Goldstone nematics

Fig. 1.2. (a) in a bulk nematic liquid crystal, the director can point in an arbitrary direction in space. This is a signature of a broken continuous rotational symmetry of the isotropic phase. The mode that restores the broken. symmetry is the Goldstone mode. It represents a homogeneous and coherent rotation of aU molecules, (b) the homogeneous surface couples to the Goldstone mode and pins the director in a certain direction in space. 

Fig. 8.3. (a) Temperature dependence of correlation lengths of 5 degrees of freedom of the nematic order in the isotropic and nematic phase. Continuations of the lines across the dotted vertical correspond to the correlation lengths in the appropriate metastable phase. Correlation lengths determine the relaxation rates of fluctuations as Pi = 1/Tj oc -t- where q is the corresponding wavevector. (b) Sketch of a typical relaxation spectra for a system with a Goldstone and soft mode, respectively. 

The correlation length of the director fluctuations is infinite in the whole range of the stable nematic phase and the director excitation with the infinite wavelength is the Goldstone mode. Fluctuations of other degrees of freedom of... 

Director modes are, as opposed to biaxial fluctuations, excited very easily in the nematic phase, where their Hamiltonian is purely elastic, whereas in the isotropic phase they are characterized by a finite correlation length. This implies that their wetting-induced behavior should be quite the inverse of that of biaxial modes. Thus, in the disordering geometry, the director modes are forced out of the substrate-induced isotropic boundary layer into the nematic core (see Fig. 8.6 bottom). The lowest mode is a Goldstone mode. In the paranematic phase a few lowest director modes are confined to the nematic boundary layer, whereas the upper ones extend over the whole sample and are more or less the same as in the perfectly isotropic phase. 

For nematic liquid crystals, the synunetry is reduced and we need additional variables. The nematic is degenerate in the sense that all equilibrium orientations of the director are equivalent. According to the Goldstone theorem the parameter of degeneracy is also a hydrodynamic variable for a long distance process 0 and the relaxation time should diverge, x—>oo. In nematics, this parameter is the director n(r), the orientational part of the order parameter tensor. For a finite distortion of the director over a large distance (L—>oo), the distortion wavevector 0 and the... 

The Goldstone mode in an achiral SmC tries to restore the symmetry of the smectic A phase Cooh —> Dooh by free rotation of the director along the conical surface with the smectic layer normal as a rotation axis. Thus, like chiral molecules convert a nematic into a cholesteric, they convert an achiral SmC into chiral SmC without any phase transition. In addition, mixing left (L)- and right (R)-handed additives results in a partial or complete compensation of the helical pitch both in cholesterics and chiral smectic C. For example, the L- and R- isomers of the same molecule taken in the equal amounts would give us a racemic mixture, that is achiral SmC without helicity and polarity. 

Fig. 13.10 Comparison of the temperature dependencies of viscosity coefficients yi (nematic), (soft mode) and y (Goldstone mode) of the same chiral mixture within the ranges of the N and SmC phases [15]. Note that Yi and y curves may be bridged through the SmA phase black points) where measurement have not been made...

A novel electromechanical effect has been observed in FLCs [130]. A periodic shear flow occurred parallel to the bounding plates and perpendicular to the helical axis (FLC layers were perpendicular to the substrates). The frequency of oscillation of the shear flow was equal to that of the applied field and the amplitude was proportional to the field strength. The electromechanical effect in FLCs seems to have many common features with a backflow effect in nematic liquid crystals, as it is caused by the coupling between the Goldstone mode and flow [131]. 

The macroscopic nematodynamic equations describe the dynamics of the slowly relaxing variables, which usually are either connected with conservation laws or with the Goldstone modes of the spontaneously broken symmetries. To formulate them we wUl follow the traditional approach [65-67] rather than the one based more directly on the principles of hydrodynamics and irreversible thermodynamics [68]. In the nematic state isotropy is spontaneously broken and the averaged molecular alignment singles out an axis whose orientation defines the director n, i. e. an object that has the properties of a unit vector with n = -n. The static properties are conveniently expressed in terms of a free energy density whose orientational elastic part is given by [69]... 

The characteristic frequency dependence is a direct result of the gapless Goldstone mode type nature of the director fluctuations. The constants C[ and C2 depend on the magnitude of the nematic order parameter, the viscoelastic constants, the molecular geometry of the spin positions and the orientation of the director with respect to the external magnetic field. 

In a dielectric relaxation experiment, the linear response of a sample to an oscilating and spatially uniform q=0) electric field is measured. The bulk nematic (non-polar) collective excitations with a nonzero wave-vector q thus cannot contribute to the linear response of a nematic. The exception is here a trivial coupling of an external electric field to the =0 (Goldstone) mode, which represents an uniform rotation of a sample as a whole and is as well non-polar. This leads to the conclusion that the dielectric relaxation experiment will measure the response of individual nematic molecules to an applied electric Beld. Similar to the case of an isotropic liquid, the relaxation rates of the individual molecular motion are expected to be observed in the 100 MHz to GHz frequency region. 

Figure 78 shows a plot of the soft mode and Goldstone mode rotational viscosities measured on either side of the phase transition between the smectic A and SmC. It can be seen that, except in the vicinity of the phase transition, the viscosity seems to connect fairly well between the two phases. The activation energies of these two processes are, however, different. This result may be compared to results obtained by Pozhidayev et al. [148], referred to in Fig. 67. They performed measurements of y beginning in the chiral nematic phase of a liquid crystal mixture with corresponding measurements in the SmC phase, and have shown the viscosity values on an Arrhenius plot for the N and SmC phases. Despite missing data of y in the smectic A phase they extrapolate the N values of y down to the smectic C phase and get a reasonably smooth fit. Their measurements also show that y is larger than y, and this is universally the case.

Olmsted P (1994) Rotational invariance and goldstone modes in nematic elastomers and gels. J Phys II 4 2215-2230... 

It has to be noted that Kremer et al conducted their studies on purposely unoriented samples. They did so because the presence of the optically active end groups in the side mesogen chains usually leads to chirality of the mesophase. As a result, polymers 31, 37, 39, 41 and 43 exhibit the cholesteric (chiral nematic) phase, and polymer 40, 41 and 43 the chiral smectic C phase.Since the smectic C shows ferroelectric-ity, in order to separate the molecular rotational modes from the ferroelectric Goldstone and soft modes (cf. Section 4.3), samples should be unoriented. On the other hand, measurements performed on the oriented sample of the chiral C phase of polymer 43 led Vallerien et al to the observation of ferroelectric modes. 

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