Phase equilibrium in a molecular model of a fluid confined in a disordered porous material

Computational investigation of hysteresis and phase equilibria of n-alkanes in a metal-organic framework with both micropores and mesopores

Adsorption hysteresis is a phenomenon related to phase transitions that can impact applications such as gas storage and separations in porous materials. Computational approaches can greatly facilitate the understanding of phase transitions and phase equilibria in porous materials. In this work, adsorption isotherms for methane, ethane, propane, and n-hexane were calculated from atomistic grand canonical Monte Carlo (GCMC) simulations in a metal-organic framework having both micropores and mesopores to better understand hysteresis and phase equilibria between connected pores of different size and the external bulk fluid. At low temperatures, the calculated isotherms exhibit sharp steps accompanied by hysteresis. As a complementary simulation method, canonical (NVT) ensemble simulations with Widom test particle insertions are demonstrated to provide additional information about these systems. The NVT+Widom simulations provide the full van der Waals loop associated with the sharp steps and hysteresis, including the locations of the spinodal points and points within the metastable and unstable regions that are inaccessible to GCMC simulations. The simulations provide molecular-level insight into pore filling and equilibria between high- and low-density states within individual pores. The effect of framework flexibility on adsorption hysteresis is also investigated for methane in IRMOF-1.

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.

Three-Dimensional Structure of a Simple Liquid at a Face-Centered-Cubic (001) Solid Surface Interface

A liquid in the vicinity of a solid-liquid interface (SLI) may exhibit complex structures. In this study, we used molecular dynamics simulations demonstrating for the first time that the liquid adjacent to the SLI can have a two-level structure in some cases: a major structure and a minor structure. Through a time-averaging process of molecular motions, we identified the type of the liquid structure by calculating positions of the maximum liquid density in three spatial dimensions, and these positions were found to distribute in many dispersed zones (called high-density zones (HDZs)). The major structure appears throughout the SLI, while the minor structure only occurs significantly within the third layer. Instead of the previously reported body-centered cubic (BCC) or face-centered-cubic (FCC) types, the major structure was found to show a body-centered tetragonal (BCT) type. The adjacent HDZs are connected by specific junctions, demonstrating that atoms diffuse along some particular high probability paths from one HDZ to another. By considering the three-dimensional liquid density distribution from the continuum point of view, more complete details of the structure and diffusive behavior of liquids in the SLI are also possible to be revealed.

Hysteresis of liquid adsorption in porous media by coarse-grained Monte Carlo with direct experimental validation

The effects of path-dependent wetting and drying manifest themselves in many types of physical systems, including nanomaterials, biological systems, and porous media such as soil. It is desirable to better understand how these hysteretic macroscopic properties result from a complex interplay between gasses, liquids, and solids at the pore scale. Coarse-Grained Monte Carlo (CGMC) is an appealing approach to model these phenomena in complex pore spaces, including ones determined experimentally. We present two-dimensional CGMC simulations of wetting and drying in two systems with pore spaces determined by sections from micro X-ray computed tomography: a system of randomly distributed spheres and a system of Ottawa sand. Results for the phase distribution, water uptake, and matric suction when corrected for extending to three dimensions show excellent agreement with experimental measurements on the same systems. This supports the hypothesis that CGMC can generate metastable configurations representative of experimental hysteresis and can also be used to predict hysteretic constitutive properties of particular experimental systems, given pore space images.

Finite-size scaling study of the vapor-liquid critical properties of confined fluids: Crossover from three dimensions to two dimensions

We perform histogram-reweighting grand canonical Monte Carlo simulations of the Lennard-Jones fluid confined between two parallel hard walls and determine the vapor-liquid critical and coexistence properties in the range of sigma<or=H<or=6sigma and 10sigma<or=L(x),L(y)<or=28sigma, where H is the wall separation, L(x)=L(y) is the system size and sigma is the characteristic length. By matching the probability distribution of the ordering operator, P(M), to the three-dimensional (3D) and two-dimensional (2D) Ising universality classes according to the mixed-field finite-size scaling approach, we establish a "phase diagram" in the (H,L) plane, showing the boundary between four types of behavior: 3D, quasi-3D, quasi-2D, and 2D. In order to facilitate 2D critical point calculation, we present a four-parameter analytical expression for the 2D Ising universal distribution. We show that the infinite-system-size critical points obtained by extrapolation from the apparent 3D and 2D critical points have only minor differences with each other. In agreement with recent reports in the literature [Jana et al., J. Chem. Phys. 130, 214707 (2009)], we find departure from linearity in the relationship between critical temperature and inverse wall separation, as well as nonmonotonic dependence of the critical density and the liquid density at coexistence upon wall separation. Additional studies of the ST2 model of water show similar behavior, which suggests that these are quite general properties of confined fluids.

Mode-coupling behavior of a Lennard-Jones binary mixture upon increasing confinement

Molecular dynamics simulations are performed on a Lennard Jones binary mixture confined in off lattice matrices of soft spheres with increasing radius. We focus on dynamics upon supercooling and in particular on testing the mode coupling theory properties of the confined mixture. Parameters of mode coupling theory in going from bulk to weak confinement, and from weak to strong confinement are extracted from simulations and analyzed. We focus on the study of the behavior of the single particle density correlators. We find that the mode coupling theory retains its validity also in the case of strong confinement, with a reduction of range of validity. The role of hopping is discussed in relation with the differences between the results obtained from the diffusion coefficients and the mode coupling theory predictions.

Mode coupling behavior of a Lennard-Jones binary mixture: a comparison between bulk and confined phases

We present a quantitative comparison at equivalent thermodynamical conditions of bulk and confined dynamical properties of a Lennard-Jones binary mixture upon supercooling. Both systems had been previously found to display a behavior in agreement with the mode coupling theory of the evolution of glassy dynamics. Differences and analogies of behavior are discussed focusing, in particular, on the role of hopping in reducing spatially correlated dynamics in the confined system with respect to the bulk.

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.

Replica Ornstein-Zernike theory of adsorption in a templated porous material: Interaction site systems

Molecular templating offers the possibility of porous materials whose selectivity rivals the molecular recognition observed in nature. The design of templated materials requires a molecular understanding of the templating effect on the material structure and performance. We present here a theoretical description of adsorption in a model templated porous material. Our model material is a quenched, equilibrated mixture of template and matrix molecular species where the template component has been subsequently removed. We propose a set of site-site [i.e., reference interaction site model (RISM)] replica Ornstein-Zernike equations relating the correlation functions of template, matrix, and adsorbing fluid molecules. To test this approach, we focus here on systems interacting via hard-sphere site-site potentials and employ a Percus-Yevick closure. We consider chain and cluster species composed of up to five spheres and observe a range of effects associated with template structure, including higher affinity toward, and enhanced templating by, compact cluster molecules. We assess these effects by grand canonical Monte Carlo simulation and discuss their implication to the design of templated molecular recognition materials.

Effect of temperature on the adsorption of water in porous carbons

We report experimental and simulation studies to investigate the effect of temperature on the adsorption isotherms for water in carbons. Adsorption isotherms are measured by a gravimetric technique in carbon-fiber monoliths at 378 and 423 K and studied by molecular simulation in ideal carbon pores in the temperature range 298-600 K. Experimental adsorption isotherms show a gradual water uptake, as the pressure increases, and narrow adsorption-desorption hysteresis loops. In contrast, simulated adsorption isotherms at room temperature are characterized by negligible uptake at low pressures, sudden and complete pore filling once a threshold pressure is reached, and wide adsorption-desorption hysteresis loops. As the temperature increases, the relative pressure at which pore filling occurs increases and the size of the hysteresis loop decreases. Experimental adsorption-desorption hysteresis loops are narrower than those from simulation. Discrepancies between simulation and experimental results are attributed to heterogeneities in chemical composition, pore connectivity, and nonuniform pore-size distribution, which are not accounted for in the simulation model. The hysteresis phase diagram for confined water is obtained by recording the pressure-density conditions that bound the simulated hysteresis loop at each temperature. We find that the hysteresis critical temperature, i.e., the lowest temperature at which no hysteresis is detected, can be hundreds of degrees lower than the vapor-liquid critical temperature for bulk model water. The properties of confined water are discussed with the aid of simulation snapshots and by analyzing the structure of the confined fluid.

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.

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.

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.

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.

Slow dynamics of a confined supercooled binary mixture. II. Q space analysis

We report the analysis in the wave vector space of the density correlator of a Lennard-Jones binary mixture confined in a disordered matrix of soft spheres upon supercooling. In spite of the strong confining medium the behavior of the mixture is consistent with the mode-coupling theory predictions for bulk supercooled liquids. The relaxation times extracted from the fit of the density correlator to the stretched exponential function follow a unique power law behavior as a function of wave vector and temperature. The von Schweidler scaling properties are valid for an extended wave vector range around the peak of the structure factor. The parameters extracted in the present work are compared with the bulk values obtained in literature.

Slow dynamics of a confined supercooled binary mixture: direct space analysis

Dynamical properties of a Lennard-Jones binary mixture embedded in an off-lattice matrix of soft spheres are studied in the direct space upon supercooling by molecular dynamics simulations. On lowering the temperature, the smaller particles tend to avoid the soft sphere interfaces and correspondingly their mobility decreases below one of the larger particles. The system displays a dynamic behavior, consistent with the mode coupling predictions. A decrease in the mode coupling crossover temperature with respect to the bulk is found. We however find that the range of validity of the theory shrinks with respect to the bulk. This is due to the change in the smaller particle mobility and to a substantial enhancement of hopping processes well above the crossover temperature upon confinement.

Lattice model of adsorption in disordered porous materials: mean-field density functional theory and Monte Carlo simulations

We present mean-field density functional theory calculations and Monte Carlo simulations for a lattice model of a fluid confined in a disordered porous material. The model is obtained by a coarse graining of an off-lattice model of adsorption of simple molecules in silica xerogels. In some of our calculations a model of a porous glass is also considered. The lattice models exhibit behavior that is qualitatively similar to that of their off-lattice counterparts but the computations required are much more tractable and this makes it feasible to investigate the effects of porous material microstructure at longer length scales. We focus on exploring in detail the behavior in the adsorption/desorption hysteresis region for these models. In agreement with recent results for a model that uses a random distribution of solid sites on the lattice [Kierlik et al., Phys. Rev. Lett. 87, 055701 (2001)] we show that the disorder of the solid matrix induces multiple metastable states within the hysteresis region, which are evident in both the mean-field theory calculations and the Monte Carlo simulations. These multiple metastable states can be connected by scanning curves that are very similar to those seen in experimental studies of adsorption hysteresis. The results from mean-field theory predict that while there is hysteresis in the adsorption/desorption isotherms it is not possible to locate a condition of phase equilibrium that satisfies thermodynamic consistency. A wider significance of these results is discussed.

Adsorption of a diatomic molecular fluid into random porous media

Structural and thermodynamic properties of a homonuclear hard dumbbell fluid adsorbed into a disordered hard sphere matrix are studied by means of integral equation techniques and computer simulation. In particular, we have rewritten the replica Ornstein-Zernike equations to deal with orientational degrees of freedom and we have solved them in two different approaches: the hypernetted chain equation and a semiempirical extension of Verlet's approximation. We have also derived direct expressions to calculate the chemical potential in these approximations. Comparison with grand canonical Monte Carlo results shows that both theoretical treatments describe adequately the physical behavior of the system, Verlet's approach being, however, clearly superior in accordance with previous findings for equilibrated hard core mixtures.

Effect of templated quenched disorder on fluid phase equilibrium

Templating offers a means to direct the structure of quenched disorder. We show here that changes in phase equilibrium due to the presence of quenched disorder can themselves be altered by templating. We calculate the phase diagram of a fluid in a collection of template-directed, quenched particles by solving a set of replica Ornstein-Zernike equations within the mean spherical approximation and show templating to enhance phase behavior, that is, shift the phase envelope upward from its location for a nontemplated system of identical available volume. This enhancement is due to an augmented number of fluid-fluid interactions.

Computer simulations of phase equilibrium for a fluid confined in a disordered porous structure

We present calculations of the phase diagrams of a Lennard-Jones 12-6 fluid confined in a disordered porous structure made up of a dispersion of spherical particles, following up on an earlier work on the same system. In particular we present additional calculations using more realizations of the matrix and we investigate the applicability of the Gibbs-Duhem integration method to the calculation of phase equilibrium in these systems. The essential picture of disordered and inhomogeneous coexisting vapor and liquid phases, which emerged in the earlier work, is confirmed by the new calculations. However, a second phase transition associated with the wetting of the porous material by the fluid is found to be more sensitive to variations of the matrix realization. While for the present model this transition appears for particular realizations of the matrix, it does not seem to survive averaging over realizations.

Structure and thermodynamic properties of a binary liquid in a porous matrix: the formalism

Using the replica trick we derive a formalism to describe the structure and the thermodynamic properties of a binary liquid in equilibrium with a porous medium. We present the replica Ornstein-Zernike equations for the general case of a k-component liquid inside a porous matrix; besides the usual liquid-state closure relations, we consider in particular the optimized random phase approximation (ORPA) restricting ourselves at present to hard-core potentials exclusively. We present furthermore several thermodynamic relations: the Gibbs-Duhem equation, the compressibility, and the viral equation. Within the framework of the ORPA (mean spherical approximation), closed expressions for the perturbation contribution to the free energy and the chemical potentials can be presented. Finally, we offer suggestions for numerical implementations.