Supercooling and superheating effects in heterogeneous systems

In the model for phase transitions in binary systems based on Fourier’s and Fick’s laws, the interface equilibrium condition $\theta = w$, relating the temperature $\theta$ and the chemical activity $w$, is here replaced by a relaxation dynamics for the liquid concentration $\chi$: \[ \frac {{\partial \chi }}{{\partial t}} + {\tilde H^{ - 1}}\left ( \chi \right ) = \beta \left ( {\theta ,w} \right )\] (${\tilde H^{ - 1}}$: inverse of the Heaviside graph); here $\beta \in {C^0}\left ( {{R^2}} \right )$ and $sign\beta \left ( {\theta ,w} \right ) = sign\left ( {\theta - w} \right )$. In the case of a single dimension of space, with an interface $x = s\left ( t \right )$, a different dynamics can be considered: \[ s’\left ( t \right ) = \beta \left ( {\theta \left ( {s\left ( t \right ),t} \right ),w\left ( {s\left ( t \right ),t} \right )} \right )\].