\documentstyle[12pt]{article}
\def\d{\displaystyle}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\ord{\phi (\vec{r},t)}
\def\stat{\phi^s (\vec{r})}
\def\stp{\phi^s (x)}
\def\s0{\phi_0}
\def\p{\partial}
\def\nvec{\vec{\nabla}}
\def\n2{\nabla^2}
\def\la{\langle}
\def\ra{\rangle}
\def\ls{L(t) \sim t^{\frac{1}{3}}}
\def\sd{L(t) \sim t^{\frac{1}{4}}}
\def\st{S(k,t)}
\def\mom{\la k \ra}
\begin{document}
\setcounter{page}{1}
\baselineskip=14pt \parskip=0pt plus2pt
\begin{center}
\begin{Large}
Phase Separation Kinetics in a Model with Order-Parameter Dependent Mobility
\end{Large}
\vskip0.5cm
by \\
Sanjay Puri$^{1,2,3}$, Alan J.Bray$^1$ and Joel L.Lebowitz$^4$
\end{center}
1. Department of Theoretical Physics, The University, \\
\ \ \ Manchester M13 9PL, U.K. \\
2. Isaac Newton Institute of Mathematical Sciences, Cambridge University, \\
\ \ \ Cambridge CB3 0EH, U.K. \\
3. School of Physical Sciences, Jawaharlal Nehru University, New
Delhi -- 110067, INDIA. \\
4. Departments of Mathematics and Physics, Rutgers University,
New Jersey 08903, U.S.A. \\
\vskip0.5cm
\begin{abstract}
We present extensive results from 2-dimensional simulations of phase
separation kinetics in a model with order-parameter dependent
mobility. We find that the time-dependent structure factor exhibits
dynamical scaling and the scaling function is numerically
indistinguishable from that for the Cahn-Hilliard (CH) equation, even
in the limit where surface diffusion is the mechanism for domain
growth. This supports the view that the scaling form of the structure
factor is "universal" and leads us to question the conventional wisdom
that an accurate representation of the scaled structure factor for the
CH equation can only be obtained from a theory which correctly models
bulk diffusion. (Figures available upon request from
lebowitz@math.rutgers.edu).
\end{abstract}
\pagebreak
\section{Introduction}
When a two-phase mixture in a homogeneous phase is quenched below the
critical coexistence temperature, it becomes thermodynamically
unstable and evolves towards a new equilibrium state, consisting of
regions which are rich in one or the other constituent of the
mixture. The dynamics of this evolution is referred to as "phase
ordering dynamics" and constitutes a well-studied problem in
nonequilibrium statistical mechanics \cite{jdg}. As a result of these
investigations, there is now a good understanding of many aspects of
phase ordering in pure and isotropic binary mixtures. Thus, it is
generally accepted that the coarsening domains are characterised by a
unique, time-dependent length scale $L(t)$, where $t$ is
time. {}Furthermore, the nature of the phase ordering process depends
critically on whether or not the order parameter is conserved. {}For
systems characterised by a nonconserved order parameter, e.g.,
ordering of a ferromagnet, growing domains obey the
Lifshitz-Cahn-Allen (LCA) growth law $L(t) \sim t^{\frac{1}{2}}$
\cite{jdg}. {}For systems with a conserved order parameter but no
hydrodynamic effects, e.g., segregation of a binary alloy, the
characteristic domain size obeys the Lifshitz-Slyozov (LS) growth law
$\ls$ \cite{jdg}. {}For systems with a conserved order parameter and
hydrodynamic effects, e.g., segregation of a binary liquid, there
appear to be various regimes of domain growth, depending on the
dimensionality and system parameters \cite{sig, kog}.
As far as the analytic situation is concerned, there is a reasonable
understanding of the nonconserved case for pure and isotropic
systems. In particular, the LCA diffusive growth law has been derived
in some exact models \cite{exa}. In addition, Ohta et al. and Oono and
Puri \cite{ojk} have proposed an analytic form for the time-dependent
structure factor which is in good agreement with numerical results,
though the quality of this agreement has recently been questioned by
Blundell et al. [5]. {}For the conserved case, the situation is less
satisfactory. There is some understanding of the growth exponents and
one has a good empirical form for the scaled structure factor -- at
least without hydrodynamics \cite{fra}. However, this functional form
is analytically derivable only in the limiting case where one of the
components is present in a small fraction \cite{jdg}. An outstanding
theoretical problem in this field is the calculation of the scaled
structure factor for the conserved case when the two components of the
mixture are present in an equal proportion, viz., the so-called
critical quench \cite{oht}. Our results in this paper provide some
interesting insights on this problem, as we will discuss later.
In this paper, we study a model for phase separation dynamics in
systems where the mobility is order-parameter dependent. We will
present detailed numerical results from a simulation of this model.
This paper is organised as follows. In Section 2, we briefly discuss
our model and its static solution. In Section 3, we present numerical
results obtained from our model. Section 4 ends this paper with a
summary and discussion.
\section{Model for Phase-Separating Systems with Order-Parameter
Dependent Mobility}
The dynamics of phase separation is usually described by the
phenomenological equation \beq \frac{\p \ord}{\p t} = \vec{\nabla}
\cdot \bigg [ M(\phi) \vec{\nabla} \bigg ( \frac{\delta
H[\ord]}{\delta \ord} \bigg ) \bigg ] , \eeq where $\ord$ is the order
parameter at point $\vec{r}$ and time $t$ and is a measure of the
local difference in densities of the two segregating species, say A
and B. In (1), $M (\phi)$ corresponds to the mobility, which is
dependent on the order parameter, in general. The free-energy
functional is usually chosen to be of the standard $\phi^4$-form,
viz.,
\beq H [\ord] = \int \mbox{d} \vec{r} \bigg [ -\frac{1}{2} \ord
^2 + \frac{1}{4} \ord ^4 + \frac{1}{2} (\vec{\nabla} \ord)^2 \bigg ] ,
\eeq
where we assume that all variables have been rescaled into
dimensionless units; and the system is below the critical temperature.
The dynamics of Eqs. (1)-(2) drives the order parameter to the local
fixed point values $\phi_0 = \pm 1$, corresponding to (say) A- and
B-rich phases, respectively. The temporal evolution described by
Eq. (1) also satisfies the conservation constraint that $\int \mbox{d}
\vec{r} \ord$ is constant in time.
There have been many studies of Eq. (1) in the limiting case of the
Cahn-Hilliard (CH) equation \cite{ch}, where the mobility is constant,
viz., $M(\phi) = 1$ (in dimensionless units). Numerical studies of the
CH equation and equivalent Cell Dynamical System (CDS) models
\cite{op} demonstrate that late-stage domain growth obeys the LS
growth law we have quoted earlier (i.e., $\ls$). These studies also
clarify the functional form of the scaled structure factor which
characterises the morphology of the coarsening domains.
{}For deep quenches, it has been pointed out by Langer et al. and
Kitahara and Imada \cite{lan} that a more realistic model for phase
separation should explicitly incorporate an order-parameter dependent
mobility of the form
\beq
M(\phi) = 1 - \alpha \phi^2 ,
\eeq
where $\alpha$ parametrises the depth of the quench. At the physical
level, this form of the mobility can be understood as follows. Deep
quenches result in enhanced segregation in that A-rich (or B-rich)
domains are purer in A (or B) than in the case of shallow
quenches. Thus, if one presumes that phase separation occurs by
exchanges of neighbouring A- and B-atoms, the probability of such an
exchange in the bulk is drastically reduced for deep quenches. This
can be mimicked by the order-parameter dependent mobility in (3) with
$\alpha \rightarrow 1$. At the mathematical level, Kitahara and Imada
\cite{lan} have shown that an order-parameter dependent mobility
arises naturally if one attempts to obtain a coarse-grained model for
phase separation from a master equation description of an appropriate
microscopic model, viz., the Ising model with Kawasaki spin-exchange
kinetics \cite{bin}.
The physical effect of the order-parameter dependent mobility is that,
as $\alpha \rightarrow 1$ (which happens for temperature $T
\rightarrow 0$), bulk diffusion is substantially suppressed because
the mobility $M(\phi_0) \rightarrow 0$. Therefore, the effects of
surface diffusion are relatively enhanced. The surface-diffusion
mechanism for domain growth has an associated growth law $\sd$
\cite{fur}, in contrast to the evaporation-condensation mechanism
which drives asymptotic growth in the CH equation and gives rise to
the LS growth law. Therefore, as $T \rightarrow 0$, one expects an
extended regime of $t^{\frac{1}{4}}$ growth in the dynamics of
Eqs. (1)-(3). This model has been studied numerically by various
authors \cite{lac} and we will remark on their results
shortly. {}Furthermore, Bray and Emmott \cite{be} have analytically
studied phase separation in models with order-parameter dependent
mobility in the limit where one of the components is present in a
vanishingly small fraction. In passing, we should also point out that
an order-parameter dependent mobility as in (3) has proven to be a
useful way of incorporating the effects of external fields which vary
linearly with distance, e.g., gravity. However, we will not go into
this here and merely refer the interested reader to Reference
\cite{koj}.
In recent work, there was proposed a novel dynamical equation for
phase separation in binary mixtures -- using the master equation
formulation for an Ising model with Kawasaki spin-exchange kinetics
\cite{ppd}. This equation was first obtained in the context of phase
separation in a gravitational field but does not reduce to the CH
equation in the absence of gravity. As a matter of fact, it takes a
form similar to that of Eq. (1), i.e.,
\beq
\frac{\p \ord}{\p t} = \nvec \cdot \bigg [ \big ( 1 - \ord^2 \big ) \nvec
\bigg ( \frac{\delta H[\ord]}{\delta \ord} \bigg ) \bigg ] ,
\eeq
with the free energy
\bea
H[\ord] & = & \frac{T}{T_c - T} \int \mbox{d} \vec{r}
\frac{1}{2} \bigg [ (1+\ord) \ln (1 + \ord) \nonumber \\
& & \ \ \ \ \ \ \ + (1-\ord) \ln (1-\ord)
- \frac{T_c}{T} \ord^2 \nonumber \\
& & \ \ \ \ \ \ \ + \frac{T_c - T}{T} (\nvec \ord)^2 \bigg ] .
\eea
Eqs. (4)-(5) have been cast in a dimensionless form by a rescaling of
the space and time variables analogous to that for the CH equation
\cite{ppd}. (Clearly, this rescaling is not appropriate in the
vicinity of the critical temperature $T_c$.) It is difficult to put
Eqs. (4)-(5) in a parameter-free form because of the additional term
in comparison to the CH equation and the nature of the static
solution, which we discuss below. The first two terms under the
integral sign in (5) are recognised as the entropy of a noninteracting
binary mixture and the next two terms correspond to the interaction
part \cite{zia}.
Eqs. (4)-(5) have the pleasant feature that they explicitly contain the
mean-field static solution $\stat$, which is the solution of
\beq
\stat = \tanh \bigg [ \frac{T_c}{T} \stat + (\frac{T_c}{T} - 1)
\n2 \stat \bigg ] ,
\eeq
where it should be kept in mind that the space variable has been
rescaled. However, we do not expect our model to be in a different
dynamical universality class from Eqs. (1)-(3). In our model, as $T
\rightarrow 0$, the saturation value of the order parameter $\phi_0
\rightarrow \pm 1$. This reduces the bulk diffusion because of the
order-parameter dependent mobility and enhances the time-regime in
which one observes surface-diffusion mediated growth. In the case
where surface diffusion is predominant, we follow the terminology
established by Hohenberg and Halperin \cite{hoh} and refer to our
model as "Model S", where S refers to surface diffusion. In the
classification of Hohenberg and Halperin, the CH equation is referred
to as Model B. {}For shallow quenches, the saturation value of the order
parameter $\phi_0$ is considerably less than 1 and the mobility
$M(\phi) ( = 1 - \phi^2)$ is not significantly reduced in the bulk. In
this limit, the dynamics of our model is in the same dynamical
universality class as Model B or the CH equation.
In this paper, we present detailed numerical results from a simulation
of (4)-(5). The purpose of this paper is two-fold. {}Firstly, our
numerical results improve substantially upon existent results
\cite{lac} for models with order-parameter dependent
mobility. Secondly, we believe that our results may be of some
relevance to an outstanding theoretical problem of phase separation
dynamics, viz., the computation of the scaling form of the
time-dependent structure factor.
Before we present numerical results, we would like to briefly discuss
the interfacial profile in our model. {}For this, we need the solution
of the 1-dimensional version of (6), viz.,
\beq
\frac{d^2 \stp}{d x^2} = -\frac{T_c}{T_c - T} \stp + \frac{T}{T_c -T}
\tanh^{-1} (\stp) .
\eeq
Multiplying both sides by $2 (d \stp / dx)$, we can trivially integrate this
equation to get
\bea
\frac{d \stp}{dx} & = & \bigg [ \frac{2T}{T_c - T} \stp \tanh^{-1}
(\stp) + \frac{T}{T_c - T} \ln \bigg ( \frac{1-\stp^2}{1-\s0^2} \bigg ) -
\nonumber \\
& & \frac{T_c}{T_c - T} (\stp^2 + \s0^2) \bigg ]^{\frac{1}{2}} ,
\eea
where we focus on the profile which goes from $-\s0$ at $x=-\infty$ to
$\s0$ at $x=\infty$. A second integration is only possible numerically
and we show the resultant profiles for $x > 0$ in {}Figure 1(a) for four
different values of $T/T_c$. This solution has the form $\stp = \s0
f(x/\xi)$ , where $f(y)$ is a sigmoidal function and $\xi$ measures
the correlation length or interface thickness in dimensionless
units. An estimate of $\xi$ is obtained as the distance over which
$f(x/\xi)$ rises from 0 to (say) $1/\sqrt{2}$ of its maximum
value. The profiles as a function of the scaled distance $x/\xi$ are
shown in {}Figure 1(b). They do not exhibit a universal collapse because
of a weak dependence of $f(y)$ on the parameter $T/T_c$. In any case,
our interest in the correlation length is primarily from a numerical
standpoint in that the discretisation mesh size in space should not
exceed the interface thickness, which is approximately $2 \xi$.
\section{Numerical results}
We have conducted extensive 2-dimensional numerical simulations of
(4)-(5) for the parameter values $T/T_c = 0.2, 0.4, 0.5$ and 0.8,
corresponding to $\s0 \simeq 0.9999, 0.9857, 0.9575$ and 0.7105,
respectively. We implement a simple Euler discretisation of (4)-(5) on
a lattice of size $N \times N$. The Laplacian and divergence operators
in (4)-(5) are replaced by their isotropically discretised
equivalents, involving both nearest and next-nearest neighbours. The
discrete implementation of our model with order-parameter dependent
mobility has the unpleasant feature that it is unstable for $\phi > 1$
and numerical fluctuations which cause $\phi$ to become larger than 1
give rise to unphysical divergences. (This property is common to all
such models \cite{lac}.) {}For $T/T_c = 0.2 (\s0 \simeq 0.9999)$, this
causes a numerical problem because of the proximity of the saturation
value to $\pm 1$. We circumvent this problem by using a very fine
mesh size ($\Delta t = 0.001$ and $\Delta x = 0.5$) and by setting the
value of $\phi$ equal to $\s0$ (or $-\s0$) whenever it exceeds $\s0$
(or becomes less than $-\s0$). We have confirmed that this procedure
does not cause any appreciable violation of order parameter
conservation for the extremely fine mesh we have used. {}For the higher
values of $T$ studied here, we use the coarser mesh sizes $\Delta t =
0.01$ and $\Delta x = 1.0$ and this suffices for our purposes.
Periodic boundary conditions are applied in both directions of our
lattice. {}For all simulations described here, the initial condition for
the order parameter consists of a uniformly distributed random
fluctuation of amplitude 0.025 about a zero background. This mimics a
critical quench from high temperatures, at which the system is
homogeneous but has small thermal fluctuations.
Apart from evolution pictures and profiles, the statistical quantity
of experimental interest is the time-dependent structure factor
\beq
S(\vec{k},t) = \la \phi (\vec{k},t) \phi (\vec{k},t)^* \ra ,
\eeq
which is the {}Fourier transform at wave-vector $\vec{k}$ of the order
parameter correlation function. In (9), $\phi (\vec{k},t)$ is the
{}Fourier transform of $\ord$ and the angular brackets refer to an
averaging over an ensemble of initial conditions. In our discrete
simulations, the wave-vector $\vec{k}$ takes the discrete values
$\frac{2 \pi}{N \Delta x} (n_x,n_y)$, where $n_x$ and $n_y$ range from
$-N/2$ to $(N/2) - 1$. We present here structure factor data obtained
on $512 \times 512$ systems as an average over 60 independent initial
conditions. The order parameter profiles are hardened before computing
the structure factor, viz., the values of $\phi > 0$ are set equal to
1 and $\phi < 0$ are set equal to -1. The structure factor is
normalised as $\sum_{\vec{k}} S(\vec{k},t)/N^2 = 1$. All results
presented below are for the spherically averaged structure factor
$S(k,t)$.
Experimentalists are typically interested in whether or not the
structure factor exhibits dynamical scaling \cite{bs}, viz., whether or
not the time-dependence of the spherically averaged structure factor
has the simple scaling form
\beq
S(k,t) = L(t)^d F(kL(t)) ,
\eeq
where $d$ is the dimensionality and $F(x)$ is a time-independent
master function. The interpretation of dynamical scaling is that the
coarsening pattern maintains its morphology but the characteristic
length scale $L(t)$ increases with time. There are many equivalent
definitions (upto prefactors) of the characteristic length scale. We
use what is perhaps the most commonly-used definition, viz., the
inverse of the first moment of the spherically averaged structure
factor $S(k,t)$. Thus, we have $L(t) = \mom ^{-1}$, where \beq \mom =
\frac{\int_0^{k_m} \mbox{d} k k \st}{\int_0^{k_m} \mbox{d} k \st} .
\eeq In (11), we take the upper cut-off $k_m$ as half the magnitude of
the largest wavevector in the Brillouin zone. At these large values of
the wavevector, the structure factor has decayed to approximately zero
and the value of $\mom$ is unchanged even if we increase the
cutoff. Of course, one could also define a length scale using higher
moments of the structure factor or zeroes of the correlation
function. However, in the dynamical scaling regime \cite{bs}, these
definitions are all equivalent.
{}Figure 2 shows evolution pictures from a disordered initial condition
for the parameter value $T/T_c = 0.2$ (or $\phi_0 \simeq 0.9999$) and
a lattice size $256 \times 256$. This low value of temperature
corresponds to a situation in which there is almost no bulk diffusion
once the order parameter saturates out to its equilibrium values. In
this case, domain growth occurs via surface diffusion and has an
associated growth law $\sd$ \cite{fur}. Notice that the domain
morphology in this case is considerably different from the morphology
in the usual CH case with the bicontinuous domains being more
serpentine and intertwined in the present case. {}Figure 3 shows the
corresponding evolution pictures from a $256 \times 256$ lattice for
$T/T_c = 0.5$ (or $\s0 \simeq 0.9575$). These pictures are more
reminiscent of the CH morphology. {}Figure 4 shows the variation of
order parameter along a horizontal cross-section at the middle of the
lattice for the evolution pictures of {}Figure 2. Figure 5 shows the
order parameter profiles corresponding to the evolution depicted in
{}Figure 3. These profiles provide a qualitative measure of the
thinning out of defects (viz., interfaces) as the coarsening proceeds.
In {}Figure 6(a), we superpose data from different times for the scaled
structure factor $\st \mom^2$ vs. $k/\mom$. The parameter value is
$T/T_c = 0.2$, corresponding to growth mediated by surface diffusion
(i.e., Model S). The structure factor data collapses neatly onto a
master curve, exhibiting the validity of dynamical scaling in this
system. The solid line refers to the scaled structure factor for the
CH equation obtained with the same system sizes and statistics as
described previously. On the scale of this figure, the scaled
structure factor for Model S is coincident with that for the CH
equation except for the first two points after $k=0$, which exhibit
violation of scaling because of finite-size effects. A similar
observation has also been made for the real-space correlation function
by Lacasta et al. \cite{lac}. However, we should stress that the
structure factor is a more sensitive characteristic of phase ordering
dynamics than the correlation function. {}Furthermore, our present data
(obtained on $512 \times 512$ systems with 60 independent runs and
$\Delta t = 0.001, \Delta x = 0.5$) constitutes a considerable
improvement over that of Lacasta et al. \cite{lac}, who used a $120
\times 120$ system with 10 independent runs and $\Delta t = 0.025,
\Delta x = 1.0$.
Before we proceed, two further remarks are in order. {}Firstly, it is
interesting that the structure factors for Model S and the CH model
are numerically indistinguishable, even though the morphologies are
different and domain growth is characterised by different power
laws. Clearly, the time-dependent structure factor (which is the
{}Fourier transform of the equal-time correlation function) is not a
sufficiently good measure of the morphology to discriminate between
these two situations and perhaps one needs to invoke other tools like
two-time correlation functions or higher-order structure factors
\cite{blu}. Nevertheless, the structure factor is an experimentally
relevant quantity and the computation of its analytic form for the CH
equation has been an outstanding problem to date. {}Furthermore, it has
been believed that a "correct" theory for the scaling form of the
structure factor must properly account for the bulk diffusion and the
LS growth law \cite{oht, maz}. However, our numerical results
demonstrate that the scaling form of the structure factor for the
conserved case is considerably robust and is not affected by the
growth exponent or the underlying growth mechanism, at least for the
model we have studied.
The second remark we wish to make concerns the dashed line in {}Figure
6(a), which is obtained from a naive application of the theory of
Mazenko \cite{maz}, who developed a Gaussian closure for the CH
equation. The naive Mazenko theory predicts that the asymptotic
growth law is $\sd$ rather than the numerically observed LS law, viz.,
$\ls$. Because of the lower growth exponent, it is presumed that the
naive Mazenko theory describes the surface-diffusion growth regime of
the CH equation. In the light of our present results, it is clear that
the form of the scaled structure factor is largely independent of the
mechanism of domain growth. Unfortunately, as is clear from {}Figure
6(a), the analytic form obtained from the naive Mazenko theory is not
correct in most respects and only gets right the approximate width of
the scaling function. We are presently investigating a Gaussian
closure of (4) to see whether it gives better results for the scaling
function.
{}Figure 6(b) plots the data of Figure 6(a) on a log-log scale and
reconfirms the coincidence of the CH and Model S scaling functions,
including the Porod tail $\st \sim k^{-3}$ for large $k$. At small
values of $k$, the scaled structure factor for Model S exhibits a
$k^4$-behaviour as in the CH case \cite{yeu}, except for the first
couple of values of $k$, which are probably affected by finite-size
effects. Again, the dashed line is from the naive Mazenko theory and
has the wrong behaviour for small values of $k$, viz., $\st \sim
k^{2}$ rather than $\st \sim k^4$. The analytic form matches the
numerical results in the Porod tail but this may be entirely
fortuitous. {}Figure 6(c) plots the data of Figure 6(a) on a Porod plot,
viz., $k^4 \st /\mom^2$ vs. $k/\mom$, which highlights features of the
Porod tail. In this case, our data is not reliable for $k/\mom \geq
2.5$. However, upto that point, the scaled form factors for the Model
S and CH cases are again indistinguishable, including the first valley
after the peak \cite{oht}.
Similar results for the scaled structure factor are found for higher
values of temperature $T$ also. This is not surprising as the morphology
for our model goes over to that for the CH equation at higher values of the
temperature (see {}Figure 3). For the sake of brevity, we do not show
structure factor data for higher values of $T$.
{}Figure 7(a) shows the time-dependent length scale $L(t)$ as a function
of dimensionless time $t$ for four different values of temperature
($T/T_c = 0.2, 0.4, 0.5$ and 0.8) in our model. Recall that surface
diffusion effects are enhanced as $T$ is lowered because $\s0
\rightarrow 1$ as $T \rightarrow 0$. {}For purposes of comparison, we
have also included the length scale data for the CH equation. {}Figure
7(b) is a log-log plot of the data in Figure 7(a). We use a fitting
routine to fit a straight line to the data. The resultant exponents
(denoted as $x$) for the CH equation and the case with $T/T_c = 0.8$
are identical, viz., $x=0.33$. On the other hand, for $T/T_c = 0.2$,
we again get a straight line but the associated growth exponent is
0.25, which is associated with domain growth via surface diffusion
\cite{fur, lac}. {}For intermediate values of $T/T_c$ (viz., 0.4 and
0.5), we do not get a good linear fit as the length scale is in a
transition regime between $\sd$ and $\ls$.
\section{Summary and Discussion}
Let us end this paper with a brief summary and discussion of
our results. We have presented detailed results from an extensive
numerical simulation of a model with order-parameter dependent
mobility. We expect this model to be in the same dynamical
universality class as other models with order-parameter dependent
mobility \cite{lan, lac} but it has the additional pleasant feature
that it explicitly contains the mean-field static solution.
Because of the large system sizes and extensive averaging
employed by us, we are able to obtain the best numerical results on
such systems to date. The salient features of our results are as
follows. In the parameter regime where surface diffusion drives domain
growth, the morphology of evolving patterns is more serpentine than
that in the CH equation. However, the scaling form of the
time-dependent structure factor for surface-diffusion mediated growth
appears to be numerically identical to that for the CH equation,
including the Porod tail and the small-$k$ behaviour. This numerical
result casts doubts on the conventional wisdom that a "correct" theory
for the scaling form of the CH structure factor must contain the
correct growth law and properly model the bulk diffusion field. As a
matter of fact, we are led to speculate that the scaling form for the
conserved case may be dictated by more general considerations, e.g.,
domain-size distributions, etc. This is an approach we are presently
pursuing in an attempt to obtain a better understanding of the
functional form of the structure factor for the conserved case.
We are also interested in examining other models of phase separation
to see whether they give rise to similar results for the scaled
structure factor. In particular, Giacomin and Lebowitz \cite{gia} have
recently studied an Ising model on a cubic lattice with Kawasaki
spin-exchange kinetics which satisfies detailed balance. The spins
interact via a long-ranged Kac interaction potential of the form
$V(r_{ij}) = \gamma^d J(\gamma r_{ij})$, where $r_{ij}$ is the
distance between spins $i$ and $j$; $\gamma$ is a parameter; and $d$
is the dimensionality. In the limit $\gamma \rightarrow 0$, Giacomin
and Lebowitz rigorously obtain an exact nonlinear evolution equation
for phase separation. Their model is of the same form as Eqs. (4)-(5)
but contains a nonlocal interaction term, instead of the gradient
square term in (5). They argue that this exact equation gives results
for interface motion which are similar to those obtained from the CH
equation. We are interested in examining whether or not this exact
equation is in the same dynamical universality class as the CH
equation.
{}Finally, we should point out that the difference in morphologies
between Model S and the CH equation must show up at some level, e.g.,
two-time correlation functions or higher-order structure factors
\cite{blu}. This is another question we are presently interested
in. Nevertheless, this possible difference in two-time correlation
functions or higher-order structure factors does not detract from the
relevance of the fact that the scaled form of the conventional
structure factor is very robust. After all, the conventional structure
factor is the primary quantity of experimental, numerical and
theoretical interest.
\section*{Acknowledegments}
SP is grateful to Alan Bray for inviting him to Manchester, where most
of the numerical calculations described in the text were completed.
He is also grateful to the Newton Institute, Cambridge, for its
generous hospitality during a period over which this work was
completed. {}Finally, he would like to thank A.-H.Machado, C.Yeung and
R.K.P.Zia for useful discusions and A.-H.Machado for sending him
copies of relevant papers. JLL and SP thank G.Giacomin for useful
discussions. JLL was supported by NSF Grant NSF-DMR 92-134244-20946.
\newpage
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\newpage
\begin{center}
{\bf {}Figure Captions}
\end{center}
\begin{itemize}
\item[{}Figure 1 :] (a) Static wall solutions of the model described in the
text (Eqs. (4)-(5)). The solutions are obtained by numerically solving
(8). We plot the profile $\stp/\s0$ vs. $x$ for $x > 0$ (where $\s0$
is the saturation value)
for four values of the temperature $T$, viz., $T/T_c = 0.2, 0.4,0.5$ and
0.8. \\
(b) Same as (a) except the distance $x$ is scaled by a
correlation length $\xi$, which is defined as the distance over which
the wall profile rises to $1/\sqrt{2}$ of its maximum value.
\item[{}Figure 2 :] Evolution pictures from a disordered initial
condition for an Euler-discretised version of (4)-(5) on a $256 \times 256$
latice. Regions with positive order parameter are marked in black and
those with negative order parameter are not marked.
The parameter value is $T/T_c = 0.2$, corresponding to a situation
in which surface diffusion is the primary mechanism of domain
growth. The discretisation mesh sizes are $\Delta t = 0.001$
and $\Delta x = 0.5$. Periodic
boundary conditions are applied in both directions. The initial
condition consists of uniformly-distributed random fluctuations of
amplitude 0.025 about a zero background, corresponding to a critical
quench. The evolution pictures are shown for dimensionless times 1000, 2000,
4000 and 10000.
\item[{}Figure 3 :] Similar to Figure 2 but for the parameter value
$T/T_c = 0.5$.
\item[{}Figure 4 :] Order parameter profiles for the evolution
depicted in {}Figure 2. The profiles are measured along a horizontal
cross-section at the centre of the vertical axis.
\item[{}Figure 5 :] Order parameter profiles for the evolution
depicted in {}Figure 3. The cross-section is the same as that for Figure
4.
\item[{}Figure 6 :] (a) Superposition of scaled structure factor
data from a simulation of (4)-(5) with $T/T_c = 0.2$, corresponding to the
surface-diffusion case. We plot $\st \mom^2$ vs. $k/\mom$ for data
from dimensionless times 2000, 3000, 4000 and 10000. The structure
factor is computed on a $512 \times 512$ lattice as an average over 60
independent initial conditions. It is normalised as described in the
text and then spherically averaged. The first moment of $\st$ is
denoted as $\mom$ and measures the inverse of the characteristic
length scale. The solid line is a scaled plot of structure
factor data from the CH
equation at dimensionless time 10000. {}Finally, the dashed line is an
analytic form obtained from a naive application of
Mazenko theory \cite{maz}, which yields
the domain growth law $\sd$. \\
(b) Plot of data from (a) on a log-log scale. The Porod tail
is extracted by hardening the order parameter field before computing
the structure factor. \\
(c) Porod plot (viz., $k^4 \st /\mom^2$ vs. $k/\mom$)
for the data from (a). This plot
highlights the features of the Porod tail. Unfortunately, our data in
this plot exhibits large fluctuations for $k/\mom \geq 2.5$.
\item[{}Figure 7 :] (a) Characteristic domain size $L(t)$ plotted as a
function of dimensionless time for our model in (4)-(5)
with $T/T_c = 0.2, 0.4,
0.5$ and 0.8. {}For comparison, we also present length scale data from a
simulation of the CH equation. The length scale is obtained as the
inverse of the first moment of the structure factor $\mom$. \\
(b) Data from (a), plotted on a log-log scale. We use a
fitting routine to fit a linear function to the length scale data.
The resultant fit (wherever reasonable) is shown on the appropriate
data set as a solid line and the corresponding exponent (denoted as
$x$) is specified on the figure.
\end{itemize}
\end{document}