The influence of electronic transfer on friction properties of hexagonal boron nitride

Abstract

A fundamental understanding of the influence of electronic transfer with varying sliding positions and loads on friction plays a vital role in elucidating the tribological properties and applications of a material in nanomechanics. We get the absorption energies of different planar distances and the potential energies of different sliding positions. Then we obtain the relationship of the friction coefficient dependence on normal force. The results suggest that the friction coefficient decreases and then increases with increasing normal force. It is attributed to the properties that h-BN is an ionic covalent compound and gives rise to electronic transfer when the normal force and sliding distance are changed.

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1. Introduction

Advances in atomic scale friction have brought friction into the nanoscale, which is of fundamental importance for developing novel nanomaterials with expected mechanical properties.^1,2 As we all know, the macroscopic friction law named Amonton’s law explains that friction force increases linearly with increasing normal force. However, many studies have shown that Amonton’s law doesn’t generally apply to the nanoscale, where the friction force is usually a nonlinear function of the normal force. For example, the diversity studies of friction behavior between bi-layer graphenes demonstrate a diversity of friction behavior, from linear, sub-linear and over-linear behavior to a negative coefficient of friction.³ For h-BN with a layered structure, the intra-layer network consists of a strong sp² covalent bond which is very similar to graphene but the electronic properties are radically diverse. The partial ionic character of h-BN turns the system into an insulator instead of a conductor for graphene.⁴ h-BN is widely used in many fields and has attracted the interest of many researchers around the world for years because of its unique insulation properties, thermal conductivity and chemical stability as well as good lubricity at high temperatures.^5–10 One important consequence is the fact that the layers may slide on top of each other to overcome relatively small energetic barriers. Oded Hod¹¹ studied interlayer commensurability and superlubricity in rigid layered materials. The results showed that the h-BN system presents a pattern very similar to that obtained for graphene, where at 0° and 60° high friction is obtained and at intermediate angles superlubricity is obtained. A. Nigues and his coworkers¹² investigated the friction performance of multi-walled boron nitride nanotubes and observed ultrahigh interlayer friction in the system. Wall–wall interactions due to the ionic character of BN bonds were proposed to induce some structural reorganization of the BNNT layer, resulting in ultrahigh interlayer friction. Jussi O. Koskilinna and his coworkers¹³ studied the friction coefficient for hexagonal boron nitride surfaces from ab initio calculations. They calculated friction coefficients of both the h-BN cluster models and a periodic model with the B3LYP method and showed how they depend on the size of the models. Wang Gao and Alexandre Tkatchenko¹⁴ studied sliding mechanisms in multilayered hexagonal boron nitride and graphene including: the effects of directionality, thickness, and sliding constraints. Although the diversity of the friction behaviors of h-BN has been demonstrated in many studies, little attention has been paid to the appropriate understanding of the dependence of the electronic transfer with the changing of the normal force and the sliding position on the friction of h-BN. Therefore, the understanding of the influence of the electronic transfer with varying load and interlayer sliding on h-BN is important, both for elucidating its fundamental tribological properties and for applications of this material in nanomechanics. In the present paper, the influence of electronic transfer with the changing of load and sliding position on the friction properties of h-BN has been studied using density functional theory based on first-principles. We calculate the interlaminar interaction of bi-layer BN under different normal forces and the potential energies at different relative sliding positions. The relationship of the friction coefficient depending on the normal force is obtained. It shows the key role of electron transfer in the friction process of h-BN, which will be helpful for understanding its various friction behaviors.

2. Computational details and model

2.1 Computational details

Calculations were carried out using Castep of Material Studio based on density functional theory.^15–22 Inter-planar bonding is very poor via van der Waals forces, so we do not consider it in this paper. The calculation is performed in a Fast-Fourier-Transform grid. Local density approximation (LDA)^23–26 and generalized gradient approximation (GGA)^27,28 are the most common methods to be used to investigate the properties of h-BN. Many literature works^29–31 report that the equilibrium interlayer distance and lattice constant are overestimated by the GGA method in comparison with the experimental values.³¹ In general, calculations using LDA produced properties slightly closer to the experimental values than calculations with GGA. We also calculated the lattice constant using LDA and GGA. The results showed that LDA is more appropriate than the GGA method. Therefore we used the CA-PZ local density approximation as the exchange–correlation energy to perform all calculations. A norm-conserving pseudopotential serves as the plane wave base.³² Pulay stress of the BFGS (Broyden–Flecher–Glodfarb–Shanno) algorithm is used to address electron relaxation. In the first place, we build a crystal structure of h-BN so that the number of the space group is 194 and lattice parameters are a = b = 2.504 Å, c = 6.652 Å.^4,33,37 The fractional coordinates of the boron and nitrogen atoms are (0.3333, 0.6667, 0.25) and (0, 0, 0.25), respectively. Then, geometry optimization is taken for the crystal structure to obtain the most stable structure in the local area. In the process of geometry optimization, the cutoff energy is 540 eV and the k-point is 12 × 12 × 4 after the convergence test.

2.2 Computational model

The interlayer friction properties of h-BN mainly depend on the bi-layer h-BN’s potential energy difference in different sliding positions. In our calculations, the models of the bi-layer h-BN and the sliding path are as shown in Fig. 1. Fig. 1(a) shows the original sliding position, each boron and nitrogen atom is located in the same position in the upper layer and sub-layer, the blue atoms are nitrogen atoms, the others are boron atoms. Fig. 1(b) shows the final sliding position, BN relatively slides toward the x axis. The boron (or nitrogen) atoms in the upper layer are located in the middle of two boron (or nitrogen) atoms in the sub-layer. The sliding path is fairly divided into six points, and the distance between adjacent locations is 0.2504 Å. The adsorption energies of the different sliding positions and the potential energies under different normal forces were calculated. Finally, the friction coefficients were obtained.

Fig. 1 The models of calculation and the sliding path. (a) The original position and (b) the final position. The red arrow shows the sliding path along the x direction. The blue atoms are nitrogen atoms, the others are boron atoms.

3. Results and discussion

3.1 Interaction energies

Although both the lattice vibration and the interaction among atoms are significant with regard to h-BN friction analysis, in this article we mainly consider the changing of friction caused by electronic interaction among atoms. Adsorption energy is used to indicate the interaction between the bi-layer BN. The method to calculate the adsorption energy is as follows:

E_ab = E_total − 2E_BN, (1)

where E_ab is the adsorption energy, E_total is the system’s total energy and E_BN is the energy of monolayer BN. At the six positions, the planar distance of the bi-layer BN is gradually compressed from 3.5 Å to 1.3 Å along the normal direction on the surface of BN, the interval step is 0.2 Å. Then, the adsorption energy of each point is systematically calculated in the process of compression as shown in Fig. 2. In Fig. 2, 1–6 show the six points. We can find that the adsorption energy gradually increases when BN is compressed from 3.5 Å to 1.3 Å and has the minimum value at 3.5 Å. So this distance is the equilibrium separation of two BN layers. Because a large distance leads to little change in repulsive force and poor attraction, the change in the adsorption energy is very small from 3.5 to 3.1 Å. The attraction and the repulsive force increase at different levels after being compressed sequentially. When the adsorption energy is 0 eV at about 3 Å, the attractive force is equal to the repulsive force. After further compression, the attraction becomes strong at different degrees at the different positions when the repulsive force becomes stronger. Finally the adsorption energy rapidly increases with decreased planar distance. The adsorption energy is gradually decreasing from the first position to the last position at the same interlayer distance. Because the repulsive force is the largest, nitrogen (or boron) atoms of the bi-layer BN are at the same interlayer distance at the first position. Although the repulsive force is large, the attraction gradually increases which results in a smaller adsorption energy with sliding along the x axis. At the last position, the adsorption energy is smaller than at other positions.

Fig. 2 The changing of the adsorption energy of different positions with decreasing planar distance, 1–6 show the six points.

Differentiating the fit adsorption energy function with respect to the planar distance gives the normal load (F_N):^21,34

F_N = −∂E_ab(z)/∂z, (2)

where F_N is the normal load and the applied normal force is equal to the compression of the planar distance of bi-layer BN. The planar distance of the different normal forces and the different positions along the sliding path are shown in Fig. 3. The planar distance has a maximum value at the original position and has a minimum at the final position which is attributed to the different repulsive forces of bi-layer BN at different positions. At the original position, the distance of the nitrogen (or boron) atoms between the upper layer and the lower layer is smaller at the same interlayer distance, resulting in the repulsive force being larger. At the final position, the distance of the nitrogen (or boron) atoms between the upper layer and the lower layer is larger, resulting in a smaller repulsive force. Meanwhile, the distance between the nitrogen (or boron) atoms of the upper layer and the boron (or nitrogen) atoms of the sub-layer is smaller, which results in larger attraction. When the normal force is larger, the planar distance is smaller under the constant sliding distance.

Fig. 3 The planar distance of the different normal forces and the different positions along the sliding path. The sliding distance ranges from 0–1.252 Å and the normal force ranges from 1–9 nN.

3.2 Friction coefficient

The relative potential energy of the different positions:

V(x,F_N) = E_ab(x,z(x,F_N)) − V₀(x,F_N), (3)

where V(x,F_N) is the potential energy of the different positions and different normal forces, E_ab(x,z(x,F_N)) is the adsorption energy of the different positions under the normal force, and V₀(x,F_N) is the minimum potential energy of the sliding path. Fig. 4 shows the potential energy of each point between 1–9 nN. The trend of potential energy in general is gradually decreasing and then increasing from the original position to the final position. When the normal force is small (1–4 nN), the change in potential energy is very small. The curve of potential energy is almost a straight line except for the last point at 5 nN, corresponding to a smaller adsorption energy which ranges from 0.1 to 2.5 eV. However, the normal force increases, the planar distance of the bi-layer BN decreases and the interaction is stronger. Then the potential energy difference becomes more and more apparent.

Fig. 4 The relationship of potential energy depending on the sliding position and normal force. The sliding position ranges from 0 to 1.252 Å and the normal force ranges from 1–9 nN.

The frictional force is calculated using the potential energy difference between the minimum and maximum (ΔV) divided by the sliding distance between the minimum and maximum orientations:

ΔV_max(F_N) = V_max(F_N) − V_min(F_N), (4)

where V_max(F_N) is the maximum of the potential energy of the whole sliding path, V_min(F_N) is the minimum of the potential energy of the whole sliding path, ΔV_max is the barrier that bilayer BN has to overcome for relative sliding to occur. The work required to conquer the friction force is as follows:

ΔE_f = 〈f_f〉Δx = ΔV_max, (5)

where ΔE_f is the work required to conquer the friction force, 〈f_f〉 is the average friction force, and Δx is the distance between the positions of the maximum and minimum of the potential energy. At the end, the average friction coefficient under the normal force is calculated by the average friction force:and according to eqn (6), it is easy to obtain the relationship between the normal load and friction coefficient as shown in Fig. 5.

Fig. 5 The relationship between the normal load and friction coefficient.

We can find that the friction coefficient decreases and then increases. It has a minimum of 0.025 at 5 nN which agrees with the smallest potential energy difference and the smaller adsorption energy. The friction coefficients range from 0.025 to 0.13 with normal load values from 1 to 9 nN. There is no constant friction coefficient value for h-BN because of the structure of the surface, environments, computational methods and so on. The experiments of Saito et al. suggest that the friction coefficient of h-BN ranges from 0.23–0.25 in dry air and from 0.06–0.12 in water.³⁵ Jussi studied h-BN cluster models and a periodic model in five orientations using hybrid density functional B3LYP calculations. The research suggests that the friction coefficient of h-BN ranges from 0.07 to 0.26 at low load.¹³ Our friction coefficient at low load is 0.13 which is in very close agreement with theirs. Lee and his coworkers studied the frictional characteristics of atomically thin sheets. Their experimental results demonstrated that the changing regular of frictional coefficient of h-BN is consistent with graphene, for which friction increases with decreasing thickness.³⁶ And also, the changing regular of our friction coefficient which decreases and then increases with increasing normal force is similar to that of graphene.²¹ In order to further explain the trend of the friction coefficient, we calculate the electron density of some special points (F_N = 1, 2, 4, 5, 6, 8 nN) and analyze the total electronic density as shown in Fig. 6 and 7.

Fig. 6 The total electronic densities at F_N = 5 nN. (a)–(f) are the electronic densities of the six points of the sliding path from the origin to the end, their planar distances are 2.5347 Å, 2.5014 Å, 2.4348 Å, 2.3348 Å, 2.2218 Å, and 2.0678 Å, respectively. (g) The slice shows the changing electron density. The electronic density is gradually stronger from blue to red color.

Fig. 7 The total electronic density of the different normal forces when the upper layer slides to the fourth point along the x direction. (a)–(f) The normal force is 1, 2, 4, 5, 6, and 8 nN, respectively. (g) The slice shows the changing electron density. The electronic density is gradually stronger from blue to red color.

When the normal force is 5 nN, the friction force is the smallest. From Fig. 6(a)–(f), the electrons transfer apparently from the upper layer to the lower layer and the locality of the nitrogen atoms is strengthened when the layers relatively slide along the x direction. This suggests that the sliding distance contributes to electronic transfer resulting from the electrostatic interaction between the bi-layer which gives rise to the changing of the friction force when the normal force (the interlayer distance) is constant. It is the same trend of electronic transfer as for F_N = 1, 2, 4, 6, 8 nN. This is in good agreement with the literature³⁷ which shows that once stacking is established, the sliding energy profile at a fixed interlayer distance is governed by electrostatic interactions resulting from the polar nature of the B–N bond. However, electrons of the lower layer transfer toward the upper layer with increasing normal force when the sliding distance is constant as shown in Fig. 7(a)–(f). This phenomenon demonstrates that changing the normal force (electrostatic interaction) also induces electronic transfer.

The research results illustrate that the normal force and the sliding position have an effect on the electron transfer between the upper and lower layer. When the sliding position and normal force are small, repulsive force plays a vital role. The electronic transfer from the upper layer to the lower layer is less than from the lower to the upper layer, which results in the friction coefficient being large. The repulsive force and attraction become strong at different levels with the sliding distance and the normal force gradually increasing, resulting in more electronic transfer. Comprehensive results of electronic transfer lead to the friction coefficient gradually decreasing. When the normal force reaches 5 nN, the potential energy difference is the smallest, which makes the net electronic transfer between the bilayer BN the smallest so that the friction coefficient has a minimum value. After this point, attractive force plays a dominant role and the electronic transfer from the upper to the lower layer is more than from the lower to the upper layer so that the friction coefficient gradually increases in the present paper.

4. Conclusion

The relationship of the friction coefficient depending on normal force is obtained by calculating the interlaminar interaction of bi-layer BN under different normal forces and their potential energies at different relative sliding positions along the x direction. The results indicate that the friction coefficient decreases and then increases with increasing normal force and has a minimum of 0.025 at 5 nN. The friction coefficients range from 0.025 to 0.13 with normal load values from 1 to 9 nN. This trend is very similar to graphene. The experimental results of Lee also demonstrated that the frictional coefficient of h-BN is consistent with graphene, for which friction increases with decreasing thickness. On the other hand, our friction coefficient at low load is 0.13 which is in very close agreement with the result of Jussi. Further analysis of electron density suggests that the electrons transfer apparently from the upper layer to the sub-layer and the locality of the nitrogen atoms is strengthened when the upper layer relatively slides keeping the normal force constant, while the electrons of the sub-layer transfer into the upper layer with increasing normal force keeping the sliding distance constant. The synthesis results of the electronic transfer lead to the friction coefficient decreasing and then increasing with the changing of normal force and sliding position.

Acknowledgements

The work was supported by the National Nature Science Foundation of China (grant 21373249 and 51322508).

References

1. M. Salmeron, Surface Diagnostics in Tribology: Fundamental Principles and Applications, edited by K. Miyoshi and Y. W. Chung, 1993, World Scientific Publishing.
2. A. Vanossi, N. Manini, M. Urbakh, S. Zapperi and E. Tosatti, Rev. Mod. Phys, 2013, 85, 529–552.
3. Z. Liu, Nanotechnology, 2014, 25, 075703.
4. X. Liu, Master thesis, Ocean University of China, 2010.
5. S. Larach and R. E. Shrader, Phys. Rev., 1956, 102, 582.
6. S. Larach and R. E. Shrader, Phys. Rev., 1956, 104, 68.
7. R. Geick, C. H. Perry and G. Rupprecht, Phys. Rev., 1966, 146, 543.
8. P. J. Gielisse, S. S. Mitra, J. N. Plendl and R. D. Griffis, Phys. Rev., 1967, 155, 1039.
9. R. Ahmeda, F. -e-Aleema, S. J. Hashemifarb and H. Akbarzadeh, Phys. Rev. B, 2007, 297.
10. R. Narahari, Materials Science and Engineering Committee, 2000, vol. 5, p. 54.
11. O. Hod, Phys. Rev. B, 2012, 86, 075444.
12. A. Nigues and A. Siria, Nat. Mater., 2014, 10, 688.
13. J. O. Koskilinna, M. Linnolahti and T. A. Pakkanen, Tribol. Lett., 2006, 1, 37–41.
14. W. Gao and A. Tkatchenko, Phys. Rev. Lett., 2015, 114, 096101.
15. Y. Qi and L. G. Hector, Phys. Rev. B, 2004, 69, 235401.
16. S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. I. Probert, K. Refson and M. C. Payne, J. Crystallogr., 2005, 220, 567.
17. M. C. Huang, Physic Progress, 2000, 20, 199.
18. W. Kohn and L. J. Sham, Phys. Rev., 1965, 137, A1697.
19. P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, 864.
20. W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133.
21. W. Zhong and D. Tomanek, Phys. Rev. Lett., 1990, 64, 3054.
22. N. Ooi, A. Rairkar and J. B. Adams, Carbon, 2006, 44, 231.
23. J. P. Perdew and Y. Wang, Phys. Rev., 1992, 45, 13244.
24. N. Ooi, A. Rairkar, L. Lindsley and J. B. Adams, J. Phys.: Condens. Matter, 2006, 18, 97.
25. J. V. Barth and L. J. Hedin, J. Phys. C: Solid State Phys., 1972, 5, 1629.
26. O. Gunnarsson, B. I. Lundqvist and S. Lundqvist, Screening in a spin-polarized electron liquid, Solid State Commun., 1972, 11, 149.
27. J. P. Perdew and Y. Wang, Phys. Rev. B, 1986, 33, 8800; J. P. Perdew, in Electronic Structure of Solids ’91, edited by P. Ziesche and H. Eschrig, Akademie Verlag, Berlin, 1991.
28. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
29. M. L. Hu and Y. Zhizhou, Comput. Mater. Sci., 2012, 54, 165–169.
30. A. Janotti, S. H. Wei and D. J. Singh, Phys. Rev. B, 2001, 64, 174107.
31. R. S. Pease, Nature (London), 1950, 165, 722.
32. M. H. Lee, PhD Thesis, Cambridge University, 1996.
33. J. Robertson, Phys. Rev. B, 1984, 29, 2131.
34. D. Tomanek, W. Zhong and H. Thomas, Europhys. Lett., 1991, 15, 887.
35. T. Saito and F. Honda, Wear, 2000, 237, 253.
36. C. Lee and Q. Li, et al., Science, 2010, 328, 76–80.
37. N. Marom and J. Bernstein, Phys. Rev. Lett., 2010, 105, 046801.

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