Order fluctuation

The second-order fluctuating rate-of-strain tensor is real and symmetric. Thus, its three eigenvalues are real and, due to continuity, sum to zero. The latter implies that one eigenvalue (a) is always positive, and one eigenvalue (y) is always negative. In the turbulence literature (Pope 2000), y is referred to as the most compressive strain rate. 

Using the velocity profile given by Eq. (161) to solve the local equations, we obtain the leading-order fluctuations <+i as... 

Other Contributions to the Kerr Constant. In the case of naturally anisotropic molecules, the deviation tensor df in the expansion (195) does not vanish, causing a cross-effect as well as higher-order fluctuational effects to appear, in addition to the previous angular correlation effect (I77a). The cross-effect appears owing to co-operation between molecular angular correlations-the first term of equation (195) and the translational fluctuations described by the second term of (195). We shall consider this effect in the next subsection. 

A14N NMR study of order fluctuations in the isotropic phase of liquid crystals has been reported. (209) The experimental data for the isotropic phases of -azoxyanisole and of diethylazoxy benzoate are accounted for in terms of short range order fluctuations of the nematic and of the smectic types respectively. 

Proton, deuteron and carbon spin relaxation measurements of liquid crystals have provided detailed information about the molecular motions of such anisotropic liquids (anisotropic rotation and translation diffusion of individual molecules), and about a peculiar feature of liquid crystalline phases, namely collective molecular reorientations or order fluctuations. Spin relaxation in liquid crystalline mesophases has challenged NMR groups since the early 1970s, shortly after the publication of theoretical predictions that order fluctuations of the director (OFD, OF), i.e. thermal excitations of the long-range orientational molecular alignment (director), may play an important unusual role in nuclear spin relaxation of ordered liquids. Unique to these materials, which are composed of rod-like or disc-like (i.e. strongly anisotropic molecules), it was predicted that such thermal fluctuations of the director should, at the frequencies of these fluctuation modes, produce rather peculiar Ti(p) dispersion profiles. For example in the case of uniaxial nematic... 

Ihese FC Txd results lead to two important conclusions by contrasting the data with pertinent theoretical expressions. Making use of the well-established finding that order fluctuations (OF) of the nematic director dominate the Zeeman relaxation time, Tiz, in the kHz region up to typically 1 MHz, one can simplify equations (lb), (3a) and (4a) for frequendes <10 kHz to the spedal forms ... 

So far, the discussion has been restricted to isolated motions of single molecules. In LCPs, however, collective motions of a lar number of molecules may occur. For the latter mechanism, known as order director fluctuations, a broad distribution of correlation times is sedicted [%, 37]. In contrast to the isolated modes, discusred above, director order fluctuations are expected to occur only in the mesophase of LCPs, but should be completely absent in the solid and glassy state of these systems. 

Molecular motions in LCs may occur as isolated or collective modes (see Fig. 4). For the latter mechanism, known as order director fluctuations, a continuous distribution of correlation times is expected [36, 37, 170-173]. Recent protean Tjz dispersion measurements of the LCPs 4 and corresponding LMLCs 7 and 8, carried out over a frequency range of five orders of magnitude (10 Hz < t0o/27t < 3 x 10 Hz), clearly show that collective order fluctuations contribute to the relaxation process only at extremely low frequencies in the kHz regime, whereas the conventional MHz range is dominated reorientations of individual molecules [174]. 

For nematic LCs, theory predicts a characteristic dispersion law Tiz(coo) oc [36, 37]. This is exactly what we observe for the monomeric 8 and dimeric systems 7. Although a somewhat higher exponent is evaluated for the polymers 4, there is no doubt that collective order fluctuations occur in these systems, likewise [174]. 

It is considerably larger in the confined liquid crystals above Tni than in the bulk isotropic phase. The additional relaxation mechanism is obviously related to molecular dynamics in the kHz or low MHz frequency range. This mechanism could be either order fluctuations, which produce the well-known low-frequency relaxation mechanism in the bulk nematic phase [3], or molecular translational diffusion. Ziherl and Zumer demonstrated that order fluctuations in the boundary layer, which could provide a contribution to are fluctuations in the thickness of the layer and director fluctuations within the layer [36]. However, these modes differ from the fluctuations in the bulk isotropic phase only in a narrow temperatnre range of about IK above Tni, and are in general not localized except in the case of complete wetting of the substrate by the nematic phase. As the experimental data show a strong deviation of T2 from the bulk values over a broad temperature interval of at least 15K (Fig. 2.12), the second candidate, i.e. molecular translational diffusion, should be responsible for the faster spin relaxation at low frequencies in the confined state. 

F.T. AreccM, A. Berne, P. BulamaccM, High-order fluctuations in a single mode laser field. Phys. Rev. Lett. 88,32 (1966)... 

Here the R 1 values provided in the Supplementary Content (Leftin and Brown 2011) for the (CH2) segments of DLPC and the liquid hydrocarbon -dodecane are compared. Whereas scaling of the relaxation rates for the lipid depends on segmental motion, anisotropic molecular motion, and collective membrane motion, the relaxation rate of the alkane depends on isotropic, fast segmental, and molecular motions only. The frequency dispersion for the liquid is linear with a slope nearly equal to zero at all frequencies. However, for DLPC the slopes of the dispersion depend significantly on temperature. This shows that the phase behavior of the membrane contributes to the structural dynamics observed, and that the rate of the acyl chain motion becomes more like the isotropic alkane with increasing temperature, thereby highlighting the contribution of order fluctuations to... 

FIGURE 4.38. Relaxation times associated with nematic order fluctuations n and with molecular relaxation (t2) as functions of temperature (Kerr effect in 5CB) [225]. 

Thus higher order fluctuations of the population in any interval or property associated with it can be calculated from the product densities. 

To describe short-range nematic order fluctuations in isotropic phases of nematics, the terms in Eq. (6.69) are retained up to quadratic in Q to give... 

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