*3.3. The E*ff*ect of Strain Rate on Grain Growth*

In NC metals, the large amounts of GBs due to the small grain size make them unstable. The grain growth (GG) in NC with small grain size is popular at annealing processes, even in rapid annealing [48], such as laser annealing. Besides the GG in high temperatures, it is also possible for GG to appear at low temperatures [49], especially at the deformation processes [50]. Thus, it has an influence on the mechanical properties of NC. In the processes of deformation, the local shear strain induced by applied tensile strain will facilitate the grain coalescence [38]. In Figure S4, the initial atomic structure and that under the strain of 4.5% with strain rate of 5 ns−<sup>1</sup> for NC gold with grain size of 3.8 nm are shown. As per the circles in Figure S4a,b, the configurations of GBs are modulated due to the realignment of atoms at GBs under the local stress and thus the size of the grain changes, accompanied by the stacking faults emission through grain interior.

We have noticed that the atomic realignment at GBs leads to the change of grain size in the processes of tensile strain. We have analyzed statistically the distribution of grain size at different strains by the methods mentioned above. We used the NC gold with average grain sizes of 3.8 nm and 4.5 nm under applied strain with a strain rate of 5 ns−<sup>1</sup> as examples. We checked the change of the ten largest grains in the simulated cell. It was found that for the sample of average grain of 3.8 nm under the strain of 4.5%, the sizes of the three largest grains increase from 5.67, 5.07 and 4.79 nm in initial the structure to 5.83, 5.58 and 5.34 nm, respectively. Clearly, the three largest grains have been grown significantly relative to the initial structure, accompanied by decrease of other smaller grains, like the Ostwald ripening. Interestingly, in the NC gold of 4.5 nm with strain rate of 5 ns<sup>−</sup>1, it isn't found that the grains grow up.

In Figure 7a–c, we show the atomic configurations of 6 nm NC gold under the tensile strain of 7% with the strain rate of 0.1 ns−<sup>1</sup> and 1 ns−1. It can find that the grain grows up under a strain rate of 0.1 ns−<sup>1</sup> and doesn't grow under a strain rate of 1 ns<sup>−</sup>1, as indicated by the circles in Figure 7b,c. Thus, it is proposed that there is a critical grain size for each strain rate. Under a fixed strain rate, it is possible to make the GG appear when grain size is less than the corresponding critical grain size. We analyzed the critical grain size by the statistical method of grain size mentioned above for the cases of different initial grain sizes under different strain rates. The details are listed in Figures S5–S8. In Figure 7d, we show the relation of critical grain size and strain rate. We found that the critical grain size became large, by following the decrease of strain rate. By the extrapolation, we can propose the critical grain size is about 25 nm under the strain rate of 10−<sup>4</sup> s−<sup>1</sup> (which is the regular strain rate used in experiments) if the main mechanism of GG isn't changed. This is consistent with the previous observations in experiments about other NC metals [51–55]. For example, in NC Pt thin film, the GG appears with grain sizes for a dozen, even tens of, nanometers during the tensile deformation test under a strain rate of 3 <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>s</sup><sup>−</sup>1.

**Figure 7.** (**a**) The initial structures of grain size of 6 nm, atomic configurations of it at tensile strain of 7% under the stain rate of (**b**) 0.1 ns−<sup>1</sup> and (**c**) 1 ns<sup>−</sup>1, and (**d**) critical grain size for grain growth as a function of strain rate. In (**a**–**c**), blue, red and green represent grain interiors with fcc, stacking faults with hcp, and atoms at grain boundaries, respectively. In (**d**) experimental results from [46–50] are provided for comparison.

For the GG during the tensile strain, the change of GBs' configuration is an important way as mentioned above. One of the important mechanisms is grain rotation, in which the dependence of GB's energy on misorientation between two nearby grains is the driving force. The local shear stress will rotate the grain to form low energetic GBs, as indicated by the previous simulations [38,56]. Generally, the GB migration and grain rotation derived by the local large stress near GBs with the assistance of dislocation take the main rule. Thus, the growth up of single grain or/and the coalescence between grains to form a larger grain appear in the tensile deformation processes. As an example, the coalescence between grains in 7 nm NC gold under tensile strain is shown in Figure S9.
