schu¨th, F.; Unger, K. K. Langmuir 1995, 11, 4765. (15) Koch, K.; v. Szombathely, M.; Neugebauer, N.; Bra¨uer, P. In Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer: Boston, 1996; p 921. (16) Sosin, K. A.; Quinn, D. F. J. Porous Mater. 1995, 1, 111. (17) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1990, 94, 6061. (18) Gusev, V. I.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815. (19) Gusev, V. I.; O’Brien, J. A. Langmuir 1997, 13, 2822. (20) Aukett, P. N.; Quirke, N.; Riddiford, S.; Tennison, S. R. Carbon 1992, 30, 913. (21) Quirke, N.; Tennison, S. R. Carbon 1996, 34, 1281.

Microporous Solids

PSDs of microporous carbons by using DFT to analyze nitrogen adsorption and then used MC simulation20 and DFT21 to predict methane adsorption. They found that the methane adsorption was substantially overpredicted. Quirke and Tennison21 attributed this discrepancy to the existence of a specific interaction between the quadrupole of the nitrogen molecule and polar sites on the adsorbent surface, whereas this interaction is not present for methane. Taking these investigations together, there is at least tentative support for the realism of PSDs obtained using MC-simulation-based methods. Unlike the PSD, which is a statistical representation of a property of individual pores, the pore network connectivity is a collective property of the pores within the solid. The analysis of connectivity thus involves two length scales. As in the PSD analysis the connection must be made between molecular interactions and single-pore adsorption behavior. In addition, the relationship between the topology of the pore network and the adsorption behavior of the solid as a whole must be made; this is the province of percolation theory.22 Seaton and co-workers23-25 have developed a method for the determination of the pore network connectivity of mesoporous solids, by the analysis of nitrogen adsorption and desorption isotherms. In their method, the PSD is obtained using as the single-pore isotherm function either the Kelvin equation24,25 or an isotherm function calculated from DFT.23 (In general, any classical or statisticalmechanical model that describes capillary condensation could be used for this purpose.) An estimate of the mean coordination number, Z, is obtained by using percolation theory to analyze the hysteresis between the adsorption and desorption isotherms. As this method is based on capillary condensation, it cannot be applied to microporous solids. In this paper, we propose a new approach to probing the pore structure of microporous solids. Monte Carlo simulation is used to analyze data from a sequence of adsorptive experiments with different adsorptives. Each experiment yields an estimate of the PSD for that solid. By considering these estimates together, it is possible to obtain a more complete picture of the true PSD than is possible with a single adsorptive. In addition, provided that at least one of the adsorptive species is of a size to experience a significant degree of molecular sieving, it is also possible to extract connectivity information from these PSD estimates. The physical basis of the connectivity analysis is illustrated in Figure 1, which shows schematically the relationship between connectivity and the PSDs measured with different adsorptives. There are four pores in this simple, model adsorbent. The smaller adsorptive species probes all the pores, and its adsorption isotherm yields the complete PSD. The larger species is excluded from the smaller pores and also from the larger pore that is “shielded” by the smaller pores. So, the PSD obtained using the larger species is (i) zero for pores that are smaller than the molecules of that species and (ii) smaller than the PSD for the smaller species above this pore size. The extent to which shielding occurs in a real pore network depends on the connectivity of the network, with the shielding effect being more pronounced for less well connected networks (i.e. networks with lower values of the mean coordination number Z). (22) Sahimi, M. Applications of Percolation Theory; Taylor and Francis: London, 1994. (23) Seaton, N. A. Chem. Eng. Sci.. 1981, 46, 1895. (24) Liu, H.; Zhang, L.; Seaton, N. A. Chem. Eng. Sci. 1992, 47, 4393. (25) Liu, H.; Seaton, N. A. Chem. Eng. Sci. 1994, 49, 1869.

Langmuir, Vol. 13, No. 16, 1997 4437

Figure 1. Schematic illustration of the effect of pore network connectivity on adsorption in microporous solids. Table 1. Lennard-Jones Parameters for the Fluid-Fluid Interactionsa

a

species

σ (nm)

/kB (K)

CH4 CF4 SF6

0.381 0.470 0.551

148.2 152.5 200.9

kB is Boltzmann’s constant.

Recently, Jagiełło et al.26 have presented data for the adsorption of methane, carbon tetrafluoride, and sulfur hexaflouride on a microporous carbon, “Carbosieve G”, which shows molecular sieving. We use this data as a prototype for the demonstration of our method. Monte Carlo Simulation We have used the Grand Canonical Monte Carlo (GCMC) simulation method27 to generate single-pore isotherms for the adsorptive species of interest (CH4, CF4, and SF6) in the pores of a model activated carbon. The independent thermodynamic variables in a GCMC simulation are the temperature, the chemical potential, and the system volume; this corresponds to the set of variables that is specified in an adsorption experiment (the chemical potential being determined by the experimental pressure), and for this reason the GCMC method is widely used for adsorption problems. The pore model and the fluid-fluid and solid-fluid interactions are inputs to the simulation. Our pores are slits, bounded by infinite layers of graphite. This model is a physically plausible simplification of the structure of real microporous carbons, and it has been widely used in the characterization of carbons.7,8,10,12,15,16,19-21 Other models have been proposed for activated carbons;3,28,29 the effect of the chosen pore model on the calculated PSD of porous carbons is considered by Davies et al.30 As CH4, CF4, and SF6 are all approximately spherical, we describe their interactions by the Lennard-Jones 12-6 potential:

νff(r) ) 4ff

[( ) ( ) ] σff r

12

-

σff r

6

(2)

where r is the molecule-molecule separation and σff and ff are the molecular diameter and the potential well depth, (26) Jagiełło, J.; Bandosz, T. J.; Putyera, K.; Schwarz, J. A. J. Chem. Eng. Data 1995, 40, 1288. (27) Adams, D. J. Mol. Phys. 1974, 28, 1241. (28) Segarra, E. I.; Glandt, E. D. Chem. Eng. Sci. 1993, 49, 2953. (29) Bojan, M. J.; Steele, W. A. Surf. Sci. 1988, 199, L395; Langmuir 1989, 5, 625. (30) Davies, G. M. D.; Seaton, N. A. Submitted to Carbon.

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Lo´ pez-Ramo´ n et al.

respectively. The potential parameters, obtained from second virial coefficient data,31 are shown in Table 1. The solid-fluid interaction potential vsf for a fluid molecule interacting with one of the pore walls is given by Steele’s 10-4-3 potential:32

νsf ) 2πsfFsσ2sf∆

[( ) ( ) 2 σsf 5 r

10

-

σsf r

σ4sf

4

-

3∆(z + 0.61∆3)

]

(3)

Here, z is the distance of a fluid molecule from the graphite surface (strictly, from the nuclei of the carbon atoms in the surface graphitic plane), Fs is the solid density, and ∆ is the separation between graphite layers. Steele obtained this potential by summing the contributions of the graphitic plans in the pore wall, assuming each plane to be structureless. Thus, the pore walls are smooth in our model pore. The solid parameters [carbon molecular diameter (σss ) 0.340 nm), well depth potential (ss/kB ) 28.0 K), solid density (Fs ) 114 nm-3), and intermolecular distance between graphite layers (∆ ) 0.335 nm)] were those of Steele.32 The solid-fluid Lennard-Jones parameters (σsf and sf ) were calculated using the usual Lorentz-Berthelot combination rules. The full solid-fluid potential, including the contributions of both pore walls, is given by

νsf total(z) ) νsf(z) + νsf(w-z)

(4)

where the pore width, w, is defined as the distance between the nuclei of the carbon atoms on opposing pore walls. Single-pore isotherms were simulated for CH4, CF4, and SF6 over a range of pore sizes for the experimental temperatures and pressures used by Jagiełło et al.26 The chemical potential was calculated from the pressure using the Peng-Robinson equation of state, with parameters chosen to match the experimental critical temperature and pressure. To minimize small-system effects, the usual periodic boundary conditions are employed; a pore that is essentially infinite in the plane of the pore walls is formed by an assembly of images of the simulation cell. The linear dimension of the simulation cell, in the plane of the pore wall, was chosen to be ten times the Lennard-Jones molecular diameter, σff. The potential cutoff radius, beyond which the fluid-fluid interactions are ignored, was 2.5σff. The initial configuration for an isotherm point was, for the first point on the isotherm, generated by placing molecules at random in the simulation cell, checking that they did not overlap. For subsequent points on the isotherm, at progressively higher pressures, the final configuration generated at the previous (i.e. slightly lower) pressure was used as the initial configuration. The Monte Carlo steps involved in a GCMC simulation are moves, creations, and destructions; we used a ratio of two attempted creations and two attempted destructions for each attempted move. For each isotherm point, the system was allowed to equilibrate over 10 000 steps (for CH4 and CF4) or 20 000 steps (for SF6). After equilibration the simulation continued for a further 100 000 steps, each representative of thermodynamic equilibrium, over which data were taken. The choice of run length warrants some discussion. The length of a Monte Carlo simulation should be sufficiently long that the mean values of the quantities of interest, (31) Hirschfelder, J. O.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. (32) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974.

Figure 2. Simulated adsorption isotherms for CH4 in slitshaped pores of various widths at (a, top) 258, (b, middle) 275, and (c, bottom) 295 K. The symbols are the simulation data and the curves the fits of the HdB equation.

averaged over the simulation, are as accurate as necessary for the purpose at hand. In our case, the amount adsorbed at each pressure and temperature has to be known sufficiently accurately that the PSD obtained using these simulation data is insensitive to the statistical error in the data. For our simulations, 100 000 steps per isotherm point was found to be sufficient. With this run length, the computer time for a single isotherm was between 1 h and 1 day on a Sun Sparc 10 workstation, with the time increasing with the number of molecules in the simulation cell (which itself depends on the adsorptive species, the pore size, the pressure, and the temperature). The computer time is also sensitive to the cutoff radius; this is chosen in the same spirit as the run length. Below our chosen value of 2.5σff, the amount adsorbed begins to depend significantly on the cutoff radius; above 2.5σff, the amount adsorbed does not change significantly, to within the accuracy of our simulation results. Figures 2-4 show simulated single-pore isotherms for CH4, CF4, and SF6, respectively, at the experimental temperatures studied by Jagiełło et al.26 (The statistical error is smaller than the symbols used in these figures.)

Microporous Solids

Langmuir, Vol. 13, No. 16, 1997 4439

Figure 3. Simulated adsorption isotherms for CF4 in slitshaped pores of various widths at (a, top) 258, (b, middle) 275, and (c, bottom) 296 K, respectively. The symbols are the simulation data and the curves the fits of the HdB equation.

Figure 4. Simulated adsorption isotherms for SF6 in slitshaped pores of various widths at (a, top) 267, (b, middle) 275, and (c, bottom) 296 K, respectively. The symbols are the simulation data and the curves the fits of the HdB equation.

As expected, Figure 2 shows that the Henry’s constant (the initial slope of the isotherm) for CH4 decreases as the pore width increases and as the temperature increases. At these temperatures and pressures, adsorption of methane occurs on the pore walls themselves; multilayer adsorption does not occur, even for pores large enough to accommodate it. In this figure, and in Figures 3 and 4, very little adsorption occurs in pores smaller than the smallest width shown: 0.75 nm for CH4, 0.85 nm for CF4, and 0.94 nm for SF6. Figure 3 shows simulated isotherms for CF4. The stronger adsorption of this species, compared with that of CH4, is reflected in the more rapid pore filling. There is one qualitative difference between the CF4 and CH4 isotherms. The CF4 isotherms for pore widths of 1.06 and 1.41 nm cross at 258 K (Figure 3a). This may be understood as follows. The 1.06 nm isotherm has a smaller Henry’s constant and so starts off below the 1.41 nm isotherm. However, the smaller pore can accommodate only one layer of CF4 molecules, whereas the larger pore can accommodate two layers. Above about 0.2 bar, the greater capacity of the larger pore outweighs the stronger

energetic interaction of the smaller pore, and the two isotherms cross. This crossing of isotherms is not seen in the simulation results at higher temperatures (Figure 3b and c), although it would be at sufficiently high pressure. (Indeed, this packing effect arises with any adsorptive for the right pore sizes and at sufficiently high pressure and low temperature; Gusev et al.18 observed it for CH4 at pressures above 7 bar.) Figure 4 shows simulated isotherms for SF6. As expected, SF6 shows stronger adsorption than either CH4 or CF4. Again some of the isotherms cross due to packing effects; for pore widths above 1.27 nm a second layer of SF6 molecules is accommodated, and pores wider than 2.02 nm are able to accommodate a third layer of molecules. In order to evaluate the integral in eq 1, the single-pore isotherm results are correlated with an analytical function. From our data analysis we found that the simulated isotherms for single pores can be well represented by the Hill-deBoer (HdB) isotherm:33 (33) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York, 1964.

4440 Langmuir, Vol. 13, No. 16, 1997

[

F(P,w) ) a0(w) 1 +

(

Lo´ pez-Ramo´ n et al.

K0T1/2 F(P,w) exp P a0(w) - F(P,w)

)]

ω(w) F(P,w) q(w) RT a0(w)RT

-1

(5)

Here, q(w) is the adsorption energy, ω(w) is the interaction energy between adsorbed molecules, a0(w) is the limiting density at high pressure, and K0T1/2 represents the preexponential factor of the Henry’s constant. q is independent of temperature but depends on the pore size. In our application of the HdB isotherm, K0 is independent of pore size and temperature; its value was obtained for each adsorptive by simultaneously fitting to the simulation data for all pore sizes and temperatures. The other parameters (ω and a0) are, like the single-pore isotherm itself, functions of both pore size and temperature. (The temperature dependence is omitted from the variables in eq 4 for clarity.) Note that the HdB equation is implicit in F(P,w). Figure 5 shows the variation of q with pore size, obtained by fitting the HdB isotherm to simulation data. The very slight scatter among different temperatures confirms the assumption that q is to a very good approximation temperature independent. It is of interest to observe that the variation of q versus pore size closely resembles the variation of the minimum of Steele’s 10-4-3 potential, eq 3 (also shown in Figure 5), suggesting that the HdB isotherm accurately represents the energetics of adsorption in this solid, as well as providing a sufficiently flexible functional form. (The relationship between the minimum in the Lennard-Jones potential and the pore size has been used to evaluate micropore size distributions from the distributions of the adsorption energies.34,35) Figure 6 shows the variation of the scaled limiting 2 density, a* 0 ) a0ωNAσ (where NA is Avogadro’s number), in physical terms the limiting density per unit projected molecular area, obtained by fitting to the simulation data. Provided the HdB is physically realistic, a*0 should be approximately equal to the number of molecular layers which the pore can accommodate. The initial value of a*0 for the smallest pores in all cases is close to 1, which is consistent with the fact that these pores can accommodate only one layer of molecules. In the case of SF6 and CF4 a plateau corresponding to two adsorbed layers (one on each pore wall) is observed for larger pores corresponding to monolayer adsorption on each of the walls. Weak temperature dependence is observed for these adsorptives. In the case of CH4 this quantity grows faster and exhibits a stronger dependence on temperature; this behavior cannot be interpreted physically, indicating that as a physical model (rather than simply a correlating function) the HdB isotherm does not well describe CH4 adsorption except in the smallest pores. The variation of the fitted interaction parameter ω with pore width for CF4 and SF6 is presented in Figure 7. The maximum in the plot corresponds to the formation of two adsorbed layers, where the effective interaction energy is enhanced due to the contribution of the interaction between the molecules adsorbed on opposite walls. The fitted value of ω for CH4 was found to vary only very slightly with pore size; for convenience, we set ω ) 2.95 for CH4. The fit of the HdB equation to the simulation data is shown in Figures 2-4. Determination of the PSD To solve eq 1 to obtain the PSD, we applied a numerical method, SAIEUS, developed by Jagiełło.36 The PSD is (34) Jagiełło, J.; Schwarz, J. A. Langmuir 1993, 9, 2513. (35) Jagiełło, J.; Bandosz, T. J.; Putyera, K.; Schwarz, J. A. J. Chem. Soc. Faraday Trans. 1995, 91, 2929. (36) Jagiełło, J. Langmuir 1994, 10, 2778.

Figure 5. Variation of q with pore size for CH4 (squares), CF4 (circles), and SF6 (triangles). Data are included for all the temperatures simulated. The lines represent the minimum of the Steele’s 10-4-3 potential for CH4 (- - -), CF4 (s), and SF6 (- - -).

Figure 6. Variation of the reduced limiting density, a*0, with pore size for CH4 (squares), CF4 (circles), and SF6 (triangles) at 258 K (s), 275 K (- - -) and 296 K (- - -).

Figure 7. Variation of the interaction parameter ω with pore size for sizes for CF4 (circles) and SF6 (triangles) at 258 K (s), 275 K (- - -), and 296 K (- - -).

represented in this method by a linear combination of B-spline functions with equally spaced knots in the range of integration. The adsorbate densities in the model pores are interpolated in two steps. First, the values of F(P,w) are obtained for all experimental pressures in each pore using the HdB equation. The interpolation of adsorbate densities between different pore sizes at a given pressure is then performed using natural spline functions. Numerical integration of eq 1 leads to a system of linear equations where the spline coefficients are unknown. Since such a system of equations is ill-conditioned, the solution is stabilized by regularization. The smoothing is combined with an additional stabilizing effect, which is achieved by imposing nonnegativity constraints on the solution. We considered two criteria for determining the optimal degree

Microporous Solids

Figure 8. Pore size distributions obtained using CH4 (diamonds), CF4 (circles), and SF6 (triangles) at three different temperatures: (a, top) 258 K for CH4 and CF4, and 267 K for SF6, (b, middle) 275 K, and (c, bottom) 295 K for CH4 and 296 K for CF4 and SF6.

of regularization. One is based on the analysis of a measure of the effective bias introduced by the regularization procedure and a measure of the uncertainty of the solution.36 The other utilizes the so-called L-curve,37 which relates the roughness of the solution to the error in the fit. As, for our data, the latter approach usually prescribed a higher degree of smoothing, giving a more robust solution, we used it to obtain the PSDs we report here. We have analyzed experimental adsorption isotherms for CH4 and CF4, and SF6 given by Jagiełło et al.26 for Carbosieve G. Isotherms are reported at three temperatures for each adsorbent: 258 K, 275 K, and 295 K for CH4; 258 K, 275 K, and 296 K for CF4; and 267 K, 275 K, and 296 K for SF6. We thus have a total of nine experimental isotherms, from each of which we can extract a PSD by solving eq 1. The PSDs obtained are shown in (37) Hansen, P. C.; O’Leary, D. P. SIAM J. Sci. Comput. 1993, 14, 1487.

Langmuir, Vol. 13, No. 16, 1997 4441

Figure 9. Fit of the simulated isotherms, integrated over the PSDs shown in Figure 8 (curves), to the experimental isotherms (symbols): (a, top) 258 K for CH4 and CF4, and 267 K for SF6, (b, middle) 275 K, and (c, bottom) 295 K for CH4 and 296 K for CF4 and SF6.

Figure 8. Figure 9 shows the corresponding fits to the experimental isotherms, which are excellent in all cases. Clearly, there are significant differences in the PSDs obtained using the various adsorptives. These differences have two possible origins: (i) due to differences in molecular size, the adsorptives may be exploring different regions of the pore space; (ii) the differences may be due to inadequacies in our model, such as the choice of pore shape, the neglect of surface heterogeneity and asphericity in the shape of the molecules, and uncertainty in the fluidfluid and fluid-solid interaction parameters. In the first case, the use of multiple adsorptives provides additional information on the pore structure of the solid; in the next section we show how it can be used to extract an estimate of the mean coordination number of the pore network. In the second case, we have a test of the realism of our modeling. (Strictly speaking, this is a test of consistency rather than directly a test of realism; however,

4442 Langmuir, Vol. 13, No. 16, 1997

a high degree of consistency is strong evidence for realism.) In the remainder of this section we investigate the consistency of the PSDs obtained using the three adsorptives and obtain the best estimate of the real PSD of Carbosieve G. We begin by making a comparison between the PSDs obtained at a single temperature. The PSDs show the same features at all the temperatures studied; we focus on the PSDs obtained at the lowest experimental temperatures (258 K for CH4 and CF4, 267 K for SF6) shown in Figure 8a. The lower limit of the PSD for each adsorptive is set by the smallest pore that the adsorptive molecule can penetrate to a significant degree: 0.75 nm for CH4, 0.85 nm for CF4, and 0.94 nm for SF6. The CH4 PSD has two peaks, the first of which (counting from the left) is below the smallest pore accessible to CF4, the next smallest adsorptive molecule. The second CH4 peak is within the range of the CF4 PSD but strikingly inconsistent with it. Although at first sight this discrepancy suggests a drastic failure of the model, this is not the case. In fact there exists an effective upper limit to the PSD obtained with a given adsorptive, beyond which that PSD is unreliable; the second peak in the CH4 PSD turns out to be beyond this limit. The origin of this upper limit is as follows. Figure 2a shows that, at this temperature and for the experimental pressure range, the CH4 single-pore isotherms are nearly linear (i.e. close to Henry’s law) above about 0.9 nm. In the PSD analysis, it is not possible to distinguish between pores of different sizes in which Henry’s law is obeyed exactly; one pore with a large Henry’s constant is indistinguishable from several pores with small Henry’s constants. As the pore size increases, the isotherms become more linear. The susceptibility of the analysis to random errors in the experimental and simulation data increases, and the PSD becomes less reliable. In mathematical terms, as the single-pore isotherms become more linear functions of pressure, the contributions of the various isotherms to the amount adsorbed become more linearly dependent. As a result the set of linear equations used to calculate the PSD (each of which represents the evaluation of eq 1 at a single pressure) becomes more ill-conditioned. Although we are able to obtain the regularized solution, the solution is not reliable in this range of pore sizes. Thus, the reliable range of the CH4 PSD is a “window of reliability” bounded on the left by the smallest accessible pore and on the right by the pore size at which adsorption, for the experimental temperature and pressure range, becomes substantially linear. (The precise location of the right hand boundary is of course arbitrary, as the reliability of the PSD decreases continuously as the pore size increases.) This observation, which has also been made by Gusev et al.,18 is applicable to supercritical adsorption (i.e. adsorption in the absence of capillary condensation) in porous solids in general. The pore width at which the isotherm becomes close to linear, over the experimental pressure range, is an increasing function of the maximum experimental pressure.18 Thus, increasing the maximum experimental pressure enlarges the PSD window for a given adsorptive. The second peak of the CH4 PSD at 258 K (Figure 8a) is largely above the upper limit of the window for this adsorptive. The shape and location of the second peak is therefore arbitrary, except to the extent that its contribution to the total adsorption is required to fit the experimental isotherm. An almost equally good fit to the experimental data can be obtained with a CH4 peak in any pore size range above 0.9 nm. Thus, the discrepancy between the second CH4 peak and the first CF4 peak does not indicate an inadequacy in our adsorption model but

Lo´ pez-Ramo´ n et al.

rather a practical limitation to the determination of PSDs in microporous solids. Figure 3a shows that, at 258 K, the upper limit of the window of reliability for CF4 is about 3 nm. From Figure 4a, we see that the upper limit for SF6 at the lowest experimental temperature for this adsorptive, 267 K, is above the largest pore size we simulated (3 nm) and, it seems, well above the size of the largest pore present in this sample. We now compare the first peaks of the CF4 and SF6 PSDs. If each species had free access to the whole pore network, these peaks would, in principle, be identical above the smallest pore size accessible to SF6, 0.94 nm. As the model used to generate the single-pore isotherms is an approximate representation of reality, in terms of pore shape and the molecular interactions, one should not expect exact agreement. Nevertheless, the pore volumes corresponding to the peaks (i.e. the areas under the PSD curves) should at least be similar. In fact, the SF6 peak has a significantly smaller volume than the CF4 peak (0.197 cm3 compared with 0.293 cm3), indicating that some of the pores that are large enough to accommodate SF6 if they could reach those pores are not in fact accessible to the SF6 molecules. The difference between these peaks is the basis of our analysis of network connectivity, which we present in the next section. The second peaks of the CF4 and SF6 PSDs are broadly consistent. They appear in the same location and correspond to similar pore volumes, although the SF6 peak is sharper. In fact, the pore volume of the SF6 peak is slightly larger than that of the CF4 peak. This is unphysical, as the pore volume accessible to SF6 (the larger molecule) should be at most as large as that accessible to CF4 and not larger. The source of this discrepancy lies in the simulated extent of adsorption of CF4 in pores within the second PSD peak. For CF4, the amount adsorbed in a pore of about 2 nm in width is, even at high pressure, only about half the amount adsorbed in the smallest pores. As a result, a substantial change in the second CF4 peak has only a small effect on the total adsorption. The fitting procedure therefore tends to offset quite large changes in the second peak against small changes in the first peak, so that the uncertainty in the second CF4 peak is relatively high. (This observation does not apply to SF6, for which the amounts adsorbed in the small and large pores are similar.) Thus this discrepancy does not indicate a fundamental flaw in our analysis. In summary, once the limitations inherent in the use of adsorptives having varying strengths of adsorption are taken into account, there is at least reasonable agreement between the PSDs obtained with the three adsorptives at a given temperature. It is instructive to compare the PSDs obtained at different temperatures (Figure 8). (Note that a slightly different set of temperatures was studied for each adsorptive.) The peaks are in similar locations at each temperature and correspond to broadly similar pore volumes. Nevertheless, the agreement between the CF4 and SF6 PSDs becomes poorer as the experimental temperature increases. The first CF4 peak remains in approximately the same location as the temperature increases while the first SF6 peak moves perceptibly to the right. A possible explanation for this is that the solidfluid interaction energy parameter (sf) for SF6 in our simulation is too small, i.e. smaller than in a real pore. If this is so, it follows that the isosteric heat for a given pore size is too small and (via the Clapeyron equation) that the simulated adsorption for this pore size will decrease too slowly with temperature. Thus, the model solid having a PSD that fits the experimental isotherm at the lowest temperatures (258 K for CH4 and CF4;

Microporous Solids

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Figure 10. Overall PSD of Carbosieve G.

267 K for SF6) overpredicts adsorption of SF6 at the higher temperatures. In order to maintain a good fit to the experimental isotherm, the PSD fitting program compensates for this effect by reducing the amount adsorbed in each pore, which it does by moving the SF6 PSD (and particularly the first peak) to the right. The window of reliability for each adsorptive shrinks as the temperature increases, as the single-pore isotherms approach Henry’s law at smaller pore widths at increased temperatures. The effect of this can be seen by comparing the second CF4 peaks at the three experimental temperatures. At the highest temperature (296 K for CF4) this peak has moved significantly from its location at lower temperatures; Figure 3c shows that the CF4 single-pore isotherms are nearly linear in this pore size range. We now turn to the combination of the PSDs obtained using the three adsorptives, each of which gives only a partial description of the full PSD, to give an estimate of the underlying function that is being probed by these adsorptivessthe overall PSD of the real solid. We do this by combining the portions of the individual PSDs that are within the window of reliability for each adsorptive. The overall PSD therefore includes the first CH4 peak and the first CF4 peak (rather than the first SF6 peak which, compared with the CF4, demonstrates molecular sieving). The overall PSD is completed by including either the second CF4 peak or the second SF6 peak. Arguments can be made for both these choices: the strength of the SF6 interaction is suspect but, on the other hand, the window of reliability of SF6 covers this peak at all the experimental temperatures, whereas this is not the case for CF4 at the highest temperature. On balance, we prefer to use the second CF4 peak. Figure 10 shows the estimates of the overall PSD, fo(w), over the experimental temperature range. We have already noted the presence of molecular sieving in comparing the PSDs for CF4 and SF6. It is possible that such sieving also exists between CH4 and CF4, although as the windows of reliability of these adsorptives do not overlap, it cannot be detected by comparing the PSDs for these species. Strictly speaking, therefore, the first peak of the combined PSD of Figure 10 has a different status than the others. The first peak is an estimate of the size distribution of the pores accessible to CH4, whereas the second and third peaks are lower bounds to this size distribution. (Of course, our analysis tells us nothing about pores inaccessible to CH4 either because they are too small to accommodate this species or because they are isolated by constrictions that do not permit its passage.) Connectivity Analysis In this section, we analyze the SF6 PSD, which we label fs(w), and the overall PSD, fo(w), to obtain an estimate of

Figure 11. Schematic illustration of the connection between adsorption and percolation in a pore network.

the mean coordination number of the pore network. In terms of the schematic illustration of molecular sieving shown in Figure 1, the smaller species represents CH4 or CF4, and the larger, SF6, the “probe species”. The PSD on the left corresponds to fo(w), and the PSD on the right corresponds to fs(w). The pore network of the real adsorbent is, of course, much larger than the small network shown in Figure 1. The shielding shown in Figure 1 becomes a percolation phenomenon22-25 on the much larger network found in real adsorbents. Figure 11 shows the penetration of CF4 and SF6 into a two-dimensional pore network. (The real network is, of course, three-dimensional and almost certainly much larger than this; in their analysis of connectivity in mesoporous solids, Liu et al.24 found pore networks with linear dimensions of between roughly 20 and 120. This figure, which is the number of pores spanning the pore network of a typical microparticle within the adsorbent, is likely to be larger for a microporous solid where the pores are presumably shorter.) Figure 11a shows the pores accessible to SF6 molecules penetrating from the external surface of the pore network; these are

4444 Langmuir, Vol. 13, No. 16, 1997

Lo´ pez-Ramo´ n et al.

Table 2. Values of X, A, and Z Obtained at the Three Experimental Temperatures

a

tempa

X

A

Z

a b c

0.788 0.799 0.811

0.648 0.691 0.725

2.28 2.30 2.33

The letters refer to the temperatures listed in Figure 9.

the pores that appear in the SF6 PSD. Similarly, Figure 11b shows the pores that appear in the CF4 PSD. As in our discussion of Figure 1, in the Introduction, we are interested in comparing the number of pores accessible to the probe species SF6 (the entire SF6 PSD) with the number of pores large enough to accommodate SF6 (that part of the CF4 PSD that is above the size of the SF6 molecule), which is shown in Figure 11c. Comparison between Figure 11a and c identifies three pores that are large enough to accommodate SF6 (and which appear in the CF4 PSD) but which are not actually accessible to SF6 (and therefore do not appear on the SF6 PSD). Percolation theory allows us to interpret this molecular sieving phenomenon at the network level. For a given network, defined by a PSD and a mean coordination number, Z, percolation theory provides the relationship between two variables: the fraction of pores in a network that are large enough to accommodate the probe species (SF6, in our case) (the “bond occupation probability”, X) and the fraction of pores that are actually accessible to this adsorptive (the “accessibility” A). In other words, X is the fraction of pores that SF6 would enter in an infinitely connected network where each pore had direct access to the adsorptive and the deviation of A from X reflects the finite connectivity of the pore network. In terms of our schematic illustration of percolation, the pores of Figure 11c contribute to X, and the pores of Figure 11a contribute to A. Percolation theory deals with the number of pores, rather than the pore volume, in a given size range. In terms of probability density functions, we need g(w) ) dN/dw (where N is the pore number) rather than f(w) ) dV/dw. Assuming that the length and breadth of the pores are uncorrelated with their width,23

go(w) )

fo(w) w

(6)

and

It is known that the percolation behavior of a network is “dimensionally invariant”, i.e. similar for most networks of a given dimensionality (three, in this case).22 Dimensional invariance implies that the relationship between A and X depends strongly on the mean coordination number, Z, and the network size, L, and only weakly on other aspects of the topology of the network. (Dimensional invariance does not apply if the sizes of neighboring pores are significantly correlated; this may be the case in some solids, but as such correlations are not readily measured experimentally for microporous solids, we do not consider this possibility in our analysis.) In adsorption terms, L is a measure of the size of the microporous regions, bounded by macropores, in a typical adsorbent. In the method of Seaton and co-workers23-25 for the measurement of the connectivity of mesoporous solids, the analysis of the sorption hysteresis loop provided several (X, A) data points for each sample which could be used to fit both Z and L. From our experiments we have only a single data point. (The values calculated at different temperatures shown in Table 2 are estimates of single variablessthe true values of X and A for the adsorption of SF6 in this solidsrather than (X, A) pairs.) Therefore, we restrict ourselves to the determination of Z and assume that the pore network is essentially infinite. (We return to this assumption below.) Our analysis is based on the simulation results of Zhang and Seaton38 for percolation on the simple cubic lattice, for which Z ) 6. Their results for the accessibility of an essentially infinite system are well described by:

A ) 0, X < 0.249 A ) 0.4567(X - 0.249)0.41 + 3.153(X - 0.249) 20.88(X - 0.249)2 + 51.58(X - 0.249)3, 0.249 < X < 0.45 A ) X, 0.45 < X

(10)

Dimensional invariance implies that the number of accessible pores per pore intersection, ZA, is a general function of the number of “occupied” pores (i.e. pores large enough to accommodate the probe molecule) per intersection, ZX. Equation 10 can thus be generalized to represent a network of arbitrary Z by replacing X by ZX/6 and A by ZA/624 to give the following function:

ZA ) 0, ZX < 1.494 fs(w) gs(w) ) w

(7)

X is the normalized integral of go(w) over the range of pore sizes that can accommodate the probe molecule:

∫w go(w) dw ∫0∞go(w) dw

ZA ) 1.314(ZX - 1.494)0.41 + 3.153(ZX - 1.494) 3.480(ZX - 1.494)2 + 1.433(ZX - 1.494)3, 1.494 < ZX < 2.7 ZA ) ZX, 2.7 < ZX

(11)

∞

X)

s

(8)

where ws is the width of the smallest pore that can accommodate SF6: 0.94 nm. The accessibility A is the normalized integral of gs(w):

∫w∞gs(w) dw A) ∞ ∫0 go(w) dw s

(9)

Clearly, both X and A are decreasing functions of ws. Table 2 shows the values of X and A for our sample calculated using eqs 7-9, at the different experimental temperatures.

which is plotted in Figure 12. Below ZX ) 1.5, the percolation threshold, no pores are accessible. Probe molecules that are so large that ZX < 1.5 will not penetrate the network beyond the first few pores near the surface of the microporous regions. Above ZX ) 1.5, the network becomes increasingly accessible as the size of the probe molecule decreases and X increases. A f X for very small probe molecules. The relationship between connectivity and molecular sieving is shown schematically in different coordinates, more directly related to the physical process, in Figure 13. The ratio A/X is the “scaled accessibility”, the fraction of those pores that are large enough to accommodate the probe molecule that are actually accessible to that species; equivalently, 1 - A/X is the (38) Zhang, L.; Seaton, N. A. Chem. Eng. Sci. 1996, 51, 3257.

Microporous Solids

Figure 12. Generalized accessibility, ZA, as a function of the generalized bond occupation probability, ZX, for a general threedimensional network.

Figure 13. Schematic diagram of the effect of Z on the accessibility of the pore network to molecules of different sizes.

fractional reduction in adsorption due to the finite connectivity of the network. The probe molecule size is related to X and the PSD (which is arbitrary in this figure) by eq 8. For a given value of Z, A/X decreases from unity as the size of the probe molecule increases, until the probe becomes so large that the percolation threshold is reached and that species has no access to the pore network. Increasing the value of Z moves the percolation threshold to the right, allowing larger species to enter the pore network. For all species, the accessibility increases with Z. The value of Z for our sample is estimated by fitting the (X, A) values given in Table 2 to eq 11 for each temperature. The results, also shown in Table 2, are very consistent, reflecting the good agreement between the PSDs at different temperatures; Z ) 2.3, to two significant figures, at all temperatures. This estimate of Z requires some qualification. Strictly speaking, it corresponds to only those pores that are large enough to admit the adsorption of the smallest adsorptive used in the experimentssCH4 in our case. If smaller pores are present, the values of X calculated using eq 8 are too high and Z is too low. (For our sample, there is evidence from hydrogen adsorption that smaller pores are present.39) On the other hand, as discussed in the previous section, the second and third peaks of the overall PSD may be too small (being strictly a lower bound on the PSD of pores accessible to CH4), (39) Lo´pez-Ramo´n, M.-V.; Jagiełło, J.; Bandosz, T. J.; Seaton, N. A., in preparation. (40) Boulton, K. L.; Lo´pez-Ramo´n, M.-V.; Davies, G. M.; Seaton, N. A. Paper presented at the 4th International Symposium on the Characterisation of Porous Solids, 1996.

Langmuir, Vol. 13, No. 16, 1997 4445

which would imply that the value of X is too low and the estimate of Z too high. This is an inherent limitation of the use of different adsorptives to build up the PSD. A further limitation of our analysis is that we have assumed that the network is essentially infinite. The effect of finite L is to make the percolation transition more diffuse and to increase A for a given value of X (i.e. for a given probe molecule size).24 Although the function A(X) is affected in quite different ways by Z and L, an analysis based on a single data point is incapable making the distinction. Thus, a single (X, A) data point can be fitted by a finite network with a low value of Z or an infinite network with a larger value of Z. In this respect, our value of Z is an upper bound on the true value. Overall, it is difficult to quantify the uncertainty in our analysis. The estimates of Z obtained by this method must therefore be interpreted cautiously. Summary In this paper, we have demonstrated how Monte Carlo simulation can be used to extract detailed PSD information for a microporous solid. By comparing the PSDs obtained with the various adsorptives, and at different temperatures, we were able to (i) assess the internal consistency of the method; (ii) validate the use of our microscopic model of adsorptionsLennard-Jones molecules in a smooth, slitshaped graphitic poreswithin a PSD analysis; and (iii) identify the “window of reliability” of each adsorptive. The overall PSD obtained is more detailed than can be obtained with a single adsorptive. We have proposed a new method for the determination the coordination number, Z, of microporous solids and have estimated this quantity for one sample. As with the PSD, there is good consistency between different temperatures. Until now, this information had not been accessible experimentally. In contrast with the earlier method of Seaton and co-workers23-25 for the measurement of the connectivity of mesoporous solids, this method is much more involved experimentally. Nevertheless, this analysis is likely to be worthwhile for applications in which pore network connectivity is important, such as in the design of carbon molecular sieves. There is one respect in which our results could have been improved. Our results suggest that the PSDs (and hence the estimate of Z) could be improved by using a slightly larger value of the solid-fluid interaction energy for SF6. A more sophisticated approach would have involved adjusting this parameter until the best agreement was obtained between the various PSDs. (In general, the potential parameters of all the adsorptives would be candidates for tuning in this way.) Nevertheless, it is worth noting that a high degree of consistency was obtained by simply using fluid-fluid interaction parameters from second virial coefficient data, and the standard Lorentz-Berthelot combination rules. The effect of fluidfluid and solid-fluid interactions on PSDs obtained by Monte Carlo simulation is considered in detail by Boulton et al.40 Acknowledgment. M.V. L. gratefully acknowledges a grant from the Spanish Ministry of Education and Science. Acknowledgement is made to the donors of The Petroleum Research Fund, administered by the ACS, for partial support of this research. LA970197H