Topology and Morphology of Self-Deforming Active Shells

We present a generic framework for modeling three-dimensional deformable shells of active matter that captures the orientational dynamics of the active particles and hydrodynamic interactions on the shell and with the surrounding environment. We find that the cross talk between the self-induced flows of active particles and dynamic reshaping of the shell can result in conformations that are tunable by varying the form and magnitude of active stresses. We further demonstrate and explain how self-induced topological defects in the active layer can direct the morphodynamics of the shell. These findings are relevant to understanding morphological changes during organ development and the design of bioinspired materials that are capable of self-organization.

Figure 1

The sphericity $\mathrm{\Psi }$ (black squares) and the number of defects $N_A$ (red diamonds) on an active deformable shell as a function of the dimensionless activity $\mathcal{Z}$. For extensile activities (positive $\mathcal{Z}$) defects are spontaneously created and annihilated at activities where the shell does not yet significantly deform. For contractile activities (negative $\mathcal{Z}$) the surface becomes less spherical while the defect number remains four. For large extensile and contractile activities protrusions are created. The snapshots indicate representative shapes at different activities.

Figure 2

Deformation of an active nematic shell ($R=12$) attached to a surface, for (a) small activity, $\mathcal{Z}=0.034$; (b) intermediate activity, $\mathcal{Z}=8.5$; and (c) large activity, $\mathcal{Z}=20$. For small activity the two $+1/2$ defects are driven gradually closer together, until the active force is balanced by the elastic force. For intermediate activity the two $+1/2$ defects come into contact and lead to a single protrusion. For large activity motile defects can drive formation of long “tentacles,” which will retract again due to surface tension. The directors on the shell are colored by the magnitude of the order (from red for disordered to yellow to white for fully aligned) and the surface is colored by the magnitude of the curvature (dark blue for strongly negative to white for strongly positive).

Figure 3

Schematic of a $+1/2$ topological defect approaching the tip of a protrusion (projection onto the plane). (a) The motile defect is moving over the surface of the protrusion, causing it to grow. (b) Closer to the tip the gradients will become smaller and the active stress will decrease. (c) When the defect has reached the tip of the protrusion, it combines with another $+1/2$ defect from the back side of the projected plane, giving the winding number along the yellow line a topological charge of $+1$. The gradients in the director field are small, and the active flow will therefore be small as well.

Figure 4

Normalized histogram of the mean curvature for negative (red) and positive (blue) topological charges on a half-sphere at $\mathcal{Z}=20$. Negative topological charges are more likely found in regions of small mean curvature, whereas positive topological charges move towards and generate their own strongly curved surfaces.