Deformation of a droplet in an electric field: Nonlinear transient ...

Journal of Colloid and Interface Science 318 (2008) 463–476www.elsevier.com/locate/jcisDeformation of a droplet in an electric field: Nonlinear transient responsein perfect and leaky dielectric mediaGraeme Supeene, Charles R. Koch, Subir Bhattacharjee ∗Department of Mechanical Engineering, 4-9 Mechanical Engineering Building, University of Alberta, Edmonton, AB, T6G 2G8, CanadaReceived 19 July 2007; accepted 12 October 2007Available online 18 October 2007AbstractDeformation of a fluid drop, suspended in a second immiscible fluid, under the influence of an imposed electric field is a widely studiedphenomenon. In this paper, the system is analyzed numerically to assess its dynamic behavior. The response of the system to a step change in theelectric field is simulated for both perfect and leaky dielectric systems, exploring the influence of the fluid, interfacial, and electrical propertiesonthe system dynamics. For the leaky dielectric case, the dynamic build up of the free charge at the interface, including the effects of convection alongthe interface due to electrohydrodynamic circulation, is investigated. The departure of the system from linear perturbation theory is explained usingthese dynamic simulations. The present simulations are compared with analytic solutions, as well as available experimental results, indicating thatthe predictions from the model are reliable even at considerably large deformations.© 2007 Elsevier Inc. All rights reserved.Keywords: Droplet deformation; Electric field; Navier–Stokes equations; Finite element; Perfect dielectric; Leaky dielectric1. IntroductionA fluid drop suspended in another immiscible fluid will deformwhen subjected to an electric field. Applications of thisphenomenon encompass spraying [1], aerosols, inkjet printing,coalescence of droplets for de-emulsification purposes [2–4],and electrowetting-based droplet manipulation in microfluidicsystems [5–7], to name a few. The problem of droplet deformationunder the influence of an imposed electric field has beenextensively studied from different perspectives. The mathematicalfoundation of the subject has been rigorously established,and several aspects of the equilibrium and transient behavior ofsuch systems have been explored.Fig. 1 schematically depicts the fundamental parametersgoverning the extent of droplet deformation under the influenceof an electric field. As noted in the figure, the key propertiesdictating the nature and extent of deformation are the viscosity,dielectric permittivity, and electric conductivity of the twofluids. The interfacial tension is unique for a given fluid–fluid* Corresponding author. Fax: +1 (780) 492 2200.E-mail address: subir.b@ualberta.ca (S. Bhattacharjee).interface. The deformation of a drop in an electric field is characterizedby a balance of interfacial tension, hydrodynamic, andelectrical stresses at the droplet interface. The electrical stressescause the interface to distort, while interfacial tension tends torestore the original shape. Viscous stresses and fluid pressuregradients due to the flow fields can also alter the deformationsubstantially. The mathematical description of this system requiressimultaneous solution of the fluid mechanical equationsand the equations of electrostatics.The first analytic result predicting the deformation of a dropin an electric field was derived by O’Konski and Thacher [8]for perfectly insulating (dielectric) drops in perfectly insulatingmedia. Subsequently, Allan and Mason [9] performed aforce balance over the surface of a dielectric drop in a dielectricmedium. Their result was equivalent to the O’Konski andThacher result. Using the notations of Fig. 1, the steady statedeformation, d, of the droplet predicted by the O’Konski andThacher/Allan and Mason (OTAM) expression isd ∞ = R 0ɛ e |E 0 | 2γ9(S − 1) 216(S + 2) 2 ,where |E 0 | is the magnitude of the electric field vector alongthe z coordinate, and the parameter S (=ɛ i /ɛ e ) represents the(1)0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2007.10.022

464 G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476Fig. 1. Schematic representation of the deformation of a droplet suspended ina second fluid in the presence of an electric field, E 0 , acting along the z axisof the cylindrical coordinate system. The shaded circle represents the sphericalnondeformed droplet before application of the field. The possible steady-statedeformed shapes, prolate and oblate spheroids, are also depicted. The nondeformeddroplet diameter is R 0 . For the deformed spheroidal shapes, the equatorialradius (a) and polar (b) semi-axis are directed along the r and z coordinates,respectively. The relevant properties of the droplet and suspending medium arethe viscosity, μ, dielectric permittivity, ɛ, and conductivity, σ . The subscriptsi and e are used to represent the droplet and the suspending fluid, respectively.The interfacial tension, γ , constitutes the restoring force.ratio of dielectric permittivities of the droplet and the suspendingmedium. The deformation parameter, d, is defined asd = b − a(2)b + a ,where a is the equatorial radius of the spheroid and b is the polarhalf-axis. The symbol d ∞ is used in Eq. (1), and in the rest ofthe paper to indicate that the deformation is evaluated at steadystate. According to Eq. (1), the deformation must always bepositive, i.e., the resultant deformed shape is always a prolatespheroid, elongated in the direction of the electric field. Allanand Mason [9], however, observed discrepancies betweenthe predictions of Eq. (1) and their experiments on electricaldeformation of droplets. In some of their experiments oblatedeformation (compression in the axis of the applied field) wasobserved, which was clearly beyond the scope of Eq. (1).Taylor [10] and later Melcher and Taylor [11] developed anelectrohydrodynamic model for the deformation of conductingdroplets suspended in a conducting medium under the influenceof an imposed electric field. This model, generally referred toas the “leaky dielectric model,” has become a cornerstone ofthe theory of electrical drop deformation. The expression forthe steady-state deformation in this model isd ∞ = R 0ɛ e |E 0 | 2 9γ 16(2 + H)[2 ( )] 2 + 3M× H 2 + 1 − 2S + 3(H − S), (3)5 + 5MwhereS = ɛ i, H = σ i, M = μ i.ɛ e σ e μ eEquation (3) successfully predicted the oblate deformations observedin the experiments of Allan and Mason [9]. The leakydielectric model, however, suffers the limitation that it is invalidfor perfect dielectric systems. This is immediately apparentfrom Eq. (3), where setting σ i = σ e = 0 does not yield theOTAM expression (Eq. (1)).The leaky dielectric model has been elaborated upon in severalsubsequent works, notably [12–17], which specifically focuson the transport modes of the free charge carriers (ions).A comprehensive review of the theoretical developments in thisarea was given by Saville [15]. Zholkovskij et al. [16] provideda solution for the electrokinetic problem that resolved the disparitybetween the OTAM and Taylor results in the limit of zeroconductivity. More recently, an extension of the Taylor modelwas proposed [17].All the above mentioned approaches pertain to steady-stateanalysis of small deformations under the influence of smallelectric fields. These results do not apply to the dynamic problem,and cannot address large deformations or breakup ofdroplets. The transient electrohydrodynamic problem addressingthe deformation of droplets is also an extensively studiedsubject [18–24].Since the theoretical treatments of drop deformation in anelectric field are limited to small deformations, or large deformationswith assumptions placed on the shape, there hasbeen a significant and continuing interest in finding numericalsolutions to these problems. The pioneering computationalstudies on transient deformation of droplets appeared in theearly 1970s [25,26]. These studies deal with conducting dropswith constant surface potential, both isolated and in pairs, andisolated charged drops. A finite perturbation technique was employedto converge to the correct force balance for this system,yielding the steady-state result, which for relatively large deformationsshowed a small deviation from the spheroidal shape.The irrotational, inviscid fluid mechanics equations were solvedto provide dynamic analyses of drop deformation and contactbetween drop pairs.Sherwood [13] analyzed the leaky dielectric model by meansof a boundary integral method. Viscosity was included in themodel, but the droplet and medium were assumed to be equallyviscous. The momentum term in the Navier–Stokes equationswas neglected. Large deformations were obtained by stepwiseincreases in the applied field. Sherwood’s results indicatethat when the permittivity ratio ɛ i /ɛ e is sufficiently high, thedrop develops the pointed ends characteristic of tip streaming,whereas a high conductivity ratio tends to produce a bulbousendedbreakup mode. The steady-state perfect dielectric problemwas also solved using Newton’s method with stepwisechanges in applied field [27].Finite element analysis of the shape patterns and stability ofa charged drop in an electric field was conducted by Basaranand Scriven [28,29]. Their steady-state analysis assumed infinitedrop conductivity, so that the final state did not involveflow, and hence the fluid mechanics equations were not necessary.Haywood et al. [30] developed a numerical techniqueto predict the transient deformation of a perfect dielectric system.Differences between the stability limits predicted fromsteady-state analysis and from their fully dynamic model wereobserved. Tsukada et al. [31] performed finite element cal-

G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476 465culations using Taylor’s leaky dielectric model and comparedtheir numerical results with experiments. These numerical resultsshowed a substantial improvement in predictive powerover the analytic formulae. Feng and Scott [22] employed theGalerkin finite element technique in a comprehensive steadystateanalysis of the leaky dielectric model. The effect of chargeconvection was also treated [24] where a parameter called electricReynolds number was introduced. This is a ratio of the timescales of charge convection and conduction, and if it is nonzero,the final interfacial charge distribution and droplet deformationcan be altered significantly.The method of Tsukada et al. [31] was extended to deal withthe full dynamic problem of a leaky dielectric droplet subjectedto a step jump in applied field of arbitrary magnitude [32]. Thedynamic response of the resulting model was compared with ahighly oscillatory solution obtained in an earlier work [33], inwhich droplet oscillations were studied in the absence of electricfields. Excellent agreement was observed in the predictionsfrom the two approaches. Charge convection and finite conductiontimes were not considered.In this paper we present a numerical scheme based on thefinite element method incorporating a moving boundary implementationfor simulation of the transient deformation of adroplet in an electric field. The method is capable of simulatingperfect dielectric fluids or fluids with finite conductivity, aslong as the charge buildup occurs in a layer of negligible thicknesscompared to the droplet radius. The time dependence of theelectrical conduction can also be taken into account, along withfinite charge convection effects on the drop interface. Largedeformations are simulated up to the point of droplet disintegrationin some cases. The steady-state deformations are comparedagainst Eqs. (1) and (3) for the perfect and leaky dielectric systems,respectively, under small deformations, and deviationsfrom these limiting cases at higher imposed fields have beenstudied. The transient deformation behavior when the chargebuildup dynamics are explicitly incorporated in the model isalso explored. Finally, a comparison with experimental resultsavailable in literature is presented.2. TheoryIn this section, we will first present the general governingequations for the droplet deformation under the influence of anelectric field. Following this, the simplified transient equationsfor the perfect dielectric and the leaky dielectric cases will bepresented. In most parts, the general formulation is based on theapproaches detailed elsewhere [15,34].2.1. General electrokinetic modelReferring to Fig. 1, we will consider two sets of governingequations pertaining to the droplet and the suspendingfluid phases. The governing equations for each phase are theNavier–Stokes equation (modified by including the electricalbody force), the Poisson equation, and the Nernst–Planck equationsdescribing the transport of the free charge carriers.The fluid flow in the system will be governed by the Navier–Stokes equation∂u i,eρ i,e + ρ i,e u i,e ·∇u i,e∂t(=−∇p i,e +∇·μ i,e ∇ui,e +∇u T ) fi,e + ρi,e E i,e (4)along with the continuity equation∇·u i,e = 0.In writing these equations, we have neglected the effect of gravity,included an electric body force, ρ f E, where ρ f is thevolumetric free charge density, and assumed incompressiblefluids. The subscripts i and e correspond to the droplet and thesuspending medium, respectively. u is the fluid velocity vectorand p is the pressure. Finally, Brownian motion and anytemperature and buoyancy effects arising from the electricalheating of the fluids are neglected.The electric field distribution, E =−∇ψ, is governed by thepotential (ψ) distribution according to the Poisson equation∇·ɛ i,e ∇ψ i,e =−ρ f i,e ,where the volumetric free charge density isn∑ρ f i,e = z k en k i,ek=1with k representing the kth ionic species.The ionic fluxes, j k , are governed by the Nernst–Planckequationsj k i,e = nk i,e u i,e − Di,e k ∇nk i,e − zk en k i,e Dk i,e∇ψ i,e .k B THere, n k denotes the number concentration of the kth ionicspecies. The Nernst–Planck equation describes the ion transportdue to convection, diffusion and electromigration. The ionicfluxes appear in the ionic species conservation equations, writtenas∂n k i,e=−∇·j k i,e (9)∂t.While Eqs. (4)–(9) constitute the fundamental governingequations for the electrohydrodynamic problem in each fluidphase, certain additional conditions apply to these systems,which are related to the electric current. The current densityin the system is given byn∑i i,e = e z k j k i,e .(10)k=1After substituting Eq. (8) in (10), one obtainsi i,e = eu i,e∑z k n k i,e − e ∑ D k i,e zk ∇n k i,e − σ i,e∇ψ i,e ,whereσ i,e =e2 ∑(zk ) 2 Dkk B Ti,e n k i,e(5)(6)(7)(8)(11)(12)

466 G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476is the solution conductivity. Note that in a homogeneous electroneutralliquid phase, Eq. (11) yields Ohm’s law, i =−σ ∇ψ.Applying conservation of charge principle, one can multiplyEq. (9) by z k e and sum over all ionic species to writee ∂ ∂t∑z k n k i,e =−e∇ ·∑ z k j k i,e =−∇·i i,e(13)in which, the final term on the right-hand side utilizes Eq. (10).Noting that the term e ∑ z k n k on the left-hand side of Eq. (13)represents the volumetric free charge density, ρ f , one can usePoisson’s equation (6), to write]∂∇·[(14)∂t (ɛ i,e∇ψ i,e ) =∇·i i,e .Utilizing E =−∇ψ, one can simplify the above equation to[ ]∂(ɛi,e E i,e )∇·+ i i,e = 0(15)∂twhich is the generalized statement for the continuity of currentin each phase. At steady state, the displacement current (firstterm) vanishes from Eq. (15), yielding ∇·i i,e = 0.These equations apply to the two phases (droplet and thesuspending medium) independently. Two of the primary variablesof the governing equations are strictly continuous at thedroplet–suspending medium interface. These include the fluidvelocity and the electric potential. The current is also continuousacross the interface. The ion concentrations and pressureare, however, discontinuous at the interface.The overall force balance at the interface can be written byconsidering the viscous, electrical (Maxwell), and interfacialtension stresses. This relationship takes the form of an interfacialboundary condition at the droplet surface. The net stressbalance at the interface can be written as[ = [−pe I +μe ∇ue +∇u T )] [ = (e · n − −pi I +μi ∇ui +∇u T )]i · n+[ɛ e E e E e − 1 ]2 ɛ =eE e · E e I· n−[ɛ i E i E i − 1 ]2 ɛ =iE i · E i I · n − 2γXn = 0.(16)Here, = I is the unit tensor, γ is the interfacial tension, and Xis the mean curvature of the interface, with the convention thatthe curvature is positive for a convex surface. The electrical andinterfacial stresses can be considered as applied forces on theinterface. The fluid stresses constitute the reaction force. Thefluid stresses in turn drive the bulk flow in each phase. Equation(16) indicates that in the absence of electrical and viscousstresses p i − p e = 2γXfor an undeformed drop. Other boundaryconditions for the problem are relatively straightforward,and will be discussed later in the context of the perfect and leakydielectric representations.An additional relationship between the individual ionicspecies concentrations in the two phases must be provided tosolve these equations. This is problematic, since an unequivocalclosure cannot be obtained unless the detailed chemistry ofthe system is invoked. Considering the state of the system beforeapplication of an external electric field, one might assumethat the individual ionic species are equilibrium partitioned betweenthe two phases, with a partition coefficient defined byΦ k = nk in k .e(17)While this definition seems adequate, it raises several concernsabout the electrical properties of the system [16]. The key issuesare related to the definitions of electrostatic boundary conditionsat the droplet surface. As stated earlier, a complete pictureis not attainable unless the detailed chemistry of the system isinvoked. These problems have often led to attempts at simplifyingthe governing equations by making suitable assumptionsabout the system. A simple assumption pertains to the casewhen both the fluids are perfect dielectrics, which leads to theperfect dielectric (OTAM) model. The second assumption relatesto a system where at least one, or both of the phases(droplet and suspending fluid) are conducting, leading to theso-called leaky dielectric (Taylor) model. We now describe thesimplified equations dealt with in these two models.2.2. Perfect dielectric modelWhen we consider two perfect dielectric liquids, the governingequations for electrostatics simplify to the Laplace equationfor both phases∇·ɛ i,e ∇ψ i,e = 0.(18)Since there is no free charge in the fluids, the electric bodyforce term vanishes from the Navier–Stokes equation, leadingto∂u i,eρ i,e + ρ i,e u i,e ·∇u i,e∂t(=−∇p i,e +∇·μ i,e ∇ui,e +∇u T i,e).(19)The interfacial stress condition, Eq. (16), simplifies in this case,since the electrostatic stresses will only act normal to the interface.Thus, the interfacial boundary condition in this casebecomes(p i − p e ) + [ (μ e ∇ue +∇u T ) (e · n − μi ∇ui +∇u T ) ]i · n · n+ 1 2(ɛi |E i | 2 − ɛ e |E e | 2) + (ɛ e E e E e · n − ɛ i E i E i · n) · n− 2γX= 0(20)combined with continuity of tangential viscous stresses[ (μe ∇ue +∇u T ) (e · n − μi ∇ui +∇u T ) ]i · n · tj = 0. (21)Here, t j (j = 1, 2) represent the unit vectors along the two orthogonalcoordinates tangential to the surface of the drop. In anaxisymmetric system, it is sufficient to consider only the tangentialvector in the r–z plane, so that t j becomes t. The othertwo boundary conditions for the Navier–Stokes equation are:= (−p e I +μe ∇ue +∇u T )e = ¯0 at infinity,(22a)u i = u eat interface.(22b)

G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476 467The boundary conditions for the electrostatic problem are asfollows:∇ψ e =−E 0ψ i = ψ eat infinity,at interface,ɛ i E i · n i + ɛ e E e · n e = 0at interface.(23a)(23b)(23c)Equation (23a) represents the application of a uniform electricfield, E 0 , far from the drop. Equations (23b) and (23c) representcontinuity of electric potential and displacement at the interface,respectively. The displacement is continuous when theinterface contains no free charge. Note that n i =−n where nis the unit outward normal to the drop surface.2.3. Leaky dielectric modelIn the case of Taylor’s leaky dielectric model [11], ion diffusionis ignored, and the electric double layer is assumed to be ofinfinitesimal thickness and confined to the droplet–suspendingfluid interface. This reduces the net free charge to a boundarycondition, and once again, the ionic free charge density, ρ f ,iszero in the bulk fluid phases.There are two ways in which the charge transport may be described.First, if the boundary charge is assumed to be in staticequilibrium, the Laplace equation may be written for each ofthe bulk phases in accordance with the principle of continuityof current, Eq. (15), as follows:∇·i i,e =∇·[−σ i,e ∇ψ i,e ]=0.(24)Here σ i,e denotes the local conductivity of the fluid, which isuniform in each phase. The Navier–Stokes equation in this caseassumes the same form as in Eq. (19) in each phase.The formulation of the Laplace equation in Eq. (24) allowsa natural interfacial boundary description for the electrostaticequation, in which the continuity of current across the interfaceis described by the use of a Neumann continuity condition∑ [ ]∇(σj ψ j ) · n j = 0.(25)j=i,eThe mathematical formulation employing the above expressionwill be referred to as the static boundary leaky dielectric model.A more general form is an explicit treatment, where conductionand convection of charge carriers are modelled directly. Forsimplicity, following Zholkovskij et al. [16], it is assumed thatno solute adsorbs on the interface, so that the ion layer is dueto electrical effects. Since the surface charge density q representsa discontinuity in electric displacement, the correspondingboundary condition can be written as∑ [ ]∇(ɛj Ψ j ) · n j = q.(26)j=i,eIf convection/diffusion are taken into account, or if the timescale of charge migration is important, an extra equation to describethe dynamics of charge transport to and from the boundaryneeds to be provided. Saville [15] gives the expression∂q(27)∂t + u ·∇ sq = qn · (n ·∇)u −‖σ E‖·n,where the use of ‖···‖ indicates the jump from inside to outside,n is the outward normal in that direction, and ∇ s is thetangential or surface gradient. The second term on the left isthe convective flux along the boundary, and the first term on theright is the change in concentration due to dilation of the interface.The term on the far right is the current discontinuity, orcharge buildup due to migration.The fluid dynamic boundary conditions, Eqs. (22a) and(22b), also apply for the leaky dielectric case. The interfacialstress condition, however, requires an additional tangentialstress arising from the interfacial charge. The conditions at theinterface are given by(p i − p e ) + [ (μ e ∇ue +∇u T ) (e · n − μi ∇ui +∇u T ) ]i · n · n+ 1 2(ɛi |E i | 2 − ɛ e |E e | 2) + (ɛ e E e E e · n − ɛ i E i E i · n) · n− 2γX= 0(28)and[ (μe ∇ue +∇u T ) (e · n − μi ∇ui +∇u T ]i · n · tj+ (ɛ e E e E e · n − ɛ i E i E i · n) · t j = 0.)(29)Here, t j are the tangential unit vectors. As before, since theproblem form employed in this paper is axisymmetric, the onlyrelevant tangential vector is the one in the r–z plane, so thisexpression reduces to a single equation. The Taylor leaky dielectricmodel employing Eqs. (26) and (27) will be denoted asthe transient charge boundary leaky dielectric model.2.4. Initial conditionsThe present problem pertains to the evaluation of the transientdeformation of an initially spherical droplet of radius R 0in response to a step change in the electric field. Prior to the applicationof a field, E 0 at t = 0, it is assumed that no electricfield is present. In all cases, the system is assumed to be stationaryand locally electroneutral at t = 0. The resulting initialconditions are expressed as followsu = 0 and p = 0 in suspending medium,u = 0 and p = 2γ in droplet,q = 0R 0on interface,(30a)(30b)(30c)where γ is the interfacial tension.Equations (30a) and (30b) specify fluid quiescence over theentire domain. It is not necessary to specify an initial conditionfor the electric field, since the Laplace equation is static. However,in the leaky dielectric case, local electroneutrality must beenforced at t = 0, and Eq. (30c) corresponds to this condition.3. Numerical solutionThe solution of the coupled system of equations presentedin Section 2 yields a time-domain solution for the deformationof a perfect or leaky dielectric fluid droplet in an immiscibleperfect or leaky dielectric medium. It is subject to the assumptionthat the electromagnetic relaxation of the system is much

468 G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476faster than the fluid relaxation, and also that the interfacial tensionis uniform and constant. For the leaky dielectric model tobe valid, the electrostatic double layer thickness must be infinitesimalcompared to the droplet radius.3.1. Computational domainFig. 2a shows the geometry used in the numerical solution.An axisymmetric cylindrical coordinate system was used. Thedeforming droplet system in Fig. 1 was assumed to be symmetricwith respect to the polar axis of the droplet, parallel to theelectric field. In the present work, it is further assumed that thefluid mechanics obeys symmetry across the equatorial plane ofthe droplet, and that the electrical problem follows a form of antisymmetrywhereby opposite charges fulfill the same roles onopposite sides of the equator. This allows the problem domainto be further reduced, so that modeling takes place on only onequadrant of the droplet, and eliminates the possibility of driftingof the droplet’s center of mass.Meshing of the computational domain was performed usinga Delaunay triangulation algorithm [35], which generatesan unstructured triangular mesh on the 2D problem domain.Quadratic Lagrange elements were used for the calculation ofall variables, except pressure, which was approximated employinglinear triangular elements. Fig. 2b shows the mesh, with aninset detailing the refinement at the droplet surface. The numberof elements in this problem is approximately 2000, due mostlyto this interfacial refinement. Both the electrostatics problemand the fluid mechanics problem use the same geometry andmesh.3.2. Scaling variables for the governing equationsElectric field, potential, and charge were scaled with respectto the applied field, E 0 , R 0 E 0 , and ɛ 0 E 0 , respectively, where R 0is the initial spherical droplet radius, and ɛ 0 is the permittivityof vacuum. Length was scaled with respect to R 0 , time with respectto the characteristic timescale for oscillation of an invisciddroplet, given by [36]√(2ρ e + 3ρ i )R03 τ =(31)24γand pressure was scaled with respect to the Maxwell stress at aplanar interface separating the two fluids under consideration( )p 0 =|E 0 | 2 ɛ e − ɛ2 e.(32)ɛ iFinally, the velocity was scaled usingv 0 = 2R 0d ∞,(33)τwhere d ∞ is the final steady state deformation predicted by anappropriate analytic expression such as Eq. (1) or (3). It shouldbe noted that this characteristic velocity is simply chosen forconvenience, and does not influence the numerical solution atlarge deformations, where the analytic expressions (1) or (3)may not be valid themselves. The above scalings lead to thedefinitions of the Reynolds, Ruark, and Strouhal numbers as,respectively,Re = ρv 0R 0(34)μ , Q= ρv2 0, and St = τv 0.p 0 R 0In the above, the density and viscosity assume the values correspondingto the medium (droplet or surrounding fluid) in whichthe parameters are evaluated.3.3. Dynamic solution structureFig. 2. (a) Two-dimensional axisymmetric geometry for the numerical solution.(b) Typical meshed geometry. Dimensions are scaled with respect to the dropradius. The inset shows the details of the mesh near the droplet boundary.The nondimensionalized governing equations were solvedin a sequential manner for the above geometry and initial mesh.Further details of the nondimensionalization and mathematicalformulation are available elsewhere [37]. Here we briefly describethe solution methodology. The finite element model was

G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476 469Fig. 3. Flowchart showing the structure of the dynamic finite element solution.solved using a fixed time step with a moving mesh. The methodis multistage, and electrostatics is solved separately. Fig. 3 detailsthe steps involved in the method. Parameter initializationis performed at the beginning of the simulation. During the firsttime step, the geometry in Fig. 2 is used. For subsequent steps,the convected geometry from the previous step is used.After defining the geometry at a given time step, the Navier–Stokes problem is defined. At this stage, in the first time step,the mesh is generated, and in subsequent steps the convectedmesh from the previous step is analyzed and replaced if necessary.The Navier–Stokes setup includes equations and boundaryconditions, as well as initial conditions for the first time step. Insubsequent steps, the initial condition is derived from the solutionat the end of the previous step. In order to allow a pressurediscontinuity at the interface, separate Navier–Stokes problemsare defined in the droplet and continuous phase, and continuityis maintained by coupling at the droplet surface boundary.To complete the fluid mechanics problem, it is necessaryto solve the interfacial stress condition. The boundary stresscalculator obtains the interfacial tension stress, and solves theelectrostatics problem using a static linear finite element solution.It then combines the interfacial tension stress withthe electrical stress to produce a usable boundary condition.Once the interfacial stress calculation is completed, the initialconditions for the solution are defined. For the first timestep, they are calculated from the initial conditions specifiedduring the Navier–Stokes setup phase. In subsequent steps, theinitial conditions are generated from the previous step’s solutionby interpolation. Since the full Eulerian Navier–Stokesequations are used to describe the fluid mechanics, simplymoving the solution with the nodes is inadmissible because itwould introduce spurious convection. Re-initialization is thereforedone in place, obtaining the solution from an unmovedcopy of the finite element structure and interpolating it onto thenew nodes.Solution of the fluid mechanics problem over one timestep is done by using the nonlinear time-dependent solverDASPK [38]. Since DASPK uses its own timestepping schemeto produce a high-order result, a midpoint solution is obtainedto allow second-order convection of the geometry. This permitslarger time steps than would be possible with a zero- or firstorderconvection scheme. DASPK takes several small backwardEuler steps at the beginning of its run in order to obtain a consistentinitial condition [39]. This provides smoothing of anyartifacts that may be present after the previous time step’s solutionis mapped to the new geometry.After the solution, the complete finite element data structureis copied into two reference structures for mesh movingand re-initialization purposes. The mesh in one of the referencestructures is moved based on the average velocity over the timestep, according to the method in Section 3.4. The other referencestructure is left intact, and is used for re-initialization ofthe solution. During the rest of the postprocessing phase, thedrop volume is checked, the deformation parameter is calculated,and important variables such as the deformation parameterare added to master data lists. The global simulation time isthen incremented, and the entire calculation repeated over thenext time step. The iterations are continued until the global timereaches a preset stop point.The entire algorithm was implemented in Matlab, with callsmade to the subroutines from the finite element package Femlab2.3 (Comsol AB) [35] for mesh generation, finite elementdata structure generation, and certain postprocessing operations.Solution of the matrix equations and the numerical solutionin the time domain were conducted by calling the appropriatesolver routines (such as DASPK) via Femlab’s high-levelsolver functions. Sections of the code, such as the interfacialstress calculation, solution of the interfacial boundary condition,and movement of the mesh at successive time steps, werecoded in Matlab.3.4. Moving meshSince the deformation of the droplet is the parameter understudy, it is necessary to formulate a moving-boundary problem.The method used is a Lagrangian moving mesh, where at theend of each time step (t), the mesh is convected accordingto the local velocity. The distance moved by each mesh point,denoted by the vector x(t + t) − x(t), where x is the positionvector of a node, is calculated approximately using a quadratic

470 G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476averaging technique, taking output times at the beginning (t),middle (t + t/2), and end (t + t) of the time step:x(t + t)= x(t) + t[ 16 u(t) + 2 (3 u t + t )+ 1 ]2 6 u(t + t) .(35)Second-order averaging is possible because the fluid mechanicstime step is handled by a nonlinear solver that produces intermediateoutput, thus making a midpoint solution available.Since the finite elements deform as the fluid moves, they canbecome unacceptably strained during large deformations. Tominimize numerical errors, the code checks the minimum elementquality before each time step, and re-meshes the geometryif this value is too low.3.5. Convergence testingThe convergence of the transient solution to a correct steadystatecan be tested by comparing against the steady-state analyticalresults for small deformation for both perfect and leakydielectric cases. The results presented here are representativeand limited to a single physical case, which will henceforth becalled the base case. It approximates a water droplet in oil witha radius of 1 µm, subject to a 1 MV/m applied electric field.Table 1 details the base case parameters for perfect dielectricfluids. Deformation in the base case is small; the perfect dielectricresult predicts a steady-state deformation parameter of3.9926 × 10 −4 ,cf.Eq.(1).Fig. 4 shows the effect of altering the boundary refinementat the droplet surface across a range of values. The parameterH max is a mesh size specification in the code, which indicatesthe spacing between the successive nodes along thespecified edge of the geometry. Here it is used on the interfacialboundary, with the bulk fluid and all other boundaries leftat the default resolution. The electrostatics problem is solvedon the same mesh as the fluid mechanics. It was noted duringtesting that refinements further away from the interface have noappreciable effect on the fluid mechanics solution.Table 1Simulation parameters for the perfect dielectric base caseApplied electric field, E 01MV/mDroplet radius, R 0 1µmDroplet permittivity, ɛ i80 × 8.854 × 10 −12 C/VmSuspending medium permittivity, ɛ e3 × 8.854 × 10 −12 C/VmInterfacial tension, γ0.03 N/mDroplet density, ρ i 1000 kg/m 3Medium density, ρ e 1000 kg/m 3Droplet viscosity, μ i0.001 Pa sMedium viscosity, μ e0.001 Pa sWith 9 nodes on the interface (H max = 0.2), the error in thesolution is significant, though not large. Increasing the resolutionto 17 nodes (H max = 0.1) results in a reversal of the error,which suggests that multiple factors, such as error in the electrostaticssolution, are contributing to the error in the first case.This implies that, despite the moderately low error, the results ingeneral are not trustworthy at such low interfacial resolutions.At a resolution of 33 nodes on the boundary (H max = 0.05), theresults are effectively converged. The steady-state is approximately0.02% greater than the analytic perfect dielectric result(Eq. (1)). The 54-node case (H max = 0.03) is virtually indistinguishablefrom the 33-node case. The global mesh in the case ofthe 33-node boundary contains 1424 elements, and in the caseof the 54-node boundary, it contains 1960 elements. For optimumcomputation, the base case is best solved with an H maxof 0.05, or 33 interface nodes. Simulations of larger deformationsdisplaying nonlinearity or instability sometimes require aninterface mesh size of 0.03 (54 nodes), but refinement beyond54 nodes has little effect on the solution for parameter rangesexplored in this study.It is critical that the electrostatics solution be accurate atthe interface, since the interfacial electric field drives the fluidmechanics problem. It was observed that the Maxwell stresseswere most accurate when the mesh used for solving the fluidmechanics problem was used when solving the electrostaticproblem.Fig. 5 shows the effect of scaled time step on the convergenceof the base case perfect dielectric problem. Time wasFig. 4. Variation of dynamic response for increasing mesh refinement at dropinterface. The parameter varied is mesh size specification in the finite elementcode, applied here to the interfacial boundary only. The nondimensional timestep is 0.06.Fig. 5. Convergence of dynamic response with time step, for a 33-node interface.

G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476 471scaled in the problem employing the characteristic time for oscillationof an inviscid droplet, Eq. (31) [36]. The time step isexpressed in radians, where 2π radians signifies one completecycle of oscillation of the drop as predicted by inviscid theory.Three time steps were tested: 0.06, 0.12, and 0.15 radians. Thefirst two cases demonstrate good agreement with one another,with a peak value difference of less than 1%. The 0.15 radiantime step, on the other hand, is too large to maintain stability,and the pole of the drop oscillates in an unsteady fashion,eventually compromising the geometric integrity of the mesh.For all results presented in this study, convergence tests wereconducted to ensure that the mesh and time stepping led to anunambiguous steady state solution.4. Perfect dielectric modelA parametric study of the perfect dielectric response to smallapplied fields leading to relatively small droplet deformationswas presented in an earlier work [40], and hence, the focus ofthis section will be to present representative results from simulationsunder high applied fields, eventually leading to destabilizationand breakup of the drop. The physical parameters areas listed in Table 1, and the electric field is varied.Fig. 6 shows four results for the perfect dielectric case. Thedeformation parameter, d, in the vertical axis is normalized bythe analytically predicted steady-state deformation, d ∞ ,givenby Eq. (1), so that ideally each numerically obtained dynamicresponse should converge to 1 at large times. It is evidentthat for the lower two values of the electric field, the numericalresult does converge almost exactly to the analytic result.The 3 MV/m case is slightly offset, which indicates the onsetof nonlinearity—the droplet becomes nonspherical and theassumption of infinitesimal deformation embedded in the analyticalexpression begins to fail. The deformation parameter inthis case is approximately 0.0036, which is relatively small. The10 MV/m case with d ∞ = 0.04 is significantly nonlinear, andillustrates well the characteristic increase in elongation other researchershave observed in large prolate deformations [27,41].Fig. 7a shows the dynamic response for fields ranging fromthe base value of 1 MV/m to a maximum value of 20 MV/m.All other parameters are those of the base case in Table 1. Theequilibrium shape of the droplet corresponding to 10 MV/mapplied field, and the final snapshot of the deformed droplet underan applied field of 20 MV/m areshowninFigs. 7b and 7c,respectively. Increasing the field past 10 MV/m in the basecase results in substantial nonlinearity. The electrical stress isproportional to the square of the local field, so deformation increasesrapidly with applied field. In particular, the deformationin the 20 MV/m case appears to have no steady-state value;it develops pointed tips as shown in Fig. 7c, and the numericalmethod breaks down due to a singularity in the boundarycurvature description. This occurs at a deformation parameterof 0.593, corresponding to an aspect ratio of 3.91. In physicalsystems, the development of the pointed tip precedes the emissionof fluid strands and/or satellite droplets from the pole of themain droplet [9,23]. This phenomenon is known as tip streaming,and occurs when the drop permittivity is large relative tothe continuous phase [13,41,42].It may be observed from the transient behavior of the deformationthat the oscillations in the dynamic response becomedamped as the deformation increases. At large applied fields,the deformation of the droplet appears to be almost monotonic.This may be attributed to the nonlinearity being such that theFig. 6. Normalized dynamic response for variation in electric fields using theperfect dielectric model. The simulation results for the 333 kV/mand1MV/mare virtually indistinguishable in the figure. The horizontal dashed line (atd/d ∞ = 1) is the steady state deformation according to Eq. (1).Fig. 7. (a) Normalized dynamic responses for high applied fields using the perfectdielectric model. (b and c) Deformed droplet shape at end of run for anappliedfieldof10and20MV/m, respectively.

472 G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476electrical forcing increases with the elongation of the droplet.The initial stages of the deformation follow similar dynamicsunder all applied fields. As the deformation increases, theelectric forcing increases, while the viscous damping and theinterfacial tension (and local curvature) based restoring forcestend to decelerate the deformation. When the destabilizing electricalforces increase faster than the restoring forces and theviscous damping, the droplet continues a monotonic accelerationtoward breakup as shown for the 20 MV/m case.5. Leaky dielectric modelThe leaky dielectric case in general is characterized bytangential electrical stresses on the drop surface, resulting insteady-state flow along the interface and persistent circulatoryflow patterns inside the droplet. For small deformations, thestatic-boundary leaky dielectric model shows very similar dynamicsto the perfect dielectric model, with the exception thatthe deformation can be negative, or zero. The full leaky dielectricmodel can show strong effects of the boundary chargedynamics and convective transport of charge along the interface,which can alter the transient response of the droplet to animposed electric field, as well as the steady state deformation.The parameters used in the leaky dielectric model are thesame as in Table 1, with the addition of a pair of conductivities.These are calculated from Eq. (12), given a set of ion characteristicscombined with a temperature and partition coefficient(Eq. (17)). The relevant ion characteristics are the absolute concentrationin the continuous phase (n e ), valence of each ion(z 1 , z 2 ), and diffusion coefficient (Di 1, D2 i , D1 e , D2 e ). This allowsa fully physical description, as in the perfect dielectriccase. This is done in both the static and full leaky dielectricmodels, even though the static model only requires a conductivityratio. Table 2 gives the full parameter set. Here, we firstdescribe simulation results using the static boundary condition,Eq. (25) for the leaky dielectric model. Following this,the simulation results based on the dynamic charge buildupequation (27), the so-called full leaky dielectric model, will bepresented.5.1. Static boundary leaky dielectric modelFig. 8 shows the effects of increasing the applied field in thebase case. The field is varied from 1 MV/m in the base case toamaximumof20MV/m. In Fig. 8a, the deformations are normalizedby the analytical expression for the deformation basedon the leaky dielectric model, Eq. (3). In this figure, it can beseen that some of the same effects are present as in the perfectdielectric case, seen in Fig. 7. The increase in applied fieldproduces a nonlinear response, accompanied by a decrease inovershoot. However, it is notable that in this case the nonlinearityis in the opposite direction from that observed in theperfect dielectric base case of Fig. 7. The high-field deformationsare less extreme than predicted by Taylor’s result, and allof them have steady-state values. Part of the reason for the reduceddeformation at high fields is that the analytic deformationparameter in these cases exceeds −1. This is a physical impossibility,since it requires a negative aspect ratio. The 20 MV/mcase has a theoretical deformation parameter of −1.9885 basedon Eq. (3), but the numerical result gives only −0.898, whichcorresponds to an aspect ratio of 0.0539. The actual deformationsmay be seen more clearly in Fig. 8b, which is a plot of theTable 2Simulation parameters for the leaky dielectric base caseApplied electric field, E 01MV/mDroplet radius, R 0 1µmDroplet permittivity, ɛ i80 × 8.854 × 10 −12 C/VmSuspending medium permittivity, ɛ e3 × 8.854 × 10 −12 C/VmInterfacial tension, γ0.03 N/mDroplet density, ρ i 1000 kg/m 3Medium density, ρ e 1000 kg/m 3Droplet viscosity, μ i0.001 Pa sMedium viscosity, μ e0.001 Pa sMedium ionic concentration, c e 0.04 mol/m 3 (4 × 10 −4 M)Ion partition coefficient, Φ = (n i /n e ) 1.0Positive ion valence, z 1 1Negative ion valence, z 2−1Droplet +ve ion diffusion coefficient, Di 1 10 −9 m 2 /sDroplet −ve ion diffusion coefficient, Di 2 10 −9 m 2 /sMedium +ve ion diffusion coefficient, De 1 10 −9 m 2 /sMedium −ve ion diffusion coefficient, De 2 10 −9 m 2 /sSystem temperature, T298 KFig. 8. Effect of high applied fields on the static leaky dielectric response.(a) Deformations scaled with respect to the Taylor expression for thesteady-state deformation, d ∞ ,givenbyEq.(3). (b) Actual variation of the deformationparameter with scaled time. Other simulation parameters are given inTable 2.

474 G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476Fig. 12. Dynamic contribution from charge relaxation on interface for variousfluid conductivities in the transient charge boundary leaky dielectric model.Incorporation of Eq. (27) along with different values of fluid conductivitieschanges the dynamic response compared to the static boundary leaky dielectricresponse.Fig. 11. Deformed droplet shape at end of run, for (a) Φ = 0.1, (b) Φ = 0.2and(c) Φ = 0.5, all obtained for an applied field of 10 MV/m.The conductivity ratios in Figs. 11b and 11c are much closerto the crossover point between the sharp-edged and blunt-diskmodes of deformation. Fig. 11b shows a third type of behaviorthat appears near the transition, in which a toroidal segmentpinches off from the main drop. Fig. 11c shows a casewith a larger partition coefficient of 0.5, and it shows most ofthe characteristics of the blunt-disk droplets. A vestige of thetoroidal ring can nevertheless be seen in the widening of thisdroplet toward its edge.Transitions between these modes of deformation are herefacilitated by changing only a single parameter, the partitioncoefficient. It is not unlikely that the observed effect is actuallypart of a compound effect involving the fluid parameters,as well as the permittivity ratio, which is known to affect thecharacter of perfect dielectric deformations [13,27,42].5.2. Transient charge boundary leaky dielectric modelThe full leaky dielectric model allows for finite-in-timecharge buildup on the interface, as well as steady-state convectioneffects. This can have a profound effect on the overallresponse of the droplet to an applied electric field, both in termsof the initial transient and the steady-state deformation.Fig. 12 shows the effect of varying the ionic concentrationin the leaky dielectric base case, starting with the static modeland demonstrating the effect of lowering the absolute conductivity.In the base case, the conductivities of the two fluids areequal. The dynamic charge simulations were done with mono-valent ionic concentrations of 2 × 10 −4 and 4 × 10 −5 M, leadingto conductivity values of approximately 1.5 × 10 −3 and3 × 10 −4 S/m, respectively. The horizontal dashed line indicatesthe analytic deformation from Taylor’s result.The dynamic response of the static leaky dielectric modelis very similar to that of the perfect dielectric model. However,the inclusion of charge dynamics in the form of Eq. (27)increases the complexity of the response significantly. The dynamicboundary cases here show non-minimum-phase behavior;their initial motion is away from the steady-state deformationthe droplet will eventually attain. In addition, the lowestconductivitycase shows a significant secondary time scale,slower than the droplet’s oscillation frequency, by which relaxationto the final state occurs. The reason for this behavior isthe finite charge buildup time. Saville [15] gives the conductiverelaxation time constant as ɛ/σ. Initially, the boundary isuncharged, so the dynamic response begins identically to theperfect dielectric case. As the charge builds up, the droplet’s deformationchanges direction toward the leaky dielectric steadystatelimit. If the charge relaxation is significantly slower thanthe drop oscillation, the transition to steady-state deformationis governed by the charge buildup. This effect is first-orderand resembles an exponential, which is shown in the lowestconductivitycase in Fig. 12. In an extreme case with very lowconductivity, the complete perfect dielectric transient oscillationmight occur, followed by a slow, monotonic transition tothe leaky dielectric steady-state limit.Fig. 13 shows the complete dynamic response of a leaky dielectricdrop using the dynamic interfacial boundary condition.The parameters are those of the leaky dielectric base case, withan ionic concentration in both fluids of 4 × 10 −5 M, leadingto a conductivity of 3 × 10 −4 S/m. This result demonstratesthe effect of charge convection along the drop interface on thesteady-state deformation. The steady-state value predicted byTaylor’s analytic result, Eq. (3), is shown by the dashed line;clearly, the steady-state value reached by the simulation is differentfrom the analytic prediction. The charge distribution on

G. Supeene et al. / Journal of Colloid and Interface Science 318 (2008) 463–476 475Fig. 13. Steady-state offset from Taylor limit resulting from charge convectionin the transient charge boundary leaky dielectric model. The horizontal dashedline shows the steady-state deformation obtained using Eq. (3).The value of the interfacial tension used in the simulations was0.0089 N/m, as reported in Lu [43].In Fig. 14, the open and solid circles depict two sets ofexperimental data obtained from [43]. The dashed line representsthe steady state deformation predicted using Eq. (3). Theopen square symbols are the numerical deformation results atsteady state, while the solid line is the best fit to the numericaldata. For small values of We < 0.05, the numerically predictedsteady-state deformation seems to be in good agreement withthe deformation predicted by Eq. (3). However, at larger valuesof We, the numerical results seem to indicate a nonlinearincrease in deformation with We. A similar trend is observedin the experiments, and the numerical predictions seem to bein good agreement with the experimental data. The nonlineardeparture from Taylor’s analytic result observed in the numericalsimulations for this case is gentle enough that it providesa possible explanation for the observed trend in the experimentaldata. The agreement between the numerical simulations andexperiments seen here is encouraging.7. Concluding remarksFig. 14. Steady-state deformation parameter plotted against Weber number fora water droplet suspended in decyl alcohol: Comparison of numerical simulationswith experimental data [43] and analytic steady-state deformation basedon Taylor model.the interface is modified by the steady-state tangential flows associatedwith the leaky dielectric system. This effect alters theelectrical stress distribution and can modify the steady-state deformation.As noted by Feng [24], the result in the case of oblatedeformation is a reduction of the steady-state deformation relativeto the analytic result.6. Comparison with experimentLu [43] reported experiments in which drops of water suspendedin organic solvents were subjected to DC electric fields.One of the control tests was a series of measurements of thedeformation of a drop of water in decyl alcohol. This experimentwas duplicated numerically in this study employing theleaky dielectric model, and Fig. 14 shows the results, wherethe steady-state deformation of the droplet is plotted against theWeber number, defined asWe = ɛ eR 0 E02 .γ(36)Numerical simulations of transient droplet deformation underthe influence of applied electric fields were presented forperfect dielectric and leaky dielectric systems. The finite elementmoving boundary model tracks the droplet deformationin time, and predicts the steady state deformation accurately.Excellent agreement with analytic expressions for steady-statedeformations are observed for low applied fields, both for perfectand leaky dielectric systems. The influence of higher appliedfields, leading to a nonlinear response, have also beendescribed. For the leaky dielectric system, a wide range ofdeformed droplet shapes are attainable by simply varying theconductivity ratio between the two fluids. Finally, the transientcharge buildup boundary condition, when incorporated in theleaky dielectric model, can show a much slower relaxation ofthe droplet to a steady-state.A representative set of results obtained from the numericalsimulations, including a comparison with experimental results,indicate that the present model can provide a generalized procedurefor assessing droplet deformation under the influence ofhigh applied electric fields. The simulations can easily be extendedto other types of electrical forcing, such as oscillatingfields, and will allow exploration of a wide range of microscopicdroplet dynamics that are being studied in conjunctionwith electrical de-emulsification or in microfluidic systems.AcknowledgmentsFinancial support for this research from National Sciencesand Engineering Research Council of Canada (NSERC) andCanada Research Chairs (CRC) program is gratefully acknowledged.References[1] A.G. Bailey, Electrostatic Spraying of Liquids, Wiley, New York, 1988.

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