FR40 · Structural superlubricity

Superlubric Flakes: How a Twist Cuts Atomic Friction

Why does a twisted flake glide while an aligned one sticks? Because its atoms stop pulling in step. On an aligned flake every atom sits in a hollow at once and the barriers add up. Twisted, the atoms sit everywhere at once and their pushes cancel except near the edge, so the flake no longer sticks. Friction drops a long way but not to zero, and vacancies or rough edges bring some of it back.

Twist a crystal flake on a matching crystal and watch its atoms stop pulling together. The barrier collapses, stick-slip stops, and only defects and edges bring friction back.

Twist angle $0^{\circ}$

Flake corrugation per atom $1\,V_0$

Stick-slip parameter

\eta

$11.1$

Friction over viscous drag $245$

A crystal flake on a matching crystal

Twist angle

0^{\circ}

Flake size

N=127

Lattice ratio

b/a_s

1

Missing atoms

0\%

Flake outline

Drag the yellow knob to twist the flake. One colour: atoms pull in step and it sticks. Mixed colours: pulls cancel and it glides.

Advanced: drag spring and heat

Drag spring

200\,V_0/a_s^{2}

Thermal energy

0\,V_0

A softer spring lets the weak barriers of a twisted flake with defects pin it. Heat lets the flake hop over small barriers early.

Model, units and sources

$V(\mathbf r)=-\frac{2V_0}{9}\sum_{l=1}^{3}\cos(\mathbf G_l\cdot\mathbf r),\qquad U(\mathbf R)=\sum_{i=1}^{N}V(\mathbf R+\mathbf r_i)$

Each atom sees a corrugation $V_0$ between its deepest and highest site. The flake energy is the sum over its atoms, computed exactly through the flake structure factor at the three substrate wavevectors. The flake is dragged at 0.003 substrate spacings per time unit with damping 3 per time unit; both are model parameters. The friction readout drags a separate copy of the flake at the shown twist, rounded to 0.2° during a sweep, and averages the spring force over four spacings after two of warm-up: while the twist changes, the landscape moves under the flake, so its spring force is not friction.

de Wijn (2012), Phys. Rev. B 86, 085429 · Vanossi et al. (2013), Rev. Mod. Phys. 85, 529

Follow the derivation in the course · Compare with friction on a rope around a post

Model assumptions

The flake is a rigid crystal of hexagonal, triangular, disk-shaped or rough outline on a rigid triangular substrate potential, following the model of de Wijn (2012). Every flake atom feels the same three-cosine potential, the flake is pulled by a spring at its centre and loses energy through a damping term, and its twist is held fixed. All quantities are in reduced units: substrate spacing, single-atom corrugation, atom mass and the matching time. Without stick-slip the friction left is the model damping, which stands in for phonon and electronic losses that are not modelled; real flakes can also rotate back into registry, which this model leaves out.

Barrier against twist angle

Barrier against flake size at 10°

Drag force on the flake

Physics tutorial

When atoms stop pulling together

BackgroundA single tip feels the full corrugation of the surface under it. A flake of many atoms feels the sum of all their corrugations. If the flake fits the substrate, every atom sits in a hollow at the same time and the barriers add. If it is twisted, the atoms sit at all possible places at once and most of their pushes and pulls cancel.

Why it mattersExperiments with graphite flakes on graphite found high friction with stick-slip only near the angles where the two lattices match, and friction below 50 pN at most other angles, as reviewed by Vanossi et al. (2013). This structural superlubricity is one of the few known routes to very low dry friction.

Start with the essentials

Focus question: Why does a twisted flake glide while an aligned one sticks?
One-sentence intuition: The flake energy is a sum of waves, one per atom, with phases set by where each atom sits. Aligned atoms share one phase and the waves add to N times the single-atom barrier. Twisted atoms spread their phases evenly across the flake, so the interior cancels and only an edge term is left. With the barrier gone the spring no longer stores energy, stick-slip stops and only the damping of the model is left.

Core mathematical model

Flake energy as a structure factor

U(\mathbf R)=-\frac{2V_0}{9}\sum_{l=1}^{3}\mathrm{Re}\!\left[S_l\,e^{i\mathbf G_l\cdot\mathbf R}\right],\qquad S_l=\sum_{j=1}^{N}e^{i\mathbf G_l\cdot\mathbf r_j}

Summing the substrate potential over the atoms of a rigid flake gives this exact form, the one de Wijn (2012) uses. Aligned, $|S_l|=N$ and the corrugation is $NV_0$ . Twisted, the phases of $S_l$ wind around the flake and $|S_l|$ stays of the order of the edge.

Width of the friction peak

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

Vanossi et al. (2013) estimate the angular width of the friction peak from the flake diameter $D$ in lattice spacings. For disk flakes the model puts the first zero of the corrugation at 6.01°, 3.01° and 1.51° for $D=10,\,20,\,40$ , so the width halves when the diameter doubles.

How the barrier grows with area

\Delta U\propto A^{\beta},\qquad \beta=1\ \text{aligned},\quad \beta=0\ \text{twisted regular},\quad \beta=\tfrac14\ \text{twisted rough}

These are the exponents de Wijn (2012) derives for a twisted flake whose outline is regular or irregular, and for an aligned flake. Fitting the model at 10° gives $\beta=1.000$ , $0.12\pm0.25$ and $0.21\pm0.03$ . Random vacancies give $0.43\pm0.04$ , close to the $\tfrac12$ of a random sum.

Common difficulties

Superlubric means frictionless

Typical misconceptionA twisted flake slides with zero friction.

Better mental modelTwisting removes the stick-slip channel only. In this model the friction left is the damping drag, which stands for sound and electronic losses; Vanossi et al. stress that real superlubric friction is reduced, not zero.

Bigger flakes always rub harder

Typical misconceptionFriction grows with contact area, so a larger flake must stick more.

Better mental modelThat holds for an aligned flake, whose barrier grows as N. A twisted flake with a regular outline keeps a barrier of edge size that does not grow on average; only defects and rough edges make it grow, and more slowly than the area.

Run the experiment

01
Watch the sweep
Leave Twist sweep running and watch the force chart while the twist passes a few degrees.
What to observe: At 0° the 127-atom flake sticks and slips with friction about 245 times the viscous drag. By 5° the sawtooth is gone and the ratio is about 1. A weak sawtooth returns between about 6° and 8.5°, near 10 times the drag at 7°, and is gone again beyond 9°.
02
Predict, then change the lattice
Guess whether a flake with a different lattice can stick at 0°, then choose Gold on graphite.
What to observe: With a lattice ratio of 1.17 the atoms are out of step even without a twist, and friction stays at about 1.7 times the viscous drag.
03
Explain the defects
Compare Twisted flake glides with Vacancies bring friction back and Rough edges bring friction back.
What to observe: On the softer spring, the perfect 1027-atom flake slides at 0.97 times the viscous drag, while 10% vacancies raise it to about 3.0 times and a rough edge to about 3.1 times. The cancellation is broken and stick-slip returns.

Superlubric Flakes: How a Twist Cuts Atomic Friction

When atoms stop pulling together

Core mathematical model

Flake energy as a structure factor

Width of the friction peak

How the barrier grows with area

Common difficulties

Superlubric means frictionless

Bigger flakes always rub harder

Run the experiment

Watch the sweep

Predict, then change the lattice

Explain the defects

Related experiments and articles

From one sticking tip to a twisted flake that glides