Surface Contact Potential Patches and Casimir Force Measurements

0.25 MB·English

by  Woo-Joong Kim

Surface Contact Potential Patches and Casimir Force Measurements
W. J. Kim∗,1 A.O. Sushkov,1 D. A. R. Dalvit,2 and S. K. Lamoreaux1
1Yale University, Department of Physics, P.O. Box 208120, New Haven, CT 06520-8120, USA
2Theoretical Division MS B213, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Dated: January 7, 2010)
We present calculations of contact potential surface patch effects that simplify previous treat-
ments. It is shown that, because of the linearity of Laplace’s equation, the presence of patch po-
tentials does not affect an electrostatic calibration of a two-plate Casimir measurement apparatus.
Usingmodelsthatincludelong-rangevariationsinthecontactpotentialacrosstheplatesurfaces, a
numberofexperimentalobservationscanbereproducedandexplained. Forthesemodels,numerical
calculations showthat ifavoltageis appliedbetween theplateswhich minimizes theforce, aresid-
ualelectrostaticforcepersists,andthattheminimizingpotentialvarieswithdistance. Theresidual
0
forcecanbedescribedbyafittoasimpletwo-parameterfunctioninvolvingtheminimizingpotential
1
anditsvariation withdistance. Weshowtheorigin of thisresidualforce byuseofasimpleparallel
0
2 capacitor model. Finally, theimplications of aresidual force that variesin a mannerdifferent from
1/d on the accuracy of previous Casimir measurements is discussed.
n
a
PACSnumbers: 31.30.jh12.20.-m42.50.Ct,12.20.-m,78.20.Ci
J

7
I. INTRODUCTION potential established between the two conductors, equal
]
tothedifferencebetweentheirworkfunctions. Wethere-
h
p It is often assumed that the surface of a conductor is foreexpectthatthe electrostaticpotentialalongachem-
- icallycleanmetalsurfacevariesonthelengthscaleofthe
anequipotential. Whilethiswouldbetrueforaperfectly
t
n clean surface of a homogeneous conductor cut along one typical size of surface crystallites, which can vary from
a sub-micron to millimeter or larger scales.
of its crystalline planes, it is not the case for any real
u
surface. Potentialpatchescanbecausedby,forexample,
q
[ oxide films or some other films adsorbed on the surface Thegenerationofanattractiveforcebetweenconduct-

3 (less than a monolayer is required), strains in the sur- ing surfaces due to metallic contacts has long been rec-
face, and chemical impurities within the surface. Such
v ognized as a possible systematic limitation to a Casimir
patches effectively create a surface dipole layer, which
1 force measurement [7, 8]. Recently, Speake and Trenkel
2 alters the potential above the surface. Various types of [9] have performed a formal treatment of the effect of
4 monopolar charge disorder could be also present, signifi-
random, zero-average, patch potentials. We reproduce
3 cantlyalteringthepropertiesofasurfacefromthatofan
here their result using a slightly different formalism,and
. ideal conductor [1]. Even for chemically unreactive no-
5 show that the result can be simply obtained, expressed
0 ble metals, such as gold and copper, carefully prepared in terms of the surface potential autocorrelation func-
9 in an ultra-clean environment to minimize such “dirt”
tion. Furthermore,weprovideasimplemodelthatshows
0 films, experiments show that typical surface potential
the origin of a distance-dependent minimizing potential,
v: variationsareonthe orderofatleastafew millivolts[2– andthatthisdependenceleadstoaresidualelectrostatic
Xi 4]. Similar effects were found in a more recent measure- forcethat canhaveacomplicatedcharacter. The goalof
ment on a pair of metallic plates employed in the Laser
this paper is not to present a full rigorous mathematical
r Interferometer Gravitational Wave Observatory (LIGO)
a derivation of these effects, but to illustrate their funda-
project [5]. The cause of surface potential variations is
mental nature and motivate functional forms that have
most likely local changesin surface crystalline structure,
been observed in experiments in a straightforward man-
giving rise to varying work functions and hence varying-
ner. More importantly, if the patch potential is actually
potentialpatches. Itiswellknownthattheworkfunction
measured, it would be possible to perform an exact nu-
of a metal surface depends on the crystallographicplane
merical calculation of the excess force. Our recent work
alongwhichitlies;asanexample,forgoldtheworkfunc-
[10]indicatesvariationsofcontactpotentialsonthelevel
tionsare5.47eV,5.37eV,and5.31eVforsurfacesinthe
of 10 mV, which is a challenging level for Kelvin probe
h100i, h110i, and h111i directions, respectively [6]. This
techniques with the appropriate spatial resolution [11].
variationis mostlikelydue to the differenteffective elec-
Thus, the full experimental description of these effects
tron masses, and the resulting different Fermi energies
remains an open challenge.
along the corresponding directions.
When two conductors of different work functions are
broughtintocontact,electronsflowuntilthechemicalpo- In this paper we define a positive force as an attrac-
tential(i.e., the Fermienergy)inboth conductorsequal- tion between the surfaces under consideration. There-
izes. Asaresult,thereisanetdipoledistributioncreated fore,attractiveelectrostaticandCasimirforcesaretaken
attheinterface,whichgivesrisetoa“contact”or“Volta” as positive.
2
II. ATTRACTIVE ELECTROSTATIC FORCE Inordertoproceed,wewillassumethattheelectrostatic
DUE TO SMALL SURFACE PATCHES patches are stochastic, uncorrelated between the differ-
ent plates, and for a given plane the 2-point correlation
In this section we will consider the effect of random function is diagonal in the k-basis. That is
surface patches on the electrostatic interaction between
metallic plates, for the plane-plane and plane-sphere ge- hVa,ki=hVb,ki=hVb,kVa,k′i=0;
ometries. We will assume that the typical patch area is hVa,kVa,k′i=Ca,k δ2(k−k′);
much smallerthan the effective areaof the surface inter- hVb,kVb,k′i=Cb,k δ2(k−k′), (6)
actions, defined for each of the geometries below.
whereh...imeansstochasticaverage. Notethat,asin[9],
weassumezerocross-correlationbetweenthe plates,and
A. Parallel planes geometry thatthecorrelationfunctionofeachplateisindependent
ofthepositionoftheotherplate(i.e.,independentofthe
Consider two plane parallel metallic surfaces at z = 0 distance d).
and z = d. Let the electrostatic potential at z = 0 be It is then easy to calculate each of the terms in the
V(x,y,z = 0) = Va(x,y) and at z = d be V(x,y,z = expression for Upp:
d) = V (x,y). The solution of Laplace’s equation in the
regionb0<z <dcanbeeasilyfoundseparatingvariables h ∂V 2i= d2k kx2sin2(kxx)cos2(kyy)
in cartesian coordinates, V(x,y,z) = X(x)Y(y)Z(z), ∂x (2π)2 4sinh2(γd)
(cid:18) (cid:19) Z
where
×{2Cb,k[cosh(2γz)−1]+2Ca,k[cosh(2γ(z−d))−1]};
1 d2X 1 d2Y 1 d2Z
=−α2; =−β2; =γ2, (1)
X dx2 Y dy2 Z dz2 ∂V 2 d2k ky2cos2(kxx)sin2(kyy)
h i=
and γ2 = α2 + β2. The general solution of Laplace’s (cid:18)∂y(cid:19) Z (2π)2 4sinh2(γd)
equation in this geometry can be written as ×{2Cb,k[cosh(2γz)−1]+2Ca,k[cosh(2γ(z−d))−1]};
∞
V(x,y,z)= dαdβ[cos(αx)+Aαsin(αx)] ∂V 2 d2k γ2cos2(k x)cos2(k y)
−∞ h i= x y
×[cos(βy)+ZAβsin(βy)] Bγ+eγz+Bγ−e−γz . (2) (cid:18)∂z (cid:19) Z (2π)2 4sinh2(γd)
×{2Cb,k[cosh(2γz)+1]+2Ca,k[cosh(2γ(z−d))+1]}.
(cid:2) (cid:3)
It is convenient to expand the boundary conditions in a
The x and y integrations are trivial, since
cosine Fourier series:
L−1 Lxdxsin2(k x) = L−1 Lxdxcos2(k x) = 1/2
d2k anxd 0the same fxor y. Fxor0the z intexgration we
Va(x,y)= (2π)2Va,kcos(kxx)cos(kyy), (3) use R ddz[cosh(2γz) ± 1] =R sinh(2γd)/2γ ± d and
Z 0
d
dz[cosh(2γ(z −d))±1] = sinh(2γd)/2γ ±d. Finally
and similarly for V (x,y). Imposing the boundary con- 0 R
b the electrostatic parallel-plate energy is
ditions we get Aα = Aβ = 0, α = kx, β = ky (hence R
γB−==q(kVx2+ekγy2d)−, BVγ+ )=/2(Vsibn,γh(−γdV)a.,γTeh−eγrde)f/o2res,inthhe(γgde)nearnadl Upp = 1ǫ06 (d2π2k)2γssininhh2((2γγdd))[Ca,k+Cb,k]. (7)
γ a,γ b,γ Z
solution of Laplace’s equation with the given boundary
In the specialcase of anisotropic patch distribution, the
conditions on each plate is
correlation functions depend only on k = |k|, that is
d2k cos(k x)cos(k y) Ca,k =Ca,k and Cb,k =Cb,k. Then
x y
V(x,y,z)= (4)
Z (2π)2 2sinh(γd) ǫ0 1 ∞ k2sinh(2kd)
× eγz Vb,k−Va,ke−γd +e−γz Va,keγd−Vb,k . Upp = 162π 0 dk sinh2(kd) [Ca,k+Cb,k]. (8)
Z
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
Now we calculate the electrostatic energy between the For the plane-plane geometry, the effective area of in-
plates. The electrostaticenergy density is u= ǫ20|E|2, so teraction Aeff is the same as the total area of the plate
the total energy Upp per unit area A=LxLy is Aeff = A. In this case, the small surface patch limit
corresponds to k2A ≫1, which basically means that we
ǫ0 1 Lx Ly d neglect finite-size effects in the computation of the elec-
U = dx dy dz
pp 2 A trostatic energy.
Z0 Z0 Z0 Let us analyze different limiting cases of Eq.(8). For
2 2 2
∂V ∂V ∂V
× + + . (5) surface potential patches small with respect to the effec-
"(cid:18)∂x(cid:19) (cid:18)∂y(cid:19) (cid:18)∂z (cid:19) # tive interaction area (k2A ≫ 1) but large with respect
3
to the plates separation (kd ≪ 1 or d ≪ λ, where λ is B. Sphere-plane geometry
a characteristic length of a potential patch) the energy
scales as 1/d, which is the same as in the usual electro- Inordertocomputethepatcheffectontheforceinthe
static casefor fixed(non-stochastic)potentialdifference. sphere-planeconfigurationwe make use of the proximity
This limit kd→0 is essentially the “proximity force ap- force approximation. Just as in the case of roughness in
proximation” (PFA), to be discussed more fully in the Casimirphysics,onemustdistinguishbetweentwoPFAs:
next Section, applied to the electrostatic problem with one is for the treatment of the curvature of the sphere
in-plane potential variations. In this approximation the (valid when d≪R, where R is the radius of curvature),
net energy and force is calculated by considering the at- andtheotheroneisthePFAappliedtothesurfacepatch
tractionbetweenpairedinfinitesimalsurfaceelementson distribution (valid when kd ≪ 1). We assume that we
each plate, and for large surface patches, the energy is are in the conditions for PFA for the curvature, but we
then the additive sum of the usual 1/d plane-plane en- keep kd arbitrary.
ergies. Indeed, if we rewrite Eq. (8) defining the root For the sphere-plane geometry we define an effective
mean square (rms) potential fluctuations Vr2ms as areaofinteractionbycalculatingalongthe planesurface
the distance r from the point of closestapproach(r =0,
∞ ∞
1
V2 = dk k(C +C )≡ dk kS(k), (9) correspondingtothe minimalsphere-planeseparationd)
rms a,k b,k
8π 0 0 where the separation between the surfaces doubles. For
Z Z
a given r, this latter separation is given by d(r) = d+
one obtains in the limit kd→0, R(1−cosθ), withsinθ =r/R. Inthe limitθ ≪1(which
corresponds to R ≫ d(r)) we obtain d(r) = d+r2/2R.
U = ǫ0Vr2ms. (10) The condition for the surface separation to be double
pp
2d that of the closest-approach distance is d(reff) = 2d =
d+r2 /2R, and the effective area is then
eff
In the opposite limit (surface potential patches small
with respect to the effective interaction area, k2A ≫ 1, Aeff =πre2ff =2πRd. (12)
and small with respect to the plates separation, kd ≫
For a sphere of radius R = 15 cm separated by a plane
1) Eq.(8) has an asymptotic behavior independent of
by a distance d = 1 µm, the effective distance above
the distance d. This is an artifact of the calculational
method, that has included the self-energy of each plate. defined is reff = 0.05 cm and the effective area is Aeff =
0.009cm2. The small patch limit corresponds to surface
Following [9], we remove from the above the potential
energy at infinite separation, in order to have an ex- patchesofareamuchsmallerthanthiseffectiveareaAeff,
pression for the interaction energy only. Using that that is k2Aeff ≫1.
Intheproximityforceapproximation,theelectrostatic
sinh(2kd)=2sinh(kd)cosh(kd), we get
force in the sphere-plane case is F (d) = 2πRU (d),
sp pp
∞ namely
U = ǫ0 dk k2 2cosh(kd) −2 S(k)
pp = ǫ40 Z0∞dk k2e(cid:18)−ksdinSh((kkd).) (cid:19) (11) Fsp =πǫ0RZ0∞dkskin2he(−kkdd)S(k). (13)
2 sinh(kd)
Z0 There are a number of models that can be used to de-
scribe the surface fluctuations. The simplest is to say
Therefore, in the limit kd → ∞ the interaction energy
that the potential autocorrelation function is, for a dis-
vanishes exponentially. The reason is that in this case
tance r along a plate surface,
the patches are so small and change sign so rapidly that
there is no net electric field at a large distance d from a R(r)=V2e−r2/λ2. (14)
0
given plate, hence there is no interaction with the other
plate. Then, by the Wiener-Khinchintheorem,the powerspec-
Some remarks are in order. First, we have assumed tral density S(k) can be evaluated as the cosine two-
thatthe boundaryconditions onthe twoplates hadonly dimensional Fourier transform of the autocorrelation
stochasticcomponentsfluctuatingaround0. Whenthere function, which in our notation is [12]
is an external fixed potential difference V between the
two plates, the energy is the sum of the usual V2 term S(k)=2·V2λ2e−πλ2k2, (15)
0
plusthepatchcomponentcalculatedabove(basicallythis 2
is due to the linearity of Laplace’s equation). The cross- where the factor of two reflects the statistically-
termsVV1,k andVV2,k cancelupontakingstochasticav- independent contributions from the two plate surfaces
erage. Second,whenthecross-correlationshV1,kV2,k′iare (C1,k +C2,k). The plane-sphere force is then given by,
notzero,thereisapossibilitythattheinteractionenergy using k =u/λ,
dependsontherelativelateralpositionofthetwoplates,
and hence it is possible to have a lateral force between V2 ∞ e−πu2
the plates due to stochastic patches. Fsp =2πǫ0R λ0 du u2e2ud/λ−1. (16)
0
Z
4
The limit of small potential patches kd → 0 (and also where V(r,ϕ) is the net potential difference between the
small with respect to the effective area of interaction, surfacedifferentials,locatedat(r,ϕ)relativetothepoint
k2Aeff ≫1), the force is ofclosestapproach(r =0). WritingV inthisformallows
the possibility that there can be a slow (coherent) varia-
F ≈ ǫ0RV02, (17) tionacrossthesurface,asopposedto,andinadditionto,
sp d innumerablesmallrandompatches. ThePFAworkshere
for two reasons. One is that the lines of electric force do
suggestingthat V2 =V2/π. For the large kdlimit, the
rms 0 notcrosseachother,thesecondisthattheradiusofcur-
forcebecomesexponentiallysmall,justasintheparallel-
vatureislarge,sotheangulardeviationsofthefieldlines
plates geometry.
aresmallwhentheplatediameterD satisfiesD ≪R. In
Another possible model for the patch distribution is
this limit, the assumption that each surface differential
the one used in [9]. Assuming that C = C = V˜2 =
a,k b,k 0 element interacts only with a single element in the other
const for kmin <k <kmax and zero otherwise, we get an plate is a good approximation.
expression for V2 (similar to Eq. (12) in [9])
rms A slow variation in potential across the plate surfaces
will manifest itself as a distance variation in the poten-
1 ∞ V˜2
V2 = dkk(C +C )= 0 (k2 −k2 ) tialthat minimizes the electrostaticattractive force,i.e.,
rms a,k b,k max min
8π Z0 8π Vm = Vm(d). Specifically, if we define the force with
∞
some externally applied voltage V0 to be
= dkkS(k). (18)
Z0 ǫ0 2π R (V(r,ϕ)+V0)2
We then obtain F(d,V0)= 2 dϕ rdr (d+r2/2R)2 , (22)
Z0 Z0
V˜2 2V2 the minimized force at a fixed distance determines the
S(k)= 4π0 = k2 −rmks2 , (19) minimizing potential,
max min
pfolrankemfionrc<e fkro<m kEmqa.x(1a3n)dthzeerroefoorteheisrwise. The sphere- 0= ∂F∂(dV,0V0)(cid:12)V0=Vm =ǫ0Z02πdϕZ0RrdrV(d(+r,ϕr2)/+2RV)m2.
(cid:12)
Fsp = k42πǫ0V−r2mks2R kmaxdkskin2he−(kkdd), (20) Tonhidsiestqaunacteio,nV(cid:12)(cid:12)mimp=lieVsma(dm)i.niNmoiztiengthpaott,einntiathledeidpeeanldizeendt
max min Zkmin case of an equipotential surface, i.e., V(r,ϕ) = const,
V would be independent of d, and the minimized elec-
whichistheidentical(apartfromanoverall,conventional m
sign) to Eq. (14) of [9]. trostatic force F(d,V0 = Vm) vanishes. Incidentally, the
second derivative of F(d,V0) with respect to V0 can be
usedtodeterminethedistanceatwhichthemeasurement
is being made,
III. ATTRACTIVE FORCE DUE TO LARGE
SURFACE PATCHES ∂2F(d,V0) R r 2πRǫ0
∂V2 =2πǫ0 dr(d+r2/2R)2 ≈ d ,
When the surface patches are larger than the effective 0 Z0
area of interaction Aeff defined in the previous section, where the finite size effects are neglected (upper limit of
the force between the plane and the sphere due to elec- the r integration is set to infinity, which is a very good
trostaticpatchescanstillbecalculatedusingthemethod approximationwhend≪R). Theimportantimplication
describedabove,buttheaveragepotential,asinferredby isthatthepatchpotentialsdonotinterferewiththeelec-
measuringthevoltageatwhichaminimumintheattrac- trostaticcalibration,thatisthefundamentalbasisofour
tive electrostatic force occurs, will vary with distance. experiment [10], and of all Casimir force experiments.
This canbe thoughtof as a finite-size effect; if the patch It is worth emphasizing a couple of points. First, that
size is roughly the diameter ofthe plates, then there can the origin of the distance dependence of the minimizing
beanon-zeroaverageoverthesurface. Alternatively,this potential V (d) is an interplay between the curvature of
m
problem can be addressed by assuming a slowly varying thesurfacesandavariationoftheelectrostaticpotentials
average potential across the plate surfaces, as developed V (i=a,b)alongthesurfaces(possiblyduelargesurface
i
below. patches). We have shown above how this effect arises in
In the PFA, the plate surfaces are divided into differ- the context of the sphere-plane geometry, but of course
ential areas,andthe attractive force, giventhe potential it can be easily generalized to any geometry involving
difference between the plates and the derivative of the non-planar surfaces. It also follows from the above that
capacitancebetweenthem,iscalculated. Specifically,for for the parallel plates geometry one should expect that,
the sphere-plate geometry, evenwhenthesurfacepotentialsV =V (x,y)varyalong
i i
the plane surfaces,there should be no interplaywith the
ǫ0 2π R V2(r,ϕ) (infinite) curvature of the planes, and thus the minimiz-
F(d)= dϕ rdr , (21)
2 0 0 (d+r2/2R)2 ing potential Vm should be distance-independent [13].
Z Z
5
implying a residual electrostatic force
Freels(d) = F(d,V0 =Vm(d))
C′2 V2(d)
= − C′ + a m
a C′ 2
(cid:20) b (cid:21)
C′C′ V2
= − a b c . (26)
C′ +C′ 2
(cid:20) a b(cid:21)
′
Itiseasytotakeacaseofparallelplatecapacitors(C =
FerIaGti.o1n:oAf atodyismtaondceel-dilelupsetnrdateinntgmthineimmiezcihnagneislemctfroorsttahteicgpeno-- −ǫ0A/d2 andCb′ =−ǫ0A/(d+∆)2,whereAistheareaaof
tential Vm(d) and electrostatic residual force Freels(d). each of the upper plates in Fig. 1, assumed to be equal;
hence, the lower continuous plate has area 2A) and to
show that there is a residual electrostatic force at the
minimizing potential. Indeed, in such case,
Second, distance dependence of the electrical potential d2
V (d) = −V , (27)
minimizing the force between the plates has been ob- m cd2+(d+∆)2
served in a number of experiments in the sphere-plane
geometry[14–16], as well as in our own work [10], with Fel (d) = ǫ0A Vc2 . (28)
further investigations under way. res 2 d2+(d+∆)2
It has been suggested that the variation of the min-
imizing potential with distance can cause an additional Alternatively, in terms of Vm(d) (up to V1, see below),
the force is
electrostatic force F(d,V0 = Vm(d)), and an estimate
was made for the possible size of the effect [17], where
thevaryingcontactpotentialisconsideredinasystemof Fel (d)= ǫ0A Vm2(d)[d2+(d+∆)2]. (29)
res 2 d4
plates connectedinseries. The analysispresentedin[17]
does not reproduce the effects seen in our experimen-
Experimentally, V (d) must include a distance-
tal work, and we were unable to develop a fundamen- m
tal theory of a plate-plate interaction that could cause independent offset V1 which arbitrarily depends on the
sum of contact potentials in the complete circuit be-
a varying contact potential. Nonetheless, we have seen
tween the plates. Therefore, the force due to large
above that within the PFA a coherent variation of the
patches considered in this section should be written as
surface potential along the non-planar surface does im-
ply a distance-dependent minimizing potential. Now we proportional to (Vm(d)+V1)2, instead of simply Vm2(d),
present a simple model that produces not only varying where V1 is determined by a fit to experimental data.
Note that when the two upper capacitor plates are at
contact potentials, but also the corresponding residual
the same distance from the lower capacitor plate, i.e.
electrostatic force, consistent with our observations in
when ∆ = 0, Eq.(27) predicts a V independent of dis-
[10]. The model is depicted in Fig. 1. In this figure, m
tance, and Eq.(29) predicts a residual electrostatic force
the two capacitors (short distance, C (d), long distance,
C (d+ ∆)) create a net force on thae lower continuous Freels(d) ∝ (Vm +V1)2/d2. Since no residual force is ex-
b
pected in this case, the minimizing potential must be
plate (setting V1 =0 initially),
Vm =−V1.
Inordertoanalyzethesphere-planegeometry,onecan
F(d,V0)=−12Ca′V02− 12Cb′(V0+Vc)2, (23) dinivtihdee tphreoxsipmhietryefionrtcoeinapfipnritoexsiimmaatliopnla)n,aeracahrewasit(hasadroanne-
dom potential. In this picture, one can think of the two
uppercapacitorplatesinFig. 1asoneofthoseinfinitesi-
where
malpartsofthewholesphericalsurface,andthedistance
∆ being a local distance ∆(r)=d+R(1−cosθ) reflect-
∂C (d) ∂C (d+∆)
′ a ′ b ing the effect of the curvature of the spherical surface.
C = ; C = , (24)
a ∂d b ∂d Inthis case,Ca′(d)=−2πǫ0R/d,andthedenominatorof
Eq. (29) becomes d2. Integrating the force on the lower
and V0 can be varied, with Vc a fixed property of the planar plate over the whole spherical surface to get the
plates. The force is minimized when netforceleadstoafurtherreductionofthepowerofdin
the denominator, leaving the sphere-plane residual elec-
′ trostatic force proportional to (Vm(d)+V1)2/d. Again,
∂F(d,V0) =0 ⇒ V (d)=− CbVc ,(25) V1 is a fit parameter that represents a sort of surface
∂V0 (cid:12)V0=Vm m Ca′ +Cb′ average potential, plus circuit offsets.
(cid:12)
(cid:12)
(cid:12)
6
IV. TOTAL ELECTROSTATIC FORCE variation in minimizing potential due to random surface
RESIDUALS potential distributions by use of Eq. (22). The physical
basis of Eq. (30) is nonetheless quite sound.
We are now in a position to compute the total elec-
trostatic residual force at the minimizing potential. On
the one hand, the presence of surface patches small with V. DISTANCE CORRECTION DUE TO
respect to the effective area of interaction in the sphere- RESIDUAL FORCES AND ITS IMPLICATION
plane geometry leads to an attractive electrostatic force ON THE INTERPRETATION OF PREVIOUS
givenby Eq.(13), which was derivedusing the proximity DATA
force approximation (d ≪ R) to treat the curvature of
the spherical plate. This component of the force due to In previous work [8], the minimizing potential was as-
stochasticpotentialpatchesis clearlyindependent ofthe sumed to be independent of distance and the absolute
applied voltageV0 betweenthe plates, andtherefore will separation between the sphere-plane plates in a Casimir
be present even when the applied voltage is set at the experimentwasdeterminedbyfittingabackgroundelec-
minimizing potential, V0 = Vm(d). As we have seen in trostatic force (at distances where the Casimir force is
SectionII,theexactdependenceofthisforceondistance relatively small and negligible) to a function
variesfordifferentmodelsforthestatisticalpropertiesof
the two-point correlation functions of the surface poten- B
tials. However,inthe limit ofpatches muchsmallerthat Freels,fit(d)= d+fd0f, (31)
the sphere-plane separation(kd≪1), all models predict
adependencyoftheformVr2ms/d,whereVrms denotesthe where Bf and d0f are fitting constants. Unfortunately,
rms voltage fluctuations.
in[8],apossibledistancevariationofthe minimizing po-
On the other hand, the presence of surface patches
tential was not measured. However, a relatively large
largewithrespecttotheeffectiveareaofinteraction,cor-
fixedpotentialwasconstantlyappliedtogivea1/dforce
responding to a coherentvariationofthe potential along
in the data, and this was used to determine the absolute
the spherical surface V(r,ϕ), leads both to a distance-
separation.
dependent minimizing potential V (d) and to a residual
m One can estimate a systematic error for [8] (and pos-
component of the electrostatic force Freels(d)=F(d,V0 = sibly for otherexperiments where V wasassumedto be
m
Vm(d)), even when the external potential V0 is set at constant) by considering a possible distance-dependence
the minimizing potential. As seen in Section III, this
ofthe minimizing potential, andtaking our Ge measure-
forceis due to the interplaybetweensurfacepatches and
mentresults[10]as“typical”. Intheprevioussectionwe
the curvature of the non-planar surface, has the form
notedthat the residuallongrangeforcein the Ge exper-
(Vm(d)+V1)2/d,andisinadditiontotheV0-independent iment is well described by Eq.(30) in terms of the mea-
force due to small potential patches.
sured distance-dependent contact potential V (d). If,
m
By adding the two contributions of small and large
instead, we had mistakenly assumed V to be indepen-
m
surface patches (Section II and III, respectively), we can
dentofdistanceandneglectedanyVrms contribution,the
determine the form of the residual electrostatic force at
same experimentaldata for the residuallong range force
the minimizing potential. By numerical modeling of dif-
in[10]isalsowelldescribedbyapower-lawfitoftheform
ferent patch sizes and distributions using Eq. (22), we
1/de,withexponente=0.72±0.2(asmentionedinfoot-
found that the experimental observations should be de-
note [24] in [10]). Note that this is the residual force at
scribed by a relationship of the form
the minimizing potential, so if the applied voltage is not
Freels(d)=πRǫ0 (Vm(d)+V1)2+Vr2ms , (30) bseuttpwreecinseeglyletcot Vthma(td)h,etrhe.ereIncatnhebefoallnowadindgitiwoenawliflolracse-,
d
(cid:20) (cid:21) sume that a power-law form for the long-range residual
wherewehavespecificallyassumedthesmallkdlimitfor force also applies to the Au experiment [8], and we will
the small patches. Both V1 and Vrms are constants de- take a nominal value of 0.8 for the exponent to assess
termined by fitting to the observed force at large plate thepossiblesystematicerrorintroducedin[8]. Thisesti-
separations, where the Casimir force is relatively small. mate is entirely heuristic, and giventhat the minimizing
In [10] we have measured the dependency of the min- potential was not measured as a function of distance in
imizing potential with distance, and we have applied [8], it is the best that canbe done in a post-analysisand
Eq.(30) to fit the observed residual electrostatic force in represents a reasonable range.
aGesphere-planeCasimirexperimentfordistanceslarge An error in the distance determination is introduced
enough (d > 5µm) to neglect possible contributions due if an improper function 1/dis used instead of the “true”
to Casimir forces. With only two adjustable parameters function,whichwetaketobe1/d0.8. Generally,wemight
V1 and Vrms, very good fit (χ2 of order unity) between expect contributions from both, as
theexperimentaldataandEq.(30)waspossible. We em-
phasizethat Eq. (30) wasobtainedin aheuristic wayby Fel (d)= B1 + B2 , (32)
numerically analyzing the distance-dependent force and res,true d+d0 (d+d0)0.8
7
where the first term represents a patch potential force mental results, and by a theoretical analysis [9]. Alter-
andaninexact cancelationofthe averagecontactpoten- natively, if a residual force that appears to have a 1/d
tial(V1),andthesecondtermrepresentstheforcedueto characterisremovedfromCasimirdata,apossibleback-
the variation in Vm(d). We assume that d0 is fixed and ground force as considered here will persist as a direct
known; our goalis to determine the distance errorin de- systematic.
termining d0 by fitting to a function of the wrong form.
We can define an effective χ2 and minimize its value to
findtheoffsetduetothepossibilitythatthewrongfitting VI. CONCLUSIONS
function, 1/(d+d0) alone, was used in [8]. A form that
allows simple numerical calculation is as follows, substi-
We have derived in a straightforward and heuristic
tuting x=d/d0 manner several important results pertaining to the ex-
2 cess electric force between plates that results from ran-
χ2 = xmaxdx Bf − α − (1−α) , dom surface patch potentials. These results have been
x+(1+ǫ) (x+1)0.8 x+1 cast in terms of the surface autocorrelation function, or
Z0 " #
alternatively the two dimensional spatial Fourier power
(33)
spectrum. Our recent measurement [10] of short-range
where d0f = d0 + ǫd0, α parameterizes the relative
amounts, at d = 0 (or x = 0) of the 1/d and 1/d0.8 forces with Ge plates in the sphere-plane configuration
has shown the importance of assessing surface patch po-
forces in the “true” (and assumed known) function, and
tentials when measuring Casimir force residuals. A re-
xmax = dmax/d0 is the largest (dimensionless) separa-
centCasimirexperimentintheplane-planeconfiguration
tion to which measurements are taken. When α=0 the
[13] has also found a large residual (non-Casimir) force
“wrong” fitting function is equal to the “true” function,
that is also probably related to electrostatic patch resid-
and ǫ = 0. For simplicity, we choose to define α as the
ual forces.
relativecontributionofthetwoforcesatd0 becausetheir
relative size is distance-dependent. Furthermore, we have shown that long-range surface
In [8] the relative contributions from an applied fixed correlations in the sphere-plane geometry can lead to a
voltage 1/d force, and the possible residual unaccounted distance dependence of the electrostatic force minimiz-
1/d0.8 force implied by our recent results [10], are in the ingpotential. This effectisdue to the dependence ofthe
ratio approximately 10/1, so α≈0.1, at distances of or- netsurfaceaveragingareaontheseparationbetweenthe
der1µm. TheparametersB andǫarethendetermined platesandhasbeendescribedbyuseofasimplecapacitor
f
by minimizing χ2. Although the integral in Eq.(33) ap- model. The model clearlyreproducesthe generaleffects,
pears as elementary, its evaluation is quite cumbersome. providesanexplanationofthe originofthe varyingmin-
Results of numerically minimizing χ2 as a function of α imizing potential, and demonstrates that even when the
and Bf, with xmax =10, show that forceis minimized, a residualelectrostaticforceremains.
The results here shouldbe comparedto earlierwork [17]
ǫ=0.65α. (34) where it was assumed that the variation in minimizing
potential was due to a voltage in series with the plates,
Therefore, for α = 0.1 we obtain ǫ = 0.065. Applying withthatvoltagevaryingwithdistancebysomeunknown
this result to Fig. 4 in [8] shows that the x-axis needs mechanism;thisanalysiscouldnotdescribetheobserved
to be shifted to the left (towardthe origin)by 0.065µm. (non-Casimir)forceinthe germaniummeasurementsde-
That is, the distance scale is offset by 0.065 µm. With scribed in [10]. We note that the effect is expected to
this displacement, the Casimir force F at the true loca- exist whenever there are surface potential patches and
c
tion is better described by the theoretical result due to non-perfectly-parallel surfaces; the model shown in Fig.
B¨ostrom and Sernelius [18], which can be seen from the 1 requires only the existence of patches with different
fractionalchangeinforce,δF /F =−3δd/d,whichgives absolute distances between the surfaces. Such distance
c c
a 20% effect and brings the measured Casimir force into differentials can be created by surface roughness and/or
agreement with the predicted result in [18] for distances lack of parallelism.
around 1 µm; however, this result should be considered Finally, we have shown that determining the distance
as preliminary and as a rough estimate of the correction between a sphere and a plane in a Casimir force exper-
magnitude that is possible. Because patch potential ef- iment can be subject to systematic effects arising from
fects are sample dependent, it is not possible to say con- residual electrostatic forces. The magnitude of the error
clusively that the effects described here contributed to inthedistancedeterminationislargeenoughtobringthe
the result, although, not having tested for such possible results presented in [8] into agreement with the calcula-
effects, an additional systematic error could be ascribed tion that takes into account properly the low frequency
to the result in [8]. permittivity ofmetals[18]. The relevanceofthe analysis
Other possible background forces that deviate from a presented in this paper to precision Casimir force mea-
1/dcharacterwillleadtocorrectionstothetruedistance surements and their possible systematic contamination
when a fit to 1/(d−d0) is performed. For the discussion should not go unnoticed. We are currently performing
here, we chose a form that is motivated by our experi- new measurements using Au coated plates in the appa-
8
ratus used for Ge measurements, and will revisit these project under SPAWAR contract number N66001-09-1-
systematic effects. 2071. D.A.R.D.’s work was funded by DARPA/MTO’s
∗Present address: Dept. of Physics, Seattle University, Casimir Effect Enhancement project under DOE/NNSA
901 12th Avenue, Seattle, WA 98122 Contract DE-AC52-06NA25396. He is grateful to R.
Onofrio for insightful discussions.
Acknowledgments
S.K.L. and A.O.S.’s work was funded by Yale Univer-
sity, and DARPA/MTO’s Casimir Effect Enhancement
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