Electrospinning and Electrically Forced Jets. I. Stability Theory Moses M

PHYSICS OF FLUIDS VOLUME 13, NUMBER 8 AUGUST 2001

Electrospinning and electrically forced jets. I. Stability theory Moses M. Hohman The James Franck Institute, University of Chicago, Chicago, Illinois 60637 Michael Shin Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Gregory Rutledge Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Michael P. Brennera) Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ͑Received 15 February 2000; accepted 10 May 2001͒ Electrospinning is a process in which solid fibers are produced from a polymeric fluid stream ͑solution or melt͒ delivered through a millimeter-scale nozzle. The solid fibers are notable for their very small diameters ͑Ͻ1 ␮m͒. Recent experiments demonstrate that an essential mechanism of electrospinning is a rapidly whipping fluid jet. This series of papers analyzes the mechanics of this whipping jet by studying the instability of an electrically forced fluid jet with increasing field strength. An asymptotic approximation of the equations of electrohydrodynamics is developed so that quantitative comparisons with experiments can be carried out. The approximation governs both long wavelength axisymmetric distortions of the jet, as well as long wavelength oscillations of the centerline of the jet. Three different instabilities are identified: the classical ͑axisymmetric͒ Rayleigh instability, and electric field induced axisymmetric and whipping instabilities. At increasing field strengths, the electrical instabilities are enhanced whereas the Rayleigh instability is suppressed. Which instability dominates depends strongly on the surface charge density and radius of the jet. The physical mechanisms for the instability are discussed in the various possible limits. © 2001 American Institute of Physics. ͓DOI: 10.1063/1.1383791͔

I. INTRODUCTION the onset of electrospinning always corresponds with the whipping of the jet. Electrospinning is a process that produces a highly im- The goal of this series of papers is to develop a theoret- permeable, nonwoven fabric of submicron fibers by pushing ical framework for understanding the physical mechanisms a millimeter diameter liquid jet through a nozzle with an of electrospinning, and to provide a way of quantitatively electric field.1–7 In the conventional view,1 electrostatic predicting the parameter regimes where it occurs. The theory charging of the fluid at the tip of the nozzle results in the consists of two components: a stability analysis of a cylinder formation of the well known Taylor cone, from the apex of of fluid with a static charge density in an external electric which a single fluid jet is ejected. As the jet accelerates and field, and a theory for how these properties vary along the jet thins in the electric field, radial charge repulsion results in as it thins away from the nozzle. By analyzing various insta- splitting of the primary jet into multiple filaments, in a pro- bilities of the jet, we show that it is possible to make predic- cess known as ‘‘splaying.’’1 In this view the final fiber size is tions for the onset of electrospinning that quantitatively determined primarily by the number of subsidiary jets agree with experiment. formed. The formalism we describe here also has application to We have recently demonstrated8 that an essential mecha- electrospraying9–14 for which it has long been known that nism in electrospinning involves a rapidly whipping fluid jet. there are many different operating modes, originally docu- The whipping of the jet is so rapid under normal conditions mented by Cloupeau and Prunet-Foch.15,16 A recent review that a long exposure photograph ͑e.g., Ͼ1ms͒ gives the en- article17 classifies the different modes as ͑a͒ dripping, when velope of the jet the appearance of splaying subfilaments. A spherical droplets detach directly from the Taylor cone; ͑b͒ systematic exploration of the parameter space of electrospin- spindle mode, in which the jet is elongated into a thin fila- ning ͑as a function of the applied electric field, volume flux, ment before it breaks into droplets; ͑c͒ oscillating jet mode, and liquid properties͒ revealed that for PEO/water solutions, in which drops are emitted from a twisted jet attached to the nozzle, and the ͑d͒ the precession mode, in which a rapidly

͒ whipping jet is emitted from the nozzle, before it breaks into a Author to whom all correspondence should be addressed. Present address: Division of Engineering and Applied Sciences, Harvard University, Cam- droplets. The last two modes are qualitatively similar to the bridge, Massachusetts 02138. Electronic mail: [email protected] whipping mode we ﬁnd in electrospinning.

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A series of papers in the electrospraying community has metric instability and an oscillatory ‘‘whipping’’ instability focused on the properties of the jet of fluid emitted from the of the centerline of the jet; however, the quantitative charac- Taylor cone.18–25 These papers demonstrate that ͑in certain teristics of these instabilities disagree strongly with experi- regimes of parameter space͒ the current–voltage ments. As an example, a water jet with a radius of order that 26 relationship can be rationalized using the asymptotic prop- used in Taylor’s 1969 experiments29 is predicted to have a erties of a thinning fluid jet, assuming the jet remains a con- most unstable wavelength between 1/27–1/2 of a jet radius,33 tinuous stream. The types of calculations described in the in direct contradiction with the pronounced long wavelength present papers would allow these jet solutions to be extended instability found in the experiments. Recent work36,37 has to predict which operating mode occurs in electrospraying as included surface charge in special limits, although the afore- a function of parameters. mentioned discrepancies with experiments remain unre- The idea of the analysis is straightforward: first we will solved. develop a stability theory for a cylinder of a fluid ͑with fixed The organization of this paper is as follows: in Sec. II we surface tension, conductivity, viscosity, dielectric constants, etc.͒ in an axial applied electric field ͑i.e., pointing along the revisit the stability of an axisymmetric jet. We derive an axis of the jet͒ with an arbitrary charge density. We then asymptotic model for the dynamics of the jet by modifying 38 39 apply this stability analysis to the relevant solution for the previous theories of nonelectrical and dielectric jets to jet’s shape, charge density, etc. The predicted mode of insta- include free charges. The linear dispersion relations extend bility that is excited turns out to depend on the fluid proper- those previously obtained in the nonelectrical, dielectric and ties and the process variables ͑e.g., applied flow rate and other special parameter limits. The simplicity of the formulas electric field͒ in a manner that can be predicted quantita- allows straightforward physical interpretation of the two tively. As will be seen in a companion paper,8 treatment of main instability modes; the Rayleigh mode, which is the the fluid as Newtonian is adequate for the range of fluids and electrical counterpart of the surface tension driven Rayleigh operating conditions explored here. instability, and a conducting mode, which involves a purely The basic principles for dealing with electrified fluids electrical competition between free charge and the electric were developed in a series of papers written by Taylor in the fields. At high fields/charge densities the classical Rayleigh 27–30 1960s. Taylor discovered that it is impossible to account instability is completely stabilized and all instabilities are for most electrical phenomena involving moving fluids under purely electrical; surface tension does not affect the stability the seemingly reasonable assumptions that the fluid is either characteristics of a strongly forced electrical jet. a perfect dielectric or a perfect conductor. The reason is that Section III develops a theory for the whipping instability any perfect dielectric still contains a nonzero free charge of the jet, which proceeds from a view of the jet as a fiber density. Although this charge density might be small enough forced by electrical and surface stresses. We apply the for- to ignore bulk conduction effects, the charge will typically 40 live on interfaces between fluids. If there is also a nonzero malism for deriving the equations of elasticity to find a electric field tangent to the interface then there will be a dynamical equation for the fiber; by properly computing the nonzero tangential stress on the interface. The only possible electrical stresses, we recover previous stability results for force that can balance a tangential stress is viscous; hence, the bending of the fiber. Because of the simplicity of this under these conditions the fluid will necessarily be in mo- theory, it is again easy to develop a simple physical interpre- tion. This idea has become known as the ‘‘leaky dielectric tation for the various mechanisms causing instability, and model’’ for electrically driven fluids. Its consequences have also to extend previous stability results to the parameter re- been successfully compared to experiments on neutrally gimes important in the experiments ͑i.e., finite viscosity, fi- buoyant drops elongated by electric fields. A detailed discus- nite conductivity, and finite surface charge͒. sion and derivation of the assumptions behind the model are Section IV examines the competition between the two 31 described in the recent review by Saville. electrical instabilities and describes which happens first. We Despite the general acceptance of the leaky dielectric find that in the absence of surface charge the axisymmetric model, a review of the literature reveals that no theory for the mode is more unstable than the whipping mode, so that an destabilization of an electrically forced jet has ever been experiment without surface charge should see a predominant quantitatively compared with experiments. The linear stabil- axisymmetric instability. Experiments on electrically forced ity analysis of an uncharged jet in an electric field was per- jets dating back to Taylor29 observe predominantly a whip- formed by Saville in the early 1970’s.32,33 The reason that ping jet; this is explained by the demonstration that when quantitative tests have not been undertaken are partially explained in Saville’s33 original papers; the stability analysis surface charge is included in the stability analysis the relative for a finite conductivity cylinder in a tangential electric field stability of these two modes can change, so that whipping neglects two important effects, the presence of surface can be more unstable. This is indeed what occurs when rea- charge on the jet and the thinning of the radius. Similarly, the sonable estimates for the surface charge density in a typical analysis34,35 of the stability of a jet with a constant surface electrospinning experiment is used. charge density assumed infinite conductivity, zero tangential The second paper in this series applies these results to electric field, and a constant radius jet. Saville’s analyses explain quantitatively our experiments on electrospinning, predict qualitative characteristics consistent with the and also shows how these results can be useful for under- experiments—most notably the presence of both an axisym- standing the origin of the different modes of electrospraying.

much longer for electrical equilibration to occur through a copper wire. The reason for this difference is that the global relaxation of charge occurs via charge neutral currents in the bulk of a medium, which depend in general on the geometry of the material in question ͑e.g., in a copper wire the RC constant is the relevant time for relaxing imposed potential differences͒. This point is especially important to the study ␶ of electrospinning and electrospraying; e is sometimes em- ployed without taking the care to make this important dis- tinction. We will assume throughout this paper that free charge in the bulk has ample time to relax to its equilibrium value ͑i.e., ␶ ͒ that e is irrelevant , so that all free charge resides on the FIG. 1. Experimental setup. surface of the jet. In a typical experiment a current, I, flows through the jet due to its bulk conductivity and the advection of charge along the surface. This current is experimentally II. ASYMPTOTIC THEORY OF AXISYMMETRIC JETS determined to be a function of the fluid parameters, Q,Eϱ , We now define and develop the physical model of the and the geometry of the nozzle. experimental system shown in Fig. 1. A thin metal tube guides the fluid perpendicular to and through the center of a A. Derivation of axisymmetric equations broad capacitor plate and opens into free space. The tube The equations for the jet follow from Newton’s Law and opening protrudes slightly from the surface of the plate. The the conservation laws it obeys, namely, conservation of mass jet then flows out of the tube and downward ͑with gravity͒ and conservation of charge. Because of viscous dissipation between the top plate and another identical plate below it. and external forcing by both gravity and the electric field, the The bottom plate sits in a glass collecting dish. The two isolated system of the jet does not conserve energy or mo- capacitor plates are positioned perpendicular to the flow so mentum. We also employ Coulomb’s integral equation for that the externally imposed electric field is very nearly uni- the electric field. form and axial in the region of the jet. The experimentally The first step in the derivation assumes that the jet is a adjustable parameters ͑aside from material parameters of the long, slender object and then uses a perturbative expansion jet fluid͒ are the electrostatic potential difference between the in the aspect ratio. This idea has been recently used in mod- plates, V, the separation between the plates, d, and the con- eling jets without electric fields ͑for a comprehensive review, stant rate at which fluid flows from the tube, Q. We will see Ref. 38͒. This approximation method rests on expanding denote the constant field far from the nozzle as Eϱ ; later in the relevant three-dimensional fields ͑axial velocity, v , ra- the paper, it will become apparent that the fringe fields z dial velocity, v , radial electric field, E , and axial electric around the nozzle are also important. r r field, E ͒ in a Taylor series in r, e.g., The external fluid is air, which is assumed to have no z ͑ ͒ϭ ͑ ͒ϩ ͑ ͒ ϩ ͑ ͒ 2ϩ ͑ ͒ effect on the jet except to provide a uniform external pres- vz z,r v0 z v1 z r v2 z r ¯ . 1 sure. This assumption holds until the jet becomes so thin that We then substitute these expansions into the full three- air drag and air currents become important. The internal fluid dimensional equations and keep only leading order terms. is assumed to be Newtonian and incompressible. This as- This leads to a set of equations that is actually rather intui- sumption holds for dilute solutions under low shear and in tive. The conservation of mass equation is the absence of high degrees of extension. The fluid param- ␲h2v͒ϭ0, ͑2͒͑ ץ␲h2͒ϩ͑ ץ eters for the internal fluid are: conductivity, K, dielectric con- t z stant, ⑀, mass density, ␳, and kinematic viscosity, ␯. The dy- where h(z) is the radius of the jet at axial coordinate z and namic viscosity is ␮ϭ␳␯. The coefficient of interfacial v(z) is the axial velocity, which to leading order is constant ͓ ͑ ͒ ϭ ͔ tension between the fluid and the air is ␥. The dielectric across the jet cross section i.e., Eq. 1 with v0 v . constant of the air is assumed to be ¯⑀. In general, overbarred Similarly, conservation of charge becomes ͒ ͑ ␲ ␴ ϩ␲ 2 ͒ϭ ץ␲ ␴͒ϩ ץ quantities will refer to the region outside the jet. t͑2 h z͑2 h v h KE 0, 3 To proceed, we must consider the relaxation of free where ␴(z) is the surface charge density, and E(z) is the charge in the system. The classical relaxation time is, in electric field in the axial direction. There is now an extra Gaussian units, component to the current besides advection due to bulk con- ⑀ duction in the fluid. ␶ ϭ . e 4␲K Momentum balance, or the Navier–Stokes equation, becomes This is the timescale for the local relaxation of bulk charge v2 1 2␴E 3␯ ͒ ͑ ͒ ץ2 ͑ ץ ϩ ϩ ϩ ץ ϭϪ ͪ ͩ ץϩ ץ density within a conducting medium. This process is usually tv z ␳ zp g ␳ 2 z h zv , 4 much faster than the global equilibration of charge on the 2 h h ␶ surface of a conductor. For example, for copper e where p(z) is the internal pressure of the fluid. The hydro- Ϸ10Ϫ19 s, whereas it is common experience that it takes dynamic terms are familiar.38 In addition there are electro-

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TABLE I. Table of symbols used in this work.

Q Volume flux I Current Eϱ Applied electric field ⑀ Fluid dielectric constant ¯⑀ Air dielectric constant K Liquid conductivity ␯ Kinematic viscosity ␳ Liquid density ␥ Surface tension ␮ Dynamic viscosity g Gravitational acceleration FIG. 2. Gaussian surface, S1 , lies just inside the interface. h Radius of jet ␴ Surface charge density V Voltage drop between the capacitor plates d Distance between capacitor plates computed with the model would ever violate this assumption E Electric field parallel to axis of jet the solution would break down. In every case we have stud- v Fluid velocity parallel to axis of jet ied the assumption has held uniformly in both space and p Pressure in the fluid ␭ Linear charge density along the jet time. P Dipole density along the jet Now we turn to the derivation of the equation for the ␤⑀/¯⑀Ϫ1 electric field. ͓The philosophy of this derivation was inspired M Internal moment of bent fiber by a comment of E. J. Hinch. See the Appendix of Ref. 39.͔ N External torque on bent fiber The electric field outside the slender jet can be written down K External force on bent fiber ͑ F Tension in bent fiber as if it were due to an effective linear charge density incor- r Coordinate of centerline of bent fiber porating both free charge and polarization charge effects͒ of ␴ D Dipolar component of free charge density charge ␭(z) along the z-axis. To find this linear charge den- K* Dimensionless conductivity sity, we perform a customary Gauss’ law argument twice. ␯* Dimensionless viscosity ⍀ The first time we pick a cylindrical Gaussian surface, S1 , 0 Dimensionless external field strength ␴ coaxial with the jet, of length dz and radius just less than 0 Dimensionless background free charge density h(z) ͑see Fig. 2͒. This surface contains no charge ͑and no effective charge, since we are inside the cylinder͒. Thus, denoting Er and Ez the radial and axial components of the field static terms in the pressure and a tangential stress term, inside the jet evaluated at the jet surface, we obtain 2␴E/␳h. The latter accelerates the jet due to the presence of both surface charge and a tangential component of the elec- ϭ Ͷ ϭ͓͑␲ 2 ͒Јϩ ␲ ͔ tric field at the jet surface. 0 E•nˆdA h Ez 2 hEr dz, S1 The pressure is so that ⑀Ϫ¯⑀ 2␲ pϭ␥␬Ϫ E2Ϫ ␴2 , ͑5͒ ␲ ⑀ ␲ ϭϪ ␲ 2 ͒ 8 ¯ 2 hEr ͑ h Ez Ј. ␬ ͑␬ where is twice the mean curvature of the interface Now we pick a new Gaussian surface, S , in the same ϭ ϩ 2 1/R1 1/R2 , where R1 and R2 are the principal radii of position as S with the same length but a radius just bigger ͒ 2 1 curvature . The electrostatic term proportional to E is the than h(z) ͑see Fig. 3͒. Since this surface surrounds the entire difference across the interface of the energy density of the jet volume, it contains some effective charge, in particular it ⑀ 2 ␲ field ( E /8 ) if we neglect contributions from the asymp- contains ␭(z)dz. Denoting fields outside the jet with an ͑ ͒ totically small Er the radial field inside the jet . The term overbar, proportional to ␴2 is just the radial self-repulsion of the free charges on the surface. ͑This term can also be interpreted as the difference in the energy density in the radial electric field, ͒ where again we neglect terms proportional to Er . When we Ϸ ϷϪ evaluate R1 and R2 we obtain R1 1/h and R2 hЉ, where the prime refers to differentiation in z. hЉ is higher order than 1/h, but it is also the stabilizing part of surface tension, so in certain cases we retain this term ͑Table I͒. The one important difference between the derivation of this equation and that for nonelectric jets is that an additional assumption must be made for the expansion in Eq. ͑1͒ to be asymptotically valid; namely, the tangential stress caused by the electric field ␴E must be much smaller than the radial viscous stress, ␮v/h. Whether or not this is true in a given FIG. 3. Gaussian surface, S2 , lies just outside the interface and thus con- situation needs to be determined a posteriori; if a solution tains charge.

both ␭(z), and its image charges. ͑See the second paper in ␲␭͑ ͒ ϭ ϭ͓͑␲ 2 ͒Јϩ ␲ ¯ ͔ ͑ ͒ 4 z dz Ͷ E nˆdA h Ez 2 hEr dz. 6 this series.͒ S • 2 The aforederived equations are nondimensionalized by ͑ ͒ For a slender jet we know that choosing a length scale r0 e.g., the nozzle diameter , a time ϭͱ␳ 3 ␥ ⑀ 4␲␴ scale t0 r0/ , an electric field strength E0 ¯ Ϸ¯ ϭ ϩ ͑ ͒ ϭͱ␥ ͓ ⑀Ϫ⑀ ͔ ͱ␥⑀ Er En ⑀ En ⑀ . 7 / ( ¯)r0 and a surface charge density ¯/r0.We ¯ ¯ ⍀ ϭ ͑ denote the dimensionless asymptotic field 0 Eϱ /E0 . For So, ease of comparison, we note that the relationship between our ⍀ and Saville’s parameter E is ⍀2ϭ4␲␤E.͒ Four di- ␤ 2␲h␴ 0 0 ␭͑ ͒ϭϪ ͑ 2 ͒Јϩ ͑ ͒ mensionless parameters characterize the material properties z h Ez ⑀ , 8 4 ¯ of the jet: ␤ϭ⑀/¯⑀Ϫ1, the dimensionless viscosity ␯* ϭͱ ͑ ϭ␳␯2 ␥ ͒ where l␯ /r0 with the viscous scale l␯ /( ) , the dimen- *ϭ ␳ 2 ␥ ⑀ sionless gravity g g r0/ , and the dimensionless conduc- ␤ϭ Ϫ tivity K*ϭKͱr3␳/(␥␤). Typical values of the viscous ⑀ 1. 0 ¯ length scale are l␯ϭ100 Å for water, and l␯ϭ1 cm for glyc- Now we can plug this effective linear charge density into erol. Whether viscous stresses are important depends on the Coulomb’s Law for the electrostatic potential outside the jet. ratio of this length scale to the jet radius. A conductivity of 1 We will assume that ␭ varies over a length scale, L, much ␮S/s ͑at the upper range for our experiments͒ corresponds to larger than the radius, and approximate the integral in that a relaxation time of Ϸ1 ␮s; K* compares this to a capillary Ϫ3 case, time for a 0.1 mm jet of about 10 s. Hence, when the jet is thick, it is a very good conductor indeed. ␭͑zЈ͒ The nondimensional equations are then ⌽¯ ͑z,r͒ϭ⌽¯ ϱϩ ͵ dzЈ ͒ ͑ 2͒ϩ 2 ͒ ϭ ץ ͱ͑ Ϫ Ј͒2ϩ 2 z z r t͑h ͑h v Ј 0, 11 K* Ј ͒ ͑ ␴ ͒ϩͩ ␴ ϩ 2 ͪ ϭ͑ ץ 1 Ϸ⌽¯ ϱϩ␭͑z͒ ͵ dzЈ t h hv h E 0, 12 ͱ͑zϪzЈ͒2ϩr2 2 1 E2 Ј 2␴E *vϩvvЈϭϪͩ ϪhЉϪ Ϫ2␲␴2 ͪ ϩ ϩg ץ r Ϸ⌽¯ ϱϪ2ln ␭͑z͒ t ␲ L h 8 ͱ␤h r ␤ 4␲ 3␯* ϭ⌽¯ ϩ ͩ ͑ 2 ͒ЈϪ ␴ͪ ͑ ͒ ϩ ͑h2vЈ͒Ј, ͑13͒ ϱ ln h E h . 9 L 2 z ⑀ h2 Elementary electrostatics tells us that the tangential compo- 1 ␤ EϪln ͫ ͑h2E͒ЉϪ4␲ͱ␤͑h␴͒Јͬϭ⍀ . ͑14͒ nent of the electric field is continuous across the interface, so ␹ 2 0 ˜ Ϸ¯ ϭ Ϸ Ez Et Et Ez . This yields a single equation for the tangential field inside the jet, B. Stability analysis 1 ␤ 4␲ Now we present and discuss the local linear stability Ϫ ͩ ͪͫ ͑ 2 ͒ЉϪ ͑ ␴͒Јͬϭ ͑ ͒ E ln h E h Eϱ , 10 analysis of axisymmetric perturbations to an electrified jet. ␹ 2 ¯⑀ There are three equilibrium states around which one could ϭ ␹Ϫ1 where E Ez . The variable is the local aspect ratio, imagine perturbing ͑see Fig. 4͒; there are two perfectly cy- which is assumed the be small. lindrical states, one in which there is an applied field but no If the fluid takes a conical shape, there is only one pos- initial surface charge density, and another ͓Fig. 4͑b͔͒ with no sible choice for ␹, the slope of the cone. This fact allowed39 applied field but an initial charge density ͑so that the field us to calculate asymptotic equilibrium cone angles at the tips inside is zero, initially͒. Neither of these are relevant for our of electrified, perfect dielectric fluid drops that agreed very electrospinning experiments. A more realistic state for ex- well with the exact theory for infinite cones. This simplifica- periments thins due to the tangential stress at the interface tion also will occur subsequently for the sinusoidally per- from the interaction between the static surface charge density turbed, perfect cylinders of linear stability analysis, where ␹ and the tangential electric field. We remark that there are must be proportional to the wave number k of the perturba- experimental configurations where Fig. 4͑b͒ is the appropri- tion. This fact will allow us to calculate dispersion relations ate limit; in particular in the ‘‘point plate’’ geometry, the for electrified jets that agree well with previous results ͑e.g., electric field decays away from the nozzle. Far enough of Saville͒, which are derived from the full three- downstream, the surface charge density can bet the external dimensional equations. For generic situations where ␹ is less field. This is the geometry used in most electrospraying easily defined, we must either disregard the factor of ln(1/x), experiments.41 since it is a logarithm and will not vary too much, or use The third state is the most general of the three; as long as some local estimate. To achieve quantitative agreement with the wavelength of the perturbation is much smaller than the experiments near a nozzle, it will turn out that it is necessary characteristic decay length of the jet, the stability character- to employ the full integral equation for the field caused by istics can be approximated by considering perturbations to a

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FIG. 5. Comparison of Saville’s ͑Ref. 33͒ dispersion relation for a zero viscosity fluid as a function of field strength, for ␤ϭ77. The solid lines are ⍀2 Saville’s results for infinite conductivity, with field strengths, 0 ϭ0,0.97,1.93,2.90,3.8 from the uppermost to the lowermost, respec- ⍀2ϭ tively. The dashed line is for zero conductivity and 0 4.83. The dotted lines are from Eq. ͑19͒.

FIG. 4. Jet equilibria around which linear stability may be performed. ͑a͒ 4␲K*⌳ 4␲K*⌳ and ͑b͒ were approached by Saville, ͑c͒ is more realistic for comparing with ␻3ϩ␻2ͫ ϩ3␯*k2ͬϩ␻ͫ 3␯*k2 experiments. ␦ͱ␤ ␦ͱ␤

2 L ⌳ ⍀2 k 8 0 ϩ ͑k2Ϫ1͒ϩ2␲␴2k2ͩ Ϫ1ͪ ϩ k2ͬ 2 0 ␦ ␦ 4␲ charged cylinder of constant radius. Since this state encom- passes the other two we will consider it alone in the follow- ␲ ⌳ 2 ␦ ⍀2 4 K* k 0 ing. Because of the simplicity of the model, it will be pos- ϩ ͫ ͑k2Ϫ1͒ϩ2␲␴2k2ϩ k2 ␦ͱ␤ 2 0 ⌳ 4␲ sible to understand the stability of the jet as a function of all fluid and electrical parameters. E ␴ 1 To determine the linear stability, we solve for the dy- ϩ 0 0 ͩ Ϫ ͪͬϭ ͑ ͒ ik L 4 0, 19 namics of small perturbations to the constant radius, electric ͱ␤ ϭ field and charge density solution using the ansatz h 1 where Lϭln(1/␹). Since this equation is cubic, there will in ϩ ␻tϩikx ϭ ϩ ␻tϩikx ϭ⍀ ϩ ␻tϩikx ␴ h⑀e , v 0 v⑀e , E 0 e⑀e , and general be three different branches of the dispersion relation; ϭ␴ ϩ␴ ␻tϩikx ␴ 0 ⑀e , where h⑀ ,v⑀ ,e⑀ , ⑀ are assumed to be an instability occurs if any of these branches has Re ␻Ͼ0. ͑ ͒ ͑ ͒ small. Substituting this into Eqs. 11 – 14 yields Although Eq. ͑19͒ is complicated, we wish to emphasize that ͑a͒ it is vastly simpler than dispersion relations that have 2␻h⑀ϩikv⑀ϭ0, ͑15͒ previously been derived for this problem; for example, the 33 ͑ K* analogous formulas in Saville for the special case of a jet ␻␴ ϩ ⍀ ϩ ͒ ϩ ␴ ϭ ͑ ͒ ͒ ⑀ ͑2 0h⑀ e⑀ ik ik 0v⑀ 0, 16 with no net charge density involves solving a cubic equation 2 with coefficients that involve determinants of matrices of Bessel functions, whose arguments include the growth rate, ⍀ 2⍀ ␻ 37 ␻ ϭ͑ Ϫ 3͒ ϩ 0 ϩ ␲ ␴ ␴ ϩ 0 ␴ . Mestel has considered the stability of a jet in a field with v⑀ ik ik h⑀ ike⑀ 4 ik 0 ⑀ ⑀ ͑ 4␲ ͱ␤ nonzero surface charge density in the very viscous limit liquid density ␳→0͒. Describing the calculation, he states ‘‘the 2␴ e⑀ ␴ ⍀ algebra is exceedingly nasty, even with the help of Math- 0 0 0 2 ϩ Ϫ2 h⑀Ϫ3␯*k v⑀ϭ0, ͑17͒ ͑ ͒ ͑ ͒ ͱ␤ ͱ␤ ematica.’’ b Equation 19 works for arbitrary values of the conductivity, dielectric constant, viscosity, and field strength, as long as the condition that the tangential electrical stress is ␦ 4␲i⌳ 4␲i␭␴ ϩ⌳⍀ ϩ ␴ 0 ϭ ͑ ͒ smaller than the radial viscous stress is satisfied. Technically, e⑀ 0h⑀ ⑀ h⑀ 0, 18 2 ͱ␤k ͱ␤k the formula is asymptotically correct in the k→0 limit. It therefore extends the previous results to regimes which have where ⌳ϭ␤ ln(1/␹)k2, ␦ϭ2ϩ⌳. These equations have a heretofore not been examined; importantly, this includes the nontrivial solution only if the determinant of the coefficient regime of most experiments. matrix vanishes; this requirement yields the dispersion rela- To demonstrate the asymptotic validity of these formu- tion, las, we first compare the results of different limits of Eq. ͑19͒

Downloaded 17 Jul 2001 to 140.247.53.216. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. I. 2207 with those of previous authors. First, we compare to Saville’s Figure 5 compares the zero viscosity and infinite con- results: Figs. 5 and 6 show comparisons in two of the three ductivity limit from Eq. ͑19͒͑dotted lines͒ to Saville’s results ͓ ͑ ͒ ⍀ parameter regimes reported in Ref. 33. We note that there is solid lines , for several values of the applied field, 0 .In one small freedom in our dispersion relation, namely, the this regime, increasing the applied electric field stabilizes the ␹Ϫ1 relationship between the local aspect ratio of the jet and jet. The agreement is excellent, and is exact at low k. Also ␹Ϫ1ϭ the wave number k. Clearly Ak. We determine A by shown in this figure is a comparison in the zero conductivity expanding Saville’s dispersion relation ͓Eq. ͑32͒ of Ref. 33͔ limit. The agreement between Saville’s relation ͑the dashed in the low viscosity limit at low k, and then demanding that line͒ and Eq. ͑19͒ is exact at low k. the value of A in his calculation agrees with ours. This im- ͑ ͒ plies that Aϭexp(␥)/2ϭ0.89..., where ␥ is Euler’s gamma. The dispersion relation 19 also yields a simple analytic We use this value of A in all subsequent comparisons, so that expression that gives the growth rate in the infinite conduc- A is independent of all fluid and control parameters.͔ tivity, zero viscosity limit,

1 ⍀2 2 ⍀ ␴ i ␻ϭ ͱ ͑ Ϫ 2͒Ϫ ␲␴2Ϫ 0 ͩ ϩ ϩ 0 0 ͪ ͑ ͒ k 1 k 2 0 1 2 . 20 2 4␲ ␤k ln͑Ak͒ ͱ␤k

␴ ⍀ When 0 , 0 are zero, this formula gives the classical Ray- This critical field is comparable to typical experimental val- ⍀ leigh instability. When the field 0 is increased, the Rayleigh ues: fora1mmjetofwater in an electric field of 2 kV/cm, ⍀ Ϸ instability is suppressed. At a critical field strength, the in- the nondimensional applied field, 0 2, so that the local stability completely goes away. In the limit of large ␤ with field in the thinning jet is close to the critical value. For a 10 ␴ ϭ ␮ ⍀ 0 0, this critical field is given by the formula m jet, 0 is approximately 0.2. The consequence of this is that it demonstrates that at high fields neither surface tension ⍀ Ϸͱ2␲ϭ2.5 . ͑21͒ 0crit ¯ nor the classical Rayleigh instability is important for the In dimensional units, the criterion is breakup of electrified jets until the jet becomes very thin. Figure 6 shows a comparison at finite conductivity, so 2␲␥ 2 ⑀ ͑⑀Ϫ¯⑀͒Eϱϭ . ͑22͒ that the electrical relaxation time /K is finite, and infinite r0 viscosity. In this limit, the growth rate, ␻, can become com- This formula has a simple physical interpretation; when the plex, corresponding to oscillatory modes. Figure 6 shows the electrical pressure per unit length of the jet exceeds the sur- growing modes for three different values of ␶ϭ(⑀/K) ϫ ␥ ␮ ⍀ ϭ face tension pressure per unit length, the instability is sup- ( /(r0 )) at an imposed electric field 0 1.56. The solid pressed. It turns out that this critical field strength has the lines and the dashed lines are Saville’s results; the dashed same value even when the conductivity or viscosity is finite. lines represent oscillatory modes, which are unstable at small r. Our dispersion relation is superimposed on Saville’s with no discernable difference.

FIG. 6. Comparison of Saville’s ͑Ref. 34͒ dispersion relation for an infinite viscosity fluid with different finite electrical relaxation time ␶ ϭ⑀ ␥ ␮ ϭ /K /( r0) 0.01, 1, and 100, respectively. The rightmost lines show nonoscillatory modes ͑i.e., Im ␻ϭ0͒ and the leftmost lines show oscillatory FIG. 7. Field dependence of the instability growth rate ␻ for a fluid with modes. The result from the dispersion relation ͓Eq. ͑19͔͒ ͑dashed lines͒ is ␯ϭ140 cP, ␥ϭ65 dyn/cm, conductivity Kϭ5.8␮ and ⑀ϭ78. As the field is superimposed on Saville’s solution ͑solid lines͒ with no discernable differ- increased ͑direction indicated by arrows͒ the ‘‘Raleigh mode’’ is suppressed ence. and the conducting mode becomes dominant.

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These plots demonstrate that the asymptotic theory re- produces the previous results. Before proceeding further, it is pedagogical to first examine how the growth rate varies with electric field, in order to develop intuition for how the field interacts with the fluid. Figure 7 shows the dependence of ␻ on k and Eϱ for a fluid with ␯ϭ140 cP, ␥ϭ65 dyn/cm, conductivity Kϭ5.8 ␮S, and ⑀ϭ78. The arrows indicate what happens when the external field increases. The zero field- result is the dashed curve; as the field increases this ‘‘Ray- leigh mode’’ has a positive growth rate for a smaller and smaller range of wave vectors. At the critical field the Ray- leigh mode is completely suppressed. However, the jet is still unstable, because a new ‘‘conducting mode’’ becomes larger and larger as the field is increased. This conducting mode is FIG. 8. Physical picture of an unstable perfect dielectric jet; the perturbation oscillatory ͑the imaginary part of its growth rate is nonzero͒, in the velocity pushes surface charge into the bulging regions. and is the root cause of the oscillatory growth rates observed by Saville.33 ͑We will see below that when surface charge is C. Physical intuition with surface charge included, both the Rayleigh mode and the conducting mode The presence of surface charge modifies the characteris- are modified in important ways; however, the basic distinc- tics of both the Rayleigh and the conducting modes in a tion between them still exists, and is important for interpret- fashion that is not ͑at first glance͒ intuitive. To explain how ing experiments.42͒ this works, we first present further description of how the The physics of this conducting mode is a consequence of various instabilities described above operate ͑to aid intu- the interaction of the electric field with the surface charge on ition͒, and then explain how a static surface charge changes the jet; surface tension is an irrelevant parameter for this the mechanics. mode. The mechanism for the instability is that a modulation 1. Perfect dielectrics in the radius of the jet induces a modulation in the surface It will be helpful for our intuitive exploration to discuss charge density ͑which is induced because at low frequencies the forms of the perturbation of the velocity, charge density, the charges try to cancel out the field inside the jet͒. The and electric field in terms of the perturbation to the radius, nonzero surface charge results in a tangential stress, which h⑀ . These follow from Eqs. ͑15͒–͑18͒. For a perfect dielec- accelerates the liquid and causes instability. The oscillatory tric, the equations reduce to character of the instability results because the timescale for 2i␻ v⑀ϭ h⑀ , ͑23͒ the fluid response is different than the timescale for the axial k surface charge rearrangements. In contrast to the ordinary ␴ ϭ ␴ ͑ ͒ Rayleigh instability ͑in an electric field͒, the conducting ⑀ 2 0h⑀ , 24 mode is always unstable at long wavelengths ͑low k͒ when 2⌳ 24␲ͱ␤ the applied field is nonzero. The maximum growth rate of the ϭϪͩ ⍀ ϩ ␴ ͪ ͑ ͒ e⑀ ␦ 0 ik ␦ 0 h⑀ , 25 ␻ ϳ⍀ ͱ conducting mode scales like conduct 0 K*. An important consequence of these results ͑cf. also with the momentum response given by ͑17͒. Saville34͒ is that they imply that models for electrically The velocity perturbation, v⑀ ,is␲/2 out of phase with forced jets which either assume perfect conductivity or a the radius perturbation, and for a growing mode it pushes perfect dielectric are qualitatively incorrect. The existence of fluid into the bulging regions and draws fluid from the nar- the conducting mode occurs because when the conductivity rowing regions. This motion pushes not only fluid but sur- is finite the equation for the growth rate is cubic, so that there face charge as well, so surface charge piles up in the bulging ␴ ͑ is an extra root. For models without finite conductivity this regions as well as fluid, hence ⑀ is in phase with h⑀ see Fig. 8͒. The electric field response is more complicated. The mode does not exist; hence such models have no chance at part of e⑀ proportional to ⍀ is in phase with h⑀ ; it is due to capturing the physics in this regime. 0 the polarization of the fluid resisting the penetration of the As a final check on the stability formulas, we note that in ⍀ ͑ external field, 0 , into the bulging regions hence it is nega- the limit where the surface charge is nonzero but the external tive, see Fig. 9͒. The part proportional to ␴ is the electric electric field vanishes, our results also reduce to those found 0 field due to the perturbed charge density ␴⑀ , and is therefore previously in the appropriate limits. As an example, Saville ␲/2 out of phase with h⑀ . considered the stability of a perfectly conducting, charged ␴ If 0 is nonzero, then there are two additional competing cylinder in the viscous limit without a tangential electric effects. The pressure due to the self-repulsion of charge (p ␻ϭ␥ ␳␯ Ϫ1 2 field. His dispersion relation gives (6r0 ) (1 ϰ␴ ) destabilizes the growing mode, because, as we have Ϫ ␲ ␴2␥⑀ Ϫ1 ͑ ␴ ͒ 4 r0 ¯) with the charge density on the cylinder . noted above, advection pushes charge into the bulging re- This agrees exactly with the prediction of Eq. ͑19͒ in the k gions, so the repulsion becomes stronger in these regions →0 limit. leading to instability. On the other hand, the tangential stress

FIG. 11. Physical picture of an unstable perfectly conducting jet. FIG. 9. Physical picture of an unstable perfect dielectric jet; the perturbation

in surface charge density causes the electric field perturbation, ea ; the polarization charge (spol) tries to cancel out the penetration of the applied field, face is cancelled out. The electric field flux within the jet Eϱ , and this causes the electric field perturbation, eb . before perturbation is conserved throughout the instability, and the field lines simply get closer together in the narrowing ␲ stabilizes the jet by applying a line tension on the interface regions and farther apart in the bulging regions. Thus, e⑀ is ͑ ͒ parallel to the flow. At long wavelengths, tangential stress out of phase with h⑀ see Fig. 11 . stabilization beats charge self-repulsion; however, at small The charge density response is more complicated and is wavelengths ͑but still within the range of validity of this dictated by the requirement that there be no change in the ⍀ theory͒ the repulsion wins. These effects balance at the turn- electric flux within the jet. The term proportional to 0 , over wave number k ϳ1/ͱ␤. Figure 10 shows this compe- denoted ea , excludes the penetration of applied field lines c ͑ ͒ tition. into the bulging fluid see Fig. 12 . This term is analogous to the term in the electric field response in the dielectric case ⍀ 2. Perfect conductors that is proportional to 0 , except in that case it is the polar- Here the velocity response is the same as before, and the ization that resists the field line penetration. The term eb , ␴ response of the two other fields is proportional to 0 , is negative, contrary to the case for perfect dielectrics. In order to counter the additional electric ⍀ 0 field within the jet due to a constant surface charge density ␴⑀ϭϪͩ ␴ ϩ ͪ ⑀ ͑ ͒ 0 i h , 26 2␲ͱ␤Lk on the perturbed interface, the charge density must decrease in the bulging regions and increase in the narrowing regions. ϭϪ ⍀ ͑ ͒ e⑀ 2 0h⑀ . 27 Thus the repulsion becomes stronger in the narrowing regions and acts to stabilize the instability. In a perfectly conducting jet, charge flows ͑instanta- There are two terms that involve the background surface neously͒ to the interface until the field normal to the inter- charge density. The first we have already discussed, and is just the suppression of the Rayleigh mode due to charge repulsion ͑Fig. 13͒. The second term involves a coupling ⍀ between the externally applied field, 0 , and the surface ͑ ␴ ͒ charge density the part that is proportional to 0 . Because this term is imaginary, it always destabilizes the jet regard- less of its sign. It also causes the frequency, ␻, to be complex, so that the modes are oscillatory.

FIG. 10. For a perfect dielectric, the axisymmetric mode is destabilized by charge density ͑especially at high wave numbers͒. Comparison of the dis- 2 persion relation for a 0.1 cm diameter jet with ␯ϭ1cm/s, ⑀ϭ78, and Eϱ ϭ ͑ ͒ ␴ ϭ ͑ ͒ V/cm with no surface charge solid line , and 0 0.75 dashed line . ϭͱ ␤Ϸ The turnover at kc 6/ 0.27 representing the competition between tangential stress stabilization and charge repulsion is evident. FIG. 12. Physical picture of an unstable perfectly conducting jet.

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stresses are much smaller than radial viscous stresses. How- ever, it should be noted that this regime is also outside the regime of any experimental configuration that we are aware of. Mestel argues that this limit is relevant by noting that the ϭͱ ␥ ␳␯2 Reynolds number ReS r0 / is large for a thin jet of water, and hence there is a boundary layer near the surface of the jet. However, this Reynolds number ͑which compares the timescale for capillary contraction to the timescale for viscous diffusion across the jet radius͒ is irrelevant for the present problem; as has been emphasized above, tangential electrical stresses are much more important than capillarity. The correct ratio to consider whether a boundary layer can form near the surface of the jet is the ratio of the tangential stress to the radial viscous stress. For a mode of wavelength k, this ratio is ͑in dimensional units͒ 2 2 3 2 FIG. 13. For a perfect conductor, the axisymmetric mode is stabilized by ␴E/h KEϱkr /␻ KEϱr k ϳ 0 ϳ 0 charge density. Comparison of the dispersion relation for a 0.1 cm diam jet 2 2 2 , 2 ␮v/h ␮ ␻ ␮ ␻ with ␯ϭ1cm/s, ⑀ϭ78 and Eϱϭ2/3 kV/cm with no surface charge ͑solid /r0 /k ͒ ␴ ϭ ͑ ͒ line , and 0 0.75 dashed line . where we have used a typical surface charge density ␴ ϳ ␻ ϳ␻ ␻ϳ 2/3 Eϱr0k/ and a typical velocity v /k. Since k at ␴ small k, this quantity is negligible as k→0, and hence in this The response of the electric field to 0 is reflected in the limit, a boundary layer does not exist. formula for the growth rate of the mode given in formula 37 ͑20͒. On one hand the Rayleigh mode is suppressed by the The strongly viscous limit investigated by Mestel can ␴2 ␴ ⍀ in principle occur experimentally, and our results compare surface charge 0, while on the other the imaginary 0 0 term causes an unstable oscillatory instability. Hence, the well with Mestel’s. In the low k limit, both of us find that the ␻2ϩ ␻ϩ existence of a static surface charge substantially modifies the growth rate obeys an equation of the form a b c ϭ → ␶ ͑ ␶ ‘‘Rayleigh mode’’ we discussed above. The mode that re- 0. We both find the same limiting formula a 3 where ͒ →Ϫ ␶ duces to the classical Rayleigh in the low ⍀ ,␴ limit is is the conducting time defined above , and b 1/2 (1 0 0 ϩ ␲␴2 ͑ always unstable even at very high fields. The transition rep- 16 0 log(k)); we differ slightly in the formula for c we →⍀2 →⍀2 ϩ␴2␶ ͒ resented by formula ͑21͒ does still exist, in that above this find c 0 vs Mestel’s c 0(1 0 ) . threshold field the mechanism for the instability is entirely electrical ͑and does not depend on the surface tension of the III. ASYMPTOTIC THEORY OF A JET WITH A CURVED jet͒. CENTERLINE Finally, we consider the new modes that are due to finite With these results in hand, we now turn towards an conductivity. These modes are in general oscillatory. The asymptotic theory for the bending modes of a jet, which are reason follows from the leading order behavior for long also important for experiments. We assume that the cross wavelength oscillatory modes, which implies section of the jet remains circular, and we will ignore the ␻Ϸ͑Ϫ ͒1/3͑ ⍀2͒1/3 2/3 ͑ ͒ coupling between the axisymmetric distortion of the jet and 1 K* 0 k , 28 the bending of the centerline. This limits this theory’s use to Ϫ 1/3 where we have explicitly written ( 1) to remind the linear stability analysis. Nevertheless, we present the general reader that there are three roots, two complex and one real. method here, as the equations are readily generalized to the The two complex roots are oscillatory with positive growth nonlinear case. ͑ ͒ rate. Formula 28 represents the balance between inertia and There are two pieces of intuition that need to be devel- ϳ␴⍀ ץ␳ tangential electrical stress, tv 0 /r0 , in the momen- oped before we delve into the calculation itself. The first ␴ϳ ⍀ Ј ץ tum balance equation, t 2K 0r0h in the charge advec- involves the calculation of the electric field of the jet. The ϳϪ Ј ץ tion equation, and th r0v in the mass conservation reader will recall that when we derived the electric field equation. Hence, qualitatively the growing conducting mode equation for an axisymmetric jet, we expressed the potential arises from a phase lag between the free charge oscillation outside the jet as if it were due to a linear monopole charge and the oscillation of the fluid. density spread along the jet’s axis,

D. Relation to previous work ␭͑zЈ͒ ␾¯ ͑x͒ϭ␾ϱ͑x͒ϩ ͵ dzЈ , ͉xϪxЈ͉ The stability of a charged jet in an electric field has been previously considered by Mestel in two different limits: the where we have assumed that the capacitor plates that supply ‘‘weakly viscous limit’’36 in which the tangential stress on the electric field are far away from the region of interest. We the jet is confined to a thin boundary layer near the surface of will retain this assumption in the present analysis, for in the jet, and the strongly viscous limit.37 The weakly viscous experiments the region of nonaxisymmetric instability and limit occurs outside the regime of validity of the present breakup is always far removed from the boundaries of the theory, since we have assumed that tangential electrical system. The major difference from the previous ͑axisymmet-

Downloaded 17 Jul 2001 to 140.247.53.216. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. I. 2211 ric͒ section is that when the jet bends the charge density along the jet is no longer uniform across the cross section of the jet, but in addition contains a dipolar component. ͑In general, if one were to develop an asymptotic theory for higher order nonaxisymmetric distortions of the jet, one would need to keep quadrapolar and higher terms in the charge density as well.͒ Hence, the equation for the electric potential will have the form, FIG. 14. Coordinates used in this calculation. The unit vector yˆ points out of the plane of the paper along the dashed centerline. ␭͑sЈ͒ ␾¯ ͑x͒ϭ␾ϱ͑x͒ϩ ͵ dsЈ ͉xϪr͑sЈ͉͒ tional motion is then just torque balance ͑here we have ne- Ϫ P͑sЈ͒ ͑x r͑sЈ͒͒ glected rotational accelerations since these are higher order ϩ ͵ dsЈ • , ͑29͒ ͉xϪr͑sЈ͉͒3 terms͒, MϩN͒͑ץ where P is the linear dipole density along the centerline, and ϩtˆϫFϭ0, ͑32͒ sץ the centerline of the jet is located at r(s), parametrized by arc length. If the centerline is restricted to oscillations in a where M ͑N͒ is the moment of internal ͑external͒ stresses on plane, then by symmetry the dipole density’s direction must the jet, and the reader will recognize the last term as the also be restricted to that plane. familiar ␶ϭrϫF. In a perfect string M and N are zero, and The second piece of intuition is based on an idea of Fϰtˆ. The idea of Mahadevan43,44 is that in a fluid jet, viscous Mahadevan43,44 concerning the dynamics of a bending fluid effects enter into the bending moment M, since this is where jet. We will consider only long wavelength modulations of resistance to bending occurs. We will see below that in the the jet, so that the radius of curvature of the centerline, R,is present problem, the bending moment also includes a contri- much greater than the radius of the jet. In this limit, the bution from the polarization of the jet, due to the tendency of equations of motion will then be structurally very similar to a polarized jet to align along the external field. the equations for a slender, elastic rod under slight bending.40 In writing this equation, we have assumed that there are The equation of motion follows from considering both force no axisymmetric motions coupled to our bending jet, so that balance and torque balance on the fiber, whose cross section h(s) is a constant. This implies inextensibility of the jet; is assumed to be both circular and constant in time. because we are not allowed to stretch in the radial direction, To arrive at the form of the equations of motion, con- to preserve volume we cannot stretch in an axial direction sider the force and torque balance on a small element of the either. For this reason F should be viewed as a Lagrange jet of radius h and length ds. We will refer to the cylindrical multiplier associated with the inextensibility constraint. We portion of its surface as its tube, and to the circular ends as also remark that if the cross section of the fiber is circular ͑as ˆ its caps. The centerline’s tangent vector, t, normal to one cap assumed͒, then the bending strain of the centerline is not is tˆ(s) and to the other tˆ(sϩds). Thus we allow for infini- coupled to twist ͑i.e., precession of the principal axes of tesimal curvature of the centerline of this cross section. inertia of the cross section of the jet͒, so this can be safely Let F(s) be the resultant of the internal forces within the neglected. jet at s, and let K(s) be the resultant of all surface forces and To use these equations of motion, we must compute the external body forces per unit length. The equation for trans- torques and forces caused by the fluid and electrical stresses, lational motion is then a task to which we now turn. ͒ ץ F͑s A. Coordinate system ␳Ar¨͑s͒ϭ ϩK, ͑30͒ sץ The coordinate system we will use is pictured in Fig. 14. where Aϭ␲h2 is the cross-sectional area. The quantity, ␳A, We form an orthogonal, right-handed coordinate system xˆ, yˆ, is the linear mass density, ␭ . If this were a simple string and zˆ, such that zˆ points in the direction of the unperturbed m flow. We will allow the jet’s centerline to deform in the xz- with constant tension, T, then FϭTtˆ and plane ͑ignoring effects related to the torsion of the center- ,F͑s͒ line͒. Define tˆ to be the tangent vector to the jet’s centerlineץ ϭ␬Tnˆ, ͑31͒ ␰ˆ s and to be a vector normal to the centerline that, for anץ unperturbed jet, points in the same direction as xˆ. ͑Note that where nˆ is the vector normal to the axis of the jet. If we this is slightly unconventional; the normal vector to a sinu- restrict this string to move in the xz-plane, then ␬ϭXЉ(s) soidal curve usually flips direction; so that ␰ˆ is well-defined and we recover the familiar dispersion relation for waves on at all points on the curve, we will bury the change of sign in ␻ ϭͱ ␭ ͒ a string, /k T/ m. our definition of the curvature of the centerline. The binor- If the tangent vector varies slowly along the jet (R mal always points in the direction of yˆ, so we retain that ӷh), then the jet is approximately in rotational equilibrium notation. We define the curvature such that, in this coordinate everywhere at every instant, and the equation for the rota- system,

Downloaded 17 Jul 2001 to 140.247.53.216. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp 2212 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al. tˆ 1 B. Electric field equationץ ϭ␬ ␰ˆϭ ˆ␰, s jet R We now need to derive an asymptotic approximation toץ the integral Eq. ͑29͒ for the electric field. The derivation will -␰ˆ 1 proceed in much the same way as the axisymmetric derivaץ ϭϪ␬ tˆϭϪ ˆt, s jet R tion: after approximating the integrals, we will express bothץ the effective charge density ␭ and the dipole density P in ץ where s is the derivative with respect to arc length of the centerline. The angle ␪ will be the angle measured in the terms of the free charge density and the electric field itself ␰y-plane from the ␰-axis in a counterclockwise direction using Gauss’s Law. Putting everything together will give an equation for the field. For the present purposes it is only with respect to tˆ. necessary to carry out this derivation to lowest order in the The oscillation of the centerline will be parametrized by distortion of the centerline from straight. a function X(z,t), so that its locus is described in Cartesian The dipole density P is caused by the distortion of the coordinates by centerline of the jet. This implies that ͑a͒ P(s) points in the ␰ r͑z,t͒ϭzzˆϩX͑z,t͒xˆ. ͑33͒ direction, and ͑b͒ the variation in P(s) occurs on the length ͑ We can then calculate the tangent and normal vectors and the scale of the modulation of the jet. It then follows that to ͒ curvature, lowest order in the distortion of the centerline P͑sЈ͒ ͑xϪx͑sЈ͒͒ 1 1 ͵ dsЈ • ϷxP͑z͒ ͵ dzЈ tˆ͑z,t͒ϭ ͓zˆϩXЈ͑z,t͒xˆ͔ ͑34͒ ͉xϪx͑sЈ͉͒3 ͑͑zЈϪz͒2ϩr2͒3/2 ͱ1ϩXЈ2 xP͑s͒ ϷzˆϩXЈxˆ, ͑35͒ Ϸ , ͑40͒ r2 1 ␰ˆ͑z,t͒ϭ ͓ϪXЈ͑z,t͒zˆϩxˆ͔ ͑36͒ where at this order sϷz and ␰ˆϷxˆ. ͱ ϩ Ј2 1 X There is also a correction to the potential from the cur- ϷϪXЈzˆϩxˆ, ͑37͒ vature of the centerline. If the local radius of curvature of the jet is R, then 1 XЉ ␬ ͑z,t͒ϭ ϭ ϷXЉ. ͑38͒ ͉ ͑ ͒Ϫ ͑ Ј͉͒ϭ͉␰␰ˆϩ Ϫ ͑ Ј͉͒ jet R 1ϩXЈ2 x s x s yyˆ x s In general we will assume that the jet is only slightly curved, ␰ 1/2 ϷͫsЈ2ͩ 1Ϫ ͪ ϩ␰2ϩy 2ͬ . ͑41͒ that is, that the radius of curvature of the centerline, R,is R much larger than the radius of the jet, h, which is equivalent ␬ Ӷ Thus, to jet 1/h. As previously mentioned, we constrain the jet cross section to remain a circle with a constant radius, h. ␭͑sЈ͒ ␭͑sЈ͒ ͵ dsЈ ϭ ͵ dsЈ We can write down the mapping, which is not one-to- ͉xϪx͑sЈ͉͒ ␰ 1/2 one, from local (␰,y,s) coordinates to Cartesian ͑x,y,z͒ coor- ͩ sЈ2ͩ 1Ϫ ͪ ϩr2 ͪ dinates. To determine this mapping, we first compute s in R ͑ ͒ terms of z along the centerline call this s0 , L ␰␭͑s͒ L Ϸ2␭͑s͒ ln ϩ ln , ͑42͒ z r R r ͑ ͒ϭ ͵ Јͱ ϩ Ј2 s0 z dz 1 X . 0 where L is the characteristic axial length scale ͑determined in

This function has an inverse, z0(s), which is the z-coordinate practice by the shape of a jet as it thins away from the of a point on the centerline at arclength s. We can then com- nozzle͒. The leading order term is familiar from the axisym- pute the desired mapping by following r along the curve to metric calculation, and in addition there is a nonaxisymmet- ␰ˆ ric correction due to the curvature of the centerline. Putting it z0(s), then following the normal vectors and yˆ by the proper amount, together, we now have that ␰␭ ␰ ͑␰ ͒ϭ ͑ ͑ ͒͒ϩ ϩ␰␰ˆ L ͑s͒ L P͑s͒ ,y,s r z0 s yyˆ ␾¯ ͑␰,y,s͒ϭ␾ϱϩ2␭͑s͒ln ϩ ln ϩ , ͑43͒ r R R r2 ␰ ϭz ͑s͒zˆϩXxˆϩyyˆϩ ͑ϪXЈzˆϩxˆ͒ where ␰ is the coordinate in the principal normal direction, 0 ͱ 2 1ϩXЈ and s is the arc length. With this equation for the field outside the jet, we can ␰XЈ ␰ ϭ ͩ ͑ ͒Ϫ ͪ ϩ ϩ ͩ ϩ ͪ compute the equation for the field inside the jet by demand- zˆ z0 s yyˆ xˆ X , ͱ1ϩXЈ2 ͱ1ϩXЈ2 ing that the electrical boundary conditions at the interface are satisfied, namely that the tangential component of the field is where X is understood everywhere to be evaluated at z0(s) and t. For small curvatures, this is approximately equal to continuous across the interface and the jump in the normal ⑀E is given by the free surface charge. The electric field ͑␰ ͒Ϸ ͑ Ϫ␰ Ј͒ϩ ϩ ͑ ϩ␰͒ ͑ ͒ ,y,s zˆ z X yyˆ xˆ X , 39 inside the jet is the sum of a piece directed along the princi- where X is evaluated at z and t. pal normal, i.e., along the direction of polarization, and a

Hence, in the absence of free charge, ␭ is of order 1/R2. Furthermore, ␭Ј is of order 1/R3 even with surface charge, ␴ since 0 is assumed constant in this derivation. Thus, if we are only interested in leading order terms in the centerline curvature, we can rewrite E1 as 2 2␲ h 1 ϭ ͩͫ ␰ˆϪ ␴ ͬϪ ␲ ␴ ͩ ͪͪ ͑ ͒ E Eϱ 2 ln 50 1 ␤ϩ2 • ¯⑀ D R 0 ␹

and P as

␤h2 4␲h2 ␤ h␴ 1 ϭ ␰ˆϩ ͩ ␴ Ϫ 0 ͩ ͪͪ P ͑␤ϩ ͒ Eϱ• ⑀͑␤ϩ ͒ D ln ␹ . FIG. 15. Picture of the cross section of a jet. tˆ points out of the plane of the 2 ¯ 2 2 R ␴ ␪ ͑ ͒ page. The asymmetric part of the surface charge density, D( ) 51 ϭ␴ ␪ ␴ Ͼ D cos ,isshownfor D 0. Physically, the field E1 perpendicular to the centerline of the jet is caused by the fact that the dipolar charge distribution piece directed along the tangent to the centerline, which we does not completely cancel out the applied field inside the jet ͑ ͒ denote by terms in parentheses , and a ‘‘fringe’’ field results from ␭ ␴ curving the line charge . For a perfect conductor with 0 ϭ ␰ˆϩ ˆ ͑ ͒ ϭ E E1 Ett. 44 0 the cancellation of E1 inside the jet leads to a simple ␲␴ ⑀ The free charge density is composed of both a monopole relation between the dipolar density and the field, 2 D /¯ ␴ϭ␴ ϩ␴ ␪ ␪ ϭEϱ ␰ˆ. and a dipole contribution 0(s) D(s)cos , where is • the angular coordinate around a given cross section of the jet ͑see Fig. 15͒. Applying the boundary conditions yields, after C. Forces and torques some algebra, Now that we understand the electrostatics, we can pro- 1 ceed with calculating the electrical stresses to get the equa- ͒ ␾¯ ϭ ˆϪ ͩ ͪ ␭Ј͑ ץϭ¯ ϭϪ Et Et s Eϱ•t 2ln ␹ s tions of motion of the centerline. We need to compute the moments (M), torques (N), and forces (K) on the jet. The 1 h͑s͒␭Ј͑s͒cos ␪ PЈ͑s͒cos ␪ internal moments are obtained by integrating the torque pro- Ϫlnͩ ͪ Ϫ , ␹ R h͑s͒ duced by the tangent–tangent component of the stress tensor, ͑45͒ ⑀ ϭϪ ϩ ␮ˆ ͑͑ˆ ͒ ͒ϩ ͑ 2Ϫ 2Ϫ 2͒ ͑ ͒ Ttt p 2 t• t•“ u ␲ Et E␰ Ey 52 1 ␭ P 8 ϭ ␰ˆϪ ͩ ͪ Ϫ E1 Eϱ ln ␹ 2 • R h over a cross section, S. The moment 2 2␲␴ 2 ␭ 1 ϭ ␰ˆϪ D Ϫ ͑ ͒ ͩ Eϱ ͪ lnͩ ͪ , 46 ϭ ͵͵ ͑ ͒ ␤ϩ2 • ¯⑀ ͑␤ϩ2͒ R ␹ M ␰ TttydA 53 S and ϭ ͑ ͒ ␤ ␲ 0 54 Pϭ h2E ϩ h2␴ . ͑47͒ 2 1 2¯⑀ D vanishes, since the stress tensor is even in y. The other moment, however, is nonzero. There are two contributions to There is also a correction to the equivalent line charge this moment, arising from electrical and viscous forces. After density ␭ from the curvature of the centerline. From the axi- some straightforward algebra, the electrical moment is symmetric calculation, we know ␤ 2␲h␴ 2 EϭϪ͵͵ E␰ ͑ ͒ h E ͒Јϩ . ͑48͒ M y Ttt dA 55͑ ץ ␭͑s͒ϭϪ 4 s t ¯⑀ S

If we assume that h is constant and that Et is given by the ⑀ 2 axisymmetric ͑␪ independent͒ part of Eq. ͑45͒, then we have ϭ PЈh Eϱ . ͑56͒ 16 t ␤ ␲ ␴ 2 h 0 E ϩ ץ␭͑s͒ϭϪ h2 4 s t ¯⑀ This moment arises from the fact that a polarized jet will try to align with the applied electric field. ␤ 1 2␲h␴ 2 0 A moment also arises from the viscous response of the Ϫ ͩ ͪ ␭Ј͑ ͒ͪ ϩ 43,44 ͩ ץ ϭϪ h Eϱ tˆ 2ln s 4 s • ␹ ¯⑀ fluid. Intuitively, in an elastic fiber, elastic forces resist the bending of the jet, and this resistance gives rise to a ␤ 1 1 2␲h␴ 40 ϭ 2ͩ Ϫ ␰ϩ ͩ ͪ ␭Љ͑ ͒ͪ ϩ 0 ͑ ͒ classical elastic bending moment, which is just the curva- h Eϱ ˆ 2ln s . 49 4 R • ␹ ¯⑀ ture of the fiber multiplied by its bending modulus. Viscosity

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plays an analogous role in a bent viscous jet; however in- 1 ⑀ KEϭ ͵ dCds͑C͒ͳ ͑͑E2ϪE2ϪE2͒nˆ ϩ2E ͑E ˆt stead of the moment being proportional to the curvature, it j ds 8␲ n t ␪ j n t j turns out to be proportional to the time derivative of the curvature.43 The reason for this is that the viscous stress is ϩ ␪ˆ ͒͒ proportional to the rate of strain, whereas in a solid the elas- E␪ j ʹ tic stress is proportional to the strain. Thus we anticipate that F F M ϰ␮k˙ ͑and M is the ﬂuid component of the moment͒. 2 y jet 4␲ h␴ 1 ¯⑀ EϱPЈ͑s͒ For this formula to be dimensionally correct, we must mul- ϭ␲ ␰ˆͫ Ϫ 0 ͩ ͪ ϩ ␴ Ϫ ͬ h ln 2Eϱ . ⑀ ␹ ␰ 0 ␲ tiply it by h4, yielding ¯ R 4 h ͑60͒

FϭϪ ␮ 4␬ ͑ ͒ M y A h ˙ jet , 57 Putting everything together, we obtain

ϭ Eϩ F where the sign reflects the fact that viscosity resists bending, K K K and the coefficient A must be calculated. It turns out that A 34 2 ϭϪ3/4␲. 4␲␴ h 1 ␥ ¯⑀␤ EϱPЈ ϭ␲ ␰ˆͫ Ϫ 0 ͩ ͪ ϩ ϩ ␴ Ϫ ͬ h ln 2Eϱ . The external torque N from surface forces is simpler to R¯⑀ ␹ R ␰ 0 4␲ h derive and involves only electrostatic effects. By symmetry ͑61͒ ϭ N␰ 0, so we must only compute Ny . In order to do this, we ϫ ͑ ͒ integrate r F across the boundary denoted by brackets of The first two terms represent line tensions ͑or compressions͒ the jet, where F is the external surface stress. Straightforward that are nonzero only when the jet is curved. Surface tension algebra yields pulls on the ends of our piece of jet, so it stabilizes the jet against curvature. The self-repulsion of charge, on the other hand, pushes on the ends of the jet, so it is destabilizing. The ϭϪ͵ ͑␪͒␰͗ E ˆ ͘ Ny dCds nˆ iTijt j third term is the force of the applied electric field normal to C the jet on the axisymmetric surface charge density ͑again, ␴ ͒ Eϱ 0h PЈ␴ only possible for curved jets . The final term is the force due Ϸ ␲ 2ͫ t ϩ 0Ϫ ␴ ͬ ͑ ͒ ds h Eϱ . 58 to the nonaxisymmetric portion of the electrostatic pressure, R h t D E2/8␲. This term tries to align the surface of the jet with the curved local field lines. This moment is entirely caused by the electrical tangential Combining these results gives the equation for the mo- ␴ stress Et . The first two terms are the moment due to the tion of the centerline. From torque balance we obtain the stress on the axisymmetric part of the charge density, which tension in the fiber to be produces a moment because the jet is curved. The third term ץϪ ץϭϪ is the moment due to the tangential stress on the asymmetric Fx sM y sNy portion of the charge density, on which the field pushes on one side of the jet and pulls on the other side. 3␲ ⑀ ͬ Ϫ ␮ 4␬ Ϫ Ј 2ͫ ץFinally we compute the external forces K on the jet. ϭ s h ˙ jet P h Eϱ Note that internal forces ͑e.g., from viscosity͒ are absorbed 4 16 t in the tension F. There are two external forces, surface ten- ␴ Eϱ 0h PЈ␴ sion and an electrical force. ϩ␲ 2ͫ t ϩ 0Ϫ ␴ ͬ ͑ ͒ h Eϱ D . 62 The surface tension force arises because when the jet is R h t bent, a segment of rest length ds has more surface area farther from the center of curvature of the jet. The surface ten- Using this in the force balance equation gives our final result sion force is therefore larger farther from the center of curvature, which tends to reduce the bending of the jet. A 3␲ ⑀ ͬ Ϫ ␮ 4␬ Ϫ Ј 2ͫ ץ␳␲ 2 ¨ ͑ ͒ϭ calculation gives the net force h X s ss h ˙ jet P h Eϱ 4 16 t ␴ Eϱ 0h PЈ␴ ͬ t ϩ 0Ϫ ␴ ͫ ץ2␲ h cos ␪ ϩ␲ 2 F h Eϱ K ϭϪ␥ ͵ d␪ͩ 1Ϫ ͪ ͑␰ˆ cos ␪ϩyˆ sin ␪͒ s R h t D 0 R ⑀␤ Ј ␲ ␥ ␲2 2␴2 ␲ 2 ¯ P Eϱ h 4 h 0 1 h Ϫ ϩ Ϫ lnͩ ͪ ϭ␰ˆ␥ . ͑59͒ 4 R ¯⑀R ␹ R ϩ ␲ ␴ ͑ ͒ 2 hEϱ␰ 0 . 63 The electrical force follows from integrating the unbal- anced electrical force over the interface. After some algebra, We now nondimensionalize according to the usual scheme, this gives with h scaled by r0 ,

3 ¯⑀͑␤ϩ1͒ Ϫ ␯*␬ Ϫ ⍀ Јͬ ͫ ץϭ͒ ͑ ¨ X s ss ˙ jet 0 P 4 16␲ͱ␤ t ⍀ ␴ 0t 0 ͬ ϩ¯⑀␴ PЈϪ⍀ ␴ ͫ ץϩ␤Ϫ͑1/2͒ s R 0 0t D

¯⑀ͱ␤⍀ ␲␴2 0t 1 4 0 1 Ϫ PЈϩ Ϫ lnͩ ͪ 4␲ R R ␹ ϩ ␤Ϫ͑1/2͒␴ ⍀ ͑ ͒ 2 0 0␰. 64

D. Conservation of charge ␴ To close the problem, we still need an equation for D . This equation comes from the general charge conservation ␪ FIG. 16. Comparison of our inviscid whipping mode formula ͑dashed͒ with equation for a stripe of charge of width hd and length ds at ͑ ͒ ϭ ⍀ Saville’s exact formula solid for K* 0.7 and for field strengths 0 an angle ␪ from the principal normal, ␰ˆ, ϭ2.2,4.4,9.8. ͑The maximum growth rate increases with the field strength.͒ Our asymptotic formula deviates from Saville’s when kϷ1, be- ͒ ͑ ␪͒ ␪␴͒ϭ ␪ ␪ ץ␪␴͒ϩ ץ t͑hd s͑v͑s, hd hd cos KE1 . 65 cause the approximations underlying it break down. Neglecting the advection velocity, after some algebra we obtain 2 4␲ h 1. Electric field without surface charge ͒ ͑ ͪͪ ␴ ϭ ͩ ␰ˆϪ ͩ ␴ ϩ ␴ ץ K Eϱ . 66 t D ␤ϩ2 • ¯⑀͑␤ϩ2͒ D R 0 We pause at this stage to study the dynamics when there is an external electric field but no surface charge. We first We can nondimensionalize this to obtain compare our formula to Saville’s formula33 for inviscid jets ␴ 2 4␲ͱ␤ h without surface charge. We set C and 0 to zero, and obtain ␴ ϭK*ͩ ⍀ Ϫ ͩ ␴ ϩ ␴ ͪ ͪ . ͑67͒ ץ t D ␤ϩ2 0␰ ␤ϩ2 D R 0 ␤ϩ ͒ 2 ␤⍀2 1 ͑ 1 k 0 ␻2ϭA␯*␻k4Ϫk2Ϫͩ Ϫ ͪ k2. ͑70͒ The two terms on the left-hand side are the convective de- 4␲ 8␲␤ ␤ϩ2 rivative of the ‘‘dipole’’ charge density, ␴ . The right-hand D In the limit of vanishing fluid viscosity this is side is the conduction current density due to the nonaxisym- ␤ ⍀2 2 ␤ϩ ͒ metric field within the jet. 0k ͑ 1 Ϫ␻2ϭk2ϩ Ϫ ⍀2k4. ͑71͒ 4␲ ␤ϩ2 8␲͑␤ϩ2͒ 0 E. Linear stability This is precisely the same as the long wavelength limit of It is now a simple matter to calculate the dispersion re- Saville’s33 formula for perfect dielectrics. lation for nonaxisymmetric modes. We describe the center- We can also compare this formula for perfect conduc- line in nondimensional coordinates by r(z,t)ϭzzˆ ␻ ϩ tors, i.e., in the infinitely conducting limit. In this limit, the ϩ␧e t ikzxˆ, and ␴ by ␴ (z,t)ϭCX(z,t), where C can in D D normal field is completely cancelled out inside the jet ͑i.e., general be complex. After much algebra we arrive at the E ϭ0͒ so dispersion relation 1 ⍀ ␴ ⍀ 0k 3 2 0 0 CϭϪi . ͑72͒ ␻2ϭϪ ␯*␻k4Ϫ4␲␴2k2 ln͑k͒Ϫik ␲ͱ␤ 4 0 ͱ␤ 2 This implies that without surface charge we have ⍀ ik 0 1 Ϫ ͑␴ k2ϩC͒Ϫk2ϩͩ ⍀ ͫϪ ␤ϩ ␤ϩ ͱ␤ 0 0 4␲ 2 2 Ϫ␻2ϭk2ϩ ⍀2k2Ϫ ⍀2k4, 4␲␤ 0 16␲␤ 0 ␤ϩ ͒ 2 ␴ ͱ␤ ͑ 1 k ik 0 ϩ ͬϩ ͪ ϫ ͓k2ͱ␤⍀ ϩik͑4␲C so there is no instability. This formula exactly agrees with 16␲␤ ␤ ␤ϩ2 0 Saville’s result33 in the long wavelength limit. For both per- ϩ ␲␤␴ 2 ͒ ͑ ͒ 2 0k 0 ͔, 68 fect dielectrics and perfect conductors, the electrical properties actually enhance the stabilization of the jet over the ef- with C given by fect of surface tension acting alone; this is analogous to the ⍀ ␲ͱ␤ axisymmetric instability described in the previous section. 2K* 0 4 K* ik ϩ ␴ k2 It should be remarked that these dispersion relations for ␤ϩ2 ␤ϩ2 0 CϭϪ . ͑69͒ both dielectric jets and perfectly conducting jets show that 4␲ͱ␤K* ϩ␻ the fiber responds exactly like a string under tension. The ␤ϩ2 velocity for waves on a perfectly conducting jet is

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The physical mechanisms behind these results can be uncovered by rewriting the dispersion relation in the limit of very large ͑but not inﬁnite conductivity͒. We neglect surface charge, and will work in dimensionful units in this subsec- tion to more clearly expose the mechanisms. We ﬁrst rewrite Eq. ͑67͒ as

Eϱ ␰ˆ ␤ϩ2 ␴ ϭ¯⑀ • Ϫ¯⑀ ␴˙ , ͑74͒ D 2␲ 4␲K D ␴ Ϫ1 and then iteratively solve for D to leading order in K . This gives ͑to leading order͒ ␰ˆ ͑␤ϩ ͒ ⑀ Eϱ• 2 ¯ Eϱ ␰ˆ͒. ͑75͒͑ ץ ⑀¯␴ ϭ¯⑀ Ϫ D 2␲ 8␲2 K t •

FIG. 17. The dependence of the growth rate on viscosity for Eϱ Using this approximation we can evaluate the forces and ϭ ϭ ␮ ␤ϭ ϭ ␥ ␳ϭ 1 kV/cm, K 0.01 S/cm, 80, r0 0.05 cm, and / 73. The upper- torques that act on the jet in the high conductivity limit, to most solid curve shows the dispersion relation for an inviscid ﬂuid, the expose the basic mechanisms. The acceleration of the center- middle curve has ␮ϭ0.01 P, and the lowermost curve has ␮ϭ1 P. Even a very small viscosity shifts the most unstable wavelength into the kϽ1 long line can be expanded in the form, wavelength regime, and greatly suppresses the growth rate. 1 1 1 76͒͑ , ץ ץ ϪA ץ ␳h2␲X¨ ϭA ϩA 1R 2 tR 3 t ssR

2 where the Ai are given below. If the signs of the Ai are ␥ ͑␤ϩ2͒Eϱ 2ϭͩ ϩ⑀ ͪ ͑ ͒ positive ͑as defined above͒ then each of the terms listed c ␳ ¯ ␲ . 73 r0 4 above are stabilizing. The term A1 corresponds to a tension, When the electrical conductivity is finite, the whipping and results in a wave-like response. The terms A2 and A3 of the jet becomes unstable; as in the case of the axisym- correspond to damping of this wave. The form of these co- metic modes, the instability is caused by an imbalance in the efficients in the limit of high conductivity and zero net sur- tangential stress on the interface, caused by the interaction of face charge density is the induced surface charge density and the tangential electric h2E2 ͑␤ϩ2͒ ϭ␲ ␥ϩ ϱ ͑ ͒ field. A1 h , 77 Figure 16 compares our predicted dispersion relation in 4 the low viscosity limit, with Saville’s corresponding formu- 2 Eϱ¯⑀͑␤ϩ1͒ ¯⑀ las. As expected, the formulas agree at low wave numbers; ϭϪ 2 ͑ ͒ A2 ␲ h , 78 the long wavelength theory deviates from the exact formulas 4 K when kϾ1. 3 ¯⑀ ¯⑀ 4 4 2 The reader may at this point be disturbed by the fact that A ϭ ␲␮h ϩ h Eϱ . ͑79͒ 3 4 32␲ K Fig. 16 predicts a short wavelength instability, which—if true experimentally—would invalidate the long wavelength as- The tension A1 is the same form as for a perfect conduc- sumption underlying this entire analysis. Indeed, Saville’s tor, and represents the enhancement of the tension in the jet analysis predicts that for a jet of water the wavelength of the due to the applied electric field. The leading order effect of instability is much smaller than the wavelength of the fiber. finite conductivity is given by A2 and the second term in A3 . However, we remind the reader that ͑a͒ Saville disregarded The A2 term is destabilizing and is the reason for the finite both viscosity and surface charge in his plotted growth conductivity instability shown in Fig. 16. This instability can rates;33 and ͑b͒ experiments29 show demonstrably in all cases be traced back to the external torque N on the jet due to the reported that the instability is long wavelength relative to the tangential electric stress. This instability is stabilized by the radius of the jet. two physical effects contained in A3 ; these correspond to the Figure 17 shows the influence of viscosity on a finite viscous moment, and ͑part of the͒ the electrical moment. conductivity jet without surface charge. Even a very small When the viscous part dominates the maximum growth rate viscosity shifts the most unstable wave number into the k of the instability, Ͻ1 long wavelength regime; also, a tenfold increase in the Ϫ ␻ ϳh2E2 ͑␮K͒ 1. viscosity decreases the growth rate by a factor of 10. This max ϱ demonstrates that the prediction of a short wavelength insta- This explains the trends shown in Fig. 17. Also note that at bility is corrected when viscosity is included. The attentive fixed field strength, the growth rate of the instability decays reader will at this point, note, however, the existence of an- with increasing conductivity, while the wave number remains other problem, namely, the growth rate for the whipping the same. mode for water is so suppressed that now the axisymmetric When the viscous moment is smaller than the electrical ͑ ␮Ͻ⑀2 2 ␲2 ͒ mode is predicted to go unstable before the whipping mode, moment cf. the two terms in A3 , ¯ Eϱ/(24K ) the contradicting experiments. maximum growth rate occurs when kϾ1, outside the range

ϭ0.01, Kϭ0.01 ␮S͒ in a 1 kV/cm external field. Upon increasing the dimensionless surface charge from 0 to 0.1, the most unstable wave number increases from kϷ0.1 to k ϭ0.8, and moreover the growth rate of the instability increases by about four orders of magnitude! To develop a physical understanding of this result, we follow the same procedure as above for the electrical case: The equation for the force now takes the form ͑to leading order͒, 1 1 1 1 ץ ϩA ץ ϩB ץ ␳␲h2X¨ ϭ͑A ϩB ͒ ϩB 1 1 R 2 2R 3 ssR 2 tR 1 ϩ2␲h␴ Eϱ ␰, ͑81͒ ץ ץ͒ Ϫ͑A ϩB 3 4 ss tR 0 • FIG. 18. The dependence of the growth rate on surface charge for a r0 ϭ100 ␮m water jet (Kϭ0.01 ␮S/cm, ␤ϭ80, ␯ϭ0.01͒ in an electric field where the Ai are the electrical coefficients, and the Bi are the ϭ Eϱ 1 kV/cm. The lowermost solid curve shows the dispersion relation for modifications due to surface charge. Note the equation also ␴ ϭ an uncharged fluid, the middle curve has 0 0.01 and the solid curve has contains a term that does not depend directly on the curva- ␴ ϭ0.1. 0 ture, and is the force the electric field exerts on the surface charge. The form of the Bi’s are summarized below: of the long wavelength theory. For a high field of 10 kV/cm ␲2 2␴2 4 h 0 with a modest conductivity 1 ␮S/cm, the critical viscosity B ϭ log kϪ1, ͑82͒ Ϫ 1 ⑀ 10 3 cm2/s. Hence even water is sufficiently viscous to ¯ damp the instability! This is the essential reason why our 2 Ϫ1 B ϭ␲h ͑␤ϩ1͒Eϱ␴ h log k , ͑83͒ results differ so markedly from Saville’s analysis in the limit 2 0 of vanishing viscosity. When viscosity does not damp the ␴2␲2 4 0 h instability, there is a breakdown of the long wavelength B ϭϪh log kϪ1, ͑84͒ 3 ¯⑀ theory, because the instability is peaked at high k. This represents a real physical effect: the jet will try to bend on a ␲h4␴2h ϭϪ 0 Ϫ1 ͑ ͒ wavelength smaller than its radius, which could correspond B4 log k . 85 to forming a kink. Note that for fixed fluid viscosity, there is K a critical electric field depending on viscosity and conductiv- The contribution to the tension (B1) is negative and repre- ity above which this will occur. sents the self-repulsion effect denoted above. This instability 2. Surface charge is stabilized by B3 . Both the term B2 and the term propor- ␰ tional to Eϱ• represents the interaction between the static We will now study the effect of surface charge on the jet charge density and the static electric field. Both of these stability. First, we remark that our formulas reduce to those forces are out of phase with the other terms. By themselves, found previously for perfectly conducting charged cylinders these represent an oscillatory instability. Finally, the term B4 in no external electric field. For example, in the limit of high competes with the viscous stabilization discussed above. viscosity, our formula reduces to The consequences of including a static charge density 4 1 are thus seen to be the following: ͑a͒ if the charge density is ␻ϭϪ ͑1ϩ4␲␴2 log k͒ ͑80͒ 3␯* k2 0 large enough that B1 exceeds A1 , the jet is unstable. The growth rate of this instability is not small even when the 34,35 as found previously by Saville and Huebner. Surface conductivity is very large, in marked contrast to the case charge destabilizes the jet at small wave numbers, due to described above where there is no free charge. The instability plain old charge repulsion. The divergence represented in can therefore be much more violent when significant free this formula as k→0 is unphysical and reflects the fact that charge is present. ͑b͒ Surface charge fights against viscous this balance is asymptotically inconsistent; at long wave- stabilization. If the surface charge is large enough the insta- lengths, inertia is always important. For small surface charge bility will be peaked at high k, which signals the breakdown densities, the jet is a string with a tension ͑in dimensional of the long wavelength theory ͑and a possible kinking of the units͒ jet͒. ͑c͒ If the normal electric field produced by the surface ␲2 2␴2 charge is of order the tangential field, the situation is com- 4 h 0 ␲␥hϩ log͑k͒. plicated. Not only are all of the aforementioned effects act- ¯⑀ ing, but there are also effects coupling the static surface ͑ When the surface charge density beats surface tension this charge to the tangential electric field term B2 , and the nor- results in an instability. mal force term͒. Both of these effects are out of phase with Figure 18 shows the effect of surface charge on the axi- the others, and could themselves dominate if conditions were symmetric instability for a 100 ␮m water jet ͑␤ϭ80, ␯ right.

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FIG. 19. Comparison of the axisymmetric mode ͑solid line͒ to the whipping mode ͑dashed line͒, for the conditions of Taylor’s 1969 experiment except the jet has no surface charge, for two different ﬁeld strengths, 1 kV/cm and 5 kV/cm ͑right͒. At 1 kV/cm both the Rayleigh and the axisymmetric conducting mode are stronger than the whipping mode. At 5 kV/cm the Rayleigh mode is suppressed but the axisymmetric conducting mode still dominates the whipping mode. This contradicts the primary observation from the experiment, where the whipping mode is observed to strongly dominate.

IV. COMPETITION BETWEEN WHIPPING MODE AND conducting instability that occurs strongly depends on the AXISYMMETRIC MODE fluid parameters of the jet ͑viscosity, dielectric constant, con- ͒ To summarize the results, we have identified three dif- ductivity . ferent instability modes of the jet: ͑1͒ the Rayleigh mode, In this section we analyze the competition between the which is the axisymmetric extension of the classical Ray- axisymmetric and the whipping conducting modes as a func- leigh instability when electrical effects are important; ͑2͒ the tion of parameters and determine where in parameter space axisymmetric conducting mode, and ͑3͒ the whipping con- each of them dominates. We will see that the dominant in- ducting mode. ͑The latter are dubbed ‘‘conducting modes’’ stability depends strongly on the static charge density of the because they only exist when the conductivity of the fluid is jet. In the next paper we will determine what sets this charge finite.͒ Whereas the classical Rayleigh instability is sup- density. Finally, we will put this all together in a way that is pressed with increasing field and surface charge density, the useful for deducing the implications for experiments. conducting modes are enhanced. For this reason, at an appre- To begin, we consider the competition between the vari- 29 ciable field strength characteristic of experiments, the con- ous modes for Taylor’s 1969 experiment, which prompted ducting modes dominate. The character of the whipping con- Saville’s original calculations. Taylor considered a jet of wa- ducting mode depends strongly on whether the local electric ter emanating from a nozzle of 0.053 cm, under a range of field near the jet is dominated by its own static charge den- electric fields from 1 kV/cm to ϳ5 kV/cm. Saville’s analysis ␴ sity ( 0), or the tangential electric field Eϱ . In the latter demonstrated that in the limit of zero viscosity and no sur- case, for a high conductivity fluid the instability is fairly face charge density at sufficiently high fields the oscillatory suppressed ͑the growth rate is O(KϪ1)͒, whereas in the whipping mode is more unstable than the axisymmetric in- former it can be quite large. In general, the nature of the stability; this result was consistent with Taylor’s experiment,

FIG. 20. Comparison of the axisymmetric modes ͑solid͒ to the whipping mode ͑dashed͒, for the same conditions as Fig. 19, except the jet has a small surface ␴ ϭ charge 0 0.1. At low ﬁelds, the axisymmetric modes still dominate the whipping mode; however at 5 kV/cm the whipping mode is stronger than both of the axisymmetric modes.

Downloaded 17 Jul 2001 to 140.247.53.216. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. I. 2219 in that Taylor also observed that at fields above approximately 2 kV/cm the jet began to whip. Figure 19 compares the two modes for a range of different field strengths; this figure is the same as Saville’s comparison ͑i.e., surface charge is neglected͒, except that we are including the correct viscous effects as well. Interestingly, we find that when viscosity is included, the ordering of the modes is reversed: namely, the analysis predicts that the axisymmetric mode is much more unstable than the whipping mode; at an electric field of 5 kV/cm, the maximum growth rate of the whipping mode is four orders of magnitude lower than that for the axisymmetric mode! Based on the scaling arguments presented above, the reason for this is clear. Including viscosity has two principal effects. First, it decreases the most unstable wavelength for the whipping mode to low wave numbers. Second, it switches the relative domi- nance of the axisymmetric and whipping modes. The first FIG. 21. Contour plot of the logarithm ͑base ten͒ of the ratio of the growth effect is consistent with the experiments, which observe a rate of the most unstable whipping mode to the most unstable varicose long wavelength instability. The second effect, however, con- mode, for a polyetheleneoxide–water mixture ͑␯ϭ16.7 cm2/s and K tradicts experiments which clearly demonstrate a pronounced ϭ120 ␮S/cm͒. The ratio is cut off at 2 and 1/2 so that the detail of the whipping instability. transition can be seen. In all cases, the axisymmetric instability prevails at low charge density and the whipping dominates at high charge density. The The only ingredient available to resolve this conundrum solid line shows the borderline above, and to the right, of which the system is surface charge density. We now plot the same set of dis- is unstable to the Rayleigh whipping instability. The dashed line shows the persion relations, but this time with an experimentally rea- borderline below and to the right of which the Rayleigh varicose instability sonable surface charge included. The surface charge corre- is not suppressed. sponds to that which would be present if there were a current ͑ ͒ of a few nanoamperes passing through the jet Fig. 20 . modes as a function of ␴ and h. We saturate this ratio at 2 At low fields, both the axisymmetric modes are stronger and 1/2 so that the detail of the transition can be seen. In the than the whipping mode; at the highest field 5 kV/cm the light regions, a whipping mode is twice or more times as whipping mode is stronger than both of the axisymmetric unstable as all axisymmetric modes; in the dark regions, the modes! The reason for this is that the electric field normal to axisymmetric is more unstable. As anticipated from the ar- the jet produced by the surface charge is actually slightly guments above, the whipping modes are more unstable for larger than the tangential electric field. For reasons that have ͑ ͒ 21 higher charge densities to the right , whereas the axisym- been previously discussed in the electrospraying literature metric modes dominate in the low charge density region ͑to and will be discussed in the second paper of this series, a the left͒. small current passing through the jet results in a rather large The solid line in the figure shows the Rayleigh charge surface charge. Hence, when both surface charge and viscos- ϭ threshold for the whipping mode, when A1 B1 . To the left ity are included, the stability analysis is qualitatively consis- of the line, charge repulsion beats surface tension and thus tent with the experiments. To achieve quantitative agreement, causes instability. As seen in the figure, the whipping modes it is necessary to determine quantitatively the surface charge becomes dominant when the charge density is about an order along the jet. of magnitude smaller than this threshold, because of the the Before closing this paper, we first summarize the com- interaction of the static charge density with the external elec- petition between the axisymmetric and whipping modes tric field. The solid dashed line shows the curve above where through a phase diagram. This will be useful for interpreting ␲␴2 the axisymmetric Rayleigh instability is suppressed 2 0 experiments. Since both the surface charge density and the ϩ ␲ Ϫ1⍀2ϭ (4 ) 0 1/2. Note that the axisymmetric Rayleigh in- jet radius vary away from the nozzle, the stability character- stability mode is operative at low radii and low charge den- ͓ istics of the jet will change as the jet thins. The electric field sities, while the whipping Rayleigh charge instability domi- also varies along the jet; however, in the plate–plate experi- nates at high radii and high charge densities. The combining ͑ ͒ mental geometry that we use Fig. 1 the field everywhere of these two effects explains why whipping dominates at along the jet is dominated by the externally applied field high charge density in high conductivity fluids. The growth produced by the capacitor plates, so it is safe to neglect the rate of the conducting modes vanishes with increasing con- field’s variance. To apply the present results to experiments ductivity and thus when one of the classical ͑conductivity in the point–plate geometry, it would be necessary to include independent͒ Rayleigh modes is present, it dominates. the variation of the field along the jet as well.͔ Figure 21 shows an illustrative result for a fluid with viscosity 16.7 P, V. CONCLUSIONS conductivity 120 ␮S/cm in an external electric field 2 kV/cm. The phase diagram shows the logarithm of the ratio The main task of this paper has been performing a com- of the maximum growth rate of the whipping conducting plete stability analysis of a charged fluid jet in a tangential mode to the maximum growth rate of all axisymmetric electric field as a function of all fluid parameters. The stabil-

Downloaded 17 Jul 2001 to 140.247.53.216. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp 2220 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al. ity analysis relied on an asymptotic expansion of the equa- 15M. Cloupeau and B. Prunet-Foch, ‘‘Electrostatic spraying of liquids in tions of electrohydrodynamics in powers of the aspect ratio cone-jet mode,’’ J. Electrost. 22, 135 ͑1989͒. 16 of the perturbations, which were assumed to be small. The M. Cloupeau and B. Prunet-Foch, ‘‘Electrohydrodynamic spraying func- tioning modes: A critical review,’’ J. Aerosol Sci. 25, 1021 ͑1994͒. analysis extends previous works to the parameter regimes 17A. Jaworek and A. Krupa, ‘‘Classification of the modes of end spraying,’’ important for experiments. The second paper in this series J. Aerosol Sci. 30,873͑1999͒. uses the stability analysis, in conjunction with an analysis of 18J. R. Melcher and E. P. 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