# Diffusion of heat during phase transition

When one substance and one phase is present, the diffusion of temperature is given by the heat equation

$$\partial_t T = \nabla\cdot(\kappa\nabla T)$$

Can the heat equation be modified to accommodate phase transitions? If so, how?

• Can't you consider kappa as a function of local temperatue and position, depending in which state is the system at that point of space? – Victor Sep 20 '16 at 21:19
• @Victor Having $\kappa(T)$ is definitely necessary for modeling phase transitions, but I feel there has to be more too it. In particular, a model of temperature diffusion for multiple phases must also include some way to capture the energy of the phase change. – eepperly16 Sep 20 '16 at 21:26
• Of course. Then I guess you have to express T in the equation as T(E) and rewrite the whole equation in terms of energy and energy flow, with $\Delta T=C_v \Delta E$. But it also quite a turbulent scenario, there is no exact solution. I found a paper that addreses this problem called:FAST IMPLICIT FINITE-DIFFERENCE METHOD FOR THE ANALYSIS OF PHASE CHANGE PROBLEMS that analises this issue – Victor Sep 22 '16 at 15:19

If we look at the transition from liquid to solid you can think of the problem as a Stefan Problem or moving boundary problem. Assuming a constant diffusion coefficient in the different phases and working with some dimensionless temperature we get the 2 heat equations in the bulk regions $$D_i \nabla^{2} T = \partial_{t} T$$ Where $i = s,l$ for solid or liquid. In order to describe heat conservation due to the phase transition at the boundary you get the interface boundary condition $$v_n = [D_i \partial_n T ]^{s}_{l}$$ Where $v_n$ is the normal velocity of the interface and $\partial_n$ is the normal derivative. The brackets means the difference of the gradients evaluated on the both sides of the interface. Finally to include surface tension and kinetic cooling the boundary condition for the temperature becomes $$T|_{\Gamma} = - d_0 \kappa - \beta v_n$$ Where $d_0$ is the capillary length, $\kappa$ is the curvature and $\beta$ is some kinetic coefficient. For references see for example James Langers Instabilities and pattern formation in crystal growth. For a discussion on how to numerically solve the equations using phase fields you could take a look at the relevant chapters in Phase-Field Methods in Materials Science and Engineering by Elder and Provatas.