8/3/2019 O. Kenneth, I. Klich, A. Mann and M. Revzen- Repulsive Casimir forces

1/4

arXiv:quant-ph/0202114v1

20Feb2002

Repulsive Casimir forces

O. Kenneth, I. Klich, A. Mann and M. Revzen

Department of Physics,Technion - Israel Institute of Technology, Haifa 32000 Israel

(Dated: February 2002)

We discuss repulsive Casimir forces between dielectric materials with non trivial magnetic suscep-tibility. It is shown that considerations based on naive pair-wise summation of Van der Waals and

Casimir Polder forces may not only give an incorrect estimate of the magnitude of the total Casimirforce, but even the wrong sign of the force when materials with high dielectric and magnetic responseare involved. Indeed repulsive Casimir forces may be found in a large range of parameters, and wesuggest that the effect may be realized in known materials. The phenomenon of repulsive Casimirforces may be of importance both for experimental study and for nanomachinery applications.

It is well known that the fluctuations of electromag-netic fields in vacuum or in material media depend onthe boundary conditions imposed on the fields. This de-pendence gives rise to forces which are known as Casimirforces, acting on the boundaries. The best known exam-ple for such forces is the attractive force experienced byparallel conducting plates in vacuum [1]. Casimir forcesbetween similar, disjoint objects such as two conduct-ing or dielectric bodies are known in most cases to beattractive [2] and are sometimes viewed as the macro-scopic consequence of Van der Waals and Casimir-Polderattraction between molecules.

In view of the dominance of the Casimir forces at thenanometer scale, where the attractive force could leadto restrictive limits on nanodevices [3, 4], the study ofrepulsive Casimir forces is of increasing interest.

Repulsive Van der Waals forces are known to be pos-sible if the properties of the intermediate medium areintermediate between the properties of two polarizable

molecules [5]. In such cases the Hamaker constant be-comes negative, a property which was successfully em-ployed to explain the wetting properties of liquid helium[6]. How can one get a repulsive behavior when the in-termediate substance is vacuum? A partial answer canbe obtained from the observation that a purely magnet-ically polarizable particle repels a purely electrically po-larizable particle [7]. Motivated by this result, Boyer, fol-lowing Casimirs suggestion, studied inter plane Casimirforce with one plate a perfect conductor while the otheris infinitely permeable. He showed that in this case theplates repel [7]. This problem was reconsidered since in[11, 12]. However, one must note that for most molecules

the magnetic polarizability is negligible compared to theelectric polarizability. Indeed, in many of the treatmentsof the subject it is assumed that the Van der Waals in-teraction is dominated by the dielectric behavior of thematerials. If one then considers the pair-wise summa-tion of Van der Waals and Casimir-Polder forces as anapproximation to the force between materials, the resultis generally attractive.

Calculations of the interaction between macroscopicbodies by summation of pair - interactions is based on

the assumption of additivity of the interatomic interac-tion energies, which is only justified within second orderperturbation theory [8]. It was pointed out by Axilrodand Teller [9] that many-particle interactions may leadto substantial corrections to the so called additive re-sult. Sparnaay [10] estimated the corrections for somesimple many-body systems to be as large as 30%. Thesecorrections are usually taken to affect the magnitude ofthe force but not its sign.

In this Letter we emphasize that for materials withhigh magnetic susceptibility pair-wise summation is nolonger a good approximation for the macroscopic Casimirforce, due to collective response of the material. Indeed,we show that these considerations may not only give anincorrect estimate of the magnitude of the force, but insome cases even of its sign! Thus restrictions imposedon the sign of the force from pair wise consideration canbe misleading, and repulsive forces can be expected in awider range of dielectric-magnetic materials. We study

the Casimir force between materials with general permit-tivity and permeability and show that for large perme-ability and permittivity, the transition between attractiveand repulsive behavior depends only on the impedanceZ =

. In addition we show that at high tempera-tures there is always attraction, and thus in some casesthe force changes sign as the temperature is increased.

We start by examining the pair interaction. TheCasimir-Polder potential between two polarizable parti-cles A and B is given by [13, 14]:

U(r) = c4r7 [23(AE

BE +

AM

BM) 7(

AE

BM +

AM

BE)](1)

where E , M are the electric and magnetic polarizability

of the particles. From this equation it is immediate thatrepulsion between particles can be obtained. For exam-ple, a purely electrically polarizable particle will repel apurely magnetically polarizable particle. What does thistell us about materials described by a given permittivityand permeability?

As an illustration for the subtle character pair wisesummations may have we consider two materials withpermeability and permittivity i, i (i = 1, 2). Whenone considers two polarizable balls in vacuum [15] the

http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1http://arxiv.org/abs/quant-ph/0202114v1

8/3/2019 O. Kenneth, I. Klich, A. Mann and M. Revzen- Repulsive Casimir forces

2/4

2

coefficient of the force can be read from (1) to be of theform

FC 23111+2

212+2

23111+2

212+2

+ (2)

7 111+2

212+2

+ 7 212+2

111+2

.

To simplifies the last expression we set 1 = 1. In thiscase

FC 111+2

23 21

2+2 721

2+2

(3)

It can easily be shown that for 2 >3716 the force (3)

is negative for any 2, and thus two such balls attract.Thus if one regards the two materials to be made of suchballs, and use as an approximation to the Casimir forcesummation of pairs of these, one comes to the conclusionthat there is attraction whenever 2 >

3716 and 1 = 1.

Next, however, looking at the whole Casimir energy wedemonstrate that this last statement is wrong.

The Casimir interaction between two polarizable mate-rials can be conveniently expressed in terms of the reflec-

tion coefficients at the boundaries ([3, 16]). The Casimirenergy per unit area of two infinite slabs separated by adistance a is given by:

EC =12

d3k(2)3

ln(1 r(1, 1)r(2, 2)e2a|k|) + (4)

ln(1 r(1, 1)r(2, 2)e2a|k|)

where the two terms on the right correspond to TE andTM modes. Here r(, ) for the TE mode is given by [23]:

r(,,) =

cos2() + sin2()

cos2() + sin2() +

(5)

and by a similar expression with for the TM mode(i.e. rTM(, ) = rTE(, )). Hence the energy is givenby:

EC =182

0

d sin (6)0 dk k

2

ln(1 r(1, 1)r(2, 2)e2ak) +

The k integration can be done by expanding the loga-rithm and integrating term by term, to obtain

EC =1

322a3

0 Li4[r(1, 1, )r(2, 2, )]sin d (7)

+ ,

where Li4(x) =

n=1xn

n4is a polylogarithmic function.

In figure (1) we show positive and negative domains ofthe Casimir energy (7). Note that the condition 2 1(i = 1, 2). Thus even if at low temperature we had re-pulsion the force will change sign as we heat the sys-tem. However, the sign change is at temperature scalesofBT

c4a which at 100nm is of the order of 1000

K.This phenomenon can be qualitatively explained as fol-lows: The dominant contribution to the free energy at

high temperatures is due to static configurations (zeromodes), since contributions from other Matsubara fre-quencies are exponentially suppressed. In particular themagnetic-electric interactions are non-static by nature(there is no static interaction between a magnetic dipoleand an electric dipole). As a result the magnetic-electricpart of the interaction which are responsible for repulsivebehavior (see eq. (1)) will vanish in the high temperaturelimit.

In view of the growing interest and possibilities of mea-suring the Casimir effect [3, 4, 21, 22] we wish to pointout some advantages that repulsive Casimir forces might

have for actual measurements: First, there is no problemof stiction, i.e. even for very close separations, the mate-rials dont collapse on each other, in contrast to the usualeffect where it can sometimes be difficult to hold themapart, a property which might be crucial for constructionof nanomachines. Moreover it might be an easier task toalign two materials in parallel, since this will be theirnatural tendency (in places where the two materials getcloser the repulsion is stronger). We would also wish tonote that although it is true that for many materials themagnetic response is negligible, there are classes of mate-rials with high permeability, such as ferrites and garnets(notably YIG) which may be suitable for constructing a

demonstration of repulsive Casimir forces.We wish to thank J. Genossar for discussions. The

work was supported by the fund for promotion of researchat the Technion and by the Technion VPR fund.

Electronic address: kenneth@physics.technion.ac.il;klich@tx.technion.ac.il

[1] H. B. G. Casimir, Proc. Koninkl. Ned. Akad. Wet. 51,793 (1948).

[2] O. Kenneth and S. Nussinov, preprint hep-th/9912291.[3] M. Bordag, U. Mohideen and V. M. Mostepanenko, Phys.

Rept. 353 (2001) 1-205.[4] E. Buks and M. L. Roukes Phys. Rev. B 63 (2001)

033402.[5] Jacob N. Israelachvili, Intermolecular and surface forces

(Academic Press, London, 1992).[6] I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii.

Adv. Phys. 10, 165-209 (1961).[7] T. H. Boyer, Phys. Rev. A 9 2078 (1974).[8] F. London, Z. Phys. Chemie. (B) 11 222 (1930).[9] B. M. Axilrod and E. Teller, J. Chem. Phys. 11, 299

(1943).

mailto:kenneth@physics.technion.ac.il;%20klich@tx.technion.ac.ilmailto:kenneth@physics.technion.ac.il;%20klich@tx.technion.ac.ilmailto:kenneth@physics.technion.ac.il;%20klich@tx.technion.ac.ilhttp://arxiv.org/abs/hep-th/9912291http://arxiv.org/abs/hep-th/9912291mailto:kenneth@physics.technion.ac.il;%20klich@tx.technion.ac.ilmailto:kenneth@physics.technion.ac.il;%20klich@tx.technion.ac.il

8/3/2019 O. Kenneth, I. Klich, A. Mann and M. Revzen- Repulsive Casimir forces

4/4

4

[10] M. J. Sparnaay, Physica 25, 353 (1959).[11] D. T. Alves, C. Farina, A. C. Tort, Phys. Rev. A 61,

034102 (2000)[12] J. C. da Silva, A. Matos neto, H. Q. Placido, M Revzen

and A. E. Santana, Physica A 292, 411 (2001).[13] V. B. Berestetskii, E. M. Lifshitz and L. P.

Pitaevskii, Quantum Electrodynamics (Course of Theo-retical Physics, vol. IV ), (Pergamon, Oxford, 1982).

[14] G. Feinberg and J. Sucher, Phys. Rev. A 2, 2395 (1970).

[15] V. M. Mostepanenko and N. N. Trunov, The CasimirEffect and its Applications, (Oxford University Press,1997).

[16] O. Kenneth, preprint hep-th/9912102.[17] I. Brevik and H. Kolbenstvedt, Ann. Phys. (N.Y.) 143,

179 (1982).[18] T. D. Lee, Particle Physics and Introduction to Field

Theory (Harwood Academic Publishers, Chur 1980).[19] J. D. Jackson, Classical Electrodynamics, 2nd ed. (John

Wiley, New York, 1975), Ch. 16.[20] I. Klich, A. Mann and M. Revzen, Phys. Rev. D65 (2002)

045005.[21] H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop

and F. Capasso, Phys. Rev. Lett. 87, 211801 (2001).[22] F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M.

Mostepanenko, preprint quant-ph/0201087.[23] r is related to the usual reflection coefficient at the in-

terface between vacuum and a medium with given and (see for example eq.7.39 in [19]) by substitutingk

2 = 2

k

2 and wick rotating

ikt. The angle

in (5) is defined by cos = ktkt

2+k2

[24] The UVL condition implies = const. It was introducedby T. D. Lee [18] in an effective description of gluon fieldsand used by Brevik and Kolbendsvet [17] in the contextof electromagnetic Casimir forces.

[25] This is a simple result of the Kramers-Kronig relations.Note, however that if the intermediate vacuum is to befilled with another substance the relevant properties arerelative to the intermediate substance, and may yield re-pulsive behavior.

http://arxiv.org/abs/hep-th/9912102http://arxiv.org/abs/quant-ph/0201087http://arxiv.org/abs/quant-ph/0201087http://arxiv.org/abs/hep-th/9912102