An atomistic and non-classical continuum field theoretic perspective of elastic interactions between defects (force dipoles) of various symmetries and application to graphene

Force multipoles are employed to represent various types of defects and physical phenomena in solids: point defects (interstitials, vacancies), surface steps and islands, proteins on biological membranes, inclusions, extended defects, and biological cell interactions among others. In the present work, we (i) as a prototype simple test case, conduct quantum mechanical calculations for mechanics of defects in graphene sheet and in parallel, (ii) formulate an enriched continuum elasticity theory of force dipoles of various anisotropies incorporating up to second gradients of strain fields (thus accounting for nonlocal dispersive effects) instead of the usual dispersion-less classical elasticity formulation that depends on just the strain (c.f. Peyla, P., Misbah, C., 2003. Elastic interaction between defects in thin and 2-D films. Eur. Phys. J. B. 33, 233-247). The fundamental Green's function is derived for the governing equations of second gradient elasticity and the elastic self and interaction energies between force dipoles are formulated for both the two-dimensional thin film and the three-dimensional case. While our continuum results asymptotically yield the same interaction energy law as Peyla and Misbah for large defect separations (∼1/rⁿ for defects with n-fold symmetry), the near-field interactions are qualitatively far more complex and free of singularities. Certain qualitative behavior of defect mechanics predicted by atomistic calculations are well captured by our enriched continuum models in contrast to classical elasticity calculations. For example, consistent with our atomistic calculations of defects in isotropic graphene, even two dilation centers show a finite interaction (as opposed to classical elasticity that predicts zero interaction). We explicitly find the physically consistent result that the self-energy of a defect is equivalent to half the interaction energy between two identical defects when they “merge” into each other. The atomistic, classical elastic and the enriched continuum predictions are thoroughly compared for two types of defects in graphene: Stone-Wales and divacancy.