A geometrically controlled rigidity transition in a model for confluent 3D tissues

The origin of rigidity in disordered materials is an outstanding open problem in statistical physics. Previously, a class of 2D cellular models has been shown to undergo a rigidity transition controlled by a mechanical parameter that specifies cell shapes. Here, we generalize this model to 3D and find a rigidity transition that is similarly controlled by the preferred surface area: the model is solid-like below a dimensionless surface area of s0 ≈ 5.413, and fluid-like above this value. We demonstrate that, unlike jamming in soft spheres, residual stresses are necessary to create rigidity. These stresses occur precisely when cells are unable to obtain their desired geometry, and numerical data lead us to conjecture that there is a well-defined minimal surface area possible for disordered cellular structures. This minimal surface may be relevant for biological tissues and foams, and helps explain why cell shapes are a good structural order parameter for rigidity transitions in biological tissues.