Atomistic understanding of diffusion kinetics in nanocrystals from molecular dynamics simulations

Understanding the grain size effect on diffusion in nanocrystals has been hampered by the difficulty of measuring diffusion directly in experiments. Here large-scale atomistic modeling is applied to understand the diffusion kinetics in nanocrystals. Enhanced short-circuit diffusivity is revealed to be controlled by the rule of mixtures for grain-boundary diffusion and lattice diffusion, which can be accurately described by the Maxwell-Garnett equation instead of the commonly thought Hart equation, and the thermodynamics of pure grain-boundary self-diffusion is not remarkably affected by varying grain size. Experimentally comparable Arrhenius parameters with atomic detail validate our results. We also propose a free-volume diffusion mechanism considering negative activation entropy and small activation volume. These help provide a fundamental understanding of how the activation parameters depend on size and the structure-property relationship of nanostructured materials from a physical viewpoint.

Figure 1

Short-circuit diffusion in NC copper. (a) Upper: NC copper sample with $3\times 3\times 3$ grains and grain size $d=15$ nm. Lower: 3D view of fast diffusion channels; see Supplemental Material for a movie with a 360${}^{\circ }$ perspective (Ref. 32). Only those atoms that have traveled farther than 0.6 nm after 1 ns are displayed. (b) MSD vs time at different temperatures with $d=10$ nm. (c) Effective diffusion coefficient vs $d$ at different temperatures. Red solid lines and blue dash lines denote those fits to the Maxwell-Garnett equation and the Hart equation, respectively. $\delta D_{\mathrm{gb}}$ from fitting is displayed for each temperature.

Figure 2

Arrhenius plot of effective self-diffusion in NC copper with grain size of (a) 7.5 nm and (b) 15 nm. Open and solid symbols stand for the present modeling and experimental data, respectively. The latter presented in (b) are reduced to the case of grain size of 15 nm according to the Maxwell-Garnett equation [Eq. (6)]. Linear fits are based on experimental data along with the modeling data above 800 K.

Figure 3

Size dependence of activation parameters. (a) Effective activation energy, (b) pre-exponential factor, and (c) activation entropy. The dashed line in (a) indicates an average value of the modeling data. The shaded area in (b) denotes the scattered range of reported experimental data, and the solid line implies the magnitude of the prefactor from other modelings on a general high-angle GB with the same EAM potential.

Figure 4

Pressure dependence of effective GB self-diffusion (on a natural logarithmic scale) of NC copper with grain diameter of 10 nm at 800 K. The slope leads to an activation volume of the order of 0.15 atomic volume.