Droplet Free Energy Functional for the Morphology of Martensites

Martensites are metastable phases, possessing a characteristic morphology, usually formed during a fast quench accross a structural transition. We attempt to understand these morphological features using a coarsegrained free energy functional ${\cal F}[\epsilon ; \Phi]$ which contains, in addition to the usual strain fields $\epsilon_{ij}$ (the `` order parameter'' for the transition), the ``vacancy'' field $\phi$ which arises due to the geometric mismatch at a parent-product interface. The relaxation of this mismatch is slow compared to typical front propagation times and hence $\phi$ is essentially frozen in the reference frame of the growing martensite front. Minimisation of ${\cal F}$ then automatically yeilds typical martensite morphologies. We demonstrate this in two dimensions for the square to rhombus transformation and obtain internally twinned martensites, which grow as thin strips, for ``hard'' martensites (e.g., Fe-based alloys) or with a `single-interface', for ``soft'' martensites (e.g., In-Tl alloys).