J. Aerosol Sci. Vol. 30, No. 7, pp.

863}872, 1999
 1999 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
PII: S0021-8502(98)00780-0 0021-8502/99/$ - see front matter

THE SURFACE CHARGE IN ELECTROSPRAYING:

ITS NATURE AND ITS UNIVERSAL SCALING LAWS

Alfonso M. Gan aH n-Calvo

Escuela Superior de Ingenieros, Universidad de Sevilla Camino de los Descubrimientos s/n,
41092 Sevilla, Spain

(First received 6 June 1998; and in ,nal form 12 November 1998)

Abstract*The electrospraying of liquids in steady cone-jet mode follows a well-de"ned EHD

mechanism described and quanti"ed in this work using a hybrid experimental}numerical technique:
a collection of emitted microjet shapes corresponding to several liquids and di!erent #ow rates have
been digitized and introduced in a quasi-one-dimensional analytical model. A universal value of the
surface charge on the liquid microjet and the resulting charged droplets, independent of their size
and of the liquid permittivity, has been found. The surface charge is shown to be always in
equilibrium, being the liquid bulk quasi-neutral. From these "ndings, we "nally present a consistent
general scaling of all EHD variables involved which is experimentally veri"ed. In this scaling, the
electric current I and the characteristic microjet radius R are both proportional to the square root

of the emitted #ow rate, Q, and independent of the liquid permittivity e .  1999 Elsevier Science

Ltd. All rights reserved

1 . I NT RO D UC T IO N
The electrohydrodynamic (EHD) atomization of liquids is one of the most intriguing free
surface phenomena for its &&magic'' combination of singular geometries (cone-jet surfaces),
surface forces and physicochemical processes. Among many other applications, the rel-
evance of the EHD spraying in chemical analysis and, in particular, in mass spectrometry
has been unsurpassed by any other sample supplying technique, owing to its merits as
a perfect source of large charge-to-mass ions of large biomolecules (Fenn et al., 1989) when
the spraying is operated in the so-called cone-jet mode (Cloupeau and Prunet-Foch, 1989).
To support any one of the di!erent existing theories for the EHD spraying of liquids,
there is a lack of conclusive experiments providing measurements of the EHD variables
involved. Among other EHD variables, the surface charge on the liquid (from the beginning
of the micro-ligament formation to its breakup into charged droplets) is probably the most
important and controversial one. The di!erent models presented in the literature by some
authors range from the ones based on hypotheses of non-equilibrium surface processes
(Turnbull, 1989; de la Mora and Loscertales, 1994) to the ones assuming quasi-electrostatic
conditions (Gan aH n-Calvo, 1997a).
In this work, using a recently proposed experimental technique (Gan aH n-Calvo, 1997b), we
report measurements of most EHD variables involved in real cone jets. Major results from
this study are (i) the quanti"cation and universal scaling of the surface charge distribution
along the emitted jet, in accordance with the surface charge encountered in the emitted
drops, and (ii) this surface charge is shown to be in equilibrium. This has important
implications in the liquid surface physicochemistry, ion evaporation threshold, etc. Other
EHD variables such as the electric "elds in the liquid, surface charge advection, liquid
velocity distribution, etc. are also quanti"ed. Six di!erent liquids and many experimental
spraying conditions have been employed in this study, and a general scaling of the whole
phenomenon has been achieved. As a conclusion, an expression which gives the value of the
maximum surface charge in the jet, and the one on the resulting droplets, is obtained. It is
also shown that this surface charge is in equilibrium and that the whole EHD spraying in
steady cone-jet mode is a quasi-equilibrium process.
As shown in this work, there are two main features of the EHD spraying of liquids in
steady cone-jet mode which allow the use of a quasi-electrostatic quasi-one-dimensional

863
864 A. M. Gan aH n-Calvo

model: (i) The inner electric displacement e E is very small as compared to the outer one,

e E [i. e. the surface charge layer is in equilibrium (Russel et al., 1989) and liquid bulk is
 
quasi neutral], where E and E are the normal outer and inner electric "elds on the jet's
 
surface, respectively, and e , e are the vacuum and liquid electric permittivities, respectively;

(ii) For most liquids and EHD spraying conditions, the radial variations of the liquid
velocity are very small compared to the average velocity.

2. E XPERI ME NT AL }N U ME R ICA L S C HE ME
Since the transversal section of the cone region is very large compared to the one of the
emitted microjet, the bulk Ohmic electric conduction is dominant in the cone with very
small inner electric "elds and liquid velocities compared to the ones in the microjet
(Gan aH n-Calvo, 1997a, b). Therefore, the cone region can be considered perfectly electro-
static (Taylor, 1964; Pantano et al., 1994; Gan aH n-Calvo, 1997a).
The important mechanisms that actually characterize EHD spraying arise at the region
where the cone turns into a slender cusp, i.e. where the tip streaming takes place. As will be
shown, owing to the slender geometry of this region and that of the whole jet, the use of
a quasi-one-dimensional model is justi"ed, and the errors associated to neglect the axial
derivatives of the jet radius in the surface tension stress equation are small. Liquid motions
inside the meniscus are irrelevant to the EHD spraying mechanism except for some limiting
marginal cases using very small liquid viscosities and electric conductivities (Barrero et al.,
1998).
The steady quasi-one-dimensional model employed here comprises the averaged conser-
vation of axial momentum (Eggers, 1993), charge, and the normal stress equilibrium
(Melcher and Warren, 1971; Gan aH n-Calvo, 1997b):



d 1 2q 3k d dv
P# ov "  # m , (1)
dz 2 m m dz dz
2Qe
I"  E #nmKE , (2)
m  X


c 1#mQ  e
!mG "P#  [(E )#(b!1) E!b(E )], (3)
(1#mQ ) m 2  X 

where z, m, P, v, E , E , E and q "e E E are the axial coordinate, jet's radius, liquid}gas
  X    X
pressure jump, liquid velocity, normal outer and inner electric "elds on the jet's surface, the
surface electric "eld in the axial direction, and the tangential surface stress (Melcher and
Warren, 1971; Gan aH n-Calvo, 1997b), respectively. c, K, o, k and Q are the liquid}gas surface
tension, liquid electric conductivity, density, viscosity and emitted #ow rate, respectively.
b is the ratio of the liquid to the vacuum permittivities b"e /e . mQ and mG denote "rst and

second derivatives of the jet radius in the axial direction, respectively. Finally, I is the total
emitted electric current.
In this work, our one-dimensional model is applied along the jet from the region where it
joins the cone and the "rst derivative of its radius is smaller than 0.3. Neglecting the radius
derivatives in equation (3), we have observed that the relative errors in the surface tension
equation are of the order of 10% or smaller.
On the other hand, as discussed in Gan aH n-Calvo (1997b) and references therein, for
a su$ciently high concentration of charge carriers in the liquid with large enough electrical
mobility compared to other velocities of the problem, these charge carriers form a quasi-
equilibrium charge layer in the liquid at the liquid}gas interface. This charge distribution in
the direction normal to the surface is given by the electrochemical equilibrium of the charge
carrier species (represented by the well-known Poisson}Boltzmann equation, Probstein,
1989, p. 100). The thickness of this layer is of the order of Debye's length, which is many
times smaller than the jet radius except for very high-conductivity liquids (such as non-
puri"ed water and liquid metals). Therefore, the liquid bulk is quasi-neutral, and the inner
 The surface charge in electrospraying 865

electric "eld can be calculated since it is divergence-free (

) E "0) and the liquid domain is
slender [O(z)<O(m) ], which gives
1 *
E "! (mE ) (4)
 2m *z X

for a given tangential "eld E in the surface. Furthermore, the normal and tangential electric
X
"elds E and E on the jet surface must satisfy
) E"0 in the outer domain. After the
 X
solution of the equations of our model is obtained for the previous simpli"cations, we will
verify that E can be e!ectively neglected in equations (1)}(3).

Based on the previous simpli"cations that can be well justi"ed, an experimental}numer-
ical scheme can be implemented to solve all EHD variables if, instead of solving
) E"0 in
the outer domain to close system (1)}(3), the jet's shape m is taken from experimental
measurements (see Fig. 1). Writing
v"Q/(nm) (5)
and taking into account previous simpli"cations, one may reduce system (1)}(3) to a single
"rst-order ODE for E , as



dE c 2(b!1)e I 12(b!1) Qe
" !  !  (E )
dz m nKm nKm 


10(b!1) QeI 2oQ dm
#  E #
nKm  nm dz



2e E 2Qe 6kQ d 1 dm
#   I!  E !
nmK m  nm dz m dz

   
2QeI(b!1) 4(b!1) Qe
 ! e#  E , (6)
nKm  nKm 

where the term b(E ) has been neglected since, as it will be shown, it is negligible in all

cases. From the region where the jet joins the cone, the jet's shape can be digitized and "tted
using a hyperbolic regression (an 8th-order regression has been used). Thus, m(z) and its
derivatives can be obtained in a closed form.

Fig. 1. Di!erent charged microjets issuing from Taylor's cones used in this study.
866 A. M. Gan aH n-Calvo

Table 1. Liquids used and some of their physical properties at 24.53C (K: S m\; o: kg m\,
c: N/m, k: centipoise). Also given, the values of d .
I
Liquid K b o c k d
I
Dodecanol (1) 8.2e-7 6.5 830 0.0275 12.5 0.926
Dodecanol (2) 2.3e-6 6.5 830 0.0275 12.5 0.662
Dodecanol (3) 2.5e-6 6.5 830 0.0275 12.5 0.649
Dodecanol (4) 7.5e-6 6.5 830 0.0275 12.5 0.441
1-Octanol 8.0e-6 10 827 0.0235 8.1 0.813
Propyl. glycol 1.2e-5 31.2 1026 0.036 41.8 0.161

The digitizing process involves an image equalization and edge recognition process which
has been optimized performing several image processing runs of a 50 km stainless-steel
calibrated wire and a 10 km calibrated tungsten wire. The major sources of errors are (i) the
non-perfect homogeneity of the light background (we note a maximum of 5% in gray level
inhomogeneity along the picture, located close to the upper and lower edges), and (ii) the
pixel size when scanning the images, which for our B&W high-resolution video camera and
optical set-up results in 0.7 km per pixel. Since we actually digitize the diameter of the jet
and promediate to obtain the jet radius, the total errors are divided by two. When dealing
with the thinest jet (about 5 km), we have found a maximum error of a 15% close to the
lowest (thiner) part of the jet in the photograph. Although this error seem to give rise to
non-natural deviations of the normal electric "eld from the expected trend of the same
relative order of magnitude (15 to 25%), the errors become important in the calculations of
E since it involves several indirect calculations of intermediate variables. In conclusion, in

order to keep the maximum predictable errors in the calculation of the inner electric "eld
E bounded, we accepted maximum errors of the order of :5% in the measurement process

of the jet radius, which made us to dismiss many runs or to reduce the usable portion of the
picture of a jet to a 40}50% of its length in some cases.
In addition to the jet's shape, we take from the experiment the measured value of the total
emitted current I for each particular run. Then, equation (6) can be integrated in the
upstream direction from a guessed value of E at a certain z point downstream (usually,

close to the lower end of the pictures). The actual solution is encountered after a short
distance upstream since it is a strong attractor of the ODE (Gan aH n-Calvo, 1997b).
Six di!erent liquids have been employed in this study, and their properties are given in
Table 1. Three stainless-steel feeding needles have been used, with outer diameters 1.2, 0.8
and 0.6 mm. The distances from the needle's border to the electrode ranged from 25 to
30 mm in all experiments. In each experiment, the electric potential di!erence was adjusted
so that the cone elongation (distance from the needle's mouth to the cone's tip) was equal to
the needle's diameter. This procedure guarantees an almost constant cone's geometry, and
that the charge at the cone's surface along various experiments is maintained approximately
constant for a given liquid, independently of the emitted #ow rate (Pantano et al., 1994). It is
worth mentioning that within the stability region, the electric potential cannot be changed
more than typically a 15%, with similar variations in the EHD variables (Cloupeau and
Prunet-Foch, 1989; Gan aH n-Calvo, 1997a).

3 . SC A LI N G O F T HE EQ U AT IO NS
The following observations can be withdrawn after the integration of system (1)}(3) for
several experiments:
(i) The point where the surface convection current
I "2Qe E /m (7)
  
equals the Ohmic bulk conduction current
I "nmKE (8)
 X
is located close to the jet's origin (Gan aH n-Calvo, 1997b).
 The surface charge in electrospraying 867

(ii) The pressure jump p across the jet's surface is almost balanced by the normal
electrostatic stress e (E )/2 alone, being the surface tension stress c/m smaller than the
 
electrostatic stress.
(iii) The polarization stress e (b!1) E/2 is negligible in all cases.
 X
(iv) The kinetic energy per unit volume of the liquid becomes of the order of the
electrostatic stress once the jet is developed (approximately tens of jet diameters down-
stream from the jet's origin).
(v) The gradient of the kinetic energy in the axial direction is mostly balanced by the
tangential electric stress resultant on the jet's surface.
(vi) The viscous extensional term in equation (1) is negligible in many cases. Assuming
that the cone is in electrostatic-capillary equilibrium (Taylor, 1964; Pantano et al., 1994), the
axial electric "eld E should be of the order of the normal electric "eld at the cone,
X
E :(c/(e ¸ )).
X  
From all the above given observations, one may obtain the characteristic values of the
jet's radius (R ), the axial length (¸ ), the normal electric "eld (E ), and the tangential one
  
(E ), assuming:
X
(a) e (E ):oQ/R , (9)
  
(b) oQ/(R¸ ):eE E /R , (10)
   X 
(c) e QE /R :RKE . (11)
    X
Using the parameters +o, K, c, e , it is useful to de"ne (Gan aH n-Calvo, 1997a)

A #ow rate: Q "ce /(oK),
 
An electric current: I "eco\
 
A distance: d "(n\ceo\K\),
 
An electric "eld: E "(2ce\d\). (12)
  
Let us introduce the non-dimensional #ow rate Q"Q/Q . In order to satisfy conditions

(a)}(c), one can "nally de"ne
R "d Q, ¸ "d Q, E "(2c/(e d )), E "E Q\, (13)
       X 
to write system (1)}(3) as



d 1 4e e 6 d 1 df
#p "   ! , (14)
dx 2f  f Qd f  dx f dx
I
e f /2#ne /f"I, (15)
 
( fQ)\"p#e#(b!1) e/Q , (16)
 
where
f"m/R , x"z/¸ , I"I/(8IQ),
  
e "E /E , e "E /E , p"2P/(e E ),
L    X X  
and Qd is a Reynolds number such that d is given by
I I
d "[(coe )/(nkK)]. (17)
I 
From system (14)}(16), one may conclude that as long as the #ow rate Q is large enough
as compared to unity, the liquid kinetic energy is almost entirely built from the surface stress
term 4e e /f. The reason for which this scaling does not exactly coincide with the one given
 
in Gan aH n-Calvo (1997a) is that the asymptotic analysis in that previous work involves very
large Q values to have large L"log Q values, being L\ the small number in the
asymptotic analysis. In the present experimental conditions, L\:O(1) in many cases.
868 A. M. Gan aH n-Calvo

4 . U NI VER S AL EXP RE SSI O N O F T HE SU RF ACE CH ARG E

The non-dimensional values of + f, e , e , e , obtained from 20 experimental conditions
  
and digitized jet's shapes in this study for which the #ow rate Q can be deemed large enough
to neglect the viscous stress and the surface tension stress are given in Fig. 2. Observe that
since the electrostatic stress e is almost constant, its action against the #ow (de/dx(0) is
 
negligible. The position x"0 has been arbitrarily selected at a point at the cone's apex such
that the point where the non-dimensional surface convection current ne /f becomes equal to

the bulk conduction current e f /2 is located in all cases at xK10.

The collapse of digitized shapes and, most conspicuously, the normal electric "eld on the
jet's surface, is very good. Furthermore, e is almost constant along the jet, with a maximum

value e K0.53 at xK10.7, i.e., close to the jet1s origin (or, in other words, close to the
 
cone's cusp). From this result, one may write a universal expression for the surface charge at
this point as

q Ke E "0.53;2n(e coK), (18)

    
independent of the #ow rate Q, the jet's radius, and other parameters of the problem. This
value decreases further downstream up to the breakup point, where the normal electric "eld
e ranges from 0.27 to 0.32 in our experiments.

Expression (18) can be used to assess how close is a certain liquid to develop gas discharge
e!ects when it is electrosprayed.

4.1. ;niversal scaling of the emitted current and droplet diameter

In order to assess the value of the surface charge on the resulting droplets, their diameter
d has been measured using a laser Doppler PDA (from Invent and TSI Corp.) simulta-
neously to the jet's shape acquisition. In Figs 3 and 4, the droplet diameter d/d and the

current I/I are represented vs the non-dimensional #ow rate Q, respectively. Also given,

experimental measurements of I/I and d/d as functions of Q from other authors (FernaH n-
 
dez de la Mora and Loscertales, 1994; Chen et al., 1995).
If the jet radius and the droplet diameter are approximately related by Rayleigh's
expression (dK3.78m , where m is the jet radius close to the breakup point), as earlier stated



by Cloupeau and Prunet-Foch (1989), and if m /R does not deviate signi"cantly from 1, the


droplet diameter should scale as the characteristic jet radius. This scaling is con"rmed by
the experiments, within the experimental errors. Thus, as suggested by the scaling proposed

Fig. 2. Non-dimensional jet's shape f (x), outer normal electric "eld e (x), tangential one e (x), and
 
the inner normal one e (x). Also given, the non-dimensional surface current ne /f.
 
 The surface charge in electrospraying 869

Fig. 3. Droplet diameter d/d as a function of the non-dimensional #ow rate Q"Q/Q .
 

Fig. 4. Electric current I/I as a function of the non-dimensional #ow rate Q"Q/Q .
 

in this work, the droplet diameter can be "tted by

d/d "2.9 Q (19)

and the total emitted current (in the absence of gas discharges) by:
I/I "2.6 Q. (20)

Assuming that the droplet's bulk is quasi-neutral, these expressions give an approximate
universal expression for the average surface charge on the (main) droplets for b values up to
80.1 (water) as
q "Id/(6Q)K0.59(e coK). (21)
  
Considering the value of the surface charge on the jet at the breakup point,
q :0.62(e coK), (22)
 
this result shows that the breakup does not a!ect the surface charge (Gan aH n-Calvo et al.,
1994). Other conclusions are outlined in the following.

5 . L IQ UI D V EL OC ITY : VAL I D IT Y OF T HE AVE RAG E D M O ME NT U M

AN D SU RF ACE CO NV ECT I O N CU RRE NT EQ U AT IO NS
Another interesting result is that the liquid velocity is, like the surface charge, indepen-
dent of the liquid #ow rate,


nKc  1
v"Q/(nm)" . (23)
eo f

870 A. M. Gan aH n-Calvo

In a slender geometry such as the jet, the transversal viscous di!usion of momentum from
the surface if very e!ective (Melcher and Warren, 1971), and the velocity pro"le can be
considered almost parabolic, with
*vKe E E m/(4k)"(cf )/(2k). (24)
  X
Therefore, the ratio of the maximum deviation *v to the average v is given by
*v/vK0.5d f e e (25)
I  
with a maximum value at the jet, from data in Fig. 2, given by
(*v/v) K0.244d . (26)
 I
This yields a maximum deviation for Dodecanol 1 of the order of 22% at xK12.5, which is
consistent with the averaging process (Eggers, 1993) used to obtain equation (1).

6. S UR FAC E CH AR GE EQ U IL I BR I U M ( QU AS I- N EU TR AL B UL K)
From data in Fig. 2, one can conclude that the surface charge is in equilibrium
(e E ;e E , or be /Q;e , Russel, 1989) and that the liquid bulk is quasi-neutral (the
    
Ohmic liquid conductivity is constant) at the whole cone-jet liquid domain, for
Q<be /e (large enough #ow rates or liquid conductivities). Note from Fig. 2 that the
  
point where this condition is more compromised is located around the maximum of e , at

the point xK11.5 [i.e. at the jet, not at the cone's apex as it has been speculated in the
literature, FernaH ndez de la Mora and Loscertales, 1994). Upstream from the point xK11,
both the inner and the outer electric "elds decrease, being the decrease of e very sharp from

this point. Therefore, the cone can be considered as a perfectly electrostatic system from the
needle's border down to the region where the cone turns into a cusp.
Furthermore, close to the stability limits, it can be shown under which conditions the
surface charge remains is in equilibrium. Equations (14)}(16) show that there are several
mechanisms by which the jet can be destabilized (i.e., when forces acting in the opposite
direction of the #ow become of the same order than those providing positive momentum):
(i) by extensional viscous stresses, when 6/(d Q):O(1), (ii) by surface tension stresses, when
I
Q:O(1), (iii) by polarization stresses when (b!1)/Qe :O(e ). In addition, surface
   
charge relaxation phenomena might as well destabilize the jet when be :(Qe ). The
   
maximum experimental values of the inner normal electric "eld e are plotted as function
 
of d Q in Fig. 5.
I
As the viscous extensional force increases, which provokes a -attening of the non-
dimensional jet pro"le f, there is a sharp decreasing of e . The data show
 
e K0.016(d Q) . Close to the stability limit for all liquids tested, the smallest value
  I
of d Q reached is K1.6. Thus, one has: (i) a maximum ratio of electric displacements
I

Fig. 5. Maximum value of the inner normal electric "eld e as function of d Q.

  I
 The surface charge in electrospraying 871

be /(Qe )K0.025bd which shows that the inner displacement is, at least, 7 times
    I
smaller than the outer displacement for all liquids used, (ii) a maximum polarization stress
of the order of 15% (max.) of the normal electrostatic stress for all liquids, (iii) a non-
negligible surface tension and extensional viscous stresses close to the stability limits. In
addition, b(e )/Q is always very small compared to e (even for water, b"80.1), which
 
justi"es the neglect of b(E ).

Therefore, in our experiments (b up to 31.2) the limiting value of the viscous term
[6/(d Q):1] or the surface stress [( fQ)\:O(1)] is reached well before the inner electric
I
displacement becomes comparable to the outer one, and the mechanism by which the cone
jet becomes destabilized for small #ow rates is the surface tension or the viscous extensional
stresses acting against the #ow, not because the di!use surface charge layer became
non-relaxed or the bulk not quasi-neutral. For very large b values, close to the stability
limits of the cone-jet mode, the quantity 0.03bd might become of the order 1 in certain
I
cases. Nevertheless, since large b values imply large electrical conductivities K as well, the
values of d are in those cases very small, and the surface charge equilibrium condition is
I
achieved as a general feature of the cone-jet electrospraying within its limits of stability.

7 . CO N CLU SI ON S
In this work, we present an experimental}numerical approach to the EHD atomization
of liquids in steady cone-jet mode which allows the calculation of all electrohydrodynamic
variables involved in the problem along the emitted charged jet. From the experimental
results obtained, a universal scaling of all the variables and, in particular, of the surface
charge is "nally proposed. The collapse of the di!erent dimensional values from a series of
experimental measurements into well de"ned, universal functions validate the scaling
proposed, for low and moderate electrolytes and liquid permittivities up to 31.2 times e .

It is shown that the EHD atomization of liquids is actually an electrochemical quasi-
equilibrium process in the sense that the liquid bulk is quasi-neutral and the free charges
form a quasi-equilibrium charge layer at the liquid surface. The inner electric "eld in the
liquid necessary to drive the bulk Ohmic current is some orders of magnitude smaller than
the external one. Therefore, the EHD atomization of liquids is independent of the liquid
permittivity within the parametric window explored in this work. It is shown that the the
ratio of the inner to the outer electric displacements, as well as the ratio of the tangential to
the normal electric "elds on the surface increase as the liquid #ow rate decreases.
The total emitted current I and the resulting average main droplet diameter d are shown
to scale as
I/I "k (Q/Q ) (27)
 ' 
and
d/d "k (Q/Q ), (28)
 B 
where I "eco\ and d "(n\ceo\K\). k and k are constants (that show a
    ' B
very slight dependence on the needle-to-electrode potential di!erence as well as on the
needle radius) that may be obtained from experimental values. Su$ciently accurate values
within the experimental uncertainties are k "2.6 and k "2.9. Like the rest of EHD
' B
variables, both I and d are independent of the liquid permittivity within the parametric
window explored in this work.
Finally, the surface charge on the jet and on the emitted droplets is shown to have
a universal value independent of the jet size, and the liquid #ow rate. It is only slightly

For example, in the case of pure water, bK80, one has K"3.045;10\ S/m calculated from the mobilities of
[H]> and [OH]\ at 203C, but this number is easily increased up to KK1 S/m with a small contamination of any
salt or impurity; on the other hand, for dodecanol, bK6.5, one has KK10\ S/m. For commercial formamide
(bK111, kK3.8 K qm\ s\) one has values of d ranging from 3 to 10.
I
872 A. M. Gan aH n-Calvo

dependent on the axial coordinate along the jet, and its maximum is located close to the
cone's apex, with an approximate universal value given by:
Ke E "0.53;2n(e coK).
q
    
This value can be used to assess the possibility of gas discharge (local ionization) e!ects
when electrospraying a given liquid, which will "rst appear close to the cone's apex. As
a general result, it may be shown that this value is always above the one corresponding to
the &&charge evaporation'' surface electric "eld threshold for very strong electrolytes and
liquid metals.

Acknowledgements*This work is supported by the Spanish ComisioH n Interministerial de Ciencia y TecnologıH a,

Project number PB96-1341.

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