Tutorial

To illustrate how to use morpho, we will solve a problem involving nematic liquid crystals (NLCs), fluids composed of long, rigid molecules that possess a local average molecular orientation described by a unit vector field \(\mathbf{n}\). Droplets of NLC immersed in a host isotropic fluid such as water are called tactoids and, unlike droplets of, say, oil in water that form spheres, tactoids can adopt elongated shapes.

The functional to be minimized, the free energy of the system, is quite complex,

$$ \begin{equation} F= \underbrace{\frac{1}{2}\int_{C}K_{11}\left(\nabla\cdot\mathbf{n}\right)^{2}+K_{22}(\mathbf{n}\cdot\nabla\times\mathbf{n})^{2}+K_{33}\left|\mathbf{n}\times\nabla\times\mathbf{n}\right|^{2}dA}_\text{Liquid crystal elastic energy}\label{eq:free} \end{equation} $$

$$ \begin{equation*} \quad + \underbrace{ \sigma\int dl }_\text{s.t.} \end{equation*} $$

$$ \begin{equation*} \quad + \underbrace{\frac{W}{2}\int\left(\mathbf{n}\cdot\mathbf{t}\right)^{2}dl}_\text{anchoring} \end{equation*} $$

where the three terms include liquid crystal elasticity that drives elongation of the droplet, surface tension (s.t.) that opposes lengthening of the boundary and an anchoring term that imposes a preferred orientation at the boundary. We need a local constraint, \(\mathbf{n}\cdot\mathbf{n}=1\), and will also impose a constraint on the volume of the droplet. For simplicity, we'll solve this problem in 2D. The complete code for this tutorial example is contained in the examples/tactoid folder in the repository.