Hypernetted Chain Theory Research Articles

A Gaussian field approach to the solvation of spherical ions in electrolyte solutions.

In this work, the electrostatic response of an electrolyte solution to a spherical ion is studied with a Gaussian field theory. In order to capture the ionic correlation effect in concentrated solutions, the bulk dielectric response function is described by a two-Yukawa response function. The modified response function of the solution is solved analytically in the spherical geometry, from which the induced charge density and the electrostatic energy are also derived analytically. Comparisons with results for small ions in electrolyte solutions from the hyper-netted chain theory demonstrate the validity of the Gaussian field theory.

Molecular dynamics and integral equation study of the structure and dynamics of solid and liquid magnesium phosphide

ABSTRACTThe static structure and self-ionic transport in solid and molten magnesium phosphide (Mg3P2) are investigated by means of the molecular dynamics simulation and the hypernetted-chain theory of liquids using a newly developed semiempirical pairwise potential. Parameters of the potential were fitted to the lattice constant and bulk modulus, and then it was tested in NVE ensemble simulation at 300 K at which X-ray powder diffraction pattern was correctly reproduced. The static structure and the dynamics of self-ion transport were investigated in NPT simulations between 300 and 1500 K. The temperature evolution of the radial distribution functions, coordination numbers, mean square displacements, self-diffusion coefficients and solid–liquid transition were established at solid and liquid phases that will be informative for the thermoelectronic, optoelectronic and energy storage applications of the magnesium phosphide.

Microphase versus macrophase separation in the square-well-linear fluid: A theoretical and computational study.

Due to the presence of competing interactions, the square-well-linear fluid can exhibit either liquid-vapor equilibrium (macrophase separation) or clustering (microphase separation). Here we address the issue of determining the boundary between these two regimes, i.e., the Lifshitz point, expressed in terms of a relationship between the parameters of the model. To this aim, we carry out Monte Carlo simulations to compute the structure factor of the fluid, whose behavior at low wave vectors accurately captures the tendency of the fluid to form aggregates or, alternatively, to phase separate. Specifically, for a number of different combinations of attraction and repulsion ranges, we make the system go across the Lifshitz point by increasing the strength of the repulsion. We use simulation results to benchmark the performance of two theories of fluids, namely, the hypernetted chain (HNC) equationand the analytically solvable random phase approximation (RPA); in particular, the RPA theory is applied with two different prescriptions as for the direct correlation function inside the core. Overall, the HNC theory proves to be an appropriate tool to characterize the fluid structure and the low-wave-vector behavior of the structure factor is consistent with the threshold between microphase and macrophase separation established through simulation. The structural predictions of the RPA theory turn out to be less accurate, but this theory offers the advantage of providing an analytical expression of the Lifshitz point. Compared to simulation, both RPA schemes predict a Lifshitz point that falls within the macrophase-separation region of parameters: in the best case, barriers roughly twice higher than predicted are required to attain clustering conditions.

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.

Minimum-modified Debye-Hückel theory for size-asymmetric electrolyte solutions with moderate concentrations

A minimum-modified Debye-Hückel (DH) theory for electrolytes with size asymmetry is developed. Compared with the conventional DH theory, the minimum-modified DH theory only introduces an extra surface charge density to capture the electrostatic effect of the size asymmetry of the electrolytes and hence facilitates a boundary element method for electrostatic potential calculation. This theory can distinguish the electrostatic energies and excess chemical potentials of ions with the same sizes but opposite charges, and is applied to a binary primitive electrolyte solution with moderate electrostatic coupling. Compared with the hyper-netted chain theory, the validity of this modified DH theory demonstrates significant improvement over the conventional DH theory.

A cavity formation energy formula for hard spheres in simple electrolyte solutions.

A formula for cavity formation energy of a hard sphere in restricted primitive electrolyte solutions is derived based on the integral equation theory. Specifically, the contact values of radial distribution functions between the hard sphere and the ionic species, determined analytically from the first-order mean spherical approximation theory, are used to evaluate the cavity formation energy. In the large solute-size limit, the scaling relation of the cavity formation energy further leads to an analytical expression for the surface tension of the electrolyte solution near a curved interface. Our theory is applied to hard spheres immersed in restricted primitive electrolyte solutions, where the good agreement of the cavity formation energy with the hyper-netted chain theory demonstrates the accuracy of our theory.

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.

Verification of the Stokes-Einstein relation in liquid noble metals over a wide range of temperatures

The validity of Stokes-Einstein (SE) relations in liquid noble metals namely Cu, Ag and Au over a wide temperature range has been studied using Dzugutov's scaling scheme and Faber's hard-sphere (HS) theory of transport coefficients. Basic ingredients of those theories are the temperature-dependent effective HS diameter and excess entropy. To determine them, we have applied variational modified hypernetted chain theory in conjunction with effective interionic interaction derived from embedded atomic method (EAM) with both temperature-dependent and independent adjustable parameters α and b. Obtained ingredients using α(T) and b(T) are close to the available experimental data than those obtained using fixed α and b with an empirical relation. Calculated transport coefficients are in good agreement with experimental data when scaling scheme with α(T) and b(T) has been considered. Simplified SE relations from transport coefficients hold when scaling scheme has been applied with α(T) and b(T) and lies around the slip boundary line.

Solvation effects on diffusion processes of a macromolecule: Accuracy required for radial distribution function to calculate diffusion coefficient.

We investigate the dependence of the diffusion coefficient of a large solute particle on the solvation structure around a solute. The diffusion coefficient of a hard-sphere system is calculated by using a perturbation theory of large-particle diffusion with radial distribution functions around the solute. To obtain the radial distribution function, some integral equation theories are examined, such as the Percus-Yevick (PY), hypernetted-chain (HNC), and modified HNC theories using a bridge function proposed by Kinoshita (MHNC) closures. In one-component solvent systems, the diffusion coefficient depends on the first-minimum value of the radial distribution function. The results of the MHNC closure are in good agreement with those of calculation using the radial distribution functions of Monte Carlo simulations since the MHNC closure very closely reproduces the radial distribution function of Monte Carlo simulations. In binary-solvent mixtures, the diffusion coefficient is affected by the larger solvent density distribution in the short-range part, particularly the height and sharpness of the first peak and the depth of the first minimum. Since the HNC closure gives the first peak that is higher and sharper than that of the MHNC closure, the calculated diffusion coefficient is smaller than the MHNC closure result. In contrast, the results of the PY closure are qualitatively and quantitatively different from those of the MHNC and HNC closures.

Temperature-dependent structure of molten eutectic Cu37Sb63

Quoted measurements of viscosity and our results of electrical resistivity experiments indicate two anomalous changes of physical properties take place in molten eutectic Cu37Sb63. Effective pair inter-ionic potentials evaluated using the hyper-netted chain theory further confirms the presence of two discontinuous structural transitions within the ranges from 1273K to 1073K and from 1073K to 903K, respectively. Moreover, the result of electrical resistivity experiments suggests the two structural transitions for molten Cu37Sb63 are reversible.

Osmotic pressure and transport coefficient in ultrafiltration: A Monte Carlo study using quantum surface charges

The calculation of the Osmotic Pressure and the Diffusion Coefficient, characterizing the cake layer developing during the Ultrafiltration (UF) of Bovine Serum Albumin (BSA), through a Multiscale Model based on Quantum Mechanics (QM) and Monte Carlo methods (MC) was the aim of this work. From the ab-initio results described in previous works, the distribution of the BSA surface charges was used. A home-made Metropolis MC algorithm, aimed at simulating the formation of BSA loose or concentrated layers during membrane operations, was also implemented. In such a MC algorithm, a DLVO energy calculation methodology of the adsorbed system was developed. Different MC simulations and Hypernetted Chain theory (HNC) calculations were performed so to determine both the Osmotic Pressure and the Diffusion Coefficient of BSA according to the Chun and Bowen approach, thus allowing a comparison between the calculated values of osmotic pressure and a set of experimental data taken from the literature. Such a comparison showed a good agreement between the simulated and the experimental results.

Ordering and diffusion in liquid magnesium antimonide (Mg3Sb2) from hypernetted-chain theory and molecular dynamics simulation

The static structure and ionic transport in molten Mg3Sb2 was investigated by means of the hypernetted-chain theory of liquids (HNC) and the molecular dynamics simulation (MD) using a semi-empirical pairwise potential together with the liquid-state density that was estimated from the procedure of finding minimum energy. Magnesium sublattice melts into a more disordered liquid state with a higher mobility compared to antimony. Highly mobile magnesium ions indicate the cluster distribution over a range. Tendency of the pre-peak at low k region (k ≅ 1.7A-1) in the pair structure factor SNN(k) may be the result of the one of the n-fold Voronoi coordination polyhedral clusters that dominates over the other clusters. The results also suggest that the pair structure factor of the less mobile ion influences the low- k peak in SNN(k) that might be a characteristic feature of the systems whose components have markedly different self-ionic diffusion.

On the structure, property, and phase behaviour of the symmetric Yukawa mixtures: testing of the consistent integral equation theories

ABSTRACTYukawa mixtures are important microscopic models of liquids. In order to obtain their properties, we must have accurate and consistent theories, without the issues arising from the multi-valued pressures and chemical potentials due to inconsistencies in the theory. This work tests two self-consistent integral equation theories (IETs): the modified hypernetted-chain (MHNC) theory and the zero-separation (ZSEP) based theory. These equations, by construction, automatically satisfy the pressure consistency the Gibbs-Duhem relation, as well as the zero-separation theorem. To verify, new Monte-Carlo (MC) simulations are carried out for temperatures 0.7 < T* < 1.1 and densities 0.4 < ρ* < 0.5. The Yukawa interactions uij are symmetric (u11 = u22). The unlike u12 is made weaker than the like-type by a factor α < 1. The structures, exemplified by the radial distribution functions, are accurately reproduced by both MHNC and ZSEP. The internal energies are predicted to within 1% of the MC data, the pressures to within 4%; and the chemical potentials to less than 3%. The vapour-liquid phase envelope is determined for a typical mixture. In this case the performance of both IETs is also shown to be satisfactory.

Pair correlation functions of strongly coupled two-temperature plasma

Using molecular dynamics simulations, we perform the first direct tests of three proposed models for the pair correlation functions of strongly coupled plasmas with species of unequal temperature. The models are all extensions of the Ornstein–Zernike/hypernetted-chain theory used to good success for equilibrium plasmas. Each theory is evaluated at several coupling strengths, temperature ratios, and mass ratios for a model plasma in which the electrons are positively charged. We show that the model proposed by Seuferling et al. [Phys. Rev. A 40, 323 (1989)] agrees well with molecular dynamics over a wide range of mass and temperature ratios, as well as over a range of coupling strength similar to that of the equilibrium hypernetted-chain (HNC) theory. The SVT model also correctly predicts the strength of interspecies correlations and exhibits physically reasonable long-wavelength limits of the static structure factors. Comparisons of the SVT model with the Yukawa one-component plasma (YOCP) model are used to show that ion-ion pair correlations are well described by the YOCP model up to Γe≈1, beyond which it rapidly breaks down.

Theory and computer simulation of hard-core Yukawa mixtures: thermodynamical, structural and phase coexistence properties

We report extensive calculations, based on the modified hypernetted chain (MHNC) theory, on the hierarchical reference theory (HRT), and on Monte Carlo simulations, of thermodynamical, structural and phase coexistence properties of symmetric binary hard-core Yukawa mixtures (HCYM) with attractive interactions at equal species concentration. The obtained results are throughout compared with those available in the literature for the same systems. It turns out that the MHNC predictions for thermodynamic and structural quantities are quite accurate in comparison with the MC data. The HRT is equally accurate for thermodynamics, and slightly less accurate for structure. Liquid-vapor (LV) and liquid–liquid (LL) consolute coexistence conditions as emerging from simulations, are also highly satisfactorily reproduced by both the MHNC and HRT for relatively long ranged potentials. When the potential range reduces, the MHNC faces problems in determining the LV binodal line; however, the LL consolute line and the critical end point (CEP) temperature and density turn out to be still satisfactorily predicted within this theory. The HRT also predicts with good accuracy the CEP position.The possibility of employing liquid state theories HCYM for the purpose of reliably determining phase equilibria in multicomponent colloidal fluids of current technological interest, is discussed.

Application of Fermi hypernetted-chain theory to spin-polarized higher-order fractional quantum Hall states

We apply Fermi hypernetted-chain theory to study the spin polarization of higher-order fractional quantum Hall (FQH) states at filling factors in between the primary FQH sequences, $\ensuremath{\nu}=p/({q}_{e}p\ifmmode\pm\else\textpm\fi{}1)$, where ${q}_{e}$ is an even integer and $p$ is a nonzero integer. The filling factors related to the higher-order FQH states include $\ensuremath{\nu}=3/8$, $4/11$, $5/13$, $5/17$, $4/13$, $6/17$, $7/11$, and so on. We use a model of strongly interacting fermions with different spin degrees of freedom to explain the states beyond primary FQH sequences. We calculate the correlation energy, the radial distribution function, as well as the static structure function associated with the Halperin wave function adopted for the mixture states of fermions with different spins. The results are comparable with those from the residual interaction between composite fermions.

Comparative study of the LOCV and the FHNC approaches for the nucleonic matter problem

The nucleonic matter problem is investigated by comparing the lowest order constrained variational (LOCV) method with the Fermi hypernetted chain (FHNC) theory, emphasizing the role of the LOCV correlation functions. In this way, the central correlation functions are used in the LOCV formalism, for the Bethe homework problem. It is shown that the LOCV computations reasonably agree with those of FHNC. Moreover, the FHNC calculations are performed with the LOCV correlation functions. It is found that, assuming the LOCV or the parametrized correlation functions, the FHNC computations do not change significantly. So, one may conclude that the mentioned consistencies refer to the choice of the LOCV correlation functions. Because, the contribution of the many-body cluster terms can be ignored, if the LOCV correlation functions satisfy the normalization constraint. Then, using the AV 18 interaction, the operator-dependent (OD) correlation functions are employed in the LOCV calculations. Note that the LOCV OD correlation functions are obtained by averaging over the states. It turns out that the overall behaviour of the LOCV OD correlation functions are similar to those of FHNC. Although, due to the many-body effects which are considered in the FHNC calculations, the LOCV results fairly differ from those of FHNC. Finally, it is worth mentioning that, unlike the recent FHNC calculations, the spin-orbit-dependent correlation functions are included in the LOCV approach.

Phase diagram of two-dimensional binary Yukawa mixtures

ABSTRACTWe have used Ramakrishnan–Yussouff (RY) density functional theory (DFT) to explore the topology of the phase diagram of two-component charge stabilised colloidal suspensions confined to a two-dimensional plane. The particles of the system interact via purely repulsive soft core Yukawa potential. Pair correlation functions (PCFs) used as input informations in DFT were calculated by solving both the hypernetted chain (HNC) and Percus–Yevick (PY) integral equation theories. To test the relative performance of the HNC and PY theories in the context of phase transitions, we have also studied the corresponding one-component systems. We found that RY DFT with HNC PCFs does not stabilise solid in both the one- and two-component cases, whereas the PY theory does. By considering the freezing into the substitutionally disordered triangular solid, we found that the temperature-composition phase diagrams of the binary mixture are narrow spindles whose thickness depends on the symmetry of the mixture components and the value of the screening constant of the Yukawa potential. Although the phase diagram obtained by RY DFT with structural inputs calculated by the PY theory is found to be shifted to higher temperature region in the temperature-composition plane, however, it captures qualitatively all the essential features of the phase diagram. Our results are in principle verifiable through computer simulations and experiments.

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.

Aggregate formation in a model fluid with microscopic piecewise-continuous competing interactions

We use the hypernetted chain theory – complemented in a few selected cases by Monte Carlo simulations to assess its accuracy – to study a model fluid composed by hard spheres surrounded by an attractive square-well, followed by a linearly decreasing repulsive ramp. The proposed competing interaction allows us to study individually the influence of all the defining parameters on the formation of aggregates out of the homogeneous fluid, as signalled by the development of a local peak at low wavevectors in the static structure factor. We document an accurate linear scaling of the temperature whereupon aggregates develop (at constant density) as a function of both the square-well width and the height of the repulsive barrier, over a wide range of such properties; a much drastic dependence is instead observed on the repulsion range, well described by an exponential increase. We show that typical correlation distances among aggregates, in early conditions for their development, fall approximately at a constant fraction of the width of the repulsive barrier. We analyse different possibilities in which attraction or repulsion prevails in the overall balance of microscopic interaction: we document, as expected, that for sufficiently weak repulsion barriers, or sufficiently large attractive wells, the structure factor shows a trend to increase in the limit of vanishing wavevectors, heralding a standard liquid–vapour phase separation at low temperatures. At the opposite end, the development of aggregates seems to be hardly favoured, for the present model, if attraction is completely neglected.