\documentstyle[twoside,bezier]{article}
\def\qbezier{\bezier{200}}
\pagestyle{myheadings}
\markboth{\hfil Interfacial Dynamics \hfil EJDE--1995/16}%
{EJDE--1995/16\hfil P. C. Fife \& O. Penrose\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1995}(1995), No.\ 16, pp. 1--49. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu (147.26.103.110)\newline
telnet (login: ejde), ftp, and gopher access:
ejde.math.swt.edu or ejde.math.unt.edu}
\vspace{\bigskipamount} \\
Interfacial Dynamics for Thermodynamically Consistent Phase--Field Models
with Nonconserved Order Parameter
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35K55, 80A22,
35C20. \newline\indent
{\em Key words and phrases:} phase transitions, phase field equations,
order parameter,\newline\indent
free boundary problems, interior layers.
\newline\indent
\copyright 1995 Southwest Texas State University and University of
North Texas.\newline\indent
Submitted September 26, 1995. Published November 27, 1995.} }
\date{}
\author{Paul C. Fife \\ Oliver Penrose}
\maketitle
\begin{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
$$
\begin{array}{rcl}
u_t + \gamma w(\phi)_t & = & \nabla^2u,\\
\\
\alpha \epsilon^2\phi_t & = & \epsilon^2\nabla^2\phi + F(\phi,u),\\
\end{array}
$$
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.
\end{abstract}
\newcommand\be{\begin{equation}}
\newcommand\ee{\end{equation}}
\newcommand\bea{\begin{eqnarray}}
\newcommand\eea{\end{eqnarray}}
\def\e{{\epsilon}}
\def\P{{\Phi}}
\def\p{{\phi}}
\def\a{{\alpha}}
\def\A{{\nabla}}
\def\b{{\beta}}
\def\g{{\gamma}}
\def\G{{\Gamma}}
\def\k{{\kappa}}
\def\s{{\sigma}}
\def\S{{\Sigma}}
\def\d{{\delta}}
\def\D{{\Delta}}
\def\i{{\psi}}
\def\z{{\xi}}
\def\o{{\omega}}
\def\O{{\Omega}}
\def\sU{{\tilde{U}}}
\def\sP{{\tilde{\Phi}}}
\def\sm{{\tilde{m}}}
\def\sD{{\tilde{D}}}
\def\inf{{\infty}}
\def\bbr{{{\Bbb{R}}}}
\def\cN{{\cal{N}}}
\def\cD{{\cal{D}}}
\def\cS{{\cal{S}}}
\def\hu{{\hat{u}}}
\def\hD{{\hat{D}}}
\def\hE{{\hat{\ell}}}
\def\ba{{\bar{\alpha}}}
\def\bx{{\bar{x}}}
\def\bw{{\bar{w}}}
\def\bt{{\bar{t}}}
\def\be{{\bar{e}}}
\def\bp{{\bar{p}}}
\def\bq{{\bar{q}}}
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\def\bv{{\bar{v}}}
\def\bd{{\bar{d}}}
\def\bg{{\bar{\gamma}}}
\def\bk{{\bar{\kappa}}}
\def\bs{{\bar{\sigma}}}
\def\bE{{\bar{\ell}}}
\def\na{{\nabla}}
\def\ra{{\rightarrow}}
\def\da{{\downarrow}}
\def\ua{{\uparrow}}
\section{Introduction}
\label{1}
In [PF1], the authors gave a thermodynamically consistent formalism for
developing models of phase--field type for phase transitions in which the
only
two field variables are temperature and an order parameter. The present
paper
develops in some detail the laws governing the motion of phase interfaces
which
are implied by these models and their generalizations, in the case of both
first and second order phase transitions. (The latter are defined here to be
those transitions in which the internal energy is the same in the two phases
at constant temperature.) These
laws are obtained by a formal reduction of the models in [PF1] to free
boundary
problems. Such a reduction is obtained by the use of systematic formal
asymptotics based on the smallness of a parameter $\e$, a dimensionless
surface tension. (This identification of $\e$ is shown in (24) and Sec. 12,
although its definition comes, via the coefficient $\k_1$ in (3), from the
gradient term in a postulated entropy functional introduced in [PF1].) This
was
the procedure first followed in [CF] for the traditional phase field
equations. We consider only models in which the
density is constant and the order parameter is not a conserved quantity.
Within these restrictions, our treatment here is in many respects
more complete and general than that given in [CF], [C1], [WS]
and in other papers. For example, in allowing the thermal diffusivity $D$ to
depend on the order parameter, we may include the case when this diffusivity
is enhanced within the interfacial region; the interface condition
expressing energy balance then includes an extra term involving
(in two dimensions) the second
derivative of the curvature with respect to arc length along the interface
and
representing lateral diffusion within that region.
We also explore other implications of the dependence of both $D$ and the heat
capacity $c$ on the order parameter, and generalize the procedure
to the case when there are several order parameters. This latter case is
frequently encountered in modeling phase transitions, and leads naturally to
anisotropies in the interface conditions.
It leads to some interesting mathematical problems involved with finding a
heteroclinic orbit for a special kind of Hamiltonian system.
The contrasting nature of phase interfaces for first and second order
transitions is brought out. In the latter case, we derive a forced
motion-by-curvature problem.
The conditions leading to a free boundary problem of Mullins--Sekerka
type are elucidated and contrasted with those usually postulated
within the framework of the traditional phase field model. In particular,
the
time evolution from Stefan--type motion into Mullins--Sekerka motion is
discussed (Sec. \ref{13}).
The relation between the interface thickness, the surface
tension, and the Gibbs--Thompson law, is discussed, and our
viewpoint
corroborated by known physical data.
Finally, the asymptotic procedure here used is developed and discussed with
great
care, and certain first order terms in the interface conditions
are derived here for the first time.
The phase field models developed in [PF1] were based on certain postulated
forms for the internal energy, free energy, and entropy of the system. These
are made precise in Sec. 3 of this paper. Other assumptions, of a
mathematical nature, are made in the paper, particularly in Sec. 5. These
latter are
assumptions about the nature of the layered solutions we are investigating,
and are made in order to carry out a matched asymptotic expansion.
Assumptions of this type are in fact nearly always made in formal
asymptotic treatments of applied problems, but are rarely made explicit. We
strive to spell them out completely.
There has been good progress in rigorous justification of the type of formal
asymptotics used here, which means proving the existence of solutions for
which our assumptions hold. See [CC], [St], [St2] for such a
justification in
the case of the traditional phase field model. (Such progress has been even
more
impressive in the case of the Allen-Cahn and Cahn-Hilliard models.)
Other thermodynamically consistent models have been developed in recent
years;
see [T], [UR], [AP1], [AP2], [WS], and the references given there (note also
[K], described from a thermodynamically consistent point of view in [WS]).
In
many cases
they are more complicated than ours, due to the inclusion of effects such as
variable density.
\section{The main ideas and results}
\label{1.5}
As in [PF1] we start from a Helmholtz free energy function of the form
$$
f(\p , T) = \bar{w}( \p ) - T \bar{s}_0 ( \p ) - cT \log{T},
$$
where $\p$ is the order parameter, $T$ the absolute temperature,
$\bar{w}(\p)$
and $\bar{s}_0$ are the temperature-independent parts of the energy density
and entropy density, and $c$ is the heat capacity at constant $\p$, which for
the time being we take to be constant. The internal energy is then
$$
\bar{e} ( \p , T) = \frac{ \partial (f/T)}{\partial (1/T)} = \bar{w}(\p) + cT
\eqno\mbox{(1)}
$$
and the kinetic equations, (3.8) and (3.6) of [PF1], can be written
in the form of the following equations, the main object of study in this
paper:
$$
cT_{\bt} + \bw(\p)_{\bt} = \na \cdot D(\p,T)\na T,
\eqno\mbox{(2)}
$$
$$
\k_0(\p, T)\p_{\bt} = \k_1 \na^2 \p - \frac{1}{T} \frac{\partial f(\p,
T)}{\partial \p}.
\eqno\mbox{(3)}
$$
Here $\bx \in R^2$ or $R^3$ and $\bt \in R$ are space and time variables,
$\na$ denotes vector differentiation with respect to $\bx$, $D$ is the heat
conduction coefficient, $\k_1$ measures the contribution to the entropy and
free energy
made by gradients in $\p$, and $\k_0$ is a relaxation time for $\p$. (The
coefficient $\k_0 $ is called $ K^{-1}_1$ in [PF1].) Note that
$\bw$ and $f$ are related to $\bar{s}_0(\p)$:
$$
- \frac{1}{T} \frac{\partial}{\partial \p} f(\p,T) = \bar{s}'_0(\p) -
\frac{1}{T}
\bw' (\p).
\eqno\mbox{(4)}
$$
where the primes indicate differentiation. In Sec. \ref{17}, the order
parameter $\p$ is generalized to have several
components, in
which case (3) becomes a vector equation, $\k_0$ becomes a matrix,
the first term on the
right of (3) becomes a more general second order operator,
and the last term becomes $-\frac{1}{T} \na_{\p}f(\p,T)$.
As in [PF1], the function $\bw$ will be postulated to be concave (in
fact,
quadratic), and at fixed $T$ the function of $\p$ on the left of (4)
has the form of a ``seat function'' of $\p$
with three zeros, at values $\p =h_-(T), h_0(T)$, and $h_+(T)$
(see Figure 1).
Fig. 1(a) illustrates the possibility (used in [WS]),
that one or more of these
functions `$h$' may be constants. Moreover, we
postulate the existence of a temperature $T_0$ (the melting
temperature if the transition is of first order) such that $f(h_-(T_0),T_0) =
f(h_+(T_0),T_0)$. More specific
assumptions on our functions are given in Section \ref{2}.
\begin{figure}
\begin{picture}(300,200)(-30,0)
\thicklines
\put(30,10){\line(1,0){70}}
\put(30,10){\line(2,3){50}}
\put(10,85){\line(1,0){70}}
\thinlines
\put(5,45){\vector(1,0){90}}
\put(55,0){\vector(0,1){95}}
\put(20,90){$h_+(T)$}
\put(60,0){$h_-(T)$}
\put(70,57){$h_0(T)$}
\put(90,35){$T$}
\put(59,73){$\phi$}
\put(0,0){$(a)$}
\thicklines
\qbezier(175,195)(185,100)(220,150)
\qbezier(220,150)(255,200)(265,105)
\thinlines
\qbezier(175,185)(185,90)(220,140)
\qbezier(220,140)(255,190)(265,95)
\put(175,150){\vector(1,0){100}}
\put(220,105){\vector(0,1){100}}
\put(225,190){$-f_\phi$}
\put(270,140){$\phi$}
\put(185,157){-1}
\put(255,157){1}
\thicklines
\qbezier(22,150)(35,100)(70,150)
\qbezier(70,150)(105,200)(118,150)
\thinlines
\qbezier(22,150)(35,90)(82,150)
\qbezier(82,150)(105,180)(118,150)
\put(10,150){\vector(1,0){130}}
\put(70,100){\vector(0,1){100}}
\put(75,190){$-f_\phi$}
\put(128,140){$\phi$}
\put(116,140){1}
\put(22,160){-1}
\thicklines
\qbezier(150,100)(270,80)(230,50)
\qbezier(230,50)(190,20)(280,0)
\thinlines
\put(230,5){\vector(0,1){95}}
\put(170,50){\vector(1,0){100}}
\put(235,90){$\phi$}
\put(260,40){$T$}
\put(170,0){$(b)$}
\put(160,80){$h_+(T)$}
\put(260,11){$h_-(T)$}
\end{picture}
\caption{ Two possible functions
$-f_{\p}(\p,T)$ plotted for $T = T_0$ (thick)\newline and $T > T_0$ (thin),
and their null sets (with the functions $h_i(T)$) : \newline
(a) $-f_{\p}(\p,T) = (\p^2 - 1)(\p_c(T-T_0) - \p)$\newline
(b) $-f_{\p}(\p,T) =\p(1 - \p^2) - T + T_0$}
\end{figure}
As indicated in [PF1] and (especially) in [PF3], the traditional phase--field
model of Langer [L] and Caginalp, in which $\bw$ is linear, can be put into
this general framework, but corresponds to cases where $s_0(\p)$ is a
nonconcave function.
The special case of (2), (3) studied in [PF3] for purposes of illustration is
revisited here in Section \ref{3}. In that section, we relate our parameters
to
various physical constants in order to gauge their orders of magnitude.
In this same vein, we relate the interface thickness $\e$ to the surface
tension
$\s$. (The parameter $\e$ is defined below in Section \ref{2}, in terms of
$\k_0$
and
$\k_1$.)
If the parameters $\k_0$ and $\k_1$ in (3) are small, in the sense to be
explained below in Section \ref{2}, then solutions can be constructed (in the
manner of
formal asymptotics) which depict spatial configurations of two distinct
phases. More precisely, at any instant of time, space is divided into
regions
$\cD_+$ and $\cD_-$, with a thin mobile layer separating them. In many
typical
cases, the order parameter $\p$ is approximately a constant, say $\p_{\pm}$,
in
$\cD_{\pm}$. In the thin layer between $\cD_+$ and
$\cD_-$,
$\p$ makes a transition from near $\p_-$ to near $\p_+$.
Our purpose is to study these layered solutions in detail. Our focus is on all
solutions of this type, rather than on solutions satisfying specific boundary
or initial conditions.
We are mainly concerned with first-order phase transitions.
A matched asymptotic analysis for this case is given in Sections \ref{4} --
\ref{11}. Only the two-dimensional case is considered, but the method
is easily extended to three dimensions. Our analysis relies on the
smallness of $\e$, a parameter (actually a dimensionless surface tension)
related to $\kappa_1$ which will be given
later. We assume
that the dimensionless width of the layers is $O(\e))$ and that their
internal
structure scales with $\e$ in a way to be
defined more carefully in Sec. \ref{4}.
Under these assumptions, the
analysis allows one to deduce further information of a detailed nature about
the layered
solutions.
For example, it provides approximate information about how the interphase
regions move. It is this property of engendering further information which
lends the assumption its credibility.
The result of the analysis is that the layered solutions can be formally
approximated
at the macroscopic level by the solution of a free
boundary
problem, the interphase layer being approximated by a sharp interface.
The free boundary problem,
set out in Section \ref{10}, consists of
heat equations in each of the two single-phase domains, coupled
through their common domain boundary (the interface) by means of two specific
relations. One of them is a Stefan--like condition, and the other is a
condition relating the temperature
there to the velocity, curvature, and surface tension . These
approximations, valid away from the layer, are supplemented by fine structure
approximations, valid in the vicinity of the layer, which give
information about the
phase and temperature profiles within the interphase region.
The analysis reveals some new effects: $(a)$ to order $\e$, the
temperature may be discontinuous at the interface; $(b)$ the effect of
interface stretching is accounted for by an extra term in the Stefan
condition; and $(c)$ there is in general a small extra normal derivative
term as well as a curvature term in
the other interface condition.
In Section \ref{12}, we consider the analogous question of phase boundary
motion
in
the case of second order phase transitions. The previous development is
easily
adapted to this case, but the results are strikingly different. The model
considered by Allen and Cahn [AC] is a particular case.
The basic free boundary problem obtained in Section \ref{10} has many
particular
limiting cases when certain order of magnitude assumptions are made on the
parameters of the problem; a few of these possibilities are explored in
Sections
\ref{13} and \ref{14}. As opposed to previous derivations of similar
limiting
cases,
we show that the various alternative free boundary problems
obtained in [CF] and [C2] as formal approximations for small $\e$ when
certain
parameters are taken to depend on $\e$, appear here as corollaries of our
basic
results. The same is true for the classic motion--by--curvature problem.
Thus
one general analysis does it all. In the case of curvature--driven free
boundary problems of various
kinds, we elucidate in Section \ref{14} the physical conditions under which
they
are valid
approximations. These conditions are distinctly different from those which
have been suggested in the past, and are motivated by thermodynamic
considerations.
In Section \ref{15}, the implications of allowing the coefficients to depend
on
$\p$
and $T$ are explored. Section \ref{16} is devoted to the interesting case,
not
considered before, when the thermal diffusivity is
enhanced within the interphase zone. Again, the analysis in
Sections \ref{4} -- \ref{11}
can be adapted. The most significant new feature is the appearance, in the
Stefan interface condition, of an extra term representing diffusion within
the
zone. This term involves the second tangential
derivative (or, in three dimensions, the surface
Laplacian) of the curvature of the interface. An analogous
result has been derived by Cahn, Elliott, and Novick--Cohen
[CEN],
in the case of Cahn--Hilliard type equations. They show that enhanced
mobility
within the interfacial zone results in a limiting free boundary problem in
which the motion is driven by the Laplacian of the curvature. See [CT] for a
materials scientific theory of a class of surface motions depending on the
Laplacian of curvature.
In Section \ref{17} the generalization, important in some applications, is
made
to
multi-compo\-nent order parameters. As we shall see, this provides a
possible
basis for treating the motion of anisotropic interfaces. Free boundary
problems of the same general type as before are obtained, but the
coefficients of the curvature and velocity in the free boundary conditions
are more complicated.
A systematic matched asymptotic analysis of moving layer problems of this
sort
was first carried out in [CF]; see also [C2] and [F2]. Similar problems were
treated using related techniques in [P] and in [RSK]. To an extent, our
conclusions are analogous to those in [CF] and [C2], but there are
many important
differences, as was mentioned above.
\section{The basic model and hypotheses for first order phase transitions;
nondimensionalization.}
\label{2}
The models considered here consist of field equations (2), (3) for a
temperature function $T(\bar{x},\bar{t})$ and an order parameter function
$\p(\bar{x},\bar{t})$. In the main part of the paper we shall assume that
$c,
\k_0$, and $D$ are positive constants, and that $\p$ is a scalar function.
Some
assumptions about the function $f$ will also be needed.
These depend somewhat on whether the
phase transition is of first or second order. We begin with the case
of a first order transition (sections \ref{4}--{\ref{11}),
for which the assumptions are set out below.
Second order transitions are treated in Sec. \ref{12}.
\medskip
\paragraph{A1.} $f(\p,T)$ is twice continuously differentiable in both variables.
Considered as a function of $\p$ at fixed $T$ for any $T$ in some interval
$T_-
< T < T_+, f(\p,T)$ has two local minima $\p = h_-(T)$ and
$\p = h_+(T)$, which we order so that
$$
h_-(T) < h_+(T).
$$
It also has a single intermediate local maximum at $\p = h_0(T)
\in
(h_-(T), h_+(T)$).
Thus $-f_{\p}$ typically has one of the forms shown in Figure 1. The two
numbers $h_-(T)$ and $h_+(T)$ are the values of $\p$ for which uniform
phases can exist at
temperature $T$. In general these two minima correspond to different values
of
the free energy. If that is the case, one of the phases is stable and the
other is metastable, so that they cannot coexist at equilibrium. Only if
they
correspond to the same free energy density,
$$
f(h_-(T), T) = f(h_+(T),T),
\eqno\mbox{(5)}
$$
can the two phases coexist at equilibrium.
\medskip
Our second assumption (which holds only for first order transitions)
will be that phase equilibrium is possible only at a single
temperature $T_0$ (the melting temperature, in the case of solid--liquid
transitions):
\paragraph{A2.} Equation (5) is satisfied if and only if $T = T_0 \in (T_-,T_+)$.
\medskip
Our third assumption strengthens the local minimum condition on
$f(\p,T)$ at $\p = h_{\pm}(T)$ in A1, to
\paragraph{A3.}
$$\frac{\partial^2f}{\partial \p^2} (h_{\pm}(T),T) > 0.
\eqno\mbox{(6)}
$$
\medskip
Our fourth assumption concerns the latent heat. To write it simply,
we denote $\p_{\pm} = h_{\pm}(T_0)$; $\p_c = h_0(T_0)$.
Since we are considering the case of a first order phase
transition, the energies of the two phases are different :
$w(\p_+) \neq w(\p_-)$. The more ordered phase (with the
order parameter $\p$ near $\p_+$) will have the lower energy, and so the
latent heat is
$$
\bE = \bw(\p_-) - \bw(\p_+),
\eqno\mbox{(7)}
$$
satisfying
\paragraph{A4.}
$$
\bE > 0. \eqno\mbox{(8)}
$$
\medskip
In view of (1) and (7), A4 is equivalent to the condition
$$
\frac{\partial}{\partial T} \int^{\p_+}_{\p_-} \frac{\partial f(\p,
T)/\partial \p}{T} d \p
> 0~~ \mbox{at}~~ T = T_0.
$$
Using A2, we see that this is
equivalent to
$$
\frac{d}{dT} \left[f(h_-(T),T) - f(h_+(T),T) \right] |_{T = T_0} > 0,
\eqno\mbox{(9)}
$$
so that if (5) holds for $T = T_0$, it cannot hold for $T \neq
T_0$, and hence A4 implies the ``only if'' part of A2.
It will be convenient to recast our equations (2), (3) in dimensionless form.
Recall that $\bx$ and $\bt$ are physical variables. We
define
dimensionless space and time by
$$
x = \bx/L;~~~t = \bt D/cL^2.
\eqno\mbox{(10)}
$$
where $L$ is a characteristic macrolength for our system. For
example, we may choose it to be he diameter of the spatial domain of
definition of
our
functions $\p$ and $T$ or the minimum radius of curvature of the initial
interface, defined to be the curve $\{\p
= \p_c\}$.
Each term of Equation (2) has the dimensions of energy density per unit time,
and the terms in (3) have dimensions of energy density per unit temperature.
We divide (2) by $\frac{DT_0}{L^2}$ and (3) by $c$, to make each term
dimensionless.
To simplify the notation further, we use a new temperature variable $u =
\frac{T}{T_0} - 1$, where $T_0$ is given in A2 above, and define
%eq 12
$$
\begin{array}{c}
w(\p) \equiv \frac{\bw(\p)}{\g cT_0},~~~\ell \equiv \frac{\bE}{\g cT_0},\\
\\
F(\p,u) = - \frac{1}{\g cT(u)} \frac{\partial}{\partial \p} f(\p,T(u)),\\
\end{array}
\eqno\mbox{(11)}
$$
where $\g$ is a dimensionless parameter chosen so that
%eq 13
$$
\frac{\partial F}{\partial \phi} (\p_c,0) = 1.
\eqno{(12)}
$$
(We are assuming that $\p_+ - \p_-$ is of
order 1.) This is our method of normalizing the seat function $F$. But we
have also
incorporated $\g$ into the definitions of the dimensionless $w$ and $\ell$
above;
this is natural since $f$, $\bw$ and $\bE$ are related by (1), (4), and (7)
and constitute an important point of departure from previous phase-field
models (see Sec. \ref{14}). The use of $\gamma$ allows us to obtain
approximate but simpler forms of the laws of interfacial motion when
$\gamma$ is small (Sec. 14).
Clearly, (7) continues to hold with the overbars removed. Since $\bw$ and
$w$
are only defined up to an arbitrary additive constant, we are free to choose
that constant so that
$$
w_{\pm} \equiv w(\p_{\pm}) = \mp \frac{\ell}{2} .
\eqno\mbox{(13)}
$$
With these representations, (2) and (3) become
%eq 15
$$
u_t + \g w(\p)_t = \na^2 u,
\eqno\mbox{(14)}
$$
%eq 16
$$
\a \e^2 \p_t = \e^2 \na^2 \p + F(\p,u),
\eqno\mbox{(15)}
$$
where $\nabla$ now denotes differentiation with respect to $x$
and we have set $\e^2 = \k_1/L^2\g c$ and $\a = \k_0 D/\k_1 c$. We
expect $\a$ to be $O(1)$ but, as we shall see in Sec. \ref{3} , $\e^2$ is
typically very small. Equations (14) and (15) form the basis of
the remainder of the paper.
Our
assumptions A1 -- A4 can now be reexpressed in terms of the
new notation:
\paragraph{Equivalent of A1:} For each small enough $u$, the function $F(\p,u)$
is
bistable in $\p$; that is, it has the form of a seat function of $\p$, as
exemplified by the graphs in Fig. 1. (Again, we denote the outer zeros of
$F$ by $h_{\pm}(u)$.)
\paragraph{Equivalent of A2:}
%eq 17
$$
\int_{h_-(u)}^{h_+(u)} F(\p,u)d \p = 0~~~\mbox{if and only if}~~u = 0.
\eqno\mbox{(16)}
$$
\paragraph{Equivalent of A3:}
%eq 18
$$
\frac{\partial F}{\partial \p} (h_{\pm}(u),u) < 0.
\eqno\mbox{(17)}
$$
It follows from (1), (5), (7), and (11) that
%eq 19
$$
\ell = - \int_{\p_-}^{\p_+} F_u(\p,0)d \p.
\eqno\mbox{(18)}
$$
We therefore have:
\paragraph{Equivalent of A4:}
$$
\frac{d}{du} \int_{h_-(u)}^{h_+(u)} F(\p,u)d \p < 0~~\mbox{when}~~u = 0.
\eqno\mbox{(19)}
$$ Again, note the relation between this and the ``only if'' part of (16).
As a first consequence of these assumptions, we note the fact,
which is guaranteed (see [F1] and its
references, for instance) by (16) and (17), that the
boundary value problem
$$
\i^{''} + F(\i,0) = 0,~~~z \in (- \inf, \inf);~~~\i(\pm \inf) =
\p_{\pm},~~~\i(0) = \p_c.
\eqno\mbox{(20)}
$$
has a unique solution $\i(z)$. Changing the
integration
variable in (19) from $\p$ to $z$ by the relation $\p = \i(z)$, we see that
(19), and hence A4, is in turn equivalent to
$$
\int_{-\inf}^{\inf} F_u(\i(z),0) \i'(z)dz < 0.
\eqno\mbox{(21)}
$$
\section{Example; numerical values for the parameters.}
\label{3}
A simple free energy function modeling liquid--solid phase transitions was
considered in the appendix of [PF3]. In dimensional form, it is
%eq 22
$$
f = f_0 \left[\frac{T}{4T_0} (\p^2 - 1)^2 + \left(\frac{T}{T_0} - 1 \right) a
(\p + 1)^2 \right] + cT \mbox{ log } \frac{T}{T_0},
\eqno\mbox{(22)}
$$
$$
\bw (\p) = - f_0 a (\p + 1)^2,
$$
where $f_0$ is a parameter with dimensions of energy density, and
$a$ is dimensionless.
To relate some of the constants in (22) to measurable quantities
we note first that, by (7), the latent heat is
$$
\bE = \bw(-1) - \bw(1) = 4af_0.
$$
Another measurable quantity giving information
about the parameters of the model is the surface tension $\bs$. It is equal
to
the excess free energy
per unit area in a plane interface, which for $T = T_0$ is ([CA])
%eq 23
$$
\bs = \int_{- \infty}^{\infty} \left[ f(\i(\bar{r}/\e), T_0) + \frac{1}{2}
\k_1T_0
\left( \frac{d \i}{d \bar{r}} \right)^2 \right] d \bar{r}.
\eqno\mbox{(23)}
$$
According to (22), the function $F(\p,0)$ is
$- \frac{f_0}{\g cT_0} \p(\p^2 - 1)$, so
that
the definition (12) of $\g$ gives $F_{\p}(0,0) = \frac{f_0}{\g cT_0} = 1$
and
$$
f_0 = \g cT_0.
$$
Since now $F(\p,0) = - \p(\p^2 - 1)$, we have from (20)
$$
\i(z) = \tanh \frac{z}{\sqrt{2}}.
$$
Using the relations $z = r/\e = \bar{r}/\e L$, $\p(\bar{r}) = \i(z) = \tanh
(z/
\sqrt{2})$, and $f(\p,T_0) = \frac{1}{4} f_0(\p^2 - 1)^2 =
\frac{1}{4} f_0 \mbox{sech}^4(z/\sqrt{2})$, we can simplify (23) to
%eq 24
$$
\begin{array}{rc}
\bs & = \e L \int_{-\infty}^{\infty} \left[\frac{1}{4} f_0 \mbox{sech}^4
\left(\frac{z}{\sqrt{2}} \right) + \frac{\k_1T_0}{4 \e^2L^2} \mbox{sech}^4
\left(\frac{z}{\sqrt{2}}\right) \right] dz\\
\\
& = \e L \g c T_0 \frac{2 \sqrt{2}}{3}~~\mbox{since}~~f_0 = \g
cT_0~~\mbox{and}~~\e^2 = \frac{\k_1}{\g cL^2}.\\
\end{array}
\eqno\mbox{(24)}
$$
Therefore we can think of the product $\e \g$ as a measure for the
magnitude of the surface tension.
The surface tension at a solid--liquid interface
can be deduced from the value of the Gibbs--Thompson coefficient
$$
G = \frac{\bs T_0}{\bE} = \frac{\bs T_0}{4af_0}.
$$
From (24) and this we obtain for the width of the
interface
$$
\e L = \frac{3 \bs}{2 \sqrt{2} \g cT_0} = \frac{3 \sqrt{2} a G}{T_0}.
$$
The value of $a$ can be estimated as follows: first, if the entropy is to be
a
concave function of $\p$, then, as shown in [PF3], we must have $a >
\frac{1}{2}$; secondly, if the liquid can be supercooled to a temperature
$T_-$
then $f$ must have a local minimum at $\p = -1$ when $T = T_-$, which with
(22) implies $T_-/T_0 > a/(1 + a)$ i.e. $a < T_-/(T_0 - T_-)$. For example
in the case of the ice-water transition we might take $T_0=273$K, $T_- =
233$K,
giving $a < 5.8$. We shall take the value $a = 1$ to be typical.
Typical values of $G$ and $T_0$ are $10^{-5}$ cm-deg and $300$K,
respectively.
Using them, we obtain
$$
\e L \approx 1.5 \times 10^{-7} ~~\mbox{cm},
$$
which is of the order of a few lattice spacings of an ice
crystal,
a not unreasonable interface thickness. If $L$ equals, say, $10$ cm., then
$\e$
is less than $10^{-7}$ and the approximations to be developed in this paper
should be very accurate.
\section{The approximation scheme.}
\label{4}
Our procedure is based on assumptions which have been used implicitly in
various earlier studies of similar problems ([CF], [RSK], [P], etc.) and have
been rigorously justified, under certain conditions, in the analogous cases
of the
Cahn--Allen equations [MSc], [Chen1] and the Cahn--Hilliard equation [ABC].
We spell them out completely. Their plausibility rests in large part on the
fact that they lead to a succession of reasonable formal approximations. For
simplicity we consider only the two
dimensional case.
The core assumption is that there exist
families of solutions ($u(x,t;\e)$,
$\p(x,t;\e)$) of (14), (15), defined for all small $\e > 0$, all $x$ in a
domain $\cD \subset R^2$, and all $t$ in an interval $[0,t_1]$, with
``internal
layers.'' This concept is defined precisely in the form of assumptions $(a)
- (e)$ below, as follows.
For such a family, we assume that, for all small $\e \ge 0$,
the domain $\cD$ can at each time $t$ be divided into two
open regions $\cD_+(t;\e)$ and
$\cD_-(t;\e)$, with a curve $\G(t;\e)$ separating them. This curve
does not intersect $\partial\cD$. It is smooth, and depends
smoothly on $t$ and $\e$. In particular, its curvature and its velocity are
bounded independently of $\e$. These regions are related to the family of
solutions as follows.
$(a)$ Let $\Omega$ be any open set of points $(x,t)~ \mbox{in}~\cD \times
[0,t_1]$ such
that \newline
dist$(x,\G(t;0))$ is bounded away from 0. Then
for
some $\e_0 > 0,~u$ can, we assume, be extended to be a smooth (say, three
times
differentiable) function of the three variables $x,~t,~\mbox{and}~\e$
uniformly for $0\le \e < \e_0,~(x,t)~\mbox{in}~\Omega$. The same is assumed
true of $\p(x,t;\e)$.
It follows in particular that the functions $u_k(x,t) \equiv
\frac{1}{k!}\partial_{\e}^k u |_{\e=0} ,~k=0,1,2,3$, are defined in
all of
$\cD \setminus \G (t;0)$. A similar statement holds for
$\partial_{\e}^k\p |_{\e=0} $.
It also follows from (15) that for $(x,t)$ in any region $\Omega$ as
described above, $F(\p,u)=O(\e^2)$. This implies, by the definition of
$h_{\pm}$, that $\p$ is close either to
$h_+(u)$ or $h_-(u)$. (The third possibility would be $\p$ near $h_0(u)$;
but in view of the instability of this constant solution of (15) (for fixed
$u$), we assume there are no extended regions where $\p$ is close to this
value.)
$(b)$ For $\Omega$ in $\cD_{\pm}(t;0) \times [0,t_1]$, we assume that
$\p$ is close to $h_{\pm}(u)$. Our
interpretation is that the material where $\p$ is close to $h_-(u)$ is in
``phase I'' (the less ordered phase, since $\p_- < \p_+$) and that where $\p$
is near $h_+$ is in phase II, the more ordered phase.
Much of our analysis will refer to a local orthogonal spatial coordinate
system $(r,s)$ depending parametrically on $t$ and $\e$, defined in a
neighborhood of
$\G(t;\e)$, which we define precisely as the set where $\phi = \phi_c$. We
define $r(x,t;\e)$ to be the signed distance
from $x$ to
$\G(t;\e)$, positive on the $\cD_+$ side of $\G(t;\e)$. Then for small
enough $\d$, in a neighborhood
$$
\cN (t;\e) = \left\{x: r(x,t;\e) < \d \right\},
$$
we can define an orthogonal curvilinear coordinate system $(r,s)$
in
$\cN$, where $s(x,t;\e)$ is defined so that when $x \in \G(t;\e)$,
$s(x,t;\e)$ is the
arc length along
$\G(t;\e)$ to $x$ from some point $x_1(t;\e) \in \G(t;\e)$
(which always moves normal to $\G$ as $t$ varies).
Transforming
$u$
and
$\p$ to such a coordinate system, we obtain the functions
$$
\hat{u}(r,s,t;\e) = u(x,t;\e), ~~\hat{\p}(r,s,t;\e) = \p(x,t;\e).
$$
Let $\hat{u}_k(r,s,t),~~\hat{\p}_k(r,s,t)$ be defined in the same
way as $u_k$ and $\p_k$ above, in terms of derivatives at $\e = 0$. They
exist, by virtue of $(a)$ above.
$(c)$ For each $t \in [0,t_1],$ we assume that the $\epsilon$-derivatives
$u_k,~k \le 3$, restricted to the
open domain $\cD \cap \{r>0\}$, can be extended to be smooth functions on the
closure $\bar{\cD} \cap \{r\ge 0\}$ (on $\G$, they no longer signify the
derivatives
indicated above). Similarly, we assume that the restrictions to
$\cD \cap \{r<0\}$ can be extended to be smooth functions on $\bar{\cD} \cap
\{r\le 0\}$ and that the analogous
statements are true of the $\epsilon$-derivatives of $\p$.
$(d)$ Let $z = r/\e$, and let $U(z,s,t;\e) = \hat{u}(r,s,t;\e)$ in the
neighborhood of $\G$ introduced above.
Then for any positive $\e_0\mbox{ and } z_0$, we assume that
$U(z,s,t;\e)$ can be extended to be a smooth
function of the variables $(z,s,t;\e)$, uniformly for $0\le\e<\e_0,~ |z|
< z_0,~ 0 \le t \le t_1, \mbox{ all}~s$. The analogous statements for
$\hat{\p}$ in place of $\hat{u}$ are also assumed to hold.
It follows that the functions $U_k(z,s,t) =
\frac{1}{k!} \partial_{\e}^kU(z,s,t;\e)|_{\e=0}$ are well defined.
We now have that for any $r_0>0,~ z_0>0,$ the Taylor series approximations
%
$$
u(x,t;\e) = u_0(x,t) + \e u_1(x,t) + o(\e),
\eqno\mbox{(25a)}
$$
$$
\hat{u}(r,s,t;\e) = \hat{u}_0(x,t) + \e \hat{u}_1(x,t) + o(\e),
\eqno\mbox{(25b)}
$$
$$
U(z,s,t;\e) = U_0(z,s,t) + \e U_1(z,s,t) + o(\e),
\eqno\mbox{(25c)}
$$
together with their differentiated versions, are valid
for all sufficiently small $\epsilon$ : in the case
of (25a,b), uniformly for dist$(x,\G(t;0)) > r_0 > 0$ and in the case of
(25c), for
%\newline
dist$(x,\G(t;\e)) < \e z_0$. Similar statements hold for
$\p, ~\hat{\p},~\P$. Truncated series as in (25a) and (25b) will
constitute our `outer' approximation; ones like
those in (25c) will constitute the `inner' approximation.
$(e)$ The approximations in (25b,c) above are assumed to hold simultaneously
in a suitable region: for some $0 < \nu < 1$, we assume that (25b) holds for
$$
\mbox{dist}(x,\G(t;\e) > \e^{\nu},
$$
and (25c) holds for
$$
\mbox{dist}(x,\G(t;\e) < 2\e^{\nu}.
$$
Differential equations for the functions $u_k$, $\p_k$, etc. can be obtained
by substituting (25a) and its analog into (14) and (15) and equating
coefficients of powers
of $\e$. For this purpose, the only conclusion from (15) which will be
needed is
the relation
%eq 26
$$
F(\p,u) = 0(\e^2),
\eqno\mbox{(26)}
$$
which provides algebraic equations relating $u_k$ and $\p_k,~~k\le
1.$ For example,
%eq 27
$$
\p_0 = h_{\pm}(u_0)~~\mbox{in}~~ \cD_{\pm}.
\eqno\mbox{(27)}
$$
In the same way, we get from (14) that
%eq 28
$$
\begin{array}{l}
\partial_te_0 = \na^2u_0~~\mbox{in}~~ \cD_{\pm},\\
\\
e_0 = u_0 + \g w(\p_0).\\
\end{array}
\eqno\mbox{(28)}
$$
Our object will be to find free boundary problems satisfied by the outer
functions $u_k$, $\phi_k$. For this, we need not only
differential equations and algebraic
relations such as (27) and (28) holding in $\cD_{\pm}$,
but also extra conditions
on $\G$. These extra conditions will be obtained by finding the
inner functions
$U_k,~\P_k$ and using assumption (e) to obtain matching conditions
relating them to the outer
functions $u, \phi$. And to find these inner functions,
we shall in turn need to relate the surface $\G(t;\e)$ to the
family $\p$ precisely and to specify a curvilinear coordinate system
near $\G$. This will be done in the next section.
\section{The $r,s$ and $z,s$ coordinate systems; matching relations.}
\label{5}
Recall that our definition of $\Gamma$ will be the level surface
%eq 31
$$
\G(t;\e) = \left\{x: \p (x,t;\e) = \p_c \right\},
\eqno{(29)}
$$ and the $(r,s)$ coordinate system is attached to $\Gamma$.
To go from Cartesian to $(r,s)$ coordinates we transform derivatives as
follows:
%eq 35
$$
\begin{array}{l}
\partial_t~~\mbox{is replaced by}~~\partial_t + r_t \partial_r + s_t
\partial_s;\\
\\
\na^2~~\mbox{is replaced by}~~\partial_{rr} + |\na s|^2 \partial_{ss} +
\na^2r
\partial_r + \na^2 s \partial_s.\\
\end{array}
\eqno\mbox{(30)}
$$
Here, we have used the fact that $|\na r| \equiv 1$.
The derivatives of $r$ and $s$ in these expressions can be written
in terms of kinematic and geometric
properties of the interface $\G$. The details of the calculation
are given in the Appendix; we quote only the results here.
Let $v(s,t;\e)$ denote the normal velocity
of $\G$ in the direction of $\cD_+$ at the point $s$, and let $\k$
denote its curvature, defined by $\k(s,t;\e) = \nabla^2 r(x,t;\e)|_{\G}$.
Then the time derivatives of $r$ and $s$ can be written
%eq 36a
$$
r_t(x,t;\e) = - v(s,t;\e),~~s_t = - \frac{rv_s}{1 + r\k},
\eqno\mbox{(31a)}
$$
where, here and below, the arguments of $v$ and $\k$ are
$(s(x,t;\e),t;\e)$, and subscripts on $v$ and $\k$ denote differentiation.
The corresponding expression for the space derivatives of $r$ and $s$ are
%eq 36b
$$
\na^2r(x,t;\e) = \frac{\k}{1 + r \k},
\eqno\mbox{(31b)}
$$
%eq 36c
$$
\na^2s(x,t;\e) = \frac{r \k_s}{(1 + r \k)^3},~~~|\na s|^2 = \frac {1}{(1 + r
\k)^2}.
\eqno\mbox{(31c)}
$$
To obtain the equations for the inner approximation
we first define, in accordance with (1) and (11), the nondimensional
internal energy
%eq 37
$$
e = u + \g w(\p).
\eqno\mbox{(32)}
$$
As in (25b), we shall denote by $\hat{e}$ the
same quantity $e$ expressed as a function of $r,s,t, \epsilon$. In view of
(30)
and (31), our basic equations (14) and (15) then become
%eq 38
$$
\begin{array}{c}
\partial_t\hat{e} - v \partial_r \hat{e} - \frac{rv_s}{1 + r \k} \partial_s
\hat{e} =\\
\\
\hat{u}_{rr} + \frac{\k}{1 + r \k} \hat{u}_r + \frac{1}{(1 + r \k)^2}
\hat{u}_{ss} + \frac{r
\k_s}{(1 - r \k)^3} \hat{u}_s,\\
\\
\a \e^2 (\hat{\p}_t - v \hat{\p}_r - \frac{rv_s}{1 + r \k} \hat{\p}_s) =\\
\end{array}
\eqno\mbox{(33)}
$$
%eq 39
$$
\e^2 \left(\hat{\p}_{rr} + \frac{\k}{1 + \k r} \hat{\p}_r + \frac{1}{(1 + r
\k)^2}
\hat{\p}_{ss}
+ \frac{r \k_s}{(1 + r \k)^3} \hat{\p}_s \right) + F(\hat{\p},\hat{u}).
\eqno\mbox{(34)}
$$
To obtain the inner expansion we follow the procedure set out in
Sec. \ref{4}(d), defining in $\cN (t;\e)$ the stretched normal coordinate
%eq 40
$$
z = r/\e
\eqno\mbox{(35)}
$$
\setcounter{equation}{35}
\noindent and the functions
$$
U(z,s,t;\e) = \hat{u}(r,s,t;\e), ~ \P(z,s,t;\e) = \hat{\p}(r,s,t;\e), ~
E(z,s,t;\e) = \hat{e}(r,s,t;\e).
$$
To obtain differential equations for $U$ and $\Phi$,
we substitute (35) into (33), (34), obtaining :
\begin{eqnarray}
U_{zz} + \e \k U_z + v\e E_z - \e^2 \k^2 U_z + \e^2 U_{ss} - \e^2E_t =
O(\e^3),
\label{37}
\\
\P_{zz} + F(\P,U) + \e \k\P_z + \a\e v\P_z = O(\e^2),
\label{38}
\\
E = U + \g w(\P).
\label{39}
\end{eqnarray}
The functions $U, \Phi, E$ can be expanded in powers
of $\epsilon$ as in (25c) :
\begin{equation}
\left.
\begin{array}{ll}
U(z,s,t;\e) = U_0(z,s,t) + \e U_1(z,s,t) + o(\e)~~~(\e \ra 0), \\
\P = \P_0 + \e \P + o(\e)\\
\end{array}
\right\}
\label{36}
\end{equation}
and so on. By the regularity assumptions in Sec. 4, $v(s,t;\e)$
and $\k(s,t;\e)$ can also be expanded in series like (\ref{36}).
The differential equations satisfied by the functions $U_0$, $\P_0$, etc. are
obtained by substituting (\ref{36}) into (\ref{37}) and (\ref{38}).
This will be done in the next section, but first we formulate the matching
conditions (e.g. [F2]) obtained by requiring that the inner and outer
expansions represent
the same function in their common domain of validity (which exists by
Assumption $(e)$ of the previous section). They are the following, where we
have omitted the carets from the symbols $u$ and $\p$.
$$
\lim_{r\ra 0\pm }u_0(r,s,t) = \lim_{z \ra \pm\infty}U_0(z,s,t);
\eqno\mbox{(40)}
$$
$$
\lim_{r \ra 0\pm} \partial_ru_0(r,s,t) = \lim_{z \ra \pm\infty}
\partial_zU_1(z,s,t).
\eqno\mbox{(41)}
$$
If $U_1(z,s,t) = A_{\pm}(s,t) + B_{\pm}(s,t)z + o(1)$ as $z \ra
\pm
\infty$, then
%eq 34
$$
A_{\pm}(s,t) = u_1(0\pm ,s,t);~~B_{\pm} =
\partial_ru_0(0\pm ,s,t),
\eqno\mbox{(42)}
$$
and so on. Similar relations apply, connecting $\p_0,~\p_1$ to
$\P_0,~\P_1$, etc. Finally if $\partial_z U_2 = A_{\pm}^{\ast}(s,t) +
B_{\pm}^{\ast}(s,t)z + o(1)$, then
$$
A_{\pm}^{\ast}(s,t) = \partial_r u_1(0\pm,s,t).
\eqno\mbox{(43)}
$$
\section{The zero-order inner approximation.}
\label{6}
We substitute (35) and (\ref{36}) into (\ref{37}) and (\ref{38}) to
obtain a series expansion in powers of $\e$ for each side of the latter. By
equating the coefficients
of each power of $\e$ we obtain a sequence of
equations for the various terms $U_i$ and $\P_i$ . The first couple of them
are
analyzed as follows :
\noindent { \boldmath $O$}{\bf{(1) in (\ref{37}):}}
$$
U_{0zz} = 0.
$$
We want $U_0$ to be bounded as $z \ra \pm \infty$, because of
(40);
so $U_0$ is independent of $z$ :
$$
U_0 = U_0(s,t).
$$
{ \boldmath $O$}{\bf{(1) in (\ref{38}):}}
%eq 42
$$
\P_{0zz} + F(\P_0,U_0) = 0.
\eqno\mbox{(44)}
$$
By (29), we have $\P_0(0,s,t) = \p_c$. By the equation for $\Phi$
analogous to (40), we seek a solution
$\P_0$ which approaches distinct
finite limits as $z \ra \pm \infty$, and it is clear from (44) that these
limits must
be roots $\P$ of $F(\P,U_0) = 0$. Moreover, it can be seen by multiplying
(44) by $\P_{0z}$ and integrating from $- \infty$ to $+ \infty$ that
the integral in (16) with $u$ replaced by $U_0$ must vanish.
By A2 (16), this implies
%eq 43
$$
U_0 \equiv 0.
\eqno\mbox{(45)}
$$
Therefore $\P_0$ must satisfy the
differential equation in (20), and by the definition
of $r$ it satisfies the other conditions in (20) as well,
so that it must actually be the function defined in (20):
$$
\P_0(z,s,t) \equiv \i(z).
\eqno\mbox{(46)}
$$
satisfying the condition
$$
\i(\pm \infty) = h_{\pm} (0) \equiv \p_{\pm}.
\eqno\mbox{(47)}
$$
(Notice that $\Phi_0$ does not depend on $s$ or $t$.)
The matching condition (40) now gives, by (45), (46) and (47),
the following boundary condition on the
lowest-order outer solution :
%eq 47
$$
u_0 \left|_{r = \pm 0} = 0;~~\p_0 \right|_{r = \pm 0} = \p_{\pm}.
\eqno\mbox{(48)}
$$
At this point, we are in a position to define and evaluate, to lowest order,
interfacial free energy and entropy densities at $T = T_0$.
The total free energy in the
system is ([PF1, eqs. 3.9 and 3.12])
$$
\bar{\cal F} [\p ,T(u)] = \int_{\bar{\O}} (f(\p, T(u) ) + \frac{1}{2} \k_1
T(u)
|\na
\p|^2) d \bar{x},
$$
$T(u) = (u + 1)T_0$.
Its dimensionless form follows from our previous
nondimensionalization procedure:
$$
{\cal F} [\p ,u] = \bar{\cal F} [\p ,T ]/\g cT_0L^2 = \int_{\O} \left(
\frac{f(\p, T )}{\g cT_0} + \frac{T(u)}{2T_0} \e^2 |\na \p|^2 \right) dx.
$$
The (dimensionless) surface tension $\sigma$ is the interfacial free energy
per
unit length of interface. It can be calculated, at $T = T_0$ ($u=0$)
by subtracting the free energy density of a uniform phase,
which is $f(\phi_{\pm},T_0)$, from the function $f$ in the integrand and then
integrating with respect to $z = r/\e$ from $-\inf$ to $\inf$.
To lowest order in $\e$ we may use the approximation
$\p = \i (r/\e ), ~T = T_0$ in this integral, obtaining
$$
\s = \e \int_{-\inf}^{\inf}
\left( \hat{f}(\i (z) ) + \frac{1}{2} (\i '(z))^2
\right) dz,
\eqno\mbox{(49)}$$
where we have set
$$
\hat{f} (\i ) = \frac{f( \i ,T_0) - f(\phi_{\pm},T_0)}
{\g cT_0} ,
\eqno\mbox{(50)}
$$
the dimensionless bulk free energy density.
But since from (44), (45), and (46)
$$
\i '' = -F(\i,0) = \frac{d}{d \i} \hat{f}(\i),
$$
it follows that
$(\i ')^2 = 2\hat{f} (\i)$, so that the
contributions of the two terms in the integral for $\s$ are equal. We
therefore
have
$$
\s = \e \int_{-\inf}^{\inf}(\i '(z))^2dz = \e
\int_{-1}^{1}\sqrt{2[\hat{f}(\p) -\hat{f}(\p_{\pm})]}d\p
\equiv \e\s_1.
\eqno\mbox{(51)}
$$
This is a standard formula (see e.g. [AC]). We shall
call $\s_1$ the scaled dimensionless surface tension.
\section{The jump condition at the interface.}
\label{7}
In this section, we show how the energy balance equation (33),
applied to the inner approximation, leads to a jump condition
for the outer solution at the interface, from which the
velocity of the interface can be determined once the outer
solution is known. We first
calculate this to lowest order, and then to order $\e$.
In view of (\ref{36}), (45), and
(46), we may set
$$
U = \e \sU,~~\P = \i + \e \sP,
$$
where $\sU,~\sP = O(1)$. We then have by (\ref{39})
$$
E = \e \sU + \g w (\i + \e \sP).
$$
Since $\i$ does not depend on $t$, it follows
that $E_t = O(\e)$, and hence from (\ref{37}) that
%eq 48
$$
\sU_{zz} + vE_z + \e \k \sU_z = O(\e^2).
\eqno\mbox{(52)}
$$
Integrating (52), we get
%eq 49
$$
\sU_z + v E + \e \k \sU + C_1(s,t,\e) = O(\e^2)
\eqno\mbox{(53)}
$$
for some integration constant $C_1 = C_{11} + \e C_{12}$.
Since by (\ref{36}a) $\sU = U_1 + \e U_2 + O(\e^2)$, the lowest order
approximation
in
(53) yields
%eq 50
$$
U_{1z} = - (vE)_0 - C_{11} = - \g v_0w(\i(z)) - C_{11},
\eqno\mbox{(54)}
$$
where in the second equation we have used the fact that the
expansions of $U,~v,$ and $\P$
induce
an expansion of $vE$ in powers of $\e$, with $(vE)_0 = \g v_0w(\i)$
being the lowest order term. Similarly induced expansions will be used
below.
To obtain the lowest order jump condition for the outer approximation, we
apply (40) and (41) to the left and right sides of (54). We thus obtain
%eq 51
$$
\partial_ru_0 \left|_{r = 0\pm} = - v_0e_0 \right|_{r = 0\pm} - C_{11} = \pm
\g
v_0 \ell/2 - C_{11},
\eqno\mbox{(55)}
$$
using (48) and (13). By subtraction we get the jump relation
%eq 52
$$
\left[ \partial_ru_0 \right] = - v_0 \left[e_0 \right] = \g v_0 \ell,
\eqno\mbox{(56)}
$$
where the square brackets indicate the limit
from the right
$(r
= 0+)$ minus the limit from the left $(r = 0-)$.
We now derive the jump condition to order $\e$ analogous to (56). Consider
the terms of order $\e$ in (53). Since $\sU_1 = U_2$,
these terms are
%eq 53
$$
U_{2z} = - (vE)_1 - \k_0 U_1 - C_{12}.
\eqno\mbox{(57)}
$$
To evaluate $U_1$, we integrate (54):
$$
U_1 = -\g v_0 p(z) - C_{11}z + C_2,
\eqno\mbox{(58)}
$$
where
$$
p(z) = \int_{z_0}^z w(\i(s))ds,
\eqno\mbox{(59)}
$$
$z_0$ will be chosen later, and $C_2$ is another integration
constant, unknown at this stage, but depending on $z_0$. Hence from (57), we
have
$$
U_{2z} = -(vE)_1 + \g \k_0 v_0 p(z) - C_{12} + \k_0C_{11}z - \k_0C_2.
$$
Applying the matching relations (43) and (40), we obtain:
$$
\partial_ru_1|_{r=0\pm} = -(ve)_1|_{r=0\pm} + \g\k_0v_0P_{\pm} - C_{12} -
\k_0C_2,
\eqno\mbox{(60)}
$$
where the $P$'s are defined by the relation
$$
p(z) = w(\phi_{\pm})z + P_{\pm} + o(1) ~~ (z \ra \pm\infty),
\eqno\mbox{(61)}
$$
i.e.
$$
P_{\pm} = \int_{z_0}^{\pm\infty}\left(w(\i (z)) - w(\phi_{\pm})\right)dz.
$$
For the sake of symmetry in notation, we choose the lower limit
$z_0$ so that
$P_+ = -P_- \equiv P$.
Thus we obtain (60) with $P_{\pm} = \pm P$.
Subtracting, we obtain
$$
\left[ \partial_ru_1 \right] = - \left[(ve)_1 \right] +
2\g \k_0v_0P.
\eqno\mbox{(62)}
$$
Combining (56) with (62), we get, to order $\e$,
%eq 55
$$
\left[ \partial_ru \right] = - \left[ve \right] + 2\e \g \k vP + O(\e^2).
\eqno\mbox{(63)}
$$
This relation can be used to determine $v$ to order $\epsilon$
once the outer solution is known to this order.
Physically, eqn (63) expresses the conservation of energy at the interface.
The left
side represents the net flux of energy into $\G$ per unit length; the first
term on the right represents the portion of that energy which is taken up
with
phase change. The second term on the right of (63), which is a higher
order term not usually displayed, represents the effect of $\G$ stretching or
contracting as it evolves; its presence is necessary to ensure conservation
of
energy to this order.
\section{The zero-order outer approximation.}
\label{8}
The zero-order outer approximation to our layered family
of solutions consists of a curve $\G_0(t)$ dividing
$\cD$ into two subregions $\cD_+(t)$ and $\cD_-(t)$,
and functions
$u_0,~\p_0$, continuous in each of $\cD_{\pm}$.
These can now be determined:
we obtain $u_0$ and $\G_0$ by solving a Stefan problem $\cS_0$,
defined below, and then we obtain $\phi_0$ from (27) by
taking $\p_0 = h_{\pm}(u_0)$.
$(a)$ In $\cD_{\pm},$ $u_0$ is to satisfy (27), (28)
$$
\partial_t e_0^{\pm} = \na^2 u_0,
\eqno\mbox{(64)}
$$
where
%eq 57
$$
e_0^{\pm} = u_0 + \g w(h_{\pm}(u_0)).
\eqno\mbox{(65)}
$$
Note that in the case where $ h_{\pm}(u)$ are independent of $u$,
(64) is
the usual linear heat equation for $u_0$. This was the case for the liquid
phase in the density functional model in [PF1], and for both phases in [WS].
$(b)$ On $\G_0(t)$ we have the interface condition (48)
%eq 58
$$
u_0 = 0
\eqno\mbox{(66)}
$$
and the Stefan condition (56)
%eq 59
$$
\left[\partial_r u_0 \right] = \g v_0 \ell.
\eqno\mbox{(67)}
$$
$(c)$ Our focus has been on the properties of layered solutions in general,
without reference to initial or boundary conditions. But we are now led to a
free boundary problem for which it is natural to specify these extra
conditions. In fact, there may be boundary conditions
for the temperature $u$, and hence for $u_0$,
at $\partial \cD$ (exclusive of $\G_0$),
and initial conditions $u_0(x,0)$, $\G_0(0)$.
If $u_0(x,0)$ is nonpositive in $\cD_+$ (the solid) and nonnegative in
$\cD_-$ (liquid), $\cS_0$ is the classical Stefan problem, and for smooth
initial conditions has a unique classical solution for a small time interval.
Our basic assumptions about the families $(u,\p)$ in Sec. \ref{4} imply that
in
fact this is true for all $t \in [0,t_1]$, the interval mentioned in Sec. 5.
If $u_0(x,0)$ has signs opposite from those, however, then it is generally
believed that $\cS_0$ is an ill-posed problem, in which case our assumptions
in Sec. \ref{4} will hold only in very special circumstances, such as when
the
domain and all data have radial symmetry. This ill-posed problem would
correspond to a model for crystal growth into a supersaturated liquid with no
account taken of curvature or surface tension effects. We shall see in
Sec. \ref{13} that if $u_0(x,0)$ is very small in magnitude, then the
assumptions in Sec. \ref{4} become reasonable again; in fact the lowest-order
free boundary problem
then contains regularizing curvature terms.
\section{The first-order interface condition.}
\label{9}
To obtain a more accurate outer solution we must calculate $u_1$, and for
this we need expressions for $u_1$ on the interface $\G$.
First, we apply the matching condition (42) to (58), taking into
account (61) and the fact that $P_{\pm} = \pm P$, to obtain
$$
u_1 \displaystyle|_{\G \pm} = \mp \g v_0 P + C_2.
\eqno\mbox{(68)}
$$
Here the subscript $\G_{\pm}$
means the limit as $\G$ is approached from the $+$ side or the $-$ side. To
determine these limits, we must now find the constant $C_2$. It turns out
that this can be done by examining the $O(\e$) terms in (\ref{38}). By using
(46),
one can put those terms into the form
$$
L \P_1 = - F_u(\i(z),0)U_1 - \k_0 \i '(z) - \a v_0 \i '(z),
\eqno\mbox{(69)}
$$
where $L$ is the operator defined by $L\P \equiv \P_{zz} +
F_{\p}(\i(z),0)\P$.
We know that the operator $L$ has a nullfunction $\i '(z)$ which decays
exponentially as $z \ra \pm\inf$, obtained by differentiating (44) with
respect to $z$ and setting
$ \P_0~=~\i $.
We are seeking a solution of (69) which grows at most as fast as a polynomial
at $\inf$. Multiplying (69) by $\i '$ and integrating, we see that the
equation
(69) has such a solution
only if the right side is orthogonal to $\i '$:
$$
\int_{- \infty}^{\infty} F_u(\i(z),0)U_1(z,s,t)\i '(z)dz + \k_0(s,t)\s_1 +
\a
v_0(s,t)\s_1 = 0,
$$
where (recall (51))
$$
\s_1 = \int_{- \infty}^{\infty} (\i '(z))^2dz.
$$
Substituting from (58), we obtain
%eq 66
$$
\int_{-\infty}^{\infty} F_u(\i(z),0)(- \g v_0p(z) - C_{11}z + C_2)
\i '(z)dz + (\k_0 +
\a v_0)\s_1 = 0.
\eqno\mbox{(70)}
$$
>From (18) and (20), the coefficient of $C_2$ here is seen to be $- \ell$.
On
the
basis of Assumption A4, we may therefore solve (70) for $C_2$ as
$$
C_2(s,t) = \left(\a \tilde{\s} + \g \tilde{p} \right) v_0(s,t) + \tilde{\s}
\k_0(s,t) + \tilde{q}
C_{11},
\eqno\mbox{(71)}
$$
where
$$
\tilde{p} = - \frac{\int p(z) \rho (z)dz}{\ell},~~\tilde{q} = - \frac{\int z
\rho
(z)dz}{\ell},~~\tilde{\s} = \frac{\s_1}{\ell},
\eqno\mbox{(72a)}
$$
with
$$
\rho (z) = F_u(\i(z),0)\i '(z).
\eqno\mbox{(72b)}
$$
Substituting (71) into (68), we find
$$
u_1 \displaystyle|_{\G \pm} = \left(\a \tilde{\s} + \g \tilde{p} \mp \g P
\right)
v_0
+ \tilde{\s} \k_0 +
\tilde{q} C_{11}.
\eqno\mbox{(73)}
$$
Finally, the constant $C_{11}$ may be found from (55):
%eq 60
$$
C_{11}(s,t) = - \partial_ru_0 |_{r = \pm 0} \pm \frac{1}{2} \g \ell v_0(s,t).
\eqno\mbox{(74)}
$$
In the classical case when the Stefan problem (64) - (67) is well
posed, it can be used to determine the quantities on the
right of (73) from
the initial and boundary conditions for $u_0$ and $\G_0$. In (74), either
sign may
be chosen; the right side is independent of the choice. We
shall use the upper sign in $\cD_+$ and the lower one in $\cD_-$.
Set $(\partial_ru_0)_{\pm} \equiv \partial_ru_0 |_{r = \pm 0}$. Then
substituting (74) into (73), we have
%eq 67
$$
u_1 \displaystyle|_{\G \pm} + \tilde{q}(\partial_ru_0)_{\pm} = m_{\pm}v_0 +
\tilde{\s} \k_0,
\eqno\mbox{(75)}
$$
where
%eq 68
$$
m_{\pm} = \a \tilde{\s} + \g(\tilde{p} \mp P) \pm \frac{1}{2} \g \ell
\tilde{q}.
\eqno\mbox{(76)}
$$
If $m_+ \ne m_-$, it is clear from (75) that the outer temperature
distribution $u$ will undergo a discontinuity of the order $\e$ across the
interface.
\paragraph{Example:} Consider the particular case when $F_u(\p,0)$ is an even
function of $\p$,
and
$\i '(z)$ is even. Then from the above, we have $\tilde{q} = 0$, so that
(75)
becomes
$$
u_1 \displaystyle|_{\G \pm} = \left( \a \tilde{\s} + \g(\tilde{p} \mp P)
\right)
v_0 +
\tilde{\s} \k_0.
$$
If, in addition, (as in [PF1])
$$
w(\p) = A \p - B \p^2 + const,
$$
then $P$ vanishes whenever $B$ does. In this
case, then, the possibility that $u_1$ is discontinuous across $\G$ is
associated with the presence of quadratic terms in $w$. The
case
when they are absent is the one treated, in the context of the traditional
phase field model, in [CF] and [F2].
\section{The first-order outer solution.}
\label{10}
We can now formulate the procedure for determining the first order
approximation to the outer solution. This approximation can be determined by
solving the following modified Stefan problem $\cS_{\e}$, which
generalizes the problem $\cS_0$ defined in section \ref{8}:
$(a)$ In $\cD_{\pm}$, $u$ is to satisfy (14)
$$
\partial_t e = \na^2u,
\eqno\mbox{(77)}
$$
where
$$
e = u + \g w(\p),
$$
$$
\p = h_{\pm}(u)~~\mbox{in}~~\cD_{\pm}~~\mbox{by (26)}.
\eqno\mbox{(78)}
$$
$(b)$ On $\G_{\pm}$ we have, from (66), (75), and (63), the conditions
%eq 70
$$
(u + \e \tilde{q} (\partial_r u)) |_{\G \pm} = \e m_{\pm} v + \e \tilde{\s}
\k ;
\eqno\mbox{(79)}
$$
%eq 71
$$
\left[ \partial_ru \right] = - v[ e ] + \e \g \k vP,
\eqno\mbox{(80)}
$$ where the coefficients are given by (72a), (76).
$(c)$ In addition, boundary conditions, to hold on $\partial \cD$, and
initial data
are to be prescribed.
The coefficients in (79), (80) are the same as in (75) and
(62). As mentioned before, when $h_{\pm} $ are constants,
(77) is the
heat equation with constant coefficients. Even when
$h_{\pm}$ are not constants, the heat equation
is a reasonable approximation in typical cases
(see Sec. \ref{13} ).
The term in $\partial_r u$ in (79) appears to introduce a singular
perturbation into the problem, but this is not likely to be true. We consider
a model problem consisting of (77) and (79) on the half line $\{r > 0\}$ with
the right side of (79) taken to be a known constant. The potential effect of
such a singular perturbation can be ascertained from the inner equations
associated with stretching the variable $r$. In the model problem it is
readily seen to be a regular perturbation.
What we have shown so far is that under the assumptions in
Sec. \ref{4} , the exact solution family
$(u(x,t;\e),\p (x,t;\e))$
satisfies (77) - (80) except for error terms of the order $\e^2$.
Let us now suppose that $\cS_{\e}$ is a well-posed problem, and let
%\newline
$(\tilde{u}(x,t;\e), \tilde{\p}(x,t;\e), \tilde{\G} (t;\e))$ denote its
solution when the conditions in $(c)$ are the same as those of the exact
family. Thus $(\tilde{u}, \tilde{\p}, \tilde{\G})$ satisfy the same
equations and initial conditions as $(u, \p , \G)$, except that the $O(\e^2)$
terms are discarded. It is natural to expect the following assertion, which
is basic to the paper, to hold:
\paragraph{Expectation:} $|u(x,t;\e) - \tilde{u}(x,t;\e)|,~ |\p(...) -
\tilde{\p}(...)|,~|\G(t;\e) - \tilde{\G}(t;\e)| = O(\e^2)~~(\e \ra 0)$,
uniformly in $\cD \times [0,t_1]$.
\section{Discussion; surface tension.}
\label{11}
As in previous phase field models, a Gibbs--Thompson term $ \e \tilde{\s}
\k$
and
a kinetic undercooling term $\e m_{\pm} v$ appear on the right of
(79). In addition, there appears an $O(\e)$ normal derivative term on
the left, which can be important for second order transitions, as we
shall discover.
The last term in (79) may be compared with the thermodynamic formula for
the Gibbs--Thompson effect, which can be written
$$
\left(T - T_0\right) |_{\G} = \frac{\bs T_0}{\bE} \bk
$$
where $\bs$ is the surface tension, $\bE$ the latent heat and
$\bk$
the curvature in physical units. The corresponding formula in our
dimensionless
units is
%eq 73
$$
u |_{\G} = \frac{\s \k}{\ell}
\eqno\mbox{(81)}
$$
where $\s = \bs/\g c T_0 L$. From (51) and (72a),
$$
\s = \e \int_{- \infty}^{\infty} \i ' (z)^2 dz = \e \s_1 = \e \ell
\tilde{\s}.
$$
Thus (81) simplifies to $u|_{\G} = \e \tilde{\s} \k$, which indeed agrees
with
the
relevant terms in (79).
The problem (77)--(80) differs from the
modifications of the Stefan problem obtained in [CF] in three respects:
(a) The $O(\e)$ correction to the value of the temperature at the interface
involves a flux term (on the left of (79)), so that it is perturbed into
a Robin boundary condition. In the example given in
section \ref{9}, of course, $\tilde{q}$ vanishes, making this correction
zero.
(b) The value of $u$ will in general be discontinuous at the interface
because
of the first term on the right of (79) if $m_+ \ne m_-$. Again in the above
example, when
$B = 0$, the discontinuity disappears. The amount of the discontinuity will
be
$O(\e)$.
(c) the Stefan condition (80) involves a small correction term due to
the stretching of the interface. A term like this was noted in [UR];
otherwise, all these effects were absent
in
previous models of phase field type.
It will be shown in Sec. \ref{13} that there are circumstances when the
free
boundary problem (77)--(80) can be approximated, in a formal sense,
by other (generally simpler) free boundary problems. There are a number of
possibilities here; they include different types of curvature-driven
interfacial motion.
%Section \ref{12}
\section{Second-order transitions.}
\label{12}
By a second-order transition we mean one where
the internal energy is the same in the two
phases, for each fixed value of the temperature in the interval $[T_-,T_+]$.
A good example is the case when $w$ is an even function of $\p$ and $F$ is
odd
in $\p$. For second order transitions,
there is no unique transition temperature $T_0$, contrary to postulate A2.
Instead,
we define $T_0$ to be some other characteristic temperature of the problem,
for
example the average of the system's initial temperature distribution.
In the notation of (1), we have $\bar{e}(h_+(T),T) = \bar{e}(h_-(T),T)$, and
in that of (65),
$$e_0^+(u) = e_0^-(u) \mbox{ for each } u.
\eqno\mbox{(82)}
$$
In view of (1), this implies that the quantity
$\frac{d}{dT}\left[\frac{1}{T}f(h_{\pm}(T),T)\right] $ is the same for either
choice of sign. Thus $\frac{d}{d T}
\int_{h_-(T)}^{h_+(T)}\left[\frac{1}{T}f_{\p}(\p ,T)\right]d\p = 0$. In
nondimensional terms (11), we have
$$
\frac{d}{d u}\int_{h_-(u)}^{h_+(u)}F(\p ,u)d\p = 0.
\eqno\mbox{(83)}
$$
Therefore in place of Assumption A4, the inequality sign in (19)
becomes an equality for all $u$, hence the latent heat $\ell = 0$, and
similarly (21) becomes
$$
\int_{- \infty}^{\infty} F_u(\i(z),u)\i '(z)dz = 0~~\mbox{for
all}~~u~~\mbox{in the range of interest}.
\eqno\mbox{(84)}
$$
In dealing with second order transitions, our formal assumptions will simply
be A1 and (84).
Following the asymptotic development in sections 4--11, we see that the
following changes are necessary.
The conclusion (45), hence also the left part of (48), no longer hold.
In fact, the value of $u_0$ on the interface is no longer determined a
priori.
Therefore in the lowest order outer problem (64)--(67), (66)
is to be replaced by $[u_0] = 0$, and the right side of (67) is replaced by
zero. Thus $u_0$ and its derivative are continuous across $\G_0$. In view
of (82), we see that $u_0$ is determined as the solution of
the
heat equation (64) in all of $\cD$, {\it with no reference to} $\G$
(appropriate boundary and initial conditions may be
prescribed).
The interface's location is found independently, as we now describe.
We pass to (70), which by virtue of (84) and (74)
with $\ell = 0$ becomes
%eq 80
$$
m_1v_0 + \k_0 = m_2 \partial_ru_0,
\eqno\mbox{(85)}
$$
where
$$
m_1 = \a - \frac{\g}{\s_1} \int_{- \infty}^{\infty} p(z) \rho (z) dz,~~~ m_2
= -
\frac{1}{\s_1}\int_{-
\infty}^{\infty} z \rho (z)dz,
$$
the functions $p$ and $\rho$ being defined in (59) and (72b).
To find the interface $\G(t)$, then, $u_0$ is first determined by (64), and
then $\G$ is found from the ``forced'' motion--by--curvature
problem
(85), with known forcing term $m_2 \partial_ru_0$ dependent on position and
on time.
In Sec. \ref{15} , we shall examine the more realistic case when the thermal
diffusivity
$D$ is different in the two phases; then it can be checked that the problem
for $u_0$ can no longer be
decoupled from that for $\G$.
The interface condition (85) is similar to the
motion-by-curvature law given by the
Cahn--Allen theory of isothermal phase transitions ([AC], [MSc]), but there
is
now an extra term proportional to the temperature gradient (which is
continuous
across the interface). For a physical interpretation of
this term, suppose that
the surface tension (excess free energy of the interface) decreases with
temperature. Then the interface will tend to move so as to increase its
temperature. This tendency is borne out by (85) in the typical case that
$m_1$
and $m_2$ are positive.
There is an analogous forcing term in
the corresponding equation (79) for
first--order transitions, but in that context its effect is relatively small.
When the temperature deviation $u$ is small ($\d << 1$ in the context of
Sec. \ref{13} ), then the forcing term can be neglected, and the interfacial
motion
follows the classical motion--by--curvature law. This case was noted in
[C3],
and is the law of motion found for a simpler model in [CA] and [AC]. (In
[C3], it was erroneously implied that our model gives only second order
transitions; see [PF2].)
\section{The transition from Stefan to Mullins-Sekerka evolution. Other free
boundary problems.}
\label{13}
There are several special circumstances in which our basic free boundary
problem (77) - (80) can be approximated formally by simpler free boundary
problems.
For example, the nonlinear diffusion equation (77) can typically be
approximated by a linear one. In fact, the left side can be written as
$$
\hat{c}_{\pm}(u)u_t,~\mbox{where}~\hat{c}_{\pm}(u) \equiv 1 +
w'(h_{\pm}(u))h_{\pm}'(u),
$$
and the specific heat
functions $\hat{c}_{\pm}(u)$ can be approximated by
constants when the functions $h_{\pm}(u)$ are constants (as in [WS] and for
one case in [PF1]), nearly constant,
and/or when $u$ is small enough. In such cases, they may by replaced by
$\hat{c}_{\pm}(0)$. In the following, we shall assume that this
approximation is valid. It should be noted that the assumption $u \ll 1$ is
entirely reasonable in many cases.
It
simply says
$$
|T - T_0|
\ll T_0.
\eqno\mbox{(86)}
$$
In the case of water, for example, it means that the temperature
range (in Centigrade degrees) in the phenomenon under consideration is much
smaller than 273.
Anticipating that the simpler problems to be examined may involve dynamics on
a longer time scale and hence slower speed, we proceed formally by rescaling
$u$ and $t$. We set
$$
u = \d\bar{u}, ~~t' = \b t, ~~v = \b \bar{v},
$$
where the parameters $\d$ and $\b$ are $\le 1$ and may be small.
We make these substitutions in (77) - (80) with the error terms $O(\e^2)$
appended, and divide by $\d$ to obtain
$$
\b\hat{c}_{\pm}\bar{u}_{t'} = \nabla^2\bar{u} + O(\e^2/\d)
\mbox{ in }\cD_{\pm},
\eqno\mbox{(87a)}
$$
$$
(\bar{u} + \e \bar{q} (\partial_r\bar{u}))|_{\G\pm} =
\frac{\e\b}{\d}m_{\pm}\bar{v} + \frac{\e}{\d}\tilde{\s}\k + O(\e^2/\d) \mbox{
on }\G,
\eqno\mbox{(87b)}
$$
$$
\left[\partial_r\bar{u}\right] = \frac{\b}{\d}\bar{v}\g\ell +
\frac{\e\b}{\d}\g\k\bar{v} P + O(\e^2/\d) \mbox{ on }\G.
\eqno\mbox{(87c)}
$$
To show how one kind of evolution may develop into a different kind at large
times, we consider first the ``normal'' case when $\d = 1$. Then if we set
$\b = 1$ as well (so no rescaling actually occurs) and disregard terms of
orders $\e$ and higher order in (87), we get the classical Stefan problem
$$
\hat{c}_{\pm}u_t = \nabla^2 u \mbox{ in }\cD_{\pm},
\eqno\mbox{(88a)}
$$
$$
u = 0 \mbox{ on }\G,
\eqno\mbox{(88b)}
$$
$$
\left[\partial_r u\right] = v \g\ell \mbox{ on }\G.
\eqno\mbox{(88c)}
$$
Let $\cD$ represent a bounded
vessel containing the material under consideration, and suppose it is
thermally
insulated, so that the normal derivative $\partial_r u = \partial_{\nu} u =
0$ on $\partial
\cD$. Suppose that $u > 0$ in the liquid phase, $u < 0$ in the solid.
Consider layered
solutions with interface $\G(t)$ evolving according to (88). Then it
can be checked that
$$
\frac{d}{dt}\left[ \hat{c}_+\int_{\cD_+} u^2dx + \hat{c}_-\int_{\cD_-} u^2dx
\right] =
-2 \int_{\cD} |\na u|^2 dx
$$
which in view of (88b) is strictly negative as long as $u$ does
not vanish identically.
Suppose
that $\G(t)$, the solution of (88), exists for all $t$ and does not intersect
$\partial \cD$. Then it is
natural to conjecture that as $t \ra \infty$, $u(x,t) \ra 0$ uniformly in
$\cD$ and that $\G(t)$ approaches a limiting configuration $\G_{\infty}$.
Eventually, then, $u = O(\e)$ and the quantity $\e\tilde{\s}\k$ on the right
of
(79) and (87b) can no longer be neglected on $\G$ relative to $u$. (This
term indicates, in fact, that the temperature $u$ can in general never
achieve smaller orders of magnitude than $\e$.) When $u$ achieves this order
of smallness, we set $\d = \e$ in (87) and observe that if we
select $\b = \e$ as well, and drop higher order terms, we obtain a reasonable
problem of ``Mullins-Sekerka'' type [MS]:
$$
\na^2 \bu = 0 ~~\mbox{in}~~ \cD_{\pm},
\eqno\mbox{(89a)}
$$
$$
\bu = \tilde{\s} \k~~\mbox{on}~~\G,
\eqno\mbox{(89b)}
$$
$$
\left[\partial_r \bu \right] = \g \bv \ell~~\mbox{on}~~\G.
\eqno\mbox{(89c)}
$$
An existence theory for the solution $(\bar{u},\G)$ of (89) has
recently
been given by Chen [Ch2]; for the Hele-Shaw problem, which bears some
similarity, see [CP].
In short, when the temperature becomes small enough, the evolution
according
to (88) is conceptually replaced by the much slower evolution according to
(89).
The evolution (89) is well known to decrease the length of $\G (t)$ and to
preserve the area inside it, so it is expected that
under the slow process, $\G$ will typically evolve from $\G_{\infty}$ (or
something
near it) into a circle with the same area.
Other free boundary problems can be obtained by assuming $\g$ and/or $\a$ are
small, i.e. that the latent heat is small or the relaxation process for $\p$
is quick. For example, if $\e ^2 \ll \d = \g \ll \e$,
we may set $\b = 1$ to obtain the
classical motion-by-curvature law
$$
\a v = - \k~~\mbox{on}~~\G,
\eqno\mbox{(90)}
$$
coupled to a heat equation with prescribed jump condition (88c) on
$\G$ (known from solving (90)).
A number of other possibilities can occur; we leave them for the reader to
discover.
\section{Comparison with other models and limiting arguments.}
\label{14}
The free boundary problem (89) and its companion with (89a) replaced by
$$
\bar{u}_t = \A^2\bar{u},
$$
have been derived by asymptotic methods from other phase field
models, most notably in [C2] and [WS]. Here we compare both the model in
[WS] and its asymptotic development with ours; similar considerations hold
for the Langer model in [C2].
The model in [WS] has the property that our functions introduced in A1 are
constant: $h_-(T) \equiv 0,~h_+(T) \equiv 1$. Thus the order parameter in
the purely liquid or solid phase does not depend on $T$. On the other hand,
the latent heat $\bar{\ell}(T)$ may depend on $T$. To compare the model in
[WS] to ours, we must replace $\phi$ by $1 - \phi$ (distinguishing ``order''
from ``disorder'' parameters). In our notation, their evolution problem
(their
eqns. 40, 41) corresponds to our (14), (15) with
$$
F(\phi,u(T)) = \frac{1}{\g c}\left(\frac{1}{T_0} -
\frac{1}{T}\right)\bar{w}'(\phi) - 4\phi (\phi - 1)(\phi - \frac{1}{2})
\eqno\mbox{(91)}
$$
and $\bar{w}(\phi) = \bar{\ell}p(\i)|_{\i = 1 - \p},$ where $p(\i)$
is a
function with $p(0) = 0,~p(1) = 1$, whose first and second derivatives
vanish at $\phi = 0\mbox{ and }1$.
Note that this model provides no theoretical limit to the extent of
supercooling of the liquid or superheating of a solid, unlike the equations
depicted in our Fig. 1 and the example in Sec. \ref{3} . In fact, at each
value
of $T$ the free energy $F$ has local minima (in $\phi$) at $\phi = 0\mbox{
and }1$, representing stable liquid and solid phases. If the temperatures
under consideration are kept fairly near to $T_0$, this should not be an
important deficiency.
The temperature-independent part of the
correponding dimensionless bulk entropy density
given by (4), $s_0 ' =
\frac{1}{\g c}\bar{s}_0 '$, satisfies
$$
s_0'(\phi) = -4\phi(\phi - 1)(\phi - \frac{1}{2}) + \frac{\bar{w}'(\phi)}{\g
cT_0},
$$
and since $\bar{w}''(\frac{1}{2}) = 0$, we have
$s_0''(\frac{1}{2}) = 1$, showing that the entropy in this model is not a
concave function of $\phi$.
Further notational comparisons are the following: the parameters
$\tilde{\e},~m$, and $a$ in [WS] correspond to our parameters $\e/2,
\a^{-1}$, and $1/4\g c$, respectively.
Their asymptotics is based on the assumption that the first term in (91) is
$O(\e)$. We rewrite the assumption as:
$$
\frac{\bar{\ell}}{\g cT_0} \left(1 - \frac{T_0}{T}\right) p'(1-\phi) = O(\e).
$$
Let us assume that the dimensionless quantity
$\frac{\bar{\ell}}{cT_0} =
O(1)$; this is reasonable in typical scenarios. Since $p'(\phi)$ is also
$O(1)$,
this implies that
$$
\frac{1}{\g}\left|1 - \frac{T_0}{T}\right| = O(\e).
$$
This relation can be guaranteed, for example, by requiring either
$(a)~~\g = O(\e^{-1})$, or
$(b)~~\frac{T - T_0}{T_0} = O(\e)$.\
Physical meaning can be given to both of these cases. In case $(a)$, the
implication is that
$$
T\frac{\partial F}{\partial T} \approx \frac{\bar{\ell}}{\g cT_0}p(1 - \p) =
O(\g^{-1}) = O(\e),
$$
whereas
$$
\frac{\partial F}{\partial\phi} = O(1).
$$
The meaning is that the free energy depends much more (by a factor
$\e^{-1}$) sensitively on $\phi$ than
on the temperature $T$.
In case $(b)$, we have, in our notation, $u = O(\e)$, which is exactly the
assumption $(\delta = \e)$ under which we have derived (89). This says
simply that the temperature is close, as measured by $\e$, to the melting
temperature $T_0$.
\section{Variable $c$, $D$, and $\a$.}
\label{15}
All of the preceding can be extended in a straightforward way to the case
when
$c$, $D$, and $\a$ are given functions of $\p$ and $T$, and $w$ depends on
$T$ as well as on $\phi$. This allows these first three
physical parameters to differ in the different phases. It was
observed by Chen [Chen3] that if $c$ differs in the two phases, then $\ell$
must depend on $T$. It is
often the case that the temperature variation in the problem under
consideration is small enough that it does not by itself induce a significant
variation in the values of these physical constants. For simplicity, we
assume this is the case, i.e. that $c$ and $D$ depend on
$\phi$ but not on $D$.
The results under this generalization are quite analogous to those obtained
before, and so are not
surprising. We record them here for the sake of completeness.
The first observation is that the expression (4) for
a term in the right side of (3)
must be supplemented by the extra term $- c ' (\phi)\log{T}$.
By way of notation, we set
$$
D(\p) = D_1d(\p),~~c(\p) = c_1k(\p),
$$
where $D_1$ and $c_1$ are defined to be the minimal values of $D$
and $c$, respectively. We assume that the dimensionless functions $d$ and
$k$
are $O(1)$.
In the definitions of dimensionless variables given in (10) and (11),
we now replace the symbols $D$ and $c$
by $D_1$ and $c_1$. The basic equations (14) and (15) now become
%eq 81
$$
\partial_t(k(\p)u + \g w (\p)) = \na \cdot d(\p) \na u,
\eqno\mbox{(92)}
$$
%eq 82
$$
\a(\p) \e^2\p_t = \e^2 \na^2 \p + F(\p,u).
\eqno\mbox{(93)}
$$
All of the analysis in the previous sections has its analog in the present
more
general context. The resulting free boundary problem
(77)--(80) takes the following form:
\noindent In $\cD_{\pm}$,
%eq 69'-83
$$
\partial_t e_{\pm}(u) = \na \cdot d_{\pm}(u) \na u,
\eqno\mbox{(77)' = (94)}
$$
$$
\begin{array}{c}
e_{\pm}(u) = k(h_{\pm}(u))u + \g w(h_{\pm}(u)),~~d_{\pm}(u) =
d(h_{\pm}(u)),\\
\\
\p = h_{\pm}(u)~~\mbox{in}~~\cD_{\pm}.\\
\end{array}
\eqno\mbox{(78)' = (95)}
$$
On $\G_{\pm}$,
%eq 70'-84
$$
u + \e n_{\pm}(d \partial_ru) |_{\G \pm} = \e m_{\pm} v + \e \tilde{\s} \k
;
\eqno\mbox{(79)' = (96)}
$$
%eq 71'-85
$$
\left[d(u) \partial_ru \right] = - \left[ ve \right] + \e \g \k v P.
\eqno\mbox{(80)' = (97)}
$$
Here the constants $m_{\pm}$ and
$n_{\pm}$ have to be defined as follows. Let
$$
d_0(z) = d(\i(z)),~~~p(z) = \int (w(\i(z))/d_0(z))dz,~~~q(z) = \int
(1/d_0(z))dz,
\eqno\mbox{(98)}
$$
and let the constants $P$ and $Q$ be such that
$$
\begin{array}{rcll}
p(z) & = & \frac{w_{\pm}}{d(\p_{\pm})} z \pm P + o(1), & z \ra \pm \infty,\\
\\
q(z) & = & \frac{1}{d(\p_{\pm})} z \pm Q + o(1). &\\
\end{array}
$$
In the following, $\rho,~\ell$ are the same as before. Let
$$
m_1 = \int \a(\i(z),0)(\i '(z))^2dz > 0,
$$
$$
\tilde{p} = \frac{\int p(z)\rho(z)dz}{\ell},~~\tilde{q} = \frac{\int
q(z)\rho(z)dz}{\ell}.
$$
Then
$$
m_{\pm} = - m_1 + \g (\tilde{p} \mp P) \pm \frac{1}{2} \g \ell(\tilde{q} \mp
Q),
$$
$$
n_{\pm} = \tilde{q} \mp Q.
$$
For future reference, we give here the modified versions of
relations
(33) and (52):
%eq 38'-88
$$
\begin{array}{c}
\partial_t\hat{e} - v \partial_r \hat{e} - \frac{rv_s}{1 + r \k}
\partial_s\hat{e} = \\
\\
(d\hat{u}_r)_r + d \frac{\k}{1 + r \k} \hat{u}_r + \frac{1}{(1 + r \k)^2}
(d\hat{u}_s)_s + d
\frac{r \k_s}{(1 + r \k)^3} \hat{u}_s,\\
\end{array}
\eqno\mbox{(33)' = (99)}
$$
%eq 48'-89
$$
(d\sU_z)_z + vE_z + \e \k d \sU_z = O(\e^2).
\eqno\mbox{(52)' = (100)}
$$
Referring to Sec. \ref{13} , we obtain the following generalizations of the
examples in (88)--(90).
\paragraph{The ``normal'' case (88):}
$$
\hat{c}_{\pm}\partial_t u = \na \cdot d_{\pm}(u) \na u
\eqno\mbox{(88a)' = (101a)}
$$
$$
u = 0~~~\mbox{on}~~\G,
\eqno\mbox{(88b)' = (101b)}
$$
$$
\left[d \partial_r u \right] = v \g\ell~~\mbox{on}~~ \G,
\eqno\mbox{(88c)' = (101c)}
$$
where a different but obvious definition is given for
$\hat{c}_{\pm}$.
\paragraph{Motion by curvature (90)} ($\d = \g \ll \e,~~
\b = 1$).
$$
m_1v = -\s_1 \k ~~\mbox{on}~~\G.
\eqno\mbox{(90)' = (102)}
$$
\paragraph{The ``Mullins-Sekerka'' case (89)} $(\b = \delta =
\e)$.
$$
\na \cdot d_{\pm} \na \bu = 0~~\mbox{in}~~\cD_{\pm},
\eqno\mbox{(89a)' = (103a)}
$$
$$
\bu = \tilde{\s} \k~~\mbox{on}~~\G,
\eqno\mbox{(89b)' = (103b)}
$$
$$
\left[d_{\pm} \partial_r \bu \right] = \g \bar{v} \ell~~\mbox{on}~~\G.
\eqno\mbox{(89c)' = (103c)}
$$
Similar considerations hold for second order transitions. The coupling of
the generalization of (85) with that of (64) is particularly interesting.
\section{Enhanced diffusion in the interface.}
\label{16}
In some materials science connections ([CT], and references therein), it is
important to take into account increased material diffusivity in surfaces.
In
our models it is heat rather than material which is diffusing. Nevertheless,
there is clearly an analogy with material diffusion, and so it may be
interesting
to adapt our results to the case when diffusivity is much greater in the
interfacial region than it is elsewhere. Specifically, we treat the case
when this ratio is of the order $\e^{-1}$. We show, among other things, that
an effect of this is the presence, in (89c) or (103c), of an extra term on
the right taking the form of the second derivative of the curvature $\k$ with
respect to arc length along the interface. A similar result was obtained by a
different route for a Cahn-Hilliard model in [CFN].
Our basic assumption is that the function $D$ can be written as the sum of
two terms as follows. Let $D_1 = ~\mbox{Min}\left[D(\p_+), D(\p_-)
\right]$. Then
%eq 93
$$
D = D(\p;\e) = D_1 \left(\e^{-1} \hD(\p) + \sD(\p) \right) \equiv D_1d(\p
;\e),
\eqno\mbox{(104)}
$$
where $\hD$ and $\sD$ are $O(1)$ functions, and $\hD(\p)$ vanishes
when $\p = \p_{\pm}$. In fact, we assume that for some positive number $\o$,
%eq 94
$$
\hD(\p) > 0~~\mbox{for}~~\p_- + \o < \p < \p_+ - \o;~~\hD(\p) =
0~~\mbox{otherwise}.
\eqno\mbox{(105)}
$$
The function $d(\p ; \e)$ is analogous to the function $d(\p)$
used
in
Sec. \ref{15} , so in particular $D_1$ is used again in the definitions of
the
nondimensional variables.
We shall direct attention to the modifications in the previous treatment
occasioned by (104). They begin in Sec. \ref{5} . Concentrating on the
first
term
in (\ref{37}), we see that it must be replaced by $\left[(\e^{-1}\hat{D} +
\tilde{D})(U_0 + \e U_1 + ...)_z \right]_z$. Therefore the terms of orders
$O(\e^{-1})$ and $O(1)$ in the revised version of (\ref{37}) (analog of
(100))
are:
$$
\partial _z \left[(\e^{-1}\hD (\P_0(z)) + \sD(\P_0(z)) +
\hD '(\P_0(z))\P_1)U_{0z} + \hD(\P_0(z))
U_{1z}
\right] = 0.
$$
Integrating this with use of the fact that $U_{0z}(z) =
\hat{D}(\P_0(z)) = 0\mbox{ at }z = \pm\inf$, we obtain
%eq 95
$$
(\e^{-1}\hD(\P_0(z)) + \sD(\P_0(z)) + \hD '(\P_0(z)) \P_1)U_{0z} + \hD
U_{1z} =
0.
\eqno\mbox{(106)}
$$
The $O(\e^{-1})$ term in this equation tells us that
%eq 96
$$
\hD(\P_0(z))U_{0z} = 0.
\eqno\mbox{(107)}
$$
Now let $I = \left\{z: \P_0(z) \in \left[\p_- + \o, \p_+ - \o \right]
\right\}$.
It follows from (105) and (107) that
$$
U_{0z} = 0,~~~z \in I.
$$
Moreover since $\hD '(\P_0(z)) = 0$ for $z \not\in I$, we see
that
the third term in (106) vanishes. Hence
$$
\sD(\P_0(z))U_{0z} + \hD(\P_0(z))U_{1z} = 0.
$$
This tells us $(a)$ that for $z \not\in I$, $U_{0z} = 0$, so that
in fact for all $z$,
$$
U_{0z} \equiv 0;
$$
and therefore $(b)$ that
%eq 97
$$
U_{1z} = 0,~~~z \in I.
\eqno\mbox{(108)}
$$
In Sec. \ref{6} , (44)--(48) remain unchanged. In particular, $\P_0(z) =
\i(z)$
as before.
Recall (99), which shows the way that our $d$ enters into the energy balance
equation. In view of that and (104), equation (52)' = (100) must be
corrected to
$$
\partial_z \left(d\sU_z \right) + vE_z + \e \k d \sU_z - \e \k^2 \hD z\sU_z +
\e\partial_s\left(\hD \sU_s \right) = O(\e^2),
\eqno\mbox{(52)'' = (109)}
$$
and the new version of (53) is
$$
\begin{array}{c}
d \sU_z + vE + \e \k \int d \sU_z dz - \e \k^2 \int \hD z \sU_z dz + \e \int
\partial_s \left(\hD \sU_s \right) dz\\
\\
+ C_1 (s,t,\e) = O(\e^2).\\
\end{array}
\eqno\mbox{(53)' = (110)}
$$
Here $C_1 = C_{11} + \e C_{12}$. As before (54), $(vE)_0 = \g
v_0w(\i(z))$. Therefore the $O(\e^{-1})$ and $O(1)$ terms in (110) are
$$
\left(\e^{-1}\hD(\i(z)) + \hD '(\i(z) ) \P_1 + \sD(\i(z)) \right) U_{1z} +
\hat{D}(\i(z))U_{2z} =
\newline
- \g v_0 w(\i(z)) - C_{11}.
$$
But we know from (108) that $\hD U_{1z} = \hD 'U_{1z} = 0$,
so in fact our alternate version of (54) is:
$$
\hat{D}(\i(z))U_{2z} + \tilde{D}(\i(z)) U_{1z} = - \g v_0 w(\i(z)) - C_{11}.
\eqno\mbox{(54)' = (111)}
$$
Let $\chi(z)$ be the characteristic function of $I'$, i.e. $\chi(z) = 0$ for
$z \in I$, and $= 1$ otherwise. In view of (108), multiplying (111) by $\chi
(z)$ does nothing to the term in $U_{1z}$, but annihilates the term in
$\hat{D}$. We so obtain
$$
U_{1z} = -\g v_0\frac{\chi(z) w(\i(z))}{\tilde{D}(\i(z))}
- C_{11} \frac{\chi}{\tilde{D}},
\eqno\mbox{(112)}
$$
$$
U_1 = -\g v_0 \int(\chi w/\sD)dz - C_{11}\int(\chi/\sD)dz + C_2,
\eqno\mbox{(113)}
$$
$C_2$ being an integration constant and the arguments of
$w\mbox{ and }\sD$ being $\i(z)$.
>From (69), which continues to hold as written, and (113), we have the other
condition
$$
\begin{array}{rcl}
0 & = & \int_{-\inf}^{\inf} F_u(\i(z),0)\i '(z)U_1(z) + (\k_0 + \a v_0)\s_1\\
\medskip
& = & \int F_u\i '(z)\left[-\g v_0 \int\frac{\chi w}{\sD}dz - C_{11}
\int\frac{\chi}{\sD}dz + C_2\right] + (\k_0 + \a v_0)\s_1,
\end{array}
$$
hence solving for $C_2$, we get
$$
C_2 = (\a \tilde{\s} + \g \bar{p}^*)v_0 + \tilde{\s}\k_0 + \bar{q}^*C_{11},
$$
where now
$$
\begin{array}{rcl}
p^*(z) & = & \int_{z_0}^z\frac{\chi w}{\sD}dz = \frac{w_{\pm}}{\sD_{\pm}}z
\pm P^*/2 + o(1) ~~(z \ra \pm\inf),\\
q^*(z) & = & \int_{z_1}^z\frac{\chi}{\sD}dz = \frac{1}{\sD_{\pm}}z
\pm Q^*/2 + o(1) ~~(z \ra \pm\inf),\\
\bar{p}^* & = & -\frac{\int p^*\rho dz}{\ell},\\
\bar{q}^* & = & -\frac{\int q^*\rho dz}{\ell},
\end{array}
$$
and $\rho, \tilde{\s}, \ell$ are the same as before (see (72a),
(72b), (18)).
>From (113), (42), we have
\begin{eqnarray}
u_1|_{\G\pm} & = & \mp \g v_0 P^*/2 \mp C_{11}Q^*/2 + C_2
\nonumber\\
& = & \mp \g v_0 P^*/2 \mp C_{11} Q^*/2 + (\a\tilde{\s} + \g\bar{p}^*)v_0 +
\tilde{\s}\k_0 + \bar{q}^*C_{11}
\nonumber\\
& = &_0(\a\tilde{\s} + \g \bar{p}^* \mp \g P^*/2) + \tilde{\s}\kappa_0 +
(\bar{q}^* \mp Q^*/2)C_{11}.
\nonumber
\end{eqnarray}
Also from (112), (41), we have
$$
\tilde{D}\partial_r u_0|_{\G\pm} = \pm\frac{\g v_0\ell}{2} - C_{11},
$$
$$
C_{11} = \pm \frac{\g v_0 \ell}{2} - \sD \partial_r u_0 |_{\G\pm}.
$$
Hence
$$
u_1|_{\pm} + n_{\pm}(\sD \partial_r u_0 )|_{\G\pm} =
m_{\pm}^*v_0 +
\tilde{\s}\k_0,
$$
$$
m_{\pm}^* = \a\tilde{\s} + \g (\bar{p}^* \mp P^*/2) \pm
\frac{1}{2}\g\ell(\bar{q}^* \mp Q^*/2), n_{\pm} = \bar{q}^* \mp Q^*/2.
$$
This equation provides the interface condition on $\G$ analogous to (75).
Hence the analog of (79) is
$$
u + \e n_{\pm}(\tilde{D}\partial_r u)_{\pm} = \e m_{\pm}^*v + \e\tilde{\s}\k.
$$
We
now construct the analog of (80). For this purpose, we write down the
$O(\e)$
terms in (110) (recall $\tilde{D}U_{1z} = 0$):
$$
(dU_z)_2 + (vE)_1 + \k_0\int\left(\hat{D}U_{2z} + \tilde{D}U_{1z}\right)dz +
\int\partial_s \left(\hat{D}U_{1s}\right)dz + C_{12} = 0.
\eqno\mbox{(114)}
$$
We apply (43) to obtain an equation for $\left(\tilde{D}\partial_r
u\right)_1|_{\G\pm}$, which is given in terms of the asymptotic behavior of
the first term in (114) according to (43). For this, we have to write each
of the other 4 terms
in (114) in the form $A_{\pm}^*(s,t) + B_{\pm}^*(s,t)z + o(1)~~(z \ra
\pm\inf).$ Only the
terms
$A_{\pm}^*$ will be relevant (see (43)). The contribution of $(vE)_1$, by
(42), is $(ve)_1|_{\G\pm}$, and the last integral in (114) is
bounded by virtue of the compact support of $\hat{D}$. By (111), the first
integral can be expressed by means of (59) and (61) as
$$
-\g v_0\int w(\i(z))dz - C_{11}z = -\g v_0\left(w_{\pm}z \pm P/2\right) -
C_{11}z + o(1)~~~(z \ra \pm\inf).
$$
Using all of these facts, we
obtain
$$
\left(\tilde{D}\partial_r u\right)_1|_{\G\pm} + (ve)_{1\pm} \mp \g\k_0v_0P/2
+
\int_{z_0}^{\pm\inf}\partial_s\hat{D}(\i(z))\partial_s U_1 (z,s,t) dz +
C_{12} = 0.
$$
We take the difference between the upper and the lower signs and
recall that $\hat{D} (\i(z))$ does not
depend on $s$, to obtain
$$
\left[(\tilde{D} \partial_r u)_1\right] + [(ve)_1] - \g\k_0v_0P +
\int_{-\inf}^{\inf}\hat{D} (\i (z))\partial_s^2 U_1(z,s,t)dz = 0.
\eqno\mbox{(115)}
$$
>From (113) and the definitions of $p^*,~q^*$, we may write
$$
U_1 = v_0M_{1\pm}(z;\g,\ell,\a) + \tilde{\s}\k_0 +
M_3(z)(\tilde{D}\partial_ru_0)|_{\G\pm},
$$
$$
M_{1\pm}(z) = -\g(p^*(z) - \bar{p}^*) \mp \frac{1}{2}\g\ell(q^*(z) -
\bar{q}^*) + \a\tilde{\s},
$$
$$
M_{3}(z) = q^*(z) - \bar{q}^*.
$$
Bear in mind that the $M$'s do not depend on $s$, but that
$v_0,~\k_0,~\mbox{and }u_0$ do. We therefore have, from this and (115)
$$
\left[ \left(\sD \partial_r u \right)_1 \right] = - \left[ (ve)_1 \right] +
\g
\k vP - a_{\pm}v_{ss} - b \k_{ss} - c \partial_s^2 \left(\sD
\partial_r
u \right)
|_{\pm 0},
\eqno\mbox{(116)}
$$
where
$$
a_{\pm} = \int_{-\inf}^{\inf}\hat{D}(\i(z))M_{1\pm}(z)dz,~~b =
\tilde{\s}\int_{-\inf}^{\inf}\hat{D}(\i(z))dz,~~c =
\int_{-\inf}^{\inf}\tilde{D}M_3dz,
$$
and we have dropped the subscripts ``0'' on the right of (116). On
the right side of (116), either sign may be chosen.
The corrections to the next order Stefan problem due to enhanced diffusion,
therefore, are as follows. Equations (77)'=(94) and
(79)'=(96) are unchanged except for replacing $d$ by $\sD$ and a slightly
different definition of $m_{\pm}$. But
(80)'=(97) becomes
%eq 71``-106
$$
\left[\sD \partial_ru \right] = - [ve] + \e \left(\g \k vP - a_{\pm} v_{ss} -
b
\k_{ss} - c\partial_s^2 \left( \sD \partial_r u_0 \right) |_{\pm
0}\right)
\eqno\mbox{(80)'' = (117)}
$$
Thus, $O(\e)$ corrections to the Stefan condition are found, depending on the
second derivatives of the velocity and curvature.
The new form of the special limit Mullins-Sekerka problem in Sec. \ref{13}
is
perhaps
more interesting, and constitutes the main point of this section.
The interface condition (89c) is changed to
$$
\left[\sD \partial_r\bu \right] = \g \bar{v} \ell - b
\k_{ss}~~\mbox{on}~~\G.
\eqno\mbox{(89c)'' = (118)}
$$
As mentioned before, the extra term represents the diffusion of
heat
within the interface itself. Thus the Mullins-Sekerka problem (89) is now
modified by this additional term in the Stefan condition.
A comment about the qualitative behavior of this free boundary problem is in
order. It is well known that solutions of (89) or (89)'=(103)
for which the interface encloses a region $\cD_-$ whose closure is contained
in
a bounded domain $\cD$, and for which $\bu$ satisfies zero Neuman conditions
on
the outer boundary $\partial \cD$, have the curve--shortening and
area--preserving properties. Thus if $L(t)$ is the length of $\G(t)$ and
$A(t)$ is the area of $\cD_-(t)$, we have
%eq 108
$$
\frac{dL}{dt} \leq 0;~~~A(t) = const.
\eqno\mbox{(119)}
$$
It is easily shown that our present revision of this problem with
(118) replacing (103c) has the same
properties (119). In fact
$$
0 = \int_{\cD} \bu \na \cdot \sD_{\pm} \na \bu = - \int_{\cD} \sD_{\pm}
\left|\na \bu \right|^2 - \int_{\G} \bu \left[ \sD \partial_r \bu\right]
$$
$$
\leq \int_{\G}\bu \left(\g v \bE + b \k_{ss} \right) = \tilde{\s} \int_{\G}
\k
\left( \g v \bE + b \k_{ss} \right)
$$
$$
= \tilde{\s} \g \bE \int_{\G} \k v + \tilde{\s}b \int_{\G} \k \k_{ss} =
\tilde{\s} \g
\bE
\frac{dL}{dt} - \tilde{\s}b \int_{\G}(\k_s)^2
$$
$$
\leq \tilde{\s} \g \bE \frac{dL}{dt},
$$
hence
$$
\frac{dL}{dt} \leq 0.
$$
The other relation in (119) is derived in the same manner.
\section{Multicomponent order parameter.}
\label{17}
It is commonplace to describe the local microscopic or macroscopic state of a
crystalline material by means of a multicomponent order parameter (see, e.g.,
[IS] and recent papers [Lai], [BB]).
Going further, we remark that in some density--functional
theories (e.g. [Ha]), the microscopic probability distribution of atoms at any
location
is
described by a
number density function. This function provides detailed information about
the
degree to which, and sense in which, the material is ordered at that
location.
It is therefore an infinite--dimensional generalization of a scalar order
parameter. One way to extract a scalar order parameter $\p$ from such a
theory
was described in [PF1], namely artificially to restrict the allowed density
functions to a one--dimensional subspace of function space. Then the scalar
$\p$ serves to designate locations on that $1$--D subspace, which can be
pictured as a straight line. In the resulting phase--field model, the
transition from liquid to solid across an interface corresponds to the
density
function changing while restricted to that line, whereas in the unrestricted
model, it may change along some other curve in function space. In principle,
this discrepancy could be
partially remedied by restricting to a higher dimensional subspace, in which
case we would be dealing with several order parameters.
We sketch now how our analysis of interfacial motion can be extended to the
case when the order
parameter has $m$ components,
%eq 109
$$
\p = (\p_1,\cdots \p_m).
\eqno\mbox{(120)}
$$
To avoid compounding complications, we treat $D$, $c$, and $\a$ as
constants, and continue to operate in 2-dimensional physical space. We
consider only first order transitions.
As mentioned in Sec. \ref{1} , the system (2), (3) was derived in [PF1] as a
gradient system with respect to
the entropy functional displayed following (3). Thus, the right side of (3)
is
$\frac{\d S}{\d
\p}$.
$\left(\mbox{Note~that} - \frac{1}{T} \frac{\partial f(\p, T)}{\partial
\p} =
\frac{\partial \bar{s}(\p,\be)}{\partial \p}. \right)$
The gradient term $-\frac{1}{2}\k_1|\nabla\p|^2$ in that functional was
chosen as the simplest negative definite
quadratic function of
$\na \p \equiv (\partial \phi/\partial x_1,
\partial \phi/\partial x_2)$. In the case when $\p$ has several
components, the
analogous term must still be a negative definite function of $\na \p$, and we
still choose it to be a quadratic form. We denote it by $- \frac{1}{2} Q(\na
\p)$. Thus the more general entropy functional is
$$
S \left[\p,\be \right] = \int_{\O} \left\{s(\p(x),\be(x)) - \frac{1}{2} Q(\na
\p) \right\} dx.
\eqno\mbox{(121)}
$$
We have considerable latitude in choosing $Q$, including the possibility of
making it an\-iso\-trop\-ic. Even with one order parameter, anisotropy can
alternatively be
modelled [MW] by making $\k_1$ depend on the direction $\theta$ of
$\nabla\p$, but if
this dependence is not such that $\k_1(\theta)|\nabla\p|^2$ continues to be
a
quadratic form, the Laplacian in (3) is replaced by a quasi-linear second
order partial differential operator whose coefficients depend discontinuously
on $\nabla\p$ at places where $\nabla\p = 0$.
The most general form $Q$, in the case of $m$ components, is
$$
Q(\na \p) = \k_1 \S_{ijk\ell} a_{ijk\ell}\partial_i \p_k \partial_j
\p_{\ell},
$$
where $\partial_i = \partial/\partial x_i$ and the $a \mbox{'} s$
form a positive definite array and are symmetric
in
the pair $(i,j)$ (which run from 1 to 2) as well as the pair $(k,\ell)$
(which run from 1 to $m$). The normalizing parameter $\k_1$ is chosen so
that
$$
\mbox{Min}_X \S a_{ijk \ell} \z_i^{k} \z_j^{\ell} = 1,
$$
where $X = \left\{ \z_i^{k}: \sum_{k,i} |\z_i^{k}|^2 = 1
\right\}$.
Our new assumptions A1 -- A4 are analogous to those given before in section
2:
A1'. As before, $f(\p,T)$ has two and only two local minima with respect to
$\p$, which are now m-tuples $\p$ denoted as before by $h_{\pm}(T)$. We can
no
longer order them or speak of a single intermediate maximum. (The
following can be partially extended to the case when there are more than two
minima;
we
do not pursue this.)
A2'. Again, for a first--order transition, equation (5) holds if and only
if $T = T_0 \in (T_-,T_+)$.
Set $\p_{\pm} = h_{\pm}(T_0)$.
The ``inner layer'' equation generalizing (20) and (44) is the system
of equations
$$
A \frac{d^2\p}{\partial z^2} - \frac{1}{\g cT_0} \na_{\p} f(\p,T_0) = 0,
\eqno\mbox{(122)}
$$
where $A$ is a positive definite symmetric $m \times m$ matrix
obtainable from $Q$, which in general depends on the orientation of the
interface (see the definition of $A$ below), and $\na_{\p}$
means $(\partial/\partial\phi_1,\dots,\partial/\partial\phi_m)$.
A3'. For all such matrices $A$, (122) has a solution $\i(z)$ satisfying
$$
\i(\pm \infty) = \p_{\pm},
\eqno\mbox{(123)}
$$
which is unique except for translation.
Moreover, we assume the solution $\i(z)$ approaches its limits exponentially.
(There are important and interesting cases then uniqueness fails; our
analysis can be partly extended to many of those cases.)
Note $(a)$ in case $m = 1$, A3' is known to be guaranteed if
(5) and (6) hold. Also $(b)$ by taking the $L_2^m$ inner product of (122)
with the vector $\partial \i / \partial z$, it is seen that (5) is a
necessary condition for A3' to hold.
Existence theorems for boundary value problems such as (122), (123) have been
proved in [S] and [Chm]; in the
latter paper examples were given in which uniqueness does not hold.
A4'. the same as before.
We nondimensionalize as before, except that $\g$ in (11) is chosen so that
$$
\mbox{Max}_{\p} |\na_{\p}F(\p,0)| = 1,
$$
which replaces (12).
The changes in the previous analysis are as follows.
In the generalization of (15), $\a$ becomes a positive diagonal matrix,
$F(\p,u) $ is now defined to be $-
\frac{1}{\g c T} \na_{\p} f(\p,T(u))$, and $\na^2\p$ is replaced by the
partial differential operator $E \p$ defined by
$$
(E \p)_{k} = \sum_{ij\ell} a_{ijk \ell} \partial _i \partial_j \p_{\ell}.
$$
The equation analogous to (\ref{38}) is
$$
\left[ A\p_{zz} + \e B\k\p_z + \e C\p_{zs} + \e\a v \p_z + O(\e^2 ) \right] +
F(\p,u) = 0,
\eqno\mbox{(124)}
$$
where the matrices $A$, $B$, and $C$ depend on the angle
$\theta$ of
orientation of $\G$ at the point $s$ as follows. We represent the unit
normal to $\G$ pointing into $\cD_+$ by $\nu = (\cos{\theta},
\sin{\theta})$, and the unit tangent vector obtained by rotating it
through an angle $\pi/2$ in the positive direction by $\tau =
(-\sin{\theta}, cos{\theta}) $. Then
$$
(A(\theta))_{k\ell} = \sum_{ij}a_{ijk\ell}\nu_i\nu_j,
$$
$$
(B(\theta))_{k\ell} = \sum_{ij}a_{ijk\ell}\tau_i\tau_j,
$$
$$
(C(\theta))_{k\ell} = \sum_{ij}a_{ijk\ell}\left(\nu_i\tau_j +
\nu_j\tau_i\right).
$$
Notice that
$$
dA(\theta) /d\theta = C(\theta), ~~~dC(\theta) /d\theta = 2(B(\theta) -
A(\theta))
\eqno\mbox{(125)}
$$
From (124), we see that the inner profile is governed by the following
system
of differential equations to replace (44):
$$
A(\theta)\P_{0zz} + F(\P_0,0) = 0, ~~~\P_0(\pm\inf) = \p_{\pm}.
\eqno\mbox{(126)}
$$
This is (122), (123) with $A$ depending on $\theta$; we again
denote the solution by $\i(z,\theta)$.
Also, in place of (69), we obtain the following system from the $O(\e)$ terms
in (124); here we use the fact that $\partial_s = \k\partial_{\theta} $, and
let primes denote $\partial/\partial z $:
$$
L\P_1 = -F_u(\i(z,\theta),0)U_1 - \k_0 B(\theta) \i '(z,\theta) -
\k_0 C(\theta)\partial_{\theta} \i '(z,\theta) - v_0 \a
\i '(z,\theta),
\eqno\mbox{(127)}
$$
$L$ is the self-adjoint ordinary differential operator
defined by
$$
L\P~\equiv~A(\theta)\P '' - G(z,\theta)\P,
$$
and the symmetric matrix function $G(z,\theta)$ is the Hessian
of the function $-\frac{1}{\g c T_0} f(\p,T_0) $ with respect to $\p$,
evaluated at $\p = \i(z,\theta)$.
Differentiating (126) with respect to z, we find that
$$
L\i ' = 0;
\eqno\mbox{(128)}
$$
taking the $(L_2)^m $ scalar product of (127) with $\i ' $, we
find a necessary condition for solvability:
$$
\langle F_u(\i(z,\theta),0)U_1, \i ' \rangle +
\omega(\theta)\k + v \langle \a \i ',\i '\rangle = 0,
\eqno\mbox{(129)}$$
where
$$
\omega(\theta) = \langle B \i ', \i ' \rangle - \langle C \partial_{\theta}
\i , \i '' \rangle.
\eqno\mbox{(130)}
$$
From this point on the analysis proceeds as before. The basic first order
outer approximation satisfies a free boundary problem like (77) - (80) with
different coefficients of $\k$ and $v$ in the $O(\e)$ terms. These
coefficients depend on $\theta$. For example, in place of $\tilde{\s}$ in
(79), we have $-\frac{\omega (\theta)}{\ell}$, and in place of $\a$, we have
$\langle \a \i ',\i ' \rangle/\s_1 \equiv
\tilde{\a} (\theta)$, where now
$$
\s_1 = \s_1 (\theta) = \langle A (\theta) \i ', \i ' \rangle.
\eqno\mbox{(131)}
$$
The various limiting problems are obtained as before. Of special interest is
the Mullins-Sekerka problem (89), in which the coefficient $\tilde{\s}$ of
$\k$ in (89b) is replaced by the $\theta$-dependent coefficient given above.
A similar statement is true of the motion-by-curvature problem (90), in
which the present analog will have a coefficient of $\k$ proportional to
$\omega$. Very probably the sign of this coefficient, which is governed by
the sign of $\omega$, determines whether these problems are well posed. For
this reason, it is of some interest to obtain a simpler expression for
$\omega$.
In their treatment of a different phase-field model with scalar order
parameter but
interfacial energy a given arbitrary function of $\theta$, McFadden et al
[MW] obtained an expression, which in our notation would be
$$
\omega(\theta) = \s_1(\theta) + \s_1 '' (\theta).
\eqno\mbox{(132)}
$$
This also holds in our framework, of course, when $m = 1$, as can
be verified by a calculation based on our notation. However, this
calculation depends very much on the matrices $A,~B,~C$ being able to commute
with one another. It appears doubtful that the expression (132) will
continue to hold in the multi-component case.
\bigskip
\noindent{\bf{Appendix. Derivation of (31)}}
\medskip
Let us represent points on the curve $\G(t)$ by the position vector
$R(\z,t)$,
where $\z$ is a parameter on $\G$ defined as follows. When $t = 0$ it is
arc length from some chosen point. When $t$ increases, the point $R(\xi,t)$
(for fixed $\xi$) has
trajectory normal to $\G(t)$ at each $t$. We will denote the unit tangent to
$\G(t)$ by $T(\z,t)$ and the unit normal in the direction of increasing $r$
by
$N(\z,t)$.
The normal velocity $v(\z,t)$ is defined by
$$
R_t = v(\z,t)N(\z,t).
\eqno\mbox{(133)}
$$
Let $\s$ be arc length on $\G$, and
$$
\a (\z,t) = \frac{\partial \s}{\partial \z} (\z,t).
\eqno\mbox{(134)}
$$
At $t = 0$, we have chosen $\z = \s$, so that
$$
\a(\z,0) = 1.
\eqno\mbox{(135)}
$$
In all of the following, we suppose $\G(t)$ is regular; and in particular
that
its radius at curvature is bounded away from zero.
Consider the point $x(r,\z,t)$ represented by
$$
x(r,\z,t) = R(\z,t) + rN(\z,t).
\eqno\mbox{(136)}
$$
For fixed $r$ and $t$ the curve $x(r,\z,t)$ will be ``parallel''
to
$\G(t)$ at a distance $r$. On this new curve, we can find the relation
between
arc length (which we continue to denote by $\s$) and the parameter $\z$.
First,
taking the differential of (136) with $t$ and $r$ fixed, we
obtain:
$$
dx = (R_{\z} + rN_{\z})d\z.
$$
But by definition we also have $dx = Td\s$. Therefore
$$
Td\s = (R_{\z} + rN_{\z})d\z.
\eqno\mbox{(137)}
$$
We also have from (134)
$$
R_{\z} = \a T,
\eqno\mbox{(138)}
$$
and since $T \cdot N = 0$,
$$
T_{\z} \cdot N + T \cdot N_{\z} = 0.
\eqno\mbox{(139)}
$$
The curvature $\k(\z,t)$ of $\G(t)$ is given by
$$
\k N = - T_{\s} = - T_{\z} \frac{\partial \z}{\partial \s} (\z,t) - -
\a^{-1}T_{\z}.
\eqno\mbox{(140)}
$$
Hence from (139), (140) and the fact that $N_{\z}$ is in
the direction of $T$ (note that $\frac{\partial}{\partial \z} |N|^2 = N_{\z}
\cdot N$), we have
$$
N_{\z} = \a \k T.
\eqno\mbox{(141)}
$$
From (137), (138), and (141) we now obtain
$$
\frac{\partial \s}{\partial \z} = \a (2 + r \k).
\eqno\mbox{(142)}
$$
Since $R_t$ and $R_{\z}$ are in the directions $N$ and $T$ respectively, we
have
$$
R_t(\z,t) \cdot R_{\z}(\z,t) = 0.
$$
Differentiating this equation with respect to $t$, we obtain
$$
R_{tt} \cdot R_{\z} + R_t \cdot R_{\z t} = 0.
\eqno\mbox{(143)}
$$
But from (133)
$$
R_{tt} = v_tN + vN_t,
\eqno\mbox{(144)}
$$
$$
R_{\z t} = v_{\z}N + vN_{\z};
\eqno\mbox{(145)}
$$
So from (143)--(145), (133), (138), we have
$$
N_t = - \a^{-1} v_{\z}T.
\eqno\mbox{(146)}
$$
If we set $dx = 0$ in (136) and use (138), (133),
(141), (146), we find:
$$
\frac{\partial r}{\partial t} = - v,
\eqno\mbox{(147)}
$$
and
$$
\frac{\partial \z}{\partial t} = \frac{r v_{\z}}{\a^2(2 + r \k)}.
\eqno\mbox{(148)}
$$
Now recall the coordinate system $r(x,t)$, $s(x,t)$ defined in section 4 (we
suppress $\e$--dependence). This $r(x,t)$ is that in (136). At $t = 0$,
this $s$ coincides with the previous $\z$. So at $t = 0$, we may replace
$\z$
with $s$ in (142), set $\a = 1$, and obtain
$$
|\na s(x,0)| = \frac{\partial s}{\partial \s} = (2 + r \k(s,0))^{-1}.
$$
Howerver, if $t = t_0$ is any other value of $t$, we can reset the clock in
the
original coordinate system $(r,\z,t)$ so that time starts at $t_0$, and make
the same conclusion. Therefore in general we will have
$$
|\na_xs(x,t)| = (2 + r \k (s,t))^{-1}.
\eqno\mbox{(149)}
$$
In the same way, we find from (147), (148)
$$
\frac{\partial r}{\partial t} (x,t) = - v(s(x,t),t),
\eqno\mbox{(150)}
$$
$$
\frac{\partial s}{\partial t} (x,t) = \frac{rv_s(s,t)}{1 + r \k (s,t)}.
\eqno\mbox{(151)}
$$
In these expressions, we have taken $\a = 1$ because of (135)
and the change of time setting.
Along with (149), we also have the obvious relation
$$
|\na_xr(x,t)| = 1.
\eqno\mbox{(152)}
$$
We have established (31a) and part of (31c). Let us now calculate
$\na^2r$ and $\na^2s$. We have, for any domain $\O$,
$$
\int_{\O} \na^2_xr(x,t)dx = \int_{\partial \O}\partial_nr(x,t)d \ell,
\eqno\mbox{(153)}
$$
where $\partial_n$ is the normal derivative and $d \ell$ is
arc length on $\partial \O$.
Let $\O$ be the curvilinear rectangle (in the $r,s$ coordinate system) shown
in the diagram,
bounded by sides $L_1 - L_4$.
\begin{figure}
\begin{center}
\begin{picture}(200,100)
\thicklines
\qbezier(10,100)(140,80)(160,0)
\put(175,20){\line(5,2){20}}
\put(166,40){\line(5,3){18}}
\qbezier(166,40)(170,33)(175,20)
\qbezier(184,51)(190,43)(195,28)
\put(170,8){$(r_0,s_0)$}
\put(184,55){$(r_0+dr,s_0+ds)$}
\put(175,30){$\Omega$}
\put(25,110){$\Gamma(t)$}
\put(157,26){$L_1$}
\put(168,50){$L_4$}
\put(195,39){$L_3$}
\end{picture}\end{center}
\caption{The domain $\O$}
\end{figure}
As $\partial_nr = 0$ on $L_2$ and $L_4$, we get:
$$
\int_{L_1\cup L_3} \partial_n rd \ell = |L_3| - |L_1|.
$$
On $L_1$ and $L_3$ we have (149), so that
$$
\begin{array}{c}
|L_1| \simeq (2 + r_0 \k(s_0,t))ds,\\
\\
|L_3| \simeq (2 + (r_0 + dr) \k (s_0,t))ds,
\end{array}
$$
so that
$$
\int_{\O} \na_x^2r(x,t)dx \simeq \k (s_0,t) drds.
$$
But the left side of this equation, to lowest order is
$\na^2r(s_0,t)$ times the area of $\O$, and this area is approximately
$$
|\O| \simeq |L_1|dr = (2 + r_0 \k(s_0,t))drds.
$$
Thus
$$
\na^2r(x,t) = \frac{\k(s,t)}{1 + r \k(s,t)}.
\eqno\mbox{(154)}
$$
In a similar way, we have
$$
\int_{\O} \na_x^2s(x,t)dx = \int_{L_4} (2 + r \k(s_0+ds,t))^{-1}dr -
\int_{L^2}(2 + r \k(s_0,t))^{-1}dr.
$$
Now writing $\k(s_0 + ds,t) = \k(s_0,t) + d\k$ and integrating
between $r_0$ and $r_0 + dr$, we obtain to lowest order,
$$
\na^2s(x,t) = \frac{r\k_s}{(2 + r \k)^3}.
$$
This establishes (31b,c).
\paragraph{Asknowledgements.}
Supported by National Science Foundation under Grants DMS 8901771 and
9201714,
and by the
Science and Engineering Research Council. Part of this research was done
during
a workshop sponsored by the International Centre for Mathematical Sciences at
Heriot--Watt University. The authors are grateful to John Cahn for
informative
discussions, particularly about the possibility of terms involving the
Laplacian of the curvature, as in Section \ref{16}, and to Cecile Schreiber
for
carefully checking the details of an earlier version.
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\end{thebibliography}
\bigskip
\noindent
\parbox{6cm}{{\sc Paul C. Fife\newline
Mathematics Department\newline
University of Utah \newline
Salt Lake City, UT 84112\newline
U.S.A. }\newline
E-mail: fife@math.utah.edu }
\qquad
\parbox{6cm}{{\sc Oliver Penrose\newline
Mathematics Department\newline
Heriot-Watt University \newline
Riccarton, Edinburgh, EH14 4AS\newline
U.K.}\newline
E-mail: oliver@cara.ma.hw.ac.uk}
\end{document}