The Saupe ordering matrices for solutes in uniaxial liquid crystals

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Abstract
A theory is described for U(θ2), the potential of mean torque of rigid solutes at infinite dilution in a uniaxial liquid crystal phase. The general form of U(θ2), is an infinite expansion of modified spherical harmonics CLn (θ2), and truncation at the second rank terms produces a practical form for U(θ2) which is used to calculate Sxx-Syy and Szz , the principal elements of the Saupe ordering matrix of a biaxial solute. The theory predicts that the dependence of Sxx-Syy on Szz for a particular solute-solvent mixture is determined entirely by λ, a parameter describing the departure from cylindrical symmetry of the potential of mean torque, and which is independent of temperature. Furthermore, if U(θ2) is determined entirely by dispersion forces then λ is predicted to be independent of the solvent and to depend entirely on the anisotropy of the electric polarizability tensor of the solute. These predictions are tested by determining the values of Sxx-Syy and Szz by analysing proton N.M.R. spectra of a solute, para-dinitrobenzene (PDNB), in three liquid crystalline solvents: 4,4′-di-n-heptylazoxybenzene (HAB), Phase 5 and E5. The variation of Sxx-Syy with Szz for PDMB in each of these three solvents corresponds closely to predictions of the theory for constant values of λ. There is, however, in each case a small but significant variation of λ with temperature. The values of λ which reproduce the variation of Sxx-Syy with Szz most closely are strongly dependent on the nature of the solvent being 0·20, 0·27 and 0·58 for HAB, Phase 5 and E5 respectively. This observation can be explained only if dispersion forces do not make the dominant contribution to the potential of mean torque for these particular solute-solvent mixtures.

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