#  Phase separation kinetics of block copolymer melts confined under moving
parallel walls: a DPD study

Ashish Kumar Singh Awaneesh Singh<EMAIL_ADDRESS>
###### Abstract

We use dissipative particle dynamics (DPD) simulations to study the effect of
shear on domain morphology and kinetics of microphase separating critical
diblock copolymer (BCP) bulk melts. The melt is confined within two parallel
solid walls at the top and bottom of the simulation box. The shear is induced
by allowing the walls to move in a direction with a specific velocity. We
explore the following cases: (i) walls are fixed, (ii) only the top wall
moves, (iii) both walls move in the same direction, and (iv) both walls move
in opposite directions. After the temperature quench, we monitor the effect of
shear on evolution morphology, the scaling behavior of the system, and the
characteristic length scale and growth. The characteristic length scale
follows typical power-law behavior at early times and saturates at late times
when both walls are fixed. The length scale changes significantly with shear
caused by wall velocities. The usual lamellar morphology, which is not
achieved for case 1 within the considered simulation time steps, is noticed
much earlier for the nonzero wall velocity cases. Specifically, it is seen
much before in case 4 than in the other cases. We find that the shear
viscosity decreases (shear-thinning) with wall velocity (shear rate) for all
the cases at a given coarsening time. Overall, we report the influence rule of
shear rates on microphase separation kinetics of BCP melts. This study can
provide a scheme to anticipate and design anisotropic microstructures under
the application of externally controlled wall shear that may further guide in
producing the various composite materials with superior mechanical and
physical properties.

##  1 Introduction

Microphase separation of block copolymer (BCP) melts [1, 2, 3, 4, 5] has
continued to be an active area of research for decades [6, 7, 8, 9, 10]. BCP
is a long linear chain molecule ($A_{n}B_{m}$) composed of two covalently
bonded incompatible sub-chains of $n$ $A$\- and $m$ $B$-monomer beads. When a
homogeneous BCP melt in its high-temperature homogeneously mixed stable phase
is quenched below the critical temperature, the system becomes
thermodynamically unstable. The small fluctuations in the density field grow,
and the phase separation occurs, forming $A$-rich and $B$-rich phases [11, 12,
13]. Since the chemically bonded immiscible sequences cannot be separated
indefinitely, their sub-chain lengths control the spatial scale. Thus,
coarsening gives rise to morphologies at a microscopic length scale, i.e.,
micro-domains rich in either of the two components are formed. Different
morphologies like lamellae, gyroids, cylinders, and spheres depend on the
relative compositions ($n:m$) of sub-chains [1, 2, 3, 4].

Domain coarsening is a well-known scaling phenomenon where the characteristic
length scale, $R(t)$ follows a power-law equation, $R(t)\sim t^{\phi}$
asymptotically [11, 12, 13]. Here, $\phi$ is the growth exponent, which
depends on the relevant coarsening mechanism [12]. BCP melts coarsen into the
well-ordered structures differently than macrophase separation of polymer
blends.[14, 15, 16, 17]. Numerous theoretical [1, 2, 3, 4] and simulation [6,
18] efforts have demonstrated that the variation of characteristic length
scale with time initially follows a power-law with $\phi\sim 1/4$ in the weak-
segregation regime[1] and $\phi\sim 1/3$ in the strong-segregation regime [2,
3, 18, 6]. The segregating BCP melts evolve into frozen microstructures with
various morphologies in the asymptotic limit. It is further demonstrated that
the effects of hydrodynamics only accelerate the growth kinetics at pre-
asymptotic stages compared to diffusive dynamics [19, 20, 21, 22, 23]. Similar
frozen morphologies are observed at late times, as in the diffusive case [19,
20, 21, 22, 23].

The self-assembled BCP morphologies are the topic of vital interest due to
their rapidly increasing applications in various fields such as advanced
materials [24], lithography [24, 25, 26], drug delivery [24, 27, 28], and
porous materials [24, 29]. The latest applications of BCPs come from their
self-assembly on the nanoscale [30]; these properties rely on the long-range
order of BCP melts [31, 32]. The change in morphology considerably affects the
final properties of multiphase materials. Therefore, coarsening domain
morphology can significantly influence the final material properties.

The control over evolved morphology at the molecular scale can help develop
new and robust materials with great industrial applications. Various methods
such as shear [33], electric field [34], chemically patterned substrate [35],
etc., have been proposed in the recent past to speed up BCP melts’
segregation. Using either steady or oscillatory shear [36] is the more
straightforward approach. Their application to BCPs plays a dominant role in
determining their self-assembled structure. Using these methods, spherical,
cylindrical, and lamellar morphologies have been illustrated successfully [37,
38, 39, 40, 41, 42]. Some recent works on BCP thin films [43, 44] have
improved the alignment of cylindrical and spherical domain evolution using
shear; it is believed that microphases try to minimize the effect of shear by
reorienting themselves towards shear flow [36].

Nevertheless, most of these studies were focused mainly on either $2d$
surfaces[33, 36] such as thin substrates/films[43, 44] or studying the
equilibrium properties[45, 46] in BCP melts for a small system sizes. [37, 38,
39, 40, 41, 42] Thus, a detailed DPD simulation study of the kinetics of
microphase separation of BCP bulk melts under the influence of shear is still
lacking. In contrast to any coarse-grained models, which use uncontrolled
approximations to model the velocity field, the DPD approach has the
considerable advantage of naturally incorporating flow fields and the
hydrodynamic effects in the system.[6] Furthermore, the influence of physical
boundary and the competition between shear and phase separation kinetics in
BCP melts are crucial in understanding the morphology and related properties
changes [36]. Thus, a more detailed study on growth kinetics, particularly the
growth exponent, scaling properties, and structural anisotropy due to
variation in shear rate, are of paramount importance for a BCP melt under
confinement. The shear is introduced by moving parallel walls confining the
system from the top and the bottom. Herein, we choose a symmetric BCP melt
where each chain has two incompatible sub-chains of equal length. The solid
walls have the same interaction with each sub-chain; thus, both phases
experience an equal amount of shear caused by the wall velocity in a
particular direction. Therefore, it is natural to inquire about the shear
effect on the microphase separation kinetics in BCP melts. We ask the
following questions: does shear stimulate or delay the segregation process in
the bulk system? Further, do micro-scale morphologies vary with low and high
wall velocities? Finally, what would be the effect on scaling functions and
growth laws?

In the next section, we briefly describe the DPD methodology that captures the
effect of shear on the phase separation kinetics of BCP melts. Then, the
simulation results are discussed in Section 3. Finally, we conclude the paper
in Section 4.

##  2 Methodology

###  2.1 Dissipative particle dynamics

DPD is a powerful simulation technique to study the kinetic behavior of
complex soft mesoscale systems [16, 47, 48, 49]. In DPD, a cluster of
particles or molecules is modeled as a single bead, which makes DPD a more
thriving numerical tool to simulate the system over a higher length and time
scale than a traditional molecular dynamics (MD) simulation technique [50, 51,
52]. We integrate Newton’s equation of motion for the system’s time evolution:

$\frac{d\vec{p}_{i}}{dt}=\vec{f}_{i}(t)=\sum_{i\neq j}\vec{F}_{ij}$ (1)

where $\vec{p}_{i}=m_{i}\vec{v}_{i}$, $\vec{v}_{i}=d\vec{r}_{i}/dt$, and
$\vec{r}_{i}$ denote the momentum, velocity, and position vectors of $i^{th}$
bead. The force $\vec{f}_{i}(t)$ on $i^{th}$ particle by all other $j$
particles consist of three pairwise additive components: (i)
$\vec{F}_{ij}^{C}$ represents the conservative force, (ii) $\vec{F}_{ij}^{D}$
denotes the dissipative force, and (iii) $\vec{F}_{ij}^{R}$ imply the random
force such that
$\vec{F}_{ij}=\vec{F}_{ij}^{C}+\vec{F}_{ij}^{D}+\vec{F}_{ij}^{R}$.

The conservative force describes the pairwise interaction between the beads.
The most common and simple choice for $\vec{F}^{C}_{ij}$ is[47]

$\vec{F}_{ij}^{C}=\begin{cases}a_{ij}\left(1-{r_{ij}}/{r_{c}}\right)\hat{r}_{ij},&r_{ij}<r_{c}\\\
0,&r_{ij}\geq r_{c}\end{cases}$ (2)

which is a linear soft repulsive interaction up to a cut-off distance $r_{c}$.
Here, $a_{ij}$ denotes the maximum repulsion of the conservative force,
$\vec{r}_{ij}=\vec{r}_{j}-\vec{r}_{i}$, $r_{ij}=\lvert\vec{r}_{ij}\rvert$ and
$\hat{r}_{ij}=\vec{r}_{ij}/r_{ij}$. The dissipative force $\vec{F}_{ij}^{D}$
has the following mathematical form

$\vec{F}_{ij}^{D}=-\gamma\omega_{D}(r_{ij})(\hat{r}_{ij}\cdot\vec{v}_{ij})\hat{r}_{ij}$
(3)

where $\gamma$ is the friction coefficient or strength of viscus dissipation,
$\omega_{D}(r_{ij})$ is the weight function of the dissipative force, and
$\vec{v}_{ij}=\vec{v}_{i}-\vec{v}_{j}$. The effect of thermal fluctuation is
described by[47]

$\vec{F}_{ij}^{R}=\sigma\omega_{R}(r_{ij})\sqrt{\Delta t}\xi_{ij}\hat{r}_{ij}$
(4)

where $\sigma$ is the strength of the random force, $\omega_{R}(r_{ij})$ is
the corresponding weight function, and $\xi_{ij}$ denotes the Gaussian random
variable with zero mean $\left\langle\xi_{ij}(t)\right\rangle=0$ and unit
variance
$\left\langle\xi_{ij}(t)\xi_{kl}(t^{\prime})\right\rangle=(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\delta(t-t^{\prime})$.[47,
48]. The symmetry relations $\vec{F}_{ij}=-\vec{F}_{ji}$, $\xi_{ij}=\xi_{ji}$,
and the force $\vec{F}_{ij}^{D}$ and $\vec{F}_{ij}^{R}$ act along the line
joining bead centres ensure the momentum conservation.[47, 48].

To attain the correct canonical thermodynamic equilibrium state of a system at
a temperature $T$, the strength of the random $(\sigma)$ and dissipative
$(\gamma)$ forces must follow the fluctuation-dissipation relation[48, 49]
$\sigma^{2}=2\gamma k_{B}T$ where $k_{B}$ is the Boltzmann constant; their
corresponding weight functions must coupled as
$\omega_{D}(r_{ij})=\left[\omega_{R}(r_{ij})\right]^{2}$. The common choice
for the weight function is[47]
$\omega_{D}(r_{ij})=\left(1-{r_{ij}}/{r_{c}}\right)^{2}$ for $r_{ij}<r_{c}$.
However, we can choose other forms of weight functions given that the above
two conditions are satisfied.

###  2.2 Model parameters and other details

The system evolution is followed by integrating the equations of motion shown
in Eq. (1) using a modified velocity-verlet algorithm.[53] In DPD simulations,
cut-off radius $r_{c}$ is commonly considered as a characteristic length
scale, and $k_{B}T$ as a characteristic energy scale.[47] Each DPD bead is
considered to have equal mass $m_{i}=m$. The simulation results are presented
in reduced DPD units with the parameters $r_{c}$, $m$, and $k_{B}T$, all set
to $1.0$.[47] Following the dimensional analysis, the characteristic time
scale is defined as $\tau=\left(mr_{c}^{2}/k_{B}T\right)^{1/2}$, which gives
$\tau=1.0$. We chose the integration time step $\Delta t=0.02$.[17] The
friction parameter value is set to $\gamma=4.5$, which is suitable for the
system to reach the equilibrium temperature more rapidly and provide numerical
stability for the specific time steps.[47]

We consider the total number density of beads $\rho=3$ appropriate for the DPD
simulation of liquids; this choice of $\rho$ also ensures that the system is
far away from the gas-liquid transition.[47, 48] The characteristic length and
time scale in the actual units are estimated to be $r_{c}=0.97nm$ and
$\tau=8.3ps$, respectively.[17, 16] We set the value of repulsive interaction
strength $a_{ij}=25$ for the interaction between chemically compatible BCP
beads (i.e., $a_{AA}=a_{BB}=25$); this choice is made to reproduce the
compressibility of water by coarse-graining ten water molecules into one
bead.[47, 48, 49, 50] The interaction parameter, $a_{ij}=60$ for chemically
incompatible beads (i.e., $a_{AB}=60$). The interaction of $A$ and $B$ beads
with the wall beads is set to $a_{ij}=45$. In our DPD simulation, we quench
the system at reduced temperature $T=1$, which we see in a short while that
the chosen temperature is well below the critical temperature for the
corresponding phase separation to take place.[17, 16]

We simulate BCP chains as bead-spring model where the beads of each sub-chains
are connected by harmonic bond potential[54, 55],
$E_{b}=\frac{k_{b}}{2}(r-r_{0})^{2}$ where $k_{b}=128$ is the elastic
constant, and $r_{0}=0.5$ is the equilibrium bond length.[16, 52] The
stiffness of the polymer chain is handled by the angle potential,
$E_{a}=\frac{k_{a}}{2}(\cos\theta-\cos\theta_{0})^{2}$, where $k_{a}=5$ is the
strength of the angle potential. The angle between two successive bonds along
the chain is $\theta$. We have taken the equilibrium value
$\theta_{0}=180$.[51, 52, 54, 55, 53] Since DPD utilizes softcore interaction
potential between the beads, there could be an unphysical bond crossing
between the BCP chains. To mitigate that, we used modified segmental repulsion
potential (mSRP) [56]. In this formulation, pseudo beads are considered at the
center of all bonds. These beads interact with a softcore repulsive
interaction[56],
$\vec{F}_{ij}^{mS}=k_{mS}\left(1-{r_{ij}}/{r_{c}^{mS}}\right)\hat{r}_{ij}$,
for $r_{ij}<r_{c}^{mS}$ where $k_{mS}=100$ (in reduced DPD units) is the force
constant. The cutoff distance for the mSRP interaction is taken as
$r_{c}^{mS}=0.6$. We use the LAMMPS simulation package[53] with mSRP code[56]
to integrate the equations of motion.

The box size used in our DPD simulation is $L_{x}\times L_{y}\times
L_{z}=64\times 64\times 64$; thus, the total number of DPD beads in the
simulation box is $N=\rho\times L^{3}=7,86,432$. The period boundary
conditions are applied in $x$\- and $y$\- directions, whereas two solid walls
bound the simulation box in the transverse $z$-direction. The height of each
wall is set to $h=1.0$. The walls are also made of DPD beads with a number
density $\rho_{w}=3$; therefore, the total number of wall beads is
$N_{w}=24,576$, so the volume fraction is $\phi_{w}=3.125\times 10^{-2}$. The
bounce-back boundary condition is applied at the fluid-wall interface[57] to
prevent the penetration of fluid beads into the walls.[58] For the simulation,
we consider a critical BCP melt $(n:m=1:1)$ with chain length $N_{p}=32$
(degree of polymerization). Total number of $A$\- and $B$-type beads is
$N_{A}=N_{B}=3,80,928$ ($\phi_{A}=\phi_{B}=4.843\times 10^{-1}$), i.e.,
$\rho_{A}=\rho_{B}=1.453$. The total number of BCP chains in the simulation
box is $N_{BCP}=23,808$.

To shear the phase separating BCP melt, we let the walls move in $x$-direction
with a constant velocity,
$\vec{v}_{w}\in\left(v_{wx},v_{wy}=0,v_{wz}=0\right)$. When walls are fixed,
corresponding velocity components $v_{wx}$, $v_{wy}$, and $v_{wy}$ are set to
zero (case 1). We consider three other cases depending on the direction of the
wall’s motion: (i) top wall moves in the positive $x$-direction (case 2) with
$v_{wx}=0.1$, $0.5$, and $1.0$, while the bottom wall is fixed. (ii) The top
and the bottom walls move in the same direction (case 3) with the same wall
velocities as in case 2. (iii) Both walls move in the opposite directions, the
top wall moves in the positive $x$-direction, and bottom wall in the negative
$x$-direction; we set this condition as case 4.

To demonstrate the effect of shear, we compute shear viscosity, $\eta$ using
the technique developed by Muller-Plathe [59, 60]. In this technique, momentum
flux is imposed on the system by the exchange of momentum of atoms along a
particular direction (say $z$-direction). The velocity profile is generated as
a response to momentum flux. Shear viscosity can be calculated from the slope
of velocity profile and imposed momentum flux using the equation:
$j_{z}(p_{x})=-\eta\left({\partial v_{x}}/{\partial z}\right)$, here
$j_{z}(p_{x})$ is the momentum flux, and ${\partial v_{x}}/{\partial z}$ is
the velocity gradient or shear rate. Further, we quantitatively evaluate the
anisotropic degree of evolved morphology by calculating the anisotropy
parameter $D_{xy}(t)$ as [61, 62];

$D_{xy}(t)=\frac{\sum(k_{x}^{2}-k_{y}^{2})/k^{2})S(k_{x},k_{y},k_{z})}{\sum
S(k_{x},k_{y},k_{z})}.$ (5)

Here, $k_{x}$ and $k_{y}$ are the components of wave vector $\vec{k}$ in the
$xy$-plane for a fixed $k_{z}$ where $k^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}$,
and $S(k_{x},k_{y},k_{z})$ denotes the structure factor at a point
$\left(k_{x},k_{y},k_{z}\right)$ in Fourier space. Similarly, we calculate the
anisotropy parameter, $D_{yz}(t)$, and $D_{xz}(t)$ in the $yz$ and $xz$
planes, respectively. Note that, for a perfect isotropic system, $D_{ij}\simeq
0$ and for a fully anisotropic system, $D_{ij}\rightarrow 1$.

###  2.3 Morphology characterization and scaling functions

The two most desirable and experimentally relevant functions to characterize
the evolution morphology are the correlation function and its Fourier
transform, the structure factor.[12, 13] We introduce the two-point equal-time
correlation function as

$C(\vec{r},t)=\left\langle\psi\left(\vec{r}_{1},t\right)\psi\left(\vec{r}_{2},t\right)\right\rangle-\left\langle\psi\left(\vec{r}_{1},t\right)\right\rangle\left\langle\psi\left(\vec{r}_{2},t\right)\right\rangle$
(6)

where $\psi(\vec{r},t)$ denotes the order parameter. The structure factor is
defined as

$S(\vec{k},t)=\int d\vec{r}\exp\left(i\vec{k}\cdot\vec{r}\right)C(\vec{r},t)$
(7)

where $\vec{k}$ is the scattering wave vector.

The order parameter $\psi(\vec{r},t)$ is obtained as the local density
difference of the constituents located at $\vec{r}$ at a given time $t$[15,
14],

$\psi(\vec{r},t)=\frac{n_{A}(\vec{r},t)-n_{B}(\vec{r},t)}{n_{A}(\vec{r},t)+n_{B}(\vec{r},t)}$
(8)

where $n_{A}(\vec{r},t)$ and $n_{B}(\vec{r},t)$ represent the local number of
$A$ and $B$ types of beads calculated as follows: we divide the simulation box
into non-overlapping unit cubes and then count the number of $A$ and $B$ types
of beads in each unit cube. Thus, from Eq. (8) $\psi(\vec{r},t)\in(0,1)$ for
$n_{A}(\vec{r},t)>n_{B}(\vec{r},t)$, and $\psi(\vec{r},t)\in(0,-1)$ for
$n_{A}(\vec{r},t)<n_{B}(\vec{r},t)$. However, for
$n_{A}(\vec{r},t)=n_{B}(\vec{r},t)$, we assign $\psi(\vec{r},t)\in(0,1)$ or
$\psi(\vec{r},t)\in(0,-1)$ with equal probability [15].

The number of peaks in the structure factor or the oscillatory behavior of the
correlation function after zero-crossing signifies the periodicity of evolving
morphology in the phase separating BCP melts. In our simulation, we
spherically average $C(\vec{r},t)$ and $S(\vec{k},t)$ for better statistics.
The corresponding quantities are denoted by $C(r,t)$ and $S(k,t)$,
respectively. The characteristic length scale, $R(t)$ is defined as the
distance over which $C(r,t)$ decays to a fraction of its maximum value
($C(r,t)|_{r=0}=1$). There are a few other definitions of $R(t)$, but they are
equivalent in the asymptotic limit; they differ only by some constant
multiplicative factors in the scaling regime.[63, 64] Herein, the distance
over which the correlation function decays to $0.2$ provides a good measure of
$R(t)$.[63, 65]

The basic assumption of scaling is the existence of a unique characteristic
length scale $R(t)$. The dynamical scaling form of the correlation function
and the structure factor has the following form[12, 13]:

$C(r,t)=f(r/R(t))$ (9) $S(k,t)=R(t)^{d}g(kR(t))$ (10)

where $f(r/R(t))$ and $g(kR(t))$ are the corresponding scaling functions. The
characteristic length scale follows the power-law growth:
$R(t)=R_{0}+at^{\phi}$ where $R_{0}$ is the average domain size in the
transient growth regime, which is taken as an offset value, therefore, we
define $R(t)\equiv R(t)-R_{o}$[17], and $a$ is a growth rate constant. To find
the asymptotic growth exponent ($\phi$), we plot $R(t)$ versus $t$ on a
logarithmic scale and estimate the slope of the curve. Further, we follow
another intuitive way to verify the same by computing the effective growth
exponent as

$\phi_{eff}=\log_{\alpha}\left[R\left(\alpha t\right)/R\left(t\right)\right]$
(11)

and plot it against inverse of domain size $1/R(t)$ where we set the log-base
$\alpha=2$.[66]

##  3 Numerical results

This section will discuss the numerical results obtained for microphase
separating symmetric BCP melt ($A_{n}B_{n};n=16$) confined within two parallel
solid walls in the transverse $z$-direction. At the onset of the DPD
simulation, we first equilibrate the initial system for $t=2\times 10^{5}$ at
a high-temperature $T=5$ (in reduced DPD units) to prepare a homogeneous
mixture. We then reset the time at $t=0$, quench the system at a lower
temperature $T=1$, and monitor the evolution pattern at different times.

###  3.1 Both walls are fixed

To begin with, we first study case 1 with both walls fixed (no shear) to put
the rest of the discussions in proper context. The evolution morphology at
$t=120$ and $t=5400$ are shown in Fig. 1(a); $A$-type beads are in red,
$B$-type beads are in yellow, and the olive color marks the wall beads. We
plot the corresponding radial distribution function (RDF), $g_{AB}(r)$ vs.
radial distance, $r$ in Fig. 1(b) at different time steps as denoted by the
various symbols. The extension in peak height and the shift to higher $r$ with
time indicate the growth of $A$ and $B$ clusters. The black curve ($t=120$)
shows a single lower peak at a small $r$, which saturates at larger $r$
values; this demonstrates an early-time evolution of homogeneous BCP melt into
tiny domains. As time progresses, $g_{AB}(r)$ begins to develop secondary
peaks as displayed by the red ($t=360$), green ($t=1200$), and blue ($t=5400$)
curves, respectively; this indicates the coarsening of BCPs into periodic
structures. Further, the local number density profile of $A$-type beads,
$\rho_{A}(y)$ along the $y$-direction is displayed in Fig. 1(c) for the same
time sets as in Fig. 1(b). The solid black line shows the actual number
density of $A$ beads ($\rho_{A}=1.453$). The increasing density profile
amplitude demonstrates the growth of phase separating domains, and the
oscillatory variation ascertains the periodic structure formation. The density
profiles in other directions are not shown here due to brevity.

Figure 1: The system is confined within fixed walls at the top and bottom, as
indicated by the olive beads. (a) Evolution morphology of BCP melt at $t=120$
and $t=5400$. (b) Comparison of radial distribution function, $g_{AB}(r)$
versus $r$ at different evolution times. (c) Average number density
distribution, $\rho_{A}(y)$ of $A$-beads along $y$-direction for the same time
steps as in (b).

We plot the scaled correlation functions, $C(r,t)$ vs. $r/R(t)$ in Fig. 2(a)
at four different times. The correlation functions at early times, $t=120$
(black symbol) and $t=360$ (red symbol), deviate from the scaling. Note that
early-time domain evolution in BCP melt is analogous to spinodal decomposition
in polymer blends.[20] An excellent data overlap of the correlation functions
at late times, $t=1200$ and $5400$ (denoted by the green and blue symbols),
justify the dynamical scaling regimes. A more oscillatory form of $C(r,t)$
after the zero crossing characterizes the development of periodic structures
in the late times. The solid black line denotes the zero-crossing of $C(r,t)$.
Notice that we do not observe a clear lamellar morphology within the given
simulation period as displayed in Fig. 1(a) for the symmetric BCP melt with
chain length $N_{p}=32$ ($A_{16}B_{16}$). Since diffusion coefficient, $D\sim
1/N_{p}$, the formation of lamellar morphology is observed for symmetric BCP
melts with shorter chains within the same simulation period, as displayed in
Fig. S1(a) for $N_{p}=8$ ($A_{4}B_{4}$) and in Fig. S1(b) for $N_{p}=16$
($A_{8}B_{8}$). Therefore, we need longer steps to observe lamellar morphology
for BCPs with chain length $N_{p}=32$. The scaled structure factor,
$S(k,t)R(t)^{-3}$ vs. $kR(t)$ in Fig. 2(b) is displayed for the same times as
in Fig. 2(a). As expected, we observe a crossover in the scaling functions.
The structure factor data deviates at early times (indicated by the black and
red symbols). Notice the narrowing peak in $S(k,t)$ and the appearance of the
shoulder (at $kR(t)\simeq 4$) in the green and blue curves at $t=1200$ and
$t=5400$; this verifies the formation of periodic morphology at the late
stages of coarsening. The structure factor shows a power-law decay,
$S(k,t)\sim k^{-(d+1)}$ at large wave vector ($k\rightarrow\infty$) regimes
where $d$ is the system’s dimension; this is known as Porod’s law[67] results
from scattering off sharp interfaces.

Figure 2: Spherically averaged scaled functions and length scale for the
evolution displayed in Fig. 1. (a) $C(r,t)$ versus $r/R(t)$ at different times
for the fixed walls. The solid line is a guide for the zero-crossing of
$C(r,t)$. (b) $S(k,t)$ versus $kR(t)$ on a logarithmic scale for the same
times as in (a). The solid line with slope $-4$ represents the Porod’s law:
$S(k,t)\sim k^{-4}$ at $k\rightarrow\infty$. (c) Characteristic length scale,
$R(t)$ versus $t$ for the same data sets as in (a) and (b). The solid line
shows the diffusive growth law $(\phi\sim 1/3)$ at early times, which
gradually saturates later, illustrating usual microphase separation kinetics
in BCP melt. (d) Presents effective growth exponent $\phi_{eff}$ versus
$1/R(t)$. The solid line shows the reference value of $\phi_{eff}\sim 1/3$ for
early diffusive growth.

Next, to study the domain growth law for the evolution shown in Fig. 1(a), we
plot the characteristic length scale, $R(t)$ vs. $t$ in Fig. 2(c), on a log-
log scale. BCP melt follows the diffusive growth (i.e., $\phi\sim 1/3$) in an
early spinodal time window up to $t_{sp}\simeq 800$ such that $R(t_{sp})\sim
4$ (equivalent to the average size of BCP sub-chains). The length scale then
crosses over to saturation when the constraints imposed by the topological
connection become evident. The saturating $R(t)$ marks the freezing microphase
morphology for symmetric BCPs. A more precise way of computing the growth
exponent is to plot $\phi_{eff}$ vs. $R(t)^{-1}$ from Eq. (11) and extract
$\phi_{eff}$ from data extrapolation in the asymptotic limit. Such an exercise
is shown in Fig. 2(d), which demonstrate that in the limit
$R(t)^{-1}\rightarrow 0$, the growth exponent converges to
$\phi_{eff}\rightarrow 0$.

###  3.2 Top wall moves

Let us return to the phase separation of BCP melt under the influence of shear
arising due to moving walls. First, we focus on case 2, where the top wall
moves in the positive $x$-direction, and the bottom wall is fixed. Any
variation in the wall velocity can alter the shear rate, $\dot{\gamma}\propto
v_{wx}$; here we choose three different wall velocities, $v_{wx}=0.1$, $0.5$,
and $1.0$.

The evolution morphology for case 2 is displayed in Figs. S2. It is apparent
that with the increase in $v_{wx}$, domains of $A$ and $B$ sub-chains adjacent
to the top wall appear to flow more along the shear direction. Recall that in
case 1, the system could not develop lamellar morphology up to simulation time
$t=5400$. Herein, lamellar slabs appear earlier with increasing shear rates;
the reason could be the flow and orientation of microdomains in the shear
direction. Therefore, due to the competition of unidirectional shear and
simultaneous microphase separation kinetics, BCP chains segregate
perpendicular to the shear direction to form a more stable lamellar morphology
than in any other adjustments. In other words, by adjusting normal to the
shear direction, $A$ and $B$ domains try to overcome the effect of shear.
Since only the top wall moves, the low shear rate ($v_{wx}=0.1$) primarily
influences a small upper region of the simulation box. Thus, lamellar
formation begins closer to the moving wall along the flow direction and
perpendicular to the $yz$-plane. A few more layers in the simulation box are
affected by the increase in wall velocity, $v_{wx}=0.5$ and $v_{wx}=1.0$, and
therefore, we get more ordered lamellar strips within the same time interval.
The unidirectional shear enhances the tendency of lamellar pattern formation
for a phase separating symmetric BCP melt.

Figure 3: Top wall is moving with velocity $v_{wx}=0.5$. (a) Plots for
$g_{AB}(r)$ versus $r$ at different times, indicated by various symbols. (b)
$\rho(y)$ of $A$-beads along $y$-direction for the same time steps as in (a).
(c) $g_{AB}(r)$ versus $r$ for different top wall velocities for case 2 at a
fixed $t=5400$. (d) $\rho(y)$ versus $y$ for the same set of data as in (c).

To characterize the evolution morphology illustrated in Fig. S2, we plot
$g_{AB}(r)$ vs. $r$ in Fig. 3(a) at different time steps for $v_{wx}=0.5$. The
black curve showed early time data when a few small microdomains evolved via
spinodal decomposition. Thus, a single peak is observed at a shorter distance
$r\simeq 5.0$. The main peak strength of $g_{AB}(r)$ increases with time and
shows a relatively higher amplitude at $r\simeq 9.25$ at the late time (see
the blue curve) than any other curves at earlier times. A secondary peak in
$g_{AB}(r)$ is also clearly visible in the blue curve at a distance $r\simeq
24.0$. The main reason is the coarsening of lamellar morphology. Corresponding
local number density profile, $\rho_{A}(y)$ is plotted against the
$y$-direction in Fig. 3(b). A much lower density profile amplitude at early
times verifies tiny microdomain formation (see the corresponding black and red
curves). However, at late times, a more considerable amplitude and oscillatory
behavior of $\rho_{A}(y)$ are observed due to larger and periodic domain
coarsening.

Further, we compare $g_{AB}(r)$ vs. $r$ in Fig. 3(c) to characterize the
evolved morphologies at a late time for different wall velocities. All the RDF
curves present an excellent data overlap closer to the first peak; however,
the main peak width slightly narrows as its position shifts to smaller
$r\simeq 9.6$, $9.6$, $9.25$, and $9.0$ with increasing shear rates.
Subsequently, the secondary RDF peaks, which are developed due to the
formation of periodic morphology, also shifted to its left at $r\simeq 25.0$,
$24.32$, $24.0$, and $22.4$ for $v_{wx}=0.0$, $0.1$, $0.5$, and $1.0$,
respectively. Overall, these plots suggest increasing periodicity and the
thinning of domain sizes with increasing shear rates. The much lower density
profile ($\rho_{A}(y)$) amplitude at smaller shear rates ($v_{wx}=0.0$,
$0.1$), indicated by the back and red symbols in Fig. 3(d), imply that the
periodic structures are not a fully developed lamellar$-$many short and
randomly oriented domain stripes are still present in the system. In contrast,
$\rho_{A}(y)$ gains significant height and periodicity at higher shear rates
($v_{wx}=0.5$, $1.0$), suggesting the evolution of appreciable lamellar
morphology.

To see the effect of various shear rates on the dynamic scaling functions for
case 2, we plot $C(r,t)$ vs. $r/R(t)$ in Fig. 4(a) and $S(k,t)$ vs. $kR(t)$ on
a log-log scale in Fig. 4(b) for different wall velocities at $t=5400$. The
oscillatory $C(r,t)$ curves in Fig. 4(a) characterize the developing periodic
morphology at a late time; the appearance of the distinct shoulder at large
$kR(t)$ for each curve in Fig. 4(b) verify the same. Notice the slight
decrease in amplitude of oscillation in $C(r,t)$ and the broadening in
$S(k,t)$s’ main peak for $v_{wx}\neq 0.0$. Since the first moment of $S(k,t)$
is inversely proportional to average domain size ($\langle k\rangle\sim
R(t)^{-1}$) [12, 13], the broadening of main $S(k,t)$ peak implies an increase
in $\langle k\rangle$ with shear rate justifies shear-thinning at late times.
Further, the shoulder in $S(k,t)$ shows up at smaller $kR(t)\simeq 2.67$ for
$v_{wx}=0.0$ (black curve), whereas it appears at larger $kR(t)\simeq 3.06$ in
the blue curve. Overall, these observations indicate the shear-thinning of
morphologies. We notice that the black curves in Figs. 4(a-b) slightly deviate
from the scaling. However, for $v_{wx}\neq 0.0$, the scaling function data
illustrates a good overlap; hence, they belong to the same universality class.
A solid line with slope $-4$ in Fig. 4(b) shows that the scaled structure
factor tail follows Porod’s law for all cases.

Figure 4: Comparing scaling function, length scale, and effective growth
exponent at $t=5400$ for different wall velocities for case 2 as indicated by
the various symbol types. $C(r,t)$ versus $r/R(t)$ is displayed in (a), and
corresponding $S(k,t)$ versus $kR(t)$ on a logarithmic scale in (b). (c)
$R(t)$ versus $t$ on a log-log scale at different wall velocities for the
evolution shown in Fig. (S2). The variation of growth exponent $\phi_{eff}$ as
a function of $1/R(t)$ is in (d).

Next, we compare the characteristic length scale, $R(t)$ vs. $t$ on a
logarithmic scale in Fig. 4(c) and the corresponding effective growth exponent
in Fig. 4(d) for $v_{wx}\neq 0.0$ values. The black curve for $v_{wx}=0.0$ is
plotted for reference. Note that domain coarsening and their flow along the
shear direction are two physical processes occurring simultaneously. The lower
shear rate ($v_{wx}=0.1$, red curve) seems to complement the early-time domain
growth ($R(t)\sim t^{1/3}$). Thus, we observe a larger average domain size up
to the time, $t_{sp}\simeq 600$, indicating lower shear-thinning than other
higher shear rates. The topological constraint becomes apparent for
$t>t_{sp}$. We notice a dip in the growth scale for a certain period
($600<t<1000$) where $A$ and $B$ domains could be adjusted in a shear
direction to minimize its effect. The length scale grows again and then
crosses over to saturation, indicating the freezing in microdomain growth to a
smaller length scale than in case 1 ($v_{wx}=0.0$), which marks the shear-
thinning effect. The shear effect becomes prominent with increasing wall
velocity, which causes domain thinning from the early times. Thus, the overall
growth rate decreases. Early-time growth exponent reduces from a pure
diffusive growth $R(t)\sim t^{1/3}$, then crosses over to frozen microphase
morphology at the late stages. The effective growth exponent, $\phi_{eff}$ vs.
$1/R(t)$ curves in Fig. 4(d) again illustrate the same; $\phi_{eff}\rightarrow
0$ in the limit $1/R(t)\rightarrow 0$.

###  3.3 Both walls move in the same direction

Now, we turn our attention to case 3, where we shear the segregating BCP melt
from both sides in the same direction and monitor its effect on the evolution
morphology (displayed in Fig. S3) and the growth kinetics. The result of shear
on the morphology is more prominent at $v_{wx}=0.5$ and $1.0$. The system
begins to develop periodic structures from the top and bottom region of the
simulation box, particularly closer to the walls where shear is more effective
than in other areas. A more explicit lamellar morphology seemed to form herein
than in case 2 at the same wall velocities. For lower velocity $v_{wx}=0.1$,
the effect of shear on the evolving morphology emerges almost similar to what
it was in case 2.

Figure 5: Case 3: Both walls are moving with the same velocities in
$x$-direction. (a) $g_{AB}(r)$ versus $r$, and (b) average number density
profiles of $A$-type beads in $y$-direction at $t=5400$ for different walls
velocities are presented with various open symbols; related morphologies are
shown in Fig. (S3)

We plot $g_{AB}(r)$ for different wall velocities in Fig. 5(a) at $t=5400$.
Similar to the previous cases, $g_{AB}(r)$ becomes more oscillatory with
increasing shear rate, signifying increasing periodicity. The peak positions
of $g_{AB}(r)$ move to their left with increasing shear rates (denoted by
various symbols); the primary peak positions are at $r\simeq 9.6$, $9.5$,
$9.0$, $9.0$, and the secondary peaks are at $r\simeq 25.0$, $24.32$, $23.04$,
$22.4$, respectively. This indicates the thinning in average domain size with
shear rates. The local density variation in Fig. 5(b) illustrates the equal
distribution of $\rho_{A}(y)$ around its average value ($\rho_{A}=1.453$).
Their enhanced amplitude than previous cases signifies the formation of a more
well-formed lamellar morphology, perpendicular to the $y$-axis; see the green
and blue curves at $v_{wx}=0.5$ and $v_{wx}=1.0$, respectively. Notice the red
curve at $v_{wx}=0.1$, showing an apparent enhancement in $\rho_{A}(y)$ than
in case 2; thus, we get a more ordered structure in case 3 than in case 2 with
the same shear rates.

The late time dynamic scaling functions, $C(r,t)$ and $S(k,t)$ in Fig. 6(a)
and Fig. 6(b) exhibit apparent deviation at $v_{wx}=0.1$ (red symbol) from the
reference case (black curve) including the high shear cases, as displayed in
green and blue symbols. However, $C(r,t)$ and $S(k,t)$ curves at higher shear
rates illustrate excellent data overlap, thus, exhibiting dynamical scaling.
The structure factor shows Porod’s tail ($S(k,t)\sim k^{-4}$) for all the
shear rates for $kR(t)\rightarrow\infty$, indicating the sharp domain
interface between $A$ and $B$ phases. The oscillation in the correlation
function and the emergence of the shoulder at large $kR(t)$ in the structure
factor represent the formation of the periodic structures. After the zero
crossing, the smaller amplitude of oscillation in $C(r,t)$ and the broadening
of the prominent peak in $S(k,t)$ with shear rates attribute to the shear-
thinning of evolving morphologies.

Figure 6: (a) Comparison of $C(r,t)$ versus $r/R(t)$ at different wall
velocities for the morphologies illustrated in Fig. (S3). Corresponding
$S(k,t)$ versus $kR(t)$ curves are in (b) on a logarithmic scale. The average
domain size, $R(t)$ versus $t$, and related effective growth exponents
$\phi_{eff}$ versus $1/R(t)$ are displayed in (c) and (d), respectively.

Similar to case 2, the low shear rate complements the early time diffusive
growth, and we observed a slightly larger average domain size up to
$t_{sp}\simeq 600$ (see the red curve in Fig. 6(c)). Then, a flattening of
length scale data (i.e., negligible domain growth) is observed up to $t\simeq
1000$ due to the competing process of shear and phase separation kinetics.
Corresponding growth exponent ($\phi_{eff}$) shows a significant dip (see the
red curve in Fig. 6(d)). The BCP sub-chains are most likely arranged normal to
the flow direction in this period to minimize the shear effect. Beyond this,
domains begin to grow again with a lower growth rate due to shear-thinning
than the reference case during the same period. The length scale tends to
saturate at a slightly larger value than the reference case (see the red
curves in Figs. 6(c-d)). At higher wall velocities, the shear effect is
prominent from the beginning; the growth exponent ($\phi_{eff}\simeq 0.2$) is
smaller, as displayed by the green and blue curves in Figs. 6(c-d). The length
scale tends to saturate to the lower values at late times, justifying the
shear-thinning. Again, the shear-thinning is more prominent for the higher
shear rate. Thus, in the limit of $R(t)^{-1}\rightarrow 0$, the effective
growth exponent $\phi_{eff}\rightarrow 0$.

Figure 7: Plot of $S(k_{x},k_{y},k_{z})$ versus $k_{y}$ (black curve) and
$k_{z}$ (red curve) at $v_{wx}=0.5$ to illustrate the structural anisotropy
(a) for the top wall motion along $x$-direction, and (b) when both walls move
in same direction. The green and blue curves in the inset demonstrate
$S(k_{x},k_{y},k_{z})$ versus $k_{y}$ along diagonal and cross-diagonal
directions respectively in $xz$-plane.

Recall that $C(r,t)$ and $S(k,t)$ denote the spherically averaged correlation
function and the structure factor. The critical BCP melts evolve to form
lamellar morphology in the asymptotic limit. Thus, it is essential to manifest
the anisotropy developed in the system at late times. In this regard, Figs.
7(a) and 7(b) display the structure factor, $S(k_{x},k_{y},k_{z})$ against the
wave vector components, $k_{y}$ (black curve) and $k_{z}$ (red curve) for case
2 and case 3 at $v_{wx}=0.5$ for $t=5400$. The black curve shows sharper and
higher amplitude peaks in $S(k_{x},k_{y},k_{z})$ along $k_{y}$ (normal to the
lamellar). The red curve shows a single low amplitude peak along $k_{z}$
(parallel to the lamellar). Thus, both these curves justify the structural
anisotropy in the system. The larger peak amplitude of the black curve for
case 3 (Fig. 7(b)) explains a higher structure anisotropy than in case 2 (Fig.
7(a)). Notice the snapshots in Figs. S2-S3; they illustrate that lamellar
slabs of each phase are coarsening nearly parallel to the $xz$-plane; thus,
the variation of $S(k_{x},k_{y},k_{z})$ along $k_{x}$ and $k_{z}$ is similar
(comparison is not shown here due to brevity). However, excellent overlapping
in green and blue curves in the insets of Fig. 7 affirms the symmetric
arrangement of lamellar slabs in the $yz$-plane. The $S(k_{x},k_{y},k_{z})$
data is averaged over sixty-four ensembles.

###  3.4 Both walls move in opposite directions

In case 4, wall motion generates equal and opposite shear in the
$x$-direction. The evolution patterns in Fig. S4 suggest that the system forms
lamellar morphology much earlier than in cases 1-3. The reason could be the
opposite shear directions facilitate a more rapid readjustment of evolving
microdomains; it is evident that well-developed lamellar patterns are formed
for $v_{wx}=0.5$ and $v_{wx}=1.0$. However, the system comprises a few shorter
patches of $A$ and $B$ domains, elongated in the $x$-direction, at
$v_{wx}=0.1$.

Figure 8: Case 4: both walls are moving opposite in $x$-direction with the
same velocities. Comparison of (a) $g_{AB}(r)$ versus $r$, (b) $\rho(y)$
versus $y$ of $A$-type beads, (c) $C(r,t)$ versus $r/R(t)$, and (d) $S(k,t)$
versus $kR(t)$ at $t=5400$ for different wall velocities as shown with various
open symbols. (e) The time dependence of average domain size $R(t)$ for the
evolution is displayed in Fig. (S4). Corresponding effective growth exponents
are plotted in (f).

As discussed earlier for other cases, $g_{AB}(r)$ vs. $r$ in Fig. 8(a) shows a
similar late time behavior for all velocities. The peaks get more oscillatory
and sharper with increasing shear rates due to the evolution of a more ordered
and periodic lamellar morphology. A tiny shift in the first peak positions (at
$r\simeq 9.5$, $9.0$, $8.32$) and a relatively larger change in secondary peak
positions (at $r\simeq 24.96$, $22.40$, $21.76$) towards lower $r$ indicates
the shear-thinning of morphologies with shear rate. Figure 8(b) illustrates
that the amplitude of $\rho_{A}(y)$ vs. $y$ increases with the shear rate. A
significant increase in the amplitude of $\rho(y)$ and much sharper domain
interfaces signify a well-separated morphology at high shear strength
($v_{wx}=1.0$), as displayed by the blue curve in Fig. 8(b). Nearly equal
periodicity and amplitude of $\rho(y)$ profile along the $y$-direction
manifests the formation of lamellar morphology normal to it at higher wall
velocities; see the green and blue curves in Fig. 8(b) at $v_{wx}=0.5$ and
$1.0$.

The scaled $C(r,t)$ in Fig. 8(c) and the corresponding $S(k,t)$ in Fig. 8(d)
demonstrate notable deviation from the dynamic scaling function in case 4.
This is indifferent to case 2 and case 3, where we have observed the scaling
in $C(r,t)$, at least up to the first zero crossing (at smaller $r$) and in
$S(k,t)$ at larger $kR(t)$. Furthermore, $C(r,t)$ becomes more oscillatory at
larger $r/R(t)$ with the increase in $v_{wx}$. In $S(k,t)$, we observe
prominent shoulders at larger $kR(t)\simeq 2.67$, $2.84$, $2.86$, and $2.93$
for $v_{wx}=0.0$, $0.1$, $0.5$, and $1.0$, respectively. In addition, however,
we notice the development of another shoulder at $kR(t)\simeq 4.68$ and $4.95$
for $v_{wx}=0.5$ and $1.0$. These results verify the formation of a more
periodic lamellar morphology with an increasing shear rate than for other
cases. Moreover, a fully evolved periodic lamellar morphology noted at
$t=5400$ for $v_{wx}=1.0$.

The time evolution of characteristic length scale, $R(t)$ vs. $t$ in Fig.
8(e), and the corresponding growth exponent, $\phi_{eff}$ vs. $1/R(t)$ in Fig.
8(f) demonstrate the coarsening kinetics under the influence of shear for case
4. The black curve represents the reference case, $v_{wx}=0.0$. Similar to the
earlier cases, for $v_{wx}=0.1$, the length scale follows diffusive growth:
$\phi\rightarrow 1/3$ up to $t_{sp}\simeq 600$ (red curve); the growth rate is
higher within this period for $v_{wx}=0.1$. Beyond this time, $R(t)$ deviates
significantly. We find a gradual dip in the length scale up to $t\simeq 1400$
(see the red curve in Figs. 8(e-f); the reason could be the domination of
shear rate over the diffusive growth, which is already slowing down due to the
topological constraint of BCP chains. Therefore, as noted earlier, BCP chains
in microdomains are possibly rearranging perpendicular to the shear direction
to reduce its effect may cause the dip during this period. The green and blue
curves at $v_{wx}=0.5$ and $1.0$ also illustrate the similar dip in growth at
much earlier times. After the dip in growth, $R(t)$ begins to grow further and
in contrast to cases 1-3, it follows diffusive growth ($\phi\sim 1/3$) for all
$v_{wx}\neq 0.0$ for a longer period before saturating. The length scale fully
saturates at a lower $R(t)$ for $v_{wx}=1.0$ (see the blue curves in Figs.
8(e-f)), which is evident from Fig. S4(c) exhibiting a well-formed lamellar
morphology. However, the length scales for $v_{wx}=0.1$ and $0.5$ seem to
saturate at the black curve (reference case) at late times. The effective
growth exponent in Fig. 8(f) demonstrates the saturation in length scale in
the asymptotic limit. Overall, the effect of shear, rendered by the horizontal
parallel walls moving in opposite directions, is significant in obtaining the
typical lamellar morphology much earlier in the microphase separating critical
BCP melt.

Figure 9: The structure factor, $S(k_{x},k_{y},k_{z})$ versus $k_{y}$ (black
curve) and $k_{z}$ (red curve) for (a) $v_{wx}=0.0$, (b) $v_{wx}=0.1$, (c)
$v_{wx}=0.5$, and (d) $v_{wx}=1.0$ to display the structural anisotropy in the
system. The green and blue curves in the inset demonstrate
$S(k_{x},k_{y},k_{z})$ versus $k_{y}$ along diagonal and cross-diagonal
directions respectively.

Recall that non-overlapping of the structure factor, $S(k_{x},k_{y},k_{z})$
along different directions, $k_{x}$, $k_{y}$, and $k_{z}$ represent the
structural anisotropy in the system. Here, we discuss the anisotropy in the
evolution morphology at a late time $t=5400$ for $v_{wx}=0.0$, $0.1$, $0.5$,
$1.0$, as displayed in Figs. 9(a-d). The black and red curves represent the
variation of $S(k_{x},k_{y},k_{z})$ vs. $k_{y}$ (normal to the stripes) and
$k_{z}$ (along with the stripes). For $v_{wx}=0.0$, lamellar patterns are not
fully developed in any particular direction. However, $S(k_{x},k_{y},k_{z})$
data along $k_{y}$ (black curve) shows a slightly larger peak and also a
developing shoulder at larger $k_{y}$ than in the red curve; this indicates
that lamellar patterns may evolve normal to the $y$-direction at much longer
times. Further, notice the stripes for $v_{wx}=0.1$ in Figs. S4(a), which are
nearly parallel to the $xy$-plane (i.e., normal to the $z$-axis). Thus, we
notice a moderately higher amplitude of structure factor peak at a smaller
$k_{z}$ (red curve) than in the black curve against $k_{y}$. However, an
emerging shoulder is apparent in both curves for the large $k_{y}$ and
$k_{z}$.

As noted earlier from Figs. S4(b-c), more distinct lamellar stripes are formed
at higher shear rates, i.e., at $v_{wx}=0.5$ and $1.0$, normal to the
$yz$-plane along the $x$-direction. Therefore, we notice multiple sharp, high
amplitude peaks in the structure factor along the $y$-direction (see the black
curves in Figs.9(c-d); these peaks depict the periodic $A$ and $B$-type
stripes. In contrast, the structure factor peak seen in the red curve at lower
shear rates diminishes at $v_{wx}=0.5$ and $1.0$. The reason is that the
stripes get more and more parallel to the $xz$-plane with the increase in
shear rate along the $x$-axis. Hence, these results verify the presence of
anisotropy in the $y$-direction. Furthermore, the green and blue curves in the
inset of Figs.9(a-d) exhibit the variation in $S(k_{x},k_{y},k_{z})$ along the
diagonals in the $yz$-plane versus $k_{y}$. The excellent data overlap for all
shear rates justifies the structural symmetry in the $yz$-plane along the
diagonals. Also, notice the structure factor peaks shown by the green and blue
curves; the main peaks get sharper and higher in amplitude with enhanced
shear, indicating more ordered lamellar morphology. We average the
$S(k_{x},k_{y},k_{z})$ data over sixty-four ensembles.

Figure 10: Anisotropy parameters, $D_{xy},D_{yz}$, and $D_{xz}$ are plotted as
a function of time for case 1 ($v_{wx}=0.0$) in (a) and for case 2 in (b-d)
for different wall velocities, $v_{wx}=0.1$, $0.5$, and $1.0$. Time variations
of $D_{yz}$ are compared in (e) and (f) for case 3 and case 4 at different
wall velocities depicted by various symbol types.

The time variation of the anisotropy parameter, $D(t)$, is plotted in Fig. 10
as denoted by various symbol types. Usually, due to structural constraints,
phase separating BCP melts have inherent anisotropy. Therefore, for case 1, we
have $D(t)\sim 0.4$ (see Fig. 10(a)), i.e., $D_{xy}$, $D_{yz}$, and $D_{xz}$
values are approximately same in this case. Thus, at our simulation’s time and
length scale, the typical lamellar morphology for a symmetric BCP melt is not
fully evolved yet for case 1 (reference case, see Figs. 1(a) and S1). For case
2, due to top wall shear in the $x$-direction, microdomains begin to reorient
themselves in forming lamellar slabs parallel to the $xz$-plane (as in Fig.
S2). As a result, $D(t)$ decreases in the $xz$-plane (green curve) and grows
$yz$-planes (red curve) with time. Furthermore, we notice that the differences
in $D(t)$ in different planes become more prominent at late times,
particularly for higher wall velocities. Thus, with the increase in wall
velocity, anisotropy in the $x$-direction (i.e., in the $yz$-plane) increases
more due to more ordered evolved morphology and decreases in the $xz$-plane.
At a low shear rate, $v_{wx}=0.1$, a decrease in $D_{xy}$ (black curve in Fig.
10(b)) suggests that only small and local slabs of both the phases are
developed in the $xy$-plane without any specific orientation. However,
$D_{xy}$ increases with higher shear rates. In Fig. 10(e) and (f), we compare
the time variation of $D_{yz}$ for case 3 and case 4 at different shear rates.
In both cases, we observe that evolved morphologies are more anisotropic
compared to case 2, as depicted in Fig. 10(b)-(d). Nevertheless, as discussed
earlier, we find a clearer lamellar morphology for case 4; thus, the
corresponding anisotropy parameter, $D_{yz}$, is also relatively higher than
in cases 2 and 3. The corresponding evolution snapshots are presented in Figs.
S2, S3 and S4, respectively.

Figure 11: (a) The variation of shear viscosity, $\langle\eta\rangle$ against
$v_{wx}$ at $t=5000$. The different symbol types at the top represent
different cases studied here. The inset depicts the time variation of
$\langle\eta\rangle$ for case 4 at different wall velocities indicated by
various symbols in the legend. (b) Average velocity profile, $\langle
v_{x}\rangle$ is plotted along the $z$-direction at $v_{wx}=1.0$ for all the
cases.

Finally, to understand the effect of shear on the phase separating BCP melt in
bulk, we plot the variation of shear viscosity $\langle\eta\rangle$ with
$v_{wx}$ in Fig. 11(a) for cases $2$-$4$ as denoted by the black, red, and
green symbols at a late time of evolution, $t=5000$. The symbol
$\langle\cdots\rangle$ denotes the average over ten ensembles. For each case,
the variation of $\langle\eta\rangle$ follows a similar pattern. i.e., with
the increase of shear rate, a gradual decrease in viscosity is noted,
illustrating the shear-thinning, which further saturates at higher shear rates
($v_{wx}$). Thus confirming the smaller average domain size, $R(t)$ at higher
$v_{wx}$ for cases 2-4 discussed above. The shear viscosity for case 4 (green
curve) is the lowest, indicating that the effect of shear is more prominent in
this case than the cases 2 and 3. In the inset of Fig. 11(a), we illustrate
the variation of $\langle\eta\rangle$ as a function of time at $v_{wx}\neq 0$
depicted by various symbol types for case 4. After an early decrease in shear
viscosity, it saturates quickly to an average value with tiny statistical
fluctuations. However, for a higher shear rate, we observed lower shear
viscosity.

Since the walls move along the $x$-direction, $v_{x}$ plays a dominant role in
the velocity profile. Therefore, we generate the velocity profiles, $\langle
v_{x}\rangle$, along the $z$-direction for all the cases at $v_{wx}=1.0$, as
displayed in Fig. 11(b). For this, we divide the system into thin slabs of
unit thickness along the $z$-direction and calculate the average velocity
components $\langle v_{x}\rangle$, $\langle v_{y}\rangle$, and $\langle
v_{z}\rangle$ over all the beads in a particular slab.[59, 60] The $y$ and $z$
components of the average velocity profile remain unaffected by the shear;
thus, we get $\langle v_{y}\rangle=\langle v_{z}\rangle=0$ for all the cases
(results are not shown here). The solid black line demonstrates the velocity
profile for case 1, where both walls are fixed, and hence, $\langle
v_{x}\rangle=0$. This is plotted as a reference curve for the other cases
studied in Fig. 11(b) at $v_{wx}=1.0$. In case $2$ (black symbol), as the
distance from the top wall increases, $\langle v_{x}\rangle$ gradually reduces
to zero. It reduces further and attains negative values with distance, which
gradually approaches zero close to the bottom wall. The velocity profile for
case $3$ (red curve) decreases smoothly with increasing distance from both the
walls and reaches a negative minimum value at $z\simeq L/2$. Finally, in case
$4$ (green curve), since both walls are moving in opposite directions,
$\langle v_{x}\rangle$ varies linearly with distance along the $z$-axis by
crossing zero near $z\simeq L/2$.

Overall, the symmetric variation of the velocity profile within positive and
negative values (i.e., the momentum flow in opposite directions) ensures
momentum conservation in the system. The momentum exchange further ensures the
drainage of excess heat generated by friction.[59] Thus, we observe only tiny
enhancements in the instantaneous temperature, $T_{i}\in(0.98-1.095)$ around
the quench temperature, $T=1.0$, but within the statistical error at various
shear rates in different cases, as displayed in Fig. S5.

##  4 Summary and conclusions

We utilize the DPD simulation technique to simulate the phase separation
kinetics of the critical BCP bulk melts under three unidirectional shear
conditions. The shear is applied by moving the parallel solid walls with a
constant velocity in a particular direction only. The walls confine the cubic
simulation box at the top and the bottom. The shear is allowed to affect the
system in three ways: (i) only the top wall moves (case 2). (ii) Both walls
move in the same direction (case 3). (iii) In the last case, both walls move
in opposite directions (case 4). We discuss the effect of shear rendered in
the system on evolution morphology, scaling functions, length scale, and
growth law for different cases. Further, we compare the anisotropy induced in
the system from the structure factor variation in different directions and
justify it further by computing the anisotropy parameter. All the shear-
influenced results are compared with no shear case, i.e., when both the walls
are fixed (case 1). Finally, we present the variation in shear viscosity and
velocity profiles for all the cases with wall velocity, which is proportional
to the shear rate.

On the application of shear, microdomains start flowing along the shear
direction. We notice that the lamellar formation always begins in the region
close to the moving wall. One of our significant observations is that
segregating BCP chains, regardless of chain size, begin to adjust
perpendicular to the shear direction and along the wall plane to overcome its
effect at moderate to high shear rates. This eventually leads to a periodic
lamellar pattern much earlier than case 1 (without shear). However, a fully
grown lamellar morphology is still not observed at the lower shear rate up to
the simulation period used in this work; nonetheless, our results suggest they
are more periodic than in case 1. The corresponding radial distribution
function, the local density profile variation, and the comparison of structure
factors in different directions perfectly characterized the morphology under
various shear conditions. The late time correlation function and the structure
factor offer a reasonably good scaling for all the shear rates in case 2 and
case 3. However, in case 4, where the shear effect is more pronounced on the
evolution morphology, the scaled correlation function and the structure factor
deviated from the scaling for all the velocities. The oscillatory nature of
the correlation function and the emerging shoulder at larger $kR(t)$ values in
the structure factor further justify the segregating periodic morphology.
Moreover, broadening of the structure factor main peak at various shear rates
signifies the shear-thinning of domain size in the asymptotic limit.

The competition between the microphase separation kinetics and the shearing
through wall motion leads to some fascinating behavior of characteristic
length scale. In particular, at lower shear rates, the flow of early small
domains and individual BCP chains in the shear direction seems to complement
the domain segregation. As a result, the average domain size becomes more
prominent for all the cases at early times. The variation of average shear
viscosity with shear rates (wall velocities) justifies that at lower shear
rates, shear viscosity is higher (i.e., low shear-thinning) than at moderate
and high shear rates. Interestingly, near the length scale crossover to
saturation for case 1, the length scale at the lower shear rates also slows
significantly (the length scale became almost flat) for case 2 and case 3
within a specific time interval.

On the other hand, we observe negative growth for case 4 for a slightly larger
period due to the shear effect dominating the diffusive growth during this
time interval. As a result, the BCP chains are merely adjusting themselves to
minimize the shear effect. Again, the length scale grows with time following
the diffusive growth and then begins to saturate at a length scale reasonably
lower or nearly equal to case 1 at late times for different cases.
Furthermore, moderate and large shear rates dominate the diffusive growth from
the beginning for almost all cases. Thus, the length scale growth rate, lower
from early times, saturates to a lower length scale at late times, indicating
the shear-thinning. Again, the variation of average shear viscosity with shear
rates justifies the same.

Overall, this work illustrates the influence of unidirectional shear induced
by moving solid walls on phase separation kinetics of the critical BCP bulk
melts. Besides, this work can guide quickly producing bulk composite materials
with diverse anisotropic morphology and various physical properties. Given the
different scientific and technological importance of morphologies obtained
from segregating BCP melts, these results will enable additional experimental
and theoretical interest in this problem. Further, we hope this work will
offer a broad framework to simulate the morphology growth in BCP melts and
polymer blends under such an external response.

## Conflicts of interest

There are no conflicts of interest to declare.

## Author Contributions

Conceptualization, AS, and AKS.; Methodology, AKS; Validation, AS; Formal
analysis and Investigation, AKS, and AS; Resources and Data curation, AKS;
Writing-original draft preparation, AKS; Writing-review and editing, AKS;
Visualization, AKS; Supervision, AS; Funding acquisition, AS and AKS. All
authors have read and agreed to the published version of the manuscript.

## Acknowledgement

A.K.S. is grateful to the CSIR fellowship (File No: 09/1217(0074)/2019-EMR-I),
New Delhi, for the financial support. A.S. acknowledges the financial support
from Grant No. ECR/2017/002529 by the Science and Engineering Research Board,
Department of Science and Technology, New Delhi, Government of India.

## Data Availability

Data will be made available on request.

## References

  * [1] Ludwik Leibler.  Theory of microphase separation in block copolymers.  Macromolecules, 13(6):1602–1617, 1980.
  * [2] Takeji Hashimoto, Mitsuhiro Shibayama, and Hiromichi Kawai.  Domain-boundary structure of styrene-isoprene block copolymer films cast from solution. 4. molecular-weight dependence of lamellar microdomains.  Macromolecules, 13(5):1237–1247, 1980.
  * [3] Takao Ohta and Kyozi Kawasaki.  Equilibrium morphology of block copolymer melts.  Macromolecules, 19(10):2621–2632, 1986.
  * [4] Frank S Bates and Glenn H Fredrickson.  Block copolymer thermodynamics: theory and experiment.  Annu. Rev. Phys. Chem., 41(1):525–557, 1990.
  * [5] IW Hamley.  Ordering in thin films of block copolymers: Fundamentals to potential applications.  Prog. Polym. Sci., 34(11):1161–1210, 2009.
  * [6] Awaneesh Singh, Raishma Krishnan, and Sanjay Puri.  Kinetics of microphase separation in block copolymers: A molecular-dynamics study.  EPL, 109(2):26006, 2015.
  * [7] Alexey A Gavrilov, Yaroslav V Kudryavtsev, and Alexander V Chertovich.  Phase diagrams of block copolymer melts by dissipative particle dynamics simulations.  J. Chem. Phys., 139(22):224901, 2013.
  * [8] Matthias Nebouy, Julien Morthomas, Claudio Fusco, Guilhem P Baeza, and Laurent Chazeau.  Coarse-grained molecular dynamics modeling of segmented block copolymers: impact of the chain architecture on crystallization and morphology.  Macromolecules, 53(10):3847–3860, 2020.
  * [9] Mihaiela C Stuparu, Anzar Khan, and Craig J Hawker.  Phase separation of supramolecular and dynamic block copolymers.  Polym. Chem., 3(11):3033–3044, 2012.
  * [10] TM Beardsley, RKW Spencer, and MW Matsen.  Computationally efficient field-theoretic simulations for block copolymer melts.  Macromolecules, 52(22):8840–8848, 2019.
  * [11] K Binder.  Phase transitions in polymer blends and block copolymer melts: some recent developments.  Springer, 1994.
  * [12] Alan J Bray.  Theory of phase-ordering kinetics.  Adv. Phys., 51(2):481–587, 2002.
  * [13] Sanjay Puri and Vinod Wadhawan, editors.  Kinetics of Phase transitions.  CRC Press, Boca Raton, Fla., 2009.
  * [14] Awaneesh Singh, Sanjay Puri, and Chandan Dasgupta.  Kinetics of phase separation in polymer mixtures: A molecular dynamics study.  J. Chem. Phys., 140(24):244906, 2014.
  * [15] Awaneesh Singh and Sanjay Puri.  Phase separation in ternary fluid mixtures: a molecular dynamics study.  Soft matter, 11(11):2213–2219, 2015.
  * [16] Ashish Kumar Singh, Avinash Chauhan, Sanjay Puri, and Awaneesh Singh.  Photo-induced bond breaking during phase separation kinetics of block copolymer melts: a dissipative particle dynamics study.  Soft Matter, 17(7):1802–1813, 2021.
  * [17] Amrita Singh, Anirban Chakraborti, and Awaneesh Singh.  Role of a polymeric component in the phase separation of ternary fluid mixtures: a dissipative particle dynamics study.  Soft Matter, 14(21):4317–4326, 2018.
  * [18] Fong Liu and Nigel Goldenfeld.  Dynamics of phase separation in block copolymer melts.  Phys. Rev. A, 39(9):4805, 1989.
  * [19] NM Maurits, AV Zvelindovsky, GJA Sevink, BAC Van Vlimmeren, and JGEM Fraaije.  Hydrodynamic effects in three-dimensional microphase separation of block copolymers: Dynamic mean-field density functional approach.  J. Chem. Phys., 108(21):9150–9154, 1998.
  * [20] Robert D Groot, Timothy J Madden, and Dominic J Tildesley.  On the role of hydrodynamic interactions in block copolymer microphase separation.  J. Chem. Phys., 110(19):9739–9749, 1999.
  * [21] Robert D Groot and Timothy J Madden.  Dynamic simulation of diblock copolymer microphase separation.  J. Chem. Phys., 108(20):8713–8724, 1998.
  * [22] Y Shiwa.  Hydrodynamic interactions in microphase separation of block copolymer films: Stability and spirals.  Phys. Rev. E, 61(3):2924, 2000.
  * [23] Aiguo Xu, G Gonnella, A Lamura, G Amati, and F Massaioli.  Scaling and hydrodynamic effects in lamellar ordering.  EPL, 71(4):651, 2005.
  * [24] Hongbo Feng, Xinyi Lu, Weiyu Wang, Nam-Goo Kang, and Jimmy W Mays.  Block copolymers: Synthesis, self-assembly, and applications.  Polymers, 9(10):494, 2017.
  * [25] Mark P Stoykovich and Paul F Nealey.  Block copolymers and conventional lithography.  Mater. Today, 9(9):20–29, 2006.
  * [26] Christopher Harrison, John A Dagata, Douglas H Adamson, et al.  Lithography with self-assembled block copolymer microdomains.  Wiley Chichester, 2004.
  * [27] Annette Rosler, Guido WM Vandermeulen, and Harm-Anton Klok.  Advanced drug delivery devices via self-assembly of amphiphilic block copolymers.  Adv. Drug Deliv. Rev., 64:270–279, 2012.
  * [28] Rachel K O’Reilly, Craig J Hawker, and Karen L Wooley.  Cross-linked block copolymer micelles: functional nanostructures of great potential and versatility.  Chem. Soc. Rev, 35(11):1068–1083, 2006.
  * [29] Tian-Yi Ma, Lei Liu, and Zhong-Yong Yuan.  Direct synthesis of ordered mesoporous carbons.  Chem. Soc. Rev, 42(9):3977–4003, 2013.
  * [30] Blair Fraser, Colin Denniston, and Martin H Muser.  On the orientation of lamellar block copolymer phases under shear.  J. Chem. Phys., 124(10):104902, 2006.
  * [31] Sufi R Ahmed, Satish B Ogale, and Peter Kofinas.  Magnetic properties and morphology of block copolymer templated ferrimagnetic cofe/sub 2/o/sub 4/nanoparticles.  IEEE Trans. Magn., 39(5):2198–2200, 2003.
  * [32] T Thurn-Albrecht, J Schotter, Gi A Kastle, N Emley, T Shibauchi, L Krusin-Elbaum, K Guarini, CT Black, MT Tuominen, and TP Russell.  Ultrahigh-density nanowire arrays grown in self-assembled diblock copolymer templates.  Science, 290(5499):2126–2129, 2000.
  * [33] Ian W Hamley.  Structure and flow behaviour of block copolymers.  J. Condens. Matter Phys., 13(33):R643, 2001.
  * [34] TL Morkved, M Lu, AM Urbas, EE Ehrichs, HM Jaeger, P Mansky, and TP Russell.  Local control of microdomain orientation in diblock copolymer thin films with electric fields.  Science, 273(5277):931–933, 1996.
  * [35] Sang Ouk Kim, Harun H Solak, Mark P Stoykovich, Nicola J Ferrier, Juan J De Pablo, and Paul F Nealey.  Epitaxial self-assembly of block copolymers on lithographically defined nanopatterned substrates.  Nature, 424(6947):411–414, 2003.
  * [36] Gaurav Arya, Jorg Rottler, Athanassios Z Panagiotopoulos, David J Srolovitz, and Paul M Chaikin.  Shear ordering in thin films of spherical block copolymer.  Langmuir, 21(24):11518–11527, 2005.
  * [37] Kristoffer Almdal, Kurt A Koppi, and Frank S Bates.  Dynamically sheared body-centered-cubic ordered diblock copolymer melt.  Macromolecules, 26(15):4058–4060, 1993.
  * [38] Glen A McConnell, Min Y Lin, and Alice P Gast.  Long range order in polymeric micelles under steady shear.  Macromolecules, 28(20):6754–6764, 1995.
  * [39] Diane B Scott, Alan J Waddon, Ye Gang Lin, Frank E Karasz, and H Henning Winter.  Shear-induced orientation transitions in triblock copolymer styrene-butadiene-styrene with cylindrical domain morphology.  Macromolecules, 25(16):4175–4181, 1992.
  * [40] Kurt A Koppi, Matthew Tirrell, and Frank S Bates.  Shear-induced isotropic-to-lamellar transition.  PRL, 70(10):1449, 1993.
  * [41] Karen I Winey, Sanjay S Patel, Ronald G Larson, and Hiroshi Watanabe.  Interdependence of shear deformations and block copolymer morphology.  Macromolecules, 26(10):2542–2549, 1993.
  * [42] Z-R Chen, AM Issaian, JA Kornfield, SD Smith, JT Grothaus, and MM Satkowski.  Dynamics of shear-induced alignment of a lamellar diblock: a rheo-optical, electron microscopy, and x-ray scattering study.  Macromolecules, 30(23):7096–7114, 1997.
  * [43] Dan E Angelescu, Judith H Waller, Douglas H Adamson, Paru Deshpande, Stephen Y Chou, Richard A Register, and Paul M Chaikin.  Macroscopic orientation of block copolymer cylinders in single-layer films by shearing.  Adv. Mater., 16(19):1736–1740, 2004.
  * [44] Dan E Angelescu, Judith H Waller, Richard A Register, and Paul M Chaikin.  Shear-induced alignment in thin films of spherical nanodomains.  Adv. Mater., 17(15):1878–1881, 2005.
  * [45] Arash Nikoubashman, Richard A Register, and Athanassios Z Panagiotopoulos.  Simulations of shear-induced morphological transitions in block copolymers.  Soft Matter, 9(42):9960–9971, 2013.
  * [46] Brandon L Peters, Abelardo Ramirez-Hernandez, Darin Q Pike, Marcus Muller, and Juan J de Pablo.  Nonequilibrium simulations of lamellae forming block copolymers under steady shear: A comparison of dissipative particle dynamics and brownian dynamics.  Macromolecules, 45(19):8109–8116, 2012.
  * [47] Robert D Groot and Patrick B Warren.  Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation.  J. Chem. Phys., 107(11):4423–4435, 1997.
  * [48] Pep Espanol and Patrick Warren.  Statistical mechanics of dissipative particle dynamics.  EPL, 30(4):191, 1995.
  * [49] Pep Espanol and Patrick B Warren.  Perspective: Dissipative particle dynamics.  J. Chem. Phys., 146(15):150901, 2017.
  * [50] Robert D Groot.  A local galilean invariant thermostat.  J. Chem. Theory Comput., 2(3):568–574, 2006.
  * [51] Petri Nikunen, Mikko Karttunen, and Ilpo Vattulainen.  How would you integrate the equations of motion in dissipative particle dynamics simulations?  Comput. Phys. Commun., 153(3):407–423, 2003.
  * [52] Petri Nikunen, Ilpo Vattulainen, and Mikko Karttunen.  Reptational dynamics in dissipative particle dynamics simulations of polymer melts.  Phys. Rev. E, 75(3):036713, 2007.
  * [53] Steve Plimpton.  Fast parallel algorithms for short-range molecular dynamics.  J. Comput. Phys., 117(1):1–19, 1995.
  * [54] Christoph Junghans, Matej Praprotnik, and Kurt Kremer.  Transport properties controlled by a thermostat: An extended dissipative particle dynamics thermostat.  Soft Matter, 4(1):156–161, 2008.
  * [55] Hsiao-Ping Hsu and Kurt Kremer.  Static and dynamic properties of large polymer melts in equilibrium.  J. Chem. Phys., 144(15):154907, 2016.
  * [56] Timothy W Sirk, Yelena R Slizoberg, John K Brennan, Martin Lisal, and Jan W Andzelm.  An enhanced entangled polymer model for dissipative particle dynamics.  J.chem. phys., 136(13):134903, 2012.
  * [57] ACC Esteves, K Lyakhova, LGJ Van Der Ven, RATM Van Benthem, and G De With.  Surface segregation of low surface energy polymeric dangling chains in a cross-linked polymer network investigated by a combined experimental–simulation approach.  Macromolecules, 46(5):1993–2002, 2013.
  * [58] Ya Liu, Gerald T McFarlin IV, Xin Yong, Olga Kuksenok, and Anna C Balazs.  Designing composite coatings that provide a dual defense against fouling.  Langmuir, 31(27):7524–7532, 2015.
  * [59] Florian Muller-Plathe.  Reversing the perturbation in nonequilibrium molecular dynamics: An easy way to calculate the shear viscosity of fluids.  Physical Review E, 59(5):4894, 1999.
  * [60] Manish S Kelkar, Jake L Rafferty, Edward J Maginn, and J Ilja Siepmann.  Prediction of viscosities and vapor–liquid equilibria for five polyhydric alcohols by molecular simulation.  Fluid Phase Equilibria, 260(2):218–231, 2007.
  * [61] Akira Onuki and Masao Doi.  Flow birefringence and dichroism of polymers. i. general theory and application to the dilute case.  The Journal of Chemical Physics, 85(2):1190–1197, 1986.
  * [62] Gang Zhang, Zhongyao Qin, Yu Qian, and Jiaming Zhu.  Microstructural evolution and kinetics of phase separation in binary polymer blends under electric fields.  Computational Materials Science, 213:111659, 2022.
  * [63] Yoshitsugu Oono and Sanjay Puri.  Computationally efficient modeling of ordering of quenched phases.  PRL, 58(8):836, 1987.
  * [64] Awaneesh Singh, A Mukherjee, HM Vermeulen, GT Barkema, and Sanjay Puri.  Control of structure formation in phase-separating systems.  J. Chem. Phys., 134(4):044910, 2011.
  * [65] Yoshitsugu Oono and Sanjay Puri.  Study of phase-separation dynamics by use of cell dynamical systems. i. modeling.  Phys.Rev. A, 38(1):434, 1988.
  * [66] David A Huse and Christopher L Henley.  Pinning and roughening of domain walls in ising systems due to random impurities.  PRL, 54(25):2708, 1985.
  * [67] G Porod.  Small angle X-ray scattering.  Academic Press, New York, 1982.

## Supplementary Information

Figure S1: Comparison of late stage morphology of phase separating critical
BCP melt for (a) $N_{p}=8$ and (b) $N_{p}=16$. The second right frame shows
the corresponding isosurfaces.

Figure S2: Morphology evolution of BCP melt when the top wall is moving in
positive $x$-direction with different velocities (a) $v_{wx}=0.1$, (b)
$v_{wx}=0.5$, and (c) $v_{wx}=1.0$ . The first column snapshots are at an
early time ($t=120$), and the second column images are at a late time step
($t=5400$). The magenta arrows show the shear direction. The third column
displays the isosurfaces for the snapshots in the second column.

Figure S3: The evolution snapshots at two time steps $t=120$ and $t=5400$ when
top and bottom walls are moving in the same direction with the same velocities
(a) $v_{wx}=0.1$, (b) $v_{wx}=0.5$, and (c) $v_{wx}=1.0$, respectively. The
third column displays the isosurfaces for the snapshots in the second column.
The magenta arrows show the shear direction.

Figure S4: Evolution morphologies at two-time steps for wall velocities (a)
$v_{wx}=0.1$, (b) $v_{wx}=0.5$, and (c) $v_{wx}=1.0$ when both walls are
moving with the same velocity but in the opposite direction; the top wall is
moving in positive $x$-direction and the bottom wall is moving in negative
$x$-direction. The third column displays the isosurfaces for the snapshots in
the second column. The magenta arrows show the shear direction.

Figure S5: The first column ((a), (c), and (e)) compares the time variation of
average bond length, $\langle l_{b}\rangle$ at different shear rates,
$\dot{\gamma}\propto v_{wx}$, as denoted by various symbols in the legend for
three cases displayed at the extreme right. These curves illustrate the
negligible bond stretching with increased shear rates caused by wall
velocities. The second column ((b), (d), and (f)) compares the instantaneous
temperature variation with time for the same cases as in the first column.
Again, we observe tiny enhancements in $T_{i}\in(0.98-1.095)$ around the
quench temperature, $T=1.0$, but within the statistical error at various shear
rates in different cases. In short, these results certainly disapprove any
ambiguity that may cause due to the high shear rates, mainly due to excess
bond stretching or temperature enhancement.