Non-linear screening of spherical and cylindrical colloids: the case of 1:2 and 2:1 electrolytes

From a multiple scale analysis, we find an analytic solution of spherical and cylindrical Poisson-Boltzmann theory for both a 1:2 (monovalent co-ions, divalent counter-ions) and a 2:1 (reversed situation) electrolyte. Our approach consists in an expansion in powers of rescaled curvature $1/(\kappa a)$, where $a$ is the colloidal radius and $1/\kappa$ the Debye length of the electrolytic solution. A systematic comparison with the full numerical solution of the problem shows that for cylinders and spheres, our results are accurate as soon as $\kappa a>1$. We also report an unusual overshooting effect where the colloidal effective charge is larger than the bare one.Comment: 9 pages, 11 figure

Guest charges in an electrolyte: renormalized charge, long- and short-distance behavior of the electric potential and density profile

We complement a recent exact study by L. Samaj on the properties of a guest charge $Q$ immersed in a two-dimensional electrolyte with charges $+1/-1$. In particular, we are interested in the behavior of the density profiles and electric potential created by the charge and the electrolyte, and in the determination of the renormalized charge which is obtained from the long-distance asymptotics of the electric potential. In Samaj's previous work, exact results for arbitrary coulombic coupling $\beta$ were obtained for a system where all the charges are points, provided $\beta Q<2$ and $\beta < 2$. Here, we first focus on the mean field situation which we believe describes correctly the limit $\beta\to 0$ but $\beta Q$ large. In this limit we can study the case when the guest charge is a hard disk and its charge is above the collapse value $\beta Q>2$. We compare our results for the renormalized charge with the exact predictions and we test on a solid ground some conjectures of the previous study. Our study shows that the exact formulas obtained by Samaj for the renormalized charge are not valid for $\beta Q>2$, contrary to a hypothesis put forward by Samaj. We also determine the short-distance asymptotics of the density profiles of the coions and counterions near the guest charge, for arbitrary coulombic coupling. We show that the coion density profile exhibit a change of behavior if the guest charge becomes large enough ($\beta Q\geq 2-\beta$). This is interpreted as a first step of the counterion condensation (for large coulombic coupling), the second step taking place at the usual Manning--Oosawa threshold $\beta Q=2$

Description beyond the mean field approximation of an electrolyte confined between two planar metallic electrodes

We study an electrolyte confined in a slab of width $W$ composed of two grounded metallic parallel electrodes. We develop a description of this system in a low coupling regime beyond the mean field (Poisson--Boltzmann) approximation. There are two ways to model the metallic boundaries: as ideal conductors in which the electric potential is zero and it does not fluctuate, or as good conductors in which the average electric potential is zero but the thermal fluctuations of the potential are not zero. This latter model is more realistic. For the ideal conductor model we find that the disjoining pressure is positive behaves as $1/W^3$ for large separations with a prefactor that is universal, i.e. independent of the microscopic constitution of the system. For the good conductor boundaries the disjoining pressure is negative and it has an exponential decay for large $W$. We also compute the density and electric potential profiles inside the electrolyte. These are the same in both models. If the electrolyte is charge asymmetric we find that the system is not locally neutral and that a non-zero potential difference builds up between any electrode and the interior of the system although both electrodes are grounded.Comment: 16 pages, 5 figures, added a new appendix B and a discussion on ideal conductors vs. good conductor

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Equation of state in the fugacity format for the two-dimensional Coulomb gas

We derive the exact general form of the equation of state, in the fugacity format, for the two-dimensional Coulomb gas. Our results are valid in the conducting phase of the Coulomb gas, for temperatures above the Kosterlitz-Thouless transition. The derivation of the equation of state is based on the knowledge of the general form of the short-distance expansion of the correlation functions of the Coulomb gas. We explicitly compute the expansion up to order $O(\zeta^6)$ in the activity $\zeta$. Our results are in very good agreement with Monte Carlo simulations at very low density

Renormalized Surface Charge Density for a Strongly Charged Plate in Asymmetric Electrolytes: Asymptotic Exact Results in Poisson Boltzmann Theory

The Poisson-Boltzmann equation for a strongly charged plate inside a generic charge-asymmetric electrolyte is solved using the method of asymptotic matching. Both near field and far field asymptotic behaviors of the potential are systematically analyzed. Using these expansions, the renormalized surface charge density is obtained as an asymptotic series in terms of the bare surface charge density.Comment: 11 pages, 4 eps figure

The interplay between screening properties and colloid anisotropy: towards a reliable pair potential for disc-like charged particles

The electrostatic potential of a highly charged disc (clay platelet) in an electrolyte is investigated in detail. The corresponding non-linear Poisson-Boltzmann (PB) equation is solved numerically, and we show that the far-field behaviour (relevant for colloidal interactions in dilute suspensions) is exactly that obtained within linearized PB theory, with the surface boundary condition of a uniform potential. The latter linear problem is solved by a new semi-analytical procedure and both the potential amplitude (quantified by an effective charge) and potential anisotropy coincide closely within PB and linearized PB, provided the disc bare charge is high enough. This anisotropy remains at all scales; it is encoded in a function that may vary over several orders of magnitude depending on the azimuthal angle under which the disc is seen. The results allow to construct a pair potential for discs interaction, that is strongly orientation dependent.Comment: 13 pages, 19 figure

On the fluid-fluid phase separation in charged-stabilized colloidal suspensions

We develop a thermodynamic description of particles held at a fixed surface potential. This system is of particular interest in view of the continuing controversy over the possibility of a fluid-fluid phase separation in aqueous colloidal suspensions with monovalent counterions. The condition of fixed surface potential allows in a natural way to account for the colloidal charge renormalization. In a first approach, we assess the importance of the so called ``volume terms'', and find that in the absence of salt, charge renormalization is sufficient to stabilize suspension against a fluid-fluid phase separation. Presence of salt, on the other hand, is found to lead to an instability. A very strong dependence on the approximations used, however, puts the reality of this phase transition in a serious doubt. To further understand the nature of the instability we next study a Jellium-like approximation, which does not lead to a phase separation and produces a relatively accurate analytical equation of state for a deionized suspensions of highly charged colloidal spheres. A critical analysis of various theories of strongly asymmetric electrolytes is presented to asses their reliability as compared to the Monte Carlo simulations

Collective modes and correlations in one-component plasmas

The static and time-dependent potential and surface charge correlations in a plasma with a boundary are computed for different shapes of the boundary. The case of a spheroidal or spherical one-component plasma is studied in detail because experimental results are available for such systems. Also, since there is some knowlegde both experimental and theoretical about the electrostatic collective modes of these plasmas, the time-dependent correlations are computed using a method involving these modes.Comment: 20 pages, plain TeX, submitted to Phys. Rev.