DOI 10.1140/epje/i2010-10556-8 Regular Article Eur. Phys. J. E (2010) T HE EUROPEAN P HYSICAL JOURNAL E Orientational order in liquid crystals exhibited by some binary mixtures of rod-like and bent-core molecules B. Kundu, R. Pratibha a , and N.V. Madhusudana b Raman Research Institute, C.V. Raman Avenue, Bangalore 560 080, India Received 26 June 2009 and Received in final form 3 December 2009 c  EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2010 Abstract. We report measurements of the temperature variations of the optical birefringence in the ne- matic (N) and partial bilayer SmA (SmA d ) phases in 4–n –octyloxy 4 ′ –cyanobiphenyl made of rod-like (R) molecules and five mixtures of this compound with 1,3–phenylene bis[4–(3–methylbenzoyloxy)] 4 ′ –n – dodecylbiphenyl 4 ′ –carboxylate, made of bent-core (BC) molecules. The birefringence decreases with the concentration x of the BC molecules but the macroscopic order parameter initially decreases upto 11 mol% of BC molecules and subsequently increases with x. This is attributed to the possible formation of polar clusters of BC molecules. Orientation of BC molecules changes between the N and SmA d phases and the birefringence data in the two phases imply that the kink angle of the BC molecules is ∼ 90 ◦ rather than ∼ 110 ◦ as obtained from calculations which minimize the energy of the molecule. IR spectroscopic mea- surements on the mixture with 11 mol% of BC molecules have been used to estimate the molecular order parameter S of the R molecules, and to provide additional support for a relatively small kink angle of BC molecules. 1 Introduction Thermotropic liquid crystals are made of organic molecules with shape anisotropy [1]. The molecules usually have an orientational order, in addition to other types of order. The uniaxial nematic has only an orientational or- der which is a second-rank tensor, resulting in anisotropies of various physical properties [1]. Indeed these anisotropies can be used as macroscopic measures of the order in the medium. For example, the refractive index develops an anisotropy Δn = n e − n o , where the subscripts refer to light polarization along the optic axis which corresponds with the director, and an orthogonal direction, respec- tively. On the other hand, as the orientational order arises from anisotropic distributions of the principal molecular axes about the director, we can define a “molecular” order parameter as well. For an arbitrarily shaped molecule the tensor order parameter [2] is given by S αβ ij = 1 2 〈 3i α j β − δ αβ δ ij 〉, (1) where α, β = x, y, z in a laboratory fixed coordinate sys- tem and i, j = ξ , η, ζ in a molecule fixed coordinate sys- tem with ζ along the long axis of the molecule. i α , j β are the projections of the unit vectors of i and j along α and β, respectively. δ αβ and δ ij are Kronecker deltas and the a e-mail: pratibha@rri.res.in b e-mail: nvmadhu@rri.res.in angular brackets signify a statistical average. The orienta- tion of the molecule can be determined by the three Euler angles θ, φ and ψ (fig. 1), where θ is the angle between the ζ and z axes, ψ is the angle between the ξ axis and normal to the z-ζ plane, and describes a rotation of the molecule around its long axis. φ is the angle between the x axis and the normal to the z-ζ plane. This describes a rotation of the whole molecule around the director (z axis). If the molecules are rigid cylindrical rods, it is enough to define S = S ζζ = 1 2 〈 3 ζ 2 z − 1 〉 = 1 2 〈 3 cos 2 θ − 1 〉. (2) In reality, even rod-like organic molecules are not rigid, but are usually made of relatively rigid aromatic cores and relatively flexible aliphatic chains. In this case, the order parameter has to be defined for each molecular seg- ment which can be considered to be rigid. Even if the entire molecule is rigid, another geometrical aspect con- tributes to the orientational order, viz. the deviation of the molecule from cylindrical symmetry. When such “bi- axial” molecules form a uniaxial nematic (N) phase, the orientational order in the medium requires two indepen- dent parameters [3]: i) the order parameter of the long axis as given by eq. (2) and ii) the difference between the “or- der parameters” of the inequivalent transverse axes of the molecules. The latter will be obviously zero for cylindri- cally symmetric molecules. The second-order parameter is