L4 · Chapter 7

Twisted flakes and structural superlubricity

Sandbox Physics · 2026-10-30

An aligned graphite flake on graphite sticks and slips; twist it a few degrees and it glides almost freely. This is structural superlubricity: the two lattices no longer match, and the forces on the atoms cancel. This chapter first looks at the experiment and the model the review uses, then computes how wide the friction peak is and what remains beside it, and finally asks when this low friction cannot be kept.

The experiment: a graphite flake on graphite

Vanossi and co-workers review the experiments of Dienwiebel, Verhoeven and co-workers: a graphite flake attached to the tip of a friction force microscope slides on an atomically flat graphite surface, and the friction is measured as a function of the twist between the two. The course has not read these original papers; the numbers below come from the review.

For most orientations the surfaces are mismatched and the friction is below 50 pN. Only in narrow ranges of twist that correspond to matching contacts does stick-slip appear, with high friction, typically 250 pN.

Fig. 4 of the review plots these data together with a generalized Prandtl–Tomlinson model: the graphite flake is treated as a rigid finite lattice, a symmetric flake of 96 atoms, and its twist is assumed not to change while it is dragged. The computed mean friction has narrow peaks near the matching angles, separated by wide ranges of ultra-low friction. The course’s flake Lab uses the same kind of model, with graphite replaced by the three-cosine potential of chapter 6.

Source: A. Vanossi

How wide is the friction peak

At small twist, turning the flake changes the phase of each atom by $\theta|\mathbf G|$ times its offset from the centre measured perpendicular to $\mathbf G$ . Treat the flake as a continuum disk and the structure factor becomes the Airy-type function of diffraction by a circular hole, with its first zero at $\theta_1=1.056/D$ radians, where $D$ is the diameter in substrate lattice spacings.

For lattice disks of diameter 10, 20 and 40 the course model puts the first minimum at 6.008°, 3.008° and 1.510°; the continuum disk gives 6.052°, 3.026° and 1.513°, within 0.8%. The half width at half maximum times the diameter is 0.61 radians for all three sizes.

Vanossi and co-workers write that the angular width of a friction peak obeys $\tan\Delta\Phi=1/D$ , so that the peak width can be used to estimate the flake diameter. It gives 5.711°, 2.862° and 1.432°: about 6% below the first zero and about 1.6 times the half width. The review does not say which width $\Delta\Phi$ means; put a measured half width into the rule and the diameter comes out about 1.6 times too large.

\frac{|S_l(\theta)|}{N}\approx\left|\frac{2J_1(u)}{u}\right|,\qquad u=\frac{|\mathbf G|D\theta}{2},\qquad \theta_1=\frac{2\times3.832}{|\mathbf G|D}=\frac{1.056}{D}

The structure factor of matching lattices at small twist.

J_1

is the Bessel function of first order, and

|\mathbf G|=4\pi/(\sqrt3\,a_s)

\tan\Delta\Phi=\frac{1}{D}

The peak-width rule of Sec. II.E of Vanossi and co-workers, with the diameter in lattice spacings.

Corrugation per atom of rigid lattice disks on a matching hexagonal lattice against twist angle, for diameters of 10, 20 and 40 lattice spacings, computed with the course model. Short ticks on the axis: the first zero of the continuum disk,

\theta_1=3.83/(|\mathbf G|D/2)

. Larger flakes lose their registry at smaller angles.

Source: A. Vanossi · A. S. de Wijn

Dragging a twisted flake

Twist the 127-atom hexagon of chapter 6 to different angles and drag it, as before, with a spring of stiffness $k=200\,V_0/a_s^{2}$ at zero temperature, reduced speed 0.003 and damping 3. The mean friction (in $V_0/a_s$ ) and the stick-slip parameter are: 280.4 at 0° ( $\eta=11.14$ ), 193.2 at 2° (8.29), 108.2 at 3° (5.43), 28.9 at 4° (2.51), 1.155 at 5° (0.16), 11.23 at 7° (1.67), and 1.141 at both 10° and 30°.

The rise at 7° comes from the first side lobe of the structure factor. Scanned in steps of 0.02°, $\eta>1$ holds in two ranges: 0° to 4.60° and 5.76° to 8.56°, the second peaking at $\eta=1.67$ at 7°. So a finite flake still sticks weakly beside the main peak; only beyond 9° does it slide smoothly all the way.

At 10° and 30° the remaining friction is the viscous drag at the same speed, $N\gamma v=1.143$ , to within 0.2%. The damping $\gamma$ is a model parameter chosen by the course, not a measured value; a rigid flake on a rigid substrate has no phonons, so the model has no other way to dissipate energy. Friction falling to the viscous drag therefore shows only that the stick-slip channel has closed; the friction of a real flake also depends on dissipation channels the model leaves out.

Source: A. S. de Wijn · A. Vanossi

Superlubricity is not zero friction

Vanossi and co-workers note that the term superlubricity has been criticized, because it suggests zero resistance by analogy with superconductivity and superfluidity. Mismatch removes only one channel of dissipation, the stick-slip instability at low speed; other dissipative processes, such as the emission of sound waves, remain, so even a completely mismatched contact has nonzero net kinetic friction. They still expect friction to be substantially reduced compared with a similar matching contact.

The low-friction state may also not last. The review cites experiments of Filippov and co-workers in which flake superlubricity disappears because the flake turns back into the matching orientation. de Wijn writes that mismatched orientations usually have a higher mean energy than matching ones and can be destroyed by large enough thermal fluctuations; in experiments with small graphite flakes on graphite the mismatched orientations decayed, while larger crystals were found theoretically to rotate more slowly, so their mismatched orientations survive. Like the model in the review, the course Lab locks the twist and cannot show this decay.

Defects also bring friction back. At 10° and on a softer spring, a perfect flake of about 1000 atoms has 0.97 times the viscous drag; a flake missing 10% of its atoms at random has 3.0 times; a flake with rough edges has 3.1 times. The size scaling of chapter 6 shows that the corrugation from this kind of disorder grows with area.

Source: A. Vanossi · A. S. de Wijn

Worked example: how far must the 127-atom hexagon turn to stop sticking

Problem. The aligned hexagon of chapter 6, $N=127$ , $\eta(0)=11.14$ . Using only the continuum-disk formula, estimate the twist at which stick-slip vanishes and whether the side lobe sticks again, then compare with the course model.

Step 1, the stick-slip parameter shrinks with the structure factor. The substrate is fixed, so twisting changes only $|S|$ . The hexagon is centrosymmetric and its three structure factors are equal, so both the corrugation and $\eta$ scale with $|S|/N$ . Stick-slip vanishes when $|S|/N<1/11.14=0.0898$ .

Step 2, replace the hexagon by a disk of equal area. Each atom occupies $(\sqrt3/2)\,a_s^{2}$ , so the flake area is $110.0\,a_s^{2}$ and the disk of equal area has diameter $D=11.83$ .

Step 3, solve. The first root of $2J_1(u)/u=0.0898$ is $u_c=3.458$ , which gives $\theta_c=2u_c/(|\mathbf G|D)=4.61^{\circ}$ ; the first zero is at 5.11°.

Step 4, the side lobe. The first negative extremum of $2J_1(u)/u$ is $-0.132$ at $u=5.136$ . Times 11.14 this gives a side-lobe $\eta=1.47$ , above 1, at 6.85°; $\eta>1$ from 5.85° to 8.05°.

Comparison with the model. In the course model $\eta$ falls to 1 at 4.60° and the first minimum of the corrugation is at 5.09°; the side lobe peaks at $\eta=1.67$ at 7.0° and spans 5.76° to 8.56°. The disk formula gets the main peak to within 1%; the side lobe is more sensitive to shape, because a hexagon is not a disk. Conclusion: this flake stops sticking beyond 4.6°, but weak stick-slip returns near 7°.

\eta(\theta)=\eta(0)\,\frac{|S(\theta)|}{N},\qquad \frac{2J_1(u_c)}{u_c}=\frac{1}{\eta(0)}

The stick-slip parameter of a twisted flake, and the equation

u

satisfies when stick-slip vanishes.

Source: A. S. de Wijn · A. Vanossi

Try it in the Lab

Find where stick-slip stops, and the side lobe

Predict first: at roughly what twist does the 127-atom hexagon stop sticking? What happens near 7°?
Choose Aligned flake sticks, raise the twist step by step to 4°, 5°, 7° and 10°, and at each step check whether the Drag force on the flake chart still shows a sawtooth and read the friction over viscous drag.
In the Barrier against twist angle chart, find the first minimum and the side lobe, compare them with the 5.09° and 7° of this chapter, and read $\eta$ at 7°.

Bigger flakes, narrower peaks

Predict first: if the flake grows from 6 shells (127 atoms) to 12 shells (469 atoms), roughly where will the first minimum move?
Choose Twist sweep and watch where the friction rises as the twist runs from 0° to 30°; then set the flake size to 469 atoms and read the position of the first minimum in the Barrier against twist angle chart.
Check the reading with $\theta_1=1.056/D$ and the equal-area diameter, then compute the width from $\tan\Delta\Phi=1/D$ and say whether it is closer to the first zero or to the half width.

Exercises with explained answers

For a disk 40 lattice spacings across, where is the first zero of the friction peak? The course model gives a half width of 0.874°; put it into $\tan\Delta\Phi=1/D$ . What diameter do you get?
Show the answer
$\theta_1=1.056/40=0.0264$ radians, or 1.51°. $D=1/\tan(0.874^{\circ})=65.6$ , 1.64 times the true diameter. To estimate a diameter from a peak width, first say which width was measured.
Why is the friction of the course model at 10° and 30° almost exactly the viscous drag? Can this be used to predict the friction of a real flake at those angles?
Show the answer
At these angles the corrugation is only about $0.5\,V_0$ and $\eta\approx0.04$ , so the flake follows the support at nearly constant speed. The substrate potential returns to the same value every period and its average effect almost vanishes; what is left is the damping $N\gamma v$ put into the model. A real flake also has phonons, sound emission, rotation and edge effects, and $\gamma$ is assumed, so no.
The course Lab locks the twist of the flake. Name two effects it therefore cannot show, and where each is reported.
Show the answer
First, the flake turning back into the matching orientation, which ends superlubricity: experiments of Filippov and co-workers, as reported by Vanossi and co-workers. Second, thermal fluctuations destroying mismatched orientations: de Wijn notes that mismatched orientations usually have higher mean energy, and large enough fluctuations let the flake turn to a matching orientation, raising friction.

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.
A. S. de Wijn, (In)commensurability, scaling and multiplicity of friction in nanocrystals and application to gold nanocrystals on graphite, arXiv:1208.1395v2; Physical Review B 86, 085429 (2012)
Version 2, 15 August 2012. Eq. (1): the hexagonal potential of one gold atom on graphite, a sum of three cosines with a prefactor of two ninths, and a single-atom corrugation of about 50 meV. Sec. II: graphite with 1.42 Å bonds and Au(111) with 2.885 Å spacing, a ratio of 1.354 in his length unit. Sec. IV: friction of a perfect crystal scales with contact area to the power zero, one half or one (Eq. 21). Sec. V: irregular outlines give powers of one quarter, three quarters or one (Eq. 31), and amorphous crystals the power one half. We built rigid flakes with this potential and fitted the exponents; we did not digitise his figures.