Rotation-induced grain growth and stagnation in phase-field crystal models

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Rotation-induced grain growth and stagnation in phase-field crystal models

Mathias Bjerre, Jens M. Tarp, Luiza Angheluta, and Joachim Mathiesen

Phys. Rev. E 88, 020401(R) – Published 15 August 2013

Abstract

We consider grain growth and stagnation in polycrystalline microstructures. From the phase-field crystal modeling of the coarsening dynamics, we identify a transition from a grain-growth stagnation upon deep quenching below the melting temperature $T_{m}$ to a continuous coarsening at shallower quenching near $T_{m}$ . The grain evolution is mediated by local grain rotations. In the deep quenching regime, the grain assembly typically reaches a metastable state where the kinetic barrier for recrystallization across boundaries is too large and grain rotation with subsequent coalescence or boundary motion is infeasible. For quenching near $T_{m}$ , we find that the grain growth depends on the average rate of grain rotation, and follows a power-law behavior with time, with a scaling exponent that depends on the quenching depth.

Received 27 May 2013

DOI:https://doi.org/10.1103/PhysRevE.88.020401

Authors & Affiliations

Mathias Bjerre¹, Jens M. Tarp¹, Luiza Angheluta², and Joachim Mathiesen¹

¹Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
²Physics of Geological Processes, Department of Physics, University of Oslo, Oslo, Norway

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Vol. 88, Iss. 2 — August 2013

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Images

Figure 1
Time snapshots of grain growth in a polycrystalline material taken at simulation time steps (a) $t = 400$ , (b) $t = 12000$ , (c) $t = 36 000$ , and (d) $t = 225 000$ . The system size is 1024 by 1024, which corresponds to approximately 100 by 100 atom units, and the simulation has been run with the temperature controlling parameter set to $a_{2} = - 0.05$ (corresponding to a relatively cold system). The individual grain colors denote the lattice orientation in radians. In the last frame the grain growth has reached a fixed state where even relatively high levels of thermal noise cannot reactivate grain-boundary migration or internal rearrangement.Reuse & Permissions
Figure 2
(a) shows a phase diagram with points indicating the two simulation protocols used. In protocol I, simulations were carried out along the coexistence line, and in protocol II, simulations were carried out for a fixed mean density $ψ_{0}$ . The average grain area in the stagnated state for the two protocols is shown as a function of $a_{2}$ in (b). We observe that for values of $a_{2}$ approaching from below $0.03$ , the grain areas become of the order of the system size before growth stagnates. For simulations according to protocol II, the average change in grain orientation vs the average grain area is shown in (c). At low temperatures (low values of $a_{2}$ ), grain rotation and grain growth quickly stagnates. Close to the melting temperature grain rotation goes down while the grain growth remains fast.Reuse & Permissions
Figure 3
Mean grain size as a function of time for different quenching depths. We observe a transition from fast stagnation at large quenching depths (low values of $a_{2}$ ) to a power-law scaling of grain size at shallow quenching depths (larger values of $a_{2}$ ). In the latter case, i.e., for high temperatures, the scaling exponent is approximately $α = 1 / 2$ . Each line is averaged over five simulations of a system of size $L = 1024$ . For $a_{2} = 0.03$ , we have further included simulations of a system of size $L = 2048$ . Two lines corresponding to power laws with exponents $1 / 4$ and $1 / 2$ have been added as guides to the eye.Reuse & Permissions
Figure 4
Individual grain trajectories in a lattice orientation and grain area diagram ( $a_{2} = 0.03$ , protocols I and II). The areas of the individual grains have been normalized by the average size to a given time. The time evolution along the trajectories is indicated by the color legend. Note that the big grain separated from all the smaller grains is formed by consecutive alignment and merging of smaller grains.Reuse & Permissions
Figure 5
Grain rotation over area change $| Δ θ / Δ A |$ in a fixed time step $Δ t = 1500$ is shown as a function of the area divided by the mean area (at a given time step) $A / 〈 A 〉$ ( $a_{2} = 0.03$ ). We observe a systematic decrease in the rotation per area as the grain area is increased. The colors of the individual points, indicated by the color legend, represent the time at which a data point was observed. We note that there seems to be no clear difference in this plot between the early- and late-stage dynamics. The local average value of $| Δ θ / Δ A |$ is shown together with the sample variation (standard deviation), marked by error bars in black, and the line on top is a best fit with a power law. The best fit yields an exponent $β = 1.25 \pm 0.06$ .Reuse & Permissions

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