Phase transitions of one-component fluids adsorbed in random porous media: Monte Carlo simulations

Nanoscale thermodynamics needs the concept of a disjoining chemical potential

Disjoining pressure was discovered by Derjaguin in 1930's, which describes the difference between the pressure of a strongly confined fluid and the corresponding one in a bulk phase. It has been revealed recently that the disjoining pressure is at the origin of distinct differential and integral surface tensions for strongly confined fluids. Here we show how the twin concept, disjoining chemical potential, arises in a reminiscent way although it comes out eighty years later. This twin concept advances our understanding of nanoscale thermodynamics. Ensemble-dependence (or environment-dependence) is one hallmark of thermodynamics of small systems. We show that integral surface tension is ensemble-dependent while differential surface tension is not. Moreover, two generalized Gibbs-Duhem equations involving integral surface tensions are derived, as well as two additional adsorption equations relating surface tensions to adsorption-induced strains. All the results obtained in this work further evidence that an approach alternative of Hill's nanothermodynamics is possible, by extending Gibbs surface thermodynamics instead of resorting to Hill's replica trick. Moreover, we find a compression-expansion hysteresis without any underlying phase transition.

Legendre-Fenchel transforms capture layering transitions in porous media

We have investigated the state of a nanoconfined fluid in a slit pore in the canonical and isobaric ensembles. The systems were simulated with molecular dynamics simulations. The fluid has a transition to a close-packed structure when the height of the slit approaches the particle diameter. The Helmholtz energy is a non-convex function of the slit height if the number of particles does not exceed that of one monolayer. As a consequence, the Legendre transform cannot be applied to obtain the Gibbs energy. The Gibbs energy of a non-deformable slit pore can be transformed into the Helmholtz energy of a deformable slit pore using the Legendre-Fenchel transform. The Legendre-Fenchel transform corresponds to the Maxwell construction of equal areas.

Thermodynamics of interfaces extended to nanoscales by introducing integral and differential surface tensions

As a system shrinks down in size, more and more molecules are found in its surface region, so surface contribution becomes a large or even a dominant part of its thermodynamic potentials. Surface tension is a venerable scientific concept; Gibbs defined it as the excess of grand potential of an inhomogeneous system with respect to its bulk value per interface area [J. W. Gibbs, "The Collected Works" in Thermodynamics (1928), Vol. 1]. The mechanical definition expresses it in terms of pressure tensor. So far, it has been believed the two definitions always give the same result. We show that the equivalence can break down for fluids confined in narrow pores. New concepts of integral and differential surface tensions, along with integral and differential adsorptions, need to be introduced for extending Gibbs thermodynamics of interfaces. We derived two generalized Gibbs adsorption equations. These concepts are indispensable for an adequate description of nanoscale systems. We also find a relation between integral surface tension and Derjaguin's disjoining pressure. This lays down the basis for measuring integral and differential surface tensions from disjoining pressure by using an atomic force microscope.

Connect the Thermodynamics of Bulk and Confined Fluids: Confinement-Adsorption Scaling

A fluid (a gas or a liquid) adsorbed in a porous material can behave very differently from its bulk counterpart. The advent of various synthesized materials with nanopores and their wide applications have provided strong impetus for studying fluids in confinement because our current understanding is still incomplete. From a large number of Monte Carlo simulations, we found a scaling relation that allows for connecting some thermodynamic properties (chemical potential, free energy per particle, and grand potential per particle) of a confined fluid to the bulk ones. Upon rescaling the adsorbed fluid density, the adsorption isotherms for many different confining environments collapse to the corresponding bulk curve. We also reveal the intimate connection of the reported scaling relation to Gibbs theory of inhomogeneous fluids and morphological thermodynamics. The advance in our understanding of confined fluids, gained from this study, also opens attractive perspectives for circumventing experimental difficulty for directly measuring some fluid thermodynamic properties in nanoporous materials.

Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in Random Porous Media: TPT of Wertheim and Scaled Particle Theory

Extension of Wertheim's thermodynamic perturbation theory and its combination with scaled particle theory is proposed and applied to study the liquid-gas phase behavior of polydisperse hard-sphere square-well chain fluid confined in the random porous media. Thermodynamic properties of the reference system, represented by the hard-sphere square-well fluid in the matrix, are calculated using corresponding extension of the second-order Barker-Henderson perturbation theory. We study effects of polydispersity and confinement on the phase behavior of the system. While polydispersity causes increase of the region of phase coexistence due to the critical temperature increase, confinement decreases the values of both critical temperature and critical density making the region of phase coexistence smaller. This effect is enhanced with the increase of the size ratio of the fluid and matrix particles. The increase of the average chain length at fixed values of polydispersity and matrix density shifts the critical point to a higher temperature and a slightly lower density.

Fluids in porous media. IV. Quench effect on chemical potential

It appears to be a common sense to measure the crowdedness of a fluid system by the densities of the species constituting it. In the present work, we show that this ceases to be valid for confined fluids under some conditions. A quite thorough investigation is made for a hard sphere (HS) fluid adsorbed in a hard sphere matrix (a quench-annealed system) and its corresponding equilibrium binary mixture. When fluid particles are larger than matrix particles, the quench-annealed system can appear much more crowded than its corresponding equilibrium binary mixture, i.e., having a much higher fluid chemical potential, even when the density of each species is strictly the same in both systems, respectively. We believe that the insight gained from this study should be useful for the design of functionalized porous materials.

The Structural Properties and Diffusion of a Three-Dimensional Isotropic Core-Softened Model Fluid in Disordered Porous Media. Molecular Dynamics Simulation

The microscopic structure and dynamic properties of an isotropic three-dimensional core-softened model fluid in disordered matrices of Lennard-Jones particles have been studied. Molecular dynamics computer simulations in Grand Canonical ensemble were used as the methodological tools. It was shown that the microscopic structure of the fluid is characterized by anomalies similar to those found in a bulk model, but that it is affected by the fluid-matrix interactions. The dynamic properties also exhibit anomalous dependence on fluid density, but the magnitude of these anomalies is suppressed in comparison to the bulk fluid model. The anomalous behaviour of the diffusion coefficient is attributed to structural changes in the first coordination shell of a given fluid particle. It seems that the anomalies can only be suppressed at matrix densities which are higher than those studied in the present work.

Effect of disorder on condensation in the lattice gas model on a random graph

The lattice gas model of condensation in a heterogeneous pore system, represented by a random graph of cells, is studied using an exact analytical solution. A binary mixture of pore cells with different coordination numbers is shown to exhibit two phase transitions as a function of chemical potential in a certain temperature range. Heterogeneity in interaction strengths is demonstrated to reduce the critical temperature and, for large-enough degreeS of disorder, divides the cells into ones which are either on average occupied or unoccupied. Despite treating the pore space loops in a simplified manner, the random-graph model provides a good description of condensation in porous structures containing loops. This is illustrated by considering capillary condensation in a structural model of mesoporous silica SBA-15.

Phase separation of binary nonadditive hard sphere fluid mixture confined in random porous media

I analyze the fluid-fluid phase separation of nonadditive hard sphere fluid mixture absorbed in random porous media. An equation of state is derived by using the perturbation theory to this complex system with quenched disorders. The results of this theory are in good agreement with those obtained from semi-grand canonical ensemble Monte Carlo simulations. The contact value of the fluid-fluid radial distribution functions of the reference which is the key point of the perturbation process is derived as well, the comparison against Monte Carlo simulations shows that it has an excellent accuracy.

A Monte Carlo study of crowding effects on the self-assembly of amphiphilic molecules

In this work, lattice Monte Carlo was used to study the effects of crowding on the self-assembly of surfactants. Simulation results show that crowding strongly shifts the critical micelle concentration (CMC) of surfactants from the bulk value. Two effects originated from crowding are found to govern the CMC shift: one is the depletion effect by crowding agents and the other is the available volume for micelle formation. The depletion effects inevitably result in the enrichment of surfactants in crowding-free regions and cause the decrease in CMC. On the other hand, the appearance of crowding agents decreases the available volume for micelle formation, which reduces the conformational entropy and impedes the micelle formation. Three factors, including the radius of crowding agents, the arrangement of crowding agents, and the volume fraction of crowding agents, are considered in this work to study the crowding effects. The trends of CMC shifts are interpreted from the competition between the depletion effects and the available volume for micelle formation.

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.

Colloid-polymer mixtures in the presence of quenched disorder: a theoretical and computer simulation study

We use theory and computer simulation to study the structure and phase behavior of colloid-polymer mixtures in the presence of quenched disorder. The Asakura-Oosawa model (AO) (Asakura and Oosawa 1954 J. Chem. Phys. 22 1255) is used to describe the colloid-colloid, colloid-polymer, and polymer-polymer pair interactions. We then investigate the behavior of this model in the presence of frozen-in (quenched) obstacles. The obstacles will be placed according to two different scenarios, both of which are experimentally feasible. In the first scenario, polymers are distributed at positions drawn from an ideal gas configuration. In the second scenario, colloidal particles are distributed at positions drawn from an equilibrium hard sphere configuration. We investigate how the unmixing transition of the AO model is affected by the type of quenched disorder. The theoretical formalism is based on the replica method of Given and Stell (1994 Physica A 209 495). Our foremost aim is to test the accuracy of three common closures to the replica Ornstein-Zernike equations, namely the hypernetted chain, the Percus-Yevick, and the Martinov-Sarkisov equations. The accuracy is determined by comparison with grand canonical Monte Carlo simulations. We find that, for quenched polymer disorder, all three closures perform remarkably well. However, when quenched colloid disorder is considered, i.e. the second mentioned scenario, the predictions of all three closures worsen dramatically.

Fluids in porous media. I. A hard sponge model

The morphology of many porous materials is spongelike. Despite the abundance of such materials, simple models which allow for a theoretical description of these materials are still lacking. Here, we propose a hard sponge model which is made by digging spherical cavities in a solid continuum. We found an analytical expression for describing the interaction potential between fluid particles and the spongelike porous matrix. The diagrammatic expansions of different correlation functions are derived as well as that of grand potential. We derived also the Ornstein-Zernike (OZ) equations for this model. In contrast to Madden-Glandt model of random porous media [W. G. Madden and E. D. Glandt, J. Stat. Phys. 51, 537 (1988)], the OZ equations for a fluid confined in our hard sponge model have some similarity to the OZ equations of a three-component fluid mixture. We show also how the replica method can be extended to study our sponge model and that the same OZ equations can be derived also from the extended replica method.

Capillary phase transitions of linear and branched alkanes in carbon nanotubes from molecular simulation

Capillary phase transitions of linear (from C(1) to C(12)) and branched (C(5) isomers) alkanes in single-walled carbon nanotubes have been investigated using the gauge-cell Monte Carlo simulation. The isotherm at a supercritical temperature increases monotonically with chemical potential and coincides with that from the traditional grand canonical Monte Carlo simulation, whereas the isotherm at a subcritical temperature exhibits a sigmoid van der Waals loop including stable, metastable, and unstable regions. Along this loop, the coexisting phases are determined using an Maxwell equal-area construction. A generic confinement effect is found that reduces the saturation chemical potential, lowers the critical temperature, increases the critical density, and shrinks the phase envelope. The effect is greater in a smaller diameter nanotube and is greater in a nanotube than in a nanoslit.

Prediction of phase behavior of nanoconfined Lennard-Jones fluids with density functional theory based on the first-order mean spherical approximation

The recently proposed first-order mean spherical approximation (FMSA) [Y. Tang, J. Chem. Phys. 121, 10605 (2004)] for inhomogeneous fluids is extended to study the phase behavior of nanoconfined Lennard-Jones fluids, which is consistent with the phase equilibria calculation of the corresponding bulk fluid. With a combination of fundamental measure theory, FMSA provides Helmholtz free energy and direct correlation function to formulate density functional theory, which implementation is as easy as the mean-field theory. Following previous success in predicting density profiles inside slit pores, this work is focused specially on the vapor-liquid equilibrium of the Lennard-Jones fluids inside these pores. It is found that outside the critical region FMSA predicts well the equilibrium diagram of slit pores with the sizes of 5.0, 7.5, and 10 molecular diameters by comparing with available computer simulation data. As a quantitative method, FMSA can be treated as an extension from its bulk calculation, while the mean-field theory is only qualitative, as its bulk version.

Effects of confinement on freezing and melting

We present a review of experimental, theoretical, and molecular simulation studies of confinement effects on freezing and melting. We consider both simple and more complex adsorbates that are confined in various environments (slit or cylindrical pores and also disordered porous materials). The most commonly used molecular simulation, theoretical and experimental methods are first presented. We also provide a brief description of the most widely used porous materials. The current state of knowledge on the effects of confinement on structure and freezing temperature, and the appearance of new surface-driven and confinement-driven phases are then discussed. We also address how confinement affects the glass transition.

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.

Capillary Condensation in a Geometrically and a Chemically Heterogeneous Pore: A Molecular Simulation Study

A computer simulation study has been carried out, using an extended Gibbs ensemble Monte Carlo technique, to examine the influence of so-called geometric and chemical disorder on the thermodynamic behavior of simple fluids confined in porous media. The technique allows the equilibrium coexistence of gas and liquid phases to be calculated in a single run. The phase diagram of Lennard-Jones fluid has been calculated in a perfectly cylindrical pore as a reference. Some disorder is then introduced in the porous material, first by spatially modifying the external potential of the initially cylindrical pore, to imitate the geometric disorder of a more realistic pore (undulation, constrictions, etc.) and second by modulating the amplitude of the same initially cylindrical potential to reproduce the energetic disorder of realistic pores due to chemical variations along it. It is shown that the chemical disorder has a much stronger effect on the phase diagram of the confined fluid. The complete adsorption/desorption isotherms are also calculated to help in understanding the large effects of chemical disorder.

Isotropic-nematic transition of hard rods immersed in random sphere matrices

Using replica density functional theory and Monte Carlo computer simulations we investigate a system of annealed hard spherocylinders adsorbed in a matrix of quenched hard spheres. Theoretical predictions for the partition coefficient, defined as the ratio of density of rods in the matrix and that in a reservoir, agree well with simulation results. Theory predicts the isotropic-nematic transition to remain first order upon increasing sphere packing fraction, and to shift towards lower rod densities. This scenario is consistent with our simulation results that clearly show a jump in the nematic order parameter upon increasing the rod density at constant matrix packing fraction, corresponding to the isotropic-nematic transition, even for sphere matrix packing fractions < or approximately equal to 0.3.

Integral equation study of a Stockmayer fluid adsorbed in polar disordered matrices

Based on replica integral equations in the (reference) hypernetted chain approximation we investigate the structural features and phase properties of a dipolar Stockmayer fluid confined to a disordered dipolar matrix. The integral equations are applied to the homogeneous high-temperature phase where the system is globally isotropic. At low densities we find the influence of dipolar interactions between fluid (f) and matrix (m) particles to be surprisingly similar to the previously investigated effect of attractive isotropic (fm) interactions: the critical temperature of the vapor-liquid transition decreases with increasing (fm) coupling, while the critical density increases. The anisotropic nature of the dipolar (fm) interactions turns out to play a more dominant role at high fluid densities where we observe a pronounced sensitivity in the dielectric constant and a strong degree of local orientational ordering of the fluid particles along the local fields generated by the matrix. Moreover, an instability of the dielectric constant, which is a precursor of ferroelectric ordering occurring both in bulk Stockmayer fluids and in fluids in nonpolar matrices, is observed only for very small dipolar (fm) couplings.

Water in nanopores. I. Coexistence curves from Gibbs ensemble Monte Carlo simulations

Coexistence curves of water in cylindrical and slitlike nanopores of different size and water-substrate interaction strength were simulated in the Gibbs ensemble. The two-phase coexistence regions cover a wide range of pore filling level and temperature, including ambient temperature. Five different kinds of two-phase coexistence are observed. A single liquid-vapor coexistence is observed in hydrophobic and moderately hydrophilic pores. Surface transitions split from the main liquid-vapor coexistence region, when the water-substrate interaction becomes comparable or stronger than the water-water pair interaction. In this case prewetting, one and two layering transitions were observed. The critical temperature of the first layering transition decreases with strengthening water-substrate interaction towards the critical temperature expected for two-dimensional systems and is not sensitive to the variation of pore size and shape. Liquid-vapor phase transition in a pore with a wall which is already covered with two water layers is most typical for hydrophilic pores. The critical temperature of this transition is very sensitive to the pore size, in contrast to the liquid-vapor critical temperature in hydrophobic pores. The observed rich phase behavior of water in pores evidences that the knowledge of coexistence curves is of crucial importance for the analysis of experimental results and a prerequiste of meaningful simulations.

Molecular simulation of the vapor-liquid phase behavior of Lennard-Jones mixtures in porous solids

We present vapor-liquid phase coexistence curves for binary fluid mixtures in a disordered porous solid. The porous material is modeled as a collection of randomly dispersed hard spheres. A variant of the Monte Carlo Gibbs ensemble method [J. K. Brennan and W. Dong, J. Chem. Phys. 116, 8948 (2002)] is used to simulate Lennard-Jones fluid mixtures at several porosities: 0.9, 0.95, and 0.975. Effects based on the size and the energetics of the mixture components are studied. Pressure-composition and pressure-density phase diagrams at reduced temperatures of 0.75 and 1.0 are reported. Compared to the bulk fluid behavior, dramatic shifts in the phase envelope were found for even highly porous structures. Both the Lennard-Jones size and energy mixture parameters were found to strongly influence the resulting shape of the phase envelope.