**1. Introduction**

To explain the properties of crystalline aggregates, such as crystal plasticity, Taylor [1,2] provided a theoretical construct of line defects in the atomic scale of the crystal lattice. With the use of the electron microscope, the sample presence of dislocations validated Taylor's theory. However, the questions remained on the source of dislocations and the conditions required for their generation and migration in the crystal. While Frank and Reed [3] presented a mechanism to show the generation of many dislocations when the crystal is subjected to an applied stress, electron microscopic observations provided the support for the proposed mechanism. The rare observation of the Frank-Read (FR) source could not be justified for the large density of dislocations seen in crystals.

Li [4] was the first to recognize this issue and put forth the concept of grain boundary ledges as sources for dislocations. Li proposed the emission of dislocations from grain boundary ledges of the type shown in Figure 1. He assumed that the density of ledges is about the same in the grain boundary, which implies that fine-grained materials have a greater dislocation density when they yield. Since then, there have been numerous confirmations of grain boundary dislocation sources under an applied stress or fluctuation of temperature. Direct evidence of these ledges has been found using transmission electron microscopy (TEM).

**Citation:** Li, J.C.M.; Feng, C.R.; Rath, B.B. Emission of Dislocations from Grain Boundaries and Its Role in Nanomaterials. *Crystals* **2021**, *11*, 41. https://doi.org/10.3390/cryst11010041

Received: 26 October 2020 Accepted: 29 December 2020 Published: 31 December 2020

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**Figure 1.** Grain boundary ledge model. (Schematic diagram) From Li [4].

Hall-Petch (HP) relation and the pile-up model were originally developed for iron and used the concept of pile-up of dislocations, originating from the Frank-Read source. Pile-up model has been frequently quoted to explain the HP relationship in other materials. However, it is somewhat ironic that the Frank-Read source and dislocation pile-ups were rarely, if ever found in iron. Li proposed the grain boundary source model. He has considered that dislocations are generated at grain boundary (GB) ledges and all of these dislocations migrate to dislocation forests in grain interiors. Li assumed that the density of ledges is about the same in the grain boundary, so the dislocation density will scale inversely with grain size. Dislocation density also scales as square of yield stress. The two relations, when combined, give the well-known Hall-Petch relation. In fact, Li compared the magnitude of stress in his model with that of the pile-up model with a reasonable agreement.

Ashby [5] described polycrystals as plastically non-homogeneous materials in which each grain deforms by different amounts, depending on its orientation and the constraints imposed by neighboring grains. If each grain were forced to undergo the same uniform strain, voids would form at grain boundaries. To avoid the creation of voids, and ensure strain compatibility, dislocations are introduced. Voids can be corrected by the same process by dislocations of opposite signs. Ashby [5] showed that the number of geometrically necessary dislocations (GNDs) generated at the boundaries and pumped into the individual grains is roughly De/4b, where D is the grain diameter, e is the average strain, and b the magnitude of the dislocation Burgers vector. In both two- and three-dimensional arrays of grains, this leads to a density of geometrically necessary dislocations. These geometrically necessary dislocations can be the origin of the grain size dependence of work-hardening in polycrystals. If it is assumed that hardening depends only on average dislocation density, then this approach yields a σ ~ D−1/2 law for grain boundary hardening, where σ is the yield stress. The concept of generating geometrically necessary dislocations to ensure strain compatibility in polycrystalline materials eliminates the necessity to consider dislocation pile-ups to rationalize the flow of the stress-grain size relationship. It suggests that grain interiors behave essentially like single crystals of the same orientation deforming by single slip while the regions on either side of the grain boundaries undergo lattice rotation and secondary slip. This, in fact, has been observed experimentally by Essmann, et al. [6]. Emission of geometrically necessary dislocations occurs from the grain boundary, which must act as a major source of dislocations.

A comprehensive review of Li's model, its experimental verification, and its comparison with Frank-Read (FR) source models, especially pile-up models, have been given by Murr [7]. A more recent review of Li's model and its implications, vis-a-vis Hall-Petch relation, is also given by Naik and Walley [8]. They reviewed some of the factors that control the hardness of polycrystalline materials with grain sizes less than 1 μm, especially the fundamental physical mechanisms that govern the hardness of nanocrystalline materials. For grains less than 30 nm in size, there is evidence for a transition from dislocation-based plasticity to grain boundary sliding, rotation, or diffusion as the main mechanism responsible for hardness. However, we disagree with their conclusion that the evidence surrounding the inverse Hall-Petch phenomenon is inconclusive, and can be due to processing artefacts, grain growth effects, and errors associated with the conversion of hardness to yield strength in nanocrystalline (NC) materials. The inverse Hall-Petch phenomena is now well established and subject of numerous theoretical and experimental studies. For a review, see reference [8,9]. There were some doubts about its validity in early years when the specimens tested were not fully dense or had other defects. The role of dislocations in fine grain and nanomaterials has renewed interest in the concept of GB as source of complete or partial dislocations. It was found that for most metals with grain sizes in the nanometer regime, experiments have suggested a deviation away from the HP relation relating yield stress to average grain size (It is not clear if such deviations are a result of intrinsically different material properties of nanocrystalline (NC) systems or due simply to inherent difficulties in the preparation of fully dense NC samples and in their microstructural characterization. For a detailed discussion on classical HP and Inverse HP models see the review by Pande and Cooper [9] Therefore, we will not go into the details of Li's model, except for the case of fine-grained materials, where GB sources may play a prominent role.

The use of large-scale molecular dynamics (MD) in the study of the mechanical properties of NC materials provides a detailed picture of the atomic-scale processes during plastic deformation at room temperature. Using an MD model performed at room temperature, it is suggested that the GB accommodation mechanism can be identified with GB sliding, triggered by atomic movement and to some extent stress assisted diffusion [10].

We will briefly consider these issues in subsequent sections.
