Continuous-time random-walk theory of interfering diffusion and chemical reaction with an application to electrochemical impedance spectra of oxidized Zr-1%Nb

A microscopic theory is developed for the interplay of diffusion and chemical reaction and the results are compared with electrode impedance measurements on an oxide electrode. The theory is based on the ideas of continuous-time random walk and accounts for the interference of diffusion and recombination of the charge carriers in the oxide. The treatment results in a dispersive diffusivity with two time constants, one of them corresponding to the random walk, the other to the reaction. Combining this diffusivity with the Warburg electrode admittance expression, which refers to cases where the rate-limiting step is diffusion in a semi-infinite medium bounded by a plane, an admittance function is obtained. The phase angle is found to be higher than 45 degrees distinguishing it from the Gerischer impedance which was developed for a related problem. The oxides were produced by hydrothermal oxidation of Zr-l%Nb alloy, a metal used as cladding material for nuclear fuel elements. The electrode impedance spectra of Zr/Zr-oxide electrodes in aqueous SO(3) (2-) solutions were taken at various anodic voltages between 1 Hz and 100 kHz and temperatures between 278 and 333 K. The theoretical admittance functions could be successfully compared with the observed spectra. Both the functional forms and the fitted parameter values support our theory which is also in keeping with Macdonald's point-defect model.