Global and local energy minimizers for a nanowire growth model

Abstract

We consider a model for vapor-liquid-solid growth of nanowires proposed in the physical literature. Liquid drops are described as local or global volume-constrained minimizers of the capillarity energy outside a semi-infinite convex obstacle modeling the nanowire. We first address the existence of global minimizers and then, in the case of rotationally symmetric nanowires, we investigate how the presence of a sharp edge affects the shape of local minimizers and the validity of Young’s law. π 2 > θ λ , and max { π 2 , θ λ } < θ < π . In the first two cases, we show that S θ is a strict local minimizer. For a precise formulation we refer to the statements of Theorems 4.4 and 4.8 below. The case max { π 2 , θ λ } < θ < π is more delicate, and we are only able to show strict local minimimality of S θ with respect to admissible sets that coincide with S θ in a neighborhood of the north pole (see Theorem 4.9). The proofs of these theorems rely on calibration techniques and on the construction of foliating families of rotationally symmetric surfaces with constant mean curvature.