Grand Potential Phase Density Parabolic

GrandPotentialPhaseDensity_Parabolic is the simplest GrandPotentialPhaseDensity implementation: the per-phase grand-potential density is a quadratic function of the component chemical potentials. It is the natural choice when calibrating against linearised thermodynamic data from a CALPHAD-style fit and for method-development studies. It sits alongside the ideal-gas, compressible-fluid, and implicit-parabolic variants used by the Grand Potential Solver.

Model

The parabolic grand-potential density reads

$$\begin{array}{}\text{(1)}& {\omega }_{\alpha }=-\sum _i\frac{1}{2}{ϵ}_{i\alpha }{\mu }_i^2+c_{0i\alpha }{\mu }_i,\end{array}$$

where ${ϵ}_{i\alpha }$ is the parabolic energy coefficient (units J·m³/mol²) and $c_{0i\alpha }$ is the equilibrium concentration (units mol/m³) of component $i$ in phase $\alpha$.

The phase concentration is the negative first derivative of the density with respect to the chemical potential:

$$\begin{array}{}\text{(2)}& c_{\alpha i}\equiv -\frac{\partial {\omega }_{\alpha }}{\partial {\mu }_i}={ϵ}_{i\alpha }{\mu }_i.\end{array}$$

The susceptibility is the negative second derivative:

$$\begin{array}{}\text{(3)}& {\chi }_{\alpha i}\equiv -\frac{{\partial }^2{\omega }_{\alpha }}{\partial {\mu }_i^2}={ϵ}_{i\alpha }.\end{array}$$

Usage

Input

Defined inside the @GrandPotentialDensity block. One $ϵ$ and one $c_0$ per (phase, component-name) pair, with <alpha> the phase index (zero-based) and <ComponentName> the component string declared in @Settings.

@GrandPotentialDensity

$EPS_0_FE   Parabolic energy coefficient (phase 0, Fe)  : 0.10
$EPS_1_FE   Parabolic energy coefficient (phase 1, Fe)  : 0.08
$C0_0_FE    Equilibrium density (phase 0, Fe)           : 1
$C0_1_FE    Equilibrium density (phase 1, Fe)           : 140984

Output

No dedicated output file. The parabolic phase-density state is read by the Grand Potential Solver; the resulting chemical-potential evolution and derived composition are written through the solver's VTK output.

Example

#include "GrandPotential/GrandPotentialSolver.h"
// GrandPotentialPhaseDensity_Parabolic is selected at the solver's
// construction time based on the @GrandPotentialDensity block.

GrandPotentialSolver GPS(OPSettings, InputFile);

for(RTC.tStep = RTC.tStart; RTC.tStep <= RTC.nSteps; RTC.IncrementTimeStep())
{
    GPS.Solve(Phi, Cx, Tx, BC, RTC.dt);

    if (RTC.WriteVTK())
    {
        GPS.WriteVTK(OPSettings, RTC.tStep);
    }
}

Dependencies

• GrandPotentialPhaseDensity — base class.
• GrandPotentialSolver — consumer.
• GrandPotentialDensity — the mixture density built from the per-phase densities.
• Composition — component registry.