FR40 · An interactive course
Nanofriction: atomic stick-slip, thermal creep and superlubric flakes
Drag the tip of an atomic force microscope across a salt crystal and the lateral force does not stay steady. It climbs while the scanner moves one lattice spacing, then drops at once as the tip jumps to the next site, and the pattern repeats. Twist a graphite flake on graphite by a few degrees and the same jumps can vanish, with friction falling more than fivefold. This course derives both from one model, a tip on a spring over a corrugated surface, then adds thermal noise and finally sums the model over the atoms of a flake.
From one tip to a twisted flake
The first chapter sets the scale: forces of a fraction of a nanonewton, lattice spacings of 0.4 nm and energies of a few tenths of an electronvolt, and asks which of the classroom friction laws survive at one contact. The second builds the Prandtl–Tomlinson model and finds where the tip lets go. The third follows the work done by the scanner into heat and checks the simulation’s energy books. The fourth adds thermal noise and derives why friction grows with the logarithm of speed. The fifth looks at the very slow limit, where the tip hops back and forth, and at where simple rate theory stops working. The sixth sums the model over the atoms of a flake. The seventh twists the flake and finds how fast the stick-slip disappears, and what friction is left. The last chapter is a guided reading of the 2013 colloquium by Vanossi and co-workers.
You need Newton’s second law with a spring and a damping force, a Taylor expansion, the Boltzmann factor and reading a semi-log plot. The Langevin equation, the Kramers rate and the structure factor of a lattice are introduced when they are first needed. FR40
01 · L1
How large friction is at a single atomic contact
Set the scales of atomic friction, fractions of a nanonewton, 0.4 nm and tenths of an electronvolt, and see which Amontons laws one tip can break.
02 · L2
How the Tomlinson model explains stick-slip
Pull a mass on a spring over a cosine landscape: derive the stick-slip parameter, the fold where the valley vanishes and its force, and check the NaCl fit.
03 · L2
Friction as the energy lost in each slip
Each slip turns the scanner’s work into heat. Friction at low speed is the energy drop at the fold; damping decides one- or two-site jumps. Check the books.
04 · L2
Why thermal activation makes friction grow with the logarithm of speed
Thermal noise lets the tip hop the barrier before its valley vanishes. Derive the log-speed law of Sang and co-workers and test it by simulation at 300 nm/s.
05 · L3
Thermolubricity and the limits of rate theory
Scan slowly enough and the tip hops back and forth, so friction is proportional to speed and vanishes. Estimate when, then test where rate theory holds.
06 · L3
How the friction of a crystal flake adds up from its atoms
Replace the tip with a chain of atoms or a crystal flake: the Frenkel–Kontorova chain, the Aubry transition, flake structure factors and scaling with area.
07 · L4
Twisted flakes and structural superlubricity
Twist an aligned flake a few degrees and stick-slip vanishes, leaving only damping. Compute the peak width, find its side lobe, see what superlubricity omits.
08 · L4
Reading the 2013 friction-modelling review by Vanossi and co-workers, section by section
Read the eight sections of this review the course uses, in order, recompute its numbers with the course models and find the problems it leaves open.
Experiments along the way
All three experiments run the models the chapters derive; none of them fits or redraws published curves. The two tip experiments use the NaCl parameters of Sang and co-workers and draw the zero-temperature path or the rate-theory prediction beside the simulated tips. The flake experiment works in reduced units, the single-atom corrugation and the substrate lattice spacing, which are model parameters; only the gold-on-graphite lattice ratio comes from de Wijn’s paper.
For friction between everyday objects, where the classroom laws do hold, see which of two stacked blocks slips first and a rope wrapped around a post.
Sources and what we recomputed
- A. Vanossi, N. Manini, M. Urbakh, S. Zapperi and E. Tosatti, Colloquium: Modeling friction: from nanoscale to mesoscale, arXiv:1112.3234v4; Reviews of Modern Physics 85, 529 (2013)
Version 4, 3 April 2013. Sec. I: the Coulomb–Amontons laws. Sec. II.A: the Prandtl–Tomlinson model (Eq. 1), the stick-slip parameter, the fold where a slip starts, the barrier that shrinks as the three-halves power of the distance to it, multiple slips when that parameter exceeds 4.604 and the energy-drop rule for low-speed friction. Sec. II.C: the Langevin equation with its noise (Eqs. 2 and 3), the ratio of pulling rate to hopping rate, thermolubricity (Eq. 4), the logarithmic law (Eq. 5) and the disagreement with low-temperature experiments. Sec. II.D: the Frenkel–Kontorova chain (Eq. 6) and the Aubry transition. Sec. II.E: superlubricity, the criticism of the term, the graphite-flake experiments it reports (below 50 pN in most orientations, about 250 pN near the matching ones, and a flake turning back into the matching orientation), Figs. 4 and 5 and the rule that the angular width of a friction peak is about one over the flake diameter. Sec. III.A: thermostats and the damping rate that no one knows how to choose. Sec. III.B: the times and sizes a simulation can reach and the gap of up to six orders of magnitude between experimental and simulated stick-slip speeds; its printed figure of one microsecond per simulation day is about nine times smaller than its own arithmetic gives. Sec. VI: conclusions. We recomputed the stick-slip parameter, the fold force and the angular widths with our own models; we did not digitise any figure.
- Y. Sang, M. Dubé and M. Grant, Thermal effects on atomic friction, arXiv:cond-mat/0109168v1; Physical Review Letters 87, 174301 (2001)
Version 1, 10 September 2001. Eq. (1): ramped creep, in which friction falls with the two-thirds power of temperature times the two-thirds power of the logarithm of speed over temperature. Eqs. (2) and (3): the Langevin equation and the tip potential, a cosine with its minima on the lattice sites. Eq. (4): the fold position. The parameters fitted to the NaCl data of Gnecco et al.: lattice period 0.4 nm, effective stiffness 0.86 N/m, corrugation 0.27 eV, damping rate 8.9 times 10 to the 5 per second, tip mass 8.7 times 10 to the minus 12 kg; footnote 21 gives a spring constant of 0.93 N/m. The text prints the spring resonance as 52 MHz; with this mass and spring it is 52.0 kHz, so we read MHz as a typo. Eq. (5): the Kramers rate. Eqs. (6) and (7): curvature and barrier near the fold. Eq. (8): the slip distribution. Eq. (9): the scaled speed. Eqs. (10) to (12): the closed form of the mean friction and its constants, quoted as 0.55 nN in theory against 0.54 nN, with an uncertainty of 0.01 nN, from simulation and experiment. We recomputed the scaled speed, the constants and the mean friction with our model; we did not digitise Figs. 1 to 3.
- E. Gnecco, R. Bennewitz, T. Gyalog, Ch. Loppacher, M. Bammerlin, E. Meyer and H.-J. Güntherodt, Velocity dependence of atomic friction, Physical Review Letters 84, 1172 (2000); abstract record in the University of Basel repository
Abstract only; we did not read the full letter. The abstract states that the measurement is between a friction force microscope tip and NaCl(100), that the mean friction grows logarithmically with velocity at low velocities, and that the data are interpreted with a modified Tomlinson model based on reaction-rate theory. These are the data Sang and co-workers fitted. We quote no number or figure from this paper.
- A. S. de Wijn, (In)commensurability, scaling and multiplicity of friction in nanocrystals and application to gold nanocrystals on graphite, arXiv:1208.1395v2; Physical Review B 86, 085429 (2012)
Version 2, 15 August 2012. Eq. (1): the hexagonal potential of one gold atom on graphite, a sum of three cosines with a prefactor of two ninths, and a single-atom corrugation of about 50 meV. Sec. II: graphite with 1.42 Å bonds and Au(111) with 2.885 Å spacing, a ratio of 1.354 in his length unit. Sec. IV: friction of a perfect crystal scales with contact area to the power zero, one half or one (Eq. 21). Sec. V: irregular outlines give powers of one quarter, three quarters or one (Eq. 31), and amorphous crystals the power one half. We built rigid flakes with this potential and fitted the exponents; we did not digitise his figures.
- D. A. Sivak, J. D. Chodera and G. E. Crooks, Using nonequilibrium fluctuation theorems to understand and correct errors in equilibrium and nonequilibrium simulations of discrete Langevin dynamics, arXiv:1107.2967v5; Physical Review X 3, 011007 (2013)
Version 5, 30 January 2013. Eq. (3): a Langevin step split into deterministic and heat-bath substeps. Eq. (5): the change in energy over a step equals heat plus protocol work plus shadow work, the error that a finite time step adds. We used this split to keep the energy ledger of the stick-slip simulation and checked that the shadow work shrinks fourfold when the step is halved.