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Measurement of the Casimir force in a gas and in a
liquid
Anne Le Cunuder, Artyom Petrosyan, Georges Palasantzas, Vitaly Svetovoy,
Sergio Ciliberto
To cite this version:
Anne Le Cunuder, Artyom Petrosyan, Georges Palasantzas, Vitaly Svetovoy, Sergio Ciliberto.
Mea-surement of the Casimir force in a gas and in a liquid. 2018. �hal-01830823v2�
Anne Le Cunuder1,∗, Artyom Petrosyan1, Georges Palasantzas2, Vitaly Svetovoy2,3 and Sergio Ciliberto1
1Laboratoire de Physique, CNRS UMR5672,
Université de Lyon, École Normale Supérieure, 46 Allée d'Italie, 69364 Lyon, France
2 Faculty of Sciences and Engineering, University of Groningen,
Nijenborg 4, 9747 AG Groningen, The Netherlands and
3 A. N. Frumkin Institute of Physical Chemistry and Electrochemistry,
Russian Academy of Sciences, Leninsky prospect 31 bld. 4, 119071 Moscow, Russia We present detailed measurements of the Casimir-Lifshitz force between two gold surfaces, per-formed for the rst time in both gas (nitrogen) and liquid (ethanol) environments with the same apparatus and on the same spot of the sample. Furthermore, we study the role of double-layer forces in the liquid, and we show that these electrostatic eects are important. The latter contributions are subtracted to recover the genuine Casimir force, and the experimental results are compared with calculations using Lifshitz theory. Our measurements demonstrate that carefully accounting for the actual optical properties of the surfaces is necessary for an accurate comparison with the Lifshitz theory predictions at distances smaller than 200nm.
Introduction. As devices enter the submicron range,
Casimir forces [112] between neutral bodies at close
proximity become increasingly important. As Casimir
rst understood in 1948 [2], these forces are due to the
con nement of quantum uctuations of the electromag-netic (EM) eld. Indeed Casimir proved that when two parallel, perfectly re ecting plates, are introduced in vac-uum, they impose, on the EM eld, boundary conditions which select only the uctuations compatible with them. As a result, an attractive force between the plates is pro-duced, which depends only on fundamental constants, on the distance d between the surfaces and on their area A:
Fc(d) = −
π2
A~c
240 d4 (1)
with ~ the Planck constant and c the speed of light.
Fol-lowing Casimir's calculation [2], Lifshitz and co-workers
in the 50's [3] considered the more general case of real
dielectric plates by exploiting the uctuation-dissipation theorem, which relates the dissipative properties of the plates and EM uctuations at equilibrium. Furthermore, for real surfaces, roughness and material optical
proper-ties can strongly alter the Casimir force [13,14].
Lifshitz formalism describes the Casimir force in a gen-eral case, where the medium between the plates need not be vaccum. According to this formalism, the force can be tuned from attractive to repulsive with a suitable choice of the interacting materials. These predictions boosted Casimir experiments to test the possibility of repulsive
forces [16]. In liquids, the determination of the Casimir
force is more complex than in a gas because of the pres-ence of additionnal eects, as e.g. the Debye screening. The Casimir-Lifshitz force has been measured between
two gold surfaces immersed in ethanol [17]; in this
exper-iment, electrostatic forces are found to be negligible as sodium iodide (NaI) was added to ethanol, decreasing the Debye screening length. However, the role of electrostatic forces and their screening by the Debye layer are
impor-É DES SURFACES
eux surfaces s’exercent à une hauteur moyenne comprise ent des aspérités. Il est donc indispensable de déterminer cette
connaître la distance de séparation effective pendant la m s surfaces rugueuses nécessite une approche statistique. L a ontact d0 est déterminée à partir de la distribution en ha
Thi s con di t i on can be con si der ed as an equat i on for t he h ei gh t of t h e h ei ght asper i t y due t o a shar p expon en t i al beh avi or of t he di st r i but i on
d0
smooth
a h1(x,y) h2(x,y)
orsque les deux surfaces rugueuses entrent en contact mé ts des aspérités de chacune des surfaces qui se touchent. L urs séparées par une distance de contact d > d0> 3w
ation la plus critique, le pic le plus élevé sur la surface de la encontre le pic le plus élevé (d0,ech) de la surface de l’éch ale de séparation est donc donnée par : d0,max= d0,ech+d0,sp.
e de se produire, sauf si les surfaces ont des rugosités comp la sphère est bien plus rugueuse que l’échantillon On con
d
t d
0est déterminée à partir de la distribution en hauteur de la
Thiscondition can beconsidered asan equation for theheight of the ght asperity due to a sharp exponential behavior of the distribution
d
0h
a
h
1(dxddd,y)
h
2(x,y)
dpiezo - h1 h2 d0=h1+h2 piezoelectric element dFIG. 1. Experimental setup. A gold-coated polystyrene bead is glued on the cantilever tip which measures the sphere-plane interaction force at distance d. The de ection of the cantilever is detected by an interferometric technique: two lasers beams, orthogonally polarized, are focused on the can-tilever, the reference one is re ected by the static base and the second one by the cantilever free end. When the cantilever is bended the optical path dierence δ between the two beams is measured through an interferometer [15].
tant and one has to consider carefully their contributions
during force measurements in liquids [18,19]. In order to
clarify the interplay of the Casimir force and additionnal eects in liquids, we have performed measurements of the Casimir force in a nitrogen atmosphere in the rst place, and then, using the same system and sample, in ethanol. The contact area is the same in both measurements. We observe that electrostatic forces, screened over the Debye length, are of the same magnitude as the Casimir force, in the 50-200nm distance range. After subtracting the electrostatic force, we obtain a Casimir force in
quanti-tative agreement with Lifshitz theory [3]. Furthermore,
2 the importance of accurately characterizing the optical
properties of the samples before any meaningful compar-ison with theory.
Experimental setup. We use an atomic force micro-scope (AFM) to measure the Casimir force between metallic surfaces. In order to measure the force with a good accuracy, the cantilever displacement is measured
with a home-made quadrature phase interferometer [15],
whose operating principle is sketched on Fig.1.
The experiment is performed in a sphere-plane geom-etry to avoid the need to maintain two at plates per-fectly parallel. Thus, a polystyrene sphere of radius
R = (75.00 ± 0.25)µm (Sigma-Aldrich) is mounted on
the tip of the cantilever with a conductive glue and then the whole probe is coated by a gold lm whose thickness is about 100nm. The plates have been gold
coated using cathodic sputtering by ACM, at the
LMA-CNRS. The diameter of the sphere has been determined
from Scanning Electron Microscopy. We use a cantilever (size 500µm × 30µm × 2.7µm, NanoAndMore) of stiness
κ = 0.57 ± 0.03N/m. The precise value of κ is
deter-mined using equipartition, ie. < δ2 >= kBT
κ , where kB
is the Boltzmann constant and T the temperature. The resonance frequency of the sphere-cantilever ensemble is
fo= 2271Hzin vacuum.
The sphere faces a glass at plate which is coated by
a gold lm of a thickness of about 100nm [20].
Accord-ing to [21], the layer is su ciently thick to be considered
as a bulk-like lm. The plate is mounted on a piezo actuator (PZ38, Piezojena) which allows us to control the plane-sphere distance. During the experiment, the plate is moved continuously towards the sphere and the induced de exion of the cantilever is detected by the in-terferometer.
In air, because of water vapor, the capillary force far exceeds the Casimir force. Therefore, our measurement in a gas was performed after lling our cell with nitrogen. Calibrations. The total force between the surfaces is
the sum of the Casimir force Fcas(d)and additionnal
con-tributions:
Ftotal= Fcas(d) + Fel(d) + FH(d, v) . (2)
Electrostatic forces Fel(d)are due to a potential
dier-ence between surfaces, owing to dierdier-ences between the work functions of the materials used, and the possible
presence of trapped charges [22]. Hydrodynamic forces
FH(d, v) are due to the motion of the uid during the
approach of the plate towards the sphere, and depend
on their relative velocity v [23]. These hydrodynamic
ef-fects are negligible in a nitrogen atmosphere, where the
viscosity is γ = 1.76 10−6Pa s, but have to be
consid-ered in ethanol where the viscosity is 1000 times higher
(γ = 1.2 10−3Pa s).
There are two main requirements for a precise deter-mination of the Casimir force. Firstly, additionnal forces
must be measured with accuracy and subtracted from the total measured force. Secondly, because the force has a strong dependance on the distance between sur-faces, an independant measure of the distance is nec-essary, which becomes di cult when the separation ap-proaches nanometer scales. The di culty originates prin-cipally from surface roughness: when the two surfaces come into contact, the highest asperities of each surface touch each other and the surfaces are still separated by
a distance upon contact d0 [24].
The piezo actuator includes a position sensor which
gives us the displacement of the plate: dpiezo. We
de- ne the origin of dpiezo as the position of contact of the
highest peak of the sphere roughness with the surface of the plate, as the sphere is much rougher than the plate. The eective separation distance which appears in the
expression of the force can be written as (see Fig.1):
d = dpiezo+ d0− δ (3)
where d0 is the distance upon contact due to surface
roughness and δ is an additional correction which results from the static de exion of the cantilever in response to the total force Ftotal.
We determined the separation upon contact d0 from
hydrodynamic calibration, performed in ethanol. Imme-diatly after measuring the Casimir force in nitrogen, we injected carefully ethanol into the cell, and we performed calibrations and measurements of the Casimir force in ethanol. As the horizontal drift of our system is negli-gible, the contact area and the separation distance upon
contact d0 are the same in each measurement. This
as-sumption is further justi ed a posteriori: our experimen-tal curves all superimpose on top of each other and on top of the theoretical curve after shifting the distance by
the same value of d0. The hydrodynamic calibration is
presented in next section, while topographic analysis is
presented in [20]. The value of d0obtained from
hydrody-namic calibration is comparable with the value obtained from roughness analysis.
Hydrodynamic calibration in ethanol. The theoretical expression of the hydrodynamic force, for non-slip
bound-ary conditions, is given by [23]:
FH= −
6πηR2
d v (4)
where η is the uid viscosity, R is the radius of the sphere,
and v = ∂d
∂t is the relative velocity between the plate
and the sphere. Indeed, in our case, the slippage can be neglected as the mean free path is in the order of inter-molecular distances (e.g a few Angstroms), and negligible in comparison with the roughness of the surface (e.g a few
tens of nanometers) [27]
As is clear from Eq. (2), among the dierent forces
100 150 200 250 300 350
Distance [nm]
140 120 100 80 60 40 20 0 20Casimir Force [pN]
(a)
100 150 200 250 300 350Distance [nm]
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00(F
exp.F
th.)/|
F
th.|
(b)
FIG. 2. Measurement of the Casimir force between two Au-surfaces in a nitrogen atmosphere. (a) Blue points correspond to the mean measured force. Blue circles correspond to a single measurement of the Casimir force. Orange solid curve corresponds to the Lifshitz theory in which the dielectric function is evaluated from measured optical data of a real gold lm [25]. For the green dashed-dotted line, is evaluated using the handbook optical data [26]. Red dashed line corresponds to the theory in the case of ideal conductors. (b) Relative dierence between the theoretical and experimental Casimir forces. Same colors as in (a). The error bars include the statistical and the systematic errors due to uncertainties on the separation upon contact d0= (31 ± 2)nm(using the hydrodynamic estimation), on the stiness κ = (0.57 ± 0.03)N/m and on the diameter
of the sphere D = (150.0 ± 0.5)µm.
is the only one which depends on velocity. Thus we performed two force measurements, moving continuously the plate towards the sphere: a rst one at velocity
v1 = 348nm s−1, and a second one at a velocity v2 =
5109nm s−1. By taking the dierence between these two
measurements, we cancelled all the velocity-independent
forces and from Eq. (4) we obtained FH measured at
v = v2− v1= 4742nm s−1 :
Ftotal(d, v20) − Ftotal(d, v01) = FH(d, v) . (5)
Here, v0
2 and v01 are the relative velocities between the
sphere and the sample, which are not exactly the piezo
velocities v2 and v1 because the cantilever is de ected
when the plate is moved towards the sphere. v0
2 and v01
were determined precisely measuring the de ection of the cantilever.
Measurements of the hydrodynamic force are presented
in [20]. Comparing the measured hydrodynamic force
with the theoretical expression of Eq. (4), we determined
the separation distance upon contact d0= (31 ± 2)nm.
Electrostatic forces. Even if the surfaces are as clean as possible, there always remain electrostatic forces be-tween them. First, an electrostatic potential dierence
Vc still exists between clean, grounded, metallic surfaces
owing to dierences between the work functions of the
materials used [28]. Second, electrostatic forces can
re-main due to the presence of trapped charges. In liquids, these trapped charges induce double-layer forces, due to the rearrangement of ions in solution, screening the elec-trostatic interactions.
When d << R, the expression of the electrostatic force is [29]: Fe= − π0dR λD Vc2 exp(−d/λD) (6) The term Vc2
d is the contribution of the contact
poten-tial Vc between the surfaces and the term exp(−d/λD)
represents the double-layer force, screened over a distance
λD (the Debye length)[30].
As there is no free charge in nitrogen, the Debye length is in nite and the electrostatic interaction is not screened, consequently there are no double-layer forces. In
ni-trogen, the contact potential was calibrated to Vc =
(87 ± 2)mV, and was compensated by an applied voltage
dierence during the measurement of the Casimir force. In contrast, in ethanol, the contact potential is strongly screened by the ions constituting the Debye layer. More-over, applying an electrostatic potential in a polar
liq-uid can yield a transcient current [31] and consequently,
charges accumulate on surfaces, preventing us from
ap-plying the method suggested in [32] to cancel the Debye
force. Therefore, we simply subtracted the contribution of electrostatic forces from force measurements, after
de-termining λD = (46 ± 6)nm and Vc = (63 ± 13)mV
[20]. In ethanol Vc is lowered because the dissociation
of molecules at the surface leads to the formation of a rst very thin screening layer of a few nm.
Measurement of the Casimir force in nitrogen. Static measurements of the Casimir force were carried out in a nitrogen atmosphere, between a gold-coated sphere and a gold-coated plate. We subtracted the vertical thermal drift by tting linearly each force curve measurement be-tween 300nm and 1µm and subtracting it from each mea-sured curve. All force curves were shifted in distance
cor-responding to the separation upon contact d0= 31nm.
The measured Casimir force is shown on Fig.2(a) for
separations ranging from 90nm to 370nm, averaging 28 independent measurements. For theoretical calculations, thermal corrections are negligible as the thermal energy
4 100 200 300 400 500 600
z [nm]
327.0 327.5 328.0 328.5 329.0 329.5F
totF
Hydr o[n
N]
(a)
1 measurement linear fit (thermal drift)50 100 150 200 250 300
z [nm]
200 0 200 400 600 800 1000 1200 1400 1600F
totF
Hydr oF
drift[p
N]
(b)
1 measurement fit Vc exp( z/ D) avg. 8 meas. 50 100 150 200 250 300z [nm]
1400 1200 1000 800 600 400 200 0 200Casimir Force [pN]
(c)
FIG. 3. Measurement of the Casimir force in ethanol. (a) Single force measurement (blue points) after subtracting the hydrodynamic force. The distance has been shifted by d0= 31nm. The red line corresponds to the linear t of the thermal drift
in the range 300nm - 600nm. (b) Single force measurement (red circles) and mean force measurement (blue points) (averaged over 8 measurements) of Fcas+ FDebye obtained after subtraction of the thermal drift and the hydrodynamic force. The mean measured force is tted by an exponential in the range d > 120nm. In this distance range, double-layer forces remove largely the Casimir force. The blue line corresponds to the exponential t, yielding the Debye length λD= (46 ± 6)nmand the surface
potential ψ0adj = (63 ± 13)mVthrough Eq. (6). (c) Measurement of the Casimir force between two Au-surfaces in a ethanol,
where blue cross correspond to the mean measured force. The orange solid line corresponds to the Lifshitz theory where the dielectric function is evaluated from measured optical data of a real gold lm [25]; for the green dashed-dotted line, is evaluated using the handbook optical data [26].
kBT is too small to populate the mode of lowest energy
~c/λT, as the separation distance d satis es
d < 370nm λT = ~c
kBT
≈ 7µm , at 300 K . (7)
We compared our experimental result with theoretical predictions of the Casimir force, based on optical prop-erties of Au taken from: 1) handbook of tabulated data
(green dashed-dotted line)[33], and 2) measurements on
Au samples presenting the same roughness and
prepara-tion condiprepara-tions as ours (orange solid line)[25].
The deviation from Lifshitz theory based on dielectric properties of real samples is less than 5 pN at closest sep-arations, while it reaches 10 pN at closest separations for calculations based on data from handbook. As the signal to noise ratio degrades with increasing distance, the devi-ation of the measurements from Lifshitz theory increases at larger distances. However, they remain compatible within the error bars, including systematic and statisti-cal errors (dominant at large distances). This demon-strates that surfaces must be carefully characterized for high precision measurements of the Casimir force. In or-der to make this argument more quantitative, we present
the dierence (Fexp − Fth)/Fth on Fig. 2(b), showing
the dierences between the theoretical and experimen-tal Casimir forces, for calculations based on data from handbook and calculations based on dielectric properties measured on lms with the same morphology as our lms. Measurement of the Casimir force in ethanol. In a liq-uid, the scenario is richer than in a gas because of the presence of additional eects, namely the hydrodynamic force and the Debye screening of the electrostatic inter-actions.
Measurements in ethanol were performed with the same apparatus, immediatly after the measurement in ni-trogen, so that the contact are be the same, as explained previously. During the measurement of the Casimir force, the approach velocity was chosen in order to compromise
between the hydrodynamic force FH we wanted to
mini-mize and vertical drift, limiting the time of measurement. The results presented in this paper were obtained with
100nm s−1.
In order to average the data collected from consec-utive runs, 8 data sets were acquired. First, all force curves were shifted in distance corresponding to the
sep-aration distance upon contact d0 = 31nm. Second, the
hydrodynamic force FH was subtracted from each
mea-surement independently. As the FH dependence on
dis-tance is accurately known [20], it can be safely subtracted
from the measured force. Third, to remove vertical ther-mal drift from force measurement, each force curve was
tted linearly between 300nm and 600nm. Fig. 3 (a)
shows a single force measurement in ethanol after sub-tracting the hydrodynamic force. Blue points correspond to the raw measurement, and the red line corresponds to the linear t of the thermal drift, which is then
sub-tracted from each measurement. In Fig.3 (b), we
rep-resent both a single force measurement (blue) and the average measured force (red), after subtracting the ther-mal drift and the hydrodynamic force. The remaining force shows the presence of repulsive forces at separa-tion distances larger than 60nm. These repulsive forces are attributed to the presence of ions in solution and on metallic surfaces. Then, the remaining curve
and the mean curve was tted by an exponential
func-tion A exp(−d/λD) in the distance range d > 120nm.
A and λD are adjustable parameters. In this distance
range, we assume that the Casimir force is negligible in comparison to the double-layer force (as predictions of the Lifshitz theory indicate). We obtained a Debye
length λD = (46 ± 6)nm consistent with measurements
reported by [34] and [35]. The exponential t is also
used to determine the electrostatic potential at the gold
surface ψ0 from the expression of the double-layer force
in a sphere-plane geometry: F = 4π0Rψ02/λD e−d/λD.
We evaluated the surface potential ψ0adj = (63 ± 13)mV.
After subtracting the measured double-layer force from each force measurement, the mesured Casimir force is obtained.
The measured Casimir force is presented on Fig.3(b).
The experimental data are compared to Lifshitz's theory for a gold sphere of radius R = 75µm and a gold plate separated from a distance d in ethanol. Finally the dier-ences between the theoretical predictions and the
mea-sured data are plotted in Fig.3(c). In spite of the rather
large error bars we can distinguish the two theoretical predictions: Casimir force measurements are in better agreement with Lifshitz theory based on optical proper-ties of real Au lms, presenting the same morphology as ours.
Conclusion. In conclusion, we have presented mea-surements of the Casimir force performed both in gas (nitrogen) and liquid (ethanol) environments with the same apparatus and on the same spot of the sample. The force measurements yield experimental evidence of the importance of electrostatic eects in ethanol. These eects were properly measured and subtracted, in order to determine the genuine Casimir force. Furthermore, these measurements demonstrate that the Casimir force is sensitive to changes in the optical properties of gold at distances of less than 200nm mostly in the gas en-viroment where the force is the strongest. Notably, to the best of our knowledge, this is the rst time that this in uence was measured experimentally at this range of separations. Our measurements are of signi cant interest given the fundamental implications of the Casimir force in the search of new hypothetical forces, technology appli-cations of Casimir forces for micro/nano device actuation
[1,36], and the very timely nature of our measurements
[37] .
APPENDIX
This work has been supported by the ERC contract Oute ucoop. We thank Irénée Frérot for helpful discus-sions.
∗ Present address of A.Le Cunuder :
Department of Condensed Matter Physics, University of Barcelona,
Marti i Franques, 1, 080028 Barcelona, Spain
ROUGHNESS ANALYSIS
Because the Casimir force is sensitive to optical
prop-erties of gold lms [25], we characterized carefully the
topography of the surfaces. Indeed, it is commonly ac-cepted that these properties can be taken from the hand-books tabulated data. In fact, optical properties of de-posited lms depend on the method of preparation. An
interesting study [25] reported a signi cant variation of
5 − 15%in Casimir force calculations, due to changes in
optical properties of Au lms.
After coating, the surface morphology of both the sphere and the plate was determined using a commer-cial AFM (Bruker). It is important to stress that these analysis were performed directly on the surfaces used in the measurement of the Casimir forces. The AFM
im-ages of a (1 × 1 µm2)sample of both surfaces are shown
in g.4. The roughness probability distribution of both
surfaces are well approximated by a Gaussian. In g.5a)
we plot the probability distribution of the sphere whose
rms roughness is wsph= (11.8 ± 0.8)nm, where the error
takes into account the AFM accuracy and the statisti-cal error based on a correlation length of about 50nm,
i.e. 400 statistical independent points on the (1 × 1 µm2)
measured surface.! " y b) x !" ! " Texxte a) Teyxte
FIG. 4. a - AFM image of 1 × 1 µm2 sample of the sphere
surface after the deposition of a 100nm thick gold lm. b AFM image 1 × 1 µm of the surface of the glass plate with a 100nmthick gold lm
The correlation length, estimated from the
Height-height correlation function plotted in g.5 b) is about
50nm. The rms roughness of the plate is wp =
(1.3 ± 0.2)nm.
This morphology analysis is then used in order to com-pare our force measurements with computations where optical properties are taken from real lms, with a simi-lar topography.
From morphology analysis, we also evaluated
approxi-matively the separation upon contact d0,rough. However,
6 101 100 101 102 r 102 101 100 101 102 H H r2a a = 0.94 ± 0.01
FIG. 5. a) Height distribution of the sphere roughness b) Height-height correlation function of the sphere surface in log-log scale. The height-height correlation function is de ned as: H(r) =< [h(r) − h(0)]2 >, where h(r) is the surface height. The roughness exponent α = 0.940 ± 0.001 is extracted from the slope of the linear t and the correlation lenght ξ = 50nm is determined from the intersection between the linear t and the saturation line.
precisely d0. This analysis just helps us to check that we
nd a separation distance upon contact d0,rough of the
same order of magnitude as d0 obtained from
hydrody-namic calibration. The AFM images4 indicate that the
sphere is much rougher than the plate. Consequently, the separation distance upon contact can be evaluated
as: d0,rough = d0,sph+ wp, where wp is the rms
rough-ness of the plate and d0,sph is the highest peak of the
sphere within the contact area [38]. One estimates that
on a surface of 1µm2, there is in average only one
as-perity larger than 2.8wsph and less than one with height
larger than 3wsph, which is statistically coherent with
the image of g.4a) and the tails of the distribution in
g.5. Thus, in a contact area of about 1 µm2 one
ex-pects to nd d0,sph = 2.8wsph = (33.6 ± 2.4)nm. Notice
that this value is statistically signi cant because the area involved in the force measurements are of the order of
1µm2) at d < 100nm (see ref.[24]). For the plate, the
rms roughness is wp = (1.3 ± 0.2)nm. Thus from the
topography analysis we evaluate that the maximum is
d0 < (34.9 ± 2.4)nm on an area of 1 µm2). This value
is, within error bars, statistically coherent with the hy-drodynamic calibration discussed in section 2. It is im-portant to stress that the hydrodynamic calibration is performed directly on the surfaces used in the measure-ment of the Casimir forces. Indeed, calibration were per-formed in ethanol immediatly after measurements of the Casimir force. As the liquid was introduced very car-refully in the cell after the measurement in nitrogen and as the horizontal drift of the sample is negligible, the contact are is the same during both measurements and calibration. This assumption is further justi ed a pos-teriori: our experimental curves all superimpose on top of each others and on top of the theoretical curves after
shifting the distance by the same value of d0. In contrast
the topography study is done on surfaces with the same statistical properties but not on the same position of the
contact area used in the experiment. Thus we use for do
the value obtained from the hydrodynamic calibration,
i.e. d=(31 ± 2)nm.
HYDRODYNAMIC CALIBRATION
The measured hydrodynamic force is plotted as a
func-tion of d in g.6 a) where it is compared to the
the-oretical force expressed in eq.(4) of main text. In the gure the measured values have been shifted
horizon-tally by do = 31nm which corresponds to the
separa-tion distance upon contact. In g.6 b), the plot of the
200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 2500 3000 3500 4000 4500 distance [nm] Hy drodynamic force [ pN] 0 500 1000 1500 0 0.5 1 1.5 2 2.5 3 3.5x 10 −3 distance [nm] 1/Force [N -1]
FIG. 6. a) Hydrodynamic force in ethanol measured at v = 4.742µm s−1.b) Inverse of the hydrodynamic force as a function of d.
inverse of the force as a function of d con rms that
the value of do is good since the curve crosses the
ori-gin. The slope m of this curve is in good agreement
with the theoretical value mthdetermined from eq.(4) of
main text. Speci cally we nd 1/m = 1.98 10−15Jand
1/mth= 6πηR2v = 1.84 10−15J, which are in agreement
within the error bars of do, of R and of η . Thus, from
hydrodynamic calibration, we get do= (31 ± 2)nm.
CALIBRATION OF THE CONTACT POTENTIAL DIFFERENCE
An electrostatic potential dierence Vc exists between
the sphere and the plate, even if the surfaces are coated with gold and they are both electrically grounded. In-deed, there can exist a large potential dierence between clean, grounded, metallic surfaces owing to dierences between the work functions of the materials used and the
cables used to ground the metal surfaces [28]. A small
potential dierence around ten mV is su cient to over-helm the Casimir force so the contact potential dierence has to be measured and the experiment has to be carried out with a compensating voltage present at all times.
Following a procedure described in [39], we measure
the contact potential dierence Vc between the sphere
and the plate by applying an oscillating potential V =
0 200 400 600 800 1000 −100 −95 −90 −85 −80 −75 Distance [nm]
Vc
[mV]FIG. 7. Value of the potential V2which minimizes the
excita-tion at frequency ω1. We measure a constant contact potential
dierence between 110nm et 1.1µm.
When d << R, the expression of the electrostatic force
induced by the voltage potential dierence V and Vccan
be approximated by: Fe= − π0dR d (V1cos(ω1t) + V2− Vc) 2 = −π0dR 2d [V1cos(2ω1t) + + 4V1(V2− Vc) cos(ω1t) + 2(V2− Vc)2+ V12] (8)
Because of the existence of a contact potential dierence
Vc, the system oscillates both at 2ω1 and at ω1. We
determine Vc adding a constant potential V2 until the
excitation at the frequency ω1 disappears. Indeed, when
V2= Vc, the system no longer oscillates at the frequency
ω1 (see eq.8) .
We measure the contact potential Vc as a function of
d between 1µm and 110nm. In practice, we move the
plate towards the sphere by discrete displacements. At
each separation distance dn, we measure the potential V2
which minimizes the amplitude of the oscillation at ω1.
The result of the measurement is plotted in g.7, where
we see that in our experiment Vc is constant as it is
the-oretically expected.
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