Transient response of an electrolyte to a thermal quench

We study the transient response of an electrolytic cell subject to a small, suddenly applied temperature increase at one of its two bounding electrode surfaces. An inhomogeneous temperature profile then develops, causing, via the Soret effect, ionic rearrangements towards a state of polarized ionic charge density $q$ and local salt density $c$. For the case of equal cationic and anionic diffusivities, we derive analytical approximations to $q, c$, and the thermovoltage $V_{T}$ for early ($t\ll\tau_{T}$) and late ($t\gg\tau_{T}$) times as compared to the relaxation time $\tau_{T}$ of the temperature. We challenge the conventional wisdom that the typically large Lewis number, the ratio $a/D$ of thermal to ionic diffusivities, of most liquids implies a quickly reached steady-state temperature profile onto which ions relax slowly. Though true for the evolution of $c$, it turns out that $q$ (and $V_{T}$) can respond much faster. Particularly when the cell is much bigger than the Debye length, a significant portion of the transient response of the cell falls in the $t\ll\tau_{T}$ regime, for which our approximated $q$ (corroborated by numerics) exhibits a density wave that has not been discussed before in this context. For electrolytes with unequal ionic diffusivities, $V_{T}$ exhibits a two-step relaxation process, in agreement with experimental data of Bonetti et al. [J. Chem. Phys. 142, 244708 (2015)].