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-Je Analy ical Ex ension o Taylo ’s Elec os a ic Solu ion and he Asymp o ic Uni e sal Scaling Laws in Elec osp aying Al onso M. Gañán-Cal o Escuela Supe io de Ingenie os, Uni e sidad de Se illa, Reina Me cedes syn, 41012 Se illa, Spain (Recei ed 21 Ma ch 1997) An analy ical cone-je solu ion o he elec ohyd odynamic a omiza ion o liquids has been ound o an asymp o ic model assuming an in ini ely long and hin emi ed je . Uni e sal exp essions o he emi ed elec ic cu en , je shape, cha ge dis ibu ion, su ace cha ge, and o he essen ial elec ohyd odynamic quan i ies a e ob ained as unc ions o he liquid p ope ies and he emi ed liquid low a e. The ag eemen wi h published expe imen s is good. [S0031-9007(97)03566-7] PACS numbe s: 47.65.+a, 47.55.– In his Le e we epo a success ul analy ical app oach o he complex and mul idisciplina y p oblem o he elec ohyd odynamic sp aying o liquids. Fo su icien ly la ge elec i ica ion le els [1,2], his sp aying me hod uses he s ong, local elec ic o ces appea ing a a cha ged liquid-gas in e ace o p o oke liquid ejec ions in he o m o capilla y liquid h eads om he ip o conelike p o usions, gi ing ise o ex emely ine cha ged ae osols wi h as applica ions in science and echnology. He e we conside he mos ele an egime, he so-called cone-je mode [3] (see Fig. 1), achie ed when a conical liquid meniscus is o med a he ip o an elec i ied capilla y needle o a ange o elec ic po en ials [1–4]. A s eady, hin bu obus liquid h ead wi h a ypical diame e om se e al mic ons o nanome e s (which e en ually b eaks up in o ema kably homogeneous size d ople s) is emi ed om he apex o he conical meniscus when a s eady cons an liquid low a e Qis supplied h ough he needle. B ie ly ou lined, ou model app oach consis s o a basic elec os a ic conical solu ion plus an in ini ely long and hin je issuing om i s apex. The elec ohyd odynamic p ocess o liquid and cha ge emission is sol ed, and he emi ed elec ic cu en , je shape, cha ge dis ibu ion, e c. a e, o he i s ime, analy ically de e mined. Excep o he no a ion, he eade mos ly in e es ed in applica ions may skip he ollowing up o he Resul s pa ag aph. Basic elec os a ic solu ion.—In he absence o liquid emission (no je ), Taylo [2] ob ained an exac , in ini e elec os a ic solu ion in sphe ical coo dina es sR,ud(see Fig. 1) consis ing o an in ini e, pe ec ly conical equilib- ium shape wi h semiangle uT0.860 274 32 . . . . The elec ic po en ial ou side his elec i ied cone is gi en by FTsR,ud√2g ´0!1y2 D0Q1y2sudR1y2,(1) whe e D0 ansuTdQ0 1y2suTdg21y2,(2) g,´0, and Q1y2s and o he liquid-su ounding gas su ace ension, elec ical pe mi i i y o acuum, and Legend e modi ied unc ion o o de 1y2, espec i ely. The symbol 0means de i a i e espec o u. The liquid dielec ic cons an ´iis absen in his esul since i deals wi h an elec os a ic solu ion o a pe ec ly conduc ing liquid, o which he inne elec ic ield is null. Elec ic cone-je model.—Conside now a e y hin and e y long cha ged liquid je issuing om he ip o Taylo ’s cone. E en in he case o a leaky dielec ic (no pe ec ly conduc ing) liquid [5], i i mo es slowly enough o allow o elec ic elaxa ion, i.e., in ime scales la ge as compa ed o he elec ic elaxa ion ime e´iyK (whe e Ks ands o he liquid elec ical conduc i i y), he liquid bulk is quasineu al and he cha ges s ay a he su ace [5–8]. The elec ic ield in he liquid Eiis hen e y small compa ed o he ou e one E. We call his a “quasielec os a icype ec ly conduc ing” (QEPC) limi , o which he su ace cha ge ss´0En2´ i E i ncan be exp essed simply as ss.´0En, whe e Enand Ei na e he no mal componen s o Eand Ei, espec i ely, a he liquid su ace. The ole o ´iis hen negligible. In ou QEPC limi , Taylo solu ion FT o he ou e ield is modi ied by he appea ance o a new e m, say FG, owing o he je . As long as ou geome y allows supe posi ion, su icien ly a away om he poin R0, we can sea ch o a solu ion o he p oblem as F FT1F G . We will use he na u al ep esen a ion o he elec ic ield in sphe ical coo dina es, gi en in e ms o FIG. 1. S eady cone-je con igu a ion and sphe ical coo di- na es sys em. 0031-9007y97y79(2)y217(4)$10.00 © 1997 The Ame ican Physical Socie y 217 VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997 Legend e unc ions, o w i e FGas a se ies o he o m FGsR,udX n Dn Qnsud1CnPnsudgRn,(3) whe e Qnand Pns and o he Legend e unc ions o o de n, and hnjis a ce ain in ini e sequence o numbe s which sa is ies he equi emen s o comple eness o he se ies. This sequence will be consis en ly ound, and Dnand Cnwill be sol ed as pa o he p oblem. The ep esen a ion in sphe ical coo dina es is use ul o de i e ela ions a he cone. Le us w i e he cone su ace as uuT1z, whe e zis a unc ion o R ep esen ing he depa u e om Taylo ’s conical shape uuTowing o he p esence o he je (space cha ge e ec ). Fi s , he po en ial decay along he cone su ace is se e al o de s o magni ude smalle han he one a he je , owing o he la ge ans e sal sec ion o he cone [8,9]. The e o e, i s su ace can be conside ed equipo en ial. Second, su ace ension mus be balanced by he elec os a ic o ce. Assuming zøuT, one can linea ize hese wo condi ions a ound uuTand ob ain, a e some algeb a, he exp ession z2X n√´0 2g!1y2 Dn QnsuTd1CnPnsuTd Q0 1y2suTdR21y21n, whe e Cn2 2Q0 nsuTd anuT2 sn21y2dsn13y2d1bgQ n su Td 2P 0 n su Td anuT2 sn21y2dsn13y2d1bgP n su T d and b11 an2uT an2uT12Q00 1y2suTd anuTQ0 1y2suTd 0.259 648 3 . . . . (4) Fu he mo e he je su ace, loca ed a ound he axis up, can be ep esen ed as jsp2udR, whe e j s ands o he local adius o he je assuming jøR (Fig. 1). The local geome y o he je is hen a e y slowly a ying cylinde . This sugges s he use o mo e amilia exp essions o he elec ic ields a he je su ace [i.e., when u!pin exp ession (3)]. In ac , one may de ine An2µcossnpd12Cn psinnp∂Dn,(5) Bn2µcossnpd12Cn psinsnpd∂Dn¡∑ gE1Csn11d2ln2gµcossnpd12Cn psinsnpd∂2p 2sinsnpd 1Cncossnpd∏(6) in e ms o Dnand Cn(whe e gEand Ca e he Eule cons an and he digamma unc ion, espec i ely). Thus, he se ies AsRdPnAnRnand BsRdPnBnRnallow one o w i e exp essions o he elec ic ields sEn,Esda he je su ace (as in cylind ical coo dina es) simply as EnAyj,(7) EsET1d dR AlnsjyRd1Bg,(8) o jøR, whe e ET√2g ´0!1y2pD0 4R21y2.(9) Func ions AsRdand BsRd ep esen a cha ge dis ibu ion loca ed a he axis o symme y and he nonsingula pa o he co ec ion o Taylo ’s solu ion, espec i ely. Je hyd odynamics.—Since he e is a angen ial elec- ic ield Espoin ing in he axial di ec ion, he e is a mo- men um exe ed by he elec ic s ess on he su ace wi h alue sssEs. In he limi o an “in ini ely” hin je , his momen um is apidly di used in he adial di ec ion by iscous s esses h oughou he je ans e sal sec ion, and he eloci y p o ile becomes almos la wi h alue yQyspj2d[7,8,10]. In his limi , he liquid momen- um balance can be exp essed as d dR ∑P11 2 Q2 p2j4∏2 s j,(10) whe e Ps ands o he liquid p essu e. Since he p essu e jump ac oss he je su ace is in a la ge ex en balanced by su ace ension (in addi ion o he elec os a ic and he pola iza ion o ces [8]) i s alue is o he o de o gyj. The e o e, P,gyjcan be neglec ed s he kine ic ene gy Q2ys2p2j4d o e y hin je s j!0[8]. The capilla y equa ion can be hence o h excluded om he analysis. Finally, he elec ic cu en is d i en by bo h he su ace cha ge mo ion and he bulk elec ic conduc ion [7,8]: I2pjssQ pj21Kpj2Es.(11) No ice ha while he i s e m becomes dominan down- s eam along he je , whe e he je adius jbecomes e y small, he second one is dominan close o he cone apex, whe e he je is hicke . In pa icula , bulk elec ic con- duc ion is se e al o de s o magni ude la ge han he su - ace elec ic con ec ion a he cone, whe e a e y small adial elec ic ield in he liquid bulk p o okes he cha ge mig a ion owa ds he apex [9]. 218 VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997 Uni e sal scaling.—Fi s , using he pa ame e s h ,K,g,´0jone can de ine se e al cha ac e is ic quan- i ies: a cha ac e is ic low a e Q0 Kg21´21 0,an elec ic cu en I0´1y2 0g 21y2, a dis ance d0 sp22g´2 0 21K22d1y3, and an elec ic ield E0 s2g´21 0d21 0d1y2. Fo a gi en liquid, i has been shown ha Q0,d0, and I0a e ac ually o he o de o he minimum low a e, je diame e , and emi ed elec ic cu en , espec i ely, ha can be ob ained by elec osp aying [6,11,12]. In oducing he nondimensional low a e QQyQ0, and he numbe L0lnQ1y2, le us de ine (i) a ypical je adius R0d0Q1y2, (ii) an axial dis ance L0d0QL0, (iii) a ypical alue o he no mal ield a he je su ace En0E0L21y2 0, (i ) he same o he axial ield, Es0E0sQL0d21y2, and ( ) a ypical po en ial a ia ion along he je F0E0L0. No e ha in he asymp o ic limi Q¿1, he cha ac e is ic axial leng h o he je L0is e y la ge as compa ed o he je adius R0. In addi ion, L0becomes a la ge numbe . Wi h hese de ini ions, one can w i e Eqs. (8), (10), and (11) as a a he simple sys em o nondimensional uni e sal equa ions in e ms o he nondimensional quan i ies zRyL0, jyR0,aAysEn0R0d AysEs0L0L21 0d,esEsyEs0, and es0sET1Ù BdyEs0, wi h he nondimensional cu en IIy≥8I2 0QyL0¥1y2: eses02Ù a,Ù 22a 3es, es 2 21pa 2 I.(12) Te ms o he o de L21 0and smalle ha e been neglec ed in sys em (12). When z!`(i.e., close o he je b eakup), su ace cha ge ad ec ion becomes dominan (i.e., pay 2!I), since he liquid je mus e en ually b eak up and he liquid domain is no longe con inuous. In addi ion, he angen ial elec ic ield a he su ace esapp oaches es0 since he po en ial decay in he adial di ec ion om he su ace, o he o de En0R0, is negligible as compa ed o he po en ial decay owing o he ex e nal ield, o he o de Es0L0. When such condi ions a z!`a e imposed, he asymp o ic solu ion o sys em (12) is gi en by `z21y8 s4ID 0d 1y4,a`I1y2z21y4 2pD1y2 0 , es`p 4D 0z21y2.(13) The solu ion o sys em (12) o z,Os1dis now consis- en ly ound in e ms o powe se ies o he o m kk`√11 ` X n1 anz23ny4!,(14) whe e ks ands o ,a,o e s . The po en ial FG is hen gi en by exp ession (3) wi h hnjh1y22 3ny4jsn1,2,...,`d. Sys em (12), oge he wi h exp essions (4) allow he comple e and consis en calcula ion o he se ies by sol ing he e ms hanj ,a,esas unc ions o he eigen alue I, he nondimensional elec ic cu en . Elec ic p oblem a ound z0.—Se ies (14) a e abso- lu ely con e gen ou side a sphe e o adius Rp(which is a unc ion o I) a ound he o igin z0, and di e ges in- side. The cha ge dis ibu ion aszda he axis o symme y inside ha sphe e, uni ocally de e mined om he alues o he elec ic ield and po en ial a he con e gence a- dius, can be ep esen ed in e ms o posi i e powe s o z such as aP` n0gnzn. This se ies is sol ed using he o e lapping egion o absolu e con e gence o bo h se- ies. Linea izing he exp ession FFT1F Ga ound uuTand in oking (3) and (4), one ob ains ha he alue o he po en ial a he cone su ace is equal o 0. Since he solu ion a he je ’s side is egula a zRp, while he cone u ns in o a cusp close o he con e gence adius o he ou e se ies [see Fig. 2(a)], Eqs. (12) a e s ill alid inside he sphe e om he je ’s side up o he poin whe e !`. Thus, he solu ion o he p oblem is gi en by he alue o I o which he po en ial becomes 0 when goes o `inside he sphe e. One inds I.1.5, a e FIG. 2. (a) Cone-je shape, and cha ge dis ibu ion aszda he axis. Also plo ed, he Taylo cone (---). (b) Je shape, su ace ad ec ion cu en , and po en ial decay along he je . 219 VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997 FIG. 3. (a) Scaling o he measu ed cu en o h ee ep esen- a i e liquids om [6], and da a om [11] o QyQ0$30.0. The asymp o ic uni e sal scaling is gi en by a solid line. (b) Scaling o he expe imen ally measu ed d ople diame e o he same liquids om [6], and o wa e -suc ose solu ions, gi en in [12]. The heo e ical uni e sal scaling o he d ople diame e gi en by a solid line. in eg a ing Eqs. (12) wi h a con e gence adius o he ou - side se ies Rp.12.5. The inne solu ion gi es he an- si ion om he je o he cone, and shows how he elec ic cu en changes om dominan Ohmic conduc ion a he cone o su ace cha ge ad ec ion a he je [8]. Resul s.—The nondimensional cha ge dis ibu ion a he axis o symme y aszdand he shape o he cone a e gi en in Fig. 2(a). O he esul s o in e es , o example, he je shape , he su ace ad ec ion cu en , and he po en ial decay a he je axis, a e plo ed in Fig. 2(b). In e ms o physical quan i ies, he o al elec ic cu en and d ople diame e gi en by ou model a e I4.25"QKgyln√Q Q0!1y2#1y2 4.25√QKg L0!1y2 , d231.89R0 b3.78p22y3Q1y2√ ´0 gK!1y6 b, (15) bbeing henondimensional adius o he je a he b eakup poin . 1.89 s ands o he Rayleigh mos p obable je o d ople ela ionship, alid o elec osp ay [3]. Howe e , d ople size canno be exac ly ob ained om his analysis since he e is no b eakup egion in he asymp o ic model. In eali y, he je b eaks up in mos cases a a poin lo- ca ed oughly om z,10 o z,100 depending on he liquid iscosi y. Since he je shape changes as slowly as z21y8, one may es ima e an a e age alue b.0.6 wi hin maximum e o s o he o de o 25%, e en below he expe imen al unce ain ies in some cases. These uni- e sal asymp o ic scalings a e compa ed wi h esul s om some expe imen al s udies [6,11,12], using many di e - en liquids wi h pe mi i i ies spanning om ´i1.9´0 o ´i111.0´0. The elec ic cu en and d ople diame- e a e gi en in Figs. 3(a) and 3(b), espec i ely. The p esen analy ical esul s sugges ha bo h he elec ic cu en and he d ople diame e a e independen o he liquid pola i y. Liquids wi h la ge pola i ies usually p esen la ge conduc i i ies oo, esul ing in la ge expe imen al QyQ0 alues, which migh ha e led o a ibu e o he liquid pola i y he ole ac ually played by L0lnQyQ0in he expe imen al co ela ions [6,11,12]. The p esen analy ical esul s also explain why he cha ge- o-mass a io QyIshows a powe law app oxima ely in e sely p opo ional o d, no ed by many au ho s (see [12]) bu unexplained be o e. Finally, his analy ical model se es as a local solu ion close o he apex o a eal elec i ied cone, whe e he emission akes place. While his egion is local enough, he in luence o he needle-elec ode po en ial di e ence in he emi ed cu en and d ople size is small, as shown in mos published expe imen s [6,11,12]. This wo k is suppo ed by he Spanish Comisión In- e minis e ial de Ciencia y Tecnologı ´a, P ojec No. PB93- 1181. [1] J. Zeleny, Phys. Re . 10, 1 (1917). [2] G.I. Taylo , P oc. R. Soc. London 280, 383 (1964). [3] M. Cloupeau and B. P une -Foch, J. Elec os . 22, 135 (1989). [4] C. Pan ano, A.M. Gañán-Cal o, and A. Ba e o, J. Ae osol Sci. 25, 1065 (1994). [5] D.A. Sa ille, Annu. Re . Fluid Mech. 29, 27 (1997). [6] A.M. Gañán-Cal o, J. Dá ila, and A. Ba e o, J. Ae osol Sci. 28, 249 (1997). [7] J.R. Melche and E.P. Wa en, J. Fluid Mech. 47, 127 (1971). [8] A.M. Gañán-Cal o, J. Fluid Mech. 335, 165 (1997). [9] D.P.H. Smi h, IEEE T ans. Ind. Appl. 22, 527 (1986). [10] A.J. Mes el, J. Fluid Mech. 274, 93 (1994). [11] J. Fe nández de la Mo a and I.G. Losce ales, J. Fluid Mech. 260, 155 (1994). [12] D.-R. Chen, D.Y.H. Pui, and S.L. Kau man, J. Ae osol Sci. 26, 963 (1995). 220