L2 · Chapter 2

How the Tomlinson model explains stick-slip

Sandbox Physics · 2026-10-30

The last chapter estimated that the spring has to stretch by almost two lattice periods before it can match the strongest pull of the surface. This chapter turns that estimate into a model with one coordinate: the tip is a point mass, the surface is a cosine energy landscape, and a spring joins the tip to a scanner moving at constant speed. Prandtl and Tomlinson proposed the model in the 1920s, and it is still where atomic friction starts.

A point mass on a spring

Vanossi and co-workers describe the Prandtl–Tomlinson model like this: a point tip of mass $m$ is dragged over a one-dimensional sinusoidal potential that stands for its interaction with a crystalline substrate; a spring of stiffness $k$ joins the tip to a support driven at constant speed $v$ .

The total energy has two parts: the periodic potential of the substrate plus the elastic energy of the spring. We write the cosine as $-U_0\cos(2\pi x/a)$ , so the valleys sit on the lattice sites $x=na$ , as in Eq. (3) of Sang and co-workers; Eq. (1) of Vanossi and co-workers uses $+U_0$ , which only shifts the origin by half a period.

The model makes no attempt to describe how energy really flows into the substrate; all dissipation is replaced by a viscous force $-m\gamma\dot x$ . The instrument reads the spring force $F=k(X-x)$ , and kinetic friction is its time average.

U(x,X)=-U_0\cos\frac{2\pi x}{a}+\frac{k}{2}(x-X)^{2},\qquad X=vt,\qquad m\ddot x+m\gamma\dot x=-\frac{\partial U}{\partial x}

The support position

X

advances as

vt

;

\gamma

is the damping rate. This chapter keeps the temperature at zero; chapter 4 adds thermal noise.

The Prandtl–Tomlinson picture. A tip at position

x

sits in a cosine landscape of period

a

and amplitude

U_0

; a spring of stiffness

k

joins it to the scanner at

X=vt

. The force the instrument reads is

F=k(X-x)

. Original schematic, not to scale.

Source: A. Vanossi · Y. Sang

The stick-slip parameter: one valley or several

The tip rests at a minimum of the total energy. Whether there is only one minimum depends on the curvature. The curvature of the substrate potential swings between $\pm4\pi^2U_0/a^2$ ; the curvature of the spring is always $k$ . Their ratio is the stick-slip parameter $\eta$ .

If $\eta<1$ , the total curvature is positive everywhere and there is only one minimum for any support position. The tip follows the support continuously, with no jumps, and friction vanishes as the speed goes to zero. Vanossi and co-workers note that experiments at low load have indeed shown this smooth sliding with ultralow friction.

If $\eta>1$ , the total curvature turns negative over a band of positions, and the landscape holds two or more valleys at once. The tip sits in one valley until it disappears, then jumps to the next: that is stick-slip. $\eta=1$ marks the change from smooth sliding to slips of one lattice site.

\frac{\partial^{2}U}{\partial x^{2}}=k\left[1+\eta\cos\frac{2\pi x}{a}\right],\qquad \eta=\frac{4\pi^{2}U_0}{k a^{2}}

The bracket swings between

1-\eta

and

1+\eta

. The quantity

\Omega_k^2

used by Sang and co-workers is exactly

1/\eta

Source: A. Vanossi · Y. Sang

The fold: where the valley disappears

As the support advances, the valley holding the tip gets shallower and more tilted. At the moment it disappears, the minimum merges with the maximum ahead of it, and the first and second derivatives vanish together. Vanossi and co-workers call this a fold catastrophe; Eq. (4) of Sang and co-workers states the same condition.

A vanishing second derivative gives the tip position at the fold, $x_c$ ; putting it into the vanishing first derivative gives the spring force there, $F_c$ , and the support position $X_c$ . For $\eta\gg1$ , $x_c\to a/4$ and $F_c$ approaches the bound $2\pi U_0/a$ of chapter 1.

While the tip sticks, it creeps near the bottom of its valley as the support moves. The spring and the curvature of the valley bottom, $k_s$ , act in series, so the slope of the force–position trace is smaller than $k$ . Sang and co-workers used exactly this slope, 0.86 N/m, as their experimental input.

\cos\frac{2\pi x_c}{a}=-\frac{1}{\eta},\qquad F_c=\frac{2\pi U_0}{a}\sqrt{1-\eta^{-2}},\qquad X_c=x_c+\frac{F_c}{k}

Tip position, spring force and support position at the fold, all measured from the centre of the starting valley.

\frac{dF}{dX}=\frac{k\,k_s}{k+k_s}=\frac{k\,\eta}{1+\eta},\qquad k_s=\frac{4\pi^{2}U_0}{a^{2}}

The slope with which the force climbs while the tip sticks: two springs in series.

Source: A. Vanossi · Y. Sang

Worked example: where the tip of Sang and co-workers slips

Problem. $U_0=0.27\,\mathrm{eV}$ , $a=0.4\,\mathrm{nm}$ , $k=0.93\,\mathrm{N\,m^{-1}}$ , zero temperature. Find $\eta$ , the tip position, spring force and support position at the fold, and the slope while the tip sticks.

Numbers. $\eta=4\pi^2\times4.33\times10^{-20}/(0.93\times1.6\times10^{-19})=11.48$ . $\cos(2\pi x_c/a)=-0.0871$ , so $x_c=0.264\,a=0.106\,\mathrm{nm}$ . $F_c=0.680\times\sqrt{1-0.0076}=0.677\,\mathrm{nN}$ . $X_c=0.106+0.677/0.93=0.833\,\mathrm{nm}=2.08\,a$ . The sticking slope is $0.93\times11.48/12.48=0.855\,\mathrm{N\,m^{-1}}$ .

Comparison. The slope Sang and co-workers read from the experiment is 0.86 N/m, which agrees with 0.855 to two significant figures; their footnote 21 also says that with $U_0=0.27\,\mathrm{eV}$ this slope corresponds to $k=0.93\,\mathrm{N\,m^{-1}}$ .

Checks. The course’s zero-temperature simulation drags the tip at 10 nm/s; the tip crosses the midpoint between valleys when the support is at $X=2.086\,a$ , only 0.0026 periods after the fold. At 100 nm/s and 1000 nm/s the delay grows to 0.0125 and 0.059 periods: the tip needs time to leave a valley that is disappearing, and each tenfold increase in speed lengthens the delay about 4.8 times. That is an observation from our simulation, not a result from the papers.

Explanation. Starting from rest, the support must travel 2.08 periods before the first slip; after that every slip happens when the support has reached the same position relative to the new valley, so the sawtooth repeats with period exactly $a$ .

Source: Y. Sang · A. Vanossi

Try it in the Lab

Between smooth sliding and stick-slip

Predict first: with the corrugation lowered to 0.02 eV, what is $\eta$ ? Is the sawtooth still there?
Choose the Smooth sliding preset, read $\eta$ and the mean friction, and count the valleys in the Energy the tip feels now chart.
Drag the corrugation slider up slowly, note the reading of $\eta$ when the sawtooth first appears, and explain it with whether the total curvature can turn negative.

Pull the tip to the fold yourself

Predict first: as you drag the scanner forward slowly, how will the valley holding the tip change in the energy chart?
Choose Cold tip slips at the fold, grab the scanner in the scene and drag it forward slowly (the arrow keys also work), watching the barrier ahead value in the energy chart.
Note the reading just before it reaches zero, and explain why the slip waits for the moment the valley disappears.

Exercises with explained answers

Swap in a spring with $k=3\,\mathrm{N\,m^{-1}}$ and keep everything else. What are $\eta$ and the fold force? How stiff must the spring be for stick-slip to disappear?
Show the answer
$\eta=11.48\times0.93/3=3.56$ , so stick-slip remains. $F_c=0.680\times\sqrt{1-1/3.56^2}=0.652\,\mathrm{nN}$ . At $k=4\pi^2U_0/a^2=10.7\,\mathrm{N\,m^{-1}}$ , $\eta=1$ ; any stiffer and there is no stick-slip.
The Smooth sliding preset lowers the corrugation to 0.02 eV. What is the stick-slip parameter, and what happens to friction as the speed goes to zero?
Show the answer
$\eta=11.48\times0.02/0.27=0.85<1$ . The total curvature is positive everywhere, the tip always has a single resting place and follows the support continuously. Only the viscous drag $m\gamma v$ remains, and it vanishes as the speed goes to zero.
Why is the slope of the force–position trace while the tip sticks always smaller than the cantilever spring constant? What sets the ratio?
Show the answer
While the tip sticks, each small advance of the support also moves the tip a little near the valley bottom, so two springs act in series: the cantilever spring $k$ and the curvature of the valley bottom $k_s$ . The slope over $k$ is $\eta/(1+\eta)$ , 0.92 for the parameters of Sang and co-workers.

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.