The Energy Landscape of 2D Cellular Matter: A Statistical Description

Cellular matter, consisting of space-filling domains, comprises foams, emulsions, and many granular and confluent tissue systems. The ground state, and in general the energy landscape, of cellular matter is difficult to determine, consisting of many metastable states separated by energy barriers. Using foam as a prototype, we show that the energy landscape of a large class of 2D cellular systems can be mapped using measures of geometric and topological statistics only. In the limit of monodisperse foam, the system energy is linearly correlated with the defect density. In polydisperse foams, the defect density and the cross-correlation between size and topology are necessary to reliably predict the energy. Surprisingly, metastable states of a larger class of systems with energy functionals more complex than that of foam are found to lie on a quantitatively equivalent energy surface. Thus, statistical information, obtained from visual information, is sufficient to evaluate the relative proximity of the cellular structure to its ground state. This work suggests that information on the mechanical state of cellular matter can be obtained by its morphology only, which makes statistics an important diagnostic tool in the study of engineering materials as well as of biological tissues.