Evaluation of Methods for Determining the Pore Size Distribution

The pore size distribution (PSD) and the pore-network connectivity of a porous material determine ... problem, as this will allow the more effective d...
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Langmuir 2007, 23, 8430-8440

Evaluation of Methods for Determining the Pore Size Distribution and Pore-Network Connectivity of Porous Carbons Q. Cai, A. Buts, M. J. Biggs, and N. A. Seaton* Institute for Materials and Processes, UniVersity of Edinburgh, Kenneth Denbigh Building, Mayfield Road, Edinburgh EH9 3JL, United Kingdom ReceiVed March 10, 2007. In Final Form: May 18, 2007 The pore size distribution (PSD) and the pore-network connectivity of a porous material determine its properties in applications such as gas storage, adsorptive separations, and catalysis. Methods for the characterization of the pore structure of porous carbons are widely used, but the relationship between the structural parameters measured and the real structure of the material is not yet clear. We have evaluated two widely used and powerful characterization methods based on adsorption measurements by applying the methods to a model carbon which captures the essential characteristics of real carbons but (unlike a real material) has a structure that is completely known. We used three species (CH4, CF4, and SF6) as adsorptives and analyzed the results using an intersecting capillaries model (ICM) which was modeled using a combination of Monte Carlo simulation and percolation theory to obtain the PSD and the pore-network connectivity. There was broad agreement between the PSDs measured using the ICM and the geometric PSD of the model carbon, as well as some systematic differences which are interpreted in terms of the pore structure of the carbon. The measured PSD and connectivity are shown to be able to predict adsorption in the model carbon, supporting the use of the ICM to characterize real porous carbons.

1. Introduction Porous carbon materials, due to their highly developed porosity and relative inertness, are widely used in industry as adsorbents, catalyst supports, battery electrodes, capacitors, gas storage, and biomedical engineering applications.1-5 Understanding the porous structure and its relationship to the performance of the material is a scientifically and technologically important problem, as this will allow the more effective design of these materials and increase the range of their application. Both the geometrical (the pore shape and the pore size distribution) and topological (the way in which the pores are connected together) characteristics of the porous structure play important roles in the applications of nanoporous carbons. The pore size distribution (PSD) has a significant influence on the transport and equilibrium of molecules in the pore structure.6 The connectivity of the pore network, usually quantified in terms of the mean coordination number Z (which is defined as the number of pores intersecting at a junction), influences the passage of molecules in the intersecting pores and governs the reaction and transport properties.7 The application of porous carbon materials depends on the ability to measure the pore structure of the carbon. This is often done by analyzing adsorption measurements, as adsorption is * To whom correspondence should be addressed. Fax: +44 131 6506551. E-mail: [email protected]. (1) Takashi, K. Control of pore structure in carbon. Carbon 2000, 38, 269. (2) Sircar, S.; Golden, T. C.; Rao, M. B. Activated carbon for gas separation and storage. Carbon 1996, 34, 1. (3) Heinen, A. W.; Peters, J. A.; Bekkum, H. Competitive adsorption of water and toluene on modified activated carbon supports. Appl. Catal., A 2000, 194195, 193-202. (4) Qiao, W. M.; Korai, Y. I.; Hori, M. Y.; Maeda, T. Preparation of an activated carbon artifact: factors influencing strength when using a thermoplastic polymer as binder. Carbon 2001, 39, 2355-2368. (5) Minamisawa, M.; Minamisawa, H.; Yoshida, S.; Takai, N. Adsorption Behavior of Heavy Metals on Biomaterials. J. Agric. Food Chem. 2004, 52, 5606-5611. (6) Hu, X.; Qiao, S. Z.; Do, D. D. Multicomponent Adsorption Kinetics of Gases in Activated Carbon: Effect of Pore Size Distribution. Langmuir 1999, 15, 6428-6437. (7) Liu, H.; Zhang, L.; Seaton, N. A. Determination of the connectivity of porous solids from nitrogen sorption measurements s II. Generalisation. Chem. Eng. Sci. 1992, 47, 4393-4404.

sensitive to both the geometry and topology of the network. The use of adsorption to characterize the material is particularly appropriate when the application of interest involves adsorption, as both the characterization method and the application are based on the same processes at the molecular scale. The standard approach to determining the PSD and the pore-network connectivity involves using (explicitly or implicitly) an “intersecting capillaries model” (ICM), which is used to approximate the microstructure of nanoporous materials. In this simplified model, the pores are assumed to be an ensemble of pores with the same shape (usually a parallel-sided slit in the case of carbons) but different sizes, which are connected in some way. The pore walls are usually assumed to be smooth, but variations include atomic8 and larger scale9 microtextural heterogeneity in the pore walls, chemical heterogeneity (e.g., polar oxygen-containing sites on the surface of the pore wall),10 different pore shapes,11-13 and pore walls of varying thickness.14 More physically realistic models of porous carbons than the ICM have been proposed (see ref 15 for a brief review of these). Gubbins et al.16-18 used reverse Monte Carlo (RMC) simulation (8) Nicholson, D. A Simulation Study of Energetic and Structural Heterogeneity in Slit-Shaped Pores. Langmuir 1999, 15, 2508-2515. (9) Turner, A. R.; Quirke, N. A grand Canonical Monte Carlo study of adsorption on graphitic surfaces with defects. Carbon 1998, 36, 1439-1446. (10) Jorge, M.; Schumacher, C.; Seaton, N. A. Simulation Study of the Effect of the Chemical Heterogeneity of Activated Carbon on Water Adsorption. Langmuir 2002, 18, 9296-9306. (11) Davies, G. M.; Seaton, N. A. The effect of the choice of pore model on the characterization of the internal structure of microporous carbons using pore size ditributions. Carbon 1998, 36, 1473-1490. (12) Bojan, M. J.; Steele, W. A. Computer simulation in pores with rectangular cross-sections. Carbon 1998, 36, 1417-1423. (13) Pantatosaki, E.; Psomadopoulos, D.; Steriotis, T.; Stubos, A. K.; Papaionnou, A.; Papadopoulos, G. K. Micropore size distributions from CO2 using grand canonical Monte Carlo at ambient temperatures: cylindrical versus slit pore geometries. Colloids Surf., A 2004, 241, 127-135. (14) Nguyen, T. X.; Bhatia, S. K. Characterization of Pore Wall Heterogeneity in Nanoporous Carbons Using Adsorption: The Slit Pore Model Revisited. J. Phys. Chem. B 2004, 108, 14032-14042. (15) Biggs, M. J.; Buts, A. Virtual porous carbons: what they are and what they can be used for. Mol. Simul. 2006, 32, 579-593 (16) Thomson, K. T.; Gubbins, K. E. Modeling Structural Morphology of Microporous Carbons by Reverse Monte Carlo. Langmuir 2000, 16, 5761-5773.

10.1021/la7007057 CCC: $37.00 © 2007 American Chemical Society Published on Web 06/29/2007

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to reconstruct the carbon structure by fitting the radial distribution function of the carbon atoms determined by experiment. Biggs et al.15 have used a more top-down approach to build model carbon structures (“virtual porous carbons” or VPCs) that match experimental quantities such as the porosity and average d002 spacing obtained from X-ray scattering. Such models are, in principle, much more realistic than the ICM, as they incorporate a much more detailed picture of the real material. These model carbons are built based on an experimental understanding of real nanoporous carbons and are able to produce adsorption isotherms and heats of adsorption that are broadly consistent with experimental values (though not tuned to describe a particular real material).19 However, such models are computationally demanding and, in their current form, difficult to adjust to give quantitatively accurate representations of experimental data. So, while they give useful insights into the structure of real carbons, they are thus not suited to routine use in the characterization of porous carbons. In this paper, we use the VPC approach indirectly, not to analyze the structure of real carbons but to validate more practicable methods of doing so. We do this by producing computer-generated VPCs, having the important characteristics of real porous carbons, and then analyzing them using an ICM. In other words, we treat the VPCs as representative of real materials but with the advantage that we have a full description of their actual structure. In doing this, we are following an earlier work of Biggs and co-workers using the “absolute assessment methodology”,19 in which the conventional characterization methods are applied to characterize the VPCs and an assessment of the characterization methods is undertaken by comparing estimates obtained from them against the corresponding exactly known structure. Biggs and co-workers19,20 have used this absolute assessment methodology to evaluate the validity of the Rs method,21 the subtracting pore effect (SPE) method,22 the Langmuir method, and the Brunauer-Emmett-Teller (BET) method23 for determining the pore volume and surface area of nanoporous carbons. In this work, we are using the same methodology to evaluate the validity of the characterization methods using the ICM for determining the PSD and the porenetwork connectivity. Several methods23-29 based on physical adsorption have been used to determine the PSD of porous carbons. Over the past

decade or so, methods based on statistical mechanical analyses of adsorption (relating a molecular picture of adsorption in the ICM to experimental measurements of adsorption on the real carbons) have become standard,28-34 except for the case of subcritical adsorption in sufficiently large mesopores where classical methods based on the Kelvin equation continue to be used.35,36 The statistical mechanical methods include the use of density functional theory (DFT) and grand canonical Monte Carlo (GCMC) simulation to describe adsorption in individual pores, among which the GCMC simulation method gives more accurate results for micropores and has become widely used. Previous study has shown that different molecular probes give different, partial information about the pore structure. This feature arises because (1) the molecular size of the probe and the strength of its interaction with the adsorbent influences the adsorption strength and (2) the real nanoporous carbons contain pores of different sizes which are connected together in a pore network, and the presence of constrictions in the pore network inhibits the passage of one or more adsorbed species. Lo´pez-Ramo´n et al.37 have shown that the partial PSDs obtained using different adsorptives (in their case, CH4, CF4, and SF6) can be combined to produce a more complete PSD of the material. However, they did not check how this overall PSD would perform when used to predict the adsorption for all these gases, and so the consistency of this approach was not fully investigated. Lo´pez-Ramo´n et al.37 also used a comparison of the partial PSDs to estimate the accessibility of the pore network to molecules of different sizes and hence, using percolation theory, the mean coordination number of the pore network, Z. A related approach was used by Ismadji and Bhatia,38 who investigated the connectivity of porous carbons by comparing the PSD from the adsorption of esters with the more complete PSD from the adsorption of argon, and Navarro et al.,39 who studied the evolution of the connectivity of a porous carbon during activation by analyzing ethane and phenanthrene adsorption against nitrogen adsorption at 77 K. Their studies demonstrated the usefulness of percolation theory to analyze the PSDs for obtaining the pore network connectivity in terms of the mean coordination number Z. In this work, we evaluate the ability of the ICM to characterize two VPCs, using GCMC simulation to simulate adsorption in individual pores within the ICM. We use the approach of Lo´pez-

(17) Pikunic, J.; Clinard, C.; Cohaut, N.; Gubbins, K. E.; Guet, J.-M.; Pellenq, R. J.-M.; Rannou, I.; Rouzaud, J.-N. Structural Modeling of Porous Carbons: Constrained Reverse Monte Carlo Method. Langmuir 2003, 19, 8565-8582. (18) Pikunic, J.; Llewellyn, P.; Pellenq, R.; Gubbins, K. E. Argon and Nitrogen Adsorption in Disordered Nanoporous Carbons: Simulation and Experiment. Langmuir 2005, 21, 4431-4440. (19) Biggs, M. J.; Buts, A.; Williamson, D. Absolute Assessment of AdsorptionBased Porous Solid Characterization Methods: Comparison Methods. Langmuir 2004, 20, 7123-7138. (20) Biggs, M. J.; Buts, A.; Williamson, D. Absolute assessment of adsorptionbased porous solid characterisation methods. II: surface area determination by the Langmuir and BET methods. In preparation. (21) Carrott, P. J. M.; Roberts, R. A.; Sing, K. S. W. Adsorption of nitrogen by porous and non-porous carbons. Carbon 1987, 25, 59-68. (22) Kaneko, K.; Ishii, C.; Ruike, M.; Kuwabara, H. Origin of superhigh surface area and microcrystalline graphitic structures of activated carbons. Carbon 1992, 30, 1075-1088. (23) Gregg, S. J.; Sing, K. S. W. Adsorption, surface area and porosity, 2nd ed.; Academic Press: London, 1982. (24) Rodrı´guez-Reinoso, F.; Linares-Solano, A. Micro-porous structure of actiVated carbons as reVealed by adsorption methods; Marcel Dekker: New York, 1988. (25) Stoeckli, H. F. A generalization of the Dubinin-Radushkevich equation for the filling of heterogeneous micropore systems. J. Colloid Interface Sci. 1977, 59, 184-185. (26) Dubinin, M. M. Progress in surface and membrane science; Academic Press: New York, 1975; pp 1-70. (27) Horva´th, G.; Kawazoe, K. Method for the calculation of effective poresize distribution in molecular-sieve carbon. J. Chem. Eng. Jpn. 1983, 16, 470. (28) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. A new analysis method for the determination of porous carbons from nitrogen adsorption measurements. Carbon 1989, 27, 853-861.

(29) Gusev, V. Y.; O’Brien, J. A.; Seaton, N. A. A Self-Consistent Method for Characterization of Activated Carbons Using Supercritical Adsorption and Grand Canonical Monte Carlo Simulations. Langmuir 1997, 13, 2815-2821. (30) Ravikovitch, P. I.; Vishnyakov, A.; Russo, R.; Neimark, A. V. Unified Approach to Pore Size Characterization of Microporous Carbonaceous Materials from N2, Ar, and CO2 Adsorption Isotherms. Langmuir 2000, 16, 2311-2320. (31) Sweatman, M. B.; Quirke, N. Characterization of porous materials by gas adsorption at ambient temperatures and high pressure. J. Phys. Chem. 2001, 105, 1403-1411. (32) Sweatman, M. B.; Quirke, N.; Zhu, W.; Kapteijn, F. Analysis of gas adsorption in Kureha active carbon based on the slit-pore model and MonteCarlo simulations. Mol. Simul. 2006, 32, 513. (33) Ohba, T.; Omori, T.; Kanoh, H.; Kaneko, K. Cluster Structures of Supercritical CH4 Confined in Carbon Nanospaces with in Situ High-Pressure Small-Angle X-ray Scattering and Grand Canonical Monte Carlo Simulation. J. Phys. Chem. B 2004, 108 (1), 27-30. (34) Davies, G. M.; Seaton, N. A. Development and Validation of Pore Structure Models for Adsorption in Activated Carbons. Langmuir 1999, 15, 6263-6276. (35) Sing, K. S. W. The use of gas adsorption for the characterization of porous solids. Colloids Surf. 1989, 38, 113-124. (36) Do, D. D.; Nguyen, C.; Do, H. D. Characterization of micro-mesoporous carbon media. Colloids Surf., A 2001, 187-188, 51-71. (37) Lo´pez-Ramo´n, M. V.; Jagiełło, J.; Bandosz, T. J.; Seaton, N. A. Determination of the Pore Size Distribution and Network Connectivity in Microporous Solids by Adsorption Measurements and Monte Carlo Simulation. Langmuir 1997, 13, 4435-4445. (38) Ismadji, S.; Bhatia, S. K. Investigation of Network Connectivity in Activated Carbons by Liquid Phase Adsorption. Langmuir 2000, 16, 9303-9313. (39) Navarro, M. V.; Seaton, N. A.; Mastral, A. M.; Murillo, R. Analysis of the evolution of the pore size distribution and the pore network connectivity of a porous carbon during activation. Carbon 2006, 44, 2281-2288.

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Ramo´n et al. to obtain the overall PSDs and the coordination numbers Z of the two VPCs by analyzing the adsorption of CH4, CF4, and SF6 at three different temperatures. The overall PSDs are then used to predict the adsorption of the three species at the three temperatures to check the consistency of the overall PSDs obtained by adsorption. Finally, the pore structure information obtained is compared with the known structure of the model carbons, allowing us to evaluate the correctness of the characterization method. It is worth emphasizing that we are using the terms “consistency” and “correctness” in particular ways here.46 We view the ability of the overall PSD to predict adsorption as evidence of consistency and reserve the term “correctness” for a comparison of the overall PSD with independently obtained structural information. Finally, we reiterate that our objective in carrying out this analysis is to use the VPCs as representative of the structure of real carbons (sufficiently realistic to capture the important characteristics of real porous carbons and with the advantage that their structure is completely known) to evaluate the use of adsorption, analyzed using the ICM, to characterize real carbons.

2. The Model Carbon 2.1. Overview of Model Carbons. There is substantial experimental evidence40 to suggest that nanoporous carbons are built up from domains of two-dimensional (2D) short-range order that may be reasonably represented by small polyaromatic molecules or similar structures. These domains assemble in a roughly aligned manner with out-of-plane spacing somewhat greater than that of graphite to form nanoscale regions of local molecular orientation (LMO). The size of these regions of LMO may be as small as the regions of 2D short-range order in highly microporous nongraphitic carbons, through to micrometers for graphitic carbons. The regions of LMO combine to form mesoscopic structures. Biggs and co-workers have used this conceptual view of carbons as a basis for model carbons.15 Two model carbons, termed Carbon 1 and Carbon 2, generated by this approach have been used in the work reported here. The essential difference between these two carbons, as will be seen later, is that the first has a substantially smaller porosity and contains porosity that becomes inaccessible to SF6, the largest adsorptive we study, as the temperature is decreased from 296 to 258 K. Note that, in a model carbon, a pore space is defined as inaccessible (closed) to the adsorbate molecules when the latter’s activation energy is insufficient to overcome the energy barrier at the pore mouth. Both physically closed pores and kinetically closed pores exist in disordered nanoporous carbons; the former has a too high energy barrier for adsorbate molecules to overcome even at the highest temperature, while the latter becomes open at a higher temperature at which adsorbate molecules gain sufficient energy to overcome the energy barrier at the pore mouth. A detailed description of this can be found in a recent paper of Nguyen and Bhatia.41 The nature of the porosity in the carbons is illustrated in Figure 1. This figure shows that the pore space is complex in character with a wide variety of pore shapes, sizes (length and breadth as well as width), and surface textures. Previous work19 has shown that simulation of adsorption on these models can yield a wide range of isotherm shapes that are seen experimentally as well (40) Bandosz, T. J.; Biggs, M. J.; Gubbins, K. E.; Hattori, Y.; Iiyama, T.; Kaneko, K.; Pikunic, J.; Thomson, K. T. Models of porous carbons. Chem. Phys. Carbon 2003, 28, 41. (41) Nguyen, T. X.; Bhatia, S. K. Determination of Pore Accessibility in Disordered Nanoporous Materials. J. Phys. Chem. C 2007, 111, 2212-2222.

Figure 1. Isoenergy map of a microporous region of Carbon 2. Red and yellow indicate regions of solid and pore, respectively.

as the experimentally observed decrease of heat of adsorption with increasing loading. 2.2. Geometric Analysis of the Porosity of Model Carbons. The pore space of the model carbons is analyzed directly to determine the porosity and the “geometric” pore size distribution. The pore space is first subdivided into individual pores using an algorithm similar to that of Thovert et al.42 A cubic lattice is superimposed on the model solid, and the energy of fluid-solid interaction is calculated at each node of the lattice; the lattice cell size is chosen to be substantially less than the characteristic size of the interaction. Nodes of the lattice where the interaction energy is greater than some limit are designated as lying in the solid phase; all other nodes are considered to be part of the pore volume. Here, we are interested in the pore space that is large enough to accommodate CH4, the smallest adsorptive molecule used in our analysis of the pore structure by adsorption. The limiting energy was thus set to the maximum energy accessible to methane molecules during a GCMC simulation of methane adsorption at 40 bar and 258 K, where the pore space is essentially saturated with methane. Nodes that belong to the pore volume are subject to cluster analysis43 to identify those nodes that belong to the percolating cluster, which are accessible to methane and, if they exist, isolated clusters, which are not accessible. The accessible pore volume is then split into separate pores by removing the outer layer of the clusters until only convex hulls remain; this may very well lead to an original cluster breaking up into multiple convex hulls. Each convex hull is considered to be the kernel of a pore. The exact extent of each pore is then determined by re-expanding all the kernels layer by layer until they can expand no further due to their coming into contact with other adjacent, reexpanding kernels. The geometrical size of a pore identified using this decomposition algorithm is equal to the cube root of the volume of the pore. The pore size distribution functions of the two model carbons determined in this way are given in section 5. (42) Thovert, J. F.; Salles, J.; Adler, P. M. Computerized characterization of the geometry of real porous media: their discritization, analysis and interpretation. J. Microsc. (Oxford) 1993, 170, 65-79. (43) Hoshen, J.; Kopelman, R. Percolation and cluster distribution. 1. Cluster multiple labeling technique and critical concentration algorithm. Phys. ReV. B 1976, 14, 3438.

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Table 1. Lennard-Jones Parameters Used in GCMC Simulations molecule

/κB (K)

σ (Å)

CH4 CF4 SF6 C

149.92 152.5 200.9 28.0

3.7327 4.70 5.51 3.4

Two alternative methods were also investigated for determining the geometric pore size distribution. In the first, the size of each pore in the decomposed pore space was equated to the size of the largest sphere that could be inserted into the pore, while the second approach was based on a box-counting method independent of the pore-space decomposition described above.44 While both these alternatives yielded geometric PSDs that differed in detail from those reported in this paper, the differences were not substantial.

3. Analysis of the Model Carbons Using the ICM to Determine the PSD The pore size distribution, f(w), is obtained by solving the adsorption integral equation:

N(T,P) )

∫0∞F(w,T,P) f(w) dw

(1)

where N(T,P) is the experimentally determined adsorption at temperature T and bulk pressure P and F(w,T,P) is the isotherm for the same fluid in a single slit pore of width w (the “single-pore isotherm”). The solution of eq 1 is strongly dependent on both the single-pore isotherms and the experimental isotherms. There are four stages to obtain the PSD of a real carbon or, in this case, a realistic model carbon: (1) determine a molecular model for gas-gas and gas-solid interaction, (2) obtain the experimental isotherms for the carbon of interest, (3) generate a database of model-pore isotherms using GCMC simulation, and (4) invert the adsorption integral eq 1 to obtain the PSD. Once the PSD of the carbon is obtained, the experimental adsorption on the same material, N(T, P), can be predicted using eq 1, given the database of single-pore isotherms, F(w, T, P), at the desired temperature and pressure. The consistency of the obtained PSD can be examined by comparing the predicted adsorption with the real experimental measurement of adsorption on the material. 3.1. Molecular Models. Three different gases (CH4, CF4, and SF6) are used as adsorptives to probe the pore size distributions of the model carbons. As CH4, CF4, and SF6 are all approximately spherical, we describe their interactions by the Lennard-Jones 12-6 potential:45

[( ) ( ) ]

Uff(r) ) 4ff

σff r

12

σff r

6

(2)

where σ and  are the molecular size and energy parameters, respectively. These parameters for the three gases are listed in Table 1. In the simulations, the potential is truncated at 15 Å, beyond which the fluid-fluid interactions are ignored. The interaction between a fluid molecule and the carbon is calculated by Steele’s 10-4-3 potential:46

Usf(z) )

[(

2πFssfσsf2∆

) ( )

2 σsf 5 z

10

]

σsf4 σsf 4 (3) z 3∆(0.61 + z)3

Figure 2. Isotherms for the adsorption of CH4 (9), CF4 (b), and SF6 (2) on the two carbons at 258 K (s), 275 K (- - -), and 296 K (‚ ‚ ‚).

where ∆ ) 0.335 nm is the distance between the two graphite layers, Fsf ) 114 nm-3 is the carbon number density of the pore walls, z is the perpendicular distance between the adsorbate molecule and the pore wall surface, and σsf and sf are the solidfluid Lennard-Jones (LJ) parameters, which are determined using the Lorentz-Berthelot combination rules with the parameter values given in Table 1. Steele’s potential (a standard in this type of work) is based on the assumption that the pore is bounded by two semi-infinite graphitic blocks. The overall adsorbate-adsorbent interaction is given by

UΣsf(z) ) Usf(z) + Usf(w - z)

(4)

where the pore width, w, is defined as the distance between the nuclei of the surface carbon atoms of the opposing pore walls. 3.2. Adsorption on the Model Carbons. The isotherms for the model carbons, taking the role of “experimental isotherms”, are simulated by GCMC simulation on the percolating clusters only, as the nonpercolating clusters are not accessible to a gas permeating from the surface of the material (though they would be to a GCMC simulation without this constraint). Details of the simulation algorithm may be found in ref 19. The isotherms, which are shown in Figure 2 for both carbons, are reported for (44) Mandelbrot, B. B. The fractal geometry of nature; Freeman: San Francisco, 1982. (45) Frenkel, D.; Smit, B. Understanding molecular simulation: from algorithm to applications; Academic Press: London, 1996. (46) Steele, W. A. The interaction of gases with solid surfaces; Pergamon: Oxford, 1974.

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three temperatures (258, 275, and 296 K) for each adsorptive. These isotherms show “experimentally” the development of adsorption with increasing temperature and the difference between different adsorptives. When comparing the isotherms from the two carbons, it shows clearly that the adsorption on Carbon 1 is less than that on Carbon 2, consistent with its lower porosity. On Carbon 2, the adsorption of the three species decreases with the increase of temperature, while on Carbon 1, the adsorption of SF6 shows a slightly different behavior, giving the least adsorption at the lowest temperature of 258 K. This different behavior of Carbon 1 arises from part of the pore space becoming inaccessible to SF6 at 258 K because the molecules do not have sufficient energy to overcome the activation-energy barrier at the entrance to that part of the pore space, which is thus disconnected from the percolating cluster. As the GCMC simulation is only carried out on percolating clusters, the adsorbed amount of SF6 in Carbon 1 at this temperature is less. 3.3. Adsorption in the Intersecting Capillaries Model (ICM). The GCMC simulation method is used to generate the singlepore isotherms, F(w,P,T), for the three adsorptives in a series of slit pores of width w ) 6-42 Å. The initial molecular 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 pressure was used as the initial configuration. For each isotherm point, the system was allowed to equilibrate over 5 × 105 steps, where a Monte Carlo step involves one random creation/ destruction attempt and a move. After equilibration, data were collected over a further 106 steps. The adsorption isotherms in a series of selected pores are given in Figure 3 for CH4, Figure 4 for CF4, and Figure 5 for SF6. As very little adsorption occurs in pores smaller than w ) 7 Å for CH4 and in pores smaller than w ) 8 Å for CF4 and SF6, the adsorption isotherms in these pores are not shown here. For all three gases, a decrease in adsorption as the temperature increases can be seen. Figure 3 shows that the adsorption of CH4 decreases as the pore width increases and that the isotherms follow Henry’s law when the pore width is greater than w ) 9 Å. Figure 4 shows simulated isotherms for CF4. Since the adsorption of CF4 is stronger than that of CH4, its isotherms reflect a more rapid pore filling. The adsorption of CF4 also decreases as the pore width increases. SF6 shows the strongest adsorption, compared with CH4 and CF4, as shown in Figure 5. The adsorption decreases from w ) 8 Å to w ) 11 Å and then increases to w ) 14 Å. The isotherm reaches the highest plateau at w ) 14 Å, from where the adsorption begins to decrease with increasing pore width. This behavior is the result of the correlation between the pore size and the molecule size. In pores wider than w ) 11 Å, a second layer of molecules begins to form and is complete when the pore width reaches w ) 14 Å. Pores wider than w ) 20 Å become able to accommodate a third layer of SF6. Some of the isotherms for CF4 and SF6 cross. This is due to the competition between the strength of adsorption and packing effect in pores of different sizes. The three gases are subcritical in pores of these sizes, so the single-pore isotherms (and the isotherms for the model porous carbons) do not show evidence of capillary condensation. 3.4. PSDs and Their Predictive Ability. We invert the adsorption integral, eq 1, using the method of Davies and Seaton,34 to obtain a PSD using each of the three adsorptives. We then obtain an overall PSD by the three separate PSDs. The reliability

Cai et al.

Figure 3. Isotherms for the adsorption of CH4 in slit-shaped pores of various widths at the three temperatures considered; the pore width is defined as the distance between the center of the surface carbon atoms of the opposing pore walls.

of the overall PSD is examined by testing its ability to predict adsorption. Nine partial PSDs are obtained in this way for Carbon 1: one for each of the three adsorptives at three temperatures. As shown in Figure 6, there are differences in the PSDs using the various adsorptives at different temperatures. This reflects the difference in the size and the adsorption energy of the molecules involved, which leads to only parts of the PSDs shown in Figure 6 being reliable. Figure 3 shows that, in the pressure range studied, the CH4 single-pore isotherms become linear above approximately w ) 10 Å at 258 K and w ) 9 Å at the two higher temperatures. As the single-pore isotherms become more linear functions of pressure, the contributions of the various isotherms to the amount adsorbed become linearly dependent and these isotherms carry essentially no additional information about adsorption within the pores.37 As a result, the calculated PSD from eq 1 in this pore size range is not reliable and so should be discarded. The reliable pore size range defines the “window of reliability” which is

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Figure 4. Isotherms for the adsorption of CF4 in slit-shaped pores of various widths at the three temperatures considered; the pore width is defined as the distance between the center of the surface carbon atoms of the opposing pore walls.

Figure 5. Isotherms for the adsorption of SF6 in slit-shaped pores of various widths at the three temperatures considered; the pore width is defined as the distance between the center of the surface carbon atoms of the opposing pore walls.

bounded on the left by the smallest accessible pore size and on the right by the pore size at which adsorption becomes substantially linear. The reliability of the PSD is improved when a more strongly adsorbing species is used. For example, the CF4 PSD is reliable up to w ) 18 Å at 258 K, w ) 16 Å at 275 K, and w ) 14 Å at 296 K, as shown in Figure 4, that the adsorption isotherms in bigger pores are linear, while the SF6 PSDs include pores up to at least w ) 36 Å, as shown in Figure 5. The window of reliability can also be extended by measuring the isotherm at lower temperatures (so that, in this work, the most reliable information about larger pores is obtained at 258 K) or extending the measurements to include adsorption at higher pressures. The window of reliability is taken into account in obtaining the overall PSDs; that is, only the reliable pore size ranges are used. It is obvious that the small pore size ranges of the CH4 PSDs (up to w ) 10 Å at 258 K, w ) 9 Å at 275 and 296 K, as shown in Figure 6) should be taken. When considering the pore size ranges probed by CF4 and SF6, we notice that the CF4

PSDs and SF6 PSDs substantially overlap except that the SF6 PSDs have much smaller peaks, which means that some pores accessible to CF4 are not accessible to SF6. This indicates that the SF6 PSDs in Figure 6 should not be used directly to build up the overall PSDs. Here, we propose two ways to utilize the CF4 and SF6 PSDs. The first is to use “corrected” SF6 PSDs, while the second is to use the first peaks of the CF4 PSDs and the second peaks of the corrected SF6 PSDs. The corrected SF6 PSDs are obtained by increasing the original SF6 PSDs in Figure 6 by a factor that describes the fraction of pores accessible to CF4 that are actually accessible to SF6. This fraction is calculated as part of the connectivity analysis, as described in section 4. Figure 7 shows the two sets of the overall PSDs at the three temperatures. There are two aspects of the PSDs shown in Figure 6 which are unphysical. First, SF6 seems to detect smaller pores (smaller than w ) 8 Å) than the smaller CF4 molecule. Second, the height

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Figure 6. Pore size distributions of Carbon 1 obtained using the CH4 (9), CF4 (b), and SF6 (2) adsorption isotherms in Figure 2.

of the first peak in the PSD is smaller for CH4 than for CF4, whereas the smaller molecule should, in principle, detect at least as many pores as the larger molecule at all pore sizes. Both these cases reflect the fact that the fitting of the PSD is insensitive to features that have little effect on the overall adsorption calculated by eq 1. In the first case, there is very little adsorption in pores smaller than 8 Å for both CF4 and SF6, as shown in Figures 4 and 5, so the objective function is insensitive to the pore volume in this range for both species. In the second case, the objective function is sensitive to the approximate shape of the PSD but not to the precise shape (particularly for CH4 as the least strongly adsorbed species), so it is possible for the values of the PSD obtained using CH4 to be less than those of the PSD obtained with CF4, though of course the pore volume associated with each peak must not be less. So as to assess which of the overall PSDs is best, all have been used to predict the adsorption of the three adsorptive species. It

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Figure 7. Overall pore size distributions of Carbon 1 constructed by combining the CH4, CF4, and “corrected” SF6 pore size distributions (]) and the CH4 and “corrected” SF6 pore size distributions ([).

is interesting that these two sets of overall PSDs, though different in shape, give similar predictions that are in good agreement with the “experimental” isotherms. Figure 8 (top) shows an example of such behavior at 258 K. This indicates that both approaches to obtaining an overall PSD are viable. Moreover, it suggests that several “good” PSDs, in terms of their predictive ability, can exist which represent the porous structure of a carbon (though of course the real material has a unique PSD). As the two overall PSDs have similar predictive ability, only the predictions from the first overall PSD (i.e., CH4 + corrected SF6) are fully shown in Figure 8. It is clear that at all three temperatures there is very good agreement between the experimental and predicted isotherms for the three adsorptives. To test the predictive ability of the overall PSD at other temperatures, the overall PSD obtained at 258 K is used to predict adsorption at the two higher temperatures. Again, the predicted isotherms,

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Figure 9. Pore size distributions of Carbon 2 obtained using the CH4 (9), CF4 (b), and SF6 (2) adsorption isotherms at 258 K in Figure 2.

Figure 10. The overall PSD of Carbon 2 obtained from combining the CH4 and corrected SF6 PSDs at 258 K.

Figure 8. Comparison of predicted isotherms (lines) with the “experimental” isotherms for CH4 (9), CF4 (b), and SF6 (2) on Carbon 1. At 258 K, the black lines are derived using the overall PSD obtained from combining the CH4 and corrected SF6 PSDs at 258 K, while the gray lines are derived using the overall PSD obtained from combining the CH4, CF4, and corrected SF6 PSDs at 258 K. At the higher temperatures, the solid lines are derived using the overall PSD obtained from combining the CH4 and corrected SF6 PSDs at the corresponding temperatures, while the dotted lines are derived using the overall PSD obtained from combining the CH4 and corrected SF6 PSDs at 258 K.

indicated by dotted lines, are in good agreement with the experimental isotherms for CH4 and CF4, while SF6 is underestimated by ∼15%. This underestimation can be explained as follows. As shown in Figure 2, the “experimental” adsorption of SF6 at 258 K is less than that at 275 and 296 K, because one more cluster is not accessible to SF6 at 258 K. This results in the missing of some pores when using this isotherm to extract

the SF6 PSD, which leads to underprediction when using this pore structure information to predict the adsorption of SF6 at 275 and 296 K. However, the missing of some pores by the SF6 PSD obtained at 258 K does not lessen the accuracy of the prediction of the adsorption of CH4 and CF4 at 275 and 296 K, mainly because these pores have no significant adsorption to CH4 and CF4 at these two temperatures and in this pressure range. The above observation suggests that if all the species used to probe the pore structure give more adsorption at lower temperatures, as observed with Carbon 2 (Figure 2), the overall PSD obtained at the lowest temperature would give good predictions for all three species at higher temperatures. The PSDs for Carbon 2, extracted from adsorption of the three species at 258 K, are given in Figure 9. Clearly, more pores are accessible to SF6, compared with Carbon 1 (see Figure 6), indicating that the pore network of Carbon 2 presents fewer constrictions narrow enough to limit penetration of this species. The overall PSD (shown in Figure 10) is obtained by picking the reliable pore size range of the CH4 PSD (up to w ) 10 Å) and the pore size range of w ) 10-35.5 Å of the “corrected” SF6 PSD. (The “corrected” SF6 PSD is almost the same as the uncorrected PSD, as the fraction of pores accessible to CF4 is almost the same as that for SF6.) When this overall PSD is used to predict the adsorption of all three species at all three temperatures, there is general agreement between the predicted and the “experimental” isotherms, as shown in Figure 11 (though we note a tendency to overpredict the adsorption of CF4 and SF6, particularly at low pressure). The results based on both Carbon

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Figure 12. Schematic illustration of the pore network accessibility. The blue and red disks represent small and large molecules, respectively (CH4 and SF6, respectively, in the present case). “1” indicates pores which are too small to accommodate either CH4 or SF6; “2” indicates pores in which both CH4 and SF6 can reside; “3” indicates pores that are big enough to accommodate CH4 but not SF6; and “4” indicates pores which are large enough to accommodate SF6 but are shielded by smaller pores and so accessible only to CH4.

In this work, we chose SF6 as the probe species to obtain an estimate of the mean coordination number of the pore network, using the method of Lo´pez-Ramo´n et al.37 We are interested in comparing the number of pores accessible to SF6 (the entire SF6 PSD obtained using this species, which is denoted here by fs(w), with the number of pores large enough to accommodate SF6 in the overall PSD, denoted by f0(w). Since percolation theory deals with the number of pores of different sizes, we transform the PSD f(w), which is in terms of pore volume, to the function g(w), which is in terms of pore number. Assuming that the length and breadth of the pores are uncorrelated with their width

Figure 11. Comparison of predicted isotherms (lines) with the “experimental” isotherms for CH4 (9), CF4 (b), and SF6 (2) on Carbon 2.

g0(w) )

f0(w) w

(5)

gs(w) )

fs(w) w

(6)

For a given network, X is the normalized integral of g0(w) over the range of pore sizes that are large enough to accommodate the probe molecule:

∫w∞ g0(w) dw X) ∞ ∫0 g0(w) dw s

1 and Carbon 2, as discussed above, are very strong evidence that the overall PSDs obtained in this way are consistent.

4. Analysis of the Pore-Network Connectivity For a specific probe species in a given pore network, percolation theory provides the relationship between two variables: the fraction of pores in a network that are large enough to accommodate the probe species (the “bond occupation probability” X) and the fraction of pores that are actually accessible to this adsorptive (the “accessibility” A). The distinction between these two variables arises because some of the pores in the network, although large enough to accommodate the probe molecules, are connected only through smaller impenetrable pores and are thus not accessible to the probe molecules, as depicted in Figure 12. The deviation of A from X reflects the finite connectivity of the pore network.

(7)

where ws is the width of the smallest pores that can accommodate SF6. A is the normalized integral of gs(w) over the same range of pore sizes.

∫w∞ gs(w) dw A) ∞ ∫0 g0(w) dw s

(8)

The relation between A and X was correlated by a simple expression by Zhang and Seaton47 for cubic lattices (having (47) Zhang, L.; Seaton, N. A. Simulation of catalyst fouling at the particle and reactor levels. Chem. Eng. Sci. 1996, 51, 3257-3272.

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Z ) 6). Lo´pez-Ramo´n et al.37 generalized this correlation to a network with an arbitrary value of Z

{

0

for ZX e 1.494

1.314(ZX - 1.494)0.41 + 3.153(ZX - 1.494) ZA ) 3.48(ZX - 1.494)2 + 1.433(ZX - 1.494)3 for 1.494 < ZX < 2.7 ZX

(9)

for ZX g 2.7

The value of Z for the model carbon is estimated by fitting the (X, A) values given in Table 2 to eq 9 for each temperature. The mean coordination numbers of both Carbon 1 and Carbon 2, obtained using the method described above, are shown in Table 2. The values of X for Carbon 1 are quite close to that for Carbon 2, while the values of A for Carbon 1 are much smaller than that for Carbon 2. This indicates that the two carbons have very similar fractions of pores big enough to accommodate SF6 molecules, but because of the connectivity effect, more pores are inaccessible to SF6 in Carbon 1. This is in agreement with the observation made in section 3 that the PSDs show that Carbon 2 has a higher accessibility. For Carbon 1, the values of Z at different temperatures are very consistent (∼1.6). This reflects the good agreement between the PSDs at different temperatures. The mean coordination number for Carbon 2 is slightly higher at 2.1. Some remarks need to be made considering the connectivity information obtained here. First, at the most basic level, the approach provided in this paper measures the accessibility of the pore network to molecules below a certain size, that is, the smallest pores that the smallest adsorptive used can probe. In heterogeneous materials such as nanoporous carbons, the pores are connected randomly and the probe molecules will only penetrate and fill those pores that are connected through pores larger than the size of the probe molecule. Second, the mean coordination numbers reported in Table 2 are averages over values with a wide variation in the local coordination number; some nodes are connected to three or more pores, while others are at the end of dead-end pores, as illustrated in Figure 12. This method provides no information about the distribution of coordination numbers within the pore network. Third, the estimation of Z in this method involves the calculation of X and A, which in turn depends on the adsorptives used (and the particular way they interact with the material) and the process for obtaining the PSD.

5. Comparison of the ICM-Derived Pore Structure with the Structure of the Model Porous Carbons Figure 13 compares the overall PSD obtained from the ICMbased analysis in section 3, hereafter denoted as ICM-PSD, with the PSD obtained from the geometric analysis described in section 2, which is denoted henceforth as g-PSD. As the g-PSD is a direct measurement of the distribution of pore volumes of different size, we can view it as a datum against which to evaluate the Table 2. Values of X, A, and Z Obtained at the Three Temperatures T (K)

X

258 275 296

0.983 0.982 0.972

258

0.935

A

Z

0.517 0.584 0.605

1.65 1.67 1.70

0.856

2.06

Carbon 1

Carbon 2

Figure 13. Comparison of the overall pore size distribution derived from the ICM-based analysis at 258 K (- - -) with that derived from the geometric analysis of the model carbons (s).

ICM-derived PSD. The ICM-PSDs shown here are those of Figures 7 and 10 re-expressed in terms of the “accessible pore size”, wa ) w - 3.4 Å, as the distance between the surface of the pore surface carbons forms the basis for defining the pore sizes in the geometric analysis. Figure 13 shows that the ICMbased analysis positions much of the pore volume at approximately the correct pore sizes for both carbons. There are also, however, some significant discrepancies between the ICM-PSD and g-PSD for both carbons: the ICM-based analysis underpredicts the volume associated with pores whose size falls below wa ) 5 Å, especially for Carbon 1, while it does not predict at all well the nature of the porosity beyond wa ∼ 10 Å, including inclusion of pore sizes that do not exist and Vice Versa. The appearance of a second, spurious peak at high pore sizes in the ICM-PSDs is consistent with earlier work on the analysis of adsorption in individual pores using slit-shaped model pores. Davies and Seaton11 simulated adsorption in pores of rectangular cross sectionssimpler than the complex pore network of the model porous carbons, but nevertheless reflecting the basic rectangular geometry of the individual pores in the model porous carbons. They calculated the PSD of pores with rectangular cross section using single-pore isotherms obtained using slit-shaped pores as the kernel of the adsorption integral equation (eq 1), and they found that individual pores of rectangular cross section generated separate peaks in the PSD, reflecting the fact that the slit-porederived PSD interprets adsorption in different regions of a rectangular pore (the corners versus the sides, for example) in terms of slit-shaped pores of different sizes. More generally, the differences between the ICM-PSD and the g-PSD may be understood in terms of local convexity and concavity in the pore space of the VPCs and, by extension, in real carbons. This is illustrated in Figure 14. In regions of local

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values computed directly on the VPC. Agreement is excellent, supporting the validity of this approach to the pore-network connectivity from the point of view of transport properties.

6. Conclusions

Figure 14. Illustration of (a) regions of local convexity and (b) regions of local concavity. The molecule is shown with its field of substantial interaction with the solid.

convexity, an adsorbate molecule interacts less strongly than it would in a slit pore, so that a narrow, convex element of the pore space is registered as a larger slit pore in the ICM. Similarly, regions of locally concave porosity appear as smaller pores. As a separate effect, thinner or less dense regions of carbon in the VPC (reducing the strength of the adsorbate-solid interaction relative to Steele’s model) will also appear in the ICM-PSD as slightly larger pores. We expect these conclusions to apply to real carbons, so that the ICM is likely to calculate a PSD that is broader than the true PSD of the material. There is a serious, fundamental difficulty in making a similar comparison for the ICM and “true” values of the mean coordination number of the pore network. Unlike in the case of the PSD, where no significance attaches to the precise way in which the pore space is divided into discrete pores, the coordination number of the VPC depends strongly on how this is done. For example, a progressively narrowing section of the pore space might reasonably be represented as a single pore or as two or more pores of different width. It is thus impossible to make a useful comparison of the ICM and true coordination numbers. Fortunately, other, less direct comparisons are possible. We saw in section 3.4 that reduced accessibility of the pore space to SF6, reflected in the calculated value of Z for the ICM, is needed to give good predictions of adsorption using the ICM, suggesting at the very least that this approach gives practically useful results for adsorption equilibrium. The transport property of a pore network is a stronger test of the way in which connectivity is handled. We report elsewhere48 our computation of the diffusion coefficients of the two VPCs using the ICM, and the corresponding

We have investigated the applicability of the ICM, analyzed using GCMC simulation and percolation theory, for the characterization of porous carbons. This approach was applied to characterize two model carbons whose adsorption behavior resembles that of real carbons. A database of adsorption isotherms of three different gases (CH4, CF4, and SF6) at three temperatures (258, 275, and 296 K) was obtained by GCMC simulation for a series of slit-shaped pores. PSDs were obtained by comparing these adsorption isotherms with the “experimental” isotherms from the model carbons, using the adsorption integral equation. A more complete picture of the PSDs was obtained by combining the partial PSDs probed by the three gases. The predicted adsorption isotherms for the three gases, at the three temperatures, from the overall PSDs, are in good agreement with those generated from the model carbons. This indicates the predictive power of these overall PSDs when used in a consistent manner and supports the use of the ICM as a tool for modeling adsorption in real carbons. Comparison of the overall PSDs derived from the ICM-based analysis with those obtained from the geometric analysis of the model carbons shows that while the distribution of the volume associated with pores of accessible width below wa ) 10 Å is reasonably well predicted, there are significant discrepancies for larger pores, including the omission of pore sizes that are known to exist and Vice Versa. These discrepancies can be understood as arising from locally convex or locally concave porosity. The comparison between the PSD using the ICM with the true structure of the model carbons thus demonstrates the limitations of using the ICM, with its assumptions of pores of constant cross section and pore intersections of negligible volume to describe real carbons. The estimate of the mean coordination number, Z, of the pore network was obtained by comparing the PSD obtained from the adsorption of SF6 with the overall PSD obtained by combining those of the three adsorptives with the aid of percolation theory. We see good agreement between the values of Z determined at different temperatures, supporting the consistency of this approach. The mean coordination numbers for the two different carbons show the difference between them; that is, Carbon 1 has lower accessibility than Carbon 2, which is in agreement with the evidence from the PSDs. Acknowledgment. Q.C. gratefully acknowledges an ORS award from Universities U.K. The U.K. Engineering and Physical Sciences Research Council (GR/R87178/01) is acknowledged for financial support of this work. LA7007057 (48) Cai, Q.; Buts, A.; Seaton, N. A.; Biggs, M. J. Pore Network Modeling of Diffusion in Nanoporous Carbons: Validation by Molecular Dynamics Stimulation, in preparation.