  Liquid crystal droplets are of great interest from physics and applications.
Rigorous mathematical analysis is challenging as the problem involves harmonic
maps (and in general the Oseen-Frank model), free interfaces and topological
defects which could be either inside the droplet or on its surface along with
some intriguing boundary anchoring conditions for the orientation
configurations. In this paper, through a study of the phase transition between
the isotropic and nematic states of liquid crystal based on the Ericksen model,
we can show, when the size of droplet is much larger in comparison with the
ratio of the Frank constants to the surface tension, a $\Gamma$-convergence
theorem for minimizers. This $\Gamma$-limit is in fact the sharp interface
limit for the phase transition between the isotropic and nematic regions when
the small parameter $\varepsilon$, corresponding to the transition layer width,
goes to zero. This limiting process not only provides a geometric description
of the shape of the droplet as one would expect, and surprisingly it also gives
the anchoring conditions for the orientations of liquid crystals on the surface
of the droplet depending on material constants. In particular, homeotropic,
tangential, and even free boundary conditions as assumed in earlier
phenomenological modelings arise naturally provided that the surface tension,
Frank and Ericksen constants are in suitable ranges.
