Electrochemical characteristics of ideal polarizable interfaces with limited number of charge carriers

Recent progress in material chemistry and surface engineering has led to emergence of new electrode materials with unique physical and electrochemical properties. Here, we introduce a physical model describing charging of ideal polarizable electrode-electrolyte interface where the electrode is characterized by a limited capacity to store charge. The analytical model treats the electrode and electrolyte phases as independent nonlinear capacitors that are eventually coupled through the condition of equality of the total stored electrical charge opposite in sign. Gouy-Chapman and condensed layer theories applied to a general $1:n$ valent electrolyte are used to predict dependencies of differential capacitance of the electrolyte phase and surface concentration of the electrical charge on the applied potential. The model of the nonlinear capacitor for the electrode phase is described by a theory of electron donors and acceptors present in conductive solids as a result of thermal fluctuations. Both the differential capacitance and the surface concentration of the electrical charge in the electrode are evaluated as functions of the applied potential and related to the capacity of the electrode phase to accumulate charge and its ability to form electron donors and acceptors. The knowledge of capacitive properties of both phases allows to predict electrochemical characteristics of ideal polarizable interfaces, e.g., current responses in linear sweep voltammetry. The coupled model also shows significant potential drops in the electrode comparable to those in the electrolyte phase for materials with low charge carrier concentrations.

Figure 1

Electrode-electrolyte interface.

Figure 2

PB model. Dependencies of the surface concentration of electric charge on the surface potential.

Figure 3

PB model. The dependence of the differential capacitance on the surface electrode potential. The filled circles depict the limit capacitance values given by Eq. (10).

Figure 4

CL model. Dependencies of the surface concentration of electric charge on the surface potential. The dashed lines represent the predictions of PB model.

Figure 5

CL model. The dependence of the differential capacitance on the surface electrode potential. The dashed lines represent the predictions of PB model.

Figure 6

Structure of the electrode-electrolyte interface. Electric current flows with the density $i$ through the system during the polarization process. Electric charge emerging on the electrolyte side is balanced with the charge in the electrode phase.

Figure 7

Dependencies of the surface electric charge (top) and electrode capacitance (bottom) on the potential drop in the electrode. $K_{\mathrm{s}}=1$.

Figure 8

Dependencies of the surface electric charge (top) and electrode capacitance (bottom) on the potential drop in electrode. $c_{\mathrm{tot}}=2\times {10}^{-5}\mathrm{mol}{\mathrm{m}}^{-2}$.

Figure 9

Dependencies of the current density (left) and relative voltage drop in the electrode phase (right) on the applied voltage. $n=-1,c_{\mathrm{tot}}=2\times {10}^{-5}\mathrm{mol}{\mathrm{m}}^{-2},K_{\mathrm{s}}=1$.

Figure 10

Dependencies of the current density (top) and a relative voltage drop in the electrode phase (bottom) on the applied voltage. $c_{\mathrm{tot}}=2\times {10}^{-5}\mathrm{mol}{\mathrm{m}}^{-2},K_{\mathrm{s}}=1$. Predictions given by the CL model, solid lines; PB model, dashed lines.

Figure 11

Dependencies of the current density (top) and relative voltage drop in the electrode phase (bottom) on the applied voltage. $n=-2,K_{\mathrm{s}}=1$. Predictions given by the CL model, solid lines; PB model, dashed lines.

Figure 12

Dependencies of the current density (top) and relative voltage drop in the electrode phase (bottom) on the applied voltage. $n=-2,c_{\mathrm{tot}}=2\times {10}^{-5}\mathrm{mol}{\mathrm{m}}^{-2}$. Predictions given by the CL model, solid lines; PB model, dashed lines.