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Electrowetting on functional fibers - Supplementary Informations
R. Dufour,a,bA. Dibao-Dina,a,bM. Harnois,a,bX. Tao,a,cC. Dufour,a,bR. Boukherroub,a,dV. Seneza,b
and V. Thomy∗a,b
1 Movies
•Movie M1 shows EWOD actuation of a 1
µ
L water droplet for Vrms = 48 and 96 V.
•Movie M2 shows EWOD actuation of a 1
µ
L water droplet for Vrms = 160 V. At this hight voltage,
hydrodynamic instabilities are observed (emission of satellite droplets).
•Movie M3 show the operation of an EWOD controlled fluidic diode, consisting in a junction between
two fibers. Actuation sequence is off-on-off-off-on-on.
2 Contact angle and drop volume measurement
2.1 Contact angles measurement
Fig. 1 SEM observation of a hanging NOA droplet. Right image depicts the continuous variation of liquid-air interface from a microscopic
equilibrium angle to the macroscopic shape.
Measurement of contact angles of clam-shell drops hanging on horizontal fibers is not straightforward
compared to classical studies on smooth or textured surfaces.1,2 Because of the complex drop shape, the
interface profile continuously varies from a microscopic contact angle to the macroscopic drop profile.
aUniversity Lille Nord de France, Villeneuve d’Ascq, France
bInstitute of Electronics, Microelectronics and Nanotechnology (IEMN), UMR CNRS 8520, Villeneuve d’Ascq, France; E-mail: vincent.thomy@iemn.univ-
lille1.fr
cLaboratory for Textile Material Engineering (GEMTEX), National Graduate School of Textile Engineering (ENSAIT), 59056 Roubaix, France
dInterdisciplinary Research Institute (IRI), USR CNRS 3078, Parc de la Haute Borne, Avenue de Halley, 59658 Villeneuve d’Ascq, France
∗Corresponding author: vincent.thomy@iemn.univ-lille1.fr
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Accurate evaluation of
θ
is particularly tricky for wetting liquid as depicted in figure 1 for a hanging NOA
droplet. Low surface tension of NOA (≈40mN/m) results in
θ
≈40◦and a very high magnification is
necessary to capture the microscopic angle. The distortion is less significant as far as hydrophobic fibers
are concerned. Indeed as shown in figure 2, the local apparent contact angle can be properly measured
within a 200
µ
mwindows around the contact line, the standard error over all measurement is ≈ ±2◦.
Fig. 2 Measurement of contact angles for water drops on fibers under EWOD actuation for VRM S = 0, 48, 96 and 160 V. Magnifications show
the measured contact angles.
Another usual error in measured contact angle results from a displacement of the drop along the axis of
the camera (i.e. rotation around the fiber). For that reason, it is necessary to use a second camera recording
a lateral view of the drop. Computing the gravity center displacement during the experiment enables to
make sure that the drop only moves upward.
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2.2 Drop volume estimation
Concerning the measurement of maximal drop volumes (Ωc), there is no direct method to compute the vol-
ume from side view of barrel-shaped drops. Consequently, the volumes were average assuming a rotational
symmetry and computed with DSA3 drop shape analysis software. The so-called ‘pendant drop’ method
was used to capture the drop shape and compute Ω. The accuracy of this approximation was estimated as
follow : (i) Drops of volume Ω≤Ωcwere simulated with Surface Evolver for a fiber radius r=117
µ
m
and 80 <
θ
<120. (ii) Side-views were created from simulation results and loaded in DSA3 software. (iii)
Drop volume was computed using rotational symmetry approximation and compared to real drop volume.
Results are shown in figure 3 and show that this approximation leads to less than 10 % error.
2 4 6 8 10
2
4
6
8
10
Computed volume [µL]
Real volume [µL]
Computed volume
y = x
0
20
40
60
80
100
Error
% Error
Contact angle [°]
80 90 100 110 120
Fig. 3 Estimation of rotational symmetry approximation accuracy for the measurement of drop volumes. The top/bottom xaxis corresponds
to simulated volumes close to Ωcassociated with contact angles ranging from 0 to 180 ◦. The left yaxis corresponds to volumes measured
from side views of the simulated droplets. Measurement error is shown on the right yaxis
It is to be noted that for drop volumes much lower than Ωcthis approximation no longer holds and
measurement error quickly increases above 10%.
3 Variation of drop contact length upon EWOD actuation
Figure 4 shows the variation of drop contact length on the fiber as a function of the applied voltage. Variation
of contact length appears to be very similar to variation of contact angle. In a first time the droplet quickly
spreads. Then starting from V2≈64 V, spreading continues but more slowly and we do not observe a clear
saturation as it is the case for planar EWOD. Once voltage is switched off (from 160 to 0 V), contact length
recover its initial value, as do the contact angle.
4 Relationship between maximum drop volume ΩMand applied voltage V
A first step toward analytical calculation of ΩMwas previously performed by E. Lorenceau et al.3. However
they considered barrel shaped droplets on wetting fibers (
θ
=0) and focused on the variation of ΩMwith
respect to the fiber radius. In their analysis they assumed that ‘ the force resulting from the horizontal fiber
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0 10000 20000 0
200
250
300
Contact length [ m]
V
RMS
2
[V
2
]
Fig. 4 Variation of drop contact length as a function of applied voltage.
is equivalent to the force generated by two similar fibers [ ... ] pointing radially.’, which is represented
in figure 5. Following this approach and considering a fiber for which
θ
=0, the maximal capillary force
exerted by the two fibers as
α
→
π
2is F
C,MAX =4
π
b
γ
. Balancing this force with the drop weight leads to
the following expression for ΩM:
ΩM≈4
π
b
γ
ρ
g
Fig. 5 Point of view of E. Lorenceau et al. concerning the force exerted by a fiber on a barrel shaped droplet. The force exerted by the
horizontal fiber is equivalent to the force exerted by two fibers pointing radially 3.
As far as our fibers are concerned, it is not rigorous to use such an analytical approach. Indeed in
the aforementioned work, the oil droplets exhibit a barrel configuration (liquid wraps around the fiber, so
contact line is separated into two roughly circular parts). Such configuration makes possible the assumption
of figure 5 (capillary force exerted by the fiber on the drop is focused at two contact points, and thus is
analogous to the force generated by two fibers pointing radially).
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On the other hand, our water droplets on hydrophobic fibers exhibit a barrel shape with a single continu-
ous contact line. In that case the capillary force exerted by the fiber is distributed all along the perimeter of
drop fiber contacting surface and is not likely to be expressed in such a way. It is to be noted that a simple
correction factor to the capillary force such as would here leads to a downward force since
θ
>90◦on the
hydrophobic fiber.
Thereby it is not straightforward to analytically predict the scaling law relating ΩMand
θ
. All we can
eventually do is to use numerical simulation results to display the ΩM=f(
θ
)relationship, which is shown
on the graph of figure 6. In the range of computed parameters (80◦<
θ
<120◦), a linear behavior is clearly
observed between the two quantities, with R2>0.99. For the 117
µ
min diameter fiber, slope is 9.15
µ
L−1
and intercept equals 7.11
µ
L(corresponding to
θ
=
π
2). Thereby from this point we can assume that the
maximal hanging drop volume follows the scaling law ΩM∝cos
θ
. It is to be noted that rigorous derivation
of this result is still to be done, and that this linear behavior is here demonstrated only for a limited contact
angle range.
-0,6 -0,3 0,0 0,3
3
6
9
M
[ L]
cos( )
Equation y = a + b*x
Adj. R-Square 0,99698
Value Standard Error
V_max Intercept 7,11573 0,02975
V_max Slope 9,15216 0,11262
Fig. 6 ΩM=f(
θ
)relationship obtained from numerical simulations of a water droplet on a 117
µ
min diameter fiber.
In a second time, using the Young-Dupr relationship, we could expect cos
θ
∝V2. However, it is clearly
observed in the case of EWOD on fiber, that the EWOD response hardly follows the Young-Dupr approxi-
mation (it is the case only for small Vvalues, i.e. up to Vrms ≈64 V, cf. figure 4 in the manuscript). As a
consequence, there is no reason for ΩMto linearly increase with V2over the whole voltage range.
In order to gibe to the reader an idea about the amplitude of the modulation (in
µ
L.V−1or
µ
L.V−2),
figure 7 represents linear curve fitting for ΩM=f(Vrms)and ΩM=f(V2
rms)relationships. The second case
exhibits a better correlation with R2=0.87 instead of R2=0.70 for the former case.
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Fig. 7 Linear curve fitting for ΩM=f(Vr ms)and ΩM=f(V2
rms)relationships.
5 Surface evolver simulations
The Surface Evolver is an interactive program for the study of surfaces shaped by surface tension and other
energies. A surface is implemented as a union of triangles and evolved toward minimal energy by a gradient
descent method.
In the case of a drop hanging on a fiber, the total energy Eresults from interfacial and gravitational
contributions which are computed using surface and line integrals. Moreover the surface is subjected to
volume and boundary constraints.
5.1 Initial geometry and constraints
Taking advantage of symmetries, only a quarter of the drop is modelled. Initial geometry is represented
in figure 8 (The fiber aligns with the x axis). It consists of 7 vertices, 16 edges and 6 faces, the following
constraints and symmetries are applied :
•Symmetry with respect to yOz plane (edges 4,11)
Equation : x =0
•Symmetry with respect to xOz plane (edge 16)
Equation : y =0
•Confinement to fiber surface (edges 9,14)
Equation : y2+z2=R2, with Rthe fiber radius
5.2 Volume and energies computation
Interfacial energies and contact angleThe energy ELV corresponding to the liquid - air interface is directly
calculated from the surface of facets (This is automatically done by Evolver when assigning an interfacial
energy
γ
LV to the facets).
The solid - liquid interfacial tension
γ
SL is obtained from
γ
LV and
θ
though the Young relationship :
γ
SL =
γ
LV .cos
θ
(the solid - vapor interface is not represented so
γ
SV =0). Since the liquid - fiber contacting
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Fig. 8 The initial drop geometry consists of 7 vertices, 16 edges and 6 faces. A first symmetry constraint is applied to edges 1 and 4 (Oyz
plane). A second symmetry constraint is applied to edge 16 (Oxz plane). The liquid - fiber contacting surface is not represented and the
corresponding energy is taken into account using a line integral along the contact line (edges 9 and 14 in red).
surface is not modelled, the corresponding energy ESL is obtained from a line integral along the contact line.
Considering cylindrical coordinates (r,
θ
,x),ESL can be written :
ESL =
γ
SL ∫C
R.x.d
θ
(1)
Where curve Ccorresponds to the contact line of the drop on the fiber. Since R=√y2+z2, this integral
can be re-written :
ESL =
γ
SL ∫C
y2+z2
R.x.d
θ
=
γ
SL ∫C
x.y
R.y.d
θ
+x.z
R.z.d
θ
(2)
Finally, from y=R.cos
θ
and z=R.sin
θ
, we introduce the following relationships :
dy=z.d
θ
(3)
dz=y.d
θ
(4)
Which enables to compute ESL in Evolver cartesian coordinate system through the following vector
integral:
ESL =
γ
SL ∫C
x.y
R.dz+x.z
R.dy=
γ
SL ∫C
e.
dl(5)
The vector integrand ise=
γ
SL.
0
x.y
R
x.z
R
Drop volumeDuring computation, the liquid drop is constrained to a fixed volume V. Since Evolver only
models surfaces, Vhas to be calculated from the drop facets. This is done using the divergence theorem:
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y
V
div
F.dV={
S
F.
dS(6)
Since the drop volume is tVdV, we need to integrate on the surface a vector
Fwhich satisfy div
F=1.
The solutions
F=y.eyor
F=z.ezare not correct because it will take into account volume of the wetted
fiber (this is because the drop - fiber interface is not represented). Thus we need to use
F=x.exand the
volume is obtaine through the following vector integral over the drop surface:
V={
S
x.ex.
dS(7)
Gravitational energySince gravitational energy EGis related to the drop volume, it is also computed using
the divergence theorem (equation 6). We now have to solve
∇.
F=
ρ
.g.z. Once again the general form
F=1
2
ρ
.g.z2.ezor the projection on yaxis with
F=
ρ
.g.y.z.eycannot be used because it will partially take
into account the fiber volume. As a consequence we use
F=
ρ
.g.x.z.exand gravitational energy is obtained
from the following vector integral over the drop surface :
V={
S
ρ
.g.x.z.ex.
dS(8)
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5.3 Model
/ / h a n g i n g c l a m s h e l l . f e
/ / By R . D u f o ur , J a n u a r y 2 0 1 2
VIEW TRANSFORM GENERATORS 2
−1 0 0 0 / / x m i r r o r
0100
0010
0001
1000 / / y m i r r o r
0−1 0 0
0010
0001
/ / PARAMETERS −MKS s y s t e m u n i t −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
p a r a m e t e r r o d r = 1 . 1 7 0E −004 / / f i b e r r a d i u s
p a r a m e t e r a n g l e = 8. 0 0 0E+ 00 1 / / c o n t a c t a n gl e w i t h f i b e r
p a r a m e t e r TENS = 7 . 28 0 E−002 / / L i q ui d s u r f a c e t e n s i o n
p a r a m e t e r v o l = 5 . 00 0 E−010 / / d r op v ol ume ( c ub ic m i c r o l i t e r s )
p a r a m e t e r d e n s = 998 / / d e n s i t y o f l i q u i d
#define r o d e (−TENS∗c os ( a n g l e ∗p i / 1 8 0 ) ) / / l i q u i d −r o d c o n t a c t e n er g y
#define h t v ol ˆ ( 1 / 3 ) / / i n i t i a l d r o p d i m e n s i o n
#define wd v o l ˆ ( 1 / 3 ) / / i n i t i a l d r o p d i m e n s i o n
gravity constant 9.81 / / g r a v i t y i s 9 .8 1 m/ s ˆ 2
g a p c o n s t a n t 0 .0 0 1 / / t o a v o i d w r a p p in g a r ou n d f i b e r
s c a l e l i m i t 0 .5 / / s c a l e f a c t o r .
/ / VOLUME AND GRAVITATIONAL ENERGY −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
/ / vo lum e
q u a n t i t y f v i n t f i x ed = 0 .2 5∗v ol metho d f a c e t v e c t o r i n t e g r a l g l ob a l
vector integrand :
q1 : x
q2 : 0
q3 : 0
/ / G r a v i t a t i o n a l e ne rg y
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q u an t i t y g ra v x en er gy mod u lu s d en s m eth o d f a c e t v e c t o r i n t e g r a l g l ob a l
vector integrand :
q1 : G∗x∗z
q2 : 0
q3 : 0
/ / CONSTRAINTS −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
c o n s t r a i n t 1 c on vex / / F i b e r s u r f a c e
f o r m u l a : y ˆ 2 + z ˆ 2 = r o d r ˆ 2
e n e r g y :
e1 : 0
e2 : −r o d e ∗x∗z / r o d r
e 3 : r o d e ∗x∗y / r o d r
constraint 2 / / V e r t i c a l sy mme try p l an e ( x m i r ro r )
f o r m u l a : x = 0
c o n s t r a i n t 3 no n ne ga t iv e / / F i b er s u r f a c e ( t o p r e v e n t c a v i n g i n )
f o r m u l a : y ˆ 2 + z ˆ 2 = r o d r ˆ 2
constraint 4 / / V e r t i c a l sy mme try p l an e ( y m i r ro r )
f o r m u l a : y = 0
/ / IN I TI AL GEOMETRY −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
vertices
1 0 −wd 0 constraint 2 / / e q u a t o r i a l v e r t i c e s
2 0 0 −wd constraint 2,4
3 h t −wd 0 / / u pp e r o u t e r c o r n e r s
4 h t 0 −wd constraint 4
5 0 −r od r 0 c o n s t r a i n t 1 , 2 / / l ow er v e r t i c e s on r od
6 h t −r od r 0 c o n s t r a i n t 1 / / u ppe r v e r t i c e s o n rod
7 h t 0 −r od r c o n s t r a i n t 1 , 4
edges
1 1 2 c o n s t r a i n t 2 / / e q u a t o r i a l e d ge s
4 5 1 c o n s t r a i n t 2
5 3 4 / / u p p er e d g e s
9 6 7 c o n s t r a i n t 1
10 6 3 / / u p p e r e d g e
11 1 3 / / v e r t i c a l e x t e r n a l e dg e s
12 2 4 c o n s t r a i n t 4
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14 5 6 c o n s t r a i n t 1 / / v e r t i c a l i n t e r n a l e dg es
16 4 7 c o n s t r a i n t 4 / / c u t t i n g t o p pl an e
f a c e s
1 1 12 −5−11 t e n s i o n TENS c o n s t r a i n t 3 / / s i d e f a c e s
4 4 11 −10 −14 t e n s i o n TENS c o n s t r a i n t 3
6 5 16 −9 10 t e n s i o n TENS c o n s t r a i n t 3
bodies
1 1 4 6
5.4 Critical volume computation
To obtain the maximal volumes of hanging drop, the above model was run from a C+ + program and di-
chotomy method was used to get Ωcwith a accuracy of 0.1
µ
L. For each volume, an evolving procedure
was used (not shown here), consisting in a sequence of evolution and refinement steps. Progressive refine-
ment along zaxis was used because of important difference deformation length-scale (typically ≈10
µ
m
mesh element size was used around the fiber and ≈100
µ
mwas used for the bottom part of the drop).
We assumed equilibrium was achieved when reaching a 10−10 % variation in energy, in that case the
probe volume was increased. Otherwise divergence was observed by mean of drop detachment from the
fiber.
It is to be noted that, from a general point of view, the drop fiber system can exhibit two different stable
configurations, corresponding to the barrel or clamshell drop shape. For the range of contact angle investi-
gated in this paper (
θ
ranging from 80◦to 120◦), the system can only present a barrel configuration (Energy
phase diagram and stability were previously studied by Chou et al.4, also achievement of a clamshell drop
is possible only for small contact angles, typically
θ
<50◦.
Thereby we expect only two cases: the droplet reaches the barrel configuration, which corresponds to a
local minimum of energy, or it falls from the fibers, and in that case the total energy keeps decreasing and
never reaches a minimum. These two cases are represented from an energetically point of view in figure 9.
System energy is represented as a function of the drop fiber contact length. Depending on the starting point
of the simulation, the numerical method can converge either to the barrel configuration or to the falling off
of the droplet. Because the numerical method uses energy gradient descent, starting points B and C in the
example of figure 9 will converge to the barrel drop while starting point A will result in drop fall, without
finding the metastable barrel state.
To avoid this problem of bi-stability, it is preferable to get the initial drop configuration close to point C,
so that if a metastable barrel shape exists, it will not be missed during energy minimization. For this reason
the initial drop shape used for the simulations presents an important drop fiber contact length (shown
below), which corresponds to point C in figure 9. Thereby during shape evolution it will converge to
the barrel configuration if the latter exists, otherwise the contact length decreases down to 0, leading to a
spherical droplet disconnected from the fiber (shown below).
References
1 R. Dufour, M. Harnois, Y. Coffinier, V. Thomy, R. Boukherroub and V. Senez, Langmuir, 2010, 26, 17242–17247.
2 R. Dufour, M. Harnois, V. Thomy, R. Boukherroub and V. Senez, Soft Matter, 2011, 7, 9380–9387.
3 E. Lorenceau, C. Clanet and D. Quere, J. Colloid Interf. Sci., 2004, 279, 192–197.
4 T. H. Chou, S. J. Hong, Y. E. Liang, H. K. Tsao and Y. J. Sheng, Langmuir, 2011, 27, 3685–3692.
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E
L
ABC
Fig. 9 Energy landscape for the drop on fiber system. Depending on the starting point A, B or C, the numerical method (energy gradient
descent) can result in the barrel configuration or falling off of the droplet.
Fig. 10 Top: initial drop shape before evolution. The initial drop fiber contact length is important so that the numerical method cannot miss
the metastable barrel configuration. In this example the contact angle is
θ
=100◦. For a volume Ω=5.3
µ
L, the numerical method converges
to the barrel shape (bottom left) while for a slightly larger volume Ω=5.6
µ
L, contact length decreases down to 0 and drop falls down
(remaining attached to the fiber by a single edge.
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