Phase-field approximation of sharp-interface energies accounting for lattice symmetry

TL;DR

This work develops a rigorous phase-field approximation for sharp-interface grain-boundary energies in polycrystals where grain orientations live in the symmetry-reduced manifold ${O}(d)/\mathcal{G}$. By combining metric-space valued Sobolev/BV theory with a lifting result, the authors define diffuse functionals that converge, in the Gamma sense, to a sharp-interface energy with a Read–Shockley type small-angle scaling $\theta|\log\theta|$. The limiting energy is characterized via a 1D cell problem and is invariant under lattice symmetries, ensuring physically meaningful grain boundaries and avoiding spurious interfaces. The framework supports both SBV$_{\mathcal{G}}$-based order parameters and metric-space BV formulations, enabling applications to grain-growth simulations and boundary reconstruction from imaging data, with a Gamma-convergence guarantee that minimizers converge to the sharp-interface minimizers.

Abstract

We present a phase-field approximation of sharp-interface energies defined on partitions, designed for modeling grain boundaries in polycrystals. The independent variable takes values in the orthogonal group $\mathrm{O}(d)$ modulo a lattice point group $\mathcal{G}$, reflecting the crystallographic symmetries of the underlying lattice. In the sharp-interface limit, the surface energy exhibits a Read-Shockley-type behavior for small misorientation angles, scaling as $θ|\logθ|$. The regularized functionals are applicable to grain growth simulation and the reconstruction of grain boundaries from imaging data.

Figures (2)

• Figure 1: Sketch of a possible configuration. Left: the dots represent the local lattice orientation in a polycrystal with five grains, labeled $(a)$, $(b)$, $(c)$, $(d)$, $(e)$. Around each interface, a possible orientation of the two basis vectors is illustrated; each interface has a change of orientation of 18 degrees. However, these changes accumulate, and if one starts with a given orientation (interface $(a)-(b)$) then the orientation after five interfaces differs from the initial one by 90 degrees, leading to a fictitious interface, for istance across the dashed red line in grain $(a)$. On the right, the explicit rotation matrices are indicated (angles in degrees).
• Figure 2: Sketch of the construction in the upper bound. The black lines represent the facets $F_l$, and the orange subset is $E^l_{\varepsilon}$. The dashed green circles with radius $2M{\varepsilon}$ around $\partial'F_l$ represent the set $\{v^0_{\varepsilon}=0\}$, the two red ones with radius $M{\varepsilon}\pm \rho_{\varepsilon}$ denote the area where the two interpolations in \ref{['eq-beta-finale']} take place. The right panel shows a blow-up of a part of the set with some labels.

Theorems & Definitions (49)

• Theorem 1.1
• Remark 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Lemma 2.5
• Lemma 2.5
• proof
• Lemma 2.6
• proof