Power-law forces between particles in a nematic

Macroscopic particles of size a immersed in liquid crystalline solvents (e.g. polystyrene spheres in a lyotropic nematic) will exert long-range forces on each other by distorting the order-parameter field around them. As a first step towards a full theory of this novel class of suspensions, we show how to calculate the energy of interaction as a function of the separation r for r much greater than a between particles of various shapes and orientations in a nematic liquid crystal. Our approach can be understood (i) as a continuum description of the concentration field of the particles, with symmetry-allowed couplings between concentration gradients and director fields or (ii) in terms of an electrostatic analogy due to Brochard and de Gennes, in which the far-field transverse components of the director distortion are treated as a two-component Coulomb field of which particles of various shapes and orientations are elementary multipolar sources. The results with the greatest potential relevance to the experiments of Raghunathan et al. are for spheres: these act like quadrupoles, and the interaction is proportional to Ka(6)/r(5), and attractive for r oriented in a range around 50 degrees to the nematic axis (n) over cap, although it is repulsive for r normal or parallel to (n) over cap. The error in a recent calculation in the literature, claiming an always repulsive, r(-3) interaction, is pointed out.