Liquid Crystals

Liquid crystals are materials made of molecules with shapes that allow them to take on intermediate phases between liquid and solid hence the name liquid crystal. [1]

Liquid crystals are the technology behind liquid crystal displays or LCD screens which use electricity to change the orientation of a small layer of liquid crystals in your screen to direct light for making images. Other interesting technology that uses liquid crystals includes: medical biosensors for detecting disease[2], microscopic robots [3], adaptive 3D printing materials[4], smart fabric [5], and more.

Liquid Crystals Mathematical Modeling

Below is a summary of the summary that we provided in this paper on modeling nematic liquid crystals using the Landau-de Gennes \(\mathbf{Q}\) tensor theory, and all references are included there.

\(\mathbf{Q}\) Tensor

The Landau-de Gennes model of liquid crystals is popular because it can describe a large class of material deformations. The model uses a tensor order parameter \(\mathbf{Q}\) to describe the local average orientation of the nematic molecules and in one state, called the uniaxial state, it is written

$$ \mathbf{Q} = s \left( \mathbf{n}\otimes\mathbf{n} - \frac{1}3\mathbb{I} \right), $$
where \(\mathbf{n}\) is the dominant eigenvector of \(\mathbf{Q}\), \(s\) is a scalar order parameter related to the largest eigenvalue, and \(\mathbb{I}\) is the identity matrix. Because of this we have that \(\mathbf{Q}\) is symmetric and traceless and therefore contains only five degrees of freedom. Additionally, the eigenvalues satisfy \(-1/3 \leq \lambda_i \leq 2/3\), and \(\sum_i \lambda_i = 0\), and so we can write the tensor in its eigenframe as
$$ \mathbf{Q} \,=\, s_1 \left( \mathbf{n}_1\otimes\mathbf{n}_1\right) + s_2 \left( \mathbf{n}_2\otimes\mathbf{n}_2\right) - \frac{1}3 (s_1 + s_2) \mathbb{I} $$
where \(\mathbf{n}_i\) are the orthonormal eigenvectors, and
$$ \lambda_1 = \frac{1}{3}(2s_1 - s_2), \quad \quad \lambda_2 = -\frac{1}{3}(s_1 + s_2), \quad \quad \mbox{and} \quad \quad \lambda_3 = \frac{1}{3}(2s_2 - s_1). $$
This later form is more general, and can describe molecular orientations in both the uniaxial and biaxial phases.

Landau-de Gennes Energy

Let \(\Omega\subset\mathbb{R}^d, d=2,3\), be a bounded domain with polyhedral boundary \(\partial\Omega \). The Landau-de Gennes free energy functional is commonly written

$$ \mathcal{E}(\mathbf{Q}) \, = \, \int_\Omega \mathcal{W}(\mathbf{Q} , \nabla\mathbf{Q} ) + \frac{1}{\varepsilon} \Psi(\mathbf{Q}) d\mathbf{x} $$
where \(\mathcal{W}(\mathbf{Q} , \nabla\mathbf{Q} ) \) is the elastic energy, and \( \Psi(\mathbf{Q}) \) is the thermotropic contribution.

One way to write the elastic energy is

$$ \mathcal{W}(\mathbf{Q} , \nabla\mathbf{Q} ) \, = \, \frac{1}{2} \left( L_1 \mid\nabla\mathbf{Q}\mid^2 + L_2 \mid\nabla \cdot \mathbf{Q}\mid^2 + L_3 (\nabla \mathbf{Q})^T \,\vdots\, \nabla \mathbf{Q} + L_1 q_0 \nabla \mathbf{Q} \,\vdots\, (\epsilon\cdot\mathbf{Q}) \right) $$
where \(L_i \) and \(q_0 \) are material parameters, and \(\epsilon\) is the Levi-Civita permutation symbol. It is common, and for the rest of this summary, to have\(L_2,L_3,q_0 = 0\) and \(L_1=1\) to get the one-constant approximation.

The thermotropic energy is written as

$$ \Psi (\mathbf{Q}) \, = \, \frac{A}{2} \texttt{tr}(\mathbf{Q}^2) - \frac{B}{3} \texttt{tr}(\mathbf{Q}^2) + \frac{C}{4} \texttt{tr}(\mathbf{Q}^2)^2 $$
where \(A,B,C\) are material parameters with \(B,C \geq 0\).

Gradient Flow

System dynamics can be modeled by an \(L^2\)-gradient flow:

$$ \mathbf{Q}_t = -\gamma\frac{\partial \mathcal{E}}{\partial\mathbf{Q}} $$
where \(\gamma>0\) is a relaxation constant and the variational derivative of the energy is given by
$$ \frac{\partial \mathcal{E}}{\partial\mathbf{Q}} \, = \, -\Delta \mathbf{Q} + \frac{1}{\varepsilon} \left( A\mathbf{Q} - B\left(\mathbf{Q}^2 - \frac{\texttt{tr}(\mathbf{Q}^2)}{3}\mathbb{I} \right) + C\texttt{tr}(\mathbf{Q}^2)\mathbf{Q} \right) $$

Energy Law

A useful property of the PDE is the following energy law

$$ \frac{d}{dt}\mathcal{E}(\mathbf{Q}) + \frac{1}{\gamma} \| \mathbf{Q}_t \|^2 = 0 $$
which implies that the energy is always decreasing in time.

Designing Numerical Schemes

Several challenges arise when designing numerical schemes for this PDE:

  1. The problem is very non-linear. Linear schemes are much faster than iterative non-linear schemes, and so it is useful to find a way to approximate the problem with linear terms.
  2. The unknowns in \(\mathbf{Q}\) are coupled by several terms. This means that we have to solve a very large linear system which is computationally expensive especially if we want to simulate anything in three dimensions. Therefore, it is useful to decouple the unknowns in the problem.
  3. To recover accurate system dynamics. By this I mean that the numerical scheme should try to model the continuous energy law as closely as possible without adding too much artificial numerical dissipation into the system.
  4. To have energy stability. This means that the numerical scheme will have a decreasing energy law at the discrete level similar to the continuous problem. It is especially nice to have a scheme that satisfies the discrete energy law for any choice of discrete time step, \(\Delta t\), since the numerical dissipation from #3 depends on \( \Delta t\) as well.