Ornstein-Zernike Equation Research Articles

Thermodynamic consistency by a modified Perkus–Yevick theory using the Mittag-Leffler function

Closure relations that satisfies thermodynamical consistencies are very important, for it implies physical consistent Ornstein–Zernike equation solutions. The method proposed by Percus (1962), which uses the functional Taylor expansion to obtain closure relations, is presented along with the particular generating functional considered to retrieve the Percus–Yevick (PY) approximation. The main assumption for this particular case, which takes into account the low density limit, is discussed. Based on this argument, it was proposed a modified generating functional with no prior considerations about the system density. As an example, the two-parameters Mittag-Leffler function was used. Since it generalizes the exponential function, the parameters determined at low densities retrieves the PY approximation. For higher densities it was possible to obtain parameters which lead to pressure consistent results, in good agreement with those found in literature.

Construction of a composite-sphere model for molecules of tetrahedral symmetry

We propose a ‘composite sphere’ model for molecules of tetrahedral symmetry. In particular, we choose as an example the hydrocarbon neopentane. We disassemble the molecule into five moieties: four methyl groups (species A) and one carbon atom (species B). By imparting pair potentials of spherical symmetry to the A–A (hard sphere type), B–B (hard sphere type), and A–B (square-well type) pairs, we are able to reconstitute the pentamer structure via Monte Carlo simulation. This is demonstrated for four density states: ρσ 3 = 0.03, 0.10, 0.15, and 0.20. We apply two criteria in assessing the per cent formation of pentamers: namely the proximity rule and the energy rule. The proximity rule gives 57% formation of pentamers, while the energy rule is able to give 99% pentamers. The structures are also calculated via the spherical Ornstein-Zernike integral equations with the Percus-Yevick (PY) closure. It is found that the PY closure is inadequate for predicting quantitatively the MC radial distribution functions. The root cause of the lack of agreement is attributed to the improper application of the isotropic version of the Ornstein-Zernike integral equations to a fluid that has become anisotropic.

Scaling of Phase Diagram and Critical Point Parameters in Liquid-Vapour Phase Transition of Metallic Fluids

The first objective of this paper is to investigate the scaling behavior of liquid-vapor phase transition in FCC and BCCmetals starting from the zero-temperature four-parameter formula for cohesive energy. The effective potentials between the atoms in the solid are determined while using lattice inversion techniques as a function of scaling variables in the four-parameter formula. These potentials are split into repulsive and attractive parts, as per the Weeks–Chandler–Anderson prescription, and used in the coupling-parameter expansion for solving the Ornstein–Zernike equation that was supplemented with an accurate closure. Thermodynamic quantities obtained via the correlation functions are used in order to obtain critical point parameters and liquid-vapor phase diagrams. Their dependence on the scaling variables in the cohesive energy formula are also determined. An equally important second objective of the paper is to revisit coupling parameter expansion for solving the Ornstein–Zernike equation. The Newton–Armijo non-linear solver and Krylov-space based linear solvers are employed in this regard. These methods generate a robust algorithm that can be used to span the entire fluid region, except very low temperatures. The accuracy of the method is established by comparing the phase diagrams with those that were obtained via computer simulation. The avoidance of the ’no-solution-region’ of the Ornstein-Zernike equation in coupling-parameter expansion is also discussed. Details of the method and complete algorithm provided here would make this technique more accessible to researchers investigating the thermodynamic properties of one component fluids.

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.

Contrasting effect of 1-butanol and 1,4-butanediol on the triggered micellar self-assemblies of C16-type cationic surfactants.

The self-assembly in aqueous solutions of three quaternary salt-based C16-type cationic surfactants with different polar head groups and identical carbon alkyl chain viz., cetylpyridinium bromide (CPB), cetyltrimethylammonium tosylate (CTAT), and cetyltriphenylphosphonium bromide (CTPPB) in the presence of 1-butanol (BuOH) and 1,4-butanediol (BTD) was investigated using tensiometry, 2D-nuclear Overhauser enhancement spectroscopy (2D-NOESY) and small angle neutron scattering (SANS) techniques. The adsorption parameters and micellar characteristics evaluated at 303.15 K distinctly showed that BuOH promotes the mixed micelle formation while BTD interfered with the micellization phenomenon. The SANS data fitted using an ellipsoid (as derived by Hayter and Penfold using the Ornstein-Zernike equation and the mean spherical approximation) and wormlike micellar models offered an insight into the micelle size/shape and aggregation number (Nagg) in the examined systems. The evaluated descriptors presented a clear indication of the morphology transition in cationic micelles as induced by the addition of the two alcohols. We also offer an investigation into the acceptable molecular interactions governing the differences in micelle morphologies, using the non-invasive 2D-NOESY technique and molecular modeling. The experimental observations elucidated from computational simulation add novelty to this work. Giving an account to the structural complexity in the three cationic surfactants, the molecular dynamics (MD) simulation was performed for CPB micelles in an aqueous solution of alcohols that highlighted the micelle solvation and structural transition, which is further complemented in terms of critical packing parameter (PP) for the examined systems.

Integral equation theory for mixtures of spherical and patchy colloids. 2. Numerical results.

Thermodynamic properties and structure of binary mixtures of patchy and spherical colloids are studied using a recently developed theory [Y. V. Kalyuzhnyi, et al., Soft Matter, 2020, 16, 3456]. The theory is based on a solution of the multidensity Ornstein-Zernike equation and provides completely analytical expressions for the structure factors of these systems and for all their major thermodynamical quantities. The considered mixtures are made up of particles of different size and with a different number of patches. A set of molecular simulation data has been generated to enable a systematic comparison and to access thus accuracy of the theoretical predictions. In general, the predictions of the theory appear to be in good agreement with computer simulation data. For the models with a lower number of patches (np = 1, 2) the theoretical results show very good accuracy. Less accurate are the predictions for the four-patch versions of the model. While theoretical results for the radial distribution functions are, generally, relatively accurate for all the models, results for thermodynamics deteriorate with increasing concentration of the spherical colloids. Possible ways to improve the theory are briefly outlined.

Radial Distribution Function for a Hard Sphere Liquid: A Modified Percus-Yevick and Hypernetted-Chain Closure Relations

Establishment of the radial distribution function by solving the Ornstein-Zernike equation is still an important problem, even more than a hundred years after the original paper publication. New strategies and approximations are common in the literature. A crucial step in this process consists in defining a closure relation which retrieves correlation functions in agreement with experiments or molecular simulations. In this paper, the functional Taylor expansion, as proposed by J. K. Percus, is applied to introduce two new closure relations: one that modifies the Percus‑Yevick closure relation and another one modifying the Hypernetted-Chain approximation. These new approximations will be applied to a hard sphere system. An improvement for the radial distribution function is observed in both cases. For some densities a greater accuracy, by a factor of five times compared to the original approximations, was obtained.

Integral equation study of the effects of rotational degrees of freedom on properties of the Mercedes–Benz water model

We developed an analytical theory for studying of rotational degrees of freedom of a simple model of liquid water. Wertheim's integral equation theory (IET) for associative liquids was applied to the Mercedes-Benz (MB) model, which is among the simplest models of water. The MB water molecules are modeled as 2-dimensional Lennard-Jones disks with three hydrogen bonding arms arranged symmetrically, resembling the MB logo. IET is based on the orientationally averaged version of the Ornstein-Zernike equation. In varying of rotational degrees of freedom we use different averaging then used before. By holding one of the temperatures constant and varying the other one, we investigate the effect of faster motion in the corresponding degrees of freedom on the properties of the simple water model and how well IET reproduce computer simulation results. The pair correlation function of the model water at high rotational or translation temperatures is correctly predicted. IET results are in good agreement with the Monte Carlo values of the pressure, energy, heat capacity. A major advantage of these theories is that they require orders of magnitude less computer time than the Monte Carlo simulations.

Density functional theory of fluid-solid phase transition in a two-dimensional system of superparamagnetic colloids in tilted magnetic field

We have used density functional theory (DFT) of freezing to investigate the fluid-solid phase transition in a system of super-paramagnetic colloidal particles confined to a two dimensional plane and exposed to an external magnetic field which is tilted from the direction parallel to the plane. Magnetic field induces magnetic moment in the particles which are directed along the field. Because of the tilted induced magnetic moments, particles interact via anisotropic repulsive dipole-dipole interaction potential. We solve the Ornstein-Zernike(OZ)equation within the hypernetted chain (HNC) approximation to obtain the pair-correlation functions (PCF), which are used as structural input in DFT. All the pair functions were expanded in a suitable orthogonal basis and the expansion coefficients were determined by using the method of Fries and Patey [32] which leads to an unphysical negative contact value of radial distribution functions. We have employed a method similar to that of Hansen and Hayter [35], to resolve this issue. The accuracy of the resulting PCFs are tested by performing NVT Monte Carlo simulation. We found that for the tilt angle of the external field in the range 900 to 750, the fluid freezes into triangular solid, beyond which DFT fails to stabilize the solids considered in this work.

Indirect Solution of Ornstein-Zernike Equation Using the Hopfield Neural Network Method

Microscopic information, such as the pair distribution and direct correlation functions, can be obtained from experimental data. From these correlation functions, thermodynamical quantities and the potential interaction function can be recovered. Derivations of Ornstein-Zernike equation and Hopfield Neural Network method are given first, as a theoretical background to follow the present work. From these two frameworks, structural information, such as the radial distribution (g(r)) and direct correlation (C(r)) functions, were retrieved from neutron scattering experimental data. The problem was solved considering simple initial conditions, which does not require any previous information about the system, making it clear the robustness of the Hopfield Neural Network method. The pair interaction potential was estimated in the Percus-Yevick (PY) and hypernetted chain (HNC) approximations and a poor agreement, compared with the Lennard-Jones 6-12 potential, was observed for both cases, suggesting the necessity of a more accurate closure relation to describe the system. In this sense, the Hopfield Neural Network together with experimental information provides an alternative approach to solve the Ornstein-Zernike equations, avoiding the limitations imposed by the closure relation.

Integral equation theory of thermodynamics, pair structure, and growing static length scale in metastable hard sphere and Weeks-Chandler-Andersen fluids.

We employ the Ornstein-Zernike integral equation theory with the Percus-Yevick (PY) and modified-Verlet (MV) closures to study the equilibrium structural and thermodynamic properties of metastable monodisperse hard sphere and continuous repulsion Weeks-Chandler-Andersen (WCA) fluids under density and temperature conditions where the system is strongly overcompressed or supercooled, respectively. The theoretical results are compared to crystal-avoiding simulations of these dense monodisperse model one-component fluids. The equation of state (EOS) and dimensionless compressibility are computed using both the virial and compressibility routes. For hard spheres, the MV-based virial route EOS and dimensionless compressibility are in very good agreement with simulation for all packing fractions, much better than the PY analogs. The corresponding MV-based predictions for the static structure factor are also very good. The amplitude of density fluctuations on the local cage scale and in the long wavelength limit, and three technically different measures of the density correlation length, are studied with both closures. All five properties grow in a roughly exponential manner with density in the metastable regime up to packing fractions of 58% with no sign of saturation. The MV-based results are in good agreement with our crystal-avoiding simulations. Interestingly, the density dependences of long and short wavelength quantities are closely related. The MV-based theory is also quite accurate for the thermodynamics and structure of supercooled monodisperse WCA fluids. Overall our findings are also relevant as critical input to microscopic theories that relate the equilibrium pair correlation function or static structure factor to dynamical constraints, barriers, and activated relaxation in glass-forming liquids.

Electrostatic interactions and stability of dusty plasmas and the multicomponent Ornstein–Zernike equation

Using the Ornstein–Zernike integral fluid equation for multi-component plasma, the dielectric properties and thermodynamical stability of dusty plasmas are studied. For the most non-ideal dust plasma subsystem, a transition to the one-component approximation is carried out. It is shown that the effective pseudopotential for determining the correlation functions in the selected subsystem should not include the contribution of this subsystem to the screening constant but also take into account the condition of total plasma quasineutrality. It is demonstrated that when the coupling parameter of the dust subsystem is smaller than unity, Γ00 < 1, the interaction potential between the charged plasma particles is fairly well described by the Debye potential with a full screening constant. For Γ00 > 1, the static dielectric function in the long wavelength domain becomes negative, and this domain expands when Γ00 increases. This leads to the appearance of attraction of particles with charges of the same sign and repulsion of particles with charges of the opposite sign. In this case, both the total pressure and the isothermal compressibility in the entire studied range of the coupling parameter Γ00 < 250 remain positive, but the isothermal compressibility of the dust subsystem becomes negative at Γ00 ≈ 2 within the studied range of variation of the plasma parameters. The sign of the derivative of the chemical potential with respect to the total number of dust particles, the positiveness of which is the third condition for the thermodynamic stability, is shown to coincide with the sign of the isothermal compressibility of the dust subsystem. Therefore, it is concluded that the equilibrium dusty plasma at Γ00 > 2 is thermodynamically unstable.

Temperature expansions in the square-shoulder fluid. I. The Wiener-Hopf function.

We investigate the spatial structure of dense square-shoulder fluids. To this end, we derive analytical perturbative solutions of the Ornstein-Zernike equation in the low- and high-temperature limits as expansions around the known hard sphere solutions. We then discuss the suitability of perturbative approaches in relation to the Ornstein-Zernike equation. Our analytical expressions are shown to reproduce reasonably well numerical data in the appropriate regimes.

Temperature dependence of the specific volume of Lennard-Jones potential and applying in case of polymers and other materials

Based on solutions of the Ornstein–Zernike equation, we derive analytical formula for the specific volume of polymers interacting through Lennard-Jones potential. We use mean spherical approximation assuming that the system is of low density, homogeneous, isotropic and composed of one component. The specific volumes of some polymers in addition to the minimum volumes of Lennard-Jones potential and the specific volumes of proteinogenic amino acids: A, R, N, D, C, Q, E, G, H, I, L, K, M, F, P, S, T, W, Y and V, are calculated according to that formula. We show that the simple formula we derive is reliable and agrees well with results obtained from experimental and fitting reported in other studies. We believe that it can be used for many systems described by Lennard-Jones potential such as biological systems, soft matter systems, large molecules such as polymers and inert gases fluids like liquid argon.

Electrostatic contribution to colloidal solvation in terms of the self-energy-modified Boltzmann distribution.

Electrostatic interactions make a large contribution to solvation free energy in ionic fluids such as electrolytes and colloidal dispersions. The electrostatic contribution to solvation free energy has been ascribed to the self-energy of a charged particle. Here we apply a variational field theory based on lower bound inequality to the inhomogeneous fluids of one-component charged hard-spheres, thereby verifying that the self-energy is given by the difference between the total correlation function and direct correlation function. Based on the knowledge of the liquid state theory, the self-energy specified in this study not only relates a direct correlation function to the Gaussian smearing of each charged sphere, but also provides the electrostatic contribution to solvation free energy that shows good agreement with simulation results. Furthermore, the Ornstein-Zernike equation leads to a set of generalized Debye-Hückel equations reflecting the Gaussian distributed charges.

Unusual geometric percolation of hard nanorods in the uniaxial nematic liquid crystalline phase.

We investigate by means of continuum percolation theory and Monte Carlo simulations how spontaneous uniaxial symmetry breaking affects geometric percolation in dispersions of hard rodlike particles. If the particle aspect ratio exceeds about 20, percolation in the nematic phase can be lost upon adding particles to the dispersion. This contrasts with percolation in the isotropic phase, where a minimum particle loading is always required to obtain system-spanning clusters. For sufficiently short rods, percolation in the uniaxial nematic mimics that of the isotropic phase, where the addition of particles always aids percolation. For aspect ratios between 20 and infinity, but not including infinity, we find reentrance behavior: percolation in the low-density nematic may be lost upon increasing the amount of nanofillers but can be regained by the addition of even more particles to the suspension. Our simulation results for aspect ratios of 5, 10, 20, 50, and 100 strongly support our theoretical predictions, with almost quantitative agreement. We show that a different closure of the connectedness Ornstein-Zernike equation, inspired by scaled particle theory, is as least as accurate in predicting the percolation threshold as the Parsons-Lee closure, which effectively describes the impact of many-body direct contacts.

Quantum statistical approach for ionization potential depression in multi-component dense plasmas

Theoretical modeling of ionization potential depression (IPD) and the related ionization equilibrium in dense plasmas, in particular, in warm/hot dense matter, represents a significant challenge due to ionic coupling and electronic degeneracy effects. Based on the dynamical structure factor (SF), a quantum statistical model for IPD in multi-ionic plasmas is developed, where quantum exchange and dynamical correlation effects in plasma environments are consistently and systematically taken into account in terms of the concept of self-energy. Calculations for IPD values of different chemical elements are performed with the electronic and ionic SFs. The ionic SFs are determined by solving the Ornstein–Zernike equation in combination with the hypernetted-chain closure relation. As a further application of our approach, we present results for the charge state distribution of aluminum plasmas at several temperatures and densities through solving the coupled Saha equations.

Long-range correlations in the statistical theory of fluid criticality

Using the approach formulated in the previous papers of the author, a consistent procedure is developed for calculating non-classical asymptotic power terms in the total correlation function hc(r) of a “critical” fluid. Analyzing the Ornstein-Zernike equation with taking account of the contribution ˜hc4(r) in the direct correlation function cc(r) allows us to find, for the first time, the values of transcendental exponents n′=1.73494... and n″=2.26989...which determine the asymptotic terms next to the leading one in hc(r). It is shown that already the simplest approximation based on only two asymptotic terms, ˜r−6/5(it was found earlier) and ˜r−n′, leads to the functions hc(r) and cc(r), which agree (at least, qualitatively) with the corresponding correlation functions of the Lennard-Jones fluid (argon) in the near-critical state. The obtained results open a way for consistent theoretical interpretation of the experimental data on the critical characteristics of real substances. Both the theoretical arguments and analysis of published data on the experimentally measured critical exponents of real fluids lead to the conclusion that the known assumption of the similarity of the critical characteristics of the Ising model and the fluid in the vicinity of critical point (the “universality hypothesis”) should be questioned.

Microscopic determination of correlations in the fluid interfacial region in the presence of liquid-gas asymmetry.

In a recent article, we showed how the properties of the density-density correlation function and its integral, the local structure factor, in the fluid interfacial region, in systems with short-ranged forces, can be understood microscopically by considering the resonances of the local structure factor [A. O. Parry and C. Rascón, Nat. Phys. 15, 287 (2019)NPAHAX1745-247310.1038/s41567-018-0361-z]. Here, we illustrate, using mean-field square-gradient theory and the more microscopic Sullivan density functional model, and how this approach generalizes when there is liquid-gas asymmetry, i.e., when the bulk correlation lengths of the coexisting liquid and gas phases are different. In particular, we are able to express the correlation function exactly as a simple average of contributions arising from two effective Ising-symmetric systems referred to as the symmetric gas and symmetric liquid. When combined with our earlier results, this generates analytical approximations for the correlation function and the local structure factor, which are near indistinguishable from the numerical solution to the Ornstein-Zernike equations over the whole range of wave vectors. Our results highlight how asymmetry affects the correlation function structure and describes the crossover from a long-ranged Goldstone mode to short-ranged properties determined by the local density as the wave vector increases.

Analytical formula of heat capacity in soft matter materials using Lennard-Jones potential

We derive an analytical formula of heat capacity based on the solutions of Ornstein–Zernike equation of Lennard-Jones potential within the Mean Spherical Approximation (MSA) assuming that the system is of low density, homogeneous, isotropic and composed of one component. We apply the derived formula to find the heat capacity at constant volume and the heat capacity at constant pressure for a number of polymers namely: Polycarbonate, Polymethylmethacrylate, Polystyrene, Polyacrylonitrile, Polyvinyl acetate, Polyvinyl propionate, Polybutylacrylate and Polyisobutylmethacrylate polymers. We also calculate the heat capacity at constant volume and the heat capacity at constant pressure for some hydrocarbons.