Charge inversion of colloids in an exactly solvable model

– We study a two-dimensional model for a long cylindrical stiff charged macroion immersed in a charge-asymmetric electrolyte with charge ratio +2/−1. The model is integrable and it allows an exact analytical determination of the effective charge of the macroion, which characterizes the electrostatic potential at large distances (compared to the screening length) from the macroion. At at high coulombic coupling, this model predicts charge inversion: for a highly negatively charged macroion, the effective charge could become positive, indicating an overscreening of the macroion by the divalent counterions. By studying the behavior of the coions and counterions density profiles close to the macroion, we show that the counterion condensation threshold is shifted to a lower value in absolute value. This plays an important role in the charge inversion phenomenon. The determination of the effective interactions between charged macroions immersed in electrolyte solutions is a central topic in colloidal science [1]. Based on the linear Debye– Hückel (DH) theory, for a very long and thin (zero radius) cylindrical macroion, with linear charge density e/l, the reduced effective potential at a distance r is y(r) = 2λK0(κr), where λ = lB/l, K0 is the modified Bessel function of order 0, y = eψ/(kBT ) with ψ the electric potential, e the elementary charge, T the temperature and kB is Boltzmann constant. Here, λ is the reduced linear charge density, expressed in units of the inverse of the Bjerrum length lB = e /(kBTε), where ε is the electric permittivity of the solvent. The Debye screening length is κ = (4πlB ∑ α n b αv 2 α) , where nα is the bulk density of the microions of the electrolyte of type α and vα their valence. For highly charged macroions, the linear approach is inappropriate. A first improvement over the linear theory is to use the mean field nonlinear Poisson–Boltzmann (PB) equation. Under this approximation, the effective potential, at large distances, is again a screened potential: y(r) ∼ 2λeffK0(κr) for κr ≫ 1, but the prefactor λeff is not anymore the bare charge λ of the colloid, but it is known as the effective or renormalized charge [2, 3]. PB approach is adequate [4,5] if the coulombic couplings between microions are small, for example for a two-component electrolyte: v 1Γ ≪ 1, v 2Γ ≪ 1, |v1v2|Γ ≪ 1 where Γ = 2lB/a, with a the average distance between microions in the electrolyte. For large coupling, new phenomena can occur that cannot be explained by the mean field PB approach. The most striking one is the phenomenon of charge inversion: a large fraction of counterions can condense into the macroion and the resulting dressed macroion can have a charge of opposite sign than