We propose a model for the calculation of renormalized charges and osmotic properties of mixtures of highly charged colloidal particles. The model is a generalization of the cell model and the notion of charge renormalization as introduced by Alexander et al. [J. Chem. Phys. 80, 5776 (1984)]. The total solution is partitioned into as many different cells as components in the mixture. The radii of these cells are determined self-consistently for a given set of parameters from the solution of the nonlinear Poisson-Boltzmann equation with appropriate boundary conditions. This generalizes Alexanders's model where the (unique) Wigner-Seitz cell radius is solely fixed by the colloid packing fraction. We illustrate the technique by considering a binary mixture of the colloids with the same sign of charge. The present model can be used to calculate thermodynamic properties of highly charged colloidal mixtures at the level of linear theories, while taking the effect of nonlinear screening into account.

We study the statistical mechanics of classical Coulomb systems in a low coupling regime (Debye-Hückel regime) in a confined geometry with Dirichlet boundary conditions. We use a method recently developed by the authors which relates the grand partition function of a Coulomb system in a confined geometry with a certain regularization of the determinant of the Laplacian on that geometry with Dirichlet boundary conditions. We study several examples of confining geometry in two and three dimensions and semi-confined geometries where the system is confined only in one or two directions of the space. We also generalize the method to study systems confined in arbitrary geometries with smooth boundary. We find a relation between the expansion for small argument of the heat kernel of the Laplacian and the large-size expansion of the grand potential of the Coulomb system. This allow us to find the finite-size expansion of the grand potential of the system in general. We recover known results for the bulk grand potential (in two and three dimensions) and the surface tension (for two dimensional systems). We find the surface tension for three dimensional systems. For two dimensional systems our general calculation of the finite-size expansion gives a proof of the existence a universal logarithmic finite-size correction predicted some time ago, at least in the low coupling regime. For three dimensional systems we obtain a prediction for the perimeter correction to the grand potential of a confined system.

We study effective colloidal interactions in de-ionized colloidal mixtures through sedimentation-diffusion equilibrium. We derive a coarse-grained effective model ͑EM͒ and compare its density profiles with those of the computationally much more expensive primitive model ͑PM͒ of colloids and counterions in gravity. The EM, which contains not only standard pairwise screened-Coulomb interactions, but also explicit many-body effects by means of a so-called volume term, can quantitatively account for all observed sedimentation phenomena such as lifting of colloids to high altitudes, segregation into layers in mixtures, and floating of heavy colloids on top of lighter ones. Without the volume term there is no quantitative agreement between the PM and EM, even in the present high-temperature limit of interest, showing that de-ionized colloidal suspensions cannot be described by a pairwise Yukawa model. DOI: 10.1103/PhysRevE.77.031402 PACS number͑s͒: 82.70.Dd, 64.10.ϩh Sedimentation of suspended colloidal particles in Earth's gravity field is often a nuisance that is to be avoided, e.g., by careful density matching the colloidal particles with the solvent or by sending samples to outer space ͓1͔. However, it has also become clear over the last decade or so that the study of the colloidal density profile ͑z͒ ͑with z the vertical height͒ in sedimentation-diffusion equilibrium can efficiently give quantitative information about the ͑osmotic͒ pressure ⌸͑͒ of a bulk colloidal suspension at density and hence about the effective colloidal interactions ͓2-4͔. In pioneering studies it was shown that this scheme, which is based on hydrostatic equilibrium, can quantitatively reproduce the known equation of state of hard spheres in the fluid phase ͓2,3͔, and recent extensions show that this method also works for hard-sphere crystals and for sticky spheres ͓4͔. In this paper we exploit the close relationship between effective colloidal interactions and sedimentation-diffusion equilibrium for the case of ͑mixtures of͒ index-matched charged colloidal spheres suspended in a dielectric solvent with monovalent salt ions, as, e.g., studied experimentally in Refs. ͓5,6͔. Sedimentation of charged colloids has turned out to be interesting in its own right, given the rich phenomenology that was found for such systems during the last few yearse.g., the existence ͑in a conducting medium͒ of a macroscopic electric field that pushes the colloids up to relatively large heights ͓5-8͔ and the "colloidal Brazil nut effect" such that the heavier colloids can float on top of a layer of lighter ones ͓9-11͔. In the regime of relatively high salt concentrations, where the ionic strength is dominated by the background electrolyte, these sedimentation phenomena have been explained by two different types of models: ͑i͒ by the multicomponent primitive model ͑PM͒ of mesoscopic charged colloids with subnanometer-sized cations and anions in a continuum solvent ͓7,9͔ and ͑ii͒ by the Yukawa model ͑YM͒ of colloids interacting solely by a pairwise repulsive scre...

Sedimentation-diffusion equilibrium density profiles of suspensions of charge-stabilized colloids are calculated theoretically and by Monte Carlo (MC) simulations, both for a one-component model of colloidal particles interacting through pairwise screened-Coulomb repulsions and for a three-component model of colloids, cations, and anions with unscreened-Coulomb interactions. We focus on a state point for which experimental measurements are available [C. P. Royall, J. Phys.: Condens Matter 17, 2315 (2005)]. Despite the apparently different picture that emerges from the one- and three-component models (repelling colloids pushing each other to high altitude in the former, versus a self-generated electric field that pushes the colloids up in the latter), we find similar colloidal density profiles for both models from theory as well as simulation, thereby suggesting that these pictures represent different viewpoints of the same phenomenon. The sedimentation profiles obtained from an effective one-component model by MC simulations and theory, together with MC simulations of the multicomponent primitive model are consistent among themselves, but differ quantitatively from the results of a theoretical multicomponent description at the Poisson-Boltzmann level. We find that for small and moderate colloid charge the Poisson-Boltzmann theory gives profiles in excellent agreement with the effective one-component theory if a smaller effective charge is used. We attribute this discrepancy to the poor treatment of correlations in the Poisson-Boltzmann theory.

We extend the classical Gouy-Chapman model of two planar parallel interacting double layers, which is used as a first approximation to describe the force between colloidal particles, by considering the finite thickness of the colloids. The formation of two additional double layers due to this finite thickness modifies the interaction force compared to the Gouy-Chapman case, in which the colloids are semi-infinite objects. In this paper we calculate this interaction force and some other size-dependent properties using a mean-field level of description, based on the Poisson-Boltzmann (PB) equation. We show that in the case of finite-size colloids, this equation can be set in a closed form depending on the geometrical parameters and on their surface charge. The corresponding linear (Debye-Hückel) theory and the well-known results for semi-infinite colloids are recovered from this formal solution after appropriate limits are taken. We use a density functional corresponding to the PB level of description to show how in the case where the total colloidal charge is fixed, it redistributes itself on their surfaces to minimize the energy of the system depending on the aforementioned parameters. We study how this charge relaxation affects the colloidal interactions.