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Journal ArticleDOI

Electrowetting: from basics to applications

TL;DR: In this paper, the authors 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.
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 in this 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-related microfluidic devices.

Summary (4 min read)

Historical record

  • The study of electromotive forces of contact has up to now provided a complete chapter distinct from the study of capillary phenomena with variable results.
  • They are linked to motion of mercury electrodes.

A.1.1 Definitions

  • The name capillary constant or interfacial tension has been given to the coefficient A of the Laplace’s equation (A.1): p = A ( 1 R + 1 R′ ) (A.1) p is the normal pressure in any point of the surface; R and R′ are the principal radii of curvature in this point.
  • Electrical equilibrium being established, the electrical potential inside the mercury is known to have a unique value V ; likewise the electrical potential inside the body of water (or acidic water) has a uniform value V0.
  • I will call this difference electrical difference at the water-mercury surface.
  • This being defined, experiments show that if the apparatus is set such that x takes a fixed value x0, A takes a value A0 perfectly defined, i. e. the perturbations observed with the ordinary settings do not occur.
  • If x is set to another value x1, A takes another value A1, value perfectly determined and without perturbations.

A.1.2 First method

  • A vertical glass capillary GG’ communicates at its bottom with a mercury reservoir A (see Fig. A.1).
  • This mercury penetrates into the capillary tube where it undergoes a depression.
  • The same apparatus has been used to demonstrate that if the electrical difference takes a new value x1, the capillary constant takes a new value A1 which remains constant as long as x1 does.
  • The microscope is left in place and the meniscus is moved back to its original position, given by the horizontal mark of the retic- 8NOTE:.

A.2 Second Law

  • When this surface is mechanically deformed, in order to increase its area, the value of the electrical difference at the surface increases.
  • One can set the experiment has follow: two glass beakers containing both mercury covered by acidic water are set side by side; they are set in electrical communication together using a coton wick or some filter paper.
  • One will notice besides that the First Law and the Second Law govern two reversed series of phenomena.
  • Inversely, when the surface is increased, the electrical difference increases; this is the Second Law.
  • When a liquid surface is deformed the electrical difference varies in such a way that the interfacial tension developed under the terms of the First Law opposes to the continuation of the motion.

A.3.1 Notations

  • One can graphically represent the state of the surface of contact for each time as follow.
  • When the state of the surface changes, P moves along a certain curve.
  • The external work dT supplied during a deformation infinitely small of the surface has for expression dT = −A× dS This work is thus graphically represented by the area generated by the displacement of the ordinate A. dq being the quantity of electricity flowing through the surface from water to mercury, one can write: dq = XdS +.
  • Y Sdx X represents the quantity of electricity flowing through the surface when the surface S varies of 1 mm2, x being kept constant.
  • One can obtain this condition by two different ways.

A.3.2 Conservation of energy

  • I consider first the case where the representative curve is closed.
  • When the quantity of electricity dq flows through the surface from water to mercury, the charge of the water volume (whose potential is V0) decreases of a quantity dq; the electrical energy of the free-surface of water decreases of V0 × dq.
  • After having followed a closed cycle, the state of the surface is the same as in the origin, the external work supplied is thus necessarily equal to the decrease of the electrical energy; one has then for any closed cycle∫ xdq = − ∫ AdS, or ∫ (xdq + AdS) = 0.
  • Let φ be the total inner energy of the surface unit water - mercury.
  • The hydrogen has not been given off neither.

A.3.4 Consequences: Expression of the two electrical

  • Capacities of a surface Eq. A.4 and Eq. A.2 determines X and Y : X = −dA dx (A.5) and Y = −d 2A dx2 (A.6) These equations can be translated in everyday words: 1. The electrical capacity of the surface unit at constant electromotive force is equal to the opposite ot the first derivative of the interfacial tension.
  • The electrical capacity of the surface unit at constant surface is equal to the opposite of the second derivative of the interfacial tension.
  • One will notice that the signs are in agreement with the experiments.
  • The first of these consequences has already been demonstrated using different arguments (see Chapter A.2.3)11.

A.4 Electrocapillary motor

  • Description and func- tioning of the machine A mercury surface running on a closed cycle is a device able to transform an indefinite quantity of electricity into mechanical work, or inversely, without undergoing any changes, some electrical work is supplied or absorbed depending on the direction of the motion of the point P on the cycle (clockwise or anticlockwise).
  • The apparatus having been built up symmetrically, the vertical thrust undergone by the bundles BB and transmitted to the arcs UU are in equilibrium at then end of HH, like the weights of the plates of a balance are in equilibrium at the end of the beam.
  • The electrical motors have the same efficiency as the electromagnetic motors [Verdet13].
  • When the mercury surface is allowed to contract it executes a mechanical work; in the same time its electrical difference decreases, as well as its capillary constant which depends on it, unless a current keeps these quantities constant.
  • There is then absorption of electrical work.

A.5 Measurement of the capillary constant of

  • Necessity to measure the capillary constant keep- ing the electrical difference constant The apparatus of the Fig. A.1 allows to keep the electrical difference constant and to measure in the same time the capillary constant.
  • One sees that the numbers of the column V do not differ from their average more than 1/10 mm.
  • These variations are the source of the perturbations.
  • Let us assume for instance that at the bottom of the beaker containing the water a large drop of mercury is introduced using a pipette in order to measure its height and to deduce from that the value of the capillary constant, method used by Mr. Quincke.

A.6.1 Description of the apparatus

  • Experiments have shown that the laws stated in the Chapter A.1 can be applied to the measurement of electromotive forces.
  • In the tube A, a mercury column is poured, sufficiently high so that the mercury penetrates by its own pressure in the capillary tip (750 mm for example).
  • The microscope is then moved to put the zero of the micrometer tangent to the meniscus; the microscope will then always keep this position.
  • Measurement of the electromotive force of a Daniell cell loaded with acid.
  • The capillary electrometer has been used for a large number of measurements;.

A.6.4 Properties of the electrometer. Invariability. Ac-

  • The invariability of the indications of the capillary electrometer has always been perfect.
  • When the apparatus is charged the mercury level goes out of the origin; when it is discharged by connecting α and β with a metal wire the mercury comes back immediately and perfectly at zero.
  • The divisions were sufficiently separated to estimate a fraction of this interval, for example 1/6.
  • In both experiments one can check directly using an auxilliary electrometer that the interuption of the motion is due to the electrization that the motion produces following the Second Law.

A.7.1 Historical background

  • Henry from Manchester has observed in 180019 that the mercury he was using as an electrode changed its shape while tarnishing.
  • Draper24 in 1845 observed that the depression of the mercury column in a capillary tube filled with water decreases with the flow of an electrical current from water to mercury; this physicist admitted as an explanation that water acquires the property to wet the mercury during the flow of the current.
  • Assuming a stationary state, if one considers lines of equal electrical differences that could be drawn on the surface, these lines would be in the same time lines of equal capillary tension (First Law, Chapter A.1); thus each point of the surface will then have a tangential velocity perpendicular to these lines of constant tension.

Summary

  • Two distincts laws have been demonstrated by the experiments.
  • The First Law (Chapter A.1) links the capillary constant to the electrical difference.
  • These two laws, established separately from the experiments are linked together by a crucial theoretical argument.
  • The electrometer described in the Chapter A.6 is the most accurate of all the electrometers known up to know.
  • The Chapter A.7 gives the explanations of Gerboin’s whirls.

A. – The junction angle

  • The junction angle between glass and mercury under pure or acidic water is equal to zero at the condition that the water wets the glass.
  • This condition is fulfiled with water only for a short time; with acidic water for several hours or even days.
  • When the motor described above is left a long time without working it is wise to raise the bundles above the mercury to wet them again with the acidic water.
  • The same condition holds for the electrometer after any deviation in the instrument.
  • 28The hypothesis which assigns the variation of the electrical difference at the electrodes to the chemical action of the current (which leads to the terms polarization by hydrogen, by oxygen) has neither been called nor discussed in the present work; I will try to show that the chemical action and the electrical phenomenon can be produced separately.

B. – Precaution when checking the First Law

  • In the Chapter A.1 and A.6 it has been assumed all the time that the wire α is connected to the negative pole of the electromotive force included between α and β.
  • And when the electromotive force x0 stops being constant the value of x cannot be determined anymore (Chapter A.1); the zero of the apparatus has been changed.

C. – Measurement of the coefficient X

  • Measurements performed to determine the numerical value of X (Chapter A.3) have shown that X is independent on s; this value has been determined with only a low accuracy and by default.
  • The reason is that the values of x are getting closer to a constant value x0 with time and this faster when they are much different than x0; this kind of loss of charge has been exactly compensated during the verification of the First Law.

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Electrowetting: from basics to applications
Frieder Mugele, Jean-Christophe Baret
To cite this version:
Frieder Mugele, Jean-Christophe Baret. Electrowetting: from basics to applications. Journal
of Physics: Condensed Matter, IOP Publishing, 2005, 17 (28), pp.R705-R774. �10.1088/0953-
8984/17/28/R01�. �hal-02148730�

1
J. Phys. Condensed Matter: TOPICAL REVIEW
Electrowetting:
from Basics to Applications
Frieder Mugele
1,*
and Jean-Christophe Baret
1,2
1: University of Twente; Faculty of Science and Technology; Physics of Complex
Fluids; P.O. 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)
*: corresponding author
phone: ++31 / 53 489 3094; fax: ++31 / 53 489 1096; email: f.mugele@utwente.nl

2
Abstract. Electrowetting has become one of the most widely used tools to manipulate
tiny amount of liquids on surfaces. Applications range from lab-on-a-chip devices to
adjustable lenses or new types of electronic displays. In the present article, we review the
recent progress in this 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 the
contact angle reduction as well as 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 – is 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 not
only give rise 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, fiber optics, and biotechnology-related microfluidic devices.

3
1. Introduction
Miniaturization has been a technological trend for several decades. What started
out initially in the microelectronics industry has long reached the area 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. One prominent example from the area of fluid mechanics are Lab-on-a-
Chip systems for applications such as DNA- or 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 or 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 nanoliters 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 well as in microfluidcs. For liquid droplets of
submillimeter dimensions, capillary forces dominate [11, 12]. The control of interfacial
energies has therefore become an important strategy to manipulate droplets at surfaces
[13-17]. Both liquid-vapor and solid-liquid interfaces have been influenced in order to
control droplets, 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-vapor 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 and the substrate topography
together provide boundary conditions within which the droplets adjust their morphology
to reach the most energetically favorable configuration. For complex surface patterns,
however, this is not always possible as several metastable morphologies may exist. This

4
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 topographical patterns 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 by 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 value of the
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 [23].
Electrocapillarity, the basis of modern electrowetting, was first described in detail
in 1875 by Gabriel Lippmann [24]. This ingenious physicist, who won the Noble prize in
1908 for the discovery of the first color photography method, found that the capillary
depression of mercury in contact with electrolyte solutions could be varied by applying a
voltage between the mercury and electrolyte. He formulated not only a theory of the
electrocapillary effect but 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 in French up to now, available to a broader
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 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

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Abstract: Wetting phenomena are ubiquitous in nature and technology. A solid substrate exposed to the environment is almost invariably covered by a layer of fluid material. In this review, the surface forces that lead to wetting are considered, and the equilibrium surface coverage of a substrate in contact with a drop of liquid. Depending on the nature of the surface forces involved, different scenarios for wetting phase transitions are possible; recent progress allows us to relate the critical exponents directly to the nature of the surface forces which lead to the different wetting scenarios. Thermal fluctuation effects, which can be greatly enhanced for wetting of geometrically or chemically structured substrates, and are much stronger in colloidal suspensions, modify the adsorption singularities. Macroscopic descriptions and microscopic theories have been developed to understand and predict wetting behavior relevant to microfluidics and nanofluidics applications. Then the dynamics of wetting is examined. A drop, placed on a substrate which it wets, spreads out to form a film. Conversely, a nonwetted substrate previously covered by a film dewets upon an appropriate change of system parameters. The hydrodynamics of both wetting and dewetting is influenced by the presence of the three-phase contact line separating "wet" regions from those that are either dry or covered by a microscopic film only. Recent theoretical, experimental, and numerical progress in the description of moving contact line dynamics are reviewed, and its relation to the thermodynamics of wetting is explored. In addition, recent progress on rough surfaces is surveyed. The anchoring of contact lines and contact angle hysteresis are explored resulting from surface inhomogeneities. Further, new ways to mold wetting characteristics according to technological constraints are discussed, for example, the use of patterned surfaces, surfactants, or complex fluids.

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TL;DR: In this paper, the authors discuss the nature and properties of liquid interfaces, including the formation of a new phase, nucleation and crystal growth, and the contact angle of surfaces of solids.
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"Electrowetting: from basics to appl..." refers background or methods in this paper

  • ...The Thermodynamic and Electrochemical Approach Lippmann’s classical derivation of the electrowetting or electrocapillarity equation is based on general Gibbsian interfacial thermodynamics [32]....

    [...]

  • ...) The local ion concentration and the electrostatic potential φ are coupled via the PoissonBoltzmann equation [32]...

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Journal ArticleDOI
TL;DR: In this paper, the authors present an attempt towards a unified picture with special emphasis on certain features of "dry spreading": (a) the final state of a spreading droplet need not be a monomolecular film; (b) the spreading drop is surrounded by a precursor film, where most of the available free energy is spent; and (c) polymer melts may slip on the solid and belong to a separate dynamical class, conceptually related to the spreading of superfluids.
Abstract: The wetting of solids by liquids is connected to physical chemistry (wettability), to statistical physics (pinning of the contact line, wetting transitions, etc.), to long-range forces (van der Waals, double layers), and to fluid dynamics. The present review represents an attempt towards a unified picture with special emphasis on certain features of "dry spreading": (a) the final state of a spreading droplet need not be a monomolecular film; (b) the spreading drop is surrounded by a precursor film, where most of the available free energy is spent; and (c) polymer melts may slip on the solid and belong to a separate dynamical class, conceptually related to the spreading of superfluids.

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TL;DR: In this article, a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations.
Abstract: Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

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TL;DR: The first € price and the £ and $ price are net prices, subject to local VAT as discussed by the authors, and prices and other details are subject to change without notice. All errors and omissions excepted.
Abstract: The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. P.-G. de Gennes, F. Brochard-Wyart, D. Quere Capillarity and Wetting Phenomena

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"Electrowetting: from basics to appl..." refers methods in this paper

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    [...]

Frequently Asked Questions (16)
Q1. What are the contributions mentioned in the paper "Electrowetting: from basics to applications" ?

Mugele et al. this paper presented an English translation of the basic paper on electrowetting, from basics to applications, which was published by the French physicist and later Nobel prize winner Gabriel Lippmann. 

Since the liquid is also coupled capacitively to the substrate, the liquid-filled channel behaves as an electric transmission line. 

A major obstacle to broader applications waselectrolytic decomposition of water upon applying voltages beyond a few hundred millivolts. 

Temperature gradients as well as gradients in the concentration of surfactants across droplets give rise to gradients in interfacial energies, mainly at the liquid-vapor interface, and thus produce forces that can propel droplets making use of the thermocapillary and Marangoni effects. 

One challenge of dip-pen lithography is to deposit a sufficiently large amount of liquid onto the pen in a controlled fashion, in order to maximize the number of spots that can be written without refilling the pen. 

The field and charge distribution are found by solving the Laplaceequation for an electrostatic potential φ with appropriate boundary conditions. 

With basic fluid manipulation techniques being established, the next step towardsbiotechnological applications of electrowetting-based devices is to demonstrate the biocompatibility of the materials and procedures. 

The entire droplet (of millimeter size) was mixed within a few seconds, more than 100 times faster than purely diffusive mixing [92]. 

The chemical contribution σsl to the interfacial energy, which appeared previously in Young’s equation (eq. ( 3)) is assumed to be independent of the applied voltage. 

They are deposited onto a hydrophilic carrier by bringing the latter close enough to the hydrophobic surface such that the droplet is transferred by capillary forces. 

They chose an iterative numerical procedure, which involved a finite element calculation of the field distribution for a trial surface profile followed by a numerical integration of eq. ( 17) to obtain a refined surface profile. 

For a relatively rough Teflon surface (contact angle hysteresis ≈50°), the authorsdeduced a contact line friction coefficient ξ ≈ 4 Pa⋅s. 

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. 

One of the interesting properties of this electrowetting-based attenuator is its low power consumption (<1mW) along with the fact that no power is required to hold the droplet in either position after the switching process. 

Except for a few simple geometries, the morphologies of liquid droplets onpatterned surfaces have to be computed numerically by minimizing the functional in eq. ( 28) (under the constraint of constant volume). 

For sufficiently thin oil layers, the free energy (per unit area) of the oil film in a van der Waals system is given by [93])1( 212)( 2 2 d dUc d AdFoiloilddoil owsooil