Charge renormalization and phase separation in colloidal suspensions

– We explore the effects of counterion condensation on fluid-fluid phase separation in charged colloidal suspensions. It is found that formation of double layers around the colloidal particles stabilizes suspensions against phase separation. Addition of salt, however, produces an instability which, in principle, can lead to a fluid-fluid separation. The instability, however, is so weak that it should be impossible to observe a fully equilibrated coexistence experimentally. Colloidal suspensions present an outstanding challenge to modern theories of statistical mechanics. In spite of the work extending all the way to the beginning of previous century, our understanding of these complex systems remains far from complete. Even such basic questions as what is the form of the interaction potential between two colloidal particles inside a solution, still remains controversial [1]. The suspensions most often studied experimentally consist of polystyrene sulphate spheres with diameter in the range of 10 − 10 cm, and 10 − 10 ionisable surface groups. The typical solvent is water at room temperaure. The main stumbling blocks are the presence of long range Coulomb interactions and the tremendous asymmetry existing between the polyions and their counterions. The ratio of the bare charge of a colloidal particle to that of its counterion can be as high as 10000:1. This large asymmetry completely invalidates most of the methods of liquid state theory, which have proven so successful in the studies of simple molecular fluids. A particularly interesting question that has provoked much controversy over the last two decades concerns itself with the possibility of fluid-fluid phase separation in charged colloidal suspensions. A naive argument, based on the theory of simple molecular fluids, suggests that Typeset using EURO-TEX 2 EUROPHYSICS LETTERS fluid-fluid (or liquid-gas) coexistence is only possible in the presence of sufficiently long ranged attractive interactions. Thus, it has been proposed by some authors that a phase separation, or even existence of voids in charged colloidal lattices requires attraction between the polyions [2]. Although appealing intuitively, this point of view is difficult to justify within the framework of statistical mechanics. The fundamental observation is that colloidal suspension is a complex fluid for which many body effects play the fundamental role. It is, therefore, erroneous to confine attention to pair interactions between the colloidal particles while ignoring the significantly larger contributions to the free energy arising from the presence of counterions [3]. This point has also been emphasized by van Roij and Hansen (RH) [4] who demonstrated, in the context of the linearized density functional theory, existence of “volume” terms, which can drive phase separation even for pairwise repulsive interactions [5]. The prediction of a liquid-gas phase separation in an aqueous solution of like-charged colloidal particles seems, however, to be contradicted by the recent simulations of Linse and Lobaskin [6], who did not find any indication of phase transition in suspensions with monovalent counterions. This apparent discrepancy between the simulations and the density functional theory suggests that a closer look at the mechanism of phase separation is worth while. Since the first simulations were performed in the absence of salt, as a starting point, we shall concentrate our attention on this regime. Our model consists of Np = ρpV spherical polyions of radius a, inside a homogeneous medium of volume V and dielectric constant D. Each polyion carries Z ionized groups of charge −q uniformly distributed over its surface. A total of ZNp monovalent counterions of charge +q are present in order to preserve the overall charge neutrality of solution. In the absence of salt, the counterions can be treated as point-like. All the thermodynamic properties of colloidal suspensions can be determined given the free energy. Unfortunately due to complexity of these systems, no exact calculation is possible and approximations must be used. We construct the total free energy as a sum of the most relevant contributions: electrostatic, entropic, and hard core, f = F/V = fel+fent+fhc. The electrostatic free energy, fel, is the result of polyion-counterion, fpc, and the polyion-polyion, fpp, interactions. Interacions between the monovalent microions are insignificant for aqueous solutions and can be ignored [3]. The polyion-counterion contribution to the total free energy can be obtained in the framework of Debye-Hückel theory [3, 7]. Fixing one colloidal particle at the origin, it is possible to show that the electrostatic potential in its vicinity satisfies the Helmholtz equation∇ψ = κψ, where κa = (4πZρ∗p/T ), and the reduced temperature and density are T ∗ = kBTq D/a and ρ∗p = ρpa , respectively. The electrostatic free energy can be obtained from the solution of the Helmholtz equation followed by the Debye charging process, yielding