Pattern selection in core-shell spheres

Curvature-induced symmetry-breaking pattern formation and transition are widely observed in curved film/substrate systems across different length scales such as embryogenesis, heterogeneous micro-particles, dehydrated fruits, growing tumors and planetary surfaces. Here, we find, both experimentally and theoretically, that morphological pattern selection of core-shell spheres, upon shrinkage of core or expansion of surface layer, is primarily determined by a single dimensionless parameter Cs which characterizes the stiffness ratio of core/shell and geometric curvature of the system. When the core remains relatively soft (Cs < 1.3), the core-shell sphere usually experiences subcritical buckling behavior with local dimples at the critical threshold. With a stiffer substrate (1.3 < Cs < 15), the system morphs into periodic buckyball patterns. With the continuous increase of the core stiffness (Cs > 15), symmetry-breaking disordered patterns involving polygon and labyrinth modes appear to be energetically favorable. With extremely large Cs ~ 1000, the core-shell sphere approximates to a planar film/substrate system and thus checkerboard patterns with grain boundaries are observed. Moreover, we find that the transition from subcritical to supercritical bifurcations can be quantitatively characterized by this parameter. Pattern selection based on this single key factor remarkably agrees with our experimental observations on oxidized polydimethylsiloxane (PDMS) microspheres in the entire validity range. Our results not only provide fundamental understanding of pattern selection in spherical film/substrate systems, but also pave a promising way to facilitate the design of morphology-related functional surfaces by quantitatively harnessing such curvature-modulus co-determined pattern formation.