Large Hard Spheres Research Articles

Pair correlation functions for a hard sphere mixture in the colloidal limit

Previously, we have proposed an approximation for the contact values of the radial distribution functions of a hard sphere mixture. For most situations, this approximation is similar to earlier approximations. However, our approximation is quite different in the colloidal limit, where one species is large in size but whose concentration is very dilute. We called this approximation the ad hoc approximation to emphasize that this new approximation was untested. In this paper, we seek to test this ad hoc approximation by computing, by means of a Monte Carlo simulation, the pair correlation functions for a binary hard sphere mixture. We are able to consider mixtures in which the large hard sphere is up to 10 times the size of the small hard sphere and concentrations of the large sphere as small as 1%. While not the colloidal limit, these results are informative. The ad hoc approximation is found to be in good agreement with the simulation results.

Density profiles and pair correlation functions of hard spheres in narrow slits

A hard-sphere fluid confined by hard, structureless, and parallel walls is investigated using a certain version of the weighted density-functional theory. The density profile, the excess coverage, the finite-size contribution to the free energy, the solvation force, and the total correlation function are determined as functions of the slit width L for various bulk number densities ${\mathrm{\ensuremath{\rho}}}_{\mathrm{b}}$. In quantitative agreement with rigorous results, the present version of density-functional theory yields a constant and large but finite number density profile for the limiting case that L is reduced to the diameter of the hard spheres. Within the Derjaguin approximation, the results for the slit geometry allows us to obtain the solvation force between two large hard spheres immersed in a fluid of much smaller hard spheres.

Theory of the depletion force due to rodlike polymers

The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. In this paper, we consider a system of two parallel plates suspended in a semidilute solution of long thin rods of length L and diameter D. By numerically solving an integral equation, which is exact in the “Onsager limit” (D≪L), we obtain the depletion force between the plates. The second integral of this determines (via the Derjaguin approximation) the depletion potential between two large hard spheres of radius R, immersed in a solution of hard rods (satisfying D≪L≪R). The results for this potential are compared with our previous second order perturbation treatment [Y. Mao, M. E. Cates, and H. N. W. Lekkerkerker, Phys. Rev. Lett. 75, 4548 (1995)], as well as with newly computed third order perturbation results. There is good agreement at low and intermediate densities (which validates our numerical procedures for the integral equation) but the gradual failure of the perturbative treatments is revealed as the density increases. From the numerical results, we conclude that for typical colloidal sphere/rod mixtures, the repulsive barrier in the depletion interaction is likely to be less than the thermal energy kBT throughout the semidilute concentration range of the rods. This reverses our previous conclusion, based on second order perturbation theory, that this barrier could easily lead to kinetic stabilization of the mixture. However, it is possible that terms of order D/L (neglected here) could further modify the results. Within the present theory, the main modification to the well-known (attractive) first order interaction, as the density is raised, is by changing the depth and range of the primary attractive well.

Integral equation approaches to mixtures of atomic and molecular fluids

A recent extension to mixtures of Verlet’s closure is applied in conjunction with the Ornstein–Zernike relation to solve the structure and thermodynamics of mixtures of hard-spheres and homonuclear hard-dumbbells. This integral equation (IE), which is proven to be very accurate when compared with simulation data, is used to explore the possibility of phase separation in an asymmetric mixture. While our results do not show evidence of such phase separation in the asymmetric binary hard-sphere mixture studied by Biben and Hansen [T. Biben and J. P. Hansen, Phys. Rev. Lett. 66, 2215 (1991)], an equivalent mixture of hard-dumbbells and large hard-spheres seems to exhibit a certain tendency to phase separate as far as the integral equation results are concerned. Finally, given the ability of this integral equation to reproduce the hard-core system, we have incorporated these results into a previous Reference Hypernetted Chain scheme to treat a mixture of N2 and Ar modeled by means of site–site Lennard-Jones potentials. In consonance with the results for pure fluids, the parameterization of a hard-core reference system with the same molecular shape leads to excellent results both for the structure and thermodynamics of real systems.

Depletion Force in Polydisperse Systems

The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. We calculate the interaction between two flat plates or large hard spheres (of radius R) in a solution of small and polydisperse hard spheres (of average diameter $\bar{\sigma} \ll R$) to second order in small sphere concentration. We find that the effect of polydispersity is to generally smoothen out the depletion force while maintaining the recently predicted repulsive barrier; but for high polydispersities, the repulsion could be eventually wiped out.

Adsorption of fluids on colloidal particles: a density functional approach

Density functional results are presented for adsorption of simple fluids on colloidal particles. Two models of colloidal particles are considered: first a study of adsorption on a large hard sphere, and second the application of the theory to describe adsorption on a large particle interacting via Lennard-Jones type attractive-repulsive forces. In both cases the theoretical predictions are compared with Monte Carlo data. It was found that the accuracy of the theory becomes better when the size of the colloidal particle increases. A study is also reported of wetting and its dependence on the size of the colloidal particles. As expected, the wetting temperature increases with decreasing colloidal particle size.

Density Profiles of Chemically Reacting Simple Fluids in Contact with a Spherical Impenetrable Colloid Particle

Density profiles of chemically reacting hard spheres near a large hard sphere, mimicking a colloid, are studied by means of Percus-Yevick approximation. The model of Cummings and Stell is used for a bulk fluid. We investigate the dependence of the profiles on the curvature of the surface, the bulk density, and the degree of association.

The direct correlation functions and bridge functions for hard spheres near a large hard sphere

The recent Monte Carlo data of Degrève and Henderson for the density profiles of hard spheres near a large hard sphere are used to obtain direct correlation functions and bridge functions for this system both directly, using the Ornstein–Zernike relation and an approximation, due to Verlet, for the bridge function.

Monte Carlo study of the adsorption layers of hard spheres near large hard spheres of varying radii

Monte Carlo simulation results are presented for the fluid density profiles of the inhomogeneous system formed by a large colloidal hard sphere of variable radius immersed in a hard sphere solvent. The behavior of this system should give insight into the contribution of hard core exclusion effects to the structure of colloids. Such exclusion effects are believed to be the source of the oscillatory short range forces, seen in recent experiments, which are not predicted by classical theories. These profiles are reported as a function of the reduced density for inverse radii between zero and unity (in units of the solvent radius). Periodic boundary conditions and the minimum image condition were used only for the solvent hard spheres. They are not used for the large hard sphere because we wish to examine the case where the large sphere is present in zero concentration. Some error is introduced by our ad hoc treatment of the large sphere. We attempt to minimize this error by taking the number of solvent hard spheres in the unit cell to be as large as is practicable. The reliability of this method is tested by comparison of our results with hard sphere radial distribution functions obtained from conventional bulk simulations by considering the case where the large hard sphere has the same diameter as the solvent hard spheres. We also test our method by comparison of the contact values of the density profiles with interpolated values between the known exact values for the case where the colloidal hard sphere has a diameter equal to that of the solvent hard spheres and the case where the colloidal hard sphere is infinitely large. Both tests indicate that our method gives satisfactory results.

The contribution of dipole alignment of the force between macroions in an ion-dipole fluid

The mean spherical approximation integral equation is solved for a pair of large charged hard spheres in an ion-dipole mixture. In this work, the effect of the molecular nature of the solvent is taken into account approximately. Extra terms, not included in the DLVO theory, are found and are significant. The effect of dipole alignment near the macroions is repulsive at large separation but can be attractive at short separations.

Formulas for the solvation force between colloidal particles obtained from the Ornstein–Zernike relation

By considering the Ornstein–Zernike relation for two large spheres in a fluid, in the limit where these spheres are exceedingly large, a formula for the solvation force between the large spheres is obtained. The result conforms with Derjaguin’s approximation. This result is extended to Coulombic systems by plausible arguments. These formulas are applied to large hard spheres in a hard sphere fluid and to charged hard spheres in a fluid of point charges. The results agree with those obtained by other methods.

An explicit expression for the solvent contribution to the force between colloidal particles using a hard sphere model

An important contribution to the force between colloidal particles, which is missing in the DLVO theory, is the effect of volume exclusion among the solvent molecules. If the colloidal particles and solvent molecules are approximated as large and small hard spheres, this volume exclusion force can be calculated from the radial distribution function for a pair of large spheres. Using the Percus—Yevick approximation, an explicit expression for this force, valid for all separations, is botained. In addition, earlier results for the Percus—Yevick hard-sphere radial distribution function and hard-spher/hard-wall interfacial density profile are extended to arbitary distances.

The partial freezing of hard-sphere mixtures

The recently developed density functional theory of freezing is applied to the two-phase coexistence between a fluid of large and small hard spheres and a solid of large spheres embedded in a fluid of small spheres. The presence of such a host fluid of small spheres is shown to be an important aspect of colloidal crystals and of super-ionic-conductor-like phases.

A Monte Carlo study of semi-dilute hard sphere mixtures

We present Monte Carlo results for the high density equation of state and the partial pair distribution functions of semi-dilute solutions of large hard spheres in a solvent of smaller spheres. The simulation data for diameter ratios of 2 : 1 and 3 : 1 agree reasonably well with the predictions of Percus-Yevick theory and its semi-empirical modifications, except for the solute-solute pair distribution function. A free volume analysis of the equation of state data points to the existence of a well defined, concentration dependent, amorphous close packed density which is significantly higher than for a pure hard sphere fluid.

Low-Temperature Ion Mobility in Interacting Fermi Liquids

The temperature dependence of the mobility $\ensuremath{\mu}$ of a heavy ion in a neutral interacting Fermi liquid is found to be of the form $\frac{1}{\ensuremath{\mu}}=R\ensuremath{-}\frac{S{T}^{2}\mathrm{ln}1}{T}+\mathrm{O}({T}^{2})$ for low $T$ where $R$ and $S$ are constants. The ${T}^{2}\mathrm{ln}T$ term is a consequence of the Friedel density oscillations around the ion and would be absent for the noninteracting Fermi liquid investigated by other authors. The coefficients are calculated for the case of a large hard sphere. The pressure dependence of the coefficients and the predicted temperature dependence are in reasonable agreement with recent experimental data for negative ions in ${\mathrm{He}}^{3}$.