FR40 · Thermally activated slips

Thermal Friction: Why Faster Scans Rub Harder

Why does atomic friction grow as the scan gets faster? Because heat gets less time to help the tip escape. A stuck tip can hop the barrier early when a thermal kick is large enough; slow scans give it more tries, so it slips at a lower force. In a model of a tip on salt at room temperature, friction rises from 0.25 to 0.46 nN between 10 and 3000 nm/s.

Six identical tips slide side by side, and heat makes each one slip at a different moment. Faster scans give heat less time, so the tips slip later and friction grows.

Simulated mean friction $0.351\,\mathrm{nN}$

Rate-theory friction $0.345\,\mathrm{nN}$

Spread of slip forces $0.0455\,\mathrm{nN}$

Slips recorded $28$

Six tips, one scanner, independent heat

Temperature

293\,\mathrm{K}

Scan speed

300\,\mathrm{nm\,s^{-1}}

Corrugation

U_0

0.27\,\mathrm{eV}

Pink flashes mark slips. Drag the scanner bar to pull all six tips at once.

Advanced: damping

Damping rate

8.9\times10^{5}\,\mathrm{s^{-1}}

The rate theory assumes heavy damping. Raise the damping and the simulated friction moves onto the rate-theory curve.

Model, units and sources

$r(X)=\frac{\Omega^{2}}{2\pi\gamma}\exp\!\left[-\frac{\Delta E(X)}{k_BT}\right],\qquad \frac{dP_{\text{stuck}}}{dX}=-\frac{r(X)}{v}P_{\text{stuck}}$

The escape rate is Eq. 5 of Sang, Dubé and Grant (2001), with the barrier and curvatures computed from the exact landscape at every scanner position. Each speed point averages three replicas of thirty slips; error bars are two standard errors. The closed-form curve uses their Eqs. 7 to 12 and is hidden where their scaled speed reaches 1.

Sang, Dubé & Grant (2001), PRL 87, 174301 · Vanossi et al. (2013), Rev. Mod. Phys. 85, 529

Follow the derivation in the course · Where the thermal noise comes from

Model assumptions

Each lane is the same Prandtl–Tomlinson tip as the stick-slip Lab, with the parameters Sang, Dubé and Grant (2001) chose for NaCl, and independent thermal noise. Slip forces are compared with a rate theory: the heavily damped escape rate of their Eq. 5, integrated over the exact barrier as the scanner advances. Their closed forms Eqs. 7 to 12 are shown as a second reference. The rate theory assumes heavy damping; at the fitted damping the simulated friction runs about 5% above it, and at ten times the damping the two agree within 1%.

Where the tips slip

Friction against scan speed

Force in each lane

Physics tutorial

Heat decides when the tip slips

BackgroundAt absolute zero a stuck tip waits until its well disappears. At room temperature the atoms around it shake, and every so often a kick is large enough to carry the tip over a barrier that has not yet vanished. The slip comes earlier and at a smaller force, and the moment is random.

Why it mattersFriction force microscopes on NaCl found that friction grows by a nearly fixed amount for every tenfold increase in scan speed. Sang, Dubé and Grant (2001) explained this with a thermally activated Tomlinson tip whose barrier shrinks as the scanner ramps it, and Vanossi et al. (2013) review the same mechanism as thermolubricity at its weak-corrugation end.

Start with the essentials

Focus question: Why does atomic friction grow as the scan gets faster?
One-sentence intuition: A thermal escape needs time. A slow scan gives the tip many attempts while the barrier is still high, so it escapes early at a low force. A fast scan pushes the tip close to the fold before the first successful attempt, so the slips happen at higher force and the mean friction rises with the logarithm of the speed.

Core mathematical model

Escape rate over the shrinking barrier

r=\frac{\Omega^{2}}{2\pi\gamma}\exp\!\left(-\frac{\Delta E}{k_BT}\right)

This is Eq. 5 of Sang, Dubé and Grant (2001), the heavily damped Kramers rate. $\Delta E$ and $\Omega$ change with the scanner position, and the barrier vanishes at the fold. The Lab computes both from the exact landscape.

Probability of still being stuck

P_{\text{stuck}}(X)=\exp\!\left[-\int^{X}\frac{r(X')}{v}\,dX'\right]

The scanner moves at speed $v$ , so the rate per unit distance is $r/v$ . Slower scans accumulate more escape probability at each position, which moves the slip-force distribution down.

Logarithmic growth with speed

\langle F\rangle\approx F_c-c\,T^{2/3}\left|\ln\frac{v}{T}\right|^{2/3}

Near the fold the barrier shrinks as the three-halves power of the distance to it, which gives this ramped-creep form, Eq. 1 of Sang et al., with units absorbed into the constants. Over 10 to 3000 nm/s the model friction rises from 0.254 to 0.462 nN at 293 K, still below the zero-temperature value of 0.533 nN.

Common difficulties

Slips happen at the same force

Typical misconceptionThe tip always slips when the force reaches one critical value.

Better mental modelThat is only true at zero temperature. At 293 K and 300 nm/s the slip force spreads by about 0.04 nN around 0.53 nN, well below the 0.68 nN fold force.

Rate theory is exact

Typical misconceptionThe simulated points should sit on the rate-theory curve.

Better mental modelThe rate formula assumes heavy damping. At the fitted damping the simulated friction runs about 5% above it; with ten times the damping the two agree within 1%. The gap is a known limit of the formula, not noise.

Run the experiment

01
Read the slip forces
Leave Room-temperature creep running until the histogram fills.
What to observe: The slips spread around 0.53 nN, close to the rate-theory distribution, and none reaches the fold line at 0.68 nN.
02
Predict, then cool the tips
Guess which way the histogram moves at 100 K, then choose Cold tips slip late.
What to observe: The distribution narrows and moves toward the fold; the mean friction rises toward the zero-temperature value.
03
Explain the speed dependence
Compare Slow scan rubs less with Fast scan rubs harder and read the friction-speed chart.
What to observe: Each tenfold increase in speed adds roughly 0.07 to 0.09 nN. The points follow a straight-ish rise on the logarithmic axis, as ramped creep predicts.

Thermal Friction: Why Faster Scans Rub Harder

Heat decides when the tip slips

Core mathematical model

Escape rate over the shrinking barrier

Probability of still being stuck

Logarithmic growth with speed

Common difficulties

Slips happen at the same force

Rate theory is exact

Run the experiment

Read the slip forces

Predict, then cool the tips

Explain the speed dependence

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