Field-Dependent Ionic Conductivities from Generalized Fluctuation-Dissipation Relations

We derive a relationship for the electric field dependent ionic conductivity in terms of fluctuations of time integrated microscopic variables. We demonstrate this formalism with molecular dynamics simulations of solutions of differing ionic strength with implicit solvent conditions and molten salts. These calculations are aided by a novel nonequilibrium statistical reweighting scheme that allows for the conductivity to be computed as a continuous function of the applied field. In strong electrolytes, we find the fluctuations of the ionic current are Gaussian, and subsequently, the conductivity is constant with applied field. In weaker electrolytes and molten salts, we find the fluctuations of the ionic current are strongly non-Gaussian, and the conductivity increases with applied field. This nonlinear behavior, known phenomenologically for dilute electrolytes as the Onsager-Wien effect, is general and results from the suppression of ionic correlations at large applied fields, as we elucidate through both dynamic and static correlations within nonequilibrium steady states.

Figure 1

Fluctuations and response for the 0.1 M solution of NaCl with ${\epsilon }_s=78.5$ (left) and ${\epsilon }_s=10$ (middle), and the 25.3 M molten NaCl. (a) Characteristic snapshots of the NaCl systems considered, with green and yellow spheres representing ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$, respectively. (b) Scaled log probability of the time integrated current as computed from histogram reweighting. Error bars are 1 standard deviation of the mean as computed from bootstrapping analysis. The dashed lines represent Gaussian distributions with the same mean and variance. (c) Field dependent ionic conductivities relative to their values at $E=0.1\mathrm{V}/\AA$. Lines are computed from reweighting $p(J,Q)$, and symbols are computed from finite differences of $⟨J{⟩}_E$ versus $E$. Error bars are 1 standard deviation of the mean.

Figure 2

Time correlation function $G_E(t)=C_{jj}(t)+C_{jq}(t)$ for the field dependent conductivity of 0.1 M NaCl with ${\epsilon }_s=10$ (left) and molten salt (right). (a) Correlations functions for $E=0$ and $E=0.1\mathrm{V}/\AA$. (b) Decomposition of $G_E(t)$ into current-current $C_{jj}(t)$ and current-frenesy $C_{jq}(t)$ correlations at $E=0.1\mathrm{V}/\AA$. The solid lines in (b) are $G_E(t)$.

Figure 3

Pair distribution functions, $g_{a,c}(\mathbf{r}|E)$, between ${\mathrm{Cl}}^{-}$ and ${\mathrm{Na}}^{+}$ with increasing field, in the cylindrical coordinates for the (a) dilute solution 0.1 M, ${\epsilon }_s=10$ and (b) molten salt.