Interfaces in ferroelastics: fringing fields, microstructure, and size and shape effects

We develop a strain-based approach to study the transformation of a finite martensite domain within an austenite host matrix. Analytical and numerical solutions are obtained for the fringing fields in the austenite and in the martensite and we test how well the stress and strain matching conditions are obeyed at the habit planes. We investigate the scaling of the energy of the fringing fields and show how simulations on relaxed microstructures corroborate the 1/|ky| behavior for the energy in Fourier space. Our results show that the functional form F=F0+aL1ξ+bLL1/ξ for the total elastic energy provides an excellent fit to the numerical simulations, thus demonstrating that ξ~√L, where ξ is the twin width for a martensite region L×L1 with length of the habit plane L1 and where aL1ξ, bLL1/ξ, and F0 are the energies of the decaying strain field at the habit plane, twin-boundary energy, and energy of a single martensite variant, respectively. However, the result is only true for sufficiently large L and we provide insight into the breakdown of the ξ~√L scaling at the nanoscale. Our approach allows us to investigate the effect of varying the finite distance between habit planes, L, and our key finding is that there is a minimum length, Lmin, for the nucleation of the twinned martensite structure which depends on temperature. As the temperature is lowered, Lmin decreases, and at temperatures close to the stability limit of the austenite phase a lattice martensite structure in which the parent and product phases spatially alternate in a checker-board pattern is stable in a narrow region of the temperature versus L phase diagram. Such patterns have been seen at the nanoscale in lithium-based perovskites and inorganic spinels, as well as in coherent decomposition of precipitates in Co-Pt alloys. Finally, we show how the nature of the fringing fields due to an inclusion within an austenite matrix sensitively depends on its shape, size, and orientation and determines whether twinning or lattice martensite are the stable structures