Note: Renormalized jellium model for charged colloids revisited

Abstract

Recently, it has been demonstrated that electrostatic interactions between charged colloids in contact with a symmetric electrolyte can accurately be described by using the so-called renormalized jellium (RJ) model proposed by Trizac and Levin.1 This mean-field approximation, based on the Poisson–Boltzmann equation (PBE), requires that the charge of the background around a tagged colloid be renormalized self-consistently to coincide with the colloid effective charge; such a requirement leads to precise thermodynamic properties even in highly charged colloidal suspensions.2–5 In this note, we revisit the renormalized jellium approximation and, in particular, provide a simple explicit recipe that facilitates the renormalization procedure being easy of implementing in different situations. We here apply such recipe in the case of charged spheres and discuss its extension to the case of charged rods. The system under consideration is a charged colloidal suspension of volume fraction η composed of, obviously, charged colloids of radius a, and counter ions in contact with a symmetric (1:1) salt reservoir of concentration 2cs , with cs the density of positive or negative salt ions; the solvent is included through the dielectric constant e. The RJ model assumes that the charge of Nc − 1 colloidal particles around a tagged macroion is smeared out in the whole suspension to form a homogeneous background with charge Zbacke, e being the elementary charge. This background charge is Zback = Zbare, with Zbare the colloidal bare charge, and not known a priori, but it is determined self-consistently to be equal to the effective charge, Zeff, of the tagged macroion.1 An extraordinary advantage of the RJ model is that Zeff is directly associated to the system osmotic pressure, the screening parameter, and the effective pair interaction between colloids (when it is explicitly considered at the level of the Yukawa approximation).1, 2 Nevertheless, since it is an explicit function of the system state, i.e., Zeff = Zeff(η, Zbare, cs), its evaluation depends on the specific conditions of the system and, currently, the entire charge renormalization procedure is carried out through an iterative process.6 This iterative procedure demands the construction of a function of the form Zeff = Zeff(Zback, Zbare). The selfconsistently condition is reached in the intersection of the function with the straight line Zeff = Zback. However, it is still possible to reformulate this scheme and avoid the full iterative procedure to gain clarity in the way in which the RJ can be applied and extended to study the physical properties of charge-stabilized colloidal suspensions. The complete original procedure can then be reformulated as follows. We start with the requirement of selfconsistency from the beginning. This means that the condition Zback = Zeff must be explicitly incorporated into the PBE. Our main assumption is that there exists a unique Zeff for a given Zbare; this avoids completely the inclusion of Zback and facilitates drastically the renormalization scheme. Additionally, one should also rephrase the typical boundary conditions (BCs) to solve the PBE with simple BCs at one point [see Eqs. (3) and (4)]. To achieve this, we use the fact that the farfield solution of the PBE has a Yukawa-like form that depends on Zeff. Therefore, in the case of charged spherical colloids, the PBE now reads d2φ dr2 + 2 r dφ dr = −3η ZeffλB a − ρ+(∞)e−φ + ρ−(∞)e, (1)