Phase behavior of patchy colloids confined in patchy porous media

A simple model for functionalized disordered porous media is proposed and the effects of confinement on self-association, percolation and phase behavior of a fluid of patchy particles are studied. The media are formed by randomly distributed hard-sphere obstacles fixed in space and decorated by a certain number of off-center square-well sites. The properties of the fluid of patchy particles, represented by the fluid of hard spheres each bearing a set of the off-center square-well sites, are studied using an appropriate combination of the scaled particle theory for the porous media, Wertheim's thermodynamic perturbation theory, and Flory-Stockmayer theory. To assess the accuracy of the theory a set of computer simulations have been performed. In general, predictions of the theory appeared to be in good agreement with the computer simulation results. Confinement and competition between the formation of bonds connecting the fluid particles, and connecting fluid particles and obstacles of the matrix, gave rise to a re-entrant phase behavior with three critical points and two separate regions of the liquid-gas phase coexistence.

Behaviour of the model antibody fluid constrained by rigid spherical obstacles: Effects of the obstacle-antibody attraction

This study investigates the behaviour of a fluid of monoclonal antibodies (mAbs) when trapped in a confinement represented by rigid spherical obstacles that attract antibodies. The antibody molecule is depicted as an assembly of seven hard spheres (7-bead model), organized to resemble a Y-shaped object. The model antibody has two Fab and one Fc domains located in the corners of letter Y. In this calculation, only the Fab-Fab and Fab-Fc attractive pairs of interactions are effective. The confinement is formed by the randomly distributed hard-spheres fixed in space. The spherical obstacles, besides the size exclusion, interact with beads of the antibody molecules via the Yukawa attractive potential. We applied the combination of the scaled particle theory, replica Ornstein-Zernike equations, Wertheim's thermodynamic perturbation approach and the Flory-Stockmayer theory to calculate: (i) the phase diagram of the liquid-liquid phase separation and the percolation threshold, (ii) the cluster size distributions, and (iii) the second virial coefficient of the protein fluid distributed among the obstacles. All these quantities were calculated as functions of the strength of the attraction between the monoclonal antibodies, and the monoclonal antibodies and obstacles. The conclusion is that while the hard-sphere obstacles decrease the critical density and the critical temperature of the mAbs fluid, the effect of the protein-obstacle attraction is more complex. Adding an attractive potential to the obstacle-mAbs interaction first increases the wideness of the T*-ρ envelope. However, with the further increase of the obstacle-mAbs attraction intensity, we observe reversal of the effect, the T*-ρ curves become narrower. At some point, depending on the obstacle-mAbs interaction, the situation is observed where two different temperatures have the same fluid density (re-entry point). In all the cases shown here the critical point decreases below the value for the neat fluid, but the behaviour with respect to an increase of the strength of the obstacle-mAbs attraction is not monotonic. Yet another interesting phenomenon, known in the literature as an approach toward the "empty liquid" state, is observed. The stability of the "protein droplets", formed by the liquid-liquid phase separation, depends on their local environment and temperature.

Isotope effects observed in diluted D2O/H2O mixtures identify HOD-induced low-density structures in D2O but not H2O

Normal and heavy water are solvents most commonly used to study the isotope effect. The isotope effect of a solvent significantly influences the behavior of a single molecule in a solution, especially when there are interactions between the solvent and the solute. The influence of the isotope effect becomes more significant in D₂O/H₂O since the hydrogen bond in H₂O is slightly weaker than its counterpart (deuterium bond) in D₂O. Herein, we characterize the isotope effect in a mixture of normal and heavy water on the solvation of a HOD molecule. We show that the HOD molecule affects the proximal solvent molecules, and these disturbances are much more significant in heavy water than in normal water. Moreover, in D₂O, we observe the formation of low-density structures indicative of an ordering of the solvent around the HOD molecule. The qualitative differences between HOD interaction with D₂O and H₂O were consistently confirmed with Raman spectroscopy and NMR diffusometry.

Equation of state for confined fluids

Fluids confined in small volumes behave differently than fluids in bulk systems. For bulk systems, a compact summary of the system’s thermodynamic properties is provided by equations of state. However, there is currently a lack of successful methods to predict the thermodynamic properties of confined fluids by use of equations of state, since their thermodynamic state depends on additional parameters introduced by the enclosing surface. In this work, we present a consistent thermodynamic framework that represents an equation of state for pure, confined fluids. The total system is decomposed into a bulk phase in equilibrium with a surface phase. The equation of state is based on an existing, accurate description of the bulk fluid and uses Gibbs’ framework for surface excess properties to consistently incorporate contributions from the surface. We apply the equation of state to a Lennard-Jones spline fluid confined by a spherical surface with a Weeks–Chandler–Andersen wall-potential. The pressure and internal energy predicted from the equation of state are in good agreement with the properties obtained directly from molecular dynamics simulations. We find that when the location of the dividing surface is chosen appropriately, the properties of highly curved surfaces can be predicted from those of a planar surface. The choice of the dividing surface affects the magnitude of the surface excess properties and its curvature dependence, but the properties of the total system remain unchanged. The framework can predict the properties of confined systems with a wide range of geometries, sizes, interparticle interactions, and wall–particle interactions, and it is independent of ensemble. A targeted area of use is the prediction of thermodynamic properties in porous media, for which a possible application of the framework is elaborated.

Empty liquid state and re-entrant phase behavior of the patchy colloids confined in porous media

Patchy colloids with three and four equivalent patches, confined in an attractive random porous medium, undergo re-entrant gas–liquid phase separation with the liquid phase density approaching zero at low temperatures. The (bonding) colloid–colloid interaction causes the liquid–gas phase separation, which is modulated by the presence of the randomly distributed hard-sphere obstacles, attracting the colloids via Yukawa potential. Due to this interaction, a layer of mutually bonded colloids around the obstacles is formed. The network becomes nonuniform, with colloid particles locally centered on the obstacles. Features described in this article may open possibilities to produce equilibrium gels with predefined nonuniform distribution of particles and indicate how complicated the phase behavior of biological macromolecules in a crowded environment may be.

Aggregation, liquid-liquid phase separation, and percolation behaviour of a model antibody fluid constrained by hard-sphere obstacles

This study is concerned with the behaviour of proteins within confinement created by hard-sphere obstacles. An individual antibody molecule is depicted as an assembly of seven hard spheres, organized to resemble a Y-shaped (on average) antibody (7-bead model) protein. For comparison with other studies we, in one case, model the protein as a hard sphere decorated by three short-range attractive sites. The antibody has two Fab and one Fc domains located in the corners of the letter Y. In this calculation, only the Fab-Fab and Fab-Fc attractive pair interactions are possible. The confinement is formed by the randomly distributed hard-sphere obstacles fixed in space. Aside from size exclusion, the obstacles do not interact with antibodies, but they affect the protein-protein correlation. We used a combination of the scaled-particle theory, Wertheim's thermodynamic perturbation theory and the Flory-Stockmayer theory to calculate: (i) the second virial coefficient of the protein fluid, (ii) the percolation threshold, (iii) cluster size distributions, and (iv) the liquid-liquid phase separation as a function of the strength of the various pair interactions of the protein and the model parameters, such as protein concentration and the packing fraction of obstacles. The conclusion is that hard-sphere obstacles strongly decrease the critical density and also, but to a much lesser extent, the critical temperature. Also, the confinement enhances clustering, making the percolating region broader. The effect depends on the model parameters, such as the packing fraction of obstacles η0, the inter-site interaction strength εIJ, and the ratio between the size of the obstacle σ0 and the size of one bead of the model antibody σhs; the value of this ratio is varied here from 2 to 5. Interestingly, at low to moderate packing fractions of obstacles, the second virial coefficient first slightly decreases (destabilization), and the slope depends on the observation temperature, but then at higher values of η0 it increases. The calculated values of the second virial coefficient also depend on the size of the obstacles.

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.

Isotropic-Nematic Transition and Demixing Behavior in Binary Mixtures of Hard Spheres and Hard Spherocylinders Confined in a Disordered Porous Medium: Scaled Particle Theory

We develop the scaled particle theory to describe the thermodynamic properties and orientation ordering of a binary mixture of hard spheres (HS) and hard spherocylinders (HSC) confined in a disordered porous medium. Using this theory, the analytical expressions of the free energy, the pressure, and the chemical potentials of HS and HSC have been derived. The improvement of obtained results is considered by introducing the Carnahan-Starling-like and Parsons-Lee-like corrections. Phase diagrams for the isotropic-nematic transition are calculated from the bifurcation analysis of the integral equation for the orientation singlet distribution function and from the conditions of thermodynamic equilibrium. Both the approaches correctly predict the isotropic-nematic transition at low concentrations of hard spheres. However, the thermodynamic approach provides more accurate results and is able to describe the demixing phenomena in the isotropic and nematic phases. The effects of porous medium on the isotropic-nematic phase transition and demixing behavior in a binary HS/HSC mixture are discussed.

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.

Vapor-liquid phase behavior of a size-asymmetric model of ionic fluids confined in a disordered matrix: The collective-variables-based approach

We develop a theory based on the method of collective variables to study the vapor-liquid equilibrium of asymmetric ionic fluids confined in a disordered porous matrix. The approach allows us to formulate the perturbation theory using an extension of the scaled particle theory for a description of a reference system presented as a two-component hard-sphere fluid confined in a hard-sphere matrix. Treating an ionic fluid as a size- and charge-asymmetric primitive model (PM) we derive an explicit expression for the relevant chemical potential of a confined ionic system which takes into account the third-order correlations between ions. Using this expression, the phase diagrams for a size-asymmetric PM are calculated for different matrix porosities as well as for different sizes of matrix and fluid particles. It is observed that general trends of the coexistence curves with the matrix porosity are similar to those of simple fluids under disordered confinement, i.e., the coexistence region gets narrower with a decrease of porosity and, simultaneously, the reduced critical temperature T_{c}^{*} and the critical density ρ_{i,c}^{*} become lower. At the same time, our results suggest that an increase in size asymmetry of oppositely charged ions considerably affects the vapor-liquid diagrams leading to a faster decrease of T_{c}^{*} and ρ_{i,c}^{*} and even to a disappearance of the phase transition, especially for the case of small matrix particles.

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.

Two- and three-phase equilibria of polydisperse Yukawa hard-sphere fluids confined in random porous media: high temperature approximation and scaled particle theory

We have studied the phase behavior of polydisperse Yukawa hard-sphere fluid confined in random porous media using extension and combination of high temperature approximation and scaled particle theory. The porous media are represented by the matrix of randomly placed hard-sphere obstacles. Due to the confinement, polydispersity effects are substantially enhanced. At an intermediate degree of fluid polydispersity and low density of the matrix, we observe two-phase coexistence with two critical points, and cloud and shadow curves forming closed loops of ellipsoidal shape. With the increase of the matrix density and the constant degree of polydispersity, these two critical points merge and disappear, and at lower temperatures the system fractionates into three coexisting phases. A similar phase behavior was observed in the absence of the porous media caused, however, by the increase of the polydispersity.

Vapour-liquid phase diagram for an ionic fluid in a random porous medium

We study the vapour-liquid phase behaviour of an ionic fluid confined in a random porous matrix formed by uncharged hard sphere particles. The ionic fluid is modelled as an equimolar binary mixture of oppositely charged equisized hard spheres, the so-called restricted primitive model (RPM). Considering the matrix-fluid system as a partly-quenched model, we develop a theoretical approach which combines the method of collective variables with the extension of the scaled-particle theory (SPT) for a hard-sphere fluid confined in a disordered hard-sphere matrix. The approach allows us to formulate the perturbation theory using the SPT for the description of the thermodynamics of the reference system. The phase diagrams of the RPM in matrices of different porosities and for different size ratios of matrix and fluid particles are calculated in the random-phase approximation and also when the effects of higher-order correlations between ions are taken into account. Both approximations correctly reproduce the basic effects of porous media on the vapour-liquid phase diagram, i.e. with a decrease of porosity the critical point shifts towards lower fluid densities and lower temperatures and the coexistence region gets narrower. For the fixed matrix porosity, both the critical temperature and the critical density increase with an increase of size of matrix particles and tend to the critical values of the bulk RPM.

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.

Phase Behavior and Percolation Properties of the Patchy Colloidal Fluids in the Random Porous Media

The lack of a simple analytical description of the hard-sphere fluid in a matrix with hard-core obstacles is limiting progress in the development of thermodynamic perturbation theories for the fluid in random porous media. We propose a simple and highly accurate analytical scheme, which allows us to calculate thermodynamic and percolation properties of a network-forming fluid confined in the random porous media, represented by the hard-sphere fluid and overlapping hard-sphere matrices, respectively. Our scheme is based on the combination of scaled-particle theory, Wertheim's thermodynamic perturbation theory for associating fluids and extension of the Flory-Stockmayer theory for percolation. The liquid-gas phase diagram and percolation threshold line for several versions of the patchy colloidal fluid model confined in a random porous media are calculated and discussed. The method presented enables calculation of the thermodynamic and percolation properties of a large variety of polymerizing and network-forming fluids confined in random porous media.