POST-PROCESSES IN TUBULAR
ELECTROSPUN NANOFIBERS
A. Arinstein1,2, and E. Zussman1
1
Department of Mechanical Engineering,
TechnionIsrael Institute of Technology, Haifa, Israel;
2
Department of Physics, Bar-Ilan University, Ramat-Gan, Israel
The post-processes taking place in tubular electrospun polymer nanofibers are
discussed. The high-speed solvent evaporation during the electrospinning of a fibers
results in quick formation of shell of a tubular nanofiber, so the state of polymer
molecules in such fiber is non-equilibrium one. However, considerable amount of
solvent is remaining inside of tubular nanofibers and evaporation of this solvent,
continuing of several minutes, accompanies by further evolution of the nanofibers
resulting in modifications of both micro- and macro-state of the nanofibers. In this
paper the possibility of modification macro-state of the nanofibers is examined, i.e.
buckling of the nanofibers in their cross-section. The theoretical model describing
kinetics of solvent evaporation in good agreement with experimental observations,
allows one to estimate the physical parameters of the system in question and
determine the conditions of fiber shell instability resulting in buckling of the tubular
nanofibers, as well as the type (mode) of this buckling
1. INTRODUCTION
Electrospinning is a common process for fabricating polymer based nanofibers
presence of a sufficiently strong electric field jetting sets in at its tip forming a jet. It
allows fabricating nanofibers with a diameter in the range of 100 to 1000 nm in a
single stage by massive thinning and rapid solvent evaporation (less than 10 ms). Due
to a huge elongation of the jet, the polymer molecules inside the jet are highly
stretched while the rapid solvent evaporation results in fixing of such stretched nonequilibrium state of the polymer matrix. The mechanical properties and the
morphology of the collected fibers are commonly related just to these properties of
the electrospinning. The interest of many researches is focused on the examination of
conditions of electrospinning, assuming that emphasis in improving fibers properties
are in general put in tailoring polymer rheology [5], in controlling the electrostatic
field, as well as a configuration of set-up [6], however studies on the effect of the
evaporation rate on physical features of the electrospun fibers was not consider in
detail [
The very fast process of rapid evaporation is very hard problem for experimental
investigation, however the theoretical analysis and computer simulations can clarify
some aspects of the problem in question. For example, the role the high evaporation
rate plays in the fabrication of polymeric electrospun nanofibers was already
discussed by Koombhongse et al [12], and treated quantitatively by Guenthner et al
[13]. In particular, it was demonstrated that whenever the evaporation is very fast, the
polymer density at the fiber/vapor interface is found to increase sharply, creating a
polymer density gradient which acts as a barrier (skin) for further solvent evaporation.
9-18
This outcome is in good agreement with our point of view according which in
spite of the fact of fast solvent evaporation the collected electrospun nanofibers
contain a minor amount of solvent. The solvent presence, as well as the slow
evaporation of it, results in relaxation of formed nanofibers, i.e., the post-processes
take place in the systems in question. This statement is of great importance for
understanding the mechanisms of formation of nanofiber mechanical properties.
2. THE MODEL FOR SOLVENT EVAPORATION
The solvent evaporation results in decrease of a fluid quantity inside of a fiber
and this fact can be visible owing to meniscus motion. Therefore just studying of
meniscus motion can allow to understand and to describe the processes accompanying
the solvent evaporation. The typical regularities of meniscus motion are displayed in
the Fig. 1 (the experimental data were obtained in [14]). First a meniscus moves with
constant rate (this initial regime one can see very well in the insert on the Fig. 1),
thereafter its velocity sharply slows down, and finally, when the distance between left
and right meniscuses of a liquid is small, meniscus slowly stops.
L (mm)
0.30
Vm(mm/s)
0.06
0.25
L (mm)
Vm(mm/s)
0.8
0.030
0.6
0.022
0.15
0.4
0.015
0.2
0.007
0.10
0.0
0.20
0
5
0.000
10 15 20 25 30 t (s)
0.05
0.00
0.05
0.04
0.03
0.02
0.01
0
2
4
6
8
10 t (s)
Fig. 1. The meniscus displacement, L(t )  L0  Lcap (t ) (left axis) and its rate (right
axis) vs. time. The dots display the experimental data [14] and the lines display their
fitting on the base of the theoretical model. The insert displays another set of
experimental data for which the initial regime is dominant.
First of all, dominant processes determining the evaporation kinetics are to be
fixed. The evaporation of liquid solvent can occur through the wall of capillary as
well as from the meniscus surfaces. At first glance just last channel is to be dominant,
since the filtration of solvent through the capillary wall is limited by very low
diffusion coefficient. But taking into account ratio of surfaces areas of meniscus and
capillary wall, we have to conclude that the contrary situation takes place: the solvent
evaporation from the meniscus surfaces has no noticeable influence on the
evaporation kinetic, and just solvent filtration through the capillary wall controls the
process in question. Indeed, the simple estimation shows that the evaporation time
through meniscus is about tm  4104 sec that exceeds (in two and a half order of
magnitude) the experimentally observed time of evaporation process1. Whereas the
1
Note that the above estimation is much understated. The reason is that the evaporated gas is
assumed to being removed without impediments. In reality, outgas can occur only through the
9-19
evaporation time through the fiber shell is about tw  102 sec that is in good agreement
with the experimentally observed time of evaporation process. Therefore, we can
conclude that the dominant process determining the rate of meniscus moving is the
infiltration through the wall of a capillary, and follow-up evaporation of liquid solvent
(see Fig. 2 with schematic description of the process in question). And just this
process is now of our interest.
evaporation
Vm
Vl
vapor
liquid
Vl
vapor
Vm
evaporation
Fig. 2. The scheme of dominant processes determining the evaporation kinetic.
According to experimental observations no deformations of most of capillaries
accompany the solvent evaporation. Therefore a pressure inside of capillary is
decreasing, and this gradient of pressure results in a flux of solvent directed from
meniscus to the center of liquid column, compensating the solvent evaporation inside
of capillary.
Since the length of capillary is much larger than its radius, locally we can use the
simplest Poisson approximation for the solvent flux, V P x , where the coefficient
2
of proportionality equals  Rcap
8 (accurate within multiplier of about 1, and  is
the solvent viscosity),
2
Rcap
P( x, t )
V ( x, t )  
.
x
8
(1)
The second equation describing the process in question corresponds to the mass
conservation law which takes into account the solvent evaporation,
V ( x, t )
2 D 1  a P( x, t ) P( Lcap (t ), t )


.
x
Rcap d w
1 a
(2)
The key point of this equation is: a flux of liquid through the capillary wall is
inhomogeneous one along the capillary since a decrease of pressure inside of a
capillary hampers the infiltration of the liquid through the wall of capillary (the
pressure variation is also inhomogeneous along capillary). And this fact was taken
into account by simplest way, using the linear dependence with one parameter a,
which equals to 1 when x = Lcap(t), i.e. in the region of meniscus.
The boundary conditions for equations (1) and (2) are
P ( x, t ) x   L
cap ( t )
 Pat 
L (t )
2
 Pb , V ( x, t ) x   L (t )   cap , Lcap (t  0)  L0 ,
cap
Rcap
t
(3)
here Pat = 105 N/m2 is the atmosphere pressure and  is the surface tension coefficient.
tube shell, and accounting of this fact results in dramatically increase of evaporation time, and
if the concentration of evaporated gas achieves the saturated one, the process can stop at all.
9-20
Assuming   5010–3 N/m and Rcap  210–5 m, and 2 /Rcap  0.510–6 N/m2,
we see that that an influence of the surface tension on the process kinetics is
negligible.
3. THE KINETICS OF SOLVENT EVAPORATION
The above equations (1) and (2) together with boundary conditions (3) allow one
to describe all process features being of largest interest for us. In particular, the
pressure distribution gives information about elastic energy being stored in the fibers
during evaporation as a result of their compression in cross-section. Just this elastic
energy can be reason for additional phenomena being observed in experiments, for
example, formation of gas bubbles inside of fibers, or modification of fiber state, i.e.
fiber collapse.
At the same time, the velocity distribution obtained as solution of above equation
system, allows one to describe the rate of meniscus moving being observed
experimentally, and the comparison of experimental and theoretical dependences
being a good proofing for the theoretical model, results in definition of physical fiber
parameters.
3.1. The pressure and velocity distribution
The solution of equations (1) and (2) can be obtained by following way. After the
differentiation of equation (2) and substitution of P x in equation (1) we get the
equation for function V(x, t),
 2V ( x, t )
 V ( x, t )  0 ,
x 2
(4)
V ( x, t )   A(t ) sinh x   ,
(5)
2
solution of which is
3
here   14 (1  a) Rcap
d w P aD .
The pressure, P(x, t), can be found using equation (1) as follows,
here   Rcap d w
P( x, t )  1 Pb  1  (1  a)  A(t ) cosh x   ,
a
2D .
(6)
Assuming P  105 Pa, D  10–11 m2/sec and   50 Pasec we see that for
capillaries with radius about Rcap  20 10–6 m and wall thickness about dw  0.5 10–6
m the above parameters of the model are   0.210–3 m and   1 sec. The fitting of
experimental data obtained for the tubular nanofibers with above radius and wall
thickness (similar to the fits displayed in the Fig. 1) results in the following typical
values of above parameters of the model,   0.110–3 m and   1.5 sec, thus
theoretical estimations are in good agreement with experimental data.
Using the first boundary conditions (8) we find that
A(t ) 

 cosh Lcap (t )  
9-21
.
(7)
Thus, we find that the velocity distribution of liquid flux along a capillary is
V ( x, t )  
 sinh x  
 cosh Lcap (t )  
.
(8)
and the distribution of pressure decrease along a capillary is

cosh x   
P( x, t )  Pat  P( x, t )  1  a Pb 1   
,
a
cosh Lcap (t )   

2
here   a
 0.05
1  a Rcap Pb
(9)
3.2. The meniscus moving
In order to find the rate of meniscus moving, the second boundary conditions
(3) is to be used which results in equation determining the above rate
Lcap (t )
 sinh Lcap (t )  
(10)

.
t
 cosh Lcap (t )  
Solution of this equation (15) is
sinh Lcap (t )    sinh L0    exp  t   .
(11)
The obtained solution (16) (see Fig. 1) contains both cases of above asymptotic
behavior.
On initial stage, if Lcap (t )   1 , one can use approximation sinh x  0.5 exp( x) , so
the linear time-dependence of meniscus moving takes place
L0  Lcap (t )   t  .
(12)
In opposite case, if t   1, i.e. Lcap (t )   1 , equation (11) get the form
Lcap (t )   sinh L0    exp  t   .
(13)
which results in the exponential decrease of rate of meniscus moving
Vm (t ) 
Lcap (t )

  sinh L0    exp  t   .
t

(14)
3.3. The elastic energy of tubular nanofibers within evaporation process
The pressure difference inside and outside of capillary gives rise to a tangential
stress being inhomogeneous along the capillary and varying with time (see Fig. 3)
  Rcap d w  P.
(15)
9-22
Pat


Pin
dw
Fig. 3. The scheme of
tangential stress occurring
because of pressure
difference inside (Pin) and
outside (Pat) of capillary
resulting in elastic
compression of capillary
shell.
Rcap
This stress being inhomogeneous along the capillary and varying with time,
compensates the pressure difference inside and outside of capillary and results in
elastic compression of capillary shell. The density of elastic compression energy,
depending on coordinate x along capillary and time, is P( x, t )2 2E , where E is
Young's modulus of the capillary shell under compression. Thus, the time-dependent
common elastic energy of compressed capillary is
Wel (t )  2Rcap d w  1
E
Lcap ( t )
 P( x, t )
2
dx 
0
Lcap ( t ) 
 Wel( 0)
 1    cosh( z) cosh( L
cap

(16)
2
(t )  ) dx
0
3
Pb2 Edw .
here Wel( 0)  1 1 a  2Rcap
After integration we get the expression for elastic energy as function of length of
column of liquid for each moment of time
2
 L (t ) 


3 tanh  Lcap (t )  ,
1
Wel (t )  Wel( 0)  cap 1 


  
2



   2 cosh  Lcap (t )    2
here Lcap(t) is determined by equation (11).
(17)
4. THE POST-PROCESSES IN ELECTROSPUN TUBULAR NANOFIBER
It is clear that the system, the energy of which is increased, tends to find another
state in order to decrease its energy. However, kinetic barriers can substantially
influence upon the system choice. On this reason it is necessary to examine various
possibilities of system energy decrease.
4.1. Formation of gas bubbles
One of the possible ways to decrease the elastic energy of fiber is to replace one
very long column of liquid by several ones separated by area filled both by vapor of
solvent and by air penetrating inside of a fiber through its shell. Indeed, such
transformation results in decrease of elastic energy of a fiber and a gain of energy in
9-23
case of a splitting of the long column of liquid into two ones (the length of each new
column is the half of the initial length), is
Wel ( L)  Wel ( L)  2Wel ( L 2) 

(18)

 
1
1
 Wel( 0) 3 tanh  L   3 tanh  L   L 

.


2
2
 2  2
   2  cosh  L   cosh  L 2   

This gain of energy has to exceed the energy required for creation of two new
2
meniscuses, 2Rcap
 (here   5010–3 N/m is the surface tension coefficient).
Therefore, the energy gain, Wel (t ) , is to compare with surface tension energy,
2
2Rcap
 , and if the first is greater than the second, the bubble formation is possible.
The energy gain, Wel (t ) , can be presented as
Wel ( L)  Wel( 0) f L  ,


(19)
here f ( x)  3tanh( x / 2)  tanh( x) 2  cosh 2 ( x)  cosh 2 ( x 2)  x 2 . This function
tends to constant value 3/2 as f ( x)  1.5  2 x exp(  x) , if x >> 1, and tends to zero as
f ( x)  2 x 2 15 , if x << 1 (see Fig. 5). Thereafter, the dimensionless function f(x) is to
2
compare with the dimensionless ratio 2Rcap
 Wel(0)  Edw Rcap Pb2  0.2 (here E is
assumed about 0.5 GPa). This numerical estimation results in the fact that the bubble
formation (splitting of the one column into two ones) is possible if the length of liquid
cr )
column exceeds L(cap
 1.5  0.15  103 m .
(0)
Wel (L)/Wel
1.5
P(L)/Patm
1.0
0.5
0.0
0
2
4
6
8
L/
Fig. 4. The reduced gain of energy Wel ( L) Wel( 0) (solid line) and the reduced
pressure difference in the center of column of liquid P( L) Patm (dashed
line) vs. reduced column length L  .
On the initial stage of evaporation when the liquid column is very long this one
can break to many (n) pieces being also enough long. The energy gain allows one to
produce 2(n – 1) new meniscuses
n
Wel( n ) ( L)  Wel ( L)  Wel ( Li )
i 1
n
here
L
i 1
i
2
 3 (n  1)Wel( 0)  2(n  1)Rcap
,
2
L 1
i
 L  1 .
9-24
(20)
This fact agrees with experimental observation that in a few seconds after
fabrication of tubular nanofibers the liquid solvent inside these nanofibers is split onto
many long regions separated by gas bubbles.
4.2. Buckling of tubular nanofibers
As mentioned above, the solvent evaporation results in decrease of a pressure
inside of tubular nanofibers. This fact means that under the action of appeared
difference of external and internal pressures the tubular fiber can modify its shape. It
is well-known that for hollow cylindrical surfaces such shape modification is
negligible if the above pressure difference is enough small, however the shape varies
dramatically if this pressure difference exceeds some critical value: the tubular fiber is
losing the shape stability and fiber buckling takes place. This critical pressure can be
estimated with help of well-known condition taking into account both geometrical and
mechanical properties of a collapsing tube,
Pcr 
2
Ed w3 n 2  1 108  0.5  10 6 3 n 2  1



 1.3  102  n  12 Pa,

3
2
6 
2
12  20  10  1  
12 Rcap 1  
1 
(21)
here  is the Poisson's ratio which can be taken equal to 0.5 and n = 2, 3,… is
azimuthal wave-number of buckling mode. Therefore, if the pressure difference, P
exceeds 6102 Pa, the buckling of fibers has to occur.
The estimation basing on the equation (9) results in the fact that the pressure
difference in the center of column of liquid of length 2L exceeds the above value of
critical pressure difference if L > 1.110–2  1.110–6 m, and this length is less than
fiber radius. That means that the conditions for the shape instability take place during
the all time of solvent evaporation, and conditions for the fiber buckling are more
favorable on initial stage of the process of solvent evaporation when the fiber regions
filled by solvent, are enough long. Nevertheless, fiber buckling was being observed
only on the final stage of evaporation, whereas the initial stage was not accompanied
by such phenomena. Such behavior has a simple reason.
The point is that, at least on the initial stage of the process in question, the tubular
fibers are filled by solvent which prevents the fiber buckling. Indeed, any small
deformation of a fiber results in decrease of internal volume of a fiber. This volume
decreasing is possible only after moving off of a portion of solvent filling a fiber and
hindering a buckling, i.e. the flux having direction opposite to initial one, is to arise.
But the flux can arise only in the case of increase of local pressure in region of
deformation that means that the difference of external and internal pressures in zone
of fiber deformation sharply decreases and no further deformation or buckling of a
fiber will take place.
Patm
Vl
Pm
Vl
Pin
Pm
Patm
Fig. 5. The scheme of fiber buckling in case of relative short regions filled by solvent.
3
Since Pin  Patm  Ed w3 12Rcap
 n 2 1 1  2  Pm  Patm  2 Rcap the pressure
difference arises, that results in solvent flux decreasing the volume of solvent in the
buckling region.

 

9-25

Such a situation when no buckling is possible in filled tubular fiber takes place if
the column of liquid is enough long and meniscus does not influence on the pressure
inside of a fiber. However if the column of liquid is enough short the forces of surface
tension disturb the above stability mechanism of the form of filled fiber, and the fiber
buckling is possible. Indeed, the forces of surface tension decrease the pressure of
solvent in meniscus region ( Pm  Patm  2 Rcap ) and such decrease allows one to
retain the critical difference of external and internal pressures in zone of fiber
deformation ( Pcr  Patm  Pin , Pcr is given in equation (21)) and in the same time
allows one to get the necessary pressure gradient along a fiber Pcap  ( Pin  Pm ) Lcr
for flux arising removing a solvent from a buckling zone with the rate exceeding the
rate of meniscus moving (see Fig. 5). Thus, the fiber buckling will take place if the
following condition is true
Pin  Patm 
Ed w3 n 2  1
2

 Pm  Patm 
.
3
2
R
12 Rcap 1  
cap
(22)
Such a picture of fiber buckling demonstrates that this phenomenon can take
place only on the final stage of evaporation when the columns of solvent are enough
short, and allows one to estimate this critical length of columns of solvent. With help
of Poisson approximation we get that
2
2
Rcap
Pcap Rcap
R 2  2
Pin  Pm
Ed w3 n 2  1  

 1 cap 


 ,
3
8 Lcr
8
Lcr
Lcr 8 Rcap 12 Rcap
1   2  

which results in following value of critical length of columns of solvent
Vl 
2
 2
Rcap
Ed w3 n 2  1 


Lcr 


 0.07  10 3 m .
3
2 

 8  Rcap 12 Rcap 1   
a
(23)
(24)
b
100 m
Fig. 6. Micrographs of fiber buckling: the buckling zones (a) and the form of
cross-section (b) of the fiber after evaporation.
Just such scenario was observed in experiments: on initial stage of evaporation no
fiber buckling occurs, however with solvent evaporation the regions containing no
solvent constitute the noticeable part of the fiber and liquid solvent can be ejected into
such regions, offering no resistance to fiber buckling, in so doing such buckling takes
place when columns of solvent are amounting about 100200
or less (see Fig. 6).
9-26
5. CONCLUSIONS
The above analysis demonstrated that despite the high evaporation rate during
fabrication, co-electrospun micron-size polymer tubular structures were found to
contain a significant amount of solvent. The extended period in which solvent
evaporation takes place, results in additional changes in the nanofiber structure.
Experimental observations reveal that the sol-vent accumulated within the core of the
fibers forms long slugs. As the remaining solvent evaporated, these slugs shortened
and eventually disappeared. Direct observation showed that this process was
associated with displacement of the menisci, i.e., the approximation of the ends of
these slugs. The theoretical model that describes the displacement of the solvent's
menisci allows one to find the correlation between the evaporation rate, the physical
parameters of the nanofibers and the observed rate of meniscus displacement. The
combined experimental data results in values of physical parameters of the system
that are shown to be reasonable from a physical standpoint. This fact, as well as the
explanation of further possible fiber evolution (bubble formation or fiber buckling), is
a good verification of the proposed model and allows one to better understand the
physical processes associated with solvent evaporation.
Moreover, we found that the mechanism used to explain and predict the structure
evolution of the tubular nanofibers can be applied to solid-core nanofibers, as well.
Indeed, the system we examined can be considered as the extreme case of the solidcore system, i.e., one with a sharp polymer density gradient (in the solid nanofiber,
the polymer density near the fiber surface is high, while the polymer density in the
fiber core is significantly lower [13]). Therefore, the kinetics of solvent evaporation in
solid-core nanofibers is seen to be similar to that in tubular fibers. As would be
expected, the rate of the process in solid-core nanofibers will be slower due to a
noticeable increase in the effective viscosity of the solvent inside the fiber, which can
be approximated in the initial stage of the process by a porous medium. Nevertheless,
the qualitative features of solvent evaporation in solid-core nanofibers remain
inherently the same, implying that the post-processes in fabricated nanofibers, in
particular, and the polymer matrix relaxation process can significantly affect the
mechanical properties of polymeric nanofibers.
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