Morphology and Mechanics: States of Cellular Matter from Tilings to Tissues
University of Ilinois
The term Cellular Matter denotes materials consisting of separate neighboring domains that fill space in two or three dimensions. Whether ordered or disordered, the domain structure is a crucial factor in understanding overall material properties ranging from mechanical elasticity to long-time aging. Important examples include foams, emulsions, polycrystals, and biological tissues. In all cases, the discrete domain structure poses challenges to both the statistical description of the morphology and to the mechanical description of equilibrium (stable) or non-equilibrium (metastable) material states. The talk presents an overview of our work exploring the relations between domain geometry, domain statistics, and mechanical energy of the cellular system, predominantly in the context of foams and biological tissue, but developing a broader view of systems with more general energy functionals dominated by interfacial terms. It is shown that simple modeling ideas on local structure can analytically explain long-standing empirical correlations between moments of size and topology distributions in disordered cellular matter, such as Lewis’ Law. Combined with a leading order model of the mechanical energy contributions of the cellular interfaces, a connection emerges between the mechanical properties of the tissue, the statistical description of its structure, and the geometry of individual domains (cells). In epidermal tissue, the equilibrium state found in experiments corresponds to mechanically relaxed configurations of this material. In ordered epithelia, the presence of defects provides a sensitive measure for long-range spatial patterns, leading to the discovery of new organizational features in the eye of Drosophila. Extending this model by adding of one additional statistical moment allows for the quantification not just of near-equilibrium, but of metastable non-equilibrium states for an entire class of interfacial energy functionals. Potential applications of such analytical tools include the non-invasive diagnostics of pathological tissue changes as well as of tissue morphogenetic development, whether in vivo or in regenerative medicine.