Orientational order tensors

For a measure of orientational order, one defines an orientational order tensor S ... 

In eq. (4.25), is the orientational order parameter (relative to the external magnetic field) of the vector mn, and < > indicates averaging over internal vibrations. The coefficients sfp in eq. (4.26) are Saupe orientational order tensor elements relative to the external magnetic field in a molecular-fixed frame. Consequently, it is necessary to know the tensor in order to be able to determine experimentally the direct dipolar contribution from... 

The orientational order tensor is sometimes called the Sanpe matrix. It relates the orientation of a vector in the molecular frame (x, y, z) to that in the reference frame of the director. The Sanpe matrix takes the form... 

SO that the anisotropy in % depends on the mixed second rank orientational-conformational order parameters and the pure second rank orientational ordering tensor it is independent of the pure conformational order parameters. The variation of the magnetic anisotropy with the mixed order parameters appears somewhat involved. The extent of this complexity is detomined by the numb of terms needed in the Fourier expansion of the tenstnr K ( limiting case when ( (pj ) is indepaKlent of the molecular crmformation, that is when the only contribution to ( q>j ) originates from the rigid sub-unit used to define the molecular fl me. Then... 

The orientational order for the nematic is now defined in terms of the orientational ordering tensors for each conformational state. The conformational order is defined by the singlet distribution function, fconi( (Pi X which as we have seen can be expanded in a basis of Fourior functions, but we shall not pursue this aspect here. 

Equations (8.20) are not sufficiently specific for practical purposes, so it is important to consider special cases leading to simpler relations. When the pore orientations are isotropically distributed, the second order tensors k, 3 and y are isotropic and are therefore scalar multiples of the unit tensor. Thus equation (8.20) simplifies to... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

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

Any or all of these forces may result in local stresses within the fluid. Stress can be thought of as a (local) concentration of force, or the force per unit area that bounds an infinitesimal volume of the fluid. Now both force and area are vectors, the direction of the area being defined by the normal vector that points outward relative to the volume bounded by the surface. Thus, each stress component has a magnitude and two directions associated with it, which are the characteristics of a second-order tensor or dyad. If the direction in which the local force acts is designated by subscript j (e.g., j = x, y, or z in Cartesian coordinates) and the orientation (normal) of the local area element upon which it acts is designated by subscript i, then the corresponding stress component (ay) is given by... 

Eqs. (46)—(51) can be used in Eq. (5) to give the average Hamiltonian in the laboratory frame in terms of the order parameter tensor. The NMR spectrum arising from Eq. (5) can provide a means to determine the orientational ordering of molecules in uniaxial LC. 

Particles subject to Brownian motion tend to adopt random orientations, and hence do not follow these rules. A particle without these symmetry properties may follow a spiral trajectory, and may also rotate or wobble. In general, the drag and torque on an arbitrary particle translating and rotating in an unbounded quiescent fluid are determined by three second-order tensors which depend on the shape of the body ... 

Fig. 4. For axially symmetric alignment, a single measured residual dipolar coupling is consistent with orientations of the corresponding internuclear vector, r, which deviate from the principal (z) axis of the order tensor by an angle , thus describing the surface of a cone, and its inverse. If alignment departs from axial symmetry, then the cone of permissible orientations will require further description in terms of an azimuth angle and exhibit some distortion along the x and directions.

In Section 2 it was established theoretically that, relative to some molecule-fixed reference axes, a molecular order (alignment) tensor with five independent parameters was sufficient to describe the molecular orientation upon which the observed dipolar couplings depend. Based on knowledge of the order tensor and the molecular structure, it is possible to predict the corresponding RDCs. From this relationship, one might anticipate that... 

For the description of the order tensor of an isolated molecule or domain, this ambiguity has no practical significance. However, it becomes an important consideration when establishing the orientation of two or more domains, as will be discussed in Section 4. This degeneracy can be lifted by considering a small number of NOE derived constraints, stereo chemical considerations, or through acquisition of RDCs in an additional independent aligning medium.55,56... 

L.133 Using two sets of backbone RDC data, collected in bacteriophage Pfl and bicelle media, they obtained order tensor parameters using a set of crystallographic coordinates for the structural model. This allowed the refinement of C -C bond orientations, which then provided the basis for their quantitative interpretation of C -H RDCs for 38 out of a possible 49 residues in the context of three different models. The three models were (A) a static xi rotameric state (B) gaussian fluctuations about a mean xi torsion and (C) the population of multiple rotameric states. They found that nearly 75% of xi torsions examined could be adequately accounted for by a static model. By contrast, the data for 11 residues were much better fit when jumps between rotamers were permitted (model C). The authors note that relatively small harmonic fluctuations (model B) about the mean rotameric state produces only small effects on measured RDCs. This is supported by their observation that, except for one case, the static model reproduced the data as well as the gaussian fluctuation model. 

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