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Langmuir 1998, 14, 4880-4886
Surface Roughness Effects in Molecular Models of Adsorption in Heterogeneous Porous Solids T. Vuong and P. A. Monson* Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003 Received January 6, 1998. In Final Form: June 12, 1998 We present an investigation of the influence of surface roughness on the interior pore walls upon adsorption equilibrium in heterogeneous porous materials using molecular models of methane adsorbed in a microporous silica gel. The molecular models treat the adsorbent as a disordered matrix of particles. Three different models of the surfaces of the matrix particles are considered where (i) the adsorbent particles exhibit smooth surfaces, (ii) the particle surfaces have a single layer of interaction sites, and (iii) atomic details of the matrix particles are incorporated. The effects of surface roughness in these models are most appreciable on the isosteric heat at low density. The effect on the adsorption isotherm comes principally at higher pressure due to an increase in the effective void space produced by the surface roughness. A comparison with experimental data is also presented.
1. Introduction In recent work we have examined the role of heterogeneous pore structure upon the adsorption isotherms and heats of adsorption using molecular models which attempt to address the three-dimensional disorder of the void space in the porous material,1-3 focusing particularly on silica xerogels. In these studies the porous material is treated as a matrix of spherical particles with some predetermined microstructure (e.g. an equilibrium hard sphere configuration). The model used in our earlier work can be viewed as a simplified version of a detailed molecular model of silica xerogel developed by MacElroy and Raghavan.4,5 In this model the matrix particles or microspheres making up the adsorbent are treated as aggregates of discrete interaction sites. This gives rise to roughness on the surface of the microspheres and allows adsorbate molecules to penetrate the outer surface to some degree. The model of Kaminsky and Monson1 is an integrated version of the MacElroy-Raghavan (MR) potential in which the interaction sites are continuously distributed over the microspheres. The surface of the microspheres is then smooth and cannot be penetrated by adsorbate molecules. An advantage of this model is that the potential between an adsorbate molecule and a matrix particle can be written analytically. The ease of computation for this potential has allowed extensive studies of how the structural arrangement of matrix particles influences adsorption equilibrium.1-3 Application of approximate integral equation theory from statistical mechanics6-9 to the model10 has also been possible. * To whom correspondence should be addressed. (1) Kaminsky, R. D.; Monson, P. A. J. Chem. Phys. 1991, 95, 2936. (2) Kaminsky, R. D.; Monson, P. A. Langmuir 1994, 10, 530. (3) Vuong, T.; Monson, P. A. Langmuir 1996, 12, 5425. (4) MacElroy, J. M. D.; Raghavan, K. J. Chem. Phys. 1990, 93, 2068. (5) MacElroy, J. M. D. Langmuir 1993, 9, 2682. (6) Madden, W. G.; Glandt, E. D. J. Stat. Phys. 1988, 51, 537. Madden, W. G. J. Chem. Phys. 1992, 96, 5422. (7) Given, J. A.; Stell, G. J. Chem. Phys. 1992, 97, 4573. (8) Rosinberg, M. L.; Tarjus, G.; Stell, G. J. Chem. Phys. 1994, 100, 5172. (9) Kierlik, E.; Rosinberg, M.; Tarjus, G.; Monson, P. A. J. Chem. Phys. 1997, 106, 264. (10) Vega, C.; Kaminsky, R. D.; Monson, P. A. J. Chem. Phys. 1993, 99, 3003.
In a recent paper3 we described a study of the influence of adsorbent microstructure on the isosteric heat of adsorption for simple molecules adsorbed in silica gel using the simplified model. A comparison with experiment for methane in silica gel indicated that the model was able to capture the density dependence of the isosteric heat at low adsorbed density reasonably accurately. However, at high density, the experimental isosteric heat showed a stronger dependence on density than that in the molecular model. Moreover, changes in the spatial arrangement and connectivity of the matrix particles were not sufficient to change this density dependence significantly. One possible explanation for this is that the surface roughness effects which are neglected in the simplified model of Kaminsky and Monson play an important role in the density dependence of the heat of adsorption. Other studies have indicated that for single pores the effects of surface roughness can be considerable,11,12 and such effects were also seen in some earlier calculations for the MR model.4,5 The purpose of the present paper is to explore this issue in more detail. We do this by examining three molecular models: (i) the model of Kaminsky and Monson where the matrix particles exhibit smooth surfaces; (ii) the model where matrix particle surfaces have a single layer of interaction sites; (iii) the MR model where atomic details of the matrix particles are incorporated. We compare several properties calculated for these different models including the adsorption isotherms, heats of adsorption, and radial distribution functions. 2. Molecular Models and Simulation Techniques As noted earlier, the interaction between an adsorbate molecule and one of the matrix particles is treated at three levels of approximation. The first model we consider is the integrated model of Kaminsky and Monson,1 which we refer to as the composite sphere (CS) potential. The potential energy between an adsorbate molecule and a matrix particle is written as (11) Bojan, M. J.; Vernov, A. V.; Steele, W. A. Langmuir 1992, 8, 901. Bojan, M. J.; Steele, W. A. In Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer: Dordrecht, The Netherlands, 1996; p 17. (12) Maddox, M. W.; Olivier, J. P.; Gubbins, K. E. Langmuir 1997, 13, 1737.
S0743-7463(98)00033-X CCC: $15.00 © 1998 American Chemical Society Published on Web 07/30/1998
Surface Roughness Effects in Adsorption
16πgsFsR3 × 3 (d6 + 21/5d4R2 + 3d2R4 + 1/3R6)σ12gs
Langmuir, Vol. 14, No. 17, 1998 4881
ucs(d) )
[
(d2 - R2)9
-
σ6gs
]
(d2 - R2)3
(1)
where d is the distance from the center of the fluid molecule to the center of the matrix particle, Fs is the density of interaction sites in the matrix particles, R is the matrix particle radius, and σgs and gs are the collision diameter and well depth for the 12-6 potential between the fluid molecule and a matrix particle interaction center. In the second model we use the CS potential with a reduced diameter but with a single layer of discrete 12-6 interaction sites randomly distributed on its surface. The potential energy between an adsorbate molecule and a matrix particle is the sum of the interactions between the fluid particle and the composite core and that between the fluid particle and each site. It is written as Nsite cs
u(ri) ) u (d) +
∑j u12-6(|rj - ri|)
(2)
where d is the distance between the adsorbate center and the composite core center, ucs(d) is the CS potential, eq 1, u12-6(|rj - ri|) is the Lennard-Jones 12-6 potential between an adsorbate molecule at ri and oxygen site j whose position is rj on the composite core surface, and Nsite is the number of surface interaction sites per matrix particle. The third model is the detailed model of silica gel developed by MacElroy and Raghavan.4 In this model, a matrix particle is treated as a collection of discrete interaction centers, one for each oxygen atom (the interactions with the silicon atoms are incorporated into an effective potential between the adsorbate molecules and the oxygen interaction sites13,14). The potential energy between an adsorbate molecule and a matrix particle is written as Noxygen
u(ri) )
∑j
u12-6(|rj - ri|)
(3)
where Noxygen is the number of oxygen atoms per matrix particle. Figure 1 shows a pictorial representation of the three models of the matrix particles. The parameters for the three models are listed in Table 1. The adsorbent microstructure for all the above models consists of 32 matrix particles at a volume fraction of 0.386 in a configuration from a canonical ensemble Monte Carlo of an equilibrium hard sphere (EHS) system. For the CS with rough surface model the dimensions of the composite core and the oxygen atom diameter are required. With the aid of the oxygen density profile reported by MacElroy and Raghavan, we can determine the size of the composite core. We chose the size to be at the point where the oxygen density ceases to be constant (see Figure 3a of ref 4). The number of sites reported in this work was obtained by integrating the oxygen density profile from the point where the profile begins to decrease (a linear decrease was assumed) to the surface of the microsphere. This gives an Nsite ) 158. The size of the oxygen atom is that reported by MacElroy and Raghavan.4 The configuration of the surface sites was achieved as follows. Sites were placed on the surface in rings to achieve an approximate close (13) Bezus, A. G.; Kiselev, A. V.; Lopatkin, A. A.; Pham, Q. D. J. Chem. Soc., Faraday Trans. 2 1987, 74, 367. (14) Kiselev, A. V.; Lopatkin, A. A.; Shulga, A. A. Zeolite 1985, 5, 261.
Figure 1. Pictorial representation of microsphere models used in this work: (a) CS model; (b) CS model with rough surface; (c) MR model. Table 1. Parameters Used for the Different Modelsa CS R (nm) σgg (nm) σgs (nm) gg/k (K) gs/k (K) σox (nm) σsi (nm) Nsite Nox Nsi
1.3465 0.39 0.33 154.8 349
CS with rough surface 1.215 0.39 0.33 154.8 229 0.285
MR 0.39 0.33 154.8 176 0.3 0.076
158 565 245
a N b site: number of sites per composite core. Nox: number of oxygen atoms per matrix particle. c Nsi: number of silicon atoms per matrix particle.
packing of sites. From this configuration sites were then removed at random until the required number of sites remained. The remaining sites were then moved on the surface of the sphere in a Monte Carlo simulation identical to that used to study hard spheres confined to the surface of a sphere.15 One configuration from this simulation was then used to provide the coordinates of the surface sites for each matrix particle in the adsorbent. The matrix particle in the MR model was constructed by using a set of oxygen and silicon atom coordinates from one of the silica gel microspheres in the work of MacElroy and Raghavan.4 In their work, the coordinates of the oxygen and silicon atoms were obtained by simulating a bulk glassy silica and cutting out spheres of specified size. It should be anticipated that finite size effects should make the surface of the microspheres different from those (15) Kratky, K. W. J. Comput. Phys. 1980, 37, 205.
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Figure 2. Adsorbate-adsorbent potential energy distributions using different molecular models for methane adsorbed in silica gel (from left to right): CS model; CS model with rough surface; MR model without thermal relaxation; MR model with thermal relaxation.
Figure 3. Comparison of adsorption isotherms obtained using different molecular models at T* ) 0.8 for methane adsorbed in silica gel: 4, CS model; 0, CS model with rough surface; ], MR model without thermal relaxation.
obtained by cutting spheres from the bulk silica. Thus in addition to studying the original MR model, we have investigated the effects of thermal relaxation of the internal structure of the microspheres. Such a relaxation should model the finite size effects in the particle formation during the silica xerogel preparation. To accomplish the thermal relaxation we carried out a Monte Carlo simulation of an individual microsphere using the potential suggested for bulk silica by Fueston and Garofalini16 on the basis of earlier work by Stillinger and Weber17 on silicon. The potential used is the same as that considered by MacElroy and Raghavan4 in their simulations of bulk silica. In these simulations the temperature of the microsphere was initially raised to 1000 K. After equilibration at high temperature the microsphere was gradually cooled to 300 K. The principal effect of this annealing process is to slightly broaden the site distribution in the (16) Feuston, B. P.; Garofalini, S. H. J. Chem. Phys. 1988, 89, 5818. (17) Stillinger, F. H.; Weber, T. A. Phys. Rev. B 1985, 31, 5262. (18) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, U.K., 1987.
Vuong and Monson
cluster near the surface and as we shall see to make the microsphere somewhat more permeable to the adsorbate molecules. We have followed the standard procedures for the grand canonical Monte Carlo simulation technique in our work.18,19 For the study of adsorption isotherms, we started from an empty matrix and then carried out a series of simulations with successively increasing values of activity. Each simulation was started from the configuration at the end the a previous simulation at a lower activity. Simulations at the lowest activity were run for a total of 4 × 106 configurations, half of which was devoted to equilibration. Simulations at the highest activity were run for a total of 18 × 106 configurations with 9 × 106 configurations for equilibration. A configuration consists of an attempted translation of a molecule followed by an attempted creation or destruction (chosen with equal probability) of a molecule. It is necessary to take into account that the microspheres in the MR model exhibit isolated cavities that might not be accessible by adsorbate molecules via diffusion but are accessible via the insertion moves in the grand canonical ensemble Monte Carlo method used in this work. Therefore, it is necessary to map out the topology within the microspheres to locate the isoslated cavities. In this work, we have estimated the distance from the center of the microsphere within which isolated cavities were detected. Adsorbate molecules were prevented from entering this region via moves in the Monte Carlo simulations. For the MR model without thermal relaxation, this radius was estimated to be 2.5σgg and 2.7σgg for the MR model with thermal relaxation. In this study the parameters used in the above models are those originally used by Kaminsky and Monson1 with the adjustment of the adsorbate-adsorbent well depth2 to yield a Henry’s constant in agreement with experimental data20 at T ) 298 K. The Lennard-Jones parameters for the adsorbate-adsorbate interactions were chosen such that the bulk fluid critical temperature and pressure are the same as that for methane.2 Table 1 summarizes all the parameter values used in this work. The adsorbate-adsorbate Lennard-Jones interaction was truncated at a separation of 2.5σgg. For the CS potential the adsorbate-adsorbent was truncated at 4.0σgs beyond the solid surface. The same truncation was also employed for the adsorbate-composite core in the CS with rough surface model while 2.5σgg was used for the adsorbate-site interactions. In the MR model, the adsorbate-oxygen interaction was truncated at 3.32σgg (1.296 nm), a cutoff distance that was reported by MacElroy and Raghavan. For the bulk properties we have utilized the LennardJones 12-6 equation of state due to Johnson et al.21 with a correction for the effect of the truncation of the 12-6 potential.22 In presenting our results, we use the reduced units defined in terms of the collision diameter, σgg, and the well depth, gg, of the adsorbate-adsorbate LennardJones 12-6 potential. The reduced units are defined as follows: activity, ζ* ) ζσgg3; configurational energy, U* ) U/Ngg density, F* ) Fσgg3; heat of adsorption, qst* ) qst/ (19) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (20) Masukawa, S. A Study on Two Phase Equilibria by Use of the Elution Gas Chromatographic Technique: The Methane-Ethane-Silica Gel System and the Methane-Normal Octane System; Rice University: Houston, TX, 1967. Masukawa, S.; Kobayashi, R. J. Chem. Eng. Data 1968, 13, 197. (21) Johnson, J.K.; Zollweg, J. A.; Gubbins, K. E. Mol. Phys. 1993, 78, 591. (22) Finn, J. E.; Monson, P. A. Phys. Rev. A 1989, 39, 6402.
Surface Roughness Effects in Adsorption
Figure 4. Comparison of adsorption isotherms obtained using different molecular models at T* ) 1.926 for methane adsorbed in silica gel. 4, CS model; 0, CS model with rough surface; ], MR model without thermal relaxation; O, MR model with thermal relaxation.
Langmuir, Vol. 14, No. 17, 1998 4883
Figure 5. Adsorbate-adsorbent radial distribution function in the Henry’s law regime at T* ) 1.926: - - -, MR model without thermal relaxation; s, MR model with thermal relaxation.
gg; temperature, T* ) kT/gg. In the comparisons with experiment, real units are used. 3. Results 3.1. Potential Energy Distribution. One indication of the impact of surface roughness is to look at the potential energy distribution for a single adsorbate within the adsorbent.1 The potential energy was randomly sampled and averaged over 2.5 × 107 samples for each model. Figure 2 shows the distribution of potential energy in the form of fractional volume of the adsorbent with attractive energy.1 This attractive energy is normalized by the CS potential minimum. It is clear from this figure that, in going from the composite sphere model to the MR model, the potential energy distribution is broadened. Thermal relaxation of the matrix spheres in the MR model seems to lead to a slightly broader energy distribution compared to the nonrelaxed matrix structure, but the overall distributions are quite similar in the two cases. 3.2. Adsorption Isotherms and Adsorbate Microstructure. Figures 3 and 4 show the adsorption isotherms for the molecular models at two temperatures, one above the Lennard-Jones bulk critical temperature and the other below the bulk critical temperature. From the figures, it is evident that as we go from the CS model to the CS with rough surface model we see that adsorption increases, especially at high activities.23 This increase is further enhanced as we go to the MR model. The reason for this increase is that surface roughness has created more accessible void space within the matrix particle, and as the degree of roughness increases, more accessible void space is created. Figure 4 also shows the comparison for the adsorption isotherms from the MR model using the relaxed and nonrelaxed matrix particles. We see that adsorption for the relaxed structure is slightly lower than that for the nonrelaxed structure. In the relaxed structure the oxygen site distribution is slightly broader than that (23) At the higher temperatures the low activity adsorption for all models will be similar because gs for each model was adjusted to reproduce the experimental the Henry’s constant at this temperature. At the lower temperature the same parameters are used and Henry’s constants for the models differ. The adsorption in the CS model is then somewhat lower than for the models with rough surfaces at low activity also.
Figure 6. Adsorbate-adsorbent radial distribution function at T* ) 1.926 and F* ) 0.42: - - -; CS model; - ‚ -, CS model with rough surface; s, MR model without thermal relaxation; ‚ ‚ ‚, MR model with thermal relaxation.
in the unrelaxed structure. There are two competing effects which follow from this. First, there is somewhat more accessible void space in the interior of the matrix particles created by the relaxation. Second, the effective diameter of the matrix particles is slightly increased by the relaxation, reducing the void space exterior to the matrix particle surface. A key effect of surface roughness in these systems is to alter the adsorbate density distribution near the surface of the matrix particles. This is evident from Figure 5, which shows the adsorbate-adsorbent radial distribution function in the Henry’s law regime for the different models. MacElroy and Raghavan4 have previously shown the effect of surface roughness on the adsorbate-adsorbent radial distribution function in the Henry’s law limit using a slightly different integrated potential. Figure 6 shows the adsorbate-adsorbent radial distribution functions for a supercritical temperature and at a high adsorbed density. The adsorbate-adsorbent radial distribution function in the neighborhood of the matrix particle surface is broadened and the peak height lowered as we go from the CS model to the MR model, reflecting the penetrability of the
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Vuong and Monson
Figure 7. Adsorbate-adsorbate radial distribution function at T* ) 1.926 and F* ) 0.42: - - -, CS model; - ‚ -, CS model with rough surface; s, MR model without thermal relaxation; ‚‚‚, MR model with thermal relaxation.
Figure 8. Comparison of isosteric heat of adsorption obtained using different molecular models at T* ) 0.8 for methane adsorbed in silica gel: 4, CS model; 0, CS model with rough surface; ], MR model without thermal relaxation.
matrix particle surface. Also shown in these figures are the results from the MR model using the relaxed matrix particles. It is evident that the interior void spaces of the relaxed matrix structure are relatively more penetrable by adsorbate molecules compared to the nonrelaxed structure. Figure 7 shows the adsorbate-adsorbate radial distribution functions, at the same conditions as those in Figure 6. The results are quite similar except that the peaks heights decrease in moving from the CS model to the rough surface models. This may be largely accounted for by the increase in the void volume as the roughness increases so that the adsorbate at a fixed total density is at an effectively lower density normalized by the void volume. 3.3. Heat of Adsorption. We have considered the influence of surface roughness on the isosteric heat by examining the results from the different molecular models. The equation for the isosteric heat that we used is the same as that from our previous work.3 We have
qst ) H(b) -
( ) ∂U
(a)
∂N(a)
(4)
T,V(a)
where H(b) is the bulk enthalpy per molecule and is obtained from the Lennard-Jones 12-6 equation of state, U(a) is the configurational of the total internal energy of the adsorbed phase which we obtained directly from simulation, and V(a) is the volume of adsorbent (void + solid). The energy derivative was obtained by numerically differentiating the configurational internal energy as was done in our previous work.3 Figures 8 and 9 show the isosteric heats as a function of adsorbed density for a subcritical and a supercritical temperature, respectively. From these figures we see that the isosteric heat of adsorption from each model decreases as a function of adsorbed density, except at medium to high densities in the supercritical temperature case. The decrease of the isosteric heat with increasing density is the kind of behavior usually associated with heterogeneous adsorbents and is often interpreted as a decrease in the availability of regions of large negative adsorbateadsorbent potential energy as adsorbate density increases. This effect is somewhat more significant in the rough surface models especially at low density since the surface
Figure 9. Comparison of isosteric heat of adsorption obtained using different molecular models at T* ) 1.926 for methane adsorbed in silica gel: 4, CS model; 0, CS model with rough surface; ], MR model without thermal relaxation; O, MR model with thermal relaxation.
roughness and penetrability of the matrix particles give rise to a wider distribution of regions of large negative potential energy as shown in Figure 2. The high density behavior for the higher temperature is somewhat more complex, particularly the occurrence of a maximum for the MR model with structural relaxation. The increase in the isosteric heat in this region comes about from two sources: the contribution from the adsorbate-adsorbate interactions and the behavior of the bulk enthalpy at supercritical conditions. These separate contributions were analyzed in detail in our previous work.3 Figures 10 and 11 show plots of the isosteric heats and the contributions from the two terms in eq 4 for the MR model with and without structural relaxation. We see that the two terms have qualitatively similar behavior for each model but the interplay between them gives rise to a maximum in one case. We should note, however, that to achieve the high densities where this happens at this supercritical temperature requires very high pressures not normally accessible in adsorption experiments.
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Langmuir, Vol. 14, No. 17, 1998 4885
Figure 10. Isosteric heat of adsorption and its component contributions for methane adsorbed in silica gel using the MR model without thermal relaxation at T* ) 1.926: b, qst*; 0, -∂U/∂N; 4, H*(b).
Figure 12. Adsorption isotherm of methane adsorbed in silica gel at T ) 25 °C: b, experimental data; 4, CS model; 0, CS model with rough surface; ], MR model without thermal relaxation; O, MR model with thermal relaxation.
Figure 11. Isosteric heat of adsorption and its component contributions for methane adsorbed in silica gel using the MR model with thermal relaxation at T* ) 1.926: b, qst*; 0, -∂U/ ∂N; 4, H*(b).
Figure 13. Isosteric heat of adsorption of methane adsorbed in silica gel at T ) 25 °C: b, experimental data; 4, CS model; 0, CS model with rough surface; ], MR model without thermal relaxation; O, MR model with thermal relaxation.
3.4. Comparison with Experiment. As in our earlier work, we have compared the Monte Carlo simulation results with Masukawa’s20 experimental data for methane adsorbed in silica gel at T ) 298 K. The silica gel used by Masukawa was grade 15 from Davison Chemical with specific surface area of 803.5 m2/g. Figure 12 shows the comparison of the adsorption isotherms predicted by the different models and that from experimental data. It is clear that the effect of surface roughness on the adsorption isotherm is not significant under these conditions. Figure 4 shows differences in adsorption isotherms for the different models but at much higher activities corresponding to very high pressure. Figure 13 shows the comparison of the isosteric heat of adsorption from the molecular models and that from experiment. The experimental results were calculated from the adsorption isotherm data of Masukawa20 via the Clapeyron expression (eq 15 of ref 3). Surface roughness in these models has a greater influence on the isosteric heat at low densities compared to that at high densities. Notice that the MR model with thermal relaxation gives significantly im-
proved agreement with experiment at low density. Indeed, increasing the surface roughness progressively improves the agreement with experiment in this region. However, none of the molecular models correctly describes the density dependence of the isosteric heat at higher densities. 4. Conclusions In this work we have investigated the effect of surface roughness on the adsorption isotherms and isosteric heats of adsorption for molecular models of methane adsorbed in silica gel. Our conclusions are as follows: (i) The effect on the adsorption isotherm is primarily seen at high pressure where the surface roughness increases the maximum density attainable in the adsorbent. Agreement with experiment is quite good for all the models considered under the conditions where experimental data are available. (ii) The isosteric heat is affected by surface roughness most significantly in the low-density regime, associated with broadening of the
4886 Langmuir, Vol. 14, No. 17, 1998
distribution of the adsorbate-adsorbent potential energy. Agreement with experimental results for the isosteric is improved in this region by the addition of surface roughness to the model but not at higher densities. (iii) The adsorbate density distribution near the surface of the matrix particles is progressively broadened by the inclusion of surface roughness in the model. (iv) In the MR model thermal relaxation of the microspheres leads to a broadening of the oxygen density distribution near the microsphere surface. This allows more substantial penetration of the microspheres by the adsorbate. One interesting feature of the present results is the overall similarity of the behavior of the MR and composite sphere models except for the adsorbate density distribution near the microsphere surface and the isosteric heat at low density. Evidently, closer agreement between results from these kinds of models and experiment for the isosteric
Vuong and Monson
heat at higher density will come from including effects other than the surface roughness. Some possibilities include a more detailed model of the adsorbate via an interaction site model of methane, using a distribution of matrix particle sizes, or allowing for sintering in the adsorbent. Acknowledgment. This work was supported by the National Science Foundation (Grant Nos. CTS-9417649 and CTS-9700999). The authors are grateful to Dr. J. M. D. MacElroy for providing us with an initial set of coordinates of the atoms making up the silica gel microspheres in his model and for suggesting the exploration of structural relaxation in the microspheres. LA980033G
