Charging Dynamics of Overlapping Double Layers in a Cylindrical Nanopore

The charging of electrical double layers inside a cylindrical pore has applications to supercapacitors, batteries, desalination and biosensors. The charging dynamics in the limit of thin double layers, i.e., when the double layer thickness is much smaller than the pore radius, is commonly described using an effective RC transmission line circuit. Here, we perform direct numerical simulations (DNS) of the Poisson-Nernst-Planck equations to study the double layer charging for the scenario of overlapping double layers, i.e., when the double layer thickness is comparable to the pore radius. We develop an analytical model that accurately predicts the results of DNS. Also, we construct a modified effective circuit for the overlapping double layer limit, and find that the modified circuit is identical to the RC transmission line but with different values and physical interpretation of the capacitive and resistive elements. In particular, the effective surface potential is reduced, the capacitor represents a volumetric current source, and the charging timescale is weakly dependent on the ratio of the pore radius and the double layer thickness.

Figure 1

Problem setup and results from DNS. (a) A cylindrical pore of radius $a$ and length ${\ell }_{\mathrm{pore}}$ with an applied potential ${\psi }_D$. The pore interacts with the bulk through a stagnant diffusion layer with thickness ${\ell }_{\mathrm{SDL}}$ and cross sectional area $A_{\mathrm{SDL}}$. (b) Plot of dimensionless potential $\mathrm{\Psi }(R=0,Z,\tau =0.14)$ at the center of the geometry for different $a/\lambda$. (c) Plot of dimensionless potential $\mathrm{\Psi }(R\to 1,Z,\tau =0.14)$ near the surface of the pore for different $a/\lambda$. Plots are presented for ${\mathrm{\Psi }}_D=0.4$, $\lambda /{\ell }_{\mathrm{pore}}={10}^{-3}$, $A_{\mathrm{SDL}}/(\pi a^2)=4$ and ${\ell }_{\mathrm{SDL}}/{\ell }_{\mathrm{pore}}\approx 0.5$.

Figure 2

Effective circuit model is identical for the thin and overlapping DL limits. However, the values of ${\mathcal{C}}_{\mathrm{pore}}$, ${\mathcal{R}}_{\mathrm{pore}}$, ${\mathcal{R}}_{\mathrm{SDL}}$, and ${\psi }_{D,\mathrm{eff}}$ are different for the two limits. A summary of these values is provided in Table 1.

Figure 3

Comparison of effective circuit models with the results from DNS. (a) Potential at the center of the pore ${\mathrm{\Psi }}_m(Z,\tau =0.07,0.2,0.4)$ and (b) local charge at the center of the pore ${\rho }_m(Z,\tau =0.07,0.2,0.4)$ for $a/\lambda =0.5$. (c) ${\mathrm{\Psi }}_m(Z,\tau =0.07,0.2,0.4)$ and (d) ${\rho }_m(Z,\tau =0.07,0.2,0.4)$ for $a/\lambda =1$. (e) Radial variation of charge $\rho (R,Z,\tau )$ for $Z=0$, 0.2, 0.4, 0.6, 0.8 and $\tau =0.07$ for $a/\lambda =0.5$.

Figure 4

Dimensionless current $I(\tau )$ at the mouth of the pore for different $a/\lambda$. The predictions from the thin DL and overlapping DL models are also provided.