Page 64 - IJB-5-1

Mechanisms and modeling of electrohydrodynamic phenomena
When η equals to one and r equals to r*, the minimum and it shows that the emitted current and diameter of jet
flow rate for the stable cone-jet structure is found, and mainly depend on flow rate and liquid properties. They
it is also known as the characteristic flow rate [31,32] . The argue that the jet radius is independent of the boundary
minimum flow rate is expressed as: condition and applied potential since the typical jet
γε ε radius is sufficiently smaller than the local size of the tip
Q ~ Q ~ r 0 (18)
0 min ρ K region. Table 1 summarizes scaling laws that describe the
dependence of jet diameter and magnitude of current on
The characteristic distance, d , can be obtained from working parameters.
0
the characteristic flow rate, Q , through the equation However, Hohman et al. obtained a different scaling
0
below: law from electrospinning experiments, and they found
γε ε (19)
r 0
Q ~ Q ~ current not only depends on flow rate and material
0 min ρ K 2 properties but also applied voltage, shape, and material of
When the dynamic pressure 1 ÁQ 4 becomes nozzle . Discrepancies in scaling laws of electrospray
[73]
2 r
comparable with surface tension, γ/r, the diameter of jet and electrospinning may be attributed to the following
scale as characteristic length, R* . reasons. Solutions from steady-state equations for
[31]
ρ Q 2 / 13 electrospray are a function of imposed flow rate, current,
*
D ~ R = ( ) (20) and voltage drop between two electrodes. Boundary
j γ condition on a surface charge is that jet shape is matched
The intermediate region which is analyzed in the to a perfectly conducting nozzle. However, the current is
model of Ganan-Calvo provides consistent for matching determined dynamically in electrospinning experiment,
the cone to transition region . Interestingly, when and flow rate and field strength are only independent
[26]
two characteristic distances (r* and R*) are multiplied parameters. Hohman et al. found that the radius of the
together, the product of these two characteristic distances jet has already decreased substantially before entering
is similar with the expression of jet diameter for IE or IP the asymptotic regime. Thus, current is determined
scaling. In the liquid meniscus, charges are transmitted by more than just the behavior in the asymptotic regime.
convection and conduction at the charged interface and Numerical steady solutions have only been obtained
only by conduction through the bulk . In this model, for low conductivity material (10 S/m) at the high
[31]
−6
the surface conduction term is ignored since it is small electric field after counting fringe fields of the nozzle and
compared to surface convective term, and the surface considering small surface charge density as a boundary
convection current is then given by : condition at the nozzle . The electric field around nozzle
[31]
[73]
I =2πru σ (21) and boundary condition become key factors to influence
s
scv
Where u is the liquid speed at the interface and the shape and charge densities on the liquid jet. It is helpful
s
free surface charge density, σ, equals to  E . At the for understanding this influence to explore what the
o
0
n
point where charge relaxation become non-negligible and predominant charge transport mechanism near the nozzle
the current I which jet carries is given : is. Differences in geometry setup are also a possible
[31]
γ KQ ε reason to lead different scaling law.
I ~( )(ε ) . 05 ~ ηγ ( 0 ) . 05 (22)
f
ε ρ
r 3.2.3 Pulsating Cone-jet Regime
Where f(ε), as obtained from experiment, is a function
of ε. Scaling law expressed in equation (22) is similar Although most attention is focused on the “continuous
with scaling law of current in IE regime of Ganan- regime” where a jet is continuously ejected from its
Calvo model, and actually f(ε) become very small when apex, a continuous regime only happens for certain
the dielectric constant is small. When η is equal to one, ranges of the operating parameters. Changes in liquid
characteristic current, I , is closely related to current, I, properties can also cause instabilities [30] . The D-O-D
0
and can be obtained from: pulsate printing as produced by an external voltage
ε pulse has become more popular due to low reagent
I ~(γ 0 05 . (23) consumption and high ion transmission efficiency
)
0
ρ at small flow rate [28] . The frequency of the lowest
γ t e / 23
The parameter Z which equals to ( ) is defined to excitation mode for a negligible amount of charge is
µQ / 13 expressed as [30] and can be calculated by equation (24)
measure the radial variation in the axial velocity profile of which is derived from the frequency spectrum for the
the jet . The electrical shear stresses which tend to make capillary waves on the surface of a charged droplet
[31]
the liquid velocity larger at the surface than at the core, while proposed by Rayleigh [12] .
viscosity tends to make the axial velocity radially uniform. 2 γ
In conclusion, the scaling model presented above has f 2, 2 z 0= = (24)
been confirmed by asymptotically self-similar solutions, π 2 r ρ 3
10 International Journal of Bioprinting (2019)–Volume 5, Issue 1