L1 · Chapter 1

How large friction is at a single atomic contact

Sandbox Physics · 2026-10-30

Push a book across a table and the force you need hardly depends on how fast you push or how large the book is. Those rules describe millions of tiny contacts averaged together. A friction force microscope removes the average: its tip touches the surface through one contact a few atoms wide, and the force it reads is a fraction of a nanonewton. This chapter sets the scales and lists the classroom laws you should expect to fail.

Three classroom laws

Vanossi and co-workers open their colloquium by summarising the Coulomb–Amontons laws in three statements: friction does not depend on the apparent area of contact; friction is proportional to the normal load; and kinetic friction, the force that keeps sliding at constant speed, does not depend on the sliding speed and is smaller than static friction.

These laws rest on a mass of data and on an average over the many asperities that touch between two rough surfaces. The same colloquium warns that even for the best-studied nanoscale systems, such as a tip sliding on graphite or NaCl, the microscopic mechanism of friction is still not settled, and the speed and temperature dependence are usually explained with simplified models that carry empirical parameters.

This course tests the laws one by one. Chapters 4 and 5 test speed: the friction of a single tip grows with the logarithm of the scan speed. Chapters 6 and 7 test area: the friction of a perfect crystal flake can grow in proportion to its area or not grow at all. Load enters the model through the depth of the corrugation, which grows as the tip is pressed harder.

Source: A. Vanossi

One asperity

Sang and co-workers write in their 2001 letter that the tip of a friction force microscope is in essence a single asperity, which removes the complication of averaging over many. The tip sits at the end of a cantilever; the scanner moves the cantilever sideways, the cantilever twists when the surface holds the tip back, and the twist times the stiffness is the lateral force the instrument reads.

Vanossi and co-workers point out that the spring in the model stands for both the torsional stiffness of the cantilever and the lateral stiffness of the contact itself. The data Sang and co-workers modelled come from measurements by Gnecco and co-workers on NaCl(100): lattice constant $a=0.4\,\mathrm{nm}$ , normal load $0.65\,\mathrm{nN}$ , and a slope of $0.86\,\mathrm{N\,m^{-1}}$ in the force–position trace while the tip sticks.

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: Y. Sang · E. Gnecco · A. Vanossi

Three scales

The length scale is the lattice period $a$ . The energy the tip feels rises and falls with position; its amplitude is the corrugation $U_0$ . Sang and co-workers fitted $U_0=0.27\,\mathrm{eV}$ and a spring constant $k=0.93\,\mathrm{N\,m^{-1}}$ to the NaCl data.

The force scale comes from the steepest slope of the corrugation. With the energy written as $-U_0\cos(2\pi x/a)$ , the slope is at most $2\pi U_0/a$ : that is the largest force with which the surface can hold the tip back.

The energy scale has to be compared with thermal energy. At 293 K, $k_BT=0.0252\,\mathrm{eV}$ , about a tenth of the corrugation. Crossing the full barrier by thermal motion alone is very rare; but chapter 4 shows that the barrier becomes small just before a slip, and then thermal motion matters a great deal.

F_{\max}=\frac{2\pi U_0}{a},\qquad \frac{U_0}{k_BT}

Left: the largest lateral force the surface can exert. Right: the corrugation over the thermal energy. With the lattice period

a

, these set every scale of atomic friction.

Source: Y. Sang · A. Vanossi

Worked example: how hard can a salt crystal hold a tip

Problem. Take the NaCl parameters of Sang and co-workers: $U_0=0.27\,\mathrm{eV}$ , $a=0.4\,\mathrm{nm}$ , $k=0.93\,\mathrm{N\,m^{-1}}$ , at 293 K. What is the largest lateral force the surface can exert? How far must the spring stretch to supply it? What fraction of the corrugation is the thermal energy?

Model. Treat the surface as a cosine of amplitude $U_0$ and period $a$ ; its steepest slope is the largest lateral force. The spring force is stiffness times stretch.

Numbers. $0.27\,\mathrm{eV}=4.33\times10^{-20}\,\mathrm{J}$ , so the largest lateral force is $0.680\,\mathrm{nN}$ . The spring must stretch $0.731\,\mathrm{nm}$ , about 1.83 lattice periods. The thermal energy over the corrugation is $0.0252/0.27=0.093$ : the corrugation is 10.7 times the thermal energy.

Explanation. The spring has to stretch by almost two lattice periods before it can match the strongest pull of the surface. Until then the tip cannot follow the scanner and stays where it is; once the spring force exceeds what the surface can supply, the tip jumps. That is the stick-slip the next chapter makes precise. The largest lateral force, 0.680 nN, is about the same as the 0.65 nN normal load of the experiment.

Checks. Units: energy over length is force. Size: fractions of a nanonewton are exactly where the simulations of Sang and co-workers and the measurements of Gnecco and co-workers lie. Chapter 2 finds a fold force of 0.677 nN, just under this upper bound.

F_{\max}=\frac{2\pi\times0.27\,\mathrm{eV}}{0.4\,\mathrm{nm}}=0.680\,\mathrm{nN},\qquad \frac{F_{\max}}{k}=\frac{0.680\,\mathrm{nN}}{0.93\,\mathrm{N\,m^{-1}}}=0.731\,\mathrm{nm}=1.83\,a

The largest lateral force and the stretch the spring needs to supply it.

Source: Y. Sang · A. Vanossi

Try it in the Lab

Read the force scale

Predict first: at zero temperature, how close will the peaks of the force sawtooth come to the 0.680 nN bound? About how far does the force fall at each drop?
Choose the Cold tip slips at the fold preset and read the peak and the force after each drop in the Lateral force: the sawtooth chart.
Explain why the force falls suddenly at the peak instead of easing off: the energy valley holding the tip has just disappeared.

Change the corrugation and watch the force follow

Predict first: if the corrugation is halved from 0.27 eV, roughly where will the peaks of the sawtooth move?
Drag the corrugation slider to about 0.135 eV and read the new peak and the stick-slip parameter $\eta$ .
Explain the reading with the largest lateral force being proportional to the corrugation, then say why the mean friction falls by more than the peak does.

Exercises with explained answers

The corrugation is halved to 0.135 eV, with the same lattice period and spring. What is the largest lateral force the surface can exert, and how many lattice periods must the spring stretch to supply it?
Show the answer
The largest force is proportional to the corrugation, so it becomes $0.340\,\mathrm{nN}$ . The stretch is $0.340/0.93=0.365\,\mathrm{nm}$ , about 0.91 periods. The spring can pull the tip free before it stretches a whole period; the next chapter shows that this means weaker slips.
How hot would the crystal have to be for the thermal energy to equal the 0.27 eV corrugation?
Show the answer
$T=U_0/k_B=4.33\times10^{-20}/1.38\times10^{-23}\approx3.1\times10^{3}\,\mathrm{K}$ . That is far above room temperature, so at room temperature thermal motion almost never carries the tip over the whole corrugation; its role appears only when the barrier has become small.
At 293 K the model gives a mean friction of 0.324 nN at a scan speed of 100 nm/s and 0.462 nN at 3000 nm/s. If the third Amontons law held, what would the friction be at 3000 nm/s? What does the ratio tell you?
Show the answer
The law would predict 0.324 nN again. The model gives $0.462/0.324=1.43$ : thirty times the speed, 43% more friction. The kinetic friction of a single contact depends on speed, so friction divided by load is not a material constant here. Chapter 4 derives this logarithmic growth.

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.