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

## TL;DR

This paper extends a 2D cellular model to 3D, revealing a rigidity transition controlled by cell surface area, with residual stresses playing a key role, and highlights implications for biological tissues and foams.

## Contribution

The study introduces a 3D cellular model demonstrating a rigidity transition governed by preferred surface area, with residual stresses necessary for rigidity, differing from particulate matter.

## Key findings

- Rigidity transition occurs at a critical surface area s_0*≈5.413.
- Residual stresses are essential for rigidity in the model.
- Shear modulus scales linearly with the control parameter at transition.

## Abstract

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 $s_0^\ast\approx5.413$, 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 we conjecture that there is a well-defined minimal surface area possible for disordered cellular structures. We show that the behavior of this minimal surface induces a linear scaling of the shear modulus with the control parameter at the transition point, which is different from the scaling observed in particulate matter. The existence of such a 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.