Conical tips inside cone-jet electrosprays

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Conical tips inside cone-jet electrosprays

Article  in  Physics of Fluids · April 2008

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Conical tips inside cone-jet electrosprays

Álvaro G. Marín, Ignacio G. Loscertales, and A. Barrero

Citation: Phys. Fluids 20, 042102 (2008); doi: 10.1063/1.2901274 View online: http://dx.doi.org/10.1063/1.2901274

View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v20/i4 Published by the American Institute of Physics.

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Conical tips inside cone-jet electrosprays

Álvaro G. Marín,1Ignacio G. Loscertales,2and A. Barrero1

1

Escuela Técnica Superior de Ingenieros, Universidad de Sevilla, Avenida de los Descubrimientos s/n, 41092 Sevilla, Spain

2

Escuela Técnica Superior de Ingenieros Industriales, Universidad de Málaga, Plaza El Ejido s/n, 29013 Málaga, Spain

共Received 13 June 2007; accepted 19 February 2008; published online 10 April 2008兲

In coaxial jet electrosprays inside liquid baths, a conductive liquid forms a cone-jet electrospray within a bath containing a dielectric liquid. An additional dielectric liquid is injected inside the Taylor cone, forming a liquid meniscus. The motion of the conductive liquid that flows toward the vertex cone deforms the inner dielectric meniscus until a liquid jet is issued from its tip. Both the conductive and inner dielectric liquid jets flow coaxially and, further downstream, they will eventually be broken up by capillary instabilities. Coaxial jet electrosprays inside liquid baths is a useful technique to generate fine simple or double emulsions. However, in certain circumstances, we have observed that the dielectric menisci present extremely sharp tips that can be stabilized and made completely steady without mass emission. In this paper, we will first explore the parametrical range of liquid properties, mainly viscosities and surface tensions, under which these sharp tips take place. In addition, a simplified analytical model of the very complex electrohydrodynamical flow is presented for a more complete approach to the phenomena. © 2008 American Institute of Physics. 关DOI:10.1063/1.2901274兴

I. INTRODUCTION

Free surface flows frequently exhibit regions of ex-tremely high curvature which are commonly appealed

singu-larities or sharp tips. Such singusingu-larities are apparently

ob-served at macroscopic length scales in the form of conical points, cusps, or corners, but they become rounded and smoothed when observed within the microscopic-nanoscopic range. Breakup of liquid jets or coalescence between droplets is a typical example of instantaneous singular points, but the case now reported here must be included in the group of steady conical points that appear in forced interfaces with non-negligible surface tension. For classical examples of this type, we must first refer to Taylor’s four-roll mill experiment,1where he found steady conical points in drops under critical values of the strain rate exerted by two pairs of rotating cylinders. Using only rotating cylinders, cuspidal sheets have also been observed in air-liquid interfaces.2More recently, Courrech du Pant and Eggers3 carried out an “in-verted” selective withdrawal experiment where they could observe steady axisymmetric air-liquid cusps with extremely high curvature radius. Electrical forces can also generate very sharp menisci in liquid interfaces, as in the case of the so-called Taylor cones4and electrosprays in general.

These sharp menisci may become unstable and emit mass either through small threads, as in the tip-streaming phenomenon found by Taylor, or through thin laminar sheets, as in the two-roller experiment. Recently, Suryo and Basaran5 have shown numerical simulations where a liquid meniscus develops a sort of “tip streaming” in a simple co-flow regime in the absence of any of the following phenom-ena: surfactants,6geometrically induced focusing,7or electri-cal fields.8 Recently, the existence of the regime reported in

Ref. 5 has been experimentally confirmed in simple coflow systems.9

In all these experiments, a liquid meniscus is forced by electrical or mechanical means to reach extremely high cur-vatures in its vertex. Then, if the meniscus is then slowly fed with a certain flow rate, it may emit tiny droplets with diam-eters in the micrometric range. Consequently, these regimes have a high practical interest due to their ability to generate micrometric or even nanometric particles for several scien-tific or industrial applications.10

The phenomenon reported in this paper was observed during experiments with coaxial jet electrosprays,8which has proved its suitability for applications as diverse as capsule production,11 simple or double emulsions,12 and hollow nanofibers.13 In coaxial jet electrosprays, a certain flow rate of a conducting liquid is forced through the annular gap be-tween two coaxial needles. In this case, the needles were immersed in a host dielectric liquid. As is well known, when an appropriate voltage is set between the needles and a downstream electrode, the conducting liquid meniscus adopts a conical shape from whose vertex a thin jet is issued. Another immiscible liquid is then injected through the inner needle in such a way that its meniscus is deformed by the electrohydrodynamic共EHD兲-driven conical sink flow of the conducting liquid forming the conejet. For sufficiently small values of the surface tension of the innermost interface,␥i,

the inner interface and the inner meniscus soon adopt a spout shape, thus forming a coaxial jet with the outer liquid. On the contrary, for larger values of the interfacial tension, which is the case under discussion in the paper, we have observed the formation of a conical tip in the inner meniscus with no jet emanating from it共see Fig.1兲.

In this paper, we experimentally explore the range of

PHYSICS OF FLUIDS 20, 042102共2008兲

values of liquid properties, mainly viscosities and surface tensions, for which a pointed meniscus is formed. Moreover, to gain insight into this extremely complicated EHD flow, we have developed a simplified analytical model that still retains the main features of the system. The model relates the cap-illary number of the flow and other flow characteristics to the tip angle. The paper is organized as follows. Section II is dedicated to the description of the experimental features; the analytical model is discussed in Sec. III, results from the model are given in Sec. IV; a brief discussion of the charge transport mechanisms in the conducting layer follows in Sec. V. Finally, the results are summarized in Sec. VI.

II. THE EXPERIMENT

The experimental setup was already employed for the production of simple and double emulsions;12it consisted of a small transparent cell made of polymethyl methacrylate with a glass window for visual observation. The cell was filled with a light dielectric liquid, hexane in this case. The coaxial needle was immersed from the upper part of the cell; its external metallic needle had 0.5 mm of inner diameter and 0.8 mm of outer diameter. The inner needle was initially a metallic needle of 0.4 mm of outer diameter and 0.2 mm of inner diameter; however, in latter experiments, to enhance the stability in the lines, silica tubes of smaller diameters were needed. For these lines, the outer diameter of the silica tubes were always 360␮m while the inner ones varied de-pending on the viscosity of the liquid: from 20␮m for the least viscous samples共5 and 20 cS兲 to 180␮m for the most viscous ones 共1000 cS and above兲. The conducting liquid 共glycerine in all cases, with an electrical conductivity of 2 ⫻10−6_{S/m兲 was always forced by a syringe pump 共Harvard}

Instruments兲 to flow within the gap in the coaxial needle, while the inner liquid was forced by either syringe pump or compressed air.

For appropriate values of both the flow rate of the con-ducting liquid and the applied voltage, the meniscus of the conducting liquid adopts a conical shape; a very thin charged jet issues from the cone vertex. If a second liquid, immis-cible with the conducting one, is injected through the inner needle, the meniscus that forms at the needle end is dragged by the EHD-driven conical sink flow of the conducting liq-uid. Note that contrary to the case of the outer interface, electrical stresses play no role in the inner interface whose shape is exclusively determined by the mechanical

equilib-rium between hydrodynamic and surface tension forces. Therefore, a dimensionless capillary number may be defined to account for the relative importance of the viscous and capillary forces in the equilibrium of the inner interface: Ca=␮oU/␥iwhere␮ois the viscosity of the conducting

liq-uid and U is its characteristic liqliq-uid velocity. When ␥i is

sufficiently small共high capillary number兲, the inner menis-cus soon adopts a spout shape, thus forming a coaxial jet with the outer liquid. For larger values of the interfacial ten-sion 共smaller capillary numbers but still close to unity兲, which is the case under discussion in the paper, the inner meniscus developed conical tips during experiments on the dripping to jetting transition. These conical points turned steady when the deforming forces are carefully controlled and the inner liquid injection is stopped.

However, they are still extremely sensitive to any pertur-bation. For example, any residual pressure difference in the inner liquid feeding line results in instabilities in the menis-cus, so much care has to be taken when choosing the feeding line diameters, lengths, and working pressures. Also, clean-ness has been found to be a critical factor for steadiclean-ness; usually, in liquid samples accidentally exposed to any kind of dust, the menisci became randomly unstable, emitting small threads of liquid. Surfactants are obviously incompatible with the formation of this kind of steady sharp tips due to either the subsequent reduction of the surface tension at the interface or to the strong reduction of surface tension that can occur at the vicinity of the tip region by the accumulation of surfactant by convection. Also, a high increase of the outer electric field can turn the inner tip unstable, transforming it into a jet, due to the increment in electrical shear stress.

Once these conical points are properly stabilized, they remain steady for times of order of hours with no detectable loss of mass by neither dripping nor jetting. Note that the emission of mass of the inner liquid would require capillary numbers, ␮oU/␥i, of order unity 共shear stress overcoming

surface tension兲 and therefore the emitted flow rate would be

Q⬃UR2⬃␥iR2/␮o, which is of order of nanoliters per

sec-ond for typical values of the interfacial tension␥i, viscosity

of the conductive liquid, ␮o, and radius of curvature at the

apex, R 共of order of 4␮m in our experiments兲. Since the volume of the inner liquid meniscus is of the order of nano-liters 共for needle diameters around 200␮m or less兲, any emission of mass from the meniscus would be detected for such flow rates in times of order of seconds.

In this experiment, we have explored different values of the viscosity and interfacial tension ratios. It was observed that the presence of surfactants in any of the liquids was incompatible with steady sharp tips. For this reason, the most successful experiments regarding the stability of conical points were achieved with pure liquids. In the experiments, we found that the existence of sharp steady tips only occurs for a restricted range of viscosity ratios 共inner to outer兲. Therefore, the viscosity of the liquids was varied starting with low viscosity ratios共␭⬇0.005 was the minimum兲, and it was increased until no steady tips were observed, which approximately occurred at ␭⬇1,2. Therefore, we conclude that the maximum viscosity ratio compatible with the forma-tion of steady tips must be around unity. In TablesIandII,

FIG. 1.共a兲 Sketch of the coaxial needle system. 共b兲 Magnified picture of the

inner conical meniscus共enhanced online兲.

we give some of the relevant properties of the liquids em-ployed; note that viscosity ratios vary in three orders of mag-nitude. The interfacial tension cannot be varied in such a wide range since the sharp interfaces only appeared in clean interfaces 共mostly above ␥⬃20 mN/m兲. Other parameters such as the volume of the anchored droplet have been also explored. For this purpose, we varied the diameters of the capillaries from 200␮m to 1 mm共which approximately cor-responds to drop volumes ranging from ⬃2 nl to 0.3␮l兲 without any substantial change on the stability of the conical point. Some other parameters cannot be accurately controlled in the experiments, for example, the velocity of the conduct-ing liquid or the contact angle of the inner meniscus and the capillary. However, both of them were approximately con-stant in all the experiments: the velocity takes values which vary from tenths of millimeters per second near the injection needle to almost meters per second in the electrified jet; the wetting angle resulted to be almost constant with a value close to 100°.

The optical system used to visualize the conical tips con-sisted of a digital high resolution camera attached to a mi-croscope Edmund VZM 1000. The system was able to give a maximum resolution of order of 10 pixels/␮m. It should be pointed out that the presence of a host liquid medium per-mitted a better visualization of the inner meniscus compared to other experiments performed in air.

Pictures of the steady menisci taken from different ex-periments are given in Fig. 2. Six different liquid couples with different viscosity ratios have been employed. The con-trast of the images in Fig.2has been enhanced as much as possible to improve their quality since the small length scale of the experiment makes difficult to take photographs even with optical equipments of good quality. A striking aspect in the pictures is that the cone angles of the menisci are close to the Taylor cone angle,4 that is, the angle adopted by a elec-trified conical meniscus of a perfectly conducting liquid in electrostatic equilibrium.

III. ANALYTICAL MODEL

The EHD problem to be solved is that sketched in Fig.3. The conductive liquid flowing between the two interfaces is mainly driven by the action of the electrical stress at the outer surface. This stress is diffused by viscosity through the conducting liquid layer which, thus, sets the inner insulating liquid into motion. Since no mass is supplied to the inner meniscus, the streamlines are closed: the insulating liquid moves toward the vertex along the generatrix and away from it along the axis.

Due to the values of the viscosities of the liquids, the small length scales of the flow in the experiment, and the low flow rates required to operate a steady state electrospray, we will make use of the Navier–Stokes equations in the low Reynolds number limit for both the inner and outer flows, so that the equations governing the velocity and pressure fields in steady state flow are

ⵜជ· uជj= 0, 共1兲

−ⵜជpj+␮jⵜ2uជj= 0ជ, 共2兲

where j stands for i 共inner兲 or o 共outer兲. We will assume axisymmetric flow and will employ spherical coordinates 共r,␪,␾兲; r is the radial coordinate,␪is the polar angle, and␾ is the azimuthal angle around␪= 0. The boundary conditions at the external interface␪=␪e, which separates the external

dielectric medium e from the outer conducting one o, are

TABLE I. Physical properties of the inner dielectric conical meniscus.

␮i共cP兲 ␭=␮i/␮o ␥i共mN/m兲

Inner dielectric liquid Silicone oil 5 4.56 0.005 22.4

Silicone oil 20 19 0.02 22.6 Silicone mix 1 130 0.1 22.1 Vaseline oil 167 0.2 28.7 Silicone mix 2 387 0.4 22.1 Silicone mix 3 520 0.5 22.15 Castor oil 850 0.9 13.6 Silicone 1000 963 1.0 21.5 Silicone 5000 5000 5.3 20

TABLE II. Physical properties of the conductive liquid and the host liquid.

␮o共cP兲 ␧/␧air ␥e共mN/m兲

Conductive liquid Glycerine 950 42.5 28.3

Host liquid Hexane 0.3 1.9

FIG. 2. Six liquid samples with different viscosity ratios giving rise to sharp

tips:共a兲 ␭=0.005, 共b兲 ␭=0.02, 共c兲 ␭=0.1, 共d兲 ␭=0.2, 共e兲 ␭=0.5, and 共f兲 ␭

⬇1. The classical Taylor angle 共⬇98.6°兲 is also represented through dis-continuous lines as a reference.

FIG. 3. Scheme of the flow in the inner and outer menisci and system of coordinates.

1 2␧eE␪ e2₊1 2共␧o−␧e兲Er 2_−共p e− po兲 − 2 ␮o r

冋

⳵u_␪o ⳵␪ + ur o

册

=␥oⵜ · nជ, 共3兲 ␴Er=␮o 1 r ⳵ur o ⳵␪ . 共4兲

Expressions 共3兲 and 共4兲 account, respectively, for the normal and the tangential equilibrium of the forces acting at the outer interface; nˆ is the normal unit to the surface, E_␪and

Er are the normal and tangential components of the electric

field, peand poare, respectively, the pressure of the external

and conductive liquids, and finally,␴is the electrical charge density at the surface.

The boundary conditions at the inner interface, ␪=␪i,

which separates the conducting liquid from the inner dielec-tric one, are

ur i = ur o , 共5兲 −共po− pi兲 + 2

再

␮o r

冋

⳵u_␪o ⳵␪ + ur o

册

−␮i r

冋

⳵u_␪i ⳵␪ + ur i

册

冎

=␥iⵜ · nជ, 共6兲 ␮i 1 r ⳵ur i ⳵␪ =␮o 1 r ⳵ur o ⳵␪. 共7兲

Equation共5兲accounts for the continuity of the velocity field while the equilibrium of both the normal and tangential stresses at the inner interface are given by expressions共6兲 and共7兲, respectively, pi− pe being the pressure jump across

the inner interface. We have assumed that free charges are located at the outer liquid interface and that the electrical effects inside the liquid bulk may be considered negligible, that is, ␧oE_␪o2共␪e兲⬃␧oE_␪o2共␪i兲Ⰶ␧eE_␪e2共␪e兲; ␧e and ␧o are the

electrical permittivities of the external medium and the outer liquid, respectively; the ratio␧o/␧ewill be referred as ␧rin

the following.

The electric field satisfies the Laplace equation:

ⵜជ· Eជe= 0, 共8兲

and the boundary conditions at the outer interface␪=␪e:

␧eEn e −␧oEn o_{⯝ ␧} eEn e =␴ and Et e = Et o . 共9兲

The rest of the boundary conditions for the electric field de-pend on the particular needle-electrode geometry used in each experiment.

Solving numerically problems共1兲–共9兲is a quite complex task. The two interfaces of the problem are unknown and their shape must be consistently found throughout an

iterative numerical scheme for the pressure, velocity, and electrical field, which are in turn affected by the shape of the interfaces. An additional difficulty comes from the existence of two very disparate length scales: the millimetric diameters of the capillaries and the micrometric diameter of the jets. In spite of these difficulties, some numerical approaches have been carried out recently.14

Here, instead of numerically solving this EHD problem, we build up an approximate analytical solution. For example, the almost conical shape of the inner meniscus suggests the use of a well-known family of low Reynolds conical flows.15–17Note that since capillary forces are inversely pro-portional to the distance r to the origin, self-similarity re-quires that the electrical field varies as r−1/2, just as in Tay-lor’s field, while pressure and viscous stress fields must vary as r−1关see Eqs.共3兲and共6兲兴. Then, an approximate model of the EHD problem has been developed based on the following main assumptions that satisfy the above requirements on the

r dependence:共a兲 the shape of the meniscus is conical, 共b兲

the flow inside the meniscus is driven by a conducting liquid layer, which is assumed to be very thin, and共c兲 the electrical field acting at the outer interface should not differ substan-tially from one that would act on a conducting conical me-niscus共Taylor’s solution兲. Thereby, the electric field problem becomes decoupled from the hydrodynamic one.

Taylor found a hydrostatic exact solution for the electric field due to the conical meniscus of a perfectly conductive liquid共zero tangential stress兲, the cone semiangle of this so-lution being␣⯝49.3°. Since the movement is driven by the electric tangential stresses acting on the surface, we will con-sider a meniscus with a cone semiangle smaller than that of Taylor’s, so that the resulting electric field has the same mathematical expression as the Taylor solution but, in this case, the tangential共radial兲 component of the electric field is nonzero and negative共pointing toward the vertex兲. In fact, the Legendre function of order1₂ which appears in the Taylor solution vanishes at the Taylor angle and takes negative val-ues for conical semiangles smaller than the Taylor one, and thus the radial component of the electric field gives negative values共see Fig. 4兲. The electrical potential found by Taylor takes the value␾= Ar1/2P1/2共x兲, where A is unknown in the

problem, P_1/2共x兲 is the Legendre function of order 1₂ and x = cos␪.

Then, the electrical shear stress at the outer surface can be easily calculated: ␶ជe=␴Eជr⬇ ␧eE␪Eជr= −␧e A2 2r

冑

1 − xo 2_P 1/2共xo兲P1

⬘

/2共xo兲eជr. 共10兲 To describe the flow inside the inner meniscus, we will use self-similar solutions. The assumption of self-similarity comes from the observation of an almost conical shape in the inner meniscus. Self-similarity has been usually assumed to describe flows near singularities due to the loss of typical length scales; the validity of these solutions is restricted to an intermediate region which excludes both the region including the vertex and the meniscus anchorage. In particular, conical self-similar solutions have been employed several times in

many other phenomena16 and more frequently employed to describe the flow inside Taylor cones.17 In each case, the radial dependence of the stream function must be carefully chosen in order to scale the forces that appear in the problem. The flow inside the inner meniscus that satisfies Stokes equations may be more easily calculated by using the axi-symmetric Stokes stream function⌿, which in spherical co-ordinates satisfies E2_E2_{⌿ = 0, E}2₌ ⳵ 2 ⳵r2+ 1 − x2 r2 ⳵2 ⳵x2. 共11兲

Since the viscous shear stress and pressure terms vary with the radial distance as r−1, the r dependence of the stream function on r must be as r2. Then, assuming separated vari-ables, the stream function reads

⌿共r,␪兲 = r2_{f共x兲. 共12兲}

Introducing the stream function in Eq.共11兲, we arrive at the fourth order ordinary differential equation:

共1 − x2_兲fIV_{共x兲 − 4xf}III_{共x兲 = 0, 共13兲}

where f is a continuous function that must exhibit regularity and symmetry at the axis, together with the rest of the boundary conditions:

f共− 1兲 = 0, f共xo兲 = 0, f

⬘

共xo兲 = Ui. 共14兲

The two first conditions in Eq.共14兲imply that both the axis 共x=−1兲 and the interface 共x=xo兲 are streamlines while the

third one gives the velocity value at the interface. Then, we can solve f共x兲 in terms of Uiand xo:

f共x兲 = Ui

冉

− x2 1 xo+ 1 + xxo− 1 xo+ 1 + xo xo+ 1

冊

. 共15兲

Then, the velocity components are

ur共x兲 = Ui

冉

2x 1 xo− 1 −xo− 1 xo+ 1

冊

, 共16兲 u_␪共x兲 = Ui − 2

冑

1 − x2

冉

− x 2 1 xo− 1 + xxo− 1 xo+ 1 + xo xo+ 1

冊

. Once the stream function and the velocity field have been calculated, expression 共2兲 can be integrated to obtain the pressure field: Pi共r兲 = Pe+ 2 Ui␮i r xo− 1 xo+ 1 . 共17兲

To find the flow magnitudes in the conductive layer requires numerically solving a quite complex hydrodynamic problem. However, this problem becomes extraordinarily simplified if the conductive layer is assumed to be thin. It should be no-ticed that the existence of such a thin layer is not a necessary condition to have sharp tips共although thin layers are usually observed in our experiments兲. If the conductive liquid layer surrounding the inner meniscus is assumed to be very thin, that is, h/rⰆ1 共where h is the thickness of the liquid layer兲, then ur⬃u␪r/hⰇu␪, as required by continuity equation. On

the other hand, momentum equations lead to the fact that pressure variations across the layer, ⌬_␪p, are much smaller

than those along the layer, ⌬rp. In addition, viscosity is

dominant in the layer and the shear stress is perfectly dif-fused through it.

IV. RESULTS OF THE MODEL

As indicated above, the assumption of a thin layer leads to an extraordinary simplification of the problem since it permits to directly relate the values of the stresses at the external medium to those at the inner liquid. Certainly, by adding up Eqs.共3兲and共6兲and also Eqs.共4兲and共7兲, together with Eqs.共10兲and共15兲, one arrives at an algebraic system of two equations involving the three unknowns A, Ui and xo.

After some algebra, one arrives at

A ¯2_{= A}2 ␧e 2共␥i+␥o兲 =

_冑

− xo 1 − xo2 1

关

共1 − xo 2_兲P 1/2

⬘

2_共x o兲 + 1 4共␧r− 1兲P_1/22 共xo兲 + 共xo+ 1兲P1/2共xo兲P_1/2

⬘

共xo兲

兴

, 共18兲 U¯i= Ui 2␮i 共␥i+␥o兲 =

_冑

− xo 1 − xo2 共1 + xo兲P1/2共x兲P_1/2

⬘

共x兲

关

共1 − xo 2_兲P 1/2

⬘

2_共x o兲 + 1 4共␧r− 1兲P_1/22 共xo兲 + 共xo+ 1兲P1/2共xo兲P_1/2

⬘

共xo兲

兴

. 共19兲

FIG. 4. Electric field components in Taylor’s solution as a function of the semiangle.

Figures5 and6show values of A and Uiin

nondimen-sional form as functions of the cone angle xo. Note that the

dimensionless velocity U¯i can be interpreted as an effective

capillary number. An interesting point of these results is that neither A nor Ui depend on the viscosity ratio 共i.e., on the

outer viscosity␮o兲. This is reasonable since in the thin layer

approximation, the viscous terms are dominant and the layer behaves as a solid moving at velocity Ui and pulling the

inner interface toward the vertex with a shear stress ␶e.

Therefore, the dependency on the outer viscosity ␮o

disap-pears when adding up Eqs.共3兲and共6兲, and also Eqs.共4兲and 共7兲. Furthermore, as shown in Eqs. 共18兲 and共19兲, the only dependence on surface tensions is through the effective sur-face tension 共␥i+␥o兲, which is clearly a result of the thin

layer approximation.

As shown in Figs.5and6, both A and Uireach a

maxi-mum within the valid range of cone angles. Note that we have only considered values of the cone semiangle smaller than 49

⬘

29°共corresponding to Taylor’s hydrostatic solution兲 since any solution above this value is physically unaccept-able due to the sign of the electrical shear stress. When the cone angle equals Taylor’s value, the electrical stress is nor-mal to the liquid interface. Consequently, both electrical shear stress ␶e and velocity Ui vanish at ␣= 49

⬘

29°. As

shown in Fig. 2, cone semiangles in our experiments are close to Taylor’s value and therefore the analytical model should be restricted to semiangles close to this value.

Once the velocity at the interface has been calculated, the radial dependency of the thickness of the thin layer can be obtained from the mass conservation equation:

Q = 2␲rh sin␪oUi, 共20兲

where Q is the flow rate injected through the conductive layer. Therefore, the thin layer thickness h depends on the radial distance as h⬃r−1_{, i.e., the layer thickness increases as}

we approach the vertex in order to maintain a constant flow rate. This condition sets limits to the assumed thin layer hy-pothesis 共h/rⰆ1兲, and thus this is only valid in a limited range of r far enough from the cone vertex where the layer thickness is small enough.

In addition to these results, we are also able to model the pressure gradients in the thin layer if one assumes a radial dependence compatible with self-similarity:

Po共r兲 = Pe+ f共r兲 = Pe+ B/r, 共21兲

B being a dimensional constant depending on xo that can be

calculated, together with the normal viscous stresses in the thin layer, from Eqs.共3兲and共6兲. By solving the system, we obtain B ¯ = 2 共␥o+␥i兲 B =

_冑

− xo 1 − xo2

冋

冉

共1 − xo2兲P1

⬘

/22共xo兲 + 关共␧r− 1兲/4兴P21/2共xo兲 + 2xoP1/2共xo兲P1

⬘

/2共xo兲 共1 − xo2兲P1

⬘

/22共xo兲 + 关共␧r− 1兲/4兴P21/2共xo兲 + 共1 + xo兲P1/2共xo兲P1

⬘

/2共xo兲

冊

− 1

册

, 共22兲 T ¯ = 4 共␥o+␥i兲 ␮o

冋

⳵u_␪o ⳵␪ + ur o

册

=

_冑

− xo 1 − xo2

冋

冉

共1 − xo 2_兲P 1/2

⬘

2_共x o兲 + 关共␧r− 1兲/4兴P_1/22 共xo兲 − xoP1/2共xo兲P_1/2

⬘

共xo兲 共1 − xo 2_兲P 1/2

⬘

2_共x o兲 + 关共␧r− 1兲/4兴P_1/22 共xo兲 + 共1 + xo兲P1/2共xo兲P_1/2

⬘

共xo兲

冊

+␥i−␥o ␥i+␥o

册

.

As shown in Fig. 7, B always takes negatives values, which is in accord with the progressive thickening of the liquid layer described in the previous paragraph. It is also interesting to observe how the pressure gradients vanish at Taylor’s angle. As mentioned above, in this limit, the liquid

conductive layer would be at rest, with no applied shear stress, and therefore with no velocity or pressure gradients in it.

Please note that it is not possible to maintain the conical shape of the inner meniscus if the conductive thin layer is at

FIG. 6. Plot for the dimensionless inner interfacial velocity U¯i= Ui␮i/共␥i

+␥o兲 vs the cone angle.

FIG. 5. Graphic for A¯2_{= A}2_␧

e/共␥i+␥o兲 in terms of the cone semiangle␣

= arccos共␪o−␲兲.

rest. On the contrary, the equilibrium of the conical charged external interface is indeed possible in a hydrostatic case since the normal electrical stress balances the capillary pres-sure.

V. CHARGE TRANSPORT AND CHARGE CONSERVATION EQUATIONS

Let us now check the consistency and validity of the analytical solution given here from the point of view of the charge conservation equations. As is well known, the electri-cal current at the conductive layer is the result of a current surface Is and the Ohmic conduction IB throughout the bulk

in the conductive liquid layer. The current surface results from two different mechanisms of charge transportation: charge convection and ionic mobility,

Is= 2␲r sin␣关␴共Ui+␯兩Eជr兩兲兴. 共23兲

␯兩Eជr兩 represents the relative velocity of the ions with respect

to the liquid flow, where␯ is the ion mobility of a particular ionic species. The convection current decays as we approach the cone vertex as the square root of the distance, while that due to mobility remains constant for all distances. In order to avoid unlimited values of the surface current, the conical region must be restricted to radius

r
Rmax=

冉

␯A

Ui

冊

2

. 共24兲

This radius represents the maximum distance from the vertex for which mobility dominates over convection. On the other hand, the Ohmic surface current through the thin layer is

IOhmic= 2␲r sin␣hK兩Eជr兩. 共25兲

I_Ohmicgrows inversely proportional to the square root of the distance to the origin, so that, to avoid unlimited values,

r Rmin=

冉

K ␧e 1 ␯A Q Ui

冊

2 . 共26兲

This radius represents the minimum distance from the vertex for which mobility is dominant over conduction. Therefore, the conduction and convection mechanisms are negligible within the range RmaxⰇrⰇRmin.

Note that this substantially differs from the typical charge transport mechanism in regular electrosprays. There, convection current becomes dominant due to the strong in-crease of velocity that takes place as we approach the cone vertex. However, in our model, surface velocity is required to remain constant for all radial distances to maintain the self-similarity between the shear stresses, normal stresses, and surface tension. Consequently, convection decreases with the distance to the origin and the majority of the charge is transported by ion mobility along the surface. This de-crease of charge convection is actually in balance with the conduction current throughout the conductive layer.

VI. CONCLUSIONS

In this paper, we have experimentally explored the range of viscosities and surface tensions for which pointed menisci are formed in the framework of compound electrosprays. In short, the experiment consisted in the formation of a cone-jet structure of a conductive liquid within a bath containing a dielectric liquid. A meniscus of a third liquid, also dielectric, was formed inside the Taylor cone. Then, the EHD-driven conical sink flow of the conducting liquid forming the cone-jet electrospray deformed the dielectric meniscus in such a way that it adopted a spout shape to form a coaxial jet with the conductive liquid. The formation of a conical tip at the dielectric meniscus, with no jet issuing from it, was observed when liquid injection was stopped in liquids with sufficiently high interfacial tension. A remarkable experimental result was that the semiangle at the tip of the meniscus comes rather close to the Taylor angle and almost independent of which were the values of the viscosities of the liquids em-ployed and the values of the interfacial tensions.

These experimental facts resemble the behavior of the steady humps reported in selective withdrawal18 and other similar entrainment experiments3 in which a liquid is with-drawn through a tube placed above a liquid-liquid interface at a certain distance, producing a hump on the lower liquid whose curvature increases with the suction flow rate; when a critical value of the flow rate is reached, the hump transforms into a jet. Both the critical tube height and the critical me-niscus curvature have been found to be independent of the viscosity of the lower liquid. In fact, they only depend on a capillary number based on the interfacial tension and on the characteristic external shear stress of the flow. Analogously, Courrech du Pont and J. Eggers3did not find any dependency with the viscosity ratio in the range they explored, although the behavior of the cuspidal tip in their system presents an unexpected stability for all capillary numbers. In our case, the possible solutions obtained from the model are also in-dependent of the outer viscosity due to the thin layer hypoth-esis; however, a more detailed numerical resolution of the problem would retain this dependency.

A simplified analytical model has been developed to de-scribe this complex EHD flow. The model employs self-similar solutions to describe the recirculatory flow inside the

FIG. 7. Graphic for B¯ =B/共␥o+␥i兲 vs the cone semiangle.

inner meniscus and a thin layer approximation for the outer conductive flow. The analytical model predicts some param-eters of the problem as a function of the cone angle xo. In the

experiments, we have found that the cone semiangles are close to the Taylor value共␣= 49

⬘

29°兲 for a broad range of ␭ values. These values are found to be compatible with the analytical solutions of the model.

Charge conservation equation sets a limit to the conical domain in which the analytical solution presented here is valid. In effect, electrical current is transported by convec-tion and ion mobility through the charged interface and by Ohmic conduction throughout the bulk of the conductive layer. The analysis of the three different mechanisms shows that the current remains constant in a conical region defined by RmaxⰇrⰇRmin, with Rmax=共␯A/ue兲2 and Rmin

=关QK/共␧e␯Aue兲兴2, where ion mobility is the dominant

mechanism of charge transport.

The regime found in our experiments is similar to others recently reported.5,9 These works have pointed out that micron-sized droplets can be certainly obtained with no need of electrical fields or surfactant-driven mechanisms if a proper mechanical deformation is applied to a liquid inter-face. Although in our case the forcing is electrical, the defor-mation of the inner meniscus is, as in the other cases, due to the shear viscous stress. Note that the liquid properties and capillary numbers are identical to those reported in viscous coflow experiments9in which the authors proved the genera-tion of micron-sized droplets in identical regimes as those reported here.

Let us finally point out that Taylor, in his latter years, was especially concerned with these types of conical points that he himself discovered in two completely different ex-periments, one regarding electrified liquid interfaces and nonelectrified ones in the other.19 The curious presence of both of them in the same experiment is a phenomenon that he would have probably enjoyed.

ACKNOWLEDGMENTS

The authors are indebted to F. J. Higuera for numerous discussions, suggestions, and corrections during the elabora-tion and writing of the paper. This work has been supported by the Spanish Ministry of Education and Science under Project No. DPI2004-05246-C04-01.

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