Ornstein-Zernike Integral Equation Research Articles

Nonequilibrium static growing length scales in supercooled liquids on approaching the glass transition

The small wavenumber k behavior of the structure factor S(k) of overcompressed amorphous hard-sphere configurations was previously studied for a wide range of densities up to the maximally random jammed state, which can be viewed as a prototypical glassy state [A. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E 86, 021505 (2012)]. It was found that a precursor to the glassy jammed state was evident long before the jamming density was reached as measured by a growing nonequilibrium length scale extracted from the volume integral of the direct correlation function c(r), which becomes long-ranged as the critical jammed state is reached. The present study extends that work by investigating via computer simulations two different atomic models: the single-component Z2 Dzugutov potential in three dimensions and the binary-mixture Kob-Andersen potential in two dimensions. Consistent with the aforementioned hard-sphere study, we demonstrate that for both models a signature of the glass transition is apparent well before the transition temperature is reached as measured by the length scale determined from the volume integral of the direct correlation function in the single-component case and a generalized direct correlation function in the binary-mixture case. The latter quantity is obtained from a generalized Ornstein-Zernike integral equation for a certain decoration of the atomic point configuration. We also show that these growing length scales, which are a consequence of the long-range nature of the direct correlation functions, are intrinsically nonequilibrium in nature as determined by an index X that is a measure of the deviation from thermal equilibrium. It is also demonstrated that this nonequilibrium index, which increases upon supercooling, is correlated with a characteristic relaxation time scale.

Fluctuations and micro-heterogeneity in mixtures of complex liquids

The role of concentration fluctuation and micro-heterogeneity in aqueous methanol mixtures are examined by two different methods: computer simulations of two systems with sizes of 2048 and 16 384 molecules, and site-site Ornstein-Zernike integral equation theory with two different closures. The focus is on the short and long range behaviour of the correlation functions, their running integrals and the structure factors. It is found that the micro-heterogeneous nature of these mixtures does not allow a proper stabilisation of the long range tail of the correlations as obtained through computer simulations, even over 6 ns runs. Similarly, approximate closures provide poor estimates of the concentration fluctuations and micro-structure of the mixtures. Interestingly, the analysis of the inadequacies of both these studies provide interesting insights about the nature and interplay of the concentration fluctuations and the micro-heterogeneity in these mixtures.

On the accurate direct computation of the isothermal compressibility for normal quantum simple fluids: Application to quantum hard spheres

A systematic study of the direct computation of the isothermal compressibility of normal quantum fluids is presented by analyzing the solving of the Ornstein-Zernike integral (OZ2) equation for the pair correlations between the path-integral necklace centroids. A number of issues related to the accuracy that can be achieved via this sort of procedure have been addressed, paying particular attention to the finite-N effects and to the definition of significant error bars for the estimates of isothermal compressibilities. Extensive path-integral Monte Carlo computations for the quantum hard-sphere fluid (QHS) have been performed in the (N, V, T) ensemble under temperature and density conditions for which dispersion effects dominate the quantum behavior. These computations have served to obtain the centroid correlations, which have been processed further via the numerical solving of the OZ2 equation. To do so, Baxter-Dixon-Hutchinson's variational procedure, complemented with Baumketner-Hiwatari's grand-canonical corrections, has been used. The virial equation of state has also been obtained and several comparisons between different versions of the QHS equation of state have been made. The results show the reliability of the procedure based on isothermal compressibilities discussed herein, which can then be regarded as a useful and quick means of obtaining the equation of state for fluids under quantum conditions involving strong repulsive interactions.

Liquid theory with high accuracy and broad applicability: Coupling parameter series expansion and non hard sphere perturbation strategy

Thermodynamic and structural properties of liquids are of fundamental interest in physics, chemistry, and biology, and perturbation approach has been fundamental to liquid theoretical approaches since the dawn of modern statistical mechanics and remains so to this day. Although thermodynamic perturbation theory (TPT) is widely used in the chemical physics community, one of the most popular versions of the TPT, i.e. Zwanzig (Zwanzig, R. W. J. Chem. Phys. 1954, 22, 1420-1426) 1st-order high temperature series expansion (HTSE) TPT and its 2nd-order counterpart under a macroscopic compressibility approximation of Barker-Henderson (Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 2856-2861), have some serious shortcomings: (i) the nth-order term of the HTSE is involved with reference fluid distribution functions of order up to 2n, and the higher-order terms hence progressively become more complicated and numerically inaccessible; (ii) the performance of the HTSE rapidly deteriorates and the calculated results become even qualitatively incorrect as the temperature of interest decreases. This account deals with the developments that we have made over the last five years or so to advance a coupling parameter series expansion (CPSE) and a non hard sphere (HS) perturbation strategy that has scored some of its greatest successes in overcoming the above-mentioned difficulties. In this account (i) we expatiate on implementation details of our schemes: how input information indispensable to high-order truncation of the CPSE in both the HS and non HS perturbation schemes is calculated by an Ornstein-Zernike integral equation theory; how high-order thermodynamic quantities, such as critical parameters and excess constant volume heat capacity, are extracted from the resulting excess Helmholtz free energy with irregular and inevitable numerical errors; how to select reference potential in the non HS perturbation scheme. (ii) We give a quantitative analysis on why convergence speed of the CPSE in both the HS and non HS perturbation schemes is certainly faster than that of the HTSE and the HS perturbation scheme. (iii) We illustrate applications of the CPSE TPT in both the HS and non HS perturbation schemes in calculating thermodynamic properties of various coarse-grained potential function models and as input information of other liquid state theories such as a classical density functional theory (DFT), and also discuss, in the framework of classical DFT, the potential of our CPSE scheme in several typical problems of chemical physics interest. (iv) Finally, we consider several topics which are possibly expected to be settled in the immediate future and possible integration with other liquid state theory frameworks aiming to solve problems in complex fluids in both bulk and inhomogeneous states.

A study for structure and inter-diffusion coefficient of liquid K1− x Cs x metal alloys

Inter-diffusion coefficients of liquid K1− x Cs x metal alloys are calculated using scaling law proposed by Samanta et al. [A. Samanta, Sk.M. Ali, and S.K. Ghosh, Phys. Rev. Lett. 87, 245901 (2001)] following Dzugutov [M. Dzugutov, Nature 381, 137 (1996)], which express the possible relationship between the excess entropy and diffusion coefficient. The interatomic interactions are described from the individual version of the electron–ion potential proposed by Fiolhais et al. [C. Fiolhais, J.P. Perdew, S.Q. Armster, J.M. McLaren, and M. Brajczewska, Phys. Rev. B 51, 14001 (1995)]. The partial pair distribution functions and structure factors are calculated from the solution of Ornstein–Zernike integral equation with Rogers–Young closure. The evaluations with the composition of static structure and the inter-diffusion properties are discussed.

Non-hard sphere thermodynamic perturbation theory

A non-hard sphere (HS) perturbation scheme, recently advanced by the present author, is elaborated for several technical matters, which are key mathematical details for implementation of the non-HS perturbation scheme in a coupling parameter expansion (CPE) thermodynamic perturbation framework. NVT-Monte Carlo simulation is carried out for a generalized Lennard-Jones (LJ) 2n-n potential to obtain routine thermodynamic quantities such as excess internal energy, pressure, excess chemical potential, excess Helmholtz free energy, and excess constant volume heat capacity. Then, these new simulation data, and available simulation data in literatures about a hard core attractive Yukawa fluid and a Sutherland fluid, are used to test the non-HS CPE 3rd-order thermodynamic perturbation theory (TPT) and give a comparison between the non-HS CPE 3rd-order TPT and other theoretical approaches. It is indicated that the non-HS CPE 3rd-order TPT is superior to other traditional TPT such as van der Waals/HS (vdW/HS), perturbation theory 2 (PT2)/HS, and vdW/Yukawa (vdW/Y) theory or analytical equation of state such as mean spherical approximation (MSA)-equation of state and is at least comparable to several currently the most accurate Ornstein-Zernike integral equation theories. It is discovered that three technical issues, i.e., opening up new bridge function approximation for the reference potential, choosing proper reference potential, and/or using proper thermodynamic route for calculation of f(ex-ref), chiefly decide the quality of the non-HS CPE TPT. Considering that the non-HS perturbation scheme applies for a wide variety of model fluids, and its implementation in the CPE thermodynamic perturbation framework is amenable to high-order truncation, the non-HS CPE 3rd-order or higher order TPT will be more promising once the above-mentioned three technological advances are established.

Effective potentials between nanoparticles in suspension

Results of molecular dynamics simulations are presented for the pair distribution function between nanoparticles in an explicit solvent as a function of nanoparticle diameter and interaction strength between the nanoparticle and solvent. The effect of including the solvent explicitly is demonstrated by comparing the pair distribution function of nanoparticles to that in an implicit solvent. The nanoparticles are modeled as a uniform distribution of Lennard-Jones particles, while the solvent is represented by standard Lennard-Jones particles. The diameter of the nanoparticle is varied from 10 to 25 times that of the solvent for a range of nanoparticle volume fractions. As the strength of the interactions between nanoparticles and the solvent increases, the solvent layer surrounding the nanoparticle is formed which increases the effective radii of the nanoparticles. The pair distribution functions are inverted using the Ornstein-Zernike integral equation to determine an effective pair potential between the nanoparticles mediated by the introduction of an explicit solvent.

Structural and thermodynamic properties of the restricted primitive model electrolyte in a mixture with uncharged hard spheres: a grand canonical Monte Carlo simulation and integral equation study

A restricted primitive model electrolyte in a mixture with uncharged hard spheres was studied at room temperature using grand canonical Monte Carlo computer simulation and Ornstein–Zernike integral equation theory in the hypernetted chain approximation (HNC). The mean spherical approximation results are also presented for a few cases. We obtained the pair distribution functions of species of the system, the dependencies of the total fluid density and the ionic fraction on the chemical potentials, the excess internal energy and the heat capacity at constant volume for a wide range of chemical potentials of the species from the simulations and HNC theory. In the majority of cases, good agreement between the theoretical predictions and simulation data is obtained. The composition of the mixture is determined by the chemical potentials of both species. The pair distribution functions have a Debye-like shape at low densities for various values of the ion fraction. By increasing the chemical potential of the uncharged component, weak trends for structuring of the solution are observed with the formation of ion-hard sphere-ion complexes. At high densities, a tendency for in-phase oscillations of ion–ion functions is observed similar to the pure electrolyte in the restricted primitive model. We analysed the chemical potential–density and the chemical potential–ion fraction projections of the equation of state in detail. Also, the heat capacity at constant volume has been calculated for the first time. The model and the results are useful for the development of the theory of inhomogeneous fluid mixtures.

Fluid of hard spheres with dipolar-like patch interaction and effect of adding an isotropic adhesion

We compare two fluid models of spherical molecules with anisotropic, purely surface interactions. Both models admit an analytical solution of the molecular Ornstein–Zernike integral equation, within the Percus–Yevick approximation plus orientational linearization. In the first model, the molecular surface corresponds to a unique non-uniform patch, with a potential obtained by truncating a long-ranged dipolar interaction exactly at the contact distance between two hard sphere particles. In the second model, a further isotropic adhesion is added to the intermolecular potential. The study is focused on the local orientational ordering. Differences and similarities with respect to hard spheres with full long-ranged dipolar forces are analysed in detail. The effect of the competition between anisotropic patch interaction and isotropic adhesion is investigated through the pair correlation function as well as via two novel anisotropic order parameters.

Microscopic structure and thermodynamics of a core-softened model fluid from the second-order integral equations theory

We have studied the structure and thermodynamic properties of isotropic three-dimensional core-softened fluid by using the second-order Ornstein-Zernike integral equations completed by the hypernetted chain and Percus-Yevick closures. The radial distribution functions are compared with those from singlet integral equations and with computer simulation data. The limits of the region of density anomaly resulting from different approximate theories are established. The obtained results show that the second-order hypernetted chain approximation can be used to describe both the structure and the density anomaly of this model fluid. Moreover, we present the results of calculations of the bridge functions.

STRINGENT VERIFICATION OF THIRD ORDER + SECOND ORDER PERTURBATION DENSITY FUNCTION THEORY: BASED ON SHORT-RANGE SQUARE WELL POTENTIAL

A recently proposed third order + second order perturbation density functional theory (DFT) approach is made self-contained by using a virial pressure from the Ornstein–Zernike integral equation theory as input to determine the numerical value of an associated physical parameter. An exacting examination is formulated by applying the self-contained perturbation DFT approach to a short-range square well fluid of low temperatures subject to various external fields and comparing the theoretical results for density profiles to the corresponding grand canonical ensemble Monte Carlo simulation results. The comparison seems favorable for the third order + second order perturbation DFT approach as a self-contained and accurate predictive approach. It is surprisingly found that this self-contained third order + second order perturbation DFT approach is displayed outstandingly even if a deep SW perturbation term is being accounted for by a second order perturbation expansion. A discussion is presented about potential opportunity for this perturbation scheme.

Effect of critical fluctuations on adsorption of van der Waals fluid in a spherical cavity

A recently proposed non-uniform fifth-order thermodynamic perturbation theory (TPT) is employed to investigate the adsorption of a hard core attractive Yukawa (HCAY) fluid in a spherical cavity. Extensive comparison with available simulation data indicate that the non-uniform fifth-order TPT is sufficiently reliable in calculating the density profiles of the HCAY fluid in the highly confining geometry, and generally is more accurate than a previous third-order + second-order perturbation density functional theory. The non-uniform fifth-order TPT is free from numerically solving an Ornstein–Zernike integral equation, and also free of any adjustable parameter; consequently, it can be applied to both supercritical and subcritical temperature regions. The non-uniform fifth-order TPT is employed to investigate critical adsorption of the HCYA fluid in a single spherical cavity – it is disclosed that the critical fluctuations near the critical point induce depletion adsorption – quantitative theoretical calculation on relationship between the critical depletion adsorption, parameters of coexistence bulk phase and the responsible external field is in agreement with qualitative physical analysis.

Structural properties of effective potential model by liquid state theories

This paper investigates the structural properties of a model fluid dictated by an effective inter-particle oscillatory potential by grand canonical ensemble Monte Carlo (GCEMC) simulation and classical liquid state theories. The chosen oscillatory potential incorporates basic interaction terms used in modeling of various complex fluids which is composed of mesoscopic particles dispersed in a solvent bath, the studied structural properties include radial distribution function in bulk and inhomogeneous density distribution profile due to influence of several external fields. The GCEMC results are employed to test the validity of two recently proposed theoretical approaches in the field of atomic fluids. One is an Ornstein-Zernike integral equation theory approach; the other is a third order + second order perturbation density functional theory. Satisfactory agreement between the GCEMC simulation and the pure theories fully indicates the ready adaptability of the atomic fluid theories to effective model potentials in complex fluids, and classifies the proposed theoretical approaches as convenient tools for the investigation of complex fluids under the single component macro-fluid approximation.

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.

Investigation of surface properties of liquid transition metals: Surface tension and surface entropy

In the present study, surface properties namely surface tension and surface entropy of liquid transition metals have been reported. The surface entropy of liquid Fe, Co and Ni metals has been investigated using the expression derived by Gosh et al. [R.C. Gosh, A.Z. Ziauddin Ahmed, G.M. Bhuiyan, Eur. Phys. J. B 56 (2007) 177]. To describe interionic interaction the pseudopotential approach has been used and radial distribution functions have been determined from the solution of Ornstein–Zernike integral equation. The calculated values of surface tension and surface entropy agree well with experiment. The present study shows that the expression derived by Gosh et al. leads to a good estimation for the surface entropy.

New free energy density functional and application to core-softened fluid

A new free energy density functional is advanced for general nonhard sphere potentials characterized by a repulsive core with a singular point at zero separation. The present functional is characterized by several features. (i) It does not involve with dividing the potentials into hard-sphere-like contribution and tail contribution in sharp contrast with usual effective hard sphere model+mean field approximation for tail contribution. (ii) It has no recourse to the use of weighted density and is computationally modest; it also does not resort to an equation of state and/or an excess Helmholtz free energy of bulk fluid over a range of density as input. Consequently, all of input information can be obtained by numerical solution of a bulk Ornstein-Zernike integral equation theory (OZ IET). Correspondingly, despite the use of bulk second-order direct correlation function (DCF) as input, the functional is applicable to the subcritical region. (iii) There is no any adjustable parameter associated with the present functional, and an effective hard sphere diameter entering the functional can be determined self-consistently and analytically once the input information, i.e., the second-order DCF and pressure of the coexistence bulk fluid, are obtained by the OZ IET. The present functional is applied to a core-softened fluid subject to varying external fields, and the density distributions predicted by the present functional are more self-consistent with available simulation results than a previous third-order+second-order perturbation density functional theory.

Multidensity integral-equation theory for short diblock hard-sphere–sticky-hard-sphere chains

The multidensity Ornstein-Zernike integral equation theory is applied to study a simple model of hard sphere/sticky hard sphere diblock chains. The multidensity integral equation formalism has been successfully used to model the equilibrium structure and thermodynamic properties of homonuclear chains and shorter dimer fluids; to our knowledge it has not been applied to model diblock chains. In this work, a diblock chain fluids is represented by an m-component equal molar mixture of hard spheres with species 1,2,...,mh and sticky hard spheres with species mh+1,mh+2,...,m. Each spherical particle has two attractive sites A and B except species 1 and m, which have only one site per particle. In the limit of complete association, this mixture yields a system of monodisperse diblock chains. A general solution of this model is obtained in the Percus-Yevick, Polymer Percus-Yevick and ideal chain approximations. Both structural and thermodynamic properties of this model are investigated. From this study, a microphase separation is predicted for relatively short diblock symmetric and asymmetric chains. This microphase separation is enhanced at lower temperature and higher density. When chain length increases, the phase transition changes from a microphase level to a macrophase level. The size of microdomain structure is found to be dependent on total chain length, relative ratio of block lengths, temperature, and density.

Virial coefficients and equation of state of the penetrable sphere model

We study the penetrable sphere (alias square mound) model in the fluid phase by means of the virial expansion, molecular dynamics simulations, and Ornstein-Zernike integral equation. The virial coefficients up to B(8) are expressed as polynomials in the Boltzmann factor with the coefficients calculated by a Monte Carlo integration. New data for pressure and internal energy are obtained by molecular dynamics simulations with attention paid to finite-size errors and properties of the Andersen thermostat. The data and virial coefficients are correlated by a formula for the Helmholtz free energy. We also propose a new closure for the Ornstein-Zernike equation and test several other closures.

Structure and Inter-Diffusion Coefficients of Liquid Na x K1−x Alloys

Structure and atomic transport properties of liquid Na-K alloys are reported. Inter-diffusion coefficients of liquid NaxK1−x, alloys are calculated using scaling law proposed by Samanta et al. following Dzugutov which express the possible relationship between the excess entropy and diffusion coefficient. The interatomic interactions are described from the individual version of the electron-ion potential proposed by Fiolhais et al. The partial pair distribution functions and structure factors are calculated from the solution of Ornstein-Zernike integral equation with Rogers-Young closure. The evaluation with the composition of static structure and inter-diffusion properties are discussed.

Inquiry into thermodynamic behavior of hard sphere plus repulsive barrier of finite height

A bridge function approximation is proposed to close the Ornstein-Zernike (OZ) integral equation for fluids with purely repulsive potentials. The performance of the bridge function approximation is then tested by applying the approximation to two kinds of repulsive potentials, namely, the square shoulder potential and the triangle shoulder potential. An extensive comparison between simulation and the OZ approach is performed over a wide density range for the fluid phase and several temperatures. It is found that the agreement between the two routes is excellent for not too low temperatures and satisfactory for extremely low temperatures. Then, this globally trustworthy OZ approach is used to investigate the possible existence or not of a liquid anomaly, i.e., a liquid-liquid phase transition at low temperatures and negative values of the thermal expansion coefficient in certain region of the phase diagram. While the existence of the liquid anomaly in the square shoulder potential has been previously predicted by a traditional first-order thermodynamic perturbation theory (TPT), the present investigation indicates that the liquid-liquid phase transition disappears in the OZ approach, so that its prediction by the first-order TPT is only an artifact originating from the low temperature inadequacy of the first-order TPT. However, the OZ approach indeed predicts negative thermal expansion coefficients. The present bridge function approximation, free of adjustable parameters, is suitable to be used within the context of a recently proposed nonhard sphere perturbation scheme.