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
në0
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].
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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.
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(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