FR40 · Prandtl–Tomlinson model

Atomic Stick-Slip: How an AFM Tip Feels Friction

Why does a tip dragged across a crystal move in jumps? Because the spring holding the tip is too soft to stop it catching in the hollows between atoms. The tip waits in one hollow while the spring stretches, then the hollow vanishes and it falls into the next. The energy released in each fall becomes heat, and that heat per atom spacing is the friction. With the parameters fitted to salt, the slow, cold tip slips at about 0.68 nN and averages 0.53 nN.

Drag a spring-held tip across a salt crystal and watch it stick, then jump one atom ahead. The sawtooth force and the heat each jump leaves are the friction.

Stick-slip parameter

\eta

$11.5$

Mean friction force $0.324\,\mathrm{nN}$

Heat per lattice period $0.809\,\mathrm{eV}$

Mean slip length $1\,a$

A tip on a spring over a salt crystal

Corrugation

U_0

0.27\,\mathrm{eV}

Temperature

293\,\mathrm{K}

Scan speed

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

Spring constant

0.93\,\mathrm{N\,m^{-1}}

Drag the purple scanner left or right, or press the arrow keys on it. The tip sticks, then jumps one atom.

Advanced: damping

Damping rate

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

Damping sets how fast the tip loses the energy of a slip. Ten times lighter damping than the NaCl fit lets the tip overshoot into the next well.

Model, units and sources

$M\ddot x=-\partial_x E(x;X)-M\gamma\dot x+\xi(t),\quad \langle\xi(t)\xi(t')\rangle=2M\gamma k_BT\,\delta(t-t')$

$a=0.4\,\mathrm{nm},\quad M=8.7\times10^{-12}\,\mathrm{kg},\quad \gamma=8.9\times10^{5}\,\mathrm{s^{-1}}$

The equation of motion is Eq. 2 of Sang, Dubé and Grant (2001), integrated with a Langevin splitting step of one tenth of the well period. The dashed reference is the zero-temperature quasi-static path, computed from the energy landscape alone. The scene is schematic and slowed about a hundred million times.

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

Follow the derivation in the course · Compare with sliding friction on stacked blocks

Model assumptions

The tip is the one-dimensional Prandtl–Tomlinson model: a point mass on a spring above a sinusoidal substrate potential, with Langevin damping and thermal noise. The default parameters are those Sang, Dubé and Grant (2001) chose to match NaCl measurements: lattice period 0.4 nm, corrugation 0.27 eV, spring 0.93 N/m, mass 8.7e-12 kg and damping 8.9e5 1/s. Wear, contact-area changes, the second lateral direction and the elasticity of the cantilever beyond one spring are left out. The motion on screen is slowed by about eight orders of magnitude.

Energy the tip feels now

Lateral force: the sawtooth

Where the work goes

Physics tutorial

Friction one atom at a time

BackgroundAn atomic force microscope drags a sharp tip across a surface while a soft cantilever holds it. On a clean crystal the measured lateral force is not steady. It rises slowly, drops suddenly, and repeats with the spacing of the surface atoms. That sawtooth is friction seen at its smallest scale.

Why it mattersThe Prandtl–Tomlinson model explains the sawtooth with only a spring and a wavy potential. It says when a tip sticks, how much force it needs before it jumps, and where the work done by the scanner goes. It is the starting point for almost every later model of atomic friction.

Start with the essentials

Focus question: Why does a tip dragged across a crystal move in jumps?
One-sentence intuition: While the spring is soft compared with the curvature of the substrate potential, the tip has more than one place to rest. It stays in its well while the spring stretches, until that well disappears and the tip falls into the next one. The energy released in the fall turns into heat, and that heat per period, divided by the lattice period, is the friction force.

Core mathematical model

Energy of the tip

E(x;X)=-U_0\cos\frac{2\pi x}{a}+\frac{k}{2}(x-X)^{2}

The tip at $x$ feels the substrate with amplitude $U_0$ and period $a$ , and a spring of stiffness $k$ ties it to the scanner at $X$ . This is Eq. 1 of Sang, Dubé and Grant (2001).

When the tip can stick

\eta=\frac{4\pi^{2}U_0}{k a^{2}}>1

With $U_0=0.27\,\mathrm{eV}$ , $k=0.93\,\mathrm{N\,m^{-1}}$ and $a=0.4\,\mathrm{nm}$ the value is $\eta=11.5$ . Below 1 the energy has a single minimum for every scanner position, so the tip follows smoothly and the zero-temperature friction vanishes.

Friction from the energy ledger

\langle F\rangle=\frac{Q_{\text{period}}}{a}

Over many periods the spring and the tip return to the same state, so all work done by the scanner leaves as heat $Q_{\text{period}}$ . At room temperature and $100\,\mathrm{nm\,s^{-1}}$ the simulated tip gives $0.324\,\mathrm{nN}$ , so each period leaves about $0.81\,\mathrm{eV}$ .

Common difficulties

The jump is the friction

Typical misconceptionFriction is the force the tip feels while it rubs against each atom.

Better mental modelThe substrate force averages to zero over a period. What remains is the energy that the slip releases and the damping removes; without slips, a slowly dragged tip returns all the work it receives.

Heat makes friction larger

Typical misconceptionA warmer tip should rub harder.

Better mental modelThermal kicks let the tip hop over the barrier before the well vanishes, so it slips at a smaller force. In this model room temperature lowers the friction from 0.53 nN to about 0.32 nN at 100 nm/s.

Run the experiment

01
Watch one slip
Choose Cold tip slips at the fold and watch the force chart and the energy chart together.
What to observe: The force climbs to about 0.68 nN, the barrier ahead shrinks to zero, and the tip falls one lattice period. The mean force is about 0.53 nN, on the $T=0$ reference.
02
Predict, then soften the substrate
Guess what happens when $\eta$ drops below 1, then choose Smooth sliding.
What to observe: At 0.02 eV, $\eta=0.85$ : the sawtooth disappears, the tip follows the scanner and the mean friction falls to almost nothing.
03
Explain longer jumps
Choose Long slips, where the damping is ten times lighter, and read the slip length.
What to observe: The tip overshoots and lands two wells ahead. Because it releases less energy per period, the mean friction drops to about 0.37 nN.

Atomic Stick-Slip: How an AFM Tip Feels Friction

Friction one atom at a time

Core mathematical model

Energy of the tip

When the tip can stick

Friction from the energy ledger

Common difficulties

The jump is the friction

Heat makes friction larger

Run the experiment

Watch one slip

Predict, then soften the substrate

Explain longer jumps

Related experiments and articles

From one sticking tip to a twisted flake that glides