Several synthetic and fabrication methods have been proposed to produce polymer nanofibers as one dimensional nanostructure. Among these is electrospinning, which has gained much more attention in the last few years as a low cost, straightforward, and efficient technique. It offers advantages like control over fiber diameters and morphology, tunable porosity, high material elongation rate, and a high surface to volume ratio [1–4]. It is an electrostatic drawing method which provides long nanofibers with uniformity in diameter from diversified compositions. When compared to similar commercial methods in which a mechanical force is essential, electrospinning involves electrostatic repulsions between surface charges which continuously reduce the diameter of a viscoelastic jet. In this manner, the process can generate much thinner fibers with diameters in the range of 10–1000 nm [5–7].
Electrospinning set-up consists of a high voltage power supply, a spinneret, and a collector. A polymer solution (or melt) is fed to the spinneret at a controllable manner by utilizing a precise pump (Figure 1). The applied high voltage across the electrode needle and the collector makes the liquid drop deform to a conical object called a Taylor cone. This phenomenon is the consequence of the interaction of two major forces: electrostatic forces between surface charges and Coulombic forces exerted by the external field. As the applied voltage exceeds a threshold value, a jet is ejected. This is due to the electrostatic forces that surpass the surface tension of the polymer solution. The jet undergoes a whipping process, leading to the formation of a long thin fiber. This continuous elongation accompanied with the solvent evaporation usually forms a fiber with a diameter in the range of submicrons to microns, which deposits in a random orientation on the collector. Initially, the jet travels a relatively short straight path, then begins to revolve or whip unpredictably; this is called jet instability [8, 9]. This bending instability has a two-edged role: it provides the possibility to have thinner fibers and it leads to an irregular deposition of fibers.
It has been illustrated that the nanofibers generated in the electrospinning technique have a wide variety of applications in tissue engineering and bioassay [10–13], sensors fabrication , textiles , filtration , electrodes , and catalyst supports . In most of these products, alignment and arrangement of the fibers are critical to achieve the desired tasks. Particularly, it plays an important role for the efficient release of proteins and growth factors in tissue engineering [19–21]. Several methods have been utilized to improve the controllability of the nanofibers deposition in recent years. It has been shown that tailoring of resultant fibers and target governing is somehow possible through the optimization of electrospinning conditions and utilizing special designed collectors and patterned electrodes [22–26]. In another study, aligned fibers were fabricated by embedding magnetic nanoparticles in a polymer solution . Despite these improvements, it is highly desirable to develop a potent technique that not only generates well-aligned nanofibers, but also provides convenient and predefined deposition of the fibrous matrices onto the surfaces of solid substrates.
Many investigations have been performed to understand the physical mechanism underlying the jet motion in the electrostatic spinning process based on experimental observations and electrohydrodynamic theories [28–34]. Mathematical modeling can explain the effects of different parameters and can make predictions possible concerning behavior of the process encountered to new conditions. To accurately describe electrospinning mathematically, the conservation of mass, momentum, and free charges as well as Maxwell’s equations should be involved in the governing equations. The complexity in solving the developed nonlinear partial differential equations causes consideration of some simplifications without losing the consistency of the model . The most recognized fitting model, which described the whole process, was proposed by Reneker et al. . They thought about the charged jet as a system of connected viscoelastic elements and the trajectory of the jet path was calculated using the linear Maxwell equation. Although the model cannot provide required details to control the electrospinning process and possible formation of beads, it accurately describes the bending instability and gives information about ultimate fiber features, which were inconsistent with related experimental observations [32, 38].
Based on the abovementioned model and previous work , magnetic electrospinning in this study is proposed in which two distinct magnetic fields (MFs) are utilized to control the jet path and the deposition target. A discrete mathematical model for the electrospinning process was established and the moving behavior of the jet was analyzed numerically using the Runge-Kutta method. All simulations were performed using MATLAB software (MATLAB 8.0, The MathWorks, Inc., Natick, MA, USA).
2.1 Mathematical modeling
To describe a magnetic electrospun jet, we did some changes to Reneker’s model  in a way that the MF effect could be calculated. The jet was simulated as a system of beads possessing charge e and mass m connected by viscoelastic elements as shown in Figure 2.
The momentum equation for motion of the beads was modified by adding the Lorentz force, which makes the calculation of MF possible:
where ri is the position in the Cartesian coordinate system (x, y, z) of bead i,
FC is the net Coulomb force
where N is the number of beads and Rij is the distance between bead i and bead j, and e is the electric charge.
FE is the electric field force, which is constant for every bead:
V0 and h are the applied voltage and distance from nozzle to collector, respectively.
Fve is the viscoelastic force for ith bead:
where aui and adi are the filament radii, σui is the stress which pulls i back to i-1.
Segment stress/strain is:
where l is the length of ideal rectilinear jet, G is the elastic modulus and μ is the viscosity.
Surface tension force (Fcap) is described by:
where α is the surface tension coefficient, ki is the curvature of jet segment.
The vector of MF is:
And FB is the Lorentz force for ith bead with charge of e and velocity Vi:
It is obvious by omitting FB, the momentum equation is similar to Reneker’s one .
3 Results and discussion
3.1 Description of the model
Formation of nanofibers in common electrospinning can be affected by various parameters. The variables can be categorized into three groups. The first group takes account of polymer solution features including type, molecular weight, and concentration of the polymer; type, vapor pressure, and diffusivity in air of the solvent, and solution properties such as rheological behavior, viscosity, surface tension, relaxation time, conductivity, and dielectric permittivity. The second category is about processing conditions involving strength and geometry of applied electric field, solution feed rate, orifice diameter, and distance between orifice and collector. The final group includes ambient conditions such as relative humidity, temperature, pressure, and type of atmosphere [25, 40].
To evaluate the performance and the validity of the model, actual experimental parameters of polyethylene oxide (PEO)/water solution were considered which had been used by Reneker et al. . Same data seem to be reasonable to serve as a standard set to compare all different scenarios. Table 1 shows the input variables of the developed model. The volumetric charge density, distance from nozzle to collector, initial jet radius, and relaxation time had the largest influence on the resulting electrospun fiber diameter. The model did not involve volumetric flow rate, which was believed to be related to bead formation, charge densities, and viscosity alteration in the process. The absence caused the model to become very sensitive to initial jet radius and volume charge density, which were not wholly consistent with real experimental data. However, the model demonstrated appropriate behaviors in response to variation of nozzle-to-collector distance and relaxation time. For ordinary values of solution surface tension, the model predicted a trivial effect of this parameter on jet radius profile or final jet radius.
The jet trajectory obtained from the numerical solution of the mentioned equations is depicted in Figure 3. Here, MF intensity was set to be zero (B=0) in order to resemble the jet trajectory of Yarin’s findings. As the beads (jet) traveled downwards toward the collector, the perturbation added to the XY plane began to grow. At a point in which it started bending instability, the loops noticeably grew outward. It should be noted that the loops did not form due to jet “spiraling”, which is evident when following the path of a single bead at several successive times over the course of its travel. From Figures 4 and 5, it is evident that the bead did not follow a spiral, but rather continued to move away from the centerline as it maintains a downward motion. This type of behavior corresponds to a visually observed path of a physical jet section during electrospinning [1, 38].
3.2 Applying focusing MF
Non-zero external MF [Bz≠0, By=Bx=0 in Eq. (10)], exerts a new force called Lorentz force on every bead, which causes the whipping radius of the jet to decrease. As the intensity of MF increases, the radius decreases, as depicted in Figures 6 and 7. In all of the electrospinning configurations, the spinneret or input point of the jet and the collector are located at Z=20 cm and Z=0 cm, respectively.
It is clear from Figure 6 that for low MF intensity, the whipping radius has a significant variation. However, the radius decreases as the intensity increases, which is illustrated in Figure 7. In Figure 8, the cross-sectional radius of the jet for different MF intensities is shown. It is evident that this parameter increased by raising MF intensity. This behavior could be the consequence of shorter trajectory for fibers and reduced evaporation time.
It is noticeable that the jet cross-sectional radius slightly increased in the beginning of the process, and when it approached the onset point of the bending instability (at z=14 cm), the jet radius decreased very remarkably in a few centimeters. It is believed that the strong repulsion between electric charges with the same sign is the cause of initial increase in jet cross-sectional radius . A precise study revealed that focusing MF caused the path of every single bead to convert into a cylindrical shape with an individual central axis. As the direction of the resulting magnetic force was perpendicular to the path of the movement, the Lorentz force for every single bead was toward the center of its path. Figures 9 and 10 depict that by increasing the intensity of MF, not only the radius of the cylindrical decreased, but also its axis approached the z-axis.
In comparison with the common electrospinning process, relatively low intensity MFs (up to 15 G) had a significant effect on the whipping radius of the jet. Further increase in magnetic strength did not make any substantial alteration on this parameter, but the cylindrical radius became smaller. It seems a reciprocal Coulombic force between two adjacent beads with a same sign charge had an adverse consequence at the jet path. Increasing the MF intensity to more than 100 G caused Coulombic forces to increase, which in turn produced a distortion on the jet path. This increase did not have a significant effect on the radius of the jet. Therefore, it is not possible to predict jet behavior in high MF intensity. Figure 11 shows the jet path in the presence of MF intensities of 120 G and 150 G.
3.3 Controlling MF
After squeezing the whipping jet radii as much as possible by focusing MF, another MF called the controlling MF was exerted on the last segments of the jet path. This new MF was perpendicular to focusing MF and jet direction (Figure 12). Adjusting the intensity of this new MF makes position control of the jet possible. Figure 13 illustrates how a magnetic force affects the jet orientation. In general, the final position of the jet on the collector was not a single point, so it was a good idea to study the center of the circle of deposition to understand the behavior of the jet and to compute the deposition location with maximum probability. It is evident from Figure 14 and Table 2 that the position of the jet contact with the collector was linearly adjusted by the intensity of MF.
For better understanding of the effects of controlling MF, the path of every single bead was studied. Figure 15A shows the section of squeezed jet by concentrating MF. In Figure 15B, the behavior of the beads is demonstrated in response of exerted controlling MF. This controlling MF equals and is applied on the jet path in a distance 10–0 cm above the collector.
Bending instability in electrospinning under different MF intensities was investigated by using a discrete mathematical model. The charged jet was considered as a chain of interconnected electrified particles, joined by viscous flexible materials. In this mathematical model, the charged electrospinning jet path is completely consistent and every motion of charges depends on the motion of the jet. The effective forces on the system are the external electrical force, Coulombic force, viscoelastic force, surface tension force, and the magnetic force just exerted on moving charges. In this simplified but practical model, some parameters, including vapor diffusivity and solvent vapor pressure, were deliberately ignored due to their negligible contributions. This exclusion made the calculations more workable. Additionally, it was presumed that gravity, aerodynamic and surface tension forces, and jet evaporation had insignificant effects in the process. Despite these simplifications, the input variables of the model included almost all of the possible parameters that could be adjusted. Therefore, different scenarios could be run to determine their effects by observing the trends of the jet properties. The jet path was studied in three steps: firstly without any external MF, secondly with only concentrating MF that is parallel with the central axis of the jet path, and finally both concentrating and controlling MF were applied. In all simulations, the input variables were the same as the real values which had been reported by Thompson et al. .
Initially, the jet path was straight and then the added perturbation to each element began to grow, which introduced the jet into the bending instability. Meanwhile, the loops continued to grow outward, and the jet moved downward. It was assumed that the jet behaved similarly to a perfect dielectric material during the whipping phase. In other words, the movement of charges was the consequence of the jet traveling . The results from the model without applying MF had an appropriate similarity with the previously reported experimental data [40, 42].
It is worthwhile mentioning that capillary instability developed by surface tension amplifies perturbations in the jet, which could lead to breakup of the thin ligament between two adjacent beads. It is generally believed that electromechanical and viscoelastic stresses which are insensitive to polymer concentration resist the capillary breakup process. Accordingly, several studies approximated the alteration of these two stresses as a function of the diameter of stretching filament. To do so, a lower bound of the Newtonian viscosity was associated with the critical entanglement concentration to sufficiently resist the capillary breakup. However, there are several evidences that the capillary instability can be suppressed even for solutions with concentrations lower than critical entanglement concentration . Among different intricate models, a relatively workable one has been introduced by Helgeson et al. to address the issue . They believed that the polymer concentration and ensuing viscoelastic stresses increased more quickly than capillary pressure, due to evaporation in the whipping phase. In other words, they neglected the transient concentration increase during jet bending by introducing a limiting concentration at which viscoelastic stresses could stabilize the filament. Therefore, a maximum bounded extensional viscosity was taken into account to approximate the underlying physics in the bending part. This helps to perform a scaling analysis to find the minimal necessary polymer concentration required to determine the spinnability of a given polymer solution. This approach interestingly addressed the effect of the molecular weight distribution of polymers on their spinnability and resulting morphology of the produced fibers .
A centripetal force was generated by the concentrating MF towards the center point. As a consequence of this phenomenon, the swing scope of the jet in the magnetic electrospinning became considerably smaller . By amplifying the intensity of MF to 5 G, 8 G, and 10 G, the swing scope decreased by about 24%, 55%, and 69%, respectively. At higher intensities, the squeeze of the jet became less significant. A precise study on a single path of any bead revealed that with a proper intensity of a concentrating MF, every single path was modified to a progressive cylindrical path with a unique central axis. The applied MF was perpendicular to the jet direction, so the developed Lorentz force for every single bead was toward the center of the spinning path, rather than the z-axis. As MF intensity increased, the radius of cylindrical path was reduced and its axis approached the z-axis. When MF intensity exceeded about 100 G, the path of the jet became distorted. These irregularities appeared when the Coulombic forces between near electrified particles exceeded the applied external magnetic force.
A study on jet cross-sectional diameters in different MF intensities revealed that stepping up MF intensity caused an increase in the thickness of the jet in a nonlinear manner. The increase on the jet cross-sectional diameter between 0 G and 30 G was more significant. It is believed that the reduction in spinning radius was the main cause of this behavior. In the designed arrangement for PEO/water solution, the fiber diameter increased from its initial value of about 0.6 μm (without MF) to a final diameter of about 2 μm in the presence of 30 G MF. In most tissue engineering applications in which the porosity of substrate is the main parameter, this alteration in cross-sectional is acceptable.
Finally, an additional MF perpendicular to jet direction was exerted on the last segments of the jet to study the control possibility of polymer deposition on the collector. This MF was applied after maximum achievable squeezing of the spinning radius. It can be concluded from the path of any single bead that by individually adjusting the intensities of x and y components of the MF, the deposition of the jet on the collector can be fully controllable.
The bending instability was simulated by a discrete mathematical model in the presence of different intensities of focusing and controlling MF. The results showed that the focusing MF could significantly decrease the whipping radius of the jets at appropriate field intensity, and that the controlling MF could largely determine the target of the polymer jet on the collector.
This study would improve understanding of the magnetic electrospinning. Several experimental and theoretical studies should be performed to extend our information regarding the elongational viscosities, relaxation times of polymer solutions, and contribution of other parameters. Verification of this simulation experimentally would not be a straightforward task, but worthwhile to evaluate the accuracy of the model. It can be concluded that the magnetic approach is an efficient and promising technique to control some stochastic behaviors in electrospinning.
The authors would like to thank Professor A. Esteki, Department of Medical Physics and Biomedical Engineering, Shahid Beheshti University of Medical Sciences for his invaluable advice. The paper is prepared from the MSc thesis of Modeling the path and destination of nanofibers produced by electrospinning process in presence of a controlled magnetic field.
Conflict of interest statement: None of the authors has any conflict of interest.
He JH, Wan YQ, Yu JY. Int. J. Nonlinear Sci. Numer. Simul. 2004, 5, 253–262.Google Scholar
Wan YQ, Guo Q, Pan N. Int. J. Nonlinear Sci. Numer. Simul. 2004, 5, 5–8.Google Scholar
Jin J, Liu Y, Qin C, Wang J, Dai L, Yamaura K. J. Polym. Eng. 2010, 30, 149–160.Google Scholar
He JH, Wan WQ. Int. J. Nonlinear Sci. Numer. Simul. 2004, 5, 243–252.Google Scholar
Liu Y, He JH, Xu L, Yu JY. J. Polym. Eng. 2008, 28, 55–66.Google Scholar
Fernández de La Mora J. Annu. Rev. Fluid Mech. 2007, 39, 217–243.Google Scholar
He JH, Wu Y, Pang N. Int. J. Nonlinear Sci. Numer. Simul. 2005, 6, 243–248.Google Scholar
Kowalewski T, Nski S, Barral S. Tech. Sci. 2005, 53, 385–394.Google Scholar
Shamsi F, Janmaleki M, Fatouraee N. Iranian J. Biomed. Eng. 2010, 3, 265–274.Google Scholar
About the article
Published Online: 2015-01-17
Published in Print: 2015-08-01