Molecular Simulation Evidence for Solidlike Adsorbate in Complex

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Molecular Simulation Evidence for Solidlike Adsorbate in Complex Carbonaceous Micropore Structures Mark J. Biggs,* Alex Buts, and David Williamson Institute for Materials and Processes, University of Edinburgh, Kenneth Denbigh Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, U.K. Received December 3, 2003. In Final Form: March 17, 2004 Adsorption of a model nitrogen vapor on a range of complex nanoporous carbon structures is simulated by grand canonical Monte Carlo simulation for a single subcritical temperature above the bulk freezing point. Adsorption and desorption isotherms, heats of adsorption, and three-dimensional singlet distribution functions (SDFs) were generated. Inspection of the SDFs reveals significant levels of solidlike adsorbate at saturation even in the most complex of the microporous solids considered. This strongly suggests that solidlike adsorbate will also occur for simple subcritical vapors adsorbed on real noncrystalline solids such as microporous carbons at temperatures above the bulk freezing point, supporting indirect experimental observations. The presence of significant levels of solidlike adsorbate has implications for characterization of microporous solids where adsorbate density is used (e.g., determination of pore volume from loading). Detailed consideration of the SDF at different loadings for a model microporous solid indicates solidlike adsorbate forms at distributed points throughout the pore space at pressures dependent on the nature of the local porosity. The nature of the local porosity also dictates the freezing mechanism. A local freezing/ melting/refreezing process is also observed. Introduction of mesoporosity into the model causes hysteresis between the adsorption and desorption isotherms. Adsorption in the hysteresis loop occurs by a series of local condensation events. It appears as if the presence of adjacent microporosity and/or adsorbate within it affects the pressure at which these events occur. Reversal of the condensation during desorption occurs throughout the mesoporosity at a single pressure; this pressure is unaffected by the presence of adjacent microporosity or the adsorbate within it. It is also shown that the empirical concept of “pore size” is not consistent for describing adsorption in the complex solids considered here. A new concept is, therefore, proposed that seeks to account for the factors that affect local adsorption energy: local geometry, microtexture, surface atom density, and surface chemistry.

1. Introduction Adsorption in nanoporous solids, which we define as those containing micro- and mesopores as defined by IUPAC,1 is relevant across science, engineering, and beyond. Some applications of particular relevance here are nanoporous solid characterization,2 natural gas storage,3,4 and porous silicon nanotechnologies.5-9 In all these applications, knowledge of the phase behavior as a function material and the thermodynamic parameters is importantsjust two examples are the use of the adsorbate density to obtain pore volumes from measured loadings and the influence of HF-solution phase behavior on the manufacture of porous silicon. It is well-known that confinement in nanosized domains causes significant deviations from bulk phase behavior including shifts in the critical and phase transition points and the occurrence * To whom correspondence should be addressed. Fax: +44-131650-6551. E-mail: [email protected]. (1) Rouquerol, J.; Avnir, D.; Fairbridge, C. W.; Everett, D. H.; Haynes, J. H.; Pernicone, N.; Ramsay, J. D. F.; Sing, K. S. W.; Unger, K. K. Pure Appl. Chem. 1994, 66, 1739-1758. (2) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (3) Menon, V. C.; Komarneni, S. J. Porous Mater. 1998, 5, 43-58. (4) Lozano-Castello´, D.; Alcan˜iz-Monge, J.; de la Casa-Lillo, M. A.; Cazorla-Amoro´s, D.; Linares-Solano, A. Fuel 2002, 81, 1777-1803. (5) Canham, L. T.; Groszek, A. J. J. Appl. Phys. 1992, 72, 15581565. (6) Faivre, C.; Bellet, D.; Dolino, G. Eur. Phys. J. B 1999, 7, 19-36. (7) Schechter, I.; Ben-Chorin, M.; Kux, A. Anal. Chem. 1995, 67, 3727-3732. (8) Bettotti, P.; Cazzanelli, M.; Dal Negro, L.; Danese, B.; Gaburro, Z.; Oton, C. J.; Vijaya Prakash, G.; Pavesi, L. J. Phys.: Condens. Matter 2002, 14, 8253-8281. (9) He´rino, R. Mater. Sci. Eng., B 2000, 69-70, 70-76.

of novel phases.10 As revealed by the recent reviews of Gelb et al.10 and Christenson,11 this behavior and the importance of understanding how it is affected by material and thermodynamic parameters have prompted considerable experimental and theoretical work. Of particular interest here is the work concerned with the phase behavior of simple fluids such as nitrogen in solids containing micropores and smaller mesopores, where a variety of experimental data suggests the existence of solidlike adsorbate at temperatures above the bulk freezing point. Some of the earliest evidence comes from Breck and Grose,12 who used accessible volumes calculated from the known crystal structures of a variety of zeolites to show that the density of nitrogen adsorbed by the zeolites at 77 K was 20% greater than that of the bulk saturated liquid. A similar conclusion was reached by Mu¨ller and Unger,13 who attributed a low-pressure hysteresis between the 77 K nitrogen adsorption and desorption isotherms on large crystals of zeolite ZSM-5 (the use of such crystals removed capillary condensation) to a liquid-solid phase change, where the density evalu(10) Gelb, L. V.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573-1659. (11) Christenson, H. K. J. Phys.: Condens. Matter 2001, 13, R95R133. (12) Breck, D. W.; Grose, R. W. A correlation of the calculated intracrystalline void volumes and limiting adsorption volumes in zeolites. In Molecular Sieves; Meier, W. M., Uytterhoeven, J. B., Eds.; American Chemical Society: Washington, DC, 1973; p 319. (13) Mu¨ller, U.; Unger, K. K. Sorption studies on large ZSM-5 crystals: The influence of aluminium content, the type of exchangeable cations and the temperature on nitrogen hysteresis effects. In Characterization of Porous Solids; Unger, K. K., Rouquerol, J., Sing, K. S. W., Kral, H., Eds.; Elsevier: Amsterdam, The Netherlands, 1988; p 101.

10.1021/la036269o CCC: $27.50 © 2004 American Chemical Society Published on Web 05/07/2004

Complex Carbonaceous Micropore Structures

ated from the saturation loading and an independently determined pore volume were found to be equal to those of the bulk solid phase. Goulay et al.14 more recently confirmed the conclusions of these earlier studies and showed nitrogen in a variety of zeolites at saturation to be consistently denser than that in the bulk phase for the temperatures they considered between 78 and 125 K. By observing viscosities of a simple fluid between two separated flat mica sheets at decreasing separations, which is commonly termed the surface force apparatus (SFA),15 Klein and Kumacheva16 showed that freezing can occur in fluid films ∼7σ or smaller in width at temperatures above the bulk freezing point, where σ is the size of the fluid molecule. Proof for elevated freezing and associated solidlike phases in noncrystalline nanoporous solids such as activated carbons is more difficult to obtain. One of the first efforts for such solids was that of Kaneko et al.,17 who used understanding of adsorption on graphite surfaces to argue that a step in the 77 K nitrogen isotherms of a number of activated carbon fibers and molecular sieves in the region P/P0 ∼ 0.005, P0 being the saturation pressure, was most probably due to a liquid-solid phase transition and that a second step in the region P/P0 ∼ 0.0085 was possibly indicative of a solid-solid phase transitionstheir arguments must, however, be considered tentative at best. Carrott and co-workers compared the takeup at saturation for a selection of adsorbates on a number of activated carbons to argue that the density of nitrogen at 77 K exceeds substantially that of its bulk phase counterpart18,19 and later estimated it to be ∼111% and ∼87% of the bulk liquid and solid phase densities, respectively.20 Most recently, differential scanning calorimetry (DSC) experiments undertaken by Kaneko and co-workers21,22 indicated significant elevation of the freezing point for CCl4 and benzene in activated carbon fibers and small enthalpies of phase change compared to those of the bulk, prompting them to suggest the transition involves the adsorbate passing from an ordered liquid to a disordered solid. Several groups have reported a range of theoretical studies concerned with elevated freezing and associated solidlike phases in nanopores; these studies were reviewed by Gelb et al.10 and more briefly and recently by Vishnyakov and Neimark.23 The vast majority of these studies is based on simple slit or cylindrical pores of infinite extent. Despite this, they have shown that the strength of the fluid-solid interaction relative to that of the fluid-fluid interaction is an important factor in explaining why freezing points can be depressed or raised above that of the bulk,24 that stable hexatic phases may exist over wide temperature ranges in microporous media,25 and that a (14) Goulay, A. M.; Tsakiris, J.; Cohen de Lara, E. Langmuir 1996, 12, 371-378. (15) Kumacheva, E. Prog. Surf. Sci. 1998, 58, 75-120. (16) Klein, J.; Kumacheva, E. Science 1995, 269, 816-819. (17) Kaneko, K.; Suzuki, T.; Kakei, K. Langmuir 1989, 5, 879-881. (18) Carrott, P. J. M.; Roberts, R. A.; Sing, K. S. W. A new method for the determination of micropore size distribution. In Characterization of Porous Solids; Unger, K. K., Rouquerol, J., Sing, K. S. W., Kral, H., Eds.; Elsevier: Amsterdam, The Netherlands, 1988; p 89. (19) Carrott, P. J. M.; Freeman, J. J. Carbon 1991, 29, 499-506. (20) Carrott, P. J. M. Carbon 1995, 33, 1307-1312. (21) Kaneko, K.; Watanabe, A.; Iiyama, T.; Radhakrishnan, R.; Gubbins, K. E. J. Phys. Chem. B 1999, 103, 7061-7063. (22) Watanabe, A.; Iiyama, T.; Kaneko, K. Chem. Phys. Lett. 1999, 305, 71-74. (23) Vishnyakov, A.; Neimark, A. V. J. Chem. Phys. 2003, 118, 75857598. (24) Miyahara, M.; Gubbins, K. E. J. Chem. Phys. 1997, 106, 28652880. (25) Radhakrishnan, R.; Gubbins, K. E.; Sliwinska-Bartkowiak, M. Phys. Rev. Lett. 2002, 89, 076101.

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shear-induced adsorbate freeze/melt cycle is the origin of the stick-slip behavior observed in the SFA.26,27 These studies have also shown that pore geometry plays an important role in freezing28 and that fluids confined in simple pore geometries demonstrate a rich phase behavior.23,29 The restriction to such simple pore geometries reflects, of course, the difficulties faced in modeling and interpreting results for more realistic pore structures. However, it is recognized that simple models are not entirely satisfactory,30 and extension to more complex models that include elements of reality are necessary.31 Some scientists have considered the phase behavior of simple fluids in more complex pore spaces and have found that pore system topology and heterogeneity play an extremely important role in dictating the phase behavior. Of particular relevance here is the observation that the vapor-liquid phase transition in complex pore spaces proceeds by passing through a series of morphologymediated metastable states, leading to what appears at the macroscopic scale to be a smooth filling process.32 Few of the studies in more complex systems consider freezing transitions and associated solidlike phases.10 All those that have focused on such phenomena have done so using ‘perturbations’ on the slit pore model of infinite extent. Some of the earliest work in this regard is that of Curry et al.,33 who modeled the dynamics of a simple fluid in a narrow (one to three molecular layers) slit pore defined by structured rigid crystalline walls of fluid atoms inscribed with shallow (one molecular layer) periodic grooves. Despite the weak fluid-wall interactions, these scientists found coexisting solid and liquid phases with the fraction of each varying with the registry of the opposing walls; they also observed two freezing mechanisms, a slower morphology-mediated process and a second more rapid freezing ‘seeded’ by an existing frozen region. Using grand canonical molecular dynamic (GCMD) simulation, Maddox et al.34 also observed morphology-mediated freezing for a simple fluid in a system defined by two slit pores of two adsorbate molecules in width separated by a slit pore 2-3 times wider. Vishnyakov et al.35 studied the effect of chemical and surface heterogeneity on adsorption in slit-shaped pores of 4-8σ in width. Interestingly, these scientists observed freezing only in the widest pores and only following capillary condensation. They found that heterogeneity in the cases they considered substantially reduced the freezing temperature relative to that predicted by the simple slit pore model and also induced the freezing temperature to increase with the pore width, contrary to both classical thermodynamic theory and previous simulations on simple slit pores; their results must be considered tentative, however, due to the small size of the systems simulated. Most recently, Ne´meth (26) Schoen, M.; Rhykerd, C. L.; Diestler, D. J.; Cushman, J. H. Science 1989, 245, 1223-1225. (27) Thompson, P. A.; Robbins, M. O. Science 1990, 250, 792-794. (28) Maddox, M. W.; Gubbins, K. E. J. Chem. Phys. 1997, 107, 96599667. (29) Ghatak, C.; Ayappa, K. G. J. Chem. Phys. 2002, 117, 53735383. (30) Radhakrishnan, R.; Gubbins, K. E.; Watanabe, A.; Kaneko, K. J. Chem. Phys. 1999, 111, 9058-9067. (31) Radhakrishnan, R.; Gubbins, K. E.; Sliwinska-Bartkowiak, M. J. Chem. Phys. 2002, 116, 1147-1155. (32) Kerlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.; Tarjus, G. Phys. Rev. Lett. 2001, 87, 055701. (33) Curry, J. E.; Zhang, F.; Cushman, J. H.; Schoen, M.; Diestler, D. J. J. Chem. Phys. 1994, 101, 10824-10832. (34) Maddox, M. W.; Quirke, N.; Gubbins, K. E. Mol. Simul. 1997, 19, 267-283. (35) Vishnyakov, A.; Piotrovskaya, E. M.; Brodskaya, E. N. Adsorption 1998, 4, 207-224.

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and Lo¨wen36 compared the behavior of a fluid in smooth and rough 3.5σ spherical cavities and found that surface roughness affects the adsorbate structure and the solidification behavior. We report here for the first time grand canonical Monte Carlo (GCMC) simulations of adsorption from the subcritical vapor of a simple fluid onto complex carbonaceous micropore spaces. We find substantial levels of solidlike adsorbate at saturation, supporting the idea that elevated freezing is likely to occur for strongly interacting simple fluids in real noncrystalline solids such as activated carbons despite their significant microtextural, geometrical, and chemical heterogeneity. The paper begins by describing the basis and the general details of the solid model used. This is followed by the specifics of the fluid and solid models considered in this paper as well as the GCMC simulations. The variation of isotherm character with model parameters is first considered in some detail, and as a consequence, a new concept for describing porosity is proposed that goes beyond the empirical concept of “pore size” by considering the various factors that dictate the adsorption energy of a local region of porosity. Despite its restrictive nature, the universality of the empirical concept of pore size demands that we establish a relationship between it and the model parameters, and this is therefore done next. This is followed by a detailed consideration of the adsorbate structure in a number of model solids to demonstrate the existence of a solidlike phase and its development with loading. We conclude with a statement regarding further work that is underway and to be reported elsewhere. 2. Model Details 2.1. Solid Model. We use a model carbon solid that captures qualitatively as much of the essence of the real carbon molecular structure as is currently understood and which is described by a minimal set of parameters that have a basis in experimental quantities widely recognized by the community. While we do not claim that the model used here reflects accurately the structure of real nanoporous carbons, it does possess a complex pore space that one would expect of such solids and, as we shall show, it is capable of yielding a rich adsorption isotherm and enthalpy behavior that mirrors that seen experimentally for carbons. The Essence of Nanoporous Carbon Molecular StructuresThe Basis of the Solid Model. The foundations of the solid model used here rest on the ideas of Oberlin and co-workers,37,38 who proposed that nanoporous carbons are built up from domains of two-dimensional short-range order that may be reasonably represented by small polyaromatic molecules or similar structures, a view supported by a vast array of experimental data. These domains assemble in a roughly aligned manner with outof-plane spacing somewhat greater than that of graphite to form nanoscale regions of local molecular orientation (LMO)39 (Figure 1a),40 which then combine to form mesoscopic structures, as illustrated in Figure 1b.41 It is the larger out-of-plane spacing and the misalignment between the domains within the regions of LMO that are (36) Ne´meth, Z. T.; Lo¨wen, H. Phys. Rev. E 1999, 59, 6824-6829. (37) Oberlin, A.; Bonnamy, S.; Rouxhet, P. G. Chem. Phys. Carbon 1999, 26, 1-148. (38) Oberlin, A. Chem. Phys. Carbon 1989, 22, 1-143. (39) Oberlin, A. Carbon 1984, 22, 521-541. (40) Oberlin, A.; Boulmier, J. L.; Villey, M. Electron microscopic study of kerogen microtexture. Selected criteria for determining the evolution path and evolution stage of kerogen. In Kerogen; Durand, B., Ed.; Technip: Paris, 1980; Chapter 7. (41) Oberlin, A.; Villey, M.; Combaz, A. Carbon 1980, 18, 357-353.

Figure 1. Schematic a shows a region of local molecular orientation (LMO) of in-plane extent, La, and out-of-plane extent, Lc, made up of domains of two-dimensional short-range order (shaded multiring structures) with out-of-plane spacing, d002, and in-plane extent, l (after Oberlin et al.40). These are combined to form mesoscale structures, as illustrated in schematic b (adapted from ref 41; copyright 1980 Elsevier).

responsible for the vast majority of the microporosity and the microtextural heterogeneity in nanoporous carbons. An understanding of how the regions of LMO are assembled to yield mesoscopic structures is not clears Oberlin and co-workers observed both random41 and gradually varying42 arrangements of LMO, while more recent computer analysis of high-resolution transmission electron microscopy images by Shim et al.43 suggests there may be some relationship between the degree of ordering of the regions of LMO and the nature of the precursor and processing conditions. Much of the mesoporosity in carbons is likely to be found between regions of LMO. Recently, Harris and Tsang44 proposed that certain porous carbons may contain significant levels of five- and seven-membered rings and, therefore, take on a fullerenelike structure; the existence of such rings and, therefore, local curvature certainly seems feasible, given the ease with which some fullerenes and other curved carbonaceous structures can be formed. As suggested by Figure 1a, the (42) Oberlin, A.; Oberlin, M. J. Microsc. (Oxford) 1983, 132, 353363. (43) Shim, H. S.; Hurt, R. H.; Yang, N. Y. C. Carbon 2000, 38, 29-45. (44) Harris, P. J. F.; Tsang, S. C. Philos. Mag. A 1997, 76, 667-677.


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Figure 2. Basis of the carbon model used here. Regions of local molecular orientation (LMO) of identical size are arranged on a regular rectangular lattice of uniform spacing. The nonporous region of LMO shown here consists of parallel finitesized graphene domains (example shown in black) arranged, once again, on a regular rectangular lattice of uniform spacing. Microporosity is predominately introduced to the model via modification of the regions of LMO using the algorithm of Figure 3. Mesoporosity is predominately introduced via inter-LMO pores.

view of Oberlin et al. on porous carbon structures does not preclude the presence of non-six-membered rings and, therefore, the curvature they allow. The small polyaromatic domains could (and are likely to) contain non-sixmembered rings, and the orientational mismatch between these domains implies non-six-membered rings (as well as heteroatoms)sthe difference between these two means of including non-six-membered rings simply reflects the distance over which curvature arising from their inclusion significantly manifests itself. In light of this interpretation, structures such as those sketched by Harris and Tsang could correspond to regions of LMO that are of the same order in size as the polyaromatic domains themselves, a possibility that appears to be supported by experimental measurements of domain and LMO extent (see below). The LMO extents, La and Lc, the domain extent, l, the domain out-of-plane spacing, d002, and to a lesser extent the domain misorientation, R, have all been measured by a variety of experimental methods (see Oberlin39). While these measurements show these parameters depend on the precursor and preparation conditions of the carbon,37 typical values for nanoporous carbons are La ∼ 10-40 Å, Lc ∼ 10-30 Å, l ∼ 5-15 Å, d002 ∼ 3.5-6.5 Å, and R ∼ (15°.38-43,45-47 Description of the Nanoporous Carbon Model. The carbon model used here is based on the foregoing view of nanoporous carbon structure and the well understood parameters of domain extent, domain out-of-plane spacing, and LMO extents, as well as tilt and rotation of the domain groups. As illustrated in Figure 2, regions of LMO of identical extent La × Lb′ × Lc were arranged on a regular rectangular lattice of uniform spacing (La + da) × (Lb′ + db′) × (Lc + dc)sthis spacing provided the means of

Figure 3. Algorithm for creating the first variant of a microporous region of LMO from a nonporous region of LMO consisting of Na × Nb′ × Nc two-dimensional graphene domains. The set of θm (θ-1 ) 0) defines the cumulative probability distribution of removing m successive out-of-plane domains in the xc direction and is, therefore, directly related to the intraLMO d002 distribution. Ran[0,1] is a uniformly distributed random number between 0 and 1. The other two variants of microporous regions of LMO are derived from the first, as described in the main text.

including mesoporosity without undue complexity. Basic nonporous regions of LMO, which are denoted by ‘NP’, were built by arranging Na × Nb′ × Nc ) La/la × Lb′/lb′ × Lc/lc parallel finite-sized graphene domains of size la × lb′ on a regular rectangular lattice of uniform spacing la × lb′ × lc, where lc is the minimum domain out-of-plane spacing. The nonporous region of LMO was used as the basis for three variants of microporous regions of LMO. The first, which is denoted by ‘P’, was generated by applying the algorithm of Figure 3 to the regions of LMO. The second variant was generated by tilting superdomains (these superdomains have a counterpart in the experimental literature termed the basic structural unit (BSU)42), which

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Figure 5. Isoenergy map of an infinite microporous region of LMO of the second variety built using the algorithm detailed in section 2.1 with tilting of superdomains 15° about both centroidal in-plane axes (P(15,15)). Black and blue indicate regions of inaccessible and accessible volumes, respectively. Note the irregularity of the cross section of the basal pores. Such irregularity for pores with other surfaces were introduced by rotating the superdomains about their normal (i.e., xc) axis.

Figure 4. Isoenergy map of a finite microporous region of LMO of the first variety (P) built using the algorithm detailed in section 2.1 showing the nature of the microporosity and identifying just a few of the points of access to this microporosity: (a) whole region of LMO where the blue surface encapsulates the inaccessible volume; (b) the region of LMO with its corner cut away along the broken lines indicated in part as black and gray indicate regions of inaccessible and accessible volumes, respectively.

are defined as a stack of successive domains lc apart, (ζi∆Ri about their in-plane centroidal axes, a and b′, and eliminating any solid atoms that overlap, where Ri are the maximum possible tilt angles for each axis and ζi are uniformly distributed random numbers between 0 and 1. These models are denoted by ‘P(∆Ra,∆Rb′)’, where the angles are expressed in degrees. The third variant is generated from the second variant by rotating the superdomains about their normal centroidal axis by (ζc∆Rc; results for this variant are not presented here, as they qualitatively differ little from those of the second. As illustrated in Figures 4 and 5, the algorithms just described yield complex pore structures that contain a wide variety of pore shapes, extents, and surfaces. This mixture of different surfaces and geometries decouples the rigid link between pore size and energy that exists in the slit pore model. As will be shown briefly below, this decoupling allows a wide range of isotherm shapes to be obtained as well as the experimentally observed decrease of the heat of adsorption with increasing loading. We will also show that pore extent and coupling between intraLMO microporosity and inter-LMO mesoporosity significantly influence adsorption behavior.

Geometric Analysis of Carbon Models and Exact Measures for Pore Volume and Surface Area. To allow evaluation of the overall adsorbate density for the model from its GCMC simulation results, exact accessible pore volumes were determinedsthis was done by two methods and then, cross-checked for consistency. The first method was based on iterative refinement of a cubic tessellation. A tessellation was laid down on the volume of the solid, ignoring the solid atoms. The cubes were then separated into three sets: ‘solid’ (S), ‘void’ (V), and ‘interface’ (I). Sets S and V contained all those cubes having all and none of their vertexes in the solid phase, respectively; the remainder of the cubes were allocated to set I. A vertex was considered to be located in the solid phase if a test fluid molecule located at the vertex exceeded, under periodic boundary conditions, a potential energy threshold, φs, which is defined below. The position of the solid-void interface within the cubes of set I was located by a variant of the tetrahedral decomposition method.48 The pore volume was equated to the volume of the cubes in set V and the tetrahedra in set I that were judged to lie in the pore space. The final pore volume for the model was determined by iteratively subdividing the cubes and repeating the above process until the evaluated volume no longer changed. The second method used to determine exact pore volume was based on iterative refinement of a Delaunay tessellation.49 The details of identifying the interface between the solid and the void space were similar to those for the cubic tessellation approach. Iterative refinement of the tessellation was done by increasing the number of seed points until the calculated volume ceased to change. The final pore volume, V e, determined from the two tessellation-based methods agreed well for all models. (45) Emmerich, F. G. Carbon 1995, 33, 1709-1715. (46) Kaneko, K.; Ishii, C.; Kanoh, H.; Hanzawa, Y.; Setoyama, N.; Suzuki, T. Adv. Colloid Interface Sci. 1998, 76-77, 295-320. (47) Rouzaud, J. N.; Clinard, C. Fuel Process. Technol. 2002, 77-78, 229-235. (48) Doi, A.; Koide, A. IEICE Trans. Commun. 1991, 74, 214-224. (49) Watson, D. F. Comput. J. 1981, 24, 167-172.


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The energy threshold, φs, was chosen to be that which yielded an accessible pore width (i.e., in which the center of a molecule could be placed) of 2.5σff - 2σfs for an infiniteextent graphite basal plane slit pore of width 2.5σff, where this width is defined as the distance between the centers of the first row of carbon atoms in each wall of the pore and σff and σfs are the length parameters of the fluidfluid and fluid-solid Lennard-Jones interactions, respectively (see section 2.2). 2.2. Fluid Model. The fluid is modeled by a spherical Lennard-Jones (LJ) molecule. Both the fluid-fluid and fluid-solid atom interactions are modeled with a truncated and shifted pair potential

φ(r) )

{

φLJ(r) - φLJ(rc) for r < rc for r g rc 0

(1)

with

[( ) ( ) ]

φLJ(r) ) 4ij

σij r

12

σij r

6

(2)

where r is the distance between the pair of interacting centers, rc is the cutoff radius, and ij and σij are the LJ energy and length parameters, respectively, for interaction between species i and j. 3. Study Details 3.1. Monte Carlo Simulation Details. Adsorption was simulated by the cavity biased grand canonical Monte Carlo (GCMC) method of Mezei,50 where chemical potential, temperature, and volume are constants. Points on the adsorption and desorption isotherms are generated in this method by changing the chemical potential, which is related to the bulk phase pressure by standard thermodynamic relations51 acting on a suitable equation of state for the bulk fluid (see section 3.2). The simulation for each point on the adsorption and desorption isotherms was started using the final state of the previous point, with the first point of the adsorption and desorption isotherms starting from an empty pore structure and the last point of the associated adsorption isotherm, respectively. Each point was determined using 50 × 103 equilibration steps per molecule, where a step is one attempted move and one attempted insertion/ deletion, followed by at least max(106, 50 × 103 steps per molecule, 10 insertions/deletions per molecule) production steps; the first of these applies at low loadings, the second, at moderate loadings, and the last, at high loadings where insertion and deletion are difficult. In the case of suspected phase transitions and other special cases, up to 50 times this number of production steps were used. The maximum number of molecules varied from approximately 1000 to 7000 depending on the model. Simulations were undertaken in reduced units, indicated here by a superscripted asterisk, where the fluidfluid interaction parameters, σff and ff, are used in the reductions. Examples of reduced units include the reduced length, x* ) x/σff, volume, V* ) V/σ3ff, temperature, T* ) kbT/ff, number density, F* ) Fσ3ff, and energy, φ* ) φ/ff, where kb is the Boltzmann constant. Isotherms were generated at a reduced temperature of T* ) 0.8, which is equal to 76.16 K. The isotherms were generated for the relative pressure range Pl e P/P0 e 1.0, where the lower pressure, Pl, was selected such that the (50) Mezei, M. Mol. Phys. 1980, 40, 901-906. (51) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice Hall: Englewood Cliffs, New Jersey, 1969.

number of molecules at the pressure was approximately three; this requirement meant Pl ∼ O(10-8-10-7), depending on the solid. The total number of adsorbate molecules, n ) 〈N〉, and the adsorbate molecule distribution throughout the pore volume, n(r) ) 〈N(r)〉sthe singlet distribution function (SDF)swere both generated during the production runs, with the latter evaluated at a resolution of 0.25σ. The isosteric heat of adsorption was also determined as a function of coverage from52-54

qst ) kbT -

|

∂〈φ〉 ∂〈N〉 T

(3)

These values were checked53,54 against the isosteric heat of adsorption obtained from

|

〈Nφ〉 - 〈N〉〈φ〉 ∂φ ) ∂n T 〈N2〉 - 〈N〉2

(4)

and found to be consistent. 3.2. Interaction Potential Details. The single-site LJ nitrogen molecule of Walton and Quirke55 was used; the nitrogen-nitrogen (ff) and nitrogen-carbon (fs) interaction parameters are σN2-N2 ) 3.75 Å and N2-N2 ) 95.2kb J and σN2-C ) 3.57 Å and N2-C ) 51.6kb J, respectively. A cutoff radius of r/c ) 2.5 was used for both the fluid-fluid and fluid-solid interactions to make the computation times more reasonable. The bulk phase behavior of this model fluid is described by the BenedictWebb-Rubin equation of state of Nicolas et al.56 as modified by Smit.57 The reduced density of the bulk phase liquid at saturation was estimated at F* ) 0.73. 3.3. Carbon Model Details. Over 30 carbon models were considered in order to explore the role of the model parameters and phenomena observed. A subset of models that spans this set in terms of their behavior are used here. As shown in Table 1, four solid model parameters were varied systematicallysthe LMO extents, Li, the interLMO spacing, di, and both the mean, d h /002, and the width of the intra-LMO distribution of out-of-plane domain spacing, where the latter is defined in terms of the cumulative probability distribution of removing m successive domains in the xc direction, {θ0, θ1, ..., θM}. The inter-LMO pore widths were selected on the basis of the number of adsorbate layers, hence, the different widths for the armchair (a), fjord (b′), and basal (c) pores. Models 1P-3P, which have no inter-LMO pores, differ in terms of their intra-LMO d002 distribution mean and width. Model 1P(15,15) is the first of these models with a superdomain tilt imposed about both the xa and xb′ axes. Models 4P-7P, which have the same region of LMO as model 3P, differ only in their inter-LMO pore width. The nonporous models 4fNP and 6fNP differ only in terms of their inter-LMO pore length-to-width ratio, Li/di, for a given inter-LMO pore width, where the superscript f denotes the approximate fraction of the largest ratio simulated for the class of model. (52) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (53) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 148-154. (54) Myers, A. L.; Calles, J. A.; Calleja, G. Adsorption 1997, 3, 107115. (55) Walton, J. P. R. B.; Quirke, N. Mol. Simul. 1989, 2, 361-391. (56) Nicolas, J. J.; Gubbins, K. E.; Streett, W. B.; Tildesley, D. J. Mol. Phys. 1979, 37, 1429-1454. (57) Smit, B. J. Chem. Phys. 1992, 96, 8639-8640.

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Table 1. Details of Carbon Models Reported Here and the Predicted Adsorbate Density at Saturation, G, Relative to the Bulk Saturated Liquid Density, Gla LMO extents

inter-LMO spacing

out-of-plane spacing distribution within LMO

model

L/a

/ Lb′

L/c

d/a

/ db′

d/c

θ0

θ1

θ2

θ3

θ4

d h /002

F/Fl

1P 1P(15,15) 2P 3P 4P 41NP 41/2 NP 41/4 NP 5P 6P 61NP 61/2 NP 7P 71NP

∞ ∞ ∞ ∞ 15.72 15.72 7.86 3.93 15.72 15.72 15.72 7.86 15.72 15.72

∞ ∞ ∞ ∞ 13.61 13.61 9.08 3.40 13.61 13.61 13.61 9.08 13.61 13.61

∞ ∞ ∞ ∞ 14.31 14.31 8.95 3.58 14.31 14.31 14.31 8.95 14.31 14.31

2.17 2.17 2.17 2.17 3.17 4.67 4.67 4.67 7.67 7.67

2.29 2.29 2.29 2.29 3.29 4.79 4.79 4.79 7.79 7.79

2.50 2.50 2.50 2.50 3.50 5.00 5.00 5.00 8.00 8.00

0.094 0.094 0.000 0.250 0.250 1.000 1.000 1.000 0.250 0.250 1.000 1.000 0.250 1.000

0.250 0.250 0.250 0.438 0.438 0.000 0.000 0.000 0.438 0.438 0.000 0.000 0.438 0.000

0.750 0.750 0.750 1.000 1.000 0.000 0.000 0.000 1.000 1.000 0.000 0.000 1.000 0.000

0.906 0.906 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

2.685 2.685 2.685 2.070 2.070 0.895 0.895 0.895 2.070 2.070 0.895 0.895 2.070 0.895

1.39 1.26 1.45 1.57 1.24 1.12 1.08 1.05 1.20 1.13 1.06 1.01 1.05 1.02

a

/ The carbon model parameters that have not been varied here are l/b ) 2.619, lb′ ) 3.403, and l/c ) 0.895

4. Results and Discussion 4.1. Overview of Adsorption Behavior and Influence of Model Parameters. Adsorption isotherms and, where different, desorption isotherms for the solid models are shown in Figure 6, where the loading axis is plotted in terms of the density, F ) n/Ve. Models 1-5 inclusively all yield unambiguous type-I isotherms, as illustrated in Figure 6a-c. Introduction of inter-LMO pores of ∼5σff (see model 6 in Figure 6d) causes the occurrence of what appears to be a small discontinuous change in the region P/P0 ) 0.5, which inspection of the singlet distribution function (not shown here) shows is due to condensation in the prismatic and cubical pore spaces defined by the edges and corners of neighboring regions of LMO; no hysteresis is observed. The sharpness of the knee of the type-I isotherms is affected by a number of model parameters to varying extents. The width of the intra-LMO d002 distribution has only a mild effect, as illustrated in Figure 6a, which shows that the isotherm knee softens slightly as the width of the distribution increases. Comparison of parts a and b of Figure 6 shows the presence of the inter-LMO pores has a substantially greater effect on the knee region for similar average pore widths (d h /002 ) 2.685 for models 1P and 2P / compared to dc ) 2.5 for models 4fNP). Figure 6b shows that the softness of the knee increases as the extent of the inter-LMO pores, Li, decreases relative to their width, di. This effect is due to the increasing importance of the lowenergy volume of the junction of the inter-LMO pores as the ratio Li/di decreases. The width of the inter-LMO pores has an even greater effect for the same reason, as illustrated in Figure 6c, which shows isotherms for models that differ only with respect to their inter-LMO pore width. Parts e and f of Figure 6, which are for models 7P and 7NP, respectively, show that type-H1 hysteresis loops may be obtained by using inter-LMO pores wider than 5σff. Inspection of the singlet distribution function (not shown here) shows that adsorption within the hysteresis loop progresses by successive capillary condensation events at increasing pressure, starting with the basal inter-LMO slit pores, followed by the armchair inter-LMO pores, fjord inter-LMO pores, the prismatic pores defined by the four edges of adjacent regions of LMO, and finally the cubical pore defined by the eight corners of adjacent regions of LMO. This sequence of filling events is, of course, due to the differences in the characteristic adsorption energies of these pores, which even differ for pores of the same nominal shape, size, and surfaces due to local variations

in the microtexture of the surfaces. Any condensation event can be seen in the adsorption isotherm when simulations are performed at both ends of the event, which has been done for a few of the transitions, as circled in Figure 6e and f. Interestingly, the transitions in model 7P are delayed compared to the corresponding transitions in model 7NP (i.e., the transition in the armchair pore occurs at a lower pressure in model 7NP compared to model 7P, for example), indicating that the presence of the intra-LMO adsorbate and/or different intra-LMO structure significantly influences the inter-LMO adsorption process. Unlike the adsorption process, reversal of the condensation occurred throughout the inter-LMO porosity at a single pressure that appears to be almost the same for the two models. 4.2. New Tentative Proposal for Describing the Porosity of Complex Microporous Solids. The analysis associated with Figure 6a-c makes it clear that significantly different type-I isotherms can be obtained for models having micropores of the same nominal width, indicating the inconsistency of the empirical concept of ‘pore size’. (“Pore size” (and similar) is placed in single quotes when referred to in the experimental sense. Absence of quotes implies reference to the physical pore size in the model unless otherwise stated explicitly.) The inconsistency arises because adsorption energy is not a function of pore size alone but also of local microtextural and geometrical character and solid surface atom density and type. We propose here a new way of describing the porosity of complex solids that seeks to address this inconsistency by acknowledging that it is a balance between these factors that decides the final adsorption energy associated with a region of porosity. The junction between the inter-LMO pores represents a region of low adsorption energy relative to the intraLMO porositysthe low energy arises because the volume is defined by the external corners of the regions of LMO. We term this type of porosity locally-convex because any line of length O(2σ) (the use of 2σ here is motivated by the distance over which van der Waals potentials are known to be intensified by opposing solid interfaces, and this distance will clearly change if longer-range forces play an important role, but this does not affect the basic concept proposed here) drawn from one part of the solid defining the porosity will not intersect significantly other parts of the solid defining the porosity either energetically or volumetrically, as illustrated in Figure 7a. Regions of high adsorption energy that take up adsorbate at low pressures are created by what we term locally-concave porosity,


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Figure 6. Adsorption (filled symbols) and, where different, desorption (open symbols) isotherms for a range of solid models to demonstrate the influence of various solid model parameters: (a) models 1P (circle right solid), 1P(15,15) (circle top solid), and 2P (solid dot) to demonstrate the effect of the intra-LMO d002 distribution width and the tilting of superdomains; (b) models 41/4 NP (solid triangle 1 down), 41/2 NP (solid triangle left), and 4NP (solid triangle up) to demonstrate the effect of the ratio of inter-LMO pore length-to-width ratio, Li/di; (c) models 3P (solid square), 4P (solid pentagon), and 5P (solid hexagon) to demonstrate the effect of the inter-LMO pore width; (d) model 6P illustrating occurrence of capillary condensation following introduction of inter-LMO pores of width ∼5σff; (e and f) models 7P (solid triangle right) and 7NP (solid diamond), respectively, illustrating occurrence of hysteresis following introduction of inter-LMO pores of a size that would be termed mesoporessthe circled discontinuous jumps are the condensation events in the indicated parts of the inter-LMO pore space (BSP and ASP are the inter-LMO basal and armchair slip pores, respectively) that have been resolved (other condensation events would be seen in the isotherm if further points were simulated in this region; see text). Lines are a guide to the eye.

which is defined by solid surfaces that can be intersected either energetically or volumetrically in a significant way by a line of length O(2σ) (see Figure 7b). These definitions allow, for example, the possibility of pores of widths greater than 2σ to take up fluid at pressures normally (i.e., empirically) reserved for such sized pores by acknowledging that microtextural and local geometric factors such as surface roughness (e.g., in armchair pores) and local (concave) defects play an important role in determining the local adsorption energy. The complexity of the pore space within the porous regions of LMO modeled here means they are dominated

by locally-concave porosity, even for models 1P and 2P, where many of the intra-LMO pores are greater than 2σ in size and possess surfaces of lower atom densities compared to those of basal slit pores. It is because locallyconcave porosity effectively behaves like basal slit pores of 1-2σ width that broadening of the intra-LMO d002 distribution has only a small effect on the softening of the knee of the type-I isotherms. The extent of the inter-LMO pores, on the other hand, means lower surface atom density cannot be offset by increased volumetric interaction, and they, therefore, very quickly start to behave like wider ‘micropores’ or even small ‘mesopores’ despite, geo-

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Figure 8. Experimental isotherm activated mesophase microbeads of 4% (+) and 40% (×) burnoff (after Kaneko et al.60). Lines are a guide to the eye. Figure 7. Examples of (a) locally-convex porosity; (b) locallyconcave porosity. The molecule is shown with its field of substantial interaction with the solid.

metrically, still being at the bottom end of the empirically defined micropore range. The empirically defined mesopores are clearly of locallyconvex porosity. It should, however, be remembered that the microtextural character of the surfaces of these pores can be important, especially at the bottom of the empirically defined mesopore range where, for example, high degrees of microtextural disorder may observably manifest themselves as ‘micropores’ (i.e., locally-concave porosity). 4.3. Relationship between the Model Parameters and ‘Pore Size’. In the absence of any heteroatoms, the geometric and microtextural character and surface atom density all play a role in determining the adsorption behavior of the models here. However, given the universality of the empirical concept of “pore size”, it is useful to relate it to the model components despite the restrictive nature of the concept. Those models that possess no inter-LMO pores all yield type-I isotherms with sharp knees. These solids would, therefore, normally (i.e., empirically) be interpreted58 as having a narrow ‘micropore size distribution’. The extent of the 2-3.5σ inter-LMO pores compared to that of those located within the regions of LMO means they fill at higher pressures than the latter despite being of similar (or even smaller) width. The differences in the surface atom density of the various inter-LMO slit, prismatic, and cubical pores mean they also fill across a range of pressures. These two effects act to soften the type-I isotherm knee. It can therefore be concluded that one major action of the inter-LMO is to broaden the ‘micropore size distribution’ beyond that arising from the locally-concave dominated intra-LMO porosity,58 with broadening being enhanced by smaller Li/di ratios, greater widths, and less dense pore surfaces. Inter-LMO pores beyond ∼4σff in width lead to deviations from the classical type-I isotherm and, if sufficiently larger, hysteresis. Such inter-LMO pores, therefore, (58) Rodrı´guez-Reinoso, F. Preparation and characterization of activated carbons. In Carbon and Coal Gasification; Figueiredo, J. L., Moulijn, J. A., Eds.; Martinus Nijhoff Publishers: Dordrecht, The Netherlands, 1986.

Figure 9. Isosteric heat of adsorption variation with loading for models 1P (circle right solid) and 2P (solid circle). Lines are a guide to the eye.

provide a means of imposing further distinct ‘pore sizes’ from the middle of the micropore size range upward. 4.4. Brief Qualitative Comparison with Experimental Data. While fitting of the solid model to data for a specific carbon is beyond the scope of the current work, comparison of the isotherms generated here with those from the literature shows that the model can reproduce isotherm shapes seen experimentally for a wide range of microporous carbons59-61, especially when comparison is done using the much more revealing log(P/P0) scalesthe isotherm behavior of models 4P and 5P, for example, corresponds well with that of the activated mesophase microbeads of Kaneko et al.60 at 4 and 40% burnoff, respectively (see Figure 8). As illustrated in Figure 9, it is also able to reproduce the well-known decreasing trend between isosteric heat and loading (e.g, refs 62-64). This is in marked contrast to the standard distributed-width infinite-extent basal slit pore (DW-IE-BSP) modelswhere the solid is described by a distribution of different width pores defined by the space between the basal surfaces of two opposing semi-infinite blocks of graphiteswhich appears to not provide a basis for defining simultaneously a unique isotherm/heat of adsorption set (see the discussion associated with Figure 13 of ref 65). (59) Atkinson, D.; McLeod, A. I.; Sing, K. S. W. J. Chim. Phys. Phys.Chim. Biol. 1984, 81, 791-794. (60) Kaneko, K.; Ishii, C. Colloids Surf. 1992, 67, 203-212.


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4.5. Intra-LMO Adsorbate Structure. Adsorbate Structure at Saturation. The effective density of the adsorbate at saturation, F ) n(P/P0 ) 1)/Ve, is given in Table 1 for each model relative to that of the bulk saturated liquid, Fl (see section 3.2). The effective adsorbate density was evaluated by dividing the GCMC-determined loading at saturation by the total accessible volume of the pore space as determined from the tessellation methods described in section 2.1. Adsorbate densities at a saturation of 126-157% of the bulk saturated liquid for models 1P-3P suggest the adsorbate may be solidlike for these models. The complexity of the pore space means that, if the adsorbate is solidlike, it is likely to be randomly packed. With adsorbate densities of up to 93% of the bulk random close-packing density of 1.22,66 this possibility appears reasonable, especially if exclusion effects67 are accounted for. In any case, it is clear that the adsorbate for these models is not akin to the bulk saturated liquid. A number of workers have recently used the pair correlation function, structure factor, and order parameters when studying solidlike adsorbate structures and freezing/melting transitions in simple slit pores (e.g., refs 23 and 31). Unfortunately, the complexity of the models used in the current study prevents the accumulation of sufficient statistics for such an analysis here. However, the singlet distribution function (SDF), F(x,y,z), is sufficient to confirm the existence or otherwise of a solidlike adsorbate and, if it does exist, to understand its formation at least qualitatively. Figure 10 shows three closely but unevenly spaced xaxc planes through the SDF for model 3P at saturation, where a log contour scale has been used to facilitate the presentation of solidlike and fluid densities in a single plot. The existence of localized F* ∼ 5 peaks of ∼1 molecular diameter, especially in the log scale where reduced densities down to F* ∼ 0.1 are shown, clearly indicates the adsorbate for model 3P is solidlike at saturation. The shading in Figure 10 indicates the inaccessible regions of the planes evaluated by the tessellation methods of section 2.1. This shows that the accessible area of each plane is well filled by the solidlike adsorbate, with the vast majority of the ‘holes’ or lower-density peaks in one plane being correlated with solidlike peaks in another (e.g., at point A). While the irregular pore structure prevents any longrange ordering of the solidlike adsorbate, there is still considerable local order. Solidlike adsorbate occurs even in the largest pores of these planes, which are approximately three molecular diameters in ‘size’ (e.g. point A), and pores of all surface types. There is also occasional less well localized regions of adsorbate such as point B in this set of planes, indicating the existence of isolated regions of liquidlike adsorbate coexisting with the solidlike phase. All these observations suggest that local adsorbate structure is dictated by a complex interplay between pore system geometry, shape, and microtexture and that any (61) Suzuki, T.; Kaneko, K.; Setoyama, N.; Maddox, M.; Gubbins, K. Carbon 1996, 34, 909-912. (62) Carrott, P. J. M.; Sing, K. S. W. Assessment of microporosity. In Characterization of Porous Solids; Unger, K. K., Rouquerol, J., Sing, K. S. W., Kral, H., Eds.; Elsevier: Amsterdam, The Netherlands, 1988; p 77. (63) Kakei, K.; Ozeki, S.; Suzuki, T.; Kaneko, K. J. Chem. Soc., Faraday Trans. 1990, 86, 371-376. (64) Carrott, P. J. M.; Ribeiro Carrott, M. M. L.; Roberts, R. A. Colloids Surf. 1991, 58, 385-400. (65) Nicholson, D. Langmuir 1999, 15, 2508-2515. (66) Torquato, S.; Truskett, T. M.; Debenedetti, P. G. Phys. Rev. Lett. 2000, 84, 2064-2067. (67) Carrott, P. J. M.; Roberts, R. A.; Sing, K. S. W. Chem. Ind. 1987, 24, 855-856.

Figure 10. Three closely but unevenly spaced xa-xc planes through the singlet distribution function (SDF) for model 3P at / saturation: (a) xb′ ) 3.729 (note that the patterns seen in the shading for this plane arise from fjord edges at lower y* coming / / into the range of the plane); (b) xb′ ) 4.083; (c) xb′ ) 4.314. A log color scale has been used to facilitate presentation of solidlike and fluid densities in a single plot. The solid atoms within (0.5σff of the plane are shown in red, and the inaccessible regions of each plane as determined from the tessellation-based methods (see section 2.1) are shaded to demonstrate that all the accessible volume is occupied. Note that many of the unfilled spaces in one plane are matched by highly localized adsorbate in adjacent planes. See text for discussion pertaining to the circled regions.

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attempt to describe adsorption in terms of a subset of these such as ‘pore size’ is unlikely to be adequate. Comparison of the adsorbate densities for models 1P3P indicates the density decreases as the mean and width of the intra-LMO d002 distribution increase. Inspection of the SDF for the first of these models (see Figure 11) clearly shows that one principle reason for this is the absence of solidlike adsorbate in pores of approximately four to five molecular diameters in ‘size’, such as at points A and B in the planes shown. The fraction of these sized pores decreases from model 1P to 2P, reflecting the larger adsorbate density for the latter. Comparison of the adsorbate densities for model 1P with those of model 1P(15,15) clearly shows that the density decreases as the level of intra-LMO structural disorder increases. Figure 12, which presents three nearby but unevenly spaced xa-xc planes through the SDF for this model at saturation, clearly shows that this decrease is due to poorer packing in the solidlike phase (e.g., point A) and a reduction in the fraction of the volume occupied by this phase in favor of less well localized high-density (e.g., point B) and liquidlike (e.g., point C) adsorbate. Despite the reduction in the level of solidlike adsorbate due to increased microstructural complexity, it is still considerable, suggesting it is reasonable to expect significant levels of solidlike adsorbate to exist in real noncrystalline nanoporous solids such as microporous carbons. The decrease in the effective adsorbate density following the introduction of the inter-LMO pores is caused by two factors. The first is the liquidlike character of much of the adsorbate within these pores, the 2.5σ basal pores of larger extent being the exception, with the fraction of liquidoccupied volume increasing with inter-LMO pore width (compare models 41NP, 51NP, and 71NP) and with decreasing extent (compare models 4fNP or fNP). The second factor is the adsorbate structuring caused by the armchair and fjord pore surfaces, which possess deep local minima separated by distances significantly greater than those of solidlike structures. Development of Adsorbate Structure with Loading. The complexity of the three-dimensional pore spaces simulated here precludes at this stage a detailed pore-level analysis of the adsorbate structure as a function of loading. Instead, the variation of the singlet distribution function (SDF) with loading for a single part of the pore space is considered and shown to exhibit two different types of freezing transitions. The region of particular interest here, point A in Figures 13-15, is a complex three-dimensional void in model 2P that extends approximately three molecular diameters in the xc direction and much further in the plane normal to this. As inspection of Figure 13a-c indicates, this part of the pore space begins to take up significant levels of adsorbate around P/P0 ∼ 10-6. The pressure at which solidlike adsorbate forms varies with position in the cavity. Three stages in the development of the lowest layer in the cavity, L1, are shown in Figure 14. This figure indicates the transition to the solidlike structure at P/P0 ∼ 10-4 is a gradual almost homogeneous one, reflecting the energetic homogeneity of this part of the pore space. Consideration of the SDF at the same pressures under desorption conditions (not shown here) suggests this process is reversible. Layer L2 develops in a different manner, as indicated by the four stages in this development shown in Figure 15. The formation of a solidlike structure occurs over a much wider pressure range by propagating in the first instance from the seed region indicated, which is located

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Figure 11. Three closely but unevenly spaced xa-xc planes through the singlet distribution function (SDF) for model 1P at / / / saturation: (a) xb′ ) 7.719; (b) xb′ ) 8.250; (c) xb′ ) 8.719. The other details are as per Figure 11. Note that many of the unfilled spaces in one plane are matched by highly localized adsorbate in adjacent planes. See text for discussion pertaining to the circled regions.

in a pore of ∼1σ width. Interestingly, this figure also shows that the solidlike structure in this initial seed region


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same pressures under desorption conditions (not shown here) suggests once again that these processes are reversible. A similar cooperative ‘freezing’ also occurs in the lower two layers at point B in Figure 13, except the initial seed of the layers below the graphitic crystallite at point B is formed due to the solidlike adsorbate above the crystallite. Discussion of Observed Freezing Behavior and Comparison with Previous Simulations. As indicated in the Introduction, the vast majority of the previous theoretical studies of elevated freezing in nanoporous systems has focused on simple pore geometries such as slit and cylindrical pores of infinite extent. It is reasonable to ask how the symmetry of these pore models affects the onset of freezing and the solidlike structures obtained. The few studies that have gone beyond these simple pores have not answered this question because either they have retained a significant degree of symmetry33,34 or they have been inconclusive.35 The work presented here clearly shows that even in the most highly symmetry-breaking of the micropore spaces simulatedsmodel 1P(15,15)sthere are significant levels of solidlike adsorbate present above the bulk freezing temperature, suggesting that such freezing is likely to occur in real noncrystalline nanoporous solids such as microporous carbons. The results here also clearly show that freezing occurs nonuniformly throughout the pore space in what appears at low loadings to be a morphology-mediated process akin to that observed in simulations on complex mesoporous solids;32 this differs from the freezing process in simpler micropores where freezing tends to occur either en-masse or layer-by-layer for smaller and wider micropores, respectively.30 We observed two freezing mechanisms in agreement with the findings of Curry et al.33 for an inscribed slit micropore model. The first involves uniform freezing in small cavities of relative energetic homogeneity. The second mechanism involves the growth of small seeds of frozen adsorbate with loading which eventually interact in a complex manner that appears to allow for melting and refreezing; the latter has never been observed in other simulation studies as far as we are aware (the melting/ refreezing seen by Curry et al.33 is induced by relative motion of the pore walls, something that is clearly not occurring here). 5. Conclusions

Figure 12. Three closely but unevenly spaced xa-xc planes through the singlet distribution function (SDF) for model 1P(15,15) / / / at saturation: (a) xb′ ) 5.207; (b) xb′ ) 5.901; (c) xb′ ) 6.820. The other details are as per Figure 11. Note that many of the unfilled spaces in one plane are matched by highly localized adsorbate in adjacent planes. See text for discussion pertaining to the circled regions.

partially melts at P/P0 ∼ 6 × 10-3 before refreezing into a different structure. Consideration of the SDF at the

Results have been presented for grand canonical Monte Carlo simulations of adsorption of a model nitrogen vapor on model complex carbonaceous porous solids at a single subcritical temperature above the bulk freezing point. The model solids are based on experimental understanding of real nanoporous carbons and are able to produce adsorption isotherms and heats of adsorption behavior that at least reflect experimental observations qualitatively. The simulations revealed significant levels of solidlike adsorbate even in the most complex of micropore spacess to our knowledge, this is the first such observation for complex pore structures. The fraction of the volume occupied by solidlike adsorbate appears to decrease with increasing disorder and pore sizes (not ‘width’ alone) beyond approximately four to five molecular diameters. For pores below this size, the nature of the adsorbate appears to be dictated by a complex interplay between pore system geometry, shape, and microtexture. The presence of significant levels of solidlike adsorbate in even the most complex of pore spaces strongly suggests that solidlike adsorbate will also occur in real noncrystalline nanoporous solids such as microporous carbons, supporting indirect experimental observations.

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/ Figure 13. The xb′ ) 4.595 plane through the singlet distribution function (SDF) of model 2P for a number of loadings: (a) P/P0 ) 6.7 × 10-8; (b) P/P0 ) 2.9 × 10-6; (c) P/P0 ) 1.2 × 10-4; (d) P/P0 ) 9.0 × 10-4; (e) P/P0 ) 6.7 × 10-3; (f) P/P0 ) 4.9 × 10-2. See text for discussion pertaining to the circled regions and planes L1 and L2.

At least two types of local freezing have been observed at the local pore levelsa homogeneous freezing and a freezing that propagates from a seed region. A local freezing/melting/refreezing process was also observed. All these processes appeared to be reversible at a local level under loading-unloading.

In addition to freezing phenomena, hysteresis between the adsorption and desorption isotherms was observed for model solids containing mesopores. Adsorption within the hysteresis loop occurred by localized condensation events in different parts of the mesoporosity. Reversal of the condensation during desorption occurred throughout


Figure 14. L1 plane (xc* ) 15.029) through the singlet distribution function (SDF) of model 1P for a number of loadings: (a) P/P0 ) 2.7 × 10-5; (b) P/P0 ) 1.2 × 10-4; (c) P/P0 / ) 6.7 × 10-3. The broken line indicates the position of the xb′ ) 4.595 plane (see Figure 13). See text for discussion pertaining to the circled regions.

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Figure 15. L2 plane (xc* ) 14.299) through the singlet distribution function (SDF) of model 1P for four different loadings: (a) P/P0 ) 2.9 × 10-6; (b) P/P0 ) 1.2 × 10-4; (c) P/P0 ) 6.7 × 10-3; (d) P/P0 ) 4.9 × 10-2. The broken line indicates / the position of the xb′ ) 4.595 plane (see Figure 13). Only the top portion of each plane is shown for space reasons. See text for discussion pertaining to the circled regions.

5800


the mesoporosity at a single pressure. The condensation process in the mesoporosity simulated here was found to be significantly influenced by the presence of adjacent microporosity and/or the adsorbate within it. It is clear from the analysis presented here that the empirical concept of pore size does not provide a basis for consistent description of adsorption in complex pore spaces such as those simulated. A new concept for describing the porosity locally has therefore been proposed which seeks to reflect the fact that the adsorption energy of a region of porosity is a function of geometrical, microtextural, and chemical factors. This report represents our initial efforts aimed at improving the understanding of the phase behavior of fluids in complex nanoporous solids in which solidlike phases form. Further work is underway, and more is planned in order to improve our understanding of these

Biggs et al.

phenomena both qualitatively and quantitatively, the latter being especially challenging due to the problems faced in accumulating sufficient statistics for the complex pore spaces simulated. While we believe the results presented here for the highly symmetry-breaking model solids provide strong evidence that elevated freezing is likely to occur in real noncrystalline nanoporous solids such as microporous carbons, it is recognized that it would be desirable to obtain results for solid models that are more strongly linked to real solids both structurally and chemicallyswork in this regard is, therefore, also planned. Acknowledgment. We thank the Engineering and Physical Science Research Council of the U.K. for support of this research (GR/M89539). LA036269O

Molecular Simulation Evidence for Solidlike Adsorbate in Complex

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