On the general scaling theory for electrospraying

J. Fluid Mech. (2004), ol. 507, pp. 203–212a. c 2004 Camb idge Uni e si y P ess DOI: 10.1017/S0022112004008870 P in ed in he Uni ed Kingdom 203 On he gene al scaling heo y o elec osp aying By ALFONSO M. GA ˜ N´ AN-CALVO G upo de Mec´ anica de Fluidos, E.S.I, Uni e sidad de Se illa, Camino de los Descub imien os s/n 41092 Spain (Recei ed 13 No embe 2003 and in e ised o m 1 Ma ch 2004) A sys ema ic dimensional a ionale is p oposed he e o analyse he elec ohyd o- dynamic equa ions go e ning liquid elec osp aying phenomena in he well-known s eady cone-je mode wi h no ambien discha ges. As a esul , a gene al, unified de- sc ip ion o he comple e pa ame ical space o he emi ed cu en and d ople size is gi en. Fou main dis inc subspaces, hei ele an bounda ies and co esponding scal- ing laws a e iden ified. Laws al eady p oposed fi in hei app op ia e egion, and p e iously unknown laws a e ound. A closed solu ion o he elec ic cu en Iwhen ine ia and pola iza ion o ces domina e is ob ained, in ag eemen wi h published expe imen al esul s. 1. In oduc ion The use o elec ohyd odynamic o ces o disin eg a e liquids om he mic on down o he nanome e ange in an o de ly way, e.g. by so-called cone-je elec osp aying (Zeleny 1917; Taylo 1964; Cloupeau & P une -Foch 1989), has g ea ele ance in he field o liquid a omiza ion, wi h housands o publica ions pe yea and comme cial de ices making use o i . Fu he mo e, since he d ople s p oduced a e highly cha ged, i has been applied wi h much success o he mass spec ome y o la ge biomolecules. Al hough elec osp ay is a obus and con ollable phenomenon, many aspec s emain no comple ely unde s ood, s i ing much con o e sy. This wo k aims o p opose a gene al dimensional desc ip ion o he en i e wo king pa ame e space o s eady cone-je elec osp aying. As a esul , we ha e es ablished a pa ame ical wo-dimensional ‘cha ’ wi h ou dis inc pa ame ical egions and co esponding scaling laws o he d ople size and he emi ed elec ic cu en , o guide elec osp ay use s o any gi en liquid and wo king condi ions. Al hough wo o hem ha e been al eady iden ified, wo a e new. These laws a e compa ed wi h a ailable published expe imen s o show hei alidi y. Some nume ical solu ions we e ecen ly p esen ed (Higue a 2003) o desc ibe he comple e ansi ion egion be ween an infini e Taylo cone and an infini e asymp o ic je (Ga˜ n´ an-Cal o 1997), which sol e he eigen alue p oblem o he emi ed elec ic cu en as a unc ion o he liquid p ope ies and he emi ed flow a e. Howe e , a comple e sys ema ic pa ame ical s udy o he phenomenon, including he asymp o ic limi s and egions o in e es , has ne e been heo e ically a emp ed. This wo k is ocused on he physical and ma hema ical modelling o malism o he cone-je elec osp aying phenomenon in o de o in es iga e whe he a comple e pa ame ical desc ip ion o iden i y all physically possible egimes and asymp o ic limi s can be es ablished (Ba enbla 1987, 1996). 2. Analysis a ionale Conside he cone-je configu a ion o figu e 1. A cone-like meniscus is a ached o a eeding ube wi h diame e D, om whose apex a hin liquid je is emi ed. The h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms. 204 A. M. Ga˜ n´ an-Cal o ~Lo ~L D z z 2ξ θ 0 Figu e 1. The cone-je geome y and coo dina es. p oblem a iables z, ,ξ, ,En,Ei n,Es,andτs=εo(En−βEi n)Esa e espec i ely he axial coo dina e along he je , adial coo dina e, he je adius, liquid eloci y, no mal ou e and inne elec ic fields on he je su ace, he su ace elec ic field in he axial di ec ion, and he angen ial su ace s ess (Melche & Wa en 1971; Ga˜ n´ an-Cal o 1997, 1999; Hohman e al. 2001b). The p oblem pa ame e s σ, K, ρ, µ and Qa e he liquid–gas su ace ension, liquid elec ic conduc i i y, densi y, iscosi y and emi ed flow a e, espec i ely. βis he a io o he liquid o acuum pe mi i i ies β=εi/εo. Using Coulomb’s law, one may exp ess he po en ial Φ( , z) due o he cone-je cha ged su ace as ha gi en by a cha ge line dis ibu ion A(z) a he axis: Φ(z, )=∞ −∞ A(z)dz 4[(z−z)2+ 2]1/2(2.1) whe e Enand Esmus be equal o he nega i e o he no mal and angen ial pa ial de i a i es o Φ, espec i ely, a he cone-je su ace gi en by =ξ(z)(Hohmane al. 2001b). The elec ic p oblem so s a ed is le unde e mined unless he app op ia e bounda y condi ions a e gi en, which include he ups eam applied elec ic po en ial a he liquid eeding ube, and he downs eam sp ay s uc u e o elec ode geome y ahead o he issuing je , oge he wi h he app op ia e chain o elec os a ic ‘images’ om −∞ o ∞(Hohman e al. 2001b). In a i ual p oblem wi h no liquid emission (Pan ano, Ga˜ n´ an-Cal o & Ba e o 1994), and he e o e wi h an equipo en ial cone su ace, each applied elec ic po en ial wi hin a na ow ange would gi e a pa icula cone-like meniscus geome y sa is ying all bounda y condi ions. The local s uc u e o he elec ic field in he icini y o he cone ip (cha ac e is ic leng h Lo, figu e 1) is shown o be Taylo ’s (Taylo 1964; Pan ano e al. 1994), whe e elec os a ic and su - ace ension o ces alone balance. Locally, and assuming z= 0 a he cone ip, Taylo ’s elec ic field is equi alen o ha gi en by he ollowing cha ge line dis ibu ion: A(z)=(σ/εo)1/2Do21/2(−z)1/2(2.2) o nega i e z alues, whe e Do=[ an(θT)Q2 1/2(θT)]−1/2,andθTand Q 1/2a e he Taylo angle in he absence o emission (see Pan ano e al. 1994; Ga˜ n´ an-Cal o 1997) and he de i a i e o he Legend e unc ion o o de 1/2, espec i ely. No e ha his exp ession h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms. On he gene al scaling heo y o elec osp aying 205 is independen o he applied po en ial. Imagine now ha liquid emission is in he o m o a s eady, ex emely hin liquid je o adius ξ, and ha he local size Loo he ip egion is sufficien ly la ge compa ed o he ypical je adius. In his case, he p oblem will be nea ly independen o he ou e , a bounda y condi ions and he applied po en ial a he scale D( he small influences o he applied ol age and he p esence o he cha ged sp ay a e no deal wi h in his wo k). Thus, ollowing p e ious s udies (Higue a 2003; Ga˜ n´ an-Cal o 1997), we will assume ha he cone-je ansi ion wi h ypical dimension Lis sufficien ly local (L.Lo) o neglec he ole o he applied po en ial a he scale D, as long as his po en ial is sufficien o main ain he s eady cone-je egime. To desc ibe he s eady cone-je elec osp ay phenomenon, we make he common, well-es ablished assump ion o a ‘leaky dielec ic’ (Sa ille 1997), which allows bulk ee cha ges o elax o he liquid su ace in imes esmalle han any o he cha ac e is ic ime ho he p ocess, i.e. e=εi/K  h=d2LQ−1,(2.3) hus defining a quasi-s eady s a e (Sa ille 1997; Ga˜ n´ an-Cal o 1997), whe e dand La e he cha ac e is ic ans e sal and axial dis ances, espec i ely (figu e 1). This condi ion will subsequen ly be e ified and can be exp essed as e/ h=βεoQ(Kd2L)−11. Unde his condi ion, βEi nEn, and hus he su ace cha ge can be exp essed as σe≃εoEn. The je slende ness (Melche & Wa en 1971; Egge s & Dupon 1994; Ga˜ n´ an-Cal o, D´ a ila & Ba e o 1997; Ga˜ n´ an-Cal o 1997, 1999; Hohman e al. 2001b) also allows some impo an simplifica ions. Fi s , owing o he smallness o he je diame e , aking he limi o (2.1) o O( =ξ)O(z) one can w i e En≃A/ξ (2.4) a he ou e je su ace. Secondly, since he su ace s ess is eadily diffused in o he whole liquid je sec ion, we can assume a plug-flow-field axial eloci y w i en by con inui y as =Q(πξ2)−1.(2.5) Thus, making use o cylind ical coo dina es cen ed a he cone-je necking (see figu e 1), he slende app oxima ion o he liquid momen um equa ion in he z- di ec ion can be w i en as: d dzσ ξ+1 2π2 ρQ2 ξ4+6µQ πξ2 d dzξ−1dξ dz=2εoEnEs ξ+εo 2 d dzE2 n+(β−1)E2 s.(2.6) The h ee e ms on he le -hand side s and o he axial esul an o he su ace ension o ce, he liquid ine ia, and he esul an o he iscous esis ance in he axial di ec ion. The wo e ms on he igh -hand side a e he axial componen o he angen ial elec os a ic su ace s ess, and he axial esul an o he no mal elec os a ic su ace s ess (comp ising he elec os a ic ‘suc ion’ and he pola iza ion o ce). The no mal and angen ial dynamical condi ions a he liquid su ace a e included in (2.6). Fu he mo e, he o mal asymp o ic bounda y condi ions a z→∞ equi e dξ/dz→0 (assuming ha he je b eakup zone is a away), while dξ/dz∼O(1) o z<0 (conical egion), wi h dξ/dz<0 in he whole zdomain. Finally, he cha ge con inui y can be exp essed as I=2Qεo ξEn+πξ2KEs(2.7) h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms. 206 A. M. Ga˜ n´ an-Cal o whe e Is ands o he o al emi ed elec ic cu en , which esul s o a gi en se o liquid p ope ies and emi ed flow a e. Fo mally, he emi ed elec ic cu en Iis an eigen alue o he p oblem (Ga˜ n´ an-Cal o 1997). In o de o sys ema ically sea ch o possible asymp o ically sel -simila solu ions confi ming he scalings sough , we will sea ch o gene alized affine ans o ma ions o he p oblem equa ions wi h asymp o ic in a iance espec o he go e ning pa ame e s iden ified (Ba enbla 1987, 1996). To ca y ou his cen al ask o his wo k, we in oduce he fi e cha ac e is ic dimensions L,d,E1,E2, associa ed wi h z, ξ, En,E s, espec i ely, and I. We emphasize cen al because we seek gene alized sel -simila solu ions o he equa ions (o in a ian solu ions wi h espec o he pa ame e s) in he in e media e egion be ween he cone and he de eloped je , since i is in his in e me- dia e egion whe e he eigen alue o he p oblem ( he emi ed elec ic cu en ) is fixed. 2.1. Dimensional a gumen s and de i a ion o he asymp o ic scales We emphasize he e ha he in e media e egion unde analysis is whe e he ansi ion om a dominan elec ic bulk conduc ion o a dominan su ace cha ge con ec ion akes place. Thus, om equa ion (2.7), one can consis en ly define Id QεoE1 =1,I d2KE2 =1.(2.8) I is also essen ial o no e ha he dominan pa o he in eg al in (2.1) is due o he p esence o he conical meniscus (2.2). Thus, om (2.1) and (2.2) one can also define E2=σ εoL1/2 .(2.9) The condi ion ha he sel -induc ion elec ic field o he je is ne e dominan leads, om (2.4), o he condi ion σL εo1/2 &E1d. (2.10) Finally, he momen um equa ion es ablishing a global balance be ween applied o ces (mo o s) and esis ance o ces p o ides he wo closing dimensional a gumen s o find he fi e cha ac e is ic dimensions (L, d, E1,E 2,I). A consis en analysis o his balance in ol es he ollowing wo domain ex emes: 2.1.1. De eloped je In his egion ex eme, I→2QεoEn/ξ, Es→(σ/εo)1/2Do21/2πz−1/2/4.(2.11) Since dξ/dz<0 e e ywhe e, he only posi i e mo o le in his egion is he axial componen o he angen ial elec os a ic su ace s ess (no e ha bo h E2 nand E2 s dec ease wi h z) wi h a limi ing alue 2εoEnEs ξ→σ εo1/2I QDo21/2πz−1/24.(2.12) The e o e, his e m mus be always dominan in he analysis o ou ansi ion scale L. Thus, o compa e su ace ension, ine ia and iscous o ces o he dominan mo o (2.12), we define he non-dimensional numbe s Rσ=ρQ3ε1/2 o Iσ1/2d4L1/2,R ρ=µQ2ε1/2 o d2L3/2Iσ1/2,R µ=σ1/2Qε1/2 o IdL1/2.(2.13) h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms. On he gene al scaling heo y o elec osp aying 207 The e a e h ee possibili ies: (I) Dominance o su ace ension o ce. Defining Rσ= 1, one mus ha e Rρ.1and Rµ.1. (II) Dominance o ine ia. Defining Rρ= 1, one mus ha e Rσ.1andRµ.1. (III) Dominance o iscous o ce. Defining Rµ= 1, one mus ha e Rσ.1and Rρ.1. 2.1.2. Cone-je necking In his opposi e egion ex eme, I→πξ2KEs. Since in his egion he angen ial elec os a ic su ace s ess decays as e han he axial componen o he no mal elec os a ic s ess (no e ha he cone is e en ually a pu e balance be ween su ace ension and no mal elec os a ic s ess), he elec os a ic suc ion o he pola iza ion o ce may domina e. Thus, i one seeks he egion joining he wo ex emes, one is le wi h wo possible defini ions: I Q=β−1 Lσεo L1/2 o I Q=1 Lσεo d1/2 ,(2.14) ep esen ing he dominance o ei he he elec os a ic suc ion o he pola iza ion o ce, espec i ely (ob iously, in ei he case he al e na i e exp ession mus be a limi ing condi ion consis en ly wi h sub-dominance). The alue o he liquid pola i y pa ame e βwill de e mine which scena io p e ails o a gi en Q. 2.2. The six undamen al asymp o ic scales The six possible condi ion pai s be ween he h ee possibili ies desc ibed in §2.1.1 and he wo desc ibed in §2.1.2, oge he wi h equa ions (2.8) and (2.9), p o ide he six diffe en se s o pa ame ical equa ions defining he six candida e gene alized affine ans o ma ions o he p oblem equa ions which may yield asymp o ically sel -simila o in a ian solu ions wi h espec o he p oblem pa ame e s, when he limi ing condi ions a e conside ed asymp o ically, i.e. assuming o al dominance o he e ms conside ed. To sa e space, we will only gi e he esul ing exp essions o he elec ic cu en Iand je diame e d. I is impo an o no e ha he non-dimensional equa- ions esul ing om he p oposed scalings (affine ans o ma ions) a e asymp o ically independen o all go e ning pa ame e s, and he e o e hei solu ions (should hey exis ) a e gene alized sel -simila . 2.2.1. IE-scaling: dominance o ine ia and elec os a ic suc ion This is he mos common pa ame ical egime encoun e ed in he elec osp ay li e a u e, which yields an in a ian o mula ion whose scaling o he elec ic cu en and je diame e is I=(σKQ)1/2,d=ρεoQ3 σK 1/6 .(2.15) Defining he wo non-dimensional pa ame e s αρ=ρKQ σεo ,α µ=K2µ3Q ε2 oσ3,(2.16) he alidi y limi s o his asymp o ic in a ian o mula ion gi en by all he limi ing condi ions may be finally exp essed as αρα1/4 µ,α ρ/(β−1) 1.(2.17) h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms. 208 A. M. Ga˜ n´ an-Cal o This scaling was o iginally p oposed by Ga˜ n´ an-Cal o (1999) and Ha man e al. (1999) wi hou men ioning i s limi s o alidi y. Since i has been widely e ified in nume ous expe imen al wo ks (e.g. Ga˜ n´ an-Cal o 1999; Ha man e al. 1999; Game o-Cas a˜ no & H uby 2002) and ex ensi ely calcula ed by Higue a (2003), we omi any u he analysis. The scaling p oposed in Ga˜ n´ an-Cal o (1997) lies wi hin his egion, bu in ha wo k he asymp o ic analysis was closed in he neck egion by an app oxima e pa ching ha missed he physical de ails in ha egion desc ibed by he comple e, non-dimensional equa ions s udied he e. Al hough he scaling o he je diame e was co ec , ha app oxima ion esul ed in he appea ance o a loga i hm o δρin he scaling o he emi ed cu en no sa is ac o ily suppo ed by expe imen s. 2.2.2. IP-scaling: dominance o ine ia and pola iza ion o ces In his case, we ob ain I=ρK2Q2 (β−1)εo1/2 ,d=ρεoQ3 σK 1/6 (2.18) wi h equi ed condi ions exp essed asymp o ically as 1αρ (β−1) αµ (β−1)4.(2.19) This new asymp o ic scaling holds o pola liquids wi hin he limi s gi en abo e. No e ha he exp ession o he je diame e is he same as in he p e ious IE-scale. Thus, d ople size measu emen s should be in a ian wi h espec o whe he elec os a ic o pola iza ion o ces a e dominan . Fu he mo e, educing his o mula ion o i s limi s, he momen um equa ion yields a beau i ul, uni o mly alid equali y a he cone-je necking (whe e I≃πξ2KEs) o sufficien ly la ge liquid pola i ies: d dz1 2π2 ρQ2 ξ4≃εo 2 d dz(β−1)E2 s=⇒I≃ρK2Q2 εo(β−1)1/2 .(2.20) Fo consis ency and o he specialized eade , i is impo an o men ion he e ha ou ansi ion egion in his egime is p eceded by a slende egion a ound z= 0 (figu e 1) wi h cha ac e is ic diame e d∗=(ρQ2/σ )1/3and leng h L∗=(β−1)d∗d∗whe e one can easily e i y ha , gi en he cha ac e is ic axial elec ic field p opo ional o Taylo ’s (E∗ 2=(σε−1 oL∗−1)1/2), hen (i) he elec ic conduc ion is dominan (I= πξ2KEs), (ii) he elec os a ic suc ion is o he o de o he pola iza ion o ce, o ine ia, and o he su ace ension o ce, i.e. σ/d∗∼ρQ2d∗−4∼εoE2 n∼εo(β−1)E2 s, and (iii) he e m 2εoEsEnξ−1is no dominan . This in e media e egion p o ides he consis en g ounds o ma ching he cone o ou ansi ion egion downs eam. In figu e 2, we ha e plo ed he elec ic cu en , I/IG(whe e IG=(σKQ)1/2), e sus he pa ame e αρ/(β−1), om se e al a ailable da a se s. We ha e used Fe n´ andez de la Mo a & Losce ales’s (1994) da a o o mamide, wa e , and oc anol. Oc anol da a ag ee wi h IE-scaling (ho izon al line), while he ends o wa e and o mamide end o ma ch he IP-scaling solu ion (inclined s aigh line) wi hin i s limi s o alidi y. We ha e also plo ed a se o in e es ing da a o wa e (L´ opez-He e a e al. 2004), which oge he wi h Fe nandez de la Mo a’s da a illus a e wo possible solu ions (‘long’ and ‘sho ’ cones), al eady epo ed in Chen, Pui & Kau man (1995) o wo e y sligh ly diffe en applied ol ages (less han 0.1%). These wo expe imen al b anches sugges he exis ence o a solu ion bi u ca ion (in ac , a bi-s able solu ion is expe imen ally ound), whe e he b anch wi h la ge cu en migh in ol e local gas ioniza ion effec s h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms. On he gene al scaling heo y o elec osp aying 209 I IG 1 0.1 1 10 100 αρ(β – 1)–1 Figu e 2. Compa ison o expe imen al da a wi h IE- and IP-scalings o he elec ic cu en made non dimensional wi h IG. Ho izon al line: IE scaling. Line o slope 1: IP scaling. Fe nandez de la Mo a’s da a: 䊊, o mamide; 䊐, e hylene glycol; 䉫,wa e ;䉮, oc anol. L´ opez-He e a’s da a: 䉲. a ound he cone ip (L´ opez-He e a e al. 2004). No e ha e/ h1 co esponds he e o αρ1, which is e ified o all published expe imen al da a wi hin his egion (β1). 2.2.3. VE-scaling: dominance o iscous o ce and elec os a ic suc ion This egime is ound in elec ospinning condi ions when he liquid has a sufficien ly la ge elec ic conduc i i y. I s cha ac e is ic emi ed elec ic cu en and je diame e scale as I=(σKQ)1/2,d=µε2 oQ3 σK21/8 (2.21) wi h alidi y limi s αρα1/4 µ,αµ (β−1)41.(2.22) He e, e/ h1 consis en ly gi es αρα1/4 µ. This scaling, which holds in a a ie y o expe imen al si ua ions o high- iscosi y liquids, was ecen ly and independen ly p oposed by Higue a (2003). Figu e 3 shows many diffe en esul s om he li e a u e (Game o-Cas a˜ no & H uby 2002; Chen e al. 1995; Ga˜ n´ an-Cal o e al. 1997) which ag ee wi h IE scaling (αρα−1/4 µ>1) o he d ople diame e s, while he ecen esul s by Ku e al. (2001) using glyce ol wi h diffe en elec ical conduc i i ies confi m he end o his VE-scaling (αρα−1/4 µ1). The plo ed s aigh lines illus a e he slope o he wo je diame e s om IE- and VE-scalings. Howe e , ca e is equi ed when using he VE-scaling o he je diame e , since high- iscosi y liquids usually yield e y long je s, wi h a s ong hinning a downs eam o ou ansi ion Lscale. One could end up wi h almos any d ople size dependence (inc easing o dec easing) on he liquid iscosi y (see o example Jayasinghe & Edi isinghe 2002), depending on he loca ion o he b eakup poin . Typically, he e is a la ge je - o-d op diame e a io in a iscous b eakup compa ed o a nea ly in iscid one, bu his p oblem is ou side he scope o he p esen wo k. h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms. 210 A. M. Ga˜ n´ an-Cal o dg dIE 1 10 0.01 0.1 101 100 αραµ –1/4 Figu e 3. Da a o he le (䉱), d ople diame e s dg om Ku e al. (2001) using one liquid only, compa ed o he VE-scaling o he je diame e (con inuous line). Da a o he igh , d ople diame e s aken om he li e a u e (Game o-Cas a˜ no & H uby 2002; Chen e al. 1995; Ga˜ n´ an-Cal o e al. 1997) o diffe en liquids, compa ed o he IE-scaling (ho izon al line). Da a made non-dimensional wi h d om he IE scaling. 2.2.4. VP-scaling: dominance o iscous o ce and pola iza ion o ce This scaling is gi en by I=µ3K3Q2 (β−1)4σ2ε2 o1/2 ,d=µQ (β−1)σ1/2 (2.23) wi h consis en alidi y limi s αρ (β−1) αµ (β−1)41.(2.24) The equi emen e/ h1 now gi es αµ1. This is also a new scaling which holds o e y iscous pola liquids wi hin he condi ions gi en abo e. Howe e , he au ho has no ound published expe imen s unde hese condi ions epo ing he exis ence o a capilla y conical meniscus, i.e. all si ua ions belonging o his asymp o ic egime show a e y elonga ed meniscus shape which ha dly esembles a well-defined cone egion. 2.2.5. Ma ginal scalings: dominance o su ace ension o ces The dominance o su ace ension is a ma ginal si ua ion ha some au ho s ha e al eady heo e ically conside ed (Che ney 1999; Higue a 2003). We men ion i he e o o mal comple eness, al hough we ha e ne e obse ed a well-defined ange o expe i- men al da a belonging o a egime whe e su ace ension domina es in absolu ely s able condi ions, and he e o e we omi a de ailed analysis. The e a e wo possibili ies: (i) Dominance o elec os a ic suc ion: I β−1 is small enough and su ace ension o ce domina es, he esul ing in a ian o mula ion gi es I=(σKQ)1/2,d=εoQ K1/2 (2.25) which is alid unde condi ions αρ.1,α µ.1,(β−1) .1. This ob iously holds o h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms. On he gene al scaling heo y o elec osp aying 211 0.01 0.01 0.1 1 0.1 1 10 100 αρ / (β – 1) αµ 1/4 /(β – 1) Fo mamide Wa e Wa e -2 E hylene Glycol Oc anol Glyca ol + Nal Glyca ol Figu e 4. The ou main pa ame e subspaces: whi e (IE-scaling), da k g ey (IP-scaling), ligh g ey (VE-scaling) and medium g ey (VP-scaling). No e ha he da a in he IP egion and close o i s e ical bounda y consis en ly end o ollow he IP-scaling in figu e 2 o he cu en , while da a in he VE egion end o ollow he VE-scaling in figu e 3 o he je size. non-pola liquids only, in he admisible limi s o s abili y. In his scaling d=L,which is a limi ing si ua ion (dmus be d.L). (ii) Dominance o pola iza ion o ce: I βis la ge enough and su ace ension o ce domina es, he esul ing asymp o ic in a ian o mula ion yields I=σKQ β−11/2 ,d=εoQ K1/3 ,(2.26) unde condi ions αρ.1,(β−1) &1,α µ.(β−1)3. This las scaling was o iginally p oposed by Fe n´ andez de la Mo a & Losce ales (1994), and subsequen ly s udied by Che ney (1999). Howe e , expe imen al da a epo ed in Fe n´ andezdelaMo a& Losce ales (1994) belong o any o he o he ou pa ame e subspaces (IE, IP, VE, VP), and do no seem o ollow hei p oposed ma ginal scaling (see figu e 2). To summa ize, he pa ame e space esul ing om his analysis is plo ed in figu e 4. We show how he dimensional a iables should scale wi h he p oblem pa ame e s wi hin each egion o asymp o ic alidi y. In each o he ou pa ame e subspaces o in e es , he se o esul ing non-dimensional equa ions is, sufficien ly a om he bounda ies, independen o he p oblem pa ame e s. We ha e p oposed a closed solu ion o he elec ic cu en Iwhen ine ia and pola iza ion o ces domina e (IP-scaling) in ag eemen wi h published esul s. Finally, i should be emphasized ha he absence o applied ol age diffe ence be ween he liquid and he elec ode in his analysis is applicable in expe imen al condi ions when he e is an almos conical s a ic meniscus shape held by elec os a ic o ces. This is ypical when IE- and IP-scalings hold. When iscous o ces a e do- minan (mos elec ospinning si ua ions) he cone de o ms by iscous o ces a la ge ups eam dis ances om he je , making he cone ha dly dis inguishable om he je in some cases (Hohman e al. 2001a; Feng 2002), which may in alida e ou assump ions. h ps://doi.o g/10.1017/S0022112004008870 Downloaded om h ps://www.camb idge.o g/co e. Uni e sidad de Se illa, on 05 No 2020 a 16:44:39, subjec o he Camb idge Co e e ms o use, a ailable a h ps://www.camb idge.o g/co e/ e ms.