Model of gel maturation

The model of gel maturation is an example of phase-field model for two phases: solid and liquid and three compositions (ternary system). In section [lbm], the kernel is activated with the keyword problem=ternary_GP_mixt. In the rest of this section, the mathematical model and input parameters are described for model option model=two_dilute.

Mathematical model

In this model we are concerned with the diffusion of three components \(A\), \(B\) and \(C\) of compositions \(C_A\), \(C_B\) and \(C_C\). The compositions are conserved and the following relationship holds:

(107)\[C_{A}+C_{B}+C_{C}=1\]

Phase-field equation

The phase-field model is derived from the grand-potential functional with dilute assumption for composition equations. The solid phase is indicated by \(\phi=0\) and the liquid phase by \(\phi=1\). The phase field equation writes:

(108)\[\tau\frac{\partial\phi}{\partial t}=W^{2}\boldsymbol{\nabla}^{2}\phi-\omega_{dw}^{\prime}(\phi)-\lambda p^{\prime}(\phi)\Delta\omega(\boldsymbol{\mu})\]

where \(\tau\) is the relaxation time, \(W\) is the interface width, \(\omega_{dw}^{\prime}(\phi)\) is the derivative of the double-well potential with respect to \(\phi\) which is defined by

(109)\[\omega_{dw}(\phi)=8\phi^{2}(1-\phi)^{2}\]

\(\lambda\) is the coupling parameter, \(p^{\prime}(\phi)\) is the derivative of an interpolation function with respect to \(\phi\):

(110)\[p(\phi)=\phi^{2}(3-2\phi)\]

and \(\Delta\omega(\boldsymbol{\mu})\) is defined by

(111)\[\Delta\omega(\boldsymbol{\mu})=\]

Let us notice that the parameters \(\tau\) can be expressed by thermodynamic properties (e.g. height of double-well and coefficient in factor of \(\delta\Omega/\delta\phi\)). Finally in the .ini input file, the usual parameters \(W\), \(\lambda\) and \(M_{\phi}\) are set.

Composition equations

(112)\[\frac{\partial C_{A}}{\partial t}=\boldsymbol{\nabla}\cdot\left[D_{A}(\phi)C_{A}\boldsymbol{\nabla}\mu_{A}\right]\]
(113)\[\frac{\partial C_{B}}{\partial t}=\boldsymbol{\nabla}\cdot\left[D_{B}(\phi)C_{B}\boldsymbol{\nabla}\mu_{B}\right]\]

where for each specie, the diffusion coefficient is interpolated by the bulk coefficients of each phase (solid and liquid):

(114)\[D_{\alpha}(\phi)=\phi D_{\alpha}^l+(1-\phi)D_{\alpha}^s\]

for \(\alpha=A,B\).

The last composition is obtained by Eq. (107)

(115)\[C_{C}=1-C_{A}-C_{B}\]

Chemical potentials

(116)\[\mu_{A}=\log\left[\frac{C_{A}}{\phi e^{-\epsilon_{A}^{l}}+(1-\phi)e^{-\epsilon_{A}^{s}}}\right]\]
(117)\[\mu_{B}=\log\left[\frac{C_{B}}{\phi e^{-\epsilon_{B}^{l}}+(1-\phi)e^{-\epsilon_{B}^{s}}}\right]\]

List of input parameters in .ini file

One example of input file for that model is settings_ternary_instantaneous_diffusion.ini. We describe below the content of that file where the sections [lbm], [params], [ccparams] and [init] must be wisely set.

1. Section [lbm]

To simulate that mathematical model, put the following options in section [lbm]

  • use the keyword problem=ternary_GP_mixt

  • use the keyword model=two_dilute

2. Section [params]

The list of parameters are summarized in Table below.

Phase-field parameters

Math symbol

Parameter name

Equation

.ini file

\(M_{\phi}\)

Mobility of interface

Eq. (336)

Mphi

\(W\)

Interface width

Eq. (336)

W

\(\lambda\)

Coupling parameter

Eq. (336)

lambda

Diffusion parameters

Math symbol

Parameter name

Equation

.ini file

\(\epsilon_{A}^{l}\)

Bulk energy of liquid phase for specie \(A\)

Eq. (116)

elA

\(\epsilon_{A}^{s}\)

Bulk energy of solid phase for specie \(A\)

Eq. (116)

esA

\(\epsilon_{B}^{l}\)

Bulk energy of liquid phase for specie \(B\)

Eq. (117)

elB

\(\epsilon_{B}^{s}\)

Bulk energy of solid phase for specie \(B\)

Eq. (117)

esB

\(\epsilon_{C}^{l}\)

Bulk energy of liquid phase for specie \(C\)

Eq.

elC

\(\epsilon_{C}^{s}\)

Bulk energy of solid phase for specie \(C\)

Eq.

esC

\(D_{A}^l\)

Bulk diffusion of specie \(A\) in liquid phase

Eq. (114)

DA1

\(D_{A}^s\)

Bulk diffusion of specie \(A\) in solid phase

Eq. (114)

DA0

\(D_{B}^l\)

Bulk diffusion of specie \(B\) in liquid phase

Eq. (114)

DB1

\(D_{B}^s\)

Bulk diffusion of specie \(B\) in solid phase

Eq. (114)

DB0

3. Section [ccparams]

cc means composantes connexes. In this section six options can be set:

  • use_connected_components=: yes or no

  • print_cc_trace= : yes or no

  • CC_phi_threshold= : real value

  • apply_virtual_volume= : yes or no

  • virtual_volume= : real value

  • virtual_volume_boundary= : real value

4. Section [init]

Example of initialization for init_type=perco

  • init_type=: keyword. Here perco

  • initClA=: positive real value

  • initClB=: positive real value

  • initCsAA=: positive real value

  • initCsAB=: positive real value

  • initCsBA=: positive real value

  • initCsBB=: positive real value

  • read_data_phi=: yes or no

  • file_data_phi=: filename.dat (format i,j,k,value). See example init_data_perco.dat