Elastic Behaviour of Nematic Liquid Crystals

The nematic phase has properties that resemble those of a solid yet also those of a fluid. A very successful approach to modelling the behaviour of liquid crystals is based on fluid dynamics. These theories are based on the concept of the Leslie-Erickson theory of hydrodynamic flow. ° The theory considers the use of a series of coefficients, the rotations between the coefficients describing the liquid. The state of a fluid at point r where r = (x, y, z) at a given time is defined by the fluid velocity v(r, i) and any two thermodynamic variables, such as pressure P(r, t) or density p(r, i). The fluid obeys the conservation laws  

Thus Aij is the stretching tensor or rate of strain and cuy is the rotation or vorticity tensor and are illustrated below. 

The above is a quick summary of the relevant hydrodynamic equations for a simple fluid. The behaviour of a nematic liquid crystal is more complex in that the stress tensor is now non-symmetric. Another variable that is introduced is the director, , defined by a unit vector, where n —Allied to these constraints, one defines the rate of rotation of the director with respect to the background fluid by 

Applying the conservation laws summarized above gives 

Conservation of angular momentum oByi njnk = Kj + eijktkj + Hjj (3.22) 

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