Shear-Coupled Grain Growth Statistics

Abstract

Grain growth (GG), driven by grain boundary (GB) migration, is a fundamental mechanism of microstructural evolution in polycrystalline materials. GB migration is frequently accompanied by a relative shear displacement of grains meeting at GBs, a phenomenon known as shear coupling. This coupling induces internal stresses within the microstructure, which recent studies have shown to play a decisive role in dictating the evolution of microstructure and GG pathways. This work provides a detailed characterization of the statistical features of two-dimensional GG in the presence of GB shear coupling through continuum modeling of GB migration that incorporates fundamental microscopic mechanisms and diffuse-interface simulations. We demonstrate that incorporating shear coupling produces a more heterogeneous, less equiaxed microstructure than conventional curvature-driven GG, while yielding topological and geometric properties consistent with experimental and atomistic observations. We further demonstrate that as grain grows, internal stress relaxes. Highly stressed grains shrink faster, and lightly stressed grains grow faster than other grains. These findings demonstrate that internal stress, an intrinsic feature of GG, profoundly changes essential features of GG microstructure and kinetics, consistent with experiments and atomic-scale simulations.

Figures (17)

• Figure 1: (a) Mean grain size (area) $\langle A \rangle$ vs. time $t$ during grain growth. Each curve is averaged over 10 independent PF simulations with and without internal stress ($\sigma^{\rm int}$) and applied external stress ($\sigma^{\rm ext}$). The shaded regions indicate the standard deviation from the 10 initially random microstructures. (b) Initial microstructure (with a white GB network and colored orientation $\theta$ field) of a typical 1000-grain polycrystal. All microstructures are shown at the same mean grain area $\langle A \rangle \approx 1\times10^5$, which evolved from the initial microstructure in the cases (c) of mean curvature flow ($\gamma$), (d) with $\sigma^{\rm int}$ and no $\sigma^{\rm ext}$, (e) with $\sigma^{\rm int}$ and applied external shear stress $\sigma^{\rm ext}_{\rm 12}$, and (f) with $\sigma^{\rm int}$ and external tensile stress $\sigma^{\rm ext}_{\rm 11}$.
• Figure 2: Probability distribution of growth rates of individual grains in the cases (a) of curvature flow $\gamma$ (grey diamond) and with internal stress $\gamma,~\sigma^{\rm int}$ (red circles), and with additional applied external (b) shear stress $\gamma,~\sigma^{\rm int},~\sigma^{\rm ext}_{12}$ (blue squares) and tensile stress $\gamma,~\sigma^{\rm int},~\sigma^{\rm ext}_{11}$ (green star).
• Figure 3: Probability distributions of number of neighbors ($N$) of individual grains during GG described by (a) $\gamma$ (curvature flow), where the solid and dashed lines represent the results from the present and previous front-tracking modeling work mason2015geometric, respectively; and (b) $\gamma, \sigma^{\rm int}$, where the orange circles, green squares, blue crosses, and the red line are the results from GG experiments in an Al thin film fradkov1985experimentalbarmak2013grain, PFC simulation la2019statistics and the present work, respectively.
• Figure 4: Geometric properties of microstructures. probability distributions of (a)-(c) reduced grain size $\mathcal{R}/\langle \mathcal{R} \rangle$ and (d)-(f) grain roundness ${\rm \bigcirc}$. The grey, red, green, and blue lines correspond to our PF results for the case of curvature flow with internal stress, external tensile stress, and external shear stress, respectively. The thick grey dotted lines in (a) and (d) are the results obtained by the front-tracking method mason2015geometric. Green squares, blue crosses, and yellow circles are the results from experiments in an Al thin film barmak2013grain and PFC simulations la2019statisticsbackofen2014capturing. Note that each probability distribution is the averaged result based on microstructures with a mean grain area $\langle A \rangle > 5\times 10^4$.
• Figure 5: Evolution of the average (a) GB degree of faceting $\langle\Delta\Phi\rangle/\phi_{\rm c}$, (b) grain aspect ratio $\langle{\rm AR}\rangle-1$ and (c) deviation from a circle $\langle \Delta \bigcirc\rangle$ vs. mean grain area in the cases of $\gamma$ (curvature flow, black circles); $\gamma,~\sigma^{\rm int}$ (red circles); $\gamma,~\sigma^{\rm int},~\sigma^{\rm ext}_{11}$ (green star); and $\gamma,~\sigma^{\rm int},~\sigma^{\rm ext}_{12}$ (blue squares).