| Popis: |
We study certain approximate solutions of a system of equations formulated in an earlier paper (Physica D 43, 44-62 (1990)) which in dimensionless form are $$u_t + gamma w(phi)_t = abla^2u,,$$ $$alpha epsilon^2phi_t = epsilon^2abla^2phi + F(phi,u),,$$ where $u$ is (dimensionless) temperature, $phi$ is an order parameter, $w(phi)$ is the temperature--independent part of the energy density, and $F$ involves the $phi$--derivative of the free-energy density. The constants $alpha$ and $gamma$ are of order 1 or smaller, whereas $epsilon$ could be as small as $10^{-8}$. Assuming that a solution has two single--phase regions separated by a moving phase boundary $Gamma(t)$, we obtain the differential equations and boundary conditions satisfied by the `outer' solution valid in the sense of formal asymptotics away from $Gamma$ and the `inner' solution valid close to $Gamma$. Both first and second order transitions are treated. In the former case, the `outer' solution obeys a free boundary problem for the heat equations with a Stefan--like condition expressing conservation of energy at the interface and another condition relating the velocity of the interface to its curvature, the surface tension and the local temperature. There are $O(epsilon)$ effects not present in the standard phase--field model, e.g. a correction to the Stefan condition due to stretching of the interface. For second--order transitions, the main new effect is a term proportional to the temperature gradient in the equation for the interfacial velocity. This effect is related to the dependence of surface tension on temperature. |
