### Fyzikální principy tvorby nanovláken 7. Teorie elektrostatického

```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  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,  )  Rr S ( )
where Rr   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,  )  Rr S ( )
  2 Rr   Rr   
S   
S    r

 sin 
0
r 
r  sin   
 
7
  2 Rr   Rr   
S   
S    r

 sin 
0
r 
r  sin   
 
Thus, multiplying both sides by
form as given below:
1
one obtains the
Rr S  
1   2 Rr  
1
 
S   
r

 sin 
0
Rr  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 Rr  
r
K
Rr  r 
r 
Rr   Ar
n
Suggested solution
1  2
1
n 1
n
r Anr
 n n  1nr  n  1n  K
n
Ar r
r


Laplace pressure
E
1
pc 
r
E
Er
Electric pressure
1
1
2
pe   0 E 
2
r
Er

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
Rr
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  2R cos
  49.2901o
 2h 
V  4 ln 2R cos (0.09)
R
 2h 
2
Vc  4 ln 1.3R (0.09)
R
2
c

16
 2h 
V  4 ln 1.3R (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
```