Mathematical Simulation of Water and Benzene Penetration in Thin
Polystyrene Films and Experimental Investigation of the Mass Transport
Processes by FEWS-FTIR Spectroscopy Procedure
Avigdor Sheshnev, Edward Bormashenko, Joseph Reiclin & Roman Pogreb
The College of Judea and Samaria, Ariel, Israel,
Eugene Shulzinger & Abraham Katzir, Tel-Aviv University, Tel Aviv, Israel
Abstract
This work presents the new approach to the investigation of mass transport process in polymers.
The Fiber Optic Evanescent Wave Spectroscopy has been used for the real time investigation of
diffusion processes in glassy polymers. Unclad AgClBr fibers of 0.9 mm diameter were dip coated by
polystyrene layers of 1-30 μm thickness. The transmission of the fibers in the mid – IR was measured
using a Fourier Transform Infrared (FTIR) spectrometer. The penetration of liquids into these layers
gave rise to significant changes in the measured spectrum. These changes were used for diffusion
studies in situ. The mathematical model, which allows realizing the quantitative treatment of
experimental data, was developed. The model deals with processes which take place at two interfaces:
polymer/liquid and polymer/optical fiber. It was established that the initial stage of diffusion is of the
strictly Fickian character. The model permitted us to calculate the coefficient of diffusion of water in
polystyrene with a high accuracy. The huge amount of experimental points, which could be obtained by
our method, allows calculation of the equilibrium concentration of penetrating liquid with an extremely
high precision. This advantage of FEWS procedure offers a high accuracy of calculations of parameters
of diffusion. It must be emphasized that the thickness of the film that could be defined with the least
precision doesn’t affect on final results. The final stage of diffusion is non-Fickian.
The sensitivity of the method was sufficient for studying the penetration of small concentrations
(up to 5 ppm) of organic liquids through polymer layers. This fact allows using the polymer coated
AgBrCl fibers as sensor elements for identifying organic liquids in water. It must be emphasized that the
cylindrical symmetry of our fibers renders them especially suitable for the diffusion study and allows us
to obviate the difficulties caused by end effects.
Keywords: water, diffusion, polymer, infrared, spectrum, ATR, fiber, absorption, organic, Fickian, real
time.
Introduction
The diffusion of organic liquids through polymers has been of great interest in polymer physics.
Diffusivity in polymers is obvious from a consideration of diverse uses of polymer films as barrier
coating materials, in packaging applications, membranes, etc. At the same time the full understanding
of mass transport processes is not obtained because of the complicated structure of polymer materials.
The processes of mass transport in polymers have a mutual dependence on the presence of void and
defect structures and on the nature of polymer chain segmental motions. Different investigators have
revealed Fickian and non-Fickian types of diffusion. The concentration dependence of diffusion
coefficient and the role played by boundary conditions are debated in scientific literature. That is why
novel experimental methods that allow the precise experimental methods, which allow the real time
244
diffusion measurements, are of great importance. Liquids penetrating polymer layers give rise to changes
in absorption in the middle and far IR. This may provide useful information about the kinetics of the
penetration of liquids into the polymer bulk. We have selected polystyrene as a model polymer because
of its strictly amorphous structure.
Peppas and Jabbari (1,2) investigated interdiffusion processes by the classic attenuated total
reflection (ATR) technique, making use of an FTIR spectrometer. Turner (3) studied adsorption of
sodium dodecyl sulfate to a polystyrene/water interface by the infrared ATR spectroscopy procedure.
Authors who used traditional ATR-FTIR methods mentioned that the fingerprint region of the spectrum
from 1800 cm-1 to 400 cm-1 was uninterpretable, because the signal to noise ratio was too high. Krska
(4) used silver halide fibers coated by low-density polyethylene for the detection of organic compounds
in water. Traditional plane ATR-FTIR elements were used by Yarwood for the investigation of water
penetration in amorphous and crystalline polymers (5) . An interrelation between diffusion process and
parameters of optical measurements in such plane elements was analyzed mathematically by Fieldson
and Barbari (6) .
The novelty of this work is that we used polystyrene-coated silver halide fibers for the water and
benzene diffusion study. The advantage of the method is that one can get results with high spectral
resolution in a relatively short time. We were thus able to obtain highly accurate results.
FEWS is a technique, which is based on attenuated total reflection (ATR). IR radiation transmitted
through a segment of infrared transmitting optical fiber undergoes total internal reflection. There is an
evanescent wave that decays exponentially in the medium surrounding the optical fiber. If the medium is
absorbing, energy is absorbed at specific wavelengths, for which the surrounding exhibits absorption.
FEWS meauserements were analyzed thoroughly from the optical point of view by Gupta and Sharma
(7).
If the transmitted power is measured at a different wavelength, using, for example, a FTIR
spectrometer, the measurement reveals the absorption spectrum of the surrounding medium. The IR
fibers used in this work have been extruded from silver halide crystals. These fibers are highly
transparent in the mid IR, and they are flexible, non-toxic and non-hygroscopic. Thin layers of polymers
can be applied on the surface of the fibers. Such layers could be useful for chemical protection of the
fibers, for example when used for biomedical studies. In previous works we have already carried out
FEWS measurements on polymer coated AgClBr fibers (8,9). Experimental investigation of water
diffusion in thin PS layers was first discussed by our group in (10). Authors have improved significantly
the mathematical treatment of experimental results, and our phenomenological is presented in this
article.
Experiments
The experimental FEWS set up, shown schematically in the Figure 1, is similar to the one used by
Gotshal et al (11). It consisted of a FTIR spectrometer (Nicolet, model 5PC) with a special cell designed
for liquid measurements. A silver halide fiber of 0.9 mm diameter, coated by a polymer layer ran
through the cell, and the liquid around it acted as the absorbing medium. The length of fiber was 10.5
cm. The radiation from the IR source of the spectrometer was focussed on the fiber input end, using a
ZnSe lens with focal length of 25 mm and a diameter of 25 mm. Radiation was collected from the
output end of the fiber with a similar lens, which focussed the radiation on the DGTS detector of the
spectrometer. The spectral resolution was 4 cm-1. The spectrometer was kept at a constant temperature
and was purged with dry nitrogen to eliminate water vapor absorption.
245
The measurement procedure was as follows: when water penetration was under investigation, the
polymer-coated fiber was inserted in the dry cell. The measured spectrum (i.e. the absorption spectrum
of the polymer) was considered as background. Distilled water was then introduced into the cell. Water
penetrated into the polymer layer, and gave rise to noticeable changes in the measured spectrum.
Measurements were taken every 5 seconds and were divided by the spectrum of background.
When the penetration of organic liquids was studied, the cell was dried, and distilled water was
simply replaced by the organic liquid. As one example we studied the penetration of benzene into
polymers.
Applying of polymer layers
Polymer layers were applied to the surface of AgClBr fibers by the dip-coating procedure as it is
schematically shown in the Figure 2. The liquid solution of a polymer (polystyrene, in our case) adheres
to the fiber surface and solidifies through the evaporation of the solvent (xylene). The principles of the
dip-coating process were theoretically established by Landau (12) and experimentally proved by Brinker
et al.(13). The dip-coating procedure permits us to control the thickness of applied layers by varying two
of the parameters in the process: the viscosity of the solution and the speed of the drag V0. Solutions of
5-20% polystyrene (Polyrex) and 95-80% xylene were used, and the drag speed of the fiber from the
solution was 0.5-50 cm/min. Polystyrene (PS) layers of a thickness of 1-30 μm were thus applied to
AgClBr fibers.
Experimental study of water diffusion processes and mathematical treatment of obtained
results.
Figure 3 illustrates the changes in absorption (per cent) measured by an AgClBr fiber coated by a
polystyrene (PS) layer of 13μm thickness, when the coated fiber was immersed in the distilled water. As
mentioned, the spectrum of the coated fiber in a dry cell was taken as the background. Spectra were
measured every 5 seconds. We can see a rise of absorbance in the 3150-3600 cm-1 band, which is
characteristic for the valence vibrations of the H2O molecule, and in the 1600-1680 cm-1 band, which is
characteristic for the deformational vibration of water molecules. The changes in absorbance are
unambiguously related to water penetration in the PS layer.
Information which is obtained by one experiment consists of a set of spectra measured every 5
seconds. For the purposes of quantitative interpretation of obtained results, we selected the absorbance
at a fixed wavenumber of 1644 cm-1. These peaks were approximated for every spectrum by Gaussian
curve:
2   2  xw 2x0 
e
 W
2
y  y0 
(1)
The quantity y0 is not related to the peak under discussion and has to be eliminated as a
background. It was established that the peak’s location x0 doesn’t vary for different spectra, and with
very high accuracy could be determined as 1644 cm-1.
246
Absorption cell
Evanescent field
E
ZnSe
Lens
ZnSe
Lens
SAMPLE
*
IR fiber
IR SOURCE




IR DETECTOR
SAMPLE
Z
PS coating
Fig. 1. Schematic drawing of the FEWS system for diffusion study
V0
solid film
fiber
liquid
Fig. 2. Dip-coating procedure.
247








8
7
t=20min
t=5min
t=1min
5
4
3
2
,
%
,
Absorbance
6
1
0
3500
3000
2500
2000
1500
1000
Wavenumbers , cm -1
Fig. 3. Absorbance changes during water penetration through a polystyrene
layer of 13m thickness
The half-width of the peaks W depends slightly on spectrum and varies accidentally within 2.5
percent of its average quantity. We supposed that the parameter which describes the water diffusion
process is the peak’s area – A (see equation (1)). According to our suggestion, A is proportional to the
total water content in the polymer layer that is under investigation. The mathematical treatment of
experimental results is based on the diffusion equation (2). For simplicity we assume that the water
penetration process takes place in a plane layer of 2δ thickness.
c
2c
(2)
D 2
t
x
Where, c represents water concentration (mol-1), D diffusion coefficient. The plane geometry can
be used because of the small thickness of layer δ as compared to the fiber’s radius R: δ/R~10-2. The use
of cylindrical coordinates gives rise to unessential changes in coefficients and doesn’t affect the general
results of mathematical treatment. Boundary conditions are given by equations (3), (4):
c
0
(3)
x
c
D
 k c  c 
(4)
x
Equation (3) expresses the absence of diffusion on the interface fiber/polymer. c  represents the
equilibrium concentration of penetrating liquid (water) at given external conditions, and essentially is
the thermodynamic parameter of the diffusion system being studied. It is worth noting most of the
above cited researchers used boundary condition c  c  instead of equation (4). This assumption can
give different diffusion coefficients established at final and initial phases of the penetration process. The
same mistake occurs when the same materials are investigated under different conditions. Solution of
system of equations (1-4) could be written as:

2
x


cx, t   c  1    n c n cos  n e   n Fo 
(5)

 n 1

248
Where, Fo 
Dt
2
is the diffusion criterion Fourier, c n and  n depend on criterion Bio:
k

(6)
D
Where,  n is the root of an equation   tg  Bi . The peak’s area is proportional to the total
mass M of absorbed liquid, so equation (5) has to be transformed, and M could be calculated as:
M t    c  x, t dx
(7)
Bi 

Now perform the integration over the volume of the polymer layer, which absorbed liquid. Let
the depth of radiation penetration in liquid  eff be less than the thickness of the polymer film. Figure 4
illustrates different possibilities of relationships between δ and δeff
The integration of equation (7) gives us:

 eff   n2 Fo 
 
M t    eff c  1 
c n sin  n
e



  eff n 1

If  eff   we receive:



M t    eff   c  c 1   cn sin  n e  n2 Fo 
 n 1

(8)
(9)

eff
Evanescent field
Water
Polymer
coating
Water
molecules
Fig. 4. Stages of water penetration in PS layers
Where: c is the water concentration in the outer medium. It is clear that M   eff   c is
constant under experiment conditions. It can be established at the initial stage of the experiment with the
assumption 0  0 . For reasonably long times the optical signal arrives to fixed value, suggesting
that concentration came to the equilibrium value c∞. In this situation  eff   , M     eff c . The
249
peak’s area at the equilibrium concentration is a constant of a particular experiment and could be
established with very high accuracy, because it is determined as a result of averaging over huge amount
of points (~102). This fact comprises the main advantage of our method. That is why the dependence:

  eff

At 
M t 

(10)
 mt  
 1   cn
sin 
 n e  n2 Fo
A
M 
 eff

n 1


has a general nature and analysis of coefficients gives the exhaustive information about diffusion
coefficient D and mass exchange coefficient K. The aforesaid is true in the case of  eff   as well. The
only difference is the definition of mt  . In this case we imply that mt  is defined as:
M t   M
(11)
m( t ) 
M    M
It is clear that we have to set  eff   in the equation (10). The accuracy of our treatment is
obviously decreased in this situation because the error of  value far exceeds the error of  . At
not-too- small values of t the experimental curves are approximated adequately by relationships of the
type:
(12)
mt   1  ae  bt
Where coefficients a and b could be calculated according to minimum square’s method. The
relationship (12) corresponds obviously to the holding of only the first term in equation (10). The
typical diffusion curve calculated according to (11) is demonstrated in figure 5.
m
(t
)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
2000
2500
3000
t, s
Fig. 5. The typical diffusional curve treated according to relationship (11). The thickness of PS
film is 20 μm.
At the initial stage of diffusion process we have the relationship ( Bi   )
2 D
(13)
mt  
t
  eff
250
0.2
y = 0.0102x0.999
R2 = 0.9933
mt 
0.1
0
0
2
4
6
8
10
12
14
Fig. 6. The initial stage of diffusion process, related to coordinates m(t), t
The exponential (final) stage of diffusion experiment is described by function
mt   1  0.562e 1.25103 t
16
18
t , sec1/2
(14)
D
2 D
 0.0225, sec . Correlating initial and final
 1.25  10 3 s-1, or
2

4 
stages of the diffusion process, we obtain:  eff  0.5   . It can be seen that the initial stage of diffusion
1

2
It is clear now that
process is strictly Fickian. We calculated the diffusion coefficient (we used polymer layers of 20 μm
thickness) and obtained D = 1.8 10-9 cm2/s. The obtained value correlates well with D values, measured
by other methods. Most likely the final stage of the diffusion process is non-Fickian. This stage is now
under our investigation.
Study of benzene diffusion process.
It is clear that polymer coated AgBrCl fibers could be used for the identification of organic
compounds in water. Solutions of different concentrations (5-200 ppm) of benzene in water were
prepared. These solutions were used for the investigation of diffusion processes in polystyrene layers.
AgClBr fibers were dip-coated by layers of 1-30μm of polymer. In this case the absorption spectrum of
distilled water was taken as background. The absorption spectra of the benzene solutions were taken
every 5 seconds. Figures 7, 8 demonstrate the time dependence of the absorbance spectra for a solution
of 50ppm of benzene in water and for a polystyrene layer of 13μm thickness. Transmittance (per cent)
dependence vs. wavenumbers is presented
Figure 7 illustrates the increase of the peak at 680 cm-1 caused by C-H deformational vibrations. It
should be mentioned that benzene could be well identified by these two characteristic peaks. As in the
case of water, the initial stages of the diffusion process are Fickian, whereas the final stage is nonFickian. Krska et. al. (4) noted a similar behavior when they studied the diffusion of chlorinated
hydrocarbons into polyethylene.
It was established that the smallest concentration of benzene in water that brings to existence
changes in the absorbance spectrum which can be revealed by the FEWS method is 5 ppm. We came to
the conclusion that polymer coated AgBrCl fibers can be used as sensor elements intended for
identification of organic compounds in water. It has to be emphasized that the applying of polymer
251
layers improves the sensitivity of the experimental procedure. The uncoated fibers are capable to reveal
the minimal concentration of 200 ppm of benzene in water, evanescent wave spectroscopy method is
based on relatively weak changes in the absorbance spectrum. The diffusion growth of characteristic
absorbance, as it was demonstrated on our experiments, allows identification of the organic compound
under investigation with very high accuracy.
t=60
t=20
t=5
t=0
100
95
85
80
75
,
%
Transmittance
90
70
65
692
687
682
677
667
672
662
657
652
Wavenum bers, cm-1
Fig.7. Transmittance changes during benzene penetration through a PS layer of 13μm thickness.
The reversibility of such sensing elements has been studied. Once a given compound was
identified, the sensing element was dried by gaseous nitrogen during 1 min. Thereafter the measurement
was repeated. We performed 10 cycles of identifying of benzene in water, and the full reversibility of the
method was proved. Fig. 8 presents mathematical treatment of spectroscopic data according our
diffusional mathematical model. It can be unambiguously seen that benzene interacts with polystyrene
layer during penetration process. The mathematical theory of diffusion of organic compounds in
polymer layers, which will give us the quantitative method of identifying organic compounds in water
solutions, is in progress now.
252
Hight
0.078
0.076
0.074
0.072
0.07
0.068
0.066
0.064
0.062
0.06
0.058
0.056
0.054
0.052
0.05
0.048
0.046
0.044
0.042
0.04
0.038
0
20
40
60
80
100
120
140
160
180
200
220
240
Tim e,1=17sec
Fig. 8. Diffusion of benzene in polystyrene film treated according relationship (11).
Conclusions
Our work demonstrated that the FEWS procedure is very valuable for the real-time study of
diffusion processes in polymers. This is a fast method with high spectral resolution, which makes it
possible to investigate the initial stages of the penetration of liquids into polymers, with a high accuracy.
The cylindrical symmetry of the sensor element (AgClBr fiber) makes it possible to interpret the spectral
data, ignoring the end effects. The mathematical model that describes the penetration process was
developed. It was proved that the initial stage of diffusion has a Fickian character. The model allows us
to calculate the diffusion coefficient with high accuracy. The FEWS method will allow us to
investigate, in the future, diffusion processes in a wide range of polymers and organic liquids, in situ and
in real time, and to elaborate sensors intended for the identifying of small quantities of organic liquids in
water.
Acknowledgements
The authors wish to thank the Israeli Ministry of Absorption for their generous support of this
work, and Mr. Arie Levite for preparing the fibers used.
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