The Saupe ordering matrices for solutes in uniaxial liquid crystals

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 C_Ln (θ₂), and truncation at the second rank terms produces a practical form for U(θ₂) which is used to calculate S_xx-S_yy and S_zz , the principal elements of the Saupe ordering matrix of a biaxial solute. The theory predicts that the dependence of S_xx-S_yy on S_zz 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 S_xx-S_yy and S_zz 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 S_xx-S_yy with S_zz 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 S_xx-S_yy with S_zz 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.