Diffusion–reaction modelling of modulated hydrogen loading with internal traps

A diffusion–reaction model is presented for modulated hydrogen loading in planar geometry taking into account the trapping at and detrapping from defects in the bulk as well as the reaction rate of hydrogen insertion into the electrode. By solving this diffusion–reaction model for the steady state, the modulation of the hydrogen concentration inside the sample as well as the hydrogen flux is obtained. The modulation provides valuable information on the kinetics owing to the phase shift that arises between the modulated hydrogen concentration imposed at the surface and the loaded hydrogen concentration in the bulk. The phase shift is, on the one hand, determined by the ratio between reaction- and diffusion-limitation and, on the other hand, by the ratio between trapping and detrapping rate. With respect to analysis, modulation has the inherent advantage that also for the general case of exhaustible trapping, the diffusion–reaction model for the modulated part can be reasonable well approximated by analytical solutions, as compared to the case without modulation. The modulation behaviour is determined by characteristic rates associated with diffusion, surface reaction, and trapping, which can directly be related to the characteristic mean time of unloading. The presented model holds not only for electrochemical loading but also for other types of loading (e.g. from the gas phase) and is, therefore, suitable for the analysis of both electrochemcial impedance spectroscopy and classical defect-specific measuring techniques in materials science (e.g. dilatometry or resistometry).