Bridge function for the dipolar fluid from simulation

The exact bridge function of the Lennard-Jones dipolar (Stockmayer) fluid is extracted from Monte Carlo simulation data. The projections g(mnl)(r) onto rotational invariants of the non-spherically symmetric pair distribution function g(r, Ω) are accumulated during simulation. Making intensive use of anisotropic integral equation techniques, the molecular Ornstein-Zernike equation is then inverted in order to derive the direct correlation function c(mnl)(r), the cavity function y(mnl)(r), the negative excess potential of mean force lny|(mnl)(r), and the bridge function b(mnl)(r) projections. b(r, Ω) presents strong, non-universal anisotropies at high dipolar coupling. This simulation data analysis may serve as reference and guide for approximated bridge function theories of dipolar fluids and is a valuable step towards the case of more refined, nonlinear water-like geometries.     

‘Exact’ integral equation theory and local formulation for excess thermodynamic properties of hard spheres

A hybrid hard sphere bridge function is proposed, which, in combination with the standard Ornstein–Zernike integral equation, can predict extremely accurately hard sphere compressibility, virial pressure, and correlation function. Second, a local formulation for determination of excess chemical potential is derived out, which, in combination with the present hybrid hard sphere bridge function and OZ integral equation, can predict the excess chemical potential also extremely accurately. The resultant excess entropy is in excellent agreement with that from the Carnahan–Starling equation of state. The present formalism performs excellently over the whole density range, i.e. from zero to freezing density, and is largely superior to a formalism available in the literature.     

The equation of state of hard hyperspheres in nine dimensions for low to moderate densities

The equation of state of hard hyperspheres in nine dimensions is calculated both from the values of the first ten virial coefficients and from a Monte Carlo simulation of the pair correlation function at contact. The results are in excellent agreement. In addition, we find that the virial series appears to be dominated by an unphysical singularity or singularities on or near the negative density axis, in qualitative agreement with the recently solved Percus-Yevick equation of state in nine dimensions.     

Virial series for fluids of hard hyperspheres in odd dimensions

A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E 76, 051202 (2007)] to obtain the exact solution of the Percus-Yevick equation for a fluid of hard spheres in (odd) d dimensions is used to investigate the convergence properties of the resulting virial series. This is done both for the virial and compressibility routes, in which the virial coefficients B(j) are expressed in terms of the solution of a set of (d-1)/2 coupled algebraic equations which become nonlinear for d>/=5. Results have been derived up to d=13. A confirmation of the alternating character of the series for d>/=5, due to the existence of a branch point on the negative real axis, is found and the radius of convergence is explicitly determined for each dimension. The resulting scaled density per dimension 2eta(1/d), where eta is the packing fraction, is wholly consistent with the limiting value of 1 for d-->infinity. Finally, the values for B(j) predicted by the virial and compressibility routes in the Percus-Yevick approximation are compared with the known exact values [N. Clisby and B. M. McCoy, J. Stat. Phys. 122, 15 (2006)].     

Small angle neutron scattering studies on non-aqueous dispersions of calcium carbonate. Part III. Concentrated dispersions

Small angle neutron scattering has been used to examine calcium carbonate dispersions in toluene over the w/w concentration range 5 to 40 %. The particles were stabilised by a combination of a surface active agent and a linear polymeric molecule. Analysis of the experimental data gave results for the structure factorS(Q) and the pair correlation functiong(r).Theoretical computations of the structure factor were made using a hard sphere model and the mean spherical approximation approach. These analyses indicated that the interaction was essentially soft at low volume fractions but became hard at the higher volume fractions.     

Second virial coefficient for the dipolar hard sphere fluid

The dipolar hard sphere fluid is a useful model for a polar fluid. Some years ago, the second virial coefficient, B(2), of this fluid was obtained as a series expansion in the inverse temperature or (dipole strength) by Keesom. Little work on this problem seems to have been done since that time. Using a result of Chan and Henderson for the spherical average of the Boltzmann factor of this fluid, more complete results are obtained for B(2). The more complete results are more negative than the Keesom series, as one would expect, but his expansion is remarkably accurate. This method can be used to obtain the second virial coefficient of the dipolar Lennard-Jones (Stockmayer) or dipolar Yukawa fluids.     

Integral equation theory for symmetric nonadditive hard sphere mixtures

An integral equation theory is presented for the pair correlation functions and phase behavior of symmetric nonadditive hard sphere mixtures with hard sphere diameters given by sigma(A)(A)() = sigma(BB) = lambdad and sigma(AB) = d. This mixture exhibits a fluid-fluid phase separation into an A-rich phase and a B-rich phase at high densities. The theory incorporates, into the closure approximation, all terms that can be calculated exactly in the density expansion of the direct correlation functions. We find that the closure approximation developed in this work is accurate for the structure and phase behavior over the entire range of lambda, when compared to computer simulations, and is significantly more accurate than the previous theories.     

一个新的桥泛函及其在非均一流体密度泛函理论中的应用

基于对OZ 方程的渐近行为与Taylor级数展开的分析，提出了一个新的桥泛函，桥泛函被表达为间接相关函数的函数，Taylor级数展开的重整化导致了一个可调参数，通过将所提出的桥泛函与一个最近提出的密度泛函理论方法学，以及单个硬墙的sum 规则结合，可以确定可调参数.所提出的桥泛函能预言如下非均一流体的密度分布：硬球流体接近一个硬墙与在球形空隙内，Lennard-Jones 流体与缔合硬球流体在两个硬墙之内.理论预言与文献所报导的模拟数据符合很好.     

Scaled particle theory for hard sphere pairs. II. Numerical analysis

We use the extension of scaled particle theory presented in the accompanying paper [F. H. Stillinger et al., J. Chem. Phys. 125, 204504 (2006)] to calculate numerically the pair correlation function of the hard sphere fluid over the density range 0< or =rhosigma(3)< or =0.96. Comparison with computer simulation results reveals that the new theory is able to capture accurately the fluid's structure across the entire density range examined. The pressure predicted via the virial route is systematically lower than simulation results, while that obtained using the compressibility route is lower than simulation predictions for rhosigma(3)< or =0.67 and higher than simulation predictions for rhosigma(3)> or =0.67. Numerical predictions are also presented for the surface tension and Tolman length of the hard sphere fluid.     

Approximative "one particle" bridge function B(1)(r) for the theory of simple fluids

New properties for the one particle bridge function B(1)(r), which are necessary to the calculation of the excess chemical potential betamue), are derived for the hard sphere fluid. The method, which only requires the knowledge of the bridge function B(2)(r), is based on an investigation of the correlation function dependence on the Kirkwood charging parameter. In this framework, the unavoidable question of topological homotopy is addressed. As far as B(2)(r) is considered as exact, this work provides useful information on B(1)(r) in the well identified dynamical regimes of the hard sphere fluid. Signatures of the transitions between these regimes are identified on the trends of B(1)(r). This approach provides self-consistent results for betamue) that agree very well with simulation data.     

Pair diffusion, hydrodynamic interactions, and available volume in dense fluids

We calculate the pair diffusion coefficient D(r) as a function of the distance r between two hard sphere particles in a dense monodisperse fluid. The distance-dependent pair diffusion coefficient describes the hydrodynamic interactions between particles in a fluid that are central to theories of polymer and colloid dynamics. We determine D(r) from the propagators (Green's functions) of particle pairs obtained from molecular dynamics simulations. At distances exceeding ~3 molecular diameters, the calculated pair diffusion coefficients are in excellent agreement with predictions from exact macroscopic hydrodynamic theory for large Brownian particles suspended in a solvent bath, as well as the Oseen approximation. However, the asymptotic 1/r distance dependence of D(r) associated with hydrodynamic effects emerges only after the pair distance dynamics has been followed for relatively long times, indicating non-negligible memory effects in the pair diffusion at short times. Deviations of the calculated D(r) from the hydrodynamic models at short distances r reflect the underlying many-body fluid structure, and are found to be correlated to differences in the local available volume. The procedure used here to determine the pair diffusion coefficients can also be used for single-particle diffusion in confinement with spherical symmetry.     

Structure of nonuniform hard sphere fluids from shifted linear truncations of functional expansions

Percus showed that approximate theories for the structure of nonuniform hard sphere fluids can be generated by linear truncations of functional expansions of the nonuniform density rho(r) about that of an appropriately chosen uniform system. We consider the most general such truncation, which we refer to as the shifted linear response (SLR) equation, where the density response rho(r) to an external field phi(r) is expanded to linear order at each r about a different uniform system with a locally shifted chemical potential. Special cases include the Percus-Yevick (PY) approximation for nonuniform fluids, with no shift of the chemical potential, and the hydrostatic linear response (HLR) equation, where the chemical potential is shifted by the local value of phi(r). The HLR equation gives exact results for very slowly varying phi(r) and reduces to the PY approximation for hard core phi(r), where generally accurate results are found. We show that a truncated expansion about the bulk density (the PY approximation) also gives exact results for localized fields that are nonzero only in a "tiny" region whose volume V(phi) can accommodate at most one particle. The SLR equation can also exactly describe a limit where the fluid is confined by hard walls to a very narrow slit. This limit can be related to the localized field limit by a simple shift of the chemical potential, leading to an expansion about the ideal gas. We then try to develop a systematic way of choosing an optimal local shift in the SLR equation for general phi(r) by requiring that the predicted rho(r) is insensitive to small variations about the appropriate local shift, a property that the exact expansion to all orders would obey. The resulting insensitivity criterion (IC) gives a theory that reduces to the HLR equation for slowly varying phi(r) and is much more accurate than HLR both for very narrow slits, where the IC agrees with exact results, and for fields confined to "tiny" regions, where the IC gives very accurate (but not exact) results. However, the IC is significantly less accurate than the PY and HLR equations for single hard core fields. Only a small change in the predicted reference density is needed to correct this remaining limit.     

A consistent calculation of the chemical potential for dense simple fluids

A general method to calculate the excess chemical potential betamuex, that is based on the Kirkwood coupling parameter's dependence of the correlation functions, is presented. The expression for the one particle bridge function B(1)r is derived for simple fluids with spherical interactions. Only the knowledge of the bridge function B(2)r is required. The accuracy of our approach is illustrated for a dense hard sphere fluid. As far as B(2)r is considered as exact, B(1)r is found to be, at high densities, the normalized bridge function -B(2)rB(2)(r=0). This expression ensures a consistent calculation of the excess chemical potential by satisfying implicitly the Gibbs-Duhem constraint. Only the pressure-consistency condition is necessary to calculate the structural and thermodynamic properties of the fluid.     

Kohn–Sham potentials by an inverse Kohn–Sham equation and accuracy assessment by virial theorem

In the recent study, the authors have proposed an integral equation for solving the inverse Kohn–Sham problem. In the present paper, the integral equation is numerically solved for one-dimensional model of a He atom and an H₂ molecule in the electronic ground states. For this purpose, we propose an iterative solution algorithm avoiding the inversion of the kernel of the integral equation. To quantify the numerical accuracy of the calculated exchange-correlation potentials, we evaluate the exchange and correlation energies based on the virial theorem as well as the reproduction of the exact ground-state electronic energy. The results demonstrate that the numerical solutions of our integral equation for the inverse Kohn–Sham problem are accurate enough in reproducing the Kohn–Sham potential and in satisfying the virial theorem.     

An exact formalism to study the thermodynamic properties of hard-sphere systems under spherical confinement

This paper presents a modified grand canonical ensemble which provides a new simple and efficient scheme to study few-body fluid-like inhomogeneous systems under confinement. The new formalism is implemented to investigate the exact thermodynamic properties of a hard sphere (HS) fluid-like system with up to three particles confined in a spherical cavity. In addition, the partition function of this system was used to analyze the surface thermodynamic properties of the many-HS system and to derive the exact curvature dependence of both the surface tension and adsorption in powers of the density. The expressions for the surface tension and the adsorption were also obtained for the many-HS system outside of a fixed hard spherical object. We used these results to derive the dependence of the fluid-substrate Tolman length up to first order in density.     

A perturbation method for the Ornstein-Zernike equation and the generic van der Waals equation of state for a square well potential model

We calculate the generic van der Waals parameters A and B for a square well model by means of a perturbation theory. To calculate the pair distribution function or the cavity function necessary for the calculation of A and B, we have used the Percus-Yevick integral equation, which is put into an equivalent form by means of the Wiener-Hopf method. This latter method produces a pair of integral equations, which are solved by a perturbation method treating the Mayer function or the well width or the functions in the square well region exterior to the hard core as the perturbation. In the end, the Mayer function times the well width is identified as the perturbation parameter in the present method. In this sense, the present perturbation method is distinct from the existing thermodynamic perturbation theory, which expands the Helmholtz free energy in a perturbation series with the inverse temperature treated as an expansion parameter. The generic van der Waals parameters are explicitly calculated in analytic form as functions of reduced temperature and density. The van der Waals parameters are recovered from them in the limits of vanishing density and high temperature. The equation of state thus obtained is tested against Monte Carlo simulation results and found reliable, provided that the temperature is in the supercritical regime. By scaling the packing fraction with a temperature-dependent hard core, it is suggested to construct an equation of state for fluids with a temperature-dependent hard core that mimicks a soft core repulsive force on the basis of the equation of state derived for the square well model.     

Extended pressure—consistent equation for simple fluids

Abstract

A previous generalization of the Percus-Yevick (PY) and hypernetted chain (HNC) equations for simple fluids, involving a density- and temperature-dependent coefficient m, is extended by including a spatial dependence in m. The new approximation yields an exact fourth virial coefficient and, by further requirement, a consistent equation of state from both the virial and compressibility forms. Comparison of calculated results for the hard sphere potential shows an improvement over the PY, HNC, and previous pressure-consistent equations     

Theory of correlated two-particle activated glassy dynamics: general formulation and heterogeneous structural relaxation in hard sphere fluids

We generalize the nonlinear Langevin equation theory of activated single particle dynamics to describe the correlated motion of two tagged spherical particles in a glass- or gel-forming fluid as a function of their initial separation. The theory is built on the concept of a two-dimensional dynamic free energy surface which quantifies the forces on two particles moving in a cooperative manner. For the hard sphere fluid, above a threshold volume fraction we generically find two relaxation channels corresponding largely, but not exclusively, to a center-of-mass-like displacement and a radial separation of the two tagged particles. The entropic barriers and mean first passage times are computed and found to systematically vary with volume fraction and initial particle separation; both oscillate as a function of the latter in a manner related to the equilibrium pair correlation function. A dynamic correlation length is estimated as the length scale beyond which the two-particle activated dynamics becomes uncorrelated in space and time, and is found to modestly grow with increasing mean relaxation time. The theory is also applied to a simplified model of cage escape, the elementary step of structural relaxation. Predictions for characteristic relaxation times, translation-relaxation decoupling, and stretched-exponential decay of time correlation functions are obtained. A novel mechanism for understanding why strong decoupling emerges in the activated regime, but stretched nonexponential time correlation functions do not change shape as the mean relaxation time grows, is presented and favorably compared with experiment. The theory may serve as a starting point for constructing a predictive model of multiple correlated caging and hopping (forward and backward) events of a pair of tagged particles.     

Coupled Influence of Colloidal and Hydrodynamic Interactions on the RSA Dynamic Blocking Function for Particle Deposition onto Packed Spherical Collectors

The influence of electrostatic double-layer and hydrodynamic interactions on random sequential adsorption (RSA) of colloidal particles onto packed spherical collectors was investigated using inverse analysis of colloid breakthrough data obtained from well-controlled particle deposition experiments. Deposition experiments were carried out using monodisperse aqueous suspensions of positively charged latex colloids and packed columns of negatively charged uniform glass beads for different combinations of ionic strength, particle size, and approach velocity. From the experimental particle breakthrough data, the initial particle deposition rates and the virial coefficients of the dynamic blocking function based on RSA mechanics were determined. The magnitudes of the virial coefficients were observed to increase from the hard sphere values with increasing flow rates and decreasing ionic strengths of the background electrolyte. Particle size also plays a significant role in governing the deposition dynamics. The deviation from the hard sphere RSA behavior becomes more prominent for larger particles. Copyright 2000 Academic Press.