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Numerical simulation of electrified jets: An application to electrospinning
D. Borzacchiello, S. Vermiglio, F. Chinesta, S. Nabat, and K. Lafdi
Citation: AIP Conference Proceedings 1769, 050004 (2016);
View online: https://doi.org/10.1063/1.4963432
View Table of Contents: http://aip.scitation.org/toc/apc/1769/1
Published by the American Institute of Physics
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Numerical Simulation of Electrified Jets: An Application to
Electrospinning
D. Borzacchiello1,a) , S. Vermiglio1 , F. Chinesta1 , S. Nabat2 and K. Lafdi2
2
1
Ecole Centrale de Nantes, GeM UMR 6183, 1 rue de la Noe, F-44300 Nantes, France
Chemical and Materials Engineering, University of Dayton,300 College Park, Dayton, Ohio, USA
a)
Corresponding author: domenico.borzacchiello@ec-nantes.fr
Abstract. This paper concerns the numerical simulation of electrified jets with application to the electrospinning process for the
fabrication of fibers with controllable size, diameter, and cross section shape. Most numerical models used to simulate electrospinning rely on the Upper Convected Maxwell model (UCM) which is fit to model polymer melts. However, in most electrospinning
processes the fluid is a polymer solution with a Newtonian solvent that evaporates after the fiber is deposited on the collector. In
this work we propose to describe the fluid rheology using Giesekus model, which predicts the properties of polymer solutions more
accurately, and show the impact of the rheological model on the prediction of the fiber radius and size.
INTRODUCTION
The physics of electrically driven jets is extensively investigated in the seminal work of Taylor [1]. Experimental
evidence shows that rectilinear jets are unstable under small bending disturbances developing a so called whipping
instability. The reason for the observed instability is attributed to the repulsive Coulombic forces between the electrical
charges in the fluid and can be explained by Earnshaw’s theorem [2]. In an ideally rectilinear electrified jet, the
net force exerted on a single charge by the remaining fluid acts along the jet axial direction. The jet is constantly
elongated due to the repulsive interaction between the charges. However, if the axial symmetry is perturbed due to
small amplitude disturbances, the net electrostatic force has a lateral component. If Coulomb interactions are strong
enough to counteract the stabilizing effect of the surface tension, the initially small disturbances are magnified and the
instability develops. A more detailed description of the instability mechanism can be found in [3].
The main interest in this phenomenon is the possibility to use whipping instabilities in electrified jets to produce
a variety of continuous polymeric fibers by the process of electrospinning. Fibers are formed by the creation and
elongation of an electrified fluid jet. The advantage is that, by using an adequate voltage difference of tens of kV, the
jet can be elongated to degree that the resulting fibers have a thinner diameter (from nanometer to micrometer) than
fibers produced with other spinning techniques.
The basic apparatus, schematically depicted in figure 1, consists of a high voltage power supply, a spinneret and
a grounded collecting plate. A electrically charged fluid, often a polymer solution, is accelerated trough a straight
capillary or a tapering cone and is cast towards the collector of opposite polarity. Different types of collectors can
be employed, including conductive papers, conductive cloths, wire meshes, parallel or grided bars, rotating rods and
rotating wheels. Other setup configurations are reported in [4]. As the free jet is stretched by the electrical field, the
elongation results into a diameter reduction, therefore the final diameter can be predicted and controlled by adjusting
the distance traveled by the jet, the voltage source, the spinneret outlet section diameter and the flow rate. Additional
constituents can be included in the fluid like chemical reagents, other polymers or dispersed fillers, therefore allowing
for a fine tuning not only of the fiber geometry but also of the mechanical, thermal and electrical properties.
Electrospun fibers are used in various field of application like tissue engineering, filtration, protective clothing,
drug delivery, energy generation and cosmetics [4, 5, 6].
Several mathematical models have been proposed to describe bending instabilities in electrified jets in an attempt
to predict the final fibers geometry through numerical simulation and select optimal process parameters.
The physics of electrified jets is governed by the following equations :
ESAFORM 2016
AIP Conf. Proc. 1769, 050004-1–050004-6; doi: 10.1063/1.4963432
Published by AIP Publishing. 978-0-7354-1427-3/$30.00
050004-1
HC
End of capillary
Electrified jet
Collector
R
V
FIGURE 1. Schematics of the electrospinning process.
•
•
•
•
Conservation of mass
Conservation of electric charge
Balance of linear momentum
Evolution of the stress tensor
Numerical difficulties associated to the solution of these equations are mainly due to nonlinear coupling, moving
meshes and the fact that the characteristic filament length is by several orders greater than the diameter, which would
either require extremely stretched elements or very fine meshes. Therefore, some simplifying assumptions are in order
to efficiently treat the problem.
Several simplified models have been proposed for both rectilinear [7, 8, 6, 9, 10] and whipping jets [11, 12, 13]
for both linear stabilities studies and direct numerical simulations. In particular, the models that deals with direct simulation of the bending instabilities, like [11] are classically based on the hypothesis of one dimensional elongational
flow along the jet axis, while the variations of the flow field in the cross section are neglected. Further simplifications
are made considering the jet as formed by discrete rectilinear cylinders with constant radius with both mass and electric charge lumped at the two ends. In this way the continuous equations are reduced to a set of discrete equations for
beads connected through spring-dashpot elements modeling the viscoelastic behavior of the fluid. In the majority of
works, viscoelasticity is taken into account with an Upper Convected Maxwell model (UCM). Polymer solutions can
be modeled more accurately with Oldroyd-B model which considers the total viscoelastic stress as the sum of a contribution coming from a Maxwellian fluid and a contribution coming from a Newtonian solvent. In fast deformation
with strong elongational rate, the elongational viscosity of both Maxwell’s and Oldroyd-B models tends to infinity.
The Giesekus model is a generalization of Oldroyd-B model based on the concept of deformation dependent mobility
[14]. This allows to have a bounded elongational viscosity as the elongational rate increases. In this paper we consider
the bending instability of an electrified jet of a Giesekus fluid. First, the governing equations are reviewed and the
characteristic numbers arising from dimensional analysis are presented. The adopted numerical method is explained
in detail and simulation results for electrified jets using both UCM and Giesekus model are reported.
GOVERNING EQUATIONS
With reference to figure 2, the electrified jet is modeled as a chain of beads connected trough rectilinear elements,
as in [11]. Each element is considered cylindrical with constant radius ai and length li . For incompressible fluids, the
conservation of mass can be expressed by saying that the volume of the element is constant, hence
πa2i li = πa20 l0
050004-2
(1)
600
mi+1 , ei+1
500
G, α
400
μ s li
μp
300
200
z
100
x
y
mi , ei
0
15
10
5
0
-5
-10
-15
-15
-10
-5
0
5
10
15
FIGURE 2. Discrete model for the electrified jet. Leftmost panel: the polymer solution is modeled, according to the Giesekus
model, using a nonlinear spring and dashpot element in parallel with a second dashpot to account for the Newtonian solvent
viscosity. Rightmost panel: Example result of a jet instability simulation using the proposed model.
in which the a0 and l0 are the radius and length of the initial pending drop. Hence, given the length
li = (xi+1 − xi )2 + (yi+1 − yi )2 + (zi+1 − zi )2 ,
the element radius is readily available:
ai =
a20 l0
,
li
(2)
(3)
The position of the beads ri = xi i + yi j + zi k evolves according to the equation
dri
= vi ,
dt
(4)
where vi is the equation of the i-th bead. The equilibrium of forces on each bead reads as:
mi
dvi
= FCi + FEi + FVi + FTi ,
dt
where
FCi =
i j
ei e j κ
ri − r j
|ri − r j |3
(5)
(6)
is the net of the Coulomb forces exerted by all beads on bead i,
is the electrostatic force,
V
FEi = − k
h
(7)
⎡ 2
⎤
a2 σi−1
⎢⎢ a σi
⎥⎥
(ri − ri−1 )⎥⎥⎦
FVi = π ⎢⎢⎣ i (ri+1 − ri ) − i−1
li
li−1
(8)
050004-3
is the viscoelastic force and
FTi = −
πγ (ai + ai−1 )2 Ki
1/2 ri · (i + j)
4 xi2 + y2i
(9)
is the surface tension. In the previous expressions mi and ei are respectively the mass and electric charge of the i-th
bead, κ is the Coulomb constant, V is the intensity of the voltage source, h is the distance between the spinneret tip and
the collector plate, γ is the surface tension, Ki is the local curvature of the jet and σ is the viscoelastic stress modeled
with the Giesekus model [14].
μ s dli
+ τi
li dt
dτi
αλ 2 μ p dli
λ
= −τi −
τ +
dt
μp i
li dt
σi =
(10)
(11)
The viscoelastic stress is split into a Newtonian contribution due to a solvent of viscosity μ s and an extra stress
τ due to the polymer. This is characterized by relaxation time λ viscosity μ p . The coefficient α takes into account the
non isotropic mobility of the molecular chains and varies between 0 and 1. Note that for α = 0 the Oldroyd-B model
is recovered and for α = 0 and η s = 0 the model is equivalent to a Maxwellian fluid.
Dimensional analysis
Dimensionless groups can be formed on the basis of the following reference quantities:
μ = μs + μ p
,
t∗ = λ
,
L ∗ = a0
v∗ =
,
a0
λ
,
σ∗ =
μ
.
λ
The problem can be therefore described in terms of the following dimensionless numbers:
Re =
m
a0 λμ
,
V=
eVλ2
ha0 m
,
Q=
e2 λ2
a20 m
,
A=
γπλ2
m
,
β=
μs
,
μ
where Re is the Reynolds number, V, Q and A express the ratio of the electrostatic, Coulombic, and surface tension
forces with respect to the inertial forces, and β is the solvent fraction.
NUMERICAL SOLUTION
The governing equations are integrated using an explicit first order Euler scheme. Using a temporal step Δt, the time
is discretized as t = nΔt with n = 0, 1, . . . , Nt and Nt being the number of time steps. The superscript n is used to
indicate that a certain quantity is evaluated at time tn . Given the beads positions rni and velocities vni at time tn , first
the beads position are updated using the following equation:
The velocities are then updated
rn+1
= rni + Δtvni .
i
(12)
vn+1
= vni + mΔt FCi + FEi + FVi + FTi n
i
(13)
In which all the forces are evaluated using the formulas 6,7,8 and 9. Finally the stress is updated:
lin+1 − lin
αλ n 2
n
n
τ
+
2
=
τ
−
Δt
τ
−
τn+1
i
i
i
μp i
lin+1 + lnI
(14)
μ s lin+1 − lin
+ τn+1
.
i
lin+1 Δt
(15)
σn+1
=
i
The time stepping is started with a single segment of length l0 and diameter a0 , that is a pending drop. Every
time that the length of the pending drop is doubled the first element is split in two and a new bead with mass m and
050004-4
1
UCM
Giesekus = 0.4 = 0.1
0.9
0.8
a/a 0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
200
400
600
800
1000 1200 1400 1600 1800
s/a 0
FIGURE 3. Jet radius a as a function of the curvilinear coordinate s along the jet centerline: comparison between Upper Convected
Maxwell and Giesekus models. The two fluids have the same zero shear viscosity.
charge e is added. In this way the bead chain grows throughout the simulation. The position of the newly added bead
is perturbed by adding a small lateral displacement of the form
xi = 10−3 l0 sin(ωt)
yi = 10−3 l0 cos(ωt)
(16)
(17)
This is needed for the onset of the instability. This procedure is iterated until the tip of the jet reaches the bottom. The
evolution of the jet radius can be computed in a post-processing phase, once the numerical integration is over. In this
paper simulations are run using reference values for the dimensionless groups from [11]. More specifically:
Re = 0.083
,
V=2
,
Q = 12 ,
A = 0.9
,
h/a0 = 629
,
l0 /a0 = 1
,
ωλ = 100 .
In addition a Giesekus fluid with identical zero shear viscosity and relaxation time is considered. For this the solvent
fraction is chosen as β = 0.4 and the mobility α = 0.1. Figure 3 reports the evolution of the dimensionless jet radius
as a function of the curvilinear coordinate along the jet centerline. It is evident the the jet of Giesekus fluid is far
more stretched than the one of Maxwellian fluid. This is because, as mentioned before, the elongational viscosity
of the UCM model grows indefinitely as the elongational rate increases, whereas it remains bounded in Giesekus
model. Therefore a higher stretching is required in case of the Giesekus fluid in order to balance the Coulombic and
electrostatic forces. As a consequence, the magnitude of the lateral swinging is amplified and the radius reduced in
the case of Giesekus model.
CONCLUSIONS
In the present paper we modified the multi-bead model presented in [11], to include a more realistic constitutive law for
the evolution of the stress tensor. The Giesekus model is known to be able to predict the behavior of polymer solutions
more accurately than the UCM model. The results presented show how the different rheological model significantly
affects the radius of the jet, the magnitude of the lateral swings due to bending instabilities and ultimately the jet
length.
REFERENCES
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G. I. Taylor, Proc. R. Soc. Lond. A: Math. Phys. Sci. 313, 453–475 (1969).
050004-5
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050004-6
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