Whereas disclination defects are energetically prohibitive in two-dimensional flat crystals, their existence is necessary in crystals with spherical topology, such as viral capsids, colloidosomes or fullerenes. Such a geometrical frustration gives rise to large elastic stresses, which render the crystal unstable when its size is significantly larger than the typical lattice spacing. Depending on the compliance of the crystal with respect to stretching and bending deformations, these stresses are alleviated by either a local increase of the intrinsic curvature in proximity of the disclinations or by the proliferation of excess dislocations, often organized in the form of one-dimensional chains known as scars. The associated strain field of the scars is such to counterbalance the one resulting from the isolated disclinations. Here, we develop a continuum theory of dislocation screening in two-dimensional closed crystals with genus one. Upon modeling the flux of scars emanating from a given disclination as an independent scalar field, we demonstrate that the elastic energy of closed two-dimensional crystals with various degrees of asphericity can be expressed as a simple quadratic function of the screened topological charge of the disclinations, both at zero and finite temperature. This allows us to predict the optimal density of the excess dislocations as well as the minimal stretching energy attained by the crystal.
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