### Re: PF simulation of Eutectic solidification

Posted:

**Thu Jan 07, 2021 3:34 pm**Dear Omid,

Wish you also a Happy New Year.

I guess that your question refers to the sum rule for entropy of transformation ΔS, which I wrote some time ago in this forum. In principle, this is a quite general property of scalar fields (like Gibbs energy) and their derivatives: There is a general path independence of the path integral over the gradient of the scalar field, i.e. the integral only depends on its initial and end point, and an integral over a closed path must be zero. You can find it e.g. on Wikipedia (https://en.wikipedia.org/wiki/Line_integral).

For the driving forces as "gradient" of the Gibbs energy in the quasi-equilibrium approach, this can be made obvious:

It can be easily seen that ΔG

However, the path is only closed, if you derive the linearisation parameters for a common quasi-equilibrium, i.e. all tangents to the Gibbs energies are parallel. If you calculate the linearisation parameters for the different phase interactions at different places in the RVE (with different conditions), then there might be deviations from the sum rule. But for a global linearized phase diagram this rule should always hold in order to be consistent.

PS: Do you agree if I divide this thread and put your question with my answer as a new thread with title "Sum rule for entropy of transformation"?

Best wishes

Bernd

Wish you also a Happy New Year.

I guess that your question refers to the sum rule for entropy of transformation ΔS, which I wrote some time ago in this forum. In principle, this is a quite general property of scalar fields (like Gibbs energy) and their derivatives: There is a general path independence of the path integral over the gradient of the scalar field, i.e. the integral only depends on its initial and end point, and an integral over a closed path must be zero. You can find it e.g. on Wikipedia (https://en.wikipedia.org/wiki/Line_integral).

For the driving forces as "gradient" of the Gibbs energy in the quasi-equilibrium approach, this can be made obvious:

It can be easily seen that ΔG

_{αβ}+ΔG_{αγ}=ΔG_{βγ}. And if this is true for ΔG, then it also must be true for ΔS = d(ΔG)/dT.However, the path is only closed, if you derive the linearisation parameters for a common quasi-equilibrium, i.e. all tangents to the Gibbs energies are parallel. If you calculate the linearisation parameters for the different phase interactions at different places in the RVE (with different conditions), then there might be deviations from the sum rule. But for a global linearized phase diagram this rule should always hold in order to be consistent.

PS: Do you agree if I divide this thread and put your question with my answer as a new thread with title "Sum rule for entropy of transformation"?

Best wishes

Bernd