**4. Conclusions**

We have explored the effect of strain rate on the elasticity, plastic deformation and grain growth of NC gold with small grain size by using molecular dynamics simulations. By considering the models of NC gold with grain sizes of 6 nm and 18 nm, we found that that the apparent Young's modulus of NC gold decreases by following the strain rate decreases. By comparing with that of single crystalline and amorphous gold, it was found that this couldbe attributed to the decrease of effective Young's modulus from GBs with the decrease of strain rate. It was noticed that under a low strain rate of less than 0.5 per ns, the value of apparent Young's modulus of amorphous gold oscillates around 3.19 GPa. The GBs had similar atomic arrangements to amorphous structure and thus small effective Young's modulus under low strain rate. Thus, by following the decrease of grain size the apparent Young's modulus of NC decreased due to the contribution of larger atomic fraction of GBs. The low apparent Young's modulus at low strain was due to the atomic rearrangement around GBs even under small applied tensile strain.

For the response of NC gold to strain rate, flow stress has similar laws to apparent Young's modulus. Following the decrease of strain rates of less than 1 per ns, the decrease of flow stress became slow, especially for larger grain size (such as 18 nm). From the relation between flow stress and strain rate, we found that strain rate sensitivity decreased and activation volume increased with the increase of strain rate. The larger the grain size, the quicker the decrease of strain rate sensitivity. Thus, under lower strain rate, the strain rate sensitivity of larger grain size was smaller. When the strain rate was below 0.1 per ns, the strain rate sensitivities of 18 nm NC gold was just 0.016. Temperature had little effect on the strain rate sensitivity and thus activation volume of 6 nm NC gold. However, there was an obvious effect on NC gold with a grain size of 18 nm. We found that the activation volume of 18 nm NC at 300 K was about 18.69 b<sup>3</sup> under a strain rate of 0.01 ns−1. This indicates that the larger flow stress of 18 nm NC is due to the dislocation pile-up against GBs as the main deformation mechanism under applied strain. For the smaller grain size (such as 6 nm) and high temperature, the smaller activation volume is due to GB slipping and/or change of atomic configuration near GBs. We found that grain growth in NC was related to strain rate in the process of tensile strain. There is a critical grain size about grain growth for each strain rate and the critical grain size increases by following the decrease of strain rate. We propose the critical size for grain growth is about 25 nm under the strain state used in the usual experiments.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2073-4352/10/10/858/s1, Figure S1: both modes of amorphous gold, Figure S2: Strain-stress curves of 6 nm and 18 nm grain sizes under different strain rates, Figure S3: The loading and unloading processes for the case of grain size of 6 nm, Figure S4: Atomic configurations of 3.8 nm grain size under strains, Figures S5–S8: analysis of critical size for grain growth, Figure S9: atomic configuration of 7 nm grain size.

**Author Contributions:** Conceptualization, X.F. and W.Z.; methodology, X.F.; software, J.L.; validation, X.F., J.L., Y.S., and D.J.S.; formal analysis, X.F., J.L.; investigation, X.F., J.L.; resources, X.F.; data curation, X.F., J.L.; writing—original draft preparation, J.L.; writing—review and editing, X.F., Y.S., D.J.S., W.Z.; visualization, J.L., X.F.; supervision, X.F.; project administration, X.F.; funding acquisition, X.F., W.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Key R&D Program of China, grant number 2016YFA0200400, and the National Natural Science Foundation of China, grant number 51627805. The APC was funded by the National Key R&D Program of China.

**Conflicts of Interest:** The authors declare no conflict of interest.
