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Physical principles of nanofiber
production
Theoretical background (3)
Electrical bi-layer
D. Lukáš
2010
1
Electric bi-layer
Boltzmann equation
Poisson Equation
Debye‘s length
2
Electric bi-layer is another object with nano-dimension in
electrospinning. Referring to second part of (Figure 3.2), one may
consider a plane surface of polymer solution, containing ions both in
polymer macromolecules and their solvent. Let the ion valence be
considered as one for simplification and e denotes the elementary
electric charge. In electrospinning, electric potential, 0, at the liquid /
polymer solution surface is generated by the electrostatic field in
between two electrodes.
Fig. 3.2:A cloud of ions (3) in a
polymer solution (4) is induced
by an electrostatic field between
a collector (5) and an electrode
(6). The thickness of the ionic
atmosphere at the vicinity of the
solution surface is called the
‘Debye’s length’, D.
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Figure 3.2: A charged colloid particle - (1): A positively charged
colloid particle is surrounded by a cloud of negative ions - (2).
Potential j is quenched with increasing distance, x, from the surface.4
5
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An electrospinner, from the point of view of ion distribution,
resembles the situation in the vicinity of a charged colloid particle.
The similitude is indicated in (Figure 3.2), where the collector plays
the role of organised groups of charges on colloid particles. The only
difference between colloid particle and electrospinner is the gap
coined by the space between the collector and liquid surface.
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+
D
+
+
+
+
Prodleva!!!
‘Debye’s length‘
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colloid particle
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The electric potential, , having the value, o, at the solution
surface, decreases with the depth in the solution as quenched by the
ion distribution in the solution surface layer, i.e. as quenched by
induced charges that shield the electric field towards the bulk of the
solution.
10
For the sake of simplification relative parallel placement of both the
collector and the solution surface will be considered henceforth.
The parallel configuration gives rise to simple symmetric
equipotential surfaces that are parallel to them too. So, the
electrostatic potential (x) can be considered as the function of the
only variable x, which is the distance measured along the axis,
perpendicular to the collector and solution surfaces, with its origin
located at the solution surface, pointing to the polymer solution
bulk.
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N… Number of space cells
n… Number of particles

x  0,1
Configuration vector

x (0,0), (1,0), (0,1), (1,1)
Boltzmann
equation
n 1 n
p (1,0) 
N N
Non-interacting free particles
n
p (1) 
N
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P (1)  ?

 E x  


p( x )  exp  
 k BT 
x

Ex   e x 
13
Analogie se
zemskou
atmosférou
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
p( x)  exp  mgx / kT 
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Fe 
1 q1 q2
4 r
1 nm
2
 10
36
Fg 
1 m1 m2
4
100 km
r
2
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To derive - x relationship, one has to start with a rule that governs
the distribution of ions acted upon by the external electrostatic field
as well as the field generated by ions themselves. The probability,
p(x), of finding an ion at a particular depth, x, in the solution depends
on its energy
e x 
through the Boltzmann equation
p( x)  exp  e ( x) / kBT 
where kB is the Boltzmann constant, and T being the absolute
temperature in Kelvin.
Kittel and Kroemer [29]
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The electrolyte for this moment comprises two kinds of ions of
opposite charge +e and –e. Their volumetric concentrations are:
n  x 
p( x) 
no
n x   no exp  e ( x) / k BT 

n  x   no exp  e ( x) / k BT 
Where n0 is the concentration of charges, when the solution is not
affected by any external electrostatic field. The concentration n0 is
universal for both the ions as the solution is electro-neutral as a
whole.
 ( x)  0  n x   n x   no


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The charge distribution is also governed by the one-dimensional
Maxwell’s first law of electrostatics, generally expressed in the form
of Poisson Equation with the potential gradient
d ( x) / dx
having the nonzero component along the x axis only. Mathematically,
the particular shape of the Equation (3.7) for this case becomes:
d  x 
 x 

2
dx

2
(3.10)
   e  x  
  e x  
  exp 

 x   en x   en x   no e exp 
 k BT  
  k BT 


(3.11)
Nonlinear differential equation!!!
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Debye and Huckel [32] Linearization
   e x  
  e x  
  exp 

 x   no e exp 
 k BT  
  k BT 
2
3
4
2
3
4
x
x
x
x
x
x
x
x
e x  1      ....., e  x  1      .....
1! 2! 3! 4!
1! 2! 3! 4!
x  1  e x  1  x, e  x  1  x
 e x 
e x 
e x 
e x   k BT   x   no e1 
1 
  no e
k BT
k BT 
k BT

d  x 
2  x 
 2no e
2
dx
k BT
2
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Debye and Huckel [32] Linearization
d  x  2no e

 x 
2
dx
k BT
2
2
2
2no e
 
1 0
k BT
2
d 2 x 
2


,
2
dx
2
2no e

k BT
d x 
  ,  x   1
dx
2


2n0e
x

 x   C e  0 exp
 k BT


x


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‘Debye’s length‘
2


2n0e

 x   0 exp
 k BT

D
D
k BT
2n0e
2

 x

x  0 exp  

 D

D
several units or
tens of
nanometres
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D
k BT
2n0e 2
„Salting (Vysolování)“
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Conductive body (liquid body)

E

E 0
+
Intensity inside conductive body is zero.
Intensity vectors (on the surface of conductive body) are perpendicular
to body surface.
Surface of body is equipotentials.
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25
Electric layer
+
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With respect to electrospinning, one can underline that Debye’s
length is the thickness of the ion cloud at the vicinity of the liquid
surface.
As this thickness in conductive liquids is generally not more than
several units or tens of nanometres, the external electrostatic field is
able to influence directly only the molecules that are close to the
liquid surface.
It is convenient to underline on this place that the electrostatic field
grasps during electrospinning preferably the surface layer of the
liquid where net charges are enormously concentrated.
The surface husk of the liquid is transmitted into a jet that is hence
supposed to be highly charged too.
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