Theory of the elastic strain energy due to crystallographic shear plane arrays in reduced rutile (TiO2)

Abstract

Calculations of the elastic strain energy due to an infinite ordered array of crystallographic shear (CS) planes lying on {132} and {121} planes in reduced rutile (Ti${\mathrm{O}}_2$) have been made by using the Fourier-transform treatment. The strain energy per unit area per CS plane and per unit volume for an infinite ordered array have been evaluated as a function of $n$ in ${\mathrm{Ti}}_n{\mathrm{O}}_{2n-1}$. It was found that, though the strain energy per unit area per {132} CS plane is less than that per {121} plane, there is a crossover in the strain-energy curves per unit volume for the {132} and {121} CS planes. These quantitative results clarify the experimental observations that initial reduction of Ti${\mathrm{O}}_2$ results in randomly distributed CS planes upon {132} planes and the change from {132} to {121} shear planes takes place as the oxygen deficiency is increased. The stability of members of the homologous series of oxides, ${\mathrm{Ti}}_n{\mathrm{O}}_{2n-1}$, has been discussed by comparing the strain energy in a crystal of an oxide ${\mathrm{Ti}}_n{\mathrm{O}}_{2n-1}$ with that in a crystal containing two phases, ${\mathrm{Ti}}_{n-1\mathrm{}}{\mathrm{O}}_{2(n-1\mathrm{})-1\mathrm{}}$ and ${\mathrm{Ti}}_{n+1\mathrm{}}{\mathrm{O}}_{2(n+1\mathrm{})-1\mathrm{}}$. The calculated results agree well with the microstructures of CS-plane arrays observed in practice and it was found that the elastic strain energy plays a significant role in controlling the microstructure of a crystal containing CS planes.