Orientational order parameters in biaxial nematics: Polymorphic notation

Abstract

In this paper, a wealth of notations introduced in the past 30 years to denote second‐rank orientational order parameters for biaxial nematics are compared, stressing sources of possible confusion. A unifying, intrinsic treatment of the second‐rank orientational order parameters is also presented, which does not suffer from the redundancy of the Saupe matrix and is independent of the way in which rotations are parametrized.

Acknowledgements

This work has been made possible by the Royal Society of London through the project Biaxial Liquid Crystals: Mathematical Models and Simulation. It is a pleasure to thank Prof. G.R. Luckhurst for careful reading of a preliminary version of the manuscript and for enlightening comments. I also thank Prof. E.G. Virga and Prof. S. Romano for useful comments and discussions during various stages of this work. I express my gratitude to Professor J.P. Straley for his comments on the choice of the order parameters made in 19.

Notes

1. The notation used here departs from that adopted by Zannoni 24 where is used for ⟨·⟩ _o and ⟨·⟩ is used for ⟨·⟩ _po .

2. Precisely, Collings et al. 71 72 consider ⟨·⟩ as a time average, instead of an ensemble average, since NMR measurements involve precisely time averages. The two averages are conceptually quite different from one another and an ergodic hypothesis should be invoked – and proved – to be sure that they produce the same results. Mettout 27 claims that his averages coincide with those employed in 71, and excludes the factor 5π²/16 from orientational averages. Here, we consider both ⟨·⟩ and as equivalent notations for ensemble averages.

3. Precisely, the parameters Q ₀ and Q ₂ defined here are proportional to that introduced by Freiser, who did not use the normalizing factors in L ₀ and L ₁. A similar remark holds for the order parameters A ₀ and A ₂ introduced by Remler and Haymet 76.

4. In this respect, a discrepancy exists between figure 2 and table 1 of 58. In fact, in figure 2 of 58 and in the matrix a _αβ of equation (3) on p. 88 of 58, ϕ denotes the proper rotation and ψ is the angle of precession. On the other hand, the angle of proper rotation is denoted by ψ in table 1 on p. 95 of 58, otherwise it would not be possible to define the order parameter P, since the angle of proper rotation is meaningless for uniaxial molecules in a biaxial phase (see Section 3).