Triple junctions in phase field models
Phase field models are nowadays one of the most important methods to predict microstructure formation and the kinetics of phase transformations. In polycrystalline materials triple and higher order junctions naturally apear, which require an accurate treatment in phase field models.
The evolution of microstructures is of fundamental importance for material properties, and its simulation is therefore of highest priority for development of novel materials, as well as industrial and academic research activities. Nowadays, phase field models are considered as the method of choice for simulating interfacial pattern formation processes and the kinetics of moving boundary problems which are the basic processes that lead to the well-known and diverse microstructures. In these methods, the complicated tracking of interfaces is replaced by evolution equations for the ‘‘phase fields’’, which serve as order parameters for distinguishing between different phases and grains. A localized spatial change of the order parameters therefore determines the positions of interfaces, and e.g. at triple junctions three order parameters vary simultaneously rather rapidly in space. When a phase field model is applied to simulate the coarsening in multi-grain structures and grain growth, the corresponding sharp interface model should fulfill Young’s law, which states that at a junction of multiple interfaces the sum of the surface tension induced forces sums up to zero. This implies that specific angles between the interfaces have to be established at these junctions, which are determined by the values of the interfacial energies. Physically, this equilibration can be considered as the result of minimization of the interfacial energy. Since the balance of forces at these multiple junctions is not contained explicitly in phase field formulations, the models have to be developed and checked with care to ensure that they do indeed satisfy this important and fundamental law. In particular for heterogeneous nucleation, where a new phase appears in contact with a third phase, the proper geometric adjustment of triple lines is important, and therefore it is essential to use consistent models.
We investigate the properties of the multi-order parameter phase field model of Steinbach and Pezzolla with respect to the behavior in triple and higher order junctions. From the structure of this model, it was speculated that ‘‘dynamical’’ solutions may exist in the triple junction, which could lead to a violation of Young’s law. Here we have confirmed analytically recent numerical simulations showing that such dynamical states do not exist, and that an equilibrium solution therefore does indeed correspond to a minimum of the free energy; this implies that Young’s law must be satisfied in the framework of the model. We have shown that Young’s law is a consequence of the interface kinetic equilibrium and not due to a mechanical force balance, in agreement with earlier predictions.