*3.1. The E*ff*ect of Strain Rate on the Young's Modulus*

The behavior of grain size dependence of mechanical properties in the elastic region is very different from that in the plastic region. In the plastic region, the flow stress and/or yield stress increase and then decrease by following the grain size decrease from hundreds of nanometers to several nanometers, as indicated by the Hall–Petch rule and inverse Hall–Petch relation. For an example, as the observation in the simulation of Cu and Au NC systems, the max value of flow stress appeared at the grain sizes of 10–15 nm under a strain rate of 0.1 ns−<sup>1</sup> and 10–18 nm under 1 ns−<sup>1</sup> [9,38], respectively. However, the max value of elasticity modulus, such as Young's modulus, appears in polycrystalline systems. For instance, the Young's modulus of Au polycrystalline is about 78 GPa from the simulation [39]. From our simulation, the Young's moduli of grain sizes of 6 nm and 18 nm were about 39.54 and 51.67 GPa at a strain rate of one per ns, respectively. The early experimental measurements in different materials, such as Fe and Pd, also indicated that the Young's modulus of NC was smaller than that of corresponding polycrystalline [40,41]. The low value of Young's modulus in NC is considered to be due to GBs. It is also possible that it is related to the pores and cracks which are unavoidable in experimental samples [40,41]. Based on the model of crystalline grains with grain-boundary fixed phase, the effective Young's modulus can be analyzed to decrease following the decrease of grain size by fixing the boundary thickness [42].

How do the GBs affect the Young's modulus? Here we expect to modulate the strain rate to check the response of Young's modulus. The stress–strain curves of grain size of 6 nm in the elastic region for five different strain rates are shown in Figure 2a (stress–strain curves under larger strain in Figure S2). We uniformly take the slope of the strain from 0.1% to 1% on the stress–strain curve as the Young's modulus. The Young's modulus of grain sizes of 6 nm increases following the increase of strain rate. The value changes from 30.90 to 58.20 GPa as the strain rate is from 107 s−<sup>1</sup> to 1010 s<sup>−</sup>1. The change of modulus for grain size of 18 nm has a similar rule, though the value is larger than that of a grain size of 6 nm at fixed strain rate in Figure 2b. We checked the Young modulus of single crystalline gold under the different strain rate, such as tensile strain along [100] and [111] directions. It is clear that the Young's modulus doesn't change following the strain rate in single crystalline. This can be understood from the elastic theory based on change of potential. The modulus is the mechanical response to the small deviation of atoms in lattice sites from equilibrium positions under applied strain, and thus has nothing to do with strain rate. Therefore, the reason is GBs which are a response to the change of Young's modulus with strain rate and grain size.

**Figure 2.** (**a**) Stress-strain curves for nanocrystalline gold simulations with grain size of 6 nm under different strain rates from 0.01 ns−<sup>1</sup> to 10 ns<sup>−</sup>1; and (**b**) Young's modulus as the function of strain rate for amorphous gold with two models (Amorphous, NC-1 nm), nanocrystalline gold with a grain size of 6 nm (NC-6 nm-m1 from strain–stress method, NC-6 nm-m2 from method 2 mentioned in the text), and grain size of 18 nm (NC-18 nm-m1, NC-18 nm-m2) and single crystal under strain long [100] direction (SC-[100]-m1, SC-[100]-m2) and [111] directions (SC-[111]-m1, SC-[111]-m2).

In the beginning of the deformation process under small strain less than 1%, NC's response to applied strain being rate sensitive implies that the system is non-elastic in the strict traditional view of elastic deformation of single crystalline. However, in this region of deformation with strain less than 1%, there is no generation of dislocations and stacking faults. The "plastic stage" is considered to begin by following the increase of dislocation density and/or stacking faults from zero. In Figure 3, the strain is typically more than 3% for the case of NC gold. Here we can call this region with strain less than 1% as the quasi-elastic region. The obtained Young's modulus is called as apparent Young's modulus (AYM) to distinguish the traditional view about elastic deformation and Young's modulus. In Figure S3, the loading and unloading processes are performed under strain rates of 10 ns−<sup>1</sup> and 1 ns−<sup>1</sup> for the sample with grain size of 6 nm. We have considered two cases, including one loading with a max strain of 0.4% and the other with a max strain of 4%. Under the smaller loading strain (0.4%), the structure can be very close to the initial state after the unloading process. For the larger loading strain (such as 4%), it is clear that the structure cannot restitute to the initial state after the unloading. This is because of the generation of dislocations under larger strain. From this view, the quasi-elastic region under the small strain is reasonable.

**Figure 3.** (**a**) Atomic configuration of grain size of 6 nm after a strain of 1% in 0.1 ps (blue and green represent grain interiors with fcc and atoms at grain boundaries, respectively); (**b**) atomic configuration after the structural relaxation of 850 ps under a strain of 1% whose initial structure is the structure in (**a**); and (**c**) distribution of atomic weight as a function of atomic displacement for grain size of 6 nm under stain of 1% (the structure in (**a**) is the initial structure) at different relaxation times. In (**a**), the red arrows in each atom represent the displacement size of atoms from the initial structure to the configuration after the relaxation of 850 ps under a strain of 1%. In (**b**), the black represents the atoms of grain interiors with fcc whose initial positions are disordered and belong to grain boundaries, and the red represents the atoms of grain boundaries whose initial positions are ordered with fcc and belong to grain interiors in (**a**).

We have found that the value at higher strain rate is larger. Is it possible that under higher strain rate, the stretching process is too fast and thus the response of atoms at GBs is dull? Thus the GBs become stiff and the effective Young's modulus from the contribution of GBs is larger. In order to consider the response of GBs to small strain applied, we have considered the second method to measure the AYM. We tested this method in single crystal Au and the results were consistent to that from the strain–stress test under fixed strain rate, as shown in Figure 2b.

We check the evolution of stress on NC models over time. The results have a similar rule to that from strain–stress curves for the different strain rate, though the measured stress is a little lower than that from strain–stress. For example, for the model of grain size of 6 nm, the AYM decreases quickly with the decrease of strain rate down to 2.5 ns<sup>−</sup>1, and then does not obviously change under the strain rate of 2.5 ns<sup>−</sup>1–0.5 ns−1. For the strain rate less than 0.5 ns−1, the AYM has a weak decrease trend and is difficult to converge. Similarly, for a grain size of 18 nm, the AYM remains constant under the strain rate of 1 ns−1–0.4 ns−1. Then, it also has a trend of decrease with the decrease of strain rate further, but it seems that the value of AYM in larger grain size (18 nm) is easier to converge than that in smaller size (6 nm). It may be understood that the contribution of GBs becomes weaker following the increase of grain size. We have proposed two amorphous models of gold (Supplementary Materials Figure S1), including the structure with a grain size of 1 nm (NC-1 nm in Figure 2b, atomic fraction of GBs is 88.3%) and one typical amorphous structure (Amorphous in Figure 2b). Under a strain of 1%, the AYM decreased by following the strain rate decreasing down to 0.5 per ns for both models. Then it began to oscillate around a small value (3.19 GPa). From these results, we can confirm that the decrease of AYM in nanoscale is basically due to GBs and not from others, such as pores. Grain boundaries become soft with low effective Young's modulus under the low strain rate. Thus, it is the time-dependent deformation mechanism related to GBs affecting the AYM under different strain rates.

We analyzed the evolution of atomic structures after the applied strain. As shown in Figure 3a,b, after the deformation of 1%, the atomic structures are relaxed for 0.1 ps and 850 ps (corresponded with a strain rate of 0.012 ns−1) for the model of grain size of 6 nm. The red arrow in each atom

in Figure 3a represents the displacement of atoms from the configuration at 0.1 ps to that at 850 ps. We noticed that the atomic displacements at GBs are much larger than that in the grain. In Figure 3c, we show the distribution of atomic weight as the function of atomic displacement at different times. Before the time of 1 ps (strain rate of 10 ns−1), the change of distribution is very large and indicates the system is unstable and tries to response to the applied strain. After 100 ps (strain rate of 0.1 ns<sup>−</sup>1), the change of distribution is almost indistinguishable. This indicates that the strain rate being set to 1 ns−1–0.1 ns−<sup>1</sup> in the usual simulation of deformation process is reasonable. It is known that the distance between the nearest neighbor atoms in Au lattice is 0.286 nm. We can see in Figure 3c that the atomic displacements are larger than the 1% of this value. This is due to the thermal movement of atoms and local larger displacement at GBs (Figure 3a). Due to the larger displacement, we found that the configuration of GBs had been changed (Figure 3b) even at the small strain of 1% with small strain rate. The arrangement of some atoms at GBs became ordered with fcc lattice and some near GBs became disordered. The time-dependent mechanism includes the rearrangements of GBs (in Figure 3) probably even in the elastic regime. Thus, at the very low strain rate in experiments, the AYM of NC is smaller than that of polycrystalline due to the special response of GBs.
