Cone-Jet Analytical Extension of Taylor’s Electrostatic Solution and the Asymptotic Universal Scaling Laws in Electrospraying

VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997 Cone-Jet Analytical Extension of Taylor’s Electrostatic Solution and the Asymptotic Universal Scaling Laws in Electrospraying Alfonso M. Gañán-Calvo Escuela Superior de Ingenieros, Universidad de Sevilla, Reina Mercedes syn, 41012 Sevilla, Spain (Received 21 March 1997) An analytical cone-jet solution for the electrohydrodynamic atomization of liquids has been found for an asymptotic model assuming an infinitely long and thin emitted jet. Universal expressions for the emitted electric current, jet shape, charge distribution, surface charge, and other essential electrohydrodynamic quantities are obtained as functions of the liquid properties and the emitted liquid flow rate. The agreement with published experiments is good. [S0031-9007(97)03566-7] PACS numbers: 47.65.+a, 47.55.–t In this Letter we report a successful analytical approach to the complex and multidisciplinary problem of the electrohydrodynamic spraying of liquids. For sufficiently large electrification levels [1,2], this spraying method uses the strong, local electric forces appearing at a charged liquid-gas interface to provoke liquid ejections in the form of capillary liquid threads from the tip of conelike protrusions, giving rise to extremely fine charged aerosols with vast applications in science and technology. Here we consider the most relevant regime, the so-called cone-jet mode [3] (see Fig. 1), achieved when a conical liquid meniscus is formed at the tip of an electrified capillary needle for a range of electric potentials [1–4]. A steady, thin but robust liquid thread with a typical diameter from several microns to nanometers (which eventually breaks up into remarkably homogeneous size droplets) is emitted from the apex of the conical meniscus when a steady constant liquid flow rate Qis supplied through the needle. Briefly outlined, our model approach consists of a basic electrostatic conical solution plus an infinitely long and thin jet issuing from its apex. The electrohydrodynamic process of liquid and charge emission is solved, and the emitted electric current, jet shape, charge distribution, etc. are, for the first time, analytically determined. Except for the notation, the reader mostly interested in applications may skip the following up to the Results paragraph. Basic electrostatic solution.—In the absence of liquid emission (no jet), Taylor [2] obtained an exact, infinite electrostatic solution in spherical coordinates sR,ud(see Fig. 1) consisting of an infinite, perfectly conical equilibrium shape with semiangle uT0.860 274 32 . . . . The electric potential outside this electrified cone is given by FTsR,ud√2g ´0!1y2 D0Q1y2sudR1y2,(1) where D0ftansuTdQ0 1y2suTdg21y2,(2) g,´0, and Q1y2stand for the liquid-surrounding gas surface tension, electrical permittivity of vacuum, and Legendre modified function of order 1y2, respectively. The symbol 0means derivative respect to u. The liquid dielectric constant ´iis absent in this result since it deals with an electrostatic solution of a perfectly conducting liquid, for which the inner electric field is null. Electric cone-jet model.—Consider now a very thin and very long charged liquid jet issuing from the tip of Taylor’s cone. Even in the case of a leaky dielectric (not perfectly conducting) liquid [5], if it moves slowly enough to allow for electric relaxation, i.e., in time scales large as compared to the electric relaxation time te´iyK (where Kstands for the liquid electrical conductivity), the liquid bulk is quasineutral and the charges stay at the surface [5–8]. The electric field in the liquid Eiis then very small compared to the outer one E. We call this a “quasielectrostaticyperfectly conducting” (QEPC) limit, for which the surface charge ss´0En2´ i E i ncan be expressed simply as ss.´0En, where Enand Ei nare the normal components of Eand Ei, respectively, at the liquid surface. The role of ´iis then negligible. In our QEPC limit, Taylor solution FTfor the outer field is modified by the appearance of a new term, say FG, owing to the jet. As long as our geometry allows superposition, sufficiently far away from the point R0, we can search for a solution of the problem as F FT1F G . We will use the natural representation of the electric field in spherical coordinates, given in terms of FIG. 1. Steady cone-jet configuration and spherical coordinates system. 0031-9007y97y79(2)y217(4)$10.00 © 1997 The American Physical Society 217 VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997 Legendre functions, to write FGas a series of the form FGsR,udX n DnfQnsud1CnPnsudgRn,(3) where Qnand Pnstand for the Legendre functions of order n, and hnjis a certain infinite sequence of numbers which satisfies the requirements of completeness of the series. This sequence will be consistently found, and Dnand Cnwill be solved as part of the problem. The representation in spherical coordinates is useful to derive relations at the cone. Let us write the cone surface as uuT1z, where zis a function of Rrepresenting the departure from Taylor’s conical shape uuTowing to the presence of the jet (space charge effect). First, the potential decay along the cone surface is several orders of magnitude smaller than the one at the jet, owing to the large transversal section of the cone [8,9]. Therefore, its surface can be considered equipotential. Second, surface tension must be balanced by the electrostatic force. Assuming zøuT, one can linearize these two conditions around uuTand obtain, after some algebra, the expression z2X n√´0 2g!1y2 Dn QnsuTd1CnPnsuTd Q0 1y2suTdR21y21n, where Cn2 2Q0 nsuTd tanuT2fsn21y2dsn13y2d1bgQ n su Td 2P 0 n su Td tanuT2fsn21y2dsn13y2d1bgP n su T d and b11tan2uT tan2uT12Q00 1y2suTd tanuTQ0 1y2suTd 0.259 648 3 . . . . (4) Furthermore the jet surface, located around the axis up, can be represented as jsp2udR, where j stands for the local radius of the jet assuming jøR (Fig. 1). The local geometry of the jet isthen a very slowly varying cylinder. This suggests the use of more familiar expressions of the electric fields at the jet surface [i.e., when u!pin expression (3)]. In fact, one may define An2µcossnpd12Cn psinnp∂Dn,(5) Bn2µcossnpd12Cn psinsnpd∂Dn¡∑fgE1Csn11d2ln2gµcossnpd12Cn psinsnpd∂2p 2sinsnpd 1Cncossnpd∏(6) in terms of Dnand Cn(where gEand Care the Euler constant and the digamma function, respectively). Thus, the series AsRdPnAnRnand BsRdPnBnRnallow one to write expressions for the electric fields sEn,Esdat the jet surface (as in cylindrical coordinates) simply as EnAyj,(7) EsET1d dR fAlnsjyRd1Bg,(8) for jøR, where ET√2g ´0!1y2pD0 4R21y2.(9) Functions AsRdand BsRdrepresent a charge distribution located at the axis of symmetry and the nonsingular part of the correction to Taylor’s solution, respectively. Jet hydrodynamics.—Since there is a tangential electric field Espointing in the axial direction, there is a momentum exerted by the electric stress on the surface with value tsssEs. In the limit of an “infinitely” thin jet, this momentum is rapidly diffused in the radial direction by viscous stresses throughout the jet transversal section, and the velocity profile becomes almost flat with value yQyspj2d[7,8,10]. In this limit, the liquid momentum balance can be expressed as d dR ∑P11 2rQ2 p2j4∏2ts j,(10) where Pstands for the liquid pressure. Since the pressure jump across the jet surface is in a large extent balanced by surface tension (in addition to the electrostatic and the polarization forces [8]) its value is of the order of gyj. Therefore, P,gyjcan be neglected vs the kinetic energy rQ2ys2p2j4dfor very thin jets j!0[8]. The capillary equation can be henceforth excluded from the analysis. Finally, the electric current is driven by both the surface charge motion and the bulk electric conduction [7,8]: I2pjssQ pj21Kpj2Es.(11) Notice that while the first term becomes dominant downstream along the jet, where the jet radius jbecomes very small, the second one is dominant close to the cone apex, where the jet is thicker. In particular, bulk electric conduction is several orders of magnitude larger than the surface electric convection at the cone, where a very small radial electric field in the liquid bulk provokes the charge migration towards the apex [9]. 218 VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997 Universal scaling.—First, using the parameters hr,K,g,´0jone can define several characteristic quantities: a characteristic flow rate Q0rKg21´21 0,an electric current I0´1y2 0gr21y2, a distance d0 sp22g´2 0r21K22d1y3, and an electric field E0 s2g´21 0d21 0d1y2. For a given liquid, it has been shown that Q0,d0, and I0are actually of the order of the minimum flow rate, jet diameter, and emitted electric current, respectively, that can be obtained by electrospraying [6,11,12]. Introducing the nondimensional flow rate QQyQ0, and the number L0lnQ1y2, let us define (i) a typical jet radius R0d0Q1y2, (ii) an axial distance L0d0QL0, (iii) a typical value of the normal field at the jet surface En0E0L21y2 0, (iv) the same for the axial field, Es0E0sQL0d21y2, and (v) a typical potential variation along the jet F0E0L0. Note that in the asymptotic limit Q¿1, the characteristic axial length of the jet L0is very large as compared to the jet radius R0. In addition, L0becomes a large number. With these definitions, one can write Eqs. (8), (10), and (11) as a rather simple system of nondimensional universal equations in terms of the nondimensional quantities zRyL0,fjyR0,aAysEn0R0d AysEs0L0L21 0d,esEsyEs0, and es0sET1Ù BdyEs0, with the nondimensional current IIy≥8I2 0QyL0¥1y2: eses02Ù a,Ù f22af3es, esf2 21pa f2 I.(12) Terms of the order L21 0and smaller have been neglected in system (12). When z!`(i.e., close to the jet breakup), surface charge advection becomes dominant (i.e., payf2!I), since the liquid jet must eventually break up and the liquid domain is no longer continuous. In addition, the tangential electric field at the surface esapproaches es0 since the potential decay in the radial direction from the surface, of the order En0R0, is negligible as compared to the potential decay owing to the external field, of the order Es0L0. When such conditions at z!`are imposed, the asymptotic solution of system (12) is given by f`z21y8 s4ID 0d 1y4,a`I1y2z21y4 2pD1y2 0 , es`p 4D 0z21y2.(13) The solution of system (12) for z,Os1dis now consistently found in terms of power series of the form kk`√11 ` X n1 anz23ny4!,(14) where kstands for f,a,ore s . The potential FG is then given by expression (3) with hnjh1y22 3ny4jsn1,2,...,`d. System (12), together with expressions (4) allow the complete and consistent calculation of the series by solving the terms hanjf,a,esas functions of the eigenvalue I, the nondimensional electric current. Electric problem around z0.—Series (14) are absolutely convergent outside a sphere of radius Rp(which is a function of I) around the origin z0, and diverges inside. The charge distribution aszdat the axis of symmetry inside that sphere, univocally determined from the values of the electric field and potential at the convergence radius, can be represented in terms of positive powers of z such as aP` n0gnzn. This series is solved using the overlapping region of absolute convergence of both series. Linearizing the expression FFT1F Garound uuTand invoking (3) and (4), one obtains that the value of the potential at the cone surface is equal to 0. Since the solution at the jet’s side is regular at zRp, while the cone turns into a cusp close to the convergence radius of the outer series [see Fig. 2(a)], Eqs. (12) are still valid inside the sphere from the jet’s side up to the point where f!`. Thus, the solution of the problem is given by the value of Ifor which the potential becomes 0 when fgoes to `inside the sphere. One finds I.1.5, after FIG. 2. (a) Cone-jet shape, and charge distribution aszdat the axis. Also plotted, the Taylor cone (---). (b) Jet shape, surface advection current, and potential decay along the jet. 219 VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997 FIG. 3. (a) Scaling of the measured current for three representative liquids from [6], and data from [11] for QyQ0$30.0. The asymptotic universal scaling is given by a solid line. (b) Scaling of the experimentally measured droplet diameter for the same liquids from [6], and for water-sucrose solutions, given in [12]. The theoretical universal scaling for the droplet diameter given by a solid line. integrating Eqs. (12) with a convergence radius of the outside series Rp.12.5. The inner solution gives the transition from the jet to the cone, and shows how the electric current changes from dominant Ohmic conduction at the cone to surface charge advection at the jet [8]. Results.—The nondimensional charge distribution at the axis of symmetry aszdand the shape of the cone are given in Fig. 2(a). Other results of interest, for example, the jet shape f, the surface advection current, and the potential decay at the jet axis, are plotted in Fig. 2(b). In terms of physical quantities, the total electric current and droplet diameter given by our model are I4.25"QKgyln√Q Q0!1y2#1y2 4.25√QKg L0!1y2 , d231.89R0fb3.78p22y3Q1y2√r´0 gK!1y6 fb, (15) fbbeing thenondimensional radius of the jetat the breakup point. 1.89 stands for the Rayleigh most probable jet to droplet relationship, valid for electrospray [3]. However, droplet size cannot be exactly obtained from this analysis since there is no breakup region in the asymptotic model. In reality, the jet breaks up in most cases at a point located roughly from z,10 to z,100 depending on the liquid viscosity. Since the jet shape changes as slowly as z21y8, one may estimate an average value fb.0.6 within maximum errors of the order of 25%, even below the experimental uncertainties in some cases. These universal asymptotic scalings are compared with results from some experimental studies [6,11,12], using many different liquids with permittivities spanning from ´i1.9´0 to ´i111.0´0. The electric current and droplet diameter are given in Figs. 3(a) and 3(b), respectively. The present analytical results suggest that both the electric current and the droplet diameter are independent of the liquid polarity. Liquids with large polarities usually present large conductivities too, resulting in large experimental QyQ0values, which might have led to attribute to the liquid polarity the role actually played by L0lnQyQ0in the experimental correlations [6,11,12]. The present analytical results also explain why the charge- to-mass ratio QyIshows a power law approximately inversely proportional to d, noted by many authors (see [12]) but unexplained before. Finally, this analytical model serves as a local solution close to the apex of a real electrified cone, where the emission takes place. While this region is local enough, the influence of the needle-electrode potential difference in the emitted current and droplet size is small, as shown in most published experiments [6,11,12]. This work is supported by the Spanish Comisión Interministerial de Ciencia y Tecnologı ´a, Project No. PB93- 1181. [1] J. Zeleny, Phys. Rev. 10, 1 (1917). [2] G.I. Taylor, Proc. R. Soc. London 280, 383 (1964). [3] M. Cloupeau and B. Prunet-Foch, J. Electrost. 22, 135 (1989). [4] C. Pantano, A.M. Gañán-Calvo, and A. Barrero, J. Aerosol Sci. 25, 1065 (1994). [5] D.A. Saville, Annu. Rev. Fluid Mech. 29, 27 (1997). [6] A.M. Gañán-Calvo, J. Dávila, and A. Barrero, J. Aerosol Sci. 28, 249 (1997). [7] J.R. Melcher and E.P. Warren, J. Fluid Mech. 47, 127 (1971). [8] A.M. Gañán-Calvo, J. Fluid Mech. 335, 165 (1997). [9] D.P.H. Smith, IEEE Trans. Ind. Appl. 22, 527 (1986). [10] A.J. Mestel, J. Fluid Mech. 274, 93 (1994). [11] J. Fernández de la Mora and I.G. Loscertales, J. Fluid Mech. 260, 155 (1994). [12] D.-R. Chen, D.Y.H. Pui, and S.L. Kaufman, J. Aerosol Sci. 26, 963 (1995). 220