Report

Physical principles of nanofiber production 7. Theory of electrospinning Taylor cone and critical tension for needle spinner D.Lukáš 2010 1 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 2 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. 3 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 49.3 o . 4 The problem has axial symmetry along the cone axis. The Maxwell equation Laplace operator 0 (3.7) r , Equation (3.7), for the axially symmetric electrostatic potential in spherical coordinate system r , , sounds as: 1 2 1 1 r 2 sin 0 2 r r r r sin where r is the radial distance from the origin and θ is the elevation angle, viz (Figure 3.4). 5 It is supposed further that the origin of the coordinate system is located in the tip of the cone. Let us consider the 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 (r, ) Rr S ( ) where Rr r n R and S are separately sole functions of r and θ respectively. 2 1 r sin 0 r r sin 6 2 1 r sin 0 r r sin (r, ) Rr S ( ) 2 Rr Rr S S r sin 0 r r sin 7 2 Rr Rr S S r sin 0 r r sin Thus, multiplying both sides by form as given below: 1 one obtains the Rr S 1 2 Rr 1 S r sin 0 Rr r r S sin 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 8 1 2 Rr r K Rr r r Rr Ar n Suggested solution 1 2 1 n 1 n r Anr n n 1nr n 1n K n Ar r r Laplace pressure E 1 pc r E Er Electric pressure 1 1 2 pe 0 E 2 r Er 1 2 E sphere 9 E r 1 1 , , gradient r r r sin 1 2 E r 1 r n S n 1 S r r n 1 r 1 2 1 n 1 2 Rr 1 2 1 n 2 10 1 1 S n sin ( n 1 ) n 2 S sin K=-3/4 where solution of S ( ) is the fractional order Legendre function P1 / 2 cos of the order ½ S P1/ 2 cos 11 o Ar 1/ 2 P1/ 2 (cos ) 0 const. 130.7099o 12 Moreover, from the graph it is evident that is finite and positive on the interval 0 o , P1 / 2 (cos ) and it is infinite at 180.o Thus the only physically reasonable electric field that can exist in equilibrium with a conical fluid surface is the one that spans in the angular area of space where the potential is finite and so the half the cone’s apex angle is 180o 130.7099o 49.2901o The angle is called as the “semi-vertical angle” of the Taylor cone. 13 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. 14 Taylor coun D.H. Reneker, A.L. Yarin / Polymer 49 (2008) 2387-2425. 15 Fe Fc 2 V Fe 4 ln(2h / R) Fc 2R cos 49.2901o 2h V 4 ln 2R cos (0.09) R 2h 2 Vc 4 ln 1.3R (0.09) R 2 c 16 2h V 4 ln 1.3R (0.09) R 2 c 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 17 Fig. 3.7. Critical voltages for needle electrospinner and for liquid surface tension of distilled water, 72 m N/ m . Curves represent Vc dependence on a distance, h, between the needle tip and collector for various values of needle radii, R. 18 R 19