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thesis entitled

MODELING THE SORPTION OF WATER VAPOR
BY A SEMICRYSTALLINE POLYAMIDE:
APPLICATION OF A NEW INTERPRETIVE SORPTION MODEL

presented by

Kuniyoshi Ohashi

has been accepted towards fulfillment
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MSU to An AIIIrmdive ActiorVEcpel Opportunity Indltmion
cWflIS-D.‘

MODELING THE SORPTION OF WATER VAPOR
BY A SEMICRYSTALLINE POLYAMIDE:
APPLICATION OF A NEW INTERPRETATIVE SORPTION MODEL

By

Kuniyoshi Ohashi

ATHESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

School of Packaging

1991

ABSTRACT

MODELING THE SORPTION OF WATER VAPOR
BY A SEMICRYSTALLINE POLYAMIDE:
APPLICATION OF A NEW INTERPRETATIVE SORPTION MODEL
By
Kuniyoshi Ohashi

Experimental water sorption isotherm on nylon-6 was determined at
5°C, 23°C and 42°C. The sorption values were fitted by the modified
dual-mode sorption model recently proposed by Hernandez et al. (1991).
This model describes very well the data over a range of water activities
from zero to one. The oxygen permeability behavior was studied as a
function of polymer moisture content and it was shown that the oxygen
permeability values decreased at low water activities and then increased
with high water activities. DSC data showed that the percent crystallinity
of nylon-6 increased with water content. The diffusion coefficient
increased at low water activities and then decreased, but the solubility
coefficient decreased at low water activities and then increased. Although
the change in S and D are not conclusive, it appears that they may be
associated with an increase in polymer crystallinity as water content

increases within the polymer.

ACKNOWLEDGEMENTS

I gratefully thank my main advisor Dr. Ruben Hernandez, whose
invaluable supports and criticisms guided me through every stage of this
research. I would also like to thank Dr. Jack Giacin and Dr. E. A.
Grulke for their precious advice, discussions, and help during this
research.

I also thank Mr. Mike Rich, mainly from whom I learned to
operate the DSC and DMA equipment, and Mr. Bob Hurwitz for helping
me to make the apparatus of equilibrium sorption isotherm. I am deeply
indebted to Suntory Limited which sent me to study in the School of
Packaging. Finally, my foremost thanks go to my wife for the love and

encouragement which made the completion of this thesis possible.

iii

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES
INTRODUCTION
LITERATURE REVIEW
Chemistry of Nylon-6
Crystal structure of Nylon-6
Crystallinity
Effect Of Crystallinity On properties
Glass Transition Temperature
Relaxation Phenomena And Physical Properties
The Association of Water With Nylon
The Permeation Mechanism For Gases Through Polymer
Solubility Characteristics
Models For Polar Systems at High Solute Activities
Diffusion Characteristics

Ideal Diffusion And Sorption

iv

Page

WONG-#03090.)

10
14
16
19
21
23

Non-Ideal Sorption And Diffusion

Modified Dual Mode Sorption Model

The Effect Of Crystallinity On Sorption

EXPERIMENTAL METHOD

Nylon-6 Film

Equilibrium Sorption Data

Oxygen Permeability Studies

Differential Scanning Calorimetry Experiments

RESULTS AND DISCUSSION

Equilibrium Sorption Isotherm

Oxygen Permeability

Differential Scanning Calorimetry (DSC) Experiments

CONCLUSION

APPENDIX

APPENDIX A

APPENDD( B

APPENDIX C

APPENDIX D

BIBLIOGRAPHY

Computer Program

For Flory-Huggins Model
Computer Program For
Langmuir-Flory-Huggins Model
Computer Program For
Diffusion And Solubility Coeff.
Differential Scanning Calorimetry
(DSC) Data

24
25
26
28
28
28
29
3 1
34
34
49
63

7O

7O

72

74

77
85

Table

P‘P'PPN?‘

10.
11.
12.
13.

LIST OF TABLES

Equilibrium sorption isotherm experimental data
Langmuir and Flory-Huggins weight fraction contribution
Langmuir and Flory-Huggins weight fraction contribution
Langmuir and Flory-Huggins weight fraction contribution
Value of CH], K and B as a function of temperature
Diffusion, solubility and permeability values

of oxygen at 5°C

Diffusion, solubility and permeability values

of oxygen at room temperature

Diffusion, solubility and permeability values

of oxygen at 42°C

The enthalpy at melting point and percent crystallinity

for nylon-6 as a function of humidity

ANOVA for the DSC data (dry vs. 33% RH samples)
ANOVA for the DSC data (dry vs. 63% RH samples)
Student t test of DSC data (dry vs. 33% RH samples)
Student t test of DSC data (dry vs. 63% RH samples)

vi

Page
35
43

45
53

56

57

58

65

828:3

LIST OF FIGURES

Figure
1. Schematic diagram of the electrobalance test apparatus
2 Oxygen permeability apparatus
3. Water sorption isotherm at 5°C, 23°C and 42°C
4 Experimental sorption values and Flory-Huggins
fitting at 23°C
5. Experimental sorption values and Flory-Huggins
fitting at low activity values
6. Sum of squares versus CHI at 5°C
7. Sum of squares versus CHI at 23°C
8. Sum of squares versus CHI at 42°C
9. Langmuir plot at 5°C
10. Langmuir plot at 23°C
11. Langmuir plot at 42°C
12. Best fit for the Langmuir-flory-Huggins model
Values of parameters are CHI=1.78, K=O.177 and B=53.5
13. Best fit for the Langmuir-flory-Huggins model
Values of parameters are CHI=1.91, K=O.152 and B=64.3
14. Best fit for the Langmuir-flory-Huggins model
Values of parameters are CHI=1.99, K=O.147 and B=74.9
15. Arrhenius plot of K of the Langmuir equation

vii

Page
30
32
36
37

38

41
42

47
48

50

51

52
54

16.
17.
18.
19.

Arrhenius plot of B of the Langmuir equation

Oxygen permeability values at 5°C, 23°C and 42°C
Diffusion coefficient values at 5°C, 23°C and 42°C
Solubility coefficient values at 5°C, 23°C and 42°C

viii

55
59
61
62

INTRODUCTION

It is well known that the presence of water molecules in a polymer
can effect the permeability of oxygen through the polymer. Recently
Hernandez et al. (1991) have studied the permeability of oxygen through
an amorphous polyamide, nylon 61/6T. This work led authors to propose
a new model to describe the sorption of water vapor by the glassy polymer.
The model is based on a Flory—Huggins and Langmuir sorption
contributions given by

V1 =V1L + VIFH (1)
where V1 is the total volume fraction of water within the polymer, and the
superscripts L and FH refer to Langmuir and Flory—Huggins water

sorption contributions, respectively. These contributions are expressed as:

 

K a1
Vfl= a)
1 + B a1
where K is the Langmuir capacity constant and B is the Langmuir affinity
constant and
a1 = exp [In W“ + (1— V1“) + x (1— V1FH)21 (3)

where a1 is water activity, and x is the Flory—Huggins interaction

parameter. Equation (3) can be written in implicit form as:
Wm=mmn m
The proposed model was successfully used to explain the depression of

oxygen permeability through the films as a function of sorbed water.

The newly developed Langmuir—Flory—Huggins model provided a
more complete description of the sorption behavior of the water vapor—
amorphous polyamide binary system. The model developed not only
describes the sorption data, but also provides a correlation between used to
evaluate the clustering function.

While the sorption of water vapor by polyamides has been reported,
there is still the need for a systematic study of the equilibrium
thermodynamics of the sorption process. This study also explains the
mechanism of water sorption and its relationship with the transport and
mechanical properties of purely amorphous and semicrystalline

polyamides.

The objectives of this study are the followings:
1. Description of the sorption process for the binary system, water and
semicrystalline polyamides nylon-6, and its relation to oxygen transport
through the polymer.
2. Application of the newly developed Langmuir—Flory—Huggins model
for interpreting the mechanism of sorption equilibrium of water vapor—

semicrystalline polyamide system.

LITERATURE REVIEW

Chemistry Of Nylon-6

Nylons, or polyamides, are melt-processable thermoplastic
materials whose chain structure features repeating amide groups. They
offer a combination of properties that includes high strength ( especially at
elevated temperature), toughness at low temperatures, stiffness, wear and
abrasion resistance, low coefficient of friction, and good chemical
resistance (Carley 1991).

Nylon-6 is polymerized from e-caprolactam (6-carbons).

CH2 - CH2 - CH2 - CH2 - CH2

| |
NH CO

to form polycaproamide or nylon-6:
H-[NH(CH2)5CO ]n-OH

 

Crystal Structure Of Nylon-6

Crystal structure deals with the organization of long chain molecules
into a microscopic array. If the anay, on the average, is highly ordered
in three dimensions, the material is said to have a high crystallinity. If the
packing of the chains, averaged over the sample, is not very regular, the

crystallinity is low. Nylons are different from many other semicrystalline

polymers such as polyethylene in that the degree of crystallinity of a given
nylon can be controlled over a wide range of crystallinity. High
crystallinity requires both parallel alignment of the chains and uniformity
in the manner in which hydrogen bonds are formed (Clark and Wilson,
1973).

Crystallinity

A numerical measure of “crystallinity” is often used as a measure of
the degree of crystalline order in a semicrystalline polymer. The term
implies the presence of a two-phase system of crystalline and amorphous
regions. This was a reasonable interpretation of the diffraction patterns of
such polymers as polyethylene and polytetrafluoroethylene. Diffraction
from these polymers resembles the superposition of a crystalline
diffraction pattern on an amorphous pattern, which in turn appears to be
an extrapolation of the diffraction pattern of the melt (Clark and Wilson,
1973).

For all the nylons there is no obvious procedure for resolution of a
diffraction pattern into crystalline and amorphous regions and the
calculation of even an empirical degree of crystallinity. There is no
obvious demarcation between crystalline and amorphous areas (Satton,
1967). An instrument such as the Du Pont 310 curve analyzer can be used
to resolve these patterns (Cambell, 1969) but the results can be quite
arbitrary. Thus for polyamides, X-ray diffraction is not commonly used
to derive a measurement of crystallinity (Clark and Wilson, 1973).

The usual methods for assessment of degree of crystallinity in nylons

are by measurement of density, differential scanning calorimetry (DSC) or

by infrared techniques. Measurement of density is perhaps most
satisfactory because it is rapid, precise, and unaffected by sample
calibration - the assumption of amorphous and crystalline densities. It also
requires close control over water content. The plasticizing effect of water
on a dry sample of low crystallinity may increase its crystallinity . It
should also be realized that massive samples are typically not uniform in
crystallinity (Clark and Wilson, 1973).

The differential calorimetry scanning method for determining the
percent crystallinity of a semicrystalline polymer is based upon the
measurement of the heat of fusion, AHf, and the reasonable assumption
that this quantity is proportional to the percent crystallinity.(Wunderlich
and Cromier, 1967). The percent crystallinity may be calculated from:

AHf
Percent Crystallinity = —— (5)
AHf"

where AHf“ is the heat of fusion for a hypothetical 100% crystalline
sample. For nylon-6, AHf" is 195 J/g (Brandrup and Immergut, 1991).
Blatz (1989) measured the amount of crystallinity in 1 mil thick cast film
of dry nylon-6, using DSC method and the percent crystallinity of that was
22%.

Infrared measurement of the intensities of crystalline and amorphous
bands has been used as a measure of crystallinity and, when carefully
done, correlates well with density measurements (Starkweather et al.
1956). For these studies, thin films were used so that the samples could be

considered homogeneous.

Effect Of Crystallinity On Properties

The effect of crystallinity on the properties of nylon-6 is
substantially the same as it is for other semicrystalline polymers. Modulus
and strength and related properties such as hardness and yield point
increase with increasing crystallinity. Measures of toughness such as
impact strength decrease, particularly in the high-crystallinity range.

However, the effect of crystallinity can hardly be discussed
independent of that of water. The properties of polyamides are as
dependent on water content as they are upon crystallinity. The plasticizing
effect of water on a dry sample of low crystallinity may increase its
crystallinity. Not only can water have an effect on crystallinity, but it also
changes physical properties independently. Once a sample has absorbed a
given amount of water at any temperature, this water can be removed and
replaced at this temperature without noticeable effect on crystallinity, but
the effects in properties will be substantial. Water acts as a plasticizer for
nylon-6 and lower the glass transition temperature and the characteristic

temperatures of mechanical relaxation (Clark and Wilson 1973).

Glass Transition Temperature

Most polymers are either completely amorphous or have an
amorphous-like component even if they are crystalline. Such materials are
hard, rigid glasses below a fairly sharply defined temperature known as
the glass transition temperature, Tg. At temperature above the glass
transition temperature, at least at slow to moderate rates of deformation,

the amorphous polymer is soft and flexible and is either an elastomer or a

very viscous liquid. Mechanical properties show profound changes in the
region of the glass transition. Many physical properties change rapidly
with temperature in the glass transition region. The importance of the
glass transition temperature, Tg, in the mass transport of a penchant-
polymer system was described by Meares (1954), and is now very well
recognized.

The glass transition temperature of any amorphous substance,
whether polymeric or not, is defined as the point where the thermal
expansion coefficient undergoes a discontinuity. In polymers, there may
be more than one discontinuity in the thermal expansion coefficient. The
largest discontinuity is usually associated with the loss of the molecular
mobility which permits configurational rearrangements of the chain
backbones; this is "the glass transition" (Ferry, 1970)

The sorption of gases above Tg indicates that the heat of solution
must include along with the heat of interaction between the diffusant and
polymer, the energy for separating the polymer chains which is
endothermic, therefore accounting for the endothermic and slightly
exothermic heat of solution. The exothermic heat of solution below Tg can
be explained by the inclusion of the exothermic heat of adsorption for the
"hole filling" in the heat of solution. The diffusion process above Tg
requires a large zone of chain activation than below Tg which is consistent
with the higher surge of activation reported above Tg (Hopfenberg and
Stannet, 1973).

The glass transition temperature is generally measured by
experiments which correspond to a time scale of seconds or minutes. If the
experiments are done more rapidly so that the time scale is shortened , the

apparent Tg is lowered. Thus, as generally measured, Tg is not a true

constant but shifts with time. (Nielson, 1974).

Blatz (1989) had measured the glass transition temperature of nylon-
6 by using differential scanning calorimetry (DSC). However, no glass
transition temperature could be determined for the nylon-6 because it is
probably hidden in the crystallinity exotherm. According to Blatz, nylon-
6 has a melting point (Tm) of 222°C and glass transition temperature (T g)
of 39°C. This Tg value of nylon-6 is reasonable from the relationship:

Tg / Tm ~ 0.6 (6)

The glass transition temperature of a nylon shifts to lower
temperature as water contents is increased. The Tg of nylon—66 decreases
from 80°C for an almost dry sample to 15°C for a sample saturated with
water (Starkwether, 197 3).

Relaxation Phenomena And Physical Properties

It is well known that increasing time and temperature have similar
effects on viscoelastic properties (T obolsky, 1960). Reliable estimates of
the long-term properties of a polymer at a given temperature can be
obtained from the short-term properties at higher temperatures. With a
hydrophilic material, like nylon, the water content is an additional
complementary variable. Increasing the water content has about the same
effect as an increase in temperature or the time scale of the experiment
(Starkweather, 1973). '

In a time-temperature superposition, a time-dependent mechanical
property, such as creep, stress relaxation, or dynamic mechanical loss is
determined at a series of temperatures. The data for the various

temperature are shifted along a log-time axis until they superimpose

(Starkweather, 1973). Quistwater and Dunell (1958, 1959) carried out a
time-humidity superposition using filaments of nylon-66. The filaments
were conditioned to various humidities, and forced longitudinal vibrations
were applied at frequencies from 2 to 30 cycles per second. At 60°C, the
maximum loss for the glass transition temperature occurred at 2% water,
at 35°C, it was at 3.5% water, and at 9°C, it was at 6 to 8% water. It was
concluded that the activation energy for the glass transition temperature is
60 to 80 kcal/mole.

Another study of time-humidity superpositions was reported by
Onogi et al. (1962). They carried out measurements of stress relaxation
on films of nylon-6 which had been treated at 150 to 155°C. Stress
relaxation experiments were conducted in tension on dry samples at
temperatures from 25 to 77°C. It was found that log-log curves of
apparent modulus, E, versus time could be superimposed by shifting
along the log-time axis for data taken at temperatures higher than 50°C.
The shift factors correspond to an activation energy of 110 kcal/mole.

In another series of experiments, stress relaxation measurements
were made at 25°C over a wide range relative humidities. At that
temperature, relaxation was most rapid at 19 to 33% R.H., and the curves
could be superimposed at those and higher humidities. For both time-
temperature and time-humidity experiments, superposition was
satisfactory only at conditions above the glass transition temperature
(Starkweather, 1973).

10

The Association Of Water With Nylon

The effect of water on the properties of nylons has often been noted.
The effect of crystallinity on the equilibrium water absorption of nylons
was reported (Starkweather et al., 1973). Lowering the amide-group
concentration and the relative humidity diminishes the effect of a change in
crystallinity. There is little effect of temperature over nylom6 and nylon-
66, but a major effect is reported for nylon-11 at 100% RH.

The dependence of equilibrium moisture content on relative
humidity for typical moldings of commercial nylons was reported (Bonner
et al., 1973). Nylon-6 reaches to equilibrium moisture content 10 wt %
under the condition of 100% RH.

The water absorption of modified composition obviously depends on
the nature of the modification. Plasticized resin such as unextracted nylon-
6 will have the same moisture content as extracted polymer but will show
lower weight gain by virtue of loss of monomer. Even unmodified resins
may contain about one percent of extractable material so that precise data
will require a moisture analysis as well as weight gain information. On the
other hand, resins containing glass or other inert additives absorb water in
proportion to the fraction of nylon percent (Bonner et al., 1973).

Water has been shown to be a very effective plasticizer in certain
amorphous and semicystalline linear polymers (Sauer and Lim, 1977).
Similarly crosslinked polymers such as structural epoxy resins undergo a
large depression in glass transition temperature when plasticized with small
quantities (1-3 wt %) of water (Moy and Karasz, 1980). Because of the
polar nature of water, and the hydrophilic nature of the polymers in which

it is soluble, specific interactions in the form of disruptive hydrogen

ll

bonding of water molecules to polar sites on the polymer chains have been
invoked to explain these large effects. Indeed, several spectroscopic
(Banks and Ellis, 1979) and calorimetric (Ellis and Karasz, 1984) studies
of epoxy resins plasticized by water have suggested that specific
interactions may be present. However, a recent NMR study of an epoxy
resin plasticized by water has clearly indicated that water molecules have a
large degree of mobility, which tends to negate any argument that water
molecules become specifically and firmly "bound" to be very good groups
in the polymer (Jelinski et. al., 1983).

The plasticization of semicrystalline polymers, such as that of nylon-
6 by water, has attracted considerable attention (Fuzek, 1980). The
nylon-6/water system also has an additional complexity in that unlike an
amorphous network, where plasticizer is assumed to be distributed
homogeneously throughout the polymer, the water will reside principally
in the noncrystalline regions. It has previously been noted (Beatty and
Karasz, 1975) that in some semicrystalline polymers, the incremental
change in heat capacity at the glass transition temperature (T g) does not
obey a simple quantitative two-phase relationship, and is substantially
reduced, as the degree of crystallinity increases.

The influence of crystallinity and crosslinking on the depression of
the glass transition temperature in nylon-6 by water was investigated by
dynamic mechanical methods (J in, X. et. al., 1984). The study had
attempted to quantitatively account, both theoretically and experimentally,
for the water-induced plasticization behavior of nylon-6 of different
crystallinity and proposed nylon-6 to be described by a three-phase model:
a fully ordered crystalline phase, a bulk amorphous phase which is not

capable of activation at Tg and is therefore unable to contribute to the

12

incremental change in heat capacity at this transition. Although it is clear
that the interpretation of plasticization studies from semicrystalline
polymers is considerably complicated by morphological factors, there
appeared to be good agreement between the theoretical and experimental
depression of the glass transition temperature in nylon-6 by water .
Regarding the analogy between the effect of crystallinity on thermal
properties and the influence of chemical crosslinking on those a network
polymer, it is probable that the influence of morphology and the
concomitant distribution of plasticizer in the amorphous region is of
greater significance in determining the plasticization behavior of this
semicrystalline polymer.

Puffr and Sabenda (1967) suggested a mechanism of water sorption
into nylon-6 at room temperature, which involved two neighboring amide
groups in water-accessible regions forming a sorption center. This
sorption center could accommodate up to 3 water molecules, involving
hydrogen bonding between adjacent amide groups. Additional sorbed
water molecules may be accommodated via a clustering mechanism.

Papir et al. (1972) investigated the relaxation behavior of nylon-6
as a function of orientation, anisotropy and moisture content by using an
inverted free-oscillating torsion pendulum and three new relaxations were
discovered. The characteristics of these new relaxations strongly depend
on the orientation anisotropy and concentration of adsorbed water in the
specimens. The results suggested that motions of the polar groups, in
addition to those of the methylene sequences, are involved and two types
of water, tightly bound and loosely bound, exist in nylon-6.

Hernandez et al. (1991) studied Fourier Transform Infrared (FTIR)
spectroscopy to detect possible charges in hydrogen bonding between N-H

13

and carbonyl group by observing the vibrational modes of the amide
group, using amorphous polyarrride. FTIR data suggest that the water does
not change the hydrogen-bonding strength of the carbonyl group but does
change the hydrogen bonding strength of the N-H group.

14

The Permeation Mechanism For Gases Through Polymers

Steady state rate of permeation of gases through a polymer is usually
described by the permeability coefficient (P). The permeability coefficient
(P) can be determined from direct measurement of the rate of transfer of a
gas through a polymer or from the relationship:

P = D*S (7)
where D and S are separately determined (Crank and Park, 1968). D is
the diffusion coefficient which describes the rate of movement of a
diffusing permeant through the polymer, and S is the solubility coefficient
of the permeant in the polymer.

Mass transport through polymers occurs by a diffusion process
rather than by a flow process such as Knudsen or Poiseuille flow that
occurs through porous materials (Lebovitz, 1966). The diffusion process
is affected by the characteristics of the polymer, diffusant gases, the
temperature and the relative humidity. Therefore, if permeation is to
occur, the permeant gas has to undergo the following processes in
succession: (i) dissolution of the penetrant molecule into the surface of the
polymer film: (ii) diffusion of the penetrant molecule through the polymer
matrix; and (iii) desorption of the permeant molecule from the other
surface of the film (Lebovitz, 1966).

The diffusion process can take place because polymer molecules have
a random kinetic agitation or heat motion. The polymer chain segments
have vibrational, rotational and translational motions that continually
create temporary “holes” in the polymer matrix. The creation of these
holes allows penetrant molecules to move trough the polymer matrix under

the influence of the concentration gradient. The amplitude and motion of

15

the polymer molecules is directly related to the temperature, chemical
composition and morphology of the polymer.

The glass transition temperature (T g) is important factor for the
mass transport of a penetrant-polymer system (Meares, 1954). The glass
transition temperature marks the transition from a “glassy” polymer state
to a “leathery” polymer state. This increases flexibility of the polymer is
causes by the unfreezing of micro brownian motion of polymer chain
segments 20-50 carbon atoms in length (Boyer, 1977). This increase in
polymer chain segmental mobility above the glass transition temperature
corresponds with an increase in permeability and diffusion.

Small diffusant molecules like the permanent gases: oxygen,
nitrogen and carbon dioxide, have almost no effect on the polymer
molecules while sorbed into the polymer matrix. Their kinetic agitations
are rapid compared to those of the polymer chains. The rate of diffusion
of these molecules is therefore controlled by their agitation which is related
to the amount of energy present in the system, as measured by the
temperature. If a concentration gradient is present across a film the
frequency of the jumps of the diffusant past the polymer chains gives a net
flux of the diffusant molecules through the film (Meares, 1965).

As stated above, permeability is related to diffusion and solubility.
The complex permeability process is best understood by considering the
effects of various factors on solubility and diffusion. This portion of the
text on the characteristics of permeation and diffusion of gases through
polymer membranes follows from the excellent review in the chapter on
Permeability and Chemical Resistance by C. E. Rogers from the book
"Engineering Design for Plastics" edited by E. Baer (1964).

l6

Solubility Characteristics

The solubility of a molecule component in a solid can be described as
the distribution of the component between two or more phases. The uptake
of penetrant by a solid is called sorption and can be considered to have
adsorption or absorption as the basic mechanism. There is no way of
distinguishing between either physical adsorption and chemisorption or
considered a composite of these various modes.

The dependence of the solubility coefficient on temperature
generally follows an Arrhenius type relationship:

5 = 50 6X1) (- AHs/RT) (8)
where AI-Is is the apparent heat of solution (Rogers, 1964). The process of
sorption by a polymer may be considered to involve two stages:
condensation of vapor onto the polymer surface followed by solution of the

condensed vapor. The heat of solution can be expressed as the sum of the

molar heat of condensation (Mg) and the partial molar heat of mixing
(AH1). The heat of mixing is always positive and the heat of condensation
can be positive or negative depending on whether the molecule is a gas or
vapor. For permeant gases, M3 is slightly positive so that S increases
slightly with temperature. However for the more condensable vapors (like
organic compounds), AHS is negative due to the relatively large heat of
condensation, thus the solubility decreases with increasing temperature.

A sorption isotherm is a curve that relates the equilibrium
concentration of the penentrant in the polymer (c), to the concentration of
penetrant (p) surrounding the solid, at any constant temperature. There

are classically four basic types of isotherms observed in polymer systems.

17

These isotherms depend on the degree of penetrant-polymer interaction.
Type I follows Henry’s law where the concentration is linearly dependent
upon pressure. Type II is characteristic in systems where only a
unimolecular layer of sorbed substance forms on the substrate showing a
concave curve towards the pressure axis. Type III sorption is
characteristic of multilayer adsorption where the attractive forces between
penetrant and solid are greater than those between the molecules of the
penetrant themselves, giving a sigmoid shaped curve. When the forces
between the penetrant and the substrate are relatively small, the sorption
process occurs essentially randomly as the results of absorption or
adsorption due to van der Waals type forces. The resulting isotherm
corresponds type IV. Type IV sorption forms a convex curve towards the
pressure axis.

Another interpretation of the type III isotherm considers that it is the
result of two simultaneously sorption process. One process, dominant at
low penetrant concentrations, corresponds to a type II isotherm due to
preferential sorption of penetrant onto a limited number of active sites in
the solid.

The other concurrent process is a type IV sorption, so the
superposition of the two curves gives the observed type III isotherm.

The solubility coefficient at a given permeant partial pressure can be
calculated from these sorption isotherms by dividing the sorbed
concentration (c) by the corresponding pressure (p) (Rogers, 1964). The
Langmuir (1918) equation for sorption limited to monolayer formation

leads to a type II isotherm. The Langmuir equation can be expressed as:

18

c C'Hp
s=——=—— (9)
p 1+bp

where b is the hole affinity constant, OH is the hole saturation constant.

The Langmuir equation works best when the energies of adsorption
are high, as in chemisorption, or when the adsorbed molecules are quite
large. In both cases, the penetrant molecules are rendered relatively
immobile by the adsorption process. However, at high concentrations or
low temperatures, discrepancies often appear in the direction of more
sorption than predicted by the Langmuir equation. This is especially
apparent when the energies of adsorption are in the range characteristic of
physical adsorption. These discrepancies may be due to the formation of
multilayers rather than momolayers.

The BET equation, which accounts for such multilayer adsorption,
was originally derived by Bnmauer et al. (1938) on the assumption that
the Langmuir equation could be applied to each layer. In the usual form of
the BET equation is different from that for the subsequent layers. The
adsorbed penetrant from the second layer on is assumed to be equal to the
heat of vaporization for the pure liquid penetrant.

The expression for the solubility coefficient in terms of readily
measured concentrations and vapor pressure is:

5 cm
8 = c/p = (10)
(p0 - p) {1 + (B—l)(p/p°)}

 

where: p0 is the permeant saturated vapor pressure

cm is the sorbed concentration of vapor

corresponding to a complete monolayer sorption

19

B = exp(Q1-QL / RT) where Q1 is the heat of
adsorption for the first layer, QL is the heat of
vaporization for the pure liquid penetrant.

The BET equation in this form can predict isotherms of types II,

III, and IV depending primarily on the relative magnitudes of Q1 and QL.
When Q1>> QL, corresponding to strong attractive forces between the
first layer and the substrate, the BET equation approximates the Langmuir
equation and type II isotherm. On the other hand, when Q1 < QL a type
IV isotherm results. Intermediates cases, Q1 > QL, leads to sigmoid
isotherm curves: it is for these type III isotherms that the BET equation
has been most used.

The magnitude of the solubility for a penetrant in a polymer
increases with : (i) the chemical similarity of the penetrant-polymer; (iii)
the larger the molar volume of the condensed penetrant; and (iv) the lower
the percent crosslinking and the crystallinity of the polymer. The sorption
of penetrant by the polymer has a profound effect on the solid properties.
In addition to the plasticizing action of the sorbed penetrant there may be
changes in the polymer structure due to swelling and distortion incurred
during sorption as well as actual chemical attack on the polymer (Rogers,
1964).

Models For Polar Systems At High Solute Activities

Solute sorbed at temperatures below their boiling point (such as
water at room temperature) may not obey simple and ideal phase equilibria
relations such as Henry’s law. For these cases, eqn. (10) does not describe

the data well over the activity range, 0 < a1 < 1, and usually

20

underpredicts the actual solubility values at high activities. This lack of
agreement has been explained as a “positive deviation” of the Henry’s law
caused by the swelling of the polymer network as the penetrant is sorbed.It
is postulated that the network swelling exposes more binding sites and
increases the sorption level of the penetrant synergistically (V ieth et al.,
1976).

Berens (1975) reported that the Flory-Huggins equation adequately
described data for vinyl chloride/poly(vinyl chloride). for high molecular
weight polymers, the Flory-Huggins expression is :

1na1=an1+V2+xV22 (11)
where X is the interaction parameter. Although eqn. (11) does not provide
the most accurate description of the thermodynamic performance of
polymer solutions, it does contain most of the essential features which
distinguish such solutions (Prausnitz et al., 1986). Eqn. (11) usually fits
equilibria data over a wider range than Henry’s law. For example,
sorption data of toluene in polyvinyl chloride is well fit by the Flory-
Huggins equation especially for 0.5 < a1 <1.0 (Berens, 1985). Sfirakis and
Rogers (1980) studies the sorption of vapors by nylon-6 but did not present
a mathematical model to describe the sorption process. However, this data
set can be fit to eqn. (11) with a x value of 1.88. The Henry’s law term
seems adequate for fitting the water/kapton data (Yang et al., 1985).
Either hole filling or chemisorption might be occurring in the polyimide
for the Langmuir contribution, as kapton is known to hydrogen bond with
water. Water is known to hydrogen bond with a variety of epoxies and the

dual mode sorption model describes some data well (Barrie etal., 1984).

21

Diffusion Characteristics

The temperature dependence of diffusion coefficients over small
temperature ranges at a constant vapor concentration can be represented by
an Arrhenious type relationship:

D=Do exp (-ED/RT) (12)
where ED is the apparent activation energy for the diffusion process. The
activation energy for diffusion is associated with the energy required for
“hole” formation in the polymer matrix plus the energy required to move
the molecule through the polymer structure. The pre-exponential factor
,(Do), can be thought of as being related to the frequency and magnitude
of the holes or “looseness” within the polymer microenvironment in the
absence of penetrant. The activation energy of the polymer increases at
temperatures above the polymer’s glass transition temperature.

The magnitude of the concentration dependence of diffusion in any
given polymer is dependent on the temperature, molecular size, chemical
similarity between penetrant and polymer and the penetrant concentration
in the polymer. The solubility coefficient is often essentially constant at
low vapor activities for the more volatile vapors while the diffusion
coefficient exhibits significant concentration dependence. The
concentration dependence of the diffusion coefficient can be represented by
the equation :

D = Do exr (v0) (13)
Where y is a characteristic constant, Do is a pre-exponential factor
representing the diffusion coefficient at zero concentration or the limiting
diffusion coefficient and c is the concentration of permeant in the polymer.

The diffusion coefficient may deviate significantly from a linear or

22

exponential dependence on c when the measurements are made over wide
temperature ranges and vapor concentration, or activity, especially with
more easily condensable vapors. Such behavior has been observed for allyl
chloride in polyvinylacetate; for n—acetates in polymethylacrylate; and for
hydrocarbons in polyethylene (Meares, 1958ab; Fujita, 1961; Rogers,
1960). The concentration dependence in theses systems can be represented
by a simple exponential frmction of vapor activity:

D = Do exp (oral) (14)
where a is a constant and a1 is the vapor activity on the high concentration
side of the film .

The magnitude of Do depends primarily on the chemical nature,
size, and shape of the penetrant and the morphology of the polymer. The
constants y and a can be interpreted as characterizing the effectiveness of
various penetrants to plasticize a polymer and to facilitate its segmental
mobility.

The addition of a plasticizer to a polymer usually, but not always,
increases the rate of gas and vapor diffusion and permeation (Boyer,
1949). An increase in transport rate is probably due to an increase in
polymer segmental mobility resulting from a decrease in the cohesive
forces between chains by replacement of polymer-polymer contacts by
polymer-plasticizer contacts. this is reflected by the observed decrease in
activation energy for diffusion with increasing plasticizer content.

Additional complications for an accurate prediction of plasticizing
action arise from the lowering of the glass transition temperature,
enhanced rates of stress relaxation, and gross morphological changes
induced by the addition of a plasticizer itself also may be an important
factor affecting the overall solubility and transport rate, especially at high

23

plasticizer contents (Boyer, 1949).

Ideal Diffusion And Sorption

Diffusion is the net transport of matter in a system by means of
random molecular motion. This results in the removal of chemical
potential differences in a system and eventually produces a uniform state of
equilibrium if the boundary conditions are not held constant for a
permeation experiment.

Frequently the rate of diffusion is proportional to the concentration
gradient although the proportionality constant known as the diffusion
coefficient may be a function of the diffusant concentration in the polymer:

0C

= -D(c) (15)

 

6x

where D is the concentration independent diffusion coefficient of the fixed
gas in the polymer, and C is the concentration of fixed gas in the polymer.
This is Fick’s first law of diffusion and is found experimentally to
represent diffusion in many systems. For gases above their critical
temperature (fixed gases) the diffusion coefficient is not a function of
concentration. Fick’s second law may be derived from Fick’s first law:

dc 6 [ 6 c J

—- = — D(C)—— (16)

dt 0x 0x
Diffusion which follows Fick’s first and second laws is termed Fickian
diffusion (Hopfenberg and Stannet, 1973)

24

Non-ideal Sorption And Diffusion

The magnitude of the negative enthalpies of sorption reported by
Meares (1958) for neon, nitrogen, oxygen and argon in glassy
polyvinylacetate and by Barrie et al. (1957) for organic vapors in ethyl
cellulose were inconsistent with the sorption theories of rubbery systems
and led Barrie to suggest a two-mode, concurrent sorption mechanism for
glassy polymers, namely ordinary dissolusion and “hole” filling
(Hopfenberg and Stannet, 197 3).

Michaels et al. (1963) also reported that the sorption of helium,
nitrogen, oxygen, argon and methane in glassy, amorphous and glassy,
crystalline polyethylene telephthalate for pressure up to 10 atmosphere
obeyed Henry’s law, but carbon dioxide at 25°C and 40°C and ethane at
25°C in the same pressure range derived from Henry’s law. The carbon
dioxide isotherms promoted Michaels et a1. (1963) to propose a dual-mode
sorption model of ordinary dissolution and adsorption in microvoids
(holes) for gas sorption in glassy amorphous polymers. It was assumed
that the total amount of solute sorbed in the polymer, C, consisted of two
thermodynamically distinct molecular populations:

C}; p

 

C=CD+CH=KDP + (17)

1 + b p
where CD and CH are the solubilities due to absorption and adsorption.
The concentration CD was represented by Henry’s law:
CD = Kb 1) (18)
where K1) is the Henry’s law constant.

The concentration CH was represented by a Langmuir isotherm:

25

CH p
CH = (19)
1+ bp

 

Modified Dual Mode Sorption Model

Hernandez et.al.(1991) have proposed to modify eqn. (17) by using
the Flory-Huggins equation to describe non-specific solution rather than
Henry’s law. The modified dual-mode sorption model represents the
sorption of water vapor by an amorphous polyamide at 23° C. This
modification allowed the model to fit over the activity range, 0 < a1 < 1.
For convenience, the modified dual mode sorption model was expressed in
terms of volume fractions and solute activity:

V1 = V1FH + V1L (1)
where V1FH refers to the Flory-Huggins contribution to the solute volume
fraction and V1L is Langmuir volume fraction contribution. Since eqn.
(11) is nonlinear, the value for VlFH was determined by numerical
methods, such as the Newton-Raphson technique and eqn. (1) became:

Ka1
V1=FH(al,x) + --—— (20)
1+Ba1

where K=C’H fbps , B=bps, p; is the saturation vapor pressure of the

solute and f is a conversion factor.

The Effect Of Crystallinity On Sorption

The effects of the degree of crystallinity on the permeability of
semicrystalline polymers was first discussed by Michaels and Parker, and

26

Michaels and Bixler have extended this study to the effect of crystallinity
on the sorption of several gases in rubbery, crystalline polyethylene, and
found that solubility constants for a particular gas were proportional the
amorphous volume fraction, that is:

k = ak* (21)
where k* is the solubility constant of a hypothetical completely amorphous
polyethylene and k is the solubility constant of partially crystalline
polyethylene. This relationship suggested that the gases sorb entirely in
quantity sorbed was directly proportional to a (Hopfenberg and Stannet,
1973).

The relationship in equation (21) was used by Michaels, Vieth and
Barrie (1963) in studying the effect of crystallinity on the sorption of
several gases in glassy crystalline polyethyleneterephthalate. Unlike the
rubbery crystalline polyethylene, it was found that the gas solubility, with
the exception of helium, is not directly proportional to amorphous volume
fraction. The decrease in solubility accompanying crystallization was
smaller than the corresponding reduction in amorphous volume fraction.

This result can be satisfactorily accounted for in terms of the “hole”
model of glassy amorphous phase. Upon annealing the polymer it is more
probable that the denser regions crystalize first, tending to remove more
of amorphous phase used for gas dissolution, therefore leaving the residual
amorphous phase with a higher concentration of “holes” than in the
previously amorphous polymer. This means that crystallization in glassy
polymers tends to increase the relative contribution of “hole” filling. Also
such an increase in concentration of “holes” with increasing crystallization
accounts for the decreases reductions in solubility (Hopfenberg and Stannet,
1973).

27

Michaels et al.(1963) generalized their original dual sorption model
to include the effects of crystallinity on the solubility of the dissolved

species:

C'pr
C: + ak*])p (22)
1+ b p

 

28

EXPERINIENTAL METHODS

Nylon-6 film

Allied-Signal Inc. P. O. Box 2332R Morristown, NJ 07963
CAPRAN 77C 1.25 MIL CAPRAN 77C is a transparent, durable, nylon—
6—type thermoplastic film made from a polymer of e-caprolactam. An all-
purpose packaging-grade film, approved in accordance with FDA

regulations.

Equilibrium Sorption Data

Equilibrium sorption data was taken on polymer films using a Cahn
Electrobalance Model 2000 (Cahn Instruments Inc. Cerritos, California).
A stream of nitrogen adjusted to specific values of water activity provided
the source of water vapor in equilibrium with the polymer films. The film
samples were dried under vacuum at 70° C before each run. The
electrobalance was maintained a constant temperature environment (5° C,
23° C and 42° C) in the Temperature Humidity Test Chamber (Thermotron
Model SM-328-SH, Thermotron Industries, Holland Michigan). The
temperature tolerance is :l:0.3°c. A sample film approximate 100 mg was
normally used for equilibrium sorption data, and the sensibility of the

apparatus was around 1 microgram. A schematic diagram of the test

29

system is shown in Figure 1. The test system allows for the continuous
collection of sorption isotherm data of water vapor by a nylon-6 film from
the initial time zero, when the film is first exposed to the vapor, to the
steady state condition, when sorption equilibrium is reached. As shown,
the nylon-6 film sample to be tested was suspended directly from one of
arm of the electrobalance and a constant concentration of penetrant vapor
was continually flowed through the sample tube (hang-down tube), such
that the sample film was totally surrounded by the vapor. A constant
concentration of permeant vapor was produced by bubbling nitrogen gas
through distilled water. This was carried out by assembling a vapor
generator consisting of a glass gas washing bottle containing the water, and
a fritted dispenser tube. The concentration obtained in this way was
adjusted to a required value by mixing it with a pure nitrogen gas stream.
Water activities were measured using Hygrometer Sensors (Hygrodynamic
Co., Silver Spring, Maryland). The vapor generator system and humidity
senser were mounted in the constant temperature chamber (Thermotron).
As shown, flowmeters were used to provided a continuous indication that a
constant rate of flow was maintained. Gas flow was regulated using needle
valves. The gain in weight of the sample due to penetrant sorption was

monitored continually until the gain was zero at equilibrium. .

Oxygen Permeability Studies

These studies were carried out on an Ox-Tran 100 Permeability
Tester (Modern Controls, Inc., Elk River, Minnesota). This apparatus
was modified to allow the two streams, oxygen and carrier gas, to be

adjusted to specific water vapor activities. An schematic of the modified

30

 

 

 

 

 

 

 

 

 

 

 

 

 

 

W
m Si

Ch A

l— ———————— 4»-
R’ t : Cu
I Tv I
o ' L] l
' B is:
I J HO

—————_—-——-

 

 

 

 

 

B - Water Bubbler RH - Humidity Sencer

Ch - Temperature Constant Ft - Rotameter
Chamber Fte - Regulator

Cu - Control Unit St - Sample film

He - Hood Si - Electric signal

N - Needle Valve T - Nitrogen Tank

Pr - Printer Tv - Three way Valve

Wm - Cahn Electrobalance

Figure 1. Schematic diagram of the electrobalance test apparatus

31

apparatus is shown in Figure 2 (Hernandez et al., 1991, IAPRI
preceedings). As shown, each stream is formed by mixing a wet and dry
gas component to obtain the required activity value. Water activities were
measured using Hygrometer Sensors (Hygrodynamic Co., Silver Spring,
maryland). Sample films were dried under vacuum at 70° C before each
run. The equilibration process between the Nylon-6 film and the gas
phase, at selected values of water activities, was carried out with the
sample film mounted into the permeation cell. The required time to reach
equilibrium was previously determined by gravimetric method, using the
Cahn electrobalance. Although time consuming, this procedure avoided
sample handling and assured correct conditions of test. The experiments
were conducted at three different temperature (5° C, room temperature
and 42° C). When the tests of 5° C and 42° C, whole system were run in

the temperature controlled chamber.

Differential Scanning Calorimetry Experiments

The Differential Scanning Calorimetry (DSC) experiments were
performed on a DuPont 9000 Thermal Analyzer (E. I. DuPont De
Nemours & Co., Inc. Wilmigton, Delaware). Film samples were placed
in small aluminum pans and vacuum dried at 70° C. Samples were then
equilibrated over salt solutions to give selected water vapor activities at
room temperature. Polyethylene closed containers with salt solution
provided selected values of water vapor activities and samples were placed
in the containers until they reached equilibrium. After reaching
equilibrium, the aluminum pans containing film samples were placed in

glass vials to maintain the conditions and then were carried to the DSC

32

   

SPECIMEN PACKAGE LOOP

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2. Oxygen permeability apparatus

33

equipment. The differential scanning calorimeter was programmed for the
heating rate of 10° C/min and the heating temperature range of room
temperature to 250 ° C with approximately 10 ~ 15 mg of sample film in

each run.

34

RESULTS AND DISCUSSION

Equilibrium Sorption Isotherm

Sorption equilibrium values of water weight fraction in the nylon-6
film at 5°C, 23°C and 42°C are presented in Table 1. Isotherm data was
obtained over a wide activity range (especially low water activity) in order
to provide a good test of the modified dual model. Figure 3 shows the
sorption isotherm for water in nylon-6 at 5°C, 23°C and 42°C. As
expected, the solubility of water in the polymer decreases when
temperature increases. The apparent sigmoid shape may be described by
the Langmuir-Flory-Huggins model, given by eqn. (20).

Figure 4 shows the experimental sorption values and Flory-Huggins
fitting at 23°C. The solid curve through the data is the Flory-Huggins
equation (eqn. 11) with X value of 1.846. The value of interaction
parameter was calculated by using computer program prepared by Dr. R.
J. Hernandez based on a Box-Kanamazu modification of Gauss method of
minimization of sum of squares for nonlinear models (Hernandez et al.
1991). This model represents the data at high water activities well, but
underestimates the data at low water activities. Figure 5 shows the
experimental sorption values and Flory-Huggins fitting at low water
activity. A linear regression method was used to determine the parameter
values estimated for eqn. (20). A computer program was used to
determine the constants of eqn. (20), x , B and K. Starting with the value

35

 

 

Table 1. Equilibrium sorption isotherm experimental data
Temperature Temperature Temperature
23°C 42°C
Water Weight Water Weight Water Weight
Activity Fraction Activity Fraction Activity Fraction
0.000 0.0000 0.000 0.0000 0.000 0.0000
0.069 0.0068 0.067 0.0055 0.040 0.0033
0.130 0.0120 0.096 0.0074 0.09 0.0069
0.182 0.0147 0.148 0.0108 0.120 0.0087
0.260 0.0202 0.182 0.0126 0.148 0.0098
0.370 0.0289 0.238 0.0162 0.23 0.0151
0.440 0.0350 0.306 0.0206 0.370 0.0235
0.650 0.0533 0.380 0.0257 0.50 0.0302
0.950 0.0893 0.470 0.0312 0.540 0.0332
0.520 0.0350 0.595 0.0369
0.650 0.0454 0.735 0.0460
0.790 0.0587 0.87 0.0616

0.840

0.0652

 

Weight Fraction

36

 

 

 

0.10
0.08 d a 5°C
0 23°C
006- . 42°C .
B
O
0.04 4
9 A
El 0
A
O
1 la
0
A
0.02- a t
'3 2
El
. A;
at
A

000 1% j , , r

0.0 0.2 0.4 0.6 0.8

Water Activity

Figure 3.

 

1.0

Water sorption isotherm at 5°C, 23°C and 42°C

Weight Fraction

37

 

0.08
‘ a Experimental(23°c)
B
0.06 - __ Flory-Huggtns(23°C)
CHI = 1.846

 

 

 

 

0.00 . r . . . , . . .
0.0 0.2 0.4 0.6 0.8 1.0

Water Activity

Figure 4. Experimental sorption values and Flory-Huggins

fitting at 23°C

Weight Fraction

38

 

'3 Experimental(23°C)

—-- FIory-Huggins(23°C)

0.02 -‘

CHI = 1.846

 

 

 

 

0.00 . . .
0.0 0.2 0.4

Water Activity

Figure 5. Experimental sorption values and Flory-Huggins
fitting at low activity values

39

of X calculated by the Box-Kanamazu method, the program calculated K
and B by a linear regression analysis of the following equation converted
from the Langmuir equation (9):

1 1 B 1

V1L K K a1
where V1L is expressed as following equation:

V1L = V“? - VF!“I (23)
where VeXP is the experimental data and VFI'I is the Flory-Huggins value.
For a series of X values, the sum of squares of the difference between the
experimental and calculated value were calculated. The values of x, B
and K were then associated with the minimum value of sum of squares.
Sum of squares were plotted as a frmction of x to find the best estimates of
the parameters. Figure 6, 7 and 8 show the sum of squares versus x at
5°C, 23°C and 42°C respectively. The x values of the minimum of each
convex curve should be the best estimate of the parameters for each
temperature.

Table 2, 3 and 4 show the contribution of Langmuir and Flory-
Huggins factors over the activity range at 5°C, 23°C and 42°C
respectively, using the parameters shown in figure 6, 7 and 8. At low
activities, the Langmuir contribution is relatively high compared to the
Flory-Huggins contribution. Figure 9, 10 and 11 present Langmuir
curves calculated by the computer program as a function of water activity
at 5°C, 23°C and 42°C respectively. The difference between experimental
and Flory-Huggins values (eqn. 23) were plotted through the Langmuir
curve. These plots are relatively close to the Langmuir curve. The

Langmuir contribution is nearly constant at activities greater than 0.1. The

Sum of Squares

40

 

 

 

 

 

2009.6
CHI = 1.782

1.80e-6 ..

K = 0.1774

8 = 53.47
1.60e-6 -
1.408-6 -
1.20e-6 -
1.00e-6 w . a .

1.77 1 78 1.79
CHI
Figure 6. Sum of squares versus CHI at 5°C

1.80

41

 

 

 

 

 

1.208-6
r0
0
5
CB
3
1.10e-6 ‘
(D
I.-
O
E
3
(I)
1.00e-6 . j . . .
1.90 1.91 1.92 1.93

CHI

Figure 7. Sum of squares versus CHI at 23°C

Sum of Squares

42

 

7.906-6

7.809-6 -

7.706-6 '-

7.606-6 '

7.50e-6 -

7.408-6 -

 

 

7.308-6

 

1.98

Figure 8.

CHI

Sum of squares versus CHI at 42°C

 

2.01

43

 

 

Table 2. Langmuir and Flory-Huggins weight fraction contributions
Temperature = 5°C
CHI = 1.782
K = 0.1774
B = 53.47
Water
Activity Langmuir F-H Total
0.0 0.0 0.0 0.0
0.069 2.610E-03 4.358E-03 6.967E-03
0.130 2.900E-03 8.361E-03 ' 1.126E-02
0.182 3.001 E-03 1.1 89E-02 1.490E-02
0.260 3.094E-03 1.742E-02 2.051 E-02
0.370 3.157E-03 2.573E-02 2.889E-02
0.440 3.182E-03 3.138E-02 3.456E-02
0.650 3.224E-03 5.043E-02 5.366E-02
0.950 3.253E-03 8.592E-02 8.918E-02

 

44

 

 

Table 3. Langmuir and Flory-Huggins weight fraction contributions

Temperature = 23°C

CHI = 1.913

K = 0.1518

8 = 64.26
Water
Activity Langmuir F-H Total
0.0 0.0 0.0 0.0
0.067 1 .916E-03 3.704E-03 5.621 E-03
0.096 2.032E-03 5.350E-03 7.382E-03
0.148 2.136E-03 8.368E-03 1.051 E-02
0.182 2.175E-03 1.039E-02 1.257E-02
0.238 2.217E-03 1.381 E-02 1.603E-02
0.306 2.225E-03 1 .812E-02 2.037E-02
0.380 2.269E-03 2.304E-02 2.531 E-02
0.470 2.286E-03 2.936E-02 3.165E-02
0.520 2.293E-03 3.306E-02 3.535E-02
0.550 2.296E-03 3.533E-02 3.764E-02
0.650 2.306E-03 4.337E-02 4.567E-02
0.790 2.316E-02 5.584E-02 5.816E-02

 

45

 

 

Table 4. Langmuir and Flory-Huggins weight fraction contributions

Temperature = 42°C

CHI = 1.993

K = 0.1467

8 = 74.90
Water
Activity Langmuir F-H Total
0.0 0.0 0.0 0.0
0.092 1 .711E-03 2.026E-03 3.494E-03
0.120 1.763E-03 6.205E-03 7.968E-03
0.148 1.797E-03 7.710E-03 9.507E-03
0.236 1.854E-03 1.260E-02 1.445E-02
0.370 1.891 E-03 2.053E-02 2.242E-02
0.505 1.908E-03 2.924E-02 3.115E-02
0.540 1 .912E-03 3.163E-02 3.355E-02
0.595 1 .916E-03 3.552E-02 3.744E-02
0.870 1.929E-03 5.780E-02 5.973E-02

 

Fraction

Weight

46

 

0.004

0.003 "

0.002 -

0.001 '1

 

0.000

0.

 

 

 

-- Langmuir

a Experim. - FH

 

 

 

 

 

 

T I I l I 1

0 0.2 0.4 0.6

Water Activity

Figure 9. Langmuir plot at 5°C

1.0

Fraction

Weight

47

 

 

 

 

 

 

 

 

 

 

 

0.003
D
D
D I:
D CI
0.002 " a
D
D
D
O
0.001 J
-— Langmuir
'3 Experim. - FH
0.000 a r ' l l
0.0 0.2 0.4 0.6 0.8

Water Activity

Figure 10. Langmuir plot at 23°C

Fraction

Weight

 

 

 

 

 

 

 

 

 

 

48
0.003
D
D
D
El
13
0.002 ~ D
D
0.001 - a
D
-— Langmuir
1 :1
D Experim. - FH
0.000 ' I v I I I e ' v
0.0 0.2 0.4 0.6 0.8 1.0

Water Activity

Figure 11. Langmuir plot at 42°C

49

points of water activity value 0.84 at 23°C and 0.735 at 42°C were
eliminated when the linear regression method was calculated, because these
points were too far from Langmuir curve. Experimental errors such as
the instrumental error of humidity sensers must have happened at these
points.

Figure 12, 13 and 14 show eqn. (9) as the solid curve of best fit for
Langmuir-Flory-Huggins model and the complete data set at 5°C, 23°C
and 42°C respectively. The Langmuir-Flory-Huggins model presented by
Hernandez et al. (1991) describes accurately the sorption of water by
nylon-6 as well as amorphous polyamide, over a broad range of water
activity. The initial Langmuir contribution suggests the presence of a
limited number of sorption sites which are immediately available to water
molecule and following broad extent may be indicative of clustering or
multi-layer formation (Kawasaki et al., 1962) at the higher water activity.

Table 5 shows the values for X, K and B at 5°C, 23°C and 42°C. K
value decreases as temperature increases, but on the other hand, x and B
value increase when temperature increases. The Arrhenius plots of K and
B are presented in Figure 15 and 16 respectively. There are good linear
relationship between 5°C and 42°C in both Arrhenius plots. The following
expression correlate the constants K and B as a function of temperature:

K = 3.54E-02 * exp(442/T) (25)
B = 957 * exp(-801fT) (26)

Oxygen Permeability

Tables 6, 7 and 8 present the values of permeability, diffusion

coefficient and solubility of oxygen in nylon-6 at 5°C, room temperature

50

 

0.10

__ . +
0.08 _ Lang PH

0 Experimental(5°C)

0.06 ‘

0.04 -1

Weight Fraction

0.02 '-

 

 

 

 

0.00 . 1 . r r 1 . r
0.0 0.2 0.4 0.6 0.8

Water Activity

Figure 12. Best fit for the Langmuir-FIory-Huggins model
Values of parameters are CHI=1.78, K=O.177 and B=53.5

1.0

Weight Fraction

51

 

0.08

— Lang. + FH e

0.06 '-

0 Experimental(23°C)

0.04 -'

0.02 -

 

 

 

 

0.00 v r r r V I m I
0.0 0.2 0.4 0.6 0.8

Water Activity

Figure 13. Best fit for the Langmuir-Flory-Huggins model
Values of parameters are CHI=1.91, K=O.152 and B=64.3

1.0

Weight Fraction

52

 

0.08

— Lang. + FH

0.06 -'

0 Experimental(42°C)

0.04 -

0.02 ‘

 

 

 

 

0.00 a 1 . 1 . 1 . f
0.0 0.2 0.4 0.6 0.8

Water Activity

Figure 14. Best fit for the Langmuir-FIory-Huggins model

Values of parameters are CHI=1.99, K=O.147 and B=74.9

1.0

Table 5. Value of CHI,

53

K and B as a function of temperature

 

 

Temperature
Parameter 5°C 23°C 42°C
CHI 1.782 1.913 1.993
K 0.1774 0.1518 0.1467
B 53.47 64.26 74.90

 

54

 

 

 

 

1 '—

. I
0.0031 0.0032

fi'

V f

l ' I 1
0.0033 0.0034 0.0035 0.0036

1/T (1IK)

Figure 15. Arrhenius plot of K of the Langmuir equation

55

 

 

 

 

0.0036

100
10 I I ' I r I I
0.0031 0.0032 0.0033 0.0034 0.0035
1 IT (1IK)
Figure 16. Arrhenius plot of B of the Langmuir equation

Table 6. Diffusion,

solubility and permeability values of oxygen

 

 

at 5°C
Water D x E09 8 x10 P
Activity (3) (b) (C)
0 0.809 0.387 11.57
0.078 1.23 0.216 9.79
0.148 0.761 0.216 6.09
0.250 0.847 0.146 4.57
0.35 0.768 0.133 3.77
0.520 0.910 0.105 3.54
0.680 0.957 0.120 4.13
0.790 0.635 0.211 4.96
0.890 0.286 0.681 7.21

 

(a) - in (cmzlsec)

(b) - in (cc Ozlcc polymer)

(0) - in (cc 02 mil/m2 day atm)

 

57

Table 7. Diffusion, solubility and permeability values of oxygen

at room temperature

 

 

Water D x E09 S x10 P
Activity (a) (b) (c)
0 1.58 0.443 25.8
0.140 2.54 0.157 14.7
0.208 1.84 0.194 13.2
0.330 2.15 0.168 13.4
0.57 0.779 0.603 17.4
0.635 0.512 1.08 20.5
0.87 0.198 4.277 31.3
0.900 0.182 5.690 38.3

 

(a) - in (cmzlsec)

(b) - in (cc 02/00 polymer)
(c) - in (cc 02 mil/m2 day atm)

 

58

Table 8. Diffusion, solubility and permeability values of oxygen

 

 

at 42°C
Water D x E09 S x10 P
Activity (a) (b) (c)
0 3.01 0.572 63.75
0.045 3.52 0.424 55.15
0.092 4.02 0.342 50.85
0.138 4.23 0.340 53.19
0.202 1.63 0.896 53.98
0.280 1.11 1.280 52.41
0.380 1.10 1.430 57.89
0.500 0.365 4.580 61.80
0.590 0.334 5.640 69.62
0.79 0.3101 9.420 104.48
0.890 0.293 11.60 125.20

 

(a) - in (cmzlsec)

(b) - in (cc 021cc polymer)

(c) - in (cc 02 mil/m2 day atm)

 

Coefficient

(cc mil l m2 day atm)

Permeability

59

 

150

100 ‘

 

 

 

 

 

 

Water Activity

Figure 17. Oxygen permeability values at 5°C, 23°C and 42°C

60

(23°C) and 42°C, respectively. Figures 17 shows the effect of temperature
on the permeability coefficient of oxygen in nylon-6, as a function of
water activity. The diffusion coefficient and solubility values of oxygen in
the polymer were also obtained from these experiments (Hernandez et al.,
1986). The diffusion coefficient was calculated from a least-squares linear
analysis of values from the unsteady state region for each permeability run.
Solubility values were calculated from the eqn. (2). The solubility of
oxygen (S) is expressed in cc 02 (STP)/cc polymer-water system.

As shown in Figure 17, oxygen permeability decreased at the water
activity levels below 0.3 and then increased at the high water activity, as a
function of water activity. Generally, it has been proposed that oxygen
permeability of nylon films, except amorphous nylon, increase as a
function of water activity.

Figure 17, 18 and 19 illustrate the effect of temperature on the
permeability coefficient, the diffusion coefficient and the solubility of
oxygen in nylon-6, as a function of water activity respectively. The
diffusion coefficients increase in the range of 0 to 0.2 water activity, and
then decrease as further increase water activity. The observed increase of
D in the region of low water activity may be attributed to a plasticization
of amide chain backbone by sorbed water. This plasticization effect would
tend to increase the mobility of oxygen molecules and polymer chain
segments within the polymer bulk phase. Plasticization may also result in
increase in the percent crystallinity (Clark and Wilson, 1973) (see
following DSC experiments). Oxygen molecules cannot diffuse through
the crystalline region, hence the diffusion coefficient tends to decrease in
higher water activities. Above the water activity 0.5, the diffusion

coefficient values at 23°C and 42°C were lower than those of 5°C. These

61

 

 

 

 

58-9
48-9 -
“E 42°C
.2
2 38-9 0
3:
3 a 23°C
0 g -
5 l;
E E 28-9 .
a v
E
a 1
18-9 -
5°C
0970 r ' r r f a r r
0.0 0.2 0.4 0.6 0.8 1.0
Water Activity
Figure 18. Diffusion coefficient values at 5°C, 23°C and 42°C

Solubility Coefficient

(00 02 I 00 polymer)

62

 

1.2

1.0-1

0.8 '-

0.6 -

0.4 -

0.2 -'

 

 

,_____.. _ _. .___

0.4

Water

42°C

23°C

5°C

 

I

0.6

Activity

 

1.0

Figure 19. Solubility coefficient values at 5°C, 23°C and 42°C

63

results are not consistent, since the intersection of the curve should not
happen. There must have been experimental errors such as the
instrumental error of humidity sensers with this study. Therefore, further
studies on the diffusion coefficient through a nylon-6 film should be
conducted to confirm the results obtained in this study. However, the
trend of values appears to be correct.

On the other hand, the solubility coefficients decrease in the range
of low water activity, and then increase as a function of water activity.
The high permeability coefficients of nylon-6 in the range of high water
activity were attributed to the higher oxygen solubility coefficient values
than the diffusion coefficient. The behavior of the solubility coefficient of
nylon-6 is opposite, compared to that of nylon 6I/6T obtained by
Hernandez et al. (1991). As mentioned above, the percent crystallinity of
nylon-6 increases with water activity values. The crystallized molecules
associated with water might be very small and disperse in the polymer.
These small crystallized molecules might make the solubility coefficient
increase with the water activity values. However, that is not fully

understood. Further studies may be conducted to confirm this hypothesis.

Differential Scanning Calorimetry (DSC) Experiments

DSC method for determining the percent crystallinity of a nylon-6
film was used in this study. Table 9 shows the enthalpy data at the melting
point and the percent crystallinity of nylon-6, as a function of water

activity. Percent crystallinity values of the samples were calculated by

using eqn. ( 1).

64

Table 9. The enthalpy at melting point and percent crystallinity

for nylon-6 as a function of humidity

 

Enthalpy at Melting Point

 

(J/g) Enthalpy Percent
Conditions Replications Mean Crystallinity
1 2 3 4 Mean
Dry 64.3 61.6 61.9 59.0 61.712.17 31.711.11

33% RH 70.1 69.2 69.710.64 35.810.33

63% RH 72.4 65.9 69214.60 35512.36

 

65

Table 10. ANOVA for the DSC data (dry vs. 33% RH samples)

 

 

 

Source Degree of Sum of Mean
of Freedom Squares Square Observed Required F
Variation df SS MS F
5V0 1%
Total 5 98.3
Varieties 1 83.8 83.8 23.1 6.94 18.00
Error 4 14.5 3.63

 

Table 11. ANOVA for the DSC data (dry vs. 63% RH samples)

 

 

 

Source Degree of Sum of Mean
of Freedom Squares Square Observed Required F
Variation df SS MS F
5% 1%
Total 5 109.2
Varieties 1 74.0 74.0 8.41 6.94 18.00

Error 4 ‘ 35.2 8.80

 

Table 12.

Student's t test of DSC data (dry vs. 33% RH samples)

 

 

 

Variable: Dry 33% RH
Mean 61.70 69.65
Std.Deviation 2.168 0.636
Observations 4 2
t-test -4.821 Hypothesis

Degree of Freedom 4 H0: 111 -= 112

Significance 0.009 Ha: 111 #112

Table 13. Student's t test of DSC data (dry vs. 63% RH samples)

 

 

 

Variable: Dry 63% RH

Mean 61 .70 69.15

Std.Deviation 2.168 4.596

Observations 4 2

t-test -2.899 Hypothesis
Degree of Freedom 4 H0: 111 = 112
Significance 0.044 Ha: 111 at 112

67

The percent crystallinity of wet samples were higher than that of dry
samples. Table 10 and 11 show the statistic analysis by ANOVA, and
Table 12 and 13 show the results of the student t test. From these statistic
results, there was a significant difference between percent crystallinity of
dry nylon-6 and wet nylon-6 with 95 % confidence, in terms of the
percent crystallinity of the dry and wet nylon-6 film.

The increase of percent crystallinity of nylon-6 might be the
plasticization effect by water. Water is assumed to be distributed
homogeneously throughout the polymer and it resides principally in the
noncrystalline region. The water in amorphous region would cause the

increase of percent crystallinity of nylon-6.

68

CONCLUSIONS

Sorption isotherm data of water vapor in nylon-6 film were obtained
over a wide range water activities at 5°C, 23°C and 42°C. The modified
dual sorption model which has been proposed by Hernandez et al. (1991)
was found to describe well the sorption of water vapor by nylon-6 film.
The dual mode sorption model is based on a Flory-Huggins and Langmuir
equation and was developed to describe water vapor sorbed by an
amorphous polyamide. Hernandez et al. used a nonlinear regression
method to determine the values of three parameters of the Langmuir
Flory-Huggins equation. In this study, a computer program was
developed and it was useful to determine the three parameters.

The oxygen permeability behavior was analyzed in terms of the
multiplicative effect of a mobility and solubility term. The oxygen
permeability coefficient for nylon-6 decreased in the range of 0 to 0.3
water activities, and then increased at the high water activities. This result
was different from general belief in which oxygen permeability of
semicrystalline nylon increases continually as a function of moisture
content of the polymer. The oxygen diffusion coefficient values appears to
decrease above 0.2 of water activity and the solubility of oxygen showed a
steadily increase at the high water activity. The combination of these two
effects resulted in the observed increase of the oxygen permeability values
above 0.3 of water activity as a function of water content in nylon-6.

There was a statistically significant increase of crystallinity of wet

69

nylon-6 film compared to the dry sample. The percent crystallinity of
samples with a water activity above 0.3 were higher than that of dry. The
change in morphology may be associated with the decrease of the oxygen
diffusion coefficients as well as the increase of the oxygen solubility in the

water / nylon-6 system.

APPENDIX

70
APPENDIX A

Computer Program For Flory-Huggins Model

100 REM BOX KANAMAZU METHOD FOR FLORY HUGGINS MODEL
105 REM PROGRAM WRTTEN BY DR. R.J. HERANADEZ, 1988
110 CLS: DIM A1 (20),V1 (20),ETA(20),X(20)

120 INPUT ”HOW MANY POINTS?", N

130 INPUT ”HOW MANY ITERATIONS?", ITE

140 INPUT ”ENTER THE FIRST ESTIMATE OF CHI", B

150 PRINT ”ENTER FIRST A1'S AND THEN V1'S"

160 A1 (1)=.069:A1(2)=.13:A1(3)=.182:A1(4)=.26:A1(5)=.37:A1(6)=.44
170 A1 (7)=.65:A1(8)=.95

190 V1 (1)=.00683:V1(2)=.012:V1(3)=.01447:V1 (4)=.0202
200 V1 (5)=.0289:V1(6)=.035:V1 (7)=.0533:V1(8)=.0893
210 PRINT ' H', "so", " S1”, " CHI”

220 RL=0I

225 RL=RL+1

230 XTX=0I :XTY=OI :SO=OI :S1=0l

240 REM EQUATION FOR THE MODEL

250 FOR K=1 TO N

255 V2=1-V1 (K)

260 ETA(K)=V1(K)*EXP(V2+B*V2*V2)

270 NEXT K

280 REM EQUATION FOR THE SENSITIVITY COEFF

290 FOR K=1 To N

295 V2=1I-V1 (K)

300 X(K)=V1(K)*V2*V2*EXP(V2+B*V2*V2)

310 NEXT K

315 REM CALCULATE 20, XTX, XTY

320 FOR K=1 To N

330 SO=SO+(A1(K)-ETA(K))*(A1(K)-ETA(K))

340 XTX=XTX+X(K)*X(K)

350 XTY=XTY+X(K)*(A1(K)-ETA(K))

360 NEXT K

370 DELTAB=XTYIXTX

380 REM CALCULATE G

390 G=DELTAB*DELTAB*XTX

400 IF (G<0!) OR (e=01) GOTO 670

410 H=1I

71

420 B=B+DELTAB*H

430 REM CALCULATE ETA'S WITH BS
440 FOR K=1 To N

445 V2=1 l-V1 (K)

450 ETA(K)=V1(K)*EXP(V2+B*V2*V2)
460 NEXT K

560 REM CALCULATE S1

570 FOR K=1 To N

580 S1=S1+(A1(K)-ETA(K))*(A1(K)-ETA(K))
590 NEXT K

600 JJ=SO-(2-(1I1.1))*G

610 IF (S1<JJ) OR (S1=JJ) GOTO 640
615 B=B-DELTAB*H

620 .H=G/(S1-SO+2!*G)

630 B=B+H*DELTAB

640 PRINT H,SO,S1,8

650 IF (RL<ITE+1) GOTO 225

660 IF (RL=ITE) GOTO 680

670 PRINT "0:",0.

680 END

72

APPENDIX B

Computer Program For Langmuir-FIory-Huggins Model

100 REM THIS PROGRAM CALCULATES K AND B FOR LANGMUIR

102 REM COMPONENT FRACTION GIVEN CHI THAT HAS BEING OBTAINED

104 REM USING FLORY.BAS IT TAKES THE DIFFERENCE BETWEEN
EXPERIMENT SORPTION VALUES

106 REM AND DO A LEAST SQUARE ANALYSIS TO GET K AND B

108 REM IT ALSO GIVES THE CORRELATION COEFFICIENT FOR

110 REM THE LINEAR CORRELATION AS WELL AS THE SUM OF SQUARE

112 REM FOR THE EXPERIMENTAL AND LANGMUIR-FLORY-HUGGINS
MOLE.

118 DIM A(30), W(30),M(30),L(30),LG(30),MFH(30)

121 A(1)=0l:A(2)=.067:A(3)=.096:A(4)=.148:A(5)=.182:A(6)=.238

122 A(7)=.306:A(8)=.38:A(9)=.47:A(10)=.52:A(11)=.55

125 A(12)=.65:A(13)=.79

126 M(1)=0!:M(2)=.0055:M(3)=.0074:M(4)=.0108:M(5)=.0126

127 M(6)=.0162:M(7)=.0206:M(8)=.0257:M(9)=.0312

128 M(10)=.035:M(11)=.0376:M(12)=.0454:M(13)=.0587

135 INPUT "ENTER VALUE OF CHI ”,CHI

140 LPRINT ”CHI VALUE USED=";CHI

145 LPRINT "ACTIVITY”; " EXPERIMENT”; " F-H"; "
LANG.” " MFH"

150 FOR I=1 TO 13

155 REM FIRST GUESS FOR VOLUME FRACTION :X=A(l)l10

160 FOR ":1 TO 4

170 Z=1-X

180 Y=Z+CHI*Z*Z

190 F=X*EXP(Y)-A(I)

200 FP=EXP(Y)*(1-X*(2*CHI*Z+1))

210 X=X-FIFP

215 W(l)=X

220 NEXT II

224 REM CALCULATE EXPERIMENT F-H TO GIVE LANG. COMP.

230 SA=0: SSA=0: SL=O:SAL=0:SSL=0

240 NEXT I

243 FOR l=2 TO 13

250 REM LINEAR REG. TO CALCULATE K&B IN LAPY

255 L(I)=M(I)-W(I)

73

260 SL=SL+(1IL(|))

270 SA=SA+(1IA(I))

280 SAL=SAL+(1/A(I))*(1/L(I))

290 SSA=SSA+(1/A(I))*(1IA(I))

295 SSL=SSL+(1/L(I))*(1IL(I))

300 NEXT l

302 PRINT 'SL="SL, 'SA="SA,

303 PRINT "SAL="SAL, 'SSA="SSA

310 REM CALCULATE K

315 K=(12*SSA-SA*SA)/(12*SAL-SA'SL)

320 REM CALCULATE BIK

325 BOK=((SL*SSA)-(SA*SAL))/((12*SSA)-SA*SA)

330 B=BOK*K

340 REM CALCULATE SUM OF SQUARE FOR EXPERIMENTAL AND
CALCULATED,SST

345 SST=0

350 FOR I=1 To 13

370 LG(|)=K*A(|)I(1+B*A(I))

380 SST=SST+((M(|)-W(I))-LG(l))"2

400 MFH(I)=W(I)+LG(I)

430 LPRINT A(l); M(l); W(l); LG(I); MFH(I)

440 NEXT I

450 SY=0:SAY=0

455 FOR l=2 To 13

460 SY=SY+(1lL(I)-BOK-(1IK)*(1IA(I)))"2

465 SAY=SAY+(1IL(l)-SLI12)"2

468 NEXT I

470 REM CALCULATE CORRELATION COEFFICIENT

475 SR=1-SY/SAY

480 LPRINT ”SUM OF SQUARE ="SST, 'K="K, "B="B, "SR="SR

490 END

74

APPENDIX C

Computer Program For Diffusion And Solubility Coeff.

100 REM THIS PROGRAM CALCULATES THE DIFFUSION COEFFICIENT
FROM PERMEABILITY

102 REM CONTINUOUS FLOW EXPERIMENTS

104 REM PROGRAM WRITTEN NY R. J. HERNANDEZ. JAN/1988.
MODIFIED FOR K. OHASHI

106 REM ON 11790

110 DIM F(50),T(50),X(50),DF(50)

120 REM THE UNITS OF TIME USED WILL DETERMINE THE UNITS IN THE
DIFFCOEFF

132 PRINT "ENTER RELATIVE HUMIDITY IN PERCENT"

133 INPUT HUM

135 PRINT ”ENTER THE RUN IDENTIFICATION NUMBER"

136 REM

137 INPUT SUN

138 PRINT "ENTER THE TEMPERATURE AT STEADY STATE"

139 INPUT W

140 PRINT ”ENTER THE NUMBER OF DATA POINTS”

150 INPUT D

160 PRINT ”ENTER THE FLOW F AND TIME T STARTING FRO ZERO”

170 FOR I=1 To D

180 PRINT ”ENTER F"

190 INPUT F0)

200 PRINT ”ENTER T"

210 INPUT T0)

220 NEXT I

230 PRINT "ENTER YOUR GUESS FOR X"

270 INPUT GUESS

280 PRINT ”ENTER INFINITE VALUE FOR X"

270 INPUT Fl

300 FOR I=1 TO D

305 DF(I)=F(I)IF|

310 A=.44313*F(I)IFI

320 X=GUESS

330 FOR J=1 TO 7

340 B=SQR(X)

350 C=EXP(-X)

75

360 L=1IB

370 H=(.5*L-B)*C

380 E=(B*C)-A

390 X=X-(EIH)

400 NEXT J

410 X(l)=X

420 GUESS=X

430 NEXT I

480 REM LINEAR REGRESSION

490 ST=0

500 SX=0

510 SXT=0

520 STSQ=0

530 SXSQ=0

540 FOR I=1 To D

545 X(|)=1IIX(I)

550 ST=ST+T(I)

560 SX=SX+X(I)

570 SXT=SXT+(X(I)*T(I))

580 SXSQ=SXSQ+(X(I)*X(I))

590 STSQ=STSO+(T(I)*T(I))

600 NEXT |

610 SLOPE=(ST*SX-D*SXT)/(ST*ST-D*STSQ)

620 DUM1=(D*SXT)-(SX*ST)

630 DUM2=(D*STSO)-(ST*ST)

640 DUM3=(D*SXSQ)-(SX*SX)

650 DUM4=SOOR(DUM2*DUM3)

660 R=DUM1IDUM4

662 LPRINT ”RUN NUMBER:' SUN

663 PRINT

664 LPRINT "TIME (MIN) ","FLOW", ”x", ”FLOW PERCENT"

670 FOR I=1 To D

700 LPRINT T(I),F(l),X(I),DF(I)

720 NEXT I

730 PRINT

735 DIFF=(4.2E-08)*SLOPE

736 LPRINT ”DIFFSION COEFF IN cm2lse0. =" DIFF

737 LPRINT

738 LPRINT ”THE PERMEABILITY COEFF IN cm3.mil/m2.day.atm ="
Fl*6.258*1.25

739 LPRINT

740 LPRINT "THE SLUBILITY IN cc 02/00 olymer ="
(.0000000000211497#)*FIIDIFF

 

76

742 LPRINT

745 LPRINT "THE CORRELATION COEFF. =" R

746 LPRINT

747 LPRINT "TEMPERATURE IN C =" W

748 LPRINT ”WATER ACTIVITY aw =" HUM/100

750 REM THE VALUE OF THE DIFFUSION COEFFICIENT IS FOR THE TIME
IN MINUTES

755 REM THE THICKNESS IS L=3.175E-03 cm (1.250 mil)

800 END

APPENDIX D

Differential Scanning Calorimetry (DSC) Data

 

 

 

77

 

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BIBLIOGRAPHY

 

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