VAPOUR SOLUTION AND DIFFUSION IN RADIATION GRAFT COPOLYMERS · 2016. 7. 15. · Graf Copolymers*...

VAPOUR SOLUTION AND DIFFUSION IN

RADIATION GRAFT COPOLYMERS

by

DAVID CHRISTOPHER TA-CHANG KUO, B.Sc., A.R.C.S.

A thesis submitted in partial fulfilment

of the.! requirements for the award of

the degree of

DOCTOR OF PHILOSOPHY

OF THE

UNIVERSITY OF LONDON

Department of Chemistry,

Imperial College of Science and Technology,

London, SW7 2AY. October, 1981.

To my Parents

3

ACKNOWLEDGEMENTS

I am greatly indebted to Dr. J.A. Barrie for his guidance and

encouragement throughout this research.

I am grateful to Dr. K. Munday for the benefit of his experience

at the initiation of the work, Mr. R.N. Sheppard for his friendship

and advice and Mr. I.E. Morrison for his experience in the field of

computer software.

I would also like to thank Dr. P.C. Clay and Mr. B. Bennett of

the Nuclear Technology Department for their cooperation in the

preparation of the graft copolymers and Dr. G.S. Parry and

Mr. N. Salpadoru of the Electron Microscopy Service for the electron-

micrographs.

I would also like to offer special thanks to Ms. M. Shanahan for

typing this thesis, and Mr. D. Webb and Dr. E. Mason for their friend-

ship and expertise in proof reading.

Finally, I would like to thank all the members of staff for their

cooperation.

A

ABSTRACT

David Christopher Ta-Chang Kuo, B.Sc., A.R.C.S.

Vapour Solution and Diffusion in Radiation Graft Copolymers.

Permeability, solubility and diffusion coefficients of ethane,

and propane have been measured in poly(dimethylsiloxane), PDMS, in the

temperature range 20 to 60°C. The transport and sorption parameters

have also been determined in a series of radiation graft copolymers

containing up to 51.A5% by weight of poly(methyl methacrylate.) , PMMA.

In addition, sorption kinetics and equilibria of methane, ethane,

propane and iso-butane in PMMA have been investigated at various

temperatures in the range 1 to 60°C. Sorption measurements were

obtained with a Sartorius electronic vacuum microbalance and permeability

and diffusivity from gas flow across a thin membrane measured using

a MKS Baratron.

Sorption of"propane and ethane in PDMS obeyed Henry's Law J the

diffusion coefficient was found to be independent of concentration.

For PMMA, sorption of methane and ethane obeyed Henry's Law, whilst the

sorption isotherms with the powder were non-linear for ethane, propane

and iso-butane and were analysed in terms of dual-mode sorption. The

absence of a Langmuir component in the sheet was interpreted in terms

of residual casting solvent preferentially occupying the microvoids.

The effect of additional trace solvent was shown to enhance the

diffusion coefficient.

5

The sorption isotherms for the graft copolymers were characteristic

of dual-mode sorption and this was attributed to the Langmuir component

in PMMA. The solubilities were found to be additive provided the prior

history of PMMA was accounted for. Permeabilities were found to be

independent of pressure and the activation energies were independent

of composition. The permeabilities were examined in terms of several

models based upon a binary phase material of which the most successful

were those of Higuchi and Bo'tcher. Variation in the post-irradiation

solvent treatment of the copolymers caused a departure from the various

models and was interpreted in terms of structural changes in the

internal distribution of the. PMMA. Supporting evidence of this was

provided by electron micrographs. Diffusion coefficients were found to

be concentration dependent and the permeation time lags were also found

to be dependent on the upstream driving pressure. These were both found

to be in good agreement with a model in which the diffusion process was

controlled by the continuous silicone phase with the PMMA heterogenieties

acting as rapid adsorption centres.

6

CONTENTS

PAGE:

LIST OF TABLES 10

LIST OF ILLUSTRATIONS 1 2

CHAPTER l: INTRODUCTION 1 5

1.1. An Historical Survey of Diffusion 2.8

1.2. Formal definition of P, D and k. 22

CHAPTER 2: THEORY

2.1. Solution of Fick's Equation! The Plane Sheet

2.1.1. Constant D System! Steady State

2.1.2. Concentration Dependent D! Steady State

2.1.3. Constant D System*. Transient State

2.1.4. Concentration Dependent D! Transient State

2.1.5. Sorption and Desorption Kinetics! The Plane Sheet

2.2. Solution of Fick's Equation! The Solid Sphere 35

2.3. The Dual-Mode Sorption Theory 38

2.3.1. The Thermodynamics of Sorption 42.

2.3.2. The Diffusion Model for Dual-Mode Sorption 43

CHAPTER 3! LITERATURE REVIEW 49

3.1. The Molecular Theories of Diffusion 50

3.2. The Dual-Mode Sorption Theory (A Review 1976-1981) 53

3.3. The Phenomenological Correlation of Transport and

Sorption Parameters 52

3.4. Transport in Heterogeneous Media 53

3.5. Review of Heterogeneous Models 79

24

24

24

26

27

7

CHAPTER 41 EXPERIMENTAL 85

4.1. Materials 85

4.1.1. Poly([c.]-dimethylsiloxane) 35

4.1.2. Poly (methyl m e t h a c r y l a t e ) s h e e t 35

4.1.3. Poly(methyl methacrylate): powder 35

4.1.4. Poly([c./.]-dimethylsiloxane-g-methyl

methacrylate) 35

4.1.5. Nomenclature of graft copolymers 39

4.1.6. Penetrant Vapour 89

4.2. Permeation 90

4.2.1. The Permeation Hardware 90

4.2.2. Permeation Technique and Measurement 94

4.3. Sorption 98

4.3.1. The Sorption Hardware 98

4.3.2. Sorption Technique and Measurement 100

CHAPTER 5: PHYSICAL PROPERTIES OF THE POLYMERS 102

5.1. Introduction 1.02

5.1.1. Glass Transition Temperatures 103

5.1.2. Densities 105

5.1.3. Dynamic Mechanical Testing 105

5.1.4. Thickness of Graft Copolymer 108

5.1.5. Stereotacticities of PMMA Samples 108

5.1.6. Electron Microscopy 109

5.2. Discussion 109

8

CHAPTER 6: RESULTS AND DISCUSSION - PDMS 117

6.1. Introduction 117

6.1.1. PDMS/Propane Equilibrium Sorption 117

6.1.2. Permeabilities and Diffusion Coefficients 120

6.2. Review of PDMS/Propane Studies 123

6.3. PDMS/Ethane 126

CHAPTER 71 RESULTS AND DISCUSSION - PMMA 129


7.1.1. PMMA/Methane 130

7.1.2. PMMA/Ethane . 134

7.1.3. PMMA/Propane 143

7.1.4. PMMA/iso-butane 155

7.2. General Remarks of the Dual-Mode Sorption Theory 162

7.3. Transport and Sorption Parameters: Correlation

Functions 168

7.4. A Critical Examination of Diffusion in PMMA

Powder and Sheet 181

CHAPTER 8: RESULTS AND DISCUSSION - PDMS-g-PMMA 184


8.1.1. Equilibrium Sorption 191

8.1.2. Steady State Permeabilities 203

8.1.3. Diffusion Coefficients 2]2

8.2. The Time Lag in Diffusion 221

9

CHAPTER 91 CONCLUSION 224

9.1 Graft Copolymers 224

9.2 Graft Copolymer Results 225

9.3 PMMA Results 227

9.4. Suggestions for Further Study 227

APPENDIX A : The Time Lag in Diffusion for a Heterogeneous

Membrane 229

APPENDIX B: Sample Calculation and Estimation of Errors 233

Bl. Sample Calcuiation of an equilibrium sorption

concentration and diffusion coefficients for

PDMS-g-46.34% PMMA/propane at 30°C 233

B2. Sample calculation of the permeability (p),

diffusion coefficient (D), and solubility

(k) from the transient and steady state permeation

of ethane through PDMS 236

APPENDIX C: Experimental Data 238

REFERENCES 266

10

LIST OF TABLES

CHAPTER 5: PHYSICAL PROPERTIES OF THE POLYMERS

5.1. Glass Transition Temperatures 104

5.2. Stereotacticities of PMMA 104

5.3. Summary of Solvent Extraction of Graft Copolymers H O

CHAPTER 6! RESULTS AND DISCUSSION - PDMS

6.1. Sorption and Transport Parameters'. Propane 124

6.2. PDMS/Propane Transport Parameters at 30°C 125

6 . 3 . Activation Energies and Heats of Dissolution 126

6.4. Sortpion and Transport Parameters: Ethane 127

CHAPTER 7: RESULTS AND DISCUSSION - PMMA

7.1. Sorption and Diffusion Parameters: Methane 132

7.2. Powder and Sheet Sorption Parameters'. Ethane 137

7.3. Powder and Sheet Diffusion Parameters! Ethane 143

7.4. Dual-Mode Sorption Parameters: Propane 145

7.5. Diffusion Coefficients'. Propane 154

CHAPTER 8! RESULTS AND DISCUSSION - PDMS-g-PMMA

8.1. Graft Copolymers: Sorption Parameters 193

8.2. Graft Copolymers*, isobar parameters at 30°C 199

8.3. Graft Copolymers'. Heats of Sorption 201

8.4. Graft Copolymers: Permeabilities at 30°C 211

11

8.5.

8 . 6 .

8.7

APPENDIX

CI

C2-C3

C4-C8

C9-C13

C14-C18

C19

C20

C21-C25

C26-C29

C30

C31

C32

C33

Graft Copolymers! Diffusion Coefficients at 30°C 214

PDMS-g-46.34% PMMA Diffusion Coefficients at 30°C 219

Dual-Mode Sorption Parameters for PDMS-g-46.34%

PMMA at 30°C 223

PDMS/Propane Sorption Isotherm Results 238

PDMS Transient Permeation Data 240/241

PMMA Sorption Isotherm Results 243/246

PMMA Transient Sorption Data 247/250

PDMS-g-PMMA Equilibrium Sorption Data 251/255

PDMS-g-24.14% PMMA*Equilibrium Sorption Data 256

Membrane and Apparatus Characteristics 257

PDMS-g-PMMA Transient Permeation Data 258/260

PDMS-g-PMMA* Transient Permeation Data 260/262

Specific Free Volumes of Polymers at 25°C 262

Transport Parameters of Various gases and Vapours

in PMMA at 30°C 263

PDMS-g-46.34% PMMA Sorption Kinetics at 30°C 264

PDMS-g-46.34% PMMA Time Lag Data at 30°C 265

12

LIST OF ILLUSTRATIONS

Figure 4. 1 The Grafting Apparatus 86

Figure 4. 2 The Permeation Apparatus 91

Figure 4. ,2 (a) The Permeation Apparatus and PET Microprocessor 93

Figure 4. .3 The Diffusion Cell 95

Figure 4. 4 The Sorption Apparatus ' 97

Figure 4. 5 Beam Assembly 99

Figure 5. 1 Density of Graft Copolymers 106

Figure 5. 2 Elastic Moduli of Graft Copolymers 107

Figure 5. 3 Relative Thickness of Graft Copolymers 112

Figure 5. ,4 Electron Micrographs 115

Figure 6. 1 PDMS/Propane Sorption Isotherms 118

Figure 6. 2 Silicone Rubberl van't Hoff Plots 119

Figure 6. 3 Silicone Rubberl Temperature Dependence of P 121

Figure 6. 4 Silicone Rubber: Temperature Dependence of D 122

Figure 7. 1 PMMA (Sheet)/Methane Sorption Isotherms 131

Figure 7. 2 PMMA (Sheet)/Methane: Concentration dependence of D 133

Figure 7. 3 PMMA (Sheet)/Ethane Sorption Isotherms 135

Figure 7. 4 PMMA (Powder)/Ethane Sorption Isotherms 136

Figure 7. 5 PMMA (Sheet)/Ethane: Concentration dependence of D 141

Figure 7. 6 PMMA (Powder)/Ethane: Concentration dependence of D 142

Figure 7. 7 PMMA (Powder)/Propane Sorption Isotherms 144

Figure 7. 8 PMMA (Powder)/Propane! Alternative Sorption Isotherms 146

Figure 7. 9 Heats of Sorption for PMMA 148

Figure 7. 10(a) PMMA (Powder)/Propane". Concentration Dependence of D 150

13

Figure 7. 10(b) PMMA (Powder)/Propane: Concentration Dependence of D 153

Figure 7. 11 PMMA (Powder) /Propane*. Concentration Dependence of D 153

Figure 7. 12 PMMA (Powder)/iso-butane Sorption Isotherms 156

Figure 7. 13 Poly(methyl methacrylate): Temperature Dependence

of k D 157


of b 158


of S Li

159

Figure 7. 16 Poly(methyl methacrylate): Arrhenius plots of

(D) c=0

160

Figure 7. 17 Poly(methyl methacrylate)*. Arrhenius plot of D^ 16]

Figure 7. 18 PMMA/Ethane Langmuir Sorption 163

Figure 7. 19 PMMA/Propane Langmuir Sorption 164

Figure 7. 20 Apparent Temperature Dependence of S /k J_i jj

167

Figure 7. 21 Diffusion Coefficient vs. Molecular Diameter Correlation 169

Figure 7. 22 Activation Energy vs. Molecular Diameter Correlation 17]

Figure 7. 23 Limiting Solubility Correlations 173

Figure 7. 24 Henry's Law Solubility Correlation 174

Figure 7. 25 Heat of Sorption Correlation 175

Figure 7. 26 Propane Diffusion vs. 1/SFV at 25°C 179

Figure 7. 27 Propane Activation Energy vs_. 1/SFV 180

14

Figure 8.1 PDMS-g-10.35% PMMA Sorption Isotherms 185

Figure 8.2 PDMS-g-24.14% PMMA* Sorption Isotherms 186


Figure 8.4 PDMS-g-39.76% PMMA Sorption Iostherms 188



Figure 8.7 G O vs. Volume Fraction at 30°C 194 D m —

Figure 8 .8 —* V o l u m e f a c t i o n at 30°C 195

Figure 8.9 (S T) vs. Volume Fraction at 30°C 196 L m —

Figure 8.10 Extracted Isotherms at 30°C 198

Figure 8.11 Craft Copolymer Tsobars at 30°C 200

Figure 8.12 (AHJ _ vs. Volume Fraction 202 S c=0 —

Figure 8.13 Temperature Dependence of Permeability:

Petroleum Extracted Series 204

Figure 8.14 Temperature Dependence of Permeability:

Acetone Extracted Series 205

Figure 8.15 Permeabilities of Graft Copolymers at 30°C 208

Figure 8.16 Permeabilities of Graft Copolymers at 30°C 209

Figure 8.17 Deviation from Standard Models 210

Figure 8.18 Diffusion Coefficients of Graft Copolymers 215

Figure 8.19 PDMS-g-46.34% PMMA Diffusion Coefficients 217

Figure 8.20 PDMS-g-46.34% PMMA: Time Lag vs. Pressure Profile 222

15

CHAPTER ONE

INTRODUCTION.

The demands of technology for the greater refinement and specificity

of rubbers and plastics has led to a growing interest in the field of

heterogeneous polymers. The great case and versatility by which blending,

grafting, masticating and block-copolymerising may be performed, allows

the modification of existing polymers with comparative economy of

resources. In conjunction with the growth in the field of synthetic

polymer chemistry, there is also a great deal of interest in the varied

physical properties of these heterogenous polymer systems.

The systematic study of gas transport through polymer membranes

constitutes one aspect of polymer characterisation which not only yields

valuable information on the usefulness of the polymer as potential gas

barriers and separators but also affords a unique and detailed method

of examining and assigning the polymer morphology. The characteristic

transport parameters, namely the diffusivity, permeability and solubility

when used in conjunction with theoretical models allows a means of inter-

preting the microstructure of these systems. The gas molecule or

penetrant may therefore be viewed as a sensitive microprobe and judicious

choice of gas is an important consideration.

The present study focussed on the transport of a simple hydrocarbon

vapour, propane, in a series of radiation graft copolymers. The two

components of the graft copolymer were a synthetic elastomer, poly(dimethvl-

siloxane) and the important engineering plastic, poly(methyl methacrylate).

16

The technique of grafting has been described by one author (174) as being

" more of an art than a science" , but despite this apparent criticism

the morphology of graft systems, especial! y of the type used in the

present study, can usually be predicted intuitively. It is often found

that if grafting is effected, and relatively long side chains are

produced, complete phase separation occurs. The phase in greater

abundance is usually interconnected over extended regions and convention-

ally referred to as the " continuous phase" . Distributed within this

continuum are domains of the " dispersed phase" . On this argument,

as the proportion of the dispersed phase increases, a critical composition

is attained at which the roles of the two phases interchange. This is

known as the point of " phase inversion" . However ;in the present system,

the silicone rubber which formed the continuous phase was lightly cross-

linked and complete phase inversion is therefore restricted. Instead,

the " dispersed phase" may be expected to form a complex pattern of

interpenetrating networks.

The simplicity of the morphology of the present system prompted a

serious examination of the " additivity rule" as applied to solubility.

In a simple binary polymer system, where the individual components are

clearly discernible, the solubility which is an extensive property of the

system, should be linearly related to the composition. Often, deviations

from this simple linear dependence have been interpreted as an indication

of mixing between the phases. The present study attempted to distinguish

between the phenomenon of true interaction and any artifacts which may

be introduced, and otherwise unaccounted for.

17

The steady state permeabilities of the graft copolymers were

measured arid tested against theoretical models. The poly(methyl meth-

acrylate) " dispersion" was expected to behave as simple impermeable

fillers reducing the effective flux across the membranes. The effects

of the dispersed phase on the measurements of the diffusion coefficients

by various methods are also discussed.

An examination of the post irradiation solvent treatment on the

graft copolymers was also performed. The selective use of solvents which

show preferential dissolution in either phase of the graft copolymers

was expected to modify the morphology. The effect was demonstrated by

mechanical testing in addition to gas transport studies.

Due to the importance of poly(methyl methacrylate) in the field of

engineering and technology, an extensive study of the transport and

solutions of four simple hydrocarbon vapours in both the polymer sheet

and powder was conducted. The applicability of the dual-mode sorption

theory to describe the transport and sorption in the glassy polymer

was tested. Also phenomenological correlations were constructed which

attempt to relate the sorption and transport parameters with fundamental

gas properties.

18

1.1. AN HISTORICAL SURVEY OF DIFFUSION

In 1822, Fourier (1) developed the equation for heat conductance,

(1.1)

where K is the heat conductivity and 30/3x the temperature gradient.

Almost thirty years later, Fick (2) advanced the mathematical theory

of diffusion by recognising the analogy between heat conduction and the

random process of diffusion. His equation for uni-directional flow in

an isotropic medium is y

where J is the rate of transfer per unit area of section, c is the

concentration of diffusant and x the spatial coordinate measured normal

to the section. D, a constant of proportionality, is known as the

diffusion coefficient. If the flux J, and the concentration, c, are

both expressed in equivalent units of amount, then D simply has the

dimensions :(LENGTH) 2 x (TIME) The conventional units for D are

(cm 2 s and are used throughout this treatise.

In order that equation (1.2) be completely defined, it is not only

necessary to specify the units in which J, c and x are measured, but also

the section through which diffusion occurs. Crank (13) has discussed

in detail, several frames of reference, fixed with respect to the volume

and the mass of the components which participate in the diffusion process.

(1.2)

19

In the simplest case of the inter-diffusion of a binary mixture with

zero volume change of mixing, the diffusion process may be completely

described by a simple diffusion coefficient known as the " mutual

diffusion coefficient" . If a volume change occurs on mixing, " mass

fixed" coefficients may be defined, and again, a simple diffusion

coefficient suffices to describe the diffusion process. In the limit

of zero volume fraction of one component, the " m u t u a l diffusion coefficient"

and the coefficients measured with respect to " mass fixed" frames of

reference are identical.

In the present study, the solubilities of the penetrants in the

polymer samples were relatively low, and no significant swelling of the

polymer was discernible. The diffusion coefficient is therefore expected

to be Independent of the frames of reference.

By a consideration of mass balance for a uni-directional flow in an

isotropic medium, Fic.ks' second equation of diffusion may be derived,

3c 3 2c 3x: (1.3)

The form in which equation (1.3) appears is only valid if the

diffusion coefficient is constant. The diffusion of gases and vapours in

high polymers are usually represented by a concentration dependent diffusion

coefficient and equation (1.3) is replaced by,

f [ D ( c ) . | £ ] (1.4) dt dx 3x

Tn 1866, Graham (3) postulated that the transport across a membrane

comprised of three concerted steps!

20

(1) a rapid dissolution at the ingoing face followed

by

(2) a slow, rate controlling transport through the

film and finally,

(3) evaporation to the gaseous state at the outgoing

face.

Later in 1879, Wroblewski (4) combined the thermodynamic process of

dissolution and the kinetic process of diffusion to yield a simple

relationship, namely,

J = DkAp// (1.5)

where k was defined as the absorption coefficient (conventionally referred

to now as the solubility) and Ap the pressure difference across the

membrane of thickness,/.

Various workers (5,6,7) continued to work along these lines, reasserting

the cogency of the original postulate of the sorption-diffusion theory.

In 1920, Daynes (8) reported that if the concentration of gas at the

outgoing face of the membrane was monitored as a function of time, with

the concentration at the ingoing face held constant, then the rate of

transport increased monotonically and tended to ah asymptotic steady state

value. Linear extrapolation of this asymptote back to the time axis

yielded a positive intercept known as the time lag, 0. He further

showed that 0 was simply related to the diffusion coefficient and hence

the solubility, k, could be estimated from equation (1.5).

21

The technique was not exploited to its fullest until 1939, when

Barrer (9) elaborated on the original idea of Daynes (8) and since

then, the principle has remained largely unchanged. The attraction

of the high-vacuum technique lies not only in its simplicity, but also

because it enables the three parameters, P, D and k , which completely

define the transport process, to be obtained from a single experiment.

Furthermore, even when extremely low solubilities are encountered, the

value of k may still be determined with accuracy when the more direct

methods may prove to be ineffective.

A major contribution to the concept of diffusion was advanced by

Barrer in 1940. Barrer (10) proposed that diffusion was an activated

process and conformed to an Arrhenius type relationship, namely,

D = D q exp(-E /RT) (1.6)

where R and T have the usual significance and E^ is the activation

energy for diffusion. D q , the pre-exponential factor may be related to

the entropy of activation.

The temperature dependence of the solubility coefficient, k, was

given by the van't Hoff relation,

k = k exp(-AH /RT) (1.7) o D

where AH^ is defined as the heat of dissolution.

The temperature dependence of the permeability P, may therefore be

inferred from equations (1.6) and (1.7) since

P = k.D (1.8)

giving

P = P exp(-Ep/RT) (1.9)

22

Ep may be viewed as a "quasi" activation energy of permeation and

is related to E^ and AH^ by the expression

E p = E d + A H d (1.10)

The diffusion coefficient is an increasing function of temperature

whereas the solubility generally decreases with increasing temperature.

The permeability may therefore increase or decrease with temperature

depending upon the relative magnitude of E^ and AH^.

1.2. Formal definition of P, D and k.

Several different units have been proposed for the transport parameters

P and D (11) and it is perhaps pertinent at this stage to define in

precise terms the units in which P, D and k are to be used.

(1) the permeability coefficient, P, is the amount of gas measured

in cm 3 at (STP) , diffusing per second through unit area of membrane (1 c m 2 ) ,

of unit thickness (1 cm) when there is a centimetre of mercury difference

in pressure across it.

Thus:

P = V. //[A.t. (Pi " P2) ] (1.11)

where V = volume of penetrant [cm 3 (STP)],

t = time (s),

t = membrane thickness (cm),

A = Cross-section area of membrane (cm 2),

P i > P 2 = ingoing and outgoing pressures respectively,

(cmllg) .

23

(2) The diffusivity or more commonly the diffusion coefficient D,

is the amount of gas (cm 3 at STP), diffusing per second through unit

cross-sectional area (1 cm 2) of membrane of unit thickness (1 cm) with

unit concentration difference across it.

Thus,

D = . „ (1.12) A.t. dc/dx

where c = concentration of penetrant in the membrane

[cm 3(STP) cm" 3]

x = distance (cm).

(3) The solubility k, is the amount of gas in cm 3 at STP, dissolved

in a 1 centimetre cube of membrane when the ambient pressure is 1 cmHg.

Thus,

k = c/p (1.13)

24

CHAPTER TOO

THEORY

2.1. Solution of Pick's, equation! The plane sheet.

Exact solutions for Pick's first and second equations of diffusion

given in equations (1.2) and (1.3) are available in standard

texts (12, 13) for a variety of sample geometries. In this section

the solutions are presented for the transient and steady state flow

through the plane sheet. In addition, expressions are also presented

for the kinetics of sorption and desorption for the same geometry and

also for the solid sphere.

The conventional permeation experiment requires a polymer medium,

in the form of a membrane, of constant cross-section bounded at x = 0

and x = The membrane is initially free of penetrant and then subjected

to an instantaneous pressure, corresponding to a concentration, c Q , at

the ingoing face (x = 0). The boundary conditions are thus,

0 < x < / t = 0

x = 0 t > 0

x = / t > o (2.1)

2.1.1. Constant D system! steady state.

In the steady state, the concentration profile across the membrane

is independent of time, so,

= 0 dt

(2.2)

25

Under these conditions, Fick's second equation reduces to a simple

second-order differential equation given by,

3x' = 0

with

c = 'A TX + A;

(2.3)

(2.4)

an exact solution.

By substituting for the boundary conditions and solving for 'Ai and A 2 ,

and

A! = (c, - c )// t o

A 2 = c

(2.4a)

(2.4b)

It follows then from (1.2) and (2.4) that

J = -D. (c . - c )/{ (2.5) t o

and in the limit c- « c , r o

j = i).c ;e (2.6) o

Often, instead of ascertaining the concentrations at the two faces of the

membrane, it is more convenient to monitor the pressure. The steady

state flux, J, may then be written in the form

J = F.p /f (2.7) o

where

V = D.k (2.8)

26

with

c = k.p (2.9)

2.1.2. Concentration dependent D: steady state.

Again, the concentration profile across the membrane is independent

of time, and equation (2.2) is still applicable. Equation (1.4) therefore

reduces to

f [ D ( c ) . ^ ] - 0

dx dx

(2.10)

Since the flux across the membrane is given by

then

J = -D(c). dc dx

Jdx = - D(c).dc

(2.11)

(2.12)

and o

Jt = I D(c).dc c ,

(2.13)

If c^ « c q , tlien differentiating with respect to c yields,

f - ( J O = D(c ) dc c = c o

o (2.14)

from a practical point of view, since J is measured as a function of

pressure and dp/dc is obtained from the sorption isotherm we can write

27

( T ^ ) . ( ^ 2 ) = D(c ) (2.15) dp dc o

o o

Alternatively, from equation (2.13), and defining an average solubility

it follows that

where

k = c/p (2.16)

P = D. k (2.17)

D = JL c o

c f o

D(c).d c (2.18) o

2.1.3. Constant D system: transient state.

Prior to the attainment of the steady state, the rate of flow and

the concentration profile across the membrane are both functions of time.

Subject to the validity of the boundary conditions given in equation (2.1),

the quantity of gas which passes through the membrane up to time t, is

given by

Q 00 2

T ^ - T V - l - ' l ^ • e x p ( - D n 2 7 r 2 t / 0 (2.19) 6 TT N

o n=l

and in the limit t

Dc

Q t - - f i t - y (2.19a)

Therefore a plot of Q vs. t approaches asymptotically a straight

line with -an intercept on the time axis of 0, where

0 = — (2.20) 6D

28

It is sometimes inconvenient to await the establishment of the steady

state, so Rogers et al. (14) provided an alternative solution to equation

(1.3) in the form

2 Ak p 00 2

a ? - c - v ^ < £ > 1 [ - ( ^ ) . ( 2 m + i)=] (2.2D m=o

At small times, the summation converges rapidly to give

a 2 A k p o D * / 2

/n(t d .dp/dt) = A i [ ( — 1 - 4 ^ (2.22)

1 A semi-logarithmic plot of the composite quantity (t 7.dp/dt) against

/ 2

reciprocal time should therefore be linear with a gradient (- 4^)• They

used their expression to estimate the diffusion coefficient of helium in

pyrex glass and since then, the method has been reported extensively in

the literature (15-17).

2.1.4. Concentration Dependent D: transient state.

Exact solutions to equation (1.4) for the diffusion coefficient

varying as a function of concentration have been found only for a

limited number of situations (13). The solutions are far from general

and often contain complicated integral functions which, although

tractable, are seldom simple to evaluate numerically.

Frisch (18;19,20) however, obtained general expressions for the

time lag in terms of the D(c) dependence without recourse to explicitly

solving the diffusion equation. The procedure is outlined in detail

in the Appendix for a specific D(c) dependence. Briefly, however,

evaluation of the triple integral,

29

3c

3t dx.dx.dt =

o o x

r t

o 'x 3x

[D(c)].dx.dx.dt (2.23)

reduccs to

c (x).dx.dx s

0 = (2.24)

c r o

D(c)dc

where cg ( x ) is the concentration-distance profile in the steady state,

and may in theory be deduced from the expression

r c D(c).dc = 9

rc I)(c) .dc (2.25)

Provided the form of the D(c) is known, the variation of the time lag

may be predicted using equation (2.24). Alternatively, if D(c) may

be expressed in an analytical form, the parameters which describe the

D(c) function may be determined by curve fitting the time lag vs.

concentration profile.

It is evident from equation (2.25) that if,

then

and

D(c) = D

c = c (x - x / O s o

0 = if 2 /6T)

(2.26).

(2.27)

(2.28)

which reduces correctly to the constant D case presented in Section 2.1.3,

30

2.1.5. Sorption and Desorption Kinetics: The Plane Sheet.

The equations "which describe the rates of sorption and desorption

are available for a variety of geometric shapes in standard texts

(13). In all cases, the process of diffusion is assumed to be controlled

by a constant diffusion coefficient. In what follows, extracts are

presented, which have been derived for the plane sheet and the solid

sphere.

The boundary conditions which describe the conventional sorption by

a sheet of thickness, ^, initially free of penetrant which is brought

into contact with a vapour at concentration c , may be written as: o

0 < x < / t = 0

x = n,x = / t > 0 (2.29)

The corresponding desorption by the sheet, which has attained a uniform

concentration c is then, o

c = c 0 < x < / t = 0 o — —

c = 0 X = -f ,x = 0 t > 0 (2. 30)

Plane Sheet: Constant D.

Under the conditions for which the transport of penetrant into the

sheet is described by a diffusion coefficient which is independent of

concentration, two standard types of solutions are obtained of which one

is more appropriate for the small time domain and the other for the

long time domain.

c = 0

c = c o

31

Small Time Domain.

The sorption rate curve may be described by the equation

M \ M rr = ( - ) . [ — + 2 I ( - l ) n . i e r f c ( - ^ ) j (2.31)

/ 2 / T n=o 2 (Dt)

for which it is found that in the domain of small times the summation

becomes relatively unimportant. Thus, the reduced weight (M /M^) is

proportional to n from which D may be calculated from the gradient.

It is normally found that the linear region, conventionally referred

to as the " square root t region", persists up to approximately M / M ^ |

Long Time Domain.

The sorption rate curve may also be described by the equation

M t . 8 c 1 r D(2m + .1) 2 tt 21) , ff) — = 1 - p . I exp [ J (2.32)

00 m=o (2m + 1) 2

for which it is found that at M /M = ^ equation (2.32) reduces to the

t 00 2 aproximate expression

ti ^ - - C .Intri - (2.33) 2

from which

TT D ' 16 9 16

D o. ^ (2.34) " t x ~5~

where tx is normally referred to as the " half time" " 2 "

32

The " half time" method is extremely easy to use, though the

advantage of its simplicity has to be offset against the possibility

of gross errors incurred by determining D using a single point on the

sorption rate curve. Furthermore, the " half time" method in itself

does not provide a test for constant D.

Towards the end of sorption, the summation in equation (2.32)

converges rapidly, and only a single term (nf = o) in the summation

series is important. Thus, equation (2.32) reduces to

M R

- = 1 - -^.exp (D7T2t//2) (2.35) M

and a graph of ln(M a > - M ) against time is linear, for which the gradient

is given by

^ U n ( M „ - M J ] = - ^ ( 2 . 3 5 a )

Variable Diffusion Coefficient: The Plane Sheet.

When the diffusion coefficient is a function of concentration, the

methods of ascertaining D, as described earlier will yield an average

diffusion coefficient. The measured diffusion coefficient, denoted by

D , may, however, be related-to the true diffusion coefficient by the

methods outlined below.

The initial rates of sorption and desorption are given by the single

expression,

d(M /M ) t

" " * "l dt2"

I g 2 ) (2.36)

33

It is normally found that when D(c) increases with concentration , the

linear portion of the sorption or desorption curves may be

extended well beyond M /M = Other features of a concentration 3 t « 2

dependent diffusion coefficient in relation to the sorption and desorption

rate curves have been summarised by Crank et al., (12,13).

Average Diffusion Coefficient.

Crank and Park (21) showed that the value of D could be related to \

the differential diffusion coefficient, D(c), using the relationship,

- 1 D ^ —

— c O ' o

c o

D(c).dc (2.37)

where the limits of integration specify the upper and lower bounds

of the concentration.

Rearranging equation (2.37) and differentiating w.r.t., c gives,

D(c) ^ D + c (2.38) o d e

o

The variation of D as a function of c can be constructed and the o

differentiations w.r.t. c may be evaluated either by numerical or

graphical techniques . This gives the first approximation to the D(c) Vn,

CQ relationship. If the diffusion coefficient is only a weak function of

concentration, the first approximation is normally found to be sufficiently

accurate. However, successive approximations may be employed by an

iterative procedure until a specified degree of accuracy is achieved.

34

Conjugate Sorption-Desorption.

Prager and Long (22) found that the mean of the sorption diffusion

coefficient, (D ), and the corresponding conjugate desorption diffusion

1 r c

coefficient, (D.), was more closely approximated by — d c

o So,

° D(c).dc.

D + D . t f C° D = — - <v, -L D(c). dc (2.39)

2 — c o J o

Weighted Mean Approximation.

Crank (24) further found that the individual values of the average

diffusion coefficient measured from sorption and desorption rates were

better related to D(c) by

c r o _

1) ^ pc ~p ! cP~.D(c).dc (2.40) S — o J o

for sorption and

c^

D ^ qc ~ q [ (c - c)" 7" 1. D(c). dc (2.41) d — o 1 o J o

in the case of desorption.

The expressions were found to be well behaved for D(c) as an

increasing function of concentration of over twenty decades with

P = 1.67 and a = 1.85. Barrie and Machin (23) found that for a diffusion

coefficient decreasing with concentration, the same expressions held

but with the values of p and a interchanged.

35

2.2. Solution of Fick's Equation: The Solid Sphere.

The use of the plane sheet for determining the diffusion coefficient

of a polymer-penetrant pair becomes less attractive, when very low

diffusion coefficients are encountered. There is a practical limit

for the finite thickness of the sheet and normally this is found to be

in the range of 10-20 microns, however, the technique of emulsion

polymerisation affords a convenient method of preparing sub-micron spheres,

usually with a high degree of mono-dispersity. The net effect, therefore,

is a pronounced reduction in the effective thickness of the sample and

a corresponding reduction in the time for equilibration. In the following

section, standard solutions of Fick's equation of diffusion are presented

for the geometry of a solid sphere.

Constant PI Solid Sphere.

If diffusion is radial and controlled by a constant diffusion

coefficient, then

3)

and

J = -D. ~ (2.42)

3r

If = fe « • " >

where u = c.r (2.44)

The forms of equations (2.42) and (2.43) are similar to equation (1.2)

and (1.3) so the solutions to radial diffusion may be inferred by

analogy with linear flow in the plane sheet.

36

As found with the plane sheet, two standard types of solutions are

obtained and again, one is more appropriate for tlx? small time domain

and the other for the long time domain.

Small Time Domain.

The sorption rate curve may be completely described by the expression

M i 00

-E. = 6 ( 5 $ ) ' [ — + 2 £ ierfc ] - 3 ^ (2.45)

M v u / V /tt n=l / Dt

M r r— •, rrrr r

and

lim ierfc ( n r ) = 0 (2.45a)

t -v o / Dt

It is thus, simple to show that the rate of initial uptake of

penetrant by the sphere is

(d M /M ) , / [ -J = ^ / D ( 2 > 4 5 b )

. i t->o r / dt

Long Time Domain

The sorption rate curve may be described by the alternative expression

viz. ,

M

^ = ^ I ^ exp (-Dn 2TT 2t/r 2) (2.46) n=l

and as discussed earlier with the plane sheet, a " half time" denoted

by tj_ at M^/M = may be defined, whence equation (2.46) simplifies to

the approximate expression,

D 3.06 x 10 r 2/t i (2.47)

37

Towards the end of sorption, the summation given in equation ( 2 . 4 6 )

converges rapidly, to give

CN (1 " M j V M j = /II (6/TT 2) - DTT 2 1 / r 2 ( 2 . 4 8 )

Thus, at the latter stages of sorption, a plot of ln(l - M /M^)

vs. time is linear from which the diffusion coefficient may be evaluated

from the gradient.

Variable Diffusion Coefficient.' Solid Sphere.

The equations presented in equations ( 2 . 4 5 ) to ( 2 . 4 8 ) inclusive are

strictly applicable for the diffusion of a penetrant in a sphere when

the process of diffusion is controlled by a constant diffusion coefficient,

D. However, again by analogy with transport in the plane sheet, an

average diffusion coefficient, D, may be defined from

d( M /M ) , r = — t «> 6 / D ( 2 . 4 9 )

T~ = ~ * V dt 2 r

Crank (24) has suggested that D, is related to the differential diffusion

coefficient D(c) through the approximation

r i D ^ —

- c o o

° D(c).dc (2.50)

Thus, the methods outlined earlier for ascertaining the form of the

concentration dependence of D(c) arc still applicable, and will not be

discussed further.

38

2.3. The Dual-Mode Sorption Theory.

Below the glass transition temperature, the equilibrium sorption

isotherms of gases and the vapours in polymers show a negative deviation

from Henry's Law. The shape of the isotherm was explained by Barrer

et al., (25) to be due to two modes of sorption, namely normal dissolution

in the polymer matrix and concurrent adsorption within the microvoids.

It was further postulated that the concentration of sorbate from normal

dissolution (c^) and the concentration of adsorbate from hole filling

(c^) together comprise the equilibrium concentration c. viz.,

c = c + c (2.51) \) n

in which c. was assumed to obey the Henry's Law of dissolution,

c D = V P < 2 - 5 2 >

where k is the Henry's Law dissoltuion constant [cm 3(STP)/cm 3(polymer)

cmHg] and p the equilibrium pressure [cmHg].

The concentration c.' was accounted for by the Langmuir isotherm,

C H = °H b p / ( 1 + b p ) ( 2 ' 5 3 )

where c' is the hole saturation constant [cm 3(SIT)/cm 3(polymer) ] H

and b the hole affinity constant [(cmHg) Substitution of equation

(2.52) and (2.53) into (2.51) then gives the formal dual-mode sorption

equation,

c = k„p + i-J, bp/(1 + bp) (2.54)

39

Two important conditions prevail at the extremeties of pressure!

(i) at low pressure, when (bp « 1), equation (2.54)

reduces to

c = (kD + c'b)p ,(2.55)

The term in parenthesis may be visualised as a " psuedo" limiting Henry's

Law constant, which may be defined as

s l - p- • f k „ + <•,?> ( 2 - 5 6 )

(ii) at high pressures, when bp » 1, equation (2.54)

reduces to

c = k D p + c^ (2.57)

From equations (2.55) and (2.57) it may be inferred that at low pressures,

the processes' of hole filling and normal dissolution are competitive ,

but at higher pressures, the microvoids saturate and normal dissolution

then dominates.

The mathematical form of equation (2.54) together with the prevailing

conditions in the limits of low and higher pressures, allow a simple

analytical means of determining k^, c 1 and b. The linear asymptote at

high pressure has a gradient k^ and an intercept at the ordinate of

CJ'J . Therefore, the value of c^ as a function of p may be estimated

by rearranging equation (2.51) and substituting for c to give

(2.58)

40

The concentration of adsorbate in the microvoids c„ may thus be H

" extracted" from the total concentration,, c. Rearranging equation

(2.53) gives

• - V + 4 (2.59) C H H b C H

According to equation (2.59), a plot of p/c^ versus p should be linear

with a gradient of 1/c' and an intercept at p = 0 of 1/c' b. II H

This method of determining the parameters k , c' and b is open D H

to two severe criticisms. Firstly, it is assumed that 'the experimentally

measured sorption isotherm has been extended to sufficiently high

pressures in order that the limiting high pressure asymptote be

truely attained. This may often be untrue and will be demonstrated

in greater detail in Chapter Seven. Secondly, the mathematical form in

which equation (2.59) appears,con tains the variable, p, on both sides

of the expression. In effect, therefore, linearity of the resulting

graph may be deceptive. If the graphical method is to be employed , it

is perhaps better to rearrange equation (2.59) into the form

1 1 1 1

II H P C I !

whence a plot of 1/c^ vs. 1/p should be linear with a gradient 1/c^b

and an intercept at l/c„ = 1/c'.

M n

A technique has been proposed by Koros et al., (73) and also discussed

in detail by Barrie et al. (68), based on the more rigorous method

of non-linear regression analysis, from which k , c' and b were evaluated

using an iterative procedure on a computer. This method, has been found

during the course of the work to be very convenient and extremely reliable,

41

2.3.1. The Thermodynamics of Sorption.

There are three fundamental thermodynamic quantities which are

relevant to the dual-mode sorption theory! namely, AH^, the enthalpy

of dissolution, AH , the enthalpy of hole filling and AH , the overall II O

heat of sorption.

The enthalpy of dissolution is obtained from the temperature

dependence of the normal Henry's T.aw constant, k^, through the van't Hoff

expression

k D = k D o.exp(-AH D/RT) (2.61)

The hole filling constant, b, obeys a similar expression in which

b = b Q exp (-AH /RT) (2.62)

The overall heat of sorption is concentration dependent and can be

obtained from the temperature variation of the free energy change in

proceeding from the gas in the standard state to gas in the sorbed state,

Hence,

and since

then

AC g = RTlnp (2.63)

_ DAG / T All = ( — r — ) ( 2 - 6 4 )

S ~ 1 c 11 —

T

AH _

l TS = (2.65)

3-

42

The dual-mode sorption expression given in equation (2.54) may be

rearranged to yield an expression which is explicit in concentration,

viz: -

[ ( k D + c' - be) 2 + 4 k p b c P + (be - k D - c ; b) ( 2 > 6 6 )

P 2k Db

and substituting equation (2.66) into (2.65) then gives an analytical

expression, which may be differentiated numerically, to give the overall

heat of sorption (Alh) S c, p

Although the three thermodynamic quantitites 411^, AH and AH^ may be

determined by the methods outlined above, there is no simple relationship

between these three quantitites. However, in the limit of high pressure,

the hole filling process is expected to be completed and the enthalpy of

hole filling AH^ is therefore not expected to contribute to the overall

heat of sorption, Thus

lim (AH ) ^ A H d (2.67) C - > no

Similarly, in the limiting Henry's Law region at zero concentration,

43

2.3.2. Hie Diffusion Model for Dual-Mode Sorption.

The basic assumptions of the dual-mode sorption theory state that!

(a) there is local equilibrium between the dissolved species and the

adsorbate immobilised within the microvoids. Thus, the kinetics of

immobilisation are assumed to be rapid and the rate-determining step

is the slow diffusion of the dissolved species.

(b) the penetrant which is adsorbed according to the Langmuir

isotherm is completely immobilised.

The fulfilment of conditions (a) and (b) form the fundamental

postulates of the dual-mode sorption theory. This is now seen as a

limiting case conventionally referred to as the total immobilisation

model. Condition (b) which assumes no mobility of the Langmuir component

has been relaxed, giving rise to the partial immobilisation theory. There

is considerable evidence in support for partial mobility of the adsorbate

although in the present study, the total immobilisation model proved

to be adequate.

Total Immobilisation Model.

In the following, a brief outline of the diffusion model based on

total immobilisation is presented. A more detailed account may be found

elsewhere (26).

The overall flux may be written as

J = - D ( c ) . ( 2 . 6 9 ) Dx

where D(c) is the experimentally observed diffusion coefficient and is

concentration dependent. However, according to condition (b) , the

flux is due solely to the dissolved species, and so,

44

J " " D D & ( 2 " 7 0 )

D^ represents the diffusion coefficient of the dissolved species and

is independent of concentration. D^ is sometimes referred to as the

" true" or " effective" diffusion coefficient since transport is

believed to be controlled by the dissolved species only. However,

whether the differential diffusion coefficient,D(c), or the diffusion

coefficient of the dissolved species, D^, should be assigned the " true"

diffusion coefficient is purely subjective. For the present purpose,

neither will be referred to as the " true" diffusion coefficient in order

to avoid the possibility of misinterpretation.

Combining (2.69) and (2.70), gives

3c

D DC

The differential (Sc^/Dc) is easily evaluated from partial differentiation

w.r.t. p,

^ D = ^ D . 3p k D (2.72) 3c 3p 3c k D + c ^ b / ( l + b p ) 2

substituting (2.72) into (2.71) gives

C H H - 1 D(c) = I) [1 + w " . ,2] (2.73)

D D + bp)

In the limit of low pressures (p o)

( D ) c = n = D .k /S^ (2.74)

45

and at high pressures (p -><*)

The diffusion coefficient, D^, may theoretically, be determined at

high pressures, when the observed D(c) is identical with D^. However,

this condition may not be possible to experience practically, due to the

inordinately high pressures which are required to saturate the

microvoids.

Solutions for D^ are presented below for measurement based on sorption

kinetics using the relationship given in equations (2.36), (2.38) and

(2.73). Since

c o

c

° D(c).dc (2.76) o

Rearranging equation (2.76) and differentiating gives

D + c ( ~ - ) = D(c ) (2.77) o dc o

o

D(c^) represents the observed diffusion coefficient at concentration,C q,

and is also given by equation (2.73). Thus, substituting (2.73) into

(2.77) and rearranging gives

D = (D + i' f-)[l + ( 2 " 7 8 )

D o • dc k „ U + bp) o D

46

D C a n be determined from a graph of D vs. c and constructing tangents D o

at various points to give (dD/dc o). In order that D^ be evaluated with

considerable accuracy, a large number of points is required on the D

vs. c curve. Furthermore, the accuracy is limited by the precision — o

in which the value dD/dc may be evaluated. o

A much simpler method may be arrived at by substituting equation

( 2 . 7 1 ) directly into ( 2 . 7 6 ) giving

c ^ 1 f ° D D = — D_ .dc ( 2 . 7 9 ) c < D 3c o J o

Since D may be assumed to be constant, integrating directly gives

D D D = r ' c =c (2.80)

o °

Substituting the expression for c^ and c q given in equation (2.52) and

(2.54) respectively then gives

W (2.81) k D P + c j j b p / d + b p )

The evaluation of D^ may therefore be performed using D, k^, c^

directly at each pressure, p.

Permeability Coefficient for Total Immobilisation.

An average solubility, k, may be defined from

k = c /p (2.82) o

47

and hence the permeability follows from

P = D.k (2.83)

comparing equations (2.80), (2.82) and (2.83) gives

c c P = D.— = D . — (2.84)

P D p

and substituting (2.52) into (2.84) gives

p = V k n ( 2- 8 5>

Partial Immobilisation Model.

In later developments, the constraint of total immobilisation of the

Langmuir component to the diffusion process has been relaxed. A finite

contribution to the overall flux has been assigned to the Langmuir

component and this has been found to be more representative of diffusion

in some glassy polymers.

In the following, the notation D^ as used in the earlier section will

be retained, and in addition D^ will be used to represent the diffusion

coefficient of the hole species which is also assumed to be constant.

By a consideration of fluxes,

D 3 c D J - - D ( C ) i £ = - n n - - D H a — (2.86)

rearranging gives,

D ( c ) - ( D n ' V * 5 T + d h ( 2 " 8 7 )

48

Substituting equation (2.87) into (2.76) gives

D = — c o

°l(DD - V &

+ V

d c

and on integrating

5 - r % " V + d h o

(2.89)

Equation (2.89) predicts a linear relationship between D and from

which D^ may be evaluated from the intercept at c q / cq = 0 a n <3 the value

(D^ - D^) from the gradient.

In the limit of D^ = 0 (i.e., the condition of total immobilisation),

equation (2.89) reduces correctly to the expression given in equation

(2.80).

Permeability Coefficient: for Partial Immobilisation.

Again, an average solubility k and a permeability P may be defined

as given in equations (2.82) and (2.83) respectively. Thus, with

equation (2.89),

P = D.

and

c c — (I) - D ) + D .— p D H H p

D

(2.90)

(2.91)

The permeability coefficient, P, is then expected to be concentration

dependent due to the term (c^ - c 0)/c^. In the limit of total immobilisati

DJJ = 0, and equation (2.91) reduces simply to equation (2.8 5).

on

49

CHAPTER THREE

LITERATURE REVIEWS

In this chapter, five reviews are presented, which are relevant to

the current study. Where adequate reviews are available elsewhere, only

a brief treatment of the subject was undertaken.

The first review concerns some of the molecular theories of diffusion.

Although at first sight there appears to be a variety of models for the

transport of a penetrant molecule within a polymer medium, many

of these were found to be extensions of either the free volume theory

or the activated transition state theory and will be treated as such.

The second review e x a m i n e s the extensive study of gas and vapour

transport in glassy polymers. Over the years, the study of the transport

of small molecules in glassy polymers has been found to be of great

academic value in the understanding of the behaviour of polymers below

the glass transition temperatures. In addition, the recognition of the

fundamental importance of these polymers as potential substitutes for

natural resources, namely wood, glass and steel, requires yet, a still

greater knowledge of polymers in the glassy state.

A means of ascertaining a priori the usefulness of polymers in the

field of packaging (27,28,29) prompted a serious examination of some

phenomenologic.al correlations between the transport and sorption

parameters and fundamental gas properties. It was found that correlation

functions have been suggested from as early at 1949, but although many

50

were founded on sound theoretical arguments, a large number are

either empirical or semi-empirical and should be used with reservations.

Heterogeneous polymers, today, constitute a large proportion of the

commercial rubbers and plastics. In conjunction with the growth in the

field of heterogeneous polymers, a growing interest in the transport

properties of gases in these polymers was found in the literature.

The concept of the small penetrant molecule, behaving as a microprobe

has been used extensively in an attempt to examine and elucidate the

microstructure of these systems. However, the interpretation of the

results, by many -investigators was found to be only semi-empirical and

serious comparisons with theoretical models for transport in heterogeneous

media was found to be more the exception than the rule.

Finally, a brief discussion of the models derived for transport in

heterogeneous media is included.

3.1. The Molecular Theories of Diffusion.

Since excellent reviews are presented elsewhere (30,31) it is felt

that a brief qualitative treatment of some of the more pertinent under-

lying assumptions connected with each model will suffice.

.'•Zone Theory (32,33,34,35).

The premise that diffusion of a molecule in a liquid or solid was an

activated process had been recognised as early as 1936 by Eyring (36).

Several other models followed and it was found that all the expressions

deduced for the diffusion coefficient considered the diffusing particle

to be vibrating and moving to successive positions of equilibrium when

51

sufficient activation energy was acquired by the system. The analogy

with diffusion in a polymer was drawn by Barrer (32). Calculations

presented by Barrer indicated that the energy of activation was

distributed over a number of degrees of freedom shared between the

diffusing molecule and the surrounding polymer segments, so called

" hot zones" .

Free Volume Theory (37).

The free-volume theory was based on the concept of density fluctuations

in liquids. Regions of low density being approximated in the crudest of

treatments to " holes" . Diffusion of a molecule is then governed by

the probability that some critical element of free-volume is made

available to the molecule for a diffusion step to occur.

The free-volume theory and activated zone theory form the two fundamental

concepts of diffusion in polymers. A notable difference between the two

theories is that in the zone theory, the diffusion coefficient is

exponentially related to temperature whereas in the free-volume theory

the diffusion coefficient is exponentially related to the average free

volume.

Several theories followed which may be treated largely as extensions

to the free volume and zone theories. Brandt (38) modified the zone

theory by accounting for the molecular structure of the polymer-penetrant

system. The activation energy for diffusion, E^, was assigned to

(i) E.. : the intermolecular energy required to separate

adjacent polymer chains in order to create a

space for the penetrant,

52

the intramolecular energy required to bend

two neighbouring chains and finally,

the thermal energy.

Brandt was able to calculate K without recourse to adjustable

parameters and he (106) also proposed that E should vary linearly

with the square of the penetrant diameter and intercept the abscissa

at between 5 and 8 X 2 .

DiBenedetto [39(a) -4() ] proposed a volume fluctuation theory which

has since been adapted by Pace and Datvner [41(a),(b),(c)] . Pace,et al.,

considered a spherical penetrant, contained within a bundle of polymer

chains. The effective motion of the gas molecule was then assumed to

occur in two separate manners , namely,

(a) Along the axis of the polymer b u n d l e s -

This required the lesser energy to effect, and

the only energy involved was the energy required

for the shortest jump step.

(b) Perpendicular to the axis:-

For the penetrant molecule to translate into an

adjacent chain configuration, the polymer segments

must bend in a concerted manner. The energy involved

is that required for intramolecular chain separations.

Since process (a) requires the lesser energy, it begs the question

of the need for process (b) ever to occur. In practice, the model

postulates that process (a) predominates until further longitudinal motion

is hindered, either through strictures in the chain, or embedded obstacles

such as crystal 1ites. For continued motion, the molecule must translate

to an adjacent " tube" through process (b) before continuing again

with process (a).

(ii)

(iii) E th

53

3.2. The Dual-Mode Sorption Theory (A Review 1976-1981).

The dual-mode sorption theory for gases was apparently first suggested

by Meares (42) in 1958. Based upon his earlier work (43,44) on the

transport of gases through poly(vinyl acetate). Meares proposed that the

sorption of gases in polymers below the glass transition temperature

(T ) led to a sorbate with two distinct modes of mobility. Barrer et al.,

(25,46,47) found that the equilibrium sorption isotherms of hydrocarbon

vapours in ethyl cellulose below T t deviated negatively from Henry's

Law. They discussed the possibility of sorption into pre-existing gaps

formed within the polymer. A tentative proposal that adsorption into

these gaps may be described by a Langmuir isotherm (50) was also suggested.

A formal mathematical model was not proposed, however, until 1963 (48);

a model for diffusion followed several years later (26) . Since then,

the dual-mode sorption theory lias met with considerable success in the

interpretation of the transport and sorption of gases in glassy polymers.

An excellent review of the subject has been presented by Vieth, Howell

and Hsieh (51). Their treatise embraced a complete examination of

theoretical and experimental aspects of the subject from the embryonic

stage. It is felt that any attempt to reiterate their critique would

be tautological and the present review, therefore, is concerned primarily

with material presented after the year 1976, although reference to earlier

publications will be cited when felt to be necessary.

In 1978, Koros and Paul (52) commenced their studies on the sorption

of C0 2 in poly(ethylenetcrcphthalate), PET. Earlier, studies on a

similar system had been reported by Michael et al., (53). There was

close agreement between the two independent studies, despite the

54

anomalous dilatometric effects of C0 2 which expands the cavity component,

and also the uncontrolled annealing in the earlier study.

Koros, et al., in particular showed the effect of temperature on

the Langmuir component to the sorption; at sequentially higher temperatures,

the magnitude of the Langmuir sorption relative to the Henry's Law sorption

b / k D ' decreased, a n c* eventually reduced to zero at TO. A quantitative

relationship, between the Langmuir capacity (c' ) and the differential H

thermal expansion coefficient of the polymer above and below T^ was

discussed. The volume coefficient of expansion of a polymer is greater

above T than below. Accordingly, in a hypothetical situation, where g

the coefficient of expansion of the polymer, above T^ could persist

below the T^, the difference at any temperature T (where T < T ) between

the actual volume of the glassy polymer and that of the hypothetical

rubber was postulated to be a measure of c' . Experimental evidence was H

provided which showed a tacit agreement with their postulate.

In a later study (54), the transient and steady state transport of

C0 2 in PET using conventional permeation techniques were reported.

Permeabilities and time lags were found to decrease with increasing

upstream driving pressures in accordance with their model for partial

immobilisation (55). It is interesting to note that their time lags

calculated using the partial immobilisation model have a tendency to lie

above the experimentally determined points. This observation was also

reported by Toi (56) who studied a similar system. Toi used a curve

fitting procedure to analyse the experimental time lags under the

constraints of the dual-mode sorption theory to arrive at the parameters

C H ' n n c 1 1 , c f° m K l tbnt the hole affinity constant, b, determined

by this method differed from the equilibrium sorption isotherm but offered

110 explanation for the apparent discrepancy.

55

The energetics of C0 2 sorption in PET was discussed by Koros,

Paul and Huvard (57). It was found that when the isosteric heats

of sorption, AH^., were plotted against concentration, a minimum was

observed,which is in common with results on other systems (68,79).

In a lengthy, thermodynamic treatment, a theoretical expression for AH^.

as a function of concentration was derived (58). Their analytical

expression included the term AH*, which was assigned to the apparent

enthalpy, characterising the temperature dependence of c' . It was H

pointed out, however, that AH* had no real significance.

In 1974, studies on the transport of vinyl chloride monomer

through poly(vinyl chloride), PVC, powder was commenced by Berens (59)!

the innovative use of submicron sized spheres allowed systems with

very low diffusion coefficient to be studied. In a later study (60),

the effect of the non-uniformity of particle sizes on the kinetics of

sorption was reported. It was observed that although emulsion poly-

merisation produced essentially mono-dispersed samples, normally in

the range of 0.1 to 1 microns, suspension and mass polymerised samples

showed a tendency to agglomerate, although primary particles in the

range 1 to 5 microns were still discernible. The kinetics of sorption

indicated the rate-controlling factor in these latter samples to be

the dimensions of the primary particle rather than the larger agglomerates.

Later (61-65) the analysis of the complex behaviour of heavy vapours

in polymer powders was reported. A means of separating Fickian diffusion

from the slower relaxational process was discussed and an overall model

proposed.

56

The use of microspheres in the study of hexane diffusion in

polystyrene was also explored by Enscore et al. (66). These workers

showed that marked differences in the sorption behaviours could be

introduced simply through varying the dimensions of the polymer sample.

The diffusion of heavy vapours in small samples (microspheres) corresponded

to Tick's equation with a heavily concentration dependent diffusion

coefficient whereas with larger samples (sheet), Case II diffusion was

observed. It is beyond the scope of this review to include the

phenomenon of Case II. diffusion but enlightening reviews of the subject

may be sought elsewhere (45,49).

Polystyrene, PS,in both film and powder form was studied by

Barrie, W M l i a m s and Monday (68) using some simple hydrocarbon vapours.

The results for the diffusion of methane in PS were in good agreement with

an earlier study by Victh, Tarn and Michaels (69). A systematic study

of propane sorption in both the polystyrene sheet and powder showed

large differences between the two samples ; it was observed that

although the Henry's Law component (k^) to the sorption process was

comparable in both cases, the powder showed an inherently higher

Langmuir component. This phenomenon was reported again by Huvard

et al. (70) for C0 2 sorption in poly(acrylonitrile), PAN. In this

case, it was found that annealing the microspheres, at a temperature

just below the T t effectively reduced the sorptive capacity of the f-

powder, but still, the magnitude of the sorption was higher than that

of the sheet. A tentative explanation based upon surface sorption

effects was proposed. Barrie et al., considered that differences in

the relative microvoid cent ent may be responsible.

57

Williams (71) studied the transport of propane in polycarbonate,

PC. He showed that the immobilisation of the adsorbate associated with

the Langmuir component was only partial and the contribution by this

component to the overall flux increased as the temperature was reduced;

the contribution by the Langmuir species was estimated to be in the

order of 10-20%. Koros, Paul and Rocha (72) found the Langmuir

contribution to be approximately 10% for C0 2 in polycarbonate*, this

estimate was revised later (73) to 7.8%. Assink (74) studied the

transport of ammonia in polystyrene by a novel pulsed NMR method",

mobility of the Langmuir species was estimated to be ^ 5%.

The transport and sorption of C0 2 in PC was studied in exhaustive

detail by Koros et al. (72). Permeabilities and time lags were reported

to decrease with increasing i n g o i n g pressures, as was found for C0 2

in PET, reasserting the model for partial immobilisation. The transport

of four additional gases, CI-U, Ar, N 2 and He was later studied by Koros,

Chan and Paul (73); their results for Ar were found to be at variance

with an earlier study by Norton (76). In addition, all the gases with

the exception of He were well described by the partial immobilisation

theory, the results for He were anomalous and indicated a Langmuir

contribution of over 133%.

It xcas also proposed in their study that since c' was associated with

t h e 1 " void" content in the polymer, it may be used in estimating the

proportion of free volume that was effectively "frozen" . The main

difficulty that they encountered was in the assignment of the correct

density of the C0 2 sorbed in this state. A similar treatment was suggested

earlier by Eilenberg and Vieth (77).

58

Included in Koros' (73) paper was an interesting discussion on

the statistical accuracies associated with estimating the dual-mode

sorption parameters. In essence, it was shown that to calculate c' H

and b as separate entities incurred much larger error than if the

product c' b was considered. H

The effect of prior exposure of PC to C0 2 on the sorption and

transport parameters was re-examined by Wonders and Paul (78). The

solubility of this gas in glassy polymers is sufficiently high at

elevated pressures to disturb the internal distribution of unrelaxed

volume. At high pressures of C 0 2 , the enhanced sorption dilates the

polymer through small scale re-adjustments of the chains, in order to

accommodate the C0 2 molecule. On desorption, the polymer expansion is

not recovered immediately and this is reflected by an increase in the

Langmuir sorptive capacity in subsequent sorption measurements at

lower pressures. Wonders and Paul examined in particular, the effect

of uniform and vectored conditioning on the transport through the PC

membrane. Uniform conditioning could be interpreted easily using the

dual-mode sorption theory, however, vectored conditioning could not be,

explained as well. Unlike uniform conditioning,in vectored conditioning,

only one surface was exposed to C 0 2 , whilst the other face of the membrane

was maintained at vacuum. The distribution of the sizes of the microvoids

therefore varied as a function of distance, establishing in effect,

a graded membrane.

The transport of gases through poly(acrylonitrile), PAN,was first

examined by Allen et al. (79). The barrier properties of PAN to seven

gases nns examined in detail. One of those gases, C 0 2 , was later (70)

59

the focal point of further examination. In this later study, the

effects of annealing on the sorption of C0 2 in PAN powder and sheet

was reported. It was shown that slow controlled cooling of a sample

held at a temperature just below the T ^ effectively reduces the sorptive

capacity. Furthermore, annealing was found to be effective on one

cycle only, and repeated annealing produced no significant changes to

the isotherm. Their studies of the transient state permeation through

the PAN sheet reflected pronounced differences between the experimental

time lags and those predicted from theory.

Ethyl cellulose was first studied by Barrer, Barrie and Slater (25).

It was these early studies that led to the speculation of pre-existing

sorption sites in the glassy polymer and the proposed use of the dual-

mode sorption isotherm to describe the sorption. Although these investiga-

tors did not analyse their results in terms of the dual-mode sorption

theory, Chan, Koros and Paul (80) later re-examined their data to find

it consistent with the proposed theory. Chan et al., also provided

additional evidence in support of partial immobilisation of the Langmuir

component.

Poly (methyl me'thacryla te), PMMA, has recently been studied with

renewed enthusiasm. One of the earliest reports of gas transport through

PMMA was by Barker (81) who studied 0 2 diffusion by a novel optical

technique. The value of the activation energy of diffusion which he

calculated was in good agreement with a later study by Patel, Patel,

Patel and Patel (82). This group of investigators also reported on

the transport of four other gases, He, H 2 , N 2 and C0 2 through the same

polymer membrane. It was found during the course of the present study that

60

the results presented by Patel et al. are prone to errors and a

careful scruntiny of their data is to be recommended.

Koros, Smith and Stannet (83) have recently reported on the

sorption of C0 2 in both PMMA and poly (ethyl methacrylate) , PEMA. A

comparison of the results indicates that the sorption of C0 2 in these

two polymers is fairly typical of that normally found in glassy polymers.

However, enthalpic results for the Langmuir sorption were negative

for PMMA but positive for PEMA. Koros et al. attributed this anomaly

to residual acetone found in the latter which was used as the casting

solvent in the preparation of the film.

In this final section, a brief resume of some theoretical considerations

related to the dual-mode sorption theory are x>resented. The basic

assumptions of the theory have already been outlined in Section (2.3.'2)

in relation to the diffusion model proposed by Vieth and Sladek (26).

There appears to be considerable evidence in support for only partial

immobilisation of the Langmuir species and Petropoulos (84) in particular

critically examined the validity of this fundamental constraint; lie"

showed that slow hole filling kinetics resulted in a time-dependent

diffusion coefficient, and pointed out that the effect of only partial

immobilisation resulted in a pressure dependent permeability coefficient,

coupled with only a weak dependence of the time lag on the upstream driving

pressure. A complete summary has been presented by Paul et al. (54,55,

85).

Tshudy and von Frankenberg (86) also relaxed the postulate of

rapid equilibrium between the two penetrant populations. Their theory

61

is not confined to the dual-mode sorption process only, but is equally

applicable to systems in which the rate of diffusion is comparable

in magnitude to the rate of immobilisation.

Bhatia and Vieth (87) questioned the applicability of a combined

Henry's Law and a Langmuir isotherm in characterising the dual-mode

sorption isotherm. They introduced a novel idea in which both the

mobile and immobile species were characterised by two separate Langmuir

isotherms.

Grzywna and Podkowka (88) also challenged the cogency of the

dual-mode sorption equation. They replaced the Langmuir isotherm by

the Freudlich isotherm but obtained only limited agreement with their

experimental data.

These last two contributions to the dual-mode sorption theory

exemplify the main pit-fall of the theory. A three-parameter equation

of the type used to describe the dual-mode sorption isotherm, is capable

of fitting a wide number of systems, although the goodness of the fit

should not be treated as concrete proof for the correctness of the model.

However, the consistency of the results reported over the years in

relation to the sorption of gases and vapours in glassy polymers certainly

suggests a basis for further examination into the subject.

62

3.3. The Phenomenological Correlations of Transport and Sorption

Parameters.

In the study of the sorption and transport processes of gases

through polymers, it becomes increasingly more important that some

meaningful correlations of the transport parameters with the fundamental

molecular structural variables of the penetrant exist . Such phenomeno-

logical correlations are not only important from the point of view of

predicting unknown parameters from existing information, but also

advance the understanding of basic diffusional processes! It

should be pointed out that numerous correlations have been suggested

in the literature, but although some are based upon sound theoretical

premises, there are in addition, many empirical correlations which are

peculiar to specific systems and may not be generally useful. These

latter correlations should therefore be treated with caution.

The correlation of the dissolution constants (k^) to the properties

of the gases was apparently first suggested by Gee (89). Gee considered

the dissolution of a gas in an elastomer as being a process of

condensation followed by mixing between the sorbate and the polymer

medium. He arrived at a simple relationship between k^ and the boiling

point of the penetrant (T^).

- I n k 4.5 +ii - 10T /T (3.1)

He showed that p was not of critical importance and could be set to

zero. Equation (3.1) then reduces simply to

at 30°C.

1 nk_. -4.5 + 0.033 T. I) — b

(3.2)

63

Experimental verification of equation (3.2) was provided immediately

by Barrer and Skirrow (90) and later by van Amerongen (125); the success

prompted further examination into other possible correlations. The

premise that a polymer matrix could be likened to a low molecular weight

liquid suggested that k ^ could also be successfully correlated to

the Leonard-Jones force constant (e/k). This remains one of the most

promising correlations, since ("e/k) is felt to be a more fundamental

thermodynamic property than the boiling point T^, for measuring the

proclivity of approximately spherical gas molecules to condense in a

liquid like matrix (91). Michaels and Bixler (93) developed this idea

and later derived an expression for predicting k^ for a hypothetically,

completely amorphous polyethylene, viz.,

lnk D ~ "3.66 + 0.026 i c/k) (3.3)

Experimental verification of equation (3.3) has been reported in the

literature (92,94).

Stern, Mullhaupt and Garesis (95) applied the theory of corresponding

states to the process of equilibium sorption. They found a smooth

relationship existed between lnk^ and the quantity T c/T> where T c

denotes the critical temperature of the gas. They modified their

expression to yield a linear relationship between Ink^ and (T c/T)2.

The heat of dissolution, AH^, by similar argument, should correlate

with the properties of the gas since the heat of dissolution may be

treated as a composite quantity comprising the heat of condensation and

tli? heat of mixing. Michaels et al. , (93) studied the sorption of twelve

gases in linear polyethylene and they found the expression,

64

A H D = A X - A 2 (c/k) ( 3 . 4 )

to hold true, in which A 2 , the gradient, was assigned the value 0.0653.

Barrer and Skirrow (90) found empirically that AH^ was proportional

to the number of carbon atoms in a homologous series of n-paraffins

for sorption in natural rubber. Reid, Pransnitz and Sherwood (9 6)

have recently found that T^, T^ and c / k are all interdependent.

Correlation functions pertaining to gas permeabilities have not been

met with the same degree of enthusiasm. The reason for this probably

lies in the more complex nature of the permeability coefficient. Despite

this, Stannett and Szwarc (97) in their examination of the effect of

irradiation on the permeability found the ratio of the permeabilities of

a number of gas polymer systems to be uncannily constant. In a similar

vein, Allen et al., (79) found the ratio of P /P to be approximately gas u 2

equal to 60 for most polymers. Helium, however, was found to be anomalous

and yielded a ratio of 2000. Hammon, Ernst and Newton (99) in their

quest for the elucidation of an effective radon barrier studied the

permeability of three inert gases through a number of polymer films.

They made a notable attempt to correlate the permeability against

(1) cohesive energy density, (2) relative density and (3) refractive

index of the polymer. They also tried to relate the permeability to the

diameter of the gas molecule, d, and also to d2,* none of which were

entirely convincing. However, they found that linearity existed between

InP and the product of the cohesive energy density and the relative

density, but offered no satisfactory explanation for their findings.

Another empirical correlation was proposed by Salame (98) in which a

correlation between the polymer structure and the gas permeability of

65

polymeric materials was suggested. Recently, a theoretical treatment

has been advanced by Lee (100), which predicts a linear relationship

between log(Permeability) and (1/ specific free volume). A modified

version of their correlation was used in the present study, and the

results are indeed promising.

The correlation of diffusion data has generated the greatest

controversy. The manner of the correlation reflects to a certain degree

the mechanism of the diffusion process and since there are a variety

of molecular theories for the process of diffusion it is not surprising

that there are also numerous correlation relationships suggested in the

literature.

Barrer et al., (113) suggested a correlation between ln(D^) and

E^/T ; the remarkable quality of their correlation was argued on the

basis of the small range of values which In (D ) spanned compared with

t Ep. Barrer et al., also showed a linear correlation between AS , the

entropy of activation,with E /T. Recently, Chen and Edin (101) have

i shown that ln(D ) is also related to AS .

o

Meares (.44) assumed that the process of diffusion occurred along

a cylindrical volume of value, ( Tid 2A/41 , where the diameter of the

cylinder was equivalent to the diameter of the penetrant, d, and A is the

length of the diffusional step. The activation energy of diffusion was

given by:-

E = (CED) . (ird2A/4) (3.5)

66

This predicted a linear relationship between E^ and d 2 . Van Amerongen

(103) however, showed that E^ was linearly related to d rather than

d 2 . The free volume theory prompted Kumins and Roteman (104,105) to relate

the semi-log f r e G volume element to the gas diameter; again linearity

was found. Brandt and Anysis (106) found that E^ was not directly

proportional to d 2 as suggested by Meares (44), but instead a small

2 o

positive intercept at d ^ 5 to 8A was observed. They asserted that

the intercept at the abscissa was related to the free distance between

chain molecular surfaces.

With the advent of the dual-mode sorption theory, numerous attempts have

been reported on the correlations of k , S , and b to T, , T and e/k. These D L b c

correlations followed along the same lines as k , the solubility coefficient

in rubbery polymers mentioned earlier. The overall solubility, S , a ij

composite quantity of no mini sorption and hole filling was found to-be

related to c/k through the integrated CIausius-Clapevron equation giving

lnS T = 0.026(~/k) - 1.68 AH + I at 25°C (3.6) L m

where AH^ is the heat of mixing and I a system constant. The validity

of equation (3.6) has been reported for the solution of inert gases in

poly(ethyleneterephthalate) (48), common atmospheric gases in poly(methyl

methacrylate) (82), and simple gases in poly(sulphones) (107).

The hole affinity constant, b, represents the ratio of the rate

constants for competing processes of adsorption and desorption in the

microvoids. Since r/k is a measure of the condensibility of a gas, or, more

generally, the propensity for gas-polymer interactions, a relation between

b and 7/k should be expected. Vieth, Tarn and Michaels (108) established

67

a relationship,

i In (c /c )

lnb = 0.026 ( r/k) + , , - I* (3.7) C H / C H

where I' is the system constant.

The agreement between experimental and theory was excellent. The

relationship suggests that the more condensible gases with a higher e/k

value, possess an inherently higher b value. Hence, the equilibrium

sorption isotherms of these gases would show marked deviations from

Henry's Law even at low pressures. Conversely, the least condensible

gases with a lower r/k value would have an extremely small b value and

the isotherms would therefore be linear over a protracted pressure range.

Similar results have also been reported by Koros, Chan and Paul (73).

c\ represents the hole saturation constant; assuming monolayer

coverage of the hole, c' was found to be inversely proportional to the H

surface area occupied by one sorbate gas molecule (108), viz.,

• - R T S 1 O ON c u "TO— • 7T (3.8) H N p A 2 a

The diffusion coefficients for hole and dissolved species were

correlated to the Leonard-Jones collision diameter (d-^j)• In the case

of D , a monotonic decrease with increasing gas diameter was found. D U H

however, also decreased with increasing size of penetrant but showed a

region.of inflection over which the diffusion coefficients was' independent

of djj.The authors (108) believed this to be indicative of some critical

dimension associated with the Langmuir nature of the polymer.

68

3.4. Transport in Heterogeneous Media.

In this section, an examination of the literature, pertaining to

the transport of gases and vapours in heterogeneous polymeric membranes

is presented. The subject matter in this review has been divided into

four general categories, namely: membranes containing inert fillers,

membranes containing active fillers , crystalline polymers and finally,

copolymer membranes. This demarcation is purely subjective and is

intended to reflect the general progression and understanding of the

subject over the years.

The transport of simple gases and vapours through rubber membranes

containing inert fillers have been studied by several workers. It was

found that a clear distinction between the " inert" and the " active"

filler was seldom recognised by many workers, with respect to possible

gas/filler interactions. As a result, the interpretation by many of the

earlier authors was found to be clouded, especially, when the ubiquitous

time lag method was used to evaluate the transport parameters.

In theory, the time-lag, 0, and the effective diffusion coefficient,

D^, is related by the expression:

D = / 2/66 (3.9)

where / is the thickness of the membrane containing the dispersion of

inert fillers. D^. is the diffusion coefficient of the filled rubber and

is related to D^, the diffusion coefficient of the corresponding unfilled

rubber by the simple expression,

(3.10)

69

where k is a structure factor and is a measure of the more tortuous path

taken by the probe molecule due to the presence of the filler.

Natural rubber containing up to 22% (by volume) of mineral and

carbon fillers was studied by van Amerongen (111). The transport

of simple gases at low activities was' described by a concentration

independent diffusion coefficient and the activation energies of

permeation were found to be independent of both filler composition

and type', indicative of transport being controlled by the continuous

phase only. Solubilities were determined by both the time lag method

and also by equilibrium sorption -studies. Differences were found between

the two methods and in a later review of the work (112), the apparent

anomalies were interpreted in terms of the poor adhesion at the inter-

face of the filler and the polymer.

An interesting aspect of the work by van Amerongen (111) was the

pronounced dependence of k on the geometric shape of the filler. It was

found that the lamella type fillers reduced the effective diffusion

coefficient by a larger proportion than the corresponding spherical filler.

Barrer and Chio (113) investigated the diffusion of n-butane in

a Santocel filled silicone rubber membrane, K was estimated from the

diffusion coefficients measured during the early time,from the time lag

and in steady state. In all three cases, the results were in good

agreement suggesting that the structure factor inhibiting the flow of

penetrant, was the same for both the steady state and the transient

state. However, in an earlier study, Barrer, Barrie and Rogers (114)

studied propane diffusion in a zinc oxide filled natural rubber. In this

case, quite definite differences were encountered between the steady

state diffusion coefficient and that measured from the time lag.

70

Kwei and Kumins (115) studied the sorption of organic vapours in

poly(vinyl. acetate) filled with an epoxv polymer. A decrease in the

solubility was interpreted in terms of reduced chain mobility in the

vicinity of the filler. In a later study (116), of the diffusion of

oxygen and argon in poly(vinyl acetate)filled with 12% titanium oxide,

diffusion coefficients measured for the filled rubber indicated a

reduction of over 70% compared witli the unfilled rubber. A detailed

discussion based on the reduction in the internal pressure and volume

of activation was presented. Their premise that the diffusion coefficient

may be derived directly from the time lag using equation (3.9) is

almost certainly incorrect since the adsorption of penetrant in the

fillers was not properly accounted for.

Polymers containing a dispersion of active fillers have been reported

extensively in the literature. In addition to increasing the effective

path length through the structure factor, k, the time lag is also

protracted due to the finite amount of penetrant adsorbed within the

fillers. If the equilibrium concentration of penetrant in the filler.is

assumed to be directly proportional to pressure (i.e., Henry's Law)

and a similar relationship is obeyed by the continuous rubber phase,

then the solution obtained by Frisch's procedure yields,

D f " S 5 [ 1 + ( k ) D / ( k ) c ] ( 3 " n )

where (k) and (k) are the Henry's Law solubility coefficients defined 1J v

per unit volume of the dispersed and continuous phases respectively.

Finger et al., (117) derived their expression for the influence

of Langmuir adsorption on the effective time lag and in the limit of

low activity (i.e., Henry's Law), their expression reduced to

71

(3.12)

where K' is the distribution coefficient of the penetrant between the

two phases.

Flynn and Roseman (118) and later Cooper (109,110) pointed out that

the expression derived by Finger was quantitatively incorrect and did not

reduce to the correct expression for the case of an inert filler. Flynn

et al., derived their expression by comparing the ratio of the concentration

gradients for the filled and the unfilled membranes. Their expression

is given by

They used their expression in the interpretation of the diffusion of

p-aminoacetophenone in a silica filled membrane. A cursive examination

of their results suggests that the fillers used behaved as highly

adsorbing fillers and the concentration of penetrant in the fillers

should thus be characterised by a Langmuir type isotherm, rather than

simple Henry's Law.

Most (119) studied the diffusion of p-aminobenzoate through silicone

rubber membranes containing up to 12.8% (by volume) of silica. Time

lags were reported to be increased by a factor of ^ compared with

the unfilled rubber, whilst the steady state permeabilities decreased

by 25%. The results again suggest that the fine dispersion of silica

were behaving as highly adsorbing fillers.

(3.13)

72

Higashide et al., (120) dispersed mica flakes in poly(methyl

methacrylate) sheets. Oxygen transport through the films indicated a

pronounced effect of the permeability on the pre-treatment -of the glass

flakes; the transport of oxygen in the unfilled membrane was, however,

not reported.

Paul and Kemp (121) measured the diffusion time lag for a silicone

rubber membrane containing a highly adsorptive molecular seive. Sorption

by the zeolite, was characterised by the classical Langmuir isotherm

and they were able to verify an earlier postulate by Paul (123). In a

later paper, the kinetics of sorption were reported (122). It was

demonstrated that for h e t e r o g e n e o u s systems in which the morphology was

simply defined, a single effective diffusion coefficient was applicable

for both the transient and steady state conditions, and any apparent

anomalies could be explained, provided the effect of the disperse phase

is properly accounted for.

The transport of low activity penetrants in crystalline polymers may

be interpreted using equation (3.11) with k^ = 0. Dissolution in the

crystalline regions is negligible and the diffusion coefficient is also

expected to be low; the crystallites may therefore be viewed as ideal

impermeable fillers. Van Amerongen (125) showed the effect of crystalli-

sation in gutta percha on the diffusion coefficient, solubility and

permeability.

Barrie and Piatt (126) studied the solubility of several isomeric

hydrocarbons in stretched rubbers. Th e permeabilities measured over a

duration of a hundred days indicated a slow monotonic decrease; part of

the reduction in the permeability was accounted for by a slow increase in

73

the degree of crystallinity, the remainder was attributed to partial

re-orientation and redistribution of the crystallites as the system

tended towards a more stable morphology.

Barrer and Chio (113) studied the diffusion of n-butane in a filled

silicone rubber above and below the crystallisation temperature. In the

absence of crystallinity diffusion coefficients measured from both the

time lag and the steady state were in close agreement however, at the

onset of crystallisation, the diffusion coefficients measured by the

two methods diverged. There is no apparent reason for this discrepancy

if dissolution in the crystalline regions is assumed to be a forbidden

process. It is thus possible that slow re-ordering of the crystals may

have introduced a form of general time dependence which was not accounted

for.

Crystallinity in polyethylene was studied in detail by Michaels and

Parker (127). The degree of crystallinity in the polymer was estimated

from the densities, assuming polyethylene to be a simple binary composite

of crystalline and amorphous materials', the solubilities were found to be

in direct proportion to the volume fraction of amorphous material.

Diffusion in the crystalline polymer was believed to be reduced by two

impedance factors, namely T and 3. T, the geometric impedance factor

accounted for the increased path length which the probe molecule must

take, in order to negotiate the crystallites and is analogous to -the

reciprocal of the detour ratio proposed by Klute (128). 3 was a chain

immobilisation factor and accounted for the reduction in the chain

segment mobility in the vicinity of the crystallites. The product, t3»

74

was estimated for several crystalline polyethylenes, by assuming the

equivalence between diffusion in vulcanised natural rubber with a

hypothetically, completely amorphous polyethylene. Separate values

of t and 3, could not, however, be measured directly since the mean

intercrystalline spacing was an indeterminant parameter.

Brown, Jenkins and Park (129) studied the diffusion of four volatile

liquids, namely dichloromethane, benzene, chloroform and 2,2,4-trimethyl-

pentane in amorphous and crystalline poly(butadiene). The onset of

crystallisation as a function of the trans- content in the 1,4-poly-

(butadiene) was manifested by a discontinuity in the solubility, and

a corresponding large increase in the concentration dependence of the

diffusion coefficient. The activation energy of diffusion, however,

remained invariant, suggesting that the predominant change was a structural

effect, whilst the basic mechanism for diffusion remained essentially

unaltered.

It is evident from the early parts of this review that diffusional

studies., in conjunction with an understanding of the transport phenomenon

has far reaching applications in the interpretation of polymer morphology.

In the remaining section, transport through copolymers with this concept

in mind, are reviewed.

Copolymers prepared by the method of grafting generally lead to

binary phase systems, provided high molecular weight branches are formed.

Munday (92) studied the transport of alkane vapours in a series of

polystyrene/ si 1icone rubber graft copolymers. Activation energies of

permeation were constant and equal in magnitude to that for silicone rubber

indicative of transport- controlled by the I'DMS phase. Further evidence

of two distinct components was provided by electron micrography and glass

transition temperature measurements.

75

In a series of papers by Huang et al., (131-135) the permeation

of several simple gases through polystyrene-polyethylene grafts were

reported. Grafting onto semi-crystalline polymers occurs mainly in

the amorphous regions, however, as the level of grafting increases,

the degree of crystallinity may be affected, due to disruption of the

crystallites. Huang reported that permeabilities decreased as the

polystyrene content increased, up to 30%, and beyond this composition,

the permeabilities began to increase sharply. They attributed the

initial reduction in permeability to the decrease in the free-volume

of the amorphous phase. It is difficult to understand why they chose

not to interpret the initial reduction in permeability by an increase

in the path length caused by the presence of the inert filler. The

subsequent increase in the permeability was felt to be caused by changes

in the amorphous-crystalline ratio.

Toi et al., (137) and Myers (136) also studied the effect of styrene

grafting on the diffusion and solubility of simple gases in high density

polyethylene. The results were consistent with the earlier study by

Huang et al., and supported the observation of the increased permeability

beyond 20% grafting.

Sorption and diffusion in asymmetric membranes was reported by

Rogers (138). Polymer sheets with a gradient of inhomogeniety were

prepared by initiating the grafting of vinyl monomers onto polyethylene

before complete equilibrium in the system was acheived. The difference

in the permeation constants evaluated from the steady state flow in the

unfilled and the graded membranes was demonstrated* when the grafted

regions acted simply as inert fillers, the rate of transport was depressed

without any significant trends as to directional effects. However, when

76

the penetrant showed preferential dissolution, or swelling characteristics,

in either phase, penetration rates showed significant directional effects.

In a later paper (139) a mathematical model was derived which expressed

a quantitative description of directional transport processes. Stannett

et al., (140) introduced inhomogeniety into styrene-2-vinylpyridene and

polyethylene-2-vinylpyridene grafted films. Graded films in this case

were prepared by a conformational gradient technique, induced by solvent

treatment and also by quatemisation of pyridene by chemical reaction with

methyl bromide; it \</as shown that under suitable conditions, vectored

flow could be introduced.

The transport properties of gases in block copolymers have only

recently been reported in the literature. Robeson et al., (142)

synthesised a series of poly(sulphone)/poly(dimethylsiloxane) block

copolymers, of the (AB)^ type, and studied both the physical and transport

properties of the system. A Maxwell (160) and Kerner (143) model of

spheres embedded in a continuous matrix was found to be an adequate

representation at the extremeties of the composition range. At intermediate

compositions, however, deviations from this model were apparent, and by

the use of a weighted Maxwell and Kerner model, phase separation was

estimated at 0.51 and 0.53 volume fraction of polvsulphone respectively.

Barnabeo et al., (144) studied the block and random copolymers of

styrene and methacrylonitrile. Semi-logarithmic plots of permeability

versus volume fraction were found to be sigmoidal in character for the

blocks but linear for the random copolymer.

77

Odani, Taira, Nemoto and Kurata (94) reported on the sorption and

diffusion of inert gases in styrene-butadiene-styrene block copolymers.

Morphologies corresponding to two general types was recognised! (i) poly-

styrene rods dispersed in a polybutadiene matrix and alternating

lamellas of styrene and butadiene components. Permeabilities of type

(i) were well characterised by the parallel model, viz.,

P = cjvPi + <{,2p2 (3.14)

but discrepanc i os wi lili this model were encountered with the lamella

type morphology and permeabilities calculated from the series model were

also in disagreement. However, the apparent incapability of the parallel

model to predict correctly the permeabilities of the lamella type system

was surmounted by introducing two additional parameters, namely t and 3

which have l>*»'en discussed earlier in relation to crystalline holymers.

Tn a later paper (1461 equilibrium solubilities were reported.

Ziegel (147) studied the transport of five gases in four thermo-

plastic polyurethanes, of varying hard and soft segments. A model

corresponding to the " hard segments" dispersed in random spatial

orientation of the soft-phase elastomer appeared to be consistent with

their transport results.

Williams (71) studied the transport of propane in a series of alternating

block copolymers of silicone rubber and poly(bisphenol-A-carbonate).

Incomplete microphase separation was suggested and was interpreted by a

novel adaptation of the Higuchi model combined with the Lichtenecker

Rule.

78

Polymer blends represent the final category of materials in this

review of transport through heterogeneous membranes. A study into

compatible polymer blends was made by Stallings et al., (149). Poly-

blends of polystyrene, (PS) and poly(phenyleneoxide), (PPO) at three

compositions were prepared by casting the polymers from a common solvent.

Evidence of compatibility was provided by the detection of a single

glass transition temperature which increases monotonically with the

composition of PPO. Diffusion coefficients of Ne, Ar, and Kr in these

membranes were measured by the time lag method,all of which showed

minima when plotted against composition. Incomplete phase separation

was discounted by the authors who preferred to visualise the anomaly in

terms of a redistribution of microvoid content in the polymer.

Polyblends were also examined by Ranby et al., (150-154). Five

systems were reported, all of which were based on poly(vinyl chloride).

In their first study, the effect of vinyl acetate in poly(ethylene-vinyl

acetate) on the physical blends with PVC were reported. The study was

extended to a critical examination of varying acrylonitrile content

in poly(acrylonitrile-butadiene) copolymer in a series of PVC/NBR

blends. In their penultimate report, the effect of chlorine content in a

series of chlorinated polyethylene on PVC/CPE blends was examined*, it was

found that compatibility was enhanced at increasing chlorine content.

Finally, blends of PVC and poly(acrylonitrile-butadiene-styrene) terpolymers

were characterised. A model in which a two phase system composed of a soft

butadiene phase and a rigid PVC/styrene-acrylonitrile phase was proposed.

It is unfortunate that the transport parameters were not obtained at more

than one temperature, since a means of estimating the activation energies

would reflect the correctness of their model.

79

A complete review of the chemistry of compatible polymers has

been presented elsewhere (148,130); the physical properties of these

systems are also available in standard texts (145). Furthermore,

proposals for future studies have been suggested by various investigators

(124,141) and it is hoped that these intended examinations will lead to

a greater understanding of the morphology of these systems.

3.5. Review of Heterogeneous Modej_.

Numerous theories related to the electrical and thermal properties

of heterogeneous media have been expounded. If the close analogy between

thermal and electrical transference with gas transport in heterogeneous

membranes is accepted, then the models which have been derived on the

basis of thermal conductivity and dielectric properties should be equally

applicable to the mass transport of gases in these systems.

The intention here, is not to delve into the mathematical derivations

of each model since adequate reviews on the subject are available else-

where (155,156). The validity of the models have been shown to be

correct (157,201) and it is felt that a brief resume^of the implicit

assumptions associated with each model would be of greater benefit.

The models which have been selected for discussion are confined to the

general type where a dispersed and continuous phase co-exist. (Laminates

and capillaries will not be included).

It was found that in some reviews of the subject that the discussions of

the models were not confined to the permeabilities. Instead, the

diffusion coefficient was used with the partition coefficient set to

unity. Throughout this section, the intensive property of the system is

the permeability since this is felt to be the more pertinent property.

80

In the following section, the notation adopted is given byl-

m The permeability of the composite media

The permeability of the continuous phase

The permeability of the dispersed phase

Volume fraction of the continuous phase

Volume fraction of the dispersed phase

One of the simplest, empirical models was first suggested by

Lichtenecker (158) and is given as follows.

- - V - ^D P = P . P (3.14a) m C D

The Lichtcnecker rule has been found most useful for compatible

polymer systems and also for copolymers in which the individual permeabilities

are within an order of magnitude of each other. Although the Lichtenecker

model has been criticised for its mathematical inexactitude (157), its

simplicity renders it extremely advantageous for quick approximative

calculations.

In 1891, Maxwell (160) derived his expression for a mixture of

spheres embedded in a continuum. At infinite dilution of the dispersed

phase, the permeability of the medium is given by

_ _ % + 2P - 24, (P - P ) P = P„. — — (3.15) m C

v ' P

D + 2 P

C :+V

pc - V

Implicit in the derivation was that the geometry of the dispersed

phase was spherical, with each sphere separated far enough from its nearest

neighbour in order to avoid any mutual perturbation of fluxes. It has

81

been found (157) that the Maxwell expression is valid for up to 10%

dispersion and a fair approximation at even 50%. The Maxwell expression

as given in equation (3.15) reduces to two limiting expressions when

the permeabilities of the two phases are vastly different from each

other.

Thus, when P » ? C D

2 (f> , P m = P • t - T (3.16)

c and when P^ » P^

_ 3-2* P ™ = V —T"^ ( 3 - 1 7 )

m C * c

Many of the models which ensued (159,168,161,169), were broadly,

extensions of the Maxwell model. As such, many of these models reduce

to equation (3.15) under the correct conditions.

Bruggeman (161) expanded on the Maxwell model in order to account

for the interactions of the fields caused by the presence of the dispersed

phase. Bruggeman !s model also assumes the disperse phase to comprise

a wide range of particle sizes and his expression is given by,

1 P r /P

* r = (3-18) (P D - P C)/(P D - P m )

The expression simplifies in the limit of P » P^, to give

3/2 P = v v m (, C (3.19)

and also in the limit P >> P D C

P = P p/(1 - A ) (3.20) M U D

82

The dispersed phase within polymeric systems normally comprises

particles of much the same size, or at most, the particles differ in

size by only an order of magnitude. Meredith and Tobias (162) modelled

this system and the permeability of a membrane representative of such

a structure is given by

(2 + 2 X O 2 + (2x - D<J>n P = P . — _ (3 21) *m V (2 - X f D ) ' 2 - (X + 1 H d

where

(P D/P r " 1)

X = _ _ (3.22) ( P D / P r + 2 )

Again, in the limit of P » P D

!m 8 ( 1 - y ( 2 - V

P c (4 + <frD) (4 - 4»d) (3.23)

and the limit of P R » P(1

5a ( 1 + v ( 2 + v - " (i - • ) ( 2 - y

c:

(3.24)

Lord Rayleigh (163) considered the specific case of identical spheres

arranged in a simple cubic lattice. He was able to extend Maxwell's equation

in this way to obtain

P m ( 2 P C * V / ( P C - V - 2 * P - g [ 3 ( P C - V P ( A P C + 3 P P ) ] C 0 / 3

Pc ( 2 P C + P D ) / ( P C ~ V + " a [ 3 ( * V . " P D ) P ( 4 P C + 3 P D ) ] ( j ) D 1 0 / 3

(3.25)

83

a Ts a system constant and was assigned the value of 1.65

by RayleighJ Runge (164) and de Vries (165) obtained values of 0.523

and 1.31 respectively. De Vries (166) also showed that the mode of

packing was an important factor and calculated values of a to be

0.129 and 0.752 for b.c.c. and f.c.c. packing respectively.

Botcher (16 7) derived his expression for ellipsoidal particles;

the expression for the more general case of spherical particles reduces

to!

pm " pr: " 3 P m V p n " p c ) / ( p u + 2 P J ( 3 " 2 6 )

The expression is unique in that the same equation is applicable even

if the roles of the dispersed phase and the continuous phase are inter-

changed.

Finally, Higuchi (168,169) considered a system composed of randomly

distributed uniform isotropic spheres. Each particle interacts with

the flux creating a perturbation flux of its own and a neighbouring

particle reacts to this perturbation to create an excess flux. The

expression is given by

_ (P + 2P - X) - *n(2P " 2P - x) P = P — - £ V - — ^ - (3.27)

" ( ? D + 2 k C " + V ^ C " + X )

where

X = k" <P d - P (,)2/(2P (, + P D ) (2.28)

84

The parameter k" involves the radial distribution function of

random spheres which is generally unknown. However, k" = 0.78 was

found to be a satisfactory representation for a large number of systems

if the particle shapes did not deviate too far from a spherical geometry,

In the limit where particle-particle interactions are absent, infinite

dilution is acheived and k" = 0. In this limit, equation (3.27) reduces

to the Maxwells expression as given in equation (3.15). Equation (3.27)

reduces to two limiting expressions when P » P^

3.22* P m = V 6 - 2.786 ( 3 ' 2 9 >

and when P » P U L»

85

CHAPTER FOUR

EXPERIMENTAL

4.1. Materials.

4.1.1. Poly([c.]-dimethyl si 1oxnne).

Sheets of nominally 1 mm thickness were supplied by Dow Corning Ltd.

The PDMS gum was press cured with 2,4-dichlorobenzoyl peroxide (1% by

weight) at 115°C for five hours and post cured at 200°C for four hours.

Samples of the desired dimensions were cut from the parent sheet and

refluxed with petroleum ether (40/60) in a soxhlet for 24 hours before

use. Outgassing was carried out in a vacuum dessicator connected to a

Morvac water pump.

4.1.2. Poly(methyl methacrylate) - sheet.

High molecular weight glassy beads of PMMA were obtained commercially,

from BDH Chemicals Ltd. A solution of PMMA in toluene (1% by weight)

was filtered through a millipore assembly and slowly poured onto a clean

mercury surface. Evaporation of the solvent took three days in a

covered Monax dish. Outgassing was acheived in a vacuum dessicator

connected to a high vacuum pumping system.

4.1.3. Poly (methyl mothacrvl ate) - Powder.

(i) Preparation of monomer.

Methyl methacrylate (30 cc) was purified by washing with sodium

hydroxide (3 x 10 cc) and extracted with distilled water. Drying was

effected over anhydrous calcium sulphate.

86

(ii) Preparation of polymer.

The technique employed was of a standard emulsion polymerisation

found in most texts (170). Distilled water (100 cc) was placed in

a 250 cc three-necked round-bottom flask, equipped with a thermometer,

addition funnel, stirrer-bar and condenser. Lauryl sulphate (0.3 g)

was added and the mixture, stirred and maintained at 85°C in a silicone

oil bath. To the solution, the purified monomer was added and allowed

to equilibriate to ambient temperature. Polymerisation was initiated

with hydrogen peroxide (10 cc) and stirred for a further 10 hours. The

mixture was filtered and washed with distilled water*, drying was

carried out at 110°C in a vacuum oven for 10 hours. The product showed

a tendency to coagulate into macro-particles although the physical

cohesion could be broken by gentle agitation.

The purity of the polymer was checked by IR and NMR spectroscopy.

The mean particle diameter was estimated from electron microscopy to

be 0.15 microns.

4.1.4. Poly([c.l.j-dimethylsiloxane-g-methyl methacrylate).

PDMS sheets and methyl, methacrylate monomer were prepared as

described in section 4.1.1. and 4.1.3(i). The PDMS sheet, usually of

about 1 g in weight rested on four silica discs (no. 3 porosity) at the

base of the grafting jar (Figure 4.1.) and outgassed for a minimum of 24

hours. Mothvl inethacrylatc monomer was degassed by employing several

" freeze-pump-thaw" cycles, and finally distilled under vacuum into

the grafting jar. The jar and its contents were cooled to liquid nitrogen

u

MMA fig, 4.1.

the grafting apparatus

• PDMS

88

temperature [to reduce the vapour pressure of the monomer (171) ] and

sealed under vacuum. A duration of 24 hours was allowed for the

equilibrium swelling of the PDMS sheet at an ambient temperature of

from a C o 6 0 generator operating at a nominal dose-rate of 1 Mrad/hour.

Samples were removed immediately from the jar and pumped under

vacuum to remove traces of unreacted monomer and low molecular weight

polymers. Higher molecular weight homopolymer was extracted with

either petroleum ether (40/60) or acetone by refluxing in a Soxhlet

for 48 hours. Grafting efficiencies of better than 70% were always

achieved. In the preparation of copolymers with grafting compositions

of greater than 40%, a two stage grafting procedure was employed. In

this case, the graft copolymer was re-swollen with monomer and the

grafting procedure repeated.

Composition of the graft copolymers were calculated from the

relative increase in weight of the PDMS sheet, i.e.,

where w^ and w^ are the initial and final weights of the sheet respectively.

It is often more convenient to refer to the composition in terms

of volume fraction rather than weight percentages. In this case

40°C, after which grafting was initiated with y r a d i a t i o n (1 Mrad)

w _ w . Composition (% by weight) , x = — x 100 (4.1)

Volume fraction of PMMA, (j) x/P»

(4.2) D x/Pi + CI 00 - x) /P z

where x is defined in equation (4.1) and Pi and P 2 are the densities

of PMMA and PDMS respectively.

89

4.1.5. Nomenclature of graft copolymers.

The notation adopted in this treatise to represent the composition

of the graft copolymers is self-explanatory, i.e.,

PDMS - g - 46.34% PMMA

indicates a graft copolymer of PMMA content (by weight) of 46.34%

calculated using equation (4.1). Due to the different post-irradiation

solvent treatments of the graft copolymers, the samples extracted

in acetone are further marked with an asterisk (*) , i.e.,

PDMS - g - 24.14% PMMA*

4.1.6. Penetrant vapour.

Methane and propane were supplied by Matheson Gas Products Ltd., in

the form of " lecture cylinders" . Glass bulbs were filled from these

cylinders by first evacuating the bulbs and subsequently withdrawing the

required amount of gas into the reservoir. Propane was then purified by

successive " freeze-pump-thaw" cycles. Methane due to its much higher

vapour pressure at liquid nitrogen temperature was used as received.

Ethane and iso-butane were supplied by the National Physical Laboratories.

The purity of the gas was 99.9 and 99.97 moles per cent respectively,

and was used as received.

90

4.2. Permeation.

4.2.1. The permeation hardware.

The permeation apparatus shown in Figure 4.2, was of the conventional

type designed by Bnrrer (9). The high vacuum side (T 6.- T X 1 ) and the

high pressure side (T 2 - Tr>) formed the two essential portions of the

apparatus. A diffusion cell (C), with the ensealed membrane separated

the two sides.

The vacuum line was serviced by a rotary oil-pump (Edward's ES 20)

which provided the backing vacuum for a mercury diffusion-pump. A cold

trap positioned between the pumping system and the rest of the line

formed an important protection against contamination by mercury vapour.

The high pressure side, conventionally referred to as the " upstream

side" comprised a gas reservoir (R), buffer volume (Vi) and a pressure

transducer (PT). The gas reservoir contained the charge of gas at a

pressure of about 60 cmHg (at normal temperature). A " cold finger"

attached to the side of the reservoir facilitated purification and

recovery of the penetrant. At regular intervals, the gas was frozen

into the side arm and the solidified vapour subsequently pumped to

remove any traces of air.

The buffer volume (V x) has a capacity of about 1000 cm 3 and was

requisite in maintaining the ingoing pressure constant during the course

of a permeation run. It was found that the eventual reduction in pressure

at the culmination of a permeation experiment was never more than 1% of

the total ingoing pressure.

To Vacuum Pump

V. PT

t 5

U t 2

FIG. q.2.

THE PERMEATION APPARATUS

92

The pressure transducer (PT) was of the strain—gauge type (model

4-327-0003, Bell and Howell Ltd.). A constant 10 volts (DC) was

applied to the transducer, and the output, in the range of 0 to 40

mvolts was monitored on a digital voltmeter (Excel XL35), which had

a resolution of 0.01 mvolts; a zeroing network enabled nullification

of the DVM at essentially zero pressure. The pressure-voltage relation-

ship was ascertained by calibration against a mercury manometer and

it was found that the output voltage was in direct proportion to the

pressure. Furthermore, the constant of proportionality remained

essentially invariant at 1.216 cmHg/mV over the three years.

The high vacuum side, also known as the " downstream side" consisted

of a buffer volume (V 2) and connections via Cajon couplings to the

Baratron head (MKS. Type 90H-3E). The capacity of the buffer volume

was chosen so as to maintain a discriminatingly small pressure when

compared with the upstream driving pressure; it was found that a half

litre bulb was sufficient to sustain a pressure of less than 1% compared

with the ingoing pressure.

The Baratron is a capacitance manometer with a full scale deflection

-4

of 3 mmHg and a resolution of 10 mmHg. Fundamentally, the design

comprises a metal diaphragm tensioned between two fixed sensor electrodes

maintained at a constant temperature of 50°C. The diaphragm deflects

when a pressure difference is applied across it, changing the capacitance

between the diaphragm and electrode. The change is sensed by a self-

balancing capacitance bridge, directly calibrated in pressure units.

Fig . 4 . 2 (a) . The permeation apparatus and PET microprocessor.

94

In later experiments a microprocessor controlled data acquisition

device was interfaced directly to the BCD output of the Baratron

for the purpose of continual monitoring of pressure,[Figure 4.2(a)].

The diffusion cell (C), also shown diagramatically in Figure 4.3,

was constructed from glass. A piece of copper gauze (CG) was sealed

into the lower section of the cell to form the support for the membrane.

In order to avoid seration of the membrane, a piece of filter paper

(Whatman No. 2) (F), was inserted between the gauze and membrane (ME).

The two halves of the cell were sealed to the membrane with a silicone

based adhesive (Silastic 732 RTV) (S), and left for 24 hours to cure.

Additional reinforcement was provided by an epoxy based sealant (A),

(Araldite Rapid, Ciba Geigy) circumscribing the exterior of the

cups. The assembled cell was then connected to the vacuum line by

butting onto the two glass limbs of the main vacuum line. A tall beaker

of mercury (M) surrounded the cell which formed both an effective seal and

also a good heat transfer medium. The assembly was enshrouded by a

tall copper cylinder and immersed in a constant temperature water bath

(W).

4.2.2. Permeation technique and measurement.

At the start of a series of studies, all the taps were open to the

pumping system and the membrane thoroughly outgassed. The pressure

decay was monitored using the Pirani Cauge, (P) (Edward's Pirani Type

Vacuum Gauge, Model C-5, C-2). Complete outgassing normally took two

to three days at ambient temperature.

95

96

Before the start of a permeation run, the desired bath temperature

was set using the mercury contact thermometer. The two, 200 W

immersion heaters, working in conjunction with a contact thermometer,

through a mercury relay, was capable of maintaining the temperature

to within 0.1°C. The diffusion cell was allowed to equilibrate for a

minimum of 48 hours at each new temperature setting.

Before each run, the leakage on the " downstream side" was

monitored by closing taps T 7 and T i 0 , wibh the remaining taps still

open to the pumping system. A leak-rate of less than j % of the total

expected flow-rate was considered to be tolerable. At the start of

the run, taps T 2 and Tz, were closed, and a charge of gas admitted from

the reservoir (R) into the buffer volume (Vi); the pressure of gas was

pre-adjusted according to the output on the digital voltmeter. Taps

T 5 , T 7 and T x o were then shut, Tz, opened smartly and the starting time

(t = 0) was noted. The output on the Baratron was followed as a function

of time, and readings were acquired at regular time intervals in both

the steady and transient state.

At the end of a run, the propane was frozen back into the gas

reservoir, first from the downstream side, then followed by the upstream

side. Subsequently, the membrane was outgassed for a minimum period of

ten-times the estimated time lag before proceeding with the next run.

A sample calculation of the results is presented in the Appendix.

= = = T 1

To Vacuum Pump R

PT

CW

WB

98

4.3. Sorption.

4.3.1. The sorption hardware.

The vacuum line and the balance assembly is shown schematically in

Figure 4.4. The pumping system, gas reservoir (R) , Pirani Gauge (P),

and the pressure transducer (PT) were in essence, similar in construction

to those described in Section 4.2.

The balance (Sartorius Electronic Microbalance 4102) operates on

the principle of automatic torque compensation, of which the important

features of the beam and the coils are shown in Figure 4.5. The balance

was mounted on wall brackets in an enclosed cabinet for the purpose of

zero-point stability and temperature constancy. The tabular quartz

beam (B) is always maintained in a null position, and as soon as the

beam is displaced from the horizontal, an error signal is induced in

the rotating coil (RC) from the oscillator coils (OC). This is an

analogue AC voltage of which the amplitude and phase are functions of

the load. The error signal is rectified and regulated in the amplifier

and finally fed back into the coil to provide the counter-torque.

At the extremes of the beam, two aluminium stirrups rest on diamond

pins; platinum hangdown wires were attached to the underside of these

stirrups, onto which the sample (S) and counter-weight (CW) were affixed.

Two Pyrex hangdown tubes, which were coated on the internal surface with

stannous oxide were connected to steel cones with silicone grease to

complete the vacuum seal.

The length of the hangdown tubes were immersed in a water bath (WB)

and temperature control was provided by a Churchill thermocirculator.

Above ambient temperature, a conventional " Churchill" was employed!

100

below this, the refridgerator model, primed with " Blucol" antifreeze

was used. In both cases, the temperature could be controlled to

within 0.2°C of the desired. Air temperature was maintained with three,

500W strip—heating elements (]{) (Bray Chromalox^ which could be operated

independently. A mercury contact thermometer operating in series with

an electronic relay (Gallenkamp, FFP-700-C) was used as control.

Two fans (Fi, F 2 ) were mounted at right angles to each other provided

adequate air circulation.

A.3.2. Sorption technique and measurement.

The sample (S) was first weighed in air then suspended from the

right hand hangdown wire. A target weight of approximately 0.3 g was

normally found to be adequate for the study. With the poly(methyl

methacrvl ate) sheet, it was necessary to coil the sheet around copper

spacers in order to attain a sufficient weight of sample. In the case

of the powder, a thin silica bucket was specially constructed and a

piece of aluminium foil was used as a lid preventing agitation of the

powder from the sudden in-rush of gas. A copper counter-weight was

suspended from the left-hand hangdown wire.

The system was evacuated and allowed to attain thermal equilibrium.

It was found that a steady drift in the zero point of the balance was

observed if the air and bath temperatures were not correctly matched.

Also, this phenomenon was accentuated with samples of larger surface

areas. In all cases, however, the drift could be almost eliminated by

regulating the temperature gradient between the bath and the surrounding

enclosure. It must he emphasised that any attempt to eradicate drifting

was only made when the sample was believed to be completely outgassed;

otherwise this may lead to erroneous results.

101

The leak rate into the system was measured by closing T 2 and

monitoring the accumulation in pressure using the Pirani Cauge (P).

When a vacuum of less than 10 3 mmHg was acheived, tap T 2 was closed

and a charge of gas admitted into the balance chamber. The weight

gain as a function of time was registered on a x-t pen recorder

(Bryans 28000) and the pressure was measured on a pressure transducer

(Bell and Howell 4-327-0003). Equilibrium was reached when no further

increase in weight was observed over a protracted period. At this point,

the gas was either frozen back into the gas reservoir, or alternatively,

a further charge of gas admitted into the chamber and another equilibrium

sorption measured. The latter, interval sorption technique, was normally

preferred for the purpose of economy of time. A sample caclulation is

presented in the Appendix.

102

CHAPTER FIVE

PHYSICAL PROPERTIES OF THE POLYMERS.

5.1. Introduction.

Some fundamental physical properties of the graft copolymers are

presented in this chapter! these include the glass transition temperatures,

densities, dynamic moduli and stereotacticities. Also included are

some selected electron micrographs which illustrate the distribution

of the dispersed phase in the graft copolymers.

The examination of the physical properties of copolymers forms a

major pre-requisite to the study of gas diffusion, especially if the

intention of the study is an examination of the polymer morphology.

For example, glass transition temperatures indicate whether complete

phase separation is present; two distinct T s in a binary graft

demonstrate the clear presence of two independent phases. Density

measurements may also demonstrate the compatibility of the components

which constitute the copolymer system. In the simplest case, a linear

relationship between the density and volume fraction should exist

(assuming additivity of volume). Deviations from this, have been

interpreted by other investigators as an indication of mixing (71),

and anomalies at interfacial phase boundaries (114, 133).

A cursive study on the effect of grafting and the subsequent post-

irradiation solvent treatment on the physical properties of the copolymers

was undertaken. It was found that the graft copolymers which were

extracted in petroleum ether were more pliable than the corresponding

103

acetone extracted samples. In order to demonstrate this quantitatively,

the elastic moduli of the samples are presented; it was beyond the

scope of the present study to examine the results in terms of standard

models, though this does not inhibit in any way the appreciation of

the differences between the petroleum ether and acetone extracted samples.

5.1.1. Glass transition temperatures.

Glass transition temperatures were measured by both differential

scanning calorimetry and mechanical loss. The former results were

obtained on a DuPont 990 Thermal Analyser, operating at a heating rate

of 10°C/min and are tabulated in colums 2 and 3 of Table 5.1. Dynamic

mechanical testing was performed on a Toyo Baldwin Rheovibron and

the results are presented in columns A and 5 of the same table.

The presence of separated phases in the graft copolymers was

manifested by two discernible glass transition temperatures, corresponding

to PDMS (-126°C) and PMMA (120°C) as measured by DSC. The higher

values, of approximately 20 centigrade degrees across the board, as

determined by dynamic mechanical measurements are a consequence of

the higher operating frequency of 11 Hz.

The scatter in the values of the T^ corresponding to the PMMA phase

as measured by DSC is felt to be a reflection of the accuracy of the

technique rather than any true measure of mixing between the phases.

It was found that even at the highest percentage of grafting, the

magnitude of the measured signal corresponding to the dispersed phase

104

TABLE 5.1. Glass Transition Temperatures (°C).

SAMPLE DSC RIIEOVIBRON

T T T T E g g I

(low) (high) (low) (high)

PDMS -126 -114

PDMS-•g-10.35% PMMA -124 100-200 -113 -

PDMS-•g-24.14% PMMA* -124 105-110 -114 -

PDMS- g-30.47% PMMA -124 103-114 -113 142

PDMS-•g-32.28% PMMA* -121 105-110 -111 144

PDMS--g-39.76% PMMA -125 105-120 -113 144

PDMS--g-46.34% PMMA -124 105-115 -114 138

PDMS-•g-51.46% PMMA -124 105-115 -116 141

PMMA (Sheet) 107-119 130

PMMA (Powder) 119 -

TABLE 5.2. Stereotacticities of PMMA

Method of Preparation 1% H% S%

y-Radiation 6 30 64

Emulsion Polymerisation 8 33 60

Commercial 12 30 58

105

was considerably smaller than that of the continuous phase, when it

would be intuitively expected that the signals be at least comparable

in size. This made the measurements of the graft copolymers at lower

PMMA content difficult, and the interpretation of the thermograms are

likely to be subject to larger uncertainties. Dynamic mechanical

testing did not appear to suffer from this limitation, though severe

softening of the sample at high temperatures, did not allow for

easy testing.

5.1.2. Densities.

The relative densities of PDMS and the graft copolymers were

measured by the method of water displacement; the accuracy of the

results is within 1 part in 1000. The density of the pure PMMA was

calculated from the absolute weight of a uniform sheet of pre-determined

cross-sectional area and thickness. The results are illustrated in

Figure 5.1.

5.1.3. Dynamic mechanical testing.

The Toyo-Baldwin Rheovibron was also used to determine the dynamic

modulus (G 1) of the graft copolymers and PDMS at 25°C. The value of

G' was estimated from the expression

2L 9 G' = rrr-r . cos 6. 10 Dynes/cm 2 (5.1)

t Un

where L is the length of the sample

F the amplitude factor

/[ the c r o s s - s e c t i o n a l area

D the dynamic force «

and 6 the phase lag

1.18

1.18 4-

1.14 4-

1.12 4-

- ^ I . i o 4-o

i i

so

CO 1.08 4-s 0)

1.04

1.02 4-

1.00

.08

* Extraction in Petroleum Ether

Y Extraction in Acetone

.0 .1 .2 .3 .4

Volume Fraction PMMA

FIG 5.1 DENSITY OF GRAFT C O P O L Y M E R S

107

9.20 +

8.90 +

8.60 4-

O ^qe.so

^ 8 . 0 0

O *

o I g 7 . 7 0

7.40 +

7.10 +

6.80 -f

MOONEY ( K = 1 )

MOONEY ( K = 2 ) t

SMALL^OOD AND GU

/ y

/ v

/ EINSTEIN

t /x

H

- . 0 .1 .3 .4 .5 .6 .7


FIG 5.2 ELASTIC MODULI OF GRAFT COPOLYMERS

108

The results are presented in Figure 5.2 and the included curves

were derived from some standard models of moduli vs. volume fraction

relationships (172).

5.1.4. Thickness of the graft copolymers.

The absolute thickness of the samples was . measured using a

standard micrometer (Moore and Wright) reading to 1 part in 100.

A total of 10 measurements were determined at random positions

across the sheet and statistically averaged', the variance was never

more than 1%.

5.1.5. Stereotacticities of PMMA samples.

The stereogularity of PMMA prepared by emulsion polymerisation,

y-radiation and the commercially obtained was checked by nuclear

magnetic resonance (Bruker 250 MHz High Resolution NMR Spectrometer).

The ratios of the Isotactic (I): Syndiotactic (S): Heterotactic (H) content

were estimated from the relative intensities of the interacting diads

(173, 188).

The grafted PMMA was prepared for NMR analysis by exhaustive

extraction in chloroform. It was found that prolonged exposure (100

hours) of the graft copolymer to hot chloroform caused severe swelling

and eventual rupture of the crosslinking. The bulk of the chloroform

was removed under vacuum and finally the PMMA was precipitated in 10

fold of methanol. The results of the diad analysis are presented in

Table 5.2.

109

5.1.6. Electron Microscopy.

o

Thin sections 800A) of the graft copolymer were prepared by

ultra-microtoming at -140°C. Specimens were cut from both the

longitudinal (L/S) and cross sectional (C/S) directions, and examined

on a Joel JEM-100B electron microscope

The electron micrographs are presented in Figures 5.4(a)-(d).

The contrast in the electron micrographs is a result of the scattering

of electrons by the heavier Si atom in PDMS; the lighter regions

correspond to PMMA.

5.2. DISCUSSION.

The grafting of vinyl monomers onto preformed rubber matrices, by

the method of y-radiation has been examined on many occasions (174).

In particular, methyl methacrylate and styrene have been studied more

extensively (175-179) primarily because of their fundamental importance

in industry.

Alexander, Charlesby and Ross (180), found that under high energy

radiation, PMMA degrades. A similar observation was reported by

Thompson (181). Gardner and Toosi (182) found that changes in the

refractive index and density were induced in PMMA when the polymer was

subjected to radiation influences. As a consequence of this, only

the minimal dosage was used in the preparation of the present series

of graft copolymers. A similiar procedure was adopted by Barrie and

Rogers (183) in their pioneering work on PDMS - g - PMMA graft

copolymers.

110

Since much of the literature on graft copolymers concentrated

on the effects of radi.-ition dosage on grafting, it was felt to be

more informative to study one aspect of post irradiation. . treatment.

The systematic study of solvent extraction was chosen, and altogether,

nine solvents were examined. The results are summarised briefly in

Table 5.3.

TABLE 5.3. Summary of solvent extraction on graft copolymers.

Solvent e b.p. Comments

PET ETHER/40'. 60 1.8 69 No effect

xylene 2.0 138 very slight toughening

benzene 2.3 80 slight whitening

toluene 2.4 111 slight whitening

chloroform 4.8 61 severe swelling

ethyl acetate 6.0 77 severe swelling

MEK 18.5 80 toughening

acetone 20.7 56 severe toughening

It was found that the solvents used in the extraction process

induced interesting morphological changes in the samples'. In Table

5.3, columns 2 and 3 list the relative dielectric constants (e) and

the boiling points (b.p.) of the solvents used respectively, and column

4 summarises briefly the effect of the solvent treatment on the samples.

Ill

Two of the solvents at the extremes of Table 5.3, namely acetone and

petroleum ether, were selected for a more rigorous study. These two

solvents were the complete antithesis of each other, with respect to

their solvation and swelling powers for the individual phases

comprising the graft copolymer. Petroleum ether was found to be a

good solvent for the silicone rubber, but a virtual non-solvent for

PMMA. Acetone, however, is a powerful solvent for PMMA but a comparatively

weak solvent for PDMS.

One of the parameters which appeared to be affected significantly

was" the thicknesses of the graft copolymers. The post-irradiation

extraction in acetone tended to partially annul the isotropic swelling

which accompanied grafting, though extraction in petroleum had no

measurable effect. The results are presented diagramatically in Figure

5.3. The solid line represents the least-squares constrained fit of

the data for the petroleum extracted samples and is given by

A theoretical relationship between the relative increase in

6 £

thickness ( — a n d the volume fraction of dispersed phase based

on isotropic swelling is included below. Although the proof is

straightforward, it is nevertheless presented here for completeness.

6 t = 0.4 <f>. € D

(5.2)

Consider a cubic expansion from (V. = €3) to V = (€ + 6 0 f

3 (5.3)

112


FIG 5.3 RELATIVE THICKNESS OF GRAFT COPOLYMERS

113

The net increase in volume is simply

Sv = (V f - V±) = (/ + 6/) 3 - / 3 (5.4)

Taking the first term of the binomial expansion and dividing through

by v f gives

Sv 35/ V^ — ~F~ ( 5 ' 5 )

By defini tion

* = ^ (5.6) D V . K '

Hence,

-0.3* (5.7)

The theoretical prediction of = 0-3 is in good agreement

with the experimental value of 0.4 calculated from the petroleum ether

treated samples. Extraction in acetone tended to restrain the expansion

in the thickness of the sample, though whether this was compensated for

in the plane of the membrane was not directly verifiable.

The results of dynamic mechanical testing are shown schematically

in Figure 5.2. The dynamic moduli, G', were measured in the x-y plane

and were used without any further corrections. A discernible trend was

found amongst the petroleum ether extracted samples, with G' increasing

monotonically with increasing volume fraction of PMMA", the results also

appeared to agree well with the Mooney model. The values of G* for the

graft copol ymcrs extracted in acetone deviated from this simple relation-

ship and their greater stiffnesses • are reflected in their much larger

G' values.

114

The effect of temperature on the stereoregularity of PMMA prepared

by free radical polymerisation was first reported by Fox in 1956 (185).

Later studies (184,186) suggested that the syndiotactic form was the

more thermodynamically favoured configuration so the '.probability that

neighbouring asymmetric carbon atoms are in the syndiotactic placement

increases as the temperature of polymerisation is reduced. The present

results are in agreement with this.

The effects of solvent treatment in block copolymers have been

reviewed by Dawkins (187). A direct comparison between block copolymers

and the graft copolymers prepared in the current study, however, should,

be treated with caution. The silicone rubber which formed the elasto-

meric matrix was lightly crosslinked and pliase separation would therefore

be considered a forbidden process. However, it is still conceivable

that solvents which show preferential dissolution in either phase may

induce structural changes in the graft. Merret (189) discussed the

effect of benzene and hexane extraction on natural rubber grafted with

PMMA. The results suggested that benzene treatment extended the PMMA

chains, conferring to the graft copolymer a greater tensile strength;

hexane extraction produced a soft flabby material. Kantz and Huang

(132) also found that post irradiation solvent treatment on their

poly(ethylene-g-styrene) graft copolymers induced interesting morphological

changes; the films washed in methanol were found to be brittle, whilst

the benzene extracted films remained flexible. However, their process

appeared to be reversible, and flexibility of the embrittled samples

could be re-introduced by simply immersing in benzene. This was not

found with the present PDMS-g-PMMA copolymers.

Fig. 5.4(a): PDMS- g- 10.35% PMMA Fig. 5.4(b): PDMS- g- 51.45% PMMA

Fig . 5. 4 (c): PDMS- g- 24.14% PMMA (L/S) Fig. 5 • 4 (d) : P DMS- g - 2 4 . 1 4% PMMA ( C / S )

116

The tendency for discrepancies between the observed and predicted

values for the densities of heterogeneous polymers have been reported

on many occasions. It should be pointed out that close agreement

between theory and experiment depends greatly upon the accuracy in

determining the densities of the two pure components. The results

presented in Figure 5.1. suggest that although there is a tacit

agreement with a simple two phase model, the lower densities of the

acetone extracted samples cannot be ruled out completely.

These preliminary results tend to suggest that extraction of the

graft copolymers in petroleum ether retains the morphology in which the

PMMA exists as submicron sized spherical domains within a silicone rubber

continuum. Extraction in acetone appears to structure the copolymer

with some preferred orientation in the x-y plane. The precise nature

of the structuring is not obvious though a morphology corresponding

to the PMMA existing as.lamella type strands within PDMS is suggested

by the electron-micrographs. Crystallisation of the PMMA may be ruled

out since only stereoregular forms of PMMA show any evidence of

crystallinity*, the diad analysis for the PMMA extracted from the graft

copolymer suggest that although there are a large proportion of sydiotactic

diads, crystallisation is still unlikely.

117

CHAPTER SIX

RESULTS AND DISCUSSION! PDMS

6.1. Introducti on.

Poly(dimethylsiloxane), PDMS, is a synthetic elastomer and was

above its T at the temperatures investigated. The sorption and 8

diffusion of simple gases and vapours at low activities was expected

to be ideal with both Fick's equation of diffusion and Henry's Law

obeyed.

6.1.1. PDMS/Propane Equilibrium Sorption.

V

The equilibrium sorption isotherms for propane in a PDMS sheet

were measured at five temperatures in the range 30° to 50°C. The

isotherms as shown in Figure 6.1 were all linear from which the Henry's

Law solubility coefficients, k g , were determined by the method of least-

squares regression, constrained to intercept the origin. The values of k are tabulated in column 4 of Table 6.1.

s

The temperature dependence of k was found to be amenable to van't Hoff's

equation from which the heat of dissolution, AH^, was estimated to be

-16.9 kJ mol . The heat of dissolution in polymers may be interpreted

as a composite quantity comprising the heat of condensation of the

vapour (AH^) and the enthalpy of mixing between the sorbate and the polymer

( A H J , viz,

AHd = ahc + aHm (6.1)

117

CD UD

• 30.0 C

O 35.0 C

A 40.0 C

O 45.0 C

O 50.0 C

^ . 0 0 8.00 16.00 24.00 32.00 Pressure / (cm.Hg)

40.00

FIG 6.1

PDMS / PROPANE SORPTION ISOTHERMS

3.06 3.12 3.18 3.24 3.30 3.36 3.42

Temperature (1 /T)/10~3IC1

FIG 6.2 SILICONE RUBBER VAN'T HOFF PLOTS

120

The close agreement between AH^ found for the present system, and

AH = -20.2 kJ mol ^ for the heat of condensation of propane suggests u

that AH is almost negligible and therefore an absence of any significant M

gas-polymer interactions.

6.1.2. Permeabilities and diffusion coefficients.

The permeabilities of propane in PDMS were measured within the

temperature range 25° to 50°C using the high vacuum permeation technique

described in Section 4.2.1. The permeabilities, P, were found to be

largely independent of pressure, although slight deformation of the

membrane at elevated pressures caused apparent increases in P at all

the temperatures investigated. The values of P are tabulated in column

2 of Table 6.1.

The temperature dependence of P was analysed in terms of equation

(1.9) giving a good linear plot as depicted in Figure 6.3. The quasi-

activation energy of permeation was estimated from the gradient of the

best fit line to be -3.3 kJ mol

Due to the thinness of the sheet (^ 1 mm), the kinetics of sorption

were not attempted at any of the temperatures', the period for the half time

sorption was estimated to be only of the order of 1 minute at 30°C,

and the inherent errors were therefore expected to be severe. Diffusion

coefficients were however, measured by the " time-lag" method and also

from the coupled steady state permeabilities and equilibrium sorption

Isotherms. The background theory for these have been presented in

Sections 2.1.1 and 2.1.3. The diffusion coefficients in both cases were

found to be independent of concentration and are presented in columns

3 and 6 of Table 6.1. The time lag diffusion coefficients are denoted by

D . and the steady state diffusion coefficients are represented by D .

121

Temperature (1 /T)/1 O^KT1

FIG 6.3 SILICONE RUBBER TEMPERATURE DEPENDENCE OF P

122

Temperature (1/T)/10~3IC1

FIG 6.4 SILICONE RUBBER TEMPERATURE DEPENDENCE OF D

123

The diffusion coefficients estimated by the " time-lag" method

were found to be consistently lower than the corresponding steady state

diffusion coefficients. It is also interesting to note that the discrepancies

between each corresponding pair of diffusion coefficients are remarkably

constant at This would suggest that a systematic error was

operative rather than random experimental scatter. Barrer, Barrie and

Rogers (190) have discussed the possible errors involved due to the

" edge effect" for the transient state flow through a clamped membrane

of thickness, in a diffusion cell of effective radius, a. They

arrived at a critical value of //a £ 0.2 for the " edge effect" to be

safely neglected. The value of //a for the present study was estimated

to be 3/ 0.08 whi ch is considerably smaller than the value proposed by

Barrer et al. This would suggest that the " edge-effect" plays a more

important role than that first reported by Barrer, Barrie and Rogers.

Activation energies of diffusion were estimated from the standard

Arrhenius relationship given in equation (1.6). E^ was estimated for

both the D and D . values, and in both cases good linear plots were found s o

as shown in Figure 6.4. E^ was calculated to be 13.9 kJ mol ^ from D

D s and 13.8 kJ mol ^ from D^.

6.2. Review of PDMS/Propane Studies.

The study of the transport of propane in silicone rubber has been

reported on many occasions (71,92,191-193), and it was felt that a critical

collation of the present results with the results by these other investiga-

tors may prove informative. It should be noted that the method of

preparation of the PDMS sheet in eacli case was found to be similar, with

124

TABLE 6.1. Sorption.and Transport Parameters: Propane

o

a, E QJ H

PL, o j=L

Q

O Pl —' or.

e e .. u

oo

PL iH H O CO D,

L—' Mi Fl E (0 E Pu E

CJ CJ u ©

vO

25.0

30.0

35.0

40.0

45.0

50.0

7.64

7.40

7.26

7.04

6.96

6.92

6.07

6.60

7.26

7.81

8.55

9.36

0.101

0.090

0.0813

0.074

0.0668

0.126

0.112

0.099

0.089

0 . 0 8 0

0.074

7.33

8.07

8.66

9.35

10.42

E = -3.3 kJ mol

E = 1 3 . 9 kJ mol

AH d = 1 6 . 9 kJ mol

-1

-1

-1

with the exception of Ismail (191) (which contained 5% Varox antioxidant).

The literature results are tabulated in Table 6.2; the notation of 6 and

S to denote whether the time lag or steady state techniques respectively

were used in determining the transport parameters will be retained.

Although the solubilities are in good agreement amongst the various

investigators, the diffusion coefficients show a spread of 11% and the

permeabilities indicate a maximum variation of over 30%. The discrepancies

125

TABLE 6.2. PDMS/Propane Transport Parameters at 30 C.

QJ O Pe P o CO

bO EC B o

e 0 1 P-. o

Q

I

£ H O PL

to L w B B o o CD

/—N

PL rH H O CO PL to en N EC B B B u u CJ

CD

Expt.

Ismail

Munday

Webb

Alexopoulos

7.40

7.00

6.59

6.66

5.49

6.60

6.76

5.96

5.46

6.42

0.112

0.103

0.111

0.122

0.085

7. 33

6.34

0.101

0.104

in the diffusion coefficients are not felt to be significantly large and

may be attributed to subtle variations in the degree of crosslinking in

each of the samples. Barrer and Skirrow (90) and later Aitken and Barrer

(194) showed that the diffusion coefficients of gases in natural rubber

were reduced as the degree of crosslinking increased. It was believed

that the mobility of the polymer segments was hindered, through the

centres of crosslinking, impeding the free movement of the penetrant

molecule. It was further shown that the equilibrium solubilities were

largely unaffected and the permeability would therefore be expected to

decrease in the same manner as the diffusion coefficient. The present

results are however, at variance with this assertion since the spread

in the permeabilities is considerably larger than that found with the

diffusion coefficients.

126

TABLE 6.3. Activation Energies and Heats of Dissolution.

Source Ref. Ep/kJ mol" 1 E^/kJ mol 1 AH /kJ m o l " 1

D (E - E j / k J mol 1

P D

Expt. -3.3 13.9 -16.9 -17.2

Ismail 191 -3.3 13.3 -16.6

Webb 192 -1.4

Munday 92 -2.9 12.0 -17.0 -14.9

Alexopoulos 193 -3.9 12.9 -16.8

The activation energies of diffusion, E^, and permeation, Ep, together

with the heats of sorption, AH^ are given in Table 6.3. Ep and E^ were

found to be in excellent agreement despite the large disagreements found

in the actual values of P and D. Since both E p and E^ are only relative

quantities, any systematic errors accrued in the determination of membrane

thicknesses and" areas are likely to be eliminated through the mutual

cancellation of errors. The heats of sorption were also found to be in

excellent agreement and differed by no more than 4%.

6.3. PDMS/Ethane.

A cursory examination of the transport of ethane in a PDMS sheet was

conducted. It was felt that since the transport parameters for this

gas have never been reported,the results may be of some benefit for

future investigations. Furthermore, the transport of ethane in poly-

(methyl mothacryl ate) is reported later in this treatise, and a comparison

between the dissolution constants in the rubber and the glass may be of

some academic interest.

127

The permeability, P, diffusion coefficient D^, and solubility,

k , were determined using the high vacuum permeation technique. P, 6

D_ and kft were all independent of pressure as found with propane and 0 0

the results are summarised in Table 6.4.

The temperature dependences of D Q , P and k Q were determined using

equations (1.16), (1.19) and (1.17) respectively. These are also

illustrated in Figures 6.2, 6.3, and 6 . 4 , respectively.

TABLE 6.4. Sorption and Transport Parameters: Ethane.

Temp./

°C 1 0 7 - cm 3 (STP) cm

cm s cmHg 1 0 6 D Q / c m

2 s 1

b

25.0 4.08 9.93

30.0 4.09 10.77

35.0 4.10 11.57

40.0 4.12 12.60

45.0 4.13 13.53

50.0 4.16 14.67

k./• c m 3 (STP)

G cm 3 (polymer)cmHg

0.0411

0.0379

0.0354

0.0325

0.0304

0.0283

E p = -0.6 kJ m o l "1 , E d = 12.6 kJ m o l " 1 , A H p = -12.1 kJ mol 1

It is interesting to note that again, A H p , for ethane is in fairly

good agreement with the heat of condensation of the vapour (AH^, = -15.7).

It is also interesting to find the activation energies of diffusion for

both ethane and propane to be uncannily constant. This would indicate

that the mechanism for diffusion of these two penetrants in PDMS is similar

128

It may be inferred from this, that these elongated penetrants diffuse

through the polymer medium with their longitudinal axes oriented in

the direction of diffusion.

129

CHAPTER SEVEN

RESULTS AND DISCUSSION: PMMA

7.1. Introduction.

The sorption of four simple hydrocarbon vapours in poly(methyl

methacrylate) was studied in the temperature range of 1°C to 60°C. The

diffusion coefficients for the four paraffins at 30°C, spanned almost

a million fold, with methane the smallest of the penetrants exhibiting

- 1 0 • - 1 the largest diffusion coefficient of 6 x 10 cm 2s and iso-butane

-17 -1

the lowest at 4 x 10 cm 2s . In the case of propane and iso-butane,

the time for equilibrium sorption in a 30pm sheet was estimated to be

in the order of years. For this reason, sub-micron sized spheres were

prepared by the method of emulsion polymerisation, and in this way,

the effective " thickness" of the sample was reduced by several orders

of magnitude to bring the experimental times within tangible limits.

Within the temperature range studied, PMMA was well below its glass

transition temperature, and the dual-mode sorption phenomenon, usually

associated with glassy polymers was expected.

The results in this chapter are divided under general headings

corresponding to the various gases studied, namely, methane, ethane,

propane and iso-butane. In each section, the equilibrium and transient

state sorption results are presented and discussed in terms of dual-mode

sorption. Also included are some correlations which attempt to relate

the sorption and transport parameters of various penetrants in PMMA to

the fundamental properties of the gases and vapours.

130

7.1.1. PMMA/Methane.

The equilibrium sorption isotherms for methane in PMMA were measured

at three temperatures within the pressure range 0 to 40 cmHg. The

isotherms were linear as depicted in Figure 7.1, and according to the

dual-mode sorption theory, the effective solubility is given by

S L = c/p = k D + c^b (2.56)

The limiting solubility S T was computed by the method of least-squares li

regression, constrained to intercept the origin. Values of S^ are given

in column 2 of Table 7.1. The temperature dependence of S was obtained Lj

from the van't Hoff relationship as given in equation (2.68), and is also

illustrated in Figure 7.15. The limiting heat of sorption,(AH ) was j c— u

calculated to be -25.6 kJ mol

The limiting heat of sorption may be interpreted as a composite

quantity, comprising the heat of dissolution, AH^, and the energy

expended in the hole-filling process. The heat of dissolution, AH^,

is normally found to be comparable to AH , the heat of condensation

(in this case, AH = -8.9 kJ mol 1 for methane). Thus, the considerably VJ

more exothermic value of (AH ) for methane in PMMA suggests that the o c— u

main contribution is the energy associated with the hole-filling process.

It is also interesting to compare the present value of (AH C) for b c— u

methane in PMMA with the same gas in polystyrene (-18 kJ mol as given

by Munday (92). The two values are in fair agreement considering the

much greater errors associated with the latter since the isotherm was

obtained at two temperatures only.

131

oo

Pressure / (crruHg)

F I G 7 . 1

PMMA ( S H E E T ) / METHANE SORPTION ISOTHERMS

132

TABLE 7.1. Sorption and Diffusion Parameters: Methane.

Temp/ °C

1 Q 3 n j cm 3 (STP)

L cm 3 (polymer) cmHg 1 0 1 0 D/cm 2 s 1

30.0 4.4 5.8

35.0 3.8 7.5

40.0 3.2 10.8

(AH ) _ = -25.6 kJ mol 1 ; E^ = 49.2 kJ mol 1

S c = 0 D

The diffusion coefficients for methane in PMMA were determined

from sorption kinetics in the temperature range 30° to 40°C. No

apparent trend was found in the diffusion coefficients with respect

to concentration as shown in Figure 7.2 so the arithmetic mean of the

experimentally measured diffusion coefficients at each temperature were

calculated and tabulated in column 3 of Table 7.1.

According to the diffusion model for dual-mode sorption (26)

the diffusion coefficient in the limit of zero concentration is given

by the expression

( D ) c = 0 " W S L

Both k and S are independent of concentration and D is normally also IJ Lt L)

expected to be constant. The present results are therefore consistent

with this. The temperature dependence of (D) _ was obtained from the 0=0

133

11.4 4-

10.6 +

9.8 4-

co »» „ 9.0

o

^ 8.2 V-

o v-

7.4 4-

6.6 4-

5.8 +

40°C

35°C

30°C

5.0 1 1 1 i i i i l.

.03 .06 .09 j i i i l

.12 .15

Concentration C/(cc(stp)/cc(polymer)

FIG 7.2 PMMA/METHANE CONCENTRATION DEPENDENCE OF D

134

Arrhenius relationship given in equation (1.6) and is also illustrated

in Figure 7 . 1 6 ; ( E D ) c _ Q was estimated to be 49.2 kJ mol

7.1.2. PMMA/Ethane.

The sorption of ethane in PMMA was determined for both powder and

sheet. It was found that the equilibrium sorption isotherms for the

sheet weire linear whereas the powder isotherms were characteristic of

dual-mode sorption. Furthermore, the levels of sorption in the sheet

were significantly lower than the corresponding isotherms for the powder.

The isotherms which showed pronounced deviation from Henry's Law were

analysed in terms of the dual-mode expression using the method of non-

linear regression analysis suggested by Barrie et al. (68). The

rudiments of the techniques have been outlined previously and will not be

discussed further.

The equilibrium sorption isotherms are illustrated in Figures 7.3 and

7.4 for the sheet and powder samples respectively. The experimentally

determined solubilities are represented by the various symbols and the

'best-fit' curves are shown by the solid lines. The dashed line in

Figure 7.4 depicts the high pressure asymptote for the isotherm obtained

at the lowest temperature; it corresponds to the theoretical expression

given in equation (2.57). It is readily apparent that despite the

deceptive linearity of the isotherms at elevated pressures, the true

limiting condition in the region of high pressure was never completely

attained in the present study. This in fact illustrates one of the main

inadequacies of the graphical technique proposed by Vieth et al. (51)

which utilises the limiting high pressure gradient to estimate k .

135

Pressure / (crruHg)

fig 7.3

PMMA (SHEET) / ETHANE SORPTION ISOTHERMS

136

^ . 0 0 8.00 16.00 24.00 32.00 Pressure / (cm.Hg)

40.00 48.00

F I D 7 . 4

PMMA (POWDER) / ETHANE SORPTION ISOTHERMS

137

TABLE 7.2. Powder and Sheet Sorption Parameters*. Ethane.

SHEET POWDER

cj

a 6 (u h

1.0

10.0

20.0

30.0 0. 0180

35.0 0. 0150

40.0 0. 0121

50.0

toO EC G

PU RH H O to a. ^—' ^ PI PI

E E o u

T J

00 EC 6 o ^ ih o a

0.0384

0.0316

0.0227

0.0190

0.0172

-1 ,a—

- EC u

1.23

0.903

0.826

0.560

0. 392

toO ec 6 u

0.0879

0.0730

0.0455

0.0396

0 . 0 2 8 0

toO EC E O

S EH

RH H O CO P. v • S—'

PI PI E E O O

T J

0.147

0.0975

0.0603

0.0412

0.0282

0.0181

( A H g ) c = 0 = -31.4 kJ mol , (Ah > = -15.5 kJ mol "", (All ) = -21.1 kJ mol -1

( A H s ) c = 0 = -32.5 kJ mol -1

The sorption parameters which describe the isotherms are presented

in Table 7.2. It is interesting to note that the two temperatures, 30° C

and 40°C where the isotherms are available for both the sheet and powder,

the limiting solubility, S^, for the sheet is in close agreement with the

Henry's Law dissolution constant, k^, for the powder. It would appear that

the lower sorption encountered with the sheet is mainly at the expense of

the Langmuir component to the total sorption. It should be appreciated that

the Langmuir component associated with adsorption in mic.rovoids is

not a fixed propertv of the polymer and is strongly dependent upon the

138

prior history of the sample. It has been demonstrated that the

Langmuir component may be reduced by annealing the polymer (70,75)

and increased through dilatometric sorption (78) using gases such as

C0 2 at elevated pressures. However, the complete abscence of micro-

voids in glassy polymers under normal conditions has never yet

been reported.

The noticeable discrepancies between the equilibrium sorption in

polymer powder and sheet have been discussed by various investigators.

Clearly, the specific surface area of the powder is greater than the

corresponding sample in sheet form. Nevertheless, the increased sorption

found with powders cannot be altogether accounted for in terms of surface

sorption effects alone. Allen et al. (79) found that the sorption of

C0 2 in poly(acrylonitrile) powder was considerably larger than the

sorption in PAN sheet. Munday (92) also found pronounced differences

between propane sorption in polystyrene sheet and powder sample and the

present results are therefore consistent with these earlier findings.

However, it is proposed here that the complete absence of the Langmuir

component to the sorption in the sheet is due to the presence of residual

casting solvent in this membrane. The PMMA sheet was prepared by the

conventional casting techniques using toluene as the solvent. It is

now realised that the diffusion coefficient of this solvent at low

activities is expected to be small ( ^ 1 0 cm 2/s at 30°C) and it is

conceivable that within the time scale of the study, complete removal

of toluene from the sheet may not have been fully effected.

139

Anomalies associated with residual casting solvent in polymer

sheets have been discussed by Koros et al. (83,196). It was shown

that the presence of trace solvent with a high affinity constant, b,

(as would be expected for toluene in PMMA) affects the Langmuir mode

to the total sorption, with the Henry's Law component, remaining

essentially unchanged. Some independent studies were performed in

relation to the present system in which trace quantitites of toluene

were introduced into the powder sample in an attempt to simulate the

conditions found with the sheet. A more detailed discussion of this

is presented in the latter portion of this chapter.

The temperature dependence of k , b and S were evaluated from the

i) l standard van't Hoff expression given in equations (2.61), (2.62), and

(2.68) and are also presented in Figures 7.13 to 7.15 inclusive. The

values of Ah^, Ah.. and (aH ) are presented at the foot of table D H B C —(J

7.2.

There is no simple relationship which exists between AH , AH and D H

(AHg) c_Q due to the pronounced temperature dependence of c R . However,

if c' is assumed to be invariant with temperature, the partial differential H

of equation (2.56) yields simply

^ V c - o = W ^ d + c h b / s l - n i ( 7 - »

It was found in the present study that the magnitudes of k R and the

composite quantity, » a r e almost comparable, whence

( A H S ) C = Q <\, (AH D + AH r)/2 (7.2)

140

It is evident from equation (7.2) that s ^ 0 1 1 ^ between

the limits of A H p and AH^, however, it is normally found that (AHg) c_g

is considerably more exothermic than either AH or AH and this indicates D H

the strong dependence of c' on temperature. H

Diffusion coefficients were measured for both the sheet and powder

by the method of sorption kinetics. In both cases the sorption processes

were well represented by a diffusion coefficient which was largely

independent of concentration as illustrated in Figures 7.5 and 7.6. The

inherent errors in the diffusion coefficients for both samples are likely

to be significant due to the enormously long times encountered with the

sheet (3 2 days for equilibration at 30°C) and at the other extreme,

the dimunitively small times found with the powders (3 1 minute for

equilibration at 30°C). However, in both cases, more accurate kinetics

are expected, when the time scale for equilibration occurs within

reasonable limits.

The diffusion coefficients are presented in Table 7.3 and represent

arithmetical averages of the experimentally measured diffusion coefficients

over the concentration range examined. The Arrhenius plots of the

diffusion coefficients are illustrated in Figure 7.16 and the activation

energies for diffusion are included at the foot of Table 7.3.

It is readily apparent that there are striking differences between

the two samples with respect to the diffusion coefficients and also the

activation energies for diffusion. Tt is unlikely that the errors

mentioned earlier in relation to the times for equilibration could affect

the diffusion coefficients to such a large extent. Tt is believed that

the discrepancies are again due to the presence of residual solvent in

141

7.4 4-

6.6

5.8 4-

co * . 5.0 4-

+ o

3.4 4-

2.6 4-

1.8

1.0 .0 .1

h h

40°C

35°C

30°C

.3 .4 .5

Concentration /(cc(stp/oc(polymer))

FIG 7.5 PMMA (SHEET)/ETHANE CONCENTRATION DEPENDENCE OF D

142

14.5

13.5 - - 30°C

12.5 - -

11.5 - • •

10.5 —

9.5 - -

co * > 8.5 4-

o

7.5 - -

O 6.5 - -

5.5 - -

4.5 —

3.5 - -

20°C

2.5 — 10°C

1.5 - -

£ 1°C

- . 0 0 .30 .60 .90 1.20 1.50 1.80

Cone entration/(cc(stp)/cc(po lymer) )

FIG 7.6 PMMA (POWDER)/ETHANE CONCENTRATION DEPENDENCE OF D

143

TABLE 7.3. Powder and Sheet Diffusion Parameters: Ethane.

SHEET POWDER

Temp./ °C

1 0 1 1 D/cm 2 s 1 1 0 U D/cm 3 s 1

1.0 0.9(9)

10.0 2.2

20.0 4.8

30.0 2.2 12.9

35.0 3.3

40.0 6.1

( V c = o = 8 1 k J m o l _ 1 ( V c = o = 5 7 k J m o 1 1

the s h e e t c a u s i n g a " plasticisation" effect." A more detailed account is

presented in the latter part of this chapter.

7.1.3. PMMA/Propane.

The sorption of propane in PMMA powder was studied at five temperatures

in the range 20°C to 60°C inclusive at 10°C intervals. The isotherms

were all concave to the abscissa, characteristic of dual-mode sorption

normally found with other glassy polymers and are illustrated in

Figure 7.7. The significance of the dashed and solid lines have already

been mentioned in relation to the sorption isotherms for ethane and will

not be discussed further; suffice to say that the agreement between theory

and experiment is good. The best-fit parameters k , r' and b together D H

with the derived quantity, S , are tabulated in Table 7.4. It should be

144

o CO

CO

o CsJ

• 2 0 . 0 C

O 3 0 . 0 C

^ . 0 0 8.00 16.00 2 4 - 0 0 3 2 . 0 0

Pressure / (crruHg) 4 0 . 0 0 4 8 . 0 0

F I O 7 . 7

PMMA (POWDER) / PROPANE SORPTION ISOTHERMS

145

appreciated that the sorption isotherm at 60°C showed only slight

curvature and the " best fit" parameters are susceptible to larger

errors. The derived quantity, S , however, is less prone to analytical li

errors since the two parameters and b occur as a product and the

inherent inaccuracies are likely to be eliminated through the mutual

cancellation of errors.

The temperature dependences of k , b and S are presented in Figures D L

7.13 to 7.15. Good linear plots allowed AH_, AH„ and (AH_) _ to be D H b c=U

measured with accuracy from the van't Hoff expressions given in equations

(2.61), (2.62) and (2.68). Since the dual-mode sorption parameters

at 60°C were felt to be less accurate, they were not utilised in the

analysis of the various heats of sorption.

TABLE 7.4. Dual-Mode Sorption Parameters: Propane.

u

(X n qj h

20.0

30.0

40.0

50.0

60.0

bO X 6 o

pl h co

E u q

0.053,

0.046, i

0.328

0.025,

0.027,

pl h co

- pa u

1.000

0.699

0.548

0.523

0.111

A H p = -19.9 kJ mol -1

b0

1 o x>

0.196

0.200

0.125

0.081

0.176

bO pc E o

pl h co

0.249

0.187

0.101

0.068

0.049

= -35.2 kJ mol -1

AH„ = -24.0 kJ mol H

-1.

7.0

6.0 +

146

t o—4.00 o - U O o"3J00 o-2js0

o-ZOO

o-1jj0

o—1.00

© p l

e

3.0 o~0J»

o-cl23

o-Q.10

e-a06

-1.0 4-

-2.0 2.9 3.0 3.1 3.2 3.3 3.4

Temperature (1 /T)/10"3K~1

FIG 7.8 PMMA (POWDER) / PROPANE ALTERNATIVE SORPTION ISOTHERMS

147

The overall heat of sorption, AH^, is concentration dependent, and

was thus derived from the Clausius-Clapyeron expression given in

equation (2.65). Graphs of vs. at constant concentrations were

constructed from which AH was determined. Typical plots are presented d

J

in Figure 7.8 for eleven concentrations', all the curves indicate a

break in the temperature range 20°C to 30°C. The main feature of

Figure 7.8 indicates that the isochores over the entire temperature range

of 20°C to 60°C cannot be represented by a single straight line, and

furthermore, if the isochores in the high temperature region were

extended to below 30°C, it would suggest an increase in the sorptive

capacity of the polymer in this latter temperature region. A second

order 3 transition has been reported for PMMA in this temperature range

(206) and has been assigned to the motion of the methacrylate side-group.

It is conceivable that below this temperature, the " freezing out" of

the motion of this pendant group creates additional microvoids within

the polymer leading to a proportionally larger microvoid component to

the sorption.

The overall heat of sorption may also be estimated from the analytical

expressions given in equation (2.65) and (2.66). (AH„) as a function of S c

concentration was generated in this manner for both ethane and propane

using the parameters of k^, c. and b as given in Tables 7.2 and 7.4 and

are illustrated in Figure 7.9. Also included in Figure 7.9 is the overall

heat of sorption for methane in PMMA.

It is interesting to note that AH^ for the least condensible of the

three gases, CHi,, shows no concentration dependence, but the magnitude

of the variation with concentration increases in the order CHz. < C 2 H 6 < C 3 H 8 .

148

Cone / cc(stp)/cc(poly)

FIG 7.9 HEATS OF SORPTION FOR PMMA

149

Furthermore, with propane, a small mimimum is observed which is in

common with other investigations (68,70,71,92). The significance of

the minimum is still unclear, although it has been shown that the

apparent temperature dependence of cl may be a contributing factor.

The diffusion coefficients of propane in PMMA were measured from

sorption kinetics using the powder sample only. Attempts to study

kinetics for the plane sheet were not successful owing to the low

diffusion coefficients which extended the time scale of

/ T well

beyond the range of practical limits.

The experimentally measured diffusion coefficients, D, were found

to be a weak function of concentration and are presented in Figures

7.10(a) and 7.10(b), as the solid symbols. According to the dual-mode

sorption theory, D may be expressed in analytical form as follows.

k pD c

d = , t—lu u n = — • k (2.81) D P C H b P / ( 1 + b p ) c 0

The value of D^, which represents the diffusion coefficient of the

dissolved species was estimated from each D using the appropriate values

of k , C', b and p. D was found to be largely independent of concentration d h d

which lends support to the total immobilisation model. Values of D^ and

(D) ^ are tabulated in Table 7.5. c=0

The theoretical variation of D with concentration is also illustrated

in Figures 7.10(a) and 7.10(b) as the solid curves. Again, the close

correspondance with the total immobilisation model may be inferred by the

close agreement between the experimental D and the theoretical expression.

150

Concentration/( cc(stp )/cc(polymer))

FIG 7.10(a) PMMA (POWDER) / PROPANE _ CONCENTRATION DEPENDENCE OF D

151

Concentration/( cc(stp)/cc(polymer))

FIG 7.10(b) PMMA (POWDER) / PROPANE CONCENTRATION DEPENDENCE OF D

152

The experimental results were also analysed in terms of the

partial immobilisation model. The theoretical expression in this

case is given by

D = (D d - D )X + D h (7.3)

where X is an integral function and is given by

X = rc r [ cr ^ p - k D ( r - l )

c r ° r-2

c P dc ] (7.4)

A plot of D vs X is then linear, with a gradient (D - D ) and intercept d h

D . In the limit r = 1, equation 7.4 reduces to the simple h

expression

- p d n d h c h b c d D • - t ' [ k D - c"d % - v + °H ( 7 ' 5 )

Plots of D vs Cp/Cq are illustrated in Figures 7.11 for the three lowest

temperatures. The experimental points are again depicted by the solid

symbols and the solid lines represent the best fit relationship through

the data. A small but finite intercept at the ordinate was found and

this theoretically represents D , the diffusion coefficient of the h

immobilised species. Values of D^ and (^) c_q a r e given in Table

7.5.

153

FIG 7.11 PMMAfPOWDER) / PROPANE CONCENTRATION DEPENDENCE OF D

154

TABLE 7.5. Diffusion Coefficients: Propane.

TOTAL IMMOBILISATION

PARTIAL IMMOBILISATION

i—i tH I i

V) M e u.

iH

w M

V) N e o

tH 1

CO

u o^

cx e <u h

o ii u

iq MD tH o tH

e u q

o m tH o rH

o ii u

io vD rH o rH

N e u o Q

m rH

O rH

tH 1

W

N e u

Ed o

20.0 6.4 2.9 7.9 2.3 0.38

30.0 18.6 7.4 21.6 6.8 0.62

40.0 63.1 19.5 68.6 18.1 1.45

50.0 172.3 45.4 189.8 38.7 6.90

60.0 430.1 73.5 496.3 63.28 35.4

( E d ) c = q = 86.5 kJ mol -1

( E d ) c = q = 84.9 kJ mol -1

( E d ) d = 67.5 kJ mol -1

(E d) d = 43.7 kJ mol -1

A critical examination of the results presented in Table 7.5 indicates

there is little to choose between either model and this may be a

reflection of the accuracy of the present results rather than the

lack of cogency of the partial immobilisation model.

The temperature dependences of (E a n <^ were evaluated from

the Arrhenius equation given in equation (1.6) and are also depicted in

Figures 7.16 and 7.17; the activation energies for diffusion are

included at the foot of Table 7.5.

155

According to the total immobilisation model presented in Section

2.3.2, (D) c_q may be expressed as follows

( 5 ) c = o " W S L ( 2"7 A )

it then follows that

( V c = 0 " < V D + AHD " ( A V c = 0 ( 7"6 )

Thus, (E r) c_Q depends not only upon the value of but also on the

two thermodynamic quantities AH and (AH ) . The validity of d o u

equation (7.6) is demonstrated by the close agreement of (E R) c_Q given

at the foot of Table 7.5 with the calculated value of 82.8 kJ mol \

7.1.4. PMMA/iso-butane.

The sorption of iso-butane in PMMA was studied at 40°C and the

isotherm is presented in Figure 7.12. As with propane, dual-mode

sorption behaviour was observed and the " best-fit" isotherm parameters

are given as k = 0.038^ cm"3 (stp)/cm 3 (polymer) cmHg, c^ = 1.30 c m 3 (stp)/

cm 3(polymer), b = 0.061 cmHg" 1 and S T = 0.119 cm 3(STP)/cm 3(polymer)cmHg.

6 li

Immense experimental difficulties were encountered with this gas

due to the extremely low diffusion coefficients of the polymer/penetrant

pair. The times for equilibration were in the order of a week and in

view of this, balance drifting effects could not be ruled out completely.

Furthermore, with such inordinately long equilibration times, it was

necessary to remove the water bath due to excessive evaporation. These

two factors are likely to detract from the overall accuracy of the

measurements.

156

o co

F I D 7 - 1 2

PMMA (POWDER) / i s o - B U T A N E SORPTION ISOTHERMS

157

3.0 3.1 3.2 3.3 3.4 3.5 3.6

Temperature (1 /T)/10~3K~1

FIG 7.13 POLY(M ETHYL METHACRYLATE) TEMPERATURE DEPENDENCE OF kD

158

3.0 3.1 3.2 3.3 3.4 3.5 3.6


FIG 7.14 POLY(M ETHYL METHACRYLATE) TEMPERATURE DEPENDENCE OF b

159

- . 5 0

-1.10

- 1 . 7 0

- 2 . 3 0

- 2 . 9 0

- 4 . 1 0

- 4 . 7 0

- 5 . 3 0

- 5 . 9 0

3.0 3.1 3.2 3.3 3.4 3.5 3.6

Temperature (1/T)/10~9IC1

fig 7.15 poly(methyl methacrylate) temperature dependence of s l

160

-20.0

-22 .0

- 2 4 . 0

- 2 6 . 0

-28 .0 © n ©

1^-30.0

^ 3 2 . 0

- 3 4 . 0

- 3 6 . 0

- 3 8 . 0

- 4 0 . 0

3.0 3.1 3.2 3.3 3.4 3.5 3.6


FIG 7.16 POLY(METHYL METHACRYLATE) ARRHENIUS PLOTS OF (D)**

+

161

- 2 9 . 5

- 3 0 . 0

- 3 0 . 5

- 3 1 . 0

*31.5

>-4-32.0

- 3 2 . 5

- 3 3 . 0

- 3 3 . 5

- 3 4 . 0

3.0 3.1 3.2 3.3 3.4 3.5 3.6

Temperature (1 /T)/10~SK~1

FIG 7.17

162

Diffusion coefficients were measured from the kinetics of sorption

at four concentrations. The measured diffusion coefficients were found

to be, surprisingly, independent of concentration and given as

according to the dual-mode sorption theory, the diffusion coefficient

would be expected to be a weak function of concentration as found with

propane..

7.2. General Remarks of the Dual-Mode Sorption Theory.

The mathematical expression which describes the equilibrium sorption

isotherms for systems exhibiting the general dual-mode sorption behaviour

assumes there is one component which corresponds to normal dissolution and

a second component which is characterised by the Langmuir isotherm. The

agreement between experimental data and the isotherm equation does not

necessarily imply the correctness of the model and the use of non-linear

regression analysis, or any other method of determining the sorption

parameters,does not offer a critical test of the Langmuir mode.

It has already been pointed out in section 2.3 that equation (2.60)

may be used as an alternative graphical technique of determining c'

and b. It has also been argued that this method is open to severe errors,

however, equation (2.60) serves as a convenient and direct method of

examining the applicability of the Langmuir isotherm in the present study,

viz,

1.23 x 10 cm 2/s. The reason for the invariance of D is unclear and -16

c 1

H (c ' b) >

1 1 (2.60)

i

163

1/p cmHg 1

FIG 7.18 PMMA/ETHANE LANGMUIR SORPTION

164

.2 .3 .4 .5 .6 .7 .0 1.0

1 /p cmHg -1

FIG 7.19 PMMA/PROPANE LANGMUIR SORPTION

165

The Langmuir components to the total sorption for both ethane and

propane are presented in Figures 7.18 and 7.19 respectively. The solid

symbols represent the experimentally determined concentrations,and the

straight lines were constructed using the parameters determined from

the curve-fitting procedure. The agreement is excellent.

In a strict Langmuir model, c' represents the maximum monolayer

coverage of a solid surface by an adsorbate and should, therefore, be

constant. It is evident from Figures 7.18, and 7.19 that c„ (given by

the reciprocal of the intercept at the ordinate) increases as the

temperature is lowered. Thus, the analogy between c' for sorption in H

glassy polymers and that of the ideal Langmuir model should be treated

with caution. Furthermore, since c' is considered to be a measure H

of the " frozen free volume" of the glassy polymer, the variation of

c' with temperature is unexpected. h

Three arguments are presented here which offer a tentative explanation

for the increase of c' at sequentially lower temperatures:-

h Firstly, the premise that c' represents the maximum monolayer coverage

h

should be examined. In the ideal Langmuir (5Q). model, the number of

sites available for sorption on the solid surface is fixed and at

saturation, all the sites are occupied with one adsorbate per site. The

adsorbate is futher assumed to be immobilised, although vibrational modes

in the plane of the surface are not precluded. The vibration of the adsorbate

effectively increases its occupied volume,reducing the number of sites

and at lower temperatures, this vibrational mode is expected to reduce,

allowing for a closer packing arrangement of adsorbate molecules. On

this argument, c^ should then represent the maximum monolayer coverage

at each temperature.

166

Secondly, the approach suggested by Paul et al. (52) is considered.

The reader is referred to Section 3.2.for a detailed account. The

argument presented by Paul et al., would imply that the large scale

motions of the polymer chains which exist above T persist at temperatures s

below the glass transition temperature. The phenomenon of c' is then

h

explained as the difference in volume between the glass and the

corresponding rubber state which would be obtained on an infinite time

scale.

Finally, it is conceivable that the major proportion of c' is due

to the frozen free-volume associated with the reduced motion of the main

polymer chains below T . However, it is plausible that small scale

motions still persist below T , and the proportion of this free-volume

may be better visualised as a mobile free volume, somewhat analogous to

that in the rubbery state, though of a different magnitude. At sequentially

lower temperatures, more mobile free-volume will be frozen in, which

could explain the increase in c'. According to this argument there must H

exist a hypothetical temperature below which no mobile free-volume can

exist and at which the Henry's Law mode to the total sorption must then

tend to zero. This, however, has not been observed and may be impossible

to verify experimentally.

At the other extreme, as the temperature of the glassy polymer is

raised, the Langmuir component to the sorption should reduce until the

T is attained when no Langmuir sorption should be evident. Although it g

was impractical to demonstrate this in the present study, other studies

confirm that this is the case.

167

2.5

2.0 4-

1.5

l.o 4-

cj _

£ 0.0

- . 5 4-

- l . o 4-

- 1 . 5 4-

-2 .0

propane

ethane

j u

2.5 2.7 2.9 3.1 3.3 3.5 3.7


FIG 7.20 APPARENT TEMPERATURE DEPENDENCE OF S^k, ,

168

In Figure (7.20), the magnitude of the composite quantity S /k Lj iJ

as a function of temperature is demonstrated . As the temperature is

increased, the Langmuir contribution (c'b) to the quantity (S /k ) H L D

reduces until the T is reached when (S /k n) = 1. For both propane g L u

and ethane, the theoretical lines through the data points were constrained

to intercept at T^ and it is evident that the agreement between theory

and experiment is good.

7.3. Transport and Sorption Parameters: Correlation Functions.

In this section, some of the transport and sorption parameters for

gases and vapours in PMMA are presented in the form of correlation graphs.

The main source of data was the present study although collation with

results reported by various other investigators will also be included.

It became evident when attempting the correlations that the physical

properties of the gases were often ill-defined. This was especially true

when the diameters of non-spherical gas molecules were considered. Various

methods for calculating gas- diameters have been reported in the

literature; Hirschfelder, Curtiss and Bird (199) gave estimates of the

molecular diameters based on measurements of gas viscosities. Molecular

diameters have also been estimated from heat conductivities, liquid

densities and computer simulated molecular models. The data provided

by these diverse techniques were found to be incomplete, and from the

point of view of generality of comparison, it was felt that the gas

diameters should be estimated from a single source only.

169

Molecular Diameter (d) /A

FIG 7.21 DIFFUSION COEFFICIENT vs MOLECULAR DIAMETER

170

The van der Waal's equation of state as given in equation (7.7)

contains the parameter 3> which is a measure of the incompressability

of the gas due to the finite volume of the molecules. An extensive

(P + a/v 2)(v - 3) = RT (7.7)

list of this parameter is available in standard texts (200) which covers

all the gases used in the present correlations. The diameter of the

gases may thus be estimated from the simple expression

3

d = /FTn (7.8) a

where N is Avagadro's number and d the gas diameter. It should be noted Si

that, d, determined by this method assumes spherical geometry of the

molecule which may often not be truely representative when the elongated

molecules such as propane and ethane are considered.

The variation of the diffusion coefficient, (D) with molecular e=0

diameter is shown in Figure 7.21 and the solid line represents the best

fit relationship. Although the correlation suggests an exponential

variation of the diffusion coefficient upon linear diameter, it should

be appreciated that logarithmic relationships of this type may be

deceptive. For example, a deviation in absolute terms of a hundred fold

would only be manifested as four fold error in logarithmic designation .

171

90.0 -

QJH 8/#

80.0 - • C 2 H 7

I 270.0 -

"~Q60.0 -"~Q60.0 -

/ * C 2 H e

^50.0 - / c h 4

0) si #002

H 2 O • / g40.0 -o •c* •to / # N a

330.0 --to 0 / #0 2

20.0 - / # H e

10.0 -

1 1 1 1 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Molecular Diameter (d)/ A

FIG 7.22 ACTIVATION ENERGY vs MOLECULAR DIAMETER CORRELATION

172

The variation in the activation energies of diffusion (E ) as a D c=0

function of gas diameters is presented in Figure 7.22. Again, a

pronounced dependence of E^ on the gas diameter is apparent. A

relationship of the form

< V c - 0 " k d 2 ( 7 - 9 )

was assumed where k was an arbitrary system parameter. The data was

constrained to fit equation (7.9) and is represented as the solid curve.

The fit appears to be good although it may be argued that a linear

relationship is equally representative. In view of the sparsity of data

o

below the critical diameter of ^ 2.5A, it is impossible at this stage to

elaborate on the justification of either relationship without recourse

to wanton speculation.

In this penultimate section, correlations are presented for the

parameters k , S and (AH ) It should be appreciated that the L) L b C—U

solubility coefficients in glassy polymers, particularly S^, are heavily

dependent upon the prior history, of the sample. Thus, any serious attempts

to compare these quantities without consideration of this additional

factor could be misleading. Furthermore, since the data presented here

are not confined to the present study, only a simplistic, semi-quantitative

treatement of the results will be attempted. It has already been discussed

in section 3.3 that r/k, the Leonard-Jones force constant,has often

been used as a fundamental correlation parameter. Furthermore, since the

boiling point and the critical temperatures are both related to s/k, only

correlations to this more meaninglful parameter will be considered.

173

—1.70 +

-2.30 4-

-2.90 4-

o o 00-3.50 4-

^ 4 . 1 0 4-

-4.70 4-

-5.30 4-

-5.90 4-

0 3

co2

c 3 h |

i—C 4 H 10

ch 4

0 2

c0 2

c2he(powderb

cah6(sheet)

.0.60 | , i i i i [ i i i i ) i i i i j » i < i i i i i i ] i i i i

0.0 50.0 100.0 150.0 200.0 250.0 300.0

Leonard Jones Well Depth (e/k)

FIG 7.23 LIMITING SOLUBILITY CORRELATION

174

-2.00

-2.30 +

- 2 . 6 0 +

-2.90 +

c j o o 0>3.20 -f-

so c3

^ 3 . 5 0 +

Si

-3.80 -f

•4.10 4-

-4.40

-4.70 +

i - C 4 H 1 0

j i ' l. • » i • ' j l

100.0 130.0 160.0 190.0 220.0 250.0 28070


FIG 7.24 HENRYS LAW SOLUBILITY CORRELATION

175

0.0 50.0 100.0 160.0 200.0 250.0 300.0


fig 7.25

heat of sorption correlation

176

. The limiting solubilities S are presented in Figure 7. 23 and l

indicate a general trend of increasing S with increasing e/k. The « l

value of S for C0 2 presented by Koros, Smith and Stannett (83) appears to be i_i

considerably larger than the norm, though this may be attributed to the

technique employed by these investigators. Koros et al., subjected

the PMMA sample to high pressures of C0 2 in order to pre-condition the

polymer. This has the effect of dilating the microvoids, and consequently

enhancing the sorptive capacity of the Langmuir component. The larger

S value is thus to be expected. Li

The variations of k^ with e/k is shown in Figure 7. 24. Only four

data points were available and these appear to be linearly disposed

yielding a best fit line given by

^ n k D = 0.02 (e/k) - 7.4

The gradient of this line is in good agreement with the value of 0.026

predicted by Michaels et al. (93) for rubbery polymers.

Correlations between (AH ) _ and e/k are presented in Figure 7.25.

S c=(j

Michaels and Bixler (93) have shown that a linear relationship exists

between AH^ and e'/k and the present results as depicted in Figure 7.25

suggest that there may also exist a linear correlation between (AH„) S c=0

and e/k.

The use of propane in the systematic study of glassy polymers has been

reported by at least three other investigators (71,92, 7 5 ) . If the list

is extended to include polymers above their T , then many more (191,192,

193,202,203, vide supra Chapter 6) may be appended to this growing

catalogue. The attraction of propane as «i microprobe lies in its high

solubility in many polymers, and even at sub-atmospheric pressures,

177

significant levels of sorption are normally achieved. The advantage of

the high solubility coefficient must, however, be offset against the

lower diffusion coefficients encountered with this vapour. With this

view in mind, a correlation is presented between the diffusion coefficient

and a fixed polymer variable, the specific free volume.

The direct comparison between the diffusion coefficient for a given

penetrant in various polymer environments is a non-trivial exercise.

However, a semi-quantitative picture may be formulated on a modified

volume-fluctuation treatment.

The diffusion coefficient according to Bueche (198) may be related

to the jump frequency and 6, the average jump step. Hence,

To a crude approximation, the average jump step for a given penetrant

molecule in different polymers may be assumed to be comparable in magnitude .

Thus, the large variation in the diffusion coefficients which are normally

experienced may be attributed to the different jump frequencies. On a

molecular level, the diffusion coefficient may be associated with the

availability of a " hole" into which the penetrant may transfer. In

a pre-emptive study, Lee (100) suggested that the free-volume of the

polymer is a good measure of the " hole" content and proceeded to define

a unit weight free-volume parameter, namely the specific free volume (SFV)

from

d - | •« 2 (7.10)

( S F V ) = V /M F

(7.11)

178

where V p is the molar free volume of an amorphous polymer (cm 3 mol

and M the molecular weight of a " mer" unit (g mol .

Fujita (204) defined a mobility parameter, viz,

m = A exp [-B /(SFV)] (7.12) a

where A and B are system constants and m is related to the diffusion a

coefficient by

D = RT.m (7.13) a

combining (7.12) and (7.13) then gives

/nD = /n(RTA) - B/(SFV.) (7.14)

According to equation (7.14), /nD is inversely proportional to SFV

provided A and B are not functions of the polymer. Lee (100) proceeded to

introduce the solubility coefficient into equation (7.14) in an attempt

to relate the SFV to the permeability. It is felt that this additional

step complicates the correlation unnecessarily, and for the present

purpose, equation (7.14) will suffice.

Values of the SFV at 25°C for five different polymers were obtained

from the literature (100) and the correlation is illustrated in graphical

format in Figure 7.26. The agreement is remarkable yielding a correlation

coefficient of better than 0.99. It should be appreciated that of the

five polymers considered, three (PS, PC, PMMA) are below their glass

179

2.0 4.0 6.0 8.0 10.0 12.0

1 /(Specific Free Volume) / gcc'1

fig 7.26 PROPANE DIFFUSION vs 1/SFV at 25°C

180

90.0

80.0 4-

70.0 4-

60.0 4-

50.0 4-o d

_ o

3 40.0 4-

30.0 4-

20.0 4-

IO.O 4-

0.0 0.0 2.0 4.0 6.0 8.0

1 /(Specific Free Volume) gcc

FIG 7.27 PROPANE ACTIVATION ENERGY vs 1/SFV

181

transition temperatures at 25°C whereas the other two (PE, NR) are well

above their T . This then begs the question as to whether the mechanism

for diffusion above T is identical to that in the glassy state. Diffusion

in rubbers is believed

to occur via the cooperative motions of the

main polymer chain, but below T , such large scale motions are believed s

to be effectively " frozen out" . The quality of the correlation for both

rubbers and glass on the same curve may be explained by the fact that

the diffusion coefficient of a given polymer above and below T differs 8

at most by a factor of two or three. On an extensive logarithmic scale,

as used in Figure 7.26, such small variations would not be noticed.

Combining equation (7.14) with (1.6) yields a relationship between

the activation energy for diffusion with the specific volume element,

The validity of equation (7.15) is shown in Figure 7.27.

7.4. A Critical Examination of Diffusion in PMMA Powder and Sheet.

The apparent dichotomy of results found with the sorption and diffusion

parameter for ethane in PMMA powder and sheet begs the question as to

which result is correct. Several facets of the experimental technique

were examined in turn and the postulate of residual casting solvent in

the sheet proved to be the most probable explanation. To recapitulate,

the diffusion coefficient at 30°C for ethane in the 36.6 I'm sheet was

-11 -13 determined to be 2.2 x 10 cm 2/s compared with 1.29 x 10 cm/s for

the 0.15 I'm spheres at: the same temperature.

D (7.15)

o

182

The first problem to be addressed, therefore, was the possible error

incurred in the determination of the individual particle diameter of the

powder. If it is argued that the diffusion coefficient determined using

the sheet is correct, then the minimum diameter of the powder must be in

the order of ^ 2 p. Furthermore, since the temperature dependences of

the diffusion coefficients for both the powder and sheet are different,

this would moreover indicated that the particle diameters vary as a

function of temperature.

Another aspect which was considered was the premise that the process

of diffusion in the powder was controlled by transport through the

" sample bed" rather than into the individual particles. To examine

this postulate, the " sample bed" thickness was varied and the diffusion

coefficient subsequently measured. It was found on increasing the volume of

sample from 0.2 to 0.7 cm 3 that the effect on the diffusion coefficient was

— 1 5 — 15 negligible, 9.9 x 10 and 10.1 x 10 cm 2/s respectively, for ethane at 1 C.

Thirdly, the possibility of a real difference between the polymer

samples was considered. The PMMA sheet and powder were obtained from

different sources with the former obtained commercially and the latter

prepared by emulsion polymerisation; it is conceivable that the two forms

may differ in their stereoregular composition. The stereotacticities

of PMMA prepared by free-radical techniques are now well reported (184-

186, 210) with the syndiotactic form favoured at lower temperatures. NMR

analysis on the two samples indicate slight differences in the stereo-

tacticities (vide supra, Table 5.2) although this is insufficient to

explain the pronounced differences in their sorption and diffusion

characteristics (210).

183

Finally, the postulate that residual casting solvent in the sheet

may have accelerated the sorption kinetics was examined. Since no

technique has yet been found which unambiguously demonstrates the

presence or absence of solvent in the sheet, trace solvent was therefore

added to the powder for comparison. To achieve this, the polymer powder

was suspended on the microbalance and outgassed. Toluene vapour

(p/p Q 0*05) at 1°C was introduced to the dry sample and the weight

uptake 0*05 w/w at equilibrium)was followed as a function of time J

complete equilibration was acheived after two weeks. Sorption kinetics

for ethane were then measured for the toluene-loaded powder and a diffusion

-12 -15 coefficient of ^ 10 cm 2/s obtained, compared with a value of 9.9 x 10

-13

cm 2/s for the vigin-powder and of 7 x 10 cm 2/s for the sheet (extra-

polated value).

The implications of the results are quite clear and suggest that even

at low levels of residual solvent, adequate plasticisation of the polymer

has occurred which facilitates the transport of penetrant molecules.

The precise nature of the plasticisation is obscure since the T^ of the

PMMA is in close agreement with the powder albeit encompassing a broader

range. If plasticisation effects are operative then it must be assumed

that the main polymer chains are largely unaffected and the solvent

molecules selectively plasticise smaller scale motions. If this

premise is true, then it would suggest that below T , the diffusional 8

process is due to the cooperative motion of short sections of the chain

or side chains.

184

CHAPTER EIGHT

RESULTS AND DISCUSSION PDMS-g-PMMA

8.1. Introduction.

The graft copolymers were prepared by pre-swelling a silicone rubber

matrix with methyl methacrylate monomer, followed by subjection to

y-radiation. The effect of the y-radiation is the generation of free-

radical sites in the rubber matrix from which initiation and subsequent

propagation of the grafts occur. Provided low radiation dosages are

employed, high molecular weight side chains are expected with little or

no homopolymer produced. Evidence of high molecular weight grafts and

other physical properties of the membranes have already been discussed in

Chapter Five.

In this chapter, the sorption of propane in the graft copolymers is

reported and the validity of the additivity rule as applied to solubilities

is examined. Permeabilities of the graft copolymers are presented and

examined in relation to some selected models presented earlier in Section

3.5. Finally, diffusion coefficients measured by various techniques are

described and a means of relating the different diffusion coeffients are

discussed.

The nomenclature suggested in Chapter Four will be retained here

and in addition, the subscripts C and D will be introduced and refer to

the " continuous" (silicone rubber) and the " dispersed" (PMMA) phases

respectively. The subscript m refers to the composite membrane.

185

o to

o o

o to en

CO

CM

CM

o

• 3 0 . 0 C

© 4 0 . 0 C

A 5 0 . 0 C

^ . 0 0 8.00 16-00 24.00 32.00 Pressure / (crruHg)

40.00 48

F I G 8 . 1

PDMS—g—24.14%PMMA* SORPTION ISOTHERMS

185

o in

• 3 0 . 0 C

Q 3 5 . 0 C

A 4 0 . 0 C

^ . 0 0 8.00 16.00 24.00 32.00 40.00

Pressure / (crruHg)

F I G 8 . 2


187

o ID

o o

• 3 0 . 0 C

Q 3 5 . 0 C

a 4 0 . 0 C

^ . 0 0 8.00 16.00 24.00 32.00 Pressure / (crruHg)

40.00

F 1 0 8 . 3


o CO

Pressure / (crruHg)

F IG 8.4

PDMS—g—39.76%PMMA SORPTION ISOTHERMS

189

o in

£ o o

1

to

o o \

o o

o in

• 30.0 C

© 35.0 C

A 40.0 C

O 45.0 C

O 50.0 C

^ . 0 0 8.00 16.00 2 4 . 0 0 3 2 . 0 0

Pressure / (crruHg) 4 0 . 0 0 4 8 . 0 0

F IG 8 . 5


190

o co

3 o v s o §

o

\

o

• 30.0 C

© 35.0 C

8.00 Pressure / (crruHg)

F IO 8.6


191

8.1.1. Equilibrium sorption.

The equilibrium sorption isotherms of nrooane in the graft copolymer

concave to the abscissa was observed which was attributed to the Langmuir

component in PMMA. The level of curvature was less severe at low volume

fractions of PMMA but became more noticeable at sequentially higher levels

of grafting as illustrated in Figures 8.1 to 8.6. The solid lines

represent the best fit relationships and the dashed lines depict the high

pressure asympote for the isotherm measured at the lowest temperature;

the best-fit parameters are presented in Table 6.1. For lower volume

fractions of PMMA, curvature in the isotherm was barely discernible and

only the overall Henry's Law solubility, (S_) , is given. L m

It should be appreciated that (k^) represents the Henry's Law

dissolution constant expressed per unit volume of membrane and is related

to the Henry's Law dissolution constants of the pure components by,

Similarly, (c') is defined as the concentration of propane in the H m

microvoids at saturation expressed per unit volume of membrane. If the

Langmuir component is attributed solely to PMMA, then,

were measured in the temperature range of 30°C-50°C. Significant curvature

(8.1)

(8.2)

(S ) , t h e o v e r a l l H e n r y ' s Law s o l u b i l i t y c o n s t a n t may b e w r i t t e n L m a s

(8.3)

192

and since b, the hole affinity constant is independent of composition,

substituting (8.1) and (8.2) into (8.3) then gives

( V r a • W c + V [ ( k D > D + ( c H V ] S V ( V C + •D-(SL)D 8- 3 ( a )

Equations (8.1) to (8.3(a)) express mathematically the additivity of

solubility and are also illustrated by the solid lines in Figures 8.7 to

8.9. The variation of (lO with volume fraction is depicted in Figure L; m

8.7J as expected , the Henry's Law component to the sorption decreases

as the volume fraction of PMMA is increased. Furthermore, the agreement

between the experimental points and the linear relationship is good.

The variations of (cT'T) and (ST) with d> are illustrated in Figures H m L m D 8.8 and 8.9 respectively. The solid lines were calculated from equations

(8.2) and (8.3(a)) using the values of (k ) , (c')_ and b given in Table D D H D

7.4. It is evident that both (c') and (ST) do not conform to the H m L m

" additivity rule" and the deviation from the expected behaviour is more

pronounced at sequentially higher volume fractions of PMMA. The direct

implication of this is a non-proportional increase in the microvoid content

of the graft copolymers with composition. If the concept of c' is h

extended to embrace adsorption in the voids formed at interfacial boundaries,

then this may offer a plausible explanation for the anomaly. However, it

is proposed here that the deviation of c' may be due to a real variation H

in the microcavity content of the PMMA dispersions. In the present study, the

graft copolymers were prepared by the polymerisation of methyl methacrylate

monomer in situ.and towards the end of polymerisation substantial quantities

of monomer vapour are still present. Post-irradiation treatment then included

193

TABLE 8.1. PDMS-g-PMMA Sorption Parameters,

a 6 to CO

u

pu B q) h

60 60

- ps u

60 ps B o

CO

60 PC B u ^ rh o pl

CO

PDMS-g-10.35% PMMA

PDMS-g-30.67% PMMA

PDMS-g-39.76% PMMA

PDMS-g-46.34% PMMA

PDMS-g-51.45% PMMA

PDMS-g-28.14% PMMA*

30.0 40.0 50.0

30.0 35.0 40.0 45.0 50.0

30.0 35.0 40.0 45.0 50.0

30.0 35.0 40.0 45.0 50.0

30.0 35.0 40.0 45.0 50.0

30.0 35.0 40.0 45.0 50.0

0.0891 0.0785 0.0721 0.626

0.0578

0.0784 0.0707 0.0641 0.580 0.0557

0.0845 0.0702 0.0629 0.0555 0.0546

0.0741 0.0652 0.0595 0.0577 0.0508

0.918 0.0797 0.0728 0.0691 0.0613

0.18

0.20(6)

0.12(9) 0.20(5) 0.11(7)

0.36(7) 0.37 0.42 0.28

0.21

0.45(9) 0.32 0.30 0.33(8) 0.21(8)

0.62(7) 0.54(5) 0.52 0.30(6) 0.27(7)

0.167 0.249 0.153 0.051 0.082

0.26

0.14(9) 0.22

0.11

0.15(8)

0.19(7) 0.13(8) 0.089(6) 0.13 0.10

0.24 0.21 0.18 0.10(6) 0.14

0.163 0.13(6) 0.12 0.16(7) 0.14(9)

0.266

0.113 0.158 0.554 0.179

0.120 0.080

0.069

0.137 0.109 0.102 0.085 0.076

0.151 0.122 0.102 0.096 0.079

0.195 0.139 0.118 0.091 0.085

0.176 0.139 0.122 0.109 0.094

0.136 0.108 0.097 0.0974 0.076

194


FIG 8.7 (kD)m vs VOLUME FRACTION AT 30°C

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0


FIG 8.8 (C„')m vs VOLUME FRACTION AT 30°C

196


FIG 8.9 (SL)m vs VOLUME FRACTION AT 30°C

197

evacuation followed by solvent extraction. Such severe treatment of.

the grafts is likely to disturb the microvoid content of the glassy

polymers in view of the high solubility of both the monomer and the

solvents in PMMA. Furthermore, the small PMMA domains which are formed

facilitate rapid desorption of the vapours and this is also expected

to enhance the cavity component. A more critical test of the " additivity rule" and one which is

without recourse to establishing a sorption model for the dispersed phase

is presented as follows!

The total sorption in the copolymers may be written in the form,

c ° c c + c l ) = V ( V c p S ( 8 : A )

where c is the total concentration of penentrant in the membrane and the

quantities c^ and cR represent the concentration af penetrant in the

continuous and dispersed phases respectivley, expressed per unit volume

of membrane.

If [ c J is defined as the concentration of penentrant in the PMMA D m domains, expressed per unit volume of dispersed phase, then

[cD] = [g " V ^ D V ^ D (8'5)

Values of a t 30°C were extracted from the graft copolymers and

are presented in Figure 8.10 as a function of pressure. The solid line

represents the sorption isotherm determined from the powder sample. It is

clear that the extracted Isotherms cannot be represented by a single line but

is perhaps better characterised by a sorption hand; the reason for this

may lie in the non-uniform history of the various samples.

198

2.70 - -

C q)

S

o

o vo \ 1 . 8 0

so

0 1 . 5 0 o

gl.20 o

so

£ .90 e o o e o .60

.30 - -

Pressure / (cmHg)

FIG 8.10 EXTRACTED ISOTHERMS AT 30°C

199

An alternative test of the additivity rule may be performed,in which

the total concentration is expressed in the form

c = c c + C D = * c [ c c ] + *D[CD] ( 8' 5 a )

(8.5b)

(8.5c)

where tc J and [cR] now refer to the concentration of penetrant expressed

per unit volume of continuous and dispersed phases respectively.

Values of c as a function of at 30°C were constructed and the

isobars are illustrated in Figure 8.11 at four pressures. The parameters

characterising each isobar, namely the gradient and the intercept at the

ordinate are tabulated in Table 8.2. Also included in column 4 of Table

8.2, are the ratios of the intercept values to the ambient pressure. It

is evident from equation (8.5(c)) that these represent the Henry's Law

solubility coefficients for PDMS.

TABLE 8.2. Graft Copolymer:Isobar Parameters at 30.0°C.

p/cmHg [c ] - [ c i / c m 3 ( S T P ) L V c cm3 (polymer)

rc i /Cm3 (STP) c cm3(polymer)

[Gc]/Cm3(STP)

P cm3(poly)cmHg

10.00

20.00

30.00

40.00

40.353

-0.820

-0.642

-1.244

0.960

1.997

3.048

4.104

0.096

0.099

0.101

0.102

and since,

then 1

c = V l c D ] ~ ^ ( T + t c c '

200


FIG 8.11 GRAFT COPOLYMER ISOBARS

201

It is gratifying to find that these estimates of the Henry's Law

solubility coefficients are constant and in good agreement with the

value of 0.101 cm3(STP)/cm3(poly)cmHg presented earlier (vide supra.

Chapter 6) for silicone rubber

The temperature dependences of G O and (ST) were determined from d m l m

the van't Hoff relationship and the values of AHp and (AHg)c_q are

presented in Table 8.3. Also included in Table 8.3 are the thermodynamic

quantities determined for PDMS and PMMA.

TABLE 8.3. Graft Copolymers: Heats of Sorption.

Sample AHD/kJ mol"1 (AHg)c=0/kJ mol 1

PDMS -16.9

PDMS-g-10. 35% PMMA -16.1

PDMS-g-24.14% PMMA -15.5 -20.1

PDMS-g-30. 67% PMMA -17.8 -23.1

PDMS-g-39. 76% PMMA -14.4 -25.5

PDMS-g-46.34% PMMA -18.1 -34.1

PDMS-g-51.45% PMMA -14.3 -24.3

PMMA -19.9 -35.0

AHp, was found to be largely independent of composition and by

considering equation (8.1), it can easily be shown that

ah (8.6) d < | ) c ( k ) c + < t > d ( k ) C d D

202


FIG 8.12 ( a h j c - o v s VOLUME FRACTION

203

Furthermore, since for PDMS and PMMA

(AHJ ~ (AH ) (8.7) D C " D D '

then equation (8.6) reduces to the relatively simple relationship of

(AHD) ^ (AHD)C * (AHD)D (8.8)

which is indeed consistent with the experimental findings.

An expression similar to equation (8.6) may be derived for the

dependence of (AHg)c_Q on composition, viz.,

® s > c o " y v c + y v d [ ( s b d ) c " ^ s d ' c - o 1 + ^ s l p c - o ( 8 - 9 )

However, in this case (AH ) 4 (AH ) _ n and the overall heat of sorption U J D C U

is expected to be a function of composition. The theoretical variation

of (AHg)c_Q with <}>p is illustrated in Figure 8.12 as the solid line.

The symbols represent the experimentally determined heats; the agreement

between experimental and theory is fair.

8.1.2. Steady State Permeabilities.

The permeabilities of the graft copolymers were measured directly

using the high vacuum permeation technique described in Section 4.2. For

all the samples, a direct proportionality existed between the steady state

flux and the ingoing pressure indicating a constant permeability", the

permeabilities P^ at 30°C are tabulated in column 3 of Table 8.4.

•14.2 4-

•14.3 4-

•14.4 4-

-14.5 4-

-14.6 4-

-14.7 4*

•14.8 4-

- 1 4 . 9 4-

- 1 5 . 0 4-

- 1 5 . 1 4-

- 1 5 . 2 4-

- 1 5 . 3 4-

PDMS—G—10.35%PMMA





\—'—1—f 3.00 3.06 3.12 3.18 3.24 3.30

Temperature (1/T)/10~*KT1

FIG 8.13 TEMPERATURE DEPENDENCE OF PERMEABILITY PETROLEUM EXTRACTED SERIES

205

- 1 4 . 5 0 +

- 1 4 . 5 8 4-

- 1 4 . 6 6 - f

- 1 4 . 7 4 +

- 1 4 . 8 2 +

L4.90 +

- 1 4 . 9 8 4-

- 1 5 . 0 6 4-

- 1 5 . 1 4 4-

- 1 5 . 2 2 4-

PDMS1—G—24.14%PMMA


j i i i i i i i i i i l j 1 l

3.06 3.12 3.18 3.24 3.30 3.36 3.42

Temperature (1/T)/KT1

FIG 8.14 TEMPERATURE DEPENDENCE OF PERMEABILITY ACETONE EXTRACTED SERIES

206

The temperature dependences of P^ are illustrated in Figures 8.13

and 8.14 for the petroleum ether and acetone extracted samples respectively.

Good linear plots were obtained with the petroleum ether extracted samples

which allowed ER to be determined with accuracy. The temperature dependence

of P for the acetone extracted samples was unusual and indicated a m

marked reduction at L 40°C. The technique employed with the acetone extracted

samples was identical to the petroleum ether extracted samples; the membrane

was mounted in the diffusion cell and subsequently outgassed. The water

bath thermostat was adjusted to 30°C and the sample allowed to equilibrate

for several days. The permeation run was then commenced and no anomalous

effects were observed at this stage. After several runs were performed,

and a constant P with pressure was verified, the thermostat was raised m

to 35°C and the procedure repeated; again no anomalies were encountered.

However, when the bath temperature was further raised to 40°C, the

permeability of the membrane was reduced markedly compared with the

preceeding runs at the lower temperatures. Furthermore, the drop in

the permeability was thermally irreversible and recovery to the original

value could not be achieved in this way. However, the original state of

the membrane could be restored by re-extraction in acetone.

The activation energies of permeation, E r, were estimated using

equation (1.9) and are tabulated in column 4 of Table 8.3. The two values

for the acetone extracted samples represent the E 1s before and after the

apparent " transition" at 40°C. In general, the activation energies of

permeation were found to be almost independent of composition and are also

in good agreement with the value for silicone rubber. The invariance of

207

E with composition may be inferred, a priori, from equations (3.16),

(3.19), (3.23) and (3.29) provided separation into the pure components

is complete. In the limit that P^ » Pp, these equations may be written

in the form

and by taking logs, followed by differentiation w.r.t. (1/T), it can

be shown that

The close agreement of the Ep's with that for PDMS thus confirms the

premise that the continuous phase is PDMS and furthermore, the rate

controlling step for transport in the graft copolymer membranes is the

slow diffusion in the silicone rubber matrix! the PMMA domains are then

simply impermeable fillers.

A more critical examination of the models outlined in Section 3.4

is shown in Figures (8.15) and (8.16). A total of six models are

illustrated, albeit only two were found to be of any real consequence!

the Lichtenecker model is included for academic interest.

The permeabilities of the graft copolymers were well represented by

the Higuchi and Bb'tcher models and the actual deviations from the various

models are futher illustrated in Figure 8.17. The applicability of the

various models in interpreting the permeabilities of composite membranes

(8.10)

(8.10(a))

208

0.9 L V

0.8 L

o {L 0-7

^ 0.6 £ *o a

0.5 L q) cl

q)

0.4 a k1 q) a;

0.3 L

0-2 L

0.1 L

v \ A

A

Higuchi

Maxwell

— — — Bruggeman \

w \\

W \ \\

\

\\ w

n \ w

a \ \ •

\ s

v \ \

X No

0 .1 0 -2 0 .3 0.4 0 .5 0 .6 0-7 0.8 0.9


FIG 8.15 PERMEABILITY OF GRAFT COPOLYMERS AT 30°C

209

o

so £

s a

q) cl

q> s o

a 0)

a;

0-9

o - 8 a

0-7

0 - 6

0-5 L

0-4 L

0-3 L

0-2 L

0-1 L

Botcher

Lichteneckerl

Meredith Sc

Tobias

\

0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9


FIG 8.15 PERMEABILITY OF GRAFT COPOLYMERS AT 30°C

210

I

.ib +

.15 +

.12 +

.09 4-

£ .06

g o so a

.03 ?> 0)

-.06 +

HIGUCHI

BRUGGEMAN

MAXWELL

+ BOTCHER

-.00 4 *

- . 0 3 4-

.1 .2 .3 .4 .5


FIG 8.17 DEVIATION FROM STANDARD MODELS

211

TABLE 8.4. Graft Copolymers: Permeabilities at 30°C.

Sample 107r ^cm3(STP)cm m cm2 s cmHg E /kJ mol 1 P

PDMS 0 7.40 -2.6

PDMS-g-10. 35% PMMA 0.0875 5.66 -3.1

PDMS-g-24. 14% PMMA* 0.2090 4.85', 3.55 - 2 . 7 ; 0.0

PDMS-g-30. 67% PMMA 0.2687 4.18 -2.1

PDMS-g-32. 28% PMMA* 0.2836 3.15; 2.54 -2.0; 0.0

PDMS-g-39. 76% PMMA 0.3541 3.20 -1.9

PDMS-g-46. 34% PMMA 0.4177 2.94 -2.2

PDMS-g-51. 45% PMMA 0.4681 2.54 -1.8 PMMA 1 ^ 24

have been reported-by various investigators. Munday. (92) found that the

models of Higuchi and Bruggeman described well transport through a series

of PDMS-g-PS membranes. Sheer (205) also found that the Bruggeman, Higuchi

and B'otcher models represented the permeabilities of his PDMS-g-PVAc

membranes. It would appear that there is little to choose between the

various models and the eventual selection of a particular model which is

felt to be reflective of the system is then somewhat subjective. It should

also be appreciated that the forcefulness of the models is greatly dependent

upon the accuracy by which P , the permeability of the continuous phase,

may be determined. It has already been pointed out in Chapter Six that

the permeability of silicone rubber to propane is not a well

established parameter and indicates a spread of approximately 30%. Adjusting

212

the present results by this factor would satisfy almost all the models

presented in Figures 8.15 and 8.16 (barring the Lichtenecker Rule). In

view of the apparent success of the Higuchi model found by other

investigators, and the good agreement with the present results, it will

be used for the remaining analysis in this treatise.

The anomalous behaviour of the acetone extracted samples with

respect to the tensile strength and relative thickness have been mentioned

in Chapter Five. The irregularities in the permeabilities have also

been discussed briefly in the earlier parts of this chapter and these are

further demonstrated in Figures 8.15 and 8.16. The total incapability

of any of the theoretical models to represent the permeabilities of these

membranes, coupled with the fact that the permeabilities are significantly

lower than for the petroleum ether extracted samples suggest major

morphological differences between the samples. The lower permeabilities

suggest the likelihood of " lamella" type structure which is in tacit

agreement with the micrographs presented in Figure 5.

8.1.3. Diffusion Coefficients.

In this section, the diffusion coefficients of the graft copolymers

determined by various techniques are reported. A means of inter-relating

and interpreting these different diffusion coefficients is discussed.

Steady State Diffusion Coefficients.

The steady state diffusion coefficients for the graft copolymers were

estimated from equation (2.14). Since the steady state fluxes (J/) were

directly proportional to pressure , p , i.e., a constant permeability,

it then follows that

213

- = J/ = ( 8 > n ) m p op

Also, since the equilibrium sorption isotherms may be represented

adequately by the dual-mode sorption equation then

ac c H b

giving

d ( c ) = itt • ^ = p -£ kn + c n b / ( 1 + b p ) 2 l 1 ( 8 - 1 3 )

9 p dc m D H

A typical example of equation (8.12) is illustrated in Figure 8.19

this curve has been labelled D^g in the diagram. Diffusion coefficients

determined by various other techniques are also included hut more will

be said about these later.

In order to compare the diffusion coefficients of the graft copolymers

satisfactorily, it is convenient to establish a reference concentration

and the diffusion coefficient in the limit of zero concentration is

normally chosen. Thus from equation (8.13)

( D W V ( k D + C H b ) ( 8 - 1 3 ( a ) )

where (D)c represents the diffusion coefficient in the absence of any

concentration effects.

Alternatively, the diffusion coefficient in the limit of high pressure

may be defined, whence

214

( d ) c = » " v kd (8.14)

From an analogy with the diffusion model for dual-mode sorption

in glassy polymers, (D)c relates the flux to the total Henry's Law

population within the membrane.

In Table 8.5, values of (D) _ and (D) at 30°C are tabulated c=U c=°°

for the graft copolymer membranes. Both (D) _ and (D) obeyed the c=U c=0°

Arrhenius expression and the values of (E^) _ and (E^) are included D c=0 D c =°°

in Table 8.5.

TABLE 8.5. Graft Copolymer: Diffusion Coefficients at 30°C.

Sample 106 < ° W 2. "I cm s

106(D) / c=°°

2 "I cm s

( V c - 0 k J mo 1 ^

( V c " kJ mol ^

PDMS 7.33 7.33 13.9 13.9

PDMS-g-10. 35% PMMA 5.57 5.57 13.0 13.0

PDMS-g-24. 14% PMMA* 3.56, 2.61 5.28, 3.26 12.2,15.5

PDMS-g-30. 67% PMMA 3.06 4.69 21.0 15.7

PDMS-g-39. 76% PMMA 2.13 4.08 23.6 12.6

PDMS-g-46. 34% PMMA 1.54 3.48 31.7 18.0

PDMS-g-51. 45% PMMA 1.44 3.42 23.0 12.6

The applicability of the Higuchi theory as given in equation (3.29)

in the interpretation of the permeabilities of the graft copolymers has

already been discussed earlier. As a further illustrative test of this

model, the relative diffusion coefficients [(D) _n/D ] as a function of

215

- . 0 .1 .2 .3 .4 .5 .6 .7 .8 .9


FIG 8.18 DIFFUSION COEFFICIENTS OF GRAFT COPOLYMERS

216

composition are depicted in Figure 8.18. The solid line was calculated

from the Higuchi model and the additivity rule; fairly good agreement

was found.

It is perhaps pertinent at this stage to examine critically one of

the main assumptions of the Higuchi model. Implicit in the Higuchi

expression, is the condition that the rate determining step for transport

through the heterogeneous medium, is the slow diffusion in the continuous

phase. Consequently, the kinetics of dissolution in the PMMA dispersed

phase must be rapid. Provided this critical condition is upheld, no

anomalous time effects are expected.

In view of the low diffusion coefficients encountered in the PMMA/

propane system (vide supra, Chapter 7) it was felt that a serious examination

of this condition was necessary; the graft copolymer selected for the

study was PDMS-g-46.34% PMMA. The diffusion coefficients of propane in

this sample were examined from (a) sorption and .conjugate desorption

kinetics (b) time lag and compared with the values estimated from the

steady state.

It has already been pointed out in Section 2.1.5 that when the

diffusion coefficient is concentration dependent, equation (2.36) yields

an average diffusion coefficient D. . Furthermore, the arithma'tic mean of

the sorption and conjugate desorption diffusion coefficients is given by the

expression:

5 = i ( d s + d d ) i f o

c f °

D(c)dc

The diffusion coefficients D and D, were measured at six c d concentrations from which the D(c) dependence was constructed by employing

a polynomial curve fitting technique. The experimental diffusion

217

Concentration / cc(stp)/cc(polymer)

FIG 8.19 PDMS—G—46.34%PMMA DIFFUSION COEFFICIENTS

218

coefficients together with the derived D(c) dependence are included in

Figure 8.19. The agreement with the steady state diffusion coefficient

is excellent.

It has already been shown that the equilibrium sorption isotherms

for the graft copolymers are quantitatively consistent with the model

for dual-mode sorption. Again, by analogy with the diffusion model

presented in Section 2.3.2, a concentration independent diffusion

coefficient, hereby designated D*, may be defined from

J = - D*. (8.15) dX

where c* is the concentration of the total Henry's Law dissolved

population.

Since the flux is also given by

J = -D(c). (8.16) dX

then by the consideration of fluxes

D(c) = D*. / (8.17) dC

It should be noted that D* theoretically represents the"diffusion coefficient

in the limit of high pressure and is equivilent to the steady state difussion

c o e f f i c i e n t i n t h i s same l i m i t [ i . e . , (D^ = P c=<« ni d

Substituting equation (8.17) into equation (2.27) then gives,

- 1 D — — c o

c ° 9r* c* D*. .dc = — D* (8.18) dC C

o o

219

In Table 8.6, the experimentally measured values of D are tabulated

in column 2. Values of D*, calculated using equation (8.18) are presented

in column 3. As expected, D* was found to be largely independent of —6 —1

concentration giving a mean value of 3.28 x 10 cm2 s

TABLE 8.6. PDMS-g-46.34% PMMA Diffusion Coefficients at 30°C.

conc/cm3(STP)/cm3(poly) 106D/cm2s~1 106D*/cm2s 1 106Df/cm2s 1

0.147 1.42 3.05 4.29

0.508 1.97 3.32 4.67 0.544 1.95 3.28 4.62 0.749 2.08 3.25 4.58

1.132 2.47 3.38 4.76

2.236 2.80 3.39 4.78

<D*> = 3.28 x 10 6 cm2s 1 ; <Df> =4.62 x 10"6cm2s"1

It should be stressed that the analogy between diffusion in the graft

copolymers exhibiting dual-mode sorption characteristics and diffusion

in conventional glassy polymers should be treated with caution. In the .

simple limiting model for diffusion in glassy polymers, the concentration

of the mobile species, and the concentration of penetrant dissolved in

accordance with Henry's Law, are treated as synonymous quantities. However,

the permeability results indicate the diffusion process in the graft

copolymer membranes is controlled solely through the continuous silicone

rubber phase. This appears to be paradoxical and begs the question of which

of the diffusion coefficients is then representative of the diffusion process.

220

If the diffusion coefficient is recognised as one which controls

the net rate of transfer across the membrane, then it should be defined

as the flux per unit concentration gradient within the continuous phase,

viz.,

3c J = -D_. (8.19)

f 3x

Thus, from a consideration of fluxes,

,* 9 t c j D(c) = D*. = D . — — (8.20) 3c f 3c

and an analogous expression to equation (8.18) may be derived, giving

[ c i D = ~ Df (8.21)

Values of D^ were calculated using equation (8.21) and are tabulated

in column 4 of Table 8.6. D^ was found to be independent of concentration

as expected.

A theoretical value of D . was calculated using equation (3.10); k

the structure factor was esimated from the Higuchi expression. Thus,

P D m = k = -4 (8.22)

p <f> d c c R

The theoretical value of D ., (4.8 x 10 cm2 s ) is in good agreement - 6 2 - 1 with the experimental value of 4.6 x 10 cm s

221

8.2. The Time Lag in Diffusion.

The use of the diffusion time lag in the examination of heterogeneous

polymer systems is not a widely practiced technique. This assertion is

especially true when the solubility of the penetrant in the media deviates

from simple Henry's Law. In this case, the time lag assumes a dependence

on the ingoing pressure (i.e., concentration) which invalidates: the

simple relationship (D = /2/60). However, Paul and Kemp (121) have

shown that provided the roles of the continuous and dispersed phases are

properly accounted for, it is still possible to use the diffusion time

lag constructively.

In the Appendix, an expression for the time lag in diffusion,represent-

ative of the present graft copolymers is derived. In Figure 8.20 the

experimental time lags for the PDMS-g-46.34% PMMA membrane are represented

as the solid symbols; the best fit relationship to equation (A.16) is

shown by the solid line, using three adjustable parameters, namely k , c' D H

and b. The three parameters are also presented in Table 8.7 and for the

purpose of comparison, k , c' and b estimated from the equilibrium sorption D H

isotherm are also included. The agreement is excellent.

The close agreement between the theoretical and experimentally

determined time lags is further illustrated by the dashed curve in —6

Figure 8.9! this was generated using equation ( A.16) with D^ = 4.8 x 10 cm2/s, k = 0.101 cm2(STP)/cm3(poly)cmHg, and k , c' and b from the sorption c D H

isotherm . Although the theoretical curve lies above the experimental

points, this is possibly due to the uncertainty in assigning the value

of D_.

Pressure / crnHg

FIG 8.20 PDMS—G—46.34%PMMA TIME LAG vs PRESSURE PROFILE

223

TABLE 8.7. Dual-Mode Sorption Parameters for PDMS-g-46.34% PMMA/

at 30°C .

Technique (k ) D m H m b

Equilibrium Sorption 0.0845 0.459 0.241

Time Lag 0.0790 0.413 0.269

224

CHAPTER NINE

CONCLUSION

9.1. Graft Copolymers

The present investigation into the transport properties of PDMS-

g-PMMA copolymers represents the third in a series of studies of graft

copolymers in which a glassy polymer (at ambient temperature) is

dispersed within a PDMS matrix. In common with the earlier studies

(92,205), the method of preparation based on 11 initiation by y-radiation"

proved to be both a convenient and effective route to the graft copolymers.

It was found empirically that for graft copolymers of greater than 50%

by volume of PMMA, the sheet structures were not uniform and were

characterised by a laminated structure in which the PDMS-g-PMMA

copolymer was " sandwiched" between a hard crusty exterior of PMMA.

In marked contrast to the earlier studies, the PDMS-g-PMMA copolymers

were translucent rather than opaque, as found with PDMS-g-PS and

PDMS-g-PVAc copolymers suggesting that the PMMA domains are considerably

smaller than the wavelength of light. However, when the samples were

illuminated by a source of intense white light, a " reddish-brown"

tinge was observed. Quantitatively, the intensity of the structural

colour, S, of a heterogeneous medium is given by

S = — — / ( A n ) * n o

225

where r is the particle radius and X the wavelength of the incident

radiation. Provided the refractive indices of the continuous phase

(rio) and the dispersed phase (n 1) are different and the domains are

comparable to the wavelength of light, scattering is observed. Further-

more, the scattering is stronger at the shorter wavelength giving rise

to the " reddish-brown" colouration. The apparent paradox of trans-

lucency, suggesting domains smaller than the wavelength of light, and

scattering suggesting domains larger than the wavelength of light may

be explained by the formation of " domain-clusters" which was also

proposed by Munday (92).

9.2. Graft Copolymer Results.

The examination of polymer morphology by the method of gas transport

proved to be most successful. The steady state permeabilities of the

graft copolymers were in agreement with the Higuchi (168,169) and

Botcher (167) models in which the PMMA domains behaved as simple

impermeable fillers. The deviation from other models was never severe

and in view of the uncertainty in establishing the permeability of pure

PDMS, there was little to choose between the various models. The

eventual selection of the Hignchi model to interpret the permeability

results and diffusion data was a subjective decision.

The permeability of the acetone treated graft copolymers were markedly

lower than the corresponding petroleum ether treated samples and were also

at variance with all the simple models. The lower permeabilities provide

further evidence of possible " lamella" formation of PMMA within the

PDMS matrix as suggested by the electron micrographs.

226

The prima facia case for the " additivity of solubilities" in the

graft copolymers was sustained provided the prior history of the PMMA

domains was accounted for. In this respect, the dual-mode sorption

analysis on the equilibrium sorption isotherms proved enligthtening

and provided a means of separating the history-dependent Langmuir

component from the history-independent Henry's Law species. An alternative

test of the additivity rule, and one which was without recourse to

establishing a sorption model for the dispersed phase was .promising; in

this way, the problem of sample history was surmounted.

Diffusion coefficients determined from the steady state and sorption/

desorption kinetics were in good agreement and suggest the absence of any

time effects. It was hoped that in view of low diffusion coefficients

experienced with the PMMA/propane system that a means of arresting rapid

equilibrium between the continuous phase species and dispersed phase

species may be effected. In this, way, a time-dependent diffusion coefficient

could be introduced. However, the PMMA domains were sufficiently small

to facilitate rapid equilibrium.

The time lag in diffusion formed an integral part of the study and

demonstrated the usefulness of the technique in arriving at the dual-

mode sorption parameters by curve-fitting the time-lag versus pressure

profile. The close agreement between the results obtained by this method,

and the more conventional techniques using the sorption isotherm is a

further indication of the absence of anomalous time effects.

227

9.3. PMMA Results.

The transport and sorption of four simple hydrocarbon vapours in

PMMA formed the second major study in this treatise. The original

intention to study the sorption of propane in a PMMA sheet was impeded

by the low diffusion coefficients encountered with this system. To

surmount the problem, microspheres were prepared which offered a

practical solution. In order to confirm the cogency of. the results

determined by this method, the transport and sorption of ethane in both

the sheet and powder, was studied. Unfortunately, marked differences in

both the diffusion coefficients and equilibrium sorption parameters

were found which cast serious doubts on the propane results. The first

problem to be addressed therefore, was the.possible errors involved in

determining the microsphere dimensions, but this was ruled out with

confidence. Secondly, the balance response time, bed effects and

balance drift were examined using methane and iso-butane. It became

increasingly apparent that the dichotomy between sheet and powder results

was real and that residual solvent in the sheet may be responsible. This

postulate was examined and shown to be true.

9.4. Suggestion for Further Study.

It was found during the course of the present study that several

areas are worthy of further investigation and these are presented below.

A closer examination of the Higuchi model when diffusion into the

domains form the rate-determining step is certainly warranted in the near

future. The PDMS-g-PMMA/iso-butane system may offer a good medium for the

228

study in view of the low diffusion coefficient of iso-butane in PMMA.

The model incorporating reversible immobilisation in the domains proposed

by Tshudy and von Frankenberg (86) may also be useful in the interpretation

of transient sorption and permeation data.

The effect of solvent treatment on graft copolymers in relation to

mechanical and transport properties also deserve further examination.

It has already been shown that " lamella" formation exists and. it would

be enlightening to examine the preferrential orientation of the hetero-

geniety, if any.

Finally, the effect of residual solvent in glassy polymers should be

examined in greater detail. The results in the present treatise

demonstrate that trace solvent in glassy polymers cause sufficient

plasticisation to enhance small scale motions without affecting the

main chain motions. Studies into this area would be of great benefit in

increasing the understanding of polymers below the T .

229

APPENDIX A

The Time Lag in Diffusion for a Heterogeneous Membrane.

The effective time lag for a membrane comprising a continuous phase,

obeying Henry's Law, and a dispersed phase showing characteristic dual-

mode sorption behaviour is presented.

According to the additivity of solubilities, the total concentration

of penetrant per unit volume of membrane is given as follows,

° = CC + CD = kC P + [kDP + < cH ) , b p / ( 1 + b p ) ] (A'1)

where the subscripts C and D refer to the continuous and dispersed phases,

respectively. The concentration c^ and c^, dissolution coefficients,.k^,

and k', and the hole saturation constant (c1)' are defined w.r.t. unit D H

volume of the membrane.

It is also.assumed that the penetrant in the dispersed phase is

effectively immobilised, and the total flux is due solely to the continuous

phase, hence

|f - f (D f. ^ ) (A.2)

3t dx f dx

where D^ is the effective diffusion coefficient of the continuous phase

due to the inclusion of the dispersion.

Integrating equation (A.2) w.r.t. x from x to and w.r.t. t from

0 to t gives

t ft . 3c , 7- .dx.dt = d t X

t 3c V i * c V d t -

o

t 3c D (-7T- ) dt (A. 3) f 3x x o

230

The total flow of gas, through the outgoing face of the membrane

up to time t in the steady state is given by

Q(t) = -ft Dc'

D f h s V d t (A. 4)

Hence, combining (A.3) and (A.4) gives

c(x,t).dx = -Q(t) -9 c c (—7T" ) • dt f Dx X

(A. 5)

Integrating (A.5) w.r.t. x from 0 to / gives

c(x,t).dx.dx = . Q(t) -rfrt Dc'

D _ ( - ) dt.dx , f Dx x o' o (A. 6)

Rearranging,

Q(t) = - c(x,t).dx.dx

but

J - -Df. '!fs Dx

(A. 7)

(A.8)

Thus,

Q(t) = Jt - ? rt

c(x,t).dx.dx (A. 9) o

However, the steady state flux is also given by

0(t) = J(t - 0) (A.10) t-x°

231

where 0 is the effective time lag.

Comparing the equations (A.9) and (A.10), then gives

ft o

c (x).dx.dx x S (A.11)

V c c \ = o

The concentration profile across the membrane is

(cJJ. (x) = (c£)(l - x//) (A. 12)

Since the concentration of penetrant in the continuous phase obeys

Henry's Law, then

(c') (x) = k' P (1 - x//). = k'.p'(x) (A.13) C s G o C

where p'(x) = Pq(1 - x//) (A.14)

Equation (A.14) gives the analogous, hypothetical, pressure profile

across the membrane.

Thus, from equation (A.l)

c (x) = k'p' + k'p' + (ciVbp'/Q + bp') (A.15) s c D H

Substituting (A.15) into (A.11), and integrating gives

0 =-k; ) [ 1 + i k f n y • f w i ( a - 1 6 )

232

where f (y) = y2 + y - (1 + y) + y) ] (A. 17) y ^

and y = bp (A.18) J o

233

APPENDIX B

SAMPLE CALCULATION AND ESTIMATION OF ERRORS

Bli Sample Calculation of an Equilibrium Sorption Concentration and

Diffusion Coefficients for PDMS-g-46.34% PMMA/Propane at 30°C.

Standard experimental parameters:-weight of sample Wl (g) : 0. 2977 + (0.05%)

thickness of sample •e (cm) : o. 108 <+ 1%)

volume of sample Vi (cm3) : o. 2785 (+ 0.5%)

density of sample . pi (g/cm3) : l. 069 (+ 0.1%)

weight of hook 0J2 (g) : o. 0947 (+ 0.1%)

weight of counterweight U>3 (g) : o. 3897 (+ 0.05%)

density of copper p2 (g/cm3) : 8. 980

volume displacement, AV (cm3) = — + — - — = 0.2456 (+ 0.5%) P i P 2 P 2 ~

„ , . AV. x Po x 273.2 x 44.097 x 10 byoyancy correction, B (yg) = .22414 x (273.2 + T) x 76

= 1737p/(273.2 + T)

Experimental conditions:-

charge pressure

ambient temperature

buoyance corrections

Po (cmHg)

T (K)

B (yg)

I 9.79 (+ 0.1%)

: 303.2 (+ 0.1%)

: 56 (+ 1%)

Initial weight reading, R (mg): 15.000 (+1 yg)

Final weight reading, H^ (mg): 15.564 (+ 1 yg)

234

Total weight uptake, M^ (pg) = Rf - R + B = 620 (+ 2p) .

Concentration, c [cm3(STP)/cm3(polymer)]

M x 22414 00

= 44.097 x Vx

= 1.132 (+ 1%)

Solubility, k [cm3 (STP)/cm3(polymer)cmHg]

= c/Po = 0.116 (+ 1%).

The main source of error was found to be the ambient temperature and

better thermostatting control should improve the uncertainties of the

results. The absolute error in c and k was + 1%, though the reproducibility

was found to be better than + %.

Sortpion/desorption kinetics.

The rates of sorption and desorption were followed as a function

of time from which (M/m ) vs. / T and (1 - M. /M ) vs. / T respectively t 00 t CO

were constructed. The gradients for the sorption process and

for the conjugate desorption process are given as

I (s"T) = 3.46 x 10"2 (+1%) s — and

-- -3 IJ (s 2) = 3.13 x 10 (+ 1%) d —

from which

D (cm2/s) = I2 . tt x /2/16 s s = 2.74 X lO-6 (+ 4%)

235

and D (cm2/s) = I 2 . TT X /2/16 d d

= 2.24 x 10"6 (+ 4%)

Thus,

D = (D + D ) /2 s d

= 2.49 x 10~6 (+ 6%)

Alternatively, the time for which I^/M^ =0.5 was determined directly,

from which

t± sorption = 222 (+ 3%) s 2

and t± desorption = 244 (+ 3%) s 2

and D or D, = 0.04919 /2/tt s d —

giving Ds = 2.59 x 10"6 (+ 5%)

and D, = 2.35 x 10~6 (+ 5%) d —

from which

D = (D +DJ/2 s d = 2.47 x 10"6 (+ 7%)

Although the absolute uncertainties in the diffusion coefficients are

large, the reproducibility was often better than + 3%. The larger errors

experienced with the half-time method are mainly due to the uncertainties

in determining the diffusion coefficient using a single point on the

sorption rate curve.

236

B2. Sample Calculation of the Permeability (P), Diffusion Coefficient, (D),

and Solubility (k) from the Transient and Steady State Permeation of

Ethane through PDMS.

Standard experimental parameters!-

Thickness of membrane , / (cm) : 0.09 (+ 1%)

Cross-sectional area of cell , A (cm2) ! 4.39 (+ 2%)

Out-going volume , V (cm3) I 676.4 (+ 0.5%)

Experimental conditions'.

Cell temperature , T (K) ! 303.2 (+ 0.2)

Charge pressure , p0 (cmHg) 10.15 (+ 1%)

Room temperature , T (K) : 295.2 (+ 0.3)

The out-going volume was monitored as a function of time, yielding a rate

of flow in the steady state of,

dp/dt (mmHg/s) = 4.17 x 10~A (+ 0.5%)

Steady state flux, J [cm3 (STP)/cm2s] = ^ . 7 d t /oU RT A

=8.36 x 10~5 (+ 4%) j f>

Permeability, P [cm3(STP)cm/cm2s cmHg)] = — L -po

= 7.42 x 10"7 (+ 5%)

Extrapolation of the steady-state flow rate to (t = 0) gives

237

0 (s) = 255 (+ 2%)

/ 2 -6 and the diffusion coefficient, D (cm2/s) = — = 6.67 x 10 (+ 4%) 0 60 —

The solubility coefficient, k [cm3(STP)/cm3(poly)cmHg]

« P/D0

= 0.111 (+ 7%)

The absolute errors in P, D and k were + 5%, + 4% and + 7%, respectively.

The reproducibility in all three quantities were found to be better than

+ ,2%.

238

APPENDIX C

EXPERIMENTAL DATA

Equilibrium Sorption Data for PDMS/Propane.

The isotherms were found to be linear and were constrained to fit

Henry's Law.

The notations used in Table CI are as follows:

T : Temperature (°C)

p : pressure (cmHg)

c I concentration [cm3(STP)/cm3(polymer)]

c/p= k ! solubility [cm3(STP)/cm3(polymer)cmHg]

Ip.c <k> =

I p 2

Membrane characteristics!

weight : 0.3694 g

thickness ! 0.094 cm

density ! 0.980 g/cm3

TABLE CI. PDMS/Propane Sorption Isotherm Results.

T = 30.0 T = 35.0

P c k P c k

5.02 0.506 0.101 4.98 0.448 0.090 9.98 1.004 0.101 10.01 0.899 0.090

15.29 1.537 0.101 14.99 1.348 0.090 29.95 3.015 0.101 ' 19.98 1.800 0.090

<k> = 0.101 <k> = 0.090

239

TABLE CI (continued).

T = 40.0 T = 45.0 T = 50.0

p c k P c k d c k

4.99 0.403 0. 081 4.99 0.366 0. 073 5.02 0.332 0. 066 10.00 0.805 0. 081 10.00 0.730 0. 073 9.99 0.663 0. 066 15.00 1.212 0. 081 15.02 1.102 0. 073 15.01 0.996 0. 066 20.01 1.621 0. 081 29.81 2.199 0. 074 20.01 1.339 0. 067 30.02 2.440 0. 081 30.02 2.001 0. 067

<k> = 0.081 <k> = 0.073 <k> = 0.067

Transient permeation results for ethane and propane through PDMS are

presented. The notation used in Table C2 and C3 are as follows.

T p

9

p

D k

Temperature ( C) pressure (cmHg) time lag (s) permeability coefficient cm3(STP).cm/cm2.s.cmHg diffusion ceofficient (cm2/s) solubility coefficient [cm3(STP)/cm3(polymer)cmHg]

Membrane and apparatus characteristics.

Area 4.39 cm2

thickness : 0.099 cm outgoing volume*. 676.4 cm3

TABLE C2. Transient Permeation Data for PDMS/Ethane

T P 0 107 P 106 D k

25.0 5.06 152.4 4.02 9.85 0.0408 25.0 10.02 152.4 4.08 9.85 0.0414 25.0 19.96 148.6 4.14 10.1 0.0410 30.0 5.03 140.0 4.03 10.7 0.0375 30.0 10.01 140.3 4.08 10.7 0.0381 30.0 19.97 138.0 4.16 10.9 0.0381 35.0 5.05 138.9 4.06 11.6 0.0348 35.0 10.01 131.6 4.09 11.4 0.0358 35.0 19.96 127.9 4.17 11.7 0.0355 40.0 5.03 118.5 4.06 12.6 0.0320 40.0 10.01 119.8 4.11 12.5 0.0327 45.0 5.01 111.7 4.10 13.4 0.0305 45.0 10.01 110.7 4.11 13.5 0.0302 45.0 19.97 109.5 4.18 13.7 0.0304 50.0 5.05 101.0 4.12 14.8 0.0277 50.0 9.98 104.0 4.15 14.5 0.0286 50.0 19.94 102.0 4.21 14.7 0.0286

TABLE C3. Transient Permeation Data for PDMS/Propane.

T P 0 107 P 106 D k

25.0 1.99 254 7.46 5.90 0.126 25.0 5.00 250 7.53 6.02 0.125 25.0 9.95 246 7.65 6.10 0.125 25.0 19.99 241 7.92 6.25 0.126 30.0 1.98 226 7.38 6.63 0.111

30.0 5.10 230 7.41 6.25 0.113 30.0 10.15 225 7.42 6.67 0.111

35.0 2.02 215 7.21 6.99 0.103 35.0 5.05 205 7.16 7.30 0.098 35.0 9.98 208 7.23 7.22 0.100 35.0 19.97 199 7.44 7.55 0.098 40.0 1.97 196 7.00 7.66 0.091 40.0 5.02 192 7.02 7.81 0.089 40.0 9.98 188 7.09 7.96 0.089 45.0 2.01 179 6.93 8.37 0.082 45.0 5.03 174 6.89 8.62 0.079 45.0 10.02 173 6.96 8.66 0.080 50.0 4.96 152 6.94 9.84 0.070 50.0 10.01 166 6.91 9.06 0.076 50.0 14.98 162 6.95 9.23 0.075 50.0 19.95 161 7.07 9.33 0.075

242

Equilibrium sorption data:

PMMA (sheet)/methane, ethane,

PMMA (powder)/ethane, propane, iso-butane

The isotherms were analysed in accordance with the dual-mode sorption

theory using three adjustable parameters, namely k , c' and b. In d h

cases where no significant curvature of the isotherm was evident, the

equilibrium sorption data were constrained to fit Henry's Law.

The following notation are used in Tables C4-C8.

T

P

c

c/p = k

'H

<k> :

temperature ( C)

pressure (cmHg)

concentration [cm3(STP)/cm3(polymer)]

solubility [cm3(STP)/cm3(polymer)cmHg]

i p-c

Z p 2

Henry's Law solutility coefficient

[cm3(STP)/cm3(polymer)cmHg]

hole saturation constant

[cm3(STP)/cm3(polymer)]

hole affinity constant [(cmHg) x]

Sheet Characteristics.

Area

Thickness

Density

87.95 cm2 -3

3.66 x 10 cm

1.18 g/cm:

Powder Characteristics.

Diameter

Density

0.15 x 10 A cm

1.18 g/cm:

243

TABLE C4. PMMA (Sheet)/Methane Sorption Isotherm Results.

T = 30.0 T = 35.0 T = 40.0

p c k P c k P c k

10.48 0.0484 0. 0046 11.06 0.0396 0. 0036 10.03 0.0274 0. 0027 15.03 0.0655 0. 0044 14.97 0.0580 0. 0039 14.97 0.0469 0. 0031 19.97 0.0866 0. 0043 20.01 0.0749 0. 0037 19.93 0.0578 0. 0029 25.02 0.1088 0. 0043 24.99 0.0956 0. 0038 20.08 0.0593 0. 0030 34.99 0.1562 0. 0045 29.99 0.1165 0. 039 24.97 0.0826 0. 0033 40.84 0.1825 0. 0045 34.99 0.1330 0. 0038 26.15 0.0858 0. 0033

27.54 0.0868 0. 0032 34.97 0.1140 0. 0033

<k> = 0.0044 <k> = 0.0038 <k> = 0.0032

TABLE C5. PMMA (Sheet)/Ethane Sorption Isotherm Results.

T = 30.0 T = 35.0 T = 40.0

p c k

10. 02 0. 1646 0. 0164 20. 03 0. 3708 0. 0185 29. 97 0. 5415 0. 0181 32. 07 0. 5727 0. 0179

<k> = 0.0180

p c k

10.01 0.1489 0.0149 20.02 0.2886 0.0144 25.04 0.3886 0.0155

<k> = 0.0151

p c k

10. 08 0. 0996 0. 0099 20. 03 0. 2368 0. 0118 29. 97 0. 3834 0. 0128 31. 08 0. 3691 0. 0119

<k> = 0.0121

244

TABLE C6. PMMA (Powder)/Ethane Sorption Isotherm Data.

T = 1.0 T = 10.0 T = 20.0

P c c P c c P c c (expt) (calc) (expt) (calc) (expt) (calc]

0. 99 0.129 0.137 1. 00 0.095 0.094 2. 19 0.123 0.125 2. 22 0.285 0.287 2. 24 0.203 0.198 5. 00 0.267 0.267 4. 95 0.565 0.564 5. 04 0.403 0.405 8. 06 0.405 0.405 9. 68 0.948 0.939 6. 65 0.500 0.505 10. 03 0.484 0.487 15. 04 1.271 1.280 9. 73 0.677 0.682 15. 21 0.690 0.683 20. 07 1.554 1.558 10. 15 0.714 0.705 19. 73 0.841 0.839 20. 08 1.588 1.558 15. 55 0.974 0.972 19. 73 0.841 0.839 25. 07 1.773 1.811 20. 01 1.167 1.171 22. 83 0.932 0.940 30. 02 2.065 2.047 25. 04 1.381 1.380 29. 99 1.160 1.158

k = 0.0384; c' = 1.23; u h

b = 0.0879

k = 0.0329; c' = 0.840; k = 0.0227; c' = 0.826; u h u h

b = 0.077 b = 0.0455

T = 30.0 T = 40.0 T = 50.0

P c (expt)

c (calc)

P c (expt)

c (calc)

P c (expt)

c (calc)

2. 12 0.081 0.084 2. 00 0.058 0.055 10.60 0.2008 0.0189 5. 48 0.208 0.204 5. 22 0.141 0.040 19.82 0.364 0.018 7. 11 0.257 0.258 7. 62 0.201 0.200 22.80 0.415 0.018 9. 99 0.346 0.348 9. 95 0.253 0.257 29.99 0.532 0.0177 15. 19 0.499 0.499 15. 33 0.383 0.381 20. 25 0.633 0.633 20. 06 0.489 0.486

25. 00 0.591 0.591 29. 99 0.691 0.695

kD = 0. 0190;c^ = 0.560; kD = 0.' .017; c^ = 0.392; <k> = 0.018 b = 0.0396. b = 0.028

245

TABLE C7: PMMA (powder)/Propane Sorption Isotherm Results.

T = 20.0 T = 30.0 T = 40.0

P c c (expt) (calc)

0. 99 0. 213 0. 216 2. 03 0. 415 0. 395 3. 02 0. 554 0. 536 5. 00 0. 740 0. 766 7. 02 0. 939 0. 959 9. 99 1. 208 1. 203

15. 00 1. 563 1. 557 20. 02 1. 892 1. 878 25. 00 2. 188 2. 180 29. 90 2. 454 2. 468

k = 0.0538; c' = I.OOO; d h

b = 0.196

P c c (expt) (calc)

1. 03 0. 172 0. 167 1. 66 0. 273 0. 252 2. 04 0. 288 0. 298 3. 34 0. 426 0. 436 4. 99 0. 567 0. 582 7. 27 0. 771 0. 754

10. 05 0. 927 0. 937 12. 65 1. 105 1. 093 16. 64 1. 313 1. 316 20. 01 1. 484 1. 495 25. 07 1. 770 1. 756 29. 97 1. 995 2. 001

k = 0.0468; cl 0 0.699! d h

b = 0.200

P c c (expt) (calc)

1. 14 0. 105 0.106 2. 77 0. 231 0.232 5. 65 0. 406 0.412 9. 95 0. 617 0.630 11. 41 0. 718 0.697 15. 11 0. 854 0.854 15. 15 0. 881 0.856 20. 01 1. 024 1.048 24. 97 1. 228 1.235 29. 97 1. 429 1.416

k = 0.0328', c' = 0.548! l/ h

b = 0.125

T = 50.0

p c c (expt) (calc)

1. 23 0. 083 0.079

2. 04 0. 128 0.127

3. 54 0. 206 0.208

7. 03 0. 870 0.371

10. 30 0. 489 0.493

12. 06 0. 564 0.570

15. 04 0. 696 0.676

20. 40 0. 836 0.841

24. 10 0. 958 0.968 29. 92 1. 149 1.143

kD = 0.0258; c = 0.523; » n o '> —

T = 60.0

p c c (expt) (calc)

1. 56 0. 068 0. 067

3. 00 0. 122 0. 121

5. 00 0. 188 0. 189

s- 98 0. 309 0. 315

12. 03 0. 411 0. 406

(—» ui

03 0. 525 0.522

20. 46 0. 644 0. 649

24. 36 0. 760 0. 759

k = 0.0275; c i = o . i n ; l) h

v ... n i ~>r.

246

TABLE C8. PMMA (Powder)/iso-butane Sorption Isotherm Results.

T = 40.0

1.29 0.144 0.145 3.21 0.356 0.338 5.14 0.502 0.511 7.03 0.634 0.664 9.87 0.889 0.872 12.04 1.016 1.018 15.26 1.257 1.218 20.41 1.473 1.511 29.97 2.007 1.999

k = 0.0385; c' = 1.30; d n

b = 0.0616

Transient sorption data:

PMMA (Sheet)/methane, ethane

PMMA (Powder)/ethane, propane iso-butane.

The diffusion coefficients of the various penetrants in PMMA were

determined from the kinetics of sorption (and desorption).

The notations adopted in Tables C9-C13 are as follows.

P c c (expt) (calc)

D, s 1 diffusion coefficient as determined by

kinetics of sorption (cm2/s)

D : Arithmetic mean of sorption/desorption

diffusion coefficients (cm2/s)

< D >

c

i y D. (cm2/s) ni=l 1

concentration [cm3(STP)/cm3(polymer)]

T temperature (°C)

247

D I diffusion coefficient of mobile species m

(D) _ I diffusion coefficient at zero concentration c=0

TABLE C9. PMMA (Sheet)/Methane Transient Sorption Data.

T = 30.0 T = 35.0 T = 40.0

c (expt) 101Q D c (expt) 1Q10 D c (expt) 1Q10 D

0.0484 6.0 0.0580 7.8 0.0469 10.78

0.0655 5.9 0.0956 7.8 0.964 10.77

0.1562 5.5 0.1165 7.0

<D> = 5.8 x 10~10 <D> = 7.5 x 10~10 <D> = 10.78 x 10~10

TABLE C10. PMMA (Sheet)/Ethane Transient Sorption Data.

T = 30.0 T = 35.0 T - 40.0

c (expt) 1011 D e (expt) 1Q11 D c (expt) 1Q11 D

0.1646 3.0 0.1489 3.0 0.2368 6.4

0.3562 1.6 0.2886 3.6 0.3834 5.8

0.5415 2.0 0.3886 3.4

<D> = 2.2 x 10"11 <D> = 3.3 x 10"11 <D> = 6.1 x lO"11

248

TABLE Cll. PMMA (Powder)/Ethane Transient Sorption Data.

T = 1.0 T = 10.0 T = 20.0

c (expt) 1015 D c (expt) 1014 D c (expt) 1014 D s s s

0.3053 7.18 0.683 1.99 0.279 4.49

0.8391 10.15 1.163 2.37 0.352 5.40

0.9372 8.87 0.840 4.49

1.555 10.56

< D > = 9.86 x 10~15 < D > = 2.18 x 10"14 < D > = 4.79 x 10"14 s s s

T = 30.0

c (expt) 1013 5

0.176 1.42

0.201 1.15

0.234 1.15

0.622 1.42

<D r> = 1.29 x 10" 1 3

u

249

TABLE CI2. PMMA (Powder)/Propane Transient Sorption Data.

T = 20.0 c (expt)

0.257 0.420 0.658 0.870 1.242 1.486 1.873 2.353

1 0 1 6 5

9.3 8.1

9.2 11.6

14.4 12.9 15.8 14.9

D = 2.9 x 10 m

-15

(D) n = 6.4 x 10 c=U -16

T = 30.0 c (expt) 1 0 1 5 5

0.168 2.1 0.252 3.3 0.298 2.1 0.368 2.8 0.522 2.9 0.720 3.4 0.788 3.7 0.933 3.5 1.197 3.9 1.513 4.7

D = 7.4 x 10"15, m

(D) n = 18.6 x 10 c=U -16

T = 40.0 c (expt)

0.105 0.340 0.406 0.617 0.811 0.718 0.854 1.055

1015 D

7.4 8.5 9.2

10.6 11.1 8.9

11.1 12.5

D = 19.5 x 10 m

-15

(D) _ = 63.1 x 10 c=U -16

T = 50.0 T = 60.0

c (expt) 1014 D c (expt) 1Q14 Dn s o 0.066 1.9 0.068 5.2 0.163 2.1 0.309 6.1 0.206 2.2 0.525 5.8 0.356 2.1 0.644 5.8 0.449 2.4 0.644 2.4 0.799 2.6 1.150 2.9

D =45.4 x 10~15, D = 73.5 x 10~15, m m

(D) . = 172.3 x 10"16 (D) _ = 430.1 x 10"16 c=U C=U

250

TABLE CI3. PMMA (Powder)/iso-butane Transient State Sorption Data,

T = 40.0

conc (expt)

0.215

0.502

0.862

1.468

<D > = 1.23 x 10 s

1 0 1 6 d

1.25

1.20 1.20 1.27

-16

Equilibrium Sorption Data for PDMS-g-PMMA/Propane.

The isotherms were analysed in accordance with the dual-mode sorption

theory, using three adjustable parameters, namely k , c' and b. Where d h

no significant curvature of the isotherms was evident, the data was

analysed in terms of Henry's Law.

The notation adopted in Tables C14-18 are as follows,

p ! pressure (cmHg)

c I concentration [cm 3(STP)/cm 3(polymer)]

k^ : combined Henry's Law dissolution constant,

[cm 3(STP)/cm 3(polymer)cmHg]

hole saturation constant [cm 3(STP)/cm 3(polymer)]

£ = k p

<k>

hole affinity constant (cmHg

solubility coefficient [cm 3(STP)/cm 3(polymer)cmHg]

i p.ca p 2

volume fraction PMMA

251

TABLE CI4. PDMS-g-10.35% PMMA Equilibrium Sorption Data (<(» = 0.0875).

T = 30.0 T = 40.0 T = 50.0

p c k P c k P c k (expt) (expt) (expt)

2. 30 0.247 0.107 1.27 0.107 0. 084 1.57 0.107 0.068

3. 69 0.375 0.102 2.57 0.212 0. 082 3.48 0.235 0.068

5. 92 0.614 0.104 4.34 0.355 0. 082 3.80 0.265 0.069

8. 37 0.621 0.103 5.18 0.424 0. 082 4.39 0.305 0.069

11. 60 1.190 0.103 5.91 0.481 0. 081 8.45 0.575 0.068

16. 53 1.691 0.102 10.36 0.848 0. 082 11.79 0.788 0.067

20. 37 2.079 0.102 15.03 1.210 0. 08 19.21 1.335 0.070

23. 36 2.377 0.109 20.06 1.609 0. 080 29.69 2.032 0.068

24. 88 2.537 0.102 25.02 2.002 0. 080 35.94 2.447 0.068

29. 24 2.983 0.102 34.03 2.716 0. 079

<k> = 0.102 <k> = 0.080 <k> = 0.069

TABLE C15. PDMS-g-30.67% PMMA Equilibrium Sorption Data (<J> = 0.2687)

T = 30.0 T = 35.0

P c c P C c

(expt) (calc) (expt) (calc)

1.31 0.161 0.163 1.57 0.163 0.162

2.61 0.304 0.307 3.57 0.351 0.351

3.77 0.430 0.327 6.64 0.622 0.623

5.90 0.637 0.637 9.56 0.873 0.871

7.07 0.755 0.749 13.02 1.157 1.158

7.22 0.756 0.763 18.04 1.567 1.566

9.42 0.977 0.970 20.28 1.746 1.746

10.66 1.084 1.085 24.71 2.099 2.101

19.15 1.847 1.859 26.95 2.282 2.280

25.02 2.395 2.389

30.9 2.844 2.844 •

k D = 0.0731, c' = 0.129

H k D = 0. 626, c^ = 0.205,

b = 0.222 b = 0.111

252

TABLE C15 (continued).

T = 40.0

p c c (expt) (calc)

1. 03 0. 098 0. 099

2. 39 0. 226 0. 219

4. 69 0. 405 0. 409

5. 52 0. 473 0. 475

7. 07 0. 592 0. 596

8. 09 0. 675 0. 674

11. 12 0. 903 0. 905

14. 30 1. 144 1. 143

16. 02 1. 282 1. 272

20. 59 1. 607 1. 611

22. 99 1. 785 1. 788

27. 90 2. 151 2. 150

k = 0.0731, c' = 0.129 u h

b = 0.222

T = 45.0

p c c (expt) (calc)

1. 09 0. 088 0. 090

3. 27 0. 265 0. 259

5. 19 0. 405 0. 400

7. 06 0. 516 0. 532

9. 07 0. 673 0. 670

11. 74 0. 855 0. 851

14. 21 1. 020 1. 015

18. 09 1. 250 1. 296

21. 23 1. 481 1. 472

23. 58 1. 636 1. 624

29. 97 2. 025 2. 033

36. 01 2. 419 2. 417

k = 0.0626, c' = 0.205 L) H

b = 0.111

T = 50.0

p c c (expt) (calc)

1. 21 0. 093 0. 089

2. 39 0. 164 0. 170

4. 20 0. 291 0. 289

8. 75 0. 578 0. 574

10. 26 0. 663 0. 666

12. 00 0. 770 0. 770

20. 31 1. 262 1. 264

25. 00 1. 541 1. 539

33. 13 2. 014 2. 014

k = 0.0578, c' = 0.117,

D H

b = 0.158

TABLE C16. PDMS-g-39.76% PMMA Equilibrium Sorption Data (<|> = 0.3541).

T = 30.0 T = 35.0 T = 40.0

(expt) (calc)

1. 01 0.144 0.140

2. 05 0.265 0.266

5. 01 0.574 0.575

8. 02 0.852 0.854

12. 01 1.205 1.200

15. 01 1.449 1.452

20. 02 1.863 1.863

25. 42 2.296 2.300

29. 99 2.673 2.666

34. 47 3.021 3.024

k = 0.0784, c' = 0.367, D H

b = 0.197

(expt) (calc)

1. 51 0.179 0.717

2. 65 0.291 0.287

4. 02 0.415 0.417

5. 47 0.546 0.547

7. 03 0.675 0.681

9. 66 0.895 0.896

15. 13 1.320 1.322

20. 26 1.714 1.707

27. 20 2.220 2.218

35. 87 2.844 2.847

k = 0.0707, cl = 0.372, d h

b = 0.138

(expt) (calc)

1. 19 0.123 0.117

2. 45 0.238 0.233

5. 19 0.464 0.467

8. 99 0.759 0.765

12. 27 1.005 1.008

17. 26 1.367 1.363

21. 29 1.641 1.642

22. 08 1.702 1.696

30. 07 2.232 2.235

k D = 0 .064, c^ = 0.422

b = = 0. 090

T = 45.0

p c c (expt) (calc)

2.02 0.176 0.177

4.73 0.381 0.384

8.80 0.662 0.663

11.98 0.877 0.869

16.57 1.161 1.56

20.00 1.354 1.365

30.02 1.963 1.967

36.05 2.329 2.325

k.. = 0.058, c' = 0.282 d rl

b = 0.134

T = 50. 0

P c c (expt) (calc)

1. 44 0.110 0.108

3. 05 0.223 0.221

5. 00 0.355 0.352

5. 00 0.350 0.352

8. 75 0.585 0.589

11. 37 0.749 0.749

15. 04 0.966 0.968

18. 49 1.165 1.171

23. 45 1.470 1.458

34. 99 2.113 2.116

k_ = 0.056, c' = 0.215 1j h

b = 0.103

254

TABLE C17. PDMS-r-46.34% P M M A Equilibrium Sorption Data (* = 0.4177).

T = 30.0 T = 35.0 T = 40.0

p c c (expt) (calc)

0.81 0.147 0.143

1.93 0.312 0.309

3.57 0.508 0.514

3.83 0.544 0.544

5.67 0.749 0.744

7.23 0.906 0.902

9.79 1.132 1.149

14.01 1.541 1.538

20.04 2.082 2.073

21.84 2.236 2.231

29.97 2.928 2.935

k = 0.085, cl = 0.459 u

b = 0.241

T = 45.0

p c c (expt) (c ale)

1. 36 0.119 0. 118

3. 71 0.300 0. 301

4. 86 0.382 0. 384

7. 09 0.540 0. 538

11. 72 0.838 0. 838

14. 88 1.037 1. 033

21. 95 1.451 1. 455

24. 41 1.597 1. 599

29. 99 1.925 1. 922

k = 0.056, c' = 0.302 l) h

b = 0.183

p c c (expt) (calc)

0. 61 0.079 0.080

1. 52 0.191 0.185

2. 90 0.322 0.326

4. 92 0.507 0.510

8. 28 0.788 0.786

10. 09 0.931 0.928

15. 24 1.316 1.316

24. 26 1.969 1.972

29. 90 2.379 2.377

k D = 0. 070, c R = 0.321

p c c (expt) (calc)

2. 48 0.189 0.192

3. 07 0.233 0.233

4. 78 0.350 0.348

7. 03 0.494 0.492

10. 03 0.681 0.675

15. 07 0.953 0.970

18. 11 1.147 1.145

20. 01 1.258 1.253

25. 51 1.570 1.562

29. 83 1.798 1.804

k = 0.' 055, c' = 0.022, D H

b = 0.140

p c c (expt) (calc)

0. 71 0.080 0.079

1. 69 0.172 0.178

3. 74 0.364 0.358

4. 95 0.452 0.455

9. 04 0.756 0.757

12. 16 0.977 0.973

15. 44 1.191 1.194

20. 47 1.525 1.526

24. 50 1.786 1.788

29. 99 2.143 2.142

k = 0.063, c' = 0.302

d n b = 0.183 b = 0.213

T = 50.0

255

TABLE C18. PDMS-g-51.45% PMMA Equilibrium Sorption Data (j> = 0.4682)

T = 30.0 T = 35.0 T = 40.0

(expt) (calc) (expt) (calc) (expt) (calc)

1. 01 0.176 0.164 1.00 0.136 0.130 1. 01 0.120 0.117

3. 01 0.431 0.430 3.01 0.350 0.354 3. 11 0.329 0.327

5. 28 0.680 0.682 5.00 0.546 0.546 5. 03 0.490 0.496

7. 03 0.853 0.856 5.01 0.545 0.547 7. 09 0.660 0.663

10. 06 1.126 0.135 7.02 0.735 0.724 10. 00 0.890 0.880

15. 02 1.559 1.558 10.00 0.956 0.966 15. 03 1.232 1.231

17. 17 1.734 1.734 15.01 1.349 1.345 24. 95 1.881 1.877

20. 01 1.964 1.962 20.22 1.716 1.718 29. 99 2.193 2.193

24. 99 2.365 2.335 25.02 2.052 2.053

29. 71 2.722 2.720 29.99 2.398 2.394

36. 03 3.197 3.205 36.03 2.800 2.803

k = 0.074, c' = 0.627, k = 0.065, c' = 0.545 D H D H

b = 0.163 b = 0.136

k = 0.0595, c' = 0.524 D H

b = 0.120

T = 45.0 T = 50.0

P c c P c c (expt) (calc) (expt) (calc)

1.01 0.104 0.102 0.99 0.087 0.087

3.02 0.271 0.277 3.01 0.231 0.242

5.02 0.433 0.429 5.07 0.384 0.381

6.20 0.512 0.514 7.00 0.504 0.502

10.94 0.831 0.830 9.99 0.692 0.679

15.00 1.094 1.085 15.00 0.953 0.960

20.36 1.410 1.412 20.0 1.225 1.230

25.02 1.680 1.692 24.97 1.492 1.494

29.99 1.987 1.987 29.99 1.757 1.757

36.01 2.347 2.347 36.01 2.074 2.070

k p = 0 .058, c' h

= 0.306 k D = 0 •051, c' = 0.287

b = 0. 167 b = 0. 149

256

TABLE CI9. PDMS-g-24.14% PMMA* Equilibrium Sorption Data.

T = 30.0

p c c (expt) (calc)

0. 99 0.125 0.126

2. 02 0.243 0.244

5. 00 0.556 0.554

6. 99 0.751 0.750

9. 99 1.041 1.039

15. 00 1.513 1.511

20. 00 1.975 1.977

25. 00 2.435 2.441

30. 02 2.909 2.905

34. 95 3.362 3.361

k = 0.092, cA = 0.167

d h

b = 0.266

T = 45.0

T = 35.0

p c c (expt) (calc)

0. 99 0.106 0.104

2. 02 0.209 0.207

5. 02 0.492 0.490

7. 02 0.665 0.670

10. 00 0.930 0.929

15. 04 1.355 1.356

20. 01 1.773 1.768

22. 66 1.986 1.986

29. 97 2.576 2.582

32. 12 2.761 2.756

k D = 0 .079, c'

n = 0.249

b = = 0. 113

T = 50.0

T = 40.0

p c c (expt) (calc)

0. 99 0.092 0.093

2. 00 0.183 0.182

5. 09 0.436 0.438

7. 03 0.593 0.592

10. 02 0.823 0.823

15. 02 1.201 1.201

20. 03 1.575 1.574

25. 00 1.939 1.941

30. 02 2.311 2.311

34. 99 2.677 2.676

k = 0.073, c' = 0.153 D H.

b = 0.158

p c c (expt) (calc)

1. 00 0.086 0.087

2. 00 0.162 0.165

5. 05 0.392 0.387

7. 00 0.528 0.524

9. 96 0.732 0.731

10. 00 0.736 0.734

19. 94 1.408 1.424

24. 99 1.773 1.774

30. 02 2.136 2.122

34. 85 2.453 2.456

k D = 0 .069, c^ = 0.051

b = ; 0. 554

p c c (expt) (calc)

1. 00 0. 074 0. 074

2. 00 0. 141 0. 144

5. 02 0. 347 0. 347

6. 77 0. 469 0. 460

9. 98 0. 657 0. 664

15. 00 0. 977 0. 979

20. 01 1. 287 1. 291

25. 00 1. 611 1. 600

34. 95 2. 210 2. 214

k d = 0.061, c^ = 0.082,

b = Q.,179

257

Transient permeation results for propane through PDMS-g-PMMA are

presented . The notation used in Tables C19 to C27 are as follows.

T *. Temperature (°C)

p : pressure (cmHg)

0 : time lag (s)

P : permeability coefficient

[cm 3(STP)cm/cm 2 s cmHg]

D ! diffusion coefficient (cm 2/s)

k ! solubility coefficient

TABLE C20. Membrane and Apparatus Characteristics.

Sample Area/ Thickness/ Outgoing cm 2 mm volume/cm 3

PDMS-g-10.35% PMMA 4.39 0.94 678.7

PDMS-g-30. 67% PMMA 4.15 1.02 680.3

PDMS-g-39.76% PMMA 4.39 1.04 677.9

PDMS-g-46.34% PMMA 4.39 1.08 680.7

PDMS-g-51.45% PMMA 4.39 1.07 683.9

PDMS-g-24.14% PMMA* 4.39 0.94 673.1

PDMS-g-32. 28% *

PMMA 4.39 0.91 680.5

TABLE C21. PDMS-g-10.35% PMMA Transient Permeation Data.

7 - 6 -T P •0 10 P 10 D k

30.0 1.14 391 5.50 3.77 0.1A6

30.0 11.67 358 5.72 A.11 0.139

35.0 6.29 338 5.A8 A. 35 0.126

35.0 11.2A 329 5.58 A.A8 0.125

AO. 0 2.3A 331 5.A3 A.A5 0.122

AO. 0 20.56 297 5.62 A.96 0.113

AA.5 5.99 29A 5.30 5.01 0.106

AA.5 10.66 286 5.34 5.1A 0.10A

50.0 2.22 282 5.20 5.23 0.099

50.0 9.7A 26A 5.23 5.A9 0.096


7 - 6 -T P 0 10 P 10 D k

30.0 2.31 699 A.18 2.A8 0.168

30.0 11.52 601 A. 22 2.89 0.1A6

35.0 2.66 603 A.11 2.87 0.1A3

35.0 11.16 538 A.15 3.22 0.129

AO. 0 A.25 532 A. 05 3.26 0.12A

AO. 0 15.25 A8A A. 08 3.59 0.114

50.0 3.28 A39 3.98 3.95 0.101

50.0 11.23 A16 3.96 A. 16 0.095


T P 0 10 7 P 10 6 D k

30.0 4.13 915 3.19 1.97 0.162

30.0 12.60 816 3.22 2.21 0.146

35.0 2.47 880 3.12 2.05 0.152

35.0 6.09 803 3.15 2.25 0.140

45.0 5.42 657 3.06 2.74 0.111

45.0 20.03 573 3.14 3.15 0.099

50.0 5.30 584 3.04 3.08 0.098

50.0 13.31 547 3.06 3.29 0.093


T P 0 10 7 P 10 6 D k

30.0 5.07 1034 2.89 1.88 0.154

30.0 18.20 823 2.98 2.36 0.127

35.0 5.74 902 2.86 2.16 0.133

35.0 12.22 804 2.89 2.42 0.119

40.0 5.35 815 2.83 2.39 0.119

40.0 17.54 685 2.86 2.84 0.101

45.0 11.25 659 2.81 2.95 0.095

45.0 14.10 638 2.81 3.05 0.092

50.0 6.27 629 2.77 3.09 0.089

50.0 11.51 591 2.77 3.29 0.084

260


T P 9 10 7 P 10 6 D k

30.0 5.00 1291 2.51 1.47 0.169

30.0 14.99 1059 2.58 1.80 0.143

35.0 5.03 1138 2.49 1.68 0.149

35.0 9.42 1032 2.51 1.85 0.136

40.0 4.97 971 2.48 1.97 0.126

40.0 15.08 848 2.49 2.25 0.111

45.0 4.95 901 2.45 2.12 0.116

45.0 15.03 765 2.46 2.49 0.100

50.0 5.00 784 2.44 2.43 0.100

50.0 15.47 693 2.43 2.75 0.088

TABLE C26. PDMS-g-24.14% PMMA* Transient Permeation Data (Unannealed).

, 7 - 6 -T P 0 10 P 10 D k

20.0 4.97 698 4.91 2.11 0.232

20.0 10.02 607 4.91 2.42 0.202

25.0 4.97 603 4.71 2.44 0.193

25.0 9.98 553 4.77 2.66 0.179

30.0 4.94 563 4.72 2.61 0.181

30.0 11.95 497 4.76 2.96 0.161

35.0 9.96 450 4.65 3.27 0.142

35.0 14.94 428 4.68 3.43 0.136

40.0 9.97 406 4.58 3.63 0.126

40.0 14.96 387 4.59 3.81 0.121

261

TABLE C27. PDMS-g-24.14% PMMA* Transient Permeation Data (Annealed at 50°C).

T P 0 io 7 P 10 6 D k

30.0 1.88 668 3.55 2.20 0.161

30.0 10.09 606 3.56 . 2.43 0.147

35.0 5.06 590 3.58 2.49 0.143

35.0 14.97 547 3.57 2.69 0.133

40.0 5.01 519 3.49 2.84 0.123

40.0 14.99 487 3.55 3.02 0.118

45.0 4.99 469 3.48 3.14 0.111

45.0 15.02 432 3.57 3.41 0.105

50.0 5.08 430 3.57 3.43 0.104

50.0 10.12 364 3.49 4.04 0.086

TABLE C28. PDMS-g-32.28% PMMA* Transient Permeation Data (Unannealed).

T P 0 10 7 P 10 6 D k

30.0 4.99 769 3.12 1.79 0.174

30.0 9.91 716 3.09 1.92 0.161

30.0 14.99 662 •3.18 2.09 0.152

35.0 5.00 702 3.01 1.97 0.152

35.0 10.00 650 2.86 2.12 0.182

35.0 14.99 631 2.81 2.19 0.128

262

TABLE C29. PDMS-g-32.28% PMMA* Transient Permeation Data (Annealed at 50°C).

T P 0 10 7 P 10 6 D k

30.0 5.05 886 2.34 1.56 0.150

30.0 10.01 817 2.35 1.69 0.139

35.0 4.85 780 2.45 ]. 77 0.138

35.0 10.00 721 2.46 1.91 0.129

40.0 5.03 672 2.39 2.05 0.117

40.0 9.98 648 2.47 2.13 0.116

40.0 14.93 576 2.41 2.40 0.100

45.0 5.00 584 2.34 2.36 0.992

45.0 10.02 557 2.39 2.47 0.968

50.0 5.02 504 2.41 2.73 0.883

50.0 10.04 488 2.42 2.82 0.858

TABLE C30. Specific Free Volume of Polymers at 25°C.

Polymer - L /

SFV _ g cm 3

( 5 W 2 -1

cm^ s

< V c - o '

kJ mol 1

Ref.

Poly(ethylene) 4.5 6.7 x 10" 7 55.8 203

Natural Rubber 4.5 1.96 x 10" 7 39.9 202

Polystyrene 6.1 1 x 10" 1 1 55.0 92

Poly(carbonate) 6.7 -12

2.5 x 10 66.6 71

Poly(methyl methacrylate) 7.9 1.9 x 10~ 1 7 86.5

TABLE C31. Transport Parameters of Various Gases and Vapours in PMMA at 30°G.

Gas Diameter/A e/k L n ( 5 ) c = 0

kJ mol

LnS Li

Lnk D (AH ) _ s c=0 Ref.

He 2.65 10 -17.1 21.3 -3.54 +1.72 82

o 2 2.92 120 -19.1 25.6 -4.79 +1.97 81,82

N 2 3.15 95 -21.1 32.9 -2.48 -6.9 82

H 2 O 3.10 -17.4 42.4 207,208

C0 2 3.23 189 -19.2 44.1 -2.02 -4.10 -30 82,83

CHI,A 3.81 148 -21.3 49.2 -5.43 -25.6

C 2 H 6 4.38 220 -24.5 81 -4.02 -31.4

C 2 H 6 4.38 220 -29.7 57 -3.18 -3.96 -32.5

C 3H8 5.24 255 -33.9 86.5 -1.68 -3.06 -35.2

CZ,H10 ' 5.82 271 -36.6 -2.13 -3.3

a

b

c

measurements determined on the plane sheet

measurements determined on the solid sphere

results at 40°C only

264

TABLE C32. PDMS-g-46.34% PMMA Sorption Kinetics at 30.0°C.

p/cmHg ,cm3(STP)

conc/ o f _ cm 3(polymer)

10 6 D/cm 2 s" 1

0.81 0.147 1.42

3.57 0.508 1.97

3.83 0.544 1.95

5.67 0.749 2.08

9.79 1.132 2.49

21.84 2.236 2.80

TABLE C33. PDMS-g-46.34% PMMA Time Lag Data at 30.0°C.

P/cmHg 0/min

0.47 20.2

0.99 19.7

1.85 19.0

2.35 17.9

3.00 17.6

3.95 16.9

5.07 16.2

6.52 15.3

7.93 15.1

8.73 14.9

8.99 14.6

11.07 14.4

12.94 13.8

14.14 13.4

16.00 13.1

18.20 12.7

20.98 12.8

25.44 11.8

29.50 11.5

(kJ = 7.91 x 10 2 , (c') = 4.13 x 1 0 _ 1 , b = 0.268 D m H m

266

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