L3 · Chapter 5

Thermolubricity and the limits of rate theory

Sandbox Physics · 2026-10-30

The logarithmic law of the last chapter assumes the tip only jumps forward. If the scanner is slow enough that the tip can hop back and forth many times per period, it stays close to thermal equilibrium, friction becomes proportional to speed, and it vanishes as the speed goes to zero. Vanossi and co-workers, citing work by Krylov and others, call this thermolubricity. This chapter estimates the speeds at which it sets in, shows it on a weakly corrugated tip in the Lab, and then tests when the rate theory of the last two chapters holds.

Two regimes and the number that separates them

Vanossi and co-workers introduce a dimensionless ratio $\delta$ : the pulling rate $v/a$ divided by the rate of thermally activated hops over the barrier. For $\delta\ll1$ the tip has time to hop back and forth across the barrier, and friction is proportional to speed, their Eq. (4); for $\delta\gg1$ thermal fluctuations only occasionally help the tip over the barrier early, which is the stick-slip of the last chapter.

They write that thermolubricity appears at very low speed or high temperature, typically $v<1\,\mathrm{nm\,s^{-1}}$ at room temperature. Sang and co-workers give a stricter bound for their parameters: linear creep only appears below 0.1 nm/s.

Whether the tip can hop back and forth depends on the barrier when the two valleys have equal energy. That happens with the scanner halfway between two sites, $X=a/2$ . The course model gives this barrier $\Delta E_{1/2}$ , and the Kramers rate then estimates the speed scale of back-and-forth hopping.

\delta=\frac{v/a}{\omega_0\,e^{-\Delta E/k_BT}},\qquad F_k=\alpha(T)\,v\quad(\delta\ll1)

Eq. (4) of Vanossi and co-workers and the condition for it.

\omega_0

is an attempt frequency and

\alpha(T)

is the equilibrium damping the tip feels, independent of the ad hoc

\gamma

of the model.

v_{\mathrm{hop}}\approx a\,\frac{\Omega^{2}}{2\pi\gamma}\,e^{-\Delta E_{1/2}/k_BT}

Each hop moves the tip by one period; below this speed scale the tip has time to hop back and forth.

Source: A. Vanossi · Y. Sang

Seeing thermolubricity in the Lab

The Lab’s Thermolubricity preset lowers the corrugation to 0.06 eV and drags at 10 nm/s. $\eta=2.55$ , so at zero temperature the tip still sticks, with a quasi-static friction of 0.046 nN and a fold force of 0.139 nN. But when the two valleys have equal energy, the barrier is only 0.0385 eV, or 1.52 $k_BT$ .

Over 80 periods, the course simulation records the tip’s valley every 50 time steps: 3288 forward crossings and 3208 backward, about 40 round trips per period. The mean friction is $(0.2\pm3.5)\times10^{-4}\,\mathrm{nN}$ , zero within the statistical error and more than two orders of magnitude below the zero-temperature value.

At higher speed the round trips cannot keep up and friction appears: 0.0044, 0.0098 and 0.027 nN at 100, 300 and 1000 nm/s, with errors of about 0.001 nN. At 1000 nm/s it is already about six tenths of the zero-temperature value, and the tip is returning to stick-slip.

The Kramers formula cannot give an accurate hopping rate here: it requires a barrier well above $k_BT$ , and with a barrier of 1.52 $k_BT$ the tip leaves the valley before it has equilibrated there. The formula gives about $8\times10^{3}\,\mathrm{s^{-1}}$ , which is good only as an order of magnitude.

Source: A. Vanossi · Y. Sang

Where rate theory holds

Damping. Eq. (5) of Sang and co-workers is the high-damping form, and they note that it corresponds to the limit $\gamma\gg\Omega$ . But their damping, $8.9\times10^{5}\,\mathrm{s^{-1}}$ , is below the angular frequency of the valley bottom, $1.15\times10^{6}\,\mathrm{s^{-1}}$ . The course raises the damping tenfold and lowers the speed tenfold, keeping $v^{*}$ fixed: the simulated friction is 0.3437 nN and the rate integral 0.3448 nN, 0.3% apart. At Sang’s damping, the simulation at the same $v^{*}$ gives 0.362 nN, 5% above the rate integral; the excess comes from inertia.

The closed form. In both cases the closed form gives 0.358 nN, because it depends only on $v^{*}$ . Under heavy damping it is 4% above the exact rate integral, the price of keeping only the leading logarithmic term. So the 1.1% agreement of the last chapter comes partly from two errors cancelling.

The distribution of slip forces. Compared by Kolmogorov distance at Sang’s damping, 400 slips at 30 nm/s lie 0.29 from Eq. (8) and 0.21 from the rate integral, far above the 1% critical value of 0.08. With ten times the damping, 300 slips lie 0.039 from the rate integral, below the critical value of 0.094, and 0.102 from Eq. (8), just above it. Sang and co-workers say the distribution in their Fig. 3 agrees very well; we have not digitised that figure and cannot tell which conditions it shows.

The time step. At 1000 nm/s the default time step gives 0.4125 nN and half the step gives 0.4115 nN, a difference smaller than their statistical errors.

The upper speed limit. Footnote 18 of Sang and co-workers says ramped creep holds only for speeds below the order of $F_c/(M\gamma)$ , about 70 micrometres per second for their parameters.

Source: Y. Sang · A. Vanossi

Worked example: how slowly must the tip of Sang and co-workers move to be thermolubric

Problem. $U_0=0.27\,\mathrm{eV}$ , $T=293\,\mathrm{K}$ , other parameters as before. Estimate the scan speed at which the tip starts to hop back and forth, and compare it with the bounds of Sang and co-workers and of Vanossi and co-workers.

Step 1, the barrier at the symmetric point. With the scanner at $X=a/2$ , the course model gives $\Delta E_{1/2}=0.433\,\mathrm{eV}=17.2\,k_BT$ .

Step 2, the hopping rate. The prefactor is $2.15\times10^{5}\,\mathrm{s^{-1}}$ , $e^{-17.16}=3.5\times10^{-8}$ , and the rate is $7.6\times10^{-3}\,\mathrm{s^{-1}}$ . A barrier of 17 $k_BT$ is high enough for the Kramers formula to be reliable.

Step 3, the speed scale. $v_{\mathrm{hop}}\approx0.4\,\mathrm{nm}\times7.6\times10^{-3}\,\mathrm{s^{-1}}=3\times10^{-3}\,\mathrm{nm\,s^{-1}}$ .

Comparison. That is about 30 times below the bound of 0.1 nm/s from Sang and co-workers, and far below the typical 1 nm/s of Vanossi and co-workers. There is no contradiction: the other two are an upper bound and a typical order of magnitude, not estimates for this parameter set. Sang and co-workers simulated speeds from 5 nm/s upwards, all far above this scale, so their tips always stuck and slipped.

Explanation. Each $k_BT$ taken off the barrier raises this speed scale about 2.7 times. With the corrugation lowered to 0.06 eV, the barrier is only 1.5 $k_BT$ and the same estimate gives micrometres per second, so at the Lab’s 10 nm/s the tip hops back and forth many times per period.

Source: Y. Sang · A. Vanossi

Try it in the Lab

A tip that hops back and forth

Predict first: with 0.06 eV corrugation, 10 nm/s and room temperature, will the mean friction be near the zero-temperature 0.046 nN, or near zero?
Choose the Thermolubricity preset, watch the tip shuttle between two valleys in the energy chart, and read the mean friction force.
Drag the temperature to 0 K and watch again, then explain why the same corrugation gives almost no friction at room temperature but a sawtooth at zero temperature.

Raise the damping and see when rate theory is accurate

Predict first: with ten times the damping, will the simulated friction move towards the rate-theory curve or away from it?
Open Advanced: damping, move the damping rate from $8.9\times10^{5}$ to about $8.9\times10^{6}\,\mathrm{s^{-1}}$ , lower the scan speed to a tenth, and read the simulated and rate-theory friction.
Compare their difference with the default setting, and use inertia to explain why rate theory is more accurate at high damping.

Exercises with explained answers

Raise the temperature to 373 K. What does the speed scale for back-and-forth hopping of the tip of Sang and co-workers become? ( $\Delta E_{1/2}=0.433\,\mathrm{eV}$ , prefactor unchanged.)
Show the answer
$k_BT=0.0321\,\mathrm{eV}$ , so the barrier is $13.5\,k_BT$ . The rate is $2.15\times10^{5}\times1.4\times10^{-6}=0.30\,\mathrm{s^{-1}}$ and the speed scale $0.4\,\mathrm{nm}\times0.30\,\mathrm{s^{-1}}=0.12\,\mathrm{nm\,s^{-1}}$ , 40 times larger than at 293 K.
Why can the Kramers formula not give an accurate hopping rate for the tip with 0.06 eV corrugation?
Show the answer
The Kramers formula assumes the tip first reaches thermal equilibrium in the valley and only occasionally crosses a high barrier, which requires a barrier well above $k_BT$ . 0.0385 eV is only 1.52 $k_BT$ ; the tip diffuses almost freely between two shallow valleys, and the formula gives only an order of magnitude.
The closed form gives 0.358 nN both for damping $\gamma$ at 300 nm/s and for damping $10\gamma$ at 30 nm/s, while the simulations give 0.362 and 0.344 nN. Why is the closed form the same, and what does the difference between the simulations show?
Show the answer
The closed form depends on damping and speed only through $v^{*}\propto\gamma v$ , which is the same in both cases. The difference between the simulations comes from inertia: at low damping the tip keeps some inertia in its valley, and the high-damping prefactor of rate theory loses accuracy.

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.