Langmuir 1999, 15, 533-544
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A Molecular Model for Adsorption of Water on Activated Carbon: Comparison of Simulation and Experiment C. L. McCallum,† T. J. Bandosz,‡ S. C. McGrother,† E. A. Mu¨ller,§ and K. E. Gubbins*,⊥ School of Chemical Engineering, Cornell University, Ithaca, New York 14853, Chemistry Department, City College of New York, Convent Avenue at 138th Street, New York, New York 10031, Departamento de Termodina´ mica y Feno´ menos de Transferencia, Universidad Simo´ n Bolı´var, Caracas 1080-A, Venezuela, and Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27695 Received May 19, 1998. In Final Form: September 28, 1998 Experimental and molecular simulation results are presented for the adsorption of water onto activated carbons. The pore size distribution for the carbon studied was determined from nitrogen adsorption data using density functional theory, and the density of acidic and basic surface sites was found using Boehm and potentiometric titration. The total surface site density was 0.675 site/nm2. Water adsorption was measured for relative pressures P/P0 down to 10-3. A new molecular model for the water/activated carbon system is presented, which we term the effective single group model, and grand canonical Monte Carlo simulations are reported for the range of pressures covered in the experiments. A comparison of these simulations with the experiments show generally good agreement, although some discrepancies are noted at very low pressures and also at high relative pressures. The differences at low pressure are attributed to the simplification of using a single surface group species, while those at high pressure are believed to arise from uncertainties in the pore size distribution. The simulation results throw new light on the adsorption mechanism for water at low pressures. The influence of varying both the density of surface sites and the size of the graphite microcrystals is studied using molecular simulation.
1. Introduction The adsorption behavior of water on activated carbons is qualitatively different from that of simple fluids, such as nitrogen, carbon dioxide, or hydrocarbons. This major difference arises from two sources: (a) for water the fluidfluid interaction is much more strongly attractive than the interaction of water with the carbon surface, in contrast to the situation for simpler fluids, where the reverse is the case, and (b) the adsorption behavior for water is largely controlled by the formation of H-bonds with oxygenated groups on the surface. Thus, the pore-filling mechanism for water in activated carbons is distinctly different from that for most fluids. For simple fluids pore filling occurs via the formation of a fluid monolayer on the surface, often followed by a second and possibly further layers, prior to capillary condensation and pore filling. By contrast, water molecules first adsorb onto oxygenated surface sites, and these adsorbed water molecules then act as nuclei for the formation of larger water clusters; eventually these clusters connect, either along the surface or across the pore, and pore filling usually occurs. When the density of oxygenated sites on the surface is appreciable, the pore filling seems to occur by a continuous filling process without capillary condensation. In this paper we address the problem of modeling this adsorption process at the molecular level. For graphitic carbons, in which all oxygenated groups have been removed from the surface through heat treatment or * To whom correspondence should be addressed. † Cornell University. Current addresses: C. L. McCallum, Intel Corporation, 5000 W. Chandler Blvd., Chandler, AZ 85226. S. C. McGrother, Department of Chemical Engineering, North Carolina State University, Raleigh, NC 27695. ‡ City College of New York. § Simon Bolivar University. ⊥ North Carolina State University.
reduction, only source (a) above is important, and the modeling is relatively straightforward. Such systems have been successfully modeled using both point charge and off-center square well models of water in slit graphitic pores.1-4 Agreement with experiment is good.4 In these cases there is almost no adsorption of water at low-tomoderate relative pressures, but there is an onset of pore filling that occurs suddenly at high relative pressures. The adsorption behavior is of type V in the IUPAC classification.5 Such type V behavior can be described even with simple Lennard-Jones interactions,6 and requires only a fluid-fluid interaction that is strongly attractive compared to the fluid-solid interaction. Activated carbons, in which a significant density of oxygenated sites are present on the carbon surfaces, are considerably harder to model and to characterize. Macroscopic phenomenological models have been proposed7,8 which give a reasonable fit to the experimental data outside the low-pressure region, but these give little insight into the underlying molecular mechanisms of adsorption and are mainly useful for data interpolation. A limited number of attempts to model water on activated carbons at the molecular level through molecular simulation have been reported. Segarra and Glandt9 modeled the activated carbon as made up of graphite platelets having dipoles (1) Antonchenko, V. Y.; Davidov, A. S.; Ilyin, V. V. Physics of Water; Naukova Dumka: Kiev, 1991. (2) Ulberg, D. E.; Gubbins, K. E. Mol. Simul. 1994, 13, 205; Mol. Phys. 1995, 84, 1139. (3) Maddox, M.; Ulberg, D. E.; Gubbins, K. E. Fluid Phase Equilibr. 1995, 104, 145. (4) Mu¨ller, E. A.; Rull, L. F.; Vega, L. F.; Gubbins, K. E. J. Phys. Chem. 1996, 100, 1189. (5) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouque´rol, J.; Siemineiewska, T. Pure Appl. Chem. 1985, 57, 603. (6) Balbuena, P. B.; Gubbins, K. E. Langmuir 1993, 9, 1801. (7) Dubinin, M. M.; Serpinsky, V. V. Carbon 1981, 19, 402. (8) Talu, O.; Meunier, F. AIChE J. 1996, 42, 809. (9) Segarra, E. I.; Glandt, E. D. Chem. Eng. Sci. 1994, 49, 2953.
10.1021/la9805950 CCC: $18.00 © 1999 American Chemical Society Published on Web 12/15/1998
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smeared out around the edges of the platelets to represent the surface functional groups. More recent work10 suggests that this model of the surface sites is insufficiently inhomogeneous to predict the correct trends in lowpressure adsorption and in heats of adsorption. Maddox et al.3 and Mu¨ller et al.4 have used models in which the pores are approximated by slits with graphite walls, on which are embedded surface sites. In the work of Maddox et al. the sites were assumed to consist of -COOH groups and were modeled using optimized potentials for liquid simulations (OPLS),11 while water was modeled using a TIP4P potential model;12 these potential models consist of Lennard-Jones terms for the dispersion and overlap interactions, plus Coulomb interactions between discrete charges placed in the molecules to mimic the H-bonding. Surface sites were placed in a regular array on the graphite surface. The adsorption isotherms of water at 298 K in a 2 nm pore with various site densities were obtained. A more detailed study was carried out by Mu¨ller et al.4 for a wide range of surface site densities at 300 K. In their work the H-bonding between water molecules and between water and a surface site was modeled using off-center square well potentials in place of charges. This approach had two advantages. First, the square well potential is short-ranged, thus eliminating the need for time-consuming Ewald lattice sums to correct for long-range Coulomb interactions; this greatly speeds up the simulations. Second, the square well model allows for highly localized attractive interactions, which may more realistically model H-bonding. Results were obtained for both regular arrays of surface sites, and for randomly placed sites. Cooperative adsorption effects were found to be very important, with water clusters forming readily in surface regions where two or more sites were placed at a separation suitable for both water-site and water-water bonding. This model captures the effects of the strongly orientation-dependent, short-range H-bonds, as well as the heterogeneous nature of the surface. However, the intermolecular potential parameters used by Mu¨ller et al. were not optimized for water/activated carbon systems, and no comparison with experimental data was made. This was due in part to a lack of appropriate data. Although a considerable amount of data on water/carbon systems has been published, there are little data in the critically important low-pressure region for well-characterized materials. In this paper we (a) report new experimental data for water adsorption at low pressure on a carefully characterized carbon, (b) describe a molecular model of water on activated carbons that we believe is more realistic than the ones3,4,9 used previously, and (c) carry out molecular simulations for this model, and compare these model results with the experimental ones. It should be emphasized that the complexity of the system is such that some simplifications in the model are essential at this stage. Complexity in the actual physical system includes poorly characterized materials, a fairly wide range of pore sizes and shapes, and a lack of precise knowledge of the species and location of the surface groups. Added to these problems are the difficulty of the molecular simulations in a system in which complex arrays of H-bonds are formed; exceptionally long runs on large systems are needed to overcome ergodic problems and ensure results that are representative of a macroscopic system. Such studies are at the limits of capability of current supercomputers. In the current (10) Gordon, P. A.; Glandt, E. D. Langmuir 1997, 13, 4659. (11) Briggs, J. M.; Nguyen, T. B.; Jorgensen, W. L. J. Phys. Chem. 1991, 95, 3315. (12) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D. J. Chem. Phys. 1983, 79, 926.
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Figure 1. Two possible mechanisms for water adsorption onto activated carbon walls, for an intermediate site density. Scheme A corresponds to sites regularly arranged, and therefore spaced well apart. Scheme B shows a random array at the same site density. Clearly cooperative bonding is more prevalent in the latter scheme. The schematic adsorption isotherm shown for Scheme A assumes that the water-surface site bonding energy HB is greater than the water-water value, i.e., HB sf > ff . If the HB HB reverse holds, i.e., ff > sf , the slope in region I will be less than that in region II for scheme A. Shaded circles are water molecules, with newly adsorbed molecules shaded darker; open circles are surface-active sites.
work we have therefore tried to capture the essential features of hydrogen bonding of water with the surface sites and with other water molecules, the strongly heterogeneous nature of the surfaces, and the distribution of pore sizes. Simplifications introduced include the use of slit-shaped pores, the assumption that the surface sites can be represented as being of a single species with a single H-bonding energy, the use of a single graphite microcrystal size, and the use of simulations for a few pore widths, together with interpolation between these, to represent the continuous pore size distribution of the real carbon. Before describing the methods and results, it is useful to speculate on the possible adsorption mechanisms for this system, particularly at low pressure. For simplicity, we assume that only one water molecule can adsorb at a given site. Two possible scenarios are illustrated in Figure 1, for the case of an intermediate site density. In the simplest scenario, A, it is assumed that the surface sites (open circles) are arranged in a regular array on the surface. At the lowest pressures single water molecules will adsorb on individual surface sites (stage I), with bonding energy HB sf . As the bulk gas pressure is increased, more surface sites are occupied, until at a pressure corresponding to stage II all sites are occupied by a single water molecule. The pressure region up to stage II is the Henry’s law region for this heterogeneous system, and the adsorption isotherm will be linear with a slope characterized by the value of HB sf . As the pressure is
Adsorption of Water on Activated Carbon
increased further, water molecules will adsorb by forming H-bonds to preadsorbed water molecules (stage III); the bonding energy will be HB ff , producing a second linear region of the adsorption isotherm, with a slope somewhat different from that of the initial Henry’s law region. A further pressure increase will lead to the formation of larger water clusters. As these clusters grow, the opportunities for cooperative adsorption effects will increase; that is, a water molecule adsorbing onto these preexisting clusters will often be able to position itself so that it forms two or more H-bonds with neighboring adsorbate molecules. In this stage (IV) the adsorption energy will be much greater, leading to a steep rise in the adsorption isotherm. This stage is followed by a rapid increase in water adsorption with pressure, with bridging between adsorbed clusters on the same pore wall, or on opposite walls. Soon after this the pore fills with water molecules (stage V). In scenario B the surface sites are placed randomly on the edges of the graphite microcrystals. In this case some sites will be relatively isolated from others; for such sites the adsorption mechanism will be similar to that for scenario A. However, some sites will be close together and can participate in a cooperative adsorption mechanism at certain pressures. At the lowest possible pressures water molecules will bond singly to individual sites, as in scenario A. However, once a few such water molecules are adsorbed (stage I), there will be opportunities for ensuing water molecules to engage in cooperative bonding (i.e., to bond simultaneously to a surface site and to a preadsorbed water molecule on a neighboring site). Such adsorption produces HB an energy release of (HB sf + ff ) and will result in a steep increase in the slope of the isotherm (stage II). Eventually, locations where such cooperative bonding can occur will be saturated, and further adsorption will be by waterwater bonding as in scenario A (stage III). Subsequent stages in pore filling would be the same as those in scenario A. More complex scenarios can be imagined and are likely to be more typical of real physical systems. Thus, in practice several species of oxygenated groups are usually present, and these may have different values of HB sf . In addition, some of these groups can H-bond to more than one water molecule. The initial pressure at which cooperative bonding occurs, as well as the range of pressures where this phenomenon is observed, will depend on the surface density of sites. For higher site densities cooperative bonding is expected to commence at lower pressures. 2. Experimental Method and Characterization Materials. As an adsorbent Norit activated carbon was chosen (Sorbonorit 2-A5998); this material is referred to as N2 in this (and earlier) papers. The precursor for this material was a peat moss. It was oxidized with 30% hydrogen peroxide at 323 K for 2 h. The details of the procedure applied were described elsewhere.13-15 We also considered the use of nitric acid as an alternative oxidizing agent. However, this strong oxidizer was found to lead to a dramatic reduction in the total pore volume, and so this approach was not pursued. Sorption of Nitrogen. The nitrogen adsorption isotherm was measured at 77 K using a Micromeritics ASAP 2010 sorptometer. Prior to the adsorption measurements the sample was heated at 393 K and outgassed at this temperature to a constant vacuum of 10-6 Torr. The isotherm was used to calculate the specific surface area using the BET method and the micropore volume (13) Bandosz, T. J.; Jagiello, J.; Schwarz, J. A. Anal. Chem. 1992, 64, 891. (14) Jagiello, J.; Bandosz, T. J.; Schwarz, J. A. Carbon 1992, 30, 63. (15) Bandosz, T. J.; Jagiello, J.; Schwarz, J. A.; Krzyzanowski, A. Langmuir 1996, 12, 6480.
Langmuir, Vol. 15, No. 2, 1999 535
Figure 2. Pore size distribution of the Norit activated carbon N2, as determined by nitrogen adsorption at 77 K using nonlocal density functional theory. The vertical dashed lines divide the PSD into three regions, which form the basis for choosing discrete pore sizes for study by simulation. using the Dubinin-Radushkevich (DR) method.16 These were found to be SBET ) 860 m2 g-1 and Vmic ) 0.426 cm3 g-1, respectively. The pore size distribution (PSD) was determined from the adsorption isotherm using the density functional theory method (DFT),17,18 which has been shown to be more reliable for small pores than semiempirical methods.19 The PSD determined in this way is shown in Figure 2 and displays sharp peaks at approximately 0.65 and 1.35 nm. There is a long tail to the distribution, which extends to 8.0 nm in Figure 2; pore widths greater than 8.0 nm account for less than 2.6% of the total pore volume. Surface Characterization. The densities of acidic and basic surface sites were determined using Boehm20 and potentiometric21,22 titration. The methods and full details of these determinations have been described previously15 and are not repeated here. The total site density was found to be 0.675 sites/nm2, with the groups being mainly basic in nature; the oxidation method used resulted in only small changes in the acidity of the initial Norit sample. The species present on the surface of the N2 carbon are characterized by pKa values of 3.3, 4.8, 7.4, and about 10. Sorption of Water. The water sorption isotherm was obtained at 298 K using a Micromeritics ASAP 2010 sorptometer equipped with a vapor sorption kit. Temperature was controlled by a Fisher Isotemp thermostatted system. Before the measurement the sample was heated at 393 K and outgassed to a constant vacuum of 10-6 Torr. HPLC grade water was used as an adsorbate. Prior to the experiment dissolved gases were removed by consecutively melting and freezing the water several times, followed by outgassing. Each point of the isotherm was recorded after sufficient time to ensure that equilibrium had been reached. The initial dose amount was 0.1 cm3. It took several days to measure the sorption at the lowest pressures (P/P0 e 0.02). The desorption process in this pressure range was not determined because of the extremely slow equilibration. Thus, it took about 55 h to desorb water from a relative pressure of 0.02 to 0.01.
3. Model Pore Size Distribution. The activated carbon is modeled as being made up of noninterconnected slit pores (16) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd edition; Academic Press: New York, 1982. (17) Olivier, J. P.; Conklin, W. B. Determination of pore size distributions from density functional theoretical models of adsorption and condensation with porous solids. Presented at the 7th International Conference on Surface and Colloid Science, Compiegne, France, 1991. (18) Lastoskie, C. M.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (19) Lastoskie, C. M.; Quirke, N.; Gubbins, K. E. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1996; p 745. (20) Boehm, H. P. In Advances in Catalysis; Academic Press: New York, 1966; Vol. 16. (21) Bandosz, T. J.; Jagiello, J.; Contescu, C.; Schwarz, J. A. Carbon 1993, 31, 1193. (22) Jagiello, J.; Bandosz, T. J.; Schwarz, J. A. Carbon 1994, 32, 1026.
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having a distribution of pore widths given by the experimentally determined PSD shown in Figure 2. The PSD from experiment gives information on the effective, or “available” pore width w. This quantity is assumed to be related to H, the distance separating the planes through the centers of the first layer of C atoms on opposing walls, by
w ) H - σc
N)
∫0 f(w) N(w) dw
(2)
where f(w) is the PSD and N(w) is the amount adsorbed in the pores of width w at the given pressure. In principle, it is possible to calculate N by carrying out simulations to determine N(w) for a large number of fixed pore widths H covering the range of widths shown by the experimentally determined PSD. The total adsorption can then be determined from eq 2, using the experimentally determined PSD, f(w). At low pressures a few pore widths suffice for such a scheme, since the isotherm changes slowly with pressure. For higher pressures a much larger number of pore widths is needed since pore filling occurs, and the adsorption changes rapidly. Since the simulations are lengthy, even on the fastest supercomputers, it is not feasible at present to carry out such calculations for a large number of pore widths, and some interpolation scheme must be devised that enables the overall adsorption to be calculated using results for a small number of pore widths. We have employed two such schemes, one suitable for low pressures and the other for higher pressures. In the low-pressure region (0 e P/P0 e 0.02) we approximate the integral of eq 2 by a finite sum over discrete, but representative, pore widths:
N ) c1N(w1) + c2N(w2) + ... + cnN(wn)
(3)
where wi are the selected pore widths and ci are weighting factors indicative of the total pore volume represented by the individual pore widths. On the basis of the PSD shown in Figure 2, three values were chosen for the pore widths to be studied: H ) 0.99, 1.69, and 4.5 nm. These widths represent regions 1, 2, and 3 in Figure 2, respectively. The weighting factors, ci, for these three pore sizes are taken to be proportional to the area under these three regions of the PSD: their values are c1 ) 0.518, c2 ) 0.371, and c3 ) 0.111, respectively. As mentioned above, this weighted-averaging procedure breaks down in the pore-filling region because pores of different width fill at different pressures. Weighted averaging using a few pore widths necessarily leads to a stepped isotherm. For these higher pressures, therefore, we use the semiempirical equation of Talu and Meunier23 to interpolate among the adsorption isotherms for the pore widths studied. This three-parameter equation was developed specifically for associating systems. It is
P)
H0Ψ exp(Ψ/Nm) 1 + KΨ
(4)
where K is a parameter related to the equilibrium constant for dimer formation, Nm is the saturation capacity of the (23) Talu, O.; Meunier, F. AIChE J. 1996, 42, 809.
Ψ)
-1 + (1 + 4Kξ)1/2 2K
(5)
NmN Nm - N
(6)
and
(1)
where σc is the effective diameter of a carbon atom. The amount of fluid adsorbed at a given pressure may be written ∞
pore (and thus varies with pore width), H0 ) limPf0 dP/ dN is the Henry’s law constant,
ξ)
where N is the amount adsorbed at pressure P. To improve this interpolation scheme, a fourth pore width, H ) 0.79 nm, was studied for the entire pressure range 0 < P < 1. The four simulated adsorption isotherms, H ) 0.79, 0.99, 1.69, and 4.5 nm, were fitted to eq 4 to yield values of the parameters H0, K, and Nm. The resulting smooth fits showed good agreement with the simulation data in the steeply rising region of the isotherms; the main discrepancy between eq 4 and the simulations occurred in the low-pressure region, where the equation gave adsorbed amounts that were too low. The parameters H0, K, and Nm varied smoothly with w and were fitted to simple relations. Using these relationships, together with eq 4, enabled us to construct adsorption isotherms for any pore width; these, together with the experimental PSD, made it possible to carry out the integration of eq 2. Surface Groups. Groups such as hydroxyl, carbonyl, carboxylic, phenolic, carbonyl, lactonic, chromene, and pyrone have been identified on active carbon surfaces. The relative and absolute amounts of these groups depend on the carbon precursor material and on the activation conditions employed in the creation of the porous sample. X-ray diffraction studies show that these oxygen-containing surface groups are located around the edges of the carbon microcrystallites.24 The experimental characterization of surface chemistry can be achieved using a variety of chemical and physical techniques. Frequently, different methods lead to quite disparate results, presumably because of the complexities of the carbon structure. The carbon used here has a total estimated surface group density of 0.675 sites/nm2, with about 3 times as many basic groups (e.g., pyrone) as acidic surface sites (e.g., carboxylic); both can bind water, since water can act as either a Lewis acid or base. To simplify our model, we assume that the mixture of surface groups can be modeled by an effective surface in which all oxygenated groups have the same binding energy, HB sf , with water molecules; moreover, we assume that each surface group can only bond with one water molecule. We refer to this as the “effective single group” model. In reality, there will exist a mixture of groups of different species on the surface, with sites which have somewhat different binding energies and some with more than one binding site. We postulate that our model can represent this surface reasonably well with a value of HB sf that is a suitably weighted mean of all these site-water interactions. In our model, surface groups are represented as OH groups and consist of a Lennard-Jones (LJ) sphere to represent an oxygen atom, and a single square well site placed on this that represents the H atom and can interact with similar sites on water molecules. Surface groups are placed randomly on a square lattice that is superimposed on the carbon surfaces, with the restriction than no two groups can be closer than 21/6σ, where σ is the LJ diameter (24) Bansal, R. C.; Donnet, J. In Carbon Black; Donnet, J., Bansal, R. C., Wang, M., Eds.; Marcel Dekker: New York, 1993.
Adsorption of Water on Activated Carbon
Langmuir, Vol. 15, No. 2, 1999 537 Table 1. Maximum Site Density n for Different Microcrystal Edge-Lengths L, for a Random Placement of Sites L/nm
n/nm-2
L/nm
n/nm-2
1.0 1.5 2.0
9.66 6.89 5.33
2.5 3.0 10.0
4.34 3.67 1.15
Table 2. GEMC and NPTMC Simulation Results and Experimental Data for Coexisting Liquid Water and Water Vapor at 298 K (The Accuracy Is the Simulation Value as a Percentage of the Experimental Value)
Figure 3. Model of the water molecule used in the simulation. The oxygen is represented by a Lennard-Jones sphere of diameter σLJ; the association sites (hydrogen atoms and lonepair electrons) are tetrahedrally arranged square well sites of HB diameter σHB ff and well depth ff .
for the O atom. The lattice lines define the edges of the graphite microcrystals, and the length L of a microcrystal edge is retained as a variable. Values of L between 1 and 10 nm have been reported, on the basis of diffraction studies; however, most studies have shown values of L in the range 1-3 nm. Experimental site densities range from 0 to 2.65 sites/nm2. We have chosen L to be 2.0 nm for most of the calculations in our study (the effect of varying site density is considered in section 5). The arrangement of groups on the surface is kept fixed as H is varied, to eliminate differences due to site location. Intermolecular Potentials. Water is modeled as a LJ sphere, with four tetrahedrally arranged square well sites4,25 (Figure 3). Two sites represent hydrogen atoms, and two mimic the lone pairs of electrons. Bonding only occurs between unlike sites. The sites are located at a distance of 0.42σ from the center of the LJ sphere; this placement is consistent with the observation of a peak in the O-H pair correlation function for water at 0.19 nm, as found from neutron diffraction experiments.26 Thus, the fluid-fluid pair intermolecular potential energy between two water molecules can be expressed as
[( ) ( ) ]
uff(rω1ω2) ) 4ff
σff r
12
-
σff r
6
+ uHB ff (rω1ω2) (7)
where r is the vector joining the centers of the two water molecules, ωi denotes the orientation of water molecule i, ff and σff are the usual LJ well depth and molecular diameter parameters, and the H-bond term is given by HB HB uHB ff (rω1ω2) ) -ff if rRβ < σff
) 0 otherwise
(8)
where rRβ is the distance between the center of a square well (SW) site R on molecule 1 and a SW site β on molecule 2. The LJ parameters, and also the HB size parameter, were taken from previous studies:4 σff ) 0.306 nm; ff/k ) 90.0 K; σHB ff ) 0.0612 nm. The calculations are sensitive to the value of the strength of the H-bond, HB ff , and we therefore carefully fit this parameter to best reproduce the gas-liquid coexistence properties of bulk water. A series of Gibbs ensemble and isothermal-isobaric simulations were undertaken at 298 K for this water model for (25) Mu¨ller, E. A.; Gubbins, K. E. Ind. Eng. Chem. Res. 1995, 34, 3662. (26) Soper, A. K.; Phillips, M. G. Chem. Phys. 1986, 107, 47.
property
experiment
simulation
accuracy
Fvapor Fliquid (mmol/m3) P0 (atm)
1.2798 55389 0.03125
1.267 ( 0.004 54400 ( 600 0.0307 ( 0.0006
99.1 ( 0.3% 98 ( 1% 98 ( 2%
(mmol/m3)
a range of values of HB ff . These simulations yielded the vapor pressure and coexisting densities of the gas and liquid phases. The value of HB ff was then chosen to give the best agreement with experimental data for these properties. This procedure gave a value of HB ff /k ) 3800 K; agreement with experiment was within 2% for each of the coexistence properties (see Table 2). The interaction of a water molecule with the walls is of the form
usf(r1ω1) ) uCf(z) + uLJ(r) + uHB sf (r1ω1)
(9)
where uCf is the interaction energy of the water with the carbon atoms, uLJ is the LJ interaction between the LJ center in the water molecule and the LJ centers that represent the O atoms in the OH groups attached to the walls, and uHB sf is the H-bond interaction between the water molecule and a site on the wall; r1 and ω1 denote the location and orientation of the water molecule, z is the distance of the center of the water molecule from the nearest point on the wall (taken to be the plane through the centers of the carbon atoms forming the first layer), and r is the distance between the centers of the LJ sites in the water molecule and the OH group on the wall. The interaction between water and the carbon atoms is modeled using the 10-4-3 potential of Steele:27
uCf(z) ) 2πFCCfσCf2∆
[( ) ( ) 2 σCf 5 z
10
-
(
σCf z
4
-
σCf4
)]
3∆(z + 0.61∆)3
(10)
where the density of the carbon is FC ) 114 nm-3, the separation of the graphite planes is ∆ ) 0.335 nm, and Cf and σCf are the LJ parameters for the interaction between carbon atoms and water. The carbon-carbon potential parameters are taken as27 Cf/k ) 28 K and σCf ) 0.340 nm. Fluid-solid cross parameters are calculated using the usual Lorentz-Berthelot rules:
Cf ) (CCff)1/2 σCf ) (σCC + σff)/2
(11)
The H-bond term in eq 9 is the same square well function given by eq 8, but with subscripts ff replaced by sf. The active sites on the carbon surface are represented by LJ (27) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974.
538 Langmuir, Vol. 15, No. 2, 1999
Figure 4. Representation of the activated carbon pore used in the MC simulations. The structureless walls are decorated with active sites that are taken to be OH groups; these are represented as Lennard-Jones oxygen atoms (large open circles) and square well hydrogen atoms (small filled circles). The pore width H is the distance between the plane through the centers of the carbon atoms in the first layer on opposite walls of the pore. The lower figures (A and B) show typical arrangements of sites on the surface for n ) 0.675 sites/nm2 for L ) 2.0 (A) and 1.0 (B) nm.
spheres with an attached single square well bonding site (Figure 4); these represent hydroxyl groups. The LJ sphere represents the O atom of the OH group and its center is placed at a distance of 0.1364 nm above the plane through the centers of the first-layer carbon atoms. The LJ parameters for this LJ sphere are the same as those for the LJ site in the water molecule, i.e., ) ff ) 90.0 K and σ ) σff ) 0.306 nm. The center of the square well site is placed at a distance of 0.42σ from the center of the LJ sphere and is placed as far from the wall as possible (i.e., on the normal to the wall which passes through the center of the LJ sphere). The wall-site square well interaction has the same range as those on the water molecules (i.e., σHB sf ) 0.0612 nm) and mimics a hydrogen atom, so that it is only able to form bonds with the lone-pair electron sites on the water molecules. Determination of the appropriate strength of the wall-fluid hydrogen bond is discussed below. 4. Molecular Simulation Method Calculations were carried out using the grand canonical Monte Carlo (GCMC) method.28 This method is convenient for adsorption studies, since the chemical potential µ, temperature T, and volume V are specified and kept fixed in the simulation. Since µ and T are the same in the bulk and adsorbed phases at equilibrium, the thermodynamic state of the bulk phase is known in such simulations. Three types of molecular moves are attempted: molecular creation, molecular destruction, and the usual Monte Carlo translation/rotation moves. Each of these three moves is attempted with equal frequency. The type of move is chosen randomly to maintain microscopic reversibility. The probability of successful creations or destructions is strongly dependent on the density of the system. The maximum allowable rotation and displacement of a molecule are adjusted so that the combined move has an acceptance probability of about 40%. This value should ensure the most efficient probing of the phase space distribution. It is noteworthy that the values of the (28) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987.
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maximum displacement and rotation are very small compared to values encountered for nonbonding systems. In these simulations the number of adsorbed molecules fluctuates during the simulation. Calculation of the average of this number for a range of chemical potentials enables the adsorption isotherm to be constructed. The walls of the slit pores lie in the x-y plane. Normal periodic boundary conditions, together with the minimum image convention, are applied in these two directions. For low pressures, P/P0 e 0.02, the length of the simulation cell in the two directions parallel to the walls was maintained at 10 nm for each of the pore widths studied, to maintain a sufficient number of adsorbed molecules. For higher pressures where more water molecules were present, the minimum cell length in the x and y directions was 4 nm. A potential cutoff of 5σ was used. For the system studied HB ff /kT ) 12.8. While this value is not high enough to require biased sampling methods,4 long runs are needed to ensure ergodicity. Associating water molecules tend to remain in energetically favorable configurations for many MC steps, and the system thus requires many steps to reach a true equilibrium state. In our runs 500 million MC steps were used for equilibration, followed by a further 500 million steps for property averaging. Shorter runs than this were not adequate to sample desorption events. The average number of water molecules in the simulation cell varied from a few molecules at the lowest pressures to a few hundred or thousands of molecules when the pores were full; filled pores contained about 320 molecules for a pore width of 0.79 nm, 460 at 0.99 nm, 830 at 1.69 nm, and 2100 at H ) 4.5 nm. Calculations were carried out on the Cornell Theory Center IBM SP2. In determining the adsorption isotherm, we commenced with the cell empty; a value of the fugacity corresponding to a low pressure was chosen and the average adsorption determined from the simulation. The final configuration generated at each stage was used as the starting point for simulations at higher fugacities. The pressure of the bulk gas corresponding to a given chemical potential was determined from the ideal gas equation of state. Gasphase densities corresponding to the range of chemical potentials studied were determined by carrying out simulations of the bulk gas. These were found to agree with those calculated from the ideal gas equation within the estimated errors of the simulations. 5. Results Low-Pressure Region (P/P0 e 0.02). At sufficiently low pressures the adsorbate molecules interact solely with the pore walls, and fluid-fluid interactions can be neglected. The simulations show that this region lies at relative pressures P/P0 below about 0.003, and this seems to be confirmed by the experimental results. Experimental measurements were carried out at relative pressures down to 0.002, corresponding to 6.3 × 10-5 bar (0.048 Torr), which is approaching the limit of sensitivity of the sorptometer. At relative pressures above 0.003 the experimental adsorption isotherm (Figure 5) shows a sharp increase in slope. This steeper region persists to relative pressures of about 0.01, after which the slope decreases again. The quantity plotted in Figure 5 is the excess adsorption, Γexcess ) (N - Nbulk)/A, where N is the number of molecules adsorbed, Nbulk is the number of molecules in a volume of bulk gas equal to that of the pore at the same temperature and chemical potential, and A is the surface area. The simulation results are sensitive to the value chosen for HB sf /k. We first made a rough estimate of this param-
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Figure 5. Adsorption of water at 298 K at low relative pressure (P/P0 e 0.02). Open circles are the experimental data (the solid line is a guide to the eye), and solid points are simulation HB results: squares, HB sf /k ) 4800 K; circles, sf /k ) 4900 K; HB triangles, sf /k ) 5000 K. Data points are the weighted average of simulation results for the three individual pore widths. Error bars denote one standard deviation of the simulated values.
eter. Typical H-bond strengths obtained by fits of simulations to bulk thermodynamic properties are HB sf /k ) 3643 K for H-bond donor group/water and 2673 K for H-bond acceptor group/water, respectively. These values were obtained for acetic acid-water complexes.11,29 These values are less than the value of 3800 K for water-water interactions. However, the HB sf /k value for our model is expected to be larger than this, since it only allows for one group-water H-bond for each group, whereas the real carbon surface will have groups that can form multiple H-bonds with water molecules. Carboxyl groups, for example, can H-bond with two water molecules, one through the H-bond donor OH site and the other with the H-bond acceptor site dO. Similarly, we expect two H-bond acceptor sites for lactonic groups, one H-bond donor site for phenolic groups, and so on. In our “effective single group” model, the appropriate value of HB sf /k will be a weighted average of the total bonding energies for the various surface groups for the real carbon. Using the experimental values of group densities for the various species present, we estimate the value of the average H-bond strength to be HB sf /k ) 4949 K. This estimate is in agreement with our initial simulations, which showed that a value in the range HB sf /k ) 4800-5000 K gives the best fit to the low-pressure data. We have carried out simulations in the pressure range 0 e P/P0 e 0.02 for three pore widths, H ) 0.99, 1.69, and 4.50 nm, and for three values of bonding strength, HB sf /k ) 4800, 4900, and 5000 K. The site density was fixed at 0.675 sites/nm2, and the length of the graphite microcrystals was 2 nm in all systems studied. For each of the three bonding strengths we calculated the isotherm for the PSD of Figure 2, using eq 3. These results are shown in Figure 5, along with the experimental data. None of the three simulated isotherms reproduce the data over the full pressure range. With HB sf /k ) 5000 K the lowest pressure part of the isotherm is correctly reproduced, and the sharp increase in slope is obtained at the correct pressure (P/P0 ≈ 0.004). However, the higher slope persists to too high of a pressure, the slope being reduced only for pressures above about P/P0 ≈ 0.03-0.04, whereas this reduction in slope occurs at P/P0 ≈ 0.01 in the (29) Gao, J. J. Phys. Chem. 1992, 96, 6432.
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experimental system. The simulations for HB sf /k ) 4800 and 4900 K show similar results, but with less adsorption, and with the increase in slope delayed to higher pressures. Snapshots of typical molecular configurations in the pore reveal that all adsorbed water molecules are bonded onto a surface site for the pressure range shown in Figure 5; however, at the higher pressures shown such adsorbed water molecules may also be H-bonded to other water molecules, where two surface sites are spaced a suitable distance apart. For pressures below P/P0 ≈ 0.003 water molecules are adsorbed onto single sites, with no waterwater bonding (Figure 6a). Since all water-surface site bonds have the same energy in our effective single group model, this very low pressure region corresponds to Henry’s law regime, and the isotherm is linear. The experimental data in this region is sparse and it is not possible to detect with confidence a linear region. We believe that even in this low-pressure region the experimental isotherm is likely to show significant nonlinearity, since different chemical groups on the carbon surface will interact with water with different bonding energies. At pressures somewhat above P/P0 ≈ 0.003 the snapshots show that cooperative adsorption effects are important. This corresponds to the region of the isotherm with increased slope. At these pressures newly adsorbed water molecules seek surface sites on which they can simultaneously bond to both the surface site and a preadsorbed water molecule, thus releasing an amount of energy HB sf + HB ff . An example of this behavior is shown in Figure 6b. Thus, the adsorption mechanism in our model is similar to that shown in stages I and II of scheme B of Figure 1. At still higher pressures, beyond those shown in Figure 5, snapshots of the simulations indicate that cooperative bonding of the type shown in Figure 6b is complete, and adsorption involves H-bonding of additional water molecules to either isolated surface sites or to preadsorbed water molecules through water-water bonds. The most noticeable difference between the simulation results and the experimental ones is the persistence of the steep region of the isotherm to too high of a pressure in the case of the model. The effective single group model apparently provides too many possibilities for cooperative bonding. A possible reason for this defect may be that on the real carbon surface many groups have more than one bonding site, which is not accounted for in our model. In addition, steric hindrance is likely to limit cooperative bonding to a greater extent in real carbons than in our model. Higher Pressures (0 e P/P0 e 1.0). On the basis of the results at low and moderate pressures, we adopted a value of the surface bonding strength of HB sf /k ) 4800 K, since this seemed to give the best overall agreement with experiment for relative pressures above 0.01. Simulations were conducted for pressures over the range 0 e P/P0 e 1 for pores of width H ) 0.79, 0.99, 1.69, and 4.50 nm; site density and microcrystal length were kept fixed at 0.675 sites/nm2 and 2.0 nm, respectively. The number and arrangement of the sites were the same for the different pore widths. The simulation results for the four pore widths are shown in Figure 7. Since the pore capacity expressed as Γexcess varies widely with pore width it is more convenient to display the results in Figure 7 as the excess density in the pore, defined as
Fexcess )
Γexcess w
(12)
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Figure 6. Snapshots of typical configurations for one of the activated carbon surfaces for a pore of width H ) 1.69 nm for HB sf /k ) 4800 K: at low pressures, top figure, P/P0 ) 0.003; lower figure, P/P0 ) 0.017. The gray background is the carbon surface; the black spheres embedded on this surface are the O atoms of the OH surface groups, the attached small white sphere being the square well bonding site (or H atom). Water molecules are represented as a large white sphere (O atom) with four small attached spheres arranged tetrahedrally, representing the two H atoms (white spheres) and two lone-pair electrons (gray spheres).
Figure 7. Simulated adsorption isotherms at 298 K for water into activated carbon pores of various widths: H ) 0.79, 0.99, 1.69, and 4.50 nm. Points denote simulated values, and lines are the fits to the simulated data using the expression of Talu and Meunier (eq 4).
where w is the effective pore width given by eq 1. As expected, the isotherms are of type V in the usual IUPAC
classification,5 and the pore filling pressure and pore capacity depend strongly on pore width. The narrowest
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Figure 8. Snapshots of typical configurations for a pore of width H ) 1.69 nm for higher pressures than those shown in Figure 6: upper figure, P/P0 ) 0.20 (showing one of the carbon surfaces); lower figure, P/P0 ) 0.575 (showing both carbon walls). Key as in Figure 6.
pores, H ) 0.79 and 0.99 nm, display much greater adsorption at low pressures than the larger pores, and pore filling occurs at a low relative pressure of P/P0 ≈ 0.2. In these smaller pores water molecules are able to form “bridges” between opposing walls at lower pressures than for larger pores; the establishment of such a link between the pore walls is quickly followed by pore filling, with the bridging water molecules acting as the nuclei for further adsorption. Snapshots of typical configurations are shown for the pore of H ) 1.69 nm in Figure 8 for an intermediate pressure of P/P0 ) 0.20 and a pressure near to pore filling, P/P0 ) 0.575. At the lower pressure of 0.20 we see water molecules bonding to preadsorbed water molecules, in addition to water molecules adsorbed to surface sites and the cooperative bonding seen in Figure 6b. This corresponds to stage III of scheme B of Figure 1. At a reduced pressure of 0.575 the water clusters have grown and bridged across the pore walls (stage IV of scheme B in Figure 1). Although large water clusters of liquidlike density are present, there is still considerable empty space in the pore at this stage. At this stage additional adsorption of water molecules occurs very easily, with new molecules forming several water-water bonds on adsorption with molecules already previously adsorbed. Thus, the pore fills rapidly, and at a pressure slightly above 0.575 the pore is full. For smaller pores the principal difference is that the formation of bridges of water molecules between opposing walls can occur more easily, and so appears at lower pressures. This is seen in the series of snapshots for a pore of width 0.99 nm in Figure 9, where bridging between walls occurs at a pressure of P/P0 ) 0.05, and pore filling has occurred at a reduced pressure of about 0.25.
To calculate the adsorption isotherm for the experimentally determined PSD of Figure 2, the semiempirical equation of Talu and Meunier23 was used to interpolate between the simulated isotherms for the four pore widths shown in Figure 7, as described in section 3. The constants in this equation were fitted for each of the pore widths, and were found to vary smoothly with H. These fitted curves are included in Figure 7. By interpolating between these fitted values of the constants K, Nm, and H0 it was possible to estimate adsorption isotherms for a wide range of pore widths, and to then carry out the integration of eq 2 using the experimentally determined PSD. A comparison of the isotherm determined in this way from the simulation results with the experimental one is shown in Figure 10. The overall agreement is quite good. The pore filling pressure is predicted well, although the simulated curve is somewhat steeper than the experimental one in this region. The predicted adsorption is too low at low pressures, and the predicted adsorption for the filled pore is also somewhat too low. The underestimation of the adsorption at low pressure, and the predicted steepness in the pore-filling region, are primarily a result of the use of the Talu-Meunier equation, as seen from the comparison of this equation with the simulation results shown in Figure 7. Errors in the experimentally determined PSD could also contribute to discrepancies in the pore-filling region. At the highest pressures, near P/P0 ≈ 1, the simulated adsorption is approximately 6% too low. The Talu-Meuinier equation describes this region well. Since the model used for water predicts the bulk density accurately, the discrepancy in this region most likely results from uncertainty in the PSD for large pores; a
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Figure 9. Snapshots of typical configurations for a pore of width H ) 0.99 nm: top figure, P/P0 ) 0.010; middle, P/P0 ) 0.050; bottom, P/P0 ) 0.30. Both carbon walls are shown.
Figure 10. Adsorption of water from experiment and simulation at 298 K. The solid line and open circles represent the experimental data; the dashed line and filled circles show the simulated isotherm, obtained from eq 2 using simulated isotherms for discrete pore widths together with the experimental PSD.
small increase in the fraction of large pores would lead to a significant increase in the adsorption capacity here. Influence of Microcrystal Size on Adsorption. For a constant site density, reducing the length of the microcrystals leads to an increase in the average spacing between active sites (see Figure 4). With sites further apart, we expect less cooperative bonding and less bridging between adjacent sites. To investigate the importance of such effects, we carried out simulations for two microcrystal lengths, L, of 1.0 and 2.0 nm at 298 K. In these calculations the site density was fixed at 0.675 sites/nm2, the pore width was H ) 1.69 nm, and the fluid-wall bonding strength was chosen to be HB sf /k ) 5000 K. The simulation cell length was fixed at 10.0 nm, and calculations were made for the pressure range P/P0 ) 0 to 0.10. The results are shown in Figure 11. As expected, there
Figure 11. Influence of microcrystal length, L, on low-pressure adsorption, for a pore width H ) 1.69 nm, site density ) 0.675 sites/nm2, T ) 298 K, and HB sf /k ) 5000 K. Circles show results for L ) 1.0 nm and squares are for L ) 2.0 nm.
is a substantial decrease (by up to 45%) in the adsorption as the crystal length is reduced from L ) 2.0 to 1.0 nm. The study of typical molecular configurations shows that cooperative bonding of the type shown in Figure 6 is much less for a microcrystal length of 1.0 nm. For example, if we consider a point on each isotherm with the same amount of adsorption, N ≈ 30, the percentage of sitebonded molecules participating in cooperative bonding is about 60% for L ) 2.0 nm compared to 33% for L ) 1.0 nm. This observation is part of the reason for the greater amount of adsorption at a given pressure in the case of the larger microcrystal. Thus, the microcrystal length has a relatively large impact on the adsorption at low pressures, since it strongly affects the relative distances between sites. Effect of Site Density on Adsorption. To study the effect of site density on adsorption, we carried out a series of simulations in which the number of active sites on the pore walls was varied. For a pore having a width H ) 1.69
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Figure 12. Effect of active site density on water adsorption at 298 K in a pore of width 1.69 nm, with HB sf /k ) 5000 K and a microcrystal length of 1.0 nm. Three site densities are shown: 0.3 (circles), 0.675 (squares), and 1.2 sites/nm2 (diamonds). For the filled pore (not shown) the adsorption was about 3.3 × 10-2 mmol/m2 for all three cases.
nm and HB sf /k ) 5000 K, three site densities were investigated: 0.30, 0.675, and 1.20 sites/nm2. The microcrystal length used for this study was 1.0 nm, and the simulation cell length was held fixed at 3.0 nm. Adsorption isotherms at 298 K were generated for the full range of relative pressures, P/P0 ) 0 to 1. Adsorption isotherms for the three site densities are shown in Figure 12 for pressures up to the pore-filling region. Three conclusions can be drawn. First, the porefilling pressure depends strongly on the site density, occurring at a lower pressure for the higher site densities. Second, the systems with higher site densities (0.675 and 1.20 sites/nm2) exhibit a gradual transition to pore filling, while the system with a low site density shows a sharp transition, which appears to be capillary condensation. Finally, an increase in site density leads to an increase in adsorption in the region prior to pore filling. This agrees with experimental observations that adsorption is enhanced by an increase in the degree of oxidation of active carbons (up to some limiting level, after which further oxidation begins to destroy the porous structure and leads to a reduction in adsorption capacity). Each of these observations can be explained by the adsorption mechanism, as described below. An increase in adsorption with site density occurs naturally because there are more sites available. In addition, at the higher site densities of 0.675 and 1.20 sites/nm2, cooperative bonding will come into play and lead to significant increases in adsorption. Since the possibilities for cooperative bonding are similar for these two higher site densities, the adsorption curves are very similar for relative pressures up to 0.10. For higher pressures adsorption of water onto surface sites (without cooperative bonding) continues up to pressures just below pore filling, since the wall sites have a higher bonding energy than the water-water H-bond. Since the pore with the site density of 1.20 sites/nm2 has more of these sites available, adsorption is higher in this relative pressure region (0.10-0.43). As a result of this increased adsorption, bridging of adsorbed water molecules between the two pore walls occurs more readily for the highest site density. Such bridging provides many opportunities for cooperative water-water bonding (newly adsorbed water molecules bonding to more than one preadsorbed molecule), and leads to lower pore-filling pressures. Snapshots of typical configurations for the three site densities at pressures up to pore filling show a qualitative difference in behavior at the lowest site density, 0.3 sites/ nm2 versus the two higher densities, 0.675 and 1.20 sites/ nm2 (Figure 13). At 0.3 sites/nm2 single water molecules
Figure 13. Snapshots of typical configurations just prior to pore filling for a pore of width H ) 1.69 nm and HB sf /k ) 5000 K. Results are shown for three site densities: n ) 0.30 (top figure) at P/P0 ) 0.81, 0.675 (middle) at P/P0 ) 0.63, and 1.20 (bottom) at P/P0 ) 0.45.
bond to surface sites, and there is almost no cooperative bonding. Pore filling appears to occur by capillary condensation, the transition being from a state in which only a few water molecules are adsorbed onto active sites (top snapshot in Figure 13) to one in which the pore is filled at a liquidlike density. At the higher site densities, however, adsorption of single water molecules onto active sites at low pressure is followed by the buildup of clusters (lower two snapshots in Figure 13), formation of bridges between pore walls, and finally pore filling. This mechanism of pore filling is clearly different from that at the low site density, and the pore filling is a continuous process. The results suggest that there may be a limiting site density above which capillary condensation no longer occurs, somewhere between n ) 0.3 and 0.675 sites/nm2 for this system. 6. Conclusions While the experimental isotherm is of class V as expected, the low-pressure measurements exhibit considerable sensitivity to the surface chemistry, which manifests itself in changes in slope at particular pressures. Although the model shows a very short Henry’s Law region (for relative pressures P/P0 e 0.003), it seems unlikely that a significant Henry’s law region exists for the
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experimental system, since a variety of surface groups are present having different adsorption energies. The model calculations show good overall agreement with the experimental data, and we believe they are significantly more realistic than previous attempts to model this complex system at the molecular level. However, the effective single group model as applied here still has significant defects. In particular it fails to predict the very low pressure data accurately. While it describes the increase in slope (seen experimentally at P/P0 ≈ 0.003) and the later decrease in slope, it predicts too large of a pressure range for this cooperative bonding region. We believe this may be due to a neglect of groups that have multiple bonding sites, and/or to a neglect of groups having different bonding energies. A further possible cause may be that the microcrystal length used (L ) 2.0 nm) is inappropriate. Another unsatisfactory feature is the method used here to describe the pore size distribution at the higher pressures. Because the semiempirical equation used to interpolate between results for different pore widths does not describe the low-pressure region well, the resulting calculated isotherm for the experimental PSD is inaccurate in this region and also has too steep of a slope in the pore-filling region. This is not a defect of the model itself, but reflects the difficulty of simulating a sufficient number of pore widths to cover the PSD for this system, which is broad. It could be diminished by the use of a carbon with a narrower PSD.
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Future experimental work on these systems should focus on carbons which have a narrow PSD and are wellcharacterized. In particular, the use of neutron or X-ray diffraction to determine the average length (or distribution of lengths) of the microcrystals would be valuable, as would the use of spectroscopic techniques such as FTIR to determine the chemical nature and density of surface groups. Further work on the molecular model should incorporate the possibility of groups having multiple bonding sites, as well as sites of different bonding energies, to better describe the low-pressure adsorption region. Finally, these models should be tested against a variety of activated carbons, having differing surface chemistry and site density. Acknowledgment. This work was supported by the Department of Energy (Grant No. DE-FG02-88ER13974). The simulations reported here were performed on the IBMSP2 at the Cornell Theory Center, with the support of the National Science foundation through a Metacenter grant (MCA93S011). S.C.M. also thanks the NSF for a CISE Postdoctoral Fellowship. International cooperation was made possible by a NSF/CONICIT U.S.-Venezuela Cooperative Research grant (No. INT-9602960). LA9805950
