17 Liquid crystals

Learning outcomes

Understand the emergence of orientational order and its description in terms of tensorial and scalar order parameters.
Construction of mean-field theories: phenomenological (Landau-de Gennes) and microscopic (Maier-Saupe).
Describe large-scale distortions of the director field with Frank elasticity, and characterise topological defects by their charge.

Ordered phases of matter are typically characterised by spatial correlations, which can be positional or orientational in nature.

Positional order is the regular arrangement of particle positions in space, often forming a repeating lattice structure. Orientational order refers to the alignment of the particles’ axes, influenced for example by the anisotropy of the particle itself (e.g. rods tending to align with other rods).

In crystalline phases, both positional and orientational order are long-ranged, meaning that the arrangement and orientation of particles are correlated over macroscopic distances. In contrast, simple liquids exhibit only very short-range (positional and orientational) correlations.

Liquid crystals represent an intermediate case, where orientational order can persist over long distances while positional order remains short-ranged or absent.

17.1 Anisotropic particles

The decoupling between orientational and translational degrees of freedom is favoured by individual microscopic units (particles) that already have preferential axis of symmetry, i.e. anisotropic particles. Anisotropic particles have shapes that are not spherically symmetric, such as rods, ellipsoids, or plates. Their interactions depend not only on the distance between particles but also on their relative orientations.

There are many kinds of anisotropic particles. The table below contains a non-exhaustive list.

Particle Shape	Description	Example Materials	Typical Applications
Rods	Cylindrical, length > width	Gold nanorods, Tobacco virus	Photothermal therapy, plasmonics
Ellipsoids	Elongated or flattened spheroids	Hematite ellipsoids	Anisotropic optics, directed self-assembly
Plates/Discs	Flat, disk-like shapes	Graphene oxide, clay platelets	Barrier materials, viscosity control
Polyhedra	Multi-faceted, highly symmetric shapes	Gold nanocubes, silica polyhedra	Photonic crystals, catalysis

Various assemblies of anisotropic particles. Left to right: Golden nanorods (from Stanford Advanced Materials), tobacco virus rods (adapted from Knapp and Lewandowski (2001)), ellipsoidal silica-coated hematite particles (adapted from Sánchez-Ferrer et al. (2010)), hard platelets in the isotropic phase (from Atashpendar et al. (2020)), and an arrangement of octahedra simulated using hard particle Monte Carlo with hoomdblue

17.2 Liquid crystal phases

The additional orientational degrees of freedom opens up the possibilities of phases that do not normally exist for spherical, isotropic particles. We progress from the most disordered to the most ordered phase:

Isotropic fluid: this phase is not very different from the ordinary fluid phase of spherical particles. There is no long-range positional or orientational order and all correlations decay rapidly and on short lengthscales.
Nematic phase: still without translational order, the constituents arrange themselves spontaneously in a preferential, average direction, called the director. Since there is no positional order, the transition from isotropic to nematic can only be detected by taking into account the relative orientations between the constituents, as the individual centres of mass are as disordered in the nematic as they are in the isotropic. If the particles are themselves chiral (i.e., the particle differs from its mirror image) the director tends to form a helix due to the propensity of the molecules to align at some angle between each other. This leads to so called chiral nematic phase (also known as cholesteric).
smectic phase: in this phase, particles not only align along a common direction (as in the nematic phase) but also exhibit partial translational order. The centers of mass of the particles tend to form well-defined layers, with orientational order within each layer. However, within a layer, the positions of the particles remain disordered, similar to a liquid. Smectic phases can be further classified (e.g., smectic A, smectic C) depending on the relative orientation of the director with respect to the layer normal.
Columnar phase: here, anisotropic particles (often disc-like) stack into columns, which then arrange themselves into a two-dimensional lattice. There is long-range positional order in two directions (within the plane perpendicular to the columns) and orientational order along the column axis.
Crystal: the most ordered phase, where both positional and orientational order are long-ranged in all directions, forming a true three-dimensional periodic lattice.

Phase	Positional Order	Orientational Order
Isotropic fluid	No	No
Nematic	No	Yes (long-range)
Chiral nematic	No	Yes (helical)
Smectic	Yes (1D: layered)	Yes (within layers)
Columnar	Yes (2D: columns)	Yes (along columns)
Crystal	Yes (3D: lattice)	Yes (long-range)

17.3 Orientational correlations and the isotropic/nematic transition

While for the spherical colloids we focused on positional correlation functions like the radial distribution function, for anisotropic particle we need to quantify orientational correlations.

Suppose we have \(N\) rod-like particles, each with an orientation angle \(\theta_i\) (in 2D) or a unit vector \(\mathbf{u}_i\) (in 3D). We want to determine the director, i.e. the average direction of the long molecular axes of all molecules in the liquid crystal.

The rods are head-tail symmetric, so, we cannot truly distinguish which one is the head or the tail (e.g. an orientation of \(\theta\) in 2D is the same as an orientation of \(\theta+\pi\)). We cannot therefore just take the average of the vectors to extract the common orientation of the various vectors, otherwise we would incur into a cancellation.

Instead, we construct a second moment tensor called the alignment tensor. There are different notations in the literature. The most general form is

\[ \mathbf{Q}=\dfrac{d}{d-1}\left\langle\mathbf{u}_i \otimes \mathbf{u}_i-\dfrac{1}{d}\mathbf{I}\right\rangle \]

where \(d\) is the dimensionality. The removed \(\dfrac{1}{d}\mathbf{I}\) term ensures that the tensor is traceless and does not include isotropic components.

The analysis of the tensor yield the main characteristics of the orientation of the system. For example

the largest eigenvalue of \(\mathbf{Q}\) is the scalar nematic order parameter \(\mathcal{S}\)
the eigenvector corresponding to the largest eigenvalues is called the director \(\mathbf{n}\) and corresponds to the main collective orientation of the system.

Notice that \(\mathcal{S}\) is not the entropy \(S\)! These commonly used symbols should not be confused.

In three dimensions, measuring \(\theta_i\) from the director, this gives

\[ \mathcal{S}=\dfrac{1}{2} \left\langle 3\cos ^2 \theta_i-1\right\rangle \]

(the expression for the director in 3d requires explicit diagonalisation).

Derivation of 2D director

For the purpose of illustrating how the calculations for the director are carried out, we provide a detailed example in two-dimensions.

In two dimensions, each particle has an orientation unit vector

\[ \mathbf{u}_i = \begin{pmatrix} \cos \theta_i \\ \sin \theta_i \end{pmatrix}, \]

To start constructing Q, we take the outer (tensorial) product of the unit vector with itself:

\[ \mathbf{u}_i \otimes \mathbf{u}_i = \begin{pmatrix} \cos^2 \theta_i & \cos \theta_i \sin \theta_i \\ \cos \theta_i \sin \theta_i & \sin^2 \theta_i \end{pmatrix}. \]

The resulting matrix is clearly symmetric and has obviously trace equal to 1 because \(\cos^2 \theta_i + \sin^2 \theta_i = 1\). Angled brackets \(\langle \cdot \rangle\) denote the average over all particles in the system. The alignment tensor is then defined as \[ Q = 2\left\langle \mathbf{u}_i \otimes \mathbf{u}_i \right\rangle -\mathbf{I} \] where we used the identity matrix to zero out the trace. The factor 2 is a geometric normalisation, and we see in the following that it is convenient when looking at the magnitude of Q.

For convenience, let’s write \(\mathbf{Q}\) in terms of its components:

\[\mathbf{Q} = \begin{pmatrix} Q_{xx} & Q_{xy} \\ Q_{yx} & -Q_{xx} \end{pmatrix}\]

where we enforced that the trace is zero. The components are given by

\[Q_{xx} = \langle 2\cos^2 \theta_i - 1 \rangle = \langle \cos 2\theta_i \rangle\]

\[Q_{xy} = \langle 2\cos \theta_i \sin \theta_i \rangle = \langle \sin 2\theta_i \rangle\]

where we used the double angle trigonometric identities \(\cos 2\theta = 2\cos^2 \theta - 1\) and \(\sin 2\theta = 2\sin \theta \cos \theta\).

The magnitude of Q is denoted by \(\mathcal{S}\) and defined as the largest eigenvalue of Q, and the corresponding eigenvector is the director. The eigenvalue problem is formally

\[ \mathbf{Q} \mathbf{n} = \mathcal{S} \mathbf{n} \]

By taking the determinant we have

\[ \operatorname{det}(\mathbf{Q}-\lambda \mathbf{I})=0, \] which gives the characteristic polynomial in terms of the components of Q:

\[-Q_{xx}^2-Q_{xy}^2+\lambda^2=0\]

So the eigenvalues are \(\lambda = \pm \sqrt{Q_{xx}^2 + Q_{xy}^2}\), and the largest eigenvalue is the positive one, which we define as the nematic order parameter

\[\mathcal{S} = \sqrt{Q_{xx}^2 + Q_{xy}^2}\]

The director \(\mathbf{n}\) is the eigenvector of the alignment tensor \(\mathbf{Q}\) corresponding to the largest eigenvalue \(\mathcal{S}\).

Since \(\mathbf{Q}\) is symmetric and traceless, it can be diagonalised using a rotation that aligns the director with the x-axis.

\[ \mathbf{Q}=\mathbf{R}(\psi)\left(\begin{array}{cc} \mathcal{S} & 0 \\ 0 & -\mathcal{S} \end{array}\right) \mathbf{R}(\psi)^T \]

where

\[ \mathbf{R}(\psi)=\left(\begin{array}{cc} \cos \psi & -\sin \psi \\ \sin \psi & \cos \psi \end{array}\right) . \]

is a rotation of angle \(\psi\) that aligns the director with the x-axis. The angle \(\psi\) is the orientation of the director!

Carrying out the matrix multiplication explicitly yields the components in terms of the director:

\[ \begin{aligned} &Q_{x x}=\mathcal{S}\left(\cos ^2 \psi-\sin ^2 \psi\right)=\mathcal{S} \cos (2 \psi)\\ &Q_{x y}=\mathcal{S}(2 \sin \psi \cos \psi)=\mathcal{S} \sin (2 \psi) \end{aligned} \]

Taking the ratio of the two components gives the angle of the director: \[\tan(2\psi) = \frac{Q_{xy}}{Q_{xx}}\]

Notice that the double angle comes out naturally, and reflects the head tail symmetry.

This lends itself to a complex plane representation of the director and the order parameter:

The director is simply \(\mathbf{n} = e^{i\psi}\)
the order parameter Q can be mapped to a complex number \(Q = Q_{xx} + i Q_{xy} = \mathcal{S} e^{i2\psi}\) where the 2 in the exponent reflects the head-tail symmetry of the nematic phase.

17.4 Landau-de Gennes mean field theory

Let us consider the uniaxial 3D case, which is the most commonly discussed case (this focuses on nematic order, excluding cholesteric and smectic order) and attempt to construct a Landau theory centred on the order parameter in the spirit of what we saw in Section 5.5. If the system is uniaxial, then the order is characterized by a single director and the \(\mathbf{Q}\) must be symmetric and traceless with a preferred axis \(\mathbf{n}\).

For historical reasons, in three dimensions, the alignment tensor is often defined as

\[ \mathbf{Q}=\mathcal{S}\left(\mathbf{n} \otimes \mathbf{n}-\frac{1}{3} \mathbf{I}\right) \]

for some scalar value \(\mathcal{S}\) (the order parameter, between 0 for the isotropic phase and 1 for perfect alignment). Notice that this is slightly different from the general definition above, but it is equivalent up to a multiplicative factor. We will work with this definition because it consistently produces the standard Landau-de Gennes free energy functional without further manipulations.

If we choose now the \(z\) axis along \(\mathbf{n}\) we can write this explicitly as

\[ \mathbf{Q}=\mathcal{S}\left(\begin{array}{ccc} -\frac{1}{3} & 0 & 0 \\ 0 & -\frac{1}{3} & 0 \\ 0 & 0 & \frac{2}{3} \end{array}\right) \]

with \(\mathcal{S}=\frac{3}{2}\left\langle(\mathbf{u} \cdot \mathbf{n})^2-\frac{1}{3}\right\rangle\) as defined earlier.

The Landau free energy density for the isotropic-nematic transition is constructed by scalar combinations of the tensor \(\mathbf{Q}\). The idea is to expand the free energy density \(f\) around the transition and write it as

\[ f=f_0+\frac{A}{2} Q_{i j} Q_{j i}-\frac{B}{3} Q_{i j} Q_{j k} Q_{k i}+\frac{C}{4}\left(Q_{i j} Q_{i j}\right)^2 \]

or in more compact form

\[ f=f_0+\frac{A}{2} \operatorname{tr} \mathbf{Q}^2-\frac{B}{3} \operatorname{tr} \mathbf{Q}^3+\frac{C}{4}\left(\operatorname{tr} \mathbf{Q}^2\right)^2 \] where we stop the construction to terms of fourth order in \(\mathbf{Q}\) and \(A,B,C\) are temperature dependent coefficients.

We note that given our definitions above the following identities hold \[ \operatorname{Tr}\left(\mathbf{Q}^2\right)=\frac{2}{3} S^2 \quad,\quad \operatorname{Tr}\left(\mathbf{Q}^3\right)=\frac{2}{9} S^3 \]

so that

\[ f=f_0+\frac{A}{3} S^2-\frac{2 B}{27} S^3+\frac{C}{9} S^4 \]

We can expand \(A,B,C\) around the reference temperature \(T^\ast\) as \[ A(T)=a\left(T-T_*\right)+\cdots \] and at zeroth order for both \(B\) and \(C\) as \(B(T)= b+\dots\) , \(C(T)=c+\dots\), with \(a,b,c>0\): \(c>0\) keeps the free energy bounded, while \(b>0\) favours \(\mathcal S>0\) (rods aligned along \(\mathbf n\)).

We can follow the scheme used for Landau phase transitions and obtain the so-called Landau-de Gennes theory result

\[ f-f_0=\frac{a}{3}\left(T-T^{\ast}\right) S^2-\frac{2 b}{27} S^3+\frac{c}{9} S^4 . \]

This has the form of standard Landau free energy (combinations of power of S up to the power of 4 with real coefficients that depend on the temperature). The cubic term is allowed because, unlike the magnetisation of a magnet, \(\mathcal S\) and \(-\mathcal S\) are not equivalent: \(\mathcal S>0\) describes rods aligned along \(\mathbf n\), while \(\mathcal S<0\) describes rods lying in the plane perpendicular to \(\mathbf n\). The cubic term makes this a first order phase transition: in a range of temperatures the free energy has two minima, the isotropic one (\(\mathcal{S}=0\)) and the nematic one (\(\mathcal{S}>0\)). There is no critical point, because the isotropic and nematic phases have different symmetry. It can be shown that the transition temperature is

\[ T_{N I}=T^{\ast}+\frac{b^2}{27 a c} \] where \(T^{\ast}\) is a material dependent temperature, identifying where the quadratic term of the free energy vanishes, and where the isotropic phase becomes unstable (spinodal).

17.5 Maier-Saupe theory

The Landau-de Gennes theory illustrates how the simple existence of a suitably defined nematic order is compatible with the construction of a theory that predicts a first order phase transition.

We can also construct a more microscopically motivated theory. We can start by linking the scalar order parameter in terms of the probability distribution \(p(\Omega)\) of finding a rod with polar angles \(\Omega=(\theta,\phi)\) in a coordinate system with its polar axis along the director \(\mathbf{n}\). The infinitesimal probability is \(p(\Omega)d\Omega\).

The nematic phase has full cylindrical symmetry around the director, meaning \(p(\theta, \phi)\) is independent of the azimuthal angle \(\phi\) and depends only on the deviation \(\theta\). Accounting for head-tail symmetry via the first non-trivial Legendre polynomial, \(P_2(\cos\theta)\), the scalar order parameter is defined as:

\[\mathcal{S}=\dfrac{3}{2}\left\langle(\mathbf{u} \cdot \mathbf{n})^2-\frac{1}{3}\right\rangle = \dfrac{1}{2}\langle 3 \cos^2\theta-1\rangle=\langle P_2(\cos\theta)\rangle = \int_0^{\pi}P_2(\cos\theta)p(\theta)\sin\theta d\theta\]

A complementary approach to the macroscopic free energy picture of Landau-de Gennes is to consider the minimal physical ingredients that lead to the transition itself. As in the case of the colloids, we have essentially two classes

attractive interactions that can favour the new symmetries (e.g. the nematic order)
entropic contributions that make it easier to pack the rods in the new symmetry (i.e. when they are aligned)

Phenomenologically, we can put these two together assuming a form for the energetic and entropic contributions to write down a free energy of the following form

\[f = f_0 - u\dfrac{S^2}{2}+k_BT \int_0^\pi p(\theta) \ln (2 p(\theta)) \sin\theta d\theta\]

where:

\(u\) is the coupling strength parameter that quantifies the energetic preference for orientational alignment (higher \(u\) favors stronger nematic order),
the factor \(2\) ensures that the entropic term vanishes for the isotropic distribution \(p(\theta)=1/2\) (with \(p(\theta)\) normalised as \(\int_0^\pi p(\theta)\sin\theta d\theta=1\)), and \(\sin\theta d\theta\) is the usual Jacobian.

The two terms above identify two simple contributions:

an energetic contribution that is a simple quadratic function of the order parameter (more order, lower energy, with the isotropic phase paying the highest cost). Near the isotropic phase the free energy needs to be even due to the head-tail symmetry.
a generic entropic contribution of the form \(k_B\int p(\Omega)\ln p(\Omega) d\Omega\) using the Gibbs entropy formula.

The minimization of the free energy with respect to \(p(\theta)\), under the normalization constraint \(\int_0^\pi p(\theta) \sin\theta d\theta = 1\), is performed using a Lagrange multiplier \(\lambda\). The variational problem is:

\[ \delta \left[ f + \lambda \left( \int_0^\pi p(\theta) \sin\theta d\theta - 1 \right) \right] = 0 \]

Taking the functional derivative with respect to \(p(\theta)\) and setting it to zero yields:

\[ \frac{\delta f}{\delta p(\theta)} + \lambda \sin\theta = 0 \]

Plugging in the expression for \(f\) and simplifying, we obtain the solution:

\[ p(\theta) = \frac{1}{Z} \exp\left( \frac{u S}{k_B T} P_2(\cos\theta) \right) \]

where \(\mathcal{S}\) is the nematic order parameter to be determined self-consistently, and \(Z\) is the normalization constant (partition function):

\[ Z = \int_0^\pi \exp\left( \frac{u S}{k_B T} P_2(\cos\theta) \right) \sin\theta d\theta \]

This is the Maier-Saupe self-consistent equation for the orientational distribution in the mean-field theory of nematic liquid crystals. In the code below, we can see how this leads to a non trivial energy profile which corresponds to the emergence of a ordered state (a minimum at high \(\mathcal{S}\)) corresponding to the nematic phase.

The transition is weakly-first order, as two separate basins of stability (separated typically by a small barrier) are formed.

These profiles are example of simple free energy landscapes. We will see more about this later in Section 18.1.

17.5.1 Hard rods: ordering from entropy alone

The Maier-Saupe theory relies on the first ingredient listed above: without the coupling \(u\) there is no transition. However, the second ingredient alone is sufficient: even hard rods, with no attractions at all, undergo a transition from the isotropic to the nematic phase at high enough density. Since there is no interaction energy, the transition is purely entropic (temperature only sets the scale, as for hard spheres). Two entropic contributions compete:

orientational entropy favours the isotropic phase: aligning the rods reduces the number of available orientations;
translational entropy favours the nematic phase: aligned rods exclude less volume to each other, so each rod has more free volume to move in.

At low density the first contribution wins and the rods are isotropic. At high density the gain in translational entropy outweighs the loss of orientational entropy, and the nematic phase has the higher total entropy.

This mirrors the freezing of hard spheres: there too the ordered phase wins on entropy. The spheres lose the disorder of their positions, but each sphere gains free volume to vibrate around its lattice site.

17.5.2 Lattice model: the Lebwohl-Lasher model

The Lebwohl-Lasher model is a lattice model designed to capture the orientational ordering of anisotropic particles, such as those found in nematic liquid crystals and is a lattice version of the Maier-Saupe mean field model, in the same way as the Ising model is a lattice model for magnets. In this model, each lattice site \(i\) is associated with a unit vector \(\mathbf{n}_i\) representing the local orientation (the “director”) of a particle at that site.

The Hamiltonian of the Lebwohl-Lasher model is given by:

\[ H = -\epsilon \sum_{\langle i, j \rangle} \left[ \frac{3}{2} (\mathbf{n}_i \cdot \mathbf{n}_j)^2 - \frac{1}{2} \right] \]

where: - \(\epsilon > 0\) is the coupling constant favoring alignment, - the sum \(\langle i, j \rangle\) runs over all pairs of nearest-neighbor sites, - \((\mathbf{n}_i \cdot \mathbf{n}_j)^2\) measures the degree of alignment between neighboring directors.

This Hamiltonian favors parallel (or antiparallel) alignment of neighboring directors, capturing the essential physics of the isotropic-nematic transition in liquid crystals.

In two dimensions, the Lebwohl-Lasher model is similarly defined, but the directors \(\mathbf{n}_i\) are restricted to lie in the plane. Each director can be represented by a unit vector \(\mathbf{n}_i = (\cos\theta_i, \sin\theta_i)\), where \(\theta_i\) is the orientation angle at site \(i\).

The Hamiltonian in 2D becomes:

\[ H = -\epsilon \sum_{\langle i, j \rangle} \left[ \frac{3}{2} \cos^2(\theta_i - \theta_j) - \frac{1}{2} \right] \]

where the sum is over nearest-neighbor pairs on a 2D lattice. This model captures the essential features of orientational ordering in two-dimensional systems. Note however that in two dimensions the continuous rotational symmetry of the director cannot be spontaneously broken at finite temperature (Mermin-Wagner theorem): the low temperature phase only has quasi-long-range orientational order: correlations decay as a power law rather than exponentially. Since \(\frac32\cos^2\Delta\theta-\frac12=\frac34\cos2\Delta\theta+\frac14\), the model is the XY model for the angle \(2\theta\); its transition is driven by the unbinding of pairs of topological defects (Kosterlitz–Thouless transition).

Activity

Modify the script above to check the following

The high temperature regime (\(T\gg 0.5\)) is mostly disordered (isotropic)
The low temperature regime forms larger and larger patches with coherent orientations
Define and plot a global nematic order parameter. Τhe global nematic order parameter S is defined as:

\[ S = \langle \cos(2(\theta - \psi))\rangle\]

where \(\psi\) is the director (average orientation), which in 2D is

\[\psi = \dfrac{1}{2} {\rm atan2}(\langle \sin(2\theta)\rangle, \langle \cos(2\theta)\rangle)\]

17.6 Splay, twist and bend

In the continuum limit, the nematic liquid crystal can be characterised in terms of the deformation of a vector field \(\mathbf{n}(\mathbf{r})\) which represents the director at every point \(\mathbf{r}\) (a local director, resulting from the average over a large number of particles in a single volume, another example of coarse-graining).

The Frank free energy density for distortions of the director field is

\[ f = \frac{1}{2} K_1 (\nabla \cdot \mathbf{n})^2 + \frac{1}{2} K_2 [\mathbf{n} \cdot (\nabla \times \mathbf{n})]^2 + \frac{1}{2} K_3 [\mathbf{n} \times (\nabla \times \mathbf{n})]^2 \]

where: - \(K_1\) is the splay elastic constant, - \(K_2\) is the twist elastic constant, - \(K_3\) is the bend elastic constant.

Each term in the Frank free energy penalizes a specific type of distortion: splay, twist, or bend. The total elastic free energy is found by integrating \(f\) over the entire sample volume. This formulation is named after Charles Frank, a pioneer in the study of crystal dislocations (which are defects in crystalline solids) and what we call today soft matter at the University of Bristol, for whom the Frank lecture theatre is named.

The formulation allows one to model the behaviour of liquid crystals with finite element methods on much larger scales than the typical size of the elementary constituents. It also allows to phenomenologically add more deformation modes, to include for example the description of chirality.

17.7 Topological defects

The continuum limit description of liquid crystals allows us to enhance their description with the notion of topological defects.

Topological defects are singularities in the director field that can not be removed by a continuous deformation of the director field: they are topologically protected. We can see this with a few examples in 2d by drawing lines of the director (remember that the head-tail symmetry means that the lines do not have any preferential direction but indicate local orientation):

Code

import numpy as np
import matplotlib.pyplot as plt

def plot_half_defect(ax, charge, title):
    # Grid
    x = np.linspace(-1, 1, 21)
    y = np.linspace(-1, 1, 21)
    X, Y = np.meshgrid(x, y)

    # Polar coordinates
    theta = np.arctan2(Y, X)

    # Director angle for nematic disclination
    # Only ±1/2 defects
    phi = charge * theta  # charge = +0.5 or -0.5

    # Director vectors (head-tail symmetry)
    U = np.cos(phi)
    V = np.sin(phi)
    mask = U < 0
    U[mask] *= -1
    V[mask] *= -1

    # Plot director field as short lines
    ax.quiver(
        X, Y, U, V,
        pivot='middle', color='black',
        scale=18, headwidth=0, headlength=0, headaxislength=0
    )

    # Defect core
    ax.plot(0, 0, 'o', color='orange', markersize=12)

    ax.set_title(title, fontsize=12)
    ax.set_aspect('equal')
    ax.set_xlim(-1.1, 1.1)
    ax.set_ylim(-1.1, 1.1)
    ax.set_xticks([])
    ax.set_yticks([])
    ax.axis('off')

# Plotting
fig, axs = plt.subplots(1, 4, figsize=(8, 3))
charges = [0.5, -0.5, +1, -1]
titles = ["+1/2 defect", "-1/2 defect", "+1 defect",  "-1 defect"]

for ax, charge, title in zip(axs, charges, titles):
    plot_half_defect(ax, charge, title)

plt.tight_layout()
plt.show()

These topological defects are places where the orientation changes discontinuously and are also called disclinations.

In two dimensions, the disclinations are characterized by a topological charge (or strength) \(s\), defined by the total rotation of the director around a closed loop enclosing the defect:

\[ \Delta \theta = 2\pi s \]

Common disclination strengths in nematic liquid crystals are \(s = \pm \frac{1}{2}\) and \(s = \pm 1\). For example, a \(+1/2\) disclination corresponds to a director field that rotates by \(+\pi\) as one encircles the defect, while a \(-1/2\) disclination rotates by \(-\pi\).

When two disclinations of opposite charges meet, they annihilate. This means the defects can cancel each other out, restoring uniform orientational order in the region. The annihilation of defect pairs is a key mechanism by which nematic liquid crystals relax toward equilibrium after being disturbed (compare with the coarsening of domains after a quench, where interfaces rather than point defects disappear over time). This is because the ideal equilibrium state should in principle be free of any such defects, as they cost energy: they deform the director field on long distances, leading to a deformation cost, and they also cause a complete loss of orientational order at the singularity point, which costs energy by itself.

In three dimensions, disclinations are line defects rather than point defects. These line defects represent regions where the orientational order of the director field is singular along a curve, rather than at isolated points. The topology and dynamics of disclination lines in 3D nematic liquid crystals are richer and more complex than in 2D, allowing for phenomena such as defect loops, entanglements, and reconnections. Disclination lines can form closed loops, terminate at surfaces, or interact with other defects, and play a crucial role in the response of liquid crystals to external fields, boundary conditions, and during phase transitions.

In 3D there is only one type of stable disclination line: half-integer lines are all equivalent, while integer lines can “escape into the third dimension” and disappear. Point defects (hedgehogs) also appear, with an integer charge. Defects therefore include lines, closed loops, hedgehogs and complex textures.

You can read more about liquid crystal’s defects and their coupling to the material properties in Jones (2002).

17.8 Summary

The table collects the main results of this chapter. Follow the links to go back to the relevant part.

Topic	Key result	Remember
Phases	isotropic, nematic, cholesteric, smectic, columnar, crystal	Orientational order without (full) positional order
Alignment tensor	\(\mathbf Q=\frac{d}{d-1}\left\langle\mathbf u_i\otimes\mathbf u_i-\frac1d\mathbf I\right\rangle\)	Head-tail symmetry: \(\langle\mathbf u\rangle=0\), so we need a tensor
Order parameter, director	\(\mathcal S\) = largest eigenvalue, \(\mathbf n\) = its eigenvector; in 3D \(\mathcal S=\langle P_2(\cos\theta)\rangle\)	\(\mathcal S=0\) isotropic, \(\mathcal S=1\) perfect alignment
Landau-de Gennes	\(f-f_0=\frac a3(T-T^*)\mathcal S^2-\frac{2b}{27}\mathcal S^3+\frac c9\mathcal S^4\)	Cubic term: first order, \(T_{NI}=T^*+\frac{b^2}{27ac}\)
Maier-Saupe	\(p(\theta)\propto\exp\left(\frac{u\mathcal S}{k_BT}P_2(\cos\theta)\right)\)	Self-consistent; weakly first order
Hard rods	No attractions	Orientational entropy lost, translational entropy gained
Lebwohl-Lasher	\(H=-\epsilon\sum_{\langle i,j\rangle}P_2(\mathbf n_i\cdot\mathbf n_j)\)	Lattice Maier-Saupe; in 2D only quasi-long-range order
Frank elasticity	\(f=\frac12K_1(\nabla\cdot\mathbf n)^2+\frac12K_2[\mathbf n\cdot(\nabla\times\mathbf n)]^2+\frac12K_3[\mathbf n\times(\nabla\times\mathbf n)]^2\)	Splay, twist, bend
Topological defects	\(\Delta\theta=2\pi s\), \(s=\pm\frac12,\pm1\)	Topologically protected; annihilate in \(\pm\) pairs

References

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