Dynamics of Ion Transport in Ionic Liquids

A gap in understanding the link between continuum theories of ion transport in ionic liquids and the underlying microscopic dynamics has hindered the development of frameworks for transport phenomena in these concentrated electrolytes. Here, we construct a continuum theory for ion transport in ionic liquids by coarse graining a simple exclusion process of interacting particles on a lattice. The resulting dynamical equations can be written as a gradient flow with a mobility matrix that vanishes at high densities. This form of the mobility matrix gives rise to a charging behavior that is different to the one known for electrolytic solutions, but which agrees qualitatively with the phenomenology observed in experiments and simulations.

Figure 1

Schematic of the system under consideration: cations and anions on a lattice of lattice constant $a$.

Figure 2

(a) The total density, $C_{+}+C_{-}$, and (b) charge density $C_{+}-C_{-}$, as functions of distance $X=x/L$ from the electrode and time after an applied step voltage $V=40V_T$ with $V_T=k_{B}T/e$. Here, $\gamma =0.25$, $l_c/l_D=10$, and $L/l_D=100$, red and blue curves denote the first and second charging regimes, and the arrow shows the increasing density deficit in the first charging regime. Numerical solution of Eqs. (11)–(12) is performed using pdepe in matlab.

Figure 3

The total charge as a function of time for different values of $l_c/l_D$ with $L/l_D=100$, $\gamma =0.25$ fixed. (Inset) The equilibrium relaxation time ${\tau }_{\text{relax}}$ as a function of $l_c$ and $l_D$ (obtained by fitting numerical results to an exponential decay).

Figure 4

The voltage drop across the system evolves nonmonotonically under constant current conditions, Eq. (18). Here $L/l_D=100$, $l_c/l_D=10$, and $\gamma =0.25$, and the total charge $Q=J\tau$. Inset: simulation data from Ref. [45] for which $\approx 400\mathrm{kA}{\mathrm{cm}}^{-2}$ corresponds to dimensionless $J=2$.