## Abstract

The review of the theoretical models, which describes mechanisms of deformation twinning in nanocrystalline and ultrafine-grained materials, is presented. Realization of special mechanisms of nanoscale deformation twin generation at locally distorted grain boundaries (GBs) in nanocrystalline and ultrafine-grained materials is observed. In particular, the micromechanisms of deformation twin formation occur through (1) the consequent emission of partial dislocations from GBs; (2) the cooperative emission of partial dislocations from GBs; and (3) the generation of multiplane nanoscale shear at GBs. The energy and stress characteristics of the deformation nanotwin generation at GBs in nanocrystalline and ultrafine-grained materials are calculated and analyzed. Competition between the twin generation mechanisms in nanocrystalline and ultrafine-grained materials is discussed.

### Keywords

- nanocrystalline and ultrafine-grained materials
- nanotwins
- plastic deformation
- grain boundaries
- dislocations

## 1. Introduction

At present, the study of the plastic behavior of nanostructured solids is one of the most important and rapidly developing directions in the mechanics of deformed solid and in the physics of condensed state. Nanostructured solids have unique physical, mechanical and chemical properties and are of great interest, both for fundamental and applied research [1]. For example, the strength and hardness of nanostructured materials are several times higher than those of conventional coarse-grained analogues of the same chemical composition. However, most nanocrystalline materials show low tensile ductility, which are highly undesirable for their practical applications. Increasing the ductility and fracture toughness of nanomaterials is a very important task, the solution of which can significantly expand the field of their application. With a large volume fraction occupied by GBs which act as effective obstacles for lattice dislocation slip (the dominant deformation mechanism in conventional coarse-grained polycrystals), the conventional lattice dislocation slip is hampered in nanostructured materials. At the same time, the specific features of the structure of nanocrystalline materials provide the action of specific deformation mechanisms, and the effect of which in coarse-grained materials was not observed or was insignificant. Identification of these specific mechanisms of plastic deformation is a key problem for understanding the nature of ductility and fracture toughness of nanostructured solids. According to modern concepts of plastic flow processes, the following specific mechanisms of plastic deformation act in nanocrystalline and ultrafine-grained materials: GB sliding [2, 3], rotational deformation mode [4, 5, 6], GBs migration [7, 8, 9] and deformation twinning [10, 11, 12, 13]. The analysis of experimental investigations of deformation mechanisms allows us to formulate the main difference between nanocrystalline materials exhibiting low and high ductility. The point is that each nanocrystalline sample consists of a number of structural elements: grains of different sizes, GBs of various types and misorientations. In this case, several mechanisms of plastic deformation can act simultaneously in a nanocrystalline sample under mechanical loading. In general, different mechanisms of plastic deformation dominate in neighboring grains of different sizes and adjacent GBs. In nanocrystalline materials with low ductility, different mechanisms of deformation act independently of each other, which lead to a substantial inhomogeneity of plastic deformation and can cause the nucleation and evolution of nanocracks. At the same time, in nanocrystalline materials exhibiting high ductility, different mechanisms of plastic deformation effectively interact with each other. Intensive crossovers occur between different deformation mechanisms which accommodate the inhomogeneities of plastic deformation. One of the main specific deformation modes which contribute greatly to plastic flow in nanocrystalline and ultrafine-grained materials is considered deformation twinning mechanism. Following numerous experimental data, computer simulations and theoretical models [10, 11, 12, 13, 14, 15, 16, 17], nanoscale twin deformation effectively operates in nanomaterials with various chemical compositions and structures. In doing so, in contrast to coarse-grained polycrystals where deformation twins are typically generated within grain interiors, in nanomaterials under mechanical load, twins are often generated at GBs; see [12] and references therein. In order to explain this experimentally documented fact indicative of specific deformation behavior of nanomaterials, it was suggested that nanoscale deformation twinning occurs through consequent emission of partial dislocations from GBs [10, 11, 12, 13]. However, in this situation, partial dislocations should exist on every slip plane or be transformed from pre-existent GB dislocations which is hardly possible in real materials [13]. In order to avoid the discussed discrepancy, Zhu and coworkers [13] suggested new micromechanism of partial dislocation multiplication which realized due to successive processes of dislocation reactions and cross-slips providing existence of the partial dislocations at a GB on every slip plane. Thus, further consequent emission of such partial dislocations from GB can provide a nanoscale formation at GB [13]. At the same time, this approach operates with dislocation reactions each transforming a partial dislocation into two dislocations: a full dislocation and another partial dislocation. Such reactions are specified by very large energy barriers (being around the energy of a full dislocation), and thereby they are hardly typical in real materials. In order to respond to these questions, in theoretical works [14, 15], alternative mechanism of nanoscale twin formation at locally distorted GBs in deformed nanomaterials was suggested. According to results of the theoretical works [14, 15], GB dislocation can exist at locally distorted GB on every slip plane due to preceding plastic deformation and thereby cause nanoscale twin formation at GB. Taking this approach into account [14, 15], micromechanisms of deformation nanotwin formation can occur through (1) the consequent emission of partial dislocation from locally distorted GBs; (2) the cooperative emission of partial dislocations from locally distorted GBs; and (3) the generation of multiplane nanoscale shear at locally distorted GBs. Realization of these mechanisms is discussed in the next sections.

## 2. Mechanisms of deformation twin generation at locally distorted grain boundaries in nanocrystalline and ultrafine-grained materials

In this chapter, theoretical description of deformation twin generation mechanisms is based on results of the following theoretical papers [14, 15, 16, 17]. According to these papers [14, 15, 16, 17], generation of nanotwins occurs at locally distorted GB segments (GB segments being rich in GB dislocations) which were produced due to either events of consequent trapping of extrinsic lattice dislocations by GB and their splitting transformations into a wall of climbing GB dislocations (Figure 1a–d) or GB deformation processes involving slip and climb of GB dislocations (Figure 1e–h). The splitting of extrinsic dislocations at high-angle GBs is a well experimentally documented process [18] resulting at its initial stage in the formation of several closely located GB dislocations (Figure 1a). These processes allow GB dislocations to exist on almost every slip plane and thereby form a nanowall of GB dislocations (Figure 1d). In this situation, under action of external shear stress, a head dislocation of a pile-up is trapped by GB and splits into the GB dislocations (Figure 1a and b). After this process, the second lattice dislocation of the pile-up moves to and is trapped by GB where this dislocation splits into new GB dislocations (Figure 1c). In this case, after the splitting of the head dislocation of the pile-up, its second dislocation can reach the GB where this extrinsic dislocation splits into new GB dislocations (Figure 1b and c). Thus, consequent events of the splitting transformations of the head lattice dislocations forming pile-up into GB dislocations and climbing of these GB dislocations along GB can form a nanowall of GB dislocations located on almost every slip plane (Figure 1d). Both the transformation of GB dislocation into partial dislocations and emission of partial dislocation into grain interior are capable of producing a deformation nanotwin (for details, see below).

As follows from works [14, 15], formation of local distorted segments of GBs can be associated with GB plastic deformation processes. First, a nanostructured specimen is deformed by GB sliding that produces pile-ups of GB dislocations stopped by triple junctions of GBs (Figure 1e). Under the action of the external shear stress, the head GB dislocations of the pile-up split at triple junction and climb along GB (Figure 1f–h). As a result, a wall of climbing GB dislocations located on every (or almost every) slip plane is formed (Figure 1h). In general, local GB fragments being rich in GB dislocations can be formed at GBs “globally” distorted by plastic deformation. Such GBs are typical structural elements of bulk nanostructured materials fabricated by severe plastic deformation methods, and they can contain nanoscale fragments with GB dislocations located on every slip plane.

Thus, micromechanisms of nanotwin formation at locally distorted GB segments represent: (1) the consequent emission of partial dislocations from GBs; (2) the cooperative emission of partial dislocations from GBs; and (3) the generation of multiplane nanoscale shear at GBs. The former two micromechanisms of nanoscale twins generation occur through splitting of the GB dislocations into immobile GB dislocations and mobile partial dislocations (Figures 2 and 3). Consequent (Figure 2) or cooperative (Figure 3) gliding of the mobile dislocations along neighboring slip planes in a grain interior results in formation of a nanotwin.

Note that an energy barrier specifying the transformation of a GB dislocation at a local distorted GB segment into another GB dislocation and a partial dislocation (Figures 2 and 3) is around the energy of a partial dislocation. Thus, this barrier is lower than the barrier required for multiplication of partial dislocations (being around the energy of a full dislocation) considered by Zhu and coworkers [13]. In these circumstances, the splitting transformation (Figures 2 and 3) is more energetically favored as compared to the multiplication reaction.

The third mechanism for nanotwin formation at a locally distorted GB is multiplane nanoscale shear (Figure 4) firstly defined in Letter [19]. Following [19], a multiplane nanoscale shear is an ideal (rigid body) shear occurring simultaneously along several neighboring crystallographic planes within a nanoscale region—a three-dimensional region having two or three nanoscopic sizes—in a crystalline solid (this notion is based on that of multiplane ideal shear in infinite crystals [20]). The multiplane shear is characterized by the shear magnitude *s* (which is identical at any time moment, for all the planes where the shear occurs) gradually growing from 0 to the partial dislocation and geometric sizes of the nanoscale region where the shear occurs. For certain value of

### 2.1. Nanotwin formation due to consequent emission of partial dislocations

Figure 2 illustrates geometric features of nanotwin formation at locally nonequilibrium GBs in nanomaterials in the situation where local GB fragments with extra GB dislocations are formed due to both GB sliding and stress-driven climb of GB dislocations. As a result, a nanoscale wall configuration *AB* of GB dislocations is formed at GB *AA’* (Figure 2a). In the framework of the model, the GB dislocations forming the wall of climbing dislocations transform into immobile GB dislocations (staying at the GB *AA’*) and mobile partial dislocation which are emitted from the GB *AA’* and can move along neighboring slip planes {111} in an adjacent grain (Figure 2b–d). In terms of the continuum approach, the emission of partial dislocations can be represented as formation of dislocation dipoles with Burgers vectors

The angle *AA’* plane (Figure 2a). The magnitude of Burgers vectors of partial dislocations is equal to *i*th dislocation moves in the grain interior (Figure 2b–d), a stacking fault of the length *τ*. The first partial dislocation is emitted from the triple junction *A* (Figure 2b) and moves across the grain interior toward the opposite GB (Figure 2b) when the shear stress reaches its critical value of *τ*, reaches the opposite GB or stops in the grain interior moving over some distance

Also, the discussed trends come into play during emission of other partial dislocations due to the effects of previously emitted dislocations. That is, the critical stress for emission of the

To analyze the suggested model, we consider the energy characteristics of the nanoscale twin generation due to consequent emission of partial dislocations from locally nonequilibrium GBs (Figure 2). First, define the conditions which are necessary for the energetically favorable emission of the first partial dislocation, which can be represented as formation of a dipole *AD* of partial dislocations with Burgers vectors

where *AB* of GB dislocations; *OA* of GB dislocations; *AD.*

Detailed description of all the terms figuring on the right-hand side of Eq. (1) is given in the theoretical paper [14, 15]. Using Eq. (1), we calculated the dependences of the energy change *G* = 73 GPa, ^{2} [21] and Cu: *G* = 44 GPa, ^{2} [21]. The dependences

Now let us consider the energy characteristics of emission of the *n*th partial dislocation, for *n* and

where *AB* of GB dislocations; *AB* of GB dislocations; *OA* with (*n* − 1 and n dislocation dipoles, respectively; *d* increases and/or the nanotwin thickness *h* decreases. For instance, for *d* = 50 nm, the generation of a nanotwin having the thickness *h* = 3 nm occurs in Cu at the critical shear stress

### 2.2. Nanotwin formation due to cooperative emission of partial dislocations

The second micromechanism of nanotwin formation is realized through cooperative emission of partial dislocations from locally distorted GBs in deformed nanomaterials. As in the previous case, the initial defect configuration represents a nanoscale wall of GB dislocations *AB* located on every (or almost every) slip plane (Figure 3a). In this situation, the GB dislocations cooperatively emit from GB and move together along neighboring slip plane forming a nanotwin (Figure 3b and c). This mechanism in the situation where the GB dislocations are located on every slip plane has been considered in theoretical papers [14, 15]. As a result, the nanotwin *ABCD* crosses the grain and joins two opposite GBs (Figure 3c). Such nanotwins have been experimentally observed in nanocrystalline nickel (Ni) [11] (Figure 7).

However, cooperative emission of partial dislocations from GBs also can occur in situation where the GB dislocations in the initial wall configuration *AB* are located on not all of slip planes. In this case, there are some gaps in arrangement of the GB *AB.* Transformations of the GB *AB* (Figure 3). Nevertheless, a nanotwin can be generated through cooperative emission of partial dislocations from such a GB fragment, if partial dislocations are generated on slip planes where the initial GB dislocations are absent.

Analyze the energy characteristics of the twin formation due to cooperative emission of dislocations from GB in nanomaterials (Figure 3). The cooperative emission process (Figure 3) is characterized by the energy difference

where *AB* of GB dislocations; *OA* with

Calculation of all the terms figuring on the right-hand side of Eq. (3) is given in the theoretical paper [14, 15]. With the help of Eq. (3) for the energy change

Note that, if the inequalities *AB* of *AA′* and move across the grain over the distance

In another case, the inequalities *AB* of the nanotwin *ABCD* (Figure 3b).

Dependences of the critical shear stress *OA* (see Figure 3).

### 2.3. Nanotwin formation due to generation of nanoscale multiplane shear

The third mechanism of nanotwin formation is realized through the generation of nanoscale multiplane shear at locally distorted GBs in deformed nanomaterials (Figure 4). As it has been noted previously, nanoscale multiplane shear is defined in work [19] as a multiplane ideal shear occurring within a nanoscale region, a three-dimensional region having two or three nanoscopic sizes. For instance, nanoscale multiplane shear can occur and produce a twin within a nanoscale internal region of a grain of a deformed nanomaterial (Figure 4). More precisely, in the model situation, a deformation twin is produced under the action of a shear stress *AB* and *CD*, on adjacent {111} planes. The Burgers vectors *AB*, the noncrystallographic dislocations at the GB fragment *AB* merge with these preexistent GB dislocations. As a corollary, during the nanotwin formation process, evolution of the noncrystallographic dislocations at the GB fragment *AB* manifests itself in evolution of the GB dislocations at this fragment (Figure 4b–d).

In the case of fcc metals (in particular, copper (Cu)), the generated dipoles of noncrystallographic partial dislocations are formed in adjacent slip planes {111} assumed to be normal to the grain boundary fragments *AB* and *CD*. The region *ABCD* (a rectangle with sizes *AB* length = *CD* length = *AD* length = *BC* length =

Analyze energetic characteristics of the generation of nanoscale twins through nanoscale multiplane shear initiated at locally distorted GBs in nanocrystalline materials (Figure 4). In this case, in terms of the dislocation theory, a deformation twin is produced under the action of a shear stress

In general, the energy change

Here *AB* of GB dislocations with *ABCD*; *AD* and *BC*, between the sheared region *ABCD* and the neighboring material; and *ABCD*.

Calculations of all the terms figuring on the right-hand side of Eq. (4) are given in the theoretical paper [14, 15]. Based on these calculations in the exemplary cases of nanocrystalline Cu, we revealed dependences of the energy change

## 3. Comparison of critical shear stress for nanotwin generation at grain boundaries due to various deformation mechanisms

In this section, we compare critical shear stresses for the considered mechanisms of nanotwin generation at locally distorted GBs in deformed nanomaterials. The dependences of the critical shear stresses

According to results presented in Figure 10, the consequent emission of partial dislocations from locally distorted GBs is not favored. However, the critical stresses at which nanotwin generation mechanisms occur are highly sensitive to material parameters and the initial state of a locally distorted GB. Therefore, in the situations with other initial states and/or other materials, the consequent emission of partial dislocations from locally distorted GBs may be favored.

## 4. Conclusions

Thus, new specific micromechanisms of nanotwin generation at GBs in deformed nanocrystalline and ultrafine-grained materials were developed and analyzed. These micromechanisms describe the formation of deformation nanotwins at locally distorted GBs that contain segments being rich in GB dislocations produced by preceding plastic deformation. The micromechanisms of deformation twin formation occur through (1) the consequent emission of partial dislocations from GBs (Figure 2); (2) the cooperative emission of partial dislocations from GBs (Figure 3); and (3) the generation of multiplane nanoscale shear at GBs (Figure 4). It is found that the deformation twinning mechanisms (Figures 2–4) can operate in nanocrystalline and ultrafine-grained materials at rather high, but realistic levels of the stress (Figures 6 and 8–10). The suggested representations on generation of nanotwins at locally distorted GBs (see also [14, 15]) logically explain numerous experimental observations [11, 13] of generation of nanoscale twins at GBs in nanocrystalline and ultrafine-grained materials. These deformation twinning mechanisms illustrate complicated interactions between different deformation modes such as deformation twinning, GB sliding, and GB dislocation climb.

## Acknowledgments

This work were supported by the Russian Fund of Basic Research (grant 16-32-60110) and Russian Ministry of Education and Science (task 16.3483.2017/PCh).