The phase eld model is a general name for a class of diuse interface models used to study a wide variety of materials phenomena. It has several advantages over other interface tracking approaches, therefore it has been used to model general multi-phase systems with features much larger than the real interface thicknesses.
The phase-eld method, as presented here, grows out of the work of Cahn, Hilliard and Allen. It is used for two general purposes:
• to model systems in which the diuse nature of interfaces is essential to the problem, such as spinodal decomposition and solute trapping during rapid phase boundary motion;
• as a front tracking technique to model general multi-phase systems.
Generally, we speak about two types of phase eld models. In the rst, called Cahn-Hilliard, the phase is uniquely determined by the value of a conserved eld variable, such as the concentration C, e.g. if C ≤ C1, then we are in one phase, if C ≥ C2 then the other. These models were rst applied to understand spin- odal decomposition, and are now used for a wide range of phenomena. In the second, called Allen-Cahn, the phase is not uniquely determined by concentra- tion, temperature, pressure, etc., so we add one or more extra eld variable(s) sometimes called the order parameter φ which determines the local phase. This class of models is widely used to study solidication and solid-state phase trans- formations in metals.
The phase-eld method is a xed-grid method; it diers from other methods in that the interface is diuse in a physical rather than numerical sense. Thus, it is also known as the diuse-interface model. More precisely, the diuse inter- face is introduced through an energetic variational procedure that results in a thermodynamic consistent coupling system. The basic idea was derived from the consideration that the two components, though nominally immiscible, does mix in reality within a narrow interfacial region. A phase-eld variable φ can be thought of as the volume fraction, to demarcate the two species and indicate the location of the interface. A mixing energy is dened based on φ which, through
a convection-diusion equation, governs the evolution of the interfacial prole. The phase-eld method can be viewed as a physically motivated level-set method. When the thickness of the interface approaches zero, the diuse-interface model becomes asymptotically identical to a sharp-interface level-set formulation. It also reduces properly to the classical sharp-interface model in general.
From the statistical (phase eld approach) point of view, the interface represents a continuous, but steep change of the properties (density, viscosity, etc.) of two uids. Within this "thin" transitional region, the uid is mixed. The mixing is determined by molecular interactions between the two species, and can be described by a stored mixing energy, which represents the balance between the competing phobic/philic relation