Casimir-Lifshitz interaction between ZnO and SiO2 nanorods in bromobenzene: retardation effects turn the interaction repulsive at intermediate separations

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by  Mathias Boström

Casimir-Lifshitz interaction between ZnO and SiO nanorods in bromobenzene:
2
retardation effects turn the interaction repulsive at intermediate separations
Mathias Boström and Bo E. Sernelius∗
Division of Theory and Modeling, Department of Physics,
Chemistry and Biology, Linköping University, SE-581 83 Linköping, Sweden
Gustavo Baldissera and Clas Persson†
Dept of Materials Science and Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Barry W. Ninham
Department of Applied Mathematics, Australian National University, Canberra, Australia
2
1 We consider the interaction between a ZnO nanorod and a SiO2 nanorod in bromobenzene. Us-
0 ing optical data for the interacting objects and ambient we calculate the force — from short-range
2 attractive van der Waals force to intermediate range repulsive Casimir-Lifshitz force to long range
n entropically driven attraction. The nonretarded van der Waals interaction is attractive at all sep-
a arations. We demonstrate a retardation driven repulsion at intermediate separations. At short
J separations (in the nonretarded limit) and at large separations (in the classical limit) the interac-
1 tion is attractive. These effects can be understood from an analysis of multiple crossings of the
1 dielectric functions of thethreemedia as functions of imaginary frequencies.

] PACSnumbers: 42.50.Lc,34.20.Cf;03.70.+k
h
p
- Casimirpredictedalreadyin1948[1]theattractionbe- in correlationbetween the charge density fluctuations at
t
n tween a pair of parallel, closely spaced, perfect conduc- large distances; new propagating solutions to Maxwell’s
a tors a distance apart. Almost 50 years later Lamore- equations appear that bring correlations between cur-
u
aux[2–4]performedthe firsthighaccuracymeasurement rentdensityfluctuations. Whenthesepropagatingmodes
q
of Casimir forces between metal surfaces in vacuum. An dominate we call the force Casimir force; when the sur-
[
interesting aspect of the Lifshitz-Casimir force is that face modes[14]dominate we callthe forcevander Waals

1 according to theory it can also be repulsive[5–7]. The force. Theclassicalliteraturesaysthatretardationisdue
v
aim of this brief report is to demonstrate nonmonotonic to the finite speed of lightthat weakensthe correlations.
6
distance dependence in the Lifshitz interaction between This is misleading. It is due to the quantum nature of
9
nanorods in solution. The force may go from attrac- light[18,19];thisbecomesobviousatthelargeseparation
2
2 tive to repulsive to attractive as the separation is in- limitwhere the weakeningis independentofthe speedof
. creased. Early theoretical and experimental efforts in- light.
1
directly demonstrating repulsive Casimir-Lifshitz forces
0 Much interest has been focused on nanowires and
2 were reviewed in a recent communication [8]. Mun-
nanorods [20, 21] and their potential applications in
1 day,Capasso,andParsegian[9]performedthefirstdirect
nanotechnology. For instance ZnO [22, 23] and SiO
: forcemeasurementsthatdemonstratedthattheLifshitz- 2
v [24]nanorodsareinterestingaspotentialbuilding blocks
Casimir force could be repulsive by a suitable choice of
i in future nanomechanics and in life science applications
X interacting surfaces in a fluid. They speculated that this
[25]. Noruzifar et al. [26], and much earlier Davies et
r effect could allow quantum levitation of nano-scale de-
a al. [27],investigatedLifshitzforcesbetweenmetalwires.
vices[9].
Theinteractionbetweenconductingparticlescanbesur-
Whentwo objectsarebroughttogethercorrelationsin
prisingly long range [27]. The interaction between rod-
fluctuations of the charge and current densities in the
like particles was investigated by different groups [28–
objects or fluctuations of the fields usually result in an
34]. Ninhamandco-workersinparticularinvestigatedin
attractive force[1, 5, 10–17]. At short distances this is
greatdetailtheLifshitz-Casimirinteractionbetweenthin
the van der Waals force[11]; at large distances the finite
dielectric rods [12, 35]. We will exploit their result [35]
velocity of light becomes important (retardation effects)
obtainedforthinrodswithdiametersmuchsmallerthan
and the result is the Casimir force[1, 15]. With retar-
the rodseparation. We presentcalculations for a system
dation effects we here mean all effects that appear be-
with a silica (SiO ) nanorod (2 nm diameter) [36] inter-
cause of the finite speed of light; not just the reduction 2
actingwithazinc oxide(ZnO)nanorod(2nmdiameter)
acrossBromobenzene (Bb)[9]) anddemonstrate that at-
tractiveandrepulsiveLifshitz forcesareproducedinone
and the same system.
∗ [email protected]
† DepartmentofPhysics,UniversityofOslo,P.Box1048Blindern, The Casimir force can be calculated if the dielec-
NO-0316Oslo,Norway tric functions (for imaginary frequencies, iω) are known.
2
101 10-8
Retarded
10-10
Nonretarded
ZnO
SiO2 10-12 n = 0
Bromobenzene
10-14
m)
g/c 10-16
"(i) U )(er 10-18
! s( 10-20
b
a
10-22
10-24
10-26
I II III
10-28
1 10 100 1000 10000
100 R (nm)
1013 1014 1015 1016 1017
" (s-1) Figure 2. (Color online) The Casimir-Lifshitz interaction en-
ergy per unit length between ZnO and SiO2 nanorods (each
with2nmdiameter)inbromobenzene. Thefullyretardedfree
Figure 1. (Color online) Thedielectric function at imaginary
energyisattractiveinregionsIandIIIbutturnsrepulsivein
frequencies for SiO2 (silica)[36], Bb (bromobenzene)[9], and
region II. The nonretarded energy and the n=0 contribution
ZnO (zinc oxide).
are both attractive and they decay ∝R−5.
and
These functions are shown in Fig.1. The dielectric func- G(ωn,k)=−a24b2(ε3−ε1ε)(23ε3−ε2)γ34K02(γ3R)
tion of ZnO on the imaginary frequency axis was ob- −a2b2 ε3 − ε2 (ε3−ε1)
tained using a modified[14] Kramers-Kronig dispersion 2 n(cid:16)γ32 γ22(cid:17)ε3(ε3+ε2)
rfuelnacttiioonnaopnptlhieedretaoltahxeisfwunacstcioanlcuolnattehdeurseinagl aaxpisa.rtiTalhlye −a2b2 +ε3(cid:16)−γε332ε1− γε112ε(cid:17)3ε−3(ε(3εε3−2+ε2ε1))oγ36γK3412(γ3R)
sfuelnfc-ctoionnsimsteenthtoGdWwaaspbparsoeadcho[n37t,he38l]o,cwalhdereenstihtye aGprpereonx’s- 2+n(cid:16)γε3223γ12γγ2212(cid:17)(cid:16)γ321 −γ221(cid:17)(ε3+1ε2−)(ε31+ε1) (3)
imation within the density functional theory. There is a +(ε3ε+3ε−2)(εε31+ε1)1(cid:16)γ−32 1 γ12(cid:17)(cid:16)εγ2332γ12γ22γ22(cid:17)
acqrnuodesnsBcinybgaabntedatwahneigeonhtheterhrfercecrquousrsveinenscgyf.obreTtZwhniesOeonaptnehndesBcuubpravftoesraftolhorewSpifoOres2-- +(cid:16)(cid:16)γγε32332 −γγε12222(cid:17)×(cid:17)(cid:16)(cid:16)γγγ132324−Kγγ212212((cid:17)(cid:17)γ((εRε33+)+εε+ε2322γ))(K12(εεγ33222++(εεγ11))Ro) .
sibility of a transition of the interaction, from attraction 3 2 3 0 3
(cid:8) (cid:2) (cid:3)(cid:9)
to repulsion and back to attraction.
The momentum dependence is found in the functions
The interaction of material 1 (SiO2) with material 2 γα = k2+εα(ωn/c)2. The parameters a and b are
q
(ZnO) across medium 3 (Bb) results in a summation of the radii of the two rods. Note that positive values
imaginary frequency terms[12, 35] of F correspond to repulsive interaction. We recently
investigated the interaction between a planar gold sur-
faceinteractingwithaplanarsilicasurfaceinbromoben-
zene [8]. There the nonretarded van der Waals inter-
∞
2πn action was attractive while both the long-range Casimir
F = ′g(ω ); ω = , (1)
n n ~β asymptote and the long-range entropic asymptote were
nX=0 found to be repulsive. In contrast in the present system
shown in Fig.2 it is only for intermediate separations
that retardation turns the interaction repulsive. The re-
versalof the sign of the Casimir energy with retardation
where
for intermediate separations as compared to without re-
tardation is due to a subtle balance of attractive (high
andlowfrequencies)andrepulsive(intermediatefrequen-
∞ cies) contributions. When the separation increases from
1 dk
g(ω )= G(ω ,k), (2) its lowest value the attractive contributions at high fre-
n n
β Z 2π
quencies are weakened by retardation at a higher rate
0
3
100 retarded van der Waals force while only low frequencies
contribute in the long range retarded Casimir regime.
The ultimate long-range asymptote, the entropic term,
only includes the zero frequency mode. The zero fre-
10-1
d quency contribution is attractive and follows the same
e
d
ar power law as the nonretarded asymptote. It is seen to
et be possible to via the optical properties tune the mate-
nr
o 10-2 rials used such that they give repulsion at specific sep-
N
d/ arations. In this case only at intermediate separations.
e
ard There is also a, to our knowledge not yet experimentally
et observed, maximum of both the repulsive and attractive
R 10-3
parts of the Lifshitz-Casimir interaction energy.
We show in Fig.3 the absolute value of the ratio be-
I II III tween the retarded and nonretarded interaction free en-
10-4 ergy. The aimofthis figureisto illustratethe deviations
1 10 100 1000 10000 from the simple attractive nonretarded R−5-power law.
R (nm) Itisclearthatretardationchangesquitedramaticallythe
interaction between silica and zinc oxide nanorods.
Usingopticalmeasurementsonthe interactingobjects
Figure 3. (Color online) The absolute value of the ratio be-
tween the retarded and nonretarded interaction free energy and liquid makes it possible to predict the force — from
per unit length (the nonretarded energy is attractive and short-range attractive van der Waals force to intermedi-
∝ R−5). The fully retarded free energy is attractive in re- ate range repulsive Casimir force to long range entrop-
gions I and III but turnsrepulsive in region II. ically driven attraction. It should be noted that at 1
µm separation the thermal entropic effect starts to be-
comeimportant. Thisissimilartowhatwefoundforthe
Casimir force between realmetalsurfaces in vacuum[39]
thantherepulsivecontributionsatintermediatefrequen- and between gold and silica in bromobenzene [8].
cies. At large separations the low frequencies dominate
giving again attraction. The distance-dependent Bessel
ACKNOWLEDGMENTS
functions, K (Rγ ) = K R k2+ε (iω )(ω /c)2 ,
i 3 i Bb n n
(cid:20) q (cid:21)
inthe Lifshitz expressionfortheCasimirenergy,Eq.(3), The research was sponsored by the VR-contract
has the effect that all frequencies contribute to the non- No:70529001which is gratefully acknowledged.
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