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LIBRARY
Michigan §tate
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I] F‘

94117’997 ______,

This is to certify that the
dissertation entitled

NONLINEAR RHEOLOGY AND STRAIN RECOVERY
OF SHORT CHAIN BRANCHED
POLYOLEFIN ELASTOMERS AND
THERMOPLASTIC OLEFIN BLENDS

presented by

BHASKAR PATHAM

has been accepted towards fulfillment
of the requirements for the

Chemical Engineering and

Doctoral degree 'n Materials Science

 

 

 

 

 

 

 

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NONLINEAR RHEOLOGY AND STRAIN RECOVERY
OF SHORT CHAIN BRANCHED POLYOLEFIN ELASTOMERS
AND THERMOPLASTIC OLEFIN BLENDS

By

BHASKAR PATHAM

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Chemical Engineering and Materials Science

2004

ABSTRACT
NONLINEAR RHEOLOGY AND STRAIN RECOVERY
OF SHORT CHAIN BRANCHED POLYOLEFIN ELASTOMERS
AND THERMOPLASTIC OLEFIN BLENDS
By

BHASKAR PATHAM

Polyoleﬁn elastomers are random copolymers having a polyethylene backbone
with the higher olefinic comonomer incorporated as short-chain branches. These
random copolymers are widely used as polymer modiﬁers for thermoplastic materials
such as polypropylenes, resulting in thermoplastic oleﬁn (TPO) blends. This thesis
addresses the nonlinear rheological behavior of the elastomers and then of the TPO
blends.

The effects of varying short chain branch density (SCB) on the melt rheology of
three ethylene-octene random copolymers have been investigated. In particular, the
strain-hardening behavior in extensional flow and strain recovery following nonlinear
shear creep has been evaluated. The zero-shear viscosity followed trends in the backbone
molecular weight closely. While the three copolymers were indistinguishable in linear
viscoelastic creep and recovery, recovery following nonlinear shear creep decreased
progressively with increasing SCB density. This reveals that the extent of rapid chain
equilibration that occurs over Rouse time scales at higher strains was progressively lower
with increasing SCB density. Strain hardening in uniaxial extensional flow was observed
for all three copolymers. At strain rates below the primitive chain equilibration rates,

strain hardening increases progressively with increasing SCB density. At higher rates,

......__________L_

upon onset of primitive chain stretch, the strain hardening behavior for the three melts
merges.

Two thermoplastic oleﬁn (TPO) blends were characterized in the context of
injection molding; the surface morphology of injection molded tensile bars with these
materials showed surface defects or flow marks to different extents. The ﬂow marks
were traced to different degrees of strain recovery in the dispersed phases of the two
blends. This recovery occurred over injection molding timescales of the order of a few
seconds. Strain recovery after shear creep was higher in the blend that displayed more
severe ﬂow marks in injection molding; the corresponding elastomer by itself also
showed a greater extent of creep recovery. The quick strain recovery in the elastomer
must be associated with elastic stresses rather than interfacial tension.

A new experimental apparatus was assembled to study strain recovery after
elongation of a model elastomer suspended in a less viscoelastic medium at room
temperature. Small drops of well-characterized elastomers suspended in a matrix ﬂuid
of lower viscosity and elasticity, are stretched rapidly without wall effects in this setup;
the stretch ratio of the drops is then recorded over time. Preliminary results conﬁrm
signiﬁcant recovery over a few seconds. Exploration of the full range of parameters with

this setup is left for future work.

  

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ACKNOWLEDGMENTS

I would like to thank my advisor and mentor Dr. Krishnamurthy Jayaraman for
exposing me to a wealth of knowledge and ideas, and for continuously pushing me in
more challenging directions in research. I have benefited from his guidance and keen
insight at all stages of my doctoral work. I have always fed off Dr. Jays’ enthusiasm for
research. Especially during the ﬁnal stages of my Ph. D., I have immensely enjoyed
working with him. I am leaving MSU with a fresh outlook and enthusiasm to tackle

more difﬁcult problems, and I owe this to Dr. Jay.

I would like to thank Dr. Petty for his continued interest in my research problem.
I would also like to thank Dr. Mackay and Dr. Steffe for kindly agreeing to serve on my
doctoral committee. I would like to thank Dr. Andre Lee for being a good friend, and Dr.

Robert Ofoli for recruiting me and being there for me in my initial days at MSU.

Among my lab-mates, special thanks are due to Dr. Darrell Marchant, who taught
me to use almost every piece of equipment in our laboratories. In many ways, the pace of
my research was catalyzed by the resourcefulness of Dr. Giri Kolli, who helped optimize
several experimental techniques I worked on, and who taught me that there is always
scope for improvement in things. Thanks are also due to Paul Papworth whose research I
have immensely drawn from, and John Mills, whose enthusiasm drove me through some
difﬁcult experiments. Giri, Darrell, Sharad, Paul, Dr. Kang, and John have ensured that I

will always look back at my years with J aygroup with pleasure.

 

I would like to thank Nancy Albn'ght, who always brightened my mood whenever
I visited the ofﬁce. I must also thank Rob Selden for helping me with my machining

needs and Roy Bailiff for giving me free access to the engineering workshop.

Outside the university, my life was enriched by a wealth of good friends, who
were my family in East Lansing. Amit, Prajakt, Kiran, Tejas, Yogi, and Freddy made the
odd apartment a happy place to live in over the years. My early chemical engineering
days at MSU gave me three great friends in Lavanya, Sachin and Sunder. Amit,
Lavanya, Giri, and Kiran have always cared more about me than I did myself. Prajakt
and Sachin are part of some of the best memories I have of my initial years at MSU. Our
extended circle of friends, especially, Prema, Mukta, Soumya, Amol, Kanchan(a),
Advait, Kanchan(y), and Carrie have also contributed to many great times. I still feel as
young as I felt ﬁve years ago when I ﬁrst came to MSU, and the credit goes to this

motley group.

Finally, I would like to thank my parents, who have been the driving force behind
me and my sister, always setting higher goals for us than we would have ourselves. Their
love and sacriﬁce have made us what we are today. I must thank my sister, who has been
my best friend and who has always ensured that I never get complacent. Their love and

faith in me have always inspired me to achieve better things.

vi

 

 

TABLE OF CONTENTS

 

 

 

 

 

LIST OF TABLFQ vi
LIST OF FIGUREQ vii
NOMENCLATURE “i
CHAPTER 1

Problem Statement 1
CHAPTER 2

General Introduction and Background 11
2.1. LINEAR AND NON-LINEAR VISCOELASTICITY ............................... 11

2.2. RHEOLOGY AND MOLECULAR ARCHITECTURE OF POLYOLEFINS. . ..17

2.2.1. Molecular architecture effects on linear viscoelasticity and theoretical
framework ................................................................................... 19

2.2.2. Prior experimental work in linear and non-linear rheology of polyoleﬁns. . . . . . ...22

2.2.3. Rheology of short-chain branched polyoleﬁn copolymers ........................... 24
2.3. FLOW INDUCED DEFORMATION OF DROPS AND ITS RECOVERY ...... 25
2.3.1. Parameters ................................................................................... 27
2.3.2. Experimental Studies on drop deformation and recovery ............................. 29
2.3.2a. Drop Deformation .......................................................................... 29
2.3.2b. Drop Breakup Considerations ............................................................ 31
2.3.2c. Viscoelastic Eﬂects in Drop Deformation and Breakup .............................. 32
2.3.3. Drop recovery .............................................................................. 33

2.3.3a. Quiescent recovery of ﬁbers and disks without prior defamation history... ......33

vii

A

 

2.3. 3b. Drop Recovery following ﬂow-induced deformation .................................. 34

 

CHAPTER 3
Measurement of Extensional Viscosity of Melts at High Strain Rates .................. 46
3.1. METHOD ................................................................................... 46
3.1.1 Test protocol ................................................................................ 48
3.1.Ia. Pressure transients during lubricated ﬂow through the semi-hyperboloidal
die ............................................................................................. 49
3.1.1b. Lubricated and unlubricatedﬂow of test materials .................................... 50
3.1.10. Correlation for extensional viscosity .................................................... 51
3.1 .1 d. Strain hardening parameter (I) .......................................................... 52
3.2. SAMPLE PREPARATION ............................................................... 52
3.2.1. LLDPE skin samples ....................................................................... 54
3.2.2. Thermoplastic core samples ............................................................... 55
3.2.3. Elastomer core samples .................................................................... 55
3.3. VALIDATION OF EXTENSIONAL FLOW MEASUREMENTS ................. 56
3.3.1. Measurements with varying core / skin viscosity ratio ................................ 56
3.3.2. Measurements with varying skin thickness ............................................. 57
3.3.3. Comparison with RME Data .............................................................. 58
3.3.4. Comparison with Cogswell’s Estimates ................................................ 59
CHAPTER 4
Melt Rheology of Ethylene-Octene Random Copolymer Elastomers: Effects of Short
Chain Branching Density 78
4.1. INTRODUCTION ......................................................................... 78
4.1.1. Objectives ................................................................................... 79
4.2. BACKGROUND ........................................................................... 80
viii

A

 

4.3. EXPERIMENTAL ......................................................................... 83
4.3.1. Materials .................................................................................... 83
4.3.2. Shear Rheometry ........................................................................... 84
4.3.3. Extensional viscosity ...................................................................... 86
4.4. RESULTS AND DISCUSSION ......................................................... 87
4.4.1. Assessment of molecular architecture using linear viscoelasticity .................. 87
4.4.2. Stress Relaxation and Tube-Model Parameters ........................................ 92
4.4.3. Trends in Nonlinear Shear Creep-Recovery ............................................ 95
4.4.4. Extensional Viscosity and Strain Hardening ........................................... 97
4.4.5. Constitutive analysis of non-linear shear creep recovery ............................. 98
4.4.6. Damping functions ........................................................................ 101
4.5. CONCLUSIONS ......................................................................... 103
CHAPTER 5

Flow Lines in Injection Molding of Thermoplastic Olefin Blends: Surface
Microstructure and Nonlinear Rheology 133
5.1 INTRODUCTION ........................................................................ 134
5.2. EXPERIMENTAL ........................................................................ 137
5.2.1. Materials ................................................................................... 137
5.2.2. Injection-molding ........................................................................ 138
5.2.3. Morphological characterization ......................................................... 138
5.2.4. Rheological characterization ............................................................ 139
5.2.4a. Extensional viscosity ...................................................................... 140
5.3. RESULTS AND DISCUSSION ........................................................ 142
5.3.1. Flow mark frequency and blend rheology ............................................. 142
5.3.2. Surface morphology of injection molded short shots ................................ 144

ix

A

5.3.3. Rapid retraction and elastomer rheology .............................................. 147
5.4. CONCLUSIONS ......................................................................... 149
CHAPTER 6

Recovery of Elastomeric Drops in Suspension following high-rate Extensional

Flew:

 

 

 

6.1. INTRODUCTION ........................................................................ 174
6.2. EXPERIMENTAL ........................................................................ 176
6.2.1. Experimental Setup ....................................................................... 176
6.2.] 0. Flow kinematics ........................................................................... 179
6.2.2. Materials ................................................................................... 180
6.2.3. Determination of Interfacial Tension ................................................... 183
6.2.4. Typical Parameter Ranges in the Experiment ......................................... 186
6.2.4a. Drop breakup Considerations ........................................................... 187
6.3. RESULTS AND DISCUSSION ........................................................ 188
6.4 CONCLUSIONS ......................................................................... 190
CHAPTER 7

Conclusions and Rn ’ " 209
7.1. CONCLUSIONS ......................................................................... 209
7.2. FUTURE DIRECTIONS ................................................................ 214
APPENDIX

FORTRAN code for solution of the integral equation for prediction of strain during
shear creep and recovery using K-BKZ constitutive _ " 21
LITERATURE CITED ............................................................................ 232

 

Table 3.1.

Table 4.1.
Table 4.2.

Table 4.3.

Table 4.4.

Table 4.5.

Table 5.1

Table 5.2.

Table 6.1.

Table 6.2.

LIST OF TABLES

Comparison of extensional viscosities of two polypropylenes measured at
1.0 s‘1 using lubricated skin-core flow through the semi-hyperboloidal die

 

e = 2
with 712(6" = 1.0s‘l , t = H . ) measured using Rheometrics Melt
e
Extensiometer (University of Minnessota), both at 180°C .................. 62
Material speciﬁcations and physical properties .............................. 104
Molecular architectural parameters for the E0 copolymers ............... 105

Zero-shear viscosities and relaxation times (at 150°C) for the E0
copolymers ........................................................................ 106

Discrete relaxation spectra of the E0 copolymer melts at 150°C.........107

Damping function ﬁtting parameters for the E0 copolymer melts ....... 108

Composition of the reactor made blends and component properties at
200°C .............................................................................. 151

Results of morphology analysis on injection molded TPO short-shots (see
Fig. 2 for coordinate labels) .................................................... 152

Typical material and ﬂow parameters involved in rapid processing ﬂows
of TPO blends comprising an elastomeric dispersed phase in a
thermoplastic matrix ............................................................ 192

Materials and properties for room temperature drop studies ............... 193

xi

Figure 1.1.

Figure 1.2.

Figure 2.1.

Figure 2.2.

Figure 2.3.

Figure 2.4.

Figure 2.5.

Figure 2.6.

Figure 2.7.

Figure 2.8.

LIST OF FIGURES

Short-chain branching in polyolefin random copolymers ..................... 9
Research synopsis ................................................................. 10
Typical ﬂow history imposed on the polymer melt and its response during
shear stress-relaxation ............................................................ 38
Flow history imposed on the polymer melt and its response during shear
creep recovery ...................................................................... 39
(a) Shear thinning and (b) Linear and non-linear startup shear

responses ............................................................................ 40

Typical linear and non-linear responses of the polymer melt during
extensional ﬂows .................................................................. 41

Dependence of deformation on capillary number for Newtonian and
viscoelastic drops of high viscosity ratio in a Newtonian matrix ........... 42

Quiescent recovery of drop deformation driven by interfacial tension: (a)
imbedded ﬁber retraction [Cohen et al. (1989)], (b) imbedded disk
retraction [Rundqvist et al. (1995)] ............................................. 43

Drop relaxation following imposition of instantaneous shear strain
[Yamane et a1. (1998)]. Drop relaxation proceeds in two stages - the ﬁrst
stage involves shape relaxation to an axisymmetric spheroid, followed by
retraction in length. The long time transient may be modeled using
Carriere-Cohen (1989) type interfacial-drag force balances ................ 44

Recovery of a viscoelastic drop suspended in a Newtonian matrix
following extensional deformation. The curves are predictions obtained
by ﬁnite element solution of Navier—Stokes equations, with the
viscoelasticity of the drop phase incorporated using Oldroyd-B model
[Hooper et a1. (2001)] ............................................................ 45

Figure 3.]. Modiﬁed capillary rheometer setup to measure the extensional viscosity of
polymer and elastomer melts .................................................... 63

Figure 3.2. Semi-hyperboloidal proﬁle converging die to achieve constant extensional
strain rate ........................................................................... 64

Figure 3.3. Shear viscosities of some of the thermoplastic polymer core melt samples
compared with that of the LLDPE skin, at 200°C ............................ 65

Figure 3.4. Shear viscosities of some of the elastomeric polymer core melt samples
compared with that of the LLDPE skin, at 200°C ............................ 66

Figure 3.5. Schematic representing skin-core flow of the test material through the
semi-hyperboloidal converging die, with optimum lubrication. The skin
thickness, b, has been exaggerated in this ﬁgure for purpose of clarity.
Typically, the skin thickness is about 14% of the reservoir (barrel) radius
Rb .................................................................................... 67

Figure 3.6. Skin-core sample for achieving lubricated ﬂow through the semi-
hyperboloidal proﬁle converging die .......................................... 68

Figure 3.7. Typical startup pressure transients during the measurement of extensional
viscosity using skin—core lubricated flow through the semi-hyperboloidal
die ................................................................................... 69

Figure 3.8. Steady state pressure drops obtained at several piston velocities using
lubricated skin-core ﬂow compared with the pressure drops obtained using

unlubricated flow of the test material through the same die ................. 70
Figure 3.9. Setup for preparation of thermoplastic core samples ......................... 71
Figure 3.10. Setup for preparation of elastomeric core samples ........................... 72

Figure 3.11. Startup pressure transients for lubricated skin-core ﬂow through the semi-
hyperboloidal die. Measurements with two different core / skin viscosity
ratios, carried out at a strain rate of 1 s‘1 and at 200°C are ahown. The test
material is a branched polypropylene, grade SD613 (Basell Polyoleﬁns
Inc.). Time has been scaled by the ratio sample-length / ram-rate. . . . . ....73

Figure 3.12. Startup pressure transients for lubricated skin-core flow through the semi-
hyperboloidal die. Measurements shown above have been carried out with
two different skin thicknesses resulting in core / skin volume ratios of
70/30 and 63/37 respectively. The measurements were carried out at a
strain rate of l s'1 and at 200°C, for a linear polypropylene grade Pro-fax
6523 (Basell Polyoleﬁns Inc.). Time has been scaled by the ratio sample-
length / ram-rate. d/ Rb, is the ratio of skin thickness and the capillary
barrel
radius ................................................................................ 74

xiii

A

Figure 3.13. Bagley plot for obtaining ends pressure drop for shear ﬂow through
straight capillaries of diameter 1.0 mm. The material tested was an
ethylene-propylene random copolymer elastomer — DOWLEX C0034 at
150°C. The entrance pressure drops were used for estimation of
extensional viscosity using Cogswell analysis ................................ 75

Figure 3.14. Comparison of estimates of extensional viscosity obtained from skin-core
flow through the semi—hyperboloidal profile converging die and that from
Cogswell analysis of entrance pressure data obtained from Bagley plots.
The data were obtained on an ethylene—propylene random copolymer
elastomer — DOWLEX C0034 at 150°C. The semi-hyperboloidal die
offered a Hencky strain of 4.88 while the straight capillary die of 1.0 mm
diameter effectively offers a Hencky strain of 4.5 ............................ 76

Figure 4.1. Shear creep compliance curves for E03 at 150°C; nonlinearity is observed
at 120 Pa ........................................................................... 109

Figure 4.2. Elastic shear compliance during creep J E (t) = J (t) — t/ 770 at 5 Pa ( I )
and recoverable compliance J r (t’) ( El ) for E01 at 150°C ............. 110

Figure 4.3. Elastic shear compliance during creep J E (t) = J (t) — t/ 770 at 5 Pa (A)
and recoverable compliance J r (t’) ( A) for E02 at 150°C ............... 111

Figure 4.4. Elastic shear compliance during creep J E (t) = J (t) — t/ ’70 at 5 Pa ( O )
and recoverable compliance J r (t') ( o ) for E03 at 150°C ............. 112

Figure 4.5. Comparison of linear shear creep recovery curves for the three E0
copolymer melts at 150°C after creep at 5 Pa ................................ 113

Figure 4.6. Dynamic shear moduli of E01. Data obtained at 120°C (Cl), 150°C (I)
and 200°C (+) have been shifted to the reference temperature of 150°C
using time-temperature superposition. The solid lines are ﬁts obtained
with the discrete relaxation spectrum ......................................... 114

Figure 4.7. Dynamic shear moduli of E02. Data obtained at 120°C (A), 150°C (A),
and 200°C (X) have been shifted to the reference temperature of 150°C
using time-temperature superposition. The solid lines are ﬁts obtained
from the discrete relaxation spectrum ......................................... 115

Figure 4.8. Dynamic shear moduli of E03. Data obtained at 120°C (0), 150°C (0),
and 200°C ()K), have been shifted to the reference temperature of 150°C

A

Figure 4.9.

Figure 4.10.
Figure 4.11.
Figure 4.12.

ﬁwn4m.

Figure 4.14.

Figure 4.15.

Figure 4.16.

Figure 4.17.

Figure 4.18.

Figure 4.19.

Figure 4.203.

nymama

using time-temperature superposition. The solid lines are ﬁts obtained
from the discrete relaxation spectrum ......................................... 116

Dynamic viscosity curves for the E0 copolymers at 150°C ............... 117
Shear stress relaxation curves for E01 at several strains and 150°C.....118
Shear stress relaxation curves for E02 at several strains and 150°C.....119
Shear stress relaxation curves for E03 at several strains and 150°C. ....120

Recovery transients y, (t’) at 150°C for the three E0 copolymers following
nonlinear shear creep at (a) 10,000 Pa up to a maximum shear strain
deformation 70m“) = 25 and (b) 30,000 Pa up to a maximum shear

deformation y(tmax) = 75 ..................................................... 121

Steady recoverable compliance vs. the maximum strain imposed on the
sample during creep stage, at several stresses and at 150°C. The three sets
of data shown in this ﬁgure correspond to creep stresses of 5 Pa
(maximum strain ~ 0.3), 10,000 Pa (maximum strain ~25) and 30,000 Pa
(maximum strain ~75) ........................................................... 122

Extensional viscosity vs. extensional strain rate for the copolymer melts
measured at a Hencky strain of 4.9 and at 150°C ........................... 123

Elongational viscosity E01 measured at 150°C and Hencky strain 4.9,
plotted at equivalent times for several strain rates. The linear viscoelastic

startup shear viscosity 776L was measured using step shear at 0.005 s"..124

Elongational viscosity of E02 measured at 150°C and Hencky strain 4.9,
plotted at equivalent times for several strain rates. The linear viscoelastic

startup shear viscosity 773 was measured using step shear at 0.005 s".. 125

Elongational viscosity of E03 measured at 150°C and Hencky strain 4.9,
plotted at equivalent times for several strain rates. The linear viscoelastic

startup shear viscosity 773 was measured using step shear at 0.005 s'l..126

Effect of SCB density on strain hardening parameter evaluated at 150°C
and at £3 = 4.9 .................................................................... 127

Comparison of the KBKZ model predictions for E01 with experimental
creep and creep recovery data at 150°C with creep stress =30,000 Pa... 128

Comparison of the KBKZ model predictions for E02 with experimental
creep and creep recovery data at 150°C with creep stress =30,000 Pa... 129

XV

A

Figure 4.20c.

Figure 4.21.

ﬁmm4n.

Figure 5.1.

Figure 5.2.

Figure 5.3.

Figure 5.4.

ﬁwmii

Figure 5.6.

Figure 5.7.

Figure 5.8.

Comparison of the KBKZ model predictions for E03 with experimental
creep and creep recovery data at 150°C with creep stress =30,000 Pa...130

Shear strain damping functions ﬁtted to nonlinear shear creep recovery
data with the K-BKZ equation ................................................. 131

Extensional strain hardening parameter [(5), = 4.9) vs. Deborah number
based on the Rouse relaxation time for the three copolymer melts at
150°C .............................................................................. 132

An unstable, advancing melt front during injection molding with ﬂow
marks, showing asymmetric path lines and an offset stagnation
point ............................................................................... 153

Various regions marked on a short shot for SEM image analysis

1. Out-of-ﬂow—mark (glossy) surface, x-y plane

2. Flow-mark (dull) surface, x-y plane

3. Near-wall cross section beneath ﬂow-mark region, x-z plane

4. Near—wall cross section beneath out-of—ﬂow-mark region, x-z plane

5. Cross section of core upstream of ﬂow front, x-z plane ............... 154

Proﬁle of the semi-hyperboloidal converging die, designed to provide a
uniform extensional strain rate and a Hencky strain of
4.9 ................................................................................. 155

Comparison of tiger stripes or flow marks on tensile bars of two different
blends. (a) Blend B1: high surface contrast between glossy (out-of-ﬂow
mark) and dull (ﬂow mark) regions; (b) Blend B2: lower surface contrast.
Gray portions represent dull regions .......................................... 156

Shear viscosity curves of the reactor-made blends B1 and B2 measured at
200°C .............................................................................. 157

Linear shear creep recovery curves for the two blends at 200°C 158

Three times the linear shear viscosity growth as a function of time at low
strain rate (0.005 s") for the two blends; and nonlinear strain averaged
extensional viscosity of blends B1 ( I ) and B2 ( A ) at Hencky strain
6” = 4.9 and 200°C, plotted as a function of the residence time in the semi-

hyperboloidal die ................................................................ 159

Extensional strain hardening parameter 2' for the TPO blends evaluated at

a Hencky strain 8” = 4.9 and 200°C .......................................... 160
xvi

A

Figure 5.9.

Figure 5.10.

Figure 5.11.

Figure 5.12.

Figure 5.13.

Figure 5.14.

Figure 5.15.

Figure 5.16.

Figure 5.17.

Figure 5.18.

Figure 5.19.

Figure 5.20.

Figure 6.1.

x-z plane section near the ﬂow-front in the core region of blend B1....161

x-y plane sections from out-of—ﬂow mark and ﬂow mark surfaces in blend
Bl .................................................................................. 162

Near-wall sections in the x-z plane beneath the out-of—flow mark and flow
mark surfaces of blend Bl ...................................................... 163

x-z plane section near the ﬂow-front in the core region of blend B2. ....164

x-y plane sections from out-of-ﬂow mark and ﬂow mark surfaces in blend
B2 .................................................................................. 165

Near-wall sections in the x—z plane beneath the out-of—flow-mark and
ﬂow-mark surfaces of blend B2 ............................................... 166

Projected area distributions for the dispersed elastomer particles in various
regions (a) for B1 and (b) for BZ ............................................. 167

Aspect ratio distributions for the dispersed elastomer particles in various
regions (a) for B1 and (b) for B2 ............................................. 168

Dynamic shear viscosities of the PP matrices and EP dispersed phases of
the blends at 200°C ............................................................. 169

Strain averaged extensional viscosity of the matrix PP (open symbols) and
dispersed EP elastomer phases (ﬁlled symbols) in blends B1 and B2,
evaluated at Hencky strain 8” = 4.9 at 200°C
..................................................................................... 170

Total strain and recoverable strain during shear creep at 3000 Pa followed
by constrained recovery for the EP elastomer phases of the blends at
200°C ............................................................................. 171

Recovered strain y,(t’= 1 s) relative to the linear equilibrium recoverable
strain corresponding to the creep stress during constrained recovery
following shear creep at several stresses for the EP elastomer phases of the
two blends, and at 200°C ...................................................... 172

Typical matrix and dispersed phase properties of TPO blends. Blends B1
and B2 are same as those studied in Chapter 5. (a) Extensional viscosities
of of PP matrices (open symbols) and EP elastomer dispersed phases
(ﬁlled symbols) evaluated at Hencky strain E": 4.9 at 200°C (b)
Dynamic storage moduli of the PP matrices (open symbols) and EP
elastomer dispersed phases (ﬁlled symbols) of blends B1 and B2 at
200°C .............................................................................. 194

xvii

A

Figure 6.2.

Figure 6.3.

Figure 6.4

Figure 6.5.

Figure 6.6.

Figure 6.7.

Figure 6.8.

Figure 6.9.

¥

Mechanism for imposing rapid ﬁnite extensional strains on a viscoelastic
drop suspended in a matrix of lower viscosity and elasticity, and observing
the recovery behavior after stoppage of ﬂow.

1. Flow of the viscous suspending ﬂuid through the semi-
hyperboloidal fixture establishes the high rate extensional ﬂow
ﬁeld

2. The elastomer drop, transported separately in an inner stream of

the suspending ﬂuid is introduced to the extensional flow ﬁeld
established by the bulk ﬂow just upstream of the convergence

3. Semi-hyperboloidal converging proﬁle fixture offers a constant
extensional strain rate over its entire length; the Hencky strain of
the ﬁxture is the upper limit of the strain attained by the drop

4. Flow is instantaneously stopped once the elastomer drop exits the
semi-hyperboloidal ﬁxture and this removes the extensional ﬂow
ﬁeld; the subsequent recovery of the stretched drop is recorded in
the downstream portion with a hi gh-speed camera ................ 195

Growth of shear viscosity for the drop phase PDMS during step shear
conducted at 0.05 s", at 25°C ................................................... 196

Dynamic moduli for the drop phase PDMS at 25°C ........................ 197

Frequency dependent complex viscosity of the drop phase PDMS at
25°C 9

Shear stress relaxation curves for the drop phase PDMS at 25°C and two
different strains as indicated within the ﬁgure ............................... 199

Shear creep at 10 Pa, and subsequent recovery for the drop phase PDMS
measured at 25°C ................................................................ 200

A photo sequence following quiescent recovery of a short ﬁlament made
with the polybutene - chlorparafﬁn oil mixture imbedded in the high-
viscosity PDMS reservoir ...................................................... 201

Estimation of interfacial tension from quiescent recovery of a short
ﬁlament of polybutene — chlorparafﬁn oil mixture imbedded in a high
viscosity PDMS reservoir. The points ( I ) are obtained from
measurements of the photographs of the recovering drop at regular time
intervals (Figure 6.8). The line is obtained from the shape evolution
equation laid out in Section 6.2.3 for the force balance between interfacial
tension forces and drag forces — interfacial tension is the ﬁtting parameter
to obtain the best ﬁt over experimental data, while the drag force is
calculated from known viscosities of the two ﬂuids ........................ 202

xviii

Figure 6.10.

Figure 6.11.

Figure 6.12.

Figure 6.13.

Figure 6.14.

Figure 6.15.

Typical drop shape recovery transient following low-rate deformation.
Comparison of the experimentally observed transient and the predictions
of interfacial tension -drag force balance proposed by Cohen et al. (1989)
for quiescent interfacial tension driven recovery ............................ 203

Wall effects during deformation of the drop leading to “space-shuttle”
shaped drops ..................................................................... 204

A typical photo sequence for drop shape recovery transient following
high-rate deformation through the extension ﬂow cell ..................... 205

Evolution of aspect ratio during recovery following high-rate deformation
through the extension ﬂow cell ................................................ 206

Comparison of the experimentally observed recovery transient following
high-rate deformation and the predictions of interfacial tension —drag force
balance proposed by Cohen et al. (1989) for quiescent interfacial tension
driven recovery .................................................................. 207

Recovery transients following high-rate deformations: effect of
extensional strain rate on the extent of elastic recovery in the initial
stage ............................................................................... 208

Dstretch
De

E (t)

Ea

G (t)

GUJ’)

12
1(1)

1.0)

%

NOMENCLATURE

Time-temperature shift factor

Minor axis of the drop

Finger strain tensor

Capillary number

Drop deformation parameter deﬁned for small deformations
Stretch ratio of the drop deﬁned for large deformation
Deborah number

Elastic modulus

Flow activation energy

Relaxation modulus

Strain dependent nonlinear stress relaxation modulus
Dynamic storage modulus

Dynamic loss modulus

Linear viscoelastic stress relaxation modulus

weighting factor for discrete relaxation spectrum

Plateau modulus

First invariant of Finger tensor
Second invariant of Finger strain tensor

Shear compliance

Elastic compliance

XX

 

J inst

Jr(t)

Linear equilibrium recoverable compliance

Compliance associated with instantaneous deformation during creep
Recoverable compliance

Drop / matrix viscosity ratio

Drop / matrix elasticity ratio

Pre-exponential factor for power-law dependence of shear viscosity on the
shear rate

Major axis of the drop
Length of the semi-hyperboloidal converging die

Molecular weight

Molecular weight of the branch

Critical molecular weight for entanglements

Entanglement molecular weight

Number average molecular weight

Weight average molecular weight

Exponent for power—law dependence of shear viscosity on the shear rate
radial direction

Instantaneous radius of the drop during deformation or recovery
Radius of the rheometer barrel

Exit radius of the semi-hyperboloidal converging die
Equilibrium spherical radius of drop

Time

Time at which creep stress is removed

'72.!

VP

NI

Greek

Residence time of the melt within the serni—hyperboloidal die

Piston velocity for testing in the pressure driven rheometer
Number average x-direction length of the dispersed phase
principal axis

Entanglement density in a polymer chain

Backbone entanglement density

Extensional strain hardening parameter

Drop shape factor or drop / matrix interaction parameter
Hencky strain

Extensional strain rate

Shear strain

Shear rate

Shear rate at long time viscous response during shear creep
Instantaneous shear strain imposed during stress relaxation
Recoverable shear strain

Interfacial tension between dispersed phase and matrix
Linear startup shear viscosity

Dynamic viscosity

Dynamic complex viscosity

Zero-shear viscosity

xxii

m- -

 

 

”at

775

#°(o
g;(t)

1(t)

Tr

T11 — 722
7d

TR

Special

G(t.r)
h(r)

AP“

Complex viscosity

Effective viscosity of a two phase drop / matrix system or emulsion or
blend

Extensional viscosity

Discrete relaxation times in the relaxation spectrum

Linear viscoelastic memory function
Stress tensor
Shear stress
Characteristic relaxation time determined from shear creep experiments
Extensional stress
Tube disengagement or reptation time
Equilibration or Rouse time
Wall shear stress

Frequency

Strain dependent nonlinear stress relaxation modulus

Shear strain damping function
Pressure drop

Entrance pressure drop

xxiii

Chapter 1.

PROBLEM STATEMENT

MOTIVATION

Blending of polymers to improve material properties offers an effective
alternative to synthesis of new materials. Rubbers are blended with therrnoplastics to
improve the impact strength and low temperature properties of the thermoplastic material.
Crosslinked rubber however is difﬁcult to process and not amenable to recycling. A
special class of materials known as thermoplastic elastomers offers an attractive
alternative to rubbers. Thermoplastic elastomers are not crosslinked, but instead have
reversible bonding, in the form of hydrogen bond forces, which makes them rubber-like
at room temperature, while the absence of cross links imparts thermoplastic-like
characteristics in the melt state [Morris (1979), Coran (2001)]. Such reversible bonding
is usually achieved via copolymerization. Block copolymers as well as random

copolymers have been used widely as polymer modiﬁers.

The advances in the ﬁeld of metallocene catalysis have brought about the advent

of a new class of random copolymers of polyethylene with a higher oleﬁn comonomer

A

such as butene (C4), or hexene (C-6), or octene (08). The metallocene catalyst
structure facilitates homogeneous incorporation of the comonomer throughout the
polymer chain, and a narrow distribution of comonomer type [Sehanobish et al. (1994)].
The higher-oleﬁn comonomers are incorporated as short-chain branches (SCB) on a
polyethylene backbone, as shown schematically in Figure 1.1. Depending on the short—
chain branch (SCB) density (1. e., comonomer content), these random copolymers show a
whole range of structures, ranging from LLDPE at low SCB densities to elastomeric at

high SCB densities [Bensason et al., (1996), Shan and White (2004)].

Such polyethylene copolymer elastomers (having high SCB density) or polyoleﬁn
elastomers are a new class of thermoplastic elastomers, and are very effective polymer
modiﬁers. For example, ethylene-octene (E0) random copolymer elastomers have
quickly established themselves as polymer modiﬁers of choice for polypropylene (PP).
Blends of polypropylene (PP) and ethylene-octene (E0) elastomers are now widely used
for internal as well as external automotive applications such as dashboards and bumpers -
and are processed via high-speed injection molding and thermoforrning. Blends
containing polyoleﬁn elastomers dispersed in a thermoplastic matrix are known as
thermoplastic oleﬁn (TPO) blends. Polyoleﬁn elastomers are also employed “neat” (i.e.,
by themselves, without adding them to another polymer) for several blow-molding and

therrnoforming applications.

Given the wide range of applications of these elastomers, their melt rheology is

obviously of great interest. Nonlinearities in rheological behavior are very sensitive to

A

the polymer chain architecture. The effects of polymer molecular weight distribution,
and sparse as well as profuse long-chain branching on the rheology of polymer melts
have all received different degrees of attention by several researchers in the past
[Graessley (1992), Ferry (1980), Kasehagen and Macosko (1998), Gabriel and Munstedt
(2003)]. A less understood aspect of polyoleﬁn random copolymers that warrants further
investigation is the effect of short-chain branching density on the melt rheology of these

copolymers, especially at high SCB densities.

The effectiveness of these polyethylene-copolymer elastomers as polymer
modiﬁers is in great part governed by the microstructure of the resulting thermoplastic
oleﬁn (TPO) blends. Microstructure of the blend is especially important when the TPO
blends are employed for non-painted external automotive parts such as car bumpers. The
evolution of such microstructure is governed by the properties of the thermoplastic
matrix, the elastomeric dispersed phase, the interfacial tension, the deformations and the
rates of deformation imposed on the blend during processing. The ﬁnal blend
microstructure as well as physical properties of the product are governed by recovery
following such deformations. It is therefore important to understand the factors driving

blend recovery in these systems, especially under high-rate processing conditions.

An interesting problem involving rapid evolution of TPO blend microstructure
has been studied in our laboratory [Patham et al. (2004a), Papworth (2001)] over the last
few years. TPO parts injection molded at high-rates develop surface gloss defects called

ﬂow-marks. This phenomenon is associated with a ﬂow transition at the advancing melt

A

front from stable symmetric fountain ﬂow to unstable, oscillating, asymmetric ﬂow.
Frequency of ﬂow-marks on the surface of the TPO part is therefore governed by how
unstable the melt front is. Recently conducted analyses of surface microstructure
associated with the ﬂow-mark regions in our laboratory reveal high degrees of recovery
of extensional strains in the elastomeric dispersed phase. Greater extents of recovery
result in greater surface contrast, and more severe gloss defects. Such recovery usually
occurs within the injection—molding timescales of the order of a few seconds, and cannot
be attributed to interfacial tension alone. It is therefore essential to relate the frequency
and severity of these surface defects to the nonlinear rheology of the TPO blends and the

components.

The ﬂow-mark phenomenon presents the need to systematically investigate the
role of component rheology and ﬂow kinematics in the manifestation of elastic effects
during recovery. While interfacial tension driven recovery in drops and blends has been
widely studied [Tucker and Moldenaers (2002), Gramespacher and Meissner (1995,
1997), Vinckier et al. (1999a, 1999b), Cohen and Carriere (1989)], the understanding of
elastic effects during recovery is not very well developed. In particular, model systems in
which the drop phase is much more viscous and elastic as compared to the matrix are

attractive due to their similarity to TPO blend systems.

This thesis addresses three speciﬁc research problems motivated by the scenario
described above. These problems are associated with polyoleﬁn elastomers and TPO

blends and focus on nonlinear rheology and strain recovery.

RESEARCH PROBLEM I

To investigate the interrelationships between short-chain branching density in

ethylene- a-oleﬁn elastomers and their nonlinear melt rheology

In this study the effect of short chain branching density on the nonlinear melt
rheology of polyoleﬁn elastomers has been investigated with three carefully chosen
metallocene based ethylene-octene random copolymer elastomers of low polydispersity.
In particular, the elasticity of the melts has been assessed using nonlinear shear creep-
recovery measurements, and the strain hardening behavior in extensional ﬂows has been
studied. Trends in linear viscoelasticity and nonlinear rheology have been carefully
evaluated to contrast them with those obtained in the presence of long-chain branching or

high molecular weight tails.

RESEARCH PROBLEM 2

To study the effect of polyoleﬁn elastomer and TPO blend rheology on the

evolution of blend morphology in high-rate processing operations.

In this study, two thermoplastic oleﬁn (TPO) blends have been theologically
characterized in the context of the ﬂow-marks associated with high-speed injection

molding. The two TPO blends displayed different frequencies and severity of ﬂow-

marks when molded at the same rate. The different degrees of surface defects or ﬂow
marks were traced to different degrees of strain recovery in the dispersed phases of the
two blends. Surface microstructure following injection molding of TPO blends has been
analyzed based on the melt rheology of the blends as well as their components. The
nonlinear melt rheology of the blends as well as their components, as assessed from
nonlinear shear creep-recovery and extensional strain hardening, are found to be

important parameters governing the blend recovery behavior.

RESEARCH PROBLEM 3

To evaluate the ﬂuid and flow parameters governing elastic eﬁects during

recovery of elastomeric drops in suspension

The ﬁnal aspect of this work aims at gaining an understanding of the effect of
drop and matrix viscosities and elasticities as well as ﬂow conditions on the extent of
recovery driven by elastic effects. A new experimental apparatus has been assembled in
our laboratory to study strain recovery after elongation of a model elastomer suspended
in a less viscoelastic medium at room temperature. Small drops of well-characterized
elastomers suspended in a matrix ﬂuid of lower viscosity and elasticity, are stretched
rapidly without wall effects in this setup; the stretch ratio of the drops is then recorded
over time. Preliminary results conﬁrm signiﬁcant recovery over a few seconds.

Exploration of the full range of parameters with this setup is left for future work.

Figure 1.2. summarizes the main thrusts of this work. Some aspects of this work,
such as the phenomenological modeling of recovery behavior are still in the formative

stages, and will be carried out in future.

ORGANIZATION OF THESIS

In Chapter 2, a detailed background is presented, in which several aspects
mentioned in the foregoing are elaborated, to place the current study in perspective. The
background broadly covers two aspects. Firstly, it summarizes the prior studies on
rheological characterization of polyoleﬁns, and the current understanding of the
relationships between rheology and molecular architecture. The second aspect involves a
summary of experimental studies and theoretical analyses of deformation and recovery

behavior of drops suspended in a sea of matrix.

One of the important methods of characterization of several materials involved in
these studies was extensional viscosity at a high extensional strain and high strain rates.
For this purpose we applied a technique involving lubricated skin-core flow of the test
material through converging dies of special proﬁles. The experimental aspects of these
measurements and other rheological characterization techniques employed are laid out in

Chapter 3 [Patham and J ayaraman (2002a)]

The experimental results on interrelationships between the molecular architecture,

particularly short-chain branching and the linear and nonlinear rheology of the ethylene-

octene elastomers are presented in Chapter 4, along with interpretation of the trends in

the light of reptation model for polymer melts. [Patham and Jayaraman (2004b, 2004c)]

In Chapter 5, thermoplastic olefin (TPO) blend microstructures associated with
high-speed injection molding are interpreted in terms of the rheology of the blends as

well as their components. [Patham et al. (2002b, 20043)].

In Chapter 6, the experimental methodology - for recording and identifying the
elastic phase of recovery of an elastomeric drop suspended in a matrix of lower viscosity
and elasticity following rapid high strain extensional deformations — is laid out.
Preliminary results from this experiment are encouraging and offer new areas to be

investigated in future.

The principal conclusions and recommendations for future work are laid out in

Chapter 7.

SCB in ethylene-octene
(E0) elastomers

Figure 1.1. Short—chain branching in polyolefin random copolymers

Research Synopsis

Figure 1.2.

 

 

 

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10

Chapter 2.

GENERAL INTRODUCTION AND BACKGROUND

2.1. LINEAR AND NONLINEAR VISCOELASTICITY

When a polymeric melt is subjected to a step strain, the stress relaxes in an
exponential fashion. In contrast, the stress instantaneously relaxes to zero in a viscous
liquid, while an elastic solid shows no relaxation. Such time-dependent response is
known as viscoelasticity. The stress relaxation data may be interpreted in the form of a

relaxation modulus deﬁned as shown below.

G(t) =

E (2.1)
7

As shown in the log-log plot of Figure 2.1, for very small strains (below a critical strain),
relaxation modulus data for all strains fall on the same curve. Such linear dependence of
stress on the applied strain (stress relaxation modulus independent of strain) is known as
linear viscoelasticity [Ferry (1980), Macosko (1994)]. The time dependence may be

characterized by a linear viscoelastic memory function as deﬁned below.

11

 

0 d
#(t) = —$

d (2.2)

In the linear viscoelastic regime, the relaxation modulus (and hence the memory
function) depend only on time, and therefore any larger deformation (below the critical
strain) results by a summation of all small deformation. This results in the general linear

viscoelastic model, as shown below

t o t
1(1): jy(t—t’)y(t’)dt’= [G(t—t’)y(z’)dt’ (2.3)

-00 —00

The exponential decay _of the relaxation modulus may be modeled using a series of

relaxation times /i.,- and weighting constants G,- as shown below,

G(t) = Z G,exp(—t//t,-) (2.4)

The set (1,- ,Gi) constitutes the discrete relaxation spectrum of the polymer melt. The

area under the stress relaxation curve yields the zero-shear viscosity of the melt.

770 = [0(1) dt (2.5)
0

 

 

Alternatively, when loaded in shear or extension, polymer melts exhibit an
instantaneous deformation similar to an elastic solid, followed by a continuous time-
dependent deformation known as creep. Creep data may be interpreted using a creep

compliance, as deﬁned below.

J (t) = M (2.6)
T

0

The creep response for melts is schematically shown in Figure 2.2. At long times, a
steady rate of straining 7,, is attained. In the linear viscoelastic regime, since strain is
linear with stress, the strain vs. time data collapse onto a single J (t) curve. The response
of the melt to creep can be decomposed into three portions: an instantaneous deformation

J ins, , an elastic portion and a viscous portion that grows with time, as shown below.

J(t) = Jim, + J, (t) + L (2.7)
770

Jim, is usually negligible as compared to the elastic and viscous portions. Thus during

the steady linear creeping regime,

1(1) = 12 + i (2.8)

Upon removal of the stress, the polymer melt recovers the elastic portion of the imposed

deformation. The recoverable compliance is deﬁned as shown below.

7r (I)

1,0) =
2'0

(2.9)

In the linear viscoelastic regime, if steady state is achieved in creep, then the equilibrium

creep recovery is a direct measure of J 2 .

Linear viscoelastic or small strain material functions are also assessed using
dynamic testing, in which sinusoidal oscillations with small strain amplitude are imposed

on the melt sample.

y = 70 sin wt (2'10)

1:10 sin(a)t+6) (2.11)

Such sinusoidal data are analyzed by decomposing the stress into two components — an
elastic component in phase with the strain and a viscous component 90° out of phase with
strain. This results in two dynamic moduli - an in-phase elastic or storage modulus G'
and an out—of—phase viscous or loss modulus G'. The elasticity of the melt is assessed

using the phase tangent.

G
t 6:— 2.12
an 6' ( )

In addition, a complex viscosity may be deﬁned, with its magnitude is given by,

 

 

. 2 , 2 “2 G
I”.|=(”,2+n.2)1/2 :|:(G_) *3) ] = __ (2.13)
(0 (I) 0)

Another distinguishing characteristic of the linear viscoelastic regime of
deformation is that normal stresses in shear are not of importance. Also, the type of
deformation, e.g., shear versus uniaxial extension, is not important. For tensile

deformations, a tensile modulus may be deﬁned as shown below.

E(t) = M2 (2.14)
8

For small strains, however,

E(t) = 3G(t) (2.15)

and by consequence, the viscosity in extension would be three times the zero-shear

viscosity.

27,; = jE(z)dz = [30mm = 3770 (2.16)
0 0

At strains larger than a critical strain, the relaxation modulus is no longer

independent of strain.
G(t.r) =3? (2.17)

Such strain dependence of the relaxation modulus at high strains is also schematically

shown in Figure 2.1. This behavior is known as nonlinear viscoelasticity.

In shear, nonlinear behavior is characterized by shear thinning and normal
stresses. Nonlinear effects are also evident in the time dependent startup shear viscosity,

deﬁned as shown below.

77%.?) =Ul—ZS—Q (2.18)

At the lowest shear rates, 77+ (t, 7) is independent of shear rate, and shows a monotonic

increase to the linear viscoelastic zero—shear viscosity. As the shear rate increases,
77+ (1,?) decreases at comparable times, and an overshoot appears in the transient.

Even at high shear rates, however, 77+ (t,}'/) at small times does not depart from the

A

linear viscoelastic response. Thus nonlinear effects occur only when both the strain rate

y and the strain y = 7t are large. These effects are schematically shown in Figure 2.3.

In extension, strain hardening, schematically shown in Figure 2.4, is observed at
high strains. Extensional strain hardening is the upward deviation from the Trouton’s

rule,

7220.1?) =

+ 3 (2.19)
no (t)

Such deviations typically occur above extensional strains of 1.0.

2.2. RHEOLOGY AND MOLECULAR ARCHITECTURE OF POLYOLEFINS

The advent of Dow’s INSITE o constrained geometry catalyst technology has
made it possible to synthesize linear low-density polyethylenes (LLDPE) with fairly
narrow molecular weight distributions [Sehanobish et al. (1994)]. These catalysts also
facilitate incorporation of higher oleﬁn comonomers in a highly homogeneous fashion
resulting in ethylene-a—oleﬁn copolymers with a fairly narrow comonomer distribution,
independent of the molecular weight. The higher oleﬁn comonomers are incorporated in

the form of short-chain branches (SCB) — butene (C4) comonomer is present in the form

of ethyl (2—carbon) branches, octene (C8) comonomer is present in the form of hexyl (6-

carbon) branches, and so on.

Such ethylene- a—oleﬁn random copolymers offer a broad variety of structures and
physical properties depending on the comonomer content or short-chain branch (SCB)
density, and this has been widely investigated. With increase in comonomer content,
there is progressive decrease in the density, crystallinity and melting point of the
copolymer [Bensason et al. (1996)]. More importantly, increasing comonomer content
results in fundamental changes in the crystalline microstructure of the copolymer. Four
distinct microstructures have been identiﬁed with increasing comonomer content
[Bensason et al. (1996), Minick et al. (1995)] - (Type IV) Materials with densities
greater than 0.93 g /cc (very low comonomer content) exhibit lamellar morphology with
well developed spherulites typical of high density polyethylenes (HDPE); (Type III)
Copolymers with densities between 0.93-0.91 g /cc have thinner lamellae and smaller
spherulites, typical of conventional linear low density polyethylenes (LLDPE); (Type II)
Copolymers with densities between 0.91-0.89 g/cc have a mixed morphology of small
lamellae and bundled crystals or very small spherulites; and (Type I) copolymers with
densities less than 0.89 g /cc (high comonomer contents) have no lamellae or spherulites,
and may have bundles crystals or granular nonlamellar morphology. The combined
observations from melting behavior, microstructure assessment, and tensile testing show
the transition from semicrystalline behavior typical of conventional LLDPEs at low

comonomer content to an essentially elastomeric behavior at high comonomer contents,

l8

corresponding to densities lower than 0.9 g /cc [Bensason et al. (1996), Minick et al.

(1995), Shan and White (2004)].

The versatility of structures achievable with ethylene- a—oleﬁn copolymers has led
to their broad applicability in polymer processing — both in “neat” form as well as in the
form of polymer modifiers for therrnoplastics such as polypropylenes. Therefore the

rheology of these copolymers is of great interest.

2.2.1. Molecular architecture effects on linear viscoelasticity and theoretical

framework

Molecular structure has profound consequences on the melt rheology in the linear
as well as nonlinear regimes of deformation. In this context, experimental studies have
primarily focused on the effects of polydispersity (Mw / MN), profuse long-chain
branching (LCB), and sparse LCB on the deviation of rheological behavior in comparison
with a perfectly monodisperse linear polymer. Limited investigations on the effect of
short-chain branching form a very small subset of this literature and will be discussed

later in this section.

The behavior of linear monodisperse polymers forms the base case with which the
behavior of polydisperse and branched melts is compared. The zero shear viscosity of

linear entangled melts shows a 3.4 power law dependence on the molecular weight of the

polymer (770 cc M 3'4 ). In linear polymer melts with molecular weights much higher than

the critical entanglement molecular weight (M >> M c; M c z 2M 6), the linear
equilibrium recoverable shear compliance J 3 is independent of the molecular weight,

and so is the plateau modulus GR, [Ferry (1980), Graessley (1992)].

The molecular basis to understand the viscoelastic response of entangled, linear
and nearly monodisperse polymer melts is described well by the Doi-Edwards [(1978,
1986), Graessley (1980)] constitutive theory. This theory is based on the idea that
transverse motions of the polymer chain are impeded because of the constraints posed by
the network formed by the entanglements with other chains. Doi and Edwards represent
these constraints by modeling the polymer as a random coil of molecular weight M and
mean square end-to-end distance R2, conﬁned within a tube of diameter a and contour
length L. Following deformation, stress relaxation proceeds ﬁrst by a rapid equilibration
of the chain within the distorted tube, then by a relatively slow disengagement of the
chain from the tube brought about by the “reptation” of the chain parallel to its own
contour. The rapid equilibration stage of stress relaxation is characterized by a Rouse
timescale 73 while the slower reptation stage is characterized by a tube disengagement
time 2;). These timescales are strong functions of the polymer chain entanglement density
Z, deﬁned as shown below [see Larson et al. (2003) for a discussion of the various

conventions of deﬁning the tube entanglement parameters],

 

(2.20)

20

Then,
I'd 0C 323 (221)

TR cc 22 (2.22)

This theory gives correct qualitative results with regard to the dependence of zero
shear viscosity on molecular weight (770 cc M 3 ) and the molecular weight independence

of J,0 for entangled linear systems. Deviations from such limiting behavior are observed

with broader molecular weight distribution or in presence of branching.

Polydispersity does not signiﬁcantly alter the dependence of 770 on the molecular

weight, the latter being a weight average molecular weight (770 cc M {if ). Jeo, however, is

extremely sensitive to the molecular weight distribution [Struglinski and Graessley

(1985)]. even in linear polymers. 1,.0 has been shown to steeply increase with the

presence of high molecular weight fractions [Graessley (1992)].
Branching profoundly affects the rheology of the polymers. From studies with

model star and comb branched melts [Roovers and Toporowski (1987)], it is clear that

the zero-shear viscosity shows an exponential dependence on the branch entanglement

21

density (M b, / M e) and the recoverable compliance increases in direct proportion with

M b, . The long-time relaxation processes are also governed by the branch relaxation.

In the absence of branch entanglement (M b, < M c), the Rouse theory [Pearson

and Raju (1982)] predicts that both 770 and 12 would be smaller for branched chains

than for linear chains possessing the same total molecular weight.

2.2.2. Prior experimental work in linear and nonlinear rheology of polyolef'ms

In a detailed study on shear creep-recovery on the widely characterized IUPAC
low-density polyethylenes (LDPE), Agarwal and Plazek (1977) reported that higher
polydispersity Mw / MN results in greater nonlinearity in the recovery behavior, and
higher equilibrium recoverable compliances. For the three LDPE studied, the differences
in recoverable compliances were more enhanced at lower creep stresses and shear strain
rates. Random, profuse long-chain branching (LCB, branches much longer than the
entanglement length, M b, >> M e) as observed in low-density polyethylenes (LDPE)
however makes it difﬁcult to separate the effects of LCB and polydispersity. Higher
recoverable compliance with increasing LCB density was also recorded in random long-
chain branched polybutadienes [Kasehagen and Macosko (1996)] — however, the samples

with higher LCB density also possessed broader molecular weight distribution.

22

Extensional behavior, on the other hand, has been widely studied for several types
of polyolefins — LLDPE [Munstedt et al. (1998), Bin Wadud and Baird (2000)], HDPE,
[Munstedt and Laun (1981)], LDPE [Munstedt and Laun (1981)], as well as
polypropylenes [Kurzbeck et al. (1999), Lanfray and Marin (1990), Fulchiron and Vemey
(1993)], and polybutadienes [Kasehagen and Macosko (1998)]. A recent paper by
Gabriel and Munstedt (2003) provides a good summary of several prior works with
polyethylenes and polypropylenes. Extensional strain hardening is very pronounced in
the presence of sparse or profuse LCB. Presence of high molecular weight tail or high
polydispersity also result in strain hardening. Linear polybutadienes studied by
Kasehagen and Macosko (1998), as well as HDPE studied by Munstedt and Laun (1981)

exhibited pronounced extensional strain hardening.

Kasehagen et al. (1998) also demonstrated that nonlinearity in extension is not
governed by the degree of sparse LCB alone. They observed no correlation between the
degree of sparse LCB and the degree of strain hardening. This observation is even more
signiﬁcant in the light of the fact that the polymers with higher degree of LCB also had
greater polydispersity. Kasehagen and Macosko (1998) demonstrated that nonlinearity in
extension is not governed by the degree of LCB alone. They conclude that addition of

small amounts of a high molecular weight fraction can lead to greater strain hardening.

23

2.2.3. Rheology of short-chain branched polyolefin copolymers

Characterization of polyolefin random copolymers with short-chain branches has
gained more importance with the advent of metallocene catalysts which produce
polyoleﬁn copolymers with a narrow comonomer distribution. Metallocene catalysts are
also known to incorporate sparse degrees of long—chain branching [Dekmezian et al.
(2002), Lohse et al. (2002), Beigzadeh et al. (2001), Kokko et al. (2000)]. Of course,
sparse LCB in such copolymers cannot be detected by the standard chromatography or
NMR techniques [Wood-Adams and Dealy (2000), Janzen and Colby (1999), Shroff and
Mavridis (1999)] and only indirect evidence has been obtained from rheological
techniques. Most commonly, very high ﬂow activation energies which are close to that
of LDPE melts [Malmberg et al.(2002), Bin Wadud and Baird (2000), Vega et a] ( 1998),
Villar et a1 (2001)], high zero-shear viscosities [Gabriel and Munstedt (2003), Chai
(2000), Shroff and Mavridis (1999), Janzen and Colby (1999)], and thermorheological
complexity [van Gurp and Palmen (1998), Vega et al. (1998)] in LLDPE melts with SCB
has been attributed to the presence of sparse LCB. Using these rheological tools, Walter
et al. (2000) and Villar et a1 (2001) have concluded that the possibility of incorporation of
sparse LCB increases with decreasing comonomer content, especially at comonomer
contents below 14 mo] %. 0n the other hand, Vega et al. (1998) have shown with
ethylene-hexene copolymers with exclusively short-chain branching that the ﬂow

activation energies increase with increasing SCB density.

24

Kalyon and Yu (1988) studied the effect of comonomer type or short branch
length on the melt rheology of ethylene- a—oleﬁn copolymers, by keeping the comonomer
content more or less constant. With increasing SCB length, they observed a slower decay
of normal stresses following cessation of steady shear ﬂow, and a weaker dependence of
strain damping functions on shear strain. The nonlinear rheology of copolymers with
varying SCB density is not understood partly because variations in SCB density are often
linked with changes in some other important molecular parameter such as polydispersity

or sparse LCB [Gabriel and Munstedt (1999)].

In Chapter 4, we have carefully chosen three ethylene-octene random copolymer
elastomers, with varying SCB density. The SCB densities have been chosen so as to
obtain elastomeric materials. The molecular weights are similar, and the elastomers are
all of a low polydispersity. Nonlinear shear creep-recovery and extensional strain

hardening measurements have been then related to the SCB density.

2.3. FLOW INDUCED DEFORMATION OF DROPS AND ITS RECOVERY

The effectiveness of these polyethylene-copolymer elastomers as polymer
modiﬁers is in great part governed by the microstructure of the resulting thermoplastic
oleﬁn ('TPO) blends. The evolution of such microstructure is governed by the properties
of the thermoplastic matrix, the elastomeric dispersed phase, the interfacial tension, the

deformations and the rates of deformation imposed on the blend during processing. The

25

ﬁnal blend microstructure as well as physical properties of the product are governed by
recovery following deformations. This recovery may be driven by both interfacial

tension as well as elasticity.

The blend systems of greatest relevance are obviously polymer blends in which
both components are viscoelastic. But the best available connections between the theory
and experiments exist for blends of Newtonian components [Tucker and Moldenaers
(2002), Doi and Ohta (1991), Vinckier et al. ( 1999a, 1999b)], or phenomena in which
non-Newtonian effects of the components are not strongly manifested [Palieme (1990),
Gramespacher and Meissner (1997), Okamoto et al. (1999)]. The behavior of semi-
concentrated blends is described well by droplet models [Taylor (1932, 1934), I-Iinch and
Acrivos (1980), Stone (1994), Maffettone and Minale (1998), Maffettone and Greco
(2004)]. In order to understand blend behavior, single droplet studies are an effective
method, because they offer precise control of variables, and the ﬂuid and ﬂow parameters
may be easily varied. The drop deformation and recovery behavior offers valuable
insight into blend behavior. In fact for semioconcentrated blends, in which there are not
sufﬁcient mutual drop interactions, the understanding obtained from single drop may be

directly extended.

In most processing operations, recovery is observed immediately following some
kind of deformation. While the rate of deformation is immaterial when it comes to
interfacial tension driven recovery, it would play an important role in viscoelastic

recovery, if and when it comes into play. The type and extent of deformation are of

26

course important factors coming into play during both interfacial tension driven recovery
as well as elastic recovery. Therefore, to study the relaxation of ﬂow induced
deformation, it is imperative to understand the various ﬂuid and ﬂow factors governing

drop deformation behavior.

2.3.1. Parameters

The major variables that come into play during deformation of a Newtonian drop
in a Newtonian matrix are: the equilibrium spherical size of the drop r0 , viscosity of the
drop 77d , the viscosity of the matrix I)", , the interfacial tension between the drop and the

matrix ﬂuids F , the principal rate of deformation G, and the non-dimensional strength of
ﬂow a. a = 0 for simple shear ﬂow, and a=1when the ﬂow is purely irrotational. In

addition to the parameter a , the drop deformation may then be characterized by two

non-dimensional parameters. The ﬁrst is the viscosity ratio

k = 171 (2.23)
77m

The other non-dimensional parameter is the capillary number Ca, which is the ratio of

viscous stresses to interfacial stresses,

nG
F/ro

Ca=

 

(2.24)

27

At small deformations, the drop shape may be defined by the deformation parameter

shown below [Taylor (1932)]

D = L—B
L+B

 

(2.25)

In the above equation, 2L and 23 are the major and minor axis lengths of the drop. D = 0
for a spherical drop, and asymptotically reaches 1 at large deformations. At large

deformations, it is more convenient to deﬁne a stretch ratio,

L
Dstretch = g (2.26)

Viscoelastic effects bring in additional ﬂuid constitutive parameters that may
govern the drop deformation behavior. While comparing the matrix and drop
viscoelasticity, it is common to deﬁne a ratio called elasticity ratio K. K may be deﬁned
as the ratio of some characteristic relaxation times of the drop and matrix respectively

[Mighri et al. (1997)].

K = ._ (2.27)

Alternatively, K has also been deﬁned based on the drop and matrix dynamic storage

moduli (6’) evaluated at some characteristic frequency a) [Levitt and Macosko (1996)]

28

K = (2.28)

In addition, a dimensionless number, E, is involved, which may be called an
elastocapillary number - the ratio of elastic and interfacial timescales [Greco (2002),

Maffettone and Greco (2004)].

Elastic Timescale (2,29)

 

= Interfacial Timescale

2.3.2. Experimental Studies on drop deformation and recovery

2. 3.2a. Drop Deformation

Several experimental studies and theoretical analyses have attempted to establish
correlations between D, Ca, k and a. Among the experimental studies, the most
comprehensive studies with Newtonian ﬂuids were performed by Rumscheidt and Mason
(1961), Grace (1971), and Bentley and Leal (1986). With the use of a computer
controlled four-roll mill, Bentley and Leal were able to precisely control the ﬂow strength
aall the way from simple shear to pure extensional ﬂow, in addition to investigating a

wide range of Ca, and k.

29

For low viscosity ratios (k << 1), the largest steady shapes are long and slender,
provided the deformation rates (or capillary numbers Ca)are very large. The flow
strength a is not a key parameter here, because the drop remains oriented closer to the
extensional axis of the shear flow [Rumscheidt and Mason (1969), Bentley and Leal
(1986)]. When drop viscosities are much smaller than that of the matrix, the drop
deformation was in fact observed to be greater than that of the macroscopic deformation
experienced by the matrix [Delaby et al. (1994), Yamane et a1. (1998)]. Yamane et al.

observed ratios Dd / Dm z 5/4 for k = 0.067 in instantaneous simple shear deformations.

For high viscosity ratios (k>>1), the ﬂow strength a is a key parameter
governing the deformation of the drop. For aSO.4 (high rotational component in ﬂow),

it is very difﬁcult to deform the drop, and almost impossible to fragment it in ﬂow.
Flows with stronger extensional components however are very effective in deforming the
drop at low capillary numbers Ca [Bentley and Leal (1986)]. In contrast to the low
viscosity scenario, the drop deformations were observed to be less than the macroscopic

deformations imposed on the matrix [Delaby et a1 (1995)].

For k of 0(1), the trends with abare the same as those observed for k >>1 but no

steady shapes are observed.

30

2. 3.2b. Drop Breakup Considerations

Breakup of stretched filaments may occur via two mechanisms — “end pinching”
or through capillary instabilities. For viscosity ratios less than 1.0, end pinching has been
observed at intermediate aspect ratios (length / diameter) of 10-20 [Vinckier et al. (1998),
Stone et a1. (1986)] and breakup via capillary instabilities occurs in highly slender drops
of aspect ratio greater than 50 [Vinckier et al. (1997)]. For viscosity ratios greater than
1.0, however, retraction without either of these instabilities has been observed up to

aspect ratios of 10 [Stone et al. (1986)].

Breakup via end-pinching of the stretched ﬁlament

Based on boundary element simulations of drop breakup in viscous systems,

Stone and Leal (1989), found that breakup via end pinching occurs over timescales t pm”, ,

given by the relationship

tpinch
77m R0
—— 1+k
I‘ { }

 

'4 25 —75 (2.30)

Higher magnitudes of the ratios correspond to higher viscosity ratios.

31

Breakup via Rayleigh instabilities

As pointed out by Stone et al. (1986), breakup of stretched ﬁlament through
Rayleigh instabilities occurs at progressively higher ﬁlament aspect ratios with increasing
drop / matrix viscosity ratio. When the viscosity of the drop phase is greater than 1.0,

drop breakup via such instabilities is not seen up to an aspect ratio of 15.

2.3.2a. Viscoelastic Effects in Drop Deformation and Breakup

Tretheway and Leal (1999) observed that for similar drop / matrix viscosity ratios,
the deformation of a viscoelastic drop relative to that of a Newtonian drop was lower.
They also observed higher critical capillary numbers for breakup and larger critical

deformations in the viscoelastic drop as compared to the Newtonian drops.

This is shown in the Figure 2.5 for a Newtonian and viscoelastic drop of viscosity
ratio 14.5. Thus, as the viscoelastic stresses in the drop increase, its deformation

decreases at a given capillary number.

The drop deformation is seen to be progressively lower relative to the global
deformation imposed on the matrix ﬂuid with increasing elasticity ratios K almost
independent of viscosity ratios, as observed by Mighri et a1 (1997). In fact, the latter

studies were performed with viscosity ratios k of 0(1) or less, for which experimental

32

results with Newtonian ﬂuids show the droplet deformation being greater than that of the

matrix.

2.3.3. Drop recovery

2. 3.3a. Quiescent recovery of ﬁbers and disks without prior deformation history

In the absence of any excess stresses within either the matrix or the drop phase,
the recovery of an elongated drop is purely driven by the need to reduce the interfacial
area between the two phases. The rate at which this equilibration takes place is governed
by the interfacial tension I‘ between the two phases (which drives the recovery) and the
viscous forces (which offer resistance to recovery). In the absence of ﬂow, the viscous
forces can be represented by the Stokes law, with the viscosity being the linear zero-shear
viscosity of the phases. Such quiescent recovery driven by interfacial tension is a slow
process, especially in melt systems in which the zero-shear viscosity of the phases can be
quite substantial. Two such instances are shown in Figure 2.6. Figure 2.6 (a) shows
retraction of an imbedded ﬁber [Cohen et al. (1989)], and 2.6 (b) shows retraction of an
imbedded disk [Rundqvist et al. (1996)] of a polymer melt in another immiscible polymer
matrix. It should be noted that in both instances, the initial shape of the imbedded
polymer at the commencement of recovery (t = 0) was not brought about by ﬂow. The

inclusions of that initial shape were simply “embedded” inside the matrix — the ﬁber was

33

made by extruding the polymer and the disk shape was obtained by compression
molding. Consequently, at the commencement of recovery, there were no ﬂow-induced

forces in either the matrix or dispersed phase.

Such quiescent recovery of ﬁbers and disks in the absence of any excess stresses
can be represented using a balance between interfacial tension forces and drag forces.

One such phenomenological model, shown below, was suggested by Cohen et al. (1989)

dL dA

In the above equation, R and L are the instantaneous radius and length of the drop
respectively, and A is the interfacial area. I is a hydrodynamic interaction parameter
between the drop and the matrix and is a function of the instantaneous drop shape, and

774; is an effective viscosity given by

: 77m +1.777d

”elf 2.7 (2.32)

2.3.3b. Drop Recoveryfollowingﬂaw-induced deformation

While many studies have focused on drop deformation and breakup, drop

recovery following ﬂow-induced deformation is a less widely investigated aspect. The

34

experimental studies by Bentley and Leal (1986) [also Stone (1994), Tretheway and Lea]
(2001)] establish the limits for drop breakup —- in terms of strain rates, stretch ratios and
viscosity ratios — in systems of Newtonian drops suspended in Newtonian ﬂuids during
deformation. Within these limits, if deformation is stopped, the drop will recover to its
original spherical shape, provided it does not undergo Rayleigh instabilities. Of course,
drop recovery in case of Newtonian systems is not a function of the rate of deformation.
In viscoelastic systems however, excess elastic stresses develop in the ﬂuid depending on
the rate of deformation and the strains imposed. These elastic stresses can be substantial
at deformation rates comparable to the polymer relaxation timescales, and at high strains.

Studies of recovery following such rapid hi gh-strain deformations are lacking.

One instance in which recovery was studied following high rate and high strain
deformation was recovery of polyisobutylene droplet in a polydimethylsiloxane matrix
following a step shear strain [Yamane et al. (1998)]. In this study, shear strains of the
order of 4-5 were imposed on the system within a few seconds. The Drop / matrix (zero-
shear) viscosity ratio was much less than 1 (60/ 900 = 0.067). The matrix was not only
more viscous, but was also substantially more viscoelastic (G ’d/G ’m = 6 / 2300 @ a): 10

8").

During such rapid deformation, the drop was deformed into an ellipsoid ﬂattened
in the gradient direction [Also Levitt and Macosko (1996)]. Also, by virtue of the very
low viscosity ratio, the drop deformed more than the macroscopic matrix deformation.

During recovery, two stages were obvious, as shown in the graph reproduced in Figure

35

2.7. During initial stages of recovery, for up to about 10 s, the stretch ratio is almost
constant. Thus the ﬁrst stage involved shape change without large contraction in major
axis. The second stage involved large contraction at long times. The times required for
complete recovery of deformed drops to their original spherical shape became longer as
the imposed strain increased. Reduction of droplet strain during the ﬁnal stages of
recovery could be expressed by a unique time constant given by the emulsion model of

Palierne; this time constant is independent of the initial drop stretch.

Recovery of drops following extensional deformation was studied by
Gramespacher and Meissner (1997) in semi-concentrated blends on polystyrene in
PMMA. Both PS and PMMA are highly viscoelastic, and have long relaxation times,
and show substantial creep recovery. Following extensional deformation at 0.1 s",
Hencky strains as high as 3.5 were completely recovered in the drop phase — the recovery

was shown to be driven primarily by interfacial tension and required long times.

In the above two studies, even though the individual phases were viscoelastic,
elastic effects apparently did not come into play during recovery. This could be because
of several reasons. In the Yamane et al. (1998) study, the matrix was much more viscous
and viscoelastic as compared to the drop phase. In the Gramespacher and Meissner
(1997) study, the deformations were brought about at low rates, and even the matrix and
dispersed phases were of comparable viscosity, which means there was substantial
viscous resistance to the recovery. Also in these studies, the ﬂuid parameters were not

varied to investigate the effects of viscosity and elasticity ratios on the recovery behavior.

36

A ﬁnite element study by Hooper et al. (2001) offers some valuable clues to the
factors governing drop recovery in viscoelastic systems. In this study, viscoelasticity was
implemented using the Oldroyd-B constitutive equation. From the simulations, it was
concluded that viscoelastic drops deformed in Newtonian matrices showed a rapid elastic
recovery at timescales shorter than the interfacial timescale, followed by a slower
interfacial tension driven recovery at long times. This has been shown schematically in
Figure 2.8. As seen in this ﬁgure, the viscoelastic drop with a drop Deborah number Ded
= 1.0 and with drop / matrix viscosity ratio p = 2.0 (dotted line) shows rapid initial recoil
at timescales t, = t / tr << 1.0, implying that elastic effects result in retraction at times
much shorter than the capillary timescales. This initial recovery of the viscoelastic drop
is much faster than a Newtonian drop with Ded = 0.0 (solid line) in spite of the fact that
the Newtonian drop has much lower viscosity (p = 0.5). In Chapter 5, rheological
parameters governing rapid recovery of blend microstructure are investigated. Factors
governing elastic effects during recovery of suspended elastomeric drops are studied in

Chapter 6.

37

 

 

 

 

 

 

 

 

 

75>}?

> c
74 7 71:72

y 73>}? LogG(t)
7'3
72<7c 751
ri<rc 75
time Logoime)

Figure 2.1. Typical ﬂow history imposed on the polymer melt and its response during
shear stress-relaxation

38

Creep Recovery Creep Recovery

 

T J (t)
J90

 

   

_------ ----------

 

 

 

 

time time

Figure 2.2. Flow history imposed on the polymer melt and its response during shear
creep recovery

39

Loan

 

 

Figure 2.3.

7f (t)

 

 

 

 

40

Increasing
shear rate

(a) Shear thinning and (b) Linear and nonlinear startup shear responses

Increasing extension rate

 

4

775+

3 770+

 

 

Log (time)

Figure 2.4. Typical linear and nonlinear responses of the polymer melt during
extensional ﬂows

41

Figure 2.5.

 

k = 14.5

Newtonian

 

Viscoelastic

 

Ca

Dependence of deformation on capillary number for Newtonian and
viscoelastic drops of high viscosity ratio in a Newtonian matrix

42

(a)

O

 

is
117603,

(b)

Side Top
view view

 

Figure 2.6. Quiescent recovery of drop deformation driven by interfacial tension: (a)
imbedded ﬁber retraction [Cohen et al. (1989)], (b) imbedded disk
retraction [Rundqvist et a]. (1995)]

43

1O

 

 

 

C
.2
up
('3
a:
.C
0
no.0
0
h
(I)
D.
O
h
D
1 l l lllllll l l lllllll l l llllll 4 I 111111.
1.0E-I-00 1 .OE+02 1 .0 E104
t [8]
Figure 2.7. Drop relaxation following imposition of instantaneous shear strain

[Yamane et al. (1998)]. Drop relaxation proceeds in two stages — the ﬁrst
stage involves shape relaxation to an axisymmetric spheroid, followed by
retraction in length. The long time transient may be modeled using
Carriere-Cohen (1989) type interfacial-drag force balances

.5
0|

2 1.4
‘5
a:
.c 1.3
.9.
9.’
<75 1.2
a
2
a

d
'
d

..L
D

Figure 2.8.

 

 

— Newtonian Drop

_ - - - Viscoelastic Drop

 

 

0 1 2 3 4 5
Recovery of a viscoelastic drop suspended in a Newtonian matrix

following extensional deformation. The curves are predictions obtained
by ﬁnite element solution of Navier—Stokes equations, with the
viscoelasticity of the drop phase incorporated using Oldroyd-B model
[Hooper et al. (2001)]

45

Chapter 3.

MEASUREMENT OF EXTENSIONAL VISCOSITY
OF MELTS AT HIGH STRAIN RATES

The extensional viscosity of the melts was determined at high extensional strain
rates by employing lubricated “skin-core” flow through semi-hyperboloidal dies ﬁtted in
a pressure driven rheometer (Dynisco Kayeness 6052), following an approach similar to
that used by Collier et a]. [Kim et al. (1994), Pendse and Collier (1996), Collier et a].
(1998)]. Originally proposed by Cogswell (1978), this method was ﬁrst adapted to piston

dn'ven rheometers by Everage and Ballman (1978).

3.1. METHOD

A schematic of the experimental setup is shown in Figure 3.1. The semi-

hyperboloidal dies have a proﬁle r2 (z+a)=b , designed to impose a ﬁxed extensional

(Hencky) strain, 6‘” on the melt ﬂowing through them.

46

R123
511 = UK?) (3.1)

e

In the above equation, R3 is the inlet radius of the semi-hyperboloidal die, which is the
same as that of the rheometer barrel, and Re is the exit radius of the die. For all the
extensional viscosity measurements reported in the subsequent chapters, we employed a

semi-hyperboloidal die designed to offer a Hencky strain of 4.88. The proﬁle of the die

is shown in Figure 3.2.

The semi-hyperboloidal die proﬁle is designed so as to provide a constant

extensional strain rate (8 ), over its entire length (L) for a given piston velocity (Vp).
V
.9 = Tpexp(8ﬂ ) (3.2)

The length of the die was 10 mm.

In order to eliminate shear gradients (due to the presence of die walls) in the test
material, the flow of test material melt was lubricated with a low viscosity “skin” made
with a high MFR polyethylene (LLDPE, DOWLEX 2503). In Figures 3.3 and 3.4, the
shear viscosities of some of the test “core” materials — both thermoplastics (Figure 3.3)
and elastomers (Figure 3.4) have been compared with the skin material. As seen from

these ﬁgures, the viscosities of the core are substantially higher than that of the skin.

47

This means, that with proper lubrication, most of the shear gradients and the associated
pressure drop would be conﬁned to the low viscosity skin. Consequently, almost the
entire contribution to the pressure reading measured upstream of the die entrance - using
a ﬂush mounted pressure transducer - would come from the plug ﬂow in the high-
viscosity core, representative of extension in the test material. This scenario with
optimum lubrication has been schematically shown in Figure 3.5 [also refer to Kim et al.

(1994)].

Prefabricated skin-core billets, with a core/skin volume ratio of 70/30 to ensure
adequate lubrication, were used for measurements. Skin-core billets were fabricated by
inserting test polymer core rods of optimum diameter in drilled LLDPE skin rods. The
skin—core billet is shown schematically in Figure 3.6. More details on sample preparation

are given in Section 3.1.2.

3.1.1 Test protocol

The hyperboloidal die was mounted in the Dynisco Capillary Rheometer. Then
the sample (two-layered skin-core billet) was inserted from the top end of the capillary
barrel at 110°C. The temperature was then raised to 200°C over the next 20 minutes and
the sample was allowed to melt and stabilize for an additional 10 minutes. The plunger
ram was then introduced into the capillary barrel, and the sample was extruded through

the hyperboloidal die at pre-programmed ram speeds. In lubricated flow tests, only one

48

ram speed was selected for every billet, while in unlubricated ﬂow tests, several ram
speeds could be run with a billet. During a test, the transient pressure was monitored in

addition to the quality of the extrudate.

3.1.1a. Pressure transients during lubricated ﬂow through the semi-hyperboloidal die

A typical transient plunger force curve for a single core/skin billet extruded at one
ﬂow rate through the hyperboloidal die is shown in Figure 3.7. The transient pressure
values are calculated by dividing the transient force with the cross-sectional area of the
capillary barrel (= IZRBZ, where R3 = 0.00955 m). When the core/skin billet is melted and
then pushed by the plunger ram, initially only the LLDPE melt that is situated at the
entrance of the die ﬂows out. Once this supply of LLDPE is exhausted, the PP/LLDPE
core/skin sample is pushed out of the die. Concurrently there is a sharp rise in force
(hence, pressure) to a peak or a plateau. However, the pressure transients are not
monotonic, as with time, there is a gradual or steep drop in force readings, which ﬁnally
stabilize at a second plateau, where the pressure reading ﬂuctuates about until all the core
material is exhausted. When the core material is nearly exhausted a mixture of core and
skin LLDPE ﬂows out, and when all core material is out, a precipitous drop in pressure
occurs as only some leftover LLDPE is ﬂowing out. Thus, the lubricated ﬂow takes
longer times to reach steady state (hence only one ram rate imposed per sample billet),
and also shows a non-monotonic behavior. The force readings are considered relatively

stable when the range of variation is less than 3-5% over time, and this is labeled as

49

steady state. The force readings over this relatively stable portion of the curve are

averaged for subsequent calculation of extensional viscosity.

3.1.1b. Lubricated and unlubricatedﬂow of test materials

Better insight is obtained regarding the pressure drop truly representative of
extension, when pressure drops of the test material with and without lubrication are
compared. As shown in Figure 3.8, the startup pressure transients for skin-core ﬂow
measured using a ﬂush mounted pressure transducer, located just upstream of the die
entrance, were much longer than those observed for single phase unlubricated ﬂow, and
also non-monotonic. The initial peak obtained is similar in magnitude to the pressure
recorded for unlubricated ﬂow. This means it takes a ﬁnite amount of time for the skin-
core ﬂow to establish a steady interface through the length of the semi-hyperboloidal die.
The initial peak or plateau at high pressure-drops is therefore is indicative of shear
contamination. Thus, long times are required to establish steady skin-core ﬂow through
the die. Therefore, it is the steady state at lower pressure drops which is truly
representative of extension of the test polymer. Such long and non-monotonic transients
were observed with the presence of slip at the capillary wall [Hatzikiriakos et a1. (1997)].
Shaw (1978) also observed drop in force readings when there was elimination of wall
shear stresses with the presence of a lubricant grease. The whole skin-core billet was
therefore used for a single piston velocity run, to account for these long and non-

monotonic pressure transients during testing. The transient force signals took longer

50

times to reach near steady values at low piston velocities as compared to those at higher

velocities.

When compared to the unlubricated ﬂow, the steady, “lower plateau” pressure
drops were lower in skin-core ﬂows, typically by 25 to 30%. The pressure drops in the
unlubricated case is sometimes as high as 50% more than that obtained in the lubricated
case. Therefore, contrary to the conclusions of Collier et al. (1996, 1998), lubricated
skin—core ﬂow is the only way to measure extensional viscosity in semi-hyperboloidal

dies. Unlubricated ﬂow has massive shear contamination.

3.1.1e. Correlation for extensional viscosity

As shown by Pendse and Collier (1996), at high Hencky strains, the following
expression provides a reasonable estimate of extensional viscosity for these

measurements.

AP
8 ,6" =— 3.3
775( H ) £8” ( )

AP is the pressure drop over the die length (with lubrication, AP is primarily due to

extension in the test polymer). Extensional viscosities of the materials were thus

measured at 6" ranging from 0.5 - 25 3". At any given strain rate 5', the duration of

51

imposition of extensional ﬂow-field t is equal to the residence time of the melt within the

semi-hyperboloidal die, as deﬁned below.

8 24.88
t,“ = —”T—- (3.4)

3.1.1d. Strain hardening parameter ()9

The primary aim of these measurements was to understand the strain hardening
behavior of the elastomer melts. The strain hardening parameter (1) in extension was

calculated with respect to the threefold linear shear stress function at matching times

[Kurzbeck et al. (1999)], as shown below.

. 8
175 (e.r=—’:’-)

1(8H,é)= + 8 (3.5)
327 (r)

 

3.2. SAMPLE PREPARATION

Due to the presence of walls in the test geometry, the contribution of shear to the
pressure drop across the die is inevitable. In order to achieve pure uniaxial extensional

ﬂow through the die, shear gradients must be eliminated from the test material by

52

lubricating its ﬂow with a lubricant skin composed of a very low viscosity polymer. This
can be ensured in the capillary rheometer using a two-layered billet prepared by
completely encapsulating the core polymer (the polymer to be tested) with a low-
viscosity polyethylene skin [Pendse and Collier (1996)]. The preparation of a two-
layered sample billet consisting of "core" polymer (the polymer to be tested) and a low-
viscosity LLDPE polymer "skin" (which provides lubrication) is outlined below. Ideally
the samples have to be prepared in the form of concentric cylindrical billets with the
sample polymer placed within the skin polymer of uniform wall thickness. Several

protocols were attempted for producing the two-layered (skin/core).

In one method, LLDPE rods about 9.55 mm in diameter were molded in the
capillary barrel and cored with a 1/4th-inch drill (6.37 mm), before stufﬁng with ﬂakes of
the core sample. The ﬂakes were obtained by ﬁrst compression molding neat core
polymer pellets into thin sheets and cutting them into approximately 1 mm by 1 mm bits.
Air voids due to entrapment of air between the ﬂakes could not be avoided. When the
LLDPE skin/"stuffed" core was melted and pushed through the hyperboloidal die,
LLDPE-rich melt and coated core melt ﬂowed alternatively out of the die. When coated
core melt ﬂowed out, there was a sharp increase in pressure, but this was followed up by
sharp decreases in pressure as soon as the LLDPE ﬂowed out next. This pattern resulted
in huge ﬂuctuations of pressure and as such no steady value could be obtained. Surging

and discontinuity in ﬂow out of the die were other problems that were experienced.

53

In the second method, a compression-molded sheet was rolled up and inserted
into the LLDPE cavity. Rolling did not produce very good results for the cores. The
transient pressure signals ﬂuctuated wildly in this case too, due to alternating ﬂow of

core-rich material and lubricant-rich material out of the die.

Finally, the billets were prepared from extruded rods of the two materials as laid

out below.

3.2.1. LLDPE skin samples

A high MFR LLDPE, DOWLEX-2503 (purchased from Dow, MFR = 105) was
employed as skin in the skin-core tests to measure extensional viscosity. In order to
make LLDPE skins, the LLDPE was melted and conveyed through a single screw
extruder (Brabender) at 1350C, and simply collected in hollow polycarbonate rods of
appropriate internal diameter. This simple method could be employed due to the low
melt temperatures and very low melt viscosity. Upon cooling, the LLDPE melt was
allowed to solidify and shrink within the polycarbonate molds, yielding solid rods of
diameter 9.4:t0.l cm. These rods were cut to a length of 111 i 1.5 mm and cored from
one end to a diameter of 8.1 mm and a depth of 85 x 5 mm, leaving a 25 mm length of
material uncored at the other end. Thus, a wall thickness of 0.7-0.9 mm was obtained and
this ensured a core: skin volume ratio of 70:30, which is considered ideal for stable core-

skin ﬂow.

54

3.2.2. Thermoplastic core samples

The core polymers and elastomers are much more viscous as compared to
LLDPE, and also possess a higher melting point. Therefore they could not be simply
collected into hollow rods, given the high pressure-drops and temperatures involved.
Therefore a simple ﬁxture was machined for attachment to the extruder outlet. It is
shown in Figure 3.9. A hollow aluminum rod ﬁxture with a ﬂange on one end was
attached to the extruder outlet, and tightened using a faceplate to prevent leaks. The
thermoplastic melt at 180°C was then made to ﬂow through this ﬁxture at very low ﬂow
rates in order to prevent imposition of strong shear histories. When the mold was ﬁlled
with the melt, it was cooled with water on the outside, ensuring no contact between the
melt and water. Once cooled, the solid rod was pushed out of the mold by forcing
additional melt into the mold. This way, solid PP rods 7.8 mm in diameter and a length

of 95 mm were obtained.

3.2.3. Elastomer core samples

Unlike the thermoplastic samples that are rigid solids at room temperature, the

elastomers are very ﬂexible and compressible. Therefore the arrangement described

above for thermoplastic samples could not be used for elastomers, since pushing solid

55

elastomer rod out of the mold is not feasible. Therefore an alternative arrangement was
devised. An aluminum mold was made with two faces which could be attached together,
each face containing a hemi-cylinder of the appropriate core radius, as shown in Figure
3.10. The two faces were attached, and the resulting mold was screwed into the extruder
outlet with matching threads. The mold was then heated using a propane torch, and
elastomer melt at 150°C was allowed to ﬂow into it at very low ﬂow rates to prevent any
substantial shear deformation. When the melt had ﬁlled the mold, it was unscrewed from
the extruder outlet, and allowed to cool down to room temperature. Once cooled, the
screws holding the two faces of the mold were removed and the mold was dismantled to
obtain an elastomer rod. During this process care was taken to discard samples which
showed evidence of thermal degradation, e.g., a drastic increase or decrease in torque

while being conveyed through the extruder screw, or darkening of color etc.

3.3. VALIDATION OF EXTENSIONAL FLOW MEASURENIENTS

3.3.1. Measurements with varying core I skin viscosity ratio

Using numerical simulations of core-skin ﬂow through a planar converging semi-
hyperbolic channel, Kim et al. (1994) concluded that in order to effectively reduce the
pressure drop due to shear, the core / skin viscosity ratios should be ideally greater than

30 - but any viscosity ratio greater than 10 was considered to be effective in eliminating

56

the shear gradients in the test polymer core resulting in plug ﬂow through the core [of
Figure 3.5]. This implies that with proper lubrication, the pressure drop through the
planar semi-hyperbolic (or in this study, axisymmetric semi-hyperboloidal) convergence
should be independent of the viscosity ratios. To test the effectiveness of lubrication, two
skin materials were employed. The viscosities of the skin with respect to the test polymer
core at the measured deformation rates were about 1/250 and 1/600 respectively. Figure
3.11 shows the transient pressure drop growth through the semi-hyperboloidal die for
skin-core ﬂow at the same ram rate with the two skin materials. The two curves are not
very different from each other, showing that the lubrication is effective in reducing the
shear pressure drop through the core and the measured steady state pressure drops are

indeed representative of extension in the test polymer.

3.3.2. Measurements with varying skin thickness

In core skin ﬂow through any geometry, the interface between the core and skin
attains a proﬁle which is a function of the volumetric ﬂow rates within the skin and core
and the viscosity ratios [Kouris and Tsamopoulos (2001)]. If the core / skin volume ratio
is less than 60 / 40, this would result in an unstable interface and improper lubrication
through the die [Joseph et al. (1997)]. However, in addition to ensuring proper
lubrication, we also have to ensure that the pressure drops may be unambiguously
attributed to extension in core - or in other words they are not contaminated by shear

ﬂow of skin. In order to ensure this, extensional viscosity was estimated using skin-core

57

ﬂow with two different skin thicknesses — resulting in core / skin volume ratios of 70 / 30
and 63/ 37 respectively. As seen from the startup pressure transients in Figure 3.12, for
both thickness values, stable ﬂow was attained, and the steady pressure drop was less
than that obtained with unlubricated ﬂow. But more importantly, the two steady state
pressure drops are the same. This means that the pressure drop in skin-core ﬂow is truly
representative only of the core and is not contaminated by the pressure drop due to shear
ﬂow in the skin. For all the measurements reported in the subsequent chapters, the
thinner skin thickness was employed to ensure enough core volume so as to avoid any

core-skin instabilities.

3.3.3. Comparison with RME Data

In Table 3.1, the extensional viscosities measured using a semi-hyperboloidal die
offering a Hencky strain 2 are compared with the transient extensional viscosity
measured using a Rheometrics Melt Extensiometer [e.g., Schweizer (2000)]. The values
obtained from the two methods are compared for two polymers at an extension rate of 1.0
s'1 and at a temperature of 180°C. For comparing the data, the viscosity value
corresponding to a Hencky strain of 2 was extracted from the transient viscosity growth
curves. For both melts, viscosity estimated from the two methods are similar in
magnitude, thus giving us greater conﬁdence in our data. While the methodology was
validated using a Hencky strain 2 die, all the measurements reported in this thesis have

been conducted using a, semi—hyperboloidal die offering a Hencky strain 4.88.

58

Measurements with the Hencky strain 2 die are more difficult to perform given the low
force readings due to the large exit diameter of the die. However, measurements with the
Hencky strain 4.88 die were more difﬁcult to validate with the RME data because in
these rheometers, it is very difﬁcult to obtain such high strains. On the other hand, we
could not obtain polymers for which extensiometer data existed at such high strains.
Therefore, these measurements were checked with an approximate method based on

entrance pressure drop measurements as described below.

3.3.4. Comparison with Cogswell’s Estimates

The extensional viscosities measured using lubricated ﬂow of melts through semi-
hyperboloidal die offering a Hencky strain of 4.88 were compared with estimates of
extensional viscosity obtained from entrance pressure drop data for ﬂow through
capillary dies using the analysis of Cogswell [Cogswell (1972, 1978), Padmanabhan and
Macosko ( 1997), Mackay and Astarita (1997)]. For measuring entrance pressure drops,
three capillary dies of diameter 1.0 mm and UD 10, 20 and 30 respectively were

employed. The pressure drops obtained for a given strain rate 7 for ﬂow through the

three IJD dies were extrapolated to L I D = 0 to obtain the entrance pressure drop, as

shown in the Bagley plots of Figure 3.13.

Cogswell (1972, 1978) developed the relationships laid out below by suggesting

that the entrance pressure drop for at the abrupt contraction at the entrance of the

59

capillary die from the melt reservoir has both shear and extensional contributions. The
shear viscosity was assumed to obey power law, and the extensional viscosity was
assumed constant through the length of the contraction. In the converging region the
ﬂow was assumed to be locally fully developed. In our measurements of entrance

pressure drop,AP

(’1’

this sudden contraction at the die entrance was equivalent to a

Hencky strain of 4.5.

 

 

. 2 2
Drameterof Barrel J = 1119.55 ___ 4.5 (3.6)

E H ,Capillary Die Entry = ln[ 1 02

Diameter of Capillary Die

This equivalent Hencky strain is similar in magnitude to the Hencky strain of 4.88
offered by the semi-hyperboloidal Die. Cogswell (1972) suggested the following
approximate relationships to evaluate extensional viscosity from entrance pressure drop
data obtained at various strain rates. For an abrupt contraction as in a die entrance, the
ﬂuid was assumed to deﬁne its own convergence proﬁle corresponding to a minimum
pressure drop. Based on this criterion, the extensional strain rate Upstream of the die

entrance was given to be

. 4 7w ,
8 entrance = 3 (n +1) A Pen 7 apparent

 

(3.7)

In the above equation, 1w is the wall shear stress, and n is the power law index of the

shear viscosity ( 7] = K 7 "—1 ). The extensional stress is estimated as

60

3
T“ —1'22 =-8-(n+1)APe,, (3.8)

The results of this approximate analysis and the extensional viscosities measured using
lubricated skin core ﬂow through the semi-hyperboloidal die are compared in Figure
3.14. As can be seen, the two measures of extensional viscosity overlap well with each
other. Cogswell type analysis has been used in the past by other authors to validate their
extensional viscosities measured using melt extensiometers [Laun and Schuch (1989),

Hingmann and Marczinke (1994)].

61

Table 3.1. Comparison of extensional viscosities of two polypropylenes measured at
1.0 s'1 using lubricated skin-core ﬂow through the semi-hyperboloidal die

. + . ..1 EH = 2 . .
With 775 (8 =1.0s , t = ) measured usrng Rheometrics Melt

8
Extensiometer (University of Minnesota), both at 180°C

 

 

 

 

Polymer RME MSU Die
PPA 7.0 x 104 Pa-s 7.41 x 104 Pa-s
Pro-fax SD613 1.1 x 103 pa-s 8.35 x 104 Pa-s

 

 

 

 

 

62

Piston

    
  

thick polymer core
Heated rheometer

. / barrel

Low-viscosity

skin layer \

 

 

 

Pressure SBMi- .
transducer hyperbolordal
profile die

Figure 3.1. Modiﬁed capillary rheometer setup to measure the extensional viscosity of
polymer and elastomer melts

63

2R
r— 9.55 rim ‘3

    

 

 

 

23a
0.831 mm

 

Figure 3.2. Semi-hyperboloidal proﬁle converging die to achieve constant extensional
strain rate

1 .0E+05

Ill

 

 

O PP-1
1_ . . .
+ . ' 0 . A PP-2
o
' 0 . - PP-3
I
1.05104 5 o o . . o O . o PP-4
' I
H b ‘ ‘ I A 0 *Skin
tn r A
u- l. I I . . . I ‘ A ‘ ‘O : ~ 0 .
E... i I I I A o
3‘ I : A O .
._ 1.05003 5 . ~ ‘ .
3 : I ‘ ‘0
u P I - ‘.
'2 I A.
> __ . . .
1 DE 02 '
. '1' i'
= 5km W
1.0E+o1 ' l l Itllll t I r rrrrnl n n n nnLLAL 1 J 1 trim 1 1 111111!
0.01 0.1 1 10 100 1000
Shear Rate [113]

Figure 3.3. Shear viscosities of some of the thermoplastic polymer core melt samples
compared with that of the LLDPE skin, at 200°C

65

1.0E+06

 

 

mews D n
D o
D r:

7.. D

. O 0
£195.04 038883” 39
> A A'A. g o
E IIIIIIIIIIIUIISDODOO
3199.03 ' ' ' ' I- 0'01
.. ' I D
> I

1.05102 Skin - L - -

V' V .. - :ﬁ.
1mw1 1 1 1111111 1 1 1 lLLALl 1 P111111] 1 1 1 111111 1 1 1111111
0.01 0.1 1 10 100 1ND
ShearRate[1ls]

Figure 3.4. Shear viscosities of some of the elastomeric polymer core melt samples
compared with that of the LLDPE skin, at 200°C

66

Skin Core b .— Core skin billet within
heated rheometer
Db —_- 2Rb reservoir (barrel)

 

 

 

 

 

 

 

 

 

 

 

 

\ X ’ Flush mounted
¥
’ pressure transducer
Shear gradients Core-skin interface
confined to low V V"
viscosity skin
*— Semi-hyperboloidal die
Plug flow
within core

Figure 3.5. Schematic representing skin-core ﬂow of the test material through the
semi-hyperboloidal converging die, with optimum lubrication. The skin
thickness, b, has been exaggerated in this ﬁgure for purpose of clarity.
Typically, the skin thickness is about 14% of the reservoir (barrel) radius
Rb

67

95mm I73mm

 

4 1L

‘

7"
10mm 85mm

‘7

Figure 3.6. Skin-core sample for achieving lubricated ﬂow through the semi-
hyperboloidal proﬁle converging die

68

 

 

_ 3-0 * Initial Plateau
a; or Peak
2 Final Steady
9 State
a 2-0 ’ p+ee..,..
1.. ’ O W.
:3 ‘ r1
(0 ' \
«n . c ‘ ,
9 i Skin-coreFlow ""+”“'*‘9”""""“'
a. .
1.0 _ .‘ begins
’WPure LLDPE
1' (33.9)
0.0 1 L 1 1 1 J
0 50 100 150 20C

Time [s]

Figure 3.7. Typical startup pressure transients during the measurement of extensional
viscosity using skin-core lubricated ﬂow through the semi-hyperboloidal

die

69

2.5

 

    
 
    

 

 

1500035" 1 - - 1 1 Unlubricated
g 2.0 -
"' Skin-Core
g- 1.5
h
o
I; 1_o Steady state pressure drop
:3 representative of extension
2
n. 0.5 -
0.0 1 1 J
0 50 100 150
Time [s]

Figure 3.8. Steady state pressure drops obtained at several piston velocities using
lubricated skin-core ﬂow compared with the pressure drops obtained using
unlubricated ﬂow of the test material through the same die.

70

836$ 28 05385.55 .6 2283qu He mBom

EEmm

 

7r—

..2 2:5

 

 

 

 

EE mN

 

 

3.50 .2558 pm 8:028 208 E35832

2298mm

 

\

 

 

 

 

525cm

71

338mm 28 otoEBmm—e mo 538535 c8 aBom .3.m unaut—

 

 

 

 

 

 

 

 

 

EE ms 85 mm
w 1
a: ........... E ......... E--.
r--3 ........... .3. ......... .m-..m---.l
E E rl...
.825
32> 85 985 .motnc__>o-_Emc £3 85 some

6.9: E:c_E:_m mﬁmcoﬂmu coomh-o>>._.

72

 

 

 

 

400 ~
2
o 300 -
o
b
O
u.
b
a.
5 200 -
n.
100 - +Viscosity Ratio = 627
+Viscosity Ratio = 250
o l 1 M l l J
0 0.1 0.2 0.3 0.4 0.5 0.6

Scaled Time

Figure 3.11. Startup pressure transients for lubricated skin-core ﬂow through the semi-
hyperboloidal die. Measurements with two different core / skin viscosity
ratios, carried out at a strain rate of 1 s’1 and at 200°C are shown. The test
material is a branched polypropylene, grade SD613 (Basell Polyoleﬁns
Inc.). Time has been scaled by the ratio sample-length / ram-rate.

73

0.14 skin

  
 

 

 

0.085 skin

Plunger Force [N]

S

+d/Fib = 0.14 (Thick Skin)

+ d/Rb = 0.085 (Thin Skin)

 

_1

Sealed Time

Figure 3.12. Startup pressuretransients for lubricated skin—core ﬂow through the semi-
hyperboloidal die. Measurements shown above have been carried out with
two different skin thicknesses resulting in core / skin volume ratios of
70/30 and 63/37 respectively. The measurements were carried out at a
strain rate of l s'l and at 200°C, for a linear polypropylene grade Pro-fax
6523 (Basell Polyoleﬁns Inc.). Time has been scaled by the ratio sample-
length / ram-rate. dl Rb, is the ratio of skin thickness and the capillary
barrel radius.

74

  
 

 

 

7.0E+07 -
6.0E+07 ~

g 5.0E+07 —

Q.

2 4.0E+07 .

a

d)

*5 3.0E+07 — ’ 24321/3

g - 14591/s

5 2-°E+°7 ‘ . 8761/3
1.0E+07 — x 316 i/s
0.0E+00 W l r r l l u

o 5 1o 15 20 25 30 35

Figure 3.13.

LID

Bagley plot for obtaining ends pressure drop for shear ﬂow through
straight capillaries of diameter 1.0 mm. The material tested was an
ethylene-propylene random copolymer elastomer — DOWLEX C0034 at
150°C. The entrance pressure drOps were used for estimation of
extensional viscosity using Cogswell analysis

75

 

 

 

 

 

 

1.0E+07 DUTRAL c0034, EP EIastomer,150°C
: Hencky Strains
’ I Semi-hyperboloidal die = 4.88
1,0E+06 : I Orifice (d = 1mm) ~ 4.5
_ I
_ I
'3 : "
d: 0 .
E. 1.0E+05 5
: 0
§ : 0
F
1.0E+04 E
E o Entrance Pressure Drop - Cogswell Analysis
+- I Skin-Core Flow Semi-hyperboloidal Die
1.0E+O3 1 1 1111111 1 1 111111l 1 1 111111L 1 1111111]
0.1 1 10 100 1000

Extensional Strain Rate [115]

Figure 3.14. Comparison of estimates of extensional viscosity obtained from skin-core
flow through the semi-hyperboloidal proﬁle converging die and that from
Cogswell analysis of entrance pressure data obtained from Bagley plots.
The data were obtained on an ethylene-propylene random copolymer
elastomer - DOWLEX C0034 at 150°C. The semi-hyperboloidal die
offered a Hencky strain of 4.88 while the straight capillary die of 1.0 mm
diameter effectively offers a Hencky strain of 4.5

76

Chapter 4.

MELT RHEOLOGY OF ETHYLENE-OCTENE
RANDOM COPOLYMER ELASTOMERS:
EFFECTS OF SHORT-CHAIN BRANCHING DENSITY

SUMMARY

The effects of varying short chain branch density (SCB) on the melt rheology of
random copolymers have been investigated. In particular, the strain-hardening behavior
in extensional flow and strain recovery following nonlinear shear creep have been
evaluated for three different ethylene octene copolymers with octene content ranging
from 20 to 40 wt % and having similar molecular weights. The rheological trends have
been carefully analyzed to contrast them with those obtained in the presence of sparse
long chain branching or a small high molecular weight fraction. The zero-shear viscosity
followed trends in the backbone molecular weight closely, which decreases with
increasing SCB. While the three copolymers were indistinguishable in linear viscoelastic
creep and recovery, recovery following nonlinear shear creep decreased progressively
with increasing SCB density. This reveals that the extent of rapid chain equilibration that

occurs over Rouse time scales at higher strains was progressively lower with increasing

77

SCB density. Strain hardening in uniaxial extensional flow was observed for all three
copolymers over strain rates of 1 to 7 s’I with the help of lubricated flow in semi-
hyperboloidal dies. At strain rates below the primitive chain equilibration rates, strain
hardening increases progressively with increasing SCB density. At higher rates, upon

onset of primitive chain stretch, the strain hardening behavior for the three melts merges.

4.1. INTRODUCTION

Random copolymers are of commercial importance for use as compatibilizers and
as elastomers. They can be linear or branched. Among branched copolymers, when
chain entanglements are present along both the backbone and the branches, these are
referred to as long chain branched; short branches are not entangled. Ethylene-a-oleﬁn
copolymers are random copolymers comprised of a polyethylene backbone and a higher
oleﬁn comonomer incorporated in the form of short-chain branches (SCB) - e. g., butene
(C4) comonomer is present in the form of ethyl (2-carbon) branches, octene (C8)
comonomer is present in the form of hexyl (6-carbon) branches, and so on. Increasing
comonomer content or short-chain branching (SCB) density leads to progressive
reduction in density, crystallinity and melting point of these copolymers. Elastomeric
characteristics are enhanced with increasing SCB density, and become especially
prominent with the loss of spherulitic structure at a speciﬁc gravity of 0.9 or lower

[Bensason et al. (1996), Minick et al. (1995), Shan and White (2004)].

78

The advent of Dow’s INSITE © constrained geometry catalyst technology has
made it possible to synthesize such random copolymers with a fairly narrow molecular
weight distribution [Sehanobish et al. (1994)], and also a narrow comonomer distribution.
The versatility of structures achievable with ethylene- a—oleﬁn copolymers has led to their
broad applicability in polymer processing -— both in “neat” form as well as in the form of
polymer modiﬁers for therrnoplastics such as polypropylenes. Therefore the rheology of
these copolymers is of great interest. Recent work [] has brought up evidence of rapid
recovery of the elastomer phase in injection molding of oleﬁn blends. Hence there is a
need to study the nonlinear rheology of ethylene-a-oleﬁn copolymers having high

comonomer CODICDIS.

4.1.1. OBJECTIVES

The objectives of this work were to investigate the effect of varying short chain
branch density ranging from 25 to 50 branches per 1000 carbons in random copolymers
on their nonlinear melt rheology including shear creep recovery from high strains and the
strain hardening behavior in extensional flow. Since other molecular features such as
sparse long chain branching and a small high molecular weight fraction can affect the
nonlinear rheology signiﬁcantly, the presence of these features has been checked with

linear viscoelastic tests before studying the nonlinear rheology.

79

4.2. BACKGROUND

Metallocene catalysts can be used to produce copolymers with a narrow
comonomer distribution and polydispersity indices of about 2. This process is also known
to incorporate sparse long-chain branching [Dekmezian et al. (2002), Lohse et al. (2002),
Beigzadeh et al. (2001), Kokko et al. (2000)]. Sparse LCB cannot be detected by
standard chromatography or NMR techniques [W cod-Adams and Dealy (2000), Janzen
and Colby (1999), Shroff and Mavridis (1999)] and only indirect evidence has been
obtained from rheological techniques. Most commonly, very high flow activation
energies which are close to that of LDPE melts [Malmberg et al.(2002), Bin Wadud and
Baird (2000), Vega et al (1998), Villar et a] (2001)], high zero-shear viscosities compared
to a linear polymer of similar molecular weight [Gabriel and Munstedt (2003), Chai
(2000), Shroff and Mavridis (1999), Janzen and Colby (1999)], and thermorheological
complexity [van Gurp and Palmen (1998), Vega et al. (1998)] in LLDPE melts with SCB
have been attributed to the presence of sparse LCB. Using these rheological tools, Walter
et al. (2000) and Villar et al (2001) concluded that the possibility of incorporation of
sparse LCB increases with decreasing comonomer content, especially at comonomer

contents below 14 mo] %.

The steady state recoverable shear compliance .12 in the linear regime serves as a

rheological indicator of both differences in molecular weight distribution [Perry (1980),

Graessley (1992)] and the presence of small high molecular weight fractions which are

80

not apparent in GPC curves [Gabriel et al. (1998)]. In a detailed study of shear creep-
recovery on the widely characterized IUPAC low-density polyethylenes (LDPE),
Agarwal and Plazek (1977) reported that higher polydispersity Mw I MN also leads to
greater nonlinearity in strain recovery. Random, profuse long—chain branching (LCB) as
observed in low—density polyethylenes (LDPE) however makes it difﬁcult to separate the
effects of LCB and polydispersity. Higher recoverable compliance with increasing LCB
density was also recorded in random long-chain branched polybutadienes [Kasehagen et
al. (1996)] - however, the samples with higher LCB density also possessed broader

molecular weight distribution.

The extensional viscosity of several types of polyoleﬁns has been reported —
LLDPE [Munstedt et al. (1998), Bin Wadud and Baird (2000)], HDPE, [Munstedt and
Laun (1981)], LDPE [Munstedt and Laun (1981)], polypropylenes [Kurzbeck et al.
(1999), Lanfray and Marin (1990), Fulchiron and Vemey (1993)], and polybutadienes
[Kasehagen and Macosko (1998)]. A recent paper by Gabriel and Munstedt (2003)
provides a good review of the elongational flow characteristics of polyethylene and
polypropylene. Pronounced strain hardening during extensional ﬂow has been reported
for melts with sparse or profuse LCB. Greater strain hardening has also been reported for
systems with a high molecular weight tail or higher polydispersity. Linear
polybutadienes studied by Kasehagen and Macosko (1998), as well as HDPE studied by
Munstedt and Laun (1981) exhibited pronounced extensional strain hardening.

Kasehagen and Macosko (1998) demonstrated that nonlinearity in extension is not

81

governed by the degree of LCB alone. They conclude that addition of small amounts of a

high molecular weight fraction can lead to greater strain hardening.

Kalyon and Yu (1988) studied the effect of comonomer type or short branch
length on the melt rheology of ethylene- a—oleﬁn copolymers, by keeping the comonomer
content more or less constant. With increasing SCB length, they observed a slower decay
of normal stresses following cessation of steady shear flow, and a weaker dependence of
strain damping functions on shear strain. The nonlinear rheology of copolymers with
varying SCB density is not understood partly because variations in SCB density are often
linked with changes in some other important molecular parameter such as polydispersity
or sparse LCB [Gabriel and Munstedt (1999)]. While investigating the rheological trends

with respect to SCB density, it is therefore important to choose the copolymers carefully.

In this study, we have characterized three carefully chosen metallocene based
ethylene-octene random copolymer elastomers, with varying SCB density. The SCB
densities have been chosen so as to obtain elastomeric materials. The molecular weights
are similar, and the elastomers are all of a low polydispersity. Nonlinear shear creep-
recovery and extensional strain hardening measurements have been then related to the

SCB density.

82

4.3. EXPERIMENTAL

4.3.]. Materials

Three ethylene-octene (E0) random copolymer elastomers from Dupont - Dow
were studied here; these were manufactured with INSITE© metallocene catalysts. The
molecular weights of these copolymers are nearly the same and the octene content varies
from 20% to 40%. The copolymer speciﬁcations and the physical properties are listed in
Table 4.1. It is clear that with increasing comonomer content, the density and the melting
point evaluated from differential scanning calorimetry (DSC) tests, decrease. The
molecular weights and polydispersity indices for the copolymers are listed in Table 4.2
along with the short chain branch density or SCB, which was evaluated as follows. The
available total weight average molecular weight was converted to the number of
methylenes (CH2) in the entire polymer chain. It might be noted here that there is one
methyl (CH3) at the termination point of the short-chain branch corresponding to every

methyne (CH), at the start of the short—chain branch point along the polymer backbone.

 

M
#CH2 in entire chain = W (4.1)
14.027

The number of short chain branches was then calculated using the molecular weight of

octene (112.22) as shown below

83

M W (wt% comonomer)
112.22

#SCB =

 

(4.2)

The SCB density in terms of number of SCB / 1000 Carbons in the entire polymer chain

was evaluated as shown below.

SCB Density = # SCB X 1000 ' (4.3)

# CH 2 in entire chain

 

The E0 copolymers in this study have SCB density in the range 25-50 SCB I 1000
carbons in the polymer. These SCB densities are substantially higher than the values for
copolymers studied by Kalyon and Yu (1988), around 8—12 SCB / 1000 carbons, or for

the LLDPEs studied by Gabriel and Munstedt (1999).

4.3.2. Shear Rheometry

The elastomers were characterized in several shear ﬂows on a TA Instruments
Advanced Rheometer AR-2000, using 25 mm parallel disk geometry. All results in this
study are reported at a reference temperature of 150°C. Dynamic testing was conducted
at three temperatures, 120°C, 150°C and 200°C and a frequency range of 0.01 to 100 5".
Strain sweeps were conducted in different frequency intervals to identify the highest

strain within the linear viscoelastic regime for each frequency and obtain good signals.

84

Master curves were obtained at 150°C using time-temperature superposition over the

entire frequency range.

Linear shear creep measurements y(t S tmax) were carried out at low shear
stresses for times (tmax) long enough to obtain a pure viscous response (a slope of 1.0 on
a log-log plot of creep compliance J (t) versus time (t)); the stress was then released and
the subsequent recovery 7, (t'=t —tmax) was recorded. Nonlinear shear creep was
conducted at two shear stresses — 10,000 and 30,000 Pa. At 10,000 Pa, shear strains of
the order of 25 were imposed in times of about 100 s. This led to a steady shear rate of

about 0.5 5". At 30,000 Pa, shear strains of about 75 were imposed in times of about 35

s, which led to a steady shear rate of about 2 3". After removal of stresses, recovery was

observed for about 1.5 hr.

Stress relaxation was recorded following imposition of shear strains ( y) ranging
from 0.05 — 5. For tests at high strains, a 10% overshoot in strain was observed, and the
intended strain was attained following a delay, which was never more than 0.075 8.
Stress relaxation data were corrected for non-homogeneous shear in parallel plate
geometry following Soskey and Winter (1984). Relaxation modulus G ( t, y) data at high
strains were used to verify time-strain separability. In addition, the linear shear stress

growth coefﬁcient, 775' (t) was measured for the elastomer melts at a shear rate of 0.005

s".

85

4.3.3. Extensional viscosity

The extensional viscosity of the blend components was evaluated by measuring
the pressure drop in lubricated “skin—core” ﬂow through semi-hyperboloidal dies ﬁtted in
a piston driven Dynisco Kayeness rheometer. This approach was ﬁrst used with piston

driven rheometers by Everage and Ballman (1978) and more recently by Pendse and

Collier (1996). The semi-hyperboloidal dies have a proﬁle given by r2 (z+a)=b; this is

designed to impose a ﬁxed extensional (Hencky) strain, 8” on the melt ﬂowing through

the die.

R123
8” = Ln(-IE—2- (4.4)

In the above equation, R3 is the inlet radius of the semi-hyperboloidal die, which is the

same as that of the rheometer barrel, and R e is the exit radius of the die. In this study, we

employed a semi-hyperboloidal die designed to offer a Hencky strain of 4.9. The semi-
hyperboloidal die proﬁle is designed so as to provide a constant extensional strain rate
(6" ), over its entire length (L) for a given piston velocity (Vp). The length of the die was

10 m.
V
e’ = fexpwy ) (4.5)

86

The ﬂow of test material melt was lubricated with a low viscosity “skin” of a high
melt flow polyethylene (LLDPE, DOWLEX 2503). Prefabricated skin-core billets, with
a core/skin volume ratio of 70/30 were used for measurements. The whole skin-core
billet was used for a single piston velocity run, to allow for long and non-monotonic
pressure transients during testing. As shown by Pendse and Collier (1996), at high
Hencky strains, the following expression provides a reasonable estimate of extensional

viscosity for these measurements.

. AP
77E (8H ,5) : -.—— (4.6)
88”

AP is the pressure drop over the die length (with lubrication, AP is primarily due to

extension in the test polymer). Extensional viscosities of the blend components were thus

measured at 6" ranging from 0.5 - 7 s".

4.4. RESULTS AND DISCUSSION

4.4.1. Assessment of molecular architecture using linear viscoelasticity

The primary focus of this study was to evaluate the effect of varying short-chain

branching density on the nonlinear shear creep recovery and extensional strain hardening

87

behavior of the short-chain branched ethylene-octene copolymers. As noted earlier,
extensional strain hardening and enhanced recovery have been widely noted in polymers
with profuse random long-chain branching or with broad polydispersity indices or with
sparse long-chain branching or with small amounts of high molecular weight fraction.
The latter two molecular features that are particularly relevant to metallocene catalyzed
copolymers cannot be detected with standard GPC based techniques but are evident in
linear viscoelastic tests. Therefore it is important to examine the linear viscoelastic data
carefully before attributing the trends in nonlinear rheology to variation in SCB density.
The random copolymers in this study all have polydispersity indices close to 2.0. The
remaining molecular features, sparse LCB and small amounts of a high molecular weight
fraction may be identiﬁed from a pronounced increase of dynamic viscosities at low

frequencies, high flow activation energies, enhanced linear steady state recoverable

compliances J 2 , and thermorheological complexity.

The shear creep compliance curves for E03 melt at three different creep stresses
ranging from 5 -— 120 Pa are plotted in Figure 4.1. The creep compliance curves at 5 and
25 Pa match, but there is a noticeable departure in the creep compliance curve at 120 Pa.
This indicates that linear response in creep is restricted to very low stresses for these
materials. A purely viscous response is also clear at long times from a slope of 1.0 on the

log-log plot. Furthermore, Figures 4.2 to 4.4 show that the elastic compliance curves

J E (t) = J (t) — t/ 170 and the recoverable compliance curves J r (t’) match for a creep

stress of 5 Pa conﬁrming linear behavior at this stress level for all the three copolymer

melts. Therefore linear viscoelastic parameters reported in this paper were obtained for

88

all the three elastomers from creep and recovery curves at 5 Pa. The steady shear rates
attained in the linear viscous portion of the creep stage were 1x10'4 5'1 for E01, 1.3x104

s'I for E02 and 1.4x10“t s'l for E03. The zero-shear viscosities 770 of the melts at 150°C

are listed in Table 4.3.

The variation in zero shear viscosity may be understood from calculations of the
molecular weight of the polyethylene backbone as shown below. First the number of
pendant methylene (CH2) in branches was calculated by taking into account the fact that

for every octene, two carbons are incorporated in the backbone, thus resulting in a hexyl

(six-carbon) branch.
# pendant CH2 in branches = 6 X (# SCB) (4.7)
M bb = [# CH 2 in entire chain - # pendant CH 2 in branches] X 14.027 (4.8)

The backbone entanglement density was then calculated using the molecular weight

between entanglements for polyethylene

= M bb (4.9)

2
b” 2100

Since the three copolymers have similar overall molecular weights, increasing SCB leads

to lower backbone molecular weights. In fact, the zero-shear viscosity follows the same

89

trend as the backbone molecular weight. In other words, the zero-shear viscosity appears
to be primarily governed by the backbone-backbone entanglements. This is not

surprising, given the fact that the short branches do not participate in the entanglements.

The recoverable compliance curve J , (t’) following linear creep at 5 Pa is plotted

in Figures 4.2, 4.3 and 4.4 for E01, E02 and E03 respectively. The corresponding
elastic compliances obtained from the creep curves of the melts after subtracting the
viscous portion from the creep compliance are also plotted in these ﬁgures. From these
ﬁgures, it is also obvious that steady recoverable compliances are obtained within times
comparable to the duration of creep stage. This means that creep has been carried out to
times long enough to ensure that the resulting recoverable compliances are independent

of duration of creep.

The linear recoverable compliance curves for the copolymer melts have been
compared in Figure 4.5 and are very close together at all times. The retardation times

required to attain the steady state recovery are also similar. The equilibrium recoverable

compliances 12 , listed in Table 4.3 are similar in magnitude. The similarity of 12 for

the copolymers in this study therefore rules out differences in the high molecular weight
fractions among the materials. This trend is different from reported trends with random
long chain branched polybutadienes, where increasing LCB fraction led to signiﬁcantly
larger recoverable compliance in the linear viscoelastic regime [Kasehagen et a1. (1996)].

This reﬂects the additional long time relaxation of the entangled long chain branches.

90

Dynamic shear moduli evaluated at three different temperatures — 120, 150 and
200°C were used to generate master curves plotted in Figures 4.6, 4.7 and 4.8 for the
three different materials with the help of time temperature superposition. The success of

time-temperature superposition conﬁrms the absence of sparse long-chain branching in
all three materials [cf. Vega et al. (1998)]. The shift factors aT were evaluated by shifting

over the entire frequency range rather than just the zero-shear viscosity. The dynamic
loss modulus data were used to estimate discrete relaxation spectra with 11 relaxation
times which are listed in Table 4.4. This was done with the help of the NLREG code
developed by Honerkaamp and Weese (1993). The discrete relaxation spectra provide
good ﬁts to the dynamic data for all the three melts as shown in Figures 4.6 to 4.8.
Terminal behavior may be detected at the lowest frequencies on the dynamic moduli
plots for E02 and E03. Among the dynamic viscosity curves presented in Figure 4.9, it
is possible to estimate zero-shear viscosities from the curves for E02 and E03; theses
values are close to those determined from linear shear creep and listed in Table 4.3.
Although the dynamic viscosity plot for E01 too appears to level at lower frequencies,
the zero shear viscosity could be estimated only from linear shear creep data which cover

shear rates of 0.005 s’1 that were not accessible in oscillatory shear.

Flow activation energies (Ea) were calculated from the shift factors assuming an

Arrhenius relationship.

 

(IT = E0[ 1 -—1—] (4.10)

91

Flow activation energies range from 37 kJ/mol to 40 kJ/mol for the three materials Vega
et al. (1998) have reported similar values for a series of ethylene-hexene copolymers
without LCB; they also reported substantially higher activation energies for copolymers
with sparse long-chain branching. Further conﬁrmation of the absence of sparse LCB is
obtained by evaluating an index of LCB with the approach of Janzen and Colby (1999).
The resulting values of the relative LCB fraction are: 6.4 x 10'7 LCB/ 10,000 Carbons for
E01 and 1.0 x 10'7 LCB / 10,000 Carbons for E02. For E03, an LCB fraction could not
be evaluated. These values are similar to values obtained, for example, by Malmberg et

a1. (2002), for LLDPE without sparse long chain branching.

4.4.2. Stress Relaxation and Tube-Model Parameters

Stress relaxation experiments were conducted on the copolymer melts to estimate
characteristic time scales and establish the range over which their nonlinear rheological

behavior displays time-strain separability, given by the equation below.

60.?) =G°<t>h<y> (4.11)

In the above equation, Go(t) is the time dependent linear (strain independent)

viscoelastic relaxation modulus and h ( y) is the strain dependent damping function.

92

In the framework of tube models for polymers without branch entanglements [cf.
Doi and Edwards (1978, 1986,) Graessley (1980)], there are two different mechanisms of
polymer melt stress relaxation with widely separated characteristic timescales. Following
very small deformations, the relaxation of stresses occurs over long times via a slow
disengagement of the polymer chain from the tube brought about by “reptation” of the

chain parallel to its own contour. This slow mode of relaxation is characterized by a tube

disengagement or reptation timescale rd. The decay of stress following linear

deformations may be modeled using the relaxation spectrum of the polymer (G,- ,/i,-) as

the surmnation of exponentials shown below.

00(1) =20,- exp(—:/2,-) (4.12)

Following large deformations however, a rapid equilibration (or retraction) of the

polymer chain occurs within the distorted tube, before onset of the slower reptation
process. The rapid equilibration stage is characterized by a Rouse timescale TR. Thus,
within times TR, there is a ﬁnite relaxation of stress resulting in a lower relaxation
modulus at the start of reptation. After the onset of reptation, the relaxation modulus
decays with time in the same fashion as G0 (t) , leading to time-strain separability
(equation 4.11) for times greater than the equilibration timescale TR [Takahashi et a1.

(1993)]. Hence the strain damping function h (7) is a direct measure of the fraction of

93

stress relaxed via the rapid equilibration of the primitive chain within the distorted tube at

a given strain.

The shear stress relaxation curves for the melts have been plotted in Figures 4.10
through 4.12. Linear behavior was observed for the melts at strains of about 0.05. The

linear viscoelastic relaxation modulus was also calculated from the relaxation spectra

provided in table 4.4 with equation 4.12. The predicted curves of G0 (t) are also plotted

in the respective ﬁgures along with the experimental data. The predicted curves match
well with the experimentally obtained linear viscoelastic relaxation modulus curve at low
strains for all the three copolymer melts. Other relaxation modulus curves G(t,y)
plotted in Figure 4.10 -4.12 were obtained at strains ranging from 1 to 5. The curves at
higher strains as well as the linear viscoelastic modulus curve are all parallel on the log-
log plot, establishing time-strain separability of nonlinear response of these melts for
times greater than ls. This would also indicate that the rapid equilibration time scale is

less than 1 s for each of the copolymer melts at 150°C.

The reptation time 2;; was determined from the long-time slope on a plot of
lnG(t,y) vs. I, following Takahashi et al. (1993). For each copolymer, the plots of
lnG(t,y) vs. t for different strains were parallel at long times, leading to a single

estimate for the reptation time of a given copolymer melt at 150°C. The reptation times
have been listed in Table 4.3 along with estimates of the Rouse time scales. The

reptation times also follow trends in the backbone molecular weight. Hence, an

94

approximate value for the Rouse time TR was estimated from rd and be with the

following relation [cf. Larson et al. (2003)].

 

7d
2' = 4.13
R 3be ( )

The maximum strains imposed in the stress relaxation experiments were about 4-
5; it is well known that the sample edge is distorted at higher strains in stress relaxation
experiments functions [Takahashi et al. (1993), Kasehagen and Macosko (1998)] leading
to large errors. Since much higher strains could be reached without sample distortion in
nonlinear shear creep recovery experiments, strain damping functions were estimated
from the nonlinear shear creep-recovery data with the K—BKZ integral constitutive

equation.

4.4.3. Trends in Nonlinear Shear Creep-Recovery

The transients of recoverable strains following creep at 10,000 Pa and 30,000 Pa
are compared for the three copolymer melts in Figure 4.13. During creep at 10,000 Pa,
shear strains of the order of 25 were imposed in times of about 100 s. This was
equivalent to a steady state creep shear rate of about 0.53'1. At 30,000 Pa, shear strains of
about 75 were imposed in times of about 35 s, which was equivalent to steady state shear

rate of about 25". These shear rates are substantially higher than the frequencies at which

95

terminal behavior is obtained for the copolymer melts. In contrast, during linear creep at

" were obtained, and the maximum strains

5 Pa, steady shear rates of about 1x10“4 3
imposed were less than 0.5. As seen in Figure 4.13, following nonlinear creep at both
stresses, the copolymer with the lowest SCB density (E01) shows the greatest initial

recovery at short times and also the greatest overall recovery. The recoverable strains

decrease progressively with increasing SCB density.

These trends offer a clear contrast to the recovery behavior of these melts

following linear creep. While 12 values for the three copolymers are close, clear trends

with respect to SCB density can be seen in steady state recoverable compliances
following nonlinear creep. In Figure 4.14, the steady state recoverable compliances
obtained at long times during recovery have been plotted against the maximum strain

imposed in the corresponding creep stage. The steady state recoverable compliances

obtained following nonlinear creep are lower than J 2 for all three melts. Among the

three copolymers, E01 displays the least nonlinearity in terms of variation of steady state
recoverable compliance with respect to creep stress or maximum imposed strain. E03
displays the greatest nonlinearity. Variations in SCB density therefore are signiﬁcant in
governing the recovery following large shear deformations brought about at higher strain

rates.

96

4.4.4. Extensional Viscosity and Strain Hardening

The range of extensional strain rates covered in this study bridge the gap between
the range obtained with the RME rheometer, and the range obtained with the approximate
Cogswell type die entrance pressure-drop measurements. The extensional viscosities of
the copolymer melts, obtained at several extension rates and at a ﬁxed Hencky

strain 8” = 4.9 , are compared in Figure 4.15. The error bars in this ﬁgure indicate the

variation between experimental runs at a given strain rate, and are based on triplicate
measurements at the two lowest rates and duplicate measurements at the higher rates.
The extensional viscosity decreases with increasing extension rates for all the copolymer
melts. The extensional viscosity is progressively higher with increasing density of short-

chain branches, the differences being more pronounced at lower rates.

The strain hardening parameter (1) in extension was calculated with respect to the

threefold linear shear stress function 3776 at matching times [Kurzbeck et a1. (1999)]. as

shown below.

. EH
,7 8":—
E( 6" )

371+(t)

 

Z(€H.é‘)= (4-14)

The ﬁxed Hencky strain of 4.9 was translated to the residence time within the semi-
hyperboloidal die at each strain rate as seen in the above equation. Thus, at every strain

rate, one point along the transient extensional viscosity growth curve at t = 4.9/ e was

97

obtained. Extensional viscosity data in this form, along with the linear viscoelastic curve

3773 (t) have been plotted for E01, E02 and E03 melts in Figures 4.16, 4.17 and 4.18
respectively. The linear viscoelastic startup shear viscosity 773 (t) was determined at

0.005 s‘1 for each melt. Strain hardening relative to 3775L (t) is evident at all strain rates

for all the three copolymers.

The strain hardening parameter 1 (equation 4.14) corresponding to e H = 4.9
has been plotted as a function of SCB density at several strain rates in Figure 4.19 for the
three copolymers. Pronounced strain hardening ( 1>1 ,) that declines with increasing
strain rate is clear for each copolymer. More importantly, at low extension rates, 2 is a

strong function of SCB density. Strain hardening is progressively higher with increasing

comonomer content. The differences in z for the three copolymers are less pronounced

at higher rates. These trends in nonlinear rheology with SCB density are examined with

the analysis that follows.

4.4.5. Constitutive analysis of nonlinear shear creep recovery

Nonlinear shear creep recovery data were analyzed using the K-BKZ integral
constitutive equation [Kaye (1962), Bernstein et al. (1963,) and Wagner (1978)] to
estimate the damping functions over a large range of shear strain. Indirect estimates of

damping functions reported in the literature [Fulchiron et a1. (1993), Kasehagen and

98

Macosko [1998)] have been obtained with startup shear experiments. The K-BKZ

integral equation for stress g (t) was used as shown below. .

I
go) = j non—K)h{11(z,z’),12(t,z’)}glo’) dt' (4.15)

—00

with the time dependent linear viscoelastic memory function
0 _ I _ Gi _ _ I A
[I (I I)— 21‘— exp[ (I I )l i] (4.16)
1' .

involving the relaxation spectrum of the melt (refer to Table 4.4). The Finger strain

tensor g :1 (t') in case of shear deformation (y) is deﬁned as shown below.

(4.17)

The strain damping function h(Il , 12) is a function of the first and second invariants of
the Finger strain tensor. The form of the strain damping function in shear may be chosen
from the Doi Edwards form or the sigmoidal form of Fulchiron and Vemey (1993) given
below. The sigmoidal form offers a wider range with two adjustable parameters a and m

and provided a better ﬁt to our data.

99

h(y) = —-—l—— (4.18)

1+ ayzm

Using equations 4.16, 4.17 and 4.18, the shear stress (equation 4.15) is given as shown

below

 

‘ G~ , {rm-70)} ,
02(1): {Z—Lexp[—(t—t)//t,~]} dt
1 (I) i 41‘ 1+a{r(t)-r(t')}2’"

(4.19)
7(1)

+ 00(1) 2
1+a{r(t)} ’"

 

During the creep stage, the stress is ﬁxed at a ﬁnite value

(0'12 (tStmax )=0'c,eep ) while during recovery stage it is set to 0 (0'12 (t> tmax )=0 ).

Equation 4.19 is a nonlinear Volterra integral equation of the second kind in the shear

strain; this was solved for y(t) using successive integration, with solutions for all past
times y(t') known [cf. Wagner and Laun (1978)]. In Figure 4.20 (a—c), the K-BKZ

model predictions for creep and recovery at a creep stress of 30,000 Pa have been
compared with experimental creep and creep recovery data. As seen from the ﬁgure, this
procedure provides very good ﬁts to the experimental data for all the copolymers. For
each copolymer, the same magnitudes of the ﬁtting parameters a and m gave equally
good ﬁts to creep and recovery data at a stress of 10,000 Pa. Thus this analysis gives a

reliable way of obtaining damping functions at high strains.

100

4.4.6. Damping functions

The dependence of damping function on strain is plotted for the three copolymers
in Figure 4.21; the damping function ﬁtting parameters a and m are listed in Table 4.5.
Upon inspection of this ﬁgure, it is clear that at high strains, the dependence of damping
function on strain becomes progressively weaker as the SCB density (comonomer
content) of the copolymers increases. The exponents m, are progressively lower with
increasing comonomer contents. A weaker strain-dependence in the damping function
may be interpreted as increased resistance to the equilibration of the extended primitive
chains. In other words, the polymer chains become progressively stiffer as the SCB

density increases.

The trends in nonlinear recoverable compliance for the ethylene-octene
copolymers, together with those in strain dependence of damping function are interesting
in that they offer a striking contrast to those obtained with random long-chain branched
polymers. In this study, with increasing SCB density the damping function shows a
weaker dependence on strain, a trend similar to that observed with increasing LCB
fraction. However, the recoverable compliance progressively decreases with increasing
SCB density. This ﬁrmly establishes that the nonlinear rheological trends for these
copolymers are solely due to variations in SCB density and not due to sparse LCB. It is

well understood that LCB profoundly affects the long-time relaxation phenomena of the

101

melt [Roovers and Toporowski (1987), Yurasova et al. (1994)] and it is these effects that
must govern the increase in recoverable compliances. In the case of short-chain branched

copolymers in the current study however, the long time relaxation processes are not

affected by the short-chain branches, and the absence of any trend in J 2 further rules out

differences in long time linear relaxation modes. Therefore the progressive decrease in
recoverable compliances following nonlinear creep with increasing SCB density must be
a consequence of the different degrees of relaxation brought about in the rapid

equilibrium stage of relaxation.

It remains to relate chain equilibration phenomena to the extensional ﬂow
behavior of these copolymers. The range of strain rates employed in the extensional flow

tests includes rates that are faster and slower than 1/ TR , of the copolymer melts. The rate
of equilibration (or retraction) of the primitive polymer chain within the deforming tube
is given by 7;]. When the deformation is imposed at rates higher than 71;] , this results
in the stretching of the primitive chain [Takahashi et al (1993)]. In Figure 4.22, the strain
hardening parameter [(8,] = 4.9) is plotted against a Deborah number based on the
Rouse time scale De 5 5711' The strain hardening behavior for the three copolymers
merges for De >1, while at lower Deborah numbers, the strain hardening follows SCB
density. Greater strain hardening is observed with increasing SCB at De <1 because the
melt is stretched slower than the rate of chain equilibration and the chain which relaxes

more during this time scale will strain harden to a lesser extent. On the other hand, when

the melt is stretched faster than the rate of equilibration of the primitive chain, it does not

102

retract within the tube, and this results in affine deformation of the chain irrespective of
SCB density. This explains the merging of strain hardening curves for the three

copolymers at De~ 0(1) or greater.

4.5. CONCLUSIONS

Three random ethylene-octene copolymers with degrees of short-chain branching
(SCB) ranging from 25 to 50 per 10 carbons and with similar molecular weights display a
strong dependence of non linear shear creep recovery and elongational strain hardening

on SCB. The absence of even sparse long-chain branching was conﬁrmed with several

linear viscoelastic tests. The similarity of 12 for the three copolymers rules out

differences in any high molecular weight fractions among them. Although the steady
recoverable shear compliance values are close for strains < 0.5, they decrease
progressively with increasing SCB density at higher strains of 25 to 100. This trend must
be a consequence of the different degrees of primitive chain relaxation in the rapid
equilibration stage obtained at large strains with different SCB. The extensional viscosity
was measured with the help of lubricated skin-core flow through semi-hyperboloidal dies
over a range of strain rates broad enough to include rates that are higher and lower than

1

the rate of equilibration of the primitive polymer chain. . At strain rates below 2' R , the

strain hardening increases progressively with increasing SCB density. At higher rates,
upon onset of primitive chain stretch, the strain hardening behavior for the three melts

merges.

103

Table 4.1.

Material speciﬁcations and physical properties

 

 

 

 

 

 

 

 

Copolymers E01 E02 E03
Octene content [wt. %] 20 30 40

Specific gravity 0.902 0.885 0.870
Melting point [°C] 98 83 48

 

104

 

 

 

 

 

 

 

 

 

Table 4.2. Molecular architectural parameters for the E0 copolymers
Copolymers E01 E02 E03
Octene content [wt. %] 20 30 40
Mw [g/mol] 129,900 127,300 127,900
Mw/ MN 2.17 2.01 1.95
SCB Density
[# SCB/ 1000 C] 25 37'5 50
M» [g/mol] 110,400 98,700 89,500
be 52.1 46.7 42,4

 

 

 

 

105

 

Table 4.3. Zero-shear viscosities and relaxation times (at 150°C) for the E0

 

 

 

 

 

 

 

 

copolymers
Copolymers 1301 1302 E03
[20 wt. % C8] [30 wt. % C8] [40 wt. % C8]
be 52.1 46.7 42.4
770 [Pa-s] 40,300 31,900 28,100
1,0 [pa-'1 5.71 x 10‘4 4.57 x 10“ 5.71 x 10“1
r. = 770 1,." [s] 23 14.6 16
2;, [s] 46.6 34.8 29.7
m [s] 0.3 0.25 0.23

 

 

 

 

106

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4.4. Discrete relaxation spectra of the E0 copolymer melts at 150°C.
, 01' [Pa]

’1' [8] E01 E02 E03
1.064 x 10'3 3.095 x 105 2.510 x 105 1.918 x 10y
3.342 x 10T 2.096 x 105 2.188 x 103 1.750 x 105
1.050 x 10‘2 1.167 x 10y 1.640 x 105 1.349 x 105
3.299 x 10'2 6.685 x 10F 9.353 x 104 9.387 x 10“
1.037 x 10“ 2.978 x 10‘r 3.755 x 104 4.413 x 10‘
3.257 x 10'1 1.294 x 104 1.374 x 104 1.921 x 10“
1.023 x 100 5.766 x 10?f 4.978 x 103 5.064 x 103
3.216 x 100 1.997 x 103 1.547 x 103 1.640 x 103
1.010 x 10I 7.445 x 102 5.280 x 102 4.001 x 102
3.175 x 10I 1.455 x 102 1.099 x 102 8.743 x 101
9.976 x 101 5.687 x 10' 1.824 x 101 1.525 x 101

 

107

 

Table 4.5. Damping function fitting parameters for the E0 copolymer melts

 

 

 

 

E01 E02 E03
a 0.045 0.2 0.525
m 1.125 0.7 0.5

 

 

 

 

108

 

1.0E-01

 

 

Eeoa,1so°o
,
[XSPa
-- : +120Pa + 1
w“ L +
a “ a a
v Z R
+
5 _ , 8
+ 53
1.0E-04 ; + 8
: + R
+ X
+ X
4F x
1.0E_05 1 1 1 111111 1 1 1 111111 1 1 1 111111 L 4 111111]
0.1 1 10 100 1000
Time [s]

Figure 4.1. Shear creep compliance curves for E03 at 150°C; nonlinearity is observed
at 120 Pa.

109

1 .0E-03

 

 

E01,150°C _ . 3 ED,
: “I D
H h I .
'7 E.
(B I
E 1015-04 5 . '
{N : H i
h ..
L D
5 _ 1:1 I
n I
C 1.0E-05 E
5 E
1.0E_06 1 1 1111111 1 11111111 1 1 1111111 #1111111] 1 1 111111]
0.1 1 10 100 1000 10000
Time [s]

Figure 4.2. Elastic shear compliance during creep J E (t) = J (t) — t/ 770 at 5 Pa ( I )
and recoverable compliance J, (t') ( Cl ) for E01 at 150°C.

110

1 .0E-03

 

 

Eeoz,1so°c
1- A A A
H b A A A ‘ ‘ 2 ‘ 8
'7“ ~ 1 A
A A
5 1015-04 1 A A
:‘ C 5 ‘
3 I A
g . 1
3 _ R
a A
f: 1.0E-05 15
~51" E
f
1.0E_06 1 11111111 1 1 1111111 1 1 1111111 1 1 1111111 1 L111111]
0.1 1 10 100 1000 10000
Time [s]

Figure 4.3. Elastic shear compliance during creep J E (t) = J (t) — t/ 770 at 5 Pa (A)
and recoverable compliance J, (t') ( A ) for E02 at 150°C.

lll

1 .0E-03

l

 

 

§E03,150°c .0. .6.
- ‘3
o O
H O
'1'! 0 2°. '
E 1015-04 5 o .
A : 9 .
7*- : e
"2 . . -
5 . .
" O
f: 1.0E-05 E
‘4" s
1.0E_06 1 1 1111111 1 11 111111 1 1 1111111 1 1 1 111111 1 11111111
0.1 1 10 1 00 1000 10000
Time [s]

, Figure 4.4. Elastic shear compliance during creep J E (t) = J (t) - t/ 770 at 5 Pa ( o )

and recoverable compliance J, (t') ( o ) for E03 at 150°C.

112

1 .0E-03

 

 

o
: 150 C A00 0 1:1 Bil
E a CD A A A
. Do 6‘
'7“ 1.0E-04 g a o
a. - 9 °
1—1 E O
a: . 8 °
5 1.0E-05 ;
10E_06 1 11 11 l 1 1111111 1 1 1111111 4 1 l llllll 1 1 1.11111
0.1 1 10 100 1000 10000
Time [s]
Figure 4.5. Comparison of linear shear creep recovery curves for the three EO

copolymer melts at 150°C after creep at 5 Pa

113

1.E+06 £01

 

 

E
'0
Q’1.E+02
1.E+OO 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1111111
0.001 1000

0.1 10
a1 a) [rad/s]

Figure 4.6. Dynamic shear moduli of E01. Data obtained at 120°C (:1), 150°C (I)
and 200°C (+) have been shifted to the reference temperature of 150°C

using time-temperature superposition. The solid lines are ﬁts obtained
with the discrete relaxation spectrum

114

1 .E+06

 

E02
F;1.E+04
E
is
q’1.E+02
1.E+00 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1LL111]
0.001 0.1 10 1000

a1 a) [rad/s]

Figure 4.7. Dynamic shear moduli of E02. Data obtained at 120°C (A), 150°C (A),
and 200°C (X) have been shifted to the reference temperature of 150°C

using time-temperature superposition. The solid lines are ﬁts obtained
from the discrete relaxation spectrum.

115

1.E+06 £03

 

 

F551.E+04
E
is
‘51.E+02
1.E+00 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1111111 1 1 111111L 1 1 11L111]
0.001 0.1 10 1000

aT w [rad/s]

Figure 4.8. Dynamic shear moduli of E03. Data obtained at 120°C (0), 150°C (0),
and 200°C ()K), have been shifted to the reference temperature of 150°C
using time-temperature superposition. The solid lines are ﬁts obtained
from the discrete relaxation spectrum.

116

 

 

1.0E+05 :
; 150°C
00 o
[38%;000 1:03
1.0E+04 E [11:9 630535)
1. DO A)
'1? ”$31.
a Ciao
1" 915m
a
5A
1.015+03 51A
013
A10
1.05+02 1 1 1 .1. 1 1 1
0.001 0.1 10 1000
an a) [rad/s]
Figure 4.9. Dynamic viscosity curves for the E0 copolymers at 150°C.

117

1.0E+04 1 50°C

 

 

 

G°(t)
2 _
51.09102 5
(5
3.4
_ 5.0
1.0E+00 . 4 ~14!“ . 1 eeeeee J
1 10 100
11$]

Figure 4.10. Shear stress relaxation curves for E01 at several strains and 150°C

118

 

 

 

150°C
’12
‘31.05102 \
£5
3.6
1,015+00 1 1441-“ 1 ...1111
1 10 100

t [s]

Figure 4.11. Shear stress relaxation curves for E02 at several strains and 150°C

119

 

1.05 04
+ (50 (t) 150°C
\~ 0.05
32
t: 1.0E+02
{5
2.4
3.9
1.0E+00 1W 1......1
1 10 100
t [s]

Figure 4.12. Shear stress relaxation curves for E03 at several strains and 150°C

120

1.5 -
.(a) Creep stress = 10,000 Pa

 

£01

£02
23; 503
k

 

 

1111 1 1 1111111

1 000 10000

 

 

 

 

 

 

 

1' [S]
2 ,-
- (b) Creep stress = 30,000 Pa
: E01
r' " E02
:'~ E03
:5
g
0 1 1 1111111 1 1L111111 1 4 1 111111 1 1 1 111111 1 1 111111]
0.1 1 10 100 1000 10000

t' [s]

Figure 4.13. Recovery transients y, (t’) at 150°C for the three E0 copolymers following
nonlinear shear creep at (a) 10,000 Pa up to a maximum shear strain
deformation yum“) = 25 and (b) 30,000 Pa up to a maximum shear
deformation yam“) = 75

121

1 .0E-03

 

 

E J?
” C
: 'A

3 t

s

1 I

'0 _ _.

53 1.0504 : ‘

a _

s“ i ' '
1 .501 .‘
~ 1502
- oEO3

1.05-05 ' ‘ . W

0.1 1 10 100

Maximum Strain

Figure 4.14. Steady recoverable compliance vs. the maximum strain imposed on the
sample during creep stage, at several stresses and at 150°C. The three sets
of data shown in this ﬁgure correspond to creep stresses of 5 Pa
(maximum strain ~ 0.3), 10,000 Pa (maximum strain ~25) and 30,000 Pa
(maximum strain ~75).

122

1 .7E+06

 

 

~150°c
? 1.3E+06 ~
In L.
9;. _
<- 9.0E+05 — . 502
ll - 0 E03
sf _ i i
+m 5.0E+05 — 3
§ r ‘
g I
1.0E+05 l 1 1 I'JL‘I 1 I 111111]
0.1 1 10

Extension Rate [s"]

Figure 4.15. Extensional viscosity vs. extensional strain rate for the copolymer melts
measured at a Hencky strain of 4.9 and at 150°C

123

1.0E+07

 

 

 

E .501,150°c
* 0.5 s'1
.... i 2 2 1.1
m _ - I
9:. t 6 7 5.5 I
1— . - I . I
+2
§1DE+05 :
1.0E+O4 11 1 . . . . ~44
0.1 1 10
t [8]

Figure 4.16. Elongational viscosity E01 measured at 150°C and Hencky strain 4.9,
plotted at equivalent times for several strain rates. The linear viscoelastic

startup shear viscosity 776' was measured using step shear at 0.005 s'1

124

1 .OE+O7

 

 

 

E 502,150°c
r
- 0.5 s"
w E 1.1 ‘
$1.0 +06 _' 2.2 ‘
E. 67 4.4 ‘
' A
2': A
I”
I“1.0E-I-05 .-
§ .-
1.0E+04' . . 1W u
0.1 10

1
t [s]

Figure 4.17. Elongational viscosity of E02 measured at 150°C and Hencky strain 4.9,
plotted at equivalent times for several strain rates. The linear viscoelastic

startup shear viscosity 773 was measured using step shear at 0.005 s'1

125

 

 

 

 

 

1.0E+07 E
g 503, 150°C
_ 0.5 s'1
1.1
H 2.2 O
3 1.0E-i-06 : 4.4 . Q
E E 6.7
s: * ' '
V b-
+Il.l
§1.0E+05 - 4,.
3’70+
1.0E+04 4 ‘ ' ' ‘ ‘
1 10

t [s]

Figure 4.18. Elongational viscosity of E03 measured at 150°C and Hencky strain 4.9,
plotted at equivalent times for several strain rates. The linear viscoelastic

startup shear viscosity 773 was measured using step shear at 0.005 s‘1

126

 

 

 

 

20 —
_ 150°C 503
0.5 s'1
_ £02
15 —
3 ' 501
.3: 1.1
a
10 _ 4o 2'2
L
Ff/A , 6.7
5 1 1 l l
20 30 4o 50 60
#SCB/1000C

Figure 4.19. Effect of SCB density on strain hardening parameter evaluated at 150°C
and at a” = 4.9

127

 

 

 

10° 501:30,ooo Pa y(t)
$2?
-; 1o _
k
E
k
E 7,(t')
.a 1
a
a:
0.1 1 1 1 111141 1 1 1 111111 11]
OJ 1 10 100
Time [s]

Figure 4.20a. Comparison of the KBKZ model predictions for E01 with experimental
creep and creep recovery data at 150°C with creep stress =30,000 Pa.

128

10° eozzsopoo Pa 7“)

r, (t')

Strain: y (t), y, (t')

 

1 L111

 

1 11L11l 1 1

1 10 100
Time [s]

Figure 4.20b. Comparison of the KBKZ model predictions for E02 with experimental
creep and creep recovery data at 150°C with creep stress =30,000 Pa.

129

10° 503: 30,000 Pa
~ 7(t)

 

Strain: y (t), y, (t')

 

Time [s]

Figure 4.20c. Comparison of the KBKZ model predictions for E03 with experimental
creep and creep recovery data at 150°C with creep stress =30,000 Pa.

130

ITITIIIII I IIII

I I I IIIIII

 

I IIIIIIII

 

 

00001 1 14 1 4 1 1 4 1 1 1 1 1 1 l 1 1 1 1 1

Figure 4.21. Shear strain damping functions ﬁtted to nonlinear shear creep recovery
data with the K-BKZ equation.

131

 

 

1 ' 150°C
A
I 0
~ A
a _ .A
V-
II 2 _ I I ‘ I
g _ O o l I
N - IEO1
_ AE02
oEOS
1 1 1 L 1 l 1 1 1 L l 1 1__1_ 1 _L 1 1_ 1 1 _l
0 05 1 15 2
De

Figure 4.22. Extensional strain hardening parameter I (8” = 4.9) vs. Deborah number
based on the Rouse relaxation time for the three copolymer melts at 150°C

132

Chapter 5.

FLOW LINES IN INJECTION - MOLDING OF
THERMOPLASTIC OLEFIN BLENDS:
SURFACE MICROSTRUCTURE AND NONLINEAR RHEOLOGY

This chapter has been accepted for publication in Journal of Applied Polymer Science,
with co-authors P. Papworth, K. J ayaraman, C. Shu, and M. D. Wolkowicz.
Morphology analyses of the blends, reported in this chapter for completeness [Table 5.2,

and Figures 5.15 and 5.16,] were performed by P. Papworth (2001)

SUNINIARY

This chapter reports an investigation of asynchronous flow marks on the surface
of injection-molded parts and short shots made from two different blends of
polypropylene and ethylene-propylene random copolymer elastomers. Flow marks were
observed on the surface with both blends; the spatial frequency of ﬂow marks on the
surface was greater in the blend Bl which also exhibited a greater contrast between the

surface regions. The same blend was distinctly faster in the linear viscoelastic tests of

133

shear creep recovery and shear viscosity growth. The degree of contrast between the
flow mark regions and the out of flow mark regions was traced to different degrees of
strain recovery in the dispersed phases of the two blends. This recovery occurred over
injection molding timescales of the order of a few seconds. Strain recovery after shear
creep was higher in the blend that displayed more severe flow marks in injection
molding; the corresponding elastomer by itself also showed a greater extent of creep
recovery. The quick strain recovery in the elastomer must therefore be associated with
elastic stresses rather than interfacial tension. Hence the nonlinear elastic recovery of the
elastomer phase at short times is an important factor in governing the microstructure

evolution associated with flow marks in blends of polypropylene and oleﬁnic elastomers.

5.1 INTRODUCTION

Surface defects known as tiger stripes have been observed on injection molded
parts with a variety of thermoplastic materials including neat polymers [Yokoi et al.
(1994), Heuzey et al. (1997), Monasse and Mathieu (1999),] ﬁlled polymers [Mizutani et
al. (2000), Hobbs (1996),] and polymer blends [Chang (1996), Hamada and Tsunasawa
(1996), Grillet et al. (2002), Jayaraman et al. (2002), Papworth (2001), Edwards and
Choudhary (2004)]. Tiger stripes are alternating glossy and dull bands that form
perpendicular to the flow direction. In many cases, the bands are asynchronous or out of
phase on opposite surfaces of the part, with a glossy region on one surface of the part

located across from a dull region on the opposite surface. Asynchronous ﬂow marks

134

differ from other defects in that as the injection speed is raised, they appear at shorter
flow lengths while higher melt and mold temperatures delay the onset [Mizutani et a1
(2000), Hobbs (1996), Chang (1996)]. The dull region is termed the flow mark while the
glossy region is termed “out-of-flow mark”. The contrast between these regions is an
indication of the severity of the ﬂow mark. The present chapter reports on an
investigation of the microstructural differences between these regions and the material
characteristics associated with both the onset and the severity of flow marks in high-
speed injection molding of thermoplastic oleﬁn (TPO) blends. These blends are
dispersions of a random ethylene-propylene copolymer (EP) elastomer in a
polypropylene (PP) matrix, providing toughened materials for various automotive parts

including bumpers, where a uniform glossy surface ﬁnish is required.

Visual observations reported by several researchers [Yokoi et al. (1994), Monasse
and Mathieu (1999), Hamada and Tsunasawa (1996), Grillet et al. (2002)] have
demonstrated that flow marks are caused by a flow transition at the advancing melt front
from stable symmetric fountain ﬂow to unstable, oscillating, asymmetric flow. This non-
ideal, perturbed melt ﬂow in the mold cavity is characterized by a shift of the stagnation
point from the mid plane of the channel towards one of the mold walls as shown in
Figure 5.1. This also leads to ﬂow paths of different lengths from the stagnation point to
the opposing walls with the melt experiencing different strain histories on the two flow
paths. This could have more noticeable effects on ﬁlled systems. Indeed, Mizutani et al.
(2000) and Hobbs (1996) reported that larger ﬁller particles and higher concentrations of

the ﬁller enhanced the contrast between the ﬂow mark and out-of-flow mark regions.

135

Hamada and Tsunasawa (1996) studied the morphology of injection-molded tensile bars
made with blends of polycarbonate (PC) and acrylonitrile-butadiene-styrene (ABS) and
reported a similar flow transition with the flow of these PC/ABS blends too. With the
help of chromic acid etching to remove ABS particles at the surface and scanning
electron microscopy, they reported different phase concentrations at the surface in
different regions with these blends: the glossy out-of—flow mark region was
polycarbonate rich at the surface while the flow mark region displayed both components
at the surface. Edwards and Choudhary (2004) have also examined variations in surface
morphology of injection molded plaques from a PC/ASA (acrylonitrile-styrene-acrylate)
blend and conﬁrmed the presence of a polycarbonate rich skin in glossy regions; they
attributed this in part to a lower surface energy for the PC phase. However, they

observed a different type of defect that improves with increasing injection rates.

The effect of melt rheology on this phenomenon has been studied experimentally
with neat melts and rigid particle ﬁlled melts by Mizutani et al. (2000). In studies with
single phase polypropylene melts, they reported that higher melt viscosity and higher
number average molecular weight led to more severe flow marks. In studies with ﬁlled
polymers, Mizutani et al. (2000) reported that the matrix with high polydispersity led to
delayed onset of the instability and to lower contrast between glossy and dull surfaces.
This effect of increasing matrix polydispersity or matrix viscoelasticity was conﬁrmed
even with polymer blends by Chang (1996), who ignored the characteristics of the
dispersed phase. It is worth noting here that recent numerical analysis of single-phase

flow stability near the advancing melt front by Bogaerds et al. (2004) predicts that the

136

onset of flow marks is delayed in melts that display greater strain hardening in
extensional flow. Hence there is a need to characterize the molten polymer blend and the
components in extensional flow and understand the effects of the blend rheology and the

dispersed elastomer rheology on the phenomenon of asynchronous flow marks.

The objectives of this study were to investigate with thermoplastic oleﬁn blends,
(l) the disperse phase morphology in ﬂow mark regions relative to that in out-of—ﬂow
mark regions and (2) the effects of the blend rheology and of the elastomer rheology on
the severity of ﬂow marks in two different TPO blends. It will be demonstrated in this
paper that the disparities in dispersed phase microstructure between the ﬂow-mark and
out-of-flow-mark regions are greater when the elastomer phase displays a greater extent

of recovery on the time scale of injection molding.

5.2. EXPERIMENTAL

5.2.1. Materials

Two reactor-made TPO blends labeled Bl and B2 were studied. The matrix
phase in B1 is a copolymer of polypropylene with 3 wt. % ethylene while the matrix in
B2 is a PP homopolymer. The dispersed elastomer phase in B1 is a random ethylene-

propylene copolymer with 35 wt% propylene while the dispersed elastomer phase in B2

137

 

is a random ethylene-propylene copolymer with 45 wt% propylene. The composition of
these blends and the component properties are summarized in Table 5.1. The
components of the reactor made blends were separated by solvent extraction of the
polypropylene matrix with boiling xylene as the solvent. The EP phase has a small

xylene soluble fraction that is extracted with the PP in this process.

5.2.2. Injection-molding

A standard tensile bar mold cavity (16 cm x 1.8 cm x 0.3 cm) was used with a
mold wall temperature of 24°C, a melt injection temperature of 202°C and an 8 second
ﬁll time. In addition to making full tensile bars for comparing the extent of tiger striping
with the two blends, short shots of roughly half the mold volume were also made with an
injection time of 4 3. Stroke lengths of 1.32 cm and 1.35 cm respectively were used with
blends B1 and B2 (3 stroke length of 2.3 cm corresponds to a full shot). Short shots
allow us to evaluate the two—phase morphology near the advancing melt front and
interpret the dispersed phase deformation observed in the ﬂow mark and out—of—flow-

mark surfaces.

5.2.3. Morphological characterization

Scanning electron microscopy (SEM) specimens were prepared by etching with

methyl-cyclohexane to dissolve the EP elastomer phase which leaves voids that appear

dark in the micrographs. Sections were microtomed from the two different surface

138

regions (x-y plane), two near wall regions (x-z plane) and the core region near the flow
front (x-z plane) of the short shot, as shown in Figure 5.2. The SEM images were
analyzed with the help of image analysis software packages from Sigma Scan (Jandel
Scientiﬁc, Sigma Scan Pro 3.0) and Scion Image (Scion Corporation, Scion Image

Version 3b).

5.2.4. Rheological characterization

Test specimens for rheological characterizations were made by compression
molding powder or pellets into disks with a 10 ton force over 5 minutes at 200°C. The
blends and the component materials were subjected to oscillatory shear on a Rheometrics
Mechanical Spectrometer RMS-SOO at 200°C using a 50 mm parallel plate conﬁguration.
Frequencies ranging from 0.02 - 200 rad/s were employed. A strain sweep was
conducted at several frequencies to identify the maximum strain for testing in the linear
viscoelastic range. The blends were also characterized in steady shear at shear rates
ranging from 0.1 —200 s". The linear viscoelastic transient viscosity growth 770* was
measured for these materials at a shear rate of 0.005 s'1 in a TA Instruments AR 2000

rheometer with 25 mm parallel plates.

Shear creep and recovery curves were recorded by operating the TA Instruments
AR 2000 rheometer in the controlled stress mode with 25 mm parallel plates. The zero-

shear viscosity 770 and the characteristic relaxation time 10 of the blend components at

139

200°C were obtained from linear shear creep measurements at low stresses (ranging from
5 — 50 Pa). Other nonlinear viscoelastic tests were conducted at 180°C where creep was
recorded up to a strain of 10 at shear stresses ranging from 100-3000 Pa; these curves
were shifted by time-temperature superposition to 200°C. The lower temperature was
used to minimize degradation at the long times involved. Upon attainment of the desired
strain, 7m, (at time t = t1), the stress was set to 0 and the strain recovery was recorded at
subsequent times (t ’= I — 1,.) The instantaneous strain during recovery y(t) is subtracted

from rm to obtain the recoverable strain y, (t ).

r. (t’)=rmax -7(t’) (5.1)

5.2.4a. Extensional viscosity

The melt extensional viscosity (775) was measured with lubricated skin-core ﬂow
through a semi-hyperboloidal die [Cogswell (197 8), Everage and Ballman (1978), Pendse
and Collier (1996)] ﬁtted in a Dynisco capillary rheometer. The die proﬁle is given by r2
(z + 20) = constant, as shown in Figure 5.3. The die chosen here offers a ﬁxed Hencky

strain of 4.9 based on the entrance and exit cross sectional areas as deﬁned below.
A (5.2)
8H :1“ entrance
A .

140

The inlet diameter of the die is same as that of the rheometer barrel. The die profile is

designed to provide a uniform extensional strain rate (8' ), for a given piston velocity (VP).
V
e = {em 5,, (5.3)

where L is the length of the die. The flow of the test polymer was lubricated with an
immiscible, low-viscosity LLDPE “skin” (made from Dowlex LLDPE 2503, MFR =
105). Prefabricated skin-core billets, with core/skin ratio of 70/30 by volume to ensure
proper lubrication, were used for measurements. With adequate lubrication, the pressure

drop AP over the die length can be used to evaluate a “strain averaged” extensional

viscosity 7],; as follows [Everage and Ballman (1978), Pendse and Collier (1996)].

7715 (3,611 )=AP/é5H (5-4)
Extensional viscosities of the blend components were thus measured at strain rates
ranging from 0.5 - 25 s". The pressure transient in each run was carefully monitored for
stable flow before recording the pressure drOp. A strain hardening parameter (1) in

extension was evaluated with respect to three times the linear viscoelastic startup shear

viscosity growth function as shown below [Kurzbeck et al.(l999)].

may .é)=nZ~ (mm/3216 (1:81, lé) (5.5)

141

The linear viscoelastic shear viscosity transient in the denominator was evaluated at a
time equal to the residence time of the melt in the semi-hyperboloidal die for a chosen

strain rate.

5.3. RESULTS AND DISCUSSION

5.3.1. Flow mark frequency and blend rheology

Flow marks are observed in tensile bars molded from both blends; the difference
between the ﬂow mark patterns obtained with blends B1 and B2 is illustrated in Figure
5.4. The flow marks are spaced closer together in the bars from B1; the flow-mark
spacing (distance between the centers of two neighboring ﬂow-marks) is 15 mm in the
B1 tensile bars and 20 mm in the B2 tensile bars. It is clear from the greater number of
flow-marks on bars molded from Bl that the instability is more pronounced in blend B1.
The ﬂow front must oscillate with greater frequency during injection molding of blend
Bl to produce the observed surface patterns. It is worth noting here that the melt

injection rate was not varied in this work.

Of several rheological tests conducted on the two blends -- steady shear viscosity

curves, nonlinear extensional viscosity in ﬂow through a semi-hyperboloidal die, linear

viscoelastic shear creep recovery and linear transient viscosity growth -- the latter two

142

tests showed the most difference between the two. The two blends have very similar
steady shear viscosity curves as shown in Figure 5.5. The linear shear creep recovery
curves plotted in Figure 5.6 show clearly that the recovery is faster in B1 than in B2. The
linear viscoelastic limit of extensional viscosity growth is 3 times the linear viscoelastic
shear viscosity growth obtained at very low strain rates and is plotted in Figure 5.7 for the
two blends. These curves are noticeably different for the two blends, with B1 having the
faster response. The nonlinear strain averaged extensional viscosity 77,;~+ at a total Hencky

strain of 4.9 is also plotted in Figure 5.7 as a function of the residence time t=4.9/ if for
several strain rates. Relative to the magnitude of 3 770+(t) at comparable times (t =4.9/ 8' ),

the nonlinear extensional viscosities of both blends are higher at all the strain rates. Thus
both blends exhibit extensional strain hardening. Although the extensional viscosity
curves are close for the two blends, the strain hardening relative to the linear viscosity
growth is lower in B1, especially at high strain rates, as shown in Figure 5.8. From
Figure 5.8, it is also clear that the strain hardening parameter (1, cf. equation 5.5,) shows
a steeper decline with increasing strain rate for blend B1 than for blend B2. The higher
frequency of flow marks in blend Bl goes with the lower extent of strain hardening in
this blend. Hence trends in the degree of strain hardening of the blends and the frequency

of flow marks appear to be consistent with the predictions of Bogaerds et al.(2004).

143

5.3.2. Surface morphology of injection molded short shots [Papworth (2001)]

The contrast or difference in gloss between the out-of-flow-mark and ﬂow-mark
regions is also more striking in the bars molded from Bl than in the bars from B2.
Representative SEM micrographs of these regions and of the core near the flow front in
short shots are presented in Figures 5.9-5.11 for blend B1 and in Figures 5.12-5.14 for
blend BZ. Two orthogonal views were obtained for the ﬂow mark and out-of-ﬂow mark
regions; the core region of the flow front was viewed only in the x-z plane. The surface
morphology in the x-y plane is presented in a zoom view as well in Figures 5.10 and
5.13. These ﬁgures show that for the TPO blends of this study, the disperse phase is
present in both regions, in contrast to observations of Hamada and Tsunasawa (1996)
with PC/ABS blends. The following analysis will also show that the particle
concentration is the same in these regions. The near wall micrographs in the x-z plane
(Figures 5.11 and 5.14) reveal that the surface morphology extends to a depth of 20—30
pm. It is apparent from combining the surface micrographs with the near-wall
micrographs that in the case of B1, the elastomer particles in the out-of-ﬂow mark region
are cylindrical with axes oriented along the flow direction while the particles in the ﬂow
mark region are elliptical disks with the disk thickness aligned with the z-coordinate
along the mold gap. In the case of B2, the elastomer particles are uniformly cylindrical in
the out-of-flow mark region but in the ﬂow mark region, the strands are shorter and
thicker on one end. In both blends, the particles are much more stretched and oriented in
the out-of-flow mark regions than in the ﬂow mark regions; the difference in particle

shapes between regions is very prominent in the case of B1.

144

The change of particle shape between the surface regions could be caused by a
combination of retraction, breakup and coalescence of the dispersed phase. In order to
identify the dominant mechanism among these, a quantitative comparison of particle
sizes and aspect ratios in the two types of surface regions was carried out with several
micrographs for each case. The total micrograph area analyzed was the same for both
out-of-flow-mark and flow mark regions and the results are listed in Table 5.2. The
particle size distributions in the various regions including the core are presented in
Figures 5.15 (a) for B1 and in Fig 5.15 (b) for B2. In both cases, the particle sizes are
more nearly uniform in the core region than in the surface regions; and the particle size
distributions in both types of surface regions are broad and bimodal. For Bl, the ﬁrst
peak of these distributions for smaller particles is at the same size for the two surface
regions and also matches that for the core. The area under this peak is also the smallest
for the flow mark region. These trends indicate that there was no drop breakup in the
transition between the two surface regions. For B2, the ﬁrst peak for both surface regions
is again at the same size and this size is bigger than the corresponding size for the core;
the area under the ﬁrst peak is also smallest for the ﬂow mark region. These trends

eliminate drop breakup as a cause of the change between the two surface regions.

The mean particle projected area in different regions is listed, along with mean
dimensions in Table 5.2. It is worth noting that the mean x and y dimensions for the
surface regions were based on a large number of particles but the z-dimension was not;

the number of particles in the near-wall x-z micrographs was much less than in the x-y

145

surface micrographs. The mean particle area and the characteristic z-dimension have
been put together to evaluate a representative particle volume for each region, with a
speciﬁed shape; this is also tabulated in Table 5.2. Comparison of the mean particle
volume appears to indicate some coalescence, especially in the case of B1. The total
particle volumes associated with different regions were very close, demonstrating that the

volume fraction of particles was the same in both types of surface regions.

The differences in degree of stretch between the surface regions may be examined
ﬁrst by evaluating the particle aspect ratios in the x-y plane. The aspect ratio
distributions for the dispersed phase in various regions are presented in Figure 5.16 (a)
for BI and in Fig 5.16 (b) for B2. The aspect ratio recorded in the x-y projection is a true
aspect ratio because a high degree of particle alignment is observed in both orthogonal
planes (x-y and x-z) for all particle projections near the wall. The aspect ratio
distribution is very broad for the out-of-ﬂow mark region in both cases. Compared to the
aspect ratio distribution for the core region which is narrow with a mean of about 2 in
both cases, this signiﬁes a broad range of stretch for the particles in the out-of-ﬂow mark
regions. The two blends are quite different in the aspect ratio distributions of the flow
mark regions. The aspect ratio distribution is quite narrow for the flow mark region in
the case of B1 with a mean of 2.1. Compared to the mean aspect ratio of 11.5 for
particles in the out-of—flow mark region in B1, this would indicate that the particles are
retracting to a more isotropic shape. The aspect ratio distribution in the ﬂow mark region
for B2 is broader than for B1 and has a mean of 8.1. Compared to the mean aspect ratio

of 12.5 in the out-of-flow mark region for B2, this would indicate incomplete retraction

146

as is evident from the images themselves. As noted earlier, the retraction is evident in the
strands with one of the ends becoming thicker. A further indication of the extent of
retraction may be obtained by comparing stretch ratios for the two regions. A stretch

ratio may be deﬁned as Tr/ 2R the ratio of the particle dimension along the flow (x-)

eq ’
direction to the diameter of the equivalent sphere (2Req). As seen in Table 5.2, for BI,

the stretch ratio declines from 4 for the out-of-flow mark region to 1.6 for the flow mark
region. For B2, The stretch ratio declines from 5.1 for the out-of-flow mark region to 3.4
for the flow mark region. This establishes retraction of stretched elastomer particles as a
dominant mechanism for the change of shape. Hence the contrast between flow marks
and other regions on the surface is governed by the extent of recovery from stretching of

the elastomer phase.

5.3.3. Rapid retraction and elastomer rheology

It remains to relate the rheology of components in the blends to observed
differences in the extent of recovery of the dispersed phase in the two blends. It is
important to note that the timescales available during injection molding operations for the
observed recovery are of the order of seconds. Over such times, the elastomer recovery
must be driven by elastic stresses rather than interfacial tension. Interfacial tension
driven recovery is much slower than injection rates and is usually completed over several
thousand seconds particularly under quiescent no-ﬂow conditions as reported by Cohen

and Carriere (1989a and 1989.b, 1997), Rundqvist et al. (1996), and Gramespacher and

147

Meissner (1995, 1997). The matrix viscosity and relaxation time will affect this rate as
well (Hooper et al. (2001), Tretheway and Leal (2001)]. The shear viscosity curves
presented in Figure 5.17 as well as the extensional viscosity curves presented in Figure
5.18, of the dispersed EP elastomers in B1 and B2 are much higher than the
corresponding curves of the matrix polypropylenes (PP) which have very similar
viscosity curves in both shear and extension. The characteristic relaxation times of the
EP elastomers (see Table 5.1) are also higher than those of the matrix polypropylenes so
that ﬂow induced stresses would decay faster in the matrix PP phase than in the dispersed
EP phase. Hence it is appropriate to examine the recovery of the elastomer phase in the

two blends at short times. This was done with nonlinear shear creep and recovery tests.

The nonlinear creep curves y(t) have been recorded up to a shear strain of 10 at
200°C and stresses ranging from 100 Pa to 3000 Pa; the stress is then released and the
creep recovery transients y, (t’) are recorded. These curves have been plotted together
for the highest tested shear stress 1': 3000 Pa in Figures 5.19 (a) and (b) for the dispersed
EP elastomers in B1 and B2 respectively. As shown by Munstedt and coworkers (1979,
1998), each point 7, (t’) on the recovery curve also represents the recoverable strain in the
elastomer melts when deformed to a total strain 7(t) that is less than the maximum strain
of 10 under the imposed stress. Figure 5.19 shows that the transient recoverable strain is
higher in the EP phase of B1 than in B2. The ratio of recovered strain relative to the

imposed strain at times of 1 to 2 s is relevant because these are the timescales of interest

in rapid processing operations such as injection molding. At times of the order of 1-2 s,

148

the recoverable strain for the elastomer in B1 closely follows the total strain, while the

recoverable strain of the elastomer in B2 is signiﬁcantly lower than the imposed strain.

The transient recoverable strain at l s from the nonlinear creep recovery
experiments conducted at 200°C is plotted against the creep stress (2') for the elastomers
in B1 and B2, in Figure 5.20. In this ﬁgure, the recoverable strains at t’ = ls during
recovery have been scaled by the linear equilibrium recoverable strain (1,01)
corresponding to that stress. This ﬁgure shows that the short time recovery of the
elastomer in B1 is consistently higher in the entire range of stress. The trends in shear
creep recovery are consistent with the extents of recovery observed in the dispersed EP
phases in flow mark regions of the injection molded articles. Similar trends are to be
expected in recovery following extensional flow at the advancing flow front.
Experiments for investigating rapid recovery of suspended elastomer drops following

controlled extensional ﬂow are underway in our laboratory.

5.4. CONCLUSIONS

Asynchronous flow marks on the surface of tensile bars injection molded from
different reactor-made blends of polypropylene and ethylene—propylene copolymer
elastomers in the same proportions have been found to appear with a greater frequency
for the blend Bl which also exhibited a greater contrast between the surface regions. The

linear shear creep recovery and the linear viscoelastic shear viscosity growth of Bl were

149

faster than for B2. The major difference between surface regions was found to be in the
degree of stretching and orientation of the dispersed elastomer phase. The dispersed
phase was highly stretched to cylindrical strands with a mean aspect ratio of about 12 in
the glossy surface regions of both blends while it was retracted in the dull regions to
different extents in the two cases. The retracted shape in the dull surface region was that
of a disk for blend B1 and the contrast between the regions was most striking. In the
other blend B2, the partially retracted shape was that of shorter strands with a mean
aspect ratio of 8 which were thicker at one end, and the contrast between the regions was
less severe. A comparison of the particle size distributions in different regions
established that rapid retraction of the suspended elastomer phase was the dominant cause
of changes in particle shape between surface regions. The two matrix polymers in B1
and B2 had very similar shear viscosity curves as well as extensional viscosity curves but
the elastomers in the two blends showed large differences in these curves. Testing of the
two elastomer components in nonlinear shear creep and creep recovery showed that at a
time of 1 s, the fractional strain recovery of the elastomer in B1 was much higher than
that of the elastomer in B2. Hence the nonlinear elastic recovery of the elastomer phase
at short times is an important factor in ﬂow mark formation with blends of polypropylene

and oleﬁnic elastomers.

150

Table 5.1

Composition of the reactor made blends and component properties at

 

 

 

 

 

 

200°C
Blend B 1 B2
Matrix speciﬁcations
. . Polypropylene copolymer Polypropylene
Composrtron with 3wt. % ethylene homopolymer
MFR [dg/min] 31 25
770 [Pa—s] 920 1010
40 = 770 Jeo [S] 48 7

 

Dispersed phase speciﬁcations

 

Ethylene-propylene

Ethylene-propylene

 

 

 

 

 

 

 

Composrtron random copolymer (EP) random copolymer (EP)
MFR 6.4 8.0
Ethylene / propylene wt. ratio 65 / 35 55/ 45
170 [Pa-s] 69400 15750
1,0 [pa-l] 0.39 x 10‘2 1.15 x 104.
10 [s] 273 18 l
Wt. % EP in blend 32 35

 

 

 

151

 

Table 5.2. Results of morphology analysis on injection molded TPO short-shots (see
Fig. 2 for coordinate labels)

 

 

 

 

 

 

 

 

 

 

 

 

 

Property Blend B1 Blend B2
Core Out-of- Flow- Core Out-of- Flow-
flow-mark mark ﬂow-mark mark
Tm” ““E’angzgaph area 8245 8229 8280 8244 8229 8280
# “Pamc'es ‘“ 1373 493 228 2068 433 350
rmcrograph
M?“ ”016““; 1.25 2.8 4.2 0.9 3.8 3.0
particle area [p.m ]
Mean x -dim [pm] 1.4 5.6 2.9 1.4 7.2 5.4
Mean y —dim um] * 0.6 1.5 * 0.6 0.7
Charac‘ensm 2 ‘mm 0.8 0.6 0.7 0.5 0.6 0.9
[um]
Mean aspect ratio 1.8 11.5 2.1 2.6 12.5 8.2
Mean part1cl3e volume * 1.3 2.9 * 1.7 20
[um 1
Equivalent sphere * *
radius Req. [ll-ml 0.7 0.9 0.7 0.8
Stretch ratio=x/(2Regj * 4 1.6 * 5.1 3.4

 

 

 

 

 

 

 

* Only x-z plane micrographs were available for the core region.

152

 

 

Stagnation point

  
  
 

z, gap direction

Centerline

x, flow direction

 

Figure 5.1. An unstable, advancing melt front during injection molding with flow
marks, showing asymmetric path lines and an offset stagnation point.

153

z. Gap Direction

y, Transverse Direction

x. Flow Direction

1 2

 

Figure 5.2. Various regions marked on a short shot for SEM image analysis
1. Out-of-flow-mark (glossy) surface, x-y plane

2. Flow-mark (dull) surface, x-y plane

3. Near-wall cross section beneath ﬂow-mark region, x-z plane

4. Near-wall cross section beneath out-of-ﬂow-mark region, x-z plane
5. Cross section of core upstream of ﬂow front, x-z plane

154

 

 

 

7

 

 

 

 

 

‘‘‘‘ r

 

 

 

Figure 5.3. Proﬁle of the semi-hyperboloidal converging die, designed to provide a
uniform extensional strain rate and a Hencky strain of 4.9

155

 

Figure 5.4. Comparison of tiger stripes or flow marks on tensile bars of two different
blends. (a) Blend B1: high surface contrast between glossy (out-of-ﬂow
mark) and dull (flow mark) regions; (b) Blend B2: lower surface contrast.
Gray portions represent dull regions.

156

1 .E+04

 

 

E :1 Blend 31

: A Blend 82

r @lélg
Fa ~ Egmmom
"3 “a
m “In
11.1.1303 : “on
1_1 : B
it “a

“a
Q
1-E+02 1 1 1111111 1_ 1_1 1_11111 1 1 1 1 11111 1 1 1 1_1111J
0.1 1 10 100 1000
Shear Rate [s"]

Figure 5.5. Shear viscosity curves of the reactor-made blends B1 and B2 measured at

200°C

157

 

 

 

0.06 :
0.05 E
: - - B1
0.04 : —32
-~ i
'11: 0.03 i
a
C"
:3 0.02 -
.5“
0.01 ~
0 - 1 11111111 J 1111.111
0.001 0.01 0.1 1 10 100 1000 10000

Time [s]

Figure 5.6. Linear shear creep recovery curves for the two blends at 200°C

158

5.0 : +

    
    

 

775 .
r I
4.8 : A 044
: . 2.2 -
79‘ 4-5 : 1: 5.5
to :5
fl. 4.4 — 1‘
*1. :22 s'1
5 4.2 3
O) .
3 .
4.0 : Blend Bl
3.8 i

 

Blend 82

 

3.6 1 1 l 1 1 1 1 1 l 1 1 1 l
-0.7 -O.5 -0.3 -O.1 0.1 0.3 0.5 0.7
Log (time, s)

Figure 5.7. Three times the linear shear viscosity growth as a function of time at low
strain rate (0.005 s") for the two blends; and nonlinear strain averaged
extensional viscosity of blends B1 ( I ) and B2 ( A ) at Hencky strain

8” = 4.9 and 200°C, plotted as a function of the residence time in the semi-
hyperboloidal die.

159

[(611 = 4.9, 8')

Figure 5.8.

 

 

A
I A
7 _
A A Blend 32
I
5 — I I Blend B1
3 _
1 4 g 11111 l 1
1 10 100
e [5”]

Extensional strain hardening parameter 1 for the TPO'blends evaluated at
a Hencky strain 8" = 4.9 and 200°C

160

 

Figure 5.9. x-z plane section near the ﬂow-front in the core region of blend Bl

161

y — transverse direction

x — ﬂow direction

 

(b) Detail

 

30 p.111

 

(c) Flow-mark ((1) Detail

Figure 5.10. x-y plane sections from out—of-ﬂow mark and ﬂow mark surfaces in blend
B1

162

 

 

z — gap direction

x — flow direction

 

 

 

(b) Flow-mark region

Figure 5.11. Near-wall sections in the x-z plane beneath the out-of-flow mark and ﬂow
mark surfaces of blend B1

163

 

30 um

 

Figure 5.12. x-z plane section near the ﬂow-front in the core region of blend BZ

 

 

y — transverse direction

x — flow direction

   

(b) Detail

 

(c) Flow-mark ((1) Detail

Figure 5.13. x-y plane sections from out-of—ﬂow mark and ﬂow mark surfaces in blend
B2

165

Z — gap direction

x — flow direction

 

(b) Flow-mark region

Figure 5.14. Near-wall sections in the x-z plane beneath the out-of-ﬂow-mark and
ﬂow-mark surfaces of blend B2

166

 

 

 

 

 

  

 

 

0.18 - ,
0.16 - .' ‘, - - Core Flow Front
0 14 _ ' 1 - 'Out of Flow Mark Surface
1: ' .' ', —F|ow Mark Surface
,2 0.12 -. .
0 _| 0
g 0.1 , ‘
“- 0.08 -,' x
g 0.06 -; ‘
0.04
0.02
0 ' )1!
0 5 1O 15 20 25 30
Projected Particle Area (umz)
(a) Blend B1
0.1
0.09 - - Core Flow Front
0.08 — Out of Flow Mark Surface
8 0.07 —Flow Mark Surface
8 0.06
E 0.05
0.04
1.. / " \
< 0.03 / \
0.02 \ \
0.01 . \ — " — §
0 ~ A L l a ~ g
0 10 20 3O 4O 50
Projected Particle Area (tuna)
(b) Blend B2

Figure 5.15. Projected area distributions for the dispersed elastomer particles in various
regions (a) for BI and (b) for B2 [Papworth (2001)]

167

 

 

 

0.09 r
- - Core Flow Front

0'08 P :. —F|ow Mark

0.07 P IN. — Out of Flow Mark
5 0.06 r . ;
N 0.04 r : ,
£0.03 - : '.

0.02 - - ' A

I / _\ J A
w- ¥_\_/ \
0 / ‘ 1 1 1 1 1‘ ‘
0 5 10 15 20 25 30
Aspect Ratio
(3) Blend B1
0.12 -
0.1 _ . — Out Of Flow Mark Surface
1: 1' '_ —F|ow Mark Surface
0 . " ' I I
“.3 0 08 - . - - Core Flow Front
E 0.06 - '.
a . ‘
2 .
< 1
1

 

 

 

Aspect Ratio

(b) Blend B2

Figure 5.16. Aspect ratio distributions for the dispersed elastomer particles in various
regions (a) for BI and (b) for B2 [Papworth (2001)]

168

 

 

 

 

1.0E+05 ; 31: EP
III-..III
I
32 EP "-.._
AA“ I
_1.0E+04 g ““4““ III.
? " ““A‘ III
E - A“‘ II
:‘ _ ‘A .I
,1 32: PP 1‘
1.0E+03 E Aééééééééééééééééééé A .
E B1:PP 93 96
age
1.0E+02 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1111111
0.01 0.1 1 10 100 1000

a) [rad/s]

7.

Figure 5.17. Dynamic shear viscosities of the PP matrices and EP dispersed phases of
the blends at 200°C.

169

 

 

1.0E+07 E
E [:1 B1:PP A 32: PP
_ C
$105106 5 -I-B1:EP +32: EP
6.: :
q- .
II t
I
f°v10£+05 5
“I 1-
§ 1. B
r 9 23 a
83
1.0E+04 1 1 1411111 1 4 1111111 ¥;1_1114_1J
0.1 1 1 10 100
818']

 

Figure 5.18. Strain averaged extensional viscosity of the matrix PP (open symbols) and
dispersed EP elastomer phases (ﬁlled symbols) in blends B1 and B2,

evaluated at Hencky strain 8;; = 4.9 at 200°C

170

 

10

0.1

Strain: y (t), y, (t')

0.01

0.01 0.1

10

Strain: y (t), y, (t')

0.01

E(a) B1: EP

r
)—

    

 

 

0.1 _

E(b) 32: EP
T

F

  

 

0.01 0.1 1 10 100

Time [s]

Figure 5.19. Total strain and recoverable strain during shear creep at 3000 Pa followed
by constrained recovery for the EP elastomer phases of the blends at

200°C.

171

 

0.1

IIIYIY]

 

1;: ‘ B1: EP

6 ..

31

\t. 001 — 82: EP

7a“ r— *—

Jl ' \
E -

 

0001 1 1 111111. 1 1 111111]

1 00 1 000 1 0000
Creep Stress, 2' [Pa]

 

Figure 5.20. Recovered strain y,(t’= l 3) relative to the linear equilibrium recoverable
strain corresponding to the creep stress during constrained recovery
following shear creep at several stresses for the EP elastomer phases of the
two blends, and at 200°C

172

 

Chapter 6.

RECOVERY OF ELASTOMERIC DROPS IN SUSPENSION
FOLLOWING HIGH-RATE EXTENSIONAL FLOWS

SUMNIARY

A new experimental apparatus has been assembled to study strain recovery after
elongation of a model elastomer suspended in a less viscoelastic medium at room
temperature. The rheological parameters and the flow conditions, which would result in
a greater role of elastic forces in driving the dispersed phase recovery, have been
investigated in this model ﬂow setup designed for single drops. Small drops of well-
characterized elastomers suspended in a matrix ﬂuid of lower viscosity and elasticity, are
stretched rapidly without wall effects in this setup; the stretch ratio of the drops is then
recorded over time. Preliminary results conﬁrm signiﬁcant recovery over a few seconds
The times over which such initial recovery is observed are signiﬁcantly shorter than the
interfacial timescales for the ﬂuid system investigated. Exploration of the full range of

parameters with this setup is left for future work.

173

 

6.1. INTRODUCTION

Recovery of stretched drops can be brought about by interfacial tension between
the drops and the matrix as well as elastic stresses within the drop. Interfacial tension is
the only restoring force when both the drop as well as the matrix are Newtonian [see, for
example, Tucker and Moldenaers (2002), Doi and Ohta (1991), Vinckier et al. (1997a,
1997b, 1999a, 1999b), ]. On the other hand, strain recovery in many viscoelastic systems
can also be modeled as solely driven by interfacial tension [see, for example, Palieme
(1990), Gramespacher and Meissner (1997), Okamoto et al. (1999)]. The apparent
insigniﬁcance of elastic effects during drop recovery in these viscoelastic systems is the
result of several factors — summed up into the four scenarios described below - which

may prevent the manifestation of nonlinear elastic effects

1. Recovery following very small deformations [Palieme (1990), Gramespacher and

Meissner (1992), Graebling et al. (1993)]

2. Recovery in the absence of prior ﬂow history — quiescent recovery of “imbedded”
ﬁbers and disks [Carriere et al. (1989), Cohen and Carriere. (1989a and 1989b),

Rundqvist et al. (1996), Carriere and Silvis (1997)]

3. Recovery following very slow deformations [Gramespacher and Meissner (1997 )]

174

 

4. Recovery of a drop in a matrix of comparable viscosity and elasticity [Gramespacher
and Meissner (1997)] or in a matrix of substantially higher viscosity and elasticity

[Yamane et al. (1998), Okamoto et al. (1999)]

In all the above four scenarios, elastic stresses within the drop would either not be
substantial (due to low strains and low strain rates) or would not be very effective in
driving the recovery (due to high matrix viscosity or strong matrix elastic effects which

keep the drop deformed [Hooper et a1. (2001), Tretheway and Leal (2001)].

In contrast, as seen in Chapter 5, elastic effects within the dispersed phase, as
assessed by nonlinear shear creep recovery, were key in driving the recovery of the blend.
The rheology of the matrix and dispersed phases of these blends is summarized in Figure
6.1, and the key ﬂuid and ﬂow parameters are listed in Table 6.1. It is clear that the
dispersed phase is more viscous and possesses greater elasticity than the matrix. Also,
the blends are subjected to signiﬁcant strain rates and high deformations during
processing. This scenario presents the need to understand in a systematic fashion the
various factors that enhance or hinder the role of elastic stresses in driving the recovery.
In other words, the problem being addressed in this chapter, and in greater body of work
in progress is - what are the conditions which are conducive to the manifestation of
elastic effects during recovery. The obvious spectrum of parameters would include the
elastomer and matrix rheology, the rate of deformation, 6" , the maximum strain imposed

during the deformation, prior to start of the recovery, 80 , the drop I matrix viscosity ratio,

k, and the drop / matrix elasticity ratio, K.

175

 

 

To investigate these aspects in a controlled fashion, an experimental apparatus has
been assembled to record recovery behavior in elastic drops suspended in a less viscous
and elastic ﬂuid, immediately following rapidly imposed high-strain ﬂow-induced

extensional deformations.

6.2. EXPERIMENTAL

6.2.1. Experimental Setup

The important ﬂow features of the current experiment are summarized in the

schematic shown in Figure 6.2. This experimental setup offers the following features

1. Imposition of calibrated rapid high-strain extensional deformations via ﬂow on a

viscoelastic drop suspended in a matrix of lower viscosity and elasticity

2. Control of the rate of deformation, and the extent of deformation
3. Instantaneous removal of the extensional ﬂow ﬁeld upon attaining a ﬁnite
deformation

176

 

 

4. Setup to observe and record the recovery of the drop immediately following

deformation

As shown in Figure 6.2, the extensional ﬂow ﬁeld is established by ﬂow of the

suspending ﬂuid through a semi-hyperboloidal converging ﬁxture with a proﬁle

r2 (z+zo )=C, and offering a Hencky strain EH = 2. The ﬁxture, by virtue of its

proﬁle offers a constant extensional ﬂow rate throughout its length. The drop is
transported in a separate stream of the suspending ﬂuid through the center and is
introduced in this well-established extensional ﬂow ﬁeld just upstream of the
convergence. Extensional strains are imposed on the drop as it ﬂows through the
convergence — it is ensured that the drop is small as compared to the exit of the
converging ﬁxture in order to avoid wall effects and any additional uncontrolled
deformation. Once the stretched drop exits the ﬁxture, the ﬂow is stopped
instantaneously by activating solenoid valves. Recovery of the drop is recorded from this
point onwards using a high-speed camera (KODAK camera with CCD connected to an

EKTAPRO high speed processor) capable of grabbing 50 frames per second.

The bulk ﬂow of the matrix is served from a reservoir. The matrix ﬂuid is
conveyed using a screw pump which can operate up to pressure drops of about 250 Pa.
The screw conveying mechanism of the pump offers high rate transport of ﬂuids while
ensuring minimal shear degradation. The feed from the pump is sent to a high pressure
manifold, where the ﬂow is distributed into ﬁve streams. Four streams feed the bulk ﬂow

of the suspending ﬂuid through the ﬂow cell, while the ﬁfth stream is used exclusively

177

 

 

for conveying the drop. The four streams feeding the bulk ﬂow enable faster
equilibration of ﬂow and quick establishment of the extensional ﬂow ﬁeld. The ﬁfth
stream is connected to the ﬂow cell via a T-ﬁtting, which can be opened to introduce the
drop and then re-closed. The ﬂow cell is essentially an axisymmetric cylinder made of
pyrex high clarity glass. One glass tube is placed upstream of the semi-hyperboloidal
converging ﬁxture for introducing the ﬂuid streams, and one tube is ﬁxed downstream of
the convergence and serves as the observation port to record recovery. The semi-
hyperboloidal converging ﬁxture is made stainless steel, and was machined to 0.001 mm
tolerance at the Physics and Astronomy Workshop at Michigan State University using an

automated milling machine.

The entire ﬂow cell is submerged in a transparent rectangular trough made of
clear polycarbonate walls ﬁlled with the same matrix ﬂuid as that ﬂowing inside the ﬂow
cell. This arrangement leads to obscuring of the glass tube surface, and therefore

eliminates undesired optical effects arising from the curvature of the glass tube.

The screw pump is controlled by a regulator which allows adjustment of the
screw rpm thus offering precise control of ﬂow rates, and consequently extensional strain
rates. The ﬂow through the screw pump can be stopped instantaneously using a solenoid
valve. The ﬂow downstream of the ﬂow cell also passes through the solenoid valve

which can be instantaneously closed when activated.

178

6.2.1a. F low kinematics

The parameters that may be precisely controlled in any experimental run in this
setup are: (l) the initial size of the spherical drop, Ro , (2) the macroscopic extensional
strain imposed on the sample, which is ﬁxed by the strain offered by the semi-

hyperboloidal converging ﬁxture, 8mm“ = 2.0, and (3) the global extensional strain

rate, which is ﬁxed by the bulk ﬂow rate through the semi-hyperboloidal ﬁxture:

(6.1)

where C is the geometrical constant which governs the die proﬁle

(r2 (z + 20 ) = C = 0.3913cm3 ). em remains constant throughout the length of the semi-

hyperboloidal convergence. With this pump, ﬂow rates of up to 40 ml / s can be obtained

for a ﬂuid of viscosity 1 Pa-s. This means that em up to 35 5'] may be achieved in the

current experimental setup. With these parameters ﬁxed, the capillary number of the

ﬂow is also known at the beginning of each run

=77m R05.
F

Ca (6.2)

The matrix ﬂuid experiences the 8mm,“ and the em imposed during the experiment.

The global stretch ratio in the matrix is the same as that imposed by the geometry

179

 

 

Ammax : exp (8 H ,max ) (6-3)

The drop however does not experience the same strain rate or the macroscopic
deformation imposed. The global stretch ratio in the drop is evaluated by comparing the
longest dr0p dimension (which turns out to be aligned in the ﬂow direction for strong

flows such as the one imposed in this experiment) Lmax , immediately after the drop exits

the semi-hyperboloidal ﬁxture to that of the diameter of the initial spherical drop

rid , max = :2: <’lm.max (6.4)

 

It should also be noted that by virtue of the current setup, the strain rate and
Hencky strain imposed on the drop cannot be independently controlled. This is because,
as the strain rate increases, so does the capillary number - and since the macroscopic
deformation does not vary because of the constant Hencky strain offered by the semi-

hyperboloidal die - the deformation of the drop increases with increasing strain rate.

6.2.2. Materials

The selection of materials was primarily governed by the need to have two

immiscible ﬂuids which have similar densities, have different refractive indices to offer

180

 

proper contrast between the drop and the matrix phase, and which can be easily tailored
to obtain a range of properties from viscous Newtonian to highly viscoelastic —especially

for the drop phase.

The drop phase was chosen to be polydimethylsiloxane (PDMS). PDMS is
available in a range of molecular weights, starting from low viscosity Newtonian ﬂuids to
high molecular weight elastomers. This makes it possible to blend PDMS of different
molecular weights - this way obtaining deformable gels with a broad molecular weight
distribution. One such gel was prepared for the current set of experiments by mixing four
PDMS of widely different molecular weights. The matrix phase was chosen to be a
mixture of polybutene and a chlorinated parafﬁn oil -— formulated primarily to match the

density with the PDMS drop ﬂuid.

The mixture of polybutene and chlorinated parafﬁn oil is a Newtonian ﬂuid, with

a constant shear viscosity of 780 mPa-s at 25°C.

The drop ﬂuid has a zero-shear viscosity of 20 Pa-s at 25°C. The viscoelasticity
of the drop ﬂuid is not as pronounced as some of the elastomers or even therrnoplastics
characterized in Chapters 4 and 5. The drop phase appears to be weakly elastic as seen in
the startup shear viscosity curve plotted in Figure 6.3. As seen in the ﬁgure, the startup
viscosity does not instantaneously reach the steady value of 20 Pa-s, indicating the non-
Newtonian behavior of the drop phase. However, the transient is not very long and

steady viscosity is reached within 1 s.

181

 

Dynamic data for the drop phase PDMS are presented in Figure 6.4 and 6.5. The
storage moduli for the drop phase are measurable and have ﬁnite positive values at the
lowest frequencies measured indicating the viscoelastic nature of the drop ﬂuid. Over the
ﬁrst decade of the tested frequencies, terminal region may be seen based on the slopes of
the storage and loss moduli curves with respect to frequency. The complex viscosity
plotted in Figure 6.5 also remains constant at the zero-shear viscosity of 20 Pa-s in the
ﬁrst decade of frequency tested. At higher frequencies however, weak shear thinning is

obvious.

The strongest evidence of viscoelasticity of the drop phase is obtained from stress
relaxation and shear creep recovery experiments. The stress relaxation curves measured
at strains of 0.05 and 1.0 are shown in Figure 6.6. Strong nonlinear damping can be
observed at a strain of 1.0 with respect to the curve at 0.05. The curves at the two strains
are parallel, indicating time-strain separability. The duration over which the stress
relaxes is striking. In contrast to the startup shear viscosity and dynamic data which
indicate small relaxation times for the PDMS, the stress relaxation experiments point to
the presence of long relaxation times. The tube disengagement or “reptation” time
assessed from a plot of LN { G (t) } vs t plots [Takahashi (1995), Tobolski (1959), also
see Section 4.4.3b], turns out to be 9.8 s, which is indicative of strong viscoelastic
character. This long relaxation time may be the consequence of the presence of small
amounts of the high molecular weight (2,000,000 CSt) silicone elastomer in the PDMS

gel.

182

Strong elastic effects are also evident in shear creep recovery experiments. As
seen in Figure 6.7, following nonlinear shear creep at 10 Pa up to a strain of about 45, the

PDMS shows substantial recoverable compliance, and along retardation time.

6.2.3. Determination of Interfacial Tension

The interfacial tension between the two ﬂuids was measured by adopting the
“imbedded ﬁber retraction” method used by Cohen et al. (1989) [Carriere and Silvis
(1997)] to determine interfacial tension between polymer melts. To measure the
interfacial tension, a short thread of the matrix ﬂuid was vertically formed within a
reservoir of PDMS using a thin syringe. While forming the thread, the ﬂuid was released
very slowly and without application of too much pressure to ensure absence of any ﬂow
induced stresses. During recovery, the thread ﬁrst recovered at one end showing some
fore-aft asymmetry forming a tear shaped drop as shown in Figure 6.8. But soon the
fore-aft symmetry was restored and the thread evolved into a short ellipsoid and
ultimately into a sphere. The evolution of drop shapes was recorded at regular intervals
using a digital camera, and some of the pictures are shown in the collage in Figure 6.8.
The evolution of the drop size during the later stages of its recovery starting from the
ellipsoidal shape was modeled using a balance between interfacial tension forces and

drag forces following Cohen et al. (1989)

183

 

dL dA
6nzneﬂ R-dT + FE = 0 (6.5)

In the above equation, R and L are the instantaneous radius and length of the drop
respectively, and A is the interfacial area. Z is a hydrodynamic interaction parameter
between the drop and the matrix and is a function of the instantaneous drop shape, and

77,17 is an effective viscosity given by

_ 7]," +1.7ﬂd
"e” _ 2.7

 

(6.6)

Along with the drop sizes, the viscosities of the two ﬂuids were also known, leaving only
the interfacial tension as the ﬁtting parameter. Carriere and Cohen (1989, 1997) modeled
the drop shape as a cylinder with hemispherical end caps, and ﬁxed the shape factor in
Equation 6.5, as 1.0. Then the solution of the differential equation yields the following

analytical expression for the evolution of drop size.

 

 

—r

2 1/2 1/2
f(r)=31n[(l+'+') ]+33’2arctan[3 r]-r—8r-2 (6.7)

In the above equation, r is the instantaneous radius of the thread normalized by the radius

of the equivalent sphere

184

r = — (6.8)

The parameter r is evaluated at regular intervals throughout the retraction process. The

value of f (r) is related to t through the equation below

f(r)=/1;‘t+f<r.> (6.9)

In the above equation, AF is an interfacial timescale, and r, is the normalized ﬁber radius

at the commencement of recovery

R
Ar = 152/:79 (6.10)

Re
re :76 (6.11)

The best ﬁt to the experimental data with this analytical expression was obtained for an
interfacial tension of 1.8 mN /m. The ﬁt of expression to experimental data is good as
seen in Figure 6.9. The experimental transient is smooth and monotonic, just like the
theoretical curve, and without any abrupt transitions in slope - this encourages us to
believe that the recovery observed is truly interfacial tension driven and any surfactant or
stress effects are absent in this measurement. Thus, 1.7 mN/m was taken to be the

interfacial tension between the two ﬂuids. The value is slightly lower than the magnitude

185

 

 

of 2.0 mN / m reported between polybutene and PDMS, and this difference must be due

to the presence of chlorparafﬁn oil in the polybutene.

6.2.4. Typical Parameter Ranges in the Experiment

The parameters that may end up affecting the fraction of recovery brought about
by elasticity, will also affect the deformation of the drop. For example, high Deborah
number deformations will also be associated with high capillary numbers, because both
parameters are proportional to the strain rate of deformation 5'. Thus a drop deformed at
high rates will also be deformed to higher strains. Alternatively, when deformed at the
same strain rate, a smaller dr0p will be deformed to a lesser extent than a larger drop. All

these factors will come into play in governing the recovery.

In this experiment viscosity ratio between the drop and matrix ﬂuids was chosen
so as to simulate the actual blend setups. The typical ranges for the various parameters
are given in Table 6.2. As shown in the table, the magnitudes of viscosity ratios, strain
rates, capillary numbers, Reynolds numbers and the interfacial time constants for the
current experimental setup and ﬂuids are in the same range as those observed for actual

TPO blend systems, as laid out in Table 6.1.

186

 

 

6.2.4a. Drop breakup Considerations

In addition to the absence of inertial effects ( as evident by the low Reynolds
numbers), the choice of the ﬂuids as ﬂow parameters has been made so as to ensure the
absence of breakup of the stretched drops. Our operating ranges involve viscosity ratios
ranging from 5-50. For simple shear ﬂows with viscous drops, Grace has shown that
drop breakup is ruled out if the viscosity ratios are greater than 1.0. Extensional ﬂow on
the other hand is a stronger ﬂow, and drop breakup may occur at any magnitude of

viscosity ratio.

Breakup of stretched ﬁlaments may occur via two mechanisms — “end pinching”
or through capillary instabilities. The typical end pinch time constants for the current
experimental ﬂuids are around 200 5 (refer to Equation 2.30). Elastic recovery on the
other hand, if detectable, is expected to occur over timescales of 5-10 8. Breakup via
end-pinching is therefore ruled out within timescales at which elastic recovery occurs.
Stone et al. pointed out that when the viscosity ratio is greater than 1.0, drop breakup via
Rayleigh instabilities is not seen up to an aspect ratio of 15. The macroscopic stretch
ratios offered by the semi-hyperboloidal convergence are not greater than 8 (the drop
phase was always stretched to lesser values, given the higher drop viscosity and elasticity
as compared to the matrix). Based on above considerations, Rayleigh instabilities are
also ruled out. Breakup is further ruled out because of the elasticity of the drops. Indeed,
in these experiments, we did not observe any drop breakup with the experimental range

of parameters.

187

 

 

 

6.3. RESULTS AND DISCUSSION

In Figure 6.10, the retraction of drop stretch ratio has been plotted during
recovery driven solely by interfacial tension and in the absence of any elastic effects.
This ﬁgure is an alternative way of representing Figure 6.9. The ordinate is the
instantaneous drop length normalized by the drop length at maximum stretch (at the
commencement of recovery). This method of presenting recovery data is suitable when
recovery transients starting from different initial stretch ratios are to be compared. The
ﬁgure also shows the results of the Carriere-Cohen drag and interfacial tension force
balance described earlier. The experimental data and the predictions agree reasonably
well. This transient for purely interfacial tension driven recovery is the base case with
which elastic effects, if present, shall be compared. The smooth and monotonic recovery
transient and the absence of any abrupt slope changes are indications that only one

mechanism of restoration of drop shape and size is at work.

In Figure 6.12, the photo sequence shows some of the key length transitions
during recovery of the PDMS drop following extension to a stretch ratio of 2.8, brought
about at a macroscopic stretch rate of 12 s". As can be seen the drop is ellipsoidal to
begin with and remains ellipsoidal at all times until it ﬁnally reaches a shape close to a
sphere. Care was taken to make sure that the drop emerging from the semi-hyperboloidal
convergence was always ellipsoidal. There are two reasons for this. First of all,

ellipsoidal drop shapes are considered ideal for phenomenological modeling of drop

188

 

 

deformation and recovery phenomena. Secondly, shapes other than ellipsoidal are
indications of non-homogeneous deformation of the drop. For example, ﬂattening of the
drop along one of the minor axes would indicate high shear gradients. Loss of fore-aft
symmetry would indicate non-homogeneous extension of the drop. With these additional
features, it would be difﬁcult to unambiguously relate the recovery behavior to the
extension rate or other parameters. For example, when drops were chosen to be bigger
than 10 microL, the drop size was such that wall effects came into play, resulting in the
loss of fore-aft symmetry as shown in Figure 6.11. Such “space-shuttle” type shapes for
drops has been reported in the ﬁnite element simulations by Singh, in the presence of
wall effects. While it is clear from Figure 6.11 that even in the absence of fore-aft
symmetry, the drop recovery behavior is quantiﬁable - but because of modeling
considerations and in order to avoid accounting for spurious experimental effects and
uncontrolled deformation, only spheroidal shapes were considered for evaluation of

recovery.

Considering once again, the photo sequence for the recovery of spheroidal drop in
Figure 6.12, there is clearly visible recovery in the ﬁrst 1 s, the extent of which is similar
to that observed over the next 9 3. Obviously, there seems to be a faster mechanism at
work in the ﬁrst few seconds. This is clearer when the drop stretch ratios are plotted at
several times during recovery as shown in Figure 6.13. The transient clearly shows two
steps, indicating the presence of two mechanisms of recovery. In the ﬁrst stage, the
aspect ratio shows a precipitous drop within 5 s, followed by a slower relaxation to

sphere over the next 75 s.

189

 

 

In Figure 6.14, the same recovery transient, in terms of normalized instantaneous
length, is compared to the prediction of drag - interfacial tension force balance with the
same matrix and drop viscosities and drop size as the experiment. Clearly the initial
recovery is much faster than interfacial tension driven recovery at comparable times.

This recovery must be brought about by elastic effects.

Another test was conducted at a lower extension rate. The initial recovery
transient for the two runs are compared along with purely interfacial tension driven
recovery in Figure 6.15. This Figure clearly establishes the role of deformation rate in
bringing about elastic effects during recovery. Compared to the drop deformed at 12 s",
the drop deformed at 10 8'1 shows lesser extent of initial recovery via elastic effects.
Clearly, higher Deborah number gives rise to greater nonlinear effects resulting in faster

initial recovery via elastic effects.

6.4 CONCLUSIONS

The new experimental setup allows the study of recovery of viscoelastic drops
suspended in a less viscous and elastic matrix, immediately following rapid ﬂow induced
deformations. Creep recovery tests on the drop ﬂuid reveal substantial elasticity in the
drop phase. Following high rate extensional deformations to stretch ratios of about 2, a

substantial portion of the imposed strain are recovered within 2-5 3, times much shorter

190

 

than the interfacial timescale. The recovery transient thus shows two stages, an initial
rapid recovery followed by a slower relaxation to sphere. The extent of recovery in the
initial stage relative to the total deformation is higher following deformation at a higher
rate — conﬁrming the elastic origins of the phenomenon. An investigation of the full
range of parameters including viscosity ratios, initial strains and ﬂuid rheological

parameters will be continued in future.

191

Typical material and ﬂow parameters involved in rapid processing ﬂows

 

 

 

 

 

 

 

 

 

 

Table 6.1.
of TPO blends comprising an elastomeric dispersed phase in a
thermoplastic matrix
Shear rates 7 50 100 5’1
Extension rates ti" 5 10 s'1
Shear viscosity ratios k =91 10 — 50
’7»:
. . . 77 E, d
Extensronal vrscosrty k E = 5 _ 20
ratios 77 5’,"
GI
Elasticity ratios K =G—7I 5 - 40
m a) = 10s—1
2 .
R 13
Reynolds numbers R e = O '0 0
77m
R .
Capillary numbers Ca =ﬂn—FL8— lO - 100
Interfacial time /1 :77»: R0 (19k+16)(2k+3-2¢(k‘1)) ~ 2 _ 25 s
constant 4 F 10 (k +1) — 2¢ (5k + 2)

 

 

 

 

 

192

 

Table 6.2. Materials and properties for room temperature drop studies

 

 

 

 

 

 

 

 

 

 

Property Flow Parameter

Viscosity ratio k = 26

Interfacial tension [N/m] 1.7 x 10'3
Drop size [m] 0.0025

Extensional Strain Rate 5 — 20 3’1

Drop Reynolds Number 0.04 — 0.15

Capillary Numbers 5 - 20
Interfacial Time Constant ~ 100 s

 

 

193

 

Figure 6.1.

'1? (811 = 43) [Pa-s]

 

 

 

1.0E+07 E (a)
: 1:1 131: PP A 32: PP
1.0E+06 E {B1: EP +82: EP
E
1
1.0E+05 ;
. n a a
e
66
1.0E+04 i 1 1111M 1 14111111 1 1 1111.11
0.1 1 . 4 10 100
8 [S l
(b)
1.UE+06
...1.01:+04
C
E.
‘3 1.015+02
1.013+00
0.1 1 10 100 1000 0.01 0.1 1 10 100 1000
aims] 011/8]

Typical matrix and dispersed phase properties of TPO blends. Blends B1
and B2 are same as those studied in Chapter 5. (a) Extensional viscosities
of of PP matrices (open symbols) and EP elastomer dispersed phases
(ﬁlled symbols) evaluated at Hencky strain 8,, = 4.9 at 200°C (b)
Dynamic storage moduli of the PP matrices (open symbols) and EP
elastomer dispersed phases (ﬁlled symbols) of blends B1 and B2 at 200°C.

194

Semi-hyperboloidal Profile
r2 (z + 20 ) = C

 

 

 

 

—> Q\

—’ u

—’ b u \®‘

—> O - G)

——>

—9

Figure 6.2. Mechanism for imposing rapid ﬁnite extensional strains on a viscoelastic

drop suspended in a matrix of lower viscosity and elasticity, and observing
the recovery behavior after stoppage of ﬂow.

1.

Flow of the viscous suspending ﬂuid through the semi-
hyperboloidal ﬁxture establishes the high rate extensional ﬂow
ﬁeld

The elastomer drop, transported separately in an inner stream of
the suspending ﬂuid is introduced to the extensional ﬂow ﬁeld
established by the bulk flow just upstream of the convergence

Semi-hyperboloidal converging proﬁle ﬁxture offers a constant
extensional strain rate over its entire length; the Hencky strain of
the ﬁxture is the upper limit of the strain attained by the drop. The
current setup offers a Hencky strain of 2.0. The inlet diameter of
the ﬁxture is 20.00 mm, and the exit diameter is 7.38 mm. The
length of the converging ﬁxture is 25 mm.

Flow is instantaneously stopped once the elastomer drop exits the
semi-hyperboloidal ﬁxture and this removes the extensional ﬂow
ﬁeld; the subsequent recovery of the stretched drop is recorded in
the downstream portion with a hi gh-speed camera

195

01
O

IIUIIITTTIIUUTFVjIU]

01)
O

N
O

..L
0

I111]

Startup Shear Viscosity [Pa-s]

 

111111 1 1 1111111 1 1 llLLlj_; 1 1 111111]

0.01 0.1 1 10 100
Time [s]

O
r

Figure 6.3. Growth of shear viscosity for the drop phase PDMS during step shear
conducted at 0.05 s", at 25°C

196

1 0000

1000
"is" 100
9:.
it; 10
"5 1
0.1
0.01
Figure6.4

 

10
w [rad l s]

Dynamic moduli for the drop phase PDMS at 25°C

197

L 4 LL

LLJ

100

é

00000000000000000
0

Complex Viscosity [Pa-s]

 

10 1 1 1 1 1 1 1 L1 1 1 1 1 1 1 1 1 I
1 10 100
a) [rad/s]

 

Figure 6.5. Frequency dependent complex viscosity of the drop phase PDMS at 25°C

198

 

 

1 .0E+05

 

1.0E+04
O
_1.OE+03 '
N O
a. - .
€1.0E+02 . .I . .
(5 .I . ' .
1.0E+01 'I . 905
I o
I ..
1,015+00 " .1-0
I
I
1.0E'01 1 1 1111111 1 1 1111111 1 1 1111111 1_1 L111111 1M
0.001 0.01 0.1 1 10 100
Time [s]

Figure 6.6. Shear stress relaxation curves for the drop phase PDMS at 25°C and two
different strains as indicated within the ﬁgure.

199

 

 

 

1.0E+01 o
10 Pa, 25 C

 

 

 

 

 

J (t)
g 1.0E-O1
0
O
C
.5!
'5.
E mos-03
8 .—
J r (t')
1 .OE_05 _l L 1— 1 1 l 1 1 1 I l
o 25 50 75 100

Time [s]

Figure 6.7. Shear creep at 10 Pa, and subsequent recovery for the drop phase PDMS
measured at 25°C

200

 

0 9 15 20

Time during
recovery [3]

 

27 40 56

Figure 6.8. A photo sequence following quiescent recovery of a short ﬁlament made
with the polybutene - chlorparafﬁn oil mixture imbedded in the high-
viscosity PDMS reservoir

     

 

 

1 -
I
0.95 -
o i
a:
\ i’
h u
I Experiment
0'9 — Carriere-Cohen
0.854ll'l"“"“+Lll-Ilnnn1|
O 100 200 300 400 500
Time [s]

Figure 6.9. Estimation of interfacial tension from quiescent recovery of a short
ﬁlament of polybutene — chlorparafﬁn oil mixture imbedded in a high
viscosity PDMS reservoir. The points ( I ) are obtained from
measurements of the photographs of the recovering drop at regular time
intervals (Figure 6.8). The line is obtained from the shape evolution
equation laid out in Section 6.2.3 for the force balance between interfacial
tension forces and drag forces — interfacial tension is the ﬁtting parameter
to obtain the best ﬁt over experimental data, while the drag force is
calculated from known viscosities of the two fluids.

 

 

202

 
    

I Experiment

— Interfacial

 

 

0.6 4 1 1 1 l 1 1 1 1 L 1 1 1 1 J

 

Time [s1

Figure 6.10. Typical drop shape recovery transient following low-rate deformation.
Comparison of the experimentally observed transient and the predictions
of interfacial tension -drag force balance proposed by Cohen et al. (1989)
for quiescent interfacial tension driven recovery

 

203

 

Unstretched
drop :
1 . . '

  
       

1.2 2.2
Time during Recovery [5]

Figure 6.11. Drop shapes obtained with a 29 microL drop with RolRe = 0.55,

deformed at 10 s'l. Wall effects are apparent from the fore-aft asymmetry
of the drop.

204

 

 

 

0.02 s 1.02 3 10.72 s 41 s

 

Figure 6.12. A typical photo sequence for drop shape recovery transient following
hi gh-rate deformation through the extension ﬂow cell.

205

 

2.2
Global strain rate during stretch = 12 s'1
o
33 1.8
a: ’ ‘x
.c » ‘.
1‘3 ‘.
a 1.4 - ‘x
o a
h \
Q ~. .
Q . .- . .
1.0 1 1 1 L 1 1 L 1 1 l 1 1 1 1 l 1- . 1. . 2 I I

 

 

 

O 25 50 75 100
Time During Recovery [5]

Figure 6.13. Evolution of aspect ratio during recovery following high-rate deformation
through the extension ﬂow cell

 

206

1.0

 

    
 

 

0.8
g Interfacial
~l \.
\ P I.
g . -.
3 0.6 -
.1 _ ‘ ~.. .
12 s1
0.4 4* L 1 l L l J J
0 25 50 75 100 125 150
Time During Recovery [5]
Figure 6.14. Comparison of the experimentally observed recovery transient following

high-rate deformation and the predictions of interfacial tension —drag force
balance proposed by Cohen et a1. (1989) for quiescent interfacial tension
driven recovery

207

 

 

1.0

    
 

 

 

Interfacial
I. ‘-~
g . ‘.~
E ‘ 1
‘1. _ ‘ \\
\ " \
f: ' ‘~ -1
E .~~. ‘ ......... 10.58
.1 ""'.~
' 12 s'1
O ............. s
0.8 J 1 1 1 1 l 1 ‘V1 1 J
0 5

1 0
Time During Recovery [s]

Figure 6.15. Recovery transients following high-rate deformations: effect of
extensional strain rate on the extent of elastic recovery in the initial stage

208

Chapter 7.

CONCLUSIONS AND RECOMMENDATIONS

7.1. CONCLUSIONS

In this thesis, the nonlinear melt rheology of polyoleﬁn random copolymer
elastomers and thermoplastic oleﬁn blends has been investigated. The relevance of
component and blend rheology in strain recovery following nonlinear deformations in
such systems has been analyzed. Three primary research problems were deﬁned in this

context. The principal conclusions are as shown below.

RESEARCH PROBLEM 1.

To investigate the interrelationship between of short-chain branching density in

ethylene- a-oleﬁn elastomers and their nonlinear melt rheology

Three random ethylene-octene copolymers with degrees of short-chain branching

(SCB) ranging from 25 to 50 per 10 carbons and with similar molecular weights display a

strong dependence of non linear shear creep recovery and elongational strain hardening

209

on SCB. The absence of even sparse long-chain branching was confirmed with several

linear viscoelastic tests. The similarity of J? for the three copolymers rules out

differences in any high molecular weight fractions among them.

Zero-shear viscosity of these copolymers follows the trends in backbone
molecular weight closely. Although the steady recoverable shear compliance values are
close for strains < 0.5, they decrease progressively with increasing SCB density at higher
strains of 25 to 100. This trend must be a consequence of the different degrees of
primitive chain relaxation in the rapid equilibration stage obtained at large strains with

different SCB.

Extensional viscosity of the elastomer melts was measured at several strain rates
and at a Hencky strain of 4.9 using lubricated skin-core flow through semi-hyperboloidal
dies. The range of extensional strain rates was broad enough to cover rates which are

higher as well as lower than the rate of equilibration of the primitive polymer chain,

given by the inverse of the Rouse timescales of the melts, 2']; 1.

Extensional strain hardening, which decreases with increasing strain rate, is

observed in all the three elastomer-melts, at all the rates tested. At low extension rates,

below 1:, extensional strain hardening progressively increases with increasing SCB

density. At higher rates, upon onset of primitive chain stretch, the strain hardening

behavior for the three melts merges.

210

RESEARCH PROBLEM 2.

To study the eﬂect of polyoleﬁn elastomer and TPO blend rheology on the

evolution of blend morphology in high-rate processing operations.

The flow mark problem associated with high-speed injection molding was studied
in two thermoplastic oleﬁn (TPO) blends. The blends had polypropylene (PP) matrices
and ethylene-propylene (EP) elastomer dispersed phases. The composition of the blends
was the same — the PP phases were very similar, but the EP dispersed phases differed in
terms of SCB density. When injection molded under same conditions, blend B1
displayed greater frequency of ﬂow marks and greater contrast between glossy and dull

regions.

Melt rheology of the blends revealed that blend Bl, which was more unstable
during injection molding was also less strain hardening in extension as compared to blend
B2. The trends in strain hardening in the blends followed those in strain hardening of the
corresponding EP elastomers by themselves. The differences in degree of contrast
between glossy and dull regions were traced to different degrees of strain recovery in the
dispersed phases of the two blends. This recovery occurred over injection molding
timescales of the order of a few seconds. Strain recovery after shear creep was higher in
blend B1, that displayed more severe flow marks in injection molding; the corresponding

elastomer by itself also showed a greater extent of creep recovery.

211

Thus the quick strain recovery observed in the elastomeric dispersed phases must
be associated with elastic stresses rather than interfacial tension. In both blends, the
dispersed elastomer phase was substantially more viscous and elastic as compared to the
PP matrix. Thus the enhanced signiﬁcance of elastic stresses in driving recovery must
also be a consequence of the absence of any signiﬁcant resistance to such recovery from

the matrix.

RESEARCH PROBLEM 3

To evaluate the ﬂuid and ﬂow parameters governing elastic effects during

recovery of elastomeric drops in suspension

An experimental apparatus was assembled to study strain recovery after
elongation of a model elastomer suspended in a less viscoelastic medium at room
temperature. In this apparatus, small drops of well-characterized elastomers suspended in
a matrix ﬂuid of lower viscosity and elasticity, are stretched rapidly without wall effects;
the stretch ratio of the drops is then recorded over time. Preliminary studies were
performed with a viscoelastic silicone elastomer drop suspended in a viscous Newtonian
liquid. Creep recovery tests on the drop ﬂuid revealed substantial elasticity in the drop

phase.

212

Following high rate extensional deformations to stretch ratios of about 2, a
substantial portion of the imposed strain were recovered within 2-5 5, times much shorter
than the interfacial timescale. The recovery transient thus showed two stages, an initial
rapid recovery followed by a slower relaxation to sphere. The extent of recovery in the
initial stage relative to the total deformation was higher following deformation at a higher

rate — conﬁrming the elastic origins of the phenomenon.

213

 

7.2. FUTURE DIRECTIONS

THE DROP RECOVERY EXPERIMENT

It is clear that the experimental drop deformation-recovery setup is promising in
that recovery immediately following controlled hi gh-rate ﬂow-induced deformations may
be recorded for well controlled drop -— matrix systems. In the preliminary results reported
in Chapter 6, deformation rate was chosen as a means to tailor the elastic effects.
Indeed, the rate of deformation had signiﬁcant a signiﬁcant effect on the extent of initial

recovery brought about by elastic effects.

As pointed out earlier, this experiment offers a very flexible setup, where the drop
properties as well as the matrix properties can be easily varied to investigate the entire
spectrum of materials - Newtonian to elastomeric. Therefore with carefully chosen
variables, there is a wealth of knowledge to be gained from this setup regarding the
factors which bring about or hinder elastic recovery of the deformed drop. The variables

that may be conveniently tailored are

9 Extent of initial deformation imposed on the drop: as pointed out earlier, with
the global Hencky strain offered by the semi-hyperboloidal convergence
ﬁxed, the extent of deformation imposed on the drop can be controlled by
varying the capillary number or the extensional strain rate. On the other

hand, another ﬁxture offering a greater Hencky strain may be employed.

214

 

 

 

 

O Viscosity ratio

0 Elasticity ratio: The suspending fluid in the current study reported in Chapter
6 was a Newtonian fluid. It would be interesting to increase the elasticity of
the matrix phase — during the deformation stage, this would result in
imposition of greater strains on the drop phase resulting in enhanced elastic
effects, but at the same time the deformed drop would also face greater

hindrance during recovery from the elastic matrix.

0 Molecular architecture: Studies should be carried out in this setup with
polymers having different molecular features — linear, sparse long-chain
branched and profusely long-chain branched. For example, polyisobutylene
elastomers with widely varying architecture are readily available and are

suitable for room temperature drop studies

9 Deborah numbers Deg: Another interesting aspect that should be pursued is to
understand the signiﬁcance of the magnitude of the deformation rate relative
to the Rouse timescale of the material in governing the recovery. For
example, it is known from reptation considerations that even as the
deformation is being imposed, the primitive polymer chain tends to retract
within the Rouse timescales thus relaxing the ﬂow induced elastic stresses.

However, if deformations are imposed at rates much higher than the retraction

215

rate of the primitive chain, un-relaxed elastic stresses would be substantially
higher at the commencement of recovery leading to greater signiﬁcance of

elastic effects.

0 Blends and Melts: Once these aspects are investigated in some detail, it may
be worthwhile to study the behavior of semi-concentrated blends in the melt

state.

MODELING OF DROP RECOVERY WITH ELASTIC EFFECTS

Another encouraging outcome of the current experiment is the possibility of
obtaining ellipsoidal drop shapes. There exists a class of semi-empirical drop models
known as ellipsoidal models [Maffettone and Minale (1998), Maffettone and Greco
(2004)], which assume ellipsoidal drop shapes at all times. These semi-empirical
phenomenological models are very easy to navigate and modify to include experimental
variables. The only constraint is the drop shape which should remain ellipsoidal at all
times. With the possibility of obtaining ellipsoidal drops conﬁrmed with the current
setup, the ellipsoidal models of Maffettone and Greco (2004) may be modiﬁed to
incorporate stronger nonlinear rheological effects such as extensional strain hardening
and the utility of these models may be extended to studying rapid elasticity driven

recovery of elastomers.

216

APPENDIX

FORTRAN code for solution of the integral equation for prediction of strain during shear

creep and recovery using K-BKZ constitutive equation

217

program ode

C ___________________________________________________________________
C WSCCRStGD_SPL_MOD.f

Wagner Shear Creep Creep Recovery Strain Growth Decay . f
SPLiced damping function

MODified relaxation spectrum

Growth of shear strain during shear creep followed by recovery
Use of K-BKZ Model (Wagner)

00000

Wagner, JNNFM, 3(4), 39 (1978)

00

Stress assumed constant

Gives rise to a nonlinear Volterra integral equation of the second
kind in shear strain (t0)

Upon use of quadratures, this equation can be reduced to a non-
linear algebraic equation which can be solved by iteration

00000

Wagner, Laun, Rheol Acta, 17(2), 138 (1978)

0

0

n : Number of steps

mode : number of relaxation mode

Dt : time step [5]

t_n : Current Time [s]

k_step : global step counter

ipause : local step counter

Stress : Shear stress (tau_12) [Pa]

g[i] : Weighting factor for relaxation spectrum [Pa/s]
Lambda [1] : Relaxation spectrum time constants [s]
Rel_Mod : Relaxation modulus [Pa]

i,j : local step counters

F : Shear strain at any time t (Flt))

W_j : Quadrature weights

t_j : time for network building, past time, t' [s]
Mem_Func : Memory function [Pa]

000000000000000

Implicit Double Precision (a—h,o-z)
Double Precision Lambda(12),g(12),F(1000000)
Double Precision a,m1,m2,Stress,Dt,t_n

common / DampF / a,m1,m2,p1,p2,cl,c2
common / Funcl / Stress,Dt,t_n
common / Funcz / mode,Lambda,g,F
common / Func3 / ipause

218

C

C _____________________________
Discrete Relaxation Spectrum
C _____________________________

C
C

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0

0 0

constants

Stress = 3.0D+O4

****************************

Relaxation Spectrum: TPO EPs
****************************
Bl: CA45 EP
Lambda(l) = 1.167D-02
Lambda(Z) = 3.246D—02
Lambda(3) = 9.032D-02
Lambda(4) = 2.513D-01
Lambda(S) = 6.992D-01
Lambda(6) = 1.945D+00
Lambda(7) = 5.412D+00
Lambda(8) = 1.506D+01
Lambda(9) = 4.190D+01
Lambda(lO) = 1.166D+02
Lambda(ll) = 3.243D+02
9(1) = 2.682D+07
g(2) = 3.723D+06
9(3) = 7.667D+05
9(4) = 1.441D+05
g(S) = 2.853D+O4
9(6) = 4.187D+03
g(7) = 7.042D+02
9(8) = 1.062D+02
9(9) = 1.392D+01
9(10) = 1.139D+00
9(11) = 2.898D-02
mode = 11
BZ: CA53 EP
Lambda(l) = 7.884D—03
Lambda(Z) = 2.194D-02
Lambda(3) = 6.103D—02

219

c Lambda(4) 1.698D-01
c Lambda(S) = 4.725D—Ol
c Lambda(6) — 1.314D+00
c Lambda(7) = 3.657D+00
c Lambda(8) = 1.018D+Ol
c Lambda(9) = 2.831D+Ol
c Lambda(lO) = 7.877D+01
c Lambda(ll) = 2.192D+02
c 9(1) = 2.304D+O7

c 9(2) = 9.8280+05

c 9(3) = 6.734D+05

c g(4) = 7.101D+O4

c 9(5) = 1.270D+04

c 9(6) = 1.359D+O3

c g(7) = 2.051D+02

c 9(8) = 2.402D+01

c 9(9) = 4.396D+00

c 9(10) = 2.603D-01

c g(ll) = 3.547D—O3

c mode = 11

C ************************

c Relaxation Spectrum: EPs

C ************************

c Lambda(l) 9.9983D1
c Lambda(2) = 3.7270D1
c Lambda(3) = 1.3893D1
c Lambda(4) = 5.1788D0
c Lambda(S) 1.9305D0
c Lambda(6) = 7.1961D-1
c Lambda(7) = 2.6824D-1
c Lambda(8) 9.9992D—2
c Lambda(9) = 3.7273D-2
c Lambda(lO) = 1.3894D-2
C ***********************

c Weights: DUTRAL-C0034

C ***********************

c 9(1) = 1.3714D—01

c 9(2) = 2.9821D+00

c 9(3) 2.8170D+01

c 9(4) 1.7070D+02

c 9(5) = 1.0437D+03

c 9(6) 7.5208D+03

c 9(7) = 5.6163D+04

c 9(8) = 4.9957D+05

c 9(9) = 1.8195D+06

c 9(10) = 2.5797D+O7

c mode = 10

C ************************

c Weights: EPR 306 Polysar

220

 

C ************************

0000000000

0

0

9(1)
9(2)
9(3)
9(4)
9(5)
9(6)
9(7)
9(8)
9(9)
9(10)

mode = l

NI—‘WU‘Iml—‘NUIQH

O

.067OD0
.2800D0
.6766D1
.9225D2
.9044D3
.9214D3
.5146D4
.4576D5
.3807D6
.8612D7

************************

c Relaxation Spectrum: EOs
C ************************

C ________________
c E01: ENGAGE 8480
C ________________
Lambda(l) = 1.064D-03
Lambda(2) = 3.342D-03
Lambda(3) = 1.050D-02
Lambda(4) = 3.299D—02
Lambda(S) = 1.037D-01
Lambda(6) = 3.257D—01
Lambda(7) = 1.023D+00
Lambda(8) = 3.216D+OO
Lambda(9) = 1.010D+01
Lambda(lO) = 3.175D+01
Lambda(ll) = 1.500D+02
9(1) = 2.910D+08
g(2) = 6.272D+07
g(3) = 1.112D+07
9(4) = 2.026D+06
9(5) = 2.873D+05
9(6) = 3.974D+04
g(7) = 5.634D+03
9(8) = 6.210D+02
9(9) = 7.368D+01
g(lO) = 4.583D+00
9(11) = 5.701D-01
mode = 11
C ................
c 802: ENGAGE—8003
C ________________
c Lambda(l) = 1.064D—03
c Lambda(2) = 3.342D-03
c Lambda(3) = 1.050D-02
c Lambda(4) = 3.299D-02

221

 

0000000

00000000000

0

000

00000000000

00000000000

0

0

Lambda(S) = 1.037D-01
Lambda(6) = 3.257D-Ol
Lambda(7) = 1.023D+OO
Lambda(8) = 3.2160+OO
Lambda(9) = 1.010D+01
Lambda(lO) = 3.175D+Ol
Lambda(ll) = 9.976D+Ol
9(1) = 2.3598D+08

9(2) = 6.5469D+07

g(3) - 1.5614D+O7

9(4) = 2.8348D+06

9(5) = 3.6222D+05

9(6) = 4.2188D+O4

9(7) = 4.8640D+03

9(8). = 4.8092D+02

9(9) = 5.2253D+01

9(10) = 3.4604D+00

9(11) = 1.8281D-01

mode = 11

ENGAGE—8100

Lambda (1) = 1.0636D-O3
Lambda (2) = 3.3418D-03
Lambda (3) = 1.0500D-02
Lambda (4) = 3.2992D-02
Lambda (5) = 1.0366D-01
Lambda (6) = 3.2572D-01
Lambda (7) = 1.0234D+00
Lambda (8) = 3.21580+00
Lambda (9) = 1.0104D+01
Lambda (10) = 3.1748D+01
Lambda (11) = 9.9755D+01
9(1) = 1.8037D+08

g(2) = 5.2380D+07

g(3) = 1.2850D+07

9(4) = 2.8451D+O6

9(5) = 4.2568D+05

9(6) 5.8966D+04

g(7) = 4.9478D+03

9(8) = 5.0999D+02

9(9) = 3.9597D+01

9(10) = 2.7538D+00

9(11) = 1.5290D-01

mode = 11

‘k**********************************

Damping Function Fitting Parameters
***********************************

*************

222

 

000

0 0000000

00

0000000

0

CA45 EP Phase

*************

l

a

m

.75

O.

l

*************

CA53 EP Phase

*************

a

O
m = l

6
0

****************

E01: ENGAGE 8480
****************
a = 0.045
ml = 2.25
m2 = 1.0
pl = 1.0
p2 = 0.0
cl = 1.0
c2 = 0.0

****************

E02: ENGAGE 8003

****************

a = 0
m1 =
m2 =
p1=
p2=
c1 =
c2 =

.2

or~<aidrdra

0000015

****************

E03: ENGAGE 8100

****************

ml =
m2 =

p2=
c1 =
c2 =

**********

Time Steps

**********

C>hic>hihih‘-

000000N

 

Dt = 0.01 !time step
n = 7212 !total # of steps
n_str_rem = 3606 !# of timesteps after which stress

lis removed
C ________
c prepare
C ________

open (1,file="PLOTDAT")

t_n = 0.0 ! initial time

11
0

kstep ! global step counter

ipause = 0.0 ! batch step counter

C ********************'k**********************************

c Calculation of initial shear strain F[1]
c Use of iterative algorithm to solve nonlinear algebraic
c equation developed in the function Func_t_0 below

2 1.2155351;;ESL';£;;;’;E;;EJ
C _______________________________
F(1) = 0.0001
eps = 0.001
C ____________________________________________

c Newton iterative method
c to solve above nonlinear algebraic equation
c defined in Func_t_0 defined below

C ____________________________________________
call newton_1_3_t0 (1000,F(l),eps)
C _____________________
c Output initial values
C _____________________
write (6,100) kstep,t_n,F(1)
write (1,100) kstep,t_n,F(l)
c ________________________________________________________

C ********************************************************

97 Continue tStarting point for each new
ttime or strain

224

 

00000

0

Increments

kstep = kstep + 1

ipause = ipause + 1

t_n = ipause*Dt
Setting up the value of stress
Stress field is changed, particularly, stress is set to zero
at a definite time; in this program, at exactly half the
simulation time (n_str_rem)

---------—--—------~----——’--—-——.-———-----—-—----—---——————-—

IF (ipause.ge.n_str_rem) THEN
Stress = 0.0
END IF

F(ipause+1) = F(ipause)
eps = 0.0001

.---m----—-—----————————--————-—————--——‘——---.

Newton iterative method
to solve the nonlinear algebraic equation
defined in Func_t_n_creep or Func_t_n_recovery

write (6,100) kstep,t_n,F(ipause+1)
write (1,100) kstep,t_n,F(ipause+1)

99 Continue

write (1,100) null
close (1)

100 Format (1x,16,lx,f10.5,1x,f15.10)

225

 

Stop
End
C ____________________________________________________________________
C ____________________________________________________________________
c Subroutine to calculate the initial strain
c main program calls the nonlinear algebraic equation solver newton1.f
c newtonl.f, in turn calls the subroutine defined below to get the
c nonlinear equation that needs to be solved
C ____________________________________________________________________
c ____________________________________________________________________
subroutine Func_t_0 (x,Eqn)
C ____________________________________________________________________
Implicit Double Precision (a-h,o-z)
Double Precision Lambda(12),g(12),F(1000000)
Double Precision a,m1,m2,Stress,Dt,t_n
C _____________________________________________________________________

common / DampF / a,m1,m2,pl,p2,c1,c2
common / Funcl / Stress,Dt,t_n
common / Func2 / mode,Lambda,g,F
common / Func3 / ipause

c Evaluation of relaxation modulus
c and the nonlinear equation expression

Rel_Mod = 0.0D0

DO i = 1,mode
Rel_Mod = Rel_Mod + g(i)*Lambda(i)*EXP(-1*t_n/Lambda(i))
END DO

DFl c1/(((a*((ABS(X))**m1))+1)**p1)
DF2 c2/(((a*((ABS(x))**m2))+1)**p2)
Damp_Func = DFl + DF2

Eqn = Stress — (Rel_Mod*Damp_Func*x)
Return

End

Subroutine to calculate the instantaneous strain during creep

main program calls the nonlinear algebraic equation solver newton1.f
newton1.f, in turn calls the subroutine defined below to get the
nonlinear equation that needs to be solved

0000

226

 

C ____________________________________________________________________
C ____________________________________________________________________
subroutine Func_t_n_creep (x,Eqn)
C ____________________________________________________________________
Implicit Double Precision (a-h,o-z)
Double Precision Lambda(12),g(12),F(1000000)
Double Precision a,m1,m2,Mem_Func,Stress,Dt,t_n
Double Precision A2,Inc_Strain,h_tn_tj,h_tn,A3
C —————————————————————————————————————————————————————————————————————

common / DampF / a,m1,m2,p1,p2,c1,c2
common / Funcl / Stress,Dt,t_n
common / Func2 / mode,Lambda,g,F
common / Func3 / ipause
C __________________
c Relaxation modulus
C __________________
Rel_Mod = 0.0D0
DO i = 1,mode
Rel_Mod = Rel_Mod + g(i)*Lambda(i)*EXP(—t_n/Lambda(i))
END DO
C _______________________________________________________

c Application of quadrature rule to evaluate the integral
c in order to calculate the instantaneous stretch ratio

DO j = 0,ipause

C _____________________________
c Setting up quadrature weights
C _____________________________
Remainder = j - (INT(j/2))*2
IF (j.eq.0 .OR. j.eq.ipause) THEN
W_j = 0.33333333333333
ELSE IF (Remainder.eq.0) THEN
W_j = 0.66666666666667
ELSE
W_j = 1.33333333333333
END IF
C ________________________________________
c Past Time, time for network building, t'
C ________________________________________

 

t j = j*Dt
C ______________________________________
c Memory function
C ______________________________________
Mem_Func = 0.0D0
DO i = 1,mode
Mem_Func = Mem_Func + g(i)*EXP(-(t_n — t_j)/Lambda(i))
END DO
c _______________________________
c Evaluation of other expressions
C _______________________________
Inc_Strain = x — F(j+l) !Incremental
!strain
DF1 = cl/(((a*((ABS(x - F(j+l)))**m1))+1)**pl)
DF2 = c2/(((a*((ABS(x - F(j+1)))**m2))+1)**p2)
h_tn_tj = DF1 + DF2 !h(t,t')
A2 = A2 + Dt*w_j*Mem_Func*h_tn_tj*Inc_Strain !Term
!within
lintegral

END DO

DF1 = c1/(((a*((ABS(x))**ml))+l)**pl)

DF2 = c2/(((a*((ABS(x))**m2))+1)**p2)

h_tn = DF1 + DF2 !h(t)

A3 = Rel_Mod*h_tn*x !Term
!Outside
lintegral

Eqn = Stress — A2 — A3

Return

End

C ____________________________________________________________________
C ____________________________________________________________________

c Subroutine to calculate the instantaneous strain during recovery

c main program calls the nonlinear algebraic equation solver newtonl.f
c newtonl.f, in turn calls the subroutine defined below to get the

c nonlinear equation that needs to be solved

228

 

Implicit Double Precision (a-h,o—z)

Double Precision Lambda(12),g(12),F(lOOOOOO)
Double Precision a,m1,m2,Mem_Func,Stress,Dt,t_n
Double Precision A2,h_tn_tj,Inc_Strain,h_tn,A3

common / DampF / a,m1,m2,p1,p2,c1,c2
common / Funcl / Stress,Dt,t_n
common / Func2 / mode,Lambda,g,F
common / Func3 / ipause
C __________________
c Relaxation modulus
C __________________
Rel_Mod = 0.0D0
DO 1 = 1,mode
Rel_Mod = Rel_Mod + g(i)*Lambda(i)*EXP(-t_n/Lambda(i))
END DO
C _______________________________________________________

c Application of quadrature rule to evaluate the integral
c in order to calculate the instantaneous stretch ratio

A2 = 0.0D0

C ****‘k‘k*‘k**************************************************

c Calculation of damping function
G The strain damping function h(t,t') is set to the minimum
c value attained by h(t,t') between times t'(t_j) and t(t_n)

c Damp_Func = 1/((a*((ABS(x))**m1))+1)

c Do j = 1,ipause

c h_tn_tj = 1/((a*((ABS(x - F(j+1)))**ml))+l)

c IF (h_tn_tj.lt.Damp_Func) THEN

c Damp_Func = h_tn_tj

C END IF

c END DO

C __________________________________________________________

C **********************************************************

DO j = 0,ipause

o
m
m
('7
Ff
P.
a
m
c
'U
m
9.
H
m
n
c
H
m
S
m
P.
(Q
5*
n
m

Remainder = j - (INT(j/2))*2

229

 

IF (j.eq.0 .OR. j.eq.ipause) THEN
W_j = 0.33333333333333

ELSE IF (Remainder.eq.0) THEN
W_j = 0.66666666666667

!(or decremental)

ELSE
W_j = 1.33333333333333
END IF
C ————————————————————————————————————————
c Past Time, time for network building, t'
C ————————————————————————————————————————
t j = j*Dt
C ______________________________________
c Memory function
C ______________________________________
Mem_Func = 0.0D0
DO i = 1,mode
Mem_Func = Mem_Func + g(i)*EXP(—(t_n — t_j)/Lambda(i))
END DO
c _______________________________
c Evaluation of other expressions
C ———————————————————————————————
Inc_Strain = x - F(j+l) !Incremental
!strain
C _________________________________________

DF1 = cl/(((a*((ABS(X - F(j+1)))**ml))+l)**pl)
DF2 = c2/(((a*((ABS(x — F(j+l)))**m2))+l)**p2)
Damp_Func = DF1 + DF2

A2 = A2 + Dt*W;j*Mem_Func*Damp_Func*Inc_Strain

END DO

DF1 = cl/(((a*((ABS(x))**m1))+1)**p1)
DF2 = c2/(((a*((ABS(x))**m2))+1)**p2)
h_tn = DF1 + DF2 !h(t)

A3 = Rel_Mod*h_tn*x

Eqn = Stress — A2 -A3

230

!Term
!within
!integral

!Term
!outside
!integral

Return

End

Include 'Newton—tO—1202.f'

Include 'Newton—tn—1202.f'

231

 

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