An accurate and robust level-set formulation for multiple junction kinetics

TL;DR

This work addresses the challenge of modeling curvature-driven grain boundary migration in polycrystals with strongly heterogeneous interface properties. It introduces a simple yet robust level-set formulation that preserves the classical mean-curvature mechanism while encoding heterogeneity through a per-grain auxiliary field $\Lambda_i$ in a right-hand-side source term, yielding $\partial \psi_i/\partial t - \mu \gamma \Delta \psi_i = \Lambda_i \mu \left(1 - \sum_{j} H(\psi_j)\right)$. Validation against Garcke's analytical quasi-static triple junction solutions in 2D (and extension to 3D) demonstrates accurate dihedral angles, migration velocities, and TJ profiles across a wide range of energy ratios, with a consistent mapping between $R_\gamma$ and $R_\lambda$. The approach further confirms its applicability to arbitrary TJ configurations via a triangular test and Young's equation, highlighting its potential for realistic, heterogeneous polycrystal simulations. Overall, the method offers a simple, efficient, and scalable framework to study highly heterogeneous interface networks and TJ kinetics, with planned extensions to multiple and higher-dimensional junctions and spatially varying GB properties.

Abstract

The front-capturing Level-Set (LS) method is widely employed in academia and industry to model grain boundary (GB) migration during the microstructure evolution of polycrystalline materials under thermo-mechanical treatments. During capillarity-driven grain growth, the conventional mean curvature flow equation, $\vec{v} = - μγκ\vec{n}$, is used to compute the GB normal migration velocity. Over recent decades, extensive efforts have been made to incorporate polycrystalline heterogeneity into this framework. However, despite increased complexity and computational costs, these approaches have yet to achieve fully satisfactory performance. This paper introduces a simple yet robust LS formulation that accurately captures multiple junction kinetics, even with extreme GB energy ratios. Validation against existing analytical solutions highlights the method's accuracy and efficiency. This novel approach offers significant potential for advancing the study of highly heterogeneous interface systems.

Figures (11)

• Figure 1: Schematic illustration for (a) Garcke's analytical case with (b) zoom on TJ point.
• Figure 2: Numerically obtained quasi-static TJ top dihedral angle $\xi_0$ and migration velocity $v_{TJ}$ along with their projections on the analytical solution line: $v_{TJ} = \pi-\xi_0$.
• Figure 3: (a) Relation between $R_\gamma$ and $R_\lambda$ with (b) zoom on low to medium heterogeneity range.
• Figure 4: Deviation of the numerical solutions $\{\xi_0,v_{TJ}\}$ from the analytical solution line $v_{TJ} = \pi-\xi_0$.
• Figure 5: Comparison between 2D (Cyan) numerical and (Black) analytical TJ profiles in quasi-static regimes (see Eq. \ref{['Garcke_Profile_dimensionless']}).