Universal cubic equation of state and contact values of the radial distribution functions for multi-component additive hard-sphere mixtures

A universal cubic equation of state (UC EOS) is proposed based on a modification of the virial Percus-Yevick (PY) integral equation EOS for hard-sphere fluid. The UC EOS is extended to multi-component hard-sphere mixtures based on a modification of Lebowitz solution of PY equation for hard-sphere mixtures. And expressions of the radial distribution functions at contact (RDFC) are improved with the form as simple as the original one. The numerical results for the compressibility factor and RDFC are in good agreement with the simulation results. The average errors of the compressibility factor relative to MC data are 3.40%, 1.84% and 0.92% for CP3P, BMCSL equations and UC EOS, respectively. The UC EOS is a unique cubic one with satisfactory precision among many EOSs in the literature both for pure and mixture fluids of hard spheres.     

Contact values of highly asymmetric hard sphere mixtures from Fundamental Measure Density Functional Theory

The White Bear version of Fundamental Measure Theory (FMT-WB) has been tested in binary mixtures of hard spheres in the vicinity of the colloidal limit, where the size ratio of the two species is exceedingly large and the large sphere mole fraction is infinitely low. Contact values of large–large sphere radial distribution functions have been calculated and compared with molecular dynamics simulations and previously proposed theoretical formulas. In contrast to the failure of BMCSL (Boublik, Mansoori, Carnahan, Starling, Leland equation of state) predictions, FMT-WB gives good agreement with simulation for a range of species size ratios and mole fractions. The performance of BMCSL is qualitatively related to one of its model parameters, which could indicate the reliability of the BMCSL result. Our results confirm the accuracy of FMT-WB in the colloidal limit for the first time and suggest that BMCSL contact values must be applied carefully to account for chain connectivity when studying certain cases with classical Density Functional Theories.     

Molecular dynamics results for the radial distribution functions of highly asymmetric hard sphere mixtures

Molecular dynamics (MD) results for the radial distribution functions of mixtures of large and small hard spheres are reported for size ratios whose (large/small) values are 1, 2.5, 5, 7.5, and 10 in the region where the concentration of the large spheres is very small. The MD contact values of these functions are compared with formulae due to Boublik, Mansoori, Carnahan, Starling, Leland, Grundke, and Henderson, Viduna and Smith, Henderson, Trokhymchuk, Woodcock, and Chan, as well as new formulae that are considered here. The new formulae give good agreement for the large–small contact values and reasonably good agreement for the large–large contact values. The Viduna–Smith formula is satisfactory for the small–small contact value and quite reasonable for the small–large contact value. Undoubtedly, further improvements are possible. These results give insight into what may be called the colloidal limit, where the size ratio is exceedingly large while the concentration of the large spheres is exceedingly small, and into the passage to this limit.     

H-theorem for the hard-sphere gas

The approach to equilibrium of the hard-sphere gas is discussed from the master-equation point of view. AnH-theorem is established, which is valid for arbitrary initial conditions.     

Extension of Compressibility-Route Cubic Equations of State and the Radial Distribution Functions at Contact to Multi-Component Hard-Sphere Mixtures

The equation of state(EOS) for hard-sphere fluid derived from compressibility routes of Percus-Yevick theory(PYC) is extended. The two parameters are determined by fitting well-known virial coefficients of pure fluid.The extended cubic EOS can be directly extended to multi-component mixtures, merely demanding the EOS of mixtures also is cubic and combining two physical conditions for the radial distribution functions at contact(RDFC) of mixtures.The calculated virial coefficients of pure fluid and predicted compressibility factors and RDFC for both pure fluid and mixtures are excellent as compared with the simulation data. The values of RDFC for mixtures with extremely large size ratio 10 are far better than the BGHLL expressions in literature.     

Extension of Compressibility-Route Cubic Equations of State and the Radial Distribution Functions at Contact to Multi-Component Hard-Sphere Mixtures

The equation of state (EOS) for hard-sphere fluid derived from compressibility routes of Percus-Yevick theory (PYC) is extended. The two parameters are determined by fitting well-known virial coefficients of pure fluid. The extended cubic EOS can be directly extended to multi-component mixtures, merely demanding the EOS of mixtures also is cubic and combining two physical conditions for the radial distribution functions at contact (RDFC) of mixtures. The calculated virial coefficients of pure fluid and predicted compressibility factors and RDFC for both pure fluid and mixtures are excellent as compared with the simulation data. The values of RDFC for mixtures with extremely large size ratio 10 are far better than the BGHLL expressions in literature.     

Analytical representation of the density derivative of the Percus–Yevick hard-sphere radial distribution function

ABSTRACT

Explicit analytical expressions are presented for the density derivative, ?g_HS(R; ρ)/?ρ, of the Percus–Yevick approximation to the hard-sphere radial distribution function for R ≤ 6σ, where σ is the hard-sphere diameter and ρ = (N/V)σ³ is the reduced density, where N is the number of particles and V is the volume. A FORTRAN program is provided for the implementation of these for R ≤ 6σ, which includes code for the calculation of g_HS(R; ρ) itself over this range. We also present and incorporate within the program code convenient analytical expressions for the numerical extrapolation of both quantities past R = 6σ. Our expressions are numerically tested against exact results.     

Oscillation of the radial distribution function

We prove that the radial distribution function oscillates at low density in a system with a short-range nonnegative potential and investigate the branching of the solutions of an approximate equation of state.     

The hard-sphere fluid: New exact results with applications

A theorem for convolution integrals is proved and then applied to extend the second zero-separation theorem to the bridge functionb(r) and direct-correlation tail functionsd(r). This theorem allows us to exactly relateb(r)/r andd(r)/ratr=0 for the hard-sphere fluid to the contact value of the radial distribution functiong(r) atr= ⁺. From this we obtain immediately the exact values of b(r)/r and d(r)/r atr=0 through second order in number density . Using our results to compare the exact and Percus-Yevick (PY) bridge function, we find that they differ significantly. After obtaining the bridge function and tail function and their derivatives atr=0 andr= through, we suggest new approximations forb(0) andd(0) as well as an analytical integral-equation theory to improve the PY approximation in the pure hard-sphere fluid. The major deficiency of that approximation has been its poor assessment of the cavity function inside the hard-core region. Our theory remedies this defect in a way that yields ay(r) that is self-consistent with respct to the virial and compressibility relations and also the two zero-separation relations involvingy(r) and its spatial derivative atr=0.     

Hydrodynamic correlation functions of hard-sphere fluids at short times

The short-time behavior of the coherent intermediate scattering function for a fluid of hard-sphere particles is calculated exactly through ordert ⁴, and the other hydrodynamic correlation functions are calculated exactly through ordert ². It is shown that for all of the correlation functions considered the Enskog theory gives a fair approximation. Also, the initial time behavior of various Green-Kubo integrands is studied. For the shear-viscosity integrand it is found that at densityn³=0.837 the prediction of the Enskog theory is 32% too low. The initial value of the bulk viscosity integrand is nonzero, in contrast to the Enskog result. The initial value of the thermal conductivity integrand at high densities is predicted well by Enskog theory.     

Second-order Percus-Yevick theory for a confined hard-sphere fluid

A fluid of hard spheres confined between two hard walls and in equilibrium with a bulk hard-sphere fluid is studied using a second-order Percus-Yevick approximation. We refer to this approximation as second-order because the correlations that are calculated depend upon the position of two hard spheres in the confined fluid. However, because the correlation functions depend upon the positions of four particles (two hard spheres and two walls treated as giant hard spheres), this is the most demanding application of the second-order theory that has been attempted. When the two walls are far apart, this calculation reduces to our earlier second-order approximation calculations of the properties of hard spheres near a single hard wall. Our earlier calculations showed this approach to be accurate for the single-wall case. In this work we calculate the density profiles and the pressure of the hard-sphere fluid on the walls. We find, by comparison with grand canonical Monte Carlo results, that the second-order approximation is very accurate, even when the two walls have a small separation. We compare with a singlet approximation (in the sense that correlation functions that depend on the position of only one hard sphere are considered). The singlet approach is fairly satisfactory when the two walls are far apart but becomes unsatisfactory when the two walls have a small separation. We also examine a simple theory of the pressure of the confined hard spheres, based on the usual Percus-Yevick theory of hard-sphere mixtures. Given the simplicity of the latter approach the results of this simple (and explicit) theory are surprisingly good.     

Generalized equilibrium concentration of polyvacancy: case study for trivacancy in hard-sphere crystals

The equilibrium concentration of polyvacancy in crystal is formulated by modifying the grand canonical ensemble. The generalized equation can be applied to obtain the concentration of vacancy in any order and shape in crystal. In this work, a Widom-like particle insertion method is adopted implemented with molecular dynamics simulation to obtain individual formation free energies for constituent vacancies in polyvacancy. As a case study, equilibrium concentrations and formation free energies were obtained of four and nine trivacancies identified in face-centred-cubic (FCC) and hexagonal-close-packed (HCP) hard-sphere crystals, respectively. The result is in excellent agreement with literature data available for the FCC. Further, stabilities of equilateral triangular trivacancies are elucidated and their relative stabilities in FCC and HCP hard-sphere crystals are reported. Overall trivacancy concentration is higher in the HCP HS crystal.     

Perturbation and variational approach for the equation of state for hard-sphere and Lennard-Jones fluids

The present work uses the concept of a scaled particle along with the perturbation and variation approach, to develop an equation of state (EOS) for a mixture of hard sphere (HS), Lennard–Jones (LJ) fluids. A suitable flexible functional form for the radial distribution function G(R) is assumed for the mixture, with R as a variable. The function G(R) has an arbitrary parameter m and a different equation of state can be obtained with a suitable choice of m. For m = 0.75 and m = 0.83 results are close to molecular dynamics (MD) result for pure HS and LJ fluid respectively.     

Perturbation and variational approach for the equation of state for hard-sphere and Lennardben Jones fluids

The present work uses the concept of a scaled particle along with the perturbation and variation approach, to develop an equation of state (EOS) for a mixture of hard sphere (HS), Lennard-Jones (LJ) fluids. A suitable flexible functional form for the radial distribution function G(R) is assumed for the mixture, with R as a variable. The function G(R) has an arbitrary parameter m and a different equation of state can be obtained with a suitable choice of m. For m= 0.75 and m= 0.83 results are close to molecular dynamics (MD) result for pure HS and LJ fluid respectively.     

带电粒子形成胶体晶体的有效硬球模型判据的计算机模拟验证

在对胶体晶体的研究中,带电粒子胶体晶体的形成机理比硬球胶体晶体更加复杂,对其形成条件目前还缺少有效的判断依据. 有效硬球模型判据提出以有效直径作为判断参数. 为了验证该判据的有效性,利用布朗动力学模拟研究了不同有效直径下带电粒子胶体晶体的特性. 为了更加定量地研究单因素对带电胶体晶体形成的影响,取有效直径为2.8至0.8,并对一定的有效直径,研究了粒子几何直径和排斥力不同情况下的结晶行为. 在布朗动力学模拟过程中,采用径向分布函数和键序参数方法检测体系的结构变化,并分析所形成的晶体结构. 结果表明,在判断带电粒子胶体体系能否形成有序结构方面,有效硬球模型判据有一定的合理性. 但是,并不能将有效直径作为唯一的判别参数,而是需要综合其他参数的影响,这显示出该判据的片面性. 关键词： 布朗动力学模拟 带电胶体晶体 有效硬球模型     

Interplay between elastic fields due to gravity and a partial dislocation for a hard-sphere crystal coherently grown under gravity: driving force for defect disappearance

We previously observed that an intrinsic staking fault shrunk through a glide of a Shockley partial dislocation terminating its lower end in a hard-sphere crystal under gravity coherently grown in ?001? by Monte Carlo simulations [Mori et al., Molec. Phys. 105, 1377 (2007)]; it was an answer to a one-decade long standing question why the stacking disorder in colloidal crystals reduced under gravity [Zhu et al., Nature 387, 883 (1997)]. Here, we present an elastic energy calculation; in addition to the self-energy of the partial dislocation [Mori et al., Prog. Theor. Phys. Suppl. 178, 33 (2009)] we calculate the cross-coupling term between elastic field due to gravity and that due to a Shockley partial dislocation. The cross-term is an increasing function of the linear dimension R over which the elastic field expands, showing that a driving force arises for the partial dislocation moving toward the upper boundary of a grain.     

Binary nonadditive hard-sphere mixtures at high dimension

At high spatial dimension, a suitably scaled classical system of interacting particles truncates at second virial terms. A binary mixture of nonadditive hard spheres with sufficiently repulsive interaction between unlike particles decomposes at sufficiently high density into two coexisting phases. The region around the critical density behaves classically.     

Non-polytropic effect on shock-induced phase transitions in a hard-sphere system

By adopting a simplified model of a non-polytropic hard-sphere system where heat capacity depends on the temperature, we demonstrate the importance of non-polytropic effect on the shock-induced phase transitions. We show explicitly that with the increase of the shock strength the perturbed temperature (the temperature after a shock) increases and the vibrational modes are gradually excited, and as a result, shock-induced phase transitions are qualitatively and quantitatively different from the phase transitions observed in a simple polytropic model. The effect on the admissibility (stability) of a shock wave is also analyzed.     

Inequalities of the electron density at the nucleus and radial expectation values of the ground state for the lithium isoelectronic sequence

The electron density at the nucleus, ρ(0), and the radial expectation values, (-2≤n≤10), of the ground state for the lithium isoelectronic sequence are calculated with a full core plus correlation (FCPC) wavefunctions. By using these obtained expectation values, the accurate inequalities of the electron density at the nucleus and the radial expectation values derived by Gálvez and Porras for these systems are examined and verified. The final results show that FCPC wavefunctions used in this work can give satisfactory results in full configuration space.