Symmetry breaking chiral models

Figure 8.20 Structure and phase sequence of prototypical bent-core mesogen NOBOW (8) are given, along with space-filling model showing one of many conformational minima obtained using MOPAC with AMI force field. With observation by Tokyo Tech group of polar EO switching for B2 smectic phases formed by mesogens of this type, banana LC field was bom. Achiral, polar C2v layer structure, with formation of macroscopic spontaneous helix in polarization field (and concomitant chiral symmetry breaking), was proposed to account for observed EO behavior.

In his interesting paper Professor Nicolis raises the question whether models can be envisioned which lead to a spontaneous spatial symmetry breaking in a chemical system, leading, for example, to the production of a polymer of definite chirality. It would be even more interesting if such a model would arise as a result of a measure preserving process that could mimic a Hamiltonian flow. Although we do not have such an example of a chiral process, which imbeds an axial vector into the polymer chain, several years ago we came across a stochastic process that appears to imbed a polar vector into a growing infinite chain. 

From this point we can then treat the action of the second Higgs field on this group in a manner described in Ref. 15. If we set the second Higgs field to have zero-vacuum expectation (symmetry breaking mechanism effectively collapses to this formalism, which is similar to the standard SU(2) x 1/(1) model Higgs mechanism. We can the arrive at a vector electromagnetic gauge theory 0(3), where p stands for partial, and a broken chiral SU(2) weak interaction theory. The mass of the vector boson sector is in the A 3 boson plus the W and Z° particles. 

Besides the effect of strong chiral amplification in the Soai reaction there is the experimental observation of spontaneous generation of ee from entirely achiral starting conditions. Fig. 4 shows that the kinetic model can give rise to mirror-symmetry breaking. 

Abstract Theoretical models and rate equations relevant to the Soai reaction are reviewed. It is found that in production of chiral molecules from an achiral substrate autocatalytic processes can induce either enantiomeric excess (ee) amplification or chiral symmetry breaking. The former means that the final ee value is larger than the initial value but is dependent upon it, whereas the latter means the selection of a unique value of the final ee, independent of the initial value. The ee amplification takes place in an irreversible reaction such that all the substrate molecules are converted to chiral products and the reaction comes to a halt. Chiral symmetry breaking is possible when recycling processes are incorporated. Reactions become reversible and the system relaxes slowly to a unique final state. The difference between the two behaviors is apparent in the flow diagram in the phase space of chiral molecule concentrations. The ee amplification takes place when the flow terminates on a line of fixed points (or a fixed line), whereas symmetry breaking corresponds to the dissolution of the fixed line accompanied by the appearance of fixed points. The relevance of the Soai reaction to the homochirality in life is also discussed. 

Early theoretical models on the feasibility of stochastic mirror symmetry breaking at prebiotic conditions have been successfully realized under laboratory conditions, particularly in studies in crystal and surface science, asymmetric auto catalysis and polymer chemistry. The first step, common in all these scenarios, is the self-assembly of non-chiral or chiral molecules to form diastereoisomeric supramolecular architectures that display different physico-chemical properties. 

To date, no chemical reaction has produced chiral asymmetry in this simple manner. However, symmetry breaking does occur in the crystallization of NaC103 [9, 10] in far from equilibrium conditions. The simple model however leads to interesting conclusions regarding the sensitivity of bifurcation discussed below. 

Many different scenarios have been suggested for the possible origins of biomolecular handedness. An extensive review can be found in the literature [19]. Note that, even if one is considering a process of chiral asymmetry generation after life arose, equations of the type (19.3.17) can still be used to describe the symmetry-breaking process, but this time the model will contain as reactants the self-replicating unit of life. 

In order to explain the optical activity of living matter, Frank [19] has proposed a kinetic model which describes the accumulation of one enantiomer due to spontaneous symmetry breaking and autocatalytic enhancement of its concentration. Although the energy difference between D- and L-enantiomers is extremely small, Frank s model has been extended in order to discuss whether a very small systematic chiral perturbation rather than an accidental event can determine which enantiomer is formed [20]-[23]. For example, Kondepudi and Nelson [23] considered the reaction scheme... 

The perfect shape of crystals reflects their inner regular construetion from a highly ordered array of molecirles. The structure of a crystal determines thermodynamic properties and can make the erystal as a whole chiral, even if it is built from achiral components. In this section, some special properties of crystals will be looked at which allow for mirror symmetry breaking or amplification, focussing on phase behavioiu studies. Most models discussed here work by imequally distributing enantiomers into different phases. Thus, the net amoimt of both enantiomers stays constant but enrichment in one phase can provide a local enviromnent of high asynunetry. 

Taking the Nambu-Jona-Lasinio (NJL) model as a simple but nontrivial model [27], we explicitly demonstrate that quark matter becomes unstable for a formation of DCDW above a critical density the NJL model has been originally presented to demonstrate a realization of chiral symmetry in the vacuum, while recently it been also used as an effective model embodying spontaneous breaking of chiral symmetry in terms of quark degree of freedom [28] 6... 

It is useful to realize the differences between chiral and nonchiral molecules. The molecular tilt breaks the axial symmetry around n for both chiral and nonchiral molecules but there are differences concerning the shape of the resulting rotational potential. In a microscopic model [21], the azimuthal potential V of the rotation around the long axis can be described for a single chiral molecule as a sum of two terms describing polar and quadrupolar ordering... 

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