Self-assembly of protruding islands on spherical substrates by surface instability

Through strain-induced morphological instability, protruding patterns of roughly commensurate nanostructures are selfassembled on the surface of spherical core/shell systems. A three-dimensional (3D) phase field model is established for closed substrate. We investigate both numerically and analytically the kinetics of the morphological evolution, from grooves to separated islands, which is sensitive to substrate curvature, misfit strain and modulus ratio between core and shell. The faster growth rate of surface undulation is associated with the core/shell system of harder substrate, larger radius or misfit strain. Based on a Ag core/SiO₂ shell system, the self-assemblies of protruding SiO₂ nano-islands are explored experimentally. The numerical and experimental studies herein could guide the fabrication of ordered quantum structures via surface instability on closed and curved substrates.