Spontaneous reflection symmetry breaking

Spontaneous Reflection Symmetry Breaking in Liquid Crystals... 

Spontaneous reflection symmetry breaking in achiral LCs is also well known, driven by specific boundary conditions. A very simple example of this type of chiral domain formation is illustrated in Figure 8.11. Suppose we start with two uniaxial solid substrates, which provide strong azimuthal anchoring ... 

The starting system is achiral (plates at 90° with isotropic fluid between), but leads to the formation of a chiral TN structure when the fluid becomes nematic. In this case, enantiomeric domains must be formed with equal likelihood and this is precisely what happens. The size of these domains is determined by the geometry and physics of the system, but they are macroscopic. Though the output polarization is identical for a pair of heterochiral domains, domain walls between them can be easily observed by polarized light microscopy. This system represents a type of spontaneous reflection symmetry breaking, leading to formation of a conglomerate of chiral domains. 

Many other interesting examples of spontaneous reflection symmetry breaking in macroscopic domains, driven by boundary conditions, have been described in LC systems. For example, it is well known that in polymer disperse LCs, where the LC sample is confined in small spherical droplets, chiral director structures are often observed, driven by minimization of surface and bulk elastic free energies.24 We have reported chiral domain structures, and indeed chiral electro-optic behavior, in cylindrical nematic domains surrounded by isotropic liquid (the molecules were achiral).25... 

In all of these cases the symmetry breaking is derived from the manipulation of surface forces. Prior to 1997, spontaneous reflection symmetry breaking had never been reported in any thermodynamic bulk fluid phase. In the LC field, this empirical fact led to the generally believed assumptions that any LC phase composed of achiral or racemic compounds must be achiral, and... 

It is interesting to point out here that with all of the theoretical speculation in the literature about polar order (both ferroelectric and antiferroelectric) in bilayer chevron smectics, and about reflection symmetry breaking by formation of a helical structure in a smectic with anticlinic layer interfaces, the first actual LC structure proven to exhibit spontaneous reflection symmetry breaking, the SmCP structure, was never, to our knowledge, suggested prior to its discovery. 

So far we have considered the formation of tubules in systems of fixed molecular chirality. It is also possible that tubules might form out of membranes that undergo a chiral symmetry-breaking transition, in which they spontaneously break reflection symmetry and select a handedness, even if they are composed of achiral molecules. This symmetry breaking has been seen in bent-core liquid crystals which spontaneously form a liquid conglomerate composed of macroscopic chiral domains of either handedness.194 This topic is extensively discussed in Walba s chapter elsewhere in this volume. Some indications of this effect have also been seen in experiments on self-assembled aggregates.195,196... 

The problem of the particle masses is resolved with the Higgs mechanism [77,78]. The principle idea of the Higgs mechanism is that the vacuum state does not reflect the full symmetry of the underlying Lagrangian. This concept is quite similar to spontaneous symmetry breaking for chiral molecules. 

An interesting situation also came to light in the limit of normal incidence. This case was impossible to analyze in the framework of the approximate model, as the modes become large quickly and violate the initial assumptions. It turned out that for a = 0 (which is a peculiar case, since the external symmetry breaking in the x direction vanishes), another stationary instability precedes the secondary Hopf bifurcation that spontaneously breaks the reflection symmetry with respect to x. It is shown by point A in Fig. 18. It is also seen from this figure, that the secondary pitchfork bifurcation is destroyed in tbe case of oblique incidence, which can be interpreted as an imperfect bifurcation with respect to the angle a [43]. 

The most important consequence of the breakdown of the B-O approximation is indisputably the Jahn-Teller (J-T) effect [7], where the structure defined on the basis of this approximation does not in fact hold. The important role of the J-T effect is emphasized in Bersuker s book [8] Moreover, since the J-T effect has been shown to be the only source of spontaneous distortion of high-symmetry configurations, we come to the conclusion that the J-T effect is a unique mechanism of all the symmetry breakings in condensed matter. It is of course well known that problems related to the definition of crystallic structure in solid-state physics are mostly ignored assuming only BO structures. Nevertheless it is often criticised by scientists dedicated to studies of the J-T effect. For instance the proper understanding of superconductors should be evidently based on a solution of the non-adiabatic problem but the impact on the crystallic structure is neither reflected in the Frohlich Hamiltonian [9,10] nor in the BCS theory [11]. 

At moderate dilutions of the system, the membrane in the L3 domain separates the phase into two statistically equivalent subvolumes this simply reflects the intrinsic local symmetry of the membrane with respect to its midsurface. At high dilutions of the system, however, the global symmetry could well break spontaneously. Below the corresponding critical concentration of surfactant, the membrane disconnects progressively and finally ends up in the form of a random dispersion of vesicles and/or micelles [219]. 

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