Relaxation Dynamics Of Two Interacting Electrical Double-Layers In A 1d Coulomb System

We consider an out-of-equilibrium one-dimensional model for two electrical double-layers. With a combination of exact calculations and Brownian dynamics simulations, we compute the relaxation time (tau) for an electroneutral salt-free suspension, made up of two fixed colloids, with N neutralizing mobile counterions. For N odd, the two double-layers never decouple, irrespective of their separation L; this is the regime of like-charge attraction, where tau exhibits a diffusive scaling in L (2) for large L. On the other hand, for even N, L no longer is the relevant length scale for setting the relaxation time; this role is played by the Bjerrum length. This leads to distinctly different dynamics: for N even, thermal effects are detrimental to relaxation, increasing tau, while they accelerate relaxation for N odd. Finally, we also show that the mean-field theory is recovered for large N and moreover, that it remains an operational treatment down to relatively small values of N (N > 3).