Evolution equations for phase separation and ordering in binary alloys

Abstract

We explore two phenomenological approaches leading to systems of coupled Cahn-Hilliard and Cahn-Allen equations for describing the dynamics of systems which can undergo first-order phase separation and order-disorder transitions simultaneously, starting from the same discrete lattice free energy function. In the first approach, a quasicontinuum limit is taken for this discrete energy and the evolution of the system is then assumed to be given by gradient flow. In the second approach, a discrete set of gradient flow evolution equations is derived for the lattice dynamics and a quasicontinuum limit is then taken. We demonstrate in the context of BCC Fe−Al binary alloys that it is important that variables be chosen that accommodate the variations in the average concentration as well as the underlying ordered structure of the possible coexistent phases. Only then will the two approaches lead to roughly the same continuum descriptions. We conjecture that in general the number of variables necessary to describe the dynamics of such systems is equal toN ₁ +N ₂−1, whereN ₁ is given by the dimension of the span of the bases of the irreducible representations needed to describe the symmetry groups of the possible equilibrium phases andN ₂ is the number of chemical components.N ₁ of these variables are nonconserved, and the remaining are conserved and represent the average concentrations. For the Fe−Al alloys this implies a description of one conserved order parameter and one nonconserved order parameter. The resultant description is given by a Cahn-Hilliard equation coupled to a Cahn-Allen equation via the lower-order nonlinear terms. The rough equivalence of the two phenomenological methods adds credibility to the validity of the resulting evolution equations. A similar description should also be valid for alloy systems in which the structure of the competing phases is more complicated.