Potential of mean torque

The non-collective motions include the rotational and translational self-diffusion of molecules as in normal liquids. Molecular reorientations under the influence of a potential of mean torque set up by the neighbours have been described by the small step rotational diffusion model.118 124 The roto-translational diffusion of molecules in uniaxial smectic phases has also been theoretically treated.125,126 This theory has only been tested by a spin relaxation study of a solute in a smectic phase.127 Translational self-diffusion (TD)29 is an intermolecular relaxation mechanism, and is important when proton is used to probe spin relaxation in LC. TD also enters indirectly in the treatment of spin relaxation by DF. Theories for TD in isotropic liquids and cubic solids128 130 have been extended to LC in the nematic (N),131 smectic A (SmA),132 and smectic B (SmB)133 phases. In addition to the overall motion of the molecule, internal bond rotations within the flexible chain(s) of a meso-genic molecule can also cause spin relaxation. The conformational transitions in the side chain are usually much faster than the rotational diffusive motion of the molecular core. 

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

The potential of mean torque, parameterized by the local interaction tensors Xa and Xc (assume to be cylindrical) for the aromatic core and the C C segment, respectively, can be mapped out at each temperature by fitting the observed quadrupolar splittings in the mesophase. Furthermore, the order matrix of an average conformer of the molecule can also be calculated.17 Now pn is needed to describe the internal dynamics. 

The three-state RIS model of conformer statistics is used to analyze the 16 independent dipole coupling constants measured in a proton NMR study of n-hexane in a nematic liquid crystal solvent. The orientational ordering of the n-hexane molecule is treated in the context of the modular formulation of the potential of mean torque. This formulation gives an accurate description of alkane solute orientational order and conformer probabilities in the nematic solvent. Consequently, substantially more accurate calculated diplar couplings are obtained, and this is achieved without the need to resort to unconventionally high values of the trans-gauche energy difference E(g) in the RIS model. 

J. W. Emsley, S. K. Heeks, T. J. Home, M. H. Howells, A. Moon, W. E. Palke, S. U. Patel, G. N. Shilstone and A. Smith, Multiple contributions to potentials of mean torque for solutes dissolved in liquid-crystal solvents. A comparison of the orientational ordering of anthracene and anthraquinone as solutes in nematic solvents, Liq. Cryst., 9 (1991) 649-660. 

Counsell, C. J. R., Emsley, J. W., Heaton, N. J., and Luckhurst, G. R., Orientational ordering in uniaxial liquid crystals the temperature dependence of the potential of mean torque for 4-n-alkyl-4 -cyanobiphenyls, MoZ. Phys., 54, 847-861 (1985). 

The Maier-Saupe theory [85] posits a simple potential of mean torque that originates from an average over the interactions a given mesogen experiences because of its (oriented) neighbors - the mean field ... 

V Q) is the potential of mean torque experienced by a single molecule and is defined by Eq. (3.16). Indeed, the potential of mean torque is responsible for making the molecules preferentially align parallel to each other and to the director. The mean field approximation enables a model for F(fi) to be constructed. 

The potential of mean torque V(cos 0) = — l Pl) Fl(cos 6) must now be altered, since for biaxial molecules their orientation is specified by the... 

It is assumed that the solvent molecules are rigid. The potential of mean torque for a biaxial solute at infinite dilution in a uniaxial solvent can be written [3.20] using a second-rank (L = 2) solute-solvent interaction tensor... 

Now using Stone s formalism [3.15], the case when the symmetry of the orientational distribution of the fi2 vector is lower than spherical is considered. In particular, possible contributions to potential of mean torque by electrostatic and dispersion interactions are briefly examined. A general expansion of the pair potential [Eq. (3.20)] as a product of a distance dependent and an angular dependent function is [3.15]... 

To derive a potential of mean torque from Eq. (3.51) first the average over the orientations of ri2 is made. If this vector is assumed to be distributed... 

The potential of mean torque Uext( >, ) is responsible for the alignment of a conformer and originates from the molecular field of its neighbors. It depends on the conformational state n of the molecule and on the polar... 

A general potential of mean torque Uext for a chain of A" + 1 carbon atoms is... 

The AP method uses Eq. (4.41) with W = 0 and corresponds to alignment of the alkyl chain through aligning of individual C-C bonds in an uncorrelated fashion. The chord model (W Wq / 0) provides a physical realization of Marcelja s [4.26] ad hoc choice of the chord segment (joining the midpoints of the pair of adjacent C-C bonds) to construct a potential of mean torque for lipid membranes. Rewriting the P2(x) in terms of the chord vector P (Fig. 4.8)... 

The model has also been applied to study the orientational order profiles of the deuterated chain attached to mesogens nCB and nOCB [4.28]. Figure 4.9 reproduces [4.29] the fit to the orientational order profile [4.29] of 8CB in the nematic phase at Tc — T = 4° C using the chord model (Wo = Wi). In this case, the potential of mean torque C/ext( 5 ) has to... 

It has recently been suggested that two main ordering mechanisms are responsible for the orientational ordering of the solutes in liquid crystals. The potential of mean torque may be approximated by... 

Collective motions are elastic deformations in liquid crystalline samples, but can also exist in their isotropic phases with a finite coherence length, giving rise to pretransitional phenomena [6.2]. These motions are perceived as hydrodynamic phenomena and are influenced by molecular properties such as elastic constants and viscosities of the liquid crystalline medium. At best, director fluctuations can only provide indirect information on the anisotropic intermolecular interactions. On the contrary, motion on a molecular level must reflect the shape of the instantaneous potential of mean torque on each molecule. Both both molecular rotation and translation are expected to be sensitive to the nature of anisotropic interactions, which determine the formation of various liquid crystalline structures. 

The potential of mean torque for the bent conformer depends on the polar angle, p, and the azimuthal angle, /, which the director makes in the principal molecular axis system (see Fig. 16) these angles are denoted collectively by O). This potential is assumed to take the form proposed by Luck-hurst et al. [65], that is... 

The second rank orientational order parameters of the two conformers can be obtained by taking the Boltzmann average using the appropriate potential of mean torque. For example, for the linear conformer the order parameter for the molecular symmetry axis is... 

Our aim here is not to make a detailed comparison of the various parametrizations which have been proposed for the potential of mean torque. Instead we wish to illustrate the nature of the results which can be obtained with models which include all of the conformations for the dimer, suitably weighted [78]. The calculations proceed in an analogous manner to the generic model for example to determine the N-I transition temperature it is necessary to determine when the molar Helmholtz free energy of the isotropic phase... 

Because the potential of mean torque has become dependent on the internal motion, then so too do the order parameters that is. Equations [18] and [19] [20] become... 

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