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This is to certify that the

dissertation entitled

THE MELT RHEOLOGY 0F A-B BLOCK COPOLYMERS WITH
SPHERICAL MICRODOMAINS

presented by

Ekong A. Ekong

has been accepted towards fulfillment
of the requirements for

 

Ph.D. degree in Chemical Engineering

é/«s2 {/83 W

 

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

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THE MELT RHEOLOGY OF A-B BLOCK COPOLYMERS WITH
SPHERICAL MICRODOMAINS

By

Ekong A. Ekong

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Chemical Engineering

1983

ABSTRACT

THE MELT RHEOLOGY OF A-B BLOCK COPOLYMERS WITH
SPHERICAL MICRODOMAINS

By
Ekong A. Ekong

A kinetic network model for polymeric melts that contain
spherical microdomains is presented and compared with experimental
results of poly(styrene-b-butadiene), (Mn = 232,000-10,000, % wt
PS = 94.1). A novel form for the segment distribution in the matrix
with a constraint at the point of attachment to the domains is
developed. Consistent expressions are developed for the rate of
creation and destruction as a function of deformation in flows. A
key parameter in this development is the degree of repulsion between
segments in the interfacial region.

Transient and steady stresses are derived for uniaxial
extensional flows and compared with an ABS melt data in the litera-
ture. At low Hencky strain rates (made dimensionless with a character-
istic relaxation time) an apparent yield stress is predicted dependent
on the range of repulsion parameter which correlates with the com-
position of the rubbery component.

Computations were done with this model also to obtain steady
and transient stresses in uniaxial shear flows. These predictions

were compared with melt rheology data gathered in this work over a

Ekong A. Ekong

temperature range of 120°C to 175°C. The shear viscosity data above
150°C indicate homopolymer-like behavior; the data at 130°C indicate
the presence of a two-phase structure. The dynamic shear viscosity
as well as the steady shear viscosity data show trends similar to
those reported by Ghijsels and Raadsen (1980) with triblock copolymer
melts at low strain rates. The observed stress growth curves show a

1 with a strain

stress overshoot at strain rates as low as 0.01 sec-
at peak stress of about 0.5. Estimation of parameters in the theory
and the sensitivity of predicted stress behavior to different para-
meters is discussed. While the theory is able to predict observed
low strain rate behavior in steady and dynamic testing, it does not

predict an overshoot in stress growth curves at such low strain rates.

In the evergreen memory
of

Kokomma Bassey-Ubong Ekong

Archibong Akpan Udoh Ekong
and

Phillip Akpabio Ekong

ii

ACKNOWLEDGMENTS

I would like to express my gratitude and appreciation to
Dr. Krishnamurthy Jayaraman for his guidance and direction throughout
the course of this work. My special thanks also go to

Dr. Eric A. Grulke, Professor Donald K. Anderson, and
Professor Dennis Heldman for their many helpful suggestions and
guidance.

Dr. Lu Ho Tung of Dow Chemical Company, Midland, Michigan,
for the preparation and characterization of the copolymer sample
used in this work.

The Department of Chemical Enginnering, Michigan State
University, for providing me the financial support in the course of
this work.

Miss Carole Nicolas for her friendship and assistance in the
experiment and data analysis and importantly all my friends, espe-
cially Dr. Mohammad Jefri, Dr. Kai Sun, Dr. Chien-Pin Chen, and

Mr. Barry Colley, for their towering friendship and moral support.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES .

NOMENCLATURE .

Chapter

I.

II.

III.

IV.

INTRODUCTION

1.1 Material Background

1.2 Evolution of the Microstructure
1.1.1 Gaussian Network Theory.
1.1.2 Non-Affine Motion Assumption
1.2.3 The Yamamoto Network Theory
1.2.4 The Reptation Theory .

BLOCK COPOLYMER MELT PROPERTIES AND THEORIES .

2.1 Previous Rheological Studies
2.2 Optical Studies . .

A TRANSIENT NETWORK MODEL FOR POLYMERIC MATERIALS
WITH SPHERICAL MICRODOMAINS . .

3.1 Objectives .
3.2 The Rate Terms
3. 3 The Macroscopic Stress Tensor

PREDICTED STRESS BEHAVIOR IN EXTENSIONAL FLOWS
4.1 Uniaxial Steady Extensional Flow
4. 2 Results . .
4.2.1 Steady State Stress
4.2.2 Stress Transients .
PREDICTED STRESS BEHAVIOR IN SIMPLE SHEAR FLOWS .

5.1 Simple Steady Shear
5.2 Results

iv

Page

vi

xi

o—J

32
32
32
4o
41
41
42
42
52

52
55

Chapter
5.3

VI. SAMPLE CHARACTERIZATION AND EXPERIMENTAL TECHNIQUES.

010101010101
(33019me

Oscillatory Shear Flow

Material and Sample Preparation

Electron Microscopy

Morphology . .
The Modified Weissenberg Rheogoniometer .
Sample Loading and Temperature Control
Rheometric Testing . . .
6.6.1 Oscillatory Shear Experiments

6. 6. 2 Steady Simple Shear Experiments

VII. RESULTS AND DISCUSSION

\1 \l\l\l
4> wNH

Introduction .

Phase Transition Temperature . .

Viscoelastic Behavior Above the Transition

Temperature .

Viscoelastic Behavior Below the Transition

Temperature . . .

7.4.1 Estimation of Model Parameters . .

7.4.2 EXperimental Evaluation of the Trans-
ient Network Model . .

7.4.3 Steady State Predictions

7.4.4 Transient Predictions

VIII. CONCLUSION AND RECOMMENDATION .

8.1
8. 2

APPENDICES
REFERENCES

Conclusion . .
Recommendations for Further Study .

Page

101

107
116

121
121
125

134

134
136

142
158

Table
2.1
2.2
4.1
6.1
6.2
7.1
7.2

LIST OF TABLES

Viscosity of block copolymers vs. homopolymers .

585 samples

Model parameter "a" from data of Munstedt

Block copolymer characterization

Property of glassy continuous phase

Phase separated block copolymer melt properties

Material constants from experimental data

vi

Page
19
21
46
75
75

107
118

Figure

1.1

LIST OF FIGURES

Variation of block copolymer morphology with compo-
sition . . . . . . . . . . . . . .

Reduced dynamic viscosity and elastic modulus vs.
reduced frequency for SBS 7-43-7

A polymer network with rubber domains .
Normalized non-Gaussian distribution functions .

Normalized extensional viscosity vs. dimensionless
strain rate. Effect of repulsion parameter "a"
with e = 0.01 . . . . .

Extensional viscosity vs. dimensionless strain
rate. Effect of the destruction parameter "c"
with a = 0.05 . .

Normalized transient extensional viscosity vs.
dimensionless time as a function of strain rate.
Effect of the destruction parameter "a" with a = 0

Normalized transient viscosity vs. dimensionless
time as a function of strain rate. Effect of the
repulsion parameter ”a," with P = 0.1 and c = 0.01

Normalized transient viscosity vs. dimensionless
time. Effect of the repulsion parameter "a" with
T = 1 and e = 0.1 . . .

Normalized shear viscosity as a function of dimension-
less shear rate. Effect of the destruction parameter,
"Elnsa = 0 . . . . . . . .

Normalized shear viscosity function and normalized
first normal stress difference functions. Effect of
repulsion parameter "a," e = .005, E = 0.05 .

Normalized shear viscosity and first normal stress
difference functions. Effect of the slip factor "5"
a = 0.03, e = 0.005 . . . . . . . .

Page

26

33

38

44

47

49

50

51

6O

61

6.6a

Normalized transient viscosity function. Effect of
the repulsion parameter a, e = 0.005, g = 0.05 .

Normalized transient viscosity function. Effect of
the repulsion parameter "a," y = 3, e = 0.005, E =
0.05 . . . . . . . . . . . . . . .

Normalized transient first normal stress difference
function a = 0.0, e = 0.005, E = 0.05 .

The normalized complex dynamic and steady viscosity as
functions of frequency and shear rate respectively

w = y . . . . . . . . .
Typical EM micrograph of ultra- thin section of poly
(styrene- b- butaiene) specimen at x 50,000

TYPical EM micrograph of ultra-thin sections of
poly(styrene-b-butadiene) specimen at x 150,000

Weissenberg Rheogoniometer internal (Sangamo Controls
Ltd.) . . . . . . . . . . .

Cone and plate platen

Combined cylindrical and cone and plate platen
(Mooney type) . . . . . . .

Non-sinusoidal waveform of oscillatory shear stress
at strain amplitude of 0.15 .

Picture showing test material extruding from gap after
a shearing for 8 min; 9 = 0.43 seC'l. Shear insta-
bility is due to stress fracture. T = 130°C

Quenched sheared materials after a shearing time of
8 min. Left hand specimen sheared at y= 0.096 _1
sec-1. Right hand specimen sheared at y= O. 43 sec

T = 130° C . . . . . . . . .

Shear stress growth function of S- B at 130° C using
the Mooney Platen . . . . . . .

Dynamic viscosity and storage modulus vs. frequency
of 8-8 at various temperatures . . . . .

Complex and steady viscosity of 8-8 at 150°C as a
function of frequency and shear rate, respectively

viii

Page

63

64

65

72

77

78

81
82

82

88

9O

9O

92

96

98

Figure

7.3

7.3a

Complex and steady viscosity of S—B at 130°C as a
function of frequency and shear rate, respectively

Dynamic and steady viscosity of 8-8 at 130°C as a
function of frequency and shear rate, respectively

Steady shear viscosity function of S-B at 150°C

Normalized transient shear stress function of S-B at
150°C . . .

Shear stress relaxation of S-B at 150°C

The steady shear viscosity as a function of shear
rate of 3-8 at 130°C

The Steady shear viscosity as a function of shear
rate of 3-8 at 124°C

Normalized transient shear stress function of 3—8
at 130°C

Normalized transient shear stress of S-B at 130°C .

Normalized shear stress relaxation as a function of
time of S-B at 130°C

Normalized shear stress relaxation as a function of
time of S-B at 130°C

Normalized shear stress relaxation as a function of
time of S-B at 124°C

Loss modulus and loss factor as functions of frequency
at 130°C

Evaluation of model parameters using linear visco-
elastic functions (dimensionless) . .

Evaluation of model parameters using linear visco—
elastic functions (dimensionless) . .

Comparison of steady shear viscosity with transient
network model .

Comparison of steady shear viscosity with model

Comparison of stress growth results at 130°C with
model. . . . . . . .

xi

Page

99

100
102

104
105

108

109

111
112

113

114

115

119

120

122

123
124

126

 

Figure Page

7.20 Comparison of stress growth at 130°C with model
A = 1.25; a = 0.55, g = 0.05, O = data;

Model . 127
7.21 Comparison of stress growth at 130°C with model of
A = 4 sec . . . . . . . . . . . . . 128
O
7.22 Comparison of stress growth at 130°C with model of
A = 1.25 . . . . . . . . . . . . . 129
O
7.23 Comparison of stress growth at B0°C with model of
A = 4 sec . . . . . . 130
O
7.24 Comparison of stress growth at 130°C with model of
. . . . . . . . . . 131

A = 1.25 sec .
O

7.25 Strain at the stress overshoot in transient shear
curves at 130°C . . . . . . . . . . . . . 132

A(B,N)

A..
13

DH

9)?

NW

(13.0

0 ZIO-
A.

llm RU

(t',t)

—+,

f<g.t)

fe(0.N)
G(R,N)

(D G) G) G)!
A 200 (.4.

NOMENCLATURE

Helmoltz free energy

As defined in Equation (1.16)
Adjustment constant (Acierno model)
Constant (in Johnson and Segalman model)

The repulsion range parameter (in non-Gaussian) network
model of this work

Finger strain tensor for the deformation t' + t, the
components are given by Brc = 2(axr/3x%)(axc/8xj)
where x; and xi are rectangular cartesian coordinates
of places occupied at times t' and t by a typical
particle of the macroscopic continuum

Configuration tensor (Giesekus, 1982)

Constant creation rate coefficient, sec—1

The deformation rate tensor, (V! + VVT)
Strain measure (Johnson and Segalman)
Distribution function of an i-type strand with n links.

Distribution function of the "most probable" network
segment

Segment distribution function at equilibrium
Rate of creation function = Lj (R,N)

Modulus (Dynes/cmz), Acierno model

Modulus of rubber elasticity, nokT

Plateau modulus, dynes/cm2

Relaxation modulus, dynes/cm2

xi

Material relaxation spectrum
The effective Hookean spring constant
Non-linear spring modulus (Yamomoto theory)

Scaler damping function in (Wagner model) where 11
and I2 and first and second invariant of the finger
strain tensor

Factorized memory damping function of the N G
Newwork model

Identity matrix

Integral kernel's defined in Appendix A
Strand complexity index

/:T (in Chapter V)

Steady-state recoverable compliance
Boltzman's constant

Normalized constant

Deformation gradient, VV

Effective deformation gradient VV - EQ
Creation rate of the ith strand withn links
Molecular weight of the primitive chain (DE theory)
Weight average molecular weight

Number average molecular weight

Memory function (Lodge's Network Theory)

Number of subunits in the "most probable" network
segment with statistical length, 1

Number of beads in Kramer's chain

Total concentration of j-segments at time t (Jongshaap)

xii

"S

70? |DU

302

"—1

Equilibrium value of Nj
Steady and transient first normal stress difference

Number of equivalent random links (Gaussian network
model)

Initial concentration of network segments, C/BO
Extra stress tensor = S

Ensemble averaged end-to-end vector for a typical
network segment

Velocity of the network segment

End-to-end vector of macromolecule at equilibrium
End-to-end vector of an active network segment
(defined in Chapter III)

r/Nl, where N1 is the extended length of the segment
Radius of cone-and-plate platens

Radius of inner cylinder (Mooney bob)

Radius of outer cylinder (Mooney cup)

Extra stress tensor = :(t)

Tangential transient shear stress during shear growth
Steady state tangential shear stress

Relaxation stress after the cessation of shear
Structural parameter (Acierno model)

Macroscopic material coordinate (Cartesian)
Temperature

Transformation matrix equation (6.2)

Present time

xiii

u = sin wt

V(x,t) Polymer velocity at the position vector x and time t

x,y,z Cartesian components of the dimensionless end-to-end
segment vector, R

Z(R,N) Configuration partition function

Greek Symbols

 

B(R,N) Destruction coefficient of the most probable network
segment

80 Destruction rate constant, sec'l(=l/AO)

BC Chain constraint exponent

v0 Oscillatory strain amplitude

y Magnitude of shear rate

2 Rate of strain tensor

i Dimensionless shear rate y 80

F Magnitude of strain rate

f Dimensionless strain rate == F/BO

c Destruction rate constant (N - G . Model)

8B Link bension coefficient

c, 2 Drag and tensorial drag coefficient

n Steady shear viscosity, Poise

n; Non-dimensional elongational viscosity

nE+ Non-dimensional transient elongational viscosity

no Zero shear viscosity, poise

ln*| Complex viscosity

n Normalized steady shear viscosity, Sxy/nokT§

xiv

2(psnaz)

1NI, 0

o» 02' El
Awe)

07
fl

Dimensionless dynamic and loss viscosities respectively
Dimensionless complex viscosity

Gap angle

Characteristic relaxation time

Destruction coefficient of the ith strand with n links

Destruction coefficient due to thermal motion
(Wagner Model)

Slip coefficient (Phan-Thien and Tanner)
Microscopic cartesian coordinates (Johnson and Segalman)

Transformed coordinate frame where R = T . Q

~
~

Dimensionless present time, Bot
Dimensionless elapsed time 80(t-t')
Torque generated in the shearing gap

Loss coefficient associated with the survivability of
strand at the elapsed time of deformation

Relaxation time of the Acierno model

Phase shift

Zero shear first normal stress difference. Dynes/cm2
Frequency of oscillation

Normalized frequency of oscillation

Partial time derivative
Contravariant time derivative E-é%( ) - vy . ( ) - ( )

- va

XV

 

CHAPTER I

INTRODUCTION

1.1 Material Background

 

In recent years there has been an increasing attention to
formulation of composite polymeric materials which combine desirable
properties of their components. However, it is often difficult to
blend polymers because of the incompatibility of polymeric chains of
different polymeric molecular structure. This problem has been over-
come by sequentially polymerizing different constituents to obtain
block or graft copolymers. Since its first theoretical conception by
Mark (1953), and its first commercial production by Shell in 1965
(Holden et al., 1969), the possible practical applications of block
copolymers have been many and varied. Block copolymers such as poly
(styrene-butadiene-styrene), SBS undergo a phase separation consist-
ing of the thermoplastic polystyrene block linked by its ends to an
elastomeric butadiene block to generate a polymeric network structure
in which the endblocks serve as physical reversible multifunctional
crosslink sites.

These systems, without vulcanization have rubber-like proper-
ties akin to rubber vulcanizates, but are moldable at temperatures
above the glass transition of the thermoplastic component (Van Breen

and Vlig, 1966; Bishop and Davison, 1969). High resilience, high

tensile strength, highly reversible elongation and abrasion resistance
may be obtained in triblock copolymer solids with careful choice of
monomers and block length. Diblock copolymers are finding increas-
ing use as a ternary component to mix highly incompatible homopolymers
of low molecular weight, by emulsification (Ramos and Cohen, 1977).
They may be used also for recovery and reuse of polymeric waste
products such as polyolefin mixtures.

Styrene block polymers can be manufactured by anionic poly-
merization reaction. Styrene-butadiene, SB, diblock, Styrene-
butadiene-styrene, SBS and butadiene-styrene-butadiene, BSB, triblocks
are produced by the reaction of styrene molecules in the presence of
a lithium catalyst to form polystyrene-lithium complex which disinte-
grates on further addition of the component butadiene to form a
diblock of 8—8 or a triblock S-B-S if more styrene is added into the
mixture. Phase separation occurs between the blocks leading to
the formation of microdomains that are responsible for the specific
properties of the block copolymer. As illustrated by Matsuo et al.
(1968), (see Figure 1.1) with SBS samples, the molecular weights of
the different blocks determines the overall morphology of this
polymeric system. Uniform spherical domains are formed when the
low M.W. component is 20 wt% or less. As the amount of this component
increases, the spheres do not grow in diameter beyond a certain
size, but instead, are transformed to uniform cylinders. At a still
higher fraction of elastomer, the cylinders become platelets, while

at midrange compositions (40-60 wt%) of each component, the material

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(b,c) SBS-3 (40%8); (d,e) SBS-5

Polymers cast from toluene solution, and

SBS-1 (20%8);

(60%B).
black and the polystyrene phase is white (Matsuo et al.,

stained with 0504, so that the polybutadiene phase is
1968.

(a)

Figure 1.1.-—Variation of block copolymer morphology with composition

consists of alternate lamallar of styrene and butadiene. As the per-
centage of elastomer is increased still further (not shown in the
figure), the phase structure goes through the same changes in reverse
--with the elastomer now constituting the continuous phase.

Theories based on statistical thermodynamics of microphase
separation have been proposed to predict the effect of molecular
parameters such as block length and thickness of interphase between
the blocks, on copolymer structure; these have been reviewed by
Folkes and Keller (1973). The present study deals with the mobility
of segments in block copolymer melts. To data no rheological theories
have been advanced to predict highly non-linear behavior observed in
block copolymer melts. Such studies are essential in establishing
processing conditions for these materials.

A number of rheological characterization studies have been
done on block copolymeric systems (Arnold and Meier, 1970; Kraus and
Gruver, 1967; Holden et al., 1969; Chung and Gale, 1976; and Ghijels
and Raadsen, 1980). The major results of these works are: (1) block
copolymer melts, especially triblocks with glassy domains, show an
apparent yield stress at shear rates, 9 << 1; (2) at low shear rates
or stresses, the viscosity of block copolymers is greater than that
of either homopolymer; (3) the Cox-Merz rule does not hold, in
general, for block, copolymer melts. In general, it is observed
that the complex viscosity, |n*l is higher than the steady shear
viscosity at w = v; (4) at large deformation rates, the dynamic shear

moduli, steady shear viscosity, and first normal stress difference

values approach those of their homopolymeric counterparts with com—
parable overall Mn. This complex rheological behavior is mainly
attributed to the persistence of the two-phase structure of block
copolymers into the melt state.

In this study we seek to extend the transient network theory
framework described in the following section, to block copolymers by
regarding regions of the dispersed phase as temporary junction sites
that impart a three-dimensional structure to block copolymers. This
allows us to develop a constitutive equation for Viscoelastic flow
properties of block copolymers. Despite an inherent weakness of
failing to use all available structural information obtainable from
non-rheological techniques of characterization (e.g., MW and MWD),
network theories are useful phenomenological models taking into
account the evolution of the microstructure. It is in describing
this evolution of microstructure that most of the available network
models differ; we proceed to discuss the description of microstruc-

ture.

1.2 Evolution of the Microstructure

 

1.1.1 Gaussian Network Theory

 

Following the theoretical formulation of constant connectiv-
ity of network models for elastomeric materials by Green and Tobolsky
(1946), Lodge (1954, 1956, 1968) and Yamamoto (1956, 1957, 1958)
extended the network theory to include polymer melts and concentrated

solutions visualizing them as a temporary network formed by transient

 

junctions or entanglements. All the Gaussian network theories stem
from the following two equations.

Stress Tensor:

P(t) = ZanZi<BB>in (1 1)

Strand distribution function fin:

 

.n (at) - fin/mm (1.2)
The distribution function, fin(B’t) is defined such that f}n(R,t) dR
if the concentration at time t of strands of complexity i and com-
posed of n equivalent random links (of length 1) with ensemble-
averaged end-to-end vectors with the range R to B + dB. The term

Hn = 3KT/nl2 is the effective Hookean spring constant of an n-link
strand, such that HnR can be interpreted as a force on the strand.
The terms Lin are the strand creation rates and fi/Ain denote strand

destruction rates, with a strand destruction coefficient given as

Aih‘ Angular brackets indicate an average value calculated with
respect to fin‘ The success of rheological constitutive equations
have been mainly based on how well the terms B, Lin and l/Ain approxi—
mate the true microstructure dynamics occurring in the polymeric
medium. The original ideas on how these terms may be modelled were
laid down by Lodge (1954), in deriving the Lodge rubber-like model.
In this review it will be useful to state them, in general, and focus
on how several researchers have modified these assumptions to achieve

useful constitutive equations.

 

Assumption 1: Ensemble-average positions of junctions move
affinely and can be identified with particles of the
equivalent macroscopic continuum. In particular, if
the melt is given a time-dependent homogeneous deforma-
tion, we have

3:13 - W (1.3)

where 3 denotes an ensemble-average strand end-to-end
vector, and V(x,t) denotes the polymer velocity at the
place of the position vector x and time, t. The superior
dot denotes a time derivative.

Assumption 2: At any instant t, the set of network strands
in a unit volume may be regarded as mutually exclusive,
mutually independent subsets. The probability per unit
time that any strand shall leave the network is a function
1/Ain(t) say at t, i, and n.

Assumption 3: (i,n) strands are created with spherically
symmetric distribution of R vectors, i.e., at a rate
which can be expressed as a function Lin(Bat) of i,
n,t and the magnitude B alone. Furthermore, at the
instant of creation,all (in,) strands have the same
distribution as that of a set of free n-Gaussian
strands.

Using equations (1.1) to (1.3), the constant volume condition
(V - V = 0) and Lin(B) expression based on the Gaussian chain assump-
tion, a general constitutive equation may be written of the form:

t
P(t) = im(t,t')B(tt')dt‘ (1.4)

~
~

-CX)

where the memory function m(t,t') is given by

A ll
m(t,t') = kltzLin(t') exp(-(t.dt"|iin(t")) (1.5)
1n

and

Here B(t,t') is the Finger strain tensor for the kinematic deforma—
tion from past time t' to the present time t.

If all the creation and loss rates are constant, i.e., all
strands have the same complexity, Lodge's "rubberlike liquid model”
results. This model predicts a frequency dependent dynamic shear
moduli, but fails to show the dependence of steady shear viscosity
on the shear rate or a non-zero second normal stress difference.

In order to correct these imperfections, several workers,
as will be shown in this section, have proposed empirically different
choices of the creation and loss rates, but leave intact the assump—
tion that the microstructure flows affinely.

If creation and loss rates are functions of instantaneous

values of strain rate invariants, various equations including those

 

of Meister (1971) and Careau (1972) are obtained. If the creation
and loss rates are functions of the instantaneous values 0f.§££§§§
invariants, we obtain the equation of Kaye (1966). These and other
related equations have been tabulated elsewhere in a common notation
(Lodge, 1974). Most of these equations, usually characterized by
many adjustable parameters, predict steady shear viscosity dependency
on the shear rate and show a second normal stress difference. How-
ever, they fail to reduce to the appropriate constitutive equation

of linear Viscoelasticity at low deformation rates.

The next integral constitutive equations are the strain-
dependent (K-BKZ type) equations in which the memory function
includes a scalar function of strain depending on the elapsed time,
t' + t as a factor. Recent step-strain data have given compelling
evidence for such a "strain/time" factorization (at least in the
terminal zone of the relaxation spectrum) (Osaki et al., 1971;

Laun, 1978). Out of this class of equations is the Wagner model
(Wagner, 1979a; Wagner and Stephenson, 1979b) with a memory function

of the form

A

m(t,t') = KtZLjh(I

. t',t), 12(t',tDexp(t'-t)/A. (1.6)
J

1( J

where E is written as an abbreviation for E2.

J 1n
In this model,assumption (2) is replaced by two independent mechan-
isms for strand loss, one due to thermal motion with constant loss
probabilities l/Ag and the other the survivability of strand at the
elasped time of deformation denoted by l/Td(t',t).

Since thermal motions determine Aj and not td(t',t), then

A? and E. would depend on the microstructure of the material, but the

J
Td would be structure independent. The loss process is thus given by

1 -1 1
73-(tist) - 49 + Td(t'at) (17)
J

Equation (1.6) is obtained by combining equation (1.7) and (1.5)

and by taking

 

10

tl
h(I ,I ) = exp dtlu
1 2 m (1.8)
’1'.

The damping function h is chosen empirically to fit stress relaxation
data for single-step strain experiments and stress growth data in
step-function elongation rate experiments. The resulting h—expression
with two adjustable parameters gave a good description of data from
a variety of experiments in shear and elongation. A functional of
the h-factor was further proposed by Wagner and Stephenson in order
to better predict recovery following elongation at constant rates.
One major drawback as to the use of two times in A(t',t) in
the Wagner model is that it is not in general possible to find an
equivalent differential form for the constitutive equation. For
some applications, it appears helpful to have a differential equation
for the stress tensor.
A fairly successful constitutive equation for polymer melts
and concentrated polymer solutions proposed by Acierno et al. (1976)
expressed the creation and loss processes as functions of structural
variables that describe how far the microstructure deviated from
equilibrium. This structural variable is governed by an independent

kinetic equation of the form

d'. %

T°——3= —‘.-". P. ’. 1.9
j dt 1 xJ axJ(tr:J/2GJ) ( )

where P5 is the non-equilibrium part of the jth contribution to the

extra stress tensor, g given by

11
t
Pi = Ej - J m.(t,t')dt'I (1 10)

gj is computed from a Maxwellian-type constitutive equation

g./G. + T. §%-(= ) = 2T.y (1 11)

Interconnection between this model and the fundamental balance law
was made clear by Jongschaap (1981) who noted that the segment

loss probability function l/Aj in this network model is given by

L.1fi/”% -1.” 1
A. T- 25- R. ('12)

J J J J

Both sides of Equation (1.12) are multiplied by xj. The result is
combined with Equation (1.9), xj is replaced by Nj/Njo and the

result multiplied by'Njo to obtain

dN. N. N.
Bil=ii’7i (1-13)
J J

Here Nj = ij(R,t)d3R is the total concentration of j- segments at

time t, and Njo is the equilibrium value of Nj‘ If in Equation (1.13)

Nj/Tj is identified with the creation rate Lj(t) = JLj(R,t)d3R, then

12

dNJ. .. N.
—=L.-?} (1.14)
1

dt 3

which is the integral of Equation (1.2) over all configuration
space. Thus the differential equations forthe structural variable
of Acierno et al. are directly related to the fundamental balance
law of the Network theory. The Acierno model is seen to allow for
the segment creation and loss rates to depend on the deformation
through the trace of the non-equilibrium part of the stress tensor.
In the context of the Network theory, it is not evident why the
particular form of the destruction process was chosen and why it is

successful.

1.2.2 Non-Affine Motion Assumption

 

The Network model of Phan-Thien and Tanner (1977) and Phan-
Thien (1978) also allow the function creation and loss rates to
depend on trgj', but in a more logical manner. More importantly,
the Thien and Tanner model altered affine motion assumption of Network
theory (see Assumption 1) allowing the network junction to "slip"
with respect to an equivalent continuum specified by the macroscopic
velocity gradient VV. In so doing, Phan Thien and Tanner introduced
an empirical "slip tensor" to describe non-affine motion of the net-
work functions and postulated it to be a linear function of the
T).

rate of deformation tensor 0 E %(VV + VV Consequently, Equation

(1.3) is reformulated as

R= (W - :2) -B (1.15)

13

in which the parameter E is the slip coefficient.

At the same time, Johnson and Segalman (1977) developed a
continuum theory of viscoelasticity which allows non-affine deforma-
tion. Two deformation histories were defined. One was the deforma—
tion history g(t) observed at macroscopic level; the other, E(t), a
history of microstructure deformation was allowed to be non-affine
with the macroscopic motion. A relationship between these two motions,

Xi and E1 at the present time t, in Cartesian coordinates was given

by

DJ?

‘1)' (1.16)

+ (‘2‘ xj,i

‘1,j
where a is a constant. They then defined a strain measure E(t,t')

governed by

GE . _ .
5em: ) - {youm ) (1.17)

KI
~

EU'J') =1

and substituted this measure of strain into the Lodge network expres-

sion to obtain

~
~

t
P(t') = { m(t,t') g(t,t') §(t,t')Tdt' (1.18)

00

As with the Thien and Tanner model, the Johnson and Segalman model
predicts a variety of non-linear rheological behavior well, particu-

larly, the viscosity is found to decrease with the shear rate. The

 

14

Phan Thien and Tanner model contains two dimensionless constants e
and E that are determined through elongational flow and viscometric
flow experiments respectively. For shear flows, Phan Thien showed
that the Thien and Tanner model was identical to Johnson and Segal-
man's if E = 1-5.

The choice of the range of "a (0 < a < 1) as reported by the
authors through comparison with experiment was not easily perceived
until the work of Lau and Schowalter (1980). They explained the
fundamental basis of both models by pointing out that these were
objective constitutive equations that can be formulated with a strain
measure derived from appropriate linear combinations of the rate of
change of material coordinates in the material fixed (corotational)
reference and the space—fixed reference (code formational) frames.
They chose e1 strain measure related to the combination ¢ expressed

in component form as

Vj,i

e
_l
11
A
H
1
on
v
<
1
min

"1'71 2

Then a strain tensor (x, t, t') was defined by

3ij (5’ t9t') = 9(59t) E()_(9 tat.)

urn I

and E(x,t',t')

ll
IIH

If c = (1 - a), the Johnson and Segalman model is obtained while the
Thien and Tanner's model results when c = E. Such a rate of deforma-

tion measure can also be used to construct anisotropic fluid models

 
 

15

associated with dilute solutions (Gordon and Showalter, 1972). A
weakness in both models is that they predict damped oscillations

in shear stress at large deformation rates.

1.2.3 The Yamamoto Network Theory

 

Yamamoto (1956, 1957, 1959) presented a more fundamental
network theory (cf. Lodge's theory) for concentrated polymer solu-
tions and melts. The general form of the microstructure dynamics
equation (Equation [1.2]) was originally proposed in the first of
three papers in which the creation rate function and chain breakage
coefficient are functions of the end-to-end distance and orientation
of the segments in the flow field. Unlike Lodge's theory, the net-
work is considered as non-Gaussian with the result that the free
energy of the network segment is a function of the end-to-end dis-

tance. Thus Equation (1.1) can be written as

E = Z {HN(R)BB 1‘ (B.t)dB (1.19)

It is to be noted that the spring modulus HN(R) is allowed to depend
on the deformation of the segment so that non-linear springs may be

conceived. Yamamoto has shown that physically plausible assumptions
about the segment creation rates and loss probabilities lead to vis-

cosity that decrease with shear rate, a negative second normal stress
coefficient, and an elongational viscosity that first increases with
the elongational rate, goes through 11 maximum and then decreases at

higher elongation rates. If the destruction coefficient is made

 

 

16

independent of the segment extension, the ensuing strain measure in
steady elongational experiments is an exponential that increases
with time in the orientation of the chains. At a critical rate of
strain, the chains are elongated infinitely without breakage leading
to an infinite elongational viscosity. Yamamoto then argued that in
actual systems, the chains will break down at finite elongations and
the destruction coefficient should be a function of the segmental
extension. In this lies the germ of ideas behind recent network
models which avoid an infinite elongational viscosity by assuming
deformation dependent destruction coefficients.

Further studies on the Yamamoto theory, especially the non-
Gaussian aspect, have been minimal with regard to modelling visco-
elastic fluids. Generally, the theory does not give constitutive
equations in an explicit form devoid of summations and integrations
over molecular variables. However, non-Gaussian network models are
receiving increasing attention in the study of rubber elasticity
(Chompff, 1977). Recently, Fuller and Leal (1981) have evaluated a
form of non-Gaussian distribution function obtained by a Kuth and
GrUhn type perturbation of the Gaussian distribution function. They
reported no trend in their results different from those of a Gaussian
network model. In the present work, a non-Gaussian distribution
function will be presented that yields strikingly different predic-
tions.

The Yamamoto network theory offers clearly a direction in
formulating Viscoelastic models of various polymeric systems if an

accurate description of its segment distribution function is found.

 

 

17

In the Lodgean theory one has no choice but to assume that the
Gaussian distribution of the chains prevails. This has been success-
ful for homopolymeric melts especially at small deformation rates,
confirming the theory that homopolymeric entanglements are a result
of weak secondary forces between primary chains, and occupy a length
scale of the order of a statistical subunit. This distribution does
not represent the microphase structure that determines copolymer melt

properties at small deformations.

1.2.4 The Reptation Theory

 

Failure to incorporate molecular variables into the network
theory still stands out as one of the major weaknesses of the several
versions of the model posed above. Recently, the entanglement con-
cept has been viewed in quite a different light by Doi and Edwards
(1978a, 1978b, 1978c). The idea that entangled chains rearrange
their conformations by reptation, i.e., curvilinear diffusion along
their own contours was first introduced by DeGennes (1971). Doi and
Edwards have formulated a theory (DE), relating the dynamics of
reptating chains to mechanical properties in concentrated polymer
liquids. They assumed that reptation would be the dominant motion
in a medium of linear long chains. Employing equations from the
theory of rubber elasticity, they calculated the contribution of
individual chains to the stress following a step strain and related
the subsequent relaxation of stress to conformational rearrangement
via reptation (1978b). Without further assumptions, notably the

”independent alignment approximation." 1AA, they arrived at a

18

constitutive equation of the BKZ type good for aribtrary deformation
histories. Irlparticular, they showed that for monodisperse entangled
linear chain polymer liquids, the plateau modulus, zero-shear viscos—
ity and steady state recoverable compliance were functions of chain
properties as

O O
GN a M

03
no a M

a: a M° (1.20)

where M0 is the molecular weight of the primitive chain. These rela-
tions agree fairly well with observed data (Graessley, 1980). The
only parameters present in this theory are the reptation tube
diameter ”a" and a monemeric friction coefficient. Due to the con-
straining nature of domains in the block copolymer systems, it is

not very evident how the reptation theory can be applied to block

copolymer rheology.

 

CHAPTER II

BLOCK COPOLYMER MELT PROPERTIES AND THEORIES

2.1 Previous Rheological Studies

 

In this chapter we wish to examine in detail data collected on
the melt rehological properties of block copolymers and rubber modi-
fied polymers to identify molecular variables affecting their behav-

ior.

TABLE 2.1.—-Viscosity of block copolymersa vs. homopolymers

 

 

Polymerb Percent S ViscosityC
80B 0 3.2
65-818-65 13 13
103-538-105 27.5 29
165-528-163 39 118
19S-3lB-193 53 36.5
243-258-245 65 31
33S-18B-33S 8O 28

838 100 5.5

 

aNote that at 175°C a lot of the domains have been destroyed
(Chung and Gale, 1976).

bMolecular weights of blocks in thousands

CAt shear stress of 2 x 105 dynes/cm2 and a temperature of
175°C (Holden et al., 1969).

19

 

 

20

Table 2.1 summarizes the steady shear melt viscosity data at
a constant stree, reported by Holden et al. for several different
samples of S-B-S triblock copolymer as well as the homopolymers,
polystyrene, and polybutadiene with the same order of overall mole-
cular weight. It is readily seen that the styrene content affects
the melt viscosities of the triblocks. 0n the other hand, it has
been shown by Matsuo that the M.W. of the individual blocks affects
the morphology of the block copolymer system. Holden explained the
large viscosities exhibited by the block copolymers as due to the
two phase structure persisting into the melt. Looking at Figure 1.1
we note that randomly distributed cylinders of polystyrene domains in
a polybutadiene matrix is the morphological structure of SBS with
39%S content which has an anamolously large viscosity.. Again cylin-
drical domains of polybutadiene is the projected morphology for the
SBS with 65%5 content, but has a lower viscosity. It can, there—
fore, be concluded using Holden's data that viscosity of block
copolymer melts is strongly dependent on the morphology of the
respective blocks, block length (M.W.) of the thermoplastic block
and chemical nature of the center block.

Arnold and Meier (1970) presented the dynamic Viscoelastic
data for various samples of SBS melts at low frequencies as shown
in Table 2.2. We note that the 22-50 sample has an S content of
about 35% by weight while the 14-50 sample has about 31%S. They
deduced that the difference of the slope d log n'/d log w between

the two samples was due to the presence of semicontinous domain phase

 

 

 

21

TABLE 2.2.--SBS Samples

 

 

SBS Nominal block Slope of log n'
Sample mol. wt.a vs log w
10-50 10-50-10 -O.36
14-50 14—50-14 -0.40
22-50 22-50-22 -0.66
14-60 14-60-14 -O.36
14-70 14-70-14 -O.36
MOPS/97b 97 0

 

aIn thousands

bMonodispersed polystyrene, M.W. = 97,000

of polystyrene in the former sample as opposed to "dispersed poly-
styrene domains" in the latter case. They further proposed a quali-
tative rheological theory for block copolymers system, stating that
at very low deformation rates, the molecular network is essentially
intact. At intermediate deformation rates, the three-dimensional
network will be disrupted and the system behaves as large star-
shaped aggregates. Finally, at high deformation rates, these aggre-
gates will, in turn, be disrupted and the system will behave as an
assemblage of individual non-aggregated molecules.

While Arnold and Meier's dynamic data agree fairly with
those of Holden et al., it is to be noted that method of sample
preparation used in their study, crumbs may have affected the
results. Ghijsels and Raadsen have found that the use of crumbs leads
to less reproducible results, especially at low deformation rates than

the use of compression moulded samples.

22

They also observed that |n*(w)| > n(i)|?=w for all these
block copolymers the disparity being greater for block copolymers
terminating in polystyrene. They attributed these to the disruption
of the domain network structure which must occur in steady flow, but
not necessarily in small amplitude oscillations. A further explana-
tion of the phenomenon observed above is that the presence of domains
in block copolymers disallows some conformation, which would have
deen available to chains through entanglement slippage. This, then,
tends to increase the elastic free energy of the chains as well as
the resulting modulus.

The two phase structure can also be manifested in block
copolymer solutions depending on the choice of the solvent (Kotaka
and White, 1973). When a good solvent for both components is used,
triblock and diblock copolymers solutions behave as homopolymeric
solutions. When a poor solvent for one component is used, 9.9.,

SBS or SB in decane, a two-phase structure of insoluble PS in a
solution of PB in decane results. The observed rheological behavior
is, however, different for triblock and diblock copolymers. In SBS,
the PB component dissolved is connected at both ends to the insoluble
PS component thus creating a three-dimensional network structure even
at a low concentration of the copolymer. In the diblock, there is

no formation of a three-dimensional network, but rather a micelle
structure in which the PS segments form a rigid core. Upon increas-
ing the polymer concentration, the number of such micelles increases

and eventually they would be arranged in a regular three-dimensional

 

23

array. The morphology of such mesormophic structures have been
revealed by electron microscopy studies of Gallot (1978). From
Kotaka and White's findings, these mesomorphic structures can be
classified as elastic gels that can undergo a complete breakdown in
structure by continuous shearing.

Another strong influence on Viscoelastic properties of block
copolymers is the interphase region existing at domain boundaries
containing segments of both blocks. Statistical thermodynamic theor-
ies of Meier (1974) and Leary and Williams (1973) indicate that the
volume fraction occupied by the interphase and, therefore, the degree
of compatibility increase with decreasing molecular weight. With
increasing temperature, a continuous increase in miscibility would
also be anticipated involving growth of the interphase at the expense
of the two pure phases, subsequent complete disappearance of the
domain phase and then the continuous phase and, ultimately, complete
homogeneity. Such predictions have been confirmed experimentally by
Chung and Gale (1976) through rheological studies. Using moderate
M.W. samples of SBS with spherical polystyrene domains, they noted
that at high temperatures, the melt experiences a transition from a
multiphase structure to a homogeneous structure. The flow behavior
above this temperature is characterized by a Newtonian viscosity
at low deformation rates and by low elasticity. Such behavior has
been observed also by Kraus (even with high M.W. diblocks and
Holden et al. using triblocks).

Kraus and Rollman (1976) have predicted the volume fraction

of the mixed interlayer for various M.W. triblock copolymer samples,

 

 

24

using the theory of Meier. They then correlated dynamical mechani-
cal moduli as a function of temperature with the results of Meier.

The composition, a of the domain phase segments changes continuously
from zero to unity within the range of the interlayer. It was
assumed by Meier that the volume fraction of domain phase segments
follows a symmetric profile over the interlayer, thus fixing an
average composition of the interlayer by domain phase segments at 0.5.
This enables one to compute the normalized volume distribution func-

tion, V(¢) of domain phase content in the interlayer. The planar
*

E
obtained by applying the principle of volume additivity as:

complex moduli, E of the composite for lamellar morphology was

* ‘k * 1* _.
EE(t) = v8 58(1) + vSES(T) + 61L J 58(1') V(¢)d¢ (2.1)
0

Where VB, Vs and VIL are the volume fractions of pure PB, pure PS,

and mixed interlayer respectively; EE, E: are the complex moduli for
pure PB and pure PS respectively. Kraus and Rollman, on the other
hand, assumed the mole fraction of domain phase segments follows a
symmetric profile over the interlayer. They were able to correlate

the dynamical mechanical moduli better. Both of these arguments have
no factual basis and were formulated for the sake of mathematical
convenience. Thus a complete understanding of block c0polymer mechani-
cal and rheological behavior will be dependent on the development of

a statistical thermodynamic theory for the precise mathematical form

of the interlayer composition profile.

25

Gouinlock and Porter (1977) working with SBS samples identical
with that of Chung and Gale generated master curves of linear visco-
elastic functions using the frequency-temperature superposition prin-
ciple as shown in Figure 2.1. Each curve (reduced dynamic viscosity,
né and reduced dynamic storage modulus, Gp) has two branches at
certain reduced critical frequencies. The low temperature data fall
on the upper branches and signify the prevalence of the two phase
structure. The high temperature results occur on the lower branch
suggesting a homogeneous structure. It is further observed that the
critical reduced frequency where branching occurs in G6 data are
larger than the critical frequency for UP. It is to be noted,
therefore, that modification of the elastic property by domain struc-
ture is considerably more pronounced than the effect on dynamic
viscosity. Moreover, experiments indicate in contrast to the deforma-
tion theory of Meier presented earlier, that domain disruption
increases with decreasing frequency. In 'Lufiit of this, the extrac-
tion of segments from the domains would be expected to involve long—
range configurational rearrangements accompanied by long relaxation
times. They then concluded that domain disruption in dynamics measure-
ments as in steady state deformation should depend principally on
the strain, i.e., strain amplitude, not on frequency, and that it
should occur preferentially, if at all, at lower reduced frequencies,
where an effect on the dynamic properties attributable to the domain

structures as such is alone inferred to exist. Another significance

of the results of Gouinlock and Porter is that the relaxation time

26

 

 

 

    

 

 

 

 

 

 

‘ Q n T ' 1 l 1 1 1 1
1o 1— ‘ nous - ‘- —1 107
1. _. .. '.. .
1 ° .0.
3 v ‘ °_ it» PM 07 .
- g o “\ ‘3
2 1o , 2, 0 00 030. — 10‘ .1
:1 - A {o o 100 31.7 3
r - , ‘5 D 111 0.0 g
g .1 ~ A 123 z.»
0; ‘ l. ‘o 3% name-0 130 1. g;
'0 P- 2'- ‘ V 140 .94 -1 '0’ 1n 2
1.00 0 9 103 300 f) 2
- .40 0 4 170 .100 "‘ 3
> ‘ o o u 5
9 10’ ° . 3 9
C 4
a — 10 z
y 'I.“ S
o
o B
u U
10' )— -1 10’ 8
a:
u o
a 0...“) a :00 non
.0 1 I l 1 l L l 1 .02
1o" 10" 1 10 10’ 10’ 10‘ 10‘

neoucco FREQUENCY, 1.1,- n,u (RAD/S)

Figure 2.1.--Reduced dynamic viscosity and elastic modulus vs.
reduced frequency for SBS 7-43-7. Reference tem-
perature is 138°C. Data of Gouinlock and Porter

(1977).

27

associated with long-range motion of the chains of these block copoly-
mers is not characterized by the peak of the loss modulus G"(w) but
at a lower critical frequency where domain phase behavior dominates.

Perhaps the most detailed account of rheological study per-

formed on block copolymers is the IUPAC commission study of SBS melt
compiled by Ghijsels and Raadsen. Steady, dynamic, creep and elonga-
tional flows were conducted. The SBS specimen under study consisted
of cylindrical polystyrene domains (18% wt) dispersed in the poly—
butadiene matrix. The effects of pressure, temperature, and time
between measurements on material properties were also tested. Their
results can be summarized as follows.

1. The melt viscosity of the triblock copolymer is much
higher than that of otherwise similar random copolymers
of same composition and molecular weight.

2. The viscosity at low shear is very sensitive to shear
history.

3. In the low shear region, the complex viscosity is as
much as three times higher than the steady-shear
viscosity at equal values of frequency and shear
rate.

4. A residual shear stress depending on previous shear
conditions is observed in shear stress relaxation
experiments.

Similar flow behavior, especially at low shear rates has been

reported by Cogswell and Hansen (1975) with ethylene polypropylene

 

 

 

28

copolymer melt and Mundstedt (1981) with ABS graft copolymer

melt.

2.2 Optical Studies

 

Electron microscopy and x-ray difraction have become invalu-
able tools in structure elucidation of block copolymer systems. Since
the method of sample preparation is known to affect rheological
results, a brief review will be outlined on how various workers have
utilized the above techniques to identify factors affecting the mor-
phology property relationships of block copolymers.

Pedemonte et al. (1975a and b) have performed a detailed
study of the dependence of their morphology and stress properties
on the preparation of samples. For Kraton 1101 (SBS with 33%S), they
have compared the original copolymer with films cast from toluene
solution at two different evaporation rates (ca. 20 and 0.5 cm3/h),
compression moulded films, and extruded and extruded-annealed speci—
mens. From annealing studies, it has been concluded that the original
material contains rod-like polystyrene domains. From the comparison
of the electron micrographs and stress-strain curves of both extruded
and extruded-annealed samples, the following conclusions have been
drawn. The high values of the Young modulus are caused by a high
degree of orientation of the polystyrene rods along the extrusion
axis; the yield point is explained by the presence of many disloca-
tions and thin ties which link consecutive cylinders. In the case of
solution cast films, the morphology of samples prepared at a high

evaporation rate does not show any regular arrangement of the

29

polystyrene which seem to have a rod-like shape, while for low rates,
a morphology similar to that of'Uwaoriginal annealed samples is
observed. In moulded films, the polystyrene chains form rod—like
domains in a rubber matrix, but no particular orientation of the
cylinders exists. But Lewis and Price used X-ray diffraction and
electron microscopy to compare two Kraton 1101 samples--one, prepared
by compression-moulding and another, a film cast from dilute benzene
solution. They observed an anisotropy of mechanical properties with
the former samples and an isotropy for the latter samples.

Kawai et al. (1968, 1969) have studied films of SI copolymer
of different composition obtained by evaporation of about 5% toluene
solution. Electron micrographs of sections perpendicular to the
film surface have revealed five types of morphology: (1) spheres of
PI randomly distributed in a PS matrix for a PS content of 73 wt%;
(2) cylinders of PI randomly distributed in a PS matrix for a PS
content of 65%; (3) a rather disordered lamellar structure for PS
content of 49% and 43%; (4) cylinders of PS randomly distributed in
a PI matrix for a ps content of 33%; and (5) spheres of PS randomly
distributed in a PI matrix for the PS content of 18%. The authors
have also studied the effect of the nature of the solvent using one
good solvent for polystyrene (MEK) and four good solvents of poly-
isoprene (cyclohexane, CCl3, n-hexane and iso-octane). With the
copolymer in such solvents, electron microscopy has revealed dis-
ordered structures. These results contradict those obtained by slow

evaporation of the solvent (MEK, dimethyl ketone and toluene) from

30

mesophase of SI and SIS copolymers (Gallot et al., 1969) of both
lamellar and cylindrical type. A possible explanation of the dis-
ordered structure observed by Kawai would be a too high evaporation
rate fixing the disordered structure in the dilute solution.

Kawai et al. (1968) have also tried to relate the composition
of SIS copolymers to their morphology and mechanical properties.
Polystyrene spheres were found dispersed in a polyisoprene matrix
for a polystyrene content of 9.5%, slightly curved PS rods arranged
nearly parallel in the PI matrix for an S content of 23%, a rather
disordered lamellar structure for a PS content of 47%, PI domains of
various shapes and orientations in a PS matrix for a PS content of
72%. Kawai et al. have also observed a systematic change in the
stress-strain behavior with the copolymer composition, a change rang-
ing from the behavior of a soft rubber vulcanizate to that of a
carbon-filled rubber vulcanizate and finally to that of a hard, but
toughened, plastic exhibiting a well defined yield phenomenon when
the PI content of the copolymer increases.

To explain the existence of three types of domain structures
(spherical, rodlike, and lamellar) in SI, SIS, and 131 block copoly-
mers cast from dilute solution, Kawai et al. (1969, 1977) have assumed
the formation of micellar structures at a critical concentration
during solvent casting. They have proposed an analysis of forma-
tion of three types of domain structure and the size of the domains
taking into account thermodynamic and molecular parameters such as

incompatibility between the PS and PI blocks, total chain length and

31

weight fraction composition of the copolymer, solvation of the blocks
and temperature. They conclude that the block segments are preferen-
tially oriented along the direction perpendicular to the interface
between the two phases and they postulate that the micelles formed

at a rather low concentration maintain their structure in the solid
state without reorganization. During evaporation, the micelles
shrink in the direction perpendicular to the interface between the
domains. Spherical micelles shrink isotropically while rodlike and
lamellae micelles shrink anisotropically. In rheological eXperiments
increasing attention to sample preparation and morphological char-
acterization are being given as attested by the works of Kraus and
Rollman, Gouinlock and Porter and Ghijsels and Raadsen.

All these studies tend to illustrate the basic feature of
block copolymers, i.e., the additional complication that arises from
the constraints that restrict the components to separate regions in
space. A more complex picture is further introduced by the geometry
of these domains which may contribute to anisotropic deformation. We
avoid the latter difficulty by choosing a block copolymer with

spherical domains and treat them as elastic barriers.

CHAPTER III

A TRANSIENT NETWORK MODEL FOR POLYMERIC MATERIALS
WITH SPHERICAL MICRODOMAINS

3.1 Objectives

 

0n the basis of rheological experimental observations presented

in the previous chapter, we undertook to formulate and test a kinetic
network model based on network theory for block copolymer melts with
spherical microdomains, incorporating realistic and tractable rate
terms for attachment, and detachment of segments (flexible sub—chain)
and domains. The resulting segment distribution is non-Gaussian so
that a general expression proposed by Yamamoto is required to calcu-
late the macroscopic stress. In the following chapters, this model
will be tested fortufiaxial extensional, simple shear, and oscillatory
flows in both steady and unsteady conditions. Next rheometric data
shall be presented on a well characterized diblock copolymer sample
--poly(styrene-b-butadiene) whose morphological structure is known

and the material functions will be compared with model results.

3.2 The Rate Terms

 

Figure 3.1 depicts spherical, rubbery domains uniformly dis—
tributed in a soft, continuous phase. The position p-is referred to
a fixed origin while the position_r is referred to the end of a seg-

ment which may or may not be at a domain; R_denotes the nondimensional

32

33

 

 

1:0
:1:

 

'0—

Figure 3.1.--A polymer network with rubbery domains.

34

position r/Nl where Nl is the extended length of the segment with N
subunits. An active network segment in this representation is a
flexible strand bridging rubbery domains and/or entanglement junctions
in the soft phase. A segment distribution function f(R,N,t) may be
defined such that fdiR is the number of elastic segments in the
network with an end to end vector in the range R toIR + dB at time

t and composed of N subunits. This function obeys the evolution

equation of Yamamoto

31;— + v - (_R_f) = 6(3.N) - B(B,N)f (3.1)
where G(R,N) and B(R, N) denote the rate of creation and the coeffi-
cient of destruction of segments with N subunits; R_denotes the
velocity of such segments which may be expressed following Phan-Thien

and Tanner as

B=(

Ill"—

- a9) '5 (3.2)

where E is the velocity gradient and Q the deformation rate tensor in
the fluid; E is a slip coefficient.

A flexible segment in this representation may be constrained by
impenetrable barriers at one or both of its ends, as in a diblock or
triblock copolymer melt. Hesselink (1971), Napper et al. (1975), and
Edwards and Dolan (1975) have derived one dimensional equilibrium dis-
tribution functions for such segments, taking the presence of these

barriers into account by imposing the boundary condition

35

r = 0,11) = o (3.3)

at the domain boundary. With spherical domains, the spherically
symmetric form satisfying Equation (3.3) proposed by Chompff may be

used.

T-- a2 —- 3 r2
fe (r,N) m exp exp - §-——— (3-4)
~ rZ/Nt2 N22

This distribution is originally attributed to Reiss (1967) and

 

Yamakawa (1968) who proposed a general expression for the total
potential energy, E, of the configuration of a free polymer chain as
N 1

E = 2 u
v=1

1

. . +-—Z V (r..) (3.5)
1,1+1 21¢j 13

Here the monomers constituting the chain are treated as hard spheres
distinguishable by their positions in the sequence constituting the
polymer and are held 1n place by ass1gned potentials ui,i+1(ri,i+1)‘
The spherically symmetric interaction potential between monomer i
and monomer j is represented by V(rij) where rij is the distance
between the centers of monomers i and j and N represents the number
of monomers in the polymer. The configurational partition function

for a polymer molecule whose first monomer (segment) is fixed with

its center at the origin assumes the form

Z = J . . . J exp (- E/kT)dtzdt3 . . . dtN (3.6)

36

where (he, dT3, etc. are the volume elements for the second and third
segments, etc. The integrals extend over all space. The configura-
tional partition function for a polymer molecule whose Nth segment

has its center fixed a distance r away from the fixed first segment

assumes the form
Z(R) = J . . . J exp (~E/kT) die, dT3 . . . dTN_1 (3.7)

where in the integration it is understood that the first and last
segments are a distance r apart.

The authors then calculated for the configuration probability
of a free chain in which one end is fixed and is constrained so as

to decouple the many body problems. This can be represented as

 

2
- <1
P ~ eXP ( 23:2) 9(1) (Jig-l) (3.8)

Here the function ¢N(r) represents a spherically symmetric external
field (centered on the first segment to which the Nth segment is
subject and clearly depends on rN' The spherically symmetric form of
¢N(r) of Equation (3.4) adopted by Chompff predicted very well the
stress-strain relationship of rubber vulcanizates at high extensions.
For block copolymer systems, the parameter "a" in Equation (3.4)
describes the range of repulsion between continuous elastic segments
and the domain to which they are attached. If the number of segments
attached to a domain is small so that the range of repulsion between

segments is less than the maximum end-to-end distance of the segment,

 

37

a < 1. The creation rate expression G chosen in this study is pat-

terned on Equation (3.4) and written as

 

23 3/2 2 2
e (3,11) = c Egg—(3%) exp[- i}; - 3N2R] (3.5)

where C is a constant rate coefficient. The symbols r and R denote
magnitudes of the vectors 3 and 3 respectively. The shape of the
distribution in Equation (3.5) is shown for several values of a in
Figure 3.2. With increasing a, the peak shifts to higher values of
R, i.e., the end-to-end distance of most probable segment is
increased. From the previous discussions, the repulsion coefficient
"a" is an inverse function of temperature. The applicable region

of temperature for G is T

< T f-Tt where T and Tt are the block

9 . 9
copolymer glass transition temperature and transition temperature to
a single phase respectively. Here "a" has a range of 1 < a_: 0.

A consistent expression for the rate coefficient of destruc-

tion 8 is obtained from the relation

8 (R,N) = Bo[1+€(A(R,N) - A(O’N))/kT] (3 6)

N

where the leading term is the contribution from Browian motion and
the second term is associated with the change in entropic free energy
A of a segment in the network by flow and repulsive interaction.
(Acierno et al., 1976). Writing the configurational partition func-

tion Z in accord with Equation (3.4) as

r'
Z (R,N) = K0 exp L:-% NR2-a2/NR{] (3.7)

 

 

38

.mcowuoczm :oepznwcpmvu cmwmmzmw1coc cme_mELozu1.m.m wcsmwa

a
Ho.

80.

 

 

93an

o

 

 

[gun/,9 - z/zaus-Idxa

39
and using
A = -kT an (3.8)

where kT is the Boltzmann's temperature, we obtain

2 2
B(R.N) = 3.0 +13%— + EL) (3.9)
NR2
The rate of destruction is Bf so that at R = 0, Bf = 0, since f is
an exponential function of 1/R2 while B is a polynonnal of l/RZ.
Thus the rate expression is well behaved. The initial distribution

of segments is given by
f(B.N,t=0) = G(R,N)/B(BJN) (3.10)

The moment integrals are considerably simplified if it is assumed

following Fuller and Leal. that 8 << 1 so that
f(B,N,t=0) 3 G(R,N)/Bo (3.10a)

with e << 1 and a < I, the third term in equation (3.9) is clearly much
smaller than the other terms.

The rate expressions outlined here should be appropriate for a
block copolymer or a filled polymer melt containing spherical domains
or particles with low surface denSity of segments and high interpene-
tration in the continuous polymer phase. The elastic free energy of
the network is largely in the flexible segments of the continuous

phase.

 

40

3.3 The Macroscopic Stress Tensor

 

As already mentioned in the preceding section, the rate
expressions chosen here will lead to a non-Gaussian segment distribu-
tion f; so a general equation of Yamamoto is used to find the macro-

scopic stress S in the network

_ 1 91: (BM
2 - 'p dR .BBf(B)N9t)dR (3-11)
OY‘

_ 1 51/1 (KNEE,
é - < R dR , (3.11a)

Combining (3.7), (3.8), and (3.11a) yields

§ = 3NKT <.BB.- 2a.BB. > (3.12)
3N2R“

The validity of this model is examined in the following two chapters

with detailed stress calculations for uniaxial extensional, simple

steady shear and oscillatory shear flows.

 

CHAPTER IV
PREDICTED STRESS BEHAVIOR IN EXTENSIONAL FLOWS

4.1 Uniaxial Steady Extensional Flow

The kinematics of this flow are described by

V1=TxaV2=-%y:V3=-%Z (4.1)

where F is the magnitude of the strain rate. The steady deformational

rate tensor ;* is given by

g=r 0 ~11 0' (4.2)
0 0 -._J

where T is the magnitude of the effective strain rate experienced by
the network I = T (I-E). Since L* is a diagonal tensor independent of

R, segment evolution equation of (3.1) becomes

A

.81“: .211.

at ia: 2
3y 2 32

+Fx

_ __i.
at 8x 2

G(R,N) - B(E,N)f (4.3)

This hyperbolic first order partial differential equation has three

characteristic lines as shown:

x = x0 exp (It)
_ i

y - yo eXD (- 2t) (4 4)
_ I:

z - 20 exp (- 2 t)

41

42

Using the macroscopic equation of stress, (3.12), the primary normal

stress difference N1 in uniaxial extension then turns out to be
: _ _ 2 - 2&2 2 2a2 .
N1 ‘ Sxx Syy ' 3N“ L< X (13N2R‘*)>’ < y (1' SAP—Rip] (4'5)

The two moment integrals in (4.5) may be evaluated using transforma-
tions described in Appendix A, similar to those employed by Fuller and

Leal. Defining non-dimensional time, strain rate and elapsed time

T = Bot; T = T/Bo t'==BO(t-t') (4.6)

we may write

n kT

T 1
“1(T’ = 132a I [11(1) - 12(111e'T + ) e‘T [11(1') - 12(1')]d1'} (4.7)

 

where no E C/Bo; and 11, 12 are integrals over space in spherical polar
coordinates as noted in Appendix A. The integration over one of the
angular coordinates, w is carried out numerically, avoiding a singu-
larity at p = n/Z with a generalized Gauss-Legendre quadrature formula

of Krylov (1962), for 12.

4.2 Results

 

4.2.1 Steady State Stress

 

At steady state equation (4.7) reduces to

nokT .

N1 = TIE—a— emT (Il-Iz)dT' (4.8)

43

Both 11 and 12 depend on the two parameters a and e. If both a and e
are set to zero, Lodge's rubber like liquid model is recovered. With
a alone set to zero, equation (4.8) may be written with a damping
function h(I,T)and a strain measure B(T,t) in the form proposed by

Wagner (1979a)

N1 = noKT [de' e-Tl h(T,t) B(i,t') (a=0) (4.9)
0

with

h(I,T'_)3 (141—;a (1-e'."T'))‘2 (1+ f1: (e2fT|-1))-3/2 (4.10)

and

B(I,t')E erT' - e-fT| +-§: (2e2le + e'fT' - 3) (4.11)

Such a factorization is not possible for the case where both a and
e are nonzero, and the distribution is non-Gaussian.

The normal stress difference N1 may be scaled with nokT--a
shear modulus--t0 compute a dimensionless elongational viscosity at

steady state

* Nl/nokT N1/?
= (4.12)

 

fit: I 00

Figure (4.1) presents a comparison of elongational viscosity plots

againststrain rate calculated with a fixed value of e = .01 and several

 

44

OOH

.Ho.o u 0 saw: =m= gmpmsmgma covmngmg Co #0040“
.mpmc sweepm mmm_:owmcmswu .m> zp_moomw> chowmcmpxm umNTFmsL0211.H.¢ m;:m_a

 

 

 

 

.._
OH H H. Ho.
_ . q H
0.0
m0.
m." M
1 1 OH
we
. _ . OS

 

 

45

values of “a." With a = 0, the elongational viscosity levels off
around a dimensionless strain rate of 0.1 to a value of 3--the Trouton
ratio between the low strain rate values of extensional and shear vis-
cosities. As the value of a is increased, an upturn in viscosity is
noted in the lower range of strain rates; an apparent yield stress

may be identified at the lower strain rates on each of the plots with
a f 0. It is worthwhile to point out here that in the limit of zero
strain rate, N1 is zero and the elongational viscosity is finite; this
must be true of kinetic network models such as the one discussed in
this work. An analytical expression may be obtained for the apparent

yield stress at low strain rates by simplifying equation (4.8) for

 

i << 1.
Ny 4 a 5 ° '
nokT = §1+2a (1+ 2 8) (F << 1,1” 7' 0) (4.13)

It is readily seen from equation (4.13) that with 6 << I, the apparent
yield stress is much more sensitive to the parameter a. Recalling
that the value of a is directly related to the range of expulsion
between segments attached to a domain, this relationship between the
apparent yield stress and the parameter a is reasonable. The signifi-
cance of this parameter is further illustrated with the elongational
viscosity data reported by Munstedt (1981) on ABS block copolymers at
190°C with various concentrations of butadiene, the rubbery component.
The apparent yield stress NY from the data is tabulated against rubber

concentration in Table 4.1 along with the non-dimensional yield stress

46

Table 4.1.--Model parameter ”a" from data of Munstedt

 

 

% Butadiene Observed Yield Stress N /G*- Estimated
in ABS NY (Pa) Y 0 a
20 2.0 x 103 .004 .005
30 5.0 x 103 .010 .013
43 1.5 x 104 .030 .038

 

*G0 Plateau Storage Modulus of 0% Butadiene in ABS.

and the corresponding values of the parameter storage modulus obtained

5 Pa. This table shows

from Figure 20 of Munstedt's paper as 5 x 10
that increasing rubber concentration in the copolymer is described

by increasing values of a in the present model, so that the segment
distribution is increasingly non-Gaussian with higher concentrations
of the rubbery domains.

The effect of the other parameter e is more noticeable in the
peak elongational viscosity attained at dimensionless strain rates of
order 1. This peak is lowered and moved to lower strain rates with
increasing values of e, as shown in Figure 4.2, where plots of elonga-
tional viscosity are presented with a fixed at 0.05, but with several
values of c. This trend is understandable since 6 is a measure of the
dependencecfi function destruction cu) the deformation. Data are not
available on peak elongational viscosities for block copolymers to
verify this trend or allow a quantitative comparison. The effect of

a on the peak value is only slight; increasing a leads to a small reduc-

tion in this value as seen in Figure 4.1.

 

 

47

ooH

OH

.mo.o u m ;p_z =0: gmmemcma cowpozgumm m
*o pummem .mpmg cvmcpm mmm_co_mcmswu .m> auwmoomw> PacowmmmwMW11.m.e wcszd

OL-o

Hc.

 

moo.u

 

mo.

Ho.

 

 

ooH

48

4.2.2 Stress Transients

 

The development of stress in experiments with a sudden step
in elongational strain rate, T may be predicted with the help of
equation (407') at several values of F.. The results are plotted in
a ratio N1(T)/nokTT against T in Figures 4.3—4.5. In Figure 4.3
a is set to zero and at I = 1 and i = 10, increasing 8 leads to reduced
overshoot. Figure 4.4 presents the transient elongational viscosity
at F = 0.1 with e = 0.01 and several values of a. As a is increased,
the trnasient viscosity is increased at all times. At T = 1, however,
as shown in Figure 4.5, the transient elongational viscosity curve
changes only slightly as a is increased. The data of Lobe and White
(1979) on carbon black filled polystyrene melts at 170°C (see Figures
5-7 of their paper) show similar trends with concentration of filler
at low elongation rates of 0.0063 sec-1 and 0.02 sec-1, increasing
carbon black content leads to higher transient elongational viscosity
at all times. Once again, the value of a in the present model corre-

lates directly with the concentration of filler in the material.

49

 

100 I

c =.005

E =.OOS

'01 .01

10 -

”111+

 

 

.1 1
.1 1 8

T

 

Figure 4.3.--Normalized transient extensional viscosity vs.
dimensionless time as a function of strain
rate. Effect of the destruction parameter

c with a = 0.

50

 

 

 

 

 

Figure 4.4.--Normalized transient viscosity vs. dimensionless time

as a function of strain rate. Effect of the repul-
sion parameter "a," with F = 0.l and e = 0.01.

51

 

 

 

 

 

 

100 i
*+
”E
a-.3
0
10 - ‘
1 F ‘
l 1 7

Figure 4.5.--Normalized transient viscosity vs. dimension—
less time: Effect of the repulsion parameter
"a" with F = 1 and e = 0.1.

 

CHAPTER V

PREDICTED STRESS BEHAVIOR IN SIMPLE SHEAR FLOWS

5.1 Simple Steady Shear

 

In uniaxial steady shear flows, the effective deformation rate

tensor is given by

 

0 2-5
g = 324 «a 0 0 (51)
_0 0 0)

 

where 1 is the magnitude of steady shear rate. For convenience, this
1

tensor is diagonalized‘ by introducing a tensor T such that T- L*T = V
Where V is a diagonal and
r— . é. . _% .—
-1(2-E) 1(2-5) O
.1
: /7TT:§T
O O O
L _

 

 

Next a coordinate transformation leads to a new frame 5 = p(p,n,z)
such that R = T - p. The diagonalized tensor V is composed of the

~

eigen values of the tensor L* and expressed as

52

 

 

53

 

 

r1 0 0—
v = 12"1 0 -1 0 (5.3)
0 0 OJ
where
~ ° 4 .
m = Y[g(2'€)] s 1 : H0
The evolution equation for segment distribution becomes
.81 ~ at _ ~ .31 = " _ A
3t+'mqg- min—8.71 GWJU BQJHf (54)
where
6(9):) = (sq-9,11) (s 5)
B (p,N) = B(I-Q,N) (5 6)
In terms of the transformed coordinates
T
“=0 I'I'B (57)

R = x + y + z = p2 - 2an + n2 + 22

54

The characteristic lines for Equation 5.4 in terms of p(p,n,z)

coordinates are

= poe1mt/2
-imt/2
n hoe
z = 20 (5.8)

Applying the macroscopic equation of stress given in (3.12) for
a non—Gaussian segment distribution function f, tangential and first
normal stress difference relations can be generated in terms of

moments in cartesian coordinates as

S = 3NKT < (1 - Z§E__ ) x > (5 9)
xy 3N2R4 y '

I _ = 2 _ 2 _ 22 2 4
N1 _ Sxx Syy 3NkT [<(x y ) (1 Za/3N R )>] (5.10)

Necessary cartesian components of the stress tensor can be evaluated

from the transformed coordinates f(p, n, 2) using the expression

xx = 9 . IT(1 . 9) (5.11)

The cartesian moments are related to the moments of the transformed
frame by the multiplicative factor, det(T) and these are expressed

as:

fil__§ ‘E 2_. 2
<xy> -§§ 2 (p n > (5.12)

(1-5)

 

 

<X - y > ='——-——;§ <0 - 2(1-€)on + 02 > (5-13)

Moment integrals in transformed coordinates are solved in Appendix B,
first by evaluating Equation (5.4) through the use of transformations

prescribed by Fuller and Leal.

5.2 Results

 

Results of steady state dimensionless viscosity (gxy/i) and
first dimensionless normal stress difference, N1 obtainable from
Equations (5.9) and (5.10) can best be discussed with and without
"8" equal to zero.

With "a" equal to zero, a case where the initial distribu-
tion of segments is Gaussian the steady shear viscosity and the
first normal stress difference are obtainable from Equations (5.19)

and (5.10) as:

oo

 

 

 

s _. _ -T' dT'(Slan' - E(cosmt' - 1))
5(1) = J 2 2 e m 2 1 3 2 (5.13)
- _ +
Y (1 g) 0 (1+2—rE-fis'inm'f' - 2%— (COSTllT'-1) ' (1%)2) /
m
~ A 2 GD_T. dT'(-§(2-€))%(1+ET' - cosmt' - e/m Sin mt')
”1(i) = 1-g e 28 252 1+5t' 2 3/2 (5'14)
(1+_fi sin mt' - 2 (cosmt'-1) - (szf—) )
o m

Here n and N1 are defined as

 

 

56

- 5x /1
n5
NOkT/B0
s N1
N :
1 ‘ nokT

where 5 E y/eo is the dimensionless shear rate, n0 is the initial
concentration of network segments and 1' denotes the dimensionless
elasped time, 80(t-t').

Further, if e is set to zero in Equations (5.14 and 5.15),
viscometric material functions similar to those of the Phan-Thien

and Tanner model, [see Equations (30) and (31) of Phan-Thien and

Tanner, 1978] result

 

 

fi = (1'5) ~7~ (5.16)
1 + €(2-€)Y
NI = 311‘5)i 5,_. (5.17)
1 + 5(2-€)i

The non-linear dependence of shear viscosity on shear rate in most
polymeric systems is accounted for in this model through the slip
mechanism, E. In Figure 5.1, the effect of the destruction coeffi-
cient c on dimensionless viscosity is presented as a result of com-
puting (5.13) using a 40-point Simpson's composite formula. This
result shows that "5" does not affect the trends in viscosity vs.

shear rate, but merely changes the scaling factor, nOkT/BO. Similar

 

 

57

0.3

.o u m =0: .memsmgmq cowpochmmn mgu 4o pommmm
.mp8; cmmgm mmmFCOVmcmec 4o covpoczm a ma xpwmoomw> cmmzm vawgmscozu1.H.m wezmwa

 

 

 

To

 

 

 

 

04

CC)
I!
w

 

 

conclusions have been arrived at by Phan-Thien and Tanner as well
as Fuller and Leal. Since the value of e affects only the scaling
factor, subsequent curves of viscosity are plotted only for e < 0.01
so that nokT/B0 coincides with the zero shear rate viscosity.

For the non-Gaussian model (a f 0), steady shear viscosity
and first normal stress difference expressions are derived from

Equations (5.9) and (5.10) as

 

 

 

~(i) = (eza ))(-g(2- i e-T'(-h(: ') B (7 ")
I) Y 1+Za L (1-€)2'? YsT 10 YsT
0
66:62 7 T
+'*_—_2hl(Y’T') 811(y,t'))dt'} (5.18)
m(l-E)
where
118,10 . (82-5)): 3/2
(1 + 2 E-s'n ' - 282 (cosm '-l) - (1+ETRZ)
' m l 1111' -—m—2' T 1‘15 )
810(i,t') = sinmt' --% (cosmt'-1)

811(i,t') = 1 -E(2-E) cosmt' - (1—E)2coszmt' - mt'sinmt'
~ 1

l 2 1~l
(1+2fi'5ian' - 2E(cosmt'-1) - (ifETY32C(y,T')

 

 

3»)

 

 

 

 

 

 

 

41(1) (£3 )Tiit) dt'e- (hm ) 3208,. ) + 88 a2
O
x (1-5)h1(i,11821(i.r')) (5.19)
where
B20(i,1') = 1 - cosmt' --% (sinmt' + mr')
B21(i,t') = E:f:?§ -t' - sinmt'( (I-E)3 + (COSTEé-ll - 1)

The second term in (5.19) is an additional contribution to the

stress level of newly formed chains with respect to the degree of
their repulsion from the domains. Again using a 40-point Simpson's
composite formula Equations (5.18) and (5.19) are computed. In
Figures 5.2 and 5.3, the viscosity and first normal stress difference
are presented with e = 0.005 and various values of "a." For a i 0
Figure 5.2 shows an apparent yield stress and a quick decay of
viscosity to a "plateau" at dimensionless shear rates of order

.01. At large shear rates the model then yields the power law

behavior. The general shape of the curves in Figure 5.2

 

60

.mo.o u w
.moo. n o =.m= Lmumsmgma cowmpzqmg 4o pomw$m .mcowpocsm mocmgmmw_u
mmwgpm Peace: “mew; vawmeLoc use cowuoczw zpwmoumw> cmwgm cmNPPmEL0211.N.m mesm_a

 

 

 

 

 

 

S o; I. 5 S.
no _ _ . _ s _ _ to
1
i
...
OJ r1 \ luoé
.. \\ 1!-
. L
14
J
1
o 21 0.0 1 Ines
m5 1
m8 1 a

 

 

 

 

2C

 

 

 

61

.moo.o u a .mo.o n a :3: 250580 aw_m 5:0 to Soweto

.mcowpoczw wocmgmmewu mmmcpm Peace: pmcwe use xawmoumw> cmmzm.umepmeLoZ11.m.m mczmva

o.oH

 

 

 

o.H w Ho.o Ho.o

H.o . .Hd q a _ q _:qu3 .1-.. __.___J N . _.._nu N.
m.o 1
H.. 1
n t
mo.o u a n: 1
§ 11
o.H 1| L'snnnul1 14
12 L
N.o 1
o.oH11 H.o 11
. modnw ..
.4

 

 

 

H.o

o.H

o.oH

 

62

are in good agreement with flow curves of block copolymer melts and
even with those of triblock melts. There is sharper upturn of shear
viscosity at low 5 with increasing "a” and experimentally a sharp
upturn of shear viscosity is also noted at higher fractions of the
domain phase. Thus "a" correlates directly with the concentration
of the domain phase. The upturn in shear viscosity results is not
as drastic as those shown in the extensional flows. This is

FT for

attributed to functions controlling their strain measure e
extensional flows and sin mi' for shear flows. The effect of the
slip factor as shown in Figure 5.3 is to change the power law
behavior of the material that occurs at large deformation rates. The
model does not predict any new trend in first normal stress differ-
ence except a slight increase in magnitude at all shear rates, as
compared with the Gaussian model. The large difference with high
slip ratios at large shear rates (see Figure 5.3) are predicted even
with a = O. The trend in normal stress-shear rate relationship pre-
dicted by the model awaits further evaluation by experimental data.
However, literature is devoid of such data for block copolymers
mainly due to general difficulty in collecting reliable normal

stress data in conventional rheometers. The normal stresses of all
melts of high M.W. is difficult to measure due to the compliance of
the instrument at high shear rates. Such problems encountered also
in this study will be discussed in the experimental section. From

the results of extensional and shear flows, the contribution of the

non-Gaussian nature of chains occurs at small chain extensions. It

 

63

 

10.0

 

 

 

 

 

1 1441111] I J 1141J

 

Figure 5.4.—-Normalized transient viscosity function. Effect
of the repulsion parameter a, e = 0.005, g = 0.05.

64

cowm_:gmc mg“ mo powwow

OH

.mo.o u w .moo.o u u .m u + =.m= gmmemcma
.cowuoczw xpwmoomw> pcm?mcwcp umNPPMEcoZ11.m.m mcsmwd

 

0.0

 

 

m.o

 

 

 

H.o

o.H

 

 

65

 

 

 

 

 

_ 11k ~
Y = 10
N1 "’ 4——
=3

1.01—-

,-

i-

F

l.
OJ.

4 lJ-llllLl 4 JLJlnLi
0'1 1.0 T 10

Figure 5.6.--Normalized transient first normal stress difference
function a = 0.0, e = 0.005, E = 0.05.

 

66

is worthwhile to emphasize here that the ensuing newtwork model is
mainly applicable to the low deformation region.

Transient stresses are computed by using time dependent
moments as developed in Equation (8.9) in the stress expressions of
equations (5.9) and (5.10). The results are plotted as fi+ (T) and
N1+ (T) vs. dimensionless time T in Figure 5.4 to 5.6. Figure 5.4
shows the shear growth viscosity at low shear rate as a function of
time. The magnitude of this material function increases strongly as
the parameter “a" increases, growing monotonically with time until
it reaches the steady state value. In contrast to the IUPAC data
on SBS melts, no stress overshoot is predicted by the model until
shear rates of the order 1 as shown in Figure 5.6. In these data

1

stress overshoot was noticed at shear rates as low as 0.01 s' . The

strain at stress peak, it has average value of 3 at § ~ 0(1) and

max
increases linearily to 6 for instance at v = 30. As g approaches
0.2, §tmax stays fairly constant at 3. Many workers, Osaki et al.
(1967), Graessley et al. (1977) have correlated this data to the
total strain on the material. An experimental value of thax = 3
have been reported by the former researchers on homopolymeric melts.
Regardless of the value of a, the stress growth curve predicted is
oscillatory when the slip ratio 5 > 0.1. More recent experiments
(Osaki et al.) have discountenanced the presence of the undershoot

after an initial overshoot.

 

 

67

5.3 Oscillatory Shear Flow

 

In oscillatory shearing, L* is given by

 

F0 2-g 0‘1
Y
E* = EQ'COSwt -g 0 0 (5.20)
L o 0 OJ

 

vmerei = wYo
and w is the frequency of oscillation and yois the strain amplitude.

Exactly the same coordinate transformation used for steady

shear flows is applied here to obtain the specific evolution equation

as

3f 170 3f 1m ST A A

-5— +-—§— (coswt)(p 56) - —§9-(coswt) (n 55) = G(91N) - B(BaN) (5~21)
where

m0 = wYO/El2-g)

We next make a domain transformation in the independent variable t to
u such that u = sinwt to obtain the characteristic lines as shown in
(5.21) . This simplifies the computation of f as described in

Appendix C.

p = p eimOu/w
0
4m u/w
- o
0 oe
z = z (5.22)

 

68

An oscillatory shear stress can be obtained through the macroscopic
stress equation of (3.12) in terms of moments in the cartesian
coordinate as

S = 3NKT < (1 --—g§E——) x > (5 23)
xy 3N2R4 y '

We note the velocity gradient is varying sinusoidally with time;
hence, the shear stress varies sinusoidally after transients have

died down and may be represented as

_ Toot
SXy — Re (Soe } (5.24)

where S0 is in general a complex functionoFOr small strain amplitudes
a strain independent complex viscosity n*(w) may be defined as the

limit

Tim In* (w,y )I= lim S /wv =|n*(w)! (5'25)
+0 0 +0 0 0

YO YO

where

n*(w) = n'(w) - ln"(w)

The real part of the complex viscosity, n' (the dynamic viscosity) is
associated with energy dissipation and the imaginary part n",
wn" = G' is associated with energy storage; these are the so-called
linear Viscoelastic moduli and are related to the oscillatory

shear stress by

 

 

69

Sxy = n ($05wt + n v0 Slnwt (5.26)

Upon computing the moment integrals encountered in

equation (5.22) as shown in Appendix C, we obtain the following

 

expressions.
~ 26. _ l
n'(G,YO) = §+2a (:-:) dT'e T [EO(®,T')BO(0,T')
o
+ 6a2h1(&,vo,t') 81(@.I'{] (5.27)
n“(' ) = 92a LZ:§l_. dt'e'T' h (m 1') B (0 t') +
“’Yo 1+2a (1_€)2 __o ’ 2 ’
O
6a2h1(&3,v0,1') B3(J1,T')] (5.28)
where

0 = 00/80 n =BOn /nOKT n =Bon /n0kT

1-(Hauflotfi
2 2 . ~ . 2
1 _ (1+ET'22 + 462 (0T'(1-%?3 +‘%?'S1an )

2 2~
(1-g) mowz

C (J) ,YO’TI):

 

 

70

 

BO(&,T') = 1-cosfit' +-§—(&T' - sin01')
w
_1_
h1(E),Y :1") = (‘512-§))2 2
0 C(0,T')%(l + mT'(Z-m0) + mg sin 01')
Bl (6,1') = (Qt' COS&T' - Sin&T')(1+ gELL—2)
(l-E)
BZ(&,T') = SinQT' +-% (1 - COS&T')

so
0.)
A
82
v
1—1
23
II

(1-1111'51'115131' - cosmT')(1+ Ell—2)
(r-E)

Equations (5.26 and 5.27) involve m0 = vo/§(2:E) in their
second term making fi' and fi" dependent on the strain amplitude.
However, numerical analysis of these function at v0 < 0.1 showed no
significant difference from the linear results. In this region
then, it is assumed the linear response applies and thus compute the

complex viscosity function as
|fi*(0.io)|§lfi*(0)l = (fi'<a>2 + a"<a)2)

Figure 5.7 shows the result of numerical integration for the
dimensionless dynamic viscosity, 0‘ and dimensionless complex vis-
cosity, |fi*| as functions of dimensionless frequency. For the sake
of comparison, this figure also shows the normalized steady shear
viscosity which was calculated in Section 5.2 using the same para-

meters, 8, E, and a. This is to see how well the empirical Cox—Merz

 

 

71

rule(which states that m(v) is equal to |n*(w))&Ȥ)works in this
constitutive equation. At low dimensionless frequencies, i.e.,
a < .001, the complex viscosity is a constant and decays slowly with
the shear rate. However, beyond this value the complex viscosity
curve is higher than the steady viscosity curve. This deviation
from the Cox-Merz rule is consistent with the data on the SBS block
copolymer melt (Ghijsels and Raadsen). However, the SBS data did
now show clearly a zero frequency limit or a crossover point both of
which are seen in Figure 5.7,. but this trend appears to be the case
if more data were collected at the lower shear rate end. In the
diblock copolymer data of this study, the leveling off of In*| is
inferred at about 9 = 10'3sec'1. The experimental results, as well
as the model (extension-dependent type) calculations portray the
junction density (including segments) as being more responsive to
total strain in copolymers than to strain rate. In homopolymers, the
response in both oscillatory and steady shear modes is dependent on
the rate of strain and thus occurs over a larger range. These results
yield |n*|/(n) ~ 2 as compared with a value of 4 for the SBS data
(Ghijsels and Raadsen), pointing out model applicability with weaker
block copolymer networks such as diblocks. Comparison of model
results with data of an SB diblock shall be deferred to the Discussion
chapter.

The model results for stress relaxation after cessation

in shear are not given because they are not substantially different

from those of linear viscoelasticity. In any case since there is no

 

72

.+ u a .>_m>wuomqmmc mum; cmmgm can xucmscmgw we
mcowpucsw mm prmmum_> acmmpm vcm owsmcxu meanu ume—msgo: mgh11.n.m weaned

 

 

 

 

w .3
.3 .2 T2 1 N.2
o u u - Hoo
4 o.H
[Hollllllll rullrlll 1llll
mo.o n w
moo.o n m
m.o n m
.C IIIIIIII
TE ||I1
- n -

 

 

o.oH

L3 ‘19 ‘l‘l‘gl

73

flow (i.e., y = 0). this can easily be computed with the explicit

distribution function given as

e‘8(psnazsN)t
f(o.n.z,t) = f0(o,n,z)
A _B(010,Z:N)t
+ 9(010121N) (1'9 ) (5.29)

 

B(o,n,z.N)

 

CHAPTER VI

SAMPLE CHARACTERIZATION AND EXPERIMENTAL TECHNIQUES

6.1 Material and Sample Preparation

 

The block copolymer employed in this investigation was a
research grade poly(styrene-b-butadiene), C0326-9 (containing a small
amount of an antioxidant, Ionox)generously provided to us by Dr. Lu
Ho Tung of the Dow Chemical Company. Characterization information
for this copolymer is provided in Tables 6.1 and 6.2.

Approximately 0.2 cm thick copolymer films were prepared for
both rheological and morphological studies by the solvent casting
technique (Hashimoto et al., 1977). Thin films of the copolymer
were made by dissolving 20 gms of copolymer in 100 ml of toluene and
the solution transferred to 10 cm Petri dishes. These solutions were
then placed in a vacuum oven kept at 30°C with all port outlets
closed except one connected through a valve regulator to a hood
chamber to insure slow evaporation. The oven was periodically flushed
with nitrogen to prevent the oxidation of unsaturated bonds in the
butadiene phase. After the films were visibly dry a procedure requir-
ing five days, they were further vacuum dried at 80°C. It was
assumed that adequate drying was achieved when the decrease in weight
of the sample varied by no more than 0.005 gm. Again to prevent

sample degradation during weighing, the vacuum oven temperature was

74

 

 

 

75

TABLE 6.1.--Block copolymer characterization

 

 

 

 

 

 

 

 

Specimen T B 81°Ck 5 Block
Code ype -fi -fi -fi Ht. Percent — -— -—
N w/ N B Block “N Mw/MN
C0326'9 (S'B)1 10,000 -1.1 5.9 232,000 ~I.7
TABLE 6.2.--Property of glassy continuous phase
Molecular Weight SolubilitybParameter Glass
1 . .
Structure Between Entanglement (Cal/cm3)2 nggzigion
C ture °C
Polystyrene 33,000 (8.1) .05C 1009
aValue derived from Newtonian Viscosity data of linear polymer

(Berry and Fox, 1968).

bHashimoto. et al., (1974).

CSolubility parameter difference between PS and PB.
dKraus and Rollman.

Note: Polybutadiene Tg ~ -90°C

Block copolymer - liquid above 100°C.

76

decreased to 25°C and the sample allowed to cool in vacuum. There-
after, the oven was brought to atmospheric pressure with the nitrogen
flush. This procedure was repeated until the constant weight was
achieved. The sample was further annealed at 110°C for 24 hours. The
film samples were then placed in a vacuum dessicator and a representa—
tive sample was used for structure elucidation by electron micro-

scopy.

6.2 Electron Microscopy

 

The domain structure of the film specimen was investigated
by transmission microscopy in a Philips 201 electron microscope
operated by K. Baker of Pesticide Research Center, M.S.U. After
embedding in a Spurr resin, the film was presectioned, stained, and
fixed with Osmium tetroxide, 0504. The specimens placed on a support
were allowed to stand forabout half an hour at room temperature over
a 2% aqueous solution ofOsO4 stabilized with a Sorensen phosphate
buffer, in a small, tightly closed glass vessel. The stained films
were then cooled with liquid nitrogen to approximately -150°C and
cut on a Sorvall Porter-Blum, MT-2 Ultramicrotome with a diamond
knife. Ultra thin sections of about 800% thick were cut normal to
the film surface by the Ultramicrotome. Figurest1.and 2 show some
of the typical electron micrographs of the butadiene-styrene block

copolymer at different magnifications.

6.3 Morphology

 

The dark areas of Figure 6.1 are the polybutadiene phase

selectivity stained by 0504 while the white portion is the polystyrene

77

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specimen a

Figure 6.1.--Typical EM micrograph of ultra-thin section of
poly(styrene-b-butadiene)

78

 

Figure 6.2.--Typical EM micrograph of ultra-thin sections of
poly(styrene-b-butadiene) specimen at x 150,000.

 

79

phase. The absence of spherical bundles or lamellae structure indi-
cates only spherical microdomain structure of polybutadiene uniformly
dispersed in a matrix of polystyrene blocks present in the copolymer
specimen. The spherical domains have an average diameter of 350A
and an average interdomain distance of 500A. The thickness of the
domain boundary interphase, AR directly related to the degree of
compatibility of the blocks is indeterminable by electron microscopy,
but are known to be significant for low to moderate M.W. copolymers
such as this specimen (Leary and Williams, 1970; Krauss and Rollman,
1976). Hashimoto et al. using SAXS studies have reported AR values
for S—I samples showing an overall independence of AR on M.W. of
their samples. On the basis of a fair agreement of micrograph of
Figure (5.1 with those of Hashimoto's (1977) and a similar order of
rubber block weight fraction it is inferred that a thick domain-
boundary interphase exists in this sample.

It can be concluded, therefore, that the structure of this
particular block copolymer conforms to assumptions in theory of
spherically symmetric rubbery domains wiflilow surface coverage

uniformly dispersed in a thermoplastic matrix.

6.4 The Modified Weissenberg Rheogoniometer
The steady, dynamic, and transient material functions such
as shear viscosity, transient, and relaxation stresses and dynamic
Viscoelastic functions were measured over a range of shear rates,
frequencies, and time with a modified Weissenberg Rheogoniometer, HRG

(Model R-16). The modification involved the removal of the axial

80

force servo system and the LVDT transducers and replaced by a dynamic
piezoelectric load cell and a charge amplifier. This, along with the

utilization of a stiff torsion bar (KT = 5.8492 x 105

dyn cm/.001“
deflection)similar to those employed by Meissner (1972) were made to
increase axial and torsional stiffness and thereby diminish unwanted
motion in the platen assembly especially during dynamic and Stress
growth measurements.

Figure 6.3 is a schematic of the internal structure of the NRG.
A detailed description and operating procedure will not be given here
as they have been reported by various authors and more recently by
Cross (1983) on the MRG used in this study. A torque in the torque
bar is measured with a linear variable displacement transducer, LVDT.
The output voltage is sent through an amplification and low frequency
filter units and is recorded on the torsion transducer meter. In
event that stress histories are required, the filtered output voltage
are recorded with a Honeywell Visicorder that records transient events
on photographic paper. An additional clam-shell electric oven was
constructed for this equipment to accommodate a Mooney platen of
Diameter, D = 10 cm

Two types of plate arrangements were utilized in this study
are shown in Figure 6.4.

1. The cone—and-plate platen with cone angles of

60 = .552° and 1.982° and D = 7.5 cm and 5 cm
respectively.

2. Combined cylindrical and cone and plate platen (Mooney)

with 60 - 0.933 Outer Cylinder diameter, 00 = 10.0l cm,

 

 

81

I 1 Torque
transducer

 

Torque bar _ c

 

 

 

 

 

Air bearing———T-

 

 

 

   
 
  

Constant Temperature____;u’--1 Upper platen

oven ‘

I
I
I
I
I L -
I
I
Clutch and L—‘--J L---- Lower platen
brake

 

 

Oscillation
drive position

E 2 Lower platen holder
Worm Top bearing
I 1 1 3?

 

 

I

 

s a:

Drive box housing

I
I
I
I
1

I

 

 

 

 

 

 

 

 

 

T\‘\~Linear ball races

 

 

 

i1__.Bottom bearing

 

H
II
1 I
II
II

 

 

 

Figure 6.3.--weissenberg Rheogoniometer internal
(Sangamo Controls Ltd.).

 

 

82

 

 

L ' J
- i
I R 1 90
I

Li. . ]

 

 

 

A

Figure 6.4a.--Cone and plate platen.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
:0: ml

1 ‘2
—-—a—.
O

 

 

 

Figure 6.4b.--Combined cylindrical and cone and plate platen
(Mooney type).

 

83

inner cylinder diameter Di = 9.8195 cm. and cylinder

height = 2.533cm.
Here, the inner cylinder is formed of a conical platen at the bottom
and cylindrical side, the diameter of which is accurately machined to
allow a radial gap equal to the gap at the edge of the cone and plate
of the platen. This ensures a uniform rate of shear throughout the
sample.

Values for steady and transient shear stresses can be calcu-

lated from the torque in this arrangement by noting that for
"Couette" cylindrical platens, the tangential shear stresses arising

in the gap is given by

For coneJand-plate

3.9:
xy 2nR

 

S (6.2)

where R is the platen radius, h the cylindrical height, and 82 and 9a
are the torques developed in "Couette" cylindrical and cone-and—plate

platens respectively. Since the shear rate is uniform throughout the

gap
The total torque = 2(1 + 6h/D) (6.3)

The cone-and-plate platens were utilized to collect steady shear and

transient shear stress data with a steady shear rate range of 0.005

 

84

sec'1 to 0.1 sec—1 at 130°C or lower. At 150°C the range improved to

v :_0.3 sec-1. Beyond these shear rates ranges shear instabilities
were noticed and this will be discussed fully in the experimental
section. The Mooney platen was useful in extending shear viscosity
data up to v = 3 sec-1. Beyond this an associated error of 9-12%

was noted in the viscosity of the calibration fluid (ASTM standard) of
n(T = 25°C) = 742.1 poise. This error is attributed to inertial
effects and non-uniform shear regime in the gap commonly associated

to large size platens performing at large shear rates (Walters, 1970).
Due to the limitation of the amount of sample tested with the Mooney
platen were limited to the range 0.1 < I < 3 sec-1.

The transient and steady first normal stress difference are
important material functions normally collected with the NRC. The
transient normal stress data of various polymeric melts manifests
strong overshoots and sometimes double peaks (Huang, 1976) before
attaining a steady state with time. Unfortunately, at the time of
this study, the WRG was equipped with a dynamic piezoelectric load
cell that registers transient events, but returns to the null state
when the steady state is attained. In the light of these no reliable
normal force data were collected for the sample. In this work signals
from the piezoelectric load cell were displayed on an oscilloscope and
utilized in attaining the exact required gap separation distance beween
the platens. This was especially useful when using the Mooney platen
since it is impossible to see the inner cylinder just touching the

outer cup for gap setting purposes.

 

 

 

85

6.5 Sample Loading and Temperature Control

In measuring the material functions of SB block copolymer melt,
the residence time of the material should be kept very short in order
to minimize oxidative reaction in'Uuapolybutadiene phase. 0n the
other hand, due to the long relaxation times of polymer melts, rather
long waiting periods are required to attain the gap setting and
equilibration of the sample to a stress-free initial state.

To shorten this period and insure an initial equilibrated
uniform distribution of the domains, premolded samples by way of
solvent cast films are helpful, with dimensions which fit the cone-
and-plate geometry of the test gap. Since the melt temperature is
known to strongly affect sample morphology care was taken not to
introduce temperature inversions by using the procedure described
below. Without setting the gap, the platens are heated to a temperature
of 5°C below the desired temperature in about 1% hours. At this point
a nitrogen purge of a to 1 lb. pressure is bled into the heated chamber
until the desired temperature is attained. It was predetermined that
an N2 pressure less than l.5lb. does not affect gap separation nor the
torsion readings. After 5 minutescxiattaining desired temperature,
the gap between the platens was then set, primarily by the use of the
normal force measuring system. Then the oven was opened and the sam-
ple is quickly transferred from the evacuated dessicator used in
storing'Uwa sample to the plate making sure that no air bubbles were
trapped between. The head was then brought down and excess melt

cleaned off with a blade and the thermal chamber closed again.

 

86

The sample was then allowed to heat up to the desired temperature,
a procedure that took 45 to 60 minutes. The temperature controller

maintained the plate temperature to within i 2°C.

6.6 Rheometric Testing

 

6.6.1 OscillatorygShear Experiments

 

The measurements of visoelastic properties of poly(styrene-b-
butadiene) block copolymer melt were carried out with the cone-and-
plate platens at T = 130°C, 150°C, and 175°C. The strain applied
to the sample by the oscillation of the bottom plate, causes the
oscillation of the top cone. Oscillatory displacements are transformed
intoeulelectrical potential by the LVDT. It is then amplified and
recorded on the Visicorder. The strain sinusoidal input wave is also
recorded on the Visicorder. A phase shift and the amplitude ratio are
determined from these two waveforms to obtain the linear Viscoelastic

functions as

O

n'(u>)= —’f=Y—— sin <1
YO
So
G'(m)=-—5xe cos 0
YO

where 0 and 52y‘/io anathe phase shift and the amplitude ratio

respectively.

 

87

An applied strain amplitude range of 0.1 - 0.2 gave no dis-
trotions of sinusirdal waveforms in data and this was taken as the
linear viscoelastic range. For high sensitivity and small sample
size, the cone-and-plate platen of D = 5 cm was mainly used in
oscillatory testing. Testing was carried out with the same sample
giving from low to high frequency of oscillation. A waiting time
of 30-45 minutes between testings was implemented.

Oscillatory testing at 124°C with strain amplitude maintained
at 0.15 resulted in nonsinusoidal torsion waveforms as shown in
Figure 6.5. Such highly non-linear oscillatory behavior have been
reported by Ghijsels and Raadsen and a triblock sample and is a peculiar

feature with structurizing dispersed systems.

6.6.2 Steady Simple Shear Experiments

 

Low drifts were noted in the torsion head transducer meter
range of 0.25 x 10.3 in and 1 x 10‘3 in the gap. Therefore, a shift
torsion bar (KT = 5.8492 x 105 dynes cm/.001 in.) is utilized as it
gives the highest sensitivity at the transducer range setting of 2.5 x
10'3in. Such choice was made to restrict the movement of the torsion
head to a minimum aiding transient measurement with the chosen platen
diameter and the anticipated value of the steady viscosity of the
sample.

A steady shear rate range of 0.005 to 3 sec-1 was attainable
with the instrument using both the cone and plate and the Mooney

1

platens. Data were obtained in the range 0.005 to 0.l sec- with a

waiting time between measurements of 30 minutes and next with one hour.

88

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89

No appreciable difference in data was noted and thus the former waiting
period was implemented. An associated error of 7-10% in the material
functions occurred in this range. Beyond this range and at a tempera-
ture of 130°C or less a variation of 12-20% was noted in the stress
readings for different runs under the same conditions. Upon closer
studies it was observed that shear instabilities, e.g., stress frac-
ture developed in the material as can be determined in Figure 6.6a and
6.6b. Figure 6.6a illustrates the situation where the material is
extruded out of the gap after a shearing time of 8 minutes. In

1 and

Figure 6.6b the appearance of the material at y = 0.096 sec-
0.43 sec'1 are compared for quenched samples which experienced similar
shearing times. Non-uniform shear profile is likely to devel0p in

the sample at i = 0.43 sec"1 resulting in faulty stress readings.

The Mooney platens have a potential range of 0.1 < I < 10 sec'1 as

seen in Holden's data. The major advantage of this platen is that the
sample is prevented from leaving the shearing gap by the guard ring.
Also very little area of the material is exposed to the air minimizing
errors due to oxidative degradation. However, the bulk (D = 10 cm)

of this platen tends to increase the inertia head leading to inacuracies
mainly in transient and oscillatory measurements. As can be seen in

the viscosity flow curves (presented in Chapter 7) no appreciable error
is incurred using this platen at the shear rates prescribed as data
extends smoothly iflxmi low to moderate shear region, i.e.,

(0.1 < v < 0.4). The effect of inertia is, however, seen in

transient measurements as will be shown shortly. For steady shear

90

 

Figure 6.6a.--Picture showing test material extruding from gap after
a shearing for 8 mins; 7 = 0. 3 sec'l. Shear insta-
bility is due to stress fracture. T = 130°C.

 

\

~ “'31
(
1

 

Figure 6.6b.--Quenched sheared materials after a shearing time of
Left hand specimen sheared at Y = 0.096

8 min.
Right hand specimen sheared at T = 0.43

sec-1.
sec-1. T = 130°C.

 

 

 

91

viscosity results an associated error of 5-7% was noted using the
Mooney platen.

Using fresh samples, stress growth experiments were con-
ducted with the two platens. After the temperature of the material
has stabilized in the gap, the clutch system was quickly engaged after
the motor has been running for at least 5 minutes. Stress transients
were recorded on the Visicorder that was calibrated with the steady
state stress value obtained from the torsion head transducer meter.
The time dependent stress is normalized with the steady state value.
Using the Mooney platen the effect of inertia on transient measurements
can be seen in Figure 6.7. An overshoot in the stress build up does

not occur until at v = 0.914 sec'l.

This is in sharp contrast with
results using the cone-and-plate platen at the same temperature,
which shows an overshoot at shear rates as low as 0.027 sec-1. It is
generally observed that overshoot occurs in stress-growth at high shear
rates. It denotes the point at which the material experiences a
maximum strain.

Transient measurements are also affected by the cone angle of
the cone-and-plate arrangement. Theoretically, the assumption the
cone angle, 60 is to be chosen such that the assumption tan 00 ~ 00
is valid. This insures the existence of a constant shear rate
throughout the melt. Meissner and Huang noted systematic differences
in transient shear stress and normal stress measurements as a func—

tion of cone angle used. However, Graessley et al. (1977) found

no difference in stress growth measurements for 1°, 2°, and 4° cone.

92

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93

In this study on comparing measurements as a function of the cone
angle of 0.552° and 1.982° gave a variation of data of 1.8% which is
well within the experimental error. It is thus presumed that the
choice of 00 : 2° introduces no significant error in the transient
measurements. The stress relaxation after cessation of shear was also
collected using the Visicorder on samples used in stress growth

tests. The results of these experiments will be presented and analyzed

in the following chapter.

CHAPTER VII

RESULTS AND DISCUSSION

7.1 .Introduction

 

The material functions, dynamic viscosity, storage modulus,
steady shear viscosity, shear stress growth, and relaxation stress
after cessation of shear of a poly(styrene-b-butadiene) with 94.1
wt. % S have been collected as functions of the deformation rate and
temperature. These results suggest that there exists a melt transi-
tion temperature demarcating the prevalence of two types of block
copolymer microstructure. The occurrence of such transition tem-
perature or region will be discussed in Section 7.2, using evidence
in the experimental results. We will not use the time-temperature
superposition principle in reducing data since such two-phase struc-
ture in block copolymer melt have been established (Chung and Gale,
1976; Gounlock and Porter, 1977). In Section 7.3 the rheological
results showing the effect of deformation on the microstructure above
transition temperature shall be presented and discussed with the
view to understanding the underlying microstructure. Next, the
rheological results below transition temperature is presented and
discussed again with a view to verifying the block copolymer
microstructure. In order to test the transient network model

developed in Chapter III, two material constants are estimated using

94

95

the linear viscoelastic data. Other model parameters shall be com-
puted by fitting model predictions with these functions. The model
then will be used to predict the steady and stress growth flow
behavior of the block copolymer at T below the transition. It is
necessary to restate here that our major focus is on the rheologi-
cal behavior of the block c0polymer at low deformations. This region
yields the most differentiating features of block copolymers with
respect to their homopolymer and random block copolymer counterparts;
it also plays a crucial role in evaluating a network model based on

a more realistic chain statistic.

7.2 Phase Transition Temperature

 

The dynamic viscoelastic properties of the block copolymer
sample are shown in Figure 7.1 at T of 130°C to 175°C. The repro—
ducibility of these results is good, 5.2% at w < 0.6 sec'1 and fair,

1 with 2° cone angle and the stiff torsion

7-9% over 0.8 < w < 3 sec-
bar. As usual, increase in temperature tends to decrease the moduli.
At 150°C the dynamic viscosity levels off at about 0.1 see”1 but stor-
age modulus as a function of the frequency shows a slope of 1.3 on the
log-log scale. On the whole, such behavior is similar to those
exhibited by homopolymers where a single phase microstructure is known
to exist. At 130°C or lower, the dynamic viscosity does not level off
at the lowest frequency tested and a larger deviation of the slope of
the dynamic storage modulus vs. frequency from 2 is noted. Next, we

evaluate the two temperature regimes for homopolymeric character by

applying the Cox-Merz rule defined earlier on the viscoelastic

 

 

 

 

96

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97

properties. These results are shown in Figures 7.2, 7.3, and 7.3a.
At 150°C as given in Figure 7.2, the complex viscosity is found to be

greater than the steady shear viscosity especially at v > 0.1 sec-1.

1

However, at v < .1 sec- both functions not only level off, but appear

to be approaching each other. The steady shear results show a limit-

5P. This compares with a homo-

ing zero shear viscosity of 1.4 x 10
polymeric PS having MW = 259,000 Mw/MN = 2.35 at T = 200°C with

no = 4.25 x 105 P (Mendelson, 1980). On the whole the deviation from
Cox-Merz rule follow a similar trend often shown by homopolymers and
random copolymers. A dissimilar deviation from the Cox-Merz rule is
found when the complex viscosity, the dynamic viscosity and steady
shear viscosity results at 130°C are compared as reported in Figure 7.4
and 7.3a. In Figure 7.3 the largest deviation of the two functions

appear at I < 0.1 sec-1.

Both functions are sensitive to the deforma-
tion rate at the low deformation rate region suggesting a more complex
microstructure controlling the viscoelastic response. At w ~ 0.01
sec—1, the complex viscosity appears to be levelling off even though
more data (at w ~ 10'3 sec-1) are needed to confirm this assertion.
If this is the case, these results suggest the occurrence of a network
structure sensitive to the imposed strain history of the material.
Figure 7.3a shows the dynamic viscosity, n' to be significantly sen-
sitive to the frequency equivalent to the shear rate range 0.05 < Y

< 0.3 sec'1 where a so-called "equilibrium" shear viscosity is
attained. At w < 1.5 sec-1, n' values are higher than those of n by

18% or less. The strong dependence of n' on w seems to lessen at

98

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100

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101

'1. Further data at w < 10'2 sec'1 will be very helpful

w - 10'2 sec
in establishing whether n' levels off and how this frequency at which
this occurs compares with that suggested for |n*|.

Since the zero shear limit for 150°C is seen at shear rates
comparable to that at which homopolymer n levels off, a single phase
microstructure is suggested. For two phase structure, such a limit
may be observed only at deformation rates that are order(s) of magni-
tude lower. Attainment of Newtonian viscosity at such low shear rates
implies the prevalence of a network structure sensitive to the applied
strain history, that have been attributed to diblock copolymers
(Krauss et al., 1971). The transition temperature region for
this diblock sample occurs at 130°C < T < 150°C. This is attributed
to a weakening and/or loss of the two-phase structure due to sharp
increase in phase miscibility and/or the attainment at or above the
transition temperature of an easily disruptible dispersed phase not
controlling viscoelastic response and, therefore, leading to NeWtonian
behavior at low deformation rates. The narrowness of the transition
suggests that chain miscibility is at least the major factor since,
in the absence of suchaiphase change, the property changes would be
expected to be more gradual.

7.3 Viscoelastic Behavior Above the
Transition Temperature

 

 

Figure 7.4 shows the effect of shear rate on the steady shear
viscosity at 150°C. We note here the quick decay of viscosity from

Newtonian behavior at higher shear rate. Such behavior is often

102

.UOOmH pm m1m co cowpoczo zpwmoumm> camcm znmmpm11.¢.m wczmwe

-smwe s

A
as Os H

o H-0H

 

- 4

 

 

OH

(asiod) 0

m3

 

 

OH

 

 

103

associated with high M.W. polydisperse homopolymeric melts. The
polydispersity of the continuous PS chains in the sample under study
is 1.7. In Figure 7.5 the shear stress growth results are portrayed

as normalized values using the constant stress value S as the

1

xy/ss
shows points

normalization constant. The curve of v = 0.0108 sec-
of inflection at 8 mins. and 16 mins. that are not found in the other
curves. The associated error observed for this shear rate between
forward and backward rotation was 3-5% at t < 2 min, 12-18% at

2 < t < 12 min. and about 5% at larger times. It is judged that this
error may be caused by incomplete relaxation of the test sample in
the gap. The other results reported in Figure 7.5, as well as the
shear stress growth curves at 130°C had associated errors at 5-8%.

0n the whole, the trends in result resemble those of homopolymers.

We note, however, the occurrance of significant overshoots at much
smaller shear rates in contrast to homopolymeric melts (Graessley

et al., 1977). Furthermore, these curves depart from linear visco-
elastic behavior even at small times. The extent of this departure
may be determined by evaluating a relaxation modulus G0 at different

strain rates from the slopes of the normalized growth curves at small

times.

+/S

S
G0 = lim xy xy,ss
t->0 t
The values computed for GO ranged from 0.28 to 0.52 over the range
of shear rates studied. Figure 7.6 shows normalized stress relaxation

functions at 150°C.

104

ooomfi pm m1m co cowpoczc zygocm mmmcpm camsm umNPPmE20211.m.N mczmwe

 

 

 

 

Aommv H
mm mm «N om 0H NH w a c
_ _ a q H _ J A I
0
rs
0
iv
L m0
9
O
43
moflo.
- L..1
0
owo.o
memo.o 1 we
Hmfio.o n s

 

 

x
xs

ss/KXS/

 

105

.. A83 5 .0003 a.“ i

we cowpmxmpmc mmmcpm cmmsmii

 

.0.“ mczmwe

 

      

 

 

 

o.¢~ o.oN o.oH o.m~ o.w o.¢ o
a _ d
ammo.
memo.
ono.o 1.N.o
1: s
X
nA.
/
S
VA
.1
as m
+
m.o

 

 

 

106

These results showed a decay of stress relaxation to zero similar

to those of homopolymers. It is concluded that an entanglement
microstructure of the continuous PS chains appears to influence the
block copolymer melt at 150°C or higher; however, these results do not
exclude the existence of domains above this transition temperature
since easily disruptable domains not controlling the viscous response
might yield similar results.

The range of temperatures where a two-phase structure mani-
fests in the material has been established at T < 150°C. In keeping
with the objective of this study, viscoelastic results of 150°C will
not be compared with the transient network model. In the next sec-
tion, the material functions of 130°C shall be presented and compared
with the model having a nonzero "a." In this analysis the tangential
shear stress shall be normalized by the constant G0 = nokT, the modu-
lus of rubber elasticity, while the shear rate, v and present time, t
are normalized by a single relaxation time 10 ( = l/BO). It is worth
emphasizing here that our interest lies in the low deformation rate
region and we seek to predict the material viscoelastic behavior at
such a range. Since linear dynamic functions were not obtained at
124°C, it is not possible to predict other material functions at
this temperature using our procedure. In this model evaluation we
will deal mostly with normalized quantities, i.e., the normalized
viscosity, normalized shear rate, in line with definitions given in

Chapter V.

 

 

107

7.4 Viscoelastic Behavior Below the
Transition Temperature

 

The complex viscosity as a function of frequency, shown in
Figure 7.3 has a slope, d log |n*|/dlogw(at w < 0.1 sech) equal to
-0.26 as compared to a value of -0.5 obtained by Ghijsels and

Raadson for SBS triblocks. Table 7.1 further illustrates the results

of the slopes of n and n' vs. v and m respectively, (see Figure 7.3a).

Table 7.1.--Phase separated block copolymer melt properties

 

Sample MN(x10'3) TOC dlogn'ldlogwl)<1 dlogn/dlogi/Y<1

 

535a 11-56-11 150 -0.61 -0.68
535b 22-50-22 170 -0.66 -0.66
585b ' 14-70-14 170 -0.35 --

SBC 232-10 130 -0.43 -0.38

 

aData of Ghijsels and Raadsen (1980).
bData of Arnold and Meier (1970).

cThis work.

These slopes indicate that the triblocks have more strength than the
diblocks (even at higher melt temperatures). Even though the SB
diblock has an MIT an order of magnitude higher than the SBS triblock,
from these slopes the network structure of the triblocks are stronger.
The results of steady shear viscosity of S-B melt sample as
a function of shear rate at 130°C and 124°C are reported in Figures

7.7 and 7.8, respectively. The upturn in viscosity occurs at about

 

 

108

HoH

Afi1ommv >

OOH H1OH NioH

.ooomH pm .m1m co mam; emmcm co cowpoczw a we mnemoomw> Lamsm Avmmpm mghii.N.m mesmwe

1oH

 

 

d _ _

  

ommm.H u o memow_a
mum-Q new mcou o

O

ommm.o u o
mcmmeQ Amcooz mcvm: .

«OH

(aslod) 0

0H

 

 

 

 

109

HoH

.UO¢NH pm mim co mow; cmmgm mo cowpoczm a ma zpmeUmw> camsm aummpm osp11.m.m mczmwe

-ooH

Afliommv >
T2 FE TS.

 

 

d

 

 

OH

OH

(astod) u

coH

 

 

110

1 at both temperatures similar to an SBS melt at 150°C

0.1 sec-
(Ghijsels and Raadsen). About shear rates of 0(1) the viscosity is
no longer strongly dependent on the shear rate, but thereafter the
material seems to approach the power law region. Upon comparing
these curves with the high temperature curves (T > 150°C) we see that
the low shear rate response is that of a weak three dimensional micel-
lar network in which the polybutadiene domains acting as junction sites
solely influences the viscoelastic response.

Further evidence of the effect of two-phase microstructure
can be seen in the shear stress growth curve5(fi°Figures 7.9 and 7.10
collected at 130°C. At small times, these curves exhibit higher
transient shear stress with lower shear rates, than the corresponding
curves at 150°C. It is further observed that the magnitude of the
overshoot from the steady state level is higher (0.18) at v = 0.0272
sec"1 than at i = 0.043 sec'1 (0.12)--a feature also present in the
SBS data. On comparing these curves with the high temperature counter-
part (Figure 7.5), it is clearly evident that a more detailed micro-
structure behavior is found in such transient flows and more effort
should be applied in this area for a better understanding of the
microstructure mobility than at steady state conditions. No stress
growth responses were obtained at 124°C due to the limitation of_the
amount of sample. The next curves (Figures 7.11 to 7.13) shows the

stress relaxation functions at 130°C and 124°C. Here in contrast to

the findings on SBS data which manifests residual stresses, these

 

111

mm

mm

uoomH pm mim mo cowpocze cwzoem mmmgpm cwwcm nmNPFQEL0211.m.N mezmwe

vm

om

0H NH

 

 

 

mmooo.o

ano.o

mmmo.o n +

 

 

m.o

v.0

m.o

m.o

o.H

SS/KxS/KXS

‘1“

 

 

112

(\I
(‘0

mm

00

OMH Hm mum Lb Cowpocsuv LHBOLG mmmme mecm U¢NZ.©ELOZII.O~.N stwwn—

0N

0m

3

6mm 0

NH

0

 

_}

 

 

25.0 n s

 

 

m0

0.0

0.0

0.0

04

NA

 

113

.ooomH pm 01m 00 we?“ co cowpoczw a ma cowwmxmpmc mmmcpm cwmnm nmeFmELozii.HH.N 0030?;

140'

Aommvp
0H NH 0 a 0

 

 

 

0
58.0 a
380.0 1 s 5
.0 m
.V /
S
VA
tn
/
w
0
m.

8'0

 

 

114

 

0H

.ooOmH Pm mum Lb 9:5. *0 :0.5.U::.+ 0 mm Cowymxwrw.» mmmLHm wazm UmermEe—OZII.NH.N 9:09....

Aommvp

0H 0H NH 0H 0 0 0 N

 

—

a .
. _ v _ .
.

 

 

 

 

115

 

 

 

 

 

.00¢NH “e 01m ee mafia we eewpecee e we :ewpexepec mmeepm Leesm eeNWFeELe211.mH.N mesa-r
Aeemvp
em era 0.8 0.3 0.2 08 or. e
.. .- , q _ _ e e
2e
memes 1:
58.0 n e 1: 5
X.
”A
/
S
X
»A
/
0
es
1e.e
es

116

functions at all shear rates decay to zero, but at a much slower
rate than those of 150°C.

At higher shear rates (i > 1 sec-1), polydispersity of the
continuous phase in our sample makes it difficult to determine.whether
domain flow or entanglement disruption in the continuous phase con-
trol the viscoelastic response. Whether domains are completely dis-
rupted by shear deformation and the point to which this occurs may
be difficult to establish with rheometry alone. This may be made
possible by utilizing electron microscopy with deformed samples as
was performed in solid elasticity (Aggarwal et al., 1969). This is
outside the scope of this study.

7.4.1 Estimation of Model
Parameters

 

 

The non-Gaussian transient network model presented in Chapter
III assumes that the continuous soft phase of the block copolymer is
composed of the "most probable" network segment with N sub-units.
This demands the knowledge of a single relaxation time 10, that
is associated with the rate coefficient, 80, (10 = 1/80) of the
destruction rate process and the modulus of rubber elasticity, 60’
(G = nokT). In reality, in any polymer matrix, there is a distribution
of N and thus multiple relaxation times obtainable from the fluid
relaxation spectrum which is often constructed from functions of
linear viscoelasticity, G'(w), 6"(w), and G°(t). It is worthwhile

to emphasize that G0 and A0 are not to be considered as adjustable

parameters in the model.

117

Even in homopolymer rheology the use of a single relaxation
time in viscoelastic models can only predict data in a restricted
range. Generally, a large relaxation time characterizes long time
behavior and is applicable with low deformation rates predictions
while a small relaxation time predicts higher order deformation rate
range. Typical dynamic shear moduli of narrow M.W. distribution
samples display two sets of relaxation times corresponding to two
relaxation mechanisms separated in the time scale. One set of relaxa-
tion times associated with the transition in the high frequency
region; another set associated with the entangelment slippage in the
low frequency region which appear as a peak of G"(w). A character-
istic relaxation time associated with long-range motions of homopoly-
mers is estimated by the inverse of the frequency at which the
peak of the loss modulus, G"(w) occurs (Onogi et al., 1970). How-
ever, in polydisperse samples there is often an overlap between these
sets of relaxations so that the peak in G" appears as a plateau.
Further, the slope of 0" vs. w is close to unity on a logarithmic
scale for homopolymers. Gouinlock and Porter have identified that
the departure from 1 of this slope in block copolymer melts is due to
the domain morphology. Ghijsels and Raadsen also found the presence
of maximum in the loss factor tan 6 (= G"/G') and related this
with domain activity. These points were considered as one of the

criteria in determining GO and A0. The other criterion is based on

In

118

the point where the upturn of viscosity occurs in the experimental
steady shear viscosity. Such an upturn also occurs in the predicted
curves based on the former criteria, but they were plotted as a

function of 10?. By comparing these points 10 can be evaluated.

Figure 7.14 shows the results of G"(w) and loss factor as a
function of frequency. The deviation of the slope of G" vs w from
1 is not very discernible but the loss factor shows a pronounced
transition at 0.25 sec-1. From this we obtained the material con-
stants shown in Table 7.2. Also using the refining criteria a

second set of relaxation times are evaluated and listed in Table 7.2.

Table 7.2.--Material constants from experimental data

 

 

 

Method 1 Method 2
10(sec) 4 1.25
d nes 4b 5
G (—L—) 6.8 x 10 2.16 x 10
o 2
cm
a _
A0 — 1/00t
b :
Go - |G*(wt)|

Values of the segment repulsion range parameter "a," the
destruction rate coefficient "a" and the slip factor a for Method 1
are determined by fitting the data of complex viscosity with Equa-
tions (5.26 and 5.27). The result of the best fit with data is given

in Figure 7.15. In Method 2 it was necessary to refit the data with

 
 

 

119

tan 6

.ooomH we xeceeeec$ 0e mcewuecee me segue» mme_ use m:_:eee mme411.¢fl.w eczmwe

A
OH eeH H es

 

0H
0.0r

0.H1

¢.~i

ZwD/UKP (m)119

0.H

T

m.H

0H

 

 

 

1‘1,

 

120

 

 

.'.asa. n. ‘ . - 5......

 

10 ._
I e = 0.005
' a = 0.45
I g = 0.03
A = 4 sec.
' o
1.0..
)-
T I
LC;
0.1 1 J I I 141 l l l l J 4411 j 1 J
0.1 01A 1.0
0

Figure 7.15.--Evaluation of model parameters using linear visco-
elastic functions (dimensionless).
o = data, ——-—- = model fit Equations (5.26) and
(5.27). T - 130°C.

 

 

121

the new set of constants and the results are shown in Figure 7.16.
These results are least sensitive tO'Hueparameter "e", the range
0.001 < e < 0.007 gave practically the same results. This parameter
is best ascertained with strong flows, e.g., in uniaxial transient
extensional flows.

7.4.2 Experimental Evaluation of
the Transient Network Model

 

 

Without any further adjustments in the parameters, steady and
transient shear results are predicted by using Equation (5.17) and
portrayed on the accompanying plots as a normalized viscosity
(fi (n/Golo)) as a function of normalized shear rate, (A0?) and

normalized transient shear stress (5:y/5 ) as a function of

xy/ss
normalized time (t/AO), respectively.

7.4.3 Steady State Predictions

The model predicts correctly the overall trends of the steady
shear data as shown in Figure 7.17 and 7.18. In Figure 7.17 the
quantitative agreement between experimental results and theory is
poor to fair in the range 0.02 < i < 0.18 where a 40 - 0% deviation
is noted. The theoretical prediction of the range 0.18 < § < 12
is satisfactory with about 5% derivation.

0n the other hand, using Method 2 having the same order of
magnitude of the relaxation time as in 1 improved results signifi-
cantly at the range of interest (see Figure 7.18). In the range
0.006 < i < 0.1, the theoretical prediction of results is excellent

having under 3% deviation. At moderate dimensionless shear rates of

122

.uoomH u H .AoN.0V use AON.00 mcewpeeem
.pww Feces u .epme u 0 .Ammepcewmcesvev mcewpecey
UTHmePeeemw> Leecwp meme: meeHeEecee Feces we seepe:_e>011.0fi.s 8230?;

 

 

 

 

II
r<

omm 0N.H
00.0
000.
00.0

n n u
0» OJ re
L

 

 

0H

erI

 

 

123

.uoomH u H

 

.868 e 1 OK .me.e 1 m .me.e u w .mee.e 1 6 Fence 026256:
pcewmcecu ;a_3 xnwmeemw> geese seeeum 0e cemwceeee011.NH.N ecemwe
0H 0.H 004 H.o
1 A _ _ 4 _

 

 

H.0

0.H

0H

 

124

0a

.emm 0N.H u e& .00.0 n m .00.0 n 0 .000.0 u 6
Fence saw: xuwmeemw> geese meeepm we cemwceeee011.0fi.u ecemwe

0.H . H.0 H0.0

 

 

 

o 6 logo.

 

 

 

0H

 

 

125

0.1 to 1, the prediction is fair to poor with 3-36% deviation and
unsatisfactory at I > 1 with 40% deviation. The slope of viscosity

as a function of shear rate at v < 0.1 sec"1 is predicted very
accurately. It is concluded that the loss factor cannot serve as a
guide in obtaining a characteristic relaxation time. At the large
shear rate range, it is unreasonable to expect a good fit in the light

of the polydispersity of the PS phase (MW/Mn = 1.7).

7.4.4 Transient Predictions

 

Comparison of the model predictions with the data for stress
growth are given in Figures 7.19 to 7.24 at low shear rates using
the two procedures. Here the agreement between data and theory is
rather fair, especially if we remember that all the parameters were
determined from data of small amplitude oscillatory shear flow only.
On the whole, the model prediction with 10 = 125 sec is good at the
lowest shear rates (0 - 15% deviation) and excellent at strains
less than 9.001 (under 3% deviation). On the other hand, the high
relaxation time model appears superior at higher shear rates for
all models significant deviations occur at intermediate times. A
weakness in the model is its failure to show an overshoot at low
shear rates. Such overshoots are shown at higher shear rates,
as illustrated in Figure 7.25 on page 132. The positions of the
overshoot, tC can be correlated most directly with the total
strain as many workers have noted previously. Figure 7.25 shows
the strain at stress peak as a function of shear rate for both

data and model predictions. Overshoots are predicted by the model only

eFmeeE II
233 Iol .80 u w .000 .1. e .03 v n K .302:
spy; 06000 we mppemmc guzogm mmmgpm we :emwgeeee011.0fi.m eczmwe

126

 

NNN0.0 u 0 A

O

 

D
n

O

O

\s

 

0.0

0.0

0.H

 

 

127

 

1.0 _

    

10? = 0.00856

0.4

 

 

 

one
.5
N
._
A
1.—

t/AO

Figure 7.20.--Comparison of stress growth at 130°C with

model A = 1.25; a = 0.55, E = 0.05
0 = dat8;____ model.

 

128

.eem e u e« we Feces Law; uoomfl 0e cpzecm memepm we cemweeeee011.HN.m ecemwe

 

 

N.0

0.0

0.0

0.0

0.H

 

ox\e
01 0 0 0 v m N H o
_ q a _ _ 4 ~ 7 1
4
. J
. 1
s
4
a
i
o
O J
000H.o n 0 K e
o
D D c 1.
HI I} 0 O
a O

a O 1

- o
L

.N.H

ss/KxS/A‘xS

129

O
.0N.H n K
%e Feces new; goomfi we £02000 mmeepm we cemweee5001i.NN.m ecemwe

 

 

 

0
mm em a em 8. Se 2 e e
N u H d 1 . 4 q d -
O
emee 1 + A
0 O 0 o 0
o 0
0 0 O
0

 

 

 

 

130

1.2 F

 

 

 

Figure 7.23.--Comparison of stress growth at 130°C with
model of 10 = 4 sec.

 

131

0H

0
.emm 0N.H u <
we Feces new: ooomH 0e gpzeem emerge we cemwceeee01-.¢N.n mcemwe

0H NH 00 0 0 w

 

 

CI

emme.e 1 +60 6

0
b
O

 

 

 

fix
*8

ss/KXS/

132

OH

.111 .000.0 n o .00.0 n 0 .00.0 1 m
.weuee .eueu n 0 .0oomH we mm>L=e geese
pcewmcegu cw peegmge>e mmecpw ecu we :wecpmii.0m.n ecemwe

eem 0
e; r- 0 no

 

 

_ _ . _ .

 

 

0.0

0H

133

at v > 0.25 sec-1.

The predicted peaks occur at strains insensitive
to the shear rate and are determined by the slip factor a.

A constant value for this strain of about three has been
reported experimentally for a homopolymeric melt (Osaki et al., 1976).
Graessley et al. (1977) have studied this quantity at low shear rates
with homopolymer samples and indicated that insensitivity of strain
at stress peak, to shear rate is associated with materials that
possess a broad relaxation spectrum.

It is noted that the foregoing feature and the fact that the
magnitudes of the overshoot for transient stresses at smaller shear
rates are larger than those at large shear rates (which is not pre-
dicted by this model) presents a severe test for viscoelastic models.
This will have to be addressed with only £33; relaxation time if the

exact physics of two—phase microstructure mobility is to be compre-

hended.

CHAPTER VIII

CONCLUSION AND RECOMMENDATION

8.1 Conclusion

 

A new kinetic network model has been developed and evaluated
for the rheology of block copolymer melts and polymer composites with
spherical microdomains. This model involves in addition to the
readily determined relaxation time A and modulus GO, three parameters:
”a" describes the range of repuslion between segments of matrix
attached to spherical domains, "e" describes the dependence of junc-
tion destruction rate on the conformation of the continuous random
phase and (g) accounts for a slip between the fluid and the network
junctions. The m0del hsused to compute the material functions in
uniaxial extension, simple shear and small-amplitude oscillatory
shear flows. Experimental data on elongation are obtained from the
literature while datacrishear flows are obtained in this work.

In uniaxial extension, the model predicts the Trouton viscos-
ity at normalized strain rates, I of 0(1) if spherical domains are
absent (a = 0). This is in good accord with data of Mundstedt and
Laun (1978). If spherical domains are present (i.e., a f o), the
model predicts a non-constant elongational viscosity at the low strain
rates, but a smaller maximum viscosity at higher strain rates. Com-

parison of these calculations with data of ABS melt (Mundstedt)

134

135

reveals that the repulsion measure ”a” determines the apparent yield
stress observed at low elongation rates. The destruction rate para-
meter "8" determines the level of the maximum elongational viscosity
at steady state as well as the stress overshoot observed at higher
rates in stress growth experiments. However, no data for elongational
flows at large strain rates are available to evaluate the model suit-
ability in this region.

The viscoelastic properties of a diblock copolymer, poly-
(styrene-b-butadiene) of high thermoplastic content have been studied
experimentally in this work. Theinaterial is composed of uniform
spherical domains of polybutadiene randomly dispersed in a poly-
styrene matrix as confirmed by electron microscopy on solvent cast
samples. The melt for rheological study was obtained from carefully
annealed solvent cast samples (toluene as solvent) leading to an
associated error of 7 to 10% in material functions at low shear rates.
An associated error of 12—20% have been reported by Ghijsels and
Raadsen in melts starting from crumbs in this region.

Microphase separations appear to start as the temperature is
lowered from 150°C. At 150°C or above the material exhibits Newtonian
behavior in the steady shear viscosity andtimrcomplex viscosity at
low deformation rates and appears to obey the Cox-Merz rule. At 130°C
or below the complex viscosity is higher than the corresponding steady
viscosity, except at very low strain rates (i < 0.05).

At 130°C and 124°C a significant upturn of steady viscosity

occurs for shear rates lower than 0.1 sec-1, similar to the SBS melt;

 

 

136

however, the slope d log nld log 1

 

1+0 is much less for the present
SB melt. In contrast with homopolymers and random copolymers, a
significant transient stress overshoot is observed in the shear

growth experiments at shear rates as low as 0.02 sec-1. It is further
noted that the height of this overshoot diminishes with increasing
shear rates. Contrary to the SBS data, no residual shear stresses

are observed in the SB data in shear stress relaxation experiments
confirming the assertion that only apparent yield stresses are
exhibited by the SB melt.

All model parameters have been found by fitting data of
oscillatory shear experiments, using two procedures to obtain the
characteristic relaxation time. The overall trends in the data have
been predicted very well, in the range of interest. Quantitatively,
the predicted shear viscosity is very sensitive to the choice of the
single relaxation time at the low shear rate range. The model also
fails to show an overshoot in stress growth at shear rates less
than 0.1 sec-1. These deficiencies are largely due to assuming that

only a single relaxation time controls the entire material viscoelastic

behavior.

8.2 Recommendations forFUrther Study

 

The stress constitutive equation presented in equation (3.12)
is written for a single relaxation time. To take into account the
distribution in N, especially in polydisperse samples, we must allow
for multiple relaxation times. Since in the network theory no inter-

chain correlation is taken into account, each active network segment

 

 

137

therefore contributes to the stress additively. The overall stress

then becomes:

III/3

(9.1)

where gireplaces S, N becomes Ni’ and f becomes f1 in equation 3.12.
The consequence of this in the specific stress relations is that Go
and 10(- l/BO) are replaced by Gi and 4i respectively obtainable

using the material relaxation spectrum H(Ai) through these relations

(Phan Thien and Tanner, 1978).

I H(i)xdx
A. = (9.2)
1 Hum
H(A)dx
G1 = A (9.3)

Here the relaxation spectrum is subdivided into intervals, such that
each interval is a wedge spectrum to facilitate the numerical proce-
dure. The relaxation spectrum can be computed from the linear vis—
coelastic data G'(w), G"(w) and G(t) by the standard method (Ferry,
1961). The long-time behavior of block copolymers is of utmost
significance in gaining the Optimum relaxation spectrum, thus it is
necessary to collect G'(w) and G"(w) data at frequencies as low as
10'4 5-1. Such ranges are achievable by using the cone-and-plate

platen with several cone angles, which were unavailable at the time

of experimentation.

138

For a complete knowledge of melt rheological behavior of
block c0polymers, steady and transient normal stress data is highly
needed. Chung and Gale and Kraus and coworkers (1971) have
associated the material exuding from the shearing gap at low deforma-
tion rates with high elasticity developed in the material, but did
not report any normal stress data. In the WRG the shearing gap is
significantly influenced by the lack of vertical stiffness of the
apparatus. This lack of stiffness affects both steady and transient
normal stress response measurements of molten polymers (Huang, 1976).
Modifications to correct for this problem were given by Hansen (1974)
and is recommended for this equipment. The use of Mooney platens
of D<:5 cm along with a steady piezoelectric load cell is further
suggested.

Curtis and Bird (1981) have presented a reptation theory for
melts starting from the general phase space formalism (Bird et al.,
1977). They modeled the macromolecules as Kramers freely chain (with
N beads and N-l rods of length a) used a nonisotropic version of
Stokes law to describe the drag force on a bead as it moves through
the melt. The model contains four parameters, the number of beads,

N, a drag coefficient r, a link tension coefficient EB and a chain

constraint exponent Bc' They report that the model yields no ~ M3'13c.
6+2BC
and 1p1,0“M which cmpares well with homopolymeric data if

BC~ 0.3 - 0.5.
Modelling of polymer molecules as beads joined by elastic

or rigid, connectors is attractive for block copolymeric systems

 

 

139

with spherical domains. However, the Curtiss-Bird theory does not
allow us to compute the relevant chain segment distribution function.
A recommended route would be the concept of configuration-dependent
molecular mobility tailored by Giesekus (1982). He associated a
tensorial drag coefficient 21 with the force, fi’ experienced by an
ith bead. This drag tensor-does not depend on the actual configura-
tion of the molecule, but only on the average configuration of all
the molecules. After some manipulations with the excess stress
relation, a configuration tensor 9i can be defined which maps the
actual molecular configuration from the equilibrium configuration as
o o
1

0 0
.r. > = l 3 < r.- r.> b.

—J

This tensor may be understood to be a measure of deformation
of an elastic continuum, note in a strict sense of a material con-
tinuum, but in a statistical sense represents only the configurational
states of a polymer chain.

With this assumption one may no longer assign individual bi
and g to every position vector r1. Instead the whole set of beads
(i a 1 . . .; N) can be classified into classes (K = 1, . . . K) with
"K beads per unit volume with a common configuration tensor 2K and a
drag tensor 5k for each class. The class K = 1 leads to various

Lodgean type models with appropriate assumptions on b, but classifi-

cation of the total number of structure elements into K classes may

 

140

encompass systems such as block or graft copolymers. Here only
detailed modelling of 2K is required to generate the constitutive
equation.

In this study the major focus was on spherical domain block
copolymer systems, but as shown in Table 2.1 cylindrical and lamellar
type systems possess superior rheological properties. 0dani et al.
(1977) have studied diffusion, solution, and permeation behavior for
a series of inert gases in block copolymer films having these morpho-

logies Ihinted that they were excellent models for understanding the i

7

relationship between the morphology and transport properties hetero-
geneous polymeric media. The preparative methods of these block
copolymers have been much refined by the Dow Chemical Company, Mid-
land). It is recommended that rheological and transport studies

of block copolymers of higher block composition be undertaken.

APPENDICES

141

 

 

APPENDIX A

UNIAXIAL EXTENSIONAL FLOW--TRANSFORMATIONS
AND CALCULATIONS

142

 

 

APPENDIX A

UNIAXIAL EXTENSIONAL FLOw--TRANSFORMATIONS
AND CALCULATIONS

The solution of (4-3) using the method of characteristics is

given by:
t a A 4

f(x ,y , 2,t) = f(xo,yo,zo)expf-]B (xoexp(1‘t’),yoexp(-I‘t'/2), zoexp(-I‘ t'/2))dt'J
O

t I
o I A
+ .LG(XOe(part),yoexp(‘TtI/2),Zoexpc-IQLI/Z).N)€Xp[- at”
ti

X B (Xoexpfi‘ tfl)IYoeXp (-’I?I:’//2),Zoexp(-’I§t”/Z))Jdt' (l-A)

A two-step change of variables similar to that of Fuller and
Leal (1981), but for uniaxial extension is performed on (1-A) to intro-
duce definite limits on the integrals. First t' and t" are changed
to x' = xoexp(§t') and x” = xoexp(§t") respectively; then x' to 2
using x' = Tx + xexp(-Pt) where T = 1-exp(-§t). Further, y = y(x/x')%,
2 = z(x/x')%, 0' = x"/x to obtain a final expression for f(x,y,z,t)

as

_ A A
f(x’y’ z’t) _ f0 (xexp(-I‘t), yexp(I‘t/2), zexp(’Ft/2))

exp [’If8( [CH 99"- P (’Ft))X»Y12.IdG]

(T+0exp(-ft))

143

144

fijdGGKT-Wexpértnxly, z I
6(T+eexp(-T‘t))

 

expEJrIe G'EGXJ Béexpi FEW (T+°°XP(‘”))]de’ 6] (2-A)
rT+9exp(~l‘t) e

11 and 12 of equation (4n’7) may be written in spherical polay

coordinates as

I " '
1 = I F (w,F,T) c0520 sinw dw

TI
2 =I F (12.1.31) 511130 dd»

0

where

F(w’f"T‘T§J‘['\,—;Lf_‘g+
1+2a ( 3‘3

Afcos
I! + A ,,_s.1.nzg+III)/2 (A,cos 111 + Alsin I») (Agcos’w+ Aisin’tp )'/Z

2a2
+(A’;cos up + Xzsin ’wfi A;COSZIIJ+ Aisin 0)

l/2a2 -
(A’tosz I» + A 2sin writ/K). sinzw 4» $3111 11»)ij (5 A)

2
Y = eXP-zafl COS’W + 12.91111 1/2
W]

A1 5 [e-ZFT

+ (10.2%)]15

L

I. 21".-

42 = [e T 1‘? (e11T - 013
1‘

A3 : e-1‘T, A“ : EFT/2

T = Bot, T, = 80(t-t’) and f = F/Bo

145

By substituting w = tan 0 in the above equations, we obtain

,’

1~ :‘I =[;W(2~w1 2
I . 2 ;q31%r_jfw .
0 V + 2w 2 {23+ Alw) “3+ 3E0”) + (01+ i” A3+ kw)”

 

- U452 exp-2a 11 + 1‘ w 1/2 (7-A)
01+ 111020? 801 [8732?]
Then letting

_ 1+(X /A )Zw
Vz ’ Hui/13W ’ (M)

we obtain with integration by parts,

 

Ifis -
I = l -2a/<S 2 '/2 2 (9-A)
I T157; [8 +(I-62) 72 l (v -1) (1‘63i + 2a(I/v2-l/2))e-2<3Vdv]
I; = 1 e.2a/6 a $20 3 2 2
-a(1‘262-52V2-2/v2)dv] (lo-A)

where 5 ; Aa/A1 : [1 + §__(82PT

1)]"’2
2%

The integrand in (8-A) has a singularity at v - 1; setting r - v + 1

allows us to evaluate the integral.

 

APPENDIX 8

SIMPLE STEADY SHEAR FLOWS--TRANSFORMATIONS
AND CALCULATIONS I:

146

APPENDIX 8

SIMPLE STEADY SHEAR FLOWS--TRANSFORMATIONS
AND CALCULATIONS

The solution of eg. (5.4) by the method of characteristics,
in p (p,n,z) coordinates is given by:

~

t .
_ _ IA 1m I
f(p,1.z.t) - io(oo.noz.t)epr dt B(pOeXPI—Zt ).
0

OOEXP('IMt'/2)i ZSN)]

t a~ o~ t
IA 1m I _Im l _ II
+ J dt G(poexp(jrt ),noexp( 7ft ),z,N)exp [:Jodt
o

x 8<DOGXP(j§@ t' ').noexp(-12.@t"). 2)] (B-l)
An identical change of variable scheme as in Appendix A is next
undertaken except that these are in the transformed coordinates
g(o.n,z), i.e., t' and t" are changed to p' = poexp(ifit') and
p" = poexp(imt") respectively; then 0' to 5 using 0' = Tp +
pexp(-imt) where T = 1-exp(-ifit). Also fi==n(0/p') 2 = z, 0 = 0/5

and 0': p"/0 to obtain a generalized moment expression as

147

148

<g(o1n,Z)> = det(l)[(” dodndzg(o,n,Z)fo( eXp(-17mt) nexp(l—-"21t,z)

—oo

-2T A [:(T + 0exp (-J§1t) )510121de
0(T+0exp(-% t))

 

 

oo 00 + -m' - -
+ % [I] dpdndzg(p,n,z) d0G((T 0exp( 13101012)
-w 1 0(T+0exp(-lfllt))

 

0 im
A _ (T+0exp(--—-t)) ,
'-J 8(9 D, 0 2 2 9 Z: N) 99-61] (8'2)

1+eexp(—‘—'g— t)

 

 

All moments <p2>, <nz>, <pn>, <pZ/R”>,<n2/R“> and <pn/R“> are gen-
erated by eq.(B-2). Since in uniaxial shear flows no deformation
occurs in z-direction, z is arbitrarily set to zero, then these

moment integrals are evaluated as exemplified by <p2> integral

<02> = d6t(T) ‘89-“ 03%) 9.1: J! dpdnpzexplg 3_2N(}\2102_2wpnq+>\:n2)
0

-oo

2a2/3N
AipZ-Zan+A:vz

 

149

T
_ I I 2
J dt'e T J[ dodnEXP[3%¥(A1 02-2W4'00+A:nz)
o

-oo

 

2
2a /3N :] (B_3)

A5202 - 2Won + X1202

Where
= -imt .3; _ -mt %
11 (e + im (1 e ))
. ' 1 2
A2 _ (EImT + % (eImT_-I))2, A“ : eImT (B_4)

It is noted here that A's in 2nd term of eq(B-2) are defined as in
eq.(B-4), however, they are functions of elapsed time, T'.

Next the p(p,n) frame is transformed into cylindrical polar
coordinates h(d,0). On intergrating out the radical component d, eq..

(B-4) becomes

<QZ> = det(I)(§—)(§;) { e"T I I (1,0) sinzwdw

T,-T' .2 I _
+ dT e I(A,w)51n vdWI (B 5)

150

where

K2(U)

 

2
(32_

10,41) =
9N2 (AisinZw-Zquinwcosw+A§c052¢)(A:sin20—2 sinwcosw+AEcoszw)

K2(u) is a 2nd order Bessel function, and

2a(A§sinw—Zquinwcosw+A%coszw )%
A§sinw - 2Wsinwcosw + Ajcoszw
Since a <0(1), p is small from the A expressions. Then the series
for the Bessel function of integral order and 0f the 3rd kind is util—

ized in order to completely integrate out the coordinate variable,i.e.,

 

K ( ) = 1n§1(_1)k 10:5:11L___ + ( 1)n+1% (U/2)n+2kLInU/2-%W(k+1)
n U 2 = 1 n-2k — = I I
K o k.(p/2) k 0 k.(n+k).
-%v(n+k+1)] (B-6)

where v(.) is the Euler's psi function.

Thus applying eq. (B—6) to eq.. (5—1) we obtain

Zn
I(0,T) =-% Jd¢SIPV (

O

l
Aisin20-2quinwcos¢+X:coszw)2
a2

(Aisinzw-quinwcosw + Aficoszw)(Afisinzw-ZWsinwcosw + AEcosE)

+ 0(a4) (B—7)

151

The first term in Eq (B-7) is analytically integrable, but the

second term is integrated by the method of partial fraction to

yield a final expression of 1(1) as

A; 2a2(811§ - A1Wq)
(B-8)

1(1)=-——1—~-————r—.—
(1212 - quZI/Z xi (111% - wqu) /
12

where

A1 = ~2WIAiAE- Afiq)/C

and

c = 4112003 - (10315 + A012.) + 12) — (2103 - 0311+ 1111))

Thus the moment integral <pZ> is obtained as

 

APPENDIX C

OSCILLATORY SHEAR FLOWS—-TRANSFORMATIONS
AND CALCULATIONS

152

 

APPENDIX C

OSCILLATORY SHEAR FLOWS-~TRANSFORMATIONS
AND CALCULATIONS

The transformation to the sinusoidal domain with u = sinmt
after the characteristics of eq. (5.21) have been defined gives a

slightly different o.d.e. of the form

A

df + Bf _ G (H)

dU (1-U2)% (1-U2)%

 

The solution of this by method of characteristics is

 

 

u du'B 1m u' -m u'
f(psnazsu) = f0(po’wo’zo)eXp_ w(1_ul2 % (DOeXP(jf——),UEXP( 2 ),20
o
- -1
Sln u A .~ . .c .
du'G imou. -1mou
+ ———_ (0 eXp( )m exp(———),z )
J (1_u|2)éw O 2 O 2 0
o
u A imp" -m0uK\
x exp - du"B(pOeXp(—25——), noexp( -§5——), 20) (C-2)
l
u

Using transformations identical to Appendix B, except the
independent variable is u' instead of t'. The moment integral is

obtained as

154

~

 

 

~
~

w -Imou -im u
<9(0,n,Z)> = d8t(T) dodndzg(o,n,Z)fo [peXp( w ).neXP( : )’ZI

 

 

 

 

 

 

 

 

1' 00A III] U _ _
T B[(T+0exp(- g )0.n,Z)d9
x exp - . ~ .
(Imow -im0u 2 wln (T+eexp(lmgfl) 2 2
J -' ——‘ ———__——
1 0(T+0exp( w )) 1+u imo ( 0 )
w m A i~ou)_ _
d06((T+0exp(- 0.0.2)
+ l%— dodnd29(p,n.Z) w .
1m M00
0 _m 1 imou l+u2_g;g ((T+ °Xp( w
0(T+0expC-—73-)) imo 0
e ( >
T+0ex — o
.1 A ,- ( ———7—J3 ),z,N)de'
x exp 10;; B(6 0.0 9 .~ (C-3)
.- . 12 . 'mo“ 2 i
imou 0 (1+ 57— (ln01/e- ) )
T+eexp(- 0 w

In oscillatory shear flows, the transients areallowed to die out
consequently the first term in the moment expressions damps out.
Pertinent moments can be derived from eq. (C-3) from the specific of

choice G and B. The moment <02> will serve as an example.

155

2 = EL .EN _ -
<0 > dEt(l) 80 (Zn) [di'e T dpdnp2
o -oo
2
exp _ 3—2N(p2>\'2 _‘ qu'pn‘l’Aznz)" ‘_.l§-_/3_N_'_. ((3-4)
1 2 A202_2wpn+)\:n2
' 3

where

u im
2 1m e-L—J2 (u—u")]du"
A1 — exp - ——9 (U'U') + 25m w 1

(0 0 (1_ulI2)2
uI
u'im
_0 _ II
1110 J1. (””11 du"
= ———— - ' + ———7—
A EXP (u u ) 21m (1-u"2)%
u

2 IITIO (U‘U') 2 IIIIO
A3 — exp-TI— X1 — exp ~Zr-(u-u )

q'= 1+8T’ m = m /B t' = 80(t-t')

Utilizing transformation procedures and integration techniques
identical to those used to obtain eq. (B—9), we obtain the ocillatory
shear function as
i: " E I

kidet(f)(%1—~—) e'T 1(1) 111! (C-5)
~ -5

e23
1+2a

 

Sxy = é;

O

156

 

where
A2 - A2
1(1) . 1 2 _ 5133 01+ 1%)0113- 1.213)
I 2 2 3/2 8
(AiA ' qzwz)
.2 2 2 2 2 2
- 21 [(A1 - 1,)11 + w (A. - 13)] (C-6)

C = 41420030030 + mi + C12) - M2013 - (1315 + 11111)

In the first term

 

xi _ A3 = -2i[5in m0(sin11t - sin(t— t'/BO))
u
E SInIfiO<U‘U..) dull
+-——— _ (C-7)
me (1_ull2)2
ul

In small-amplitude oscillatory shear flows,

~

< 1 m m << 1
Yo ’ 0’ 0

and thus

sinm0(sinwt - sin(t-T'/BO)) ; m (SIPMt’SI“°It‘T'/Bo))

o
and

sinmo(u-u") ; mo(u-u") (C-8)

also

 

157
sinw(t-T'/BO) = sinwtcosit' - cosmtsinlt'
where 5 = w/BO
Upon integration of 2nd term in equation (C-4) we obtain
A§ - A3 = -2im0 { [l-cosfii + g-(Ji' - sinl i')] sinwt

+ [ —sin&i + %-(1-c050T')] cOSwt}
(L1

(C-9).

All the other terms in equation (C-6) were analyzed with the above

approximationS‘ and higher order quantities of sin wt and c050t were

discarded. From these, then, the final viscoelastic function expres-

sions of equation 5.26 and 5.27 were calculated.

 

REFERENCES

158

 

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