phasefield_phic

Description

The present model belongs to the class of diffuse interface models, where the local state of an inhomogeneous microstructure is described by a conservative concentration field \(c\) and a non-conservative field \(\phi\) [MS2], the so-called order parameter. In the phase field approach, the free energy density for an inhomogeneous system can be approximated by the Ginzburg-Landau coarse-grained free energy functional, which contains a chemical free energy density \(f_{ch}(c, \phi)\) and a gradient energy term: f(c , ) = f_ch(c ,) + . The usual specific quadratic contribution with respect to \(\nabla \phi\) is related to the interfacial energy. The state laws and the complementary evolution laws for the phase field and chemical contributions are the following:

(588)\[\begin{split}\mu = \frac{\partial f}{\partial c} = \frac{\partial f_{ch}}{\partial c} \\ \vect{\xi} = \frac{\partial f}{\partial \nabla \phi} = \alpha \ \nabla \phi\end{split}\]

(589)\[\begin{split}\vect{J} = - L(\phi) \nabla \mu = - L(\phi) \nabla (\frac{\partial f}{\partial c}) \\ \pi = - \beta \ \dot \phi - \frac{\partial f}{\partial \phi}\end{split}\]

where \(\mu\) is the chemical potential.

\(L(\phi)\)

is the Onsager coefficient, related to the chemical diffusivities \(D_1\) and \(D_2\) in both phases by means of the interpolation function \(h(\phi)\) as:

(590)\[L(\phi) = h(\phi) D_1/k_1 + (1-h(\phi))D_2/k_2\]

\(\beta\)

material parameter, which is inversely proportional to the interface mobility.

\(\alpha\)

composition gradient energy coefficient.

The evolution equations for order parameter and concentration are respectively based on the time-dependent Ginzburg-Landau and usual diffusion equations, which are:

prefix	size	description	default
dC	V	Concentration gradient	yes
J	V	Concentration flux	yes
dphi	V	Order parameter gradient	yes
xi	V	Microstress	yes
pi	S	Internal microforce	yes
C	S	the concentration	yes
phi	S	Order parameter	yes

Syntax

***behavior phasefield_phic \(~\,\) **energy <ENERGY> \(~\,~\,\) ... \(~\,\) **kinetics \(~\,~\,\) *mobility <COEFFICIENT> \(~\,\) **chemical_interpolating_function val

**energy

this option will be much more detailed in the <ENERGY> section.

**kinetics

give the \(\beta\) coefficient.

**chemical_interpolating_function

define the polynomial degree of interpolating function. Tree choices are availables.

CODE	DESCRIPTION
\(0\)	\(h(\phi) = \phi\)
\(1\)	\(h(\phi) = \phi^2 ( 3 - 2 \phi)\)
\(2\)	\(h(\phi) = \phi^3 ( 6 \phi^2 - 15 \phi + 10)\)

Example

***behavior phasefield_phic
 **energy kim
  *phase1
   c1 0.7
   b1 0.0
   k1 1.
   D1 0.1
  *phase2
   c2 0.3
   b2 0.0
   k2 1.
   D2 0.1
  *interface
   energy 1.
   thickness 0.25
   zeta 0.05
   ENER 0.5
 **kinetics
  *mobility 1.
 **chemical_interpolating_function 1.
***return