Integral equation theory for Lennard‐Jones fluids: The bridge function and applications to pure fluids and mixtures

Perturbative Expansion in Reciprocal Space: Bridging Microscopic and Mesoscopic Descriptions of Molecular Interactions

Determining the Fourier representation of various molecular interactions is important for constructing density-based field theories from a microscopic point of view, enabling a multiscale bridge between microscopic and mesoscopic descriptions. However, due to the strongly repulsive nature of short-ranged interactions, interparticle interactions cannot be formally defined in Fourier space, which renders coarse-grained (CG) approaches in k-space somewhat ambiguous. In this paper, we address this issue by designing a perturbative expansion of pair interactions in reciprocal space. Our perturbation theory, starting from reciprocal space, elucidates the microscopic origins underlying zeroth-order (long-range attractions) and divergent repulsive interactions from higher order contributions. We propose a systematic framework for constructing a faithful Fourier-space representation of molecular interactions, capturing key structural correlations in various systems, including simple model systems and molecular CG models of liquids. Building upon the Ornstein-Zernike equation, our approach can be combined with appropriate closure relations, and to further improve the closure approximations, we develop a bottom-up parameterization strategy for inferring the bridge function from microscopic statistics. By incorporating the bridge function into the Fourier representation, our findings suggest a systematic, bottom-up approach to performing coarse-graining in reciprocal space, leading to the systematic construction of a bottom-up classical field theory of complex aqueous systems.

Nonadditivities of the Particle Sizes Hidden in Model Pair Potentials and Their Effects on Physical Adsorptions

It is important to understand the mechanism of colloidal particle assembly near a substrate for development of drug delivery systems, micro-/nanorobots, batteries, heterogeneous catalysts, paints, and cosmetics. Understanding the mechanism is also important for crystallization of the colloidal particles and proteins. In this study, we calculated the physical adsorption of colloidal particles on a flat wall mainly using the integral equation theory, wherein small and large colloidal particles were employed. In the calculation system, like-charged electric double-layer potentials were used as pair potentials. In some cases, it was found that the small particles are more easily adsorbed. This result is unusual from the viewpoint of the Asakura-Oosawa theory, and we call it a "reversal phenomenon". Theoretical analysis revealed that the reversal phenomenon originates from the nonadditivities of the particle sizes. Using the knowledge obtained from this study, we invented a method to analyze the size nonadditivity hidden in model pair potentials. The method will be useful for confirmation of various simulation results regarding the adsorption and development of force fields for colloidal particles, proteins, and solutes.

Structure and thermodynamics of a 2D Lennard-Jones hexagonal fluid

The thermodynamic and structural properties of the 2D hexagonal soft-sites fluid are examined by integral equation theory benchmarked against extensive Monte Carlo simulations. Hexamers are built of six equal Lennard-Jones segments. Site-site integral equation theory is used to compute site-site correlation functions, excess internal energies and isotherms over a wide range of conditions and compared with results obtained from Monte Carlo simulations. Various approaches for computing the pressure are discussed as well. Satisfactory qualitative agreement between theory and simulations is found with details depending on the applied closure relation.

Data-driven approximations to the bridge function yield improved closures for the Ornstein-Zernike equation

A key challenge for soft materials design and coarse-graining simulations is determining interaction potentials between components that give rise to desired condensed-phase structures. In theory, the Ornstein-Zernike equation provides an elegant framework for solving this inverse problem. Pioneering work in liquid state theory derived analytical closures for the framework. However, these analytical closures are approximations, valid only for specific classes of interaction potentials. In this work, we combine the physics of liquid state theory with machine learning to infer a closure directly from simulation data. The resulting closure is more accurate than commonly used closures across a broad range of interaction potentials.

Reduced density profile of small particles near a large particle: Results of an integral equation theory with an accurate bridge function and a Monte Carlo simulation

Solute-solvent reduced density profiles of hard-sphere fluids were calculated by using several integral equation theories for liquids. The traditional closures, Percus-Yevick (PY) and the hypernetted-chain (HNC) closures, as well as the theories with bridge functions, Verlet, Duh-Henderson, and Kinoshita (named MHNC), were used for the calculation. In this paper, a one-solute hard-sphere was immersed in a one-component hard-sphere solvent and various size ratios were examined. The profiles between the solute and solvent particles were compared with those calculated by Monte Carlo simulations. The profiles given by the integral equations with the bridge functions were much more accurate than those calculated by conventional integral equation theories, such as the Ornstein-Zernike (OZ) equation with the PY closure. The accuracy of the MHNC-OZ theory was maintained even when the particle size ratio of solute to solvent was 50. For example, the contact values were 5.7 (Monte Carlo), 5.6 (MHNC), 7.8 (HNC), and 4.5 (PY), and the first minimum values were 0.48 (Monte Carlo), 0.46 (MHNC), 0.54 (HNC), and 0.40 (PY) when the packing fraction of the hard-sphere solvent was 0.38 and the size ratio was 50. The asymptotic decay and the oscillation period for MHNC-OZ were also very accurate, although those given by the HNC-OZ theory were somewhat faster than those obtained by Monte Carlo simulations.

Glass transition in hard-core fluids and beyond, using an effective static structure in the mode coupling theory

The dynamical arrest in classical fluids is studied using a simple modification of the mode coupling theory (MCT) aimed at correcting its overestimation of the tendency to glass formation while preserving its overall structure. As in previous attempts, the modification is based on the idea of tempering the static pair correlations used as input. It is implemented in this work by computing the static structure at a different state point than the one used to solve the MCT equation for the intermediate scattering function, using the pure hard-sphere glass for calibration. The location of the glass transition predicted from this modification is found to agree with simulations data for a variety of systems --pure fluids and mixtures with either purely repulsive interaction potentials or ones with attractive contributions. Besides improving the predictions in the long-time limit, and so reducing the non-ergodicity domain, the same modification works as well for the time-dependent correlators.

Number Density Distribution of Small Particles around a Large Particle: Structural Analysis of a Colloidal Suspension

Some colloidal suspensions contain two types of particles-small and large particles-to improve the lubricating ability, light absorptivity, and so forth. Structural and chemical analyses of such colloidal suspensions are often performed to understand their properties. In a structural analysis study, the observation of the number density distribution of small particles around a large particle (g_LS) is difficult because these particles are randomly moving within the colloidal suspension by Brownian motion. We obtain g_LS using the data from a line optical tweezer (LOT) that can measure the potential of mean force between two large colloidal particles (Φ_LL). We propose a theory that transforms Φ_LL into g_LS. The transform theory is explained in detail and tested. We demonstrate for the first time that LOT can be used for the structural analysis of a colloidal suspension. LOT combined with the transform theory will facilitate structural analyses of the colloidal suspensions, which is important for both understanding colloidal properties and developing colloidal products.

Number density distribution of solvent molecules on a substrate: a transform theory for atomic force microscopy

Atomic force microscopy (AFM) in liquids can measure a force curve between a probe and a buried substrate. The shape of the measured force curve is related to hydration structure on the substrate. However, until now, there has been no practical theory that can transform the force curve into the hydration structure, because treatment of the liquid confined between the probe and the substrate is a difficult problem. Here, we propose a robust and practical transform theory, which can generate the number density distribution of solvent molecules on a substrate from the force curve. As an example, we analyzed a force curve measured by using our high-resolution AFM with a newly fabricated ultrashort cantilever. It is demonstrated that the hydration structure on muscovite mica (001) surface can be reproduced from the force curve by using the transform theory. The transform theory will enhance AFM's ability and support structural analyses of solid/liquid interfaces. By using the transform theory, the effective diameter of a real probe apex is also obtained. This result will be important for designing a model probe of molecular scale simulations.

Hypernetted-chain investigation of the random first-order transition of a Lennard-Jones liquid to an ideal glass

The structural and thermodynamic behavior of a deeply supercooled Lennard-Jones liquid, and its random first-order transition (RFOT) to an ideal glass is investigated, using a system of two weakly coupled replicas and the hypernetted chain integral equation for the pair structure of this symmetric binary system. A systematic search in the density-temperature plane points to the existence of two glass branches below a density-dependent threshold temperature. The branch of lower free energy exhibits a rapid growth of the structural overlap order parameter upon cooling and may be identified with the ideal glass phase conjectured by several authors for both spin and structural glasses. The RFOT, signaled by a sharp discontinuity of the order parameter, is predicted to be weakly first order from a thermodynamic viewpoint. The transition temperature T(cr) increases rapidly with density and approximately obeys a scaling relation valid for a reference system of particles interacting via a purely repulsive 1/r(18) potential.

Metastable-state dynamics of a liquid: a free-energy landscape study

Using the time dependence of density fluctuations in a supercooled liquid obtained from the solutions of the equations of nonlinear fluctuating hydrodynamics (NFH), the evolution of the system in the free energy landscape is studied. A crossover from a continuous fluid type dynamics to that of hopping between different free energy minima is observed as the liquid is increasingly supercooled. We demonstrate that our results are also in agreement with equilibrium density functional analysis of the same system. The density field obtained in the numerical solution of the NFH equations are further analyzed to introduce complimentary density of voids in the supercooled liquid state and its static and dynamic correlations are computed. The nature of the relaxation of vacancy correlations are observed to be similar to that of the density fluctuations.

Time dependent stretching of aging dynamics in a generalized hydrodynamic model for supercooled liquids

The nonequilibrium dynamics and aging behavior of a supercooled liquid is investigated from an analysis of the correlation of density fluctuations at two different times. The dynamic correlation functions are computed by solving numerically the equations of nonlinear fluctuating hydrodynamics. The aging time dependence follows a modified stretched exponential form with a relaxation time which is dependent on the aging time. This is similar to the behavior seen in the aging data of dielectric response functions of a typical glass forming liquid.

Regarding convergence curve of virial expansion for the Lennard-Jones system

We calculate the convergence curve of the virial expansion for a Lennard-Jones system in the density-temperature plane using the approximate method based on the density expansion of the Ornstein-Zernike equation and the condition of thermodynamic consistency [J. Chem. Phys. 106, 6095 (1997)]. At subcritical temperatures, this curve is close to the binodal. At supercritical temperatures, the curve does not coincide with the freezing curve. In the latter case, the densities along the convergence line are distinctly smaller than the densities corresponding to the condensed state.

Characteristic temperatures of glassy behaviour in a simple liquid

A model for the metastable liquid in terms of holes present in the amorphous structure is considered using the classical density functional theory (DFT). For a one component Lennard-Jones liquid we obtain the temperature dependence of the free volume v(f) in the metastable state. A temperature T(0), similar to that of the characteristic transition of the free volume theory, is identified by extrapolating v(f)(T) to zero. The Kauzmann temperature T(K) is also obtained here by extrapolating the entropy difference between the supercooled state and that of the crystal to zero. We compare the temperatures T(0) and T(K) obtained in our model with other two characteristic temperatures for glassy behaviour, namely (a) the dynamic transition temperature T(c) of the mode coupling theory (MCT) and (b) the glass transition temperature T(g) which was obtained by Leonardo et al (2000 Phys. Rev. Lett. 84 6054) from studying the violation of the fluctuation-dissipation theorem. All the four temperatures, obtained from independent routes, are located with respect to the melting temperature T(m) in a manner which is in agreement with experiments.

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.

Mass dependence of shear viscosity in a binary fluid mixture: mode-coupling theory

An expression for the shear viscosity of a binary fluid mixture is derived using mode-coupling theory in order to study the mass dependence. The calculated results on shear viscosity for a binary isotopic Lennard-Jones fluid mixture show good agreement with results from molecular dynamics simulation carried out over a wide range of mass ratio at different composition. Also proposed is a new generalized Stokes-Einstein relation connecting the individual diffusivities to shear viscosity.

Depletion potential between large spheres immersed in a multicomponent mixture of small spheres

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.

Bridge functions near the liquid-vapor coexistence curve in binary Lennard-Jones mixtures

We have carried out extensive molecular-dynamics simulation studies of binary Lennard-Jones mixtures to calculate directly the bridge function at state points lying in a very narrow single fluid phase region between the vapor-liquid and solid-liquid coexistence lines [Lamm and Hall, Fluid Phase Equilib. 182, 37 (2001); 194-197, 197 (2002)]. By varying the density close to the liquid-vapor coexistence line, significant deviations are observed at intermediate distances between the simulated bridge function and two widely used approximate closures in the integral equation theory of liquids, viz. the hybrid mean spherical approximation and the Duh-Henderson closures. The overall qualitative agreement remains the same with small variation in temperature that brings the system closer to either the liquid-vapor or liquid-solid coexistence curve. We also report a comparison of the direct and indirect correlation functions obtained from our simulation studies as well as from the integral equation theory of liquids. Our results emphasize the need for developing new closures applicable to binary fluid mixtures over a wide range of thermodynamic parameters.

Scaling law of shear viscosity in atomic liquid and liquid mixtures

A scaling law relating the shear viscosity of one and two component liquid mixtures to their excess thermodynamic entropies defined through pair correlation functions is derived by approximating the mode coupling theory expressions of frictions and then combining with the Stokes-Einstein relation. Molecular dynamics simulation has been performed to generate the data of shear viscosity for one and two component liquid mixtures to test the derived scaling law. The derived scaling laws yield numerical results of shear viscosity for one component and two component liquid mixtures, which are in excellent agreement with the molecular dynamics simulation results for a wide range of density and interaction potential.

Triangle of Liquid−Gas States

We demonstrate for the first time that (a) the straight line of the unit compressibility factor (Zeno line) tends asymptotically to the liquid branch of binodal at low temperatures, (b) the straight line with a half density has to be close to the average of vapor-liquid densities along the binodal curve (rectilinear diameter), and (c) the phase coexistence curves are inscribed into the right triangle in the density-temperature plane, which is formed by the Zeno line and by the segments which this line cuts off on the axes. These statements are confirmed for model systems and for a wide group of real substances (for the first time including metals: Hg, Cs, and Cu). Critical parameters of all substances under study are located in the vicinity of the triangle median, drawn to the density axis, with a dispersion on the order of 2 in reduced units.

Molecular dynamics study of the density and temperature dependence of bridge functions in normal and supercritical Lennard-Jones fluids

A systematic study of the density and temperature dependence of bridge functions has been carried out using molecular dynamics simulation studies in one-component Lennard-Jones fluids. In deriving the liquid structure, approximate closures are generally used in integral equation theories of liquids to obtain static density correlations. In the present work, we have directly compared the simulated bridge function to two such commonly used closures, viz., hybrid mean spherical approximation (HMSA) [J. Chem. Phys. 84, 2336 (1986)] and Duh-Henderson [J. Chem. Phys. 104, 6742 (1996)] closures with thermodynamic parameters varying from the normal liquid to the supercritical fluid phase far from and near the critical point. In the normal liquid region, both closures show a qualitative agreement with the simulated bridge function, although the extent of correlation at distances sigma < r < or = 2.5sigma is generally underestimated. A similar behavior is obtained in supercritical fluids far from the critical point where critical fluctuations are no longer important. In contrast, significant deviations are observed in the bridge functions in supercritical fluids near the critical point even at densities as small as 25% or 50% of the critical density. Such behavior appears to have resulted from competing contributions to the bridge function from decreasing indirect correlations and small yet significant cavity correlations persistent even at very low densities.

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.

Nature of the entropy versus self-diffusivity plot for simple liquids

The empirical relation (D(*))(alpha) = a exp[S] between the self-diffusion coefficient D(*) and the excess entropy S of a liquid is studied here in the context of theoretical model calculation. The coefficient alpha is dependent on the interaction potential and shows a crossover at an intermediate density, where cooperative dynamics become more important. Around this density a departure from the Stokes-Einstein relation is also observed. The above relation between entropy and diffusion is also tested for the scaled total diffusion coefficient in a binary mixture.

The gas–liquid phase-transition singularities in the framework of the liquid-state integral equation formalism

The singularities of various liquid-state integral equations derived from the Ornstein-Zernike relation and its temperature derivatives, have been investigated in the liquid-vapor transition region. As a general feature, it has been found that the existence of a nonsolution curve on the vapor side of the phase diagram, on which both the direct and the total correlation functions become complex-with a finite isothermal compressibility-also corresponds to the locus of points where the constant-volume heat capacity diverges, in consonance with a divergence of the temperature derivative of the correlation functions. In contrast, on the liquid side of the phase diagram one finds that a true spinodal (a curve of diverging isothermal compressibilities) is reproduced by the Percus-Yevick and Martynov-Sarkisov integral equations, but now this curve corresponds to states with finite heat capacity. On the other hand, the hypernetted chain approximation exhibits a nonsolution curve with finite compressibilities and heat capacities in which, as temperature is lowered, the former tends to diverge.

Fluid-phase diagrams of binary mixtures from constant pressure integral equations

A new algorithm for solving integral equations of the theory of liquids at fixed pressure is introduced. Combining this technique with the Lee's star function approximation for the chemical potentials, we obtain an efficient method to investigate fluid-phase diagrams of binary mixtures. We have tested the capabilities of such technique to study symmetric and asymmetric phase diagrams in nonadditive hard spheres and Lennard-Jones mixtures. We find that the integral equation theories, although approximate, can provide a flexible tool to determine the fluid-phase diagrams whose accuracy is critically dependent on the quality of the closure and of the resulting chemical potentials.