  A non-isothermal phase field model that captures both displacive and
diffusive phase transformations in a unified framework is presented. The model
is developed in a formal thermodynamic setting, which provides guidance on
admissible constitutive relationships and on the coupling of the numerous
physical processes that are active. Phase changes are driven by
temperature-dependent free-energy functions that become non-convex below a
transition temperature. Higher-order spatial gradients are present in the model
to account for phase boundary energy, and these terms necessitate the
introduction of non-standard terms in the energy balance equation in order to
satisfy the classical entropy inequality point-wise. To solve the resulting
balance equations, a Galerkin finite element scheme is elaborated. To deal
rigorously with the presence of high-order spatial derivatives associated with
surface energies at phase boundaries in both the momentum and mass balance
equations, some novel numerical approaches are used. Numerical examples are
presented that consider boundary cooling of a domain at different rates, and
these results demonstrate that the model can qualitatively reproduce the
evolution of microstructural features that are observed in some alloys,
especially steels. The proposed model opens a number of interesting
possibilities for simulating and controlling microstructure pattern development
under combinations of thermal and mechanical loading.
