LANDAU THEORY FOR SHAPE MEMORY POLYCRYSTALS
R. Ahluwalia
T. Lookman
A. Saxena and R. C. Albers
%Theoretical Division
Los Alamos National Lab
We study theoretically the viscoelastic properties of sheared binary fluids that have strong dynamical asymmetry between the two components. The dynamical asymmetry arises due to asymmetry between the viscoelastic stresses, particularly the bulk stress. Our calculations are based on the two-fluid model that incorporates the asymmetric stress distribution. We simulate the phase separation process under an externally imposed shear and compare the asymmetric case with the usual phase separation under a shear flow without viscoelastic effects. We also simulate the behavior of phase separated stable morphologies under applied shear and compute the stress relaxation.
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%\title{LANDAU THEORY FOR SHAPE MEMORY POLYCRYSTALS}
%\author{R. Ahluwalia, T. Lookman, A. Saxena and R. C. Albers\\
%Theoretical Division, Los Alamos National Lab}
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\begin{document}
\title{VISCOELASTIC PROPERTIES OF DYNAMICALLY ASYMMETRIC BINARY FLUIDS UNDER SHEAR FLOW}
\author{Vinay Dwivedi}
\affiliation{Department of Mechanical and Environmental
Engineering, University of California, Santa Barbara California 93106}
\affiliation{Theoretical Division, Los Alamos National Laboratory,
Los Alamos, New Mexico, 87545}
\author{Rajeev Ahluwalia}
\author{Turab Lookman}
\author{Avadh Saxena}
\affiliation{Theoretical Division, Los Alamos National Laboratory,
Los Alamos, New Mexico, 87545 USA}
\date{\today}
\begin{abstract}
We study theoretically the viscoelastic properties of sheared
binary fluids that have strong dynamical asymmetry between the two
components. The dynamical asymmetry arises due to asymmetry
between the viscoelastic stresses, particularly the bulk stress.
Our calculations are based on the two-fluid
model that incorporates the asymmetric stress distribution. We
simulate the phase separation process under an externally imposed
shear and compare the asymmetric case with the usual phase
separation under a shear flow without viscoelastic effects. We
also simulate the behavior of phase separated stable morphologies
under applied shear and compute the stress relaxation.
\end{abstract}
\pacs{83.60,83.50.Ax,64.75.+9}

\maketitle

\section{INTRODUCTION}
Many fluids can exhibit viscoelastic behavior, i.e. their response
to deformation is intermediate between that of solids and fluids.
For short times, the response is elastic and the stress is
proportional to the applied strain. On the other hand, in the long time limit,
a fluid like response with stress 
proportional to the strain rate is observed. The effect of 
viscoelasticity on the morphologies of phase separating
polymer solutions and blends has been recently experimentally
studied by Tanaka \cite{soltanaka,blendtanaka}. It is now well
established that due to an asymmetry between the viscoelastic
properties of the two phases, a transient network of the more
viscoelastic phase is observed. While there is some understanding
of the influence of viscoelastic effects on phase separation, little  
 is known on
how viscoelasticity effects
phase separation when an external shear flow is applied. Although
effects of shear on newtonian (purely hydrodynamic) binary fluids
have been well studied \cite{shexp,lookman} and dynamical steady
states with stringy morphologies have been observed, it is also
important to study similar situations for binary fluids where
viscoelastic effects can dominate.

Theoretically, the coupling between the stress and concentration
fluctuations was investigated by Helfand and Fredrickson
\cite {fred} in the context of shear flow in polymer
solutions. Doi and Onuki \cite{doi} studied this coupling by
introducing a two-fluid model that considers two different
velocities for the two components. Taniguchi and Onuki
\cite{Onuki} have used the two-fluid model to study viscoelastic
phase separation. However, a two-fluid model incorporating
dynamical asymmetry due to asymmetric distribution of bulk
stresses 
\cite{araki2d,araki3d,revtanaka,china1} 
has successfully explained
many
features of the viscoelastic phase separation experiments of
Tanaka, such as the transient network formation and eventual phase
inversion. Recently, this theory has also been applied to study
the viscoelastic phase separation in diblock copolymers
\cite{china2}. The two-fluid model was also used to simulate
polymeric foams under shear \cite{grant}. In this paper, we use
this model to study phase separation under a shear flow for a
binary fluid that is characterized by a strong asymmetry between
the viscoelastic moduli of the two phases, particularly the bulk
modulus.

In addition to phase separation under shear flow, the behavior of
stable phase separated states under a deformation is also
important. Recently, Buxton and Balazas \cite{buxton} have studied
the mechanical properties of phase separated morphologies using a
block and spring model. This model can only model elastic
properties and cannot describe the viscoelastic response. Thus we
study the behavior of fully phase separated stable states under a
constant shear based on the two-fluid model. The time evolution of
mechanical stresses generated for such phase separated states is
also studied.

The paper is organized as follows. In Sec. 2, we discuss the
two-fluid model used in our simulations. Section 3 describes
results on viscoelastic phase separation under shear flow while
section 4 is devoted to the response of stable phase separated
morphologies subjected to steady shear. We conclude the paper 
with a summary and discussion in Sec. 4.
\section{TWO-FLUID MODEL}
The coupling between viscoelastic stresses and diffusion is
studied using the two-fluid model. We consider a dynamically
asymmetric mixture of a viscoelastic fluid $(A)$ and a purely
hydrodynamic (non-viscoelastic) fluid $(B)$. In this model,
different velocity fields, $\vec{v_A}(\vec{r},t)$ for the viscoelastic
phase and $\vec{v_B}(\vec{r},t)$ for the non-viscoelastic fluid are
introduced. The average velocity is given by
$\vec{v}_{average}(\vec{r},t)={\phi(\vec{r},t)}\vec{v_A}(\vec{r},t)
+({1-\phi(\vec{r},t)})\vec{v_B}(\vec{r},t)$, where $\phi(\vec{r},t)$
is the concentration of the viscoelastic fluid. The free energy of
mixing can be written in terms of the concentration
$\phi(\vec{r},t)$ as
\begin{equation}
F_{mix}=\int d\vec{r}[{\phi}^2{(1-\phi)}^2 +(C/2)({\vec{\nabla}}\phi)^2],
\end{equation}
where $C$ is the concentration gradient coefficient that is assumed to be
constant in the present work. The free energy $F_{mix}$ is a
simplified version of the Flory-Huggins \cite{flory} type free
energy.  The applied shear flow is implemented by considering an average 
velocity 
$\vec{v}_{average}=\vec{v}_{applied}+\vec{v}$, where $ \vec{v}$ is the contribution to 
the average velocity 
due to 
concentration fluctuations and $\vec{v}_{applied}$ is the contribution due to 
the external 
shear flow given as, 
\begin{equation}
\vec{v}_{applied}=s|x-x_{0}|\hat{j}.
\end{equation}
This represents an external shear (symmetric about $x_{0}$) 
applied along the  $y$ direction and $s$ is the shear rate. 
The appropriate equations of motion for incompressible
viscoelastic binary fluids under external shear flow are 
\begin{equation}
{{\partial \phi}\over{\partial t}}+\vec{v}\cdot{\vec{\nabla}{\phi}}
+s|x-{x_{0}}|
{{\partial \phi}\over{\partial y}}
={\vec{\nabla}}\cdot{ {{\phi}(1-\phi)^2}\over{\zeta} }
[\vec{\nabla}\cdot{\stackrel{\leftrightarrow}\Pi}
-\vec{\nabla}\cdot{\stackrel{\leftrightarrow}\sigma}].
\end{equation}
Here $\stackrel{\leftrightarrow}\sigma$ represents the
viscoelastic stress tensor.
The osmotic tensor
$\stackrel{\leftrightarrow}\Pi$ is defined as 
$\vec{\nabla}\cdot
\stackrel{\leftrightarrow}\Pi=
\phi\vec{\nabla}\{2{\phi}(1-\phi)^2-2(1-\phi){\phi}^2
-C{\nabla}^2{\phi}\}$. The quantity $\zeta$ is the friction
coefficient associated with the velocity $\vec{v_A}-\vec{v_B}$.
The dynamics of the average velocity field $\vec{v}$ is given by the usual
Navier-Stokes equation
\begin{equation}
\rho
{{\partial \vec{v}}\over{\partial t}}=
-\vec{\nabla}\cdot \stackrel{\leftrightarrow}\Pi+\vec{\nabla} P+
\vec{\nabla}\cdot\stackrel{\leftrightarrow}\sigma +
{\eta_s}{\nabla}^2{\vec{v}},
\end{equation}
where a pressure has been introduced to account for the
incompressibility constraint. If we use the overdamped limit
${{\partial \vec{v}}\over{\partial t}}=0$, then under the
incompressibility condition $\vec{\nabla}\cdot{\vec{v}}=0$, the
velocity can be expressed in Fourier space as
\begin{equation}
\vec{v_k}={\stackrel{\leftrightarrow}T_k}\cdot(-\vec{\nabla}\cdot\stackrel{\leftrightarrow}\Pi+
\vec{\nabla}\cdot\stackrel{\leftrightarrow}\sigma)_{k},
\end{equation}
where $()_k$ represents the Fourier transform and $T_{k}$ is a tensor defined in
fourier space as
\begin{equation}
{\stackrel{\leftrightarrow}T_k}={{1}\over{{\eta_s}{k^2}}}
\bigg[ \stackrel{\leftrightarrow}I-{{\vec{k}\vec{k}}\over{k^2}}\bigg],
\end{equation}
where $\stackrel{\leftrightarrow}I$ represents the unit tensor.
Next, we have to specify a constitutive law for the viscoelastic
stresses. We use the Maxwell model according to which the equation
of motion for the stresses is given by 
\begin{eqnarray}
{{\partial \stackrel{\leftrightarrow}\sigma_b}\over{\partial t}}
+{{\stackrel{\leftrightarrow}\sigma_b}\over{{\tau_b} (\phi)}} &=&
m_b(\phi)\stackrel{\leftrightarrow}M,
\nonumber \\
 {{\partial
\stackrel{\leftrightarrow}\sigma_s}\over{\partial t}}
+{{\stackrel{\leftrightarrow}\sigma_s}\over{{\tau_s} (\phi)}} &=&
m_s(\phi)\stackrel{\leftrightarrow}M,
\end{eqnarray}
where $\tau_b$ and $\tau_s$ are the internal molecular relaxation
times associated with the bulk and shear stresses respectively.
The matrix $\stackrel{\leftrightarrow}M$ is given as
\begin{equation}
\stackrel{\leftrightarrow}M=\left[ \begin{array}{cc} 2{{\partial
{v_A}_{x}}\over{\partial x}} & ({{\partial
{v_A}_{x}}\over{\partial y}}+ {{\partial
{v_A}_{y}}\over{\partial x}})\\
({{\partial {v_A}_{x}}\over{\partial y}}+ {{\partial
{v_A}_{y}}\over{\partial x}})& 2{{\partial
{v_A}_{y}}\over{\partial y}}\end{array} \right].
\end{equation}
The velocity $\vec{v_A}$ represents the velocity of the
viscoelastic phase defined as
\begin{equation}
\vec{v_A}=\vec{v}+\vec{v}_{applied}-
{{(1-\phi)^2}\over{\zeta}}\left[\vec{\nabla}
\cdot\stackrel{\leftrightarrow}{\Pi}
-\vec{\nabla}\cdot\stackrel{\leftrightarrow}\sigma\right].
\end{equation}
The above definition of $\stackrel{\leftrightarrow}M$ assumes that
the stress acts on the viscoelastic fluid only. 
The final stresses $\stackrel{\leftrightarrow}{\sigma_b}^{f}$ and
$\stackrel{\leftrightarrow}{\sigma_s}^{f}$ are calculated as
$\stackrel{\leftrightarrow}{\sigma_b}^{f}=
(1/2)Tr(\stackrel{\leftrightarrow}{\sigma_b})
{\stackrel{\leftrightarrow}{I}}$
and 
$\stackrel{\leftrightarrow}{\sigma_s}^{f}=
\stackrel{\leftrightarrow}{\sigma_s}-
(1/2)Tr(\stackrel{\leftrightarrow}{\sigma_s})\stackrel{\leftrightarrow}{I}$.
The total stress is calculated as
$\stackrel{\leftrightarrow}{\sigma}=
\stackrel{\leftrightarrow}{\sigma_b}^{f}+
\stackrel{\leftrightarrow}{\sigma_s}^{f}$. The model decribed in
equations $(1)$ to $(9)$ can be used to simulate the viscoelastic phase
separation under shear flow with shear rate $s$.
\section{VISCOELASTIC PHASE SEPARATION UNDER SHEAR FLOW}
We describe our results on simulations of
viscoelastic phase separation, both with and without shear. The
model described in Sec. II is discretized on a
$128 \times 128$ grid with periodic boundary conditions, using a 
spectral approach for the velocity equation and explicit, central 
difference schemes $(\Delta x=1,\Delta t= 0.005)$ for the rest. 

Following Tanaka and Araki \cite{araki2d,araki3d}, we choose
$m_b={m_b}^{0}\theta(\phi-{\phi_0})$ ($\phi_0$ is the initial
concentration before quenching), $m_s={m_s}^0{\phi}^2$,
${\tau_b}={\tau_b}^0{\phi}^2$ and ${\tau_s}={\tau_s}^0{\phi}^2$.
With this choice of the moduli and time scales, Tanaka and Araki
were able to reproduce  a number of experimental features of viscoelastic
phase separation \cite{araki2d,araki3d}.

We first simulate phase separation without an external
shear$(s=0)$. We consider a $50-50$ mix 
with initial concentration fixed at  $\phi_0=0.5$. The field
$\phi(\vec{r},0)$ is initialized by small fluctuations of order 0.001 around
$\phi_0$ and the model in Sec. II is simulated for three
different cases. Case I corresponds to hydrodynamic phase
separation in the absence of viscoelastic effects. Here we set all the
stresses to zero and solve Eqs. (3) and (5) with only one
velocity field $\vec{v}$. Case II is the full asymmetric
viscoelastic model with $\eta_s=1$, $\zeta=0.1$, ${m_b}^0=5$, ${m_s}^0=0.5$, 
${\tau_b}^{0}=10$ and ${\tau_s}^{0}=50$. To clarify the role of the bulk stress, in case III we
switch off the bulk modulus, i.e., ${m_b}^{0}=0$.

Figure 1 shows the phase separation process for the three cases.
The left column shows the non-viscoelastic (hydrodynamic) case,
middle column depicts the full viscoelastic model with asymmetric
bulk modulus and the right column represents the phase separation
for viscoelastic case without bulk modulus. This figure shows the
importance of the bulk modulus in the formation of a network of
the more viscoelastic component. For the case with the bulk modulus,
initially, holes of the non viscoelastic phase are formed in a
partially phase separated matrix of the more viscoelastic phase.
As these holes grow, the area of the viscoelastic region
decreases and a network like structure is formed. These results
are consistent with earlier results of Tanaka and Araki
\cite{araki2d}. 
For the full viscoelastic model, we do not
observe the breaking of the network within the time interval
we simulated. The case without the bulk modulus appears to be
similar to the purely hydrodynamic case and there is no network
formation. Figure 1 also shows that viscoelastic effects slow down
the domain growth, as can be inferred by comparing the domain
patterns at time $t=1000$ for all the three cases.

We have also monitored the evolution of the viscoelastic stresses
during the phase separation. Figure 2 shows the behavior of the average
normal stress $\sigma_{xx}-\sigma_{yy}$, the average bulk stress
$\sigma_{xx}+\sigma_{yy}$ and the average shear stress $\sigma_{xy}$ for
the case with bulk stress as well as that for the case without bulk
stresses. The normal and shear stresses do not decay completely in
the time simulated and  exhibit small fluctuations of order $0.001$, possibly due to the
motion of the viscoelastic phase. The behavior of the bulk stress
is worth noticing as for the case with a bulk modulus, the early
stage of phase separation is associated with significant
compressive stresses. The compressive stresses are
associated with the formation of the network of the more
viscoelastic phase. The network of partially phase separated
viscoelastic phase shrinks as the phase separation proceeds
resulting in the compressive stresses observed in the early
stages.

In Fig. 3, we display the phase separation process when an
external shear with shear rate $s=0.01$ is applied. As in the case
without shear, we consider three cases corresponding to the
hydrodynamic case without viscoelastic effects, full viscoelastic
model with bulk modulus and viscoelastic model without bulk
modulus. At this shear rate, the very early stages are not
influenced significantly by the applied shear for all the cases.
At later times, there is an underlying tendency for the
domains to align along the shear direction, although the alignment
is not complete within the time simulated $(t=2500)$. The case
with the bulk modulus exhibits network morphologies in the earlier
stages even for this case. The alignment is also slower in the
initial stages for the case with the bulk modulus.

The time evolution of viscoelastic stresses for this value of
shear is shown in figure 4 for the case with bulk stress as well as
case without bulk stress. Fluctuations due to random motion of
domains in the early stages are observed for both normal and shear
stresses and the amplitude of the fluctuations decreases at late
times as the domains tend to align along the flow direction. For
this case also, the bulk stress for the case with bulk modulus becomes
negative in the early stages, indicating the shrinking of the
network of the viscoelastic phase.

Finally, we consider phase separation under a relatively large
shear rate $s=0.1$. Figure 5 shows the phase separation at this
shear rate for all three cases. For this case, the external shear
effects dominate the phase separation and the tendency for the
domains to align along the shear to form stringy patterns is much
stronger than the earlier case. The purely hydrodynamic model
rapidly forms the stringy phase, whereas for the viscoelastic
cases, the intermediate stages are characterized by complex motion of
the domains.  For the case with the bulk modulus, the early stages
do not show the formation of a well developed network as the shear
dominates over the internal dynamics. The snapshots at $t=2500$ show
that a stringy phase seems to form at long times for the
viscoelastic cases also, although the width of the lamellae is
larger compared to the hydrodynamic case.

Figure 6 shows the time evolution of the viscoelastic stresses for
this case. After transient fluctuations, the normal stresses decay
to zero as the steady state is established. Compressive bulk
stresses are observed even for this case but the magnitude is 
smaller for this case, compared to the earlier cases. This is due
to the fact that the shear flow is so fast that the transient
network of the viscoelastic phase is not allowed to fully develop.
This is clear from the snapshot at time $t=50$ for the
viscoelastic case. The behavior of the shear stresses is
interesting. At long times, the shear stresses
saturate to non zero values. The saturation stresses are related
to the effective shear viscosities for this binary fluid.
\section{PHASE SEPARATED STABLE STATES UNDER SHEAR}
So far we have considered the effects of shear on binary
fluids undergoing phase separation. In this section we apply
shear on stable phase separated morphologies. The aim of this
section is to explore the possibilities of using this model to
study mechanical properties of polymer solutions and blends.
Recently, Buxton and Balazs \cite{buxton} have used a block and
spring model to simulate deformation of a random two-phase
morphology obtained from a Cahn-Hilliard simulation. In this
approach, there is no coupling between the morphologies and the
deformation. Such an approach can only be used to study linear
elastic behavior and does not describe the full viscoelastic
response. The morphologies cannot evolve in response to
deformation. However, the two-fluid framework used in this paper
explicitly incorporates a coupling between the stress and the
concentration and can describe deformation induced morphological
evolution.

Here, we consider a simple phase separated morphology
below the coexistence temperature. We consider a $50-50$
$(\phi_0=0.5)$ binary fluid that exists as a band of the fluid
$A$ sandwiched between layers of fluid $B$. This is a stable
phase separated configuration. Polymer blends can show such 
macro-phase separation in equilibrium and block copolymers also 
 micro-phase separate into lamellar morphologies \cite{hamley}.
We apply shear in the same way as
discussed in the previous section. We consider two different
shearing conditions. In one case the shear is normal to the $A/B$
interface and in the other case, the shear is parallel to the
interface.  Such a situation (for block copolymers) has been investigated  
experimentally 
\cite{nature} and theoretically using hydrodynamic models \cite{drolet}. 
The morphological evolution for the case of shear
normal to the $A/B$ interface is depicted in Fig. 7. All three
cases that have been discussed in the previous section are simulated for this case
also. For all the cases, initially the interface starts to move in
response to the shear and at long times there is a tendency to
form many bands aligned along the shearing direction. The bands
appear to be wider for the viscoelastic cases in comparison to the
hydrodynamic case. However, we do not find much difference between
the case with bulk modulus and the case without bulk modulus.
Interestingly, for the case when the shear is applied parallel to
the interface, the band remains stable and no morphological
evolution is observed for all the three cases. This is consistent with the 
shear induced reorientation of lamella for block copolymers observed in experiments \cite{nature} 
and theoretical models \cite{drolet}.

Figure 8 shows the evolution of stresses for the full viscoelastic
model (with bulk modulus) for both cases (shear parallel and
perpendicular to the interface). For shear parallel to the
interface, the normal and bulk stresses remain zero during the
shearing process. This is due to the fact that there are no
morphological changes and only shear stresses are generated. For
the case of shear normal to the interface,  normal as well as bulk
stresses are generated due to shear induced motion of the
interface depicted in Fig. 7. For a short interval, weak
compressive stresses are observed. The early time behavior of
the shear stress also differs from the case when the shear
is applied parallel to the interface. This figure demonstrates the
role played by the morphological evolution on the viscoelastic
properties of binary fluids. For the case without bulk modulus, very 
similar stress evolution (not shown here) is observed. However, no bulk 
stress is generated.

Finally, we study the relaxation of the morphologies after the
applied shear is suddenly removed. We remove the shear at $t=50$
corresponding to the morphologies shown in the bottom row of
Fig. 7. Fig. 9 shows the final morphologies at $t=2150$ after
removing the shear. It is clear that the morphological relaxation
is much faster for the non-viscoelastic case and the band
reappears, although it is shifted from the original position. For
the viscoelastic cases, the morphological relaxation is very slow
and the pattern does not evolve to the original band within the
time span of the present simulations.
\section{SUMMARY}
We have investigated the viscoelastic properties of
dynamically asymmetric binary fluids under shear flow. The
dynamical asymmetry arises as the binary fluid constitutes one
viscoelastic fluid and the other non-viscoelastic purely
hydrodynamic fluid. We have simulated phase separation for this
binary fluid both with and without shear. Only for the case with a
nonzero bulk modulus, there is a tendency to form a network of
the more viscoelastic phase in the initial stages of phase
separation, both with and without shear flow. However, for the
high shear rate, the network phase is short lived as the shear has
a tendency to suppress network formation. For the high shear rate,
in the long time limit, a stringy phase is observed even for the
viscoelastic cases, however the length scales associated with the
stringy phase are larger compared to the purely hydrodynamic case.
Thus, viscoelastic effects enhance the extent of phase separation
under shear. We have also studied the temporal evolution of the
effective viscoelastic stresses during the phase separation
process. Interfacial motion results in fluctuations of the shear
and normal stresses in the early stages. Transient compressive
stresses are also observed corresponding to shrinking of
viscoelastic phase in the early stages. At long times, the shear
stresses saturate to a nonzero value. This value is related to
the effective viscosity of the two-phase fluid.

We have investigated the effects of shear on stable phase
separated morphologies. A lamellar structure of the binary fluid
below the coexistence temperature is sheared, both normal and
parallel to the interface. There are crucial differences between
these two cases. For the shear parallel to the interface, no
morphological evolution is observed while for the case of shear
normal to the interface, the interface moves and splits into bands
that tend to align along the shear. The evolution of effective
stresses also depends on the direction of the interface. For
example, for the shear parallel to the interface, no  bulk and 
normal stresses are generated. On the other hand, for the shear
normal to the interface, bulk, normal and shear stresses are
generated due to interfacial motion.

This paper demonstrates the role played by viscoelasticity on
shearing of binary fluids. The simulations show that the
underlying viscoelastic properties can significantly influence the
shearing behavior. Further experiments are needed to test
predictions of our simulations. The approach used in the
present work can also be applied to other complex fluids such as
block copolymers and microemulsions.
\begin{acknowledgments}
V.D. is grateful to the LANL-UCSB CARE program for providing 
support through a summer student stipend. This work was 
supported by the US Department of energy.
\end{acknowledgments}
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\newpage
Figure Captions:

1. Phase separation process without shear flow for the purely 
hydrodynamic case, 
viscoelastic case with bulk modulus and viscoelastic case without 
bulk modulus.
  
2. Evolution of the normal, bulk and shear stresses for the 
situation depicted in Fig. 1. The stresses are calculated by averaging 
the local stresses over the entire system. 

3. Phase separation process with shear flow of shear rate 
$s=0.01$ for the purely 
hydrodynamic case, 
viscoelastic case with bulk modulus and viscoelastic case without 
bulk modulus. 
Shear is applied along the $y$ direction and periodic 
boundary conditions are assumed.
  
4. Evolution of the normal, bulk and shear stresses for the 
situation depicted in Fig. 3.

5. Phase separation process with shear flow of shear rate 
$s=0.1$ for the purely 
hydrodynamic case, 
viscoelastic case with bulk modulus and viscoelastic case without 
bulk modulus.  
Shear is applied along the $y$ direction and periodic 
boundary conditions are assumed.

6. Evolution of the normal, bulk and shear stresses for the 
situation depicted in Fig. 5.

7. Evolution of the band morphology under a shear flow of shear rate 
$s=0.1$ for the purely 
hydrodynamic case, 
viscoelastic case with bulk modulus and viscoelastic case without 
bulk modulus.  
Shear is applied along the $y$ direction and periodic 
boundary conditions are assumed.  

8. Evolution of the normal, bulk and shear stresses for the 
 case with nonzero bulk modulus (evolution depicted in the middle column of Fig. 7).

9.  Morphologies for the three cases at $t=2150$ time steps after the
 shear flow has been removed. The shear flow was removed at $t=50$ steps after  the shear was applied (corresponding to the last row in Fig. 7). 
\end{document}
\newpage
\begin{center}
PLAN OF THE TALK
\end{center}
INTRODUCTION
\begin{itemize}
\item Martensitic Transformations
\item Polycrystals
\end{itemize}
THEORETICAL MODELING
\begin{itemize}
\item Symmetry adapted strain components
\item Ginzburg-Landau elastic free-energy functional with coupling to grain orientations
\item Elastic Compatibility
\item Equations of Motion
\end{itemize}
SIMULATIONS
\begin{itemize}
\item Grain growth in the austenite phase
\item Dynamics of martensitic transformation in the polycrystal
\end{itemize}
CONCLUSIONS
\newpage
\begin{center}
INTRODUCTION
\end{center}
Martensitic Transformations:
\begin{itemize}
\item First order, displacive, solid-solid structural phase transformation. Example: Cubic to Tetragonal transition in FePd Alloys.
\item Strain accomodation results in a multi-domain twinned domains of the crystallographic variants of the product phase. Domain wall orientations are determined by the symmetry and compatibility conditions.
%\vskip3.5cm
\end{itemize}
\newpage
Polycrystals:
\begin{itemize}
\item Polycrystals have large number of grains with different crystallographic orientations, separated by grain boundaries.
\item Martensitic transformation for polycrystals is more complex than that for single crystals. Domain wall orientations vary from grain to grain. Grain boundaries play a crucial role.
\end{itemize}
\newpage
Models of grain growth:
\begin{itemize}
\item Monte-Carlo simulations of large Q Potts models (Sahni et. al.,
PRL (1983))
\item Phase-Field models (Chen et. al., PRB (1994), Morin et. al., PRL (1995))
\end{itemize}
\vskip3cm
Important Issues:
\begin{itemize}
\item Need models that couple elasticity to grain growth to see how strain fields distribute at grain boundaries.
\vskip0.1cm
\item Need to include non-linear elasticity to study martensitic transformations in polycrystals
\item Role of grain boundaries in martensitic nucleation
\end{itemize}
\newpage
\begin{center}
THEORETICAL MODELLING
\end{center}
SYMMETRY ADAPTED STRAIN COMPONENTS FOR A SQUARE TO RECTANGLE TRANSITION IN A POLYCRYSTAL
\begin{itemize}
\item Bulk strain
\vskip0.1cm
${e_{1}}=(\eta_{xx}+\eta_{yy})/{\sqrt{2}}$
\item Deviatoric strain
\vskip0.1cm
${e_{2}}=[(\eta_{xx}-\eta_{yy})\cos(2\theta)
+2\eta_{xy}\cos(2\theta)]/{\sqrt{2}}$
\item Shear strain
\vskip0.1cm
${e_{3}}=[-(\eta_{xx}-\eta_{yy})sin(2\theta)
+2\eta_{xy}\cos(2\theta)]/{2}$
\end{itemize}
Here $\eta_{ij}$ represents the strain tensor in the global coordinate system and $\theta$ represents the grain orientation.
\newpage
FREE-ENERGY FUNCTIONAL  FOR SQUARE TO RECTANGLE TRANSITION

\begin{eqnarray}
F=\int d\vec{r}  \big[f_{op} + f_{cs} + f_{grain} + f_{ss} \big] \nonumber
\end{eqnarray}

\underbar{ORDER PARAMETER}

\begin{eqnarray}
f_{op}=\tau{{e_2}^2}-
2{e_2}^4 +
{e_2}^6 +
{{a}\over{2}}{(\vec{\nabla} e_{2})}^2
\nonumber
\end{eqnarray}

\underbar{COMPRESSION-SHEAR}

\begin{eqnarray}
f_{cs}={{A_1}\over{2}}{e_{1}}^2 +
{{A_3}\over{2}}{e_{3}}^2
\nonumber
\end{eqnarray}
\underbar{GRAIN}

\begin{eqnarray}
f_{grain}={\tau_g}|Z|^{2}+|Z|^{4}+{{b}\over{2}}({Z}^N+{Z^*}^N)+
{{K}\over{2}}(\vec{\nabla}{Z})^2
\nonumber
\end{eqnarray}
$Z=|Z|exp(i\theta)$ is a complex order parameter representing the grain orientation $\theta$, $N$ is number of orientations present.
\newpage
\underbar{SURFACE STRESS}
\begin{eqnarray}
f_{ss}={\alpha}{e_1}(\vec{\nabla}{Z})^2
\nonumber
\end{eqnarray}
\underbar{ELASTIC COMPATBILITY}
\begin{itemize}
\item The strain fields $e_1$, $e_2$ and $e_3$ are related to each other by
the elastic compatibility relation
\begin{eqnarray}
{{\nabla}^2}{e_1}-({{\partial^2}\over{\partial x^2}}
-{{\partial^2}\over{\partial y^2}}){\Gamma_2}-\sqrt{8}
{{\partial^2}\over{ {\partial x}{\partial y} }}{\Gamma_3}=0 \nonumber
\end{eqnarray}
\begin{eqnarray}
\Gamma_2={e_2}\cos(2\theta)-\sqrt{2}{e_3}sin(2\theta)
\nonumber
\end{eqnarray}
\begin{eqnarray}
\Gamma_3={e_3}\cos(2\theta)+{{{e_2}sin(2\theta)}\over{\sqrt{2}}}
\nonumber
\end{eqnarray}
\end{itemize}
\underbar{LONG-RANGE INTERACTION}
\begin{eqnarray}
F_{cs}=\int d\vec{k}\left[{{A_3}\over{2}}|e_3(\vec{k})|^2 +
{{A_1}\over{2}}|C_2(\vec{k})\Gamma_2(\vec{k})
+C_3(\vec{k})\Gamma_3(\vec{k})|^{2}\right]
\nonumber
\end{eqnarray}
\begin{eqnarray}
C_{2}={{{k_x}^2-{k_y}^2}\over
{{k_x}^2+{k_y}^2}}\nonumber
\end{eqnarray}
\begin{eqnarray}
C_{3}={{\sqrt{8}{k_x}{k_y}}\over
{{k_x}^2+{k_y}^2}}\nonumber
\end{eqnarray}
\newpage
EQUATIONS OF MOTION
\begin{itemize}
\item Over damped time dependent Ginzburg-Landau equation for grain growth
\begin{eqnarray}
{{\partial Z}\over{\partial t}}=-{{\delta F}\over{\delta Z^*}}\nonumber
\end{eqnarray}
\item Equations of motion for deviatoric and shear strains, obtained using Lagrangian formalism by including inertial effects and dissipation
\begin{eqnarray}
{{ \partial^2 e_2} \over {\partial t^2} }
={\nabla}^2\left[{{\delta F}\over{\delta e_2} }+{\gamma_2}{{\partial e_2}\over
{\partial t}}\right]\nonumber
\end{eqnarray}
\begin{eqnarray}
{{ \partial^2 e_3} \over {\partial t^2} }
={\nabla}^2\left[{{\delta F}\over{\delta e_3} }+{\gamma_3}{{\partial e_3}\over
{\partial t}}\right]\nonumber
\end{eqnarray}
\end{itemize}
\newpage
\begin{center}
SIMULATIONS
\end{center}

TECHNIQUE

\begin{itemize}
\item Discretize using finite differences for both space and time variables
\item Periodic boundary conditions
\item Compute long-range term by using pseudo-spectral (fourier transform) method
\end{itemize}

SIMULATIONS
\begin{itemize}
\item We prepare system in a polycrystalline austenite phase by evolving the system of equations corresponding to a temperature in the austenite phase. We start
from a state with random distribution of orientations, with small amplitude of the grain growth order parameter.
\item After obtaining a well developed grain pattern, we quench into the martensitic phase to observe nucleation and growth of martensitic domains
\end{itemize}
\newpage
\begin{center}
CONCLUSIONS
\end{center}
\begin{itemize}
\item Elastic Stresses are localized at the grain boundaries in the austenite phase, particularly at the corners.
\item In the martensite phase, the corners act as nucleation sites for the martensite domains.
\item Domain wall orientations in the martensite phase depend on the grain orientations.
\end{itemize}

FUTURE WORK

\begin{itemize}
\item Thermal cycling of the transformations
\item Stress-Strain curves and the shape memory effect
\item Extension of current model to other ferroic systems such as
ferroelectrics.
\end{itemize}
\end{document}
\vskip3cm

\begin{abstract}
\end{abstract}

\pagebreak
\section*{Introduction}
Phase separation phenomena in Binary mixtures have been the subject of much recent
research in condensed matter physics \cite{jd}. In a typical phase separation experiment,
a binary mixture ( such as an alloy, polymer blend or a binary liquid mixture)
is quenched from it's one phase region into a region of it's phase diagram where the
constituent phases like to segregate. The subsequent dynamics consists of the formation
and growth of domains which are rich in either of the phases. It is now well
established that for
mixtures whose constituent phases have identical dynamical
properties, the domain growth satisfies the dynamical scaling hypothesis in the late stages.
According to this hypothesis, the equal time structure factor of the appropriate
order parameter satisfies the scaling law
\begin{equation}
\label{scal1}
S(\vec{k},t)=L(t)^{d}
F(kL(t)),
\end{equation}
where $F(x)$ is a scaling function and $L(t)$ is a time dependent length scale which can
be associated with the mean size of the growing domains. The dynamical scaling implies
that the evolution of domains is self-similar, i.e. domain size
grows but the overall morphology does not change with time. The other interesting aspect
is
the functional form of the length scale $L(t)$. It is now conclusively established that
for pure and isotropic systems, $L(t) \sim t^{\phi}$, where $\phi$ is the growth exponent
which crucially depends on the nature of the dynamics. For example in the case of a
binary alloy where
 there is no intrinsic dynamical asymmetry between the two phases, the growth is driven by
 surface tension and is characterized by an exponent $\phi=1/3$ \cite{jd}. This is
commonly refered
to as the Lifshitz-Slyozov law which also describes domain growth in polymer solutions and
blends for shallow quenches. For binary liquids the hydrodynamic interactions are
important and for this case, the growth exponent $\phi=1$ \cite{hyd}.

Recently, there have been some experiments investigating the role of dynamical asymmetry
between the constituent phases of the phase separating system. The dynamical asymmetry
usually arises when the characteristic relaxation times of the molecules of the coexisting
phases are different. Tanaka has studied phase separation in deeply quenched
semi-dilute polymer
solutions \cite{tanaka1} where the asymmetry arises due to the viscoelastiy of the
polymer rich domains. In another interesting experiment, Tanaka \cite{tanaka2}has
investigated domain growth in a polymer blend which is quenched to a temperature which is
close to the glass transition temperature of the minority species. The common feature of
these systems is that the time scales of molecular motion of the minority phase
are much
slower relative to the other phase. This leads to unusual phase separation which is now
commonly refered to as viscoelastic phase separation.

The main features of viscoelastic phase separation are as follows. After an initial
period in which no growth takes place, domains of the more mobile majority phase start
nucleating.These majority phase domains grow resulting in the thinning of minority phase
regions which form a spongelike percolating network (This is in contrast to usual phase
separation where the minority phase forms isolated droplets). The shrinking continues
until the network breakes and isolated droplets of the minority phase start appearing.
This is known as phase inversion.

On the theoretical side, Taniguchi and Onuki \cite{Onuki} have simulated a viscoelastic
model which incorporates the coupling between stress and diffusion for a semi-dilute
polymer solution and demonstrated the existence of spongelike
network. However, they were not able to see the phase inversion (breaking of the
network)
within the time scales of their simulations. Subsequently, Tanaka and Araki
\cite{tanaka3} simulated a
viscoelastic model with the effects of bulk stress included. Using this model, they were
able to demonstrate most of the experimentally observed features like the formation of the
minority phase network which eventually breakes down leading to phase inversion.

Although, the viscoelastic models are crucial to explain the experimental observations of
Tanaka, dynamical asymmetry can also be studied in framework of the usual Cahn-Hilliard
theory of phase separation by making use of an order parameter dependent mobility.
Sappelt and Jackle \cite{Jackle} have studied domain growth in a Cahn-Hilliard system with
a mobility function that is asymmetric about a fixed concentration \cite{Jackle}.
Although they found an
unusual growth mechanism for concentrations where the less mobile phase is the majority
phase, they could not show the existence of a spongelike network of the minority phase.

In this paper, we study dynamically asymmetric phase separation within the framework of
Cahn-Hilliard theory by choosing an appropriate mobility function. We propose a model with
an order parameter dependent mobility which can model many of the features observed in
Tanaka's experiment, atleast from the point of view of pattern formation. Unlike
the viscoelastic theories, we do not incorporate stress fields and the dynamics in our
model is driven by surface tension only. The effect of dynamical asymmetry comes from the
mobility function.

The organization of this paper is as follows. In section II, we introduce
our dynamical model. We also explain the modelling of the order parameter dependent
mobility and it's connection with the experiments of Tanaka. In section III, we give
numerical results for pattern evolution. We also show results for the domain growth law
and the time-dependent structure factor. Section IV is devoted to a discussion of the
results and the limitations of the model.
\section*{Dynamical Model}
The theory is formulated in terms of an order parameter which in context
of binary mixtures is the concentration difference between the two species. Since the
concentration difference is a conserved quantity, the time evolution of a scaled
dimensionless
order parameter $\phi(\vec{x},t)$ is described by the equation
\begin{equation}
\label{cahn}
{{\partial \phi(\vec{x},t)}\over{\partial t}}=\vec{\nabla}\cdot
\left[M(\phi(\vec{x},t))\vec{\nabla}\left(-\phi(\vec{x},t)
+{\phi(\vec{x},t)}^3-{\nabla}^2{\phi(\vec{x},t)}\right)\right],
\end{equation}
where $\vec{x}$ and $t$ are respectively are the scaled space and time variables and
$M(\phi)$ is the mobility function. This is the deterministic Cahn-Hilliard equation
which is also refered to as the Model B in the Halperin and Hohenberg system of
classification \cite{Halp}. In conventional theories of spinodal decomposition,
the mobility functiom $M(\phi)$ is usually treated as a constant. However, recently
there have been some studies where the effect of an order parameter dependent
mobility on the dynamics of phase separation has been investigated \cite{puri}.

In this paper, we consider a mobility function of the type
\begin{equation}
\label{mob}
M(\phi)={{1}\over {1+exp(\alpha\phi-\beta{\phi}^2)}},
\end{equation}
where $\alpha$ and $\beta$ are positive constants $(\beta > \alpha)$. In figure (1),
we plot this function for the case $\alpha=100$ and $\beta=160$. In the early stages of domain growth $\phi$ is small and the mobility is a sharply peaked step function
about $\phi=0$. However, as the system orders and $\phi$ grows, because of the negative
quadratic term the  asymmetry in the mobility weakens and eventually disappears
beyond a value ${\phi}^{*}=\alpha/\beta$. The negative
quadratic term provides a competing effect on the dynamical asymmetry, analogous
to the weakening of stress fields in the late stages that was observed in
Tanaka's experiments \cite{tanaka2}.

\section*{Numerical Results}
In this section, we give details of our numerical simulations of phase separation
for an off-critical quench using the above described model.
 We solve equation (\ref{cahn}) with the mobility function given in equation
(\ref{mob}) on an $N \times N $ square lattice with periodic boundary
conditions. A simple Euler discretization is used with mesh size $\Delta x=1.2$ and the
smallest time step $\Delta t=0.02$.
The initial condition are given
by
\begin{equation}
\label{init}
\phi(\vec{r},0)=\bar{\phi} +\delta\phi(\vec{r},0),
\end{equation}
where $\bar{\phi}$ is the off-criticality and $\delta\phi(\vec{r},0)$ represents
random fluctuations
uniformly distributed in the interval $[-0.005,0.005]$.
In all simulations reported in this paper, we choose $\bar{\phi}=-0.1$ which corresponds
to a minority phase concentration of $0.45$.

We first describe our results on pattern evolution.
We consider two  different quenches
corresponding
to $(\alpha=100,\beta=130)$ and $(\alpha=100,\beta=160)$. In figure 2 we display the
evolution of domains corresponding to $\bar{\phi}=-0.1$, $\alpha=100$ and $\beta=160$.
The darker contrast
regions correspond to the minority phase $(\phi > 0)$ and the brighter regions correspond
to the majority phase (The shade varies with the extent of order which is characterized
by the value of the order parameter). In the earliest time shown in the figure, domains
of the more mobile majority start appearing in a matrix of partially ordered minority
phase.
\newpage
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%\end{center}
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\newpage
\center{Figure Captions}
\begin{itemize}
\item[Figure 1:] Temporal evolution of a disordered initial condition for
the model described in the
text. The system of interest is a
binary mixture AB with either surfactants (S) or vacancies (V) on the
``links'' between the ``sites'' of A and B. The initial condition for the
AB field
$\langle\sigma_k \rangle$ consists of uniformly distributed random
fluctuations of
amplitude $0.05$ around a zero background. The initial condition for the
SV field $\langle S_{kL_k} \rangle $ has similar fluctuations about a
background $c=-0.45$. Evolution pictures are shown at times
800, 1600, 3200 and 8000, depicted on the appropriate frame. Sites with
positive $\langle \sigma_k \rangle$ are marked in black and sites with negative
$\langle \sigma_k \rangle$ are not marked. Similarly, links with positive
$\langle S_{kL_k} \rangle$
are marked by crosses.
\item[Figure 2:] Order parameter profiles for the temporal evolution shown in
Figure 1 The cross-section is taken parallel to the horizontal axis at the
centre of the vertical axis. The solid line refers to the AB field and the
dashed line refers to the SV field.
\item[Figure 3:] Analogous to Figure 1 but the SV background is $-0.40$
\item[Figure 4:] Analogous to Figure 2, for the evolution shown in Figure
3.
\item[Figure 5a:] Dynamical scaling of the structure factor for the
evolution of Figure 1, with surfactant field average $c=-0.45$. The figure
superposes data for $S(k,t){{\langle k \rangle}(t)}^2$ vs. $k/{\langle k
\rangle(t)}$ from dimensionless times 1600, 4800, 6400, and 8000 (denoted by
the symbols indicated).
\item[Figure 5b:] Data from Figure 5a, plotted on a log-log scale.
The dashed line with a slope of -3 corresponds to the Porod law $S(k,t)\sim
k^{-3}$, which characterises scattering off sharp interfaces.
\item[Figure 6a:] Superposition of the scaled structure factor data (i.e.,
plot of ${S(k,t){\langle k \rangle}(t)}^2$ vs. $k/{\langle k \rangle
(t)}$) for the pure binary mixture (at dimensionless time 4000); and
the surfactant concentrations $c=-0.40$ and $-0.45$ (at dimensionless time
8000).
\item[Figure 6b:] Data from Figure 6a, plotted on a log-log scale. The
dashed line has the
 same meaning as in Figure 5b.
\item[Figure 7a:] Characteristic domain size $L(t)$ as a function of time
t for the pure case (upto dimensionless time 4000); and the surfactant
concentrations $c=-0.40$ and $c=-0.45$ (upto dimensionless time
10000). We
have used a nonlinear fitting routine to approximate the data by a best
fit of the form $L(t)=a+bt^{\phi}$. For the pure case, the best fit is
denoted as a solid line on the appropriate data set and corresponds to a
growth exponent of $\phi =0.33 \pm 0.01$. For the cases with surfactants,
the data does not fit the power-law form.
\item[Figure 7b:] Data from Figure 7a (for the cases with surfactants)
plotted on a log-log scale. The dashed line has a slope of $1/3$ and
corresponds to the Lifshitz-Slyozov growth law.
\end{itemize}
\end{document}

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