Diffusion

The diffusion module group covers solute transport on the simulation grid and its coupling to the phase-field. Several complementary solvers share the same field layer (Composition) but differ in their thermodynamic assumptions. This page gives the overview and the shared conventions; each solver has a dedicated page (see Solvers below).

For the theoretical background of partitioning and driving force, see Thermodynamics and Diffusion.

Key Classes and Concepts

• EquilibriumPartitionDiffusionBinary: binary-alloy solver based on local equilibrium partitioning at interfaces, with an optional anti-trapping current.
• InterfaceDiffusion: adds a flux confined to the diffuse interface (grain-boundary / surface diffusion).
• Composition: the shared concentration storage; read by every solver above.

Models

Equilibrium-partition binary diffusion

Concentration evolves through a Fickian term with an optional anti-trapping correction:

$$\begin{array}{}\text{(1)}& \frac{\partial c}{\partial t}=\mathrm{\nabla }\cdot (D(\varphi )\mathrm{\nabla }c)-\mathrm{\nabla }\cdot {\mathbf{j}}_{\text{AT}},\end{array}$$

with the phase-weighted diffusivity $D(\varphi )=\sum _{\alpha }{\varphi }_{\alpha }{D}_{\alpha }$. The anti-trapping current ${\mathbf{j}}_{\text{AT}}$ is active when $EnableAntiTrapping = Yes. The driving force model is chosen via $DrivingForceModel (default STANDARD) and the spatial stencil via $DiffusionStencil (default ISOTROPIC).

Stoichiometric phases are flagged per component via $Flag_<n> and are not solved but kept at their reference composition.

Interface diffusion

InterfaceDiffusion injects a tangential flux that is active only where ${\varphi }_{\alpha }$ crosses the threshold set by $Threshold. The flux is scaled by $dSR and per-component $dIDC_<n>, and is averaged over a stencil of size $Range when $Average = Yes.

Usage

Input — Equilibrium-partition binary

Placed in the @EquilibriumPartitionDiffusionBinary block.

@EquilibriumPartitionDiffusionBinary

$RefElement                 Reference component                  : Fe
$EnableAntiTrapping         Enable anti-trapping current         : Yes
$ConsiderChemicalPotential  Use chemical-potential formulation   : No
$DrivingForceModel          Driving-force model                  : STANDARD
$DiffusionStencil           Spatial stencil                      : ISOTROPIC

$Flag_0   Stoichiometric phase 0                                 : No
$IDC_0    Interface diffusion coefficient for phase 0            : 1.0
$DC_0     Bulk diffusion coefficient for phase 0                 : 1.0e-12
$AE_0     Activation energy for phase 0                          : 150000
$EF_0     Entropy factor for phase 0                             : 0.0

$Cs_0_1   Solvus concentration (0 in 1)                          : 0.05
$Ts_0_1   Solvus temperature (0 in 1)                            : 1700
$ML_0_1   Liquidus slope (0 -> 1)                                : -500
$ML_1_0   Liquidus slope (1 -> 0)                                : 500

Input — Interface diffusion

@InterfaceDiffusion

$dSR        Interface-diffusion scaling factor                   : 1.0
$dIDC_0     Interface diffusion coefficient for component 0      : 1.0e-11
$Average    Average flux over the interface stencil              : Yes
$Range      Averaging stencil range                              : 2
$Threshold  Phase-field threshold                                : 0.05

Output

Both solvers update the Composition storage; visual output is handled by Composition::WriteVTK. Flux fields (when written) go through the same VTK pipeline.

Example

See OpenPhase-main/examples/ for a binary-alloy solidification run that constructs EquilibriumPartitionDiffusionBinary alongside Composition, PhaseField, Temperature, and InterfaceProperties.

Solvers

• Diffusion/EquilibriumPartitionBinary.md
• Diffusion/InterfaceDiffusion.md

Dependencies

• Composition — shared concentration storage.
• PhaseField — weights the bulk diffusivity.
• Temperature — controls Arrhenius diffusivity.
• InterfaceProperties — supplies the interface-mobility scaling.
• BoundaryConditions, Settings.