Page 1

Casimir spring and compass: Stable levitation and alignment of compact objects

Sahand Jamal Rahi1, ∗and Saad Zaheer2

1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

We investigate a stable Casimir force configuration consisting of an object contained inside a

spherical or spheroidal cavity filled with a dielectric medium. The spring constant for displacements

from the center of the cavity and the dependence of the energy on the relative orientations of the

inner object and the cavity walls are computed. We find that the stability of the force equilibrium

can be predicted based on the sign of the force, but the torque cannot be.

The Casimir force between atoms or macroscopic ob-

jects arises from quantum fluctuations of the electrody-

namic field [1, 2]. In all known examples it is found that

the force is attractive, as long as the space between the

objects is empty, and the magnetic susceptibility of the

objects is negligible compared to their electric suscep-

tibilities. But when space is filled with a medium with

electric permittivity ?Mintermediate between that of two

objects, ?1< ?M< ?2, the force between the two becomes

repulsive [3]. This effect has recently been verified exper-

imentally [4]. But while repulsive forces are nothing new,

they become interesting for applications when they pro-

duce stable equilibria, which, for example, the Coulomb

force cannot.

For an infinite cylinder enclosed in another, the

Casimir force has recently been shown to have a stable

equilibrium in the two directions perpendicular to the

cylinder axes, when the material properties are chosen

so that the force between two slabs of the same materials

would be repulsive [5]. If the inner and outer cylinders

have square cross sections, the direction of the torque

exerted by one cylinder on the other is found to agree

qualitatively with the predictions of the pairwise summa-

tion or proximity force approximations (PSA or PFA).

Orientation dependence of the Casimir energy has also

been studied recently for a small spheroid facing another

spheroid or an infinite plate [6], or for a small spheroid lo-

cated inside a spherical metal cavity [7]. (The smallness

of the spheroid refers to keeping only the first term in

the series expansion of the Casimir energy in the largest

length scale of the spheroid.) The preferred orientation

of the small spheroid flips when it is moved between the

inside and the outside of the spherical shell; there is no

torque if the spheroid faces an infinite metal plate, which

is the limit of an infinitely large shell.

In this paper we investigate the first configuration, de-

picted in Fig. 1, in which a compact object levitates sta-

bly due to the Casimir force alone. We characterize the

Casimir energy

E = E0+1

2ka2

R2+1

3!k3a3

R3+1

4!k4a4

R4+ ···

(1)

by the spring constant k and the coefficients knin a se-

ries expansion in a/R, where a is the magnitude of the

FIG. 1: (Color online) Summary of the configurations we con-

sider and of the results. To be concrete, we have assumed

that the small spheroid’s zero frequency permittivity satisfies

?I,0 > ?M,0 and that it is larger in the body-fixed ˆ z direction,

so αE

⊥⊥. Furthermore, the magnetic permeabilities are

all set to one. a) Direction of the force on such a spheroid in

a spherical cavity if fE

and the direction of the torque when fE

fied when either ?M,0 > ?O,0 or ?M,0 ? ?O,0. b) A finite size

sphere experiences a restoring force for the various combina-

tions of materials listed in Table I. c) Direction of the torque

in the center of a slightly spheroidal cavity if gE> 0, which

requires either ?M,0 < ?O,0 or ?M,0 ? ?O,0.

zz> αE

1 > 0, which holds when ?M,0 > ?O,0,

2 > 0, which is satis-

displacement from the center of the cavity and R the ra-

dius of the (undeformed) spherical cavity. (k has units of

energy here.) Unlike the case of infinite cylinders, where

one object can be displaced along the cylinder axes with-

out changing the energy, our case exhibits, for appropri-

ately chosen materials, true stability in all directions and

applies to realistic situations. For example, we compute

the force on a metal sphere in a spherical drop of liquid

surrounded by air. By measuring the mean square de-

viation from the center, ?a2? = 3kBTR2/k, the spring

constant k can be verified experimentally. We can es-

timate that the size of the droplet R has to be smaller

than ≈ 3µmr3

inner object, for the thermal motion to be confined near

the center of the cavity. (This length scale is obtained by

balancing 3kBT for room temperature with a rough esti-

mate of the spring constant, k ∼?c

objects nearly concentric against the gravitational force,

R has to be smaller than ≈ 1µm for the typical metal or

liquid densities considered here. A variety of applications

may benefit from our analysis; in cancer therapy new

treatments utilize nanocarriers that trap drug particles

inside R ≈ .1µm polymer or lipid membrane shells [8],

R3, where r is the typical dimension of the

R

r3

R3.) To keep the two

arXiv:0909.4510v1 [cond-mat.stat-mech] 24 Sep 2009

Page 2

2

and molecular cages are proposed as containers for the

storage of explosive chemicals [9]. Our results may guide

the search for better materials and sizes of the enclosing

cell.

Whether the center of the cavity is a point of stable

or unstable equilibrium turns out to depend on whether

the Casimir force is repulsive or attractive for two sep-

arated objects under those conditions. The direction of

the torque, on the other hand, depends on the dielectric

properties of the medium and the cavity walls in an un-

intuitive way, which cannot be predicted by the PSA or

PFA, see Fig. 1. In particular, this behavior is not due

to dispersion effects, which explain similar phenomena

reported in Ref. [5]. We calculate the torque on a small

spheroid that is displaced from the center of a spherical

cavity (Fig. 1 a)) or concentric with a slightly spheroidal

cavity (Fig. 1 c)). In the former case, the orientation

dependence manifests itself in k. In the latter case, E0

depends on the relative orientation of the spheroid in-

side the deformed cavity. (If the cavity is spherical, E0

is a finite constant that can be ignored.) By choosing

appropriate materials, the inside object, e.g., a nanorod,

can be made to align in different ways with the cavity

shape, a situation which is reminiscent of a compass nee-

dle aligning with the magnetic field of the earth.

The starting point of the analysis is the scattering the-

ory approach. The method is explained and derived in

detail in Ref. [10], where a partial overview over its pre-

cursors, e.g. [11, 12, 13], is provided. The Casimir energy

?∞

is expressed in terms of the inner object’s exterior T-

matrix, Fee

Fii

regular wave functions to outgoing waves when the source

lies at infinity. The interior T-matrix expresses the oppo-

site, the amplitudes of the regular wave functions, which

result from scattering outgoing waves from a source in-

side the object. The translation matrices Wioand Vio

convert regular wave functions between the origins of the

outer and the inner objects; they are related by complex

transpose up to multiplication by (−1) of some matrix

elements.

With uniform, isotropic, and frequency-dependent per-

mittivity ?x(icκ) and permeability µx(icκ) functions (x =

I: inner object; x = O: outer object; x = M: medium)

the T-matrix of the sphere is diagonal. It is given by

E =?c

2π

0

dκlndet(I − Fii

OWioFee

IVio) (2)

I, and the outer object’s interior T-matrix,

O. The exterior T-matrix describes the scattering of

Fee

I,lmE,lmE(icκ) = Fee

il(ξ)∂r(ril(zIξ)) −

kl(ξ)∂r(ril(zIξ)) −

for E (electric) polarization and by the same expres-

sion with

?M

I,lE(ξ) =

?I

?Mil(zIξ)∂r(ril(ξ))

?I

?Mil(zIξ)∂r(rkl(ξ))

−

(3)

?I

replaced by

µI

µM

for M (magnetic) po-

larization (not to be confused with subscript M indi-

cating the medium’s response functions).

the frequency dependence of the response functions has

been suppressed. The indices of refraction nx(icκ) =

??x(icκ)µx(icκ) of the sphere and the medium appear

ξ = nM(icκ)κr. The first equality in Eq. (3) defines an

abbreviation for the T-matrix, in which the superfluous

polarization and angular momentum (l,m) indices are

suppressed. The interior T-matrix of the spherical cavity

is obtained from the exterior T-matrix of the sphere by

inserting the outside object’s radius and response func-

tion in place of those of the inside object and exchanging

the modified spherical Bessel functions il and kl every-

where.

However, the scattering approach is not limited to

simple geometries. An array of techniques is avail-

able for calculating the scattering amplitudes of other

shapes. We employ the perturbation approach to find

the T-matrix of a deformed spherical cavity [14, 15] of

radius R + δ(1 − 3/2sin2θ).

cated in Fig. 1 c), is chosen so that the volume is un-

changed to first order in δ. We find the O(δ) correction,

F(1), to the T-matrix in a perturbation series expansion,

F = F(0)+F(1)+···, by matching the regular and outgo-

ing fields according to the Maxwell boundary conditions

along the deformed object’s surface [16]. On the other

hand, for a small object (compared to the wavelength

of the radiation), we can approximate the T-matrix to

lowest order in κ using the static polarizability tensor,

Fee

script 0 indicates the static (icκ = 0) limit and P is

the polarization label. The T-matrix elements involving

higher angular momenta l > 1 are higher order in κ. For

a small ellipsoid, in particular, the electric polarizability

tensor αEis diagonal in a coordinate system aligned with

the ellipsoid’s body axes,

In Eq. (3)

in the ratio zI(icκ) = nI(icκ)/nM(icκ) and the argument

The deformation, indi-

I,1mP,1m?P= 2/3(nM,0κ)3αP

mm?+O(κ5), where the sub-

αE

ii=V

4π

?I,0− ?M,0

?M,0+ (?I,0− ?M,0)ni,

(4)

where i ∈ {x,y,z} [17]. The larger the semi-axis in di-

rection i, the smaller the depolarization factor ni(not to

be confused with the index of refraction). The entries

of the magnetic polarizability tensor αMare obtained

by exchanging µx,0for ?x,0in Eq. (4). In the small size

limit the polarizability tensor of a perfect metal ellipsoid

is obtained by taking both ?I,0 to infinity and µI,0 to

zero.

For simplicity we specialize to a spheroid, which has

two equal semiaxes. We choose the semiaxes along ˆ x

and ˆ y to be equal, therefore, αP

direction of displacement of the spheroid from the center

of the cavity is the lab’s ˆ z axis.

between the spheroid’s and the lab’s ˆ z axes. For such

a small spheroid inside a spherical cavity of radius R,

the spring constant is obtained by expanding the log-

xx= αP

yy= αP

⊥⊥. The

θ denotes the angle

Page 3

3

determinant in Eq. (2) to first order,

kR→∞=

?c

R4nM,0

+?αE

?Tr αEfE

⊥⊥

1

(5)

zz− αE

?3cos2θ−1

2

fE

2+ E → M?,

where the material dependent functions

?∞

fE

0

9π

fE

1=

0

ξ5dξ

9π

ξ5dξ

?Fii

?4

O,1M− 2Fii

O,1E− Fii

O,2E

?,

2=

?∞

5Fii

O,1E−1

5Fii

O,2E− Fii

O,1M

?

(6)

express the rotation invariant and the orientation depen-

dent parts of the energy, respectively. fM

obtained from fE

erywhere. This result is valid for asymptotically large

R; it involves only the zero frequency (icκ = 0) response

functions and the l = 1,2 scattering amplitudes of the

cavity walls in Eqs. (5) and (6).

The behavior of the functions fP

µM,0= µO,0, is as expected: fE

when ?M,0> ?O,0, negative when ?M,0< ?O,0, and fM

always has the opposite sign of fE

a small object with ?I,0> ?M,0is levitated stably, when

fE

occurs under the same conditions as repulsion for two ob-

jects outside of one another. The opposite sign of fM

expected from equivalent expressions for the two-infinite-

slab geometry [18]. When the dielectric contrast between

the medium and the outer sphere is taken to small or

large limits, stability or instability is maximized. To ver-

1

and fM

2

are

1 and fE

2 by exchanging E and M ev-

1, depicted in Fig. 2 for

1 is monotonic, positive

1

1. When fE

1is positive,

1 is negative, ?M,0> ?I,0has to hold. Thus, stability

1

is

0.20.40.6 0.81.

1

ΕM,0

ΕO,0? 1

?3

?2

?1

1

2

3

4

f1M

f1E

FIG. 2: (Color online) fE

spring constant kR→∞, which is invariant under a rotation

of the inside object. The vertical lines indicate the values

pertaining to the configurations presented in Table I, ethanol-

vacuum (0.16), bromobenzene-vacuum (0.30), and gold cavity

walls (1). In this plot, µM,0 = µO,0.

1 and fM

1

describe the part of the

ify whether stability is observable for realistic materials

and object sizes, we evaluate Eq. (2) numerically for a

sphere of radius r inside a spherical cavity of radius R

filled with various liquids. [23] The coefficients k, k4, and

k6in the series expansion in Eq. (1) are listed in Table I.

For comparison, the asymptotic result kR→∞is also in-

cluded. If both inside and outside object are spherically

symmetric, the series in Eq. (1) does not contain terms

∼an

Rn with n odd.

TABLE I: k, kR→∞, k4, and k6 are listed for various combi-

nations of materials for the case depicted in Fig. 1 b). The

dimensionless numbers in the table have to be multiplied by

?c

R. R is given in microns [µm]. kR→∞ depends on R only

through the ratios

same for all R. The highest cutoff used was lmax = 30.

?c

Rand

r

R, so its numerical prefactor is the

Inside-Medium-Outside R r/R

Gold-Bromobenzene-

Vacuum

kkR→∞

k4

k6

0.1 1/4 4.0e-2 5.4e-2 1.2

3/4 2.2e1 1.4

1.0 1/4 6.9e-2 5.4e-2 2.6

3/4 7.0e1 1.4

0.1 1/4 5.0e-2 3.7e-2 1.6

3/4 2.7e1 9.9e-1 5.2e3 2.2e6

1.0 1/4 4.5e-2 3.7e-2 1.7

3/4 6.0e1 9.9e-1 1.8e4 1.2e7

0.1 1/4 6.1e-3 7.1e-3 1.9e-1 1.3e1

3/4 4.2 1.9e-1 8.6e2 4.0e5

1.0 1/4 1.8e-2 7.1e-3 7.3e-1 6.0e1

3/4 2.3e1 1.9e-1 5.7e3 3.1e6

0.1 1/4 1.5e-2 1.2e-2 4.6e-1 3.1e1

3/4 1.0e1 3.3e-1 1.9e3 8.4e5

1.0 1/4 1.9e-2 1.2e-2 8.2e-1 7.2e1

3/4 4.1e1 3.3e-1 1.2e4 7.6e6

8.2e1

4.2e3 1.8e6

2.0e2

1.8e4 1.0e7

Gold-Ethanol-

Vacuum

1.0e2

1.4e2

Silica-Bromobenzene-

Gold

Silica-Ethanol-

Gold

The three materials were chosen so that the sequence

of permittivities ?I, ?M, ?Oeither increases or decreases

for the imaginary frequencies that dominate in Eq. (2).

Contrary to the prediction of the PSA and PFA the force

is not symmetric with respect to exchange of the inner

and outer permittivities. In the same medium, a high di-

electric sphere is held more stably in the center of a cav-

ity with low dielectric walls than a low dielectric sphere

inside a cavity with high dielectric walls.

The asymptotic result kR→∞ yields a good approxi-

mation of k for

expect kR→∞ to grow linearly with the volume of the

inner sphere, since polarizability is proportional to the

volume, see Eq. (4). In fact, k for

times larger than for

This means that for a gold sphere in a liquid drop with

r

R=3

Casimir spring holds the particle near the center effec-

tively,

Compared to the stability conditions studied thus far,

the orientation dependence of the energy is more varied.

fE

most ratios of medium to outside permittivities, unlike

r

R=

1

4. But from Eq. (5) one would

r

R=3

4is about 1000

r

R=

1

4, instead of just 27 times.

4and R = 1µm at room temperature, indeed, the

??a2?/R < 0.1.

2 and fM

2, plotted in Fig. 3, have the same sign for

Page 4

4

fE

ranges of values, the contributions to the torque from

electric and magnetic polarizability are opposite for a

small perfect metal spheroid, for which ?I,0> ?M,0and

µI,0< µM,0. Unlike fE

sign again at

So, while the direction of the total force and the stability

of the equilibrium can be predicted based on the relative

magnitudes of the permittivities, the torque cannot be.

The second sign change of fE

with the PSA or PFA. Furthermore, it raises the question

whether calculations of the Casimir torque for infinite

conductivity metals are ‘universal’ in the sense that they

produce the correct qualitative results for real materials.

1and fM

1, which always have opposite signs. In these

1and fM

1, also, fE

?O,0

?M,0≈ 2000, respectively.

2and fM

2 change

?O,0

?M,0≈ 80 and at

2 and fM

2

does not agree

0.2 0.4 0.60.81.

1

ΕM,0

ΕO,0? 1

?0.1

0.1

0.2

0.3

f2M

f2E

FIG. 3: (Color online) fE

spring constant kR→∞, which changes with the orientation of

the inside spheroid. In this plot, µM,0 = µO,0.

2 and fM

2

describe the part of the

The orientation dependent part of the energy for the

configurations discussed thus far vanishes, of course,

when the small spheroid is in the center of the spheri-

cal cavity. If the cavity is slightly deformed, however,

the energy, E0, depends on the relative orientation of the

spheroid and the cavity. We deform the spherical cavity

as described earlier and obtain to first order in δ/R,

E0=?c cos2θ

R4nM,0

δ

R

??αE

zz− αE

⊥⊥

?gE+ E → M?,

(7)

where the orientation-independent part of the energy has

been dropped. For µO,0= µM,0, gEand gMare given by

?∞

× (z2

?∞

where the arguments of the modified spherical Bessel

functions and of zO(0) are suppressed. iland klare func-

tions of ξ, and˜klstands for kl(zO(0)ξ), where zO(icκ) =

nO(icκ)/nM(icκ) is the ratio of the permittivities of the

cavity walls and the medium. The material dependent

functions gEand gMare plotted in Fig. 4.

gE=

0

ξ4dξ

10π

i1k0+i0k1

(˜k0i1+˜k1i0zO+i1˜k1/(zOξ)(1−z2

4˜k2

O))

2

O− 1)

ξ4dξ

10π

?

1/ξ2− (˜k0+˜k1/(zOξ))2?

i1k0+i0k1

(˜k1i0+˜k0i1zO)

,

gM=

0

2(z2

O− 1)˜k2

1,

(8)

0.20.40.6 0.8 1.

1

ΕM,0

ΕO,0? 1

?0.1

0.1

0.2

0.3

0.4

0.5

0.6

gM

gE

FIG. 4: (Color online) gEand gMdescribe the dependence of

the energy E0on the relative orientation of the inside spheroid

and the deformed cavity walls.

Again, gEhas the same sign for

Fig. 4) and

?O,0

?M,0= 1, gEalso vanishes at

The rich orientation dependence of the energy is ex-

pected to collapse as the size of the inside object grows

to fill the cavity and the PFA becomes applicable. Based

on the stability analysis for finite inside spheres, though,

we expect the asymptotic results, fP

the orientation dependence for reasonably small inside

spheroids.

In reality, various corrections to the idealized shapes

have to be taken into account. A drop of liquid, which is

placed on a surface, is influenced by gravity and interac-

tions with the substrate. An asymmetric inner object, or

one that is displaced from the center, deforms the shape

of the droplet additionally by causing uneven Casimir

stresses.

This work was supported by the NSF through grant

DMR-08-03315 (SJR). We thank T. Emig, N. Graham,

R. Jaffe, and M. Kardar for fruitful discussions.

?O,0

?M,0→ 0 (left in

?O,0

?M,0→ ∞ (right). In addition to the root at

?O,0

?M,0≈ 0.46.

2and gP, to predict

∗Electronic address: sjrahi@mit.edu

[1] H. B. G. Casimir and D. Polder, Phys. Rev. 73, 360

(1948).

[2] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793

(1948).

[3] I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii,

Adv. Phys. 10, 165 (1961).

[4] J. N. Munday, F. Capasso, and V. A. Parsegian, Nature

457, 170 (2009).

[5] A. W. Rodriguez, J. N. Munday, J. D. Joannopoulos,

F. Capasso, D. A. R. Dalvit, and S. G. Johnson, Phys.

Rev. Lett. 101, 190404 (2008).

[6] T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Phys.

Rev. A 79, 054901 (2009).

[7] S. Zaheer, S. J. Rahi, T. Emig, and R. L. Jaffe (2009),

arXiv:0908.3270.

Page 5

5

[8] D. Peer, J. M. Karp, S. Hong, O. C. Farokhzad, R. Mar-

galit, and R. Langer, Nat. Nanotechnol. 2, 751 (2007).

[9] P. Mal, B. Breiner, K. Rissanen, and J. R. Nitschke, Sci-

ence 324, 1697 (2009).

[10] S. J. Rahi, T. Emig, N. Graham, R. L. Jaffe, and M. Kar-

dar (2009), arXiv:0908.2649.

[11] T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Phys.

Rev. Lett. 99, 170403 (2007).

[12] O. Kenneth and I. Klich, Phys. Rev. B 78, 014103 (2008).

[13] P. A. Maia Neto, A. Lambrecht, and S. Reynaud, Phys.

Rev. A 78, 012115 (2008).

[14] C. Yeh, Phys. Rev. 135, A1193 (1964).

[15] V. A. Erma, Phys. Rev. 179, 1238 (1969).

[16] R. F. Millar, Radio Sci. 8, 785 (1973).

[17] L. D. Landau and E. M. Lifshitz, Electrodynamics of con-

tinuous media (Pergamon Press, Oxford, 1984).

[18] O. Kenneth, I. Klich, A. Mann, and M. Revzen, Phys.

Rev. Lett. 89, 033001 (2002).

[19] I. Pirozhenko, A. Lambrecht, and V. B. Svetovoy, New

J. Phys. 8, 238 (2006).

[20] A. Milling, P. Mulvaney, and I. Larson, J. Colloid Inter-

face Sci. 180, 460 (1996).

[21] L. Bergstr¨ om, Adv. Colloid Interface Sci. 70, 125 (1997).

[22] V. A. Parsegian, Van der Waals Forces (Cambridge Uni-

versity Press, Cambridge, 2005).

[23] For the permittivity function of gold we use ?(icκ) =

ω2

p

cκ(cκ+γ), where ωp

9.27 · 1013Hz [19] as used in Ref. [4]. For the other

materials we use an oscillator model, ?(icκ) = 1 +

PN

[5], with [ωn]n=1,2 = [5.47,128.6]·1014Hz and [Cn]n=1,2 =

[2.967,1.335] for bromobenzene [20] as used in Ref. [4],

and with [ωn]n=1−4 = [0.867,1.508,2.026,203.4] · 1014Hz

and [Cn]n=1−4 = [0.829,0.095,0.798,1.098] for silica

(SiO2) [21]. A discussion of permittivity models is given,

for example, in Ref. [22].

1 +

= 1.14 · 1016Hz and γ=

n=1

Cn

1+(cκ/ωn)2, with [ωn]n=1,2 = [6.6,114]·1014Hz and

[Cn]n=1,2 = [23.84,0.852] for ethanol [20] as used in Ref.