A liquid crystal has orientational or partial translational order while retaining some fluid mobility.
A polar liquid crystal distinguishes head from tail and is described at leading order by a vector polar order parameter .
A nematic liquid crystal identifies the director with . Its rotational order is described by a symmetric traceless tensor such as .
A nematic director is a unit vector specifying the local unoriented axis of a nematic liquid crystal. Head-tail symmetry identifies with , so the order-parameter space is a Real projective space rather than a sphere.
A nematic disclination is a topological line defect of the unoriented director field. In a two-dimensional sample its point cross-section can have half-integer winding because and represent the same state.
For a continuous director angle around a closed counterclockwise circuit, the enclosed topological charge is . Since is defined modulo , elementary nematic disclinations can have .
A two-dimensional director winding can sometimes unwind when the director may tilt out of the plane. For a three-dimensional nematic the fundamental group of the order-parameter space is , so two disclination lines are topologically equivalent to none.
In the one-elastic-constant approximation, slow director distortions have a quadratic gradient energy with one modulus. For a planar angle field it reduces to up to the convention-dependent effective modulus.
Articles by others on the same topic
Liquid crystals are a state of matter that have properties between those of conventional liquids and solid crystals. In a solid crystal, the molecules are arranged in an ordered structure, while in a conventional liquid, they are disordered and free to move around. Liquid crystals, however, exhibit a unique combination of both order and fluidity.