defect structures in the cholesteric dipole.

Similar numerical method can be used to study defect structures in cholesteric
LC and blue phase under the different geometric constraints (?, ?, ?).

###  4.2 Gradient Flow Approach

Gradient flow is a dynamics driven by a free energy. There are quite a number
of works devoted to obtain the defect patterns by solving the gradient flow
equation corresponding to different LC systems (?, ?, ?, ?, ?). For the LdG
theory, the corresponding $L^{2}$-gradient flow equation can be written as

$\frac{\partial\mathbf{Q}}{\partial
t}=-\gamma\frac{\delta\mathcal{F}}{\delta\mathbf{Q}},$ (82)

where $\gamma>0$ is dissipative coefficient. An advantage of gradient flow
approach is that it also provides part information of dynamical evolution of
defect structure.

On the numerical perspective, the main challenge in developing numerical
schemes for gradient flow systems is to maintain the energy dissipation
property at the discrete level. During the last a few decades, energy stable
numerical schemes for gradient flow systems have been studied extensively,
examples include full-implicit scheme (?, ?), convex splitting method (?, ?),
stabilization method (?, ?), invariant energy quadratization (?, ?) and scalar
auxiliary variable (SAV) (?, ?).

In a recent work, ? developed a second-order unconditionally energy stable
based on SAV and Crank-Nicolson for the LdG theory, in which the LdG free
energy is given by

$\mathcal{F}[\mathbf{Q}]=\mathcal{F}_{\rm b}[\mathbf{Q}]+\mathcal{F}_{\rm
el}[\mathbf{Q},\nabla\mathbf{Q}],$ (83)

where

$\displaystyle\mathcal{F}_{\rm b}=\int f_{\rm
b}(\mathbf{Q})=\int\frac{A}{2}\mathrm{tr}\mathbf{Q}^{2}-\frac{b}{3}\mathrm{tr}\mathbf{Q}^{3}+\frac{C}{4}(\mathrm{tr}\mathbf{Q}^{2})^{2}\mathrm{d}\mathbf{x}$
$\displaystyle\mathcal{F}_{\rm
el}=\int\frac{L_{1}}{2}|\nabla\mathbf{Q}|^{2}+\frac{L_{2}}{2}Q_{ik,i}Q_{jk,j}+\frac{L_{3}}{2}Q_{jk,i}Q_{ij,k}\mathrm{d}\mathbf{x}.$

Here $C>0$, $L_{1}>0$ and $L_{1}+L_{2}+L_{3}>0$, so the total energy is
bounded from below. Due to quartic term $C(\mathrm{tr}\mathbf{Q}^{2})^{2}$,
there exists $a_{1},C_{0}>0$ such that
$f_{b}(\mathbf{Q})-a_{1}\mathrm{tr}\mathbf{Q}^{2}/2+C_{0}>0$, Let $r$ be the
scalar auxiliary variable defined by

$r(t)=\sqrt{\mathcal{F}_{1}}=\sqrt{\mathcal{F}_{\rm
b}-\int_{\Omega}\frac{a_{1}}{2}\mathrm{tr}\mathbf{Q}^{2}+C_{0}}$ (84)

and

$\mathcal{L}=a_{1}\mathbf{Q}+\frac{\delta\mathcal{F}_{e}}{\delta\mathbf{Q}},$
(85)

then the gradient flow equation can be rewritten as

$\begin{cases}&\frac{\partial\mathbf{Q}}{\partial t}=-\mu\\\
&\mu=\mathcal{L}\mathbf{Q}+\frac{r(t)}{\sqrt{\mathcal{F}_{1}}}\frac{\delta\mathcal{F}_{1}}{\delta\mathbf{Q}}\\\
&r_{t}=\frac{1}{2\sqrt{\mathcal{F}_{1}}}\int_{\Omega}\frac{\delta\mathcal{F}_{1}}{\delta\mathbf{Q}}:\mathbf{Q}_{t}\mathrm{d}\mathbf{x}.\\\
\end{cases}$ (86)

and the corresponding numerical scheme is given by

$\displaystyle\frac{\mathbf{Q}^{n+1}-\mathbf{Q}^{n}}{\Delta t}=-\mu^{n+1/2},$
(87)
$\displaystyle\mu^{n+1/2}=\frac{1}{2}(\mathbf{Q}^{n+1}+\mathbf{Q}^{n})+\frac{r^{n+1}+r^{n}}{2\sqrt{\mathcal{F}_{1}[\bar{\mathbf{Q}}^{n+1/2}]}}\frac{\delta\mathcal{F}_{1}}{\delta\mathbf{Q}}[\bar{\mathbf{Q}}^{n+1/2}],$
$\displaystyle
r^{n+1}-r^{n}=\int_{\Omega}\frac{1}{2\sqrt{\mathcal{F}_{1}[\bar{\mathbf{Q}}^{n+1/2}]}}\left(\frac{\delta\mathcal{F}_{1}}{\delta\mathbf{Q}}[\bar{\mathbf{Q}}^{n+1/2}]\right)_{ij}(\mathbf{Q}_{ij}^{n+1}-\mathbf{Q}_{ij}^{n})\mathrm{d}\bm{x}.$

For the LdG free energy with cubic term ($L_{4}Q_{kl}Q_{ij,k}Q_{ij,l}$),
although the total free energy is not bounded below (?), ? constructed an
unconditionally stable numerical scheme for 2D $\mathbf{Q}$-tensor by a
stabilizing technique, They also established unique solvability and
convergence of such a scheme. The convergence analysis leads to the well-
posedness of the original PDE system for the 2D $\mathbf{Q}$-tensor model.
Several numerical examples are presented to validate and demonstrate the
effectiveness of their scheme.

###  4.3 Machine learning approach

Over the last decade, machine learning has made a huge impact on the research
areas of materials and soft matter, showing the highly powerful and effective
performance by using the techniques of deep learning. Here, we just take a
recent work by ? as an example of the LC system to demonstrate such trend. In
?, authors investigated a problem for identifying the topological defects of
rod-like molecules confined in a square box from “images”. Unlike conventional
images with correlated physical features where supervised machine learning has
been successful, these images are coordinated files generated from an off-
lattice sampling. A single-line structure $[l,x_{l},y_{l},\theta_{l}]$ is
given for each rod-like molecule, where $[x_{l},y_{l},\theta_{l}]$ specifies
the location coordinates and angular orientation of the molecule labelled $l$,
and the labels are not related . The task is to identify which of the four
defect patterns from the off-lattice data, and two types of machine-learning
procedures, the feedforward neural network (FNN) and the recurrent neural
network (RNN), are considered. It is shown that FNN is not readily appropriate
for studying defect types in this off-lattice model. However, with a coarse-
grained position sorting in the initial data input, referred to as
_presorting_ , an effective learning can be realized by FNN. On the contrary,
RNN performs exceptionally well in identifying defect states in the absence of
presorting. Mort importantly, ? pointed out that by dividing the whole image
into small cells, an RNN approach with the data in each cell as an input can
detect the types and positions of nematic defects in each image instead of
naked eyes.

##  5 Numerical methods for computing liquid crystal hydrodynamics

In this section, we review some progress on numerical methods to study LC
hydrodynamics, including vector models, tensor models, and molecular models.

###  5.1 Vector models

A NLC flow behaves like a regular liquid with molecules of similar size, and
also displays anisotropic properties due to the molecule alignment, which is
usually described by the local director field. The Ericksen–Leslie equations
have been applied to describe the flow of NLCs and attracted many theoretical
and numerical researches. According to the macroscopic hydrodynamic theory of
NLCs established by ? and ?, ? proposed a simplied Ericksen–Leslie equations
for a NLC flow,

$\left\\{\begin{array}[]{l}\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}-\nu\Delta\bm{u}+\nabla
p+\lambda\nabla\cdot((\nabla\bm{d})^{\top}\nabla\bm{d})=\bm{0},\\\
\nabla\cdot\bm{u}=0\\\
\partial_{t}\bm{d}+(\bm{u}\cdot\nabla)\bm{d}-\gamma\Delta\bm{d}-\gamma|\nabla\bm{d}|^{2}\bm{d}=\bm{0},\\\
|\bm{d}|=1.\end{array}\right.$ (92)

Here $\bm{u}$ denotes the solenoidal velocity field, $p$ denotes the pressure,
and $\bm{d}$ denotes the molecular alignment satisfying a sphere constraint
almost everywhere. The positive parameters $\nu,\lambda,\gamma$ are
respectively a fluid viscosity constant, an elastic constant and a relaxation
time constant. The system (92) consists of the Navier-Stokes equations coupled
with an extra anisotropic stress tensor and a convective harmonic map
equation. Although simple, this system keeps the core of the mathematical
structure, such as strong nonlinearities and constraints, as well as the
physical structure, such as the anisotropic effect of elasticity on the
velocity vector field $\bm{u}$, of the original Ericksen-Leslie system. The
first energy equality, which is established under certain boundary conditions,
expresses the balance of energy in the system between the kinetic and elastic
energies.

? provided an excellent overview of the numerical methods for (92). Since an
almost everywhere satisfaction of the sphere constraint restriction is not
appropriate at a numerical level, two alternative approaches have been
introduced: a penalty method and a saddle-point method. With a Ginzburg–Landau
penalty function
$F_{\varepsilon}(\bm{d})=(|\bm{d}|^{2}-1)^{2}/4\varepsilon^{2}$ to enforce the
sphere constraint, a penalty formulation is obtained by weakening the
constraint as

$\left\\{\begin{array}[]{l}\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}-\nu\Delta\bm{u}+\nabla
p+\lambda\nabla\cdot((\nabla\bm{d})^{\top}\nabla\bm{d})=\bm{0},\\\
\nabla\cdot\bm{u}=0\\\
\partial_{t}\bm{d}+(\bm{u}\cdot\nabla)\bm{d}+\gamma(f_{\varepsilon}(\bm{d})-\Delta\bm{d})=\bm{0},\\\
|\bm{d}|\leqslant 1,\end{array}\right.$ (97)

where $f_{\varepsilon}(\bm{d})=\nabla_{\bm{d}}F_{\varepsilon}(\bm{d})$, and
the energy estimate for (97) was established by ?. Finite element methods of
mixed types play an important role when designing numerical approximations for
the penalty formulation in order to preserve the intrinsic energy estimate.
Alternatively, with a Lagrange multiplier $q$ to enforce the sphere
constraint, a saddle-point formulation reads as

$\left\\{\begin{array}[]{l}\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}-\nu\Delta\bm{u}+\nabla
p+\lambda\nabla\cdot((\nabla\bm{d})^{\top}\nabla\bm{d})=\bm{0},\\\
\nabla\cdot\bm{u}=0\\\
\partial_{t}\bm{d}+(\bm{u}\cdot\nabla)\bm{d}+\gamma(q\bm{d}-\Delta\bm{d})=\bm{0},\\\
|\bm{d}|=1,\end{array}\right.$ (102)

and the energy estimate for (102) was derived by ?. These approaches are
suitable for their numerical approximation by finite elements, since a
discrete version of the restriction is enough to prove the desired energy
estimate. The penalized Ginzburg–Landau problem (97) can be stated in a
saddle-point framework as well,

$\left\\{\begin{array}[]{l}\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}-\nu\Delta\bm{u}+\nabla
p+\lambda\nabla\cdot((\nabla\bm{d})^{\top}\nabla\bm{d})=\bm{0},\\\
\nabla\cdot\bm{u}=0\\\
\partial_{t}\bm{d}+(\bm{u}\cdot\nabla)\bm{d}+\gamma(q\bm{d}-\Delta\bm{d})=\bm{0},\\\
|\bm{d}|^{2}-1=\varepsilon^{2}q,\end{array}\right.$ (107)

which establishes a connection between (97) and (102).

The numerical approximation of the Ericksen–Leslie equations is difficult and
computationally expensive because of the coupling between the nonlinear terms
and the constraint conditions. ? dealt with the approximation of (97) for 2D
domains, and convergence of finite element approximations is established under
appropriate regularity hypotheses with dependence on the penalty parameter
$\varepsilon$. ? studied a Fourier-spectral method for (97) by proving its
convergence in a suitable sense and establishing the existence of a global
weak solution of the original problem and its uniqueness in the 2D case. The
error estimates exhibit the spectral accuracy of the Fourier-spectral method,
and a fully discrete scheme is constructed with a complete stability and error
analysis. ? proved a convergence result for a Ginzburg–Landau-type equation
where the dependence of $\varepsilon$ is of polynomial order. ? presented one
of the simplest time-stepping schemes for the 2D Ginzburg–Landau problem (97),
where the space is discretized by $H^{1}$-conforming finite elements and time
is discretized implicitly with respect to the linear terms and semi-implicitly
with respect to the nonlinear terms, while the anisotropic stress tensor is
fully explicit. This scheme reduces the computational cost significantly and
larger scale numerical simulations are allowed because only a sequence of two
decoupled linear problems for the velocity-pressure pair and the director
field need to be solved separately at each time step. ? presented an efficient
and accurate numerical scheme for (97) in a cylinder domain. The time
discretization is based on a semi-implicit second-order rotational pressure-
correction scheme and the Legendre–Galerkin method is used for the space
variable. Annihilation of a hedgehog-antihedgehog pair with different types of
transport is simulated numerically.

By introducing an auxiliary variable $\bm{w}=\nabla\bm{d}$, ? avoided using
Hermite finite elements for the approximation of the director equation in (97)
and formulated the director equation in the framework of mixed methods. They
showed how a mixed method may be used to eliminate the need for Hermite finite
elements and establish convergence of the method. ? considered another
auxiliary variable $\bm{w}=-\Delta\bm{d}$ to avoid the large number of extra
degrees of freedom and the nonlinearity of the numerical schemes, and
constructed a fully discrete mixed scheme which is linear, unconditionally
stable and convergent towards (97). With an auxiliary variable
$\bm{w}=-\Delta\bm{d}+f_{\varepsilon}(\bm{d})$, two finite-element Euler time-
stepping schemes that are implicit for the linear terms and semi-implicit for
the nonlinear have been developed. Specifically, ? proposed a fully implicit
approximation to deal with the time integration of $f_{\varepsilon}$, while ?
suggested a fully explicit one. The time step of the explicit scheme must be
quite small if the size of $\varepsilon$ is proportional to the space
parameter $h$, but their numerical experiences have demonstrated to be
optimal. ? used convenient conformal $C^{0}$ finite elements in solving the
problem in the weak form, and derived a discrete energy law for a modified
midpoint time discretization scheme. A fixed iterative method is used to solve
the resulted nonlinear system so that a matrix-free time evolution may be
achieved and velocity and director variables may be solved separately.

? presented a finite element scheme directly for the Ericksen–Leslie equations
(92), which is a more difficult task than for the Ginzburg–Landau problem (97)
because the sphere constraint $|\bm{d}|=1$ is difficult to be fulfilled at the
discrete level. ? developed a linear semi-implicit algorithm which is
unconditionally stable for both the Ginzburg–Landau problem and the
Ericksen–Leslie problem, and it does not involve nonlinear iterations. ?
proposed a two sub-step viscosity-splitting time scheme for (102), which is a
fully decoupled linear scheme from the computational point of view. The first
sub-step couples diffusion and convection terms whereas the second one is
concerned with diffusion terms and constraints. Some numerical experiments in
2D domains are carried out by using only linear finite elements in space,
confirming numerically the viability and the convergence of this scheme.

? considered numerical approximation of the flow of LCs governed by the
Ericksen–Leslie equations. Care should be taken to develop numerical schemes
which inherit the Hamiltonian structure of these equations and associated
stability properties. ? developed a first-order and a second-order, coupled,
energy stable numerical schemes for a modified Ericksen–Leslie hydrodynamic
model, which can reduce to (92) with the omission of some terms. Two ways are
presented to develop decoupled schemes for the model and the energy stability
is shown as well. Prediction comparisons of the modified model with a reduced
model are made, demonstrating quite different but more realistic orientational
dynamics in flows of NLCs. ? investigated the kinematic transports of the
defects in the NLC system by numerical experiments with difference schemes and
the semi-implicit scheme which is more efficient although it does not preserve
discrete energy relation. The presented development and interaction of the
defects are partly consistent with the observation from the experiments.

Simulating the rise of Newtonian drops in a NLC is a computational challenge
because of the numerical difficulties in handling moving and deforming
interfaces as well as the complex rheology of the NLCs. The anisotropic
rheology of the LC can be represented by the Ericksen–Leslie theory,
regularized to permit topological defects. ? simulated the rise of Newtonian
drops in a NLC parallel to the far-field molecular orientation by computing
the moving interface in a diffuse interface framework. The numerical results
revealed interesting coupling between the flow field and the orientational
field surrounding the drop, especially the defect configuration. For example,
drops with planar anchoring on the surface rise faster than those with
homeotropic anchoring due to the viscous anisotropy of the nematic. With both
types of anchoring, the drag coefficient of the drop decreases with increasing
Ericksen number as the flow-alignment of the nematic orientation reduces the
effective viscosity of the LC.

LC droplets immersed in another fluid matrix or another liquid droplet
immersed in LC matrices have many interesting technological applications (?).
Based on an energetic variational approach, ? derived a phase-field theory for
immiscible mixtures of NLCs and viscous fluids. A novel phase transition
mechanism is implemented to couple the NLC phase with the viscous fluid phase
to arrive at the dissipative hydrodynamic model for incompressible fluid
mixtures. In ?, a decoupled, linear scheme for a simplified version of the
phase field model, as well as a coupled, nonlinear scheme for the full model,
are developed and shown as unconditionally energy stable with consistent
discrete dissipative energy laws. The effectiveness of the presented numerical
examples show the effectiveness of the new model and the developed numerical
schemes.

A number of models for smectic phase LCs have been developed and studied
during the last two decades, and we take one example of smectic LC
hydrodynamics here. ? considered the numerical approximations for solving a
particular hydrodynamics coupled smectic-A model developed by ?. The model,
which is derived from the variational approach of the modified Oseen–Frank
energy, is a highly nonlinear system that couples the incompressible
Navier–Stokes equations and a constitutive equation for the layer variable,
but appears to be the minimal model of unknowns. The director field is assumed
to be strictly equal to the gradient of the layer and thus the total free
energy is reduced to a simplified version with only one order parameter.
Rather than imposing nonconvex constraint directly on the gradient of the
layer variable, the free energy is modified by adding a Ginzburg–Landau type
penalization potential, which is a commonly used technique in LC theory. Two
linear, second order time marching schemes, which are unconditionally energy
stable, are developed with the Invariant Energy Quadratization method for
nonlinear terms in the constitutive equation, the projection method for the
Navier–Stokes equations, and some subtle implicit-explicit treatments for the
convective and stress terms. Various numerical experiments are presented to
demonstrate the stability and the accuracy of the numerical schemes in
simulating the dynamics under shear flow and the magnetic field.

###  5.2 Tensor models

In recent years, more complex problems of modelling the interaction of flow
and molecular orientation in a LCs are studied. The hydrodynamic
$\mathbf{Q}$-tensor model has been applied to the flows of LCs and LC
polymers. In particular, the nondimensionalized hydrodynamic
$\mathbf{Q}$-tensor model of NLCs

$\left\\{\begin{array}[]{l}\partial_{t}\mathbf{Q}+\bm{u}\cdot\nabla\mathbf{Q}-\mathbf{S}(\nabla\bm{u},\mathbf{Q})=M\mathbf{H},\\\
\nabla\cdot\bm{u}=0,\\\ \partial_{t}\bm{u}+\bm{u}\cdot\nabla\bm{u}=-\nabla
p+\eta\Delta\bm{u}+\nabla\cdot\Sigma-\mathbf{H}\nabla\mathbf{Q}.\end{array}\right.$
(111)

can be derived from a variational approach together with the generalized
Onsager principle. The molecular field $\mathbf{H}$, which provides the
driving motion, is related to the derivative of the free energy. More details
are referred to ?, ?. ? developed the first-order and second-order coupled
energy stable numerical schemes for the $\mathbf{Q}$-tensor based hydrodynamic
model of NLC flows. The first-order coupled scheme is extended to a decoupled
scheme, which is energy stable as well. With the fully coupled schemes
implemented in 2D space and time, defect dynamics for flow of NLCs in a
channel is studied. The developed methodology also provides a paradigm for
developing energy-stable schemes for general hydrodynamic models of complex
fluids with an energy dissipation law. Furthermore, ? developed a second-order
semi-discrete scheme in time, which is linear and unconditionally energy
stable at the semi-discrete level, to solve the governing system of equations
by following the novel ‘energy quadratization’ strategy. Several numerical
examples are presented to demonstrate the usefulness of the model and the
effectiveness of the numerical scheme in simulating defect dynamics in flows
of LCs. ? described a lattice Boltzmann algorithm to simulate LC hydrodynamics
in terms of a tensor order parameter, including a molecular field that
provides the driving motion. A lattice Boltzmann algorithm was described in
detail to simulate LC hydrodynamics. Backflow effects and the hydrodynamics of
topological defects are naturally included in the simulations, as are non-
Newtonian flow properties such as shear thinning and shear banding.

? considered the NLC in a spatially inhomogeneous flow with a second-rank
alignment tensor, whose evolution is determined by two coupled equations: a
generalized Navier–Stokes equation for the flow, and a convection-diffusion
type equation for the alignments. A specific model with three viscosity
coefficients allows the contribution of the orientation to the viscous stress
to be cast in the form of an orientation-dependent force, which effectively
decouples the flow and orientation to circumvent the fully coupled problem. A
time-discretized strategy for solving the flow-orientation problem is
illustrated using the Stokes flow in a lid-driven cavity.

Dispersing colloidal particles into LCs provides an approach to build a novel
class of composite materials with potential applications. Many researches have
been devoted to the hydrodynamic equations governing their dynamical evolution
to understand the physics and dynamical properties of such colloid LC
composites. ? provided an overview of theoretical understanding of the
hydrodynamic properties of such materials from computer simulations, including
a single particle and two particles forming a dimer and dispersion. A number
of open questions in the dynamics of colloid LC composites are also raised.

###  5.3 Molecular models

For the liquid crystalline polymer (LCP) model, the Doi–Hess theory by ? and ?
is the simplest and the most studied model. Many interesting dynamics have
been found in rod-like nematic LCPs in a shear flow. ? considered the
stochastic model of concentrated LCPs in the plane Couette flow, where the
flow is one dimensional while the configuration variable of the rod is
restricted to the circle, and presented the local existence and uniqueness
theorem for the solution to the coupled fluid-polymer system. ? considered the
extended Doi model for nematic LCPs in the planar shear flow and studied the
formation of microstructure and the dynamics of defects. The Fokker–Plank
equation is discretized with the spherical harmonic spectral method and
multiple flow modes are replicated in the simulations. A comparison among
complete closure models, Bingham closure models and kinetic models is given in
? as well.

? presented a new kinetic hydrodynamic coupled model for the dilute LCP
solution for inhomogeneous flow, which accounts for translational diffusion
and density variation. The coupled kinetic hydrodynamic model is a combination
of an extended Doi kinetic theory for rigid rod-like molecules and the
Navier–Stokes equation for incompressible flow. They studied the
microstructure formation and defects dynamics arising in LCPs in plane shear
flow with mass conservation of LCPs in the whole flow region. The LCP
molecular director is restricted in the shear plane, and LCP molecules are
ensured anchoring at the boundary by an additional boundary potential. Plane
Couette flow and Poiseuille flow were studied with a second-order difference
scheme and the fourth-order Runge–Kutta method for time-stepping. Their
numerical results in plane Couette flow predicted seven in-plane flow modes,
four of which have been reported by ?. In plane Poiseuille flow, the micro-
morph is quasi-periodic in time when flow viscosity and molecular elasticity
are comparable. Different local states, such as flow-aligning, tumbling or
wagging, arise in different flow regions. To describe the microstructures and
defect dynamics of LCP solutions, ? later purposed a general nonhomogeneous
extension of the Doi’s kinetic theory with translational diffusion and
nonlocal potential. A reduced second-order moment model for isotropic long-
range elasticity was obtained as a decent tool for numerical simulations of
defect dynamics and texture evolution. Their numerical results of in-plane
rotational case show that this model qualitatively predicts complicated
nonhomogeneous director dynamics under moderate nematic potential strength,
and the translational diffusion plays an important role in defect dynamics.

##  6 Numerical methods for computing transition pathways and solution
landscape of liquid crystals

NLCs often exhibit a number of ordered phases. When an ordered phase is
metastable, phase transition proceeds via nucleation and growth under thermal
fluctuations or external perturbations. Such transition is often called as a
rare event, which can be characterized by a long waiting period around one
local metastable state and followed by a sudden jump over the transition
state, to another stable state (?). Because the transition state is an
unstable critical point of the LC free energy, i.e., an index-1 saddle point
with a single unstable direction that connects two local minima, it is of
great challenge to accurately compute the transition states and the transition
pathways.

In this section, we will first review the numerical methods to compute the
transition states and transition pathways. Then we will introduce a novel
solution landscape, which describes a pathway map consisting of all stationary
points (i.e. extrema and saddle points) and their connections.

###  6.1 Transition pathways

There are two popular approaches to find the saddles points and transition
pathways. One is the class of surface-walking methods for finding saddle
points starting from a single state, such as gentlest ascent dynamics (?) and
dimer-type method (?, ?). The other approach is categorised as path-finding
methods for computing minimal energy path (MEP) that involves two end states,
including nudged elastic band method (?) and string method(?).

In the case when the object of interest is the most probable transition path
between metastable/stable states of the smooth potential energy, it is known
that for overdamped Langevin dynamics the most probable path for the
transition is the MEP. The string method serves this purpose well, which was
first proposed by E, Ren and Vanden-Eijnden (?) and has some variants (?, ?,
?, ?). The string method proceeds by evolving a string $\phi$ with intrinsic
parametrization in the configuration space by using the steepest decent
dynamics to the MEP:

$\phi_{t}=-\nabla F^{\perp}(\phi)+\lambda\mathbf{\tau},$ (112)

where $\nabla F^{\perp}(\phi)$ is the component of $\nabla F$ normal to
$\phi$, $\mathbf{\tau}$ denotes the unit tangent vector to $\phi$, and
$\lambda$ is the Lagrange multiplier to impose the equal arc-length
constraint.

In the numerical algorithm, the string $\phi$ is discretized to $N$ nodes
$\\{\mathbf{Q}_{i}^{n},i=1,..,N\\}$, where $n$ represents the iteration step.
The string method adopts a time splitting scheme via the following two-step
procedure: Step 1: string evolution: The current path
$\\{\mathbf{Q}_{i}^{n},i=1,..,N\\}$ is updated by following the gradient
decent direction :

$\bar{\mathbf{Q}}^{n+1}_{i}=\mathbf{Q}^{n}_{i}-\alpha_{i}\nabla
F^{\perp}[\mathbf{Q}^{n}_{i}],\quad i=1,2,\cdots,N,$ (113)

where $\alpha_{i}$ is a stepsize that is chosen as a constant or determined by
the linesearch.

Step 2: string reparametrization: Redistributes the nodes
$\bar{\mathbf{Q}}_{i}^{n+1}$ to obtain a new path $\mathbf{Q}^{n+1}_{i}$ by
enforcing the equal arc length parametrization. One can calculate the arc
length corresponding to the current nodes $\bar{\mathbf{Q}}_{i}^{n+1}$ and
then use a linear or cubic spline interpolation to find $\mathbf{Q}^{n+1}_{i}$
by the equal arc length.

Parametrization by equal arc length gives a good accuracy for the MEP, but not
for the transition state. To achieve the better estimation of the transition
state and the energy barrier, one way is to use energy-weighted arc length so
that finer resolution can be achieved around the transition state (?). If the
final minima is far from the transition state and too many nodes are needed
for the whole MEP, the free-end nudged elastic band method can be used to
obtain subtle depiction around the transition state by allowing the last node
not to be a minimum (?).

To improve the accuracy of both MEP and transition state without using too
many nodes, in (?), the authors proposed a multi-scale scheme of the string
method by applying a global coarse string and a local fine string. The global
coarse string is used to obtain the whole MEP by connecting two minima, and
then the local fine string is utilized to compute the local MEP by choosing
two nodes with the highest energies on the global coarse string as the two
ends for the case of single energy barrier. Both strings follow the same two-
step procedure described above to make the implementation straightforward, and
better estimation of the transition state can be achieved by the local fine
string. They implement the multi-scale string method on numerical simulations
of the transition pathway in three-dimensional cylinder between PR and PP with
and without axial invariance, between ER and PP or ERRD for different values
of temperature and radius of cylinder.

Figure 12: (Credit: (?)) Transition pathway from ER to PP. (a-d) four states
along the MEP (e), corresponding to the ER (a), the transition state (b), an
intermediate state (the 6th node) (c), and the PP (d), respectively. In the
MEP(e), the red line with green beads is the global coarse string and the blue
line with blue beads is the local fine string.

For example, the transition pathway between ER and PP for fixed temperature
and radius of cylinder is shown in Fig. 12. A pair of +1/2 disclination lines
starts to form in the middle of the cylinder. These two disclination lines
appear to join their both ends, hence forming a disclination ring (Fig.
12(b)). The width of the ring is increasing when less than the distance
between defects of stable 2D PP layer. Finally, the ring is broken by the top
and bottom boundary and the line defect straightens to PP line defect
configuration (Fig. 12(d)). The local fine string in Fig. 12 finds a more
accurate transition state than the global coarse string.

###  6.2 Saddle points

The classical transition state theory describes a sufficiently accurate
transition process for the systems with smooth energy landscapes. On the
energy landscape, the transition state is a saddle point with the lowest
energy that connects two neighbouring local minima, i.e., index-1 saddle
point. As long as the transition state is located, one can calculate the MEP
using gradient flow from the transition state perturbed along both sides of
its unstable direction.

Extensive surface walking methods have been proposed to search saddle points.
A key character of such methods is to perform a systematic search for a saddle
point starting from a given initial state, without knowing the final states.
In this subsection, we focus on a typical class of surface walking methods,
the so-called minimum mode following methods, where only the lowest eigenvalue
and the corresponding eigenvector of the Hessian are needed and subsequently
used together with the energy gradient (often referred as the force) to
compute transition states. Besides the eigenvector-following method by ?, the
trajectory following algorithm by ?, the activation-relaxation technique by ?,
and the step and slide method by ? are also some samplers of surface-walking
methods, and we refer to ? and ? as some excellent reviews.

In ?, the authors proposed the gentlest ascent dynamics, which is a dynamical
formulation of the gentlest ascent method by ?. It refers to the following
dynamical system

$\left\\{\begin{aligned} \dot{x}&=-\nabla E(x)+2\dfrac{\langle\nabla
E(x),x\rangle}{\langle v,v\rangle}v,\\\
\dot{v}&=-\nabla^{2}E(x)v+\dfrac{\langle v,\nabla^{2}E(x)v\rangle}{\langle
v,v\rangle}v,\end{aligned}\right.$ (114)

where $E(\cdot)$ repersents the energy functional. Besides the position
variable $x$, another variable $v$ represents the unstable direction. The
dynamics (114) can drive $x$ climb out of the basin of attraction by following
the ascent direction $v$, which is the eigenvector that corresponds to the
smallest eigenvalue of its Hessian, to find the saddle point. It was proved in
(?) that the stable fixed points of this dynamical system are precisely the
index-1 saddle points.

To avoid the calculation of the Hessian, a dimer method was developed using
only first derivatives by ?. Specifically, a dimer with two points is placed
at the current position to compute the Hessian-vector multiplication with a
finite difference scheme. The dimer method proceeds by alternately performing
the rotation step for finding the lowest eigenmode and the translation step by
using the modified force with either the steepest descent algorithm or the
conjugate gradient method to search the saddle point. Later, ? proposed a
shrinking dimer dynamics by introducing a dynamic reduction of the dimer
length, and proved linear stability and convergence results for such dynamics.
To accelerate the convergence and further improve the efficient of the dimer-
type methods, ? developed an optimization-based shrinking dimer (OSD) method
by establishing a minimax optimization for the saddle searching problem and
finding the unstable direction in an optimization framework. All these methods
can be applied to the LC models for identifying the transition states
accurately (?).

Besides the investigations of the minima and index-1 saddle points on the
energy landscape of NLCs in confinement, there is substantial recent interest
in the high-index saddle points of the LdG free energy, which are stationary
points of the NLC free energy. The Morse index of a saddle point is the
maximal dimension of a subspace on which its Hessian is negative definite,
which is equal to the number of negative eigenvalues of its Hessian (?). For
example, a stable stationary point has index $0$ and a transition state is an
index-$1$ saddle point. In recent years, powerful numerical algorithms have
been developed to find multiple solutions of nonlinear equations, including
the gradient-square-minimization method (?), the minimax method (?), the
deflation technique (?), and the homotopy method (?, ?). Despite substantial
progress in this direction, it is still a numerical challenge to
systematically find saddle point solutions, especially those with high
indices, of nonlinear systems of partial differential equations such as the
Euler-Lagrange equation. In ?, the authors proposed a high-index saddle
dynamics (HiSD) to efficiently compute the saddle points of any index
(including minima) for the LdG energy (?).

For a non-degenerate index-$k$ saddle point $\hat{\mathbf{x}}$, the Hessian
$\mathbb{H}(\mathbf{x})=\nabla^{2}E(\mathbf{x})$ at $\hat{\mathbf{x}}$ has
exactly $k$ negative eigenvalues
$\hat{\lambda}_{1}\leqslant\cdots\leqslant\hat{\lambda}_{k}$ with
corresponding unit eigenvectors
$\hat{\mathbf{v}}_{1},\ldots,\hat{\mathbf{v}}_{k}$ satisfying
$\big{\langle}\hat{\mathbf{v}}_{j},\hat{\mathbf{v}}_{i}\big{\rangle}=\delta_{ij}$,
$1\leqslant i,j\leqslant k$. Define a $k$-dimensional subspace
$\hat{\mathcal{V}}=\mathrm{span}\big{\\{}\hat{\mathbf{v}}_{1},\ldots,\hat{\mathbf{v}}_{k}\big{\\}}$,
then $\hat{\mathbf{x}}$ is a local maximum on a $k$-dimensional linear
manifold $\hat{\mathbf{x}}+\hat{\mathcal{V}}$ and a local minimum on
$\hat{\mathbf{x}}+\hat{\mathcal{V}}^{\perp}$, where
$\hat{\mathcal{V}}^{\perp}$ is the orthogonal complement space of
$\hat{\mathcal{V}}$.

The HiSD dynamics for a $k$-saddle ($k$-HiSD) is given as follows:

$\left\\{\begin{aligned}
\beta^{-1}\dot{\mathbf{x}}&=-\left(\mathbf{I}-2\sum_{j=1}^{k}\mathbf{v}_{j}\mathbf{v}_{j}^{\top}\right)\nabla
E(\mathbf{x}),\\\
\gamma^{-1}\dot{\mathbf{v}}_{i}&=-\left(\mathbf{I}-\mathbf{v}_{i}\mathbf{v}_{i}^{\top}-2\sum_{j=1}^{i-1}\mathbf{v}_{j}\mathbf{v}_{j}^{\top}\right)\mathbb{H}(\mathbf{x})\mathbf{v}_{i},\;i=1,\ldots,k,\\\
\end{aligned}\right.$ (115)

where the state variable $\mathbf{x}$ and $k$ direction variables
$\mathbf{v}_{i}$ are coupled, $\mathbf{I}$ is the identity operator and
$\beta,\gamma>0$ are relaxation parameters. The $k$-HiSD dynamics (115) is
coupled with an initial condition:

$\mathbf{x}(0)=\mathbf{x}^{0}\in\mathbb{R}^{n},\quad\mathbf{v}_{i}(0)=\mathbf{v}_{i}^{0}\in\mathbb{R}^{n},i=1,\ldots,k,$
(116)

where $\mathbf{v}_{1}^{0},\ldots,\mathbf{v}_{k}^{0}$ satisfy the orthonormal
condition
$\left\langle\mathbf{v}_{i}^{0},\mathbf{v}_{j}^{0}\right\rangle=\delta_{ij}$,
$i,j=1,2,\ldots,k$. The first equation in (115) describes a transformed
gradient flow, which allows $\mathbf{x}$ to move along ascent directions on
the subspace $\hat{\mathcal{V}}$ and descent directions on the subspace
$\hat{\mathcal{V}}^{\perp}$. The second equation in (115) is used to search
for an orthonormal basis of $\hat{\mathcal{V}}$. Because the Hessian
$\mathbb{H}(\mathbf{x})$ is self-adjoint in a gradient system, we can simply
take $\mathbf{v}_{i}$ as a unit eigenvector corresponding to the $i$th
smallest eigenvalue of $\mathbb{H}(\mathbf{x})$, which can be obtained from a
constrained optimization problem,

$\min_{\mathbf{v}_{i}\in\mathbb{R}^{n}}\quad\langle\mathbb{H}(\mathbf{x})\mathbf{v}_{i},\mathbf{v}_{i}\rangle,\qquad\mathrm{s.t.}\quad\langle\mathbf{v}_{j},\mathbf{v}_{i}\rangle=\delta_{ij},\quad
j=1,2,\ldots,i.$ (117)

Then we minimize the $k$ Rayleigh quotients (117) simultaneously by solving
the second equation in (115). Furthermore, one can apply the locally optimal
block preconditioned conjugate gradient (LOBPCG) method (?) to calculate the
smallest $k$ eigenvalues and $\mathbf{v}_{1},...,\mathbf{v}_{k}$.

###  6.3 Solution landscape of liquid crystals

Solution landscape is a novel concept, which is defined as a pathway map
consisting of all stationary points and their connections, which presents a
hierarchy structure to show how minima are connected with index-1 saddle
points, and how lower-index saddle points are connected to higher-index ones,
finally to a parent state, i.e., the highest-index saddle point (?). Solution
landscape not only advances the understanding of the relationships between the
minima and the transition states on the energy landscape, but also provides a
full picture of the entire family of stationary points in both gradient
systems and dynamical systems (?).

To construct a solution landscape, one needs to apply HiSD and follow two
algorithms, downward search to find all connected low-index saddles from a
high-index saddle and upward search to find a connected high-index saddle,
which drive the entire search to navigate up and down on the energy landscape
(?).

Downward search algorithm: Given an index-$m$ saddle point $\hat{\mathbf{x}}$
and $m$ unit eigenvectors $\hat{\mathbf{v}}_{1},\ldots,\hat{\mathbf{v}}_{m}$
corresponding to the $m$ negative eigenvalues
$\hat{\lambda}_{1}\leqslant\ldots\leqslant\hat{\lambda}_{m}$ of the Hessian
$\mathbb{H}(\hat{\mathbf{x}})$ respectively, we search for a lower index-$k$
($k<m$) saddle point using HiSD dynamics (115). For the initial condition, we
choose $\mathbf{x}(0)=\hat{\mathbf{x}}\pm\varepsilon\mathbf{u}$ for
$\mathbf{x}$, where we perturb $\hat{\mathbf{x}}$ along a direction
$\mathbf{u}$ with a small $\varepsilon$. The direction $\mathbf{u}$ can be
chosen as $\hat{\mathbf{v}}_{i}$ or a linear combination of $(m-k)$ vectors in
the set of unstable directions
$\\{\hat{\mathbf{v}}_{k+1},\ldots,\hat{\mathbf{v}}_{m}\\}$, whose negative
eigenvalues have the smallest magnitudes. The other $k$ eigenvectors
$\hat{\mathbf{v}}_{1},\ldots,\hat{\mathbf{v}}_{k}$ are the initial unstable
directions $\mathbf{v}_{i}(0)$. A typical choice of initial conditions in a
downward search is
$(\hat{\mathbf{x}}\pm\varepsilon\hat{\mathbf{v}}_{k+1},\hat{\mathbf{v}}_{1},\ldots,\hat{\mathbf{v}}_{k})$.
Normally, a pair of index-$k$ saddles can be found, corresponding to the $\pm$
sign of the initial searching direction.

Upward search algorithm: We can also search a high index-$k$ saddle from a low
index-$m$ saddle $\hat{\mathbf{x}}$ ($m<k$) by using the HiSD method. Starting
at an index-$m$ saddle $\hat{\mathbf{x}}$, to search for a high index-$k$
saddle, $(k-m)$ other unit eigenvectors
$\hat{\mathbf{v}}_{m+1},\ldots,\hat{\mathbf{v}}_{k}$ corresponding to the
smallest $k-m$ positive eigenvalues of the Hessian
$\mathbb{H}(\hat{\mathbf{x}})$ are needed. The initial state $\mathbf{x}(0)$
is set as $\hat{\mathbf{x}}\pm\varepsilon\mathbf{u}$ where $\mathbf{u}$ is a
linear combination of
$\\{\hat{\mathbf{v}}_{m+1},\ldots,\hat{\mathbf{v}}_{k}\\}$, and a typical
initial condition for $k$-HiSD in an upward search is
$(\hat{\mathbf{x}}\pm\varepsilon\hat{\mathbf{v}}_{k},\hat{\mathbf{v}}_{1},\ldots,\hat{\mathbf{v}}_{k})$.

Each downward or upward search represents a pseudodynamics between a pair of
saddle points, which presents valuable insights into transition pathways
between stable and unstable solutions and the corresponding energy barriers.
By repeating downward search and upward search, we are able to systematically
find saddle points of all indices and uncover the complex connectivity of the
solution landscape.

In ?, the authors demonstrated the success of the solution landscape and
applied the LdG model to construct the pathway maps of 2D NLCs confined in a
square domain with tangent boundary conditions. One of the technical
advantages of this method is to produce the entire family tree under a parent
state.

Figure 13: (Credit: (?)) Solution landscape of NLC confined on a square at the
domain size $\lambda^{2}=50$.

For small nano-scale square domains, it is known that the well order
reconstruction solution (WORS) is the unique solution with isotropic diagonals
and is the global energy minimum in this asymptotic regime (?, ?). Moreover,
the existence of WORS can be proved in arbitrary sized nematic wells (?). By
studying the second variation of LdG free energy numerically, ? shows that the
WORS is a high-index saddle point in a large well. In Figure 14 (a), the Morse
index of the WORS increases with the domain size and the WORS is always the
parent state in the solution landscapes on a square domain. Intuitively, this
is because the length of the diagonal defect lines increases as the domain
size increases, and thus the WORS has an increasing number of unstable
directions and an increasing Morse index with the increasing square edge
length. Using WORS as the parent state (the highest-index saddle) and the HiSD
method (?), ? constructed a solution landscape on the square domain in Figure
13 and reported novel saddle point solutions with multiple interior defects,
such as $N\pm$, $M\pm$, $S\pm$, and $T\pm$, that were not reported in the
existing literature . Besides these new solutions, solution landscape provides
rich information of dynamical pathways. For example, two movies, a dynamic
downward pathway sequence and an upward pathway sequence in Figure 13, are
shown in ? to demonstrate the hidden physical processes and transitions on the
complicated energy landscape of NLCs.

Figure 14: (Credit: (?)) Comparison of the parent states of the solution
landscapes on the square (a) and the hexagon (b). The domain size
$\lambda^{2}=5,70,120$, and $600$, respectively.

Solution landscape of the NLC system can be affected by many factors, such as
material properties, external fields, temperature, boundary conditions, domain
size and shape, etc. Compared to a square domain, hexagon is a generic regular
polygon with an even number of sides. In ?, the authors investigated the
solution landscapes of NLCs confined in a hexagon with tangent boundary
conditions. The Ring solution, which is the analogue of the WORS, is a
minimizer on a hexagon when the parameter of domain size $\lambda$ is small
enough. However, the Ring solution becomes and remains as an index-$2$ saddle-
point solution for large $\lambda$, i.e., the Morse index of the Ring solution
does not increase with $\lambda$. By using the HiSD method, the parent states
in the solution landscapes of NLCs on a hexagon are found to be the Ring
solution, index-$3$ T135 solution, and index-$14$ H* solution when
$\lambda^{2}=70,120$, and $600$ in Figure 14 (b) respectively, where T135 and
H* solutions emerge through saddle-node bifurcations.

Figure 15: (Credit: (?)) (a) Solution landscape on a hexagon at
$\lambda^{2}=600$. The number in the parentheses indicates the number of
solutions without taking symmetry into account. The height of a node
approximately corresponds to its energy. (b) Solution landscape of the H
class, and the configurations of sample solutions and the corresponding plots
of $|\mathbf{Q}-\mathbf{Q}^{H}|$, where $\mathbf{Q}$ is the sample solution in
H class, and $\mathbf{Q}^{H}$ is the index-$8$ H solution. (c) Solution
landscape of the TD class, and the configurations of sample solutions and the
corresponding plots of $|\mathbf{P}-\mathbf{P}^{TD}|$, where $\mathbf{P}$ is
the sample solution in TD class, and $\mathbf{P}^{TD}$ is the index-$3$ TD
solution.

In the NLC system confined on a hexagon, the solution landscape is very
complicated at $\lambda^{2}=600$ in Figure 15(a). There are three notable
numerical findings, including a new stable T solution with an interior $-1/2$
defect, new H and TD classes of saddle point solutions with high symmetry and
high indices, and new saddle points with asymmetric defect locations. Novel
H-class solutions with interior point defects have Morse indices ranging from
$8$ to $14$, and the connectivity of these solutions is shown in Figure 15(b),
as well as the corresponding configurations and their defect profiles. The
parent state is the index-$14$ H* saddle point solution connecting to the
lowest index-$8$ saddle point solution, labelled as H. The H-class solutions
look very similar to each other at first glance and their subtle differences
are illustrated by plotting $|\mathbf{Q}-\mathbf{Q}^{H}|$, where $\mathbf{Q}$
is a critical point solution in H-class and $\mathbf{Q}^{H}$ is the index-$8$
H solution. The differences concentrate on the vertices with conspicuous red
or white points in the dark blue background (Figure 15(b)). The vertex with a
pinned +1/3 defect and a splay profile of direction is referred to as a splay-
like vertex. The vertex with a $-1/6$ defect and a bend profile of direction
is referred to as a bend-like vertex. These conspicuous points are localised
near or at the bend-like vertices. The TD-class solutions, in which TD is an
abbreviation for “triangle double”, appear to be a superposition of two Ring
solutions on a regular triangle, with two interior $-1/2$ point defects and an
interior $+1/2$ point defect. All saddle points in this class have three
defective vertices, either bend-like or splay-like, and the connectivity of
this class is shown in Figure 15(c). The lowest-index saddle point solution in
this class is the index-$3$ TD solution with no bend-like vertices.

Figure 16: (Credit: (?)) (a) The transition pathways between stable states
including two T, six M and three P solutions at $\lambda^{2}=600$. (b)
Solution landscape starting from the H solution. All local minima such as
Tleft, P36, M26, M35, P25, and Tright are connected by the index-$8$ H
solution.

A further innovative aspect of the solution landscape of NLCs is to provide
rich information on dynamical pathways, which include transition pathways with
a single transition state, multiple transition states, and dynamical pathways
mediated by high-index saddle points. For example, in Figure 16(a), the
transition pathway between Tleft and M26 is connected by single transition
state T04. On the other hand, one transition pathway between P25 and P36 can
be
P${}_{25}\leftrightarrow$M1${}_{35}\leftrightarrow$M${}_{35}\leftrightarrow$M1${}_{53}\leftrightarrow$P36,
connected by two transition states M135 and M153. The “longest” transition
pathway appears to be the pathway between the two T solutions: Tleft and
Tright, despite Tleft and Tright are two symmetric solutions related by a
$60^{\circ}$ rotation. One switching pathway between Tleft and Tright cannot
be achieved by
T${}_{\mathrm{left}}\leftrightarrow$T0${}_{4}\leftrightarrow$M${}_{26}\leftrightarrow$M1${}_{62}\leftrightarrow$P${}_{25}\leftrightarrow$M1${}_{15}\leftrightarrow$M${}_{15}\leftrightarrow$T0${}_{3}\leftrightarrow$Tright,
which indicates the transition between two energetically-close but
configurationally-far T solutions has to overcome four energy barriers.

Figure 16(b) shows how the different P, M, and T solutions are connected by
high-index saddle points. For example, the index-$1$ M1 connects stable M and
P, and the index-$1$ T0 connects stable M and T. Contrarily, two M or two P
are connected by the index-$2$ BD. Furthermore, Tleft and Tright, which are
configurationally far away from each other, are connected by an index-$8$ H
solution by following HiSD:
Tleft$\leftarrow$T135left$\leftarrow$H$\rightarrow$T135right$\rightarrow$Tright.
The index-$8$ H solution is the stationary point in the intersection of the
smallest closures of on the energy landscape, which is able to connect every
stable solution (two T, three P and six M solutions) through dynamical
pathways in Figure 16(b). In ?, the authors deduce that index-$1$ saddle
points are efficient for connecting configurationally-close stable solutions.
For configurationally-far stable states, they are generally connected by
multiple transition states, or in another way, connected by a high-index
saddle point.

##  7 Conclusion and future directions

LCs are fascinating examples of complex fluids that combine fluidity of fluids
with the directionality of solids. During the last decades, the study of LCs
has grown tremendously due to the widespread applications in the industry,
such as the display devices, the photonic devices and the biological sensors.
Modeling, analysis, and computation are indispensable tools for describing,
understanding, and predicting the physical phenomena related to the LCs. The
topics of LCs land on a number of branches in physics, materials science, and
mathematics, forming problems of fundamental importance.

Theoretical studies of LCs are primarily concerned with how topology and
geometry of the elements (macromolecules or anisotropic molecules) produce and
affect the mesoscopic structure, and how the structure determines the
properties of the materials. It is crutial to understand the LC phenomena by
building appropriate mathematical models and carry out both theoretical
analysis and numerical simulation. Scientific values of LCs include two major
aspects:

Raise new mathematical problems. The study of LCs involves calculus of
variation, PDEs, harmonic analysis and computational methods. For example,
Onsager model can be expressed as either a variation minimization problem, or
a nonlinear nonlocal integral equation or PDE problem. In general, it is
impossible to find all analytical solutions for such problem. It is because of
the problem with special physical properties, which allows us to find all the
solutions and prove that all solutions have axial symmetry. LC is prone to
defects, which corresponds to the mathematical singularities. The main reason
is because the nature of such problem is a vector field. Mathematicians often
use the LCs as the background to study singularities of harmonic maps. The
$\mathbf{Q}$-tensor theory is a regularization model for the vector model,
involving the mathematical problems such as eigenvalues degradation and
multiscale analysis, which provides direction and guidance to the relevant
mathematical study. Furthermore, because of multiscale phenomena in complex
fluids, it is desirable to study modeling and numerical methods in
computational and applied mathematics.

Provide new methodologies. The choice of order parameter in the macroscopic
model of LC and is a key to the understanding of the connections between
macroscopic theories and microscopic theories. For rod-like liquid crystal,
axially symmetric assumption is generally believed by chemists, physicists,
and scientists in materials or mechanics. The strict mathematical proof can
not only verify their intuition, but also achieve deep understanding of the
mechanism of axial symmetry. Continuum NLC theories, which assume that
macroscopic quantities of interest vary slowly on the molecular length-scale,
are typically defined in terms of a macroscopic order parameter and a free
energy; the experimentally observable states correspond to energy minimizers.
Mathematically, this reduces to analyzing and numerically computing solutions
of the associated Euler Lagrange equations which are a system of highly
nonlinear and coupled partial differential equations. Recent years have seen a
boom in analytical and numerical efforts to compute the solution landscapes of
NLC systems.

Although tremendous progress have been achieved in both theoretical and
numerical study of LCs over the last four decades, there are still many
challenges in modeling and computation of LCs. According to the contents of
this review, we list some of perspectives as below.

Symmetry of equilibrium configurations. Symmetry is the centre theme of LC
studies. The axially symmetry of equilibrium solutions of the Onsager’s
molecular energy plays fundamental role in molecular kinetic theory. However,
it is only proved for the Maier-Saupe potential. To prove analogous results
for the Onsager potential, even for the minimizers, is an important and
difficult problem. For the Landau-de Gennes energy with isotropic elasticity,
it is believed that, the minimizing solutions corresponding to the hedgehog
boundary condition in a three dimensional ball is axially symmetric.
Furthermore, all minimizers should be one of the three types: radial hedgehog,
ring disclination and split core. However, the rigorous proofs are
challenging. Similar problems can be asked for the 2D point defects, which are
closely related to the disclinations in three dimensional space. If the
anisotropic elastic constants are not zero, corresponding problems are more
complicated and difficult. These problems are related to fine structures of
the point defects and disclinations, which are of fundamental importance.

Analysis on the isotropic-nematic interface. Study on profiles and structure
of two phases interface is physically important and mathematically
challenging. One of the simplest model is the energy functional with the order
parameter being a scalar function, which has been widely studied during past
two decades. When the order parameter is a high dimensional vector, the
problem is much more difficult. ? studied the asymptotical behavior of
minimizers for the high dimensional problem. The phase transition interface
between isotropic and nematic phases is even harder because it is not only
high dimensional but also involves anisotropic elasticity. The problem on one
dimensional line has been studied in some special cases (?, ?). The general
problem, such as properties of profiles, the boundary condition on the
interface and the limit behavior of the equilibrium configuration are needed
to be further studied.

Wellposedness, blow-up and long time behavior of solutions to different
dynamical models. Although there are many works on the wellposedness of
solutions to various dynamical models, some basic questions are still
unsolved, for examples, the global existence of weak solutions to the three
dimensional Ericksen-Leslie system and the inhomogenous Doi-Onsager equation.
Constructing solutions which blow up at finite time are also of interest. In
addition, the long time dynamics would be complicated, even under a given flow
field. For examples, there are many periodic solutions which are called
kayaking, wagging, logrolling, et. al. It would be interesting to rigorously
justify the existence of these solutions.

Dynamics of point defects, disclinations and two phase interfaces. In the
Ginzburg-Landau theory for superconductors, dynamics of vortices and filaments
have been well understood. The method can not be directly applied for similar
problems in LCs, for example, the uniaxial limit of dynamical
$\mathbf{Q}$-tensor system, due to some essential differences: First, the
elasticity is usually anisotropic which will cause complex phenomenons such as
back flows; Second, the $\mathbf{Q}$-tensor can escape to biaxial, and the
minimizing manifold is isomorphic to the projective space $\mathbb{RP}^{2}$
which has different topological property with $\mathbb{S}^{2}$ or
$\mathbb{S}^{1}$; Thirdly, the hydrodynamics will play important roles. The
singularity set of the limit system can be isolated points, lines and
surfaces, which has co-dimensional three, two and one respectively. The
evolution of the singularity set is an interesting and challenging problems
which may rely on deeper understanding on the profiles near the singularity
set.

Relationship between different dynamical models. The rigorous derivation of
the Ericksen-Leslie theory from the Doi-Onsager theory are only established
for smooth solutions with isotropic elasticity ($k_{1}=k_{2}=k_{3}$). To
generalize it to the anisotropic elasticity case would be an important
problem. Moreover, as the defect solutions can only be described by weak
solutions in the vector theory, justification of convergence for weak
solutions is an important problem. However, analogous results for weak
solutions are very few.

Solution landscape of confined liquid crystals. NLCs in confinement typically
exhibit multiple stable and unstable states, which correspond to stationary
solutions of a free energy. Solution landscape of confined NLCs, a hierarchy
of connected solutions, is able to provide a rich and insightful information
of the physical properties of such multisolution problems. The HiSD is an
efficient numerical algorithm for computing saddle-point solutions for LC
systems, and has been successfully applied to construct the solution
landscapes of the NLCs confined in a square and a hexagon. Solution landscape
reveals not only transition pathways between stable solutions connected by
index-1 saddle points, but also innovative dynamical pathways mediated by
high-index saddle points. Several challenging analytic and numerical questions
remain for the solution landscape, for instance, the completion of solution
landscape. Can we obtain bounds for the Morse index of the parent state in the
solution landscape? How to estimate the number of the stationary solutions of
a given free energy? Building theoretical framework of solution landscape is a
big challenge as well as a great opportunity for applied mathematicians to
make contributions.

Finally, we would like to emphasize again that we only focus on the simplest,
NLC systems in the content of this review, and have not touched much upon the
other LC systems. There is a crucial need for exhaustive theoretical and
numerical studies of complex and unconventional LC systems, to complement
thriving experimental work. We list a few samplers in the remaining part, in
which the theory is relatively open, the energies are more complex, and the
solutions are less regular with new analytic and numerical challenges.

Phase transitions in multiphase liquid crystal systems. Recent experiments
show NLC-cholesteric (by adding chiral particles), NLC-smectic (by lowering
the temperature) phase transitions in LC systems (?). Compared with single
phase LC, these heterogeneous systems are inherently more complex with
multiple competing NLC, cholesteric/smectic and mixed solutions, novel
solutions with phase separation, and interactions between phases (?). Multiple
phases in the heterogeneous system need to be analyzed under the unified
theory framework. For instance, the complex N-S or N-C systems can be analyzed
in terms of LdG-type energies with multiple order parameters for coupled
system with smectic/cholesteric terms. The heterogeneous systems can also be
analyzed by molecular models. On the other hand, the accurate structures (e.g.
interfaces between phases) and complex defects, have multiscale properties.
The theory is relatively open with new analytic and numerical challenges.

Ferroelectric and ferromagnetic liquid crystals. LC is sensitive to external
fields such as electric field and magnetic field (?, ?, ?, ?). Due to the
photoelectric effect, LCs are applied in display devices and have
revolutionized the display industry. Besides normal LC, ferroelectric LC as a
new LC material has attracted extensive attention. Ferroelectric LC materials
have the characteristics of high response speed, high contrast, high
resolution and large capacity information display. It has a good application
prospect in the fields of display, optical interconnections and optical
information processing. One future direction is to develop suitable LC models
from molecular model to macroscopic models to study the photoelectric effect
of ferroelectric LCs and the design of voltage waveform.

Active liquid crystal systems. The non-equilibrium phenomena studied in active
matter systems are widespread in nature, especially playing an important role
in biological phenomena (?, ?, ?). The active matter system is a typical non-
equilibrium system, which is composed of a large number of active particles.
These particles obtain energy for self-propulsion by converting other forms of
energy into kinetic energy (?). Due to the interaction between particles and
particles, and particles and media, the entire system can exhibit extremely
rich dynamic phenomena macroscopically. Active LC system is a paradigm of
active systems for biologically inspired complex fluids with orientational
alignment. In active LC, the rod-like constituents endow the fluid they are
immersed in with active stresses. For instance, in cell motivity the motility
mechanism arising by the interaction of myosin and actin. Their dynamic
properties, phase transition process, and external field response are worthy
of study and exploration.

## Acknowledgement

We would like to thank Dr. Yucen Han, Dr. Yiwei Wang, and Mr. Jianyuan Yin for
providing great assistance. We also thank Professors Yucheng Hu, Yuning Liu,
Apala Majumdar, Jinhae Park, and Zhifei Zhang for fruitful discussions and
helpful comments. Wei Wang was partially supported by the National Natural
Science Foundation of China No. 11922118 and 11931010. Lei Zhang was partially
supported by the National Natural Science Foundation of China No. 11861130351
and the Royal Society Newton Advanced Fellowship, in partnership with Apala
Majumdar from Strathclyde. Pingwen Zhang was partially supported by the
National Natural Science Foundation of China No. 21790340, 11421101.

##  8 Appendix

###  8.1 Elementary identities for the high order moment

Let
$\mathbf{Q}_{2}[f]=\langle\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I}\rangle_{f},$
and define $\mathbf{Q}_{4}[f]$ as follows:

$\displaystyle Q_{4\alpha\beta\gamma\mu}[f]\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{\langle}{m}_{\alpha}{m}_{\beta}{m}_{\gamma}{m}_{\mu}-\frac{1}{7}(m_{\alpha}m_{\beta}\delta_{\gamma\mu}+m_{\gamma}m_{\mu}\delta_{\alpha\beta}+m_{\alpha}m_{\gamma}\delta_{\beta\mu}+m_{\beta}m_{\mu}\delta_{\alpha\gamma}$
(118)
$\displaystyle+m_{\alpha}m_{\mu}\delta_{\beta\gamma}+m_{\beta}m_{\gamma}\delta_{\alpha\mu})+\frac{1}{35}(\delta_{\alpha\beta}\delta_{\gamma\mu}+\delta_{\alpha\gamma}\delta_{\beta\mu}+\delta_{\alpha\mu}\delta_{\beta\gamma})\Big{\rangle}_{f}.$

Let $P_{k}(x)$ be the $k$-th Legendre polynomial and

$\displaystyle
S_{2}=\langle{P}_{2}(\mathbf{m}\cdot\mathbf{n})\rangle_{h_{\mathbf{n}}},~{}S_{4}=\langle{P}_{4}(\mathbf{m}\cdot\mathbf{n})\rangle_{h_{\mathbf{n}}},$
(119)

which only depend on $\alpha$. Firstly, we have a lemma:

######  Lemma 8.1.

$\displaystyle\mathbf{Q}_{2}[h_{\mathbf{n}}]$ $\displaystyle=$ $\displaystyle
S_{2}(\mathbf{n}\mathbf{n}-\frac{1}{3}),$ $\displaystyle
Q_{4\alpha\beta\gamma\mu}[h_{\mathbf{n}}]$ $\displaystyle=$ $\displaystyle
S_{4}\Big{(}n_{\alpha}n_{\beta}n_{\gamma}n_{\mu}-\frac{1}{7}(n_{\alpha}n_{\beta}\delta_{\gamma\mu}+n_{\gamma}n_{\mu}\delta_{\alpha\beta}+n_{\alpha}n_{\gamma}\delta_{\beta\mu}$
$\displaystyle+n_{\beta}n_{\mu}\delta_{\alpha\gamma}+n_{\alpha}n_{\mu}\delta_{\beta\gamma}+n_{\beta}n_{\gamma}\delta_{\alpha\mu})+\frac{1}{35}(\delta_{\alpha\beta}\delta_{\gamma\mu}+\delta_{\alpha\gamma}\delta_{\beta\mu}+\delta_{\alpha\mu}\delta_{\beta\gamma})\Big{)}.~{}~{}~{}~{}~{}~{}~{}~{}$

######  Corollary 8.2.

Let
$\mathbf{M}=\langle\mathbf{m}\mathbf{m}\rangle_{h_{\mathbf{n}}},~{}\mathbf{M}^{(4)}=\langle\mathbf{m}\mathbf{m}\mathbf{m}\mathbf{m}\rangle_{h_{\mathbf{n}}}$,
there holds that

$\displaystyle\mathbf{M}$ $\displaystyle=$ $\displaystyle
S_{2}\mathbf{n}\mathbf{n}+\frac{1-S_{2}}{3}\mathbf{I},$ (120) $\displaystyle
M^{(4)}_{\alpha\beta\gamma\mu}$ $\displaystyle=$ $\displaystyle
S_{4}n_{\alpha}n_{\beta}n_{\gamma}n_{\mu}+\frac{S_{2}-S_{4}}{7}(n_{\alpha}n_{\beta}\delta_{\gamma\mu}+n_{\gamma}n_{\mu}\delta_{\alpha\beta}+n_{\alpha}n_{\gamma}\delta_{\beta\mu}+n_{\beta}n_{\mu}\delta_{\alpha\gamma}$
(121)
$\displaystyle+n_{\alpha}n_{\mu}\delta_{\beta\gamma}+n_{\beta}n_{\gamma}\delta_{\alpha\mu})+\big{(}\frac{S_{4}}{35}-\frac{2S_{2}}{21}+\frac{1}{15}\big{)}(\delta_{\alpha\beta}\delta_{\gamma\mu}+\delta_{\alpha\gamma}\delta_{\beta\mu}+\delta_{\alpha\mu}\delta_{\beta\gamma}).~{}~{}~{}~{}~{}~{}~{}~{}$

###  8.2 Some identities used in Section 3.4.3.

For any constant vector $\mathbf{u},~{}\mathbf{w}\in\mathbb{R}^{3}$, and a
vector field $\mathbf{v}(\mathbf{m})$ defined on ${\mathbb{S}^{2}}$,

$\displaystyle\mathbf{u}\cdot\Big{(}\int_{{\mathbb{S}^{2}}}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I}){\mathcal{R}}\cdot(f\mathbf{v}){\rm
d}\mathbf{m}\Big{)}\cdot\mathbf{w}$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{S}^{2}}}\Big{(}(\mathbf{m}\cdot\mathbf{u})(\mathbf{m}\cdot\mathbf{w})-\frac{1}{3}\mathbf{u}\cdot\mathbf{w}\Big{)}{\mathcal{R}}\cdot(f\mathbf{v}){\rm
d}\mathbf{m}$ $\displaystyle=$
$\displaystyle-\int_{{\mathbb{S}^{2}}}{\mathcal{R}}\Big{(}(\mathbf{m}\cdot\mathbf{u})(\mathbf{m}\cdot\mathbf{w})\Big{)}\cdot(f\mathbf{v}){\rm
d}\mathbf{m}$ $\displaystyle=$
$\displaystyle-\int_{{\mathbb{S}^{2}}}\mathbf{m}\times\mathbf{u}\cdot\mathbf{v}(\mathbf{m}\cdot\mathbf{w})f+(\mathbf{m}\cdot\mathbf{u})(\mathbf{m}\times\mathbf{w})\cdot\mathbf{v}{f}{\rm
d}\mathbf{m}$ $\displaystyle=$
$\displaystyle\mathbf{u}\cdot\langle\mathbf{m}\times\mathbf{v}\mathbf{m}+\mathbf{m}\mathbf{m}\times\mathbf{v}\rangle\cdot\mathbf{w}.$

Hence, we have:

$\displaystyle\int_{{\mathbb{S}^{2}}}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I}){\mathcal{R}}\cdot(f\mathbf{v}){\rm
d}\mathbf{m}=\langle\mathbf{m}\times\mathbf{v}\mathbf{m}+\mathbf{m}\mathbf{m}\times\mathbf{v}\rangle.$
(122)

Let $\mathbf{v}={\mathcal{R}}{\mu}$, we have

$\displaystyle\int_{{\mathbb{S}^{2}}}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I}){\mathcal{R}}\cdot(f{\mathcal{R}}\mu){\rm
d}\mathbf{m}=\langle\mathbf{m}\times{\mathcal{R}}\mu\mathbf{m}+\mathbf{m}\mathbf{m}\times{\mathcal{R}}\mu\rangle.$
(123)

Let $\mathbf{v}=\mathbf{m}\times\kappa\cdot\mathbf{m}$, we have

$\displaystyle\int_{{\mathbb{S}^{2}}}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I}){\mathcal{R}}\cdot(f\mathbf{m}\times\kappa\cdot\mathbf{m}){\rm
d}\mathbf{m}$ (124) $\displaystyle=$
$\displaystyle\langle\mathbf{m}\times(\mathbf{m}\times\kappa\cdot\mathbf{m})\mathbf{m}+\mathbf{m}\mathbf{m}\times(\mathbf{m}\times\kappa\cdot\mathbf{m})\rangle$
$\displaystyle=$ $\displaystyle\langle
2(\mathbf{m}\cdot\kappa\cdot\mathbf{m})\mathbf{m}\mathbf{m}-(\kappa\cdot\mathbf{m})\mathbf{m}-\mathbf{m}(\kappa\cdot\mathbf{m})\rangle$
$\displaystyle=$ $\displaystyle
2\mathbf{D}:\langle\mathbf{m}\mathbf{m}\mathbf{m}\mathbf{m}\rangle-\mathbf{D}\cdot\langle\mathbf{m}\mathbf{m}\rangle+\mathbf{\Omega}\cdot\langle\mathbf{m}\mathbf{m}\rangle-\langle\mathbf{m}\mathbf{m}\rangle\cdot(\mathbf{D}+\mathbf{\Omega}).$

###### Lemma 8.1

For any antisymmetric constant matrix $\mathbf{\Omega}$, we have

$\displaystyle{\mathcal{R}}\cdot\big{(}\mathbf{m}\times(\mathbf{\Omega}\cdot\mathbf{m}){f}_{0}\big{)}$
$\displaystyle=(\mathbf{n}\times(\mathbf{\Omega}\cdot\mathbf{n}))\cdot{\mathcal{R}}{f_{0}},$
$\displaystyle{\mathcal{R}}\cdot\big{(}\mathbf{m}\times(\mathbf{D}\cdot\mathbf{m}){f}_{0}\big{)}$
$\displaystyle=\frac{1}{2}{\mathcal{R}}\cdot\big{(}f_{0}{\mathcal{R}}(\mathbf{m}\mathbf{m}:\mathbf{D})\big{)}.$

Proof. The lemma is a direct consequence of the following identities

$\displaystyle{\mathcal{R}}\cdot\big{(}\mathbf{m}\times(\mathbf{\Omega}\cdot\mathbf{m})\big{)}={\mathcal{R}}_{i}(\epsilon^{ijk}m_{j}\Omega_{kl}m_{l})=(\mathbf{I}-3\mathbf{m}\mathbf{m}):\mathbf{\Omega}=0,$
$\displaystyle(\mathbf{m}\times(\mathbf{\Omega}\cdot\mathbf{m}))\cdot{\mathcal{R}}{f_{0}}=(\mathbf{m}\times(\mathbf{\Omega}\cdot\mathbf{m}))\cdot(\mathbf{m}\times\mathbf{n})f_{0}^{\prime}$
$\displaystyle\quad=(\mathbf{n}\times(\mathbf{\Omega}\cdot\mathbf{n}))\cdot(\mathbf{m}\times\mathbf{n})f_{0}^{\prime}=(\mathbf{n}\times(\mathbf{\Omega}\cdot\mathbf{n}))\cdot{\mathcal{R}}{f_{0}}.$

This completes the proof. MM $\square$

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