Grain splitting is a mechanism for grain coarsening in colloidal polycrystals

Letter

Grain splitting is a mechanism for grain coarsening in colloidal polycrystals

Anna R. Barth, Maya H. Martinez, Cora E. Payne, Chris G. Couto, Izabela J. Quintas, Inq Soncharoen, Nina M. Brown, Eli J. Weissler, and Sharon J. Gerbode

Phys. Rev. E 104, L052601 – Published 16 November 2021

Abstract

In established theories of grain coarsening, grains disappear either by shrinking or by rotating as a rigid object to coalesce with an adjacent grain. Here we report a third mechanism for grain coarsening, in which a grain splits apart into two regions that rotate in opposite directions to match two adjacent grains' orientations. We experimentally observe both conventional grain rotation and grain splitting in two-dimensional colloidal polycrystals. We find that grain splitting occurs via independently rotating “granules” whose shapes are determined by the underlying triangular lattices of the two merging crystal grains. These granules are so small that existing continuum theories of grain boundary energy are inapplicable, so we introduce a hard sphere model for the free energy of a colloidal polycrystal. We find that, during splitting, the system overcomes a free energy barrier before ultimately reaching a lower free energy when splitting is complete. Using simulated splitting events and a simple scaling prediction, we find that the barrier to grain splitting decreases as grain size decreases. Consequently, grain splitting is likely to play an important role in polycrystals with small grains. This discovery suggests that mesoscale models of grain coarsening may offer better predictions in the nanocrystalline regime by including grain splitting.

Received 29 August 2021
Accepted 18 October 2021

DOI:https://doi.org/10.1103/PhysRevE.104.L052601

Physics Subject Headings (PhySH)

Crystal growth Entropy Grain boundaries Polycrystals

Colloidal crystal Hard sphere colloids

Optical tweezers

Polymers & Soft MatterStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Anna R. Barth^*, Maya H. Martinez^*, Cora E. Payne , Chris G. Couto, Izabela J. Quintas, Inq Soncharoen, Nina M. Brown, Eli J. Weissler, and Sharon J. Gerbode ^†

Department of Physics, Harvey Mudd College, Claremont, California 91711, USA

^*These authors contributed equally.
^†gerbode@hmc.edu

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Issue

Vol. 104, Iss. 5 — November 2021

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Images

Figure 1
A grain rotates to better match the orientation of its larger neighbor in a simple 2D colloidal polycrystal. (a) Grain 1 (center of the microscope image) is oriented at an angle $θ$ relative to the horizontal axis (dotted line), and is adjacent to grain 2 (shaded dark) and grain 3 (shaded light). The polycrystal is shown here at $t = 0$ , and grain boundaries are highlighted with bold white outlines. The misorientation angles between grain 1 and its neighbors are $ϕ_{1, 2}$ and $ϕ_{1, 3}$ , respectively. (b) Over time, grain 1 rotates counterclockwise to better match the orientation of grain 2. Grain 1's orientation $θ$ is plotted with black markers. Error bars indicate standard error of the mean (SEM) obtained by averaging over all particles in the grain. The rotation reduces the Read-Shockley energy per unit length (gray line and open circles, scaled GB energy) relative to its $t = 0$ value.
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Figure 2
A colloidal crystal grain splits to coalesce with its neighboring grains. (a) A central grain (grain 1) experiences opposing torques (curved arrows) and splits into two counterrotating regions. During splitting, individual granules outlined in blue and yellow rotate independently (arrows indicate particle displacements; circles indicate minimal displacements smaller than the size of an arrowhead). (b) The local crystal orientations around the particles from the two counterrotating regions (counterclockwise in blue, clockwise in yellow) abruptly change during splitting. Both regions initially rotate counterclockwise as a rigid object, and then rapidly rotate in opposite directions. This rapid rotation splits the grain, with each of the two regions coalescing with its adjacent grain. Error bars indicate SEM obtained by averaging over orientations of individual granules.
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Figure 3
Rotating granules are determined by the underlying Moiré pattern. (a) Two overlaid triangular lattices with misorientation angle $ϕ$ form a Moiré pattern with hexagonal regions. For each pattern, an ideal granule is outlined. The diameter $d$ of an ideal granule decreases with increasing $ϕ$ . (b) When each black lattice site is matched to the nearest blue lattice site, the displacements show clockwise rotation about the center of each ideal granule. (c) In the experiment from Fig. 2 at $t = 8$ min, the lattices of grains 1 and 2, which have a misorientation angle of $ϕ_{1, 2} = 9^{\circ}$ , determine ideal granules (outlined in pink). The counterclockwise-rotating granules are fragments of ideal granules that have been cut off by neighboring boundaries.
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Figure 4
The Helmholtz free energy of the grain increases before decreasing during the experimentally observed grain splitting. (a) Each particle's free space (light gray shape at the center of each particle) is shown before and after grain 1 splits. (b) A single particle is highlighted, showing that the edge of the particle's free space corresponds to the farthest position the particle could move to (dotted particle outline) without overlapping its neighbors in their current positions. (c) Helmholtz free energy over time as the grain splits, assuming that particles move along straight trajectories. Grain splitting lowers the energetic cost of the grain boundaries, but first requires an increase in energy. Here the elapsed time $δ t = t - 8$ min.
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Figure 5
Grain splitting events were simulated to determine how the free energy barrier depends on grain diameter $D$ and misorientation angle $ϕ$ . An example simulated grain splitting event with $D = 15 LC$ and $ϕ = 15^{\circ}$ is shown. A central grain (white particles) is initially neighbored by grains (light and dark gray particles) with equal and opposite misorientation angles $ϕ$ . The central grain splits via granules that rotate counterclockwise (blue) and clockwise (yellow) according to the underlying Moiré pattern. Particle displacements are shown with blue and yellow arrows.
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Figure 6
The barrier to grain splitting increases with increasing grain diameter $D$ and decreasing ideal granule diameter $d$ (increasing misorientation angle $ϕ$ ). The shaded surface plot with dashed lines represents the prediction that the energy barrier scales like $L \approx \sqrt{3} D (D / d + 1)$ , an approximation for the total length of granule boundaries. The black data points are the free energy barrier heights $Δ F / k_{B} T$ as directly computed from grain splitting simulations. The single yellow diamond data point represents the energy barrier from the experimentally observed splitting event [Fig. 4]. Shades of gray correspond to values of $Δ F / k_{B} T$ , as indicated by the color bar.
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