Dislocation screening in crystals with spherical topology

Abstract

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 either by 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 as 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, at both 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.

Faceting and Flattening of Emulsion Droplets: A Mechanical Model

“…(4) vanish upon integration over the entire surface. The optimal dislocation flux Φ can be computed, for a given surface, by minimizing the energy E S [25], although all our results still hold qualitatively under the assumption of lit-tle to no screening (i.e. Φ = 0).…”

“…Following Ref. [25], we express η in terms of a discrete set of "seed" disclinations, coupled with a continuous distribution of screening dislocations. This yields:…”

Interaction of a defect with the reference curvature of an elastic surface

“…We note that our work is in contrast with the literature where the deformed shape of the elastic sheet is predetermined and a defect configuration is sought which is compatible with the shape morphology. 4,5,[15][16][17] These frozen topographies appear when the surface is backed by a fluid substrate as is the case in the problems associated with the determination of virus shapes. 4 On the contrary, we assume that the position of the defect is fixed and allow the elastic surface to develop any arbitrary shape.…”

Cholesteric shells: two-dimensional blue fog and finite quasicrystals

“…Reasoning along similar lines, one also expects that as R increases, locally different tessellations can be patched together at only moderate cost, as the density of structural defects arising in the patching decreases with size [30]: this results in an increased likeliness of observing amorphous structures. Additionally, as in scar screening [50][51][52] in two-dimensional spherical crystals, large strains introduced by the topologically required defects could be alleviated by the proliferation of excess dislocations with null topological charge when R significantly exceeds the lattice spacing [53]. Correspondingly, by systematically starting from different initial conditions for a given set of parameters, we found that for sufficiently large R amorphous states are either metastable [30] or even thermodynamically stable, in which case their free energy is lower than that of any of the quasicrystal phases observed in our simulations (see Fig.…”