Characteristics of transition to disclination disorder on curved crystalline surfaces

In many engineering or biological contexts, two-dimensional lattices structures have to reconcile regular crystalline order with intrinsic curvature, such as the periodic tiling of ommatidia in bulging anthropod eyes or the self-assembly of protein monomers to form viral capsids. We previously described a criterion derived directly from shape properties of the surface that is an accurate predictor of whether a curved structure can be defect-free or not. In practice, however, this transition is modified by activation barriers or more favorable intermediate positions. The shape of the energy landscape determines if a transition preserves the symmetry of defect placement because an energy barrier must be overcome or breaks the symmetry due to stable intermediate positions. While backed by numerical computations, we derive and understand these findings analytically, demonstrating that the dependence of transition characteristics can be predicted a priori for general surface shapes. These results give practical insight into transitions to disorder in many ordered biological systems and inform material design considerations through geometric control, while also improving our fundamental understanding.