Fig. 1. (a) Evolution of flow stress (black), yield strength (red) and yield stress (blue) as a function of grain size. The critical grain size of the softening region in the current work is 12.3 nm, which is comparable to the value predicted by the previous results of Schiøtz and coworkers [7,17] and Lu et al. [8] (10-15 nm). The simulation cells have dimensions (Lx × Ly × Lz) of 40 × 40 × 3.65 nm3 for grain sizes of 6.4, 8.2, 10.3 and 12.3 nm and 80 × 80 × 3.65 nm3 for grain sizes of 14.6 and 17.9 nm. (b) Evolution of stress as a function of strain for the NC pure Cu ‘sample’ with a grain size of 10.3 nm. The deformation within the range of ε < 0.02 is elastic (as indicated by the red dashed line). At ε = 0.046 (point A), the grain interior is dislocation-free. At ε = 0.052 (point B), stacking faults appear (consisting of red atoms and indicated by yellow arrows); at ε = 0.060 (point C), stacking faults spread everywhere. Upon prolonging the strain to ε = 0.150 (point D), the stacking faults disappear in some grains but appear in other grains.

Fig. 2. (a) A simulated NC pure Cu ‘sample’ with columnar grains. The simulation cell has dimensions (Lx × Ly × Lz) of 40 × 40 × 3.65 nm3. The distribution of grain sizes is shown in the insert. (b) Microstructure with all solute atoms segregating at GBs. (c) Microstructure with all solute atoms forming the solid solution.

Fig. 3. Simulation results for NC ‘samples’ containing solid solutions with different solute concentrations (c = 0.01-0.20). (a) Stress-strain curves. (b) Elastic modulus and (c) yield stress as a function of solute concentration. The black dashed lines in (b) and (c) indicate the results for the pure Cu ‘sample’. The signs in (b) and (c) are connected by broken lines.

Fig. 4. Simulation results for NC ‘samples’ with different DOEs (DOE = 0-1). (a) Stress-strain curves. Solute distributions for (b) DOE = 0, (c) DOE = 0.5 and (d) DOE = 1 for c = 0.10. (e) Elastic modulus and (f) yield stress as a function of DOE. For c = 0.10, both the elastic modulus and yield stress show a drop at a larger DOE (approximately 0.9), which is attributed to the formation of solute-rich clusters and phases. The black dashed lines in (b) and (c) indicate the results for the pure Cu ‘sample’. The signs in (e) and (f) are connected by broken lines.

Fig. 5. Simulation results for NC ‘samples’ containing GB segregation with different solute concentrations (c = 0.01-0.40). (a) Stress-strain curves. (b) Elastic modulus and (c) yield stress as a function of solute concentration. (d) GB energy as a function of solute concentration. The GBs are unsaturated for c < 0.03 but supersaturated for c >0.03. The GB energies for c = 0.3 and 0.4 are not shown, as the calculated values of GB energy are not accurate because of the precipitation of the Ag-rich phase at the GBs. (e) Evolution of the system energy change as a function of strain for pure Cu (blue) and c = 0.03 (orange). The black dashed lines in (b-d) indicate the results for the pure Cu ‘sample’. The signs in (b-e) are connected by broken lines.

Fig. 6. (a) Dislocation network in NC ‘sample’ for c = 0.03 at a strain of 0.07, analysed by DXA. The atoms are not shown. The numbers 1, 2 and 3 indicate three different dislocations. (b) Microstructure of the NC ‘sample’ for c = 0.03 at a strain of 0.07. The length in the Z direction is 3Lz. The highlighted region indicates the slice. The right part shows the microstructure of stacking faults containing 2-layer, 1-layer or 4-layer atoms. (c) Microstructure of the slice. The atoms with FCC structures are deleted, leaving 3 dislocations (dislocations 1, 2 and 3 indicated in (a)), GB segments and stacking faults. Dislocation 1 intersects with GBs at points A and B. The movement directions of dislocations 1 and 3 are indicated by the red arrows. The atoms are distinguished by their centrosymmetric parameters. The color bar indicates the value of centrosymmetric parameters (varying from 0.21 to 16.17).

Fig. 7. Evolution of (a) dislocation length in grain interior and (b) total dislocation length (dashed lines) and the dislocation length at GBs (solid lines), as a function of strain, for pure Cu (green), NC alloy with c = 0.03 (red) and NC alloy with c = 0.20 (magenta). The gray region in (b) corresponds to the dislocation length in the grain interior shown in (a). The critical strain at the yield point is marked by dash-dot lines. (c) Dislocation networks at the different strains for pure Cu, c = 0.03 and 0.20. The different types of dislocations are denoted by different colours, as indicated in the bottom of the figure. For pure Cu and c = 0.03, ‘sample’ is constructed by the Cu matrix and GB, while for c = 0.20, ‘sample’ is constructed by the Cu matrix, GB, solute-rich phase (Ag matrix) and phase boundary (PB).

Fig. 8. (a) Microstructure of a grain surrounded by three different GBs, GB1, GB2 and GB3, in the NC ‘sample’. The local segments (namely, S1, S2 and S3) of the three GBs are enlarged and shown below the grain microstructure. Dislocation networks (coloured curves) and solute (gray cycles) distribution at GBs are shown for different strains and systems: (b) at ε = 0 and for pure Cu, (c) at ε = 0.046 and for pure Cu, (d) at ε = 0 and for c = 0.03, (e) at ε = 0.046 and for c = 0.03. The blue, green, yellow, purple, cyan, and red curves represent $\frac{1}{2}$<110>, $\frac{1}{6}$<112>, $\frac{1}{3}$<001>, $\frac{1}{6}$<110>, $\frac{1}{3}$<111> and the other dislocations, respectively.

Fig. 9. (a) Comparison between the strengthening effect obtained in the current work (cycles) and that predicted according to Hu et al.’s work (green curve) [31]. The yellow and red curves represent the Hall-Petch relation and reverse Hall-Petch relation, respectively. Note that the trendlines for Hu et al.’s work, Hall-Petch relation and reverse Hall-Petch relation are not calculated according to the real experimental data but play the role of auxiliary lines for convenience. For the data in the current work, the solute concentration is fixed at c = 0.03, and the DOE varies from 0 to 1. Upon GB segregation (namely, increasing the DOE), the yield stress increases. (b) Map for maximizing yield stress by optimizing the GB excess (or DOE) and the solute concentration (along the directions indicated by the black arrows). The yellow and green bars represent the results for DOE = 0 (i.e., solid solution) and 1 (i.e., GB segregation), respectively. The red and blue bars denote the results for c = 0.03 and 0.10, respectively.