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Retardation turns the van der Waals attraction into Casimir repulsion already at 3 nm PDF

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Preview Retardation turns the van der Waals attraction into Casimir repulsion already at 3 nm

Retardation turns the van der Waals attraction into a Casimir repulsion already at 3 nm ∗ Mathias Boström and Bo E. Sernelius Division of Theory and Modeling, Department of Physics, Chemistry and Biology, Linköping University, SE-581 83 Linköping, Sweden Iver Brevik Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Barry W. Ninham Department of Applied Mathematics, Australian National University, Canberra, Australia 2 CasimirforcesbetweensurfacesimmersedinbromobenzenehaverecentlybeenmeasuredbyMun- 1 0 day et al. Attractive Casimir forces were found between gold surfaces. The forces were repulsive 2 between gold and silica surfaces. We show the repulsion is due to retardation effects. The van der Waalsinteractionisattractiveatallseparations. Theretardationdrivenrepulsionsetsinalreadyat n around 3nm. To ourknowledge retardation effects haveneverbeen found at such asmall distance a before. Retardation effects are usually associated with large distances. J 5 PACSnumbers: 42.50.Lc,34.20.Cf;03.70.+k ] h Whentwo objectsarebroughttogethercorrelationsin tween a pair of parallel, closely spaced, perfect conduc- p - fluctuations of the charge and current densities in the tors a distance apart. Almost 50 years later Lamore- t n objects or fluctuations of the fields usually result in an aux[18] performed the first high accuracy measurement a attractive force[1–10]. At short distances this is the van ofCasimirforcesbetweenmetalsurfacesinvacuum. Fol- u der Waals force[1]; at large distances the finite velocity lowing Tabor and Isrealachvili, and later Lamoreaux, a q of light becomes important (retardation effects) and the new research field developed[19, 20]. [ resultisthe Casimirforce[2,8]. With retardationeffects 1 weheremeanalleffects thatappearbecauseofthe finite An interesting aspect of the Casimir force is that ac- v cordingtotheoryitcanalsoberepulsive[3,21,22]. Early speed of light; not just the reduction in correlation be- 6 experiments showing this indirectly will be discussed at tween the charge density fluctuations at large distances; 3 the end of this communication. Munday, Capasso, and 1 newpropagatingsolutionstoMaxwell’sequationsappear Parsegian[23] performed the first direct force measure- 1 that bring correlations between current density fluctua- 1. tions. When these propagating modes dominate we call ments that demonstrated that the Casimir force could be repulsive by a suitable choice of interacting surfaces 0 the forceCasimir force;when the surfacemodes[7] dom- in a fluid. They found attractive Casimir forces between 2 inate we callthe forcevan der Waalsforce. The classical 1 gold (Au) surfaces in bromobenzene (Bb). When one literature says that retardationis due to the finite speed : surface was replaced with silica (SiO ) they measured a v of light that weakens the correlations. This is mislead- 2 repulsive Casimir force. They speculated that this effect i ing. Itisduetothequantumnatureoflight[11,12];this X couldallowquantumlevitationofnano-scaledevices[23]. becomes obvious at the large separation limit where the r a weakening is independent of the speed of light. Wepresentcalculationsforthe samesystem(SiO [24] 2 Since the famous experiments of Deryaguin and interacting with Au[25] across Bb[23]) and find that at- Abrikossova[13]therehasbeenmuchinterestinphenom- tractiveandrepulsiveCasimirforcesareproducedinone ena which measure the van der Waals forces acting be- and the same system. Most notably we find that re- tween macroscopic bodies. The early experiments which tardation effects changes the sign of the Casimir force measuredtheforcesbetweenquartzandmetalplatescov- already for separations of the order of 3-5 nm. Retar- eredonlythe retardedregion. TheexperimentsofTabor dation effects usually come at much larger separations. and Winterton[14] and subsequently of Israelachviliand TheCasimirforcecanbecalculatedifthedielectricfunc- Tabor[15,16] fitted the potentialto a powerlaw of1/dn tions (for imaginary frequencies, iω) are known. These (d being the distance) where n varied from non retarded functions are shown in Fig.1. (n = 3) to fully retarded (n = 4) value. There was a There is a crossing between the curves for SiO and gradualtransitionfromnonretardedtoretardedforcesas 2 Bb. This opens up for the possibility of a transition of the separationwas increased from 12 nm to 130 nm[15]. the Casimir energy, from attraction to repulsion. A gradualtransition in the case of two goldsurfaceswas demonstrated experimentally by Palasantzas et al.[17] The interaction of material 1 (SiO ) with material 3 2 Casimirpredictedalreadyin1948[2]theattractionbe- (Au) across medium 2 (Bb) results in a summation of 2 100 as compared to without retardation is due to a sub- tle balance of attractive (high frequencies) and repul- Au sive (low frequencies) contributions. When the separa- tion increases from its lowest value the attractive con- tributions at high frequencies are weakened by retarda- 10 tion at a higher rate than the repulsive contributions ) " at low frequencies. The distance-dependent exponential, !(i Bb exp −2d q2+ε (iω)(ω/c)2 , in the Lifshitz expres- Bb (cid:20) q (cid:21) sion for the Casimir energy, Eqs.(2) and (3), has the SiO 2 effect that all frequencies contribute to the nonretarded 1 vanderWaalsforcewhileonlylowfrequenciescontribute 1015 1016 1017 inthe longrangeretardedCasimirregime. The ultimate " (rad/s) long-range asymptote, the entropic term, only includes the zero frequency transverse magnetic modes. Both Figure 1. (Color online) Thedielectric function at imaginary thelong-rangeCasimirasymptoteandthelong-rangeen- frequenciesforSiO2 (silica)[24],Bb(bromobenzene)[23],and tropic asymptote are repulsive for the systemconsidered Au (gold)[25]. here. However, as can be seen in Fig.3. there is a tran- sition region between the van der Waals region (where retardation does not influence the result) and the long imaginary frequency terms[7]: rangeCasimirregion. InthisregiontheretardedCasimir ∞ energy is attractive. This means that it is possible to ′ F = g(ωn). (1) via the optical properties tune the materials used such nX=0 that they in one region give attractive Casimir interac- tions and in another region give repulsive Casimir inter- Note that positive values of F correspond to repulsive actions. There is also a, to our knowledge notyet exper- interaction. imentally observed, maximum of the repulsive Casimir Intheretardedtreatmenttherearecontributionsfrom interaction. This maximum (which occurs around d = thetwomode-typestransversemagnetic(TM)andtrans- 5 nm) and the transition from attraction to repulsion verseelectric(TE),g(ω )=gTM(ω )+gTE(ω ),where n n n (which occurs around d = 3 nm) pose suitable experi- mental tests for the validity of the theory. Notably the gTM(ωn)= β1 R (2dπ2q)2×lnnε31(i−ωne)−γ22−γ2εd2(hiωεεn11(()iiγωω3nn))γγ22;−+εε22((iiωωnn))γγ11i caoronutrnidbuatpiopnrosxfirmomatetlrya2n6svÅer.sTehmisamgneeatnicstmhaotdeastoanreevzeerryo hε3(iωn)γ2+ε2(iωn)γ3io short separation the Casimir force comes entirely from γgTE=(ωnq)2=−βε1 R(iω(2dπ2)q)2(linωn/1c−)2.e−2γ2dhγγ22−+γγ11ihγγ22−+γγ33io; ttraarndsavtieornseiselnecetgrliecctmedo.deTsh.iTshisesaecmonosdeeqsuaenreceabosfetnhtatiftrhee- i i n n q total force in the system is very small. Our results sug- (2) gest that it is possible to measure retardation effects at Inthe nonretardedtreatmentstherearenoTEcontribu- much shorter separations than has previously been ex- tions and the result is pected. Experiments are often performed in the sphere- g(ω )= 1 d2q ln 1−e−2qd ε1(iωn)−ε2(iωn) plate geometry. Then the force in terms of the Casimir n β (2π)2 n hε1(iωn)+ε2(iωn)i (3) free energy, F (d), is[26] 2πR·F (d), where we see that R × ε3(iωn)−ε2(iωn) . the force is proportional to the energy; the maximum hε3(iωn)+ε2(iωn)io repulsive force occurs at the position of the maximum Frequencyintervalswheretheinterveningmediumhas repulsiveenergy. The attractiveCasimirenergybetween a dielectric permittivity in between the permittivity of gold surfaces in either bromobenzene or air is shown in the two objects give a repulsive contribution; other in- Fig.4. Hereboththeretardedandnonretardedenergies tervals give an attractive contribution. This is obvi- decrease monotonically with separation. ous in the nonretarded treatment because of the factor The long-range part of the interaction is shown in (ε −ε )(ε −ε ) appearing in the integrand. It Fig.5. In the 1-2 µm range the thermal entropic effect Au Bb SiO2 Bb is not so obvious in the retarded treatment where the starts to become important. This is similar to what we integrands are more complex. The decomposition of the found for the Casimir force between real metal surfaces Casimirfreeenergyintothefrequencydependentanddis- in vacuum[25]. tance dependent spectral function, g(ω ), is presented Usingopticalmeasurementsonthe interactingobjects n in Fig.2. We note as for the systems above the rever- and liquid makes it possible to predict the force — from sal of the sign of the Casimir energy with retardation short-rangeattractive Casimir force to long range repul- 3 0.0003 10-3 d = 5nm _______ retarded 10-4 0.0002 _ _ _ _ _ non-retarded 2m) all terms 2m) g/c 10-5 c0.0001 er erg/ d = 10nm gy ( 10-6 n = 0 term !) (n 0 Ener 10-7 g( mir si 10-8 -0.0001 a C 10-9 1015 1016 1017 102 103 ! (rad/s) n d (nm) Figure 2. (Color online)The spectral function, g(ωn), as a Figure 5. The fully retarded Casimir interaction free energy function of imaginary frequencies for two different separa- andthen=0entropictermforasystemwithAuinteracting tions. with SiO2 across bromobenzene. 0.005 sive Casimir force. We find similar effects of retardation when gold is replaced with silver (the difference is that 2m) 0 the crossing from attractive to repulsive Casimir force c g/ then comes around d = 5 nm). We predict strong retar- y (er -0.005 dation effects that are much more pronounced at short Energ -0.01 NReotn.-ret. aCnadsiminitrerinmteedraiactteionsepaasrcaotmionpsariendstyostienmssyswteitmhsrwepituhlsiavte- mir RReett.. TTEM tractive Casimir interaction. We stress that it should si -0.015 zero be possible to measure a maximum repulsive Casimir a C force. We suggest that this distance of maximal repul- -0.02 sion should be the optimal distance to enable quantum 2 3 4 5 6 7 8 9 10 levitationofnano-scaledevices(floatingoneachotherin d (nm) a liquid due to repulsive Casimir interactions)[23]. An- othersystemthatchangessignforaparticularseparation Figure 3. (Color online) The retarded and nonretarded wasdiscussedbyMundayetal.[27]forbirefringentdisks Casimirinteractionfreeenergybetweengoldandsilicainbro- in a fluid; however, there was little discussion regarding mobenzene. AlsoincludedarethecontributionsfromTMand TE modes. the significance of transition. We would finally like to mention a caveat that poses inevitabledifficultiesininterpretationofmeasurementof 0.0 these forces. With isotropic molecules the granularity Au-Bb-Au -0.50 of liquids near surfaces gives rise to oscillatory forces– 2m) Au-Air-Au much larger than van der Waals–extending to at least g/c -1.0 4-6 molecular layers before merging into continuum the- y (er -1.5 ory[6,10,28,29]. Withanisotropicmolecules,overlapof g surface induced order parameters gives rise to attractive er n orrepulsiveforcesofthe samerange. Ifthey donotturn mir E -2.0 __ ___ ___ __ rneotnar-dreetdarded up, the reasons may be that the surfaces used are rough si -2.5 at a molecular level. This tends to smooth oscillations. a C The minimum roughness would be say around 2 molec- -3.0 ular layers at each surface. This gives rise to an uncer- 1 2 3 4 5 6 7 8 9 10 taintyindistancethatmustbeconsideredincalculations d (nm) that aim at comparing with high accuracy experiments. Tosumup,wehaveseenthattheeffectsofretardation Figure 4. (Color online) The retarded and nonretarded turnup alreadyat veryshortdistances,of the orderof a Casimir interaction free energy between two gold surfaces in either (a) bromobenzene or (b) air. few nm. Remarkably the effect of retardation can be to turnattractionintorepulsion. Theessentialphysicalpa- 4 rameterisε(iωn),arealandpositivequantityintimately [1] F. London, Z. Phys.Chem. B 11, 222 (1930). connected with absorption (imaginary permittivity ε′′) [2] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 through the Kramers-Kronigdispersion relation (1948). [3] I.E.Dzyaloshinskii,E.M.Lifshitz, andP.P.Pitaevskii, 2 ∞ xε′′(x) Advan.Phys.10, 165 (1961). ε(iωn)−1= π Z x2+ω2 dx. (4) [4] D. Langbein, Theory of Van der Waals attraction, 0 n (Springer,NewYork,1974,)intheseriesSpringerTracts in Modern Physics. 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