Nematic tactoid population

Tactoids are pointed, spindlelike droplets of nematic liquid crystal in an isotropic fluid. They have long been observed in inorganic and organic nematics, in thermotropic phases as well as lyotropic colloidal aggregates. The variational problem of determining the optimal shape of a nematic droplet is formidable and has only been attacked in selected classes of shapes and director fields. Here, by considering a special class of admissible solutions for a bipolar droplet, we study the prevalence in the population of all equilibrium shapes of each of the three that may be optimal (tactoids primarily among them). We show how the prevalence of a shape is affected by a dimensionless measure α of the drop's volume and the ratios k_{24} and k_{3} of the saddle-splay constant K_{24} and the bending constant K_{33} of the material to the splay constant K_{11}. Tactoids, in particular, prevail for α⪅16.2+0.3k_{3}-(14.9-0.1k_{3})k_{24}. Our class of shapes (and director fields) is sufficiently different from those employed so far to unveil a rather different role of K_{24}.