  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.
