Capillary condensation in disordered porous materials: hysteresis versus equilibrium behavior

Adsorption of fluids in a pore with chemical heterogeneities: the cooperative effect

In this work, we study the cooperative adsorption of fluids in a heterogeneous pore, in which the pore walls are composed of homogeneous substrates with chemical groups (CGs) decorating them. The adsorption caused by the homogeneous substrates alone and that by CGs do not add up to the overall adsorption, indicating the existence of a cooperative effect. The cooperative effect is the source of cooperative adsorption, and is characterized in this work by the ratio of the overall adsorption to the sum of adsorption by the substrate only and that by CGs. It is found that the cooperative adsorption does not depend monotonically on the substrate or the CGs. Two different origins of the cooperative adsorption play different roles depending on which one dominates the overall adsorption. Our simulations reveal that, when the homogeneous substrate dominates the overall adsorption, weakening of the attractive fluid-substrate interaction or alternatively strengthening of the fluid-CGs interaction leads to a stronger cooperative effect and enhances the cooperative adsorption. However, when CGs dominate the overall adsorption, weakening of the attractive fluid-CG interaction or strengthening the fluid-substrate interaction results in strong cooperative adsorption. In order to investigate the effects of the distribution of CGs on cooperative adsorption, a design-test method is generalized and used in this work. Simulation results show that the overall adsorption can be significantly affected by the CG distribution.

Capillary condensation in porous materials. Hysteresis and interaction mechanism without pore blocking/percolation process

We have performed measurements of boundary hysteresis loops, reversal curves, and subloops in p+-type porous silicon, a porous material composed of straight non-interconnected pores. These data show that a strong interaction mechanism exists between the pores. The pores of porous silicon are non-independent, whereas they are not interconnected. This hysteretic behavior is very similar to that observed in porous glass, which consists of cavities connected to each other by constrictions. This questions the so-called pore blocking/percolation model developed to explain the behavior of fluid in porous glass. More generally, if we disregard the shape of the boundary hysteresis loops which depends on the porous material (H1 for MCM-41 and SBA-15, H2 for porous glass and p+-type porous silicon), the hysteretic features inside the main loop are qualitatively the same for all these porous systems. This shows that none of these systems are composed of independent pores. A coupling between the pores is always present whether they are interconnected or not and whatever the shape of the main loop is.

Liquid-vapor coexistence at a mesoporous substrate

The condensation of hexane vapor onto a mesoporous Si substrate with a pore radius of 3.5 nm has been studied by means of volumetry and ellipsometry. The filling fraction of the pores and the coverage of the substrate have been determined. The coverage of the regime after the completion of capillary condensation has been compared to recent theoretical work.

Dynamical aspects of the adsorption hysteresis phenomenon

Equilibrium and nonequilibrium transport properties of adsorbates in mesoporous Vycor porous glass have been experimentally studied using nuclear magnetic resonance techniques. With the known geometrical characteristics of porous glass and with measured self-diffusivities, transient sorption curves have been quantitatively compared to those predicted within a Fick's law model. This model correctly describes data outside a hysteresis region. In contrast, in the hysteresis region, a two-step mechanism of density relaxation is required to explain the behavior. These two mechanisms are identified as diffusion at early stages and activated density redistribution at later stages of adsorption. The latter mechanism, being intrinsically slow in nature, is anticipated to prevent the system from reaching equilibrium.

Constrained equilibrium as a tool for characterization of deformable porous media

A new method for characterizing the deformable porous materials with noncritical adsorption probes is proposed. The mechanism is based on driving the adsorbate through a sequence of constrained equilibrium states with the insertion isotherms forming a pseudocritical point or a van der Waals-type loop. In the framework of a perturbation theory and Monte Carlo simulations we have found a link between the loop parameters and the host morphology. This allows one to characterize porous matrices through analyzing a shift of the pseudocritical point and a shape of the pseudospinodals.

Mercury porosimetry in mesoporous glasses: a comparison of experiments with results from a molecular model

We present results from experiments and molecular modeling of mercury porosimetry into mesoporous Vycor and controlled pore glass (CPG) solid materials. The experimental intrusion/extrusion curves show a transition from a type H2 hysteresis for the Vycor glass to a type H1 hysteresis for the CPG. Mercury entrapment is observed in both materials, but we find that the amount of entrapped mercury depends on the chosen experimental relaxation time. No additional entrapment is found in a second intrusion/extrusion cycle, but hysteresis is still observed. This indicates that hysteresis and entrapment are of different origin. The experimental observations are qualitatively reproduced in theoretical calculations based on lattice models, which provide significant insights of the molecular mechanisms occurring during mercury porosimetry experiments in these porous glasses.

Preferred orientations and stability of medium length n-alkanes solidified in mesoporous silicon

The n-alkanes C(16)H(34), C(17)H(36), C(19)H(40), and C(25)H(52) have been imbibed and solidified in mesoporous, crystalline silicon with a mean pore diameter of 10 nm. The structures and phase sequences have been determined by x-ray diffractometry. Apart from a reduction and the hysteresis of the melting-freezing transition, we find a set of six discrete orientation states ("domains") of the confined alkane crystals with respect to the lattice of the silicon host. The growth process responsible for the domain selection is interpreted as a nanoscale version of the Bridgman technique known from single-crystal growth. Oxidation of the pore walls leads to extrusion of the hydrocarbons upon crystallization, whereas the solidified n-alkanes investigated in nonoxidized, porous silicon are thermodynamically stable.

A computational study of the reconstruction of amorphous mesoporous materials from gas adsorption isotherms and structure factors via evolutionary optimization

A general method for the three-dimensional reconstruction of mesoporous materials by evolutionary optimization against target data is developed. The method is applied specifically in reconstruction of amorphous material models using gas adsorption data, structure factor data, or a combination of both. A recently introduced lattice-gas approach is used to model adsorption in these calculations, and a high-pass limited Fourier representation is used to facilitate evolution of large-scale structures during the optimization. Reconstructions are made of several material models which mimic real materials obtained either by phase separation and etching or by sol-gel processing. Analysis of the reconstructions provides considerable insight into the type and quantity of structural information probed by gas adsorption and small-angle scattering experiments. We find that reconstructions based only on structure factors tend to underestimate the mean pore size. We also find that in many cases excellent reconstructions can be obtained using only adsorption-branch data, and that in all cases reconstructions based jointly on both types of data are superior to those based only on one, suggesting that these measures contain "complementary" information. It is also found that in most cases the use of desorption data is not warranted, and that the use of adsorption data taken at many temperatures will not improve reconstructions. The reproducibility of the method is shown to be satisfactory. The method can be computationally expensive if gas adsorption data are used, but it is easily parallelized, and therefore results can still be obtained in reasonable time. Finally, the possible application of this approach to real systems, including templated porous materials, is discussed.

Exploration of molecular dynamics during transient sorption of fluids in mesoporous materials

In recent years, considerable progress has been made in the development of novel porous materials with controlled architectures and pore sizes in the mesoporous range. An important feature of these materials is the phenomenon of adsorption hysteresis: for certain ranges of applied pressure, the amount of a molecular species adsorbed by the mesoporous host is higher on desorption than on adsorption, indicating a failure of the system to equilibrate. Although this phenomenon has been known for over a century, the underlying internal dynamics responsible for the hysteresis remain poorly understood. Here we present a combined experimental and theoretical study in which microscopic and macroscopic aspects of the relaxation dynamics associated with hysteresis are quantified by direct measurement and computer simulations of molecular models. Using nuclear magnetic resonance techniques and Vycor porous glass as a model mesoporous system, we have explored the relationship between molecular self-diffusion and global uptake dynamics. For states outside the hysteresis region, the relaxation process is found to be essentially diffusive in character; within the hysteresis region, the dynamics slow down dramatically and, at long times, are dominated by activated rearrangement of the adsorbate density within the host material.

Change in desorption mechanism from pore blocking to cavitation with temperature for nitrogen in ordered silica with cagelike pores

To verify pore blocking controlled desorption in ink-bottle pores, we measured the temperature dependence of the adsorption-desorption isotherms of nitrogen on four kinds of KIT-5 samples with expanded cavities hydrothermally treated for different periods of time at 393 K. In the samples, almost spherical cavities are arranged in a face-centered cubic array and the cavities are connected through small channels. The pore size of the channels increased with an increase in the hydrothermal treatment time. At lower temperatures a steep desorption branch changed to a gradual one as the hydrothermal treatment was prolonged. For the sample hydrothermally treated only for 1 day, the rectangular hysteresis loop shrank gradually with increasing temperature while keeping its shape. The temperature dependence of the evaporation pressure observed was identical with that expected for cavitation-controlled desorption. On the other hand, for the samples hydrothermally treated for long times, the gradual desorption branch became a sharp one with increasing temperature. This strongly suggests that the desorption mechanism is altered from pore blocking to cavitation with temperature. Application of percolation theory to the pore blocking controlled desorption observed here is discussed.

Freezing of fluids confined in a disordered nanoporous structure

Freezing of a simple fluid in a disordered nanoporous carbon is studied using molecular simulations. Only partial crystallization occurs, and the confined phase is composed of crystalline and amorphous nanodomains. This freezing behavior departs strongly from that for nanopores of simple geometry. We present a method for analyzing the freezing in such disordered materials in terms of a transition in the average size and number of crystalline clusters. The results provide a basis for the interpretation of experiments on freezing in such materials, particularly 1H-NMR and scattering experiments.

Influence of surface chemical heterogeneities on adsorption/desorption hysteresis and coexistence diagram of metastable states within cylindrical pores

Grand canonical Monte Carlo simulations are performed to determine the adsorption/desorption isotherms at different temperatures of a Lennard-Jones fluid confined within a simple model of cylindrical pores presenting chemical heterogeneities. A complex hysteresis loop is observed, showing hysteresis subloops (scanning curves). This is shown to be consistent with the existence of several metastable states (local minima in the system free energy). A recent extension to the Gibbs ensemble technique is then used to calculate the complete coexistence diagram of these local minima.

Gas adsorption and desorption in silica aerogels: a theoretical study of scattering properties

We present a numerical study of the structural correlations associated with gas adsorption and desorption in silica aerogels in order to provide a theoretical interpretation of scattering experiments. Following our earlier work, we use a coarse-grained lattice-gas description and determine the nonequilibrium behavior of the adsorbed gas within a local mean-field analysis. We focus on the differences between the adsorption and desorption mechanisms and their signature in the fluid-fluid and gel-fluid structure factors as a function of temperature. At low temperature, but still in the regime where the isotherms are continuous, we find that the adsorbed fluid density, during both filling and draining, is correlated over distances that may be much larger than the gel correlation length. In particular, extended fractal correlations may occur during desorption, indicating the existence of a ramified cluster of vapor filled cavities. This also induces an important increase of the scattering intensity at small wave vectors. The similarity and differences with the scattering of fluids in other porous solids such as Vycor are discussed.

Wetting of rings on a nanopatterned surface: a lattice model study

We perform mean-field density functional theory calculations on a lattice model to study the wetting of a solid substrate decorated with a ring pattern of nanoscale dimensions. We have found three different liquid morphologies on the substrate: a ring morphology where the liquid covers the pattern, a bulge morphology where a droplet is forming on one side of the ring, and a morphology where the liquid forms a cap spanning the nonwetting disk inside the pattern. We investigate the relative stability of these morphologies as a function of the ring size, wall-fluid interaction, and temperature. The results found are in very good agreement with experiments and calculations performed on similar systems at a micrometer length scale. The bulge morphology has also been observed in Monte Carlo simulations of the lattice model. Our results show that (i) morphologies of wetting patterns previously observed on a much larger (microm) scale can also form on a nm length scale, (ii) whether or not this happens depends crucially on the size of the wettable pattern, and (iii) the wettable ring may only be partially wet by the bulge morphology of the fluid. This morphology is a result of a spontaneously broken symmetry in the system.

Mean-field theory of liquid droplets on roughened solid surfaces: application to superhydrophobicity

We present calculations of the density distributions and contact angles of liquid droplets on roughened solid surfaces for a lattice gas model solved in a mean-field approximation. For the case of a smooth surface, this approach yields contact angles that are well described by Young's equation. We consider rough surfaces created by placing an ordered array of pillars on a surface, modeling so-called superhydrophobic surfaces, and we have made calculations for a range of pillar heights. The apparent contact angle follows two regimes as the pillar height increases. In the first regime, the liquid penetrates the interpillar volume, and the contact angle increases with pillar height before reaching a constant value. This behavior is similar to that described by the Wenzel equation for contact angles on rough surfaces, although the contact angles are underestimated. In the second regime, the liquid does not penetrate the interpillar volume substantially, and the contact angle is independent of the pillar height. This situation is similar to that envisaged in the Cassie-Baxter equation for contact angles on heterogeneous surfaces, but the contact angles are overestimated by this equation. For larger pillar heights, two states of the droplet can be observed, one Wenzel-like and the other Cassie-like.

Soft core fluid in a quenched matrix of soft core particles: a mobile mixture in a model gel

We present a density-functional study of a binary phase-separating mixture of soft core particles immersed in a random matrix of quenched soft core particles of larger size. This is a model for a binary polymer mixture immersed in a cross-linked rigid polymer network. Using the replica "trick" for quenched-annealed mixtures we derive an explicit density functional theory that treats the quenched species on the level of its one-body density distribution. The relation to a set of effective external potentials acting on the annealed components is discussed. We relate matrix-induced condensation in bulk to the behavior of the mixture around a single large particle. The interfacial properties of the binary mixture at a surface of the quenched matrix display a rich interplay between capillary condensation inside the bulk matrix and wetting phenomena at the matrix surface.

Knowledge-based reconstruction of random porous media

An evolutionary optimization technique is used to reconstruct digitized material models of 300(3) nm3 size for mesoporous two-phase systems. The models are adapted to the two-point probability (TPP) and to a volume-based pore-size distribution (PSD) which were derived from SANS and adsorption experiments and which carry statistical information about morphology and topology of the pore system. To avoid extreme update-costs, the bulk of mutations are assessed by means of a suitable approximation of the PSD; it is demonstrated that a sporadic insertion of the PSD suffices to drive the algorithm towards satisfactory models in acceptable time. Our approach is knowledge-based in the sense that (i) the mutations are restricted to expedient exchanges of phase-voxels by a heuristic rule, and (ii) the sporadic calculation of the PSD from the current state of the model, in essence, provides an efficient self-control for the evolutionary process. We applied the method to reconstruct periodic models of the xerogel Gelsil 200. Such reconstructs of real mesoporous solids could be utilized, for instance, to verify theories of adsorption and capillary condensation.

Helium condensation in aerogel: avalanches and disorder-induced phase transition

We present a detailed numerical study of the elementary condensation events (avalanches) associated to the adsorption of in silica aerogels. We use a coarse-grained lattice-gas description and determine the nonequilibrium behavior of the adsorbed gas within a local mean-field analysis, neglecting thermal fluctuations and activated processes. We investigate the statistical properties of the avalanches, such as their number, size and shape along the adsorption isotherms as a function of gel porosity, temperature, and chemical potential. Our calculations predict the existence of a line of critical points in the temperature-porosity diagram where the avalanche size distribution displays a power-law behavior and the adsorption isotherms have a universal scaling form. The estimated critical exponents seem compatible with those of the field-driven random field Ising model at zero temperature.

Does water condense in carbon pores?

Using grand canonical Monte Carlo (GCMC) simulations of molecular models, we investigate the nature of water adsorption and desorption in slit pores with graphitelike surfaces. Special emphasis is placed on the question of whether water exhibits capillary condensation (i.e., condensation when the external pressure is below the bulk vapor pressure). Three models of water have been considered. These are the SPC and SPC/E models and a model where the hydrogen bonding is described by tetrahedrally coordinated square-well association sites. The water-carbon interaction was described by the Steele 10-4-3 potential. In addition to determining adsorption/desorption isotherms, we also locate the states where vapor-liquid equilibrium occurs for both the bulk and confined states of the models. We find that for wider pores (widths >1 nm), condensation does not occur in the GCMC simulations until the pressure is higher than the bulk vapor pressure, P0. This is consistent with a physical picture where a lack of hydrogen bonding with the graphite surface destabilizes dense water phases relative to the bulk. For narrow pores where the slit width is comparable to the molecular diameter, strong dispersion interactions with both carbon surfaces can stabilize dense water phases relative to the bulk so that pore condensation can occur for P < P0 in some cases. For the narrowest pores studied--a pore width of 0.6 nm--pore condensation is again shifted to P > P0. The phase-equilibrium calculations indicate vapor-liquid coexistence in the slit pores for P < P0 for all but the narrowest pores. We discuss the implications of our results for interpreting water adsorption/desorption isotherms in porous carbons.

Capillary condensation in cylindrical nanopores

Using grand canonical Monte Carlo simulations, we have explored the phenomenon of capillary condensation (CC) of Ar at the triple temperature inside infinitely long, cylindrical pores. Pores of radius R = 1 nm, 1.7 nm, and 2.5 nm have been investigated using a gas-surface interaction potential parametrized by the well depth D of the gas on a planar surface made of the same material as that comprising the porous host. For strongly attractive situations--i.e., large D--one or more (depending on R) Ar layers adsorb successively before liquid fills the pore. For very small values of D, in contrast, negligible adsorption occurs at any pressure P below saturated vapor pressure P0; above saturation, there eventually occurs a threshold value of P at which the coverage jumps from empty to full, nearly discontinuously. Hysteresis is found to occur in the simulation data whenever abrupt CC occurs--i.e., for R > or = 1.7 nm--and for small D when R = 1 nm. Then, the pore-emptying branch of the adsorption isotherm exhibits larger coverage than the pore-filling branch, as is known from many experiments and simulation studies. The relation between CC and wetting on planar surfaces is discussed in terms of a threshold value of D, which is about one-half of the value found for the wetting threshold on a planar surface. This finding is consistent with a simple thermodynamic model of the wetting transition developed previously.

Lattice density functional for colloid-polymer mixtures: comparison of two fundamental measure theories

We consider a binary mixture of colloid and polymer particles with positions on a simple cubic lattice. Colloids exclude both colloids and polymers from nearest neighbor sites. Polymers are treated as effective particles that are mutually noninteracting, but exclude colloids from neighboring sites; this is a discrete version of the (continuum) Asakura-Oosawa-Vrij model. Two alternative density functionals are proposed and compared in detail. The first is based on multioccupancy in the zero-dimensional limit of the bare model, analogous to the corresponding continuum theory that reproduces the bulk fluid free energy of free volume theory. The second is based on mapping the polymers onto a multicomponent mixture of polymer clusters that are shown to behave as hard cores; the corresponding property of the extended model in strong confinement permits direct treatment with lattice fundamental measure theory. Both theories predict the same topology for the phase diagram with a continuous fluid-fcc freezing transition at low polymer fugacity and, upon crossing a tricritical point, a first-order freezing transition for high polymer fugacities with rapidly broadening density jump.

Hysteresis and scanning behavior of mesoporous molecular sieves

Sorption hysteresis is a widely studied phenomenon whose predicted behavior is well documented and researched. On the other hand, there is much less known about the region that lies between sorption isotherms, believed to be a metastable region. Scanning curves are a way of understanding the mechanism of hysteresis and a tool for hysteresis model validation. Scanning curves were produced for mesoporous materials: SBA-15 and MCM-41 for N(2) sorption at 77 K and Ar sorption at 87 K. A limited set of different scanning behaviors is identified. Like most hysteresis theories, it was found that a single model for scanning behavior cannot be extended to all materials under the same or different experimental conditions. Two behaviors are consistent with recent theories and simulations; however, several are not. The implications as to the characterization of pore dimensions and structure are discussed.

Modeling of the hysteresis phenomena in finite-sized slitlike nanopores. Revision of the recent results by rigorous numerical analysis

The systematic investigation of the hysteresis phenomena in finite-sized slitlike nanopores via the Aranovich-Donohue (AD) lattice density functional theory (LDFT) is presented. The new reliable quantitative modeling of the adsorption and desorption branch of the hysteresis loop, through the formation and movement of the curved meniscus, is formulated. As a result, we find that our proposal, which closely mimics the experimental findings, can reproduce a rounded shape of the desorption branch of the hysteresis loop. On the basis of the exhausted commutations, we proved that the hysteresis loop obtained in the considered finite-sized slitlike geometry is of the H1 type of the IUPAC classification. This fundamental result and the other most important results do not confirm the results of the recent studies of Sangwichien et al., whereas they fully agree with the recent lattice studies due to Monson et al. We recognize that the nature of the hysteresis loops (i.e. position, width, shape, and the multiple steps) mainly depends on the value of the energy of both the adsorbate-adsorbate and adsorbate-adsorbent interactions; however, the first one is critical for the appearance of hysteresis. Thus, for relatively small adsorbate-adsorbate interactions, the adsorption-desorption process is fully reversible in the whole region of the bulk density. We show that the strong adsorbate-adsorbent interactions produce (also observed experimentally) multiple steps within hysteresis loops. Contrary to the other studies of the hysteresis phenomena in confined geometry via the LDFT formalism, we constructed both ascending and descending scanning curves, which are known from the experimental observations. Additionally, we consider the problem of the stability of both the obtained adsorption and desorption branches of the computed hysteresis loop in finite-sized slitlike nanopores.

Water in carbon nanotubes: Adsorption isotherms and thermodynamic properties from molecular simulation

Grand canonical Monte Carlo simulations are performed to study the adsorption of water in single-walled (6:6), (8:8), (10:10), (12:12), and (20:20) carbon nanotubes in the 248-548 K temperature range. At room temperature the resulting adsorption isotherms in (10:10) and wider single-walled carbon nanotubes (SWCNs) are characterized by negligible water uptake at low pressures, sudden and complete pore filling once a threshold pressure is reached, and wide adsorption/desorption hysteresis loops. The width of the hysteresis loops decreases as pore diameter narrows and it becomes negligible for water adsorption in (8:8) and (6:6) SWCNs. Results for the isosteric heat of adsorption, density profiles along the pore axis and across the pore radii, order parameter across the pore radii, and x-ray diffraction patterns are presented. Layered structures are observed when the internal diameter of the nanotubes is commensurate to the establishment of a hydrogen-bonded network. The structure of water in (8:8) and (10:10) SWCNs is ordered when the temperature is 298 and 248 K, respectively. By simulating adsorption isotherms at various temperatures, the hysteresis critical temperature, e.g., the lowest temperature at which no hysteresis can be detected, is determined for water adsorbed in (20:20), (12:12), and (10:10) SWCNs. The hysteresis critical temperature is lower than the vapor-liquid critical temperature for bulk Simple Point Charge-Extended (SPCE) water model.

Gas Adsorption in Active Carbons and the Slit-Pore Model 1: Pure Gas Adsorption

We describe procedures based on the polydisperse independent ideal slit-pore model, Monte Carlo simulation and density functional theory (a 'slab-DFT') for predicting gas adsorption and adsorption heats in active carbons. A novel feature of this work is the calibration of gas-surface interactions to a high surface area carbon, rather than to a low surface area carbon as in all previous work. Our models are used to predict the adsorption of carbon dioxide, methane, nitrogen, and hydrogen up to 50 bar in several active carbons at a range of near-ambient temperatures based on an analysis of a single 293 K carbon dioxide adsorption isotherm. The results demonstrate that these models are useful for relatively simple gases at near-critical or supercritical temperatures.

Phase coexistence in heterogeneous porous media: A new extension to Gibbs ensemble Monte Carlo simulation method

The effect of confinement on phase behavior of simple fluids is still an area of intensive research. In between experiment and theory, molecular simulation is a powerful tool to study the effect of confinement in realistic porous materials, containing some disorder. Previous simulation works aiming at establishing the phase diagram of a confined Lennard-Jones-type fluid, concentrated on simple pore geometries (slits or cylinders). The development of the Gibbs ensemble Monte Carlo technique by Panagiotopoulos [Mol. Phys. 61, 813 (1987)], greatly favored the study of such simple geometries for two reasons. First, the technique is very efficient to calculate the phase diagram, since each run (at a given temperature) converges directly to an equilibrium between a gaslike and a liquidlike phase. Second, due to volume exchange procedure between the two phases, at least one invariant direction of space is required for applicability of this method, which is the case for slits or cylinders. Generally, the introduction of some disorder in such simple pores breaks the initial invariance in one of the space directions and prevents to work in the Gibbs ensemble. The simulation techniques for such disordered systems are numerous (grand canonical Monte Carlo, molecular dynamics, histogram reweighting, N-P-T+test method, Gibbs-Duhem integration procedure, etc.). However, the Gibbs ensemble technique, which gives directly the coexistence between phases, was never generalized to such systems. In this work, we focus on two weakly disordered pores for which a modified Gibbs ensemble Monte Carlo technique can be applied. One of the pores is geometrically undulated, whereas the second is cylindrical but presents a chemical variation which gives rise to a modulation of the wall potential. In the first case almost no change in the phase diagram is observed, whereas in the second strong modifications are reported.

Application of the Bethe-Peierls approximation to a lattice-gas model of adsorption on mesoporous materials

We calculate adsorption and desorption isotherms in models of several classes of porous materials using a lattice-gas model solved in the Bethe-Peierls (quasichemical) approximation. Isotherms and fluid density profiles from the Bethe-Peierls and Bragg-Williams approximations are compared with grand-canonical Monte Carlo simulation results. The Bethe-Peierls approximation produces both more accurate adsorption and desorption isotherms and more realistic fluid density profiles than the Bragg-Williams approximation. Details of the application of the Bethe-Peierls approximation applied to a three-dimensionally inhomogeneous system are given. We show that the numerical solution of this theory can be accomplished using a self-consistent iterator very similar to that currently used in studies employing the Bragg-Williams approximation. This iterative scheme is substantially more efficient than the numerical optimization method used in many previous studies of lattice-gas models in the quasichemical approximation. We find that use of the Bethe-Peierls approximation is only slightly more computationally demanding than the Bragg-Williams approximation, and thus recommend it for use in future work on this class of models.

Dynamic aspects of mercury porosimetry: a lattice model study

Grand canonical Monte Carlo simulations using both Glauber dynamics and Kawasaki dynamics have been carried out for a recently developed lattice model of a nonwetting fluid confined in a porous material. The calculations are aimed at investigating the molecular scale mechanisms leading to mercury retention encountered during mercury porosimetry experiments. We first describe a set of simulations on slit and ink-bottle pores. We have studied the influence of the pore width parameter on the intrusion/extrusion curve shapes and investigated the corresponding mechanisms. Entrapment appears during Kawasaki dynamics simulations of extrusion performed on ink-bottle pores when the system is studied for short relaxation times. We then consider the more realistic and complex case of a Vycor glass building on recent work on the dynamics of adsorption of wetting fluids (Woo, H. J.; Monson, P. A. Phys. Rev. E 2003, 67, 041207). Our results suggest that mercury entrapment is caused by a decrease in the rate of mass transfer associated with the fragmentation of the liquid during extrusion.

Thermodynamic Characterization of Fluids Confined in Heterogeneous Pores by Monte Carlo Simulations in the Grand Canonical and the Isobaric−Isothermal Ensembles

Materials presenting nanoscale porosity are able to condense gases in their structure. This "capillary condensation" phenomenon has been studied for more than one century. Theoretical models help to understand experimental results but fail in explaining all experimental features. Most of the time, the difficulties in making quantitative or even qualitative predictions are due to the geometric complexity of the porous materials, such as large pore size distribution, chemical heterogeneities, or pore interconnections. Numerical calculations (lattice gas models or molecular simulations) are of considerable interest to calculate the adsorption properties of a fluid confined in a porous model with characteristic sizes up to several tens of nanometers. For instance, the grand canonical Monte Carlo method allows one to compute the average amount of fluid adsorbed in the porous model as a function of the temperature and the chemical potential of the fluid. However, the grand potential, necessary for a complete characterization of the system, is not a direct output of the algorithm. It is shown in this paper that the use of the isobaric-isothermal (NPT) ensemble allows one to circumvent this problem; that is, it is possible to get in one single Monte Carlo run the absolute grand potential for any given thermodynamic state of the fluid. A simplified thermodynamic integration scheme is then used to evaluate the grand potential over the whole isotherm branch passing through this initially given point. Since the usual NPT technique is a priori limited to homogeneous pores, it is proposed, for the first time, to generalize this procedure to a pore presenting a chemical heterogeneity along its axis. The new method gives the same results as the previous for homogeneous pores and allows new predictions for chemically heterogeneous pores. Comparison with the full integration scheme shows that the proposed direct calculation is faster since it avoids multiple Monte Carlo runs and more precise because it avoids the possible cumulative errors of the integration procedure.

Water adsorption in disordered mesoporous silica (Vycor) at 300K and 650K: A Grand Canonical Monte Carlo simulation study of hysteresis

This numerical simulation paper focuses on the adsorption/desorption of water in disordered mesoporous silica glasses (Vycor-like). The numerical adsorbent was previously obtained by off lattice method, and was shown to reproduce quite well the micro- and mesotextural properties of real Vycor, as well as morphological (pore size distribution) and topological (pore interconnections) disorder. The water-water interactions are described by the SPC model while water-silica interactions are calculated in the framework of the PN-TrAZ model. The water adsorption/desorption isotherms and the configurational energies are calculated by the Grand Canonical Monte Carlo simulation method. The low pressure results compare well with experiments, showing the good transferability of the intermolecular potential. It is shown that if the hysteresis loop observed in the adsorption/desorption isotherm is considered as a true phase transition (which is actually still an open question in the case of disordered porous materials), then it is possible to calculate the grand potential by applying the thermodynamic integration scheme. The grand potential is shown to be multivalued for low (subcritical) temperature, and continuous for high (supercritical) temperature. A coexistence point is found within the hysteresis loop, actually close to the vertical desorption line. Below the equilibrium chemical potential, the gaslike branch is stable whereas the liquidlike branch is metastable. The situation is reversed above the coexistence point.

Grand Potential, Helmholtz Free Energy, and Entropy Calculation in Heterogeneous Cylindrical Pores by the Grand Canonical Monte Carlo Simulation Method

The adsorption of fluids in porous media is still an open area of research, since no model is able to explain all experimental features. The difficulties rise from the complexity of the real porous materials which present surface heterogeneities, large pore size distributions, and complex networks of interconnected pores. In parallel to experimental efforts trying to produce more ordered porous materials, theoreticians try to introduce more disorder in their models, with the help of molecular simulation for instance. This grand canonical Monte Carlo simulation study concentrates on the adsorption of a simple Lennard-Jones fluid in three porous substrates, to compare the effect of purely geometric heterogeneity (spatial deformation of the external potential) as opposed to purely chemical heterogeneity (amplitude variations of the external potential). This separation is unrealistic, since geometric fluctuations of a real pore diameter along its axis generally induce variations in the amplitude of the external potential created by the pore. However it enables one to compare both effects. In this paper, a thermodynamic integration scheme is applied to a complete set of adsorption/desorption isotherms. The grand potential, free energy, and entropy are calculated, which allows one to discuss the features of the phase diagrams. It is shown that a purely geometric deformation (undulation) of the external potential does not affect the thermodynamic characteristics of the confined fluid. On the other hand, amplitude modulation of the external potential (chemical heterogeneity) strongly distorts the phase diagram. This heterogeneity is actually able to stabilize a "bridgelike" phase which corresponds to an accumulation of molecules in the most attractive region of the pore.

Computer simulation of the phase diagram for a fluid confined in a fractal and disordered porous material

We present a grand canonical Monte Carlo simulation study of the phase diagram of a Lennard-Jones fluid adsorbed in a fractal and highly porous aerogel. The gel environment is generated from an off-lattice diffusion limited cluster-cluster aggregation process. Simulations have been performed with the multicanonical ensemble sampling technique. The biased sampling function has been obtained by histogram reweighting calculations. Comparing the confined and the bulk system liquid-vapor coexistence curves we observe a decrease of both the critical temperature and density in qualitative agreement with experiments and other Monte Carlo studies on Lennard-Jones fluids confined in random matrices of spheres. At variance with these numerical studies we do not observe upon confinement a peak on the liquid side of the coexistence curve associated with a liquid-liquid phase coexistence. In our case only a shouldering of the coexistence curve appears upon confinement. This shoulder can be associated with high density fluctuations in the liquid phase. The coexisting vapor and liquid phases in our system show a high degree of spatial disorder and inhomogeneity.

Density functional theory for general hard-core lattice gases

We put forward a general procedure to obtain an approximate free-energy density functional for any hard-core lattice gas, regardless of the shape of the particles, the underlying lattice, or the dimension of the system. The procedure is conceptually very simple and recovers effortlessly previous results for some particular systems. Also, the obtained density functionals belong to the class of fundamental measure functionals and, therefore, are always consistent through dimensional reduction. We discuss possible extensions of this method to account for attractive lattice models.

A grand canonical Monte Carlo study of capillary condensation in mesoporous media: Effect of the pore morphology and topology

We study by means of Grand Canonical Monte Carlo simulations the condensation and evaporation of argon at 77 K in nanoporous silica media of different morphology or topology. For each porous material, our results are compared with data obtained for regular cylindrical pores. We show that both the filling and emptying mechanisms are significantly affected by the presence of a constriction. The simulation data for a constricted pore closed at one end reproduces the asymmetrical shape of the hysteresis loop that is observed for many real disordered porous materials. The adsorption process is a quasicontinuous mechanism that corresponds to the filling of the different parts of the porous material, cavity, and constriction. In contrast, the desorption branch for this pore closed at one end is brutal because the evaporation of Ar atoms confined in the largest cavity is triggered by the evaporation of the fluid confined in the constriction (which isolates the cavity from the gas reservoir). This evaporation process conforms to the classical picture of "pore blocking effect" proposed by Everett many years ago. We also simulate Ar adsorption in a disordered porous medium, which mimics a Vycor mesoporous silica glass. The adsorption isotherm for this disordered porous material having both topological and morphological defects presents the same features as that for the constricted pore (quasicontinuous adsorption and steep desorption process). However, the larger degree of disorder of the Vycor surface enhances these main characteristics. Finally, we show that the effect of the disorder, topological and/or morphological, leads to a significant lowering of the capillary condensation pressure compared to that for regular cylindrical nanopores. Also, our results suggest that confined fluids isolated from the bulk reservoir evaporate at a pressure driven by the smallest size of the pore.

Modeling mercury porosimetry using statistical mechanics

We consider mercury porosimetry from the perspective of the statistical thermodynamics of penetration of a nonwetting liquid into a porous material under an external pressure. We apply density functional theory to a lattice gas model of the system and use this to compute intrusion/extrusion curves. We focus on the specific example of a Vycor glass and show that essential features of mercury porosimetry experiments can be modeled in this way. The lattice model exhibits a symmetry that provides a direct relationship between intrusion/extrusion curves for a nonwetting fluid and adsorption/desorption isotherms for a wetting fluid. This relationship clarifies the status of methods that are used for transforming mercury intrusion/extrusion curves into gas adsorption/desorption isotherms. We also use Monte Carlo simulations to investigate the nature of the intrusion and extrusion processes.

Molecular simulation evidence for solidlike adsorbate in complex carbonaceous micropore structures

Adsorption of a model nitrogen vapor on a range of complex nanoporous carbon structures is simulated by grand canonical Monte Carlo simulation for a single subcritical temperature above the bulk freezing point. Adsorption and desorption isotherms, heats of adsorption, and three-dimensional singlet distribution functions (SDFs) were generated. Inspection of the SDFs reveals significant levels of solidlike adsorbate at saturation even in the most complex of the microporous solids considered. This strongly suggests that solidlike adsorbate will also occur for simple subcritical vapors adsorbed on real noncrystalline solids such as microporous carbons at temperatures above the bulk freezing point, supporting indirect experimental observations. The presence of significant levels of solidlike adsorbate has implications for characterization of microporous solids where adsorbate density is used (e.g., determination of pore volume from loading). Detailed consideration of the SDF at different loadings for a model microporous solid indicates solidlike adsorbate forms at distributed points throughout the pore space at pressures dependent on the nature of the local porosity. The nature of the local porosity also dictates the freezing mechanism. A local freezing/ melting/refreezing process is also observed. Introduction of mesoporosity into the model causes hysteresis between the adsorption and desorption isotherms. Adsorption in the hysteresis loop occurs by a series of local condensation events. It appears as if the presence of adjacent microporosity and/or adsorbate within it affects the pressure at which these events occur. Reversal of the condensation during desorption occurs throughout the mesoporosity at a single pressure; this pressure is unaffected by the presence of adjacent microporosity or the adsorbate within it. It is also shown that the empirical concept of "pore size" is not consistent for describing adsorption in the complex solids considered here. A new concept is, therefore, proposed that seeks to account for the factors that affect local adsorption energy: local geometry, microtexture, surface atom density, and surface chemistry.