Emergence of rigid Polycrystals from atomistic Systems with general Interactions

Abstract

We investigate the formation of polycrystalline structures in a class of particle systems. The atomistic energy is modeled as a sum of particle energies that favor atoms being locally isometric to a reference lattice. The discrete frame invariant energy allows for particle configurations in which no underlying lattice is assumed a priori. We prove a discrete-to-continuum limit for configurations with finite surface-energy scaling by means of $\Gamma$-convergence. The resulting continuum theory is described by piecewise constant fields encoding the local orientation of the configuration. The limiting energy is concentrated on grain boundaries, corresponding to the interfaces between regions where the microscopic configuration has constant orientation. The associated energy density depends on the orientations of the two grains as well as on the normal to the interface. Due to our assumptions on the rigid interactions, solid-solid phase transitions with interpolating boundary layers are not energetically favorable; the energy density therefore decomposes into twice the energy density for solid-vacuum transitions.