Interface-dependent nanoscale friction of copper bicrystals: tilt versus twist

Abstract

Interfaces with different structural units have a strong impact on their microscopic deformation behavior and correlated macroscopic mechanical response. In current study, we elucidate the underlying nanoscale friction mechanisms of Cu bicrystals by means of molecular dynamics simulations. Four grain boundaries, i.e., pure tilt and twist with two misorientation angles of 7.63° and 67.38°, are considered to address the grain boundary structure dependence of the friction. While the small- and high-angle tilt grain boundaries are respectively composed of parallel edge dislocation dipoles and edge dislocations of like sign, the small- and high-angle twist ones respectively incorporate two sets of intersecting screw dislocations and a planar defect. Simulation results demonstrate that the grain boundary resistance to dislocation motion and absorption, as well as the grain boundary evolution, are significantly varied with grain boundary structural units. It is found that splitting, annihilation and generation of grain boundary dislocations are the three competing decomposition mechanisms of the well-defined grain boundaries. The anisotropic dislocation-grain boundary interactions in turn results in a strong grain boundary structure dependence of the frictional response for scratching in the vicinity of grain boundaries. These findings will not only advance our understanding of the interface-dependent nanoscale friction behavior of metals, but also provide rational design guidelines for the synthesis of advanced functional nanostructured materials with unique internal interface textures.

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Introduction

An interface is a common boundary separating two different phases and/or crystals of different chemical compositions and/or crystallographic misorientations. Interfaces have a strong impact on the chemical, physical and mechanical properties of materials particularly with high surface to volume ratio, due to the chemical and/or crystallographic discontinuity.^1,2 For instance, nanocrystalline metals with average grain sizes down into the nanometer regime present varied mechanical properties because of the presence of single-phase interface of grain boundary (GB): the strength of materials increases with decreasing grain size on the basis of Hall–Petch mechanisms and reaches its maximum at a grain size of 10–20 nm, below which the strength decreases with a further grain refinement following the inverse Hall–Petch effect, accompanied by the transition of dominated plastic deformation mechanism from dislocation slip to GB sliding.^3,4 Furthermore, the impact of GBs is heavily influenced by the structural type of GBs. While the conventional GB strengthening compromises the ductility of nanocrystalline copper, the introduction of aligned coherent twin boundaries (TBs) leads to a superior combination of ultra-high strength and intermediate ductility.^5,6 Therefore, a rational design and synthesis of nanocrystalline materials with advanced functional properties can be achieved by engineering their interface textures.

The intricate interplay between dislocation and GB is central for understanding the unique mechanical behavior of polycrystalline materials with microstructural feature sizes down to the nanometer scale. The dislocation–GB interactions, such as dislocation repulsion, absorption, transmission and remission, are strongly dependent on the structural characteristics of GBs. The structural units composed of GBs can be categorized by the axis of rotation and the angle of misorientation between neighboring grains: the rotation axis perpendicular and parallel to the GB normal respectively leads to tilt and twist GBs; and the misorientation angle less and larger than 15° respectively defines low and high angle GBs.⁷ The GB structure dependence of dislocation–GB interactions has been examined experimentally and theoretically. Chandra et al. found that the resistance of twist GB to edge dislocation absorption in Al is strongly influenced by misorientation angle.⁸ Liu et al. reported the misorientation angle dependence of dislocation–GB interactions and GB evolution in copper nanowire with twist GBs.⁹ Capolungo et al. showed that the interfacial dislocation emission from copper GB ledge requires smaller critical stress than that from planar GB.¹⁰ Schonfelder et al. demonstrated that the mobility of high and low angle twist GBs in copper are dominated by a collective shuffle mechanism and a dislocation based mechanism, respectively.¹¹ Barrales-Mora et al. found that the mobility of low angle GBs decreases with decreasing misorientation. Furthermore, grain rotation plays a crucial role in the capillarity-driven shrinkage of the isolated cylindrical grain with pure tilt GBs, while dislocation annihilation is the dominant mechanism of grain shrinkage for mixed tilt-twist GBs.^12,13 More recently, Zhang et al. reported that GB inclination angle and GB character have significant influence on the interfacial dislocation emission at the onset of plasticity in Cu bicrystals under both shear and tensile loadings.^14,15

Despite the insights provided by previous studies with uniaxial stress state, our understanding of the role of dislocation–GB interactions played in the nanoscale friction with localized multi-axis stress state is still limited. Nanoscale friction and wear, as common reasons for energy and material loss, are strongly inter- and intra-interfaces dependent.^16–19 Given the comparable size of friction asperity with average grain size, a typical friction route generally involves multiple grain boundaries composed of different structural units. In addition to GB structures, it is demonstrated that stress states also have an important influence on dislocation–GB interactions.^20,21 Although some work on the friction of nanocrystalline metals had been reported recently,^22–27 the influence of GB structure on dislocation–GB interactions and their correlations with frictional response are not well addressed. Therefore, to advance our understanding of the atomic nature of frictional behavior of nanostructured materials, in the present work we perform molecular dynamics (MD) simulations to explore the GB structure dependent friction of face-centered cubic (FCC) copper bicrystals with 〈001〉 symmetric GBs. To characterize the effect of GB structure, we consider pure tilt and twist GBs with two misorientation angles of 7.63° and 67.38°. The microscopic deformation mechanisms, in particular the dislocation–GB interactions, are analyzed in details with advanced analysis techniques for lattice defects, and are further correlated with the macroscopically observed frictional response. Based on such investigations, we demonstrate that the friction of copper bicrystal has a strong GB structure dependence.

Simulation method

Friction procedure

Fig. 1(a) illustrates the MD model of friction, which is composed of a copper bicrystal substrate and a spherical probe. The substrate has a dimension of 40 nm, 10 nm and 30 nm in X, Y and Z direction, respectively. Periodic boundary condition is only imposed in Z direction, and the bottom of the substrate is fixed in space to restrict its rigid motion during the friction. The atomic interactions in the Cu substrate are described by an Embedded-Atom Method (EAM) potential developed for Cu.²⁸ The atomic interaction between the substrates and the spherical frictionless probe with a radius of 6 nm is modeled by a strong repulsive potential.²⁹

Fig. 1 MD model of friction of bicrystal copper. (a) Schematic illustration of friction configuration; (b)–(d) show dislocation structures in GB planes of 7.63° tilt, 7.63° twist and 67.38° tilt GBs, respectively. Dislocation segments are represented by tubes: blue and green ones stand for 1/2〈110〉 full and 1/6〈112〉 partial dislocations, respectively.

Prior to the friction, the as-created Cu bicrystals are relaxed to their equilibrium configurations by following procedures: the atoms in the substrate first undergo conjugate gradient energy minimization at 0 K, and then are heated to 30 K under 0 bar for 50 ps under the isothermal–isobaric NPT ensemble. After relaxation, the equilibrium substrates are subjected to the friction in the microcanonical NVE ensemble. Fig. 1(a) indicates that the friction process is composed of two sequential stages, as first penetration and following scratching, respectively. In the penetration stage, the probe moves along negative Y direction with a constant velocity of 20 m s⁻¹ to penetrate into the surface of the left grain until a pre-determined depth of 1 nm is reached. The penetration position has a distance of 11 nm from the centered GB. Then the probe scratches a distance of 22 nm along X direction with a constant velocity of 20 m s⁻¹ in the scratching stage, in which the penetration depth is kept unchanged. It should be noted that both the penetration and scratching velocities of 20 m s⁻¹ are seven orders of magnitude higher than typical velocities utilized in nanoscratching experiments. Determined by the inherent characteristic of atomic bond vibrations, in MD simulation there is an intrinsic requirement of the integration time step to be of the order of fs, which results in significant discrepancies of time-scale between MD simulations and experiments. Therefore, to make the computational time reasonable, in current study both the penetration and scratching velocities are set as 20 m s⁻¹. All the MD simulations are performed using the LAMMPS with an integration timestep of 1 fs.³⁰

Grain boundary structure

By symmetrically rotating the neighboring two grains for an equal and opposite angle along 〈100〉 axis, a 〈100〉 symmetrical GB perpendicular to X direction is introduced in the center of the substrate. We consider two kinds of GB structure, as pure tilt and twist, respectively. Furthermore, for each kind of GB structure, two misorientation angles of 7.63° and 67.38° are considered. ESI Fig. 1† present atomic structures of the four GBs. Furthermore, Table 1 lists detailed parameters for the four GB structures. The microstructural characteristics of well-relaxed GB structures are analyzed by using the Dislocation Extraction Algorithm (DXA), which provides detailed information of dislocation type and Burgers vectors.³¹ Fig. 1(b)–(d) present the GB dislocations for 7.63° tilt GB, 7.63° twist GB and 67.38° tilt GB after the equilibration, respectively.

Table 1 Parameters for the four GBs

Boundary type	(h k l) GB plane	GB energy (mJ m^−2)	Mechanical properties
			Young's modulus (GPa)	Yield strength (GPa)
7.63° tilt	∑113 (15 1 0)	358	117.4	1.27
67.38° tilt	∑13 (3 2 0)	468	174.0	1.21
7.63° twist	∑113 (1 0 0)	219	140.5	1.34
67.38° twist	∑13 (1 0 0)	357	172.5	1.21

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It is seen from Fig. 1(b) and (d) that each tilt GB entirely consists of 1/2 〈110〉 edge dislocations. Fig. 1(d) shows that the 67.38° tilt GB is composed of an array of parallel edge dislocations with the same sign. The spacing D between dislocations is 0.65 nm, indicating the dislocation model of GB structure, i.e., sin(θ/2) = |b|/2D, where θ is the misorientation angle between two adjoining grains and |b| is the magnitude of the Burgers vector of the 1/2〈110〉 discrete dislocations, can be applied to characterize the GB characteristics.⁷ However, the dislocation model fails to address the 7.63° tilt GB that is composed of bundles of dislocation dipoles, i.e., two antiparallel edge dislocations with the same Burgers vectors, as shown in Fig. 1(b). The height of each dislocation dipole is 0.92 nm and the spacing between dislocation dipoles is 2.72 nm.

In contrast to tilt GBs composed of parallel edge dislocations, Fig. 1(c) shows that the bulk part of 7.63° twist GB mainly consists of two sets of orthogonally intersected 1/2〈110〉 screw dislocations. The spacing between parallel dislocation segments is approximately 2 nm. In addition, the dominant GB dislocations in the vicinity of upper free surface and bottom fixed surface are 1/6〈112〉 partials, which is resulted from the splitting of 1/2〈110〉 dislocations. The uncompleted splitting processes of GB dislocations are also observed in the bulk part beneath the free surface. For the 67.38° twist GB, however, there is no dislocation observed in the GB because of the interfacial defect formed.

Defect analysis

Unambiguously identifying the type of defects is crucial for elucidating microscopic deformation mechanisms of materials at the nanoscale. In this work a modified version of the bond angle distribution (BAD) method is utilized for the defect analysis, which provides a further distinguishing of the dislocation cores from the free surface atoms by considering the number of bonds between the nearest neighbor pairs.^32,33 A single hexagonal-close-packed (HCP)-coordinated layer identifies a coherent TB, two adjacent HCP-coordinated layers indicate an intrinsic stacking fault (ISF), and two HCP-coordinated layers with an FCC-coordinated layer between them represent an extrinsic stacking fault (ESF). For a clear visualization of crystal defects, atoms sitting on perfect FCC lattice sites are omitted. The coloring scheme is as follows: HCP atoms are colored in red, atoms at the surface are colored in blue, and the remaining atoms are categorized as defects which include dislocation cores and vacancies. To further explore the dislocation–GB interaction, dislocations are quantitatively characterized by the DXA. The AtomViewer^32,33 and Ovito³⁴ are jointly employed for visualization and post-processing of MD data, as well as generating MD snapshots.

Results and discussion

Friction mechanisms of 7.63° tilt GB

The friction of the substrate with 7.63° tilt GB is first studied to obtain a fundamental understanding of nanoscale friction of bicrystal Cu. There are three force components acting on the probe, herein defined as scratching force along X direction, penetration force along negative Y direction and lateral force along lateral Z direction, respectively. While the penetration is performed in the grain interior, the penetration force-penetration depth curve shown in Fig. 2 demonstrates that the penetration stage is composed of two phases that are categorized according to different deformation behavior of the material. The two subfigures in Fig. 2 respectively present defect configurations before and after the penetration. In the first phase the material undergoes pure elastic deformation, and the penetration force increases rapidly. The plasticity is initiated by dislocations gliding in {111}〈110〉 slip system after the shear stress acting on the 〈111〉 slip plane exceeds the critical resolved shear stress. The subsequent plastic deformation in the second phase is exclusively governed by dislocation slip. Due to a large proximity of 11 nm between the penetration position and the centered GB, as well as a shallow penetration depth of 1 nm, no intersection of dislocations with the centered GB is observed during the penetration stage. The fundamentals of spherical penetration of single crystalline metallic materials are also well documented elsewhere.^29,35–37

Fig. 2 Penetration force-penetration depth curve during the penetration of Cu substrate with 7.63° tilt GB. The subfigures present defect configurations, in which atoms are colored according to their BAD values, and perfect FCC atoms are omitted.

The friction coefficient, defined as the ratio of scratching force to penetration force in this work, is employed to quantitatively access the frictional properties of the bicrystal Cu. Fig. 3 plots the friction coefficient-scratching length curve during the scratching of the Cu substrate with 7.63° tilt GB, and the position of the centered GB is highlighted by a dashed line. It is seen from Fig. 3 that the frictional response can be divided into six sequential phases, as highlighted by dotted lines. In the phase I the friction coefficient increases monotonously until reaching a local maximum value of 0.180, after which it mainly decreases to a global minimum value of 0.085 in the phase II. In the phase III the friction coefficient keeps increasing with decreasing distance between the probe and the centered GB, and reaches its global maximum value of 0.211 when the probe is right on the centered GB. After the probe scratching across the centered GB, the friction coefficient decreases rapidly in the phase IV until reaching a local minimum value of 0.15. Upon further advancement of the probe, in the phase V the friction coefficient first increases slightly, followed by a further decrease to another local minimum value of 0.120. However, when the probe is approaching the GB residing in the right side of the substrate, the friction coefficient mainly increases in the final phase VI. The variation amplitude of the friction coefficient, defined as the differential value between the global maximum and minimum values during the scratching stage, is 0.126 for the Cu substrate with 7.63° tilt GB.

Fig. 3 Friction coefficient-scratching length curve during the scratching of Cu substrate with 7.63° tilt GB. The position of centered GB is highlighted by a dashed line, and the different phases of the scratching are highlighted by dotted lines.

In addition to the frictional response of the Cu substrate with 7.63° tilt GB plotted in Fig. 3, the deformation behavior of the material is also studied by performing a dynamic analysis of defect evolution during the scratching. Fig. 4 presents representative defect configurations at different scratching lengths. In addition, Fig. 5 plots variations of normalized number of atoms of different lattice structures with scratching length, and different phases of the friction coefficient are also highlighted by dotted lines. The normalized number for each atom type is defined as the ratio of the change in number to the initial number.

Fig. 4 Representative defect configurations at a scratching length of (a) 1.6 nm, (b) 6.4 nm, (c) 10.0 nm, (d) 13.2 nm, (e) 18.4 nm and (f) 21.6 nm during the scratching of Cu substrate with 7.63° tilt GB. Atoms are colored according to their BAD values, and perfect FCC atoms are omitted.

Fig. 5 Variations of normalized number of (a) FCC, (b) HCP, (c) other and (d) ISF atoms during the scratching of Cu substrate with 7.63° tilt GB.

It is seen from Fig. 5 that in the phase I the number of FCC atoms mainly decreases accompanied with increases of both HCP and Other atoms, indicating nucleation and subsequent slide of dislocations in front of the probe dominate the plastic deformation. Accompanied with increased dislocation density due to nucleation and multiplication, the resultant highly overlapped strain fields between close dislocations increase the resistance to the further motion of each one. Furthermore, Fig. 4(b) shows that the propagation of dislocations is also impeded by the centered GB. Consequently, the resultant strengthening of the material leads to a rapid increase of the friction coefficient. However, reactions of dislocation annihilation beyond the probe lead a significant rate of recovery, which corresponds to strong fluctuations in the normalized numbers of atoms. In the phase II, while dislocation pile-up is built in the vicinity of the centered GB, successive dislocations nucleate from the scratched surface to accommodate the plastic deformation, which leads to a decrease of the friction coefficient. In particular at a scratching length of 5.7 nm, a large amount of accumulated dislocations annihilate, as evidenced by the increase of FCC atoms and simultaneous decreases of HCP, Other and ISF atoms, which further lowers the frictional resistance. When the probe is approaching the centered GB in the phase III, the adsorption of dislocation in the left grain and dislocation slip in the right grain have a strong competition. In the initial period of the phase III, while the increase of FCC atoms is accompanied with decreases of both Other and ISF atoms, the increase of HCP atoms indicates the formation of mechanical TBs through deformation twinning is also one important deformation mode. Due to the high strain rate, low stacking fault energy and low temperature, friction-induced deformation twinning in nanocrystalline Cu is reported theoretically and experimentally.^23,38 The formation of mechanical twins through the dissociation of partial dislocations provides increased number of active slip systems.^39,40 After the probe scratches across the centered GB, dislocation transition through GB dominates the plastic deformation in the phase IV. Fig. 4(e) shows that in the phase V the plastic deformation is dominated by nucleation and subsequent glide of dislocations in the right grain interior, which leads to a decreased friction coefficient. In the phase VI, the interaction of propagating dislocations with the GB in the right side leads to strain hardening of the material, which consequently results in a further increase of the frictional resistance. The description of frictional response of bicrystal materials can also be found elsewhere.^25,41,42

Fig. 4 and 5 jointly demonstrate that dislocation–GB interactions have strong impacts on deformation behavior and frictional response of the material. Fig. 6 further presents typical dislocation–GB interactions. Since the height of the dislocation dipole is small, i.e. less than 4b (b is the magnitude of Burgers vector of 1/2〈110〉 dislocation for copper, 0.2555 nm), the high binding energy yields a higher stability of the dipole than that of two independent dislocations.⁴³ Consequently, the GB impedes the motion of leading dislocations, against which trailing dislocations pile up. Simultaneously, Fig. 6(a) shows that the intersection of dislocation dipoles with cutting dislocations leads to the formation of tripole and quadrupole with lower stress field than that of randomly distributed dislocations, which capture further impinging dislocations.^44,45 In addition, decomposition of dislocation dipoles is also one operating mode of dislocation–GB interactions, as shown in Fig. 7 at a scratching length of 6.0 nm. While the center segments of the first dislocation dipole are joined by junctions, the two ends are transformed into 1/6〈112〉 partials and 〈100〉 dislocations. The formation of 1/6〈112〉 partials is accompanied by the dissociation of 1/2〈110〉 GB dislocations due to the cutting of impinging 1/6〈112〉 partials, which can be described as, for instance:

1/2[−1 0 −1] − 1/6[−1 1 −2] = 1/6[−2 −1 −1]

Fig. 6 Dislocation–GB interactions during the scratching of Cu substrate with 7.63° tilt GB. Scratching length: (a) 4.8 nm, (b) 8.8 nm, (c) 14.8 nm and (d) 22.0 nm.

Fig. 7 Scratching-induced decomposition of dislocation dipoles in 7.63° tilt GB.

Due to the opposite sign of GB dislocations, above dissociation of the dislocation dipole leads to formation of partial dislocation junctions. With the increasing overlap between strain fields of intersecting dislocations, the two GB dislocations approach and annihilate each other by transforming into a lower energy configuration of 〈100〉 dislocation,⁴⁶ which can be described as:

1/2[1 0 −1] − 1/2[−1 0 −1] = [100]

With a further scratching, the decomposition of the second dislocation dipole also commences, as shown in Fig. 6(b). The short dipole debris formed in the decomposition inhibits the motion of dislocations. Accordingly, a significant strain hardening of the material occurs, which leads to an increase of the frictional resistance. However, the dipole debris will preferentially annihilate to form the 〈100〉 dislocation, which subsequently leads to the GB failure shown in Fig. 6(c). Consequently, dislocations in the left grain transmit across the broken GB to glide in the right grain, which decreases the frictional resistance due to the softening of the material.

Influence of GB structure

With the fundamental understanding of the friction mechanisms of the Cu substrate with 7.63° tilt GB, friction simulations for the other three Cu substrates are also performed to investigate the influence of GB structure. Fig. 8(a) plots variations of penetration force with penetration depth during the penetration stage for the four GBs, demonstrating significant differences in both the elastic and plastic responses between Cu substrates with different mechanical properties. Table 1 lists mechanical properties of four Cu substrates in terms of Young's modulus and Yield strength that are derived according to the Hertzian contact theory.⁴⁷ However, the GB structure has a trivial influence on deformation mechanisms of the four Cu substrates during the penetration stage, primarily due to the large proximity between the penetrated position and the centered GB, as well as the shallow penetration depth. Specifically, the mechanism of elastic–plastic transition for each Cu substrate is the same as nucleation and glide of lattice partial dislocations from the penetrated surface, and following plastic deformation is exclusively dominated by dislocation slip. Furthermore, there is no dislocation–GB interaction observed for each Cu substrate.

Fig. 8 Mechanical response in friction of Cu substrates with different GB structures. (a) Penetration force-penetration depth curves during the penetration stage; (b) friction coefficient-scratching length curves during the scratching stage.

Fig. 8(b) plots variations of friction coefficient with scratching length during the scratching stage of the four Cu substrates, and variation magnitudes of friction coefficient are also listed in Table 1. When the scratching is performed in grain interiors, it is seen from Fig. 8 that the frictional response for each GB presents similar features. For scratching in the left grain interior, nucleation and multiplication of dislocations in front of the probe increase the overlap between strain fields of each dislocation with close distance, which results in a strong dislocation repulsion. Consequently, the strain hardening increases the frictional resistance of the material, leading to a rapid increase of the friction coefficient from 0 to a local maximum value. In contrast, when scratching is performed in the right grain interior, the friction coefficient decreases from a local maximum value to a local minimum value due to dislocation avalanche. When scratching is performed in the vicinity of the centered GB, however, both frictional response and deformation behavior of the materials show a strong GB structure dependence due to different dislocation–GB interactions.

For the 7.63° tilt GB that is composed of dislocation dipoles, the major modes of dislocation–GB interactions and their respective correlations with the frictional response can be summarized as: GB blocking motion of impinging dislocations causes strain hardening of the material, thus leading to an increase of the friction coefficient; decomposition of GB dislocation dipoles and consequent dislocation transmission across the GB result in softening of the material, which lowers the frictional resistance.

For the 67.38° tilt GB that is composed of parallel edge dislocations with the same sign, Fig. 9(a) shows that the cutting of impinging 1/6〈112〉 partial dislocations leads to creation of two kinds of local structures in the GB: one is the junctions that connect GB dislocations, the other is the newly formed 1/6〈514〉 GB dislocations that are inclined to initial parallel GB dislocations, which can be described as:

1/2[1 0 1] − 1/6[−2 1 −1] = 1/6[5 −1 4]

Fig. 9 Dislocation–GB interactions during the scratching of Cu substrate with 67.38° tilt GB. Scratching length: (a) 6.0 nm, (b) 11.2 nm, (c) 13.2 nm and (d) 22.0 nm.

In addition, there is a local GB migration caused by the glide of 1/2〈110〉 GB dislocations, accompanied with the formation of 1/6〈514〉 GB dislocations. The newly formed local GB dislocation networks act as stable barriers for further motion of impinging dislocations. Subsequently, a prismatic dislocation loop separated from the defect zone beneath the probe glides towards the bottom to accommodate the plastic strain, as shown in Fig. 9(a). The prismatic dislocation loop is composed of eight 1/6〈112〉 partials with Burgers vectors that do not line in the loop planes and four parallel 1/3〈100〉 dislocations. Since the energy per unit length of the prismatic dislocation loop is much lower than that of a straight dislocation, its formation and subsequent easy glide lead to softening of the material. With a further advancement of the probe, however, Fig. 9(b) shows that the newly formed 1/6〈514〉 GB dislocations act as preferable sites of GB failure, accompanied with dislocation transmission across the GB, which consequently further lower the frictional resistance. When the probe is approaching the centered GB, the plastic deformation is governed by the glide of half-prismatic dislocation loops along the GB, as shown in Fig. 9(c). Since the Burgers vector of the dislocation loops are not in the same plane with the GB dislocations, the glide leads to an increase of the friction coefficient. Fig. 9(b) and (c) also indicate the local GB migration through shuffling. At a scratching distance of 13.2 nm, Fig. 9(c) shows that dislocations in the left grain interior completely disappear due to dislocation annihilation and absorption by the centered GB, leading to a decrease of the friction coefficient. Upon further scratching, the half-prismatic dislocation loops gliding along the GB are also absorbed by the GB, leading to a further chaos of the GB profile.

Therefore, For the 67.38° tilt GB, the major modes of dislocation–GB interactions and their respective correlations with the frictional response can be summarized as: GB blocking motion of impinging dislocations and the pinning of the glide of half-prismatic dislocation loops along the GB lead to strain hardening of the material, thus leading to increase of the friction coefficient; dislocation transmission accompanied by GB failure and dislocation absorption by the GB result in softening of the material, which lower the frictional resistance.

For the 7.63° twist GB that is composed of intersecting 1/2〈110〉 screw dislocations, the dislocation–GB interactions are significantly different from that for the tilt GBs. Prior to the reach of leading dislocations, part of 1/2〈110〉 GB screw dislocations split into 1/6〈112〉 partial dislocations that are separated by ribbons of stacking fault due to the increased overlap between strain fields of dislocations, which can be described as:

1/2[0 −1 1] = 1/6[1 −1 2] + 1/6[−1 −2 1]

The splitting of a perfect dislocation into partials reduces the elastic strain energy in the material. Fig. 10(a) demonstrates that the cutting of impinging 1/6〈112〉 partial dislocations greatly facilitate the splitting process through the dislocation reaction described as:

1/2[0 1 1] − 1/6[−1 1 2] = 1/6[1 1 2]

Fig. 10 Dislocation–GB interactions during the scratching of Cu substrate with 7.63° twist GB. Scratching length: (a) 5.2 nm, (b) 8.8 nm, (c) 12.0 nm and (d) 22.0 nm.

The aforementioned splitting not only facilitates dislocation slip, but also lowers the line energy, thus jointly lead to softening of the material accompanied with a decrease of the friction coefficient. Fig. 11 presents an enlarged view of the splitting process, demonstrating that the splitting produces a complex GB structure along dislocation cores, which acts as barrier to dislocation motion. Furthermore, the subsequent complicated cross-slip between partial dislocations also creates sessile segments to impede dislocation motion. Consequently, the strain hardening of the material results in an increase of the friction coefficient.

Fig. 11 Splitting of 1/2[110] screw GB into 1/6[112] partials during the scratching of Cu substrate with 7.63° twist GB.

Fig. 10(b) shows that at a scratching distance of 8.8 nm, a local fraction of GB dislocation network fails due to the high effective shear stress applied by the dislocations pile up, and dislocation transmission across the GB occurs. Furthermore, there are partial dislocations emitted from the GB observed. Consequently, the friction coefficient decreases when the probe is approaching the centered GB. When the probe is scratching in the right grain interior, the glide of dislocations in the right grain is pinned by the cross-slip-induced sessile segments in the centered GB, which increase the frictional resistance. However, at a scratching length of 12 nm, the complete absorption of dislocations in the left grain by the GB leads to a decrease of the friction coefficient, as shown in Fig. 10(c). Consequently, the friction coefficient reaches a plateau after the probe scratches across the centered GB due to the competition between pinning of dislocation motion and absorption of dislocations by the GB. With a further scratching, fresh dislocations start to nucleate from the free surface of the right grain, leading to a decrease of the friction coefficient.

Therefore, for the 7.63° twist GB, the major modes of dislocation–GB interactions and their respective correlations with the frictional response can be summarized as: GB blocking motion of impinging dislocations in the left grain and GB pinning of glide of transmitted dislocations in the right grain lead to strain hardening of the material, thus leading to increase of the friction coefficient; GB dislocation splitting, dislocation transmission through the GB, dislocation emission from the GB and dislocation absorption by the GB result in softening of the material, which lowers the frictional resistance.

For the 67.38° twist GB that is composed of planar defect, Fig. 12(a) shows that a significant amount of impinging dislocations are accumulated in the vicinity of the centered GB, and resulting strain hardening of the material leads to a further increase of the friction coefficient. To accommodate the plastic strain, glissile dislocation loops form to glide along the centered GB, accompanied with absorption of dislocations by the GB, which jointly lower the frictional resistance. However, the glide of dislocation loops is restricted by the fixed bottom, and the further increased dislocations density leads to more pronounced dislocation pile-up. The high effective shear stress at the head of dislocation pile up facilitates the transmitting of plastic strain to the neighboring grain. Consequently, Fig. 12(b) shows that partial dislocations nucleate and glide in the right grain interior, which lowers the frictional resistance. After the probe scratching across the centered GB, Fig. 12(c) shows that the plastic deformation of the material is mainly carried out by dislocation slip in the right grain interior, and the dislocation structure in the left grain remains stable.

Fig. 12 Dislocation–GB interactions during the scratching of Cu substrate with 67.38° twist GB. Scratching length: (a) 7.2 nm, (b) 9.2 nm, (c) 14.4 nm and (d) 22.0 nm.

Therefore, for the 67.38° twist GB, the major modes of dislocation–GB interactions and their respective correlations with the frictional response can be summarized as: GB blocking motion of impinging dislocations leads to strain hardening of the material, thus leading to an increase of the friction coefficient; glide of dislocation along the GB and dislocation absorption by the GB result in softening of the material, which lowers the frictional resistance.

The friction of metals with a permanent change in shape is a resultant of plastic deformation, which is mainly carried out by the motion of a large number of dislocations. In single crystalline metals, the long and free glide of dislocations, as well as the association and annihilation reactions between dislocations, accommodate plastic strain in the material, thus leads to a reduction of the frictional resistance. In contrast, dislocation multiplication and cross slip-induced impediment of dislocation motion strengthen the material, which in turn increases the frictional resistance.^48,49 In addition to dislocation activity that exclusively governs the plastic deformation of single crystalline metals, dislocation–GB interactions, such as dislocation repulsion, absorption, transmission and remission, play pronounced role in the friction of their polycrystalline counterparts with average grain size down to nanometer regime. By elucidating the correlation between microscopic deformation behavior and macroscopic frictional response, the GB–dislocation interactions-dependent frictional response of bicrystal metal is illustrated in Fig. 13. In particular, the variation of friction coefficient for scratching in the vicinity of GBs is strongly dependent on the dominant strengthening or softening behavior of the material. The strengthening mechanisms include repulsion and pinning of dislocation motion by GBs, and the softening mechanisms include decomposition of GB dislocations, dislocation absorption by GBs, dislocation emission from GBs and dislocation transmission through GBs. The GB structure further has a strong impact on the competition between the strengthening and softening mechanisms, which in turn results in a strong GB structure dependence of the frictional response.

Fig. 13 Schematic illustration of dislocation–GB interaction dependent friction of bicrystal metals.

First, GBs act as barriers to dislocation motion, due to not only the repulsion effect associated with the overlapped strain fields between dislocations, but also the crystal discontinuities that inhibit dislocation transmission through GBs. Since the active slip planes in neighboring grains have different orientations, a gliding dislocation cannot simply traverse through GBs into neighboring grain unless its glide plane is preserved in the neighboring grain. Alternatively, the stress concentration ahead of dislocation pile up formed in the vicinity of GBs may activate new slip planes in neighboring grain to propagate plastic deformation.^50,51 Therefore, the impediment of dislocation motion and consequent dislocation nucleation in adjacent grain are the most pronounced for the not well-defined GB, i.e. 67.38° twist GB, than that for the other three well-defined ones. As compared to the 7.63° twist GB that is composed of intersecting screw dislocations, tilt GBs composed of parallel edge dislocations are less effective in blocking dislocations due to higher GB energy. The 7.63° tilt GB is not effective in blocking dislocations due to the larger inter- and intra-spacing of dislocation dipoles, as compared to the 67.38° tilt GB with a higher GB energy.

Second, the intersecting of impinging dislocations with GB dislocations that impede their motions leads to a change of local configuration of GBs. For the small angle GBs, i.e. 7.63° tilt or twist, the decomposition of local GB by splitting of 1/2〈110〉 dislocation into 1/6〈112〉 partials is observed. In contrast, the decomposition of the high angle 67.38° tilt GB is accompanied by the formation of GB dislocation junctions and inclined 1/6〈514〉 GB dislocations, as well as GB migration through diffusion. The newly created local configuration of GBs not only acts further obstacles to dislocation motion, but also provides preferred sites for GB failure. Specifically, GBs break from the created local configuration, leading to dislocation transmission across the GBs into neighboring grain. However, there is no GB failure observed for the high angle 67.38° twist GB. However, the sessile segments in the GB act as pinning points to inhibit the subsequent glide of transmitted dislocations. In particular for the 7.63° twist GB, the newly created local GB configuration provides preferred sites for dislocation emission.

Third, the absorption of gliding dislocations by GBs is different for different GB structures. For the well-defined GBs that composed of dislocation networks, i.e. 7.63° tilt GB, 67.38° tilt GB and 7.63° twist, the absorption occurs by the dissociating of impinging dislocations into GB dislocations.⁵² Furthermore, the complete absorption of dislocations in the left grain after the probe scratches across the centered GB is observed. However, there is no dislocation transmission observed for the not well-defined 67.38° twist GB, in which the adsorption of dislocations in the left grain is accompanied with the nucleation of lattice dislocations from the GB-free surface intersection in neighboring grain.

Conclusions

In summary, we perform MD simulations to elucidate the fundamental mechanisms of Cu bicrystal under friction, with an emphasis on the influence of GB structures. Four GBs with different structural units, as 7.63° tilt with 1/2〈110〉 edge dislocation dipoles, 67.38° tilt with 1/2〈110〉 edge dislocations with the same sign, 7.63° twist with intersecting 1/2〈110〉 screw dislocations, and 67.38° twist with planar defect, are considered. Simulation results indicate that in addition to dislocation slip, dislocation–GB interactions and GB evolution play an important role in the friction: GB hindering dislocation motion-induced strengthening yields a higher friction coefficient of the material; dislocation absorption, emission and transformation, as well as decomposition of GB dislocations, results in softening of the material accompanied with lowered frictional resistance.

The dislocation–GB interactions in friction exhibit a strong GB structure dependence. The impediment of dislocation motion and consequent dislocation nucleation in adjacent grain are the most pronounced for the not well-defined 67.38° twist GB. For the well-defined GBs, the 7.63° twist GB is more effective in blocking dislocation motion than the two tilt ones. The resistance to dislocation repulsion is smaller for the 7.63° tilt GB than the 67.38° tilt one. Simultaneously, the cutting of impinging dislocations leads to significant local decompositions of the well-defined GBs. Specifically, splitting of 1/2〈110〉 dislocations into 1/6〈112〉 partials is the most common dissociation way of the GBs. However, there are additional 〈100〉 edge dislocations and 1/6〈514〉 GB dislocations formed for the 7.63° and 67.38° tilt GBs, respectively. The created local GB configurations act as preferred sites for GB failure, which is accompanied with dislocation transmission across the GB. There is no failure of the not well-defined 67.38° twist GB observed. In addition, the well-defined GBs have less resistance to dislocation absorption than the not well-defined one. The observed remarkable GB structure-dependence of friction in bicrystal metals has theoretical and practical significance for the tribology, design and manufacturing of interface-dominated advanced functional nanostructures.

Acknowledgements

The authors gratefully acknowledge support from the National Natural Science Foundation of China (51405106 and 51222504).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra10879b

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