Grand Potential Solver

The GrandPotentialSolver solves the equation

$$\begin{array}{}\text{(1)}& \dot{{\mu }_i}={\left(\sum _{\alpha }{\varphi }_{\alpha }\frac{{\partial }^2{\omega }_{\alpha }}{\partial {\mu }^2}\right)}^{-1}(\mathrm{\nabla }\left(M_i\mathrm{\nabla }{\mu }_i\right)+\sum _{\alpha }\dot{{\varphi }_{\alpha }}\frac{\partial {\omega }_{\alpha }}{\partial \mu })\end{array}$$

where ${\mu }_i$ is the chemical potential (J/mol) of the component $i$, ${\varphi }_{\alpha }$ is the phase-field of phase $\alpha$, ${\omega }_{\alpha }$ is the grand potential density (J/m${}^3$/mol) of phase $\alpha$, $M_i$ the chemical mobility of the component $i$. Further, a driving force contribution is calculated

$$\begin{array}{}\text{(2)}& {\mathrm{\Delta }}_{\alpha \beta }=(\frac{\partial \omega }{\partial {\varphi }_{\beta }}-\frac{\partial \omega }{\partial {\varphi }_{\alpha }}).\end{array}$$

Further details of the grand potential method can be found in e.g.^[1]

Dependencies

• BoundaryConditions
• GrandPotentialDensity
• InterfaceProperties
• PhaseField
• RunTimeControl
• Settings
• Temperature
• VTK

Usage

Below a minimalistic usage example is given

// ....
// Initialize solver and read input of module dependencies
Settings              OPSettings (InputFileName);
BoundaryConditions    BC         (OPSettings, InputFileName);
GrandPotentialDensity omega      (OPSettings, InputFileName);
InterfaceProperties   BC         (OPSettings, InputFileName);
PhaseField            Phi        (OPSettings, InputFileName);
RunTimeControl        RTC        (OPSettings, InputFileName);
Temperature           Tx         (OPSettings, InputFileName);
// ....
// Initialize solver and read input
GrandPotentialSolver GPS (OPSettings, InputFileName);
// ....
omega.Set(Temp.Tx,GPS.ChemicalPotential);
// ....

for(RTC.tStep = RTC.tStart; RTC.tStep <= RTC.nSteps; RTC.IncrementTimeStep())
{
    if (RTC.WriteVTK())
    {
        // ....
        // Write VTK output
        GPS.WriteVTK (OPSettings, Phase, omega, RTC.tStep);
        // ....
    }

    if (RTC.WriteRawData())
    {
        // ....
        // Write restart output
        GPS.Write(OPSettings, RTC.tStep);
        // ....
    }

    // ....
    GPS.Solve(Phase,omega,BC,IP,Tx,RTC.dt);
    // ....
}

Input

@GrandPotentialSolver

$MASS_FE        Molar Mass of Fe [kg/mol]                         : 0.055845

$CI_0_FE        Initial moles per volume of Fe in vapor [mol/m^3] : 1
$CI_1_FE        Initial moles per volume of Fe in gamma [mol/m^3] : 140984

$M0_0_FE        Chemical mobility of Fe in vapor [m^2/s]          : 1.00e-12
$M0_1_FE        Chemical mobility of Fe in gamma [m^2/s]          : 1.00e-12

$IM             Use Interface Mobility Coefficient                : No
$dMdc           Phase Mobility Concentration Coupling             : No

$Implicit       Use semi implicit Euler scheme                    : No
$MAXI           Maximum of iterations                             : 1000
$ACC            Relative Accuracy of chemical potential           : 1.0e-08

$TOC            Conserve total amount of components               : Yes
$TOCMAXI        Maximum of iterations                             : 10
$TOCACC         Relative Accuracy                                 : 1.0e-10

References

[1] Mathis Plapp (2011). Unified derivation of phase-field models for alloy solidification from a grand-potential functional. Physical Review E, 84, 031601. DOI: 10.1103/PhysRevE.84.031601 ↩