Charging dynamics of electric double layer nanocapacitors in mean-field

Abstract

An electric double layer capacitor (EDLC) stores energy by modulating the spatial distribution of ions in the electrolytic solution that it contains. We determine the mean-field time scales for planar EDLC relaxation to equilibrium, after a potential difference is applied. We tackle first the fully symmetric case, where positive and negative ionic species have same valence and diffusivity, and then the general, more complex, asymmetric case. Depending on applied voltage and salt concentration, different regimes appear, revealing a remarkably rich phenomenology relevant for nanocapacitors.

Figures (4)

• Figure 1: An ideal electric double layer capacitor (EDLC). The total amount of salt is fixed and, in the linear regime, it defines the thickness of the double layer $\lambda_\mathrm{D}$.
• Figure 2: Exponential relaxation times $\tau$ extracted from linear fits of $\log(\sigma(t))$ vs $t$, as a function of dimensionless voltage $v$. For given $\lambda_\mathrm{D}$ and $v$, two different relaxation processes are often seen in $\sigma(t)$ (see Fig. \ref{['figm:phasesym']}): filled symbols indicate the early-time process, whereas empty symbol the late-time process, when present. For $v\le1$ and $\lambda_\mathrm{D}/L\ge1$, the relaxation is purely diffusive and takes place on a scale $2L$. For $v\le1$ and $\lambda_\mathrm{D}/L\ll1$, the double layer relaxes at early times on a time ${L\lambda_\mathrm{D}}/{D}$, that extends into the nonlinear regime as $({L\lambda_\mathrm{D}}/{D})\cosh({v}/{2})$ (dotted curves). This is followed by a slower diffusive relaxation over a length $L$, signaling depletion (empty symbols). For $v\gg1$, collective ion migration causes full depletion: this early-time process is not shown here because it is non-exponential (the gray dashed line however shows its time scale for the unscreened regime of Fig. \ref{['figm:phasesym']}, where the process is linear). At late times, a fast diffusive relaxation follows (empty symbols), signaling ion rearrangement inside counterionic double layers of thickness $\mu_\mathrm{nen}$ (the Gouy-Chapman length).
• Figure 3: a) Regime diagram for the symmetric electrolyte case ($D_+=D_-=D$, $q_+=q_-=1$). Five different regimes are separated by the boundary lines discussed in the text. b-g) In black, the relative difference between the instantaneous electrode charge density $\sigma(t)$ and its equilibrium value $\sigma_\mathrm{eq}=\sigma(t\to \infty)$. In addition, in e and g, the gray curves show the relative difference between volume charge density $\rho(t)$ at contact with the electrode and its equilibrium value $\rho_\mathrm{eq}$. Time $t$ has units of $L^2/D$. In crimson, scaling of relaxation times as extracted from linear fits and confirmed analytically supp.
• Figure 4: a) Regime diagram for the completely asymmetric case ($D_+/D_-=1/10$, $q_+=1, q_-=2$). Depletion of positive ions (blue tones) is distinct from that of negative, more charged, ions (hatch patterns). b-g) As in Fig. \ref{['figm:phasesym']}. In e and g, dotted and dashed grey lines represent the charge densities at the negative and the positive electrodes, respectively; in the symmetric case, these were equal. Time $t$ has units of $L^2/D_+$.