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
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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
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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)
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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)
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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)
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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
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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.
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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
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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.
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