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Mechanisms and modeling of electrohydrodynamic phenomena
of the electrical field shielding by the large outer diameter A B
of the capillary and also a decrease of the diameter of the
accelerated jet, which behaves as a vena contracta . This
[74]
corresponds to increasing field stress at the tip of the jet in
the direction toward the counter electrode . It is difficult
[20]
to stabilize and control the trajectory of submicron jets
under an electric field and jet may have different instable
status. When the charge of the jet is not too high, liquid jet
is broken up into drops by varicose instabilities as shown
in Figure 4 and satellite droplets may be created at the
moment of the breakups . The process of a breakup may
[18]
degrade resolution on the substrate. The smaller droplets
move away from the axis of the jet, which is faster than
the larger ones . For slightly higher voltages, lateral
[18]
kink-type instabilities appear, and the jet stretches out into
fine droplets of different sizes . Thus, a big challenge
[18]
for utilizing the electrically driven cone-jet transition
as a writing device still exist in overcoming the rapid
disintegration in electrospray and controlling the whipping
of the jet in electrospinning . The common feature of
[38]
them is no boundary jetting methods, but axisymmetric
instability is applied on breaking up the jet into numerous
tiny droplets in electrospraying, and non-axisymmetric,
known as whipping action, is used to thin the liquid jet
and deposit it as a fiber filament in electrospinning.
However, both of instabilities need to be prevented from
high resolution printing process. In the following part, the
theory of the jet stabilization is discussed, and then the Figure 4. Cone-jet mode: (A) Varicose instabilities; (B) kink
stabilization of jet under an electric field is considered. instabilities . Adapted by permission from Michel Cloupeau et al.
[18]
When a liquid jet is either neutral or lower surface (1994) under the Elsevier.
charging under the condition of lower field strengths, the
breakup of a jet into droplets is attributed to the effect that tangential stress caused by the electric field must
of surface waves . Surface waves propagating along the be much smaller than the radial viscous stress . They
[75]
[79]
surface of a jet are generated by disturbances of the jet . performed a local linear stability analysis by considering
[61]
As the amplitude of surface waves of an appropriate axisymmetric perturbation to a charged cylinder of
wavelength (“varicose waves”) is enhanced by surface constant radius under an assumption that the wavelength
tension forces, they lead to the breakup of a cylindrical jet of the perturbation is much smaller than the characteristic
into droplets with radius r=1.89α . This mechanism is decay length of jet . They identified three different
[20]
[79]
well established for uncharged jets and was first described instability modes that are axisymmetric extensions of the
by Rayleigh . Rayleigh reveals that disturbances on a jet classical Rayleigh instability, and two conducting modes,
[76]
with wavelengths greater than the circumference of the which only exist when the conductivity of fluid is finite,
jet will grow, and rapid growing disturbance will control including the axisymmetric conducting mode motivated
the breakup of jet . by different time scale for fluid response and axial surface
[77]
charge arrangement, and a whipping conducting mode
3.3.1 Three Instability Modes of a Charged Jet
caused by static charge density of jet, which dominated
Saville performed qualitative analysis for the stability of in local electric field. Rayleigh instability is suppressed
an uncharged liquid cylinder in the longitudinal electric when the electrical pressure per unit length of the jet
field , and this analysis ignores the presence of surface exceeds the surface tension pressure per unit length with
[62]
charge on the jet and thinning of the radius [78,79] . Another an increase of electric field and surface charge density,
analysis of stability is for a charged cylinder in a radial and conducting mode which is raised from the interaction
electric field (without tangential field) and a constant of electric field with a surface charge on the jet makes
radius jet . Hohman et al. developed a 1D model for a jet unstable . Both the axisymmetric modes are stronger
[80]
[79]
long and slender object made of a leaky dielectric material, than the whipping mode at low fields by adding the effect
and this model is asymptotically valid in the assumption of viscosity and surface charge, but the whipping mode
12 International Journal of Bioprinting (2019)–Volume 5, Issue 1