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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