A unified 3D phase diagram of growth induced surface instabilities

Biological world metabolizes itself with germination, growth, development, and aging every second. A variety of fascinating morphological patterns arise on surfaces of growing, developing or aging tissues, organs and micro--organism colonies. The basic mechanism has been long believed to be the mechanical mismatch due to -differential growth between layers with different biological compositions. These patterns have been observed in separate systems and topologically classified as crease, wrinkle-fold, period-double, ridge, delaminated-buckle, and coexistence states. However, a general and systematic understanding of their initiation and evolution remains largely elusive. We construct a unified 3D phase diagram that predicts initially flat tissue layers can transform to various instability patterns, systematically depending on three physical parameters: mismatch strain, modulus ratio between layers, and adhesion energy on the interface. Our phase diagram matches consistently with our mimic in vitro experiments and documented data in state-of-the-art literature