Numerical Parameters).
How does the averaging method used by MICRESS® work for the calculation of the average driving force? The chemical driving force \Delta G for each interface grid point is averaged along the direction of the normal through the interface. The weight of each cell in the averaging process consists of 3 factors:
the length of the path along the normal through the given cell (i.e. cells which are just touched slightly by the normal direction have less weight). This is important to avoid possibly strong fluctuations of the driving force if the normal direction changes slightly. Locking the averaging path to the main grid directions can be avoided by an extra noise on the normal direction using the smooth
keyword.
the gradient term (i.e. effectively by the term \sqrt{\phi_1 \cdot \phi_2} for the double obstacle potential) in the cell along the gradient path.
according to the avg input which is transformed to an averaging length l = \frac{avg}{1 - avg} \cdot \eta where \eta is the interface thickness. The distance for averaging is restricted to the distance l from the interface grid point for which the averaged driving force is calculated. Inside the averaging region the corresponding weighting factor is proportionally decreasing from 1 in the centre to 0 at the edge.
For the extreme cases of avg = 0 and avg = 1 one gets no averaging at all or complete averaging over the entire interface length including the weighting according to 1 and 2. The cut-off of the driving force (max
keyword) is done before the averaging.
Using the keyword linear
in the phase diagram input, one can define a linear phase diagram like the one shown below, where the two phase lines may intersect for c = 0. In the general case, the phase can intersect at any concentration c \ne 0.
Calculation of the driving force using linearised phase diagrams
Important: The temperature T_0, for which the linearisation parameters are specified in the input file, and the corresponding equilibrium compositions are not shown here.
At the applied temperature T, the interface most probably is not in equilibrium, because the local phase fractions (phase-field parameter) inside the interface (i.e. for each interface cell) are not in accordance with the mixture composition c and the equilibrium compositions c_L^* / c_S^* for this given temperature T.
Instead, accordance can be found for another temperature T_{eq}, which can be determined easily for a binary phase diagram, but which can also be calculated straightforward for multicomponent systems.
Thus, the effective equilibrium compositions are taken at the temperature T_{eq} rather than at the real temperature T. The difference between T and T_{eq} is used to calculate the chemical driving force:
In the multiphase case, T_{eq} is determined independently for each pair-wise phase interaction. Therefore, in triple junctions each pair-wise interface can be close to or far from local equilibrium. In the case of Thermo‑Calc™ coupling as well as for linearTQ, the construction of the (extrapolated) linear phase diagram is slightly different. The diagram is constructed rather in terms of \Delta G than of T, the slopes m' of the phase lines are derived from the driving force \Delta G and related to the real slopes m:
Furthermore, a driving force offset and a temperature dependence of the equilibrium concentrations \delta c / \delta T is calculated or can be specified.