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Electrospinning and electrically forced jets. II. Applications 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. Brenner a) Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ~Received 15 February 2000; accepted 10 May 2001! Electrospinning is a process in which solid fibers are produced from a polymeric fluid stream ~solution or melt! delivered through a millimeter-scale nozzle. This article uses the stability theory described in the previous article to develop a quantitative method for predicting when electrospinning occurs. First a method for calculating the shape and charge density of a steady jet as it thins from the nozzle is presented and is shown to capture quantitative features of the experiments. Then, this information is combined with the stability analysis to predict scaling laws for the jet behavior and to produce operating diagrams for when electrospinning occurs, both as a function of experimental parameters. Predictions for how the regime of electrospinning changes as a function of the fluid conductivity and viscosity are presented. © 2001 American Institute of Physics. @DOI: 10.1063/1.1384013# I. INTRODUCTION In the previous article 1 we developed a theory for the stability of electrically forced jets. The theory predicts the growth rates for two types of instability modes of a charged jet in an external electric field as a function of all fluid and electrical parameters, generalizing previous works 2–6 to the regime of electrospinning experiments. The first type of mode considered is a varicose instability, in which the cen- terline of the jet remains straight but the radius of the jet is modulated; the second type is a whipping instability, in which the radius of the jet is constant but the centerline is modulated. It was demonstrated that there are three different modes which are unstable: ~1! the Rayleigh mode, which is the axisymmetric extension of the classical Rayleigh insta- bility when electrical effects are important; ~2! the axisym- metric conducting mode; and ~3! the whipping conducting mode. ~The latter are dubbed ‘‘conducting modes’’ because they only exist when the conductivity of the fluid is finite.! Whereas the classical Rayleigh instability is suppressed with increasing applied electric field or surface charge density, the conducting modes are enhanced. The dominant instability strongly depends on the fluid parameters of the jet ~viscosity, dielectric constant, conductivity! and also the static charge density on the jet. In particular, for a high conductivity fluid, when there is no static charge density on the jet, the varicose mode dominates the whipping mode. When there is a large static charge density, the whipping mode tends to dominate; the reason for this is that a high surface charge simulta- neously suppresses the axisymmetric Rayleigh mode and en- hances the whipping mode. The aim of this article is to apply the results of the sta- bility analysis to electrospinning and ~in less detail! to elec- trospraying. Electrospinning is a process in which nonwoven fabrics with nanometer scale fibers are produced by pushing a polymeric fluid through a ~millimeter-sized! nozzle in an external electric field. 7–13 A schematic of a typical apparatus is shown in Fig. 1. In the conventional view, 7 electrostatic charging of the fluid at the tip of the nozzle results in the formation of the well-known Taylor cone, from the apex of which a single fluid jet is ejected. As the jet accelerates and thins in the electric field, radial charge repulsion results in splitting of the primary jet into multiple filaments, in a pro- cess known as ‘‘splaying.’’ 7 In this view the final fiber size is determined primarily by the number of subsidiary jets formed. We argue, by comparing theory and experiments, that the onset threshold for electrospinning as a function of applied field and flow rate quantitatively corresponds to the excita- tion of the whipping conducting mode. 14 This rigorously es- tablishes that the essential mechanical mechanisms of elec- trospinning are those of the whipping mode. The idea is that the whipping of the jet is so rapid under normal conditions that a long exposure ~.1 ms! photograph gives the envelope of the jet the appearance of splaying subfilaments. In order to achieve the quantitative comparisons that es- tablish this conclusion, it is necessary to extend the theoret- ical discussion of Part I 1 to apply to electrospinning a! Author to whom all correspondence should be addressed. Present address: Division of Engineering and Applied Sciences, Harvard University, Cam- bridge, Massachusetts 02138. Electronic mail: [email protected] PHYSICS OF FLUIDS VOLUME 13, NUMBER 8 AUGUST 2001 2221 1070-6631/2001/13(8)/2221/16/$18.00 © 2001 American Institute of Physics Downloaded 07 Jun 2004 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

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PHYSICS OF FLUIDS VOLUME 13, NUMBER 8 AUGUST 2001

Electrospinning and electrically forced jets. II. ApplicationsMoses M. HohmanThe James Franck Institute, University of Chicago, Chicago, Illinois 60637

Michael ShinDepartment of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139

Gregory RutledgeDepartment 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 2001!

Electrospinning is a process in which solid fibers are produced from a polymeric fluid stream~solution or melt! delivered through a millimeter-scale nozzle. This article uses the stability theorydescribed in the previous article to develop a quantitative method for predicting whenelectrospinning occurs. First a method for calculating the shape and charge density of a steady jetas it thins from the nozzle is presented and is shown to capture quantitative features of theexperiments. Then, this information is combined with the stability analysis to predict scaling lawsfor the jet behavior and to produce operating diagrams for when electrospinning occurs, both as afunction of experimental parameters. Predictions for how the regime of electrospinning changes asa function of the fluid conductivity and viscosity are presented. ©2001 American Institute ofPhysics. @DOI: 10.1063/1.1384013#

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I. INTRODUCTION

In the previous article1 we developed a theory for thstability of electrically forced jets. The theory predicts tgrowth rates for two types of instability modes of a chargjet in an external electric field as a function of all fluid anelectrical parameters, generalizing previous works2–6 to theregime of electrospinning experiments. The first typemode considered is a varicose instability, in which the cterline of the jet remains straight but the radius of the jemodulated; the second type is a whipping instability,which the radius of the jet is constant but the centerlinemodulated. It was demonstrated that there are three diffemodes which are unstable:~1! the Rayleigh mode, which isthe axisymmetric extension of the classical Rayleigh insbility when electrical effects are important;~2! the axisym-metric conducting mode; and ~3! the whipping conductingmode. ~The latter are dubbed ‘‘conducting modes’’ becauthey only exist when the conductivity of the fluid is finite!Whereas the classical Rayleigh instability is suppressedincreasing applied electric field or surface charge density,conducting modes are enhanced. The dominant instabstrongly depends on the fluid parameters of the jet~viscosity,dielectric constant, conductivity! and also the static chargdensity on the jet. In particular, for a high conductivity fluiwhen there is no static charge density on the jet, the varicmode dominates the whipping mode. When there is a la

a!Author to whom all correspondence should be addressed. Present adDivision of Engineering and Applied Sciences, Harvard University, Cabridge, Massachusetts 02138. Electronic mail: [email protected]

2221070-6631/2001/13(8)/2221/16/$18.00

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static charge density, the whipping mode tends to dominthe reason for this is that a high surface charge simuneously suppresses the axisymmetric Rayleigh mode andhances the whipping mode.

The aim of this article is to apply the results of the sbility analysis to electrospinning and~in less detail! to elec-trospraying. Electrospinning is a process in which nonwovfabrics with nanometer scale fibers are produced by pusha polymeric fluid through a~millimeter-sized! nozzle in anexternal electric field.7–13A schematic of a typical apparatuis shown in Fig. 1. In the conventional view,7 electrostaticcharging of the fluid at the tip of the nozzle results in tformation of the well-known Taylor cone, from the apexwhich a single fluid jet is ejected. As the jet accelerates athins in the electric field, radial charge repulsion resultssplitting of the primary jet into multiple filaments, in a process known as ‘‘splaying.’’7 In this view the final fiber size isdetermined primarily by the number of subsidiary jeformed.

We argue, by comparing theory and experiments, thatonset threshold for electrospinning as a function of applfield and flow rate quantitatively corresponds to the exction of the whipping conducting mode.14 This rigorously es-tablishes that the essential mechanical mechanisms of etrospinning are those of the whipping mode. The idea is tthe whipping of the jet is so rapid under normal conditiothat a long exposure~.1 ms! photograph gives the envelopof the jet the appearance of splaying subfilaments.

In order to achieve the quantitative comparisons thattablish this conclusion, it is necessary to extend the theoical discussion of Part I1 to apply to electrospinning

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1 © 2001 American Institute of Physics

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2222 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al.

experiments, where both the jet radius and charge denchange away from the nozzle. To this end, this articleorganized as follows. First, we review the first articlediscussing phase diagrams, a way of summarizing the stity calculations that facilitates comparison with experimenTo make use of the phase diagrams, it is necessary to cpute the steady-state shape and charge density of the jefunction of the axial coordinate,z, a task turned to in Sec. IIIOur discussion builds on previous work in electrosprayfor the shape of the steady jet in an electrospray,15–19 butdiffers from the previous work in several important respethat were necessary to understand our experiments. Atclose of Sec. III we use this information to present scallaws for the dependence of the maximum growth rate, whping frequency, and wave numbers on the imposed elecfield and fluid parameters. Section IV shows how the cobined information can be used to rationalize a set of obvations from high-speed movies for the destabilization ofelectrospinning jet. Finally, Sec. V presents operating dgrams. These diagrams summarize the behavior of the~stable, dripping or whipping! as a function of applied electric field and volumetric flow rate. We argue that experimetal measurements of the onset of electrospinning correspquantitatively to the onset of the whipping conducting mothereby establishing this mode as the essential mechaniselectrospinning. We also study how these results depenfluid viscosity and conductivity. In the limit of low viscosityand high conductivity, the operating diagrams qualitativresemble those measured by Cloupeau and Prunet-Foch20 instudies of electrospraying.

II. STABILITY ANALYSIS

A. Phase diagrams

First, we summarize the results of the stability analypresented in Part I. Since both the surface charge densitythe jet radius vary away from the nozzle, the stability chacteristics of the jet will change as the jet thins. To captthis, we plot phase diagrams as a function of the mostportant parameters that vary along the jet: the surface chas, and the jet radius,h. @As we will see in the next sectionthe electric field also varies along the jet; however, inplate–plate experimental geometry that we use~Fig. 1! thefield everywhere along the jet is dominated by the extern

FIG. 1. The plate–plate experimental geometry.

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applied field produced by the capacitor plates, so it is safneglect the field’s variance. To apply the present resultsexperiments in the point–plate geometry, it would be necsary to include the variation of the field along the jet as we#Figures 2 and 3 show examples of phase diagrams, wgive the logarithm of the ratio of the maximum growth raof the whipping conducting mode to the maximum growrate of all axisymmetric modes as a function ofs andh forvarious external field strengthsE` . We saturate this ratio a2 and 1

2 so that the detail of the transition can be seen. Inwhite regions, a whipping mode is twice or more timesunstable as all varicose modes; in the black regions, the vcose mode is more unstable. Generically, the whippmodes are more unstable for higher charge densities~to theright!, whereas the axisymmetric modes dominate in the locharge-density region~to the left!.

Phase diagrams are shown for three different fluids: F2 shows a comparison between glycerol~high viscosity,n514.9 cm2/s, and low conductivity,K50.01mS/cm! and anaqueous solution of polyethylene oxide~PEO; high viscosity,n516.7 cm2/s, and high conductivity, K5120mS/cm!seeded with KBr. Figure 3 shows a comparison betweglycerol withK50.58mS/cm and water (n50.01 cm2/s) forthe same conductivity. For the two high-viscosity fluids, tstructure of the diagrams is qualitatively similar over tentire range of electric field strengths and for different coductivities. ~However, it should be noted that the absolumagnitude of the growth rates of both the axisymmetric awhipping modes increases with the electric field strengHence, at higher field strengths the jet is more unstablethe oscillation frequency is greater.! The phase diagram fowater differs quantitatively from those for glycerol at higfield .2 kV: when the jet radius is larger than 1022 cm, forall charge densities the whipping instability strongly domnates the axisymmetric instability. Thick water jets whmore strongly than they undergo axisymmetric instability.

B. Jet paths

The stability of the jet not only depends on the structuof the phase diagram, but also the path the jet follows indiagram. This is determined by the„h(z),s(z)… profiles,summarizing the properties of the jet at distancez from thenozzle. The types of instabilities that the jet undergoes~ei-ther whipping or breaking! strongly depend on this path, anthis path through the diagram strongly depends on bothexternal field strength and the conductivity of the fluid.

To proceed further, we therefore need a theory forjet’s shape and charge density as it thins away fromnozzle. Such theories have been described extensively inelectrospray literature;21,16–19in the next section, we reviewthis work and present the modifications that we found nessary to obtain quantitative agreement with our own expments. To anticipate the results, Ganan-Calvo22 and Kir-ichenko et al.23 independently demonstrated that when tjet is sufficiently thin, the conduction current becomes neligible and a very simple formula applies: the surface chais linearly proportional to the radius,s5Ih/(2Q) in physi-cal units. This implies that theultimatepath of a jet through

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2223Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. II.

FIG. 2. Contour plots of the logarithm~base ten! of the ratio of the growth rate of the most unstable whipping mode to the most unstable varicosecomparing glycerol~n514.9 cm2/s, K50.01mS/cm! and PEO~n516.7 cm2/s andK5120mS/cm!. The ratio is cut off at 2 and

12 so that the detail of the

transition can be seen. In all cases, axisymmetric modes prevail at low charge density and whipping modes at high charge density. An experimentn theasymptotic regime will follow a straight line of unit slope through these log–log domain plots.

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the phase diagrams is a straight line with unit slope. Tintercept of the line is determined by the ratio of current aflow rate:

log10h5 log10s1 log10 S 2Q

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Note that the asymptotic path depends strongly on the r2Q/I . When the current is higher~flow rate lower! the inter-cept of the line decreases. Qualitatively, this implies thatjet has to spend more time in a regime where the whippinstability dominates, before approaching the ultima

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axisymmetric-instability-dominated regime. A crucial quetion is, therefore: what is the currentI that flows through the

jet? We will see below that the current is a globally definquantity, depending in a nontrivial way on the jet shape aelectric field strengths. Experiments reveal that the currincreases strongly with the conductivityK, the imposed elec-tric field E` , and the flow rateQ. The paths swept by fluidsof different conductivities therefore vary substantially. Ttypical currents for low-conductivity glycerol and highconductivity PEO differ by three orders of magnitude~0–200 nA and 0–200mA, respectively!; hence, the paths the

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2224 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al.

FIG. 3. Similar contour plot to Fig. 2 comparing glycerol and water (n50.01 cm2/s), both withK50.58mS/cm.

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It will also be important to understand what the jet dobefore reaching this asymptotic regime. We will see innext section that nearer to the nozzle, before the jet reaits asymptotically thinning state, some of the current is cried by conduction. In this regions will be smaller than theasymptotic formula predicts, and the path will describecurve that lies above the asymptotic line. How far aboveline it curves greatly influences the stability properties ofjet.

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III. DETERMINATION OF THE JET SHAPE ANDSURFACE CHARGE DENSITY

The conclusion of the stability analysis is that the wayelectrically forced jet destabilizes~at high enough fields sothat the classical Rayleigh mode is suppressed! depends criti-cally on both the surface charge density and the radius ofjet. We can calculate these quantities as a function ofz ~theaxial coordinate! using the conservation of mass, charge, aforce-balance~Navier–Stokes! equations presented in thprevious article,1 together with a way of calculating the electric field strength. A more detailed discussion and derivat

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2225Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. II.

FIG. 4. Plots of the measured current as a functionexternal field, for several different flow rates~measuredin ml/min!. The upper plot shows the measured currefor a glycerol jet (K50.01mS/cm) and the lower plotsshow currents for a jet of an aqueous solutionPEO (K5120mS/cm). Data for three different vol-ume flow rates are shown for each jet. The currentcreases monotonically with applied field and applieflow rate.

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of these equations is presented in that article, as welcone-jet electrospraying studies where similar equatihave been applied.16–19,24,25We will express the equations idimensionless form: The equations are nondimensionalby choosing a length scaler 0 ~determined by the nozzle radius!, a time scalet05Arr 0

3/g, an electric field strengthE05Ag/(e2 e)r 0 and a surface charge densitys0

5Age/r 0, where hereg is the fluid surface tension,e and eare the dielectric constants of fluid and air, respectively,r is the fluid density. Denoting the radius of the jet ash(z)and the fluid velocity parallel to the centerlinev(z), thenconservation of mass gives

h2v5Q, ~1!

whereQ is the volume flow rate.Letting the surface charge bes(z), conservation of

charge gives

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of the fluid ~with b5e/ e21!, and I is the current passingthrough the jet. Force balance gives

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eration and the dimensionless viscosity isn* 5Aln /r 0 withthe viscous scale ln5n2/(rg). In these equationsQ is re-lated to the experimental flow rate,Qexp, measured in cm3/s,and I is related to the physical current measured in esu/follows:

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To these equations we must add an equation for the etric field. We previously1 introduced asymptotic equationfor the field that were valid in the slender jet. These eqtions will not be useful for our present purposes becausethe difficulty in specifying the physical boundary conditionin the approximate equations. Instead we will useasymptotic version of Coulomb’s law, specified later.

In this section we will construct solutions of these equtions and quantitatively compare them to the jet profilesour electrospinning experiments. We will then use theselutions in the subsequent sections to assess the stabilitthe jet solution. There have been several groups16,17 whohave previously compared solutions of equations similarthe above with experimental profiles of jets for the purpoof developing an understanding of current–voltage relatiin electrospraying. Ganan-Calvo has also determined theface charge along the jet in an electrospray study, by msuring the jet profile and current in an experiment, and thusing equations similar to those written above to deducesurface charge density.26 The approach that we follow herdiffers from these studies in several important respewhich were necessary to understand our electrospinningperiments.

These steady-state equations give solutions as a funcof three parameters: the imposed flow rateQ, currentI, andthe voltage drop between the plates. However, our elecspinning experiments have onlytwo independent parametersgiven an imposed flow rate and field strength, the currendetermined dynamically. As an illustration, Fig. 4 showmeasurements of the current passing through a jet as a ftion of both applied field and flow rate.

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2226 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al.

Two sets of experiments are shown: a glycerol jet, witconductivity K50.01mS/cm, and a PEO/water jet, withconductivity of 120mS/cm. In both cases, the current icreases monotonically with the electric field strength awith the volume flow rate passing through the jet. Thereapproximately a 100-fold difference between the currepassing through the two jets. Note that the current–voltcharacteristic is roughly linear, except at higher voltagesthe more conducting fluid, where it increases at a faster rThe current is also a roughly linear function of the flow raThe prediction of this current should follow from the theorIt is clear that the above equations are not capable of dothis without including additional physics. It should be rmarked that theI -V characteristics in our electrospinninexperiments disagree with those usually studied in elecspraying, where the current is both independent of the vage~e.g., Ref. 17!, and scales likeI;AQ.

For the mathematical problem, the determination ofcurrent is equivalent to specifying boundary conditionsthe equations. The current at the nozzle isI 52ph(0)s(0)1Kph(0)2E(0). If thecharge density at the nozzle is spefied, then the electric field can be determined everywherethe current is fixed. Hence, determining the current is equlent to specifying the charge density at the nozzle. Sincecharge density is fixed by the mechanisms of charge traport down the capillary to the nozzle, we believe this isphysically meaningful way of posing this problem.

Many electrospraying studies have focused on undstanding ‘‘universal’’ current–voltage relationships, whicwere first explored by de la Mora.27,21 In these studies, electrospraying from an isolated needle at high potential relato ground tends to produce a current that is independenthe voltage. To rationalize this, Ganan-Calvo aothers16–19,22have proposed that the current is determineda matching condition of the jet onto a perfectly conductinozzle. Namely, the boundary condition ons discussedabove is that the jet shape is matched to a perfectly conding Taylor cone. Ganan-Calvo showed that this leads tcurrent that independent of the voltage, which is quanttively in good agreement with experiments.17,22

We were not able to utilize these results for the presstudy because our electrospinning experiments do not shcurrent independent of the voltage drop between the captor plates. In fact, an experimental study of the factorssides the voltage drop setting the current~and hence theshape and charge density on an electrospinning jet! revealedthat detailed factors~such as the shape and material propties of the nozzle! had a strong influence on the jet behaviWe therefore needed to develop a methodology which cotake these effects into account.

Our methodology is most similar to two other studiefirst, a recent study of Hartmannet al.28 on the cone-jetshapes in electrospraying. Second~in an unpublished workwe learned of after completing this study!, Ganan-Calvo car-ried out an electrospraying study between two parallel plaand achieved good agreement with his experiments.29,30Bothstudies focus on a regime of electrospraying where therent is apparently independent of the voltage. Like thesethors, to obtain quantitative agreement with experiments

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were forced to take account of experimental details. Hoever, the way in which we fixed some of the most subproblems differs from their approaches; the major issuesnoted later in this work.

A. Asymptotics

The asymptotic behavior of steady-state solutions pvides a good estimate of the radius and surface chargesity of a thinning jet given its experimental parameters.the jet thins it better approximates a cylinder. For a cylindin an axially applied electric field, the electric field within thjet is purely axial and equal in magnitude to the applied fieThis implies the equation for the current is

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The leading order balance has the current asymptoticdominated by advection,Qs/h'I , because the conductiocurrent vanishes ash→0. The physical reason that the suface charge density decays~as h! as the jet thins is thathough the radius decreases, the velocity increases toserve volume flow rate, and the net effect stretches any gpatch of advected jet surface.

The dominant balance in the Navier–Stokes equatiobetween inertia, tangential electric stress, and gravity,

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and by Ganan-Calvo22,17,29in his universal theory of electrospraying. Sinces}h, the jet follows a line in the phasediagrams of unit slope, with an offset determined by tvalues of the current and volume flow rate.

B. Numerical solutions

We now proceed to calculate the steady states numcally, in order to obtain the path of the jet before enteringasymptotic regime. In the process of working out these cculations and comparing the resulting solutions with expements, we encountered many difficulties that had to bemounted. We will now list and discuss these issues.

To calculate the electric field near the jet, it is necessto impose the correct boundary conditions~of a constant po-tential drop! between the two capacitor plates. This requiraccounting for image charges caused by the chargingpolarization of the jet. Our equation for the electrical potetial is therefore

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2227Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. II.

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wheref` is the electrostatic potential that gives the constelectric field produced by the capacitor plates, andl(z) isthe effective charge density on the jet~see Sec. II of Ref. 1!.This cascade ensures that the surfacez50 andz5d will bethe proper equipotentials, since all integrals terms vanishthose surfaces, leaving justf5f` .

In practice we retain only the first term on each sidThis means neither surface is exactly an equipotential; hever, this is sufficient for unambiguously fixing the boundaconditions on the plate and enforcing the experimentallyposed potential drop up to a known error. Using these bouary conditions, the solutions for the electric field will ostesibly improve near the capacitor plates and thus~for theupper plate! near the nozzle. The behavior of the field nethe nozzle is crucially important, as we will see below, fobtaining reasonable solutions.

Additionally, near the plate the locality assumptions thwere used in an asymptotic approximation for the fieldRef. 1 for deriving the electric field equation break dowFor this reason, we opted to solve the full integral equatnumerically to determine the electric field. The integral eqtion that we use is

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~10!

We have developed algorithms to solve these equations@Eq.~10! for the electric field together with Eqs.~1!–~3!# numeri-cally as a function of all parameters. The numerical scheuses a standard second order in space finite differencecretization of the equations. Without an electric field, we ccalculate the solution of a purely gravity driven thinning jeusing as an initial guess the analytic solution of a perfcylinder with gravity turned off. We then make successcalculations of the shape while gradually increasing gravusing the previously calculated solution as a guess inintegration scheme. At each step of the iteration, the resulnonlinear equations are solved using Newton’s method

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turn we use the final solution of that iteration as an initguess for our solution with electric fields, slowly increasithe applied field/decreasing the flow rate and solving insimilar fashion.~We begin with a jet at high flow rate toensure a solidly stable initial guess; a gravity-driven jetlow flow rate is highly susceptible to the Rayleigh instabilitwhich causes numerical problems.! Figure 5 shows an example of how the jet shape changes with ramping fistrength.

As discussed earlier, the difficulty with this procedurethat it does not fix the current, which is mathematicaequivalent to specifying the charge density near the jet. Sithe current is a crucial factor in determining the stabilitythe jet, this conclusion implies that the detailed shapematerial properties of the nozzle affects the stability of tjet, even when the jet is very far from the nozzle. By ‘‘dtailed,’’ we do not mean something as mundane as a depdence on the nozzle radius, but we conjecture that evariations in the shape and material properties of the etrode, or the protrusion length of the nozzle through thepacitor plates have a noticeable effect. Our own experime

FIG. 5. Change in jet shape with ramping electric field strength. Aselectric field is ramped from 0 to 5 kV/cm, the jet thins substantially. Tfluid is glycerol withn514.9 cm2/s, with K50.01mS/cm.

FIG. 6. Comparison of theoretical~dashed! and experimental~solid! radialjet profiles for a glycerol jet in a uniform applied field. The flow rate isQ51 mL/ min and the applied voltage is 30 kV over 6 cm. The characteridecay lengths of the experimental and theoretical curves, normalized toinitial radius of the jet, are 1.28~3! and 3.26, respectively. See Fig. 8~c! forthe considerable improvement that results when the correct fringe fieldthe nozzle are used instead of a uniform applied field. There the two culie nearly on top of one another.

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uter rads of equ

2228 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al.

FIG. 7. Typical nozzle fringe fields: the nozzle used to compute the field is a solid metal cylinder that protrudes 7.2 mm from the plate and has an oiusof 0.794 mm. In these pictures the nozzle points upwards, and we only present the right half of it. All dimensions are in centimeters. We show linealfield strength on the top left and equipotentials on the top right. The bottom figure shows the extractedE`(z), as described in the text.

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confirm that changing the protrusion length of the nozchanges the stability properties of the jet.14

The idea that the detailed shape of the nozzle matwas previously recognized by Pantanoet al.15 and Hartmannet al.28 in carrying out similar studies for electrosprayinThis point of view is also consistent with experiments datback to G. I. Taylor’s original experiments describing tTaylor cone. In this experiment, Taylor found it necessarymachine a conducting nozzle with an opening angle ofactly the Taylor cone angle, 49.3°; presumably, his initefforts using more ordinary nozzles failed, and he ascrithis failure to the fringe fields near the nozzle. In our owexperiments, making the nozzle protrude by different dtances from the equipotential conducting plate resultedquantitative changes in the stability properties of the jet.14

In order to compute the theoretical current and jet pfiles, we need to figure out what is happening at the nozFor the long wavelength theory developed earlier, this tralates into determining a relation between the electric fiE(0) and the surface charge densitys~0! at the nozzle.Given such a boundary condition, the current is determithrough a combination of the equation for the current eqtion ~2! and the equation for the electric field~10!. The de-termination of the current is inherently nonlocal, in that t

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current depends on the shape, charge density, and elefield along the entire jet.

We do not know from first principles what this boundacondition should be, because it is not clear what is the pdominant charge transport mechanism near the nozzleorder to determine the effective boundary condition,chose to iterate between theoretical solutions for a varietdifferent boundary conditions and a carefully controlledof experiments. The philosophy we followed was to find tset of boundary conditions for which quantitative agreemwith experiments is obtained over a range of applied fieand flow rates. Using the apparatus outlined in Fig. 1,measured the steady shapes of jets of several fluids ovrange of electric field strengths. The currents from the jwere also measured independently.

We experimented with various possibilities for the effetive boundary condition at the nozzle and found that we wonly able to find steady solutions for the field strengths aflow rates observed experimentally when we tooks(0) to bevery small, so that at the nozzle the radial field is musmaller than the tangential field. Including a substanamount of initial surface charge caused the jet to ballooutwards as it exited the nozzle due to self-repulsioncharge. This caused the solutions to destabilize at mode

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2229Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. II.

FIG. 8. Comparison of theoretica~dashed! and experimental~solid! pro-files for a glycerol jet flowing betweentwo plates spaced 6 cm apart, with0.794 mm outer diameter nozzle protruding 7.2 mm from the top plate, avarious fields and flow rates. Agreement is best for low flow rates andhigh fields ~b, c, e, f!. The pair ofnumbers beneath each plot is the thoretical healing length/experimentahealing length, normalized to the outediameter of the nozzle. By healinglength we mean the axial interval ovewhich the initial radius of the jet de-creases by a factor of 1/e. The differ-ence in healing lengths between thleft and right experimental profiles isused as an error estimate.

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fields, typically far below the field strengths where steajets are observed experimentally. This difficulty did not exwhens(0)50 ~or small! for fluids of sufficiently low con-ductivity. Since our experiments never observe a destabiltion of the jet through a dramatic ballooning near the nozzour calculations imply that the charge distribution nearnozzle must be small enough so this cannot occur; hencefeel that takings(0) to be small is a physically reliablapproximation~when the fluid conductivity is sufficientlysmall!. A reason for why this is so is that a basic assumptunderlying the derivation of the equations for the jet shap

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that the time scale for charge relaxation across the crosstion of the jet is much faster than that for axial charge relaation. This assumption clearly breaks down at the nozwhere there has not been sufficient time for radial charelaxation to occur. Without radial charge relaxation, the sface charge density has not had time to build up.~We thankProfessor D. Saville for providing this argument to us.!

A typical comparison of the radius of the theoretical prfile for the jet with experiments is shown in Fig. 6, forglycerol jet in an electric fieldE`55 kV/cm, with a volumeflow rateQ51 ml/min. The agreement is terrible: the the

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ted

2230 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al.

FIG. 9. Experimental pictures that were edge detecto produce the experimental profiles of Fig. 8.

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retical profile systematically decays too slowly away frothe nozzle. The reason for this disagreement is not the chof the effective boundary condition. Extensive experimention with different boundary conditions~e.g., varying thefraction of the current carried by surface advection! showedthat the agreement shown in Fig. 6 is the best achieva@And, as described earlier, ifs~0! is increased too much fromzero, we could not even find steady solutions at the pareter values of the experiments.#

The problem instead is that we have computed this pfile under the assumption that the applied fieldE` is uniform

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in z. However, in the experiment, the metal nozzle protrud7.2 mm from the top capacitor plate, which is a small fration of the distance between the two capacitor plates. Hoever, this causes fringe fields. Near this nozzle the localE`

will be higher than the average field between the two plaExperimentally, there is strong evidence that these frinfields are important in determining the shape and stabilitythe jet: when the nozzle is pushed out or retracted, theresignificant changes in the stability characteristics. Whennozzle is pulled in, the initiation voltage for a steady jincreases, and vice versa. The fringe field of the nozzle

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2231Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. II.

therefore crucial to determine the behavior of the jet nearnozzle, which sets the behavior for the rest of the jet dowstream.

We therefore need to compute the fringe field. UsiMatlab’s PDE Toolboxwe computed the field in the vicinityof an experimental nozzle, and extracted from this solutthe variation inE`(z) away from the nozzle. The result oftypical finite element calculation is shown in Fig. 7. Thcalculation assumes that the nozzle is a perfectly conducsolid cylinder with the same protrusion and outer diametethat in the experiment. An average of the cross-sectioelectric field was extracted from the finite element calcution, asE`(z).

When we include the effects of the fringe fields nearnozzle, agreement improves markedly. Figure 8 shows nsuch comparisons at three different values of applied fi~22, 26, and 30 kV over 6 cm! and at three different flowrates ~1.0, 1.5, and 2.0 mL/min!. The photographs fromwhich the experimental curves in Fig. 8 were extractedshown in Fig. 9. The improvement from Fig. 6 to Fig. 8~c! isconsiderable. Agreement between theory and experimebest at low flow rates and high voltages. One reason foris perhaps that in these limits, the electric field is greatethe nozzle while the rate of charge advection is less, soassumption that all the current at the nozzle is carriedconduction@s(0)50# could be closer to the truth. For illustration, Fig. 10 shows the radius, surface charge density,locity, and electric field variation of the jet in Fig. 8 witQ51 mL/min and a potential difference of 30 kV.

We were not able to achieve similar comparisonstween theory and experiment for higher conductivity fluidThe difficulty is that at a sufficiently high conductivity, theffective boundary conditions(0)50 produced~long wave-length! instabilities near the nozzle, at a field strength belthe largest field strength where steady jets were observeexperiments. Despite extensive numerical experimentawe were not able to find a boundary condition that did n

FIG. 10. Plot of the jet radius~cm!, the surface charge density~esu/cm2!, thefluid velocity cm/s and the electric field~statvolt/cm! for the jet with volumeflow rateQ51 ml/min and potential difference 30 kV. Note that the surfacharge density decays according to the prediction of the asymptotic lawafter z'1 cm, about ten nozzle diameters.

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suffer such instabilities; a correct treatment therefore appto require more physical understanding into the mechanof charge transport near the nozzle.

The possible mechanisms include~a! advection of thecharge density built up in the Debye layer around the meelectrode and~b! advection of free charge density througthe liquid near the nozzle. Other difficulties may be duethe fact that the electrostatics near the nozzle are not lwavelength. Radial fields can be strong and can drive shwavelength motion~just including the axial fringe field doenot incorporate this effect!.

The relevant quantity for the stability calculations is tpath of the jet in theh2s plane. A noteworthy feature oFig. 10 is that the jet does not reach the asymptotic reguntil about ten nozzle diameters away from the nozzle.this point, the radius of the jet has already decreased substially ~by about a factor of 50!. The consequence of this ithat the stability of the jet is determined by more than juthe behavior in the asymptotic regime; the difficulties at tnozzle are expected to influence the stability of the entire

The calculations of this subsection bear some similato a recent study by Hartmannet al.,28 who calculated elec-trified jet shapes in electrospraying. Although we were uaware of this study when the research reported heredone, many of the same subtleties reported here werefound by these authors. They also found it necessary to cpute the fields for the detailed nozzle under considerationobtain quantitative agreement with experiments. They areported difficulties with the boundary conditions at tnozzle, though they dealt with it differently: they report ththey could only make their numerical calculation converge‘‘the appropriate value of@the slope# is chosen for the liquid-

ly

FIG. 11. Strobe photograph of the overall picture of an unstable jet.shutter speed is 3ms. On large scales, it appears as a whipping rope, wblob-like varicose protrusions.

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the jet awo

2232 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al.

FIG. 12. ~a! Close to the nozzle, the centerline of the jet is straight. Small axisymmetric distortions can be seen on the axis of the jet.~b! Further down thejet from the nozzle, the axisymmetric disturbances have grown into large blobs which propagate down the centerline. The two video frames showttwo different times~separated by a fraction of a second! at the same spatial location.~c! Further downstream, the centerline itself starts to whip. The tframes show the jet at two different times at the same spatial location.

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air interface at the axial position where the liquid is attachto the nozzle.’’ This is mathematically similar~though physi-cally different! to our method of fixing the charge densitythe nozzle; their calculations appear to indicate thatcharge density is very small near the nozzle, as we hfound here. Another difference between the two calculatiis that they treat a finite jet~with an end!, while our jet thinsindefinitely. Finally, the experiments reported here havedifferent current voltage relationship than those reportedHartmannet al.: whereas they present scaling laws statthe current is independent of the voltage, our experimeand simulations show a strong dependence. Presumablyis due to the differences in geometries of the two studies

To conclude this section, we have demonstrated quatative agreement between theory and experiments forshapes of low conductivity fluids. Such agreement requiaccurate computation of both the fringe fields of the nozand invention of an effective boundary condition on the sface charge density at the nozzle. Both of these featuresnificantly influenced the shape and surface charge densitthe jet. Since the surface charge strongly affects stability,clear that the properties of both electrosprays and elec

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spinning fibers will generically depend strongly on precdetails of the nozzle configuration. The methods develoin this section for determining the jet shape do not workhigher conductivity fluids. We believe that the reason for tis that our understanding of the mechanisms for charge trport near the nozzle is inadequate.

The relationship between the surface charge and raof the jet also implies scaling laws for the real and imaginaparts of the growth rate of the whipping instabilities asfunction of experimental parameters. We have numericacalculated the maximum growth rate and oscillating fquency of the whipping mode as a function of external fieand fluid parameters. We find that the whipping frequenincreases linearly with the applied field strength, increalike the square root of the conductivity, and decreases wthe square root of the viscosity. Similarly the growth raincreases linearly with the electric field strength and condtivity, and decreases inversely with viscosity. The oscillatofrequency is numerically much larger than the growth raconsistent with our experiments. The dominant forces demining these forces are the net force and torque exertedthe charged jet by the external electric field. In a future wo

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2233Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. II.

we plan on testing these relations with systematic expments.

IV. COMPARISON OF STABILITY CHARACTERISTICSWITH EXPERIMENTS

Combining knowledge of the the jet shape and surfcharge density along the jet from Sec. III with the stabilcharacteristics yields predictions for the behavior of the jea function of experimental parameters.

We will start with one such comparison, to give a flavfor how the techniques developed in this article canapplied. We have taken a close up high speed 1frames/second movie of a PEO/water jet with viscosityn516.7 cm2/s, conductivityK511mS/cm, Q55 ml/min, a

FIG. 13. Phase diagram for the PEO/water jet. The black curve is theymptote given by h52Qs/I . The jet shape starts at (h,s)5„0.1 cm,s(0)…, and will eventually end up on the black asymptote. A pawhich starts withs~0! small passes through the curved region will produa jet which transitions from axisymmetric instability to a whipping instabity, as observed.

FIG. 14. Operating diagram for glycerol jet. The shaded region shwhere the amplification factor of varicose perturbations isG>e2p, signify-ing the onset of the varicose instability. The points represent experimemeasurements of the instability thresholds for these parameters. No wping instability is present, in agreement with experiments.

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voltage drop of 9 kV over 15 cm, and a measured curren0.9 mA. Viewed on large scales far from the nozzle, thelooks like a whipping rope~Fig. 11!.

Figure 12 shows a succession of frames from the moat different times and at different spatial locations alongjet. The first frame@Fig. 12~a!# shows a segment of the jevery close to the nozzle with a slight axisymmetric excition. The second set of frames@Fig. 12~b!# shows two differ-ent pictures of the same jet a few nozzle diameters dostream, taken at the same spatial location, separated inby a fraction of a second. The axisymmetric disturbanhave grown into large axisymmetric blobs that travel dowstream. The third set of images is even farther fromnozzle; here the centerline of the jet begins to whip backforth. The axisymmetric blobs continue to propagate dothe axis of the whipping jet.

Now we want to demonstrate how these results canexplained using the formalism developed in this article. Fure 13 shows the stability diagram described in Sec. IIthe parameters of this PEO/water jet. The black region infigure denotes parameter values where an axisymmetricstability dominates, and the white region denotes wherewhipping mode dominates. The solid line denotesasymptotic laws5I /Qh, where we have used themeasuredvalue of the current to place the offset. Before reaching tasymptotic regime, the jet most likely curves upwardswardss50 at largeh, for the reasons discussed in the prvious section.

It is easy to rationalize qualitatively the sequence ofstabilities observed in the video frame from this stabildiagram: initially, the jet starts at a large radius with a smsurface charge. In this regime, the jet is predominantlystable to axisymmetric disturbances. However, in orderreach the asymptotic regime the path of the jet movesthe region of the diagram where whipping instabilities domnate. ~Note that if the current were much smaller, thasymptotic regime would not overlap the whipping regionthe stability diagram and the jet would not whip.!

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FIG. 15. Operating diagram for a PEO jet~K5120mS/cm, h516.7 P,e/ e542.7, r51.2 g/cm3, g564 dyn/cm!. The lower shaded region showwhere the amplification factor of varicose perturbations isG>e2p, signify-ing the onset of the varicose instability. The upper shaded region showonset of the whipping instability. Both instability thresholds agree with eperimental measurements for this configuration.

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thrdiagr.

2234 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al.

FIG. 16. Operating diagram for a jet with viscosity ten times lower (h51.67 P) and ten times higher (h5167 P) than the PEO jet in the above figure, withe other parameters the same. For the more viscous jet, the shaded regions correspond to amplification factors ofG5e2p. For the more viscous jet, the loweshaded region shows the onset of the varicose instability and the upper shaded region shows the onset of the whipping instability. The operatingam forthe lower viscosity fluid is more complicated, with many different possible regimes: the shaded region shows theG5e5p contour for the varicose instabilityThe solid line is theG5e2p contour for the whipping instability.

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We should remark that it is clear from the dynamicsFig. 12 that nonlinear effects are also of paramount imptance. The fact that the axisymmetric blobs saturate andnot cause the jet to fragment is a nonlinear effect, as isinteraction of these blobs with the whipping motion.

V. OPERATING DIAGRAMS

Theoretical operating diagrams are calculated by cobining the stability analysis of an electrically forced jet1 withour experimental observation that the instabilities leadingelectrospinning are convective in nature. DenoteA(t) theamplitude of a perturbation~of a given mode! to the jet andv its corresponding growth rate. Then

A

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dzlog ~A!5v, ~11!

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so that

A~z!5A~0! expS E0

z

dz8v„h~z8!,E~z8!,s~z8!…

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whereU(z)5Q/(ph2) is the advection velocity of the perturbation.~In general the advection velocity also includes tgroup velocity of the perturbation, which we neglect in theillustrative calculations.! Given the shape and charge densof a jet, this formula predicts both when and how the jet wbecome unstable. If the distance between the nozzle andgrounded plate isd, then the maximum amplitude of a peturbation as it is advected away from the nozzle is giventhe amplification factor

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FIG. 17. Operating diagram for a jet with conductivity ten times lower (K512mS/cm) and ten times higher (K51200mS/cm) than the PEO jet in the abovfigure, with the other parameters the same. All of the shaded regions correspond to amplification factors ofG5e2p except for the whipping instability of themore conducting jet, which corresponds toG5e5p. In both figures, the lower shaded region shows the onset of the varicose instability and the upperregion shows the onset of the whipping instability. Changing the fluid conductivity changes both the onset threshold for the whipping mode and thcoseinstability.

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2235Phys. Fluids, Vol. 13, No. 8, August 2001 Electrospinning and electrically forced jets. II.

G5A~d!

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Evaluating the integral requires using the properties of the~radius, charge density, etc.! as it thins from the nozzle.

An operating diagram summarizes how the amplificatfactor G depends on the external electric field and voluflow rateQ at a fixed set of fluid parameters. Such operatdiagrams are given in Fig. 14 for glycerol jets, and Fig.for PEO electrospinning jets.~These figures are reproducefrom Ref. 14.! The diagrams show the contours ofG for boththe axisymmetric and whipping modes. The general shapthe operating diagram is insensitive to the precise conchosen over the range of reasonable amplification factWe generally chooseG52p as a representative contour.

In principle, determiningG requires solving for thesteady-state jet profile and surface charge density for eexternal condition~E, Q!. As a first guess in calculating thesshapes, we have assumed that the surface charge densitthe jet radius are related by the asymptotic formulas5Ih/Q, and taken several types of approximations forjet radius, such as~1! taking the jet radius as a constant a~2! taking the jet radius predicted by the asymptotic formuof Eq. ~6!. Both procedures led to operating diagrams qutatively and quantitatively similar to those shown in Fig. 1We believe that the quantitative discrepancies betweentheory and experiment are due to the fact that we arecalculating the integral in Eq.~13! explicitly. In obtaining theagreement between theory and experiment, we have assuthe relationI;E and I;Q found experimentally.

It is also of interest to understand how the operatdiagrams change as a function of fluid parameters. Figureshows operating diagrams at the same conditions as thoFig. 15 except with viscosity ten times higher and lowrespectively. Changing the viscosity leaves the onset throld for the whipping instability unchanged, but substantiachanges the onset for the varicose instability. The reasonthis is that the viscous Rayleigh instability has a growth rof orderr 0Arg/h2 smaller than the inviscid Rayleigh instability. Most of the varicose instability occurs when the jdiameter is large enough that the growth rate is larger.the less viscous jet we plot the contourG5e5p for the vari-cose instability, because theG5e2p contour is unstablethroughout this parameter regime. Since our calculation oGdepends on the distance between the plates likeed, thismeans that the jet will break up before it hits the bottoplate. The change inG implies the jet will break up abouhalf-way between the plates~'7 cm from the top plate!.

We can also study the effect of the conductivity of tfluid on the operating diagrams~Fig. 17!. To do this, weassume~as the earlier experiments indicate! that the currentthrough the jet is proportional to the conductivity~so that thesurface charge on the jet increases linearly with the condtivity !. Changing the conductivity of the fluid moves the oset thresholds for both instabilities. Higher conductivleads to the suppression of the Rayleigh mode at lower fi

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strengths and enhancement of the whipping instabilLower conductivity enhances the varicose instability asuppresses the whipping instability. The mechanismsthese effects are that~as described in detail in part I1! surfacecharge enhances the whipping mode and suppresses thecose instability. Higher conductivity implies more surfacharge on the jet.

These results have implications for the operating dgrams of electrospraying: Cloupeau and Prunet-Foch20,31

have measured the operating regimes for an electros~corresponding to low viscosity and high conductivity!. Al-though quantitative comparisons with their results arepossible~because not all the parameters for the fluids usetheir study are given!, the operating diagram for a less viscous, highly conducting fluid bears resemblence to thmeasurements of where the ‘‘cone jet’’ mode of electrospring exists. They identify the lower threshold of their stabilidiagram with the onset of the ‘‘cone jet’’~i.e., the cessationof varicose instability!, and the upper threshold as the poiat which ‘‘multiple jets’’ appear. If we identify this latterthreshold as the onset of whipping, the shape of our opeing diagram is very similar to that reported by Cloupeau aPrunet-Foch. They also measure the dependence of theierating diagrams on liquid conductivity, and demonstrate twith increasing conductivity the critical flow rate at whicthe cone-jet mode sets in decreases. This is consistentour findings here: higher conductivity fluids have a highsurface charge density. This stabilizes the Rayleigh moand allows a stable jet at a lower flow rate than otherwpossible. All of these qualitative relations merit further stud

VI. CONCLUSIONS

The research described in this article was conducteddevelop a quantitative description of the mechanisms of etrospinning. Our experiments have demonstrated cleardence that the essence of this phenomena is a whippingThe theory demonstrates the mechanism of the whipping:charge density on the jet interacts with the external fieldproduce an instability. High surface charge densities tendsuppress the varicose instability, and enhance the whippinstability. It is the competition of these two factors that sthe operating regime where electrospinning can occur.quantitative agreement of the predicted operating diagrawith that observed experimentally indicates that the whping mode predicted by linear stability analysis correspoquantitatively with the bending of the jet observed expementally, and which appears to be the primary mechanunderlying submicron fiber formation during electrospining.

Since the surface charge~or the current passing througthe jet! is the most important parameter for determining sbility, it was necessary to understand how this charge istermined in experiments. The dependence of the currenthe external field and conductivity differs from that typicalobserved in electrospraying experiments,17 where the currentis independent of the voltage.

Eventually we hope that it will be possible to use idesimilar to those developed here to predict and correlate

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2236 Phys. Fluids, Vol. 13, No. 8, August 2001 Hohman et al.

properties of the materials produced by electrospinning.example, electrospinning produces a batch of uniform fibwhose diameter depends on the nozzle configuration andfluid used. It would be extremely useful to extend the theretical framework to predict the final fiber morphology asfunction of experimental conditions.

ACKNOWLEDGMENTS

We are grateful to L. P. Kadanoff, L. Mahadevan,Saville, and D. Reneker for useful conversations. We athank A. Ganan-Calvo for detailed comments on this mascript. M.M.H. acknowledges support from the MRSECthe University of Chicago, as well as support from NSGrant No. DMR 9718858. M.B. and M.M.H. gratefully acknowledge the Donors of The Petroleum Research Fund,ministered by the American Chemical Society, for partsupport of this research, and also the NSF Division of Maematical Sciences. M.S. and G.C.R. are grateful to thetional Textile Center for funding this research project, NM98-D01, under the United States Department of CommeGrant No. 99-27-7400.

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