Poisson-Nernst-Planck charging dynamics of an electric double layer capacitor: symmetric and asymmetric binary electrolytes

TL;DR

This work analyzes the mean-field charging dynamics of an ideal planar electric double-layer capacitor by solving the Poisson-Nernst-Planck equations for symmetric and asymmetric binary electrolytes under a sudden applied voltage. By nondimensionalizing with $\epsilon = \lambda_D/L$ and $v = \beta e V_0$, it discerns linear, purely nonlinear, and depletion regimes, deriving dominant time scales such as $\tau_0 \sim \frac{L\lambda_D}{D} - \frac{\lambda_D^2}{D}$ (large $\epsilon$) and $\tau_{PNL} = (\lambda_D L/D)\cosh(v/2)$, as well as fully depleted limits with $\tau \sim \frac{4L^2}{D v^2}$ and its generalizations to asymmetry. The study extends prior symmetric analyses to partially and fully asymmetric cases, identifying multiple relaxation hierarchies, voltage-dependent capacitances via the Grahame relation, and depletion-driven diffusive time scales $\tau' \sim L^2/\pi^2 D$. It also discusses non-idealities through the electrostatic coupling parameter $\Xi$, outlining regimes where mean-field remains valid versus where correlations or packing effects become important, thus providing a framework to map experimental conditions to dominant charging mechanisms and time scales for EDLCs. Overall, the results illuminate how geometry, ion asymmetry, and nonlinearity intertwine to shape EDLC relaxation, with practical implications for rapid charging and energy storage technologies.

Abstract

A parallel plate capacitor containing an electrolytic solution is the simplest model of a supercapacitor, or electric double layer capacitor. Using both analytical and numerical techniques, we solve the Poisson-Nernst-Planck equations for such a system, describing the mean-field charging dynamics of the capacitor, when a constant potential difference is abruptly applied to its plates. Working at constant total number of ions, we focus on the physical processes involved in the relaxation and, whenever possible, give its functional shape and exact time constants. We first review and study the case of a symmetric binary electrolyte, where we assume the two ionic species to have the same charges and diffusivities. We then relax these assumptions and present results for a generic strong (i.e. fully dissociated) binary electrolyte. At low electrolyte concentration, the relaxation is simple to understand, as the dynamics of positive and negative ions appear decoupled. At higher electrolyte concentration, we distinguish several regimes. In the linear regime (low voltages), relaxation is multi-exponential, it starts by the build-up of the equilibrium charge profile and continues with neutral mass diffusion, and the relevant time scales feature both the average and the Nernst-Hartley diffusion coefficients. In the purely nonlinear regime (intermediate voltages), the initial relaxation is slowed down exponentially due to increased capacitance, while bulk effects become more and more evident. In the fully nonlinear regime (high voltages), the dynamics of charge and mass are completely entangled and, asymptotically, the relaxation is linear in time. We finally discuss non-ideal behavior in real capacitors and provide conditions for which mean-field is expected to hold.

Figures (12)

• Figure 1: Cartoon representing a charged, planar EDLC. Red cations and blue anions are treated at mean-field level, as suggested by the color of the solution, representing charge density. Within the mean-field approximation, for large enough $L$ and sufficiently low applied voltage, the thickness of the electric double layer at equilibrium is $\lambda_\mathrm{D}$.
• Figure 2: Logarithmic plots of the quantities defined at the top left, where $\rho^{\pm L} = \rho(-L,t) = -\rho(L,t)$ is the charge density in the solution at contact with the electrode, $\sigma(t)$ is the surface charge density of the electrode, and $\rho^{\pm L}_{\mathrm{eq}}$ and $\sigma_{\mathrm{eq}}$ are their respective values at equilibrium. Here, $\epsilon=0.01$ and $v$ increases progressively, as indicated at the top right of each panel. When the curves shown are linear, their slopes correspond to exponential relaxation rates. $s_0$ corresponds to a relaxation time $\lambda_\mathrm{D} L/D = 0.01\, L^2/D$ and $s'$ of $L^2/(\pi^2D) \simeq 0.10\, L^2/D$. At $v=0.001$ (a), only the double layer formation process is visible with its rate $s_0$ (unless too close to $t=0$, where other rates $s_i$ can play a role). At $v=0.1$ and $1$ (b-c), the first slope corresponds to the build-up of the double layer, while the second slope reflects its reorganization as the bulk is depleted from ions. The system is in the purely nonlinear regime at $v=1$ and the double layer builds up in a time $\tau_\mathrm{PNL}\propto \cosh{(v/2)}$, as defined in Eq. \ref{['eq:tauNLcosh']} in Sec. \ref{['sec:DynPlanSymPurelyNL']}. At $v=10$ (d) the system is partially depleted at equilibrium, as discussed in Sec. \ref{['sec:DynPlanSymFullyDepleted']} (see Fig. \ref{['fig:DeplDuStat']}c), but continues to relax with a time $\tau_\mathrm{PNL}$. At $v=40$ and $100$ (e-f), the system is fully depleted (Sec. \ref{['sec:DynPlanSymFullyDepleted']}): the early-time curves represent ion migration (the relaxation is not yet linear in time, but not perfectly exponential either); the late-time part represents relaxation of the counterionic double layers, in a time $\sim{\mu_{\mathrm{nen}}^2}/{D}$. Insets show $\vert (\rho^{\pm L}-\rho^{\pm L}_\mathrm{eq})/\rho^{\pm L}_\mathrm{eq} \vert$ and $\vert (\sigma-\sigma_\mathrm{eq})/\sigma_\mathrm{eq}\vert$ in linear scale.
• Figure 3: (a) Mass as a function of $z$ (in units of Table \ref{['tab:units']}), at different times (here in units of ${L^2}/{D}$), for $\epsilon=0.01$ and $v=2$. In these units, the electric double layer formation occurs on a time scale $\sim 0.02$. At much shorter times (dark blue) the system has not moved yet. At later times (lighter greens) the double layer has formed already and the mass diffusion process manifests itself, with a relaxation time scale $\tau'\simeq0.1$: the sinusoid of Eq. \ref{['eq:n2ndorder']} is visible. At times much larger than $\tau'$ (dark green) the system is at equilibrium at a new value of bulk density, predicted by Eq. \ref{['eq:n2ndorderbulk']}. Dotted lines indicate the analytical predictions for later times as per Eq. \ref{['eq:n2ndorder']}, where the parameter $A$ was set to 0.006 for all curves. (b) Potential as a function of $z$, at different times (here in units of ${L^2}/{D}$), for $\epsilon=0.01$ and $v=2$. The inset is a zoom close to the left electrode, while the arrow indicates the point of null potential (see Sec. \ref{['sec:partiallyasym:pnl']}), which in the present case is $z_0=0$. (c) Points show $\hbox{Du}_n$ as a function of $v$, for different values of $\epsilon$, as extracted from numerical solutions of the Poisson-Nernst-Planck equation. Smaller $\hbox{Du}_n$ represent stronger depletion. Solid lines are a guide to the eye. The curves at $\epsilon=100$ and $1000$, not shown here, coincide with the $\epsilon=10$ curve. In dotted lines, the prediction from Eq. \ref{['eq:n2ndorderbulk']}.
• Figure 4: Ratios of $|\sigma(t)|$ (black) and $|\rho(\pm L,t)|$ (gray) versus their equilibrium values, as a function of time. Here, $\epsilon=0.1$ and $v=200$. The blue dashed lines represent the ideal linear evolution of $\rho$ and the theoretical prediction for $\sigma$ from Eqs. \ref{['eq:tstar']}--\ref{['eq:parabolicsigma']}.
• Figure 5: Relaxation times $\tau$ as a function of $v$, for the $1:2$ case (left) and the $1:10$ case (right); different colors correspond to different values of $\epsilon = \lambda_\mathrm{D}/L$. Times are extracted from linear fits in logarithmic scale of $\sigma$, similar to what shown in Fig. \ref{['fig:plotfit']} for the symmetric case. When two exponential relaxation times are visible from numerical data, one at short times and one at long times, they are both represented here (if the same $v$ and $\epsilon$ correspond to two symbols of the same color, the solid one represents early-time and the empty one late-time relaxation). The dense dotted lines represent the purely nonlinear time obtained from Eqs. \ref{['eq:Ctotqq']} and \ref{['eq:RNLqq']} and preceding ones; for comparison, we also plot the time given by Eq. \ref{['eq:tauNLcosh']}, relevant for the symmetric case and shown with sparser dots for $\epsilon=0.01$. The gray and black lines represent the times given by Eqs. \ref{['eq:tstarqq']} and \ref{['eq:tauFDNLqq']}, respectively.