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Capillary force on a micrometric sphere trapped at a
uid interface exhibiting arbitrary curvature gradients
Christophe Blanc, Denys Fedorenko, Michel Gross, Martin In, Manouk
Abkarian, Mohamed Amine Gharbi, Jean-Baptiste Fournier, Paolo Galatola,
Maurizio Nobili
To cite this version:
Christophe Blanc, Denys Fedorenko, Michel Gross, Martin In, Manouk Abkarian, et al.. Capillary force
on a micrometric sphere trapped at a uid interface exhibiting arbitrary curvature gradients. Physical
Review Letters, American Physical Society, 2013, 111, pp.058302. �10.1103/PhysRevLett.111.058302�.
�hal-00840804�
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Capillary Force on a Micrometric Sphere Trapped at a Fluid Interface Exhibiting
Arbitrary Curvature Gradients
ArticleinPhysical Review Letters · August 2013
DOI: 10.1103/PhysRevLett.111.058302·Source: PubMed
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Capillary Force on a Micrometric Sphere Trapped at a Fluid Interface
Exhibiting Arbitrary Curvature Gradients
Christophe Blanc,
1,2
Denys Fedorenko,
1,2
Michel Gross,
1,2
Martin In,
1,2
Manouk Abkarian,
1,2
Mohamed Amine Gharbi,
1,2
Jean-Baptiste Fournier,
3
Paolo Galatola,
3
and Maurizio Nobili
1,2
1
Universite
´
Montpellier 2, Laboratoire Charles Coulomb, UMR 5521, F-34095 Montpellier Cedex 5, France
2
CNRS, Laboratoire Charles Coulomb, UMR 5521, F-34095 Montpellier Cedex 5, France
3
Universite
´
Paris Diderot, Sorbonne Paris Cite
´
, Laboratoire Matie
`
re et Syste
`
mes Complexes (MSC),
UMR 7057 CNRS, F-75205 Paris, France
(Received 27 December 2012; published 2 August 2013)
We report theoretical predictions and measurements of the capillary force acting on a spherical colloid
smaller than the capillary length that is placed on a curved fluid interface of arbitrary shape. By coupling
direct imaging and interferometry, we are able to measure the in situ colloid contact angle and to correlate
its position with respect to the interface curvature. Extremely tiny capillary forces down to femtonewtons
can be measured with this method. Measurements agree well with a theory relating the capillary force to
the gradient of Gaussian curvature and to the mean curvature of the interface prior to colloidal deposition.
Numerical calculations corroborate these results.
DOI: 10.1103/PhysRevLett.111.058302 PACS numbers: 82.70.Dd, 68.03.Cd
Strong normal restoring forces due to the surface tension
are sufficient to confine solid objects at fluid interfaces.
This trapping can be cleverly used to both address funda-
mental problems [1,2] and to create new materials [3,4]. In
many studies, the interface is planar and spatially uniform,
merely providing a 2D confinement on particles. When
curved, interfaces might play a more active role, imposing
a lateral force on the particles. Usually, the curvature is
induced by the colloids themselves. This is the case of
heavy colloids that attract each other by falling in the
gravitational well they have created [5]. When the effective
weight of the trapped particles is negligible, a nonspherical
shape [6] or the pinning of the triple line [7–9] induces a
surface deformation. Lateral forces may emerge from the
coupling with such capillary induced curvature of the
interface [10]. A spectacular demonstration of this force
is the meniscus-climbing technique of the beetle larva [11].
In the presence of a curved interface, the mechanical
equilibrium conditions at the triple line larva-water-air
impose an extra surface deflection and thus a lateral force
on the larva. This force is also sufficient to drive micron-
long cylinder self-assembly on a water-air curved interface
[12]. Despite these interesting studies, a full comparison
between experiment and theory is still lacking. Theories
are restricted to spherical colloids trapped on a minimal
surface [13] or with weak mean curvature [14], and no
effects on spherical colloids have been reported so far.
In this Letter, we combine theo ry and experiment to
address the capillary force acting on a spherical colloid
placed on a curved fluid interface. We develop a new
theoretical model able to predict this force in the general
case of interfaces with arbit rary curvature. Using a built-in
inte rferometric method coup led with particle tracking,
we me asured the femtonewton forces which control the
equilibrium position of microspheres on a curved inter-
face. We found a good agreement with our theor etical
predictions.
Let us consider a spherical colloid of radius a trapped at
an interface of tension
LV
between a liquid L
and a vapor V. At equilibrium, the contact angle between
the interface and the colloid (C) surface is everywhere
equal to the Young angle with cos ¼ð
CV
CL
Þ
[15]. The interface prior to colloid deposition is assumed
to be arbitrarily deformed with typical curvatures much
smaller than a
1
.Ifa is much smaller than the capillary
length [15], the gravitational corrections to the capillary
force can be neglected [16,17]. The tangential force f
acting on the colloid can then be expressed as the gradient
of a scalar potential W, which is expected to depend only
on the local curvature tensor field prior to colloid
deposition K
ij
ðrÞ¼n D
i
t
j
, where i, j 2f1; 2g, n is the
normal to the interface, (t
1
, t
2
) is a tangential basis, and D
i
is the covariant derivative [18]. We expand W in terms of
all the scalars that can be formed by contracting the tensors
K
jk
(order 1), K
jk
K
‘m
(order 2), D
j
D
k
K
‘m
and K
jk
K
‘m
K
np
(order 3), etc. Laplace law [15] imposes the total curvature
K
j
j
to be constant (Einstein’s summation convention used).
Then, taking into account that D
i
K
jk
¼ D
j
K
ik
[18], the
force f
i
¼D
i
W reduces to f
i
/ð þ K
‘
‘
ÞD
i
ðK
k
j
K
j
k
Þþ
Oð5Þ. In terms of the principal curvatures c
1
and c
2
of
the interface, K
‘
‘
¼ c
1
þ c
2
and K
k
j
K
j
k
¼ðK
‘
‘
Þ
2
2c
1
c
2
,
and thus
f
i
¼½ þ ðc
1
þ c
2
ÞD
i
ðc
1
c
2
ÞþOð5Þ; (1)
where the constants and depend only on , a, and .
To determine and , we calculate the interface shape
in the presence of the colloid for an arbitrary asymptotic
PRL 111, 058302 (2013)
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profile. The system free energy is F ¼ A
LV
þ
CL
A
CL
þ
CV
A
CV
þ PV
L
, with A
XY
the area of the
X-Y interface, V
L
the volume of L, and P the pressure
jump across the L-V interface (see the Supplemental
Material [19]). For small deformations of the latter,
F ’
Z
S
1 þ
1
2
ðruÞ
2
þ PuðrÞ
dr
þ
Z
S
cos
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þðrhÞ
2
q
þ P½H þ hðrÞ
dr; (2)
up to a constant. Here, uðrÞ describes in the Monge gauge
the height of the interface above a reference plane parame-
trized by r, H is the height of the center of the colloid, and
hðrÞ is the height of the colloid with respect to its center
(Fig. 1). The domain S is the projection of the upper cap
on the reference plane and
S the rest of this plane. The
equilibrium configuration is obtained by minimizing F
with respect to arbitrary variations of uðrÞ, H, and the
boundary of S, parametrized by r ¼ ðÞ in polar coor-
dinates. As a result, we obtain the Laplace law r
2
u ¼ P,
the equilibrium conditions of the contact line
1 þ
1
2
ðruÞ
2
cos
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þðrhÞ
2
q
þ
1
2
u
u
r
ðu
r
h
r
Þ¼0; (3)
and of the vertical position of the colloid
Z
2
0
1
u
u
r
þ
P
2
2
d ¼ 0; (4)
plus the matching condition u½ðÞ;¼Hþh½ðÞ;.
In the above formulas, the subscripts indicate partial
derivatives, e.g., u
r
¼ @u=@r.
To deal with the free boundary problem at the contact
line, we search for a perturbative solution uðr;Þ¼u
0
ðrÞþ
u
1
ðr;Þ, ðÞ¼
0
þ
1
ðÞ, and H¼H
0
þH
1
, where
is a small parameter and the zeroth-order solution u
0
is a
parabola (constant curvature accounting for P). Inserting
this perturbative expansion into the equilibrium equations
yields r
2
u
1
¼ 0 with, at first order in , boundary con-
ditions on the fixed, unperturbed contour r ¼
0
. The most
general solution for u
1
breaking the axial symmetry can be
expanded in multipoles as
u
1
ðr; Þ¼
X
1
n¼1
ðA
n
r
n
þ A
0
n
r
n
ÞcosðnÞ
þ
X
1
n¼1
ðB
n
r
n
þ B
0
n
r
n
ÞsinðnÞ; (5)
where
u ¼ u=
0
and
r ¼ r=
0
. Note that A
n
and B
n
characterize the asymptotic profile. Matching the boundary
conditions yields, at first order in p ¼ Pa= (i.e., for c
1
þ
c
2
a
1
), A
0
n
=A
n
¼ B
0
n
=B
n
¼ðn 1Þ=ðn þ 1ÞþOðp
2
Þ.
Integrating the stress tensor [20](I is the unit tensor)
¼
1 þ
1
2
ðruÞ
2
þ Pu
I ru ru (6)
on an arbitrary contour surrounding the colloid yields a
curvature force parallel to the reference plane
F
c
¼
6
a
4
sin
4
½1 2aðc
1
þ c
2
Þcosrðc
1
c
2
Þ; (7)
which agrees with Eq. (1). Here, rðc
1
c
2
Þ is the gradient of
the Gaussian curvature, prior to colloid deposition, at the
position where it will be placed. It is thus calculated on the
unperturbed interface and depends only on the coefficients
A
2
, A
3
, B
2
, and B
3
. Equation (7) generalizes the result of
Wu
¨
rger [13] to surfaces of arbitrary shapes. To validate
our theoretical formula (7), we consider a spherical colloid
free to slide along the z axis. Imposing a deformation
uðL; Þ¼u
2
cosð2Þþu
3
cosð3Þ far from the colloid,
we calculate numerically the resulting shape uðr; Þ of
the interface and the force f
x
exerted on the colloid [17].
We find a good agreement, as shown in Fig. 1.
The value of the expected capillary force obtained from
Eq. (7) is extremely weak. Indeed, typical values of surface
tensions at fluid-air interfaces 50 mN=m, particle
radius a ¼ 1 m ‘
c
, and gradients of Gaussian curva-
ture of the order of 1=R
3
10
12
m
3
, where a R ¼
100 m, give a force of the order of femtonewtons.
Measuring such tiny forces to test the experimental validity
of Eq. (7) is rather challenging [21,22]. We use a null force
method where the known gravity force on the particle
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
1.0×10
-7
1.1×10
-7
1.2×10
-7
1.3×10
-7
¯
P
¯
F
x
y
z
u(r)
H
h(r)
ρ(φ)
S
¯
S
FIG. 1 (color online). Normalized force
F ¼ f
x
=ðLÞ vs nor-
malized pressure
P ¼ PL= for a colloid of radius a ¼ 0:2L
with Young angle ¼ 45
. The boundary deformation corre-
sponds to u
2
¼ u
3
¼ 5 10
3
L, which yields, in the absence
of the colloid, a Gaussian curvature gradient rðc
1
c
2
Þ’6
10
4
L
3
. The solid (black) line is the exact numerical result, the
dashed (red) line is our analytical result (7), and the dash-dotted
(blue) line corresponds to the analytical result without the
pressure (i.e., total curvature c
1
þ c
2
) correction [13]. The inset
shows the parameters defining the geometry of the spherical
colloid trapped at a liquid interface.
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competes with the force F
c
to be measured. Especially
devised setup and method, combining direct imaging of the
particle and interferometry, give the capillary force down
to the femtonewton scale.
In the experiment, we use designed containers formed
by two coaxial cylinders in which the liquid filling the
gap exhibits an interface with a spontaneous large gradient
of Gaussian curvature. Using standard photolithography
procedures, we patterned rings of a negative photoresist
(SU8-2025, Microchem) on a Si wafer. After polydime-
thylsiloxane (PDMS) molding, we obtained 100 m
thick cuvettes of inner and outer radii R
1
¼ 100 m and
R
2
¼ 200 m (Fig. 2). The cuvettes are filled with mineral
oil (M3516, Sigma).
The air-oil interface is accurately characterized by
means of a custom-made phase shift interferometer (PSI)
mounted on a Leica 2500 optical microscope. The cuvettes
are fixed on the stage of the microscope placed on a
vibration-isolation table (CVI). A 633 nm monochromatic
light and a 25 Mirau objective (Nikon) are used to create
interferences between the reflected beams at the reference
mirror and at the air-oil interface. The interferometer is
first vertically aligned by maximizing the fringes width at
a horizontal air-glycerin interface. With the cuvette, a set
of interference images is collected by a CCD Sony XCD-
X710 camera during the vertical scanning of the objective
over distances of a wavelength using a nanopositioner
(Nano-F, MCL). The continuous interface profile is then
reconstructed by standard phase-unwrapping algorithms.
It follows the Young-Laplace equation P ¼ ðc
1
þ c
2
Þ
that can be worked out analytically because hydrostatic
pressure is negligible at the considered scale. Using nota-
tions of Fig. 2(b), the slope angle ’ðRÞ is given by
sin’ðRÞ¼
sin’
0
R
2
R
1
R
R
2
R
1
R
; (8)
where ’
0
is the corresponding value at the container walls,
fixed by the Young’s relation at the air-oil-PDMS triple
line. The bottom of the interface should be located at
the radius R
g
¼
ffiffiffiffiffiffiffiffiffiffiffi
R
1
R
2
p
141 m, independently of ’
0
.
The fringes are indeed centered at R
g
. Moreover, the PSI
reveals that Eq. (8) well describes the actual profile
[see Fig. 2(c)], with best-fitting values ’
0
¼ 53
5
over different samples, corresponding to the measured
contact angle of a mineral oil droplet deposited on PDMS.
Silica beads of 3:93 0:44 m diameter (Bangs
Laboratories) hav e been used. Their contact angle has been
controlled by treating their surface either with oleophilic N,N-
dimethyl-N-octadecy-3-amin opropyltrimethoxysilylchloride
(DMOAP) [23] or with oleophobic perfluorooctyltrichlor-
osilane (PFOTS). After treatment, the colloids are thor-
oughly dried and deposited individually with an air pulse
at the air-oil interface. A bead rapidly sits in the vicinity of
R
g
due to its higher density (
b
2gcm
3
compared to
o
¼ 0:84 g cm
3
for oil). This localization allows the
in situ measurement of [23] by vertical scanning inter-
ferometry (VSI) [24] achieved with the PSI setup (see the
Supplemental Material [19]). We found ¼ 0
–5
for
DMAOP treated beads and ¼ 30
2
for PFOTS ones.
Once deposited, a colloid can be moved across the inter-
face with a tiny air jet. The relaxation dynamics to its
equilibrium position is recorded at 30 fps. From image
analysis, we extract data sets of the particle center of mass.
A typical trajectory is shown in Fig. 3(a) for a 0
contact
angle bead. Starting from the vicinity of the outer radius,
the particle falls down toward R
g
, the minimum height of
the profile. This behavior is expected since only gravity
and buoyancy forces act. For oleophobic beads, a different
behavior is observed. As shown in Fig. 3(b), a bead par-
tially embedded in oil overpasses its gravity potential
minimum, climbs the slope, and finally reaches an equi-
librium radius R
eq
such than R
1
<R
eq
<R
g
. In Fig. 3(c),
different experiments on the same bead and on different
beads and cuvettes are presented. A systematic radial
deviation from the minimum height location R
g
R
eq
¼
3:6 0:9 m is observed for PFOTS treated beads [see
Fig. 3(c)]. It indicates the presence of an additional inward
radial force. The dispersion of points around R
eq
for differ-
ent cuvettes and beads is roughly the same as the one for
the different experiments repeated on the same bead. It thus
may be related more to differences in the pinning of the
triple line on the bead from one experiment to another
rather than to the small differences on beads sizes and
cuvettes.
More quantitatively, the interface profile from Eq. (8)
yields, with Eq. (7), a curvature force,
(a)
(c)
(b)
FIG. 2 (color online). Experimental setup and geometric nota-
tions in (a) 3D view and (b) side view. (c) Example of an
experimental radial profile with its best fit (plain curve) obtained
from Eq. (8) and yielding ’
0
57
. Inset: Interface region
reconstructed by PSI from which the radial profile has been
extracted.
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![FIG. 1 (color online). Normalized force F ¼ fx=ð LÞ vs normalized pressure P ¼ PL= for a colloid of radius a ¼ 0:2L with Young angle ¼ 45 . The boundary deformation corresponds to u2 ¼ u3 ¼ 5 10 3L, which yields, in the absence of the colloid, a Gaussian curvature gradient rðc1c2Þ ’ 6 10 4L 3. The solid (black) line is the exact numerical result, the dashed (red) line is our analytical result (7), and the dash-dotted (blue) line corresponds to the analytical result without the pressure (i.e., total curvature c1 þ c2) correction [13]. The inset shows the parameters defining the geometry of the spherical colloid trapped at a liquid interface.](/figures/fig-1-color-online-normalized-force-f-1-4-fx-d-lth-vs-27liu68p.webp)
![FIG. 3 (color online). Example of trajectories of beads on the air-oil interface with contact angle (a) ¼ 0 and (b) ¼ 30 . The black line is the locus of the radial minimum height of the interface (Rg). The white circle corresponds to the final equilibrium radius Req of the bead. Note also the visible thermally driven fluctuations R around Req. (c) Deviation Rg Req for various experiments made with different beads with ¼30 2 and cuvettes with h R2i error bars. (d) Typical probability distribution function vs R and best fit with a normal distribution. (e) Curvature force Fc, gravity force Fg, and the resulting total force FT vs R. Fc and Fg are plotted using ¼ 28 mNm 1 [25], 0¼53 , ¼30 , R1¼100 m, R2 ¼ 200 m, a¼2 m, and mf ¼ 5:75 10 11 g.](/figures/fig-3-color-online-example-of-trajectories-of-beads-on-the-jldzxy1h.webp)
