Casimir force driven ratchets

0.2 MB·English

by  Thorsten Emig

Casimir force driven ratchets
T. Emig
Laboratoire de Physique Th´eorique et Mod`eles Statistiques,
CNRS UMR 8626, Universit´e Paris-Sud, 91405 Orsay, France
(Dated: February 6, 2008)
7
0 We explore the non-linear dynamics of two parallel periodically patterned metal surfaces that
0
arecoupled bythezero-pointfluctuationsoftheelectromagnetic fieldbetween them. Theresulting
2
Casimir force generates for asymmetric patterns with a time-periodically driven surface-to-surface

n distance a ratchet effect, allowing for directed lateral motion of the surfaces in sizeable parameter
a ranges. It is crucial to take into account inertia effects and hence chaotic dynamics which are
J described by Langevin dynamics. Multiple velocity reversals occur as a function of driving, mean

6 surface distance, and effective damping. These transport properties are shown to bestable against
2 weak ambient noise.

] PACSnumbers: 12.20.-m,05.45.-a,07.10.Cm
h
c
e TheobservationofCasimirforcesinincreasinglysmall tors [12] and vortex physics in superconductors [13], to
m
devices on sub-micron scales has generated great cur- nameafewrecentexamples. Mostoftheworksonratch-
- rent interest in exploring the role of these forces for etsconsideranexternaltime-dependentdrivingforceact-
t
a the development and optimization of micro- and nano- ing onoverdampeddegreesoffreedomto rectify thermal
t
s electromechanical systems [1, 2, 3]. These systems can noise. For nano-systems, however, it has been pointed
.
t serve as on-chip fully integrated sensors and actuators out that inertia terms due to finite mass should not be
a
with a growing number of applications. It was pointed neglectedand,actually, canhelp the ratchets to perform
m
out that Casimir forces can make an important contri- more efficiently than their overdamped companions [14].
-
d bution to the principal cause of malfunctions of these Finite inertia typically induce in Langevin dynamics de-
n devices in form of stiction that results in permanent ad- terministicchaosthathasbeenshowntobeabletomimic
o hesion of nearby surface elements [4]. This initiated in- the role of noise and hence to generate directed trans-
c
terest in repulsive Casimir forces by modifying material port in the absence of external noise [15]. Here we use
[
propertiesaswellasthegeometryoftheinteractingcom- this effect in the different context of so-called pulsating

1 ponents [5, 6]. (oreffectively on-off)ratchetswherethe strengthsofthe
v
periodic potential varies in time [10]. We consider weak
1 A complementary strategy, which will be considered
thermal noise only to test for stability of the inertia in-
4 in this Letter, is to make actually good use of Casimir
6 ducedtransport—notasthesourceofdriving. Wenote
forces between metallic surfaces and employ them to ac-
1 thatinthe absenceofinertia,finite thermalnoiseis nec-
tuate components of small devices without contact. We
0 essary for on-off ratchets to generate directed motion.
7 will demonstrate that this can be achieved by coupling
0 two periodically structured (locally) parallel surfaces by
In this Letter we demonstrate that the system de-
/ the zero-point fluctuations of the electromagnetic field
t scribed above indeed allows for directed relative motion
a between them. The broken translation symmetry paral-
m of the surfaces due to chaotic dynamics caused by the
lel to the surfaces results in a sideways force which has
lateral Casimir force. The transport velocity is stable
- been predicted theoretically [7, 8] and observed experi-
d across sizeable intervals of the amplitude and frequency
n mentally between static surfaces [9]. If at least one of of surface distance oscillations and damping. The veloc-
o the surfaces is structured asymmetrically there is an ad-
ityscaleslinearwithfrequencyacrosstheseintervalsand
c ditionalbreakingofreflectionsymmetryandthesurfaces
: isalmostconstantbelowacriticalmeandistancebeyond
v can in principle be set into relative lateral motion in the
which it drops sharply. The system exhibits multiple
i directionofbrokensymmetry. The energyforthistrans-
X currentreversalsasfunctionoftheoscillationamplitude,
port has to be pumped into the system by externaldriv-
r meandistanceanddamping. The Casimirratchetallows
a ing. This can be realized by setting the surfaces into
contact-less transmission of motion which is important
relative oscillatory motion so that their normal distance
since traditional lubrication is not applicable in nano-
is an unbiased periodic function of time. Since the side-
devices. This actuation mechanism should be compared
waysCasimirforcedecaysexponentiallywiththenormal
to other actuation schemes as magnetomotive or capaci-
distance [8], the surfaces experience an asymmetric peri-
tive(electrostatic)forcetransmission. TheCasimireffect
odic potential that varies strongly in time.
induced actuation has the advantage of working also for
This scenario resembles so-called ratchet systems [10] insulators and does not require any electrical contacts
thathavebeenstudiedextensivelyduringthelastdecade and/or external fields. It can also scale down effects of
in the context of Brownian particles [11], molecular mo- parasiticcapacitancethatreducestheefficiencyofactua-
2
tion at high frequencies [4]. It should be mentioned that (a) 15 (b)
h(x)
other applications of zero-pointfluctuation induced (van 2 1
H
derWaals)interactionstonano-deviceshavebeenexper- 10
imentally realized already to construct ultra-low friction h(x)
1 1 5
0
bearings from multiwall carbon nanotubes [16].
In the following, we consider two (on average) paral- 0
0 0
lelmetallicsurfaceswithperiodic,uni-axialcorrugations
(alongthe x -axis)that havedistance H, see inset(a)of
1
Fig. 1. To begin with, we assume that both surfaces are
at rest with a relative lateral displacement b. Then the
surface profiles can be parametrized as
∞
h (x ) = a c e2πinx1/λ1 +c.c., (1a)
1 1 n
nX=1
∞ 0
h (x ) = a d e2πin(x1−b)/λ2 +c.c. , (1b) 1
2 1 n
nX=1
FIG. 1: The lateral Casimir force acting between the two
where a is the corrugationamplitude, λ1,λ2 arethe cor- surfaces as function of the shift ˆb at time s = 0 and half
rugation wave lengths, c , d are Fourier coefficients. period s = π/ω (drawn to a larger scale by a factor 103)
n n
The Casimir energy E of this configuration is the for parameters η = 0.65, H0 = 0.1λ. Insets: (a) Surface
change ofthe groundstateenergyofthe electromagnetic profilesattheirequilibriumpositionatˆb=0.182(b)Periodic
fieldduetothesuppressionofthetangentialelectricfield variation of themaximum force atˆb=0 with time.
at the surfaces. The dependence of E on H and b causes TheCasimirpotentialofEq.(2)hastwointerestingprop-
macroscopic forces on the surfaces. For a varying sepa- erties which are useful to the construction of a ratchet.
ration H this is the normal Casimir attraction between First, it decays exponentially with H, and thus can be
metallic surfaces [17], modified by the corrugations. Be- essentially switched on and off periodically in time by
lowwewillassumeH =H(t)tobeatime-dependentdis- oscillating H. Second, the potential is not only periodic
tance that is kept at a fixed oscillation by an additional in b but acquires asymmetry from the surface profiles
external force from clamping to an oscillator. In such at small H ≪ λ and an universal symmetric shape for
setup, the surfaces can react freely only to the lateral H ≫λsincetheeffectofhigherharmonicsofthesurface
force component Flat(b,H) = −∂E/∂b. This side-ways profile is exponentially diminished [18].
force has been computed for sinusoidal corrugations to The relative surface displacement b(t) can be consid-
second order in the amplitude a, using a path integral ered as a classical degree of freedom with inertia. Its
formulation [8]. This result is readily extended to peri- equationofmotionisdescribedbyLangevindynamicsof
odic profiles of arbitrary shapes as described by Eq. (1). the form
We find thatthe corrugationlength haveto be commen-
surate,λ /λ =p/qwithintegersp,qinordertoproduce ρ¨b+γρb˙ =F [b,H(t)]+ 2γρTξ(t), (4)
1 2 lat
a finite lateral force per surface area. For the purpose of p
this work, it is sufficient to consider the case p=1. The where ρ is the mass per surface area, γ the friction co-
lateral(b-dependent)partofthe Casimirenergypersur- efficient, T the intensity (divided by surface area) of
face area can then be written as the Gaussian noise ξ(t) with zero mean and correla-
′ ′
tions hξ(t)ξ(t)i = δ(t − t) so that the Einstein rela-
2~ca2 ∞ H
E(b)= H5 nX=1(cid:16)cnd−nqe−2πinb/λ1 +c.c.(cid:17)J(cid:18)nλ1(cid:19) tniooinseisduoebetyoede.ffeTcthsisofsttoecmhapsetriactuterremanddespcrreibsseusraem. b(Aiendt-
(2) ditionalcontributions fromthermally excited photons to
to order a2. The exact form of the function J was ob- the Casimir force can be neglected at surface distances
tainedinRef.8intermsoftranscendentalfunctions. For well below the thermal wavelength ~c/(2T).) The sys-
the present purpose it is sufficient to use the simplified tem is driven by rigid oscillations of one surface so that
expression the distance H(t) = H g(t) oscillates about the mean
0
distance H with g(t) = 1−ηcos(Ωt). For simplicity,
π2 0
J(u)≃ 1+2πu+γu2+32u4 e−2πu (3) we consider now equal corrugation lengths λ1 =λ2 ≡λ.
120(cid:0) (cid:1) Wedefinethefollowingdimensionlessvariables: ˆb=b/λ,
withγ =12.4133,whichisexactforbothasymptotically s=t/τ forlaterallengthsandtimewiththetypicaltime
large and small u and approximates the exact results scale τ = (λ/a) ρH5/~c resulting from a balance be-
0
with sufficient accuracy for all u. (The maximal devi- tween inertia anpd Casimir force. Hence velocities will
ation from the exact result is ≈±0.5% around u=0.5.) be measured in units of v = λ/τ. There are five di-
0
3
mensionless parameters which can be varied indepen-
dently for fixed surface profiles: the damping γˆ = τγ,
the angular frequency ω = τΩ, the driving amplitude
η, the scaled mean distance H /λ and the noise inten-
0
sity Tˆ = (T/~c)(H5/a2). The dimensionless equation of
0
motion forˆb(s) reads
¨ˆb+γˆˆb˙ =Fˆ [ˆb,gˆ(s)]+ 2γˆTˆξˆ(s) (5)
lat
q
with the Casimir force
∞
4π H
Fˆ (ˆb,gˆ)= f cos(2πnˆb)J ngˆ 0 , (6)
lat gˆ5 n (cid:18) λ (cid:19)
nX=1
where we have chosen surface profiles with c =
n
i f /(2n), d = f /(2n) with real coefficients f in
n n n n
Epq. (1), and gˆ(s)=p1−ηcos(ωs).
We begin to analyze Eq. (5) by noting that directed
FIG. 2: Mean hhvii and standard deviation σ of the (nega-
transportis possible in certainparameter ranges even in
tive) velocity as function of the driving amplitude η for the
the deterministic case where noise is absent. However,
frequencies ω = 5.0 and ω = 4.72 (for the latter only the
to probe the robustness of transport, we consider in the
stable plateau is shown). The parameters are H0 = 0.1λ,
following primarily the limit of weak noise by choosing γˆ = 0.9, Tˆ = 10−3. Inset: Dependence of the same quan-
Tˆ = 10−3. In fact, it has been shown for underdamped tities on frequency for fixed η = 0.65. Straight dashed lines
ratchets with time-independent potentials and periodic correspond in both graphs to thevelocity ω/(2π).
driving that even an infinitesimal amount of noise can
change the rectification from chaotic to stable [14]. To about the average velocity must be small, i.e., the vari-
lookforsimilargenericbehaviorofourpulsatingratchet, ance σ2 =hhv2ii−hhvii2 must be smaller than hhvii2.
weconsideraspecificgeometryconsistingofasymmetric
Naively, one can expect directed motion of the sur-
andasawtooth-likesurfaceprofilecorrespondingtothree face profile h (x ) into the positive x -direction (ˆb˙ < 0)
harmonics with f = 0.0492, f = 0.0241, f = 0.0059 2 1 1
1 2 3
since the Casimir forcein Fig. 1 is asymmetric with neg-
and f = 0 for n > 3. Inset (a) of Fig. 1 shows these
n
ative values lasting for longer time than positive ones.
profilesin their stable positionwithˆb=0.182thatmini-
However,the actualbehaviorismorecomplicateddueto
mizestheCasimirenergy. Theresultingspatialvariation
chaoticdynamics. Fig.2showsthedependenceoftheav-
oftheCasimirforcewithˆbisplottedinFig.1forminimal
eragevelocityanditsstandarddeviationσonthedriving
(s=0)andmaximal(s=π/ω)surfacedistancewithpa-
amplitudeη andfrequencyω forH =0.1λ,γˆ =0.9. For
rameters H /λ = 0.1, η = 0.65. It can be clearly seen 0
0 a fixed frequency there is an optimal interval of driving
that the asymmetry is reduced at larger distance where
amplitudes across which the average velocity is almost
the variation of the force becomes more sinusoidal. In-
constant with hhvii ≃ −ω/(2π). Small deviations from
set(b)showstheon-off-liketime-dependenceoftheforce
the latter value result from noise as we have checked by
amplitudeatˆb=0duetotheoscillatingsurfacedistance. studying the dynamics at Tˆ = 0. At higher driving am-
The non-linear equation of motion of Eq. (5) has to
plitudesweobserveasecondnarrowerintervalwithmax-
be solved numerically. The trajectoryˆb(s) was obtained
imal hhvii which is more strongly reduced and smeared
from a second order Runge-Kutta algorithm. As initial
out from its deterministic value −2×ω/(2π) by noise.
conditions we used an equidistant distribution over the
At the plateaus of constant velocity the standard devia-
interval [−1,1] for ˆb(0) and ˆb˙(0) = 0. For each set of tion σ is substantially reduced, rendering transport effi-
parameters we calculated 200 different trajectories from cient. Outside the plateaus velocity reversals occur and
varying initial conditions and noise, each evolving over σ increases linearly with η. For fixed amplitude η, the
4 ×103 periods 2π/ω so that transients have decayed. average velocity is stable at the value −ω/(2π) over a
The averagevelocity hhvii involves two different averages sizeable frequency range (see inset of Fig. 2).
of ˆb˙(s): The first average is over initial conditions and In order to understand the observedbehavior we have
noise for everytime step, thenthe averagedtrajectoryis analyzedthedynamicsinthethreedimensionalextended
averagedoveralldiscretetimesofthenumericalsolution. phase space. There attractorsof the long-time dynamics
For an efficient directed transport it is not sufficient to canbe identified fromPoincar´esectionsusingthe period
have only a finite average hhvii. To exclude trajectories 2π/ω of the surface oscillation as stroboscopic time. To
withahighnumberofvelocityreversals,thefluctuations obtainacompactsection,thetrajectoryisfoldedperiod-
4
(a) (b) area mass density of ρ = 10g/m2 for silicon plates
with a thickness of a few microns, one obtains v =
0
~ca2/ρH5 ≈ 5.5mm/s. The actual average velocity
0
vpω/2π is of the same order for the frequencies studied
0
above. For λ=1µm, the time scale is τ =λ/v ≈10−4s
0
leading to driving frequencies and damping rates in the
kHz range for the parameters considered here.
Our results show that Casimir interactions offer novel
contact-lesstranslationalactuationschemes for nanome-
chanical systems. Similar ratchet-like effects are ex-
pected between objects of different shapes as, e.g., peri-
odicallystructuredcylinders,inducingrotationalmotion.
FIG.3: Meanhhviiandstandarddeviationσofthe(negative)
velocityasfunctionof(a)themeanplatedistanceH0 forγˆ= The use of fluctuation forces appear also promising to
0.9 and (b) damping γˆ for H0 =0.1λ. The other parameters movenano-sizedobjectsimmersedinaliquidwhereelec-
are η=0.65, ω=5.0, Tˆ=10−3. trostatic actuation is not possible. Another application
is the separation and detection of particles of differing
icallyinx ononeperiodofthe Casimirpotential. From
1 massadsorbedtothesurfaces. Forsurfacesoscillatingat
these sections we can distinguish between periodic and
very high frequencies additional interesting phenomena
chaotic orbits. As a start, we consider the deterministic
related to the dynamical Casimir effect occur [19], lead-
limit with Tˆ = 0. The plateaus around η = 0.65 and
ing to the emission of photons that could contribute to
η = 0.7 result both from periodic orbits of period one,
ratchet-like effects as well.
corresponding to a single point in the Poincar´e section.
I acknowledge useful discussions with R. Golestanian
On the right (downward) edges of the first plateaus we
and M. Kardar.
have observed period doubling, i.e., a periodic attractor
with period two. Upon a further increase of η, chaotic
orbits dominate the motion. Hence the system exhibits
a period-doubling route to chaos with enhanced velocity [1] E. Buks and M. L. Roukes, Europhys. Lett. 54, 220
fluctuations. The findings apply basically also to weak (2001).
noise(Tˆ =10−3)butthe sharppointsofthe periodicat- [2] H.B.Chan,V.A.Aksyuk,R.N.Kleiman,D.J.Bishop,
tractorsinthe Poincar´esectionsaresmearedoutleading and F. Capasso, Science291, 1941 (2001).
[3] H.B.Chan,V.A.Aksyuk,R.N.Kleiman,D.J.Bishop,
to a decreased hhvii. The transition from chaotic to peri-
and F. Capasso, Phys.Rev.Lett. 87, 211801 (2001).
odic dynamics at the beginning of the rising edge of the [4] E. Buks and M. L. Roukes, Phys. Rev. B 63, 033402
plateaus is accompanied by a velocity reversal. This is
(2001).
consistent with the earlier observation for non-pulsating [5] E. Buksand M. L. Roukes,Nature 419, 119 (2002).
potentialsthatvelocityreversalsaredue toabifurcation [6] O. Kenneth, I. Klich, A. Mann, and M. Revzen, Phys.
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The amplitude of the Casimir potential can be tuned [7] T. Emig, A. Hanke, R. Golestanian, and M. Kardar,
Phys. Rev.Lett. 87, 260402 (2001).
by varying the mean distance H . FromFig. 3(a) we see
0 [8] T. Emig, A. Hanke, R. Golestanian, and M. Kardar,
that the dynamics show a sharp transition at a critical Phys. Rev.A 67 (2003).
H /λ from efficient transport with large hhvii and small
0 [9] F. Chen, U. Mohideen, G. L. Klimchitskaya, and V. M.
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velocity fluctuations. Interestingly, below the transition [11] F.Ju¨licher,A.Ajdari,andJ.Prost,Rev.Mod.Phys.69,
hhvii is almost constant independently of H /λ. The ob- 1269 (1997).
0 [12] R.D.AstumianandI.Derenyi,Eur.Biophys.J.27,474
servedtransportbehavior is also stable againsta change
(1998).
of effective damping γˆ as shown in Fig. 3(b). Whereas
[13] C. C. deSouzaSilva, J.VandeVondel,M.Morelle, and
fluctuations increase with decreasing γˆ, there is a stable V. V.Moshchalkov, Nature440, 651 (2006).
plateau of constant average velocity across which fluc- [14] F. Marchesoni, S. Savel’ev, and F. Nori, Phys. Rev. E
tuations are diminished. In the deterministic limit, we 73, 021102 (2006).
havealsoobservedadditionalplateauswithinvertedand [15] J. L. Mateos, Phys. Rev.Lett. 84, 258 (2000).
doubled average velocity by varying γˆ and η. Remnants [16] J. Cumings and A.Zettl, Science 289, 602 (2000).
[17] H. B. G. Casimir, Kon.Ned. Akad.Wetensch. Proc. 51,
ofasecondplateauaroundγˆ =1.9,washedoutbynoise,
793 (1948).
can be seen in Fig. 3(b). [18] R. Buscher and T. Emig, Phys. Rev. Lett. 94, 133901
Finally, let us estimate typical velocities v = λ/τ.
0 (2005).
With the typical lengths H0 = 0.1µm, a = 10nm real- [19] R.GolestanianandM.Kardar,Phys.Rev.Lett.78,3421
ized in recent Casimir force measurements [9] and an (1997).