Macroscopic Description of Order Parameters

All liquid crystalline phases possess orientational order among the molecules. The amount of order among the molecules along a director h f) cannot be described by a vector order parameter. This is due to the directions n and —n being fully equivalent for alignment of the molecules. Hence, a tensor of the second rank is required to describe the orientational order of the molecules. In fact, the second-rank tensor is the first term in an infinite expansion of the orientation distribution function. A macroscopic approach is 

FIGURE 3.1. Description of a biaxial body in terms of the Eulerian angles, (a) Orientation in the space-fixed (X,F,Z) frame, (b) in a molecule-fixed x y z) frame. 

The isotropic part of Xa(3 is given by the average x = Xxx + Xyy + Xzz)-To define an order parameter that vanishes in the isotropic phase, the anisotropic part of the diamagnetic susceptibility is extracted  

Now Qq/3 is traceless by definition. In the principal axis system of the susceptibility tensor, Qa/3 can be written in terms of two order parameters Q and P, 

When both Q and P are non-zero, this corresponds to a biaxial nematic phase. In the isotropic phase, both order parameters vanish. Now P is identical to zero in a uniaxial medium, such as nematic and smectic A phases. This is because Xxx — Xyy — Xxj the component of diamagnetic 

---