Molecular biaxiality parameter

FIGURE 3 Hie dependence of the biaxial order parameter < 22 on the major cvdo- parameter predicted for a selection of values of the molecular biaxiality parameter, X. 

We now discuss the orientational order of the LCPs 3,4 and low molecular wei t analogues 6-8 in terms of the familiar oder parameters 22 and Sxx — Syy [34]. They express the ordering of the molecular axis Z and the anisotrc of the orientational order, respectively. Angular dependent lineshapes of the polymers indicate that each repeating unit can be characterized by an order tensor axialfy symnKtrk along Z [10,95], Thus, within experimental error — Syy = 0 for these tems, in contrast to similar combined LCPs, for which a small molecular biaxiality has been observed [114,115]. 

The local order in a cholesteric may be expected to be very weakly biaxial. The director fluctuations in a plane containing the helical axis are necessarily different from those in an orthogonal plane and result in a phase biaxiality . Further, there will be a contribution due to the molecular biaxiality as well. It turns out that the phase biaxiality plays a significant role in determining the temperature dependence of the pitch. Goossens has developed a general model taking this into account. The theory now involves four order parameters the pitch depends on all four of them and is temperature dependent. However, a comparison of the theory with experiment is possible only if the order parameters can be measured. 

Eq. (3.40) is equivalent to that for cylindrically symmetric molecules, and the second term results from the molecular biaxiality D 0). 2,0 and 2,2 can be written in terms of order parameters (c q q) and (do,2cos20) and expansion coefficients U2mm U202 — U220 for pure systems) which are similar to U2 in Eq. (3.29),... 

Figure 6. Reduced values xt and Xi as a function of reduced temperature plotted according to Eq. (45) using order parameters calculated using mean field theory. Open squares ( ) assume no molecular bi-axiality, so D = Q full circles ( ) are for an assumed molecular biaxiality of 0.3 and a A value of 0.3.

The values of the scaled transition temperature, l TNi/e2oo, together with the transitional values of the ord parameters ni, ni and ni are listed in TABLE 1 for several values of the biaxiality parameter, X. As the molecular biaxiality increases so the major order parameter at the transition decreases indeed when X is 0.3 ni has decreased to about half that for a uniaxial molecule. In contrast although as expected the biaxial ordo- parameter ni increases with X the value is extremely small. It would seem, therefore, that the molecular biaxiality has the greatest effect on Ni and that ni is essentially negligible. The influence of X on the second rank order parameter is mimicked by its fourth rank counterpart ni indeed when X is 0.3 this order parameter is four times smaller than that for uniaxial molecules. For X of 0.4 all of the transitional order parameters are extremely small, hinting at the approach of a second ord transition. This occurs when X is 1 / >/6 but now the transition from the isotropic phase is directly to a biaxial and not a uniaxial nematic phase, as we shall see. 

The order parameters S for all three molecular axes or alternatively, the combination S plus D describe on the level of the first relevant polynomial term the orientational distribution of a rigid, non-cylindrical molecule in the uniaxial nematic phase. Additional order parameters come into play for biaxial phases (Straley, 1974). A concise overview on the concepts from statistical mechanics relevant to order parameters was given by Zannoni (1979). 

As observed earlier, the assumption that the molecule is cylindrically symmetric is clearly not valid for real systems, and consequently the use of a single order parameter is not adequate. Most molecules are lath-shaped and have a biaxial character. Therefore two order parameters are required to describe the uniaxial nematic phase composed of biaxial molecules. If are the principal axes of the molecule (C defining the molecular long axis), it is necessary to introduce an additional order parameter... 

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