Geometrical incompatibility guides pattern selection in growing bilayer tubes

Pattern selection and subsequent morphological evolution are of remarkable significance, since they are critical for living creatures to fulfill certain biological functions and also have widespread potential applications from disease diagnosis to advanced manufacturing. Geometrical incompatibility is omnipresent in biological systems and plays a critical role in pattern selection of the growing soft biological tissues. However, how geometrical incompatibility guides pattern selection in growing soft matter remains poorly understood. Here, we present a theoretical model to investigate the influence of geometrical incompatibility on pattern selection of growing bilayer tubes. Our linear stability analysis illustrates that an increase of the geometrical incompatibility parameter provokes the instability pattern transition from a longitudinal pattern to a two-dimensional (2D) pattern and then to a circumferential pattern. Based on the theoretical model, a series of quantificational experiments and finite element simulations are implemented to study how geometrical incompatibility guides pattern selection of growing bilayer tubes and explore the post-buckling evolution of the emerging patterns. Both the numerical simulations and experimental observations agree well with our theoretical predictions. In particular, with further growth far beyond the threshold, a secondary bifurcation is observed in the post-buckling evolution of the 2D pattern. This study suggests that geometrical incompatibility can serve as an implementable experimental tool to quantificationally guide pattern selection and subsequent morphological evolution of growing soft matter, which can be used for growth self-assembly and multifunctional surface manufacturing.