Electrohydrodynamic Phenomena

J. Indian Inst. Sci. A Multidisciplinary Reviews Journal ISSN: 0970-4140 Coden-JIISAD

Aditya Bandopadhyay1* and Uddipta Ghosh2 REVIEW REVIEW ARTICLE

Abstract | This work is a review article focused on exploring the interactions between external and induced electric felds and fuid motion, in the presence of embedded charges. Such interactions are generally termed electrohydrodynamics (EHD), which encompasses a vast range of fows stemming from multiscale physical effects. In this review article we shall mainly emphasize on two mechanisms of particular interest to fuid dyna- mists and engineers, namely electrokinetic fows and the leaky dielectric model. We shed light on the underlying physics behind the above mentioned phenomena and subsequently demonstrate the presence of a common underpinning pattern which governs any general electrohydrodynamic motion. Hence we go on to show that the seemingly unre- lated felds of electrokinetics and the leaky dielectric models are indeed closely related to each other through the much celebrated Maxwell stresses, which have long been known as stresses caused in fuids in presence of electric and magnetic felds. Interactions between Maxwell Stresses and charges (for instance, in the form of ions) present in the fuid generates a body force on the same and eventually leads to fow actuation. We show that the manifestation of the Maxwell stresses itself depends on the charge densities, which in turn is dictated by the underlying motion of the fuid. We demonstrate how such inter-related dynamics may give rise intricately coupled and non-linear system of equations governing the dynamical state of the system. This article is mainly divided into two parts. First, we explore the realms of electrokinetics, wherein the formation and the structure of the so-called electrical double layer (EDL) is delineated. Subsequently, we review EDL’s relevance to electroosmosis and streaming potential with the key being the presence and absence of an applied pressure gradient. We thereafter focus on the leaky dielectric model, wherein the fundamental governing equations and its main difference with electrokinetics are described. We limit our 1 Department attentions to the leaky dielectric motion around droplets and fat surfaces of Mechanical Engineering, Indian and subsequent interface deformation. To this end, through a rigorous Institute of Technology review of a number of previous articles, we establish that the interface Kharagpur, Kharagpur, West Bengal 721302, shapes can be fnely tailored to achieve the desired geometrical charac- India. 2 Discipline of Mechanical teristics by tuning the fuid properties. We further discuss previous stud- Engineering, Indian ies, which have shown migration of droplets in the presence of strong Institute of Technology Gandhinagar, Palaj, electric felds. Finally, we describe the effects of external agents such Gujrat 382355, India. *[email protected]

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as surface impurities on leaky dielectric motion and attempt to establish a qualitative connection between the leaky dielectric model and EDLs. We fnish off with some pointers for further research activities and open questions in this feld.

Keywords: Electrokinetics, Electrohydrodynamics, Leaky dielectric model, Electrical double layer, Droplet, Electroosmosis, Streaming potential, Thin flm, Interfacial phenomena

1 Introduction polymer transport, protein manipulation etc.26– Electrokinetic phenomena refers to those pro- 29. Streaming potential phenomenon has been cesses in which the fuid fow affects and is utilized to sense fows through porous media30–37 affected by the presence of electric felds. Reuss1 and for harvesting mechanical energy into electri- was the frst to investigate the phenomenon in cal energy18–20, 38–43. which the action of an electric feld led to the fow While the above development of EDL and of liquid and a potential difference at the end of electrokinetic phenomena took place for single two columns. Based on these ideas2 showed that phase fows there was a parallel development of the process of driving a fow across a charged the area of electrohydrodynamics (EHD) which membrane leads to the generation of a poten- refers to the study of fuid fow interacting with tial across the two ends. The former case where electric felds. Historically the studies of EHD the application of an electric feld leads to the have been to study two phase fows in the pres- generation of a fow is termed as electroosmosis ence of electric felds44–46. The developments of while the later phenomenon in which the fow of the last century can be traced back to47, 48 who a fuid in contact with a charged substrate in the studied the action of electric felds at liquid inter- absence of any electric feld leads to the genera- faces and their stability in particular. Much of the tion of a potential difference is known as stream- developments of the fows between conducting ing potential. The fundamental coupling between and insulating fuids was undertaken by Melcher the charged substrate and fuid fow is possible and coworkers49. The assumption of fuids being due to the generation of the electrical double completely conducting or completely insulating layer (EDL)3–14. The idea is that the preferrential however incorrectly predicted some of the obser- attraction of counterions towards a charged sur- vations of deformation of oil droplets suspended face leads to the formation of a region with net in other oils. The discrepancy was resolved by46 charges in the vicinity of the wall. This region who considered the fuids to be leaky dielectrics with a net charge is then able to interact with in which the small yet fnite conductivity of the applied electric felds. The fundamental concepts fuids is accounted for. In this way the net charge behind the EDL were introduced by13 and later accumulated at the interface between the two fu- improved by14, who incorporated the effects of ids leads to remarkably different behaviour than thermal energy due to Brownian motion, which the analysis in which the two fuids are consid- were previously neglected in the capacitor based ered to be either conducting or insulating. The model put forward by15. The developments in leaky dielectric model has found success in a wide the theory of the EDL led to developments in variety of situations involving commonly used the linear relationship between the net mass insulating oils such as silicone oil, castor oil, corn fux and the applied electric feld for the case of oil etc.50, 51. The leaky dielectric model was then electroosmosis8, 16. On similar lines, the linear utilized extensively by Melcher and coworkers to relationship between the applied pressure gradi- address a host of problems ranging from stability ent and induced electric feld was developed17–20. of fat interfaces44 to travelling waves with conju- The development of these theories was central to gated heat transfer45. the development of colloidal science. Manipula- The infuence of AC electric felds with high tion of micron and submicron sized particles frequencies on the interface between two fu- bearing a surface charge suspended in aqueous ids was studied by52. Later on53 demonstrated solutions by means of electric felds is termed as that the instability at the surfaces arises from the electrophoresis5, 21–25. Electrophoresis is a widely action of the Maxwell stress and not due to dielec- utilized phenomenon for many lab-on-a-chip tric breakdown. Later, the infuence of a thermal applications which deal with particle separation, gradient on bulk electroconvection was studied45,

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54. The idea is that the presence of temperature link between the electrokinetic model and the gradients lead to the formation of permittiv- leaky dielectric model. Recently, efforts have been ity gradients. This allows for the electrostriction made to unveil the combined infuence of fuid term (see Appendix A) to yield a net volumetric rheology and applied electric felds in multiphase force. This mechanism acts in such a manner that systems78, 79. The fact that droplets can be coa- the temperature gradient is nullifed. The idea of lesced or sheared due to electric felds allows one utilizing electric felds to achieve an augmented to study emulsions in such systems80. We shall heat transfer has also been discussed by55. Later build upon the various aspects of this in the sub- on, there have been attempts to achieve enhanced sequent sections. rates of condensation by utilizing electric felds as At the heart of both the effects is the presence well56–58. On similar lines, there have been stud- of charges which are acted upon by electric felds. ies to understand the infuence of injected space The coupling of fuid fow and electric felds is charge on electrohydrodynamic instabilities aris- achieved through the Maxwell stress and ionic ing due to the EHD volumetric force described in advection5, 23, 24, 81–87. In this article we shall look the appendix A of59. On the basis of53, 60 analyzed into some of the fundamental aspects of electro- theoretically the problem of the action of an AC kinetic transport through narrow confnements. electric feld on a fat interface. Later61 studied Later on we shall dwell upon the leaky dielectric the effect on the similar fat interface between framework with an emphasis on the dynamics of two fuids due to the action of a transverse elec- droplets and fuid interfaces in the presence of tric feld. In an attempt to bridge the gap between high electric felds. The potential for further work electrokinetic phenomena and the electrical dou- is outlined as a conclusion. ble layer phenomena,62, 63 studied the infuence of electrical forces towards interfacial instability. 2 Electrokinetics One of the frst analyses to understand the infu- 2.1 Electrical Double Layer ence of AC felds on the dynamics of droplets was The genesis of electrokinetics lies in the devel- undertaken by64. The fundamental idea behind opment of a surface charge due to the chemical such analysis is described in the later sections. A equilibrium between the aqueous solution and combined action of acoustic waves and electric the substrate. For example in the early experi- felds has been studied to manipulate droplets at ments by1, various clay plugs were used. In such the microscale65. Later, Choi and Saveliev66 have clays, the regions in the pores where the crys- analyzed the infuence of AC electric felds on the tal face is exposed, the edge groups, which may coalescence of droplets. contain silicates, aluminates, etc. lose hydrogen The fundamental ideas behind analyzing thin ions, depending primarily on the solution pH, flm systems have been described in detail by67. as a result of which the surface has a net nega- Miksis and Ida68, 69 have described the theoreti- tive charge. Another mechanism by which natu- cal basis of the dynamics of thin flms. Schaef- rally occuring clays may acquire net charges is fer et al.70 utilized the aforementioned thin flm through isomorphic substitution where ions with equations and obtained a relationship between a certain valency, e.g. Al 3 is replaced by an ion the observed patterns on an initially fat substrate + with a lower valency, e.g. Ca 2, thereby rendering and the magnitude of the applied feld. Based + the wall negatively charged88. Similarly another on the key observation, there have been several mechanism of substrate charging is specifc ion attempts to understand and exploit such insta- adsorption88–93. In this, the silica/glass based bilities to generate desired patterns on various substrates may adsorb hydrogen ions to the sur- polymer substrates71. Based on the lubrication face (or lose the ions, depending on the pH) to analysis, there have been attempts to understand achieve a net negative charge. the fows of liquids in such electrifed environ- The charge at the wall consequently affects ments72. The action of such transverse electric the distribution of ions in the aqueous solu- felds has also been studied for the case of formation, especially in the vicinity of the wall94, 95. tion of bilayers from such thin flms73. Li et al.74 To understand this, let us consider the situation have utilized the aforementioned ideas regarding where there is no fow. If we consider the walls to unstable interfaces in such electrifed environ- possess a net negative charge, then the positively ments to obtain patterned cylinders. In recent charged ions (counterions) are attracted towards times, Thaokar and coworkers have also analyzed it. Simultaneously the negatively charged ions the instability of two fuidic layers due to AC elec- (coions) are repelled by the wall. Therefore, the tric felds75, stability of fuid layers at the limit of region near the wall has an excess of counterions. ‘small features’76, air liquid patterning77, and the

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Besides being attracted by the wall, there is ther- example, NaCl) such that z z , z z allow- + = − =− mal motion for each of the particle which causes ing us to write the above distribution as94, 95: the ions to wiggle around continuously instead zeζ of forming a single stacked layer on the charged n n0 exp ∓ . (2) ± = k T wall. The sheilding of the wall by the coions is B such that as we move away from the wall the con- In order to fnd out the distribution of the poten- ditions of the bulk are reached, i.e. the number of tial we make use of Gauss’ law (see appendix) to counterions and coions becomes equal far from write the bulk. Thus, there is a region near the wall 2 ǫ ψ ρe,f (z en z en ), (3) having a net charge. The thickness of the layer ∇ =− =− + + + − − depends on the concentration of the aqueous where ǫ ǫ0ǫ (see appendix) represents the per- = solution. This layer is called as the electrical dou- mittivity of the medium. Note that the above ble layer (EDL)4, 5, 94, 95. The original idea of such equation is written for the situation where the a sheilding was put forth by Helmholtz at a time ionic distributions are in equilibrium. This when statistical mechanics was not developed. implies that there is no fow taking place. In the Consequently the idea was that the charge at the presence of a fow, we must appropriately account wall is screened by the counterions from the bulk for the effect of the fow by solving completely for in the form of a single layer. This was the idea of a the ionic transport equation with due regard to molecular capacitor. The thermal agitation inher- the fuxes caused by fow, diffusion, and electro- ent for all systems causes such a condensed image migration. This will be discussed later. For a one to become diffuse. The idea behind the electri- dimensional planar system with uniform surface cal double layer is represented in Fig. 1.The elec- properties and the Poisson–Boltzmann distribu- trostatic attraction (or repulsion) of ions when tion may be written as94, 95: combined with the concept of thermodynamic d2ψ zeψ equilibrium defnes a Boltzmann distribution of ǫ 2n0ze sinh (4) 94, 95 dy2 = k T ions such that we have : B The above equation governs the potential distri- z eζ (1) n n0 exp − ± , bution in the fuid, ψ . The boundary conditions ± = k T B are that at y 0 , ψ ζ while at y , ψ 0. = = →∞ = where n and n defne the ionic concentrations In order to gain some insight into the equa- + − of the cations and anions respectively, n n0 rep- tion, we may linearize the sinh term to obtain the = resents the concentration which corresponds to linearized Poisson–Boltzmann equation11, 94–96: the zero potential (far from the surface), z and 2 2 2 + d ψ 2n z e z represent the valency of the ions, e represents 0 ψ (5) 2 − dy = ǫkBT the electronic charge, kB represents the Boltz- mann constant, and T represents the absolute which may be further simplifed by taking a temperature. In order to simplify the situation dimensionless electric potential, ψ zeψ/kBT = let us consider a z:z symmetric electrolyte (for and the channel half-height, H, to be the characteristic length to obtain

(a) (b) (c)

Figure 1: a Physical situation in the vicinity of the wall. The charged substrated attracts more counterions (red spheres) near the wall and repels cooins away from the wall. b The relative attraction of counterions and coions is schematically shown for a negatively charged wall for the case where the concentration of both the species tends to the bulk concentration of n0 . c The potential at the charged surface is effectively screened by the charges due to which the potential varies with the distance into the bulk. The potential at the interface of the frst immobile layer (where the ions are held frmly to the surface) and the diffuse layer is called as the zeta potential, ζ . For all practical interest, it is enough to specify the zeta potential at the surface.

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d2ψ (6) outline the strategy to obtain the ionic distribu- κ2H 2ψ κ2ψ, dy2 = = tion by accounting for the fnite volume occu- pancy of ions instead of assuming them to be 1 ǫkBT where κ− 2 2 represents the inverse of point sized ions. = 2n0z e the characteristic thickness of the EDL94, 95, signi- The free energy of a system is comprised of fying the relative penetration depth of the effect contributions from the electrostatic self energy, of the potential of the wall into the bulk. It is the chemical potential of the ions, and the entropic contributions from the various species. interesting to note that the thickness of the EDL 94, 95, 98, 99 is smaller for a larger bulk concentration. The Mathematically, this may be written as : solution for the above linearized equation yields ǫ 2 F (ψ) eψ(z n z n ) dV the solution for the electric potential away from = − 2 |∇ | + + + − − − 11 k T a wall as : B n a3 ln n a3 n a3 ln n a3 3 ion ( ion ion ( ion) + a + + + − − (7) ion ψ ζ exp κy , 3 3 3 = − (1 n aion n aion) ln(n aion dV (10) + − + − − + n n0 exp( ψ) (8) The variational minimization of the above with ± = ∓ Moreover for the case of channels having nar- respect to the electric potential, ψ , yields the rower confnements such that the centerline standard Poisson equation. The variational deriv- is not very far from the wall, the above solu- ative of the free energy with respect to the ionic densities leads to to the modifed Boltzmann dis- tion for the electric potential may be appropri- 94, 95, 98, 99 ately adapted with the following conditions. The tribution : boundary condition at the wall is still the zeta zeψ potential while the boundary condition at infnity n0 exp ∓ kBT must now be replaced by a symmetry boundary n (11) ± = 1 2ν sinh 2 zeψ condition at the channel centerline. This implies + kBT that ψ(H) ζ and ψy(0) 0 , where the center- = = line is considered to be at y 0 and the wall is = 2.1.2 Constant Permittivity considered to be at y H . Using these boundary = In the standard model the permittivity of the conditions the solution for the electric potential medium is assumed to be the permittivity of in a channel with the coordinate axis at the cen- water. However, if we consider the physical scn- 7, 96 terline as : eario painted by Eq. (8) we see that the electric feld in the vicinity of the wall is rather large. An cosh κy cosh κy (9) ψ ζ ψ ζ approximate estimate of the electric feld near the = cosh κH ; = cosh κ wall is given by ζκ where ζ 0.25 V and κ 10 ≈ ≈ Below, we note the approximations in the analysis nm. In this case the water dipoles in the vicinity and how those can be overcome. of the walls tend to be oriented along the wall instead of being randomly oriented100–104. In the 2.1.1 Point Sized Ions bulk region away from the wall, there are enough Working in the continuum framework, we have ions which have shielded the wall, thus allowing assumed the felds of the ionic density as being the water molecules to be randomly oriented. The continuous. However, in the vicinity of the wall, constrained orientation of water dipoles near the the concentration of counterions is signifcantly wall leads to the drastic fall in the permittivity in large, that too in a small volume (this is especially the region near the wall. The decreased permit- true for situations with a thin EDL). Physically, tivity is also partially attributed to the decrease however, there cannot be an arbitrarily high con- in the water concentration due to the preferren- centration of ions near the wall. The limit on the tial attraction of counterions near the wall and maximum ionic density in the vicinity of the wall an expulsion of water away from the wall. Based occurs due to the fact that in reality ions are fnite on the original model by Booth, the relative per- sized94, 95, 97. In fact the ionic size is amplifed by a mittivity decreases in regions of high electric feld solvation radius of the ions. To account for this, (since high electric felds orient dipoles). The 3 functional relationship between the permittivity one must introduce a steric factor, ν 2n0a , = ion and the magnitude of the electric feld strength, where n0 represents the bulk concentration, and dψ/dy is given by102: aion represents the size of the ion. Let us briefy | |

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2.2.1 Advection–Diffusion–Electromigration ǫ n2 n2 3 r ri 1 ri Equation ǫr,0 =ǫr,0 + − ǫr,0  dψ  � � A dy The mass fux of a system in general is given by the  � � (12) density of the particular species multiplied by the � � 82, 105 dψ 1 velocity of the species. Thus, Ji nivi . The coth A � � , = ×  dy − dψ  velocity of the species in such a case is comprised � � �� A � � dy of the background convection velocity and contri-  � � � � � � � � bution from other effective fuxes. Thus, the fux � � 94, 95 where A is defned as the solvent polarization may be split as : Ji ρiui ni(vi ui) , where = + − number given by 5 νd (n2 2) where ν repre- the latter term is regarded as the density multi- eL ri + d sents the dipole moment of water, nri represents plied by the drift velocity. The drift velocity for an the refractive index of water, e represents the elec- uncharged species is comprised of a diffusion fux tronic charge, and L represents the system length- due to the gradients in concentrations. The dif- scale. The timescale of atomic polarization and fusive component of the fux may be represented 15 18 94, 95 orientational polarization (O (10− 10− ) s as : Jd D ni . Besides the advective and dif- 11 − =− ∇ and O(10− ) s, respectively) is much faster than fusive fux, there exists a fux of species which are the typical timescales of applied frequencies acted upon by various external felds. For the case 9 (which may be as fast as 10− s). In such cases the of an electric feld, there is a electromigrative fux infuence of the rapidly changing electric feld on which may be understood as follows. The force act- the magnitude of the permittivity is solely mani- ing on an ion is given by the electric feld multiplied fested through the instantaneous variations in the by its own charge.This leads to a net migration. In electric feld. the absence of any viscous forces, the charge would The two simplifying assumptions are valid accelerate. But at equilibrium, there is a balance of for cases where the ionic concentrations are the two forces. In general such fuxes are defned in 94 em low. A low concentration implies that the EDL terms of the mobility, defned as : ωi v /Fext . = i thickness is large. A low concentration is leads Therefore, the electromigrative fux of a species is 3 95, 106 to a smaller steric factor, ν 2n0a . Addition- given by Jem niωiFext . The transport of ions = ion = ally the thicker EDL leads to a gentler gradient of an aqueous electrolyte are governed by three inside the EDL (the feld strength is proportional kinds of fuxes as explained below106. to ζκ . Therefore, while accounting for systems Ji ρivi Ja Jd Jem with larger ionic concentrations, it is worth tak- = = + + (13) ρiui Di ni niωiFext ing into account the aforementioned effects into = − ∇ + consideration. The specifcation of the mobility may be done in the following way: At equilibrium, the forces acting on an ion moving through a liquid are 2.2 Electroosmosis and Streaming given by zeE 6πηaionv where η is the viscos- Potential = ity of the solvent. Upon rearranging we obtain The development of the EDL on various sub- ω v/F 1 107, 108. However, using the strates mentioened above has signifcant impact = = 6πηaion ideas of random walks and its connection with on the processes which involve fuid fow of aque- diffusion coeffcients, Einstein proposed the ous solutions over such surfaces. Most impor- relationship (also known as the Nernst–Einstein tantly, the fact that there is a region near the relationship)106: substrate surface possessing a net charge leads to the direct conclusion that an external electric Di ωi (14) feld may cause motion in the ions. The presence = kBT of net charge in the region of fuid also leads to a using which we can write net body force acting on the fuid—thereby caus- ing or altering the fuid fow11, 82. The fuid fow in Di Ji ρiui Di ni ni Fext k T return is able to alter the ionic transport through = − ∇ + B (15) advection. In order to understand the complex Dize ρiui Di ni ni ψ interplay of ionic distribution, ionic transport, = − ∇ − kBT ∇ and fuid fow, we must frst look into the various where we have utilized the fact that Fext zeE fuxes that the ion can encounter. = and the fact that the electric feld may be represented as the gradient of a potential as E ψ , = −∇

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Figure 2: Schematic depiction of the electroosmotic fow in a narrow confnement. The channel walls have negative charge, which leads to the development of a positively charged electrical double layer (EDLs) close to the wall. These charges tend to reach equilibrium and obey the Boltzmann distribution, as discussed in (2). If an external electric feld is applied, the EDL as a whole experiences a net force (here) in the positive axial direction. This eventually manifests as a net body force on the fuid. In absence of any opposing force, the fuid would thus start fowing in the positive x-direction. This phenomena is called Electroosmotic fow and the resulting velocity profle for a typical uniform surface charge has been shown in the fgure.

where ψ represents the potential. With this notion zeψ of the ionic fux, we can defne the conserva- ρe 2zen0 sinh . (17) =− k T tion of fux at steady state with no reactions as B Ji 0 . The current density corresponding to In electrophoresis there is an electric feld applied ∇· = the mass fux due to the ith species is defned as in the longitudinal direction as well. Due to the zieJi (Fig. 2). orthogonal direction and slender nature of the channels, it is fair to assume that the longitudinal electric feld does not lead to a temporal change 2.2.2 Electroosmosis in the ionic distribution (despite the fact that The action of electric felds on ions leads to a the ions are moving). This is same as assuming preferrential movement in the solvent liquid as an x-invariance in the channel. Thus, the above discussed above (see Fig. 2 for a schematic rep- charge density is maintained in the channel. resentation). However, the EDL leads to an accu- For solving for the fuid fow, we utilize the mulation of a preferrential charge in the vicinity fact that fows in such channels are essentially of the wall and thus there is a net charge near the inertia free. Therefore the governing equation for wall. Far from the wall, the total charge in a rep- the fuid fow in the longitudinal direction may resentative control volume will be zero. Because be written as7, 82, 105, 109: of this, there is no net effect of the action of the 2 electric feld. The positive fux of one kind of ions 0 p η v ρeE (18) due to the action of the electric feld is exactly = −∇ + ∇ + which for a simple two dimensional fow, the balanced by the negative fux due to the action of momentum equation along the channel (x-direc- the electric feld on the other kinds of ions; this tion) may be written as110: fact along with the net electroneutrality condition 2 leads to a net zero fux in the bulk. Near the wall, ∂p ∂ u (19) 0 η ρeEx however, the net charge is ρe e(z n z n ) . =−∂x + ∂y2 + = + + + − − Upon the assumption of the Boltzmann distribu- where E represents the magnitude of the applied tion (Eq. (2)) and a z : z symmetric electrolyte we x electric feld along the channel. The effect of grav- obtain ity is typically neglected in such analysis as can be zeψ zeψ (16) found out from a simple order of magnitude cal- ρe zeno exp exp . = −kBT − kBT culation. Utilizing the expression for the charge density derived above the x-momentum equation The above consideration yields the total charge may be modifed to be written as110: density in a channel as94, 95:

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of this potential lies in the streaming of ions38, 40, ∂p ∂2u ∂2ψ η ǫ (20) 102, 110. The developed potential at the ends of the 0 2 2 Ex. =−∂x + ∂y − ∂y channel leads to a conduction current through The above equation may be integrated twice the system. This is essentially the electromigrative under the assumption that the pressure gradient fux of the system. The diffusive fux is neglected is constant to obtain the velocity as110: in the process because the concentrations in the

2 2 reservoirs are equal and we expect no such fux ∂p h y ǫExζ cosh κy u 1 1 , driving ions across the channel. A conceptual =−∂x 2η − H − η − cosh κH representation of Streaming Potential generated (21) from a pressure driven fow has been depicted in where we have used the no slip and symmetry Fig. 3. boundary conditions for the x-component of The starting point of the analysis is with the the velocity, i.e. u(H) 0 and uy(0) 0 . The solution for electroosmosis. For convenience, we = = frst term depicts the classical Poiseuille compo- rewrite it here17: nent of the fow and the second term depicts the ∂p H 2 y2 ǫζE ψ velocity due to the electroosmotic component. It u 1 s 1 , (22) =−∂x 2η − H 2 − η − ζ is clear that if the length of the EDL is very thin, then the electroosmotic component of the veloc- where Es represents the yet unknown stream- ity changes rapidly only near the wall. In the bulk, ing potential. The total ionic current, as per the where y 0 the value cosh κy/ cosh κH tends to discussion above, is zero. When we disregard ≈ become zero as κH , which is true for large the diffusive fux, we have for a z:z symmetric →∞ H or large κ (which implies thin EDL). electrolyte: The idea of driving fows electrically has been 2 2 z e Es exploited in recent times to achieve good mix- i zeJ zeu(n n ) (n n ) ion ion f ing111. This has been made possible by utiliz- = = + − − + + + − (23) ing patterned electrodes. Moreoever, the idea has been applied to drive a net fow solely by which upon substitution of the velocity feld and the action of AC felds by achieving a break in integrating over the channel may be written as17: symmetry through unequally sized electrodes. 2 112 H ∂p H2 1 y sinh zeψ d Boy and Storey have analyzed numerically zen0 0 2 2 y − ∂x η − H kB T Es 2 2 = n0z e H zeψ zen0ǫζ H ψ zeψ the infuence of time periodic electric felds on 0 cosh dy 0 1 sinh dy f kB T + η − ζ kB T triggering instabilities in an otherwise stable (24) confguration. 3 Electrohydrodynamics 2.2.3 Streaming Potential Electrohydrodynamics (EHD) is a more general In the electrokinetic phenomenon discussed class of phenomena, which encompasses Electro- above the prime mover for the fow was the appli- kinetics as a special case. Since we have already cation of the electric feld. However, we may ask discussed electrokinetic fows in details in the ourselves the question that can a pressure driven previous section, here we would focus our atten- fow for the case of a net charged fuid (due to the tion on a separate class of creeping electrohy- EDL) lead to the development of an electric feld? drodynamic fows, actuated by external electric The pressure driven fow causes the net advec- felds, which do not necessarily depend on the tion of ions in the downstream direction. The fact formation of EDLs and thus can generate fows that there is a preferrential counterion charge in even in poorly conducting fuids46, 106, 113. Much the vicinity of the wall leads to the advection of a of the investigations on EHD fows stem from the net charge. The advection of a net charge is iden- early work of Taylor and Melcher, who studied tifed as the component of the ionic fux due to electric feld driven motion of very poorly con- the velocity feld, i.e. the advective fux. The cur- ducting fuids. Aptly named “The Leaky Dielec- rent associated with the advection of these ions is tric Model”, Taylor’s theory deals with multiphase also referred to as a streaming current. However, fows, involving fuids with distinct permit- we cannot have a system in which there is a cur- tivities and conductivities. The motion is actu- rent in the absence of any electric feld. In order ated because of Maxwell’s Stresses acting at the to have a net zero current, there is an electric feld interface between the fuids, where a net charge induced along the length of the channel. This is accumulation occurs because of the permittiv- called as the streaming potential, since the genesis ity differences between the fuids46, 114. Note that

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Figure 3: Schematic depiction of the physical situation leading to the generation of Streaming Potential. A channel with charged wall naturally develops electrical double layers (EDLs) around it as shown in the fgure. Now, if a mechanical fow is actuated in the channel, for instance with an applied pressure gradient, it drives a current along the axial direction in the channel. For instance here, a current fows in the positive x-direction. This current is often termed as the “Streaming Current” and it leads to an effective (or, conceptual) accumulation of charge at the ends of the channel. For example, since here the current is positive, it would lead to the effective (or, conceptual) accumulation of positive charge downstream. This accumulated charge creates an electric feld of its own, which counteracts the Streaming Current and tries to make the net current at every cross section 0, in order to maintain electroneutrality. This induced electric feld is called “Streaming Electric feld” and the resulting electric potential is called the “Streaming Potential”. this kind of motion does not need the presence of leaky dielectric fuid often used in experiments is electrical double layer at the interfaces, although Castor Oil114, has O(103) times larger viscos- ∼ presence of an interface is necessary for charge ity as compared to water114, 121. Therefore, if the accumulation and unbalanced Maxwell stresses. typical length scales of the fow are in the range of The rest of the section is arranged in the follow- 1 mm–1 cm , the fow can be characterized as ∼ ing way: frst, we give a brief overview of the gov- creeping motion. On the other hand, the potential erning equations pertaining to the leaky dielectric φj varies such that the net current is conserved eve- model46, 114–116. This is followed by its applica- rywhere in the domain. At the interfaces, the usual tions to multiphase systems in order to probe the no-slip, kinematic, tangential stress and normal physics of such fows. To this end, we shall frst stress conditions are applied for the velocity, while discuss EHD motion of liquid droplets46, fol- the potential remains continuous and the jump in lowed by EHD fows near fat interfaces, pertain- the electric feld results in net charge accumula- ing to the motion of thin flms and superimposed tion therein. This interfacial charge in turn is gov- fuids70, 117–120. erned by a separate surface conservation equation. We now express the physical paradigms described above in compact mathematical forms. To this end, 3.1 The Leaky Dielectric Model we note that the fuid fow is governed by the fol- As mentioned earlier, leaky dielectric model often lowing equations, for the j-th fuid114: deals with multiphase fows, which involves two 2 or more fuids, with distinct interfaces. We assume ηj vj ∇pj 0 and ∇ vj 0. (25) that the j-th phase/liquid has viscosity η , con- ∇ − = ; · = j The electric feld in the j-th fuid is given by ductivity σj and permittivity εj . Apart from the Ej ∇φj , while the resulting current is fuid properties, the fow is mainly governed by =− 46, 114 Ij σjEj , according to Ohm’s law . In the the following variables: the velocity and pressure = absence of any net charge in the fuids, the con- feld in the j-th fuids ( vj , pj ) and the electrostatic servation of current thus leads to: ∇ Ij 0 . potentials ( φj ). Typically, the fows are governed · = Assuming the conductivities in both the fuids to by Stokes equations along with the incompress- remain constant, this leads to the following equa- ible continuity equation. This is because, the fuids tion for the potential: which exhibit leaky dielectric properties, possesses 2 (26) large viscosity and hence their fows are domi- φj 0. nated by viscous stresses. For instance, a typical ∇ =

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The above equations must be complemented (c) The stress balance across the interface: with appropriate boundary conditions. Usually, at (τ 1 τ 2) n γ ∇ n n ∇sγ (29) the boundaries of the fuid domain (these might − ·ˆ= ·ˆ ˆ − be solid boundaries as well), the values of poten- In the above, δ is the identity matrix, γ is the sur- tial and the velocity components are specifed. face tension and ∇s is the surface gradient opera- For instance, at a solid boundary located at xs , we tor, expressed as: ∇s δ nn ∇ . All of the = −ˆˆ · can enforce no-slip, no-penetration condition as above conditions are satisfed at z h(x, y, t) . 122 = follows : vj 0 at x xs , while the potential The quantities τ j are the total stresses in the fuid, = = usually has a specifed value at the solid bounda- which are expressed as follows: ries: φ(xs) φ0 . However, the most important = τ τ (m) τ (e) (30) boundary conditions for leaky dielectric models j j j are perhaps applied at the fuid-fuid interface, = + since the motion is essentially actuated by the T stresses at the interfaces. To this end, we shall con- pjδ ηj ∇vj ∇vj sider a generic interface between fuids 1 and 2, =− + + (31) 1 as shown schematically in Fig. 4. This interface is εjEjEj εj Ej Ej δ 2 described by the equation: F z h(x, y, t) 0 + − · ≡ − = (m) T and the surface normal at a given point on the In the above, τ pjδ ηj ∇vj ∇vj j =− + + interface reads: n ∇F/ ∇F . Here we have is the hydrodynamic (or, mechanical) stresses ˆ = � � (e) 1 chosen a Cartesian description of the interface, and τ εjEjEj εj Ej Ej δ is the electri- j = − 2 · although the same description also remains valid cal stresses, also known as the Maxwell stresses for other coordinate systems as well. (see Appendix-A for a detailed derivation). The At the interface, four conditions need to be imbalance in the electrical stresses in fuids 1 applied, which govern the motion of the fuids. and 2, as attributable to their permitivitty differ- Two additional conditions governing the spa- ences, naturally gives rise to a net fow through tio-temporal evolution of the potential are also the boundary condition (29). The potential, on required to complete the description. The four the other hand satisfes the following conditions boundary conditions governing the fuid motion at the interface46, 114: (a) Continuity in Potential: are as follows: φ1 φ2. (32) (a) The kinematic condition: = (b) Jump in electric feld: DF 0 (27) Dt = (ε1∇φ1 ε2∇φ2) n qs. (33) − ·ˆ=− (b) The no-slip condition: Once again, the above equations are applied at z h(x, y, t) . In (33), qs is the net charge per unit δ nn (v1 v2) 0 (28) = −ˆˆ · − = area at the fuid-fuid interface. This is a-priori

Figure 4: Shows a schematic depiction of a generic interface between fuids 1 and 2. The unit normal and the height of the interface from a given plane have been depicted in the fgure. Owing to the application of electric feld and the permitivitty differences between the fuids 1 and 2, there would be some accumulated charge at the interface, as shown in the fgure. Here we have chosen a Cartesian coordinate system to describe the interface, although any other convenient coordinate systems might also be chosen to best suit a given scenario.

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Figure 5: Deformation of an initially spherical drop into a prolate shape occurs when the system is acted upon by an electric feld. Here, the electric feld is pointing upwards. The polar accumulation of charges (a consequence of the leaky dielectric model) leads to the elongation. The electric tangential stresses are balanced by the hydrodynamic stress, thus leading to a net fow both inside and outside the droplet. Inter- estingly, the nature of deformation depends not just on the strength of the electric feld but the relative ratios of the conductivities and permittivities of the two fuids. unknown and must be determined from a gov- 3.2 Electrohydrodynamics of Drops erning equation of its own. To this end, we arrive One of the most classical applications of the leaky at the interfacial charge transport equation, dielectric model is to study EHD motion, migra- which only applies to motion of charge along the tion and deformation of dielectric liquid drop- interface. This equation can be expressed as fol- lets in another immiscible dielectric fuid106. We lows113, 114: At z h(x, y, t): present a brief overview of such phenomena in = this section. The rest of the section is arranged ∂qs ∇s (vsqs) (σ1∇φ1 σ2∇φ2) n. (34) as follows: frst we would consider the simplest ∂t + · = − ·ˆ kind of EHD fows ignoring the effects of charge In the above equation, vs is the velocity tan- convection. Subsequently, we will look into some gential to the interface, which is expressed as: important previous studies, which accounted vs δ nn v , evaluated at z h(x, y, t) . The for interfacial charge convection in studying the = −ˆˆ · = above equation basically states that the interfacial EHD fows around the droplets. Finally, we shall charge can show spatio-temporal variation owing review studies exploring the effects of other exter- to current fow from the bulk of the fuid towards nal entities (such as surfactants) on EHD fows the interface. A very important non-dimensional around dielectric droplets. A representative sche- number coming out of (34) is Ree εcvc/acσc , matic of a deformed droplet under the action = where ξc is the characteristic value of the quan- of externally imposed feld has been depicted in tity ξ . Ree is often called the Electrical Reynolds Fig. 5. Number121, which can be very small for creeping motion and for fuids with low electrical 3.2.1 Mechanics Without Charge Convection permitivitty. If, Ree 1 , the LHS in (34) can ≪ One of the earliest studies on EHD fows dates be dropped entirely and the charge conserva- back to123, who theoretically investigated the tion equation is essentially replaced by continu- deformation of a poorly conducting droplet in ity of current across the interface, which may be 46 another dielectric liquid, subject to EHD motion expressed as follows : (σ1∇φ1 σ2∇φ2) n 0 − ·ˆ= and subsequently compared the theoretical at z h(x, y, t) . Equations (25) and (26), subject = results to the experimental observations of124, to conditions (27)–(33) and (34) complete the 125. Taylor studied the deformation of a dielec- mathematical description of the leaky dielectric tric droplet, originally spherical of radius a in model. We shall now move on to some of the the presence of a steady uniform electric feld most prominent applications of the leaky dielec- ( E ) applied in the background (see Fig. 5 for a tric model discussed herein. 0

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schematic depiction). Following the notation of Here, S ε1/ε2 and M η1/η2 . The above = = the previous section, the droplet phase will be expression shows that the droplet becomes pro- denoted as liquid-2 and the outer fuid would be late if �>1 , while oblate shape is achieved when called fuid-1. Taylor used the governing equa- �<1 . Taylor’s theory successfully predicted tions discussed in the previous section along with the qualitative nature of deformation for a large the assumptions Ree 1 and small deformation, number of observations in Allan and Mason’s ≪ to deduce the fow and the electric feld around experiments. the droplet. The electrostatic potential was shown Taylor’s analysis essentially sheds light on the to be of the form46: fact that even the slightest amount of conductivity in a liquid can drastically alter the behav- 1 R a3 φ1 E0 cos θ r − , (35) ior of a droplet of that liquid in the presence of = + 2 R · r2 + an electric feld. This can be attributable to the fact that even a very small conductivity would 3E0r cos θ φ2 . (36) also drive a current in the fuid, which would = 2 R + eventually lead to net free charge accumula- Here, R σ2/σ1 , θ is the polar angle and r is the tion at the interface. This charge accumulation = distance from the origin in spherical coordinate, would in turn lead to a jump in the electric feld the origin being at the center of the droplet. The across the interface, as per Eq. (33), which would background electric feld is aligned with the posi- fnally lead to imbalance in the Maxwell stresses tive z-axis. The resulting velocity feld was solved across the interface, hence giving rise to fuid using the Stream Function approach, subject to motion. This is the most fundamental mecha- the proper interfacial conditions as outlined in nism behind the fow dynamics of a large num- the previous section. The stream functions were ber of EHD phenomena. shown to be of the form46: Later,64 deduced the deformation of a die- 4 lectric droplet owing to an applied oscillat- Aa 2 2 ψ1 Ba sin θ cos θ, (37) ing electric feld, with undisturbed strength: = r2 + E E0 cos ωt . Their analysis was carried = 3 5 out in much the same way as that of Taylor’s. Cr Dr 2 ψ2 = + sin θ cos θ. (38) Finally, it was shown that the deformation can a a3 be expressed as: D DS DT , where DS is the = + The constants A, B, C and D all have the dimen- steady deformation of the droplet and DT is the sions of velocity. Using the normal stress condi- oscillating component of the deformation. The tion, Taylor showed that the deformation of the oscillating component was shown to be of the 2iωt droplet (assuming the deformation to remain form: DT Re H e (here H may be a = ∗ ∗ small) can be analytically expressed as follows46, 106: complex quantity), where Re() is the real part 2 and i √ 1 . It should be noted that the fre- ℓ 9 aε1E0 = − D � �. (39) quency of the oscillation of the deformation is = ℓ = 16 γ ⊥ twice the frequency of the applied electric feld, as In the above, D is the deformation, defned as attributable to the quadratic non-linearity in the ratio of the length of the droplet parallel to the Maxwell stresses. The steady state deformation applied feld ( ℓ ) and the length perpendicular to was derived as: 46, 106 the same ( ℓ ). The factor is expressed as : 2 ⊥ 9 ε D a 1E¯0 2M 3 S �S (41) � S(R2 1) 2 3(SR 1) + (40) = 16γ = + − + − 5(M 1) +

R(11 14) R2 15( 1) 9(19 16) 15b2ω2(1 )(1 2Q) �S 1 ¯ + + ¯ [ + + + ]+ + + (42) = − 5(1 ) (2R 1)2 b2ω2(Q 2)2 + [ ¯ + + + ]

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account the deformed interface shape, provided In the above, E¯0 is the RMS value of the applied feld, R R 1, Q S 1, M 1 and the deformation is small enough. Recalling that ¯ = − = − = − b ε1/σ2 . Again, �S > 1 denotes prolate defor- in the regime of creeping fows the deformation = mation, �S < 1 denotes oblate deformation. is proportional to the Capillary number Ca, one More recently,126 have probed into the tran- may employ a dual perturbation series121 using sient deformation of a stationary droplet subject both Ree and Ca as gauge functions as follows: to EHD forces, constrained inside a spherical 2 ξ = ξ0 + Reeξ 1,0 + Caξ 0,1 + Re ξ 2,0 confnement of radius ac . To this end, the tem- ( ) ( ) e ( ) (45) 2 poral variations of charge at the interface was + Ca ξ(0,2) + ReeCaξ(1,1) +··· taken into account, while the effects of charge advection were neglected. Mandal et al.126 carried It is very easy to infer that at steady state, the lead- out a regular perturbation analysis by assuming ing order terms in the above asymptotic expan- ζ Oh 2 1 , where Oh is the Ohnesorge num- sions would lead to Taylor’s results. Feng113 = − ≪ ber defned as, Oh η1/√ρ1aγ , which enabled showed using numerical simulations as well as = them to omit the temporal component of the asymptotic analysis using the form in (44) that inertial forces in the Navier–Stokes equations. It the most dominant effect of charge advection is was shown that the droplet exhibits a transient to transport them from the equator to the poles, deformation of the following type: which increases the free charge density near the poles. This eventually weakens the tangen- ℓ ℓ t/τ1 2t/τ2 t/τ2 D � − ⊥ DS 1 �1e− �2e− �3e− tial component of the electric feld at the inter- ¯ = ℓ ℓ = ¯ − − − � + ⊥ face and hence hinders the so-called “base” fow (43) without charge convection. The overall result 2 9CaŴR�∗ In the above DS , where Ca η1uc/γ is that the tangential component of the veloc- ¯ = 16(R 2)2 = ¯ + 2 ity at the interface is weakened and the position is the Capillary number and Ŵ R¯ , R (R 2) +α3(R 1) = ¯ + − ¯ − of the peak of this velocity is shifted towards the where α a/ac is the ratio of the droplet radius = equator. This assertion has been verifed through to the radius of the confnement. Further, both numerical and asymptotic analysis by113. He τ ε /σ , k 1, 2 are the characteristic time k = k k = showed that the tangential velocity at the droplet constants. The constants ∗, k ( k 1, 2, 3 ) 113 = interface upto O(Ree) has the following form : have complicated expressions and may be found in the original paper. Presence of multiple time uθ r 1 2(SR 1) 2 ¯ Ree β β (7 cos θ 3) u|¯= − 1 2 scales in the transient deformation is perhaps the c =− SR 1 + + − � ¯ − � most noteworthy feature of the work by Mandal sin θ cos θ (46) et. al., which might, in special circumstances, lead to non-monotonic deformation. Here, uc is the characteristic velocity of the prob- 2 9 SR 1 ε2γ E lem, defned as: uc � ¯ − � , while the 10η2(2 R)(M 1) 3.2.2 Mechanics with Charge Convection = + ¯ + expressions for the constants β , β , etc. can be Active charge convection at the interface requires 1 2 found in the original paper of Feng113. It should one to account for Eq. (34), instead of imposing be noted that Feng assumed the droplet to remain current continuity at the interface. This leads to spherical in his analysis. Das and Saintillan127 increased non-linearity and coupling in the systems have investigated the transient deformation of of equations, that eventually creates some intrigu- a droplet under uniform applied electric feld ing physical effects. One of the standard ways to using the leaky dielectric model. They assumed account for charge convection is to apply a simple Ree O(1) , while an asymptotic analysis was regular perturbation series, with Ree (εcuc/aσc) , ∼ = carried out using δ 3CaE as the gauge func- i.e., the electrical Reynolds number as the gauge 4(1 2R)2 = + ¯ function. Physically, this simply means that the tion, where CaE is the electrical Capillary number 2 interfacial advection of the surface charge is weak aε1E0 defned as: CaE γ . As such all the variables and only slightly changes the potential distributions = 2 are expanded like: ξ ξ0 δξ1 δ ξ2 , in the fuids. A typical asymptotic expansion for any = + + +··· where ξ can represent any variable. The deformed variable (say, ξ ) would look like the following113: shape of the droplet is defned as127: 2 ξ ξ0 Reeξ1 Ree ξ2 (44) 2 = + + +··· F r (1 δf1 δ f2 ) 0 r r/a. ≡¯− + + +··· = ;¯= In the above, ξ might represent any variables like (47) pk , vk , ... etc. Further, one might also take into The functions f1, f2 are expressed as:

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2 f1 f12(t)P2(µ) (48) (1,0) 6MExEz(R Q)(3R Q 3)(2 63 45)k2 U ¯ − ¯ − + + + (55) = d,x = 35(3 2)2( 1)( 4)(R 2)2(2R 3) + + + ¯ + ¯ + 2 1 2 aε1Ec f2 f (t) f22(t)P2(µ) f24(t)P4(µ) (49) Here, M , Ec is the char. elec- =−5 12 + + = µVc tric feld, Vc is the char. velocity, 2 In the above, Pn(µ) is the Legendre Polynomial of k0 4x (1 x ), k2 4/H , with x x /H = ¯d − ¯d =− ¯d = d order n and µ cos θ , where θ is the polar angle, is the position of the droplet centroid with respect = measured from the direction of the applied elec- to the channel centerline. The O(Ca) velocity tric feld. Ordinary differential equations for the felds can be found in the original paper by116. transient deformation functions f12(t), f22(t) The above expressions clearly demonstrates that etc., can be deduced from the combination of the there is a cross migration of the droplet as a result boundary conditions as discussed in the previous of charge migration on the droplet interface. This section. The detailed expressions can be found in assertion is confrmed by the fact that the frst 127 the original paper by Das and Saintillan . migration velocity appears at O(Ree) . Note that More recently,116, using the leaky dielectric the droplet would not migrate in the absence of model have demonstrated that application of trans- the transverse electric feld, i.e., Ex , as outlined verse electric feld relative to an established back- by the proportionality of the velocity with Ex . ground parabolic fow can generate cross-stream Deformation of the droplet also leads to cross- migration. They considered the motion and defor- stream migration. The general conclusion from mation of a droplet in a Plane Poiseuille fow in a the analysis of116 is that a tilted electric feld with channel of height 2H, with applied axial ( Ez ) as well respect to the dominant direction of the fow is as transverse felds ( Ex ). Both the deformed shape of necessary for cross-stream migration. In another the interface as well as charge advection at the inter- work,121 have further shown that similar cross face have been taken into account in the said analysis stream migration also occurs for vertically set- and an asymptotic expansion of the form given in tling droplets, under the infuence of transverse (45) was employed. Mandal et al.116 applied Lamb’s and parallel electric felds. They showed through general solution128 along with the general method a perturbation analysis that for a settling droplet outlined by129–132 to evaluate the fow feld, droplet as well, the frst cross stream migration comes deformation and droplet migration velocity. It was about at O(Ree) . In the same work, they also per- shown that the migration velocity has the form: formed experiments, which verifed the theoretical predictions. (0,0) (1,0) (0,1) U U ReeU CaU d = d + d + d +··· (50) (i,j) i,j i,j where U U ez U ex 3.2.3 Mechanics with Other External Effects d = d,z + d,x Ha and Young carried out a number of pioneer- The droplet shape was of the form: ing investigations133, 134 on the effects of non-ionic r 2 surfactants on the deformation and stability of r 1 Caf10 CaReef11 Ca f20 ¯ = a = + + + +··· conducting as well as non-conducting droplets (51) in DC feld. They frst executed a series of experi- ments133, where the break-up characteristics The functions f may be expressed in terms of ij of droplets with varying degrees of conductiv- spherical harmonics as follows: ity and surfactant concentration were studied n ∞ ij ij in the presence of external DC feld. The results f L cos(mϕ) L sin(mϕ) Pm(µ) ij = nm + ˆnm n showed that even a slight non-uniformity in the n 1 m 1 = = (52) surfactant concentration can lead to a drastic change in the mechanism of droplet break-up. In the above, Pm(µ) are associated Legendre poly- n For lower surfactant concentrations, break-up nomials of degree n and m and ϕ is the azimuthal occurred through the formation of bulbous ends angle. The droplet migration velocities were giving away to multiple smaller droplets. How- shown to have the following expressions: ever, as the concentration was increased beyond (0,0) (0,0) a certain level, the bulbous break-up mechanism U k0 k2 U 0 (53) d,z = + 3 2 ; d,x = changed to “tip-streaming”, wherein, the ends of +

6M(R Q)(3R Q 3) (36 2 119 75)E2 (8 2 42 40)E2 U (1,0) ¯ − ¯ − + + + x + + + z (54) d,z = 35(3 2)2( 1)( 4)(R 2)2(2R 3) + + + ¯ + ¯ +

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the droplet became pointy and shot out several charge qs in the leaky dielectric model. In addi- smaller droplets in the process of breaking up. tion to this, the authors were also able to dem- Even higher concentration of surfactants brought onstrate that using the electrokinetic description back the bulbous-end and subsequent break-up of the interface and the surrounding EDLs, the mechanism prevalent at the lower concentration leaky dielectric equations of Sect. 3.1 can be suc- values. The break-up occurred for larger values of cessfully recovered. This analysis thus established the Electrical Capillary number, defned as CaE , in a strong link between two apparently separate agreement with previous numerical simulations branches of electrohydrodynamics. This work of of135. A subsequent theoretical analysis by134 show Schnitzer and Yariv also verifes the existence of good comparison with the experimental results. a deeper and more fundamental theory under- The main reason of changes in the shape of the neath the leaky dielectric model. In other words, droplet and its subsequent break-up was attrib- the leaky dielectric equations work as effective utable to the change in the surface tension of the “lumped” frst-approximation equations, describ- droplet owing to the fow-driven re-distribution ing the Electrohydrodynamic phenomena. of surfactants. More recently114 have developed a theoretical model for small deformation and 3.3 Electrohydrodynamics of Flat nearly spherical droplets as well as for large defor- Interfaces mation with prolate droplets using the spheroidal Apart from EHD motion of droplets, another coordinate system. Their semi-analytical solu- classic application of the leaky dielectric model tions were compared with existing numerical and has been in the realm of thin flm dynamics70, 117– experimental data, showing good agreement. 120, 138–147, with fat or nearly fat interfaces. Simi- We end our discussion on the EHD of drop- lar to the EHD of droplets, studies on EHD of lets with a brief discussion on the recent fndings flms have also investigated the deformation and by136, wherein the dynamics of a nearly spheri- the resulting fow near the interfaces. A key fea- cal droplet was considered, in the presence of a ture of EHD motion of fat interfaces is that there steady uniform electric feld. The droplet as well has to be a non-uniformity in the electric feld as the suspending phases are assumed to carry applied across or near the fat interface118–120. The ionic species, which are adsorbed into the sur- reason for such requirement can be traced back faces. However, the adsorption from the two to the boundary conditions outlined in Sect. 3.1. phases into the surface are different, which leads Since, here the interfaces are inherently fat, a to the presence of a net charge on the interface. uniform electric feld would only cause a pres- This interfacial charge eventually leads to the sure jump across the interface, while the fuids formation of electrical double layer (EDL, see will stay in equilibrium and the interface would Sect. 2) around the interface. Schnitzer and Yariv remain fat. Therefore, the only way to introduce considered the EDL to be thin, which indicates, imbalance in the tangential Maxwell stresses � D/a 1 and hence the interface along = ≪ at the interface is to introduce non-uniformity with the EDLs on the either side act as an effective in the electric feld itself, which in turn would interface—the Taylor’s interface. The magnitude actuate fuid fow and lead to interface deforma- of the applied electric feld was assumed to be low tion. It is important to note that, for EHD fow to moderate, indicated by eaE0/kT O(1) . It is ∼ around droplets, the curvature of the interface well known that 1 naturally leads to a sin- ≪ itself was a source of non-uniformity, which lead gular problem12, 82, 85, 137. The authors probed into to imbalances in the Maxwell stresses. One of the the resulting dynamics using matched asymp- most widely used ways to achieve the said non- totic expansion82, wherein the Poisson–Nernst– uniformity is to employ patterned electrodes120, Planck–Navier–Stokes model described in Sect. 2 138 on surrounding solid walls, while the exter- was employed. To this end, was used as the nal feld acts across the interface. A schematic gauge function, while the EDLs in both the fuids representation of a nearly fat interface confned were considered to be the inner layer and the bulk between two parallel plates, under the action of fuids were treated as the outer layer. The analysis externally imposed feld has been demonstrated of Schnitzer and Yariv demonstrated that in the in Fig. 6. leading order of the interfacial charge is com- Overall, the fundamental mechanism of inter- pletely screened by the EDL. However, at O(�) , face deformation and fow actuation basically there is a charge imbalance when the two EDLs remains the same as outlined mathematically and the interface are accounted for together. in Sect. 3.1 and described qualitatively in Sects. Schnitzer and Yariv argued that this O(�) charge 3.2.1 and 3.2.2. In what follows, we would briefy in the effective interface gives rise to the surface

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Figure 6: When a stable flm (may be thick or thin) is subjected to an electric feld, small initial perturba- tions may lead to charge accumulation and a net unbalanced electric stress. The consequence of this is regions of strong undulations. If the feld strength is too high then the lower fuid may eventually hit the top electrode.

review some of the selected prior studies on EHD E and ζ are not asymptotically large. In the afore- motion of flms and fat interfaces and outline mentioned study of146 assumed a zeta potential of their key features. the form: ζ(x) ζ0(n cos qx) . It was assumed = + The pioneering experimental work of138 that at the interface between the two fuids, no demonstrated that thin flms can be deformed charge accumulation occurs, which allowed the in a controlled way to form pillar like structures authors to apply the boundary condition (33) in in the presence of non-uniform electric felds, the following form: (ε1∇φ1 ε2∇φ2) n 0 . − ·ˆ= applied through patterned electrodes. Their Approximate analytical solutions were obtained study used polymeric flms and showed that the using a domain perturbation approach, where an pillar-like structures always follow the electrode expansion similar to (45), albeit only in terms of shapes/patterns. More recently, Mandal et. al.146 Ca was employed and velocity felds till O(Ca) have investigated the overall fow dynamics of and the resulting deformations till O(Ca2) were superimposed fuids in a narrow confnement deduced. Their analytical solutions showed that and probed into the transient interface deforma- the interface deformation has the following form: tion pattern and alterations to the fow dynamics 2 (57) h Cah1 Ca h1 caused by it (see Fig. 6 for a schematic depiction). = + +··· In their study, the non-uniformity in the Maxwell t/T1 h1 K1(t) sin(qx) K1(t) L1(1 e− ) stresses were provided by actuating an electroos- = ; = − ; motic fow (see Sect. 2) with non-uniform surface (58) potential. It can be shown137 that for thin EDLs, for electroosmotic fows, the no-slip boundary h2 = K21(t) cos(2qx) + K22(t) sin(qx) conditions may be replaced with slip velocity (59) + 33( ) cos( ); etc. condition, called the Smoluchowski Slip velocity. K t qx This slip velocity can be expressed as follows137: The detailed expressions for the various functions δ can be found in the original paper. Mandal et al. εζ nn E (56) v −ˆˆ · , at x xs. showed that the the deformed interface distorts =− η = the streamlines, which can be extreme for large In the above, ζ is the “Zeta Potential” at the solid- deformations. They also demonstrated that for liquid interface and δ nn E is the tangential special cases, multiple recirculation rolls might −ˆˆ · component of the electric feld to the surface, appear in the channel, which are otherwise absent while other symbols bear their usual meaning. from such fows. Mandal et al.146 verifed their The above condition may be shown12, 148 to be analytical results by comparing them to inde- valid for arbitrary variations in ζ as well as sur- pendent numerical simulations using the Phase- face geometry, provided the applied electric feld feld formalism149–151.

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Usually, formation of re-circulation rolls are a the potential would have an expansion of the common theme of EHD motion near fat inter- form: faces, with patterned electric felds. Esmaeeli and 2 120 φ φ0 ξφ1 O(ξ ) (66) Reddy have recently probed into EHD motion = + ˆ + ˆ of superimposed dielectric fuids in a narrow The other relevant variables (such as velocity, confnement, with a potential φ φ0 cos(kx) pressure etc.) might also be expanded in a similar = applied on the bottom wall. An asymptotic manner. The governing equations and the bound- analysis similar to the form given in (57) was ary conditions remain the same as outlined in employed, wherein only the leading order velocity Sect. 3.1. In the leading order, the fuid is station- feld and the O(Ca) deformation was determined. ary and the potential would vary linearly between O(ξ) The effects of the charge convection at the inter- the electrodes. However, at ˆ , the perturba- face was neglected and otherwise the equations tions to the interface shape leads to an imbalance outlined in Sect. 3.1 were assumed to be valid. in the Maxwell stresses across the interface, which The solutions were deduced in terms of stream naturally leads to a fow in the confnement. This function, with ψ for both the fuids satisfying fow and the original interface deformation can 4ψ 0 . It can be verifed that the stream func- be represented with the help of normal modes as ∇ = tion for both the fuids has the following form: follows:

(67) ψk =[ C1 + C2y cosh(2qy) + C3 + C4y (w , ξ) w (z, t), ξ(t) sin k x (60) l = ˆ l ˆ · sinh 2 ] cos 2 ; = 1, 2. ( qy) ( qx) k

On the other hand, the deformation was shown (s iα)t ∞ int w (z, t) e + W (z)e (68) to be of the form: ˆ l = n,l n =−∞ h1 a0 cos(2qx) (61) = Note that the frequency of axial variation of the (s iα)t ∞ int ξ(t) e + Zne , l 1, 2. (69) interface deformation and the velocity feld is ˆ = = n twice as that of the applied surface potential as =−∞ the leading order electric feld. Quite obviously, [117] used the above forms of the disturbance this is attributable to the quadratic non-linearity functions and converted the differential equa- inherent in the Maxwell’s Stresses. tions governing the evolution of ξ and w into ˆ ˆ l More recently,117 have investigated the dynam- an eigen value problem for s using the Floquet ics and stability of thin superimposed flms in theory152. We can easily infer that s > 0 would narrow confnements subject to potential differ- indicate instability, while s < 0 would lead to ences varying arbitrarily in time. As such, the fol- either marginally or neutrally stable states. A key lowing description for the applied potential was conclusion from the analysis of Bandopadhyay chosen: and Hardt is that the presence of viscous fuids and confnement generally dampens the unsta- φ V (t) at the wall: z h1, h2 (62) =± = ble growing modes. In addition, they also showed that the presence of viscous dissipation gener- ∞ ∞ ally shifts the critical Mason Number (defned 2 V (t) βn cos(nt) γn sin(nt) (63) ε1V = + as Ma ref , V is the char. applied poten- n 0 n 1 η ωh2 ref = = = 1 1 tial, ω is the char. time scale of variation of the The interface was subsequently subject to small applied voltage) required for instigating instabil- arbitrary deformation of the form (for a pictorial ity towards a larger value. Their analysis has also depiction, one may refer to Fig. 6): revealed that the system becomes particularly F z ξ(x, y, t) 0 (64) sensitive to the distance between the electrodes as ≡ − = the viscous forces grow stronger. ξ(x, y, t) ξ(t) sin k x (65) = ˆ · Here, k kxex kyey is the wave vector of the 4 Looking Ahead = ˆ + ˆ disturbance along the channel. Assuming the sur- A detailed account of the fundamental principles face deformation to be small, one can carry out governing electrokinetics and the leaky dielectric a regular perturbation (domain perturbation) models have been presented here. We have also ξ analysis taking ˆ as the gauge function. As such, discussed a number of prominent previous studies, which outline various interesting facets of the

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EHD motion in single and multiphase systems. can actuate fuid motion through the tagential We started from the fundamental building blocks stress balance conditions. The analysis of leaky of charge accumulation near a solid surface lead- dielectric phenomena reveals how droplets as well ing to the formation of EDLs. We established that as fat interfaces may be caused to deviate and in equilibrium the charge density follows the deform based on pure hydrodynamic effects. The Boltzmann distribution, which, upon activating mechanism of charge convection at the interface an externally applied electric feld generates fow adds further complexities to the overall dynamics in the fuid. Several improvements upon the basic of the fuids by actively displacing charges along electrical double layer theory were discussed, the interface leading to alterations in the electric including the effects of fnite sized ions and vari- felds in the process. In essence, the leaky dielec- able permittivity. We can thus conclude that fuid tric model demonstrates that the presence of fow is affected quite signifcnatly by the action of even a very small conductivity can lead to dras- electric felds. Analysis of electrokinetics allows tic changes in the fow dynamics through charge us to appreciate how microscale fows can be accumulation at the interfaces and as such one manipulated with relative ease by exploiting the would need to consider the conservation of cur- physicochemical equilibrium between the aque- rent in order to arrive at a physically consistent ous solution and the channel substrate. Moreo- mathematical model. We have further given over- ver, we have demonstrated how the fow in the views of several previous studies, which showed absence of any applied electric feld can lead to that strong electric felds may be utilized for sort- the generation of an induced electromotive force ing of droplets based on their conductivity, size, (EMF), which counteracts the streaming cur- or dielectric property. In such cases, it has been rent produced by the mechanically driven fow. well established that the infuence of dielectro- This induced potential (or, EMF) is often termed phoresis110, 162 (migration in presence of non- as the streaming potential. A common theme in uniform electric felds) may also be effective in both of the above cases is that the electric feld conjunction of electrophoresis (defned as the acts on net charge densities, usually near an inter- motion of a charge particle upon applying an face (such as the electrical double layer near the electric feld) to enhance the effciency and fne wall), which gives rise to a net volumetric or tuning of the sorting process. “Body Force” on the fuid itself. The idea behind Despite a large number of previous studies on generation of a voltage by driving fows through the topics discussed herein, there remains a num- such narrow confnements may be exploited ber of open challenges both in electrokinetics and to harvest a fraction of mechanical energy into leaky dielectric fows. For instance, electrokinet- electrical energy which may be utilized to power ics near fuid-fuid interfaces are not yet fully fow-induced low-power electronics and other understood and hence it remains an area of active small-scale energy conversion devices20, 41, 43, research136. The complete behavior of the full 153–155. Recently attention has shifted to func- Poisson–Nernst–Planck–Navier–Stokes equations tional nano- and microchannels where the walls are also poorly understood for complex systems are coated with polyelectrolyte brushes156–161. The because of the non-linearities involved82. Spatio- presence of such brushes leads to several intrigu- temporal variations in salt and charge concen- ing physical effects ranging from ion-exclusion trations with unsteady external forcing and their near the wall to a sharp variations in dielectric subsequent effect on the overall dynamics of the properties of the fuid157, which may be exploited system is still not fully resolved and remains a to achieve signifcant improvements in energy strong area of research163. The interplay between harvesting ability of such systems and for detect- other non-electrostatic ineractions (such as the ing analytes157. Yukawa potential) and the Columbic forces have Contrary to the above situation, leaky dielec- also remained largely unexplored39, 105, 164. On tric motion in poorly conducting fuids do not the other hand, as regards to the leaky dielectric require the presence of such an electrical double model, although the role of the interfacial stresses layer123. The resulting unbalanced force is only and the charge dynamics is well established, their concentrated at the interface rather than being infuence on interfacial instabilities and eventual spread-out over the volume of the fuid. It thus liquid/droplet break-up is not yet fully under- follows that to actuate leaky dielectric motion, stood135. Furthermore interactions between we must require at least one fuid-fuid interface multiple droplets and their subsequent effect on with distinct permittivity on either sides. We sub- the rheology of a suspension of droplets in the sequently discussed the governing equations and presence of uniform and non-uniform electric established how unbalanced Maxwell Stresses felds are also among the potentially interesting

218 1 3 J. Indian Inst. Sci.| VOL 98:2 | 201–225 June 2018 | journal.iisc.ernet.in Electrohydrodynamic Phenomena and active research topics114, 165. Many issues in In order to symmetrize the above expression, we the dynamics of thin flms in presence of exter- make use of the Gauss’s law of magnetism which nal electric felds also remain unresolved. For asserts us that there are no magnetic monopoles, instance effect of electric felds on the rupture of i.e. B 0 . This can then simply be added to ∇· = dielectric or, poorly conducting flms is an attrac- equation (74) to obtain tive future avenue of research, with potentially fe ǫ0(( E)E E E) new physical effects to be unraveled. Finally, we = 1∇· − ×∇× hope that the methods described in this work will (( B)B B B) (75) set the tone for future research of fuid fow and + µ0 ∇· − ×∇× ∂ its subsequent manipulation with electric felds. ǫ0 (E B) − ∂t × upon which further simplifcation is obtained by noting the vector identity E E 1 ( E 2) to yield Acknowledgements ∇ = 2 ∇ | | Prof. Suman Chakraborty is gratefully thanked for 1 2 fe ǫ0(( E)E (E )E) ǫ0 E initiating AB and UG to the area of electrokinet- = ∇· + · − 2 ∇| | ics and electrohydrodynamics. The authors also 1 1 2 (( B)B (B )B) B (76) thank Dr. Shubhadeep Mandal for his inputs. + µ0 ∇· + · − 2µ0 ∇| | ∂ ǫ0 (E B). Derivation of Maxwell Stress − ∂t × The form of the Maxwell stress may be deduced When the effects of the magnetic feld is neglected by starting with the force acting on a unit volume the above expression may be set in the form of a due to the simultaneous action of an electric feld, divergence of a tensor form as E , and a magnetic feld, B . The Lorentz force is f τ E obtained as49 e =∇· ; 1 E 2 0E E 0 E I Fe qE qv B (70) τ ǫ ǫ (77) = + × = ⊗ − 2 | | =⇒ 1 E fe ρeE J B (71) τij ǫ0EiEj ǫ0Ek Ek δij = + × = − 2 where the former is the total force and the latter is the force per unit volume. ρe depicts the where I and δij represent the identity tensor and charge density and J represents the amperic fux Kronecker delta respectively. (current). We must now appeal to the Maxwell’s While the above derivation is done for a vol- laws to relate ρe and J to the fundamental quanti- ume in vaccum, one can logically extend the same ties E and B . Gauss’s law yields ρe ǫ0 E while to a medium with a permittivity given by ǫ ǫ0ǫr 1 = ∇·∂E = Ampère’s law yields J B ǫ0 . The where ǫr represents the relative permittivity. In = µ0 ∇× − ∂t physical meaning of the above two expressions that case the body force may be evaluated by tak- is clear. Gauss’s law indicates that the fux of the ing the divergence of the Maxwell stress tensor electric feld through a surce is due to the charge 1 enclosed by that surface. Ampère’s law indicates τ E ǫE E ǫ E 2I τ E = ⊗ − 2 | | =⇒ ∇ · (78) that the magnetic feld around a contour is pro- 1 portional to the electric current, J , and the dis- ( ǫE)E (ǫE )E (ǫE E) 2 placement current, ∂E/∂t . Thus, we obtain = ∇· + ·∇ − ∇ · For negligible magnetic effects we may assume that 1 ∂E f ǫ EE B B ǫ B (72) E 0 , i.e. the electric feld is irrotatonal. We e 0 0 ∇× = = ∇· + µ0 ∇× × − ∂t × will also make use of the fact that for a medium where we can utilize the identity with permittivity ǫ , the Gauss’s law implies that ǫE ρ , where ρ represents the free charge ∂ ∂E ∂B ∇· = e,f e,f (E B) B E (73) density. Thus we obtain the force as ∂t × = ∂t × + × ∂t 1 2 (79) along with the Maxwell-Faraday equation fe ρe,f E E ǫ E ∂B to modify equation (72) to yield = − 2| |∇ ∇× =−∂t 1 ∂ fe ǫ0( EE E E) ( B B) ǫ0 (E B). (74) = ∇· − ×∇× + µ0 − ×∇× − ∂t ×

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Dr. Aditya Bandopadhyay completed his Dr. Uddipta Ghosh completed his B.Tech B.Tech and M.Tech under the Dual Degree and M.Tech under the Dual Degree pro- program at IIT Kharagpur in 2012 and his gram at IIT Kharagpur in 2012 and his Ph.D in 2015 also from IIT Kharagpur. He Ph.D in 2016 also from IIT Kharagpur. He then did his Postdoctoral Research at Uni- then did his Postdoctoral Research at Uni- versity of Rennes 1 (2015–2016) and subse- versity of Rennes 1 as a Marie Curie Fellow quently joined TU Darmstadt as a Humboldt Fellow in from 2016 till 2017. Since December 2017, Dr. Ghosh has 2017. Since August 2017, Dr. Bandopadhyay has been work- been working at IIT Gandhinagar as an Assistant Professor ing at IIT Kharagpur as an Assistant Professor in the in the discipline of Mechanical Engineering. His research Department of Mechanical Engineering. His research inter- interests include low Reynolds number fows, electro-hydro- ests include low Reynolds number fows, electro-hydrody- dynamics, and reactive transport in porous media. namics, and reactive transport in porous media.

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