Anomalous Diffusion in Driven Electrolytes due to Hydrodynamic Fluctuations

Abstract

The stochastic dynamics of tracers arising from hydrodynamic fluctuations in a driven electrolyte is studied using a self-consistent field theory framework in all dimensions. A plethora of scaling behaviour including two distinct regimes of anomalous diffusion is found, and the crossovers between them are characterized in terms of the different tuning parameters. A short-time ballistic regime is found to be accessible beyond two dimensions, whereas a long-time diffusive regime is found to be present only at four dimensions and above. The results showcase how long-ranged hydrodynamic interactions can dominate the dynamics of non-equilibrium steady-states in ionic suspensions and produce strong fluctuations despite the presence of Debye screening.

Figures (2)

• Figure 1: An electrolyte driven by an externally applied electric field $E$ produces body forces on the positive and negative ions in the fluid. These fluctuating body forces lead to the generation of long-ranged flow fields that can be measured through the stochastic trajectories of tracer particles.
• Figure 2: Summary of the different regimes of the dynamics in different dimensions. (a) In $d=1$, there is a crossover between two super-ballistic anomalous regimes occurring at $\tau^{(1)}_{\textrm{a}\times}=C_0^{2/3} D^{1/3} \lambda_{\rm e}^{-4/3}$. (b) In $d=2$, there is only one ballistic regime observed at all times. (c) In $d=3$, the dynamics shows a crossover from ballistic to a first anomalous regime at the time-scale $\tau^{(3)}_{\textrm{b}\textrm{a}}=a^2 D^{-1}$, followed by another crossover to a second anomalous regime at the time-scale $\tau^{(3)}_{\textrm{a}\times}=C_0^{2} D^{3} \lambda_{\rm e}^{-4}$. (d) In $d=4$, there is a crossover from a ballistic regime to a diffusive regime at the time-scale $\tau^{(4)}_{\textrm{b}\textrm{d}}=C_0^{1/2} \lambda_{\rm e}^{-1} a^2 \sqrt{\ln(L/a)}$. In $d>4$, the same crossover occurs from ballistic to diffusive behaviour at the time-scale $\tau^{(d)}_{\textrm{b}\textrm{d}}=C_0^{1/2} \lambda_{\rm e}^{-1} a^{d/2}$.