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A Simulation Study of Energetic and Structural Heterogeneity in Slit-Shaped Pores David Nicholson Computational and Structural Group, Department of Chemistry, Imperial College, London SW7 2AY, U.K. Received September 1, 1998. In Final Form: December 1, 1998 Adsorption of methane and carbon dioxide in model pores has been investigated using grand ensemble simulation. The adsorbent models were designed to imitate two heterogeneous effects that are generally regarded as being of distinct origin. The first of these, energetic heterogeneity, was modeled by generating thermally disordered surfaces that were roughened on an atomic scale. The second type of heterogeneity, structural heterogeneity, was modeled using smooth walled slit pores having a range of pore widths, and adsorption in these was examined as a function of pore width distribution. Particular emphasis has been placed on isotherms and isosteric heats of adsorption (qst). It is shown that the rapid decrease of qst with adsorbate density in the early stages of adsorption cannot be satisfactorily accounted for by energetic heterogeneity. On the other hand, plots of heat of adsorption against adsorbate density are shown to be particularly sensitive to structural heterogeneity. It is argued that the isosteric heat measurements could form the basis of a valuable method for analyzing micropore distribution.
1. Introduction All solids are inherently disordered above absolute zero, and it has been recognized that the surface region, being a region of broken symmetry, is particularly liable to be defective.1-4 Above about 2/3 of the melting temperature, roughening and/or surface melting can occur, depending on the crystallographic orientation of the surface and its specific chemical identity,5 so that solids prepared at high temperature may retain substantial disorder when cooled rapidly.4 Many simulation and theoretical studies have been made of adsorption in narrow pores. Surface roughening in these models is interesting because not only high-energy surface sites are created, due to vacancies, steps, etc., but also microscopic regions of variable pore width where atoms displaced from regular lattice sites are in closer than average proximity across the pore.6 Both atomic disordering and pore-narrowing result in enhanced heats of adsorption, due to the existence of strong adsorption sites. Furthermore, clustering of the adsorbate at highenergy sites can be anticipated which may also modify adsorption. Thermal disordering of a surface is one way in which adsorption heterogeneity can be manifested. A second way is through structural heterogeneity.7 It is well established that potential energy wells in pores become progressively deeper as the pore is narrowed, due to overlap, until a limiting width is reached when repulsive interactions begin to dominate. Internal pore widths that are slightly (1) Stranski, N. Naturwissenschaften 1942, 28, 425. (2) Weeks, J. D.; Gilmer, G. H. Adv. Chem. Phys. 1949, 40, 157. (3) Dunning, W. D. The Gas Solid Interface; Flood, E. A. Ed.; Marcel Dekker: New York, 1967; p 271. (4) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (5) Broughton J. Q.; Gilmer G. H. J. Chem. Phys. 1983, 79, 5105. (6) Bojan, M. J.; Vernov, A.; Steele W. A. Langmuir 1992, 8, 901. Steele W. A.; Bojan, M. J. Proceedings of the 4th IUPAC Symposium on Characterisation of Porous Solids; Royal Society of Chemistry Special Publication No. 213; McEnany, B., Mays, T. J., Rouquerol, J., RodriguezReinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Royal Society of Chemistry: London, 1997; p 49. (7) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Surfaces, Elsevier: Amsterdam, 1988. Henchal, M.; Brauer, P.; v. Szombathely, M.; Jaroniec, M. Stud. Surf. Sci. Catal. 1994, 87, 633.
larger than a given molecular probe, therefore, present very strong adsorption sites. It follows that an adsorbent, in which a wide range of pore widths is available, presents a distribution of adsorption energies. Such adsorbents need not necessarily possess a network of interconnected pores. For example a compact of spheres, or of more irregular particles, offers a range of “pore widths” to an adsorbate, ranging in size from the ultramicroporous crevices, where adsorbent particles are nearly in contact, to much wider voids and cavities. It is clear from the foregoing discussion that, since constrictions may also occur inside pores with atomically roughened surfaces, it is not easy to make a sharp distinction between energetic and structural heterogeneity. Model studies in sphere packs and related systems have been made,8-10 but although these models closely approach a realistic representation of some materials, it is not always easy to discern the underlying cause of some of the effects observed in simulations. In the present work, we have used grand ensemble simulation to examine the adsorption of two species, methane and carbon dioxide, in slit pores with thermally roughened surfaces and in smooth-walled slit pores of different widths, in an attempt to clarify the influence of different kinds of heterogeneity on adsorption in complex adsorbent materials. The two adsorptives differ in being essentially spherical and linear, respectively. A further incentive for this choice is that separation of methane/ carbon dioxide mixtures is industrially important. The experimental properties of primary interest are adsorption isotherms and isosteric heats of adsorption. The latter are generally regarded as being especially sensitive to heterogeneity. In particular, a heat of adsorption that decreases steeply during the initial stages of adsorbate loading is usually taken to indicate the existence of a few strongly adsorbing sites which are occupied at low pressure, although, conversely, it is not necessarily true that heterogeneous surfaces will always show such a decrease.4 As a particular example, it was observed11 (8) Kaminsky, R. D.; Monson, P. A. Langmuir 1993, 9, 561. (9) MacElroy, D. M.; Ragahavan, K. J. Chem. Phys. 1990, 93, 2068. (10) Vuong, T.; Monson, P. A. Fundamentals of Adsorption; LeVan, D., Ed.; Kluwer Academic Publishers: Boston, MA, 1996; p 1009. (11) Masukawa, S.; Kobayashi, R. J. Chem. Eng. Data 1968, 39, 6402.
10.1021/la981143q CCC: $18.00 © 1999 American Chemical Society Published on Web 03/04/1999
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that qst for methane on silica gel adsorbent at 300 K decreases monotonically by more than 5 kJ mol-1 from its zero coverage value of ∼11 kJ mol-1, over a range of increasing adsorbate densities below the knee of a type I isotherm. Simulation studies of heats of adsorption in pores have been reported for zeolitic adsorbents12 and for silica sphere packs.10 Various factors can contribute to heterogeneity in these models. A steep decrease in qst with density was noted for simulations of methane adsorption in model sphere packs, and the shape of the heat curve was found to vary considerably with temperature and with the specific model. 2. Methods 2.1. Preparation of the Adsorbents. The starting point for the model was the fcc lattice. The model pores were all constructed as slits having (100) surfaces with two opposite walls each containing two layers of 20 × 20 atoms. The pore width, measured as the distance between the plane of the atomic nuclei in the inner pore walls, was varied from 1.03 to 2.27 nm. The atoms of the solid were modeled as Lennard-Jones spheres with a hard sphere diameter, σss ) 0.26 nm. In the adsorption simulations, the Lennard-Jones energy parameter, ss/k, was set to 200 K for most simulations. This ensures that the two-layer adsorbent is approximately equivalent to silica. In the ordered (O) pores, the atoms were left on their original fcc sites. In the disordered (D) pores, with roughened surfaces, atoms were displaced using a MC program designed to move atoms to vacant sites on a lattice. A total of 106 moves were run at a series of values of T* (kT/ss) to produce the disordered systems. 2.2. Adsorption Simulations. The adsorbate models have been used in previous studies. Methane was modeled as Lennard-Jones spheres with /k ) 148 K and σ ) 0.3812 nm. Carbon dioxide was modeled13 as three 12-6 centers having OO/k ) 75.2K , σOO ) 0.3026 nm, CO/k ) 44.5 K, σCO ) 0.2925 nm, CC/k ) 26.3 K, and σCC ) 0.2824 nm; point charges placed on the C atom (+) and on each of the O (-) atoms to give a quadrupole of -14 C m2. Parameters for interactions between unlike species were calculated from the Lorentz-Berthelot combining rules. GCMC simulations were run for a minimum of 107 configurations with 106 rejects before sampling. Periodic boundary conditions were applied in directions parallel to the pore walls, and a cut off was applied at 1.7 nm. No long range corrections were used, since these are negligible in small pores. The number of adsorbent atoms in each simulation was kept fixed at 800, and the number of adsorbate molecules varied between a very low number at extremely low adsorptive pressures to about 600. Acceptable reproducibility was found for the very low-density runs when the run length was extended by up to 1 order of magnitude. Adsorption isotherms are reported as adsorbate number density F (nm-3) against fugacity f (bar). Fugacity relates directly to chemical potential set in the GCMC simulations and is close in magnitude to pressure, in the systems examined here. The actual pressure can be calculated from f using a suitable equation of state. The isotherms are absolute, not excess, isotherms. Isosteric heats were calculated from fluctuations and plotted against absolute density.14 Contributions from wall and adsorbate interactions were collected separately.
3. Results and Discussion 3.1. Adsorbent Properties. A measure of the extent of surface roughening is given by a plot of the number of vacancies nv against T* (Figure 1). For the ordered surfaces, nv is 2.0 in the two-layer model. It is clear that pore size has no discernible effect on the amount of disordering within the limit of reproducibility of the simulations. Figure 2 is a snapshot showing a view of an internal surface disordered at T* ) 0.8. Examination of (12) Talu, O; Myers, A. L. Fundamentals of Adsorption; LeVan, D., Ed.; Kluwer Academic Publishers: Boston, MA, 1996; p 945. (13) Hammonds, D.; McDonald, I. R.; Tildesley, D. J. Mol. Phys. 1990, 70, 175. (14) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982.
Figure 1. Mean number of vacancies per atom as a function of the reduced temperature for disordering.
Figure 2. View of a pore of width 2.27 nm, thermally disordered at T* ) 0.8.
similar pictures from several pores and from different angles leads to the conclusion that, although self-adsorbed atoms and vacancies are apparent on the disordered surfaces, steps and kinks are not evident. Experimentally and theoretically, these features are associated with crystal growth rather than thermal disorder. The subsequent adsorption studies were restricted to pore widths of 1.032 and 2.27 nm for the O-pores and to D-pores produced from these at T* ) 0.8. 3.2. Methane and Carbon Dioxide at 298 K. Figures 3 and 4 show isotherms and isosteric heats of adsorption for methane and carbon dioxide, respectively, at 298 K in pores of width 1.032 nm, wide enough to accommodate about two layers of adsorbate. The bottom panels in each figure show the separate wall and molecule components of qst. The isotherms are classical type I and sigmoid for methane and CO2, respectively, reflecting the fact that the two adsorptives are above and below their bulk critical temperatures. The inset on the isotherm plots shows the initial stages of adsorption close to the Henry law region. It is apparent that the Henry law constant, kH, is insensitive to the nature of the pore surface at this temperature. The probable explanation is that kH is essentially an average of the Boltzmann factor for adsorbate-adsorbent interactions over the adsorption space and that kinetic energy is relatively high at 298 K. At higher coverages the isotherms for D- and O-pores diverge, since molecules can pack into the O-pores in a more ordered way. This is particularly the case for CO2, where the long molecules can adopt closer packing through orientational adjustment.
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Figure 3. Adsorption of methane at 298 K in slit pores. The O-pore width is 1.032 nm, and the D-pore is produced from this by thermal disordering at T* ) 0.8. The top panel shows the isotherms for the O-pore [-O-] and D-pore [-0-]. The middle panel is the total (absolute) isosteric heat; as for the isotherms, circles and squares represent results for the O- and D-pores, respectively. The bottom panel shows the adsorbate-adsorbent contribution (qw) as open points and the adsorbate-adsorbate contribution as filled points.
The isosteric heats of adsorption for the D- and O-surfaces differ by about 0.5 kJ mol-1 (for both adsorptives). The separated heat curves show that this is entirely due to adsorbate-adsorbent interactions and also demonstrate that adsorbate interactions dominate qst. For neither adsorbate does qst decrease strongly with adsorbate density. In a much wider pore with H ) 2.27 nm, the effects of surface disordering on the isotherms at 298 K very nearly disappear, as shown in Figure 5. This confirms that
Nicholson
Figure 4. As in Figure 3 for carbon dioxide adsorption at 298 K.
adsorbate packing is the essential factor in altering the form of the isotherm in these model systems. In the wider pore space, the adsorbate can more easily adjust to the roughened surface, since there are now approximately four layers of adsorbate to accommodate the asperities generated at the surface. The heat curves in the wider pore (Figures 6 and 7) are now quite different in shape from those found in the narrower pore. For methane, as seen from the component curves in Figure 6, the intermolecular adsorbate contribution increases more slowly than in the narrow pore, and the wall contribution decreases more rapidly. Consequently, the resultant heat curve has a minimum. Moreover, the Dpore exhibits an initial decrease in qst, in contrast to the O-pore. Although this is qualitatively the effect anticipated from adsorption on roughened surfaces, the scale of the decrease is again only about 0.5 kJ mol-1, much smaller
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Figure 6. Isosteric heat curves for methane for the same pores as in Figure 5. The notation for the heat curves is the same as in Figure 3. Figure 5. Isotherms at 298 K for methane (top) and carbon dioxide (bottom) in an O-pore of width 2.27 nm and a D-pore produced from this at T* ) 0.8.
than the decrease of about 5 kJ mol-1 suggested by experimental data11 mentioned in the Introduction. For carbon dioxide, the adsorption is dominated by intermolecular adsorbate interactions, and qst increases with F. As the inset to Figure 7 shows, the difference between O- and D-surfaces at zero coverage is again about 0.5 kJ mol-1. 3.3. Adsorption of Methane at 150 K. Since no strong decrease in qst due to surface disordering is observed in any of the above systems, it is of interest to enquire whether lower temperature or stronger adsorbent interaction significantly alters the picture. Figure 8 shows isotherms and heats calculated in the same systems at 150 K in a pore with H ) 1.032 nm. The main modification to the isotherm from atomic disorder in the surface is again that due to packing. However, in contrast to the data for 298 K, the expected increase in kH is now observed at very low density as seen in the inset graph. As noted before,15 the linear region of the isotherm is restricted to extremely low adsorbate density, and this is especially the case at lower temperature. It is therefore quite difficult, experimentally, to obtain reliable values of kH from isotherm plots. Since 150 K is well below the bulk critical temperature for methane, the isotherms have an initial sigmoid shape that is only apparent at extremely high resolution. (15) Nicholson, D. Stud. Surf. Sci. Catal. 1991, 62, 1.
The isosteric heat of adsorption in the D-surface pore, shown in the middle panel of Figure 8, is now increased by about 2 kJ mol-1 in comparison to the O-surface pore, but the qst curves for both adsorbents show a steep initial rise, rather than a decrease. This low density behavior in qst can be attributed to the steep initial increase in qm caused by the clustering of the adsorbate at low adsorbate density which can be seen in snapshots taken from the simulations. The presence of favored sites on the D-surface does not modify the clustering to an extent that can be detected in qm or qst (bottom panel of Figure 8), implying that favored surface sites do not contribute to the nucleation of clusters. 3.4. Adsorption of Methane at 298 K with Ess/k ) 400 K. Pores with a strongly adsorbing surface were created by doubling ss/k to 400 K. The isosteric heat curves for these simulations are shown in Figure 9. The gap between the zero coverage heats for the Oand D-surfaces is now widened to about 1.2 kJ mol-1 at 298 K (Figure 9, top panel), and qst increases much more slowly on the D-surface. The separated molecule and wall parts of qst, plotted in the lower panel of Figure 9, reveal that the reason for this is that rate of decrease of qw with F nearly balances the rate of increase of qm with F. The slightly steeper decrease of qw with F is insufficient to cause qst to decrease as F increases. 3.5. Selectivity. Since methane and CO2 adsorption at 298 K are both modified in much the same way and to much the same extent by atomic disorder in the surface, it is not surprising to find that selectivity in mixed adsorbate systems is entirely unaltered by surface rough-
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Nicholson
Figure 7. As in Figure 6 for carbon dioxide adsorption.
ening. Figure 10 shows plots of the selectivity for 50/ 50mixtures defined in the usual way by
SCO2/CH4 )
xCO2 yCH4 xCH4 yCO2
where xi and yi are mole fractions in the adsorbate and adsorptive phases, respectively. 3.6. Adsorption of Methane in Smooth-Walled Slit Pores. In none of the above studies was it found possible to reproduce the strong decrease in qst observed in some experimental heat curves11 and in simulations with sphere pack adsorbents.8-10 This is found to be the case not only at 298 K, but also at low temperatures, when kinetic energy is relatively low or the surface is much more strongly adsorbing than the oxide surface which is the primary model used here. This suggests that energetic heterogeneity due to thermal disordering is not a major contribution to this phenomenon. Since pore models with thermally disordered surfaces do not match well to experimental observations, it is pertinent to enquire whether structural heterogeneity is capable of producing the observed behavior. To investigate this possibility, simulations of methane adsorption at 298 K were performed in a number of smoothwalled slit pores having a range of slit widths from 0.7 to 2.5 nm. The “pseudoatom” model was similar to that used previously to model oxide-like slit pores16 in which the adsorption potential varies only in a direction normal to the pore walls. The total adsorption, F(f), and isosteric heat of adsorption, q(F), in a distribution, Fi, of pore widths (16) Nicholson, D.; Gubbins, K. E. J. Chem. Phys. 1996, 104, 8126.
Figure 8. Adsorption of methane at 150 K in an O-pore of width H ) 1.032 nm and in a D-pore produced from this by thermal disordering at T* ) 0.8. The notation is the same as in Figure 3.
were then calculated from the equations
∑FiViFi ∑ViFi ∑qiFiViFi q) ∑FiViFi F)
where Fi, qi, and Vi are the density, isosteric heat, and pore volume in a pore of width Hi, at fugacity f. As before these are absolute rather than excess properties. It is of course assumed that no network or kinetic effects are operating in the models with a distribution of pores.
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Figure 9. Isosteric heat curves methane at 298 K in an O-pore of width H ) 1.032 nm and in a D-pore produced from this by thermal disordering at T* ) 0.8. The surface parameter has been doubled to ss/k ) 400 K.
Figure 10. Selectivity for a methane/carbon dioxide mixture in an O-pore of width H ) 1.032 nm and in the corresponding D-pore.
Figure 11 shows the isotherms and heat curves for the individual pores. The observed trends are similar to those discussed above, although the pore walls are now perfectly homogeneous energetically. It is interesting to note that the adsorbate-adsorbent part of the heat of adsorption now makes a dominant contribution to qst in very small pores and that the plots of qm against F are indistinguishable except for the two smallest pore widths. It should be noted that the potential well in a slit pore with H ) 0.7 nm is close to the deepest possible for the methane molecule in this geometry.
Figure 11. Isotherms (top panel) and heat curves (bottom panel) for methane adsorbed at 298 K in a set of smooth walled slit pores with oxide-like surfaces. The separate components of each qst curve are shown in the bottom panel. All the lines passing through the origin are the adsorbate-adsorbate components (qm) of qst, the remaining lines being the adsorbentadsorbate components (qw).
Composite isotherms and heat curves for three distributions are shown in Figure 12. The symmetrical distribution and the distribution skewed to high pore widths both give isosteric heat curves that are qualitatively similar to experiment and to those obtained from sphere pack simulations. It is especially interesting to note that the isosteric heats are apparently very sensitive to the extreme micropore end of the distributions. This is, of course, in keeping with the usual explanation of high initial isosteric heats which attributes these to a small concen-
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Figure 12. Composite isotherms and heat curves for methane adsorbed at 298 K in smooth-walled slit pores having the discrete distribution functions illustrated.
tration of highly energetic sites. These results suggest that such sites could be crevices of exactly the size to adsorb the probe adsorbate molecule at the lowest possible energy. Further confirmation of the importance of very small pores in determining the shape of heat curves is illustrated in Figure 13. Here the smallest pore size has been removed from the distribution. It is observed that the isotherms can no longer distinguish between the left skewed distribution and the symmetrical distribution; however, the isosteric heat curves remain quite distinct. Moreover, the initial steep fall in qst, attributed to the highest energy sites, has vanished. 3.7. Other Adsorbates. The structural heterogeneity part of this work has been restricted to methane adsorption. However, it has been noted in previous work17 that adsorptives such as nitrogen and carbon dioxide are able to occupy narrower spaces than methane. This suggests that other probes may lead to quantitatively different results.18 There are also implications for separations in materials with distributions of pore widths which remain to be more fully explored. It should be mentioned that the (17) Cracknell, R. F.; Nicholson, D.; Tennison, S. R.; Bromhead J. Adsorption 1996, 2,193. (18) Samios, S.; Stubos, A. K.; Kanellopoulos, N. K.; Cracknell, R. F.; Papadopoulos, G. K.; Nicholson, D. Langmuir 1997, 13, 2795.
Nicholson
Figure 13. As in Figure 12, but the distributions here omit the narrowest pores with H ) 0.7 nm.
models investigated here do not include any electrostatic effects. The existence of electrostatic charges, especially when these are concentrated in a particular region, as may occur in some zeolites, would be expected to greatly modify both selective adsorption and observed heterogeneous effects for adsorptives having significant permanent multipoles such as carbon dioxide or nitrogen. 4. Conclusions Simulation studies of idealized model pores have been carried out for methane and carbon dioxide adsorbates in energetically and structurally heterogeneous pores. Energetic heterogeneity was developed by using thermal disordering to generate roughness on an atomic scale, and structural heterogeneity was modeled by constructing distributions of smooth-walled pores of different width. Emphasis was placed on the characteristic behavior of isosteric heat curves, observed in many experimental systems, and also found in simulations of sphere packs, whereby qst falls by several kJ mol-1 from its zero coverage value. In the thermally disordered systems this behavior could only be found in fairly wide pores. Even then, the similarity with the aforementioned observations was qualitative, rather than quantitative, the difference between ordered and disordered surfaces being rather small. Since the surfaces studied here probably overstate the amount of atomic disorder that may be present in real
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solids, it seems unlikely that energetic heterogeneity, from this source alone, is able to account for experimental observations. On the other hand, structural heterogeneity offers a feasible rationalization of these observations. Conversely, isosteric heats prove to be highly sensitive to the presence of very small pore widths in the adsorbent, suggesting that deconvolution of isosteric heat curves might provide the basis of a valuable method for analyzing
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micropore distribution. It would appear that the possibilities for exploiting this property in this way have not yet been fully realized. Acknowledgment. I thank Denise Bergin for help with preparing the adsorbents. LA981143Q
