Derivation of a Mathematical Model for the Growth of Anodic TiO₂Nanotubes under Constant Current Conditions

Due to certain limitations of traditional models, the growth mechanism of porous anodic TiO 2 nanotubes has not been well determined currently. Herein, for the first time, a mathematical model of voltage-time transient curves under constant current conditions is derived theoretically, based on the conception of ionic and electronic currents and Ohm's law. The simulation results show high fidelity to the experimental curves, and illustrate the linear correlation between nanotube length and ionic current. Further, based on this model, the avalanche breakdown can be explained, which shows advantage over the former derivations on compact films. And this model indicates that the discrepancy between compact and porous oxide films lies on the magnitude of electronic current during anodization. Moreover, the proportion of the ionic and electronic current is then calculated during constant current anodization. It can be concluded that the ionic current contributes to the oxide growth while the electronic current gives rise to the oxygen bubble evolution which acts as the growth mould of the oxide. The present results promote the understanding of the growth kinetics of porous anodic oxides from qualitative interpretation to quantitative analyses.