Generalized coupling parameter expansion: Application to square well and Lennard-Jones fluids

The bridge function required to yield a singlet integral equation (IE) up to the second order in density expansion for the hard sphere fluid confined in a slit-pore is evaluated. The slit-fluid bridge function can be divided into wall-particle bridge diagrams with h b-bond, which were evaluated by recently proposed Transition Matrix Monte Carlo (TMMC) Mayer-sampling method. The bulk-fluid total correlation function h b(r) used in cluster integrals is determined by solution of the bulk-fluid Ornstein–Zernike (OZ) equation with a hypernetted chain closure (HNC). The calculation is performed for the reduced density of bulk fluid in equilibrium with the fluid in slit-pores from 0.3 to 0.7 with narrow slit width of 3.0σ and 4.0σ. The quantity of the slit-fluid bridge function is assessed by comparison of the density profile obtained from the singlet IE theory and the grand canonical Monte Carlo (GCMC) simulation. Good agreement between the proposed approach and the GCMC data is observed. The reduced normal pressure is also calculated, and agrees well with the simulation data at low to medium densities but becomes a little larger at high density. It is expected that the result can be improved by adding higher order bridge coefficients. The direct evaluation of the slit-fluid bridge function seems to be practical since a great improvement of the quality of the singlet IE theory has been achieved for predicting the structural and thermodynamic properties of fluids confined in narrow slit pores.

The integral equation theory is used to investigate the structural and thermodynamic properties of the Lennard-Jones fluid with a new approximate bridge function, which is based on the modification of its expansion in powers of the thermal potential. The thermodynamic consistency is achieved via the virial and compressibility routes. The problem of the renormalization of the indirect correlation function is also considered with an original prescription for partitioning the pair potential. The bridge function as well as the excess chemical potential and the excess entropy, which are calculated by means of standard formulas of statistical mechanics, are compared to simulation data. The thermodynamic properties of the supercritical Lennard-Jones fluid calculated with the present closure relation are in excellent agreement with those obtained from computer simulation.

Building upon past work on the phase diagram of Janus fluids [F. Sciortino, A. Giacometti, and G. Pastore, Phys. Rev. Lett. 103, 237801 (2009)], we perform a detailed study of integral equation theory of the Kern-Frenkel potential with coverage that is tuned from the isotropic square-well fluid to the Janus limit. An improved algorithm for the reference hypernetted-chain (RHNC) equation for this problem is implemented that significantly extends the range of applicability of RHNC. Results for both structure and thermodynamics are presented and compared with numerical simulations. Unlike previous attempts, this algorithm is shown to be stable down to the Janus limit, thus paving the way for analyzing the frustration mechanism characteristic of the gas-liquid transition in the Janus system. The results are also compared with Barker-Henderson thermodynamic perturbation theory on the same model. We then discuss the pros and cons of both approaches within a unified treatment. On balance, RHNC integral equation theory, even with an isotropic hard-sphere reference system, is found to be a good compromise between accuracy of the results, computational effort, and uniform quality to tackle self-assembly processes in patchy colloids of complex nature. Further improvement in RHNC however clearly requires an anisotropic reference bridge function.

We study the structure and phase behavior of a binary mixture where one of the components is self-propelling in nature. The interparticle interactions in the system are taken from the Asakura-Oosawa model for colloid-polymer mixtures for which the phase diagram is known. In the current model version, the colloid particles are made active using the Vicsek model for self-propelling particles. The resultant active system is studied by molecular dynamics methods and integral equation theory. Both methods produce results consistent with each other and demonstrate that the Vicsek model-based activity facilitates phase separation, thus, broadening the coexistence region.

The objective of this paper is to investigate the convergence of coupling-parameter expansion-based solutions to the Ornstein–Zernike equation in liquid state theory. The analytically solved Baxter’s adhesive hard sphere model is analyzed first by using coupling-parameter expansion. It was found that the expansion provides accurate approximations to solutions—including the liquid-vapor phase diagram—in most parts of the phase plane. However, it fails to converge in the region where the model has only complex solutions. Similar analysis and results are obtained using analytical solutions within the mean spherical approximation for the hardcore Yukawa potential. However, numerical results indicate that the expansion converges in all regions in this model. Next, the convergence of the expansion is analyzed for the Lennard-Jones potential by using an accurate density-dependent bridge function in the closure relation. Numerical results are presented which show convergence of correlation functions, compressibility versus density profiles, etc., in the single as well as two-phase regions. Computed liquid-vapor phase diagrams, using two independent schemes employing the converged profiles, compare excellently with simulation data. The results obtained for the generalized Lennard-Jones potential, with varying repulsive exponent, also compare well with the simulation data. Solution-spaces and the bifurcation of the solutions of the Ornstein–Zernike equation that are relevant to coupling-parameter expansion are also briefly discussed. All of these results taken together establish the coupling-parameter expansion as a practical tool for studying single component fluid phases modeled via general pair-potentials.

We present an analytical integral equation theory for polyelectrolyte solutions modeled as linear freely-jointed tangent hard-sphere polyanionic chains and cationic hard-sphere monomeric counterions embedded in a continuum dielectric medium. Each hard-sphere segment on the polyelectrolyte chain and hard-sphere counterion are univalent with unit diameters. The model was formulated in the context of the multi density Ornstein–Zernike integral equation theory within the mean spherical approximation. Analytical solutions for the model were obtained using the ideal chain approximation. The contact values of the radial distribution functions, internal energy, Helmholtz energy, osmotic pressure, and activity coefficient of the system were derived as a function of chain length, density, and Bjerrum length via the energy route. Predictions from the theory were compared with computer simulation data reported in the literature, and very good agreement was found.

We analyze the depletion potential between large spheres in a multicomponent mixture of dense small spheres (up to seven components) using the integral equation theory (IET), in which semiempirical bridge functions are incorporated, and the insertion approach within the framework of density functional theory (DFT). The diameters of the small spheres considered are in the range of d(S)-5d(S). The results from the IET and DFT are in close agreement with each other. The depletion potential in the mixture is substantially different from that in a one-component system of dense small spheres with diameter d(S). In comparison with the latter, the former possesses in general a less pronounced oscillatory structure, and the free-energy barrier for large spheres to overcome before reaching the contact is significantly reduced. This tendency can be enhanced as the number of components increases. In a several-component mixture of small spheres whose diameters are suitably chosen and in which the packing fractions of the components share the same value, the depletion potential is essentially short ranged and attractive and possesses a sufficiently large, negative value at the contact.

ABSTRACTIn this work, we present a simple mathematical expression for a continuous version of the square-well (SW) discontinuous potential of variable range (). This expression can be useful in theoretical statistical mechanics methods and specially in conventional molecular dynamics (MD) simulations. To illustrate the latter point, the continuous version of some SW potentials of variable range were selected and studied with continuous MD. SW single state thermodynamic and structural properties (vapour, liquid and solid), vapour–liquid phase diagrams and surface tensions accurately reproduce available simulation data obtained by Monte Carlo and discontinuous MD simulation methods. This expression can be easily implemented in popular simulation packages (GROMACS, NAMD, CHARMM, DLPOLY, EXPResSo, HOOMD, TINKER, ls1 mardyn, etc.) that can be useful to study more complex systems. Besides we also present a continuous version for the square-shoulder potential.

We show that the Zwanzig first-order perturbation theory can be obtained directly from a truncated Taylor series expansion of a two-body perturbation theory and that such truncation provides a more accurate prediction of thermodynamic properties than the full two-body perturbation theory. This unexpected result is explained by the quality of the resulting approximation for the fluid radial distribution function. We prove that the first-order and the two-body perturbation theories are based on different approximations for the fluid radial distribution function. To illustrate the calculations, the square-well fluid is adopted. We develop an analytical expression for the two-body perturbed Helmholtz free energy for the square-well fluid. The equation of state obtained using such an expression is compared to the equation of state obtained from the first-order approximation. The vapor-liquid coexistence curve and the supercritical compressibility factor of a square-well fluid are calculated using both equations of state and compared to Monte Carlo simulation data. Finally, we show that the approximation for the fluid radial distribution function given by the first-order perturbation theory provides closer values to the ones calculated via Monte Carlo simulations. This explains why such theory gives a better description of the fluid thermodynamic behavior.

In this work, integral equation theory is extended to describe the structures and surface tensions of confined fluids. To improve the accuracy of the equation, a bridge function based on the fundamental measure theory is introduced. The density profiles of the confined Lennard-Jones fluids and water are calculated, which are in good agreement with simulation data. On the basis of these density profiles, the grand potentials are then calculated using the density functional approach, and the corresponding surface tensions are predicted, which reproduce the simulation data well. In particular, the contact angles of water in contact with both hydrophilic and hydrophobic walls are evaluated.

A self-consistent density-functional approach has been employed to study the structure of nonuniform binary hard-core Yukawa mixtures as well as the structure of its uniform counterpart. The second-order direct correlation function and the bridge function of the corresponding uniform fluid mixture required as input in the self-consistent theory are obtained from integral equation theory using an accurate closure relation. The calculated density and concentration profiles of the nonuniform mixtures as well as the radial distribution function profiles of the uniform fluid mixtures are shown to compare well with available simulation results.

The integral equation theory for a model 2–2 electrolyte is studied in detail. In this model electrolyte, the ions are assumed to be the same size, and interact via a continuous potential energy which behaves as the Coulomb potential at large distances and an inverse ninth power repulsion at short distances. The ions are embedded in a dielectric continuum of fixed dielectric constant, here taken to be 78.3 ε0 in order to model water at 25 °C. The bridge function for this model is studied as a function of concentration (a) for six proposed closures, and (b) via ‘‘exact’’ inversion of data from computer simulations. A proposed closure derived from examination of the inverted bridge function yields predictions in good agreement with computer simulations. We emphasize the importance of choosing an ‘‘optimized’’ long-range potential, as opposed to the traditional Coulomb choice. A simple functional form for the bridge function results from this optimized choice of long-range potential.

The structural and thermodynamic properties of the argon fluid were investigated by means of a modification of the optimised random phase approximation (ORPA) theory and the square-well (SW) potential. The results were obtained at the SW potential values 1.25 < λ < 2, and along different isotherms at a wide range of densities. The radial distribution function, g(r), was calculated, and the results obtained were extrapolated to give the appropriate contact values g(σ), g(λσ −) and g(λσ +). The results were compared with the simulation and experimental data available in the literature. The results calculated for the Helmholtz free energy (A) and internal energy (E) are in good agreement with the experimental data at the sub- and super-critical temperatures. The ORPA version considered here provided good results for the compressibility factor Z from the Virial route with the SW attractive range λ = 1.7 at high temperatures and low densities.

The influence of equilibrium chemical reactions on vapor—liquid phase diagrams

We describe a Monte Carlo method for simulation of vapour–liquid phase coexistence in size-asymmetrical Lennard–Jones (LJ) binary mixtures. The method incorporates the Solvent Repacking Monte Carlo (SRMC) approach, which offers efficient trial moves for the exchange of a large particle for several small particles, into the Gibbs Ensemble Monte Carlo (GEMC) method. SRMC yields a significant efficiency improvement in simulation of dilute large species mixtures at low temperature compared to the original GEMC method with identity exchange moves. Vapor–liquid phase diagrams are reported for LJ mixtures with a diameter ratio of 1:2 with well depth ratios and 2, producing spindle-type, azeotrope and closed loop types of phase diagram, respectively.