A 2006 note by P.-G. de Gennes predicts that two free (unsupported) crystalline monolayers in incommensurate, weakly coupled contact are not superlubric. Bulk incommensurate crystals of layered materials slide with almost no friction, the residual viscous stress coming from weak two-phonon processes; but a two-dimensional crystal has anomalous thermal fluctuations (Jancovici's singular structure factor S(k⃗)∝(ka)−(2−m)S(\vec{k})\propto(ka)^{-(2-m)} near a Bragg wavevector), which turn the dynamic structure factor at small wavevector into a broad multiphonon central peak. Inserting this into the phonon-friction mechanism gives a scaling law σ=σ1 (cV/vth2) θm−3\sigma=\sigma_1\,(cV/v_{\mathrm{th}}^{2})\,\theta^{m-3} for the viscous friction stress, with the Jancovici exponent mm and σ1=U12/(Mc2a3)\sigma_1=U_1^{2}/(Mc^{2}a^{3}); for graphite-like parameters (θ=0.1\theta=0.1 rad, V=100V=100 micrometers per second) the estimate is σ∼102\sigma\sim 10^{2} atm - very large. The derivation is explicitly coefficient-free and untested. The open problem is to establish this fluctuation-mediated friction law quantitatively and experimentally: a controlled calculation of S(k⃗,Ω)S(\vec{k},\Omega) for a free two-dimensional crystal at small wavevector and low frequency, numerical prefactors and validity regimes for σ(θ,V,T)\sigma(\theta,V,T) (low-frequency window, decoupled-to-strong-coupling crossover, intermediate coupling, free multilayer extension), and an experimental test of the friction law on free, unstiffened monolayer contacts. No published work to date computes, tests, refutes, or reformulates the law.

Background

Layered crystals such as graphite or molybdenite (MoS2_2) combine stiff, strongly bonded atomic planes with weak van der Waals attraction between planes. When two such crystals are brought into contact through their basal planes and one is rotated by a misorientation angle θ\theta, the surface lattices are generically incommensurate (their periodicities share no common period), and the lateral forces exerted across the interface cancel to a large degree by symmetry. Friction force microscopy on graphite indeed shows that friction nearly vanishes at generic rotation angles and peaks only near the aligned, commensurate orientations (Dienwiebel et al. 2004). This ultralow-friction state of an ideal incommensurate contact is called structural superlubricity. The residual friction of such a contact is a viscous stress σ\sigma (lateral force per unit area proportional to the relative sliding velocity VV) produced by phonons: the upper crystal exerts on the lower one a weak periodic potential U1exp⁡(iτ⃗+⋅r⃗)U_1\exp(i\vec{\tau}_+\cdot\vec{r}), where τ⃗+\vec{\tau}_+ is a reciprocal lattice vector (a wavevector appearing in the Fourier decomposition of the lattice potential) of the upper layer; sliding at velocity V⃗\vec{V} modulates this perturbation at the frequency Ω=τ⃗+⋅V⃗\Omega=\vec{\tau}_+\cdot\vec{V}, which can excite lattice vibrations (phonons) and dissipate energy. For three-dimensional crystals, Sacco, Sokoloff and Widom (1979) and Sokoloff (1990) showed that one-phonon excitation is blocked by energy and momentum conservation except at isolated resonant modes, so the dominant channel is a two-phonon (Raman-like) process, which yields only very small damping. In this weak-coupling situation the companion review by Friedel and de Gennes (2007) estimates a viscous stress of order 10−410^{-4} atm for bulk graphite at θ=0.1\theta=0.1 rad and V=10−4V=10^{-4} m/s - genuinely superlubric.

The situation changes qualitatively when the sliding bodies are not bulk crystals but free monolayers - single atomic sheets, e.g. two graphene planes, with no solid support in the contact region. A two-dimensional crystal is marginal in the sense of the Landau-Peierls argument: thermal fluctuations of its displacement field diverge (logarithmically) with system size, so it has no sharp Bragg diffraction peaks. Jancovici (1967) computed the static structure factor of the harmonic two-dimensional solid - the thermal average of the squared density modulation at wavevector τ⃗+k⃗\vec{\tau}+\vec{k} near a reciprocal lattice vector τ⃗\vec{\tau} - and found a power-law singularity

S(k⃗)∝(ka)−(2−m),m=kBT (τa)22πMc2,S(\vec{k})\propto(ka)^{-(2-m)},\qquad m=\frac{k_B T\,(\tau a)^{2}}{2\pi M c^{2}},

where aa is the unit-cell size, MM the unit-cell mass, cc a sound velocity, and the exponent mm (typically ≪1\ll 1 at room temperature) is now called the Jancovici exponent. Because the displacements of a harmonic crystal are Gaussian variables, the dynamic generalization is controlled by the mean-square displacement difference, and de Gennes (2006) argued that the dynamic structure factor S(k⃗,Ω)S(\vec{k},\Omega) at fixed small kk is not a narrow one-phonon peak but a broad multiphonon central peak of width ωk=ck\omega_k=ck; for frequencies below this width, Ω<ωk\Omega<\omega_k, its value is of order S(k⃗)/ωkS(\vec{k})/\omega_k.

De Gennes (2006) then constructed the scaling theory of the friction stress between two free, clean, dislocation-free, identical monolayers misoriented by θ\theta, in the decoupled regime of weak interlayer coupling - the regime in which bulk contacts are superlubric. (At stronger coupling, or at special commensurate angles, the interface instead builds the network of screw dislocations - line defects of the displacement field with quantized Burgers vector, in the construction recounted by Friedel and de Gennes (2007) - and friction becomes solid, with a finite threshold stress of Peierls-Nabarro type - the stress needed to move a dislocation through the periodic atomic lattice.) Following Sacco, Sokoloff and Widom, the dissipation per unit contact area is TS˙∼a−2U12(τ⃗+⋅V⃗)2(kBT)−1S(k⃗,Ω)T\dot{S}\sim a^{-2}U_1^{2}(\vec{\tau}_+\cdot\vec{V})^{2}(k_B T)^{-1}S(\vec{k},\Omega) with k⃗=τ⃗+−τ⃗−\vec{k}=\vec{\tau}_+-\vec{\tau}_- and k∼τθk\sim\tau\theta; inserting the multiphonon value of S(k⃗,Ω)S(\vec{k},\Omega) and writing TS˙=σVT\dot{S}=\sigma V gives

σ=σ1 cVvth2 θm−3,σ1=U12Mc2a3,Mvth2=kBT,\sigma=\sigma_1\,\frac{cV}{v_{\mathrm{th}}^{2}}\,\theta^{m-3},\qquad \sigma_1=\frac{U_1^{2}}{M c^{2} a^{3}},\qquad M v_{\mathrm{th}}^{2}=k_B T,

a viscous stress strongly enhanced at small misorientation by the diverging structure factor. With graphite-like parameters (U1/Mc2=0.1U_1/Mc^{2}=0.1, Mc2=5Mc^{2}=5 eV, vth=300v_{\mathrm{th}}=300 m/s, θ=0.1\theta=0.1 rad, V=100V=100 micrometers per second, m≈10−3m\approx 10^{-3}) the estimate is σ1∼104\sigma_1\sim 10^{4} atm and σ∼102\sigma\sim 10^{2} atm: 'with free monolayers, we do not expect superlubricity'. The same conclusion is stated to hold for one free monolayer facing a bulk crystal. The derivation is explicitly coefficient-free ('ignoring all numerical coefficients') and rests on an interpolated S(k⃗,t)S(\vec{k},t), the harmonic Gaussian treatment of displacements, and the low-frequency window Ω<ωk\Omega<\omega_k. A monolayer supported by a substrate is stiffened and its anomalous fluctuations are suppressed, so adsorbed-monolayer systems (e.g. Müser, Wenning and Robbins (2001)) lie outside the mechanism; the paper sketches bent-membrane and rolled-cylinder setups as possible realizations of a genuinely free contact. What the note leaves open - numerical prefactors and the exact magnitude of σ\sigma for real materials, the experimental measurement of S(k⃗,Ω)S(\vec{k},\Omega) for a free monolayer at small kk and Ω\Omega (ideally by inelastic neutron scattering), the feasibility of free, clean monolayer contacts, the regime boundaries in (V,θ,T)(V,\theta,T), the intermediate interlayer-coupling regime, and the free nn-layer extension (asserted only as m↦m/nm\mapsto m/n) - is precisely the problem stated here.

Problem Statement

Establish, quantitatively and experimentally, the fluctuation-mediated viscous friction law between two free (unsupported) crystalline monolayers in incommensurate, weakly coupled contact - the geometry of de Gennes (2006), with the variant of one free monolayer facing a bulk crystal included, as the source states the same conclusion holds there. All requested results below serve this single objective: determining the actual friction stress σ(θ,V,T)\sigma(\theta,V,T) of the free monolayer contact and testing it, thereby confirming, correcting, or refuting the prediction that such contacts are not superlubric.

(1) Dynamic structure factor. Compute in a controlled way the dynamic structure factor S(k⃗,Ω)S(\vec{k},\Omega) of a free two-dimensional crystal at wavevectors close to a reciprocal lattice vector (ka≪1ka\ll 1) and frequencies below the one-phonon scale, Ω<ωk=ck\Omega<\omega_k=ck: go beyond the interpolated form and the harmonic Gaussian treatment used in the source, and state the role of anharmonicity (coupling of flexural and in-plane modes) and of any membrane tension, both of which can modify the Jancovici singularity S(k⃗)∝(ka)−(2−m)S(\vec{k})\propto(ka)^{-(2-m)}.

(2) Friction law with prefactors and validity domain. From it, derive the friction stress σ(θ,V,T)\sigma(\theta,V,T) with numerical prefactors for a realistic material (graphite or MoS2_2 monolayers), together with its regime of validity, each of the following being a required part of that domain: the low-frequency window Ω<ωk\Omega<\omega_k expressed as bounds on (V,θ)(V,\theta), the temperature dependence through the Jancovici exponent m(T)m(T), the behavior as θ\theta approaches commensurate angles and the decoupled-to-strong-coupling crossover, and the intermediate interlayer-coupling regime between the decoupled limit and the quantized-dislocation (strong-coupling) limit, which is unanalyzed even for bulk crystals (Friedel and de Gennes 2007). Where an item cannot be settled, the treatment establishes what it determines and what remains open.

(3) Multilayer extension. Derive, rather than assert, the free nn-layer generalization (the source states only that the Jancovici exponent is divided by nn), i.e. how the friction approaches the bulk-crystal value as the number of free layers grows.

(4) Experimental test. Test the law: either measure S(k⃗,Ω)S(\vec{k},\Omega) of a free monolayer at small kk and Ω\Omega in the regime relevant to the friction mechanism, or directly measure the friction stress between two free, clean, unstiffened, misoriented monolayers as a function of VV and θ\theta, with evidence that the sheets are genuinely free (anomalous fluctuation spectrum unsuppressed by any support). The test should discriminate the predicted scaling σ∝V θm−3\sigma\propto V\,\theta^{m-3} and its magnitude (of order 10210^{2} atm at V=100V=100 micrometers per second and θ=0.1\theta=0.1 rad for graphite-like parameters), or establish the corrected law if the quantitative treatment modifies the source's coefficient-free estimate.

References

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