Liquid crystals correspond to matter in non-isotropic phases, like the nematic, or smectic phases, but which don't have full crystalline order.

See more in Principles of condensed matter physics book, and de Gennes and Prost, "The physics of liquid crystals". Liquid crystal theory

Landau-de Gennes bulk free energy density

I think derived following Landau's theory of phase transition, when given an order parameter: including terms that satisfy certain symmetries.

$F_{\text{bulk}} = A Q_{ij} Q_{ji}/2 + B Q_{ij} Q_{jk} Q_{ki}/3 + C (Q_{ij} Q_{ji})^2/4$

where the order parameter, for uniaxial LCs, is:

$Q_{ij} = \frac{3q}{2} \langle n_i n_j - \delta_{ij} /3 \rangle$

where $q$ is a scalar indicating the level of ordering (i.e. the variance of individual molecule's direction from the director field $\mathbf{n}$). $\mathbf{n}$ is the director (the direction in which the molecules point in average at a given point, where the direction is considered as a ray, i.e. $\mathbf{n}$ and $-\mathbf{n}$ are physically equivalent.

Generalized elasticity of liquid crystals

The free energy of distortion (per unit volume) of a liquid crystal has the form:

$F_d=\frac{1}{2}K_1(\nabla \cdot \mathbf{n})^2+\frac{1}{2}K_2(\mathbf{n}\cdot\nabla \times \mathbf{n})^2+\frac{1}{2}K_3(\mathbf{n}\times (\nabla \times\mathbf{n})^2$

where $K_1$, $K_2$, and $K_3$ are the elastic constants corresponding to the three types of elastic deformation that alter the long-range order in liquid crystals (and are thus opposed by elastic forces):

• splay
• twist
• bend

Friederik transition

See Complex fluid dynamics for the dynamics of liquid crystals

People

P.G. de Gennes (see his book on Physics of liquid crystals)