Langmuir 1997, 13, 2815-2821
2815
A Self-Consistent Method for Characterization of Activated Carbons Using Supercritical Adsorption and Grand Canonical Monte Carlo Simulations Vladimir Yu. Gusev and James A. O’Brien* Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286
Nigel A. Seaton Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, United Kingdom Received April 29, 1996. In Final Form: March 11, 1997X Adsorption of methane was measured on BPL-6 activated carbon, and simulated using the Grand Canonical Monte Carlo method in 40 slit-shaped pores ranging in width from from 0.63 to 5.72 nm (1.65-15 methane molecular diameters). The simulations were used, in combination with a single experimental isotherm at 308 K, to extract a pore size distribution (PSD) for the carbon. This PSD was then used, in combination with higher-temperature simulations, to predict methane adsorption on the same adsorbent at two higher temperatures. The predicted isotherms show excellent agreement with experiment at 333 and 373 K for pressures of up to 3 MPa. We demonstrate that the sensitivity of the PSD to characterization using a supercritical adsorbate such as methane is bounded by a limiting pore width, which is a function of pressure, temperature, and the adsorbate used. Above this limiting pore width, adsorption may be used to characterize porous solids instead by their surface area. At 308 K, the PSD of micropores can be effectively characterized using methane adsorption at pressures up to 3 MPa. An implementation of our method is available on the World Wide Web.
Introduction Adsorption is one of the most versatile industriallyuseful methods for characterization of porous solids. The traditional thermodynamic methods for interpreting adsorption isotherms are the Gibbs-Kelvin (GK), Brunauer, Emmett, and Teller (BET), and Dubinin and Radushkevich (DR) methods and their modifications.1 These methods are based on phenomenological assumptions such as subcritical adsorbate homogeneity and incompressibility, gas phase ideality, independence of interfacial tension of liquid, γ, of its curvature (GK), identity to bulk liquid of all adsorbed layers beyond the first and absence of lateral interactions in adsorbed layer (BET), volume filling of pores and Gaussian distribution of micropores (DR), etc. (For a recent discussion of these methods see ref 2). Despite their deficiencies, these methods are in widespread use. In some cases they are known to produce inconsistent results.3 Considerable progress in the description of strongly inhomogeneous confined fluid has led in recent years to the development of new theoretical methods for pore characterization. These include the density functional theory (DFT)-based method proposed by Seaton et al.4 and the nonlocal density functional theory (NLDFT)-based method due to Lastoskie et al.5 These methods are free of the phenomenological assumptions listed above. How* Author to whom correspondence should be addressed. Current address: PTT, Inc., 630 Third Ave., New York, NY 10017. X Abstract published in Advance ACS Abstracts, May 1, 1997. (1) Gregg S. J.; Sing K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (2) Cracknell, R. F.; Gubbins, K. E.; Maddox, M.; Nicholson, D. Modeling Fluid Behavior in Well-Characterized Porous Materials. Acc. Chem. Res. 1995, 28(7), 281. (3) Russel, B. P.; LeVan, D. M. Pore Size Distribution of BPL Activated Carbon Determined by Different Methods. Carbon 1994, 32(5), 845855. (4) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. A New Analysis Method for the Determination of the Pore Size Distribution of Porous Carbons From Nitrogen Adsorption Measurements. Carbon, 1989, 27, 853.
S0743-7463(96)00421-0 CCC: $14.00
ever, they are based on a mean-field approximation of fluid-fluid attractions, which may become inaccurate for fluids confined within very small pores.6 Monte Carlo (MC) and molecular dynamics (MD) simulations model actual molecular microscopic configurations of the confined fluid using realistic intermolecular interaction potentials and, in principle, are exact for the potentials used.7 However, since they are generally believed to be theoretically and computationally more demanding,2 these methods have not been used for the purpose of direct characterization of porous solids, but rather as a benchmark for DFT methods (see, e.g., refs 5 and 8). In grand canonical Monte Carlo (GCMC) simulation, the temperature, volume, and chemical potential (which can be directly related to pressure and temperature using the bulk gas equation of state) are specified, while the number of particles and associated configurational energy are allowed to fluctuate. Hence, GCMC sampling is capable of yielding directly the amount adsorbed in arbitrary confined spaces as a function of pressure and temperature and thus presents a convenient tool for modeling adsorption in pores. Due to the considerable sensitivity of nitrogen adsorption isotherms to the pore structure in both microporous and mesoporous regimes and to its relative experimental simplicity, measurements of subcritical nitrogen adsorption at 77 K are often used to provide experimental input (5) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Pore Size Distribution Analysis of Microporous Carbons: A Density Functional Theory Approach. J. Phys. Chem. 1993, 97, 4786. (6) Peterson, B. K.; Gubbins, K. E.; Heffelfinger, G. S.; Marini Bettolo Marconi, U.; van Swol, F. Lennard-Jones fluids in cylindrical pores: Non-local theory and computer simulation. J. Chem. Phys. 1988, 88, 6487. (7) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (8) Neimark, A. V.; Gusev, V. Yu.; O’Brien, J. A.; Ravikovich, P. Enhanced DFT and GCMC Calculations of Adsorption Equilibrium in Nanopores. Proceedings of AIChE Annual Meeting, Miami, November 1995; p 17.
© 1997 American Chemical Society
2816 Langmuir, Vol. 13, No. 10, 1997
to characterization methods.9 Traditional and DFT methods of characterization were developed mostly with this system in mind. However, there are compelling arguments in favor of using other adsorbates and/or thermodynamic conditions for characterization of porous solids. The most important reasons include the need to (a) employ a specific adsorbate at specific conditions (for example, carbon dioxide at room temperature could access micropores which would present too much diffusion resistance for subcritical nitrogen) or (b) mimic the conditions of the specific industrial application (for example, it might be advantageous to use supercritical methane adsorption to characterize an adsorbent proposed for the separation of methane-containing mixtures or for adsorptive storage of methane at ambient temperatures). There are additional issues such as the long equilibration times (up to several days) and other experimental difficulties (see, e.g., ref 10) associated with measuring adsorption of subcritical gases in micropores. In the following sections we report the results of our study of methane adsorption for characterizing a heterogeneous carbonaceous adsorbent. It uses, as input information, methane experimental supercritical adsorption measurements and GCMC simulations of adsorption of this gas in a number of model slit pores of various widths. The slit-shaped nature of activated carbon granule and fiber micropores is suggested by their molecular sieving properties (see ref 12 and references therein), highresolution electron microscopy,13 neutron diffraction, and small angle neutron scattering (see ref 14 and references therein). Methane adsorption has been extensively modeled in infinite slit pores using DFT, GCMC, and MD simulations, mainly due to interest in adsorptive storage of natural gas (see refs 15-17 and references therein). The feasibility of our approach to the characterization problem has been demonstrated in a recent study10 of the pore structure of heterogeneous activated carbons, in which NLDFT-generated adsorption isotherms of methane borrowed from ref 15 were used. Consistency of GCMC simulations themselves with adsorption experiment in the case of supercritical adsorption in a well-defined uniform slit pores of activated carbon fibers was shown recently for nitrogen.14 As we intend our method to be used directly in engineering and scientific practice, we have made it available on the World Wide Web (http://www.yale.edu/ yaleche/chemeng/job/gcmc psd.htm). (9) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Reporting Physisorption Data for Gas/Solid Systems - with Special Reference to the Determination of Surface Area and Porosity. Pure Appl. Chem. 1985, 57, 603. (10) Sosin, K. A.; Quinn, D. F. Using the High Pressure Methane Isotherm for Determination of Pore Size Distribution of Carbon Adsorbents. J. Porous Mater. 1995, 1, 111. (11) Peterson, B. K.; Walton, J. P. R. B.; Gubbins, K. E. Phase Transitions in Narrow Pores: Metastable States, Critical Points, and Adsorption Hysteresis. In Fundamentals of Adsorption, Liapis, A. L., Eed.; Engineering Foundation: New York, 1897; p 463. (12) Everett, D. H.; Powl, J. C. Adsorption in Slit-like and Cylindrical Micropores in the Henry’s Law Region. J. Chem. Soc., Faraday Trans. 1 1976, 72, 619. (13) Marsh, H.; Crawford, D.; O’Grady, T. M.; Wennerberg, A. Carbons of High Surface Area. A study by Adsorption and High Resolution Electron Microscopy. Carbon 1982, 5, 419. (14) Kaneko, K.; Cracknell, R. F.; Nicholson, D. Nitrogen Adsorption in Slit Pores at Ambient Temperatures: Comparison of Simulation and Experiment. Langmuir, 1994, 10, 4606. (15) Tan, Z.; Gubbins, K. E. Adsorption in Carbon Micropores at Supercritical Temperatures. J. Phys. Chem. 1990, 94, 6061. (16) Matranga, K.; Stella, A.; Myers, A. L.; Glandt, E. D. Molecular Simulation of Adsorbed Natural Gas. Sep. Sci. Technol. 1992, 27, 1825. (17) Bojan, M. J.; van Slooten, R.; Steele, W. Computer Simulation Studies of the Storage of Methane in Microporous Carbons. Sep. Sci. Technol. 1992, 27, 1837-1856.
Gusev et al.
Experimental Section We used a custom-built volumetric1 apparatus to measure adsorption at 308, 333, and 373 K and pressures from several torr to about 23 000 Torr (approximately 3 MPa). The static volumetric method consists essentially of measurements of pressure, volume, and temperature of the gas before and after it contacts the adsorbent. This information is then used to calculate adsorption by means of the gas equation of state. The procedure is repeated at successive pressures. The pressure was measured by two MKS Baratron capacitance pressure gauges, Models 127A and 122A, having ranges of 0.1-1000 Torr and 100-25000 Torr, respectively. Pressures were read by a digital two-channel MKS read-out, Model PDR-C-2C, with an accuracy of 0.15% of reading. There are two thermostats in the system: an air thermostat, which always operates at a temperature of 35 °C, and a metal one operating at any user-chosen temperature in the range 35-200 °C. The air thermostat is a Plexiglas enclosure containing a reference volume, pressure transducers, stainless steel plumbing and valves, a heater, five fans, and four temperature sensors. The second thermostat, which contains the adsorbent within a stainless steel adsorber, is a removable aluminum cylinder with a heater and temperature sensor bonded to its external surface. The aluminum cylinder transfers the heat from the electrical heater to the adsorber more evenly than if the heater were bonded directly to the adsorber. The aluminum cylinder is wrapped in insulating cloth and inserted in a glass Dewar flask to avoid heat losses to the environment. The adsorber is connected to the volumetric system via a 1.5 mm external diameter stainless steel capillary. Inside the aluminum cylinder, the capillaries form coils to allow thermal equilibration of the gas flow before it enters the adsorber. The temperature in each of the thermostats is sensed by an Omega Model 100W platinum resistance thermometer and controlled by a digital Omega Model CN8500 PID controller. This arrangement controls temperature to better than 0.2 K. The dead volumes of plumbing, pressure sensors, and adsorber, both with and without adsorbent, were calibrated by helium expansion from the reference volume with an accuracy of about 0.5%. Because of the manner in which dead volume was measured, adsorption was measured in the Gibbs excess, rather than in the absolute sense. The accuracy of a single adsorption measurement was about 1%. For the whole pressure range, allowing a 10-15 min lag between pressure changes usually was enough to assure that equilibrium was attained. About 20 g of the BPL 6×16 microporous carbon (Calgon Carbon Corp., Pittsburgh, PA) were outgassed at 10 µm of mercury pressure and 373 K for 24 h before measuring each adsorption isotherm. The mass of the outgassed adsorbent was used in all subsequent calculations. The methane (National Compressed Gases, Inc.) had a quoted purity of better than 99.97% and was used as received.
GCMC Method Model. The interactions between fluid molecules were described by truncated Lennard-Jones (LJ) potentials
uff(r) )
{
[( ) ( ) ]
4ff 0,
σff r
12
-
σff r
6
, r < Rc
(1)
r > Rc
The LJ fluid parameters used in the simulation were taken from fits to second-virial-coefficient data18 and were σff ) 0.381 nm and ff/kB ) 1481 K. Each wall of the model graphitic slit pore was represented by a series of stacked planes of LJ atoms. The interaction energy between a fluid particle and a single pore wall at a distance z (measured between the centers of the fluid atom and the atoms in the outer layer of the solid) was described by Steele’s 10-4-3 potential18 (18) Steele, W. A. The Interactions of Gases with Solid Surfaces; Pergamon: Oxford, 1974.
Characterization of Activated Carbons
usf(z) ) 2πFssfσsf2∆
[( ) ( ) 2 σsf 5 z
10
-
σsf z
Langmuir, Vol. 13, No. 10, 1997 2817
4
4
σsf -
]
3∆(0.61∆ + z)3
Pmove ) min{1,exp[-∆Ec(r)/kT]} (2)
where ∆ ) 0.335 nm is the separation between graphite layers and Fs ) 114 nm-3 is the surface number density of carbon atoms in a graphite layer. σsf and sf are solidfluid LJ collision and well-depth parameters. The interaction of a molecule with slit-shaped pore walls is described as the interaction, usf(z), with two such planar surfaces placed a distance H (also measured between centers of the corresponding atoms) apart:
upore ) usf(z) + usf(H - z) The solid-fluid interaction parameters σsf and sf can, in principle, be determined from Henry’s constants of hightemperature gas adsorption Kexp on the uniform basal surface of graphitized carbon blacks,18 as the latter may be expressed in terms of the solid-fluid potential as
Kexp )
∫0∞(exp[-usf(r)/kBT] - 1) dr
(3)
where kB is Boltzmann’s constant and T is the absolute temperature. The Lorentz combination rule was used to estimate the value of σsf ) 0.3675 nm. The value of sf/kBT was estimated, by fitting to available experimental Henry’s law constants for methane adsorption on graphite, to be within the range of 66-71 K (Graphon19) and 44-74 K (Spheron-620). Bojan et al.17 based on their analysis of experimental data (graphitized carbon black P3321), used the value of sf/kBT ) 66 K in the 10-4 potential. The precise value is not crucial to the success of our PSD-determination procedure. Therefore, since the value of sf obtained by the Berthelot rule (sf/kBT ) 64.39 K) lies within the range of the independent estimates given above, we used it for simplicity. The agreement between the parameters of the pairwise potential calculated from Henry’s law constants and estimates based on Lorentz-Berthelot rules was previously noted for rare gases adsorbed on graphite.22 GCMC Simulations. In GCMC calculations the chemical potential of the gas phase is specified, as is its temperature (which, together, permit the pressure to be calculated using the bulk equation of state of the gas). The initial configuration of molecules (which may be one having no molecules at all) is chosen in a rectangular simulation cell with a set of associated position vectors. To generate configurations with the correct limiting probability, a Markov chain of consecutive trials is realized. A suitable scheme23 for generating grand canonical ensemble Markov chains includes trials of three types:24 moving a molecule, creating a molecule, and deleting a molecule. The probability of a move being accepted (as in the Metropolis sampling scheme25) is (19) Specovius, J.; Findenegg, G. H. Physical Adsorption of Gases at High Pressures: Argon and Methane onto Graphitized Carbon Black. Ber. Bunsenges. Phys. Chem. 1089, 82, 174-180. (20) Stacy, T. D.; Hough, E. W.; McCain, W. D., Jr. Adsorption of Methane at Temperatures to 121 °C and Pressures to 650 Atm. J. Chem. Eng. Data. 1968, 13, 74. (21) Sams, J. R. Two-Dimensional Second Virial Coefficients of Methane and Tetradeuteromethane on Graphitic Carbon. J. Chem. Phys. 1965, 43(7), 2243. (22) Steele, W. A. The Interaction of Rare Gas Atoms with Graphitized Carbon Black. J. Phys. Chem. 1978, 82, 7. (23) Adams, D. J. Chemical Potential of Hard Sphere Fluids by Monte Carlo Methods. Mol. Phys. 1974, 28(5). (24) Norman, G. E.; Filinov, V. S. Investigations of Phase Transitions by a Monte-Carlo Method. Teplofiz. Vys. Temp. 1969, 7(2), 233-240.
(4)
where ∆Ec (r) is the change in configurational energy resulting from the move. The probability of a molecule creation being accepted23 is
{
}
1 Pcreate ) min 1, exp[B - ∆Ec(r)/kT] (N + 1)
(5)
where B ≡ µ′/kBT + ln〈N〉 ) ln(fV/kBT) is Adams’ constant, µ′ is the excess chemical potential, relative to an ideal gas having the same density, 〈N〉 is the mean number of particles, V is the volume of the system, and f is the bulk gas phase fugacity. Thus, Adams’ constant B represents the influence of the bulk phase on adsorption in a pore. The probability of a molecule deletion being accepted is
Pdelete ) min{1,N exp[∆Ec(r)/kT - B]}
(6)
In order to preserve microscopic reversibility throughout the simulation, the numbers of attempted deletions and creations should be kept equal. In dense phases (such as an adsorbed phase at high pressure) the acceptance rate for creating particles can be low, so we used either ten creation attempts or ten deletion attempts at each step of the simulations (while allowing one move attempt). The simulation run for each point on the adsorption isotherm was divided in 11 equal blocks; in all but the first, ensemble averages of the number of particles 〈N〉 and potential energy 〈E〉 were calculated. GCMC configurations generated during the first block were used to relax the system from its initial nonequilibrium configuration and were then discarded. The number of moves in each block was chosen to be sufficiently high to ensure statistical independence of the averages taken in each block. After the whole run was performed, standard deviations of the energy and the particle numbers were estimated for the individual blocks. In all cases, the next isotherm point was started off from the final configuration of the previous run. Standard deviations of about 1% were considered sufficiently small. To speed up the calculations, the wall potential was tabulated at intervals of 0.01σff and Newton-Gregory forward differencing was used during the run to calculate its value. As an independent test of the GCMC sampling scheme (the results of which depend on the system size if the system is too small) we carried out independent canonical Monte Carlo simulations for the density of the highest density point in our GCMC simulations. We then determined the chemical potential in the canonical simulations at ever-increasing system sizes (but the same density) using Widom’s particle insertion method26 and compared the value with that used as input to our GCMC simulation. Convergence of the two simulation approaches indicated the smallest adequate system size for the simulations, which turned out to be surface slabs of 10σff square. The canonical Monte Carlo simulations used for sampling the chemical potential were themselves independently compared with MD simulations of methane adsorbed in carbon slit pores,17 which also used Widom’s particle insertion algorithm, and showed good agreement. Even so, if the size of simulation cell determined at high pressures were also used at low pressures, a considerable number of grand ensemble configurations (25) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys. 1953 21, 1087-1092. (26) Widom, B. Some Topics in the Theory of Fluids. J. Chem. Phys. 1963, 54, 5237-5247. Widom, B. Potential Distribution Theory and the Statistical Mechanics of Fluids. J. Phys. Chem. 1982, 86, 869-872.
2818 Langmuir, Vol. 13, No. 10, 1997
Figure 1. Methane adsorption isotherms on activated carbon BPL-6 (a, mmol/g, symbols) and excess densities simulated by the GCMC method (Fφ, mmol/cm3, lines) in slit-shaped pores of various widths at 308 K. Each isotherm is labeled with the corresponding pore width, H, in units of σff.
with no particles would have been realized, affecting microscopic reversibility. Therefore, the size of the simulation cell was increased at low pressure so that the numbers of attempted particle creations and deletions were always maintained equal. Hence, the size of the simulation cell was adjusted with regard to pressure and pore width within the range 10-1500σff. The potential cutoff Rc was, however, kept constant at 5σff. GCMC simulations were run for 5 × 105 to 1 × 107 configurations at each isotherm point, which took between 10 and 60 min on Sun Sparc or HP workstations (we note that this is on the order of the time needed to measure one point of the adsorption isotherm in our physical experiment, too!). To calculate bulk methane thermodynamic properties, a virial-type equation of state27 was used in both simulations and experiment. Results The experimental excess adsorption isotherms (Figure 1 depicts those measured at 308 K) are steeply rising curves flattening at sufficiently high pressure and are of type I according to the IUPAC classification.1 Such isotherms are characteristic of supercritical fluid adsorption on a predominantly microporous solid. GCMC adsorption isotherms were simulated in 40 pores ranging from 1.65 to 15 σff at 308.15 K at intervals of either 0.01, 0.1, 0.5, or 1 σff (see the caption of Figure 3 for details) and at pressures corresponding to those of the experimental isotherm. We have determined empirically that the pore width on the lower end of this range (1.65 σff) effectively represents the smallest pore in which adsorption of model molecules was still possible. Thus, the value of 1.65 σff was used to estimate the excluded pore width for adsorption of methane molecules at all pore sizes as 1.65 - 1 ) 0.65 σff ) 0.248 nm. In the following, the excess density of the adsorbate was determined as number of particles per unit accessible volume of the pore, less the density of the bulk phase (the latter being determined from the equation of state). Figure 1 depicts some of the simulated isotherms for different pore widths (lines, plotted as adsorbate excess number density, FΓ) and the experimental isotherm (points, plotted as Gibbs excess adsorption, a), as a function of the bulk pressure, P. The simulated isotherms, in common (27) Sychev, V. V.; Vasserman, A. A.; Zagoruchenko, V. A.; Spiridonov, G. A.; Tsymarny, V. A. Thermodynamic Properties of Methane; National Standard Reference Data Service of the USSR 1986.
Gusev et al.
Figure 2. Methane number density profiles in slit-shaped pores of various widths at P ≈ 3 MPa, T ) 308 K. Each line is labeled with the corresponding pore width, H, in units of σff.
with the experimental one, all exhibit type I behavior. In pores having widths from 1.65 to 1.9 σff, increased pore width causes increased adsorption at all pressures; these are pores where only one layer of adsorbate molecules can exist. At higher pressures the corresponding adsorption isotherms become rather flat due to strong confinement in these narrow pores. Changes in the pore size have the strongest effect on the amount adsorbed in these small pores. In pores having widths above 1.9 σff (the optimal size to accommodate exactly one layer of methane molecules) increased pore width causes isobaric adsorption instead to fall, evidently due to the decrease in the wall potential. At the same time, the flattening of these isotherms in the high-pressure region (above 0.7 MPa) becomes considerably less pronounced. This decrease in adsorption on increasing the pore width persists up to our maximum value of 15 σff at low pressures (below 0.7 MPa). At higher pressures, however, a second layer of molecules may build up in pores wider than 2.6 σff. This leads to a temporary increase in adsorption on increasing the pore width and, therefore, to a crossover of some single-pore isotherms. The pore width of 2.9 σff is optimal for two layers under the conditions simulated, and further increase in the pore size is, once again, accompanied by a decrease in adsorption. Formation of up to four distinct molecular layers is possible in pores of 3.0 to 15 σff within the studied pressure range. This process, however, does not lead to changes in the form of the isotherms, as dramatic as those caused by the first and second layer formation. Therefore, the amount adsorbed decreases monotonically on increasing the pore width for pores wider than 2.9 σff at all studied pressures. A similar oscillatory variation of the adsorption excess with changing pore size has been observed previously15 in NLDFT calculations. The differences in behavior described above are important for characterization and, in order to present their impact on the performance of the method, we use the technique of comparative plots.1 Figure 3 represents the GCMC simulated excess adsorption isotherm densities per unit surface area in pores of different widths, versus that which occurred in the largest pore studied at the same pressure, i.e., 15 σff (5.7 nm). Absolute density profiles of the adsorbate in the pores (see Figure 2) at our highest pressure (3.03 MPa) become equivalent to one another at pore widths above 6 σff in the region close to the walls, while in the centers of the pores they approach the bulk value. This suggests that the maximum pore width of H ) 15 σff studied was indeed sufficiently large to be considered as two independent walls rather than a single pore.
Characterization of Activated Carbons
Figure 3. Comparative plot of adsorption (dimensionless excess adsorption vs dimensionless excess adsorption in a 15 σff pore at the corresponding pressure) in carbon slit-shaped pores of various widths at 308 K, based on GCMC simulations. All pores are divided into groups as explained in the text. The pore widths for which isotherms are presented are (in units of σff) as follows: 1.65-1.69 in steps of 0.01; 1.7-3.4 in steps of 0.1; 3.5-4 in steps of 0.5; 5-6 in steps of 1.0; 7-10.5 in steps of 0.5; 11-15 in steps of 1.0.
As the adsorption is expressed in terms of excess per unit surface area in a comparative plot, the influence of the nonadsorbed gas in the middle region of the larger pores is excluded. Therefore, the comparative plot allows us to compare the adsorption to that on an open surface. We see from Figure 3 that all of our pores may be classified by size roughly into three groups. The first group includes the narrowest pores of 1.65-2.5 σff, where no more than one layer of molecules can be adsorbed. Adsorption in the pores of this group is qualitatively different from that on an open surface. The second group includes the pores of 2.5-6 σff, where it is possible to build up to four adsorbed layers. The influence of both walls on the adsorption is enough for the isotherms to be different from that in the pore of width 15 σff. The density profiles in 6 σff and 15 σff pores are still different from one another in the center of the pore (Figure 2) but this difference is already just enough for corresponding comparative plots to be different even considering the uncertainty of the numerical simulations. The third group of pores wider than 6 σff cannot be distinguished from the 15 σff wide pore on the comparative plot; thus the corresponding adsorption isotherms can be overlaid by uniform scaling at all pressure points. These pores become essentially equivalent in their excess adsorption properties since the wall potentials no longer overlap in the center of the pore and the cooperative effect of fluid-fluid interaction is relatively small. In this manner, we can define a limiting pore width, Hl, beyond which the pore should be considered equivalent to two open surfaces with regard to a supercritical fluid. Hl is defined as that pore width where corresponding comparative plots coincide within simulation uncertainty. At 308 K, Hl ≈ 6σff (2.3 nm). This operational definition of Hl takes into account not only the effect of the wall potential on adsorption but also the fluid-fluid cooperative adsorption increase (which, generally, is also a function of the pore size). Therefore, Hl is a function of the temperature, the pressure, and the molecular potential parameters of both adsorbing fluid and porous solid. A closer look at the comparative plot allows us to identify the limit of sensitivity of the supercritical methane adsorption to the accessible pore size H’ as a function of pressure. The pores belonging to the first group are sensitive to the pore width within any arbitrarily small subregion of the pressure range. (Note, however, that if
Langmuir, Vol. 13, No. 10, 1997 2819
Figure 4. Limiting accessible pore size, Hl′, to which the GCMC-based method is sensitive, as a function of pressure, for methane on BPL-6 carbon at 308 K. The line is drawn through the simulation-based points as a guide to the eye.
measurements were limited to the Henry’s law region alone, it would not be possible to distinguish between, say, one strongly adsorbing pore and two less strongly adsorbing ones.) Thus, in principle the characterization method at 308 K will always be effective within the pore size range covered by this pore group, i.e. below H’ ≈ 0.7 nm (H ) 2.5 σff), provided that the intervals, into which the pore size range was divided, are not too small. Adsorption in some of the pores in the second group may become insensitive to the pore size if too narrow a pressure range is employed. The pressure dependence of this sensitivity limit, as extracted from our available simulation data, is shown in Figure 4. We note that an increase in the pressure is accompanied by a relatively steep increase in the pore sizes detectable by the method for pressures of up to about 1 MPa. Therefore, the pressure range of up to 3 MPa and temperature 308 K employed in this study allow us to cover the entire microporous (up to 2 nm) region. Put another way, adsorption experiments that do not go as far as about 1 MPa for this system will not be sensitive to the entire micropore region. The wider pores (mesopores) can only be characterized by their surface area; the flat character of the sensitivity curve in Figure 4 at higher pressures suggests that a considerable increase in pressure would be required to characterize the pore size distribution of mesopores using this method. The problem of fitting simulated isotherms to an experimental one measured at the same temperature is usually considered as the least-squares sense solution to the equation
Rmn‚Vn ) Am
(Vj g 0)
(7)
Here Rmn is an (m × n) matrix of adsorbate excess densities Fijsim simulated at the ith experiment pressure in the jth slit pore, Vn is a n-vector (the solution) of volumes Vj, and Am is an m-vector of experimental data aiexp (experimental adsorption isotherm). It is recognized that the problem represented by eq 7 is ill-posed in nature, and a number of special regularization algorithms (see ref 28 for a review) have been suggested to deal with it. Non-negativity constraints, {Vj g 0}, evidently should be used in conjunction with eq 7 as a natural means of stabilizing the solution28 of the problem, as unconstrained solution leads to strong oscillations (giving unphysical negative contributions to the pore size distribution) about the pore volume axis in the pore size distribution. Coping with the ill-conditioning of the matrix Rmn is a key step in the regularization of the solution. This
2820 Langmuir, Vol. 13, No. 10, 1997
property can be characterized by the ratio of the largest of the singular values to the smallest of them, the socalled condition number.29 A matrix Rmn is singular if its condition number is infinite, and it is ill-conditioned if its condition number is too large. We found that, while building the matrix Rmn by adding to the problem pores of increasing widths one at a time, the condition number started increasing rapidly after a pore of width approximately equal to Hl was added. The matrix Rmn then became progressively ill-conditioned. This is precisely because of the fact, discussed above, that the pores with H > Hl carry essentially no additional information about adsorption within pores. This also suggests that only simulated isotherms for pores having H < Hl should be used for finding the pore size distribution, while the surface area of the pores with H > Hl can be estimated using the corresponding simulated isotherms (of course, the total surface area can also be calculated from the above). This division of the pore size range into two regions makes the whole solution to the characterization problem eq 7 more stable for a given set of the simulated isotherms. To solve eq 7 with constrains, we used the SVDNNLS method. It is based on the non-negative least squares (NNLS) technique described by Lawson et al.30 An iterative procedure is used to partition the matrix Rmn in a finite number of steps, so that the Kuhn-Tucker conditions (see ref 30) are satisfied and the resulting submatrix Rmn′ (n’ g n) yields a non-negative solution to eq (7). Further regularization is achieved by combining the NNLS method with singular value decomposition (SVD),29 which controls the singularity of the Rmn′ submatrix in the solution of system of linear equations at each step. This is especially important if more than one pore in the H > Hl region appears in the solution with a positive weight. Figure 5 shows the integral simulated adsorption isotherm fitted to the experimental adsorption isotherm of methane on BPL-6 activated carbon at 308 K. The quality of the fit is quite good over the whole pressure range, with an average relative error of about 1%. Figure 6 depicts the corresponding pore size distribution. On the basis of this PSD, the BPL-6 activated carbon is a predominantly microporous adsorbent having an overall accessible (based on the above definition of the excluded pore width) micropore volume of about 0.42 cm3/g. This is within the scatter of other estimates.3,9,31 We note, however, from our own and others’9 experience that the pore structure of the activated carbon BPL-6 will vary considerably from batch to batch. The micropores resulting in non-negative weights in the pore size distribution found for the activated carbon BPL-6 were characterized by comparable singular values due to the strong influence of confinement on adsorption in these pores. The “spikiness” of the pore size distribution depicted in Figure 6 is because the SVDNNLS method results in some of the weights being exactly zero, while the rest are non-negative. The resulting pore size (28) Mamleev, V.; Zolotarev, P.; Gladyshev, P. Heterogeneity of the Sorbents (in Russian); Nauka: Alma-Ata, 1989. Von Szombathely, M.; Brauer, P.; Jaroniec, M. J. The Solution of Adsorption Integral Equations by Means of the Regularization Method. Comput. Chem. 1992, 13, 17. Jagiello, J. Stable Numerical Solution of the Adsorption Integral Equation Using Splines. Langmuir 1994 10, 2778-2785. (29) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in Pascal. The Art of Scientific Computing; Cambridge University Press; Cambridge, 1989; p 61. (30) Lawson, C. L.; Hanson, R. J. Solving Least Squares Problems; Englewood Cliffs, NJ, Prentice-Hall: Englewood Cliffs, NJ, 1974. (31) Reich, R.; Ziegler, W. T.; Rogers, K. a. Adsorption of Methane, Ethane and Ethylene Gases and Their Binary and Ternary Mixtures and Carbon Dioxide on Activated Carbon at 212 - 301 K and Pressures to 35 atm. Ind. Eng. Chem. Process Des. Dev., 1980, 19, 336.
Gusev et al.
Figure 5. Fit, using the SVDNNLS technique, of methane experimental adsorption isotherm with a set of GCMC isotherms simulated in slit pores at 308 K. The inset graph contains the same information on a log-log plot in order to demonstrate the equally high quality of the fit at low pressure.
Figure 6. Pore size distribution, obtained from a fit (see Figure 5) of the experimental adsorption isotherm for methane on BPL-6 carbon at 308 K to a linear combination of GCMCsimulated isotherms. The PSD is presented as a histogram of five pore sizes, the form in which it was used as input to the predictive simulations.
distribution may be smoothed using, e.g. Tikhonov regularization (see ref 28). This, however, requires the use of a subjectively chosen regularization procedure and, generally, should lead to a worse fit. Instead, we focus on the possibility of using the pore size distribution obtained above for predicting the adsorption properties of the adsorbent. The outcome of this approach provides insight into the physical significance of the pore size distribution that we obtain. The pore size distribution we obtained at 308 K was a set of five (H ) 1.9, 2.3, 2.6, 2.9, and 4 σff) pores with associated weights. The GCMC simulation was used once again to calculate adsorption in these pores and to generate the integrated methane adsorption isotherm at two higher temperatures 333 and 373 K. Figure 7 shows the comparison of these predicted isotherms with our own experimental measurements of methane adsorption on BPL-6 carbon at the same temperatures. The quantitative agreement is excellent in both instances and serves as a stringent test of selfconsistency of the method. We note that even though it is possible to provide alternative definitions of accessible volume,13 this issue is not crucial when using simulation techniques to predict the adsorption properties of the same gas, since the pores in these methods are unambiguously characterized by the distance between centers of the outer atoms in the opposite walls.
Characterization of Activated Carbons
Langmuir, Vol. 13, No. 10, 1997 2821
Conclusions
Figure 7. GCMC prediction of methane adsorption on BPL-6 activated carbon using the pore size distribution derived from the 308 K adsorption isotherm: GCMC integral isotherms, lines; experiment, squares and circles.
It is important to note also that other pore models may be used for the simulation of the initial block of isotherms, such as finite-length slits32 or those having triangular cross section (to represent pores of a crushed activated carbon16). (32) Vlasov, A. I.; Bakaev, V. A.; Dubinin, M. M.; Serpinskii, V. V. Monte Carlo Modeling of the Adsorption of Argon on Activated Carbons. Dokl. Akad. Nauk SSSR 1981, 260, (4), 904-6; Dokl. Phys. Chem. (Engl. transl.) 1982, 260(4), 878-880.
We have shown that supercritical adsorption of methane, in conjunction with GCMC simulations, can be used for the self-consistent interfacial characterization of a microporous solid. We used potential parameters obtained from the bulk properties of the studied solid and fluid, and the Lorentz-Berthelot combining rules to estimate solid-fluid potential parameters. Under the studied conditions (temperature 308 K and pressures up to 3 MPa), the pore size distribution of micropores can be assessed, while larger pores should be characterized by their surface area. The range of micropore widths within which the method is effective for assessing the pore size distribution can be widened if a larger pressure range is used. The pore size distribution of the activated carbon BPL-6, determined by the method at 308 K has been successfully used to predict the adsorption of methane on the same activated carbon at temperatures more than 60 K higher. It is important to note that for both adsorption characterization of the interface properties of porous solids and prediction of adsorption, only data for one singlecomponent adsorption isotherm, along with the molecular properties of the bulk solid and fluid, were required. Acknowledgment. This work is supported in part by the U.S. National Science Foundation through Grant No. CTS-9215604. LA960421N
