11institutetext:  School of Interdisciplinary Research, Indian Institute of
Technology, Hauz Khas, New Delhi – 110016, India
Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi –
110016, India
School of Physical Sciences, Jawaharlal Nehru University, New Delhi – 110067,
India

First pacs description Second pacs description Third pacs description

# Ordering Kinetics of Canted and Uniform States in
Nematic Liquid Crystals

Nishant Birdi 11 Varsha Banerjee 1122 Sanjay Puri 1133112233

###### Abstract

We undertake a comprehensive Monte Carlo (MC) study of the ordering kinetics
in nematic liquid crystals (NLCs) in 3-dimensions $(d=3)$ by performing deep
quenches from the isotropic $(T>T_{c})$ to the nematic $(T<T_{c})$ phase. The
inter-molecular potential between the nematogens, represented by continuous
$O(3)$ spins with inversion symmetry, is accurately mimicked by the
generalised Lebwohl Lasher (GLL) model. It incorporates second and fourth
order Legendre interactions, and their relative interaction strength is
$\lambda$. For $\lambda<-0.3$, we observe canted morphologies with a
$\lambda$-dependent angle-of-tilt between the neighbouring rod-like molecules.
For $\lambda\geq-0.3$, the molecules align to yield uniform states. The
coarsening morphologies obey generalized dynamical scaling in the two regimes,
but the scaling function is not robust with respect to $\lambda$. The
structure factor tail in the canted regime follows the Porod law: $S(k,t)\sim
k^{-4}$, implying that the coarsening dynamics is due to the annihilation of
interfacial defects. This is unexpected, as the GLL model is characterised by
a continuous order parameter. The uniform regime on the other hand, exhibits
the expected generalized Porod decay: $S(k,t)\sim k^{-5}$, characteristic of
scattering from string defects. Finally, the domain growth obeys the Lifshitz-
Allen-Cahn law: $L(t)\sim t^{1/2}$ for all values of $\lambda$. Our results
for the novel canted regime are relevant for a large class of systems with
orientational ordering, e.g. active matter, membranes, LC elastomers, etc. We
hope that our work triggers-off stimulating investigations in them.

###### pacs:

nn.mm.xx

###### pacs:

nn.mm.xx

###### pacs:

nn.mm.xx

Liquid crystals (LCs) are a state of matter that is intermediate between
conventional solids and liquids [1, 2, 3, 4]. They are anisotropic in nature,
and combine the fluidity of liquids with the long-range order of solids.
Chemically, LCs are organic compounds consisting of carbon, hydrogen, oxygen
and nitrogen. Based on the structural arrangement of their constituent
elements, the LC molecules are broadly classified as calamitic (rod shaped),
discotic (disc shaped) and bent-core (banana shaped). Calamitic LCs are the
simplest in this family, with tremendous applications in electro-optic devices
- the most significant being liquid crystal displays (LCDs) [5]. At high
temperatures, they exhibit an isotropic (I) phase characterized by randomly
oriented rods (or nematogens), having translational as well as rotational
symmetry. At low temperatures, the nematogens align statistically parallel in
an arbitrary direction to yield the nematic (N) phase. It is characterized by
long-range orientational order which extends over thousands of molecules. If
the N phase is uniaxial, it is described by a sign-invariant unit vector known
as the nematic director n.

LCs are experimentally accessible systems possessing continuous symmetry, and
allow the study of topological defects which are of great interest to the
scientific community [6, 7, 8, 9, 10, 11]. A number of approaches, ranging
from continuum free energy models to off-lattice and on-lattice models have
come forth to understand the emergent phases and their optical, electrical and
magnetic responses [4]. The quest for a unified theoretical framework persists
for this fundamental and technologically rich system, to not only understand
experimental observations in conventional LCs, but to also include novel LC-
based materials such as the biaxial bentcores [12] and the ferromagnetic
nematics [13].

One of the early theoretical frameworks to understand the I-N phase transition
was provided by Maier and Saupe (MS) [14, 15]. They used an approximate
molecular field theory to obtain the pseudo-potential between two rod-like
nematogens. The MS theory did predict the I-N transition to be first order,
but the quantitative evaluations for the orientational order parameter were
far from accurate. Consequently, Humphries et al. extended the MS theory by
expressing the pseudo-potential in terms of a complete set of Legendre
polynomials $\ P_{l}(z)$ [16]. Due to the inversion symmetry of the rod-like
molecules, this expansion contains only the even order polynomials ($P_{2},\
P_{4},$ etc.), and the coefficients are found to decrease rapidly with
increasing $l$ [17]. The MS theory is recovered if the summation in the
expansion is restricted to only $l=2$. The simplest three dimensional ($d=3$)
lattice version of the MS theory is the Lebwohl-Lasher (LL) model [18]. Each
lattice site $i$ has a rod-like molecule with orientation of the long axis
defined by a three-component ($n=3$) unit vector
$\bm{S}_{i}=\left(\sin\vartheta_{i}\cos\varphi_{i},\sin\vartheta_{i}\sin\varphi_{i},\cos\vartheta_{i}\right)$
with inclination $\vartheta_{i}\in[0,\pi]$ and azimuth
$\varphi_{i}\in[0,2\pi)$. The Hamiltonian for the LL model is given by:

$H=-J\sum_{\langle ij\rangle}P_{2}\left(\cos\theta_{ij}\right),$ (1)

where $J$ is the strength of the nearest neighbour (nn) interactions,
$\theta_{ij}$ is the angle between the nn nematic rods
($\cos\theta_{ij}=\bm{S}_{i}\cdot\bm{S}_{j}$) and
$P_{2}(z)=\left(3z^{2}-1\right)/2$. In subsequent discussions, we set $J=1$
for convenience. The orientational order parameter is given by:

$\mathcal{S}=\langle
P_{2}\left(\cos\theta_{i}\right)\rangle=\left\langle\frac{3\cos^{2}\theta_{i}-1}{2}\right\rangle.$
(2)

In the above expression, $\cos\theta_{i}=\bm{S}_{i}\cdot\bm{n}$ and the
angular brackets $\langle\cdot\cdot\cdot\rangle$ imply an ensemble average.
The isotropic phase corresponds to $\mathcal{S}=0$ while a fully aligned
nematic phase has $\mathcal{S}=1$. A defect corresponds to a region of low
order or $\mathcal{S}\simeq 0$.

The LL model has certain disadvantages inherent in a lattice model. For
example, the translational motion of molecules is eliminated, so that one
cannot discriminate between a liquid crystal and a crystal. It is unclear
whether the structures seen in the LL model or its generalizations would
survive if molecular motion were permitted. Further, diffusive motion and
hydrodynamics is also absent, along with the corresponding conservation laws.
Nevertheless, it has been widely used in the literature to understand the
nematic mesophase because of it’s simplicity, analytical tractability and
computational ease. The LL model exhibits a weak first order I-N transition,
and inherits the quantitative discrepancies of the MS theory [19].

The shortcomings of the LL model have been overcome by the inclusion of higher
order long-range and short-range interactions. Scattering experiments indicate
that only the terms upto $P_{4}$ in the expansion of the pseudo-potential are
statistically significant [20]. Consequently, a reasonable approximation
representing the interaction between neighbouring nematogens is the
generalized Lebwohl-Lasher (GLL) model, described by the Hamiltonian [21, 22]:

${\color[rgb]{0,0,0}H=\sum_{\langle
ij\rangle}-\big{[}P_{2}\left(\cos\theta_{ij}\right)+\lambda
P_{4}\left(\cos\theta_{ij}\right)\big{]}\equiv\sum_{\langle
ij\rangle}E(\cos\theta_{ij})},$ (3)

where $\lambda$ is the relative strength of the $P_{4}$ interactions with
respect to the $P_{2}$ interactions. It is a measure of anisotropy in the
system and can be positive or negative. The Legendre polynomial
$P_{4}(z)=\left(35z^{4}-30z^{2}+3\right)/8$.  There are very few studies of
the GLL model, and these are limited to investigations of the effect of the
$P_{4}$ term on the isotropic-ordered phase transition [23, 21, 22, 24]. With
increasing strength of the $P_{4}$ term, the transition temperature
$T_{c}(\lambda)$ increases and the transition becomes first-order. Further,
the inclusion of the $P_{4}$ term improves agreement between experimental
observations and numerical results.

In this letter, we perform MC simulations with non-conserved order parameter
dynamics to study the kinetics of phase transition or coarsening in the $d=3$
NLCs. We study this important non-equilibrium phenomenon using the GLL model,
motivated by it’s success of accurately describing the equilibrium properties.
The phenomenon of coarsening is initiated when a system is rendered
thermodynamically unstable following a quench of an intensive thermodynamic
state variable such as temperature or pressure. The initial symmetry of the
disordered phase gets spontaneously broken at the onset of order. A slow non-
equilibrium evolution begins via the emergence of correlated regions of
competing equilibrium phases or domains separated by interfaces or defects.
The domains grow in size via annihilation of defects [25, 26, 27]. This far-
from-equilibrium progressive dynamics is usually termed as domain growth or
coarsening. The nature of the ordering dynamics depends upon several factors
such as the symmetry of the order parameter, conservation laws, relevance of
hydrodynamics, etc. The growth laws also provide important insights about the
free-energy landscape and relaxation time-scales in the system.

The main observations from our study are as follows:
(a) For $\lambda<-0.3$, the system exhibits canted ground states (GS) with a
$\lambda$-dependent angle of tilt between the nn nematic rods. For
$\lambda\geq-0.3$ on the other hand, uniform GS with a parallel alignment of
nn rods are observed.
(b) The domain growth exhibits dynamical scaling in the two regimes. The
ordering dynamics in each regime is characterized by a unique length scale,
and the morphology of the domains does not change with time apart from a scale
factor. The scaling functions in the canted and uniform regimes are distinct,
indicating that the morphology is not robust with respect to $\lambda$.
(c) The structure factor for the canted domain morphologies exhibits an
unexpected Porod decay, $S(k)\sim k^{-(d+1)}$, characteristic of scattering
from sharp interfaces. The uniform morphologies on the other hand, exhibit the
expected generalized Porod decay, $S(k)\sim k^{-(d+2)}$, due to scattering
from string defects typical of the continuous order parameter $O(n)$ models
with inversion symmetry.
(d) In both regimes, the domain growth obeys the Lifshitz-Allen-Cahn (LAC) law
$L(t)\sim t^{1/2}$, characteristic of systems with non-conserved order
parameter kinetics. The growth is faster with increasing values of $\lambda$.
The GLL model offers a simple framework to study geometric incompatibilities
arising in orientational ordering. Such systems are ubiquitous, and our
results therefore are relevant for the large class of complex systems with
orientational order.

Coarsening in $d=2$ NLCs is well studied, using coarse-grained free energy
models [28, 29, 30] as well as discrete models such as the LL [31, 32, 33, 34,
35] and the GLL[36]. These studies predict the LAC law for domain growth, with
logarithmic corrections. The primary defects in $d=2$ are vortices and anti-
vortices. There are limited studies in $d=3$ using coarse-grained free energy
models [37, 38], lattice models such as the LL [31, 39, 40] and the off-
lattice Gay Berne (GB) model [41]. The domain growth obeys the LAC law, and
strings and hedgehogs are the dominant defects.

Let us first examine the GS $(T=0)$ of Eq. (3) as a function of $\lambda$.
Fig. 1(a) shows the variation of $E(\cos\theta_{ij})$ vs. $\cos\theta_{ij}$
for three different values of $\lambda=-0.8,0.2,0.8$. Using the standard
energy minimization procedures, it is easy to see that this model has two
distinct classes of GS: (i) For $\lambda<-0.3$, the global minimum of the
energy results for
$\theta_{ij}=\theta_{c}=\cos^{-1}\left[\left(15\lambda-6\right)/\left(35\lambda\right)\right]^{1/2}$
(or $\pi-\theta_{c}$). The variation of $\theta_{c}$ for different values of
$\lambda$ is specified in Table. 1. Thus, in the GS, the nn nematic rods are
aligned at an angle $\theta_{c}$ which is unique to each value of $\lambda$.
Note that the subsequent nn rods can form rotating or zig-zag arrangements,
see the schematic in Fig. 1(b) for $\lambda=-0.8$
($\theta_{c}=36.69^{\circ}$). We refer to these as the canted GS. To the best
of our knowledge, they have not been investigated in the literature. (ii) For
$\lambda\geq-0.3$, the global minimum of the energy is observed at
$\theta_{ij}=\theta_{u}=0$ (or $\pi$), as seen in the schematic in Fig. 1(c).
We refer to it as uniform GS. For $\lambda>0.4$, there is a local minimum of
the energy at $\theta_{ij}=\pi/2$ corresponding to a perpendicular alignment
of rods which is metastable. It has consequences on the early stage of domain
growth, as will be seen in due course. It should be noted that any
configuration where the angle between nn rods is $\theta_{c}$ or $\theta_{u}$
as the case may be, is a GS. There are infinite GS corresponding to any value
of $\lambda$.

[scale=0.25]f1.pdf

Figure 1: (a) Plot of the energies of the nn pairs in the GLL model, $E(\cos\theta_{ij})$ vs. $\cos\theta_{ij}$, for different values of $\lambda$. Schematic for the (b) canted and (c) uniform ground states. In (b), all the nn rods have an angle of $\simeq 36.7^{\circ}$ ($\lambda=-0.8$) between them and the rods show a rotating or a zig-zag arrangement. Table 1: Ground state minimum in the canted regime, $\theta_{c}$ (in degrees) for different values of $\lambda$. $\lambda$ |  -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4
---|---
$\theta_{c}$ |  39.23 38.11 36.69 34.84 32.31 28.56 22.21

We now proceed to study the domain growth kinetics in the $d=3$ GLL model
after a quench from the disordered phase $(T>T_{c})$ to the ordered phase
$(T<T_{c})$. These quenches have been guided by the equilibrium phase diagram
obtained by Chiccoli et al. using cluster MC simulations [22].  We consider
simple cubic lattices of size $N=N_{L}^{3}=(256)^{3}$ with periodic boundary
conditions, and typically quench the system to $T=0.5~{}T_{c}(\lambda)$. These
deep quenches ensure that the non-equilibrium evolution is not influenced by
the properties of the phase transition. The initial configuration is chosen to
be of randomly oriented nematic rods, corresponding to the $T=\infty$ phase.
The non-equilibrium evolution is studied via MC simulations. They provide an
efficient route to the equilibrium state as compared to the alternative
Molecular Dynamics (MD) simulations [42]. A randomly chosen nematogen
$\bm{S}_{i}$ is rotated by a small amount and the new orientation is accepted
with standard Metropolis acceptance rates [43]. Each data set has been
averaged over 20 different initial conditions. Error bars are not indicated in
the figures as the standard errors in the data are smaller than the symbol
sizes.

In Fig. 2, we show prototypical domain evolution snapshots (a) in the canted
regime for $\lambda=-0.8$ and (b) the uniform regime for $\lambda=0.8$ at $t$
(in MCS) $=10^{5}$. The different colors correspond to the orientation
$(\vartheta,\varphi)$ of the nematic rods and are specified in the key. (The
range of $\vartheta$ is halved due to the inversion symmetry of the
nematogens.) For $\lambda=-0.8$, the relative orientation between the nn
nematic rods in the GS is $\theta_{c}\simeq 36.7^{\circ}$ (or
$143.3^{\circ})$, see Table I. Hence the domains in Fig. 2(a) are not
monochromatic because of the canted arrangements of the nematogens. For
clarity, we explicitly show the orientation of the nematic rods in a
horizontal ($xy$) slice taken at $z=N_{L}/2=128$ in Fig. 2(b). (Only a
$32^{2}$ corner of the entire $256^{2}$ slice is shown for a distinct view).
The morphology in Fig. 2(c) on the other hand, is characterized by
monochromatic (similar orientation) domains as the corresponding GS has
$\theta_{u}=0$ (or $\pi$). The corresponding horizontal slice in Fig. 2(d)
confirms the emergence of correlated regions of parallel rods characteristic
of the uniform regime. We mention here that canted states have been observed
in assemblies of rod-shaped particles with long-range interactions. For
instance, MD simulations of nematogens with Gay Berne interactions showed
“shape frustrations” and “rippled” (or canted) structures in the smectic phase
in concurrence with experimental observations [44, 45]. It is interesting to
note that the canted states are preserved even in these off-lattice
descriptions incorporating hydrodynamics.

[scale=0.31]f2.pdf

Figure 2: Domain evolution snapshots of the GLL model for (a) $\lambda=-0.8$
and (c) $\lambda=0.8$ at $t=10^{5}$. The lattice size is $256^{3}$. [(b) and
(d)] Orientations of the nematic rods in the horizontal ($xy$) slices (at
$z=N_{L}/2=128$) of the evolution snapshots shown in (a) and (c). For a better
and clear view, we show only a $32^{2}$ corner of the entire $256^{2}$ slice.
These rods depict the formation of canted and uniform states, for the
respective $\lambda$ values. The key explaining the angular orientations
($\vartheta,\varphi$ in degrees) of the nematic rods in the evolution
snapshots shown in (a) and (c) is given at the bottom of the figure.

Further insights on the coarsening morphologies are provided in Fig. 3 which
shows horizontal ($xy$) slices ($32^{2}$ corner of the entire slice taken at
$z=N_{L}/2=128$) with bonds and interfacial defects for $\lambda=-0.8$ (top
row) at $t=$ (a) $10^{2}$, (b) $10^{5}$ and for $\lambda=0.8$ (bottom row) at
$t=$ (d) $10^{2}$, (e) $10^{5}$. In these slices, the orientation
$\theta_{ij}$ between nn rods is evaluated. If
$\theta_{ij}=\theta_{c}(\lambda)\pm 10^{\circ}$, the midpoint of the
corresponding nn sites is coloured green (for $\lambda=-0.8$) or red (for
$\lambda=0.8$). In case of a defect (i.e.
$\theta_{ij}\neq\theta_{c}(\lambda)\pm 10^{\circ}$), the midpoint is coloured
black. Morphologies in both the regimes are isotropic and coarsen with time.
This is evident from the plot of nn angle distribution $P(\theta_{ij})$ vs.
$\theta_{ij}$ at different times shown in Fig. 3(c) for $\lambda=-0.8$ and
Fig. 3(d) for $\lambda=0.8$. As expected, the distributions for $\lambda=-0.8$
in Fig. 3(c) peak at $\theta_{ij}\equiv\theta_{c}\simeq 36.7^{\circ}$ and
$143.3^{\circ}$ as time evolves. For $\lambda=0.8$ in Fig. 3(f), the
morphologies evolve toward the GS corresponding to
$\theta_{ij}\equiv\theta_{u}=0$ (or $\pi$). Notice the secondary peak at
earlier times, due to the metastablity from $\theta_{ij}=\pi/2$. The domain
growth is faster for $\lambda=0.8$, suggested from the magnitude of the peak
in these distributions at later times. An important point of distinction
between the canted and uniform morphologies is the distinct nature of defects
- interfaces in the canted state and blobs in the uniform state. We will make
quantitative statements about them shortly.

[scale=0.25]f3.pdf

Figure 3: Horizontal ($xy$) slices (at $z=N_{L}/2=128$) of the bond evolution
snapshots for $\lambda=-0.8$ at $t=$ (a) $10^{2}$, (b) $10^{5}$ and for
$\lambda=0.8$ at $t=$ (d) $10^{2}$, (e) $10^{5}$. The interfacial defects
between the nn lattice sites are coloured in black while the bonds between the
nn sites (with a tolerance of $10^{\circ}$) are coloured in green for
$\lambda=-0.8$ and in red for $\lambda=0.8$, respectively. The white regions
are the corresponding lattice sites. For a better and clear view, we show only
a $32^{2}$ corner of the entire $256^{2}$ slice. Plot of the nn angle
distribution, $P(\theta_{ij})$ vs. $\theta_{ij}$ for (c) $\lambda=-0.8$ and
(f) $\lambda=0.8$, at different times.

Next, we quantify the domain morphologies. The usual probe is the correlation
function:
$C(\vec{r},t)=\frac{1}{N}\sum_{i}^{N}[\langle\psi(\vec{r}_{i},t)\psi(\vec{r}_{i}+\vec{r},t)\rangle-\langle\psi(\vec{r}_{i},t)\rangle\langle\psi(\vec{r}_{i}+\vec{r},t)\rangle]$,
where $\psi(\vec{r}_{i},t)$ is the order parameter, and
$\langle\cdot\cdot\cdot\rangle$ represents the thermal average. Small-angle
scattering experiments yield the structure factor
$S(\vec{k},t)=\int\\!\mathrm{d}\vec{r}~{}\exp(-i\vec{k}\cdot\vec{r})~{}C(\vec{r},t)$,
where $\vec{k}$ is the wave-vector of the scattered beam. A characteristic
length scale $L(t)$ is usually defined as the distance at which $C(\vec{r},t)$
decays to, say, $0.2$ times its maximum value. If the domain growth is
characterized by a unique length scale $L(t)$, then $C(\vec{r},t)$ and
$S(\vec{k},t)$ show the dynamical scaling property [26, 27]:
$C(\vec{r},t)=g(r/L)$; $S(\vec{k},t)=L^{d}f(kL)$.  For each value of
$\lambda$, we calculate $C(\vec{r},t)$ using the local order parameter
$\psi(\vec{r}_{i},t)\equiv P_{2}(\cos\theta_{i})$. Fig. 4(a) shows the scaled
correlation function, $C(r,t)$ vs. $r/L(t)$, at $t$ (in MCS) $=5\times
10^{4},\ 10^{5}$ for $\lambda=-0.8$ and $\lambda=0.8$. The data for different
$t$-values exhibits an excellent collapse indicating that the domain growth
exhibits dynamical scaling for both values of $\lambda$. However, the
small-$r$ behavior for the two values of $\lambda$ is distinct. The scaling
function is not robust with respect to $\lambda$ or in other words does not
exhibit super-universality.

What are the implications of the absence of super-universality in Fig. 4(a)?
Let us examine the corresponding structure factor $S(k,t)$ \- especially the
asymptotic (large-$k$) tail which contains information about the defects in
the system. Continuous $O(n)$ spin models,  whose Hamiltonians are invariant
under rotation, exhibit the generalized Porod law with the asymptotic form:
$S(k,t)\sim k^{-(d+n)}$ [25, 26, 27]. For $n=1$, the defects are interfaces,
and the corresponding scattering function is the Porod law. For $n>1$, the
different topological defects are vortices ($n=2,\ d=2$), strings ($n=2,\
d=3$), and monopoles or hedgehogs ($n=3,\ d=3$). Fig. 4(b) shows the plot of
$L(t)^{-3}S(k,t)$ vs. $kL(t)$, corresponding to the data sets in Fig. 4(a), on
a double log scale. The data for $\lambda=0.8$ has been shifted for clarity.
For $\lambda=-0.8$ (canted regime), the data exhibits Porod decay $S(k,t)\sim
k^{-(d+1)}=k^{-4}$ for $d=3$ characteristic of scattering from sharp
interfaces. This is surprising because $\mathcal{S}$ is a continuous order
parameter. For $\lambda=0.8$ (uniform regime), we find that the asymptotic
tail exhibits the generalized Porod decay $S(k,t)\sim k^{-(d+2)}=k^{-5}$,
indicating the predominance of string defects in the ordering kinetics. The
distinct dominating defect structures in the two regimes are indeed evident in
Fig. 5.

[scale=0.58]f4.pdf

Figure 4: (a) Scaled correlation functions, $C(r,t)$ vs. $r/L(t)$, for
$\lambda=-0.8$ (green coloured symbols) and $\lambda=0.8$ (red coloured
sysmbols) at different times $t$ (in MCS). These plots clearly show that the
GLL model exhibits dynamical scaling but not SU. (b) Scaled structure factors,
$S(k,t)L(t)^{-3}$ vs. $kL(t)$ on a log-log plot, corresponding to the data
sets in (a). The solid lines denote the relevant Porod tails.

The defect morphology for $\lambda=-0.8$ in Fig. 5(a) shows interfaces (black
colour) separating the canted domains (green colour). It is plotted the same
way as the slices in Fig. 3. The string defects for $\lambda=0.8$ in Fig. 5(b)
have been identified by considering all the possible square plaquettes in the
cubic lattice and checking whether the nematic rods in the plaquette rotate by
$180^{\circ}$ or traverse a “non-contractible loop” [28, 24]. The observation
of two distinct defect structures in the canted and uniform regimes is novel,
and also the most interesting result of our study.

[scale=0.305]f5.pdf

Figure 5: (a) Interfacial defects (in black colour) between canted domains (in
green colour) for $\lambda=-0.8$ at $t=10^{5}$. We show only a $16^{3}$ corner
of the entire lattice for a better and clear view. (b) String defects for
$\lambda=0.8$ at $t=10^{5}$. The system size is $256^{3}$.

Finally, we determine the domain growth laws $[L(t)$ vs. $t]$ for the canted
and uniform regimes. They reveal important details of the free-energy
landscape and relaxation time scales in the system. For example, pure
isotropic systems with non-conserved dynamics obey the Lifshitz-Allen-Cahn law
(LAC): $L(t)\sim t^{1/2}$ [46]. On the other hand, pure isotropic systems with
conserved kinetics and diffusive transport follow the Lifshitz-Slyozov (LS)
law: $L(t)\sim t^{1/3}$ [47]. These growth laws are characteristic of systems
with no energy barriers to coarsening and a unique relaxation timescale. Fig.
6 shows $L(t)$ vs. $t$ on a log-log scale for five distinct values of
$\lambda:-0.8,\ -0.6,\ -0.2,\ 0.2,\ 0.8$. The dashed line with slope 1/2 is a
guide to the eye. The system obeys the LAC law, i.e., $L(t)\sim t^{1/2}$ for
all values of $\lambda$, although the growth is via the annihilation of
different dominating defect structures in the canted and uniform regimes. As
noticed earlier in the context of Fig. 2, the domain growth is faster with
increasing values of $\lambda$. The metastability in the energy for
$\lambda=0.8$ is an initial barrier which has to be overcome before domain
growth can proceed.

[scale=0.21]f6.pdf

Figure 6: Plot of the characteristic length scale, $L(t)$ vs. $t$ on a log-log
scale, for different values of $\lambda$ in the $d=3$ GLL model. The dashed
line indicates the LAC law: $L(t)\sim t^{1/2}$.

Let us conclude this paper with a summary and discussion of our results. We
have undertaken a comprehensive MC study of the kinetics of ordering in
nematic liquid crystals (NLCs) after a deep thermal quench. They are modelled
by the generalised Lebwohl Lasher (GLL) model ($d=3,n=3$) which incorporates
the fourth order Legendre interactions $P_{4}(\cos\theta_{ij})$ in the Lebwohl
Lasher (LL). (Here $\theta_{ij}$ is the anglular orientation between
neighbouring nematogens $i$ and $j$.) Consequently, the inter-molecular
potential is mimicked accurately, as evidenced from the convergence between
experimental observations and corresponding numerical evaluations.

The ground state (GS) characteristics are found to vary with parameter
$\lambda$, which represents the relative strength of the $P_{4}$ interactions
with respect to the $P_{2}$ interactions. For $\lambda<-0.3$, we observe a
hitherto unreported canted phase, where the ground state (GS) configuration
has nearest neighbour (nn) nematogens aligned at angle:

$\theta_{c}(\lambda)=\cos^{-1}\left[\left(15\lambda-6\right)/\left(35\lambda\right)\right]^{1/2}.$
(4)

For $\lambda\geq-0.3$, we observe a uniform phase where the alignment between
nn rods is at an angle $\theta_{u}(\lambda)=0,\pi$. The coarsening
morphologies exhibit dynamical scaling, implying the presence of a unique
length scale for each value of $\lambda$. So the scaling function lacks super-
universality as it is not robust with respect to $\lambda$. The structure
factor tail in the canted regime obeys the Porod law $S(k,t)\sim k^{-4}$
characteristic of interfacial defects. This is unexpected, as the GLL model is
characterised by a continuous order parameter. The uniform regime on the other
hand, exhibits the expected generalized Porod decay $S(k,t)\sim k^{-5}$. The
coarsening dynamics in this regime is governed by the anhilation of string
defects. Finally, the domain growth obeys the Lifshitz-Allen-Cahn law
$L(t)\sim t^{1/2}$ for all values of $\lambda$, but the growth is slower in
the canted regime.

Has the canted phase been observed in NLCs? The magnitudes of the second and
fourth order Legendre order parameters ($\langle P_{2}\rangle$ and $\langle
P_{4}\rangle$) in LC samples can be obtained using Raman confocal
microspectrometry, but these experimental measurements are rather difficult
and scarce. Fortunately for us, they have been reported in uniaxial LCs
p(DR1M-co-MMA), an azobenzene copolymer [48, 49]. The authors investigated the
influence of $\langle P_{4}\rangle$ relative to $\langle P_{2}\rangle$, and
indeed observed that the angular distribution function $[P(\theta_{ij})$ vs.
$\theta_{ij}]$ exhibits a peak at angles other than 0 (or $\pi$)! However, the
possible value of the model parameter $\lambda$ is not clear from these
measurements.

What could be the consequences of the novel canted phase for other physical
settings? The simplistic LL model has been widely used in the literature to
study diverse phenomena such as ordering strategies in active matter [50],
buckling and wrinkling of membranes [51], frustrations in orientational order
due to curved geometries [52] and the elasticity of LC elastomers [53], to
name a few. With the customary discrepancies between experimental observations
and the corresponding numerical evaluations via the LL model, it is tempting
to believe that the GLL model will offer an accurate description to capture
the phenomena driven by orientational ordering. As a corollary, the
consequences of the canted phase in these systems could be intriguing. It may
also be worthwhile to expand the scope of the GLL model for the large class of
colloidal systems with rod shaped molecules which yield a variety of plastic
(or soft) crystals that exhibit parallel and crossed orientations [54, 55]. We
hope that our study initiates work in these unexplored directions.

###### Acknowledgements.

NB acknowledges UGC, India for a junior research fellowship. VB acknowledges
DST, India for MATRICS and DST-UKIERI for research grant. NB and VB gratefully
acknowledge the High Performance Computing (HPC) facility at IIT Delhi for
computational resources.

## References

  * [1] Stephen M. J. Straley J. P. Rev. Mod. Phys.461974617.
  * [2] de Gennes P. G. Prost J. The Physics of Liquid Crystals (Oxford: Oxford University Press) 1995.
  * [3] Priestly E. Introduction to Liquid Crystals (Springer) 2012.
  * [4] Andrienko D. Journal of Molecular Liquids2672018520.
  * [5] Chen H.-W., Lee J.-H., Lin B. Y., Chen S. Wu S. T. Light: Sci. Appl.7201817168.
  * [6] Chuang I., Durrer R., Turok N. Yurke B. Science25119911336.
  * [7] Pargellis A., Turok N. Yurke B. Phys. Rev. Lett.6719911570.
  * [8] Lavrentovich O. D., Pasini P., Zannoni C. Žumer S. Defects in Liquid Crystals: Computer Simulations, Theory and Experiments (Springer) 2001\.
  * [9] Wang X., Miller D. S., Bukusoglu E., de Pablo J. J. Abbott N. L. Nature Materials152016106.
  * [10] Kim D. S., Copar S., Tkalec U. Yoon D. K. Science Advances42018.
  * [11] Sandford O’Neill J. J., Salter P. S., Booth M. J., Elston S. J. Morris S. M. Nature Communications1120202203.
  * [12] Reddy R. A. Tschierske C. J. Mater. Chem.162006907.
  * [13] Mertelj A., Lisjak D., Drofenik M. Copic M. Nature5042013237.
  * [14] Maier W. Saupe A. Zeitschrift für Naturforschung A1301 Jul. 1958564 .
  * [15] Saupe A. Angewandte Chemie International Edition in English196897.
  * [16] Humphries R. L., James P. G. Luckhurst G. R. J. Chem. Soc., Faraday Trans. 26819721031.
  * [17] Sweet J. R. Steele W. A. The Journal of Chemical Physics4719673022.
  * [18] Lebwohl P. A. Lasher G. Phys. Rev. A61972426.
  * [19] Zhang Z., Mouritsen O. G. Zuckermann M. J. Phys. Rev. Lett.6919922803.
  * [20] Hamley I. W., Garnett S., Luckhurst G. R., Roskilly S. J., Sedon J. M., Pedersen J. S. Richardson R. M. The Journal of Chemical Physics104199610046.
  * [21] Zhang Z., Zuckermann M. J. Mouritsen O. G. Molecular Physics8019931195.
  * [22] Chiccoli C., Pasini P. Zannoni C. International Journal of Modern Physics B1119971937.
  * [23] Fuller G., Luckhurst G. Zannoni C. Chemical Physics921985105 .
  * [24] Priezjev N. V. Pelcovits R. A. Phys. Rev. E642001031710.
  * [25] Bray A. J. Advances in Physics512002481.
  * [26] Puri S. Phase Transitions772004407.
  * [27] Puri S. Wadhawan V. Kinetics of phase transitions (CRC Press, Boca Raton) 2009.
  * [28] Zapotocky M., Goldbart P. M. Goldenfeld N. Phys. Rev. E5119951216.
  * [29] Denniston C., Orlandini E. Yeomans J. M. Phys. Rev. E642001021701.
  * [30] Bhattacharjee A. K., Menon G. I. Adhikari R. Phys. Rev. E782008026707.
  * [31] Blundell R. E. Bray A. J. Phys. Rev. A461992R6154.
  * [32] Toyoki H. Phys. Rev. E4719932558.
  * [33] Dutta S. Roy S. K. Phys. Rev. E712005026119.
  * [34] Singh A., Ahmad S., Puri S. Singh S. EPL (Europhysics Letters)100201236004.
  * [35] Singh A., Ahmad S., Puri S. Singh S. Eur. Phys. J. E3720142.
  * [36] Singh A. Singh S. Eur. Phys. J. E362013122.
  * [37] Bray A. J., Puri S., Blundell R. E. Somoza A. M. Phys. Rev. E471993R2261.
  * [38] Wickham R. A. Phys. Rev. E5619976843.
  * [39] Zhang Z., Mouritsen O. G. Zuckermann M. J. Phys. Rev. E4819932842.
  * [40] Bradac Z., Kralj S. Zumer S. The Journal of Chemical Physics1352011024506.
  * [41] Billeter J. L., Smondyrev A. M., Loriot G. B. Pelcovits R. A. Phys. Rev. E6019996831.
  * [42] Care C. M. Cleaver D. J. Reports on Progress in Physics6820052665.
  * [43] Landau D. P. Binder K. A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge: Cambridge University Press) 2005.
  * [44] Neal M. P., Parker A. J. Care C. M. Molecular Physics911997603.
  * [45] Withers I. M., Care C. M., Neal M. P. Cleaver D. J. Molecular Physics10020021911.
  * [46] Allen S. M. Cahn J. W. Acta Metallurgica2719791085.
  * [47] Lifshitz I. Slyozov V. Journal of Physics and Chemistry of Solids19196135.
  * [48] Lagugne Labarthet F., Buffeteau T. Sourisseau C. The Journal of Physical Chemistry B10219985754.
  * [49] Labarthet F. L., Buffeteau T. Sourisseau C. Applied Spectroscopy542000699.
  * [50] Das R., Kumar M. Mishra S. Scientific Reports720177080.
  * [51] Saikia L., Sarkar T., Thomas M., Raghunathan V. A., Sain A. Sharma P. Nature Communications820171160.
  * [52] Mbanga B. L., Grason G. M. Santangelo C. D. Phys. Rev. Lett.1082012017801.
  * [53] Selinger J. V. Ratna B. R. Phys. Rev. E702004041707.
  * [54] Liu B., Besseling T. H., van Blaaderen A. Imhof A. Phys. Rev. Lett.1152015078301.
  * [55] Murphy R. P., Hong K. Wagner N. J. Langmuir3220168424.