Percus Yevick Research Articles

An excess free energy and chemical potential for hard homonuclear diatomics from integral equation approach

It has been shown that an improved prediction of the thermodynamic quantity for the hard homonuclear diatomics can be performed with an interpolation scheme that relates the thermodynamic property of the hard sphere to that of tangent hard spheres at the same density. Using the analytic expressions based on the solution of an integral equation an excess Helmholtz free energy per particle and chemical potential for hard homonuclear diatomic fluid have been computed. Calculations are carried out for hard homonuclear diatomics with reduced internuclear separations of 0.1 to 1 at reduced densities of 0.2 to 0.9. Our findings for the excess Helmholtz free energy from the Percus–Yevickand Martynov–Sarkisov approximations presents good agreement with available accurate data, having maximum deviations of 15% from it over the separation and density ranges of the calculations. The excess chemical potential from the Martynov–Sarkisov approximation shows better agreement with accurate available data than those from the Percus–Yevick approximation. For the excess chemical potential, a maximum deviation of 9.5% over the range of the calculations has been shown up for the Percus–Yevick approximation.

Spatial distribution of reduced density of hard spheres near a hard-sphere dimer: Results from three-dimensional Ornstein–Zernike equations coupled with several different closures and from grand canonical Monte Carlo simulation

We calculated the spatial distribution function, which is the one-body reduced density distribution of solvent particles around a nonspherical solute, using the three-dimensional Ornstein–Zernike equations coupled with closures. The solvent was a hard-sphere fluid, and the contact dimer of the solvent particles was the nonspherical solute. Two traditional closures, the Percus–Yevick and hypernetted-chain approximations, and three closures with bridge functions (BFs) were examined. These spatial distribution functions were compared with the results obtained by grand canonical Monte Carlo (GCMC) simulations. Three closures with BFs, such as the modified hypernetted-chain (MHNC) closure using a bridge function proposed by Kinoshita, gave precise spatial distribution functions. These results were much more accurate than those obtained by the traditional closures. The deviations from the spatial distribution function obtained by the GCMC simulation appeared only near the concave surface of the solute dimer. However, the deviations were minor compared with those between the simulation and the predictions of the two traditional closures. By contrast, the three-body correlation functions for the closures with BFs were not more accurate than those obtained by the traditional closures.

Zero shear viscosity of two polyatomic glass-forming liquids calculated by Vogel–Fulcher law and schematic mode-coupling theory

A theoretical method based on the simple Vogel–Fulcher law for estimating the zero shear viscosity of two glass-forming liquids, 1,3-bis(1-nathtyl)-5–(2-naphthyl) benzene and polystyrene, over a range of temperatures from that close to their glass transition temperatures to high temperatures was reported. One parameter of this model, i.e., the critical number density, at which particles are trapped in a cage that only segmental vibration is possible was identified using the intermolecular radial distribution function derived from the Percus–Yevick integral equation and a simplified schematic mode-coupling theory.

Radial Distribution Function of Hard Spheres at Different Densities

The model of hard sphere system has important part in modern theories of liquids. Hard sphere fluid is considered as the most intensively studied system among all model fluids. The thermodynamic and structural properties of real fluids can be studied with the help of hard spheres fluid. The exact solution of Ornstein-Zernike (OZ) equation with Percus-Yevick (PY) along with the equation of state provides an approximate analytical expression for radial distribution function (RDF). This work intends to study the radial distribution function of Hard sphere (HS) using the PY approximation for different densities ranging from low to high to find out their structural properties, packing behavior, phase transitions, and thermodynamics. We use FORTRAN program for this purpose. At low densities, the RDF has a single peak at a distance corresponding to the hard sphere diameter, indicating that the particles are well- separated and not interacting with each other like gaseous. As density increases, the peak becomes broader and shifts to smaller distances, indicating that the particles are coming into closer contact and interacting more strongly with each other. The height of the peak also increases, indicating that there is a greater probability of finding a particle at a certain distance from another particle.

Calculation of the entropy for hard-sphere from integral equation method

The Ornstein-Zernike integral equation method has been employed for a single-component hard sphere fluid in terms of the Percus-Yevick (PY) and Martynov-Sarkisov (MS) approximations. The virial equation of state, excess chemical potential, and free energy have been computed in both approximations. The excess entropy has been obtained with both thermodynamic relation and free energy expression for reduced densities of 0.1 to 0.9. It has been shown that the entropy values from thermodynamic relation in the MS approximation are better than those from the PY approximation, especially for high densities, and presents a reasonable comparison with available data in the literature, while the values from free energy expression in both approximations are not only close to each other but also comparable to accurate ones.

Simulations and integral-equationtheories for dipolar density interacting disks.

Integral equationtheories (IETs) based on the Ornstein-Zernike (OZ) relation can be used as an analytical tool to predict structural and thermodynamic properties and phase behavior of fluids with low numerical cost. However, there are no studies of the IETs for the dipolar density interaction potential in two-dimensional systems, a relevant interdomain interaction in lipid monolayers with phase coexistence. This repulsive interaction arises due to the excess dipole density of the domains, which are aligned perpendicular to the interface. This work studies the performance of three closures of the OZ equationfor this novel system: Rogers-Young (RY), modified hypernetted chain (MHNC), and variational modified hypernetted chain (VMHNC). For the last two closures the bridge function of a reference system is required, with the hard disk being the most convenient reference system. Given that in two dimensions there is no analytical expressions for the hard disk correlation functions, two different approximations are proposed: one based on the Percus-Yevick (PY) approximation, and the other based on an extension of the hard spheres Verlet-Weis-Henderson-Grundke (LB) parametrization. The accuracy of the five approaches is evaluated by comparison of the pair correlation function and the structure factor with Monte Carlo simulation data. The results show that RY closure is satisfactory only for low-structured regimes. MHNC and VMHNC closures perform globally well, and there are no significant differences between them. However, the reference system in some cases affects their performance; when the pair correlation function serves as the measure, the LB-based closures quantitatively outperform the PY ones. From the point of view of its applicability, LB-based closures do not have a solution for all studied interaction strength parameters, and, in general, PY-based closures are numerically preferable.

Simulation of Colloidal Dispersion of Spherical Particles Interacting Via a Linear Combination of Dispersive and Screened Yukawa Potential

We have considered a two-dimensional colloidal dispersion of spherical particles interacting via a linear combination of Hamaker type dispersive potential and screened Yukawa potential. We have compared the pair correlation functions (g(r)) calculated from isothermal isochoric (NVT) Monte Carlo simulation with the PCF obtained by solving PercusYevick (PY) integral equation theory, at six values of reduced densities ranging from 0.03 to 0.11, keeping the relative strength of two types of potential fixed. Structure factors are calculated by Fourier transform of total PCF (h(r)).

Effect of the varying repulsive core of Gay–Berne pair potential on the structure and freezing transition in a two-dimensional system of soft ellipses

ABSTRACT We have investigated the effect of the strength of repulsive core of Gay–Berne (GB) ( n − 6 ) pair potential on the pair correlations and isotropic-nematic ( I − N ) freezing transition in a system of soft ellipses confined to a two-dimensional plane. The power n has been varied in the range 8 ≤ n ≤ 16 at an interval of two. Percus–Yevick (PY) integral equation theory (IET) has been solved to calculate the pair correlations of the isotropic fluid, which has been then used in the density functional theory to determine the isotropic-nematic transition parameters. The pair correlations obtained by PY theory have been compared with those obtained by NVT Monte-Carlo (MC) simulation. We observe a better agreement between PY and MC correlations at lower n -values of the interaction potential. An increase in the values of n results in more intricate structures manifested in both the radial distribution part and the higher order correlations. The I − N transition parameters are found to be sensitive to the value of the exponent n . The transition density is found to decrease, whereas the corresponding orientational order parameter is found to increase with the increase in n for a given temperature. At a given temperature, we observe a cut-off of n , below which the nematic does not stabilise. The smaller value of change in density at the transition indicates that the I − N transition is of weakly first order.

Extension of Radiative Transfer Coherent Backscattering RT-CB code to dense discrete random media

The Radiative transfer coherent backscattering (RT-CB) code is extended to apply to dense discrete random media of optically soft spherical particles. This is achieved by utilizing the well-known static-structure-factor (SSF) correction derived from the Percus–Yevick approximation for sticky-hard-sphere (SHS) pair correlation function. The code is verified against the numerically exact electromagnetic method for small spherical media consisting of submicrometer-sized icy particles at optical wavelengths. The SSF-corrected RT-CB method significantly improves solution accuracy, and the solution agrees well with the numerically exact solution when the packing density is less than 20% and particles are optically soft, i.e., the refractive index of particles is close to that of the background medium.

Pair correlation functions and stability of nematic in a system of Gay-Berne ellipsoids doped with spherical colloids

Pair-structure of a binary mixture of soft-ellipsoids and spheres interacting via generalized Gay-Berne intermolecular potential using Percus–Yevick (PY) integral equation theory. The generalized Gay-Berne interaction model allows us to vary the diameter of the spheres in comparison to the width of the Gay-Berne ellipsoids. Three different cases of sphere size namely diameter of the spheres being smaller, equal and larger than the minor axis of the ellipsoids have been considered. We have analyzed the radial distribution functions and static structure factors of the mixtures and found that addition of spherical particles to a system of ellipsoids results in stronger pair-correlations between ellipsoidal particles. We observe that smaller spherical additives are more effective in enhancing the ellipsoidal pair-correlations in comparison to the bigger spheres at similar density.The static structure factors are not found to diverge in the low wavelength limit indicating a stable mixture for the set of parameters studied in this work. We have also used classical density functional theory of freezing to investigate the effect of spherical additives and their size on the isotropic-nematic phase transition. We found a stable nematic phase whose order parameter is found to increase with the increase in not only the density of spherical additives but also with increase in their size.

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.

Computation of Structure and Electrical Resistivity of Liquid Na-Rb Alloys

The structure and electrical resistivity of Na1-xRbx binary alloys (where x = 0, 0.1, 0.2,…,1) are computed using Percus-Yevick (PY) equation, hard-sphere model and Faber-Ziman formula respectively. The partial structure factors and total structure factor are computed using hard-sphere model for Na1-xRbx. In the calculation of resistivity using Faber-Ziman formula, we have employed Ashcroft empty-core pseudo-potential and Hartree dielectric screening. Calculated values of resistivity are compared with the experimental results and other theoretical values reported in literature. It is found that the electrical resistivity calculated using Faber-Ziman formula for binary alloy Na1-xRbx is in good agreement with the values reported experimentally.

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.

Precursor nuclei on the bottom of a vibrating container: The onset of granular self-assembly crystallization

Kinetic energy is transferred through collisions to the millimeter particles that are in contact with the base of a vertically vibrated 3D container. In the early stages of a vibrational annealing process where the dimensionless acceleration is kept constant, the spontaneous appearance and disappearance of unstable clusters of ordered particles near the bottom surface of the container is observed. In later stages of this vibrational annealing process, the precursor nuclei stabilize becoming stable crystal seeds which resembles a thermal phase transition. Molecular Dynamics simulations based on these experimental observations are used to study the unstable–stable transition. The Ornstein–Zernike equation using the Percus–Yevick closure is applied to the stages before and after the observation of the stable crystal seeds in order to extract the effective potentials associated with the phase transition. Both the radial distribution function and the effective potentials show a clear correspondence of the spatial correlation as the crystallization phase begins to appear.

Long-range oscillations of a single-particle distribution function for a molecular system of hard spheres near a solid surface in the Percus–Yevick approximation

The properties of molecular systems are defined by asymptotic behavior of distribution functions that have descending character with long-range oscillating order. We consider the molecular system of hard spheres near hard wall. The decrement and period of oscillation are calculated for the distribution function which defines the local density profile.

Tagged-particle motion of Percus-Yevick hard spheres from first principles.

We develop a first-principles-based generalized mode-coupling theory (GMCT) for the tagged-particle motion of glassy systems. This theory establishes a hierarchy of coupled integro-differential equations for self-multi-point density correlation functions, which can formally be extended up to infinite order. We use our GMCT framework to calculate the self-nonergodicity parameters and the self-intermediate scattering function for the Percus-Yevick hard-sphere system based on the first few levels of the GMCT hierarchy. We also test the scaling laws in the α- and β-relaxation regimes near the glass-transition singularity. Furthermore, we study the mean-square displacement and the Stokes-Einstein relation in the supercooled regime. We find that qualitatively our GMCT results share many similarities with the well-established predictions from standard mode-coupling theory, but the quantitative results change, and typically improve, by increasing the GMCT closure level. However, we also demonstrate on general theoretical grounds that the current GMCT framework is unable to account for violation of the Stokes-Einstein relation, underlining the need for further improvements in the first-principles description of glassy dynamics.

Chain-End Effects on Supramolecular Poly(ethylene glycol) Polymers.

In this work we present a fundamental analysis based on small-angle scattering, linear rheology and differential scanning calorimetry (DSC) experiments of the role of different hydrogen bonding (H-bonding) types on the structure and dynamics of chain-end modified poly(ethylene glycol) (PEG) in bulk. As such bifunctional PEG with a molar mass below the entanglement mass is symmetrically end-functionalized with three different hydrogen bonding (H-bonding) groups: thymine-1-acetic acid (thy), diamino-triazine (dat) and 2-ureido-4[1H]-pyrimidinone (upy). A linear block copolymer structure and a Newtonian-like dynamics is observed for PEG-thy/dat while results for PEG-upy structure and dynamics reveal a sphere and a network-like behavior, respectively. These observations are concomitant with an increase of the Flory–Huggins interaction parameter from PEG-thy/dat to PEG-upy that is used to quantify the difference between the H-bonding types. The upy association into spherical clusters is established by the Percus–Yevick approximation that models the inter-particle structure factor for PEG-upy. Moreover, the viscosity study reveals for PEG-upy a shear thickening behavior interpreted in terms of the free path model and related to the time for PEG-upy to dissociate from the upy clusters, seen as virtual crosslinks of the formed network. Moreover, a second relaxation time of different nature is also obtained from the complex shear modulus measurements of PEG-upy by the inverse of the angular frequency where G’ and G’’ crosses from the network-like to glass-like transition relaxation time, which is related to the segmental friction of PEG-upy polymeric network strands. In fact, not only do PEG-thy/dat and PEG-upy have different viscoelastic properties, but the relaxation times found for PEG-upy are much slower than the ones for PEG-thy/dat. However, the activation energy related to the association dynamics is very similar for both PEG-thy/dat and PEG-upy. Concerning the segmental dynamics, the glass transition temperature obtained from both rheological and calorimetric analysis is similar and increases for PEG-upy while for PEG-thy/dat is almost independent of association behavior. Our results show how supramolecular PEG properties vary by modifying the H-bonding association type and changing the molecular Flory–Huggins interaction parameter, which can be further explored for possible applications.

Shape and Structure Formation of Mixed Nonionic-Anionic Surfactant Micelles.

Aqueous solutions of a nonionic surfactant (either Tween20 or BrijL23) and an anionic surfactant (sodium dodecyl sulfate, SDS) are investigated, using small-angle neutron scattering (SANS). SANS spectra are analysed by using a core-shell model to describe the form factor of self-assembled surfactant micelles; the intermicellar interactions are modelled by using a hard-sphere Percus–Yevick (HS-PY) or a rescaled mean spherical approximation (RMSA) structure factor. Choosing these specific nonionic surfactants allows for comparison of the effect of branched (Tween20) and linear (BrijL23) surfactant headgroups, both constituted of poly-ethylene oxide (PEO) groups. The nonionic–anionic surfactant mixtures are studied at various concentrations up to highly concentrated samples ( ≲ 0.45) and various mixing ratios, from pure nonionic to pure anionic surfactant solutions. The scattering data reveal the formation of mixed micelles already at concentrations below the critical micelle concentration of SDS. At higher volume fractions, excluded volume effects dominate the intermicellar structuring, even for charged micelles. In consequence, at high volume fractions, the intermicellar structuring is the same for charged and uncharged micelles. At all mixing ratios, almost spherical mixed micelles form. This offers the opportunity to create a system of colloidal particles with a variable surface charge. This excludes only roughly equimolar mixing ratios (X≈ 0.4–0.6) at which the micelles significantly increase in size and ellipticity due to specific sulfate–EO interactions.

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.