**2. Computational Approach and Models**

All the simulations were performed with standard molecular dynamics simulation (MD) methods which were implemented in large-scale atomic/molecular massively parallel simulator (LAMMPS) [32]. The atomic interaction (*Vp*) between gold atoms was constructed under the frame of embedded-atom method (EAM) potential and composed by a pairwise potential and a many-body embedding energy, as indicated by the formula,

$$V\_p = 1/2 \sum\_{\text{i.i.j.}} \text{i.j.}\_{\text{i.i.j.}} \text{l}I\_{\text{i.j}}(r\_{\text{i.j}}) + \sum\_{\text{i.i.j.}} \text{i.j.}\_{\text{i.i.}} \text{i.p.}\_{\text{i.}} \rho\_{\text{j}}(r\_{\text{i.j}}) \tag{1}$$

where *U*i,j (*r*i,j) was the pairwise potential for the atom at ri and that at *r*j, *E*<sup>i</sup> was the embedding energy of atom at ri from the contributions of nearby atoms whose density was described by the item of \* j (i<sup>j</sup>) ρ<sup>j</sup> - *r*i,j . Here we used the parameters from Foiles et al. [33] to parameterize the EAM of gold. This potential can predict the mechanical properties of gold well. The simulated lattice constant of gold is 0.4078 nm and the stable lattice is fcc. The calculated Young's modulus of polycrystalline gold with this potential is about 78 GPa and is consistent to the value from experiments.

The NC gold models were constructed with the Voronoi method which was popularly used for the building of atomic models for polycrystalline systems. Here the Voronoi method used was implemented in the program Atomsk [34]. We constructed the models of NC gold with average grain sizes of 6 nm and 18 nm, respectively. The periodic boundary conditions were applied for the three directions. To account for statistical effects, there are more than 15 Voronoi grains in each model. Grain orientations are random and the expected mean values (6 nm and 18 nm in two models) around distribution of grain size are used, as shown in Figure 1a,b.

**Figure 1.** Atomic configurations of nanocrystalline gold samples with mean grain sizes of (**a**) 6 nm and (**b**) 18 nm. Blue and green represent grain interiors with fcc lattice and atoms at grain boundaries, respectively.

It is known that the Voronoi approach is just the method of geometric construction with atoms in discrete lattice site. In atomic model of NC, it is popular that the atoms of GBs are unstable. In order to obtain the reasonable atomic configurations at GBs, we needed to relax the unfavorable atomic configurations at GBs. Thus, before the simulations of mechanical properties were performed, the atomic models of NC were annealed at room temperature for 100 ps. All the annealing processes were carried out with an isothermal–isobaric (NPT) ensemble. Then, the NC gold samples were subjected to uniaxial tension tests along the *x*-direction with NPT ensemble. To check the effect of strain rate, the strain rate was modulated from 100 ns−<sup>1</sup> to 0.01 ns<sup>−</sup>1. The pressure in *y*- and *z*-direction was kept at zero in the process of uniaxial tension. The time interval for the step of Newton equation of motion was 1 fs. Besides the method of strain–stress, we considered the second method (method 2) to measure the Young's modulus. In this method, the model of NC is firstly stretched quickly with the strain of 1% along *x* direction under a strain rate of more than 10 ns<sup>−</sup>1. Then the strain in *x* direction is kept to be 1% and the length of the sample in *y* and *z* directions can be changed under zero pressure,

while the internal coordinates of atoms can relax and we measure the change of stress in *x* direction by following the increase of time (which implies the decrease of strain rate).

In order to visualize and analyze the simulated atomic structures, we used the visualization tool OVITO [35]. Here, the dislocation types and dislocation densities were identified with the method of dislocation extraction algorithm (DXA) [36]. The conventional common neighbor analysis (CNA) [37] was designed to characterize the local structural environment by the atomic pattern matching algorithm, which could detect and classify grain interiors (fcc), stacking faults, GBs and surfaces atoms. The atomic-level strains were analyzed on the basis of the displacement of atoms between the two nearby configurations in the process of tensile strain.

To analyze the grain growth in the process of strain, the change of grain size of each grain in the atomic structure of NC was calculated with the rule as described below. The core of each grain was firstly detected by checking the atoms with their nearest neighbors who had the fcc lattice. The cluster analysis on these cores was applied to distinguish each grain. Then the grain size was calculated from the number of atoms in each cluster with a grain skin of 0.816 nm thickness. The calculated grain size with this method is consistent with that from the Voronoi method in the initial configuration of the sample.
