Abstract
Point defects are ubiquitous in two-dimensional crystals and play a fundamental role in determining their mechanical and thermodynamical properties. When crystals are formed on a curved background, finite-length grain boundaries (scars) are generally needed to stabilize the crystal. We provide a continuum elasticity analysis of defect dynamics in curved crystals. By exploiting the fact that any point defect can be obtained as an appropriate combination of disclinations, we provide an analytical determination of the elastic spring constants of dislocations within scars and compare them with existing experimental measurements from optical microscopy. We further show that vacancies and interstitials, which are stable defects in flat crystals, are generally unstable in curved geometries. This observation explains why vacancies or interstitials are never found in equilibrium spherical crystals. We finish with some further implications for experiments and future theoretical work.
Highlights
The rich physics of the ordering of matter on planar surfaces takes on a new complexion when the ordering occurs on a curved two-dimensional manifold
For example, favors the appearance of topological defects that are energetically prohibitive in planar systems. This has been demonstrated in the case of sufficiently large spherical crystals [1, 2, 3, 4, 5, 6], toroidal hexatics [7], and both crystals and hexatics draped over a Gaussian bump [8, 9]
We provide a study of the stability of vacancies and interstitials in curved two-dimensional crystals
Summary
The rich physics of the ordering of matter on planar surfaces takes on a new complexion when the ordering occurs on a curved two-dimensional manifold. Recent experiments [12, 13] have investigated the dynamics of defects by directly visualizing colloidal particles absorbed on spherical oil-water interfaces. We show that continuum elasticity theory [1, 2] can be used to provide explicit first-principles predictions for the elastic stiffness Defects such as vacancies and interstitials are quite common in two-dimensional crystals [14]. We provide a study of the stability of vacancies and interstitials in curved two-dimensional crystals. Our analysis uses continuum theory, and the results are directly applicable to other systems such as, for example, the analysis of vacancies and their relation to failure stress, which has recently investigated in straight carbon nanotubes [20], which provides another example for curved crystals. The Thomson problem java applet used for this analysis is described in the Appendix
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