Adsorption of fluids in disordered porous media from integral equation theory

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.

Deformable hard particles confined in a disordered porous matrix

With suitably designed Monte Carlo simulations, we have investigated the properties of mobile, impenetrable, yet deformable particles that are immersed into a porous matrix, the latter one realized by a frozen configuration of spherical particles. By virtue of a model put forward by Batista and Miller [Phys. Rev. Lett. 105, 088305 (2010)], the fluid particles can change in their surroundings, formed by other fluid particles or the matrix particles, their shape within the class of ellipsoids of revolution; such a change in shape is related to a change in energy, which is fed into suitably defined selection rules in the deformation "moves" of the Monte Carlo simulations. This concept represents a simple yet powerful model of realistic, deformable molecules with complex internal structures (such as dendrimers or polymers). For the evaluation of the properties of the system, we have used the well-known quenched-annealed protocol (with its characteristic double average prescription) and have analyzed the simulation data in terms of static properties (the radial distribution function and aspect ratio distribution of the ellipsoids) and dynamic features (notably the mean squared displacement). Our data provide evidence that the degree of deformability of the fluid particles has a distinct impact on the aforementioned properties of the system.

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.

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.

Scaled Particle Theory for Multicomponent Hard Sphere Fluids Confined in Random Porous Media

The formulation of scaled particle theory (SPT) is presented for a quite general model of fluids confined in a random porous media, i.e., a multicomponent hard sphere (HS) fluid in a multicomponent hard sphere or a multicomponent overlapping hard sphere (OHS) matrix. The analytical expressions for pressure, Helmholtz free energy, and chemical potential are derived. The thermodynamic consistency of the proposed theory is established. Moreover, we show that there is an isomorphism between the SPT for a multicomponent system and that for a one-component system. Results from grand canonical ensemble Monte Carlo simulations are also presented for a binary HS mixture in a one-component HS or a one-component OHS matrix. The accuracy of various variants derived from the basic SPT formulation is appraised against the simulation results. Scaled particle theory, initially formulated for a bulk HS fluid, has not only provided an analytical tool for calculating thermodynamic properties of HS fluid but also helped to gain very useful insight for elaborating other theoretical approaches such as the fundamental measure theory (FMT). We expect that the general SPT for multicomponent systems developed in this work can contribute to the study of confined fluids in a similar way.

Dynamic arrest of colloids in porous environments: disentangling crowding and confinement

Using numerical simulations we study the slow dynamics of a colloidal hard-sphere fluid adsorbed in a matrix of disordered hard-sphere obstacles. We calculate separately the contributions to the single-particle dynamic correlation functions due to free and trapped particles. The separation is based on a Delaunay tessellation to partition the space accessible to the centres of fluid particles into percolating and disconnected voids. We find that the trapping of particles into disconnected voids of the matrix is responsible for the appearance of a nonzero long-time plateau in the single-particle intermediate scattering functions of the full fluid. The subdiffusive exponent z, obtained from the logarithmic derivative of the mean squared displacement, is essentially unaffected by the motion of trapped particles: close to the percolation transition, we determined z approximately = 0.5 for both the full fluid and the particles moving in the percolating void. Notably, the same value of z is found in single-file diffusion and is also predicted by mode-coupling theory along the diffusion-localization line. We also reveal subtle effects of dynamic heterogeneity in both the free and the trapped component of the fluid particles, and discuss microscopic mechanisms that contribute to this phenomenon.

Fluids in porous media. III. Scaled particle theory

Based on a new and consistent formulation of scaled particle theory for a fluid confined in random porous media, a series of new approximations are proposed and one of them gives equations of state with excellent accuracy for a hard sphere fluid adsorbed in a hard sphere or an overlapping hard sphere matrix. Although the initial motivation was to remedy a flaw in a previous formulation of the scaled particle theory for a confined fluid, the new formulation is not a trivial and straightforward correction of the previous one. A few conceptual and significant modifications have to be introduced for developing the present formulation.

Geometrical ambiguity of pair statistics. II. Heterogeneous media

In the first part of this series of two papers [Y. Jiao, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 011105 (2010)], we considered the geometrical ambiguity of pair statistics associated with point configurations. Here we focus on the analogous problem for heterogeneous media (materials). Heterogeneous media are ubiquitous in a host of contexts, including composites and granular media, biological tissues, ecological patterns, and astrophysical structures. The complex structures of heterogeneous media are usually characterized via statistical descriptors, such as the n -point correlation function Sn. An intricate inverse problem of practical importance is to what extent a medium can be reconstructed from the two-point correlation function S2 of a target medium. Recently, general claims of the uniqueness of reconstructions using S2 have been made based on numerical studies, which implies that S2 suffices to uniquely determine the structure of a medium within certain numerical accuracy. In this paper, we provide a systematic approach to characterize the geometrical ambiguity of S2 for both continuous two-phase heterogeneous media and their digitized representations in a mathematically precise way. In particular, we derive the exact conditions for the case where two distinct media possess identical S2 , i.e., they form a degenerate pair. The degeneracy conditions are given in terms of integral and algebraic equations for continuous media and their digitized representations, respectively. By examining these equations and constructing their rigorous solutions for specific examples, we conclusively show that in general S2 is indeed not sufficient information to uniquely determine the structure of the medium, which is consistent with the results of our recent study on heterogeneous-media reconstruction [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Natl. Acad. Sci. U.S.A. 106, 17634 (2009)]. The analytical examples include complex patterns composed of building blocks bearing the letter "T" and the word "WATER" as well as degenerate stacking variants of the densest sphere packing in three dimensions (Barlow films). Several numerical examples of degeneracy (e.g., reconstructions of polycrystal microstructures, laser-speckle patterns and sphere packings) are also given, which are virtually exact solutions of the degeneracy equations. The uniqueness issue of multiphase media reconstructions and additional structural information required to characterize heterogeneous media are discussed, including two-point quantities that contain topological connectedness information about the phases.

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.

Density functional theory model of adsorption on amorphous and microporous silica materials

We present a novel quenched solid density functional theory (QSDFT) model of adsorption on heterogeneous surfaces and porous solids, which accounts for the effects of surface roughness and microporosity. Within QSDFT, solid atoms are considered as quenched component(s) of the solid-fluid system with given density distribution(s). Solid-fluid intermolecular interactions are split into hard-sphere repulsive and mean-field attractive parts. The former are treated with the multicomponent fundamental measure density functional. Capabilities of QSDFT are demonstrated by drawing on the example of adsorption on amorphous silica materials. We show that, using established intermolecular potentials and a realistic model for silica surfaces, QSDFT quantitatively describes adsorption/desorption isotherms of Ar and Kr on reference MCM-41, SBA-15, and LiChrosphere materials in a wide range of relative pressures. QSDFT offers a systematic approach to the practical problems of characterization of microporous, mesoporous, and amorphous silica materials, including an assessment of microporosity, surface roughness, and adsorption deformation. Predictions for the pore diameter and the extent of pore surface roughness in MCM-41 and SBA-15 materials are in very good agreement with recent X-ray diffraction studies.

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.

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.

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.

Application of the Replica Ornstein-Zernike Equations to Study Submonolayer Adsorption on Energetically Heterogeneous Surfaces

Weapply the replica methodology and the replica Ornstein-Zernike equations to calculate adsorption isotherms on an energetically heterogeneous surface. The surface is modeled by quenching a two-dimensional "fluid of centers." After the quenching, we study gas adsorption using the replica Ornstein-Zernike equation with the hypernetted chain closure. The results of theoretical predictions are compared with the data obtained from the Grand Canonical Monte Carlo simulation, and the obtained agreement is satisfactory. Copyright 1999 Academic Press.

Partitioning of Polymerizing Fluids in Random Microporous Media: Application of the Replica Ornstein-Zernike Equations

We have investigated a model for a polymerizing fluid in which each of the particles has two bonding sites, such that chains can be formed via a chemical association mechanism. The fluid model is considered to be in a random quenched microporous matrix. The matrix species are assumed to be either impermeable to adsorbed fluid particles or permeable, such that the surface of the matrix particles represents a permeable membrane of finite width. We have studied the influence of the matrix species on the formation of chains due to association. The model is investigated by means of the associative replica Ornstein-Zernike equations with the Percus-Yevick closure and the ideal chain approximation. We have observed that the average chain length is longer in the presence of an impermeable matrix than in the case where the matrix is absent. Matrix is therefore conducive to the growth of the polymerizing species in micropores. There is a decrease in the average chain length with increasing permeability of matrix species. This behavior reaffirms the attenuating role of the permeable matrix species as a whole. Copyright 1999 Academic Press.

Adsorption of a Hard Sphere Fluid in a Disordered Polymerized Matrix: Application of the Replica Ornstein-Zernike Equations

A model of hard spheres adsorbed in disordered porous media is studied using the associative replica Ornstein-Zernike (ROZ) equations. Extending previous studies of adsorption in a hard sphere matrices, we investigate a polymerized matrix. We consider an associating fluid of hard spheres with two intracore attractive sites per particle; consequently chains consisting of overlapping hard spheres can be formed due to the chemical association. This is the generalization of the model with sites on the surface of Wertheim that has been studied in the bulk by Chang and Sandler. The matrix structure is obtained in the polymer Percus-Yevick approximation. We solve the ROZ equations in the associative hypernetted chain approximation. The pair distribution functions, the fluid compressibility, the equation of state and chemical potential of the adsorbed fluid are obtained and discussed. It is shown that the adsorption of a hard sphere fluid in a matrix at given density, but consisting of longer chains of overlapping hard spheres, is higher than the adsorption of this fluid in a hard sphere matrix.