Microscopic Model for Cyclic Voltammetry of Porous Electrodes

Cyclic voltammetry (CV) is a widespread experimental technique for characterizing electrochemical devices such as supercapacitors. Despite its wide use, a quantitative relation between CV and microscopic properties of supercapacitors is still lacking. In this Letter, we use both the microscopic “stack-electrode” model and its equivalent circuit for predicting the cyclic voltammetry of electric double-layer formation in porous electrodes. We find that the dimensionless combination $\omega {\tau }_n$, with $\omega$ the scan frequency of the time-dependent potential and ${\tau }_n$ the relaxation timescale of the stack-electrode model, governs the CV curves and capacitance: the capacitance is scan-rate independent for $\omega {\tau }_n\ll 1$ and scan-rate dependent for $\omega {\tau }_n\gg 1$. With a single fit parameter and all other model parameters dictated by experiments, our model reproduces experimental CV curves over a wide range of $\omega$. Meanwhile, the influence of the pore size distribution on the charging dynamics is investigated to explain the experimental data.

Figure 1

(a) Sketch of a charged supercapacitor which contains an electrolyte and two porous electrodes of thickness $H$ and a surface-to-surface distance $2L$ connected to a voltage source with a potential difference $2\mathrm{\Phi }(t)$. (b) The applied sawtooth potential $\mathrm{\Phi }(t)$ that is used in CV experiments and in the present study. (c) Sketch of transmission line model, where the nanoporous cathode (red) and anode (blue) are both modeled by a cylindrical channel of uniform diameter. (d) Sketch of stack-electrode model, where the nanoporous cathode (red) and anode (blue) are both modeled by a stack of $n\ge 1$ parallel planar electrode sheets with a fixed spacing $h$, all subject to the same dimensionless potential $\mathrm{\Phi }(t)$ (red) or $-\mathrm{\Phi }(t)$ (blue). (e) The equivalent electronic circuit for the transmission line model, where the number of circuit elements $n$ is infinite. (f) The equivalent electronic circuit for the stack-electrode model, where $R\sim L$ and $R^{\prime }\sim h$.

Figure 2

(a),(b) The cyclic voltammetric curves for $n=3$, $\kappa L=100$, $H/L=1$, and ${\mathrm{\Phi }}_0=0.001$, in (a) for low scan rate $\omega {\tau }_{\mathrm{RC}}={10}^{-4}$, and in (d) for high scan rate $\omega {\tau }_{\mathrm{RC}}=1$. The symbols and lines in (a) and (b) correspond to PNP and circuit model calculations. $C_s$ as a function of $\omega {\tau }_{\mathrm{RC}}$ (c) and as a function of $\omega {\tau }_n$ (d) and (e), for $\kappa L=100$, $H/L=1$, $n=1$, 2, 3, 5, 10, and ${\mathrm{\Phi }}_0=0.001$. Data reflect PNP calculations in the stack-electrode geometry (symbols) and its equivalent circuit (lines). (f) The CV curves for $\kappa L=100$, $H/L=1$, ${\mathrm{\Phi }}_0=0.001$, different $n=1$, 2, 3, 5, 10, and $\omega {\tau }_n=0.01$, 0.1, 1. In (f) the symbols are calculated by PNP, and the lines are guides to the eye.

Figure 3

(a) CV curves for $\kappa L=100$, $H/L=1$, $n=5$, ${\mathrm{\Phi }}_0=0.1$, 1, 2, $\omega {\tau }_{\mathrm{RC}}=0.01$ (solid lines), and $\omega {\tau }_{\mathrm{RC}}=0.1$ (dashed lines). (b) $\omega {\tau }_n$ dependence of $C_s/\cosh ({\mathrm{\Phi }}_0/4)$ for $n=1$, 2, 3 and ${\mathrm{\Phi }}_0=0.1$, 1, 2 (symbols). The lines in (b) for different $n$ are calculated with the electronic circuit model.

Figure 4

Comparison of experimental data (symbols) in Ref. [40] and the circuit model (lines) for $\omega =0.01,0.02,0.05,0.1,0.2,0.5,1,2,5{\mathrm{s}}^{-1}$ corresponding to $0.01\mathrm{V}{\mathrm{s}}^{-1}$ to $5\mathrm{V}{\mathrm{s}}^{-1}$, with ${\mathrm{\Psi }}_0=0.5\mathrm{V}$ corresponding to ${\mathrm{\Phi }}_0\simeq 20$. The direction of the arrow indicates an increase in scanning rate.

Figure 5

(a) Predicted $J_{\max }$ as a function of $\omega {\tau }_n$ for ${\mathrm{\Phi }}_0=0.001$, $\kappa L=1000$, $H/L=1$, $n=50$, and $q=1$, 0.8, and 1.1. (b) $J(t)/J_{\max }$ vs $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for ${\mathrm{\Phi }}_0=0.001$, $\kappa L=1000$, $H/L=1$, $q=1$, 0.8, 1.1, and $\omega {\tau }_{\mathrm{RC}}=7.375\times {10}^{-3}$. Here, we determined $J_{\max }$ at $\omega {\tau }_n={10}^{-3}$ from the data in panel (a). Conversely, $J(t)$ and $\mathrm{\Phi }(t)$ for the different $n$ were evaluated at $\omega {\tau }_{\mathrm{RC}}=7.375\times {10}^{-3}$. This treatment ensures that all the scaled currents in (b) vary between $-1$ and 1, similar to Fig. S8 [33].