Convection can enhance the capacitive charging of porous electrodes

Abstract

Charge transport in porous electrodes is foundational for modern energy storage technologies like supercapacitors, fuel cells, and batteries. Supercapacitors in particular rely solely on storing energy in charged pores. Here, we simulate the charging of a single electrolyte-filled pore using the modified Poisson-Nernst-Planck and Navier-Stokes equations. We find that electroconvection can substantially speed up the charging dynamics. We uncover the fundamental mechanism of electroconvection during pore charging through an analytical model that predicts the induced flow field and the electric current arising due to convection. Our findings suggest that convection is especially important in the limit of slender pores with thin electric double layers, and becomes significant beyond a certain threshold voltage that is an inherent electrolyte property.

Figures (5)

• Figure 1: Velocity and charge concentration fields as well as charge relaxation dynamics inside a pore. (a) Simulation results for $L_\mathrm{p}/r_\mathrm{p}=r_\mathrm{p}/\lambda=5$ and $\Tilde{\zeta}=1$. Scaled charge density $\Tilde{\rho}_v$, electric field $\mathbf{E}$ (left split), and axial velocity $\Tilde{w}$ (right split) directly after switching on the wall potential and at the short $\tau_\mathrm{s}$ and long timescale $\tau_\mathrm{l}$. (b) Typical relaxation of the pore charge $q$ for $\Tilde{\zeta}=10$ as predicted by models accounting for diffusion and electromigration (PNP), and additionally for electroconvection (PNP-NS). The charging times $t_{99,\mathrm{conv}}$ and $t_{99,\mathrm{ref}}$ are indicated.
• Figure 2: Simulation results in the strongly nonlinear regime $\Tilde{\zeta}=10$. (a) Heatmap of the relative deviation in charging time $\Delta\Tilde{t}$ over the scaled pore radius $r_\mathrm{p}/\lambda$ and aspect ratio $L_\mathrm{p}/r_\mathrm{p}$. Symbols mark calculated data points with linear interpolation in between. (b) $\Delta\Tilde{t}$ along vertical cut lines through the heatmap at different horizontal positions with indication of scaling exponents. (c) Analogous to (b), but using horizontal cut lines along different vertical positions.
• Figure 3: Simulation results in the weakly nonlinear regime $\Tilde{\zeta}=1$. (a) Heatmap of the relative deviation in charging time $\Delta\Tilde{t}$ over the scaled pore radius $r_\mathrm{p}/\lambda$ and aspect ratio $L_\mathrm{p}/r_\mathrm{p}$. Symbols mark calculated data points with linear interpolation in between. The influence of convection is largely independent of $L_\mathrm{p}/r_\mathrm{p}$ but shows a maximum at $r_\mathrm{p}/\lambda\approx10$. (b) $\Delta\Tilde{t}$ over the scaled pore radius $r_\mathrm{p}/\lambda$, with additional datapoints compared to (a), and maximum induced velocity $\mathrm{max}(w)_{r,z,t}$, for $L_\mathrm{p}/r_\mathrm{p}=5$.
• Figure 4: Comparison of the model of the flow field, \ref{['eq_w_axial']} (a) and \ref{['eq_w(rzt)']} (b, c) , (lines) and the simulation results (symbols) in the weakly nonlinear regime $\Tilde{\zeta}=1$. (a) Axial velocity $\Tilde{w}$, scaled to its maximum absolute value, over the scaled radial coordinate, exemplarily for $r_\mathrm{p}/\lambda=L_\mathrm{p}/r_\mathrm{p}=5, t=\tau_\mathrm{p}, z=r_\mathrm{p}$. (b) Local induced axial velocity in $m\per s$ over the scaled time for $r_\mathrm{p}/\lambda=5$ and $L_\mathrm{p}/r_\mathrm{p}=10$. (c) Maximum induced velocity at $z=r_\mathrm{p}$ throughout pore charging, scaled to its maximum absolute value, over the scaled pore radius for $L_\mathrm{p}/r_\mathrm{p}=5$.
• Figure 5: Comparison of the model of the convective current, \ref{['eq_i_zconv']}, (lines) and the simulation results (symbols) in the weakly nonlinear regime $\Tilde{\zeta}=1$. (a) Axial convective current density $\Tilde{i}_\mathrm{z,conv}$, scaled to its maximum absolute value, over the scaled radial coordinate, exemplarily for $r_\mathrm{p}/\lambda=L_\mathrm{p}/r_\mathrm{p}=5, t=\tau_\mathrm{p}, z=r_\mathrm{p}$. (b) Local convective current density in A□m over the scaled time for $r_\mathrm{p}/\lambda=5$ and $L_\mathrm{p}/r_\mathrm{p}=10$. (c) Minimum convective current density at $z=r_\mathrm{p}$ throughout pore charging, scaled to its maximum absolute value, over the scaled pore radius for $L_\mathrm{p}/r_\mathrm{p}=5$.