![Contact angle versus applied (RMS) voltage for a glycerol-salt (NaCl) water droplet (conductivity: 3 mS cm −1 ; AC frequency: 10 kHz) with silicone oil as the ambient medium. Insulator: Teflon AF 1601 (d ≈ 5 µm). Note that θ Y is almost 180 • for this system. Filled (open) symbols: increasing (decreasing) voltage. Solid line: parabolic fit according to equation (8) (reproduced from [52]).](https://www.researchgate.net/profile/Frieder-Mugele/publication/244425138/figure/fig7/AS:668688789299210@1536439239799/Contact-angle-versus-applied-RMS-voltage-for-a-glycerol-salt-NaCl-water-droplet_Q320.jpg)
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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 17 (2005) R705–R774 doi:10.1088/0953-8984/17/28/R01
TOPICAL REVIEW
Electrowetting: from basics to applications
Frieder Mugele
1,3
and Jean-Christophe Baret
1,2
1
University of Twente, Faculty of Science and Technology, Physics of Complex Fluids, PO Box
217, 7500 AE Enschede, The Netherlands
2
Philips Research Laboratories Eindhoven, Health Care Devices and Instrumentation, WAG01,
Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands
E-mail: f.mugele@utwente.nl
Received 11 April 2005, in final form 10 May 2005
Published 1 July 2005
Online at stacks.iop.org/JPhysCM/17/R705
Abstract
Electrowetting has become one of the most widely used tools for manipulating
tiny amounts of liquids on surfaces. Applications range from ‘lab-on-a-chip’
devices to adjustable lenses and new kinds of electronic displays. In the present
article, we review the recent progress inthis rapidly growing field including
both fundamental and applied aspects. We compare the various approaches
used to derive the basic electrowetting equation, which has been shown to
be very reliable as long as the applied voltage is not too high. We discuss
in detail the origin of the electrostatic forces that induce both contact angle
reduction and the motion of entire droplets. We examine the limitations of the
electrowetting equation and present a variety of recent extensions to the theory
that account for distortions of the liquid surface due to local electric fields, for
the finite penetration depth of electric fields into the liquid, as well as for finite
conductivity effects in the presence of AC voltage. The most prominent failure
of the electrowetting equation, namely the saturation of the contact angle at
high voltage, is discussed in a separate section. Recent work in this direction
indicates that a variety of distinct physical effects—rather than a unique one—
are responsible for the saturation phenomenon, depending on experimental
details. In the presence of suitable electrode patterns or topographic structures
on the substrate surface, variations of the contact angle can give rise not
only to continuous changes of the droplet shape, but also to discontinuous
morphological transitions between distinct liquid morphologies. The dynamics
of electrowetting are discussed briefly. Finally, we give an overview of recent
work aimed at commercial applications, in particular in the fields of adjustable
lenses, display technology, fibre optics, and biotechnology-relatedmicrofluidic
devices.
(Some figures in this article are in colour only in the electronic version)
3
Author to whom any correspondence should be addressed.
0953-8984/05/280705+70$30.00 © 2005 IOP Publishing Ltd Printed in the UK R705
R706 Topical Review
Contents
1. Introduction 706
2. Theoretical background 708
2.1. Basic aspects of wetting 708
2.2. Electrowetting theory for homogeneous substrates 710
2.3. Extensions of the classical electrowetting theory 715
3. Materials properties 719
4. Contact angle saturation 722
5. Complex surfaces and droplet morphologies 725
5.1. Morphological transitions on structured surfaces 725
5.2. Patterned electrodes 726
5.3. Topographically patterned surfaces 727
5.4. Self-excited oscillatory morphological transitions 730
5.5. Electrostatic stabilization of complex morphologies 731
5.6. Competitive wetting of two immiscible liquids 732
6. Dynamic aspects of electrowetting 732
7. Applications 734
7.1. Lab-on-a-chip 734
7.2. Optical applications 737
7.3. Miscellaneous applications 741
8. Conclusions and outlook 741
Acknowledgments 742
Appendix A. Relationships between electrical and capillary phenomena 743
A.1. First law 744
A.2. Second law 749
A.3. Mathematical theory 754
A.4. Electrocapillary motor 757
A.5. Measurement of the capillary constant of mercury in a conducting liquid 760
A.6. Capillary electrometer 762
A.7. Theory of the whirls discovered by Gerboin 766
References 770
1. Introduction
Miniaturization has been a technological trend for several decades. What started out initially in
the microelectronics industry long ago reached thearea of mechanical engineering, including
fluid mechanics. Reducing size has been shown to allow for integration and automation of
many processes on a single device giving rise to a tremendous performance increase, e.g. in
terms of precision,throughput,and functionality.Oneprominent examplefrom the area of fluid
mechanics the ‘lab-on-a-chip’ systems for applications such as DNA and protein analysis, and
biomedical diagnostics [1–3]. Most of the devices developed so far are based on continuous
flow through closed channels that are either etched into hard solids such as silicon and glass,
or replicated from a hard master into a soft polymeric matrix. Recently, devices based on
the manipulation of individual droplets with volumes in the range of nanolitres or less have
attracted increasing attention [4–10].
From a fundamental perspective the most important consequence of miniaturization is a
tremendous increase in the surface-to-volume ratio, which makes the control of surfaces and
surface energies one of the most important challenges both in microtechnology in general as
Topical Review R707
well as in microfluidics. For liquid droplets of submillimetre dimensions, capillary forces
dominate [11, 12]. The control of interfacial energies has therefore become an important
strategy for manipulating droplets at surfaces [13–17]. Both liquid–vapour and solid–liquid
interfaces have been influenced in order to controldroplets, as recently reviewed by Darhuber
and Troian [15]. Temperature gradients as well as gradients in the concentration of surfactants
across droplets give rise to gradients in interfacial energies, mainly at the liquid–vapour
interface, and thus produce forces that can propel droplets making use of the thermocapillary
and Marangoni effects.
Chemical and topographical structuring of surfaces has received even more attention.
Compared to local heating, both of these two approaches offer much finer control of the
equilibrium morphology. The local wettability andthe substrate topography together provide
boundary conditions within which the droplets adjust their morphology to reach the most
energetically favourable configuration. For complex surface patterns, however, this is not
always possible as several metastable morphologies may exist. This can lead to rather abrupt
changes in the droplet shape, so-called morphological transitions, when the liquid is forced to
switch from one family of morphologies to another by varying a control parameter, such as
the wettability or the liquid volume [13, 16, 18–20].
The main disadvantage of chemical and topographicalpatterns is their static nature, which
prevents active control of the liquids. Considerable work has been devoted to the development
of surfaces with controllable wettability—typically coated with self-assembled monolayers.
Notwithstanding some progress, the degree of switchability, the switching speed, the long
term reliability, and the compatibility with variable environments that have been achieved
so far are not suitable for most practical applications. In contrast, electrowetting (EW) has
proven very successful in all these respects: contact angle variations of several tens of degrees
are routinely achieved. Switching speeds are limited (typically to several milliseconds) by
the hydrodynamic response of the droplet rather than the actual switching of the equilibrium
valueofthe contact angle. Hundreds of thousands of switching cycles were performed in long
term stability tests without noticeable degradation [21, 22]. Nowadays, droplets can be moved
along freely programmable paths on surfaces; they can be split, merged, and mixed with a
high degree of flexibility. Most of these results were achieved within the past five years by a
steadily growing community of researchers in the field
4
.
Electrocapillarity, the basis of modern electrowetting, was first described in detail in
1875 by Gabriel Lippmann [23]. This ingenious physicist, who won the Nobel prize in
1908 for the discovery of the first colour photography method, found that the capillary
depression of mercury in contact with electrolyte solutions could be varied by applying
avoltage between the mercury and electrolyte. He not only formulated a theory of the
electrocapillary effect but also developed several applications, including a very sensitive
electrometer and a motor based on his observations. In order to make his fascinating work,
which has only been available inFrenchuptonow,availabletoabroader readership, we
included a translation of his work in the appendix of this review. The work of Lippmann
and of those who followed him in the following more than a hundred years was devoted to
aqueous electrolytes in direct contact with mercury surfaces or mercury droplets in contact
with insulators. A major obstacle to broader applications was electrolytic decomposition of
water upon applying voltages beyond a few hundred millivolts. The recent developmentswere
initiated by Berge [24]inthe early 1990s, who introduced the idea of using a thin insulating
layer to separate the conductive liquid from the metallic electrode in order to eliminate the
4
The number of publications on electrowetting has been increasing from less than five per year before 2000 to 8 in
2000, 9 (2001), 10 (2002), 25 (2003), and 34 (2004).
R708 Topical Review
σ
lv
ε
d
σ
sl
σ
sv
θ
Y
U
d,
Figure 1. Generic electrowetting set-up. Partially wetting liquid droplet at zero voltage (dashed)
and at high voltage (solid). See the text for details.
problem of electrolysis. This is the concept that has also become known as electrowetting on
dielectric (EWOD).
In the present review, we are going to give an overview of the recent developments in
electrowetting, touching only briefly on some of the early activities that were already described
in a short review by Quilliet and Berge [25]. The article is organized as follows. In section 2
we discuss the theoretical background of electrowetting, comparing different fundamental
approaches, and present some extensions of the classical models. Section 3 is devoted to
materials issues. In section 4,wediscussthe phenomenon of contact angle saturation, which
has probably been the most fundamental challenge in electrowetting for some time. Section 5
is devoted to the fundamental principles of electrowetting on complex surfaces, which is the
basis for most applications. Section 6 deals with some aspects of dynamic electrowetting, and
finally, before concluding, a variety of current applications ranging from lab-on-a-chip to lens
systems and display technology are presented in section 7.
2. Theoreticalbackground
Electrowetting has been studied by researchers from various fields, such as applied
physics, physical chemistry, electrochemistry, and electrical engineering. Given the various
backgrounds,different approaches were usedto describe the electrowetting phenomenon,i.e. to
determine the dependence of the contact angle on the applied voltage. In this section, we will—
after a few introductory remarks about wetting in section 2.1—discuss the main approaches of
electrowetting theory (section 2.2): the classical thermodynamic approach (2.2.1), the energy
minimization approach (2.2.2), and the electromechanical approach (2.2.3). In section 2.3,we
will describe some extensions of the basic theories that give more insight into the microscopic
surface profile near the three-phase contact line (2.3.1), the distribution of charge carriers near
the interface (2.3.2), and the behaviour at finite frequencies (2.3.3).
2.1. Basic aspects of wetting
In electrowetting, one is generically dealing with droplets of partially wetting liquids on planar
solid substrates (see figure 1). In most applications of interest, the droplets are aqueous salt
solutions with a typical size of the order of 1 mm or less. The ambient medium can be either
air or another immiscible liquid, frequently an oil. Under these conditions, the Bond number
Bo =
gρ R
2
/σ
lv
,whichmeasures the strength of gravity with respect to surface tension,
is smaller than unity. Therefore we neglect gravity throughout the rest of this paper. In the
absence of external electric fields, the behaviour of the droplets is then determined by surface
tension alone. The free energy F of a droplet is a functional of the droplet shape. Its value is
given by the sum of the areas A
i
of the interfaces between three phases, the solid substrate (s),
Topical Review R709
σ
lv
σ
sl
σ
sv
θ
Y
Figure 2. Force balance at the contact line (for θ
Y
approximately 30
◦
).
the liquid droplet(l), and the ambient phase, which we will denote as vapour(v) for simplicity
5
,
weighted by the respectiveinterfacial energies σ
i
,i.e.σ
sv
(solid–vapour), σ
sl
(solid–liquid),
and σ
lv
(liquid–vapour):
F = F
if
=
i
A
i
σ
i
−λV . (1)
Here, λ is a Lagrangian variable present to enforce the constant volume constraint. λ is
equal to the pressure drop p across the liquid–vapour interface. Variational minimization
of equation (1)leads to the two well-known necessary conditions that any equilibrium liquid
morphology has to fulfil [11, 12]: the first one is the Laplace equation, stating that p is a
constant, independent of the position on the interface:
p = σ
lv
1
r
1
+
1
r
2
= σ
lv
·κ. (2)
Here, r
1
and r
2
are the two—in general position dependent—principal radii of curvature of the
surface, and κ is the constant mean curvature. For homogeneous substrates, this means that
droplets adopt a spherical cap shape in mechanical equilibrium. The second condition is given
by Young’s equation
cos θ
Y
=
σ
sv
−σ
sl
σ
lv
, (3)
which relates Young’s equilibrium contact angle θ
Y
to the interfacial energies
6
.Alternatively
to this energetic derivation, the interfacial energies σ
i
can also be interpreted as interfacial
tensions, i.e. as forces pulling on the three-phase contact line. Within this picture, equation (3)
is obtained by balancing the horizontal component of the forces acting on the three-phase
contact line (TCL); see figure 2.
7
Note that both derivations are approximations intended for mesoscopic scales. On the
molecular scale, equilibrium surface profiles deviate from the wedge shape in the vicinity of
the TCL [26, 27]. Within the range of molecular forces, i.e. typically a few nanometres from
the surface, the equilibrium surface profiles are determined by the local force balance (at the
surface) between the Laplace pressure and the disjoining pressure, in which the molecular
forces are subsumed. Despite the complexity of the profiles that arise, these details are not
relevant if one is only interested in the apparent contact angle at the mesoscopic scale. On
that latter scale, the contact line can be considered as a one-dimensional object on which
the interfacial tensions are pulling. As we will see below, a comparable situation arises in
electrowetting.
5
Note that the ambient phase can be another liquid, immiscible with the droplet, instead of vapour.
6
Note that these conditions are only necessary and not sufficient. In addition, the second variation of F must be
positive. In the presence of complex surfaces, some morphologies may indeed be unstable although both necessary
conditions are fulfilled [13].
7
Thevertical force component is balanced by normal stresses in the stiff solid substrate.
R710 Topical Review
2.2. Electrowetting theory for homogeneous substrates
2.2.1. The thermodynamic and electrochemical approach. Lippmann’s classical derivation
of the electrowetting or electrocapillarity equation is based on general Gibbsian interfacial
thermodynamics [28]. Unlike in the recent applications of electrowetting where the liquid is
separated from the electrode by an insulating layer, Lippmann’s original experiments dealt
with direct metal (in particular mercury)–electrolyte interfaces (see the appendix and [23]).
For mercury, several tenths of a volt can be applied between the metal and the electrolyte
without any current flowing. Upon applying a voltage dU,anelectric double layer builds up
spontaneously at the solid–liquid interface consisting of charges on the metal surface on the
one hand and of a cloud of oppositely charged counter-ions on the liquid side of the interface.
Since the accumulation is a spontaneous process, for instance the adsorption of surfactant
molecules at an air–water interface, it leads to a reduction of the (effective) interfacial tension
σ
eff
sl
:
dσ
eff
sl
=−ρ
sl
dU (4)
(ρ
sl
= ρ
sl
(U) is thesurface charge density of the counter-ions
8
.) (Our reasons for denoting the
voltage dependent tension as ‘effective’ will become clear below.) The voltage dependence of
σ
eff
sl
is calculated by integrating equation (4). In general, this integral requires additional
knowledge about the voltage dependent distribution of counter-ions near the interface.
Section 2.3.2 describes such a calculation on the basis of the Poisson–Boltzmann distribution.
Fornow,wemake the simplifying assumption that the counter-ions are all located at a fixed
distance d
H
(of the order of a few nanometres) from the surface (Helmholtz model). In this
case, the double layer has a fixed capacitance per unit area, c
H
= ε
0
ε
l
/d
H
,whereε
l
is the
dielectric constant of the liquid. We obtain [29, 30]
σ
eff
sl
(U) = σ
sl
−
U
U
pzc
ρ
sl
d
˜
U = σ
sl
−
U
U
pzc
c
H
˜
U d
˜
U = σ
sl
−
ε
0
ε
l
2d
H
(U −U
pzc
)
2
. (5)
Here, U
pzc
is the potential (difference) of zero charge. (Note that mercury surfaces—like
those of most other materials—acquire a spontaneous charge when immersed into electrolyte
solutions at zero voltage. The voltage required to compensate for this spontaneous charging
is U
PZC
;seealso figure A.4 in the appendix.) The chemical contribution σ
sl
to the interfacial
energy, which appeared previously in Young’s equation (equation (3)), is assumed to be
independent of the applied voltage. To obtain the response of the contact angle, equation (5)
is inserted into Young’s equation (equation (3)). For an electrolyte droplet placed directly on
an electrode surface we find
cos θ = cos θ
Y
+
ε
0
ε
l
2d
H
σ
lv
(U −U
pzc
)
2
. (6)
For typical values of d
H
(2 nm), ε
l
(81), and σ
lv
(0.072 mJ m
−2
) we find thattheratio
on the rhs of equation (6)isonthe order of 1 V
−2
.Thecontact angle thus decreases rapidly
upon the application of a voltage. It should be noted, however, that equation (6)isonly
applicable within a voltage range below the onset of electrolytic processes, i.e. typically up
to a few hundred millivolts. As mentioned alreadyinthe introduction, modern applications
of electrowetting usually circumvent this problem by introducing a thin dielectric film, which
insulates the droplet from the electrode; see (1). In this EWOD configuration, the electric
double layer builds up at the insulator–droplet interface. Since the insulator thickness d is
usually much larger than d
H
,the total capacitance of the system is reduced tremendously.
The system may be described as two capacitors in series [29, 30], namely the double at the
8
This description ignores any deviations that may occur close to the TCL.
Topical Review R711
0 100 200 300 400
-1.0
-0.5
0.0
0.5
1.
0
0 100 200 300
400
-1.0
-0.5
0.0
0.5
1.
0
U [V]
cos
θ
Figure 3. Contact angle versus applied (RMS) voltage for a glycerol–salt (NaCl) water droplet
(conductivity: 3 mS cm
−1
;ACfrequency: 10 kHz) with silicone oil as the ambient medium.
Insulator: Teflon AF 1601 (d ≈ 5 µm). Notethatθ
Y
is almost 180
◦
forthis system. Filled
(open) symbols: increasing (decreasing) voltage.Solid line: parabolic fit according to equation (8)
(reproduced from [52]).
solid–insulator interface (capacitance c
H
)andthedielectric layer with c
d
= ε
0
ε
d
/d (ε
d
is the
dielectric constant of the insulator). Since c
d
c
H
,thetotal capacitance per unit area c ≈ c
d
.
With this approximation, we neglect the finite penetration of the electric field into the liquid,
i.e. we treat the latter as a perfect conductor. As a result, we find that the voltage drop occurs
within the dielectric layer, and equation (5)isreplaced by
σ
eff
sl
(U) = σ
sl
−
ε
0
ε
d
2d
U
2
. (7)
(Here and in the following, we assume that the surface of the insulating layer does not
give rise to spontaneous adsorption of charge in the absence of an applied voltage, i.e. we set
U
pzc
= 0.) In this equation the entire dielectric layer is considered part of one effective solid–
liquid interface [30]with a thickness of the order of d,i.e.inpractice typically O(1 µm).In
that sense, the interfacial energy in equation (7)isclearly an ‘effective’ quantity. Combining
equation (7)with equation (3), we obtain the basic equation for EWOD:
cos θ = cos θ
Y
+
ε
0
ε
d
2dσ
lv
U
2
= cos θ
Y
+ η. (8)
Here, we have introduced the dimensionless electrowetting number η = ε
0
ε
r
U
2
/(2dσ
lv
),
which measures the strength of the electrostatic energy compared to surface tension. The ratio
in the middle part of equation (8)istypically four to six orders of magnitude smaller than that
in equation (6), depending on the properties of the insulating layer. Consequently, the voltage
required to achieve a substantial contact angle decrease in EWOD is much higher.
Figure 3 shows a typical experimental example. As in many other experiments,
equation (8)isfound to hold as long as the voltage is not too high. Beyond a certain
system dependent threshold voltage, however, the contact angle has always been found to
become independent of the applied voltage [31–37]. This so-called contact angle saturation
phenomenon will be discussed in detail in section 4.
2.2.2. Energy minimization method. For EWOD, equation (8)wasfirstderivedby
Berge [24]. His derivation, however, was based on energy minimization rather than interfacial
thermodynamics: the free energy F of a droplet in an EWOD configuration (figure 1)is
R712 Topical Review
composed of two contributions—in addition to the interfacial energy contribution F
if
that
appeared already in equation (1), there is an electrostatic contribution F
el
:
F
el
=
1
2
E(r) ·
D(r ) dV. (9)
E(r) and
D(r ) = ε
0
ε(r )
E(r) denote the electric field and theelectric displacement at r ·ε(r )
is the dielectric constant of the medium at the location r .Thevolumeintegral extends over
theentire system. As in the previous section, the liquid is considered a perfect conductor;
hence surface charges screen the electric fields completely from the interior of the liquid and
the integral vanishesinsidethedroplet.
Before performing the minimization of the free energy, we have to identify the correct
thermodynamic potential for electrowetting. In electrical problems, there are two limiting
constraints: constant charge and constant voltage. The corresponding thermodynamic
potentials are related by a Legendre transformation [38]. In electrowetting, the voltage is
controlled. The thermodynamic potential corresponding to this situation is given by
F = F
if
− F
el
=
i
A
i
σ
i
−pV −
1
2
E(r) ·
D(r ) dV. (10)
Before computing the contact angle decrease, one usually introduces another
simplification: the electrostatic energy may be split into two parts. The first part arises from
the parallel plate capacitor formed bythedroplet and the electrode, with C = c
d
A
sl
.The
second part is due to the stray capacitance along the edge of the droplet. The fringe fields are
mainly localized within a small range around the contact line. Therefore, their contribution to
the total energy is negligible for sufficiently large droplets. (In section 2.3.1,wewill consider
the effect of fringe fields in more detail.) Hence, we find F
el
≈ CU
2
/2 = ε
0
ε
d
U
2
A
sl
/(2d).
Apart from this formal explanation, the negative sign in equation (10) can also be
understood intuitively by considering the entire system consisting of both the droplet and
the power supply (the ‘battery’) required to apply the voltage. Upon connecting the initially
uncharged droplet to the battery, the charge δQ flows from the battery (δ Q
b
=−δQ) to the
droplet (and to the electrode). The work done on the droplet–electrode capacitor is given
by δW = U(Q)δQ = Q/Cδ Q,with U (Q) being the charge dependent voltage on the
capacitor. The electrostatic energy stored in the droplet in the final state is E
drop
=
δW =
Q
2
/(2C) = CU
2
b
/2. The work done on the battery is δW
b
= U
b
δ Q
b
=−U
b
δ Q.Incontrast
to U(Q),thebattery voltage U
b
is constant, such that we obtain a release of electrostatic
energy of E
batt
=
δW
b
=
U
b
δ Q
b
=−
U
b
δ Q =−CU
2
b
.Hence the net electrostatic
contribution to the free energy is −CU
2
b
/2. Electrowetting is thus driven by the energy gain
upon redistributing charge from the battery to thedroplet. The contact angle decreases because
this increases C and thus allows for the redistribution of even more charge.
For homogeneous electrodes, the free energy thus reads
F = A
lv
σ
lv
+ A
sv
σ
sv
+ A
sl
σ
sl
−
ε
0
ε
d
U
2
2d
−pV. (11)
Equation (11)hasthesame structure as the free energy intheabsence of electric fields
(equation (1)). On comparing the coefficients, the electrowetting equation (8)isrediscovered.
As in the previous section, this derivation is based on the assumptions that (i) the σ
i
are voltage
independent, (ii) the liquid is perfectly conductive, and (iii) contributions from the region
around the contact line (due to fringe fields) can be neglected. An important additional insight
fromthisderivation is the fact that the energy gain in electrowetting is actually taking place
in the battery, i.e. quite remotely from the droplet itself. This illustrates again the effective
character of the definition in (equation (7)).
Topical Review R713
2.2.3. Electromechanical approach. Both methods discussed so far predict the same contact
angle reduction. However, they do not provide a physical picture of how the contact angle
reduction is achieved in mechanical terms. Such a picture can be obtained by considering the
forces exerted on the liquid by the electric field. These forces contain contributions due to
theresponse of the free electric charge density ρ
f
and the polarization density in the presence
of electric field gradients. This approach was introduced to the field of electrowetting by
Jones et al [34, 39]and recently reviewed by Zeng and Korsmeyer [4]. In the case of simple
liquids, one of the most frequently used formulations is the Korteweg–Helmholtz body force
density [38]
f
k
= ρ
f
E −
ε
0
2
E
2
∇ε + ∇
ε
0
2
E
2
∂ε
∂ρ
ρ
(12)
where ρ and ε are the mass density and the dielectric constant of the liquid, respectively. The
last term in equation (12)describes electrostriction and can be neglected in the present context.
The net force acting on a volume element dV of the fluid is obtained by a volume integration
over equation (12). As a fundamental consequence of momentum conservation, the same
force can also be obtained by an integration along the surface of dV over the momentum flux
density of the electric fields, i.e. the Maxwell stress tensor. It seems particularly appropriate
because in the perfect conductor limit of electrowetting the entire ‘body’ force actually acts
at the surface: ρ
f
is zero within the bulk, and free surface charges screen the electric field
from the interior. The second term on the rhs of equation (12), the so-called ponderomotive
force density, ∝∇ε,vanishes everywhere except at the surface. Neglecting electrostriction,
the Maxwell stress tensor consistent with equation (12)is[38]
T
ik
= ε
0
ε(E
i
E
k
−
1
2
δ
ik
E
2
). (13)
Here, δ
ik
is the Kronecker delta function, and i , k = x, y, z.Letusnow consider a volume
element dV at the liquid–air interface of a perfectly conductive liquid droplet (see figure 4).
The tangential component of the electric field at the surface vanishes and the normal component
is related to the local surface charge density by ρ
s
= ε
0
E ·n,wheren is the (outward) unit
normal vector. Furthermore,
E vanishes within the liquid. To obtainthenet force acting on
the liquid volume element, we calculate
F
i
=
T
ik
n
k
dA (14)
using the Einstein summation convention. We find that the only non-vanishing contribution is
aforceper unit surface area dA directed along the outward surface normal n:
F/d A = P
el
n =
ε
0
2
E
2
n =
ρ
s
2
E (15)
where we have introduced the electrostatic pressure P
el
= ε
0
E
2
/2 acting on the liquid surface.
P
el
is thus a negative contribution to the total pressure within the liquid.
How does this local pressure at the surface affect the contact angle of a sessile droplet?
The solution of the problem requires a calculation of the field (and charge) distribution along
the surface of the droplet. Far away from the contact line, the charge density at the solid–liquid
interface is ρ
sl
= ε
0
ε
d
U/d and the liquid–vapoursurface charge density vanishes. As the three-
phase contact line is approached, both charge densities increase due to sharp edge effects, as
first pointed out in [31]. The force arising from the charges atthesolid–liquid interface leads to
a normal stress on the insulator surface,which is balanced by the elastic stress. The forces at the
liquid–vapour interface, however, contain both a vertical and a horizontal component pulling
on the liquid. Kang [40]assumed that the droplet remains wedge shaped and calculated the net
horizontal force acting on the liquid by integrating the horizontal component of equation (15)
R714 Topical Review
l
∂ dV
n
v
E
Figure 4. Force acting on a volume element dV at the liquid–vapour interface with surface charges
(‘+’). Solid box: area for surface integral.
along the liquid–vapour interface
9
.The field and charge distribution are found by solving
the Laplace equation for an electrostatic potential φ with appropriate boundary conditions.
For the wedge geometry, an analytic solution can be obtained using conformal mapping as
first described by Vallet et al [31]inthecontext of electrowetting. Both the field and charge
distribution are found to divergealgebraically upon approaching the contact line. The resulting
Maxwell stress is thus maximal at the contact line and decays to a practically negligible value at
adistance of a few d from the TCL [40]. Integrating the horizontal component of the Maxwell
stress, we obtain the net force acting on the droplet. For the horizontal component, the result
reads
F
x
=
ε
0
ε
d
2d
U
2
= σ
lv
η. (16)
Given the rapid decay of the Maxwell stress, this force can be considered as localized at the
contact line, in a coarse grained sense—on a length scale much larger than d.Expression (16)
can thus be used in the force balance at the contact line in the spirit of Young. As a result, we
rediscover equation (8)for the third time. All three methods are thus equivalent.
It is worth pointing out that the result in equation (16) can be obtained much more easily
if we presuppose that the force is localized close to the contact line, as we did implicitly in
the previous section when we neglected the contribution of fringe fields. Adapting the ideas
of Jones et al [34, 35, 39, 41], we can calculate the net force by choosing a sufficiently large
box around the contact line (see figure 5)thatthe electric field vanishes along most sections of
the closed area .For such a box, only the section along A–B (in figure 5)contributes to the
integral in equation (14). As a result, we obtain exactly the same expression as equation (16).
This means in particular that the net force pulling on the contact line is independent of the
dropletshape. The result alsoimpliesthatthe edge of any non-deformable, perfectlyconductive
body would experience exactly the same force [4]. This is in fact not surprising since the net
force calculated by integrating the Maxwell stress tensor must be the same as the one obtained
by minimizing the energy: the gain in electrostatic energy upon moving the contact line in
figure 5 by dx is given by the increment inthesolid–liquid interfacial area. The contribution
of the fringe fields remains constant—independent of the surface profile. (A derivation of
the electrowetting equation that makes use of this argument was given in [32].) This shape
independence of the force also implies that the contact angle reduction and the force should
be regarded as independent phenomena [42].
9
We will see in section 2.3.1 [43]thatthis assumption is not correct. However, the main conclusion with respect to
the contact angle remains valid.
Topical Review R715
A
B
z
x
Σ
Figure 5. Integration box for the calculation of the net force acting on the contact line.
2.3. Extensions of the classical electrowetting theory
2.3.1. Fine structure of the triple line. In the previous section, we discussed the response of
the liquid on a mesoscopic scale. The impact of the fringe fieldsontheliquid surface in the
vicinity of the TCL was ignored. If we look at the surface profile within that range, the liquid
surface is expected to be deformed, as first noted by Vallet et al [31]. In order to calculate
the equilibrium surface profile, Buehrle et al [43]proceeded in analogy with conventional
wetting theory in the presence of molecular forces ([26]; see alsosection 2.1): in mechanical
equilibrium, the pressure p across the liquid–vapour interface must be independent of the
position on the surface. Therefore any electrostatic pressure P
el
(r) = ε
0
E(r)
2
/2closeto the
TCL must be balanced by an additional curvature of the surface such that
σ
lv
κ(r) − P
el
(r) = p = const. (17)
Compared to conventional wetting theory, there is one major difference: while the
disjoining pressure at a given position r dependsonlyonthe film thickness at that position [26],
the electric field and thus P
el
(r) depends on the global shape of the droplet. Thus the droplet
shape and the field distribution have to be determined self-consistently. Buehrle et al [43]
addressed this question for the case of droplets of infinite radius. They chose an iterative
numerical procedure, which involved a finite element calculation of the field distribution for
atrialsurface profile followed by a numerical integration of equation (17)toobtain a refined
surface profile. The calculation was a two-dimensional one, i.e. possible modulations of the
profile along the contact line were not included. The procedure was found to converge to
an equilibrium profile after a few iteration steps. The following main results were found.
(i) The surface profiles are indeed curved, as sketched in figure 6.Thecurvature of the surface
profiles and thus the electric field diverges algebraically at the TCL, as in [40](with a different
exponent, however). (ii) The asymptotic slope of the profile at the substrate remains finite and
corresponds to θ
Y
,independently of the applied voltage.This is only possible because the
divergence of the curvature is very weak. In fact, Buehrle et al [43]confirmed analytically
that P
el
∝ r
ν
with an exponent −1 <ν<0. (iii) The apparent contact angle θ is in agreement
with the electrowetting equation (equation (8)) up to the highest values of η investigated
(corresponding to θ = 5
◦
). In view of the discussion in the previous section, this result is
not unexpected. It also implies that contact angle saturation does not occur within a two-
dimensional electromechanical model—in contrast to some arguments in the literature [40].
Recently, Papathanasiou and Boudouvis repeated the same calculation for droplets of finite
size using a slightly different numerical scheme [36]. Except for small deviations from the
electrowetting equation, which may be due to the finite size of their system, they reproduced
most of the results presented in [43].
Despite the striking difference between the apparent contact angle and the local one at the
contact line, the calculations showed that the surface distortions are significant only within a
rather small region of O(d) around the TCL. From an applied point of view, this allows for
R716 Topical Review
2
1
0
0123
z
x
gas
liquid
f(z)
η
Figure 6. Equilibrium surface profiles (θ = 60
◦
; η = 0.2, 0.4,...,1.0; ε
d
= 1). Reprinted with
permission from Buehrle et al [43]. Copyright 2003 by the American Physical Society.
the comforting conclusion that the simple models, as described in section 2.2,aresufficient as
long as the phenomena of interest occur on a length scale larger than d.
2.3.2. Electrolyte properties. Typical liquids used in electrowetting are aqueous salt
solutions. They are conventionally described as perfect conductors with surface charges
perfectly screening any external electric field. Microscopically, however, external fields are
screened by an inhomogeneousdistribution of ions close to the electrolyte surface. For typical
ion concentrations,the penetration depth of the electric field is of the order of a few nanometres,
given by the Debye length κ
−1
,with κ =
(
n
b
i
q
2
i
)/ε
0
ε
l
k
B
T .(Thesumruns over all the
ionic species i,with n
b
i
and q
i
being the bulk concentration and the charge of the ith species.
k
B
is the Boltzmann constant and T is the temperature.) The local ion concentration and the
electrostatic potential φ are coupled via the Poisson–Boltzmann equation [28]
∇
2
φ =−
i
q
i
n
b
i
ε
0
ε
l
exp(−q
i
φ/k
B
T ). (18)
Within this framework, the osmotic pressure
=
n
b
i
k
B
T (exp(−q
i
φ/k
B
T ) −1) (19)
of the ions has to be taken into account as an extra contribution to the free energy, such that
the last term in equation (10)isreplaced by [44]
F
el
=
1
2
E
D +
dV =
ε
0
ε
l
2
(∇φ)
2
+
dV . (20)
In order to calculate F
el
,asolutionofthe Poisson–Boltzmann equation is required. For
two specific situations the relation between and φ can be simplified considerably: when
qφ/k
B
T 1, the Poisson–Boltzmann equation (as well as the expression for ) can be
linearized. In this case, one obtains = ε
0
ε
l
κ
2
φ
2
/2. One should note that k
B
T/e ≈ 25 mV
(e:elementary charge) at room temperature. Hence the applicability of the linearized Poisson–
Boltzmann equation is limited to situations where the potential drop within the liquid is rather
small. (While this is usually fulfilled in the centre of the droplet, deviations should be expected
close to the TCL, where the electric field strength diverges (see the preceding section).) The
second simple situation corresponds to having monovalent salt solutions with q
1
=−q
2
= ze
(z:valency) and n
b
1
= n
b
2
.Inthis case, hyperbolic sine and cosineterms appear in equation (18)
and in the expression for ,respectively.
Topical Review R717
If we consider only the contributions of these terms to the energy per unit area of the solid–
liquid interface (i.e. neglecting fringe field contributions), the problem is a one-dimensional
one. Using appropriate boundary conditions (fixed potentials on theelectrode and in the bulk
liquid), analytic expressions for both φ and F
el
/A
sl
can be obtained. The latter is a correction
to the electrostatic contribution in equation (11), which ultimately leads to a correction to the
electrowetting number η in the electrowetting equation (8). More specifically, one obtains for
the linearized Poisson–Boltzmann equation [45]
η
lin
= η ·
1
1+ε
d
λ/ε
l
d
(21)
where λ = κ
−1
.Inthecase of monovalent salts, the result is
η
mv
= η ·
(1 −φ
0
)
2
+
16κd
ν
2
ε
l
ε
d
sinh
2
(νφ
0
/4)
(22)
where ν = eU/k
B
T . φ
0
is the potential (in units of U )atthe solid–liquid interface. It is
given by the solution of the equation 1 − φ
0
− (2ε
l
κd)/(ε
r
ν) · sinh(νφ
0
/2) = 0(constant
potential boundary conditions [45]). With λ d in typical experiments, it is obvious that the
correction in equation (21)issmall. Numerical solutions for φ
0
show that the same is true for
the correction in equation (22). As already indicated in section 2.2.1,corrections due to the
double layer thus have a rather weak effect on the apparent contact angle.
Kang et al [46, 47]aswellasChou [48]went one step beyond the above calculation and
analysed specifically the contribution arising from the vicinity of the TCL. They calculated
the electrostatic contribution to the line tension τ
e
,i.e.totheexcess free energy per unit
length on the contact line. This excess energy arises from the overlap of the double layers
originating from the solid–liquid and from the liquid–vapour interfaces. In [46], analytical
results for τ
e
were obtained for wedge-shaped surface profiles within the linear approximation
of the Poisson–Boltzmann equation. τ
e
wasfound to be of the same order of magnitude as the
molecular line tension, i.e. 10
−12
−10
−10
Jm
−1
.Numerical solutions for the full non-linear
Poisson–Boltzmann equation produced similar results. Like the line tension of molecular
origin, the impact of this electrostatic line tension will hence be negligible for droplets with
adiameter of, for example, a hundred nanometres or more. This conclusion is supported by
numerical calculations of equilibrium surface profiles based on the full Poisson–Boltzmann
equation in analogy to the discussion in section 2.3.1 [45].
2.3.3. AC electric fields. The theoretical treatment of electrowetting as discussed so far was
based on static considerations. In the case of slow variations of the applied voltage, the contact
angle and droplet shape can follow adiabatically the momentary equilibrium values. If the AC
frequency exceeds the hydrodynamic response time ofthe droplet (for typical millimetre-sized
droplets at frequencies exceeding a few hundred hertz), the liquid response depends only on
the time average of the applied voltage, i.e. the RMS value has to be used in equation (8).
This statement is correct as long as the basic assumptions in the derivation of the Lippmann
equation are not violated: one of them, the assumption that the liquid can be treated as a perfect
conductor, however, breaks down upon increasing the frequency. While the dissolved ions can
follow the applied field at moderate frequencies and thus screen the electric field from the
interior of the liquid, they are not able to do so beyond a certain critical frequency ω
c
.Far
below ω
c
,the liquid behavesas a perfect conductor;far above it behavesas adielectric. (Electric
field-induced actuation of liquids beyond ω
c
is still possible. However, the forces in that range
are dielectric body forces. For a review on dielectrophoresis; see [10].) For homogeneous bulk
R718 Topical Review
l, ε
1
, σ
1
d, ε
d
, σ
d
Figure 7. Capillary bridge between bare and insulator-covered electrode (see the text for details).
(a) (b)
Figure 8. Frequency dependence of contact angle (insulator: 1 µmthermally grown Si oxide,
hydrophobized with a monolayer of octadecyltetrachlorosiloxane). Droplet: salt (NaCl) water;
conductivity: 0.2mScm
−1
;diameter: approximately 2 mm; ambient medium: silicone oil;
U
RMS
= 50 V. (a) f = 1kHz. (b) f = 20 kHz (reproduced from [83]).
liquids, the critical frequency for which ohmic and displacement currents are equal is given
by [49]
ω
c
=
σ
l
ε
l
ε
0
(23)
where σ
l
and ε
l
are the conductivity and the dielectric constant of the liquid, respectively.
Foranaqueous salt (NaCl) solution with a conductivity of 0.1Sm
−1
(≈10
−4
mol l
−1
),we
have ω
c
= O(10
8
s
−1
).Fordemineralized water (σ = 4 × 10
−6
Sm
−1
)ω
c
is as low as
4 ×10
3
s
−1
.Therelevant critical frequency in electrowetting, however, depends not only on
the intrinsic properties of the liquid but also on the geometric and electric properties of the
insulating layer. For instance, the characteristic time constant τ
c
for charge relaxation in the
configuration sketched in figure 7 is given by
10
τ
c
= ε
0
ε
d
+ ε
l
d
l
σ
d
+ σ
l
d
l
. (24)
Using, for instance, d = 1 µm, l = 1 mm, ε
d
= 2, ε
l
= 81, σ
d
= 0, and σ
l
= 0.1Sm
−1
(as above), we obtain 2π/τ
−1
c
≈ 4 × 10
7
s
−1
,whereas for demineralized water, we have
2π/τ
−1
c
≈ 1.6 ×10
3
s
−1
.
Figure 8 illustrates this breakdown of electrowetting at high frequency for a millimetre-
sized droplet of demineralized water. At high frequency, a substantial fraction of the voltage
that is applied to the wire drops within the droplet. Therefore, both the voltage at the contact
lineand thus the energy gain upon moving the latter are reduced. The continuous nature of
the transition from conductive to dielectric behaviour as a function of frequency is illustrated
in figure 9 for various salt concentrations.
The details of the contact angle response to the electric field are rather complex and
geometry dependent when the field penetrates substantially into the liquid. The transition
10
This equation can be derived using the boundary conditions forthe electric and displacement fields at the solid–
liquid interface in combination with the continuity equation [49]. An alternativederivation based on equivalent circuit
diagrams was described by Jones et al (e.g. [34]) for the specific case δ
d
= 0.
Topical Review R719
140
120
100
01020
[kHz]
[°]
θ
ν
Figure 9. Contact angle θ versus frequency (see figure 8 for experimental details). Conductivity:
1850 µScm
−1
(squares), 197 µScm
−1
(circles), 91 µScm
−1
(diamonds), and 42 µScm
−1
(triangles). θ
Y
and θ are shown as dashed and solid lines, respectively. Reprinted from [50].
from low frequency electrowetting behaviour to high frequency dielectrophoretic behaviour
is much better illustrated in experiments that measure the forces exerted by the electric fields.
Jones et al performed a series of experiments in which they studied the rise of liquid in
capillaries formed by two parallel electrodes at a distance D, each covered with an insulator
(figure 10(a)) [35, 41]. (Other examples where finite conductivity effects play a role will be
discussed in sections 5.3 to 5.5.) The authors modelled the liquid as a capacitor in parallel
with an ohmic resistor; see figure 10(a). The electric fields within the different materials can
be calculated from elementary electrostatics. Using either the Maxwell stress tensor or the
derivative of the total electrostatic energy with respect to the height of the liquid, a frequency
dependent expression for the electric force pulling the liquid upwards is obtained. Balancing
this force with gravity, Jones et al [35] obtained an expression h = K (ω)U
2
with an analytical
function K (ω).Thelowand high frequency limits are given by
h =
ε
d
ε
0
4ρ
l
gdD
U
2
; ω ω
c
(ε
l
−1)ε
0
2ρ
l
gD
2
U
2
; ω ω
c
(25)
where ρ
l
is the density of the liquid and g is the gravitational acceleration. The critical
frequency ω
c
= 2σ/(D · (2c
l
+ c
d
)) involves the capacitances per unit area c
l
and c
d
of the
liquid and of the insulating layer, respectively. Figure 10(b) shows K (ω) determined from
aseriesofexperimental height of rise versus voltage curves. Good agreement was achieved
with model calculations based on independently measured liquid properties.
In addition to this frequency dependent reduction of the rise height, Jones et al [35]also
observed a deviation from the predicted parabolic voltage dependence at high voltage. This
observation is in qualitative agreement with earlier experiments [30], in which electrowetting-
induced capillary rise was investigated using DC voltage. According to those authors, the
deviation from parabolic behaviourin h(U ) coincided with the onset of contact angle saturation
on a planar substrate made of the same material.
3. Materials properties
In classical electrowetting theory, the liquid istreated as a perfect conductor. For aqueous salt
solutions this corresponds to the limit of either high salt concentration or low frequency, as
R720 Topical Review
c
d
c
d
c
air
c
1
g
1
c
d
c
d
air
liquid
V
electrode
dielectric layer
dielectric layer
electrode
2.5
2.0
1.5
1.0
0.5
0.0
DC 10 100 1k 10k 100k 1M
Coefficient, K (10
-5
cm/v
2
)
(b)(a)
Frequency, f (Log: Hz)
DI water
Mannitol (20mM)
KCl(1mM)
Theory: DI water
Theory: Mannitol
Theory: KCl
Figure 10. Pellat experiment: electrowetting-induced capillary rise. (a) Schematic set-up and
electric equivalent circuit. (b) Frequency dependence of K (w) for DI water, mannitol, and KCl.
Reprinted with permission from [35].
discussed in the preceding section. The requirements regarding the concentration and nature
of charge carriers are not very stringent. At low frequency ( f < 1kHz),evendemineralized
water displays substantial electrowetting [35, 50](seealsofigures 8 and 10). Frequently,
experiments are performed with salt concentrations of the order of 0.01–1 mol l
−1
.Most
authors report no significant influence due to the type or concentration of the salt (see for
instance [24, 32]). However, Quinn et al [51]found systematic pH dependent deviations from
equation (8), which they attributed to specific adsorption of hydroxyl ions to the insulator
surfaces. Electrowetting was also observed for mixtures of salt solutions with other species
(e.g. glycerol [52–54], ethanol [31, 55]) without deterioration of electrowetting performance.
In particular, electrowetting also occurs in the presence of biomolecules such as DNA or
proteins [5, 21, 56, 57]and has even been demonstrated for physiological fluids [5]. One
complication with biological fluids is, however, that the performance can be affected by
unspecific adsorption of biomolecules to the surfaces [56]. Adsorbed biomolecules generally
reduce θ
Y
and increase contact angle hysteresis. Room temperature ionic liquids [55]were
also shown to display electrowetting. Electrowetting is thus a rather robust phenomenon that
depends only weakly on the liquid properties.
In contrast, the properties of the insulatinglayers are much more critical. Substantial
activities have been aimed at optimizing the properties of these layers in order to minimize the
voltage required for contact angle reduction. At the same time, the materials used should be
chemically inert and stable in order to ensurereproducibility and a long lifetime. Two main
criteria can be derived immediately from equation (8): first, the contact angle at zero voltage
should be as large as possible,in order to achieve a largetuning range and,second, the dielectric
layer should be as thin as possible. The first choice can be met by either using an intrinsically
hydrophobic insulator, such as many polymer materials, or by covering hydrophilic insulators
with athin hydrophobic top coating. One possible top coating is self-assembled monolayers
(e.g. silanes on glass or SiO
x
)[54]. More frequently, however, thin layers of amorphous
fluoropolymer(Teflon AF or Cytop) are used. These materialscan be depositedby spincoating
or by dipcoating. Depending on the solution concentration and on the deposition parameters,
layers with a thickness ranging from a few tens of nanometres to several micrometres [58] can
be produced. Apart from being hydrophobic,Teflon-like layers can be prepared as very smooth
Topical Review R721
200
150
100
50
0
0246810
required EW voltage
U
BD
d [µm]
U [V]
Figure 11. Electrowetting and dielectric breakdown voltage versus insulator thickness. Solid
line: voltage required for a contact angle decrease from 120
◦
(0 V) to 70
◦
(for ε
d
= 2; σ
lv
=
0.072 J m
−2
). Dashed line: critical voltage for dielectric breakdown (for EBD = 40 V µm
−1
).
with very small contact angle hysteresis (<10
◦
for water in air). The material is chemically
inert and resists both acids and bases. Seyrat and Hayes [58]developed a preparation protocol
that leads to very homogeneous Teflon AF layers with high dielectric strength (≈200 V µm
−1
).
Furthermore, amorphous fluoropolymer has become very popular, not only as a top coating
but also as an insulating layer [19, 20, 51, 53, 58–60].
The critical materials parameter for the insulator is its dielectric strength, or the breakdown
field strength E
BD
.Thisnumber limits the minimum thickness of the insulating layer: the
voltage required to achieve a desired variation of the contact angle cos θ is given by [58]
U( cos θ) = (dσ
lv
cos θ/ε
0
ε
r
)
1/2
. (26)
Dielectric breakdown occurs at U
BD
= E
c
d.The competition between the two effects
is illustrated in figure 11.Theintersection between the square root function and the straight
line determines the minimum insulator thickness required to obtain a certain cos θ for a
given dielectric strength—implying a corresponding minimum voltage. The latter can only be
reduced by improving the dielectric strength or by using a different material. There are two
limitations to this procedure: (i) in view of the diverging electric fields closetothecontact line,
the breakdown voltage may be exceeded locally, although U/d is still smaller than E
BD
;and
(ii) the dielectric strength of thin layers may differ from the corresponding bulk values [58].
Popular inorganic insulator materials include SiO
2
[34, 56, 61–64]andSiN[61, 65, 66].
Thin layers with a high dielectric strength can be produced using standard vacuum deposition
or growth techniques. In combination with a hydrophobic top coating, they perform well
as electrowetting substrates. Compared to Teflon AF (as an insulator), they also offer the
advantage of a higher dielectric constant, which contributes to reducing the operating voltage
further (see equation (26)). The dependence on the dielectric constant prompted Moon et al
[61]tostudy thin layers of a ferroelectric insulator (barium strontium titanate (BST)) with a
specifically high dielectric constant of ε
d
= 180. For a 70 nm BST layer covered by 20 nm
Teflon AF, they achieved a contact angle reduction of 40
◦
with an applied voltage of 15 V.
Polymer materials that were used in previous electrowetting studies include parylene-N
and parylene-C [22, 30, 32, 35, 37, 41, 67], conventional Teflon films [31, 67, 68],
polydimethylsiloxane (PDMS) [69, 70], as well as various other commercial polymer foils
of variable surface quality [31, 69, 71, 72]. Parylene films are deposited from a vapour phase
of monomers, which polymerize upon adsorption onto the substrate. The surfaces are known
to be chemically inert and robust and display a high dielectric strength (200 V µm
−1
;[30]).
R722 Topical Review
In electrowetting, Parylene is almost exclusively used in combination with hydrophobic top
coatings. One important advantage of the Parylene coatings is that the vapour deposition
process allows for uniform conformal coatings on topographically patterned substrates,
including the interior of capillaries [22].
Recently, Chiou et al [73]presentedan interesting new approach making use of
a photoconductive material, which allowed themtoswitch the electrowetting behaviour
optically—aprocessthe authors termed‘optoelectrowetting’. Theadvantageof thisapproachis
that individual addressing of electrodes in a digital microfluidic chip does not require individual
electrical connections to allelectrodes (20 000 in [73]). Electrode activation is achieved by
directing a laser beam onto the desired electrode.
4. Contact angle saturation
The parabolic relation between theobservedcontact angleand theapplied voltage(equation (8))
was shown experimentally to hold at low voltage. At high voltage, however, the contact angle
has always been found to saturate. In particular,novoltage-induced transition from partial to
complete wetting has ever been observed. (On the basis of equation(8), such a transition would
be expected to occur at U
spread
= (2σ
lv
d(1 − cos θ
Y
)/(ε
0
ε
d
)).Instead, θ adopts a saturation
value θ
sat
varying between 30
◦
and 80
◦
,depending on the system [24, 30–33, 37, 61, 71];
see alsofigure 3.) It has now become clear that the linear electrowetting models described
in section 2 cannot explain the phenomenon of contact angle saturation [36, 43]. However,
thelatter studies showed that the electric field strength diverges close to the TCL. Although
the divergence is cut off at small length scales (κ
−1
,i.e.afewnanometres), the field strength
is expected to reach very high values—several tens or hundreds of volts per micrometre. So
far, no consistent picture of contact angle saturation has emerged. Nevertheless, a number of
mechanisms have been proposed to explain various observations:
(i) Verheijen and Prins [32]found indications that the insulator surfaces were charged
after driving a droplet to contact angle saturation. They suggested that charge carriers are
injected into the insulators, as sketched in (figure 12). These immobilized charge carriers then
partially screen the applied electric field. In order to quantify the effect, they assumed that the
immobile charges are located at a fixed depth within the insulating layer and that their density
σ
T
is homogeneous within a certain range (≈d) on both sides of the contact line
11
.With these
assumptions, they derived a modified version of equation (8):
cos θ = cos θ
Y
+
ε
0
ε
d
2dσ
lv
(U −U
T
)
2
, (27)
where U
T
is the potential of the trapped charge layer outside the droplet, i.e. σ
T
= ε
0
ε
d
U
T
/d.
σ
T
and, thus, also U
T
are unknown functions of the applied voltage that depend on the (non-
linear) response of the insulator material. The authors determined these functions by fitting
equation (27)totheirexperimental data. The result was self-consistent, but it was not possible
to establish a correlation between this threshold behaviour andotherknownmaterial parameters
or a microscopic process that could be responsible for the charge trapping. Papathanasiou and
Boudouvis [36]triedtoestablish such a correlation by comparing numerically computed values
for the electric field strength at the contact line (averaged overa certain area) with the dielectric
strength of a variety of dielectric materials used in electrowetting experiments. The authors
reported good agreement with published experimental saturation contact angles. However, it
11
These assumptions seem somewhat artificial for homogeneous dielectric layers. However, it should be recalled
(see section 3)that many electrowetting experiments—including that of Verheijen and Prins [32]—are performed on
composite substrates made of a thicker main insulating layer and a thin hydrophobic top coating.
Topical Review R723
vapour
dA cos θ
liquid
V
V
insulator
dA
d
dσ
L
dσ
M
σ
M
σ
L
θ
vapour
trapped
charge
dA cos θ
liquid
insulator
dA
dd
1
d
2
dσ
L
dσ
M
σ
M
σ
T
σ
T
σ
L
θ
(a)
(b)
V
σ
M
L
Figure 12. (a)Schematic picture of the virtual displacement of the contact line in the presence
of a potential across the insulator. An infinitesimal increase in base area d A at fixed voltage V
changes the free energy of the droplet, as a result of a change in interface area and the placement
of additional charge dσ
L
and image charge dσ
M
.(b) The virtual displacement of the contact line in
the presence of a sheet of trapped charge. Now, the infinitesimal increase d A alters the free energy
not only via the charge distribution between the electrode and the liquid but also via the charge
distribution below the vapour phase.Reprinted with permission from [32].
should be noted that the agreement is sensitive to the size of the box used to average the electric
field. The specific choice of 100 nm in [36]isnot obviously related to any physical length
scale of the system (such as κ
−1
).
(ii) Vallet et al [31] observed two other phenomena that can coincide with contact angle
saturation. They found that the contact line of salt solution droplets luminesces at high voltage.
Light was found to be emitted in a series of short pulses with durations of less than 100 ns. The
wavelength of the emitted light was verified to correspond to known emission characteristics
of several ambient gas atmospheres. Simultaneously with the optical observation, the authors
measured the current in the system: a series of spikes occurred at the same time as the optical
emission pulses, indicating discrete discharge events. Within reasonable errorbars,theonset
voltage for both processes agreed with the contact angle saturation voltage. Both effects were
attributed to the diverging electric field strength close to the contact line.
(iii) In the same publication, Vallet et al [31]alsoreported another phenomenon that
occurred only for low conductivity liquids (deionized water and water–ethanol mixtures). For