Langmuir 1994,10, 1556-1565
1556
A Direct Assessment of Mean-Field Methods of Determining Pore Size Distributions of Microporous Media from Adsorption Isotherm Data R. D. Kaminsky,’ E. Maglara, and W. C. Conner Chemical Engineering Department, 159 Goessmann Lab, University of Massachusetts, Amherst, Massachusetts 01003 Received November 4, 1993. I n Final Form: January 21, 1994’ The ability of mean-field-based theories to extract pore size information from adsorption isotherms of microporous materials is assessed. These theories represent natural and simple extensions of the Kelvin equation commonly applied in mesopore analysis. Adsorption behavior for a number of model spherical micropores is accurately determined using statisticalmechanics. Using these results,pore sizes are calculated via a prototypical mean-field method proposed by Horvhth and Kazawoe and compared to the true values. Although moderately accurate, it is found that several key underlying assumptions in the method limit its quality. Most notably, simple mean-field approaches calculate polydisperse pore size distributions for systems which are in fact monodisperse in pore size. Improvements to the methodology are discussed and explored. 1. Introduction
Microporous materials, media with pore diameters of less than 2 nm, are often employed in vapor separations, enhanced vapor storage, and high surface area catalysts.’ The distribution of pore sizes is central to understanding adsorption and transport in microporous materials. Two principle experimentalmethods exist for the determination of pore size distributions in microporous materiah2 The first is X-ray diffraction. This method is very effective and has a firm foundation in theory for materials with well-defined regular structures. The second method is adsorption isotherm analysis, commonly using nitrogen or argon at cryogenic temperatures. The interpretation of adsorption data of microporous materials to obtain pore distributions is less theoretically rigorous than X-ray diffraction, although the data are significantly less expensive to obtain. Moreover, the method is better suited for the study of media with polydisperse pore sizes and of amorphous solids. The most basic method of obtaining pore size distributions of porous media from adsorption isotherms relies on the well-known Kelvin equation (see section 3) to relate changes in adsorption with pressure, i.e., the adsorption isotherm.2 Although a number of assumptions are made, the Kelvin equation approach has proven to be highly satisfactory for many porous systems, particularly for systems with pore radii from 5 to 50 nm. This approach though breaks down for microporous systems. The reason for this is that the Kelvin equation method is based on bulk liquid properties. In particular, it assumes that except for adsorbate molecules immediately adjacent to the pore walls, the adsorbate molecules have no interactions with the pore walls. While this assumption is reasonably valid for large pores, since the adsorbateadsorbent interactions tend to be short ranged (about two molecular diameters), it is quite unrealistic for small pores where the adsorbate-adsorbent potential is significantly nonzero in all regions of the pore.
* To whom correspondence should be addressed. Abstract published in Advance A C S Abstracts, April 1, 1994. (1) Ruthven, D. M.Principles ofAdsorptionand AdsorptionProcesses; Wiley: New York, 1984. (2) Gregg, S. J.; Sing, K. S. W. Adsorption, surface area, and porosity; Academic Press: New York, 1982.
In 1979 HorvAth and Kawazoe proposed an intriguing approach to treat microporous systems which is simple yet potentially widely appli~able.~ The Horvbth-Kawazoe method basically extends the Kelvin equation by modeling the adsorbed molecules as a fluid influenced by a uniform potential field. By applying statistical thermodynamics a simple mean-field scheme is proposed to describe the interaction between the pore walls and the adsorbate molecules. The term “mean field” indicates that the potential interactions between an adsorbate molecule and the adsorbent, which may in fact be strongly spatial dependent, are replaced by a uniform potential field. In the original paper HorvAth and Kawazoe3employed their approach to microporous media modeled as a collection of idealized slit pores. The application of the HorvAthKawazoe approach was later extended by Saito and Foley4 to models using cylindrical pores and by Baksh and Yang5 to models using spherical pores. Although the HorvAthKawazoe approach shows moderate success,it is not always effective. I t is the goal of this paper to assess critically and to develop a firmer theoretical basis of mean-field methods of adsorption isotherm analysis to obtain pore size distributions in microporous media. In particular we will focus on the general mean-field method proposed by HorvAth and Kawazoe (which will be referred to as the “HK method”). The HK method represents the prototypical mean-field approach for the analysis of pore size distributions of micropores. The investigation of the HK method will center around applying it to well-defined model systems. The adsorption behavior of these model systems can be nearly exactly calculated by direct application of statistical thermodynamics. By comparing the results of the HK method with the known parameters of model systems, strengths and weaknesses in the approach become apparent. Before proceeding it must be noted that although meanfield-based approaches to obtain pore size distributions from isotherm data are most commonly applied due to their computational ease, more sophisticated statistical thermodynamic approaches exist. Currently one of the most promising approaches involves density functional
@
0743-7463/94/2410-1556$04.50/0
(3) Horvhth, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983,16 (6),470. (4) Saito, A.; Foley, H. C. AZChE J. 1991, 37 (3), 429. (5) Baksh, M. S. A.; Yang, R. T. AIChE J . 1991, 37 (6), 923.
0 1994 American Chemical Society
Pore Size Distributions of Microporous Media Off
Langmuir, VoZ.10, No. 5, 1994 1557
The potential interactions used in our model are based on the 12-6 Lennard-Jones (LJ) potential. The W potential models molecules as spherical particles interacting pairwise via
($y2-(37
u(x) = 4t,[
Figure 1. Diagram of the spherical pore model used in this ~ the fluid-fluid (adsorbate work. R is the pore radius, C T is adsorbate) collision diameter, do is the closest distance an adsorbate particle center may approach the pore wall, and r is the accessible pore radius, R - do. The specific number of adsorbate particles is depicted for example purposes only.
theory: which has proven to be a powerful technique for investigating the molecular behavior of absorbed fluids. In principle, sophisticated statistical thermodynamic methods allow one to model in detail adsorption phenomena involving complex adsorbent media. However, in practice such approaches can be computationally complex and have too many experimentally unknown parameters to be practical for direct engineering use. Nevertheless, such approaches are quite important for developing detailed fundamental understanding of the physics pertaining to fluids interacting with microporous media. Despite the fact that theoretically sophisticated methods may eventually supplant mean-field approaches as the methods of choice, simple mean-field analysis of adsorption isotherms remains a powerful and useful way of thinking about microporous media. The remainder of the paper is as follows. Section 2 defines the model system of study. Section 3 details the theoretical assumptions behind the Kelvin equation and Horv6th-Kawazoe approaches for determining pore size distributions from adsorption isotherms. Section 4 evaluates the HK method by applying it to fully specified models of microporous systems. Section 5 explores how the theoretical weaknesses of the HK method may be circumvented while still maintaining computational tractability. Lastly, section 6 presents our overall conclusions. 2. The Model System
The model system of study in this work is that of a bulk vapor at a prescribed pressure and temperature in equilibrium with a single closed spherical pore. This model, with a number of adsorbed molecules for examplepurposes, is depicted in Figure 1. Although the pore is closed and hence not physically connected to the bulk vapor, this poses no problem for analysis of equilibrium adsorption. Thermodynamicssolely demands equality of temperature and chemical potential between the bulk and adsorbed fluids for equilibrium to exist regardless of kinetic limitations. The spherical pore model was chosen for several reasons. First, the model is simple yet qualitatively similar to pores, or "cages", found in certain zeolites (without the interconnecting channels of course). Second, the model allows only a finite number of adsorbate molecules inside and hence simplifies theoretical analysis as discussed below. Third, a spherical pore intuitively should to be well-defined by a single pore radius. As will be shown in the next section, the HK method does not in fact assess a single pore radius for this system.
where x: is the center-to-center distance between the two fluid particles (Le., molecules), c~ is the fluid-fluid attractive interaction strength, and a, is the fluid-fluid interaction range (or "collision diameter"). The fluidfluid (Le., adsorbate-adsorbate) interactions are directly modeled with this potential. The fluid-pore wall (Le., adsorbate-adsorbent) potential interactions are modeled by taking the pore wall to be a thin spherical shell of uniformly distributed Lennard-Jones sites,5
where z is the distance of the fluid particle from the pore center, R is the pore radius as defined from center to wall, PA is the density of adsorbent sites per unit area of the pore wall, qpis the interaction strength between a fluid particle and a pore wall W site, and afpis the interaction range between a fluid particle and a pore wall W site. This potential is simple yet at least to a first approximation qualitatively realistic. The potential is repulsive for distances less than approximately do = 0.858afpfrom the pore wall nearly independent of pore size. For all the pores considered in this work the accessible pore radius is taken in terms of that accessible to the adsorbate center, not surface, and is defined as r = R - do. This is the approximation suggested by Horviith and K a ~ a z o e .It ~ should be noted that experimentally reported pore radii, especially for molecular sieves, are very often "kinetic" pore radii, that is, values characteristic of the narrowest opening into a pore. As such, kinetic pore radii indicate the size of the largest molecule which may diffuse through porous systems. The kinetic pore radius for a molecular sieve may differ substantially from the "structural" radii of the void spaces (micropores) in which the bulk of adsorption occurs. In this work, we are concerned only with equilibrium adsorption behaviorand hence structural radii and not kinetic radii. Interaction parameters for the model were set on the basis of those suggestedby Baksh and YangS for an argon-Y zeolite system. Specifically, qf/k = 119.8 K, a, = 0.3405 nm, pAtfp/k = 2776 K/nm2, and afp = 0.3065 nm. No attempt was made to optimize these parameters to model accurately any particular experimental system. Such an effort is not the goal of this work. The goal of this work is to explore in depth mean-field methods of measuring microporous pore sizes, and hence only a moderately realistic system needs to be defined. With the model defined, adsorption isotherms were nearly exactly determined via statistical thermodynamic analysis. In particular, the isotherms were calculated via direct calculation of the grand partition function, m
(3) (6) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Langmuir 1993,9,2693.
Kaminsky and Conner
1558 Langmuir, Vol. 10, No. 5, 1994
where N is the number of adsorbate particles in the pore, A is the activity, and Z(N) is the N-particle configurational integral. By basic statistical thermodynamics the average number of particles in the pore (i.e., the adsorption) for a given chemical activity is given by
o’6m 0.5
_ _ . .,
:
I
Activity is readily related to the bulk pressure via the equation of state and for an ideal gas is trivially related to bulk pressure via X = P/(k79,where k is the Boltzmann constant and T is the system temperature. The N-particle configurational integrals are in general more difficult to obtain. These integrals are defined as
Z(N) = sv...svexp(-U(rl, ...,r N ) / k n d rl...drN (5) where U(rl, ..., rN) is the total potential energy when N particles are at position vectors rl, ..., r N . The job of evaluating the N-particle configurational integrals is made considerably easier for the model employed in this work since a closed spherical pore can accommodate essentially only a finite number of adsorbate particles. Hence, the summation in eq 3 is naturally truncated at the maximum number of particles which may effectively fit in the pore. The details of how the configurational integrals are calculated are laid out in detail by K a m i n ~ k y .Basically ~ the method, which shall be referred to as the indirect configurational integral evaluation (ICIE) method, involves Monte Carlo numerical integration and is similar to the method by Woods, Panagiotopoulos, and Rowlinson: which shall be referred to as the direct configurational integral evaluation (DCIE) method. Unlike the DCIE method, the ICIE method allows for accurate calculation of configurational integrals up to moderately high adsorbate densities. As such, very accurate adsorption isotherms over a wide range of pressures can be produced. The key to the ICIE method rests with the statistical thermodynamic relation7
Z(N) = ZW-U V(e-UN’kT)B(N-l,V,n
(6)
where U N is the total potential energy contribution of the Nth particle. The subscript on the average indicates that the averaging is done over the canonical ensemble of N 1 particles in a system of volume V at temperature T. It is noted that eq 6 is closely related to the equation derived by Widomg for evaluating the chemical potential in the canonical ensemble. The average in eq 6 is readily evaluated by performing a canonical (NVT) Monte Carlo simulation with N - 1 particles and sampling the potential energy experienced by a randomly placed test particle. Hence if Z(N-1) is known, Z ( N ) may be evaluated. To obtain the entire set of N-particle configurational integrals, a series of canonical simulations with increasing N is run starting from the trivial case of N = 0 where Z(N=O) is by definition exactly unity. This approach though becomes exceedingly computationally expensive for systems with (7) Kaminsky, R. Submitted for publication to Mol. Phys. (8) Woods, G. B.; Panagiotopoulos, A. Z.; Rowlinson, J. S. Mol. Phys. 1988, 63 (l),49. (9)Widom, B. J. Chem. Phys. 1963,39, 2808.
F,
0.0 00
05
1:O
115
4
2 . 0 ~ 1 0 ~
PiPosat (b) Figure 2. Effect of pore size on adsorption isotherms. Adsorption isotherms are plotted for pores of sizes R* = 1.5 (dotteddashedlines),2.0 (dashedlines),2.5 (dotted lines),and 3.0 (solid lines) at !P = 0.8. The isotherms are plotted on (a)semilog and (b) linear plots for clarity. The adsorbate density p is defined as the average number of adsorbed particles per pore volume (V = 4/3nRS). R* = R / U Band T* = kT/eB.
more than about 100 adsorbate particles. As such the method is best for small, fully bounded systems. Openended slit and cylindrical pores can be studied by use of periodic boundaries, but in general large systems are needed to remove artifacts introduced by these boundaries. This being the case, in this paper only systems of spherical pores are considered although a few slit and cylindrical pores were tested for comparison, and the results obtained were qualitatively similar to those of the spherical pores. Several adsorption systems were studied using the ICIE method. Figure 2 presents adsorption isotherms at T* = kT/tff = 0.8 (Le., T = 95.8 K) for single isolated spherical pores of radii R* = R/aff = 1.5, 2.0, 2.5, and 3.0 (Le., R = 0.510,0.681,0.851,and 1.022nm, respectively). The single pores studied may be alternatively considered as a system composed of independent monodisperse-sized pores. So to have similar values for the different systems, adsorption density is reported as p = ( N ) /V, where ( N )is the average number of particles in a pore and V = (4/3)7rR3 is the volume of the pore. Adsorption in Figure 2 is depicted both on a log scale and on a linear scale with respect to the reduced pressure P/PoSat,where Posatis the bulk saturation pressure. A t saturation, the four different sized pores hold approximately 4,13,30,and 61 argon atoms for the R* = 1.5, 2.0, 2.5, and 3.0 systems, respectively. Layering behavior is particularly evident in the adsorption isotherm for the largest pore size considered. For reference, the critical point of a 12-6 Lennard-Jones fluid occurs at T* = 1.31 and the saturation pressure is PBata$/eff = 0.004 69 for the temperature considered. The bulk fluid is well modeled as an ideal gas up to saturation. On the basis of multiple Monte Carlo runs with different random seeds, the accuracy of the isotherms depicted in Figure 2 is estimated to be within f l %. The effect of temperature is shown in Figure 3. Adsorption isotherms for R* = 2.0 and 3.0 are depicted for
Pore Size Distributions of Microporous Media
Langmuir, Vol. I O , No. 5, 1994 1559
0.6
0.4
0.3
Q
0.2
0.1
. ~ 1 o . ~ 10.' I o0 P/Posa' Figure 3. Effect of temperature on adsorption isotherms. Adsorption isotherms are plotted for pores of sizes R* = 2.0 (solid lines) and 3.0 (dashed lines) at P = 0.6 (left-most lines), 0.8 (centerlines),and 1.0 (right-most lines). The adsorbate density p is defined as the average number of adsorbed particles per pore volume (V = 4/3rR3). R* = RIae and P = kT/em. O-'O
1o
IO-*
200
150 7
J m co c!
2 100
.
h
0
m
E
3
50
I
i
Figure 4. Experimentaldata for argon adsorbed in Linde type Y zeolite (solid line) and silicalite (dashed line) at T = 77 K. Filled circles are the data points. Adsorption is reported in terms of an equivalent volume of argon vapor at T = 298 K and P = 1 atm.
T* = 0.6 and 1.0. From the figure it is readily apparent that decreasing temperature tends to sharpen filling transitions and to cause filling to occur effectively first at lower values of PlPpt. However, decreasing temperature does not necessarily greatly affect the qualitative shape of the isotherms. The low-temperature isotherms in Figure 3 are quite similar in shape to some experimental adsorption isotherms. Figure 4 reports experimentally measured adsorption datal0for argon adsorbed in Linde type Y zeolite and silicalite at T = 77 K (P = 0.643). Zeolite Y has roughly spherical pores of about two argon diameters in radius, although the kinetic pore size is only about half that value." Silicalite has roughly cylindrical pores of about one argon diameter in radius.'l The isotherms in Figure 4 were obtained with an Omicron Omnisorp 300 adsorption system from Coulter Instruments that had been modified to add small doses (
