NOTES TO ACCOMPANY SECOND AND THIRD ATOMIC FRICTION DEMOS

The first demo showed the force on a single atom moving along the surface. What happens when the number of sliding atoms increases?

The answer depends critically on the spacing between atoms on the two surfaces. If the ratio of the atomic spacing in the slider to the spacing on the substrate is a rational number the surfaces are called commensurate. If the ratio is irrational, the surfaces are called incommensurate. Most contacting surfaces are incommensurate. In this case the atomic forces add incoherently, leading to very small friction forces.

Demos 2 and 3 demonstrate the effect of changing the number of atoms in the slider. The slider is a small crystal of blue atoms sliding over a substrate containing red atoms. The blue atoms are attracted to each other by the Lennard-Jones potential. There are 10 atoms on the bottom of the slider that interact with the substrate atoms. The only difference between demo 2 and demo 3 is the spacing a between substrate atoms. All of the controls and windows are the same as in the first demo. However the spring constant and load controls go to values that are ten times larger than in the first demo because there are ten atoms that interact with the substrate.

The second demo shows a special commensurate case where the two surfaces have exactly the same atomic spacing. In this special case all the sliding atoms feel the same interaction from the substrate. All get stuck at potential energy minima between substrate atoms and then slip together to the next minimum when the force exceeds the static friction. You can try varying the controls and do all the activities described for the first demo. You should see that the friction force is just proportional to the number of atoms, i.e. ten times bigger than for the first demo, if the load per atom and spring stiffness per atom is the same.

This case of equal spacing is very rare. Even two surfaces of the same crystalline material only have equal spacing when they are exactly aligned. The third demo shows what happens when the atoms on the substrate are 14% closer together. Now each atom is in a different position relative to the substrate atoms. Some are near an energy minimum between substrate atoms while others are at an energy maximum over a single substrate atom. Because the forces on sliding atoms add incoherently, the net friction is small.

Even at the highest load and weakest spring the motion of the incommensurate slider is nearly smooth, leading to very low kinetic friction. The static friction is also small. While the force increased in proportion to the number of atoms for the commensurate surface, the friction on the incommensurate surface is less than that for a single atom. The peak force increases from about 2 to about 3 as the load increases from zero to 100. In contrast, the peak force for the single atom was above 2 at zero load and increased to about 6 with a load of just 10. One way of comparing the results is to define a friction coefficient µ as the change in friction divided by the change in load. For the incommensurate surface µ=0.015, about 30 times smaller than the value of µ=0.43 for a single atom or the commensurate surface.

These demos consider very small contacts, but the basic conclusions have been confirmed by much larger simulations and theoretical analysis. The friction only rises with the number of contacting atoms when the surfaces are commensurate. Even then, friction forces can be small if the ratio of spacings is not a simple fraction like 1:1 or 1:2. The friction between incommensurate surfaces is always small and rises more slowly than the number of contacting atoms. The same result is found for disordered solids, like window glass, that don’t have a constant spacing. Since surfaces are almost always incommensurate, we reach the surprising conclusion that friction should be small and rare. Experiments tell us this is wrong, so something must be missing. One possibility is discussed in the remaining demos.

Activity 5: Another Model for the Algebra Savvy

The behavior illustrated in these demos is captured by the Frenkel-Kontorova model. Like the Tomlinson model in Activity 4, the atoms can’t move up and down and the potential from the substrate is a simple sinusoidal function: v(x) = v₀cos(2πx/a), where a is the spacing between atoms. The sliding atoms are connected by springs with a natural length of b. When k is large compared to (2π)²v₀/a² the spacing between sliding atoms can’t deviate much from b. If the surfaces are incommensurate, the static friction is essentially zero. When k is small, the spacing between sliding atoms can shift to lock locally into registry with the substrate resulting in static friction. There is an “Aubry transition” between these two types of behavior at a critical value of k that depends on the ratio b/a.

Frenkel-Kontorova models apply to a wide variety of systems and exhibit lots of interesting behavior. A review is given by Braun and Kivshar, Physics Reports 306, 1-108 (1998); doi:10.1016/S0370-1573(98)00029-5. Aubry transitions in a combined Frenkel-Kontorova-Tomlinson model are discussed by Weiss and Elmer, Physical Review B 53, 7539 (1996); doi:10.1103/PhysRevB.53.7539.

Main Points to Remember

• Atoms on commensurate surfaces lie at the same position relative to substrate atoms. They move up and down in energy coherently as they move over the substrate. The friction rises linearly with the number of contacting atoms if the load per atom is kept fixed.
• The friction is large when the ratio of periods is a simple one, like 1:1 or 1:2.
• Atoms on an incommensurate surface lie at different positions relative to substrate atoms. The energy adds incoherently and the friction is small.
• Atoms on disordered surfaces also have low friction because they lie at different positions relative to substrate atoms.
• Commensurate surfaces are very rare, but friction is almost always large – Something must be added to our model.

Potentially Useful References:

Pedagogical:

· J. Krim, “Friction at the Atomic Scale”, Scientific American 275(4), 74-80 (1996).

· J. Ringlein and M. O. Robbins, “Understanding and illustrating the atomic origins of friction,” Am. J. Phys. 72, 884-891 (2004). doi:10.1119/1.1715107.

• Ugo Besson, Lidia Borghi, Anna De Ambrosis, and Paolo Mascheretti, “How to teach friction: Experiments and models,” Am. J. Phys. 75, 1106-1113 (2007). doi:10.1119/1.2779881.

A more technical review and a book:

· M. H. Müser, M. Urbakh, and M. O. Robbins, “Statistical Mechanics of Static and Low-Velocity Kinetic Friction,” Advances in Chemical Physics, 126, 187-272 (2003).

· B. N. J. Persson, "Sliding Friction: Physical Principles and Applications" (Springer, Berlin, 1998)

Some theoretical and experimental papers on the connection between incommensurate surfaces and low friction:

· M. Hirano and K. Shinjo, “Atomistic locking and friction,” Phys. Rev. B 41, 11837-51 (1990); doi:10.1103/PhysRevB.41.11837.

· M. Hirano and K. Shinjo and R. Kaneko and Y. Murata, “Observation of superlubricity by scanning tunneling microscopy”, Phys. Rev. Lett. 78, 1448-1451 (1997); doi:10.1103/PhysRevLett.78.1448.

· J. Krim and D. H. Solina and R. Chiarello, “Nanotribology of a Kr Monolayer: A Quartz-Crystal Microbalance Study of Atomic-Scale Friction,” Phys. Rev. Lett. 66, 181-184 (1991); doi:10.1103/PhysRevLett.66.181.

· M. Cieplak and E. D. Smith and M. O. Robbins, “Molecular Origins of Friction: The Force on Adsorbed Layers,” Science 265, 1209-12 (1994); doi:10.1126/science.265.5176.1209.

• M. H. Müser, L. Wenning, and M. O. Robbins, “Simple Microscopic Theory of Amontons' Laws for Static Friction,” Phys. Rev. Lett. 86, 1295-8 (2001); doi:10.1103/PhysRevLett.86.1295.
• M. Dienwiebel et al., “Superlubricity of graphite,” Phys. Rev. Lett. 92, 126101 (2004); doi:10.1103/PhysRevLett.92.126101.

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This text is copyrighted by Mark Robbins and Oleg Tchernyshyov. Non-profit educational use is permitted with appropriate acknowledgement.