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 ($\tau$) electroneutral salt-free suspension, made up fixed colloids, with $N$ neutralizing mobile counterions. For odd, double-layers never decouple, irrespective their separation $L$; this is regime like-charge attraction, where $\tau$ exhibits diffusive scaling in $L^2$ large $L$. On other hand, even $N$, $L$ no longer relevant length scale setting time; role played by Bjerrum length. This leads to distinctly different dynamics: even, thermal effects are detrimental relaxation, increasing $\tau$, while they accelerate odd. Finally, also show that mean-field theory recovered moreover, it remains operational treatment down relatively small values ($N>3$).