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Emission of Dislocations from Grain Boundaries and Its Role in Nanomaterials

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Emission of Dislocations from Grain Boundaries and Its Role in Nanomaterials crystals Review Emission of Dislocations from Grain Boundaries and Its Role in Nanomaterials James C. M. Li 1,*, C. R. Feng 2 and Bhakta B. Rath 2 1 Department of Mechanical Engineering, University of Rochester, Rochester, NY 14627, USA 2 U.S. Naval Research Laboratory, 4555 Overlook Ave SW, Washington, DC 20375, USA; [email protected] (C.R.F.); [email protected] (B.B.R.) * Correspondence: [email protected] Abstract: The Frank-Read model, as a way of generating dislocations in metals and alloys, is widely accepted. In the early 1960s, Li proposed an alternate mechanism. Namely, grain boundary sources for dislocations, with the aim of providing a different model for the Hall-Petch relation without the need of dislocation pile-ups at grain boundaries, or Frank-Read sources inside the grain. This article provides a review of his model, and supporting evidence for grain boundaries or interfacial sources of dislocations, including direct observations using transmission electron microscopy. The Li model has acquired new interest with the recent development of nanomaterial and multilayers. It is now known that nanocrystalline metals/alloys show a behavior different from conventional polycrystalline materials. The role of grain boundary sources in nanomaterials is reviewed briefly. Keywords: dislocation emission; grain boundaries; nanomaterials; Hall-Petch relation; metals and alloys 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. Citation: Li, J.C.M.; Feng, C.R.; With the use of the electron microscope, the sample presence of dislocations validated Rath, B.B. Emission of Dislocations Taylor’s theory. However, the questions remained on the source of dislocations and the from Grain Boundaries and Its Role in Nanomaterials. Crystals 2021, 11, 41. conditions required for their generation and migration in the crystal. While Frank and https://doi.org/10.3390/cryst11010041 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 Received: 26 October 2020 support for the proposed mechanism. The rare observation of the Frank-Read (FR) source Accepted: 29 December 2020 could not be justified for the large density of dislocations seen in crystals. Published: 31 December 2020 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 Publisher’s Note: MDPI stays neu- boundary ledges of the type shown in Figure1. He assumed that the density of ledges tral with regard to jurisdictional clai- is about the same in the grain boundary, which implies that fine-grained materials have ms in published maps and institutio- a greater dislocation density when they yield. Since then, there have been numerous nal affiliations. 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). Copyright: © 2020 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Crystals 2021, 11, 41. https://doi.org/10.3390/cryst11010041 https://www.mdpi.com/journal/crystals Crystals 2021, 11, 41 2 of 12 Crystals 2021, 11, x FOR PEER REVIEW 12 of 12 Figure 1. GrainFigure boundary 1. Grain ledge boundary model. ledge (Schematic model. diagram) (Schematic From diagram) Li [4]. From Li [4]. Hall-Petch (HP)Hall-Petch relation and (HP) the relation pile-up and model the pile-up were originally model were developed originally for developediron for iron and used the conceptand used of thepile-up concept of dislocatio of pile-upns, of originating dislocations, from originating the Frank-Read from the source. Frank-Read source. Pile-up model Pile-uphas been model frequently has been quoted frequently to explain quoted the to HP explain relationship the HP in relationship other materi- in other materials. als. However, However,it is somewhat it is somewhat ironic that ironic the Frank-Read that the Frank-Read source and source dislocation and dislocation pile-ups pile-ups were were rarely, if rarely,ever found if ever in foundiron. Li in proposed iron. Li the proposed grain boundary the grain source boundary model. source He has model. He has considered thatconsidered dislocations that are dislocations generated areat grain generated boundary at grain (GB) boundary ledges and (GB) all ledgesof these and all of these dislocations migratedislocations to dislocation migrate forests to dislocation in grain forests interiors. in grain Li assumed interiors. that Li assumedthe density that the density of ledges is aboutof ledges the same is about in the the grain same boundary, in the grain so boundary,the dislocation so the density dislocation will scale density will scale inversely withinversely grain size. with Dislocation grain size. density Dislocation also scales density as square also scales of yield as square stress. of The yield two stress. The two relations, whenrelations, combined, when give combined, the well-known give the Hall-Petch well-known relation. Hall-Petch In fact, relation. Li compared In fact, Li compared the magnitudethe of magnitudestress in his of model stress with in his th modelat of the with pile-up that ofmodel the pile-up with a modelreasonable with a reasonable agreement. agreement. Ashby [5] describedAshby polycrysta [5] describedls as polycrystals plastically non-homogeneous as plastically non-homogeneous materials in which materials in which each grain deformseach grainby different deforms amounts, by different depe amounts,nding on dependingits orientation on itsand orientation the constraints and the constraints imposed by neighboringimposed by grains. neighboring If each grains.grain were If each forced grain to were undergo forced the to same undergo uniform the same uniform strain, voids wouldstrain, form voids at would grain boundaries. form at grain To boundaries. avoid the creation To avoid of the voids, creation and ensure of voids, and ensure strain compatibility,strain compatibility,dislocations are dislocations introduced. are Voids introduced. can be Voidscorrected can be by corrected the same by pro- the same process cess by dislocationsby dislocations of opposite of signs. opposite Ashby signs. [5] showed Ashby that [5] showedthe number that of the geometrically number of geometrically necessary dislocationsnecessary (GNDs) dislocations generated (GNDs) at the generated boundaries at the and boundaries pumped and into pumped the individ- into the individual ual grains is roughlygrains isDe/4b, roughly where De/4b, D is wherethe grain D is diameter, the grain e diameter, is the average e is the strain, average and strain, b and b the the magnitudemagnitude of the dislocation of the dislocation Burgers vect Burgersor. In vector.both two- In both and two- three-dimensional and three-dimensional ar- arrays of rays of grains,grains, this leads this to leads a density to a density of geometrically of geometrically necessary necessary dislocations. dislocations. These These geo- geometrically metrically necessarynecessary dislocations dislocations can can be bethe the origin origin of ofthe the grain grain size size dependence dependence of of work- work-hardening in hardening in polycrystals.polycrystals. If If it it is is assumed assumed that that hardening hardening depends depends only only on on average average dislo- dislocation density, then this approach yields a σ ~ D−1/2 law for grain boundary hardening, where σ is the cation density, then this approach yields a σ ~ D−1/2 law for grain boundary hardening, yield stress. The concept of generating geometrically necessary dislocations to ensure strain where σ is the yield stress. The concept of generating geometrically necessary dislocations compatibility in polycrystalline materials eliminates the necessity to consider dislocation to ensure strain compatibility in polycrystalline materials eliminates the necessity to con- pile-ups to rationalize the flow of the stress-grain size relationship. It suggests that grain sider dislocation pile-ups to rationalize the flow of the stress-grain size relationship. It interiors behave essentially like single crystals of the same orientation deforming by single suggests that grain interiors behave essentially like single crystals of the same orientation slip while the regions on either side of the grain boundaries undergo lattice rotation and deforming by single slip while the regions on either side of the grain boundaries undergo secondary slip. This, in fact, has been observed experimentally by Essmann, et al. [6]. lattice rotation and secondary slip. This, in fact, has been observed experimentally by Ess- Emission of geometrically necessary dislocations occurs from the grain boundary, which mann, et al. [6]. Emission of geometrically necessary dislocations occurs from the grain must act as a major source of dislocations. boundary, which must act as a major source of dislocations. Crystals 2021, 11, 41 3 of 12 A comprehensive review of Li’s model, its experimental verification, and its compari- son with Frank-Read (FR) source models, especially
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