On the Gibbs Adsorption Equation for Diffuse Interface Models
Geoffrey B. McFadden, A A. Wheeler
In this paper we discuss some applications of the classical Gibbs adsorption equations to specific diffuse interface models that are based on conserved and non-conserved order parameters. Such models are natural examples of the general methodology developed by J.W. Gibbs in his treatment of the thermodynamics of surfaces. We employ the methodology of J.W. Cahn, which avoids the use of conventional dividing surfaces to define surface excess quantities. We show that the Gibbs adsorption equation holds for systems with gradient energy coefficients, provided the appropriate definitions of surface excess quantities are used. We consider in particular the phase field model of a binary alloy with gradient energy coefficients for solute and the phase field. We derive a solute surface excess quantity that is independent of a dividing surface convention, and find that the adsorption in this model is influenced by the surface free energies of the pure components of the binary alloy, as well as the solute gradient energy coefficient. We present one-dimensional numerical solutions for this model corresponding to a stationary planar interface and show the consistency of the numerical results with the Gibbs adsorption equation. We also discuss the application of the Gibbs adsorption equation in the context of other diffuse interface models that arise in spinodal decomposition and order-disorder transitions.