Chirality order parameter

It is considered that, if ideal, optically active poly(alkyl(aryl)silane) homopolymer and copolymer systems could be obtained which had stiffer main-chain structures with longer persistence lengths, it should be possible to clarify the relationship between the gabs value and the chiral molar composition. The magnitude of the chirality of the polyisocyanates allowed precise correlations with the cooperativity models.18q In the theory of the cooperative helical order in polyisocyanates, the polymers are characterized by the chiral order parameter M, which is the fraction of the main chain twisting in one helical sense minus the fraction of the main chain twisting in the opposing sense. This order parameter is equal to the optical activity normalized by the value for an entirely one-handed helical polymer. The theory predicts... 

A theoretical study of numerous chiral molecules including bridged biaryls 5 and 6 has been undertaken using a molecular Monte Carlo simulation approach coupled with calculations of molecular chirality based on a chirality order parameter. The method successfully predicts the helical twisting powers <2003JCP10280>. 

For the sake of completeness, but also for distinction, let us finally make a comment on an entirely different kind of chiral order parameter. Let us imagine that the substance we are dealing with is a mixture of two enantiomers (R) and (S). We can then define a scalar quantity... 

Importantly it was also realized by the Boulder group [6], that the combination of polar packing and the tilt of the molecular planes gives the smectic layers a chiral structure, which is usually referred as layer chirality. Since the molecules (at least those studied first) do not contain any chiral carbons, smectic layers can form two different structures that are non-superposable mirror images of each other. To distinguish between these two structures, we can define the chiral order parameter as [42]... 

The order parameters the diquark gap A, which can be seen as the gain in energy due to diquark condensation, and the mass gaps (pu,indicate chiral symmetry breaking. 

These experiments clearly confirm the theoretical concept of the helical twist of cholesteric phases developed by Goossens and Vertogen 79,86), which is based on the hindered rotation of chiral molecules around their long molecular axis and the introduction of the order parameter D. Furthermore they support the concept of the flexible spacer, described in Chap. 2.1. 

Finally, the difference of chirality enhancement in the N and SmC phases should be mentioned. As shown in Sect. 2.1, enhancement rate in SmC is about one order of magnitude larger than that in N. In the SmC chirality enhancement is attributed to two effects (1) the interaction between bent-core and chiral host molecules and (2) the coupling between ee, tilt, and spontaneous polarization. The latter effect is absent in the N phase and is an additional effect in SmC. Moreover, the chiral discrimination parameter AU is expected to be larger in SmC than in N because of a confined geometry, i.e., smectic layer. 

Vra / ft ) is the quadrupole coupling constant. The matrix of S values represents the order parameters, and they give the alignment of the compound with respect to the applied magnetic field. They can be, and usually are, defined in terms of a molecular-fixed coordinate system. S is a symmetrical 3x3 matrix, and the sum of the diagonal elements of S is zero, so that in a molecular-fixed coordinate system, the number of components of the S matrix varies from 5 for compounds with no elements of symmetry, such as chiral species, to 1 for entities with a C3 or higher axis of symmetry. 

Our interest is the dynamical behavior of ee or the chiral symmetry breaking and, in particular, as to whether ee increases asymptotically. In order to quantify the monomeric ee or the degree of chiral symmetry breaking, we define a monomeric order parameter

[Pg.103]

The evolution Eq. 13 of the order parameter has a similar form to the time-dependent Landau equation [17], which is fundamental in nonequihbrium phase transitions. The asymptotic value of the order parameter 4>i,oo is determined as the zero of the velocity 4>i- The main difference from the standard model of phase transitions lies in the time dependence in the coefficients Ait) and B(t) induced by that of the achiral concentration ait) and the total chiral concentration qft). Because the concentrations a and q are nonnegative, A(t) cannot exceed Bit) Ait) < Bit). 

In all the other cases, with either a linear or a nonlinear recycling process or with both, the coefficients A and B are no longer zero at the same time, and a definite value of the order parameter

0, B becomes nonzero since /xqi,oo > 0. If the linear recycling exists as X > 0, not all the achiral substrate transform to chiral products but a finite amount remains asymptotically as a(t = oo) > 0. Therefore, nonzero values of ko, k or k2 k 2 give contributions to the coefficients A or B. 

So far we have discussed the ee amplification by starting from an initial state with a finite enantiomeric imbalance, and the effect of spontaneous reaction is mostly neglected, ko = 0. However, reaction experiments without adding chiral substances do produce chiral species [13-15], implying that the coefficient ko of the spontaneous reaction should be nonzero. In these experiments, it is found that values of the ee order parameter 0 of many rims are distributed widely between - 1 and + 1, and the probability distribution function has double peaks at positive and negative values of 0. 

Twisting a nematic structure around an axis perpendicular to the average orientation of the preferred molecular axes, one arrives at the molecular arrangement commonly called cholesteric (Kelker and Hatz, 1980). The twisted nematic phase is optically uniaxial, however with the axis perpendicular to the (rotating) director. Such a mesophase combines the basic properties of nematics with the implications of chirality The structure itself is chiral and as a consequence, a non-identical mirror image exists as it is shown schematically in Fig. 4.6-7. Besides the order parameters mentioned before, the essential characteristics of a cholesteric mesophase are the pitch, i.e., the period of the helical structure as measured along the twist axis, and its handedness, i.e., whether the phase is twisted clockwise or anticlockwise. 

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