HOW DO ENTANGLED POLYMER LIQUIDS FLOW?
A Dissertation
Presented to
The Graduate Faculty at The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Sham S. Ravindranath
August, 2010
HOW DO ENTANGLED POLYMER LIQUIDS FLOW?
Sham S. Ravindranath
Dissertation
Approved:
Accepted:
_______________________________
Advisor
Dr. Shi-Qing Wang
________________________________
Department Chair
Dr. Ali Dhinojwala
_______________________________
Committee Member
Dr. Ali Dhinojwala
________________________________
Dean of the College
Dr. Stephen Z. D. Cheng
_______________________________
Committee Member
Dr. Mark D. Foster
________________________________
Dean of the Graduate School
Dr. George R. Newkome
_______________________________
Committee Member
Dr. Gary Hamed
________________________________
Date
_______________________________
Committee Member
Dr. Avraam I. Isayev
ii
ABSTRACT
Understanding the nonlinear flow behavior of entangled polymer liquids has
immense significance because of its direct relevance and usefulness in predicting
behavior of these liquids during processing. In startup shear, it is well known that shear
stress overshoot emerges when the rate of shearing is higher than the inverse of overall
chain relaxation time τ. For solutions with the number of entanglements per chain Z ≥ 25,
we have revealed using PTV that the shear field becomes inhomogeneous across the gap
after the stress overshoot. For solutions with Z ≥ 40, the shear stratification persists even
in steady state, indicating that different states of chain entanglement are possible
corresponding to the same shear stress.
The number of entanglements per chain,
Weissenberg number, the shear history, the type of solvent used, and polydispersity of the
system are some of the control parameters that can strongly affect the observed
nonlinearity.
PTV observations of step strain experiments revealed a great deal of information
about the physics of polymer flow. Contrary to the common perception that the
entanglement network would be strong enough so that it would not collapse after a large
step strain, macroscopic motions in the sample interior were observed after shear
cessation at strains ≥ 135%. The collapse of the entanglement network after shear
cessation suggested that the entanglement network is a fragile system of finite cohesive
iii
strength and cannot escape structural failure. In other words, the network disintegrates
when an elastic retractive force greater than the cohesive strength of the network is built
due to deformation. PTV technique also revealed the coexistence of multiple shear rates
under large amplitude oscillatory shear (LAOS) across the sample thickness in entangled
solutions with Z ≥ 25. In the Lissajous plots (shear stress vs. shear strain), distortion of
an ellipse begins to appear when the nonlinear velocity profile is first noticed. These
observations are again contrary to the conventional perception that the system undergoing
LAOS will experience homogeneous shear in each cycle so that material functions can be
introduced to analyze the nonlinear dependence of these functions on the amplitude and
frequency.
The above mentioned experimental observations in simple shear as well as in
uniaxial elongation of entangled polymers has helped us to recognize that three forces
(intermolecular locking force, retractive force and entanglement cohesive force) play
important roles during response of a deformed entanglement network. The new
theoretical considerations have further helped us to discover the striking scaling features
associated with stress overshoot in well entangled polymer solutions and melts.
iv
ACKNOWLEDGEMENTS
I wish to express my heartfelt gratitude to my Ph.D. advisor, Professor Shi-Qing
Wang, for giving me an opportunity in his group and for his counsel throughout the
course of this research. His emphasis on critical thinking, intuitive approach, and not
getting carried away by mundane ideas has been the guiding force in my graduate
research. Prof. Shi-Qing Wang’s style of research, giving priority to wisdom over book
knowledge, has worked well in our collaboration. His passion for the two S’s (science
and swimming) has made a lasting impression in me. Importantly, association with my
advisor has given me significant confidence and self belief in my abilities. I would
always remember the discussions we had during group meetings. Professor Shi-Qing
Wang’s body language, never short of facial expressions, would help me in retaining
these memories. I would like to thank Prof. Ali Dhinojwala, Prof. Mark D. Foster, Prof.
Gary Hamed, Prof. Darrell Reneker and Prof. Avraam I. Isayev for serving on my
committee and for their valuable suggestions. My association with the faculty members
of polymer science department has increased the courage within me to aim for excellence
in life rather than settle for mediocrity.
I would like to convey my special thanks to Dr. K. Guruswamy and Dr. Ashish
Lele from National Chemical Laboratory, India for their continued effort in my
professional upliftment. My success in graduate work is partly due to the intense training
v
that I received at National Chemical Laboratory. My thanks also goes to Ed Laughlin for
transforming scientific ideas into working concepts. I would like to thank Dr. Pouyan
Boukany and Dr. Prashant Tapadia for there help during my graduate studies. All the
members
of
Dr.
Wang’s
group
were
a
part
of
this
successful
journey.
Acknowledgements to my friends and members of Akron Cricket Club for adding
additional fun and laughter over the past 5 years.
Finally, I take this opportunity to extend my gratitude to my parents for there
support and love. Speaking of love and support, I would like to express my gratitude to
my wife for being a source of joy, encouragement and inspiration throughout. I end by
mentioning that words always fall short of expressing things of immense and deeper
significance.
vi
TABLE OF CONTENTS
Page
LIST OF TABLES ……...……………………………………………………………...viii
LIST OF FIGURES ……..………………………………………………………….........ix
CHAPTER
I. INTRODUCTION ……………………………...………………………………........1
II. STARTUP SHEAR……………………………….……………………...………....12
2.1 Introduction……………..…………….……...………………………………......12
2.2 Results and Discussion…………………………………………………………..13
2.2.1 Effect of Entanglement Level and Weissenberg Number………………….13
2.2.2 Effect of Solvent...........................................................................................29
2.3 Summary…………………………………………………………...…………….44
III. LARGE AMPLITUDE OSCILLATORY SHEAR (LAOS).….……………..........46
3.1 Introduction……………..…………….……...………………………………......46
3.2 Results and Discussions………………………………………………………….48
3.2.1 Banding Through Chain Orientation and Diffusion.....................................48
3.2.2 Banding Due to Catastrophic Yielding………….........................................54
3.2.3 Effect of Entanglement Density....................................................................57
3.2.4 Effect of Solvent and Roughness of Shearing Plate…….............................60
3.3 Summary……………….…………………………………………………….......65
vii
IV. STEP SHEAR…….……………………………………………………………......67
4.1 Introduction……………..…………….……...………………………………......67
4.2 Results and Discussion…………………………………………………………..69
4.2.1 Elastic Breakup After Step Deformation …..……………...…..…..............69
4.2.2 Effect of Solvent of Varying Molecular Weight ……...……………….......75
4.2.3 Effect of Roughness of Shearing Plates………………………………..…..82
4.3 Summary……………….…………………………………………………….......87
V. UNIVERSAL SCALING CHARACTERISTICS OF STRESS OVERSHOOT…...89
5.1 Introduction……………..…………….……...………………………………......89
5.2 Materials………………..…………….……...………………………………......90
5.3 Experimental Apparatus and Measurements.....……..………………………......92
5.4 Results and Discussions………………………………………………………….93
5.4.1 Scaling Behavior of Well-Entangled Solutions Without Interfacial Slip.....93
5.4.2 Less Universal Behavior in Weakly Entangled Solutions…………...……103
5.4.3 Strain Recovery………………..………..………........................................103
5.5 Summary….…………….……………………………………………………....107
VI. CONCLUSIONS......…………………………….. …………..…...………..…….109
BIBLIOGRAPHY…………………………………………………………………...….113
viii
LIST OF TABLES
Table
Page
2.1
Molecular characteristics of long chain 1,4-polybutadienes ………..…….…......14
2.2
Molecular characteristics of oligomeric BDs at room temperature …………......14
2.3
Molecular characteristics of entangled PBD solutions at room temperature ........15
3.1
Molecular characteristics of long chain PBDs.…..………………………..……..47
3.2
Molecular characteristics of oligomeric BDs at room temperature.......…….…...47
3.3
Molecular characteristics of entangled PBD solutions at room temperature..…...47
4.1
Molecular characteristics of long chain PBDs………………………..…..……...68
4.2
Molecular characteristics of various solvents at room temperature…..…..……...68
4.3
Properties of PBD solutions at room temperature...………………….…..……...68
5.1
Molecular characteristics of long chain PBDs……………………….…..………91
5.2
Molecular characteristics of oligomeric BDs at room temperature.......…….…...91
5.3
Molecular characteristics of entangled solutions at room temperature….………91
ix
LIST OF FIGURES
Figure
1.1
Page
Zero-shear viscosity, η0 versus molecular weight for a linear polymer.
Mc is critical molecular weight;η0 ~ Mw3-3.6 when Mw> Mc..…...................………2
1.2
Illustration of tube model depicting a polymer chain in a tube-like
confinement mimicking an entangled environment ………..……………………..4
1.3
Flow curve of entangled polymer in steady shear measured experimentally
(filled circles) and predicted by original tube model of Doi-Edwards (dashed
line). Three regime of linear (Newtonian), stress plateau (onset of the stress
maximum in DE model) and Rouse regime are evident...………………...………5
1.4
Steady state flow curve of entangled polymeric liquid exhibiting the
changes of shear stress and normal stress difference with shear rates at
which the various mechanisms are dominant. In Newtonian regime at
low rates, the reptation (moderated by CLF) dominates, until CCR causes
the shear stress to plateau when shear rate is faster than reptation. At high
rates faster than CLF mode, the chains start to stretch and shear stress
increased again. The inset shows the schematic picture of three mechanisms
of reptation, CLF and CCR..…...………………...………………………………..6
2.1
Particle tracking velocimetry (PTV) setup, where the upper rotating plate is
made of steel. Lower stationary steel plate with a hole is covered with a glass
x
cover slip, so that silver particles in the sample can be illuminated with a laser
and watched with a CCD camera placed at an angle.…...………………...……..17
2.2
Homogeneous shear
(a) Dynamic storage modulus G', loss modulus G" and complex viscosity |η*|
of 0.7M(5%)-2K from linear oscillatory shear (SAOS) at room
temperature (≈ 25 0C)………………………………………………………..18
(b) Particle-tracking velocimetric (PTV) measurements of the velocity
profiles at different times. A 35 mm parallel-disk cell was used
on Bholin-CVOR rheometer……………...…………………….………........19
2.3
Transient shear banding
(a) Linear viscoelastic measurements (SAOS) of 0.7M(10%)-9K at 25 0C.........20
(b) Velocity profiles of 1 s-1 at different times in a 35 mm parallel-plate on
Bohlin-CVOR. Inset shows the corresponding rheological data with arrows
indicating the time of the reported velocity profiles…………….…………...22
(c) Flow profiles of 3 s-1 at different times and inset shows the rheological
data of 3 s-1.………………………..………………………….…….…..........22
2.4
Shear banding in steady state and return to homogeneous shear
(a) SAOS of 1M(15%)-9K at room temperature………………………………..23
(b) Velocity profiles at different times for shear rate of 0.7 s-1 at the PTV
observation plane in a 25 mm parallel-plate geometry. Rheological data
is shown in inset……………………………………………………………..25
(c)-(d) Velocity profiles at different times for rates mentioned in the figures…...26
(e) Velocity profiles at different times for shear rates mentioned in the figures.
xi
(f) Restoration of linear velocity profile at a sufficiently high rate……………..27
2.5
Steady state shear banding in cone-plate geometry
(a)-(b) Velocity profiles at different times for shear rates of 2.3 s-1 and 5 s-1.
Cone-plate of 25 mm diameter and 50 were employed on Bohlin-CVOR
rheometer…………………………………………………………………....28
2.6
SAOS measurement at room temperature of the three entangled solution
with Z = 40. Oligomeric PBD of different molecular weights have been
used as solvent…………………………………………………………………...30
2.7
Cone/Partitioned Plate (C/PP) set up
(a) Side view of cone-partitioned plate (C/PP) set up coupled with in situ PTV
setup: (1) shaft connected to ARES motor (2) bottom cone (3) glass cover slip
(4) flexible plastic film (5) outer ring (6) trench (H x W= 1.5 mm x 1.3 mm)
(7) inner circular disk (8) shaft connected to ARES transducer
(9) micro-lens………………………………………………………………...32
(b) Top view of C/PP……………………………………………………………32
2.8
Validation of Cone/Partitioned Plate (C/PP) set up
(a)-(b) Comparison of rheological data at two shear rates between cone-plate
geometry (C/P) (open symbols) and C/PP geometry (closed symbols).
40 cone was used in both case. Inset of (b): PTV measurements of the
velocity profiles at different times corresponding to the two startup
shear experiments in C/PP…………………………...………………………35
(c) Comparison at 10 s-1 between conventional cone-plate (C/P) measurements
and those made with C/PP………………………………..…………………36
xii
(d) Linear oscillatory shear measurements of G' and G" at frequency ω =
10 rad/s before and 20 seconds after a startup shear of 10.0 s-1 in C/PP……36
2.9
Nonlinearity through significant failure at the interfaces
(a) Shear stress growth upon startup shear at four discrete shear rates beyond
the terminal flow region based on the C/PP device, along with one sheared in
the terminal region for reference……………………………………………..37
(b) PTV observations of flow profiles at different times at 1 s-1………………..38
(c) Normalized velocity profiles at four discrete rates upon startup shear……...39
2.10
Another example of shear banding
(a) Shear stress Vs. strain at five discrete shear rates…………………………...40
(b)-(c) PTV determination of the velocity profiles at six discrete shear rates,
where the open symbols represent the velocity of moving plate…………….41
2.11
Homogeneous shear
(a) Shear stress Vs. strain at five discrete shear rates beyond the terminal flow
region………………………………………………………………………...43
(b) PTV determination of the velocity profiles at three discrete shear rates
upon startup shear, revealing approximately homogeneous shear…………...43
3.1
Banding through chain orientation and diffusion
(a) Rheological response at room temperature for LAOS of strain amplitude
γo = 100% and frequency ω = 1 rad/s > ωc. The strain amplitude γo = 100%
is at the radial distance of 4 mm from the edge in a 25 mm parallel-plate
geometry..........................................................................................................49
(b) Lissajous plots (obtained by plotting stress voltage signal vs. strain voltage
signal) at different times……………………………………………………..50
(c) Velocity profiles at the instant of maximum plate speed for different times
xiii
in a 25 mm parallel-plate geometry. ………………………………………...51
(d) Velocity profiles at the instant of maximum plate speed for different
oscillating frequencies and strain amplitude γo = 100% …………………….52
(e) Velocity profiles show the variation in banding at the instants of 3/8th
and 5/8th of an oscillating cycle that have the same plate speed……………..53
3.2
Banding due to catastrophic yielding
(a) Rheological response at room temperature for LAOS of strain amplitude
γo = 400% and frequency ω = 1 rad/s > ωc. The strain amplitude γo = 400%
is at the radial distance of 4 mm from the edge. The inset shows the
Lissajous plot at 50 s…………………………………………………………55
(b) Velocity profiles at different times at the instant of maximum plate speed…55
(c) Velocity profiles at different oscillating frequencies for a strain amplitude
of 400% at the instant of maximum plate speed…………………………..…56
(d) Velocity profile at the instant of maximum plate speed at 350%
and 0.1 rad/s…………………………………………………………………56
3.3
Effect of entanglement density
(a) Velocity profiles of 1.8M(15%)-4K solution at different strain amplitudes
and frequency of 0.5 rad/s at the instant of maximum plate speed…………..58
(b) Velocity profiles for different oscillating frequencies at the instant of
maximum plate speed at a radial distance of 8 mm from the edge in 35 mm
parallel-plate disks…………………………………………………………...59
3.4
"Phase diagram" depicting the conditions under which banding occurs
through two different mechanisms, where ωτ = Z is the same condition as
ωτR = 1, γD ≈ 0.8 and γE ≈ 2.7. No experimental information is currently
available in the middle region although it is tempting to think that this
xiv
region is part of the "diffusive banding" region……………………………...…..60
3.5
Effect of solvent and roughness of shearing plate
(a) Velocity profiles at the instant of 5/8th cycle of 36th wave (74.6 s) of 5
repeat experiments based on 4 different loadings on smooth surfaces of
5.40, 25 mm cone-plate geometry. The applied strain γ is 175 % and
frequency ω is 3 rad/s………………………………………………………..61
(b) Lissajous plot of 36th wave of the 5 repeat experiments. The inset shows
the maximum stress vs. time data of the 5 repeat experiments as directly
given by the rheometer. ……………………………………………………...62
(c) Velocity profiles at the instant of 5/8th cycle of 36th wave (74.6 s) of 5 repeat
experiments of 4 different loading. By gluing sand paper both the surfaces
were roughened and PTV was done by placing the camera horizontally.
Inset shows Lissajous plots of 36th wave with the smooth and rough
surfaces………………………………………………………………………63
(d) Velocity profiles at the instant of 5/8th cycle of 25th wave (154.6 s) of 5
repeat experiments of 4 different loading…………………………………...64
(e) Velocity profiles at the instant of 5/8th cycle based on smooth surfaces. The
applied strain γ is 175 % and 300 % at frequency ω of 1 rad/s. The velocity
profile is basically linear at all other moments of the cycle. The measurement
was done on Ares rheometer where the bottom plate is the rotating one……64
4.1
Elastic breakup after step deformation
(a) Shear stress growth and relaxation during and after discrete step shearing
xv
of different magnitudes, where the Weissenberg number γ& τ is 930………...70
(b) Relaxation modulus evaluated from (a) to normalize the data with the
strain magnitude……………………………………………………………...70
(c) Damping function h from different tests at long times involving different
applied shear rates. Open symbols designate step strains where a stress
overshoot emerges during shear…………………………………………......72
(d) Direct measure, γh(γ), of the stress relaxation by normalizing the data
in (c) at long times with the equilibrium relaxation modulus Ge……………72
(e) PTV detection of macroscopic motions after discrete step strains where
the final positions of the tracked particles are given………………………....74
(f) PTV observation of macroscopic motions upon shear cessation beyond
stress overshoot……………………………………………………………....74
4.2
Effect of solvent of varying molecular weight
(a) SAOS measurements of storage, loss moduli (G', G") and the complex
viscosity |η*|…………………………………...……………………….….....75
(b) Shear stress growth and relaxation during and after discrete step shearing
of different magnitudes, where the Weissenberg number γ& τ is 280………...77
(c) Relaxation modulus evaluated from (b) to normalize the data with the
strain ………….……………………………………………………………...77
(d) Damping function h from different tests at long times. Open symbols
designate step strains where a stress overshoot emerges during shear……....78
(e) Direct measure, γh(γ), of the stress relaxation by normalizing the data
xvi
in (d) at long times with the equilibrium relaxation modulus Ge………….....78
(f) PTV detection of final positions of the tracked particles after discrete step
strains………………………………………………………………………...79
4.3
Effect of solvent of varying molecular weight
(a) Total displacement of tracer particles across the gap after cessation of step
deformation as observed through PTV. Both the surfaces are smooth……...81
(b) Relaxing shear stress vs. time for strains of 35% and 350%............................81
4.4
Effect of roughness of shearing plate
(a) Shear stress vs. time plot of 5 step shear experimental repeats from four
different loadings of 1M(10%)-1.5K solution………………………………..83
(b) Total displacement of tracer particles across the gap after cessation of
step deformation for 5 repeats as observed through PTV…………………....83
(c) Displacement of tracer particles across the gap at different times after
cessation of step deformation for loading-3 (L3)……………………………86
(d) Total displacement of tracer particles across the gap after cessation of
step deformation for 5 repeats of 4 different loadings. Both the surfaces were
roughened by gluing sand paper. Inset shows the rheological response of the
solution with smooth and rough surfaces……………………………………..86
5.1
Universal scaling – shear stress Vs. time
(a) Shear stress growth as a function of time at various applied shear
rates for the 1M(10%)-15K solution…………………………………………94
(b) Shear stress growth as a function of time at various applied shear
rates for the 1M(15%)-15K solution………………………………...………96
xvii
(c) Normalized peak stress σmax/Gc vs. normalized time for the three solution...96
5.2
Universal scaling – shear stress Vs. strain
(a) Shear stress σ Vs. strain γ in the elastic deformation regime ( γ& τR > 1)…....98
(b) Shear stress σ Vs. strain γ in the transitional regime ( γ& τR < 1)…………....98
(c) Shear stress σ vs. strain γ when γ& τR > 1 for 1M(15%)-15K…………...…..99
(d) Master curve showing a linear relationship between the normalized peak
stress σmax/Gc and strain γmax at stress maximum for the three solutions…..99
5.3
Master curve
(a) Master curve of stress growth in the elastic deformation regime, obtained
by normalizing the curves in Figure 5.2 (a) with peak values σmax and γmax
respectively……………………………………………………………..…..100
(b) Master curve of stress growth in the γ& τR < 1 regime for 1M(10%)-15K
solution……………………………………………………………………...100
(c) Master curve in γ& τR > 1 elastic regime for 1M(15%)-15K solution………101
(d) Super-master curve in the elastic deformation regime, obtained by
combining master curve data of the three solution………………………...101
5.4
Universal scaling – strain at max. stress Vs. Rouse Weissenberg number
(a) Master curve showing the strain γmax at stress maximum vs. γ& τR………...104
(b) The strain γmax at stress maximum vs. γ& τR, where four groups of data
from the literature on similar solutions were represented for comparison....104
xviii
5.5
Strain recovery
(a) Strain recovery experiments after shearing at γ& = 20 s-1 for durations
0.16, 0.23, 0.56 s respectively………………………………………………106
(b) Strain recovery experiments after shearing at γ& = 0.2 s-1 for durations
1.6, 6, 9.2, 14 s respectively………………………………………………...106
(c) Strain recovery experiments at three different rates ( γ& τR < 1). Recovery
was complete in each test because the durations of shear are all within τR..107
xix
CHAPTER I
INTRODUCTION
Liquids possessing unique microstructures are used routinely in day to day life.
They may be paints, lubricants, liquid crystal displays, toothpaste, soaps, ice cream, etc.
The microstructure of these liquids provides unique properties, making them suitable for
a particular application. These liquids, often known as complex fluids, may be subjected
to chaotic and complex flow conditions during their synthesis, processing or at the time
of product usage. It is a great challenge to relate the microstructural changes during flow
to its processing behavior and to product properties. Entangled polymer liquids are one
class of complex fluids wherein chain entanglements gives rise to its microstructure.
Annually, over a hundred billion pounds of entangled polymer liquids are shaped into
various products through different processing techniques. It is of enormous practical
importance to relate the structural changes during processing to properties of the polymer
product. In addition, during processing problems such as sharkskin, melt fracture, die
swell etc., lead to significant polymer wastage, energy loss and overall inefficiency.
Accurate understanding of deformation and flow behavior of entangled polymers will be
helpful in optimizing processing and in achieving desirable properties through processing.
Cone-plate, planar Couette and circular Couette shear cells, presumably capable
of producing uniform shear rate across the gap, have been widely used to generate startup
1
shear,1-12 and interpretation of rheometric measurements is conventionally made with the
assumption of homogeneous shear. Under shear deformation, entangled polymer liquids
have been studied under three modes of deformation: startup shear, large amplitude
oscillatory shear and step shear.
Startup Shear: Commercial polymeric products are mostly of high Mw and hence are
strongly entangled. Entanglements in linear high Mw polymeric liquids are universal and
independent of chemistry. In absence of flow (linear viscoelastic regime), the effect of
entanglement manifests itself in sluggish relaxation, slow diffusion and the strong Mw
dependence of the zero-shear viscosity, η0 when Mw exceeds a critical value, Mc as shown
in Figure 1.1
ηo
3 – 3.6
1
MC
Figure 1.1 Zero-shear viscosity, η0 versus molecular weight for a linear polymer. Mc is
critical molecular weight;η0 ~ Mw3-3.6 when Mw> Mc.
A central aim in polymer rheology was to obtain constitutive relation, the simplest
of which is true steady-state shear stress as a function of applied shear rate. Most
existing studies reported a monotonic relation between stress and shear rate,3, 6, 9-12 and
2
produced an impression that there was no shear banding for well-entangled polymers,
although it is unclear whether steady state measurements have ever been attained in these
studies.
Assuming that entangled wormlike micelles would behave like entangled polymer
chains, micellar solutions had been regarded for a long time as the only polymer like
system to possess true stress plateau.13, 14 Such a feature encouraged experimental efforts
to look for and discover shear banding in wormlike entangled micellar solutions.15-24 The
observed shear inhomogeneity is plausible due to the character that the "polymer chains"
are living, i.e., the chain connectivity is achieved through non-covalent bonding and may
be readily breakable in flow. Because of this possibility, we did not previously think that
shear banding in wormlike micellar solutions necessarily implies occurrence of
inhomogeneous shear in entangled polymer solutions. This partially explains the ten year
gap between the first report of shear banding in entangled micellar solutions15 and that in
entangled polymer solutions.25, 26 Although many theoretical studies of shear banding at a
continuum level using constitutive models27-29 were inspired by the experimental studies
of micellar solutions they did not lead to an examination of whether and why shear
banding should be observable in entangled polymer solutions and melts.
Rouse theory predicts a linear relation between η0 of a melt and polymer Mw as
shown in a limited range of low molecular weights (unentangled melts). In the 1940s, the
first transient network model was discovered by Green and Tobolosky to understand
linear mechanical properties of entangled polymer. deGennes proposed reptation idea for
snake-like wriggling chain confined in tube-like environment from Edwards’s model of
rubber network in 1971 to describe the strong molecular weight dependence of zero-shear
3
viscosity.30
This theory successfully gives a characteristic time,
τd for chain
disengagement from tube by center of mass diffusion and predicts that η0 varies as the
cube of molecular weight. Then, Doi-Edwards tube model was developed in the late
1970s to describe both linear and nonlinear viscoelastic properties of linear monodisperse
polymer melts.30 Figure 1.2 shows the schematic picture of entangled polymer chains
and tube idea in polymer melts.
Figure 1.2 Illustration of tube model depicting a polymer chain in a tube-like
confinement mimicking an entangled environment .
Further modification of the tube model through contour length fluctuation (CLF)
and constraint release (CR) has made it possible to obtain a good theoretical prediction of
linear viscoelastic properties of entangled liquids.30-35
The tube concept received
considerable attention in polymer rheology community due to simplicity of the idea and
successful prediction in the linear viscoelastic regime. The original version of the tube
model (up to 1993) actually encountered a shear stress maximum that would imply a nonmonotonic constitutive curve or flow instability for entangled polymers as shown in
Figure 1.3.
4
5
10
Stress plateau
4
σ (Pa)
10
Experimental
Linear
Doi-Edward
Rouse
3
10
γ& < 1/ τ d
2
1/ τ d <> γ& < 1/ τ e
γ& > 1/ τ e
10
10
-3
-2
10
10
-1
10
.
0
1
10
2
10
-1
ω,γ (s )
Figure 1.3 Flow curve of entangled polymer in steady shear measured experimentally
(filled circles) and predicted by original tube model of Doi-Edwards (dashed line). Three
regime of linear (Newtonian), stress plateau (onset of the stress maximum in DE model)
and Rouse regime are evident.
Since the implied instability was not observed experimentally, later theoretical
studies attempted to remedy the Tube theory and recover a monotonic constitutive
relation using the concepts of convective constrain release (CCR) and contour length
fluctuations (CLF).30-35 In the CCR model, the surrounding chains are convected away
and their topological constraints on the probe chains will be removed during this process
as shown in Figure 1.4. The probe chain is allowed to readjust its position and relax
stress faster. The chain stretching is also proposed in fast flow when the chains are
aligned and deformed in the direction of flow such that the tube stretches past the
equilibrium value. This stretched chain can snap back to equilibrium or disordered state,
after constraint release which can eliminate the instability in the DE model.32-35
5
This
well-known CCR mechanism has been incorporated into numerous tube models and the
results compared with experimental rheometric measurement.32-35 Previous experimental
observations25, 26 of shear banding highlight the inability of entangled fluids to maintain
uniform deformation when sheared on time scales faster than the terminal relaxation time.
A notable shortcoming of the previous studies is that the entangled fluid was sheared only
for a small number of strain units and "steady state" velocity profiles at various applied
shear rates were unknown. The present work is carried out to extensively probe the
transient and steady state velocity profiles of entangled solutions with varying
entanglement density and Mw of the solvent used.38, 39
Figure 1.4 Steady state flow curve of entangled polymeric liquid exhibiting the changes
of shear stress and normal stress difference with shear rates at which the various
mechanisms are dominant. In the Newtonian regime at low rates, the reptation
(moderated by CLF) dominates, until CCR causes the shear stress to plateau when shear
rate is faster than reptation. At high rates faster than CLF mode, the chains start to
stretch and shear stress increases again. The inset shows schematically the three
mechanisms of reptation, CLF and CCR.
6
Large Amplitude Oscillatory Shear:
Oscillatory shear deformation has been an
effective way to probe the complex fluids.
At small amplitudes, so small that the
measurement does not perturb the microstructure of a given sample, oscillatory shear
experiment has provided us structural fingerprints of a variety of viscoelastic materials.
Large amplitude oscillatory shear measurements have the flexibility of varying both the
frequency of oscillation and the magnitude of imposed strain with an additional
advantage over continuous shear experiment: Edge effects can be minimized.
During oscillatory shear, the complex fluid is subjected to a sinusoidal shear
strain of amplitude γ0 and frequency ω, such that the shear strain as a function of time is
γ& (t ) = γ0 sin(ωt)
(1)
and shear rate is
γ& (t) = γ0ωcos(ωt)
(2)
= γ& 0 cos(ωt)
If the response is linear, the resulting shear stress will be also sinusoidal as:
σ (t) = σ0 sin (ωt+ δ )
(3)
Large amplitude oscillatory shear (LAOS) protocol has been used extensively to
probe gels,40 emulsions,41, 42 surfactant solutions,43-49 suspensions,50-53 yield materials,54,
55
biopolymers,56-60 electrorheological and magnetorheological fluids,61-63 filled
polymers,64-67 polymer solutions and melts.68-75
In most cases, LAOS produces a non-sinusoidal stress response, i.e., distorted
sine waves for a variety of materials under examination. This means that the material in
question is no longer able to respond in a linear manner to the applied sinusoidal strain.
7
As sub-harmonic frequencies emerge, the material functions such as G' and G'' requires
re-definition.
Considerable efforts have been spent towards obtaining useful and
desirable material information from LAOS experiments. Graphical representations such
as Lissajous curves have been employed to describe the nonlinear response.
Interpretation in terms of Fourier analysis has been suggested. Notably, Wilhelm et al.
have extensively applied Fourier analysis for LAOS in different systems.76-82 Fourier
analysis has also been employed by Karis et al. and Sim et al. in their papers.83, 84 Cho et
al. have proposed a new method of interpretation based on the geometric aspect of
viscoelasticity.85 Here, the material properties in the nonlinear regime are extracted by
means of generalized storage modulus and generalized loss modulus. LAOS results have
also been compared with network model,84, 86, 87 separable BKZ model and molecular
stress function model,88, 89 in an attempt to relate the results to underlying physics. These
efforts formed a theoretical background concerning the origin of the observed nonlinear
behavior under LAOS, in hope that LAOS measurements could become fingerprints
detailing inherent material properties.
Using an effective particle tracking velocimetry technique (PTV), we probe the
flow response of entangled polybutadiene solutions under LAOS.90 In this work, we
focus on understanding the effect of varying entanglement density Z,91 molecular weight
of the solvent and the surface roughness of shearing plates under LAOS.92
Step Shear: Long flexible linear polymers coil up in a mutually interpenetrating manner
leading to chain entanglement. The entanglement interactions have been represented as
junctions of a temporary network.93 The simplest way to probe the effect of chain
entanglement and to learn about its strength is to carry out step shear of different strain
8
amplitudes, where a given strain γ is suddenly applied and maintained subsequently.
Presumably, how the shear stress relaxes after the step shear reveals the response of the
entanglement network to external deformation.
The most serious attempt to offer a theoretical depiction of entangled polymers
under external deformation is that of the Doi-Edwards tube theory.30 In this theory, a test
chain is placed in a tube to depict the entanglement constraints imposed by surrounding
chains and an instantaneous step shear produces affine deformation of the test chain.
Upon shear cessation, the deformed test chain was perceived to retract within the tube on
a time scale of Rouse time τR, leading to a kink-like characteristic drop of the predicted
shear stress.30 The remaining shear stress would relax through reptation of the test chain
out of the tube.94 Hence, at large step strains, the DE theory anticipate an entangled
polymer to exhibit two-step relaxation dynamics. The entanglement network is perceived
to remain intact both during and after shear. It is the quiescent chain relaxation dynamics
and diffusion within the constraints of entanglement that was thought to dictate the
observed stress decay in any previous theoretical perception including the DE theory.
Since the chain entanglement was expected to stay intact, it was regarded legitimate to
consider and calculate material functions such as a strain-dependant relaxation modulus
G(t,γ) = σ(t)/γ that normalizes the magnitude of the resulting shear stress σ with the
amplitude γ of the applied step shear. The DE theory found it more convenient to
evaluate a damping function h(γ) defined as
h(γ) = G(t, γ)/Ge(t), for t > τR,
(1)
where the equilibrium relaxation modulus Ge(t) is the relaxation modulus G(t, γ) obtained
for γ <<1. Since the long-time shear stress σ(t) was predicted to have the same time
9
dependence regardless of γ, h is introduced to show only the strain dependence of the
relaxation characteristics. Any drop of h with γ would reflect strain softening of the
entanglement network.
Initial experimental reports of step shear deformation dated back to early 70's.95-97
Since the advent of Doi-Edwards tube theory,98, 99 it has been routine to present the shear
stress relaxation data in terms of the damping function h. Most experiments are based on
either polystyrene100-105 or polybutadiene solutions.106-108 Summarizing all the rheological
observations of step shear over the past 3 to 4 decades, it can be said that only a fraction
of all studies109, 110 found agreement101 with the Doi-Edwards theoretical prediction for h
and were therefore regarded as normal. On the other hand, samples with sufficiently high
entanglement density typically showed excessive strain softening110,
111, 112
when
compared to the DE damping function h, while some experimental data could only
display a damping function above the DE prediction.106 In the literature, these three types
of experimental results have been classified as normal type A, kink-like type C and type
B behavior respectively.109 The type B behavior usually involved weakly entangled
solutions.
Well-entangled polymer solutions usually produce type C behavior.
For PS
solutions, there has appeared an artificial criterion involving the product of concentration
φ and molecular weight M. Paradoxically, these strongly entangled systems have weaker
finite-size effects and should supposedly fit the tube model description better. But they
show type C behavior and deviate more from the tube model prediction. To account for
the type C behavior, a free energy argument was put forward to indicate the possibility of
strain inhomogeneity in a largely deformed sample upon shear cessation.112 This idea of
10
inhomogeneous strain was further elaborated in a subsequent study within the framework
of the DE theory.102 No specific mechanism was put forward concerning the origin of
such inhomogeneous deformation.
Subsequently, based on crude particle tracking
observations, a delayed slip phenomenon was reported after shear cessation in a solution
that showed type C behavior.103 In many studies of type C behavior, failure of the LodgeMeissner relation was also reported.105
More recently, Archer and co-workers have shown that the same solution could
shift from type C to type A behavior depending on the surface condition of the cone-plate
fixture.105, 107 Also, using short PBD chains as solvent, the same group reported108 the
transition from type A to type C at higher levels of entanglements per chain (Z > 54).
These observations have reinforced the speculation109, 110 that type C behavior is perhaps
an experimental artifact, in part related to slip.
Applying PTV method we have tried to investigate the origin of various types of
stress relaxation behavior. PTV observations of step shear has allowed a first peek into
how
chain
entanglements
responds
11
to
built
up
retractive
forces.
CHAPTER II
STARTUP SHEAR
2.1 - Introduction
Tapadia and Wang, for the first time reported shear banding in startup shear
across sample thickness in entangled polybutadiene solution using a conventional coneplate shearing geometry.25 This was followed by reporting of shear banding in entangled
polybutadiene solutions in a linear Couette shear cell by Boukany and Wang.26 These
experimental observations of shear banding highlight the inability of entangled fluids to
maintain uniform deformation when sheared on time scales faster than the terminal
relaxation time. A notable shortcoming of these studies is that the entangled fluid was
sheared only for small strains and the "steady state" velocity profiles at various applied
shear rates were unknown. In section 2.2.1 of this chapter, we provide an extensive
quantification of the transient and steady state velocity profiles in startup shear of three
entangled solutions with the number Z of entanglements per chain ranging from 13 to
64.38 Velocity profiles in parallel-disk and cone-plate shear cells have been reported for a
range of shear rates in the stress plateau region. These solutions showed similar flow
characteristics in parallel-disk and cone-plate shear cells. In section 2.2.2, the important
role played by oligomeric butadiene solvent of different molecular weight is
highlighted.39 This is done by studying three entangled solutions having the same level
12
of entanglements (Z ≈ 40), but prepared using oligomeric butadiene solvents of 1.5K, 5K
and 15K molecular weight, respectively. The results from both 2.2.1 and 2.2.2 have been
summarized in section 2.3.
2.2 - Results and Discussion
The effect of number of entanglements per chain, the applied Weissenberg number
and molecular weight of the solvent on the flow behavior of entangled polymer solutions
are discussed in this section.
2.2.1 - Effect of Entanglement Level and Weissenberg Number
In this section, the transient and steady state velocity profiles are reported of three
entangled solutions with the number Z of entanglements per chain 13, 27 and 64,
respectively.38 For each solution, velocity profiles over a wide range of Weissenberg
number are shown. It is important to note that the two solutions with Z = 27 and 64 have
been prepared in an oligomeric butadiene solvent of 9K molecular weight.
Higher
molecular weight oligomeric solvents, such as 9K, are effective in greatly reducing
interfacial slip and the mechanism is discussed in greater detail in section 2.2.2.1.
2.2.1.1 - Materials
The molecular characteristics of the two high molecular weight parent PBD used
to make the three entangled solutions are listed in Table 2.1. The PBD sample labeled as
0.7M was provided through the courtesy of C. Robertson from Bridgestone-America and
1M PBD was synthesized at Dr. Roderic Quirk’s group at The University of Akron. The
13
entangled solutions were prepared using oligomeric butadiene solvents whose molecular
characteristics are listed in Table 2.2.
Table 2.1. Molecular characteristics of long chain 1,4-polybutadienes
Parent
Mn
Mw
1,4% / 1,2%
Mw/Mn
PBD
(g/mol)
(g/mol)
6
0.7M
1M
Source
0.74 x 10
1.01 x 106
addition
6
0.75 x 10
1.05 x 106
1.02
1.03
Bridgestone
Univ. of Akron
92 / 8
96 / 4
1,4-cis /
1,4-trans
56 / 36
68 / 28
Table 2.2. Molecular characteristics of oligomeric BDs at room temperature
Oligomeric
Mn
Mw
Solvent
(g/mol)
(g/mol)
1.5K
1500
-
2K
1800
-
5K
9K
15K
3800
8500
14020
4800
8900
15,000
ηs
Source
1,4-cis /
addition
1,4trans
0.7
-
-
97
15 / 45
5 / 10
1.8
10
36
92 / 8
-
56 / 36
-
(Pa.s)
Sigma-Aldrich
Cat. No. 20,0484
Sigma-Aldrich
Cat. No. 20,0433
(60% unsaturation,
Phenyl terminated)
Bridgestone
Goodyear
Bridgestone
1,4% /
1,2%
Three entangled solutions were prepared by first dissolving high molecular weight
PBD in toluene to which oligomeric butadiene was added. Silver coated silica particles
with an average diameter of 10 µm (Danted Dynamics S-HGS-10), first ultrasonicated in
toluene, were added to the polymeric solution such that the final loading of the particles
was 500-600 ppm. Most of the toluene was evaporated slowly in a fume hood over
14
several days and the remaining solvent was removed in vacuum condition until the
residue was less than 0.5 %. Since Tg of 1,4-PBD is ca. -100 oC, any residual toluene of
this level is not rheologically noticeable and thus of no consequence. The molecular
characteristics and linear viscoelastic properties of all the entangled solutions are briefly
summarized in Table 2.3, where the number of entanglements per chain, Z = (Mw/Me)φ1.2,
with Mw being the molecular weight of the high molecular weight parent PBD, Me the
entanglement molecular weight for PBD melts taken to be 1600 g/mol and φ being
weight fraction of the high molecular weight parent PBD.
From oscillatory shear
measurements at room temperature, the terminal relaxation time τ and zero-shear
viscosity η can be determined. 0.7M(5%)-2K solution refers to 5 wt % of 0.7M PBD
(listed in Table 2.1) dissolved in 95 wt % of 2K oligomeric butadiene (listed in Table 2.2).
The rest of the samples are also labeled in the same manner.
Table 2.3. Molecular characteristics of entangled PBD solutions at room temperature
Entangled
Me(φ)
τ
ηo
(kg/mol)
(s)
(Pa.s)
ηo/ ηs
lent
b
(nm)
(mm)
28
0.006
17
17
17
17
13
0.09
1.2
0.9
0.2
0.6
Z
solutions
0.7M(5%)−2K
0.7M(10%)−9K
1M(10%)-1.5K
13
27
40
57
27
25
40
16
15.6
21062
56911
49985
217
5691
71407
1M(10%)-5K
1M(10%)-15K
40
40
64
25
25
16
25
71
65
95618
2.5×105
5.1 x 105
53121
6944
51000
1M(15%)−9K
15
2.2.1.2 - Apparatuses
To determine the linear viscoelastic properties of the three entangled solutions,
small amplitude oscillatory shear (SAOS) frequency sweep measurements were
conducted using an Advanced Rheometrics Expansion System (ARES) at room
temperature (≈ 25 0C). All startup shear experiments in controlled-rate mode were
carried out on a Bohlin-CVOR rheometer at room temperature. For the 5 wt % and 10
wt % solutions, measurements were made on a 35 mm diameter parallel-disk cell and for
the 15 wt % solution 25 mm diameter parallel-disk was employed to stay within the
torque limit of the rheometer. The flow profiles of 1M(15%)-9K solution having Z = 64
was also analyzed using cone-plate geometry of cone angle θ = 5.4o and diameter 25 mm.
Focusing on determination of velocity profiles with PTV setup and being less accurate in
the rheological measurements, the meniscus was wrapped with a thin flexible film to
prevent sample loss.38
2.2.1.3 - Particle Tracking Velocimetry (PTV) Setup.
The silver particles embedded in the sample were illuminated by passing a laser
sheet with a cross-section of 0.2 mm × 2 mm across the gap at an angle ≈ 450 as shown in
Figure 2.1.37
The movement of the illuminated particles across the entire sample
thickness was captured with a black and white CCD camera (with a maximum speed of
30 frames per second) and recorded on videotape. The CCD camera was mounted with a
DIN objective lens (3.2X) through an adaptive tube (Edmund Optics: U54-868). During
analysis, the distance traveled by a particle can be determined by playing 1 to 3 frames
using MGI Videowave 4 software.
16
Ω
Glass cover slip
Objective lens
Laser
CC
D
Figure 2.1 Particle tracking velocimetry (PTV) setup, where the upper rotating plate is
made of steel. Lower stationary steel plate with a hole was covered with a glass cover
slip, so that silver particles in the sample can be illuminated with a laser and watched
with a CCD camera placed at an angle.
For solutions with concentrations of 5 % and 10 %, a 35 mm diameter paralleldisk cell was used and PTV observations were made at a radial distance of 7-8 mm from
the meniscus. For the solution of concentration 15 %, velocity profile determination was
made at a radial distance of 4 mm from the meniscus for either a 25 mm diameter
parallel-disk cell or a 25 mm diameter cone-plate cell.
In case of parallel-disk
measurements, the average shear rate refers to the location of the PTV observation plane.
All measurements were carried out at room temperature around 25 0C. It is worth
mentioning that the current PTV setup has an inherent error of ca. 10 % in its reading of
the sample gap that produces the same level of uncertainty in the determination of
velocity profiles. However, such a small imperfection has little consequences for our
measurements.
17
2.2.1.4 - Homogeneous Shear
The response of the least entangled solution 0.7M(5%)−2K (Z ≈ 13) to startup
shear was investigated first in the 35 mm parallel-disk on Bholin-CVOR rheometer.
Using high vinyl (45%) phenyl terminated oligomeric BD as the solvent, the chain
dynamics of the high molecular weight parent butadiene polymer are made sluggish. The
results of small amplitude oscillatory-shear (SAOS) frequency sweep measurements of
this solution are shown in Figure 2.2 (a). The SAOS measurements indicate a terminal
relaxation time of τ = 40 s for the long chains in the sluggish solvent. The sluggish
dynamics makes the entire stress plateau accessible rheologically and resolvable by our
PTV measurements.
10
5
G'
G"
10
10
10
5
10
4
10
3
10
2
|η*|
4
|η*| (Pa.s)
G', G" (Pa)
0.7M(5%)-2K
10
3
2
(a)
1
10
-3
10
10
-2
10
-1
10
ω (rad/s)
0
10
1
10
2
Figure 2.2 Homogeneous shear
(a) Dynamic storage modulus G', loss modulus G" and complex viscosity |η*|
of 0.7M(5%)-2K from linear oscillatory shear (SAOS) at room temperature (≈ 25 0C).
18
The velocity profiles of the entangled solution at a radial distance of 8 mm from
the edge as analyzed through the motions of the tracked particles for two different applied
shear rates are shown in Figure 2.2 (b).
400
800
300
200
100
.
γ = 0.25 s-1
Moving
plate (MP)
8s
30 s
160 s
600
Gap (µm)
0
400
0.7M(5%)-2K
.
γ = 3.5 s-1
200
0
3000
1s
10 s
40 s
(b)
2000
1000
Velocity (µm/s)
0
Stationary
plate (SP)
Figure 2.2 Homogeneous shear
(b) Particle-tracking velocimetric (PTV) measurements of the velocity profiles at
different times. A 35 mm parallel-disk cell was used on Bholin-CVOR rheometer.
It is to be noted that one of the rates, presented in Figure 2.2 (b), 3.5 s-1, is in the
end of the plateau, while the other rate, 0.25 s-1, is in the middle of the stress plateau. For
the two rates, velocity profiles before the shear stress overshoot, after the overshoot and
in steady state are presented. It can be seen from Figure 2.2 (b) that the velocity profiles
hardly deviated from linearity, and the sample experiences uniform shear at all times.
The entangled polymer solution at such low level of entanglements, responds to the fast
19
external deformation (rate faster than the terminal relaxation time) by flowing
homogeneously across the gap and apparently disentangling uniformly throughout.
Observations were similar for other shear rates in the stress plateau regime, when 2K
oligomeric BD was replaced by oligomeric BD solvents of different molecular weight
and in cone-plate geometry.38
2.2.1.5 - Transient Shear Banding.
The flow response of slightly more entangled PBD solution 0.7M(10%)−9K (Z ≈ 27)
is probed next using 35 mm parallel-disk on Bholin-CVOR rheometer. This solution is
based on the same parent polymer, 0.7M PBD, as used in the previous solution, but with
a higher parent polymer concentration of 10 wt %. Figure 2.3 (a) shows the linear
viscoelastic properties of this solution as measured using ARES at room temperature.
5
4
10
10
G', G" (Pa)
(a)
4
3
10
|η*| (Pa.s)
10
3
10
G'
G"
|η*|
2
10
0.7M(10%)-9K
2
-2
10
-1
10
0
10
ω (rad/s)
1
10
10
2
10
Figure 2.3 Transient shear banding
(a) Linear viscoelastic measurements (SAOS) of 0.7M(10%)-9K at 25 0C.
20
The rate of shear deformation at PTV observation plane was chosen to be faster
than the overall rate of long chain relaxation, so as to visualize the nonlinear flow
behavior of this solution. The velocity profiles at different instants for local shear rates of
1 s-1 and 3 s-1 respectively, as observed at a radial distance of 8 mm from the edge on
Bholin-CVOR rheometer are presented in Figures 2.3 (b) and (c). The inset of Figures
2.3 (b) and (c) shows the stress growth at rates 1 s-1 and 3 s-1, respectively, and arrows
indicate the times when the velocity profiles are determined.
It can be seen from Figure 2.3 (b) that the velocity profile is homogeneous at 2 s,
but exhibits nonlinearity at 8 s after shear stress overshoot and returns to a completely
linear profile in steady state at 34 s. For the rate 3 s-1 similar behavior can be observed as
shown in Figure 2.3 (c). The applied load of deformation is taken uniformly by the
sample in steady state as evident by the linear velocity profile, and the sample undergoes
shear thinning homogeneously across the gap. Thus, for modestly entangled polymers, it
is feasible for conventional rheological characterization to provide reliable information
about the constitutive behavior. For rates outside the range of 0.5 to 4 s-1, transient shear
banding was observed to be progressively weak. Similar flow behavior has been found
when 9K oligomeric BD was replaced by oligomeric BD solvents of different molecular
weight.38
21
920
(MP)
0.7M(10%)-9K
.
800
γ = 1 s-1
2s
8s
34 s
400
10
4
σ (Pa)
Gap (µm)
600
200 1000
0.1
1
0
1200
10 100
time (s)
(b)
(SP)
800
400
Velocity (µm/s)
0
920
(MP)
0.7M(10%)-9K
.
γ = 3 s-1
0.75 s
10 s
100 s
600
400
200
104
σ (Pa)
Gap (µm)
800
.
γ = 3 s -1
100
0.01 0.1
0
3000
1 10 100
time (s)
(c)
2000
1000
Velocity (µm/s)
(SP)
0
Figure 2.3 Transient shear banding
(b) Velocity profiles of 1 s-1 at different times in a 35 mm parallel-plate on BohlinCVOR. Inset shows the corresponding rheological data with arrows indicating the
time of the reported velocity profiles.
(c) Flow profiles of 3 s-1 at different times and inset shows the rheological data of 3 s-1.
22
2.2.1.7 - Shear Banding in Steady State
Finally, the flow behavior of much more entangled solution, 1M(15%)−9K (Z ≈ 64)
was studied.38 The solution was prepared by dissolving 15 wt% of 1 million molecular
weight PBD in 85 wt% of 9K oligomeric BD. The Linear viscoelastic properties at room
temperature are shown in Figure 2.4 (a), which indicates that the terminal relaxation time
is τ = 65 s.
10
5
10
6
10
5
10
4
10
3
10
2
10
10
10
4
3
G'
G"
|η*|
2
10
-3
10
-2
10
|η*| (Pa.s)
G', G" (Pa)
1M(15%)-9K
(a)
-1
10
ω (rad/s)
0
10
1
10
2
Figure 2.4 Shear banding in steady state and return to homogeneous shear.
(a) SAOS of 1M(15%)-9K at room temperature.
Owing to a higher parent polymer concentration, the 25 mm parallel-disk was
used to make PTV measurements and the location of observation plane was 4 mm from
the edge. In Figures 2.4 (b) to (f), transient and steady state velocity profiles at various
shear rates in the stress plateau region are shown.
23
The profiles just before stress
overshoot, immediately after the overshoot and in steady state are represented
respectively by closed circles, closed diamonds and closed squares.
Open triangles
indicate approximate attainment of steady state.
For all the shear rates indicated in Figures 2.4 (b) to (e), it can be observed that
the flow profile just before the stress overshoot, as represented by the closed circles, was
homogeneous and immediately after the stress overshoot, a highly nonlinear profile
develops as shown by the closed diamonds in the plots.
It can be seen that
inhomogeneity happens by occurrence of elastic recoil-like motions within the bulk of the
sample.
Buildup of significant retraction forces within the chains has caused the
inhomogeneous failure of the entanglement network and from Figures 2.4 (b) to (e), it
can be observed that for various rates the location of the fault plane was different. It is
also important to mention that the location where the high shear region emerged in the
gap was different for a given rate among the different repeats.
External deformation implemented at sufficiently high rates can cause a wellentangled polymer (that is an elastic solid on short time scales due to chain entanglement)
to undergo cohesive structural breakup.38 The inhomogeneously disintegrated “solid” is
the starting point for the subsequent flow. Naturally, the flow after the yield point could
be inhomogeneous. Indeed, the present study confirms previous reports25, 26 that transient
shear banding takes place in well-entangled polymer solutions due to the sudden startup
shear. What is surprising is that the initial shear inhomogeneity does not adjust back to a
state of uniformity even after hundreds of strain units. After the initial elastic failure, as
the shear stress drops toward a steady value the sample adjusts its velocity profile over
many strain units to settle down to a steady state, represented by closed squares in
24
Figures 2.4 (b) to (e). It is obvious from Figures 2.4 (b) to (e) that inhomogeneity in flow
persists even after several hundreds of strain units, and the sample essentially divides
itself into high-shear and low-shear regions. Comparison between closed squares and
open triangles indicates that in steady state the velocity profile hardly varies over
additional 75 strain units. The PTV observations using cone-plate geometry of 25 mm
diameter and 5.4° cone angle have been reported in Figure 2.5 (a) and (b). Figure 2.5 (a)
and (b) shows shear banding after stress overshoot for two applied rates of 2.3 and 5 s-1,
respectively.38
920
γ = 0.7 s
-1
2.8 s
14 s
428 s
646 s
600
1M(15%)-9K
400
10
200
4
σ (Pa)
Gap (µm)
800
(MP)
.
1000
0.1
0
1000
800
1
(b)
10 100 1000
time (s)
600 400 200
Velocity (µm/s)
0
(SP)
-200
Figure 2.4 Shear banding in steady state and return to homogeneous shear.
(b) Velocity profiles at different times for shear rate of 0.7 s-1 at the PTV observation
plane in a 25 mm parallel-plate geometry. Rheological data is shown in inset.
25
920
-1
γ = 1.3 s
1.5 s
3s
192 s
292 s
600
1M(15%)-9K
400
σ (Pa)
Gap (µm)
800
(MP)
.
200
10
4
1000
200
0.1
0
1
1500
10 100
time (s)
(c)
(SP)
1000
500
Velocity (µm/s)
0
980
(MP)
1M(15%)-9K
.
γ = 2.2 s-1
Gap (µm)
800
600
400
0.5 s
3.5 s
150 s
175 s
200
0
(d)
(SP)
2000
1500
1000
500
Velocity (µm/s)
0
Figure 2.4 Shear banding in steady state and return to homogeneous shear.
(c)-(d) Velocity profiles at different times for rates mentioned in the figures.
26
920
(MP)
800
Gap (µm)
600
400 1M(15%)-9K
.
-1
γ = 4.5 s
0.4 s
2.5 s
66 s
100 s
200
(e)
0
5000 4000 3000 2000 1000
Velocity (µm/s)
0
(SP)
-1000
920
(MP)
1M(15%)-9K
.
γ = 6 s-1
0.5 s
7s
41 s
80 s
600
105
400
σ (Pa)
Gap (µm)
800
10
4
200
1000
0
6000
0.1
1
10
time (s)
100
(f)
4000
2000
Velocity (µm/s)
(SP)
0
Figure 2.4 Shear banding in steady state and return to homogeneous shear.
(e) Velocity profiles at different times for shear rate mentioned in the figure.
(f) Restoration of linear velocity profile at a sufficiently high rate.
27
776
σ (Pa)
(MP)
4
10
600
Gap (µm)
1000
0.1
1 10 100
time (s)
400
.
γ = 2.3 s-1
200
0
1.3 s
5.3 s
o
144 s CP/5 /25 mm
193 s 1M(15%)-9K
(a)
(SP)
1600
1200
800
400
Velocity (µm/s)
0
800
(MP)
Gap (µm)
600
400
1M(15%)-9K
o
CP/5 /25 mm
.
γ = 5 s-1
200
0
5000
0.4 s
1.6 s
60 s
80 s
(b)
(SP)
4000
3000 2000
Velocity (µm/s)
1000
0
Figure 2.5 Steady state shear banding in cone-plate geometry.
(a)-(b) Velocity profiles at different times for shear rates of 2.3 s-1 and 5 s-1. Cone-plate
of 25 mm diameter and 50 were employed on Bohlin-CVOR rheometer.
28
2.2.1.7 - Homogeneous Shear Upon Sufficient Disentanglement at High Rates
As shown in Figure 2.4 (f), at a high applied rate of γ& h ~ 6 s-1, the sample shows a
tendency to return to homogeneous shear.38 We can estimate that the level of remaining
chain entanglement at γ& h by assuming that the corresponding relaxation time
τh = 1/ γ& h .
Thus, τ/τh ∼ (Ζ/Ζh)3.4, i.e., Zh = Z/( γ& h τ)1/3.4 ~ 7, which is a reasonable number since a
solution with Z ~ 13 does not suffer any inhomogeneous shear as shown in the preceding
subsection 2.2.1.5. Zh is the number of entanglements left at the applied shear rate of γ& h .
2.2.2 - Effect of Solvent
In this section, the effect of oligomeric butadiene solvent of varying molecular
weight has been probed.39 Three entangled solutions having the same number Z of
entanglements per chain (Z ≈ 40) in oligomeric butadiene solvent of molecular weights
1.5K, 5K and 15K respectively have been probed. 10 wt% concentration of 1 million
molecular weight PBD has been used in all three solutions. Solution preparation method
has been described in section 2.2.1.1. The molecular characteristics at room temperature
of the 1 million parent PBD, the three oligomeric BD solvents and the three entangled
solutions have been reported in Tables 2.1, 2.2 and 2.3, respectively.
The linear
viscoelastic measurement of the three similar entangled solutions has been presented in
Figure 2.6.
29
4
G', G" (Pa)
10
3
10
G'
1M(10%)-15K
G"
G' 1M(10%)-5K
G"
2
10
G' 1M(10%)-1.5K
G"
-3
10
-2
10
10
-1
0
10
ω (rad/s)
10
1
10
2
Figure 2.6 SAOS measurement at room temperature of the three entangled solution with
Z = 40. Oligomeric PBDs of different molecular weights have been used as solvent.
2.2.2.1 - Wall Slip Due to Lack of Entanglement at Sample/Wall Interface.
A Slip boundary condition pertains for the polymer flow in the presence of
containing walls when entanglement interactions are lost at the polymer/wall interface
through either chain desorption or disentanglement.
Lack of interfacial chain
entanglement leads to a great disparity between the bulk viscosity η that is high due to
chain entanglement and the interfacial viscosity ηi. The most effective way to describe
the consequence of a huge ratio of η/ηi is to introduce the Navier-de Gennes
extrapolation length b,114, 115
b = (η/ηi)a,
(1)
where thickness a of the interfacial layer free of chain entanglement can be theoretically
30
argued to be comparable to the entanglement spacing lent. For solutions, ηi can be as low
as the solvent viscosity ηs, and the entanglement spacing lent would grow upon dilution
according to
lent(φ) = lent(φ=1)φ-2/3,
(2)
that has been established both theoretically and experimentally116-119 where φ is the
polymer weight or volume fraction. The modification of a shear flow due to slip is
measured in terms of b/H (H is the gap across the shearing plates). Wall slip is a leading
form of inhomogeneous shear in polymer flow.
Interestingly, unlike the case of melts, the magnitude of wall slip for solutions can
be effectively altered by selecting different values for ηi in eq 1. In other words, a
solution does not need to be as entangled as a melt to suffer more wall slip when a
solvent of water-like viscosity is used to prepare the solution.
Conversely, the effect of wall slip can also be greatly minimized for solutions by
using a polymeric solvent of molecular weight M1 with the same chemical structure as
the "parent" polymer of molecular weight M so that the viscosity ratio upon interfacial
chain disentanglement by shear is
η/ηi = (M/M1)3.4,
(3)
which can be a significantly reduced value characteristic for melts when M1>Me
2.2.2.2 - Cone-Partitioned Plate Setup With PTV
All experiments were performed at room temperature using a TA Instruments
Advanced Rheometric Expansion System (ARES) equipped with a cone-partitioned plate
(C/PP) assembly as sketched in Figure 2.7 (a)-(b),39 as well as with a conventional cone31
plate (C/P) apparatus that was for the purpose of comparison with, validation and
calibration of C/PP. The bottom rotating cone has a 4o angle and radius of either 20 mm
or 15 mm. On the top, an inner circular disk was linked to the torque transducer of ARES
and the meniscus was formed by the bottom cone and an outer ring on the top that was
detached to the inner circular disk on the top. The gap between the inner circular disk
and the outer ring was around 0.2 mm. The top outer ring was held fixed. For the bottom
cone of R2 = 20 mm, the upper stationary surface was made of an inner circular disk of
R1 = 12.5 mm and ring of outer radius equal to 20 mm. Once the sample was loaded, the
trench (H x W = 1.5 mm x 1.3 mm) on the top was filled with 1M(10%)-15K solution to
prevent any sample leaking through the 0.2 mm gap. Since there was absolutely no
motion between the central disk and the outer ring on the upper stationary side of the
shear cell, the transducer (connected to the central disk) cannot be affected by the fixed
ring.
2R1
(7)
(6)
Laser
D
CC
(9)
(8)
(5)
(4)
(3)
(1)
(2)
2R2
Side view (a)
2R1
inner circular
disk
clearance
optical
window
outer ring
Top view of
top plate (b)
Figure 2.7 Cone-partitioned plate (C/PP) set up.
(a) Side view of C/PP set up coupled with in situ PTV setup: (1) shaft connected to
ARES motor (2) bottom cone (3) glass cover slip (4) flexible plastic film (5) outer
ring (6) trench (H x W= 1.5 mm x 1.3 mm) (7) inner circular disk (8) shaft connected
to ARES transducer (9) micro-lens.
(b) Top view of C/PP.
32
Typically, a flexible plastic film was wrapped around the outer meniscus by
gluing the film to the top stationary ring. Hence, the film remains stationary when the
bottom cone rotates. It is interesting to note that dielectric spectrometers employ a
similar strategy to eliminate the edge effect and to achieve uniform electric fields.
For PTV measurements, the laser sheet was passed at an angle of ca. 45o through a
glass window of 3 mm radius on the top stationary plate as shown in Figure 2.7 (a). To
eliminate reflections from the bottom cone, its surface was coated with a thin layer of
black enamel (Rust-oleum Corporation, specialty high heat black enamel). The PTV
observation point was around 9-10 mm from the center. The performance of PTV was
evaluated by comparison between the prescribed homogeneous shear field due to the
rotating cone and that determined from PTV for a Newtonian fluid. The current PTV
scheme of observation has an inherent error of ca. 10% in its reading of the sample gap,
resulting in a similar level of error in the velocity profiles.
2.2.2.3 - Validation of Cone-Partitioned Plate (C/PP) Setup
In this section, we take several steps to evaluate the performance of the C/PP
setup and contrast results obtained with C/PP with those with conventional C/P.39 First,
we use two materials that do not suffer edge fracture to show that the C/PP works as well
as the conventional C/P.
Figure 2.8 (a) illustrates the shear stress growth of a monodisperse 1,4polybutadiene melt of molecular weight equal to 44 kg/mol. The applied shear rates are
in the Newtonian region of this melt and comparison was made possible because the
sample does not suffer edge fracture at these shear rates of 1 and 10 s-1. Similarly, a
33
moderately entangled polymer solution such as 0.7M(5%)-4K having roughly Z = 13
entanglements per chain suffers negligible edge fracture although it exhibits a
considerable level of viscoelasticity.
Figure 2.8 (b) shows the stress growth upon startup shear at two respective
apparent shear rates of 1.0 and 5.0 s-1 in the non-Newtonian (stress plateau) region, where
linear velocity profiles were obtained from PTV observations in the C/PP as shown in the
inset. The C/P and C/PP measurements were performed on the same ARES rotational
rheometer using a 40 mm, 40 cone at the bottom and on the top a 25 mm diameter inner
plate. An outer ring on the top was used to form the meniscus. The overlapping of data
from the C/P fixture with those from the C/PP device serves to calibrate between the C/P
and C/PP measurements and thus validates rheological measurements obtained with the
C/PP device.39
The C/PP setup clearly shows its superiority over the conventional C/P fixture,
when edge fracture has an accumulating effect as in the case of a strongly entangled
solution. This is illustrated in Figure 2.8 (c), where the decline of the shear stress in C/P
arises from the decreasing contact between the sample and the fixture due to edge
fracture. Finally, we are interested to find out whether the sample in the C/PP setup is
still intact after shearing for a high number of strain units. Figure 2.8 (d) examines this
question by performing oscillatory shear measurements before and after the steady shear
experiment involving a shear rate of 10 s-1. Clearly, sound measurements can be made
even after hundreds of strain units, free of either edge instability around the meniscus that
is film-wrapped or at the gap between the disk and ring that is filled up by excess sample.
34
10
5
σ (Pa)
1, 4 PBD melt (Mw = 44 kg/mol)
10 s
10
-1
4
1s
1000
100
1000
} cone-plate (C/P)
} cone-partitioned
plate (C/PP)
0.01
5s
0.1
time (s)
10
0.7M(5%)-4K
1s
σ (Pa)
(a)
1
-1
-1
780
.
Gap (µm)
.
γ = 1 s-1 γ = 5 s-1
1s
6s
34 s
600
100
-1
0.1 s
3s
11 s
400
200
10
(b)
0
0.1
1
time (s)
4000 3000 2000 1000
Velocity ( µ m/s)
10
0
50
Figure 2.8 Validation of cone-partitioned plate (C/PP) setup.
(a)-(b) Comparison of rheological data at two shear rates between cone-plate geometry
(C/P) (open symbols) and C/PP geometry (closed symbols). 40 cone was used in
both case. Inset of (b): PTV measurements of the velocity profiles at different
times corresponding to the two startup shear experiments in C/PP.
35
4
σ (Pa)
10
10
3
Onset of
edge fracture
in C/P
.
C/P
C/PP
γ = 10 s-1
(c)
1M(10%)-1.5K
2
10 -2
10
4
10
3
-1
0
10
time (s)
10
σ, G', G" (Pa)
10
10
10
2
ω = 10 rad/s
.
γ
(d)
1M(10%)-1.5K
-1
app
- 10 s
G' - before
G" - before
G' - after
G" - after
C/PP
10
1
2
10
-1
10
0
10
time (s)
1
10
2
400
Figure 2.8 Validation of cone-partitioned plate (C/PP) set up.
(c) Comparison at 10 s-1 between conventional cone-plate (C/P) measurements and those
made with C/PP.
(d) Linear oscillatory shear measurements of G' and G" at frequency ω = 10 rad/s before
and 20 seconds after a startup shear of 10.0 s-1 in C/PP.
36
2.2.2.4 - 1M(10%)-1.5K Solution: Nonlinearity Through Failure at Interface
The shear stress growth upon startup shear at five rates, four in the stress plateau
region and one in the terminal region, have been presented in Figure 2.9 (a). As indicated
in rheological data in section 2.2.1.6, the sample takes many strain units to settle down to
a steady state defined by the unchanging value of the shear stress. At reported rates of 50
s-1 and 10 s-1, the stress undershoot was clearly noticeable beyond the overshoot.
1M(10%)-1.5K
4
.
γapp (s-1)
0.04
0.1
1
10
50
σ (Pa)
10
1000
(a)
600
0.01
0.1
1
10
time (s)
100
1000
Figure 2.9 Nonlinearity through significant failure at the interfaces.
(a) Shear stress growth upon startup shear at four discrete shear rates beyond the
terminal flow region based on the C/PP device, along with one sheared in the
terminal region for reference.
Figure 2.9 (b) shows the velocity profiles at different times during a startup shear
of an apparent rate of 1 s-1. The initial flow field was linear up to the stress overshoot at t
~ 2.0 s. After the overshoot, recoil like motion occurred near the lower sample/cone
interface as shown by the squares. Finally, the solution settled to highly inhomogeneous
37
shearing where much of the bulk experienced a low shear rate and there is an
immeasurably thin layer of high shear near the moving bottom surface.39
800
Stationary
plate (SP)
1M(10%)-1.5K
.
γapp = 1 s-1
Gap (µm)
600
0.15 s
5.5 s
47 s
174 s
400
200
(b)
0
800
600
400
200
Velocity (µm/s)
0
Moving
plate (MP)
-200
Figure 2.9 Nonlinearity through significant failure at the interfaces.
(b) PTV observations of flow profiles at different times at 1 s-1.
However, interfacial slip, i.e., disentanglement of one monolayer at the interface,
is insufficient to produce such a velocity profile. To appreciate this point, we need to
estimate the slip length b of this solution. We can evaluate the upper bound of b
according to bmax = (η/ηs)lent where the entanglement spacing lent ~ 20 nm, η can be read
to be ca. 25 kPa.s from SAOS as |G*|/ω at |G*|=2000 Pa and ηs is taken as the solvent
viscosity of 0.7 Pa.s from Table 2.3. With these values we calculate bmax ~ 0.7 mm. This
level of wall slip can only accommodate the imposed rate up to V/H = γ& c (1 + 2bmax/H) ~
0.1 s-1 << 1 s-1 where γ& c is around 1/τ = 0.06 s-1 and H is taken to be around 1 mm. Thus,
38
at this apparent shear rate of 1 s-1, for the bulk of the sample to undergo shear at a level of
γ& c , chain disentanglement must occur in the sample interior as well. In other words, the
rapidly sheared layer must have a thickness much greater than that of a monolayer
although our PTV technique does not have adequate spatial resolution to determine its
thickness. Figure 2.9 (c) summarizes steady state PTV measurements at several applied
shear rates. Clearly, the steady state profiles are nonlinear at these rates in the stress
plateau region.
(SP)
800
(c)
1M(10%)-1.5K
-1
1 s , 174 s
-1
600
2 s , 129 s
Gap (µm)
-1
4 s , 83 s
-1
7 s , 75 s
400
200
0
(MP)
1
0.8
0.6
0.4
0.2
0
V/Vmax
Figure 2.9 Nonlinearity through significant failure at the interfaces.
(c) Normalized velocity profiles at four discrete rates upon startup shear.
2.2.2.5 - 1M(10%)-5K Solution: Another Example of Shear Banding
In the case of the 1M(10%)-1.5K solution, inhomogeneous shear occurs in the
appearance of apparent slip. Appreciable bulk shear banding was only observed at higher
39
rates as shown Figure 2.9 (c). We have shown that the slip length b can be significantly
reduced by choosing an oBD of higher molecular weight.37 As a consequence, shear
banding can be more readily seen in the sample interior. Hence in this section we choose
to observe the flow bevaior of an equally entangled solution made with oBD solvent of
Mw – 5000 g/mol. Before reporting on the PTV observations, we first apply C/PP to
obtain steady state measurements of the shear stress from the startup experiments
reported in Figure 2.10 (a).
4
1M(10%)-5K
σ (Pa)
10
1000
.
γapp (s-1)
0.02
0.04
0.2
(a)
300
0.1
1
10
100
1
10
1000
γ
Figure 2.10 Another example of shear banding
(a) Shear stress Vs. strain at five discrete shear rates.
Our PTV observations reveal that the velocity profiles are sharply nonlinear even
in the steady state as shown in Figures 2.10 (b)-(c) and the tendency of the solution to slip
can be greatly reduced. The local shear rates in the fast sheared layers can be higher than
those in the slowly sheared bulk by a factor of over 10.
40
800
(SP)
-1
0.1 s , 2047 s
-1
0.2 s , 688 s
-1
0.4 s , 441 s
Gap (µm)
600
400
200
(b)
0
400
800
1M(10%)-5K
(MP)
300
200
100
Velocity (µm/s)
0
(SP)
-1
1 s , 200 s 1M(10%)-5K
-1
2 s , 146 s
-1
Gap (µm)
600
4 s , 96 s
400
200
(c)
0
3500
(MP)
2800
2100 1400
700
Velocity (µm/s)
0
Figure 2.10 Another example of shear banding
(b)-(c) PTV determination of the velocity profiles at six discrete shear rates, where the
open symbols represent the velocity of moving plate.
41
2.2.2.6 - 1M(10%)-15K Solution: A Case of Homogeneous Shear
We completely suppress interfacial wall slip by further increasing the molecular
weight of oBD solvent to 15 kg/mol, in the 1M(10%)-15K solution. Figure 2.11 (a)
shows the shear stress growth during startup shear.
Remarkably, the C/PP allows
hundreds of strain units to elapse without any difficulty before reaching steady state, as
shown in Figure 2.11 (a).39
The PTV observations reveal linear velocity profiles at all times, contrary to the
responses of the two preceding solutions of the same level of chain entanglement. Figure
2.11 (b) indicates that the velocity profile is basically linear both right after the shear
stress overshoot and after hundreds of strain units for three discretely applied rates. To
understand why this sample is capable of undergoing homogeneous shear upon startup
shear, we must ask why the other samples with the same number of entanglements per
chain (Z = 40) suffer shear banding. When sheared at a high Weissenberg number in a
startup experiment, a well-entangled solution initially undergoes elastic deformation like
a solid until the point of yield, i.e., the point of cohesive failure at the stress maximum.
This cohesive breakdown inevitably takes place through chain disentanglement, which is
a molecular event that eventually nucleates to produce a macroscopic fault plane where
the entanglement network locally breaks down. For the 1M(10%)-1.5K solution, since
the slip length b is as large as ca. 1 mm, any initial failure produced by one monolayer of
chain disentanglement yields a sufficient correction to the boundary-imposed shear field,
allowing the rest of the sample to recoil and avoid high shear.
42
.
γ (s-1)
(a)
0.05
1
2
5
13
σ (Pa)
10
4
1000
1M(10%)-15K
400
0.1
1
10
100
1000
γ
800
(SP)
1M(10%)-15K
(b)
-1
1 s , 510 s
-1
2 s , 253 s
Gap (µm)
600
-1
2 s , 10 s
-1
0.2 s , 1231 s
400
200
0
2000
(MP)
1500
1000
500
Velocity (µm/s)
0
Figure 2.11 Homogeneous shear
(a) Shear stress Vs. strain at five discrete shear rates beyond the terminal flow region.
(b) PTV determination of the velocity profiles at three discrete shear rates upon startup
shear, revealing approximately homogeneous shear.
43
Because of the high MW solvent, the ability of 1M(10%)-15K to undergo
interfacial wall slip is reduced by a factor that can estimated by the solvent viscosity ratio
equal to 36/0.7 ~ 51 (cf. Table II). A monolayer of chain disentanglement, at the
interface or in the bulk, would produce little correction to the imposed initially
homogeneous shear field. As a consequence, every other molecular layer continues to
undergo the same rate of deformation that causes the first monolayer to disentangle. In
other words, this sample is capable of achieving uniform chain disentanglement
everywhere across the sample thickness. This essentially ensures that the shear field
remains homogeneous during chain disentanglement.39
2.3 - Summary
In section 2.2.1, three entangled polybutadiene solutions {0.7M(5%)-2K,
0.7M(10%)-9K and 1M(15%)9K} of varying number of entanglements per chain 13, 24
and 64 was subjected to various apparent rates of simple shear in standard rotational
rheometry to explore their nonlinear flow behavior in terms of the corresponding velocity
profiles. Through particle tracking velocimetric technique, it was found that the least
entangled solution with Z ≈ 13 exhibited homogeneous shear at all times. The solution
with Z ≈ 27 transiently developed shear banding after the shear stress maximum before
returning to shear homogeneity in steady state.
For solutions with Z ≈ 64, shear
inhomogeneity emerges after stress overshoot. Far beyond the terminal flow region, coexistence of low and high shear-rate states survives even after hundreds of strain units of
shearing, indicating that different states of chain entanglement are possible corresponding
to the same shear stress. At sufficiently high rates, the velocity profile approximately
44
recovers linearity in steady state, where entangled solutions become sufficiently
disentangled. The present collection of PTV observations systematically examined how
shear inhomogeneity occurs and when linear velocity variation reemerges as a function of
the level of chain entanglement and the intensity of the imposed shear flow.38
Further in section 2.2.2, three entangled polybutadiene solutions {1M(10%)-1.5K,
1M(10%)-5K, 1M(10%)-15K} having the same level of entanglements but oBD solvents
of different molecular weight was studied by employing a critically improved rheometric
setup and particle-tracking velocimetry (PTV).
We have successfully overcome an
inherent limitation in traditional rheometric measurements that has previously made it
impossible to elucidate without ambiguity the nonlinear flow behavior of well-entangled
polymer solutions in continuous shear. The cone-partitioned plate setup allows us to
achieve true steady state. Startup shear appear to produce inhomogeneous shear during
transient and steady shear in 1M(10%)-1.5K and 1M(10%)-5K solutions. A comparison
between 1M(10%)-15K and 1M(10%)-1.5K solutions shows a striking difference in the
velocity profiles although both have the same level of chain entanglement. The high
molecular-weight solvent of PBD-15K greatly altered chain disentanglement allowing the
system to respond to the suddenly imposed shear in a uniform and homogenous manner.39
45
CHAPTER III
LARGE AMPLITUDE OSCILLATORY SHEAR (LAOS)
3.1 - Introduction
Observation of shear banding in startup shear of well entangled polymer
solutions,25, 26, 38, 39 naturally raises the curiosity to probe their nonlinear flow behavior in
large amplitude oscillatory shear (LAOS).
As stated in the first chapter, LAOS is
extensively used to probe the response of complex fluids to nonlinear deformation. In
subsections 3.2.1, 3.2.2, 3.2.3 and 3.2.4 of this chapter, we report investigation of LAOS
behavior of entangled polybutadiene (PBD) solutions.90-92 In section 3.2.1 and 3.2.2, the
important role of applied strain in LAOS has been highlighted in the case of well
entangled 1M(15%)-9K solution, having 64 entanglements per chain.91 In section 3.2.3,
the behavior of entangled solutions of varying entanglement density has been probed.91
In section 3.2.4, the important effect of oligomeric BD solvents of varying molecular
weight and the roughness of shearing plate in reducing interfacial failure has been
studied.92 Finally, the results of these four subsections have been summarized in section
3.3. The entangled polymer solutions were prepared according to the method described
in section 2.2.1.1 and the PTV setup is similar to that employed in section 2.2.1.4. The
molecular characteristics of the parent PBD, oligomeric butadiene and the entangled
solutions have been presented in Table 3.1, 3.2 and 3.3, respectively. All the rheological
46
measurements were made on Bohlin-CVOR rheometer at room temperature except that
of Figure 3.5 (e), for which the ARES was employed.
Table 3.1. Molecular characteristics of long chain PBDs
Parent
PBD
700K
1.0M
1.8M
Mn
Mw
(g/mol)
0.74×106
1.014×106
1.56×106
Mw/Mn
Source
1.02
1.03
1.19
Bridgestone
University of Akron
Goodyear
(g/mol)
0.75×106
1.052×106
1.86×106
Table 3.2. Molecular characteristics of oligomeric BDs at room temperature
Oligomeric
Mn
Mw
BD
(g/mol)
(g/mol)
1.5K
1500
-
-
2K
1800
-
-
4K
9K
10K
3500
8500
8900
3800
8900
10500
1.08
1.04
1.18
46K
45000
46000
1.02
ηs
Mw/Mn
Source
(Pa.s)
Sigma-Aldrich
Cat. No. 20,0484
Sigma-Aldrich
Cat. No. 20,043-3
Goodyear
Goodyear
Bridgestone
Goodyear
0.7
97
1.2
10
14
2600
Table 3.3. Molecular characteristics of entangled PBD solutions at room temperature
Entangled
solutions
Z
Me(φ)
τ
ηo
(kg/mol)
(s)
(Pa.s)
ηo/ ηs
lent
b
(mm)
0.35
0.7M(10%)−4K
27
27
10
25129
20940
(nm)
17
1M(10%)-1.5K
40
25
17
50000
71428
17
1.2
1M(10%)-10K
40
25
50
1.7 x 105
10714
17
0.2
1M(10%)-46K
40
25
75
3.9 x 105
150
17
0.002
1M(15%)−9K
64
16
65
5.1 x 105
51000
13
0.6
1.8M(15%)−4K
119
16
100
-
-
-
-
47
3.2 - Results and Discussion
In this section, the effect of applied strain, entanglement density, molecular
weight of solvent and roughness of shearing plate on the flow behavior of entangled
polymer solutions will be discussed
3.2.1 - Banding Through Chain Orientation and Diffusion
In the subsections 3.2.1 and 3.2.2, we will focus on exploring the various aspects
of LAOS of 1M(15%)-9K entangled solution on Bohlin-CVOR rheometer. So unless
specified, all figures in these two subsections are results of our study on 1M(15%)-9K
solution.
A 25 mm parallel-plate was employed in both the sections and PTV
measurements were made 4 mm from the edge. The specified strain was what the sample
experiences at the PTV observation plane. The results in cone-plate geometry are similar
to those in parallel-plate geometry and hence only the results from parallel plate
geometry have been reported. For a given oscillating frequency ω and strain amplitude
γo, the entangled network would undergo a maximum shear rate of γ& = γoω. To probe
the nonlinear behavior of entangled solutions, frequencies ω greater than crossover
frequency ωc were employed in all of our LAOS experiments. At frequencies ω > ωc, the
rate of shear deformation was faster than the overall rate of chain relaxation, ensuring
that the chains orient in the flow direction.
The 1M(15%)-9K entangled solution showed linear response under oscillatory
shear deformation of γo = 70%, over the range of frequencies tested, from 0.1 to 16 rad/s
i.e. Lissajous plots (shear stress vs. shear strain loop) were perfectly "elliptical". Our
48
PTV observations indicate that the shear deformation was homogeneous at
γo = 70% for
all the frequencies tested.91
Figure 3.1 (a) shows the rheological response at
Unlike the behavior at
γo = 100% and ω = 1 rad/s.
γo = 70%, the elastic modulus G' and shear stress proportional to
|G*| decreases within a few oscillating cycles, while the value of viscous modulus G"
increases. Under the influence of this LAOS, the entangled chains apparently were able
to orient sufficiently and arrange themselves into a new environment with reduced
topological hindrance to each other, resulting in a decrease of G'. At the end, G' and G"
are almost equal. It is likely that this is a state of less chain entanglement.
5
(a)
γo = 100%
ω = 1 rad/s
10
10
4
σo
4
G'
σo (Pa)
G', G" (Pa)
10
G"
1M(15%)-9K
10
3
10
t (s)
100
10
3
Figure 3.1 Banding through chain orientation and diffusion
(a) Rheological response at room temperature for LAOS of strain amplitude γo = 100%
and frequency ω = 1 rad/s > ωc. The strain amplitude γo = 100% is at the radial
distance of 4 mm from the edge in a 25 mm parallel-plate geometry.
49
Lissajous plots presented in Figure 3.1 (b) provide a useful picture of the response
of the entangled network. In Figure 3.1 (b), an undistorted ellipse can be noticed at 62 s,
while distorted ellipse can be seen at 125 s and beyond.
γo= 100%
(b)
stress (a.u.)
ω = 0.5 rad/s
1M(15%)-9K
25 s
125 s
62 s
201 s
strain (a.u.)
Figure 3.1 Banding through chain orientation and diffusion
(b) Lissajous plots (obtained by plotting stress voltage signal vs. strain voltage signal) at
different times.
PTV observations offered insight into the shift from linear to nonlinear response.
The velocity profile across the sample thickness at the instant of maximum plate speed at
a radial distance of 4 mm from the edge is shown in Figure 3.1 (c). Linear velocity
profile can be observed up to 75 s. Then a striking banding emerges by 94 s. With time
the entangled network evolves into regions of three different shear rates, with most of the
imposed deformation taken by the thinnest fluid layer in the center of the sample where
the local shear rate was the highest. Very little evolution of banding takes place after 150
50
s and the steady state velocity profile is represented by solid circles in Figure 3.1 (c). The
local shear rates at this moment in the lower solid region, middle fluid layer and upper
region is ca. 0.3, 5.5 and 1.2 s-1, respectively, indicating that the "liquid" layer
experiences a shear rate 18 times higher than that of the "solid" region. Note that the
averaged shear rate is
γoω = 1.0 s-1. Apparently, the entangled network could not
undergo uniform shear in steady state. During the induction period where no banding
was observed reorganization of entanglement states takes place, then an anisotropic
condition
was
eventually
met
to
allow
inhomogeneous
disentanglement) across the gap.91 Strain amplitude
yielding
(dramatic
γo = 100% was the necessary
condition to produce enough chain deformation for banding to take place for this sample.
1
(MP)
γo= 100%
ω = 1 rad/s
0.8
Y/H
0.6
1M(15%)-9K
0.4
0.2
(c)
0
0
0.2
0.4
0.6
-1
V/H (s )
75 s
94 s
132 s
144 s
183 s
0.8
(SP)
1
Figure 3.1 Banding through chain orientation and diffusion
(c) Velocity profiles at the instant of maximum plate speed for different times in a 25
mm parallel-plate geometry.
51
Unlike the observations on polydisperse samples,90 where the thickness of the
fluid layer gradually grows with time, the fluid layer in our monodisperse sample does
not grow in its thickness, but instead evolves into higher and higher shear rate values
while the solid layers settle to lower rates. In Figure 3.1 (d), the velocity profile for
different frequencies ω > ωc at
γo = 100% and at the instant of maximum plate speed
have been presented. Independent of the applied frequency, banding was first observed
at ca. 90 s, which is slightly more than the reptation time τ = 65 s. Since ω varied by a
factor of 4, for a given time duration the sample experienced 4 times as many cycles at ω
= 2 rad/s as it had at ω = 0.5 rad/s. It is surprising that the emergence of banding occurs
at the same time.
1
(MP)
Y/H
0.8
0.6
1M(15%)-9K
γo = 100%
0.4
ω (rad/s)
0.5 (201 s)
1 (183 s)
2 (201 s)
0.2
(d)
0
(SP)
0
0.5
1
1.5
-1
V/H (s )
2
Figure 3.1 Banding through chain orientation and diffusion
(d) Velocity profiles at the instant of maximum plate speed for different oscillating
frequencies and strain amplitude γo = 100% .
52
Figure 3.1 (e) shows the variations in banding at the instants of 3/8th and 5/8th of
an oscillating cycle corresponding to the same plate speed. At these long times the
system has reached "steady state", i.e., the velocity profiles are the same in every
subsequent cycle. Similar banding features were observed when strain amplitude was
increased to 125%. At
γo = 125%, banding was first observed at ca. 60 s independent of
the range of frequencies ω > ωc investigated (0.5 - 4 rad/s). Even at γo = 125%, it takes
60 s before banding is observed to take place. At γo = 250%, banding could be observed
after ca. 20 s of oscillatory shearing.91 As the applied strain amplitude
γo increases to
produce higher chain orientation and deformation, the time taken to display banding
diminishes. It is important to note that banding profiles are random and irreproducible.
1
(MP)
1M(15%)-9K
(e)
0.8
Y/H
0.6
0.4
γo = 100%
ω = 1 rad/s
0.2
181.48 s (3/8 cycle)
183.03 s (5/8 cycle)
0
(SP)
0
0.2
0.4
-1
V/H (s )
0.6
Figure 3.1 Banding through chain orientation and diffusion
(e) Velocity profiles show the variation in banding at the instants of 3/8th and 5/8th of an
oscillating cycle that have the same plate speed.
53
3.2.2 - Banding Due to Catastrophic Yielding
Unlike the behavior at strain amplitudes
γo < 250%, where nonlinear response was
observed only after some oscillation cycles, the response was highly nonlinear even
within the first oscillatory cycle at strain amplitudes
γo > 300%. The rheological
responses in terms of the apparent mechanical signals and Lissajous plot for
γo = 400%
and frequency ω = 1 rad/s is shown in Figure 3.2 (a). Experiments at amplitudes
γo =
400% were terminated at 65 s, well before any noticeable edge fracture could take place.
From the distorted ellipse, a highly nonlinear velocity profile can be expected. The
evolution of the banding profiles with time at the instant of the maximum plate speed is
shown in Figure 3.2 (b). Little change in banding profile takes place by completion of 6th
cycle (37.68 s) and the steady state profile is represented by solid circles.91
The local shear rates in the lower solid region, middle fluid layer and upper region
are ca. 0.75, 16 and 4.5 s-1 respectively, whereas the averaged shear rate is
γoω = 4.0 s-1.
Figure 3.2 (c) presents the steady state velocity profiles at the maximum plate speed at γo
= 400% for different oscillating frequencies. Figure 3.2 (d) shows the banding profile at
197 s at frequency as low as 0.1 rad/s, which is just 6 times higher than the cross over
frequency ωc, i.e., ωτR < Z. Even at such a low frequency banding developed within the
first cycle. Clearly, at
γo > 300%, the banding occurs almost immediately. It is evident
that sufficient chain deformation has produced enough elastic retraction force Fretract in
each entangled chain. This force Fretract can overcome the cohesion of the entanglement
network.113 Here Fretract is simply proportional to the amplitude γo since σ = |G*|γo.
54
4
3 10
5000
(a)
1M(15%)-9K
G"
1000
4
10
5
Normalised stress
σ
o
100
σ (Pa)
G', G" (Pa)
G'
o
γo = 400%
ω = 1 rad/s
50 s
Normalised strain
10
t (s)
2000
70
30
1
(MP)
1M(15%)-9K
(b)
Y/H
0.8
0.6
γo = 400%
ω = 1 rad/s
0.4
6.3 s
12.6 s
31.4 s
50.3 s
0.2
0
(SP)
0
1
2
-1
V/H (s )
3
4
Figure 3.2 Banding due to catastrophic yielding
(a) Rheological response at room temperature for LAOS of strain amplitude γo = 400%
and frequency ω = 1 rad/s > ωc. The strain amplitude γo = 400% is at the radial
distance of 4 mm from the edge. The inset shows the Lissajous plot at 50 s.
(b) Velocity profiles at different times at the instant of maximum plate speed.
55
1
(MP)
0.8
Y/H
0.6
(c)
γo = 400%
0.4
ω (rad/s)
1
(50 s)
0.5 (50 s)
0.25 (63 s)
0.2
1M(15%)-9K
0
(SP)
0
1
2
-1
V/H (s )
3
4
1
(MP)
1M(15%)-9K
(d)
0.8
Y/H
0.6
0.4
γo = 350%
0.2
ω = 0.1 rad/s
197 s
0
(SP)
0
0.1
0.2
0.3
-1
V/H (s )
Figure 3.2 Banding due to catastrophic yielding
(c) Velocity profiles at different oscillating frequencies for a strain amplitude of 400%
at the instant of maximum plate speed.
(d) Velocity profile at the instant of maximum plate speed at 350% and 0.1 rad/s.
56
3.2.3 - Effect of Entanglement Density
To probe the effect of degree of chain entanglement, three other entangled
polybutadiene solutions 0.7M(5%)-2K, 0.7M(10%)-4K and 1.8M(15%)-4K were studied
under LAOS.91 These samples have the number Z of entanglements per chain varying
from 13, 27 to 119. LAOS experiments on 1.8M(15%)-4K were done using 25 mm
parallel-plate disks and PTV observations was at a radial position 4 mm from the edge.
Experiments on 0.7M(5%)-2K and 0.7M(10%)-4K solutions were done using 35 mm
parallel-plate geometry and PTV observations was at a radial position of 8 mm from the
edge.
Figure 3.3 (a) shows the steady state velocity profiles at the maximum plate speed
at ω = 0.5 rad/s for different strain amplitudes γo. Banding was much stronger and could
be first observed at strain amplitude γo = 75%, since this sample is twice as entangled as
that studied in the preceding sections. The entangled solid-like layer hardly experiences
any deformation for all the applied amplitudes. The ratio of local shear rates in the
entangled to disentangled layer at γo = 400 % was as high as 9/0.15 = 60 in contrast to a
factor of 20 in 1M(15%)-9K. Apart from this, all other features are similar to the sample
investigated in the preceding subsections.
The least entangled solution 0.7M(5%)-2K solution with Z = 13 entanglement
points per chain and τ = 42 s only shows a linear velocity profile at oscillating frequency
of 0.25 rad/s and
γo = 900%, where a flexible film was wrapped around the meniscus to
prevent the possible occurrence of edge fracture at such large strain amplitudes. When
the level of entanglements per chain was increased to 27, shear banding could be
57
observed again under LAOS. To compare with results on the 0.7M(5%)-2K solution, 35
mm parallel-plate disks were used and a flexible film was wrapped around the meniscus.
Figure 3.3 (b) shows the velocity profiles at the maximum plate speed at a radial distance
of 8 mm from the edge for different frequencies at strain amplitude
γo = 400%. At
frequencies ω = 1 and 2 rad/s, two "liquid" layers could be observed by the end of the
first cycle. The nonlinearity in velocity profile was weaker when ω = 0.5 rad/s was used,
which is just 5 times the crossover frequency ωc. Nevertheless, for entangled network
with degree of entanglements greater than 25, banding can be observed at frequency ω >
5 ωc and at strain amplitudes γo > 100%.
1
(MP)
0.8
Y/H
0.6
1.8M(15%)-4K
ω - 0.5 rad/s
0.4
γo
75% (132 s)
150% (63 s)
200% (63 s)
400% (44 s)
0.2
(a)
0
0
0.5
1
1.5
-1
V/H (s )
(SP)
2
Figure 3.3 Effect of entanglement density
(a) Velocity profiles of 1.8M(15%)-4K solution at different strain amplitudes and
frequency of 0.5 rad/s at the instant of maximum plate speed.
58
1
(MP)
0.8
Y/H
0.6
700K(10%)-4K
γ = 400%
0.4
o
ω (rad/s)
0.2
(b)
0
0.5 (75 s)
1 (73 s)
2 (69 s)
(SP)
0
2
4
-1
V/H (s )
6
8
Figure 3.3 Effect of entanglement density
(b) Velocity profiles for different oscillating frequencies at the instant of maximum plate
speed at a radial distance of 8 mm from the edge in 35 mm parallel-plate disks.
We may summarize our PTV observations of LAOS in entangled polymer
solutions in terms of a diagram to highlight the significant difference, with which the
system develop non-uniform deformation. Figure 3.4 shows that shear banding may
occur via two different mechanisms, i.e., either because of an elastic breakdown of the
entanglement network at high enough strains or due to rearrangement of the state of chain
entanglement over a time scale comparable to the dominant chain diffusion time τ. For
simplicity, we call them elastic banding and diffusive banding, respectively.
59
1.0
0
kn
o
wn
Diffusive
banding
Elastic
banding
un
Z
uniform deformation
ωτ
uniform deformation
γD γE
γ
Figure 3.4 "Phase diagram" depicting the conditions under which banding occurs through
two different mechanisms, where ωτ = Z is the same condition as ωτR = 1, γD ≈ 0.8 and
γE ≈ 2.7. No experimental information is currently available in the middle region
although it is tempting to think that this region is part of the "diffusive banding" region.
3.2.4 - Effect of Solvent and Roughness of Shearing Plate
In this section, we report rheological and PTV observations of the three entangled
polymer solutions 1M(10%)-1.5K, 1M(10%)-10K and 1M(10%)-46K, respectively,
under large amplitude oscillatory shear (LAOS).92 A 5.40, 25 mm cone-plate geometry
was used throughout and PTV observation was made at a radial distance of 4 mm from
the edge. Velocity profiles at the instant of 5/8th cycle of an oscillatory wave in steady
state (when shear stress response is steady) are presented in Figure 3.5 (a) for the
1M(10%)-1.5K solution in 5 repeats. The 5 repeats come from 4 different loadings L1,
L2, L3 and L4. L1-rep refers to the repeat experiment of the 1st loading-L1. Each of the
experiments was done 3 hours after the sample loading. The applied strain γ is 175 %
and frequency ω is 3 rad/s, corresponding to a Deborah number of ωτ ~ 51. At the
60
instant of 5/8th cycle, the average shear rate across the gap is 3.7 s-1. It can be noted from
Figure 3.5 (a) that interfacial failure is observed in case of L1 and L3, while bulk banding
can also be seen along with interfacial failure in case of L1-rep, L2 and L4. Though PTV
observations of LAOS of different repeats present quite different velocity profiles, the
rheological measurements essentially overlap as shown in Figure 3.5 (b). In Figure 3.5
(b), the Lissajous plot of 36th wave of the 5 repeats has been presented along with the
inset showing peak shear stress vs. time data read from the rheometer. The strong
distortion observed in the Lissajous plot is due to the disentanglement-reentanglement
kinetics within each cycle.92
900
(MP)
(a)
800
Gap (µm)
Moving plate
velocity
600
L1
L1-rep
L2
L3
L4
1M(10%)-1.5K
th
5/8 cycle of 36th wave
γ = 175 %, ω = 3 rad/s
400
200
0
(SP)
0
1000
2000
3000
Velocity (µm/s)
Figure 3.5 Effect of solvent and roughness of shearing plate
(a) Velocity profiles at the instant of 5/8th cycle of 36th wave (74.6 s) of 5 repeat
experiments based on 4 different loadings on smooth surfaces of 5.40, 25 mm coneplate geometry. The applied strain γ is 175 % and frequency ω is 3 rad/s.
61
1M(10%)-1.5K
4
10
ο
σ (Pa)
ο
Stress (a.u.)
L1
L1-rep
L2
L3
L4
1000 0 20 40 60 80
time (s)
(b)
γ = 175 %, ω = 3 rad/s
th
36 wave
Strain (a.u.)
Figure 3.5 Effect of solvent and roughness of shearing plate
(b) Lissajous plot of 36th wave of the 5 repeat experiments. The inset shows the
maximum stress vs. time data of the 5 repeat experiments as given by the rheometer.
Figure 3.5 (c) presents the PTV observations made with two rough (sandpaper
covered) surfaces. Apparently, on rough surfaces, significant bulk shear banding can
take place during LAOS even for this solution that is inherently capable of significant
wall slip. Since our PTV does not have sufficient resolution to distinguish shear banding
with thin banding width at the interface from the wall slip phenomenon, the appreciable
shear band width produced with the rough surfaces is insightful and significant.92
The PTV observations of 1M(10%)-10K entangled solution under LAOS is shown in
Figure 3.5 (d). The applied strain is 175 % and the oscillation frequency is 1 rad/s. For
all the 5 repeats based on 3 separate loadings, strong shear banding can be observed in
the bulk. Only in the case of loading-3 (L3), some failure at the bottom interface can be
62
seen along with bulk banding.
Use of 10K PB as the solvent produces a marked
difference in the deformation field profiles during LAOS by comparison between Figure
3.5 (a) and (d). The tendency to fail at the interfaces as in the case of 1.5K solution is
effectively removed in 1M(10%)-10K.
Finally, it is most striking to show the consequence of further increasing the
molecular weight of PB solvent (PBS). Figure 3.5 (e) reveals, at all time, essentially
homogeneous LAOS in the 1M(10%)-46K solution for two values of the amplitude. This
sample is able to evolve toward its final state of chain entanglement without developing
any inhomogeneous structural change. The stress level hardly changed for γ = 175 % and
dropped no more than 10 % for γ = 300 %.92
Rough Surfaces
(MP)
800
(c)
600
1M(10%)-1.5K
th
5/8 cycle of 36th wave
γ = 175 %, ω = 3 rad/s
400
L5
L6
L7
L7-rep
L8
200
0
0
Stress (a.u.)
Gap (µm)
900
Smooth
Rough
th
36 wave
Strain (a.u.)
1000
2000
Velocity (µm/s)
3000
(SP)
Figure 3.5 Effect of solvent and roughness of shearing plate
(c) Velocity profiles at the instant of 5/8th cycle of 36th wave (74.6 s) of 5 repeat
experiments of 4 different loading. By gluing sand paper both the surfaces were
roughened and PTV was done by placing the camera horizontally. Inset shows
Lissajous plots of 36th wave with the smooth and rough surfaces.
63
800
L1
L1-rep
L2
L3
L4
Gap (µm)
600
(MP)
(d)
400
200
1M(10%)-10K
5/8 cycle of 25th wave
γ = 175 %, ω = 1 rad/s
th
0
0
200
400
600
800
Velocity (µm/s)
(SP)
1000
(SP)
800
1M(10%)-46K
ω = 1 rad/s
Gap (µm)
γ
time (s)
175 % 312 s
300 % 73 s
600
400
200
(e)
0
0
500
1000
1500
Velocity (µm/s)
(MP)
2000
Figure 3.5 Effect of solvent and roughness of shearing plate
(d) Velocity profiles at the instant of 5/8th cycle of 25th wave (154.6 s) of 5 repeat
experiments of 4 different loading.
(e) Velocity profiles at the instant of 5/8th cycle based on smooth surfaces. The applied
strain γ is 175 % and 300 % at frequency ω of 1 rad/s. The measurement was done
on Ares rheometer where the bottom plate is the rotating one.
64
3.3 - Summary
In sections 3.2.1, 3.2.2, 3.2.3 and 3.2.4, nonlinear flow behavior of entangled
polybutadiene solutions has been investigated under LAOS using a rheo-PTV technique.
The effect of applied strain amplitude, degree of chain entanglements, role of solvent and
surface roughness have been clearly highlighted.
We have shown co-existence of
multiple shear rates under LAOS across the sample thickness in entangled solutions with
Z > 25. We find in the present monodisperse solutions that at strain amplitudes
γo
between 100% - 250%, nonlinear velocity profile was seen only after some oscillating
cycles.
In the Lissajous plots, distortion of an ellipse begin to appear when nonlinear
velocity profile was first noticed. At strain amplitudes γo > 300%, liquid-like layer of less
entanglement developed instantaneously, i.e., within the first cycle of oscillation.
In section 3.2.4, the important role of the polymeric butadiene solvent (PBS) in
regulating the nonlinear rheological responses of PB solutions to large amplitude
oscillatory shear (LAOS) has also been elucidated.
Equally important was the
demonstration of the effectiveness of rough surfaces in altering the structural rearrangement in the 1M(10%)1.5K solution with strong capability to undergo wall slip.
At the same level of chain entanglement, the three solutions made with PBS of different
molecular weights show significant different reactions to external deformation. Different
repeats even produced different structural responses.
Nevertheless, the rheological
characteristics remain the same corresponding to the different deformation fields in both
step shear and LAOS.90-92
65
In short, there are five important findings from the present work. (a) Different
states of material deformation produce the same rheological properties. (b) Large
deformation produces unpredictable structural inhomogeneity. (c) Surface roughness can
effectively eliminate wall-slip-like interfacial failure allowing bulk shear banding to
prevail. (d) High molecular weight of PBS saves these entangled PB solutions from
undergoing severe interfacial failure. (e) The PBS of highest molecular weight (46
kg/mol) actually can suppress inhomogeneous structural breakdown of entangled network.
66
CHAPTER IV
STEP SHEAR
4.1 - Introduction
Observation of shear banding in startup shear25, 26, 38, 39 and LAOS90, 91, 92 made us
think about the case wherein shearing was stopped before any nonlinearity in the velocity
profile could be seen.36, 37, 92 In this chapter, the response of entangled polymer solutions
to step deformation have been presented. These step deformations, also called stress
relaxation experiments, are the simplest way of probing the viscoelastic behavior of
entangled polymers. In section 4.2.1, the behavior of 1M(10%)-15K solution to step
deformations of varying amplitude and applied at different rates have been presented.37
In section 4.2.2 and 4.2.3, respectively, the role solvents of varying molecular weight
and effect of roughness of shearing plates have been shown.37, 92 Finally, the results of
these three subsections is been summarized in section 4.3.
The entangled polymer
solutions were prepared according to the method described in section 2.2.1.1 and the PTV
setup is similar to that employed in section 2.2.1.3. The molecular characteristics of the
parent PBD, oligomeric butadiene and the entangled solutions have been presented in
Table 4.1, 4.2 and 4.3, respectively. All the rheological measurements were made on
Bohlin-CVOR rheometer at room temperature except that of Figure 4.3 (a-b), where in
67
ARES was employed. A 25 mm diameter, 5.4 0 cone-plate geometry was used in all the
case.
Table 4.1. Molecular characteristics of long chain PBDs.
Parent
polymer
Mn
Mw
(g/mol)
(g/mol)
1.014×106 1.052×106
1M
Mw/Mn
Source
1.03
University of Akron
Table 4.2. Molecular characteristics of various solvents at room temperature
Solvent
Mn
(g/mol)
Mw
(g/mol)
Mw/Mn
Source
oil (DTDP)
530
530
1
Imperial oil
1.5K
1500
-
-
15K
14020
15000
1.07
Sigma-Aldrich
Cat. No. 20,0484
Bridgestone
46K
45000
46000
1.02
Goodyear
ηs
(Pa.s)
0.2
0.7
36
2600
Table 4.3. Properties of entangled PBD solutions at room temperature
τ
τR
ηo
(s)
(s)
(Pa.s)
40
40
18.3
17
0.45
0.42
60,484
50000
40
66.6
1.66
2.5 × 105
1.87
5
Entangled solutions
Mw/Me
1M(10%)-DTDP
1M(10%)-1.5K
1M(10%)-15K
1M(10%)-46K
40
75
68
3.9 × 10
ηο/ηs
lent
(nm)
b
(mm)
302,420
71428
17
17
5.1
1.2
6,944
17
0.12
150
17
0.002
4.2 - Results and Discussion
In this section, breakup of the entanglement network after step deformation is
discussed in detail. The effect of molecular weight of the solvent and roughness of
shearing plate on the elastic breakup are also presented.
4.2.1 - Elastic Breakup After Step Deformation
Small amplitude oscillatory shear (SAOS) frequency sweep of 1M(10%)-15K
solution has been reported in section 2.2.2.
A high shear rate γ& corresponding to
Weissenberg number τ γ& = 930 was applied for different durations in each discrete step
shear experiment to produce strain amplitudes γ ranging from 0.1 to 8.4 as shown in
Figure 4.1 (a). Even though the stress relaxation does not reveal a kink-like sharp drop
except for γ = 8.4 that produced a stress overshoot, the relaxing stress level in the
superimposable region was actually lower for γ = 4.2 than for γ = 2.5. Similarly, in the
superimposable region, the relaxing stress level was lower for γ = 5.6 than for γ = 1.2.
This phenomenon was observed in spite of buildup of significantly higher stress with
increasing γ, as indicated by the filled circles. Figure 4.1 (b) plots the "relaxation
modulus" calculated from the stress data of Figure 4.1 (a) according to G(t) =
σ(t)/γ(t).36,37 Conventionally, one chooses to simplify the representation in terms of the
damping function h of eq 1, recognizing that the stress decay was eventually the same at
long times, i.e., in the superimposable region, as evident from the experimental data in
Figures 4.1 (a)-(b).
69
4
10
(a)
1M(10%)-15K
.
γτ
- 930
σ (Pa)
1000
γ
100
0.1
0.4
1.2
2.5
10
0.01
4.2
5.6
8.4
During shear
0.1
1
10
time (s)
100 500
6
10
γ
0.1
0.4
1.2
2.5
4.2
G(t,γ) (Pa)
5
10
4
10
(b)
5.6
8.4
0.1
During
0.4
shear
8.4
1000
100
1M(10%)-15K
.
γτ - 930
10
0.01
0.1
1
10
time (s)
100 500
Figure 4.1 Elastic break up after step deformation.
(a) Shear stress growth and relaxation during and after discrete step shearing of different
magnitudes, where the Weissenberg number γ& τ is 930.
(b) Relaxation modulus evaluated from (a) to normalize the data with the strain
magnitude.
70
Figure 4.1 (c) summarizes the rheological measurements in terms of the damping
function, along with the Doi-Edwards (DE) theoretical prediction (in filled circles),
where the different symbols denote results obtained using different shear rates. The
relaxation behavior would be labeled type B because it shows weaker strain softening
than the DE prediction. Unfortunately, this representation of the relaxation behavior in
term of h of eq 1 disguised a surprising character of such step strain experiment and DE
theory as shown Figure 4.1 (d): the partially relaxed stress levels in the superimposable
region are actually lower for higher values of the imposed γ. Such a non-monotonic
feature of the DE theoretical prediction has gone unnoticed until recently.36, 37 More
importantly, the experimental data reveal the same unexpected non-monotonicity. It
seemed to us that this was possible only if the entanglement network has suffered
breakup because an intact network should always resist with a higher shear stress to a
higher imposed γ.
In passing, we note that in Figures 4.1 (c)-(d) the open symbols designate those
step strain experiments where shear stress overshoot emerged before reaching the
prescribed strain amplitude. For example, at the Weissenberg number of 930, the stress
overshoot occurred before γ = 8.4 as shown in Figure 4.1 (a). Our previous PTV
observations have revealed emergence of inhomogeneous shear beyond the overshoot.25,
26, 38, 39
Therefore the PTV produces insight into why kink-like stress decline can occur at
high enough strains: stress relaxation can be speeded up in presence of a pre-existing
inhomogeneous strain field, and kink-like behavior was more likely to take place for step
strains exceeding the point of stress overshoot.
71
h(γ,t')
1
.
γτ - 930
.
γτ - 280
.
γτ - 90
0.1
Doi-Edwards
(c)
1M(10%)-15K
0.04
0.05
0.1
γ
1
10
(d)
1
e
σ(t')/G (t')
1.0
.
γτ - 930
.
γτ - 280
.
γτ - 90
0.1
Doi-Edwards
1M(10%)-15K
0.04
0.04
0.1
γ
1
10
Figure 4.1 Elastic break up after step deformation.
(c) Damping function h from different tests at long times involving different applied
shear rates. Open symbols designate step strains where a stress overshoot emerges
during shear.
(d) Direct measure, γh(γ), of the stress relaxation by normalizing the data in (c) at long
times with the equilibrium relaxation modulus Ge.
72
The non-monotonicity in Figure 4.1 (d) motivates us to carry out in situ particle
tracking velocimetric (PTV) observations and find out why the sample at higher imposed
strains displayed lower relaxed stresses. Figure 4.1 (e) depicts the non-quiescence in
terms of where some of the tracked particles moved after the shear cessation. Our PTV
measurements also confirm that there was indeed quiescent relaxation for imposed strains
as low as 120 %, at and below which it would be meaningful to characterize stress
relaxation dynamics with relaxation modulus or damping function.36, 37
Figure 4.1 (e) shows that the magnitude of macroscopic motion increases strongly
with increasing γ. At γ = 4.2, there are actually at least three "fault" planes visible where
the tracked particles moved in opposite directions. The speedup of these non-quiescent
relaxation relative to the quiescent relaxation below γ = 1.2 is indeed visible from Figure
4.1 (a). At a high strain, the sample may recoil with such a sufficient magnitude that by
the time when the macroscopic motion ceases the residual stress could already be lower
than a stress level resulting from a low imposed strain. This is how the non-monotonic
behavior is produced in Figure 4.1 (d).
Figure 4.1 (f) shows that the macroscopic motions inside the sample was more
intense at the higher strains after shear cessation than those depicted in Figure 4.1 (e).
Unlike the observations made in Figure 4.1 (e) where the step strain involved uniform
shear, stress overshoot and inhomogeneous shear have emerged before shear cessation for
the conditions examined in Figure 4.1 (f). Thus, upon shear cessation, not only the
sample could recoil along the "fault" planes already formed during shear, as shown
around Gap = 170 µm for Weissenberg number = 30 in Figure 4.1 (f).
73
750
1M(10%)-15K
.
γτ - 930
Gap (µm)
600
Fault plane-3
(MP)
γ
0.8
1.2
2.5
4.2
5.6
400
Fault plane-2
200
Fault plane-1
0
100
(e)
0
-100
-200
∆X (µm)
700
-300
.
γτ - 30, γ - 5.6
.
γτ - 15, γ - 5.6
.
600
(SP)
-400
(MP)
γτ - 930, γ - 8.4
Gap ( µm)
1M(10%)-15K
400
200
(f)
0
600
400
200
(SP)
0
-200 -400 -600
∆X (µm)
Figure 4.1 Elastic break up after step deformation.
(e) PTV detection of macroscopic motions after discrete step strains where the final
positions of the tracked particles are given.
(f) PTV observation of macroscopic motions upon shear cessation beyond stress
overshoot.
74
4.2.2 - Effect of Solvent of Varying Molecular Weight
In contrast to the 1M(10%)-15K, at the same level of chain entanglement, this
second solution is capable of massive wall slip upon replacing the slip-suppressing PBD15K with ditridecyl phthalate (DTDP), a hydrocarbon oil. From the small amplitude
oscillatory shear measurements of its linear viscoelastic properties shown in Figure 4.2
(a), we estimate the extrapolation length b of this solution to be around 5 mm as listed in
Table 4.3.92
10
4
10
5
10
4
1000
100
G'
G"
η* (Pa s)
G', G" (Pa)
1000
100
η*
(a)
1M(10%)-oil
10
0.01
0.1
1
ω (rad/s)
10
10
100
Figure 4.2 Effect of solvent of varying molecular weight.
(a) SAOS measurements of storage, loss moduli (G', G") and the complex viscosity |η*|.
In the preceding subsection 4.2.1 we observed macroscopic motions only in the
sample interior even though the sample/wall interfaces are always expected to fail first.
With b/H >> 1 in 1M(10%)-oil solution, sample recoil due to interfacial slip can occur on
75
a time scale shorter than the terminal relaxation time, thus greatly accelerating the stress
relaxation. Figures 4.2 (b) and (c) shows the actual stress σ and its normalization, i.e.,
"relaxation modulus", G(t) = σ(t)/γ(t), as a function of time.
The conventional "damping function" h in Figure 4.2 (d) indicates that the sample
shows type C behavior. Figure 4.2 (e) further indicates that even at imposed strains
below 1.0 the sample shows lower stresses than those of the 1M(10%-15K) solution.
Since there is neither bulk nor interfacial failure in 1M(10%)-15K below γ = 1.0, we can
tell from the comparison, even without performing PTV observations, that the 1M(10%oil) solution must have suffered interfacial slip at these low strains.
Our PTV observations indeed reveal sample recoil due to interfacial wall slip as
shown in Figure 4.2 (f), which occurs at strain amplitude as low as 10 %. These PTV
measurements confirm the key difference between the two solutions: The one based on
hydrocarbon oil relieves its residual stress by interfacial slip, a mode of failure that is
ineffective in the other solution (1M(10%)-15K) due to the suppression of wall slip by
the polymeric solvent PBD-15K.
Indeed, as speculated often in the past, our PTV
visualization shows that the type C relaxation behavior can occur readily in entangled
polymers that undergo massive slip upon step shear.37 Finally, we note that this type C
behavior amounts to having all the data points in Figure 4.2 (d) below that of the DoiEdwards curve. Clearly, the DE curve is arbitrary: Whether the data are above or below
(i.e., type B and type C) this curve, these samples display non-quiescent relaxation and
thus cannot be depicted by any theory that is constructed to depict quiescent relaxation
that only intact entanglement networks can exhibit.
76
10
4
1M(10%)-oil
.
γτ - 280
(b)
σ (Pa)
1000
100
γ
10
0.02
0.1
0.4
1.2
2.5
1
0.01
4.2
5.6
8.4
During shear
0.1
1
10
100 400
time (s)
10
4
1M(10%)-oil
.
γτ - 280
G(t,γ) (Pa)
1000
100
10
3
0.01
(c)
γ
0.02
0.1
0.4
1.2
2.5
0.1
4.2
5.6
8.4
During shear
1
10
time (s)
100 400
Figure 4.2 Effect of solvent of varying molecular weight.
(b) Shear stress growth and relaxation during and after discrete step shearing of different
magnitudes, where the Weissenberg number γ& τ is 280.
(c) Relaxation modulus evaluated from (b) to normalize the data with the strain.
77
h(γ,t')
1
.
γτ - 280
.
γτ
- 90
0.1
Doi-Edwards
(d)
1M(10%)-oil
0.05
0.01
0.1
γ
1
10
σ(t')/Ge(t')
1
(e)
0.1
.
γτ - 280
.
γτ
- 90
Doi-Edwards
1M(10%)-oil
0.01
0.01
0.1
γ
1
10
Figure 4.2 Effect of solvent of varying molecular weight.
(d) Damping function h from different tests at long times. Open symbols designate step
strains where a stress overshoot emerges during shear.
(e) Direct measure, γh(γ), of the stress relaxation by normalizing the data in (d) at long
times with the equilibrium relaxation modulus Ge.
78
675
Gap (µm)
540
405
γ
(MP)
1M(10%)-oil
.
0.1 γτ
- 280
0.4
0.8
1.2
2.5
270
135
(f)
0
(SP)
600
400
200
∆X (µm)
0
-200
Figure 4.2 Effect of solvent of varying molecular weight.
(f) PTV detection of final positions of the tracked particles after discrete step strains.
Further increasing the molecular weight of the polybutadiene solvent (PBS) to 46
kg/mol, which is a well entangled melts itself, we give the 1M(10%)-46K solution little
ability to take advantage of any internal or interfacial slip. Suppose a large step strain
(i.e., γ = 450 %) causes one monolayer in the sample interior to first undergo slip due to
the elastic restoring force. Let us estimate the amount of internal displacement due to
this internal slip and determine whether it would result in significant macroscopic (elastic)
recoil.
Assume the interfacial failure of the entanglement network produces a slip
velocity Vs under shear stress σ. If this slip lasts for a period of ∆t, then a displacement
resulting from the internal slip can be estimated as ∆x ~ Vs∆t = (lentσ/ηi)∆t, where σ ~
Gγ =(η/τ)γ results from a sudden step strain of γ. Thus, we have ∆x ~ bγ(∆t/τ), where
79
b = (η/ηi)lent, and τ is the terminal relaxation time. It is clear that ∆t cannot exceed τ
since even relaxation from linear response occurs on the time scale of τ, i.e., ∆t/τ < 1.
We consequently conclude for b/H << 1 that a strain ∆γs ~ ∆x/H, due to internal slip,
may be negligibly smaller than the step strain γ since ∆γs/γ < (b/H) << 1. The third
solution of 1M(10%)-46K was designed to have b ~ 2 µm << H ~ 1 mm. Therefore,
internal slip cannot affect stress relaxation. This means that all layers across the sample
thickness would be subjected to the same level of residual shear stress at any moment
after step strain and the chance for spatial inhomogeneous response during relaxation is
greatly reduced.
Figure 4.3 (a) confirms that macroscopic motion was greatly reduced after step
shear at both γ = 350 % and 450 %.92 In absence of macroscopic motions during
relaxation, the stress decrease with time is not much different from what is observed in
the linear response regime as shown in Figure 4.3 (b). In Figure 4.3 (b) the initial drop
was due to rapid stress drop associated the PBS of high molecular weight, which is a
smaller percentage of the total residual stress for γ = 350 %. The stress relaxation
following γ = 350 % is compared with the linear relaxation behavior for γ = 35 % by
matching the stress level after the solvent relaxation. The experiments on 1M(10%)-46K
were done on ARES rheometer.
80
γ
Gap (µm)
800
450 %
350 %
600
400
200
.
γτ = 400
(a)
1M(10%)-46K
0
100
50
0
-50
-100
∆X (µm)
5
10
35 %
(b)
5
10
4
35 % (Pa)
4
10
350 %
1000
350 % (Pa)
10
1000
100
100
10
0.01
0.1
1
10
100
1000
time (s)
Figure 4.3 Effect of solvent of varying molecular weight.
(a) Total displacement of tracer particles across the gap after cessation of step
deformation as observed through PTV. Both the surfaces are smooth.
(b) Relaxing shear stress vs. time for strains of 35% and 350%.
81
4.2.3 - Effect of Roughness of Shearing Plates
In this section, rheological and PTV observations of step shear deformation of
1M(10%)-1.5K solution with smooth and rough shear surfaces have been presented.92 In
case of smooth surfaces, the top plate was made up of steel and bottom plate was of glass.
PTV was done through the bottom as mentioned in section 3.2.2. In case of rough
surface measurements, both the surfaces were roughened by gluing sand paper and PTV
was done by placing the camera horizontally and viewing from the edge. Figure 4.4 (a)
shows the shear stress vs. time data of 5 step shear repeat experiments of 1M(10%)-1.5K
solution with smooth surfaces. The applied strain and Weissenberg number are 450%
and 400 respectively. For comparison with a linear response, the step shear data at 35%
strain has also been plotted. The filled symbols represent the build up of shear stress
during step deformation and the open symbols are the shear stress relaxation data after
cessation of step deformation. The 5 repeats come from 4 different loadings, named as L1,
L2, L3 and L4 respectively. The repeat experiment of the fourth loading, L4-rep, is also
reported. In case of L1, L3, and L4-rep, the solution was allowed to relax for nearly 12
hours before the step shear experiments were done and in case of L2 and L4 experiments,
the solution was relaxed for 3 hours. Not surprisingly, excellent over lapping of the 5
shear stress data was observed. It can be seen from the plot that the built up shear stress
after step deformation of γ - 450% relaxes much faster compared to the one at γ - 35%.
Careful observation reveals that faster stress relaxation happens only during the initial 10
- 15 seconds after cessation of deformation. Such faster stress relaxation is the origin for
the observation of non-linear behavior in relaxation modulus.92
82
σ (Pa)
(a)
10
1M(10%)-1.5K
4
.
= γτ = 400
Wi
γ = 450 %
1000
L1
L2
L3
L4
L4-rep
100
10
0.01
0.1
γ = 35 %
1
10
100
time (s)
900
Gap (µm)
800
(MP)
(b)
600
L1
L2
L3
L4
L4-rep
400
γ. = 450 %
Wi = γτ = 400
200
1M(10%)-1.5K
0
800
(SP)
400
0
-400
-800
∆X (µm)
Figure 4.4 Effect of roughness of shearing plate.
(a) Shear stress vs. time plot of 5 step shear experimental repeats from four different
loadings of 1M(10%)-1.5K solution.
(b) Total displacement of tracer particles across the gap after cessation of step
deformation for 5 repeats as observed through PTV.
83
Simultaneous PTV observation offered the right insight into the source of faster
stress relaxation at such non-linear strains during the initial 10 - 15 seconds after shear
cessation. Instead of remaining stationary, the tracer particles start to move immediately
after the step deformation. In Figure 4.4 (b), the total displacement of the tracer particles
across the gap after shear cessation is reported for each of the 5 repeats. It can be seen
from the plot that in case of L1, L2, L4 and L4-rep repeats, the tracer particles near the
two interfaces exhibited maximum motion, moving as much as 600 to 800 microns, after
cessation of flow.
Surprisingly, PTV observation of the L3 loading (open circles)
showed maximum displacement of the tracer particles well within the bulk. Within these
five repeats, variation in tracer particles motion can be seen across the gap but hardly any
deviation in rheological response was observed. In some of the repeats, around 20-30
microns of Y-motion of few particles was also observed. For all the repeats, most of the
tracer particle displacement happens within 10 seconds of stopping of step deformation,
which was the origin for the faster stress relaxation compared to the linear step
deformation of γ - 35%, where the tracer particles across the gap remained stationary
after shear cessation. As discussed in section 4.2.2, such motions along flow direction
are due to the elastic yielding of the entangled network after the flow is stopped.37, 92 The
entangled network was unable to hold the built in retractive forces within the chains and
relaxes the stress by undergoing inhomogeneous disintegration of the network. Choosing
the weakest plane, the network can disintegrate anywhere within the gap. Owing to the
use of 1.5K butadiene solvent, the entangled solution 1M(10%)-1.5K has a large slip
extrapolation length b ≈ 1.2 mm and because of which it is far easier for the solution to
fail at the interfaces than in the bulk.
It is highly possible that a combination of
84
interfacial and bulk failure could be occurring simultaneously across the entire sample
after cessation of step deformation in cases where the maximum tracer particle
displacement happened at the interfaces. A detailed PTV analysis of the L3 case where
internal failure happens is shown in Figure 4.4 (c) Maximum relative movements of the
traced particles are observed to take place at two locations as indicated.92
Figure 4.4 (d) reports the PTV observation made with two rough surfaces.
As
mentioned before, the two surfaces were roughened by gluing sand paper to the surfaces
and PTV observation was made through the edge. The laser was passed through the
bottom plate vertically through a small opening in the sand paper and the camera was
placed horizontal. The applied strain was 450% and the Weissenberg number was 400.
It can be clearly seen from Figure 4.4 (d) that for the 5 repeats coming from 4 different
loadings, the tracer particles within the bulk showed maximum displacement after step
deformation. This indicates the positive effect of rough surfaces in preventing surface
failure (I will add some references here). The inset shows the stress relaxation behavior
of the 1M(10%)-1.5K solution with the plates being smooth and rough. Surprisingly, it
can be seen that the relaxation behavior in both the cases are similar.92
85
900
Gap (µm)
800
600
400
200
(MP)
(c)
δ∆X
peaks
δY
Loading-3
0.5 s
1s
2s
4s
10 s
54 s
δ∆X peaks
δY
γ = 450 %
.
γτ
= 400
1M(10%)-1.5K
0
(SP)
400
200
0
-200 -400 -600
∆X (µm)
Rough surfaces
900
L5
L5-rep
L6
L7
L8
600
σ (Pa)
Gap (µm)
800
400
200
0
400
10
4
10
3
10
2
(MP)
(d)
Smooth
Rough
1
10
0.01
1
time (s)
0
100
γ = 450 %
.
γτ = 400
-400
(SP)
-800
∆X (µm)
Figure 4.4 Effect of roughness of shearing plate.
(c) Displacement of tracer particles across the gap at different times after cessation of
step deformation for loading-3 (L3).
(d) Total displacement of particles across the gap after cessation of step deformation for
5 repeats of 4 different loadings. Both the surfaces were roughened by gluing sand
paper. Inset shows the rheological response with smooth and rough surfaces.
86
4.3 - Summary
Stress relaxation behavior after a step strain is the simplest experiment to probe
any nonlinear response of an entangled polymer. Classically, we had the perception that
the entanglement network formed of linear polymer chains would be so strong that it
would not collapse after a large step strain. At a fixed entanglement density, i.e., using
the same parent polymer at 10% concentration, we show by varying the extent to which
the solutions are capable of undergoing interfacial wall slip that the non-linear relaxation
behavior upon step shear was associated with macroscopic motions in the bulk due to
network breakdown and/or at the sample/wall interface due to slip. The observed
macroscopic motions are the result of sample recoil under the influence of the residual
elastic shear stress. It was the macroscopic recoil that accelerates the stress relaxation
relative to that taking place quiescently at low strains. The different extent of motions
occurring in the bulk and/or at the interface leads to relaxations ranging from type B to
type C. As revealed using PTV, type B and A behaviors appear to be more closely
related to internal bulk failure. On the other hand, solutions with great ability to undergo
wall slip readily exhibit type C behavior. It is only in reference to the DE theoretical
prediction that the relaxation behavior has been traditionally grouped into type A, B and
C. This classification has no real meaning since none of these three types bear any
relevance to the DE prediction.37 As explained else where,93 chain disentanglement was
possible when the sufficiently high retraction force (due to the chain deformation during
the step shear) overcomes the cohesive force arising from chain entanglement. The PBS
of highest molecular weight (46 kg/mol) actually can suppress non-quiescent relaxation
after step shear.92 Also, the effectiveness of rough surfaces in altering the structural re87
arrangement in the 1M(10%)1.5K solution with strong capability to undergo wall slip
was
demonstrated
both
in
case
of
88
step
shear
as
well
as
LAOS.92
CHAPTER V
UNIVERSAL SCALING CHARACTERISTICS OF STRESS OVERSHOOT
5.1 - Introduction
Particle-tracking velocimetric (PTV) observations till now have revealed a rich
variety of flow phenomena in entangled polymer solutions under nonlinear deformation
conditions. For example, in startup shear, the shear field becomes inhomogeneous in
space after the shear stress overshoot as observed by different PTV setups.25, 26, 38, 39 For
solutions with the number of entanglement per chain Z ≥ 40, the shear stratification
persists even in steady state.38,
39
In step shear, strains of 135% and higher caused
internal macroscopic motion after shear cessation due to breakdown of entanglement
network.36,
37, 92, 120, 121
PTV also revealed shear banding under large amplitude
oscillatory deformation in entangled polymers.90, 91, 92 The new insights derived from
these PTV observations allowed us to probe well-known non-linear rheological behaviors
in a different perspective.
In this chapter, we describe some apparently universal scaling behavior127
concerning stress overshoot in startup shear of entangled polymer solutions.25,
128-134
Specifically, the characteristics associated with the shear stress overshoot during startup
shear can be defined in terms of how the coordinates of the stress overshoot (γy, σy) scale
with the applied shear rate.
We can carry out most of the present study without
89
application of the particle-tracking velocimetric (PTV) method because homogeneous
shear prevails up to the shear stress overshoot according to the recent PTV observations
and PTV observations are only made to confirm that interfacial wall slip dictates the
shear responses for well-entangled solutions with hydrocarbon oil as the solvent. Along
with a similar set of scaling phenomena observed for uniaxial extensional flow of
entangled polymer melts,122 the present results elucidate the universal origin of the wellknown stress overshoot in shear deformation of entangled polymers.127
5.2 - Materials
We have prepared five entangled polybutadiene (PBD) solutions of varying
degree of chain entanglement and ability to undergo wall slip at three concentrations with
four different solvents. The solutions are prepared according to steps described in section
2.2.1.1 of chapter 2. The solutions are based on the parent PBD that is either of Mw =
1.05×106 and Mn = 1.01×106 g/mol made in Professor R. Quirk's lab at Akron or Mw =
0.75×106 and Mn = 0.74×106 g/mol made in Bridgestone-Americas by courtesy of C.
Robertson. Molecular characteristics of long chain PBDs is given in Table 5.1. The
solvents listed in Table 5.2 are provided with courtesy by A. Halasa of Goodyear and X.
Wang of Bridgestone-Americas. Table 5.3 summarizes some essential information on
linear viscoelasticity of these five solutions.
Judging from their linear viscoelastic
properties, we know that the concentrations of 0.7M(5%)-oil and 0.7M(5%)-4K solutions
are slightly different.
Also, concentration of 1M(10%)−9K solution appears to be
slightly higher. For simplicity, we still denote them as 5 and 10% solutions. It is worth
noting that the Rouse relaxation time of these entangled solution can be estimated from
90
the terminal relaxation time τ as τR = τ[ Me(φ)/Mw], where the enlarged entanglement
spacing a relative to that, a0, of the polymer melt is related to Me(φ) as (a/a0)2 = Me(φ)/Me,
with Me being the entanglement molecular weight of the pure melt. The concentration
dependence of the entanglement spacing can be characterized by (a/a0)2 = φ1.2. The
polymeric "solvent" of high molecular weight is chosen to completely suppress
interfacial wall slip.
Table 5.1. Molecular characteristics of long chain 1,4-polybutadienes
Parent
Mn
Mw
1,4% / 1,2%
Mw/Mn
Source
PBD
0.7M
(g/mol)
0.74 x 106
(g/mol)
0.75 x 106
1.02
Bridgestone
addition
92 / 8
1M
1.01 x 106
1.05 x 106
1.03
Univ. of Akron
96 / 4
1,4-cis /
1,4-trans
56 / 36
68 / 28
Table 5.2. Molecular characteristics of oligomeric BDs at room temperature
Oligomeric
Mn
Mw
Mw/Mn
Source
BD
4K
(g/mol)
3900
(g/mol)
-
-
Goodyear
1.2
9K
8900
-
-
Goodyear
10
15K
14020
15000
1.07
Bridgestone
36
ηs (Pa.s)
Table 5.3. Molecular characteristics of entangled solutions at room temperature
Entangled
Mw
solutions
0.7M (5%)−4K
(kg/mol)
700
1.02
0.7M (5%)−oil
700
0.7M (10%)−9K
Mw/Mn
Me(φ)
Gc
Mw/Me
τ
τR
(s)
0.25
54
(Pa)
310
13
(s)
3
1.02
54
230
13
1.25
0.1
700
1.02
27
1350
27
18
0.7
1M (10%)−15K
1,100
1.05
27
1350
40
71
1.8
1M (15%)−15K
1,100
1.05
17
3900
66
83
1.25
91
5.3 - Experimental Apparatus and Measurements
Our experiments involve measurements of startup shear on two separate rotational
rheometers: an Advanced Rheometric Expansion System (ARES) and an Anton Paar
MCR 301. All tests were carried out at room temperature of around 23 oC. The range of
shear rate γ& was in the stress plateau where shear stress overshoot is observed. ARES
equipped with a cone-plate assembly of θ = 4o and diameter 25 mm or 15 mm was used to
make most of the measurements. 15 mm diameter disk was used for rates higher than 30
s-1. Start up shear experiments on 1M(15%)−15K solution was also performed on Anton
Paar MCR 301 equipped with a cone-plate assembly of θ = 4o and diameter 15 mm. All
the strain recovery experiments were performed using MCR 301 with a cone-plate cell of
diameter 15 mm and θ = 4o. To prevent edge fracture from developing to a measurable
extent, all experiments were aborted soon after the overshoot has appeared and before
any meniscus instability has had any chance to emerge. Since the imposed simple shear
is homogeneous up to the shear stress overshoot, it is unambiguous to take shear rate γ& =
Ω/θ as uniform across the gap, with Ω being the angular velocity of the rotating cone. To
supplement our study with information about the linear relaxation dynamics, standard
small amplitude oscillatory shear measurements were carried out for these samples.
Table 5.3 summarizes all the basic molecular characteristics including the crossover
modulus Gc = G'(ωc) = G"(ωc). For well-entangled monodisperse systems, there is an
empirical relationship: relating the plateau modulus Gp to Gc as Gp ≈ 3.6Gc, where Gp is
taken as G'(ωmin) [at ωmin G"(ωmin) shows a minimum]. For our 10 and 15 % solutions
such a relationship holds approximately.
92
5.4 - Results and Discussion
In this section, the universal scaling characteristics of well entangled polymer
solutions in absence of interfacial slip and the strain recovery behavior of the solution
will be discussed.
5.4.1 - Scaling Behavior of Well-Entangled Solutions Without Interfacial Slip
In this subsection, we will focus on startup shear characteristics of entangled
solutions prepared using polymeric solvents. It has been shown elsewhere that polymeric
solvent can be effectively used to prevent wall slip in entangled polymer solutions.37
Specifically, we study three polybutadiene-based solutions with the number of
entanglement per chain Z ranging from 27 to 64. The frequency dependence of storage
and loss moduli G' and G" of 1M(10%)-15K has been published37 and the key parameters
from this linear viscoelastic measurement have been provided in Table 5.3. Shear stress
overshoot typically emerges when the applied shear rate γ& is greater than the reciprocal
terminal relaxation time τ and location of the peak can be specified by its coordinates
σmax and γmax.
The shear stress overshoot can be examined for a full range of applied rates as
shown in Figure 5.1 (a). There appear to be two scaling regimes with γ& τR = 1 as the
dividing line, where the peak stress appears to satisfy
σmax ∼ (tmax)−1/2, for γ& τR > 1
(1a)
σmax ∼ (tmax)−1/4, for γ& τR < 1.
(2a)
and
93
5
10
0.08
0.1
0.3
0.5
1
1M(10%)-15K
-1/2
σ (Pa)
10
4
3
5
10
15
30
50
70
90
-1/4
(a)
1000
0.01
0.1
1
time (s)
10
50
Figure 5.1 Universal scaling - shear stress Vs. time.
(a) Shear stress growth as a function of time at various applied shear rates for the
1M(10%)-15K solution.
To further explore the universality of the observed scaling behavior, we vary the
level of chain entanglement.
The shear stress growth at various rates of the most
entangled sample of 1M(15%)-15K is shown in Figure 5.1 (b).
Again, the elastic
deformation regime exhibits the same strikingly simple scaling of Eq. (3)127
The scaling behavior observed in Figure 5.1 (a) and Figure 5.1 (b) can be
summarized in Figure 5.1 (c) to show the two regimes depicted by Eq. (1a-2a). In Figure
5.1 (c), the scaling of lower entangled solution 0.7M(10%)-9K with Z = 27 is also
included. Figure 5.1 (c) shows the emergence of the linear scaling behavior for the three
solutions of varying degree of entanglement from Z = 27 to 66 when σmax and tmax are
94
normalized by corresponding cross over modulus Gc and rouse relaxation time τR. From
Figure 5.1 (c), we have universally, in Eq. (3), Gmax = 2Gc, which offers new insight into
the significance of the crossover modulus Gc defined in the linear viscoelastic description.
The coordinates where the crossover from elastic deformation to viscoelastic response
occurs are essentially given by σmax/Gc = 5.0 and t/τR = 1.0. In other words, Eqs. (1a) and
(2a) can be rewritten as
σmax ≈ 5Gc(tmax/τR)−1/2, for γ& τR > 1
(1b)
σmax ≈ 5Gc(tmax/τR)−1/4, for γ& τR < 1.
(2b)
and
It is also instructive to represent the stress growth σ(γ) as a function of the
increasing strain γ = γ& t in both regimes. Figure 5.2 (a) shows a family of σ vs. γ curves
for γ& τR > 1, revealing that σmax is actually linearly proportional to γmax:
σmax = Gmax γmax, for γ& τR > 1
(3)
where the effective modulus Gmax at the stress peak is the slope represented by the dashed
line in Figure 5.2 (a). It is also important to note from Figure 5.2 (a) that these curves
deviate little from an initial linear growth until γc ~ 1. In other words, only after γc does
the shear stress grows more weakly than linearly. The universal features disappear in the
regime defined by γ& τR < 1. Figure 5.2 (b) reveals that Eq. (3) is no longer valid. The
scaling behavior of Eq. (3) is again observed for the more entangled solution 1M(15%)15K in the elastic deformation regime γ& τR > 1 as shown in Figure 5.2 (c).
95
10
5
-1/2
1M(15%)-15K
σ (Pa)
(b)
10
-1/4
4
0.2
0.4
0.7
1
2
4
7
1000
0.01
0.1
1
time (s)
10
20
30
40
70
100
10
100
σmax/Gc
40
Z = 27
Z = 40
Z = 64
-1/2
10
5.0
1.0
-1/4
(c)
1
0.1
1
10
30
tmax/τR
Figure 5.1 Universal scaling – shear stress Vs. time.
(b) Shear stress growth as a function of time at various applied shear rates for the
1M(15%)-15K solution.
(c) Normalized peak stress σmax/Gc vs. normalized time for the three solutions.
96
Hence, the scaling behavior shown in Figure 5.2 (a) and (c) can be summarized in
Figure 5.2 (d), wherein the lower entangled solution (Z = 27) 0.7M(10%)-9K is also
included. Figure 5..2 (d) shows the emergence of the linear scaling behavior master
curve for the three solutions of varying degree of entanglement from Z = 27 to 66 when
σmax was normalized by corresponding cross over modulus Gc. Because of the linear
relationship depicted in Eq. (3), it was tempting to construct a master curve as shown in
Figure 5.3 (a). A single master curve emerges from Figure 5.2 (a) upon normalizing the
stress and strain with their values at the peak as shown in Figure 5.3 (a). Existence of this
universal curve suggests that in this elastic deformation regime the physics concerning
the origin of the stress maximum was the same at all these seven rates. In other words,
the stress maxima attained with different imposed rates at different strains correspond to
an entanglement network whose elastic modulus is of the same value of Gmax. The
network at the stress peak was actually softer by a factor of 1.5 than its initial state as
indicated by the initial slope.127
The universal features disappear in the regime defined by γ& τR < 1.
Figure 5.2 (b) reveals that Eq. (3) was no longer valid. As a consequence, there was no
master curve like Figure 5.3 (a) in this transitional regime between the purely elastic
( γ& τR > 1) and purely viscous (terminal flow with γ& τ < 1) responses, which is shown in
Figure 5.3 (b). The behavior of the highest entangled solution 1M(15%)15K is shown in
Figure 5.3 (c). The universality in the elastic deformation regime can be summarized in
Figure 5.3 (d). All the curves from Figure 5.3 (a) and 5.3 (c) can collapse onto a single
super-master curve.
97
32000
(a)
σ (Pa)
24000
G
max
.
γ (s-1)
γc
16000
8000
3
5
10
15
30
50
70
90
.
γ τR > 1
1M(10%)-15K
0
0
2
4
6
8
γ
10
12
14
6400
(b)
σ (Pa)
4800
3200
.
γ (s-1)
1600
.
γτ
R
0.05
0.1
0.3
0.5
<1
1M(10%)-15K
0
0
1
2
γ
3
4
Figure 5.2 Universal scaling – shear stress Vs. strain.
(a) Shear stress σ Vs. strain γ in the elastic deformation regime ( γ& τR > 1).
(b) Shear stress σ Vs. strain γ in the transitional regime ( γ& τR < 1).
98
40000
(c)
σ (Pa)
30000
γc
.
20000
γ (s-1)
10000
2
4
10
20
30
40
.
γ τR > 1
1M(15%)-15K
0
0
2
4
6
γ
8
10
20
15
σmax/Gc
(d)
Z = 27
Z = 40
Z = 64
10
.
2.0
γ τR < 1
5
.
γ τR > 1
0
0
2
4
γ max
6
8
Figure 5.2 Universal scaling – shear stress Vs. strain
(c) Shear stress σ vs. strain γ when γ& τR > 1 for 1M(15%)-15K.
(d) Master curve showing a linear relationship between the normalized peak stress
σmax/Gc and strain γmax at stress maximum for the three solutions.
99
1.2
σ/σmax
(a)
1
0.8
.
γ (s-1)
0.6
3
5
10
15
30
70
90
0.4
.
γ τR > 1
0.2
1M(10%)-15K
0
0
0.4
0.8
1.2
1.6
2
γ/γmax
σ/σmax
1.2
(b)
1
0.8
0.6
.
γ (s-1)
0.4
.
γτ
0.2
R
0.05
0.1
0.3
0.5
<1
1M(10%)-15K
0
0
0.4
0.8
1.2
1.6
2
γ/γmax
Figure 5.3 Master curve.
(a) Master curve of stress growth in the elastic deformation regime, obtained by
normalizing the curves in Figure 5.2 (a) with peak values σmax and γmax respectively.
(b) Master curve of stress growth in the γ& τR < 1 regime for 1M(10%)-15K solution.
100
σ/σmax
1.2
(c)
1
0.8
.
0.6
γ (s-1)
2
4
10
20
30
40
0.4
γ. τR > 1
0.2
1M(15%)-15K
0
0
0.4
0.8
1.2
1.6
2
γ/γmax
σ/σmax
1.2
.
(d)
γ τR > 1
1
0.8
.
γ (s-1)
0.6
4
0.7M(10%)-9K, 27
100
0.4
3 1M(10%)-15K, 40
90
0.2
2
40
0
0
0.4
0.8
1M(15%)-15K, 64
1.2
1.6
2
γ/γmax
Figure 5.3 Master curve.
(c) Master curve in γ& τR > 1 elastic regime for 1M(15%)-15K solution.
(d) Super-master curve in the elastic deformation regime, obtained by combining master
curve data of the three solutions.
101
Choosing the two limiting rates from each of these sets of data, we show a supermaster curve in Figure 5.3 (d). With Equation. (3) supported by Figure 5.2 (d) and tmax =
γmax/ γ& by definition, the combination of Equations. (1b) and (2b) with Eq. (3) produces
the following quantitative scaling predictions
γmax ≈ (5/2)2/3( γ& τR)1/3, for γ& τR > 1
(4)
γmax ≈ (5/2)2/3( γ& τR)1/5, for γ& τR < 1.
(5)
and
From an experimental standpoint, the scaling exponent 1/3, is a necessary
consequence of both the scaling law of Eq. (1b) revealed in Figure 5.1 (c) and the linear
scaling of Eq. (3) disclosed in Figure 5.2 (d). From a theoretical standpoint, equating
σmax ∼ (tmax/τR)−1/2 with σmax ~ γmax truly reflects the meaning of the stress maximum: a
yield point where force imbalance takes place113 between the (entropic) elastic stress
σretract that grows linearly with time according to σ ~ γmax and the inter-chain interactions
denoted by σimg, where the subscript "img" stands for inter-molecular gripping force.
The experimental data can indeed be analyzed to reveal the scaling behavior given in Eqs.
(4) and (5), as shown in Figure 5.4 (a). A similar master curve to Figure 5.4 (a) was to
our knowledge first proposed by Menezes and Graessley129 for four oil-based PBD
solutions with Z ranging from 5 to 23. We will later compare the present results with the
literature data.
102
5.4.2 - Less Universal Behavior in Weakly Entangled Solutions
Whether the response to startup shear is sufficiently solid-like or not also depends
on the level of chain entanglement. We contrast the preceding subsection with two
solutions of 0.7M (5%)-4K and 0.7M (5%)-oil, both having only 13 entanglements per
chain. The essential difference between the two samples was in the choice of the solvent.
With a polymeric solvent of 1,4-polybutadiene of Mw = 3.9 kg/mol, interfacial slip can be
minimized further although slip length was already reasonably low (ca. 0.1 mm) even for
the 0.7M (5%)-oil sample. As seen in well entangled samples, stress overshoot starts to
occur when the externally imposed rate exceeds the terminal relaxation rate of 1/τ. Fig.
5.4 (b) depicts how the strain γmax at the stress maximum varies with the normalized rate
for both solutions as well as the literature data on solutions of a comparable level of
entanglement128-133 that uniformly employed hydrocarbon oil as the solvent. Clearly,
these modestly entangled solutions including the present two and the ones reported in the
literature no longer obey the scaling law with 1/3 exponent for γ& τR > 1, which applies to
the well-entangled samples with Z ≥ 27.
5.4.3 - Strain Recovery
There appear to be two different scaling regimes separated by the criterion
of γ& τR = 1. We would like to understand the significance of this condition. To this end
and to further probe the nature of the stress overshoot, we have conducted strain recovery
experiments. Our measurements show in Fig. 5.5 (a) that for γ& τR > 1 (i.e., γ& = 20 s-1)
strain was 100 % recoverable up to the moment tmax of the shear stress maximum.
103
20
Z = 27
Z = 40
Z = 64
10
γmax
1/3
1/5
1
50
γmax
0.1
(a)
1
.
γ τR
10
100
0.7M(5%)-4K, Z = 13
0.7M(5%)-oil, Z = 13
M-G (PB-C), Z = 14
M-G (PB-B), Z = 9
Osaki et al. (PS), Z = 14
Pearson et al. (PI), Z = 17
(b)
10
1/3
1
0.01
0.1
1
.
γτ
10
100
R
Figure 5.4 Universal scaling – Strain at maximum stress Vs. Rouse Weissenberg number.
(a) Master curve showing the strain γmax at stress maximum vs. γ& τR.
(b) The strain γmax at stress maximum vs. γ& τR, where four groups of data from the
literature on similar solutions were represented for comparison.
104
The shear deformation beyond the maximum point involves flow, providing an
additional clue about the origin of the shear stress overshoot:
For t ≤ tmax, elastic
deformation occurs in the scaling regime (i.e., when γ& τR > 1), and flow deformation
ensues for t > tmax. In other words, the elastic deformation appears to be a prerequisite for
the observed scaling laws, and this strain recovery experiment supports the notion of
associating the shear stress overshoot with the concept of yield.
For γ& τR < 1, the imposed shear deformation cannot be fully recovered unless the
deformation has only lasted for a time t ≤ τR, as shown in Fig. 5.5 (b). Thus, with γ& τR < 1,
the point of the shear stress maximum is not exactly the yield point because a certain
level of flow has already taken place. This apparently is why the features associated with
the stress overshoot produced by γ& < 1/τR cannot be described by the same scaling laws
found in elastic deformation produced by γ& > 1/τR. Nevertheless, it is remarkable that
the strain is nearly recoverable up to the stress overshoot at this rate. Finally, Fig. 5.5 (c)
shows that as long as the imposed deformation has only taken a time no longer than τR it
is elastic. So, these strain recovery experiments are rather instructive as they show that
τR is an important time scale in governing bulk deformation and flow of entangled
polymers. As we know, τR is a time scale on which significant local relaxation starts to
occur in spite of chain entanglement.127
105
10
5
γ (0.56 s)
γ (0.23 s)
γ (0.16 s)
σ
1M(15%)-15K
σ (Pa)
.
γ = 20 s-1
15
10
.
10
γ τR > 1
4
γ
5
0
(a)
3
10
-3
10
10
σ (Pa)
10
10
10
-1
0
10
10
time (s)
1
10
2
-5
10
3
5
γ (14 s)
γ (9.2 s)
γ (6 s)
γ (1.6 s)
σ
4
.
10
-2
1M(15%)-15K
.
γ = 0.2 s-1
3
γ τR < 1
3
4
2
γ
1
10
2
10
1
0
(b)
10
-3
10
-2
10
-1
0
10
10
time (s)
1
10
2
-1
10
3
Figure 5.5 Strain recovery.
(a) Strain recovery experiments after shearing at γ& = 20 s-1 for durations 0.16, 0.23,
0.56 s respectively.
(b) Strain recovery experiments after shearing at γ& = 0.2 s-1 for durations 1.6, 6, 9.2, 14 s
respectively.
106
1.2
-1
1M(15%)-15K
0.04 s , 1.6 s
-1
0.2 s , 1.5 s
-1
1s ,1s
0.8
.
γ τR < 1
γ
0.4
0
(c)
-0.4
-3
10
10
-2
10
-1
0
10
10
time (s)
1
10
2
10
3
Figure 5.5 Strain recovery.
(c) Strain recovery experiments at three different rates ( γ& τR < 1). Recovery was
complete in each test because the durations of shear are all within τR.
5.5 - Summary
The stress overshoot in startup shear of entangled polymers in the strongly shearthinning region (often referred to as the stress plateau region) has been found to show
universal scaling behavior.
Specifically, in the elastic deformation regime, we found that (a) the maximum
shear stress, or the "yield stress" σmax, scales linearly with the shear strain γmax at the yield
point, (b) γmax scales with the applied shear rate γ& , much more weakly than linearly, as
γmax ~ γ& 1/3, (c) there is a super-master curve where data from various samples of different
levels of chain entanglement at different shear rates all collapse onto a single normalized
107
stress vs. normalized strain curve, underscoring the universal meaning of the stress
maximum:
Apparently an entangled network has reached the same state of
microstructure at the stress peak as long as the system was brought to this point in the
elastic deformation limit of γ& τR > 1, (d) Our strain recovery experiments reveal that for
γ& τR > 1 the sample does not undergo any flow up to the stress maximum at tmax.127
Our new theoretical considerations113 have provided one plausible rationale to
elucidate the origin of these scaling phenomena, where the yield point (i.e., the shear
stress overshoot) can be regarded as a point of force imbalance between the force that
produced the chain deformation and the retraction force due to the chain deformation.
Finally, it is important to emphasize that yield due to chain disentanglement associated
with the phenomenon of the stress overshoot does not necessarily produce visible flow
inhomogeneity. One questions what would produce the stress overshoot if it was not
structural failure on time scales much shorter the terminal relaxation time. For weakly
entangled polymers, absence of any macroscopic heterogeneity in the flow response does
not imply that structural failure of the entanglement network did not occur smoothly.
108
CHAPTER VI
CONCLUSIONS: HOW DO ENTANGLED POLYMER LIQUIDS FLOW?
A Majority of commercially available polymer products are well entangled.
During processing of these polymers in the liquid state, intense flow fields are applied.
Understanding the response of entangled polymer liquids to these flow fields and
predicting the effect of structural changes during flow on product properties is of
enormous importance. Not surprisingly, the study of polymer flow forms one of the core
research areas in the field of rheology. Knowledge derived from polymer flow studies
can also be applied to other areas such as polymer crystallization, polymer composites,
biopolymers, associating polymers, filled systems such as tires, flow through microchannels, gels and other complex fluids.
In this work, the non-linear rheological behavior of entangled polymer solutions has been
studied utilizing conventional rotational rheometers and the particle tracking velocimetric
technique. In chapter 2, using PTV we have revealed that for entangled solutions with the
number of entanglements per chain Z ≥ 25 the shear field becomes inhomogeneous across
the gap after the stress overshoot in startup shear. For solutions with Z ≥ 40, the shear
stratification persists even in steady state, indicating that different states of chain
entanglement are possible corresponding to the same shear stress.25, 26, 38, 39 At sufficiently
high rates, the velocity profile approximately recovers linearity in steady state, where
entangled solutions become fully disentangled.
109
Observations of shear banding in
well-entangled polymers are unexpected according to the traditional perception that a
liquid is a liquid no matter how viscoelastic it is. Actually, it is not difficult to perceive a
well-entangled polymer to suffer inhomogeneous shear in response to large external
deformation that is taking place much faster than the chainlike molecules have the time
(i.e., terminal relaxation time τ) to move around. On a time scale far shorter than τ, an
entangled polymer could simply undergo cohesive breakdown if the imposed external
deformation is arbitrarily high. During its solidlike response, any local structural failure
could immediately allow other layers of the sample to avoid high shear. This is the origin
of temporary shear inhomogeneity. Here, it is indeed surprising that shear banding would
prevail at long times. We are merely at the beginning of confirming such banding in
entangled polymer solutions and developing a phenomenological understanding of why
shear banding is stable. Shear banding in entangled DNA solutions135, 136 has offered
further evidence of inhomogeneous structural disintegration during startup shear.
In chapter 3, the nonlinear rheological behavior of entangled polybutadiene
solutions with different degrees of chain entanglement has been investigated under LAOS
using a rheo-PTV technique. In this work, we have shown co-existence of different
transient shear rates under LAOS across the sample thickness in entangled solutions with
Z > 25.
This finding calls into question the objective of obtaining a rheological
fingerprint from LAOS measurements of entangled polymers and other viscoelastic
materials. We find in the present monodisperse solutions that at strain amplitudes
γo
between 100% and 250%, a nonlinear velocity profile is seen only after some oscillating
cycles. In other words, at these levels of chain deformation, the entangled chains may
first reorganize into a less entangled state through diffusion of these deformed chains and
110
then disentangle inhomogeneously to produce shear banding. In the Lissajous plots,
distortion of an ellipse begins to appear when the nonlinear velocity profile is first
noticed.
At strain amplitudes
γo > 300%, a liquid-like layer of less entanglement
develops instantaneously, i.e., within the first cycle of oscillation.
As long as the
conditions of γo > 100% and ω > ωc are met, banding would form, where the thickness of
the "liquid" layer hardly changes although the local shear rate in such a layer may rise
over time until the steady state is approached.90-92, 137
In chapter 4, the response of entangled liquids to step deformation is probed. At a
fixed entanglement density, i.e., using the same parent polymer at 10% concentration, we
show by varying the extent to which the solutions are capable of undergoing interfacial
wall slip that the non-linear relaxation behavior upon step shear can arise from
macroscopic motions in the bulk due to inhomogeneous network breakdown and/or at the
sample/wall interface due to slip. The observed macroscopic motions are the result of
sample recoil under the influence of the residual elastic shear stress. It is the macroscopic
recoil that accelerates the stress relaxation relative to that taking place quiescently at low
strains.36,
37, 92
An important aspect was that the motions after shear cessation were
observed even when the deformation field was homogeneous during shearing. PTV
observations of step strain tests have also revealed macroscropic motion in entangled
melts after shear cessation.138 Even in extensional stretching of entangled melt, the
filament after step deformation undergoes necking due to the residual elastic forces.122
The striking experimental observations revealed in chapters 2, 3 and 4 have led to the
recognition that three forces (intermolecular locking force, retraction force and cohesive
111
entanglement force) play important roles during response of a deformed entanglement
network.113 Based on the new theoretical description, we were further able to discover
striking scaling features127 associated with stress overshoot in startup shear as shown in
Chapter
5
that
are
highly
universal
112
for
well
entangled
polymer
liquids.
BIBLIOGRAPHY
1.
Huppler, J. D.; et al. Trans. Soc. Rheol. 1967, 11, 181.
2.
Lee, C. L.; Polmanteer, K. E.; King, E. G. J. Polym. Sci., Part A-2 1970, 8, 1909.
3.
Graessley, W. W. Adv. Polym. Sci. 1974, 16, 1.
4.
Osaki, K.; Fukuda, M.; Ohta, S. I.; Kim, B. S.; Kurata, M. J. Polym. Sci., Polym.
Phys. Ed. 1975, 13, 1577.
5.
Graessley, W. W.; Park, W. S.; Crawley, R. L. Rheol. Acta 1977, 16, 291.
6.
Menezes, E. V.; Graessley, W. W. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 1817.
7.
Osaki, K.; Inoue, T.; Isomura, T. J. Polym. Sci., Polym. Phys. Ed. 2000, 38, 1917.
8.
Osaki, K.; Inoue, T.; Isomura, T. J. Polym. Sci., Polym. Phys. Ed. 2000, 38, 2043.
9.
Bercea, M.; Peiti, C.; Simionescu, B.; Navard, P. Macromolecules 1993, 26, 7095.
10. Mhetar, V. R.; Archer, L. A. J. Polym. Sci., Polym. Phys. Ed. 2000, 38, 222.
11. Pattamaprom, C.; Larson, R. G. Macromolecules 2001, 34, 5229.
12. Islam, M. T.; Archer, L. A. J. Polym. Sci., Polym. Phys. Ed. 2001, 39, 2275.
13. Rehage, H.; Hoffmann. H. J. Phys. Chem. 1988, 92, 4712.
14. Rehage, H.; Hoffmann. H. Mol. Phys. 1991, 74, 933.
15. Callaghan, P. T. et al. J. Phys. II 1996, 6, 375.
16. Mair, R. W.; Callaghan, P. T. Europhys. Lett. 1996, 36, 719.
17. Britton, M. M.; Callaghan, P. T. Phys. Rev. Lett. 1997, 30, 4930.
18. Britton, M. M.; Mair, R. W.; Callaghan, P. T. J. Rheol. 1999, 43, 897.
113
19. Fischer, E.; Callaghan, P. T. Europhys. Lett. 2000, 50, 803.
20. Salmon, J. B.; Manneville, S.; Molino, F. Phys. Rev. Lett. 2003, 90, 228303.
21. Salmon, J. B.; Manneville, S.; Colin, A. Phys. Rev. E 2003, 68, 051503.
22. Becu, L.; Manneville, S.; Colin, A. Phys. Rev. Lett. 2004, 93, 018301.
23. Hu, Y. T.; Lips, A. J. Rheol. 2005, 49, 1001.
24. Lopez-Gonzalez, M. R.; Holmes, W. M.; Callaghan, P. T. Soft Matter 2006, 2, 855.
25. Tapadia, P.; Wang, S. Q. Phys. Rev. Lett. 2006, 96, 016001.
26. Boukany, P. E.; Wang, S. Q. J. Rheol. 2007, 51, 217.
27. Lu, C. Y. D.; Olmsted, P. D.; Ball, R. C. Phys. Rev. Lett. 2000, 84, 642.
Fielding, S. M.; Olmsted, P. D. Phys. Rev. Lett. 2004, 92, 084502.
28. Vasquez, A.; McKinley G. H.; Cook, P. J. Non-Newt. Fluid Mech. 2007, 144, 122.
29. Bautista, F.; Soltero, J. F. A.; Perez-Lopez, J. H.; Puig, J. E.; Manero, O. J. NonNewt. Fluid Mech. 2000, 94, 57.
30. de Gennes, P. G. J. Chem. Phys. 1971, 55, 572.
Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics, Clarendon Press: Oxford,
1986.
31. Cates, M. E.; McLeish, T. C. B.; Marrucci, G. Europhys. Lett. 1993, 21, 451.
32. Marrucci, G. J. Non-Newt. Fluid Mech. 1996, 62, 279.
33. Milner, S. T.; McLeish, T. C. B.; Likhtman, A. E. J. Rheol. 2001, 45, 539.
34. Graham, R. S.; Likhtman, A. E.; McLeish, T. C. B. J. Rheol. 2003, 47, 1171.
35. Doi, M.; Edwards, S. F. J. Chem. Soc. Faraday Trans. 2 1979, 75, 38.
36. Wang, S. Q.; Ravindranath, S. et al. Phys. Rev. Lett. 2006, 97, 187801.
37. Ravindranath, S.; Wang, S. Q. Macromolecules 2007, 40, 8031.
38. Ravindranath, S.; Wang, S. Q.; Olechnowicz, M.; Quirk, R. P. Macromolecules
2008, 41, 2663.
114
39. Ravindranath, S.; Wang, S. Q. J. Rheol. 2008, 52, 957.
40. Knudsen, J. C.; Karlsson, A. O.; Ipsen, R.; Skibsted, L. H. Colloids Surf. A:
Physicochem. Eng. Aspects 2006, 274, 56.
41. Mason, T. G.; Lacasse, M. D.; Grest, G. S.; Levine, D.; Bibette, J.; Weitz, D. A.
Phys. Rev. E 1997, 56, 3150.
42. Bower, C.; Gallegos, C.; Mackley, M. R.; Madiedo, J. M. Rheol. Acta 1999, 38, 145.
43. Watanabe, H.; Kotaka, T.; Hashimoto, T.; Shibayama, M.; Kawai, H. J. Rheol. 1982,
26, 153.
44. Watanabe, H.; Sato, T.; Osaki, K.; Yao, M. L.; Yamagishi, A. Macromolecules 1997,
30, 5877.
45. Tirtaatmadja, V.; Tam, K. C.; Jenkins, R. D. Macromolecules 1997, 30, 1426.
46. Hamley, I. W.; Pople, J. A.; Booth, C.; Derici, L.; Clerc, M. I.; Davidson, P. Phys.
Rev. E 1998, 58, 7620.
47. Daniel, C.; Hamley, I. W.; Wilhelm, M.; Mingvanish, W. Rheol. Acta 2001, 40, 39.
48. Stangler, S.; Abetz, V. Rheol. Acta 2003, 42, 569.
49. Hyun, K.; Nam, J. G.; Wilhelm, M.; Ahn, K. H. Rheol. Acta 2006, 45, 239.
50. Raghavan, S. R.; Khan, S. A. J. Colloid Interface Sci. 1997, 185, 57.
51. Yziquel, F.; Carreau, P. J.; Tanguy, P. A. Rheol. Acta 1999, 38, 14.
52. Heymann, L.; Peukert, S.; Aksel, N. J. Rheol. 2002, 46, 93.
53. Narumi, T.; See, H.; Suzuki, A.; Hasegawa, T. J. Rheol. 2005, 49, 71.
54. Karis, T. E.; Kim, C. A.; Jhon, M. S. Macromol. Mater. Eng. 2002, 287, 583.
55. Karis, T. E.; Kono, R. N.; Jhon, M. S. J. Appl. Polym. Sci. 2003, 90, 334.
56. Tariq, S.; Giacomin, A. J.; Gunasekaran, S. Biorheology 1998, 35, 171.
57. Thien, N. P.; Newberry, M.; Tanner, R. I. J. Non-Newton. Fluid Mech. 2000, 92, 67.
58. Hyun, K.; Ahn, K. H.; Lee, S. J. J. Non-Newtonian Fluid Mech. 2002, 107, 51.
115
59. Veerman, C.; Sagis, L. M. C.; Venema, P.; Linden, E. V. Rheol. Acta 2005, 44, 244.
60. Lefebvre, J. Rheol. Acta 2006, 45, 525.
61. Parthasarathy, M.; Klingenberg, D. J. J. Non-Newton. Fluid Mech. 1999, 81, 83.
62. Sim, H. G.; Ahn, K. H.; Lee, S. J. J. Rheol. 2003, 47, 879.
63. Hu, B.; Fuchs, A.; Huseyin, S.; Gordaninejad, F.; Evrensel, C. J. Appl. Polym. Sci.
2006, 100, 2464.
64. Krishnamoorti, R.; Giannelis, E. P. Macromolecules 1997, 30, 4097.
65. Krishnamoorti, R.; Giannelis, E. P. Langmuir 2001, 17, 1448.
66. Ren, J.; Silva, A. S.; Krishnamoorti, R. Macromolecules 2000, 33, 3739.
67. Ren, J.; Casanueva, B. F.; Krishnamoorti, R. Macromolecules 2003, 36, 4188.
68. Adrian, D. W.; Giacomin, A. J. J. Rheol. 1992, 36, 1227.
69. Reimers, M. J.; Dealy, J. M. J. Rheol. 1996, 40, 167.
70. Wilhelm, M. Macromol. Mater. Engr. 2002, 287, 83.
71. Debbaut, B.; Burhin, H. J. Rheol. 2002, 46, 1155.
72. Clemeur, N.; Rutgers, R. P. G.; Debbaut, B. Rheol. Acta 2003, 42, 217.
73. Schlatter, G.; Fleury, G.; Muller, R. Macromolecules 2005, 38, 6492.
74. Vinogradov, G. V.; Isayev, A. I.; Mustafaev, D. A.; Podolsky, Y. Y. J. Appl. Poly.
Sci. 1978, 22, 665.
75. Vinogradov, G. V.; Isayev, A. I.; Katsyutsevich, E. V. J. Appl. Poly. Sci. 1978,
22, 727.
76. Wilhelm, M.; Maring, D.; Spiess, H. W. Rheol. Acta 1998, 37, 399.
77. Wilhelm, M.; Reinheimer, P.; Ortseifer, M. Rheol. Acta 1999, 38, 349.
78. Wilhelm, M.; Reinheimer, P.; Ortseifer, M.; Neidhofer, T.; Spiess, H. W. Rheol.
Acta 2000, 39, 241.
79. Dusschoten, D. V.; Wilhelm, M.; Spiess, H. W. J. Rheol. 2001, 45, 1319.
116
80. Kallus, S.; Willenbacher, N.; Kirsch, S.; Distler, D.; Neidhofer, T.; Wilhelm, M.;
Spiess, H. W. Rheol. Acta 2001, 40, 552.
81. Neidhofer, T.; Wilhelm, M.; Debbuat, B. J. Rheol. 2003, 47, 1351.
82. Neidhofer, T.; Sioula, S.; Hadjichristidis, N.; Wilhelm, M. Macromol. Rapid
Commun. 2004, 25, 1921.
83. Karis, T. E.; Seymour, C. M.; Kono, R. N.; Jhon, M. S. Rheol. Acta 2002, 41, 471.
84. Sim, H. G.; Ahn, K. H.; Lee, S. J. J. Non-Newtonian Fluid Mech. 2003, 112, 237.
85. Cho, K. S.; Hyun, K.; Ahn, K. H.; Lee, S. J. J. Rheol. 2005, 49, 747.
86. Yosick, J. A.; Giacomin, A. J. J. Non-Newtonian Fluid Mech. 1996, 66, 193.
87. Yosick, J. A.; Giacomin, A. J.; Moldenaers, P. J. Non-Newtonian Fluid Mech. 1997,
70, 103.
88. Giacomin, A. J.; Jeyaseelan, R. S.; Dealy, J. M. J. Rheol. 1993, 37, 811.
89. Wapperom, P.; Leygue, A.; Keunings, R. J. Non-Newton. Fluid Mech. 2005, 130, 63.
90. Tapadia, P.; Ravindranath, S.; Wang, S. Q. Phys. Rev. Lett. 2006, 96, 196001.
91. Ravindranath, S.; Wang, S. Q. J. Rheol. 2008, 52, 341.
92. Ravindranath, S.; Wang, S. Q.; Olechnowicz, M.; Chavan, V.; Quirk, R. P. J. Rheol.
under review.
93. Green, M. S.; Tobolsky, A. V. J. Chem. Phys. 1946, 14, 80.
94. de Gennes, P. G. J. Chem. Phys. 1971, 55, 572.
95. Einaga, Y.; Osaki, K.; Kurata, M.; Kimura, S.; Tamura, M. Polym. J. 1971, 2, 550.
96. Einaga, Y.; Osaki, K.; Kurata, M.; Yamada, N.; Tamura, M. Polym. J. 1973, 5, 91.
97. Fukuda, M.; Osaki, K.; Kurata, M. J. Polym. Sci., Polym. Phys. Ed.1975, 13, 1563.
98. Doi, M.; Edwards, S. F. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1802.
99. Doi, M. J. Polym. Sci. 1980, 18, 1005.
100. Osaki, K.; Kurata, M. Macromolecules 1980, 13, 671.
117
101. Osaki, K.; Nishizawa, K.; Kurata, M. Macromolecules 1982, 15, 1068.
102. Morrison, F. A.; Larson, R. G. J. Polym. Sci., Polym. Phys. Ed. 1992, 30, 943.
103. Archer, L. A.; Larson, R. G.; Chen, Y. L. J. Rheol. 1995, 39, 519.
104. Larson, R. G.; Khan, S. A.; Raju, V. R. J. Rheol. 1988, 32, 145.
105. Sanchez-Reyes, J.; Archer, L. A. Macromolecules 2002, 35, 5194.
106. Islam, M. T.; Sanchez-Reyes, J.; Archer, L. A. Rheol. Acta 2003, 42, 191.
107. Archer, L. A.; Sanchez-Reyes, J.; Juliani Macromolecules 2002, 35, 10216.
108. Juliani; Archer, L. A. J. Rheol. 2001, 45, 691.
109. Osaki, K. Rheol. Acta 1993, 32, 429.
110. Venerus, D. C. J. Rheol. 2005, 49, 277.
111. Vrentas, C. M.; Graessley, W. W. J. Rheol. 1982, 26, 359.
112. Marrucci, G.; Grizzuti, N. J. Rheol. 1983, 27, 433.
113. Wang, S. Q.; Ravindranath, S.; Wang, Y.; Boukany, P. E. J. Chem. Phys. 2007, 127,
064903.
114. De Gennes, P. G. J. Chem. Phys. 1971, 55, 572. 94
115. Wang, S. Q. Adv. Polym. Sci. 1999, 38, 227.
116. Colby, R. H.; Rubinstein, M. Macromolecules 1990, 23, 2753.
117. Colby, R. H.; Fetters, L. J.; Graessley, W. W. Macromolecules 1991, 24, 3873.
118. Yang, X.; Wang, S. Q. Macromolecules 1999, 32, 2638.
119. Brochard, F.; de Gennes, P. G. Langmuir 1992, 8, 3033.
120. Boukany, P. E.; Wang, S. Q. Macromolecules 2009, 42, 2222.
121. Boukany, P. E.; Wang, S. Q.; Wang, X. R. Macromolecules 2009, 42, 6261.
122. Wang, Y.; Boukany, P. E.; Wang, S. Q.; Wang, X. Phys. Rev. Lett. 2007, 99, 237801.
118
123. Lin, Y. H. Macromolecules 1987, 20, 3080.
124. Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of rate processes (McGrawHill, New York, 1941.93.
125. De Gennes, P. G. J. Phys. (Paris) 1975, 36, 1199.
126. Take step shear for example. The external work per unit volume done during an
affine elastic deformation is given by w = ½σγ where σ = Gpγ with Gp given by Eq.
(3). This stored elastic energy Eel = w per unit volume is available to pay for
entropy loss on the order of kBT for each strand involved in disentanglement.
Since the number of strands per unit volume is φρNa/Me(φ) we can equate Eel to
[φρNa/Me(φ)]kBT to arrive at a condition for disentanglement: γ c2 ~ 2, i.e. γc ~ 1.4.
127. Ravindranath, S.; Wang, S. Q. J. Rheol. 2008, 52, 681.
128. Huppler, J. D.; Macdonal, I. F.; Ashare, E.; Spriggs, T. W.; Bird, R. B. Trans. Soc.
Rheol. 1967, 11, 181.
129. Menezes, E. V.; Graessley, W. W. J. Polym. Sci., Polym. Phys. Ed. 1982, 20,1817.
130. Pearson, D. S.; Kiss, A. D.; Fetters, L. J.; Doi, M. J. Rheol. 1989, 33, 517.
131. Osaki, K.; Inoue, T.; Isomura, T. J. Polym. Sci., Part B. 2000, 38, 1917.
132. Osaki, K.; Inoue, T.; Isomura, T. J. Polym. Sci., Part B. 2000, 38, 2043.
133. Osaki, K.; Inoue, T.; Uematsu, T. J. Polym. Sci., Part B. 2000, 38, 3271.
134. Pattamaprom, C.; Larson, R. G. Macromolecules 2001, 34, 5229.
135. Boukany, P. E.; Hu, Y. T.; Wang, S. Q. Macromolecules 2008, 41, 2644.
136. Boukany, P. E.; Wang, S. Q. J. Rheol. 2009, 53, 73.
137. Li, X.; Wang, S. Q.; Wang, X. J. Rheol., 2009, 57, 1255.
138. Boukany, P.; Wang, S. Q. J. Rheol. 2009, 53, 617.
119