Electron. J. Diff. Eqns.,
Vol. 1995(1995), No. 16, pp 1-49.
### Interfacial dynamics for thermodynamically consistent
phase-field models with nonconserved order parameter

Paul C. Fife & Oliver Penrose

**Abstract:**

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

,

,

where
is (dimensionless) temperature,
is an order
parameter,
is the temperature-independent part of the energy
density, and
involves the
-derivative
of the free-energy density. The constants
and
are of
order 1 or smaller, whereas
could be as small as 10^{-8}.
Assuming that a solution has two single-phase regions separated
by a moving phase boundary
,
we obtain the differential equations and boundary conditions satisfied by
the `outer'solution valid in the sense of formal asymptotics away from
and the
`inner' solution valid close to
.
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()
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
is very small, and possibly
or
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
)
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.

Show me the
PDF file (392 KB),
TEX file, and other files for this article.

Paul C. Fife

Mathematics Department,
University of Utah,

Salt Lake City, UT 84112, USA

e-mail address: fife@math.utah.edu
Oliver Penrose

Mathematics Department,
Heriot-Watt University,

Riccarton, Edinburgh, EH14 4AS, UK

e-mail address: oliver@cara.ma.hw.ac.uk

Return to the EJDE web page