L2 · Chapter 4

Why thermal activation makes friction grow with the logarithm of speed

Sandbox Physics · 2026-10-30

At room temperature the tip never stops shaking. The shaking cannot carry it over the 0.67 eV barrier it faces at rest, but the further the scanner pulls, the lower the barrier gets, and thermal noise always kicks the tip over before the barrier reaches zero. The slower the pull, the longer the tip waits, the earlier it slips and the smaller the friction. This chapter turns that picture into formulas and compares them, one by one, with simulation and the numbers of Sang and co-workers.

A tip that shakes

Eqs. (2) and (3) of Vanossi and co-workers and Eq. (2) of Sang and co-workers state the same equation: the equation of motion of chapter 2 plus a random force $\xi(t)$ . The random force and the damping come from the same fast excitations that are not modelled explicitly, such as phonons, so their strengths cannot be chosen independently: the fluctuation–dissipation relation makes the noise strength proportional to $\gamma T$ .

With that relation, a tip held in a fixed valley reaches the right thermal equilibrium. In the course’s check, the variance of the tip position differs by 0.4% from a numerical integral of the Boltzmann distribution; the standard deviation is 0.019 nm, about a twentieth of the lattice period.

Shaking of that size cannot make the tip hop between resting valleys: with the scanner at $X=0$ the barrier ahead is 0.667 eV, about 26 $k_BT$ .

m\ddot x+m\gamma\dot x=-\frac{\partial U}{\partial x}+\xi(t),\qquad \langle\xi(t)\,\xi(t')\rangle=2m\gamma k_BT\,\delta(t-t')

The random force has zero mean and no memory; its strength is fixed by

\gamma

and

T

together.

Related on this site: A bead in an optical trap

Source: A. Vanossi · Y. Sang

The barrier vanishes before the fold

As the scanner advances, the valley holding the tip gets shallower. The course model gives a barrier of 0.241 eV with the support at $X=a$ , 0.094 eV at $1.5\,a$ , and only 0.005 eV at $2\,a$ ; the fold is at $2.08\,a$ .

Near the fold the energy curve can be written locally as $Fx-x^{3}$ , and the barrier vanishes as $f^{3/2}$ , where $f$ is the fraction still to go to the fold. Eq. (7) of Sang and co-workers gives this relation; Vanossi and co-workers call it the scaling of a fold catastrophe.

The rate of hopping over the barrier comes from the Kramers formula. Eq. (5) of Sang and co-workers uses the high-damping form, whose prefactor depends only on the curvatures at the valley bottom and barrier top and on the damping. With Sang’s parameters the course finds that prefactor is about $2\times10^{5}\,\mathrm{s^{-1}}$ ; Sang and co-workers also note that this attempt frequency is small compared with typical values for surface adatoms, which is why very low barriers matter.

\tau^{-1}=\frac{\Omega^{2}}{2\pi\gamma}\,e^{-\Delta E/k_BT},\qquad \Delta E\propto f^{3/2},\qquad f=1-\frac{X}{X_c}

\Omega^{2}

is set by the curvatures at the valley bottom and the barrier top; the course’s rate integral takes the square root of their product, which matches the form of Sang and co-workers near the fold.

Source: Y. Sang · A. Vanossi

From waiting to a logarithmic law

When does the tip hop? Roughly when the rate times the remaining time is about one. Each step of the scanner lowers the barrier as $f^{3/2}$ and raises the rate exponentially. So the barrier at the slip is about $k_BT\lvert\ln v^{*}\rvert$ , where $v^{*}$ is a dimensionless speed that combines damping, temperature and speed.

The barrier is proportional to $f^{3/2}$ , so at the slip the tip is still $f\propto(T\lvert\ln v^{*}\rvert)^{2/3}$ short of the fold. The further from the fold, the smaller the force at the slip and the smaller the mean friction. That is Eq. (10) of Sang and co-workers: friction grows with speed, but as $\lvert\ln v\rvert^{2/3}$ rather than the $\ln v$ of linear creep.

Eq. (5) of Vanossi and co-workers has the same form, with the warning that $\lvert\ln v\rvert^{2/3}$ and plain $\ln v$ are very hard to tell apart in experiments and simulations. The evidence Sang and co-workers offer is the way data at different temperatures collapse onto one line under their scaling.

From their Eqs. (9), (11) and (12), Sang and co-workers compute $F_c=0.55\,\mathrm{nN}$ , $\Delta F/T^{2/3}=1.91\times10^{-3}\,\mathrm{nN\,K^{-2/3}}$ and $Tv^{*}/v=3.9\times10^{-2}\,\mathrm{K\,s\,nm^{-1}}$ . Rederiving the same formulas independently, the course gets 0.542 nN, $1.91\times10^{-3}$ and $3.92\times10^{-2}$ . The values they extract from simulated and experimental data are $0.54\pm0.01\,\mathrm{nN}$ , $(1.82\pm0.02)\times10^{-3}$ and $(4.5\pm0.2)\times10^{-2}$ .

\langle F\rangle=F_c-\Delta F\,\lvert\ln v^{*}\rvert^{2/3},\qquad v^{*}\propto\frac{\gamma\,v}{k_BT}

Eq. (10) of Sang and co-workers.

F_c

and

\Delta F

come from Eqs. (11) and (12); the full form of

v^{*}

is Eq. (9).

Mean friction of the NaCl tip of Sang et al. at 293 K against scan speed, logarithmic speed axis. Dots with bars: the course Langevin simulation, mean and standard error of four runs. Solid line: the slip rate integrated along the full landscape. Dashed line: the closed form

F_c-\Delta F\,|\ln v^{*}|^{2/3}

of Sang et al. Grey: the zero-temperature value

0.533\,\mathrm{nN}

. Numbers from the FR40 benchmark record.

Source: Y. Sang · A. Vanossi · E. Gnecco

Worked example: friction of a room-temperature tip at 300 nm/s

Problem. Parameters of Sang and co-workers, $T=293\,\mathrm{K}$ , $v=300\,\mathrm{nm\,s^{-1}}$ . Use their closed form to compute the mean friction, find the barrier at which the tip slips, and compare with simulation.

Step 1, the dimensionless speed. $v^{*}=0.03924\times300/293=0.0402$ , $\lvert\ln v^{*}\rvert=3.214$ , $3.214^{2/3}=2.178$ .

Step 2, friction. At 293 K, $\Delta F=1.91\times10^{-3}\times293^{2/3}=0.0843\,\mathrm{nN}$ . $\langle F\rangle=0.542-0.0843\times2.178=0.358\,\mathrm{nN}$ .

Step 3, the barrier at the slip. By the estimate of the last section the barrier is about $3.21\,k_BT=0.081\,\mathrm{eV}$ ; on the course’s energy landscape that puts the support at $1.55\,a$ , where the spring force is 0.525 nN, against a fold force of 0.677 nN.

Checks. Over 400 simulated slips the mean friction is $0.362\pm0.002\,\mathrm{nN}$ , 1.1% above the closed form; the tips slip on average at $1.57\,a$ . Integrating the Kramers rate along the true energy landscape gives 0.345 nN, 5% below the simulation, because the high-damping assumption does not fully hold at Sang’s damping (chapter 5).

Explanation. The tip does not wait for the barrier to vanish; it hops while the barrier is still 3 to 4 $k_BT$ , about an eighth of the barrier at rest. Higher temperature or a slower scan makes it hop earlier and lowers friction: at 100 nm/s the simulated friction is 0.448, 0.324 and 0.280 nN at 100 K, 293 K and 373 K.

Source: Y. Sang · A. Vanossi

Try it in the Lab

Where the six tips slip

Predict first: six tips are pulled by the same scanner. Will they slip at the same force? Earlier or later than the zero-temperature fold force?
With the Room-temperature creep preset, let the Where the tips slip histogram collect a few dozen slips, and read the Spread of slip forces.
Choose Cold tips slip late, compare the position and width of the histogram, and use a barrier of about $k_BT\lvert\ln v^{*}\rvert$ to explain why cold tips slip closer to the fold with a narrower spread.

Friction grows with the logarithm of speed

Predict first: if the scan speed rises from 10 nm/s to 3000 nm/s, three hundred times faster, how much will the mean friction rise?
Choose Slow scan rubs less and then Fast scan rubs harder, read the simulated and rate-theory friction, and find the matching points on the Friction against scan speed chart.
Estimate the difference in friction between the two speeds with $\lvert\ln v^{*}\rvert^{2/3}$ , compare it with the readings, and explain why every point lies below the zero-temperature value.

Exercises with explained answers

At 293 K and 10 nm/s, estimate the mean friction from the closed form of Sang and co-workers. (Hint: $v^{*}$ is proportional to speed.)
Show the answer
$v^{*}=0.0402\times10/300=0.00134$ , $\lvert\ln v^{*}\rvert=6.62$ , $6.62^{2/3}=3.52$ . $\langle F\rangle=0.542-0.0843\times3.52=0.245\,\mathrm{nN}$ . The course simulation gives $0.254\pm0.001\,\mathrm{nN}$ .
Why does atomic friction fall with temperature in this model, when the friction of many materials does not behave that way at large scale?
Show the answer
Thermal noise helps the tip over the barrier early, so the spring is stretched less at the slip, the energy drop is smaller and friction falls. Vanossi and co-workers point out that this assumes temperature changes no other surface or material parameter; measurements have found a friction peak at cryogenic temperatures that the Prandtl–Tomlinson model cannot fully reproduce.
If you replaced $\lvert\ln v^{*}\rvert^{2/3}$ with $\lvert\ln v^{*}\rvert$ , could data between 10 and 3000 nm/s tell the two apart?
Show the answer
Over this range $\lvert\ln v^{*}\rvert$ runs from 6.62 to 0.91, and both forms can be fitted roughly with suitable constants. Vanossi and co-workers also note that the two are hard to tell apart in experiments and simulations. A stronger test is to change temperature: Sang’s form requires friction to scale with $T^{2/3}$ , so data at different temperatures should collapse onto one curve.

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.