Langmuir 1992,8, 901-908
901
Simulation Studies of Adsorption in Rough-Walled Cylindrical Pores M.J. Bojan, A. V. Vernov,t and W. A. Steele* Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802 Received September 3, 1991. In Final Form: November 19, 1991 A model is developed for porous materials in which the atomic nature of the solid is explicitly included. The result is a gas-solid interaction law for moleculesin rough-walledpores of arbitrary shape. Simulations of adsorption isotherms and energies are reported for methane at 300 K in several model porous solids. The methane-solid interaction was obtained from a summation over solid atoms using parameters for the pairwise energies appropriate for methane in coal. Rough-walled cylindrical pores were considered in this work with radii equal to 6.5,10.0,and 13.4 A;for reference, adsorption on the flat adsorbent surface waa also studied. Calculations of the average gas-solid adsorption energy which show the effects of variable pore radius are shown. The average methane-methane energy on these heterogeneous surfaces is shown to be non-negligible. The isotherm data are subjected to a simple analysis that allows one to fit their dependence on pore radius in a straightforward way.
Introduction Computer simulation studies of adsorption in pores is producing a wealth of significant results. Classical theories of hysteresis in sorption isotherms have been validated;l the atomic layering structures of sorbed material has been determined?together with the effects of this layering upon thermodynamic proper tie^.^ One criticism of this work is that the pores studies are overly simple. Most often, one assumes parallel-walledslits or simple cylinderswith atomsolid interactions that are based on perfectly smooth walls. This gives potential functions dependent only upon atomwall separation distance. Such homogeneous models can produce artifacts in the computed thermodynamic properties and almost certainly yield incorrect transport properties (diffusion constants for motion parallel to the infinite dimension in particular). The exception to this generalization is simulations of sorption in zeolites.4 These are of course an extremely important class of materials; however, the fact that one is limited to experimental zeolite crystal structures means that both the experiments and the simulations tend to be directed at understanding the behavior as molecular sieves. In this paper, we develop an approach to the atomic modeling of porous materials that is at the same time realistic and highly flexible. It derives from an idea published by Bakaev5 who suggested that one could t Permanent address: Peoples' Friendship University, Chair of Physical & Colloidal Chemistry, Laboratory of Mathematical Simulation of Physical & Chemical Processes, M.-Maklaya, 6, Moscow,
USSR
117198. (1) Peterson, B. K.; Gubbins, K. E. Mol. Phys. 1987, 62, 215. Panagiotopoulos, A. Z. Mol. Phys. 1987,62,701. Peterson, B. K.; Walton, J. P. R. B.; Gubbins, K. G. Proceedings of the Zndlntermtional Conference on Fundamentala of Adsorption; Liapis, A. I., Ed.; Engineering Foundation: New York, 1987; p 463. Heffelfinger, G . S.; van Swol, F.; Gubbins. K. E. J. Chem. Phvs. 1988.89.5202. Walton. J. P. R. B.: Quirke. N. Molec. Simul. 1989,2,>61. Schoen, M.; Rhykerd,'C. L., Jr.; Cus'hmani J. H.; Diestler, D. J. Mol. Phys. 1989, 66, 1171. (2) Schoen, M.; Diestler, D. J.; Cushman, J. H.J. Chem. Phys. 1987, 87,5464. Snook, I. K.; van Megen, W. J. Chem. Phys. 1980, 72, 2907.
Peterson, B. K.; Heffelfinger, G.S.; Gubbins, K. E.; van Swol, F. J. Chem.
Phvs. (3) Magda, J. J., Tirell, M.; Davis, H. T. J.Chem. Phys. 1986,83,1888. Kaminsky, R. D.; Monson, P. A. J. Chem. Phys. 1991,95,2936. Schoen,
1990. . .~.. _ _ -,93. .. ., 679. - . -.
M.; Cushman, J. H.; Diestler, D. J.; Rhykerd, C. L., Jr. J. Chem. Phys. 1988,88,1394. Peterson, B. K.; Gubbins, K. E.; Heffelfinger, G. S.; Marconi, U. M. B.; van Swol, F. J. Chem. Phys. 1988,88, 6487. (4) Woods, G . B.; Rowhaon, J. S. J. Chem. SOC.,Faraday Trans. 2 1989,85,765. Woods,G. B.; Panagiotopoulos, A. 2.;Rowlinson,J. S. Mol. Phys. 1988,63, 49.
generate a random heterogeneous surface by utilizing the sequential addition algorithm reviewed by Finnep which produces an amorphous, close-packed hard-sphere solid. (Originally used to study metallic glasses, the basic idea was described by Bernal' and predates computers by several decades.) Of course the surface of a block of this material will exhibit random roughness. If one then allows the spheres to be oxides or any other atomic groups of interest, a pairwise summation of the gas atom-solid atom interactions will give an energetically heterogeneous adsorbent with a distribution of adsorption energy over the surface that is spatially random even for rather small samples. This model for a free surface has been and is still being intensively studied. Although the range of the energetic heterogeneity for this Bemal surface is somewhat limited by the algorithm, one can increase the geometric heterogeneity be deleting atoms from the solid surface to form a hole or a crack. This extended geometric heterogeneity will be discussed elsewhere. (Evidently, the introduction of chemical heterogeneity is also straightforward.) Here, atoms are removed from a glassy cubic block to form pores of the desired size and shape. We present the results of high-temperature high-pressure simulations of a simple gas sorbed in these pores. Although there is no practical limit to the complexity of the pore shapes that can be created in this way, the present paper is concerned with straight-walled cylindrical pores. (Simulations of adsorption in cylindrical pores of variable diameter are underway in an effort to learn more about the effects of bottlenecks upon thermodynamic and transport properties of the sorbed gas.) The physical problem that is the basis for the simulations to be presented here is to gain a better understanding of the sorption of methane in the micropore structure of coal. The temperature of primary interest is room temperature (300 K)and the pressures range up to 1000 psi (70 atm). Sorption in micropores of various radii was simulated. The thermodynamic data obtained included both adsorption heats and isotherms. Since room temperature is well above the methane critical point, hysteresis in the isotherms was neither expected nor observed. The algorithm (5) Bakaev, V. A. Surf. Sci. 1988,198, 571. (6) Finney, J. L. In Amorphous Metallic Alloys; Luborsky, F. E., Ed.; Butterworths: London, 1983; pp 42-57. (7) Bernal, J. D. Proc. R. SOC.London, A 1964,284,299. Bernal, J. D. Nature 1960,188, 910.
0743-7463/92/2408-0901$03.00/00 1992 American Chemical Society
Bojan et al.
902 Langmuir, Vol. 8, No. 3,1992 Table I. Parameters for the Pairwise Energies methane-methane methane-CH site
148.2 94.61
3.817 3.559
utilized was isokineticmolecular dynamics. Consequently, the diffusive transport of methane along these roughwalled pores was also observed. However, these results will be reported elsewhere. A principal goal of this study was to obtain sufficient information concerning the dependence of sorption behavior upon pore radius to allow one to interpolate and extrapolate the results to arbitrary pore radii ranging from a couple of atomic diameters up to the macropore range. In this way, one could calculate sorption properties in a porous sample containing pores not of a siiigle radius but the more realistic case of a distribution of pore radii where one obtains the total sorption at a given pressure by evaluating the amount of sorption in the pores of a given radius, multiplied by the fraction of such pores, and summing over all radii.
Simulations In any simulation based on summations of pair potentials, the chief criterion for realism is a proper choice for potential parameters. In sorption problems, this means both the gas-gas and the gas-solid interactions. A t the high temperature of these simulations, one can approximate the CH4 energies as those of an effectively spherical atom. For convenience, we use Lennard-Jones 12-6 functions in all cases u(r)= 4r((312-
(q
The well-depth and size parameters used for CH4-CH4 energies are listed in Table I; they are close to those obtained in previous analyses of the bulk second virial coefficient data for methane.8 In modeling the pairwise CH4-coal site energy, one should take account of the available experimental information. For the relatively low-rank coals that are of interest here, elemental analysis indicates that sufficient hydrogen is present to warrant a formula (CH), rather than the expected C,.9 Numerous simulation studies of bulk hydrocarbons have produced a reasonable consensus for the parameters of the united atom representation of the CH or CH2 interactions.1° These are listed in Table I. Lorentz-Berthelot combining rules are then used to obtain the values for the CH4-CH pairs that are also listed there. The second question is how to determine the density of the CH sites in our coal model. It is not appropriate to use spheres of radius ~TCH-CH, because this is a van der Waals’ separation, in contrast to the chemical bonding that characterizes the site-site separations in the solid. Therefore, we have assumed that the solid is made up of an amorphous packing of spherical sites with density equal to that of amorphous but nonporous carbon. Thus, we take Pcoal = 0.09 atom/A3 l1 and Paphere = 0.601/Usphere5for (8) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1954. (9) Whitehurst, D. D.; Mitchell,T. 0.; Farcasiu, M. Coal Liquefaction; Academic Press: New York, 1980. Meyers, R. A. In Coal Handbook; Meyers, R. A., Ed.; Marcel Dekker, Inc.: New York, 1981. (10) Weiner, S. J.; Kollman, P. A.; Case, D. A.; Singh, U. C.; Ghio, C.; Alagona, G.; Profeta, S.,Jr.; Weiner, P. J . Am. Chem. SOC.1984,106,765. (11) CRC Handbook of Chemistry and Physics, 59th ed.; Weast, R. C., Astle, M. J., Eds.; CRC Press, Inc.: West Palm Beach, FL, 1978. (12) Hoover, W. G. Phys. Rev. A: Gen. Phys. 1985,31,1695. Evans, D. J.; Morriss, G. P. Chem. Phys. 1983, 77, 63.
Figure 1. Computer-generatedpicturesof the adsorbent atoms used in the pairwise summations of molecule-solid energy for methane in a rough-walled pore. A view directly along the pore axis is shown in panel A and a view at a slight angle to the axis is shown in B.
amorphous close-packing of spheres of volume Usphere. If One Sets Pcoal = Psphere, the result is an average site-site spacing of 2.3 A in the adsorbent. Thus, the sequential addition algorithm was used to build up a cubic sample of 6500 randomly packed hard spheres. One virtue of this algorithm is that it generates a solid that obeys two-dimensional periodic boundary conditions. A very large sample was produced by attaching replicas of the original cube to two sides of the original. Atom matching at these sides is present, as implied by periodic boundary conditions. The sphere size was chosen to give a box which is 41.4 A X 39.8 A X 41.7 A,which thus gives an adsorbent of density of 0.094 atom/A3. A cylindrical pore of radius R can be generated in this block by the simple expedient of deleting all atoms in the cube whose centers are within a distance R of a line drawn through the centers of two opposing sides of the cube. This line is thus the pore axis and, by definition, is parallel to the z direction in the pore. Since the wall of such a pore is rough on an atomic scale, one has at best an average pore radius. Simulations of methane adsorption are reported here for approximate pore radii of 6.9,10.4, and 13.9 A;in addition, adsorption on the flat surface of this solid was also simulated. For a molecule in the pore, the gas-solid interaction is assumed to be an infinite sum over all the CH sites of the solid. However, the infinite sum is impractical and unnecessary, since the atomic structure of the adsorbent is no longer of interest at large molecule-site distances where the energy is small in any case. Therefore, only the atomic structure of roughly the first three layers of sites was taken into account; for larger distances, the energy was estimated by integrating the long-ranger+ term over the solid. This was done partly analytically and partly numerically; tabulated results were fitted to an interpolation function for use in the actual simulations. Figure 1 shows the computer-generated spherical atoms in the cylindrical shell that forms the atomically discrete pore wall for the smallest pore radius studied. This approach leaves the number of sites actually remaining in the summation at a level that is readily handled in the calculation of the methane-solid force during the simulation. For a given pore radius and at T = 300 K, isokinetic molecular dynamics12were carried out for various numbers of methane molecules in the pore. As noted above, periodic boundary conditions and the minimum image convention were employed in the z direction, so that the pore is essentially infinite in length. A timestep of 1.6 fs was employed;a run of 8000 timesteps was adequate to obtain equilibrium at this temperature, and data gathering was done for 20 000 timesteps. Averages of the gas-solid and gas-gas components of the energy per methane were evaluated. In addition, configurations of the molecules
Adsorption in Rough- Walled Cylindrical Pores
Langmuir, Vol. 8, No. 3, 1992 903
were saved at intervals and later used to calculate mean square displacements and, most importantly, the adsorption isotherms. The isotherms were calculated by evaluation of the chemical potential Clads of the methane by the particle insertion method of Widom.13 This calculation rests upon an evaluation of the average Boltzmann factor, BF, of a methane molecule at point r within the pore
63
BF(r) = (exp[-u(r)/kTl) (2) where u(r) is the total interaction energy of a molecule added at a point r (chosen randomly within the pore) to a computer-generated configuration of methanes. The brackets indicate averaging over configurations and over a large number of particle insertions. It is now well-known that
47
(3) where @) is the standard state chemical potential for an ideal gas in equilibrium with the fluid in the pore. Note that the ratio BF(r)/p(r),where p(r) is the adsorbed phase density at r, should be independent of r. Once it was established that this is actually the case, this fact was used to improve the averaging process. Before presenting the results of the simulation, it is helpful to give pictorial representations of the surface roughness and the energetic heterogeneity of the pore walls produced in this way. This can be done by locating the minima in the methane-solid energy; these will occur at distances rmin from the pore axis and will correspond to adsorption energies U m h . Both quantities will depend upon z, 4, the cylindrical coordinates that locate the radius vector along which one is calculating a minimum. Thus, z is the distance along the axis and I$ is the azimuthal angle. If the cylindrical inner surface of the pore is “rolled flat”, one can make contour plots of rmin and Umin on a surface whose coordinates are z, and 4 R m i d where R m i d is the median pore radius. Figure 2 shows lines of constant rmin for the case &id = 10.0 A. The surface roughness is clearly illustrated, with rminranging from 9.0 to 10.9 A. Since one expects that the “pockets” on this surface will correspond to strong adsorptive sites, these are shown by the black areas which indicates the regions where rmin > 10.0 A. A similar contour diagram for the adsorption energy is shown in Figure 3. The minimum energies for this heterogeneous surface range from -1.35 to -3.42 kcal/mol; the black regions indicate those areas where this energy is less than -2.6 kcal/mol. There is a reasonable, but far from perfect, correspondence between the locations of these strong adsorption areas and the low points (large rmin) on the surface shown in Figure 2. Although there is a temptation to call these regions adsorption sites, this is not appropriate. The regions are not only highly irregular in shape, size, and spacing, they are also produced by choosing an arbitrary contour line in a continuum of adsorptive energy variations over this surface. Another way of looking at the heterogeneity is to evaluate the distribution of the adsorption energies. Although the spatial distribution of the energy minima is lost in such a function, the use of “site distributions” in descriptions of heterogeneity is widespread.14 Here, the surface of Figure 3 is subdivided into a 50 X 50 set of elements and the adsorption energy at the center of each element is determined. (The element size was chosen to minimize the variation in adsorption energy within an element.) A (13) Widom, B. J. Stat. Phys. 1978,19, 563. (14) Jaroniec,M.; Madey, R. Physical Adsorption on Heterogeneous Solids: Elsevier: Amsterdam, 1988.
31 . I
s
2
16
0
Figure 2. A contour diagram of lines of constant rmb,where ris the distance (from the pore axis) of the minimum in the gassolid energy. The surface shown is that for a ore of radius 10 8, and the contour lines are drawn for every 0.2 from 9.0 to 11.0 A. The blackened regions indicate rmin> 10.0 8,.
x
histogram of the number of elements with energy between Uand U + 6U is then constructed. The results are shown in Figure 4, where it can be seen that the distributions are roughly Gaussian in U , with widths that are more or less independent O f R m i d , but peak positions that shift regularly with Rmid. It is concluded that this shift in position is due to the fact that an atom in a pore with strongly curved walls (i.e., small R m i d ) will in general be closer to more sites and thus will have a more negative U than for a flatter surface. It is important to realize that the energy distributions of Figure 4 cannot be viewed as a site distribution, since the sitewise adsorption concept is based on the idea of site occupancy in the monolayer equal to zero or to unity. Consequently, the site dimensions must be close to the adsorbate atom diameter. However, the choice of a 50 X 50 subdivision evidently gives “site” dimensions of -0.6 A-much smaller than the methane size of 3.8 A. In fact, a single methane will occupy -40 sites using this arbitrary definition. Results In addition to simulations of isotherms and adsorption energies, it is useful to carry out direct calculations of Henry’s law constant KH and the adsorption energy o(0)
Bojan et al.
904 Langmuir, Vol. 8, No. 3, 1992 I
63
I
I
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47
n
4
L
W
c4
*
31
c
I
. I
c a
r
O C - 1
-1.8
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I
-1.4 U/R (de0 K )
1
- 1.0
I
I
Figure 4. Distribution functionsfor the fractionof the surfaces characterized by a given energy value U = -u-. The surfaces of the systems studied were divided into 2500 elementa and a histogram of the minimum energy obtained for each elementwas evaluated. Curves are shown for each of the four systems considered here, with peak positions shifting from left to right for Rmid = 6.5, 10.0,13.4,and 00 A, successively.
16
Table 11. Henry’s Law Constants and Zero Coverage Limiting Energies of Adsorption Rmid
0
0
8.1
16
24
32
Z(N Figure 3. A contour diagram of lines of constant u,dR, where u- is the value of the minimum gas-aolidenergy. The surface is the same as that for Figure 2;contoursare drawn for every 400 K from -700 to -1300 K. The blackened regions indicate umh
< -1300 K.
of an isolated methane in the pore. The expressions to be evaluated are
K’,
J= ‘ AkT
Vwm {expE-u,(r)/kTl-
1) d r
(4)
Jvpmus(r) {exp[-u,(r)/kTl- 1)dr 6(0) =
Jvwjexp[-u,(r)/kTl - 1)d r
(5)
Simpson’s rule can be used to evaluate these integrals. Initial slopes of the isotherm are given by na lim - = K’HP (6) n .4 A where A is the surface area and K’H is KHIA. Also, since the initial value of the energy of adsorption is given by U(O),these numbers serve as a useful check on the accuracy of the simulations. Table I1 shows the numerical results. A feature of these high temperature studies is that a significant fraction of the total methane in the pore is unadsorbed gas. One can readily determine this by evaluating the local density of the adsorbate. In fact, we have calculated the radial density p ( r ) , where r is now the distance from the pore axis. This is done by determining the average number of molecules navein cylindrical shells
6.5 10.0 13.4 m
KH’, molecdes/(Az atm) 1.76 x 10-3 1.12 x 10-3 0.881X 0.800 x 10-3
&O), kcal/ mol -2.343 -2.057 -1.900 -1.824
of radii r - 6r/2 and r + 6r/2. The density is then given by naVe/2rrZ6r,where Z = pore length. This calculation gives another check on the simulation accuracy since the limiting density far from the surface is P/kT. (Small corrections for methane nonideality are necessary at high pressure.) In adsorption calculations, one defines n,, the molecules adsorbed, as ntot - nun,where tot and un denote total and unadsorbed molecules, respectively, in the adsorption volume. Here, nun = V p r e pun = Vpre.P/kT. In our case, Vpre is not precisely defined, but one obtains a good working definition by setting the outer radii of the pore equal to those distances where p ( r ) drops sharply toward zero. Figure 5 shows three curves of p(r) computed for different pore fillings and pore radii. The unadsorbed gas density is indicated, as are the densities associated with monolayer formation on the pore wall and multilayer formation in the interior region. The distance at which the changeover between monolayer and multilayer occurs is taken to be in the region between the density peaks; since the density does not go to zero in this region, the split is somewhat arbitrary. We have taken it to be the distance of minimum density evaluated at the highest coverage simulated for each pore radius. It is evident that all three can give significant contributions to nbt. Furthermore, the dependences of the densities of these separate contributions upon pressure are rather different, as will be shown below. In Table 111, some relevant data such as pore volume and pore area are listed for the systems studied. Because
Langmuir, Vol. 8,No. 3,1992 905
Adsorption in Rough- Walled Cylindrical Pores
1
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Table 111. Pore Size Parameters
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002
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Figure 5. Dependence of the local density of methane upon distance from the pore axis is shown for adsorption at 300 K in the three pores considered. The three panels are methane loadings arbitrarily chosen to be 60, 50, and 90 molecules for poreradiiRmid= 6.5,10.0,and 13.4A,respectively. The hatching scheme indicates that the total density curve is subdivided into unadsorbed gas (\\\\),monolayer adsorption ( X X X X ) , and multilayer adsorption (////).
of the rough walls employed, the calculations of these quantities are nontrivial. Their evaluation was done in the course of Henry's law calculation and rests upon the idea that no adsorbate atoms will be found in regions where exp[-u,(r)/krJ is very small. Thus, as one moves along
6.88 10.35 13.82
4782 10830 24830 26550
1391 2093 3593
1315 2023 3486 1649
a pore radius vector, the location of the pore wall can be taken to be at the distance where this Boltzmann factor initially becomes smaller than 1. The average of this distance over the entire pore wall is denoted by RH. A glance a t the curves in Figure 5 shows that there is a significant difference between Rmidand RH. The definition of as the median location of the potential minimum means that the position of maximum monolayer density will be found close to Rmid, but the definition of RHensures that it will correspond to a point on the steeply decreasing portion of the density curve. The pore volume V,,,, has been calculated using RH, since the integrals in Henry's law calculation should be carried out over the larger volume. However, if one wishes to evaluate monolayer adsorption per unit area, the area used should be based on &id, since this is close to most probable radius of the cylindrical surface which passes through the centers of the monolayer molecules. Table I11 indicates that there is a small but non-negligible difference between areas calculated from 2TRmidZ and ~ T R H Z . Figure 6 shows the simulated isotherms obtained for the three pore radii considered. The total amount of gas per unit pore volume VH(see Table 111)is plotted together with the three contributing terms: the unadsorbed gas (equal to P/k!i');the gas held in the first monolayer, and the multilayer gas. The boundary between multilayer and monolayer is somewhat arbitrary, but the general idea is to take it a t the end-point of the steep decrease in monolayer density, as illustrated in Figure 5. (One expects that the Ymonolayer width" will be close to the methane size-values taken here give widths ranging between 3.3 and 3.5 k)I t is evident that all three contributions to the total gas in the pore are significant. Consequently, the procedure followed here is that fits of these data to a monolayer isotherm should be based on the monolayer portion only. One can use a separate Henry's law to describe the multilayer sorption. The adsorption isotherm obtained for the flat surface is shown separately in Figure 7 because the amount of unadsorbed gas is dependent upon an arbitrarily chosen adsorption volume. In the simulation, one inserts a hard wall at a distance sufficiently near the surface to keep most atoms in the adsorbed layer, but far enough to have no significant effect upon the adsorbed layer. The actual value used here was 18.4 A. The rather large amount of unadsorbed gas indicated in Figure 7 is of course determined by this distance and could be significantly raised or lowered without affecting the sorption results. Average energies of adsorption vary with pore filling or amount adsorbed as shown in Figures 8, 9, and 10. In order to exhibit the dependence of these energies upon the surface densities of the adsorbed methane, the ordinates are molecules adsorbed per unit area of the monolayer. Figure 8 gives the average methanemethane energy; the change in the average methane-solid energy with increasing coverage is plotted in Figure 9. (Zero coverage methane-solid energies are tabulated in Table 11.) The total integral molar adsorption energies shown in Figure 10 are of course sums of the data of Figures 8 and 9 and Table 11. A t 300 K, methane is very much a supercritical fluid, both in two and three dimensions. A linear variation in
Bojan et al.
906 Langmuir, Vol. 8,No. 3, 1992 a
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-Total --Layer I
-------
0.080
Layer 2 Unads
0.064
Prrrrurc (aim) 0.010 --Layer I ---Layer 2 Unada
0
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1
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1
20
40
60
80
100
Pressure ( a h )
Figure 7. Same as Figure 6,but for the flat surface. Since the adsorption volume is arbitrary for this system, the plota shown are for molecules per unit area rather than per unit volume as in Figure 6.
t
200 Pressure (aim)
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Y
C
-
a
t
/
Pressure 1aim)
Figure 6. Isothermsobtained from the simulations for methane
in the three pore systems studied. €bsulta are molecules per unit pore volume plotted versus pressure. The four curves in each panel indicate the pressure dependence of the total gas in the pore and the amount in the monolayersand in the multilayer region (see Figure 5) and the unadsorbed gas. The data are given for &d = 6.5 A (a), 10.0 A (b), and 13.4 A (c). methane-methane energy is expected on the basis of van der Waals’ or mean field models. Of course, the slopes of these curves are sensitive to the nature of the surface heterogeneity, since the distribution of strong adsorption energies over the surface determines the number of pairs
1
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0
0.01
1
0.02 “//Amid
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1
0.03 0.04 (mOleC/&*)
1
0.05
Figure 8. Dependence of the average methane-methane interaction energyupon methane loading is shown here. Theenergy is plotted againstthe adsorbedphase surfacedensityin molecules/ A*, but both monolayer and multilayer molecules are included in this quantity (unadsorbed gas is excluded). The area is evaluated from 2?rR,&. The 8 bok for &d = 6.6 A, 0 for R d = 10.0 A, 0 for Rdd = 13.4 G d X for the flat surface have the same meaning here as in Figures 9-12.
of methane at a distance suitablefor a significantmetbanemethane interaction energy. Discussions of this problem have been given by Zgrablich et al.,I5 who emphasize the fact that one should take this lateral interaction into (16) Riccardo, J. L.; Chade, M. A.; Pereyra, V. D.; Zgrablich, Submitted for publication in Langmuir.
G.
Adsorption in Rough- Walled Cylindrical Pores I I
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Langmuir, Vol. 8, No. 3, 1992 907
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&,id,
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6.5 10.0 13.4 m
n
8
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1
0.02
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0.04
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0.05
nod/Amid ( m O l e C / A Z )
Figure 9. Change in the average methane-solid interaction energy with methane surface density, defined as in Figure 8.
Table IV. Henry’s Law Constants K H ~ molecules/ ’, (A2 atm) K H ~molecules/atm , 1.68x 10-3 0.174 1.02 x 10-3 0.186 0.831 X 0.245 0.741 X 0.375
plotted differences reflect changes in the distributions of the adsorbed atoms over the gas-solid energy distributions. In this respect, the simulations confirm the well-known argument that the preferred occupancy of the strongest area elements at low coverages becomes less marked as the surface fills. An interesting feature of Figure 10 is the fact that the (negative) energy of adsorption in the pores actually increases somewhat with increasing coverage, in contrast to that for the flat surface. Evidently, the increase in the average methane-methane lateral interaction energy at all but the lowest coverages makes a sufficiently large contribution to actually outweighthe decreasing methanesolid energy. This point is made clear by the curves of Figure 8 which show a smaller lateral interaction for the flat surface than for the curved pore walls, probably because the methanes remain closer together when distributed over a circular arc than on a straight line. (In the smallest pore, relatively large multilayer adsorption develops which involves a considerable methane-methane attractive energy.) Analysis of Isotherm Data
Figures 6 and 7 show that the simulated isotherms for these systems change with pore radius in a regular way. In an attempt to determine the specific reasons for these changes, the isotherms have been treated in an ad hoc fashion by first assuming that the monolayer and the multilayer portions can be taken to be additive and that coupling between monolayer and multilayer molecules need not be explicitly considered. The data show that na2, the number of molecules in the multilayer, is linearly dependent upon pressure. Therefore, we define a secondlayer Henry’s law constant by
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0
0.01
0.02
0.03
I
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0.04
0.05
1
02
n d / A m i d (molec / A 1
Figure 10. Average potential energies per molecule in the ad-
sorbed film are plotted versus methane surface density, defined
as in Figure 8. These energies are equal to the integral molar energies of adsorption for these systems and can be used to evalyate the isosteric heat of adsorption qat from qBt= 0 +
n.caU/an,).
account in realistic theoretical treatments of the problem-a point that has been often ignored in previous work. The decreases in average methane-solid energy shown in Figure 9 are evidently a consequence of adsorption on successively weaker elements of area as surface density increases. The fact that these curves are similar for all systems studied is not a great surprise in view of the energy distributions of Figure 4. The effects of the shift in peak position have been largely removed by evaluating the energy relative to the zero coverage value. In fact, the
na2 = K H ~ (7) Best-fit values of KHZare shown in Table IV. Note that an attempt at calculating KHZfrom an equation similar to (3) must make some assumption concerning the effects of the methane molecules in the underlying monolayer upon the overlayer isotherm. The monolayer part of the total adsorption is treated first by evaluating a Henry’s law constant K H ~from ’ eq 3 modified by integrating radially only between those distances that have been taken to be the inner and outer limits of the monolayer. Values of KHl’ calculated in this way are listed in Table IV. Plots of nal/Amid, the molecules adsorbed per unit monolayer area, versus pKHI/Amid are shown in Figure 11. It can be seen that all data fall more or less on a single curve which has the general shape of the Langmuir isotherm. The simplest approach turns out to be adequate here. Thus, we write
with l/P* = (KHl/Amid)/(nml/Amid). This equation gives a reasonably good fit to the data using a monolayer capacity nmllA = 0.038 molecule/A2,as can be seen by the curve in Figure 11. Of course, more complex theories will yield better agreement with the simulations, if only because they will contain more
908 Langmuir, Vol. 8, No. 3,1992
Bojan et al.
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Figure 11. Monolayer adsorption per unit area for all four systems studied is plotted versus pressure times Henry's law constant for the monolayer of each system. Points are the simulated data and the smooth curve showsa Langmuir isotherm (eq 8) for a monolayer capacity nml/Amid= 0.038 molecule/A2. adjustable parameters. When the various contributions to the total methane loading in the pores are summed to give the methane loading in each of the pores considered here, this oversimplified approach to isotherm analysis gives satisfactory results, as can be seen in Figure 12. Of course, the heats of adsorption implied by this model are incorrect, since the use of a homogeneous Langmuir model implies negligible methane-methane interactions as well as a coverage-independent methane-solid energy. These two errors partially cancel each other to give a total adsorption energy that does not vary much with
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Figure 12. Simulation resulte for the pressure dependence of the total methane loading at 300 K are compared with curves calculated as described in the text.
coverage-see Figure 10. There is no plausible reason why this cancellation should be generally (or even frequently) encountered for adsorption on other heterogeneous surfaces. Furthermore, a more careful analysis is required before one can conclude that the cancellation in the heats will necessarily lead to a simplified adsorption isotherm.
Acknowledgment. This work was supported by agrant from the Gas Research Institute. Many helpful discussions with Dr. Victor Bakaev are gratefully acknowledged. Registry No. CHI, 74-82-8.
