Langmuir 1993,9, 1995-1998
1995
Experimental Study of a Nanometric Liquid Bridge with a
Surface Force Apparatus
Jbr6me Crassous,t Elisabeth Charlaix,*Tt Hervb Gayvallet,? and Jean-Luc Loubet*
Laboratoire de Physique, Ecole Normale Superieure de Lyon, 46 Allke d'ltalie, 69007 Lyon,
France, and Laboratoire de tribologie et dynamique des syst&mes,Ecole Centrale de Lyon, B.P.
163,69131Ecully Cedex, France
Received November 2, 1992. I n Final Form: June 24, 1993
We use a surface force apparatus to measure the forces exerted between cobalt surfaces by a n-decane
liquid bridge formed by capillary condensation. The solid surfaces are wetted by the liquid and have an
average roughnessof 2nm. When the sphere moves toward the plane, the liquid bridge forma by coalescence
of two wetting films coating the surfaces. The static attractive force exerted by the liquid bridge is well
described by the macroscopictheory of capillaritydown to curvature radii of 8 nm. We present preliminary
results on the viscous damping induced by the liquid bridge.
Introduction
Consider two solid surfaces close to one another and in
contact with a vapor at pressure Pv. If the liquid phase
of the vapor wets the surfaces, a liquid bridge can form
spontaneously between the two surfaces and remain stable
even if the pressure Pv of the vapor is lower than the
pressure Pmtof the saturated vapor. This phenomenon
of capillary condensation is of great importance in confiied
geometries. It governs the adhesion properties of surfaces
as well as the mechanical properties of contact between
solids and thus plays an important role in the physics of
dispersed systems such as powders, aerosols,porous media,
soils, etc.
The equilibrium mean curvature K of the capillary
condensate obeys Kelvin's law
molecular weight. On the other hand, molecular dynamics
simulations have shown that the macroscopic theory of
capillarity should hold down to radius of curvature of the
order of some molecular ~ i z e In
. ~this
~ ~work we report
direct measurements of the forces exerted between cobalt
surfaces by a condensed liquid bridge of nanometric
curvature. The measurement of strong capillary attraction
is made possible by the enhanced mechanical stability of
the SFA developed by Tonck, Georges, and Loubet? in
the presence of a strong attractivepotential. Such stability
is provided by an appropriate feedback control of the
relative displacement of the surfaces and by the use of
rigid surfaces. Thus we are able to measure the capillary
attraction while varying continuously the distances between the surfaces. Moreover, we obtain the dynamical
properties of the liquid bridge by monitoring the force
response to a small oscillatory component of the surface
separation.
which expresses the equality of the chemical potentials in
the ideal vapor p b e at pressure Pv,and in the liquid
phase a t a pressure PLgiven by the Laplace equation
Experimental Section
Our system consists of a plane and a sphere of radius 1.435
mm. Both are made of fire-polishedglass coated with a 50 nm
thick cobalt layer deposited under vacuum. The roughness of
the surfaces,2nm peak to peak, is measured with an atomic force
microscope (Figure 1). The fluid is n-decane and has surface
tension y = 24 mN/m at 20 "C,shear viscosity q = 0.92 cP, and
molecular volume Q, = 0.322 rims. Liquid n-decane wets cobalt
at room temperature. This system has been studied previously
with the same apparatus under "saturated" conditions,i.e. when
the surfaces are immersed in a macroscopic liquid drop.6
The surfaces are mounted on the SFA which is placed in a
chamber in the presence of a desiccator (PzOa) and of a beaker
of n-decane. After some hours the system is fully stabilized at
room temperature and experimental results are reproducible.
An experimentalrun consistsof the following: the surfacesbeing
initially located about 50-100 nm apart, the sphere is moved
toward the plane at a constant velocity ranging from 0.1 to 1
nmh, to some nanometers beyond mechanical contact, then it
is withdrawn with opposite velocity to ita initial position. In
addition to this steady motion, a harmonic displacementof small
amplitude (b= 0.1 nm, frequency f = 38 Hz)is superimposed
on the sphere. The relative displacement of the sphere with
respect to the plane as well as the force exerted on the plane are
monitored as a function of time. The relative displacement is
measured by the mean of a capacitor senso+ with a precision of
0.1 nm for the 0-frequency component, and 10-2 nm for the
PL=Pv+yK
(2)
Here y is the liquid-vapor surface tension, U L the liquid
molecular volume, and K the sum of the two principal
curvatures of the liquid-vapor interface.
At room conditions the radius of curvature of the liquid
bridge is usually very small. Typical values of y and UL
are 1t2N/m and
m3; thus a vapor pressure lower
than Peatby a few percent gives an equilibrium radius of
the order of 10 nm. This typical size raises the problem
of the validity of macroscopic laws (Laplace law, Stoke's
law for viscous flow) when applied to the liquid bridge.
The surface force apparatus (SFA), which allows measurement of forces between surfaces separated by a few
angstroms, is ideally suited for experimental investigation
of liquid bridge formed by capillary condensation.
In early work Fisher and Israelachvili,' and later
Christenson,2have studied optically the equilibrium radius
of curvature of volatile liquids condensed between atomically smooth mica surfaces. They have established the
accuracy of Kelvin's law down to curvature radii of some
nanometers for liquids such as water and alkanes of small
+ &ole Normale Superieure de Lyon.
Ecole Centrale de Lyon.
(1)Fieher, L. R.;IeraelachviliJ. N. J.Colloid Interface Sci. 1980,80,
t
628.
(2)Christenson, H.K. J. Colloid Interface Sci. 1988, 121, 170.
(3) Koplii, J.; Banavar, J. R.; Willemsen, J. F. Phys. Reo. Lett. 1988,
60, 1282;Phys. Fluids A 1989,1, 781.
(4) Thomson,P. A.; Robbins, M. 0. Phys. Rev. Lett. 1989,63 (7),766.
(6)Georgee,J. M.; Milliot, S.;Loubet, J. L.; Tonck,A. To be published
in J. Chem. Phys.
0743-7463/93/2409-1995$04.00/00 1993 American Chemical Society
1996 Langmuir, Vol. 9, No. 8,1993
Letters
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h (nm)
0
-101
0 .1 .2 .3 .4 .5 .6 .7
horizontal distance (pm)
Figure 2. Plain line: the static force F d R measured in the
presence of decane vapor. The arrows show the direction of the
sphere motion. The x-axis value of the point of discontinuity
gives the thickness of the wetting films a t coalescence: 2e = 3.8
nm. On backward motion, the value of FaJR at h = 2e is 0.294
N/m, close to the theoretical value 4 r y = 0.3 N/m. Dashed line:
theoretical capillary force exerted by a liquid bridge of constant
volume V = 1.1 pm3. Inset: Expansion of the ordinate by a
factor 103. The plain line is the static attraction F , J R measured
in the presence of decane vapor, and the dashed line is the
attraction measured with a macroscopicliquid drop between the
surfaces.
I
Figure 1. The cobalt surfaces observed with an atomic forces
microscope. The top half images the roughness of a 0.5 m X 0.5
pm s uare of the surface. The gray scale spans from 0 (black)
to 2 0 1 (white). The bottom half plots the height profile of a cut
of the top half square along its first bisectrix (black straight
line). The height unit is A; the unit of the horizontal distance
is pm.
8:
harmonic component a t frequency f. The force is resolved in ita
0-frequencycomponent, referred to as "static force" Fdt, and in
its component at frequency f whose imaginary part Fi (out-ofphase with the harmonic displacement) measures the viscous
damping due to the medium between the surfaces.
At the end of the experiment, a macroscopic drop of liquid
decane is added between the sphere and the plane, and calibration
runs in "saturated" conditions are performed. These "saturated"
runs allow us to check the following pointss
(i) The attraction between the surfaces across the liquid phase
(Figure 2). The measured attraction compares well to the
theoretical formula for a van der Waals attraction: F w / R =
-Asrs/6h2, for distances h between the surfaces up to 20 nm, and
with a Hamaker constant As= = 1.1 X 1@l8J.
(ii) The elastic repulsion after mechanical contact. This
repulsion follows the Hertz law: FH a (-h)312where -h is the
sphere displacement after mechanical contact.
(iii) The viscous damping defined as D = F J h . The damping
compares very well with its theoretical value calculated within
the lubrication approximation with non-slip boundary conditions: D-l= h/127r27R2f.We find a value of 9 = 0.88 CPa t 23.5
"C in good agreement with the macroscopic viscosity of liquid
decane at this temperature, measured with a Ubbelhode viscosimeter.
Due to surface roughness all these macroscopic laws for the
saturated case are verified with slightly different origins of the
distance h between the surfaces. We find that those origins do
(6) Tonck, A.; Georges, J. M.;Loubet, J. L.J. Colloid Interface Sci.
1988,126,150. Tonck, A. Dbveloppement d'un appareil de mesum de
forcea de surface et de nanorhklogie; Thhe, Eole Centrale de Lyon,
France, 1989.
not differ by more than 1 nm. In the following, we choose as an
origin of the sphere-plane distance the point of mechanical
contact, i.e. the origindefined by Hertzlaw. With this convention
the 'viscous" origin in saturated conditions, i.e. the position of
the ideal surfaces where no-slip boundary conditions apply, is
located 0.25 nm away from the mechanical origin on the side of
the liquid phase.
Results and Discussion
The static force between the two surfaces in "unsaturated'' conditions is shown in Figure 2. The strong
attractive peak close to mechanical contact is about 500
times larger than the attraction of the surfaces immersed
in the fluid phase (Figure 2 inset). This extra attraction
corresponds to the capillary force exerted by the liquid
bridge condensed between the surfaces. There is an
important dissymmetry between the forward and backward motion of the sphere: the capillaryattraction appears
with a discontinuity as the sphere approaches the plane
but disappears smoothly as the sphere moves backward.
We interpret this dissymmetry, and the discontinuity of
the "forward" capillary attraction, as the formation of the
liquid bridge through coalescence of two wetting films
coating the surfaces. The average thickness of these
wetting films at coalescence corresponds to half of the
surface separation at the point of discontinuity of the static
force, i.e. 2e = 3.8 nm.
The capillary attraction between surfaces due to a small
liquid bridge has been calculated within the framework of
macroscopic theory of capillarity'
where p is the misymmetric radius of the bridge (seeFigure
3),z the meniscus height, 8 the contact angle, and Fsm the
surface attraction through the liquid phase. Assuming
that (i) z << p << R, (ii) the surfaces are coated with a film
of constant thickness e, (iii) the contact angle 8 is 0, which
should be the case in the presence of a wetting film,and
(iv) the surfaces attraction through the liquid phase is
negligible, which is verified in the "saturated" runs, the
capillary force reduces to
Langmuir, Vol. 9, No. 8,1993 1997
Letters
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I;(
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\ R
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I
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\
\
\
I
4=
t
I
I
Figure 3. Schematic illustrationof the geometry of the surfaces
and of the liquid bridge. R is the radius of the sphere. 2/2and
are the two principal radii of curvature of the meniscus; e is
the thickness of the wetting films. The dotted area corresponds
to the volume V of the condensate.
p
'3= 4ry (1 -) h - 2 e
(3)
R
Z
The macroscopic theory of capillarity thus predicts that,
when h = 2e, the capillary force is F,,/R = 4ry,whatever
the amount of condensed liquid. In our experiment we
find that at the point of discontinuity of the "forward"
static force, the value of the "backward" static force is
0.294 N/m, to be compared to the theoretical value 4 r y
= 0.3 N/m. Our results thus show very good agreement
with the Laplace law and the macroscopic theory of
capillarity.
Equation 3 also gives the radius of curvature 2/2 of the
meniscus as a function of the separation h between the
surfaces. In our experiment z is not constant with respect
to h, which means that the liquid bridge is not in
equilibrium with the vapor at pressure PV in the chamber.
However the evolution toward equilibrium is governed by
the decane flux from and to the liquid bridge and is
expected to be slow because of the low vapor pressure of
decane (P,t = 150Pa at chamber temperature). The decay
time of a small perturbation of z around its equilibrium
value zeq,due to mass exchange through the vapor phase,
- PvI2,
can be estimated as T = rRP,t In (Rlp)/2Diff(Peat
where Difi is the molecular diffusion coefficient of decane
vapor into air (seeAppendix). For vapor pressure Pv lower
thanPmtby a few percent, T is of the order of some minutes,
which is comparable to the length of a run. Besides, liquid
drainage through the wetting film, whose thickness is
comparable to the surface roughness, should not yield
important decane fluxes.
The capillary force may be expressed as a function of
the volume of liquid condensed in the bridge
where h' = h - 2e. One can see in Figure 2 that eq 4 with
a constant value of V fits well with the "backward" static
force in the vicinity of the contact. For large surface
separation, the experimental static force decreases more
rapidly than the theoretical one, which indicates that the
volume of the liquid bridge decreases. A t a distance h =
60 nm the liquid bridge disappears completely.
In the fit of eq 4 to the data, we assume that the film
thickness e is uniform and equal to the value measured
at coalescence e = 1.9 nm. One may fear that this value
overestimates the actual film thickness far from the
contact, since in the vicinity of coalescence the interaction
h
I
I
h(nm)
I
60
40
Figure 4. Plain line: the viscous damping D = F r / h measured
in the presence of decane vapor, to the power -1/5. Dashed line:
theoretical value of D-llS (eq 5) for a liquid bridge of volume V
= 1.1 pm3and of viscosity I ) = 23 cP. The noisy values well above
the dashed lines reflect the absence of liquid bridge.
of a film with the opposite film and solid surface should
increase its thickness. Apparently, effects of variation in
the film thickness are not important in our data. If the
predominant interactions are van der Waals forces, the
equilibrium thickness e* of the wetting film far from the
contact is given by
where A ~ L V
is the Hamaker constant associated to the
system cobalt/decane/air and zW/2 is the equilibrium
radius of curvature of the liquid bridge. The value PV is
not measured accurately; however a lower bound for e*
can be estimated from a lower bound of zeq. Using the
value z = 7.8 nm obtained when the surfaces are in contact
(the radius of curvature is then close to its minimum) and
V (As&vLv)'/~, with AVLV= 4
the approximation A ~ L =
X
5,' one finds e*mb= 1.6 nm, a lower bound which
does not differ significantly from the overestimated value
measured at coalescence.
The damping measured in unsaturated runs (Figure 4)
confirms the existence of the liquid bridge: the "forward"
viscous force is not measurable up to a point of discontinuity coincidingwith the discontinuity of the "forward"
static force and corresponding to the sudden formation of
a viscous continuous phase between the surfaces. The
"backward" damping remains finite in the range 0 < h <
60 nm, showing the presence of the liquid bridge, and
disappears around h = 60nm. A theoretical expression
for the damping associated to the bulk flow inside the
liquid bridge can be calculated within the lubrication
approximation. We neglect all effects due to the presence
of the interface, such as dynamic contact angle, and take
P L ( ~=)Pat,- 2y/z as boundary condition for the pressure
PL in the liquid phase. The "bulk" damping D is then
+ V/rR)'12+ 2eH
12~'~R'f (h'2
+ V/rR)'l2- h'
)
(5)
where the subscript H means that the distance is measured
from the "hydrodynamic" origin, i.e. the plane where noslip boundary condition is applied (this change of origin
does not modify h'). If h' << ( V / T R ) ~
eq/ 5~reduces to D-1
= hH/ 1 2 r 2 ~ R 2Le.
f , the expressionof "saturated" damping.
The opposite limit h' >> (V/rR)l12corresponds to the
(7) Israelachvili,J. N. Intermolecular and Surface Forces; Academic
Press: New York, 1985.
1998 Langmuir, Vol. 9, No. 8, 1993
plane-plane approximation
D-' = hH6/3fqp
(6)
Figure 4 shows D-lI6 as a function of h. The data show
a linear dependence and seems consistent with eq 6 at
first sight. However the value of q v 2 measured from the
slope is significantly larger than the expected value based
on the volume V derived from the capillary force and the
macroscopic viscosity of liquid decane. Moreover we find
that whatever the volume V of the liquid bridge, no
agreement can be obtained between the data and the
theoretical damping ( 5 ) by using the macroscopicviscosity.
Agreement can be obtained only with values of q significantly higher than the macroscopic viscosity (for instance
with the volume V derived from the capillary force the
best agreement is obtained with q = 23 CPas shown in
Figure 4). We do not fully understand the origin of this
excessive viscous damping. Clearly it cannot be due to an
offset of position of the no-slip plane, which would mainly
result in a translation of the data parallel to the x-axis.
Recent experimentaland theoretical works@have reported
a high increase of viscosity in fluid films of very small
thickness (1-2 nm) confined in atomically smooth solid
surfaces. However neither the scale at which those effects
are observed nor the roughness of the solid surface is
comparableto our experimentalconditions. Furthermore,
excessivedamping is not observed in 'saturated" data and
does not vanish at large surface separation in vapor data.
Thus confinement effects do not appear to be the cause
of the excessive damping. A possible explanation is that
the damping measured is not due to bulk viscous dissipation in the liquid bridge, but to the meniscus. Since the
capillary number in this experiment hardly exceeds lW,
this would imply an unusually large dependence of the
dynamic contact angle on the capillary number. Further
experiments are under way to precisely determine the
origin of the viscous damping and the role of the dynamic
contact angle.
(8) Thomson, P.A.; Greet, G. S.; Robbins, M.0.Phys.Reo. Lett. 1992,
68,3448.
(9) Hu,H.W.;Carson, G.A.; Granick, S. Phys. Reo. Lett. 1991, 66,
2758.
Letters
In summary, our results show that the static and
dynamical properties of a nanometric liquid bridge formed
by capillary condensation can be fully characterized with
a SFA. The static force is correctly predicted by the
macroscopic theory of capillaritydown to radii of curvature
of the order of some molecular size. This is in agreement
with earlier experimental work1g2and molecular dynamics
~imulations.~1~
The viscous damping is significantlyhigher
than expected. We have also been able to measure the
thickness of a wetting film. Further experiments are under
way, with improved vapor pressure control, in order to
reach more quantitative results on dynamical properties
of nanometric liquid bridges.
Acknowledgment. This work was made with the
financial support of Groupement de Recherche 936. We
thank J. F. Joanny and J. L. Barrat for helpful discussions.
Appendix: Estimation of the Meniscus Relaxation
Time Through Vapor Diffusion
Let Pv" be the decane vapor in the chamber, the
equilibrium geometry of the liquid bridge being charac=
terized by Kelvin's law: In Pv"lP,t = (Pv"- PMt)/Pmt
-2~y/kTz, = -2rO/z, and the geometrical relation pW2
= 2R(z, - h). Let Zm and Pm be the actual parameters of
the liquid bridge. Close to the meniscus, the equilibrium
with the vapor phase is reached very quickly. A stationnary
vapor pressure gradient between the liquid bridge and the
chamber is established after a time R2/DBof the order of
some seconds. Assuming a planeplane geometry, the mats
flux 4 toward the liquid bridge is related to the vapor
pressure radial gradient through the diffusion equation:
( 2 r p z m D d k n dPvldp = -4 with boundary conditions
Pv(pm)lP,t = 1- 2rO/zmand Pv(p = R) = Pv". The growth
of the liquid bridge is governed by the mass flux: 4 =
U L - ~ ~ Tdzddt.
R Z ~ Linearizing the solution around z,
gives an exponential relaxation with time constant 7 = yR
ln(Rlp,)P,J2DdP,t
- Pv")~.With DB le2cm2/s
40 nm one gets r 300 s, which is about the
and z,
time needed for the surfaces to move 30 nm in the slowest
runs.
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