  We introduce a diffuse-interface Landau-de Gennes free energy for
free-boundary nematic liquid crystals (NLC) in three dimensions submerged in
isotropic liquid, where a phase field is introduced to model the deformable
interface. The energy we propose consists of the original Landau-de Gennes free
energy, three penalty terms and a volume constraint. We prove the existence and
regularity of minimizers to the diffuse-interface energy functional. We also
prove a uniform maximum principle of the minimizer under appropriate
assumptions, together with a uniqueness result for small domains. Then, we
establish a sharp-interface limit where minimizers of the diffuse-interface
energy converge to a minimizer of a sharp-interface energy under the framework
of $\Gamma$-convergence. Finally, we conduct numerical experiments with the
diffuse-interface model, the findings of which are compared with existing
works.
