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)].

Figure 1

A model thermoelectric cell consisting of a 1:1 electrolyte with Debye length ${\kappa }^{-1}$ (solvent not shown) and two flat electrodes separated over a distance $2L$. At time $t=0$, the temperature of one electrode increases by a factor $1+\epsilon$.

Figure 2

Thermal and ionic relaxation at $n=1$ and $a/D=100$ in response to a thermal quench ($\epsilon ={10}^{-3}$) at $x=L$. Numerical solution (symbols) to Eq. (6) at times $tD/L^2=\{{10}^{-6}-10\}$ for $T$ (a), $\psi$ (b), $q$ (c), and $c$ (d) where obtained at $\xi =1, {\alpha }_{+}=0.5, {\alpha }_{-}=0.1, f=2\times {10}^{-3}$. (a) Also shows Eq. (8) (lines) at the same times and Eq. (9) (plusses) at $tD/L^2={10}^{-3}$. (b)–(d) Also show Eq. (15) with lines. The insets of (c) and (d) show Eq. (12) (dashed lines) at times $tD/L^2=\{{10}^{-6}-{10}^{-3}\}$.

Figure 3

Numerical (symbols) and analytical (lines) results for ${\psi }_1$ (a) and $q_1(x,t)$ (b) at $n=100$. All other parameters, colors, symbols, and line styles are as in Fig. 2.

Figure 4

The relaxation of ${\psi }_1(-L,t),q_1(\pm L,t)$, and $c_1(\pm L,t)$ (black, red, blue) from numerics (symbols) and Eq. (15) (lines) ($j\le 10$), for $n=1$ (a) and $n=100$ (b). Plotted as well are $D{\kappa }^{2}t$ (black dashed) and $2\sqrt{Dt/\left(L^2\pi \right)}$ (red dashed).

Figure 5

Predictions for $V_T\left(t\right)=\epsilon {\psi }_1(-L,t)+O\left({\epsilon }^2\right)$ from a numerical simulation of Eq. (6) (symbols) and from the analytical expression Eq. (16) (lines), on double logarithmic scales (a) and linear scales (b). The dashed line represents $-{\psi }_1(-L,t)$ at times where ${\psi }_1(-L,t)<0$. In (a), going from top to bottom, the lines represent $\xi =\{10,5,2,1,0.5,0.2,0.1\}$. All other parameters are the same as in Fig. 2.