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.

Figure 1

Surface wrinkling morphologies in various living matter with toroidal geometry. (a) Bidirectional stripes on a garlic. (b) Hexagonal-labyrinth bistable patterns on a dehydrated pepper. (c) Transverse and longitudinal wrinkles on an adult male Caenorhabditis elegans. Image obtained through SEM by Carolyn Marks and David Hall of the Hall Lab in WormAtlas [20]. (d) Wrinkles in Crown jellyfish; photo by Jason Webb. (e) Geometry of a core-shell torus.

Figure 2

Bifurcation diagrams of diverse toroidal core shell structures. (a) A cherrylike torus ($\alpha =1.1$ and $C_s=81$). Axisymmetric stripes appear in the outside region of torus, while spiral stripes emerge in the inner region. The inset in red box indicates axisymmetric stripes on a peach. (b) A cherrylike torus with a soft core ($\alpha =1.1$ and $C_s=8.1$). A hexagonal to labyrinthlike topography transition, associated with a bistable phase, is observed in the postbuckling regime. The inset in red box shows hexagons on a shrinking cherry. (c) A donutlike torus ($\alpha =5$ and $C_s=16$). Stripes emerge in the inner region initially, yet hexagonal pattern prevails on the outer surface and eventually evolves into labyrinthlike topography. The inset in red box illustrates hexagonal-labyrinth bistable patterns on a dehydrated pepper. (d) A donutlike torus with a soft core ($\alpha =5$ and $C_s=0.16$). Localized dimples are observed. The inset in red box demonstrates dimples on a dehydrated pepper. The spatial point on the core-shell surface with maximum deflection at the final incremental step is chosen for each bifurcation curve.

Figure 3

Phase diagram of wrinkling morphologies in a variety of core-shell structures, consistent with experimental observations. Localized dimples (blue region) appear in core shell systems with small $C_s<1$, while bidirectional stripe topography (orange region) and spiral and axisymmetric coexistent stripe morphology (purple region) dominate in core shell systems with larger $C_s>10$. Periodic hexagonal patterns (red region) and hexagonal-labyrinth bistable modes (green region) are favorable in between.

Figure 4

Comparison between loading and unloading stages ($\alpha =1.1$ and $C_s=8.1$). The top row shows the deflections $w_{\max }/h_f$ as a function of overall strain ${ϵ}_{th}$ within hysteresis loops: (a) smooth-hexagonal transition, (b) hexagonal-labyrinth transition. Panels (c) and (d) demonstrate the elastic energy differences between the loading and unloading states, $U_{\mathrm{lo}}-U_{\mathrm{ul}}$, as a function of overall strain ${ϵ}_{th}$, corresponding to (a) and (b), respectively. In the range ${ϵ}_{h\to s}<ϵ<{ϵ}_{s\to h}$ ($U_{\mathrm{lo}}-U_{\mathrm{ul}}=U_s-U_h$) and ${ϵ}_{l\to h}<ϵ<{ϵ}_{h\to l}$ ($U_{\mathrm{lo}}-U_{\mathrm{ul}}=U_h-U_l$), the Maxwell equal-energy conditions are defined by the points when $U_{\mathrm{lo}}-U_{\mathrm{ul}}=0$.