Hard-sphere Part Research Articles

Dependence of the condensate chemical potential on droplet size in thermodynamics of heterogeneous nucleation within the gradient DFT

A scheme of computation of the condensate chemical potential per molecule as a function of the droplet equimolecular radius for stable and critical droplets on uncharged or charged spherical particle of molecular size at heterogeneous nucleation has been considered. The scheme is based of the gradient density functional theory (DFT) with the van der Waals (vdW) and Carnahan–Starling (СS) models for the hard-sphere contribution to intermolecular interaction in liquid and vapor phases and interfaces. The particle serving as a condensation center in the case of heterogeneous nucleation has been characterized by an attractive short-range molecular potential and the long-range electric Coulomb potential. The dielectric permittivities of the droplet–vapor systems have been taken as known functions of the local condensate density and temperature for polar and nonpolar fluids. Detailed numerical calculations of the density profiles in critical and stable equilibrium droplets at water or argon nucleation in presence of the capillary, electrostatic and molecular forces have been performed. Dependence of the condensate chemical potential in the droplet on the droplet equimolecular radius has been analyzed in the case of homogeneous and heterogeneous nucleation and compared for water and argon within the vdW and СS models for the hard-sphere part of the equation of state.

The two-Yukawa model and its applications: the cases of charged proteins and copolymer micellar solutions

Charged and uncharged colloidal systems are known from experiment to display an extremely rich phase behavior, which is ultimately determined by the effective pair potential between particles in solution. As a confirmation, the recent striking observation of an equilibrium cluster phase in charged globular protein solutions [Stradner, Sedgwick, Cardinaux, Poon, Egelhaaf & Schurtenberger (2004). Nature, 432, 492–495] has been interpreted as the effect of competing short-range attractive and long-range repulsive interactions. The `two-Yukawa (2Y) fluid' model assumes an interparticle potential consisting of a hard core plus an attractive and a repulsive Yukawa tail. We show that this rather simple model can indeed explain satisfactorily the structural properties of diverse colloidal materials, measured in small-angle neutron scattering (SANS) experiments, including the cases of equilibrium cluster formation and soft-core repulsion. We apply this model to the analysis of SANS data from horse-heart cytochrome c protein solutions (whose effective potential can be modeled as a hard-sphere part plus a short-range attraction and a weaker screened electrostatic repulsion) and micellar solutions of a triblock copolymer (whose effective potential can be modeled as a hard-sphere part plus a repulsive shoulder and a short-range attraction). The accuracy of the 2Y model predictions is successfully tested against Monte Carlo simulations in both cases.

First-order correction to classical nucleation theory: A density functional approach

It is shown that the classical expression for the change in grand potential of a system on formation of a cluster of radius R is modified by a factor [1−(2w+6δT)/R], to first order in 1/R, where w is a correction due to the nonzero compressibilities of liquid and vapor (near the triple point, w is approximately equal to the product of liquid compressibility and surface tension), and δT is the coefficient in the expression relating the surface tension of the droplet, γ(R), to the planar surface tension, γ∞, i.e., γ(R)=γ∞(1−2δT/R). An expression for δT is derived involving the pair and triplet correlation functions and the density profile of the planar surface. This complements the expression for δT involving the pair distribution function derived by Blokhuis and Bedeaux; the equivalence of the two expressions in the low density limit is demonstrated. Calculations of δT and w are performed using mean-field density functional theory for the Yukawa potential and an r−6 potential, as well as using the square-gradient approximation. δT is found to be negative for all conditions investigated; its magnitude depends on the potential used, and tends to increase with increasing temperature. However, the ratio δT/w is found to be relatively insensitive to potential and to temperature, being between about −1.2 and −1.5 for the conditions investigated. The effect of using a weighted density approximation in place of the local density approximation for the hard-sphere part of the potential is estimated in a square-gradient approximation and found to be small.

Charged hard spheres in a uniform neutralizing background using ‘mixed’ integral equations

The structure and thermodynamic properties of a collection of charged hard spheres immersed in a uniform neutralizing background are studied using ‘mixed’ integral equations, wherein the Percus-Yevick approximation is used with the hard-sphere part of the potential and the hypernetted-chain approximation for the correction due to the Coulomb tail. Numerical solutions are presented along a particular isotherm and a comparison is made with the results of the Mean Spherical Model for the same system.

A Semi-theoretical van der Waals Equation for Calculation of Vapor-Liquid Equilibria at High Pressures

An equation of the van der Waals type is derived and applied to the calculation of vapor–liquid equilibria at elevated pressures. For the hard-sphere part of the equation the theoretical relation of Carnahan–Starling is used. The effect of intermolecular forces is represented by the usual van der Waals term but the "constant a" is permitted to vary with temperature and density as dictated by the experimental PVT data for argon; this variation of the "constant a" is generalized for other simple fluids through corresponding states. While this treatment is of little interest for representing the configurational properties of pure fluids, it is useful for calculating the properties of simple fluid mixtures. Illustrative vapor–liquid equilibrium calculations are performed for binary and ternary mixtures of nitrogen, argon, and oxygen at 6 and 18 atm.