Geometrical incompatibility regulated pattern selection and morphological evolution in growing spherical soft tissues

Surface morphological patterns are widely observed in natural systems, which are closely correlated to vital biological functions and inspire surface morphology designs in soft matter systems. Geometrical incompatibility widely exists in biological tissues across different length scales and plays an important role in growth-induced pattern selection and morphological evolution of soft tissues. However, the underlying physical mechanism of growth-induced pattern formation and post-buckling evolution in geometrically incompatible spherical soft tissues remain elusive. Here, the effect of geometrical incompatibility on the growth-induced pattern selection and post-buckling evolution are investigated through swelling experiment, theoretical analysis and numerical simulation. The results show that not only the instability pattern but also the instability threshold can be regulated by manipulating geometric incompatibility. Notably, when the geometrical incompatibility parameter exceeds a critical value, spontaneous instability is observed before growth. With continuous growth, the core–shell soft sphere buckles into a periodic buckyball pattern and evolves toward a bean-shaped pattern, and then undergoes a wrinkle-to-fold transition into a labyrinth topography. Our results demonstrate, both experimentally and theoretically, that geometrical incompatibility can guide the growth-induced pattern formation and morphological evolution effectively. This study not only enhances our understanding of the growth-induced pattern selection and morphological evolution in spherical soft tissues, but also provides an inspiring insight for the fabrication of morphological patterns on curved surfaces.