A mathematical analysis of phase field alloys and transition layers

A phase field approach to binary alloys is studied. The model identifies all macroscopic parameters and the interface thickness. Formal asymptotics of the system of parabolic differential equations leads to new interface relations as part of macroscopic model which arises in the limit of vanishing interface thickness. Equilibrium phase diagram can be retrieved from these interface relations. Under suitable conditions we prove that the phase field system has a unique solution which converges to the limiting macroscopic solution and therefore this provides a proof of the existence theorem for a correspondent sharp interface problem. The concentration and phase are discontinuous across the interface in the limit as interface thickness approaches zero. Transition layers in concentration are induced due to the change in phase and the change in material diffusion across the interface. Solute trapping arises as a consequence of these layers. The classical Stefan problem is retrieved from pure phase field equations under certain circumstances.