We use a diffuse-interface model with an order parameter 
and a
concentration 
as the only field variables. Moreover, the
interface we
consider is straight and represents a boundary between two grains of
the same
phase. In particular, we exclude at the outset any influence that
curvature of the
boundary or temperature variation may have on the motion. In reality,
these
boundaries
are usually curved; that additional effect has just not as yet been
included in our model.
Typically, the
concentration could represent the fraction of available lattice
sites
which are filled by solute atoms, and so we shall take its range to be
the
interval 
. Similarly 
, the two extremes 
representing the two perfect grains on either side.
As is usual with diffuse-interface models (e.g. [2]), the
postulated evolution system can be understood
as a gradient flow with respect to a functional. In this case, the
functional is
appropriately named the free energy 
, since temperature will
be
constant. It has contributions coming individually from and
, plus a small contribution resulting from the interaction between
the two
fields:
The first term will be of standard van der Waals-Cahn-Hilliard form
Here 
is the domain in space occupied by the solid material.
Rather,
assuming everything is independent of one space coordinate, we take
it to be
the strip 
, representing a 2D
cross
section of the metal film. The grain boundary will be a subregion of
consisting of a slim rectangle 
bounded on top and bottom by the
two
faces 
. The law of motion of this rectangle is our
primary
object of interest. The parameter 
is some measure of the
relative
weight attached to gradients of in the expression for the free
energy
(we shall see that 
is also a measure of the
thickness of the interface ), and is approximately the free
energy
density if were constant in space. We take it to be the
following
double-well potential with barrier at 
(see, e.g.
[1]):
For the values 
is left undefined.
We take
where 
is a convex function representing the free energy density of
the
solid due to the presence of the solute. A typical form for is
Other effects of the type we are seeking can be obtained by including
terms
involving 
in (4); we shall omit them for simplicity.
Our object will be to discover interaction terms 
which can
serve as
impetus for the motion of the interface and which have physical
interpretations. They will be of the assumed form
The small dimensionless normalizing parameter 
is
chosen so that the derivatives of 
are of order unity. Setting
,
we assume the smallness relations
Our chosen kinetic equations take the form
(the error term depends on and will not be given explicitly),
where
represents the diffusivity of the solute as a function of the
degree (measured by ) to which the material is ordered, and 
is a
relaxation time.
One interpretation and/or derivation of the system (8) involves
its
characterization as a gradient (steepest descent)
flow for the functional with respect to a certain
-dependent
scalar product, at least in the case that 
and is given by
(5).
We are mostly interested, however, in the case when 
vanishes in much of the domain of interest; the gradient nature of the
flow
then degenerates.
It is well known, in fact, that the diffusivity is
much larger in the grain boundary than in a perfect crystal.
We shall make the simplifying assumption that
in the crystals, that is when 
.
Our specific choice for is
where 
is a constant.
By the choice (3) of , the system (8) now becomes
To account for the possibility that , we extend
(10) as
a variational inequality (see [1] for an analogous
situation).
The
result is that in places where , (10) need not hold;
all that is necessary there is for 
and its gradient
to be continuous.
As boundary conditions on 
, physical considerations suggest that
where 
denotes differentiation in the
direction normal to 
, and (neglecting 
)
where 
is a given value depending on the properties of the
external
vapor.
Conditions at 
are also needed. We require
in regions to the left and right in our strip domain where
the pure
crystals reside, i.e. where 
. These regions are not known at
the outset.