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TRANSIENT SORPTION AND PERMEATION IN FLUOROPOLYMERS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Sangwha Lee, B.S., M.S.
*****
The Ohio State University
1995
Dissertation Committee: Approved by
K. S. Knaebel
S. T. Yang Advisôr J. Charmers Department of Chemical Engineering DMI Number: 9526055
UMI Microform 9526055 Copyright 1995, by DMI Company. All rights reserved.
This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103 Copyright by
Sangwha Lee
1995 (Dedication)
To My Parents ACKNOWLEDGEMENTS
I express sincere appreciation to Dr. Knaebel for his coherent guidance, patience,
and insight throughout the research. Thanks to the other members of my advisory
committee, Drs. Yang and Charmers, for their suggestions and comments. To my
parents, I thank you for understanding and supporting me all the time.
DuPont should be recognized for their financial support, and donation of fluoropolymer resins and other materials. Appreciation is also extended to Mr. D. Roy
Groldsbeny, Dr. L. W. Buxton, and Dr. Sina Ebnesajiad for their sincere help and advice.
Thanks to God !
in VITA
April 26, 1963 ...... Bom - Seoul, Korea
1986 ...... B. S., Seoul National University, Seoul, Korea
1988 ...... M. S., Seoul National University, Seoul, Korea
1989 -1990 ...... Researcher, Korean Institute of Science and Technology Seoul, Korea
PUBLICATIONS AND CONFERENCE PRESENTATIONS
Lee, S. W.. Study on Paraffin Dehvdrocvclization. M.S. Thesis, Seoul National University, Seoul, Korea, 1988.
Lee, S. W. and H. I. Lee, Paraffin Dehvdrocvclization over Bimetallic Catalyst (Pu Sn) SuvDorted by y-Alumina. Chemical Engineering Society of Korea, Fall Symposium, Pusan, Korea, Oct.1987.
FIELDS OF STUDY
Major Field: Chemical Engineering
Studies in mass transfer phenomena in fluoropolymer-penetrant system under Dr. Kent S. Knaebel.
IV TABLE OF CONTENTS
ACKNOWLEDGMENTS...... üi
VITA ...... iv
LIST OF TABLES...... ix
LIST OF FIGURES...... xi
ABSTRACT...... xvi
CHAPTER PAGE
I. INTRODUCTION...... 1
II. TRANSPORT THEORIES IN POLYMER MEMBRANES...... 8
A. TRANSPORT PROPERTIES...... 8 1. Diffusivity ...... 11 1.1. Activated state theory ...... 11 1.2. Molecular m odels ...... 12 1.3. Free volume theory ...... 14 2. Solubility ...... 20 2.1. Energetic viewpoint ...... 20 2.2. Thermodynamic sense ...... 22 3. Physicochemical nature of components ...... 25 3.1. Geometric effects of components ...... 25 3.2. Electronic factors affecting to penetrants and polymers ...... 28
B. FICKIAN DIFFUSION...... 30 1. Diffusion process ...... 30 2. General solutions ...... 32 2.1. Initial value problem ...... 32 2.2. Boundary value problem ...... 36 2.3. Fickian characteristics...... 38 C. NON-FICKIAN DIFFUSION...... 40 1. Permeation anomalies ...... 40 2. Sorption anomalies ...... 43 3. Prior mathematical models ...... 51 3.1. Model equations for continuous morphological change ...... 52 3.2. Model equations for discontinuous morphological change ...... 57
D. KINETIC TH E O R Y ...... 63 1. Assumptions and preliminary information ...... 63 2. Development of model equations ...... 66 2.1. Exponential dependence of rate equation ...... 68 2.2. Quadratic dependence of rate equation ...... 69 3. The characteristic of kinetic model ...... 72 4. Justification of the theory and its limitations ...... 80 m . FLUOROPOLYMERS...... 82
A. GENERAL PHYSICAL AND CHEMICAL PROPERTIES 82
B. MOLECULAR STRUCTURE...... 88
C. PROCESSING PROPERTIES...... 94 1. Orientation ...... 94 2. General effects of processing ...... 96
D. SPECIFIC APPLICATIONS...... 97
IV. EXPERIMENTAL METHODOLOGY...... 100
A. ORGANIC SOLVENT SELECTIONS...... 101
B. EXPERIMENTAL METHODS, APPARATUS AND CONDITIONS...... 104
C. ANALYSIS METHODS...... 114 1. Computer spreadsheet ...... 114 2. Statistical analysis ...... 115
V. RESULTS AND DISCUSSION...... 116
A. EXPERIMENTAL RESULTS...... 116 1. Sorption characteristic ...... 116 1.1. Polymer ty p e ...... 116 1.2. Solvent type ...... 120
VI 1.3. Repeated exposure ...... 123 1.4. Orientation ...... 123 1.5. Non-isotropic expansion ...... 127 1.6. Solubility ...... 132 1.7. Integral sorption with low activity of penetrant ...... 133 1..8. Thickness effect ...... 137 1.9. Presorbed samples ...... 137 2. Permeation characteristics ...... 142 2.1. Polymer and solvent type ...... 142 2.2. Effects of processing properties ...... 142 2.3. Overshoot ...... 149 2.4. Temperature and thickness effects ...... 149 2.5. Possible glass transition effects ...... 153 3. Stress-strain te st...... 155 3.1. Polymer type ...... 155 3.2. Solvent type ...... 158 3.3. Successive impregnation ...... 162 3.4. Repeated exposures ...... 162 3.5. Plasticization effects ...... 166
B. DISCUSSION OF EXPERIMENTAL RESULTS...... 169 1. Analysis of transient sorption ...... 169 1.1. Fickian sorption ...... 169 1.2. Non-Fickian sorption ...... 170 2. Analysis of permeation d ata ...... 176 3. Analysis of stress-strain tests ...... 180
C. DISCUSSION OF MATHEMATICAL MODELS...... 182 1. Fickian diffusion ...... 182 2. Non-Fickian diffusion ...... 189
D. GENERALIZATIONS...... 207
VI. CONCLUSIONS AND RECOMMENDATIONS...... 218
LIST OF REFERENCES...... 221
APPENDICES...... 229
A. PERMEATION DATA ...... 230
B. SORPTION DATA ...... 302
C. STRESS-STRAIN DATA ...... 361
Vll D. COMPUTER PROGRAM ...... 400 1. Curve fitting program 2. Crank-Nicholson method
vm LIST OF TABLES
TABLE PAGE
2.1. Permeability of polymer films to water vapor at 25 ° C ...... 29
2.2. The evaluated parameter values of quadratic type of rate equation to predict sequential differential sorption behavior ...... 74
3.1. General trends of properties in relation to fluorine content ...... 84
3.2. Comparison between fully fluorinated polymers and partially fluorinated polymers ...... 84
3.3. Transitions of PTFE ...... 89
3.4. Typical properties of T^ films ...... 95
3.5. Basic factors influencing end-product properties ...... 96
3.6. Specific applications of various fluoropolymers ...... 97
4.1. Physical and chemical properties of organic solvents ...... 102
4.2. Solubility parameters and related properties ...... 103
4.3. Matrix of experimental tests ...... 112
5.1. Equilibrium expansion of polymer films (10 mil) after immersion in penetrants ...... 130
5.2. Solubility of solvents in polymers at 25, 45, and 65 °C ...... 132
5.3. Sorption energies fiom Arrhenius-type plots ...... 133
5.4. Susceptibility of fluoropolymers to various types of solvents 143
IX 5.5. The permeation rate of aromatic liquids through fluoropolymers 149
5.6. The tensile properties of unsorbed and sorbed fluoropolymers ...... 158
5.7. The tensile properties of gradually impregnated ECTFE by toluene ...... 162
5.8. The tensile properties of repeatedly exposed ECTFE by toluene ...... 166
5.9. Activation energies from Arrhenius plots (LN(P.R.) vs. Temp.-^(K ') ) ...... 176
5.10. Comparison of permeation rate and solubility between soft (dull) side and tough (shiny) side in thin ECTFE ...... 178
5.11. Comparison the calculated values with the experimental results of FEP-benzene system ...... 183
5.12. Fitted rate parameters of penetrants in PVDF ...... 193
5.13. Fitted rate parameters of penetrants in ECTFE...... 196
5.14. Fitted parameters for sorption experiments with ECTFE ...... 204 LIST OF FIGURES
FIGURES PAGE
2.1. Mechanism of liquid permeation through a membrane ...... 9
2.2. Diffusion mechanism according to the molecular model suggested by Pace and Datyner (1979) ...... 15
2.3. Volume-temperature behavior of an amorphous polymer suggested by Vrentas and Duda (1977) ...... 19
2.4. Schematic representation of compatible and incompatible systems (Brydson, 1975) ...... 21
2.5. Diffusivities of gases and vapors in poly(vinyl chloride) at 25 °C as a function of their van der Waals volume, b (Reproduced by Rogers (1985)) ...... 26
2.6. Anomalous shaped permeation-rate curves ...... 41
2.7. Non-Fickian or anomalous sorption and desorption curves compared with Fickian-type curves (Rogers, 1965) ...... 44
2.8. Other anomalous sorption curves such as overshoot, moving front, and drastic acceleration ...... 45
2.9. Idealized concentration distribution proposed by Peterlin (1969) for concurrent Fickian and Case II sorption ...... 59
2.10. The sequence of differential sorption in methyl acetate- poly(methyl methacrylate) system at 30 °C (Kishimoto et al., 1960) ...... 73
2.11. Differential sorption sequence predicted by new kinetic model ...... 76
XI 2.12. Logarithms of relaxation time t = 2t* versus initial concentration c,- for differential sorption (Taken from Fujita, kishimoto, and Odani, 1959) ...... 78
3.1. Molecular structures of fluoropolymers ...... 88
3.2. Comparison of molecular structures (ETFE vs. PVDF) ...... 90
3.3. Applications of fluoropolymers ...... 99
4.1. Apparatus for measuring liquid permeation rates ...... 105
4.2. Apparatus for measuring liquid sorption rates ...... 107
4.3. Apparatus for measuring vapor sorption rates ...... 108
4.4. Tensile test specimen ...... 109
4.5. Schematic diagram of Tensometer ...... 110
5.1. Integral sorption and desorption of benzene in fluoropolymers at 25 °C (Film thickness is 10 m il) ...... 117
5.2. Integral sorption of benzene in ETFE (at 25 °C) fitted by Fickian solution with constant diffusivity ...... 119
5.3. Log-log plots of the kinetics of benzene sorption by fluoropolymers at 25 °C ...... 121
5.4. Integral sorption of aromatic liquids into fluoropolymers at 25 °C ...... 122
5.5. Integral sorption of polar solvents into fluoropolymers at 25 °C ...... 124
5.6. Integral sorption of toluene into fluoropolymers with different exposure times at 45 °C ...... 126
Xll 5.7. Integral sorption and resorption of toluene into ECTFE sample (length/width = 4, 14) at 25 °C and dimensional change of ECTFE sample in the machine and transverse directions ...... 128
5.8. Time dependence of integral sorption and specimen length for benzene penetration in PVDF and ECTFE at 25 °C ...... 131
5.9. Effects of temperature on solubility for aromatic solvents ...... 134
5.10. Vapor sorption of benzene into fluoropolymers at 45 °C ...... 135
5.11. Integral sorption of toluene and chlorobenzene in thick ECTFE film at 45 and 65 °C (Film thickness is 90 m il) ...... 138
5.12. Sorption curves showing effects of presorption by vapor-phase benzene on the sorption by the corresponding pure liquid ...... 139
5.13. Arrhenius plots of permeation rates of various organic liquids in fluoropolymers ...... 144
5.14. The transient permeation rate of aromatic solvents through ECTFE at 25 °C ...... 147
5.15. The comparison of permeation rates of benzene in ECTFE, when shiny side is exposed, with when dull side is exposed a t2 5 “C ...... 148
5.16. Overshoot in transient permeation of ECTFE-benzene system at 45, 75 °C ...... 150
5.17. The permeation rates of fluoropolymers with different thickness (10, 90 mil) at 45, 75 “C ...... 152
5.18. Temperature and penetrant effects on permeation rate and solubility for PFA ...... 154
5.19. Stress-strain curves of fresh fluoropolymers at room temperature .... 157
5.20. Stress-strain curves of impregnated fluoropolymers by toluene 159
5.21. Stress-strain curves of "new" PVDF samples impregnated by various organic solvents ...... 163
xin 5.22. Stress-strain curves according to the gradual increase of impregnation time by toluene ...... 164
5.23. Stress-strain curves of ECTFE samples repeatedly exposed by toluene ...... 165
5.24. Tensile stress-strain curves of ECTFE over various organic chemicals ...... 167
5.25. Tensile stress-strain curves of PVDF with various organic chemicals ...... 168
5.26. Arrhenius plots of permeation rates of aromatic solvents in fluoropolymers ...... 177
5.27. The fitted data of PFA-toluene at 25, 45, and 65 °C by simple Fickian equation ...... 184
5.28. The fitted data of ETFE-benzene at 45 °C by a concentration dependent Pick’s equation ...... 188
5.29. The fitted data of PVDF-aromatic liquid sorption at 25, 45, and 65 °C by quadratic type of rate equation ...... 190
5.30. The comparison of calculated permeation rates firom kinetic model with experimental data from PVDF-penetrant systems ...... 195
5.31. The fitted data of ECTFE-aromatic liquid sorption at 25, 45, and 65 °C by exponential type of rate equation ...... 197
5.32. The Arrhenius plot of diffusivity and rate coefficient obtained from fitting PVDF sorption data ...... 201
5.33. The Arrhenius plots of diffusivity and rate coefficient obtained from fitting ECTFE sorption data ...... 202
5.34. The fitted curves of presorbed ECTFE sample by benzene at 25 °C ...... 205
5.35. The fitted curves of repeatedly exposed ECTFE samples by toluene at 45 °C ...... 206
XIV 5.36. Dependence of penneation rate on thermodynamic (a-d) properties of penetrants for PFA polymers ...... 209
5.36. Effect on solubility and diffusivity of molar volume (e-h) and latent heat of vaporization at 25 °C ...... 210
5.37. Dependence of permeation rate on thermodynamic (a-d) properties of penetrants for ETFE polymers ...... 211
5.37. Dependence of solubility in ETFE on molar volume (e-f) and latent heat of vaporization at 25 ®C ...... 212
5.38. Dependence of permeation rate on thermodynamic (a-d) properties of penetrants for ECTFE polymers ...... 213
5.38. Effect on solubility and diffusivity of molar volume (e-h) and latent heat of vaporization at 25 °C ...... 214
5.39. Dependence of permeation rate on thermodynamic (a-d) properties of penetrants for PVDF polymers ...... 215
5.39. Effect on solubility and diffusivity of molar volume (e-h) and latent heat of vaporization at 25 °C ...... 216
XV ABSTRACT
The main goal of this study is to elucidate the chemical structure-physical property-
performance relationships among several combinations of fluoropolymers and liquid
penetrants, focusing on their diffusivity, solubility and permeability. The polymers studied
were ETFE, PVDF, ECTFE, PFA, and FEP. Both transient sorption and permeation
experiments have been conducted under various temperatures (25, 45, 65 or 75 °C),
thicknesses (10 and 90 mil), and organic solvents (nonpolar to polar).
For the transient sorption experiments, the range of responses has been Fickian to
various forms of non-Fickian behaviors, some of which exhibited acceleration during the
final stage of uptake. PFA and FEP exhibited Fickian diffusion with a concentration -
independent diffusivity. ETFE exhibited Fickian diffusion with a concentration-dependent
diffusion coefficient. PVDF and ECTFE exhibited non-Fickian behaviors which show
acceleration during final stage of uptake.
The suspected causes for the unusual sorption behavior were stmctural characteristics
due to processing (e.g. skin or orientation with respect to processing direction), intrinsic chemical structure, and morphological deformation induced by swelling. These causes appear to be characteristic of individual types of polymers in conjunction with individual liquids.
XVI A simple kinetic model was devised to interpret the transient sorption results, particularly the observed acceleration in ECTFE and PVDF. The model assumed that structural changes due to swelling and relaxation could be expressed as a volume change, which is proportional to amount of liquid sorbed. The volume increase is induced by disruption of polar intersegmental attractions within the polymer. From the model and experimental data, diffusivities and other coefficients that characterize sorption were obtained.
From stress-strain tests, it was also confirmed that the transport properties and mechanical properties are closely related in the presence of penetrants. For strong swelling agents, the diffusion was non-Fickian and a significant change of mechanical properties was observed. For weak swelling agents, the transport was nearly Fickian and a negligible change of mechanical properties was observed.
The diffusivities observed in the transient sorption experiments were combined with measured solubilities, and the products were compared with measured permeabilities. For
FEP, the products agreed well with experimental results. The permeability of PVDF evaluated from the new kinetic model was also compared with experimental data. The discrepancy was acceptable (e.g. -30%). The quadratic type of kinetic model shed light on prediction of transport properties as a quantitative basis even from anomalous sorption kinetics.
X V I 1 CHAPTER I
INTRODUCTION
Recent development of fluoropolymers and their drawbacks
Fluoropolymers are a kind of paraffinic polymers that have some or all of the
hydrogen replaced by fluorine. They can be classified into two sub-classes: (1 ) fiuorocarbon
polymers that are made from perfluoromonomers, and (2) all of the others that are made
from monomers or comonomer systems that contain hydrogen and chlorine. The commercially important fiuorocarbon polymers are Teflon® resins (PTFE, FEP, and
PFA) made by E. I. DuPont. The second group of fluoropolymers include modified
Tefzel® ETFE, Kynar® (or Hylar, Solef) PVDF, and Halar® ECTFE.
Due to their outstanding chemical resistance and low permeability, fluoropolymers are widely employed to seal and isolate materials, especially under harsh demanding conditions. The degree of inertness of fluoropolymers reflects their unique chemical structure which is simply formed from strong C-C and very strong C-F chemical bonds
(Doban et al., 1955).
While fluoropolymers demonstrated advantages in the areas of heat and chemical resistance soon after they were developed in the 1940s, mechanical properties such as creep resistance, stiffness, and toughness were improved in the 1990s.' ‘ Recently introduced fluoropolymers with improved mechanical properties, better processibility and
1 high purity are creating new application opportunities when their chemical resistance is
adequate and higher mechanical strength is needed. Fluoropolymers are used across the
chemical, pulp, and paper and related process industries, where corrosive and abrasive
environments call for superior material of construction, to the semiconductor,
pharmaceutical, and biotechnology industries, where there are critical needs to reduce all
contamination.^ *
The increase of mechanical strength unfortunately offsets the strong inherent
chemical resistance by the introduction of foreign substituents such as chlorine and
hydrogen. It is imperative to select an appropriate fluoropolymer, with respect to the
balance of economic aspects and its performance in various demanding conditions. The problem is that it is almost impossible to test the polymer samples under every demanding condition.
Permeability
Permeability is an important factor in determining the suitability of a particular polymer for specific applications such as protective coatings, packaging materials, selectively separating barriers, biomedical devices, etc. Therefore, Understanding the underlying physics and chemistry of membrane permeation is necessary to make a rational selection of polymeric materials and a membrane development.
The nature of the solution (solubility) and diffusion processes can help elucidate inherent characteristics of polymeric materials such as flexibility and conformation of chain segments, interactions, free volume change, structural and morphological features, as the so- 3
called “molecular probe" aspect. Macroscopic observations, however, indicate that the
classical diffusion theory embodied by Pick’s law cannot describe the diffusion process
adequately in all cases. Depending on the polymer and penetrant, a variety of “non-Fickian
(anomalous)" behaviors may appear, which are explained qualitatively or with ad hoc
models. This so-called non-Fickian behavior usually occurs with glassy polymers, semicrystalline polymers above the glass transition temperature, and polymers with more rigid chain conformations and higher internal viscosity, when the penetrant swells the polymer (Rogers, 1965). The penetrant absorbed by the polymer lowers the glass transition temperature and loosens the segmental motion at any given temperature. In such cases, the diffusivity may be a function of concentration, time, spatial coordinates, and history of a sample.
Considerable practical value lies in developing a generalized theory for diffusion which can account quantitatively for the non-Fickian behavior. It is also necessary to elucidate the mechanism of diffusion on a microscopic level in order to better understand the phenomenology of diffusion of small molecules in polymers.
Literature review
Since the 1950's, many types of models have been proposed to explain anomalous diffusion behavior of small molecules in polymeric materials. A promising free-volume model was developed by Fujita and his coworkers in the early 1960s. Their free-volume theory provides the quantitative explanation of the concentration and temperature dependence of diffusivity of organic vapors in amorphous polymers. The free-volume 4 approach is simple and has evolved into useful forms (Fang et al., 1975; Vrentas and Duda,
1977, 1978, 1984). One of the well known phenomenological models based on the free- volume concept is the dual sorption model (Michaels et al., 1963; Paul and Koros, 1976).
The drawback of free-volume theory lies in the difficulty of providing a precise physical definition for the free-volume parameters defined in the model.
Secondly, modified forms of diffusion equations have been suggested within Fickian framework. Crank (1951,1953) ingeniously interpreted the advancing sharp boundary and sigmoidal sorption by introducing a discontinuous diffusion coefficient and a relaxation diffusion coefficient. Most work in this category includes the effect of time on surface concentration, diffusion coefficient, and flux equation (Long and Richman, 1960;
Petropoulos and Roussis, 1974; Neogi, 1983; Cohen, 1983,1984; Duming, 1985; Ocone and
Astarita, 1987; Camera-Roda and Sarti, 1986,1990; Doghieri et al., 1993). The relaxation term, however, might be misleading and may not represent the true mechanism of the process itself; it is used to describe any time dependent process. Therefore, the approach based on relaxation mechanism actually may have slight (or no) physical meaning for some materials because the parameters are only chosen to fit anomalous sorption curves.
Thirdly, new constitutive equations were devised to interpret the anomalous diffusion process. Thomas and Windle ( 1982) first considered the effect of osmotic-pressure-driven viscous response of the polymer chain relaxation. They assumed that diffusional flux of the fluid is proportional to the local rate of expansion of the polymer matrix. Their model can predict all the essential characteristics of Case H transport. Coupled diffusion and relaxation models are also included in this category. According to Berens and Hopfenberg (1978), the 5 first-order relaxation was linearly superimposed with the Fickian diffusion. Peterlin ( 1969) proposed a model based on the linear combination of Fickian and Case II contributions to the transport process. The superposition of contributing processes is a useful approach for the qualitative explanation of anomalous diffusion behavior in glassy polymers, but the quantitative treatment in terms of a linear combination of the two contributing processes is not fully justified and may be a considerable oversimplification.
Until now, no known molecular microscopic models or comprehensive macroscopic models have performed successfully even though some theoretical models have been suggested based on thermodynamics and transport theories (Duming and Tobar, 1986;
Carbonell and Sarti, 1990; Lustig et al., 1991). In a practical sense, it is difficult to get a completely unified model because basically different diffusion-mechanisms are involved depending on the types of polymers, types of penetrants, and ambient conditions. It is worth mentioning, however, that submicroscopic voids between polymeric molecules provide space for the material absorbed without chemical reaction, and the sorbed polymer, under given conditions of vapor activity and temperature, undergoes plasticization process which appears to provide a direct measure of the time- and concentration-dependent change of available free volume in the polymer. It is, therefore, reasonable to assume that any structural changes such as disengagement of segments, microcavity formation, swelling, and phase transition, could be explicitly expressed by a volume change via free-volume creation. This change eventually leads to deviation from normal Fickian diffusion by inducing time-dependence of both solubility and diffusivity. The solubility and diffusivity of small molecules in polymeric materials are well known to be dependent on the amount of free volume in the 6 system. Diffusivity is much more dependent on the amount of free volume than is solubility
(Peterlin, 1975, 1979). Some consistent and unified explanations may be provided if a relaxation mechanism is expressed in terms of free-volume parameters.
Objectives of the research
The main goal of this research is to understand of the “chemical structure - physical property - performance” relations among polymers and organic chemicals. A possible outcome is to obtain a systematic and comprehensive view on transport properties of fluoropolymers, and furthermore, to get information about the molecular nature of the polymer chains. One way to achieve this was to relate the properties to the performance of polymeric materials via mathematical models discussed later.
Product development can be promoted by better understanding the effects of penetrants on different fluoropolymers, by systematically evaluating certain processing conditions for specific types of polymers. Thus it may be possible to adjust polymer composition and processing conditions to achieve acceptable barrier properties. Moreover, increased knowledge of the behavior could be useful in making a rational selection of polymeric materials for a range of applications with respect to engineering performance and marketing aspects, or uncover new uses for existing polymers.
Development of new kinetic model and experiments
A new kinetic model was developed based on the assumption that any stmctural change caused by coupled swelling-relaxation effects could be explicitly expressed as free 7
volume change, which is proportional to the amount of liquid sorbed. The physical
interpretation is that the creation of free volume is induced by disruption of polar
intersegmental attractions to accommodate the penetrating liquid.
The present work is focused on individual measurements of diffusivity, solubility,
and lumped permeability of some organic liquids in several fluoropolymers in order to
evaluate the systematic relationships between polymer structures and transport properties.
The scope of this work is currently limited to studies of PFA, ETFE, ECTFE, and PVDF
with a few sets of FEP. The penetrants involved are benzene, toluene, and chlorobenzene.
Other chemicals, such as phenol, methyl ethyl ketone and dichloromethane, were added to explore some specific effects. Other aspects being considered are polymer thickness,
nperature, orientation, and repeated exposures. Some mechanical properties of the fluoropolymers were measured by stress-strain tests and the results were correlated with the observed transport properties.
Finally, the new kinetic model was employed to interpret anomalous transient sorption behavior, especially to encompass the drastic acceleration observed in some fluoropolymers such as PVDF and ECTFE. For FEP films, the diffusivities observed in the transient sorption experiments were combined with measured solubilities, and the products were compared with measured permeabilities. CHAPTER II
TRANSPORT THEORIES IN POLYMER MEMBRANES
A. TRANSPORT PROPERTIES
The transport of a penetrant through a homogeneous membrane, in the absence of gross defects such as pores or cracks, is usually considered to occur by the following process: (1) solution (condensation and mixing) of the gas or vapor in the surface layers,
(2) migration to the opposite surface under a concentration (chemical potential) gradient, and (3) evaporation from that surface into the ambient phase (see Figure 2.1). Such a view of the diffusion of gases through solids was first proposed in 1866 by Graham. If the evaporation process is not the rate determining step, the constant of proportionality in the rate equation, which is called permeability, P, can be expressed as the product of solubility, S, and effective diffusivity, D:
P = DS (1)
Solubility is a thermodynamic property and diffusivity is a kinetic property.
Ideally, sorption equilibrium obeys Henry’s law and diffusion follows Pick's law. For the frequently observed ideal behavior noted in organic rubbers, the diffusion coefficient varies with temperature by an Arrhenius relationship (Barrer, 1934,1939): VAPOUR PHASE DISSOLUTION
LIQUID PHASE EVAPORATION^
& MEMBRANE! ^ '’^THICKNESS
Figure 2.1. Mechanism of liquid permeation through a membrane. 10
I> = Z)^exp(-£„//î7) (2) where is the pre-exponential factor, Ep is the activation energy for diffusion, R is the gas constant, and T is the absolute temperature. The solubility coefficient, S, varies with temperature according to the thermodynamic relationships:
S = S„expi-I^HJRT) (3) where S„ is the pre-exponential factor, and AH^ is the enthalpy change upon solution of the liquid in the polymer. Strictly idealized behavior results not only from the Fickian diffusion but additionally from the Henry’s law description of solution. For the ideal cases, the temperature dependence of permeability can be represented by:
p [r,] R Tj 7 / ( 4
The sorption energy, AH^, and the activation energy for diffusion, £p, are typically small.
Nevertheless, it is clear that two opposing factors affect the permeability as temperature is increased. The positive diffiisional activation energy is usually larger in absolute value than the negative sorption energy, AH^, so the overall permeability normally increases with the increase of temperature.
There is, however, considerable evidence which suggests that this simple model is not adequate for viscoelastic materials such as glassy polymers and semicrystalline 11 polymers at high activities of organic vapors. The major shortcomings of this simple theory is in that it cannot treat the very complex time dependence of diffusivity and solubility in mass transport coupled with structural relaxation of polymeric materials.
1. DIFFUSIVITY
1.1. Activated state theory
The diffusion process in polymer-penetrant system can be visualized as a successive sequence of unit jumps of small molecules into a transient gap which is formed by a cooperated segmental motion of polymer chains. The activated state theory assumes that the holes are continuously formed and destroyed due to thermal fluctuations. Diffusion is visualized accordingly as a thermally-activated process, hence the diffusional activation energy can be considered as the energy to loosen the polymer structure or increase the segmental mobility.
Eyring's (1936,1937,1941) hole theory offers a simple insight into physico-chemical properties’ dependence on diffusion process that is somewhat different from eq.(2). Eyring and his coworkers applied the transition-state theory of rate processes to the diffusion process and derived the following expression:
D = (2.7 kTIh) exp(A5 * / JQ exp(-£/BT) (5)
where k is the Boltzmann constant, and h is the Planck constant, and AS* is the entropy of activation, and A is an average jump length. According to Eyring's theory, it is expected that the diffusion process requires the provision of additional space enough to accommodate 12 diffusing molecules. The amount of energy required for hole formation will increase as the size and the shape of the diffusing molecules increase. Molecules with large diffusional cross-sections diffuse more slowly since there are fewer of these large holes, due to the requirement of the breakage of more secondary bonds. The pre-exponential factor also increases with the size of diffusing molecules.
The diffusion of gases through membranes such as rubbers, resins, and plastics generally satisfies an Arrhenius-type equation and hence apparently involves the passage of the gas molecules from normal to activated states (Van Amerongen, 1946, 1950).
Also, the relationship that ln(D) is linearly dependent on 1/T has been verified for many systems well above the glass transition temperature, (Crank and Park, 1951; Prager and
Long, 1951).
1.2. Molecular models
The activated state theory has been extended to provide more detailed molecular pictures involved in the diffusion process. The molecular models focus on the interpretation of the diffusion process in terms of specific postulated motions of diffusing molecules and polymer chains and take into account of the pertinent intermolecular forces.
A molecular rationalization of Ep was first suggested by Mears (1954) in terms of the product of cohesive energy density {CED) and the volume of cylindrical cavities which can accommodate the diffusing molecules. With this assumption:
E^^CBD-^d^X (6) 13 where À is the length of imaginary cylindrical length and d is the diameter of a diffusing molecule. According to Mears' theory, the activation energy for diffusion, , increases linearly with d ", and decreases as the temperature is raised since the cohesive energy density decreases rapidly with the increase of temperature.
This idea was refined in more detail by DiBenedetto (1963) and Paul (1965) as a cell model formed by the centers of four neighboring chains. The sorbed molecules reside in an equilibrium sorbed cage where it behaves as a 3-dimensional oscillator.
Thermally induced fluctuations in the volume of the unit cell produce a void around the sorbed gas, allowing a ± A diffusional jump. The length (0.42 - 0.49 nm) of the diffusional Jump in this physical model is much smaller than the length (8.4 nm) in the previous Mears' model and seems intuitively to be more physically realistic.
Brandt (1959, 1963) assumed that, in order to create a passage for a diffusing molecule, the activation energy is used in part to “bend” surrounding molecular chains
(£^) and in part to overcome the attractive forces between them (£,). According to
Brandt's theory, diffusional jumps require the cooperation of neighboring polymer segments only for penetrating molecules which are too large to pass through existing free spaces. In order to test the above theory, Brandt and Anysas (1963) studied the diffusion of CO,, CH3CI, and SFg in FIFE, PFA, and PCTFE. The results indicated that plots of
£g versus a ' (cross-sectional area « d " ) were nonlinear and exhibited a downward concavity. Finally, they found that values of £^ for a given penetrant in different polymers were not simply proportional to the cohesive energy density of the polymers in contrast with the Mears' model. 14
Pace and Datyner (1979, 1980) proposed a new diffusion theory which incorporates features of both the molecular models of DiBenedetto and Paul (1963, 1965) and Brandt
(1959,1963). This theory assumes that the diffusing molecules move through not only parallel to the chains (DiBenedetto and Paul model) but also across them when neighboring chains separate sufficiently to permit passage of the molecule (Brandt model) (see Figure
2.2). These parallel jumps occur in series with perpendicular jumps which require more energy and are therefore taken as rate limiting.
1.3. Free volume theory
Contrary to molecular models, the free-volume theory is based on an oversimplified view of molecular processes. This theory relates the diffusion coefficient to the free volume of the system on the basis of fluctuation analysis. The development of free-volume model was motivated by the interesting experimental observation of Batschinski (1913) that the fluidity of many liquids increases linearly with free volume. Fox and Flory (1950,1954) first made a critical contribution with their supposition that the glass transition can be attributed to the falling of the free volume below some critical value.
Doolittle (1951) found that the fluidity, of many simple hydrocarbon liquids could be represented by a simple exponential form of free volume, :
(if = (a) (b) Figure 2.2. Diffusion mechanism according to the molecular model suggested by Pace and Datyner (1979) (a) Proposed polymer microstructure with locally parallel chains and 4 coordination (b) Two possible motions of a spherical penetrant relative to this local structure. 16 where b is a constant of order unity. The free volume in a liquid was considered as the space seemingly arising from the total thermal expansion of the liquid without change of phase. Relative free space, /, is therefore defined as the fractional increase in volume resulting from expansion: y , Vo ® where is the volume of free space per gram of liquid at any temperature, and is the volume of liquid extrapolated to absolute zero, and V is the volume of liquid at any temperature. Williams, Landel, and Ferry (WLF) (1955) derived a universal function for temperature dependence of viscosity and mechanical relaxation in amorphous polymers and other supercooled, glass forming liquids. They showed that Doolittle's free-space equation is valid for a large number of glass formers via their (WLF) universal function. They also proposed a description of the free volume by the simple correlation: [0.025 + a ( r - r p ] where is the volume of the glass at and a = a, - is the difference between the thermal expansion coefficients of the liquid and the glass. The WLF universal function implies that rate processes involving molecular rearrangements depend on temperature primarily through their dependence on free volume. Cohen and Turnbull ( 1959) later provided a theoretical basis for Doolittle-WLF free- volume theory by introducing the principal idea that molecular transport occurs by 17 movement of molecules into voids of a size greater than some critical value, which are formed by the redistribution of the free volume. The idea is analogous to that of molecular cooperation suggested earlier by Buche (1953) and Barrer (1957), but different in that diffusion occurs not as a result of an activation in the ordinary sense, but rather as a result of redistribution of the free volume within the liquid. They finally obtained the equation for the diffusion coefficient: D =ga(D [fexp(- 0 (10) where a{V*) is roughly the diameter of the critical cage volume, V*, and g is a geometric factor, and C7 is a kinetic velocity, and 0 is an overlap factor ranging between 1/2 and 1. They further assumed that the l^may be expressed by an approximate expression: V,‘ CV,(T-T^ (11) where is the temperature at which the free volume disappears, and a, are the mean values of the coefficient of thermal expansion and molecular volume over temperature range of interest, respectively. The free-volume model was originally derived to explain the temperature dependence of the viscosity, but it has a much broader application to many physical processes. In the most widely used free volume treatment, Fujita and coworkers (1961) employed the WLF modification of the Doolittle equation to the diffusion process in the polymer. They considered that the mobility of small molecules in polymer - diluent mixtures should also be controlled by the free volume of the system and derived an equation for the diffusion 18 coefficient as a function of free volume: D = /? r4 ^ e x p (-fi^ //) (12) where Aj is a parameter which is dependent on the size and shape of the penetrating solvents, and Bj is a parameter characterizing the efficiency of use of the available free volume fraction,/, as defined by eq.(8). For relatively low penetrant concentrations (1-1.5 vol%) and constant temperature, /c a n be expressed by: / =/o + Y/* V (13) where Y/ is the free-volume sensitivity and v is the volume fraction of the penetrant in the polymer. Eq.(12) not only fits well the experimental data but also gives interesting insight into the mechanism which is responsible for the pronounced concentration- dependence of diffusion coefficients in polymer-solvent systems. The free-volume model of Fujita has been modified into several useful forms. Fang and Stem (1975) developed a more general expression for the permeability coefficient of liquids and gases through polymeric membranes. Vrentas and Duda (1977, 1982, 1992) have developed a more complex free-volume model with predictive capabilities for the determination of polymer-solvent diffusion coefficients (see Figure 2.3). Vrentas and Duda (1978) also proposed an expression for the temperature dependence of the mutual diffusion coefficient at zero penetrant concentration, and extended their model to describe the effect of the glass transition on the diffusion process. 19 HOLE FREE VOLUME EXTRA HOLE FREE FOR EQUILIBRIUM VOLUME FOR GLASS LIQUID ^ CO interIstitial FREE 'VOLUME OCCUPIED I VOLUME TEMPERATURE Figure 23. Volume-temperature behavior of an amorphous polymer suggested by Vrentas and Duda (1977): (A) Volume of equilibrium liquid, (B) Volume of non-equilibrium liquid or glass. (C) Sum of occupied volume and interstitial free volume, (D) Occupied volume. 2 0 2. Solubility 2.1. Energetic viewpoint The fundamental question underlying the phenomena of solubility is the nature and strength of intermolecular forces between solute and solvent. From a simplistic picture of intermolecular forces, a chemical will be a solvent for another material if the two molecules are compatible in the sense that the force of attraction between molecules is not less than the forces of attraction between two like molecules of either species (see Figure 2.4). Hildebrand (1919) used the latent internal energy of vaporization (d£^,^^) to provide a useful basis for some suitable measure of the attraction force holding molecules together. The theory is based on the concept of “regular solution” with an ideal entropy of mixing and a non-ideal enthalpy of mixing. The solubility parameter of a solvent, 5, was defined by the expression: 8 = (14) where is the molar volume of the solvent. This theory was initially developed for mixing of nonpolar substances in which solute and solvent do not form specific interactions. However, many of the solvents and polymers in common use are polar and form specific interactions. Later, Hansen (1969) adopted the regular solution theory to polar substances by including a polar part divided into a dipole - dipole contribution (p) and a hydrogen bonding contribution (h), and named it a cohesion parameter. 21 ^ Type A Figure 2.4. Schematic representation of compatible and incompatible systems (Brydson, 1975): (a)F^apAA, Fab^Fbb: mixture compatible, (b) or Fgg>F^B: molecules separate (F^g = Average force of attraction between dissimilar molecules; F^^, Fgg = Average force of attraction between like molecules of either species). ^^yap = (15) Combining the above equations gives: = s / - 8 / - i } <«> The solubility parameter b may be thought of as a vector in a three-dimensional ôp,ôj, 6^ space. A solvent, therefore, with given values of <5^, and 6^ is represented as a point in space, with 5 being the vector from the origin to this point. A material with a high Ô value requires more energy for dispersal than is gained by mixing it with another material having a low cohesion parameter, so immiscibility results. On the other hand, two materials with similar b values gain sufficient energy on mutual dispersion to yield perfect mixing. Solubility parameters never provide the complete answer to specific interactions between polymers and solvents, and only the enthalpy term (Aff) is considered in the solubility parameter theory even though the entropy of mixing (AS) should be considered to describe the miscibility. Despite that, solubility-parameter approaches can be good guides and have been used widely. 2.2. Thermodynamic sense The Gibbs free energy is very useful to determine the available work when the heat absorbed, TAS, exceeds the change in enthalpy, e.g., negative AG value. If applied to the isothermal mixing of molecules, the equation indicates that mixing will occur if 23 745 is greater than AH. AG = AH -7AS where 4 G is the change in Gibbs free energy in the process. Only if 4 G is negative will the solution process be thermodynamically feasible. The change in enthalpy may be positive or negative. A positive AH means that the solvent and polymer prefer their own company and solution is endothermie, that is, the pure materials are in a lower energy state, while a negative AH indicates that the solution is in the lower energy state. Negative AH (exothermic) occurs where specific interactions such as hydrogen bonds are formed between the solvent and polymer molecules. When mixing takes place, it is to be expected that the distance between polymer molecules will increase, and that will increase the movement of the polymer molecules and thus increase their degree of freedom or disorder. This means that such a mixing process is bound to cause an increase in entropy (45 > 0). A consequence of this is that the term 745 will be positive during mixing and therefore solvation will occur if the heat of mixing, AH, is zero or at least less than 745. It has been shown by Hildebrand (1950) that in the absence of specific interactions: \2 (1 8 ) 24 where ôp are identical, then AH will be zero, and so 4 G is bound to be negative, and the compounds will mix. The temperature and concentration dependence of polymer-solvent interactions are largely associated with the entropy term in the Gibbs free energy criterion for solubility, TAS 2: AH. Thus, increasing the entropy change by raising the solvent concentration and temperature can permit a greater tolerance in AH (e.g. dilute polymer solutions are more favored thermodynamically than concentrated ones). A polymer property which depends on both the solubility and diffusivity is the permeability of gases, liquids, and solutes through membranes. Closely related to the solubility of a polymer in a solvent is the corresponding solubility of the solvent in the polymer with resulting polymer swelling. The degree of swelling of the polymer is a maximum when the solubility parameters match. Hildebrand polymer parameter ranges thus may be determined experimentally by observation of degrees of swelling in a “spectrum” of solvents with known Hildebrand parameters. In addition, the electronic chemical structure of the polymer may be elucidated experimentally through the wise choice of a spectrum of solvents with known solubility parameters, e.g., cohesion parameters combined with strong and moderate hydrogen bonding capability. This approach was discussed in the context of the selection of organic solvents (in Section IV.A.) and in the context of interpretation of experimental results (in Section V.B.). 25 3. Physicochemical nature of the components In a polymer above its glass transition temperature (T^), as in simple liquids, fluctuations in density are constantly disappearing and reforming as a result of thermal fluctuations. Diffusive motion thus depends on the relative mobilities of penetrant molecules and polymeric chains as they are affected by size, shape, concentration, component interactions, temperature, and other factors which can affect polymeric segmental mobility (Rogers, 1985). Below the segmental motion is almost frozen owing to the lack of sufficiently large holes which can accommodate large-scale segmental motion. Instead, the extra hole free volume is effectively localized (or frozen) into the polymer. Therefore it is reasonable to expect that only small-scale motion is possible in the glassy state (Vrentas, 1978; Anet and Bourn, 1967). 3.1. Geometric effects of components An increase in the size of a penetrant in a series of chemically similar molecules generally leads to an increase in solubility and a decrease in diffusion coefficient. The increase in solubility is directly related to the dependence of AH^ on AH^^^ (heat of condensation) and AH^^^ (heat of mixing). In the absence of specific interactions, values of AH^.^ are small and positive as shown in eq.(18). For more condensible substances (e.g. large size of molecules), AH^ becomes negative due to the contribution of The decrease in diffusivity is a reflection of a need to create large holes corresponding to penetrant molecules (see Figure 2.5). Since the permeability coefficient is the product of these, its variation with penetrant size is relatively less affected. It was also 26 -5 \j-H* 6 \V h, * -7— Ne •^T"02 -8 -9 A r-^ A K r-I \ % ‘ fO \ I— CHjOH o» o \ O -(f \ \r-H2C*CHCI -12 \ r(CH3)2C0 CjHjOH \ I p n-C4ri|o -/3 \ 1 pn-C^HgOH "“C3 H7 OH--J 1 r - " " ^ 5 H, 2 -74 CçHç—^ n-CgH,4 -75 -76 1 J J _____ 1 1 1 1 I I .02 .04 .06 .08 .10 .12 .74 .16 .18 by liter/mol Figure 2.5. Diffusivities of gases and vapors in poly(vinyl chloride) at 25 °C as a function of their van der Waals volume, b: (a)0:Data of Berens and Hopfenberg (1982), (b)®-Data used by those authors from Tikhomirov et al. (1968) (Reproduced by Rogers (1985)). 27 found that the diffusivities of linear molecules were much higher than those of spherical molecules of similar volume or molecular weight. That implies that linear molecules move faster along their long dimension during diffusion than spherical molecules (Berens and Hopfenberg, 1982). With other factors equal, the permeation rate of penetrants can be expected to decrease as the structural symmetry and cohesive energy of the polymer increase. In a comparison of polymers such as polyethylene and polybutadiene, which have similar cohesive energies, the latter is found to be considerably looser in structure due to lack of symmetry along the chain, which leads to much larger diffusion coefficients. Again, in a comparison of two symmetrical polymers, such as polyvinylidene chloride and polybutadiene, the former is found to be more polar, which leads to much lower diffusion coefficients due to its higher cohesive energy (Rogers, 1985). The restraints of crosslinking on the segmental mobility of the polymer make the diffusion process more dependent on the size and shape of the penetrant molecules. Diffusion coefficients decrease approximately linearly with crosslink density between low and moderate degrees of crosslinking. At higher densities the decrease becomes nonlinear (Crank and Park, 1968). The presence of crystalline domains in the polymer has at least two effects on sorption and diffusion behavior. At temperatures below the melting point, crystalline regions are generally inaccessible to most penetrants. Hence, they act as excluded volumes for the sorption process and as impermeable barriers for the diffusion process (Sovolev et al.. 1957). Another effect on sorption and diffusion processes is to decrease 28 chain-segmental mobility in the amorphous phase by acting as giant crosslinking regions (Michaels et al„ 1959, 1961). 3.2. Electronic factors affecting to penetrants and polymers Chemicals that readily condense will permeate at higher rates than counterparts having similar sizes and properties that do not condense, as the same reason previously discussed in section n.A.3.1. Good solvents swell and plasticize the substrate, which leads to increased mobility of both polymer chains and penetrants. So the permeation of vapors and liquids into the polymer membrane proceeds faster than that of gases. A given penetrant will be most soluble in a polymer having similar functional groups or a similar degree of polarity. For example, nonpolar penetrant can permeate nonpolar semicrystalline polymer more rapidly rather than polar semicrystalline polymer. In most cases, nonlinear diffusion is induced by complex polymer-penetrant interaction which usually leads to structural deformation of the polymer. The permeation of water vapor varies greatly from polymer to polymer, as shown in Table 2.1, depending on the type and degree of specific interaction that is possible between the water and certain polar substituents (e.g. hydroxyl group) on the polymer chains. 29 Table 2.1 Permeability of polymer films to water vapor at 25 °C (Rogers, 1985). (cm^ at STP)(mm x 10**) P— Film I — (cm“s)(cmHg) Poly(vinylidene chloride) (Saran) 0.3-1.0 Polytetrafluoroethylene (Teflon) 0.3 Polyethylene (density 0.960) 1.2 Polyethylene (density 0.938) 2.5 Polyethylene (density 0.922) 9.0 Polypropylene (density 0.907) 5.1 Poly(vinyl chloride) 6.1 Polystyrene 12 Polybutadiene 47 Polyisoprene (natural rubber) 30 Polyamide (nylon 66) 68 Cellulose acetate 550 Cellulose acetate (15% dibutyl phthalate) 740 Ethyl cellulose 1300 Poly(vinyl alcohol) 4200 Hydroxylated polymers such as cellulose, poly(vinyl alcohol), and poly(vinyl acetals) exhibit high water absorption (P>500). Hydrocarbon polymers such as polyolefins and polystyrene have low water absorption (0.3 (40 B. FICKIAN DIFFUSION 1. Diffusion process Diffusion is the process by which matter is transported from one part of a system to another as a result of random molecular motions, e.g., having no preferred direction of motion. In a macroscopic viewpoint, random molecular motions can result in a net transfer of molecules from the region of higher to that of lower concentration. This was recognized by Pick (1855), who first put diffusion on a quantitative basis by adopting the mathematical equation of heat conduction derived some years earlier by Fourier (1822). Neglecting conveyance. Pick's first law states that the flux in the x- direction, , is proportional to the concentration gradient: K = (19) where flux is the amount of substance diffusing across unit area per unit time, and D is called the diffusion coefficient. The first law can only be directly applied to diffusion in the steady state, that is, where concentration is not varying with time. Pick's second law of diffusion describes the non-steady state. By considering mass balance on a finite volume element, the Pick's second law can be described by eq.(20), which neglects conveyance and chemical reactions, and assumes no density change, and constant temperature, and no external forces (e.g. electrical or gravitational). = —(D—) + — (D— ) + — (D— ) (20) dt dx dy V dz dz where D is defined as an effective diffusivity. In many systems, e.g., the inter-diffusion of 31 metals or the diffusion of organic vapors in high-polymer substances, D depends on the concentration of the diffusing substance. Also when the medium is not homogeneous, D varies from point to point. In general, Fickian diffusion is observed in polymeric solids well above or in glassy systems when the penetrants are simple gases, solvents with small molecular diameters, or partial solvents at very low temperatures and penetrant activities. 32 2. General solutions General solutions of the diffusion equation can be obtained via Laplace transforms, Boltzmann transformation, or separation of variables, for a variety of initial and boundary conditions, provided the diffusion coefficient is constant. Such solutions are suitable either for evaluation of diffusion behavior at non-steady state (Initial value problem) or for evaluation of diffusion behavior at steady state (Boundary value problem). In this section, only one-dimensional diffusion into a plain sheet of medium will be considered, and the thickness of film is so thin that effectively all the diffusing substances enter through plane faces and a negligible amount through the edges. 2.1. Initial value problem The transient distribution of a diffusant in a given film during sorption is governed by the one-dimensional differential equation for diffusion due to Pick, with the space coordinate taken in the direction of the film thickness. For the one-dimensional case, the diffusion equation (20) reduces to; The solution to the differential equation depends on the initial (I.C.) and boundary conditions (B.C.) taken for the diffusant concentration as well as on the nature of the diffusion coefficient. 33 B.C. : c=c^, x=0, fzO c=c#x=l, nO I.C. : c=Jix), Q If the surface concentrations are constant and initial distribution is given by fix), the solution in the form of a trigonometric series is obtained either by the method of Laplace transformation or separation of variables (Crank, 1975). 7 exp ^ I t: I n ^ I (22) exp ff[u) s in ^ ^ d u h i 0 ^ In most common occurrences, f(x) is either zero or constant so that integral is readily evaluated. Very often the problem is symmetrical about the center plane of the sheet, and the formulae are most convenient if the center and surface are taken as x = 0, ± /. Uniform initial concentration. Surface concentration equal This is the usual case of sorption or desorption by a membrane. If the region -I < X < lis initially at a uniform concentration Cg, and the surfaces are kept at a constant concentration Cj, the solution becomes (Crank, 1975): _c-c, c 34 If M, denotes the total amount of diffusing substance which has entered the sheet at time t, and denotes the corresponding quantity after infinite time, then: \ ^exp(-D(2n+l)^7C^/4/^) (24) M . n-c(2n+iyn^ Theoretical results of the time-dependent diffusion equation provide a simple expression for the linear behavior at small values of given by initial slope, from which the diffusion coefficient can be evaluated provided the thickness of the membrane I is known. An alternative evaluation method is a half-time method. D - ° ^ m T,/, is the time where M^/M = 0.5 and is called the half time. A third method of evaluating Fickian diffusion data is based on the so-called late time approximation. (27) For later stages of diffusion, the plot of ln(M,- M) vs. t approaches a straight line with a slope given by: 35 |[ln(M.-M,)] = (28) Some approximations can be applied to evaluate diffusivity if the diffusion coefficient varies with concentration. For example, the value of D obtained from the initial portion of sorption data represents the diffusivity at the lower initial solid phase concentration, while analysis of desorption data yields the diffusivity at the higher initial solid phase concentration. The average diffusion coefficient can be approximated by averaging from the initial gradient of uptake and from the gradient of release. Uniform initial distribution, surface concentrations different This is the case of permeation through a membrane . If one face x = 0 of the membrane is kept at a constant concentration c, and the other x = late,, and the membrane is initially at a uniform concentration Cg , the concentration change can be expressed as follows. The rate at which the diffusing substance emerges from unit area of the face x = / of the membrane is given by = -D(dc/dX)^^,. In most practical cases, the membrane is initially at zero concentration, and the concentration at the face through which diffusing substance emerges is maintained effectively at zero concentration (e.g. both c„ and c, are 36 zero). By integrating them with respect to r, from zero to infinity, the total amount of diffusing substance, g , , approaches the line: (30) Dc, ;2 As r-«o, This has an intercept L on the t-axis given by L = / 76D. The intercept, L, is referred to as the “time lag.” 2.2. Boundary value problem Constant diffusion coefficient Consider the case of diffusion through the membrane of thickness I whose surface concentration at x = 0, x = / are maintained at constant c,, c, values, respectively. After a time, steady state is reached in which the concentration remains constant at all points of the sheet. The diffusion equation in one dimension then reduces to, with a constant diffusion coefficient: d^c _ On integrating with respect to x, from 0 to /, one gets: X (32) Cj-Cj I The rate of transfer of the diffusing substance is the same across all sections of the membrane and is given by: 37 ^ (3 3 ) In some practical systems, the surface concentrations {c, and c, ) may not be known but only the gas or vapor pressure p,, and partial pressure p, on the two sides of the membrane. The rate of transfer at steady state is then written: = ^ (34) and the constant P is referred to as the permeability constant. If the diffusion coefficient is constant, and if the sorption isotherm is linear, the permeability coefficient is given by eq.(I), where S is described as S = c/p by the linear isotherm relation. Concentration-dependent diffusion coeffîcient If the diffusion coefficient varies with concentration, it is clear that the simple value of D deduced from the measurement of the steady rate of flow is some kind of mean value over the range of concentration involved. % Dj=—— fodc ‘■1 Integrating between c, and c,, the two surface concentrations, we have The usual approach to the boundary value problem is to guess a functional form for D{c) and employ sufficient flux and time lag values at different concentration to obtain the 38 parameters in D(c) (Frish, 1957; Crank and Park, 1968). jD(c) =Z)oexp(YflC) (37) ^ 4 exp(Yj c^) - 1 + 6xp(2 Yj (2Yg -3) (39) (Gxp(Y/)C^)-l)^ where Z)q and Yd constants, and L is time lag. This approach is discussed in Section V.C.2. in the context of data analysis. 2.3. Fickian characteristics The isothermal, Fickian diffusion of penetrants into polymers can be classified into at least two classes (Frish, 1965). Type is a Fickian diffusion observed with very small penetrant molecules, exhibiting: (1) negligible departures from Henry's law of solubility, (2) a concentration-independent diffusion coefficient, and (3) an apparent temperature-independent activation energy for diffusion (£^). Type B is a still Fickian diffusion observed generally with larger organic vapor penetrants such as benzene, exhibiting: (1) nonlinear solubility, (2) concentration-dependent diffusion coefficients, and (3) temperature-dependent activation energy for diffusion. The plot of ln(D) vs. MT is linear for Type A diffusion, while in contrast the same plot for Type B diffusion exhibits a significant downward curvature, reminiscent of the Doolittle-type temperature dependence of fluidity. Sorption curves having characteristics expected from the above-mentioned 39 assumptions for diffusivity and solubility are customarily called Fickian. The following is a summary of important Fickian sorption features that have been made available by the mathematical studies of the diffusion equation (Crank and Park, 1968). (a) Both sorption and desorption curves (Af, /M_ vs. are linear in the initial stage. For sorption, a linear region extends over 60% or more of the equilibrium amount of sorption, When D(c) increases with c, the sorption curve is linear almost up to sorption equilibrium. (b) Beyond the linear portion, both sorption and desorption curves are concave to the time axis irrespective of the form of D(c). » (c) A series of sorption and desorption curves for films of different thicknesses are superposable to a single curve if each curve is replotted in the form of a reduced curve, e.g., is plotted against t'^U. (d) The reduced sorption curve always lies above the corresponding reduced desorption curve if D exhibits an increasing function of c. The divergence of the two curves becomes more marked as D increases more strongly with c in the concentration range considered. Criteria (a), (b), and (c) are independent of the form of D as a function of c. In practical sense, a given system is often regarded as Fickian if experimentally determined sorption curves have appearances consistent with criteria (a) and (b). 40 C. NON-FICKIAN DIFFUSION The term non-Fickian has been employed to indicate the diffusion process deviated from Fickian transport characteristics. Non-Fickian transport commonly results from nonlinear dependence of transport properties on structural changes associated with the glass transition. A new term “structural diffusivity” was devised just to express the characteristic of diffusivity in polymeric system, in which a small degree of structural change can induce a drastic change of diffusivity. This behavior has been previously observed in both permeation experiments and transient sorption experiments, as described below. 1. Permeation anomalies A number of permeation anomalies have been reported in the literature and are shown in Figure 2.6. (a) Anomalous shaped permeation-rate curves: An initially high permeation rate of methylene chloride into polystyrene was observed and then decreased to a final steady state value (Park, 1951). Another case observed by Mears (1958) was an initially high flux of allyl chloride through a poly(vinyl acetate) membrane at 40 °C (-10°C above its T^. The initial permeation rate decayed relatively quickly but then increased to a steady-state value. Mears suggested that the initial high flux was due to the stress induced by the “vectored" swelling of the polymer, which caused an increase in D. This stress decayed rapidly because the polymer was 41 t t Figure 2.6. Anomalous shaped permeation-rate curves: (a) Types observed by Park (1951 ) for methylene chloride in polystyrene at 25 °C, (b) Type observed by Mears (1958) for allyl chloride in poly(vinyl acetate) at 40 °C. 42 in the rubbery state, and D returned to its characteristic value for the system at the experimental temperature. (b) Permeation time lag: The permeation time lag (L) sometimes cannot be predicted from the diffusion coefficient from steady-state permeation measurements. This may be due to a history (time)-dependence of the diffusion coefficient. In addition, the time lag is not scaled by the square of the membrane thickness I. (c) Multiple mechanisms-. Transport may occur by multiple, concurrent mechanisms due to the presence of capillaries, internal pores, micro cracks, or other interconnected defects in the polymer. 43 2. Sorption Anomalies A variety of sorption anomalies have been reported and classified empirically; pseudo-Fickian, sigmoidal, two-stage, overshoot, moving front, drastic acceleration, etc. (see Figures 2.7 and 2.8). These are briefly described here. Some are discussed in more detail later in the context of the experimental results in Section V.A. 1. (a) Pseudo-Fickian: The term pseudo-Fickian has been used to describe sorption curve that resembles Fickian curve but shows an unusually small initial linear region, e.g., significantly less than 50% linearity. The approach to final equilibrium is also very slow (Rogers, 1965). (b) Sigmoidal: Among anomalous sorption curves, this is most commonly observed. The sorption curve is sigmoidal in shape with a single inflection point often at about 50% of equilibrium sorption. The initial rate of desorption exceeds that of sorption but then becomes slower, and the curves cross (Park, 1951; Long and Kokes, 1953; Newns, 1956). (c) Two-stage: This sorption curve approaches a quasi-equilibrium after an initial rapid uptake which is a linear function of square root of time, followed by a slow approach to final equilibrium (Bagley and Long, 1955). (d) Overshoot: This sorption curve exhibits an over-saturation mass uptake before leveling off to an equilibrium concentration (Watt, 1960; Overbergh et al., 1975; Vrentas et al., 1984). 44 Fickian Pseudo-Fickian M, Sigmoid Tw o-stage f ’S -----^ Figure 2.7. Non-Fickian or anomalous soiption and desorption curves compared with Fickian-type curves (Rogers. 1965) (S: Sorption, D: Desorption). 45 1.2 1.0 0.8 (a) 0.2 - % B 20 o g o c (b) 3 1 10 o en o> I 2 0 30 Time (days) Figure 2.8. Other anomalous sorption curves: (a) Overshoot of equilibrium sorption in n-hexane-polyethylene system at 0 °C (Rogers, 1965), (b) Moving front: Linear weight gain as a function of time for 1 mm PMMA sheet swollen in methanol (Thomas and Windle, 1980) (O: 15°C, • : 10 °C). 46 1.0 0.8 0.6 8 (C) 0.4 n-Hexane P/Pq = 0.775 T = 30°C 0.2 1% Mil Polystyrène Film 200 400 600 800 1000 1200 Time (hours) Figure 2.8. (continued) (c) Sorption plot for the n-hexane vapor (activity = 0.775) - polystylene film system which shows drastically accelerating sorption behavior (Jacques et al.. 1974). 47 (e) M ovins front: The uptake rate is linearly proportional to the elapsed time during sorption. An initial induction period is sometimes observed (Alfrey, 1965; Thomas and Windle, 1980). (0 Drastic acceleration: The final approach to sorption equilibrium is so rapid that asymptotically slow approach to equilibrium is not observed (Jacques et al., 1974). These sorption anomalies have been usually interpreted based on phenomenological mechanisms and system conditions. Pseudo-Fickian diffusion was generally observed in differential sorption experiments, and it seems to reflect a transition stage, through which distinct sorption characteristics appear: sigmoidal, two-stage, and Fickian. Sigmoidal and two-stage anomalies have been observed in many systems irrespective of differential or integral soiption. Long and Richman (1960) first confirmed the change of surface concentration experimentally and then interpreted these via first- order surface relaxation kinetics. Overshoot phenomena are usually observed in sorption experiments involving solvent crystallization. The crystallized domain induced by imbibed solvent adversely affects the sorption process and eventually repels the sorbed material. Overshoot also occurs in sorption experiments involving a slow reordering relaxation process that might give rise to development of dispersed microvoids within the unrelaxed film core which "heal" after equilibrium is attained. 48 Linear sorption kinetics have been known to accompany a sharp moving boundary separating an outer swollen shell from an unpenetrated glassy core. Thus, two different phases exist during diffusion process. In general, the sorption capacity of the swollen shell is much larger than that of the glassy core, enough to neglect the contribution from the penetration of solvent (so-called precursor) into the glassy region. Furthermore, the diffusion coefficient in the glassy region is much smaller than that in the swollen region and is usually taken to be negligible. Acceleration features observed at the final stage of soiption have been interpreted as another outcome of linear sorption kinetics. A possible outcome envisaged by Jacques et al. (1974) is that more rapid relaxation in the later stages of sorption is induced by the overlap of precursors ahead of moving fronts and leads to an acceleration of the fronts. On the contrary, Gostoli and Sarti (1983) hypothesized “differential swelling stress” as the primary explanation. Namely, the loss of differential swelling stress in the swollen region as the fronts meet, can lead to an increase in the equilibrium sorption caused by the increase of specimen dimension. This explanation is supported by the observations of Thomas and Windle (1977,1981), in which the equilibrium volume of PMMA specimen was increased when the fronts met. Based on experimental observations and interpretations, some phenomenological terms have been incorporated into the respective mathematical models in order to get a satisfactory description of the non-Fickian diffusion behavior. These terms are: 49 (a) “Swelling stress (internal stress or differential stress)” related to diffusivity, solubility, chemical potential, convective contribution, strain, morphological changes such as crazing, cracking, formation of the microvoids and phase transition (Alfrey et al., 1966; Petropoulos et al., 1974, 1984 (a, b); Sarti, 1979; Neogi et al., 1986; Thomas and Windle, 1982). (b) “Chemical or physical interactions” related to immobilization, dissolution, or plasticization (Michaels et al., 1963). (c) “Processing properties” related to surface skin, orientation, or heterogeneity (e.g., crystallinity, blending) (Eby, 1964; Taraiya et al., 1993). (d) “Viscoelastic responses” related to diffusivity, solubility, diffusional flux or other convective contributions (Crank, 1953; Long and Richman, 1960; Neogi, 1983; Camera-Roda and Sarti, 1986,1990; Doghieri et al., 1993). Indeed, some parameters have proved to be successful in accommodating several non-Fickian features. Crank (1975) suggested a synthesized model including the combined effects of history, orientation, and stress on the diffusion process and surface concentration within the polymer system. Such a composite model may predict diffusion behavior under given conditions and be useful for the interpretation of experimental data. Such models, however, still appear to suffer in presenting comprehensive and perspective explanation on non-Fickian behavior, and are limited in the sense that only a few factors among so many terms are considered depending on the experimental conditions. 50 Thus, a comprehensive macroscopic or complete microscopic model should be constructed to get full insight on complex transport processes, but that is well beyond the scope of this thesis. A simplified view may, however, give some partial answers. For example, the most direct way to do this is to model how relaxational effects due to the glass transition can affect the transport boundary value problems, although they lack a molecular picture to this complex transport phenomena. In the following sections, currently known mathematical models are reviewed according to the conceptual development, and a simplified kinetic model is suggested. 51 3. Prior mathematical models Diffusion behavior of certain organic liquids in many glassy polymers cannot be described adequately by a concentration dependent form of Pick's second law with constant boundary conditions, especially when mass transfer is coupled with structural changes associated with the glass transition, induced by the plasticizing effects of solvent. Such structural changes as swelling, microcavity formation, phase transition, etc. require the rearrangement of polymeric segments, and their kinetics are dominated by relaxation phenomena. In formulation of non-Fickian diffusion mathematics, the most important factor to consider is how relaxation effects can influence the governing constitutive equation and boundary conditions. That is, relaxation parameters can be accommodated by variable boundary conditions or a modified continuity equation, or both, depending on specific systems and conditions (Frish, 1980). Three basic classes of non-Fickian behaviors can be distinguished depending on the relative contributions of diffusion and relaxation mechanisms: 1) Case I (which appears as Fickian diffusion): the rate of diffusion is much less than that of relaxation processes. 2) Case II (moving front): diffusion is very rapid compared with relaxation at the moving boundary. 3) Case III (non-Fickian or anomalous diffusion): diffusion and relaxation rates are comparable. 52 The three cases can be distinguished by the shape of their sorption curves. To a first approximation, the initial mass uptake during sorption can be represented as a following rate law. F = MJM, = kr (40) where k and n are constants. For a Fickian or Case I system, n = 0.5 and k is proportional to the diffusion coefficient over the initial half of the sorption experiment. For Case II, n = 1.0 and k is directly proportional to the velocity of moving front. For Case HI, n usually lies between 0.5 and 1.0. Based on the observed phenomena, the model equations can be broadly categorized as continuous or discontinuous. Case II sorption is classified as discontinuous and Case III can be classified as continuous or discontinuous, depending on the observed phenomena and basic assumptions of model equations. Continuous model equations encompass phenomena where the structural change takes place gradually over the whole volume of the polymer sample. On the other hand, discontinuous model equations deal with the phenomena where the morphological change appears to be abrupt. 3.1. Model equations for continuous morphological change For continuous model equations, no sharp morphological discontinuity is observed, but significant anomalies may be exhibited. The main concept employed in continuous model equations is that diffusion process in a polymeric material is limited by the characteristic times of relaxation due to stress rearrangement and morphological 53 changes. Crank (1953) originally introduced the time- and concentration-dependence of diffusion coefficient by postulating that : { d D ! d t \ = {dDJdc){dc!dt\ + a(D„-Z)) 41 ( ) where Z),(c) is the instantaneous diffusion coefficient with concentration, which slowly relaxes to a final equilibrium value, D_, with a first-order rate coefficient, a(c). In other words, one part of the diffusion process is instantaneous and the other part proceeds slowly. The instantaneous part accompanies a change in the concentration of the diffusing substance. The slow part takes place even when the concentration is not changing significantly. With a suitable choice of parameters, it was possible to predict many features of unusual sorption behaviors. This was, however, mostly data fitting, and it did not help elucidate the underlying molecular mechanisms. More recently Cohen (1983,1984) has emphasized the importance of relaxation and suggested that Crank's relaxation mechanism be replaced by: (3D/30, = (3D, I d c ) ( 3 c l 3 t \ + a ( D . - D f Clearly eq.(41) and (42) are special examples of how relaxation effects can enter phenomenologically into the diffusion coefficient. However, it is still unclear in the sense of separating the kinetic property, D, into individually self-consistent diffusivities such as an instantaneous diffusivity and a slowly changing diffusivity. 54 Long and Richman (1960) provided an important link between sorption and molecular relaxation processes. They suggested that the surface concentration, c, , associated with molecular relaxation, can be expressed as the sum of two parts: one representing initial instantaneous quasi-equilibrium, and the other representing a slower relaxation-induced response leading to final equilibrium: + (1 -exp[-pf]) (43) where c, is the initial surface concentration, and c_ is the equilibrium surface concentration, and p is a rate parameter which is assumed to be constant with respect to time and concentration. The time-dependent surface concentration was incorporated into the boundary condition of the diffusive flux equation while employing a constant diffusion coefficient. It successfully described two-stage sorption behavior. This model is discussed in Section V.C.l. in the context of data analysis. In one way the model is unsatisfactory in that it is difficult to see why a relaxation process should only apply to the surface. However, the contribution is that it introduces the idea of a relaxation control on concentration, rather than on the diffusion coefficient as in Crank's work. To get more plausible predictions, a diffusion coefficient that may vary with concentration and time should be incorporated in the diffusive flux equation. In 1978, Berens and Hopfenberg described Case in diffusion behavior by considering individual contributions of Fickian diffusion and relaxation. Their model assumes linear superposition of a first order relaxation term upon the ideal Fickian diffusion equation. It can be written as: 55 (44) " 8 ^ ^ —— -r-T Gxp(-D(2M + 1)2 %2f /4/2)] (45) «=0 (2«+l)2jr2 = ^ ^ ,[ 1 -exp(-^.r)] f=l (46) where the fraction of uptake due to Fickian diffusion is I - a, and that due to relaxation is a; il/j and k, are coefficients accounting for relative amounts of uptake and relaxation rate constant in the “ i ” partitions of relaxation sorption, respectively. In principle for a polymer structure, there could be an infinite number of partitions, but in practice two relaxation terms have been found adequate to fit even the most complex experimental sorption data. Their simple approach was based on observation of sorption curves which consisted of quick Fickian diffusion followed by slow relaxation. Even though the parameters can provide quick and simple indications on the phenomenological trends of sorption curves, it still lacks physical meaning. Neogi (1983) introduced a viscoelastic diffusive flux to solve time-dependent diffusion in a relaxing medium with both time dependence of diffusivity and solubility. He utilized the same type of boundary condition as Long and Richman’s (1960). j's = - Jp ( t - 1' ) V c (X, tO d t/ (47) 0 p ( t) 5 (t) + ^ exp=^'^ (48 ) 56 where ô(t) is the Dirac delta function, D. and D_ are the initial and final diffusivities, and t is the relaxation time. In spite of concentration-independent diffusion coefficient, the theory can predict diverse behaviors that are seen experimentally, at least in the range of variables studied. Neogi's model allows for the transition from initial diffusivity, D ,, to final diffusivity, D_ , based on the elapsed time, irrespective of the concentration level or history which is contrary to the usual physical understanding. Camera-Roda and Sarti (1986, 1990) introduced a Maxwell-type viscoelastic constitutive equation for the diffusive mass flux incorporating both a concentration dependent relaxation time and diffusivity. To reduce the complexity due to coupling of stress, concentration, and chemical interactions, the relaxation properties were lumped into the non-Fickian diffusive flux term. The results obtained by numerically solving sorption and desorption problems indicate that the model appears to be able to describe, albeit qualitatively, all the Fickian and non-Fickian behaviors observed experimentally. Their relations are: (49) /p = V(|) (50) /g = (51) where the diffusive mass flux, J, is considered as the sum of two terms: Jp = Fickian term, and Jp = relaxation term. Even though the model equations involving relaxation terms described the observed phenomena well, they still had unresolved problems of how to correlate the relaxation parameters with molecular interactions and rearrangement. Actually, relaxation terms which involve time dependent processes are only another semi- 57 empirical expression of the process. 3.2. Model equations for discontinuous morphological change Discontinuous model equations deal with phenomena in which structural changes occur abruptly: these processes often accompany a sharp boundary separating two different morphologies. Alfrey et al. (1966) have defined an extreme case of this category as Case n, for which the major characteristics are: ( 1 ) A sharp boundary separating the inner glassy core of unpenetrated polymer from the swollen, penetrated rubbery shell. (2) The sharp front advances with constant velocity. (3) There is a linear weight gain with time. These observations require that diffusion of a penetrant through the swollen shell up to the moving front should be rapid compared with the swelling-induced relaxation at the front which controls the constant penetration rate. Crank (1951) first introduced a discontinuous diffusivity to predict a sharp advancing front. Though his model can predict the sharp advancing front, the initial predicted uptake was still found to be linear with the square root of time and then becomes concave to the time axis. The resulting deceleration of the advancing front indicates that the model is not relevant to Case II behavior. 58 Frish (1969) incorporated a convective term into the diffusion equation to describe the movement of a front with constant velocity, u: In this model. Case II transport is treated as a convection process which is assumed to be caused by internal stresses arising from swelling of the polymer. While this equation accurately represents the observed Case II sorption, it lacks a firm physical basis, e.g., by which to ascertain the velocity, u, from independent measurements or properties. The previous analysis was carried out by trying to formulate the bulk effects (e.g. concentration-dependent diffusivity, the gradient of internal stress, convective flux) in an appropriate way to fit the non-Fickian behavior. The physics of the phase transition taking place at the moving boundary were not taken into consideration. The first one who explicitly considered the two-phase nature of the phenomena was Peterlin (1969). According to his model, Fickian diffusion is assumed to occur ahead of the sharp moving front. The predicted concentration distribution consists of a step concentration profile, representing the swollen polymer, with a Fickian wave extending into the glassy core, as shown in Figure 2.9. Consequently, Fickian wave proceeds the moving front. With these assumptions, the initial uptake rate can be expressed as a linear combination of the moving front rate and the Fickian diffusion rate. His model can be written as: F = kpyfDt^k^t 5 9 Penetrant-Polymer Sorption Interface Discontinuity Penetrant (Liquid) Polymer ► x=vt Figure 2.9. Idealized concentration distribution proposed by Peterlin (1969) for concurrent Fickian and Case II sorption: where C, is the concentration of the saturated polymer behind the advancing boundary, and C„ is the critical concentration of the Fickian profile preceding this boundary. 60 where F is again the fractional mass change, and and kj, are constants representing the Fickian and Case II contributions, respectively. This model was successfully adapted to some experimental results (Wang et al., 1969; Kwei et al., 1972). Interpretation of the parameters in terms of physical or chemical properties would not be justified, however, because the basis of the theory is oversimplified. For instance, even the most basic terms describing geometry (such as thickness) and conditions (such as temperature) were not part of the model. Later, Astarita and Sarti (1978) approached the two-phase concept by taking account of the kinetics of the phase transition. They used the Fick’s second law for the swollen polymer with the following boundary conditions: Ê £ = ± { r ^ ) d t dx dx B.C. : x=0, c=Cq dx d t where c is the penetrant concentration at a distance from the solvent - polymer (external) interface and at time t; Cg is the equilibrium concentration of the penetrant; A(t) is the time-dependent distance of moving front from the solvent-polymer interface. A crucial point they made was that the phenomena of swelling is a kinetic one, the rate of which depends on how much the local activity of the solvent exceeds some threshold value below which no swelling at all would occur. The basic parameters of this model are the driving force for swelling and the type of swelling kinetics. The swelling kinetics occurring 61 at moving boundary was assumed to have a form of the power-type kinetics: where r is the rate of phase transition; c* is the concentration level corresponding to the threshold activity for swelling ; the rate of phase transition is governed by the local driving force, c - c*. The mathematical model predicts, at least qualitatively, all the essential features of the relaxation-controlled transport. Astarita and Joshi (1978) have extended this model to consider also a small amount of penetrant diffusion into the glassy region of the polymer ahead of the advancing gel/glass interface. Due to the oversimplification of the real phenomena, however, it is not well understood how to relate model parameters to the physical properties of both penetrant and polymer. One of the typical oversimplifications is that the kinetics of the advancing front were guessed as the power-type with no physical basis. Thomas and Windle (1982) have modelled the viscoelastic processes occurring at the moving front in terms of the viscous action of a swelling thin layer of glass to the osmotic pressure of the penetrant. According to their model (TW), the penetrant first diffuses into the previously existing interstitial sites, and the sorbed materials start to create the osmotic pressure after the empty space is completely filled. The osmotic pressure developed in the precursor of the front creates space for further penetrant molecules to enter the element. The next penetrant, in turn, enters more rapidly due to the increase of the viscous creep rate caused by the strongly concentration-dependent 62 viscosity. This “autocatalytic” response continues until the concentration in the element (e.g. swelling thin layer) reaches an equilibrium value. Of course, most of the build-up of penetrant in an element must occur much faster than the time needed to fill the next successive element, and the thickness of the swollen, glassy surface element should be thin enough for the diffusional resistance to be neglected. They treat the rate of change of volume fraction, <#>, of penetrant, as the rate of linear viscous creep deformation driven by the osmotic pressure, n: i * = JE. (55) dt q % = -(Ær/T^)ln(4)) n ='n«exp(-« where is the partial molar volume of a penetrant molecule, and ti is an elongational viscosity of the polymer and m is a material constant. The TW model was solved numerically, and it was found that it can accurately predict all the characteristics of Case II diffusion. Recent work emphasized improving the mathematical foundations and computational convenience of the TW model (Hui et al., 1987; Fu and Burning, 1993). Even though the TW model is well supported by phenomenological observations and thermodynamic sense of diffusion process, an avoidable inconsistency exists: for example, the incorporation of viscoelastic mechanical responses such as viscosity and osmotic pressure into the purely viscous constitutive equation. 63 D. KINETIC THEORY The development of polymeric barrier materials requires not only empirical approach but also quantitatively accurate description of the diffusion of small organic molecules in high polymer systems to simulate common phenomena such as sorption and permeation. Experiments to explain those phenomena show that the classical diffusion theory embodied by Pick's law cannot describe the process accurately in all cases. Instead, a confusing variety of non-Fickian behaviors appear which are explained qualitatively or with ad-hoc models. Considerable practical value lies in developing a generalized diffusion theory which can account for diverse types of non-Fickian behaviors quantitatively as well as qualitatively. The nature of rate processes, such as viscosity, diffusion, and mechanical relaxation, is essentially the same for all supercooled systems, polymeric, non-polymeric, and mixtures, independent of molecular structure. The rate of all such processes depends on measurable physical conditions (e.g. temperature, pressure, concentration) primarily through their dependence on free volume (Williams et al., 1955). In the derivation of a new kinetic model, a relaxation mechanism coupled with diffusion was expressed in terms of free- volume parameters suggested by Fujita (1961). As a result, some consistent and unified explanations were made possible. 1. Assumptions and preliminary information Glass transition phenomena are unequivocally known to be time dependent. The time dependence of glass transition phenomena provokes complex responses such as non 64 linear response, memory effects, and a complex relaxation time coefficient. Among kinetic theories of the glass transition, only a conceptual approach will be given below, which is based on the rate of change of normalized volume fraction, <5 = (V, - VJ/V,. In such equations, it is assumed that the rate of approach toward equilibrium, dôldî, is linearly proportional to the magnitude of the departure from equilibrium, 5, with constant relaxation time. The rate equation has a form of: = -Ô /T (58) d t where ô is a measure of the relative departure of specific volume from equilibrium, and T is the average relaxation time or the reciprocal of the rate constant, , for the volume change. Equation (58) is the well known equation for the isothermal decrease of volume which has been verified by Kovacs (1963) experimentally. It is clear that the approach toward equilibrium involves molecular motion, and hence the time scale of the response (or relaxation time) depends on the viscosity of the system. It is well known that the viscosity itself depends in a very sensitive way on the volume available to the system. The relaxation time thus must depend in a complicated way on the free volume in the system, e.g., t is a function of ô. If the same microscopic movement process is assumed to underlie self-diffusion, viscosity and structural relaxation, the relaxation time for structural change (e.g. volume change) can be expressed by the WLF-Doolittle type dependence on free volume. 65 T = T^exp(5^//) (59a) = (59b) where Bj is a constant of the order of unity a n d /is the fractional free volume, as defined in eq.(12). This supposition is consistent with existing theories, and a number of experimental results indicate that the activation enthalpy for volume relaxation is generally the same as the activation enthalpy for the viscosity of a given material (Matsuoka and Blair, 1977; Hutchinson and Kovacs, 1976; Grest and Cohen, 1980). The same analogy can be applied to the analysis of structural changes coupled with mass transfer in the polymer-penetrant system. In general, this structural change leads to an increase in volume via the creation of free volume in the material. Basically, the increase in free volume causes a deviation from Fickian diffusion by inducing time- dependent diffusivity and solubility in the system. It thus can be said that the diffusion process in polymeric materials is strongly dependent on the concentration of sorbed penetrants primarily through their dependence on free volume. 6 6 2. Development of model equations The anomalous sorption behavior often observed in the polymer-penetrant system can be interpreted as a contribution of relaxation effect in addition to Fickian diffusion. It is hypothesized that any structural changes (such as swelling, microcavity formation, primary and secondary phase transitions, breakage of physical and chemical bonds) could be measured as a volume change, AV, which is proportional to the amount of material sorbed. The fractional volume change is assumed to be equal to the fractional uptake (or release) due to relaxation effects, , as follows: F . Æ = L I î . = 2 . * AF. F.-F„ C,.-C,^ f.-f, V. w here/is the fractional free volume defined in eq.(8), and v is the volume fraction of the penetrant. The unit of concentration is defined as [grams/cm^ (the mass of the penetrant/ the volume of the polymer, )]. The net mass of penetrant taken up due to relaxation (e.g. in the free volume), A/^ , is proportional to Ç - Çq where the concentration of penetrant in the new (or created) free volume at a specific time, C j, augments that of penetrant in the original free volume, Çg, and it approaches A/jj. at equilibrium. It is evident that the rate of free volume change is the same as that of the specific volume (cf. eq(13)). 67 If the amount of sorbed material is low (e.g. less than 10-15 vol%), the volume fraction of penetrant can be slightly modified into a simple form below. ^VIVo L V UÙ.VIV0 K The amount of material sorbed, , thus can be expressed in terms of a volume fraction, V , as shown in eq.(60)). It is assumed that the rate of accumulation of the penetrant due to structural change (volume relaxation) is proportional to, , the fraction that is still unoccupied, via a pseudo-first order mechanism which is basically identical to equation (58): kf 1 - F ^ - ^ F ^ (62) (63) The rate coefficient, may depend on penetrant concentration because higher concentration of penetrant produce a faster rate of structural change. Finally, the total mass uptake at any time, t, can be split into fractions as done by Berens and Hopfenberg (1978) (cf. eq.(44)). These, again, represent both Fickian diffusion in the original volume, as 1 - a, and relaxation of the polymer structure to generate free volume, as a. The distinction between their approach and the present model lies in the definition of in that the uptake due to relaxation, , is defined quantitatively rather than qualitatively. Namely, it is defined in terms of measurable parameters such as volume or concentration in the present model, while their version was 6 8 inferred from sorption curves. The relaxation rate equation, given by eq.(63), is subtle because the rate coefficient depends on the fractional relaxation, That dependence is shown below to be exponential under one set of assumptions, and linear under another set of assumptions. That leads to an overall dependence of rate equation which is exponential or quadratic, respectively. 2.1. Exponential dependence of the rate equation First, a simple exponential function is proposed, relating the diffusion coefficient and rate constant to the penetrant volume fraction. D = D^exp(Y^v“) (64) ^/=*/oexp(YitV”0 (65) When above expressions are combined, the rate coefficient can be expressed as an exponential dependence on concentration of penetrant in the free volume. kf = kf^ exp( Y* (v . Fjj )” ) (66a) */ - */.«P ( Yt V { 1 ”(D ./Z ).)/Y i,)*) When M = m = 1, the rate coefficient reduces to; & = ^<,exp(^%) , where K ' = y^ v_ (67a) & = , ^kere K = y j (67b) For simplicity, n and m values are assumed to be unity for the remainder of this work. The 69 resulting rate equations are: — »^,exp(rF,Xl-F,) ^ R , JTP = V ' ( 1 -Fr ) (6 8 b) Even though the diffusion coefficient of the Fujita's free volume theory (cf. eq.(12)) instead of eq.(64) is inserted into the original rate equation (eq.(63)), the same expression can be obtained only differing in the context of the value of K. The lumped parameter, K, is more precisely described and reasonably understood in terms of free- volume parameters. kj = ^^exp( Yjt [ Fg MPJ (69a) I °d & = kfo (P J Po w = 1. (69b) y 2.2. Quadratic dependence of rate equation The goal of this approach is to express kj. and in terms of diffusivity ratios, then to integrate between known values of initial and final diffusivities in order to determine the inherent time dependence. The WLF-Doolittle dependence on both diffusion coefficient and rate coefficient is assumed here. D = RTAjGxp{-BJf) (12) = (59b) 70 If these two equations are combined, new expression can be derived as follows. kf = kf, e x p (-5 ^ //^ (Z)/ (70) Considering original free-volume effect on rate process, an alternate expression instead of eq.(59b) can be used to interpret the transient sorption of a solute in a polymer as follows: If that alternate rate expression is used, a more simple equation can be obtained. (72) kf-kf,{piD ^ ^ ^ The diffusivity can be reduced to a simple exponential form and further to linear form if the free volume increment is very small compared with the amount of initial free volume, via a Taylor's series expansion of eq.(12) (Kulkami and Stem, 1983). D = 2)^exp(Y^v), -where ^ fo D = D^(X+ YjV), where y^,v ^0.3 (74) In this case, a new relationship can be obtained in terms of diffusion coefficients, and the rate coefficient turns out be linearly dependent on the concentration of penetrant. While an exponential concentration dependence of D was observed with many organic vapor-polymer systems, a weaker linear concentration dependence was reported for some 71 systems such as benzene-rubber (Aitken and Barrer, 1955; Hayes and Park. 1955). When eq.(76) is inserted into the original rate equation (eq.(63)) and integrated, the following analytical equation is derived. 1 - exp(-^, er) (77) 72 3. The characteristics of the kinetic model Two different types of rate equations were derived in the previous sections: a quadratic dependence type and an exponential dependence type. The latter type can predict more drastic effect of free volume on sorption behavior. A differential integrator (DIVPAG) from the IMSL library was used to solve the initial-value problem for a stiff ordinary differential equation (e.g. exponential type of rate equation) using Gear's BDF method (Gear, 1971). This kinetic model can predict the various anomalies (such as sigmoidal, pseudo-Fickian, two-stage, and drastic accelerating sorption) with arbitrarily chosen values of diffusion and relaxation parameters. Kishimoto et al. (1960) performed extensive differential sorption measurements in a variety of binary polymer-penetrant systems, and discovered a characteristic sequence in the shape of the sorption curves with increasing the penetrant concentration (see Figure 2.10). The sequence was: sigmoidal - pseudo-Fickian - two-stage - pseudo-Fickian - Fickian '------Region I ------' '------Region II ------' This sequence also can be predicted by adjusting diffusion and relaxation parameters and is well explained within the framework of our kinetic model. The analytical relaxation equation (77), which is derived from the quadratic dependence of rate expression, is useful to predict the above sequential sorption curves. 73 _ 0 15 a" a+ 6 0 50t. 0 527 E>. Fickian 0 489 • 0-504 E O) Fickian 0 467 0 489 c pHu 4o Ficklon Region II 0-433 •0 467 1 010 1 pwuda Fkkion E ^o,oO»ooo - “ ” 0-414 •0 433 O) tw o i io g t 60 o -o- «r ■■ 0-376 • 0-414 c g tw o xioge 0- 339 — 0 376 — o o oooooooooooo o L. tw o stogo - t- > 0 0 5 0 301 - 0-339 C „ o < .o o o o oooooo (U tw o slogo u 0 248 - 0 301 c o pttudo Fickian 0 - 188 - • 0-248 u ...... ptoutfo Flekjan Region I 0 094 • 0-188 a o®** ^ ^^nooooo O O sigmoidal 0 - - 0094 sigm aidol O 10 20 30 4 0 (min ) Figure 2.10. The sequence of differential sorption in methyl acetate-Poly(methyl methacrylate) system at 30 °C (Kishimoto et al., I960) (Reproduced by Duming et al., 1990). 74 To be convenient, the sequence of differential sorption curves is divided into two regions such as I and II. According to the sequential sorption curves in region I, Fickian- diffusion fraction seems to be much more influenced by the amount of sorbed material than the relaxation-induced sorption. As the concentration is increased, the separation of the two mechanisms becomes more evident due to the shift of Fickian-diffusion fraction from relaxation-induced sorption and eventually leads to the two-stage sorption curve. The reason may be that Fickian diffusion can become much more prominent even by the increase of small-scale segmental motion due to the small change of penetrant concentration. On the contrary, the penetrant concentration in region I seems not to be enough to induce large-scale segmental motion which is mainly responsible for relaxation sorption. Table 2.2 The evaluated parameter values of quadratic type of rate equation to predict sequential differential sorption behavior. Regions Sequential Adiustable Parameters of Quadratic Rate Equation Types a K, K DID. Sigmoidal 0.7 0 . 0 1 0 . 0 1 4.0 I Pseudo-Fickian 0.7 0 . 1 0 . 0 1 2 . 0 Two-stage 0.7 0.5 0 . 0 1 2 . 0 Two-stage 0.7 1 . 0 0 . 0 2 2 . 0 Pseudo-Fickian 0.5 2 . 0 0 . 1 2 . 0 II Pseudo-Fickian 0.3 3.0 1 . 0 1 . 0 Fickian 0 . 0 4.0 1 . 0 *K, = D j f 75 This interpretation is well supported by the wise selection of parameter values in the new kinetic model. As seen in Table 2.2, the sequence of differential sorption in region I was better predicted by the drastic increase of diffusivity rather than that of relaxation coefficient: the diffusivity parameter, , was increased by 1 0 times, but relaxation coefficient, , remained constant. The Fickian fraction in region I was assumed to be 0.3 which was also observed experimentally (Long and Richmann, 1960) (see Figure 2.11). On the other hand, the penetrant concentration in region II seems to be enough to disturb the integrity of polymer structure (e.g. the disruption of secondary intersegmental bonding). As a result, the relaxation process becomes more prominent due to the increase of large-scale segmental mobility (or decrease of relaxation time), and the relaxation fraction (or relaxation-induced sorption) shifts to the short time region, and finally a viscous, Fickian sorption curve appears. As seen in Table 2.2, the increase of the relaxation coefficient, k j, in region II is more drastic compared with that of diffusivity parameter, Kp. The Fickian fraction (1 - a) in region II was chosen to be gradually increasing to represent the transition from relaxation-controlled diffusion to concentration-driven difiusion (or pure Fickian diffusion) (see Figure 2.11). Odani et al. (1961) confirmed that the shift of the second-stage portion (or relaxation fraction) to the short time region when the initial concentration is significantly different from zero (e.g. above some critical value). Also they found that relaxation time scale (t), controlling second stage sorption, decreased in an exponential manner with the increase of concentration (see Figure 2.12). 76 o o 01 o CO a d o 0] d S' o Oî d o / o ^ q o 0.0 1.7 3.3 5.0 6.7 6.3 10.0 S Q R T ( T im e ) Figure 2.11. Differential sorption sequence predicted by new kinetic model (a: Sigmoidal, b: Pseudo-Fickian, c: Two-stage, d: Two -stage). 77 N O (0 m d o w d o m d o N d o y/ o y o o 0.0 0.8 1.7 2.5 3.3 4.2 5.0 S Q R T ( T im e ) Figure 2.11 (continued) (e: Two-stage, f: Pseudo-Fickian, g: Pseudo-Fickian, h: Fickian) 78 - 1 Ci X10^ g/g Figure 2.12. Logarithms of relaxation time t = It* versus initial concentration c. for differential sorption, t* is time corresponding to the second-stage inflection point (Taken from Fujita, Kishimoto, and Odani, 1959) (o: Poly(methyl methacrylate) + methyl acetate (30 °C), □: Cellulose acetate + methyl acetate (20 °C), a: Cellulose nitrate + acetone (25 °C), o: Atactic polystyrene -f benzene (25 °C), e: Isotactic polystyrene (low crystallinity) + benzene (35 °C), • : Isotactic polystyrene (high crystallinity) + benzene (35 °C), a: Regenerated cellulose + water (15 °C)). 79 From the observation and interpretation of differential sorption behavior, the sigmoidal curve was found to represent the sorption process with comparable relaxation and diffusion time scales. The pseudo-Fickian curve turned out be a transition stage in which the two time scales are about to become distinct. The two-stage curve represented the sorption process in which Fickian-diffusion fraction is distinctly separated from relaxation-controlled sorption, and the relaxation process seems to follow the first-order kinetics of concentration according to Odani et al. (1961). 80 4. Justification of the theory and its limitations According to the dual sorption model (Michaels, 1963), frozen microvoids (or localized excess free volume) exist in glassy polymers. At the initial stage of sorption, the penetrant molecules will occupy existing interstitial space between the polymer chains rather than participating in creating sorption sites. The reason is that the effective viscosity which controls the occupation of the existing sites will be much lower than that which controls the creation of new sorption sites. Therefore, it is reasonable to assume that the diffusion process is controlled by Fickian diffusion initially, if the amount of sorbed material is not enough to make a significant disruption of the polymer structure. The kinetic model, based on both simple superposition principles and continuous bulk deformation, cannot be applied to Case II sorption because the relaxation process in Case II sorption occurs only at the moving front. The elementary physical concept is same in both cases, however, in that the creep response of glassy polymers (e.g. viscoelastic relaxation process) is strongly dependent on penetrant concentration. Both phenomena (continuous and discontinuous deformation) require that highly packed glassy state move towards a structure in which a high free volume is present. The free volume concept can be applied to the glassy state because below the molecular mobility is not quite zero (Anet and Bourn, 1967; Vrentas, 1978). But, there still remains uncertainty that one type of relaxation coefficient (or relaxation time) can fit the whole spectrum of glass transition, ranging from solid-like vibration to liquid-like segmental motion. Glass transition phenomena is known to involve more than one relaxation mechanism, e.g., a distribution of relaxation times. So far in previous studies. 81 the free volume theory of Fujita could not fit the transport behavior near or below the glass transition temperature. Another limitation is that the diffusive process should be strongly coupled to mechanical response of the polymer in the sense that the rate at which the penetrant is sorbed must be compatible with the swelling rate, which is controlled by the creep deformation of the polymer matrix. If not, the assumption of linear superposition of two contributing processes may not be valid. In any case, it should be emphasized that free-volume concept is useful in describing relaxation phenomena coupled with mass transfer, and it will aid in the understanding of the structure-property relationships in the glassy state as well as in the rubbery state. CHAPTER III FLUOROPOLYMERS Since the discovery of PTFE by Plunkett in 1938, fluoropolymers successfully performed in areas that require superior resistance to heat and harsh chemicals. The unique, outstanding degree of inertness to heat and chemicals reflects the strong integrity of the intrinsic chemical structure. A. General physical and chemical properties Fluoropolymers can be broadly classified into two major categories: fully fluorinated polymers (Teflon® PTFE, PFA, and FEP) and partially fluorinated polymers (Tefzel® ETFE, (Hylar®, Kynar® or Solef®) PVDF, Halar® ECTFE). The polymeric characteristics are similar within each group, but there are important differences between these groups. Molecules of Teflon® fluorocarbon resins are formed from strong C-F and C-C inter-atomic bonds. Moreover, the carbon backbone of the linear chain is completely sheathed by the tightly held electron cloud of fluorine atoms with electronegativities balanced. This sheath shields the carbon chain from chemical attack and confers chemical inertness and stability. It also reduces the surface energy resulting in a low coefficient of friction and non-stick properties.’ The unique properties attributed to the high strength of the inter-atomic bonds, and nonpolarity lead to a high degree of crystallinity. 8 2 83 This results in beneficial properties such as insolubility, low coefficient of friction, high thermal stability, low permeability, strong chemical resistance, and mechanical toughness. When the fluorine of the fluorocarbon resins is replaced with hydrogen or chlorine, a distinct change of polarity and mechanical properties of the polymers occurs. For example, an increase of polarity is observed because the substituents (H and Cl) have different electronegativities relative to fluorine. It is believed that electronically unbalanced linear chains within their own chains prefer to align themselves in parallel arrays, maximizing the accessibility of substituent groups to permit electrostatic (polar) attraction. That results in the formation of one-interchain polar attraction which greatly enhances the formation of all possible additional bonds between two interacting chains. Fully fluorinated polymers are usually selected when chemical inertness and high- temperature service are required. Partially fluorinated fluoropolymers are generally selected when their chemical resistance is adequate and higher mechanical strength is needed. Some general trends of properties in relation to fluorine content are shown in Table 3.1. The basic differences between two groups are described in Table 3.2. 8 4 Table 3.1 General trends of properties in relation to fluorine content. Low fluorine content High fluorine content Coefficient of friction Anti-stick Thermal stability Mechanical strength Creep resistance Cohesive energy Dielectric constant Processing ease Chemical resistance Table 3.2 Comparison between fully fluorinated polymers and partially fluorinated polymers. Polymer Fully fluorinated oolvmers Partially fluorinated polymers Nonoolarity Polarity Low Interchain Forces High Interchain Forces Property High C-F and C-C Bond C-H = 5-10% weak of C-F Bond Strength C-Cl = 25% weak of C-F bond High Crvstallinity Relatively low Crvstallinity High Melting Point High Molecular Property (1.5 times Benefits stiffer than fluorocarbon polymers) High Thermal Stability Inertness to Chemical Attack Economical Processing 85 Teflon^ PTFE^ The Teflon® PTFE resin offers the best corrosion protection against virtually all industrial chemicals up to 260 °C. Processing techniques include compression molding, isostatic molding, ram or paste extrusion, and sintering. Teflon® PFA^ This melt-processible resin, introduced in 1972, combines the excellent chemical and thermal properties of Teflon® PTFE with the excellent melt processibility and stress- crack resistance with a long flex life. It can be injection molded, transfer molded, melt extruded, or roto processed. It remains flexible and tough at temperatures as low as -200 °C, and retains its strength and stiffness at temperature as high as 260 °C. Teflon® FEP^ Introduced in 1960, Teflon® FEP can be melt extruded for easy processing. This broadens the range of application and makes Teflon® FEP perfect for tubes or for dielectric insulation of wiring. Teflon® FEP provides durable service up to a maximum service temperature of 204 °C. Tefzel® ETFE^ DuPont introduced Tefzel in 1972, having much of the chemical resistant properties of Teflon®, but with greater strength, stiffness, and abrasion resistance. Tefzel® resists chemical attack better than any fluorinated plastic except Teflon®. Because Tefzel® is melt 86 processible, it has the design versatility to solve some of the industry's toughest materials problems in process equipment and components where both chemical and thermal resistance as well as good mechanical properties are needed. Tefzel® has a continuous-use temperature range from -100 °C to 155 °C. S o le f .Kvnar®. and Hvlar® PVDF*'^ PVDF is a tough engineering thermoplastic that offers a unique balance of chemical and mechanical properties. Due to its unique polarity, it can be dissolved in some polar solvents such as organic esters and amines, in addition to being readily melt- processed by standard methods of molding and extrusion. PVDF fluoropolymers have excellent resistance to creep and fatigue. In general, PVDF resin is one of the easiest fluoropolymers to process and can be easily recycled without detriment to physical and mechanical properties. High purity PVDF (Kynar®) meets the most stringent standards for ultra-pure fluid handling materials in the semiconductor, pharmaceutical, and biotechnology industries, where there is a critical need to minimize all particulate contamination. In addition, Hylar® PVDF is resistant to effects of nuclear radiation, even at the condition of 100 megarads of gamma radiation from a Cobalt-60 source at 50 °C and in high vacuum (10'* Torr). This stability to radiation effects has resulted in the successful use of Hylar® components in plutonium reclamation plants. 87 Halar^ ECTFE ^ Halar® ECTFE is a melt processible fluoropolymer from Ausimont USA. It is a tough material with excellent impact strength over its broad-use temperature range, and is one of the best fluoropolymers for abrasion resistance. Halar® fluoropolymer is a thermoplastic which can be processed by virtually any technique applicable. At elevated service temperature Halar® fluoropolymer has superior moisture vapor impermeability compared to certain other fluoropolymers at the same conditions. 88 B. Molecular Structure The molecular structures are drawn below, in Fig.3.1. F F r F F F F r-F F F F n I I I l I I I l I I (C-C)n- (C-C),2-(C-C), (C-C)ioo-(C-C), I I I l I I I l I I F F ^F F FCF3-* l-F F F O J PTFE FEP PFA r F F H H -, r F H F H-| r F F H H-, 1 1 1 1 I I I I I l I I (C-C)-(C-C) (C-C)-(C-C) (C-C)-(C-C) 1 1 1 1 I I I I I l I I L F F H H . n - F H F H- n - F Cl H H - ETFE PVDF ECTFE Figure 3.1. Molecular structures of fluoropolymers Poh( tetrafluoroethvlene ) Virgin PTFE has a crystallinity in the range of 92-98% and has an unbranched chain structure. The two types bonds involved in the PTFE structure are carbon-carbon bonds, which form the backbone of the polymer chain, and carbon-fluorine bonds (Bunn, 1955). The fluorine atoms also form a protective sheath over the chain of carbon atoms. If the atoms attached to the carbon chain backbone were smaller or larger than fluorine, the sheath would not form a regular, uniform cover. By X-ray diffraction analysis below 19 °C. the molecules pack like cylindrical rods in a nearly hexagonal arrangement with a nearest neighbor distance of 5.62 Â. At 25 °C, the separation of the chain axes is 5.66  89 (Sperati, 1961). Between 19 °C and 30 °C, the chain segments are disordered from a perfect hexagonal lattice by small angular displacements about their longitudinal axes. At the first order transition at 30 °C, the preferred crystallographic direction is lost and the molecular segments oscillate above their long axes with a random angular orientation in the lattice (Clark, 1959). PTFE transitions occur at specific combinations of temperature and at the frequency of mechanical or electrical vibrations. Table 33 Transitions in PTFE Type of Temperature Regions Affected Transition r c ) 1st Order 19 Crystalline, Angular Displacement Causing Disorder 30 Crystalline, Crystal Disordering 90 Crystalline 2nd Order -90 Amorphous, Onset of Rotational Motion (C-C Bond) -30 Amorphous 130 Amorphous PoMtetrafluoroethvlene-perfluorofalkvlvinvl ether!) copolymer This melt processible copolymer (PFA) contains a fluorocarbon backbone in the main chain and randomly distributed perfluorinated ether side chain (-OC3F7). The introduction of a perfluorovinyl ether side chain greatly reduces the crystallinity of PTFE. In contrast to HFP, only a small amount of vinyl ether is required to reduce crystallinity and develop adequate toughness. 9 0 PoIv(tetrafluoroethvlene-perfluorofalkvlvinvl etherl) conolvmer FEP resin is a copolymer of tetrafluoroethylene (TFE) and hexafluoropropylene (HFP). It retains most of the desirable characteristics of PTFE but with a melt viscosity low enough for conventional melt processing, because the perfluoromethyl side groups (-CF3) on the main polymer chain reduce crystallinity, which varies between 30 and 45 %. The a relaxation (Glass I) is a high temperature transition (157 °C), and y relaxation (Glass II) (the intermediate friction maxima) occurs between -5 °C and 29 °C. Besides a and y relaxations, one other dielectric relaxation was observed below -150 °C, which did not vary in temperature or in magnitude with comonomer content or copolymer density (Eby and Wilson, 1962). PoMethvlene-tetrafluoroethvlene) copohmer ETFE is isomeric with the homopolymer of vinylidene fluoride. The tetrafluoroethylene segments of the molecules account for - 15% of the weight of an approximately 1:1 copolymer. H H F F HFHF MM I I II -C*C“C"C* -C-C-C-C- MM I I I I H H F F HFHF Ethylene-tetrafluoroethylene Unit Poly(vinylidene fluoride) Unit Figure 3.2. Comparison of molecular structures ( ETFE vs. PVDF). 91 The molecular conformation is an extended zigzag, each molecular having four nearest neighbors with the CH, groups of one chain adjacent to the CF, groups of the next. Nearly perfect alternation of isomeric units in a ca. 1:1 monomer ratio has been confirmed by infrared spectroscopy. Bands at 733 and 721 cm ' have an intensity proportional to the concentration of (CH,)„ groups (n = 4 and < 6 , respectively) present in a copolymer containing 46 mol% tetrafluoroethylene (Wilson and Starkweather, 1973). The dynamic mechanical behavior showed that the a relaxation occurs at 110 °C, P relaxation at -25 °C, and y relaxation at -120 °C. The a, y relaxations reflect motions in the amorphous regions, whereas the p relaxation occurs in the crystalline regions. As ETFE copolymer is isomeric with PVDF, the y relaxations occur at about the same temperature. These a, y relaxations are attributed to the motion of long and short segments in the amorphous regions, respectively (Starkweather, 1973). Poh(ethvlene-chlorotrifluoroethvlene) coDohmer ^ An essentially 1:1 alternating copolymer of ethylene and chlorotrifluoroethylene (ECTFE) has been developed. The early copolymers were thermally unstable. Thermal stability is greatly improved by the addition of sterically hindered phenols and other commercially available stabilizers. Polar association occurs between fluorine and hydrogen atoms in this arrangement and may account in part for the relatively high melting point and the unique properties of the alternating 1:1 copolymer. The ECTFE copolymer is partially crystalline, e.g., 50-55%, depending on the method of preparation. X-ray diffraction of oriented film indicates that the unit cell of the 9 2 alternating copolymer is hexagonal, with a chain repeat distance of 0.502 nm. The chain repeat distance is consistent with an approximately extended zigzag chain containing one molecule of ethylene and one molecule of chlorotrifluoroethylene. The interchain spacing in the copolymer of 0.57 nm is about the same as one-half the interchain spacing in polychlorotrifluoroethylene (0.32 nm) plus one-half the interchain spacing in polyethylene (0.22 nm) (Mannon, 1964). The ECTFE exhibits three second-order transitions: an «-transition at ca 140 °C, a p- transition at ca. 90 °C, and a y-transition at ca. -65 °C. The a- and y-transitions are believed to be associated with motions of chain segments in the crystalline phase, whereas the p-transition is believed to be associated with motions of chain segments in the amorphous phase. Pol\(vinvlidene fluoride') Poly vinylidene fluoride (PVDF) is the addition polymer of 1, 1-difluoroethene, CHj = C F ,, which polymerized readily by free-radical initiators to form a high molecular weight, partially crystalline polymer. The spatially symmetrical disposition of the hydrogen and fluorine atoms along the polymer chain gives rise to unique polarity which can affect solubility, and crystal morphology. The dielectric constant is unusually high. PVDF exhibits a complex crystalline polymorphism, absent in other synthetic polymers, such as a, p, y, and ô forms (Hasegawa and Tokahashi, 1972; Richardson et al., 1983). Crystallinity can vary from about 35 to more than 70 %, depending on the method of preparation and thermo-mechanical history. Chain relaxation behavior has been studied 93 by dielectric methods, NMR, DMS, DSC, and dilatometry. All available information indicates that the glass transition of PVDF lies in the -30 to -50 °C (Miyamoto et al., 1980; Fujji et al., 1978). 9 4 C. Processing properties of fluoropolymers Processing has impact on the performance of fluoropolymers. For example, preforming pressure, sintering time, cooling rate, void content, crystallinity, and degree of orientation can have a significant effect on certain end-use physical properties, such as tensile strength, stiffness, impact strength, and permeability, and dielectric strength. Some “basic factors” that influence these end-product properties are presence of macroscopic flaws, extent of microporosity, percent crystallinity, molecular weight, degree of orientation. 1. Orientation It is well known that the orientation of polymer films significantly improves many of their properties. The orientable fluoropolymer films here fall into three basic categories: perfluorinated, ethylene containing, and chlorotrifluoro polymers. The perfluorinated class contains mostly TFE with relatively small amount of perfluorinated comonomers: FEP and PFA. In the ethylene contdning class, there is about 50 mol % of ethylene as an alternating copolymer: ETFE, ECTFE and PVDF. The chlorotrifluoro type include pure CTFE or CTFE with a minor amount of copolymer such as PVDF. If they are uniaxially oriented under the proper conditions, a unique and commercially useful product will result because of the increase of creep resistance and tensile strength. The machine-direction (MD) orientation process yields films with enhanced mechanical properties in both the machine and transverse directions. In addition, other properties are improved by the orientation, including optical clarity and the chemical 95 and moisture barrier properties (Stanley and Levy, 1990). Table 3.4 Typical Properties of T‘ Films.** ’ Property T’Tefzel® ETFE Conventional (English Units) Tefzel® ETFE ETFE fMD) ETFErTD) Shrinkage (%) 23 (200°C) 7 ('200°0 Elastic Modulus (osi) 500.000 130.000 170.000 Tensile Strength fnsi) 34.000 7.000 4.500 Elongation (nsi) 45 650 300 Continuous Service 150 150 150 Temperature f°C) * MD = Machine Direction; TD = Transverse Direction; Film thickness is 2 mil. * T" films are uniaxially oriented by a patented DuPont process that improves mechanical properties, dimensional stability, and impermeability. 96 2. General effects of processing Direct measurement of basic factors which can influence end-product properties are not usually available. Instead, a number of highly sensitive indirect tests have been devised. They are based on measurement of dielectric strength, tensile strength, specific gravity, and permeability. Table 3,5 Basic factors influencing end-product properties. 10 Basic Factors Mechanical Prooertv Electrical Prooertv Chemical Prooertv Tensile Strength Dielectric Strength Permeabilitv Microporosity T I ! Î Crystallinity T 1 N/E 1 Molecular wt. ÎI N/E N/E Orientation Î I N/E I * N/E indicates negl gible effects. 97 D. Specific applications ^ Fluorinated plastics have provided superior resistance against the attack of harsh chemicals in the chemical processing industry. Fluoropolymers generally outlast most plastics, elastomers, and super-alloys even in the most demanding containment and sealing applications such as 0 -rings and seals, lined piping systems, tank linings, coatings, and flue duct expansion joints. Today, fluoropolymers liners and coatings are rapidly replacing metal alloys in pipes, valves, vessels, columns, and other chemical processing equipment. Fluoropolymers offer numerous advantage over exotic metals, such as better corrosion protection and lower life cycle costs. In fact, chemical processing components made with exotic metals can cost up to four times more than systems made with Teflon® and Tefzel®. The specific applications are categorized in terms of their uses in Table 3.6, and illustrated in Figure 3.3. Table 3.6 Specific applications of various fluoropolymers. ' ‘ fa') Electrical and Electronic End Use ►Low dielectric constant. ►Specialty printed wiring and circuit ►Long term reliability over wide range of substrate. temperature, frequency and moisture. ►Microphone electret membrane. ►Remarkable dielectric stability in critical ►Insulation for multi-conductor flat cable. applications such as corrosive chemicals ►Electronic components for computer and and high temperature. aircraft. 98 Table 3.6 (continued) (b) Medical-Phaimaceutical-Semiconductor End Use ►High purity( absence of plastic additives). ►Thermally sealed sterilizable bags and ►Chemical inertness pouches for drug contact. ►Good barrier property and excellent ►Elastomer/film protective laminates and release property. cap liners for sealing liquids in bottles. ►Biological culture bags and liners. ►Sanitary tubing in semiconductor, biogenetic, and food and daily fluid handling svstems. tc) Chemical Process End Use ►Chemical inertness in the harsh and ►Vessels and component linings. corrosive environments. ►Anti-stick roll covers, laminate for fabric ►Withstands both high and low temperature or elastomer conveyor belting and mold extremes. release films. ►Excellent processibilty (easily cut, thermo ►Corrosion-resistant electrolytic cell covers. formed, heat sealed and welded). ►Thermoformed parts, mpture disk seals ►Superior anti-stick/low fiction property. and gaskets for high temnerature. fd) Special End Use ►Chemical inertness. ►UV resistant high intensity lamp lenses ►Optical clarity. and air sampling bags for outdoor ►High temperature resistance. ambient storage. ►Superior anti-stick properties. ►Flame resistant protective clothing items. ►Long term durability. ►Disposal protective film for laboratory furniture. ►Substrate for pressure sensitive adhesive tape. ►Metallized substrate for outer space thermal control applications. ►Environmental growth chambers, solar energy collectors and radome windows. ►Components in plutonium reclamation plants. 99 Medical-Pharmaceutical Cell Culture Baas Chemical Process 'ank Lining Electrical-Electronic Flat Cable insulation Distributor Composite Release Sheet— Manufactunna Aid Specialty Personal Safety Hood Figure 3.3. Applications of fluoropolymers: (a) Electrical and electronic end use. (b) Mechanical-pharmaceutical-semiconductor end use. (c) Chemical process end use. (d) Special end uses.' ^ CHAPTER IV EXPERIMENTAL METHODOLOGY This chapter investigates the permeation mechanism for a variety of organic chemicals through fluoropolymers over a range of conditions. Three types of experiments were conducted: permeation into air, soiption from a bath, and stress-strain tests. In each test, available standard methods were closely followed as explained in Section B below. To ensure statistical validity, 3 replicates of each experiment were completed. Permeation is observed under steady-state or quasi-steady state conditions. The analysis of permeation data is referred to as a “Boundary Value Problem.” In such experiments, the permeation rate is determined by measuring the rate of mass loss of a penetrant through a polymer film to the air from an otherwise closed vessel. The mass transfer resistances from the bulk penetrant to the polymer surface and from its outside surface into ambient air are assumed to be insignificant. Therefore, the permeability is determined directly from the measurement of the mass flux. Transient sorption experiments are conducted to determine both the effective diffusivity and solubility of the penetrant in the polymer. These involve a pre-weighed sample of polymer sheet immersed in a penetrant. The sample is removed and blotted dry. then weighed at certain intervals. The corresponding rate data, e.g., mass vs. time, is interpreted via the solution to an “Initial Value Problem.” That is, the effective diffusivity 1 0 0 101 can be obtained from transient sorption uptake or release experiments, using the solution of governing PDE (e.g. Pick's second law), along with appropriate initial and boundary conditions. Likewise, the solubility can be measured by allowing the pre-weighed sample to remain in contact with the penetrant for a long time and reweighing it once it has equilibrated. In stress-strain tests, two properties are measured: tensile stress and tensile strain. Tensile properties are the most important indicators of strength in a material. In brief, these indicate the force necessary to elongate the specimen along with how much the material stretches before breaking. The elastic modulus is essentially a measure of stiffness and is a very useful property to know because parts should be designed so that their behavior in normal use falls in the linear region in which the modulus is measured, otherwise, permanent deformation would occur. Stress-strain tests were conducted before and after immersion in various penetrants to determine whether they affected the mechanical properties of the polymers. A. Organic solvent selections The solvents selected for these experiments generally can be classified as two types, based on molecular structures: aromatics and hydrocarbons. The solvents also can be broadly categorized as three types, based on the chemical properties: nonpolar, polar, and H-bonding. For instance, benzene and toluene are nonpolar. Chloro-organics, such as chlorobenzene, dichloromethane and chloroform, are usually polar. Phenol and methyl ethyl ketone (M.E.K.) are considered to be H-bonding. With molecules containing 102 hydroxy (-0H) or carbonyl (0=C<) groups, H-bond formation results, which is a rather strong interaction compared with dipole-dipole interaction. All the organic solvents used here are in the liquid state at ambient conditions (25 °C, 1 atm). The shapes of molecules are roughly spherical except M.E.K. M.E.K. has a rather branched shape. The physical and chemical properties are described in Table 4.1. Table 4.1. Physical and chemical properties of organic solvents.*' Physical and Chemical Properties QH, QH 5CH3 C A C i C4H5OH Molecular Weight tg/moO 78.11 92.14 1 12.56 94.11 Normal Boiling Point ("“O 80.1 1 1 0 . 6 132 181.8 Snecific Gravi tv diouidl 0.879 0.867 1 . 1 1 1.058 Liquid Surface Tension CN/m') 0.0289 0.0290 0.033 0.036 Liquid Water Interface Tension (N/mt 0.035 0.0361 0.0374 0 . 0 2 Latent Heat of Vaoorization fkJ/molI 30.8 33.37 28.23 46.6 Molecular Volume (cmVmol) 89 107 1 0 2 89 Physical and Chemical Properties CH3CI2 CH3COQH5 CHCI3 Molecular Weight fg/mol') 84.93 72.11 119.39 Normal Boiling Point 39.8 79.6 61.2 Snecific Gravi tv diouidl 1.322 0.806 1.49 Liquid Surface Tension CN/m) N/A N/A 0.0271 Liquid Water Interface Tension fN/ml N/A N/A 0.0328 Latent Heat of Vaoorization fkJ/mol) 28.6 32.02 31.0 Molecular Volume (cmVmoI) 65.0 90.1 81.0 * All values are measured at 20 °C, unless specified. * N/A indicates “not available". 103 A list of liquids was compiled with Hansen solubility parameters including three sub-divisions of dispersion (ôj), hydrogen-bonding ability (5^), and polar contribution (5^) (see Section II.A.2.1). The electrostatic properties such as dipole moment (jj) and polarity (p) were also listed in Table 4.2. Gordon (1966) discussed the nature of molecular interactions in terms of the fraction of total interaction due to dipole-dipole (p), induction (z), and dispersion (d) effects such that p + i + d=l. Table 4.2 Solubility parameters and related properties.’^ Solvent <5a <5* H- P P bondine CA 18.6 18.4 0 . 0 2 . 0 DOor 0 . 0 0 0 0 . 0 CACH 3 18.2 18.0 1.4 2 . 0 poor 0 . 0 0 1 0.4 C A C i 19.6 19.0 4.3 2 . 0 poor 0.058 1 . 6 CAOH 24.1 18.0 5.9 14.9 moderate 0.057 1 . 6 CHjClj 20.3 18.2 6.3 6 . 1 poor 0 . 1 2 0 1 . 8 CH3 COCA 19.3 14.1 9.3 9.5 moderate 0.510 3.3 Units: ô (MPa’^), p (Debye), p (dimensionless). 104 B. Experimental methods, apparatus, and conditions Permeation The permeation experiments were conducted in cup-shaped vessels made of 316 stainless steel and Carpenter 20. The vessels are “vapometers,” Model 68-1, manufactured by Thwing-Albert Instrument company of Philadelphia, PA. Each test of a liquid permeating through a polymer sample into air closely follows the current ASTM method for evaluating vapor transmission of volatile liquids. A schematic diagram of the apparatus is shown in Figure 4.1. The permeation chambers measure 2 inches deep with 2.5 inch inside the diameter. The top flange is 1/8 inch thick at the outer edge, 3/16 inch thick at the inner edge, with an outer diameter of 3.625 inches and an inner diameter of 2.5 inches. The bottom flange is 1/4 inch thick. Eight 10-32 Allen screws (1/8 inch diameter) hold the flange together. Each vessel is inoculated with approximately 50 ml of an organic solvent, and the polymer disk is placed onto the cup opening, followed by an annular "Viton" gasket, and capped by the top flange. The Allen screws are tightened evenly and securely. After that, the permeation chambers are inverted so that the liquid inside is in direct contact with the polymer disk, penetrating downward. The permeation vessels are kept in a fume hood at room temperature, or in a ventilated convection oven at high temperature. The mass of the permeation chambers is usually measured every four days using a Mettler 2 0 0 0 microbalance with an accuracy of ± 1.0 mg. 105 Top Flange Gasket Polymer Disk Bottom Flange Vessel Assembled Apparatus Figure 4.1. Apparatus for measuring liquid permeation rates. 106 Blank experiments, in which the permeation vessels contained polymer disks but no liquids, were conducted to compensate for extraneous effects such as humidity and dust. Fluctuations of the mass of the empty chambers were taken into account by subtracting the mass gain of blank vessels from the total mass of the filled vessels. Sorption Integral sorption experiments were carried out, typically using pure penetrant liquids and dry semicrystalline fluoropolymers. Specifically, a pre-weighed and measured polymer disk (3 inch diameter) is suspended on a wire, then immersed in a jacketed flask filled with the liquid, of which temperature is controlled by circulating water through the jacket. At periodic time intervals, the disk is removed from the liquid to be weighed. Before weighing the disk, the liquid adhering to the surface of polymer disk is removed quickly by flushing with compressed air. A schematic diagram of the apparatus is shown in Figure 4.2. An Ohaus Galaxy 110 microbalance with an accuracy of ± 0.1 mg is used to measure the mass change of immersed polymer films. The dimensional changes of each sorbed sample were measured periodically using “digital electronic calipers” (sensitivity = ± 1 mil). Vapor sorption experiments were also performed, but only at a temperature of 45 °C. The whole system is in communication with an external degassed liquid solvent reservoir whose temperature is controlled by external water jacket in which water circulates at a set temperature. The penetrant vapor is carried by air flow. A schematic diagram of this apparatus is shown in Figure 4.3. 107 Water out Penetrant Water in i s r Polymer disk 5 » Figure 4.2. Apparatus for measuring liquid sorption rates. 108 Flow Meter 1 Flow Meter 2 Vent Needle . Polymer Valve / Disk Liquid Penetrant Air Cylinder Oven Figure 4.3. Apparatus for measuring vapor sorption rates. 109 Instron tests A series of stress-strain experiments was conducted in order to measure the change of mechanical properties of the impregnated polymer samples by organic solvents. Each specimen is prepared by cutting the sample sheet into the following shape with a sharp die, as shown in Figure 4.4. 5.5"- T i Ô i T 0.5' Figure 4.4. Tensile test specimen. A Tensometer model 20 (Monsanto, Serial No. ST20-136) measures the applied tensile force and the resulting extension of the sample in accordance with major international standards. The test specimen is held in grips between the load cell and crosshead. Both ends of the specimen are firmly clamped in the jaws of the tensile testing machine. One jaw is fixed and the other is movable. The movable jaw moves at a constant rate (19.9 inches/min). The stress is automatically plotted against strain (elongation) on graph paper (see Figure 4.5). Numerical values of stress and strain were also recorded from the digital readout of the Instron machine (Tensometer 20). 110 Figure 4.5. Schematic diagram of Tensometer (1. Load cell cover, 2. Load cell assembly, 3. Crosshead nose piece, 4. Crosshead, 5. Leadscrew, 6 . Front control panel, 7. Top gearbox cover, 8 . Timing belt drive cover, 9. Crosshead limit stops). Ill Expérimental conditions Variables that were systematically tested include: the thickness of the sample (10 and 90 mils), temperature (25, 45, and 65 or 75 °C), and the penetrants (aromatic liquids and some hydrocarbon solvents). Several fluoropolymers are tested: ETFE, ECTFE, PVDF, PEA, and FEP. The full matrix of test conditions is given in Table 4.3. 112 Table 4.3. Matrix of experimental tests Type/Polymer Permeant Thickness Temperature Tests # P/1 1-5 1 1-3 15 P/1 2-3 2-3 4 P/2 1-5 1 1-3 15 P/2 2-3 2-3 4 P/2* 1-3 1 1-3 9x2 P/3 1-5 1 1-3 15 P/3 2-3 2-3 4 P/3 1-3 1 2-3 6 P/4 1 1 1-3 3 P/5 1-5 1 1-3 15 subtotal 99 S/1 1-3 1 1-3 9 S/1 4-6 1 1 3 S/1 2-3 2 2-3 4 S/2 1-3 2 1-3 9 S/2 4-6 1 1 3 S/2 2-3 2 2-3 4 S/3 1-3 1 1-3 9 S/3 4-6 1 1 3 S/3 2-3 2 2-3 4 S/4 1 1 1-3 3 S/5 1-3 1 1-3 9 S/5 4-5 1 1 3 subtotal 63 T/1 2-3,5-6 1 1 4 T/2 ^ 2-3,5-6 1 1 4x2 T/3^ 2-3, 5-6 1 1 4x2 T/4 2-3,5 1 1 3 T/5 2-3,5 1 1 3 subtotal1 ------______26 * Permeation experiments examining possible skin effects according to surface appearance. ^ Stress-strain tests running the machine and transverse direction, respectively. _L3 KEY for Table 1: Experiment Type: P = Permeation (into air) S = Sorption (in a bath) T = Stress-strain test (in an ambient air) Note: Three replicates were performed at each test condition. Polymer Type: 1. Ethylene tetrafluoroethylene (ETFE) 2. Ethylene chlorotrifluoroethylene (ECTFE) 3. Polyvinylidene fluoride (PVDF) 4. Fluorinated ethylene-propylene (FEP) 5. Fluoroalkoxy fluorocarbon (PEA) Note: The fully fluorinated fluoropolymers (FEP and PFA) were chosen for comparison with partially fluorinated polymers in sorption and permeation behavior. Penetrant: 1. Benzene 4. Phenol 2 . Toluene 5. Dichloromethane 3. Chlorobenzene 6 . Methyl ethyl ketone Note: Other liquids such as chloroform and water were used as penetrants to support the experimental results. Temperature: Sorption Permeation Stress-strain 1.25°C 1.25°C 1.25°C 2 .45°C 2.45 °C 3. 65°C 3. 75°C Thickness: 1.10 mil = 0.0254 cm 2.90 mil = 0.229 cm Note : The thick films (90 mil) were not used in permeation experiments at room temperature because the time to reach steady state was too long. Similarly, in some sorption experiments the same problem occurred, depending on experimental conditions. The experiments for the thick films have been conducted at 45 °C or higher. 114 C. Analysis Methods 1. Computer spreadsheet The experimental data are analyzed and tabulated using Quattro Pro 4.0, a commercial spreadsheet program. The permeation rate at steady state is calculated via the linear regression option available in Quattro Pro. Only steady-state permeation data were used in the calculation, neglecting the initial transient period. Linear regression finds the best fit in which the dependent variable, total mass loss (ikf), is approximated by a linear function of the independent variable, accumulated time (r), in the form of; M = P t + intercept (78) where the coefficient P is the average permeation rate, e.g., flux multiplied by the exposed surface area of the polymer disk. This analysis calculates the value of P that minimizes the standard error of the dependent variable, M. The rate of permeation is reported in units of grams per square meter per day (grams/m^.day). A sample calculation is reported in Appendix A. The sorption data are also analyzed and tabulated using Quattro Pro 4.0, and plotted as a graph (e.g. M, !M_ vs. SQRT(Days)). According to the shape of plotted graph, the diffusion coefficient was obtained from the Pick’s diffusion equation or the new mathematical model that simulates anomalous sorption. 115 2. Statistical Analysis To ensure statistical validity, at least 3 replicates of each experiment were completed. When the three average permeation rates were within a reasonable standard deviation of their mean, the mean was calculated and reported as the permeation rate at those sets of conditions. For example, the average permeation rate of benzene in ETFE (at 25 °C) were computed by Quattro Pro to be 0.4680, 0.4829, and 0.4873 grams/m'.day. The mean is thus 0.4794 grams/m'.day. The standard deviation, s, for a small group numbers is defined as: ^ (79) n - 1 where x is the mean, and x,. is the individual value, an n is the number of samples. The standard deviation for the permeation rates is 1.7%. CHAPTER V RESULTS AND DISCUSSION A. EXPERIMENTAL RESULTS 1. Sorption characteristics For the transient sorption experiments, the range of responses spanned from Fickian, with either concentration-independent or concentration-dependent diffusivity, to various forms of non-Fickian behavior, some of which exhibited acceleration during the final stages of uptake (see Figure 5.1). A summary of the experimental data is in Appendix B. 1.1 Polymer types PFA/FEP films can be considered to represent a seemingly ideal Fickian sorption and desorption behavior with a constant diffusion coefficient because both reduced curves roughly coincide with each other over the entire range of the experimental time scale. When viewed on “fractional mass change" versus “square root of time” coordination, the data are roughly linear initially and then approach saturation asymptotically, in congruence with Fick's law. The diffusion coefficient thus can be obtained from the analytical solution of Fick's second law, namely the half time method or the late time approximation (cf. eqs.(25-28)). 116 117 1.25 1.25 FEP (65 ®C) ETFE g. 0.75- 0.75- 2 s S 0.5- s 0.25- 0.25 - 0 0.5 1.5 0 2 3 4 5 6 SQRT(Days) SQRT(Days) 1.25 1.25 PVDF ECTFE S' 0.75- 2 2 2 0.5- 0.5- 0.25- 0.25- 0 2 4 6 8 0 1 2 3 4 5 6 SQRT(Days) SQRT(Days) Figure 5.1. Integral sorption and desorption of benzene in fluoropolymers at 25 °C except for FEP (Film thickness is 10 mil) (□: Sorption, 0 : Desorption). * If not specified, the thickness of polymer samples is considered 10 mil. 118 ETFE film exhibited an slightly distorted Fickian sorption behavior because of a slight inflection at the initial sorption stage (see Figure 5.2). Additional evidence of divergence from Fick's law was that the reduced desorption curve is slightly above the sorption curve over initial half of sorption curve, and both reduced curves cross each other. The diffusion coefficient of ETFE can be considered to be more concentration- dependent than FEP/PFA because the divergence of the reduced sorption and desorption curves is more marked at the final stage of sorption for the former material. A suitable diffusivity can be obtained as an average value from initial gradient of uptake, , and initial gradient of release, D^. PVDF film produced a typical sigmoidal sorption curve with a single inflection point at M/M_ = 0.5. After a short linear period, the curve became convex with respect to time axis, indicating that uptake was accelerating, followed by an asymptotic approach to saturation. The reduced desorption curve apparently intersected the sorption curve and approached the final state very slowly. The sorption curve may be described by eq.(53) which is equivalent to that of Peterlin. ECTFE film produced even more unusual results. Sorption curves exhibited essentially no linear region, but were immediately convex with respect to time, indicating a greater acceleration of uptake than PVDF. Finally, instead of approaching saturation asymptotically, the sorption rate appeared to accelerate until saturation was attained. This accelerating feature cannot be simply described by a linear combination of Fickian and Case II relaxation effects like eq.(53). 19 10 CM O O O O 10 O'u d o o to d o to w d o • / o / q ®— o 0.0 0.5 1.1 1.6 2.1 2.7 3.2 S Q R T (d a y s ) Figure 5.2. Integral sorption of benzene in ETFE (at 25 °C) fitted by Fickian solution with constant diffusivity (cf. eq.(24)). 120 When the sorption curves are reported on logarithmic scales, it is easy to evaluate the power of time to which the mass uptake of the penetrant is proportional. Some curves exhibited a slight curvature, which indicated an acceleration in the mass uptake rate (see Figure 5.3). In case of FEP and ETFE films, according to the log-log plot, the slope over the initial half of sorption is approximately 0.5 which is a characteristic value for Fickian diffusion. However, the slope of PVDF case usually lies within the range between 0.5 and 1.0. This observation indicates that relaxation effects contribute to the total sorption uptake along with concentration-driven Fickian diffusion. For the case of ECTFE film, the slope at the final stage of uptake appeared to be about 2.2, indicating a rate dependence of sorption uptake, M/M^ « which corresponds to drastic acceleration. 1.2. Solvent types In general, nonpolar solvents penetrate faster through nonpolar fluoropolymers than polar solvents. For the same polarity a small size of penetrant is more mobile than a large one. Teflon films (FEP and PFA), which are known to be nonpolar, and modified fluoropolymers (ETFE and PVDF) followed those general rules if homologous aromatic solvents (benzene, toluene, and chlorobenzene) were used as penetrants: nonpolar benzene and toluene penetrated faster than polar chlorobenzene, and small benzene penetrated faster than large toluene (see Figure 5.4). However, polar ECTFE films were more susceptible to polar solvents. The penetration rate was reversed among the homologous series: chlorobenzene was faster than toluene. More polar solvents such as methylene chloride and methyl ethyl ketone 121 FEP ETFE O' S' s s n = 0.5 n = 0.58 0.1 0.1 0.1 10 100 0.1 10 100 Tinic(Days) Tim e(D ays) ECTFE PVDF S' S' n = 0.7 s n = 0.54 s S n = 2.2 n = I.O 0.01 0.01 0.1 10 100 0.1 10 100 Tim e(D ays) Tim e(D ays) Figure 5 J . Log-log plots of the kinetics of benzene sorption by fluoropolymers at 25 °C (n: Slope of log - log plot). 122 1.25 1.25 PFA ETFE «y 0.75- 0.75- 0.5- 0.25- 0.25- 0 1 2 3 4 0 1 2 3 4 SQ R T(D ays) SQ RT(D ays) 1.25 1.25 PVDF ECTFE , 0.75- 0.75- w s' 0.25- 0.25- 0 2.55 7.5 10 0 1 2 3 4 5 SQ R T(D ays) SQ R T(D ays) Figure 5.4. Integral sorption of aromatic liquids into fluoropolymers at 25 °C (□: Benzene. 0 : Toluene, o: Chlorobenzene). 123 (M.E.K.) penetrated much faster than less polar chlorobenzene (see Figure 5.5). For solvents with strong H-bonding ability such as M.E.K. and phenol, PVDF showed much more susceptibility than the other fluoropolymers. No apparent sorption of phenol was observed on Teflon films (FEP and PFA) and ECTFE at 25 °C. 1.3. Repeated Exposure Some of the thin fluoropolymer films were tested two (or more) times by repeating the following sequence: immersing the sample, removing and desorbing the residual solvent, and then resorbing. The residual solvent was removed at high temperature (>65 °C). Upon repeated exposure, ETFE and PVDF responded almost the same way, but for the case of ECTFE the attainment of ultimate equilibrium was delayed, possibly indicating structural rearrangement by the impregnated solvent, comparable to annealing. However, the acceleration feature during the later stages of uptake was still observed as was for fresh samples (see Figure 5.6). 1.4. Orientation Uniaxially oriented ECTFE films were cut to rectangular shapes (length/width = 4) which had a major direction, in the machine (M) or transverse (T) directions, respectively. The T-sample reached a quicker equilibration state than the M-sample. Moreover, the expansion in the major direction of the M-sample was still zero in contrast with the relatively small expansion of the T-sample during the initial stage of sorption. This result indicates that the relaxation in the machine direction is more retarded than the 124 1.25 ,« 0.75 H 0.5 H ETFE 0.25 H SQRT(D ays) 1.25 S 0.5 4 ECTFE 0.254 0 2 3 4 5 6 SQRT(D ays) Figure 5.5. Intégral sorption of polar solvents into fluoropolymers at 25 °C (□: Chlorobenzene, 0 : M.E.K., o: Phenol, A: Dichloromethane). 125 1.25 -O ^ 0.75- S 0.5- PVDF 0.25- 0 2.5 5 7.5 10 12.5 SQRT(D ays) 1.25 0.75- S" s s* 0.5- PVDF 0.25- 0 0.5 1 1.5 2 SQRT(Days) Figure 5.5. (continued) (□: Chlorobenzene, 0 : M.E.K., o: Phenol, a: Dichloromethane) 126 1.25 1.25 ETFE PVDF îT 0.75- s “ S "%. s s 0.5- 0.5- 0.25- 0.25- 00.5 1.5 2 Z5 0 1 2 3 4 SQ R T(D ays) SQ R T(D ays) 1.25 ECTFE ST 0.75- 0.5- 0.25- 0 0.5 1 1.5 SQ R T(D ays) Figure 5.6. Integral sorption of toluene into fluoropolymers with different exposure times at 45 °C (□: Original sample, 0 : Resorbed sample once, o: Resorbed sample twice). 127 transverse direction, and as a result the relaxation of the whole sample is also retarded. The discrepancies between M- and T- samples in sorption curves lessened as the ratio of length to width increased (4 - 14), and the number of repetitions of exposure increased (see Figure 5.7). 1.5. Non-isotropic expansion Thickness changes were difficult to detect accurately, due to slight variation in the original material and due to the polymer being flexible and slightly compressible. Conversely, shape changes were evident in ECTFE film due to the existence of tough, shiny skin. During sorption, the dull side expanded more than the shiny side, which caused the polymer to curl into the shiny side. For other polymers, there were no significant shape changes. It was found that the longitudinal swelling (or linear expansion) of the polymer sample, observed during immersion in aromatic solvents, was fairly small and isotropic for ETFE and PVDF (e.g. on the order of 1 -2 %). The major distinctions were that PVDF expanded more in the (inferred) machine direction than in the (inferred) transverse direction, while ETFE showed the opposite response. ECTFE, on the other hand, exhibited similar expansion (-1 %) in the transverse direction but about 3-4 % expansion in the machine direction. Results for the three polymers and three solvents are listed in Table 5.1. 128 1.25 M (Sorption) T (Sorption) M (Resorption) wer 0.75- S T (Résorption) S* 0.5- 0.25- 0 2 3 4 5 6 SQRT(D ays) 1.25 Machine Direction , 0.75- Transverse Direction 0.5- 0.25- 0 1 2 3 4 5 SQ RT(D ays) Figure 5.7. Integra! sorption and resorption of toluene into ECTFE sample (length/width = 4) at 25 '’C and dimensional change of ECTFE sample in the machine and transverse directions (L/L^ : Fractional length change). 129 1.25 0.75- O" T(4) 0.5- M(4) T(I4) 0.25- M (14) 0 2 3 4 5 6 SQRT(D ays) Figure 5.7. (continued) (length/width = 4 and 14) 130 Table 5.1 Equilibrium expansion of polymer films (10 mil) after immersion in penetrants. Temperature Stretching Expansion of Polvmer. % Penetrant rc)Direction ETFE ECTFE PVDF Transverse 1.2 0.93 1.2 25 Machine 0.93 3.5 2.2 Transverse 1.5 1.1 1.4 45 Machine 0.30 3.5 2.4 Benzene Transverse 1.4 1.2 1.3 65 Machine 0.27 4.8 2.9 Transverse 1.3 0.93 1.2 25 Machine 0.93 3.0 2 . 0 Transverse 1.3 0.87 1.1 45 Machine 0.57 3.5 2.2 Toluene Transverse 1.07 0.93 1.3 65 Machine -0.07 4.13 2.1 Transverse 1.2 1.1 0.83 25 Machine 0.08 3.2 1.5 Transverse 1.3 0.73 0.87 45 Chloro- Machine 0 . 2 3.3 1.4 benzene Transverse 1.0 1.2 1.2 65 Machine 0.23 4.5 1.9 * If not specified, the thickness of the polymer samples will be considered as 10 mil. As shown in the Table 5.1, there are practically no effects of the solvent types on the degree of expansion% of each fluoropolymer (at least in this series), and there are negligible effects of temperature. In PVDF, the expansion of longitudinal length was almost linearly dependent on the mass uptake (see Figure 5.8). This means that the swollen area of the penetrated region does not seem to be restricted by the mechanical integrity of the unpenetrated or less penetrated core region. In case of ECTFE, however, the expansion in the machine direction seems to be constrained by the mechanical integrity of the less expandable 131 1.25 PVDF 0.75- uo S' 0.5- s s' 0.25- 0 2 4 6 8 SQ RT(days) 1.25 ECTFE 0.75- u o S' s 0.5- s 0.25- 0 2 3 4 5 SQ RT(days) Figure 5.8. Time dependence of integral sorption and specimen length for benzene penetration in PVDF and ECTFE at 25 °C (□: Sorption curve, e: Expansion in the machine direction, o: Expansion in the transverse direction). 132 tough side. Above some critical fractional mass uptake, the expansion started to increase apparently by the swelling stress, possibly generated from more expandable dull side, and finally accelerated as the tough side relaxed. In addition, swelling was initially much faster in the transverse direction than in the machine direction (see Figure 5.8). 1.6. Solubility As can be seen in Table 5.2, the solubility of the penetrant mainly depends on its size and the relative polarities of the penetrant and polymer. Nonpolar benzene and toluene exhibited high solubility in PVDF and ETFE. Conversely, chlorobenzene was somewhat more soluble in ECTFE than toluene. Benzene exhibited highest solubility in fluoropolymers among the homologous aromatic solvents. Solubility increased by approximately 10-20 % with the increase of temperature from 25 °C to 45 °C and from 45°C to 65“C. Table 5.2 Solubility of solvents in polymers at 25,45 and 65 °C (moles (penetrant)/cm (polymer)). Polymer Temperature Solubilitv in nolvmer (moles/cm^) x 10^ CC) Benzene Toluene Chlorobenzene 25.0 5.21 3.99 3.80 ETFE 45.0 5.74 4.25 4.10 65.0 6.77 4.84 4.56 25.0 7.43 5.68 6.25 ECTFE 45.0 8.30 6.18 6.85 65.0 10.79 7.49 7.92 25.0 5.59 4.05 3.33 PVDF 45.0 6 . 2 0 4.53 3.79 65.0 7.28 5.16 4.36 133 The solubilities of the aromatic solvents in fluoropolymers were plotted in Figure 5.9, according to the thermodynamic relationship given by eq.(3). The energies of sorption {AH) turned out be small and positive according to the temperature dependence of solubility (see Table 5.3). Table 5 3 Sorption energies from Arrhenius-type plots. (Ln(Solubility) vs. Temperature''(K'*)) Polymer Sorption Energy, AH^ Benzene Toluene Chlorobenzene ETFE 655.08 R 480.63 R 456.89 R ECTFE 930.26 R 690.60 R 592.71 R PVDF 661.28 R 604.46 R 677.17 R * R is the gas constant (1.987 cal/mol.°K) 1.7. Integral sorption with penetrant at low concentration Benzene vapor was generated in the apparatus shown in Figure 4.3 by keeping the “degassed” liquid chamber at 30 °C and by controlling the flow rate of carrier gas (air) at a low value. The sorption behaviors of fluoropolymers under low activity of penetrants are quite different from liquid-sorption behaviors. For example, the time to reach equilibrium uptake is much longer in ECTFE and PVDF than in ETFE and PEA. The equilibrium solubility is almost same for each modified fluoropolymer, and turned out be about one six of that of liquid sorption. Most sorption curves represent seemingly Fickian diffusion behavior, even for ECTFE (see Figure 5.10). 134 IE-03 IE-02 ECTFE z IE-O-H ETFE lE-04 0.0028 0.003 0.0032 0.0034 O.TO28 0.003 0.0032 0.0034 Temperature' T em perature' I(K 'l) IE-03 IE-03 FEP PVDF IE-04 IE-04 0.0028 0.003 0.0032 0.0034 0.0028 0.003 0.0032 0.0034 Temperature' ^(K'^} T em perature' * (K '0 Figure 5.9. Effect of temperature on solubility for aromatic solvents (□: Benzene, 0: Toluene, O: Chlorobenzene). * If not specified, the unit of solubility (5) is considered to be [moeles/cm^] 135 1.25 PFA O" 0.75” 0.5- 0.25- 0 0.5 1.5 2 2.5 3 SQ R T(D ays) 1.25 ETFE er 0.75- 0.5- 0.25- 0 2 3 4 5 SQ R T(D ays) Figure 5.10. Vapor sorption of benzene into fluoropolymers at 45 °C (□: Vapor sorption, 0 : Liquid sorption). 136 1.25 PVDF g. 0.75- 0.5- 0.25- 0 2 3 4 5 SQ RT(D ays) 1.25 ECTFE S" 0.75- 0.5- 0.25- 0 2.5 5 7.5 10 SQ RT(D ays) Figure 5.10. (continued) (□: Vapor sorption. 0: Liquid sorption) 137 1.8. Thickness effects The sorption curve of the thick ECTFE film (90 mil) also exhibited anomalous sorption behavior accompanying structural deformations (e.g. curling of disk shape). The drastic acceleration, often observed in thin films ( 1 0 mil), seems to be less evident in the thick films. The discrepancy of penetration rates between toluene and chlorobenzene decreased with the increase of temperature (45 °C-65 °C) (see Figure 5.11). 1.9. Presorbed samples Equilibrated samples obtained from vapor sorption experiments were completely sorbed in liquid solvent, and the uptake rate was measured as shown in Figure 5.12. Presorbed ETFE exhibited almost the same uptake rate as a fresh sample, except that the former exhibited more Fickian-like behavior. The reason may be due to the decrease of relaxation effects because relaxation was already triggered by presorbed molecules. Presorbed PVDF exhibited a faster uptake rate throughout the whole range of sorption curve. This also indicated that presorbed solvent increased the segmental mobility of chains, resulting in the increase of uptake rate of the penetrant. Presorbed ECTFE also exhibited a faster uptake rate at the final stage of sorption, probably due to the same reason as for PVDF. The initial uptake, however, turned out be almost same for both samples. A more detailed look indicated that the sorption curve of the presorbed sample exhibited a slower start followed by a more accelerated uptake compared with the fresh sample. This experimental observation indicates that ECTFE more strictly follows the assumption of Fickian diffusion followed by relaxation- 138 1.25 45 ®C D6 D» O - •EP 0.75 H S s 0.5 H 0.25 H 2.5 7.5 SQ R T(D ays) 1.25 65 ®C 0.75 H ? S S 0.5 H 0.25 H 0.5 2.5 SQ RT(D ays) Figure 5.11. Integral sorption of toluene and chlorobenzene in thick ECTFE film at 45 and 65 °C (Film thickness is 90 mil) (□: Toluene, 0 : Chlorobenzene). 139 1.25 ETFE 0.75- s s 0.5- 0.25- 0 2 3 4 5 SQRT(Days) 1.25 PVDF o- 0.75- s s 0.25- 0 2 4 6 8 SQRT(D ays) Figure 5.12. Sorption curves showing effects of presorption by vapor-phase benzene on the sorption by the corresponding pure liquid (□: Fresh, 0 : Presorbed). 140 1.25 ECTFE 0.75- S 0.25- 2 3 4 5 SQRT(Days) 0.4 - ECTFE 0.3- ST S S 0.2- 0.1 - 0 0.5 1.5 2 2.5 SQ R T(D ays) Figure 5.12. (continued) [□; Fresh, o ; P re s o rb e d ] 141 controlled sorption, compared with PVDF. Namely, the presorbed molecules altered the structure sufficiently, e.g., by creating free volume, to allow an initial period of Fickian diffusion, followed by a subsequent sorption due to relaxation. 142 2. Permeation characteristics Compatibility of polymeric materials governs their suitability for nearly all potential applications. An aspect of compatibility that is frequently important for fluoropolymers is their ability to isolate fluids by serving as a barrier to mass transport. This property is commonly expressed as permeability. The purpose of the permeation experiments is to measure the permeability of several substances in a variety of polymers, examining a range of temperature and polymer thickness. The information gathered will lead to a quantitative basis for predicting the ability of the polymers to contain or isolate these substances. It may also reveal subtleties that are affected by processing conditions or formulation that may be a tool for product development. Depending on the types of polymers, solvents, and processing properties (e.g. skin, orientation), different permeation behaviors was observed. Below, these are explained according to the experimental conditions. A summary of the experimental data is reported in Appendix A. 2.1. Polymer and solvent types Teflon films (FEP and PFA) always showed excellent barrier properties, e.g., low permeation rates (P.R.) for organic chemicals compared with the other fluoropolymers. Their behaviors were very similar to each other, as expected from the notion that the chemical and physical structures are almost identical to each other. Contrary to the corresponding expectation of the performance of Teflon resins. 143 modified fluoropolymers (ETFE, ECTFE and PVDF) showed different permeation behavior to different types of solvents. For example, ECTFE turned out be most susceptible to polar chlorinated hydrocarbons. On the other hand, PVDF was most susceptible to hydrogen (H)-bonding organics such as phenol and methyl ethyl ketone. Tefzel ETFE exhibited intermediate resistance to polar and H-bonding organics among the modified fluoropolymers (see Table 5.4 and Figure 5.13). Table 5.4 Susceptibility of fluoropolymers to various types of solvents. High P.R. Low P.R. Nonpolar (C^Hg, CgHgCH^) ECTFE - ETFE - PVDF - FEP/PFA Polar (CgHjCl, CH^Cl^) ECTFE - ETFE - PVDF - FEP/PFA H-bonding (CgHjOH, CHjCOC^Hj) PVDF - ETFE - ECTFE - FEP/PFA For a homologous series of aromatic solvents, fluoropolymers showed the same trends of permeation behaviors according to the degree of geometric and electronic effects, respectively. Small and nonpolar benzene produced the highest P.R. among aromatic solvents. If the penetrant is not so polar, the size became a critical factor in determining the permeation rate. For example, nonpolar toluene and polar chlorobenzene exhibited almost the same P.R. due to the similar sizes. In ECTFE, however, polar chlorobenzene gave a higher P.R. than nonpolar toluene, due to polar-polar interactions (cf. Figure 5.26). 144 IE+03 lE+03 BENZENE TOLUENE 1 0 2 E+ H 1E+02H T lE+OI -| e £ iB+014 4 lE+OOH ie+oo H IE-01 H ie-o H lE-02 IE-02 0.0028 0.003 0.0032 0.0034 0.0028 0.003 0.0032 0.0034 Temperature' ^(K' *) Temperature* ^(K‘ lE+03 IE-f0 2 H 1E+0H : lE+OOH IE-014 IE-024 CHLOROBENZENE IE-03 4 — 0.0028 0.003 0.0032 0.0034 Temperature* *(K* 1) Figure 5.13. Arrhenius plots of permeation rates of various organic liquids in fluoropolymers (□: ETFE, 0 : ECTFE. o: PVDF, a: PFA). 145 IE+02 PHENOL ^ICHLOROMETHANE lE+OI- IE+03- ft. lE+00- &. IE+02- lE-01 - lE+OI - lE-02 IE+00 0.0028 0.003 0.0032 0.0034 0.0028 0.003 0.0032 0.0034 Temperature' *(ir Temperature' ^(IT IE+04 M.E.K. IE+03- • IE+02 lE+OI- IE+00 0.0028 0.003 0.0032 0.0034 Temperature' ^(K' Figure 5.13. (continued) (□: ETFE. 0 : ECTFE, o: PVDF, a : PFA). 146 It is generally accepted that modified fluoropolymers are highly resistant to penetration of small molecules due to their increased strength of polar intersegmental interactions. In liquid sorption cases, however, modified fluoropolymers gave higher permeation rates than Teflon films. 2.2. Effects of processing properties (orientation and surface skin) If one merely looks at the final permeation rate, it would not have been evident, but, as was the case for transient sorption, the most interesting permeation behavior was observed in ECTFE. The initial permeation rate was very low and it was difficult to get three consistent permeation rates. As time proceeded, however, the polymer structure relaxed due to swelling stress, and at some critical point the permeation rate increased very rapidly as shown in Figure 5.14. The ECTFE film also exhibited other unusual behavior due to the existence of a dense, shiny skin that was possibly formed during manufacturing using a take-up drawn, while the surface exposed to air became soft and dull. The two different surfaces apparently have different properties. The shiny skin seems to be more tough than the dull side. This is illustrated by the onset of rapid relaxation of different time scales, depending on which side is exposed to the penetrating solvent (see Figure 5.15). When the shiny side was in contact with the penetrant, the onset of rapid relaxation started later than if the dull side was in contact. This indicates that the shiny side is tougher and more resistant to swelling stress. The steady-state permeation rate when the shiny side was in contact with the solvent was slightly less than that of the dull side. This indicates that solubility of the shiny side is less than that of the dull side or there are sorption differences between shiny and dull sides. 147 ECTFE 1.5 4 I PL. I I « 0.5 H e e s g oH & -0.5 100 Time(Days) Figure 5.14. The transient permeation rate of aromatic solvents through ECTFE at 25 °C (□: Benzene, 0 ; Toluene, o: Chlorobenzene). 148 1.5- ECTFE OB •X» 0.5- 0 - -0.5 100 Time(Days) Figure 5.15. The comparison of permeation rates of benzene in ECTFE, when shiny side is exposed, with when dull side is exposed at 25 °C (□: Dull side, 0 : Shiny side). 149 2.3. Overshoot Overshoot was exhibited during some permeation experiments with ECTFE (see Figure 5.16). This anomalous behavior was not detected clearly at room temperature, but at 45 and 75 °C this behavior was clearly observed. After the permeation rate reached maximum, it leveled off and dropped to a lower, steady value. This phenomenon implies that the relaxation process, during transient permeation, creates a supersaturated space which can accommodate rushing penetrants into polymer matrix, but which is not thermodynamically stable. 2.4. Temperature and thickness effects Permeation rates were measured at each of three successive temperatures (25, 45, 75 °C) for various types of penetrants. Results are shown in Figure 5.13. In Table 5.5, the permeation rates only of homologous aromatic liquids are shown for the three different temperatures. The permeation behavior of all three polymers (10 mil) was highly sensitive to temperature change: a drastic increase of permeation rate occurred. Each exhibited increases of factors of 200 to 300 in P.R. over the 50 °C span. Table 5.5 The permeation rate of aromatic liquids through fluoropolymers (10 mil). Polymer Temperature Permeation Rate (srams/m'.dav) (C°) Benzene Toluene Chlorobenzene 25.0 0.4794 0.2693 0.3150 ETFE 45.0 5.1767 2.5662 2.6293 75.0 109.8720 61.3581 60.9391 25.0 1.1908 0.7897 1.4061 ECTFE 45.0 22.3765 11.3682 22.5039 75.0 693.8069 307.0620 433.7100 25.0 0.1153 0.0530 0.0468 PVDF 45.0 2.0536 0.9013 0.6268 75.0 35.1882 18.1996 17.6400 150 45 ®C î 204 :©■ 104 e o 54 40 Time(Days) 1000 ? ? 75 ®C 800 4 î 2 -o* Os I 400 H i £ 200 0 5 10 15 20 25 Time(Days) Figure 5.16. Overshoot in transient permeation of ECTFE-benzene system at 45, 75 °C (□: Sample 1, 0-Sample 2, o: Sample 3). 151 The thickness effect on permeation rate is displayed in Figure 5.17. According to Figure 5.17, the permeation data were re-analyzed by considering thickness effect, e.g., permeation rate (gram/m'.day) times sample thickness (mil). Thus, permeabilities of both the thin and thick films should be same if the ideal permeation mechanism is applied to the samples (cf. Section II.A). At 45 °C, ECTFE samples exhibited roughly the same permeation rates for both thin and thick film cases. The thick films (90 mil) of the other polymers (ETFE and PVDF) exhibited higher permeabilities than the thin films (10 mil). The thin films at 75 °C, however, exhibited much higher permeabilities than the thick films, especially for ECTFE and ETFE. The weak sensitivity of thick films to temperature may be responsible for this result, while thin films exhibited much higher sensitivity to temperature change. Generally observed trends of permeabilities for aromatic liquids in thin films were also different from those in thick films: Thin films (ECTFE > ETFE > PVDF) - Thick films (ECTFE > ETFE - PVDF). According to the permeation theory for isotropic materials, the permeation rate should be inversely proportional to the thickness of polymer sample. In a practical sense, however, thickness anomalies usually occur because the effect of film thickness is viscoelastic in nature, being due to the slow change of the polymer structure accompanying diffusion of the penetrant (Vrentas and Duda, 1975). In addition, physical properties may vary with depth in the polymer even though the polymer has a homogeneous chemical structure, e.g., the skin at one or both surfaces may be more dense 152 150 300 Toluene (45 ®C) Chlorobenzene (45 ®C) 100 - 200 K PS & & Z >J z 100 - ETFE ECTFE PVDF ETFE ECTFE PVDF 4000 6000 Toluene (75 ®C) Chlorobenzene (75 ®C) 3000- 4000 PS es g 2000- & z 2000 1000 - ETFE ECTFE PVDF ETFE ECTFE PVDF Figure 5.17. The permeation rates of fluoropolymers with different thickness at 45 and 75°C (□; 1 0 mil. ■: 90 mil). The unit of permeation rate (P.R.) is [grams.mil/m'.day]. 153 and impermeable than the core. As expected, the permeation rate of thin film did not match those of thick films in the aspect of inverse relationship. 2.5. Possible glass transition effect The temperature dependence of permeation rate of PFA deviated distinctly from Arrhenius-type of relationship, as shown in Figure 5.18. The solubility appeared almost constant through the entire temperature range. Additional evidence was that the sorption curve at 45 °C exhibited more distinct deviation from Fickian sorption curve compared with those at 25, 65 °C (cf. Figure 5.27). One possible cause is that it might cross a glass transition temperature (or some disordering of polymer structure) within the temperature range. PTFE is known to have a glass transition around 30 °C, and FEP and PFA are very similar to PTFE in terms of chemical and physical properties. Based on these and other observations, PFA is more like PTFE than is FEP. However, it still needs more clear evidence to be conclusive. 154 IE+02 1E+01- & IE+00- IE-01 - IE-02 0.0028 0.003 0.0032 0.0034 Temperature* *(K*1) IE-03 IE-04 0.0028 0.003 0.0032 0.0034 Temperature* *{K* *) Figure 5.18. Temperature and penetrant effects on permeation rate and solubility for PFA [□: Benzene, 0 : Toluene, o: Chlorobenzene]. 155 3. Stress-strain test Besides categorizing their inherent mechanical properties, the purpose of stress-strain testing is to understand the effects of penetrants on the following subjects: (1) Which penetrants affect strength, and/or toughness in the various fluoropolymers? (2) What is the range of effects? (3) What does the performance of stress-strain testing indicate about the nature of interactions between polymers and penetrants? Does this information correlate with the observation of transient sorption and permeation anomalies? These issues are addressed in sequence below, after the general discussion of mechanical properties. 3.1. Polymer types Teflon resins (PFA and PEP) and Tefzel ETFE do not exhibit a distinct yield point at room temperature. On the other hand, the other modified resins (PVDF and ECTFE) exhibit tensile stress-strain diagrams that are typical of semicrystalline polymers. There are four distinct regions: elastic, yield, draw region with neck, and post-neck region (Peterlin, 1973). Elastic deformation is observed below the yield point and the initial slope in the elastic region is referred to as Young's modulus. The yield point, which is associated with the initiation of neck formation, manifests itself as a distinct maximum or 156 a region of strong curvature approaching zero slope in the stress-strain curve. Next, the drop from the upper to the lower yield load occurs following neck formation. Therefore, in the draw region, an almost constant load propagates through the sample and transforms the original glassy structure into a highly oriented fibrous structure. After that, there is usually a strain hardening process resulting from molecular orientation, which increases the modulus and tensile strength. Finally the sample is broken, even though this final step could not be confirmed in most cases because of the limitation of stretching of the device used here. It was found that ECTFE thin film, oriented in the machine direction, did not exhibit a draw region associated with neck formation when the sample was stretched along the machine direction. Besides, FEP, PFA, and ETFE films do not exhibit significant neck formation during stress-strain testing. Also, fresh PFA resin appears to be more tough than fresh FEP (see Figure 5.19). 157 12.5 20 10- 15- S FEP 2.5- PFA 0-9 0-9 0 2.5 5 7.5 10 0 2 4 6 8 Strain Strain 40 2 0 - bs u 2 0 - *S5 0» s H 1 0 - MO-PVDF I MO-ECTFE TO-PVDF TO-ECTFE 0"V 1 I I------1-----1----- 0-9 0 0.5 1 1.5 2 2.5 3 0 2 3 4 Strain Strain Figure 5.19. Stress-strain curves of fresh fluoropolymers at room temperature (MO: The major direction of a test sample is machine-oriented, TO: The major direction of a test sample transverse-oriented). * The number in x-axis (strain) represents "inch" scale, and 1 inch indicates 20% expansion of the test sample in the stretching direction. * All the stress-strain tests were run in an ambient condition ( 1 atm. Room Temp.). 158 3.2. Solvent types The results from stress-strain tests in which polymers had been immersed in various solvents are summarized below in Table 5.6 and Figure 5.20. Table 5.6. The tensile properties of unsorbed and sorbed fluoropolymers. Polymer Mechanical No C A C H , C A C i CHjClj CHjCOCjH, Prooertv solvent FEP Elastic Strain N/Y N/YN/YN/YN/Y PFA Yield Stress ETFE Elastic Strain 0.072 N/Y N/YN/YN/Y Yield Stress 8.4 ECTFE Elastic Strain 0.09 N/Y N/Y N/YN/Y (Machine) Yield Stress 17.1 ECTFE Elastic Strain 0 . 1 0 0.30 0.33 0.3 0.32 (Transverse) Yield Stress 17.0 9.0 9.4 9.4 8.3 PVDF Elastic Strain 0 . 2 2 0.38 0.39 0.45 0 . 8 (Machine) Yield Stress 29.7 28.8 26.4 23.5 25.5 PVDF Elastic Strain 0.18 0.39 0.34 0.4 0.67 (Transverse) Yield Stress 32.0 28.5 28.1 24.3 25.6 * Elastic strain indicate the strain at yield point. * N/Y denotes “no apparent yield point.” In FEP/PFA tensile tests, there was no significant change of the stress-strain diagram when tested after immersion in various types of solvents, ranging from nonpolar to highly polar chemicals. A very slight change of elastic response (e.g. decrease of the elastic modulus at the initial strain range) was detected in sorbed PFA films. 159 FEP PFA 1 0 - 7.5- I .0- 5 - pa- B 2.5-" 2 3 4 5 2 3 4 5 S train S train ETFE 15- I 1 0 - B f- 2 3 4 5 S train Figure 5.20. Stress-strain curves of impregnated fluoropolymers by toluene (□; Fresh sample, o; Exposed sample). 160 40 40 MO-PVDF TO-PVDF 30- 30- : I 09 — a - "5B I 10- 10 - 0-9 ---- 1----- 1-----1-----1----- 1----- 0.51.5 2 2.5 0 0.5 I 1.5 2 2.5 3 Strain Strain MO-ECTFE TO-ECTFE 2 0 - 15- æ 10 - o c -o- g H 5-, 5 - 0-9- 0-9^ 0 2 3 4 0 1 2 3 4 Strain Strain Figure 5.20. (continued) (□: Fresh sample, 0 ; Exposed sample) 161 ETFE exhibited higher tensile strength than PFA and FEP resins. A barely noticeable flat inflection point exists early in tensile test of fresh ETFE sample, and the slope in this elastic region is quite steep. However, ETFE following immersion exhibited no inflection point, rather it exhibited a smooth transition between the initial elastic and the draw region. The impregnated ECTFE film exhibited a significant alteration of tensile properties. The most striking change was the disappearance of its yield point in the machine-oriented (MO) sample, and the tensile strength increased more gradually and smoothly with the increase of strain at constant rate. If the impregnated ECTFE film was stretched along the transverse direction, a significant change of stress-strain diagram was also observed: the yield stress decreased by -50 % of the original value and the strain at the yield point increased up by 300% (see Table 5.6). In PVDF tensile tests, two types of samples were tested. The first sample was one from the same lot of material which was used in sorption and permeation experiments. The other was a sample from a different lot which was only used for tensile testing and is called “new sample.” For old PVDF samples, there did not appear to be any distinct differences between machine-oriented (MO) and transverse-oriented (TO) samples except that the bell shape around the yield point is sharper in the TO-sample than in the MO- sample. The impregnated samples exhibited slight decreases of yield stress and slight increases of the elastic strain. For highly plasticizing solvents like M.E.K., the change was more significant. Especially, the bell shape almost disappeared in MO-PVDF. In TO- PVDF, this shape remained, but became much more flattened. 162 For new PVDF samples, enormous changes were observed in the TO-sample. The fresh TO-sample was so brittle that it broke as soon as tensile test started. The impregnated sample, however, exhibited much more ductile behavior, as did the old sample (see Figure 5.21). 3.3. Successive impregnation ECTFE samples were also tested in order to see the plasticization effect on mechanical properties with the increase of impregnation time. As expected, the tensile properties changed gradually during impregnation: the yield stress decreased, and the elastic strain increased (see Fig.5.22). Table 5.7 The tensile properties of gradually impregnated ECTFE by toluene. Stretching Tensile Properties Imoreenation Time / Eouilibrium Time direction 0 0.33 0.67 1 Machine Soecified Strain 0.5 0.5 0.5 0.5 Yield Strain 16.0 13.3 12.8 12.3 Transverse Elastic Strain 0.1 0.15 0.25 0.3 Yield Strain 17.0 12.5 10.9 9.0 3.4. Repeated exposures A number of ECTFE samples were run in tensile tests after sorption, desorption, and resorption to observe the effect of the repeated exposures. The yield stress and the elastic strain were increased by small amounts with the increase of repeated exposure numbers (see Figure 5.23). This simply indicated that the polymer samples became more stiff during impregnation This result may be related to the delayed attainment of ultimate 163 TO-PVDF (new) V3 -o* --0 - I ■— — 6 Strain Figure 5.21. Stress-strain curves of “new” PVDF samples impregnated by various organic solvents (□: No solvent. 0: Toluene, o: Dichloromethane, a ; M .E .K .). 164 MO-ECTFE 2 0 - IO) ,s- 0) I 0 1 2 3 4 Strain TO-ECTFE 15- 10 - I J I 0-9 0 2 4 6 8 Strain Figure 5.22. Stress-strain curves according to the gradual increase of impregnation time by toluene (□; 0, 0; 1/3. o: 2/3. A: 1). 165 TO-ECTFE 2 0 - æ 15- 5 - 0 0.25 0.5 0.75 1.25 Strain MO-ECTFE 25- 2 0 - w 15- V S 10- H 5 - 0.5 1 1.5 Strain Figure 5.23. Stress-strain curves of ECTFE samples repeatedly exposed by toluene (□: Fresh sample.^ : 1st exposure, o: 2nd exposure). 166 equilibrium in repeatedly exposed samples due to the increase of resistance to penetration of small molecules at least in the initial stage of sorption. The reason may be that during repeated exposures rearrangement of polymer chains occurs. Another potential cause is the loss of additives (e.g. plasticizer) during sorption-desorption cycles. ECTFE samples showed a significant mass loss after desorption (-0.05 wt%) beyond the initial mass. Table 5.8 The tensile properties of repeatedly exposed ECTFE by toluene. ECTFE Tensile Repeated Exposure Times Properties 0 1 2 Machine Elastic strain 0.11 0.15 0.15 Yield stress 21.2 24.4 25.4 Transverse Elastic strain 0.11 0.13 0.12 Yield stress 20.1 21.9 22.2 * Original sample was also aged at 75 °C for 5 days. 3.5. Plasticization effects For ECTFE, the plasticization effects of several chemicals were almost identical as shown in Figure 5.24. This observation may give some clue in explaining the almost identical shape of sorption curves in ECTFE film, irrespective of solvent types. To the contrary, in PVDF film a significant change in the tensile stress-strain curve was observed especially in the case of M.E.K., as shown in Figure 5.25. This change may be responsible for the generation of a more steep sorption curve in the PVDF-M.E.K. system (cf. Figure 5.5). 167 MO-ECTFE 2 0 - V) E H 1 0 - 0 -ÿ 0 2 3 4 5 Strain TO-ECTFE 15- 0) B 2 4 6 8 Strain Figure 5.24. Tensile stress-strain curves of ECTFE over various organic chemicals (□; No solvent, 0; Toluene, o; Dichloromethane, a ; M .E .K .). 168 c o . ^ , m o -p v d f 30 H I 20 4 o "w H 10- 0.5 1.5 2 2.5 Strain 30 H o TO -PV D F / i r \ -OA £ à /q 6 ...... 9^ « E H 1--- 2 S train Figure 5.25. Tensile stress-strain curves of PVDF with various organic chemicals (□; No solvent, 0; Toluene, o; Dichloromethane, M.E.K.). 169 B. DISCUSSION OF EXPERIMENTAL RESULTS 1. Analysis of transient sorption Various sorption curves in semicrystalline fluoropolymers were presented in Section V.A.l. By varying the penetrants and polymers, temperature, and external solvent concentration, a broad range of sorption behavior has been observed from ideal Fickian diffusion to relaxation controlled kinetics. 1.1. Fickian sorption Experimental evidence of Fickian diffusion in FEP and PFA films has been observed for poorly interacting solvents (e.g. those exhibiting low solubility in the polymer) and even for inherently more interacting penetrants such as M.E.K. and Dichloromethane. Moreover, the solubility of organic solvents in those resins is very low compared with the other fluoropolymers. Low solubility and Fickian sorption kinetics imply that generally accepted mechanisms, such as an activated solution-diffusion theory, could describe the transport of penetrants in those resins as discussed in Section n.B.2.2. The observed Fickian sorption in FEP and PFA resins indicates the absence of significant structural relaxation caused by plasticizing effects of organic chemicals. In this respect, these resins have good barrier properties against harsh organic chemicals, and the permeation behavior can be easily predicted by classical methods, e.g., Fick's law for diffusion kinetics and Henry's law for solubility. 170 1.2. Non Fickian sorption Partially fluorinated polymers produce various anomalous sorption behaviors. The deviation from normal Fickian sorption appears to increase as the following order: ETFE-PVDF -ECTFE. The main cause of non-Fickian diffusion in partially fluorinated polymers seems to arise from polar interchain attractions caused by substituent elements. As fluorine atoms are replaced by hydrogen, the substituted hydrogen atoms become increasingly electrophilic in nature. The reason is as follows. The fluorine atom has a strong electron- withdrawing power, leaving the carbon, to which it is attached, in an electron-deficient state. The decrease in electron density around the carbon atom is compensated, in part, by a shift of the electrons away from the hydrogen, increasing the proton character of this atom. This becomes even more apparent when one considers the relation of position and proximity of fluorine to hydrogen in reactivity toward H-bonding capable amine compounds (Bro, 1959). Thus, a 1:1 random copolymer of tetrafluoroethylene and ethylene which is considered a head-to-head type (ETFE), is much less reactive toward H-bonding capable solvents than is head-to-tail type (PVDF), yet their elemental analyses are identical. In this case, the carbon bearing hydrogen which is adjacent to only one carbon carrying fluorine (e.g. hydrogen in PVDF), is less electrophilic in character than counterparts (e.g. hydrogen in ETFE). In ECTFE, on the other hand, one fluorine atom is replaced by a chlorine atom which has less withdrawing power than fluorine. The chlorine atom disturbs the electronic balance within intrachain segments rather than increases the protonic character of neighboring hydrogens, and as a result dipole-dipole interactions between 171 interchain segments increase. In conclusion, the inter-segmental bonds in PVDF are more like H-bonding. The inter-segmental bonds in ECTFE are more like dipole-dipole interactions. The character of the inter-segmental bonds in ETFE seems to be less polar than in ECTFE, and less protonic than in PVDF. Therefore, it is expected that the increase of polar inter-segmental attraction produces a more compact packing of interchains (and a high cohesive energy density). That, in turn, increases the apparent activation energy for diffusion and thereby suppresses the diffusion coefficient. Actually, vapor sorption experiments indicate that ECTFE and PVDF can be more effective barrier membranes against gases or solvents of low activity, compared with Teflon and Tefzel resins. At high penetrant activity, however, the segmental mobility in partially fluorinated polymers increases due to the disruption of polar inter-segmental bonds, and consequently these polar polymers become more vulnerable to the attack of chemical solvents. In addition, the creation of free volume due to the disruption of inter-segmental bonds may induce the time dependence of both diffusivity and solubility. ETFE Even though a very slight inflection point in the sorption curve was detected, Tefzel ETFE could be approximated by Fickian diffusion kinetics. The degree of deviation from normal Fickian sorption was very small compared with ECTFE and PVDF. The reason is that the polymer structure appears to be more uniform in that there is no apparent skin. Moreover, the strength of interchain interaction may be so weak 172 that relaxation effects could be minimized. In summary, the relaxation-controlled structural change is not large enough to make a significant contribution to anomalous mass uptake via free volume creation, or the time scale of relaxation is so small enough to neglect relaxation effects on diffusion process. ECTFE In ECTFE, the relaxation at the final stage of uptake occurred very quickly and led to rapid uptake of solvent. Also, rapid acceleration accompanied rapid expansion in the machine direction. This feature was first reported for sorption of n-hexane in polystyrene by Jacques et al. (1974) as the Super Case II sorption. As previously mentioned in Section V.A.l.4., the sorption behavior of both types of rectangular samples exhibited rapid mass uptake during the final stages of sorption, but the uptake rate of the T-sample was a little bit faster than that of the M-sample. In addition, the expansion in the major direction of the M-sample was still zero, in contrast with the relatively small expansion of the T-sample even at initial stages of sorption. These results clearly indicate that swelling stress controls the relaxation process coupled with mass transfer. T-sample is more prone to be affected by transverse swelling stress imposed by the dull side. The discrepancy between T- and M-samples decreases as the ratio of length to width increases or the number of repetition of exposure increases. The results also indicate that diffusion process in ECTFE films is strongly coupled with structural deformation. 173 Another suspected cause of anomalous sorption is the existence of dissimilar surfaces. This factor was discussed in the context of experimental results (e.g. in Sections V.A.1.5. and V.A.2.2.). As was mentioned there, the shiny surface seems to be more dense and tough than the dull side. During sorption, the dull side expands more than the shiny side, and the dissimilar expansion results in curling towards the shiny side. During the final stage of sorption, the tough shiny side relaxes, possibly aided by the tension caused by the stress imposed by the opposite swollen dull side. From a macroscopic viewpoint, it may be said that the origin of accelerated mass uptake comes from rapid expansion in the machine direction during the final stage of sorption, and the rapid expansion provides the space for additionally incoming penetrants. The onset of rapid relaxation is possibly retarded by the more tough (shiny) side under different conditions. In general, high temperature, high concentration, and strongly plasticizing solvents produce a more rapid onset of relaxation. As mentioned previously, ECTFE has inter-segmental bonds of dipole-dipole type which is basically formed by balanced electronegativities between chains. It is believed that the stiff ECTFE chains are oriented in parallel arrays, maximizing the accessibility of substituent groups to interchain polar attraction, such that the disruption of one inter-chain polar attraction induced by one polar solvent molecule greatly facilitates the disruption of all possible bonds between two interacting chains, as in unzipping a zipper. From a microscopic viewpoint, the disruption of one interchain bond caused by one polar solvent molecule can influence on neighboring interchain bonds by inducing the disturbance of intrachain electronic balance. Ultimately, the disruption of all possible 174 bonds between two interacting chains is exponentially facilitated as the sorption proceeds. This may result in greater mobility for the polymer molecules and lead to the accelerated mass uptake during the final stage of sorption. PVDF For convenience, the solvents used in PVDF sorption experiments are divided into three types: (1) Solvent I : non-plasticizing solvent (phenol) or vapor with low concentration (benzene vapor). (2) Solvent H : slightly plasticizing solvent (benzene, toluene, and chlorobenzene) (3) Solvent IE: highly plasticizing solvent ( M.E.K. and dichloromethane). For solvents of type I, a very slight inflection point was observed in the sorption curves. The slight inflection point indicates that a slight structural change occurs during diffusion. For solvents of type n, the observed sorption curve exhibits sigmoidal shapes followed by asymptotic approach to the final equilibrium uptake. For solvents of type III, the sorption behavior resembles the sorption curve of ECTFE except for the slight tendency toward asymptotic approach to the final equilibrium uptake. In contrast with ECTFE, PVDF does not exhibit the characteristic of a tough skin even if it has a shiny surface. That is, there was no significant distortion (or curling) during sorption. It was also observed that the longitudinal expansion of PVDF was linearly 175 dependent on mass uptake, as shown in Figure 5.8. The characteristics due to orientation in the machine direction were not clearly observed in PVDF. The ratio of expansion in the (inferred) machine direction to that of the (inferred) transverse direction was, however, about 2. In contrast with the almost identical sorption curves in ECTFE irrespective of solvent types, PVDF exhibited various types of sorption curves depending on the solvent types. As mentioned previously, PVDF has more H-bonding characteristics. In general, H-bonding is stronger and more localized at the interaction sites, compared with dipole forces. Therefore, the disruption of one interchain bond may be much less influential on the neighboring interaction sites in contrast with ECTFE case. In conclusion, it can be said that the sorption behavior of PVDF is mainly determined by the degree and strength of interactions between polymers and penetrants, not by a combination of structural factors due to chemical composition and processing conditions like ECTFE. 176 2. Analysis of permeation The three aromatic solvents (benzene, toluene, and chlorobenzene) behave similarly for PFA, ETFE, and PVDF. Toluene and chlorobenzene exhibit about the same permeation rate but lower than benzene. In ECTFE, however, toluene exhibits a lower permeation rate than chlorobenzene. For highly polar solvents such as dichloromethane, ECTFE exhibits most susceptibility for permeation, followed by ETFE, PVDF and PFA. However, PVDF exhibits the most susceptibility to H-bonding capable M.E.K. and phenol, followed by ETFE, ECTFE and PFA. In any case, PFA manifests itself as a highly resistant barrier against all types of solvents. On the contrary, modified fluoropolymers respond differently depending on the types of solvents. ETFE is always somewhat more susceptible to each class of solvent. Therefore, it is again confirmed that the character of interchain bonds of PVDF are more protonic, and the interchain bonds of ECTFE have the character of dipole-dipole interactions. Finally ETFE has both characters, but the strength of interchain bonds is less than that of PVDF in protonic character, and less than that of ECTFE in dipole-dipole interactions. According to eq.(4), the apparent activation energy of permeation was obtained and summarized in Table 5.9 and plotted in Figure 5.26. Table 5.9 Activation energies from Arrhenius plots (Ln(P.R.) vs. Temp. ' (K'')). Polymer Permeation Activation Enersv (cal/mole) Benzene Toluene Chlorobenzene ETFE 234.287 R 130.890 R 129.888 R ECTFE 1.484.848 R 656.387 R 925.435 R PVDF 75.041 R 38.852 R 37.709 R * R is the gas constant (1.987 cal/mol."K) 177 IE+03 IE+03 ETFE ECTFE IE+02- IE+02- ^ lE+OI - z lE+01 - lE+00- lE+00- lE-OI lE-OI 0.0028 0.003 0.0032 0.0034 0.0028 0.003 0.0032 0.0034 Temperature' ^(K' Temperature* 1(K" %) IE+02 PVDF lE+OI - & IE+00- lE-OI- lE-02 0.0028 0.003 0.0032 0.0034 Temperature* ^(K* *) Figure 5.26. Arrhenius plots of permeation rates of aromatic solvents in fluoropolymers (□; Benzene, 0 ; Toluene, o; Chlorobenzene). 178 The obtained activation energies indicate that in ETFE and PVDF the same activated solution-diffusion mechanism is operating for aromatic solvents, in that the activation energy is about the same for both toluene and chlorobenzene, and it doubles for benzene. But, in ECTFE a dissimilarity of activation energies between toluene and chlorobenzene is apparent (see Table 5.9). This indicates that a different permeation mechanism is operating for ECTFE in comparison to PVDF and ETFE. The exposure of the tough skin to the penetrant delays the onset of rapid relaxation in comparison to the dull side. Also, it is confirmed that the shiny side is less permeable and exhibits lower solubility than the dull side (see Table 5.10). Table 5.10 Comparison of permeation rate and solubility between soft (dull) side and tough (shiny) side in thin ECTFE. Properties Temp. Benzene Toluene Chloro )enzene (oC) .. S . T S T s T Permeation Rate 25 1.1908 1.0506 0.7897 0.6907 1.4061 1.2833 (gram/m*.day) 45 22.376 17.221 11.368 9.5469 22.503 19.502 Sorbed Mass+ 25 0.0521 0.0472 0.0440 0.0429 0.0635 0.0559 (gram/gram) 45 0.0594 0.0531 0.0499 0.0459 0.0672 0.0598 * S: Soft side, T: Tough side * +: The mass change of polymer disk used in permeation experiments. Another peculiar anomaly was observed, which is called here “overshoot permeation.” This phenomenon also implies that some kind of run-away chemical reaction (or structural relaxation) occurs during penetration of a solvent into ECTFE film. The overshoot of the permeation rate cannot be explained by the mechanism of Case II sorption which often leads 179 to Super Case II sorption phenomena. The overshoot may originate from a combination of structural factors due to chemical composition and processing conditions. It may also be evidence of microvoids that initially convey material, but which shrink as the polymer swells 180 3. Analysis of stress>strain tests Stress-strain tests are probably the most widely used of ail mechanical tests. They are very important and practical ones which engineers have to understand. However, the relationship of this test to actual applications is not as clear as is generally assumed. Because of the viscoelastic nature of polymers with their sensitivity to many factors, stress-strain tests are, at best, only a rough guide to how a polymer will behave in a finished object. The action of solvents on polymers is in many ways similar to that of heat. Appropriate solvents can form secondary bonds to the polymer chains, and they can penetrate and replace the interchain secondary bonds, and thereby pull apart and dissolve linear and branched polymers. Therefore, it is quite interesting to look in detail at the deformation of the polymer below the yield point (often called the elastic region) because the plasticizing effect is predominantly exhibited in this elastic region. Some stress-strain tests have been done on some of the fluoropolymers, before and after exposure to the penetrants. A compilation of experimental stress-strain curves is in Appendix C. These tests have allowed the nature of the transport properties to be viewed in the framework of mechanical properties. Teflon resins (PFA and PEP) and Tefzel ETFE exhibit no (or a very slight) change of their stress-strain curves when they are impregnated with organic solvents. The slight inflection observed in the ETFE sorption curve may relate to this slight change of elastic modulus in the stress-strain diagram. However, these films seem to maintain their structural integrity quite well, and their transport properties are also not significantly affected 181 even though the slight change of mechanical properties did occur. ECTFE films show drastic changes of their stress-strain diagrams after they are impregnated with solvents. Especially, the yield point of machine-oriented (M0-) sample is not detected and, instead, the tensile stress continuously increases until the sample was broken. In addition, the sharp peak in the stress-strain curve of the transverse-oriented (TO-) sample become much more flattened. These behaviors are shown in Figures 5.20 and 5.24. From the stress-strain tests of ECTFE, it is concluded that the significant change of mechanical properties must be related to the significant change of transport properties in a way to the accelerated uptake at the final stage of sorption. PVDF also exhibits different trends in its stress-strain curves, depending on the types of solvents. Aromatic solvents induce slight changes of tensile properties of PVDF. The yield stress does not seem to be affected, but the elastic strain slightly increases. This change must be related to the anomalous sigmoidal sorption behavior. In particular, M.E.K. produced a significant alteration of the stress-stress curve. The yield stress was reduced by 80% of its original value and the elastic strain was increase by a factor of 4 ~ 5 (cf. Table 5.5). This drastic change is also reflected in the sorption behavior of the PVDF-M.E.K. system as shown in Figure 5.5. The degree of change becomes larger as the degree of plasticization effects become greater. These behaviors are shown in Figures 5.20 and 5.25. 182 C. DISCUSSION OF MATHEMATICAL MODELS The transport behavior of organic chemicals in semicrystalline fluoropolymers spans the range from classical Fickian to highly anomalous behavior. For the Fickian sorption behavior, sorption rate data are analyzed via the methods discussed in Section II.B. It was confirmed experimentally, however, that the structural deformation (or relaxation) of the polymer can strongly influence the transport properties of polymer- penetrant systems. Since sorptive capacity appears to be controlled by the available free volume in the polymer, the increase in mass uptake beyond the Fickian equilibrium sorption, may be taken as a direct measure of the volume increase of the polymer due to structural relaxation. In this section, the new kinetic models developed in Section n.C and n.D are applied to interpret the anomalous sorption behavior. 1. Fickian diffusion The sorption kinetics of FEP, PFA, and ETFE may well be described by simple Fickian mathematics with either a concentration-independent or a concentration-dependent diffusion coefficient. The slight inflection point in the soiption curve may indicate some surface concentration relaxation, but it may be neglected. The rate of transfer of a diffusing substance is given by eq.(33): (33) where / is the thickness of the sample and c ,, c, are the surface concentrations at each side. The surface concentration exposed to air, c, , is assumed to be zero. Thus, the 183 diffusivity from the simple Fickian diffusion equation combined with the solubility can predict the permeation rate. Some calculated values and corresponding experimental results are displayed in Table 5.11. Table 5.11 Comparison the calculated value with the experimental results of FEP- benzene system. Temperature Calculated Values Experimental CQ Solubility Diffusivity Flux Flux 25 0.0079 0.14 0.0484 0.0443 45 0.0094 1.028 0.4227 0.4717 65 0.0106 4.895 2.2698 * Units: Solubility (g/cm^), Diffusivity (cm^/day), Flux (g/m^.day). The results are in good agreement with experimental data. This means that transport behavior of organic chemicals in FEP thin films can be accurately predicted by Pick's diffusion equation. The sorption kinetics of PFA also shows a sorption curve of almost perfect Fickian shape except the case of 45 °C (see Figure 5.27). The sorption behavior of ETFE thin films exhibit concentration-dependent diffusivities and indicate slight relaxation effects. For this case, the continuity equation (eq.( 2 1 )) with a concentration-dependent diffusivity and a time-dependent boundary condition can be used to predict the sorption behavior of ETFE. The surface concentration is assumed to be controlled by first order relaxation kinetics as given by eq.(43). The adjustable parameters are then initial surface concentration, surface relaxation coefficient and plasticizing coefficient. 184 o o o (0 0) 6 o N d .S' o m d o N d O 0.0 0.5 1.0 1.5 2.0 2.5 3.0 S Q R T ( D a y s ) Figure 5.27. The fitted data of PFA-toluene at 25 °C by simple Fickian equation. [Q Benzene, Toluene, o. Chlorobenzene] 185 o o N O (0 0) 6 2§ s 2 o N d o o o o 0.0 0.2 0.3 0.5 0.7 0.8 1.0 S Q R T ( D a y s ) Figure 5.27. (continued) (45 °C ) [Q Benzene, (}: Toluene, o. Chlorobenzene] 186 o o o CO 0) d o N d o m 6 o Cl d 0 •/ 1 i l ° 0.00.1 0.2 0.3 0.5 0.6 0.7 S Q R T ( D a y s ) Figure 5.27. (continued) (65 °C ) [□: Benzene, 0: Toluene, O: Chlorobenzene] 187 = ^ s i +( ~ Q )(1 - where Cjj = instantaneous surface concentration, = equilibrium surface concentration, P = surface relaxation coefficient, and Y d = plasticizing coefficient. However, this numerical fitting is quite tedious and complex because three adjustable parameters are needed to fit data. Without consideration of surface relaxation, the sorption curve of ETFE was analyzed by Pick's equation with concentration dependent diffusivity. The employed intrinsic diffusivity (D^) was obtained from separate vapor sorption experiments, in which the concentration interval was short and sorption behavior of ETFE was nearly Fickian. The only adjustable parameter, Y d > was easily determined as 4.86. The fitted curve is shown in Figure 5.28. 188 N 0) d cr .4> (0 d CO d o d 0.0 0.1 0.3 0.4 0.5 S Q R T (D ays) Figure 5.28. The fitted data of ETFE-benzene at 45 °C by a concentration dependent Pick’s equation. 189 2. Non-Fickian diffusion The sorption behavior of PVDF and ECTFE was analyzed by kinetic model equations presented in Section II.D. The sigmoidal sorption curve of PVDF is well fitted by the quadratic type of rate equation that was developed in Section II.D.2.2. On the contrary, the acceleration feature of ECTFE is well fitted by the exponential type of rate equation that was developed in Section n.D.2.1. Quadratic type o f rate equation For the quadratic type, there exist four adjustable parameters; (1) a = Fraction of total uptake that occupies the newly created free volume. (2) Dj (cm^/day) = Initial diffusivity or intrinsic diffusivity. (3) kfg (1/day) = Rate coefficient. (4) 6 = Diffusivity ratio, (D_ - DJ/D^. The value of D^ was obtained directly from the initial slope of the sorption curve based on Fickian response fraction, a (cf. eq.(25)). So, this value should be re-evaluated considering total mass uptake: D^ = Dj (1-a)^ (Jacques et al., 1974). The values listed in Table 5.12 represent the observed sorption curves quite well, as shown in Figure 5.29. 190 o o N tH o (0 m d o N d o m d o N d o o > q ^ ° 0.0 1.4 2.9 4.3 5.7 7.1 8.6 S Q R T (D a y s) Figure 5.29. The fitted data of PVDF-aromatic liquid sorption at 25 °C by quadratic type of rate equation [□: Benzene, O: Toluene, o: Chlorobenzene]. 191 o o w o (0 m 6 o w cr 6 u o CO d 0 ■<# 01 d o o > q o 0.0 0.4 0.8 1.2 1.6 2.1 2.5 S Q R T (D ays) Figure 5.29. (continued) (45 °C) [□: Benzene, 0; Toluene, o; Chlorobenzene]. 192 o o N O © d o OJ cr d o N d o 0.0 0.2 0.4 0.6 0.8 0.9 1.1 S Q R T (D ays) Figure 5.29. (continued) (65 °C) [□: Benzene, 0: Toluene, o: Chlorobenzene]. 193 Table 5.12 Fitted rate parameters of penetrants in PVDF. Temperature Parameter Fitted Parameter Values for PVDF f 10 mil) CO Benzene Toluene Chlorobenzene a 0.699 0.672 0.756 D ^xior 25 0.016 0.011 0.006 0.013 0.006 0.011 DJD, 27.84 35.09 12.12 a 0.725 0.766 0.787 45 D^xior 0.280 0.162 0.095 0.174 0.148 0.208 DJD, 2 1 . 1 2 15.11 8.300 a 0.650 0.742 0.743 65 D „xl(r 1.915 0.973 0.806 4 . 1.200 1.257 1.220 DJD, 25.46 13.00 9.128 As temperature increases, the initial diffusivity and relaxation coefficient, , and relaxation coefficient, , increase. Regardless of temperature, the amount of uptake due to relaxation, a, is about 0.7-0.8. Even though this model equation agrees well with the observed sigmoidal sorption curve, it is premature to claim that physical understanding can be drawn from this equation. To validate the model equation, the permeation rates are calculated from fitted parameters and compared with experimental permeation data. The flux equation can be described with the exponential type of diffusivity dependence on concentration, with regards to eqs.(37) and (38): (80) Based on the above equation, the permeation rate was calculated and compared with 194 experimental data. The deviation from experimental data ranges were within ± 30% except for benzene (almost two times of experimental values) (see Figure 5.30). The result is remarkable in that permeation rate can be predicted from purely fitted parameters for sigmoidal sorption and the discrepancy is acceptably small. Thus, the quadratic type of rate equation is able to shed light on the prediction of transport properties quantitatively even from anomalous sigmoidal sorption curves. 195 lE+02 lE+02 lE+OI- lE+01- d CL z 2 lE+00" lE+00- lE-01- lE-01- 0.0028 0.003 0.00320.00340.0028 0.003 0.0032 0.0034 Temperature* ^(K* 1) Temperature* ^(K" lE+02 lE+01- w lE+00- lE-01- lE-02 0.0028 0.003 0.0032 0.0034 Temperature* *(K* Figure 5.30. The comparison of calculated permeation rates from kinetic model with experimental data from PVDF-penetrant systems. [Q calculated value, experimental result ] 196 Exponential t\pe o f rate equation The exponential type of rate equation was used to fit the rapidly accelerating sorption curves of ECTFE (see Figure 5,31). Four adjustable parameters are also needed to fit that data: (1) a = Fraction of total uptake that occupies the newly created free volume. (2 ) Dg(cmVday) = Corrected initial diffusivity. (3) kfo (I/day) = Rate coefficient. (4) Yk (cmVgram) = Sensitivity coefficient of rate parameter. Table 5.13 Fitted rate parameters of penetrants in ECTFE. Temperature Parameter Fitted Parameter values for EC'FEE no mill (°C) Benzene Toluene Chlorobenzene a 0.656 0.643 0.678 25 D ^xior 0.0245 0.0177 0.0151 0.0325 0.0201 0.0295 Y* 90 101 72 a 0.678 0.734 0.632 45 D ^xl& 0.625 0.244 0.472 & 1.06 0.609 0.771 Yt 6 8 76 58 a 0.675 0.675 0.607 65 D ,x l(r 5.28 3.094 3.83 11.4 7.21 7.59 Y* 47 59 47 197 o o o (0 03 d o N S' d o co t d o W d o o ^ q o 0.0 0.8 1.5 2.3 3.1 3.8 4.6 S Q R T (D a y s) Figure 5.31. The fitted data of ECTFE-aromatic liquid sorption at 25 °C by exponential type of rate equation [□; Benzene, 0 ; Toluene, o: Chlorobenzene]. 198 o o N O o d o w d §r O N d o o ^ o 0.0 0.2 0.3 0.5 0.6 0.8 0.9 S Q R T (D ays) Figure 5.31. (continued) (45 °C) [□; Benzene, 0: Toluene, o: Chlorobenzene]. 199 o o N O (0 05 d o w d o N d § L o 0.0 0.1 0.1 0.2 0.2 0.3 0.4 S Q R T (D ays) Figure 5.31. (continued) (65 °C) [□: Benzene, O: T o lu e n e , o: Chlorobenzene]. 2 0 0 The amount of uptake due to relaxation is about 0.6 - 0.7, and the initial diffusivity and relaxation coefficient increase by almost 20 times as temperature increase. The relaxation sensitivity coefficient, , decreased with the increase of temperature (see Table 5.13). This clearly indicates that relaxation effects decrease as the temperature increases, possibly due to diffusion-limited relaxation. Even in Case II sorption, sorption mechanism changed from relaxation-controlled diffusion to diffusion-limited relaxation (or Fickian diffusion) as temperature increased (Thomas and Windle, 1981, 1982). As seen in Figure 5.31, the acceleration feature slightly diminished as the temperature increased. Finally, the obtained physical parameters are plotted by the Arrhenius relationship (see Figures 5.32 and 5.33), in which the linear relationship was well predicted. The theoretically obtained initial diffusivity gave reasonable values ( 10 '" ~ 10'’“ cmVsec) which are typical of glassy polymers. In addition, the predicted diffusivity in PVDF increased as the size of penetrant decreased. In ECTFE, predicted diffusivity of benzene is higher than the other penetrants at any temperatures. 201 lE+Ol lE+OI TOLUENE lE+00 BENZENE IE+00 «2, IE-01- > lE-OI lE-02 Diffusivity IE-02 Diffusivity IE-03 - Rate Coefficient o IE-03 Rate coefficient IE-04 IE-05 IE-05 IE-06 IE-06 0.003 0.00320.0028 0.0034 0.0028 0.003 0.0032 0.0034 Temparature* ^(K" T em parature' ^(K" lE+Ol- lE-03 CHLOROBENZENE IE+00- PVDF là lE-OI- y& IE-02- u Diffusivity b IE-03 - 5 lE-05 O Rate coefficient z m3 Q IE-04- Benzene Z -a lE-06 IE-05- Toluene IE-06- Chlorobenzene IE-07- T------r IE-07 0.0028 0.003 0.0032 0.0034 0.0028 0.0030.0032 0.0034 Temparature* ^(K* 1) Temparature* ^(K* Figure 5.32. The Arrhenius plot of diffusivity and rate coefficient obtained from fitting PVDF sorption data. 2 0 2 lE+02 IE+01 lE+01- BENZENE TOLUENE IE+00- lE+00- « lE-01 lE-01 Diffusivity r Diffusivity U IE-02 lE-02 Rate coefficient Rate coefficient lE-03- lE-03 - Z lE-04- lE-04- lE-05 IE-05 lE-06 IE-06 0.0028 0.003 0.0032 0.0034 0.0028 0.003 0.0032 0.0034 Temperature* ^{K* *) Temperature* ^(K* lE+02 lE-03 CHLOROBENZENE lE+01- ECTFE IE+00- lE-04- lE-OI p Diffusivity Q IE-02 Z Rate Coefficient U lE-03 lE-05 Benzene z lE-04 Toluene lE-05 Chlorobenzene lE-06 IE-06 0.0028 0.003 0.0032 0.0034 0.0028 0.003 0.0032 0.0034 Temperature* ^(K* Temperature* ^(K* h Figure 5.33. The Arrhenius plot of diffusivity and rate coefficient obtained from fitting ECTFE sorption data. 2 0 3 Application of exponential type of rate equation For the exponential type of rate equation suggested in Section II.D.2.1, it is impractical to predict permeation rates from purely fitted parameter values. In particular, the parameter KCy/Yd) ^nd (D./Dg) in eq.( 6 8 b) are so closely related that these two values cannot be obtained independently by curve fitting. To get theoretical P.R. for ECTFE, the value of Yd value should be determined via a series of differential sorption experiments. Some anomalous sorption curves of ECTFE, however, can be used to validate this model equation. For example, the initial diffusion coefficient, , in repeatedly exposed samples appears to be consistently smaller than that of fresh samples, based on the experimental observations. Likewise, the relaxation fraction, a, in presorbed samples also appears to be consistently larger than that of fresh samples. The fitted values are well matched with the data and are summarized in Table 5.14. The fitted curves and data are also shown in Figures 5.34 and 5.35. 2 0 4 Table 5.14 Fitted parameters for sorption experiments with ECTFE Solvent (°C) Parameter Fitted Parameter values Fresh Samole Presorbed a 0.656 0.816 Benzene D i x l t f 0.207 0.389 (25 °C) 0.0325 0.052 tk 90 90 Solvent (°C) Parameter Fresh Samole 1st Exoosed 2nd Exoosed a 0.734 0.618 0.543 Toluene 3.45 1.13 0.680 (45 °C) 4b 0.609 0.408 0.256 76 77 79 2 0 5 o o o 0.0 0.8 1.6 2.4 :3.2 4.0 S Q R T (D ays) Figure 5.34. The fitted curves of presorbed ECTFE sample by benzene at 25 °C (^:Fresh sample, Q Presorbed sample). 2 0 6 N O CO 0) 6 o N d cr o> o CO d o N d o o > o o 0.0 0.3 0.6 0.9 1.2 1.5 1.8 S Q R T (D ays) Figure 5.35. The fitted curves of repeatedly exposed ECTFE samples by toluene at 45 °C (□:Fresh sample, 0 : 1st exposed sample, o: 2nd exposed sample). 2 0 7 D. GENERALIZATIONS It is intuitive to expect that the permeation rate, solubility, and diffusivity correlate with general physical parameters such as molar volume (see Section II.A.3.1), polarity (see Section II.A.3.2), solubility parameters (see Section II.A.2.1), and latent heat of penetrating solvents (see Section II.A.3.1). This section examines the dependence of those transport properties on the proceeding thermodynamic properties. The object is to discern whether consistent relationships exist for the various penetrant and polymer combinations. PFA It is known that Teflon PFA resin is nonpolar and highly resistant to chemical attack (cf. Section III.A and V.A.2.I). As expected, PFA exhibited clear dependence of transport properties on a geometric factor, molar volume, rather than on electronic factors such as polarity or solubility parameters, as shown in Figure 5.36(a-d). In particular, diffusivity, solubility, and permeation rate decreased in an exponential manner with the increase of molar volume, as shown in Figure 5.36(a, e, g). ETFE Among the fluoropolymers tested, ETFE did not exhibit any clear dependence of transport properties on geometric, electronic factors, or solubility parameters, as shown in Figure 5.37(a-f). Rather permeation rate and solubility showed complicated dependence or irregular dependence on penetrant properties. 2 0 8 ECTFE In contrast, ECTFE exhibited relatively clear dependence on electronic factors such as polarity (p) and H-bonding solubility parameter ( 6 %), as shown in figure 5.38(a-d). This observation is well supported by the previously mentioned interpretations on ECTFE transport behavior (see Section V.B.I.2.). There seem to exist optimal values of polarity (p) and H-bonding parameter (5%). Compared with PVDF and ETFE, the diffusivity and solubility of ECTFE seem to depend on the molar volume of the penetrant in an exponential manner, as depicted in Figure 5.38(e, g). PVDF Figure 5.39(d) shows that PVDF distinctly exhibited the dependence of P.R. on Ô,.. Compared with ECTFE, the 0 ^ value of PVDF, at which the maximum P.R. occurs, shifted from a lower (~ 6 ) to a higher value (-10). Furthermore, the P.R. of PVDF is more steeply dependent on value. Transport properties (including polymer solubilities) are expected to decrease with the increase of penetrant size. For PFA and ECTFE, the transport properties are roughly in agreement with this supposition, as shown in Figures 5.36 and 5.38. On the contrary, ETFE and PVDF does not follow this rule (see Figure 5.37 and 5.39). The reason may be that the selected penetrants do not have chemical similarity to each other. In addition, the solubility does not show any clear dependence on latent heat of vaporization of penetrants, as shown in Figures (5.36-39) (f). 2 0 9 IE+02- 16+02- 16+01- 16+01 16+00- 16+00 fi£ & 16-01 - & 16-01 - 2 I U 16-02 16-02- 16-03- 16-03 —I------1------1------1------16-04 60 70 80 90 100 110 0 0.05 0.1 0.15 (a) Molar Volume (b) Polarity 16+02- 16+02- 16+01- 16+01- _ 16+00- 16+00- ei 06 & 16-01- 1 16-01- 2 2 U 16-02- 16-02- 16-03 - 16-03- 16-04 16-04 18 19 20 21 22 23 24 25 0 2 4 6 8 10 12 14 16 (c) Delt(t) (d) Delt(h) Figure 5.36(a-d). Dependence of permeation rate on thermodynamic properties of penetrants for PFA polymers (□: 25 °C, 0 :45 °C, o; 75 °C). * Delt(t) = (5, or Ô ; Delt(h)=ô,, ; Latent heat = 2 1 0 lE-03 lE-03 ^ lE-04- ^ IE-04- L: U lE-05 lE-05 60 70 80 90 100 no 28 29 30 31 32 33 34 (e) Molar- Volume (f) Latent Heat lE-02 lE-02 lE-03 - lE-03- s Q z 9. z u o- lE-04- lE-04- D- -Q lE-05 lE-05 85 90 95 100 105 110 28 29 30 31 32 33 34 (g) Molar Volume (h) Latent Heat Figure 5.36(e-h). (continued) Effect on solubility and diffusivity of molar volume and latent heat of vaporization at 25 °C. 211 lE+03- lE+03- lE + 02- 1E+02-Y ee eê & i lE+Ol- ^ lE+Ol- z lE+00- lE+00- lE-01 lE-01 100 110 0 0.1 0.2 0.3 0.4 0.5 0.6 Polarity (a) Molar Volume (b) lE+03- lE+03- lE+02- lE+02- g lE+01 - ~ lE+01 lE+00- lE-OI lE-01 18 19 20 21 22 23 24 25 0 2 4 6 8 10 12 14 16 (c) Delt(t) (d) Delt(h) Figure 5.37(a-d). Dependence of permeation rate on thermodynamic properties of penetrants for ETFE polymers (□: 25 C, 0 ; 45 C, o; 75 C). 2 1 2 lE-02 lE-02 % lE-03- lE-03- IE-04 lE-04 60 70 80 90 100 110 28 29 30 31 32 33 34 (e) Molar Volume (f) Latent Heat Figure 537(e-f). (continued) Dependence of solubility in ETFE on molar volume and latent heat of vaporization at 25 C. 2 1 3 lE+04 lE+04 lE+03 lE+03 IE+02 lE+02 J lE+01 IE+00 lE+00 lE-01 lE-01 60 70 80 90 100 no 0 0.1 0.2 0.3 0.4 0.5 0.6 (a) Molar Volume (b) P o larity lE+04 lE+03- lE+03 IE+02- T lE+02 BS bI j IE+01- J lE+01 lE+00 lE-OI lE-01 18 18.5 19 19.5 20 20.5 0 2 4 6 8 10 (C) D elt(t) (d) Delt(h) Figure 538(a-d). Dependence of permeation rate on thermodynamic properties of penetrants for ECTFE polymers (□: 25 °C, 0:45 °C, o; 75 °C). 2 1 4 lE-02 lE-02 60 Z lE-03- lE-04 60 70 80 90 100 110 28 29 30 31 32 33 34 (e) Molar Volume (f) Latent Heat lE-03 lE-03 lE-04- lE-04 « s s z z lE-05 - lE-05- lE-06 lE-06 85 90 95 100 105 110 28 29 30 31 32 33 34 (g) Molar Volume (h) Latent Heat Figure 5.38(e-h). (continued) Effect on solubility and diffusivity of molar volume and latent heat of vaporization at 25 °C. 2 1 5 lE+04- 16+04 lE+03- 16+03 ^ lE+02- T 16+02- ee , C Û ( S lE+OI- ^ 16+01- IE+00- 16+00 lE-01 - 16-01 16-02 0 0.1 0.2 0.3 0.4 0.5 0.6 Molar Volume (b) Polarity 16+04 16+04- 16+03 16+03- 16+02- 16+02 os 0.* 16+01- k 16+01 z u 16+00- 16+00 16-01- 16-01 16-02- 16-02 18 19 20 21 22 23 24 25 0 2 4 6 8 10 12 14 16 (c) Delt(t) (d) Delt(h) Figure 539(a-d). Dependence of permeation rate on thermodynamic properties of penetrants for PVDF polymers (□: 25 °C, 0:45 °C, o: 75 °C). 2 1 6 lE-02 lE-02 ^ lE-03- ss lE-03- uZ lE-04 lE-04 0 0 0 10070 80 90 100 11060 28 29 30 31 32 33 34 (e) Molar Volume (f) Latent Heat lE-03 lE-03 lE-04- S lE-05- g Z lE-06- lE-06- lE-07 lE-07 85 95 100 105 11090 28 29 30 31 32 33 34 Molar Volume Latent Heat (g) (h) Figure 5.39(e-h). (continued) Effect on solubility and diffusivity of molar volume and latent heat of vaporization at 25 °C. 2 1 7 In conclusion, some polymers examined here exhibited clear trends that coincide with their chemical and structural characteristics. That is, their dependence on thermodynamic properties of the various penetrants followed consistent trends. The relatively inert PFA showed effects only of molecular size. Similarly inert, but structurally modified ETFE showed nearly random dependence on all of the properties. ECTFE had been characterized as having dipole-dipole bonds (see Section V.B.1.2), and that behavior was confirmed in relation to transport properties. Likewise, PVDF exhibits H-bonding (see also Section V.B.1.2), and that behavior was confirmed in relation to transport properties. The obtained trends may make it possible to extrapolate or interpolate transport properties for other penetrants. CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS Exposure of semicrystalline fluoropolymer films of varying thicknesses to various liquid and vapor phase penetrants at several temperatures and concentrations exhibited a broad range of sorption behavior, from classical Fickian diffusion to unusual anomalous sorption. From the comprehensive experimental results, it was concluded that the intersegmental bonds of PVDF are likely to be protonic, and those of ECTFE are much like dipole-dipole interactions. To the contrary, ETFE is less polar than ECTFE, and less protonic than PVDF. The increase of polar intersegmental attraction produced a high cohesive energy density and resulted in good mechanical properties. An unfortunate result, however, from the standpoint of stability of barrier characteristics was that non- Fickian diffusion was induced by the relaxation-controlled disruption of polar intersegmental bonds, apparently caused by solvent plasticization. That eventually led to the creation of free volume at a rate that was dependent on both time and concentration. This change finally induced time-dependence of both solubility and diffusivity, which was expected since they are both well known to be dependent on the amount of free volume. Based on the relaxation mechanism described above, a simple kinetic model was devised to get a consistent and unified interpretation of the observed anomalies for PVDF and 218 2 1 9 ECTFE. The rate mode! represented exponential dependence of diffusivity on relaxation. The theoretical results obtained using the derived rate equations have been shown to describe the anomalous sorption behavior in a consistent manner, in that the physical parameters from both types of equations gave reasonable values and expected general trends of temperature dependence. Quantitative agreement for PVDF was also obtained using reasonable values of parameters for the quadratic rate equation. Despite some discrepancies, the anomalous sorption behavior in the glass transition range could be described simply on the basis of volume relaxation kinetics. However, the present work opens up the possibility of more detailed and realistic interpretation of the observed phenomena by a more suitable elaboration of the current model, e.g., via correlation of model parameters with physical properties of the penetrant and polymer. Transport properties (permeation rate (P.R.), diffusivity, and solubility) are correlated with general physical properties such as molar volume, polarity, and H-bonding solubility parameter of the penetrants. The relatively inert PFA shows only the clear dependence of transport properties on molecular size of the penetrants. Similarly inert, but stmcturally modified ETFE exhibits nearly random dependence on all the physical factors. ECTFE, on the other hand, exhibits relatively clear dependence of P.R. on polarity and H-bonding parameter. Likewise, PVDF exhibits distinct dependence of P.R. on H-bonding parameter. The increase of mechanical strength of fluoropolymers by the introduction of altemate substituents (e.g. Cl and H instead of F) unfortunately offsets the strong inherent chemical resistance. A greater extent of substitution eventually leads to higher permeation 2 2 0 rates and unpredictable anomalous sorption behavior. To overcome this disadvantage, it may be possible to modify some processing conditions to promote generation of a tough skin or a greater degree of orientation. The existence of a tough skin retards the relaxation process by inducing low solubility and low diffusivity. A greater extent of orientation may reduce permeability under low solvent activity. Development of composite materials or altemate processing conditions is also needed because even thick (90 mil) PVDF and ECTFE films exhibited anomalous diffusion behavior coupled with significant structural deformation during solvent uptake. For example, modified fluoropolymers could be coated by PFA or FEP films. This composite material yields strong chemical resistance and mechanical properties because the strong mechanical integrity of partially fluorinated polymers can be maintained by avoiding anomalous diffusion behavior with the help of low solubilities of coated PFA and FEP resins. REFERENCES 1 («The • New Breeds,” Plastics Technoloty, 37, 58, March (1991). ^ “Easier - to-process fluoropolymers open up wider application range,” 64, 162 (1987). ^ “DuPont Polymers for the Chemical Processing Industry,” DuPont, H-42451, May (1992). * “Hylar™— Polyvinylidene Fluoride,” Ausimont USA, Inc., AUS-0021, April (1991). ^ “Halar® ECTFE,” Ausimont USA, Inc., 1992. ® “Kynar®,” Spotlight Vol. 19, No 1, Elf Ato Chem, May (1991). ’ Films of Teflon® PFA /f Films of Tefzel® ETFE,” DuPont, H-04310, May (1992). 8 “j 2 Films-Specialty Release Applications,” DuPont, H-04315, October (1990). ® “Fluoropolymers,” Encyclopedia of Polymer Science and Eneineerins, Vol. 16, John Wiley & Sons, Inc., New York (1986). “Teflon® PTFE-Properties Handbook,” DuPont, 220313B, November (1992). “ “Teflon® Film,” DuPont, H-04318, May (1991). “Chris Hazardous Chemical Data,” (1985). “CRC Handbook of Solubility Parameters and Other Cohesion Parameters,” A. Barton, Ed., CRC Press (1983). Aitken, A. and Barrer, Trans. Faraday 5oc.. 51, 116 (1955). Alfrey, T., Chem. Em . News. 43, 64 (1965). 2 2 1 2 2 2 Alfrey, T., E. F. Gumee and W. G. Lloyd, J. Polvm. ScL C. 12, 249 (1966). Anet, F. A. and A. J. Boum, J. Am. Chem. Soc.. 89, 760 (1967). Astarita, G. and G. C. Sarti, Polymer Ensineerine and Science, 18, 388 (1978). As tari ta, G. and S. Joshi, J. Memb. ScL 4, 165 (1978). Bagley, E. and F. A. Long, J. Am. Chem. Soc.. 77, 2172 (1955). Barrer, R. M., Trans. Faraday Soc.. 35, 628, 644 (1939). Bueche, F., J. Chem. Phys.. 21, 1850 (1953). Barrer, R. M., /. Phys .Chem., 61, 178 (1957). Barrer, R. M., J. Chem. Soc.. 278 (1934). Batschinski, A. J., Z. Phys. Chem.. 84, 644 (1913). Berens, A. R. and H. B. Hopfenberg, Polymer, 19, 489 (1978). Berens, A. R. and H. B. Hopfenberg, J. Memb. ScL. 10, 283 (1982). Boyer, R. F., J. A m i Phys.. 20, 540 (1949). Brandt, W. W., J. Phys. Chem.. 63, 1080 (1959). Brandt, W. W. and G. A. Anvsas. J. A ddL Polym. ScL. 7, 1919 (1963). Bro, M. I., J. o fA ddL Polym. ScL. 1, 310 (1959). Brydson, J. A., “PlasticsMaterials” London, Newnes-Butterworths (1975). Bunn, C. W., J. Polym. ScL 16, 332 (1955). Camera-Roda, G. and G. C. Sarti, AIChEJ.. 36, 851 (1990). Camera-Roda, G. and G. C. Sarti, Trans. Theory Stat. Physics. 15, 1023 (1986). Carbonell, R. G. and G. C. Sarti, American Chemical Society. 29, 1194 (1990). Clark, E. S., Paper presented at Symposium on Helices in Macromolecular Systems. Polytechnic Institute of Brooklyn, Brooklin, N. Y., May 16 (1959). 2 2 3 Cohen, D. S., J. Polvm. ScL Polvm. Phys. E d. 21, 2057 (1983). Cohen, D. S., J. Polvm. ScL Polvm. Phys. Ed.. 22, 1001 (1984). Cohen, M. H. and D. Turnbull, J. Chem. Phys.. 31, 1164 (1959). Crank, J., Trans. Faraday Soc.. 47, 450 (1951). Crank, J. and G. S. Park, “Diffusion in Polymers” Academic Press, London (1968). Crank, J. and G. S. Park. Trans. Faraday Soc.. 47, 1072 (1951). Crank, J., J. Polvm. ScL, 11,151 (1953). Crank, J., “Die Mathematics of Diffusion” 2nd Ed., Oxford University Press, Oxford (1975). Crank, J., Trans Faraday Soc.. 47, 450 (1951). Darby, R. A., E. I. DuPont de Nemours & Co., Inc., Private Communication. DiBenedetto, A. T., J. Polvm. ScL C. 10, 17 (1965). Doban, R. C., C. A. Sperati and B. W. Sandt, Soc. Plastics Ensrs Journal, 11, 17-21, 24, 30 (1955). Doghieri, F., G. Camera-Roda and G. C. Sarti.. AIChE J.. 39, 1847 (1993). Doolittle, A. K., J. A ddL Phvs.. 22, 1471 (1951). Duda, J. L. and J. S. Vrentas, “Encyclopedia of Polymer Science and Engineering,” 2nd Edition, Wiley, New York, 1988. Duda, J. L., J. S. Vrentas, S. T. Ju and H. T. Liu. AIChE J., 28, 279 (1982). Duming, C. J. and J. L. Spener, Journal of Polymer Science: Polymer Letters Edition. 23, 171 (1985a). Duming, C. J., Journal of Polymer Science: Polymer Physics Edition. 23, 1831 (1985b). Ocone. R. and G. Astarita, AIChE J.. 33, 423 (1987). Duming, C. J. and M.Tabor. American Chemical Society. 19, 2220 (1986). 2 2 4 Duming, C. J. and S. Mehdizadeh. AIChE J., 36, 877 (1990). Eby, R. K. and F. C. Wilson, J. A dvL Phvs.. 33, 2951 (1962). Eby, R. K., J. AddL Phvs.. 35, 2720 (1964). Eyring, H., J. O. Hirschfelder and D. P. Stevenson, J. Chem. Phvs.. 5, 896 (1937). Eyring, H., J. Chem. Phvs.. 4, 283 (1936). Fang, S. M., S. A. Stem and H. L. Frish. Chemical Eneineerine Science. 30, 773 (1975). Fick, A., Annin Phvs.. 170, 59 (1855). Fourier, J. B., “Theorie analytique de la chaleur” (1822); English Translation by A. Freeman, Dover Publ., New York (1955). Fox, T. G. and P. J. Flory, J. Polvm. ScL. 14, 315 (1954). Fox, T. G. and P. J. Flory, J. AddL Phvs.. 21, 581(1950). Frish, H. L., J. Phvs. Chem.. 61, 93 (1957). Frish, H. L., Polvm. Ens. ScL 20, 2 (1980). Frish, H. L., T. T. Wang and T. K. Kwei. J. Polvm. ScL (A2). 7, 879 (1969). Frish, H. L., J. Polvm. Sci. B.. 3, 13 (1965). Fu, T. X. and C. J. Duming, AIChEJ.. 39, 1030 (1993). Fujita, H., Fortschr. Hochpolvm. Forsch, 3, 1 (1961). Fujita, H., A. Kishimoto and K. Matsumoto, Trans. Faraday Soc.. 56, 424 (1960). Fujita, H., A. Kishimoto and H. Odani, Prog. Theoret. Phvs. (Kvoto)Suvvl.. 10, 210 (1959). Fujji, M., V. Stannett and H. B. Hopfenberg, J. Macromol Sci. Phvs.. B15, 421 (1978). Gear, C. W., “Numerical Initial-Value Problems in Ordinary Dijferential Equations, Prentice-Hall, Englewood Cliffs,” New Jersey (1971). 225 Glaslone, S., K. J. Laider and H. Eyring, “The Theory of Rate Processes,” 1st Edition, Mcgraw-Hill, New York and London (1941). Gordon, J. L, J. Paint TechnoL, 38(492), 43 (1966). Gostoli, C and G. C. Sarti. Chem. Ens. Comm.. 21, 67 (1983). Graham, T., Phil Mas., 32, 401 (1866). Grest, G. S. and M. H. Cohen. Physical Review B, 21, 4113 (1980). Hansen, C. M., Ind. Eng. Chem. Prod. Develop.. 8, 2 (1969). Hasegawa, R., Y. Tokahashi, Y. Chatami and H. Tadkoro, Polvm. J.. 3, 600 (1972). Hayes, M. J. and G. S. Park, Trans. Faraday Soc.. 51, 1134 (1955). Hildebrand, J. H., J. Am. Chem. Soc.. 41, 1067 (1919). Hildebrand, J. H. and R. L. Scott, “The Solubility o f Nonelectrolvtes.” 3rd Ed., Reinhold, New York (1950). Hui, C. Y., K. G. Wu., C. L. Ronard and E. J. Kramer, J. A w l Phys.. 61, 5129, 5137 (1987). Hutchinson, J. M. and A. J. Kovacs, Journal of Polymer Science: Polymer Physics Edition. 14, 1575 (1976). Imbalzano, J. F., Chemical Eneineerins Progress. 69 (1991). Jacques, C. H. M., H. B. Hopfenberg and V. T. Stannet, In: “Permeability of Plastic Films and Coatings,” Ed. H. B. Hopfenberg, Plenum, New York (1974). Kishimoto, A., H. Fujita, H. Odani, M. Kurata and M. Tamura, J. Phys. Chem.. 64, 594 (1960). Kovacs, A. J., Fortschr. Hochvolym. Forsch.. 3, 394 (1963). Kulkami and S. A. Stem. J. o f Polym. ScL. 21, 441 (1983). Kumins, C. A., J. Polym. ScL C. 10, 1 (1965). Kwei, T. K., T. T. Wang and H. M. Zupko, Macromolecules. 5, 645 (1972). 2 2 6 Long, F. A. and D. Richman, J. Am. Chem. Soc.. 82, 513 (1960). Long, F. A. and R. J. Kokes, J. Am. Chem. Soc.. 75, 2232 (1953). Lustig, S. R,, J. M. Caruthers and N. A. Peppas, Chemical Eneineering Science, 47, 3037 (1992). Mannon, P. J., Nuclenics, 22, 72 (1964). Matsuoka, S. and H. E. Bair, J. AdvL Phvs.. 48, 4058 (1977). Mears, P. J., Polvm. ScL. 27, 405 (1958). Meats, P., J. Am. Chem. Sac.. 76, 3415 (1954). Miamoto, Y., H. Miyah and K. J. Asai, J. Polvm. ScL Polvm. Phvs. Ed.. 18, 597 (1980). Michaels, A. S., W. R. Vieth and J. A. Barrie, J. AddL Phvs.. 34, 1,13 (1963). Michaels, A. S. and H. J. Bixler. J. Polvm. ScL. 50, 413 (1961). Michaels, A. S. and R. B. Parker. J. Polvm. ScL. 41, 53 (1959). Neogi, ?.. AIChEJ.. 29, 829 (1983). Neogi, P. and F. Adib, AIChEJ.. 33, 164 (1987). Neogi, P., M. S. Kim and Y. Yang, AIChEJ.. 32, 1146 (1986). Newns, A. C , Trans. Faraday Soc.. 52, 1533 (1956). Odani, H., S. Kida and M. Tamura, Rep. Pros. Polvm. Phvs. Japan, 4, 94 (1961b). Odani, H., J. Hayashai and M. Tamura, Bull. Chem. Soc. Japan. 34, 817 (1961a). Overbergh, N., H. Berghmans and G. Smets, Polvmer. 16, 703 (1975). Pace, R. J. and A. Datyner., J. Polvm. Sci Polvm Phvs. Ed.. 17, 437, 453 ,465 (1979). Pace, R. J. and A. Datyner., J. Polvm. ScL Polvm. Ens. ScL. 20, 51 (1980). Park, G. S., Trans. Faraday Soc.. 48, 11 (1951). 227 Paul, D. R. and W. J. Koros. J. Polvm. ScL Polvm. Phvs. Ed.. 14, 675 (1976). Paul, D. R. and A. T. DiBenedetto, J. Polvm. Sci. C. 10, 17 (1965). Peterlin, A., J. Polvm. ScL, B3, 1083 (1965). Peterlin, A., Makromol Chem., suppl. 3, 215 (1979). Peterlin, A., J. Macromol ScL, B ll, 57 (1975). Peterlin, A., “Mechnism of Deformation in Polymeric Solids in Polymeric Materials, Ed., Eric Baer, Metals Park, Ohio (1973). Peterlin, A., Makromol Chem., 124, 136 (1969). Petropoulos, J. H., J. Memb. ScL. 17, 233 (1984b). Petropoulos, J. H. and P. P. Roussis, In: “Permeability of Plastic Films and Coatings,” Ed. H. B. Hopfenberg, Plenum, New York (1974). Petropoulos, J. H.. J. Polvm. ScL Polvm. Phvs. Ed.. 22,183 (1984a). Petropoulos, J. H. and R. P. Roussis, “In: Permeability of Plastic Films,” Ed., H. B. Hopfenberg, Plenum, New York (1974). Plunkett, R. J., U. S. Pat. 2230654 (Feb. 4, 1941) ( to Kinetic Chemicals, Inc.). Prager, S. and F. A. Lone. J. Am. Chem. Soc.. 73, 4072 (1951). Richardson, A., P. S. Hope and I. M. Ward, J. Polvm. Sci. Polym. Phvs. Ed.. 21, 2525 (1983). Rogers, C. E., ^"Polymer Permeability,” Ed., J. Comyn, Elsevier Applied Science Publishers Ltd., London (1985). Rogers, C. E., “Physics and Chemistry of the Organic Solid State,” Vol II, Eds., D. Fox, M. M. Labes and A. Weissberger, Interscience Publ., New York, (1965). Sarti, G. C., Polvmer. 20, 829 (1979). Sobolev, I., J. A. Meyer, V. Stannett and M. Szwarc, Ind. Em r. Chem.. 49, 441 (1957). Sperati, C. A. and H. W. Starkweather, Fortschr. Hochpolvm.-Forsch.. 2, 465 (1961). 2 2 8 Starkweather, H. W., J. Polvm. ScL Polvm. Phvs. Ed.. 11, 587 (1973). Taraiya, A. K., G. A. Orchard and I. M. Ward. J. Polvm. ScL (B). Polvm. Phvs.. 31, 641 (1993). Thomas, N. and A. H. Windle, Polvmer, 22, 627 (1981). Thomas, N. and A. H. Windle, Polvmer. 18, 1195 (1977). Thomas, H. L. and A. H. Windle, Polvmer, 23, 529 (1982). Tikhomirov, B. K., H. B. Hopfenberg, V. T. Stannett and J. L. Williams, MakromoL Chem., 118, 177 (1968). Tool, A. Q., J. Res. Natl. Bur. Std, 42, 171 (1949). Van Amerongen, G. J., J. Applied Phvs., 17, 972 (1946). Van Amerongen, G. J.. J. Polvmer ScL, 5, 307 (1950). Vrentas, J. S. and J. L. Duda, J. AddL Polvm. ScL. 22, 2325 (1978). Vrentas, J. S., C .M. Jarzebski and J. L. Duda, AIChE J.. 21, 895 (1975). Vrentas, J. S., J. L. Duda and A. C. Hou, J. AddL Polvm. ScL. 29, 399 (1984). Vrentas, J. S. and J. L. Duda, J. Polvm. ScL Polvm. Phvs., 15, 403, 417 (1977). Wang, T. T., T. K. Kwei and H. L. Frish, J. Polvm. ScL (A2), 7, 2019 (1969). Watt, I. C, Textile Res. J.. 30, 443, 644 (1960). Williams, M. L., R. F. Landel and J. D. Ferry, J. Am. Chem. Soc.. 77, 3701 (1955). Wilson, F. C. and H. W. Starkweather, J. Polvm. ScL Phvs. E d . 11, 919 (1973). Zielinski, J. M. and J. L. Duda, AIChE J., 38, 405 (1992). APPENDICES 2 2 9 APPENDIX A PERMEATION DATA 2 3 0 ETFE-BENZENE-ROOM TEMP.-IO MIL P E R M llll 2.5 :0.4680 :0.4829 :0.4873 1" 0.5- g -0.5- pH TIME (days) ETFE-BENZENE-45 C-10 MIL PERM1121 5.3437 5.1851 5.1826 4.9953 TIME (days) ETFE-BENZENE-75 C-10 MIL PERM1131 250 :109.4062 :108.5060 -*e- 5b -.136.1433 — a — .-120.9975 6 8 TIME (days) PERMEATION RATE (grams/s.m/day) o II o o J Ch m '3! ta- ü O è - & r 0 \ KZ ETFE-TOLUENE-45 C-10 MIL PERM1221 :2.5286 :2.5143 :2.6557 :4.6848 TIME (days) ETFE-TOLUENE-75 C-10 MIL PERM1231 1 2 0 :65.1098 100- :66.4895 i 80- :60.6957 :63.1829 ON TIME (days) ETFE-CHLOROBENZENE-ROOM TEMP.-IO M: PERM1311 :0.3148 :0.3193 0.5--- :0.3180 0.3080 15 20 25 30 TIME (days) ETFE-CHLOROBENZENE-45 C-10 MIL PERM1321 1 0 I 84 :2.5427 :2.6265 I 6 -*K- 4 :2.7187 I — a — 2- :3.4499 O' 0 - 2 - 1 -4- ÇU -6 0 10 15 20 25 TIME (days) 00 ETFE-CHLOROBENZENE-75 C-10 MIL PERM1331 160 «2 140- :61.5039 :64.8320 -.80.6469 80- :74.4114 60- 20 - TIME (days) VO PERMEATION RATE (grams/s.m/day) o o Ln o" II oI Ma r O O S S *1 o r oo Ln ON 00 0 \ ON ON 0172 ETFE-PHENOL-45 C-10 MIL PERM1421 :1.6634 I 2.5- :1.5957 TIME (days) ETFE-PHENOL-75 C-10 MIL PERM1431 :5.6171 :5.8359 :6.6248 I— I P i TIME (days) ETFE-DICHLOROMETHANE-ROOM TEMP.-IO MIL PERM1511 :36.0677 :33.8514 :33.3801 B :32.1538 30 40 50 60 TIME (days) g ETFE-DICHLOROMETHANE-45 C-10 MIL PERM1521 :108.1539 'O 130 :117.0535 % 120 —X— 5 ) :115.5864 « 80- 6 8 10 12 TIME (days) ETFE-DICHLOROMETHANE-75 C-10 MIL PERM1531 1300 :1023.778 I 1200 - I :974.5942 —X— 1100— : 1039.826 è —B— I 1000— :963.1579 z I 900- 800- g Pi 700 8 10 TIME (days) ECTFE-BENZENE-ROOM TEMP.-IO MIL PERM2111 .T.1375 : 1.3505 :1.0664 :1.2090 % 0.5- -0.5 TIME (days) ECTFE-BENZENE-ROOM TEMP.-IO MIL PERM2111‘ :1.1520 :1.1356 :0.9794 :0.9353 % 0.5- 0 - -0.5 100 120 TIME (days) ECTFE-BENZENE-45 C-10 MIL PERM2121 :23.4050 :22.4563 :22.7481 20 -- -10 TIME (days) 00 ECTFE-BENZENE-45 C-10 MIL PERM2121* :16.5182 :16.9191 CO S 20- : 18.2185 15- :17.2304 TIME (days) ECTFE-BENZENE-75 C-10 MIL PERM2131 1200 :627.006 :760.184 -*e- 5b :694.229 - B :613.754 10 15 TIME (days) o ECTFE-TOLUENE-ROOM TEMP.-IO MIL PERM2211 I :0.7365 I :0.7568 :0.7164 I% 0 1 Eg pH -0.5 40 50 60 TIME (days) ECTFE-TOLUENE-ROOM TEMP.-IO MIL PERM2211* I :0.7371 :0.6441 C/3 1 0.6" :0.6907 è Ë g <’■“ z 2I 0.2- i "■ Pu -0.2- 100 120 TIME (days) » ECTFE-TOLUENE-45 C-10 MIL PERM2221 :10.8336 :11.7239 : 15.9563 15- :11.5471 1 0 - TIME (days) g ECTFE-TOLUENE-45 C-10 MIL PERM2221* :16.5182 (/} W 10- ;18.2185 : 17.2304 TIME (days) g ECTFE-TOLUENE-75 C-10 MIL PERM2231 600 :284.697 € 500- :333.925 5 400- :302.565 300- :284.697 200 - 100- TIME (days) a ECTFE-CHLOROBENZENE-ROOM TEMP.-IO MIL PERM2311 1.5474 1.3425 1.3584 % 0.5 1.3899 30 TIME (days) g ECTFE-CHLOROBENZENE-ROOM TEMP-10 MIL PERM2311* t :1.2858 a 1.5- :1.3624 I :1.4066 I :1.0783 §1 -0.5 60 TIME (days) ECTFE-CHLOROBENZENE-45 C-10 MIL PERM2321 22.1302 22.2845 23.0971 20 25 TIME (days) 00 ECTFE-CHLOROBENZENE-45 C-10 MIL PERM232P :19.1586 : 19.3654 : 19.6023 5 ) :19.8821 30 40 TIME (days) VO ECTFE-CHLOROBENZENE-75 C-10 MIL PERM2331 800 :450.416 :437.018 oh :413.696 :374.425 200 ' ...... 15 TIME (days) s PERMEATION RATE (grams/s.m/day) o Ln a o “‘ II I 1 8 o r O n O Vi U) I9Z ECTFE-PHENOL-45 C-10 MIL PERM2421 Î :0.3076 :0.3057 :0.3425 :0.2954 TIME (days) ECTFE-PHENOL-75 C-10 MIL PERM2431 :2.2178 73 :2.8027 :2.6653 :2.4239 O’ ■■ TIME (days) a ECTFE-DICHLOROMETHANE-ROOM TEMP.-IO MIL PERM2511 350 300- :156.9672 — )K - :159.0183 250- —B— :190.0194 200 - « 150- f Z I 2 g ^ 100- TIME (days) 2 ECTFE-DICHLOROMETHANE-45 C-10 MIL PERM2521 800 W 750- :574.1119 ^ 700- :630.1957 :699.4124 600- :719.5372 S 550- 500- 450- 400 TIME (days) g ECTFE-DICHLOROMETHANE-75 C-10 MIL PERM2531 7000 :3961.398 I ' 6000- :4579.703 a 5000- :4563.047 4000- :3455.303 3000- 2000 - 1000 TIME (days) PERMEATION RATE (grams/s.m/day) o o o o io b \ bo < 0 dd § 1 ^ w ii S r 00 w z PERMEATION RATE (grams/s.m/day) I-Tj < 0 td § S a. S 1 w ! U t n r w vo 8 9 2 PVDF-BENZENE-75 C-10 MIL PERM3131 :36.0522 :34.9310 :34.5816 30- :40.0166 S 25- 20 - 15- TIME (days) K) s PERMEATION RATE (grams/s.m/day) ô , , Ô , , Ô o ( 3 b ^‘ O 00 LA 4^ vo VO t—^ 00 o \ QLZ PVDF-TOLUENE-45 C-10 MIL PERM3221 I :1.0174 I :1.6991 I :0.8543 :0.8323 z o PUÏ 30 35 40 TIME (days) PVDF-TOLUENE-75 C-10 MIL PERM3231 : 18.4378 : 18.0795 : 18.0814 :18.5651 20 - S 15- 10 - TIME (days) w PVDF-CHLOROBENZENE-ROOM TEMP.-IO MIL PERM3311 :0.0728 ^ 0.6" :0.0423 CO :0.0603 :0.0379 0.2- 0 - - - 0.6 100 120 140 TIME (days) a PVDF-CHLOROBENZENE-45 C-10 MIL PERM3321 :0.6894 4- :0.5911 3- :0.6000 & I 2 - z 0 7^ ? rHH 0 - 1pH ------1------12 14 16 18 20 22 24 26 28 30 TIME (days) PVDF-CHLOROBENZENE-75 C-10 MIL PERM3331 30 ;17.3675 1 “ : 17.8292 — 5K— 20 - :17.7233 —B— I 15- : 18.2102 z 0 10- 5- 1Pm 0 10 15 20 25 30 TIME (days) a PERMEATION RATE (grams/s.m/day) < 0 1 I | § o r ô , , Ô , , Ô 1 Ô 1 N) ^3 to ) ' to t to f to to 45^ 1 O to to N) o Ln 9LZ PERMEATION RATE (grams/s.m/day) 'T 3 0 1 M d s Z S O r o r 00 t 0 \ isistLn 1-^ 00 <1ê vo LLZ PVDF-PHENOL-75 C-10 MIL PERM3431 :12.1056 CO : 12.4527 20 - :17.1360 15- :13.3627 TIME (days) 00 DICHLOROMETHANE- PERM3511 :9.4478 B 20 - :9.4544 :7.5684 :8.6430 10- 5-- TIME (days) \o PVDF-DICHLOROMETHANE-45 C-10 MIL PERM3521 80 I'TO- :35.4824 ^ # 6 0 :33.6679 5 ) 5 0 - :39.0550 H 4 0 - — —)K------X ----- § 30 4 % I 2 0 d 10 pH 0 0 8 10 12 14 16 18 TIME (days) O PVDF-DICHLOROMETHANE-75 C-10 MIL PERM3531 600 550- Î :439.463 1 500- :465.682 së 2 400- ii 350- B 300- TIME (days) K) 00 PERMEATION RATE (grams/s.m/day) o o I Cd § o I W ioi§ I y I r 00 zsz FEP-BENZENE-45 C-10 MIL PERM4121 1.4 :0.4715 I ' 1.2- :0.5019 :0.4420 0.8 - B 0.4- (4 0 .2 - TIME (days) w FEP-BENZENE-75 C-10 MIL PERM4131 :12.1355 # 25- :17.6603 20 - 15- TIME (days) g FEP-TOLUENE-75 C-10 MIL PERM4231 :7.5067 :7.6205 ^ 12- :6.2943 TIME (days) KJ\ PERMEATION RATE (grams/s.m/day) ON 00 o 9 o " 5 0 s g Q. D) -p^ N 'C ts) w 00 o “' w § W 1 L n o r 9 8 3 PERMEATION RATE (grams/s.m/day) ? w o"' § Ux~' I S o - Q. is I r o 0 \ S ON W w LSZ PERMEATION RATE (grams/s.m/day) dd Z s S ip I—^ L U t o r 88Z PFA-BENZENE-75 C-10 MIL PERM5131 :15.7816 :14.9761 20 - :13.5441 15- TIME (days) K VO PFA-TOLUENE-ROOM TEMP.-IO MIL PERM5211 0.3 :-0.0275 :-0.0297 :-0.0393 :-0.0104 NJ TIME (days) O PERMEATION RATE (grams/s.m/day) o o ? o H O .S l § L n O r \6Z PFA-TOLUENE-75 C-10 MIL PERM5231 :8.5871 IB 20- :9.2802 (A :9.3375 15- :8.0561 Z 10- TIME (days) § PFA-CHLOROBENZENE-ROOM TEMP.-IO MIL PERM5311 0.3 :-0.0341 Î 0.2 e :-0.0284 0 . 1- :-0.0262 S 5 1-0.0582 2I - 0. 1- -0.3- TIME (days) g PERMEATION RATE (grams/s.m/day) o p o " o >3 0 ^ 03 n wI t o § tn 1 4 ^ U\ o OJ Ui r f6Z PFA-CHLOROBENZENE-75 C-10 MIL PERM5331 25- 8.5871 20 - ■ W 9.2802 C/3 9.3375 & 15- - 8.0561 I - Z 10 0 5-- pH1 0- 8 10 12 14 16 18 20 TIME (days) PFA-PHENOL-ROOM TEMP.-IO MIL PERM5411 0.3 :0.0232 Î 0.2- B lO.OOOO (/> I 0. 1- :0.0223 :0.0174 1 »- • z 2 - 0. 1- I -0.3- TIME (days) PFA-PHENOL-45 C-10 MIL PERM5421 :0.2434 1.5- B :0.2743 (A :0.2441 & :0.2233 z 2 0- i I-»'- Pu, TIME (days) 8 PERMEATION RATE (grams/s.m/day) ON • - d I O" I 0 S r1 < 1 è L n n r w VO I—‘ 00 8 6 3 PFA-DICHLOROMETHANE-ROOM TEMP.-IO MIL PERM5511 3.5 :2.3208 :2.2706 2.5- :2.4577 :2.5105 1.5- 0.5 TIME (days) s PFA-DICHLOROMETHANE-45 C-10 MIL PERM5521 :9.2260 Iq 20- :11.2668 :11.1992 15- :12.6850 Z 10- w TIME (days) 8 PFA-DICHLOROMETHANE-75 C-10 MIL PERM5531 250t :67.3691 Îa 200- :88.0449 CO :80.0021 150- :59.4164 % 100- S 50- TIME (days) APPENDIX B SORPTION DATA 3 0 2 3 0 3 SORPTION EXPERIMENT S O R P l l l l POLYMER : ETFE TEMPERATURECC) : 25 PERMEANT : BENZENE T H IC K N E S S (m ü ) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.02374 0.02388 0.02353 SUMMARY OF SORPTION EXPERIMENT FRACTIONAL MASS ORANGE TIME(Days) SORTdlME) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.000 0.0000 0.0000 0.0000 0.0000 0.187 0.4320 0.1708 0.1689 0.1580 0.613 0.7829 0.3440 0.3470 0.3386 1.279 1 .1 3 1 0 0.5285 0.5320 0.5124 2.373 1.5403 0.7927 0.7968 0.7720 3.109 1.7631 0.8975 0.8995 0.8849 3.565 1.8880 0.9317 0.9338 0.9300 4.579 2.1399 0.9681 0.9680 0.9616 5.243 2.2898 0.9772 0.9703 0.9707 6.742 2.5966 0.9749 0.9772 0.9819 10.150 3.1859 0.9977 0.9932 0.9932 13.435 3.6654 1.0000 1.0000 1.0000 3 0 4 SORPnON EXPERIMENT S 0 R P 1 1 2 1 POLYMER : ETFE TEMPERATURE (“Q : 45 PERMEANT : BENZENE T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLES 0.02794 0.02418 0.02696 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS ClGANGE SAMPLE 1 SAMPLE 2 SAMPLES 0.0000 0.0000 0.0000 0.0000 0.0000 0.0188 0.1371 0.1546 0.1670 0.1802 0.0644 0.2537 0.3209 0.3521 0.3411 0.1042 0.3227 0.4266 0.4786 0.4419 0.2438 0.4937 0.7436 0.8397 0.7616 0.3234 0.5686 0.8376 0.9458 0.8643 0.4984 0.7060 0.9628 0.9616 0.9554 0.9761 0.9880 0.9804 0.9842 0.9884 2.1295 1.4593 0.9883 0.9910 0.9961 5.8958 2.4281 1.0000 1.0000 1.0000 3 0 5 SORPTION EXPERIMENT S 0 R P 1 1 3 1 P O L Y M E R ; E T F E TEMPERATURE (°C) : 65 P E R M E A N T : B E N Z E N E T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.03060 SUMMARY OF SORPTION EXPERIMENT T IM E (D a y s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0.0065 0.0807 0.2923 0.0137 0 .1 1 7 2 0.4393 0.0282 0.1680 0.6496 0.0506 0.2250 0.8906 0.0861 0.2934 0.9966 0 .1 1 7 2 0.3423 1.0034 0.1563 0.3953 0.9966 0.2648 0.5145 1 . 0 0 0 0 3 0 6 SORPTION EXPERIMENT S O R P 1 2 1 1 POLYMER : ETFE TEMPERATURECQ : 25 PERMEANT : TOLUENE THICKNESS(mü) : 10 EQUILIBRIUM VALUE ^ra m /g ram ) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.02194 0.02123 0.02219 SUMMARY OF SORPTION EXPERIMENT TEME(Days) SQRT(TIME) FRACTIONAL MASS ClHANGE SAMPLE 1 SAMPLE 2 SAMPLE 3 0.0000 0.000 0.0000 0.0000 0.0000 0.2134 0.462 0.1425 0.1234 0.1524 0.6289 0.793 0.2800 0.2648 0.2786 1.2857 1.134 0.4125 0.4036 0.4048 2.3899 1.546 0.5975 0.5861 0.5738 3.1273 1.768 0.7050 0.6992 0.7048 3.5812 1.892 0.7800 0.7738 0.7429 4.5631 2.136 0.8650 0.8663 0.8405 5.2554 2.292 0.9125 0.9126 0.8881 6.3451 2.519 0.9475 0.9486 0.9357 6.7569 2.599 0.9500 0.9512 0.9429 10.1591 3.187 0.9900 0.9923 0.9857 13.4478 3.667 1.0000 1.0000 1.0000 3 0 7 SORPTION EXPERIMENT S O R P 1 2 2 1 POLYMER : ETFE TEMPERATURE (°C) : 45 PERMEANT : TOLUENE T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.02200 0.02321 0.02287 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS Q KANGE SAMPLE 1 SAMPLE 2 SAMPLE 3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0506 0.2250 0.1965 0.2313 0.2225 0.1013 0.3182 0.3010 0.3220 0.3254 0.1678 0.4097 0.4104 0.4240 0.4306 0.3002 0.5479 0.6144 0.6032 0.6244 0.4413 0.6643 0.7662 0.7438 0.7799 0.6004 0.7749 0.8980 0.8662 0.8900 0.9819 0.9909 0.9701 0.9683 0.9737 1.2785 1.1307 0.9776 0.9796 0.9880 2.0072 1.4168 0.9851 0.9932 0.9952 5.3074 2.3038 1.0000 1.0000 1.0000 3 0 8 SORPTION EXPERIMENT S O R P 1 2 3 1 POLYMER : ETFE TEMPERATURE (°C) : 65 PERMEANT : TOLUENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.02559 SUMMARY OF SORPTION EXPERIMENT TIME (D a y s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 . 0 0 0 0 0.0000 0.0000 0.0065 0.0807 0.2312 0.0137 0 .1 1 7 2 0.3408 0.0282 0.1680 0.4868 0.0506 0.2250 0.6714 0.0861 0.2934 0.8925 0 .1 1 7 2 0.3423 0.9777 0.1563 0.3953 0.9980 1 0.2648 0.5145 1 . 0 0 0 0 3 0 9 SORPTION EXPERIMENT S 0 R P 1 3 1 1 POLYMER :ETFE TEMPERATURECQ : 25 PERMEANT ; CHLOROBENZENE THICKNESS(iml) : 10 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 SAMPLE 2 SAMPLES 0.02514 0.02526 0.02470 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS OKANGE SAMPLE 1 SAMPLE 2 SAMPLES 0.0000 0.000 0.0000 0.0000 0.0000 0.000 0.4751 0.1353 0.1392 0.1432 0.226 0.8021 0.2770 0.2869 0.2777 0.643 1.2182 0.4376 0.4325 0.4403 1.484 1.5483 0.5751 0.5717 0.5879 2.397 1.7719 0.6850 0.6835 0.6985 3.140 1.8983 0.7590 0.7532 0.7722 3.604 2.1317 0.8414 0.8481 0.8612 4.544 2.2953 0.8964 0.8882 0.9111 5.268 2.5218 0.9366 0.9388 0.9610 6.360 2.6022 0.9493 0.9515 0.9718 6.771 3.1887 0.9915 0.9895 1.0087 10.168 3.6684 1.0000 1.0000 1.0000 3 1 0 SORPTION EXPERIMENT S O R P 1 3 2 1 POLYMER : ETFE TEMPERATURE (°C) : 45 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.02728 SUMMARY OF SORPTION EXPERIMENT T IM E (D ay s) SQRT(TIM E) FRACTIONAL MASS CHANGE SAMPLE 1 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0.029 0.170 0.168 0.096 0.310 0.307 0.196 0.443 0.456 0.247 0.497 0.519 0.376 0.613 0.689 0.450 0.670 0.779 0.601 0.776 0.884 0.776 0.881 0.945 1 . 0 0 1 1 . 0 0 1 0.973 1.158 1.076 0.975 1.458 1.208 0.985 4.287 2.071 0.989 4.922 2.219 1 . 0 0 0 311 SORPTION EXPERIMENT S O R P 1 3 3 1 POLYMER :ETFE TEMPERATURE (°C) : 65 PERMEANT : CHLOROBENZENE T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 0.02931 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SORT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 . 0 0 0 0 0.0000 0.0000 0.0065 0.0807 0.1958 0.0137 0 .1 1 7 2 0.2963 0.0282 0.1680 0.4339 0.0506 0.2250 0.6067 0.0861 0.2934 0.8307 0 .1 1 7 2 0.3423 0.9436 0.1563 0.3953 0.9894 0.2648 0.5145 1 . 0 0 0 0 3 1 2 SORPTION EXPERIMENT S 0 R P 1 4 1 1 POLYMER :ETFE TEMPERATURE (°C) : 25 PERMEANT : PHENOL THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00236 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 2.2007 1.4835 0.3256 2.9715 1.7238 0.3721 5.0882 2.2557 0.4419 9.1264 3.0210 0.6744 1 1 .9 3 1 9 3.4543 0.7442 1 7 .1 1 9 4 4.1376 0.8140 21.8069 4.6698 0.9535 26.0771 5.1066 1.0000 27.5326 5.2472 0.9767 38.2097 6 .1 8 1 4 1.0000 313 SORPTION EXPERIMENT S 0 R P 1 5 1 1 POLYMER : ETFE TEMPERATURECQ : 25 PERMEANT : DICHLOROMETHANE THICKNESS(mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.04559 SUMMARY OF SORPTION EXPERIMENT T IM E (D a y s) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 . 0 0 0 0 0.0000 0.0000 0.0076 0.0874 0.1352 0.0167 0.1291 0.2064 0.0319 0.1787 0.2977 0.0451 0.2125 0.3642 0.0750 0.2739 0.5077 0.0979 0.3129 0.5991 0.1292 0.3594 0.7331 0.1549 0.3935 0.8149 0.2132 0.4617 0.9253 0.2590 0.5089 0.9371 0.4076 0.6385 0.9786 0.7708 0.8780 0.9988 1.2799 1 .1 3 1 3 1 . 0 0 0 0 3 1 4 SORPTION EXPERIMENT S O R P 1 6 1 1 POLYMER :ETFE TEMPERATURE (“C) : 25 PERMEANT : M .E.K THICKNESS (mil) : 10 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 0.02959 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0215 0.1467 0.1294 0.0417 0.2041 0.1904 0.1042 0.3227 0.3327 0.1542 0.3926 0.4270 0.2097 0.4580 0.5231 0.2917 0.5401 0.6525 0.4576 0.6765 0.8946 0.5993 0.7741 0.9464 0.8076 0.8987 0.9686 1.6965 1.3025 0.9760 3.9639 1.9910 0.9945 5.0087 2.2380 0.9963 7.3845 2.7174 1.0000 3 1 5 SORPTION EXPERIMENT S 0 R P 2 1 1 1 POLYMER :ECTFE TEMPERATUREfC) : 25 P E R M E A N T : B E N Z E N E T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.03457 0.03486 0.03485 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS Q KANGE SAMPLE 1 SAMPLE 2 SAMPLES 0.0000 0.000 0.0000 0.0000 0.0000 0.4468 0.447 0.0512 0.0423 0.0645 0.7792 0.779 0.1132 0.0937 0.1332 1.1281 1.128 0.1814 0.1495 0.1991 1.5387 1.539 0.2744 0.2372 0.2966 1.7649 1.765 0.3364 0.2931 0.3496 1.8890 1.889 0.3721 0.3278 0.3825 2.1410 2.141 0.4636 0.4139 0.4599 2.2885 2.289 0.5194 0.4607 0.5086 2.5158 2.516 0.6651 0.5695 0.6275 2.5951 2.595 0.7426 0.6088 0.6877 2.6047 2.605 0.7581 0.6118 0.6948 2.7042 2.704 0.8992 0.6873 0.8152 2.7514 2.751 0.9380 0.7266 0.8911 2.8699 2.870 1.0047 0.8882 1.0000 2.9576 2.958 1.0000 0.9834 1.0000 3.1853 3.185 0.9953 1.0000 0.9971 3.3632 3.363 0.9922 0.9924 0.9871 4.6158 4.616 0.9814 0.9819 0.9871 3 1 6 SORPTION EXPERIMENT S O R P 2 1 2 1 POLYMER :ECTFE TEMPERATURE (“C) : 45 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.04026 SUMMARY OF SORPTION EXPERIMENT nM E (D ays) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0069 0.0833 0.0621 0.0146 0.1208 0.1001 0.0340 0.1845 0 .1 7 1 1 0.0563 0.2372 0.2307 0.0756 0.2749 0.2738 0.1035 0.3217 0.3346 0.1326 0.3642 0.3967 0.1889 0.4346 0.5234 0.2181 0.4670 0.5919 0.2451 0.4951 0.6768 0.2688 0.5184 0.7833 0.2986 0.5465 0.9354 0.3229 0.5683 0.9886 0.3438 0.5863 1.0013 0.3729 0.6107 0.9987 0.4014 0.6336 1.0051 0.4681 0.6841 1.0076 0.6472 0.8045 1.0025 0.8799 0.9380 1.0000 3 1 7 SORPTION EXPERIMENT S O R P 2 1 3 1 POLYMER :ECTFE TEMPERATURE CQ : 65 PERMEANT : BENZENE T H IC K N E S S (m il) ; 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.04788 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0035 0.0589 0.1604 0.0082 0.0907 0.2802 0.0127 0 .1 1 2 8 0.3758 0.0176 0.1326 0.4791 0.0221 0.1487 0.5923 0.0273 0.1653 0.7791 0.0315 0.1774 0.9264 0.0363 0.1906 0.9824 0.0407 0.2018 0.9978 0.0449 0.2119 1.0000 0.0553 0.2352 0.9967 0.0713 0.2670 1.0000 3 1 8 SORPTION EXPERIMENT S O R P 2 2 1 1 POLYMER : ECTFE TEMPERATURE (“Q : 25 PERMEANT : TOLUENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.03190 0.03167 0.03184 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS ClGANGE SAMPLE 1 SAMPLE 2 SAMPLES 0.000 0.000 0.0000 0.0000 0.0000 0.207 0.455 0.0559 0.0592 0.0446 0.623 0.789 0.1036 0.1184 0.0843 1.469 1.212 0.1612 0.1838 0.1355 2.386 1.545 0.2188 0.2383 0.1785 3.122 1.767 0.2549 0.2788 0.2149 3.575 1.891 0.2796 0.3022 0.2413 4.567 2.137 0.3207 0.3442 0.2810 5.250 2.291 0.3553 0.3738 0.3124 6.336 2.517 0.4112 0.4174 0.3620 6.750 2.598 0.4342 0.4377 0.3884 7.326 2.707 0.4688 0.4626 0.4182 7.550 2.748 0.4819 0.4766 0.4298 8.250 2.872 0.5280 0.5140 0.4678 8.755 2.959 0.5559 0.5483 0.4942 10.153 3.186 0.6743 0.6386 0.5785 11.306 3.362 0.8503 0.7695 0.6645 12.545 3.542 0.9852 0.9657 0.8198 13.578 3.685 0.9918 0.9907 0.8992 14.345 3.787 0.9984 1.0016 0.9901 17.345 4.165 0.9984 1.0016 0.9901 21.313 4.617 1.0000 1.0000 1.0000 3 1 9 SORPTION EXPERIMENT S O R P 2 2 2 1 POLYMER : ECTFE TEMPERATURE (“0 :45 PERMEANT : TOLUENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE SAMPLE 1 0.03539 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0083 0.0913 0.0370 0.0167 0.1291 0.0607 0.0340 0.1845 0.0978 0.0563 0.2372 0.1348 0.0753 0.2745 0.1630 0.1056 0.3249 0.2059 0.1340 0.3661 0.2415 0.1910 0.4370 0.3156 0.2194 0.4684 0.3541 0.2472 0.4972 0.3822 0.2708 0.5204 0.4163 0.2986 0.5465 0.4474 0.3479 0.5898 0.5096 0.4076 0.6385 0.6030 0.4743 0.6887 0.7556 0.5229 0.7231 0.9141 0.5625 0.7500 0.9837 0.6458 0.8036 1.0030 0.7271 0.8527 1.0074 0.8778 0.9369 1.0000 1 3 2 0 SORPTION EXPERIMENT S O R P 2 2 3 1 POLYMER : ECTFE TEMPERATURE (“Q : 65 PERMEANT : TOLUENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.04147 SUMMARY OF SORPTION EXPERIMENT TTME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0042 0.0645 0.1304 0.0087 0.0932 0.2097 0.0128 0.1133 0.2711 0.0177 0.1331 0.3376 0.0219 0.1479 0.3926 0.0267 0.1635 0.4565 0.0316 0.1777 0.5243 0.0364 0.1907 0.6074 0.0403 0.2007 0.6854 0.0446 0 .2 1 1 1 0.7980 0.0494 0.2223 0.9105 0.0543 0.2330 0.9655 0.0602 0.2453 0.9872 0.0681 0.2609 1.0000 0.0957 0.3093 1.0026 0.1389 0.3727 1.0000 321 SO RPnO N EXPERIMENT S O R P 2 3 1 1 P O L Y M E R : E C T F E TEMPERATURE (°C) : 25 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.04617 0.04623 0.04590 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS ClHANGE SAMPLE 1 SAMPLE 2 SAMPLE 3 0.000 0.0000 0.0000 0.0000 0.0000 0.221 0.4697 0.0412 0.0403 0.0447 0.638 0.7985 0.0899 0.0932 0.0943 1.477 1.2152 0 .1 5 1 1 0.1537 0.1600 2.401 1.5497 0.2210 0.2229 0.2295 3.135 1.7705 0.2734 0.2783 0.2866 3.585 1.8934 0.3046 0 .3 1 1 1 0.3164 4.550 2.1331 0.3745 0.3816 0.3896 5.261 2.2937 0.4207 0.4320 0.4367 6.353 2.5205 0.5281 0.5252 0.5285 6.765 2.6009 0.5618 0.5605 0.5645 7.319 2.7054 0.6080 0.6297 0.6315 7.552 2.7480 0.6292 0.6461 0.6588 8.242 2.8709 0.7253 0.7670 0.7816 8.761 2.9599 0.8277 0.9106 0.9032 9.136 3.0226 0.9139 0.9610 0.9591 10.163 3.1880 0.9988 0.9887 0.9950 11.299 3.3615 1.0000 1.0038 1.0000 21.322 4.6176 1.0000 1.0000 1.0000 3 2 2 SORPTION EXPERIMENT S O R P 2 3 2 1 POLYMER : ECTFE TEMPERATURE (°C) : 45 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.04856 SUMMARY OF SORPTION EXPERIMENT T IM E (D a y s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 . 0 0 0 0 0.0000 0.0000 0 .0 1 1 8 0.1087 0.0663 0.0194 0.1394 0.0963 0.0375 0.1936 0 .1 5 1 1 0.0590 0.2430 0.1998 0.0764 0.2764 0.2308 0.1069 0.3270 0.2826 0.1368 0.3699 0.3344 0.1938 0.4402 0.4213 0.2229 0.4721 0.4689 0.2507 0.5007 0.5145 0.2743 0.5237 0.5559 0.3021 0.5496 0.6139 0.3493 0.5910 0.7453 0.3757 0.6129 0.8530 0.4090 0.6396 0.9451 0.4285 0.6546 0.9720 0.4708 0.6862 0.9907 0.5201 0.7212 0.9959 0.6479 0.8049 0.9990 0.8833 0.9399 1 . 0 0 0 0 32 3 SORPTION EXPERIMENT S O R P 2 3 3 1 POLYMER : ECTFE TEMPERATURE (°Q : 65 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.05707 SUMMARY OF SORPTION EXPERIMENT TTME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0049 0.0697 0.1564 0.0093 0.0962 0.2355 0.0134 0.1159 0.2999 0.0186 0.1365 0.3772 0.0228 0.1510 0.4333 0.0271 0.1647 0.5005 0.0314 0.1773 0.5768 0.0363 0.1905 0.6845 0.0401 0.2003 0.7948 0.0450 0.2121 0.9052 0.0490 0.2213 0.9568 0.0540 0.2324 0.9706 0.0597 0.2444 0.9880 0.0684 0.2615 0.9945 0.0956 0.3092 0.9972 0.1399 0.3741 1.0000 3 2 4 SORPTION EXPERIMENT S O R P 2 5 1 1 POLYMER : ECTFE T E M P E R A T U R E (“C ) : 2 5 PERMEANT : DICHLOROMETHANE THICKNESS(mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.06035 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0063 0.0791 0.0706 0.0188 0.1369 0.1465 0.0299 0.1728 0.2040 0.0472 0.2173 0.2886 0.0736 0.2713 0.4071 0 . 1 0 0 0 0.3162 0.5344 0.1278 0.3575 0.7175 0.1528 0.3909 0.9704 0 .2 1 1 8 0.4602 0.9965 0.2611 0 .5 1 1 0 1.0017 0.4069 0.6379 1.0009 0.7729 0.8792 1.0061 1 .2 8 1 9 1.1322 1 . 0 0 0 0 325 SORPTION EXPERIMENT S O R P 2 6 1 1 POLYMER : ECTFE TEMPERATURE (°C) : 25 PERMEANT : M.E.K. THICKNESS (mil) : 10 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 0.02785 SUMMARY OF SORPTION EXPERIMENT HM E(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0215 0.1467 0.0209 0.0417 0.2041 0.0304 0.1042 0.3227 0.0607 0.1542 0.3926 0.0835 0.2097 0.4580 0.1025 0.2917 0.5401 0.1290 0.4576 0.6765 0.1784 0.5993 0.7741 0.2182 0.8076 0.8987 0.2732 1.6965 1.3025 0.5123 3.9639 1.9910 1.0000 5.0087 2.2380 0.9943 7.3845 2.7174 1.0000 3 2 6 SORPTION EXPERIMENT S 0 R P 3 1 1 1 POLYMER : PVDF TEMPERATURE (°C) : 25 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLES 0.02444 0.02458 0.02456 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS Q HANGE SAMPLE 1 SAMPLE 2 SAMPLES 0.000 0.000 0.000 0.000 0.000 0.307 0.554 0.060 0.060 0.058 1.234 1 .1 1 1 0.134 0.137 0.135 2.411 1.553 0.206 0.207 0.210 3.446 1.856 0.265 0.261 0.268 4.581 2.140 0.331 0.325 0.333 7.452 2.730 0.497 0.485 0.508 9.670 3.110 0.641 0.621 0.657 11.670 3.416 0.786 0.762 0.802 15.157 3.893 0.932 0.926 0.940 17.622 4.198 0.954 0.952 0.960 24.310 4.930 0.988 0.988 0.994 30.432 5.517 0.994 0.994 0.998 43.698 6.610 1.000 1.000 1.000 49.594 7.042 1.002 1.002 1.004 3 2 7 SORPTION EXPERIMENT S O R P 3 1 2 1 P O L Y M E R : P V D F TEMPERATURECQ : 45 P E R M E A N T : B E N Z E N E T H IC K N E S S (im l) ; 10 EQUILIBRIUM VALUE SAMPLE 1 SAMPLE 2 SAMPLES 0.02733 0.02728 0.02688 SUMMARY OF SORPTION EXPERIMENT T IM E (D ay s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 SAMPLE 2 SAMPLE 3 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.0391 0.1976 0.0839 0.0823 0.0789 0.0774 0.2782 0 .1 1 9 6 0 .1 1 8 1 0 .1 1 4 0 0.2510 0.5010 0.2589 0.2576 0.2474 0.4651 0.6820 0.4071 0.4061 0.3877 0.9942 0.9971 0.8482 0.8479 0.8123 2.1049 1.4508 0.9875 0.9875 1 .0 0 0 0 5.8835 2.4256 1 .0 0 0 0 1 .0 0 0 0 1 .0 0 0 0 32 8 SORPTION EXPERIMENT S O R P 3 1 3 1 POLYMER : PVDF TEMPERATURE (“C) : 65 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.03159 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0125 0 .1 1 1 8 0.1449 0.0213 0.1459 0.1984 0.0559 0.2364 0.3795 0.0899 0.2999 0.5528 0.1222 0.3496 0.7339 0.1778 0.4216 0.9370 0.2694 0.5191 0.9921 0.3438 0.5863 0.9969 0.4681 0.6841 1.0000 3 2 9 SORPTION EXPERIMENT S O R P 3 2 1 1 POLYMER : PVDF TEMPERATUREfC) ; 25 PERMEANT : TOLUENE THICKNESS(mil) : 10 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.02126 0.02113 0.02111 SUMMARY OF SORPTION EXPERIMENT TIME (Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 SAMPLE 2 SAMPLE 3 0 .0 0 0 0 .0 0 0 0 .0 0 0 0 .0 0 0 0 .0 0 0 0.626 0.791 0.097 0.095 0.069 1.834 1.354 0.145 0.133 0.132 2.879 1.697 0.189 0.174 0.174 3.946 1.986 0.233 0.217 0 .2 1 2 6.934 2.633 0.341 0.317 0.310 10.609 3.257 0.477 0.437 0.431 14.557 3.815 0.659 0.597 0.583 16.782 4.097 0.756 0 .6 8 6 0.672 18.797 4.336 0.839 0.776 0.761 22.233 4.715 0.924 0.887 0.875 24.757 4.976 0.945 0.928 0.917 3 1 .4 1 8 5.605 0.975 0.973 0.975 37.506 6.124 0.984 0.986 0.989 50.775 7.126 0.995 0.995 0.998 56.673 7.528 1 .0 0 0 1 .0 0 0 1 .0 0 0 3 3 0 SORPTION EXPERIMENT S O R P 3 2 2 1 POLYMER :PVDF T E M P E R A T U R E (“Q : 4 5 PERMEANT ; TOLUENE THICKNESS (mü) : 10 EQUILIBRIUM VALUE ^ram /gram ) SAMPLE 1 0.02371 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0194 0.1394 0.0314 0.0396 0.1990 0.0544 0.0826 0.2875 0.0858 0.1889 0.4346 0.1506 0.3417 0.5845 0.2218 0.4802 0.6930 0.2803 0.6590 0 .8 1 1 8 0.3556 0.8941 0.9456 0.4456 1.1236 1.0600 0.5439 1.7653 1.3286 0.8243 2.1139 1.4539 0.9142 2.3688 1.5391 0.9477 2.6208 1.6189 0.9665 2.9736 1.7244 0.9770 4.2465 2.0607 0.9916 5.9500 2.4393 1.0000 331 SO RPnO N EXPERIMENT S O R P 3 2 3 1 POLYMER :PVDF TEMPERATURE (°C) : 65 PERMEANT : TOLUENE THICKNESS (mil) : 10 EQ UILIBRIU M VALUE ^ram/gram) SAMPLE 1 0.02583 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0056 0.0745 0.0635 0.0132 0.1149 0.1096 0.0326 0.1807 0.1981 0.0521 0.2282 0.2635 0.0826 0.2875 0.3635 0 .1 1 8 1 0.3436 0.4750 0.1500 0.3873 0.5808 0.2083 0.4564 0.7654 0.2597 0.5096 0.8865 0.2958 0.5439 0.9327 0.3229 0.5683 0.9538 0.4611 0.6791 0.9885 0 .5 1 1 1 0.7149 0.9885 0.5556 0.7454 0.9904 0.8806 0.9384 0.9962 1.2951 1.1380 1.0000 3 3 2 SORPTION EXPERIMENT S O R P 3 3 1 1 POLYMER : PVDF TEMPERATURE (“C) : 25 PERMEANT : CHL0R0BEN2ENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.02110 0.02110 0.02103 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS ClHANGE SAMPLE 1 SAMPLE 2 SAMPLE 3 0.000 0.0000 0.0000 0.0000 0.0000 1.604 1.2665 0.0911 0.1047 0.0979 2.793 1.6712 0.1400 0.1535 0.1435 3.965 1.9911 0.1844 0.1930 0.1868 7.058 2.6567 0.2778 0.2884 0.2756 10.723 3.2747 0.3711 0.3930 0.3713 14.705 3.8347 0.4822 0.5140 0.4852 16.850 4.1048 0.5378 0.5791 0.5444 18.876 4.3446 0.5978 0.6442 0.6082 22.332 4.7257 0.6933 0.7465 0.7039 25.299 5.0298 0.7644 0.8140 0.7790 31.560 5.6178 0.8867 0.9209 0.8929 37.598 6.1317 0.9533 0.9674 0.9590 56.770 7.5346 0.9778 0.9884 0.9863 73.409 8.5679 1.0000 1.0000 1.0000 3 3 3 SORPTION EXPERIMENT S O R P 3 3 2 1 POLYMER :PVDF TEMPERATURE (°C) : 45 PERMEANT : CHL0R0BEN2ENE T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.02375 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0289 0.1701 0.0455 0.0962 0.3102 0.0971 0.1960 0.4428 0.1508 0.2467 0.4967 0.1756 0.3762 0.6133 0.2355 0.4495 0.6705 0.2748 0.6014 0.7755 0.3326 0.7765 0.8812 0.3967 1.0014 1.0007 0.4793 1.1577 1.0760 0.5310 1.4582 1.2076 0.6612 1.9479 1.3957 0.8182 4.2872 2.0706 0.9525 4.9219 2.2185 0.9773 6.1071 2.4712 0.9876 13.9537 3.7355 1.0000 SORPTION EXPERIMENT 3 3 4 S O R P 3 3 3 1 POLYMER :PVDF TEMPERATURE (°C) : 65 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 0.02753 SUMMARY OF SORPTION EXPERIMENT TM E(D ays) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0083 0.0913 0.0815 0.0153 0.1236 0.1109 0.0354 0.1882 0.1872 0.0528 0.2297 0.2374 0.0833 0.2887 0.3224 0.1184 0.3441 0.4073 0.1500 0.3873 0.4835 0.2083 0.4564 0.6222 0.2604 0.5103 0.7383 0.2951 0.5433 0.8059 0.3229 0.5683 0.8544 0.4604 0.6785 0.9619 0.5115 0.7152 0.9757 0.5556 0.7454 0.9792 0.8806 0.9384 0.9931 1.2951 1.1380 1.0000 SORPTION EXPERIMENT 3 3 5 S O R P 3 4 1 1 POLYMER :PVDF TEMPERATURE (°C) : 25 PERMEANT : PHENOL T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 0.01434 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 2.2764 1.5088 0.1233 5.2639 2.2943 0.1967 8.2313 2.8690 0.2633 11.4396 3.3822 0.3100 14.3132 3.7833 0.3633 18.0833 4.2525 0.4333 21.0104 4.5837 0.4667 25.1306 5.0130 0.5367 29.1181 5.3961 0.5833 32.1771 5.6725 0.6333 36.1979 6.0165 0.6833 41.3299 6.4288 0.7333 43.7396 6.6136 0.7633 48.3542 6.9537 0.8000 54.3681 7.3735 0.8533 60.3736 7.7700 0.8867 70.9063 8.4206 0.9533 78.3056 8.8490 0.9667 85.1493 9.2276 0.9800 90.3299 9.5042 0.9933 95.0243 9.7480 0.9967 99.2944 9.9647 1.0000 100.7500 10.0374 1.0000 111.7396 10.5707 1.0000 3 3 6 SORPTION EXPERIMENT S O R P 3 5 1 1 POLYMER : PVDF TEMPERATURE(°C) : 25 PERMEANT : DICHLOROMETHANE THICKNESS(mil) : 10 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 0.04585 SUMMARY OF SORPTION EXPERIMENT T IM E (D a y s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.0090 0.0950 0.0257 0.0208 0.1443 0.0401 0.0340 0.1845 0.0555 0.0486 0.2205 0.0689 0.0757 0.2751 0.0946 0.1014 0.3184 0 .1 1 6 1 0.1306 0.3613 0.1377 0.1563 0.3953 0.1552 0.2139 0.4625 0.1942 0.2618 0 .5 1 1 7 0.2240 0.4097 0.6401 0.3104 0.7694 0.8772 0.6064 1.2826 1.1325 0.9589 1.6951 1.3020 0.9712 3.1167 1.7654 0.9877 5.1799 2.2759 0.9918 8.0326 2.8342 1 .0 0 0 0 10.0729 3.1738 1 .0 0 0 0 3 3 7 SORPTION EXPERIMENT S O R P 3 6 1 1 POLYMER : PVDF TEMPERATURE (°C) : 25 PERMEANT : M.E.K. THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.06403 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE I 0.0000 0.0000 0.0000 0.0076 0.0872 0.0464 0.0188 0.1371 0.0791 0.0309 0.1757 0.1134 0.0461 0.2146 0.1484 0.0584 0.2416 0.1788 0.0731 0.2703 0.2100 0.0875 0.2959 0.2412 0 .1 1 1 4 0.3338 0.3052 0.1266 0.3558 0.3470 0.1458 0.3818 0.4041 0.1707 0.4132 0.6301 0.1924 0.4387 0.8912 0.2409 0.4908 0.9429 0.3622 0.6018 0.9559 1.2000 1.0955 0.9810 3.4876 1.8675 0.9962 4.5230 2.1267 1.0091 6.9117 2.6290 1.0000 3 3 8 SORPnON EXPERIMENT S 0 R P 4 1 1 1 POLYMER :FEP TEMPERATURE (°C) ; 25 PERMEANT : BENZENE T H IC K N E S S (m il) ; 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.00347 0.00365 0.00389 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS ClHANGE SAMPLE 1 SAMPLE 2 SAMPLE 3 0.0000 0.0000 0.0000 0.0000 0.0000 0.2951 0.5433 0.1250 0.1446 0.1889 1.2510 1 .1 1 8 5 0.3750 0.3855 0.4111 2.3885 1.5455 0.5625 0.5783 0.5778 3.4311 1.8523 0.6625 0.6988 0.6778 4.5715 2.1381 0.7750 0.7831 0.7778 7.4427 2.7281 0.9375 0.9277 0.9000 9.6778 3.1109 0.9500 0.9518 0.9556 11.6778 3.4173 0.9875 0.9759 0.9667 17.6293 4.1987 0.9875 0.9759 0.9778 24.3219 4.9317 1.0000 1.0000 1.0000 3 3 9 SORPnON EXPERIMENT S O R P 4 1 2 1 POLYMER : FEE TEMPERATURE (“Q : 45 P E R M E A N T : B E N Z E N E T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 SAMPLE 3 0.00445 0.00430 0.00431 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS ClRANGE SAMPLE 1 SAMPLE 2 SAMPLE 3 0.0000 0.000 0.0000 0.0000 0.0000 0.0441 0.210 0.2039 0 .1 9 1 9 0.1800 0.0839 0.290 0.2816 0.2727 0.2600 0.2582 0.508 0.4951 0.5051 0.5100 0.4528 0.673 0.6602 0.6566 0.6600 0.9841 0.992 0.9029 0.9091 0.9100 2.1244 1.458 0.9806 1.0000 1.0000 5.9045 2.430 1.0000 1.0000 1.0100 3 4 0 SORPTION EXPERIMENT S O R P 4 1 3 1 POLYMER :FEP TEMPERATURE (°C) : 65 PERMEANT : BENZENE T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00491 SUMMARY OF SORPTION EXPERIMEN^T TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0072 0.0851 0.1316 0.0166 0.1290 0.2456 0.0304 0.1743 0.3509 0.0420 0.2048 0.4386 0.0579 0.2406 0.5263 0.0781 0.2795 0.6228 0.1700 0.4123 0.8772 0 .3 1 1 8 0.5584 0.9561 0.4348 0.6594 0.9912 1.3696 1.1703 1.0000 341 SORPTION EXPERIMENT S0RP5111 POLYMER : PFA TEMPERATURE (“Q : 25 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00480 SUMMARY OF SORPTION EXPERIMENT TIME (Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.0951 0.3084 0.1597 0.4056 0.6368 0.3782 1.9236 1.3869 0.8319 2.3882 1.5454 0.8739 3.0132 1.7359 0.9328 3.9479 1.9869 0.9496 7.1222 2.6687 1 .0 0 0 0 9.1965 3.0326 1 .0 0 0 0 3 4 2 SORPTION EXPERIMENT S O R P 5 1 2 1 POLYMER : ETFE TEMPERATURE (°C) : 45 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00524 SUMMARY OF SORPTION EXPERIMENT T IM E (D ay s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 .0 0 0 0 0.0000 0.0000 0.0139 0 .1 1 7 9 0.1395 0.0340 0.1845 0.2481 0.0799 0.2826 0.3953 0.1486 0.3855 0.6279 0.2097 0.4580 0.6977 0.2875 0.5362 0.8062 0.3569 0.5974 0.8605 0.5472 0.7397 0.9612 0.7938 0.8909 0.9922 0.9910 0.9955 1 .0 0 0 0 343 SORPTION EXPERIMENT S O R P 5 1 3 1 POLYMER :PFA TEMPERATURE (°C) : 65 P E R M E A N T : B E N Z E N E T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE I 0.00605 SUMMARY OF SORPTION EXPERIMENT TTME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0146 0.1208 0.3581 0.0292 0.1708 0.5338 0.0552 0.2350 0.7297 0.0892 0.2987 0.8784 0.1198 0.3461 0.9392 0.1802 0.4245 0.9797 0.2663 0.5161 1.0000 0.3406 0.5836 1.0000 0.4601 0.6783 1.0000 3 4 4 SORPTION EXPERIMENT S O R P 5 2 1 1 POLYMER : PFA TEMPERATURE (“Q : 25 PERMEANT : TOLUENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00421 SUMMARY OF SORPTION EXPERIMENT T IM E (D a y s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.0951 0.3084 0.1456 0.4056 0.6368 0.3204 1.9236 1.3869 0.7379 2.3882 1.5454 0.8058 3.0132 1.7359 0.8738 3.9479 1.9869 0.9223 7.1222 2.6687 1 .0 0 0 0 9.1965 3.0326 1 .0 0 0 0 3 4 5 SORPTION EXPERIMENT S O R P 5 2 2 1 P O L Y M E R ; P F A TEMPERATURE (°C) : 45 PERMEANT : TOLUENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00427 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SORT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.0146 0.1208 0 .1 1 4 3 0.0354 0.1882 0.2095 0.0833 0.2887 0.3619 0.1521 0.3900 0.5048 0.2049 0.4526 0.6095 0.2899 0.5385 0.7333 0.3618 0.6015 0.7810 0.5507 0.7421 0.9333 0.7840 0.8855 1 .0 0 0 0 1.0090 1.0045 1 .0 0 0 0 3 4 6 SORPTION EXPERIMENT S O R P 5 2 3 1 POLYMER :PFA TEMPERATURE (°C) : 65 P E R M E A N T ; T O L U E N E THICKNESS (mil) : 10 EQUILIBRIUM VALUE ^iram/gram) SAMPLE 1 0.00506 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0146 0.1208 0.3089 0.0326 0.1807 0.4634 0.0583 0.2415 0.6260 0.0919 0.3031 0.7724 0.1264 0.3555 0.8618 0.1757 0.4192 0.9350 0.2688 0.5184 0.9837 0.3340 0.5780 0.9919 0.4646 0.6816 1.0000 3 4 7 SORPTION EXPERIMENT S O R P 5 3 1 1 POLYMER : PFA TEMPERATURE (°C) : 25 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00435 SUMMARY OF SORPTION EXPERIMENT T IM E (D ay s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.0951 0.3084 0.1402 0.4056 0.6368 0.3178 1.9236 1.3869 0.7290 2.3882 1.5454 0.7944 3.0132 1.7359 0.8505 3.9479 1.9869 0.9065 7.1222 2.6687 0.9813 9.1965 3.0326 1 .0 0 0 0 3 4 8 SORPTION EXPERIMENT S O R P 5 3 2 1 POLYMER : PFA TEMPERATURE (T ) : 45 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00457 SUMMARY OF SORPTION EXPERIMENT T IM E (D ay s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.0174 0 .1 3 1 8 0.1455 0.0368 0 .1 9 1 8 0.2273 0.0837 0.2893 0.4182 0.1521 0.3900 0.5364 0 .2 1 1 1 0.4595 0.6364 0.2882 0.5368 0.7182 0.3590 0.5992 0.8182 0.5479 0.7402 0.9727 0.7813 0.8839 1.0091 0.8574 0.9313 0.9909 1.0000 1.0000 1 .0 0 0 0 3 4 9 SORPnON EXPERIMENT S O R P 5 3 3 1 P O L Y M E R : P F A TEMPERATURE (=C) : 65 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00517 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0146 0.1208 0.2857 0.0354 0.1882 0.4683 0.0611 0.2472 0.6190 0.0944 0.3073 0.7698 0.1264 0.3555 0.8492 0.1778 0.4216 0.9365 0.2708 0.5204 0.9762 0.3361 0.5798 0.9921 0.4670 0.6834 1.0000 3 5 0 SORPTION EXPERIMENT S O R P 5 5 1 1 POLYMER : PFA TEMPERATURE(°C) : 25 PERMEANT : DICHLOROMETHANE T H X C K N E S S (m ü ) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00766 SUMMARY OF SORPTION EXPERIMENT T IM E (D ay s) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0104 0 .1 0 2 1 0.1204 0 .0 2 2 2 0.1491 0.1885 0.0347 0.1863 0.2513 0.0500 0.2236 0.3089 0.0771 0.2776 0.3979 0.1028 0.3206 0.4764 0 .1 3 1 9 0.3632 0.5393 0.1569 0.3962 0.6021 0.2160 0.4647 0.7068 0.2632 0.5130 0.7644 0.4097 0.6401 0.8953 0.7736 0.8796 0.9843 1.2847 1 .1 3 3 5 1 .0 0 0 0 1.6958 1.3022 0.9948 3.1160 1.7652 1 .0 0 0 0 35 1 SORPTION EXPERIMENT S O R P 5 6 1 1 POLYMER :PFA TEMPERATURE (°C) : 25 PERMEANT : M.E.K. T H IC K N E S S (m il) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00283 SUMMARY OF SORPTION EXPERIMENT TTME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0215 0.1467 0.0571 0.0417 0.2041 0.0857 0.1042 0.3227 0.2143 0.1542 0.3926 0.2571 0.2097 0.4580 0.3143 0.2917 0.5401 0.3857 0.4576 0.6765 0.5143 0.5993 0.7741 0.6429 0.8076 0.8987 0.7286 1.6965 1.3025 0.9000 3.9639 1.9910 0.9857 5.0087 2.2380 1.0000 7.3845 2.7174 1.0000 3 5 2 SORPTION EXPERIMENT S O R P 2 2 2 2 POLYMER :ECTFE TEMPERATURE (°C) : 45 PERMEANT : TOLUENE THICKNESS (mil) : 90 EQ U ILIBRIU M VALUE (gram/gram) SAMPLE 1 SAMPLE 2 0.03495 0.03499 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 SAMPLE 2 0.000 0.0000.000 0.000 0.526 0.725 0.065 0.066 1.486 1.219 0 .1 1 5 0.116 2.450 1.565 0.152 0.153 5.689 2.385 0.255 0.256 8.700 2.950 0.333 0.335 12.819 3.580 0.429 0.431 16.823 4.102 0.521 0.524 20.747 4.555 0.623 0.627 28.065 5.298 0.902 0.909 31.696 5.630 0.985 0.986 36.602 6.050 1.002 1.001 42.863 6.547 1.005 1.004 50.646 7.117 1.003 1.003 69.728 8.350 1.000 1.000 3 5 3 SORPTION EXPERIMENT S O R P 2 2 3 2 POLYMER :ECEFE TEMPERATURE (“Q : 65 PERMEANT : TOLUENE THICKNESS (mil) : 90 EQUILIBRIUM VALUE ^ram/gram) SAMPLE 1 0.04075 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0181 0.1344 0.0282 0.0736 0.2713 0.0656 0.1583 0.3979 0.1050 0.3417 0.5845 0.1675 1.1000 1.0488 0.3447 1.3167 1.1475 0.3827 1.5319 1.2377 0.4259 2.4889 1.5776 0.6164 3.2229 1.7952 0.8029 3.5806 1.8922 0.8913 3.7958 1.9483 0.9279 4.1743 2.0431 0.9705 4.5910 2.1427 0.9834 5.7278 2.3933 0.9986 6.3896 2.5278 1.0000 3 5 4 SORPTION EXPERIMENT S O R P 2 3 2 2 POLYMER :ETFE TEMPERATURE ("C) : 45 PERMEANT : CHLOROBENZENE THICKNESS (mü) : 90 EQUILIBRIÜM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 0.04815 0.04807 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRTCTIME) FRACTIONAL]SIASS CHANGE SAMPLE 1 SAMPLE 2 0.000 0.000 0.000 0.000 0.552 0.000 0.074 0.074 1.588 0.000 0.136 0.136 3.559 0.000 0.220 0.219 5.580 0.000 0.289 0.287 7.779 0.000 0.359 0.356 10.837 0.000 0.451 0.445 14.917 0.000 0.573 0.562 18.913 0.000 0.726 0.708 22.841 0.000 0.934 0.919 30.689 0.000 1.001 1.002 33.823 0.000 0.999 0.999 38.725 0.000 1.000 1.000 45.000 0.000 1.002 1.003 52.776 0.000 1.002 1.003 58.826 0.000 1.001 1.001 1 71.862 0.000 1.000 1.000 355 SORPTION EXPERIMENT S O R P 2 3 3 2 POLYMER :ECIPE TEMPERATURE (°C) : 65 PERMEANT : CHLOROBENZENE THICKNESS (mil) : 90 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.9142/16.1715 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0.0236 0.1537 0.0366 0.0785 0.2801 0.0739 0.1632 0.4040 0 .1 1 3 5 0.3479 0.5898 0.1787 1 .1 1 1 8 1.0544 0.3681 1.3167 1.1475 0.4088 1.5319 1.2377 0.4555 1.6743 1.2939 0.4846 2.4938 1.5792 0.6765 3.2347 1.7985 0.8861 3.5806 1.8922 0.9443 3.7958 1.9483 0.9662 4.1743 2.0431 0.9861 4.5910 2.1427 0.9891 5.7292 2.3936 1.0000 6.3896 2.5278 1.0000 3 5 6 SORPTION EXPERIMENT DIFF1121 POLYMER :ETFE TEMPERATURE (°C) : 45 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 0.00406 0.00401 SUMMARY OF SORPTION EXPERIMENT TTME(Days) SQRT(TTME)FRACTIONAL]VLASS CHANGE SAMPLE 1 SAMPLE 2 0.0000 0.0000 0.0000 0.0000 1.2894 1.1355 0.5658 0.5467 2.0723 1.4396 0.7105 0.6933 3.3038 1.8176 0.8158 0.8267 4.2604 2.0641 0.8816 0.8800 6.3103 2.5120 0.9868 0.9733 11.9928 3.4631 1.0000 1.0000 1 5 .9 1 1 7 3.9890 0.9868 0.9867 1 19.2134 4.3833 1.0000 1.0000 * The term "DIFF" indicates the sorption experiment with a vapor-phase penetrant. 3 5 7 SORPTION EXPERIMENT D IF F 2 1 2 1 POLYMER :ECTFE TEMPERATURE (°C) : 45 PERMEANT ; BENZENE THICKNESS (mü) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 0.00368 0.00398 SUMMARY OF SORPTION EXPERIMENT TEME(Days) SQRTCITME) FRACTIONAL MASS CHANGE SAMPLE I SAMPLE 2 0.0000 0.0000 0.0000 0.0000 0.7938 0.8910 0.0822 0.0946 2.0543 1.4333 0.1507 0.1622 3.0977 1.7600 0.1918 0.2162 5.1772 2.2754 0.2466 0.2703 6.9986 2.6455 0.2740 0.2973 8.7721 2.9618 0.3151 0.2973 11.2098 3.3481 0.3699 0.4054 13.6058 3.6886 0.4247 0.4459 15.2802 3.9090 0.4521 0.4595 17.2802 4.1570 0.4795 0.5000 20.2802 4.5034 0.5342 0.5541 23.2802 4.8250 0.5753 0.5946 29.2802 5 .4 1 1 1 0.6712 0.7027 40.2802 6.3467 0.8082 0.8378 43.2802 6.5788 0.8356 0.8514 48.2802 6.9484 0.9452 0.9459 51.2802 7.1610 0.9589 0.9459 53.2802 7.2993 1.0137 0.9865 55.9226 7.4781 1.0000 0.9865 59.2206 7.6955 1.0137 1.0000 60.6665 7.7889 1.0000 1.0000 3 5 8 SORPTION EXPERIMENT DIFF3121 POLYMER :PVDF TEMPERATURE (°C) : 45 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 SAMPLE 2 0.00403 0.00403 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SQRT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 SAMPLE 2 0.0000 0.0000 0.0000 0.0000 1.2894 1.1355 0.2927 0.2840 2.0723 1.4396 0.3780 0.3704 3.3038 1.8176 0.5122 0.5062 4.1931 2.0477 0.5732 0.5802 6.2242 2.4948 0.7317 0.7407 7.2604 2.6945 0.7805 0.7901 9.3103 3.0513 0.8780 0.8889 11.9855 3.4620 1.0000 1.0000 1 8 .9 1 1 7 4.3488 1.0122 1.0000 22.2134 4.7131 1.0000 1.0000 3 5 9 SORPTION EXPERIMENT DBFF4121 POLYMER :FEP TEMPERATURE (°C) : 45 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00061 SUMMARY OF SORPTION EXPERIMENT FRACTIONAL MASS CHANGE TIME(Days) SQRT(TIME) SAMPLE 1 0.0000 0.0000 0.0000 0 .1 1 1 8 0.3344 0.2143 0.2507 0.5007 0.3571 0.7715 0.8784 0.7143 1.1986 1.0948 0.8571 2.0250 1.4230 1.0000 4.6549 2.1575 1.0000 7.7875 2.7906 1.0000 3 6 0 SORPTION EXPERIMENT DIFF5121 POLYMER :PFA TEMPERATURE (°C) : 45 PERMEANT : BENZENE THICKNESS (mil) : 10 EQUILIBRIUM VALUE (gram/gram) SAMPLE 1 0.00064 SUMMARY OF SORPTION EXPERIMENT TIME(Days) SORT(TIME) FRACTIONAL MASS CHANGE SAMPLE 1 0.0000 0.0000 0.0000 0 .1 1 1 8 0.3344 0.2667 0.2507 0.5007 0.4000 0.7715 0.8784 0.6667 1.1986 1.0948 0.8000 2.0250 1.4230 0.9333 4.6549 2.1575 1 .0 0 0 0 7.7875 2.7906 1 .0 0 0 0 APPENDIX C STRESS-STRAIN DATA 361 Monsanto TENSOMETER 20 aoo 20 50 TENSlOOO : 100 ISO 12 200 10 250 300 350 * 4 400 450 500 w 20 8 Monsanto TENSOMETER 20 200 ^ 0 20 50 TENS1211 16 100 ISO 200 10 250 300 350 400 450 10 14 18 20 Monsanto TENSOMETER 20 TENS1311 100 ISO 200 250 300 350 400 450 _ 500 W 20 S Monsanto TENSOMETER 20 50 350 MO 20 18 50 TENS1511 16 100 ISO 200 10 250 300 350 400 450 20 Monsanto TENSOMETER 20 35 00 20 sa. 45Û MO 50 TENS 1611 100 150 200 10 250 300 350 400 450 500 12 14 16 20 u> Monsanto TENSOMETER 20 2Sg 350 400 ^ 0 20 18 50 16 TENS2000(M) : 100 150 12 200 10 250 300 350 ; 4 400 450 w 500 16 O' Monsanto TENSOMETER 20 00 500 20 18 50 16 TENS2000(M) 100 14 ISO 200 250 300 350 400 450 10 16 18 Monsanto TENSOMETER 20 450 MO 20 18 50 16 TENS221KM) 100 1E| 150 12 200 10 250 300 350 400 450 500 2010 VO Monsanto TENSOMETER 20 [SjL 20 Jim m 18 50 TENS2311(M). 16 100 150 200 10 250 300 350 400 450 r W — •«•J 10 12 14 16 18 20 O Monsanto TENSOMETER 20 20 Sûr 350 SSO 50 j TENS2511(M) 100 14 ISO 200 10 250 300 350 * 4 400 450 U) 10 12 14 16 18 20 Monsanto TENSOMETER 20 20 4% ^ 0 18 50 100 TENS2611(M) 14 150 200 10 250 300 350 ? 4 400 450 500 w 10 14 16 18 20 Monsanto TENSOMETER 20 50 MO 20 18 60 100 14 ISO 200 10 250 300 350 400 450 500 w 10 18 20 Monsanto TENSOMETER 20 100 00 SQO 20 2 # 50 TENS2000(T) 100 14 150 12 200 10 250 300 350 400 450 500 18 20 Monsanto TENSOMETER 20 35g 50 20 aoQ. SQO 50 100 TENS2000(T) 150 200 10 250 300 350 400 450 500 w 10 12 14 16 18 20 Dî Monsanto TENSOMETER 20 20 300 50 16 TENS221KT) 100 14 150 12 200 10 250 300 350 400 450 14 16 20 Monsanto TENSOMETER 20 TENS2311(T) 500 W Monsanto TENSOMETER 20 MO 20 18 50 100 150 200 250 300 350 400 450 500 12 14 18 20 00 Monsanto TENSOMETER 20 500 20 50 TENS251KT) 100 150 200 10 250 300 350 400 450 500 ^ 20 VO Monsanto TENSOMETER 20 50 MO 20 50 TENS2611(T) 100 150 200 10 250 300 350 400 450 500 Ü» 10 12 14 16 18 20 Monsanto TENSOMETER 20 >9?:, SA 0 0 50 400 450, . 500 j Ü H s s f f i œ TTiTi ! I : I 18 TENS3000(M) 100 1 4 ; 150 200 250 300 i: -I - 350 i.i- 400 450 w 00 500 10 12 14 16 18 20 Monsar i TENSOMETER 20 50 450 MO ».0 asm 18 50 16 100 TENS321KM) 14 150 12 200 10 250 300 350 400 450 u> 500 8 10 12 14 16 18 20 Monsanto TENSOMETER 20 400 500 !0 a u au 2UU auu aau au 18 50 ' * i 100 TENS3311(M) 14 150 12 200 250 300 350 400 450 u> 00 500 w 10 12 14 16 18 20 Monsanto TENSOMETER 20 50 100 ISO 200 250 300 350 400 450 500 ^ 18 50 TENS3511(M) 16 100 14 150 12 200 iO 250 300 350 400 450 OJ 500 g 10 12 14 16 18 20 Monsanto TENSOMETER 20 00 50 400 450 500 20 18 50 TENS361KM) 16 100 14 150 12 20C 10 25C 30C iff 35C ? 4 40C 450 U) SOC uî00 10 12 14 16 18 20 Nions ito TENSOMETER 2( 100 aoo asp 500 20 18 SI 16 TENS3000(T) 1( 14 12 21 10 2 ! 3( 3! 4( I 2 4! w 51 10 12 14 16 18 2 0 Monsanto TENSOMETER 20 350 4 00 4 5 0 20 300 500 18 50 16 TENS3000(T) 100 14 15C 12 20( 10 25( 30( 35( z 4 40( 45C i i ; w ;i± h r 00 50C 10 12 14 16 18 2 0 Monsanto TENSOMETER 20 50 200 500 20 UKL a m 4 # 18 50 TENS3211(T) ; 1 16 : il. - 10C 14 15( 12 201 10 25< 30( 35( Z 4 40( 45( u> 00 50( 00 10 12 14 16 18 20 Monsaiito TENSOMETER 20 2üPi irffl ,199 190 200 250 300 350 400 450 500 ^ 18 50 16 TENS3311(T) 100 14 150 12 200 10 250 300 350 400 450 w 00 500 ^ 10 12 14 16 18 2 0 Monsaito TENSOMETER 20 so 3QQ 450 500 20 M. a a 4aa 18 t -I 50 t I li: 16 TENS3511(T) 100 14 150 12 200 10 250 300 350 400 450 w vO 500 O 10 12 14 16 18 20 Monsanto TENSOMETER 20 i^ùPi 50 100 150 200 250 300 350 400 450 500 ^ 18 50 16 TENS3611(T) too 14 150 12 200 10 250 300 350 400 450 U) vo 500 10 12 14 16 18 20 Monsanto TENSOMETER 20 200 250 a # 500 20 SSL 18 50 TENS4000 16 lo 14 is 12 20 10 25 30 35 z 4 40 - I 45 w 50 I 10 12 14 16 18 2 0 Monsanto TENSOMETER 2 50 400 500 20 s a a a a 2 s a a a a 18 TENS4211 16 14 12 10 z 4 w ÎS 10 12 14 16 18 20 Monsanto TENSOMETER 2( 2 0 p| I I M I 1 i I I 199 |1f?| .999, 950 ,490. . , . . . 4$P 500 Q 18 5( 16 TENS4311 1C 14 1{ 12 21 10 31 31 z 4 Monsanto TENSOMETER 20 ?of[ 50 100 ISO 200 250 300 350 400 450 MO ^ 18 50 16 TENS4511 101 14 15- 12 20 10 25 30 35 z 4 40 Monsanto TENSOMETER 20 450 20 50 TENS5000 100 ISO 200 10 250 300 350 400 450 u> vo 500 OV 10 14 16 18 20 Monsanto TENSOMETER 20 50 20 a m 00 18 50 TENS5211 100 14 150 200 10 250 300 350 400 450 w 500 'O 20 Monsanto TENSOMETER 20 500 20 a m 18 50 TENS5311 100 150 200 10 250 300 350 400 450 500 0 2 4 6 a 10 12 14 16 18 20 00 Monsanto TENSOMETER 20 2 ^ 5 0 20 50 TENS5511 100 150 200 10 250 300 350 400 450 500 10 20 u> APPENDIX D COMPUTER PROGRAM 1. Curve fitting program 2. Crank-Nichloson method 4 0 0 4 0 1 C PROGRAM : CURFIT.FOR C *** All numbers are double precision numbers I C C PROGRAM TO GET THE BEST F IT OF A FONCTION WHICH IS TO BE DEFINED AS C 'FUNC' BY LEAST SQUARE F IT METHOD C C The function to be used is a POLYNOMIAL OF NEGATIVE POWER C C - A MODIFICATION OF JMG'S PROGRAM CURFIT.FOR - C - A MODIFICATION OF [KIM.KKIFUN.FOR - C - MODIFIED BY LEES FOR SPECIAL PURPOSE. C C If you want to change the function for the fit and the number of C parameters, you have to make the following changes: C 1) Change the definition of the function in the external C f u n c t i o n FUNC. C 2) F in d t h e s t a t e m e n t PARAMETER(NPAR=3) a n d c h a n g e th e i n t e g e r 3 C into the desired number of parameters. C 3) Check the output format for the screen in the subroutine PRTPAR. C 4) Check the output FORMAT in the subroutine MAKEFILE12. C 5) Check if the maximum number of data points 600 in the statement C PARAMETER(NMAX=600) i s g o o d f o r y o u . C C *** Main program for nonlinear least-squares fitting C *** One ODE e q u a ti o n & u s e r - f r i e n d l y (FCN & DIVPAG) C PARAMETER(NPAR=3, MPAR=4) PARAMETER( IPAR=5, JPAR=100 ) PARAMETER(NMAX=100) C DOUBLE PRECISION X(NMAX), Y (NMAX), YFIT(NMAX),SIGMAY(NMAX) DOUBLE PRECISION SIGMAA(NPAR), A(NPAR). FLAMDA,DATA(NMAX) DOUBLE PRECISION AP(IPAR,JPAR.NPAR).API(IPAR,JPAR.MPAR) DOUBLE PRECISION A I(MPAR), CHI(IPAR,JPAR), CHISQRl,A 4,CHIMIN DOUBLE PRECISION T O L l, T 0L 2, DELTAY, P I , SGMl, SGM2, SCMA, SGMG INTEGER IREAD,IN,IC,10,ILINE,ICOL EXTERNAL CURFIT,FUNC,FDERIV,FCHISQ,PRTPAR EXTERNAL PLOT,MPRTPAR,STIFF,FCN,FCNJ,MAKEFILE12 COMMON A4 C CHARACTER*10 FOLD CHARACTER*11 FNEWl. FNEW2, FNEW3 C C **************************************************** C Introduction of USER-FRIENDLY programs C 10 CALL HOME CALL REVRSE C ILIN E=12 ICOL=38 IREAD=4 IN=5 IC=6 10=7 C W R IT E (IC ,* )' 1 . ENTER YOUR DATA' WRITEdC, *) WRITECIC,*)' 2. DO CALCULATIONS' WRITEdC, *) WRITEdC,*)' 3. PLOT FUNCTION' WRITEdC, *) WRITECIC,*)' 4. STOP' WRITECIC,*) CALL NORMAL 4 0 2 W R IT E C IC ,*)' ENTER CHOICE BY NUMBER PLEA SE:' C CALL GOTOXY( ILIN E,IC O L) READ{IN,*)IGO GOTO(100,200,300,400), IGO C C End of 'USER-FRIENDLY' program C *************************************************************** C *** NO.1 option of user-friendly program C 100 PI = 3.14159256 WRITE(6,711) 711 FORMAT(// ' Enter TOLl (a constant less than 0.1) : TOLl = ? ',$) READ(5,*) TOLl TOL2=1000000. FLAMDA=0.0300 DELTAY=1.0 C POSSIBLE MODES ARE - 1 , 0 , 1 MODE=0 C MAXIMUM NUMBER OF TRY WRITE(6,712) 712 FORMAT(/' Enter maximum number of iteration : NLOOPMAX=? ',$) READ(5,*)NL00PMAX C C *** Information for the fit C WRITE(6,311) 311 FORMAT(// ' Name of data file excluding ■.prn" ? ',$) READ(5,321) FOLD 321 FORMAT(AlO) FNEWl=FOLD//'A' FNEW2=FOLD//'B' FNEW3=FOLD//'P' W R IT E (6,331) FNEWl 331 FORMAT(/ ' * Interacting I/O w ill be recorded in ',A ll,' .DAT') WRITE(6,333) FNEW2 333 FORMAT(' * The data and calculated values in ',A ll, '.DAT') WRITE(6,335) FNEW3 335 FORMAT(' * The final parameters are w ritten in ' ,A11,' .DAT') C C *** A s s ig n t h e IN ITIA L VALUES OF PARAMETERS C WRITE(6,350) 350 FORMAT(// ' Please enter initial values of the parameters.'/) OPEN ( 1 4 , FILE= ' PARA. DAT' , STA'TUS= ' NEW' ) DO 360 I=1,NPAR WRITEdC, 365) I 365 FORMAT( / ' PARAMETER( ' , 1 1 , ' ) : ', $ ) READ (5,*) A(I) 360 CONTINUE WRITE(14,*) (A d) , I=1,NPAR) CLOSE(1 4 ) WRITE(6,713) 713 FORMAT(/' Enter intial value of final parameter : A(4)=? ',$) READ(5,*)A4 C C *** Read data file while counting the number, NDATA C OPEN(10,FILE=FOLD,STATUS:' OLD', DEFAULTFILE=' .PRN') I I = 0 371 II = II + 1 READdO, *,END=373) X (I1),Y (I1) SIGMAY(II)=DELTAY GO TO 371 373 NDATA = I l - l CLOSE(1 0 ) 4 0 3 GOTO 10 C C C •** NO.2 option oC user-friendly program C 200 WRITE(6.714) 714 FORMAT(/' Enter iteration numbers of A(3) parameter : ILOOP=? ',$) READ(5.*) ILOOP WRITE(6,715) 715 FORMAT(/' Enter iteration numbers of A(4) parameter : JLOOP=7 ',$) READ(5,*) JLOOP C A 1=A(1) A2=A{2) A 3=A(3) DA4=A4 OPEN(1 3 ,FILE=FNEW3,STATUS:'NEW') WRITE(13,199)A1,A2,A3,DA4 199 FORMAT(1X,4E10.3,2X,'IQ JQ ',2X,'CHISQl') DO I=l,ILOOP A3=A3+0.01*(DFLOAT(I)-1.0) DO J=l,JLOOP A4=DA4+0.1 * (DFLOAT(J)-1.0) A (1)=A 1 A (2)=A 2 A (3)=A 3 OPEN(14,FILE:' PARA.DAT' , STATUS:' OLD') W R IT E (1 4 ,*) ( A ( I l ) , 1 1 : 1 , NPAR) CLOSE(1 4 ) FLAMDA:0.030 C OPEN(1 1 .FILE:FNEW1. STATUS:' NEW') WRITE( 1 1 ,* ) ' IN IT IA L VALUES OF PARAMETERS' DO 1 2 : 1 , NPAR WRITEdl, *) 'A C ,12,') : ',A(I2) AI(I2)=A(I2) ENDDO W R IT E d l,* ) A C , MPAR, ') = ',A 4 AI(MPAR):A4 W RITE(6,381) NDATA 381 FORMAT(/ '*** The number of data points is now ',13) C W RITEdl, *) W R IT E d l, *) ' TRACE OF FITTING PROCEDURE' W RITEdl, *) W R IT E d l, *) ' A d ) th r o u g h A (NPAR) ' C CALL CURFIT(MODE,TOLl, T 0L 2, NLOOPMAX,FLAMDA, NDATA, X,Y,YFIT, SIGMAY,A,SIGMAA,CHISQl, NPASS) C C **•*•*••********•***••***•******•****•*•*****•*••**** C *** Final report *** PRINT 600, NPASS W RITEdl, 600) NPASS 600 FORMAT(6X,'LOOPING STOPPED AFTER ',1 3 ,' ITERATIONS') PRINT 610, CHISQl W RITEdl, 610) CHISQl 610 FORMAT( / 6X ,'F IN A L CHI SQUARE PER DEGREE OF FREEDOM : ', E 1 2 . 5 / ) C C *** Write parameters and their errors on the screen and file *11. DO 620 1 3 : 1 , NPAR WRITE(6,630) 13,A (I3),I3,SIGMAA(I3) W RITEdl,630) I3,A(I3),I3,SIGMAA(I3) 4 0 4 620 CONTINUE 630 FORMAT(' A(',13,') = '.E12.5. ' SIGMAA('.13,') = '. E12.5) CLOSE(1 1 ) CHI(I,a)=CHISQl C C *** Make file #12 in which the in itial data and calculated ones C *** are listed on the 2nd and 3rd columns respectively C OPEN(1 2 ,FILE=FNEW2, STATUS=' NEW') CALL MAKEFILE12(NDATA,A,X,Y,DATA) CLOSE(12) C C *** Make file #13 of one column which lists parameters C IQ =I JQ —J WRITE(13,299) A (l),A (2),A(3),AI(4 ),IQ,JQ,CHISQl 299 FORMAT(/1X,4E10.3,2(1X,12),E10.3) ENDDO ENDDO CLOSE(1 3 ) GOTO 10 C c •••***•*••***•*•••***•**••**•*•*•*•**••**•*••*•••**•****•**•♦••**•••♦* C *** NO.3 option of user-friendly program C 300 CALL PLOT(NDATA,A,X,Y,DATA) GOTO 10 C c ***********•**♦**•*•**•***•*•******************•*••*****•*•*•*•***•*** C *** NO.4 option of user-friendly program C 400 END C C ********* End of Main Program ********** C ======:======C c C SUBROUTINE CURFIT C SUBROUTINE CURFIT(MODE,TOLl, T 0L 2, NLOOPMAX,FLAMDA,NDATA, ► X,Y,YFIT,SIGMAY,A,SIGMAA,CHISQl,NPASS) PARAMETER(NPAR=3, N=1) PARAMETER(NMAX=100) DOUBLE PRECISION X(NMAX), Y (NMAX), SIGMAY(NMAX) DOUBLE PRECISION YFIT(NMAX), WEIGHT(NMAX),X1,X2 DOUBLE PRECISION A(NPAR), SIGMAA(NPAR) DOUBLE PRECISION ALPHA(NPAR,NPAR),B(NPAR) DOUBLE PRECISION ARINV(NPAR,NPAR), ARRAY(NPAR,NPAR) DOUBLE PRECISION DERIV(NPAR), BETA(NPAR), C(NPAR) DOUBLE PRECISION T,TEND,YY(N) EXTERNAL FUNC,FDERIV,FCHISQ,PRTPAR EXTERNAL STIFF,MPRTPAR NPASS=0 CHISQ=0. IF(NPAR.GT.NDATA) RETURN NFREE-NDATA-NPAR IF(NFREE.LT.l) RETURN IF(MODE.LT.- 1 .OR.MODE.GT.1 )RETURN DO 30 1 = 1 ,NDATA IF(MODE.EQ.O) THEN WEIGHT(I)=1.0 ELSE IF(M O D E .E Q .l) THEN 4 0 5 WEIGHT(I) =1.0/SIGMAY(I )**2. ELSE IF(MODE.EQ.-l) THEN IF(Y (I).EQ .0.0) THEN WEIGHT(I)=1.0 ELSE WEIGHT(I)=1./ABS(Y(I)) ENDIF ENDIF 30 CONTINUE C C *** Main Loop *** C 35 DO 42 J=1,NPAR B E T A (J)= 0. DO 40 K=l,J ALPHA(J,K)=0. 4 0 CONTINUE 42 CONTINUE C C INPUT COEFFICIENTS FOR ST IFF INTEGRATOR C T = 0 .0 IDO=l Y Y (1 )= 0 .0 C DO 50 I=1,NDATA X 1=X (I) TEND=X1*X1 CALL STIFF(IDO.T.TEND,YY) X2=YY{1) IF(A(2).LT.0.0) THEN TEND=1.001‘X1*X1 IDO=3 CALL S T IF F ( IDO,T,TEND,YY) RETURN ELSE ENDIF YFIT(I )=FUNC(XI,X2,A) DO 48 J=1,NPAR DERIV(J )=FDERIV(XI.A,J,I,NDATA,IDO,T,TEND.NPASS.YY > BETA(J)=BETA(J )+WEIGHT(I )* (Y (I)-Y FIT(I)) *DERIV(J) DO 46 K=1.J ALPHA(J . K)=ALPHA(J ,K)+WEIGHT(I)*DERIV(J ) *DERIV{K) 46 CONTINUE 48 CONTINUE 50 CONTINUE DO 54 J=1.NPAR DO 52 K=1,J ALPHA(K, J )=ALPHA(J . K) 52 CONTINUE 54 CONTINUE CHISQ1=FCHISQ(Y.SIGMAY.NDATA,NFREE,WEIGHT,MODE,YFIT) C C *** Do it again with larger FLAMDA *** C 70 NPASS=NPASS+1 C •* * S u b r o u t in e PRTPAR p r i n t s p a r a m e te r s on th e s c r e e n C *** and into file #11. c I F (NPASS.GT.NLOOPMAX) RETURN DO 65 I=1,NPAR B(I)=BETA(I) 65 CONTINUE DO 74 I=1.NPAR DO 72 J=1,NPAR ARRAY(I,J)=ALPHA(I ,J ) 4 0 6 72 CONTINUE ARRAY( 1 , 1 ) =ARRAY( I , I ) * ( I .+ FLAMDA) 74 CONTINUE CALL DLINRG(NPAR,ARRAY. NPAR,ARINV,NPAR) DO 75 J=1,NPAR B ( J ) = 0 . DO 73 K=1.NPAR B(J)=B(J)+BETA(K)*ARINV(J,K) 73 CONTINUE C(J)=A(J)+B(J) 75 CONTINUE C CALL MPRTPAR(NPASS,C) CALL PRTPAR(NPASS,C) C C I F CHI SQUARE INCREASED, INCREASE FLAMDA AND TRY AGAIN C C INPUT COEFFICIENTS FOR ST IFF INTEGRATOR T = 0 .0 ID0=1 Y Y (1 )= 0 .0 C DO 78 1 = 1 ,NDATA X 1= X (I) TEND=X1*X1 IF(I.EQ.NDATA) ID0=3 CALL STIFF(ID O ,T,TEN D ,Y Y ) X2=YY(1) IF(C(2).LT.0.0) THEN TEND=1.001*X1»X1 IDO=3 CALL S T IF F ( IDO,T,TEND,YY) RETURN ELSE ENDIF YFIT(I )=FUNC(XI,X2,C) 78 CONTINUE CHISQR=FCHISQ(Y,SIGMAY,NDATA,NFREE,WEIGHT,MODE,YFIT) WRITE(S,* ) 'FLAMDA,CHISQl, CHISQR' WRITE(6 ,* )FLAMDA,CHISQl, CHISQR IF(CHISQ1.LT.CHISQR) THEN FLAMDA=2. ‘ FLAMDA IF (FLAMDA.GT.T0L2) FLAMDA=0.03 GOTO 70 ENDIF C EVALUATE PARAMETERS AND UNCERTAINTIES DO 100 J=1,NPAR A (J )= C ( J ) SIGMAA(J )=SQRT(1.0/ALPHA(J ,J ) ) 100 CONTINUE FI.AMDA=FLAMDA/4. C TEST CONVERGENCE TO CHI SQUARE TOLERANCE IF ( (CHISQl-CHISQR)/CHISQR.GT.TOLl.OR.CHISQR.GT.T0L2) GOTO 35 IF (CHISQl.GT.TOLl) GOTO 35 RETURN END C C c DOUBLE PRECISION FUNCTION FCHISQ(Y,SIGMAY,NDATA,NFREE, ‘ w e ig h t ;MODE,YFIT) c PARAMETER(NMAX=100) c DOUBLE PRECISION Y(NMAX), SIGMAY(NMAX) , YFIT(NMAX) DOUBLE PRECISION CHISQ,WEIGHT(NMAX) 4 0 7 c CHISQ=0. FCHISQ b O. C ACCUMULATE CHI SQUARE DO 30 1 = 1 ,NDATA CHISQ=CHISQ+WEIGHT(I ) * (Y(I)-Y FIT (I ) )* *2. 30 CONTINUE C DIVIDE BY NUMBER OF DEGREES OF FREEDOM FREE=NFREE FCHISQ=CHISQ/FREE RETURN END C C C DOUBLE PRECISION FUNCTION FD ERIV (X I. A ,IN D E X ,I, NDATA, IDO,T,TEND,NPASS,YY) C PARAMETER(NPAR=3, N =l) C DOUBLE PRECISION X1,A(NPAR),A1,FG,FL DOUBLE PRECISION T,TOL,TEND,YY(N), XL,XG EXTERNAL MPRTPAR,FUNC,STIFF XL=1. 01*X1 XG=1.011*X1 TENDL=XL»XL TENDG=XG*XG A1=A(INDEX) A(INDEX)=0.999»A1 CALL MPRTPAR(NPASS,A) CALL S T IF F ( IDO, T , TENDL, YY) X2=YY(1) FL=FUNC(X1,X2,A) C I F ( I . EQ. NDATA. AND. INDEX. EQ. NPAR) IDO=3 C A(INDEX)=1.001*A1 c CALL MPRTPAR(NPASS,A) CALL S T IF F ( IDO,T,TENDG,YY) X2=YY(1) FG=FUNC(X1,X2,A) A(INDEX)=A1 FDERIV=(FG-FL)/( 0 .002*A1) RETURN END c c c DOUBLE PRECISION FUNCTION FUNC(X,XX,A) PARAMETER(NPAR=3) DOUBLE PRECISION X.XX,A(NPAR), D I,SUM,TSUM,SC.DF c DATA SC,DF/8.10S7D-01,18886/ c TSUM=0.0 DO 1 = 1 ,1 1 DI=DFLOAT(I)-1.DO SUM=(1.0/(2.0*DI+1.0)**2.)*DEXP(-A(2)*DF*((2.0*DI+1.0)*X)**2.) TSUM=TSUM+SUM ENDDO 4 0 8 FUNC=(1 .ODO-A < 1 ))* (1 .ODO-SC*TSUM)+A <1)*XX RETURN END c c c SUBROUTINE PRTPAR(NPASS,A) C PARAMETER(NPAR=3) DOUBLE PRECISION A(NPAR) C W R IT E (6.61) N P A S S ,(A (I),1 = 1 ,NPAR) W R IT E (11,61) N P A S S ,(A (I),1 = 1 ,NPAR) 61 FORMATdX, I3,3E15.6) RETURN END c c SUBROUTINE MPRTPAR(NPASS,C) PARAMETER(NPAR=3 ) DOUBLE PRECISION C(NPAR) OPEN(14, FILE=' PARA.DAT' , STATUS='OLD') W R IT E (14,*) ( C ( I ) , 1 = 1 , NPAR) CLOSE(1 4 ) RETURN END c c c SUBROUTINE S T IF F ( IDO,T,TEND,YY) PARAMETER(NPARAM=50, N=1) PARAMETER(MABSE=1, MBDF=2, MSOLVE=2) DOUBLE PRECISION 2 ( 1 , 1 ) , PARAM(NPARAM), T,TEND,TOL,YY(N) EXTERNAL FCN,FCNJ, DIVPAG,SSET TO L =0.001 CALL SSET(NPARAM,0 . 0 , PARAM,1) PARAM(10)=MABSE PARAM(1 2 )=MBDF PARAM(1 3 )=MSOLVE c CALL DIVPAG(IDO,N,FCN, FCNJ, Z,T,TEND,TOL, PARAM,YY) c RETURN END c c c SUBROUTINE FCN(N,X,YY,YPRIME) PARAMETER(NPAR=3) DOUBLE PRECISION X.YY(N), YPRIME(N),A(NPAR),A4 EXTERNAL ALNREL COMMON A4 OPEN(14,FILE=' PARA.DAT', STATUS='OLD') R EA D (14,*) ( A ( I ) , I = 1 , NPAR) c YPRIME(1)=A(3)*DEXP(A4*YY(1))* (1 .ODO-YY(1)) c CLOSE(1 4 ) 4 0 9 RETURN END c c c SUBROUTINE FCNJ(N,X,YY,DYPDY) DOUBLE PRECISION X.YY(N).DYPDY«N,*) RETURN END c c c SUBROUTINE MAKEFILE12(NDATA,A,X,Y,DATA) C PARAMETER(NPAR=3, N=1) PARAMETER(NMAX=100) C DOUBLE PRECISION A(NPAR),X(NMAX), Y(NMAX), DATA(NMAX) DOUBLE PRECISION TEND,T,YY(N) EXTERNAL FUNC,ST IFF C T=0 .0 ID O=l Y Y (1 )= 0 .0 C WRITE(12,815) 815 FORMAT( / 9 X , 'X ', 3 X ,'M e a s u r e d ', 2 X ,'C a l c u l a t e d ') DO 820 1 = 1 ,NDATA X 1=X (I) IF d.E Q .N D A T A ) IDO=3 TEND=X1*X1 CALL S T IF F ( IDO,T,TEND,YY) X2=YY(1) DATA( I ) =FUNC(XI, X2, A) WRITE(12,825) X (I),Y(I),DATA(I) 820 CONTINUE 825 FORMAT(2X,3(F10.5)) RETURN END c c* c SUBROUTINE PLOT(NDATA,A,X, Y, DATA) PARAMETER(NPAR=3) PARAMETER(NMAX=100) DOUBLE PRECISION X(NMAX), Y(NMAX), DATA(NMAX) DIMENSION XP(NMAX),YP(NMAX),YCP(NMAX) c CHARACTER*50 X T IT ,Y T IT ,TITLE c DO 10 1 = 1 ,NDATA XP(I)=SNGL(X(D) YP(I)=SNGL(Y(D) YCP(I)=SNGL(DATA(I)) YHIN=MIN(Y P( I ) , YMIN) YMAX=MAX(YP(I).YMAX) 10 CONTINUE C XMIN=X(1) XMAX=X(NDATA) YMAX=1.2*YMAX C CALL HOME 4 1 0 CALL NORMAL PRINT*,' PRINT*,' PLOTTING STEP ' PRINT*,' WRITE(6 ,* )'ENTER TITLE FOR X-AXIS' READ(5,100) XTIT WRITE(6 ,* )'ENTER TITLE FOR Y-AXIS' READ(5,100) YTIT WRITE(6 .* )'ENTER TITLE OF THE PLOT' READ(5,100)TITLE 100 FORMAT(A50) C NUM = NDATA CALL QP_START CALL QP_AX(XMIN,XMAX,6,1,YMIN,YMAX,5,3) CALL Q P_TTL(TITLE,X TIT,Y TIT) CALL Q P_PTS(X P,Y P,N U M ,2,6) CALL QP_PTS(XP, YCP, NUM,1 ,1 ) CALL QP_PTX CALL QP_PAUSE CALL QP_CLEAR CALL QP_END RETURN END C C *«•**»•***•»♦•*******•****•••*******•*.*** C 4 1 1 c * ...... c c c c THIS PROGRAM IS FOR THE SIMULATION OF NON-HOMOGENEOUS MEMBRANE C C SORPTION BEHAVIORS. C C C C...... g c c c C * . . . . program : MASS.FOR C ..... Numerical scheme : Crank-Nicholson method. C***** Treatment of boundary discontinuity is reffered to 'Numerical C methods and Modeling for Chemical Engineers' ed. by Mark E. Davis. C C ..... start of main progeam C PROGRAM MASS PARAMETER(NPAR=4) PARAMETER(NGRID= 31 . NMAX=1C001) C DIMENSION THETA(0:NGRID,0:NMAX),SMASS(0;NMAX),SDELT(0:NMAX), + X(NGRID),y(NGRID),DIFF(NPAR).SURF(NPAR) REAL LENGTH(0 : NGRID) , KS, KT, KKS, KKT EXTERNAL PDE,TRIDAG,MAKEFILEll COMMON N.M .L.DELT CHARACTER*10 FOLD CHARACTER*11 FNEWl, FNEW2, FNEW3 C c ...... C Information for the fit c c WRITE(5,101) 101 FORMAT(//' Name of data file excluding ".DAT" ? ',$) READ( * ,1 0 2 ) FOLD 102 FORMAT(AlO) FNEW1=F0LD//'A' FNEW2=F0LD//'B' FNEW3=FOLD//'C' W R IT E (5,103) FNEWl 103 FORMAT!/ ' * Interacting I/O w ill be recorded in ',A ll,'-DAT') WRITE(5,104) FNEW2 104 FORMAT(' * The data and calculated values in ',A ll,' .DAT') WRITE(5,105) FNEW3 105 FORMAT(' * The final parameters are w ritten in ',A ll, '.DAT') OPEN(UNIT=8, FILE=FOLD,STATUS=' OLD' , DEFAULTFICE=' . DAT') OPEN(UNIT=9, FILE=' DATA.DAT' , STATUS=' OLD') OPEN(UNIT=1 0 ,FILE=FNEW1, STATUS:' NEW') OPEN(UNIT:1 1 ,FILE=FNEW2,STATUS:' NEW') OPEN(UNIT:12, FILE:FNEW3, STATUS:'NEW') C c ...... C Assign the initial value of parameters C C WRITE(6,106) 106 FORMAT(// ' Please enter initial values of the parameters.'/) WRITE( 1 0 ,* ) ' IN ITIA L VALUES OF PARAMETERS' DO 1 : 1 , NPAR WRITE(6,107) I 107 FORMAT!/ ' DIFF( '.II,') : ',$) READ!*,*) DIFF(I) 4 1 2 WRITE (10.*) 'D IFF(M ,') = '.DIFF(I) ENDDO C DO 1 = 1 ,NPAR WRITE(6.108) I 108 FORMAT( / " SURF{ ' , 1 1 , ' ) : ', $ ) READ(*,*) SURF(I) WRITEdO, *) 'SU R F(',I,') = ' ,SURF(I) ENDDO C c ***************************************************** C Read data file while counting the nunil>er, NDATA C C READC9,♦)N.M.L.DELT 1=0 371 1=1+1 READ(8.*.END=373)X(I) .Y (I) GOTO 371 373 NDATA=I-1 DELX=l./DFLOAT(N) C C CALL PDE(D IF F .S U R F .THETA.SDELT,SMASS) C C C *** Final report *•* C C C Ma)ce files #11 in which the initial data and calculated ones C are listed on the 2nd and 3rd columns respectively C C CALL MAKEFILEl1 (NDATA. X. Y. SDELT. SMASS) C C •***•**♦**********•••*•*********•**•***•♦************♦*******< C Ma)(e file #12 of one column which lists final parameters C C DO I=;.NPAR WRITE(12.*) 'D IF F l'.I.') = '. DIFF(I) WRITEC12.*) 'SU R F('.I.') = '. SURF(I) ENDDO C CLOSE(8 ) CLOSE(9 ) CLOSE(1 0 ) C L O S E (ll) CLOSE(1 2 ) C STOP END C C *************** End of main program **************** c c ------c C***.*.************.**.**************+********.***********...... C. . . . . SUBROUTINE PDE C SUBROUTINE PDE(DIFF.SURF.THETA.SDELT.SMASS) 4 1 3 PARAMETER(NPAR=4) PARAMETER(NGRID=31, NMAX=10001) DIMENSION THETA(0;NGRID,0:NMAX), THETAP(0 :NGRID,0 : NMAX), THETAN(0 :NGRID,0 :NMAX), U (0 : NGRID), AR(0 :NGRID), BR(0 :NGRID), CR(0 :NGRID). DR(0 :NGRID), AL(0 : NGRID), BL(0 : NGRID), CL(0 : NGRID), DL(0 : NGRID) , DIFFSN(0:NGRID,0:NMAX),DIFFSP(0:NGRID.0:NMAX). D IFFTN (0 :N G RID ,0 : NMAX), D IFFT P( 0 :NGRID,0 :NMAX), SMASS ( 0 : NMAX) , SDELT ( 0 : NMAX). D IFF (NPAR) , SURF (NPAR) REAL KS.KT.KKS.KKT EXTERNAL TRIDAG,MAKEFILEll COMMON N ,M ,L,DELT C C •****•***•***•*•***•**•****•**•***•****•*****•**••*•**•**••********. C Set the numerical parameters and the initial conditions C C L l = l L2=L+1 DELX=1. /DFLOAT(N) DELXL=1. /DFLOAT(L) DELXR=1. /DFLOAT(N-L) RL=DELT/(DELXL**2.) RR=DELT/(DELXR**2.) C K S=D IFF(1) KKS=DIFF(2) K T=D IFF(3) KKT=DIFF(4) WRITE(6,• ) 'KS,KKS,KT,KKT' WRITE(6 ,* )KS,KKS,KT,KKT C S0=SURF(1) SS0=SURF(2) SN=SURF(3) SSN=SURF(4) WRITE(6 ,* )'SO,SSO,SN,SSN' WRITE(6,*)SO,SSO,SN,SSN c DO 1 = 1 ,N-1 THETA(I,0)=0.0 ENDDO THETA(0,0)=S0 THETA(N,0)=SN C C ***************************•*****************•*♦•**♦*****♦.♦*♦♦*♦**> C NUMERICAL SCHEME BY CRANK-NICOLSON METHOD* C c DO 20 J=0,M TIME DEPENDENT SURFACE CONCENTRATION THETA(0 ,J )=30+(1.-SO)* (1 .-EXP(-SSO *DFLOAT(J)*DELT)) THETA(0,J+1)=S0+(1 .-SO)•(1 .-EXP(-SSO‘DFLOAT(J+1)*DELT) ) THETA(N,J)=SN+(1 .-SN)* (1 .-EXP(-SSN*DFLOAT(J)*DELT) ) THETA(N,J+1)=SN+(1 .-SN)* (1 .-EXP(-SSN*DFLOAT(J+1)*DELT)) CONCENTRATION DEPENDENT DIFFUSION COEFFICIENT (DUBININ-ASTAKOV) DO 1 = 1 ,N -1 THETAP(I ,J )=0.S*(THETA(I ,J )+THETA(I+1,J )) THETANd, J)=O.S*(THETA(I, J)+THETA(I-1, J) ) DIFFSP(I ,J )=EXP(KS*THETAP(I ,J )**KKS) DIFFSN(I,J)=EXP(KS*THETAN(I , J )* *KKS) D IF F T P ( I , J)=EXP(KT*THETAP( I , J ) * *KKT) 4 1 4 DIFFTN(I ,J)=EXP(KT*THETAN(I , J ) **KKT) ENDDO C Set C-N coefficients in soft matrix A L (1 )= 0 .0 BL(l)=(l.+(RL/2.)*(DIFFSP(1,J)+DIFFSN(1,J))) CL(1)=-(RL/2.)*DIFFSP(1,J) DL(1)=(RL/2.)*DIFFSN(1,J)*(THETA(0,J)+THETA(0,J+1))+ + (l.-(RL/2.)*{DIFFSP(1,J)+DIFFSN(1, J)))*THETA(1. J) + + (RL/2-)*DIFFSP(l,J)*THETA(2,J) C DO 1 = 2 ,L -1 AL(I)=-(RL/2.)*DIFFSN(I,J) BL(I )=1.+RL*(DIFFSP{I ,J )+DIFFSN(I,J ))/2 . CL(I)=-(RL/2.)*DIFFSP(I,J) DL(I)=(RL/2.)*DIFFSN(I,J)*THETA(I-1,J)+ + (l.-RL/2.*{DIFFSP(I,J)+DIFFSN(I,J)))*THETA(I,J) + +(RL/2.)‘DIFFSP(I,J)»THETA(I+1,J) ENDDO C AL(L)=-(RL/2.)‘DIFFSN(L.J) BL(L)=1.+(RL/2.)* (DIFFTP(L,J)+DIFFSN(L,J)) C L (L )= 0 .0 DL(L)=(RL/2.)*DIFFSN(L,J)‘THETA(L-1,J)+ ( 1 (RL/2.)* + (DIFFTP(L,J)+DIFFSN(L,J)))‘THETA(L,J)+(RL/2.)‘ + DIFFTP(L,J)‘(THETA(L+1,J)+THETA(L+1,J)) C C *** OBTAIN NEW CONCENTRATION IN SOFT SIDE USING SUBROUTINE 'TRIDAG' CALL T R ID A G (A L ,B L,C L ,D L ,U ,L1,L 2,N ) C C “ * REPLACE OLD SOFT CONCENTRATION WITH NEW SOFT CONCENTRATION. DO 1 = 1 ,L THETA(I,J+1)=U(I) ENDDO C C “ * Set C-N coefficients in tough matrix. A R (L )= 0 .0 BR(L)=1.+(RR/2.)‘ (DIFFTP(L,J)+DIFFSN(L,J)) CR(L)=-(RR/2.)‘DIFFTP(L,J) DR(L)=(RR/2)‘DIFFSN(L.J)‘ (THETA(L-1,J)+THETA(L-1,J+D)+ + (l.-(RR/2.)‘{DIFFTP{L,J)+DIFFSN(L,J)))‘THETA(L,J)+ + (RR/2.)‘DIFFTP(L,J)‘THETA(L+1,J) C DO I=L+l,N-2 AR(I)=-(RR/2.)‘DIFFTN(I,J) BR(I)=1.+RR‘ (DIFFTP(I,J)+DIFFTN(I,J)>/2. CR(I)=-(RR/2.)‘DIFFTP(I,J) DR(I)=(RR/2.)‘DIFFTN(I,J)‘THETA(I-1,J)+ + (l.-RR/2.*(DIFFTP(I,J)+DIFFTN(I,'J) ) )‘THETA(I, J) + +(RR/2. ) ‘DIFFTPd, J)‘THETA(I+1, J) ENDDO C AR(N-1)=-(RR/2.)‘DIFFTN(N-1,J) BR(N-1)=1.+RR‘ (DIFFTP(N-1,J)+DIFFTN(N-1,J))/2. C R (N -1 )= 0 . DR(N-1)=(RR/2.)‘DIFFTN(N-1,J)‘THETA(N-2,J)+(l-(RR/2.)‘ + (DIFFTP(N-1,J)+DIFFTN(N-1,J)))‘THETA(N-1,J)+ + (RR/2.)‘DIFFTP(N-1,J)‘(THETA(N,J)+THETA(N,J+1)) C C **♦ OBTAIN NEW TEMPERATURE IN REACTAITT USING SUBROUTINE 'TRIDAG' CALL TRIDA G(AR,BR,CR,DR,U ,L,N ,N) C C “ ‘ REPLACE OLD TOUGH CONCENTRATION WITH NEW CONCENTRATION. DO I= L ,N -1 T H E T A d, J + 1 )= U (I) ENDDO 4 1 5 c c c Calculate the fractional uptake C C SMASS(0)=0.0 SDELT(0)=0.0 TMASSL=0.0 TMASSR=0.0 DO 1 = 0 ,N-1 IF(I.GE.L) THEN TMASSR=0.5 * (THETA(I.J+1)+THETA(1+1.J+ 1) ) ‘DELXR+TMASSR ELSE TMASSL=0.5 * (T H E T A (I,J+ 1 )+ T H E T A (I+ l,J+ 1 )) *DELXL+TMASSL ENDIF ENDDO FL=DFLOAT(L)/DFLOAT(N) TMASS=TMASSL*FL+TMASSR*( 1 .0-FL) SMASS(J+1)=TMASS SDELT(J+1)=SQRT(DELT*DFLOAT(J+l)) 20 CONTINUE RETURN END C C c SUBROUTINE MAKEFILEll(NDATA,X,Y,SDELT,SMASS) c c PARAMETER(NPAR=4) PARAMETER(NGRID=31, NMAX=10001) DIMENSION X(NGRID), Y(NGRID), SDELT(0 :NMAX), SMASS(0:NMAX) c WRITE(11,201) 201 FORMAT(/9X,'X',3X, 'M easured',2x,'C alculated') NSTEP=0 DO 1 = 1 ,NDATA DO J=NSTEP,NMAX IF(X (I).LE.lE-04) GOTO 100 IF(ABS(X(I)-SDELT(J)) -LE.l.OE-03) GOTO 100 ENDDO 100 WRITE(11,202)X(I),Y(I) , SMASS(J) NSTEP1=J NSTEP=NSTEP1 WRITE(6, *) 'NSTEP,X(D ,SDELT(J) ,Y (I) ,SMASS(J) '. WRITE(6,*)NSTEP,X(I ) ,SDELT(J),Y(I ) ,SMASS(J) ENDDO 202 FORMAT(2X,F10.4,F10.4,F10.4) RETURN END c c c SUBROUTINE TRIDAG(A ,B ,C ,D ,U ,N I, N2, N) DIMENSION A(0;N),B(0:N),C(0:N),D(0:N) ,U(0:N) DO 100 I=N1+1,N2-1 B(I)=B(I)-A(I)*C(I-1)/B(I-1) D(I)=D(I)-A(I)*D(I-1)/B(I-1) 100 CONTINUE U(N2-1)=D(N2-1)/B(N2-1) DO 200 K=N2-2,N1,-1 U(K)=(D(K)-C(K)*U(K+1))/B(K) 200 CONTINUE RETURN END