Driving force¶

Options¶

The driving force options in the phase-interaction input allow increasing numerical stability and (together with the interface mobility and the antitrapping model) calibrating the correct interface kinetics. They consist of parameters on averaging, limiting and (in future) noise. The optional driving force parameters avg, max and smooth are defined as follows.

`avg`¶

The averaging factor stands for the intensity of averaging or a relative length over which averaging is done. It is also meant to help stabilizing the interface. This allows the user to move continuously from no averaging to complete averaging of the chemical driving force along the interface normal. Averaging the driving force avoids spreading of the interface, which can occur if the resolution is too low and there is a strong gradient of the driving force over the interface. On the other side, averaging increases artificial solute trapping. The factor is defined as the averaging length over the sum of the interface thickness and the averaging length. The allowed values range from $0$ to $1$ (no averaging to complete averaging), a typical value that works in most cases is $0.5$ . This value is also recommended for automatic mobility calculation of diffusion-controlled growth.

`max`¶

This parameter allows specifying an upper limit for the driving force in $J/cm^3$ which increases numerical stability. One should choose a value which is above what is to be expected to be on the save side (e.g. $1000$ ). A smoothing of the cut-off is done using a tanh-function. Do not define it if this is not intended! Its application can be useful because of the following reasons:

While using TQ coupling, there is always a risk of having local numerical instabilities which can occasionally result in "astronomically" high values of the driving force. These, in turn, produce astronomical increments of the phase-field variable, etc. This is a good reason to set a (less than the astronomical) limit to the driving force!

The DP_MICRESS output of the driving force gets nicer (without the need to rescale), if the maximum driving force is in the same order of magnitude as the majority of the values.

Sometimes, (e.g. during the initial transient or after nucleation), the intermediately forming microstructure cannot be resolved properly at a given scale. Limiting the driving force gives a chance to reduce numerical trouble without having to change the phase-field mobility or to use a higher grid resolution.

`smooth`¶

The parameter smooth is a directional noise on averaging which can be helpful for reducing the effect of grid anisotropy, i.e. a directional averaging is obtained. The local phase-field gradient is rotated randomly with a maximum amplitude of the given angle (in degrees), in 3D along three axes. This procedure is done for each interface cell independently, i.e. for each cell a complete path through the interface is calculated

Averaging of the driving force¶

Averaging of the driving force is important for reducing the effect of the artificial solute trapping which is a problem for all phase-field methods when the spatial resolution is too low.

It is possible to fine-tune averaging by specification of an averaging factor avg. The effect of a high value of this factor is a strong reduction of the risk of an interface spreading (interface gets thicker than the given value of $\eta$ ), but at the expense of a reduced effective mobility (i.e. the front moves slower because the average solute undercooling is decreased) and at the expense of an increased grid anisotropy (e.g. dendrites may tend to follow the grid direction instead of the anisotropy direction). In most cases, it seems to be best to use values around $0.5$ which corresponds to an averaging length of the order of the interface thickness. For fine-tuning of the kinetics, both the interface mobility and the averaging length have to be optimized (see topic 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.

Driving force calculation using linearised phase diagrams¶

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:

$\Delta G = \Delta S \cdot \Delta T = \Delta S \cdot (T - T_{eq})$

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 $m$ :

$m'_{1/2} = \delta(\Delta G) / \delta c_1 = \Delta S_{1/2} \cdot m_{1/2}$

$m'_{2/1} = \delta(\Delta G) / \delta c_2 = \Delta S_{2/1} \cdot m_{2/1}$

Furthermore, a driving force offset and a temperature dependence of the equilibrium concentrations $\delta c / \delta T$ is calculated or can be specified.