Double-Layer Effects in Electrowetting with Two Conductive Liquids

Journal of The Electrochemical Society, 156 �1� P21-P28 �2009� P21

Double-Layer Effects in Electrowetting with Two ConductiveLiquidsCharles W. Monroe,a,*,z Michael Urbakh,b,z and Alexei A. Kornyshevc,z

aDepartment of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109–2136, USAbSchool of Chemistry, Tel Aviv University, Ramat Aviv, 69978 IsraelcDepartment of Chemistry, Imperial College London, London SW72AZ, United Kingdom

Interfaces between two immiscible electrolytic solutions �ITIESs� provide significant contact-angle variation at substantially loweroperating voltages than conventional electrowetting systems require. This voltage reduction has been justified elsewhere by atheory that rationalizes ITIES electrowetting in terms of the interfacial capacitances involved. Here, the previous theory isextended to allow more detailed descriptions of the various double-layer capacitances. Past work relied, in part, on the Gouy–Chapman theory, which incompletely describes the liquid/electrode interfaces commonly used for electrowetting. In order to assessthe impact of this incompleteness, it is shown how experimental measurements of the liquid/electrode specific capacitances can beincorporated when predicting the interfacial contact-angle variation with respect to applied voltage. The extended theory refinesestimates of the potential ranges over which contact angles vary, predicts more realistic response curves, and can aid the design ofvariable-focus lenses or display elements that exploit the low-voltage electrowetting of ITIESs.© 2008 The Electrochemical Society. �DOI: 10.1149/1.2971190� All rights reserved.

Manuscript submitted March 24, 2008; revised manuscript received June 25, 2008. Published November 10, 2008.

0013-4651/2008/156�1�/P21/8/$23.00 © The Electrochemical Society

“Electrowetting” refers to any situation where applied voltagesinduce shape variation of a fluid/fluid interface contacting a sub-strate. The magnitude of the effect varies with the strength of anelectric field, which is usually sustained by imposing a potentialdifference across the fluid/fluid interface. Several applications ofelectrowetting are in commercial development, within microfluidicdevices such as electronic display elements and variable-focuslenses.1-4

Most commercial implementations use a liquid/liquid interfaceformed between one conductive and one nonconductive liquid �thejunction between a nonconductive liquid and a conductive liquid isabbreviated here as an “n/c junction”�. In fact, many patents claimthat n/c junctions are necessary to induce a significant electrowettingeffect with liquid/liquid interfaces.1-3 Academic research has alsofocused mainly on n/c junctions for a number of years.4 Recently,however, the possibility has arisen that electrowetting systems basedon two conductive liquids—so-called biconductive systems—mayalso have favorable properties.

Theoretical studies of biconductive electrowetting have dealtwith interfaces between two immiscible electrolytic solutions�ITIESs�.5-7 There is a long history of ITIES research in electro-chemistry �for review, see Ref. 8�, but until recently, the uniqueproperties of this interface remained basically unnoticed by thephysicists, materials scientists, and electrical engineers developingelectro-optical devices. Equally, electrochemists, occupied by vari-ous alternative functionalizations afforded by ITIESs,9-11 were es-sentially unaware of their potential applicability in the field of vari-able optics.

ITIES electrochemistry.— Standard ITIESs are comprised of anaqueous electrolytic solution �for example, lithium chloride in wa-ter� and an organic electrolyte solvated by an oil �e.g., tetrabutylam-monium tetraphenylborate in nitrobenzene or 1,2-dichloroethane�.Large free energies of transfer, in the range of 25–30 kJ/mol, pre-vent ions in either phase from crossing the liquid/liquid interface.Thus, when subjected to a moderate applied voltage, two back-to-back electrical double layers form at the ITIES. Charge screening inthese double layers confines the potential drop to a region just a fewDebye lengths across. It follows that a high electric field, nearly1 V/nm, can be localized at an ITIES by applying a potential dropof just a few hundred millivolts across it.

Electrochemical studies show that traffic of ions through anITIES does not occur if the applied potential remains lower than the

* Electrochemical Society Active Member.z E-mail: [email protected]; [email protected]; [email protected]

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smallest ionic free energy of transfer in the two electrolytes �dividedby Faraday’s constant�. Although this means that voltage across theliquid/liquid interface cannot vary outside a range much larger than�250 mV from its potential of zero charge, the limitation is weakerthan it seems: ideal-polarizability windows for ITIES correspond tomuch larger voltage windows between the working and referenceelectrodes immersed in the two liquids.

Electrowetting systems should be restricted to negative voltagesas well, to avoid side reactions at the substrate/liquid boundaries.12

At positive voltages, crystal reorganization, anion adsorption, or oxi-dative processes may impede shape changes or degrade the fluids.Voltametric studies have shown that on gold substrates, someITIESs are ideally polarizable from 0 to − 700 mV vs Ag/AgCl.The analysis here shows significant interfacial shape change to bepossible within this window.

Conventional vs ITIES-based electrowetting.— Academic elec-trowetting studies typically employ a horizontal planar workingelectrode, with a liquid droplet on top of it, both of which are sur-rounded by a second, immiscible liquid, as shown in Fig. 1. �Systemgeometries for liquid lenses differ but operate on the same prin-ciple.� One liquid phase is usually organic, whereas the other isaqueous. To induce a deformation of the droplet, the potential of theworking electrode is varied. In systems based on an n/c junction, thepotential drop occurs between the working electrode and a counter-electrode immersed either in the droplet or the surrounding solution.As Fig. 1 shows, the potential in ITIES-based systems can be con-trolled relative to a reversible reference electrode.

Investigations with n/c junctions show that if the nonconductivefluid comprises the surrounding solution, tens of volts are needed toachieve significant variation of the liquid/liquid interfacial shape.4

When the nonconductive fluid comprises the droplet, this range low-ers to 1 or 2 V.13,14 Just 0.5 V, however, can achieve the same shapevariation in an ITIES-based system. This difference is significantbecause faradaic processes at the electrode/electrolyte interfacestypically become significant in the 0.5–1 V range. By reducing op-erating voltages, the use of ITIESs could improve the energy effi-ciency of electrowetting-based technologies, which is a critical re-quirement for portable applications.

Operating voltages are dictated primarily by the system capaci-tance and limited by secondary processes. Existing n/c devices mayrequire large voltages because nonconductive phases have very lowcapacitances. When a voltage is applied across an n/c junction, theresulting field is dispersed across the entire nonconductive liquidand is not intense enough to affect significant shape response unlessthe range of operating voltages is correspondingly wide. In ITIES-

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P22 Journal of The Electrochemical Society, 156 �1� P21-P28 �2009�P22

based systems, however, the localization of electric fields near thephase boundaries yields high interfacial capacitances that can reducevoltage requirements.

During device operation, excess charge density accumulates inthe liquids near the three-phase �liquid/liquid/substrate� contactline.6,15 This can encourage irreversible side reactions, which occurmore easily when liquids directly contact conductive electrodes. Toprevent electron transfer across the solid/liquid interfaces in n/c de-vices, the electrodes must be coated by insulating solids, causingunavoidable voltage losses. The operating voltages for ITIES-basedelectrowetting are small enough, and ideal-polarizability windowsare wide enough, to allow direct contact between the operating liq-uids and a conductive substrate.

Finally, n/c junctions limit the options for device designs becausetheir use disallows fine-tuning of liquid-phase mechanicalproperties.c Because the nonconductive liquids tend to be additive-free �e.g., pure hexadecane or silicone oil� and their electrochemicalproperties vary unpredictably with additive content, independentvariation of their mechanical and electrical properties may be im-possible. But with ITIESs, salt concentrations may be adjusted toalter properties of either liquid.

c For instance, one might want to equalize the densities of the two liquid phases tomake the system response independent of its orientation relative to the vertical orto tune their viscosities to optimize the response time.

Figure 1. Typical electrowetting geometries used in academic studies: �a�the droplet is organic and the surroundings, aqueous �the organic/substrateinterface pictured here has a lower surface energy than the aqueous/substrateinterface�, and �b� the liquid phases are switched �at zero voltage, �c wouldbe supplementary to the angle in �a�, as depicted�. Although the three-phasecontact angle �c differs between the images, �a� and �b� both depict crosssections of truncated-spherical droplets with equal volumes.


Structure of this work.— The basic principles of electrowettingwith ITIESs were formulated previously.5-7 Laws for the depen-dence of contact angle on electrode potential were presented, includ-ing typical contact angle vs voltage curves �referred to henceforth as“electrowetting curves”�. These electrowetting curves also rational-ized the commonly observed effects of contact-angle saturation anddroplet contraction or detachment. The analysis was based on theVerwey–Niessen �VN� theory of double-layer formation at theITIES and the classical Gouy-Chapman �GC� theory of the diffusedouble layer at the working electrode/solution interfaces.16-18

Although the VN theory has been shown to work very well forITIESs,19 the GC theory may only be considered complete when theelectrical response at the working electrode is dominated by diffuse-layer capacitance. Therefore, this paper extends the previous theoryto account for non-GC voltage response of the double-layer capaci-tances at the solution/electrode interfaces. Many theoretical papershave addressed such deviations in minute detail,20-23 but in caseswhere the specific adsorption of ions is negligible, experimental datahave been successfully reproduced by adding a potential-dependentinner-layer capacitance in series with the nonlinear GC diffuse-layercapacitance.24-27

Although there are limitations to considering inner and diffusedouble layers to be clearly distinguishable regions,23,28,29 this notionserves its purpose in the present study fairly well. All that is neededto get reliable predictions from the electrowetting theory is an ad-equate parametrization of the voltage dependence of every relevantinterfacial capacitance. Unfortunately, capacitance measurementsare unavailable for many electrode/electrolyte interfaces, particu-larly those involving organic electrolytic solutions. The tactic of thepresent study is to employ the simplest inner-layer model that rea-sonably reproduces literature data for the specific capacitance of aninterface between the electrode of interest and an aqueous electro-lytic solution, and then to use that model to estimate properties forthe relevant electrode/organic electrolyte interface.

Despite its limitations, the present approach takes the ITIES-based electrowetting theory away from qualitative, test-of-feasibilityterms and moves it toward producing more quantitative simulationsthat can rationalize the observed electrochemical reality. In particu-lar, because ITIES-based electrowetting systems require moderatevoltages �ideal-polarizability windows range up to several hundredmillivolts between the working and reference electrodes�, the incor-poration of inner-layer capacitances more precisely establishes themagnitude of the contact-angle response within experimentally sig-nificant ranges.

In principle, if measurements were available, one could usespline fits of literature data for the relevant interfacial capacitancesand solve the problem phrased in the Electrowetting Theory Over-view section to obtain electrowetting curves. It is much easier, how-ever, to employ analytical expressions for the voltage dependence ofcapacitance. As previously mentioned, VN theory describes ITIEScapacitance fairly well in terms of bulk solution properties withoutadditional fitting parameters. Solid/liquid interfaces, however, pose amore difficult problem, which is addressed in the Modeling Interfa-cial Capacitance section. This problem is solved by combining theGC diffuse-layer theory with the elementary Mott and Watts–Tobin�MWT� approximation for the inner-layer part.26 One might criticizethe use of such a primitive theory, but within the potential range ofinterest, the combined MWT/GC model is shown to fit existing ex-perimental data for 0.1 M aqueous KClO4 against polycrystallinegold,30 making any criticisms of the approximation used inessential.

The organic phase, however, raises a moot point. Experimentsare sparse; capacitances must be approximated for interfaces be-tween organic electrolytes and most metals. Still, MWT theory hasproved to model mercury/organic interfaces successfully as well.27

Thus, the gold/organic electrolyte interface is treated here with thesame MWT/GC model that was used to describe the aqueous phase,but with a pertinent modification of parameters. Although this ap-proximation is clearly an easy target for criticism, it is hoped that thepredictions in the Electrowetting Curves section will be nearly quan-


P23Journal of The Electrochemical Society, 156 �1� P21-P28 �2009� P23

titatively valid; it is hoped that they may also spark further investi-gation into the capacitances of interfaces between metals and or-ganic electrolytic solutions.

Electrowetting Theory Overview

System geometry and energy functional.— Consider the sce-nario pictured in Fig. 1, which consists of an electrolytic droplet�phase d� on a metal working electrode �phase e�, both covered byan ambient surrounding solution �phase s�. Each liquid is assumed tobe an ideal binary electrolyte. If buoyant forces can be neglected andthe droplet has an average radius of curvature greater than 1000Debye lengths,5 then the free energy change due to the presence ofthe droplet, �G, is7

�G = ��de − �se + Ede�� − �*� − Ese��Ade

+ ��ds + Eds��*��Ads + Vd�p �1�

where Vd is the droplet volume and �p is the Laplace pressure. Forthe interface between phases i and j �where i, j = e,d,s�, Aij is thesurface area of the interface, �ij is the surface tension measured at

zero applied potential, and Eij is the free-energy change per unitinterfacial area induced by the applied voltage.

The electrostatic energies, which also include effects due to en-tropy of mixing, can be obtained by solving the electrostatic equa-tions governing the three-phase system.7 They depend on various

potential drops: Ese depends on the applied voltage, �� theworking-to-reference potential drop, read by the voltmeter V in

Fig. 1�; Eds depends on the voltage partitioned to the liquid/liquid

interface, �* �the droplet-bulk-to-reference potential drop�, and Ede

depends on the voltage partitioned to the droplet/electrode interface,�� − �* �the working-to-droplet-bulk potential drop�.

ITIES constraint.— If the free energies of transfer are high forevery ion in phases d and s, then within the ideal-polarizabilitywindow �i.e., ��*� � 300 mV�, ions cannot cross the liquid/liquidinterface and the droplet phase remains electrically neutral overall.

This electroneutrality makes the Eds term in Eq. 1 significant �for

ion-permeable liquid/liquid interfaces, Eds and �* both vanish7�; italso constrains the droplet’s shape.

Net droplet electroneutrality can be expressed in terms of thesurface-charge densities at the droplet/electrode and droplet/surroundings interfaces, �de and �ds, respectively, by

Ade�de�� − �*� + Ads�ds��*

� = 0 �2�

This equation determines a self-consistent value for �*. It also re-arranges to

A��,�*� =

Ade

Ads= −

�ds��*�

�de�� − �*�

�3�

which ties the potential to liquid/liquid interfacial geometry, ex-pressed through the area ratio A.

Parameters and variables.— It is convenient to characterizeelectrolytic solution i in terms of its inverse Debye length, �i, de-fined for symmetric electrolytes as

�i = �2F2zi2ci

0iRT�4�

Here, F is Faraday’s constant, 0 is the permittivity of vacuum, R isthe gas constant, and T is the absolute temperature; zi, ci, and i arethe cation equivalent charge, the molar electrolyte concentration,and the dielectric constant of electrolytic solution i, respectively.The subscript on zi may be dropped if it is assumed that both elec-trolytes have the same equivalent charge, z = zd = zs. This allowsnotation to be simplified by introducing a dimensionless potential,


=zF�

RT�5�

For 1:1 salts at room temperature, is a potential in units of�25 mV. Several other dimensionless potentials with analogousdefinitions are used below �such as �, *, etc.�.

The electrostatic and capillary contributions to functional 1 arescaled by two dimensionless parameters, b and C, which arise natu-rally from diffuse double-layer properties in both liquids5

b = �RT

Fz20

�ds�d�s

�dsand C = � s�s

d�d�6�

Note that 0s�s and 0d�d are the diffuse-layer capacitances GCtheory predicts in the zero-voltage limit. Thus, in cases wherediffuse-layer capacitance dominates, b quantifies the relative scalesof the electrostatic and capillary energies at the liquid/liquid inter-face and C, the relative scales of the electrostatic energies stored atthe two metal/solution interfaces.

The electrostatic energies appearing in Eq. 1 are more or lessproportional to combinations of b and C. To account for this, onecan introduce dimensionless interfacial energies

Ese�� =Ese

4bC�ds, Ede�� −

*� =

CEde

4b�ds,

and Eds�*� =

Eds

4b�ds�C + �1/C��7�

which show how diffuse-layer properties scale terms in the free-energy functional.5

When inner-layer capacitance can be neglected in the relationsfor Eij, biconductive electrowetting has been shown to manifest anumber of surprising, nontrivial voltage dependences, characterizedentirely by b and C.5,7 Some of the stranger earlier predictions van-ish in the present analysis because inner-layer capacitances dampenthe potential response, as well as making b and C alone insufficientto characterize the electrical state.

Obtaining contact angles.— Application of the Euler–Lagrangeprocedure from Ref. 7 to Eq. 1 shows the droplet to take the shapeof a plane-truncated sphere. For a truncated-spherical droplet withconstant volume, A depends on the contact angle �c �cf. Fig. 1�through

A =1 + cos �c

2�8�

The minimization of Eq. 1 with respect to contact angle yields an-other explicit relationship for cos �c,

cos �c =cos �c

0 + 4b�Ese�� − Ede�� − *��

1 + 4bEds�*�

�9�

in which �c0 represents the contact angle at zero applied voltage �the

usual Young angle�

cos �c0 =

�se − �de

�ds�10�

Elimination of cos �c from Eq. 8 and 9 gives a result that can besolved for * at fixed total applied potential �

4bCEse −1

CEde + �1 − 2A��C +

1

CEds� + cos �c

0 + 1 − 2A = 0

�11�

After the dependence of * on � has been obtained, Eq. 3 and 8yield the contact angle. Implementation requires the potential depen-dences of the energies Ese, Ede, and Eds. Furthermore, the potentialdependences of the charges � and � are needed to determine A.
de ds


In principle, the unknown functions in Eq. 3 and 11 can all bededuced from experimental measurements of the interfacial specificcapacitance as a function of potential in two-phase systems �that is,three separate systems comprising interface se, de, and ds alone�. Tounderstand how more subtle double-layer properties affect the elec-trowetting response, however, it is informative to introduce theoret-ical models by which the interfacial capacitances may be param-etrized.

Modeling Interfacial Capacitance

Liquid/liquid interface.— Although it relies on the naïve as-sumption that the liquid/liquid interface is a planar, quasi-static, ion-impermeable surface, VN theory describes the capacitance ofITIESs to a first approximation and often works more effectivelythan other models. For the electrostatic energy and surface charge atthe liquid/liquid interface, VN theory yields

Eds�*� = 1 − �1 +

4 sinh2��1/4�*�

�C + �1/C��2 and �ds�*�

=2RT0

�ds�d�s sinh��1/2�*�

Fz��C + �1/C��2 + 4 sinh2��1/4�*� 1/2 �12�

The surface-charge expression has been validated experimentally fora moderately dilute ITIES when �*� � 6;19 it proves to predict thedifferential capacitance within a few percent. More detailed theoriesfor ITIES capacitance exist,31-33 but they are analytically compli-cated and contain several additional parameters. Equations 12 aresufficiently accurate at the potentials of interest during electrowet-ting that such complexity is unmerited.

Solid/liquid interfaces.— Solid/liquid interfaces necessitate moresophisticated models of the interfacial electrostatic response. Still, inthe spirit of simplicity, a classical approach can be taken in which itis first assumed that the inner layer and the diffuse double layer canbe partitioned by an equipotential dividing surface, and second, thatno ions from the bulk solution can enter the inner layer. The formerassumption is expressed as

� = �il + �dl �13�

where � is the potential drop between the solid surface and thebulk solution; �il and �dl are the potential drops across the innerand diffuse layer, respectively. The latter assumption implies

��il� = �il + 2 sinh−1 Kil��il�0�

�il

2� �14�

in which � and are the inverse Debye length and dielectric con-

stant of the electrolytic solution, respectively, and Kil ��il� is theintegral specific capacitance of the inner layer. In the absence ofnonlocal effects, the total differential specific capacitance of the in-

terface, C, relates to Kil through

1

C=

1

Cil��il�+

1

0� cosh�1

2�dl

=1

K1 +

d ln�Kil/0��d ln�il

�−1

+1

0�1 + � Kil

0�

�il

22�−1/2

�15�Here, the second expression is useful for data fitting; it expresses thetotal capacitance of the metal/solution interface through �il and

Kil ��il� only. �The Appendix outlines a derivation.�The characterization of total differential capacitance as a func-

tion of the working-to-reference potential drop is a standard electro-

chemical experiment. Such measurements of C �� can be fit di-


rectly with Eq. 14 and 15, because Eq. 14 gives �il as a function of

� and Eq. 15 yields C ��il�. Given a parametrization of

Kil ��il�, the coupled system can be solved iteratively with a com-puter to obtain the parameters that best fit experimental data.

After inner-layer capacitance data are fit, the obtained parameters

can be used to compute the specific electrostatic energy E andsurface-charge density �, which are needed to simulate electrowet-ting. At a fixed applied voltage �, these are written in terms of theinner-layer potential drop �il as

�� = −2RT0�

Fzsinh1

2�� − �il�� 16�

E�� = − 20��RT

Fz2

sinh2�1

2� −

1

2�il

� 0�

Kil��il�+ sech2�1

4� −

1

4�il� �17�

Equation 14 relates � to �il. Thus, � and E depend parametri-cally on � alone.

Inner-layer model.— A number of theories for Kil exist.34-36 TheMWT mean-field theory, a two-state model for dipole reorientationin ion-free inner layers,26 is among the simplest, givingd

Kil��il� = K 1 + �0tanh��il/�

�il/� �18�

This interpolation formula depends on three parameters: the dimen-sionless potential energy required to realign a dipole, , the integral

inner-layer capacitance in the high-frequency limit, K , and a di-mensionless parameter related to the number of dipoles in the inner

layer, �0. The three properties , �0, and K , are easily extractedfrom fits of experimental data.

With the solvent dipole moment �, experimentally obtained pa-rameters from Eq. 18 can be used to estimate fundamental inner-layer properties according to the following expressions

d =NA�

RT

RT

Fz, =

K NA�

0RT

RT

Fz, and �� =

K �0

RT�RT

Fz2

�19�

Here, d is the inner-layer thickness, is the high-frequency dielec-tric constant �arising from electronic polarizability, vibrations, andlibrations of solvent molecules�, �� is the molar surface concentra-tion of reorientable solvent dipoles, q is the elementary charge, andNA is Avogadro’s number.

Aqueous solution/gold interface.— Figure 2 shows a nonlinearleast-squares fit of the MWT/GC double-layer model �Eq. 14-18� tothe differential capacitance measured for an interface between 0.1 Maqueous potassium perchlorate and polycrystalline gold.30 Thetheory fits the data with an average error of �5% between 0 and−800 mV vs saturated calomel electrode �the range of interest for

electrowetting�. The best fit yielded parameter values of K

= 8.843 �F/cm2, �0 = 3.129, and = 11.604. Insertion of thesevalues into Eq. 19 �along with water’s bulk dipole moment, �= 6.2 � 10−30 C m� yields = 4.5, d = 4.5 Å, and �� = 9.9� 10−6 mol/m2.

The obtained is within the range expected for water.26,37,38 Toget a better physical sense of the other inner-layer properties, it is

d The differential capacitance obtained by insertion of Eq. 18 into Eq. A-5 is

C �� = K �1 + � sech2�� /��.
il il 0 il


convenient to phrase d relative to the average solvent exclusiondiameter, d0, which relates to the molar concentration of pure sol-vent, c0, by

d0 = �1

6�NAc0−1/3

�20�

For water at room temperature, d0 = 3.86 Å. Thus, the inner-layerthickness d yielded by the best-fit curve is 1.2 solvent exclusiondiameters. Also, d0 can be used to put the surface concentration interms of the fractional electrode area occupied by reorienting di-poles. This fraction, �, is

� =1

4�d0

2NA�� 21�

For the system depicted in Fig. 2, � = 0.70. That is, 70% of theavailable electrode area is covered by reorientable molecules, whilethe remaining 30% is covered by molecules with fixed alignment.

Property estimates for organic/gold interfaces.— Each of thethree parameters in Eq. 18 can be expressed in terms of , �, and� = d/d0, according to

=Fz�

NA��3 �1/6��NAc0

, �0 =2c0�NA��2�

30 RT�,

and K =0

��3 �1/6��NAc0 �22�

Properties of the organic electrolyte/gold interface can be estimatedusing these equations, with the , �, and � values extracted fromthe fit in Fig. 2. Values of c0 and � for nitrobenzene can then beinserted to obtain a first-order estimate of the effect of solvent on ,

�0, and K .As discussed earlier, the above approximations and property es-

timations should be taken with a grain of salt. The reader should bereferred to a thorough review that discusses the limitations of thisdouble-layer model.22 What matters primarily below, however, is notthe detailed physics underlying the inner-layer model, but rather themodel’s accurate interpolation between the GC-dominated capaci-tance response near open circuit and the constant capacitancereached far from the open circuit. Such saturation is typical at largenegative potentials,27 and these are of primary interest because spe-cific anion adsorption, faradaic side reactions, and electrode-surfacereconstruction occur at positive voltages, weakening the electrowet-ting response or degrading the constituent materials. As the theoret-ical curve in Fig. 2 illustrates, the combined MWT/GC model offers

Figure 2. Differential capacitance vs applied voltage for the 0.1 M aqueousKClO4/polycrystalline gold interface. Experimental data are indicated byblack points; the gray line shows a best fit with the MWT/GC theory �Eq.14-18�. Data reprinted from Stolberg et al.30


a suitable interpolation formula for capacitance across the ideallypolarizable negative-voltage regime, which is consequently em-ployed below.

Electrowetting Curves

The surface-charge density and specific electrostatic energyyielded by the combined MWT/GC theory can now be put into thegeneral electrowetting model for the ITIES described earlier. In thissection, the electrowetting response is simulated assuming that theaqueous phase is 0.1 M aqueous KClO4 and that the organic phasecomprises a 0.01 M electrolytic solution solvated by nitrobenzene.The substrate is taken to be polycrystalline gold.

Table I lists the bulk properties of the electrolytic solutions inwater and nitrobenzene. Table II summarizes the inner-layer proper-ties measured above for the 0.1 M aqueous KClO4/gold interfaceand provides estimates for the inner-layer properties of anitrobenzene-solvated organic-electrolyte/gold interface. Also rel-evant are capillary parameters, including the nitrobenzene/water sur-face tension, �ds = 0.0254 J/m2,39 and the Young angle. For a0.01 M organic droplet surrounded by a 0.1 M aqueous solution ona gold substrate �as depicted in Fig. 1a�, a typical �c

0 value is 77°, orcos �c

0 = 0.225.e If the two liquid phases are switched �as in Fig.1b�, one would expect a supplementary Young angle at zero voltage,�c

0 = 180 − 77 = 103 ° �cos �c0 = −0.225�.

Figure 3 presents model results in two cases corresponding to theconfigurations depicted schematically in Fig. 1. Predictions of theprevious GC theory,5,7 which accounts for nonlinear diffuse-layerpolarization but neglects the inner-layer capacitances of the metal/electrolyte interfaces, are shown for comparison. Also shown are

e This angle was measured by A. E. S. Sleightholme and A. R. Kucernak at ImperialCollege London. They used a 1 �L droplet comprising 0.01 M nitrobenzene-solvated tetrabutylammonium tetraphenylborate, surrounded by 0.1 M aqueousLiCl; the substrate was a thin ��20 nm� gold film, produced by sputtering at lowcurrent onto titanium-doped quartz. Contact angles at 0 V vs Ag/AgCl were ob-tained from CCD photographs. Further experimental details will be provided in alater communication.

Table I. Measured and derived bulk properties of aqueous andnitrobenzene solutions for a typical ITIES at 298.15 K. Note that1Õ� represents the Debye length.

Aqueous Nitrobenzene

78.33 34.82� �C m� 6.2 � 10−30 1.33 � 10−29

c0 �M� 55.35 9.74c �M� 0.1 0.011/� �nm� 0.96 2.03

Table II. Inner-layer properties. One column gives inner-layerproperties for the interface between 0.1 M aqueous KClO4 andpolycrystalline gold, obtained from fits of experimental data.30

The other provides estimates for K�, �0, and � based on Eq. 22and parameters from Table I, assuming that the dimensionlessproperties ��, �, and � are similar for an organic electrolyte/goldinterface.

Aqueous/gold�measured�

Organic/gold�estimated�

4.4849 4.4849� 1.1647 1.1647� 0.69745 0.69745

K ��F/cm2� 8.843 5.362

�0 3.129 2.214

11.604 9.945



predictions in the other extreme �MWT theory�, where the diffuse-layer contribution to capacitance is neglected by assuming that �dand �s are infinite. In this limit, the contact angle varies with poten-tial according to

cos �c − cos �c0 =

��2

2�ds�Kil

de�� − Kilse�� 23�

where Kilde and Kil

se are the integral inner-layer capacitances at thedroplet/electrode and surroundings/electrode interfaces. If diffuse-layer contributions are negligible, then the relative magnitudes of

Kilse and Kil

de determine whether the area of droplet/electrode contactincreases or decreases with polarization.

The inclusion of inner-layer capacitances therefore causes theo-retical predictions to change significantly, both for an organic elec-trolytic droplet submerged in aqueous electrolytic surroundings �Fig.3a, corresponding to the configuration in Fig. 1a� and for an aqueousdroplet submerged in an organic electrolyte �Fig. 3b, correspondingto Fig. 1b�.

In Fig. 3a, the electrowetting curve is qualitatively similar to theprediction of GC theory, but a larger voltage is needed to achieve agiven variation of the contact angle. Note, however, that the under-lying mechanism governing the electrowetting has changed. Be-

Figure 3. Inner-layer capacitance dampens the electrowetting response.Comparison of electrowetting simulations with the present double-layermodel �MWT/GC theory� to simulations that neglect the inner-layercapacitance5 �GC theory� or neglect the diffuse-layer capacitance �MWTtheory�. �a� C = 2.18 and �c

0 = 77 °, corresponding to the experimental con-figuration shown in Fig. 1a. �b� C = 0.46 and �c

0 = 103 °, corresponding tothe configuration in Fig. 1b. In both cases, b = 0.0086.


cause diffuse layers contribute much less to the free energy than theinner layers across the entire range of potentials, MWT theory aloneis enough to approximate the contact angle.

The dampening of the response is unsurprising; potential dropsacross inner layers lower the overall free-energy change induced byan applied voltage. The widening of the necessary operating-voltagerange, however, is not dramatic. Although inner-layer potentiallosses weaken ITIES-based electrowetting, 500 mV remains suffi-cient to deliver �50 ° of contact-angle variation, which should besufficient for practical applications of the biconductive electrowet-ting effect.

Figure 3b, for the case of an aqueous droplet, demonstrates thatthe more detailed MWT/GC theory can yield results that are quali-tatively dissimilar from either the diffuse-layer dominated GCtheory or the inner-layer dominated MWT theory. Because the dif-fuse layers and inner layers make comparable contributions to thefree energy in the potential range shown, the electrowetting curve isalmost exactly intermediate between the MWT and GC extremes.

For ITIES-based systems the GC theory always predicts thatcontact angles eventually increase as the magnitude of potentialrises.5 This occurs because the theory gives diffuse-layer capaci-tances that increase monotonically with ��. In the GC theory ofelectrowetting, the capacitance increase is unbounded at the metal/surroundings interface, whereas the ITIES constraint �i.e., the netelectroneutrality of the droplet interior� inhibits capacitance in-creases at the interfaces bounding the droplet. Even if the droplethas the highest diffuse-layer capacitance, as is typical in the con-figuration of Fig. 1b, the effect of the ITIES constraint may besufficiently strong that the droplet contracts at all potentials. This isseen in the corresponding GC curve in Fig. 3b. Such a blatant con-tradiction to the well-known “Frumkin effect,” which states thatvoltage increases cause higher-dielectric phases to cover an increas-ing proportion of electrode surfaces,40 is disquieting.

The MWT/GC theory makes electrowetting simulations meet ex-pectations based on the Frumkin effect. When potentials are large,the diffuse-layer capacitance is much smaller than the inner-layercapacitance. Because these capacitances add in series, the capaci-tance of the entire double layer reaches a limiting value as ��increases �cf. Fig. 2�. The limiting value is determined by inner-layer properties, which in turn depend on the properties of the sol-vent. Thus, the MWT/GC curves in Fig. 3a and b show that thesolvent with a higher dielectric constant covers more of the elec-trode as potential rises in magnitude. The contact angle of an organicdroplet rises, and that of an aqueous droplet falls.

Note that at low voltages, the two MWT/GC curves in Fig. 3 arenot mirror images of each other with respect to a horizontal axisthrough �c

0; that is, the electrowetting response is asymmetric withrespect to an exchange of liquid positions. This residual asymmetryowes to the ITIES constraint.

When C � 1 �that is, when the droplet has a higher diffuse layercapacitance than the surroundings�, the GC theory predicts satura-tion toward the dewetting limit when the magnitudes of appliedvoltages grow large. This effect, seen in the GC curve in Fig. 3b,occurs because the charge separation within the droplet reduces sig-nificantly if its contact area with the electrode contracts. If theMWT/GC theory applies to the droplet, then the potential drop par-titioned to the inner layer at the droplet/electrode interface does notcontribute to charge separation and the driving force for contractioninduced by * becomes much smaller.

Nonmonotonicity of electrowetting curves, which was predictedpreviously in a very limited range of diffuse-layer properties,5 is farmore significant within the parameter space of the present MWT/GC

model. The effect depends strongly on the relative magnitudes of K

for the two solvents. If one solution has a high diffuse-layer capaci-

tance and a low K while the other has a low diffuse-layer capaci-

tance and a high K �for instance, if the aqueous phase has very lowionic strength and the organic phase has high ionic strength�, thenthe system response is nonmonotonic. Inflection points in experi-



mental electrowetting curves are often interpreted as showing theonset of contact-angle saturation. It may well be that the dependenceof interfacial capacitances on voltage is more significant to contact-angle saturation than net electroneutrality of the droplet.

Conclusions and Outlook

For biconductive electrowetting systems based on ITIESs, amore realistic model of the double layers at metal/electrolyte inter-faces can substantially affect the predicted dependences of contactangles on applied voltages. The augmented MWT/GC theory con-siders these double layers to consist of inner and diffuse parts; itsimply parameterizes metal/electrolyte interfacial capacitances andwas shown to fit capacitance vs voltage curves with reasonable ac-curacy. Incorporation of the MWT/GC capacitance model may pro-vide more accurate simulations of ITIES-based electrowetting thanthe previous GC model,5,7 which neglected inner-layer capacitancesaltogether.

In the case where an organic droplet is surrounded by an aqueoussolution, the electrowetting curves yielded by both models are quali-tatively similar. The MWT/GC theory shows that the potential re-quired to reach the dewetting limit is about −800 mV, a more plau-sible result than the −200 mV predicted when inner-layercapacitances are neglected.

When the droplet is aqueous, however, the MWT/GC double-layer model is shown to yield electrowetting curves that are lesscounterintuitive than earlier predictions. The accounting for inner-layer capacitance yields electrowetting curves that are consistentwith expectations based on the Frumkin effect �that aqueous/metalcontact increases relative to organic/metal contact whenever poten-tials rise in magnitude�. Therefore, it may be interesting to investi-gate this less popular configuration experimentally, at least to testthe augmented theory.

Perhaps the most important issue to address experimentally is therarity of interfacial capacitance data for metal/organic electrolyteinterfaces. In the present discussion, the weakest assumption is thatinner-layer capacitances for gold/organic electrolyte interfaces canbe estimated based on the properties of gold/aqueous electrolyteinterfaces. Accurate characterization of the organic interface wouldbe required before the results of the MWT/GC electrowetting modelcould be definitively compared to experimental data.

From a practical perspective, it is also worth mentioning that thegeometries for modern electrowetting-based devices usually differfrom the geometry studied here and in the academic literature. Inexisting variable-focus lenses, for instance, the liquids usually formtwo essentially planar layers, which sit within a conical or cylindri-cal cavity that is closed on both ends by transparent glass. To treatthis geometry would require additional recalculation because therelationship between the contact angle and the surface curvaturediffers from the case of a truncated sphere treated in this work. Thegeneral theoretical framework in the Electrowetting Theory Over-view section was provided in hopes that it might aid extensions ofthe theory to other geometries. But such questions are already in therealm of engineering. Many laboratories are equipped to measurethe geometry discussed here, making it the most logical choice forexperimental tests of the theory.

Finally, it should be noted that the viability of the proposed elec-trowetting model strongly depends on substrate quality. In experi-ments, substrates must be made as smooth as possible; otherwise,complications such as pinning and contact angle hysteresis are in-evitable in the voltage response. No issues of this kind were consid-ered in this paper, which assumes that the system can easily reach itsequilibrium state and that substrates are ideally flat. But even single-crystal surfaces have steps and defects, which can negate both ofthese assumptions, leading to pinning of the three-phase boundaryline and consequent contact-angle hysteresis.12

Not to discourage the reader of this paper, we wish to state thatpinning and hysteresis are not insurmountable problems in ITIES-based electrowetting systems. Preliminary experiments by our grouphave shown that microrough electrodes can be “tamed” by appropri-


ate operational control schemes.41 This allows us to expect that in-terfacial electrochemistry may soon change the face of variable op-tics.

Acknowledgments

This work was supported by the Leverhulme Trust �grant no.F/07058/P� and the Israel Science Foundation �grant no. 773/05�.

The University of Michigan assisted in meeting the publication costs ofthis article.

AppendixAssume that the total potential drop is partitioned cleanly between an inner solvent

layer and the diffuse double layer. Thus, if the potential drop at the metal/solutioninterface is ��, then

�� = ��il + ��dl �A-1�

where ��il and ��dl are the inner-layer potential drop and the diffuse-layer potential

drop, respectively. The total differential capacitance of the solid/liquid interface, C, isdefined as

C = −d�

d��A-2�

where � is the charge density in the solution at the dividing surface that separates theinner layer from the diffuse layer. Rearrangement and insertion of Eq. A-2 shows that

1

C= −

1

d�/d��il−

1

d�/d��dls =

1

Cil

+1

Cdl

�A-3�

which defines separate differential specific capacitances for the inner layer and diffuse

layer, Cil and Cdl, respectively. In the absence of specific adsorption or other nonlocalinteractions between the inner and diffuse layers, these two capacitances depend only on

the potential drops across the respective regions they describe: Cil = Cil��il� and

Cdl = Cdl��dl�.If ions do not enter the inner layer, then the potential distribution within it obeys

Laplace’s equation. In the diffuse layer, where ions screen the electrode charge, anonlinear Poisson–Boltzmann equation holds. To solve these governing equations for

fixed ��, it is easiest to assume the integral specific capacitance of the inner layer, Kil,defined as

Kil��il� = −�

��il�A-4�

is a known function of ��il. Note that Kil and Cil relate through

Cil��il� = Kil��il��1 +d ln Kil

d ln ��il �A-5�

An analogous formula relates the integral capacitance of the diffuse layer, Kdl��dl�, to

Cdl��dl�.From here on, results are more compactly expressed in terms of dimensionless

potentials �il and �dl. At the dividing surface that separates the inner and diffuselayers, the electric displacement must be continuous. Thus

Kil��il��il = 20� sinh�1

2�dl �A-6�

in which is the bulk dielectric constant of the solvent and � is the inverse Debyelength of the solution. The continuity of potential at the boundary between the inner anddiffuse layers allows insertion of Eq. 13 to eliminate �dl. Algebraic manipulation thenyields Eq. 14.

Equation A-6 also shows that �il is an implicit function of the total dimensionlesspotential drop, �il = �il��. Because Eq. A-1 can be rearranged to isolate thediffuse-layer potential drop, it follows that �dl = �dl�� as well. In short, both �il

and �dl, and, consequently, Cil, Cdl, and Kil, can be considered as parametric functionsof � only.

The definition of Cil from Eq. A-3, implicit differentiation of Eq. 14, use of therelationship from Eq. A-5, and insertion of � and the dimensionless potentials give Eq.15. It is worthwhile to note that in the limit where the inner-layer contribution disap-

pears, Kil → , the inner-layer potential approaches �il → 0 and the total specificcapacitance becomes

limKil→

C = 0� cosh�1

2� �A-7�

the expression familiar from nonlinear GC theory.17,18

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Double-Layer Effects in Electrowetting with Two Conductive Liquids

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