A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets (Communications in Computational Physics, 2018)
We develop a novel finite element method for a phase field model of nematicliquid crystal droplets. The continuous model considers a free energy comprised ofthree components: the Ericksen's energy for liquid crystals, the Cahn-Hilliard energyrepresenting the interfacial energy of the droplet, and an anisotropic weak anchoringenergy that enforces a condition such that the director field is aligned perpendicularto the interface of the droplet. Applications of the model are for finding minimizersof the free energy and exploring gradient flow dynamics. We present a finite elementmethod that utilizes a special discretization of the liquid crystal elastic energy, as wellas mass-lumping to discretize the coupling terms for the anisotropic surface tensionpart. Next, we present a discrete gradient flow method and show that it is monotoneenergy decreasing. Furthermore, we show that global discrete energy minimizers Γ-convergeto global minimizers of the continuous energy. We conclude with numericalexperiments illustrating different gradient flow dynamics, including droplet coalescenceand break-up.
Keywords: Nematic liquid crystal; phase field; Ericksen's energy; Γ-convergence; gradient flow.