Correlation Functions Of Fluids Research Articles

Analytic results for the three- and four-particle correlation functions of the fluid of hard disks.

Analytic expressions for the three- and four-particle correlation functions of the hard disk fluid in terms of geometric functionals have been derived using the combined geometric/thermodynamic idea of Boublik [Fluid Phase Equilib. 316, 1 (2012)]. To assess the approximate result, extensive Monte Carlo simulations were performed to obtain the triplet correlation function, g3, over a wide range of densities. We then focussed on the rolling contact configurations that are required in the second-order thermodynamic perturbation theory of patchy colloid models. The analytic approximation is shown to perform very well, and the simulation data have therefore been parametrized using Boublik's functional form for their subsequent use.

A comparison of the correlation functions of the Lennard–Jones fluid for the first-order Duh–Haymet–Henderson closure with molecular simulations

ABSTRACTFirst-order integral equation theories are much more computationally efficient than second-order theories, but the latter are usually much more accurate for computing correlation functions of fluids. We here test the accuracy of the Duh–Haymet–Henderson (DHH) integral equation theory by comparing radial distribution, cavity correlation and bridge functions computed from DHH, first-order and second-order Percus–Yevick theories, with molecular dynamics calculations for the Lennard–Jones fluid. We find that the DHH theory is almost as accurate as the second-order Percus–Yevick theory at liquid-like densities for both sub- and super-critical temperatures. However, the accuracy of the DHH theory decreases with decreasing density. The correlation functions computed from DHH theory are very similar to those computed from first-order Percus–Yevick theory at low densities. The cavity correlation and bridge functions at low densities computed from these two theories are qualitatively different from results computed from molecular simulations. However, the radial distribution functions computed from all three methods are essentially identical at low densities, indicating that errors in the cavity correlation and bridge functions at low densities cancel out to give high accuracy in the radial distribution function.

Perturbation theory based direct correlation function for fluids with hard core potentials

Using second-order Barker–Henderson perturbation theory we are able to derive an explicit expression for the direct correlation function of fluids with hard core potentials. Using the obtained direct correlation function, one can explicitly calculate all thermodynamic properties of simple fluids with hard core potentials. Comparisons with computer simulation data show good agreement for both thermodynamic properties and the static structure factor of the hard core double Yukawa potential.

Ising-like criticality derived from the theory of fluids

It is shown that the conventional theory based on the integral equations for the correlation functions of fluid allows for a reasonable scaling analysis of the critical phenomena. The calculated critical exponents are in rather good agreement with the experimental data for real media, but some of the exponents (first of all, Fisher's exponent eta ) differ from the values predicted by the three-dimensional Ising model. The possibility to obtain Ising-like criticality from the statistical theory of fluids means that the latter does not disagree with the Kadanoff-Wilson-Fisher renormalization group approach.

Model colloidal fluid with competing interactions: Bulk and interfacial properties

Using a simple mean field density functional theory (DFT), the authors investigate the structure and phase behavior of a model colloidal fluid composed of particles interacting via a pair potential which has a hard core of diameter sigma, is attractive Yukawa at intermediate separations, and is repulsive Yukawa at large separations. The authors analyze the form of the asymptotic decay of the bulk fluid correlation functions, comparing results from DFT with those from the self-consistent Ornstein-Zernike approximation (SCOZA). In both theories the authors find rich crossover behavior, whereby the ultimate decay of correlation functions changes from monotonic to long wavelength damped oscillatory decay on crossing certain lines in the phase diagram or sometimes from oscillatory to oscillatory with a longer wavelength. For some choices of potential parameters the authors find, within the DFT, a lambda line at which the fluid becomes unstable with respect to periodic density fluctuations. SCOZA fails to yield solutions for state points near such a lambda line. The propensity towards clustering of particles, which is reflected by the presence of a long wavelength (>>sigma) slowly decaying oscillatory pair correlation function, and a structure factor that exhibits a very sharp maximum at small but nonzero wave numbers, is enhanced in states near the lambda line. The authors present density profiles for the planar liquid-gas interface and for fluids adsorbed at a planar hard wall. The presence of a nearby lambda transition gives rise to pronounced long wavelength oscillations in the one-body density profiles at both types of interface.

Static properties and time correlation functions of fluids with steeply repulsive potentials

There has been much interest in recent years in the properties of fluids composed of particles that interact through potentials with a ‘tunable’ softness, particularly for highly repulsive particles close to the hard-sphere limit. Much of this interest is driven by the granular media and colloid communities, but there are fundamental issues concerning the physics of fluids in general that can be addressed with such model systems. In this report we continue our series of investigations into the properties of an example of such a fluid, the so-called inverse power or soft-sphere fluid which is composed of particles interacting through a pair potential, ϕ (r)=ϵ (σ/r) n , where n measures the steepness or stiffness of the potential. We review the current state of our knowledge of the properties of such fluids and point out some still unresolved areas. We present the results of new computations for a range of n values and densities. Interest is focused on the time correlation functions as exemplified by the force autocorrelation function, CF (t), and the shear stress correlation function, C s(t). Powles and Heyes [POWLES, J. G. and HEYES, D. M., 2000, Molec. Phys., 98, 917.] showed that at short times the shear stress autocorrelation function fits quite closely to the analytic form, where x is a reduced time that incorporates the effects of n and temperaure, x = (T)*1/2 nt*. T* and T* are the reduced temperature and time in particle units consisting of particle mass and the pair potential parameters, ϵ and σ. We give further supporting evidence for this functional form, both for C s and CF . However, we still lack a theoretical explanation for this behaviour. A formal time expansion of CF (t) with similar approximations to those used in our previous publications for the O(t 2) coefficient predicts a Gaussian analytic form for C(t), therefore posing an unresolved paradox at present. We present a general procedure by which we can derive formally exact expressions for static properties (such as the interaction energy, pressure and infinite frequency elastic moduli) that can be conveniently reduced to relatively simple approximate expressions involving n and an equivalent hard-sphere density and equation of state, which become more accurate in the hard-sphere limit. We also derive a simple (approximate) analytic formula for the mean square force on a particle in terms of n and the hard-sphere equation of state, which is accurate in the hard-sphere limit. We examine the radial distribution functions of these fluids for various n and density, and compare their forms close to contact with the hard-sphere radial distribution functions with appropriate diameters.

Structure and freezing of a fluid of long elongated molecules

The pair correlation functions of a fluid of long elongated molecules interacting via theGay–Berne pair potential are calculated using the Percus–Yevick integral equation theory.Numerical accuracy has been examined by considering a large number of sphericalharmonic coefficients for each orientation-dependent functions for a system of moleculeshaving a length-to-breadth ratio equal to 4.4 at different densities and temperatures. Thepair correlation functions of the isotropic fluid found from the Percus–Yevick theoryhave been used in the density-functional theory to locate the isotropic–nematic,isotropic–smectic A and nematic–smectic A transitions. It is found that at lowtemperatures the fluid freezes directly into the smectic A phase on increasing thedensity. The nematic phase is found to stabilize in between the isotropic andsmectic A phases only at high temperatures and high densities. The calculatedphase diagram is in good qualitative agreement with computer simulation results.

Viscoelasticity of fluids with steeply repulsive potentials

We consider the infinite frequency moduli and time correlation functions of fluids composed of particles that interact through a steeply repulsive potential of the general analytic form, φ(r)=kBT exp[−αf(r)], where α is a measure of the steepness or stiffness of the potential. Although these potentials have different analytical forms, in the steeply repulsive limit of α→∞, the derived properties become almost identical and are only dependent on the value of α and other basic variables. All the infinite frequency moduli which we study are proportional to α and the interaction part of the pressure is only weakly dependent on α. For the force and other configurational property time functions C(t), time t can be replaced by αt, i.e., C(t)=1−T*(αt*)2+O[(αt*)4], where T*=kBT/ε, is the reduced temperature, kB is Boltzmann’s constant, where ε is a characteristic energy for the potential, and t* is a reduced time. We proved this in earlier publications for an inverse power, r−n potential (where α≡n), and show here this more general relationship. The effective hard-sphere diameter by the Barker–Henderson equation, and an alternative prescription derived here, give to first order in α−1 the same formula for the effective hard-sphere diameter for these potentials. We have carried out molecular-dynamics simulations that confirm the equivalence in the steeply repulsive limit of both the static and dynamical properties of two such potentials, which have an inverse power r−n and exponential potential exp(−κr) form. We consider that the theory for the infinite frequency shear rigidity modulus presented here could be usefully applied to predict the infinite frequency storage modulus of colloidal liquids.

Thermodynamic and structural properties of Yukawa hard chains

The thermodynamic and structural properties of Yukawa hard chains are considered in this work. We specifically examined the influence of the Yukawa potential range parameter λ on the thermodynamic properties, inter- and intrachain correlation functions of the chain fluid. The compressibility factor or pressure of Yukawa chain fluids was calculated via a first-order perturbation theory using hard-sphere chain fluid as a reference. Monte Carlo simulations were performed to calculate the thermodynamic properties and inter- and intrachain correlation functions of the fluid for λ=1.8 and 3.0. Simulation results for the compressibility were compared with calculations from the first-order perturbation theory and the Statistical Associating Fluid Theory-Variable Range (SAFT-VR) model. We found that both theories represent the simulated data very well. In particular, the SAFT-VR equation provides an excellent estimate of the properties of Yukawa hard chains over wide ranges of temperature and density. Simulated data for interchain and intrachain correlation functions reveal that the local structure of the Yukawa hard chains is strongly effected by the range parameter of the potential.

Calculation of the Effective Absorption Coefficient in Multi‐Phase Powders

A calculation of the effective absorption coefficient μeff in multi‐phase powders is presented. Characteristic in the calculation is, that the distribution function of the grains is similar to the correlation function for fluids. It is shown, that for average grain diameters comparable to the inverse of the absorption coefficients, the commonly used formulae for μeff (either a linear combination of the contituting μ′s or simply μ for a single component) do not hold.The analytical expressions derived for μeff are compared with numerical simulations, which show that, within certain limits, the former give reasonable estimates of μeff. The simulations also show, however, that shape effects can have influence on μeff which are not included (yet) in the formulae.Also shown is that surface roughness can give effects comparable to those of microabsorption, but can be much more sensitive to particle statistics.

Hydrodynamic boundary conditions, correlation functions, and Kubo relations for confined fluids.

Dynamical correlation functions of a fluid slab confined between two solid walls are computed using a phenomenologial, hydrodynamic appoach that generalizes Onsager's principle of linear regression of fluctuations to inhomogeneous systems. The phenomenological results are compared to exact molecular-dynamics simulations on simple model systems. This comparison permits a determination of the phenomenological parameters that describe the hydrodynamics of the fluid slab and especially of the boundary conditions (BC's) that account for the presence of solid walls. In most cases, the hydrodynamic BC for the tangential velocity field is found to be a no-slip BC and must be applied in a plane that is separated from the solid by about one layer of fluid atoms. A set of formal relations between the parameters that characterize the hydrodynamic BC and the equilibrium correlation functions of the inhomogeneous fluid is also derived. These relations are analogous to the usual Green-Kubo equalities for the transport coefficients of bulk fluids.

INTEGRAL EQUATIONS FOR THE CORRELATION FUNCTIONS OF FLUIDS NEAR WALLS

INTEGRAL EQUATIONS FOR THE CORRELATION FUNCTIONS OF FLUIDS NEAR WALLS

The calculation of correlation functions in fluids by a weighted density concept

An exact equation for the pair correlation function of fluids is derived by density functional methods. Comparing this result with the hypernetted chain (HNC) equation we propose a weighted density approximation for the so-called bridge function that improves the ordinary HNC theory. For a hard sphere fluid this method leads to thermodynamically more consistent results than the HNC, Percus-Yevick (PY) and Rogers-Young (RY) approximations. Adding an attractive potential to the hard sphere repulsion we obtain better correlation functions than can be found with the HNC and the reference HNC (RHNC) approximation.

A triplet correlation function for Lennard-Jones fluids

A triplet correlation function for Lennard-Jones fluids is calculated with the inhomogeneous integral equation method recently introduced by Attard for hard core fluids. The results are compared with those of molecular dynamics simulations. A simple extension of the mixed closure for the bulk fluids successfully reproduces the correlation function for the fluids near the triple point.

Integral equation theory for the adsorption of chain fluids in slitlike pores

The adsorption of hard chains in slitlike pores is studied via an integral equation theory. The theory uses the growing adsorbent model in conjunction with the polymer reference interaction site model (polymer-RISM) theory to predict the wall–fluid correlation functions; wall–fluid and fluid–fluid correlations are treated using the Percus–Yevick closure relations. The theory is compared to Monte Carlo simulation data for the density profiles of 4-mers and 8-mers in slitlike pores; the theory is fairly accurate, though it tends to overestimate the density at the wall at low densities and underestimate the density at the wall at high densities. The theory also underestimates the amplitude of the oscillations in the density profile. The theory is very accurate, however, for the adsorption isotherm of 4-mers and 8-mers.

Asymptotics of the correlation functions for fluids

An equation based on the Ornstein–Zernike theory is suggested to describe the asymptotic behavior of the two-particle distribution function. The equation is shown to be the general one. It is possible to study the long-range part of the correlation function for any interparticle interaction potential as well as for any closure of the Ornstein–Zernike equation. Calculations are performed for a hard-sphere particle system for hypernetted chain, Percus–Yevick, and Martynov–Sarkisov approximate integral equations.

Triplet correlation functions for fluids of nonspherical molecules

Computer simulations of orientationally averaged triplet correlation functions are presented for fluids of several nonspherical molecules. These specific models considered included hard prolate and hard oblate molecules as well as a site-site plus quadrupole representation of CO2. A spherical harmonic expansion of the triplet function is written down and utilized in connection with an extended superposition approximation (ESA) which is based on the spherical harmonic expansions of the orientation-dependent pair correlation functions. The ESA is evaluated for the systems studied and the results are shown to be in good agreement with the directly simulated orientationally-averaged triplet function. In addition, a series representation of the functions of r 12, r 13, r 23 (the sides of the 1, 2, 3 triangle) which had been previously suggested in connection with simulations of the triplet correlation function of spherical particles is considered here for nonspherical particles. It is shown that this approach ...

Extraction of transport coefficients from correlation functions.

In this paper we apply a new method to the calculation of transport coefficients by computer simulation of fluids. It differs from previous methods in two essential respects. Firstly, it involves the simultaneous fitting of all of the correlations among conserved hydrodynamic variables, rather than just the density or current autocorrelation function. This gives more detailed information on the hydrodynamics, and, in particular, makes it possible to separate some of the effects of viscoelasticity, which may seriously bias transport-coefficient estimates. Secondly, it is formulated in a simple way, which avoids certain complicating approximations traditionally made in computing fluid correlation functions. Results of pilot calculations for the soft-sphere viscosity agree with those of previous workers. The most conspicuous physical effect observed is a strong wave-vector dependence of the thermal diffusivity, in agreement with previous work.

Correlation functions in Couette flow from group theory and molecular dynamics

We describe group theory statistical mechanics, GTSM, which enables us to predict new non-vanishing time correlation functions in fluids at steady state subjected to planar couette flow. These are by symmetry trivially zero at equilibrium. An ensemble average is treated using the rules of group theory in the laboratory XYZ frame and in the molecule-fixed xyz frame of the point group character tables. In this paper we determine the effect of couette flow on a range of ensemble averages by establishing the symmetry of the strain rate tensor in terms of the irreducible representations of the Rh (3) rotation reflection group in the XYZ frame. This symmetry, D (0) g + D (1) g + D (2) g , is the same as the pressure tensor, P and consists of an antisymmetric vorticity term, D (1) g and a symmetric strain rate component of symmetry D (0) g + D (2) g . This allows non-zero ensemble averages of the same symmetry in the XYZ frame. Depending on the number of off-diagonal elements in the strain rate tensor, up to six...

Integral equations for continuum percolation

Continuum percolation is studied by means of integral equations for the connectedness function which can be derived from similar equations for the correlation function of fluids with the aid of a diagrammatic expansion. In particular we examine a refinement (which we call PY-d(2)) of the Percus–evick approximation first proposed by Stell and a HNC-type approximation for the connectedness function. These equations are applied to the system of randomly centered spheres, where they are solved numerically. The value of the critical density and the way in which the mean clusters size diverges are given much more accurate by the PY-d(2) approximation than by the PY approximation, while the critical exponents remain unchanged. We find that the HNC approximation does not possess a critical point where the mean clusters size diverges and that the long range behavior of the connectedness bridge function has a more important role in the percolation problem than in the thermal critical point.