Transport and deformation of droplets in a microdevice using dielectrophoresis

Pushpendra Singh1

Nadine Aubry2

1Department of MechanicalEngineering,New Jersey Institute ofTechnology,Newark, NJ, USA

2Department of MechanicalEngineering,Carnegie Mellon University,Pittsburgh, PA, USA

Received August 31, 2006Revised November 2, 2006Accepted November 23, 2006

Research Article

Transport and deformation of droplets in amicrodevice using dielectrophoresis

In microfluidic devices the fluid can be manipulated either as continuous streams ordroplets. The latter is particularly attractive as individual droplets can not only move butalso split and fuse, thus offering great flexibility for applications such as laboratory-on-a-chip. We consider the transport of liquid drops immersed in a surrounding liquid by meansof the dielectrophoretic force generated by electrodes mounted at the bottom of a micro-device. The direct numerical simulation (DNS) approach is used to study the motion ofdroplets subjected to both hydrodynamic and electrostatic forces. Our technique is based ona finite element scheme using the fundamental equations of motion for both the dropletsand surrounding fluid. The interface is tracked by the level set method and the electrostaticforces are computed using the Maxwell stress tensor. The DNS results show that the drop-lets move, and deform, under the action of nonuniform electric stresses on their surfaces.The deformation increases as the drop moves closer to the electrodes. The extent to whichthe isolated drops deform depends on the electric Weber number. When the electric Webernumber is small, the drops remain spherical; otherwise, the drops stretch. Two droplets,however, that are sufficiently close to each other, can deform and coalesce, even if the elec-tric Weber number is small. This phenomenon does not rely on the magnitude of the elec-tric stresses generated by the bulk electric field, but instead is due to the attractive electro-static drop–drop interaction overcoming the surface tension force. Experimental results arealso presented and found to be in agreement with the DNS results.

Keywords:

Dielectrophoresis / Droplets / Microfluidics DOI 10.1002/elps.200600549

644 Electrophoresis 2007, 28, 644–657

1 Introduction

An appealing approach to the issue of controlling fluids inmicrodevices is the use of droplets which can transport var-ious types of fluids and particles, and has been referred to as“digital microfluidics.” One advantage of this techniquecompared to those using fluid streams is the possibility ofprogrammable microchips aiming at bio-chemical reactionswithin individual droplets. Droplets can serve as carriers forvarious types of reactions [1], the transcription and transla-tion of single genes for the creation of new enzymes [2],therapeutic agents for targeted drug delivery and micro-fabrication of materials, e.g., the synthesis of supraparticlesinitially contained within droplets [3]. Current challenges

include the controlled production, transport, splitting, andcoalescence of droplets at a certain location and at a giventime within the same device. Our approach is to make use ofelectric fields to accomplish these tasks. Electric fields areparticularly powerful in small devices due to the fact thatsmall potentials can generate relatively large field ampli-tudes. The production of droplets in a straight microchannelwas recently achieved using an electric field and the electro-hydrodynamic instability it triggers at the interface betweentwo fluids [4, 5]. It will thus not be addressed here. Instead, inthis paper we concentrate on the simultaneous transport anddeformation of droplets, the deformation having the poten-tial to lead to drop splitting and coalescence.

For this purpose, we consider dielectric drops suspendedin a dielectric liquid and subjected to a nonuniform electricfield. In some microfluidic applications, it is required that thedrops be transported in a predictable and controlled manner,and thus the operating parameter values must be carefullyselected to prevent the drop deformation from becomingexcessive and the drops from coalescing. Yet, in other appli-

Correspondence: Professor Nadine Aubry, Department ofMechanical Engineering, Carnegie Mellon University, Pittsburgh,PA 15213, USAE-mail: [email protected]: 1412-268-3348

Abbreviations: DEP, dielectrophoretic; DNS, direct numericalsimulation; PD, point dipole; RMS, root mean square

Additional corresponding author: Professor Pushpendra Singh,E-mail: [email protected]

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com

Electrophoresis 2007, 28, 644–657 Miniaturization 645

cations, e.g. laboratory-on-a-chip or the removal of waterdroplets from oil [6–12], it may be advantageous to bring twofluids initially separated in two drops together into one singledrop, and in this case drop coalescence is sought. Drop split-ting can also find applications, e.g. when the amount of sam-ple available is small and various droplets need to be dis-pensed for multiple purposes/reactions [13]. In this regard,we recall that O’Konski and Thacker [14] and Garton andKrasnucki [15] noted that a dielectric drop placed in a dielec-tric liquid and subjected to a uniform electric field deforms.These observations were later confirmed by Taylor [16] whoconsidered the case where the drop or ambient liquid, or both,are conducting, and introduced a leaky dielectric model. Thedeformation and breakup of a dielectric drop in a dielectricliquid was analyzed analytically in refs. [17, 18]. It was shownin refs. [16, 19] that for the leaky dielectric model, the shearstress on the surface of the drop is nonzero, and the fluidinside the drop circulates in response to these shear stresses(also see refs. [20–22]). A condition under which a drop placedin a conducting liquid remains spherical was also obtained.

Lee and Kang [23] and Lee et al. [24] extended the pre-vious studies to the case where the electric field is no longeruniform, but instead varies linearly in space in the vicinityof the drop. They analytically studied the influence of thespatial variation of the electric field on the steady dropshape and on the circulating flow inside the drop. Theiranalysis, however, is relevant when the drop size is smallcompared to the length scale over which the electric fieldvaries substantially, when the drop deformation from thespherical shape is small and when the drop itself is nottranslating and held fixed at a certain location by the actionof a body force. In many instances, however, these assump-tions are not valid and there is thus a need to develop amore general approach.

In most cases, when a drop is subjected to a nonuniformelectric field, and the dielectric constants of the drop and theambient fluid are different, the electric stress acting on thedrop’s surface not only deforms the drop, but also generates anet electric force, referred to as the dielectrophoretic (DEP)force, which causes the drop to translate. If the drop shape isspherical, the DEP force is the same as that acting on a rigidspherical particle having the same dielectric constant.According to the point dipole (PD) model, which assumes thatthe gradient of the electric field is constant, the time-averagedDEP force acting on a spherical drop (or particle) in an alter-nating current (AC) electric field is given by refs. [25–29]

FDEP ¼ 2pa3e0ecbrE2 (1)

where a is the drop radius, ec is the permittivity of the fluid,e0 = 8.8542610212 F/m is the permittivity of free space and Eis the root mean square (RMS) value of the electric field.Expression (1) is also valid for a direct current (DC) electricfield where E is simply the electric field intensity. The coeffi-cient b(o) is the real part of the frequency-dependent Clau-sius–Mossotti factor given by

bðoÞ ¼ Ree�d � e�ce�d þ 2e�c

� �, where e�d and e�c are the frequency-

dependent complex permittivity of the drop and the fluid,respectively. The complex permittivity e� ¼ e� js=o, wheree is the permittivity, s is the conductivity and j ¼

ffiffiffiffiffiffiffi�1p

. Thefrequency dependence can be included in simulations byselecting an appropriate value of b. The above expression forthe force is, of course, valid only when the drop shaperemains spherical, which may not be the case when theinterfacial tension is not sufficiently large and drop defor-mation takes place.

Furthermore, when subjected to a nonuniform electricfield, dielectric drops interact with each other via the elec-trostatic particle–particle interactions that are similar to theinteractions among rigid particles. In the PD limit, anexpression for the interaction force between two dielectricspherical particles suspended in a dielectric liquid and sub-jected to a uniform electric field was given in [25, 26]. Fromthis expression it is easy to show that the electrostatic inter-action force between two particles is attractive and also thatit causes the particles to orient such that the line joiningtheir centers is parallel to the electric field direction (exceptin the degenerate case when the line joining their centers isperpendicular to the electric field, in which case they repel).Similar interactions take place between particles in a non-uniform electric field. An expression for the electrostaticforce between two spherical particles in a nonuniform elec-tric field in the PD limit was given in [27–29]. Directnumerical simulations (DNSs) conducted using thisexpression for the interaction force show that two particlessubjected to a nonuniform electric field attract each otherand orient such that the line joining their centers is parallelto the local electric field direction while they move togethertoward the location where the electric field strength islocally maximal or minimal, depending on the value of theirdielectric constant relative to that of the liquid [27–29]. Theextent of this attraction, which, if it is strong, manifestsitself in particle chaining, depends on a dimensionless pa-rameter which can also be found in the above references(see also below).

Our DNS results presented below show that the same isessentially true for a dielectric drop suspended in a dielectricliquid except that the problem of the motion of a drop iscomplicated by the fact that the drop also deforms and maycoalesce with nearby drops. Since the drop shape and thehydrodynamic and electric forces acting on the drop are notknown a priori, they have to be obtained by solving the cou-pled equations governing both the drop shape and these for-ces. Specifically, for a dielectric drop subjected to a nonuni-form electric field one must obtain the electric stress dis-tribution on the drop surface and then use the momentumand mass conservation equations to determine the drop ve-locity and deformation. The only approach available to theo-retically study such a problem is the DNS approach, which isused in this paper.


646 P. Singh and N. Aubry Electrophoresis 2007, 28, 644–657

Specifically, we use the DNS to investigate the translationand deformation of drops under the action of a nonuniformelectric field and their eventual capture at the electrodesedges. Also investigated is the mechanism of coalescence ofdrops due to the electrostatic particle–particle interactionforces. Although the former problem has been studied in theliterature, past approaches have used the small perturbationapproach which (i) provides accurate information only forsmall droplet deformations from the spherical shape, (ii)does not take into account the change in the electric field dueto the drop deformation or the presence of the other drops,which can be significant when the dielectric mismatch be-tween the drop and the ambient fluid is not very small andthe drop devices size are comparable. Our approach in thispaper relaxes these limitations.

2 Problem description and governingequations

Let us consider a drop of fluid with viscosity Zd and densityrd, placed in an ambient fluid with viscosity ZL and densityrL. The drop is assumed to be immiscible with the ambientfluid and its dielectric constant is different from that of theambient fluid. The system is subjected to a nonuniformelectric field which is generated by the electrodes mounted inthe bottom surface of the microdevice, as shown in Fig. 1.Since the dielectric constants of the drop and the ambientfluid are different, a spatially varying electric stress acts onthe interface which causes the drop to deform and translate.

Figure 1. Schematic of the microdevice used in the simulations.The electrodes, shown as grey strips, are mounted on the bottomsurface of the device. The domain dimensions in the x, y, and zdirections are 1 mm, 1 mm, and 3 mm, respectively. The width ofthe electrodes and the distance between them is 750 mm. Thedimensionless voltage applied to the electrodes is shown.

Let us denote the domain containing the liquid and the dropsby O, and the domain boundary by G. The governing equa-tions for the two-fluid system are

r � u ¼ 0 (2)

rquqtþ u � ru

� �¼ �rpþr � ð2ZDÞ þ gkdðfÞn 1 r � sM (3)

u = uL = 0 on G (4)

where u is the velocity, p is the pressure, Z is the viscosity, ris the density, D is the symmetric part of the velocity gra-dient tensor, n is the outer normal, g is the interfacial ten-sion, k is the surface curvature, f is the level set functiondefined to be the signed distance from the interface [30, 31],d is the delta function and sM is the Maxwell stress tensor.In this paper we consider only the case in which the densityand viscosity of the drop are the same as those of the ambi-ent liquid, although these assumptions could be easilyrelaxed. The bulk imposed velocity of the liquid is assumedto be zero and solid walls are placed at the domain bound-ary. No-slip boundary conditions (i.e. zero liquid velocity)are imposed there.

The level set method is used to track the interface (seerefs. [30, 31]). In this method, the interface position is notexplicitly tracked, but instead it is defined to be the zero levelset of a smooth function which is assumed to be the signeddistance from the interface.

In order to compute the electric stress acting on the dropsurface, we first obtain the electric potential V, which isobtained by solving r � erVð Þ ¼ 0 subjected to the voltageboundary conditions [32, 33]. The dielectric constant e isequal to ec for the ambient liquid and ed for the drop. Thevoltage V is prescribed on the electrodes and its normalderivative is taken to be zero on the remaining domainboundary. The electric stress is given by the Maxwell stresstensor

sM ¼ eEE� eðE �EÞI=2 (5)

where I is the identity tensor and E ¼ �rV is the electricfield (see refs. [19, 32, 33]). We will assume that both theliquid and the drop are perfect dielectrics. Our results, asnoted earlier, are also applicable to the case of AC electricfields, provided that the RMS value of the electric field isused, that b is replaced by the real part of the complex fre-quency dependent Clausius–Mossotti factor and that theforce is the time averaged force.

The governing equations (2–4) can be non-dimensionalized by assuming that the characteristic length,time, velocity, pressure, stress, and electric field scales are a,a/U, U, rU2, ZU/a, and bE0 respectively, where U is thecharacteristic fluid velocity, E0 = V0=L is the strength of theelectric field, and L is the distance between the electrodes. Itis easy to show that the nondimensional equations afterusing the same symbols for the dimensionless variables are

=u = 0

quq t

1u�ru

��= 2 =p 1

1Re

=?(2D) 11

ReCak d(f) n 1

1ReMa

aLr � eEE� eðE �EÞI=2ð Þ (6)

Notice that the spatial gradient of the electric field is non-dimensionalized using the characteristic length L. The



dimensionless dielectric constant e is obtained by dividing byec, the dielectric constant of the continuous phase. The aboveequations contain the following dimensionless parameters:the Reynolds number Re = rUa=Z, which is the ratio ofinertial and viscous forces, the capillary numberCa = UZ0=g, which is the ratio of viscous and surface tension

forces, the Mason number Ma ¼ ZU

e0ecab2E2

0

which is the ratio

of the viscous and electric forces, and the length ratio a/L.Another useful parameter which gives the relative impor-tance of the electric and interfacial tension forces, is the

electrical Weber number We =e0ecab

2E20

g. Notice that We can

also be defined as the ratio of the capillary and Mason num-bers. Here, we would also like to point out that our definitionof the Weber number differs from the standard definition,

We =e0ecaE2

0

g, by a factor b2 (which is dimensionless) and, as

discussed below, this allows for a better estimate of the elec-tric force acting on a drop.

It is worth noting that a drop moving in a microdevice alsodeforms because of the velocity field it experiences, and thusthe hydrodynamic Weber number should also be considered.However, when the drop stops moving, as is the case after it iscaptured at an electrode’s edge, the shape of the deformeddrop is determined by the balance of the electric and surfacetension forces, and the hydrodynamic forces are zero.

3 Numerical scheme

A code based on the finite element method, with featuresdescribed in Pillapakkam and Singh [31] and Singh andAubry [32–34], is used for solving the time-dependent prob-lem for the deformation of a drop in an electric field. Here,the governing (fluid and electric field) equations are solvedsimultaneously everywhere, i.e., both inside and outside thedrops in the computational domain, and the Marchuk–Yanenko operator-splitting technique is used to decouple thedifficulties associated with the incompressibility constraint,the nonlinear convection term, and the interface motion [35–37]. The electric force exerted on the drop due to the electricfield is obtained in terms of the Maxwell stress tensor com-puted directly from the electric potential [32, 33, 38-–40].

The operator splitting scheme gives rise to the followingfour subproblems: A Stokes like problem for the velocity andpressure; a nonlinear convection–diffusion problem for thevelocity; and an advection problem for the interface. The firstproblem is solved by means of a conjugate gradient (CG)method [34] and the second problem is dealt with using aleast square conjugate gradient method [34]. The third prob-lem consists of the advection of the level set function f,which is solved using a third order upwinding scheme [31,36]. The advected function f is then re-initialized to be adistance function, which, as noted in [31], is essential forensuring that the scheme accurately conserves mass.

In [34], a finite element code based on the above methodwas used to study the deformation of a drop suspended in afluid and subjected to a uniform electric field. The code wasvalidated by showing that the numerically computed resultsfor the deformation of a drop in a uniform electric field werein agreement with the analytical results which assume thatthe drop is approximately spherical. As expected, the agree-ment was found to be very good for small drop deformations,but was observed to deteriorate with increasing deformations.

4 DEP force-induced drop motion anddeformation

As noted earlier, if a drop is sufficiently small compared tothe length scale over which the nonuniform electric fieldvaries, the PD approach can be used to estimate the DEPforce from which the drop velocity can be estimated. Thedrop, however, also elongates along the direction of the localelectric field. This latter complexity is not present for the caseof rigid particles.

Even though the analysis based on the PD is notstrictly valid in a typical device used for dielectrophoresisfor which the drop size is of the same order as the devicesize [32], it is still useful for estimating the order of mag-nitudes of the forces. We are also interested in identifyingthe parameter values for which the drop translates to theminimums or maximums of the electric field withoutundergoing substantial deformation, and thus is not likelyto breakup.

The critical electric field strength below which the dropdeformation remains small can be estimated from the resultobtained by Allen and Mason [17] for the case of a drop placedin a uniform electric field. The deformed shape in their anal-ysis is determined by the balance of the surface tension force,which tends to make the drop spherical, and the force due tothe electric stress, which tends to elongate the drop. The elec-tric stress distribution on the surface of the drop is deduced byassuming that the drop remains spherical. Allen and Masonobtained the following expression for the drop deformation:

D =9ae0ecE

20b

2

8pg¼ 9

8pWe (7)

This expression implies that the deformation increases as thesquare of the electric field and the square of the Clausius–Mossotti factor. Moreover, it varies inversely with the surfacetension coefficient and is proportional to the electric Webernumber We. The deformation is defined as the parameter

D =L� BLþ B

, (8)

where L and B are, respectively, the major and minor axes ofthe drop, assuming that the shape of the latter is approximatelyellipsoidal. The deformation parameter D varies between 0and 1; for a spherical drop, D is zero and its value increaseswith increasing deformation from a spherical shape.



To ensure that the drop deformation is small, one mayassume that We is such that D is less than 0.1. From Eq. (7),this implies

We � 0:8p9� 0:28 (9)

An estimate of the characteristic drop velocity can beobtained by the balance of the DEP force and the drag actingon the drop. In this estimate, we further assume that thedrop remains spherical and that the drag is given by theStokes drag formula. In our case, since the drop viscosity isequal to the ambient fluid viscosity, the Stokes drag isFD ¼ 5paZU, where U is the drop velocity. From Eq. (1) anestimate of the DEP force acting on the drop is

FDEP ¼4pa3e0ecbE2

0

L. By equating these two forces, we obtain

the following expression for the drop velocity:

U ¼ 4a2e0ecbE20

5ZL(10)

If it is desired that the drop deformation remains small, thecapillary number based on the above velocity must alsoremain small. The latter is given by

Cadep ¼4a2e0ecbE2

0

5gL=

4ab5L

We (11)

Notice that the capillary number based on the drop velocitydiffers from the Weber number by the dimensionless factor

of4ab5L

. This factor is the same as the dimensionless group

defined to be the ratio of the particle–particle interactionforce and the DEP force [27, 29, 32] and is indicative of theextent of particle chaining [41].

Another important effect, not accounted for in the aboveanalysis, is that the presence of the drop modifies the electricfield distribution around it. This, in turn, affects the electricstress distribution on the drop surface, and thus its defor-mation. These effects are particularly important in themanipulations of drops in microdevices where the drop sizecan be of the same order as the device size [33]. The DNSresults presented in this paper include these effects, sincethe solution is obtained by solving the exact governing equa-tions.

5 Results

A typical computational domain used in this study is shownin Fig. 1. The electrodes are embedded in the bottom surfaceof the device, which is also the case for the device used in ourexperiments. The dimensions of the computational domainare 1.0, 1.0, and 3.0 mm in the x, y-, and z-directions respec-tively, and the width of the electrodes is 0.75 mm.

The viscosities and densities of the ambient fluid and thedrop are assumed to be equal to 1.0 Poise and 1 g/cm3,respectively. The dielectric constant of the ambient fluid is

held fixed and assumed to be 1.0. The interfacial tension be-tween the ambient fluid and the drop, and the strength of theelectric field are prescribed in terms of the electric Webernumber. The dependence of the DEP force, and the resultingdrop deformation, are investigated for several values of thedrop dielectric constant.

In our simulations, recall that the normal derivative ofthe electric potential on the domain side walls is assumed tobe zero, the voltage is prescribed on the electrode surfacesand the fluid velocity at the domain boundaries is taken to bezero. The initial velocities of the drop and the ambient fluidare assumed to be zero. Simulations are started by placingspherical drops at various locations within the domain, andterminated when the drops are captured at the electrodeedges and assume fixed shapes.

In this paper, we only consider the case of positive di-electrophoresis because the electric field distribution in ourexperimental device is such that for negative dielectrophor-esis the drops simply levitate and move away from the elec-trodes where the electric field strength is relatively weak, andthus remain approximately spherical. This, however, is notthe case for all devices based on dielectrophoresis, e.g., theDEP cage.

5.1 Experiments

Experiments were conducted in a device with a rectangularcross section which was made using conventional machin-ing techniques (see Figs. 2 and 3). The width of the devicewas 5.0 mm and the length was 15 mm. The width of theelectrodes was 800.0 mm and their length was equal to thedevice’s width. The device contained eight electrodes thatwere mounted in the grooves machined in the bottom sur-face. The distance between the electrodes was 800 mm. Thedepth of the ambient fluid in the device was approximately8 mm. An ac electric field in the device was generated byenergizing the electrodes such that the phase of adjacentelectrodes differed by p, and the frequency used in all theexperiments described here was 1 kHz. The electric fieldstrength was varied by changing the magnitude of the volt-age applied to the electrodes.

The drops of various sizes were formed at a small dis-tance from the electrodes by injecting a given amount offluid into the ambient fluid with a syringe. The density andviscosity of the drops were not equal to the correspondingvalues for the suspending liquid. In fact, the suspendingliquid was selected so that the drop density was larger, whichensured that the drop did not levitate and move away fromthe electrodes before the electric field was applied. Thedielectric constant of the drops was greater than that of theliquid in order to ensure positive dielectrophoresis. After theelectric field was applied, the DEP force caused the drops todeform and move towards the electrode edges, where theywere captured. Furthermore, for the two cases considered inthis paper, the drop did not touch the electrodes or the bot-tom surface which remained covered by the ambient liquid.



Figure 2. Deformation of a glycerin drop suspended in silicon oiland subjected to a nonuniform electric field. Electrodes appear asgold-colored areas (appearing darker in grey) and are located onthe bottom surface. The drop diameter is approximately 300 mm.The width of the electrodes is 800 mm and thedistance between theelectrodes is also 800 mm. The domain width is 5.0 mm. (a) Drop ofspherical shape is attached to the left electrode. The electric fieldstrength is 300 V. After the electric field strength is increased to,400 V (b), the drop stretches in the direction approximately nor-mal to the electrodes. The left end of the drop remains attached tothe left electrode and the right end is pulled towards the edge ofthe right electrode. The drop deformation increases when thevoltage applied to the electrodes is increased to 500 V (c).

Figure 3. Coalescence of two silicon drops suspended in corn oilin a nonuniform electric field. The device dimensions are statedin Fig. 2. The diameter of the right drop is ,350 mm and that of theleft drop is ,700 mm. (a) The voltage applied is 500 V. The twodrops are trapped at the electrodes edges. (b) and (c) After thevoltage is increased to 780 V, the drops stretch towards eachother and merge. The camera used was unable to capture inter-mediate shapes. The combined drop becomes spherical and istrapped at the left electrode edge, the initial location of the largersized drop.

Figure 2 shows the deformation of a glycerin drop sus-pended in silicon oil for three values of the electric fieldstrength. The dielectric constant of glycerin is 47.0 and its

conductivity 6.46106 pS/m, while the corresponding valuesfor silicon oil are 4.4 and 2.67 pS/m. The density of glycerinis 1.23 g/cm3 and that of silicon oil 0.945 g/cm3. The inter-facial tension between glycerin and silicon oil was estimatedusing a combining law to be 11.2 dynes/cm [42]. The diam-eter of the drop was approximately 300 mm. After the electricfield was switched on, the drop moved toward the nearestelectrode edge and was trapped there. As the electric fieldstrength was increased the drop elongated. The electricWeber number was estimated to be 0.12. In the top view, thedrop shape appeared to be ellipsoidal, with the major axis ofthe ellipsoid being approximately normal to the electrodes.Our experimental setup did not allow for the measurementof the drop or fluid velocities, and thus the other dimension-less parameters could not be estimated. The drop deforma-tion decreased as the electric field strength was reduced.

The merging of two silicon drops suspended in corn oildue to the electrostatic drop–drop interactions is illustratedin Fig. 3. The dielectric constant of corn oil is 2.2, its con-ductivity is 32.0 pS/m, and its density is 0.91 g/cm3. Initially,when the electric field strength was small, the two dropswere trapped at the edges of adjacent electrodes. The inter-facial tension between corn oil and silicon oil was estimatedusing the combining law to be 2.6 dynes/cm. The diameterof the smaller drop was approximately 350 mm and that ofthe larger drop 700 mm. The drops did not touch the bottomsurface of the device which remained covered by corn oil.When the electric field strength was increased, the dropsquickly stretched towards each other and coalesced. Theelectric Weber number based on the maximum electric fieldapplied was 0.096. Since the combined drop was initiallyhighly elongated, it retracted into an approximately sphericalshape under the action of the interfacial tension. After thedrop retracted, it was captured at the edge of the left elec-trode. Our experiments show that the drops strongly inter-acted with each other via the electrostatic forces, even whenthe distance between them was of the order of their radius.On the other hand, if the drops did not deform, their behav-ior would be similar to that of the rigid particles whichremain attached to adjacent electrodes edges and do notcome together.

5.2 DNS results

We begin by discussing the electric field distribution in thedomain without the drop. From Fig. 4 we note that the elec-tric field strength is maximum at the electrodes edges anddecreases with increasing distance from the electrodes. Theelectric field does not vary in the y-direction. Although thepresence of a drop modifies the electric field distribution,especially near the surface of the drop, and can even causenearby drops to coalesce, the qualitative nature of the electricfield remains unchanged in the sense that the DEP force isdirected towards the electrodes edges. As noted before, inthis paper we only consider the case of positive dielec-trophoresis, i.e., the dielectric constant of the drop is greater



Figure 4. Isovalues of the dimensionless electric field magnitudeon the domain midplane. The magnitude is maximal at the elec-trode edges.

than that of the ambient liquid, for which the DEP forceeverywhere in the domain is directed towards the electrodesedges. We will assume that the ambient liquid wets the bot-tom surface of the device which ensures that the dropremains away from the bottom surface, as it was the case inthe experiments.

We first describe the motion of two drops of radius150 mm. The initial shape of the drops is assumed to bespherical and their velocities to be zero (see Fig. 5). Thedielectric constant of the drops is 1.1. The dimensionlessparameters based on the maximum drop velocity attained areRe = 71.2, Ca = 1.89, and We = 0.86. Thus, as Ca is aroundone, the drops deform due to their motion, and since We isaround one, they also deform under the action of the electricfield. The drops begin to move towards the electrode edgesunder the action of the DEP force and deform due to thespatial nonuniformity of the DEP force (see Fig. 5). FromFig. 5b we note that the fluid velocity is larger on the dropsurfaces that are closer to the electrode which is a con-sequence of the fact that the electric force magnitude in thisregion is larger. This nonuniformity in the velocity causesthe drops to deform. It is interesting to note that in the caseof a rigid particle, the velocity is determined by the totalelectric force that acts on the particle, and the velocity of amaterial point inside the particle depends upon the linearand angular velocity of the particle.

From Fig. 5 we notice that the DEP force causes the dropsto stretch significantly, which may or may not be desirable.

The excessive deformation of the drops, however, as dis-cussed in Section 4, can be controlled by reducing the electricfield strength such that the electric Weber number We issmaller than 0.28.

5.2.1 Dielectrophoresis of a drop

Let us first describe the motion of a drop of radius 250 mm, asshown in Fig. 6, towards the nearby electrode edge where itgets trapped. The initial shape of the drop is assumed to bespherical and its velocity zero. The dielectric constant of thedrop is 1.1. The values of the dimensionless parametersbased on the maximum drop velocity attained are Re = 19.9,Ca = 0.016, We = 0.039.

The drop begins to move towards the upper electrode’sedge under the action of the DEP force and deforms as itmoves closer to the bottom surface on which the electrodesare mounted (see Fig. 6). The lower surface of the drop flat-tens as the DEP force pushes it against the bottom surface.The deformed drop continues to slide in the z-direction andstops moving when the total DEP force acting on it becomeszero. Figure 6 shows that the drop stops moving when itscenter is approximately over the electrode’s edge. Also notethat even though the drop is deformed, in the top viewshown in Fig. 6d, it appears to be approximately spherical.

Figures 7 and 8 show the equilibrium shapes and posi-tions of the drops captured at the electrode’s edge for twoother cases. The dielectric constant of the drops in Figs. 6 and7 is 2.0. The dimensionless parameters based on the max-imum drop velocity attained take the values Re = 6.31,Ca = 0.025, We = 0.096 for Fig. 7, and Re = 11.25, Ca = 0.009,We = 0.019 for Fig. 8. These figures show that the deformedshape and the drop’s equilibrium position at the edgedepend on the dielectric constant, and the other dimension-less parameters. For example, the equilibrium positions ofthe drops in Figs. 7 and 8 is slightly lower than in Fig. 6, i.e., alarger portion of these drops is over the surface not coveredby the electrode. Also notice that although in the top viewsshown in Figs. 6–8 the drops appear to be spherical, they aredeformed and the shapes in side views for the three cases aredifferent. This is an important point which must be kept inmind while analyzing experimental photographs that onlyshow the top views of the deformed droplets (since the sideviews are difficult to image).

In Fig. 9 we consider a drop of radius 250 mm anddielectric constant 10.68. The ratio of the drop and ambientliquid’s dielectric constants is the same as in Fig. 2. The dropwas released at the same distance from the electrodes as inFig. 6. The electric Weber number is 0.12 which was alsoselected to match the experimental value in Fig. 2. The otherdimensionless parameters based on the maximum drop ve-locity are Re = 38.75 and Ca = 0.031, and clearly both of theseparameters are zero after the drop is captured. It is worthnoting that even though these latter parameter values are notavailable for experiments, the final shape of the drop is



Figure 5. (a) DEP force-induced motion and deformation of twodielectric drops at various times. The dimensionless parametervalues are: Re = 71.2, Ca = 1.89, We = 0.86. For these parametervalues the deformation of the drops is large, especially when thedrops get close to the electrodes (gold-colored; dark regions ingrey plots). (a) t = 0.01, (b) t = 0.1, (c) t = 0.2, and (d) t = 0.3. (b) TheDEP force-induced velocity distribution and the drop shape onthe domain midplane are shown from our numerical simulations.Notice that the velocity on the drop surface is the largest in theregion that is closest to the electrode edge, and the velocity is inthe direction of the local electric force. (a) t = 0.01, (b) t = 0.1, and(c) t = 0.2.



Figure 6. The deformed shapeof a dielectric drop being cap-tured at the electrode edge isshown at three different times.The dimensionless parametervalues are: Re = 19.9, Ca = 0.016,We = 0.039. Side view of thedrop at times (a) t = 0, (b) t = 0.4,and (c) t = 0.8, the steady shapeof the captured drop; the dropappears to be spherical in thetop view (d) at the latest time.Notice that the final drop shapeis less deformed than in (b).

Figure 7. A dielectric drop beingcaptured at the electrode edge.The dimensionless parametervalues are Re = 6.31, Ca = 0.025,We = 0.096. Side views of thedrop at times (a) t = 0.2, (b)t = 0.6, and (c) t = 1.4, the steadyshape. Top view of the drop atthe latest time (d).

independent of the transients that depend on the initialposition of the drop. The transients also depend on the den-sity and viscosity ratios which are different from 1 in theexperiments, but these ratios do not affect the deformed dropshape in equilibrium. The drop first moves towards the edgeof the upper electrode and, as Fig. 9b shows, stretch towardsthe lower electrode. The drop maintains the shape shown inFig. 9c. The top view of the stretched shape is similar to thatseen in the experiments (see Fig. 2). The side view shows thatthe lower end of the drop is farther away from the electrodeplane, which may also be the case in Fig. 2, although the sideviews in the experiments could not be obtained. The stretch-ing of the drop is due to the electric field direction near thebottom plane, which is perpendicular to the electrodes there.This suggests that the mechanism responsible for the

stretching of the drop between the electrodes is the same asthat causing a drop to stretch in a uniform electric field.Furthermore, notice that the drop deforms even though theelectric Weber number is smaller than 0.28, which indicatesthat the electric field strength near the electrodes is strongerthan the overall strength given by the scaling argument.

In Fig. 10 we consider a drop of radius 250 mm anddielectric constant 5.0, and the drop is released at the samedistance from the electrodes as in Fig. 6. The dimensionlessparameters take the values Re = 57.49, Ca = 0.046, andWe = 0.40. The drop moves towards the edge of the upperelectrode while also stretching towards the lower electrode,as is apparent in Fig. 10b. Finally, Fig. 10c shows that thelower end of the drop reaches the edge of the lower electrode.After this substantial stretching, the drop maintains its



Figure 8. A dielectric drop being captured at the electrode edge.The dimensionless parameter values are: Re = 11.25, Ca = 0.009,We = 0.019. Side views of the drop at times (a) t = 0.4, and (b)t = 1.0. Top view of the drop at the latest time (c). Notice that thedrop is less deformed than in Fig. 7.

shape. Notice here that the electric Weber number is largerthan 0.28, and thus the electric field strength is sufficientlylarge to overcome the interfacial tension.

5.2.2 Electrostatic interaction and coalescence of

two drops

We next consider the case in which two drops with radii 250and 200 mm interact with each other via electrostatic parti-cle–particle interactions. The larger sized drop is initially at a

distance of 300 mm from the bottom surface and the smallerone is at a distance of 250 mm, as displayed in Fig. 11. Thedielectric constant of the drops is 1.1 and the dimensionlessparameters based on the maximum drop velocity areRe = 37.75, Ca = 0.03, We = 0.032.

As was the case for a single drop, the drops are attractedby their respective nearest electrode edges. The initial dis-tance of 700 mm between the drops is selected to ensure thatthe force induced due to the electrostatic particle–particleinteractions is significant. It can be seen from Fig. 12a thatthe electric field strength is modified by the presence of thedrops. The modified electric field is such that its magnitudein the region between the drops is larger than in Fig. 4 wherethe drops are absent. An increase in the electric field strengthin the region between the drops implies that the magnitudeof the electric stress acting on the drop surfaces in this regionalso increases. Furthermore, since the electric stress isattractive, the drop surfaces are pulled towards each other(see Fig. 11). The drops deform, but yet remain attached atthe electrode edges, since the attractive force is not suffi-ciently large to either dislodge them from these edges orsufficiently deform them so that the drop surfaces come incontact with each other. A comparison with Fig. 6 shows thatthe center of the upper drop has moved downwards and thedrop is stretched in the direction parallel to the electrodes.All parameters for this case are the same as for the singledrop case shown in Fig. 6 in which the center of the drop isapproximately directly over the electrode’s edge.

We next describe the results shown in Fig. 13 for whichthe dielectric constant of the drops is 2.0 and the otherdimensionless parameters are Re = 12.88, Ca = 0.013,We = 0.019. The Weber number for this case is the same asfor Fig. 8. From Fig. 12b we note that the electric fieldstrength in the region between the drops is greater than for

Figure 9. DEP force-inducedstretching of a drop. The dimen-sionless parameter values are:Re = 38.75, Ca = 0.031, andWe = 0.12. Top view of the dropat times (a) t = 0.2: the drop istrapped at the electrode edge,(b) t = 1.3: the DEP force causesthe drop to stretch, and (c)t = 2.1: the drop is stretched inthe direction perpendicular tothe electrodes. (d) Side view ofthe drop at t = 2.1.



Figure 10. DEP force-inducedstretching of a drop. The dimen-sionless parameter values are:Re = 57.49, Ca = 0.046, We =0.40. Top view of the drop attimes (a) t = 0.2: the drop istrapped at the electrode edge,(b) t = 1.4: the DEP force causesthe drop to stretch, and (c)t = 2.2: the drop is stretchedfrom one electrode to the adja-cent one. Side view of the dropat the latest time (d).

Figure 11. Interaction betweentwo dielectric drops via the elec-trostatic forces. The dimension-less parameter values are:Re = 37.75, Ca = 0.03, We =0.032. Side view of the drop attimes (a) t = 0.2: the drops arecaptured at the electrodesedges, (b) t = 0.6: the drops con-tinue to deform due to the DEPforce, and (c) t = 1.2: the cap-tured drops have reached theirsteady shape. Top view of thedrops at the latest time (d).

the case shown in Fig. 12a where the dielectric constant is1.1. This implies that the electrostatic particle–particle inter-actions are now stronger. Figure 13 shows that the drops aremarkedly less deformed in the lateral direction than inFig. 11, and as before, they do not coalesce. The interfacialtension of the drops is too large, and as a result the drops donot deform sufficiently for their surfaces to come togetherand thus for them to coalesce.

Our next step is to render the drops more deformable.For this, we reduce the interfacial tension by a factor of fivecompared to the value used in Fig. 13. All the other parame-ters are held constant. The dimensionless parameters in thissituation are Re = 14.37, Ca = 0.057, We = 0.096 and the

results are displayed in Fig. 14. The latter parameter is thesame as in Fig. 3. In this case the electrostatic particle–parti-cle interaction force between the drops’ surfaces overcomesthe interfacial tension and consequently the drops deform.The surfaces of the two drops eventually touch, the dropscoalesce, and the combined drop extends between two adja-cent electrode edges. The interfacial tension then acts to pullthe extended combined drop back into the spherical shapeand when this happens it loses contact with the lower elec-trode’s edge. The combined drop continues to move towardsthe edge of the upper electrode where it eventually attaches.It is worth noting that the net result is that the smaller dropis engulfed by the bigger drop, which is in good qualitative



Figure 12. Isovalues of the dimensionless electric field magni-tude on the domain midplane in the presence of two drops. (Thedrops are marked as circles.) The dielectric constant of the dropsin (a) is 1.1, and in (b) it is 2.0. Notice that in both cases the electricfield strength in the region between the drops is increased com-pared to the case without the drops displayed in Fig. 4.

agreement with the experimental results shown in Fig. 3.This example demonstrates that even when the electricWeber number is smaller than 0.1, in which case an isolated

drop should maintain an approximately spherical shape, thedrop–drop interactions can cause them to deform and coa-lesce.

Here, we would like to point out that in the level setmethod when the distance between two surfaces becomessmaller than the size of an element, the surfaces simplymerge. Furthermore, it is important to emphasize that ourDNS approach resolves the electric stresses acting on twosurfaces exactly (within numerical errors), but only when thedistance between them is larger than the size of a few ele-ments, and that the accuracy diminishes as the two surfacescome closer. In addition, as the distance between the twosurfaces decreases, other forces that are relevant to the prob-lem of thin film and its drainage, such as the van der Wallsforce and the double layer repulsion, which are not includedin the governing equations used here, become important [8].In other words, the DNS results presented are accurate onlyup to the point when the distance between the drops is largerthan the size of an element and after the drops have merged;during the time period in which the behavior of the thin filmis important, our simulations are only approximate.

6 Conclusions

The direct numerical simulation approach was used to studythe DEP force-induced motion and deformation of drops in amicrofluidic device for the case when both the drop and theambient fluid are perfect dielectrics. The drop interface istracked using the level set method and the electric stressacting on the drop surface is computed in terms of the Max-well stress tensor (MST). The latter is obtained by first com-

Figure 13. Interaction betweentwo dielectric drops via the elec-trostatic forces. The dielectricconstant of the drops is 2.0 andthe dimensionless parametervalues are: Re = 12.88, Ca =0.013, We = 0.019. The drops arecaptured at the electrode edges.Notice that due to the attractiveelectrostatic forces, the dropsmove slightly toward eachother, but for the parametervalues selected, they remainattached at their respective elec-trode’s edges. Top views of thedrops at times (a) t = 0.2, (b)t = 0.6, and (c) t = 1.8. Side viewof the drop at the latest time (d).



Figure 14. Coalescence of two drops due to the electrostatic particle–particle interactions. The dielectric constant of the drops is 2.0. Thedimensionless parameter values are Re = 14.37, Ca = 0.057, We = 0.096. The drops are initially captured at the electrode edges. The attrac-tive electrostatic particle–particle interactions cause them to move towards each other, as well as deform. The adjoining surfaces are pulledtoward each other and the drops coalesce. After the merger, the combined drop which is initially elongated, becomes spherical and movesto the edge of the upper electrode. These results are in qualitative agreement with the experimental results shown in Fig. 3. Top view of thedrops at times (a) t = 0, (b) t = 0.8, (c) t = 1.2, (d) t = 1.4, (e) t = 2.2, and (f) t = 3.0.

puting the electric potential distribution within the domain(accounting for the presence of the drops) from which theelectric field is determined.

Simulation results are in agreement with the experi-ments. Our key results are as follows. A drop moves to theelectrode edge just like a particle, but in addition it deforms.The drop deformation can be made small by selecting theparameters of the problem so that the electric Weber numberis smaller than 0.28. This can be done, for instance, by re-ducing the magnitude of the electric field. In this case, thetransport of drops is slower, but they remain spherical. Forhigher electric Weber number values, the drop deformationcan be significant, leading, for instance, to the bridging oftwo adjacent electrodes. Although this was not shown in thispaper, substantial stretching can also lead to the formation ofa neck in the middle of its length and eventually drop split-ting [34]. Moreover, it was demonstrated both experimentallyand numerically that a drop can coalesce with other drops.An interesting observation is that this phenomenon canoccur even when the electric Weber number is smaller than1, in which case isolated drops do not deform. However, theelectrostatic drop–drop interaction can cause the drops todeform and then merge if the latter interaction overcomesthe surface tension force.

Finally, in this paper, we have considered the simulta-neous transport and deformation of a drop as the lattertranslates. For certain applications (e.g., mixing), it may beadvantageous to use a combination of rotation (rather than

translation) and deformation, or even translation, rotation,and deformation. Such combinations should be enabled bythe phenomenon of traveling wave dielectrophoresis whoseeffect on solid particles has been recently studied numeri-cally [43, 44].

We gratefully acknowledge the support of the New JerseyCommission on Science and Technology through the New-JerseyCenter for Micro-Flow Control and the National Science Foun-dation. We also wish to thank G. Barnes, M. Janjua and S.Nudurupati for the help they provided in conducting the experi-ments.

7 References

[1] Song, H., Tice, J. D., Ismagilov, R. F., Angew. Chem. Int. Ed.2003, 42, 768–772.

[2] Tawfik, D. S., Griffiths, A. D., Nat. Biotech. 1998, 16, 652–656.

[3] Millman, J. R., Bhatt, K. H, Prevo, B. G., Velev, O. D., Nat.Mater. 2005, 4, 98–102.

[4] Ozen, O., Aubry, N., Papageorgiou, D., Petropoulos, P., Elec-tromech. Acta 2006, 51, 5316–5323.

[5] Ozen, O., Aubry, N., Papageorgiou, D., Petropoulos, P., Phys.Rev. Lett. 2006, 96, 1–4.

[6] Eow, J. S., Ghadiri, M., Chem. Eng. J. 2002, 85, 357–368.



[7] Eow, J. S., Ghadiri, M., Sharif, A. O., Williams, T. J., Chem.Eng. J. 2001, 84, 173–192.

[8] Frising, T., Noik, C., Dalmazzone, C., J. Disper. Sci. Technol.2006 27, 1035–1057.

[9] Chabert, M., Dorfman, K. D., Viovy, J.-L., Electrophoresis2005, 26, 3706–3715.

[10] Eow, J. S., Ghadiri, M., Sharif, A., Colloids Surf. A 2003, 225,193–210.

[11] Eow, J. S., Ghadiri, M., Chem. Eng. Process. 2003, 42, 259–272.

[12] Eow, J. S., Ghadiri, M., Colloids Surf. A 2003, 215, 101–123.

[13] Ahmed, R., Hsu, D., Bailey, C., Jones, T. B., First InternationalConference on Microchannels and Minichannels, April 24–25, Rochester, NY 2003, Paper No. 1ICMM2003-1110.

[14] O’Konski, C. T., Thacher, H. C., J. Phys. Chem. 1953, 57, 955–958.

[15] Garton, C. G., Krasuchi Z., Proc. Roy. Soc. A. 1964, 280, 211–226.

[16] Taylor, G., Proc. Royal Soc. Lond., A 1966, 1425, 1959–1966.

[17] Allen, R. S., Mason, S. G., Proc. Royal Soc. London, A 1962,267, 45–61.

[18] Torza, S., Cox, R. G., Mason, S. G., Phil. Trans. Royal Soc.Lond. A 1971, 269, 295–319.

[19] Melcher, J. R., Taylor, G. I., Annu. Rev. Fluid Mech. 1969, 1,111–146.

[20] Sherwood, J. D., J. Fluid Mech. 1988, 188, 133–146.

[21] Saville, D. A., Annu. Rev. Fluid Mech. 1997, 29, 27–64.

[22] Darhuber, A. A., Troian, S. M., Annu. Rev. Fluid Mech. 2005,37, 425–455.

[23] Lee, S. M., Kang, I. S., J. Fluid Mech. 1999, 384, 59.

[24] Lee, S. M., Im, D. J., Kang, I. S., Phys. Fluids. 2000, 12, 1899–1910.

[25] Pohl, H. A., Dielectrophoresis, Cambridge University Press,Cambridge 1978.

[26] Klingenberg, D. J., van Swol, S., Zukoski, C. F., J. Chem.Phys. 1989, 91, 7888–7895.

[27] Kadaksham, J., Singh, P., Aubry, N., J. Fluids Eng. 2004, 126,170–179.

[28] Kadaksham, J., Singh, P., Aubry, N., Electrophoresis 2004,25, 3625–3632.

[29] Kadaksham, J., Singh, P., Aubry, N., Mech. Res. Commun.2006, 33, 108–122.

[30] Osher, S., Sethian, J. A., J. Comput. Phys. 1988, 79, 12–49.

[31] Pillapakkam, S. B., Singh, P., J. Comput. Phys. 2001, 174,552–578.

[32] Singh, P., Aubry, N., Phys. Rev. E 2005, 72, 1–5.

[33] Aubry, N., Singh, P., Euro Physics Lett. 2006, 74, 623–629.

[34] Singh, P., Aubry, N., ASME, American Society of MechanicalEngineers, New York 2006, Paper Number FEDSM2006-98413.

[35] Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D., Int. J. ofMultiphase Flows 1998, 25, 755–794.

[36] Singh, P., Joseph, D. D., Hesla, T. I., Glowinski, R., Pan, T. W.,J. Non-Newtonian Fluid Mech. 2000, 91, 165–188.

[37] Glowinski, R., Pironneau, O., Annu. Rev. Fluid Mech 1992,24, 167–204.

[38] Wohlhuter, F. K., Basaran, O. A., J. Fluid Mech. 1992, 235,481–510.

[39] Basaran, O. A., Scriven, L. E., J. Colloid Interface Sci. 1990,140, 10–30.

[40] Baygents, J. C., Rivette, N. J., Stone, H. A., J. Fluid Mech.1998, 368, 359–375.

[41] Kadaksham, J., Singh, P., Aubry, N., Electrophoresis 2005,26, 3738–3744.

[42] Israelachvili, J., Intermolecular and Surface Forces, Aca-demic Press, London 1991.

[43] Aubry, N., Singh, P., Electrophoresis 2006, 27, 703–715.

[44] Nudurupati, S. C., Aubry, N., Singh, P., J. Phys. D 2006, 39,3425–3439.


Transport and deformation of droplets in a microdevice using dielectrophoresis

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