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J. Phys. Chem. B 2001, 105, 1338-1343
Ab Initio Studies on the Atomic-Scale Origin of Friction between Diamond (111) Surfaces Raisa Neitola and Tapani A. Pakkanen* Department of Chemistry, UniVersity of Joensuu, P.O. Box 111, FIN-80101, Joensuu, Finland ReceiVed: August 22, 2000; In Final Form: October 2, 2000
Ab initio Hartree-Fock method was used to examine the atomic-scale friction that occurs when two diamond (111) hydrogen-terminated surfaces are placed in sliding contact. The orientations of the hydrogen atoms at the surfaces and the formation of wear particles were investigated by geometry optimization, but the main interest was in the interaction energies, friction force, and the coefficient of friction. Optimized structures of the diamond surfaces showed that no wear occurred at the applied pressures when the two surfaces were placed in sliding contact. The vibrational motions of the hydrogen atoms accommodated the stress due to the pressure of the surfaces. The friction force was found to increase linearly with the normal load, indicating that Amontons’ law also holds true in the microscale world. The calculated coefficient of friction was µ ≈ 0.22, which is in agreement with the results of molecular dynamic and experimental studies.
Introduction The unique friction and wear properties of diamond make it particularly interesting for self-lubricating and wear-resistant applications and superhard coatings.1-4 Various theoretical approaches have been used to investigate diamond surfaces. Small surface models can be studied with high-level quantum mechanical methods, while large systems can be simulated with molecular dynamics. Harrison and co-workers5-19 have carried out extensive MD simulations of atomic-scale friction at different crystal faces of diamond as a function of load, temperature, sliding speed, and sliding direction. As well, they have applied MD methods to study friction and wear in (111) crystal face of diamond and related hydrocarbon systems. Their work demonstrates the applicability of MD methods to describe the physical phenomena of friction. The most recent work of this group focuses on compression and friction of n-alkanes (n ) 8, 13, 22) that are chemically bound to the diamond (111) surfaces.20-21 Besides the MD studies, a large number of theoretical studies on diamond surfaces have been carried out by ab initio methods.22-29 These have included investigations of hydrogen abstraction from diamond surfaces, calculations of atomic and electronic structures of diamond surfaces, and calculations of hydrogen atom and methyl radical associations with diamond surfaces. Few quantum mechanical studies have dealt with friction between diamond surfaces even though friction is an everyday physical phenomenon. Friction is the force, Fµ, that opposes the motion of two bodies in sliding contact. In macroscale, it obeys Amontons’ law: friction is directly proportional to the load FN and the coefficient of friction µ. Despite the paucity of studies, several different models have been developed30-33 to calculate atomic-scale friction force and the coefficient of friction. Using these models Zhong and coworkers investigated a four-layer slab of graphite and a monolayer of palladium.30,31 We tested ab initio Hartree-Fock methods with the aim of evaluating their suitability for modeling the friction between two diamond (111) surfaces placed in sliding contact. Friction is almost always examined on a macroscopic scale, even though the interactions that dominate friction take place at atomic level.
Because very little is known about friction and wear between two surfaces at the microscopic level, we chose to explore the problem at the ab initio HF/3-21G* and HF/6-31G* level of theory. We wished to know, for example, how atomic orientation and movement of surface atoms affect the friction between two surfaces. To this end we investigated the atomic-scale origins of the friction between two hydrogenated diamond surfaces that first were gradually pressed tighter against each other and then were placed in sliding contact. Additionally, we calculated the friction force and the coefficient of friction by the method of Zhong and co-workers.30-33 Theoretical Methods The calculations were carried out with Hartree-Fock method with the split valence basis sets 3-21G* and 6-31G*. Basis set superposition error was taken into account in all calculations. The interaction energy, ∆E(r), of two hydrogenated diamond surfaces A and B at a distance r was calculated as
∆E(r) ) EAB(r) - EA - EB where the total energy of the double surface system is EAB(r) and EA and EB are the energies of separate diamond surfaces.34 Diamond surface models, which are described in more detail below, were constructed and the results were visualized with the molecular modeling program Sybyl 6.2.35 Geometry optimization and all energy calculations were performed with the program system Gaussian9436 on a Silicon Graphics Origin200 R10000 4-processor workstation. Results and Discussion Surface Model. The friction phenomenon was studied for the (111) crystal face of diamond. The diamond (111) surface is important because, unlike the (110) and (100) surfaces, it does not allow reconstruction. The surface model was designed to be as large as possible, while remaining computationally feasible. In all our calculations we used a C13H22 surface model consisting of three six-membered carbon rings, as shown in Figure 1. The model was terminated with hydrogen atoms. All carbon atoms were covalently bonded to the hydrogen atoms
10.1021/jp003025t CCC: $20.00 © 2001 American Chemical Society Published on Web 01/25/2001
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J. Phys. Chem. B, Vol. 105, No. 7, 2001 1339
Figure 1. Surface model, C13H22, of the diamond (111) surface viewed from the side and the top. Dark gray spheres represent carbon atoms and light gray spheres hydrogen atoms.
Figure 3. Diamond (111) surfaces at the interaction viewed from the side and the top. Dark gray spheres represent carbon atoms and light gray spheres hydrogen atoms. The distance between the surfaces is 4.5 Å. (a) Initial system, (b) upper surface of the first system moved 0.74 Å to the right, (c) upper surface moved a farther 0.74 Å to the right.
Figure 2. Optimized hydrogen atoms.
and maintained sp3 hybridization. The surface carbons occupy sites analogous to those in cyclohexane in the chair conformation. The C-C distance was 1.545 Å and the C-H distance 1.083 Å. In accordance with the tetrahedrally coordinated structure, all bond angles were set at 109.47°. Two similar C13H22 surfaces were used to model the interactions between diamond surfaces. Only the positions of hydrogen atoms between surfaces were optimized, as shown in Figure 2. Interaction of Surfaces. The friction force between two diamond surfaces was examined for surfaces pressed closely together. The movement of hydrogen atoms and energy changes between the surfaces was of interest. Three different cases were calculated. In the first case (Figure 3a) the surfaces were set so that the hydrogen atoms were located symmetrically between the surfaces and the hydrogen atoms of the upper surface were
positioned directly over the hydrogen atoms of the lower surface. In the second case the upper surface of the first case was moved 0.74 Å to the right (see Figure 3b), so that the hydrogen atoms were no longer face to face. In the third case the upper surface was moved a farther 0.74 Å to the right. Now the hydrogen atoms were located so that they pointed directly to the center of the six-membered rings on the bottom surface (shown in Figure 3c). The distance between the surfaces was defined as the distance between the nearest carbon layers. In the beginning the distance between the carbon layers was set at 4.5 Å, so that the hydrogen atoms on the different surfaces did not touch each other. Diamond surfaces were then pressed against each other, with the upper surface moved downward in small, 0.25 Å, steps. The pressing was stopped when the distance between the surfaces was 3.0 Å. Study was first made of geometry optimizationsespecially the movement of hydrogen atoms (Figure 2). Average optimized C-H bond lengths obtained with basis set 3-21G* are shown in Figure 4. Behavior was similar in all three cases: C-H bond lengths shortened as the distance between the surfaces was decreased. Not surprisingly, the bond lengths were reduced most when the hydrogen atoms were directly opposite each other (Figure 3a). We also examined the C-C-H bond angles, and there were no marked changes in these either. Only in one instance did the bond angles deviate by a few degrees from
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Figure 4. Average C-H bond lengths as a function of the distance between surfaces calculated with the basis set 3-21G*. See Figure 3 for explanation of 3a-c.
Figure 5. Optimized structures taken from the first system of interaction of surfaces (see Figure 3a). Red spheres represent carbon atoms, and blue spheres represent hydrogen atoms. The hydrogen atoms of the upper surface are positioned directly over the hydrogen atoms of the lower surface. Distance between surfaces is (a) 4.25. (b) 3.5, and (c) 3.0 Å.
TABLE 1: Interaction Energies E(kJ/mol) at Different Distances between Diamond Surfaces Calculated at 3-21G* and 6-31G* Levels of Theorya distance 3a 3b 3c between surfaces (Å) 3-21G* 6-31G* 3-21G* 6-31G* 3-21G* 6-31G* 4.50 4.25 4.00 3.75 3.50 3.25 3.00 a
12.86 30.10 66.27 140.48 286.51 553.28 1021.12
11.08 32.86 70.28 146.48 297.60 573.24 977.87
10.25 22.42 47.68 97.43 187.94 337.51 566.32
11.39 23.98 49.82 100.02 191.40 341.29 569.47
7.31 15.37 30.49 59.09 109.38 194.69 333.46
31.54 60.46 111.75 199.03 340.22
For explanations of 3a-c, see Figure 3.
those expected. This exception can be seen in Figure 5c: the positions of the hydrogen atoms became distorted when the surfaces were in closest proximity. In general, the HF/6-31G* basis set predicted shorter bonds than did HF/3-21G*. Still, there was no significant difference between the basis sets. We also investigated the changes in interaction energies as the diamond surfaces were pressed closer against each other. The interaction energies increased as the distance between the surfaces decreased because the repulsive interactions between the hydrogen atoms increased; and when the surfaces were moved farther apart the energies slowly approached zero. The interaction energies are presented in Table 1. As can be seen, the interaction energies were highest when the hydrogen atoms were in contact with each other (3a), and the number of interactions between atoms was the greatest. Energies were lowest when hydrogen atoms pointed directly to the center of the six-membered rings at the bottom surface (Figure 3c). In this situation the hydrogen atoms had more space, so that there were fewer interactions between atoms. As the values in Table
Figure 6. Interaction energies between diamond surfaces pressed increasingly closer, as a function of distance between the surfaces, calculated with the basis set 3-21G*. For explanations of 3a-c, see Figure 3.
1 show, in all three cases calculated, the interaction energies approximately doubled with every compression. Values with basis set 3-21G* were slightly higher than those with basis set 6-31G*, with the one exception mentioned above (Figure 5c). In that case the hydrogen atoms were distorted, causing the repulsion forces between the hydrogen atoms to decrease and the value of the interaction energy to be higher than the comparable value calculated with basis set 3-21G*. Comparison of the energies in the above three cases reveals a similar pattern of behavior (see the curves in Figure 6). Evidently the ab initio method can handle well this kind of modeling of diamond surfaces. Furthermore, since the differences in the results with the two basis sets were almost negligible, we conclude that the 3-21G* basis set is accurate enough. Sliding Contact. We now focus on the interaction energies between two diamond surfaces in sliding contact. The starting configuration was similar to the situation where two surfaces were pressed together (shown in Figure 3a). Sliding was modeled by moving the upper surface to the right in 20 steps of 0.07 Å each, and in each step the geometry was optimized. Sliding of the diamond surfaces terminated with the hydrogen atoms pointing directly to the center of the six-membered rings in the bottom surface (Figure 3c). Three different configurations were examined, with distances between surfaces of 4.0, 3.75, and 3.5 Å. As before, the distance between the surfaces was defined as the distance between carbon layers. Calculations were carried out only with HF/3-21G* because differences in the results obtained with the two basis sets were negligible in the previous experiment. The interaction energies between two diamond (111) surfaces in sliding contact are presented in Table 2. Interaction energies between surface atoms were lowest when the distance between the surfaces was 4.0 Å, when the hydrogen atoms barely touched one another. At 4.0 Å the values of interaction energies varied between 30 and 70 kJ/mol. Energies were doubled when the distance between the surfaces was 3.75 Å, and a similar increase was observed when the distance between the surfaces was further decreased to 3.5 Å. Continued sliding of diamond surfaces gave us the opportunity to examine the potential energy curves for sliding contact (Figure 7). All three curves are periodic. The most interesting feature of the curves is that the minima and maxima are located at the same places. The maxima correspond to the
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J. Phys. Chem. B, Vol. 105, No. 7, 2001 1341
Figure 7. Potential energy as a function of sliding distance.
TABLE 2: Calculated Potential Energies (kJ/mol) for Three Sliding Experiments distance between surfaces sliding distance (Å)
4.0 Å
3.75 Å
3.5 Å
0.00 0.07 0.15 0.22 0.30 0.37 0.44 0.52 0.59 0.67 0.74 0.81 0.89 0.96 1.04 1.11 1.18 1.26 1.33 1.41 1.48
66.53 66.11 65.04 63.89 62.24 60.45 58.35 56.42 54.30 51.63 48.15 46.20 43.63 40.74 38.33 36.44 34.76 33.03 31.85 30.75 29.84
141.03 140.79 139.14 136.14 133.03 127.99 123.46 117.91 111.76 106.77 99.44 94.13 87.40 81.86 76.88 72.56 68.54 64.96 62.47 60.01 58.09
286.75 286.22 282.33 275.78 267.28 256.75 244.62 232.44 219.37 208.42 192.07 181.77 166.60 155.66 144.66 135.80 127.44 120.24 115.44 110.84 107.53
point of maxima hydrogen-hydrogen repulsion, when the hydrogen atoms of the upper surface were against the hydrogen atoms of the lower surface. Visualization of the optimized structures provides insight into how the interactions of the hydrogen atoms affect the energy and friction. Optimized structures for the sliding experiment where the distance between surfaces was 3.75 Å are shown in Figure 8. After two or three steps, when the hydrogen atoms were no longer positioned directly over one another (Figure 8 b-f), the potential energy was decreased. Analysis of the three cases with surfaces at different distances apart indicates that as the distance between the surfaces increases both the interactions between atoms and the interaction energies decrease. The motion of hydrogen atoms has been studied earlier by MD simulations using larger multi layer models. The conclusions drawn from the ab initio single layer model are in a good agreement with the MD work.5,6 The similarity of the results supports the choice of the single layer ab initio model for the diamond (111) surface. The sliding does not result in shearing of the hydrogen atoms from the surfaces in any of the examined cases. Although the hydrogen atoms on the opposite surfaces brushed against each other, no covalent C-H bonds were broken. At the pressures applied the stress caused by sliding and pressing was not enough
Figure 8. Optimized structures for the sliding system. The distance between diamond (111) surfaces is 3.75 Å. The upper surfaces are successively moved to the right (a) 0 Å, (b) 0.296 Å, (c) 0.592 Å, (d) 0.887 Å, (e) 1.183 Å, and (f) 1.479 Å.
to break C-H bonds. Apparently, the vibrational motions allowed the stress due to the pressure of the surfaces to be accommodated. The stress accommodation via vibrational motion and the observations of wear have been discussed earlier in the MD studies.8-10,12,16 Friction Force and the Coefficient of Friction. In the following, we present the friction force and the coefficient of the friction for the contact of two diamond surfaces as calculated by the procedure of Zhong and co-workers.30-33 First we calculated the normal load, FN, from the interaction energies (Figure 6). Figure 9 presents curves for different values of the normal load. The maxima corresponded to the situation shown in Figure 3a, and the minima to the situation shown in Figure 3c. Fluctuations in the curves became more pronounced as distances between the hydrogen atoms on opposite surfaces decreased. At the same time, the normal load was increased due to the increase in repulsive interactions between atoms. At low normal loads, the hydrogen atoms pass by one another easily. The friction force in the atomic scale under normal load was obtained by calculating the energy difference between the minimum and maximum (Figure 9) and by dividing this by the distance between the minimum and maximum. As shown in Figure 10, the friction force, Fµ, increases almost linearly with the increase in the normal load FN. Thus, Amontons’ law essentially holds for atomic-scale friction. The calculated coefficient of friction, that is, the friction force divided by the normal force, was about 0.22. These results can be compared with the MD results of Harrison et al. although their data is in
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TABLE 3: Experimental Friction Coefficients for Diamond and Other Carbon Compoundsa diamond (111) surface6 diamond (111) surface6 hard carbon37 hard carbon37 hard carbon containing hydrogen37 hard carbon containing hydrogen37 tungsten tip on a diamond (111) surface38 tungsten tip on a diamond (111) surface38 diamond tip on a polycrystallne CVD diamond film38 diamond tip on a polycrystallne CVD diamond film38 a
method
coefficient of friction µ
other information
MD MD
0.17 0.43 0.1-0.3 0.12-0.4 0.22-0.5 0.01-0.35 0.3 0.05-0.15 0.16 0.85
P ) 3.9 GPa P ) 14.2 GPa in air in vacuum in air in vacuum for loads up to 400 nN >400 nN in air in UHV
AFM AFM
Literature sources are indicated as superscripts.
friction of a hydrogen-terminated diamond (111)/tungsten carbide contact.39 Conclusions
Figure 9. Distance between the surfaces as a function of sliding distance at different normal loads.
Figure 10. Friction force as a function of normal load.
a different regime at load 0.05-0.8 nN.6 The friction coefficients of both studies are in good agreement. Our atomic-scale coefficient of friction is not readily compared with coefficients measured experimentally37,38 because there usually is no unambiguous value of the coefficient of friction. Despite the differences between theoretical and experimental methods, however, our results seem to agree rather well with those of experimental studies. Experimental values of the coefficient of friction are listed in Table 3. As can be seen, our calculated value of 0.22 fits within the range of values ∼0.01 to ∼1.35 calculated for hard carbon containing hydrogen in a vacuum.37 In addition, our value is of the same order of magnitude as coefficients of friction calculated by molecular dynamics (MD)6 and measured by friction force microscopy (AFM).38 Interestingly, our data (Figure 10) is in good agreement with UHV AFM experiments by Enachescu et al. for the
The goal of this study was to model the atomic-scale friction of a hydrogenated diamond (111) surface, C13H22, with a suitable ab initio method. Our calculations of two diamond surfaces placed in sliding contact provide information about the interaction energies and friction forces between the two surfaces, as well as about the coefficient of friction. The Observations complement previous work.5-19 The interaction energies were highest when the interaction between hydrogen atoms was the strongest, in other words, when the hydrogen atoms were face to face, and the distance between the surfaces was smallest. In addition, the optimized C-H bond length shortened when the diamond surfaces were pressed closer against each other. Both in pressing and in sliding of the surfaces, the pattern of behavior was similar for the three cases computed: the curves of C-H length (Figure 4) and energy (Figures 6 and 7) were of similar shape. Moreover, the observation that sliding to the [11 2h] crystallographic direction did not result in the breaking of bonds or formation of wear debris means that the vibrational modes allowed hydrogen atoms to move and bend a little. Evidently the ab initio method is a good choice for modeling the friction characteristics of diamond surfaces. There were no great differences in the calculated results between the basis sets 3-21G* and 6-31G*. The results based on ab initio and MD calculations are in good agreement demonstrating that the potential model used in MD studies is accurate. Ab initio methods can be useful for finding potential models for other friction systems as well. Our calculations of the friction force and the coefficient of friction showed the friction force to vary directly with the normal load. Despite this variation, however, the coefficient of friction was about 0.22 for all values of the normal load, which means that Amontons’ law is obeyed on the atomic scale. References and Notes (1) Spear, K. E.; Dismukes, J. P. Synthetic Diamond: Emerging CVD Science and Technology; John Wiley & Sons Inc.: New York, 1993. (2) Field, J. E. The Properties of Natural and Synthetic Diamond; Academic Press Limited: London, 1992. (3) Miyoshi, K. Diamond Films Technol. 1998, 8, 153-172. (4) Bhushan, B. Handbook of Micro/Nanotribology; CRC press: Boca Raton, FL, 1995. (5) Perry, M. D.; Harrison, J. A. J. Phys. Chem. 1995, 99, 99609965. (6) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner D. W. Phys. ReV. B 1992, 46, 9700-9708. (7) Perry, M. D.; Harrison, J. A. J. Phys. Chem. B 1997, 101, 13641373. (8) Perry, M. D.; Harrison, J. A. Langmuir 1996, 12, 4552-4556.
Origin of Friction between Diamond (111) Surfaces (9) Perry, M. D.; Harrison, J. A. Thin Solid Films 1996, 290-291, 211-215. (10) Harrison, J. A.; Brenner, D. W. J. Am. Chem. Soc. 1994, 116, 10399-10402. (11) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner D. W. J. Phys. Chem. 1993, 97, 6573-6576. (12) Harrison et al. Thin Solid Films 1995, 260, 205-211. (13) Harrison, J. A.; Perry, S. S. MRS Bull. 1998, 23, 27-31. (14) Harrison, J. A.; Colton, R. J.; White, C. T.; Brenner D. W. Wear 1993, 168, 127-133. (15) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner D. W. MRS Bull. 1993, 18, 50-53. (16) Perry, M. D.; Harrison, J. A. Tribology Lett. 1995, 1, 109-119. (17) Brenner, D. W.; Harrison, J. A. Am. Ceram. Soc. Bull. 1992, 71, 1821-1828. (18) Harrison, J. A.; Brenner, D. W.; White, C. T.; Colton, R. J. Thin Solid Films 1991, 206, 213-219. (19) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner, D. W. Surf. Sci. 1992, 271, 57-67. (20) Tutein, A. B.; Stuart, S. J.; Harrison, J. A. J. Phys. Chem. B 1999, 103, 11357-11365. (21) Tutein, A. B.; Stuart, S. J.; Harrison, J. A. Langmuir 2000, 16, 291-296. (22) Brown, R. C.; Cramer, C. J.; Roberts, J. T. J. Phys. Chem. B 1997, 101, 9574-9580. (23) Kern, G.; Hafner, J. Phys. ReV. B 1997, 56, 4203-4210. (24) de Sainte Claire, P.; Hase, W. Phys. ReV. B 1997, 56, 1354213555. (25) Kern, G.; Hafner, J. Phys. ReV. B 1998, 58, 2161-2169.
J. Phys. Chem. B, Vol. 105, No. 7, 2001 1343 (26) Strout, D.; Scuseria, G. J. Chem. Phys. 1995, 102, 8448-8452. (27) Page, M.; Brenner, D. J. Am. Chem. Soc. 1991, 113, 3270-3274. (28) de Sainte Claire, P.; et al. J. Phys. Chem. 1996, 100, 1761-1766. (29) de Sainte Claire, P.; Hase, W. Phys. ReV. B 1997, 56, 1354213555. (30) Zhong, W.; Toma´nek, D. Phys. ReV. Lett. 1990, 64, 3054-3057. (31) Toma´nek, D.; et al. Europhys. Lett. 1991, 8, 887-892. (32) Matsuzawa, N.; Noriyuki, K. J. Phys. Chem. A 1997, 101, 1004510052. (33) Overney, G.; Zhong, W.; Toma´nek, D. J. Vac. Sci. Technol. B 1991, 9, 479-482. (34) Scheiner, S. Molecular Interactions; John Wiley & Sons: New York, 1997. (35) Sybyl 6.2, Tripos, Inc., USA. (36) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Cioslowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; HeadGordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94, Rev. E.2; Gaussian, Inc.: Pittsburgh, PA, 1995. (37) Miyake, S.; Takahashi, S.; Watanabe, I.; Yoshihara, H. ASLE Trans. 1987, 30, 121-127. (38) Mate, C. Wear 1993, 168, 17-20. (39) Enachescu, M.; van den Oetelaar, R. J. A.; Carpick, R. W.; Ogletree, D. F.; Flipse, C. F. J.; Salmeron, M. Phys. ReV. Lett. 1998, 81, 18771880.
