Temperature Dependence of the Nematic Order Parameter

As seen from Eq. 6.11, only negative sign in front of brackets can give us Sc = 0 at Tc = Tc that is at the same characteristic temperature. Another solution of Eq. 6.12, namely, Sc = B/3C, if substituted into Eq. 6.11, results in branch S+ 

Finally, from (6.13) we find one more critical temperature = T + B l 

7AaC that is even higher than and there is no other real solutions of the stability equation. Totally, we have now three characteristic temperatures  

Within the range of Tc Tc a hysteresis in the order parameter should be observed upon the heating and cooling scans. Such hysteresis is often observed under polarization microscope in the form of two-phase textures. If a sample is placed between crossed polarizers, dark spots of the isotropic phase sharply contrasts with bright nematic backgrotmd (like in Fig. 1.3c) or vice versa. The temperature behaviour of function S(T) is shown in Fig. 6.5a. In the figure  

the universal ratio Sc/S = 2/3 is valid for any expansion up to the fourth order term and T can be found by plotting S vs temperature. By the way, in experiment, the phase transition point Tjv/ is associated with temperature Tc. 

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In general, the ellipticity coefficient is temperature dependent because of the temperatm e dependence of the correlation length (T) and the surface order parameter 5q(T). It increases by approaching the isotropic-nematic phase transition from above. By measuring ps(T) one can therefore directly determine the product T)Sq T), and, if we assume a power law dependence of (T), the temperature dependence of the nematic order parameter at the surface Sq T) can be extracted. 

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