**2. Modeling and Simulation of Dislocations from Grain Boundaries**

Computer modeling of dislocations of dislocation emission from grain boundaries has been extensively studied in recent years. Excellent work can be attributed due to Yamakov et al. [10]. Using MD simulation, they examined the nucleation of extended dislocations from the GBs in NC aluminum. They showed the length of the stacking fault connecting the two Shockley partials that form the extended dislocation, i.e., the dislocation splitting distance, *r*split, depended not only on the stacking-fault energy but also on the resolved nucleation stress. Their simulations were only for columnar grain microstructures with a grain diameter, *d*, of up to 70 nm. The magnitude of *r*split relative to *d* was found to represent critical length scale controlling the low-temperature mechanical behavior of NC materials. They, thus, confirmed that the mechanical properties of NC materials, such as the yield stress, depend critically on the grain size.

It was shown that under high stresses, needed to nucleate extended dislocations from GBs, the magnitude of *r*split can be significant, particularly in a NC fcc metal. In fact, it becomes comparable to the grain size, *d*. Thus, the splitting distance can be considered as a new length scale, in addition to *d*. A complete extended dislocation cannot be nucleated unless *d* > *r*split(σ). Therefore, it appears to be a necessary condition for the dislocation– glide mechanism to operate in a NC fcc metal. The authors have beautifully demonstrated the interplay between these two length scales and their effect on the mechanical properties of NC materials.

Recent simulations have confirmed the important role dislocations are expected to play in the deformation of NC fcc metals [11–13]. However, it should be noted that for the

rather small grain diameters (of up to 12 nm) considered in these simulations, single partials produce stacking faults as they glide through the grain and become incorporated into the GBs. These stacking faults remain in the grains as planar defects which are not recovered. This process, most likely, will only operate during the initial stages of deformation, and is quite different from the usual slip mechanism in materials with large grain sizes. The largest grain sizes considered in these simulations was of *d* = 12 nm) [13], and there, the dislocations could account for only about 30% of the total deformation, and for even smaller grain sizes, practically no dislocations were observed. No dislocations were observed in grains smaller than 8 nm. What was the dominant deformation mechanism observed? Most MD simulations involved GB-mediated processes, such as GB sliding [11–13] or GB diffusion [14–16]. Although most of the studies of inverse HP has been for fcc metals there is no reason to believe that it will not apply to other structures. The role of dislocations coming from the Frank-Read (FR) source diminish as grain size decreases, discussed in detail in reference 9 and 45. For grain sizes below 12 nm, the only dislocation source active would be at grain boundaries.

In recent years, the interaction of dislocations with GBs has been subject of both experimental and theoretical studies. The theoretical work of Li led to the conclusion that the nucleation of a dislocation on a GB requires GB ledges [4,13]. On the other hand, experimental studies of the plastic deformation of polycrystals at low temperatures suggested that GBs become the dominant dislocation sources for relatively small grain sizes. The nucleation process may be expected to be independent of grain size [13,16]. Therefore, it is expected that the GBs should provide the necessary dislocation sources at least for the low-temperature deformation of NC materials. Hence, at the relatively high stresses required for the nucleation process, the dislocation-glide mechanism should, in principle, take place even in NC materials. For details of their simulation procedure in NC metals with columnar microstructure, see reference [17].

The process briefly is: (1) at a high enough stress, only a single 1/6 [112] partial is nucleated from a GB or a triple junction. (2) This partial dislocation, while still attached to GB, glides through the interior of the grain, connected to the GB by the stacking fault left behind by the glide. (3) For large grains, dislocation is annihilated at the opposite GB before a second partial can be formed, and a stacking fault transecting the entire grain remains. (4) If the grain is large enough, the energy of the propagating stacking fault eventually becomes too high and a second partial is nucleated to terminate the stacking fault; together with the first partial, an extended 1/2 [110] dislocation is thus formed.

It should be stated that only if the grain size is large enough [16] for the slip mechanism by dislocation glide to be fully developed, conventional HP hardening should occur due to the retarding effect of pile-ups. For the formation of these pile-ups in NC materials, and their effect on the HP mechanism, see reference [18–20]. It is shown that interaction between two extended dislocations should lead to another length scale—the critical distance for a dislocation pile-up—above which one can expect to observe the HP effect. Thus, it is clearly demonstrated that at such small grain sizes even a single complete extended dislocation is not expected. All of this leads us to conclude that when grain sizes are in nanoscale, the dominant sources of the dislocations are grain boundaries and FR mechanism is not present.

An important contribution to this field is the work of Borovokov [21]. In this MD study, they demonstrate that one of the mechanisms that can play an important role in the strength and plasticity of metallic polycrystalline materials is the heterogeneous nucleation and emission of dislocations from GB. This was done by considering the dislocation nucleation from copper bi-crystal with a number of 110 tilt GBs. These GBs covered a wide range of misorientation angles. It is conclusively shown that the mechanic behavior of GBs and the energy barrier of dislocation nucleation from GBs are closely related to the lattice crystallographic orientation, GB energy, and the intrinsic GB structures. Thus, one can conclude that atomistic analysis of the nucleation mechanisms can provide the exact mechanism of this kind of nucleation and emission process, and that can help us to better understand the role of GBs as a dislocation source. For details, see also reference [22].

We should also mention, in this context, a theoretical model that has been developed by Li et al. [23]. It deals with the effect of shear-coupled migration of grain boundaries on dislocation emission in NC materials. They showed that the dislocation emission can be enhanced considerably as shear-coupled migration. They also discuss factors that can lead to an optimal dislocation emission. Their conclusions have been confirmed quantitatively by the existing MD simulations based on the results of Shiue and Lee [24].
