Entropy production and energy dissipation in symmetric redox supercapacitors

In this work we propose a theoretical model that accounts for the main features of the loading-unloading process of a symmetric redox ${\mathrm{MnO}}_2$-based supercapacitor dominated by fast electrochemical reactions at the electrodes. The model is formulated on the basis of nonequilibrium thermodynamics from which we are able to deduce generalized expressions for the electrochemical affinity, the load-voltage and the current-voltage equations that constitute generalizations of the current-overpotential and Butler-Volmer equations, and that are used to describe experimental voltagram data with good agreement. These equations allowed us to derive the behavior of the energy dissipated per cycle showing that it has a nonmonotonic behavior and that in the operation regime it follows a power-law behavior as a function of the frequency. The existence of a maximum for the energy dissipated as a function of the frequency suggests the that the corresponding optimal operation frequency should be similar in value to ${\omega }_{\max }$.

Figure 1

Schematic representation of a symmetric supercapacitor.

Figure 2

Schematic comparison between three characteristic curves of Gibbs free energy.

Figure 3

(a) Three overlapping Langevin trajectories for fixed values ${\mathcal{E}}_1=0.4\text{V}, \mathrm{\Omega }=2.56, \omega =0.2748{\text{s}}^{-1}, k_0=0.077{\text{s}}^{-1}$, and $\kappa =0.025$. (b) Three overlapping voltagrams for the same parameters.

Figure 4

Experimental data fitting of the CV with sweep rates $\nu =$ 0.03 $\mathrm{V}{\text{s}}^{-1}$ (diamons), $\nu =$ 0.05 $\mathrm{V}{\text{s}}^{-1}$ (crosses), and $\nu =$ 0.07 $\mathrm{V}{\text{s}}^{-1}$ (circles) with the parameters given in the text. (b) Comparision of a experimental data fitting of a CV with sweep rate $\nu =$ 0.07 $\mathrm{V}{\text{s}}^{-1}$ (circles) by means of a sinosoidal (dots) and triangular (continuous line) voltage oscillation.

Figure 5

Experimental data fitting for $\nu =$ 0.07 $\mathrm{V}{\text{s}}^{-1}$ (circles) with parameters: $I_{\rho }=4.95, \omega =0.2748, {\mathcal{E}}_1={\mathcal{E}}_0=0.4, \kappa =0.025, \zeta =0.025$, and $\mathrm{\Omega }=2.56$. (a) Comparison of the fitting curves by fixing all parameters and changing $\omega$. (b) Comparison of the fitting curves by fixing all parameters and changing $\mathrm{\Omega }$.

Figure 6

Normalized dissipated energy dependence for experimental values as a function of frequency. The parameters are $\mathrm{\Omega }=2.75$ (continuous line), ${\omega }_{\max }=0.302$, and $A_{\max }=0.6$; $\mathrm{\Omega }=2.62$ (dashed line), ${\omega }_{\max }=0.46$, and $A_{\max }=0.57$; $\mathrm{\Omega }=2.56$ (dots), ${\omega }_{\max }=0.52$, and $A_{\max }=0.55$. The fixed parameters are ${\mathcal{E}}_1={\mathcal{E}}_0=0.4, k_0=0.077{\text{s}}^{-1}, \kappa =0.025$, and $\zeta =0.025$.

Figure 7

Hysteresis loops for a nonconstant regular parameter ${\mathrm{\Omega }}_{\wedge }\left(V\right)$ and ${\mathrm{\Omega }}_{\vee }\left(V\right)$ compared to fixed $\mathrm{\Omega }=2.56$; all curve parameters are ${\mathcal{E}}_1=0.4, w=0.2748, \zeta =0.025$, and ${\mathrm{\Omega }}_{\max }=2.56$.