Characterization of Porous Materials by Gas Adsorption at Ambient

J. Phys. Chem. B 2001, 105, 1403-1411

1403

Characterization of Porous Materials by Gas Adsorption at Ambient Temperatures and High Pressure M. B. Sweatman and N. Quirke* Department of Chemistry, Imperial College, South Kensington, London, SW7 2AY, U.K. ReceiVed: September 16, 2000; In Final Form: December 11, 2000

Activated carbons are amorphous microporous graphitic materials formed (or activated) from a variety of organic precursers using high-temperature steam or acids. The possibility of modifying the activation process to create smaller or larger pores, from nanometers to microns in width, tailored to adsorb specific molecules or classes of molecule make activated carbons important industrial adsorbents. For the physical chemist they pose the challenge of understanding how gases adsorb in graphitic nanopores, that is, in restricted geometries, and of using that understanding to improve their characterization. One aim is to make predictions concerning the adsorption properties for a given material, i.e., a specific microstructure. In this paper we use molecular simulation methods, including Gibbs ensemble simulation, to determine new molecular models for nitrogen, methane, and carbon dioxide and grand canonical ensemble simulation (together with new experimental data for the adsorption of these gases on Vulcan at 298 K and up to 20 bar) to generate new adsorption isotherms for model carbon pores. These new data are used to calculate pore-size distributions for typical activated carbons. We find that at these temperatures the high-pressure carbon dioxide measurements reveal more micropore structure than the measurements of nitrogen and methane up to 20 bar, or carbon dioxide measurements up to 1 bar. We also investigate the ability of pore-size distributions (PSDs) obtained from one gas to predict the adsorption of the other gases at the same temperature. We find that carbon dioxide PSDs are the most robust in the sense that they can predict the adsorption of methane and nitrogen with reasonable accuracy.

1. Introduction Porous carbonaceous materials are used routinely in many industrial processes including gas separation. They are produced from a range of raw materials, such as peat, wood, lignite, anthracite, fruit pits, or shells. The organic material is converted into activated carbon using steam (temperatures above 1200 K) or acids (temperatures above 600 K). The activation process creates a very large adsorption capacity through the formation of an amorphous microporous structure and so can yield high separation factors for some adsorbate mixtures. The possibility of modifying the activation process to create smaller or larger pores, from nanometers to microns in width, tailored to adsorb specific molecules or classes of molecule make activated carbons important industrial adsorbents. They have been the focus of much research, with interest in both their characterization and their adsorption properties. For the physical chemist they pose the challenge of understanding how gases adsorb in graphitic nanopores, that is, in restricted geometries, and of using that understanding to improve their characterization. One aim is to make predictions concerning the adsorption properties for a given material, i.e., a specific microstructure. In this paper we develop accurate molecular models of gas adsorption in graphitic pores, investigate gas adsorption as a function of pore width and gas pressure (pore phase diagrams), and use this information to investigate the extent to which gas adsorption data taken at room temperature and high pressure can produce reliable descriptions of the micropore structure of typical activated carbons. Amorphous materials are usually characterized using nitrogen adsorption isotherms at 77 K taken at pressures up to 1 bar to * Corresponding author.

obtain pore size distributions. However, higher temperatures, higher pressures, and different adsorbates could also be used. In the present work we characterize three representative activated carbons using carbon dioxide, methane, and nitrogen adsorption isotherms at 298 K up to 20 bar. The use of room temperatures may alleviate the problem of achieving true equilibrium when isotherms are taken for nanoporous materials at low temperature (77 K). The characterization of porous materials usually involves the approximate solution of the adsorption integral

∫

V(P) ) A f(w)V(w,P) dw

(1)

where V(P) is the experimentally determined excess volume of adsorbate (at STP) per gram of material, f(w) is the required pore size distribution, and V(w,P) is the excess average density of adsorbate at pressure P in a pore of size w. The integral is over all pore sizes, w. Equation 1 is a Fredholm equation of the first kind, and as such it can present many difficulties. Depending on the form of the “kernel”, V(w,P), and the adsorption isotherm, V(P), there can be from zero to an infinity of solutions for f(w).1,2 The solution of eq 1 is an ill-posed problem, and where there are many solutions it indicates that the problem is under-constrained. Nevertheless, our task is to find useful solutions to eq 1 in the sense that physical properties of microporous carbons, and the adsorption of gases in particular, can be reliably predicted. For example, in this work we are content to be able to predict the adsorption of gases from a characterization performed with a particular gas. It would then be possible to predict the

10.1021/jp003308l CCC: $20.00 © 2001 American Chemical Society Published on Web 01/30/2001

1404 J. Phys. Chem. B, Vol. 105, No. 7, 2001 selectivity and separation factors of a microporous carbon from the experimental adsorption of a single gas. Several methods for solving eq 1 are known and have been used in previous studies, including best-fit methods3,4 and matrix methods.5,6 The best-fit methods are essentially trial-and-error methods where very many trial functions are tested, with the best-fit trial function taken as the solution. They can employ optimization procedures to direct the trial function selection toward better solutions. The matrix methods amount to solving a system of linear equations by matrix inversion. With both methods, additional constraints are often required to force more physically appealing or acceptable solutions, including constraints on the smoothness of the solution function and the range of w (see refs 1, 5, and 6, and references therein). In this work we use a best-fit method combined with a simulated annealing optimization method2 to choose trial functions. Any solution method for eq 1 requires that the kernel, V(w,P), is known. The first step is to identify a pore geometry and associated measure, w. We invoke the standard7 idealized carbon slit-pore model with w the chemical width between carbon atoms in the first layer of each semi-infinite wall. Thus f(w) describes a poly-disperse array of slit pores. Given a fixed geometry, the function V must be calculated for all relevant values of w and P. The conventional approach7 has been to employ macroscopic approximations such as the Kelvin equation or semiempirical methods such as the Horvath and Kawazoe correlation.8 For materials containing sub-nanometer pores these methods can be inaccurate. However, density functional theory (DFT) and molecular simulation, being fully statistical mechanical, are ideally suited to this regime. In the present study we wish to model our adsorbates as nonspherical molecules with electrostatic interactions so that their behavior in micropores is accurately captured. Given the limited accuracy of DFT in this context, we use molecular simulation to generate the required adsorption isotherms. We require accurate models of both adsorbate-adsorbate and adsorbate-solid interactions. In this work we use Gibbs ensemble simulation9 to fine-tune adsorbateadsorbate interactions to reproduce published experimental data for gas-liquid coexistence of each adsorbate.10 We also use grand canonical ensemble simulation11 to fine-tune the adsorbatesolid interaction to reproduce new experimental data12 for the adsorption of each adsorbate on Vulcan13 (the standard lowsurface-area carbon) up to 20 bar at 298 K. Finally, databases are constructed for each adsorbate using grand canonical ensemble simulation from which V(w,P) is determined by linear interpolation. The work in this paper builds upon previous work by Scaife et al.4 who used the poly-disperse slit-pore model to characterize amorphous carbons. Scaife et al. used an established nonlocal density functional theory to build the databases V(w,P) and they also employed a best-fit method to solve eq 1. They examined PSDs generated by adsorption of nitrogen and argon at 77 K and carbon dioxide at 293 K up to 1 bar. Ravikovitch et al.5 also used the poly-disperse slit-pore model to characterize amorphous carbons. They used nitrogen and argon at 77 K and carbon dioxide at 273 K as adsorbates. In essence, they used the same density-functional methods as Scaife et al. to generate V(w,P). For carbon dioxide they also used grand canonical ensemble simulations, with a published model14 for gas-gas interactions, to generate V(w,P) up to pressures of 1 bar. They found reasonable agreement between pore-size distributions determined with the different gases on various porous carbons. They also found reasonable agreement between PSDs of solely microporous carbons generated with the DFT

Sweatman and Quirke and simulation databases for carbon dioxide. Ravikovitch et al. use a matrix regularization method to solve eq 1. Other authors have investigated the possibility of predicting adsorption isotherms at various temperatures from a slit-pore PSD obtained by analysis at a different temperature. For carbon dioxide15 at a range of sub- and super-critical temperatures, Samios et al. found that reasonable agreement could be obtained for one microporous carbon, but not another suspected of having embedded polar sites. For methane,6 Gusev et al. obtained good agreement for one microporous carbon over a 65 K range of super-critical temperatures. As an alternative to fixed geometry models, some work has been published concerning the use of less constrained representations of the pore space.16 In particular, Gubbins and coworkers17 have used a reverse Monte Carlo method to generate random assemblies of idealized basal planes of graphite consistent with input carbon-carbon radial distribution functions. However, such approaches have the disadvantage of introducing many more parameters to describe the pore space. We compare the micropore characteristics of three activated carbons (samples A, B, and C where we know that sample A is AX21)18 determined within the slit-pore model using simulation techniques for three adsorbates: carbon dioxide, methane, and nitrogen at ambient temperatures. We study the importance of obtaining high-pressure adsorption measurements at these temperatures. We find the following: (1) In contrast to previous modeling work, the standard lowsurface-area carbon, Vulcan, is best described by a (micro) pore size distribution as for high-surface-area carbons. (2) At ambient temperatures the high-pressure measurements of carbon dioxide reveal micropore structure not seen with the other gases or with measurements up to 1 bar. (3) The PSDs predicted by carbon dioxide adsorption are “universal” in the sense that the carbon dioxide PSD can be used to predict the adsorption of methane and nitrogen with reasonable accuracy. The remainder of the paper is organized as follows. Section 2 describes our methods, while Sections 3, 4, and 5 describe, discuss, and summarize our results. 2. Methods The method used to obtain pore size distributions comprises four stages: (1) determine a molecular model for the gas-gas interaction, (2) determine a molecular model for the gas-solid interaction, (3) generate a database of local isotherms V(w,P), and (4) invert the adsorption integral, eq 1. Section 2.1 describes the formulation of the molecular models and Section 2.2 discusses the solution of eq 1. In all simulations the Lorentz-Berthelot combining rules have been used for Lennard-Jones interactions, i.e., σij ) (σii + σjj)/2 and ij ) (ii + jj)1/2. 2.1. The Molecular Models. Gibbs ensemble simulation has been used to predict the liquid-vapor phase equilibria of methane, nitrogen, and carbon dioxide. We have generated new molecular models for these gases by adjusting model parameters to give good agreement between simulation and experimental data. Repulsive-dispersive interactions are modelled by 3, 2 and 1 Lennard-Jones sites in carbon-dioxide, nitrogen and methane, respectively (the carbon-dioxide model is linear). The electric quadrupoles of nitrogen and carbon dioxide are represented by partial charges. A dynamic pair-potential cutoff equal to the minimum of half the instantaneous simulation box length and 1.5 nm is used in all Gibbs ensemble simulations. The long-

Characterization of Porous Materials

J. Phys. Chem. B, Vol. 105, No. 7, 2001 1405

TABLE 1: Model Parameters for Adsorbate-Adsorbate Interactions

2

VS ) 2πFc ∆gs σgs

parameter

N2

CH4

CO2

σff (nm)

0.334 (0.33)

0.37 (0.374)

ff/kB (K)

34.7 (36.0)

148.75

lx (nm)

(0.05047

0

lq (nm)

(0.0847 (0) (0.1044 (0) 0.373 (0) -0.373 (0)

0

C:0.275 (0.2757) O:0.3015 (0.3033) C:28.3 (28.129) O:81.0 (80.507) C:0 O:(0.1149 0 (0.1149 0.6512 -0.3256

q (e)

0

Bracketed numbers indicate values obtained from previous work (refs 14, 21, and 22).

TABLE 2: rms Percentage Error between Experimental Results (ref 10) and Gibbs Ensemble Simulation Results for Coexisting Properties of This Work property

N2 (90 K -115 K)

CH4 (119 K-171 K)

CO2 (250 K - 285 K)

Fl Fg Pco

1.0% 8.8% 4.0%

1.2% 10.7% 6.7%

0.5% 5.8% 6.3%

range (beyond the cutoff) dispersive contribution to the total energy of each configuration, ULR, is calculated by11

ULR ) 2π

∑i Ni∑j Fj ∫r drr2φij(r) ∞

(2)

c

for each box, where the indices sum over each Lennard-Jones site, the average density of Lennard-Jones site, j, Fj, is set equal to the experimental result for the respective coexisting phase, Ni is the number of sites of species i in the simulation box, and φij is the Lennard-Jones pair-potential. We neglect the longrange quadrupole-quadrupole contribution to the total energy since it is typically less than 0.1% of the total energy for a lattice-like initial configuration at 298 K. The Lennard-Jones parameters of the methane molecular model were chosen initially to force agreement between the location of the liquid-gas critical point determined by experiment and by the Gibbs ensemble simulations of Panagiotopoulos19 for the Lennard-Jones fluid. The parameters for nitrogen were initially set to those given by J. Delhomelle20 also obtained by fitting results of Gibbs ensemble simulations to experimental data. However, we model nitrogen’s quadrupole by the same arrangement of partial charges as in the Kuchta and Etters model.21 The model for carbon dioxide was initially set to the EPM2 model of Harris and Yung14 also obtained by Gibbs ensemble simulation. For each adsorbate, the Lennard-Jones site parameters only were adjusted until satisfactory agreement with experimental data10 for the liquid-gas coexisting densities and pressure were obtained. Table 1 reports the molecular models for methane, nitrogen, and carbon dioxide resulting from the Gibbs ensemble simulations. In Table 1, lx is the distance of each site, and lq is the distance of each partial charge, from the center-of-mass of the molecule. The bracketed numbers are the values of the parameters prior to adjustment. The root-mean-square percentage error between experimental and simulation results are given in Table 2. In this table, Fl, Fg, and Pco denote the liquid density, gas density, and pressure (of the gas phase for simulation data) of the coexisting phases. The agreement between results using the new models and experiment is good. We assume that the inner surface of a pore in the amorphous carbon samples adsorbs in the same way as a Steele potential22

(( ) ( ) 2 σgs 5 z

10

-

)

σgs4 σgs 4 z 3∆(z + 0.61∆)3 (3)

where the number of carbon atoms per unit cell, Fc ) 114 nm-3, the graphite plane spacing, ∆ ) 0.335 nm, σgs and gs are obtained from the Lorentz-Berthelot rules, σss ) 0.34 nm and ss are determined by fitting the results of grand canonical simulations for excess adsorption to new experimental data for excess gas adsorption on Vulcan up to 20 bar at 298 K.12 Vulcan is a low-surface-area carbon that has previously been used to define a reference carbon surface. Previous work assumed that the pore space in Vulcan consists of wide pores (that can be modeled as infinitely wide slit pores) and that the surface area of Vulcan is known approximately from the BET method.22 In fact, it is not possible to unambiguously define the surface area of real materials since our notion of smooth surfaces breaks down at the atomic scale. There are equivalent difficulties with the notions of pore width and volume. Thus, any microscale definition of geometry must include, and be defined with respect to, the interaction of the material with the probe. In this work, a “chemical” definition of pore width, w, is used equal to the “physical” width minus 0.24 nm.23 The physical width is the distance between atom centers in the first layers of the pore walls. We enforce this empirical definition for each gas and adsorbent.24 The BET method is popular for measuring surface area in porous materials even though its accuracy is limited. It is based on many assumptions, including the notion of the surface area occupied by a nitrogen molecule in a completely filled adsorbed monolayer at 77 K. For microporous materials the BET method can seriously over-estimate25 or under-estimate the surface area since it does not properly take account of overlapping surface fields or of cooperative adsorption phenomena, such as capillary condensation, that disguise the monolayer adsorption capacity. However, with the assumption of sufficiently wide pores and a BET surface area of 72.64 m2/g for Vulcan,4 irrespective of temperature and adsorbate, ss can be varied until simulated isotherms best fit the experimental isotherms. A different solid interaction strength, ss, is obtained for each gas, with individual gas site-solid parameters, gs, determined by application of the Lorentz-Berthelot rules. In the present work we do not account for any possible interaction between partial charges and the solid surface. Again, a cutoff of 1.5 nm was used for all gas-gas interactions, with long-range corrections applied only to dispersion interactions. Figure 1 shows the fit to experimental data. The best-fit gassolid interaction parameters are given in Table 3. It is clear that for each gas the experimental isotherms are more curved than the corresponding simulation isotherms and the fit is poor. In our previous work4 this fit had been attempted up to 1 bar only and since the adsorption up to 1 bar is almost linear the resulting fits were accurate. Given that the gas-gas interaction parameters have been demonstrated to be accurate using Gibbs ensemble simulation, we suggest that the Vulcan surface must exhibit a significant degree of inhomogeneity. That is, the Vulcan surface must present significant (1) adsorption sites with various energies, and/or (2) edge defects or embedded micropores. The fit up to 20 bar in Figure 1 attempts to include these effects in a superficial manner. To improve these fits requires a more detailed model of the Vulcan surface. Energetic heterogeneity might arise as a result of the presence of particles other than carbon embedded in the surface. This

1406 J. Phys. Chem. B, Vol. 105, No. 7, 2001

Sweatman and Quirke and surface areas of the porous carbons analyzed in this work are also likely to be correspondingly different. 2.2. Inverting the Adsorption Integral. The final stage in deriving a pore size distribution from the measured adsorption isotherms is to invert the adsorption integral (eq 1). In this work we employ a procedure similar to that described in previous papers.3,4 We assume f(w) to be a sum of log-normal functions

f(w) )

Figure 1. Fit to adsorption on Vulcan at 298 K. Lines with un-filled symbols are experiment, lines with filled symbols are the fits generated using Grand canonical ensemble simulation. The size of the symbols is greater than the error in experiment and simulation.

TABLE 3: Initial Model Parameters for Gas-Solid Surface Interactions parameter

N2

CH4

CO2

σsf (nm)

0.337

0.355

sf/kB (K)

25.0

57.2

ss/kB (K)

18

22

C:0.308 O:0.321 C:25.0 O:42.2 C:22 O:22

situation requires additional information, such as the distribution and species of non-carbon particles, and is not considered further. Alternatively, investigation of the gas-solid potential obtained by explicitly summing contributions from individual carbon atoms in a surface indicates that significant energetic heterogeneity arises automatically22 if the atoms in the surface layers are sufficiently separated. This assumes that the solid surface comprises a semi-infinite stack of identical lattice-like layers. In this case an adsorbate molecule can experience a much stronger interaction with the surface if it is positioned between surface atoms (in directions parallel to the surface) than if it is positioned directly over a surface atom. However, the distance required between carbon atoms in each surface layer is large in this case and would lead to an unstable arrangement. We have attempted to improve the model of the Vulcan surface by allowing it to be represented by a distribution of poly-disperse slit pores. Thus our treatment of Vulcan is entirely consistent with our treatment of other microporous carbons and is the simplest modification to the original model. We fix ss for carbon dioxide by requiring that the surface area calculated from the resulting PSD is approximately equal to 70 m2/g. Remembering that our goal is to be able to predict the adsorption of one gas from the adsorption measurements of a different gas, we then fix the other values of ss for methane and nitrogen by requiring that the carbon dioxide PSD for Vulcan yields predicted isotherms via eq 1 for methane and nitrogen that are in approximate agreement with experimental isotherms. Thus, each ss is consistent in the sense that a single PSD (obtained from the carbon dioxide adsorption isotherm) that approximately solves eq 1 for each gas has a surface area close to 70 m2/g. Experimental data26 supports this new model for Vulcan. Adsorption isotherms of nitrogen at 77 K on Vulcan and AX21 are very similar up to 0.001 bar if the Vulcan isotherm is scaled by a factor of 29. This indicates that Vulcan and AX21 share a similar microporous character. All PSDs and surface areas determined in this work are obtained with respect to the BET surface area of Vulcan. If, in fact, the surface area of Vulcan is smaller or larger than the BET value given above, then the PSDs

∑i Ri/(γiw(2π)1/2) exp(-(ln w - βi)2/(2γi))

(2)

and vary the free parameters Ri, βi, and γi until a satisfactory fit to the experimental isotherm via eq 1 is achieved. We find that two log-normal modes are sufficient to provide excellent fits to all the experimental data. The specification of two lognormal modes is a constraint similar to the “smoothing” constraints used by other workers.5 We use a simulated annealing optimization procedure2 to direct the search algorithm. We set the objective function to be the root-mean-square deviation (rmsd) of the isotherm calculated with eq 1 from the experimental isotherm. When the rmsd is within a given tolerance (always set equal to the rms of the experimental error), a fraction of the calculated total pore volume is added to the objective function. This allows the search algorithm to find solutions that have smaller total pore volume without violating the fit tolerance. The result is that solutions with unphysically large total pore volumes are ignored. This is similar to imposing a cutoff in the range of w as proposed by other workers.6 The optimization process is quick, typically taking a minute or less on a 350 MHz PC. However, due to the under-constrained nature of the solution of eq 1 and the stochastic nature of our optimization method, it is not possible to arrive at the same best-fit solution given different initial conditions for the optimizer. To arrive at overall best-fit solutions we repeat the optimization process 50 times with random initial configurations. Analysis of the 50 results in terms of the correlation between each free parameter (Ri, βi, γi) and the rmsd of the fit leads to overall best-fit solutions as discussed below. 3. Results 3.1. Vulcan. 3.11. Carbon Dioxide. Analysis of the 50 bestfit solutions (see Section 2.2) for carbon dioxide indicates that the overall best solution comprises two modes. The best onemode solution has a rmsd value of 0.035 cm3/g while the majority of solutions have two modes with a rmsd value between 0.022 and 0.035 cm3/g. It should be noted that most solutions have a rmsd value less than the experimental error (0.12 cm3/ g), so it is not possible to draw definite conclusions in this report. The first mode, centered near 0.91 nm dominates the adsorption capacity of Vulcan, but there is insufficient information (i.e., the problem is under-constrained) in the experimental isotherms to accurately determine the shape of the second mode, except that it contributes much less to the total adsorption. 3.12. Methane. Analysis of the 50 best-fit solutions for methane indicates that adsorption is dominated by one mode, peaking sharply near 0.86 nm. There is insufficient information in the experimental isotherm to decide the presence or shape of a second mode. Consequently, the analysis was repeated using only one log-normal mode. This mode also peaked sharply at 0.86 nm. All results are within the rms experimental error of 0.04 cm3/g. 3.13. Nitrogen. Analysis of the 50 best-fit solutions for nitrogen indicates that adsorption is dominated by one mode.



Figure 2. Best-fit pore size distributions for carbon dioxide (full line), methane (short-dashed line), and nitrogen (long-dashed line) on Vulcan.

Figure 3. Adsorption isotherms of carbon dioxide (diamonds and full line), methane (triangles and short-dashed line), and nitrogen (circles and long-dashed line) on Vulcan at 298 K. Symbols are experimental data, lines are isotherms calculated via eq 1 using the CO2 PSD (full line) in Figure 2.

However, there is insufficient information in the experimental isotherm to decide the shape of this mode or the presence of a second mode. Consequently, the analysis was repeated using only one log-normal mode which peaked sharply at 0.66 nm. All results are within the rms experimental error of 0.02 cm3/g. We note that using the carbon dioxide PSD, no value of ss for nitrogen can be found that fits the experimental data accurately. This suggests either that (i) we have omitted an important mechanism for the adsorption of nitrogen on Vulcan, or (ii) that the data for nitrogen on Vulcan, the most difficult isotherm to measure due to the small adsorption at room temperatures, is somehow inconsistent with that for the other gases. Since our methodology yields satisfactory correspondence between the isotherms and PSDs of carbon dioxide and methane (see below), we assume that the second case is true and that the experimental data for nitrogen on Vulcan is less accurate than that for CO2 and CH4. Hence, rather than choosing the value of ss for nitrogen by fitting to the experimental results, we are instead guided by the Lorentz-Berthelot rules and choose it to be the average of the values for carbon dioxide and methane, i.e., ss/kB ) 19.5 K. We performed the analysis for nitrogen above with this value. Figure 2 shows the overall best-fit PSDs for carbon dioxide, methane, and nitrogen on Vulcan. The carbon dioxide PSD has surface area of 66 m2/g. Figure 3 shows the predicted adsorption isotherms for each gas on Vulcan using the CO2 PSD (full line) in Figure 2 and eq 1. We see that the CO2 PSD is able to quite accurately reproduce the adsorption isotherm of methane but not nitrogen. We have fixed the values of ss for each gas to achieve the best fit shown. The gas-solid interaction parameters are given in Table 4. Table 5 shows the rmsd between adsorption isotherms for each gas predicted with the PSDs for each gas

Figure 4. Databases, generated from Grand canonical simulations, used in eq 1 for (a) carbon dioxide, (b) methane, and (c) nitrogen. The model parameters are given in Tables 1 and 4.

TABLE 4: Final Model Parameters for Gas-Solid Surface Interactions parameter

N2

CH4

CO2

σsf (nm)

0.337

0.355

sf/kB (K)

26.0

54.5

ss/kB (K)

19.5

20

C:0.308 O:0.321 C:23.8 O:39.2 C:19 O:19

TABLE 5: RMSD, Measured in cm3(STP)/g, between Adsorption Isotherms Predicted Using Each PSD in Figure 2 and Experimental Isotherms for Adsorption on Vulcan at 298 K adsorbate

N2 PSD

CH4 PSD

CO2 PSD

N2 isotherm CH4 isotherm CO2 isotherm

0.008 0.58 1.51

0.19 0.03 0.3

0.20 0.06 0.02

shown in Figure 2, and their experimental counterparts. We find that the carbon dioxide PSD is best able to reproduce the adsorption isotherms of the other gases, while the nitrogen PSD is least able to do this. 3.2. The Databases. Figure 4a shows the carbon dioxide database for 298 K produced by grand canonical simulation (to 20 bar, model parameters listed in Tables 1 and 4). The figure


Sweatman and Quirke

Figure 5. PSDs for AX21 obtained from measurements of the adsorption of carbon dioxide up to 1 bar at 293 K, but using different databases; full line, simulation database in Figure 4a; dashed line, DFT database used in ref 4.

shows that adsorption of carbon dioxide is substantial for narrow slit pores and for wider pores at high pressure. In narrow pores, high adsorption of carbon dioxide is caused by overlapping strongly attractive gas-solid potentials leading to a high potential energy density within the pore. For wider pores, where the carbon dioxide slit-pore potential energy density is smaller, the high adsorption at moderate pressure is caused by nearcritical capillary condensation of carbon dioxide. For wider pores still, capillary condensation of carbon dioxide is observed at sufficiently high pressure (greater than 20 bar). Figure 4, parts b and c, show the methane and nitrogen databases, respectively. They show that these gases are adsorbed in a manner consistent with supercritical adsorption. 3.3. Comparison of Simulation and Density Functional Theory. We first compare our results for the adsorption of carbon dioxide in AX21 at 293K up to 1 bar against those obtained previously4 by using density functional theory (DFT). Density functional theory can also be used to generate databases of adsorption isotherms, V(w,P). The DFT uses a spherical Lennard-Jones molecular model for carbon dioxide with parameters adjusted to fit experimental data for adsorption on Vulcan up to 1 bar. We have recalculated the PSD using the new inversion method of Section 2.2 with the same DFT carbon dioxide database used in ref 4. We fit27 to the experimental data, obtained at 293 K, with a tolerance equal to the experimental rms error of 1.2 cm3/g. Figure 5 shows predicted PSDs for carbon dioxide in AX21 using adsorption data at 293 K up to 1 bar. The PSDs shown are representative of the set of best-fit solutions. It shows that the PSD predicted using the new simulation databases is centered at about 0.56 nm. This compares to about 0.71 nm for the PSD predicted using the DFT database and the method of Section 2.2. Each PSD essentially comprises one log-normal mode. This contrasts with the two-mode PSD determined from nitrogen adsorption at 77K.4 The volume of the PSDs in Figure 5 are 0.34 and 0.50 cm3/g for the simulation PSD and new DFT PSD, respectively. These differences arise from the more accurate molecular models employed for both gas-gas and gassolid interactions in the current work. 3.4. Pore Size Distributions at 298 K. In this section we present the PSDs predicted for three typical amorphous carbons, denoted samples A (AX21), B, and C, using adsorption measurements12 for nitrogen, methane, and carbon dioxide up to 20 bar at 298 K. In each case the optimization procedure in Section 2.2 is used with, initially, two log-normal modes and a fit tolerance set to the rms error of the experimental data. Where the method of Section 2.2 was unable to decide the presence of two modes, the analysis was repeated with one mode.

Figure 6. Best-fit PSDs for samples: (a) A, (b) B, and (c) C. The lines represent PSDs for carbon dioxide (full line), methane (shortdashed line), and nitrogen (long-dashed line).

TABLE 6: Total Pore Volume Measured in cm3/g adsorbate

sample A

sample B

sample C

N2 CH4 CO2

1.02 1.13 1.58

0.53 0.66 0.72

0.42 0.56 0.65

Figures 6a, 6b, and 6c show results for the PSDs of samples A, B, and C, respectively. In each figure the carbon dioxide PSD shows two modes centered at around 0.65 to 0.75 nm and 1.4 to 2.0 nm, respectively. The total pore volume of each PSD is reported in Table 6. Sample A has approximately twice the micropore volume of B and C, and appears to have a more developed bimodal PSD at this resolution, otherwise the pore structure is similar for the three samples. Figures 7, parts a, b, and c, show the predicted isotherms for samples A, B, and C, respectively, that result when the CO2 PSDs (full lines in Figure 6, parts a, b, and c) are used to predict the adsorption of each gas. Table 7 reports the full set of results for the rmsd between isotherms for each gas predicted using the PSDs for each gas shown in Figure 6, parts a, b, and c, and the corresponding experimental isotherms. The bracketed numbers indicate the rms experimental error. We generally find that adsorption isotherms predicted by the PSDs of each adsorbate are in good agreement, although the agreement is not as good for sample A. The carbon dioxide PSD is the most robust in


J. Phys. Chem. B, Vol. 105, No. 7, 2001 1409 4. Discussion

Figure 7. Adsorption isotherms for samples (a) A, (b) B, and (c) C at 298 K. Symbols are experimental data, lines are isotherms calculated via eq 1 using the CO2 PSD in Figures 6a, 6b, and 6c. The isotherms are carbon dioxide (diamonds and full line), methane (triangles and short-dashed line), and nitrogen (circles and long-dashed line).

TABLE 7: RMSD, Measured in cm3(STP)/g, between Adsorption Isotherms Predicted Using PSDs Obtained from Different Adsorbates and Experimental Isotherms at 298 Ka adsorbate

N2 PSD

CH4 PSD

CO2 PSD


Sample Ab 0.22 (0.34) 1.79 4.6 0.3 (2.95) 15.2 15.1

3.52 10.1 0.6 (0.95)


Sample Bc 0.4 (0.18) 1.0 2.5 0.8 (0.61) 14.6 12.4

1.5 1.7 0.4 (0.23)


Sample Cd 0.5 (0.17) 1.0 3.6 1.2 (1.15) 13.8 15.5

1.7 1.8 0.6 (0.33)

a

Bracketed numbers indicate the rms experimental error. b PSDs from Figure 6a; experimental isotherms from Figure 7a. c PSDs from Figure 6b; Experimental isotherms from Figure 7b. d PSDs from Figure 6c; experimental isotherms from Figure 7c.

the sense that it predicts the adsorption of the other gases most accurately.

The critical temperature for carbon dioxide is 304.1 K.10 At ambient temperatures, carbon dioxide exhibits significant adsorption in slit pores due either to a high potential energy density in the pore space or to capillary condensation (or near critical capillary condensation) for sufficiently high pressures less than the saturation pressure (∼64 bar at 298 K). Thus, carbon dioxide adsorption isotherms measured up to high pressure in slit pores at ambient temperatures are sensitive to slit width. It follows that carbon dioxide is a sensitive probe of the PSD of porous materials if measurements are made up to the saturation pressure. The critical temperatures of methane and nitrogen are 190.6 K and 126.2 K, respectively. These adsorbates are significantly supercritical at ambient temperatures, and they are not as sensitive as carbon dioxide as probes of the microstructure. Thus inversion of eq 1 is more constrained for carbon dioxide than it is for methane and for nitrogen. Since we are using a stochastic optimization process to solve eq 1, the degree of constraint is very influential. We expect the set of best-fit carbon dioxide PSDs of our amorphous carbon samples to exhibit less variation than the methane and nitrogen PSDs. This is illustrated in Figures 2, 6a, 6b, and 6c where the methane and nitrogen PSDs are sufficiently un-controlled that they are best represented by one log-normal mode, whereas each carbon dioxide PSD is best represented by two modes. The reverse argument is also true; that since methane and nitrogen are less sensitive probes than carbon dioxide, the methane and nitrogen adsorption isotherms are less sensitive to variations in the PSD than the carbon dioxide isotherm. This is why the carbon dioxide PSD is the most robust, and can be used to predict the adsorption of methane and nitrogen with generally good accuracy. In essence, there is more information contained in the carbon dioxide isotherm. Our solution method for eq 1 also attempts to minimize the total pore volume provided the fit is within a given tolerance. Thus, the best-fit carbon dioxide PSD is more likely to have a larger total pore volume than the best-fit methane PSD (and the best-fit methane PSD is more likely to have a larger total pore volume than the best-fit nitrogen PSD) since the solution of eq 1 is less well constrained for methane than carbon dioxide (and less well constrained for nitrogen than methane). Table 5 confirms this. For the same reason, we expect the total pore volume calculated with respect to an adsorption isotherm up to 1 bar to be less than that calculated from an adsorption isotherm up to 20 bar. Comparison of the total pore volume of the PSDs in Figures 5 and 6a confirms this expectation. However, the difference in the total pore volume of the PSDs in Figure 5 alone is due entirely to the differences in the molecular models used in generating the DFT and simulation databases, i.e., the spherical nature of the DFT molecular model and the assumption of (infinitely) wide pores in the determination of ss used in the DFT databases. The carbon dioxide PSDs in Figure 6a (sample A is AX21) cannot be compared directly to the nitrogen PSD obtained from measurements on AX21 at 77 K in ref 4. The PSDs in ref 4 are calculated with respect to physical pore width, whereas in this work we use chemical width to define w. To convert a PSD determined with respect to physical width, fp(wp), to a PSD determined with respect to chemical width, now denoted fc(wc), requires application of a conversion factor.28 Thus the total pore volume calculated in ref 4 for AX21 from nitrogen adsorption at 77 K (1.58 cm3/g) is converted to 1.36 cm3/g with respect to chemical width. This value is smaller than the total pore volume, obtained with carbon dioxide up to 20 bar at 298 K, for AX21 quoted in Table 6 of this work (1.58 cm3/g). Use of a spherical


Sweatman and Quirke atures. This understanding is of considerable importance in the design of materials such as activated carbons for separation processes since it implies that one PSD may be used to predict separation factors for gas mixtures. A complete description of the pore size distribution requires carbon dioxide adsorption measurements up to its saturation pressure (57 bar10 at 293 K). The higher the resolution of the PSD, the more precisely we can hope to tailor the material, either through changes to the activation procedures or though informed choice of existing microstructures, for specific applications.

Figure 8. PSDs for AX21. The full and dashed lines are the carbon dioxide PSD of Figure 6a and the PSD obtained from nitrogen measurements at 77 K in ref 4 (shifted by 0.24 nm to allow better comparison between chemical and physical widths).

molecular model for nitrogen in ref 4 would tend to generate PSDs with greater volume, so the difference could be due in part to the assumption in ref 4 that Vulcan consists entirely of (infinitely) wide pores. We have not made this assumption in this work. Figure 8 compares the PSD obtained with carbon dioxide at 298 K of this work (full line in Figure 6a), with the PSD from ref 4 obtained with nitrogen at 77 K. We see that each PSD comprises two significant log-normal modes, with the PSD predicted by this work showing modes centered at smaller widths than in ref 4. This shift is probably due both to the more realistic diatomic nitrogen model used to generate the simulation database and the lower value for the gas-solid interaction in this work as compared to ref 4. Other workers29 have concluded that carbon dioxide at ambient temperatures is a more accurate probe of microporosity than nitrogen at 77 K. Using the Dubinin-Radushkevich theory for adsorption in micropores, Cazorla-Amoros et al. found that micropore volumes predicted from carbon dioxide measurements up to 4 bar at ambient temperatures tended to be larger than from nitrogen measurements up to 1 bar at 77 K. They interpret this as bring caused by diffusional limitations of nitrogen at 77 K in very narrow micropores. This mechanism might also be partly responsible for the larger pore volume for AX21 obtained in this work with carbon dioxide at 298 K (1.58 cm3/g) compared to that in ref 4 with nitrogen at 77 K (1.36 cm3/g). To describe the entire PSD we require experimental measurements of carbon dioxide adsorption up to the saturation pressure. Our measurements at 298 K up to 20 bar, and those in ref 29 up to 4 bar, fall short of this target. It would be interesting to repeat the above methodology and compare adsorption measurements of carbon dioxide at ambient temperatures up to its saturation pressure with measurements of nitrogen at 77 K up to 1 bar. New adsorption databases for nitrogen at 77 K based on grand canonical simulation, the accurate molecular model given in Table 1 and the methodology of Section 2.1 would be needed to afford a fair comparison. This work is in progress. 5. Conclusions We have shown that at ambient temperatures, carbon dioxide is a more sensitive probe of the microporosity of active carbons than methane or nitrogen due to near critical capillary condensation in graphitic micropores. The use of simulation methods, instead of density functional theory, to generate databases of adsorption isotherms in model slit pores has been shown to lead to more reliable pore size distributions; indeed PSDs predicted from carbon dioxide adsorption isotherms can be used to estimate the adsorption isotherms of methane and nitrogen on the same material with reasonable accuracy at ambient temper-

Acknowledgment. We thank N. Kanellopoulos, T. Steriotis, and co-workers for supplying the experimental data used in this work, and D. Nicholson, K. Travis, and T. Grey for “Simapos”, our grand-canonical ensemble simulation code. We thank EPSRC for support through the equipment grant GR/M94427. References and Notes (1) Davies, G. M.; Seaton, N. A.; Vassiliadis, V. S. Langmuir 1999, 15, 8235. (2) For example, see Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical recipes in Fortran 77: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: New York, 1992. (3) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (4) Scaife, S.; Kluson, P.; Quirke, N. J. Phys. Chem. B 1999, 104, 313. (5) Ravikovitch, P. I.; Vishnyakov, A.; Russo, R.; Neimark, A. V. Langmuir 2000, 16, 2311. (6) Gusev, V. Y.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815. (7) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by powders and porous solids; Academic Press: London, 1999. (8) Horvath, G.; Karazoe, K. J. Chem. Eng. Jpn. 1983, 16, 474. (9) Panagiotopoulos, A. Z. Mol. Phys. 1987, 61, 813. Panagiotopoulos, A. Z.; Quirke, N.; Stapleton, M.; Tildesley, D. J. Mol. Phys. 1988, 63, 527. Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: London, 1996. (10) Duschek, W.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1990, 22, 827. Duschek, W.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1990, 22, 841. Pierperbeck, N.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1991, 23, 175. Handel, G.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1992, 24, 685. Gilgen, R.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1992, 24, 1243. Nowak, P.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1997, 29, 1137; ibid. 1157. (11) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1987. (12) Steriotis, T.; Kanellopoulos, N.; and co-workers. Private communication. (13) Certified reference material No. M11-02 (Vulcan 3-Gsgraphitised carbon black) supplied by Laboratory of the Government Chemists. (14) Harris, J. G.; Yung, K. H. J. Phys. Chem. 1995, 99, 12021. (15) Samios, S.; Stubos, A. K.; Papadopoulos, G. K; Kanellopoulos, N. K.; Rigas, F. J. Colloid Interface Sci. 2000, 224, 272. (16) MacElroy, J. M. D.; Seaton, N. A.; Friedman, S. P. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; 1997. Segarra, E. I.; Glandt, E. D. Carbon 1994, 17, 2953. Biggs, M.; Agarwal, P. Phys. ReV. A 1992, 46, 3312. (17) Thompson, K. T.; Gubbins, K. E. Abstracts Carbon 99, p 466. (18) AX21 MASTsKOH-activated carbon made by Amoco supplied to us by S. Tennison of MAST. AX21 MAST is a typical high-surfacearea microporous activated carbon (SN2-BET ) 3080 ( 19 m2/g at 77 K). (19) Panagiotopoulos, A. Z. Int. J. Thermophys. 1994, 15, 1057. (20) Delhomelle, J. Ph.D. Thesis, Universite´ de Paris-Sud U. F. R. Scientifique d’Orsay, 2000. (21) Kuchta, B.; Etters, R. D. Phys. ReV. B 1987, 36, 3400. (22) Steele, W. A. The interaction of gases with solid surfaces; Pergamon: New York, 1974. (23) Kaneko, K.; Cracknell, R. F.; Nicholson, D. Langmuir 1994, 10, 4606. (24) We use a chemical definition of pore width because all experimental measurements reported here use a gravimetric method that yields excess adsorptions, V ) Va(1 - Fb/Fa) ) Va - AFbVp, where Va is the absolute adsorption, Fb is the bulk density, and Vp is the “chemical” pore volume (all dependent on temperature and pressure). A discussion of pore volume



in adsorption experiments and the necessary corrections can be found in Neimark, A., V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148. (25) Kaneko, K.; Ishii, C.; Rybolt, T. In Characterization of Porous Solids III; Rouquerol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Studies in Surface Science and Catalysis, 87; Elsevier: New York, 1994. (26) Sweatman, M. B.; Quirke, N. Work in progress. (27) Our simulation databases, V(w,P), describe adsorption at 298 K. To convert these databases to describe adsorption at 293 K the following approximation is used: V(w,P,T1) )

((

)

Va(w,P,T0) Fb(P)

T1/T0

-1

)

T1Fb(P) T0

where Va ) V + Fb is the average absolute density, Fb is the bulk density, and we set T0 ) 298 K and T1 ) 293 K.

(28) fc(wc) )

(

( ))

Fb(P) wc wc + - 1 fp(wc + 2.4A) wp V(wc,P) wp

Inspection of the carbon dioxide database, V(wc,P), shows that to within an error of 3% this can be approximated by fc(wc) )

wc f (w + 2.4A) wp p c

(29) Cazorla-Amoros, D.; Alcaniz-Monge, J.; Linares-Solano, A. Langmuir 1996, 12, 2820. Garcia-Martinez, D.; Cazorla-Amoros, D.; LinaresSolano, A. Stud. Surf. Sci. Catal. 2000, 128, 485. Cazorla-Amoros D.; Alcaniz-Monge, J.; de la Casa-Lillo, M. A.; Linares-Solano, A. Langmuir 1998, 14, 4589.

Characterization of Porous Materials by Gas Adsorption at Ambient

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