Solidification

Driving force

Near the melting temperature $T_m$ , assume that the enthalpy and entropy changes are approximately independent of temperature:

$\Delta H \approx \Delta H_m \quad \text{and} \quad \Delta S \approx \Delta S_m.$

At the melting point, the solid and liquid are in equilibrium, so

$\Delta G_m = \Delta H_m - T_m \Delta S_m = 0,$

which gives

$\Delta S_m = \frac{\Delta H_m}{T_m}.$

Therefore, at a temperature $T$ near $T_m$ ,

$\begin{aligned} \Delta G(T) &= \Delta H - T\Delta S \\ &\approx \Delta H_m - T\Delta S_m \\ &= \Delta H_m - T\frac{\Delta H_m}{T_m} \\ &= \Delta H_m\left(\frac{T_m-T}{T_m}\right) \\ &= \Delta H_m\frac{\Delta T}{T_m}, \end{aligned}$

where $\Delta T = T_m-T$ is the undercooling.

References

CFD-DivingForceForSolidification.pdf

Porter, D. A., & Easterling, K. E. (2009). Phase transformations in metals and alloys (revised reprint). CRC press.