Effect of Nanopore Wall Geometry on Electrical Double-Layer Charging Dynamics

TL;DR

This work develops a perturbation-based reduced-order model for electric double-layer charging in axisymmetric pores with gradually varying radius. By introducing the electrochemical potential of charge and deriving a one-dimensional diffusion-like equation with a radius-dependent diffusion coefficient and a pseudo-advective term, it reveals that sloped (conical) walls can accelerate charging and increase stored charge via an additional cross-sectional flux. The model yields an explicit equivalent circuit representation, including per-length resistance and capacitance that depend on local radius, and is validated against full Planck-Nernst-Poisson simulations with substantial computational savings. The framework enables integration into pore-network models, offering design principles for optimizing supercapacitor performance through geometry and entrance effects.

Abstract

Confinement strongly influences electrochemical systems, where structural control has enabled advances in nanofluidics, sensing, and energy storage. In electric double-layer capacitors (EDLCs), or supercapacitors, energy density is governed by the accessible surface area of porous electrodes. Continuum models, built on first-principles transport equations, have provided critical insight into electrolyte dynamics under confinement but have largely focused on pores with straight walls. In such geometries, a fundamental trade-off emerges: wider pores charge faster but store less energy, while narrower pores store more charge but charge slowly. Here, we apply perturbation analysis to the Poisson-Nernst-Planck (PNP) equations for a single pore of gradually varying radius, focusing on the small potential and slender aspect ratio regime. Our analysis reveals that sloped pore walls induce an additional ionic flux, enabling simultaneous acceleration of charging and enhancement of charge storage. The theoretical predictions closely agree with direct numerical simulations while reducing computational cost by 5-6 orders of magnitude. We further propose a modified effective circuit representation that captures geometric variation along the pore and demonstrate how the framework can be integrated into pore-network models. This work establishes a scalable approach to link pore geometry with double-layer dynamics and offers new design principles for optimizing supercapacitor performance.

Figures (7)

• Figure 1: A schematic of generalized nanopore, with red circles representing cations and blue circles representing anions. Although initially uncharged, at time $\tau=0$, a negative potential of $\Phi_w$ is applied to the pore wall, driving cations into the pore and expelling anions. There exist three distinct regions: the reservoir, the static diffusion layer, and the pore. The reservoir acts as an unchanging repository of charged ions, allowing ions to flow in and out freely. The reservoir remains electroneutral, serving as the reference level for potential. The static diffusion layer (SDL) serves as a bridge between the reservoir and the pore. While electrically neutral, its potential increases linearly as ions approach the pore entrance. The radius and length scale of this region are adjustable parameters, when formulated appropriately, define the dimensionless Biot number---a quantity characterizing the resistance of electrochemical flux from the reservoir region to the pore entrance. The pore region is where the EDL forms, and is the main focus of analysis. Importantly, the walls are ideally blocking, and the pore’s axial length is much larger than its radial dimension. The radius of the pore depends on the axial coordinate, given by $\alpha(Z)$.
• Figure 2: A schematic of an equivalent circuit diagram for the dimensional electrochemical potential of charge, $\mu_w^*$, inside a converging pore. The reservoir region acts as the reference potential, with the SDL region bridging the reservoir and the pore regions. This SDL region is treated as a lumped resistor $(R_\mathrm{SDL})$, which is inversely related to the Biot number. Inside the pore, each axial point has an associated resistance $(R_p)$ and capacitance $(C_p)$, which are functions of radius, Debye length, diffusivity, and electric permittivity. Each axial junction has current $i$ flowing through it, and the current $di$ is being stored in the EDL. Driving this electrokinetic flow is an applied wall potential of $\mu_w^*=2\Phi_w^*$.
• Figure 3: The electrochemical potential $\hat{\mu}(Z,\tau)$, the electric potential $\hat{\Phi}(Z,R,\tau)$, and the charge density $\hat{\rho}(Z,R,\tau)$ plotted at various times---early (a,c,e) and intermediate times (b,d,f)---against direct numerical simulations. (a-b) represent plots of the electrochemical potential as a function of axial position, and include the SDL region when $Z<0$. The radial electric potential (c-d) and charge density (e-f) profiles are plotted midway, $(Z=0.5)$, through the pore as a function of radial position. Schematics of all the pores are depicted on the right side of the figure: diverging, narrow, wide, and converging. The entrance-to-end ratio for the converging and diverging cases are 2:1 and 1:2, respectively, with corresponding equations $\alpha(Z) = 2-Z$ and $\alpha(Z) = 1+Z$, respectively. The wide $\alpha(Z)=2$ and narrow $\alpha(Z)=1$ pores represent the widest and narrowest radii for the changing radius cases. $\tau$ denotes the non-dimensional time, $Z$ the non-dimensional length along the pores, and $R$ the non-dimensional radial length.
• Figure 4: The electrochemical potential $\hat{\mu}(Z,\tau)$ and the electric potential $\hat{\Phi}(Z,R,\tau)$ plotted at various times---early (a,c) and intermediate times (b,d)---against direct numerical simulations with larger changes in slope. (a-b) represent plots of the electrochemical potential as a function of axial position, and include the SDL region when $Z<0$. The radial electric potential (c-d) profiles are plotted midway, $(Z=0.5)$, as a function of radial position. Three converging pores and one diverging pore are tested. The converging pores share the same entrance radius of 20 nm, but differing in their end radii. The 20–10 nm case corresponds to the converging pore discussed previously, while the 20–1 nm and 20–0.1 nm pores explore steeper slopes, testing the limits of the perturbation analysis. Finally, the diverging pore is also added with a radius starting of 2 nm and an ending radius of 20 nm.
• Figure 5: Contour plots of charge density ($\hat{\rho}$) throughout the pore at various times. (a-c) depict the converging geometry, and (d-f) depict the diverging geometry, described by $\alpha(Z)=2-Z$ and $\alpha(Z) = 1+Z$, respectively. The charge density contours are shown at early (a,d), intermediate (b,e), and equilibrium (c,f) times. Lighter regions correspond to higher charge densities, while darker regions represent a more neutral state. $\tau$ denotes the non-dimensional time, $Z$ the non-dimensional length along the pores, and $R$ the non-dimensional radius. The initial and system conditions are $\hat{\mu}(Z,\tau=0) = 1$, $\kappa=2$, $\ell_s^*=\ell_p^*$ and $a_s^*=4a_p^*$.