Curvature-Regulated Multiphase Patterns in Tori

Biological functions in living systems are closely related to their geometries and morphologies. Toroidal structures, which widely exist in nature, present interesting features containing positive, zero, and negative Gaussian curvatures within one system. Such varying curvatures would significantly affect the growing or dehydrating morphogenesis, as observed in various intricate patterns in abundant biological structures. To understand the underlying morphoelastic mechanism and to determine the crucial factors that govern the patterning in toroidal structures, we develop a core-shell model and derive a scaling law to characterize growth- or dehydration-induced instability patterns. We find that the eventual patterns are mainly determined by two dimensionless parameters that are composed of stiffness and curvature of the system. Moreover, we construct a phase diagram showing the multiphase wrinkling pattern selection in various toroidal structures in terms of these two parameters, which is confirmed by our experimental observations. Physical insights into the multiphase transitions and existence of bistable modes are further provided by identifying hysteresis loops and the Maxwell equal-energy conditions. The universal law for morphology selection on core shell structures with varying curvatures can fundamentally explain and precisely predict wrinkling patterns of diverse toroidal structures, which may also provide a platform to design morphology-related functional surfaces.