---
title: "How the friction of a crystal flake adds up from its atoms"
description: "Replace the tip with a chain of atoms or a crystal flake: the Frenkel–Kontorova chain, the Aubry transition, flake structure factors and scaling with area."
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language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/stories/nanofriction/from-one-tip-to-a-flake/"
last_modified: "2026-10-30"
---

[Nanofriction](https://sandboxphysics.com/stories/nanofriction/)

L3 · Chapter 6

# How the friction of a crystal flake adds up from its atoms

Sandbox Physics · 2026-10-30

Real contacts are not one atom. A nanocrystal of graphite or gold has hundreds or thousands of atoms feeling the substrate at once. Their forces can add up or cancel, depending on whether the two lattices match. This chapter first looks at a one-dimensional chain of atoms, then extends the Prandtl–Tomlinson model to a rigid flake, and finally asks how friction changes with flake size.

## Before you begin

The stick-slip parameter and energy drop of chapters 2 and 3; complex exponentials and sums of cosines. The reciprocal vectors of a two-dimensional lattice help; the chapter gives the parts it needs.

- Write the Frenkel–Kontorova model and explain what kinks and the Aubry transition each mean.
- Write the energy of a rigid flake as a sum over structure factors and explain why it differs so much between aligned and twisted flakes.
- Restate the size-scaling exponents of de Wijn and compare them with the fits from the course model.

## A chain of atoms: the Frenkel–Kontorova model

Eq. (6) of Vanossi and co-workers is the Frenkel–Kontorova model: a chain of atoms joined by springs $K$ with natural spacing $a_c$, lying on a sinusoidal potential of period $a_b$.

When the number of atoms equals the number of substrate valleys, adding one atom makes a kink: a local compression of the chain. Kinks move far more easily than atoms; the Peierls–Nabarro barrier they must cross is always smaller, often much smaller, than $U_0$. Each run of a kink from one end of the chain to the other moves the whole chain by one period.

Vanossi and co-workers use this to explain an old puzzle: Frenkel estimated the yield stress of a perfect crystal to be of the order of the shear modulus, but measured values are much smaller. The reason is that dislocations, which play the part of kinks in a real crystal, move at much smaller stresses.

$$
H=\sum_i\left[\frac{p_i^{2}}{2m}+\frac{K}{2}\left(x_{i+1}-x_i-a_c\right)^{2}+\frac{U_0}{2}\cos\frac{2\pi x_i}{a_b}\right]
$$

Kinetic energy, the springs between neighbours and the substrate potential. To probe static friction, every atom is pushed by a slowly growing force until the chain slides.

Source: A. Vanossi

## Mismatched chains and the Aubry transition

If $a_b/a_c$ is irrational, an infinite chain takes a staircase shape: regions of near match separated by regularly spaced kinks. Vanossi and co-workers write that when the chain stiffness $K$ exceeds a critical value $K_c$, the kinks move freely, the smallest force makes the chain slide and static friction vanishes; below $K_c$ the kinks are pinned and stick-slip appears.

For the golden-mean ratio, $K_c$ takes its smallest possible value, about 1.0291926 in units of $2U_0(\pi/a_b)^{2}$.

Three limits matter for the next two chapters. First, even when static friction vanishes, kinetic friction does not, because sliding at any finite speed excites phonons in the chain. Second, a chain of finite length at zero temperature is always statically pinned, because its free ends lock. Third, the model describes only how a crystalline contact starts to slide; the plastic deformation of real friction experiments lies outside it.

Source: A. Vanossi

## A rigid flake is a sum of phases

de Wijn writes the energy of one gold atom on graphite as a sum of three cosines with prefactor $2V_0/9$; the corrugation one gold atom feels, $V_0$, is about 50 meV. The course’s flake model uses the same form in units of $V_0$ and the substrate lattice constant $a_s$; in the course these are model parameters, not a specific material.

For a rigid flake, adding up the energies of all atoms can be rewritten exactly as three structure factors $S_l$. Each atom contributes a phase of unit length, $e^{i\mathbf G_l\cdot\mathbf r_j}$. When the lattices match and are aligned, all phases are equal, $|S_l|=N$, and the corrugation is $N$ times that of one atom. Twist the flake and the phases spread around the unit circle; most of them cancel.

The course checks two things: the structure-factor form differs from the atom-by-atom sum by no more than $3\times10^{-15}\,V_0$ per atom, and for matching lattices at 0° and 60° the corrugation of hexagonal, triangular and disk flakes is exactly $1\,V_0$ per atom.

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

$\mathbf R$ is the flake centre, $\mathbf r_j$ the position of each atom relative to it, and all three reciprocal vectors $\mathbf G_l$ have length $4\pi/(\sqrt3\,a_s)$.

Source: A. S. de Wijn · A. Vanossi

## How friction scales with contact area

de Wijn classifies the mismatch by the exponent $C$ with which the corrugation grows with flake diameter. For perfect crystals his Eq. (21) gives three branches, $A^{0}$, $A^{1/2}$ and $A$, and most orientations belong to $A^{0}$, which does not grow with size.

Real particles have ragged edges. de Wijn writes an irregular shape as a sum of many regular shapes with random phases and obtains his Eq. (31): the exponents become $1/4$, $3/4$ and 1. He also notes that the friction of amorphous crystals grows as $A^{1/2}$ whatever their shape.

At a 10° twist the course fits the exponent of corrugation against atom number $N\propto A$: 1.000 for aligned flakes; $0.12\pm0.25$ for twisted regular hexagons, consistent with 0, with the corrugation jumping between 0.5 and 9.3 $V_0$ with no trend; $0.21\pm0.03$ for flakes with rough edges, against de Wijn’s 0.25; and $0.43\pm0.04$ for flakes missing 10% of their atoms at random. de Wijn makes no direct prediction for the last case; random vacancies also have random phases, and the argument he uses for amorphous crystals says the exponent should approach 0.5, so we fit only flakes above 357 atoms.

Gold on graphite. de Wijn gives the lattice ratio of gold to graphite as 1.354 in his length unit, which is 1.1726 in the course’s units. At 0° the course model’s corrugation varies erratically between 0.97 and 26.4 $V_0$ as the atom number runs from 19 to 4921. de Wijn also points out that this ratio is close to $4/3$, so first-order corrections from lattice deformation can be as large as the leading terms; the course’s rigid model leaves them out.

$$
F_{\mathrm{fric}}\propto A^{C/2}\quad(C=0,1,2),\qquad F'_{\mathrm{fric}}\propto A^{\min\left((2C+1)/4,\,1\right)}
$$

Eqs. (21) and (31) of de Wijn; $F'_{\mathrm{fric}}$ is for flakes with irregular outlines. $A$ is the contact area, proportional to the atom number.

Source: A. S. de Wijn

## Worked example: an aligned flake of 127 atoms

Problem. A hexagonal flake of 6 shells on a matching lattice, aligned at 0°, dragged along $x$ by a spring of stiffness $k=200\,V_0/a_s^{2}$ at zero temperature. Find the atom number, corrugation, stick-slip parameter and quasi-static friction, and compare with simulation and with viscous drag.

Step 1, atom number and corrugation. A hexagon of $s$ shells has $3s(s+1)+1$ atoms, so $N=127$ for $s=6$. Aligned, every atom has the same phase and the corrugation is $127\,V_0$.

Step 2, reduce to one dimension. Along the line $y=0$, one of the three cosines is always 1 and the other two are equal, so the energy is a cosine of period $a_s$ and amplitude $U_0=4N V_0/9=56.4\,V_0$: exactly the Tomlinson model of chapter 2.

Step 3, the stick-slip parameter. $\eta=4\pi^{2}\times56.4/200=11.14$, close to the 11.48 of the tip of Sang and co-workers. For $\eta>1$ we need $N>9k/(16\pi^{2})=11.4$: on this spring, only the aligned flake of one shell (7 atoms) fails to stick and slip.

Step 4, friction. The energy-drop method of chapter 3 gives a quasi-static friction of $276.2\,V_0/a_s$. The course simulation, dragged at reduced speed 0.003 with damping 3, gives $280.4\,V_0/a_s$, 1.5% higher.

Explanation. The viscous drag at the same speed is $N\gamma v=127\times3\times0.003=1.143\,V_0/a_s$, so stick-slip friction is about 245 times larger. The next chapter twists this flake and watches that ratio fall.

$$
U(X,0)=-\frac{2V_0N}{9}\left[1+2\cos\frac{2\pi X}{a_s}\right],\qquad \eta_{\mathrm{flake}}=\frac{4\pi^{2}}{k a_s^{2}}\cdot\frac{4NV_0}{9}
$$

The energy of the aligned flake along the $x$ axis, and the stick-slip parameter that follows from it.

Source: A. S. de Wijn · A. Vanossi

## Try it in the Lab

### [How big must an aligned flake be to stick](https://sandboxphysics.com/labs/twisted-flake-superlubricity/)

1. Predict first: with the flake shrunk to 2 shells (19 atoms), what is $\eta$? How stiff must the drag spring be for stick-slip to disappear?
2. Choose Aligned flake sticks, drag the flake size to its minimum, read $\eta$ and the friction over viscous drag, and check whether the Drag force on the flake chart still shows a sawtooth.
3. Raise the drag spring slowly, find the value at which the sawtooth disappears, and check it with $\eta=N|\mathbf G|^{2}V_0/(3k)$.

### [Defects make the corrugation grow with size](https://sandboxphysics.com/labs/twisted-flake-superlubricity/)

1. Predict first: twisted by the same 10°, which corrugation will grow with size, that of a perfect hexagon or that of a flake missing 10% of its atoms?
2. Choose Twisted flake glides and then Vacancies bring friction back, read the flake corrugation per atom, and compare the twisted, perfect and twisted, 10% vacancies lines in the Barrier against flake size at 10° chart.
3. Then choose Rough edges bring friction back, compare the trends of the three flakes, and explain the difference with the rule that a sum of random phases grows as a square root.

## Exercises with explained answers

1. Why does the corrugation of a twisted perfect hexagonal flake not grow with atom number?

   Show the answer

   After the twist, the phases $\mathbf G_l\cdot\mathbf r_j$ sweep evenly around the unit circle inside the flake; neighbouring rows cancel, and only unpaired parts near the edge remain. Those do not grow with area and oscillate with size, so the corrugation jumps within a bounded range: the $C=0$ branch of de Wijn.

2. Use de Wijn’s random-phase argument to derive the area exponent of a mismatched ($C=0$) flake with an irregular outline.

   Show the answer

   Each step edge contributes an energy of order $V_0$ with a random phase. The number of steps $M$ is proportional to the diameter $d$, and a sum of random phases grows as $\sqrt M\propto d^{1/2}$. Since $d\propto A^{1/2}$, the corrugation grows as $A^{1/4}$, matching $(2C+1)/4$ of Eq. (31) at $C=0$.

3. A finite Frenkel–Kontorova chain at zero temperature is always statically pinned. What does that imply for the claim that mismatch means no friction?

   Show the answer

   Phase cancellation alone cannot establish zero friction. Mismatch removes only the systematic adding-up of atomic forces in an infinite contact; the edges, free ends and defects of a finite contact leave uncancelled parts that pin it. Even with zero static friction, sliding at finite speed still excites phonons.

[Previous: Thermolubricity and the limits of rate theory](https://sandboxphysics.com/stories/nanofriction/thermolubricity-and-the-limits-of-rate-theory/) [Next: Twisted flakes and structural superlubricity](https://sandboxphysics.com/stories/nanofriction/twisted-flakes-and-structural-superlubricity/)

## Sources and what we recomputed

1. [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)](https://arxiv.org/abs/1112.3234v4)

   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.

2. [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)](https://arxiv.org/abs/1208.1395v2)

   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.