This study aims to develop a model for simulating the electric double layer dynamics in CV measurements while simultaneously accounting for transport phenomena in both the electrode and the electrolyte. It also aims (i) to identify the dimensionless parameters that govern the CV measurements, (ii) to provide a physical interpretation of the shape of CV curves, and (iii) to investigate the effect of the electrode electrical conductivity on the predicted double layer capacitance. The transient double layer dynamics was simulated using the modified Poisson-Nernst-Planck (MPNP) model with a Stern layer and accounting for the presence of the electrode.
Overall, four dimensionless numbers were identified as,
Figure 2a shows the predicted current density js versus surface potential ψs (js-ψs curves) obtained from CV simulations for three cases featuring different values of T, Li, a, c∞, D, v, and ψmax. However, the dimensionless numbers for all cases were identical, namely, Π1 =19.47, Π2 =0.0038, Π3 =64, and Π4= ∞. Results were obtained by numerically solving the MPNP model with a Stern layer without electrode, i.e., Le=0 nm. Figure 2a shows that the predicted js-ψs curves were significantly different for these three cases. However, Figure 2b demonstrates that the dimensionless j*s-ψs* curves overlapped after using the scaling parameters. Overall, these results demonstrate that (i) the scaling parameters introduced here and the dimensional analysis for CV simulations were valid and (ii) the double layer charging dynamics near planar electrodes in CV measurements was governed by four dimensionless numbers, i.e.,Π1, Π2, Π3, and Π4.
Fig.2. Predicted (a) jsversus ψs curves and (b) j*sversus ψs* curves for three cases with different parameters.
Interpretation of the hump in CV curves
Figure 3a shows the predicted js-ψs curves from CV simulations for three values of potential window, i.e., Δψ =0.3, 0.4, and 0.5 V. Results were obtained by solving the MPNP model with a Stern without the electrode (Le=0 nm). It is evident that j*s reached the maximum value at about ψs=0.2 V for all three curves and then decreased for larger surface potential. Thus, a hump was observed around ψs =0.2 V typical of experimental cyclic voltammetry measurements.
To physically interpret the observed hump in js-ψs curves, Figure 3b shows the corresponding anion concentration c2 at the electrode surface x=0 nm as a function of surface potential for the same cases considered in Figure 3a. The maximum ion concentration cm=1/NAa3 due to finite ion size was also plotted in Figure 3b. It is evident that the surface anion concentration c2(x,t) increased rapidly with increasing potential up to ψs =0.2 V. This regime corresponded to the increase of current density js shown in Figure 3b where it reached a maximum at ψs =0.2 V corresponding to the crest of the hump. For ψs >0.2 V, the anion concentration asymptotically approached its maximum value cm. Then, the ion accumulation near the electrode surface became slower as the electric potential increased. This, in turn, resulted in the decrease in the current density js (Figure 3a). Overall, these results demonstrate that the hump observed experimentally in CV curves for EDLCs can be attributed to the saturation of ion concentration at the electrode surface.
Fig. 3. Predicted (a) j*s versus ψ s curves and (b) c2(x=0) versus ψs curves determined from CV simulations for three values of potential window, i.e., ψmax-ψmin =0.3, 0.4, and 0.5 V.
H. Wang and L. Pilon, 2012. Physical Interpretation of Cyclic Voltammetry Measurements, Electrochimica Acta, Vol. 64, pp.130-139. doi 10.1016/j.electacta.2011.12.118