Electronic Journal of Differential Equations,
Vol. 1995(1995), No. 16, pp 1--49.
Title: Interfacial Dynamics for Thermodynamically Consistent
Phase-Field Models with Nonconserved Order Parameter
Authors: Paul C. Fife (Univ. of Utah, Salt Lake City, UT, USA)
Oliver Penrose (Heriot-Watt Univ., Edinburgh, EH14, UK)
Abstract:
We study certain approximate solutions of a system of equations formulated
in an earlier paper (Physica D {\bf 43} 44--62 (1990)) which in
dimensionless form are
$$u_t + \gamma w(\phi)_t = \nabla^2u\,,$$
$$\alpha \epsilon^2\phi_t = \epsilon^2\nabla^2\phi + F(\phi,u)\,,$$
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.
We also consider some cases in which the temperature $u$ is very small,
and possibly $\gamma$ or $\alpha$ as well; these lead to further
free boundary problems, which have already been noted for the standard
phase--field model, but which are now given a different interpretation and
derivation.
Finally, we consider two cases going beyond the formulation in the
above equations. In one, the thermal conductivity is enhanced (to order
$O(\epsilon^{-1})$) within the interface, leading to an extra term in
the Stefan condition proportional (in two dimensions)
to the second derivative of curvature with respect to arc length. In
the other, the order parameter has $m$ components, leading naturally
to anisotropies in the interface conditions.
Submitted September 26, 1995. Published November 27, 1995.
Math Subject Classification: 35K55, 80A22, 35C20.
Key Words: phase transitions; phase field equations; order parameter
free boundary problems; interior layers.