A collapsed interface approach to resolve grain boundaries in finite element simulations of polycrystalline diffusion

TL;DR

This work tackles polycrystalline diffusion by explicitly modeling anisotropic GB transport using a collapsed-interface finite element that analytically integrates GB thickness into a 2D surface element. The method treats in-plane and through-plane diffusivity independently and employs affine parameterization to vary grain size and GB volume fraction without remeshing, enabling efficient computation of the effective diffusivity tensor $\overline{\boldsymbol{D}}$ via linear homogenization. It identifies four diffusion regimes, demonstrates the model reproduces 3D GB transport features with high fidelity, and provides insights into GB-mediated channeling and concentration jumps. Applied to Li6PS5Cl, the framework connects atomistic diffusion data to continuum predictions, offering a scalable tool for diffusion-driven design and optimization of polycrystalline materials.

Abstract

Atomic diffusion affects the properties of various engineering materials, which predominantly occur in the polycrystalline state. A rigorous description of polycrystalline diffusion must therefore account for crystallographic defects, especially grain boundaries (GBs), whose structure and volume fraction - and hence the effective grain size - govern mass transport. Experiments and atomistic simulations consistently show that GBs can accelerate diffusion by up to several orders of magnitude and that fluxes along and across the interface are generally anisotropic. Conventional mesoscale models either neglect GBs or invoke idealized analytical corrections. Fully resolved finite-element meshes are accurate but computationally infeasible when nanometer-thin GB layers are involved. We introduce a collapsed-interface finite element that integrates the GB thickness analytically and embeds the result in a two-dimensional surface element. The formulation (i) treats in-plane and through-plane diffusivity independently, (ii) couples to the surrounding grain matrix without the need for mesh manipulations, and (iii) parametrizes both grain size and GB volume fraction via simple affine scalings, allowing systematic variation without remeshing. Effective diffusivity tensors are extracted by linear computational homogenization. The new finite element reproduces three-dimensional GB transport phenomena - channeled fluxes, concentration discontinuities - at a fraction of the computational cost of explicit models. Parametric studies spanning multiple orders of magnitude in GB diffusivity reveal four distinct diffusion regimes and quantify their impact on the overall response. The framework thus connects atomistic data and continuum predictions, providing an efficient tool for diffusion-driven design and optimization of polycrystalline materials.

Figures (17)

• Figure 1: Relation between macroscopic position $\overline{{\boldsymbol{ x}}}\in\overline{\Omega}$ (mesoscale) and corresponding periodic microscopic polycrystalline sample $\Omega$ (microscale) in the context of linear computational homogenization. The quantities introduced in \ref{['eq:homogenization']}, as well as the macroscopic diffusion tensor $\overline{{\boldsymbol{ D}}}$, are indicated for reference.
• Figure 2: (a) Interface of thickness $2h$ and the split into tangential and normal coordinates (b): Atomic flux in a 2D channel example with in-/outflux on top and bottom. The concentration profile $c(\zeta_{\rm t1}, \zeta_{\rm n})$ as well as the required constant, linear and quadratic modes (dotted lines) are given. (c): Exploded view of the $\zeta_{\rm n}$-collapsed quadratic prism element: Nodes 1-6 constitute the bottom ($p=-1$), nodes 7-12 the middle ($p=0$), and nodes 13-18 the top layer ($p=1$).
• Figure 3: Limit cases of a spheroid with semi-axis lengths $a_x, a_y, a_z$: (a) sphere, (b) prolate (needle-like), and (c) oblate (flake-like).
• Figure 4: Sketch of the compared setups using a fully-resolved GB (left) and the proposed collapsed representation (right). Geometrical features and assigned material properties are given as well. For the fully resolved case, the two possible parametrizations of the junction domain (A) and (B) are shown. In this case, the GB width $2h$ is chosen to be two percent of the grain size $L_{\rm grain}$.
• Figure 5: Concentration profiles across the GB in the positions a, b, c and d highlighted in the schematic at the top: Each row represents a different parameter configuration ($D_\Vert/D_\Omega$, $D_\perp/D_\Omega$). Least-squares second-order polynomial fits for the setups (A) and (B) (see \ref{['fig:2d_sketch']}) are given by the $\times$-markers in the respective color. For the isotropic cases (top and bottom row), setups (A) and (B) coincide. The matching results for the proposed collapsed GB representation are shown for reference. For an easy comparison, the absolute concentration values were shifted such that the concentration value of setup (A) at position $-h$ resembles the zero value. Note that the scaling of the vertical concentration axes differs between the subplots.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5