Origin of Hysteresis of Gas Adsorption in Disordered Porous Media

P.O. Box 361, 57001 Thermi-Thessaloniki, Greece; and National Center for Scientific ... to explore the origin of hysteresis in disordered porous media...
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Langmuir 2003, 19, 3338-3344

Origin of Hysteresis of Gas Adsorption in Disordered Porous Media: Lattice Gas Model versus Percolation Theory E. S. Kikkinides,*,† M. E. Kainourgiakis,‡ and A. K. Stubos‡ Center for Research and Technology Hellas, Chemical Process Engineering Research Institute, P.O. Box 361, 57001 Thermi-Thessaloniki, Greece; and National Center for Scientific Research “Demokritos”, 15310 Ag. Paraskevi Attikis, Greece Received October 30, 2002 In the present study we attempt to explore the origin of hysteresis in disordered porous media by testing the validity of a recently developed lattice gas model against experimental findings obtained from SAXS studies. Hence, we study the sorption-desorption of CH2Br2 in Vycor at 300 K, by means of Monte Carlo simulations in stochastic reconstructions of Vycor porous glass. By varying the solid-fluid interaction over the fluid-fluid interaction, we are able to match the experimentally measured hysteresis loop. Comparison between the simulation results and experimental measurements of the two-point correlation functions of wet (preadsorbed with CH2Br2) Vycor, obtained from SAXS measurements, reveals that the lattice model is capable of predicting the distinct long-range correlation behavior that occurs in the pore space of the material during desorption. Visual images of the simulated fluid configurations show a clustering of the sorbed phase during desorption which is believed to be responsible for the long-range correlation. Additional simulations in two-dimensions indicate the presence of a complex density field where both pore blocking and cavitation can take place at different regions of the porous matrix.

Introduction Disordered porous materials are widely used in many processes of industrial or environmental importance, such as gas separation and purification, oil recovery, heterogeneous catalysis, and so forth. It is well-known that the equilibrium and dynamic processes that occur in the pore space of these materials are strongly influenced by the confinement and the geometrical disorder of the porous matrix.1 Improved understanding of the effect of disorder on these processes is expected to lead to significant enhancements in the performance of the industrial and environmental applications. The sorption of condensable vapors in mesoporous media (average pore size above 2 nm and below 50 nm; see, for example, refs 2 and 3) is a typical example of an equilibrium process in a confined multiphase environment. A standard sorption experiment involves the progressive increase of the pressure of the vapor above a porous sample and the subsequent measurement of the amount of adsorbate at this pressure, at equilibrium. The adsorption process is frequently followed by a desorption experiment, where the pressure is progressively reduced from its maximum value to zero, and the desorption isotherm is measured accordingly. During desorption, the isotherm is expected to exhibit hysteresis, the origin of which has been the subject of continuous research.1-4 Early theories are based on the Kelvin equation, developed for individual * To whom correspondence should be addressed. Phone: +302310-498-116. Fax: +30-2310-498-131. E-mail: [email protected]. † Chemical Process Engineering Research Institute. ‡ National Center for Scientific Research “Demokritos”. (1) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573. (2) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982. (3) Rouqerol, F.; Rouqerol, J.; Sing, K. S. W. Adsorption by Powders and Porous Solids; Academic Press: New York, 1999. (4) Burgess, C. G. V.; Everett, D. H.; Nuttall, S. Pure Appl. Chem. 1989, 61, 1845.

pores of simple geometry, where hysteresis occurs due to the geometrical differences in the shape of the liquidvapor meniscus during adsorption (condensation) and desorption (evaporation).2-6 Simulation studies based on density functional theory (DFT) and grand canonical Monte Carlo (GCMC) and molecular dynamics (MD) simulations have explored the validity of Kelvin’s equation and have shown its limitations.7-11 A different approach, which also holds for a single pore, states that nucleation is postponed beyond the equilibrium phase transition, resulting in the occurrence of metastable states analogous to the superheating or supercooling of a bulk fluid.12-14 The bulk critical temperature Tc is shifted to a lower value due to confinement.14 Hysteresis of this type may emerge from any theory of the van der Waals or mean field type. Such an approach has been exploited using density functional theory.14-18 Unfortunately “independent pore” models cannot capture the collective phenomena observed in real porous solids spanning length scales that are significantly larger than the pore correlation length of the material. Failure to account for the geometrical (5) Cohan, L. H. J. Am. Chem. Soc. 1938, 60, 433. (6) Everett, D. H. In The Solid-Gas Interface; Flood, E. A., Ed.; Marcel Dekker: New York, 1967. (7) Peterson, B. K.; Gubbins, K. E. Mol. Phys. 1987, 62, 215. (8) Heffelfinger, G. S.; van Swol, F.; Gubbins, K. E. J. Chem. Phys. 1988, 5202. (9) Bettolo Marini Marconi, U.; van Swol, F. Europhys. Lett. 1989, 8, 531. (10) Walton, J. P. R. B.; Quirke, N. Mol. Simul. 1989, 2, 361. (11) Donohue, M. D.; Aranovich, G. L. J. Colloid Interface Sci. 1998, 205, 121. (12) Hill, T. L. J. Chem. Phys. 1947, 15, 767. (13) Saam, W. F.; Cole, M. W. Phys. Rev. B 1975, 11, 1086. (14) Ball, C. P.; Evans, R. Europhys. Lett. 1987, 4, 715. (15) Evans, R.; Bettolo Marini Marconi, U.; Tarazona, P. J. Chem. Soc., Faraday Trans. 1984, 82, 1763. (16) Evans, R.; Marini Bettolo Marconi, U.; Tarazona, P. J. Chem. Phys. 1986, 84, 2376. (17) Ravikovitch, P. I.; Domhnaill, S. C. O.; Neimark, A. V.; Schuth, F.; Unger, K. K. Langmuir 1995, 11, 4765. (18) Ball, C. P.; Evans, R. Langmuir 1987, 4, 715.

10.1021/la026775y CCC: $25.00 © 2003 American Chemical Society Published on Web 02/27/2003

Gas Adsorption in Disordered Porous Media

disorder of the porous matrix often leads to misleading results on the properties of the material. One approach to address the effect of disorder on hysteresis is by employing concepts from percolation theory. According to this, the porous solid is treated as a network of interconnected pores of simple geometry (e.g. slits, cylinders, etc.) and varying size. In this respect, hysteresis is expected to occur during desorption partly by an independent single-pore thermodynamic mechanism (such as the ones described above) and partly because of inaccessibility of potentially open (supercritical) pores to the gas phase due to the porous network interconnecting structure.19-22 The simplest version of this approach lies in the concept of pore blocking, which applies in the “inkbottle” pore model.23 Hysteresis in this model is based on the assumption that evaporation of a large pore is delayed if the only available path to the bulk fluid is through a smaller pore. The above concept has been generalized to more sophisticated pore network models using elements from percolation theory.18-22 It must be emphasized, however, that, despite the incorporation of disorder or pore network effects, the thermodynamics of adsorption is still modeled at the single-pore level. Furthermore, decoupling the effect of disorder from the thermodynamics of adsorption is rather questionable, if not invalid. A more recent approach involves the simulation of sorption-desorption in a different class of representations of porous materials known as stochastic models24,25 that can effectively describe the microstructure of the porous material in a more realistic manner than that of the pore network models. In addition, the sample size of a structure generated by a stochastic model can be much larger than the pore correlation length. There are currently two different classes of methods that represent the structure of porous materials: statistical and process-based methods. In the former class, 3-D porous structures are generated by stochastic reconstruction processes, utilizing statistical information obtained from 2-D TEM images of thin sections or alternatively from small angle scattering (SAS) experiments.26 The basic underlying principle is that both the real and model structures should have identical statistical properties, such as the average porosity, , and the two-point autocorrelation function, Rz(r), which are used as input for the creation of the simulated structures under the assumption of statistical homogeneity. A typical example of a successfully generated structure employing the above method is that of Vycor porous glass.27-29 In contrast to the statistical methods, process-based models try to account for the physical processes underlying the formation of a certain microstructure. Process-based reconstruction methods, although more sound from a physical point of view, require prior knowledge of the formation process and are limited to the specific material considered. Process-based models have been successfully applied to simulate the structure (19) Mason, G. Proc. R. Soc. London 1988, A415, 453. (20) Parlar, M.; Yortsos, Y. C. J. Colloid Interface Sci. 1988, 124, 162. (21) Liu, H.; Zhang, L.; Seaton, N. A. J. Colloid Interface Sci. 1993, 156, 285. (22) Guyer, R. A.; McCall, K. R. Phys. Rev. B 1996, 54, 18. (23) McBain, J. W. J. Am. Chem. Soc. 1935, 57, 699. (24) Adler, P. M. Porous Media: Geometry and Transports; Butterworth: London, 1992. (25) Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties; Springer-Verlag: New York, 2002. (26) Levitz, P.; Ehret, G.; Sinha, S. K.; Drake, J. M. J. Chem. Phys. 1991, 95, 6151. (27) Teubner, M. Europhys. Lett. 1991, 14, 403. (28) Pellenq, R. J. M.; Levitz, P. E. Mol. Phys. 2002, 100, 2059. (29) Kainourgiakis, M. E.; Kikkinides, E. S.; Stubos, A. K. J. Porous Mater. 2002, 9, 141-154.

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of controlled porous glasses and silica gels using both offlattice30-33 and lattice-based34 approaches. A series of recent molecular simulation studies by Monson and co-workers35-37 presented mean field density functional theory calculations and Monte Carlo simulations for a lattice model of a fluid confined in stochastic representations of disordered media. Several interesting conclusions can be drawn from these studies. First of all, hysteresis is found to be a coupled combination of confinement, wetting, and disorder and is no longer associated with a simple picture of van der Waals-like metastability. Instead, the structural disorder of the porous matrix gives rise to the formation of a complex free-energy landscape, resulting in a large number of metastable states. A consequence of the above result is that hysteresis may occur with or without an equilibrium phase transition depending on whether thermodynamic consistency is maintained along the hysteresis loops. The above arguments have been complemented by the same authors using simple off lattice models, where they showed that the hysteresis loop obtained by grand canonical Monte Carlo (GCMC) simulations could be reproduced by a molecular dynamics (MD) algorithm in which adsorption and desorption occur through diffusive mass transfer.38,39 Further to these very interesting results, an additional conclusion obtained from the above studies is that hysteresis can be qualitatively predicted without the need to resort to concepts from percolation theory. In fact, Sarkisov and Monson39 provide strong evidence using MD that hysteresis in the ink-bottle model is not associated with the concept of pore blocking. Recent in situ sorption combined with small angle X-ray scattering (SAXS) studies made by our group40,41 have revealed a very interesting feature of Vycor when it is preadsorbed with CH2Br2, a condensable vapor that has a similar electron density to that of Vycor solid.40-41 From the point of view of SAXS spectra, it is quite reasonable to consider the adsorbed phase as part of the solid phase (since they both have similar electron densities) and treat the system as a biphasic one, as for the case of dry Vycor.41 Following this approach, we were able to determine the spatial correlation function of the dry and wet (preadsorbed with CH2Br2) Vycor during both adsorption and desorption and to show the existence of long-range spatial correlations in the desorption branch of the hysteresis loop. This particular behavior has been previously observed using ultrasonic attenuation and light scattering to study spatial correlations in Vycor during adsorption-desorption of hexane.42 In a further step, using stochastic reconstructions of this porous material, we were able to attribute this long-range correlation behavior during desorption to the pore accessibility of the porous matrix by employing an algorithm based on invasion percolation. Since in our (30) Gelb, L. D.; Gubbins, K. E. Langmuir 1998, 14, 2097. (31) MacElroy, J. M. D.; Raghavan, K. J. Chem. Phys. 1990, 93, 2068. (32) Kaminsky, R. D.; Monson, P. A. J. Chem. Phys. 1991, 95, 2936. (33) Page, K. S.; Monson, P. A. Phys. Rev. E 1996, 54, 6557. (34) MacFarland, T.; Barkema, G. T.; Marko, J. F. Phys. Rev. B 1996, 53, 148. (35) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.; Tarjus, G. Phys. Rev. Lett. 2001, 87, 055701. (36) Woo, H.-J.; Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7472. (37) Sarkisov, L.; Monson, P. A. Phys. Rev. E 2002, 65, 011202. (38) Sarkisov, L.; Monson, P. A. Langmuir 2000, 16, 9857. (39) Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7600. (40) Kikkinides, E. S.; Kainourgiakis, M. E.; Stefanopoulos, K.; Mitropoulos, A. Ch.; Stubos, A. K.; Kanellopoulos, N. K. J. Chem. Phys. 2000, 112, 9881. (41) Mitropoulos, A. Ch.; Haynes, J. M.; Richardson, R. M.; Kanellopoulos, N. K. Phys. Rev. B 1995, 52, 10035. (42) Page, J. H.; Liu, J.; Abeles, B.; Deckman, H. W.; Weitz, D. A. Phys. Rev. Lett. 1993, 71, 1216.

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previous work40 the issue of thermodynamics was completely ignored on the basis of a similarity concept in the correlation properties of wet Vycor during adsorption, application of percolation theory seemed to be the only choice at that point to model desorption in reconstructed Vycor. Nevertheless and despite its obvious limitations, our simple percolation-based algorithm was able to reproduce the experimentally observed long-range correlation. In the present study we attempt to provide an answer to the following issue: Is the lattice gas model35-37 (that has offered new insight in desorption hysteresis) capable of predicting the long-range spatial correlation behavior observed in the desorption branch of the hysteresis loop of Vycor? And if this is the case, then how should the fluid configuration look inside the porous matrix? Will the fluid obey the basic concepts of percolation theory (such as pore blocking for example), or do we have entirely different types of adsorbate configurations in the pore space that can still generate long-range spatial correlations? Clearly, the answer to these questions will shed some more light on the role of disorder in the formation of the hysteresis loop and will test the validity of the recently developed lattice model35-37 against a distinct feature that has been experimentally observed inside the hysteresis loop. Lattice Model Development The lattice gas model is described by the following Hamiltonian:35-37,43

∑τiZiτjZj - wmf∑[τiZi(1 - Zj) +

H ) -wff

〈ij〉

τjZj(1 - Zi)] (1) where τi ) 0, 1 and Zi ) 1, 0 denote the fluid and matrix occupancy variables, respectively, and the sums in (1) run over distinct pairs over nearest neighbor sites. The parameters wff and wmf correspond to the respective fluidfluid and material-fluid interaction strengths. The porosity  and the autocorrelation function Rz(r) can be defined by the statistical averages24-25,40:

Rz(r) )

〈(Z(x) - )(Z(x + r) - )〉  - 2

viscous flow in the reconstructed Vycor structures predicts experimentally measured values of permeability with a relative error of less than 10%. A typical 3-D image of Vycor generated by this method is shown in Figure 1. Details of the reconstruction method can be found elsewhere.24,29 Monte Carlo Simulations

〈ij〉

 ) 〈Z(x)〉

Figure 1. 3D visual image of reconstructed Vycor porous glass (le ) 1.5 nm, sample size is 192 nm).

(2a) (2b)

Note that 〈 〉 indicates spatial average. For an isotropic medium, Rz(r) becomes one-dimensional, as it is only a function of r ) |r|.24-25,40 Ideally, a representative reconstruction of a porous medium in three dimensions should have the same correlation properties as those measured on a single two-dimensional section, expressed properly by the various moments of the phase function. In practice, matching of the first two moments, that is, porosity and autocorrelation function, has been customarily pursued.24 In the present work, a standard stochastic reconstruction technique based on Gaussian random fields is employed to generate a porous material with the same porosity and autocorrelation function as those of the original Vycor porous glass. This method is well documented in the literature24 and has been successfully applied before in the reconstruction of Vycor, since it was shown that the developed structure matches higher moments of the phase function as well.29 In addition, simulation of Knudsen and (43) Kierlik, E.; Rosinberg, M. L.; Tarjus, G.; Pitard, E. Mol. Phys. 1998, 95, 341.

Grand canonical Monte Carlo (GCMC) simulations of the lattice model described in the section above have been performed. Since the grand canonical ensemble is being employed in the present study, one must subtract the term µ∑iτiZi from both sides of eq 1.44-45 The simulations have been carried out, by employing a standard Metropolis method.44-46 Cubic lattice structures with pixel sizes le of 1.5 and 3 nm and sample lengths of 192 and 300 nm, respectively, have been considered in the present study. The typical number of Monte Carlo steps per (void) lattice site was 5 × 103, out of which the first 2.5 × 103 were discarded for equilibration. These numbers were found to be sufficiently large to ensure reproducibility of the results, in agreement with similar studies in stochastic reconstructions of Vycor,37 although in several cases (inside the hysteresis loop) simulations with as many as 3 × 104 MC steps per site have been carried out. The density of the interstitial fluid can be simply calculated from the expression

Ff )

∑i〈τiZi〉t ∑i〈Zi〉t

(3)

where 〈 〉t indicates an average value in terms of GCMC steps. To mimic the actual experimental procedure, we started our simulations during adsorption from a state of low relatively pressure (directly related to the chemical potential employed in GCMC simulations) and carried out a sequence of simulations of progressively increasing pressure where the final density configuration of each (44) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: New York, 1982. (45) Binder, K. Applications of the Monte Carlo method in Statistical Physics; Springer: Berlin, 1984. (46) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987.

Gas Adsorption in Disordered Porous Media

Figure 2. Comparison between experimental and simulation adsorption-desorption hysteresis loops of CH2Br2 on Vycor at 300 K. The ratio of wmf/wff used in the simulations is 1.1 and 0.72 for pixel sizes le ) 1.5 and 3 nm, respectively. The quantity Fads corresponds to the fluid density at a certain pressure during adsorption while Fdes corresponds to the fluid density at the same pressure during desorption. The lines between the simulation points are drawn to guide the eye.

state was the initial condition of the next state. The same procedure was adopted during desorption where the sequence of the simulations started from an initially fully condensed state and subsequently pressure was progressively decreased, ensuring once again that the final density configuration of each state was the initial condition of the next state. Results and Discussion In the present study, all simulations have been performed at T/Tc ) 0.52, which corresponds to the experimental temperature of adsorption of CH2Br2 on Vycor.40,41 Obviously, since the pixel size le of reconstructed Vycor is larger than the size of CH2Br2, we cannot get quantitative predictions of the isotherm. This limitation could be overcome by using finer lattice domains;47 however, this would necessitate the resolution of two additional issues: (a) the need for exceedingly large computer memory40 and (b) the need to include long-range interactions in the fluidfluid and fluid-solid potentials. Attempting to resolve any one of these issues would essentially turn against the very purpose of using the lattice gas model, which is to study the combined effect of wetting, confinement, and disorder in realistic structures of porous media using a rather simple model that can qualitatively capture all the basic experimental observations.35-37 Therefore, in the present study we focused on matching the experimental hysteresis loop (caused by the discrepancy between the average fluid densities during adsorption and desorption at the same pressure) without matching at the same time the actual pressure of the gas. Hence, we varied the ratio of the fluid-solid to fluid-fluid interaction strengths until the width of the experimental hysteresis loop was matched. The results of the simulation are shown in Figure 2 in comparison with the corresponding experimental data. It can be seen that GCMC predicts quite well the shape and width of the experimentally observed hysteresis loop. The fact that the simulations in the lattice with the higher resolution (le ) 1.5 nm) show a sharper desoprtion “avalanche”, compared to the experimental data and the lower resolution lattice (le ) 3 nm) simulation, is attributed to the much smaller sample size of the lattice with the (47) Panagiotopoulos, A. Z. J. Chem. Phys. 2002, 112, 7132.

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Figure 3. ‘Transient” behavior of the average fluid density in Vycor obtained by the GCMC process (le ) 1.5 nm).

more refined structure. Nevertheless, we can overcome this limitation and obtain smoother isotherms by taking sample averages over a sufficient number of matrix realizations.36 The results of Figure 2 confirm recent studies showing that the proposed lattice model can capture all the basic macroscopic features of the actual hysteresis loop.35-37 In Figure 3 we show typical results from the iteration procedure of GCMC during adsorption and desorption inside the hysteresis loop. It is seen that the required “equilibration time” is much longer during desorption than during adsorption. Moreover, the “transient” behavior of the fluid density in the GCMC process is qualitatively similar to the dynamic behavior obtained from GCMD simulations on simpler systems.38 This result supports the recent suggestion that the Metropolis GCMC algorithm provides in essence a simulation of the dynamic mechanism of adsorption and desorption inside the hysteresis loop, seen in GCMD simulations.38 To investigate the spatial correlations of wet Vycor during adsorption and desorption, we used a simple levelcut technique in the density field. Since the values 0 and 1 correspond to bulk gas and liquid, respectively, a threshold value of 0.5 seems to be appropriate for level cutting. Hence, densities with values above this threshold are considered as part of the sorbed (condensed) phase while densities with values below the threshold have been removed from the pore space. The above procedure can be justified on the following grounds: First we present in Figure 4 a histogram of the whole density spectrum at a certain relative pressure. It can be seen that in general two major spikes exist in the density spectrum, corresponding to bulk vapor and bulk liquid, respectively. The actual situation is visually illustrated in Figure 5, where we present a 2-D section cut from wet Vycor during adsorption. Moreover, Figure 6 shows the effect of the threshold value on the total fluid density calculated by summing over all pixels containing the sorbed phase that has evolved from the level-cut technique. It is evident that a threshold value of 0.5 gives an average density which is in very good agreement with the respective value that is computed from the GCMC simulations during both adsorption and desorption. Following the level-cut procedure described above, we are able to calculate the two-point autocorrelation functions in wet Vycor during adsorption and desorption of CH2Br2 , inside the hysteresis loop. The calculations are performed following the definition of this property from (2b), where the sorbed phase is considered as part of the solid phase of Vycor. The results are presented in Figure

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Figure 4. Histogram of the density field inside the pore space of Vycor, determined by GCMC during adsorption at Fads ) 0.365.

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Figure 6. Effect of the threshold value on the average density of the sorbed phase resulting from the level-cut procedure during adsorption and desorption at Fads ) 0.365. The horizontal dashed lines correspond to the respective values resulting from GCMC simulations.

Figure 7. Two-point autocorrelation functions obtained from GCMC simulations in wet Vycor during adsorption and desorption at Fads ) 0.365. The corresponding data on dry Vycor are shown for comparison. Figure 5. Visual images of wet Vycor after GCMC simulations in three-dimensions during adsorption at Fads ) 0.365. The intensity of the gray color scales with the magnitude of the fluid density (le ) 1.5 nm).

7. It is seen that, during adsorption, the autocorrelation function of the wet Vycor almost coincides with the one that corresponds to the dry sample. This result has been previously observed experimentally by SAXS40 and implies the absence of long-range correlations during adsorption. One could explain this intuitively on the basis of the fact that an adsorption process is performed in a random way and thus it is not expected to alter the general features of the porous matrix. In the same figure we present the corresponding correlation function for the case of desorption at the same relative pressure. It is seen in this case that, within the region of the hysteresis loop, the autocorrelation function shows a distinct long-range behavior before diminishing at much larger distances compared to the case of wet Vycor during adsorption. This result is again in full agreement with the experimental observations from SAXS studies.40 It is thus evident that the proposed lattice gas model in refs 35-37 is capable of predicting the long-range spatial correlations that exist inside the hysteresis loop during desorption of a condensable gas (e.g. CH2Br2) in a disordered porous medium,

such as Vycor. This distinct behavior has been previously attributed to the nonrandom behavior of the desorption process and has been explained using concepts from percolation theory.40,42 However, in the present work percolation theory has not been considered and hysteresis occurs due to the existence of multiple metastable states emanating from the complex free-energy field that results from disorder. It is therefore interesting to try to explore further the issue of long-range correlation observed during desorption, with the aid of visual images of the sorbed phase during adsorption and desorption. First we compare in Figure 8 the visual configurations of the sorbed phase in the inner part of reconstructed Vycor. A clustering behavior not observed during adsorption is seen to prevail in the desorption branch of the hysteresis loop, at the same chemical potential. This clustering is a result of the GCMC simulations without the need of imposing any additional constraints, as it would have been the case when invoking percolation theory concepts.18-22,40,42 Clearly, disorder of the porous matrix is still responsible for the above behavior, but in the present case long-range correlation emanates from a different fundamental origin (as does hysteresis35-37). Indeed, as reported recently,35-37 the structural disorder of a porous medium, such as Vycor, gives rise to the formation of a complex free-energy field, which results in a large number of metastable states, giving, in turn, rise to hysteresis. To check whether several

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Figure 8. Visual images of the sorbed phase inside the porous space of reconstructed Vycor during adsorption and desorption at Fads ) 0.365 (le ) 1.5 nm).

features of percolation theory are still obeyed (such as pore blocking), we have employed the lattice gas model for a simple 2D section of reconstructed Vycor. Accordingly, we have carried out GCMC simulations in two-dimensions during both adsorption and desorption. Typical visual images of the density field are presented in Figure 9. During adsorption, the adsorbate fills preferentially the smaller pores and wets the larger cavities, which remain unsaturated (Figure 9a). On the other hand, during desorption, a much more complex fluid configuration is observed (Figure 9b). Hence, in certain regions of the pore space a behavior analogous to pore blocking is observed while cavitation of the larger cavities takes place at different regions of the pore space (i.e. large pores empty despite the nonexistence of contact with the bulk vapor phase). The latter result is in line with the recent MD findings of Sarkisov and Monson39 showing the absence of pore-blocking effects in the simple ink-bottle model. Summarizing, it must be pointed out that pore blocking during desorption is necessary to obtain clustering in the adsorbate structure and most likely the observed longrange spatial correlations of wet Vycor during desorption.40-42 On the other hand, there are regions in the pore structure where such phenomena do not occur and the local behavior changes. This implies that in these regions mass transfer from the large cavities operates efficiently even though the small pores separating the cavities from the bulk vapor remain filled with liquid. On the basis of the results of the present study, it is evident that the

Figure 9. Visual images of wet Vycor after GCMC simulations in two-dimensions (a) during adsorption and (b) during desorption. The intensity of the gray color scales with the magnitude of the fluid density (le ) 0.7 nm).

recently proposed lattice gas model35-37 can capture simultaneously entirely different and contradicting types of behavior that could not be described before by a single theoretical model. Conclusions We have studied the adsorption-desorption of condensable gases in disordered porous media, by means of Monte Carlo simulations for a lattice model of sorption in stochastic reconstructions of Vycor porous glass. By varying the solid-fluid interaction over the fluid-fluid interaction, we are able to match the experimentally

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measured hysteresis loop of CH2Br2 in Vycor at 300 K. Furthermore, comparison between the simulation results and experimental measurements of the two-point correlation functions of wet (preadsorbed with CH2Br2) Vycor, obtained from SAXS measurements, reveals that the lattice model is capable of predicting the distinct longrange correlation behavior that occurs in the pore space of the material during desorption. This is further illustrated from visual images of the simulated fluid configurations showing a clustering behavior of the sorbed phase during desorption compared to the more random configuration during adsorption. On the other hand, by performing GCMC simulations in 2-D sections of reconstructed Vycor, during both adsorption and desorption, we can envisage a complex density field where pore blocking and cavitation phenomena can take place at

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different regions of the pore space. The quantitative prediction of the hysteresis loop of CH2Br2 in Vycor, both in terms of macroscopic properties such as the shape and width of the hysteresis loop and in terms of microscopic fluid configurations in the pore space of the disordered matrix, validates several of the basic features of the recently proposed lattice gas model of adsorption in porous media. Acknowledgment. The authors wish to dedicate this work to Professor Douglas H. Everett, who has recently passed away. His pioneering work on hysteresis and characterization of porous solids has been an inspiration to all of us. LA026775Y