1

Physical principles of nanofiber production

7. Theory of electrospinningTaylor cone and critical tension for

needle spinner

D.Lukáš

2010

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.

2

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.

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 .

4

o.349

The problem has axial symmetry along the cone axis. The Maxwell equation

0

Equation (3.7), for the axially symmetric electrostatic potential in spherical coordinate system sounds as: ,,r

,r

0sinsin

1112

22

rrr

rr

where r is the radial distance from the origin and θ is the elevation angle, viz (Figure 3.4).

5

(3.7)

Laplace operator

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

)(),( SrRr

where nrrR

0sinsin

12

r

rr

6

R and S are separately sole functions of r and θ respectively.

0sin

sin2

SrR

r

rRr

rS

7

0sinsin

12

r

rr

)(),( SrRr

0sin

sin2

SrR

r

rRr

rS

Thus, multiplying both sides by one obtains the form as given below: SrR

1

0sinsin

11 2

S

Sr

rRr

rrR

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

K -K

8

Kr

rRr

rrR

21 nArrR

Knnnrnr

AnrrrAr

nn

nn

11

11 12

Laplace pressure

rpc

1

Electric pressure

rEpe

1

2

1 20 2

1

rE9

Suggested solution

ErE

E

sphereE

sin

1,

1,

rrr

Sr

Sr

rE n

n11

2

11 rr n

2

11 n

2

1n

10

2

1

rE

2

1

rR

gradient

nnS

S)1(sin

sin

1

2

1n

where solution of is the fractional order Legendre function of the order ½

)(S cos2/1P

11

cos2/1PS

K=-3/4

)(cos2/12/1 PAro

.0 const

o7099.130

12

)(cos2/1 P

,0o

o180

ooo 2901.497099.130180

Moreover, from the graph it is evident that

is finite and positive on the interval

and it is infinite at .

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

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

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

15

Taylor coun

cos2 RFc

)/2ln(4

2

Rh

VFe

ce FF

)09.0(cos22

ln42 RR

hVc

)09.0(3.12

ln42 RR

hVc

16

o2901.49

17

)09.0(3.12

ln42 RR

hVc

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

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.

mmN /72

18

19

R