Ultrathin fibres from electrospinning experiments under driven fast-oscillating perturbations
Ivan Coluzza, Dario Pisignano, Daniele Gentili, Giuseppe Pontrelli, Sauro Succi
UUltrathin fibres from electrospinning experiments under drivenfast-oscillating perturbations
Ivan Coluzza,
1, 2, ∗ Dario Pisignano,
3, 4
DanieleGentili, Giuseppe Pontrelli, and Sauro Succi CNR-Istituto per le Applicazioni del Calcolo “Mauro Picone”,Via dei Taurini 19, I-00185 Rome, Italy Faculty of Physics, University of Vienna,Boltzmanngasse 5, 1090 Vienna, Austria Dipartimento di Matematica e Fisica “Ennio De Giorgi”,Universita del Salento, via Arnesano, I-73100 Lecce, Italy National Nanotechnology Laboratory of IstitutoNanoscienze-CNR, via Arnesano, I-73100 Lecce, Italy
Abstract
The effects of a driven fast-oscillating spinneret on the bending instability of electrified jets,leading to the formation of spiral structures in electrospinning experiments with charged polymers,are explored by means of extensive computer simulations. It is found that the morphology of thespirals can be placed in direct correspondence with the oscillation frequency and amplitude. Inparticular, by increasing the oscillation amplitude and frequency, thinner fibres can be extractedby the same polymer material, thereby opening design scenarios in electrospinning experiments.
PACS numbers: 47.20.-k, 47.65.-d, 47.85.-g, 81.05.Lg a r X i v : . [ phy s i c s . f l u - dyn ] D ec NTRODUCTION
The dynamics of charged polymer jets under the effect of an external electrostatic fieldstands out as major challenge in non-equilibrium thermodynamics, with numerous applica-tions in micro and nano-engineering and life sciences as well [1–11]. Indeed, charged liquidjets may develop several types of instabilities depending on the relative strength of the var-ious forces acting upon them, primarily electrostatic Coulomb self-repulsion, viscoelasticdrag and surface tension effects. Among others, one should mention bending and “whip-ping” instabilities, the latter consisting of fast large-scale lashes, resembling the action of awhip. These instabilities are central to the manufacturing process known as electrospinning[12–17]. Some of them were analysed in the late sixties by G.I. Taylor [18]. In the subsequentyears, it became clear that the driver of such instabilities is the Coulomb self-repulsion, asit can be inferred by simply observing that any off-axis perturbation of a collinear set ofequal-sign charges would only grow under the effect of Coulomb repulsion [2, 19].In the electrospinning process, ultrathin fibres, with diameters in the range of hundredsof nanometres and below, are produced out of charged polymer jets. In addition to itsfundamental importance in the fields of soft matter physics and fluid dynamics, electrospin-ning is raising a continuously increasing interest due to its widespread application fields.For this reason, nowadays this technique is an excellent example of how applied physicsimpacts on engineering and materials science. For example, recently electrospun nanofibresand nanowires have been used for realizing organic field-effects transistors [11, 20], whosefabrication benefits from the concomitant high spatial resolution of active channels, large-area deposition, and frequently improved charge carrier mobility, which is highly relevantfor nanoelectronics. Other important applications are in the field of photonics [16], and in-clude solid-state organic lasers [21], light-emitting devices [22, 23], and active nanomaterialsfeaturing high internal orientational order of embedded chromophores, thus exhibiting polar-ized infrared, Raman [24–26] and emission spectra [27]. Electrohydrodynamics jet printinghas been recently applied in order to control the hierarchical self-assembly of patterns indeposited block-copolymer films [10], with important applications in nanomanufacturingand surface engineering. Electrostatic spinning is also a tool for medical sciences and re-generative medicine, e.g. for the construction of complex scaffolding necessary for tissuegrowth [8, 28], or for the accurate encapsulation of solutions into monodisperse drops [4].In general electrospinning is an excellent technique to transfer to a macroscopic compositematerial the properties of complex polymer solutions [5], which can have a direct impact on2he mechanical properties of resulting nanofibers [29]. In this respect, an aspect which isparticularly critical is that fibres can reach a diameter in the nanometre scale. Electrospin-ning can reach such scales, especially thanks to the occurrence of instabilities that drive thesystem to extend on the plane orthogonal to the jet axis [1–3, 6, 7, 9]. In such experiments,a droplet of charged polymer solution is injected from a nozzle at one end of the apparatus(spinneret), then it is elongated and the resulting collinear jet moves away from the dropletunder the effect of an externally applied electrostatic field. Such collinear configuration ishowever unstable against off-axis perturbations, as one can readily realize by inspecting theeffect of Coulomb repulsion on different portions of the jet. The resulting bending instabilityoften gives rise to three dimensional helicoidal structures (spirals). Due to polymer massconservation, the spiral structures get thinner and thinner as they proceed downwards, untilthey hit a collecting plate at the bottom. Based on the above, it is clear that an accuratecontrol of the effects of the bending and whipping instabilities on the morphological fea-tures of the resulting spirals is key to an efficient design of the electrospinning process. Thefundamental physics of the electrospinning process is governed by the competition betweenCoulomb repulsion and the stabilizing effects of viscoelastic drag and surface tension. Sincethe timespan of the entire process is comparable with the relaxation time of the polymermaterial, electrospinning qualifies as a strongly off-equilibrium process.A number of papers have been dealing with the theory of the electrospinning process [1,2, 15, 30–33]. Broadly speaking, these models fall within two general classes: continuumand discrete. The former treat the polymer jet as a charged fluid, obeying the equations ofcontinuum mechanics, while the latter represent the jet as a discrete collection of chargedparticles (beads), subject to four type of interactions: Coulomb repulsion, viscoelastic drag,curvature-driven surface tension and, finally, the external electric field.However, due to the complexity of the resulting dynamics and to the large number ofexperimental parameters involved in the process (related to solution, field and environmentalproperties), electrified jets are still treated by empirical approaches. For instance, eithernear-field techniques have been proposed to reduce instabilities [34, 35], or the effect of thedifferent parameters on the resulting nanofibre properties and radius have been determinedempirically through systematic campaigns [7, 36]. Indeed, to date most investigations onelectrospinning have been focussing on the experimental exploration of the various type ofpolymers that are liable to be electrospun into fibres, as well as on the processing/propertiesof the spun fibres. 3n addition, the dynamics of the jet is also sensitive to random external disturbances(noise), and particularly to erratic oscillations of the injection apparatus and of ambientatmosphere, which could act as hidden variables critically affecting the reliability of theprocess. In the sequel, we shall consider such fast mechanical oscillations of the spinneretas a driving perturbation, whose amplitude and frequency can be fine-tuned in order tominimize the thickness of the electrospun fibre. This portrays an angle of investigationof the electrospinning process that we proceed to investigate, on a systematic basis in thispaper. In particular, it is found that either increasing the perturbation amplitude, frequency,or both can lead to obtain thinner jets, hence thinner fibres in the end. More precisely, byincreasing the perturbation amplitude, spirals are seen to open up into a broader coneenvelope, hence resulting into thinner fibres. By increasing the perturbation frequency, onthe other hand, the spirals are developed with a shorter pitch, resulting again in thinner fibresat the collecting plate. To highlight the potential of driving the injected fluid to producethinner fibres in practice, we point out that increasing the perturbation amplitudes, from say1 . − cm to 1 . − cm, while keeping the other process parameters unchanged, leads toa three-fold reduction of the resulting fibre thickness for polyethylene oxide and other plasticmaterials. Similarly, varying the perturbation frequency from 10 s − to 10 s − determinesa three-fold fibre thickness decrease. MATHEMATICAL MODEL
The mathematical model used in this paper closely follows the one given in the pioneeringwork by Reneker and coworkers [2], namely the jet is described by a sequence of discretecharged particles (beads), obeying Maxwell fluid mechanics under the effects of the fourforces described above. The polymer jet is represented by means of a Maxwellian liquid,coarse-grained into a viscoelastic bead-spring model with a charge associated to each bead.This representation is justified by the earlier work of Yarin [1, 37]. In Fig. 1 we show howthe experimental set-up looks like. On the top of the figure there is the pendant drop, whichis dangling from a pipette or from a syringe. Between the pipette placed along the Z -axis ata height h and the collecting plane (not visible in the figure) at Z = 0, the electric field V /h is applied downwards along the Z direction. We shall consider the following experimentalparameters: a is the initial cross-section radius of the electrified jet, e is the charge perparticle, G is the elastic modulus, h is the distance from the drop to the collector, m is the4ass of each particle, µ is the dynamic viscosity, θ = µ/G is the relaxation time, α is thesurface tension, and V is the voltage applied between the drop and the grounded collectorplate. In what follows we will use Gaussian units for charges (CGS units). In Fig. 1, thespring represents the stress σ in the Maxwellian fluid and it is determined by the stressequation between each consecutive pairs of beads. The coupled system of Newton equationsfor the beads and the associated stress equation reads as follow: m d (cid:126)R i dt = (cid:126)F idσ i dt = l i dl i dt − σ i , (1)where (cid:126)R i = ( X i , Y i , Z i ) is the position vector of the i-th bead, l i = (cid:2) ∆ X i + ∆ Y i + ∆ Z i (cid:3) / is the distance between two bonded beads. Space and time units have been chosen as t = t phys /θ and l = l phys /L , where L = ( e /πa G ) / . In the sequel, the superscript (subscript)U (respectively D), will refer to the “up” ( i + 1) and “down” ( i −
1) beads, respectively,where i is the bead index. The coupling between the equations Eq. 1 takes place throughthe net viscoelastic force (cid:126)F VE acting on the bead i , which is given by: (cid:126)F VE = F ve (cid:20)(cid:18) a U σ Ui X i +1 − X i l Ui − a D σ Di X i − X i − l Di (cid:19) (cid:126)i + (cid:18) a U σ Ui Y i +1 − Y i l Ui − a D σ Di Y i − Y i − l Di (cid:19) (cid:126)j + (cid:18) a U σ Ui Z i +1 − Z i l Ui − a D σ Di Z i − Z i − l Di (cid:19) (cid:126)k (cid:21) (2)where in reduced units one has a U = 1 /l Ui and a D = 1 /l Di . Besides the viscoelastic force (cid:126)F VE , each bead is subject to three additional forces, namely: the driving force (cid:126)F of theexternal field (cid:126)V , acting between the drop and the collection plate, the bead-bead Coulombinteraction (cid:126)F Coul , and the surface tension force (cid:126)F
Cap , which acts as an effective bendingrigidity penalising curved jet shapes. These forces read as follows: (cid:126)F = − eµ | (cid:126)V | hLmG (cid:126)k(cid:126)F Coul = Q (cid:88) j (cid:18) X i − X j R ij (cid:126)i + Y i − Y j R ij (cid:126)j + Z i − z j R ij (cid:126)k (cid:19) (cid:126)F Cap = A k i (cid:32) − X i ( a U + a D ) (cid:112) X i + Y i (cid:126)i − Y i ( a U + a D ) (cid:112) X i + Y i (cid:126)j (cid:33) (3)5here F ve ≡ Q = πa µ LmG , A = παa µ mL G , R ij is the distance between two beads, and k i is thelocal curvature, defined as the radius of the circle going through the points i + 1, i and i − πa l = πa l (4)This determines the fibre thickness a as a function of the elongation l . At each time step,we first integrate the above stress equations Eq. 1, and then we use the updated stress termsto integrate the equations of motion which, in turn, are solved with a simple velocity Verletintegration scheme with the same integration time step ∆ t used for the stress equation. NUMERICAL INTEGRATION SCHEME
We solve Eq. 1 in an equivalent but numerically more convenient form: d ( e t σ ) dt = e t (cid:18) l dldt (cid:19) (5)using the forward Euler discrete integration scheme with an integration step ∆ t : e t σ ( t ) − e t − ∆ t σ ( t − ∆ t ) = ∆ t e t (cid:18) l ( t ) l ( t ) − l ( t − ∆ t )∆ t (cid:19) (6)which leads to the explicit time marching scheme σ ( t ) = e − ∆ t σ ( t − ∆ t ) + l ( t ) − l ( t − ∆ t ) l ( t ) (7)The Euler integration is coupled to a Verlet time integration scheme as follows, given theinitial conditions at t = − ∆ t and at t = 0 :1. Stress at time t : σ ( t − ∆ t ) → σ ( t ) from Eq. 7;2. Forces at time t : (cid:126)F ( t ) = (cid:126)F VE + (cid:126)F Coul + (cid:126)F Cap + (cid:126)F ;3. Positions at time t + ∆ t : (cid:126)r ( t + ∆ t ) = 2 (cid:126)r ( t ) − (cid:126)r ( t − ∆ t ) + (cid:126)F ( t )∆ t ;4. Velocities at time t + ∆ t : (cid:126)v ( t + ∆ t ) = ( (cid:126)r ( t + ∆ t ) − (cid:126)r ( t − ∆ t )) / (2∆ t ).We wish to point out that the 1 /r Coulomb singularity needs to be handled with greatcare, especially at the injection stage, where beads are inserted at a mutual distance shorterthan their linear size. This imposes very small time-steps, of the order of 10 − µ/G .6he effect of mechanical oscillations is mimicked by injecting the tail bead i = N ( t ); (cid:126)V i Init =0, N ( t ) being the number of beads in the chain at time t , at an off-axis position given by: X N = N s L cos(Ω t ) (8) Y N = N s L sin(Ω t ) (9) Z N = h − L ins (10)where L ins = h/I F is the insertion length and I F the insertion factor, h the vertical size ofthe apparatus, typically I F = 50000, N s (cid:28) N s as to the noise strength. Weremind that the insertion algorithm proceeds by inserting the N -th bead (polymer jet tail)once the distance between the ( N − Z = h exceeds theinsertion distance, L ins = h/I F . Thus, the jet is represented by a bead chain with the tail, i = N , at the spinneret and the head, i = 1, proceeding downwards to the collector. Theidea behind this model [2] is that, by choosing Ω (cid:29)
1, the above algorithm would generate aquasi-random sequence of initial slopes of the polymer bead chain, whose envelope defines aconical surface known as the Taylor cone. In this work, however, the expression (8-9) standsfor a deterministic, controlled source of fast oscillations, whose amplitude and frequenciescan be fine-tuned to design thinner fibres. To quantitatively explore this idea, we have runsimulations with several values of the perturbation strength N s and perturbation frequencyΩ. Boundary conditions are imposed at the two ends of the jet: at Z = h (top) and at thecollecting plate Z = 0 (bottom). The latter is treated as an impenetrable plane, at whichthe Coulomb forces are set to zero and the Z component of the bead position is not allowedto take negative values. The top boundary condition is such that the X and Y componentsof the forces acting on the pendant drop are set to zero, while the Z component must benon positive, i.e. point downwards.In what follows, we will use the scheme above to solve the time development of the jetwith a reference set of parameters pretty close, yet not exactly equal to the one given in theoriginal paper by Reneker et al. [2] as presented in Table I. For the case in point L = 0 . θ = 0 .
01 s. The resulting dimensionless parameters are given in the right panel ofTable I, where Q , F ve and A , are the strength of Coulomb repulsion, viscoelastic forces andsurface tension, respectively, K s = Ω θ is the injection frequency, H = h/L and (cid:126)F is thescaled force resulting form the external field, all in dimensionless units.7 ESULTS AND DISCUSSION
In Fig. 2(a), we show a typical shape of the jet at the end of a simulation, lasting about 10 time steps. As one can appreciate, spirals show from three to about ten turns over distancesof a few centimetres, which is consistent with experimental observation [38, 39]. While wecannot rule out the possibility that kinematic effects play a major role in the spiral formation,we must observe that the morphology of the spiral appears highly dependent on the choice ofthe initial conditions and forcing parameters. In this respect, even the kinematics alone maynot be trivial at all. We also considered a scenario whereby the perturbation is not appliedon the XY -plane, but only along the X -axis. Since the component of the force along the Y -axis is always zero, the jet does not develop any component in the Y Z -plane, resultingin a flat sinusoidal profile in the XZ -plane (Fig. 2(b)). Also these classes of planar spiralsare frequently observed in experiments, and not explored in previous theoretical papers.The extreme variability of experimentally observed spirals can therefore be accounted for byconsidering the oscillating perturbation as an independent variable affecting the resultingdynamics of the electrified jets. Effect of the perturbation amplitude
One of the environmental parameters of the model is the amplitude of the noise on thependant drop, related to the vibrations that affect the experiment at the microscale. In thepaper of Reneker et al. [2], the noise amplitude was kept fixed to N s = 10 − . In Fig. 3 weshow the comparison between the results in the amplitude range N s ∈ [5 10 − , . . . , − ].The main effect of increasing the amplitude is to increase the aperture of the spiral, basicallyin linear proportion. This indicates that the spiral keeps full memory of its initial slope,which is consistent with the deterministic nature of the model.In Fig. 3(b), we show the running draw ratio a ( Z ) /a along the polymer chain, consistingof about hundred discrete beads, for different values of the noise strength N s . The final drawratio is computed according to the standard expression resulting from mass conservationbetween the head (index 0) and tail (index f ) dimers in the chain, namely a f a = (cid:113) cL ins l final ,where c = 0 .
06 is the initial polymer concentration in the solution and l final is the elongationof the bead closer to the collecting plane at end of the simulation. As expected by massconservation, at all values of N s , the jet thickness is a decreasing function of the distancefrom the spinneret ( Z = 20). Such a decreasing trend is more and more pronounced as N s is8ncreased. For the bead closest to the collector, the draw ratio goes from a f /a = 2 . − for N s = 5 10 − to about a f /a = 7 . − for N s = 5 10 − . Thus, an order of magnitudeincrease in the perturbation amplitude leads to a factor three reduction in the fibre thickness.By inspecting the extension of the jet, it is seen that, on average, the thickness is stillreduced by a similar amount as in the case of the planar-perturbation (Fig. 4). However, atvariance with the case of planar-perturbation, the elongation shows recurrent oscillations,due to the fact that, close to the turning points, the fibre gets compressed, resulting in alocal increase of its thickness. Such a compression, which is visible in Fig. 2(b) through theblobs at the turning points, is a purely two-dimensional effect, since in three-dimensions thebead can turn back by taking a smooth round trip around the Z -axis. This two-dimensionaltopological constraint leads to the peculiar banana-like shape of the oscillation at high valuesof the perturbation amplitude, N s = 5 10 − (Fig. 4(a)), which results in large spikes of thefibre thickness (Fig. 4(b)). Such spikes are clearly undesirable in an experimental setting,since they would introduce a critical dependence of the fibre thickness on the location ofthe collecting plane. Depending on the height of the plane, or fluctuations of the jet, thefibre accumulation point on the collection plane would correspond either to a maximum ora minimum thickness. Moreover, one should also take into account the drying process ofthe fibre that could freeze the spikes into the final fibre. Thus, even though the qualitativemorphological differences induced by dimensionality do not significantly affect the time-averaged behaviour of the fibre diameter, compared to a pure horizontal driving force, theplanar perturbation is not only more realistic, but also definitely more convenient from theexperimental viewpoint.We also investigated the effect of planar noise asymmetry and vertical driving force witha range of frequencies (see Figs. 5-7). A wide variety of beautiful helices is obtained in thisway, confirming the richness of the phenomenology shown by the dynamics of electrifiedjets. We found that the planar driving force remains the best design principle. In fact, byintroducing asymmetry in the planar noise we quickly move towards the one-dimensionalscenario with sharper turns that locally thicken the fibre. We also found that an additionalvertical driving force does not provide a significant reduction of the fibre thickness. In thefuture we plan to include an additional component of the driving force along the Z -axisand/or consider an oscillating external field in the model.9 ffect of the perturbation frequency We have also explored the effect of the perturbation frequency, Ω, by running a series ofsimulations with Ω = 10 , , , s − , for both planar and three-dimensional scenar-ios. The corresponding spiral structures are shown in Fig. 8 and in Fig. 9, respectively. Fromthese figures, it is apparent that the main effect of increasing the perturbation frequencyis to produce more compact and broader structures, i.e. spirals with a larger aperture anda shorter pitch. This can be traced back to the fact that, by increasing the perturbationfrequency, the Coulomb repulsion in the XY -plane is enhanced, thus leading to larger ratiobetween the horizontal and vertical speeds, and hence a shorter pitch. Thus, increasingthe frequency leads again to thinner fibres at the collector, offering an additional designparameter, which minimizes the fibre thickness. To the best of our knowledge, such a designparameter has not been considered before. Effect of the location of the perturbation source
In most experiments the polymer jet is seen to fall down along a straight-line configu-ration, prior to the development of bending instabilities. In other words, a vertical neckprecedes the formation of spiral structures. Here, we have not been able to find any pa-rameter regime for which the neck would smoothly develop into a spiral structure: the twoconfigurations do not appear to belong together. What we have found instead is that spiralsstart precisely at the location where the perturbation is applied, which means that whenthe perturbation is applied at the spinneret, no neck is observed.To illustrate this point, in Fig. 10 we report the configuration of the spiral and thethickness of the jet, for a case where the perturbation is applied 5 cm below the pendantdrop. The jet shows a clear neck, up precisely to the point where the planar perturbation isapplied. Subsequently, the spiral forms in the very same way as it did when the perturbationwas placed at the pendant drop. A series of simulations was carried out by changing thelocation of the perturbation, always to find the same result: the spiral starts to developprecisely at the perturbation location. Furthermore, we have observed that, upon impingingon the collector, the jet forms planar coil patterns, similar to those observed in experiments,as one would expect for a thread falling on a stationary surface [40].These findings suggest that the presence of the neck might reflect a different and moreelaborate pathway to instability than just mechanical perturbation at the spinneret. Among10thers, a possibility is provided by environmental fluctuations, say electrostatic or perhapshydrodynamic ones, which may require a finite waiting time before building up sufficientstrength to trigger bending instabilities. The importance of the formation of the neck onthe development of the instability has been discussed in the work of Li et al. [41, 42] where adetailed analysis of the initial step of the spinning process was developed and linked to thephysico-chemical properties of the polymer solution jet. The results presented in the workof Li et al. suggest a strong dependence of the instability on such parameters. In the presentstudy, we have also considered a broad spectrum of different physical parameters with respectto the one that define the nondimensional quantities in Table I (results not shown). Forsome values of the parameters we could not produce helices, suggesting that the properties ofthe polymer solutions strongly affect the instability process, hence in qualitative agreementwith the results of Li et al. [41, 42]. However, even when we did detect helices, we stillcould not observe any qualitative change of the picture presented in this work, namely thatplanar oscillations driving the spinneret offer a viable route to produce much thinner fibres,regardless of the chemical physical properties of the polymer solution. A detailed analysisof the effect of the solutions used in the experiments will make the object of future work.
CONCLUSIONS
Summarizing, we have presented a computational study of the effects of a fast-oscillatingperturbation on the formation and dynamics of spiral structures in electrified jets usedfor electrospinning experiments. In particular, we have provided numerical evidence of adirect dependence of the spiral aperture on the perturbation amplitude and frequency. Asa result, our simulations suggest that both parameters could be tuned in order to minimizethe fibre thickness at the collector, by maximizing the spiral length. This might open upoptimal design protocols in electrospinning experiments. In other words, independently ofthe rheology of the polymer solution, provided that fibres can be electrospun, applying adriving perturbation oscillating at high frequency at the pendant drop can further reducethe thickness.
Acknowledgements
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10 (cm) ( c m ) z xy (b) Figure 2. Final configuration of the filament in the case of a circular driving perturbation (a) and alinear one (b). Parameters used for the simulations are V = 10 V, Ω = 5 10 s − , N s = 10 − andthe total spinneret-collector distance h = 20 cm, which resulted in the rescaled quantities L = 0 . Q = F ve = 78309 . F = 157 . A = 171 . H = 62 .
8. The linear dimensions are depicted inthe figure and correspond to the following range of the axes: X ∈ [ − ,
5] cm, Y ∈ [ − ,
5] cm and Z ∈ [0 ,
20] cm.
15 -10 -5 0 5 10 15 -15-10-5 0 5 10 15 0 10 20 N s =5 10 -4 N s =10 -3 N s =2 10 -3 N s =5 10 -3 X (cm) Y (cm)Z (cm) (a) a / a Z (cm) N s =5 10 -4 N s =10 -3 N s =2 10 -3 N s =5 10 -3 (b) Figure 3. Effect on the spiral shape of the mechanical perturbation applied to the pendant drop.(a) Final configuration. Parameters used for the simulations are as above except for N s . The plotshows the comparison of the profiles obtained with different values of the perturbation strength N s .(b) Plot of the elongation along the spiral as a function of the distance from the collector. The plotshows a monotonic decrease of the fibre thickness as a function of the distance from the pendantdrop. Finally, the plot highlights that an increase in the spiral radius induces an increase in theelongation of the electrified jet and correspondingly a reduction of the diameter of the resultingfibres. -5 -0.1 -0.05 N s =5 10 -4 N s =10 -3 N s =2 10 -3 N s =5 10 -3 X (cm)
Y (cm)
Z (cm) (a) a / a Z (cm) N s =5 10 -4 N s =10 -3 N s =2 10 -3 N s =3 10 -3 N s =3.5 10 -3 N s =4 10 -3 N s =4.5 10 -3 N s =5 10 -3 (b) Figure 4. Effect on the spiral shape of the perturbation applied to the pendant drop only along the X -axis. a) Final configuration of the filament. Parameters used for the simulations are V = 10 V, Ω = 5 10 s − and h = 20 cm. The plot compares the profiles obtained with different valuesof the noise strength N s , with the jet oscillating only in the XZ -plane. (b) Plot of the elongationas a function of the distance from the collector. The spikes, not present in the case of the circularapplied perturbation, correspond to the bending points that occur along the Z -axis. a / a Z (cm) N sy =2N sy =3N sy =4N sy =5N sy =6N sy =7 (a) a / a Z (cm) K sz =2K sz =3K sz =5K sz =10 (b) Figure 5. Plot of the fibre thickness a/a as a function of the height from the the collecting planeat Z = 0. We considered both asymmetric planar noises ( X/Y noise ratio N sy >
1) and verticalnoises along the Z -axis ( XY /Z relative noise amplitude N sz = 1 and relative frequency K sz > V = 10 V, I F = 5 10 , N s = 10 − , h = 20 cm. As forthe 1D noise scenario the elongation profile strongly fluctuates with the height from the collectingplate, making asymmetric or vertical driving forces not desirable for optimal experimental set up. Z ( c m ) Y (cm) K sz =2K sz =3K sz =5K sz =10 (a) Z ( c m ) Y (cm) N sz =.1N sz =.2N sz =.5N sz =.8 (b) Figure 6. Spiral jet generated under the presence of a vertical driving force along the Z -axis on thependant drop with a symmetric planar noise ( N sy = 1). For clarity we plotted only the projectionon the ZY -plane, but it should be noted that the spirals all keep they circular symmetry on the XY -plane. The parameters used for the simulations are V = 10 V, I F = 5 10 , h = 20 cm. (a)Spiral structures for different values of the relative load frequency K sz to the planar noise frequency K s = 5 10 , and a fixed noise strength N sz = N s = 10 − . (b) Jet profile plotted for a range ofrelative noise intensities N sz with respect to the XY -plane noise N s = 10 − at a fixed frequency K s = 5 10 . The results indicate that even large vertical noises do not affect considerably theshape of the helix and in particular they do not lead to thinner fibres. Z ( c m ) Y (cm) N sy =2N sy =4N sy =5N sy =7 (a) -25-20-15-10-5 0 5 10 15 20 25 -3 -2 -1 0 1 2 3 4 5 Y ( c m ) X (cm) N sy =2N sy =4N sy =5N sy =7 (b) Figure 7. Plot of the jet profile projected on the
Y Z -plane (a) and XY -plane (b) for variousrelative intensities N sy of the noise along the Y -axis with respect to the noise N s on the XY -plane. V = 10 V, I F = 5 10 , N s = 10 − , h = 20 cm. The greater the asymmetry in the noise, thelarger the asymmetry in the helix profile as visible in (b). Note the scales of X and Y axes aresignificantly different here. This transition is rather fast to the point that above an asymmetryratio of N sy = 7 we could not get stable numerical solutions any more, and already at N sy = 7 thehelix gets deformed towards 1D banana shape profiles (Fig. 5(a)). Ω =10 (s -1 ) Ω =5 10 (s -1 ) Ω =10 (s -1 ) X (cm) Y (cm)Z (cm) (a) a / a Z (cm) Ω =10 (s -1 ) Ω =5 10 (s -1 ) Ω =10 (s -1 ) (b) Figure 8. (a) The spiral structures for different values of the load frequency in the one-dimensionaldriving scenario along the X -axis. Parameters used for the simulations are V = 10 V, I F = 5 10 , N s = 10 − , h = 20 cm. (b) Corresponding fibre extension plot, where the characteristic spikes dueto the local bending are visible. As for the circular case, also in this case the increase in frequencyresults in thinner fibres. Ω =10 (s -1 ) Ω =5 10 (s -1 ) Ω =10 (s -1 ) X (cm) Y (cm)Z (cm) (a) a / a Z (cm) Ω =10 (s -1 ) Ω =5 10 (s -1 ) Ω =10 (s -1 ) (b) Figure 9. Comparison of the profiles obtained with different values of the perturbation frequencyΩ. Parameters used for the simulations are as above except for Ω. (a) Final configuration of thefilament under a circular perturbation. (b) Plot of the elongation as a function of the distancefrom the collector. Similarly to what observed by varying the perturbation strength (Fig. 3) alsoupon increasing the frequency Ω the radius of the spirals increases and stretches the fibre.
200 -100 -200 0 100 200 N s =5 10 -4 N s =10 -3 N s =2 10 -3 N s =5 10 -3 X (cm) Y (cm)Z (cm) -100 (a) a / a Z (cm) N s =5 10 -4 N s =10 -3 N s =2 10 -3 N s =5 10 -3 (b) Figure 10. Effect on the spiral shape of the mechanical load applied 5 cm below the pendant dropfor various values of the load strength N s . (a) Final configurations of the filament. Parameters areas above except for N s . (b) Elongation as a function of the distance from the collector. Above theheight of 15 cm, the location of the perturbation source, the fibre does not stretch appreciably. ABLES able I. Simulation parameters obtained from the experimental data in the paper of Reneker etal. [2] Experimental Parameters Simulation Parameters α = 700 g / s a = 1 . − cm e = 8 . g / cm / s θ = 0 .
01 s G = 10 gcm s h = 20 cm m = 2 .
83 10 − g µ = 10 g cm sV = 10 VΩ = 10 s − Q ≡ F ve = 78309 . F = 157 . A = 171 . K s = 100 H = 62 ..