Order matrix, Saupe

One of the characteristic features of mesogenic molecules in uniaxial mesophases is their orientational order specified by a set of order parameters, which forms a symmetric and traceless order matrix S. For rigid molecules, the elements of order matrix Sa/Y defined by Saupe,52 are given by... 

Another method to determine the magnitude and rhombicity of the alignment tensor is based on the determination of the Saupe order matrix. The anisotropic parameter of motional averaging is represented by this order matrix, which is diagonalized by a transformation matrix that relates the principal frame, in which the order matrix is diagonal,... 

Note 3 Molecules which constitute nematogens are not strictly cylindrically symmetric and have their orientational order given by the Saupe ordering matrix which has elements 5 aP = (3< a P> - 5ap)/2, where la and ip are the direction cosines between the director and the molecular axes a and P, 5ap is the Kronecker delta and a, p denote the molecular axes X, Y, Z. 

Note 4 The constituent molecules of a nematogen are rarely rigid and their orientational order is strictly defined, at the second-rank level, by a Saupe ordering matrix for each rigid sub-unit. 

J. W. Emsley, R. Hashim, G. R. Luckhurst and G. N. Shilstone, Solute alignment in liquid crystal solvent. The Saupe ordering matrix for anthracene dissolved in uniaxial liquid crystals, Liq. Cryst., 1 (1986) 437 154. 

The equation for RDC, Eq. (2), could be rewritten in the form of the expectation value for a vector in a Saupe order matrix, i.e., Eqs. (3,4) [9]. The x,y,z in Eq. (3) are directional cosines of a bond vector in an arbitrary molecular frame (Eig. la), the most convenient would be an existing PDB coordinate frame Saupe element Sjj is the averaged projection of axes of the molecular frame onto the direction of Bq with jSx.v.z (Fig. la) specifying the projection angle for each axis 8,y is Kronecker delta. The Saupe matrix is symmetric and traceless and contains five unknown variables. 

The coefficients Cj are called motional constants. They are directly related to the elements of the Saupe order matrix by the following relations, where a is the angle between the liquid crystal optic axis and the applied field direction [43, 45]. 

For asymmetric molecules the Hamiltonian may be written in an analogous form if the principal axes system of the g- and the hyperfine tensor coincide with the principal axes system of the Saupe order matrix S [65]. 

Use a computer algebra package such as Maple or Mathematica to show that the Saupe ordering matrix for a biaxial nematic can be written, in the frame of reference for which it is diagonalized, as... 

If the Saupe ordering matrix is written in terms of the laboratory axis frame, but assuming now that the molecules are uniaxial, then phase biaxiality can be described in terms of the order parameter P, which is nonzero for tilted smectic phases and other intrinsically biaxial phases. For example the diagonal ordering matrix for the molecular long axis z can be written as ... 

Although the irreducible spherical tensor approach is valuable fix the definition of the (xdo ing tensors and for their manipulation under rotation it does not always provide a ready undostanding of the physical significance of the various components. This is sometimes available from a Cartesian representation of the ordering tensor. The most familiar example is the Saupe ordering matrix which represents the orientational ordering at the second rank level [8]. It is defined by... 

Here the bar indicates an ensemble average and /(cu) is the normalized singlet orientational distribution function for finding the director at an orientation co in a molecular frame. Of this infinite set only the second rank quantities 2, can readily be measured and these five components, corresponding to m =0, 1, 2, form the ordering tensor which is the irreducible analogue of the more familiar Saupe ordering matrix. This matrix is defined by... 

The orientational order of molecules in nematics is often described in terms of the so-called Saupe matrix [1], whose elements are defined as ... 

In the molecular orientation system of axes x, y, z, the second-order Saupe matrix is diagonal... 

Equations [11] and [12] separate the information on anisotropic molecular optical properties (0(0/) and (4)0(0/) from that on orientation distribution and polarizer orientation ( D and ((4)p5T[JV ) which can be expressed through the parameters chosen to describe the orientation distribution the orientation factors [3], [4], the Saupe matrix elements [8], or the order parameters [9]. 

---