Fluctuations homophase

In this section we use the Landau-de Gennes theory to discuss thermal fluctuations in the isotropic phase of liquid crystals. For a physical system in thermal equilibrium, the instantaneous value of the order parameter will almost always be equal or close to its mean value (or equivalently, the equilibrium value). However, deviations from the mean value of the order parameter do occur, and the problem is to calculate the magnitude and the statistical distribution of these deviations, or fluctuations. We distinguish between two types of fluctuations (1) homophase fluctuations, which occur within the range of stability of a single phase and are completely described by the rms deviation of the order parameter from its equilibrium value, and (2)... 

Since homophase fluctuations in the isotropic phase involve only states close to Q(r) = 0, we can neglect the cubic and quartic terms in the expansion of If we also set H = E = 0, Eq. [43] reduces to... 

As shown in Section 2.3, Q(r) can be expressed in terms of S(r) and n(r). If we were to make this substitution in Eq. [57], the resulting expression would contain several interaction terms coupling S(r), n(r), and their spatial derivatives. Thus, in general, the problem of calculating the fluctuation spectra of S(r) and h(r) can be quite complicated because of this coupling. In the latter part of this section we will show one way to handle the coupling between S(r) and n(r), but first we want to consider a simple system that illustrates both the physics of homophase fluctuation phenomena and the mathematics involved in manipulating the formalism developed in previous sections. 

Figure 2. First order phase transition a) homophase fluctuation, b) heterophase fluctuation [2].

Fig. 40. Schematic description of unstable thermodynamic fluctuations in the two-phase regime of a binary mixture AB at a concentration cb (a) in the unstable regime inside the two branches tp of the spinodal curve and (b) in the metastable regime between the spinodal curve tp and the coexistence curve The local concentration c(r) at a point r = (x. y, z.) in space is schematically plotted against the spatial coordinate x at some time after the quench. In case (a), the concentration variation at three distinct times t, ti, u is indicated. In case (b) a critical droplet is indicated, of diameter 2R , the width of the interfacial regions being the correlation length Note that the concentration profile of the droplet reaches the other branch ini, of the coexistence curve in the droplet center only for weak supersaturations of the mixture, where cb -

The primary effect of wetting is related to the existence of a slow mode characterized by a soft dispersion of its relaxation rate, whereas the upper part of the spectrum remains more or less the same as in a homophase system (see insets of Fig. 8.5). The elementary mode of fluctuations of the degree of order is localized at the phase boundary between the wetting layer and the bulk phase and it corresponds to fiuctuations of the thickness of the central part of the slab. The next mode, which is also localized at the nematic-isotropic interface, represents fluctuations of the position of the core. The relaxation rates of these two modes are the same as long as the two wetting layers are effectively uncoupled. 

Fig. 39. Micro-EDX analysis of Ndl23 crystals grown by the modified TSSG method in low-Po, atmosphere from contamination-firee Nd Oj crucibles with different post-growth heat treatments. In all the cases final oxygenation at 340°C in oxygen was applied. The picture demonstrates (a) tweed structure formation and (b) nanoscale composition fluctuations in crystals with the anomalous peak effect on a magnetization curve. Note that the composition profile for heavy atoms (Ba/Nd ratio) is similar to wave-like fluctuations typical for demixing behavior or a spinodal homophase decomposition rather than for a heterophase decomposition with the formation of a boundary between the crystal matrix and the precipitated phase (M. Nakamura et al. 1996c).

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