Order parameters directors

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

To completely specify the orientational ordering, the complete set of orientational order parameters, P/,L = 0,2,4.. ., is required. Only the even rank order parameters are non-zero for phases with a symmetry plane perjDendicular to the director (e.g. N and SmA phases). 

An aligned monodomain of a nematic liquid crystal is characterized by a single director n. However, in imperfectly aligned or unaligned samples the director varies tlirough space. The appropriate tensor order parameter to describe the director field is then... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

This transition is usually second order [18,19 and 20]. The SmC phase differs from the SmA phase by a tilt of the director with respect to the layers. Thus, an appropriate order parameter contains the polar (0) and azimuthal ((]i) angles of the director ... 

The two singlet distribution functions are not in themselves sufficient to characterise the order in a smectic A phase because there is, in general, a correlation between the position of a molecule in a smectic layer and its orientation. We need, therefore, the mixed singlet distribution function P(z,cos ) which gives the probability of finding a particle at position z and at an orientation P with respect to the director [18,19]. At the level of description provided by the order parameters it is necessary to introduce the mixed order parameter... 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

Fig. 4a-e. Snapshots of configurations taken from the production stages of simulations of GB(3.0, 5.0, 1, 2) at scaled temperatures a 3.00 b 2.19 c 1.49 d 1.00 e 0.50. The molecules are represented hy lines which are shorter than the molecular length the thick lines show the director and their lengths are proportional to the orientational order parameter, P2, for the configuration... 

The classical scheme for dichroism measurements implies measuring absorbances (optical densities) for light electric vector parallel and perpendicular to the orientation of director of a planarly oriented nematic or smectic sample. This approach requires high quality polarizers and planarly oriented samples. The alternative technique [50, 53] utilizes a comparison of the absorbance in the isotropic phase (Dj) with that of a homeotropically oriented smectic phase (Dh). In this case, the apparent order parameter for each vibrational oscillator of interest S (related to a certain molecular fragment) may be calculated as S = l-(Dh/Di) (l/f), where / is the thermal correction factor. The angles of orientation of vibrational oscillators (0) with respect to the normal to the smectic layers may be determined according to the equation... 

All results were obtained from chains of 200 backbone atoms, averaging 100 chains to obtain M2, axial extensions, end-to-end distances, and segmental and chain order parameters and directors. 

This equation describes the orientation around a common axis called the director of the domain. For perfectly parallel orientation (P2) equals 1. The orientation of the directors in the solution is described by the order parameter PD. The overall orientational order of the anisotropic solution is given by... 

On a molecular level the director is not rigorously defined, but the molecular director is typically considered to be the average long axis of the molecules, oriented along the macroscopic director with some order parameter less than one. This type of anisotropic order is often called long-range orientational order and, combined with the nonresonant optical properties of the molecules, provides the combination of crystal-like optical properties with liquidlike flow behavior characteristic of liquid crystals. 

Many conformations were sampled by the usual MC procedure. The result is of course that there is no preferred orientation of the molecule. Each conformation can, however, be characterised by an instantaneous main axis this is the average direction of the chain. Then this axis is defined as a director . This director is used to subsequently determine the orientational order parameter along the chain. The order is obviously low at the chain ends, and relatively high in the middle of the chain. It was found that the order profile going from the centre of the molecules towards the tails fell off very similarly to corresponding chains (with half the chain length) in the bilayer membrane. As an example, we reproduce here the results for saturated acyl chains, in Figure 10. The conclusion is that the order of the chains found for acyl tails in the bilayer is dominated by intramolecular interactions. The intermolecular interactions due to the presence of other chains that are densely packed around such a chain,... 

The above spectral densities can be modified for the occurence of chain flexibility, and for the director being oriented at dLD w.r.t. the external BQ field in the L frame. For CD bonds located in the flexible chain, the effect of DF is reduced due to an additional averaging of the time dependent factor (/f g) by conformational transitions in the chain. Consequently, the spectral densities given in Eqs. (60)-(62) are modified by replacing Soc%0(Pm,q) by the segmental order parameter YCD of the C-D bond at a particular carbon site on the chain.146,147 As observed experimentally,148,149 the spectral densities in a flexible chain show a SqD dependence when DF dominate the relaxation rates. The general expression of Jm(co 0LD) due to DF in uniaxial nematic phases is given by... 

The transition moment of the dye should align well with the nematic director -the order parameter... 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

Tumbling regime At very low shear rates, the birefringence axis (or the director) of the nematic solution tumbles continuously up to a reduced shear rate T < 9.5. While the time for complete rotation stays approximately equal to that calculated from Eq. (85), the scalar order parameter S,dy) oscillates around its equilibrium value S. Maximum positive departures of S(dy) from S occur at 0 n/4 and — 3n/4, and maximum negative departures at 0 x — k/4 and — 5it/4, while the amplitude of oscillation increases with increasing T. 

Orientational Distribution Function and Order Parameter. In a liquid crystal a snapshot of the molecules at any one lime reveals that they arc not randomly oriented. There is a preferred direction for alignment of the long molecular axes. This preferred direction is called the director, and it cun be used to define- an orienlalional distribution function, f W). where flH win Vilb is proportional to the fraction of molecules with their long axes within the solid angle sinbdw. 

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