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Gao D, et al.
Table 1. A summary of scaling laws that outline the relationship between processing parameters and diameter of jet or emitting current
Name Dominant conditions Scaling of jet diameter Scaling of current Material Reference
Ganon-Calvo scaling IE-scaling (inertia stress and 3 I=(γKQ) 0.5 Non-polar liquid with low
law for high electric electrostatic suction) D = ( ρε 0 Q ) / 16 viscosity, such as Octanol
j
conductivity (Above γ K
10 S/m) IP-scaling (inertia stress and Polar liquid with low viscosity,
−4
22
polarization force) D = ( ρε γ 0 K Q 3 ) / 16 I = ( (ε ρ r KQ 0 ) . 05 such as water, and formamide
− )ε
j
1
VE-scaling (viscous force and 23 I=(γKQ)1 Ink material with high electric
electrostatic suction) D = ( µε γ 0 K Q ) / 18 conductivity, such as glycerol
j
VP-scaling (Viscous force and µ Q 33 2 Not found in published paper
KQ
polarization force) D = ( (ε r − )γ1 ) . 05 I = ( (ε µ − ) γ ε 2 ) . 05
j
42
1
De la Mora Hydrodynamic time and  r γ 0 Ethylene glycol
KQ 05
/
scaling law for electrical relaxation time D ~( Q r 0 13 I = ()ε (ε ) .
)
f
j
high electronic K r
conductivity (Above Inertia stress and surface 2 γ Glycerine solution
KQ 05
10 S/m) tension D ~( ρ Q ) / 13 I = ()ε (ε r ) .
−4
f
j
γ
Choi scaling law Electrical field force and Not mentioned in the Water (0.1 mM KBr added),
surface tension D ~ ε γ 0 d E N reference paper and Glycerine
j
Where r is the droplet radius. Marginean et al. found Chen et al. proposed a different model for the intrinsic
[30]
that equation (24) can also express the relationship pulsations frequency; this is related to the drop formation
between the square of the pulsation frequency and the rate, Q, in equation (27), (28).
cube of the anchoring radius of the menisci (the position ρ 3
of the contact line at the tip of the capillary during the f ~ Q ~ Q ~ Q c / 32 ~ d n ε ⋅ 0 E ⋅ K ( ⋅ d n ) . 05
2
pj
liquid recoil phase) for low-conductivity liquids and V pj Q ∆ t pj ( dd ) µ L εε γ
r 0
0
n 0
relatively large nozzle radius. They also found that the
effect of charge in this region is negligible. (27)
Another finding is that the capillary waves not only In conclusion, as upstream flow rates are low, the
governs transitions associated with the meniscus (dripping cone formation rate is affected by stress at the liquid-to-
mode to spindle mode) but also influences the breakup of the air interface and by viscous drag in the thin nozzle, and
liquid filament (varicose to kink) . Choi et al. proposed a Taylor cone only deforms at the tip [28] . Equation (27)
[30]
scaling law for jet diameter in the pulsating cone-jet regime, is used to obtain frequency at a low flow rate, and the
2
4
based on data from their experiments and literature . They drop formation rate scales as d E L where d is inner
−1
[64]
defined the ratio between the surface tension and the electric diameter of nozzle, E is the nominal electric field, and
field forces as the “electrical capillary number (Ca),” L is the nozzle length [28] . However, as supplied flow rate
and considered D and D as characteristic length scales is not limited by upstream conditions, the equation (24)
j
n
associated with the surface tension force and the electrical is applied to acquire frequency (Chen, 2011). Since
field force separately . The scaling law for jet diameter is: equation (25) reveals a 0.5 power dependence between
[64]
γ d the diameter of jet and nozzle, it implies this equation
d ∝ N (25) may not be employed for high conductivity liquid
ε E
−4
0 (≥10 S/m).
This scaling law is different from those in section 3.3 Theory and Analysis of Jet Stability
3.2.2; a simple reason is that Choi et al. obtained diameter
of the jet at initiation condition where the jet is the widest. In EHD inkjet printing, a continuous fine jet is generated
Choi et al. substituted equation (25) into equation (24) to at the apex of a liquid cone in the cone-jet mode and the
derive a scaling law without droplet radius. The pulsation length of jet increases with the viscosity, the resistivity
frequency read as: and flow rate of liquid . The field strength at the tip of the
[18]
β = ε ε/ 0 (26) jet increases with the length of the jet, due to a decrease
International Journal of Bioprinting (2019)–Volume 5, Issue 1 11