D.Lukáš 2010

1. Physical principles of nanofiber production7. Theory of electrospinningTaylor cone and critical tension for needle spinner D.Lukáš 2010

2. Experimental as well as theoretical results on water droplet disintegration under the action of electrical forces can be extended to a description of electrospinning onset. Experiments have shown that the elongation of the droplet ellipsoidal shape leads to a quick development of apparently conical / wedge / vertex from which appears a jet. Macro-particles

3. Particularly referring to (Figure 3.4), it may be concluded that preliminary electrostatic analysis near a wedge shaped conductor has quite a remarkable characteristic similarity with electrospraying and electrospinning of conductive liquids, where cone-like liquid spikes appear just before jetting and spraying. This analysis was carried out by Taylor [16] in 1956. Figure 3.4.

4. Figure 3.4. (a) An analysis of electrostatic field near a conical body, where the field strength varies by rn about the wedge. Variables (r, ) represent the polar coordinates in two dimensions. (b) Taylor’s analysis of field near a liquid conical conducting surface, where field varies by 1/r . The characteristic value of the cone’s semi-vertical vertex angle, α, is .

5. The problem has axial symmetry along the cone axis. The Maxwell equation Laplace operator (3.7) Equation (3.7), for the axially symmetric electrostatic potential in spherical coordinate systemsounds as: where r is the radial distance from the origin andθ is the elevation angle, viz (Figure 3.4).

6. It is supposed further that the origin of the coordinate system is located in the tip of the cone. Let us considerthe trial solution at the vicinity of the cone tip for separating the variables, r and ,θ of the potential, , in the above equation in the form of where R and S are separately sole functions of r and θrespectively.

8. Thus, multiplying both sides by one obtains the form as given below: K -K The first term is a function of r only, while the last one depends solely on θ. That is why the last Equation is fulfilled only if

9. Suggested solution Laplace pressure Electric pressure

10. gradient

11. K=-3/4 where solution of is the fractional order Legendre function of the order ½

13. Moreover, fromthegraphitisevidentthat isfiniteand positive on the interval anditisinfiniteat . Thustheonlyphysicallyreasonableelectricfieldthatcanexist in equilibriumwith a conical fluid surfaceistheonethatspans in theangular area ofspacewherethepotentialisfiniteandsothe half thecone’s apex angleis Theangleiscalled as the “semi-verticalangle” oftheTaylorcone.

14. Taylor’s effort subsequently led to his name being coined with the conical shape of the fluid bodies in an electric field at critical stage just before disintegration.

15. Taylor coun D.H. Reneker, A.L. Yarin / Polymer 49 (2008) 2387-2425.

17. where, • h is the distance from the needle tip to the collector in centimetres, • R denotes the needle outer radius in centimetres too and • surface tension, is taken in mN/m. • The factor 0.09 was inserted to predict the voltage in kilovolts. CGSe  SI

18. Fig. 3.7. Critical voltages for needle electrospinner and for liquid surface tension of distilled water, . Curves represent Vc dependence on a distance, h, between the needle tip and collector for various values of needle radii, R.

19. R