driving force with linearized phase diagrams
Posted: Tue Oct 13, 2009 10:05 pm
How the driving force delta G is calculated in MICRESS using linearized 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 (but need not) intersect for c=0 (note that the temperature T_0 for which the linearisation parameter 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 cL*/cS* for this given temperature T. Instead, accordance can be found for another temperature T_eq, which can be easily determined for a binary phase diagram, but which can also be calculated straightforward for multicomponent systems. Thus, the effective equilibrium compositions are taken rather at the temperature T_eq than at the real temperature T. The difference between T and T_eq is used to calculate the chemical driving force:
deltaG =deltaS * deltaT = deltaS (T - Teq)
In the multiphase case, T_eq is determined independently for each pairwise phase interaction. In triple junctions therefore, each pairwise interface can be more or less 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 rather in terms of deltaG than of T, the slopes m' of the phase lines are derived from the driving force deltaG and related to the "real" slopes m:
m'_1/2 =d(deltaG)/dc1 = deltaS_1/2 * m_1/2
m'_2/1 =d(deltaG)/dc2 = seltaS_2/1 * m_2/1
Furthermore, a driving force offset and a temperature dependence of the equilibrium concentrations dc/dt is calculated or can be specified.
Bernd
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 (but need not) intersect for c=0 (note that the temperature T_0 for which the linearisation parameter 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 cL*/cS* for this given temperature T. Instead, accordance can be found for another temperature T_eq, which can be easily determined for a binary phase diagram, but which can also be calculated straightforward for multicomponent systems. Thus, the effective equilibrium compositions are taken rather at the temperature T_eq than at the real temperature T. The difference between T and T_eq is used to calculate the chemical driving force:
deltaG =deltaS * deltaT = deltaS (T - Teq)
In the multiphase case, T_eq is determined independently for each pairwise phase interaction. In triple junctions therefore, each pairwise interface can be more or less 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 rather in terms of deltaG than of T, the slopes m' of the phase lines are derived from the driving force deltaG and related to the "real" slopes m:
m'_1/2 =d(deltaG)/dc1 = deltaS_1/2 * m_1/2
m'_2/1 =d(deltaG)/dc2 = seltaS_2/1 * m_2/1
Furthermore, a driving force offset and a temperature dependence of the equilibrium concentrations dc/dt is calculated or can be specified.
Bernd