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

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by  Mathias Boström

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.
Fromthisperspectiveitisthusthe couplingbetweenthe
[5] J.MahantyandB.W.Ninham,DispersionForces,(Aca-
system to a frequency-dependent heat bath, rather than
demic Press, London, 1976).
the finite speed of light, which is important. Repulsive [6] J. Israelachvili, Intermolecular and Surface Forces, 2nd
Lifshitz forces were found a long time ago in indirect Ed., (AcademicPress, London, 1991).
force measurements. One example is the work of Ander- [7] Bo E. Sernelius, Surface Modes in Physics (Wiley-VCH,
son and Sabiski on films of liquid helium on calcium flu- Berlin, 2001).
orite, and similar molecularly smooth surfaces [30]. The [8] K. A. Milton, The Casimir Effect: Physical Manifesta-
tions ofZero-Point Energy,(WorldScientific,Singapore,
films ranged from 10-200 Å[30]. The thickness of the
2001).
films could be measured to within a few percent in most
[9] V. A. Parsegian, Van der Waals forces: A handbook
cases. For the saturated-film measurements the repul- for biologists, chemists, engineers, and physicists, (Cam-
sive van der Waals potential was equal to the negative bridge University Press, NewYork,2006).
of the gravitational potential [30]. A good agreement [10] B. W. Ninham and P. Lo Nostro, Molecular Forces and
was found[21] between these experimental data and the Self Assembly in Colloid, in Nano Sciences and Biology,
results from Lifshitz theory. In another more subtle ex- (Cambridge UniversityPress, Cambridge, 2010).
[11] H. Wennerström, J. Daicic, and B. W. Ninham, Phys.
perimentHauxwellandOttewill[31]measuredthe thick-
Rev. A 60, 2581 (1999).
ness of films of oil on water near the alkane saturated [12] B. W. Ninham, and J. Daicic, Phys. Rev. A 57, 1870
vapour pressure. For this system n-alkanes up to oc-
(1998).
tane spread on water. Higher alkanes do not spread. It [13] B. V. Deryaguin and I. I. Abrikossova, Soviet Phys.–
is an asymmetric system (oil-water-air)and the surfaces Doklady 1, 280 (1956).
are molecularly smooth. The phenomenon depends on a [14] D.TaborandW.H.S.Winterton,Proc.Roy.Soc.(Lon-
balance of van der Waals forces against the vapor pres- don) A 312, 435 (1969).
[15] J. N. Israelachvili and D. Tabor, Proc. R. Soc. Lond. A
sure[22, 31, 32]. The net force, as a function of film
331, 19 (1972).
thickness depends onthe dielectric propertiesofthe oils.
[16] L. R. White, J. N. Israelachvili and B. W. Ninham, J.
As demonstrated[22] it involves an intricate balance of Chem. Soc., Faraday Trans. 1, 72, 2526 (1976).
repulsive and attractive components from different fre- [17] G.Palasantzas,P.J.vanZwol,andJ.Th.M.DeHosson,
quency regimes. When the ultraviolet and visible com- Appl. Phys.Lett. 93, 121912 (2008).
ponentsareexponentiallydampedbyretardationtheop- [18] S.K.Lamoreaux,Phys.Rev.Lett.78,5(1997);81,5475
posing (repulsive) infrared components take over[32]. (1998).
[19] A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K.
Itisinstructivefinallytocomparethiswiththetheory
Lamoreaux, NaturePhysics, 7, 230, 2011.
forsurfacemeltingofice[33]. Surfacemeltingtakesplace [20] K. Milton, NaturePhysics, 7, 190, 2011.
if the liquid phase wets the solid-vapor interface at the
[21] P. Richmondand B. W.Ninham, Solid StateCommuni-
triplepoint. Again,itisthedelicatebalancebetweenthe cations 9, 1045 (1971).
permittivities ε(iω )forthe liquidandthe solidwhichis [22] P.Richmond,B.W.NinhamandR.H.Ottewill,J.Col-
n
the main factor. While water is more polarizable than loid Int.Sci. 45, 69 (1973).
iceatopticalfrequencies,thepermittivityforiceexceeds [23] J.N.Munday,F.Capasso, andV.A.Parsegian, Nature,
that of water for frequencies higher than about 2×1016 457, 07610 (2009).
[24] A. Grabbe, Langmuir 9, 797 (1993).
rad/s. Athinlayerofwateratfirsttends tothicken,but [25] M. Boström and B. E. Sernelius, Phys. Rev. Lett. 84,
the thickening becomes gradually hampered because of
4757 (2000).
the retardation. The surface melting is thus incomplete. [26] B. V. Derjaguin, I. I. Abrikosova, E. M. Lifshitz, Quart.
One notices from Fig. 1 in [33] that the crossing point Rev., 10, 295 (1968).
between the permittivities for ice and water occurs at [27] J. N. Munday, D. Iannuzzi, Y. Barash, and F. Capasso,
about 1.5×1016 rad/s, thus about at the same point as Phys. Rev.A 71, 042102 (2005).
[28] D. Qua, G. Brotons, V. Bosio, A. Fery, T. Salditt, D.
the crossing point in Fig. 1 above.
Langevin, and R. von Klitzing, Colloids and Surfaces
The research was sponsored by the VR-contract A: Physicochemical and Engineering Aspects 303, 97
No:70529001which is gratefully acknowledged.
(2007).
[29] E. Bonaccurso, M. Kappl, H.-J. Butt, Current Opinion
in Colloid & Interface Science, 13, 107 (2008).
[30] C. H. Anderson and E. S. Sabisky, Phys. Rev. Lett. 24,
1049 (1970).
∗ [email protected] [31] F. Hauxwell and R. H. Ottewill, J. Colloid Int. Sci. 34,
5
473 (1970). [33] M. Elbaum and M. Schick, Phys. Rev. Lett. 66, 1713
[32] B.W.NinhamandV.A.Parsegian, Biophys.J. 10,647 (1991).
(1970).