On the Gibbs adsorption equation and diffuse interface models

Abstract

In this paper we discuss some applications of the classical Gibbs adsorption equation to speci c di¬use 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 de ne surface excess quantities. We show that the Gibbs adsorption equation holds for systems with gradient energy coe¯cients, provided the appropriate de nitions of surface excess quantities are used. We consider, in particular, the phaseeld model of a binary alloy with gradient energy coe¯cients for solute and the phase eld. We derive a solute surface excess quantity that is independent of a dividing surface convention, and nd that the adsorption in this model is in®uenced by the surface free energies of the pure components of the binary alloy as well as the solute gradient energy coe¯cient. 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 Gibbs adsorption equation in the context of other di¬use interface models that arise in spinodal decomposition and order{disorder transitions.

A quantitative multi-phase field model of polycrystalline alloy solidification

“…The convergence of the model was studied by conducting two-dimensional simulations of two isothermal free dendrites, grown using Eqs. (33) and (34). The anisotropy was chosen to have the standard fourfold form (33) and (34) were simulated in a dimensionless form using a finite-difference Euler time-stepping method.…”

Sharp-interface model for eutectic alloys. Part I: Concentration dependent surface tension

“…The first term of (5.20) becomes in Fourier space 22) which is the stiffest term and will be treated implicitly. However, in this form it is linear and diagonal in Fourier space and hence one only has to solve a diagonal system.…”

“…In this case, the strategy is the identification of the Gibbs adsorbtion equation, which allows likewise the calculation of the surface tension, at least in phase equilibrium. The background and strategy are carefully described in [22], where also a phase field model of a binary mixture is formulated. Despite the fact that mechanical effects are ignored here, the main basic difference to our approach is the following.…”

“…We start from the assumption that the phase field is given by the concentration itself. In [22] additionally to the concentration a phase field is introduced, to take care of the possibility that there might be no one-to-one correspondence between the present phase and the present concentration.…”

Diffuse interface model of surfactant adsorption onto flat and droplet interfaces

“…As we deal with a diffuse interface, the excess amount of surfactant is to be obtained over integration over the diffuse interface (McFadden and Wheeler 2002). Consequently, there exists no analytical expression for the equation of state, but we infer that the excess amount of surfactant is proportional to ψ 0 ; and consequently, that the interfacial tension lowering is proportional to that for a sharp interface: dσ $ Àψ 0 dμ ψ;0 : After substitution of Eq.…”