Lateral capillary forces of cylindrical fluid menisci : an experimental, analytical and numerical study.

Full text

Lateral capillary forces of cylindrical fluid menisci : an
experimental, analytical and numerical study.
M. Mastrangeli, J.B. Valsamis, C. Van Hoof, J.P. Celis, Pierre Lambert
To cite this version:
M. Mastrangeli, J.B. Valsamis, C. Van Hoof, J.P. Celis, Pierre Lambert. Lateral capil-
lary forces of cylindrical fluid menisci : an experimental, analytical and numerical study..
Journalof Micromechanics and Microengineering., 2010, 20 (7), pp.1-23. <10.1088/0960-
1317/20/7/075041>.<hal-00503889>
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Lateral capillary forces of cylindrical fluid menisci: an
experimental, analytical and numerical study
M Mastrangeli1,2,J-B Valsamis3,CVanHoof
1,4, J-P Celis2and
P Lambert3,5
1IMEC, Kapeldreef 75, 3001 Leuven (BE)
2MTM Department, Katholieke Universiteit Leuven, Kasteelpark Arenberg 44, 3001 Leuven
(BE)
3BEAMS Department, Université Libre de Bruxelles, Avenue F. D. Roosevelt 50, CP 165/56,
1050 Bruxelles (BE)
4ESAT Department, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, 3001 Leuven
(BE)
5Femto-ST, CNRS UMR 6174, 24 rue Savary, 25000 Becançon (FR)
E-mail: mastran@imec.be
Abstract. Self-assembly and self-alignment driven by capillary meniscus forces are presently
at the core of many important technological applications, including solder bonding in flip-chip
assembly and fluidic self-assembly for microelectronic packaging. Lateral capillary meniscus
forces were object of substantial theoretical and numerical modeling in recent years. Anyway,
these studies were unsatisfactorily supported by direct experimental investigations. In this
paper we present a comprehensive study of lateral capillary forces of cylindrical menisci, where
experimental, analytical and numerical analyses of the same physical system are compared. We
describe the conceptually simple experimental apparatus we designed to investigate lateral forces
arising from small perturbations of cilindrical liquid menisci. The apparatus allowed controlling
all physical and geometrical parameters relevant to the experiments. We then reproduce our
experimental data with a novel analytical model of lateral fluid meniscus forces and with finite
element simulations. The remarkable agreement between our experiments and models, while
confirming earlier reports, provides a solid foundation for all applications of lateral capillary
forces. Moreover, our experimental apparatus may be used as testbed for further experimental
investigations of confined fluid menisci.
Keywords: Capillarity, self-assembly, self-alignment, analytical modeling, finite-elements modeling
PACS numbers:
AMS classification scheme numbers:
Submitted to: Langmuir

2Lateral capillary forces
1. Introduction
Capillary and surface-tension-related phaenomena are ubiquitous in everyday life. Their
thermodynamic origin lies in the minimization of the free energy of physical systems containing
non-rigid interfaces [1, 2]. While being fundamental in bio-chemistry [3], capillary phaenomena are
presently at the core of many technological applications as well, such as e.g. metallic bonding
and soldering [4], microfluidics [5], switchable lenses [6], tightness systems [7] and precision
manufacturing [8].
With specific respect to electronic manufacturing and packaging, the self-aligning motion of
IC dies on top of substrates induced by the minimization of the interfacial energy of molten-solder
bumps was first exploited by flip-chip assembly techniques to achieve passive yet highly-accurate
die registration [9–12]. Capillarity is one powerful enabler of the three-dimensional deployment and
actuation of articulated, hinged microelectromechanical structures of unprecedented complexity
[13], and it is one of the physical mechanisms driving the growing class of packaging techniques based
on self-assembly [14]. Surface-tension-driven self-assembly was first exploited for the construction
of heterogeneous functional systems by the Whitesides Group [15–18]. It was later optimized and
adapted to specific part-to-substrate assembly tasks by Srinivasan [19], Böhringer’s group [20], Scott
[21], Koyanagi’s group [22, 23], Parviz’s group [24, 25] and Jacobs’ group [26, 27], to mention but
a few examples.
All aforementioned electronic manufacturing techniques share, at least partly, the same
underlying mechanism. Upon contact between the functional side (i.e. suitably pre-conditioned
to enable further processing steps) of the part to be assembled (hereby representing e.g. IC die,
microdevice, MEMS component) and the highly-energetic mating surface - normally composed by
a fluid, such as e.g. hydrocarbons [19], water-based solutions [22] or molten solders [28] - of the
corresponding binding site on the substrate, capillary forces - both perpendicular (i.e. axial) and
parallel (i.e. lateral) to the substrate - act on the part and drive it to self-align with the underlying
binding site [29]. The part, floating on the liquid meniscus, in this position achieves its rest (i.e.
minimal energy) configuration. The same capillary forces intervene to oppose any sufficiently small
displacement of the part from its rest position, and are therefore also referred to as restoring forces.
The accuracy and degree of registration between part and binding site are mainly function of the
lateral capillary forces, which in turn depend on many parameters, as evidenced in several studies
briefly reviewed below. Axial capillary meniscus forces, investigated in details in [8], are further
studied in a forthcoming publication [30].
Early theoretical investigations on lateral capillary forces aimed at predicting - both
analytically and numerically - the shapes and corresponding self-aligning performance of fluid drops
(representing e.g. molten-solder bumps) vertically-constrained by parallel plates and laterally-
confined by the geometrical patterns, with constrasting wetting properties, of the same plates [31–
33]. Finite elements, quasi-static numerical simulations - commonly performed using the powerful
freeware Surface Evolver (SE) [34] of demonstrated reliability [35] - of such archetypical physical
system - where the top plate is normally represented by a part of finite dimensions with all
translational and rotational degrees of freedom free - later confirmed and further illustrated the
dependency of lateral capillary forces on the physical properties of the fluid meniscus and on the
geometry of the confining, planar geometrical patterns [36, 37]. Briefly, the forces are proportional to
the surface tension of the fluid and inversely proportional to the height of the meniscus (also referred
to as the gap in the following). Moreover, the magnitude of the lateral forces follow the (a)symmetry
of the patterns of the confining sites; namely, they depend on the direction of the perturbation of

Lateral capillary forces 3
the meniscus (i.e. displacement of the top part) from its rest configuration as compared to the
sides of the confining planar patterns: forces arising from displacements along shorter sides are
stronger. Interestingly, Böhringer proposed a geometrical model that, while capturing many known
properties of confined capillary menisci, greatly simplified energy and force calculations [38]. Built
around the two-dimensional convolution of the patterns of the confining surfaces, the model easied
the theoretical investigations on geometry-dependent self-alignment performance of binding sites
[39, 40]. Nonetheless, being essentially two-dimensional, Böhringer’s model accurately matches
earlier results only for part displacements from the rest position larger than the meniscus height.
Thus it cannot be used to model the alignment performance when in close proximity to the rest
position - which is conversely of utmost importance in flip-chip-like assembly of parts with very small
interconnection pitches. Particularly, the model neglects the curvature of the surface of the fluid
meniscus, which particularly affects the capillary restoring forces for small meniscus perturbations
[36, 41]. Given fixed geometrical boundary conditions, the meniscus curvature is directly influenced
by the volume of the fluid: hence the need for accurate fluid volume control to achieve high process
reliability and reproducibility [42].
As compared to the extensive modeling literature, experimental investigations on lateral
capillary forces are up to now rare and overall unsatisfactory. Few works reported on the
determination of the alignment accuracy achievable between part and binding site by means of
capillary self-alignment, either optically [19, 43, 44] or analyzing assembly cross-sections by scanning
electron microscopy [45]. Best claimed figures are of the order of 1µm or lower, which are amenable
to the aforementioned packaging applications. Other researchers [46, 47] were able to estimate the
maximum adhesion force binding flat parts onto fluid drops by recording the velocity of the fluid
flow that caused the detachment of the parts in blowing tests [48]. Though these may be assumed
as reasonable estimates of the maximal lateral capillary restoring forces of the menisci, we remark
that in such conditions, where the direction of displacement of the floating part is not constrained to
be parallel to the substrate, the measured values may hardly be entirely attributed to lateral force
components. To our knowledge, only Zhang reported in literature experimental measurements and
numerical simulations of lateral capillary forces arising from the same physical system [49]. The
system was composed by a 450 µm x 250 µm flat silicon piece floating on a matching rectangular
site confining a thin fluid drop - all immersed in water. The measurements of lateral capillary
forces were performed in situ by means of a dedicated micromachined optical encoder, featuring
laser-illuminated, calibrated optical gratings and a horizontal probe that displaced the part from
its rest position in a direction parallel to the substrate. Anyway, though the proposed numerical
model show reasonable match with experimental data, no mention is to be found of e.g. surface
tension and volume of the fluid and meniscus height, which are pivotal parameters to reproduce and
eventually appreciate the numerical results. Moreover, while their optical apparatus is very elegant,
the way the physical system is set up is essentially stochastic, difficult to control and reproduce.
By the same token, only one measurement is presented.
In this paper we present a comprehensive study of lateral capillary meniscus forces where
experimental, analytical and numerical results of analyses on the same physical system are presented
and compared. We describe the conceptually-simple experimental setup that we designed to quasi-
statically measure the lateral restoring forces arising from fluid menisci of known physical and
geometrical properties confined between two parallel plates. While easy to use being macroscopic,
our setup still allows to control the position of the movable bottom plate with 1µm accuracy, and
it can resolve forces as low as about 1µN. By fixing the confining pads diameters and the meniscus
height beforehand, our setup allows for reproducible experiences. The experimental results obtained

4Lateral capillary forces
Figure 1. Schematic representation of the experimental setup used in this work. Relative
dimensions are out of scale for representation purposes.
for circular pads (i.e. cilindrical fluid menisci) are then supported with good accuracy by both
a simple analytical model, predicting restoring forces for small displacements of cilindrical fluid
menisci, and by a SE model of the same system. In view of our results, our setup may constitute a
referential testbed for further investigations on confined fluid menisci.
2. Experimental measurement of lateral capillary meniscus forces
2.1. The experimental setup
In designing our measurement apparatus, we tried to satisfy several requirements at once. 1)
We wanted it to be simple and easy to use, yet allowing precise spatial manipulation: we aimed at
macroscopic apparatus with micrometric spatial resolution; 2) the apparatus had to enable carefully-
controlled and reproducible experiences: all relevant meniscus parameters of each experiment had
to be reliably known (before or after the experiment itself); 3) we were interested in measuring
lateral forces of the same order of magnitude of those arising in the technological applications of
our interest, i.e. forces reportedly of the order of tens of µN [37]: this required both the use
of an upscaled version of the physical system described in Section 1 (to keep up with point 1)
that could still give rise to such forces, and the design of a correspondingly-sensible apparatus,
i.e. able to reliably resolve few µN at maximum; 4) we were interested in lateral capillary forces
only: the apparatus had to constrain the perturbation of the fluid meniscus to develop only in the
direction parallel to the top and bottom surfaces confining the meniscus, and it had to exclude
interferences from the axial component of the capillary force; 5) we were mainly interested in
sampling the capillary force-versus-lateral displacement characteristic of the fluid meniscus close to
its rest position: the imposed displacement had to amount to a small fraction of the height of the
menisci only. Finally, 6) the fluids to be used had to have low volatility, so to allow for quasi-static,
possibly long measurements without time-dependency of the fluid volume due to sustained fluid
evaporation.
A schematic diagram and the actual implementation of the experimental setup are shown in
Fig 1 and Fig. 2, respectively. The actual system under investigation was composed by two equal

Lateral capillary forces 5
Figure 2. a) The experimental setup used in this work. A closeup view of the inset is shown in
b), whose inset is further enlarged in Figure 4a.
glass cylinders of known dimensions (diameter D=2·R=9.4mm, thickness t=1.6mm) which
were used to vertically and laterally shape and confine the liquid menisci. The bottom cylinder
was rigidly attached to a stage - anchored to the underlying table - that could be manually moved
along all 3 translational degrees of freedom with micrometric precision (Newport M-562-XYZ). The
lateral position of the movable stage was tracked by a laser displacement sensor (Keyence LC-2440)
featuring a linear output characteristic for a displacement range of 3 mm within the interferometer’s
optimal working distance (3 cm from the laser source). The sensor’s controller unit (Keyence LC-
2400W) displayed calculated displacements (either absolute or relative to a prefixed position) in real
time, and allowed direct filtering and averaging operations on acquired signals. The top cylinder was

6Lateral capillary forces
Name Density Viscosity Surface Tension Supplier
[Kg/m3][Pa·s][N/m]
Water 1000 0.001 0.072 Tap water
Oil 5 934 0.0093 0.0201 Dow Corning DC200FLUID10
Oil 6 960 0.096 0.0209 Dow Corning DC200FLUID100
Oil 1 970 0.485 0.0211 Rhodorsil R47V500
Oil 7 971 0.971 0.0212 Dow Corning DC200FLUID1000
Oil 2 973 4.865 0.0211 Rhodorsil R47V5000
Table 1. Fluids used in the experiments.
rigidly attached to the bottom surface of a machined aluminum parallelepiped (hereby referred to as
the shuttle, of mass msh =12.748gincluding the top pad) featuring finely-smoothed surfaces, which
in turn was held by two, equal blades of certified dimensions (feeler gage in close grain high carbon
spring steel, Precision Brand) and mass mb=2·2.85ghanging from an overarching solid bridge
(not shown). Upon attachment of the coupled blades to the bridge, the sliding motion of the shuttle
(i.e. the bending motion of the blades) was constrained to take place exclusively along the single
(lateral) direction where the measurements of displacement were performed. This double-cantilever-
supported shuttle constituted the actual sensing device of the apparatus, and is hereby referred to
as the spring for conciseness. The lateral position of the shuttle was tracked by an independent,
dedicated laser displacement sensor (and control unit), identical to the aforementioned but pointing
in the opposite direction. The vertical position of the (shuttle, and consequently of the) top pad was
kept fixed throughout all the experiments, i.e. the hanging point and the lenght of the cantilevers
holding the shuttle were never changed after initial calibration (described in the next Section) in
order to consistently dispose of exactly the same sensing device in all experiences. Also, from this
fixed boundary condition it followed that: 1) the vertical distance between the glass cylinders,
i.e. the height hof the fluid menisci, could be precisely set at the beginning of each experiment
using only the controlled vertical motion of the bottom stage; and that 2) the axial components of
the capillary force were excluded from measurements, as desired, being balanced by the vincular
reactions of the rigid supports. To inspect and set the initial, relative pad alignment, the plane
parallelism of the surfaces of the pads and the initial profile of the fluid menisci, the positions of
the top and bottom pad were visually tracked by two cameras connected to PCs: an USB camera
pointed along the direction of lateral motion of the shuttle, and a high-speed camera (Photron
Fastcam SA3 120K) pointed along the direction perpendicular to the previous and intersecting
it in correspondence with the initial position of the bottom pad. The pads were illuminated by
a flat backlight (LDL-TP-83x75, with PD-3012 power supply unit, CCS) with very-uniform light
emission profile. This visual tracking setup easied the process of the alignment of the pads (see
next Section). Pictures of the menisci were taken with the high-speed camera at the beginning
(i.e. after complete meniscus trimming, described below) and at the end of every experiment to
check whether the volume of the meniscus eventually changed during the measurements. The 1024
pixels-wide pictures spanned a field of view of about 20mm, providing a resolution (and positioning
accuracy) of about 20µm.
All the experiments were performed in a laboratory environment; room temperature varied
between 25oand 28oC, monitored relative air humidity was about 38%. Different liquids were
used, whose properties are summarized in table 1.

Lateral capillary forces 7
The corresponding capillary lenght Lc=γ
ρg of the oils was about 1.485mm. In all
experiments, the heights of the tested cylindrical menisci were smaller than Lc, so that gravity could
be neglected as compared to capillary forces. For each fluid, we performed a series of experiments
with progressively smaller meniscus heights. In the first experiment of each series, the meniscus
height was calibrated to exactly 1mm by means of a ceramic slip gage (Mitutoyo) of known thickness.
The contact angle [1] of silicon oils θoil on glass surfaces is rather small (about 20 degrees). This put
a geometric constraint on the maximum allowed perturbation of the fluid meniscus (i.e. maximum
displacement of the top pad relative to the bottom pad) to avoid fluid overflow beyond the edges
of the bottom pad and consequent change of the volume of the fluid meniscus (see Section ?).
Nevertheless, the range of allowed top pad relative displacements was fully coherent with our focus
on small lateral meniscus perturbations. In some explorative experiences we also used water as
meniscus fluid (γ=72mJ/m2,Lc=2.72mm); as expected, its high volatility made our quasi-
static measurements problematic and difficult to reproduce and model (see Section 5). The fluids
were dispensed between the pads. Though not very precise, due to hardly-controllable tip pinch-off
effects, we could not dispose of any dispensing method other than manual pipetting from calibrated
pipettes. Because of this intrinsic dispensing imprecision, we calculated the actual volumes Vand
height hof the fluid menisci by offline MATLAB®-based post-processing of the pictures taken at
the beginning of the experiments. Our semi-automatic, numerical algorithm computed V,hand
Dfor each experiment (see [30] for more details). The exact calibration of the images was done
using the 1mm-thick ceramic gage, leading to a resolution of 14.2µm/pixel. The numerical volume
calculation was facilitated by the assumption of axisymmetric meniscus geometry - a constraint
that we experimentally enforced in every experience: for each dispensed fluid drop, we purposely
set the vertical position of the bottom pad in order to obtain a perfectly cylindrical meniscus
profile (which is also assumed in the analytical model, see Section 4), as judged by visual camera
inspection (see e.g. Fig. 4a). We made 3 numerical estimates for each experiment, and we discarded
all experiments for which we could not get a ratio between the average values of Vand hand their
standard deviation larger than 10. To give an order of magnitude, the uncertainty on the heights
was typically 2 pixels, i.e. about 28µm. Thus we kept the heights of the menisci always higher
than 280µm - which was not hard to do given the much larger diameter of the pads. Moreover, by
comparing the reconstructed value of Dto its measured value we could assess the accuracy of the
estimates (see e.g. Table 3).
We remark that, though our experiments focused on circular pads only, lateral capillary forces
arising from fluid menisci shaped by pads of arbitrary geometries can in principle be measured with
our setup - though the exact knowledge of the dispensed fluid volume may be harder to get for
non-axisymmetric menisci.
2.2. The working principle
The working principle of the sensing apparatus is sketched in Fig. 3, together with the local frame
of reference. The initial conditions of each experiment were defined by manually trimming the fluid
meniscus. The in-plane position of the bottom pad was set to be aligned with that of the top
pad as judged by visual camera inspection: complete alignment (i.e. vertical superposition of the
centers of both pads, with respective coordinate xand u) was reached when both, perpendicular
cameras showed alignment of the edges of both pads. Then the vertical position of the bottom
pad was manually set so to get a cylindrical profile, as previously discussed (Fig. 4a). In this
condition, both the fluid meniscus and the spring were in their rest configurations: consequently,

8Lateral capillary forces
Figure 3. The working principle of our sensing device, based on the balance between capillary
and elastic forces. Relative dimensions are out of scale for illustration purposes. (Actual examples
of rest and perturbed configurations are shown in Fig. .)
Figure 4. Experimental measurement of lateral capillary forces of a cylindrical meniscus. a)
Rest configuration. Top and bottom pad are visually aligned, and the height of the meniscus
(1.205µm) is trimmed to get a cylindrical profile. b) Perturbed configuration. After imposing a
displacement (x= 812µm) to the bottom pad, the lateral force balance between capillary and
elastic spring forces determines the equilibrium displacement of the top pad (y=x−u= 660µm
relative to the bottom pad).
both interferometers were reset to null relative displacement, to define the reference starting position
(i.e. x0=u0=0).
The measurements were then performed by imposing a known lateral displacement xto the
bottom pad, as tracked by the dedicated sensor (Fig. 4b). This lateral shift perturbed both the
fluid meniscus and the spring from their respective rest positions. As a consequence, two opposing
lateral forces acted on the top pad: the capillary force tending to restore the meniscus rest position,
and the elastic force tending to restore the shuttle rest position. The equilibrium of the lateral
forces - achieved after some settling time, dependent on the viscosity of the fluid and the velocity
of the movement of the shuttle - determined the actual displacement of the top pad: this was u
(as tracked by the dedicated sensor) with respect to the rest position of the spring, and y=x−u
relative to the bottom pad. Hence, because of lateral force balance, the restoring lateral meniscus
force corresponding to the net displacement uof the top pad relative to the bottom pad could be
calculated by multiplying the absolute displacement uof the top pad times the bending stiffness of
the spring - whose estimation is described in the next Section.
By imposing subsequent displacements to the bottom pad, we could experimentally sample

Lateral capillary forces 9
Estimation method K[N/m]
Auxiliary cantilever - analytical 1.1506
Auxiliary cantilever - dynamic 0.9323
Auxiliary cantilever - weighting 1.0563
Dynamic 0.9375
Analytic 0.9036
Table 2. Summary of the estimates of the spring’s bending stiffness.
the restoting force versus lateral displacement characteristic of our physical system. In all cases,
displacements were imposed in both directions around the initial aligned position, in order to
ascertain eventual asymmetries or histeretic phaenomena. From this curve, the stiffness of the fluid
meniscus could be obtained by numerical polynomial fitting.
2.3. Bending stiffness of the spring
We estimated the bending stiffness Kof our double-cantilever spring holding the shuttle at its
sliding extremity in 3 different ways, obtaining a total of 5 estimates. All geometrical parameters
of the spring were accurately known: beam lenght L= 282mm, thickness h=0.102mm, width
b=12.7mm, total spring mass (including both beams, shuttle and top pad) M=18.448g. Multiple
alternative estimates were motivated mainly by the uncertainty on the effective Young Modulus E
(standard assumed value: 210Gpa) and density ρ(standard assumed value: 7800kg/m3) of our steel
cantilevers, directly affecting our analytic estimates. A good agreement between all estimates was
obtained - as summarized in Table 2 and detailed here below. Nonetheless, we tended to attribute
higher confidence to the 2 fully-experimental estimates of K(defined below as K3and K4), which
both avoid the guesses on Eand ρ. Therefore, we assumed for the spring a bending stiffness equal
to the average of K3and K4,i.e. K=0.9969N/m, with a relative uncertainty of 5.96%.
We remark that such high sensitivity enabled both the spring’s desired high force resolution
and its correspondingly-high susceptibility to environmental perturbations, discussed in Section 5.
2.3.1. The auxiliary cantilever method The first estimate involved an auxiliary steel cantilever
of known dimensions (Precision Brand, length l=86.4mm, thickness h=0.102mm, width
b=12.7mm). The measurement principle exploited the lateral force balance between the cantilever
and the spring (Fig. 5). Starting from the initial rest position, common to both cantilever and
spring, a laser-tracked lateral displacement imposed on the cantilever induced a laser-tracked lateral
displacement on the spring. After determining the bending stiffness of the cantilever, the stiffness
of the spring was obtained from its force-versus-displacement curve (shown in Fig. 6) by polynomial
fitting.
We estimated the bending stiffness kof the auxiliary cantilever in 3 ways. Assuming the
standard stainless steel’s Young’s Modulus Eand densityρ, the bending stiffness of a cantilever k
is given analytically by [50]:
k=3EI
L3(1)
where Iis the cantilever’s second moment of inertia. We estimated I:
(i) Analytically, as I=bh3
12 . Inserting this in Eq.1 leads to k1=1.097N/m.

10 Lateral capillary forces
Figure 5. Experimental measurement of the stiffness of the spring by means of an auxiliary
cantilever using lateral force balance.
0 50 100 150 200 250 300 350 400 450
0
50
100
150
200
250
300
350
400
Applied force on the shuttle (uN)
Displacement of the shuttle (um)
k1
k2
k3
Figure 6. Displacement of the shuttle induced by the force applied on the shuttle by the auxiliary
cantilever. 3 estimates for the applied force are given for each displacement value, according to
the 3 estimated values of the cantilever stiffness k#(see text).
(ii) From the knowledge of the first resonance f1of the cantilever, as obtained by solving Euler’s
beam equation ([50], p. 273):
I=2πf1ρ
Eβ4
1
(2)
where β1=1.875. We measured the vibration period t=97ms of the cantilever analyzing
its laser-tracked displacements on a digital oscilloscope. Hence, from Eqs. 2 and 1 we got
k2=0.8889N/m.
We also experimentally estimated the bending stiffness of the cantilever by measuring its point
load-versus-point displacement characteristic (shown in Fig. 7) using several known loads. This
gave us a value of k3=1.0071N/m. We consider this the most reliable of our estimates of k.

Lateral capillary forces 11
0 200 400 600 800 1000 1200
0
200
400
600
800
1000
1200
Applied weight (mg)
Displacement of the shuttle (um)
Figure 7. The displacement versus applied load used to calibrate the auxiliary cantilever.
Finally, from the measured force/displacement curve described above we consequently got a
value of the bending stiffness K#for each value k#:K1=1.1506N/m,K2=0.9323N/m and
K3=1.0563N/m.
2.3.2. The dynamic method Knowing the natural oscillation frequency f1of the double-beam
spring, its stiffness K4can be directly estimated according to:
K4=4π2f2
1Meff (3)
where Meff is the effective spring mass, including the mass of the shuttle and the kinetic energy-
averaged mass of the cantilevers (according to Rayleigh method; see[50], p. 23, and Appendix A
for details). The oscilloscope-measured natural frequency of the spring was f1=1.266hz which
through Eq. 3 led to K4=0.9375N/m.
2.3.3. The analytic method Finally, a fifth estimate of Kwas calculated fully analytically. We
assumed that K5had 2 components: 1) the mechanical stiffness of 2 parallel, coupled cantilevers -
with their unclamped extremities constrained by the shuttle to slide along a direction perpendicular
to the cantilevers - given by material strength theory; and 2) a component due to the gravitational
potential energy, which we converted into a so-called gravitational stiffness.
The mechanical component was obtained from:
Kmech =2·12EI
L3(4)
We estimated the gravitational stiffness as (see Appendix B for details):
Kgrav =6g
5L(msh +mb
2)(5)
where gis the acceleration of gravity. All parameters being known, we got a value of
K5=0.9036/m.

12 Lateral capillary forces
Figure 8. The Surface Evolver model of the fluid meniscus, for the case of experiment 2 defined
in Table 3. The undeflected and maximal perturbation configurations (top pad displacement:
273µm) are shown.
3. Finite element model
We modeled our experiments on cylindrical fluid menisci with Surface Evolver [34] (Fig. 8). The
code was adapted from the column example wrote by SE’s developer K. Brakke‡. All geometrical
(radius of the pads R, meniscus height and volume as obtained by image post-processing for each
experiment) and physical (fluid density and surface tension) parameters reproduced those of the
experiments. We imposed the pinning of the fluid triple contact-line along all the circular edge
of both pads - a condition that we enforced at the beginning of each experiment, as already
said, and that was satisfied for all measurements of small meniscus perturbations, as judged by
visual inspection. The capillary forces were calculated by the method of virtual works [50] using
central differences. To define a reliable degree of mesh refinement, we tested our model until we
got a satisfactory match against the benchmark given by the axial force produced by a perfectly
cylindrical fluid meniscus, which is analytically given by F=−πγR. In simulating the force-vs.-
displacement curve of few of our experiences, we input the same relative displacements of the top
relative to the bottom pad that were measured in the experiments. We finally extrapolated the
simulated lateral stiffness of the fluid meniscus by polynomial fitting of the curve.
4. Analytic model
We developed an analytical model to estimate the lateral stiffness of a cylindrical meniscus confined
between two circular pads. Perfectly-cilindrical meniscus profiles were assumed for analytical
closed-form tractability, though such assumption is ideal and not always satisfied in experimental
conditions.
We firstly computed the lateral area of a tilted cylinder of height h, radius Rand relative
shift between pads u(see Fig. 9 for the definition of the geometrical parameters). In this
configuration, the cylinder axis is not perpendicular to both circles but inclined with an angle
αgiven by tan α=u/h.
The equation of this cylinder is given by:
S≡¯
OP =z¯
1z+u(z)¯
1x+R¯
1r(6)
=(z
hu+Rcos θ)¯
1x+(Rsin θ)¯
1y+z¯
1z(7)
Computing the area element dS as:
dS =¯
N(8)
‡The code is available at http://www.susqu.edu/brakke/evolver/html/column.htm

Lateral capillary forces 13
x
z
u
u(z)
z1z
u(z)1x
r1r
O
P
S
h
R
Figure 9. Geometry of a fluid meniscus confined between circular parallel pads. h, radius R.
The offset bewteen both circular pads is u.
with ¯
N=∂¯
S
∂θ ×∂¯
S
∂z , we finally find:
dS =R1+cos
2θu2
h2(9)
The lateral area is consequently equal to:
S=R
h
ˆ
0
dz
2π
ˆ
0
1+cos
2θu2
h2dθ (10)
Using the well-known approximation (1 + x)n≈1+nx for small x, we replace the square root
by 1+1
2
u2
h2cos2θ, which finally leads to:
S≈2πRh(1 + u2
4h2)(11)
Since the total (variable) energy of the system is here equal to:
E=γS (12)
(where γis the surface tension), the lateral restoring force is equal to:
F=−δE
δu =−πRγ u
h(13)
which correspond to a constant stiffness kgiven by:
k=δF
δu =πRγ
h(14)
This formulation was benchmarked using our Surface Evolver model, in the case of r=50µm,
h=70µm,γ=0.325N/m and a volume of liquid given by V=πr2h. The comparison is plotted
in figure 10. Good agreement between the models was achieved for small relative shifts (i.e. small
u/h values) - which are coherent with those induced in our experiments.

14 Lateral capillary forces
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
1
2
3
4
5
6x 10−5
Ratio Lateral Shift / Meniscus height (u/h) (no unit)
Lateral Force (N)
Approx
Surface Evolver
Figure 10. Comparison of analytical and numerical (SE) results for the benchmark configuration
withR= 50µm,h= 70µm and γ=0.325N/m.
5. Results and discussion
When performing the experiments, the magnitude of the successive displacements along the curve
was empirically determined as a compromise between the competing needs to accurately sample
the curves and to avoid as much as possible the effects of environmental noise. The spring was
indeed as sensible as to be clearly perturbed by the movement of the surrounding air. This
was the main source of noise, together with the vibration of the movable stage (induced by floor
vibrations, though partly-attenuated by an absorbing plastic layer set underneath the apparatus).
As a consequence, the spring could eventually undergo stochastic swinging movements as large
as few tens of micrometers before settling to the equilibrium position imposed by the boundary
conditions. To cope with this: 1) we spaced the successive positions of the bottom pad 50µm apart;
2) we moved the bottom pad pretty slowly in between the prefixed positions to avoid inducing
excessive air and fluid flows, and we waited up to several tens of seconds after each prefixed position
was reached to let the transient phaenomena (in the air and in the meniscus) estinguish; and 3) we
applied the maximum number of measurement averages (131072) on the incoming signals available
in the units controlling both laser sensors, to cancel out fluctuations and thus achieve the maximal
force resolution possible. On the other hand, such relatively wide spacing between sampling point,
as compared to the full range of imparted displacements, made no harm to faithfully reconstruct
the desired curves: close to the origin, the behavior of the fluid menisci was indeed expected to be
linear [36, 37] - as we experimentally confirmed it to be a posteriori.
Table 3 and Fig. 11 show a summary of results of our investigations. We performed a total
of 34 experiments; however, we report for comparison only those experiments for which 1) no fluid
overflow nor sensible evaporation took place, and 2) the estimates of volumes and heights of the
menisci were accurate (according to criteris discussed in Section 2.1). The lateral stiffness of several
cylindrical fluid menisci, as resulting from experiments and both the analytic and numerical models,
are thereby shown together with the physical and geometrical boundary conditions of each case and
the relative errors in the estimates. As an example, Fig. 12 shows the specific capillary force-versus-

Lateral capillary forces 15
Experiment Geometry Fluid K[N/m] Modeling error
height: 1.138mm Exp: 0.2466
1 volume: 85.6nL Oil 2 SE: 0.2415 2,07%
diameter: 9.186mm ±2.28% An: 0.2797 13.42%
height: 1mm Exp: 0.2827
2 volume: 74.2nL Oil 6 SE: 0.294 3.99%
diameter: 9.63mm ±2.45% An: 0.3152 11.48%
height: 0.852mm Exp: 0.3392
3 volume: 57.1nL Oil 6 SE: 0.3456 1.89%
diameter: 9.41mm ±0.11% An: 0.3698 9.01%
h: 1mm Exp: 0.2407
4 V: 65.5nL Oil 2 SE: 0.2648 10.01%
D: 9.29mm ±1.21% An: 0.3182 32.19%
h: 0.906mm Exp: 0.2962
5 V: 54.2nL Oil 2 SE: 0.3285 10.90%
D: 9.15mm ±2.64% An: 0.351 18.50%
h: 0.720mm Exp: 0.3485
6 V: 47.3nL Oil 2 SE: 0.3989 14.46%
D: 9.44mm ±0.42% An: 0.4417 26.75%
Table 3. Summary of results (Exp = experimental, SE = Surface Evolver, An = analytical).
The relative errors of the the SE and An models compared to the experimental estimates are
shown in the last column. The relative error in the reconstruction of the pad diameter by the
image post-processing algorithm was used to assess the accuracy of the estimates on meniscus
heights and volumes.
Figure 11. Comparison of lateral stiffness of 6 measured cylincrical menisci (see Table 3), as
resulting from experimental data and numerical and analytical models.
lateral displacement characteristic relative to the second experiment reported in Table 3, with both
experimental and modeling data.
The experimental results confirm that lateral capillary forces arising from cylindrical menisci
of lower height (which, because the procedure we adopted, means correspondingly lower volumes)

16 Lateral capillary forces
0 50 100 150 200 250 300
0
10
20
30
40
50
60
70
80
90
Relative pad displacement (um)
Restoring force (uN)
Figure 12. Experimental and numerical (SE) results for experiment 2 in Table 3.
are proportionally larger. Furthermore, the general trend apparent from comparison of the results
is that both SE and analytical models tend to overestimate the meniscus stiffness, with the SE
estimate closer to the experimental values. Also, the relative errors of analytical estimates (up to
32%, with an average of 18.6%) are larger than those of the numerical model (up to 14%, with an
average of 7.2%). We attribute this to several factors and sources of errors.
5.1. Sources of error
The relative error on the reconstructed value of pad diameters D(shown in the last column of Table
3) can be assumed to be an indication of the error on the estimates of volumes and heights of the
menisci. These errors in turn directly affect the geometry of the menisci, and thus both models’
estimates of their lateral stiffness. Indeed, given the relative errors δh and δR on the estimates of
hand R, respectively, the relative error on the volume of a cylinder is:
δV
V=δh +2·δR (15)
which equals 3·δR assuming equal relative errors for hand R. Considering e.g. the case of
experiment 2, with reference parameter and error values shown in Table 3, the effects of relative
errors on h,Dand Vgive a relative error on the simulated versus experimental values of meniscus
stiffness of 8.6% (augmented case) and 7.62% (dimished case), respectively. Thus, this and similar
error propagation analyses make the relative error between SE data and real data more plausible,
and may partly explain it. By the same token, we also remark that the relative uncertainty on the
experimental value of the stiffness of the spring (∼6%) is by itself close to the average relative error
of the SE model compared to experiments (7.2%).
Relative errors on estimates of Ddo also affect, though to a lesser extent, the estimates for k
resulting from the analytical model. However, with this regard we can further assume a posteriori
that the axysimmetric meniscus geometries we tried to enforce by visual inspection at the beginning
of each experiment were still not perfectly cilindrical; we think this may be the main reason of the
systematic analytical overestimates of the stiffnesses of the menisci.

Lateral capillary forces 17
0 200 400 600 800 1000 1200 1400 1600 1800
0
50
100
150
200
250
Displacement of bottom pad (um)
Displacement of top pad (um)
Figure 13. Experimental top pad-versus-bottom pad (i.e. xTvs. xB)displacement curve in
presence of hysteresis (circles: forward scan (increasing displacements), crosses: backward scan).
The large top pad displacement induced the fluid overflow over the edge of the bottom pad,
changing the volume and the profile of the meniscus, and consequently the magnitude of the
restoring force.
Furthermore, in both models we used the values of fluids’ surface tensions γgiven by the
providers. It cannot be excluded that adsorbtion of contaminants from air during the experiments
might have made the actual values different from the nominal ones, especially for long experiments.
A change in γwould proportionally affect the restoring forces, thus the stiffnesses of the menisci.
In view of all these plausible sources of errors, we consider the matching between our models
and experiments to be satisfying.
5.2. Hysteresis
Fig. 13 shows the results of one early experiment, where relatively-large top pad displacements were
induced on a oil meniscus. Comparing the forward and the backward curves, a clearly hysteretic
behaviour was seen. This was due to the overflow of the fluid beyond the edge of the bottom pad.
This happened when the angle between the surface of the meniscus on its advancing side and the
surface of the bottom pad was larger than
θmax =θ+ 180o−φ(16)
φbeing the angle between the bottom pad surface and its side surface (equal to 90oin our
case) [51]. Beyond this value of that angle, the fluid was no longer confined by the rim of the
pad and it wet the vertical side of it, overflowing (Fig. 14a). The overflow changed the residual
amount of fluid in the meniscus. The profile of the meniscus during the backward movement was
thus different compared to the forward movement, hence the difference in the resulting restoring
force. Depending on θand the actual height hof the meniscus, fluid overflow could be avoided by
keeping the displacement of the top relative to the bottom pad below ymax =h·tan(θ)(Fig. 14b).
For a typical case with h=1mm and θ=20
o, we get a limit value of 364µm.
We incidentally observe that, thanks to 1) the aforementioned geometrical relation (Eq. 16),
and 2) the possibility to exactly track in real time the value of the relative displacement between
top and bottom pad, our setup directly enables an alternative method (which we called the overflow

18 Lateral capillary forces
Figure 14. Hysteresis by large displacement of the top relative to the bottom pad (a). Beyond a
geometrically-predictable value of displacement (i.e. of the advancing angle θ+ 180 −φbetween
the fluid and the bottom pad, as shown in b) the fluid is no longer confined on top of the bottom
pad surface, and it overflows. The volume of the residual fluid confined between the pads is
decreased, changing the profile of the meniscus. Triple contact-line unpinning along the edge of
the top pad is also seen.
0 100 200 300 400 500 600 700 800
0
50
100
150
200
250
300
Displacement of bottom pad (um)
Displacement of top pad (um)
onward
backward
Figure 15. Hysteresis due to fluid evaporation during experimental measurements of relative
pads displacements. A time-varying meniscus volume affects the restoring forces during onward
and backward displacements.
method) to experimentally determine the contact angle of liquids on flat surfaces - eventually using
non-circular pads. This novel method may be particularly useful for estimating very-small CAs, as
they are “magnified” by the additional value 180o−φdue to the constrained geometry of the setup.
Interestingly, for this collateral application no thorough calibration of the apparatus is necessary,
apart from the synchronization of the laser and visual tracking system.
Hysteretical results were also seen when using water as fluid even applying only small
displacements (see e.g. Fig 15). In this case the source of hysteresis was the rapid evaporation
of water - that evidently affected the volume and thus the curvature of the meniscus - and/or the
higher susceptibility of the water surface to adsoption of surfactants from air, which sensibly affect
its surface tension.

Lateral capillary forces 19
6. Conclusions and future work
In this paper we presented a comprehensive study of lateral capillary forces arising from small
perturbances of cylindrical menisci. We experimentally measured the restoring forces by means
of a sensing apparatus of our design, operating on the principle of lateral force balance between
capillary and elastic forces. We also presented a novel analytical model describing in closed-form
the lateral stiffness of cylindrical menisci. The predictions of this model, together with those of a
Surface Evolver model of the experimental system, showed a good match with the experimentally-
measured values of the stiffnesses. We also discussed the sources of noise, error and hysteresis
possibly affecting our sensing apparatus and/or intrinsic to such quasi-static type of experiments.
The possibility of tracking in real time and with micrometric accuracy the relative
displacements of top and bottom pads of our system lets our apparatus enable the direct
measurement of the contact angle of liquids on top of flat surfaces. This novel method exploits
the sudden overflow of liquid over the edge of the confining bottom surface for sufficiently large
perturbations of the meniscus profile. The validation of such overflow method, which may be useful
particularly to measure rather small liquid contact angles, is currently being pursued.
Finally, our experimental apparatus could constitute a reference testbed to further investigate
lateral capillary forces, arising from fluid menisci of even arbitrary profiles and shapes. The meniscus
geometry can indeed be defined by the geometry of the confining pads - besides the other parameters
discussed earlier. Scaling properties may be also investigated, as well as the self-alignment dynamics
enabled by capillary forces, which is fundamental to many present-day technological applications
and indeed object of a forthcoming publication of our research group [Lambert’s dynamic paper].
Appendix A. The effective spring mass
The mass of the spring’s 2 cantilever (mb=5.7g) was not negligible compared to that of the shuttle
and top pad (ms=12.778g). Therefore, in the dynamic estimation of the spring’s stiffness we
introduced an equivalent mass for both beams meq, which would have the same kinetic energy as
the actual cantilevers for the same shuttle velocity vaccording to:
1
2meqv2=2·1
2ˆL
0
v2(z)dm=2·1
2λˆL
0
v2(ξ)dξ (A.1)
where dm=λdξ, and λhas the dimension of mass per unit length. The velocity v(z)of
each cantilever element located at a distance zfrom the clamped extremity was assumed to be
proportional to its displacement computed by material strength theory:
v(z)=q(z)
uv(A.2)
where the element q(z)is given by:
q(z)= F
EI Lz2
4−z3
6(A.3)
and u=q(L). Using Eqs. A.3, A.2, and 4 we get:
v2(z)=v29z4
L4−12z5
L5+4z6
L6(A.4)

20 REFERENCES
which, inserted in Eq. A.1, leads to:
meq =13
35mb(A.5)
Finally, the effective spring mass Mef f is given by:
Meff =msh +meq =14.8171g(A.6)
Appendix B. The gravitational stiffness
The gravitational component of the spring’s stiffness arises from the fact that an horizontal
displacement uof the shuttle is concurrent to a vertical parasitic motion pgiven by Henein ([52],
formula 5.13) as:
p≈3u2
5L(B.1)
Considering that the shuttle undergoes a pupward displacement while each beam’s mass center
undergoes a p/2vertical displacement, the gravitation stiffness Kgrav is defined as follows (mbis
the mass of the 2 cantilevers):
1
2Kgravu2=mshgp +mbgp
2(B.2)
which together with Eq. B.1 leads to Eq. 5.
Acknowledgments
This work was supported by the EU project HYDROMEL. The authors thank Bruno Tartini and
Jean-Salvatore Mele for their valuable help in manufacturing.
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