Complex Disordered Systems

Maier-Saupe Theory, Lebwohl-Lasher Model, Frank Elasticity, Topological Defects

Francesco Turci

Today

Recap: Landau–de Gennes predicts a first-order isotropic–nematic transition
Maier–Saupe: a microscopic mean-field theory
Hard rods: order from entropy alone
Lattice model, Frank elasticity, topological defects

Maier-Saupe Theory of Nematic Liquid Crystals

We saw the Landau-de Gennes free energy for nematic liquid crystals. f-f_0=\frac{a}{3}\left(T-T^{\ast}\right) S^2-\frac{2 b}{27} S^3+\frac{c}{9} S^4 .
- phenomenological (the coefficients are results of a perturbation)
- based on symmetries only (rotational invariance, head-tail symmetry)
- no molecular basis
The Maier-Saupe theory is a simple alternative microscopic mean-field theory that gives us a more physical insight.
The idea is to construct the free energy as we have done in various prior cases by identifying:
- an entropic contribution
- an energetic contribution

As for the Landau-de Gennes theory, we will focus on :

angular deviations \theta from the director and their probability distribution p(\theta)
a scalar orientational order parameter \mathcal{S}

Maier-Saupe Theory of Nematic Liquid Crystals

Energetic Contribution

We assume molecules interact with an angular potential that depends on their orientations via a the second Legendre polynomial P_2(x)=\frac{1}{2}\left(3 x^2-1\right):

V\left(\Omega_i, \Omega_j\right)=-u P_2\left(\mathbf{u}_i \cdot \mathbf{u}_j\right)

We can average over all possible orientations of molecule j to get the mean-field potential felt by molecule i: U_i^{\mathrm{mf}}(\Omega)=\rho \int V\left(\Omega, \Omega^{\prime}\right) p\left(\Omega^{\prime}\right) d \Omega^{\prime}=-u S P_2(\mathbf{u}_i \cdot \mathbf{n})

where we have defined the scalar order parameter S=\int P_2\left(\mathbf{u} \cdot \mathbf{n}\right) p(\Omega) d \Omega and identified the director \mathbf{n} and the energetic coupling u (which absorbs the density \rho).

Average over all orientations of molecule i to get the average interaction energy per molecule (and divide by 2 to avoid double counting):

U^{mf} = -\dfrac{u}{2} S^2

A potential that aligns molecules with the director, quadratic to reflect head-tail symmetry.

Maier-Saupe Theory of Nematic Liquid Crystals

Entropic contribution

The second contribution is entropic in nature.

Use orientational distribution function p(\Omega) and (as in Landau-de Gennes) and assume uniaxial symmetry around the director \mathbf{n}, i.e. p(\Omega)\sim p(\theta).

The usual Gibbs (or Shannon) entropy functional -k_B \int p(\Omega)\ln p(\Omega) d\Omega contributes to the free energy as

f_{\rm entropy }=k_B T \int_0^\pi p(\theta) \ln (2 p(\theta)) \sin \theta d \theta

where \sin \theta d \theta comes from the solid angle and the factor 2 comes from the normalisation \int_0^\pi p(\theta)\sin\theta d\theta=1. ¹

Maier-Saupe Theory of Nematic Liquid Crystals

Together we, have the Maier-Saupe free energy functional per molecule

f=f_0-u \frac{S^2}{2}+k_B T \int_0^\pi p(\theta) \ln (2 p(\theta)) \sin \theta d \theta

The only constraint is normalization of p(\theta): \int_0^\pi p(\theta) \sin \theta d \theta=1 where the \sin \theta comes from the solid angle element.

Introduce the Lagrange multiplier \lambda to enforce the constraint (see the aside on Lagrange multipliers) and minimize the functional and set the variation to zero: \delta\left[f+\lambda\left(\int_0^\pi p(\theta) \sin \theta d \theta-1\right)\right]=0 The functional derivative yields (check!):

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

Maier-Saupe Theory of Nematic Liquid Crystals

Remember that - by definition -

S=\left\langle P_2(\cos \theta)\right\rangle=\int_0^\pi P_2(\cos \theta) p(\theta) \sin \theta d \theta

So \dfrac{\delta S^2}{\delta p} =2 S P_2(\cos\theta)\sin\theta

So when calculating the partial derivative we have

\dfrac{\delta f}{\delta p} =-uS P_2(\cos\theta)\sin\theta +k_BT\left(\ln(2 p(\theta)) + 1\right) \sin \theta

and hence, after simplifying \sin \theta, our Lagrange equation reads

-u S P_2(\cos \theta)+k_B T(\ln (2 p(\theta))+1)+\lambda=0

Its solution yields

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

where the partition function Z ensures normalization.

Self-consistency in the Maier-Saupe Result

The solution is said to be self-consistent because p(\theta) depends on S, which in turn depends on p(\theta).

The self-consistency condition is

S = \int_0^\pi P_2(\cos \theta) p(\theta) \sin \theta d\theta

where p(\theta) itself depends on S:

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

This equation must be solved for S at each temperature T.

In general, this requires a numerical solution (see problem classes for a special case where we can make progress analytically).

Aside (optional): Maier-Saupe numerical solution

🪁 Play with the numerical solution

Aside (optional): Maier-Saupe numerical solution

The resulting phase transition is weakly first order with a discontinuous jump in S at the transition temperature. u/k_BT \approx 4.55.

Quick comparison Maier-Saupe vs Landau-de Gennes

Both theories predict a first-order isotropic-nematic phase transition with a discontinuous jump in the order parameter S.
Landau-de Gennes is phenomenological, Maier-Saupe is microscopic.
Maier-Saupe provides a molecular basis for the coefficients in Landau-de Gennes
Both theories can be extended to include spatial variations in the order parameter and director field, leading to descriptions of defects and elasticity in nematic liquid crystals.
Maier-Saupe requires self-consistency requirement, while Landau-de Gennes is often simpler to analyze analytically: (slightly) harder numerics.
Easier to cast Maier Saupe onto a lattice model (Lebwohl-Lasher).

Hard rods: ordering from entropy alone

Maier-Saupe needs u\neq 0. But even hard rods, with no attractions at all, go from isotropic to nematic at high density (Onsager, 1949): the transition is purely entropic.

Two entropic contributions compete:

orientational entropy favours the isotropic phase: aligning reduces the 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 high density the translational gain wins: the nematic phase has the higher total entropy.

Note

Same story as the freezing of hard spheres: the spheres lose the disorder of their positions, but each gains free volume to vibrate around its lattice site.

Aside (optional): simple lattice model, Lebwohl-Lasher

A way to construct a Maier-Saupe-like model on lattice.
Vectors on a lattice represent the orientations of molecules.
Nearest-neighbor interactions favor alignment of neighboring vectors.

In 2D

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

🪁 Demonstration in class

Aside (optional): simple lattice model, Lebwohl-Lasher

In 2D, \frac32\cos^2\Delta\theta-\frac12=\frac34\cos2\Delta\theta+\frac14: the model is the XY model (planar spins) for the angle 2\theta.

no true long-range order at T>0 (Mermin–Wagner, below)
at low T, orientational correlations decay as a power law (quasi-long-range order) rather than exponentially
the transition happens when bound \pm\tfrac12 defect pairs unbind (Kosterlitz–Thouless; defects later today)

(Mermin-Wagner theorem) Continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d \leq 2.

Here the continuous symmetry is the rotational invariance of the director field in the plane.

Continuum limit and Frank elasticity

Charles Frank (1958, of our Frank lecture theatre) developed a continuum theory to describe the elastic properties of nematic liquid crystals.

The Frank free energy density describes the cost of spatial variations in the director field \mathbf{n}(\mathbf{r}):

f_{\text{Frank}} = \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

K_1: splay elastic constant
K_2: twist elastic constant
K_3: bend elastic constant

Pure splay, twist and bend compared to equilibrium from Binder et al. J. Phys. Mater. (2020)

With this free energy, one can model large-scale deformations and defects in nematic liquid crystals.

Topological Defects in Nematic Liquid Crystals

We are working in the continuum with a director field \mathbf{n}(\mathbf{r}) that describes the local average orientation of molecules.
In this framework, defects are singularities of the field, called disclinations
Examples in 2D:

These singularities cannot be removed by any smooth deformation of the field: they are topologically protected.
To make them vanish, one needs oppositely signed defects to bring together so that they annihilate each other.
At finite temperature, defects are spontaneously created/annihilated.

Topological Charge of a Defect

The topological charge (or winding number) s of a defect quantifies how many times the director field rotates by 2\pi when encircling the defect once.

\Delta \theta=2 \pi s

For example, a charge of s=1/2 is a rotation by \pi is performed when going around the defect once.

Aside (optional): conservation of topological charge

Topological charge is conserved: the total charge in a closed system remains constant over time.
Defects can be created or annihilated only in pairs of opposite charge (e.g., +1/2 and -1/2).
The topology of the space imposes such constraints of conservation via the Poincaré-Hopf theorem, which states

The total topological charge of the defects in a field confined to a compact space is equal to the Euler characteristic of that space

In practice:

sphere (\chi=2): total charge +2 (e.g. four +\tfrac12)
torus (\chi=0): total charge 0, as in periodic boxes
disc (\chi=1), director tangent to the edge: total charge +1
This imposes constraints on the dynamics and interactions of defects.

Director fields on a sphere, on a torus and a double torus. The number of handles (the genus g) determines the Euler characteristic \chi=2-2g. From Nelson (2002)

Aside (optional): topological defects

Analogies well-beyond liquid crystals:

vortices in superfluids and superconductors
dislocations in crystals
electromagnetic field configurations
flow fields microfluidic junctions

Topological defects in microfluidic devices

Vortex anti-vortex pair in a superfluid (a) initial configuration and (b) after relaxation. From Magierski, *Eur Phys J* (2024)

Aside (optional): defects in 3 dimensions

In 3D the disclinations become lines
The director field wraps around the line defect in a similar way as in 2D.
only one type of stable line: integer lines “escape into the third dimension”
new point defects (hedgehogs), closed loops and complex textures

Aside (optional): defects and phase transitions

Defects control the stability and dynamics of nematic liquid crystals and can drive transitions between phases.

Defect formation and nematic to smectic phase transition via defect branching, Gim et al Nature Communications (2017)

Aside (optional): complex disclination patterns

Example from Kim et al Science Advances (2018)

Liquid-crystals defects created and controlled by external forces (air pillars through a substrate).

Summary: liquid crystals

Topic	Key result
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 needs a tensor
Order parameter, director	\mathcal S = largest eigenvalue of \mathbf Q, \mathbf n = its eigenvector; 3D: \mathcal S=\langle P_2(\cos\theta)\rangle
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
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	purely entropic: orientational entropy lost, translational entropy gained
Lebwohl-Lasher	lattice Maier-Saupe; in 2D only quasi-long-range order
Frank elasticity	splay K_1, twist K_2, bend K_3
Topological defects	\Delta\theta=2\pi s, s=\pm\frac12,\pm1; annihilate in \pm pairs

Full summary with links in the notes

Aside (optional): constrained optimisation with Lagrange multipliers

Many physics problems can be cast in terms of constrained optimisation

Area	Problem	Constraint
Mechanics	Minimize potential energy of a pendulum	Fixed length of the string
Thermodynamics	Maximize entropy S	Fixed total energy E
Statistical Mechanics	Maximize Gibbs entropy -\sum_i p_i\ln p_i	Normalisation \sum_i p_i=1, fixed \langle E\rangle
Liquid Crystals	Minimize Frank energy	Director normalization \\|\mathbf{n}\\|=1
Liquid Crystals	Minimize Maier-Saupe free energy f[p]	Normalisation of p(\theta)

Method: Use Lagrange multipliers \lambda to convert constrained into unconstrained problem: \mathcal{L}=f(\mathbf{x})-\lambda g(\mathbf{x}) then solve \nabla\mathcal{L}=0.

Aside (optional): geometrical understanding of Lagrange multipliers

Take a function f(x,y).
A constraint is expressed by fixing the value of a second function g(x,y)=c
We want to find the extrema of f(x,y) subject to the constraint g(x,y)=c: It has to be a curve
This means find the extremal values of f on the curve \mathcal{C}: g(x,y)=c
We could parametrise \mathcal{C} but we can also simply think in terms of its geometrical features.

Aside (optional): geometrical understanding of Lagrange multipliers

Take the example of a minimum:

at every point of \mathcal{C} there is a tangent vector \vec{t}
we want to find the points on \mathcal{C} where f is minimal. This means that changes of f along \mathcal{C} are minimal.
we can calculate such changes as the projection of the local gradient with the tangent vector and state that \vec{\nabla}f \perp \vec{t}
Notice that, by definition, \vec{\nabla}g\perp \vec{t}
So we are simply saying that

\vec{\nabla}f\parallel \vec{\nabla}g\to \vec{\nabla}f=\lambda\vec{\nabla}g

for some scalar \lambda.

Aside (optional): example

At the constrained extremum the contours of f are tangent to the constraint g=c. Nexcis, Wikimedia Commons (public domain).

Aside (optional): many variables, example of probabilities

Discrete case: Find extremum of entropy S = -\sum_i p_i \ln p_i with constraints \sum_i p_i = 1 \quad \sum_i p_i E_i = \langle E \rangle

Using Lagrange multipliers: \mathcal{L} = -\sum_i p_i \ln p_i - \lambda_1 \left(\sum_i p_i - 1\right) - \lambda_2 \left(\sum_i p_i E_i - \langle E \rangle\right)

Setting \frac{\partial \mathcal{L}}{\partial p_i} = 0: -\ln p_i - 1 - \lambda_1 - \lambda_2 E_i = 0 \quad \Rightarrow \quad p_i = \exp(-\lambda_2 E_i - \lambda_1 - 1)

This is the Boltzmann distribution: p_i \propto \exp(-\beta E_i) with \beta = \lambda_2.

Aside (optional): Functional Derivatives

Functional derivative is the extension of ordinary derivative to functionals (functions of functions)
A general formulation (that you may have already seen) starts from an action functional S[\phi]:

S[\phi] = \int \mathcal{L}(\phi(x), \phi'(x), x) dx

\mathcal{L} is the Lagrangian function: e.g. x is space, \phi(x) a field defined over space (e.g. density)

Perturb S[\phi] by an amount \epsilon with an arbitrary function \eta(x) vanishing at the boundaries

S[\phi+\epsilon\eta]=\int \mathcal{L}(\phi+\epsilon\eta, \phi'+\epsilon\eta', x) dx

Expanding to first order in \epsilon:

S[\phi+\epsilon\eta]\approx S[\phi]+\epsilon\int\left[\frac{\partial \mathcal{L}}{\partial \phi}\eta+\frac{\partial \mathcal{L}}{\partial \phi'}\eta'\right] dx

Aside (optional): Functional Derivatives

Integrating by parts the second term:

\int\frac{\partial \mathcal{L}}{\partial \phi'}\eta' dx = \left[\frac{\partial \mathcal{L}}{\partial \phi'}\eta\right]_{\text{boundary}}^{\phantom{x}}-\int\frac{d}{dx}\left(\frac{\partial \mathcal{L}}{\partial \phi'}\right)\eta dx

Since \eta(x) vanishes at the boundaries, the first term drops out, giving:

\delta S=\int\left[\frac{\partial \mathcal{L}}{\partial \phi}-\frac{d}{d x}\left(\frac{\partial \mathcal{L}}{\partial \phi^{\prime}}\right)\right] \eta(x) d x

If we want S[\phi] to be stationary with respect to arbitrary perturbations \eta(x), then \delta S=0 for all \eta(x).

This leads to the Euler-Lagrange equation:

\frac{\partial \mathcal{L}}{\partial \phi}-\frac{d}{d x}\left(\frac{\partial \mathcal{L}}{\partial \phi^{\prime}}\right)=0

which defines the functional derivative as \frac{\delta S}{\delta \phi(x)}=\frac{\partial \mathcal{L}}{\partial \phi}-\frac{d}{d x}\left(\frac{\partial \mathcal{L}}{\partial \phi^{\prime}}\right)

Aside (optional): continuous limit

Replace sums with integrals. Let p_i \to p(x) where x is a continuous variable.

Constraints become: \int p(x) dx = 1, \quad \int p(x) E(x) dx = \langle E \rangle

Entropy functional: S[p] = -\int p(x) \ln p(x) dx

Lagrangian: \mathcal{L} = -\int p(x) \ln p(x) dx - \lambda_1 \left(\int p(x) dx - 1\right) - \lambda_2 \left(\int p(x) E(x) dx - \langle E \rangle\right)

Functional derivative \frac{\delta \mathcal{L}}{\delta p(x)} = 0: -\ln p(x) - 1 - \lambda_1 - \lambda_2 E(x) = 0 \quad \Rightarrow \quad p(x) = \exp(-\beta E(x) - c)