Hypernetted Chain Integral Equation Research Articles

A comparative study on surface tension, diffusion coefficient and shear viscosity coefficient of liquid transition metals

Surface tensions of liquid transition metals of the 3d, 4d and 5d series have been calculated using modified Mayer’s formula involving hard sphere (HS) interaction of liquids derived from first and second order approximations of Percus–Yevick (PY) solutions of isothermal compressibility. Effective HS diameter and packing fraction are key ingredients of this study. To determine them, we have used effective pair potential derived from both Bretonnet–Silbert (BS) pseudopotential model and many body potential obtained from embedded atom method (EAM) in conjunction with variational modified hypernetted chain (VMHNC) integral equation theory of liquid structure. Calculated values of surface tension are used to evaluate transport coefficients such as diffusion and shear viscosity coefficients. Comparing with experimental and simulated data, calculated results suggest that expression derived from second order approximation of PY solution with the combination of EAM potential and VMHNC theory works better than any other combinations for the concerned systems.

Structures and freezing transitions in two-dimensional colloids with tunable repulsive interactions

We have used density functional theory of freezing and hypernetted chain(HNC) integral equation theory to investigate the effects of gradual softening of inverse-power repulsive interaction on the structure and fluid-triangular-solid freezing transition of a one component system of particles confined to a two-dimensional plane and interacting via core-softened potential. We have also performed Monte Carlo simulation to test the accuracy of the pair-structures obtained by HNC theory and to see the equilibrium configuration of particle in different thermodynamic conditions. The pair-structures are found to compare well with those obtained by MC results, particularly in high softness parameter region. The radial distribution given by HNC exhibits a non-monotonous behavior with the change in particle density in addition to the emergence of a new peak at shorter distance. In contrast to HNC, the RDF, obtained by MC simulation for high-softness parameter, display a splitting of the second peak at high density.

Calculation of diffusion coefficient of liquid less simple metals

Diffusion coefficient of liquid less simple metals has been studied using linear trajectory (LT) and small-step (SS) diffusion theories of liquid metals with Carnahan–Stirling (CS) approximation. Friction coefficients, the key ingredients of this study, are calculated using effective pair potentials, v(r), and static structure factors, S(q). For v(r), Brettonet–Silbert pseudopotential model has been chosen whereas for S(q), both variational modified hypernetted chain (VMHNC) integral equation and linearized Weeks–Chandler–Andersen (LWCA) perturbation theory have been employed. Calculated friction coefficients are found comparable to the available simulated data. Inclusion of CS approximation increases the value of friction coefficient obtained from hard part which dominates over rest of the coefficients. Applied LWCA theory provides slightly larger values of diffusion coefficients than those obtained from VMHNC. Comparing with literature values, calculated results suggest that the combination of LWCA theory along with LT and SS theories with CS approximation is excellent for calculation of diffusion coefficient.

A unified derivation of Percus–Yevick and hyper-netted chain integral equations in liquid state theory

A general one-parameter integral equation for liquids is derived that incorporates intrinsically both the famous Percus–Yevick and hyper-netted chain equations as special cases. To examine its accuracy and reliability, the equation is solved self-consistently to find the structure and thermodynamics of Lennard–Jones fluid successfully.

Pair-correlation functions and freezing transition in a 2D binary mixture of ultrasoft colloidal particles interacting via Hertzian potential

Density functional theory (DFT) of freezing and hypernetted chain (HNC) integral equation theory has been used to investigate the nature of pair-correlations and the freezing transitions in a binary mixture of ultrasoft colloidal particles confined to a two-dimensional (2D) plane. The particles of the system interact via Hertzian type model potential with varying softness. The accuracy of the pair-correlations obtained by HNC theory has been tested by performing NVT Monte Carlo (MC) simulation. The partial pair correlation functions of the mixture are found to exhibit non-monotonous behaviour whose nature depends on the softness of the potential and composition. Pair-correlations obtained by solving the HNC integral equation theory have been used in DFT to determine the freezing parameters of fluid-substitutionally disordered triangular solid. The constant pressure phase diagrams in the composition-temperature plane are found to be narrow spindle types.

Effects of interionic pair interactions on atomic transport properties of liquid Al

The effects of parameters used for tuning the interionic pair interaction of the Ashcroft’s pseudopotential model on the temperature-dependent atomic transport properties of liquid Al have been studied employing the Universal Scaling Laws (USLs) and Hard Sphere (HS) theories of liquid metals. Temperature-dependent effective HS diameter and excess entropy are the key ingredients of the applied theories. We have calculated them by using Ashcroft’s empty core (EMC) pseudopotential and the variational modified hypernetted chain (VMHNC) integral equation theory of liquid. Obtained results clearly suggest that the core radius in the interaction plays a major role in determining atomic transport properties of liquid Al for both USLs and HS theory. It is also observed that USLs are more sensitive than HS theories in response to variation of core radius.

Validity of the Stokes−Einstein relation in liquid 3d transition metals for a wide range of temperatures

The validity of Stokes−Einstein (SE) relations obtained from both slip and stick boundary conditions has been investigated for an extending temperature range approximately 250 K from the melting points of liquid Fe, Co and Ni using Dzugutov scaling scheme for transport coefficients. The basic ingredients of Dzugutov scaling scheme are the temperature dependent effective hard-sphere (HS) diameter, σ(T), and the excess entropy, Sex(T). Sex is calculated considering both two (S2) and many body approximations (ST) of liquid atoms. For both ingredients, we have applied variational modified hypernetted chain (VMHNC) integral equation theory in conjunction with both Brettonet−Silbert (BS) pseudopotential and many body potential obtained from embedded atom method (EAM). Calculated transport coefficients using the combination of EAM and ST are in better agreement with available literature values than any other applied combinations. We observe that SE relation holds for the studied representative systems when the combination of EAM potential and ST has been considered.

A comparative study on atomic transport properties of liquid less-simple metals

We have studied the atomic transport properties of liquid less-simple metals namely Bi, Cd, Pb, Sb, Sn and Tl using universal scaling laws and hard sphere (HS) theories of liquid. Effective interionic interaction in metals has been derived from both Brettonet–Silbert (BS) and Ashcroft’s empty core pseudopotential models. For structure calculation, variational modified hypernetted chain integral equation theory has been considered. Excess entropy and effective HS diameter are the basic ingredients of the study. Obtained ingredients are found to be close with experimental data than any other available literature values when BS model has been chosen. Comparing with experimental data, obtained results for transport coefficients suggest that Faber’s HS theory is excellent for diffusion coefficient and is reasonably good for viscosity coefficient along with BS pseudopotential than any other combinations employed in this study. The probable reason may be the hybridization of s and d electrons exists in the concerned systems.

Static and dynamic properties of multi-ionic plasma mixtures.

Complex plasma mixtures with three or more components are often encountered in astrophysics or in inertial confinement fusion (ICF) experiments. For mixtures containing species with large differences in atomic number Z, the modeling needs to consider at the same time the kinetic theory for low-Z elements combined with the theory of strongly coupled plasma for high-Z elements, as well as all the intermediate situations that can appear in multicomponent systems. For such cases, we study the pair distribution functions, self-diffusions, mutual diffusion, and viscosity for ternary mixtures at extreme conditions. These quantities can be produced from first principles using orbital free molecular dynamics at the computational expense of very intensive simulations to reach good statistics. Utilizing the first-principles results as reference data, we assess the merit of a global analytic model for transport coefficients, "pseudo-ions in jellium" (PIJ), based on an isoelectronic assumption (iso-n_{e}). With a multicomponent hypernetted-chain integral equation, we verify the quality of the iso-n_{e}prescription for describing the static structure of the mixtures. This semianalytical modeling compares well with the simulation results and allows one to consider plasma mixtures not accessible to simulations. Applications are given for the mix of materials in ICF experiments. A reduction of a multicomponent mixture to an effective binary mixture is also established in the hydrodynamic limit and compared with PIJ estimations for ICF relevant mixtures.

Structure Factor of EuCl3 Aqueous Solutions via Coupled Molecular Dynamics Simulations and Integral Equations.

Identifying the structure of an aqueous solution is essential to rationalize various phenomena such as crystallization in solution, chemical reactivity, extraction of rare earth elements, and so forth. Despite this, the efforts to describe the structure of an aqueous solution have been hindered by the difficulty to retrieve structural data both from experiments and simulations. To overcome this, first, undersaturated EuCl3 aqueous solutions of concentrations varying from 0.15 to 1.8 mol/kg were studied using X-ray scattering. Second, for the first time, the theoretical X-ray signal of 1.8 mol/kg EuCl3 aqueous solution was simulated, with precise details for the complete range of scattering vectors using coupled molecular dynamics and hypernetted chain integral equations, and satisfactorily compared with the 1.8 mol/kg experimental X-ray scattering signal. The theoretical calculations demonstrate that the experimental structure factor is dominated by Eu3+-Eu3+ correlations.

A generalized Newton iteration for computing the solution of the inverse Henderson problem

ABSTRACT We develop a generalized Newton scheme called IHNC (inverse hypernetted-chain iteration) for the construction of effective pair potentials for systems of interacting point-like particles. The construction is realized in such a way that the distribution of the particles matches a given radial distribution function. The IHNC iteration uses the hypernetted-chain integral equation for an approximate evaluation of the inverse of the Jacobian of the forward operator. In contrast to the full Newton method realized in the Inverse Monte Carlo (IMC) scheme, the IHNC algorithm requires only a single molecular dynamics computation of the radial distribution function per iteration step and no further expensive cross-correlations. Numerical experiments are shown to demonstrate that the method is as efficient as the IMC scheme, and that it easily allows to incorporate thermodynamical constraints.

Pair correlation function and freezing transitions in a two-dimensional system of model ultrasoft colloids

Density functional theory (DFT) of freezing has been used to investigate the freezing transitions in a system of colloidal particles confined to a two-dimensional plane. The particles interact via a model Hertzian type potential of varying softness. The pair-correlation functions (PCFs) needed as input structural information in DFT are calculated by solving hypernetted chain (HNC) integral equation theory. The PCFs thus obtained have been compared with those obtained through experiment and simulations and are found to be in good qualitative agreement. We found that the PCFs are sensitive to the softness of the potential: showing splitting of pair-correlation peak in the harder case and anomalous non-monotonic density dependence in the softer case. Using the common tangent construction method, we have also proposed the fluid-triangular solid phase diagrams in the temperature-density plane. We found that the phase diagram exhibit solid-fluid coexistence region whose thickness decreases with the increasing temperature as well as with increasing softness of the potential. In the temperature and density range of our calculation, DFT fails to produce any reentrance in the phase diagram.

Modeling micro-heterogeneity in mixtures: The role of many body correlations.

A two-component interaction model is introduced herein, which allows us to describe macroscopic miscibility with various modes of tunable micro-segregation, ranging from phase separation to micro-segregation, and is in excellent agreement with structural quantities obtained from simulations and the liquid state hypernetted-chain like integral equation theory. The model is based on the conjecture that the many-body correlation bridge function term in the closure relation can be divided into one part representing the segregation effects, which are modeled herein, and the usual part representing random many body fluctuations. Furthermore, the model allows us to fully neglect these second contributions, thus increasing the agreement between the simulations and the theory. The analysis of the retained part of the many body correlations gives important clues about how to model the many body bridge functions for more realistic systems exhibiting micro-segregation, such as aqueous mixtures.

Investigation of surface tension, viscosity and diffusion coefficients for liquid simple metals

Surface tension of some liquid simple metals has been investigated employing the hard sphere theories which were derived from the first and second order approximations of the isothermal compressibility. Atomic transport properties such as diffusion and shear viscosity are also studied for the same systems. The premier ingredients involved in the theories are effective HS diameter, the packing fraction associated with it, and the effective pair potential. For effective calculations, these ingredients are evaluated by using a local pseudopotential, variational modified hypernetted chain integral equation theory (VMHNC) and also using the linearized Weeks-Chandler and Andersen thermodynamic perturbation theory. Calculated results, when compared with the available experimental data and other theoretical values, reveal that the first order approximation theory along with VMHNC predicts much better results than those of the other theoretical values for all concerned systems except for Pb and Al. The cause of discrepancy in the results for Al and Pb is also discussed.

Integral equation theory based direct and accelerated systematic coarse-graining approaches.

Coarse-grained (CG) molecular dynamics (MD) simulations have become popular for investigating systems on multiple length and time scales ranging from atomistic to mesoscales. In CGMD, several atoms are mapped onto a single CG bead and the effective interactions between CG beads are determined. Iterative coarse-graining methods, such as iterative Boltzmann inversion (IBI), are computationally expensive and can have convergence issues. In this paper, we present a direct and computationally efficient theoretical procedure for coarse-graining based on the Ornstein-Zernike (OZ) and hypernetted chain (HNC) integral equation theory. We demonstrate the OZ-HNC-based CG method by coarse-graining a bulk water system, a water-methanol mixture system, and an electrolyte system. We show that the accuracy of the CG potentials obtained from the OZ-HNC-based coarse-graining is comparable to iterative systematic coarse-graining methods. Furthermore, we show that the CG potentials from OZ-HNC can be used to reduce the number of iterations and hence the computational cost of the iterative systematic coarse-graining approaches, like IBI and relative entropy minimization.

Coexistence of low and high overlap phases in a supercooled liquid: An integral equation investigation.

The pair structure, free energy, and configurational overlap order parameter Q of an annealed system of two weakly coupled replicas of a supercooled "soft sphere" fluid are determined by solving the hypernetted-chain (HNC) and self-consistent Rogers-Young (RY) integral equations over a wide range of thermodynamic conditions ρ (number-density), T (temperature), and inter-replicas couplings ε12. Analysis of the resulting effective (or Landau) potential W(ρ,T; Q) and of its derivative with respect to Q confirms the existence of a "precursor transition" between weak and strong overlap phases below a critical temperature Tc well above the temperature To of the "ideal glass" transition observed in the limit ε12→0. The precursor transition is signalled by a loss of convexity of the potential W(Q) and by a concomitant discontinuity of the order parameter Q just below Tc, which crosses over to a mean-field-like van der Waals loop at lower temperatures. The HNC and RY equations lead to the same phase transition scenario, with quantitative differences in the predicted temperatures Tc and To.

A thermodynamic self-consistent theory of asymmetric hard-core Yukawa mixtures

We perform structural and thermodynamic calculations in the framework of the modified hypernetted chain (MHNC) integral equation closure to the Ornstein–Zernike equation for binary mixtures of size-different particles interacting with hard-core Yukawa pair potentials. We use the Percus–Yevick (PY) bridge functions of a binary mixture of hard-sphere (HSM) particles. The hard-sphere diameters of the PY bridge functions of the HSM system are adjusted so to achieve thermodynamic consistency between the virial and compressibility equations of state. We show the benefit of thermodynamic consistency by comparing the MHNC results with the available computer simulation data reported in the literature, and we demonstrate that the self-consistent thermodynamic theory provides a better reproduction of the simulation data over other microscopic theories.

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.

Cluster formation in binary charge-stabilized colloidal suspensions confined to a two-dimensional plane

Hypernetted chain (HNC) integral equation theory has been used to study the structural features of binary charged stabilized colloidal suspensions confined to a two-dimensional plane. The particles interact via purely repulsive Yukawa intermolecular potential, the inverse screening length scaled by the average distance between strongly interacting components of the mixture (dimensionless screening parameter) being 1, 3 and 5. Results of HNC theory for one-component systems are found to be in very good agreement with that of simulation, in the parameter range of our study. Binary Yukawa systems with dimensionless screening parameters 1 and 3 are found to exhibit diffuse clusters of the weakly interacting particles, marked by the emergence of a cluster peak in the corresponding partial structure factor curves. No cluster peak is found in the system with the screening parameter 5. For the entire range of mixture parameters, the strongly interacting particles remain homogeneously distributed.

An investigation of the liquid to glass transition using integral equations for the pair structure of coupled replicae.

Extensive numerical solutions of the hypernetted-chain (HNC) and Rogers-Young (RY) integral equations are presented for the pair structure of a system of two coupled replicae (1 and 2) of a "soft-sphere" fluid of atoms interacting via an inverse-12 pair potential. In the limit of vanishing inter-replica coupling ɛ12, both integral equations predict the existence of three branches of solutions: (1) A high temperature liquid branch (L), which extends to a supercooled regime upon cooling when the two replicae are kept at ɛ12 = 0 throughout; upon separating the configurational and vibrational contributions to the free energy and entropy of the L branch, the Kauzmann temperature is located where the configurational entropy vanishes. (2) Starting with an initial finite coupling ɛ12, two "glass" branches G1 and G2 are found below some critical temperature, which are characterized by a strong remnant spatial inter-replica correlation upon taking the limit ɛ12 → 0. Branch G2 is characterized by an increasing overlap order parameter upon cooling, and may hence be identified with the hypothetical "ideal glass" phase. Branch G1 exhibits the opposite trend of increasing order parameter upon heating; its free energy lies consistently below that of the L branch and above that of the G2 branch. The free energies of the L and G2 branches are found to intersect at an alleged "random first-order transition" (RFOT) characterized by weak discontinuities of the volume and entropy. The Kauzmann and RFOT temperatures predicted by RY differ significantly from their HNC counterparts.