We present a phase field model for nematic liquid crystal droplets with anisotropic surface tension. Our model couples the Cahn-Hilliard equation to Ericksen's one constant model for liquid crystals with variable degree of orientation. We present a special discretization of the liquid crystal energy that can handle the degenerate elliptic part without regularization. In addition, our discretization uses a mass lumping technique in order to handle the unit length constraint. Discrete minimizers are computed via a discrete gradient flow. We prove that our discrete energy Gamma-converges to the continuous energy and our gradient flow scheme is monotone energy decreasing. Numerical simulations will be shown in 2-D to illustrate the method. This work is joint with Amanda Diegel (post-doc at LSU). I will also show 3-D simulations of the Ericksen model coupled to the Allen-Cahn equations (with a mass constraint). This work is joint with REU 2017 students: A. Morvant and E. Seal. If time permits, I will also discuss more recent work on the generalized Ericksen model (with multiple constants).