Programming Shapes with Competing Layered Patterns

TL;DR

The ability to program biologically inspired shapes, such as dental epithelium and the ventral furrow in Drosophila, using a bilayer with simple, distinct deformation gradients is demonstrated.

Abstract

Studying shape changing thick surfaces induced by differential growth helps us understand morphogenesis in biology and offers opportunities for device design. While ideal 2D differential growth maps have been well studied for both isotropic and anisotropic growth, scenarios involving gradients in thickness growth are far less explored. In this paper, we focus on a bilayer system in which the two layers undergo independent but incompatible growth. We examine how the strength of the growth patterns and the aspect ratio of the bilayer influence the resulting shapes. We first investigate the effect of global area difference in the bilayer. Next, we make one of the two layers active and program it with positive or negative, localized or uniform curvature. We then present examples involving competition between two active surfaces with opposite curvature signs or different curvature distributions and understand how the final configurations follow from the principles identified earlier. Finally, we demonstrate the ability to program biologically inspired shapes, such as dental epithelium and the ventral furrow in Drosophila, using a bilayer with simple, distinct deformation gradients.

Figures (22)

• Figure 1: (a) Schematic of a swelling pattern of a bilayer, where $\eta_a(r)$ and $\eta_b(r)$ denote the swelling functions on the apical and basal surfaces respectively. (b) Phase diagram of the equilibrated configurations under uniform shrinkage applied to the apical surface. The background colors are given by analytical predictions, while each point represents a simulation result for the corresponding parameters. Self-intersection denotes numerical breakdown, where local spring crossings occur due to incompatible deformations, leading to crumpling. (c)-(e) Representative shapes corresponding to the states in (b), colored blue-to-red from the center of the initial reference disk to its edge for ease of visualization. (c) A cylindrical shape for $\alpha = 60$ and $d = 0.15$. (d) A spherical cap for $\alpha = 25$ and $d = 0.01$. (e) An hybrid state near the phase boundary for $\alpha = 20$ and $d = 0.1$.
• Figure 2: Simulation results for competition between a passive layer and one with a saddle-shaped target curvature. (a) Phase diagram. States are characterized by the symmetry in Gaussian curvature: S - symmetric, SA - symmetric near center and asymmetric after some radius, A - asymmetric from center. State si refers to the states that have self-intersections in the final configuration, and have not been further analyzed. (b) Swelling function $\eta_\text{s-}(r)$ corresponding to different target radius of curvature of $\gamma_\text{K}R_0$. (c) Predicted extrinsic curvature induced by the swelling profile on the apical surface. The dashed line and dotted line refers to a upward and downward bending respectively. (d–f) Representative configurations for each distinct state. For each configuration: Top left: Snapshot of the 3D configuration, with a blue-to-red color gradient denoting radial distance for visualization. Top right: Normalized Gaussian curvature profile mapped onto the initial flat configuration. It is normalized by a factor $(\gamma_K R_0)^2$. Red and orange lines denote the principal axes of symmetry breaking. Bottom left: Gaussian curvature as a function of radial distance, with points along the principal axes highlighted in red and orange. Bottom right: Normalized height $z$ plotted against normalized radial position $r$ for the points along the two principal axes.
• Figure 3: Simulation results for competition between a passive layer and one with an anti-cone target curvature. (a) Phase diagram. States are characterized by the symmetry in Gaussian curvature: S - symmetric, SA - symmetric near center and asymmetric after some radius, A - asymmetric from center. (b) Swelling function $\eta_\text{a-}(r)$ corresponding to different curvature parameters $\phi$. (c) Predicted extrinsic curvature induced by the swelling profile on the apical surface. The dashed line and dotted line refers to a upward and downward bending respectively. (d–f) Representative configurations for each distinct state. For each configuration: Top left: Snapshot of the 3D configuration, with a blue-to-red color gradient denoting radial distance for visualization. Top right: Normalized Gaussian curvature profile mapped onto the initial flat configuration, normalized by $R_0^2/K$. Red and orange lines denote the principal axes of symmetry breaking. Bottom left: Gaussian curvature as a function of radial distance, with points along the principal axes highlighted in red and orange. Bottom right: Normalized height $z$ plotted against normalized radial position $r$ for the points along the two principal axes.
• Figure 4: (a,c) Swelling functions of sphere and cone patterns. (b,d) Predicted extrinsic curvature induced by the swelling profile on the apical surface. The dashed line and dotted line refers to a upward and downward bending respectively. (e–f) Top left: Snapshot of the 3D configuration, with a blue-to-red color gradient denoting radial distance for visualization. Top right: Normalized Gaussian curvature profile mapped onto the initial flat configuration, normalized by $(\gamma_K R_0)^2$. Red and orange lines denote the principal axes of symmetry breaking. Bottom left: Gaussian curvature as a function of radial distance, with points along the principal axes highlighted in red and orange. Bottom right: Normalized height $z$ plotted against normalized radial position $r$ for the points along the two principal axes.
• Figure 5: Universal scaling behavior of the normalized Gaussian curvature located at the center of the disk. Measured Gaussian curvature in the final configuration, normalized by the target curvature, is plotted against $\alpha^2|K_{\text{total}}|$ for both spherical cap and saddle patterns. Here, $K_{\text{total}}$ represents the target integrated Gaussian curvature over the pattern surface. Red and blue points denote the results for the spherical cap and saddle target patterns, respectively.