Anomalous Grain Growth in a Polycrystalline Monolayer of Colloidal Hard Spheres

Understanding grain growth is key for controlling the microstructure and the mechanical properties of most polycrystalline materials, including metals, alloys, and ceramics. However, the precise mechanisms and kinetics of grain growth remain poorly understood both at the theoretical level and experimentally as direct observation is cumbersome in atomic systems. Here, we study the grain growth process in a polycrystalline monolayer of colloidal hard spheres. We find that the bond-orientational correlation function satisfies the dynamic scaling hypothesis and has the general scaling form predicted for systems containing random domain walls. However, the associated correlation length grows slower than $\sim t^{1/2}$, which corresponds to normal curvature-driven grain growth. To understand the origin of this anomalous grain growth, we directly monitor the evolution of the grain boundary network by measuring the so-called grain boundary character distribution. We show that there is a strong annihilation of large-angle grain boundaries while small-angle grain boundaries become relatively more present. Using scaling arguments, we derive the time dependence of the correlation length and show its good agreement with the data. We conclude that the origin of anomalous grain growth is the curvature-driven coarsening of the large-angle grain boundaries at a rate that depends on their relative length in the total grain boundary network.

Popular Summary

Most solid materials are polycrystalline, which means that they are composed of many crystalline grains with random orientations. At high temperature, the interfaces between neighboring grains, known as grain boundaries, start moving and annihilating one another, which leads to an increase of the average size of the grains. This process of grain growth is of key importance for tuning the properties of materials, whose mechanical strength strongly depends on the size of the grains. While theory predicts how quickly the grain size should grow, observations reveal a slower growth rate that differs markedly from these predictions, and there is no unified theoretical framework to describe this “anomalous” grain growth. In our work, we elucidate the origin of anomalous grain growth and derive a theoretical expression that characterizes this process.

We use a polycrystalline material made of micrometer-sized plastic beads as a model for atoms in real materials. In our experiment, grain growth occurs spontaneously, and we show that this growth shares many features with other growth phenomena found in paramagnets, liquid crystals, and demixing fluids. Our system, however, exhibits anomalous grain growth. We directly monitor this process by measuring the length of each grain boundary and the difference in crystal orientations across them, called the misorientation. Grain boundaries with a high misorientation move and annihilate faster than those with a lower misorientation. We quantitatively prove that this leads to anomalous grain growth by deriving a growth law for these grain boundaries that compares well to our data.

Our work lays the foundations for a fruitful description of grain growth that combines the usual framework of generic growth phenomena with the approach of material sciences.

Figure 1

(a)–(c) Microscopy images of the colloidal polycrystalline monolayer at different times during grain growth: (a) 32 $t_B$, (b) 100 $t_B$, and (c) 1000 $t_B$. The scale bar represents $20\mu \mathrm{m}$. (d)–(f) The corresponding smoothed orientation fields (full field of view) at the same times. The scale bar represents $100\mu \mathrm{m}$.

Figure 2

(a) The bond-orientational correlation function $g_6(r,t)$ as defined by Eq. (b3) for different times. (b) Time-evolution of the bond-orientational correlation length $R_6$. The red solid line is the result of the fit to $R_6\sim {(t-t_6)}^{{\alpha }_6}$. All error bars correspond to the standard deviation of the different experimental runs. (c) The correlation function $g_6$ as a function of the distance rescaled by the correlation length $R_6$. The solid black line is the correlation function given by OJK theory; see Eq. (8). (d) A log-log plot of $1-g_6$ as a function of the rescaled distance $x=r/R_6(t)$, which scales as $x$ (black dashed line) following the prediction from Porod’s law; see Eq. (9).

Figure 3

(a)–(c) Grain boundary character distribution (GBCD), $p(\theta ,\psi ,t)$, as a function of the misorientation, $\theta$, and the inclination, $\psi$, for three different times during coarsening. (d) The $\psi$-averaged GBCD, $p(\theta ,t)$, as a function of the misorientation, $\theta$, for different times during coarsening. The LAGB regime is defined as GBs with misorientations above ${\theta }_L$. The value ${\theta }^{*}$ is a crossover point in the SAGB regime corresponding to the value $p^{*}$ of the GBCD. (e) GBCD as a function of time for different misorientations, where each curve is colored according to the misorientation as indicated by the color bar. The time dependence of the GBCD is strongly misorientation dependent below ${\theta }_L$, involving a transition from an increasing to a decreasing GBCD at the crossover value ${\theta }^{*}$, where the GBCD equals $p^{*}$. Above ${\theta }_L$, the GBCD curves decay in the same fashion, which corresponds to an average master curve denoted $p_L(t)$ (magenta line).

Figure 4

(a)–(c) View of the GBs colored according to the value of their misorientation at the exact same times as in Figs. 1 and 1. This provides a direct way to visualize the SAGBs and LAGBs during coarsening. The scale bar represents $100\mu \mathrm{m}$.

Figure 5

(a) Time evolution of the total lengths of GBs, SAGBs, and LAGBs, namely, ${\mathrm{\Lambda }}_I$, ${\mathrm{\Lambda }}_S$, and ${\mathrm{\Lambda }}_L$, in units of particle diameters. (b) Time evolution of the length scale $R_L$, obtained from $A/{\mathrm{\Lambda }}_L$, where $A$ is the area of the field of view. The quantity $R_L{\mathrm{\Lambda }}_I^{1/3}$ behaves as $t^{1/3}$ (black dashed line) at long times. The bond-orientational correlation length, $R_6$, exhibits the same scaling properties as $R_L$. The offsets of the curves are arbitrary to facilitate visualization and the differences in time range are attributable to difficulties in accessing $R_6$ ($R_L$ and ${\mathrm{\Lambda }}_I$) accurately at long (short) times.

Figure 6

(a) Microscopy image of a small area of the colloidal polycrystalline monolayer. The scale bar represents $20\mu \mathrm{m}$. (b) A particle $j$ (black disk) with $k$, one of its six nearest neighbors in the hexagonal lattice (purple disks). The angle between the bond $j-k$ and the $x$ direction is denoted $\mathrm{\Delta }{\theta }_{jk}$, and it is used for the computation of ${\psi }_6$ and the local orientation ${\theta }_6$ (see text). The smoothing of the ${\psi }_6$ field requires the particles in the second shell of nearest neighbors (white disks). (c) Voronoi tessellation, where each cell is colored according to the value of ${\theta }_6$ as indicated by the color bar. (d) Smoothed orientation field ${\tilde{\theta }}_6$.

Figure 7

(a) Local view of the smoothed orientation map ${\tilde{\theta }}_6$. The inset on the left of the snapshot shows the raw orientation map ${\theta }_6$ before smoothing. The scale bar represents $20\mu \mathrm{m}$. (b) The output of the grain detection algorithm adapted from Dillmann et al. [29]. Each grain is colored according to the average ${\tilde{\theta }}_6$ of the particles within it. (c) Delaunay triangulation of the crystal-like particles. The grain edges are colored in black, the GB connectors in red, and the GB segments in blue. The inset shows the construction of a GB segment involving three particles $i$, $j$, and $k$ (see text). (d) The output of the grain detection together with the GB segments colored according to their misorientation as indicated by the color bar.

Figure 8

(a) Definition of the GB inclination $\psi$ from the GB line (black line) in the starting lattice. (b) Construction of the GB by the rotation of grains ① and ② by $\mp \theta /2$, where $\theta$ is the misorientation. (c) Schematic view of a GB as encountered in an experiment, where $\alpha$ is the orientation of grain ① and $\beta$ the tangent direction of the GB with respect to the reference $x$-axis. (d) Similar view as in Fig. 7 with the GB segments colored according to their inclination as indicated by the color bar.