Elec o-Flow Focusing: The High-Conduc i i y Low-Viscosi y Limi
Al onso M. Gan
˜a
´n-Cal o*
Escuela Supe io de Ingenie os, Uni e sidad de Se illa, Camino de los Descub imien os, 41092 Se illa, Spain
(Recei ed 28 July 2006; e ised manusc ip ecei ed 6 Feb ua y 2007; published 30 Ma ch 2007)
Elec o- low ocusing, a echnique combining he ea u es o elec osp ay (ES) and low ocusing (FF),
p o ides a eliable ool o each pa ame ical mic oje ing anges no a ainable by ES o FF alone unde
speci ic ope a ional egimes (liquid p ope ies and low a e). In his Le e , we p o ide no only a closed
heo e ical model p edic ing he diame e o a high elec ical conduc i i y elec o- low ocused liquid
mic oje , bu also i s con ec i e o absolu e ins abili y, linked o he je ing- o-d ipping ansi ion and he
minimum liquid low a e ha can be ejec ed in s eady je ing egime, in which he smalles d ople s a e
issued. Good ag eemen is ound wi h expe imen al alues.
DOI: 10.1103/PhysRe Le .98.134503 PACS numbe s: 47.55.D, 47.20.D , 47.55.db, 47.65.d
Owing o ex ensi e echnological applica ions, on he
ack o pionee s [1,2], he physics o cone-je elec osp ay
has come unde in ense sc u iny in he las decades ([3–9],
among o he s). In addi ion, low ocusing [10,11] has
ea ned a en ion in he mic o luidic communi y owing o
i s simplici y, obus ness, and eliabili y as a means o
gene a e s eady mic oje s and nea ly monodispe se mic o-
d ople s. Nume ous applica ions exis , e.g., he p oduc ion
o high quali y ae osols and mic osphe es [12]. The com-
bina ion o elec osp ay and low ocusing (elec o- low
ocusing [13], EFF; see Fig. 1), an al e na i e o ei he
echnique on i s own, is he e analyzed in he limi s o high
elec ical conduc i i y o he ocused liquid, and low is-
cosi ies o bo h ocused and ocusing luids. Fo any pa -
icula choice o luid p ope ies and low a es, EFF leads
o ex ended mic oje ing pa ame ical anges, and he e-
o e o hinne and as e mic oje s whose inal diame e is
he e heo e ically p edic ed. Ou analysis is pa icula ly
well sui ed o high-conduc i i y aqueous solu ions and
liquid me als ocused by o he dielec ic luids (gases o
liquids, [14]). Fu he model de elopmen s could explo e
he in luence o o he pa ame e s: mode a e iscosi y, bulk
ee cha ge elaxa ion, o pola iza ion o ces.
Thus, a plu ali y o heo e ical insigh s a e he e b ough
oge he , including ee-su ace in iscid elec ohyd ody-
namic low, and spa io empo al s abili y analysis. In e ec ,
we examine he pa ame ical anges whe e elec o- low
ocused s eady je s can be achie ed in he low- iscosi y
ange. The con ec i e-absolu e (CA) ins abili y bounda y
ob ained is assumed coinciden wi h he je ing- o-d ipping
ansi ion; i is loca ed a he minimum low a e ha can be
s eadily elec o- low ocused (as obse ed in elec osp ay).
Ou model includes he key physical pa ame e s ac ing on
he CA ansi ion in a wide ange o si ua ions. We conside
liquids o low iscosi y and high elec ical conduc i i y,
e.g., aqueous solu ions and liquid me als, ocused by low-
iscosi y dielec ic luids. In addi ion, in e acial bounda y
laye e ec s a e aken in o accoun , e ealing hei in lu-
ence on he je ins abili y.
Theo e ical model. S eady je diame e .—A low a e Q1
o a liquid o densi y 1and small iscosi y 1, wi h good
elec ical conduc i i y, is ejec ed in o he ambien h ough
a small o i ice o adius R, in he o m o a liquid je o
adius RoRissuing om a la ge meniscus (see Fig. 1
and [10]). A low a e Q2o a second dielec ic luid o
densi y 2and small iscosi y 2is o ced coaxially wi h
he liquid conduc o h ough he ound exi o i ice. The 1–
2 in e ace is main ained a a cons an elec ic ol age V
( angen ial elec ic ield along he je su ace anishes)
ela i e o he o i ice pla e. Neglec ing he bounda y laye
hickness a he o i ice edge and he in luence o he hin
ocused je , he luid eloci y nea he axis a he exi o i ice
can be equa ed o he in iscid eloci y ([15], p. 1294):
U2Q2
2R2:(1)
In addi ion, since he no mal elec ic ield a he je su ace
(En) is much la ge han he one a he ups eam meniscus
su ace (Fig. 1[6]), he ollowing momen um balance
holds a he o i ice exi :
2R1
o1Q1
Ro2"oE2
n2Q2
2R2
(2)
(su ace ension , acuum pe mi i i y "o). Di iding
Eq. (2)by1Q1
Ro2leads o
H
2
R
1
z
V
1
D
()
[]
2/1
2
/1 R
Ro
o−
=
η
η
=0
Ro
FIG. 1. The elec o- low ocusing (EFF) con igu a ion. Obla e
sphe oidal coo dina es.
PRL 98, 134503 (2007) PHYSICAL REVIEW LETTERS week ending
30 MARCH 2007
0031-9007=07=98(13)=134503(4) 134503-1 ©2007 The Ame ican Physical Socie y
2
We 11
4Q2x4;(3)
whe e xRo=R,2=1, and QQ2=Q1.We 
1Q2
12R3
o1is he Webe numbe and 
2"oE2
nR4
o1Q2
11is he nondimensional je su ace
cha ge (squa ed).
Co ona o gas ioniza ion e ec s as well as ion e apo a-
ion kine ics a he icini y o he exi o i ice, ei he a he
o i ice edge o a he je su ace, will be igno ed by ou
heo y [16]. Now, he je su ace cha ge a he o i ice may
be accu a ely calcula ed by using he je ’s geome y nea
he exi (Fig. 1). Obla e sphe oidal coo dina es (OSC) (,
) p o ide an adequa e basis o he desc ip ion o he
in iscid liquid low and he elec ic ield dis ibu ion up-
s eam om he ci cula ou le ( adius R). The exp ession
o he je ’s su ace a he o i ice in OSC is hus o’1
x2=2, as long as x1. The elec os a ic ield p oduced by
he slende je can be app oxima ed by he wi e-o i ice ield
gi en by [17]:
; ’Va c anh 
a c anh o!En 1
h
@
@
121=2

@
@ ;(4)
whe e his he scale ac o o coo dina e . Thus, he
dimensionless je su ace cha ge a he o i ice exi can be
app oxima ed by
2"oE2
nR4
o1Q2
11’2x2"oR2V2
1Q2
1lnx=22;(5)
up o Ox2e o s, which combined wi h Eq. (3) and he
We;x;Q;de ini ions yield he exp ession o he je
adius Roa he o i ice exi in e ms o he expe imen al
pa ame e s.
Je ins abili y. De i a ion o he dispe sion ela ion and
pa ame ical ealm o je ing.—The spa io empo al (CA)
ins abili y o liquid je s has been he subjec o inc easing
a en ion since pionee ing wo ks [18,19]; see also
[11,20,21]. Assuming in iscid liquid low, Leib and
Golds ein de e mined he minimum liquid eloci y below
which a cylind ical je becomes absolu ely uns able. A
ha poin , je expe imen s show a ansi ion om je ing o
d ipping, unde simila ope a ional condi ions. To in es-
iga e he CA ins abili y o he issuing cha ged je , we
assume: (1) he liquid meniscus om which he je issues
is s able; (2) he je is su icien ly slende o he in ini e
cylinde heo y o hold a he icini y o he o i ice exi .
Assump ion (1) in ol es se ing he app op ia e eeding
ube–o i ice dis ance H o ensu e meniscus s abili y (see
[22] o he pu ely elec os a ic case, and [23] o a pu ely
hyd odynamic example): he e o e, ou s abili y analysis
will be limi ed o he s udy o he CA ins abili y o he
issuing liquid je . We he e o e s udy he spa io empo al
s abili y o an in ini e cylind ical liquid je wi h adius Ro,
mo ing wi h uni o m eloci y U1su ounded by a co low-
ing s eam o luid 2 wi h uni o m eloci y U2.U1and U2
will gene ally be di e en , so ha ou model will be
asymp o ically co ec i and only i iscous e ec s a e
consis en ly conside ed, con ined o he icini y o he
in e ace. E alua ing he ac ual su ace eloci y Us[24]
is o undamen al impo ance in ou spa io empo al ins a-
bili y analysis since he e ec o he con ec i e eloci y on
su ace wa es canno be neglec ed. Thus, iscous e ec s
a e only no iceable in wo adjoining bounda y laye s
whose hicknesses a e 11RR2
o1
1Q1
11=2and 2
2R31
2Q1
21=2. Axial p essu e g adien s a e mode a e
enough along he je su ace (RoR) o ensu e ha
almos -Blasius laye s de elop simul aneously om he
meniscus, p o ided 1Roand 2Ro. Thus, equa -
ing he iscous angen ial s esses 1and 2a he je
su ace, one ob ains a he exi o i ice:
11UsU131=222U2Us31=2(6)
whe e U1and U2a e na u ally he axial eloci ies o he
co e liquid and he ocusing luid away om he bounda y
laye s. This balance immedia ely p o ides an exp ession
o Us, gi en ha U1Q1=R2
oand U2Q2=2R2:
Us11=3U2=U1
11=3U1
11=3Qx2=2
11=3Q1
R2
o
(7)
whe e 2=1. We use cylind ical pola coo dina es
( ,z) along he je , and assume pe u ba ions p opo ional
o expikz ! . Wa e equency !, ime , wa e num-
be kand coo dina es ; zga e scaled wi h U1=Ro,Ro=U1,
1=Ro, and Ro, espec i ely. In ou analysis, we assume he
liquid o be an in iscid pe ec conduc o . The equa ions
exp essing mass, momen um, and elec ic cha ge conse -
a ion o bo h luids 1 (conduc o ) and 2 (dielec ic),
oge he wi h he no mal s ess balance a he je su ace
(see, o example, [25] o a simple de i a ion wi hou
dynamic e ec s o he ou e dielec ic), p o ide he dis-
pe sion ela ion be ween he wa e numbe kand i s co e-
sponding equency !. In ou spa io empo al analysis,
since he liquid mo es wi h uni o m eloci y U1,we
need o eplace he wa e equency !by !0!k
in he dispe sion ela ion. Fu he mo e, while in [25] he
elec ic ield is caused by a gi en ol age di e ence be-
ween he je and a concen ic elec ode o adius b, he
su ace cha ge is he e ela ed o he applied ol age V
h ough Eq. (5). Acco dingly, he ollowing asymp o ically
consis en dispe sion ela ion esul s:
!k!kUs
U1Iok
I1k!kUs
U1!kU2
U1Kok
K1kk21
We 1kK1k
Kokk(8)
PRL 98, 134503 (2007) PHYSICAL REVIEW LETTERS week ending
30 MARCH 2007
134503-2
whe e bo h !and ka e assumed complex numbe s; Iand
Ka e Bessel unc ions. Nex , we ollow a spa io empo al
o malism o desc ibe he CA cha ac e o axisymme ic
ins abili ies in he ou -dimensional space We;;;g.
We seek solu ions o he dispe sion ela ion 8 which sa is y
d!=dk 0and si in he lowe complex hal plane
Imk<0, wi h Im!0[18,26–29]. We choose solu-
ions whose spa ial b anches depa ing om he saddle
poin d!=dk 0o igina e om sepa a e hal es o he k
plane ([28], p. 484).
Al hough an exhaus i e explo a ion o he pa ame ical
ealm We;;;gis he e ou o ou scope, a signi ican
collec ion o esul s a e summa ized in Figs. 2(a) and 2(b),
whe e we ha e plo ed by con ou lines he CA ansi ion
su ace We;; o 0:1and 1below
(abo e) which he ins abili y is con ec i e (absolu e). The
alues o and explo ed co espond o common sol-
en s and liquid me als EFF ocused by any gas ocommon
liquid dielec ic (e.g., wa e in ai , wa e in oil [30], me -
cu y in oc ane). In e es ingly, he e is a Webe numbe
ange We We; whe e he ins abili y is always ab-
solu e (uncondi ional d ipping) independen ly o . This
a ea is ma ked below he 0isocon ou in Figs. 2(a)
and 2(b). Mo e in e es ingly, om he ends exhibi ed in
hese igu es and making We su icien ly small, one can
ob ain he densi y a io as a unc ion o below which
uncondi ional con ec i e ins abili y (uncondi ional je ing)
is ound in he absence o elec i ica ion, see Fig. 3.Fo 
alues abo e ha cu e in Fig. 3, one can ind
absolu e o con ec i e ins abili y depending on We and .
This s iking esul has a clea physical explana ion: hose
ocusing luids p o iding su ace eloci ies Us=U1>1
abo e all ups eam wa e speeds ( equi ing smalle 
wi h as la ge as possible) always pull je s om hei
ocused pa ne s. The loca ion o nine luid pai s in he (,
) plane is plo ed: h ee liquid conduc o s (me cu y, gal-
lium, saline wa e ) combined wi h h ee ocusing dielec ic
luids (ai , hep ane, silicone oil 5cP). Only hose
combina ions in ol ing me als and ocusing luids wi h
la ge kinema ic iscosi ies 22=2(namely ai
21:5105m2=s, and silicone oil 25:15 
106m2=s) lead o uncondi ional je ing in he absence
o elec i ica ion, a esul o p ac ical impo ance.
Ou CA p edic ions ha e been ca e ully es ed by ex-
pe imen . A p essu ized aluminum cubic box (50 mm long,
wall hickness 5 mm) wi h a 2 mm hole cen e ed in one o
i s aces is used. A 3 mm disk (75 m hick) wi h a cen al
o i ice ( adius R100 m) is glued o he inne side o
he 2 mm hole and aligned wi h i , co e ing i comple ely.
A s ainless s eel liquid eeding ube (OD 1 mm, ID 0.7 mm)
passes h ough he opposi e ace o he box, and a 7 mm
long silica capilla y (sha pened as in Fig. 1, wi h D1
150 m) p o udes om i s end inside he box. The ip o
said silica capilla y, coaxial wi h he o i ice, is loca ed a
H200 m om he 3 mm disk. Fil e ed d inking wa e
(measu ed elec ical conduc i i y K0:015 S=m, and
su ace ension 0:072 N=m) is supplied h ough he
ube by a Cole-Pa me pump (sy inge B-D20 cm3), ol -
age by a powe supply Be an 250B-10R, and comp essed
ai is supplied o he box a a cons an low a e Q2
23 l=h(s d). He e 0:0012 and 0:016. T ansi-
ional low a es a e ca e ully assessed by obse a ion o
he sudden issuing sp ay changes a he je ing- o-d ipping
0.1
1
10
100
0.000 01 0.001 0.1 10
ρ
We
ABSOLUTE
(DRIPPING)
µ
=0.01
0.7
Γ=0.5
1
1
2
10
5
(a)
µ
=1
0.1
1
10
100
0.01 0.1 1 10
ρ
We
ABSOLUTE
(DRIPPING)
Γ=0.5
0.7
1
2
10
5
(b)
FIG. 2. Isocon ou s o he CA ansi ion su ace We;;;g. (a) 0:01; (b) 1.
0.000 01
0.001
0.1
10
0.001 0.1 10 1000
µ
ρ
ALWAYS CONVECTIVE
(UNCONDITIONAL JETTING)
(Γ=0)
ai -Hg
ai -Ga
ai -
O
H2
hep ane-OH
2
hep ane-Hg
hep ane-Ga
S.oil(5)-OH2
S.oil(5)-Hg
S.oil(5)-Ga
FIG. 3. Densi y a io  alues below which uncondi ional je -
ing is ound: no elec i ica ion. Se e al luid pai s shown: h ee
liquid conduc o s (me cu y, gallium, saline wa e ) and h ee
ocusing dielec ic luids (ai , hep ane, silicone oil 5cP).
PRL 98, 134503 (2007) PHYSICAL REVIEW LETTERS week ending
30 MARCH 2007
134503-3
ansi ion, hus ob aining he ansi ion Q1as a unc ion o
applied ol age V, om which We and a e e alua ed
om Eqs. (3) and (5). Expe imen al ag eemen wi h heo-
e ical p edic ions (Fig. 4) is good and p omising: de ia-
ions occu only o V>1kV(We alues abo e 0.8 in
he plo ), when angen ial elec ic ields on he je su ge
owing o ini e conduc i i y e ec s, enhancing con ec i e
ins abili y. A e y signi ican d ople olume educ ion
ob ained o inc easing applied ol age in he je ing e-
gime, close o he CA ansi ion, is shown in Fig. 4(inse ).
Expe imen ally measu ed d ople size alues de ia e om
heo e ical p edic ion [using Eqs. (3) and (5), and he
ins abili y wa eleng hs gi en by he s abili y analysis]
less han 20%, wi hin he je ing pa ame ical egime.
This wo k is suppo ed by he Minis y o Science
and Technology o Spain, G an No. DPI2004-07197.
The expe imen al measu emen s ha e been aken by
M . Benjamin Blu h. Sugges ions om D . Pascual
Riesco-Chueca and ex ensi e discussions wi h D . Pio
Ga s ecki a e highly app ecia ed.
*Email add ess: [email p o ec ed]
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[29] J. M. Chomaz, Annu. Re . Fluid Mech. 37, 357 (2005).
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FIG. 4. Expe imen al (do s) s heo e ical (con inuous line) 
alues o he CA ansi ion as a unc ion o We (0:0012,
0:016). The inse gi es he heo e ical d ople olume DV
in pl (je ing egime), as a unc ion o he applied ol age; [30]
shows a simila end in a liquid-liquid expe imen al se up.
PRL 98, 134503 (2007) PHYSICAL REVIEW LETTERS week ending
30 MARCH 2007
134503-4