Universal scaling of polygonal crack patterns in dried particulate suspensions

Polygonal crack patterns exist in nature over a surprisingly wide array of length scales. Although many factors are known to influence the crack pattern, one well-known result is that the characteristic area of the polygons increases with material thickness. We have quantified this dependence in drying particulate suspensions of cornstarch and CaCO₃ particles. By varying the thickness, boundary adhesion, packing fraction, and suspending liquid, we provide a universal picture of how polygonal crack patterns are formed. We find that all polygonal crack patterns follow the same power law dependence, A ∝ h^4/3, where A is the average area of a polygon and h is the thickness. This power law is due to a simple energy balance between stress and surface energy. In both cornstarch and CaCO₃, large polygons form during the initial drying stage, with prefactors which depend on the modulus of the film and the adhesion to the surface. In cornstarch, well-known columnar polygons form at a later stage during the drying process due to a deswelling of the hygroscopic particles. In this regime, the effective material thickness is determined by the diffusion of solvent from the material, leading to a boundary between "wet" and "dry" particles.