Electrohydrodynamic patterns in macroion dispersions under a strong electric field

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Abstract
Recent reports have shown that initially homogeneous solutions of charged colloidal particles or polyelectrolytes may develop instabilities under strong electric field. In particular, striking dynamical structures forming quasi-stationary zigzag patterns have been observed, under strong ac electric field, when these macroion dispersions are confined into a slab cell. We develop in this paper the basis of a theoretical approach aimed at describing the large scale, long time electrokinetic phenomena occurring, under strong electric field, within a dispersion of macroions in a simple electrolyte of high ionic strength. We assume that the macroions’ charges can be described, at large length scales, by a smooth charge profile that merely generates some small perturbations on the already out-of-equilibrium situation of a simple electrolyte under strong electric field. This allows us to overcome the complexity of the nonlinear electrokinetic equations by expanding them around the far-from-equilibrium system with no macroion. This approach is therefore to be contrasted with the classical theory for which the perturbations of the ionic concentrations are evaluated as linear responses to a “weak” applied electric field with respect to their equilibrium distributions around a macroion at rest. We show here that the out-of-equilibrium ionic distributions in the solution are perturbed over large length scales in the vicinity of the macroions, which leads to the breakdown of (equilibrium) electroneutrality in the solution far beyond the Debye length scales. The electrical body force arising from the coupling between this large scale charge density and the applied electric field eventually triggers some electrohydrodynamic flows which, in turn, convect the very slowly diffusing macroions in the solution. Numerical resolutions of the model in two analytical limit regimes show that this process is able to select quasistationary dynamical patterns from preexisting inhomogeneous distributions of macroions, in good agreement with experimental observations. In addition, we show, using simple dynamical scaling arguments, that this nonlinear coupling between the macroion density fluctuations and the associated electrohydrodynamic flows dominates the large scale, long time stochastic dynamics of the macroion distribution, suggesting that it might also be responsible, through a noise-driven process, for the primary segregation itself.