L4 · Chapter 8

Reading the 2013 friction-modelling review by Vanossi and co-workers, section by section

Sandbox Physics · 2026-10-30

Open arXiv:1112.3234v4, published as Reviews of Modern Physics 85, 529 (2013). It is a Colloquium that runs from the nanoscale to the mesoscale; this course uses eight of its sections. After this chapter you can read those sections in order, know which chapter of the course recomputed each result, and check one of the review’s order-of-magnitude estimates yourself.

First, know what kind of paper this is

This is a Colloquium, not a complete survey. Sec. II covers minimal models, Sec. III molecular dynamics, Sec. IV multicontact models and Sec. V a few special frictional phenomena. The authors say in the introduction that they draw examples from their own experience, and Sec. III.D states that its results come mostly from their own groups and certainly do not review the whole field. Read it as a map of models with examples chosen by the authors.

Sec. I states the gap this course keeps returning to: even for the most studied nanoscale systems, such as an atomic force microscope tip sliding on graphite or NaCl, the microscopic mechanism of friction is still lacking, and the observed dependence on speed and temperature has so far been explained with simplified models that contain empirical parameters.

Source: A. Vanossi

After Sec. I: the order for the other seven sections

Sec. II.A (page 3 of the arXiv version): Eq. (1) of the Prandtl–Tomlinson model, the stick-slip parameter, the fold where a slip starts, the barrier that shrinks as a $3/2$ power near it, the multiple slips that appear at low damping when the parameter exceeds 4.604, and the rule that low-speed friction equals the energy drop. Chapters 2 and 3 of the course recompute each of these.

Sec. II.C (page 4): the Langevin equation with its noise (Eqs. 2 and 3), the dimensionless ratio that separates the two regimes, thermolubricity (Eq. 4) and the logarithmic law (Eq. 5). The section ends by admitting that recent experiments see a friction peak at low temperature, in strong disagreement with Eqs. (4) and (5), and that the Prandtl–Tomlinson model fails to reproduce the observed temperature and speed dependence or the force traces measured with atomic resolution. This matches chapters 4 and 5.

Secs. II.D and II.E (pages 5 to 8): the Frenkel–Kontorova chain, the Aubry transition, superlubricity, the criticism of the term, and Figs. 4 and 5. These match chapters 6 and 7.

Secs. III.A and III.B (pages 10 to 12): how thermostats remove Joule heat, how damping is chosen, and what times and sizes a simulation can cover. Finish with the conclusions of Sec. VI (page 21). The other sections, such as Secs. II.B, II.F, III.C, III.D, IV and V, can wait until after this course.

Source: A. Vanossi

Recompute as you read

What can be recomputed and agrees. The multiple-slip threshold of 4.604: the tip of Sang and co-workers has $\eta=11.48$ ; dragged at 10 nm/s at zero temperature with the damping cut to 0.3 of its value, each slip jumps two valleys instead of one, and friction falls from 0.534 nN to 0.367 nN, within 0.2% of the energy-drop rule. The logarithmic law of Eq. (5): between 10 and 3000 nm/s the course simulation differs from the closed form of Sang and co-workers by 0.15% to 3.5%. The width rule $\tan\Delta\Phi=1/D$ : it is about 6% below the first zero of a disk and about 1.6 times its half width (chapter 7).

What needs a condition. The review says thermolubricity typically needs speeds below 1 nm/s at room temperature. For the 0.27 eV tip of Sang and co-workers, the course estimates that the speed scale for hopping back and forth at 293 K is only about 0.003 nm/s; lower the corrugation to 0.06 eV and the tip already hops between two valleys at 10 nm/s (chapter 5). The bound depends strongly on the corrugation.

What cannot be recomputed. The course did not compute the critical stiffness of the Aubry transition, $K_c\approx1.0291926$ . The 50 pN and 250 pN of the graphite-flake experiments come from original papers the course has not read. The model curve of Fig. 4 cannot be redone either: the review says only that the potential is a sum of pairwise interactions between carbon atoms and gives no parameters.

Source: A. Vanossi · Y. Sang

Worked example: recomputing the time-scale estimate of Sec. III.B

Problem. Sec. III.B estimates that a medium-size parallel computer completes about $10^{10}$ atom-steps per second; with time steps of order a femtosecond, a simulation of $10^{5}$ atoms advances $10^{5}\,\mathrm{fs}$ per second, which it calls about $10^{9}\,\mathrm{fs}$ , or 1 microsecond, per simulation day. At the friction-force-microscope speed of about 1 micrometre per second, a typical run moves the tip only about 1 pm. Check whether these numbers agree with each other and whether the conclusion holds.

Step 1, steps per second. $10^{10}/10^{5}=10^{5}$ steps; at 1 fs per step that is $10^{5}\,\mathrm{fs}$ per second, as the review says.

Step 2, how far one day goes. A day has 86 400 seconds, which gives $8.6\times10^{9}\,\mathrm{fs}$ , about 8.6 microseconds, roughly 9 times the printed $10^{9}\,\mathrm{fs}$ . The two printed numbers cannot both hold, and the text does not say which one is meant.

Step 3, how far the tip moves. With 8.6 microseconds the tip moves 8.6 pm at 1 micrometre per second; with the review’s 1 microsecond it moves 1 pm. The lattice period of Sang and co-workers is $a=0.4\,\mathrm{nm}$ , which takes $4\times10^{-4}\,\mathrm{s}$ of simulated time: about 46 days by our arithmetic, about 400 days by the review’s.

Explanation. The printed numbers are off by about a factor of 9, but the conclusion stands: at experimental speeds a full atomistic simulation does not see even one atomic-scale slip. That is why the course uses the Prandtl–Tomlinson model, with a single degree of freedom: it can run at experimental speeds, at the price of dropping every internal degree of freedom of the substrate and the tip. The same section notes that in experiments stick-slip disappears above about 1 micrometre per second, whereas in molecular dynamics the transition happens at metres per second, a gap of up to 6 orders of magnitude.

N\,N_{\mathrm{step}}\simeq10^{10}\ \mathrm{s^{-1}},\qquad N=10^{5}\;\Rightarrow\;N_{\mathrm{step}}\simeq10^{5}\ \mathrm{s^{-1}},\qquad 10^{5}\ \mathrm{fs\,s^{-1}}\times86\,400\ \mathrm{s}=8.6\times10^{9}\ \mathrm{fs}

The chain of estimates in Sec. III.B. The last step uses 86 400 seconds per day and comes out about 9 times larger than the printed number.

\Delta x=v\,t_{\mathrm{sim}}=10^{-6}\ \mathrm{m\,s^{-1}}\times8.6\times10^{-6}\ \mathrm{s}=8.6\ \mathrm{pm},\qquad \frac{a}{v}=\frac{0.4\ \mathrm{nm}}{10^{-6}\ \mathrm{m\,s^{-1}}}=4\times10^{-4}\ \mathrm{s}

At a scan speed of 1 micrometre per second: how far the tip moves in one day of simulated time, and how much simulated time one 0.4 nm lattice period takes.

Source: A. Vanossi · Y. Sang

What the review explicitly leaves open

How to choose the damping. Sec. III.A says that in the Prandtl–Tomlinson and Frenkel–Kontorova models the damping $\gamma$ changes the kinetic and frictional properties, yet there is no clear way to choose its value; the authors call this a lamentable state of affairs common to all molecular-dynamics friction simulations. The course keeps the value $8.9\times10^{5}\,\mathrm{s^{-1}}$ fitted by Sang and co-workers. The multiple slips of the previous section and the remaining friction of the twisted flake in chapter 7 both depend directly on this choice.

The friction peak at low temperature. Sec. II.C reports analyses of the Prandtl–Tomlinson model showing that friction may peak where the motion changes from multiple slips at low temperature to single slips at higher temperature; the same paragraph admits that the model fails to reproduce the observed temperature and speed dependence. The course model cannot settle this either.

No theory of friction yet. The conclusions of Sec. VI say that the Prandtl–Tomlinson model is extremely useful despite its deceptive simplicity, but that its success tends to hide the real complexity of the phenomenon. The authors also write that we do not have a proper theory of friction, one that could calculate the frictional work quantitatively, not just simulate it, in the cases where linear response fails, which are the majority; and that molecular dynamics has so far been strictly classical, with the importance of quantum effects still to be gauged.

Source: A. Vanossi · Y. Sang

Try it in the Lab

Damping is a choice we make

Predict first: at zero temperature, if the damping is lowered, how many valleys will each slip cross? Will the mean friction rise or fall?
Choose Cold tip slips at the fold and read the mean friction force and the mean slip length. Then open Advanced: damping, lower the damping rate to about 0.3 of its value, wait for a few slips and read them again.
Compare with the Long slips preset, use the energy-drop rule of chapter 3 to explain why friction falls when each slip crosses two valleys, and say why the worry of Sec. III.A applies here.

Does friction grow exactly as a logarithm

Predict first: if friction grew exactly as $\ln v$ , how should the increase in friction change from one tenfold step in speed to the next?
Choose Room-temperature creep, set the scan speed to 10, 100 and 1000 nm/s in turn, wait each time until more than 30 slips are recorded, and read the simulated mean friction.
Compute the increase over each tenfold step, compare with the closed-form curve in the Friction against scan speed chart, decide whether the increase grows with speed, and set this against the review’s remark that the two forms are hard to tell apart.

Exercises with explained answers

With the review’s $N\,N_{\mathrm{step}}\simeq10^{10}\,\mathrm{s^{-1}}$ , how much wall-clock time does a simulation of $10^{6}$ atoms with 1 fs steps need for a tip moving at 1 micrometre per second to cross one 0.4 nm period?
Show the answer
$10^{4}$ steps per second, so $10^{-11}\,\mathrm{s}$ of simulated time per second. The crossing needs $4\times10^{-4}\,\mathrm{s}$ of simulated time, so the wall-clock time is $4\times10^{7}\,\mathrm{s}$ , about 460 days.
The course simulation gives mean friction of 0.2535, 0.3240 and 0.4113 nN at 10, 100 and 1000 nm/s, with standard errors of about 0.0014, 0.0017 and 0.0033 nN. Can these numbers tell a pure logarithm $\ln v$ from a law of the form $(\ln v)^{2/3}$ ?
Show the answer
The increases over the two tenfold steps are $0.0705\pm0.0022$ and $0.0873\pm0.0037\,\mathrm{nN}$ . A pure logarithm needs them equal; they differ by 0.017 nN, about 4 times the uncertainty of 0.004 nN. The closed form of Sang and co-workers gives 0.0737 and 0.0890 nN, which also grow with speed. So a simulation with 400 slips per speed can tell them apart; an experiment with fewer slips, calibration errors and a limited range of speeds finds it much harder.
Sec. III.A says there is no clear way to choose the damping. Which results of the course depend on that choice, and which quantity does the review say is independent of it?
Show the answer
Dependent: the slip length and friction at zero temperature (damping cut to 0.3 lowers friction from 0.534 to 0.367 nN); the remaining friction of the twisted flake at large angles, which is exactly $N\gamma v$ ; and whether the strongly damped closed form applies (chapter 5). After Eq. (4) the review states that the thermolubric coefficient $\alpha(T)$ is the equilibrium damping felt by the tip and is independent of the ad hoc $\gamma$ .

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.