Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Abstract

We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the ‘sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of $$\Gamma $$ Γ -convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, that is, on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum–solid and solid–solid phase transitions. The latter relies fundamentally on a structure result for grain boundaries showing that, due to the extremely brittle setup, interpolating boundary layers near cracks are energetically not favorable.

The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling

“…To this end, we proceed as follows. In Lemma 4.3 we show that ψ(ν) can be equivalently defined using in place of Q ν any rectangle coinciding with Q ν along the interface, but with arbitrarily small height (similar results appeared in different contexts, e.g., [14,15,23,17,24,25,29]). Hence the energy of any sequence (u ε ) admissible for (1.4) concentrates arbitrarily close to the jump set of χ ν , i.e., the interface { x, ν = 0}.…”

“…where Q ν is the square with one side orthogonal to ν, u pos ε and u neg ε are the ground states depicted in Figure 1, and ∂ ± ε Q ν are a discrete version of the top/bottom parts of ∂Q ν . Asymptotic formulas like (1.3) are common in discrete-to-continuum variational analyses and are often used to represent Γ-limits of discrete energies [6,10,13,12,8,29]. However, proving an asymptotic lower bound with the density (1.3) for this model requires additional care and is the technically most demanding contribution of this paper.…”

The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling

“…In Lemma 4.3 we show that ψ(ν) can be equivalently defined using in place of Q ν any rectangle coinciding with Q ν along the interface, but with arbitrarily small height (similar results appeared in different contexts, e.g. [16,17,19,[25][26][27]31]). Hence the energy of any sequence (u ε ) admissible for (1.4) concentrates arbitrarily close to the jump set of χ ν , i.e., the interface { x, ν = 0}.…”

“…1, and ∂ ± ε Q ν are a discrete version of the top/bottom parts of ∂ Q ν . Asymptotic formulas like (1.3) are common in discrete-to-continuum variational analyses and are often used to represent -limits of discrete energies [6,8,12,14,15,31]. However, proving an asymptotic lower bound with the density (1.3) for this model requires additional care and is the technically most demanding contribution of this paper.…”

Distribution of Cracks in a Chain of Atoms at Low Temperature

“…We also mention that, by way of contrast, much less is known in higher dimensions, still within the zero temperature regime. For specific pair potentials, basic crystallization results are established in [23,29,35,40,45] and the formation of polycrystals has been obtained in [13,19]. We also refer to [1,[20][21][22] for partial results relating atomistic models to a corresponding variational continuum Griffith functional.…”