Enhancement of non-contact friction between metal surfaces induced by the electrical double layer
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Electric double layer effect in non contact friction between metal surfaces
A.I. Volokitin ∗ Samar State Technical University, 443100 Samara, Russia
Non contact friction between a gold tip and a gold plate is calculated for different mechanisms.It was shown that in an extreme-near field ( d < nm) the contribution from the electrical doublelayer leads to the enhancement of the radiative friction by many orders of magnitude in comparisonto the result of the conversional theory which does not include this contribution. This result openthe way to detect the Casimir friction in the extefeme-near field. PACS: 44.40.+a, 63.20.D-, 78.20.CiAll bodies are surrounded by a fluctuating electromagnetic field due to quantum and thermal fluctuations. Thisfluctuating electromagnetic field is responsible for a variety of phenomena, starting with blackbody radiation, theexplanation of the properties of which, given by Planck , led to the birth of quantum mechanics. Casimir was first who realized that quantum electromagnetic fluctuations lead to an attraction forces between uncharged, perfectlyconducting plates. These forces are named as the Casimir forces when retardation is included, or van der Waals forceswhen retardation is ignored. Later this idea was generalised by Lifshitz for real materials and finite temperatures. Tocalculate the Casimir forces Lifshitz used the fluctuational electrodynamics developed by Rytov . Approximatelytwenty years later Polder and van Hove used the same theory of the fluctuating electromagnetic field to calculate theradiative heat transfer between two plates in presence of the temperature difference between them. It was predictedthat the radiative heat flux in the near field (when the distance between the bodies d < λ T = c ¯ h/k B T : at roomtemperature λ T ∼ µ m) can be by many orders of magnitude larger than the limit, which is established by Planck’slaw for blackbody radiation. These effects were observed experimentally for Casimir forces at the end of the lastcentury , and roughly ten years later for radiative heat transfer . Casimir forces dominate the interactions betweennanostructures and are responsible for such important nanoscale phenomena as adhesion, wetting, friction, and stickingin small devices such as micro- and nanoelectromechanical systems (MEMS and NEMS). As a result of the practicalimportance of the problem of fluctuation-induced electromagnetic phenomena for the design of nanoelectromechanicalsystems and the great progress in the methods of force detection, experimental and theoretical studies of Casimirforces between neutral bodies have experienced an extraordinary rise in the last few year .At present radiative heat transfer at the nanoscale level is actively studied because of its fundamentalimportance and perspective to find application in a variety of technologies ranging from thermophotovoltaicenergy converters , non-invasive thermal imaging to thermomagnetic information recording and processing and nanolithography . Over the past decade super-Planckian heat transfer has been observed for vacuum gapsbetween bodies in the interval from hundreds of nanometers to several ˚Angstr¨oms . Generally, the results of thesemeasurements turned out to be in good agreement with the predictions based on the fluctuation electrodynamicsfor a wide range of materials and geometries. However, there are still remain significant unresolved problems inunderstanding heat transfer between bodies in an extreme near field (gap size < nm) . In Refs. the heatflux between a gold coated near-field scanning thermal microscope tip and a planar gold sample in an extreme nearfield at nanometer distances of 0.2-7nm was studied. It was found that the experimental results can not be explainedby the conventional theory of the radiative heat transfer with the reflection amplitudes for surfaces in the Rytov’stheory given by the classical Fresnel’s formulas. In particular, the heat transfer observed in Ref. for the separationsfrom 1 to 10nm is orders of magnitude larger than the predictions of the conventional Rytov’s theory and its distancedependence is not well understood. These discrepancies stimulate the search of the alternative channels of the heattransfer that can be activated in the extreme near-field. One of the obvious channels is related with “phonon tunneling”which stimulated active research of the phonon heat transfer in this region . In Ref. it was shown that the heatflux in the extreme near-field can be strongly enhanced in presence of the potential difference between bodies. Thisenhancement is due to the electric field effect related with the fluctuating dipole moment induced on the surface bythe potential difference. In this Letter another mechanism, related with the fluctuating dipole moment of the electricdouble layer on metal surface, is considered. In contrast with the electric field effect, which is short range and operatesfor the separations of order ∼ nm, the double layer effect is long range and operates for the separations < nm.In Ref. it was shown that the surface dipole moment responsible for heat flux can be related not to the potentialdifference but to the “intrinsic” dipole moment of the electrical double layer on metal surface.The electrical double layer on the metal surface is associated to a redistribution of the electron density, as a resultof which the surface atomic layers 1 and 2 are charged with the surface charge density ± σ d (see Fig. 1) where forgold, according to the estimation in Ref. , σ d ≈ . Cm − .For relative motion of the bodies a fluctuating electromagnetic field leads to the Casimir friction which was calculated ____ _ _ _ __ _ _ double layer _ _ ++++ v dd _ + + + +___ ___ double layer ++++_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ + + + + ___ ___ Рис. 1: Scheme of electrical double layers on metal surfaces sliding relative to each other with velocity v in the framework of the fluctuation electrodynamics for relative sliding of two plates ? . The Casimir friction existseven at zero temperature when it is determined by quantum fluctuations and denoted as quantum friction . Casimirfriction determines limit to which the friction and, consequently, the force fluctuations can be reduced, because oflink between them determined by the fluctuation-dissipation theorem. Practical significance of the Casimir frictionis determined by its role in the ultra sensitive force registration and non contact friction. In contrast to the Casimirforces and radiative heat transfer which were measured in many experiments, the detection of the Casimir frictionis still a challenging problem for experimentalists. However, the frictional drag between quantum wells and graphenesheets, and the current-voltage dependence of nonsuspended graphene on the surface of the polar dielectric SiO , wereaccurately described using the theory of the Casimir friction . In Ref. it was shown that the Casimir frictioncan be detected mechanically using an AFM.In this article the Casimir friction is calculated between gold plates and between a gold tip and a gold plate inpresence of the surface dipole moment. This dipole moment can be due to the “intrinsic” dipole moment of the surfacedouble layer or induced by the potential difference. It was shown that in an extreme near-field the Casimir friction isenhanced by many orders of the magnitudes in comparison to the case when the surface dipole moment is not takeninto account, and it can measuring using the present state of art equipment.According to the theory of the Casimir friction for two plates separated by a vacuum gap with thickness d andsliding with relative velocity v (see Fig. 1) for d < λ T = c ¯ h/k B T ) and non relativistic velocity v ≪ c the frictional forcebetween surfaces is dominated by the contribution from evanescent electromagnetic waves and determined by ? f x = ¯ h Z ∞ dω π Z q>k d qq x (2 π ) e − k z d (cid:20) R p ( ω, q )Im R p ( ω ′ , q ) | − e − k z d R p ( ω, q ) R p ( ω ′ , q ) | [ n ( ω ′ ) − n ( ω )] + ( p → s ) (cid:21) , (1)where n i ( ω ) = [ e ¯ hω/k B T i − − , ω ′ = ω − q x v , R p and R s are the reflection amplitudes for p − and s − polarizedelectromagnetic waves, k = ω/c , k z = p q − k , q > ω/c is the component of the wave vector parallel to the surface.To linear order in the sliding velocity f x = γv where the friction coefficient at T = T = T is determined by ? γ rad = ¯ h π k B T Z ∞ dω sinh (¯ hω/ k B T ) Z ∞ dk z k z ( k z + k ) e − k z d (cid:20) Im R p ( ω, q )Im R p ( ω, q ) | − e − k z d R p ( ω, q ) R p ( ω, q ) | + ( p → s ) (cid:21) , (2)In the conventional theory of the fluctuation-induced electromagnetic phenomena (FIEP) the reflection amplitudesare determined by the classical Fresnel formulas R p = iεk z − k ′ z iεk z + k ′ z , (3) R s = ik z − k ′ z ik z + k ′ z , (4)where k ′ z = p εk − q , ε is the dielectric function of the substrate. However, Eqs.(3)-(4) do not take into accountthe dipole moments which can exist on the surfaces and give rise to dominant contribution to FIEP in the near-field.First, the potential difference between metallic surfaces will induce the surface charge densities ± σ s . The interactionof the normal component of the electric field E z of the electromagnetic wave with the charged surface will result insurface dipole moment p s = α s E z = σ s M E z , where α s = σ s M is the surface dipole susceptibility, σ s = ϕ/ πd is thecharge density induced on the surfaces by the potential difference ϕ between the surfaces, M is the surface mechanicalsusceptibility which determine the surface displacement under the action of external mechanical stress: u = M σ extzz .In a elastic continuum model M = iρc t (cid:18) ωc t (cid:19) p l ( q, ω ) S ( q, ω ) , (5)where S ( q, ω ) = "(cid:18) ωc t (cid:19) − q + 4 q p t p l ,p t = "(cid:18) ωc t (cid:19) − q + i / , p l = "(cid:18) ωc l (cid:19) − q + i / , where ρ , c l , and c t are the density of the medium, the velocity of the longitudinal and transverse acoustic waves,Second, a redistribution of the electron density will result in formation of the electrical double layer on the metalsurface when the surface atomic layer 1 is charged negatively, and the sub-surface layer 2 is positively charged (seeFig. 1). The generic mechanism for such redistribution for metal surfaces is the “spill out” of electron into vacuum.. The dipole moment of the electrical double layer p d = α d E z where the double layer susceptibility α d can be foundfrom the equations of motion for layers 1 and 2 ( − ρ d ω − iρ d ωγ d + K ) u − Ku = σ d E z , (6) u = M [ K ( u − u ) − σ d E z ] , (7)from where α d = σ d [ ω d − ( ω + iωγ d ) M K ] ρ d ω d [ ω d − ( ω + iωγ d )(1 + M K )] , (8)where ρ d and γ d are the density and damping constant for vibrations of atomic plane 1, K = Y /d is the elasticconstant for the interplanar interaction, Y is the Young modulus, d is the interplanar distance, σ d is the surfacecharge density of plane 1, ω d = p Y /ρ d d .In the presence of the surface dipole moment the reflection amplitudes for the p -polarized electromagnetic wavesare determined by R p = iεk z − k ′ z + 4 πiq α ⊥ εiεk z + k ′ z − πiq α ⊥ ε (9)where α ⊥ = α d + α s , α d and α s are the normal components of the surface and double layer susceptibilities, respectively.For s -polarized electromagnetic field the reflection amplitude R s = ik z − k ′ z ik z − k ′ z . (10)For small separation between surfaces and good conductors a fluctuating electromagnetic field in the vacuumgap is determined by the fluctuations of the surface dipole moments and can be calculated in the electrostaticapproximation . The friction force can be calculated from equation ? f x v = Q + Q (11)where Q and Q are the heat generated in surfaces 1 and 2 which are determined by the rate of the work of theelectric field in surfaces 1 and 2 in the rest reference frame of surfaces 1 and 2 Q = Z ∞−∞ dω π Z d q (2 π ) ω Im h p z E z i , (12) Q = Z ∞−∞ dω π Z d q (2 π ) ω Im h p z E z i (13)where E iz is the electric field on surface i in the rest reference frame of surface i : E z = ap z + bp − z , (14) E z = ap z + bp +1 z , (15)where p iz = p iz ( ω ) is the dipole moment in surface i in the rest reference frame of surface i , p ± iz = p iz ( ω ± ) , ω ± = ω ± q x v , a = 4 πq coth qd, b = 4 πq sinh qd . (16)According to the fluctuational electrodynamics , the surface dipole moments p z = p f z + α E z = p f z + α ( ap z + bp − z ) , (17) p z = p f z + α E z = p f z + α ( ω )( ap z + bp +1 z ) , (18)where α i = α i ( ω ) , p fi is the fluctuating dipole moment due to the thermal and quantum fluctuations in solids and p indi = α i E iz is the induced dipole moment. From (17) and (18) p z = (1 − aα − ) p f z + bα p f − z (1 − aα )(1 − aα − ) − b α α − , (19) p z is obtained from p z as a result of the index permutations ( ↔ ) and frequency change ( ω → ω + ) in the rightside of Eq.(19).According to the fluctuation-dissipation theorem, the spectral density of fluctuations of the dipole moment in surface i in the rest reference frame of this surface is determined by h| p fi | i = ¯ h Im α i ( ω, q ) coth ¯ hω k B T i . (20)From Eqs. (11), (12) - (15), and (19) f x = 4 Z ∞ dω π Z d q (2 π ) q x b Im α ( ω )Im α ( ω ′ ) | (1 − aα ( ω ))(1 − aα ( ω ′ ) − b α ( ω ) α ( ω ′ ) | [ n ( ω ′ ) − n ( ω )] , (21)In present article the surface dipole susceptibility is determined by vibrational modes thus the friction in Eq.(21) isoriginated from the phonon excitations. The phononic friction coefficient is given by γ ph = ¯ h π k B T Z ∞ dω sinh (¯ hω/ k B T ) Z ∞ dqq b Im α ( ω )Im α ( ω ) | (1 − aα ( ω ))(1 − aα ( ω ) − b α ( ω ) α ( ω ) | (22)For q ≫ | ε | ω/c and /ε ≪ πqα ≪ from Eq. (9) R p ≈ πqα and from Eqs. (2) and (22) the contribution to thefriction coefficient from the p -polarised waves γ rad ≈ γ ph ≈ h k B T Z ∞ dω sinh (¯ hω/ k B T ) Z ∞ dqq Im α ( ω )Im α ( ω )sinh qd (23)After calculations which are similar to the one conducting in Refs. for phononic heat transfer, it can be shownthat the phononic friction associated with the van der Waals interaction between surface displacements is given alsoby Eq. (21) where α should be replaced on M and a = H πd , (24) b = H π q K ( qd ) d , (25)where H is the Hamaker constant, K ( x ) is the modified Bessel function of the second kind and second order.Fig. 2 shows the friction coefficient associated to friction between two gold plates on the distance between them fordifferent mechanisms at T = 300 K. For gold c l = 3240 ms − , c t = 1200 ms − , ρ = 1 . × kgm − , Y = 79 GPa anddielectric function ε = 1 − ω p ω + iων , (26)where ω p = 1 . × s − , ν = 4 . × s − , . The susceptibility of the double layer For the damping constant inEq.(8 for the susceptibility of the double layer an estimate was used γ d = Cω (27)Such estimation was used for the damping of the surface phonon mode (the Rayleigh waves) with C = 0 . forgold , and for the 2D sheet flexural vibrational damping of graphene sheet with C =0.01. In Fig. 2 blue linesare for the radiative friction associated with p -polarized waves with taking into account the contribution from theelectrical double layer at C = 0 . (full blue line), . (dashed blue line), and (dotted blue line). Brown line for thephonic friction associated with the electrostatic interaction of the surface dipole moments induced by the potentialdifference between surfaces ϕ = 10 V. Pink line for the phononic friction associated with the van der Waals interactionbetween the fluctuating surface displacements. Green line shows the result of the conversion theory for the radiativefriction without taking into account the contribution from the surface dipole moments. In the extreme near-field( d < nm) the double layer contribution leads to the enhancement of the radiative friction by many orders of themagnitude in comparison with the conversion theory which does not include this contribution. the p-wave contributionis dominated by the contribution from the double layer and the radiative heat transfer is reduced to the electrostaticphonon heat transfer. This means that in the extreme near field, the dipole moment fluctuations of the double layergive the dominant contribution to the heat transfer, and the contribution from bulk fluctuations can be neglected. For d < nm the contribution from the double layer exceed the s-wave contribution and for the subnanometer distancesthis contribution is several orders of magnitude greater than the heat flux predicted by the conventional theory ofradiative heat transfer which does not take into account the contribution from the double layer.Fig. 3 shows the results for the friction between a gold tip with radius R = 30 nm and a gold plate. The frictioncoefficient in the tip-plane configuration can be obtained from the plane-plane configuration using proximity (orDerjyaguin) approximation Γ tip = 2 π Z R dρργ ( z ( ρ )) (28)where R is the radius of the tip, γ ( z ) is the friction coefficient for friction between two plates separated by distance z ( ρ ) = d + R − p R − ρ denoting the tip-surface distance as a function of the distance ρ from the tip symmetry. 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