ENERGY

Description

This class is used for specifying the chemical free energy density in phase field model \(f_{\rm{ch}}(c, \phi)\) and to calculate its partial derivatives \(\displaystyle{\frac{\partial f_{\rm{ch}}}{\partial c}, \frac{\partial f_{\rm{ch}}}{\partial \phi}, \frac{\partial^2 f_{\rm{ch}}}{\partial \phi^2}, \frac{\partial^2 f_{\rm{ch}}}{\partial c^2}, \frac{\partial^2 f_{\rm{ch}}}{\partial c \partial \phi}...}\)

The chemical free energy density of binary alloy is a function of the order parameter \(\phi\) and of the concentration field \(c\). In order to ensure the coexistence of both phases \(\alpha\) and \(\beta\) discriminated by \(\phi\), \(f_{\rm{ch}}\) must be non-convex with respect to \(\phi\). Consequently, \(f_{\rm{ch}}\) is split into a local homogeneous free energy \(f_0(c,\phi)\), which is built with the free energy densities of the two phases \(f_1\) and \(f_2\) and a double well potential accounting for the free energy penalty of the interface [MS2]:

(619)\[f_{\rm{ch}}(\phi,c) = f_0(c,\phi) + W g(\phi)\qquad{\rm where}\qquad g(\phi) = \phi^2(1-\phi)^2\]

where \(W\) is the height of the double-well barrier. Both material parameters \(W\) and \(\alpha\) are calculated in function of the interfacial energy \(\sigma\) and the interfacial thickness \(\delta\) as

(620)\[W = 6 \Lambda \frac{\sigma}{\delta} \qquad{\rm and}\qquad \alpha = 3 \frac{\sigma \lambda}{\Lambda} \qquad{\rm where}\qquad \Lambda = \ln[(1-\bf \zeta)/ \bf \zeta]\]

The material parameter \(\bf \zeta\) specifies the way that the interface width \(\delta\) has been defined, Assuming that the interface region ranges from \(\bf \zeta\) to \(1-\bf \zeta\).

Syntax

The basic input syntax here is:

**energy <ENERGY> \(~\,\) *phase1 \(~\,~\,\) c1 <COEFFICIENT> \(~\,~\,\) b1 <COEFFICIENT> \(~\,~\,\) k1 <COEFFICIENT> \(~\,~\,\) D1 <COEFFICIENT> \(~\,\) *phase2 \(~\,~\,\) c2 <COEFFICIENT> \(~\,~\,\) b2 <COEFFICIENT> \(~\,~\,\) k2 <COEFFICIENT> \(~\,~\,\) D2 <COEFFICIENT> \(~\,\) *interface \(~\,~\,\) energy <COEFFICIENT> \(~\,~\,\) thickness <COEFFICIENT> \(~\,~\,\) zeta <COEFFICIENT> \(~\,~\,\) ENER <COEFFICIENT>

*interface

specify the material parameters related to the phase field interface, which are the interfacial energy \(\sigma\), the interfacial thickness \(\delta\) and \(\bf \zeta\)

The following coefficients are available:

k1, k2

are respectively the curvatures of the local free energies \(f_1\) and \(f_2\) with respect to concentration.

b1, b2

are the heights of the free energies \(f_1\) and \(f_2\).

c1, c2

are the coherent equilibrium concentrations.

D1, D2

are the chemical diffusivities in both phases \(1\) and \(2\).

CODE	DESCRIPTION
kim	Polynomial formulation of homogeneous free energy
afa	Interpolating free energy densities

kim

The chemical free energy density is a quadratic function of the concentration, where the chemical free energies of the two phases are interpolated for intermediate values of \(\phi\) with a polynomial \(h(\phi)\) varying in a monotonic way between both phases:

(621)\[f_0(\phi,c)=h(\phi)f_1(c) + [1-h(\phi)] f_2(c)\]

where \(f_1\) and \(f_2\) are the chemical free energy densities of both phases, which have been described by simple quadratic functions of the concentration \(c\):

(622)\[f_i(c) = \frac{1}{2} k_i (c-a_i)^2 + b_i\]

where \(i = \{1,2\}\) denotes phase \(1\) or \(2\).

afa

This energy is obtained by a linear interpolation of the free energy parameters \(a(\phi), b(\phi)\) and \(k(\phi)\). It is summarised as:

(623)\[f_0(\phi,c) = \frac{1}{2} k(\phi)(c -a(\phi))^2 + b(\phi)\]

where

(624)\[a(\phi) = a_2 + \Delta a h(\phi),\quad k(\phi) = \frac{k_1 k_2}{k_1 + h(\phi)\Delta k} \quad {\rm and}\quad b(\phi) = b_2 + \mu_{\rm{eq}} \left(\Delta a + \frac{(\mu_{\rm{eq}})^2 \Delta k}{2\bar{k}} \right) h(\phi)\]