Orientational order parameter

In nematic and smectic A and B uniaxial phases, the average direction of orientation of the long axes of the molecules defines the director n. If the liquid crystal molecules are taken as rod-like, the degree of parallel order of the individual molecules is described by a single orientational parameter  

For rigid non-cylindrical, elongated molecules, the order is characterized by two parameters the first-order parameter pictures the average orientation of the molecular long axis with respect to the main director the second-order parameter, d = Sxx yy, reflects the difference in order- 

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The practical goal of EPR is to measure a stationary or time-dependent EPR signal of the species under scrutiny and subsequently to detemiine magnetic interactions that govern the shape and dynamics of the EPR response of the spin system. The infomiation obtained from a thorough analysis of the EPR signal, however, may comprise not only the parameters enlisted in the previous chapter but also a wide range of other physical parameters, for example reaction rates or orientation order parameters. 

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). 

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

To start the review of the PIMC results [328], we note that the detailed study of the quantum APR model (Eq. (41)) was partly motivated by the strong changes in shape of the orientational order parameter as a function of temperature as the rotational constant was increasing from its classical value 0 = 0 (see Fig. 3 in Ref. 327). For small enough 0 it was found that the order parameter decays monotonically with increasing temperature, similarly to the classical case. This is qualitatively different for larger 0, where T ) becomes a non-monotonic function of temperature. 

The orientational order parameter for a liquid crystal can be measured by first calculating the ordering tensor... 

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]. 

The distribution of the intermolecular vector is also of value in distinguishing between smectic A and smectic B phases with the latter having long range bond orientational order [23, 24]. At the local level we can define a bond orientational order parameter, PeCn) for molecule i at position q by [25]... 

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... 

Fig. 12. The distance dependence of P2+(r ), the second rank orientational order parameter for the intermolecular vector in the nematic phase formed hy GB(0.345, 0.2, 1, 2)at T of 2.6...

Fig. 18. The dependence of the second rank orientational order parameters, P2 , for the X-C bond on its position along the alkyl chain containing four (A), six ( ) and eight ( ) atoms...

Throughout the simulations, data are collected at periodic intervals. These include the radii of gyration, the lamellar thickness, the kinetic and potential energies, the single chain form factor S(q), the orientational order parameter, and free energy. S(q) is calculated as... 

B. Schartel, Y. Wachtendorf, M. Grell, D.D.C. Bradley, and M. Hennecke, Polarized fluorescence and orientational order parameters of a liquid-crystalline conjugated polymer, Phys. Rev. B, 60 277-283, 1999. 

T Damerau and M Hennecke, Determination of orientational order parameters of uniaxial films with a commercial 90°C-angle fluorescence spectrometer, J. Chem. Phys., 103 6232-6240, 1995. 

Figure 6. V ariation of the orientational order parameter 5 along the hydrocarbon chains of the lipids of DMPC lipid bilayers, according to MD simulations of Berger et al. [58]. The line is drawn to guide the eye. The spheres are experimental values obtained by Seelig and Seelig [59] using 2H-NMR spectroscopy. (Numbering of C-atoms from the head group to the CH3 terminal group). Redrawn from [58] by permission of the Biophysical Society...

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,... 

Furthermore, in all phases studied the first spectral moment Mi of the H NMR spectra can be calculated and the weighted mean splitting of the H NMR spectrum can be obtained, which is proportional to the average chain orientational order parameter of the lipid, using ... 

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