Finite-Length Linear Diffusion

In practical applications, very often diffusion is not semi-infinite. Such finite-length linear diffusion is observed, for example, for internal diffusion into mercury film deposited on a planar electrode, in deposited conducting polymers, for hydrogen diffusion into thin films or membranes of Pd or other hydrogen absorbing materials, or for a rotating disk electrode where the diffusion layer corresponds to the layer thickness. There are two cases of finite-length diffusion displayed Fig. 4.11  

---

Fig. 4.11 Concentration profiles in two cases of finite-length linear diffusion left -transmissive boundary, right - reflective boundary...

Fig. 7.8 Electrical equivalent circuits of faradaic impedance corresponding to (a) indirect, Eq. (7.81), and (b) direct, Eq. (7.83), hydrogen absorption reaction with finite-length linear diffusion of hydrogen...

As was shown earlier, the presence of the CPE of fractal impedance produces a distribution of the time constants. In addition, other elements such as the Warburg (semi-infinite or finite-length) linear or nonlinear diffusion, porous electrodes, and others also produce a dispersion of time constants. Knowledge about the nature of such dispersion is important in the characterization of electrode processes and electrode materials. Such information can be obtained even without fitting the experimental impedances to the corresponding models, which might be still unknown. Several methods allow for the determination of the distribution of time constants [378, 379], and they will be briefly presented below. 

Total electrode impedance consists of the contributions of the electrolyte, the electrode solution interface, and the electrochemical reactions taking place on the electrode. First, we consider the case of an ideally polarizable electrode, followed by semi-infinite diffusion in linear, spherical, and cylindrical geometry and, finally a finite-length diffusion. 

The total impedance complex plane plot for indirect hydrogen absorption without hydrogen evolution, including solution resistance and double-layer capacitance, is displayed in Eig. 7.9. It shows two semicircles due to the — Cdi and / ab Cp coupling followed by the finite-length reflective linear diffusion displaying a line at 45° followed by a capacitive line at 90°. 

---