Free energy landscape within the hysteresis regime for fluids confined in disordered mesoporous solids.

Adsorption/desorption and melting/freezing in structurally disordered nanoporous solids exhibit strongly non-equilibrium behavior as revealed by the formation of a hysteresis region populated by the multitude of different states. Many questions concerning the free energy spectrum of these states, including the existence of the equilibrium transition, if any, their accessibility in the experiments, and internal relaxation dynamics toward the global energy minimum, still remain poorly addressed. By using a serially connected pore model with the statistical disorder as a minimal model of the pore networks, we explore the system free energies along the solid-liquid and liquid-gas transitions in the pore systems. The rigorous results obtained with this model shed light on the occurrence and nature of the equilibrium transition line in porous solids with arbitrary pore topology. We discuss further the free energies along the experimentally measured boundary and scanning transitions and how close the equilibrium states can be approached in these experiments.

Connecting dynamic pore filling mechanisms with equilibrium and out of equilibrium configurations of fluids in nanopores.

In the present study, using dynamic mean field theory complemented by grand canonical molecular dynamics simulations, we investigate the extent to which the density distributions encountered during the dynamics of capillary condensation are related to those distributions at equilibrium or metastable equilibrium in a system at fixed average density (canonical ensemble). We find that the states encountered can be categorized as out of equilibrium or quasi-equilibrium based on the magnitude of the driving force for mass transfer. More specifically, in open-ended slit pores, pore filling via double bridging is an out of equilibrium process, induced by the dynamics of the system, while pore filling by single bridge formation is connected to a series of configurations that are equilibrium configurations in the canonical ensemble and that cannot be observed experimentally by a standard adsorption process, corresponding to the grand canonical ensemble. Likewise, in closed cap slits, the formation of a liquid bridge near the pore opening and its subsequent growth while the initially detached meniscus from the capped end remains immobilized are out of equilibrium processes that occur at large driving forces. On the other hand, at small driving forces, there is a continuous acceleration of the detached meniscus from the capped end, which is associated with complete reversibility in the limit of an infinitesimally small driving force.

Nonequilibrium Steady States in Fluid Transport through Mesopores: Dynamic Mean Field Theory and Nonequilibrium Molecular Dynamics.

We present a dynamic mean field theory (DMFT) and nonequilibrium dual control volume grand canonical molecular dynamics (GCMD) simulation study of steady-state fluid transport in slit-shaped mesopores under an applied chemical potential gradient. The main focus is on states where the bulk conditions on one side of the pore would lead to a capillary condensed state in the pore at equilibrium while those on the other side would lead to a vapor state in the pore. This choice of conditions is motivated by certain separation applications in which condensable vapors permeate through mesoporous membranes. Under these circumstances, we have found partially filled states with a liquid-like state at the high chemical potential end of the pore and a vapor-like state at the low chemical potential end. This phenomenon is accompanied by hysteresis. The existence of partially filled states has been hypothesized in previous work but the present paper reveals them as an emergent feature of the systems. We find that predictions of DMFT are in good qualitative agreement with the overall GCMD results. However, the GCMD results demonstrate that the transport is faster through the partially filled pore than through the unfilled pore, a feature not captured by DMFT.

A comparison of dynamic mean field theory and grand canonical molecular dynamics for the dynamics of pore filling and capillary condensation of fluids in mesopores.

We use results from grand canonical molecular dynamics (GCMD) to test the predictions from dynamic mean field theory (DMFT) for the pore filling and capillary condensation mechanisms of a fluid confined in slit shaped mesopores. The theory predicts that capillary condensation occurs by a nucleation process in which a liquid bridge forms between the two walls, and the pore is filled via the growth of this bridge. For longer pores, multiple bridging is seen. These mechanisms are confirmed by the molecular dynamics simulations. The primary difference between the theory and simulations lies in the role of fluctuations. DMFT predicts a single nucleation time and location, while in GCMD (and in nature) a distribution of nucleation times and locations is seen.

Dynamic density functional theory with hydrodynamic interactions: theoretical development and application in the study of phase separation in gas-liquid systems.

Building on recent developments in dynamic density functional theory, we have developed a version of the theory that includes hydrodynamic interactions. This is achieved by combining the continuity and momentum equations eliminating velocity fields, so the resulting model equation contains only terms related to the fluid density and its time and spatial derivatives. The new model satisfies simultaneously continuity and momentum equations under the assumptions of constant dynamic or kinematic viscosity and small velocities and/or density gradients. We present applications of the theory to spinodal decomposition of subcritical temperatures for one-dimensional and three-dimensional density perturbations for both a van der Waals fluid and for a lattice gas model in mean field theory. In the latter case, the theory provides a hydrodynamic extension to the recently studied dynamic mean field theory. We find that the theory correctly describes the transition from diffusive phase separation at short times to hydrodynamic behaviour at long times.

Dynamic mean field theory for lattice gas models of fluid mixtures confined in mesoporous materials.

We present the extension of dynamic mean field theory (DMFT) for fluids in porous materials (Monson, P. A. J. Chem. Phys. 2008, 128, 084701) to the case of mixtures. The theory can be used to describe the relaxation processes in the approach to equilibrium or metastable equilibrium states for fluids in pores after a change in the bulk pressure or composition. It is especially useful for studying systems where there are capillary condensation or evaporation transitions. Nucleation processes associated with these transitions are emergent features of the theory and can be visualized via the time dependence of the density distribution and composition distribution in the system. For mixtures an important component of the dynamics is relaxation of the composition distribution in the system, especially in the neighborhood of vapor-liquid interfaces. We consider two different types of mixtures, modeling hydrocarbon adsorption in carbon-like slit pores. We first present results on bulk phase equilibria of the mixtures and then the equilibrium (stable/metastable) behavior of these mixtures in a finite slit pore and an inkbottle pore. We then use DMFT to describe the evolution of the density and composition in the pore in the approach to equilibrium after changing the state of the bulk fluid via composition or pressure changes.

Dynamic mean field theory of condensation and evaporation processes for fluids in porous materials: application to partial drying and drying.

We study the dynamics of evaporation for lattice gas models of fluids in porous materials using a recently developed dynamic mean field theory. The theory yields a description of the dynamics that is consistent with the mean field theory of the thermodynamics at equilibrium. The nucleation processes associated with phase changes in the pore are emergent features of the dynamics. Our focus is on situations where there is partial drying or drying in the system, associated with weakly attractive or repulsive interactions between the fluid and the pore walls. We consider two systems in this work: (i) a two-dimensional slit pore geometry relevant to the study of adsorption/desorption or intrusion/extrusion dynamics for fluids in porous materials and (ii) a three dimensional slit pore modeling a pair of square plates in a bath of liquid as used in recent theoretical studies of dewetting processes between hydrophobic surfaces. We assess the theory by comparison with a higher order approximation to the dynamics that yields the Bethe-Peierls or quasi-chemical approximation at equilibrium.

Solid-fluid and solid-solid equilibrium in hard sphere united atom models of n-alkanes: rotator phase stability.

We present a study of the phase behavior for models of n-alkanes with chain lengths up to C(21) based on hard sphere united atom models of methyl and methylene groups, with fixed bond lengths and C-C-C bond angles. We extend earlier work on such models of shorter alkanes by allowing for gauche conformations in the chains. We focus particularly on the orientational order about the chain axes in the solid phase near the melting point, and our model shows how the loss of this orientational order leads to the formation of rotator phases. We have made extensive calculations of the thermodynamic properties of the models as well as order parameters for tracking the degree of orientational order around the chain axis. Depending on the chain length and whether the carbon number is even or odd, the model exhibits both a rotator phase and a more orientationally ordered solid phase in addition to the fluid phase. Our results indicate that the transition between the two solid phases is first-order with a small density change. The results are qualitatively similar to those seen experimentally and show that rotator phases can appear in models of alkanes without explicit treatment of attractive forces or explicit treatment of the hydrogen atoms in the chains.

Contact angles, pore condensation, and hysteresis: insights from a simple molecular model.

We discuss the thermodynamics of adsorption of fluids in pores when the solid-fluid interactions lead to partial wetting of the pore walls, a situation encountered, for example, in water adsorption in porous carbons. Our discussion is based on calculations for a lattice gas model of a fluid in a slit pore treated via mean field density functional theory (MFDFT). We calculate contact angles for pore walls as a function of solid-fluid interaction parameter, alpha, in the model, using Young's equation and the interfacial tensions calculated in MFDFT. We consider adsorption and desorption in both infinite pores and in finite length pores in contact with the bulk. In the latter case, contact with the bulk can promote evaporation or condensation, thereby dramatically reducing the width of hysteresis loops. We show how the observed behavior changes with alpha. By using a value of alpha that yields a contact angle of about 85 degrees and maintaining the bulk fluid in a supersaturated vapor state on adsorption, we find an adsorption/desorption isotherm qualitatively similar to those for graphitized carbon black where pore condensation occurs at supersaturated bulk vapor states in the spaces between the primary particles of the adsorbent.

Calculation of free energies and chemical potentials for gas hydrates using Monte Carlo simulations.

We describe a method for calculating free energies and chemical potentials for molecular models of gas hydrate systems using Monte Carlo simulations. The method has two components: (i) thermodynamic integration to obtain the water and guest molecule chemical potentials as functions of the hydrate occupancy; (ii) calculation of the free energy of the zero-occupancy hydrate system using thermodynamic integration from an Einstein crystal reference state. The approach is applicable to any classical molecular model of a hydrate. We illustrate the methodology with an application to the structure-I methane hydrate using two molecular models. Results from the method are also used to assess approximations in the van der Waals-Platteeuw theory and some of its extensions. It is shown that the success of the van der Waals-Platteeuw theory is in part due to a cancellation of the error arising from the assumption of a fixed configuration of water molecules in the hydrate framework with that arising from the neglect of methane-methane interactions.

Further studies of a simple atomistic model of silica: thermodynamic stability of zeolite frameworks as silica polymorphs.

We have applied our previously reported model of silica based on low coordination and strong association [J. Chem. Phys. 121, 8415 (2004)], to the calculation of phase stability of zeolite frameworks SOD, LTA, MFI, and FAU as silica polymorphs. We applied the method of Frenkel and Ladd for calculating free energies of these solids. Our model predicts that the MFI framework structure has a regime of thermodynamic stability at low pressures and above approximately 1400 K, relative to dense phases such as quartz. In contrast, our calculations predict that the less dense frameworks SOD, LTA, and FAU exhibit no regime of thermodynamic stability. We have also used our model to investigate whether templating extends the MFI regime of thermodynamic stability to lower temperatures, by considering templates with hard-sphere repulsions and mean-field attractions to silica. Within the assumptions of our model, we find that quartz remains the thermodynamically stable polymorph at zeolite synthesis temperatures (approximately 400 K) unless unphysically large template-silica attractions are assumed. These predictions suggest that some zeolites such as MFI may have regimes of thermodynamic stability even without template stabilization.

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.

Crystal nucleation in binary hard sphere mixtures: a Monte Carlo simulation study.

We present calculations of the nucleation barrier during crystallization in binary hard sphere mixtures under moderate degrees of supercooling using Monte Carlo simulations in the isothermal-isobaric semigrand ensemble in conjunction with an umbrella sampling technique. We study both additive and negatively nonadditive binary hard sphere systems. The solid-fluid phase diagrams of such systems show a rich variety of behavior, ranging from simple spindle shapes to the appearance of azeotropes and eutectics to the appearance of substitutionally ordered solid phase compounds. We investigate the effect of these types of phase behavior upon the nucleation barrier and the structure of the critical nucleus. We find that the underlying phase diagram has a significant effect on the mechanism of crystal nucleation. Our calculations indicate that fractionation of the species upon crystallization increases the difficulty of crystallization of fluid mixtures and in the absence of fractionation (azeotropic conditions) the nucleation barrier is comparable to pure fluids. We also calculate the barrier to nucleation of a substitutionally ordered compound solid. In such systems, which also show solid-solid phase separation, we find that the phase that nucleates is the one whose equilibrium composition is closer to the composition of the fluid phase.

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.

Modeling spontaneous formation of precursor nanoparticles in clear-solution zeolite synthesis.

We present a lattice model describing the formation of silica nanoparticles in the early stages of the clear-solution templated synthesis of silicalite-1 zeolite. Silica condensation/hydrolysis is modeled by a nearest-neighbor attraction, while the electrostatics are represented by an orientation-dependent, short-range interaction. Using this simplified model, we show excellent qualitative agreement with published experimental observations. The nanoparticles are identified as a metastable state, stabilized by electrostatic interactions between the negatively charged silica surface and a layer of organic cations. Nanoparticle size is controlled mainly by the solution pH, through nanoparticle surface charge. The size and concentration of the charge-balancing cation are found to have a negligible effect on nanoparticle size. Increasing the temperature allows for further particle growth by Ostwald ripening. We suggest that this mechanism may play a role in the growth of zeolite crystals.

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.

A study of the phase behavior of a simple model of chiral molecules and enantiomeric mixtures.

We present a study of the solid-fluid and solid-solid phase equilibrium for molecular models representative of chiral molecules and enantiomeric mixtures. The models consist of four hard sphere interaction sites of different diameters in a tetrahedral arrangement with the fifth hard sphere interaction site at the center of the tetrahedron. The volumetric properties and free energies of the pure enantiomers and binary mixtures were calculated in both fluid and solid phases using isobaric Monte Carlo simulations. The models exhibit essentially ideal solution behavior in the fluid phase with little chiral discrimination. In the solid phase the effects of chirality are much greater. Solid-fluid phase behavior involving the pure enantiomer solids and also racemic compounds was calculated. The calculations indicate that, depending on the relative sizes of the hard sphere interaction sites, packing effects alone can be sufficient to stabilize a racemic compound with respect to the pure enantiomer solids.

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.

On the mechanical properties and phase behavior of silica: a simple model based on low coordination and strong association.

We present a simple and computationally efficient classical atomistic model of silica in which the silicon and oxygen are simulated as hard spheres with four and two association sites, respectively. We have performed isobaric-isothermal Monte Carlo simulations to study the mechanical and phase behavior of this model. We have investigated solid phase structures of the model corresponding to quartz, cristobalite, and coesite, as well as some zeolite structures. For the model these phases are mechanically stable and highly incompressible. Ratios of zero-pressure bulk moduli and thermal expansion coefficients for alpha quartz, alpha cristobalite, and coesite are in quite good agreement with experimental values. The pressure-temperature phase diagram was constructed and shows three solid phases corresponding to cristobalite, quartz, and coesite, as well as a fluid or glass phase, behavior qualitatively similar to that seen for silica experimentally.

Solid-fluid and solid-solid phase equilibrium in a model of n-alkane mixtures.

Solid-fluid and solid-solid phase equilibrium for binary mixtures of hard sphere chains modeling n-hexane, n-heptane, and n-octane has been calculated using Monte Carlo computer simulations. Thermodynamic integration was used to calculate the Gibbs free energy and chemical potentials in the solid and fluid phases from pure component reference values. A multiple stage free energy perturbation method was used to calculate the composition derivative of the Gibbs free energy. Equation of state and free energy data for the fluid phase indicate ideal solution behavior. Nonideality is much more significant in the solid phase with only partial solubility of shorter chains in the longer chains and essentially no solubility at the other end of the composition range. The miscibility decreases with increasing chain length difference between the components. For the model of n-hexane/n-octane mixtures solid--solid phase separation has been observed directly in some of the simulations, with the components segregating between the layers of the solid structure. The behavior is similar to that seen in some binary n-alkane mixtures with longer chain lengths but comparable chain length ratios between the components. Such phase separation, although indicated thermodynamically, is not seen directly in the simulations of the n-heptane/n-octane mixture due to the difference in the pure component crystal structures.