Axisymmetric instability

By considering axisymmetric instabilities of highly conducting viscoeleastic solutions of PEO, in the theoretical studies, a linear stability analysis combined with a model for stable electrospun jet was used to calculate the expected bead growth rate and the bead wave number for given electrospinning conditions. 

The analysis reveals that the unstable axisymmetric mode for electrically driven, highly conducting jets is not a capillaiy mode but is mainly driven by electrical forces due to the interaction of chaiges on the jet. Both experiments and stability analysis elucidated that the axisymmetric Instability with a high growth rate can be seen in practice when the electrical force is effectively coupled with viscoelastic forces. ... 

Onset of axisymmetric instability causing bead formation in the fibers pictures taken at different distances from the needle (a) 1 cm,... 

The other instabihty mode is dependent on the conductivity of the fluid. Axisymmetric conductive instabihties can be viewed as a direct competihon between the surface charge and surface tension of the fiber as it moves. Although in the literature there are reports of formation of beads during the axisymmetric instabilities, this seems an appropriate case to describe the phenomenon of their formation. The fibers tend to decrease in diameter and form lumps in a certain frequency which depends on surface charges. According to the hterature, there is an excess surface charge in areas with bumps and a lower surface charges density in areas where the jet is thinner [22, 24]. 

Figure 3.9 An electrospinning jet of 4% solution of PHBV in chloroform imaged at different distances from the tip showing the development of axisymmetric instabilities (applied voltage 20 kV and feed rate 4 mL/h). Images (a) through (h) correspond to distances of 1, 3, 5, 7, 9, 12, 15, and 30 cm from the capillary tip. Reprinted with permission from Zuo et al. (2005). Copyright 2005. John Wiley Sons.

The authors then perform a linear stability analysis solving the dynamics resulting from small perturbations of the jet radius or charge density, or of the electrie field. These take expressions of the form, r=ro +where m indicates the growth rate of the instability, k is the disturbance wavenumber along the z direction, and the perturbation amplitude, Ar, is much smaller than yq. In this way, three instability modes are identified (for an introduction to the main axisymmetric and non-axisymmetric instabilities observed in electrospinning, see Section 2.1.3) ... 

Fig. 8 Schematic sketch of an axisymmetric instability (left) and bending instability adapted from Shin et al. [33]...

Fig. 9 Example of an operating electric field versus feeding rate diagram for electrospinning adapted from Hohman et al. [32,33]. The upper shaded area shows the theoretically predicted onset of bending instabilities the lower one shows the corresponding onset of axisymmetric instability. The lines represent experimental results on the instability thresholds for the two types of instabilities for PEO solutions for a given set of electric conductivity, viscosity, dielectric constant, and surface free energy values corresponding to the ones assumed in the theoretical treatment...

On the other hand, both axisymmetric and bending/whipping instabilities are caused by an electrically driven force due to the fluctuations in the dipolar component of the charge distribution and are essentially independent of the surface tension of the polymer solution. Axisymmetric instability often occurs at a higher electric field than Rayleigh instability whereas bending or whipping instability is not axisymmetric. 

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