Modeling capillary forces for large displacements
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Mastrangeli, M., Arutinov, G., Smits, E. C. P., & Lambert, P. (2015). Modeling capillary forces for large displacements. Microfluidics and Nanofluidics, 18(4), 695-708. https://doi.org/10.1007/s10404-014-1469-9
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10.1007/s10404-014-1469-9 Document status and date: Published: 01/01/2015 Document Version:
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R E S E A R C H P A P E R
Modeling capillary forces for large displacements
Massimo Mastrangeli• Gari Arutinov•
Edsger C. P. Smits•Pierre Lambert
Received: 19 June 2014 / Accepted: 12 August 2014 / Published online: 10 September 2014 Ó Springer-Verlag Berlin Heidelberg 2014
Abstract Originally applied to the accurate, passive
positioning of submillimetric devices, recent works proved capillary self-alignment as effective also for larger com-ponents and relatively large initial offsets. In this paper, we describe an analytic quasi-static model of 1D capillary restoring forces that generalizes existing geometrical models and extends the validity to large displacements from equilibrium. The piece-wise nature of the model accounts for contact line unpinning singularities ensuing from large perturbations of the liquid meniscus and dew-etting of the bounding surfaces. The superior accuracy of the generalized model across the extended displacement range, and particularly beyond the elastic regime as com-pared to purely elastic models, is supported by finite ele-ment simulations and recent experiele-mental evidence. Limits of the model are discussed in relation to the aspect ratio of the meniscus, contact angle hysteresis, tilting and self-alignment dynamics.
1 Introduction
Liquid bridges connecting adjacent surfaces (Broesch and
Frechette2012; Broesch et al.2013) find extensive use in
assembly (Lambert 2013) and precision
engineer-ing (Lambert2007). A common example is represented by
a liquid droplet bridging a flat (e.g., surface-mount Jacobs
et al. 2002; Knuesel and Jacobs 2010; Fukushima et al.
2012) component onto a stationary bottom surface. The position of the droplet—and hence that of the floating component—can be constrained by patterning a receptor
site onto the substrate (Mastrangeli et al. 2011). A single
state of globally minimal energy exists for this capillary
system (Berthier et al.2010), provided that the surface of
the component matches in shape and size that of the bottom site and that the volume of the droplet is sufficiently small
to avoid tilting (Scott et al. 2003; Abbasi et al. 2008). In
this equilibrium state, the component stands parallel to the substrate and its edges are aligned to those of the
site (Berthier et al. 2010). Upon formation of the vertical
liquid bridge, capillary forces (Mastrangeli et al. 2010;
Berthier et al. 2010) and torques (Takei et al. 2010;
Ber-thier et al.2010; Broesch et al.2014) act on the component.
Capillary self-alignment refers to the motion of the com-ponent toward the absorbing state across an initial lateral offset. The same capillary forces tend to restore the system into the equilibrium state against perturbations and
dis-placements (Mastrangeli et al.2011). Self-recovery of rest
position and intrinsic mechanical compliance sparked interest in droplet-based capillary systems for relevant technological applications, such as conformal joints (Patra
and Lee 1991; Lenders et al.2012; Valsamis et al.2013),
compliant handling (Lambert2007), precise registration of
components (Sato et al.2003; Sariola et al.2010) and
self-assembly (Mastrangeli et al.2011). Thereby the focus has
Electronic supplementary material The online version of this article (doi:10.1007/s10404-014-1469-9) contains supplementary material, which is available to authorized users.
M. Mastrangeli (&) P. Lambert
Department of Bio, Electro And Mechanical Systems (BEAMS), E´ cole Polytechnique de Bruxelles, Universite´ Libre de Bruxelles, Brussels, Belgium
e-mail: massimo.mastrangeli@ulb.ac.be G. Arutinov E. C. P. Smits
Holst Center/TNO, High Tech Campus 31, 5656 AE Eindhoven, The Netherlands G. Arutinov
Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
mainly been on sub-millimetric components actuated by droplets of similar size, and correspondingly on displace-ments from equilibrium of relatively small magnitude. Elastic models accurately capture the response of the capillary system to such small lateral displacements
(Ber-thier et al. 2010,2013; Mastrangeli et al. 2010; Lambert
et al.2010; Gao and Zhou2013). In this type of models, the
restoring capillary forces arise exclusively from the shear deformation of liquid interfaces pinned to bounding solid surfaces. Recently, however, capillary self-alignment was successfully demonstrated also for centimeter-sized
com-ponents (Fukushima et al. 2012; Arutinov et al. 2012)
across relatively large lateral offsets (Arutinov et al.2013,
2014). Accounting for such evidence prompts an expansion of the reach of purely elastic models through wetting arguments.
In this paper, we present a generalized quasi-static model of lateral capillary restoring forces valid for a sig-nificantly extended range of relative component displace-ments as compared to elastic models. By accounting for both liquid meniscus’ partial wetting of solid surfaces
(earlier attempted Lienemann et al.2004,2012) and
lim-ited angle hysteresis over edges, the model portraits more accurately the behavior of finitely deformable solid/liquid interfaces. For displacements larger than the elastic limit, we show that this behavior is distinctively characterized by the unpinning of triple contact lines. These wetting dis-continuities are integral to the presented model and sup-ported by recent experimental evidence (Arutinov et al.
2013,2014). We claim for the first time that the
disconti-nuities divide the capillary response of the meniscus into three sequential regimes and pose the physical justification of the piece-wise nature of the model. The proposed model coincides with the purely elastic one for small displace-ments, and it describes the system with significantly better accuracy otherwise. It thus bridges the gap with abstract
geometrical models (Lienemann et al.2004), which tend to
hold in the domain of large displacements.
An intuitive formulation of the model and its finite
element simulation are presented, respectively, in
Sects.2.1 and2.2. Comparative results are illustrated in
Sect.3, followed by a discussion of experimental support
and limitations of the model (Sect.4), and by conclusions
(Sect.5). Full derivation of the model is described in the
‘‘Appendix’’.
2 Model formulation 2.1 Analytical model
We consider the system geometry sketched in Fig.1. We
refer to the solid-bounding objects as pads, and to the
liquid bridge interchangeably as meniscus. Top (t) and bottom (b) pads have the same square shape of side length L. The meniscus has constant volume V, density q; surface tension c and height or gap h. Starting from the equilibrium
position u0¼ 0 (Fig.1a), a quasi-static horizontal
dis-placement u [ u0 of the top relative to the bottom pad is
imposed along a main orthogonal direction (Fig.1b). Null
relative tilt of the top pad is assumed for all u values considered. Lateral, front and rear sides of the meniscus, and parallel and perpendicular edges of the pads, are defined by their, respectively, parallel or perpendicular orientation with the direction of deformation (refer to
Fig.1b). The shearing perturbation u induces an
asym-metrical deformation of the meniscus. The model describes the capillary response of the system to increasing deformations.
The following simplifying model assumptions are used:
1. Quasi-static equilibrium, i.e., the system is originally
in global equilibrium and arbitrarily close to local equilibrium in every perturbed configuration. Inertial and viscous effects are therefore neglected by
assum-ing We 1 and Ca 1; respectively;
2. Constant gap hðuÞ ¼ h0¼ h; and smaller than the
capillary length Lc¼pffiffiffiffiffiffiffiffiffiffic=qg(i.e., Bo 1) to neglect
gravitational effects;
3. Ideally straight surfaces of meniscus sides (i.e.,
polyhedral liquid bridge), including negligible weight of the top pad;
4. Ideal smoothness and chemical homogeneity of solid
surfaces, i.e., null contact angle hysteresis except along edges, and smooth unpinning of triple contact lines upon dewetting;
5. No liquid overflow beyond pad edges.
Given the previous hypotheses, which define the
quasi-static model framework, contact angles h can be defined
everywhere on each solid surface (except over disconti-nuities such as edges) by the corresponding Young–Dupre´
equation (Berthier and Brakke2012):
csv csl¼ c cos h ð1Þ
where the pedexes s, l, v refer to solid, liquid and vapor phases, respectively, and * stands for t and b. Top and bottom pads are partially wetting. They have differing
interfacial energies csl and csv for generality, and all its
sides have the same surface energies. Given the contact
angles hb6¼ ht (with 0\h\p=2), we define hmin¼
minðht;hbÞ and hmax ¼ maxðht;hbÞ:
Upon relative displacement u [ 0 of the pads the meniscus deforms to accommodate the shear stress. The capillary reaction against the perturbation is mainly exerted through the front and rear sides of the meniscus. For increasing values of u, the reaction assumes one of two
sequential types depending on the degree of meniscus deformation. For each of the perpendicular sides of the meniscus, the transition between the two types of reaction is signaled by the unpinning of one contact line. At pad level, the contact line that undergoes unpinning is specifi-cally the one on the perpendicular edge subtending the smallest edge angle. The opposite edge bears no unpinning since overflow is geometrically avoided according to hypothesis 5 (and as detailed below). The elastic reaction takes place before this discontinuity. In the elastic response, the sides stretch their surfaces and incline for-ward due to edge angle hysteresis. Over the edges of the pads, the contact lines work like hinges maintaining their position fixed. Unpinning is prompted by reaching the receding value of the contact angle, here coinciding by assumption with h. Beyond this point, the shear defor-mation of the meniscus is accommodated by its unpinned
side(s) by sliding the unpinned contact line across the surface of the pad. In this sliding response, the surface area and inclination of the unpinned side(s) remain constant.
The complete response of the meniscus as function of the relative displacement u of the pads is thus divided into
three sequential regimes Rj. In the first regime R1 (full
elastic, Fig.1b), both perpendicular sides of the meniscus
enact elastic response and have the same inclination; in the
second regime R2(mixed, Fig.1c), the perpendicular sides
enact either responses; in the third and last regime R3(full
sliding, Fig.1d) both perpendicular sides are sliding. The
separation between regimes is intrinsically set by the unpinning discontinuities. The discontinuities tend to happen for the same u instead of sequentially—i.e., the second regime disappears—in the limit of equal surface energies of the pads. For each of the perpendicular sides, the elastic and sliding force components have, respectively,
Fig. 1 Geometrical sketches (upper rows, not to scale) and SE snapshots with inset views (lower rows, L¼ 1 mm; h ¼ 50 lm) of sequential regimes Rj with 0\ht\hb\p=2. a Global equilibrium,
b full elasticregime, c mixed regime, d full sliding regime. During capillary self-alignment, the regimes are traversed in the reverse order with respect to model derivation
the form (see ‘‘Appendix’’) Fel¼ cL ffiffiffiffiffiffiffiffiffiu h2þu2
p and Fsl¼
cL cos h. Hence, the capillary restoring force FðuÞ is described by the following piece-wise equation:
FðuÞ ¼ 2cL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu h2þ u2 p 0\u\u1 cLðcos hmaxþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu h2þ u2 p Þ u1\u\u2
cLðcos hmaxþ cos hminÞ u2\u\umax
8 > > > < > > > : ð2Þ
where umax is defined in ‘‘Appendix’’ and relates u to L
rather than to h. The domain boundary values:
u1¼ h cot hmax and u2¼ h cot hmin ð3Þ
can be obtained by imposing force continuity between adjacent regimes (smooth transitions, hypothesis 4). They
satisfy the geometric interpretation shown in Fig.1b, c,
respectively.
Overflow (of) avoidance (hypothesis 5) sets additional
constraints on u and h. The liquid meniscus is constrained
within the edges of top and bottom pads. Along the pad edges, the contact line is pinned and the angles formed by the meniscus with the surfaces of the pads (i.e., edge angles) can assume a multiplicity of values. Edge angle hysteresis is measured by the range of angles coexisting over the same contact line position (i.e., the canthotaxis
sector Berthier and Brakke2012, Fig.2). While the lower
limit of the edge angle coincides with the receding contact
angle hr, the higher limit hofis determined by case-specific
boundary conditions. Exceeding the higher limit of edge angles prompts overflow of the liquid bridge beyond the
edges of the pads. This evenience represents a major failure
mode for capillary self-alignment (Mastrangeli et al.2011)
and coincides with the ultimate limit of validity of the model. Meniscus confinement can be enforced chemically or topographically. In the former case, surface chemistry is tailored to make the pads more wettable than the
sur-rounding areas (Arutinov et al.2012), and hof is set by the
advancing contact angle of the surrounding area (Fig. 2b).
In latter case (Fig.2a), the canthotaxis sector is extended
by edge confinement (Tsai et al. 2007; Limatainen et al.
2013), and hof is imposed by the Gibbs’ criterion
(Mastr-angeli et al. 2010):
hof¼ hþ p / ð4Þ
/ being the slope angle of the pad’s edge. Accordingly, overflow takes place when the edge angle reaches the value of the advancing contact angle with respect to the surface of the pad’s sloped side. In the following, we consider and model the case of topographical confinement with vertical
pad sides (i.e., /¼ p=2, Fig. 2a). The analytical model for
the chemical confinement case can be similarly derived and is described in the ‘‘Appendix’’.
Consistency with the Gibbs’ criterion (Eq.4) to avoid
meniscus overflow imposes coupled constraints on the
values of hwith respect to Eq. (3). Specifically, transitions
between R1 and R2 and between R2 and R3 can take place
without overflow for corresponding ui only if:
hmin p=2 hmax R1 to R2
hmax p=2 hmin R2 to R3
ð5Þ These model boundary conditions prescribe a mutual relation between the surface energies of the pads for physical consistency. Hence, overflow avoidance implies that the full sequence of reaction regimes be conditional to
the coupled choice of h. The case of perfect pad
wetta-bility, normally assumed in elastic models, is trivially excluded in presence of edge confinement, and only pos-sible for chemical confinement. Liquid overflow due to enhanced wettability of pads with topographical edge confinement was recently evidenced in self-alignment
experiments (Arutinov et al.2014).
2.2 Finite element model
The capillary system was simulated by a quasi-static finite element numerical model in Surface Evolver (Brakke
1992) (SE) (Fig. 1). Water (Sariola et al.2010; Fukushima
et al. 2012; Arutinov et al. 2012) (c¼ 72 mN=m; q ¼
1;000 kg=m3; Lc¼ 2:7 mm) was chosen for the liquid
bridge constrained within square pads of L¼ 1 mm
(Bo¼ qgL2=c¼ 0:14), thickness of 125 lm and density of
(a)Edge confinement by topographical step.
(b)Edge confinement by wetting contrast.
Fig. 2 Meniscus confinements by edge angle hysteresis for hr6¼ ha.
For case (a) developed in the text, the quasi-static model assumes hr¼ ha¼ h and the canthotaxis spans an angular sector of p /
1;380 kg=m3. Domain constraints were used to confine the triple contact lines of the meniscus strictly in the planes and within the perimeters of the pads. The weight of the top pad (173 lg) had negligible impact on the model. The simulations proceeded by the displacement of the top pad
within the range u0\u\ulim in steps of 1 lm.
umax\ulim\L was chosen to check the limits of validity of the model within a physically realistic range of u. Values of
hju0 ¼ 25; 50 and 100 lm were simulated using
corre-sponding values of liquid volume V¼ 25; 50 and 100 nL.
They correspond to h=L aspect ratios (AR) of 1=40; 1=20 and 1=10, respectively, spanning the AR range used in
applications. Several combinations of hband htvalues were
simulated for each AR, with h f0; 10; 35; 50;
65; 80g. The case of perfect wetting (h¼ 0) was
con-sidered, in spite of its inconsistency with edge confinement
(Eq.5), as reference for the purely elastic scenario. umax
was defined for each h pair and h values according to
Eq. (12) (see ‘‘Appendix’’). Mesh refinement and geometry evolution made repeated use of the built-in Hessian of the energy function. Energy convergence was assumed to be attained at its fifth significant digit for all simulated con-figurations. The SE model complied with all assumptions
stated in Sect.2.1, except for the straight surface of the
sides of the meniscus (see Fig.1) and the constancy of
hðuÞ. After every update of u; hðuÞ was updated through a local-search routine based on Newton’s method to keep the
system in local equilibrium (Patra and Lee 1991). The
nonconstancy of hðuÞ was implemented in SE for physical consistency with (1) the timescale separation due to the dominance of vertical over lateral capillary forces for this
system (Lienemann et al. 2004), (2) the tendency of the
liquid bridge to assume the geometry locally closest to a
section of a sphere (Patra and Lee 1991; Berthier and
Brakke 2012), and (3) conservation of the meniscus
vol-ume, which upon contact line unpinning(s) induces an increase of h to partially compensate the decrease of wet pad surface(s). After setting hðuÞ, the lateral capillary force
FðuÞ was computed by the method of virtual
works (Lanczos 1970) implemented using central finite
differences.
3 Results
SE simulations evidenced the continuity and monotonicity of EðuÞ and hðuÞ, and their sensitivity to the energy of the
solid surfaces as lumped in the h pairs (see Fig.3a, b,
respectively, for h0¼ 25 lm). Higher surface energies (i.e.,
lower h) yield higher total system energies and energy gradients (i.e., forces, shown below) as well as smaller
variations in hðuÞ. As for the latter, values of h[ 0 induce
an increase of hðuÞ, whereas perfect wetting (h¼ 0)
causes a decrease.
An instance of the capillary forces predicted by the
analytical and SE models is shown in Fig.4. For this case
(h¼ 25 lm; hb¼ 65 and h
t¼ 50), all regimes are
encountered. The sequential unpinning discontinuities are seen to coincide with the sharp beginning of linear decrease
of the wet areas Aof the pads. The accuracy of the match
between proposed and SE model across all range of u can be appreciated—particularly with respect to the purely
elastic model represented by F1ðuÞ. The small deviations
0 50 100 150 200 250 0 5 10 15 20 25 30 35 u [µm] E(u)−E(0) [nJ] 0°−0° 65°−10° 65°−35° 50°−50° 65°−50° 65°−80° (a) 0 50 100 150 200 250 −10 0 10 20 30 40 u [µm] Normalized h [%] 0°−0° 65°−10° 65°−35° 50°−50° 65°−50° 65°−80° (b)
Fig. 3 SE simulations for h0¼ 25 lm ðAR ¼ 1=40) parameterized
by hb-ht pairs. a Total energy EðuÞ Eð0Þ versus u: b Normalized
gapðh h0Þ=h0versus u −140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [ µ N] 0 20 40 60 80 10090 92 94 96 98 100
Relative wet pad area [%]
SE u
1 u2 F1 F2 F3 At Ab
Fig. 4 SE and analytical predictions of FðuÞ versus u for h0¼ 25 lm
ðAR ¼ 1=40) with hb¼ 65 and ht¼ 50. Vertical lines correspond
to uiand signal contact line unpinning from pad edges by the decrease
in wet pad area (A). Analytic forces Fj, holding in respective
domains Rj, are shown across the entire domain of u for comparison.
are attributed to the nonconstancy of hðuÞ and to the hys-teresis of edge angles simulated in SE. As the gap increases with u the lateral sides of the meniscus may deform and curve, contributing elastically to the total energy and the ensuing capillary force in ways not accounted for analyti-cally. The edge angle hysteresis in SE is larger than what considered in the analytical counterpart. Such hysteresis retards the unpinning of the contact lines. Consequently,
the uipredicted by the analytical model tend to be smaller
than the simulated ones—a relatively small error over h
may produce large effects given thatouiðh;hÞoh ¼ h
sin2ðhÞ. The
force predicted, particularly for large displacements, is hence cumulatively affected by both unaccounted effects.
For instance, for the case of Fig.4, the analytical values of
u1¼ 11:7 and u2 ¼ 21 lm compare with u1¼ 13 and u2¼
24 lm obtained from SE, leading to a relative error of
about 4 % over the value of F3ðumaxÞ. Figure 5 further
exemplifies the verification of the model for other cases
with h¼ 25 lm and significant combinations of h values.
The error over F3ðuÞ remains small and bounded for larger h, while that of the purely elastic model increases. 4 Discussion
As shown by the relative errors over Fðumaxðh; hÞÞ
sum-marized in Table1, for larger AR values, the discrepancy
between analytical and SE models increases, and several effects undermine the validity of the former. As mentioned in the previous section, SE simulations show that for receding contact lines, the edge angle hysteresis tends to slightly deviate from the analytical description. This may be attributed to the finite length of the pads in the direction perpendicular to u, since the ensuing curvature of the contact line is not considered in the derivation of Gibbs’
criterion (Mastrangeli et al. 2010). Moreover, the
hypoth-esis of straight meniscus sides holds fairly well in SE
except expectedly at the pads’ corners (Fig.1b, c). This is
permitted by and consistent with the updating of hðuÞ by
timescale separation (see Sect.2.2) and, importantly, the
unpinning of the contact lines upon reaching the limits of the elastic responses. Contact line unpinning allows pre-serving the inclination of the unpinned meniscus sides independently of further increments of u—in contrast to purely elastic and perfect wetting cases. Yet the curvature and deformation of the sides of the meniscus become evidently less negligible as u and AR increase. As a result, contact line unpinning becomes less sharp and resembles unzipping, as it takes place not at once but rather gradually, starting from the extremities of the edges toward the center. This fuzzyfies the boundaries between the regimes, since
the ui can not be univocally identified. Therefore, the
model tends to loose accuracy for increasing values of AR besides of u. −150 −100 −50 0 u [µm] Force [µN] 0 20 40 60 80 10099.8 99.9 100 100.1
Relative wet pad area [%]
SE F 1 At Ab (a) −140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [µN] 0 20 40 60 80 10090 92 94 96 98 100
Relative wet pad area [%]
SE F 1 F2 F3 u2 u1 At Ab (b) −140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [µN] 0 20 40 60 80 10090 95 100
Relative wet pad area [%]
u 1 u2 SE F1 F2 F3 u1 (= u2) (c) −140 −120 −100 −80 −60 −40 −20 0 Force [µN] u [µm] 0 20 40 60 80 100 90 92 94 96 98 100
Relative wet pad area [%]
SE F
1 F2 F3 u2 u1 Ab At
(d)
Fig. 5 Capillary forces predicted for h¼ 25 lm ðAR ¼ 1=40) and various combinations of h. a Perfect wetting (no contact line
unpinning), b the three sequential regimes, first unpinning on bottom
pad, c absence of second regime for hb
(u1¼ u2¼ ud; F1ðudÞ ¼ F2ðudÞ ¼ F3ðudÞ), d the three sequential
It can additionally be argued by the Hauksbee
princi-ple (Berthier and Brakke2012) that for relative
displace-ments of the order of umax the condition of strictly
horizontal translation of the top pad (see Sect. 2.1) may
break down, as tilting of the same may intervene to acco-modate the relevant perturbation of the meniscus. Together with meniscus overflow, tilting is reportedly a significant failure mode in capillary self-alignment (Mastrangeli et al. 2011). In presence of tilting, the self-aligning process is compromised because the capillary forces are opposed by
friction between solid surfaces (Arutinov et al.2012;
Ab-basi et al.2008; Sato et al.2003). Moreover, the tilt mode
provides limited restoring torque (Berthier et al.2010), and
the system may effectively escape from the state of local energy minimum only through external agitation (Abbasi
et al.2008; Gao and Zhou2013). The incidence of tilting
can be lessened and even avoided by optimizing the
vol-ume (Scott et al. 2003; Arutinov et al. 2012) and
wet-ting (Arutinov et al. 2013) of the liquid meniscus—as
confirmed by the experimental evidence provided below and, incidentally, by the disappearance of pad tilt during
evaporation of the meniscus (Sariola et al.2010).
The analytical and numerical models, consistently with earlier instances, follow the progressive displacement of the capillary system from its state of global equilibrium opposed by restoring forces. Hereby, the work of dis-placement is used to stretch the meniscus and eventually dewet solid surfaces. Through this approach, our extended model allows to predict the parabolic regime we earlier reported in experimental high-resolution tracking of
cap-illary self-alignment dynamics (Arutinov et al.2013). The
parabolic regime ensues from a constant acceleration imparted by the relaxing liquid bridge to the top pad. The constant acceleration is consistent with the constant restoring capillary force predicted by Eq. (2) for large relative displacements.
However, in its present form, the proposed model can not entirely frame the actual experimental dynamics of capillary
self-alignment (Arutinov et al.2013). This can be illustrated
through sequential snapshots from a high-speed recording of
a capillary self-alignment experiment, shown in Fig.6. In
this realization, a transparent top pad (polyethylene
naphthalate, L = 5 mm, water contact angle ht¼ 65) was
controllably dropped onto a thin water layer (h¼ 125 lm)
precoated over the entire bottom receptor site (silicon
dioxide surface, hb¼ 10) except for its corners
(Mastran-geli et al.2011) (Fig.6a). Chemical edge confinement was
used as the bottom pad was surrounded by nonwetting areas
with water contact angle hof¼ 120(Arutinov et al.2012).
Edge confinement is exemplified elsewhere (Arutinov et al. 2014). In spite of the optimized precoating (Arutinov et al. 2012), conformal coverage of both pads by the liquid bridge is achieved along the self-alignment process only after a sequence of dynamic regimes. In the initial transient wetting
regime (Arutinov et al. 2013), the meniscus deforms and
spreads until its contact lines pin on the edges of the
pads (Berthier et al. 2011) (Fig.6b). Only then can the
capillary forces exerted by the meniscus be applied to the pads, causing their relative translation. Henceforth, the self-aligning motion starts and proceeds along a predictable spatiotemporal trajectory across the aforementioned
para-bolic regime (Fig. 6c) and subsequent underdamped
har-monic oscillations (Arutinov et al.2013) (Fig.6e). In case of
partial wetting, it is only after crossing the receptor site for the first time that the surfaces of both pads are fully wet by the
liquid bridge (Fig.6d). After transient wetting, the rear side
of the meniscus (following the definition of Fig. 1b) was
pinned throughout the process on both pads. Hence, the
mixed R2regime predicted by the model extended from the
inception of the translational motion of the top pad through
the constant acceleration regime (Fig.6c) till first crossing of
the target position. The front side of the meniscus remained unpinned, as expected from the model. The full elastic
regime R1 appeared during all subsequent overshoots
con-ducing the oscillating top pad to the final equilibrium
posi-tion in accurate alignment with the bottom site (Fig.6f).
Notably, in our consistently repeatable experiments
(Aruti-nov et al.2014), no evident tilt of the top pad was recorded
even for very large initial offsets (Arutinov et al. 2014)
(1:5 mm in the illustrated instance, i.e., about 30 % of L)— i.e., also for offsets larger than the limiting value of umax correspondingly estimated in the model. This evidence fur-ther suggests that the incidence of tilt is significantly dependent on the contingent dynamics through which the rest position is approached.
Importantly, during the actual dynamics of
self-align-ment, the reaction regimes Rj described by the proposed
model are traversed in a different, generally reverse temporal
sequence, so that R0 represents the absorbing state of the
dynamics instead of the initial state as assumed in the mod-eling. A comprehensive dynamic model of the process would therefore need to consider advancing rather than receding contact angles [dependent on line velocity (Snoeijer and
Andreotti 2013)] in the sliding regimes—together with
inertia, fluid flow and viscosity (Lambert et al.2010) and
Table 1 Relative errors for Fðumaxðh; hÞÞ
hb ht h0¼ 25 lm (%) h0¼ 50 lm (%) h0¼ 100 lm (%) 0 0 2.7 4.2 7.4 65 10 2.8 5.5 16 65 35 3.1 4.3 7 50 50 3.2 5.4 3.7 65 50 4.3 7.3 7.4 65 80 8.6 14.7 17.1
energy dissipation in the bulk at the moving contact
lines (Snoeijer and Andreotti 2013). The proposed model
provides an extended quasi-static scenario toward the
development of such as yet elusive (Lambert et al. 2010;
Arutinov et al.2013) description.
5 Conclusions
We introduced an analytic description of restoring capillary forces accounting for an extended range of lateral pertur-bations of a confined liquid bridge. The proposed model provides a simple and accurate description of the capillary system by integrating the effect of partial wetting of the bounding solid surfaces by the liquid bridge. Thanks to this, important physical and geometrical parameters of the
system—such as surface energies, pad dimensions and type of edge confinement against liquid overflow—are addi-tionally accounted for, which further distinguishes the present from earlier models. The analytic description is supported by physically consistent finite element simula-tions. The numerical simulations also highlight the physical limits of validity of the model, coherently with its stated assumptions and boundary conditions. Predictions con-cerning the effect of surface energies on meniscus
over-flow (Arutinov et al. 2014) and transiently constant
acceleration of the top pad during capillary
self-align-ment (Arutinov et al. 2013) are supported by recent
experimental evidence.
The model bridges the domain of small displacements of purely elastic models with that of large displacements. The latter is captured by computationally efficient models based gripper top pad receptor site liquid meniscus (water) non-wetting area 1 mm bottom (a)
unpinned contact line on top pad
pinned contact line top
bottom t = 0 ms
1 mm (b)
unpinned contact line
on top pad 1 mm
t = 4 ms
pinned contact line top bottom (c) 1 mm t = 7 ms (d) pinned contact line 1 mm t = 10 ms
pinned contact line top bottom (e) 1 mm t = 26 ms (f)
Fig. 6 Sequential snapshots from high-speed recording of a capillary self-alignment experiment. The transparent top square pad was released from a vacuum gripper onto a chemically patterned and shape-matching receptor site conformally precoated with water (L¼ 5 mm; h ¼ 125 lm (AR ¼ 1=40), ht¼ 65 and hb¼ 10). Top
and bottom contact lines are highlighted. Full video available in
Supporting Information. a Prealignment of top pad by vacuum gripping (no liquid bridge). b Establishment of liquid bridge and transient wetting—mixed regime. c Constant acceleration—mixed regime. d First crossing of receptor site—R0: e Overshoot—fully
on two-dimensional convolutions of the shapes of the
bounding pads (Lienemann et al.2004) (Fig.7). The
analyt-ical formulation of the model, hereby developed for identanalyt-ical square shapes of the pads, can be trivially adapted to rectan-gular pads. A generalization for polygonal pad shapes of
higher order may also be conceived (Berthier et al.2013).
Experimental benchmarking of the analytical force model may be envisioned, for instance, through the adap-tation of an earlier setup for lateral capillary force
mea-surement (Mastrangeli et al. 2010), which could notably
impose a strictly horizontal displacement of the top pad. Pivotal to the formulation of the model is the energetic description of contact line unpinning and subsequent slid-ing. Under model assumptions, this has the simple form derived from the Young–Dupre´’ equation. A similar for-mulation could be adopted to describe more realistic sce-narios involving finite contact angle hysteresis, provided substitutive descriptions of the energy and force balance at
the triple contact lines (Snoeijer and Andreotti2013).
Acknowledgments This research has been funded by the Interuni-versity Attraction Poles Program (IAP 7/38 MicroMAST) initiated by the Belgian Science Policy Office.
Appendix: Model derivation
In this section, we present the full derivation of the ana-lytical model presented in the main text.
With reference to the geometry sketched in Fig.1, we
subsume the partial wetting of the surfaces of the pads in
considering 0\ht\hb\p=2, yielding u1¼ h cot hb and
u2¼ h cot ht according to Eq. (3). The alternative case of
0\hb\ht\p=2 differs only in the sequence of contact
line unpinnings over the pads, its formulation being the
same upon mutual replacement of t with b. Figure8
pro-vides the reference geometries for the estimation of the
updated values h0and h00of hðuÞ upon transitions between
regimes under conservation of meniscus volume. The following holds under model assumptions:
1. the surface energy EðuÞ coincides with the total free
energy of the capillary system, and it is invariant under swapping of the surface energies of pads;
2. partial wettability of the surfaces of the pads
deter-mines the existence of finite relative displacements ui causing the sequential unpinning of the external contact lines (i.e., those whose vertical projection lies outside the opposite pads’ surface).
The analytical formulation of the model proceeds from the calculation of the energy EjðuÞ of the system
(Liene-mann et al. 2004) for each regime Rj determined by
sequential unpinning discontinuities. The lateral capillary force FjðuÞ and stiffness kjðuÞ of the meniscus are com-puted by subsequent partial derivatives over u of the energy function.
The energy of the global equilibrium state R0is (up to an
additional arbitrary constant):
E0 ¼ Eðu0¼ 0Þ ¼ L2ðctslþ c b slÞ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} pads þ 4V Lc |ffl{zffl} meniscus ð6Þ
For the deformed states R1 and R2:
E1¼ Eð0 u u1Þ ¼ L2ðct slþ c b slÞ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} pads þ 2V Lc |ffl{zffl} lateral þ 2cLpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2þ u2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} front & rear
¼ E0 2V Lcþ 2cL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2þ u2 p ð7Þ u linear elastic h L u2 u1 full (nonlinear) elastic mixed full sliding 2D pad shape convolution 0 umax
//
Fig. 7 Qualitative range of validity of capillary force models
(a)
(b)
Fig. 8 Sketches(not to scale) for geometrical estimation of h0and h00 under model assumptions upon transitions between adjacent regi-mes: a from R1to R2, b from R2to R3
E2¼ Eðu1 u u2Þ ¼ L2ct sl |ffl{zffl} top pad þ Lðu u1Þcb svþ L½L ðu u1Þc b sl |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} bottom pad þ 2cV L |ffl{zffl} lateral þ cL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih02þ u2 1 q |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} rear þ cLpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih02þ u2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} front ¼ E1ðu1Þ þ cL pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih02þ u2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h02þ u2 1 q þ ðcb sv c b slÞLðu u1Þ
ffi const þ cLpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2þ u2þ cLðu u1Þ cosðhbÞ
ð8Þ
using Eq. (1) and the following approximation for the
constancy of h (see Fig.8a):
h0¼ h L Luu1 2 ¼ h 1 þu u1 2L þ o2ðu u1Þ ffi h for u 2L þ u1¼ 2L þ h cot hb ð9Þ Similarly for R3: E3¼ Eðu2 u\umaxÞ ¼ 2V Lc |ffl{zffl} lateral þ cL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h002þ u2 1 q |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} rear þ cL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h002þ u2 2 q |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} front þ Lðu u1Þcb svþ L½L ðu u1Þc b sl |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} bottom pad þ Lðu u2Þct svþ L½L ðu u2Þc t sl |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} top pad
ffi E2ðu2Þ þ cLðu u2Þðcos hbþ cos htÞ
ð10Þ
using the approximation (see Fig.8b):
h00¼ h L L ðu u1þu22 Þ ffi h for u L þu1þ u2 2 ¼ L þ h 2ðcot hbþ cot htÞ ð11Þ Equation (11), more stringent than Eq. (9), sets the strict limit of validity of the model over u under the assumptions of constant V and h. This condition assumes and is
con-sistent with choices of coupled pairs of h satisfying the
condition set by Eq. (5) for overflow-less transition between adjacent capillary regimes. Equation (11) defines
umaxðh; hÞ and relates it to the pad size L rather than to h
(L h in general) as in purely elastic models. Given L ¼
1 mm; h and h, the relative errors in capillary force
esti-mates between analytical and numerical models for each of
the cases reported in Table 1 of the main text were
eval-uated for the corresponding value:
umax¼ 1 10 Lþ h 2ðcot hbþ cot htÞ : ð12Þ 0 50 100 150 200 250 0 5 10 15 20 25 30 u [µm] E(u)−E(0) [nJ] 0°−0° 65°−10° 65°−35° 50°−50° 65°−50° 65°−80° (a) 0 50 100 150 200 250 −10 −5 0 5 10 15 20 25 30 u [µm] Normalized h [%] 0°−0° 65°−10° 65°−35° 50°−50° 65°−50° 65°−80° (b)
Fig. 9 SE simulations for h0¼ 50 lm ðAR ¼ 1=20) parameterized
by hb–ht pairs. a Total energy EðuÞ Eð0Þ versus u: b Normalized
gapðh h0Þ=h0versus u −140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [ µ N] 0 50 100 150 200 250 75 80 85 90 95 100
Relative wet pad area [%]
SE F1 F2 F3 u2 u1 At Ab
Fig. 10 SE simulation and analytical fit of FðuÞ versus u for h0¼
50 lmðAR ¼ 1=20) with hb¼ 65 and ht¼ 50. The relative error
From Eqs. (7), (8) and (10) it follows, respectively: R1 F1ðuÞ ¼ oE1ðuÞ ou ¼ 2cL u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2þ u2 p k1ðuÞ ¼ oF1ðuÞ ou ¼ o2E1ðuÞ ou2 ¼ 2cL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 h2þ u2 p u 2 ðh2þ u2Þ32 ! 8 > > > > > > > > < > > > > > > > > : ð13Þ −150 −100 −50 0 u [µm] Force [µN] 0 50 100 150 200 25099.4 99.6 99.8 100
Relative wet pad area [%]
A t Ab SE F1 (a) −140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [µN] 0 50 100 150 200 25075 80 85 90 95 100
Relative wet pad area [%]
SE F 1 F2 F3 At Ab (b) −140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [µN] 0 50 100 150 200 25080 90 100
Relative wet pad area [%]
SE F 1 F2 F3 u1 At Ab (c) −140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [µN] 0 50 100 150 200 25070 80 90 100
Relative wet pad area [%]
SE F
1 F2 F3 u1 u2 At Ab
(d)
Fig. 11 Capillary forces predicted for h = 50 lm (AR = 1/20) and various combinations of hb and ht values. a Perfect wetting (no
contact line unpinning). b The three sequential regimes, first
unpinning on bottom pad. c Absence of second regime for hb¼ ht
(u1¼ u2¼ ud; F1ðudÞ ¼ F2ðudÞ ¼ F3ðudÞ). d The three sequential
regimes, first unpinning on top pad
0 50 100 150 200 250 0 5 10 15 20 25 u [µm] E(u)−E(0) [nJ] 0°−0° 65°−10° 65°−35° 50°−50° 65°−50° 65°−80° (a) 0 50 100 150 200 250 −5 0 5 10 15 20 25 u [µm] Normalized h [%] 0°−0° 65°−10° 65°−35° 50°−50° 65°−50° 65°−80° (b)
Fig. 12 SE simulations for h0¼ 50 lm ðAR ¼ 1=20) parameterized
by hb–ht pairs. a Total energy EðuÞ Eð0Þ versus u: b Normalized
gapðh h0Þ=h0versus u −140 −120 −100 −80 −60 −40 −20 0 Force [ µ N] u [µm] 0 50 100 150 200 25080 90 100
Relative wet pad area [%]
SE F
1 F2 F3 u1 At Ab
Fig. 13 SE simulation and analytical fit of FðuÞ versus u for h0¼
100 lmðAR ¼ 1=10) with hb¼ 65 and ht¼ 50. Relative error for
R2 F2ðuÞ ¼ oE2ðuÞ ou ¼ cLð u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2þ u2 p þ cos hbÞ k2ðuÞ ¼ oF2ðuÞ ou ¼ o2E2ðuÞ ou2 ¼ cLð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 h2þ u2 p u 2 ðh2þ u2Þ32 Þ ¼k1ðuÞ 2 8 > > > > > > > > > > > > < > > > > > > > > > > > > : ð14Þ R3 F3ðuÞ ¼ oE2ðuÞ ou ¼ cLðcos hbþ cos htÞ k3ðuÞ ¼ oF3ðuÞ ou ¼ o2E3ðuÞ ou2 ¼ 0 8 > > > > > < > > > > > : ð15Þ
The formulation is consistent with energy and force continuity across adjacent domains, since Eiþ1ðuiÞ ¼ EiðuiÞ
and Fiþ1ðuiÞ ¼ FiðuiÞ hold for all i ¼ 0; 1; 2. Note that for
R1j0\u\h the small displacement values F1¼ 2cL u h and
k1¼ 2cL
h of linear elastic models are recovered (Tsai et al.
2007; Mastrangeli et al.2010; Lambert et al.2010; Berthier
et al.2010). Conversely, the absence of elastic work in R3is
evidenced by the null constant value of k3. Also, ht! hb
implies a singular domain for R2as u2! u1. Particularly, for
the limiting case of full wetting of both pads—i.e., for
hb¼ ht¼ 0—u1! 1, i.e., the domain of R1 extends
indefinitely. The purely elastic regime is thus recovered,
whereby partial dewetting of the surface of the pads is not possible. In this ideal condition, further model convergence is
given by F1ju h¼ F3jhb¼ht¼0. Plots of energy, gap and
restoring force versus u for AR¼ 1=20 and 1=10 (h ¼ 50
and 100 lm, respectively) are shown in Figs.9,10,11,12,
13and14—complementing those for AR¼ 1=40 presented
in the main text.
Finally, by considering meniscus confinement within the
pads by chemical contrast (Fig.2b) rather than by
topo-graphical step, the formal derivation shown above can be adapted to account for the case of liquid bridge
over-flow (Lienemann et al.2004). Overflow is here supposed to
take place beyond the edge of the bottom pad onto an
adjacent and less wettable surface (see Fig.15). The case
of overflow beyond the top pad is energetically equivalent. We assume that the two (pad and adjacent) surfaces are at the same level and that the energetic barrier to overflow is only chemical in nature. For the less wettable surface, the validity of a specific Young–Dupre´ equation is also
assumed, yielding a contact angle hofb [ p=2 [ hb. The
overflow happens when the edge angle of the meniscus reaches the advancing value of the contact angle on the adjacent surface (hereby again assumed to coincide with its
static value hofb), prompting the unpinning of the contact
line toward the adjacent surface. Unpinning takes place for
u uofand signals the transition to a regime akin to either
the mixed (Rof
2 ) or the full sliding one (Rof3). uof can be
either larger or smaller than u1, yet not larger than u2
−140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [ µ N] 0 50 100 150 200 25099.8 99.9 100 100.1
Relative wet pad area [%]
A t A b SE F 1 (a) −140 −120 −100 −80 −60 −40 −20 0 Force [ µ N] u [µm] 0 50 100 150 200 25080 90 100
Relative wet pad area [%]
SE F 1 F 2 F 3 u 1 A t A b (b) −140 −120 −100 −80 −60 −40 −20 0 Force [ µ N] u [µm] 0 50 100 150 200 25088 90 92 94 96 98 100
Relative wet pad area [%]
A t Ab SE F1 F2 F3 u1 (=u2) (c) −140 −120 −100 −80 −60 −40 −20 0 u [µm] Force [ µ N] 0 50 100 150 200 25060 80 100
Relative wet pad area [%]
SE F
1 F2 F3 u2 u1 At Ab
(d)
Fig. 14 Capillary forces predicted for h = 100 lm (AR = 1/10) and various combinations of hb and ht values. a Perfect wetting (no
contact line unpinning). b The three sequential regimes, first
unpinning on bottom pad. c Absence of second regime for hb¼ ht
(u1¼ u2¼ ud; F1ðudÞ ¼ F2ðudÞ ¼ F3ðudÞ). d The three sequential
because in R3 the inclinations of both perpendicular sides of the meniscus remain constant. For the former case of
0\uof\u1 (Fig.15a):
Rof2 F2ofðuÞ ¼ oE of 2 ðuÞ ou ¼ cL u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2þ u2 p cos hof b k2ofðuÞ ¼ oF of 2 ðuÞ ou ¼ o2Eof2ðuÞ ou2 ¼ cL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 h2þ u2 p u 2 ðh2þ u2Þ32 ! ¼k1ðuÞ 2 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : ð16Þ
For the latter case of u1\uof (Fig.15b):
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