Stable levitation and alignment of compact objects by Casimir spring forces.
ABSTRACT 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-unlike the direction of the torque-can be predicted based on the sign of the force between two slabs of the same material.
- SourceAvailable from: Federico Capasso[show abstract] [hide abstract]
ABSTRACT: We present a scheme for obtaining stable Casimir suspension of dielectric nontouching objects immersed in a fluid, validated here in various geometries consisting of ethanol-separated dielectric spheres and semi-infinite slabs. Stability is induced by the dispersion properties of real dielectric (monolithic) materials. A consequence of this effect is the possibility of stable configurations (clusters) of compact objects, which we illustrate via a molecular two-sphere dicluster geometry consisting of two bound spheres levitated above a gold slab. Our calculations also reveal a strong interplay between material and geometric dispersion, and this is exemplified by the qualitatively different stability behavior observed in planar versus spherical geometries.Physical Review Letters 04/2010; 104(16):160402. · 7.94 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: We give a comprehensive presentation of methods for calculating the Casimir force to arbitrary accuracy, for any number of objects, arbitrary shapes, susceptibility functions, and separations. The technique is applicable to objects immersed in media other than vacuum, to nonzero temperatures, and to spatial arrangements in which one object is enclosed in another. Our method combines each object's classical electromagnetic scattering amplitude with universal translation matrices, which convert between the bases used to calculate scattering for each object, but are otherwise independent of the details of the individual objects. This approach, which combines methods of statistical physics and scattering theory, is well suited to analyze many diverse phenomena. We illustrate its power and versatility by a number of examples, which show how the interplay of geometry and material properties helps to understand and control Casimir forces. We also examine whether electrodynamic Casimir forces can lead to stable levitation. Neglecting permeabilities, we prove that any equilibrium position of objects subject to such forces is unstable if the permittivities of all objects are higher or lower than that of the enveloping medium; the former being the generic case for ordinary materials in vacuum. Comment: 44 pages, 11 figures, to appear in upcoming Lecture Notes in Physics volume in Casimir physicsLecture Notes in Physics 07/2010;
Stable Levitation and Alignment of Compact Objects by
Casimir Spring Forces
Rahi, Sahand Jamal, and Saad Zaheer. “Stable Levitation and
Alignment of Compact Objects by Casimir Spring Forces.”
Physical Review Letters 104.7 (2010): 070405. © 2009 The
American Physical Society
American Physical Society
Final published version
Tue Sep 13 15:40:32 EDT 2011
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
Stable Levitation and Alignment of Compact Objects by Casimir Spring Forces
Sahand Jamal Rahi1and 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
(Received 25 September 2009; published 19 February 2010)
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—unlike the direction of the
torque—can be predicted based on the sign of the force between two slabs of the same material.
DOI: 10.1103/PhysRevLett.104.070405 PACS numbers: 12.20.?m, 03.70.+k, 42.25.Fx
The Casimir force between atoms or macroscopic ob-
jects arises from quantum fluctuations of the electrody-
namic field . Typically, 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 susceptibilities. But
when space is filled with a medium with electric permit-
tivity ?Mintermediate between that of two objects, ?1<
?M< ?2, the force between the two becomes repulsive .
This effect has recently been verified experimentally in the
large separation (retarded) regime . But while repulsive
forces are nothing new, they become interesting for appli-
cations when they produce 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 repul-
sive . On the other hand, for a metal sphere or an
electrically polarizable atom inside an otherwise empty
spherical cavity there is no point of stable equilibrium .
In this Letter we investigate the first configuration,
depicted in Fig. 1, in which a compact object levitates
stably due to the Casimir force alone. We consider the
following cases: a finite sphere or a small spheroid inside a
spherical cavity, and a small spheroid inside a slightly
deformed spherical cavity. The Casimir energy
E ¼ E0þ1
is characterized by the spring constant k and the coeffi-
cients knin a series expansion in a=R, where a is the
magnitude of the displacement from the center of the
cavity and R the radius of the (undeformed) spherical
cavity. (k has units of energy here.) Unlike the case of
infinite cylinders, our case exhibits, for appropriately
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 determining the mean square deviation from the
center, ha2i ¼ 3kBTR2=k, the spring constant k can be
measured experimentally. We can estimate that the size
of the droplet R has to be smaller than ?3 ?mr3
is the typical dimension of the 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 tem-
perature with a rough estimate of the spring constant, k ?
thegravitational force, R has to be smaller than ?1 ?m for
the typical metal or liquid densities considered here. Our
calculations show that, for example, a sphere of gold of
radius r ¼3
R ¼ 0:1 ?m has an rms deviation
center due to thermal motion and a displacement from the
center by a ? 10?6R due to gravity. A variety of technol-
ogies may benefit from our analysis, e.g., nanocarriers
(R ? 0:1 ?m) for drug particles  or molecular cages
for explosive molecules .
Whether the center of the cavity is a point of stable or
unstable equilibrium turns out to be correlated with
R3, where r
R3.) To keep the two objects nearly concentric against
4R inside a spherical drop of ethanol of radius
ha2i¼ 0:04R from the
FIG. 1 (color online).
sider and of the results. We have assumed that the small
spheroid’s zero frequency permittivity satisfies ?I;0> ?M;0and
that it is larger in the body-fixed ^ z direction, so ?E
Furthermore, the magnetic permeabilities are all set to one.
(a) Direction of the force F on such a spheroid in a spherical
cavity if ?M;0> ?O;0, and the direction of the torque ? when
either ?M;0> ?O;0 or ?M;0? ?O;0. (b) A finite size sphere
experiences a restoring force F for the various combinations of
materials listed in Table I. (c) Direction of the torque ? in the
center of a slightly spheroidal cavity if either ?M;0< ?O;0or
Summary of the configurations we con-
PRL 104, 070405 (2010)
19 FEBRUARY 2010
? 2010 The American Physical Society
whether the Casimir force is repulsive or attractive for two
parallel plates under those conditions. The direction of the
torque, on the other hand, depends on the dielectric prop-
erties of the medium and the cavity walls in an unintuitive
way, which cannot be predicted by the pairwise summation
or proximity force approximations (PSA or PFA), see
Fig. 1. In particular, this behavior is not due to dispersion
effects, which explain similar phenomena reported in
Ref. . 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, which captures both the torque and the orienta-
tion dependence of the total force. In the latter case, E0
depends on the relative orientation of the spheroid inside
the deformed cavity. (If the cavity is spherical, E0is 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 needle aligning with the
magnetic field of the earth.
The starting point of the analysis is the scattering theory
approach. The method is explained and derived in detail
in Ref. , where a partial overview over its precursors,
the outer object’s interior T matrix, Fii
T matrix describes the scattering of regular wave functions
to outgoing waves when the source lies at infinity. The
interior T matrix expresses the opposite, the amplitudes of
the regular wave functions, which result from scattering
outgoing waves from a source inside the object. The trans-
lation matrices Wioand Vioconvert regular wave func-
tions 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 ?xðic?Þ and permeability ?xðic?Þ functions (x ¼
I: inner object; x ¼ O: outer object; x ¼ M: medium) the
T matrix of the sphere is diagonal. It is given by
IVioÞ is expressed in
terms of the inner object’s exterior T matrix, Fee
0d?ln detðI ? Fii
O. The exterior
for E (electric) polarization and by the same expression
?Mfor M (magnetic) polarization (not
to be confused with subscript M indicating the medium’s
response functions). In the above equation the frequency
dependence of the response functions has been suppressed.
The indices of refraction nxðic?Þ ¼
the sphere and the medium appear in the ratio zIðic?Þ ¼
nIðic?Þ=nMðic?Þ and the argument ? ¼ nMðic?Þ?r. 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 functions in place of those of
the inside object and exchanging the modified spherical
Bessel functions iland kl.
However, the scattering approach is not limited to sim-
ple geometries. An array of techniques is available for
calculating the scattering amplitudes of other shapes. We
employ the perturbation approach to find the T matrix of a
deformed spherical cavity [10,11] of radius R þ ?ð1 ?
3=2sin2?Þ. The deformation, indicated in Fig. 1(c), is
chosen so that the volume is unchanged to first order in
?. We find the Oð?Þ correction, Fð1Þ, to the T matrix in a
perturbation series expansion, F ¼ Fð0Þþ Fð1Þþ ???,
by matching the regular and outgoing fields according to
the Maxwell boundary conditions along the deformed
object’s surface . On the other hand, for a small object
(compared to the wavelength of the radiation), we can ap-
proximate the T matrix to lowest order in ? using the static
polarizability tensor, Fee
Oð?5Þ, where the subscript 0 indicates the static (ic? ¼
0) limit and P is the polarization label. The T-matrix
elements involving higher angular momenta l > 1 are
higher order in ?. For a small ellipsoid, in particular, the
electric polarizability tensor ?Eis diagonal in a coordinate
system aligned with the ellipsoid’s body axes, ?E
?M;0þð?I;0??M;0Þni, where i 2 fx;y;zg . The larger the
semiaxis in direction i, the smaller the depolarization
factor ni(not to be confused with the index of refraction).
The magnetic polarizability tensor ?Mis obtained by ex-
changing ?x;0for ?x;0in the expression for ?E. In the small
size limit the polarizability tensor of a perfect metal ellip-
soid is obtained by taking both ?I;0! 1 and ?I;0! 0.
For simplicity we specialize to a spheroid, which has
two equal semiaxes. We choose the semiaxes along ^ x and ^ y
to be equal; therefore, ?P
of the lab frame to be along the direction of displacement
of the spheroid from the center of the cavity. ? denotes the
angle between the spheroid’s and the lab’s ^ z axes. For such
a small spheroid inside a spherical cavity of radius R
[Fig. 1(a)], the spring constant is obtained by expanding
the log determinant in the expression for E to first order,
??. We fix the ^ z axis
??Þ3cos2? ? 1
2þ E ! M
where the material dependent functions
express the rotationinvariant and the orientation dependent
parts of the energy, respectively. fM
2by exchanging E and M everywhere. This
PRL 104, 070405 (2010)
19 FEBRUARY 2010
result is valid for asymptotically large R; it involves only
the zero frequency (ic? ¼ 0) response functions since R is
the size of the inner object. Notice that only the l ¼ 1, 2
scattering amplitudes of the cavity walls appear in Eqs. (2)
The behavior of the functions fP
?M;0¼ ?O;0, is as expected: fE
when ?M;0> ?O;0, negative when ?M;0< ?O;0, and fM
ways has the opposite sign of fE
small object with ?I;0> ?M;0is levitated stably, when fE
negative, ?M;0> ?I;0has to hold. Thus, stability occurs
under the same conditions as repulsion for two half spaces.
The opposite sign of fM
1is expected from equivalent ex-
dielectric contrast between the medium and the outer
sphere is taken to small or large limits, stability or insta-
bility is maximized.
To verify whether stability is observable for realistic
materials and object sizes, we evaluate the energy E nu-
merically for a sphere of radius r inside a spherical cavity
of radius R filled with various liquids [Fig. 1(b)] . The
coefficients k, k4, and k6in the series expansion in Eq. (1)
are listed in Table I. For comparison, the asymptotic result
kR!1is also included. If both inside and outside object are
spherically symmetric, the series in Eq. (1) does not con-
tain terms ?an
The three materials were chosen so that the sequence of
permittivities ?I, ?M, ?Oeither increases or decreases for
the imaginary frequencies that contribute most to the en-
ergy. 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
dielectric sphere is held more stably in the center of a
cavity with low dielectric walls than a low dielectric sphere
inside a cavity with high dielectric walls.
The asymptotic result kR!1yields a good approxima-
tion of k forr
4. But from Eq. (2) one would expect
kR!1to grow linearly with the volume of the inner sphere,
since polarizability is proportional to the volume. In fact, k
1, depicted in Fig. 2 for
1is monotonic, positive
1. When fE
1is positive, a
Rn with n odd.
of just 27 times. This means that for a gold sphere in a
liquid drop withr
ture, indeed, the Casimir spring holds the particle near the
Although k, k4, and k6increase by 3 orders of magnitude
in some cases, the prefactor1
n!ensures that the coefficients
in the Taylor expansion in Eq. (1) increase only by 1 order
of magnitude. Thus, for small excursions from the center,
e.g., a=R < 0:1, the higher corrections n ? 4 can be
Compared to the stability conditions studied thus far, the
orientation dependence of the energy is more varied. fE
2, plotted inFig. 3,have the same sign formost ratios
of medium to outside permittivities, unlike fE
which always have opposite signs. In these 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 ?I;0< ?M;0.
?M;0? 80 and at
4is about 1000 times larger than forr
4and R ¼ 1 ?m at room tempera-
=R < 0:1.
1, also, fE
?M;0? 2000, respectively. So, while the
2change sign again at
FIG. 2 (color online).
constant kR!1, 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, ?M;0¼ ?O;0.
1describe the part of the spring
tions of materials for the case of a spherical inner object inside a
spherical cavity, depicted in Fig. 1(b). The dimensionless num-
bers in the table have to be multiplied by@c
k, kR!1, k4, and k6are listed for various combina-
R. R is given in
microns [?m]. kR!1depends on R only through the ratios@c
R, so its numerical prefactor is the same for all R. The highest
cutoff used was lmax¼ 30. (The asymptotic result kR!1only
requires l ¼ 1, 2.)
Gold-bromobenzene-vacuum 0.1 1=4 0:040 0.054 1.2
8:2 ? 101
1:8 ? 106
2:0 ? 102
1.0 1=4 0:069 0.054 2.6
0.1 1=4 0:050 0.037 1.6
1.0 1=4 0:045 0.037 1.7
0.1 1=4 0:015 0.012 0.46
1.0 1=4 0:019 0.012 0.82
1.418000 1:0 ? 107
1:0 ? 102
2:2 ? 106
1:4 ? 102
18000 1:2 ? 107
3:1 ? 101
8:4 ? 105
7:2 ? 101
12000 7:6 ? 106
FIG. 3 (color online).
constant kR!1, which changes with the orientation of the inside
spheroid. In this plot, ?M;0¼ ?O;0.
2describe the part of the spring
PRL 104, 070405 (2010)
19 FEBRUARY 2010
direction of the total force and the stability of the equilib-
rium can be determined based on the relative magnitudes
of the permittivities, the torque cannot be. The PSA and
PFA predict the orientation with the lowest energy in the
Furthermore, the second sign change of fE
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
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 spherical
cavity. If the cavity is slightly deformed, however, the
energy, E0, depends on the relative orientation of the
spheroid and the cavity [Fig. 1(c)]. We deform the spheri-
cal cavity as described earlier and obtain to first order in
?M;0¼ 1 correctly but not when
??ÞgEþ E ! M?;
where the orientation-independent part of the energy has
been dropped. For ?O;0¼ ?M;0, gEand gMare given by
where the arguments of the modified spherical Bessel
functions and of zOð0Þ are suppressed. iland klare func-
tions of ?, and~klstands for klðzOð0Þ?Þ, where zOðic?Þ ¼
nOðic?Þ=nMðic?Þ is the ratio of the permittivities of the
cavity walls and the medium. The material dependent
functions gEand gMare plotted in Fig. 4.
Again, gEhas the same sign for?O;0
also vanishes at?O;0
The rich orientation dependence of the energy is ex-
pected to collapse as the size of the inside object grows to
ð~k0i1þ~k1i0zOþ i1~k1=ðzO?Þð1 ? z2
?M;0! 0 (left in Fig. 4)
?M;0¼ 1, gE
?M;0! 1 (right). In addition to the root at?O;0
fill the cavity and the PFA becomes applicable. Based on
the stability analysis for finite inside spheres, though, we
expect the asymptotic results, fP
orientation dependence for reasonably small inside sphe-
roids. For comparison with real experiments, of course,
various corrections to the idealized shapes considered here
have to be taken into account.
This work was supported by the NSF through Grant
No. DMR-08-03315 (S.J.R.). We thank T. Emig, N.
Graham, R. Jaffe, and M. Kardar for fruitful discussions.
2and gP, to predict the
 H.B.G. Casimir and D. Polder, Phys. Rev. 73, 360 (1948);
H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793
 I.E. Dzyaloshinskii, E.M. Lifshitz, and L.P. Pitaevskii,
Adv. Phys. 10, 165 (1961).
 J.N. Munday, F. Capasso, and V.A. Parsegian, Nature
(London) 457, 170 (2009).
 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).
 S. Zaheer, S.J. Rahi, T. Emig, and R.L. Jaffe,
 D. Peer, J.M. Karp, S. Hong, O.C. Farokhzad, R.
Margalit, and R. Langer, Nature Nanotech. 2, 751 (2007).
 P. Mal, B. Breiner, K. Rissanen, and J.R. Nitschke,
Science 324, 1697 (2009).
 S.J. Rahi, T. Emig, N. Graham, R.L. Jaffe, and M. Kardar,
Phys. Rev. D 80, 085021 (2009).
 T. Emig, N. Graham, R.L. Jaffe, and M. Kardar, Phys.
Rev. Lett. 99, 170403 (2007); O. Kenneth and I. Klich,
Phys. Rev. B 78, 014103 (2008); P.A. Maia Neto, A.
Lambrecht, and S. Reynaud, Phys. Rev. A 78, 012115
 C. Yeh, Phys. Rev. 135, A1193 (1964).
 V.A. Erma, Phys. Rev. 179, 1238 (1969).
 R.F. Millar, Radio Sci. 8, 785 (1973).
 L.D. Landau and E.M. Lifshitz, Electrodynamics of
Continuous Media (Pergamon Press, Oxford, 1984).
 O. Kenneth, I. Klich, A. Mann, and M. Revzen, Phys. Rev.
Lett. 89, 033001 (2002).
 For the permittivity function of gold we use ?ðic?Þ ¼
9:27 ? 1013Hz  as used in Ref. . For the other
1014Hz and ½Cn?n¼1;2¼ ½23:84;0:852? for ethanol 
as used in Ref. , with ½!n?n¼1;2¼ ½5:47;128:6? ?
1014Hz and ½Cn?n¼1;2¼ ½2:967;1:335? for bromobenzene
 as used in Ref. , and with ½!n?n¼1?4¼
½0:867;1:508;2:026;203:4? ? 1014Hz and ½Cn?n¼1?4¼
½0:829;0:095;0:798;1:098? for silica (SiO2) .
 I. Pirozhenko, A. Lambrecht, and V.B. Svetovoy, New J.
Phys. 8, 238 (2006).
 A. Milling, P. Mulvaney, and I. Larson, J. Colloid Interface
Sci. 180, 460 (1996).
 L. Bergstro ¨m, Adv. Colloid Interface Sci. 70, 125 (1997).
c?ðc?þ?Þ, where !p¼ 1:14 ? 1016Hz and ? ¼
0.20.4 0.60.8 1.
FIG. 4 (color online).
the energy E0on the relative orientation of the inside spheroid
and the deformed cavity walls.
gEand gMdescribe the dependence of
PRL 104, 070405 (2010)
19 FEBRUARY 2010