An accurate integral equation theory for hard spheres: Role of the zero‐separation theorems in the closure relation

Multiphase displacement manipulated by micro/nanoparticle suspensions in porous media via microfluidic experiments: From interface science to multiphase flow patterns

Multiphase displacement in porous media can be adjusted by micro/nanoparticle suspensions, which is widespread in many scientific and industrial contexts. Direct visualization of suspension flow dynamics and corresponding multiphase patterns is crucial to understanding displacement mechanisms and eventually optimizing these processes in geological, biological, chemical, and other engineering systems. However, suspension flow inside the opaque realistic porous media makes direct observation challenging. The advances in microfluidic experiments have provided us with alternative methods to observe suspension influence on the interface and multiphase flow behaviors at high temporal and spatial resolutions. Macroscale processes are controlled by microscale interfacial behaviors, which are affected by multiple physical factors, such as particle adsorption, capillarity, and hydrodynamics. These properties exerted on the suspension flow in porous media may lead to interesting interfacial phenomena and new displacement consequences. As an underpinning science, understanding and controlling the suspension transport process from interface to flow patterns in porous media is critical for a lower operating cost to improve resource production while reducing harmful emissions and other environmental impacts. This review summarizes the basic properties of different micro/nanoparticle suspensions and the state-of-the-art microfluidic techniques for displacement research activities in porous media. Various suspension transport behaviors and displacement mechanisms explored by microfluidic experiments are comprehensively reviewed. This review is expected to boost both experimental and theoretical understanding of suspension transport and interfacial interaction processes in porous media. It also brings forward the challenges and opportunities for future research in controlling complex fluid flow in porous media for diverse applications.

Bridge functions of classical one-component plasmas

In a recent paper, Lucco Castello et al. [arXiv:2107.03537] performed systematic extractions of classical one-component plasma bridge functions from molecular dynamics simulations and provided an accurate parametrization that was incorporated in their isomorph-based empirically modified hypernetted chain approach for Yukawa one-component plasmas. Here the extraction technique and parametrization strategy are described in detail, while the deficiencies of earlier efforts are discussed. The structural and thermodynamic predictions of the updated version of the integral equation theory approach are compared with extensive available simulation results revealing a truly unprecedented level of accuracy in the entire dense liquid region of the Yukawa phase diagram.

Monte Carlo simulations in the unconstrained ensemble

The unconstrained ensemble describes completely open systems whose control parameters are the chemical potential, pressure, and temperature. For macroscopic systems with short-range interactions, thermodynamics prevents the simultaneous use of these intensive variables as control parameters, because they are not independent and cannot account for the system size. When the range of the interactions is comparable with the size of the system, however, these variables are not truly intensive and may become independent, so equilibrium states defined by the values of these parameters may exist. Here, we derive a Monte Carlo algorithm for the unconstrained ensemble and show that simulations can be performed using the chemical potential, pressure, and temperature as control parameters. We illustrate the algorithm by applying it to physical systems where either the system has long-range interactions or is confined by external conditions. The method opens up an avenue for the simulation of completely open systems exchanging heat, work, and matter with the environment.

Testing the isomorph invariance of the bridge functions of Yukawa one-component plasmas

It has been recently conjectured that bridge functions remain nearly invariant along phase diagram lines of constant excess entropy for the broad class of R-simple liquids. To test this hypothesis, the bridge functions of Yukawa systems are computed outside the correlation void with the Ornstein-Zernike inversion method employing structural input from ultra-accurate molecular dynamics simulations and inside the correlation void with the cavity distribution method employing structural input from ultra-long specially designed molecular dynamics simulations featuring a tagged particle pair. Yukawa bridge functions are revealed to be isomorph invariant to a very high degree. The observed invariance is not exact, however, since isomorphic deviations exceed the overall uncertainties.

Structural and thermodynamic properties of hard-sphere fluids

This Perspective article provides an overview of some of our analytical approaches to the computation of the structural and thermodynamic properties of single-component and multicomponent hard-sphere fluids. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions linking the equation of state of the single-component fluid to the one of the multicomponent mixtures are also discussed.

Free-energy functional of instantaneous correlation field in liquids: Field-theoretic derivation of the closures

This paper presents a unified method for formulating a field-theoretic perturbation theory that encompasses the conventional liquid state theory. First, the free-energy functional of an instantaneous correlation field is obtained from the functional-integral representation of the grand potential. Next, we demonstrate that the instantaneous free-energy functional yields a closure relation between the correlation functions in the mean-field approximation. Notably, the obtained closure relation covers a variety of approximate closures introduced in the liquid state theory.

Temperature expansions in the square-shoulder fluid. II. Thermodynamics

In Paper I [O. Coquand and M. Sperl, J. Chem. Phys. 152, 124112 (2020)], we derived analytical expressions for the structure factor of the square-shoulder potential in a perturbative way around the high- and low-temperature regimes. Here, various physical properties of these solutions are derived. In particular, we investigate the large wave number sector and relate it to the contact values of the pair-correlation function. Then, the thermoelastic properties of the square-shoulder fluids are discussed.

Closure for the Ornstein-Zernike equation with pressure and free energy consistency

The Ornstein-Zernike (OZ) integral equation theory is a powerful approach to simple liquids due to its low computational cost and the fact that, when combined with an appropriate closure equation, the theory is thermodynamically complete. However, approximate closures proposed to date exhibit pressure or free energy inconsistencies that produce inaccurate or ambiguous results, limiting the usefulness of the Ornstein-Zernike approach. To address this problem, we combine methods to enforce both pressure and free energy consistency to create a new closure approximation and test it for a single-component Lennard-Jones fluid. The closure is a simple power series in the direct and total correlation functions for which we have derived analytical formulas for the excess Helmholtz free energy and chemical potential. These expressions contain a partial molar volumelike term, similar to excess chemical potential correction terms recently developed. Using our bridge approximation, we have calculated the pressure, Helmholtz free energy, and chemical potential for the Lennard-Jones fluid using the Kirkwood charging, thermodynamic integration techniques, and analytic expressions. These results are compared with those from the hypernetted chain equation and the Verlet-modified closure against Monte Carlo and equations-of-state data for reduced densities of ρ^{*}<1 and temperatures of T^{*}=1.5, 2.74, and 5. Our closure shows consistency among all thermodynamic paths, except for one expression of the Gibbs-Duhem relation, whereas the hypernetted chain equation and the Verlet-modified closure exhibit consistency between only a few relations. Accuracy of the closure is comparable to the Verlet-modified closure and a significant improvement to results obtained from the hypernetted chain equation.

Computation of virial coefficients from integral equations

A polynomial-time method of computing the virial coefficients from an integral equation framework is presented. The method computes the truncated density expansions of the correlation functions by series transformations, and then extracts the virial coefficients from the density components. As an application, the method was used in a hybrid-closure integral equation with a set of self-consistent conditions, which produced reasonably accurate virial coefficients for the hard-sphere fluid and Gaussian model in high dimensions.

Colloids and polymers in random colloidal matrices: demixing under good-solvent conditions

We consider a simplified coarse-grained model for colloid-polymer mixtures, in which polymers are represented as monoatomic molecules interacting by means of pair potentials. We use it to study polymer-colloid segregation in the presence of a quenched matrix of colloidal hard spheres. We fix the polymer-to-colloid size ratio to 0.8 and consider matrices such that the fraction f of the volume that is not accessible to the colloids due to the matrix is equal to 40%. As in the Asakura-Oosawa-Vrij (AOV) case, we find that binodal curves in the polymer and colloid volume-fraction plane have a small dependence on disorder. As for the position of the critical point, the behavior differs from that observed in the AOV case: While the critical colloid volume fraction is essentially the same in the bulk and in the presence of the matrix, the polymer volume fraction at criticality increases as f increases. At variance with the AOV case, no capillary colloid condensation or evaporation is generically observed.

A theoretical investigation on the honeycomb potential fluid

A local self-consistent Ornstein-Zernike (OZ) integral equation theory (IET) is proposed to provide a rapid route for obtaining thermodynamic and structural information for any thermodynamically stable or metastable state points in the bulk phase diagram without recourse to traditional thermodynamic integration, and extensive NVT-Monte Carlo simulations are performed on a recently proposed honeycomb potential in three dimensions to test the theory's reliability. The simulated quantities include radial distribution function (rdf) and excess internal energy, pressure, excess chemical potential, and excess Helmholtz free energy. It is demonstrated that (i) the theory reproduces the rdf very satisfactorily only if the bulk state does not enter deep into a two phases coexistence region; (ii) the excess internal energy is the only one of the four thermodynamic quantities investigated amenable to the most accurate prediction by the present theory, and the simulated pressure is somewhat overestimated by the theoretical calculations, but the deviation tends to vanish along with rising of the temperature; (iii) using the structural functions from the present local self-consistent OZ IET, a previously derived local expression, due to the present author, achieves even a higher accuracy in calculating for the excess chemical potential than the exact virial pressure formula for the pressure, and the resulting excess Helmholtz free energy is in surprisingly same with the simulation results due to offset of the errors. Based on the above observations, it is suggested that it may be a good procedure to integrate the theoretical excess internal energy along the isochors to get the excess Helmholtz free energy, which is then fitted to a polynomial to be used for calculation of all of other thermodynamic quantities in the framework of the OZ IET.

Simple relationship between the virial-route hypernetted-chain and the compressibility-route Percus-Yevick values of the fourth virial coefficient

As is well known, approximate integral equations for liquids, such as the hypernetted chain (HNC) and Percus-Yevick (PY) theories, are in general thermodynamically inconsistent in the sense that the macroscopic properties obtained from the spatial correlation functions depend on the route followed. In particular, the values of the fourth virial coefficient B(4) predicted by the HNC and PY approximations via the virial route differ from those obtained via the compressibility route. Despite this, it is shown in this paper that the value of B(4) obtained from the virial route in the HNC theory is exactly three halves the value obtained from the compressibility route in the PY theory, irrespective of the interaction potential (whether isotropic or not), the number of components, and the dimensionality of the system. This simple relationship is confirmed in one-component systems by analytical results for the one-dimensional penetrable-square-well model and the three-dimensional penetrable-sphere model, as well as by numerical results for the one-dimensional Lennard-Jones model, the one-dimensional Gaussian core model, and the three-dimensional square-well model.

Approximate analytical structure and thermodynamics for penetrable sphere fluids using effective hard spheres

A penetrable sphere fluid is modeled using the effective hard sphere fluid and its known structure and thermodynamics. It is observed that the obtained equation of state and the radial distribution function are in fairly good agreement with computer simulation data both at low temperatures and/or low densities.

Optimized theory for simple and molecular fluids

An optimized closure approximation for both simple and molecular fluids is presented. A smooth interpolation between Perkus-Yevick and hypernetted chain closures is optimized by minimizing the free energy self-consistently with respect to the interpolation parameter(s). The molecular version is derived from a refinement of the method for simple fluids. In doing so, a method is proposed which appropriately couples an optimized closure with the variant of the diagrammatically proper integral equation recently introduced by this laboratory [K. M. Dyer et al., J. Chem. Phys. 123, 204512 (2005)]. The simplicity of the expressions involved in this proposed theory has allowed the authors to obtain an analytic expression for the approximate excess chemical potential. This is shown to be an efficient tool to estimate, from first principles, the numerical value of the interpolation parameters defining the aforementioned closure. As a preliminary test, representative models for simple fluids and homonuclear diatomic Lennard-Jones fluids were analyzed, obtaining site-site correlation functions in excellent agreement with simulation data.

Low-temperature and high-temperature approximations for penetrable-sphere fluids: comparison with Monte Carlo simulations and integral equation theories

The two-body interaction in dilute solutions of polymer chains in good solvents can be modeled by means of effective bounded potentials, the simplest of which being that of penetrable spheres (PSs). In this paper we construct two simple analytical theories for the structural properties of PS fluids: a low-temperature (LT) approximation, that can be seen as an extension to PSs of the well-known solution of the Percus-Yevick (PY) equation for hard spheres, and a high-temperature (HT) approximation based on the exact asymptotic behavior in the limit of infinite temperature. Monte Carlo simulations for a wide range of temperatures and densities are performed to assess the validity of both theories. It is found that, despite their simplicity, the HT and LT approximations exhibit a fair agreement with the simulation data within their respective domains of applicability, so that they complement each other. A comparison with numerical solutions of the PY and the hypernetted-chain approximations is also carried out, the latter showing a very good performance, except inside the core at low temperatures.

Corridor of existence of thermodynamically consistent solution of the Ornstein-Zernike equation

We obtain the exact equation for a correction to the Ornstein-Zernike (OZ) equation based on the assumption of the uniqueness of thermodynamical functions. We show that this equation is reduced to a differential equation with one arbitrary parameter for the hard sphere model. The compressibility factor within narrow limits of this parameter variation can either coincide with one of the formulas obtained on the basis of analytical solutions of the OZ equation or assume all intermediate values lying in a corridor between these solutions. In particular, we find the value of this parameter when the thermodynamically consistent compressibility factor corresponds to the Carnahan-Stirling formula.

Approximative "one particle" bridge function B(1)(r) for the theory of simple fluids

New properties for the one particle bridge function B(1)(r), which are necessary to the calculation of the excess chemical potential betamue), are derived for the hard sphere fluid. The method, which only requires the knowledge of the bridge function B(2)(r), is based on an investigation of the correlation function dependence on the Kirkwood charging parameter. In this framework, the unavoidable question of topological homotopy is addressed. As far as B(2)(r) is considered as exact, this work provides useful information on B(1)(r) in the well identified dynamical regimes of the hard sphere fluid. Signatures of the transitions between these regimes are identified on the trends of B(1)(r). This approach provides self-consistent results for betamue) that agree very well with simulation data.

Radial distribution function of penetrable sphere fluids to the second order in density

The simplest bounded potential is that of penetrable spheres, which takes a positive finite value epsilon if the two spheres are overlapped, being zero otherwise. In this paper we derive the cavity function to second order in density and the fourth virial coefficient as functions of T* identical with k(B)T/epsilon (where k(B is the Boltzmann constant and T is the temperature) for penetrable sphere fluids. The expressions are exact, except for the function represented by an elementary diagram inside the core, which is approximated by a polynomial form in excellent agreement with accurate results obtained by Monte Carlo integration. Comparison with the hypernetted-chain (HNC) and Percus-Yevick (PY) theories shows that the latter is better than the former for T* < or similar to 1 only. However, even at zero temperature (hard sphere limit), the PY solution is not accurate inside the overlapping region, where no practical cancellation of the neglected diagrams takes place. The exact fourth virial coefficient is positive for T* < or similar to 0.73, reaches a minimum negative value at T* approximately 1.1, and then goes to zero from below as 1/T(*4) for high temperatures. These features are captured qualitatively, but not quantitatively, by the HNC and PY predictions. In addition, in both theories the compressibility route is the best one for T* < or similar to 0.7, while the virial route is preferable if T* > or similar to 0.7.

Phase diagram of the hard-core Yukawa fluid within the integral equation method

In this study, the integral equation method proposed recently by Sarkisov [J. Chem. Phys. 114, 9496 (2001).], which has proved accurate for continuous potentials, is extended successfully to the hard sphere potential plus an attractive Yukawa tail. By comparing the results of thermodynamic properties, including the liquid-vapor phase diagram, with available simulation data, it is found that this method remains reliable for this class of models of interaction often used in colloid science.

Evaluation of bridge-function diagrams via Mayer-sampling Monte Carlo simulation

We report coefficients of the h-bond expansion of the bridge function of the hard-sphere system up to order rho(4) (where rho is the density in units of the hard-sphere diameter), which in the highest-order term includes 88 cluster diagrams with bonds representing the total correlation function h(r). Calculations are performed using the recently introduced Mayer-sampling method for evaluation of cluster integrals, and an iterative scheme is applied in which the h(r) used in the cluster integrals is determined by solution of the Ornstein-Zernike equation with a closure given by the calculated clusters. Calculations are performed for reduced densities from 0.1 to 0.9 in increments of 0.1. Comparison with molecular simulation data shows that the convergence is very slow for the density expansion of the bridge function calculated this way.

Structure of penetrable-rod fluids: Exact properties and comparison between Monte Carlo simulations and two analytic theories

Bounded potentials are good models to represent the effective two-body interaction in some colloidal systems, such as the dilute solutions of polymer chains in good solvents. The simplest bounded potential is that of penetrable spheres, which takes a positive finite value if the two spheres are overlapped, being 0 otherwise. Even in the one-dimensional case, the penetrable-rod model is far from trivial, since interactions are not restricted to nearest neighbors and so its exact solution is not known. In this paper the structural properties of one-dimensional penetrable rods are studied. We first derive the exact correlation functions of the penetrable-rod fluids to second order in density at any temperature, as well as in the high-temperature and zero-temperature limits at any density. It is seen that, in contrast to what is generally believed, the Percus-Yevick equation does not yield the exact cavity function in the hard-rod limit. Next, two simple analytic theories are constructed: a high-temperature approximation based on the exact asymptotic behavior in the limit T--> infinity and a low-temperature approximation inspired by the exact result in the opposite limit T--> 0. Finally, we perform Monte Carlo simulations for a wide range of temperatures and densities to assess the validity of both theories. It is found that they complement each other quite well, exhibiting a good agreement with the simulation data within their respective domains of applicability and becoming practically equivalent on the borderline of those domains. A comparison with numerical solutions of the Percus-Yevick and the hypernetted-chain approximations is also carried out. Finally, a perspective on the extension of our two heuristic theories to the more realistic three-dimensional case is provided.

Structure of inhomogeneous Lennard-Jones fluid near the critical region and close to the vapor-liquid coexistence curve: Monte Carlo and density-functional theory studies

Results for the density profiles of the Lennard-Jones (LJ) fluid subjected to diverse external fields are presented for the Monte Carlo simulations within the grand canonical ensemble and for the third order and second order perturbation density-functional approximation (DFA). In all cases, the bulk LJ fluid in the particle reservoir to which the nonuniform fluid under consideration is connected, is at the conditions situated at "dangerous" regions of the phase diagram, i.e., near the critical temperature or close to the gas-liquid coexistence curve. It is found that the previously investigated third order and second order perturbation DFA for hard core attractive Yukawa fluid [J. Chem. Phys. 122, 064503 (2005)] can perform successfully also for the nonuniform LJ fluid only on the condition of high accuracy of the required bulk second order direct correlation function. The present report further indicates that the proposed third order and second order perturbation DFA is efficient and suitable for both supercritical and subcritical temperatures.

Phase diagram of complex fluids using an efficient integral equation method

We present an adaptive technique for the determination of the phase diagram of fluids within the integral equation theory. It enables an efficient and accurate systematic mapping of the thermodynamic space in order to construct the binodal and spinodal lines. Results are obtained with the thermodynamically consistent integral equation proposed by Sarkisov [J. Chem. Phys. 114, 9496 (2001)] within the tangent linear technique that yields an exact differentiation of correlation functions. The generality of the numerical approach is assessed by determining both the liquid-vapor coexistence and the critical parameters of the generalized Lennard-Jones (n,6) potentials with varying repulsive part, including the hard-sphere limit.

Chemical potentials and phase equilibria of Lennard-Jones mixtures: a self-consistent integral equation approach

We explore the vapor-liquid phase behavior of binary mixtures of Lennard-Jones-type molecules where one component is supercritical, given the system temperature. We apply the self-consistency approach to the Ornstein-Zernike integral equations to obtain the correlation functions. The consistency checks include not only thermodynamic consistencies (pressure consistency and Gibbs-Duhem consistency), but also pointwise consistencies, such as the zero-separation theorems on the cavity functions. The consistencies are enforced via the bridge functions in the closure which contain adjustable parameters. The full solution requires the values of not only the monomer chemical potentials, but also the dimer chemical potentials present in the zero-separation theorems. These are evaluated by the direct chemical-potential formula [L. L. Lee, J. Chem. Phys. 97, 8606 (1992)] that does not require temperature nor density integration. In order to assess the integral equation accuracy, molecular-dynamics simulations are carried out alongside the states studied. The integral equation results compare well with simulation data. In phase calculations, it is important to have pressure consistency and valid chemical potentials, since the matching of phase boundaries requires the equality of the pressures and chemical potentials of both the liquid and vapor phases. The mixtures studied are methane-type and pentane-type molecules, both characterized by effective Lennard-Jones potentials. Calculations on one isotherm show that the integral equation approach yields valid answers as compared with the experimental data of Sage and Lacey. To study vapor-liquid phase behavior, it is necessary to use consistent theories; any inconsistencies, especially in pressure, will vitiate the phase boundary calculations.

Hard-sphere radial distribution function again

A theoretically based closed-form analytical equation for the radial distribution function, g(r), of a fluid of hard spheres is presented and used to obtain an accurate analytic representation. The method makes use of an analytic expression for the short- and long-range behaviors of g(r), both obtained from the Percus-Yevick equation, in combination with the thermodynamic consistency constraint. Physical arguments then leave only three parameters in the equation of g(r) that are to be solved numerically, whereas all remaining ones are taken from the analytical solution of the Percus-Yevick equation.

A consistent integral equation theory for hard spheres

The standard integral equation approach is used to extract the bridge function and other correlation functions of hard spheres fluid. To achieve this, we first use a recent consistent closure relation proposed by Bomont et al. [J. Chem. Phys. 119, 2188 (2003)] that has already proven to be accurate to describe the Lennard-Jones fluid properties. Second, we take advantage of the coherent scheme derived by Bomont [J. Chem. Phys. 119, 11484 (2003)] to calculate the excess chemical potential, the entropy and some relative transport properties. Very good agreement is obtained for structural quantities and thermodynamic properties as compared to exact data at densities ranging from 0.1 to 0.9.