Characterization of Porous Materials by Gas Adsorption - American

Amorphous materials are usually characterized using nitrogen adsorption isotherms at 77 K taken at pressures up to 1 bar to obtain pore size distribut...
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Characterization of Porous Materials by Gas Adsorption: Comparison of Nitrogen at 77 K and Carbon Dioxide at 298 K for Activated Carbon M. B. Sweatman and N. Quirke* Department of Chemistry, Imperial College, South Kensington, London, SW7 2AY, United Kingdom Received February 28, 2001. In Final Form: May 9, 2001 Amorphous materials are usually characterized using nitrogen adsorption isotherms at 77 K taken at pressures up to 1 bar to obtain pore size distributions. Activated carbons are amorphous microporous graphitic materials containing pores which can range from nanometers to microns in width and which can, in principle, be tailored to adsorb specific molecules or classes of molecule by changing the method of preparation (the activation process). 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. In this paper, we compare pore size distributions of an ultrahigh surface area activated carbon (AX21) determined from nitrogen adsorption measurements up to 0.6 bar at 77 K with those determined from carbon dioxide adsorption measurements up to 20 bar at 298 K. Our analysis employs grand canonical and Gibbs ensemble Monte Carlo simulations together with accurate site-site interaction models of the adsorbates. We find that the calculated pore size distributions for each adsorbate are quite different, and the adsorption of one gas can be estimated from the adsorption of the other gas to within an error of 25% at the highest pressures only. At lower pressures, we speculate that large errors are due to the behavior of nitrogen in carbon micropores in which diffusion is severely limited. To substantiate this speculation, we have calculated the self-diffusion coefficient for nitrogen at 77 K and carbon dioxide at 298 K in carbon slit pores using equilibrium molecular dynamics. The results suggest that nitrogen is diffusionally limited, and possibly frozen, in such pores whereas carbon dioxide remains mobile. We conclude that room-temperature carbon dioxide adsorption isotherms up to the saturation pressure could provide a more accurate characterization of carbon microstructure than nitrogen isotherms at 77 K up to 1 bar.

Introduction There has been steady progress in the analysis of gas adsorption in porous materials for over a century. Traditional methods1 involving fitting semiempirical isotherms or “characteristic curves” have recently been superseded by the advent of density functional theory (DFT) and Monte Carlo simulation techniques. The traditional methods can be grouped into those with an underlying physical basis, such as the Langmuir and Brunauer-Emmett-Teller (BET) isotherms, and those that are empirical, such as the Dubinin-Radushkevich (DR) and Dubinin-Stoekli (DS) methods. The essential difference between these methods is that in the low-density limit the physically based methods reduce to the correct Henry’s law limit, while the empirical methods do not. All of these techniques require some a priori knowledge of the porous character of the material in question. For example, the Langmuir isotherm is restricted to analysis of an adsorbed monolayer, the BET isotherm is derived with the assumption of a possibly infinite number of adsorbed layers, and different versions of the DR or DS methods have been proposed for different kinds of porosity. Most of these methods can be viewed as little more than recipes for finding empirical parameters (such as surface areas) specific to each material and method. Consequently, they are unlikely to be satisfactory for analysis of the * Corresponding author. (1) For example, see: Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984. Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: New York, 1991. Yang, R. T. Gas Separation by Adsorption Processes; Imperial College Press: London, 1997. Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: San Diego, CA, 1999.

adsorption of different adsorbates in materials with a wide range of pore sizes. Further, they are unlikely to allow prediction of detailed physical properties of these materials, including thermodynamic properties, with respect to different adsorbates. Indeed, as far as we know, these techniques have never been used in this way. Conversely, simulation and DFT methods have proved instrumental in recent decades in gaining an understanding of the physics involved in a range of gas adsorption phenomena, including wetting, capillary condensation, layering, and so forth.2 These methods are grounded in statistical mechanics. As such, they are ideally suited to aid the task of characterizing, and then predicting the properties of, nanoporous materials. In this work, we examine the accuracy with which the adsorption properties of an activated carbon can be determined from a single characterization analysis. Activated carbons are amorphous microporous graphitic materials formed by treating a wide range of organic precursors including coal, tar, nutshells, and particular polymers. The treatment can include further carbonization (depending on the degree of initial carbonization), grinding, and activation. Activation typically consists of hightemperature treatment with steam or carbon dioxide. Acidic or basic treatments are also common. The resulting pore network can contain pores that range from nanometers to microns in width and exhibit a surface area, as seen by gas adsorbates, in excess of 1000 m2/g. In addition, (2) For example, see: Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Dekker: New York, 1992. Molecular Simulation and Industrial Applications: Methods, Examples and Prospects; Gubbins, K. E., Quirke, N., Eds.; Gordon and Breach Science Publishers: Australia, 1996.

10.1021/la010308j CCC: $20.00 © 2001 American Chemical Society Published on Web 07/17/2001

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small amounts of noncarbonaceous material (ash) can remain within the material. In principle, the pore network can be tailored to preferentially adsorb specific molecules or classes of molecule by changing the method of preparation. For the physical chemist, they pose the challenge of understanding adsorption processes in amorphous microporous materials and of using that understanding to improve their characterization. We focus on the accuracy with which the adsorption of carbon dioxide up to 20 bar at 298 K is predicted from an analysis of the adsorption of nitrogen up to 0.6 bar at 77 K,3 and vice versa. We use nitrogen at 77 K because it is the most commonly used probe of porous materials. Carbon dioxide at room temperature has been suggested as an alternative to nitrogen at 77 K.4 The comparison of nitrogen adsorption at 77 K and carbon dioxide adsorption at 298 K represents a considerable test for the techniques used. In previous work,5 we examined the correspondence between characterizations based on nitrogen, carbon dioxide, and methane, all at 298 K. We found that carbon dioxide was the most sensitive of these adsorbates and that adsorption of the other gases (at 298 K) on three different activated carbons could be predicted to within an error of 10% using the carbon dioxide characterization. The methods and techniques employed in this work are based on Monte Carlo simulation and the polydisperse slit pore model of activated carbon. The grand canonical (GCEMC) and Gibbs (GEMC) ensemble Monte Carlo simulation techniques are well established and described in detail elsewhere.6 GCEMC simulates a fluid phase at specified chemical potential, µ, temperature, T, and volume, V, while GEMC simulates coexisting phases at the same µ, T, and pressure, P, without simulating the interface between the two phases. The polydisperse slit pore model has found numerous applications.7-10 It approximates a real pore network in terms of a distribution, f, of slit pores characterized in terms of pore width, w. Thus, the adsorption volume, Va, is given by the adsorption integral for a given T

Va(P) )

∫dw f(w) v(P,w)

(1)

where v is the average excess density of fluid in the pore. This is clearly a gross simplification for many reasons. It ignores any effect of pore curvature, junctions, finite pore length, or pore connectivity. Also, an idealized geometric characterization cannot properly account for energetic surface inhomogeneity; that is, the polydisperse slit pore model can confuse more energetic adsorption sites with narrower pore widths. Scaife et al.8 also used the polydisperse slit pore model to characterize amorphous carbons, but they used an (3) The nitrogen adsorption measurements at 77 K were obtained with a different technique on different samples of Vulcan and AX21 to the carbon dioxide measurements at 298 K. Nevertheless, we expect each experimental method to be sufficiently accurate, and each sample to be sufficiently similar, so that our work is valid. (4) See, for example: Garcia-Martinez, J.; Cazorla-Amoros, D.; Linares-Solano, A. Stud. Surf. Sci. Catal. 2000, 128, 485 and references therein. (5) Sweatman, M. B.; Quirke, N. J. Phys. Chem. B 2001, 105, 1403. (6) For example, see: Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1987. Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: San Diego, CA, 1996. (7) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (8) Scaife, S.; Kluson, P.; Quirke, N. J. Phys. Chem. B 1999, 104, 313. (9) Ravikovitch, P. I.; Vishnyakov, A.; Russo, R.; Neimark, A. V. Langmuir 2000, 16, 2311. (10) Samios, S.; Stubos, A. K.; Papadopoulos, G. K.; Kanellopoulos, N. K.; Rigas, F. J. Colloid Interface Sci. 2000, 224, 272.

established nonlocal density functional theory to determine v(P,w). They examined pore size distributions (PSDs) generated by adsorption of nitrogen and argon at 77 K and carbon dioxide at 293 K up to 1 bar. They found that carbon dioxide at 293 K produced a PSD that was shifted to smaller pore widths compared with nitrogen and argon at 77 K. They speculated that this was caused by freezing of nitrogen and argon at 77 K in the smallest pores which blocked access to the full pore network. Ravikovitch et al.9 also used the polydisperse slit pore model to characterize amorphous carbons from adsorption measurements of nitrogen and argon at 77 K and carbon dioxide at 273 K. In essence, they used the same densityfunctional methods as Scaife et al. to generate v(P,w). For carbon dioxide, they also used grand canonical ensemble simulations, with a published model for gas-gas interactions, to generate v(P,w) 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 and simulation databases for carbon dioxide. Other authors10 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 dioxide at a range of sub- and supercritical temperatures, it was found that reasonable agreement could be obtained for one microporous carbon but not for another suspected of having embedded polar sites. For methane, good agreement was obtained on one microporous carbon over a 65 K range of supercritical temperatures. Some work has been published concerning pore network representations that are less constrained than idealized geometric models. Recently, Gubbins and co-workers11 have used reverse Monte Carlo to generate graphitic pore networks consistent with input carbon-carbon radial distribution functions. However, while appealing, such approaches inevitably introduce many more unknown parameters than the polydisperse slit pore model. The paper is organized as follows. The next section describes our simulation and characterization methods in detail. This is followed by the results section. It demonstrates that nitrogen and carbon dioxide PSDs of our activated carbon (AX21) are quite different, and the predicted adsorption of one gas from the other gas is accurate to within an error of 25% at the highest pressures only. We then discuss possible reasons for this difference. Method Our method consists of three main steps for each gas: 1. Definition of accurate molecular models for each adsorbate using Gibbs ensemble Monte Carlo simulation (GEMC). 2. Calibration of the gas-surface interaction using grand canonical ensemble Monte Carlo simulation (GCEMC). 3. Characterization of the activated carbon using simulated annealing optimization and prediction of adsorption isotherms. Each of these steps is detailed in the following subsections. Molecular Models. We have improved the site-site interaction models of nitrogen and carbon dioxide so that the liquid-gas phase diagrams obtained by GEMC agree (11) Thompson, K. T.; Gubbins, K. E. Abstracts Carbon 99; page 466.

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nated adsorption in Vulcan. Thus, our calibration process must allow for microporosity. In other studies (for example, see refs 8 and 13), it is assumed that Vulcan consists of wide pores, and the gas-surface interaction is calibrated with respect to infinitely wide pores. Since, for the same gas-surface interaction, significant adsorption in micropores occurs at much lower pressures than monolayer completion on an isolated surface and these phenomena are sensitive to the strength of the gas-surface interaction, these studies would all tend to overestimate the strength of gas-surface interactions. We model the interaction of a Lennard-Jones site in a gas molecule with the carbon surface by a Steele potential14 Figure 1. Comparison of experimental adsorption isotherms for nitrogen on AX21 (solid line) and Vulcan (dotted line) at 77 K. The Vulcan isotherm has been scaled by a factor of 29.

well with experimental data.12 Our model for carbon dioxide is identical to that in our previous work.5 In our simulations, we have 500 molecules distributed between two cubic boxes each of initial side length 3.5 nm. The initial configuration is set to that of a lattice filling each box. Site-site interactions are modeled by the LennardJones potential for repulsive-dispersive interactions. Partial charges are used to model the electrostatic quadrupole of each gas. We use a cutoff in all interactions of 1.5 nm and account for long-range dispersive interactions by assuming that the pair-correlation function for each phase is unity beyond the cutoff. Long-range electrostatic interactions are omitted since their contribution to the total energy is typically less than 0.1% for the initial configuration. Our simulations are performed using three types of Monte Carlo move: intrabox moves, interbox moves, and volume moves. The step sizes for intrabox and volume moves are chosen to ensure that about 1/3 of all such moves are accepted. Similarly, we manipulate the probability for attempting each type of move so that the number of accepted interbox moves is about 1/10 of the number of accepted intrabox moves and that there are about N/2 (N is the number of molecules, i.e., 500) times as many attempted intrabox moves as volume moves. These choices seem to ensure rapid and stable phase separation. A total of 10 million attempted moves are performed in each simulation, and the first 5 million of these are counted as equilibration moves. Gas-Surface Calibration. Vulcan is considered to be a “reference” carbon material and has been used in some studies8,13 of porous carbons to calibrate gas-surface interactions. Figure 1 compares the adsorption at 77 K of nitrogen on Vulcan and AX21. The adsorption of nitrogen on Vulcan has been scaled by a factor of 29. Up to pressures of 0.001 bar, adsorption of nitrogen on Vulcan and AX21 is very similar. Because we know that AX21 is microporous (it is a super high area adsorbent) and that micropores will fill with nitrogen before larger pores as pressure is increased, Figure 1 indicates that Vulcan and AX21 share a similar microporous character. This agrees with our previous work5 in which it was concluded that curvature of carbon dioxide, methane, and nitrogen adsorption isotherms on Vulcan could not be explained without the assumption that Vulcan’s porosity partly consisted of micropores. Indeed, we found that microporosity domi(12) 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. Gilgen, R.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1992, 24, 1243. Nowak, P.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1997, 29, 1137, 1157. (13) Cracknell, R. F.; Nicholson, D. Adsorption 1996, 2, 193.

(( ) ( )

V1(z) ) 2πFc∆gsσgs2

2 σgs 5 z

10

-

σgs z

4

σgs4

)

3∆(z + 0.61∆)3

(2)

with parameters Fc ) 114 nm-3 and ∆ ) 0.335 nm, and z is the distance of the gas site from the plane of the carbon atom centers in the first layer of the surface. The crossparameters are obtained from the Lorentz-Berthelot rules

σgs ) (σgg + σss)/2

gs ) xgg × ss

(3)

with σss ) 0.34 nm. The remaining parameter, ss, is fixed for each gas by calibrating to Vulcan as follows. We characterize the porosity of Vulcan, treating it as just another porous carbon. We calibrate ss for nitrogen by choosing it such that the calculated surface area (see the following characterization subsection) of Vulcan agrees with the BET value, which is approximately 70 m2/g for nitrogen on Vulcan at 77 K.8 The value of ss for carbon dioxide is obtained by choosing it such that the PSD resulting from the nitrogen calibration reproduces the adsorption isotherm of carbon dioxide on Vulcan.3 We use this method since we expect that nitrogen at 77 K up to 1 bar is a more sensitive probe of pore width than carbon dioxide at 298 K up to 20 bar. Thus, the nitrogen PSD is more likely to be accurate than the carbon dioxide PSD. Of course, there is an inconsistency in this approach since the BET method assumes that adsorption is on a single surface and that there are no cooperative adsorption effects due to micropores. It is not possible to say whether the BET method is likely to over- or underestimate the surface area of microporous materials since this depends on the detail of the microporosity. If Vulcan predominantly consists of very narrow micropores (w , 2σgg), then we should expect the BET method to underestimate its surface area since less than two adsorbed layers can form in such pores. However, If Vulcan predominantly consists of larger micropores (w g 2σgg) then we should expect the BET method to overestimate its surface area due to cooperative adsorption phenomena. Also, and from a different perspective, at the microscopic level the concept of surface area is not absolutely defined and, generally, any estimate should be expected to be different for different adsorbates. Thus, we cannot expect the microporous surface area experienced by carbon dioxide to be identical to that experienced by nitrogen. Nevertheless, and in the absence of any more accurate reference data, we require the nitrogen PSD to yield a surface area close to 70 m2/g for Vulcan and we use this PSD to calibrate the carbon (14) Steele, W. A. The interaction of gases with solid surfaces; Pergamon: New York, 1974.

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dioxide-surface model. In this sense, our method is a calibration since other reference estimates of the surface area of Vulcan would lead to different values of ss for each gas. Characterization. We assume that the pore surface of AX21 is identical to Vulcan’s. We characterize each porous carbon by the polydisperse ideal slit pore model, eq 1. An ideal slit pore of width H is constructed by two opposing, parallel ideal surfaces. The external potential generated is

Vext(z) ) V1(z) + V1(w - z)

(4)

Each surface is modeled by a Steele potential (eq 2) with parameters determined by the method described in the previous subsection. To obtain the PSD, f(w), from eq 1, we first need to calculate v(P,w). This is achieved by linear interpolation of a database of v. The database for each gas, for sufficient range and resolution of values of P and w, is calculated by GCEMC simulation. To determine ss for each gas, as described in the previous subsection, several databases with different values of ss are required. In our simulations, we use identical models for gas-gas interactions as in the GEMC simulations, except that a cutoff in all gas-gas interactions of 1.5 nm is employed with no long-range corrections. There is no cutoff in the range of gas-surface interactions. The cutoff of 1.5 nm resulted from some preliminary simulations in which it was found that larger cutoffs made no significant difference to the databases. The minimum box length and height (the directions parallel to the slit pore surfaces) are set to 3 nm. The step sizes for intrabox moves are chosen to ensure that about 1 /3 of all such moves are accepted. We attempt each type of move (intrabox, creation, and destruction) with equal probability. A total of 2 million attempted moves are performed in each simulation. We reject the first 1 million attempted equilibration moves. Our definition of v must be consistent with the experimental technique used to obtain the reference adsorption isotherms. The experimental data we use are consistent with chemical width, wc. The chemical width describes the width occupied by adsorbate molecules, which is typically about σss less than the physical width, w, defined by the distance between and normal to carbon atom centers in the first layer of the opposing slit walls. Because the volume occupied by adsorbate molecules is not absolutely defined (it depends on the nature of the gas-surface interaction), wc is not absolutely defined. In this work, we follow others15 and arbitrarily set wc ) w - 0.24 nm. Thus, each database entry is calculated as

v(P,wc) )

〈N〉 - Fb Awc

(5)

where 〈N〉 is the average number of molecules in each simulation with volume V ) Awc and Fb is the bulk gas density (calculated from separate GCEMC simulations of the bulk phase for each P). With (5), the adsorption integral becomes

V(P) )

∫dwc f(wc) v(P,wc)

(6)

Both (1) and (6) are examples of Fredholm equations of (15) 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.

the first kind.16 Hence, the inversion of (6) to obtain f(wc) is an ill-posed problem.17 That is, there can be from zero to an infinite number of solutions for f depending on V and v. Also, the solution can be unstable in that small changes in v or V can result in large changes in solutions for f. Despite these well-documented difficulties, previous work has used a number of methods to obtain possible solutions for f. These include best-fit methods and matrix methods. The best-fit methods are essentially trial-anderror 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.16 With both methods, additional constraints are often required to force more physically appealing or acceptable solutions,17 including constraints on the smoothness of the solution function and the range of pore widths. We use a best-fit method similar to that used in our previous work5 to find f from (6). We assume f to be a sum of log-normal functions

f(wc) )

∑i Ri/(γiwc(2π)0.5) exp(-(ln wc - βi)2/(2γi))

(7)

and vary the free parameters Ri, βi, and γi until a satisfactory fit is achieved to the experimental isotherm via eq 6. We find that two log-normal modes are sufficient to provide reasonable fits to all the experimental data. Specification of two log-normal modes is a constraint similar to the “smoothing” constraints used by other workers. We use a simulated annealing optimization procedure16 to direct the search algorithm. We set the objective function to be the sum of the root-mean-square deviation (rmsd) of (i) the isotherm and (ii) the gradient of the isotherm, calculated with eq 6 from the experimental isotherm and experimental isotherm gradient. When the rmsd of the isotherm 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 wc as proposed by other workers. The optimization process is quick, typically taking a minute or less on a 350 MHz PC. However, because of the nature of the solution of eq 6 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 25 times with random initial configurations. The surface area, S, is calculated from the PSD by

S)

∫dwc f(wc)/wc

(8)

Results Molecular Models. The molecular models for nitrogen and carbon dioxide are presented in Table 1. With these molecular models, the coexisting liquid-gas phase diagrams predicted by GEMC simulations are compared in (16) 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. (17) Gusev, V. Y.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815.

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Figure 2. Liquid-gas coexisting densities of (a) nitrogen and (b) carbon dioxide. Lines are experiment (ref 11), and points and error bars are the results of Gibbs ensemble calculations in this work (model parameters in Table 1). Table 1. Model Parameters for Gas-Gas Interactions parameter

nitrogen

σff (nm)

0.333

ff /kB (K)

34.4

lx (nm)

(0.0505

lq (nm)

(0.0847 (0.1044 0.373 -0.373

q (e)

carbon dioxide C: 0.275 O: 0.3015 C: 28.3 O: 81.0 C: 0 O: (0.1149 0 (0.1149 0.6512 -0.3256

Table 2. Model Parameters for Gas-Surface Interactions Such That the Surface Area of Vulcan Is Calculated To Be Approximately 70 m2/g parameter σsf (nm)

nitrogen 0.337

sf/kB (K)

28.7

ss/kB (K)

24

carbon dioxide C: O: C: O: C: O:

0.308 0.321 20.6 34.9 15 15

Figure 2a,b against experiment. In this figure, where error bars are not shown, statistical errors are always less than indicated by the symbol size. These molecular models are clearly quite accurate over the range of temperatures simulated, which almost encompass the temperatures of interest in this work. Gas-Surface Calibration. The gas-surface interaction parameters, calibrated to yield approximately 70 m2/g for the surface are of Vulcan as described in the previous section, are shown in Table 2. The corresponding databases, v, are shown in Figure 3a,b. The nitrogen database clearly demonstrates how pore width influences the

Figure 3. Databases, v(P,wc), for (a) nitrogen at 77 K and (b) carbon dioxide at 298 K. Model parameters are given in Table 2. 〈Nex〉 is the excess (compared to bulk gas) number of adsorbed molecules per cubic nanometer.

pressure at which significant adsorption occurs. For very narrow micropores, the pore is essentially filled at all pressures shown; for larger micropores, the pore filling pressure is very sensitive to pore width. For wide pores, a monolayer forms at about 0.001 bar with pore filling, again, sensitive to pore width. Capillary condensation is evident for wc > 1.3 nm (∼4σgg) whereby the pore fills from an adsorbed monolayer. Below this width, packing effects are evident in the filled pore at pressures approaching saturation and capillary condensation appears to end or be disrupted. A confined fluid phase transition is again evident for 0.7 < wc < 1.0 (for 2σgg < wc < 3σgg), whereby the pore fills from zero adsorbed layers. This phenomenon has previously been called a “bilayer” transition18 but could also be called “re-entrant” capillary condensation with packing effects disrupting capillary condensation for 1.0 < wc < 1.3 and enhancing capillary condensation for 0.7 < wc < 1.0. The carbon dioxide database is less complex and is typical for a fluid in the region of the critical temperature. It is possible that capillary condensation could occur for pressures higher than shown in Figure 3b. Characterization. The best-fit PSD for Vulcan determined by nitrogen adsorption up to 0.3 bar at 77 K is shown in Figure 4. This PSD is composed of two distinct modes, at about 0.9 and 1.9 nm, and adsorption on Vulcan is predicted to be dominated by adsorption in micropores. The pore volume and surface area of the nitrogen PSD are (18) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693.

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Figure 4. Best-fit PSD for Vulcan from nitrogen adsorption measurements up to 0.3 bar at 77 K using the model parameters in Tables 1 and 2.

Figure 5. Fit to experimental adsorption isotherms of (a) nitrogen at 77 K and (b) carbon dioxide at 298 K, on Vulcan using the PSD for nitrogen in Figure 4. Symbols are experiment, and solid lines are best-fit isotherms.

0.04 and 69 m2/g, respectively. Using this value of ss and the PSD for nitrogen, the value of ss that produces the closest fit for carbon dioxide adsorption on Vulcan is determined. All the gas-surface model parameters are given in Table 2. Parts a and b of Figure 5 show adsorption isotherms for nitrogen at 77 K and carbon dioxide at 298 K, respectively, on Vulcan. The solid lines in these figures show the bestfit isotherms; the nitrogen and carbon dioxide fits both have rmsd values of 0.35 cm3(STP)/g. The rms experimental error is 0.23 and 0.11 cm3(STP)/g, respectively.

Sweatman and Quirke

Figure 6. Best-fit PSDs for AX21 from nitrogen adsorption measurements at 77 K up to 0.3 bar (solid line) and carbon dioxide at 298 K up to 20 bar (dashed line) determined with the model parameters in Tables 1 and 2.

The best-fit PSDs for AX21 determined by nitrogen adsorption up to 0.6 bar at 77 K and carbon dioxide adsorption up to 20 bar at 298 K are shown in Figure 6. As with Vulcan, the nitrogen PSD is composed of two distinct modes, at about 0.85 and 1.7 nm. The carbon dioxide PSD also predicts distinct, but much broader, modes at 0.6 and 1.85 nm. Figure 6 indicates that adsorption of nitrogen at 77 K in AX21 is limited in pores with Hc < 0.8 nm. A similar but less pronounced feature is also observed with Vulcan in Figure 4. The pore volume and surface area of the nitrogen and carbon dioxide PSDs are 1.43 and 2.0 cm3/g and 2250 and 3580 m2/g, respectively. The PSD obtained from nitrogen at 77 K on Vulcan in Figure 4 is very different to the corresponding PSD obtained from carbon dioxide adsorption in ref 3. The nitrogen PSD in this work is presumed to be more accurate because the nitrogen isotherm at 77 K up to 0.3 bar contains more information than the carbon dioxide isotherm (see Figure 5) at 298 K up to 20 bar. The PSD obtained from nitrogen at 77 K on AX21 in Figure 6 is qualitatively similar to the corresponding PSD in ref 8. The quantitative difference is a result of the calibration method used to determine model parameters in this work (we allow Vulcan to be microporous) and the use of accurate simulation methods rather than the less accurate density functional methods in ref 8. The PSD obtained from carbon dioxide at 298 K on AX21 in Figure 6 is similar to the corresponding PSD in ref 5. However, the carbon dioxide PSD in Figure 6 is amplified with respect to the corresponding PSD in ref 3 (which has a volume of 1.58 cm3/g as opposed to 2.0 cm3/g in this work) because the carbon dioxide-surface interaction strength is less in this present work. This is because we have used the nitrogen PSD in Figure 4 to calibrate this parameter. Parts a and b of Figure 7 show adsorption isotherms for nitrogen at 77 K and carbon dioxide at 298 K, respectively, on AX21. The solid lines in these figures show the overall best fits; the nitrogen and carbon dioxide fits have rmsd values of 13.5 and 1.09 cm3(STP)/g, respectively. The rms experimental error is 5.2 and 0.9 cm3(STP)/g, respectively. The dashed lines show adsorption isotherm predictions using the PSD of the other gas. The predicted isotherms are in poor agreement with experiment, with rmsd values just under 200 and 100 cm3(STP)/g for the predicted adsorption of nitrogen and carbon dioxide, respectively. Specifically, the carbon dioxide PSD overpredicts nitrogen adsorption at 77 K, and the nitrogen PSD underpredicts

Characterization of Porous Materials

Figure 7. Fit to experimental adsorption isotherms of (a) nitrogen at 77 K and (b) carbon dioxide at 298 K, on AX21 using the PSDs for nitrogen and carbon dioxide in Figure 6. Symbols are experiment, solid lines are best-fit isotherms, and dashed lines are predicted isotherms using the PSD for the other gas.

carbon dioxide adsorption at 298 K. The agreement is very poor at the lowest pressures, improving at the highest pressures to obtain predictions within 25% of experiment. Discussion The most significant error in the predicted adsorption isotherms for AX21 in Figure 7a,b occurs at low pressure where the predicted isotherms are totally inaccurate. The rmsd of the predicted isotherms is significantly greater than the rms experimental error. A similar, but less clear, error is evident at low pressures for the carbon dioxide isotherm on Vulcan in Figure 5b. Indeed, the isotherm for carbon dioxide has a rmsd from the experimental isotherm in excess of the rms experimental error (0.35 cm3(STP)/g compared to 0.11 cm3(STP)/g). Clearly, this is unsatisfactory, and we are forced to search for an explanation. Since there are many potential sources of error in our approach, we shall attempt to list them all and determine which are the most significant. Significant errors could potentially arise because of the following: 1. the gas-gas interaction model 2. the slit pore model, that is, lack of pore curvature, junctions, energetic heterogeneity, and network connectivity 3. the Steele potential, that is, its functional form and σgs 4. the calibrated strength of gas-surface interactions, that is, gs 5. the assumption that the surface of Vulcan is like that of other microporous carbons

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6. the assumption that porous carbons are rigid and inert 7. the assumption that thermodynamic equilibrium is achieved in experiments Given the accuracy with which our molecular models reproduce experimental data for liquid-gas coexistence, we discard point 1. Our previous work5 indicates that the polydisperse slit pore model is sufficiently accurate at ambient temperatures for nitrogen, methane, and carbon dioxide adsorption up to 20 bar. However, nitrogen adsorption at 77 K up to 1 bar is likely to be more sensitive to pore geometry and impurities (ash) than carbon dioxide at 298 K because the external potential (modeled by Vext) is effectively stronger. This is clearly indicated by Figure 3a,b in which, in the narrowest slits at the lowest pressures, the density of nitrogen at 77 K is much greater than the density of carbon dioxide at 298 K. So, the effects of pore curvature, junctions, and energetic inhomogeneity are likely to be greater. Although we do not expect this factor alone to cause the very significant errors seen in Figure 7a,b, we cannot yet discard this point. It is unlikely that pore network connectivity can solely be responsible for these errors because they are not apparent in our previous work involving carbon dioxide and nitrogen at 298 K. However, there is speculation4 that very narrow constrictions (with chemical width less than or similar to the diameter of a nitrogen atom) in pore networks significantly reduce the diffusion of nitrogen at 77 K but not carbon dioxide at ambient temperatures. It is also interesting to note the rather unphysical nature of all the nitrogen PSDs generated in this work. It is reasonable to expect all PSDs to be composed of overlapping modes; that is, regions where f is zero between two log-normal modes are unexpected. This suggests either that the polydisperse slit pore model and Steele potential we have used are insufficiently accurate for nitrogen at 77 K or that the mechanism that causes the serious errors seen in Figure 7a,b concerns the behavior of nitrogen at 77 K in carbon slit pores and this behavior corrupts the PSD analysis. A great deal of theoretical work has relied upon the Steele potential. It is derived from theoretical considerations, and it is easy to show that it is an accurate approximation to the potential arising from summing individual atom-atom pair potentials in a graphitic surface. Although some microporous surfaces might consist of only one graphitic layer (as opposed to the semiinfinite number assumed in construction of the Steele potential), differences can be effectively accounted for by appropriate calibration of ss. The Steele potential was sufficiently accurate in our previous work at ambient temperatures. So, we discard point 3. We have calibrated the gas-surface interaction strength for nitrogen to yield a surface area of approximately 70 m2/g for Vulcan. We have then used the resulting PSD to calibrate the gas-surface interaction strength for carbon dioxide. As mentioned previously, this calibration is inconsistent in the sense that we model Vulcan as a microporous carbon whereas the reference BET value8 of 72 m2/g for the surface area assumes (infinitely) widely separated surfaces. So, it is quite possible that our calibrated values for ss are incorrect. However, although adopting a different value for the reference surface area of Vulcan would change our results quantitatively, we do not expect that it would affect the serious disagreement seen in Figure 7a,b. To justify this point, we have repeated our analysis for a range of values for the nitrogen ss. For each value, we vary the PSD until the best possible fit to experimental nitrogen adsorption on Vulcan is obtained.

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Figure 8. Variation of the rmsd of the best-fit isotherm for nitrogen adsorption on Vulcan (triangles) and Vulcan surface area (diamonds) with nitrogen-surface interaction strength. The lines are a guide to the eye.

Sweatman and Quirke

carbons. In every case, the isotherms for each gas are in good agreement at low pressures if scaled by appropriate factors. The simplest explanation for these data is that the surfaces of all these porous carbons are similar. However, Figure 9 indicates that the carbon dioxidesurface interaction is stronger for AX21 than Vulcan. This could be due either to a greater proportion of micropores in AX21 or the presence of more energetic sites in AX21 that are available for carbon dioxide only. Taken together, these figures indicate that nitrogen at 77 K cannot access some of the micropores in AX21 accessible to carbon dioxide at 298 K or that there are of more energetic sites in AX21 that are available for carbon dioxide only. Thus, we cannot yet discard point 5. It is conceivable that porous carbons are not inert and are able to “swell” upon adsorption of gas. If this happens, the degree of swelling is likely to depend on pressure and the particular gas adsorbed. However, this factor was not observed to be significant in our recent work5 concerning gas adsorption on microporous carbons at ambient temperatures. So, we discard point 6. It has been speculated previously that nitrogen freezes,8 or is at least diffusionally limited,4 in very narrow carbon pores, leading to blocking of a proportion of the pore network. This phenomenon could account for the results of this work. To investigate this possibility, we have investigated the self-diffusion of nitrogen at 77 K and carbon dioxide at 298 K in ideal slit pores for a range of widths and pressures. The diffusion of a substance in a slit pore can be expressed as20

∂ ln f ∇µ J ) -kBTD ∂ ln F Figure 9. Comparison of experimental adsorption isotherms for carbon dioxide on AX21 (solid line) and Vulcan (dotted line) at 298 K. The Vulcan isotherm has been scaled by a factor of 45.

As before, this PSD is then fixed and the carbon dioxidesurface interaction strength is adjusted until the best fit to experimental carbon dioxide adsorption data is obtained. Figure 8 shows the variation in the fit to nitrogen adsorption on Vulcan and the resulting Vulcan surface area with gas-surface parameter strength. As expected, we obtain the same qualitative error seen in Figure 7a,b for each value of ss in Figure 8. This result is primarily a consequence of the following observation. While the nitrogen adsorption isotherms at 77 K on Vulcan and AX21 (shown in Figure 1) are similar if the Vulcan isotherm is scaled by a factor of 29, the corresponding factor for the carbon dioxide isotherms is 45 (shown in Figure 9). Thus, once surface-gas interaction parameters are calibrated to Vulcan, it becomes impossible to find a PSD for AX21 that scales nitrogen adsorption at 77 K by a factor of 29 and simultaneously scales carbon dioxide adsorption at 298 K by a factor of 45. So, we discard point 4. Figure 1 indicates that Vulcan and AX21 share a similar microporous character. Alternatively, Figure 1 could be explained by supposing that the surfaces of Vulcan and AX21 are quite different (geometrically or energetically) and that the agreement is coincidence. However, Figure 1 is by no means an isolated example. Scaife19 presents isotherms for the adsorption of nitrogen and argon at 77 K on graphite, on Sterling and Vulcan (which are low surface area carbons previously thought to be nonporous), and on AX21, Supersorb, Pica, and Norit microporous (19) Scaife, S. Ph.D. Thesis, University of Wales, Bangor University, 2000. Kluson P. Unpublished work.

(9)

where the flux vector, J, is equal to the chemical potential gradient (in directions parallel to the pore) multiplied by the Darken factor and the total diffusion coefficient, D. The coefficient D comprises several terms,20 the most significant of which is generally the self-diffusion coefficient, Ds. This coefficient can be determined for a fixed number of particles by equilibrium molecular dynamics (EMD) simulation from the velocity autocorrelation function,

Ds )

1

N

∑∫ 〈uxi(0) uxi(t) + uyi(0) uyi(t)〉 dt 2Ni)1 0 ∞

(10)

where N is the number of particles in the EMD simulation; uxi is the x-component of the velocity of the ith particle, and so forth; the ensemble average, 〈 〉, denotes an average obtained by choosing a series of time (denoted t) origins. We have calculated this coefficient for nitrogen at 77 K and carbon dioxide at 298 K in slit pores by EMD simulation. The nitrogen and carbon dioxide models are identical to those in Table 1 except that partial charges and long-range corrections are omitted. The nitrogensurface interaction strength is set to ss/kb ) 30.0 K, which corresponds to the best possible fit to Vulcan data as shown in Figure 8 (the corresponding carbon dioxide-surface interaction strength is set to 24.0 K). In addition, we assume specular reflection20 at pore walls to simulate thermal motion of atoms in the pore walls. We use a temperature thermostat, and all initial configurations (and hence the fixed number of particles) are obtained from GCEMC simulations. We use the same gas slit pore model in these GCEMC simulations and choose a configuration (20) For example, see: Nicholson, D. Carbon 1998, 36, 1511 and references therein.

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Table 3. Self-Diffusion (See Text) in mm2/s of Nitrogen at 77 K in Carbon Slit Poresa wc (nm) P (bar)

0.46

0.76

1.26

1.76

0.0001 0.01 1

0.12 0.10 0.07

0.12 0.11 0.08

0.5 0.24 0.15

0.55 0.26 0.18

a The self-diffusion of liquid nitrogen at 77 K and 1 bar is about 0.75 mm2/s.

from this ensemble with a number of molecules equal to the ensemble average. The maximum trial displacement of molecules is manipulated so that the acceptance ratio is about 50% for all GCEMC simulations. For some GCEMC simulations of nitrogen, the problem of quasiergodicity21 is apparent; that is, in these simulations the probability of successful creation and deletion moves becomes extremely small because of the unlikely occurrence of a “hole” in dense nitrogen. Where quasi-ergodicity is a problem, we perform two simulations: one initialized from an empty box and another initialized from a “solid” configuration. The solid configuration is generated by first simulating with an enhanced potential (e.g., with ss increased by several orders of magnitude) and then relaxing this potential to the true value. These simulations are continued until the number of molecules reaches an apparent equilibrium over the course of 1 million attempted simulation moves. The ensemble average number of molecules is then taken as the average of these two quasi-ergodically limited simulations. A configuration with this average number of molecules is obtained by further simulation and manipulation of the gas-surface potential. Finally, each EMD simulation is integrated for 4500 time steps, the first 500 of which are rejected for the purpose of calculating Ds. Each time step is 0.4 fs. By using similar methods, we have also calculated the self-diffusion of liquid nitrogen at 77 K and 1 bar and carbon dioxide gas at 298 K and 20 bar. The results of these calculations for nitrogen, shown in Table 3, indicate that at 77 K nitrogen is diffusionally limited in carbon micropores with wc < 1 nm for sufficiently high (undersaturated) pressure. For wc > 1.0 nm or at lower pressures, nitrogen’s diffusivity in these slit pores increases. The self-diffusion coefficient of liquid nitrogen at 77 K and 1 bar is calculated to be 0.75 mm2/s, which is larger than all the values in Table 3. The same pattern of behavior is observed for carbon dioxide at 298 K, but the magnitude of the self-diffusion coefficient is generally an order of magnitude larger for carbon dioxide than nitrogen. This EMD study is not a rigorous study of the diffusivity of nitrogen in carbon slit pores because we have obtained the self-diffusion coefficient for a fixed number of particles, and the self-diffusion coefficient is only part of the total diffusion coefficient. However, it does provide a good indication of the poor diffusivity of nitrogen in very narrow carbon micropores. Likewise, it cannot be decided from these results whether nitrogen actually freezes in carbon micropores at 77 K. Rigorous studies of the freezing of Lennard-Jones fluids in ideal slits have been published recently, notably by Dominguez et al.22 and Radhakrishnan and co-workers.23,24 Dominguez et al. employed a NPzT (21) Valleau, J. P.; Whittington, S. G. Statistical Mechanics; Plenum: New York, 1986. (22) Dominguez, H.; Allen, M. P.; Evans, R. Mol. Phys. 1999, 96, 209. (23) Radhakrishnan, R.; Gubbins, K. E. Mol. Phys. 1999, 96, 1249. (24) Radhakrishnan, R.; Gubbins, K. E.; Sliwinska-Bartkowiak, J. J. Chem. Phys. 2000, 112, 11048.

ensemble in which the normal pressure in a slit is held constant and found first-order freezing of a Lennard-Jones fluid in a strongly attractive slit pore by using a thermodynamic integration route. Radhakrishnan and coworkers observed similar phenomena by using GCEMC simulation and a Landau free-energy approach to determine phase coexistence. The consensus of this body of work is that the freezing temperature of Lennard-Jones fluids in slit pores is influenced by pore width and gassurface interaction strength. For hard-walls, the freezing temperature is lowered with respect to the bulk freezing temperature, while for strongly attractive fluid-surface interactions it is increased. Generally, the magnitude of the shift in freezing temperature with respect to bulk freezing increases with smaller slit widths. For sufficiently attractive fluid-surface interactions (relative to fluidfluid interactions), the fluid layer in contact with the surface can freeze independently of the remainder of the fluid in the pore at a pressure which is lower than that at which pore freezing occurs. Finally, pore freezing can be accompanied by significant hysteresis. Radhakrishnan et al.24 predict that, for nitrogen in carbon slit pores, the pore freezing temperature is raised considerably with respect to bulk freezing and relatively more than for any other system they consider. However, this prediction is based on different molecular models for gas-gas and gas-surface interactions to those used in this work. At 77 K, nitrogen is 14 K above its triple-point. It is not clear whether nitrogen actually freezes in carbon slit pores at undersaturated bulk pressures at this temperature. In addition to slit pore width limited diffusion, diffusion of nitrogen at 77 K will also be reduced by pore junctions. Maddox and co-workers25 performed hybrid grand canonical molecular dynamics (GCMD) simulations of a model of nitrogen at 78.3 K in model graphite pore junctions and found that freezing and hysteresis occurred for particular junction configurations when they did not occur for comparable straight slit pores. Such effects are likely to also occur with nitrogen at 77 K in real microporous carbons. This behavior could be enhanced by the presence of impurity-induced high-energy adsorption sites. All the above work indicates a mechanism by which the low-pressure adsorption behavior in Figure 7a,b can possibly be explained. Essentially, carbon dioxide at 298 K is able to access more of the microporous volume than nitrogen at 77 K. This effect is particularly pronounced in AX21 but is also likely to occur with Vulcan if we accept that significant adsorption in Vulcan occurs in micropores. This diffusion limitation effect invalidates the calibration method used in this work, which is based on the assumption that each adsorbate experiences a very similar PSD. We propose that the poor diffusivity of nitrogen at 77 K in carbon micropores is the most significant of the above factors, and this is in general agreement with previous speculation.4,8 Although factors 2 and 5 above (the polydisperse slit pore approximation and neglect of carbon dioxide specific adsorption sites, respectively) might have some impact on the accuracy of our results, they are not required to explain the poor agreement in Figure 7a,b. Conclusions These results support previous claims that nitrogen is diffusionally limited in microporous carbons4,8 and indicate that connectivity can significantly affect the adsorption (25) Maddox, M. W.; Quirke, N.; Gubbins, K. E. Mol. Simul. 1997, 19, 267.

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of nitrogen at 77 K in these materials. This has important consequences for all characterization studies of microporous carbons that employ nitrogen at 77 K, including the BET method for calculating surface area. Similar consequences are likely for other materials and adsorbates where gas-surface interactions are as strong (relative to gas-gas interactions) as those for nitrogen at 77 K in porous carbons. For example, ref 24 provides an indication that adsorption experiments with nitrogen, carbon dioxide, methane, and argon on microporous carbons and silica should be performed at temperatures well above the corresponding bulk triple-point temperatures. We expect that measurement of nitrogen adsorption above 90 K or

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carbon dioxide adsorption above 260 K, but below their respective bulk critical temperatures and up to their respective saturation pressures, will provide more accurate characterization data than nitrogen at 77 K for porous carbons. Acknowledgment. Many thanks to David Nicholson and co-workers for assistance with the simulation code. We also thank EPSRC for support through the equipment grant GR/M94427. LA010308J