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\begin{document}
\vspace*{2cm}
\begin{center}
{\large \bf DIGM: MOTION RESULTING FROM DIFFUSION WITHIN AN INTERFACE}

\vspace{1cm}
{\sc J. W. CAHN, P. C. FIFE, and O. PENROSE}
\end{center}
\vspace{3cm}

{\bf Abstract.}  A well-known but little understood experimental phenomenon
called diffusion
induced grain boundary motion is given a theory based on a phase-field model
with order parameter and concentration of a diffusing solute as field
variables.  Suggested physical mechanisms for producing the motion are
investigated and assessed in the framework of the model.
\vspace{1.5cm}\\

\section{Introduction}

Interfaces separating crystalline grains in polycrystalline solids, such as
metal and ceramics, are sometimes observed to
migrate, causing some grains to grow at the expense of others.  When this is
motion by curvature, it is called grain growth.  But in the presence of
chemical composition gradients, there is another cause for the motion that
has
been found to be strong enough to overwhelm motion by curvature,
when curvatures are less than approximately $10^6 m^{-1}$.
%may or may not be
%accompanied by changes in the chemical composition; in any case, it
%can have important practical consequences.
This
phenomenon, called diffusion-induced grain boundary motion (DIGM),
was first discovered by Hillert and Purdy in 1977 \cite{HP}.  A particularly
%interesting type
%of boundary motion is sometimes induced, or greatly accelerated,
simple geometry for observing this phenomenon is a
thin metallic film containing transverse planar grain boundaries, immersed in a
vapor containing a
substance that
%which
can be incorporated in the crystals as a solute in a solid solution.
%!
Although
DIGM has been studied extensively
and various explanations have been
proposed, the true mechanism underlying this type of boundary migration
remains something of a mystery.

It is the purpose of this note to report on progress made on a theoretical
basis for
the observed coupling of the motion of such a grain boundary with the solute
diffusion field within the boundary itself.  We
%will
consider a phase-field model for an abstract diffuse interface
that
%will have
has many of the attributes of a grain boundary.  We take the boundary to
%It will
be a thin mobile zone
of disorder (grain boundary) in which the solute diffuses rapidly, between
two
highly ordered regions (crystalline grains) with low diffusivities.  The
ordered regions
support stress, but this stress is not transmitted through the zone
where the
ordering is low.  Since grain boundaries are surfaces between grains of the
same phase, differing only in the orientation of the crystal axes, the
properties of both crystalline grains are represented by only the solute
concentration and the magnitude of an order parameter.

Two coupling mechanisms are considered within the context of our phase
field
model: the dependence of the stress in the ordered grains on the
concentration of the
diffusing
element (Sec. \ref{cbse}) and the dependence of the interfacial free
energy on
this
same concentration (Sec. \ref{cdife}).  Both effects are found to produce
boundary
motion under
certain conditions, and the velocity is estimated in terms of the
other
parameters and data entering the model, such as the diffusivity, the initial
concentration in the crystal, and that in the
surrounding vapor.

\section{A phase field approach} \lbl{kebc}

We use a diffuse-interface model with an order parameter $\d$ and a
concentration $u$ as the only field variables.  Moreover, the
interface we
consider is straight and represents a boundary between two grains of
the same
phase.  In particular, we exclude at the outset any influence that
curvature of the
boundary or temperature variation may have on the motion.  In reality,
these
boundaries
are usually curved; that additional effect has just not as yet been
included in our model.

Typically, the
concentration $u$ could represent the fraction of available lattice
sites
which are filled by solute atoms, and so we shall take its range to be
the
interval $[0,1]$.  Similarly $-1 \le \d \le 1$, the two extremes $\pm
1$
representing the two perfect grains on either side.

As is usual with diffuse-interface models (e.g. \cite{fife}), the
postulated evolution system can be understood
as a gradient flow with respect to a functional.  In this case, the
functional is
appropriately named the free energy $F[\d,u]$, since temperature will
be
constant.  It has contributions coming individually from $\d$ and
$u$, plus a small contribution resulting from the interaction between
the two
fields:
\be
F [ \phi, u ]  = F_1 [ \phi ] + F_2[ u ] + \epsilon F_{12} [ \phi, u
].
\lbl{F}
\ee

The first term will be of standard van der Waals-Cahn-Hilliard form
\be
F_1 [ \phi]  = A\int_{\Omega}
[ w( \phi) + \frac{1}{2} \w^2 | \nabla \phi |^2 ] dv.
\lbl{F1}
\ee
Here $\L$ is the domain in space occupied by the solid material.
Rather,
assuming  everything is independent of one space coordinate, we take
it to be
the strip $\{-\B < x < \B\}\times \{-h < y < h\}$, representing a 2D
cross
section of the metal film.  The grain boundary will be a subregion of
$\L$
consisting of a slim rectangle $\L_0$ bounded on top and bottom by the
two
faces $\{y = \pm h\}$.  The law of motion of this rectangle is our
primary
object of interest. The parameter $\w^2$ is some measure of the
relative
weight attached to gradients of $\d$ in the expression for the free
energy
(we shall see that $\w$ is also a measure of the
thickness of the interface $\L_0$), and $w$ is approximately the free
energy
density if $\d$ were constant in space.  We take it to be the
following
double-well potential with barrier at $|\d| = 1$ (see, e.g.
\cite{elliott}):
\bea
w(\phi) & = & {1 \over 2} (1 - \phi^2) \hbox{~~if~~} | \phi| < 1
\nonumber \\
\hbox{and} & = & + \infty \hbox{~~if~~} |\phi| > 1
\lbl{w}
\eea
For the values $|\phi| = 1, ~w(\phi)$ is left undefined.

We take

\be
F_2[u] = \int_{\Omega}W(u) dv,
\lbl{F2}
\ee
where $W$ is a convex function representing the free energy density of
the
solid due to the presence of the solute.  A typical form for $W$ is
\be
W(u) = \mbox{const}\cdot [u \ln u + (1-u) \ln (1-u)].
\lbl{W2}
\ee

Other effects of the type we are seeking can be obtained by including
terms
involving $\A u$ in (\ref{F2}); we shall omit them for simplicity.

Our object will be to discover interaction terms $F_{12}$ which can
serve as
impetus for the motion of the interface and which have physical
interpretations.  They will be of the assumed form
\be
F_{12} = \epsilon A \int_{\Omega} p(\phi, u) dv.
\lbl{F12}
\ee
The small dimensionless normalizing parameter $\epsilon$ is
chosen so that the derivatives of $p$ are of order unity.  Setting
$\bar{\e}
= \frac{\e A}{W''(1/2)}$,
we assume the smallness relations
\be
|\epsilon| \ll 1;~~|\bar{\e}| \ll 1.
\lbl{eps}
\ee


Our chosen kinetic equations take the form
\be
\t\frac{\M\d}{\M t} = - \frac{\w F}{\w \d},~~\frac{\M u}{\M t} =
\A\cdot
D(\d)\A u(1 +
O(\bar{\e}))
\lbl{ev}\ee
(the error term depends on $p$ and will not be given explicitly),
where
$$\frac{\w F}{\w \d} = -\w^2\W\d + w'(\d) + \e A \frac{\M p}{\M\d},
$$
 $D(\d)$ represents the diffusivity of the solute as a function of the
degree (measured by $\d$) to which the material is ordered, and $\t$
is a
relaxation time.

One interpretation and/or derivation of the system (\ref{ev}) involves
its
characterization as a gradient (steepest descent)
flow for the functional $F[\d,u]$ with respect to a certain
$D$-dependent
scalar product, at least in the case that $D > 0$ and $W$ is given by
(\ref{W2}).

We are mostly interested, however, in the case when $D(\d(x,t))$
vanishes in much of the domain of interest; the gradient nature of the
flow
then degenerates.
It is well known, in fact,  that the diffusivity is
much larger in the grain boundary than in a perfect crystal.
We shall make the simplifying assumption that
$D(\phi) = 0$ in the crystals, that is when $|\phi| = 1$.
Our specific choice for $D$ is
\be
D(\phi) = D(0)(1-\phi ^2),
\lbl{Dphi}
\ee
where $D(0)$ is a constant.

By the  choice (\ref{w}) of $w$, the system (\ref{ev}) now becomes
\be
\tau \frac{\partial \phi}{\partial t}  =
 \phi +  \w^2 \nabla^2 \phi - \epsilon \partial p/ \partial \phi
~~\hbox{  if  }~~| \phi| \ne 1,
\lbl{dphi/dt}
\ee
\be
\frac{\partial u}{\partial t} =   \A\cdot  D(\d) \A u (1 + O(\bar{\e})).
\lbl{du/dt2}
\ee
To account for the possibility that $|\d| = 1$, we extend
(\ref{dphi/dt}) as
a variational inequality (see \cite{elliott} for an analogous
situation).
The
result is that in places where $| \phi| =
1$,  (\ref{dphi/dt}) need not hold;
all that is necessary there is for $\phi$ and its gradient
to be continuous.


As boundary conditions on $\M\L$, physical considerations suggest that
\be
\frac{\partial \phi}{\partial n} = 0 {\rm ~on~} \partial \Omega,
\lbl{phibc}
\ee
where $\partial / \partial n$ denotes differentiation in the
direction normal to $\partial \Omega$, and (neglecting $\bar{\e}$)
\be
u = u^* \mbox{ on }\M\L,
\lbl{vapor}
\ee
where $u^*$ is a given value depending on the properties of the
external
vapor.

Conditions  at $x =  \pm \B$ are also needed.  We require
\be
 \d = \pm 1\mbox{  and  } \frac{\M u}{\M t} = 0
\lbl{inf}
\ee in regions to the left and right in our strip domain $\L$ where
the pure
crystals reside, i.e. where  $D = 0$.  These regions are not known at
the outset.


\section{Travelling wave solutions} \lbl{tws}

To model a moving grain boundary we look for a leftward travelling
wave solution of the equations, in which $\phi$ and $u$ depend
on $x$ and $t$ only through the combination $x+ct$, where the positive
velocity
$c$ is
to be determined.
We shall use the notation $z=x+ct$ and look for solutions
of (\ref{dphi/dt}) - (\ref{inf})
in the
form
$\phi=\phi(z,y),~ u=u(z,y)$.  We also prescribe the resident value
$u_0$ of
$u$ in the eroding grain on the left.  In all, we get, in addition to
(\ref{phibc}) and (\ref{vapor}),

\bea
c\tau \phi_z & = &  \phi + \w^2(\phi_{zz}+\phi_{yy})
 - \epsilon (\partial p / \partial \phi ),
\lbl{twphi}\\
cu_z & = & \A\cdot D(\d)\A u(1 + O(\bar{\e}));
\lbl{twu}
\eea
\bea
u(z,y) = u_0,~~\d(z,y) = -1,~~~~z\mbox{ large negative },
\nonumber \\
\d(z,y) = 1,~~~~z\mbox{ large positive }
\lbl{infbc2}
\eea

Assuming that a travelling wave exists, we see that (\ref{twphi}) can
be used
to obtain an expression for
$c$ in
terms of $p(\d,u)$, hence in terms of $u$, which will be useful in the
following sections.
We set

\begin{equation}
g(z,y) = -\frac{\partial p}{\partial \phi}[\d (z,y), u(z,y)],
\lbl{ar}
\end{equation}
multiply (\ref{twphi}) by $\d_z$, integrate with respect to $z$ and
$y$
and use the boundary
conditions (\ref{phibc}), (\ref{infbc2}) to obtain

\begin{equation}
c\t\int\int \d_z^2 dzdy =
\epsilon \int\int\d_zgdzdy.\lbl{ao}
\end{equation}

Physically, this is surprising: when the coupling $\e g$ is absent, no
traveling wave motion is possible,
even though there would be a decrease in the total free energy
if the boundary
were to
move.



\section{An approximate solution}
$\lbl{as}$

We formally simplify the problem (\ref{twphi}), (\ref{twu}),
(\ref{phibc}),
(\ref{infbc2}) by utilizing the smallness of $\e$ and $\bar{\e}$.
First,
(\ref{twphi}) with $\epsilon = 0,
~c=0$ gives  the following lowest order approximation for $\phi$ :
\be
\phi^{(0)} = S(x/\delta),
\lbl{phi0}
\ee
where the function $S(x)$ is defined to equal $\sin x$ for $|x| <
\pi/2$, and
is continued to equal $\pm 1$ for $x$ outside that interval.

One can now obtain
the
lowest order approximation to $u$ in terms of the unknown velocity $c$
(which
will be determined later). We use (\ref{phi0}) in (\ref{twu}) with
the error term neglected.
By  (\ref{phi0}) and (\ref{Dphi}),
\bea
 D(\phi(z)) \approx D(\phi^{(0)}(z)) & = & D(S(z/\delta))
\nonumber \\
& = & \left\{
\begin{array}{l}
D(0) \cos^2(z/\delta) \hbox{ if } -\pi\w/2 < z < \pi\w/2 \\
0 \hbox{ otherwise }. \\
\end{array}
\right.
\lbl{D1}
\eea

The equation to be solved is
\be
cu_z = (D(z)u_z)_z + D(z)u_{yy},
\lbl{mueqn}
\ee
where $D(z)$  stands for $D(\phi^{(0)}(z))$.

The domain $\L$ is split into three parts as shown:

\vskip 40 pt

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\put(110,0){\line(0,1){80}}

\put(95,40){$\Omega_0$}
\put(0,40){$\Omega_-$}
\put(170,40){$\Omega_+$}
\put(85,85){$y = h$}
\put(80,-10){$y = -h$}
\put(115,40){$z > \pi\w/2$}
\put(30,40){$z < -\pi\w/2$}

%the coordinate axes
%\thinlines
%\put(-20,40){\line(1,0){240}}
%\put(100,-20){\line(0,1){120}}
%\put(220,40){\line(-1,1){8}}
%\put(220,40){\line(-1,-1){8}}
%\put(230,40){$z$-axis}
%\put(100,100){\line(1,-1){5}}
%\put(100,100){\line(-1,-1){5}}
%\put(900,105){$y$-axis}

\end{picture}

Fig. 1: The regions $\Omega_-, \Omega_0, \Omega_+$
\vskip 20 pt

In the left grain  $\Omega_-,~D = 0$, so that  $u_z=0$, and from
(\ref{infbc2}),
\be
u(z,y) = u_0 \hbox{ in } \Omega_-
\lbl{uinOm-}
\ee
In the region $\Omega_+ $
we again have $u_z=0$, and hence
\be
u(z,y)=u(\pi\w/2,y) \hbox{ in } \Omega_+
\lbl{uinOm+}
\ee
where $u(\pi\w/2,y)$ is not yet known (this is the concentration left
in the
wake of the moving boundary).
In the remaining region, $\Omega_0 = \{-\pi\w/2<z<\pi\w/2,-h<y<h\}$,
the full equation (\ref{mueqn}) is to be solved.
The boundary conditions, determined on $\{y=\pm h\}$ by (\ref{vapor})
and on $\{z = \pm \pi\w/2 \}$ by the condition that $u$ and its
gradient
should be continuous inside the metal, are
\bea
u = u* &  &~~~ (y=\pm h)
\nonumber \\
u = u_0 & \hbox{~and~}u_z=0 & ~~~ (z=-\pi\w/2)
\nonumber \\
%u = u(+\infty,y)  \hbox{~and~}
u \hbox{ continuous; } & u_z=0 & ~~~ (z=\pi\w/2)
\lbl{bc0}
\eea
The number of boundary conditions appears to make the problem
overdetermined,
but in fact some of them follow automatically from others.  This
circumstance
is due to the very singular nature of (\ref{mueqn}): on the two
lateral sides
of
$\L_0$, $D$ and its derivative both vanish.

It turns out that a useful approximation to the solution of
(\ref{mueqn}),
(\ref{bc0}) can be obtained under the assumption that the following
parameter
$\h$ is small:
\be
\h ~\X~ \sqrt{\frac{\w c}{D(0)}} \ll 1.
\lbl{eta}
\ee

The approximation is
\be
u(z,y) \approx u^{(0)}(z,y) =
u_0 + (u^* - u_0)f(z) \frac{\cosh y\h/\w}{\cosh h\h/\w},
\lbl{v}
\ee
where $f(z)$ is a positive function with the properties
\bea
f(z)       & \approx &  e^{-\h^2 \tan{|z|\over\delta}}
~~(-\pi\w/2 \le z \le 0)
\nonumber \\
\hbox{and} & \approx & 1~~~(0 \le z \le \pi\w/2).
\lbl{fapprox}
\eea

We shall use this approximate expression for $u$ in all of the
following.


Note that if
$h\h/\w \gg 1$, then for most values of $z \in (-\pi\w/2,\pi\w/2)$ the
function  $u^{(0)}(z,y)$ as given by (\ref{v})
changes abruptly from
$u^*$ at the top and bottom ($y = \pm h$) to near $u_0$ within a
distance of the order of $\w/\h$.
On the other hand if $h\h/\w \ll 1$, then the profile is approximately
parabolic :
\be
u(z,y)~\Z~u^* + \frac{\h^2}{\w^2}(u_0 - u^*)(h^2 - y^2).
\lbl{parab}
\ee

Finally, with knowledge of an approximation $u = u^{(0)}$ obtained
this way, a leading order approximation for $c$ may
also be obtained.  In (\ref{ao}) we set $\d = \d^{(0)}$ as given by
(\ref{phi0}) and $u = u^{(0)}$ as given by (\ref{v}).
Using the fact that   $\int_{-\infty}^{\infty} (\phi^{(0)}_z)^2 \,dz =
 \pi/2\w$ by (\ref{phi0}),
we obtain
\bea
%c & \approx &
%\frac{\epsilon
%\int_{-\pi\w/2}^{\pi\w/2} (1/\delta)\cos{z \over
%\delta}\bar{g}(z)\,dz}
%{\tau\pi/2\w}
%\nonumber \\
c \approx \frac{2 \epsilon }{\pi \tau}
\int_{-\pi\w/2}^{\pi\w/2} \bar{g}(z) \cos{z\over \delta}\,dz,
\lbl{c}
\eea
where $\bar{g}(z)$ is the $y$-average of $g(z,y)$.
This formula is the main tool in our search for mechanisms to make $c
> 0$.
In the following sections
it will be applied to different interaction functions $g$.


\section{Coupling by a concentration-dependent interfacial free
energy}
\label{cdife}

It was suggested by Meyrick \cite{Meyrick} that interface motion in
some contexts could be caused by an
interaction in which the interfacial free energy
depends on the concentration of solute atoms.  This interaction
is analogous to the Marangoni effect \cite{marangoni,scriven},
in which an interface moves because the surface tension of a
fluid depends on concentration or temperature.

In our context, the interfacial free energy should be interpreted as
the term
$F_1$ in (\ref{F}).  A simple way to model this mechanism is to make
the
coefficient
of $w(\phi)$ in the formula (\ref{F1}) depend linearly on
$u$. This is equivalent to
adding a term of the form (\ref{F12}) in which
\be
p =  uw(\d).
\lbl{Mey}
\ee  so that, by (\ref{w}),
$g(u,\phi) = -\partial p/ \partial \phi = u\phi$ for $|\d| < 1$.

The coefficient $\epsilon$ in (\ref{F12})
can be  of either sign. If $ \e > 0$,
the interaction free energy density $\epsilon A p$
increases with $u$ at fixed $\phi$, and so it is expected that the
interface
will tend to move towards smaller values of $u$. If $\epsilon < 0$,
it would tend to move the other way.

With $\phi^{(0)}$ given by (\ref{phi0}) and $u^{(0)}$ by (\ref{v}),
formula (\ref{c}) now yields
\be
c \approx {\epsilon \w \h^2 \over 2 \tau}(u^*-u_0)
{\tanh { \sqrt{2/\pi} h \h/\w} \over  { \sqrt{2/\pi} h \h/\w}}.
\lbl{c3M}
\ee
Our setup requires $c \ge 0$.  For the Meyrick mechanism,
the boundary can therefore move only if $ u^*-u_0 $ has the same sign
as
$\epsilon $.  Even if the signs are right, there is no guarantee that
(\ref{c3M}) can be solved for $c$.  The right side also depends on $c$
through $\h$ (\ref{eta}), resulting in an implicit equation for $c$.  If a
solution with $c>0$ exits, one factor ``$c$'' may be eliminated from both
sides.  The equation for $c$ so obtained has a solution only if $(u^* -
u_0)\e\w^2/\t D(0)$ is large enough.

However, experiments \cite{handwerker} have shown
that for many alloy systems, it is possible for the grain boundary to
move
for either sign of
$ u^*- u_0 $ (The direction of motion of the boundary depends on
the sign of $ u^*-u_0 $). So a term of the form
(\ref{Mey}) cannot provide the sole driving force for DIGM.


\section{Coupling by stress energy.}
\label{cbse}

In this model, we depend on the assumption that the travelling wave is
fully developed, so that
the dissolving grain, occupying the region $\{z<-\pi\w/2\}$, has a
given
limiting
concentration $u_0$ as $z \ra -\B$, and the grain being formed in the
wake of
the moving interface has
a limiting concentration profile $u_+(y)$ as $z \ra +\B$, which is
unknown at
the moment.

At any location, the equilibrium lattice spacing of a
perfect crystal is generally determined by the solute concentration $u$, and
therefore a
spatial
variation of $u$ in either grain will induce stress, whose energy should be
incorporated into the free energy functional $F$.
In the dissolving grain, the stress energy density in our
approximation will
take the form
$\epsilon A P_-(u - u_0)^2$, with $\e A P_-$ positive.  In the
advancing grain, which we suppose to be
disconnected as explained below from the other grain, it is $\epsilon A P_+(u
- u_+)^2$.
In the interface where any crystalline structure is imperfect, the
transmission of stress due to concentration differences is weakened.  This
fact is the source of a coupling between the two fields $u$ and $\d$, and
hence of a stress-induced term $F_{12}$.

We model the ability to transmit stress in this way as an all-or-none
phenomenon.  No stress can be transmitted to
neighboring
portions of the metal across a sufficiently disordered layer, say when
$|\d| < \d_1$ for some $\d_1 \in (0,1)$, but  stresses
due to nonconstant $u$ in the dissolving (advancing) grain occur as
though it
were a perfect crystal occupying the entire region $\{z : \d(z) <
-\d_1~(> \d_1)\}$.

Since there will be no stress energy in the region
$\{z : | \d(z)| < \d_1\}$, the scaled coupling free energy
density
$p$ (see  (\ref{F12})) can be written
\begin{equation}
p(z,y) =
P_-(u -u_0)^2 H(-\d -\d_1) + P_+(u - u_+)^2H(\d - \d_1),
\lbl{fa}
\end{equation}
where $H$ is the Heaviside function (unit step function).

Expressing the derivative of $H$ as a delta-function, we obtain
\begin{equation}
\bar{g}(z) = -\overline{\frac{\w p}{\w \d}} =
\mbox{ \boldmath $ \delta$}(\d + \d_1)P_-\overline{(u -u_0)^2} -
\mbox{ \boldmath $ \delta$} (\d - \d_1)P_+\overline{(u -u_+)^2}
\lbl{fb}
\end{equation}
where the bars indicate averages over $-h<y<h$ at fixed $z$.

After some tedious calculations using (\ref{phi0}), (\ref{v}), and
(\ref{c}), we find under certain assumptions on $\d_1$ and the other
parameters that
\be
c \approx  \frac{2\e\w }{\pi \tau }P_-(u_0 - u^*)^2.
\lbl{c2p}
\ee

Other expressions are obtained from different assumptions, but in all
cases,
$c$ is an even function of  $(u^* - u_0)$, so that a reversal of the sign of
$(u^* -
u_0)$ has no effect on the velocity.  This appears to conform to some
experiments, but more investigation is needed.  We conclude that this
effect
is a feasible impetus for the motion.

A possibly more realistic case is that in which both effects in this
and the
last
section
occur, with different values of $\e$.  Then an expression for $c$
depending
on these two values can easily be obtained.

\section{Acknowledgements}  We thank Gary Purdy for many stimulating
discussions which caught the attention of PF and OP during an ICMS
workshop
at Heriot-Watt University, and for his continued interest and critical
comments during this research. The workshop was supported by the SERC,

the European Community and the London Mathematical Society.
JC is grateful for the invitation to
come to Heriot-Watt and for support from ARPA.  PF is grateful to
Heriot
Watt.  His research was also supported by NSF Grant 9201714.


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\end{thebibliography}

\vspace{.4in}
\noindent J. W. Cahn

\noindent NIST, Bldg 223, Rm A153,

\noindent Gaithersburg, MD 20899, USA
\vspace{.25in}

\noindent P. C. Fife

\noindent Mathematics Department, University of Utah

\noindent Salt Lake City, UT 84112, USA
\vspace{.25in}

\noindent O. Penrose

\noindent Mathematics Department, Heriot-Watt University

\noindent Riccarton, Edinburgh EH14 4AS, U.K.

\end{document}
