NONLINEAR RHEOLOGICAL CHARACTERIZATION AND MODELING OF
COMPLEX FLUIDS UNDER LARGE AMPLITUDE OSCILLATORY SHEAR (LAOS)
By
Christopher J
oseph
Hershey
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Chemical Engineering
Doctor of Philosophy
201
8
ABSTRACT
NONLINEAR RHEOLOGICAL CHARACTERIZATION AND MODELING OF
COMPLEX FLUIDS UNDER
LARGE AMPLITUDE OSCILLATORY SHEAR (LAOS)
By
Christopher J
oseph
Hershey
Dynamic oscillatory shear tests have historically been one of the most common ways a
rheologist probes the material response for complex fluid systems including neat polymer melts
a
nd solutions, blends, and composites. In small angle oscillatory shear (SAOS) testing, the
material functions
and
describe the linear viscoelasticity of the complex fluid and may be
related to the morphological changes occurring in the system. H
owever, typical processing
conditions occur at fast flow rates and generate large deformations resulting in a strain dependence
on the rheological properties.
In this research, the nonlinear viscoelastic behavior of polypropylene (PP) nanocomposite
melts
and oligomer modified polyamide (PA) blends under large amplitude oscillatory shear
(LAOS) flows was investigated using Fourier transform (FT) rheology and stress decomposition
(SD) techniques. With the development of high performance data acquisition (DA
Q) cards in
recent years, raw voltages of angular displacement (strain) and torque (stress) from the rheometer
These higher order harmonics are strongly correlate
d to the chain dynamics and morphological
changes in a polymer system.
Polypropylene

clay nanocomposites were produced using concentrations of 3 and 5 wt%
of silane treated nanoclay to ensure that
the system was dilute, and
the filler

network contribution
was negligible. To promote particle

polymer interactions, the silane treated clay was reacted with
a maleated polypropylene compatibilizer. The nonlinear intensity ratio
of the third order
harmonic to the first order harmonic of the shear stress as w
ell as the zero

strain limit nonlinearity
parameter
were determined through FT rheology experiments. To describe the trends in
and
for polymer nanocomposite systems, a nonlinear viscoelastic differential model was
developed for LAOS type flow
s.
Blends consisting of a PA6/PA66 copolymer in an 80:20 mixture by weight were melt
mixed with varying concentrations (5 and 10 wt%) and molecular weights (
750 and 1000)
of an elastomeric polyisobutylene succinic anhydride (PIBSA) oligomer and
tested under LAOS
conditions. The low molecular weight PIBSA acts as a plasticizer on the matrix blend, reducing
its shear stress with increasing concentration. The SD technique separates the elastic (
) and
viscous (
) contributions of the stress w
aveform, much like
and
in SAOS flows. It was
found that by increasing the concentration of PIBSA, the normally viscous response of the PA
blend matrix transitioned to an elastic response. By combining SD with FT rheology, it was found
that the
ratio were nearly identical for both PA blends with 0wt% and 5wt% PIBSA, while
much larger values of
were identified for 10wt% PIBSA blends at lower strains.
Copyright by
CHRISTOPHER
JOSEPH
HERSHEY
2018
v
DEDICATED
TO
MY PARENTS
& GRANDMA
vi
ACKNOWLEDGMENTS
I have been very fortunate during my time here at Michigan State University for the
education that I received during both my
undergraduate and graduate studies, the lifelong friends
that I have made and the memories that we have shared. There have been many individuals that
have impacted my life in such a tremendous way, for whom without I would not be where I am
today. I must
first thank my academic advisor, Dr. K. Jayaraman, for encouraging me to pursue
my ideas and challenging me to be a better researcher. If I had not met him during my time as an
undergraduate research assistant, my life may have taken me in a direction aw
ay from the polymer
sciences, a field which I can say I truly love. Thank you, Dr. Jay, for taking me on as your student
then and for sticking with me to the very end.
I would like to thank my committee members, Dr. A. Lee, Dr. R. Averill, and Dr. D. Liu
,
for their suggestions and teachings during my graduate career. I would also like to take the time
to thank my colleagues, Weijie, who took the time to help me get acclimated to the research group,
and Xinting and Xing, whom provided me with valuable ins
ight through the discussions we have
shared. I would like to thank Mike Rich for his counsel and sharing with me many wonderful
stories. I am especially thankful to the CHEMS office staff, who keep the department running
smoothly. I am also very gratefu
l for my closest friends, Jacob, Steve, Mike and Joel, whom I
have memories with that will last a lifetime and stories that will last two.
I want to give my deepest love and thanks to my family. My Mom, Dad and Grandma
have always instilled the value of
education in me. They have always pushed me to be the best
person that I can be and have supported me in all of my endeavors. It is to them that I dedicate
this dissertation as a symbol of my love and appreciation for everything they have done.
vii
TABLE OF
CONTENTS
LIST OF TABLES
................................
................................
................................
.........................
ix
LIST OF FIGURES
................................
................................
................................
.........................
x
CHAPTER 1
................................
................................
................................
................................
....
1
INTRODUCTION
................................
................................
................................
..........................
1
1.1 Research Background and Motivation
................................
................................
.................
1
1.2 Dynamics of Entangled Polymer Chains
................................
................................
.............
3
1.3 Dynamics of Polymer Composites
................................
................................
.......................
6
1.4 Dynamic Shear Rheology
................................
................................
................................
....
8
1.5 Research Objectives
................................
................................
................................
...........
14
1.6
Scope of Thesis
................................
................................
................................
..................
15
CHAPTER 2
................................
................................
................................
................................
..
18
NONLINEAR CHARACTERIZATION TECHNIQUES USING
LARGE AMPLITUDE OSCILLATORY SHEAR FLOW
................................
.....................
18
2.1 Introduction
................................
................................
................................
........................
18
2.2 Data Acquisition
................................
................................
................................
................
18
2.3 Pre

Processing
................................
................................
................................
....................
24
2.4 Post

Processing
Representative Figures
................................
................................
.........
27
2.4.1
Lissajous

Bowditch Curves
................................
................................
......................
28
2.4.2 Fourier Transform Rheology
................................
................................
....................
29
2.4.3 Stress Decomposition
................................
................................
................................
32
2.5 Conclusions
................................
................................
................................
........................
33
CHAPTER 3
................................
................................
................................
................................
..
3
4
DYNAMICS OF ENTANGLED POLYMER CHAINS WITH
NANOPARTICLES ATTACHMENT UNDER
LARGE AMPLITUDE OSCILLATORY SHEAR (LAOS)
................................
....................
3
4
3.1
Introduction
................................
................................
................................
........................
3
4
3.2 Theory
................................
................................
................................
................................
3
7
3.2.1 Model for Polymer Nanocomposites
................................
................................
........
3
7
3.2.2 LAOS Simulation Scheme
................................
................................
........................
4
0
................................
................................
.............
4
1
3.3 Results and Discussion
................................
................................
................................
......
4
3
3.3.1 Linear Viscoelasticity
................................
................................
...............................
4
3
3.3.2 LAOS Strain Sweeps
................................
................................
................................
4
5
Asymptotic Solution
................................
................................
...................
5
6
3.4 Conclusions
................................
................................
................................
........................
6
2
viii
CHAPTER 4
................................
................................
................................
................................
..
6
5
FOURIER TRANSFORM RHEOLOGY OF
POLYPROPYLENE

LAYERED SILICATE NANOCOMPOSITES
................................
...
6
5
4.1 Introduction
................................
................................
................................
........................
6
5
4.2 Experimental
................................
................................
................................
......................
6
7
4.2.1 Materials
................................
................................
................................
...................
6
7
4.2.2 Sample Preparation
................................
................................
................................
...
6
8
4.2.3 Linear Rheology
................................
................................
................................
........
6
8
4.2.4 N
onlinear Rheology
................................
................................
................................
..
69
4.2.5 Fourier Transform Rheology
................................
................................
....................
69
4.3 Results and Discussion
................................
................................
................................
......
7
2
4.3.1 Dynamic Frequency Sweep Tests
................................
................................
.............
7
2
4.3.2 Fourier Transform Rheology
................................
................................
....................
7
3
4.4 Conclusion
s
................................
................................
................................
........................
8
2
CHAPTER 5
................................
................................
................................
................................
..
8
3
EFFECTS OF REACTIVE OLIGOMER ADDITIVES ON
MELT RHEOLOGY OF
NYLONS: FOURIER TRANSFORM RHEOLOGY
...................
8
3
5.1 Introduction
................................
................................
................................
........................
8
3
5.2 Experimental
................................
................................
................................
......................
8
4
5.2.1 Materials
................................
................................
................................
...................
8
4
5.2.2 Blend Preparation
................................
................................
................................
......
8
4
5.2.3 Dynamic Shear Rheology
................................
................................
.........................
8
4
5.2.4 Nonlinear Analysis
................................
................................
................................
....
8
5
5.3 Results and Discussion
................................
................................
................................
......
8
5
5.3.1 Linear Viscoelasticity
................................
................................
...............................
8
5
5.3.2 Non
linear Viscoelasticity
................................
................................
.........................
8
6
5.4 Conclusion
s
and Recommendations
................................
................................
..................
9
1
CHAPTER 6
................................
................................
................................
................................
..
9
2
CONCLUSIONS AND RECOMMENDATIONS
................................
................................
.....
9
2
6.1 Conclusions
................................
................................
................................
........................
9
2
6.2 Recommendations
................................
................................
................................
..............
9
5
BIBLIOGRAPHY
................................
................................
................................
..........................
9
7
ix
LIST OF TABLES
Table 1.1
Voltage ranges for 2K FRTN1 torque transducer
equipped to ARES rheomete
r
..
21
Table 3.1
Relationship between
and
for each entanglement pair
............................
4
3
Table 3.2
Definition of
for each free and attached chain entanglement pair
...................
5
7
Table 4.1
Formulations for Different Nanocomposites
................................
.........................
6
8
x
LIST OF FIGURES
Figure 1.1
Schematic of reptation theory showing test chain in a tube entangled with matrix
chains
................................
................................
................................
.......................
4
Figure 1.2
Schematic of stress relaxation test and relaxation regimes with corresponding
relaxation timescales
................................
................................
................................
5
Figure 1.3
Dynamic
shear tests using (a) frequency sweeps and (b) strain sweeps
..................
8
Figure 1.4
Methods to quantify nonlinearities in (a) strain and stress waveforms using (b)
Lissajous

Bowditch curves (c) FT rheology and (d) stress decomposition
techniques
................................
................................
................................
..............
10
Figure 1.5
S
chematic showing trends found in Lissajous

Bowditch curves
...........................
11
Figure 2.1
Illustration of a typical rheometer designed for testing LAOS flows
....................
19
Figure 2.2
Rear panel connections for TA Instruments ARES

Classic rheometer
.................
21
Figure 2.3
Raw voltages o
f torque and angular displacement for an entire LAOS test
..........
24
Figure 2.4
Schematic illustrating the effect of oversampling number on a noisy waveform
..
25
Figure 2.5
Oversampled and clipped stress and strain waveforms of polypropylene
homopolymer at
=
1.56 and 1 rad/s
................................
................................
...
27
Figure 2.6
Lissajous

Bowditch curves for polypropylene at 1 rad/s showing effect of strain on
viscoelasticity and steady

state behavior
................................
...............................
28
Figure 2.7
Fourier intensity spectrum for polypropylene at 1 rad/s and
=1.56 showing (a)
correct (b) low and (c) high oversampling numbers
................................
..............
30
Figure 2.8
FT rheology parameters for polypropylene at 1 rad/s showing (a)
and (b)
31
Figure 2.9
Elastic and viscous stress waveforms relative to total stress waveform for
polypropylene at 1 r
ad/s and
=1.56
................................
................................
....
33
Figure 3.1
Asymptotic solutions for linear viscoelast
ic storage modulus with
(
a
)
varying
strength of attachment
and fixed
= 0.1
;
(
b
)
varying
volume fraction of attached
chains
and fixed
= 20
................................
................................
....................
4
4
xi
Figure 3.2
Numerical
predictions of
dynamic storage modulus
with
strain amplitude and
=
= 1 for
(
a
) matrix
;
(
b
) composite
with
= 0.1 and
= 5;
(
c
) composite
with
= 0.1 and
= 20
................................
................................
................................
4
7
Figure 3.3
Numerical predictions of
with
strain amplitude and
=
= 1 for
(
a
) matrix
;
(
b
) composite
with
= 0.1 and
= 5;
(
c
) composite
with
= 0.1 and
= 20
.
48
Figure 3.4
Numerical predictions of
with
strain amplitude and
=
= 1 for
(
a
) matrix
;
(
b
) composite
with
= 0.1 and
= 5;
(
c
) composite
with
= 0.1 and
= 20
.
5
0
Figure 3.5
Effect of uniform CCR rates for composites with
= 0.1 and
= 20 at De = 0.2
on
(
a
)
;
(
b
)
.
Stretch parameters:
=
= 0.01 and
=
= 100
...........
5
1
Figure 3.6
predicted with (a)
>
and (b)
<
for nanocomposites with
= 0.1
and
= 20 at De = 0.2
................................
................................
............................
5
3
Figure 3.7
Predicted
with
<
and
>
for composites with
= 0.1 and
= 20 at
De = 0.2
................................
................................
................................
..................
5
4
Figure 3.8
Component stresses with increasing strain amplitude and
>
for composites
with
= 0.1 and
= 20 at De = 0.2 showing (a) normalized storage modulus; (b)
third harmonic ratio
................................
................................
................................
5
5
Figure 3.9
Asymptotic solutions of
showing effect of
for
= 0.1 and
=
= 1.
Results of simulations are also plotted for comparison
................................
.........
58
Figure 3.10
Comparison of
plots obtained from linear averaging (Equation 3.1) and double
reptation (Equation 3.4) mixing rules for composites with
=
0.1,
= 20 and
=
= 1
................................
................................
................................
.................
59
Figure 3.11
Asymptotic solution of
for
= 20 and (a) uniform CCR rates,
=
= 1 and
(b) independent CCR rates,
= 1,
= 5
................................
...........................
6
0
Figure 3.12
The low De maximum of Q
0
relative to the high De plateau of Q
0
for
= 20 with
(a) uniform CCR rates, varying
=
and (b) independent CCR rates, varying
with fixed
= 1
................................
................................
..............................
6
2
Figure
4.
1
(a) Storage modulus and (b) loss modulus of nanocomposites with varying
loading
and the matrix
................................
................................
................................
........
7
3
Figure
4.2
Dynamic storage modulus for composites and matrix at 200
°
C for (a) 1 rad/s with
corresponding (b)
parameter (c) Q parameter and (d) relative intensities
acquired using FT rheology at
= 0.8. Relative intensities shift factors are 1, 10,
100 for PP, PPNC3, and PPNC5, respectively
................................
......................
7
4
xii
Figure
4.3
Dynamic storage modulus against strain amplitude at various frequencies for (a)
matrix (b) 3wt% clay nanocomposite and (c) 5wt% clay nanocompo
site
.............
7
6
Figure
4.4
Relative third harmonic ratio against strain amplitude at various frequencies for (a)
matrix (b) 3wt% clay nanocomposite and (c) 5wt% clay nanocomposite
.............
7
8
Figure
4.5
Nonlinear parameter Q against strain amplitude at various freque
ncies for (a) matrix
(b) 3wt% clay nanocomposite and (c) 5wt% clay nanocomposite
........................
8
0
Figure
4.6
Frequency dependence of
=0.5) for nanocomposites and matrix
........
8
1
Figure 5.1
Dynamic storage modulus comparing effect of oligomer addition
.......................
8
6
Figure
5.2
Lissajous curves with corresponding elastic stresses for varying oligomer
concentration and strain amplitude at (a)
= 0.1 rad/s, and (b)
= 1 rad/s
........
8
7
Figure 5.3
Elastic and viscous stress waveforms at
=1.14 and
=1 rad/s for (a) nylon
co
polymer matrix (b) 5 wt% oligomer and (c) 10 wt% oligomer
..........................
88
Figure 5.4
Normalized elastic stress against strain amplitude for
= 1 rad/s
........................
89
Figure 5.5
Effect of oligomer addition on the nonlinear
parameter for
= 1 rad/s
........
9
0
1
C
HAPTER
1
INTRODUCTION
1.1
Research Background and Motivation
Rheology describes the flow and deformation of a material under an applied mechanical
force.
It is
the branch of physics which acts as a
bridge connecting fluid and solid continuum
mechanics
.
This encompasses a breadth of materials from liquids
to solids
particularly
to
complex
fluids
such as polymer melts
.
Understanding
the rheology of a material
proves
invaluable
for the
manufactur
ing
everyday goods
.
The manufacturing of plastic parts th
rough polymer processing involve a variety of
methods
such as
injection molding, extrusion,
compression molding, film blowing, film casting
and thermoforming.
Processing temperature,
s
peed and
part
size
are
some of the most important
parameters controlling the final quality of the part.
Polymer melts are highly sensitive to
processing temperature through their viscosit
y
. With increasing temperatures, viscosity decreases
resulting in
less resistance
to flow
and
faster
flow rates.
Speed and dimension are directly related
to
the rate of deformation (strain rate) and deformation amplitude (strain), respectively.
Polymer melts are viscoelastic fluids
,
meaning that their stress response is
both viscous
and elast
ic in nature.
In a
purely viscous response
,
stress grows proportionally to the applied strain
rate;
a
purely elastic response show
stress growth proportional with strain amplitude.
When a
material
undergoes a deformation
, the molecules or atoms making up
that material need time to
conform
.
In other words, each material has a characteristic structural relaxation time.
For
Newtonian fluids, such as water,
the relaxation time is much faster than the
time scale
associated
with the strain rate
leading t
o a pu
rely viscous response.
Conversely f
or elastic solids, such as
2
metal
s
,
the relaxation time is very long
leading to a purely elastic response
. Due to the molecular
weight, topology
and entanglements in polymer
melts, the
structural relaxation times are on
the
order of the deformation
rate.
Purely viscous and elastic responses are classified as linear
responses since the stress grow proportional to the strain rate and strain, respectively. The same
linear response
behavior is observed in viscoelastic mater
ials when both the strain rate and strain
are small
. However
, when the strain deformation or strain rate
is large enough to disrupt the
equilibrium structure of the polymer chains, then a
deformation dependent material response is
observed: the nonlinear
viscoelastic region
[1]
.
The lin
ear viscoelastic region
may be
probed by a rheologist through various
shear flow
tests
on a rheometer
w
hen the deformation and rate are kept small
. These include
steady
tests
such
as
steady shear,
stress relaxation
and
creep
,
as well as dynamic tests
(oscillatory shear)
such as
frequency
and strain sweep tests
.
Material functions such as
the
plateau modul
us,
,
shear stress
relaxation modulus, G(t),
creep compliance, J(t), zero

shear viscosity,
and the dynamic storage,
and loss,
moduli
are all obtainable through at least one of these tests.
These parameters
may
then be used to determine
features such as the polymer characteristic relaxation times, molecular
weight and even polydispersity
[2]
.
Dynamic
shear
strain sweep t
ests
operate
by
compress
ing a material
between two parallel
plates
or similar shear geometry in a rheometer. One plate is then subjected to an excitation
deformation
at a set frequency, while the resulting stress is measured
on the other plate.
From the
stress

strain relationship, the dynamic moduli are obtained
and a
s the deformation increases from
small strains to large strains
, th
ese
moduli become dependent on the strain amplitud
e marking the
onset of nonlinearity and hence
the nonlinear region
.
This
particular test
probes what
is
now
known as
large amplitude oscillatory shear (LAOS)
flows
[3, 4]
.
Over the past 20 years, rheologists
3
have begun to investigate the nonlinear region
through LAOS
[5]
.
Before LAOS, the nonlinear
region was only described
qualitatively with an increase/decrease in
described as strain
stiffening
/softening and an increase/decrease in
as strain
th
ickening/thinning. Now
quantitative interpret
ations of
LAOS flows
through
Fourier transform (FT) rheology
[6

14]
and
stress decomposition (SD) methods
[15

18]
have
been developed
. These methods have
frequently
cited in the literature
for systems including:
neat polymers
[12]
, branch

type polymers
[8, 9, 12]
,
blends
[19

21]
,
and
composites
[6, 7, 20, 22

24]
.
However,
interpretations of LAOS fl
ows using FT rheology
and SD methods are still in
their early stages.
While
the method development has been refined extensively in the literature
for
various types of systems, the quantity and variety of these systems is lacking. Particularly the area
of
polymer nanocomposites
, a class of viscoelastic materials
highly utilized in industry.
What
literature
that
does
exist are concerned primarily with highly loaded nanocomposites
[6, 7, 25]
,
where the nonlinear response is dominated by particle

particle interactions.
Therefore, this
research focuses on dilute nanocomposites where particle

polymer interactions govern the
rheology and nonlinear response.
This
also
requ
ires the development of nonlinear viscoelastic
constitutive model
s
for polymer nanocomposites to
relate the nonlinear response to particle

polymer chain dynamics
.
1.2
Dynamics of
Entangled
Polymer Chains
The dynamics associated with entangled polymer melt
s
are
considered here
.
The dynamics
of polymer
s
are well represented by reptation

based tube models
[26]
.
Reptation of polymers,
originally proposed by de Gennes
[27]
and later refined by Doi and Edwards
[28

31]
,
was
4
series of obstacles. These obstacles, made up of other polymer chains, form entanglements with
the primitive chain to create a temporary cage
1.1
.
Figure 1.1
Schematic of
reptation theory showing test chain in a tube
entangled with matrix
chains.
The number of entanglements and rate at which the
y are removed dictate the stress
relaxation behavior
of the polymer chain.
For linear, enta
ngled monodisperse
polymer chains,
several stress relaxation mechanisms occur in the linear viscoelastic region:
Rouse motions,
reptation and constraint release
[26]
.
The effect that these relaxation
mecha
nisms have on the stress
relaxation is best presented using a steady shear
stress relaxation master curve shown in Fig
ure
1.2
.
5
Figure 1.2
Schematic of stress relaxation
test and relaxation regimes
with corresponding
relaxation timescales.
When an
instantaneous strain deformation is applied to the sample
,
an increase in the stress
(shear modulus)
is observed and the
relaxation is recorded over time
.
At very short times, polymer
chains are frozen and behave as a glass
for which this region is
appro
priately named the glassy
region
.
The glass transition region
marks the onset of stress relaxation through
Rouse
motion
s
of
chain segments
, or fast thermal vibration
s
,
having a
distribution of very short
Rouse
relaxation
time
s
.
A
t
the equilibration
time
which is twice the longest Rouse time,
=
,
test chain
theory resulting in the
aptly named plateau
region
, characterized by the plateau modulus
for which all polymer chains ar
e now hi
ghly
entangled with adjacent polymer chains.
Given
enough time
, polymers
enter
the terminal region
whereby the chains
diffuse out of their tube
through large thermal motions
. This relaxation
mechanism is reptation and
is characterized by the
reptation
or
disengagement
time
,
,
until all
remaining stress
is relaxed.
An additional linear mechanism, constraint release
(double reptation)
,
considers both the reptation of
the primitive chain as well as the surrounding matrix
chains which
6
form the
entanglements.
With constraint release, the reptation
timescale is reduced
resulting in
faster stress relaxation and
an earlier onset in terminal behavior.
When
the strain deformation is
large
, additional nonlinear relaxation mechanisms are
present:
conve
ctive constraint release
and
chain retraction
after
stretching
.
Due to the velocity
gradient in simple shear flows
,
tube
entanglements
flow at different rates due to their position
along neighboring streamlines.
The inclusion of convective
constrain
t
rel
ease (CCR)
reduces the
polymer stress by
remov
ing
these
entanglements at a rate proportional to the shear rate
[32]
.
The
result is a plateau in the shear stress with increasing shear rate
, unlike the original Doi

Edwards
model which predicted that stress would go through a maximum
leading to excessive s
hear
thinning. In extensional flow, CCR
is less significant since there is no gradient along the
streamlines
[26]
. Instead, chains are subjected to
stretching
,
leading to
their stress buildup.
Only
after chains have retracted to their equilibrium length are they then able to reptate out of
their tube.
Large amplitude shear flows are able to stretch chains to some degree
, though not as severely as
extensional flows.
In comparison with Fig
ure
1.
2
,
the faster relaxing chain retraction mechanism
occurs on the order of Rouse times, while the CCR effect would be observed in the terminal region
.
1.3
Dynamics of Polymer Composites
Polymer composites are heterogenous mixtures
consisting
of
at least
a po
lymer matrix and
a
filler
material
.
Common fillers include
fibers (one

dimensional), platelet particles (two

dimensional) and spherical particles (three

dimensional).
The filler
type, concentration, size,
shape
,
and chemical compatibility with the polyme
r matrix are all factors which affect the rheology
of composites
[33]
.
Composites are desired
in commercial products for
their increased toughness
and
large strength

to

weight ratio
as well as their flame retardancy and improved barrier
properties
[34]
.
7
Th
ree types of interactions are present in polymer composites
which
lead to their desired
mechanical properties: particle

particle, particle

polymer, and polymer

polymer
[33, 35]
.
Particle

particle interactions are prominent in high
ly loaded systems where the filler content is high and
the interparticle spacing between particles are
small leading to a percolated filler network
[36]
.
These interactions also exist if the interparticle spacing is small enough to allow
attachment of
polymer chains between two separate particles i.e. bridgin
g.
Once a filler network forms, a sharp
liquid

to

solid transition
is
present
,
forcing
the
viscoelastic properties
to
become elastic

dominant.
In dynamic
shear rheology, this
effect is marked
by an increase in the storage modulus (elastic
response) at lo
w frequencies
relative to the unfilled polymer matrix
[33]
.
The chain confinement
o
f the polymer matrix between the filler
particles
lead to hindered mobility and thus longer
relaxation times
resulting in a loss of the
observable terminal region.
Since
the inverse of time is
frequency, the long

time terminal behavior for steady shear in Fig.
1.2
corresponds with the low
frequency terminal behavior in dynamic shear tests.
Dilute polymer composites
are able to have a similar
rheological response
to
filler
network
systems
through strong
particle

polymer interactions. Surface modification
of the filler surface
can create
attachment sites for the polymer matrix
resulting in
particle

attached chains with
reduced mobility
.
Some filler types, such as montmorill
onite clay, exist naturally with
free oxygen
and hydroxyl groups
which provide the basis for chemical attachment. Further s
ilylation of
these
layered

silicate fillers
have been shown to react both covalently
with the
hydroxyl groups
and
through hydrogen b
onding
with the
oxygen groups
on the clay edges and faces, respectively
[37,
38]
.
Improvements in nanocomposite properties have
been shown u
sing
an amine functionalized
silane
[38, 39]
,
which are able to
further react to long chain maleated polypropylene result in
attached chains with hindered mobility
as characterized by an increase in the low frequency
8
dynamic moduli.
In
a
ddition to the reduced ch
ain mobility, the large molecular weight of the
maleated polypropylene compatibilizer
created entanglements with the free polypropylene matrix
chains.
Reduced relaxation dynamics in f
iller networks and entanglement
networks
between
particle

attached chains
and free chains
are also accompanied by
a smaller linear viscoelastic
processing window.
At large deformations and deformation rates,
the networks break down
resulting in strain softening (decreased elastic response)
of the material. This is known as
the
Payne effect
for the breakdown of filler networks, though more recently it has been used
to
describe the breakup of particle

attached entanglement networks
[40]
.
1.4
Dynamic Shear Rheology
Dynamic
(oscillatory)
shear rheology
determines
both the viscous and elastic material
response
using an oscillatory excitation force at a desired strain amplitude
and frequency
,
(
1.1
)
T
he most common tests include frequency sweeps
(constant strain amplitude, varied frequency)
and strain sweeps (constant frequency, varied strain amplitude).
Illustrative examples of these
tests are shown in Fi
gure 1.3
.
Figure 1.3
Dynamic shear tests
using (a) frequency sweeps and (b) strain sweeps.
9
Frequency sweeps
shown in Fig
ure 1.3
(a)
are useful in identify
ing
the relaxation rates
associated with
reptation mechanisms.
As mentioned
earlier,
the disengagement time,
, is the
timescale by which st
ress is relaxed through reptative
motions.
In steady shear relaxation
, for
times greater than the reptation time, the terminal regime is observed.
Similarly, this terminal
region is observed in dynamic shear tests at low frequencies i.e. the inverse of l
ong scale reptation
times.
The terminal region is easily identified by the
quadratic scaling of storage modulus,
with
frequency (i.e.
) and the linear scaling of the loss modulus,
, with frequency (i.e.
).
While frequency sweep test
s occur over a range of frequencies, the strain amplitude is kept
small ensuring the sample is tested in the linear region.
The linear region is identified through
strain sweep tests depicted in Fi
gure 1.3
(b).
At a fixed frequency
,
the material undergoes a series
of increasing deformations.
The linear region is associated with a strain independent behavior in
the viscoelastic moduli.
At a critical strain
, the moduli
eventually show a strain dependence and
either decrease or increase
in magnitude, depending on the type of material.
This strain marks the
onset of nonlinearity and it
is a useful measure to separate out the linear and nonlinear regimes.
Recent literature in dynamic shear rheology concerning strain sweep tests
have
coin
ed the term
small angle oscillatory shear (SAOS) and large amplitude oscillatory shear (LAOS)
for linear and
nonlinear testing
, respectively
[5]
.
During
both SAOS and LAOS testing, oscillatory deformations following Eq
uation
1.1
are
applied and the resulting shear stress is determined,
(
1.2
)
Where
and
are the Fourier transform amplitude and phase angles for the
harmonic. In the
linear viscoelastic regime (SAOS) only the first harmonic (
= 1) is present. During LAOS flows,
the nonlinear relaxation mechanisms described in the previous sections
lead to
increase
s
in
the
10
higher order harmonics which are quantif
ied by FT rheology
using
Eq
uation 1.2
.
Before the
designation of LAOS flows,
measurements of the nonlinear response were limited to
a
rheometer
effectively reporting only
the first harmonic in stress across all strain amplitudes
. This
restrict
ed
the rheo
logist to
only
qualitatively identify
strain stiffening/softening
(
increase/decrease)
and
shear thickening/thinning
(
increase/decrease)
behavior
as well as
determ
in
e
the onset of
nonlinearity
for a given material
.
Relations to the structural morp
hology of the system were
limited
. However, in recent years, LAOS flows are accompanied by
qualitative,
quantitative,
and
semi

quantitative
techniques such as
Lissajous

Bowditch
analysis,
Fourier transform (FT)
rheology and stress decomposition.
Examples
of theses analyses are given in Fi
gure 1.4
.
Figure 1.4
Methods to quantify nonlinearities
in
(a) strain and stress waveforms using (b)
Lissajous

Bowditch curves (
c) FT rheology and (d) stress decomposition techniques.
A brief overview concerning the t
ypes of
LAOS techniques is presented here,
with a more
complete dis
cussion
given in Chapter 2.
An essential component
for any LAOS
test
are the
strain
11
and stress waveforms from the rheometer during testing
depicted in Fig
ure 1.4
(a).
Obtaining
these
wavef
orms
from
a
rheometer using
the
raw voltages of torque and angular displacement
through
a
high

speed data acquisition
(DAQ) car
d has been the standard
implementation
in literature
[11,
13, 14]
. In more recent years, commercial
rheometers provided by
companies such as
TA
instruments and Anton

Paar
, have incorporated LAOS test procedures directly into their software
.
Plotting the normalized
stress waveform against the normalized strain waveform yield the
qualitative Lissajous

Bowd
itch curves in Fig
ure 1.4
(b).
The
shape of these
curves are useful in
determining the type of response
the material undergoes during deformation
.
They can go from
completely circular (purely viscous) to
ellipsoidal (viscoelastic) to
a straight line (pure
ly elastic)
as seen in Fig
ure 1.5
.
Figure 1.5
Schematic showing trends
found
in Lissajous

Bowditch curves.
An equivalent measure of viscoelasticity is provided
in
SAOS tests
by
using
, the first
harmonic phase angle.
Additional nonlinearities at large deformations d
ue to microstructural
changes
[8, 12, 41]
,
entanglement and filler network breakup
[25, 42]
, or even
systematic errors due
to wall slip and edge fracture are all picked up through Lissajous

Bowditch curves.
A requirement
for additional LAOS analyses, such as FT rheology, is the acquisition of oscillatory
data at steady
12
state.
Lissajous

Bowditch curves provide a useful measure in the steady

state respons
e
since a
transient response (i.e. decaying
amplitude with time) lead to spiral curves instead of close
d loops.
Fourier transform rheology
is the interpretation of higher order
harmonics in the stress
waveform
at large deformations.
Fourier transformation translates the time

dependent stress
response into a frequency

dependent stress response at each strain amplitude.
The result are
intensities or stress
amplitudes
corresp
onding to the
higher order harmonics, which are some
integer multiple of the fundamental testing frequency i.e. the frequency imposed by the rheometer.
An example of the intensity plot from FT rheology is given in Fig
ure
1.4
(c).
Stress is an odd
function
of strain
(i.e.
) resulting in the observance of only odd harmonics.
By
normalizing the intensities with respect to the first harmonic, a relative intensity at each harmonic
is obtained with the largest corresponding to the third harmonic,
.
This parameter is a
characteristic measure of FT rheology and is the basis for all other nonlinear parameters such as
) and the zero

strain intrinsic nonlinearity
[10]
. More details are provided in
Chapter 2 regarding these parameters.
The onset of nonlinearity has been loosely
defined
in FT rheology
to occur
at
the strain
amplitude where
the value of
is greater than 0.5% of
stress response (i.e.
)
[5]
.
This is
the direct result of a major limitatio
n in FT rheology: electronic noise.
By definition, the
SAOS regime should
only have first harmonic contributions to the stress
when the strain amplitude
is kept low.
F
or most polymer
melts and solutions
these low strains lead to low stresses (i.e. low
voltages) which are more susceptible to electronic noise generated by the torque transducer.
Fourier transformation of the noise can lead to erroneous data
and an incorrect interpretation in the
LAOS behav
ior.
Thus
,
a requirement for FT rheology is the oversampling of
the raw voltage
waveforms
[11]
. Oversampling is an averaging technique used in combination with high

speed
13
DAQ cards and multiple
oscillation cy
cles per strain. By averaging several thousand data points
per second
or more depending on the testing frequency and DAQ rate, the signal

to

noise (S/N)
ratio
can be significantly improved after Fourier transforming the stress waveform
. An S/N ratio
of 1
0,000 is observed in Fig
ure 1.4
(c), which is typical for polymer melts
, while polymer solutions
can have S/N ratios as high as 100,000
[13]
.
Stress decomposition
is the final LAOS technique discussed
in this section
.
In SAOS
dynamic testing,
the stress waveform is decomposed into
the storage modulus (elastic response)
and the loss modulus (viscous res
ponse).
These responses can be attributed to the magnitude of
the phase lag
since
and
where
is the complex
modulus.
In a purely elastic
material,
an instantaneous
or in

phase
stress response is observed
upon deformation (i.e.
) while in a purely viscous material the stress
response is completely
out

of

phase with the strain deformation (i.e.
)
.
Thus, the elastic and viscous responses
are dependent on the relative
position between the stress and strain waveforms.
In the nonlinear
regime, the stress waveform is no longer sinusoidal
resulting in phase angles at higher order
harmonics
.
Stress decomposition techniques utilize
this nonlinear phase lag
to decompose
the
stress waveform into elastic and viscous contributions as shown in Fig
ure
1.4
(d). This was first
done geometrically by Cho
et. al.
[15]
and later a mathematical derivation using orthogonal
Chebyshev pol
ynomials was applied by Ewoldt et. al.
[17]
.
T
he linear viscoelastic moduli are
recovered via SD
techniques;
however a more accurate representation of these moduli
are
calculated in the nonlinear regime.
The nonlinear viscoelastic moduli
also hold the same physical
meaning as their
linear viscoelastic counterparts
.
Additionally,
the example in Fig
ure 1.4
(d)
show
a total
stress waveform
and viscous waveform
being nearly
identical in amplitude and phase,
suggesting that the material response is viscous dominant.
T
he elastic waveform
has a much lower
14
amplitude and takes on an exotic shape allowing the rheologist to classify the nonlinearity as an
elastic or solid

like response.
The LAOS techniques introduced in this section
are considered the most useful measures
of nonlinear rheology
developed
to date. While each have their own unique advantages,
associating
the nonlinear
viscoelastic behavior
with
polymer
architecture
,
morphological changes
and chain dynamics
still prove
s
challenging due to the
relative newness of this field.
Therefore,
mo
re
variety in the systems tested as well as relationships between
the nonlinear parameters to
chain dynamics through viscoelastic models are much needed.
1.5
Research Objectives
The motivation of this work stems from the lack of
experimental
and theoretical
research
concerning
large amplitude oscillatory shear flows. Particularly in the area of polymer
nanocomposites
, where the current research is concerned primarily with highly loaded systems
where particle

particle interactions govern the r
heology.
Dilute nanocomposites with strong
particle

polymer interactions have been
researched extensively in linear viscoelasticity, though
no
such research exists concerning nonlinear viscoelasticity
under LAOS
. This also implies that the
dynamics assoc
iated with th
ese systems are not well understood in LAOS flows
. This research
seeks to accomplish the following objectives:
(1)
To
design and implement
LAOS
functionality
in the current rheometer setup
and
develop a numerical framework for interpreting LAOS flows
using
techniques such as
Lissajous

Bowditch
curves, Fourier transform rheology, and stress decomposition
.
(2)
To develop a nonlinear viscoelastic model
for polymer nanocomposites and relate
the
dynamics associated with free and attached chain entanglements to the
nonlinear trends
observed in nonlinear rheology particularly those concerning FT rheology.
15
(3)
To
test
dilute polymer nanocomposites
under LAOS flows to determine the
effect
of
entanglem
ent network
breakdown on the nonlinear rheology and relate these
effects to dynamics
associated with model predictions.
(4)
To use
FT rheology and stress decomposition methods
to determine the elastic
effect imparted by low

molecular weight oligomers in po
lyamide blends
.
1.6
Scope of Thesis
This thesis is concerned with the nonlinear viscoelasticity of complex fluids. Particularly,
how complex fluids behave under large amplitude oscillatory shear (LAOS) flow.
In recent years,
LAOS flows have been successf
ully utilized to
relate the observed nonlinear
ity of polymers,
blends and composites to
their respective
chain
dynamics,
molecular topology,
and
structural
morphology.
Interpretations of LAOS flows are
made possible through qualitative and
quantitative
me
thods.
The first portion of this thesis (Chapter 2) describes
the
framework necessary for
interpreting LAOS flows. This
chapter
is applicable to both LAOS flows tested experimentally
and those simulated numerically using viscoelastic
constitutive models. Firstly, experimental tests
require an
initial
step in the data acquisition
of the torque
and
angular displacement at discrete
points in time during
strain sweep testing using raw voltages from the rheometer
instrument
. After
data ac
quisition, the resulting
voltages are then
converted to their corresponding stress and strain
values
.
This pre

processing step
is then followed by post

processing used for both experiments
and simulations. Post

processing involve several techniques
to in
terpret LAOS flows
for
which
the algorithms are discussed in detail.
Plots of shear stress versus shear strain, known as Lissajous

Bowditch plots, offer a graphical
way to evaluate the viscoelasticity of a material as a function of
strain and frequency.
Additionally, the stress decomposition (SD) technique can further
classify
16
the nonlinear mechanism b
y decomposing the shear stress component into
its
viscous and elastic
contributions
. Finally, the most sensitive method to interpret LAOS flows is
by utili
zing Fourier
transform (FT) rheology. In FT rheology, the time

dependent stress waveform is Fourier
transformed into higher order harmonics
, giving a quantitative value to the nonlinear behavior.
The second portion of this thesis (Chapters 3

4)
is
concern
ed with the dynamics of
polymer
nanocomposites under LAOS flows.
Graphical methods like the Lissajous

Bowditch plots are
excellent tools for quickly interpreting the viscoelasticity of a material, however they do little in
describing the
dynamics of polym
er systems.
In polymer nanocomposites,
the chain dynamics
associated with
polymer

particle attached chains
increase the solid

like (elastic) behavior of the
polymer matrix
as well as the nonlinearity at large strain amplitudes.
This effect is captured by
the
higher order harmonics in FT rheology.
In an effort to relate the nonlinear parameters
in
FT
rheology to the chain dynamics of particle

attached chains, a nonlinear viscoelastic constitutive
model was developed. This model, which accounts for
entang
lement networks formed by free
polymer chains
and
particle

attached chains, is subjected to LAOS flows
using both numerical
simulations and asymptotic solutions.
Chain
relaxation mechanisms
such as double reptation,
convective constraint release, chain stretch, and finite extensibility are all accounted for in the
model
and
are explained
in further detail in the next section.
Chapter 4
further expands the
experimental understanding of polym
er layered silicate nanocomposites tested under LAOS
flows.
With surface treated montmorillonite clay nanofillers
reacted to maleated polypropylene chains,
particle

polymer chain dynamics are investigated
using FT rheology. Furthermore, where most
resear
ch is concerned with the effect of particle

particle interactions, the
systems presented here
consist of primarily particle

polymer and polymer

polymer interactions.
17
The final
portion
(Chapter 5) investigates
the formulation of
polyamide blends
reacted wit
h
a
functionalized
elastomeric oligomer
under LAOS flows.
The oligomer of interest is
polyisobutylene succinic anhydride (PIBSA).
For this research both
FT rheology
and
stress
decomposition
are
utilized to
in the
nonlinear analysis.
In viscoelastic mate
rials,
stress relaxation
is dependent on both its
viscous (energy dissipation) and elastic (energy storage)
behavior
.
In
nonlinear rheology, the less dominant component is typically associated with the nonlinear
behavior. For polyamides, which are viscou
s dominant,
the degree of nonlinearity is probed by
added varying concentrations of the elastic dominant PIBSA. Even
with a low molecular weight,
variation in
PIBSA
concentration
show
distinct transitions in the
elastic
nonlinear behavior
suggesting the possibility of phase separation
indicating suitable homogenous blend formulations.
18
C
HAPTER
2
NONLINEAR
CHARACTERIZATION TECHNIQUES USING
LARGE AMPLITUDE OSCILLATORY SHEAR FLOW
2.1 Introduction
The subject of this th
esis is the nonlinear characterization of complex fluids under large
amplitude oscillatory shear (LAOS) flows.
As the name suggests, LAOS tests
involve dynamic
testing of a material in a shear rheometer using large amplitude shear deformations. While
all
dynamic shear rheometers are capable of imposing large strains onto a material, special hardware
and software is needed to interpret the
nonlinear response.
The implementation of the
required
hardware (i.e.
high

speed
data acquisition card) and
the
desig
n of the
necessary
software
are
explained
in detail here.
The
software developed for this research was written using MATLAB
.
Similar
packages to interpret LAOS flows
include
the MITLaos
package developed by Ewoldt
[43]
as well as a LabVie
w implementation designed by Wilhelm
[13]
and are freely available upon
request.
2.2 Data Acquisit
ion
The rheometer used in this study was an ARES

Classic manufactured by TA instruments.
It is classified as a separated motor transducer (SMT)
rheometer, where the two platens which
compress the sample
rotate independently from one another
;
the bottom pl
ate applies
the
deformation
(i.e. strain)
while the top plate measures the torque
(i.e.
stress
)
.
A
2K FRT
N1
force
rebalance
torque transducer is equipped capable of measuring torques up to 2000 g

cm a
s well as
simultaneously measuring normal forces during
testing.
A schematic of the rheometer test setup
19
as suggested by Wilhelm
[13]
for LAOS flows is given in Fig
ure 2.1
to help facilitate the
discussion.
Figure 2.1
Illustration of a typical
rheometer designed for testing LAOS flows.
A parallel plate geometry is depicted in Fig
ure 2.1
, though
any shear
geometr
y
may be
used for testing LAOS flows. Couette
geometries are useful for quantify low viscosity fluids such
as dilute polymer
solutions
. Torsion bars test solid samples that are below the glass transition
temperature
or at elevated temperatures below the melting point of the sample. Cone and plate
and parallel plate geometries are
used for the more viscous polymer melts, with the latter
being
the subject of this research.
Giacomin et. al.
[44]
compared the FT rheology results between
cone
and plate and parallel plates and found that the nonlinear ratio
for parallel plates needed to be
multiplied by
3/2 to match the nonlinear results of cone and plates.
L
inear viscoelasticity needs
no correction,
though
care must be taken when comparing
nonlinear results.
20
In order to probe the nonlinear rheology
of a sa
mple under LAOS flows using one of the
techniques introduced in the previous chapter (i.e. Lissajous

Bowditch, FT rheology and stress
decomposition)
,
raw voltages of the
angular displacement (i.e. strain) and torque (i.e. stress)
waveforms must first be ac
quired during testing from the rheometer.
Several components are
required for the acquisition of these waveforms:
(1) A rheometer
capable of outputting voltages of angular displacement and torque through
B
NC (Bayonet Neill

Concelman)
connectors
(2)
Double
shielded BNC cables to prevent electronic noise
(3)
An analog

to

digital (A/D) BNC adapter
(4)
High

speed data acquisition (DAQ) card
(5) A computer with sufficient
random

access
memory (RAM) installed
(6) Software for communicating with rheometer,
DAQ ca
rd and for post

processing LAOS
data
For the ARES
used in this study, several BNC connect
ions
are available
on the back

side
of the rheometer depicted in Fig
ure 2.2
.
21
Figure 2.2
Rear panel
connections
for
TA Instruments
ARES

Classic rheometer
.
Doubled shielded BNC cables
were
connected to the Torque
, Strain and Normal outputs in
Fig
ure
2.1
.
The
output voltages for each of these connections are summarized in Table
1.1
.
Table 1.1
Voltage ranges for
2K FRTN1 torque transducer equipped to ARES
rheometer
Connector
Voltage Range
Strain
(angular displacement)
0 Volts = 0 radians
,
± 5 Volts
= ± 0.5 radians
Low
Torque Calibration
0 Volts = 0 g

cm, ±5 Volts = ± 200 g

cm
High Torque Calibration
0 Volts = 0 g

cm, ±
5 Volts = ± 2000 g

cm
Normal Forc
e
0 Volts = 0 gmf, ±10 Volts = ±
2000 gmf
Identifying the
range of voltages is an important step in the acquisition of accurate LAOS
data
as these can vary depending on the rheometer and transducer installed.
The torque voltages
listed in Table
1.1
have
two regimes
for the 2K FRTN1 transducer
: low torque and high torque
.
The rheometer automatically switches to the appropriate calibration depending on the measured
22
stress
during testing. This will be made more clearly in the next section,
though
it should be
recognized that this transition
can create a challenge in separating out the waveforms at individual
strains
during
LAOS post

processing.
An additional input in Fig
ure
2.2
can be used to input a raw voltage corresponding
to a strain waveform
, adhering to the voltages in
Table
1.1
.
Klein et. al.
[45]
reconstructe
d stress data by
mathematically
superimpos
ing
several
strain
waveforms in the form
of sine, rectangular, triangular and sawtooth
shapes
which
were
found to
correspond to
the linear
response
, strain thinning, shear thickening and
wall slip
characteristics
,
respectively.
By fitting these responses to the stress waveform
,
they were able to
quantify
each linear and nonlinear contribution
based on the harmonics from FT rheology.
Experimentally, this could be validated by
testing these exotic waveforms using th
this feature was not explored for the work presented in this thesis.
Data acquisition was achieved using
a 16

bit resolution high

speed DAQ card
(PCIe

6341
X series) by National Instruments with a 100 kS/sec/channel
sampling rate
(kS = kilo Samples)
.
Resolution
determines the number of discrete voltages that can be
measured
from the rheometer.
For a 16

bit resolution, there are
measurable voltages
for the
ranges
listed in Table
1.1
.
Sampling rate controls the n
umber of voltages
recorded
over time
.
Early high

speed data
acquisition of
LAOS
flows
by
van Dusschoten and Wilhelm
[11]
used
a
16

bit resolution
DAQ
card with a
33 kS/sec/channel sampling rate
.
The sampling
rate
they suggested needed to be large
enough to allow for oversampling of the
data
, taking
several thousand data points
(time, stress,
strain etc.)
and averaging them into a single
data point
. T
hus a 100 kS/sec sampling rate with an
oversampling number o
f 1000
would generate 100 data points per second.
Oversampling serves
two purposes: increasing
the
S/N ratio
after Fourier transformation and to decrease file sizes.
The
former is more important than the latter
since
large digital storage capacity has become more
23
affordable in recent years.
O

the

during
testing the data is
averaged according to a fixed oversampling number.
With larger storage
devices an
d the fact that oversampling and S/N ratios are
correlated, it is suggested here that
averaging of the waveforms should occur in the pre

processing step outlined in the next section
and not during the data acquisition step
.
The ARES rheometer
is controlled
by the TA Orchestrator software
while data acquisition
was accomplished via
MATLAB using the built

in DAQ module. This module is able to
communicate directly with the National Instruments DAQ card.
Strain sweep tests are
defined
with
the following param
eters
: frequency, initial strain, final strain
, points per decade and cycles
before measurement.
The
total experiment runtime is dependent on f
requency, points per decade
and the number of cycles
as follows,
(
2.1
)
Where
is the testing frequency, P is
the number of points per decade and C is the number of
cycles before measurement.
The cycles
before
measurement represent a delay before the final
measurement cycle
i.e. the cycle used for determining the viscoelastic modul
i.
The testing time
should be evaluated before every test to ensure that
the material will remain thermally stable.
In
addition, file sizes generated from high

speed data acquisition m
ay also be estimated
in megabytes
,
(
2
.2
)
Where N is the number of channels (i.e. time, stress, strain),
is the DAQ sampling rate and the
prefactor 8
are the number of
bytes in a double

precision
floating point number.
Clearly an
incr
ease in the number of strain amplitudes and cycles
bef
ore measurement lead to larger file sizes,
though these offer an increase in resolutio
n and S/N ratios, respectively, for nonlinear analyses.
The size of the file may be reduced
significantly in the pre

processing stage after oversampling.
24
2.3
Pre

Processing
The input for pre

processing is a single file containing
the angular displacement, torque
,
and normal force voltages
as well as time from a single LAOS test over several strain amplitudes.
The pre

processing stage serves three purposes: ove
rsampling the
raw data, converting angular
displacement and torque into strain and stress, and c
ropping the
waveforms to their corresponding
strain amplitudes.
An
example of the raw voltage waveforms for a polypropylene homopolymer
at 1 rad/s
is
shown in F
ig
ure
2.3
.
Figure 2.3
Raw voltages of torque and angular displacement for an entire LAOS test.
I
nspection of the raw data show two
trends that are typical in
data collected during LAOS
tests. The first is a
short dead time
which is attributed to the ti
me between starting data collection
and beginning the
strain sweep
test
from
the
TA
Orchestrator software. The time at which testing
begins is manually recorded and the
dead time is trimmed for post

processing.
The second feature
is
a
sharp voltage drop
in torque
at 220 sec
. This corresponds to the
rheometer switching between
25
low and high torque calibrations, which is unavoidable.
In addition to the voltage drop, the
rheometer
re

tests the strain amplitude which includes delaying for a set number of
cycles
,
prolonging the experiment
.
Post

processing is unaffected if the data is trimmed
at the point where
the high torque calibration occurs. Hence, for raw voltage waveforms
similar to Fi
gure 2.3,
the
low torque and high torque calibrations are post

p
rocessed individually.
As it was mentioned earlier, oversampling improves the S/N ratio
by averaging over
several thousand data points to create a smooth waveform. This is
illustrated in Fig
ure 2.4,
for
a
simple sinusoidal waveform
generated
statistical
n
oise
for different oversampling numbers,
,
Figure 2.4.
Schematic illustrating the effect of oversampling
number on a noisy waveform.
All three cases in Fig
ure 2.4
have features characteristic of a sine wave regardless of the amount
of noise present
.
For
= 1,
where
there is
no oversampling
,
the noise
would contribute to higher
order harmonics
after Fourier transformation
preventing the reconstruction of the original sine
wave
. This noise is
almost completely
reduced
for
= 1
000, where
the original sine wave is
recovered
.
26
When interpreting real data, the choice in oversampling number is not arbitrarily made as
in Fig
ure 2.4
.
Instead, the oversampling number
is
calculated based on the DAQ rate and
excitation frequency,
(
2.3
)
Where
is the DAQ sampling frequency,
is the excitation frequency
and
is the
maximum observable harmonic in the nonlinear spectrum.
Due to the
harmonic dependence on
the oversampling number, the nonlinearity
must be approximated before testing if oversampling

the

a range of
oversampling numbers can be tested to identify which harmonic
generates the greatest S/N ratio.
After oversamplin
g the raw voltages, the data is
clipped to individual strain amplitudes
and converted from
angular displacement and torque to strain and stress. For parallel plates, the
conversion
of angular displacement
follows
to strain
amplitude
follows
,
(2.4)
Where
R is the plate radius and H is the gap height. Converting torque
to shear stress
follows,
(2.5)
Where
is the gravitational constant
.
Using
the example polypropylene shown in Fig
ure 2.3
, the converted strain and stress
waveforms at
=1.56
and 1 rad/s
are
shown in Fig
ure 2.5
.
27
Figure 2.5
Oversampled and clipped stress and strain waveforms
of polypropylene homopolymer
at
= 1.56 and 1 rad/s.
The LAOS test
conducted to generate these
waveforms w
as
implemented using a 4

cycle
delay before measurement
with the
rheometer
run
ning
an extra cycle after delay resulting in 5
cycles per strain
shown in Fig
ure 2.
3
.
T
ransitions
between increasing strain amplitudes result in
the occasional flow
instability. Thus
,
the first cycle is always
neglected in LAOS analysis which
explains why Fig
ure 2.5
has 4 cycles
. Once the waveforms are
appropriately scaled with
Equations
2.4

2.5
and clipped as in
Fig
ure 2.5,
each strain is
ready for LAOS analysis th
us concluding the
pre

processing step.
2.4 Post

Processing
Representative Figures
The MATLAB code developed for this research
focuses on the
three primary analyses for
interpreting LAOS flows
:
Lissajous

Bowditch plots
, FT rheology and stress decomposition.
This
section details the type of curves
obtained through the MATLAB code written for this thesis
.
A
full review of these methods is available elsewhere
[5]
.
28
2.4.1 Lissajous

Bowditch Curves
Lissajous

Bowditch plots are
easily prepared by plotting the no
rmalized stress waveform
against the strain waveform. They
offer a quick qualitative interpretation into the viscoelasticity
of a system.
Lissajous

Bowditch plots for polypropylene at 1 rad/s at several strains are shown in
Fig
ure 2.6.
Figure 2.6
Liss
ajous

Bowditch curves for polypropylene at 1 rad/s
showing effect of strain on
viscoelasticity and steady

state behavior.
Each strain presented in Fig
ure 2.6
occurs in the LAOS region
for
this particular
polypropylene. From
to 1.56
the ellipsoidal
shape
begins to transition toward a more
spherical behavior suggesting a viscous or liquid

like response associated with the
disentanglement of chains
at large strains. A secondary purpose for Lissajous

Bowditch plots is
identifying
the steady state beha
vior of the polymer during testing. Each Lissajous

Bowditch
curve in Fig
ure 2.6
is the superposition of 4
deformation cycles. At large strains where
= 3, it
is clear that a transient behavior is observed.
This transient may be due to melt edge fracture or
sample
drooling
and
should be neglected in a LAOS analysis such as FT rheology, since the
transient behavior will
affect the higher order harmonic contributions to stress.
29
2.4.2 Fourier Transform Rheolog
y
Fourier transform rheology
was first introduced nearly two decades ago with the
introduction of high speed data acquisition cards
and pioneered by Wilhelm
.
The LabView
program written by Wilhelm implemented a discrete Fourier transform (DFT) algorithm
t
o
determine the Fourier coefficients and phase angles in
Equation
1.2.
Similarly, the MATLAB
code presented here
computes the
transform (FFT) algorithm
,
(2.6)
Wher
e
is
a
generalized discretized time

domain
vector of length
,
is the corresponding
imaginary frequency

domain Fourier spectrum
vector
.
Any
imaginary number
including
are
generalized to the form
and the complex magnitude of this
number yield the corresponding
Fourier intensities,
(2.7)
With corresponding phase angles,
(2.8)
To determine the
true viscoelastic
phase angle, the strain waveform must also be Fourier
transformed and the resulting phase angle subtracted from the stress phase angle.
With the
Fourier intensities determined
from
Equation 2.7,
the corresponding intensity
plots can be constructed as a
function of the higher order harmonics.
The
S/N ratio is evaluated
from these figures and are
sensitive to the choice in oversampling number.
The effect of
oversampling number on the
Fourier transform intensities are compared for the
polypropylene
sample
at
and 1 rad/s
in Fig
ure 2.7
.
30
Figure 2.7
Fourier intensity spectrum for polypropylene at 1 rad/s and
=1.56 showing
(a)
correct (b) low and (c) high oversampling numbers.
An example of an ideal oversampling number is
shown in Fig
ure
2.7
(a), where the S/N
ratio is 10,000:1, typical of polymer melts.
When the oversampling number is too low, as in Fig
ure
2.7
(b), the S/N ratio is still quite high
, however the presence of many higher harmonics results
from the Fourier transform
fitting noise
instead of the true
stress waveform. Conversely, when the
oversampling number is too large, as in Fig
2.7
(c), the
S/N ratio is too
low
and no
distinguishable
harmonics are present.
The third harmonic intensity is the largest nonlinear
contribution making
it an ideal
measure for nonlinear rheology. This
is the motivation for using the
parameter as a means to
quantify structural morphologies
and polymer topology.
For many neat polymer
s,
has been
shown to increase quadratically with strain ampl
itude. This feature is also predicted by all
31
constitutive models
. Due to the quadratic nature of
,
Hyun
and Wilhelm
formulated a nonlinear
parameter Q
[10]
,
(2.9)
Since the quadratic dependence
of
is normalized in Q, a low strain plateau region is expected
resulting in qualitative features similar to the linear viscoelastic moduli (i.e. strain
stiffening
/softening). By taking the limit of
Q at low strains, the strain dependence can be
completely removed resulting in a frequency dependent nonlinear parameter known as the zero

strain intrinsic nonlinearity
,
(
2
.
1
0)
The nonlinear parameters
, Q and
are plotted for polypropylene at 1 rad/s
in Fig
ure
2.8
.
Figure 2.8
F
T rheology parameters for polypropylene at 1 rad/s showing (a)
and (b)
.
Higher order harmonic contributions are by definition zero in the linear viscoelastic limit.
The
nonlinear harmonic contribution in Fig
ure 2.8
(a) at low strains is the result of electronic noise
from the rheometer
. With increasing strain amplitude, the n
onlinearity grows quadratically with
strain amplitude indicating the
onset of nonlinearity and the start of the LAOS regime at
=
0
.4.
32
Similarly,
at this strain a plateau is observed in Q which can be extrapolated to the limit of zero

strain to get a val
ue of the
parameter.
2.4.3 Stress Decomposition
The final LAOS analysis is a semi

qualitative technique known as stress decomposition
where the shear stress is decomposed into
elastic and viscous contributions, similar to the linear
viscoelastic moduli
except applicable to the nonlinear regime
.
The original description of this
technique was done by Cho et. al.
[15]
and
later
refined by Ewoldt
et. al.
[17]
through
the use of
Chebyshev polynomials
to calculate the elastic and viscous stresses
.
An equivalent metho
d for
determining these stresses is by using the Fourier coeff
icients and phase angles from FT
rheology
[16]
,
(2.11)
(2.12)
These stresses are compared to the total stress for polypropylene at
= 1.56 at 1 rad/s in
Fig
ure 2.9
.
The shape and magnitude of the
elastic and viscous stresses relative to the overall shear
stress is useful in determining the type of nonlinearity. In Figure 2.9, the overall stress is most
closely related to the viscous stress suggesting a liquid like response and the resulting nonlin
earity
is due to the elastic stress. For more complex systems such as polymer nanocomposites where
particle

particle interactions dominate, the elastic and viscous stresses are highly sensitive to the
breakup of any filler network that forms
[18]
.
33
Figure 2.9
Elastic and viscous stress waveforms
relative to total stress waveform for
polypropylene at 1 rad/s and
=1.56.
2.5
Conclusions
This chapter focuses on the necessary framework for
testing the nonlinear rheology of
complex fluids under large amplitude oscillatory shear flows. A rheometer capable of outputting
raw voltages of stress and strain has the capability of testing LAOS flow
s so long as
a high

speed
data acquisition card is used
to collect the data. Faster sampling rates provide a greater S/N ratio
which is essential to determine the nonlinearity present in a system. These nonlinearities are
interpreted through several techniques such as Lissajous

Bowditch plots, Fourier transfo
rm
rheology and stress decomposition.
The most sensitive method, FT rheology, has been recently
cited in the literature as being able to detect structural morphologies, polymer topology such as
long chain branching, and even percolation thresholds in poly
mer nanocomposites.
34
C
HAPTER
3
DYNAMICS OF ENTANGLED POLYMER CHAINS WITH
NANOPARTICLES ATTACHMENT UNDER
LARGE AMPLITUDE OSCILLATORY SHEAR (LAOS)
3.1 Introduction
The rheology of polymer melts reinforced with nanoparticles is of continued interest
[36,
39, 40, 46

49]
becaus
e of the broad range of potential applications for these materials. In addition
to varying the loading of nanoparticles, particle

particle interactions and polymer

particle
interactions may be varied producing a variety of structural features with associat
ed rheological
signatures.
For particle volume fractions above the percolation threshold, a filler network may be
formed by direct particle

particle interactions or indirectly through bridging chains be
tween
nanoparticle surfaces. The nonlinear viscoelastic behavior of such polymer nanocomposites has
been attributed largely to the breakup of the filler networks. Polymer

particle interactions may lead
to a shell of adsorbed chains around the particles tha
t have greatly reduced mobility.
[50

54]
Furthermore, the attached polymer in this shell may be highly stretched.
[52]
Entanglement
networks between free polymer chains and polymer chains attached to nanoparticles are also
present.
[55]
Since polymer chains may
be attached to nanoparticles with varying levels of
interaction using different coupling agents and at different sites
[38, 56, 57]
like edges and faces of
nanolayers, it is important to understand the contribution that ent
angled chains attached to
nanoparticles make to the rheology of the nanocomposite melts.
When a polymer nanocomposite melt is subjected to
large amplitude oscillatory shear
(LAOS), the filler network is the first to breakdown leading to a strain amplitude
dependence of
35
the viscoelastic moduli known as the Payne effect.
[36, 40, 46]
The Payne effe
ct has also been
reported in nanocomposites with very low volume fractions of nanoparticles
[58]
where polymer

particle interactions dominate.
The stress response in large amplitude oscillatory flows may be
analyzed using Fourier transform (FT) rheology
[11, 13, 14]
and stress decomposition
[15

17, 20,
59]
(SD). In FT rheology, the combination of fast data acquisition
[14]
rates and oversampling
[11]
allow detection of higher harmonic content in the torque response

in particular the ratio
of
the third harmonic to the base harmonic. Hassanabadi et al.
[7]
have noted that the intensity ratio
increases progressively with volume fraction of nanoparticles in an EVA melt; the power law
exponent of the intensity ratio against strain amplitude decr
eases progressively from the value of
2 observed with the unfilled melt. An important point to note here is that the strain amplitude range
for LAOS tests on polymer melts in rotational rheometers is limited by the onset of edge fracture.
In another paper
comparing the responses of polyethylene nanocomposites with carbon nanotube
loading below and above the percolation threshold, Ahirwal et al.
[6]
report different trends in the
intrinsic nonlinearity parameter Q
0
with frequency.
The dynamics of entangled polymer chains are well represented by reptation

based tube
models.
[26]
One such model is the Marrucci

Ia
nniruberto
[60]
constitutive equation. Reptation of
poly
mers, originally proposed by de Gennes
[27]
and later refined by Doi and Edwards,
[28

31]
a series of obstacles. These obs
tacles, made up of other polymer chains, form entanglements with
to diffuse out of the tube is referred to as the reptation time or disengagement time. The
primitive
chain and tube are progressively longer with increasing molecular weight, resulting in more
entanglements and hence longer reptation times. Since the inception of the tube model, additional
36
relaxation mechanisms have been introduced. Specifical
ly, in the single mode Marrucci

Ianniruberto model, these are: double reptation, convective constraint release (CCR), chain stretch,
and finite extensibility effects.
The dynamics
of
a mixture of free polymer chains and
polymer chains attached to
nanoparti
c
les has been modeled recently by Sarvestani.
[61]
T
he stress was obtained
by linear
averaging over
volume fractions
(
) of the two types of chains
(
3.
1)
denoted by the indices
and
. The stress contribution for each type of chain was
predicted with
a
single mode Marrucci

Ian
niruberto constitutive equation
used to describe the chain dyn
amics
.
The relaxation times of the nanoparticle

attached chains were taken to be much greater than those
of the free chains. Results of computations were
presented to show that disentanglement of the
attached chains by convective constraint release (CCR), a mechanism relevant to fast flows, could
lead to strong nonlinear viscoelastic effects including the strain softening of the dynamic moduli.
The present
work seeks to address two limitations of the mixture model presented by
Sarvestani.
[61]
First,
linear averaging over volume fraction is inconsistent with the double
reptation formulation of the Marrucci

Ianniruberto model requir
ed for representing the mixture of
cha
ins
. Furthermore,
the CCR parameter which was assumed to be the same for both types of
chains in that work, will in general be different for the two types of chains, as explained in the
following sections.
The ob
jective of this paper is to present
a differ
ent version of the
Sarvestani
[61]
model
with FT rheology results from numerical computations as well as an asymptotic analysis for the
zero

strain intrinsic nonlinearity parameter
. This model
incorporates
(a)
an averaging scheme
consistent with the double reptati
on formulation for a mixture of different types of entangled
37
chains
based on types of entanglements
and (b) different CCR parameters
for
the two types of
chains in
describing the no
nlinear viscoelastic response to
large amplitude oscillatory shear of
nanop
article filled polymer melts.
3.2 Theory
3.2.1
Model for Polymer Nanocomposites
The entanglement network in polymer nanocomposites arises from a mixture of two chain
types: free matrix chains and particle

attached chains. A proper mixing rule is necessary to model
the entanglement contribution to the stress. The mixing rule used here
for such bi

disperse blends
is given in
Equation
2.
(3.
2)
A bilinear averaging scheme weighted with the volume fraction of chains
is used for the stress
tensor
where the indexing corresponds to the i

ed with the j

chain. Test chains are able to remove entanglements with tube chains by a process known as
reptation at a time scale
, the reptation time. Similarly, tube chains undergo the same reptative
relaxation mechanism resulting in an
additional loss of entanglements with the test chains. The
combined loss of entanglements between both test and tube chains through reptation is known as
double reptation.
[26]
The timescale associated with the loss of entanglements through dou
ble
reptation is given as,
(3.
3)
For
polymer nanocomposites, the mixing rule in
Equation
2 is expanded as follows.
(3.
4
)
38
where the superscripts
have been replaced with
to denote free and attached chains,
respectively. The free chain volume fraction is taken as
, where
is the volume
fraction of attached chains.
The stress tensor,
obeys the relation given by Marrucci
and Ianniruberto,
[60]
(3.
5
)
where
is the plateau modulus determined from linear viscoelasticity and
is the coupled
orientation

stretch tensor, unique to the Marrucci

Ianniruberto model. For large amplitude flows,
chain stretch is likely to occur and thus a complete mode
l should incorporate a finite extensibility
parameter
[62]
,
(3.
6)
Here,
is comparable to the square of the maximum chain stretch of the i

th chain.
The coupled orientation and chain stretch tensor,
, is solved using the single mode
differential
Marrucci

Ianniruberto model,
(3.
7)
w
here
is the velocity gradient tensor. The last two terms on the right

hand side of
Equation
7
correspond to the reptation and chain stretch relaxation mechanisms, respectively. The
characteristic relaxation times are given as the orientation time,
and the chain stretch relaxation
time,
. By taking the trace of
Equation
7
, the ch
ain stretch equation takes the form,
(3.
8)
From the derivation of the coupled orientation

stretch tensor
, the trace of
is equivalent to
the square of chain stretch.
[60]
For
which it is clear that the rate of chain stretc
h is controlled by
the stretch relaxation time.
39
As
a
consequence of the mixing
rule
in
Equation
4
, the orientation time originally derived
by
Marrucci and Ianniruberto [60]
is modified to take the form,
(3.
9)
w
here
is the
stretch relaxation
time and
is the CCR parameter for the tube chains.
For small amplitude oscillatory shear (SAOS) flows, the speed and deformation are not
sufficient in magnitude to cause chains to stretch (i.e.
) forcing the orientation time in
Equation
3.
9 to follow the double reptation timescale shown in
Equa
tion 3.
3. For LAOS flows,
chain stretch can occur and the CCR effect, the second term on the right

hand side of
Equation
3.
9, decreases the orientation time through the removal of entanglements. The rate at which these
entanglements are removed is contro
lled by the CCR parameter.
I
t should be noted that in SAOS
flows
, where chain stretch does not occur, there is symmetry between free

attached and attached

free entanglements (i.e.
). This is not
necessarily
true, however,
for LAOS flows
as t
he
CCR effect
and chain stretch relaxation time
in
Equation 3.
9
may
reduce the orientation time at
different rates for each entanglement.
In this model characteristic relaxations times were defined as a set of
dimensionless quantit
ies,
(3
.
10)
(3.
11)
(3.
12)
(3.
13)
Here it should be noted ratio of reptation times,
in
Equation 3.
10
, was originally described by
Sarvestani [55]
and
is referred to in this work as the relative strength of attachment, with strong
40
attachment yielding greater values of
. The quantities in
Equations 3.
11

3.
12 are referred to as
the relative stretch times and are related by the parameter
, defined in
Equation 3.
13.
3.2.2
LAOS Simulation Scheme
The shear stress components,
,
in
Equation 3.
4
for simple shear flows are calculated using
the component forms of
Equation 3.
7 for each entanglement pair,
(3.
14)
(3.
15)
(3.
16)
where the orientation time,
is defined in
Equation
3.
9.
Two types of simulations are performed: strain sweep tests and frequency sweep tests. Strain
sweep tests are conducted to investigate trends at large strains. For low and moderate strains, an
asymptotic solution is derived and discussed in the nex
t section. Frequency sweep tests are
simulated to validate the asymptotic solution. For both tests, the shear stress component in
Equation 3.
5 is solved for each entanglement pair by first solving the coupled ordinary differential
equations in
Equations
3.
14

3.
16 using MATLAB. The result are four shear stresses, each
corresponding to a different entanglement pair, which are combined into a total shear stress
according to the mixing rule in
Equation 3.
4.
p
ackage, the total shear stress is evaluated using
a Fourier series expansion,
[3]
(3.
17)
41
where
and
are the Fourier coefficients and phase angles for the
harmonic, respectively.
Representing the stress as a Fourier series as in
Equation 3.
17 begins the analysis known as Fourier
transform (FT) rheology.
[13]
Using the first harmonic (
), the linear viscoelastic storage modulus,
and loss
modulus
are obtained,
(3.
18)
(3.
19)
A charact
eristic measure for FT rheology is the relative third harmonic ratio,
, defined as,
(3.
20)
The relative third harmonic ratio is reported to scale quadratically with strain,
[5, 10]
a trend
predicted by all constitutive models
[63]
including the one presented in this work, as will be
discussed in a later section.
Noting the quadratic dependence of
with strain, Hyun and Wilhelm
[10]
derived a
nonlinear parameter
,
(3.
21)
At low strains,
becomes independent of strain leading to a new nonlinear parameter
,
(3.
22)
where
is now referred to as the
zero

strain intrinsic nonlinearity
, a frequency dependent
parameter.
3.2.3
Asymptotic Analysis for
Since
is a zero

strain limit parameter, a low strain asymptotic solution is derived by
expanding the tensor
using a power series expansion of the strain amplitude,
,
42
(3.
23)
and the chain stretch, which is equ
ivalent to the trace of
, is written as
(3.
24)
Substitution of
Equations
3.
23

3.
24 into
Equations
3.
7

3.
8 yields a set of differential equations
which are solved analytically and substituted into
Equation 3.
5 to obtain the low strain asymptote
of shear stress,
for each entanglement pair.
The real,
and imaginary,
Fourier c
omponents are determined from the stress
waveform for the
harmonic as follows,
(3.
25)
(3.
26)
The real and imaginary Fourier components in
Equations
3
.
25

3.
26 are then combined using the
mixing rule in
Equation 3.
2 for each

th harmonic,
(3.
27)
(3.
28)
where the storage and loss moduli for each harmonic take the form,
(3.
29)
(3.
30)
Finally, the asymptotic expression for
is obtained as,
(3.
31)
43
3.3 Results and Discussion
3.3.1
Linear Viscoelasticity
Linear viscoelastic frequency sweeps are first investigated using an asymptotic analysis as
outlined in the previous section. By integrating the shear stress using
Equations
3.
25

3.
26, the real
and imaginary first harmonic intensities for the entanglement
pairs are obtained as
,
(3.
32)
(3.
33)
where, for generality, the Deborah number,
, is defined using the characteristic
timescale for double reptation given in
Equation 3.
3. It is more convenient and practical to express
these numbers in terms of the Deborah number defined with the free chain reptation time,
. The relationship between
and
is given in Table
3.
1.
Table 3.1
Relationship
between
and
for each entanglement pair
Combining
Equation 3.
32 with
Equations 3.
27 and
3.
29, the linear viscoelastic storage modulus is
given as,
(3.
34)
An expression for the loss modulus may also be obtained in the same fashion. The storage
modulus, which scales quadratically with frequency in the terminal region, is most sensitive to
relaxation phenomena and is therefore i
nvestigated in this section.
44
It is evident from Table
3.
1 and
Equation 3.
34 that two parameters govern the trends in the
linear viscoelastic storage modulus:
the strength of individual chain attachment
and the volume
fraction of attached chains
.
The effects of these parameters on the storage modulus are shown
in Figure
3.
1.
Figure 3.1
Asymptotic solutions for linear viscoelast
ic storage modulus with
(
a
)
varying
strength
of attachment
and fixed
= 0.1
;
(
b
)
varying
volume fraction of attached chains
and fixed
= 20
.
It is seen from
Figure 3.
1(a) that with increasing values of
at a fixed value of
an
increase in the modulus is observed, particularly at lower frequencies.
The increase in modulus for
nanocomposites, particularly at low frequencies, is well known for systems where polymer

particle
interactions occur. Surface treatment of the filler, which increases the number of active sites for
attachment with the matrix, is one method to promote poly
mer

particle interactions.
Figure 3.
1(b)
45
shows that an increase in
with a fixed value of
= 20, also increases the magnitude of the
storage modulus at low frequencies.
With increasing values of
and
ge modulus is
observed at low Deborah numbers, starting at De=2/
. This
corresponds to the normalized inverse of the attached chain relaxation time, where the factor of 2
comes from double reptation.
In the next section, the nonlinear viscoelastic regime
is discussed where additional
nonlinear relaxation mechanisms relax the stress: convective constraint release, chain retraction
and finite extensibility.
3.3.2
LAOS Strain Sweeps
Strain sweep simulations conducted using 24 logarithmically spaced strain am
plitudes
ranging from 0.01 to 10 are tested over varying Deborah numbers. For each strain, 1000 cycles
with 2
16
data points per cycle are simulated. The number of data points per cycle is chosen to
improve the signal

to

noise ratio during Fourier transfo
rm rheology calculations.
[9]
The choice
in cycle number is necessary to reduce
numerical error
[8]
and validate numerical s
imulations with
asymptotic solutions.
Strain sweep simulations are used to probe the nonlinear behavior of the model.
At large
amplitudes, several nonlinear relaxation mechanisms occur: CCR, chain stretch and finite
extensibility effects. In slow to mode
rate flows (i.e. De <
), the CCR relaxation has a
pronounced effect on the nonlinearity. Uniform CCR rates having
=
, for both free and
attached chains are presented first and followed by independently varying CCR rates,
.
Chain
stretch relaxation and finite extensibility parameters are taken from Marrucci and
Ianniruberto
[60]
as
=
= 0.01 and
=
= 100, respectively, and remained constant for the
46
analysis of this work. This is a reasonable choice since chains become fully stretched only for
flows with De >
, w
hich are not discussed in this work.
Four types of entanglements are present in the system: those in an attached chain
environment (i.e. attached and free test chains entangled with attached tube chains) and those in a
free chain environment (i.e. attached and free test chains entangled with
free tube chains). For
uniform CCR rates where
=
, the stress magnitudes follow:
>
=
>
where
the larger stresses have slower relaxation rates. The equality for mixed chain entanglements may
be inferred from orientatio
n time in
Equation 3.
9 when the reptation, CCR and chain stretch
relaxation modes are equivalent. Since these entanglements are indistinguishable, they contribute
equally to the overall system stress and nonlinearity.
The strain dependence of the storage
modulus for the matrix (
= 0) and two
nanocomposites (
= 0.1) having both weak (
= 5) and strong (
= 20) polymer

particle
interactions with for
=
= 1 is shown in
Figure 3.
2.
For all cases, De =
1 has an onset of
nonlinearity near
= 1 and demonstrates the most intense strain softening effect. For De = 0.1,
the composites onset are
= 2 and 0.5 for
= 5 and
20, respectively, where the matrix onset
occurs near
= 4. The decrease in onset strain amplitude with increase in par
ticle

polymer
attachment is similar to the Payne effect, however for dilute systems the effect is associated with
the breakdown of entanglement networks and not a filler network. At low Deborah numbers (De
= 2/
), the strain softening is due to the break
down of attached

attached networks. At high
Deborah numbers (De > 2), all entanglements are lost resulting in the greatest degree of strain
softening. To further investigate the nonlinearity in Figure 3.2, the accompanying third harmonic
ratio
from
FT rheology is presented in Figure 3.3.
47
Figure 3.2
Numerical predictions of
dynamic storage modulus
with
strain amplitude and
=
= 1 for
(
a
) matrix
;
(
b
) composite
with
= 0.1 and
= 5;
(
c
) composite
with
= 0.1 and
= 20.
48
Figure
3.3
Numerical predictions of
with
strain amplitude and
=
= 1 for
(
a
) matrix
;
(
b
)
composite
with
= 0.1 and
= 5;
(
c
) composite
with
= 0.1 and
= 20.
49
In all cases, the
is shown to scale quadratically with strain amplitude. Deviation from
this scaling is considered the onset of nonlinearity,
[5]
shown in
Figure 3.
2. Comparing the matrix
and
= 5 cases, a monotonic increase in the
mag
nitude with increasing Deborah number is
observed, with the value for the composite being slightly greater at all Deborah numbers. A clear
reversal in the
trends at De = 0.1 and 0.2 for
= 20 is shown in
Figure 3.
3(c). Here, the
nonlinearity goes
through a minimum at De = 0.2 from De = 0.1 to 1 for the stronger attachment.
The Deborah number where this minimum occurs corresponds to the low De plateau observed
from the linear viscoelastic frequency sweeps in
Figure 3.
1. For De = 0.2 in
Figure 3.
3
(c), the
nonlinearity deviates from a quadratic scaling with strain amplitude and exhibits a greater power
law exponent with strain near
= 1 than the matrix and
= 5 cases, which still show a power
law exponent of 2. The contrast between c = 20 and
c = 5 is more strongly evident in the plots of
another nonlinear parameter
, defined in
Equation 3.
21 and presented in
Figure 3.
4.
With increasing strain amplitude, a decrease in
is observed for all cases except
= 20,
where instead an overshoot is
observed for De = 0.2. This overshoot in
is reported
for comb
polymers with highly entangled arms experimentally by Hyun and Wilhelm
[10]
and
numerically
predicted by Hyun et al.
[9]
using the pom

pom model. In comb polymers and branched polymers,
to which the pom

pom model applies, the backbone can relax substantially only after the branches
relax.
[27]
This
is not the case in polymer nanocomposites, where both the
attached chain
entanglements and free chain entanglements are able to relax simultaneously. Only decreases in
with strain amplitude have been reported for nanocomposites

possibly because the LAOS
testing was not done in the low frequency region. At
this time, more research in the area of
nanocomposites is needed to verify trends in
.
50
Figure 3.4
Numerical predictions of
with
strain amplitude and
=
= 1 for
(
a
) matrix
;
(
b
)
composite
with
= 0.1 and
= 5;
(
c
) composite
with
= 0.1 and
= 20.
51
Figure 3.5
Effect of uniform CCR rates for composites with
= 0.1 and
= 20 at De = 0.2 on
(
a
)
;
(
b
)
.
Stretch parameters:
=
= 0.01 and
=
= 100.
The effect of varying CCR on the overshoot in
is examined first with uniform
CCR rates:
=
and next with independently varying CCR rates where
. Two cases of uniform
CCR rates are shown in
Figure 3.
5.
An increase in
from 1 to 5 leads to increases in
and
. The qualitative trends in both parameters are preserved with an increase in the CCR
parameter: an increase in the power law exponent of
with strain amplitude and the presence
of an overshoot in
. An increase in nonlinearity with increasing CCR
parameter is also reported
by Sarvestani
[61]
in his model computations with respect to the dynamic storage modulus at large
strains.
52
The choice of independently varying CCR rates, where
, is proposed to control the
rate of release of entanglements during f
low for each network. The motivation for this stems from
the reported configurations of attached chains especially close to the nanoparticle surface. Holt et.
al.
[52]
also simulated
particle

polymer interactions of varying strengths and molecular weights
and found that for stronger chain attachment through covalent bonding, particle

attached chains
are stretched near the particle surface, while still entangled with the bulk matrix far
away from the
surface. Senses and Akcora
[64]
derived a mechanistic model to predict the elastic stress buildup
in polystyrene/silica nanocomposites under LAOS flows and found that the model matches well
with experiments when chains near the filler surface are stretched. These results imply that a
d
istribution of stretching occurs along the attached chain segments during flow. Stretched chains
are unable to form entanglements until fully retracted,
[26]
thus a distribution of stretching in the
attached chains would lead to fewer entangleme
nts than with unstretched chains. Hence we have
chosen to represent this effect with a higher CCR parameter for the attached chains; i.e.
>
.
For large
,
for the free

attached entanglements approaches
resulting in faster rel
axation
than the attached

free entanglements. This faster relaxation leads to the following ordering of
component stresses from different types of entanglements:
>
>
>
, where now the
component stresses of mixed chain entangleme
nts are no longer equal as they are with uniform
CCR rates allowing each entanglement to uniquely contribute to the stress and nonlinearity.
Conversely, it follows that attached chains convected at a slower rate will require
<
in which
approaches
and the stresses follow:
>
>
>
. It is the effect of CCR on the
chain environment (i.e. attached and free chains entangled with attached tubes) and the mixing of
these entanglement networks which show differe
nces
in the nonlinearity for
>
and
<
.
53
Figure 3.6
predicted with (a)
>
and (b)
<
for nanocomposites with
= 0.1
and
= 20 at De = 0.2.
The third harmonic ratio predicted with
is presented in
Figure 3.
6(a) and the
result predicted with
<
is presented in
Figure 3.
6(b), both at De = 0.2. The
=
= 1
case from
Figure 3.
5(a) is plotted for comparison.
T
he increase in logarithmic slope obtained with
=
is not observed in either case with unequal CCR parameters. For
>
in
Figure
3.
6(a), a decrease from the quadratic scaling is observed near
= 0.6. While for
<
in
Figure 3.
6(b),
follows very nearly a quadratic scaling with st
rain amplitude for larger strains.
54
Figure 3.7
Predicted
with
<
and
>
for composites with
= 0.1 and
= 20 at
De = 0.2.
The effect that varying CCR rates have on the attached and free chain environments is more
clearly depicted in a plot of
, presented in
Figure 3.
7. A weak overshoot is observed in
Figure
3.
7 for the
<
while similar low strain asymptote is
obse
rved when
compared with
=
= 5 in
Figure 3.
5(b). The
plot for
>
however, clearly shows no overshoot and a
progressive decrease in magnitude with increasing strain amplitude. The disappearance of the
overshoot may be attributed to
an increase in nonlinearity in the low strain region. This indicates
that the nonlinearity is more sensitive to the dynamics of the slower relaxing attached chains and
the entanglements they participate in. It has already been mentioned that only a decre
ase in
with
increasing strain amplitude for nanocomposites are reported in the literature. Therefore, we focus
the discussion now on nonlinear trends caused by the removal of entanglements by CCR for the
case
>
.
55
For
>
, the component stresses
,
,
are evaluated at varying strain
amplitudes and plotted in
Figure 3.
8(a). This figure illustrates the order of breakup of the
entanglement networks and their contribution to the nonlinearity.
Specif
ically, the onset of
nonlinearity in
Figure 3.
8(a) follows the order:
. Similar trends in the
parameter in
Figure 3.
8(b) confirm that attached chain entanglements contribute greatest to the
nonlinearity at each strain am
plitude.
Figure 3.8
Component stresses with increasing strain amplitude and
>
for composites with
= 0.1 and
= 20 at De = 0.2 showing (a) normalized storage modulus; (b) third harmonic ratio.
56
3.3.3
Asymptotic Solution
A low strain asymptotic solution of
is derived. The asymptotic shear stress for each
entanglement pair network is integrated using
Equations
3.
25

3.
26 for the third harmonic (
).
The real third harmonic intensity for each entanglement is given as,
(3.
35)
and the imaginary third harmonic inten
sity,
(3.
36)
As with the first har
monic ratios in
Equations
3.
32

3.
33, the Deborah number
is defined in
Table 3.
1. The scaled stretch time
appearing in
Equations
3.
35

3.
36 is defined as,
(3.
37)
and the scaled stretch time is rewritten in terms of dimensionless groups in
Table 3.
2.
57
Table 3.2
Definition of
for each free and attached chain entanglement pair
The third harmonic intensities for each entanglement may be separated into three terms
arising from the following effects: double reptation with chain stretch (DRCS), convective
constraint release (CCR) and finite extensibility (FE). Since both
and
are finite and non

zero, the DCRS term will always remain non

zero. However, if CCR is neglected (i.e.
) or
the chains are considered Gaussian with infinite extensibility (i.e.
), the CCR and FE terms
will respectively drop out of
Equations
3.
35

3.
36. Combining the third harmonic contributions
from various entanglement types into the mixing rule given in
Equations
3.
27

3.
28 yields the third
harmonic intensity. Finally, combining the total first and third harmonic intensities into
Equation
3.
31 yields the asymptotic solution for
, which is discussed here.
The zero

strain intrinsic nonlinearity
is a frequency dependent parameter. Unlike the
linear viscoelastic moduli which depend on
and
alone,
is sensitive to the
CCR parameters.
The effect of
on
for the matrix and composites with
= 0.1 is shown in
Figure 3.
9 using the
asymptotic solution. Results of numerical simulations using FT rheology are presented with the
asymptotic solutions.
58
In
Figure 3.
9, fo
r weak attachment with
= 5,
is found to be higher for the composite
than for the matrix up to De = 1. Above De = 1, the two curves merge onto a high Deborah number
plateau, similar to the linear viscoelastic modulus in
Figure 3.
1(a). For strong at
tachment with
= 20, a peak appears in
at De = 0.08, with a minimum near De = 0.2. The peak is due to the
relaxation of entanglements with attached chains, whereas
the minimum results from the
destruction of these entanglements.
Figure 3.9
Asymp
totic solutions of
showing effect of
for
= 0.1 and
=
= 1. Results
of simulations are also plotted for comparison.
The minimum is observed only for well separated relaxation times

when the particle

polymer interactions are strong. However, this is not unique to the double reptation averaging
scheme used in this work.
59
Figure 3.10
Comparison of
plots obtained from linear averaging (
Equation 3.
1) and double
reptation (
Equation 3.
4) mixing rules for composites with
= 0.1,
= 20 and
=
= 1.
A comparison of
between double reptation averaging used in this work and the linear
aver
aging presented by Sarvestani
[61]
is made for
= 0.1 and c = 20, shown in
Figure 3.
10. Both
linear averaging and double reptation generate a peak in
. The nonlinearity predicted by linear
averaging
in
Equation 3.
1 is higher than by double reptation
in
Equatio
n 3.
4 because the weighting
is higher in
Equation 3.
1.
In an effort to illustrate the differences between having uniform CCR rates and independent
CCR rates, plots of
are compared for increasing various volume fraction of attached chains,
with
fixed at 20. These plots are shown in
Figure 3.
11 for nanocomposites with uniform CCR
rates (
=
= 1) in
Figure 3.
11(a) and independent CCR rates (
= 5,
= 1) in
Figure 3.
11(b).
The
curves in
Figure 3.
11 rise in magnitude with increas
ing volume fraction of attached chains
regardless of the chosen CCR parameters.
60
Figure 3.11
Asymptotic solution of
for
= 20 and (a) uniform CCR rates,
=
= 1 and
(b) independent CCR rates,
= 1,
= 5.
For
=
in
Figure 3.
11(a), all curves merge to a high De plateau above De = 1. This
is consequence of
=
resulting in equal nonlinear contributions to
.
For
>
, the
plateau at large De is also raised in magnitude with increasing volume fraction of
attached chains,
due to an increase in nonlinearity from
entanglements shown in
Figure 3.
8(b). The increased
relaxation of
entanglements due to an increase in the CCR rate in the attached chain
environment shifts the nonlinearity
to lower Deborah numbers, resulting in an increase in the
plateau.
Hassanabadi et. al.
[7]
report that
for EVA nanocomposites with low loadings of
clay was greate
r than
for the matrix at larger Deborah numbers (ca.
). Based on the
61
results of
Figure 3.
11(b), only when the attached chain CCR rates are faster than CCR rates for
free chains, can the nonlinearity of the composite be greater than that of th
e matrix
.
This shows
that for
>
, faster flows, indicated by higher Deborah numbers, have a greater convective
effect on the nanoparticles resulting in the particle

attached chains contributing more to the
nonlinearity for all Deborah numbers.
T
o further illustrate that the nonlinear behavior in nanocomposites is more sensitive to
independent CCR rates than uniform CCR rates, the peak magnitude
at low Deborah numbers
relative to the
plateau at high Deborah numbers is calculated for
over varying
values.
This ratio is plotted in
Figure 3.
12(a) against values of
=
and in
Figure 3.
12(b) against
varying
with
fixed at 1.
For all choices of
=
presented in
Figure 3.
12(a), the relative
nonlinearity is independent of the CCR rate, showing only an increase with increasing
.
However,
Figure 3.
12(b) shows an increase in the relative nonlinearity with increasing
which
is amplified by an i
ncrease in
.
By incre
asing
and the CCR rate of the attached chain
environment, the
and
entanglement networks dominate the nonlinear response.
62
Figure 3.12
The low De maximum of Q
0
relative to the high De plateau of Q
0
for
= 20 with (a)
uniform CCR rat
es, varying
=
and (b) independent CCR rates, varying
with fixed
=
1.
3.4 Conclusions
A new nonlinear viscoelastic model is developed in this work for entangled polymer chain
networks with attachments to nanoparticle surfaces. The stress contributions of different types of
entanglements are averaged in a double

reptation framework with inde
pendent convective
constraint release parameters for particle

attached chains and free chains. The nonlinearity here
may be attributed to the breakup of entanglements between particle

attached and free chains.
Entanglements with the particle

attached chai
ns lead to the greatest extent of nonlinearity; this is
63
further intensified by slower relaxation dynamics and an increase in the fraction of attached chains
present in the system.
Numerical simulations of strain sweep tests under LAOS flows are performed
and analyzed
using Fourier transform rheology to compare the nonlinearity between nanocomposites and
unfilled polymers. Several measures are used to describe the nonlinearity: the onset of strain
softening in the elastic modulus as well as the magnitude a
nd strain dependence of
and
.
The trends in the zero

strain intrinsic nonlinearity
are also investigated using a low

strain
asymptotic solution of the model.
The onset of strain softening in the low frequency plateau region is seen to be parti
cularly
sensitive to the strength of polymer

particle attachment
or c
. The predicted nonlinearity for cases
where the particle

chain attachment is strong (large
) displays several distinct features. When the
CCR parameters are identical for attached cha
ins and free chains, the nonlinearity including strain
softening and a strain overshoot in Q are predicted. Specifically, there is an increase in the power
law exponent of
with strain amplitude corresponding to an overshoot in
at low Deborah
numbe
rs (De
2/
) when the CCR parameters are chosen to be equal for both chain types. When
the CCR parameter for attached chains is larger than for the free chains, the predicted nonlinearity
displays no overshoot in Q.
The choice of higher CCR parameter
for attached chains is motivated by reports in the
literature that the attached chains are stretched near the particle surface during flow, leading to
fewer entanglements.
[52]
This ef
fect is modeled here using a faster CCR rate for attached chains
than for free chains. The latter aspect leads to a higher
for the composite compared to the
matrix at all Deborah numbers. However, when the CCR rates are equal, the same degree of
nonl
inearity is predicted for the composite and the matrix at high Deborah numbers. Hence in
64
LAOS flows, faster convective constraint release of attached chain entanglements leads to quicker
breakup of the entanglement network resulting in higher nonlineariti
es predicted for
nanocomposite systems.
65
C
HAPTER
4
FOURIER TRANSFORM RHEOLOGY OF
POLYPROPYLENE

LAYERED SILICATE NANOCOMPOSITES
4.1 Introduction
Polymer layered

silicate nanocomposites have been of practical interest for reinforcing the
polymer matrix le
ading to an increase in both stiffness and toughness of manufactured
products.
[65]
Economic factors and commercial availability make clay and other layered

silicates
an attractive choice for both industry and
academia. The abundance of literature on the processing
of these materials also offer a means to fine tune the desired final composite properties by tailoring
the interactions between the polymer and filler phases.
[38, 66, 67]
Additional interactions between
particles are also present with high filler
concentrations resulting in a percolated filler network.
[68

70]
Though the filler network dominates the rheological properties
, it can often lead to a brittle
material due to early breakup of the network at large deformations i.e. the Payne effect.
[40, 46,
71]
Similar improvements in the rheology have been shown at low lo
adings when improved
dispersion of the nanofiller.
[38, 72]
In general, two criteria are needed for good dispersion: intercalation of the polymer chains
and delamination of the silicate layers.
[49, 69, 73]
Montmorillonite clay, a 2:1 phyllosilicate, is
one example of a layered silicate filler used in making nanocomposites. Montmorillonit
e exists
naturally in stacks of nanolayers having a dimension of 1 nm thickness and an aspect ratio around
200. To promote intercalation of the polymer matrix, the gallery spacing is first increased through
a cation exchange process, substituting the sodi
um cations for larger molecular weight
surfactant.
[57, 74]
Surface treatment of the nanoclay through sila
ne addition then provide
66
chemically reactive sites for the polymer matrix or an energetically similar compatibilizer resulting
in particle

attached chains.
[38, 66]
These particle

attached chains have reduced mobility due to
their attachment and with
sufficient molecular weight, are free to entangle with the matrix chains
to form an entanglement network which demonstrates a liquid

to

solid transition in the rheological
response.
[33, 37, 47]
The linear viscoelastic effect of particle

attached entanglements on the melt rheology for
layered

silicate nanocomposites has been characterized extensively. Most notably is the observed
increase in the dynamic shear moduli at low frequenci
es due to the slower dynamics of attached
chains.
[33, 37]
When exposed to increasing deformations, these entanglements also lead to an
earlier observed nonlinear response similar to the Payne effect for percolated filler networks.
[58]
In recent years, the nonlinear response has been probed using large amp
litude oscillatory shear
(LAOS) flows in order to relate the nonlinear behavior to the dynamics and morphology of various
complex fluids, including nanocomposites.
[4]
Much of the literature concerning LAOS flows of nanocomposites is focused
on loadings
above the percolation threshold.
[6, 7, 25]
Arguably the most sensitive method to quantify the
nonlinearities in LAOS flows is through Fourier transform (FT) rheology, whereby the non

sinusoida
l stress waveform in the nonlinear regime is Fourier transformed into higher order
harmonics.
[5, 13, 14]
Normalizing the third harmonic intensity with the first harmonic yields
,
a characteristic measure in FT rheology. H
assanabadi et. al.
[7]
used FT rheology to determine the
nonlinear
ratio for ethyl vinyl acetate (EVA) composites using bo
th carbon nanotubes (CNT)
and montmorillonite clay and found that with increasing filler loading, particle

particle
interactions raised the nonlinearity in magnitude. The quadratic scaling of
with strain
amplitude observed for the matrix, a trend fo
und in many unfilled polymer systems, was no longer
67
observed for composites where particle

particle interactions were dominate. Similar trends in FT
rheology for highly filled nanocomposites are reported confirming the particle

particle effect on
the nonl
inear response.
[6, 25]
The aim of this work is to quantify the nonlinear response of particle

attached
entanglements in dilute polypr
opylene montmorillonite clay nanocomposites under LAOS flows
using FT rheology. This study also tests LAOS flows over a range of frequencies to characterize
both the strain and frequency dependence of the nonlinear FT rheology parameters. At the
frequenc
ies chosen, the nonlinear response due to entanglement breakup of both particle

polymer
and polymer

polymer interactions are explored.
4.
2 Experimental
4.
2.1 Materials
Polypropylene (PP) homopolymer matrix (trade name: PP4792E1) with a melt flow rate
(MFR)
of 2.7 g/10 min (ASTM D1238, 230
°
C, 2.16kg load) was obtained from ExxonMobil. A
maleic anhydride grafted polypropylene

polyethylene (PP

g

MA) copolymer (trade name:
PO1015) with an MFR of 150 g/10 min (ASTM D1238, 230
°
C, 2.16kg load) and a maleic
anhydr
ide (MA) content of 0.42% was purchased from ExxonMobil and used as a compatibilizer.
The polypropylene homopolymer and compatibilizer were then melt mixed with an organically
modified montmorillonite (OMMT) clay. The OMMT (trade name: Nanomer I.44P) was
obtained
from Nanocor. The OMMT is produced by the manufacturer with a quaternary onium surfactant
with two C

18 tails through a cation exchange process to increase interparticle spacing between
the clay galleries and increase exfoliation during processi
ng. To improve compatibility of the clay
with the matrix, a vapor

phase silylation technique was employed to treat the clay surfaces with
1% by weight of aminoalkyldimethoxysilane.
[75]
The combined effect of silylation and
68
compatibilizer lead to the formation of particle

attached chains which have reduced chain
dynamics relative to the matrix chains.
4.
2.2 Sample Preparation
Nanocomposites were prepared with both 3 and 5wt%
of clay with a ratio of PP

g

MA to
OMMT held at a 1:1 compatibilizer to clay by weight. A summary of the materials used in this
study is found in Table
4.
1.
Table 4.1
Formulations for Different Nanocomposites
Sample
Polypropylene
(PP4792E1)
Maleic anhyd
ride
grafted PP
(PO1015)
Organically modified
montmorillonite clay
(I.44P)
PP
100 wt%


PPNC3
96 wt%
3 wt%
3 wt%
PPNC5
90 wt%
5 wt%
5 wt%
Polymer nanocomposites were melt blended in a Leistritz
twin screw extruder. A
masterbatch of 54:23:23 PP:PP

g

MA:OMMT was first extruded at 200 RPM and 180
°
C. The
resulting extrudate was then cooled and pelletized. Masterbatch pellets were then let

down with
PP and extruded to achieve the desired clay conc
entrations. Polymer nanocomposite pellets were
compression molded into 75x75x1 mm plaques at 200
°
C using a Wabash compression molding
machine. The plaques were then used for rotational rheometry testing.
4.
2.3 Linear Rheology
Steady and dynamic shear tes
ts were performed in an ARES rotational rheometer with a
2K FRTN1 torque transducer from TA instruments. Tests were conducted using 25 mm parallel
plates in a forced convection oven under a nitrogen atmosphere at 200
°
C. Before each test, samples
were loa
ded into the rheometer using a gap

closing test procedure controlled by the rheometer
software to remove loading effects and achieve reproducible data. Plates had an initial gap setting
69
of 2.05 mm where samples were loaded and allowed to melt for 2 min.
After melting, the gap was
closed to 1.05 mm uniformly over 500 sec. The sample was then
trimmed,
and the plates were
then further closed to 1 mm over 25 sec at which point a 10 min annealing stage began. The total
sample loading time with annealing was
approximately 20 min. Dynamic time sweep tests at 1
rad/s and 1% strain confirmed that the transients in modulus were minimized and reproducible for
each nanocomposite tested.
The linear region was determined through small angle oscillatory shear (SAOS) s
train
sweep tests at 1 rad/s from strains of 0.1

100% with 7 points per decade. For all samples,
frequency sweep tests at a strain amplitude of 1% were conducted from 0.01

100 rad/s with 7
points per decade at 200
°
C.
4.
2.4 Nonlinear Rheology
Large
amplitude oscillatory shear (LAOS) tests used the same loading procedure as for the
linear testing of polymer nanocomposites. Several low frequencies ranging from 0.1

10 rad/s were
tested from 30

300% strains with 7 points per decade with 5 cycles per str
ain. This cycle number
was chosen to prevent thermal degradation for low frequencies, where the testing time was much
longer. Additionally, the deformation history was kept constant across all frequencies. In all
strain sweep tests, the total testing t
ime was kept under 1 hour which was verified through time
sweep tests as a thermally stable operating window.
4.
2.5 Fourier Transform Rheology
To interpret LAOS results, Fourier transform (FT) rheology was employed.
[13]
Raw
voltages of torque (stress), angular displacement (strain) and force were acquired from the
rheometer using a 16

bit resolution h
igh

speed data acquisition (DAQ) card (PCIe

6341 X series,
70
National Instruments) with a 100 kS/sec/channel sampling rate. A lab

written MATLAB code for
data acquisition and post processing was used.
The advantage of FT rheology is in its ability to determ
ine nonlinearities that develop in
the stress waveform. To reduce systematic noise from the rheometer, oversampling of the raw
voltages acquired through the DAQ card was necessary to improve the signal

to

noise (S/N) of the
Fourier transformed stress wave
form.
[11, 14]
The oversampling number is the number of
raw
(
4.
1)
where
is the DAQ sampling frequency,
is the testing frequency and
is the maximum
observable harmonic. Wilhelm et. al.
[14]
suggested that the maximum harmonic is doubled in the
oversampling number as to prevent oversampling
of the nonlinear stress response. It should be
stress waveform was performed during data aquisition. In this work, we have eliminated the need

the

averaging and instead oversample the data after testing has completed. This then
allows for an optimum oversampling number to be determined during post

processing so as to
achieve large S/N ratios.
A full review on FT Rheology and LAOS techniques is avail
able elsewhere.
[5]
In this
work, three nonlinear parameters are of particular importance: the relative third harmonic ratio
, the nonlinear parameter
and the zero

strain intrinsic nonlinearity
.
The ARES rheometer used in this study is equipped with a separated motor

transducer
(SMT), meaning the bottom plate displaces the sample and the top plate measures the resulting
torque. The deformation app
lied is sinusoidal,
71
(
4.
2)
where
and
are the strain and strain amplitude, respectively. The resulting stress is
mathematically represented by a Fourier series,
(
4.
3)
where
and
are the Fourier coefficients and phase angles, respectively. In the linear regime,
only the first harmonic (
= 1) is observed. In the nonlinear regime, higher order harmonics appear
resulting in a non

sinusoidal stress waveform. It is the increase in
the third harmonic (
= 3) from
which the nonlinear FT rheology parameters are derived. The relative third harmonic is defined
as,
(
4.
4)
It has been shown mathematically throughout the literature
[5, 9, 76]
that the
harmonic
of stress, when expanded as a power series in strain, is related to
. Therefore, the relative third
harmonic ratio
scales quadratically with strain. These trends were also observed
experimentally in neat systems.
[10, 12]
However, deviations from a slope of 2 have been reported
for composites,
[7]
strain hardening systems,
[77]
strain stiffening suspensions,
[78]
and long chain
branching.
[79]
In an effort to develop a frequency dependent nonlinear parameter to remove the strain
dependence of
, Hyun and Wilhelm
[10]
first developed a new nonlinear parameter,
(
4.
5)
At low strains a monotonic value is observed resulting in the final nonlinear p
arameter
,
(
4.
6)
72
Where the zero

strain intrinsic nonlinearity
is now a frequency dependent parameter which is
sensitive to both system morphology and relaxation dynamics. Experimentally, the
was shown
to be sensitive
to the percolation threshold in composites when the filler loading was increased,
and particle

particle interactions dominated the rheology.
[6, 7]
In addition, well separated maxima
in
at low and high frequencies were obs
erved for comb

type polymers which corresponded to
the polymer backbone and arm relaxation times.
[10]
The same sensitivity in
to relaxation
dynami
cs was later shown numerically using constitutive models such as the model.
[9]
4.
3 Results and Discussion
4.
3.1 Dynamic Frequency
Sweep Tests
Ren et. al.
[38]
compounded PP/OMMT composites using the same organically modified
filler and compatibilizer used in this study, with a different polypropylene matrix having a higher
melt flow rate. It was proposed that two attachment sites are available for the dimethoxy groups:
covalent linkages at the clay edge and hydrogen bonding
in the clay galleries. The resulting imide
linkage between the silylated clay and PP

g

MA compatibilizer lead to the formation of particle

attached chains. Entanglements with particle

attached chains generate an increased linear
viscoelastic response in
both the storage and loss moduli at low frequencies for PP/OMMT
nanocomposites compared with the PP matrix as shown in Fig
ure 4.
1.
73
Figure
4.
1
(a) Storage modulus and (b) loss modulus of nanocomposites with varying loading
and the matrix.
For highly
filled systems, particle

particle interactions have been reported to dominate the
linear viscoelasticity at low frequencies due to the formation of a filler network.
[40, 69, 70]
The
composites studied here have low concentrations where no filler network is present, though the
possibility for chain bridging exist
s. However, the increase in storage modulus in Fig
ure
4
.1
(a) at
low frequencies is primarily attributed to the entanglements with particle

attached chains.
4.
3.2 Fourier Transform Rheology
The entanglement network formed by particle

attached chains was tested under LAOS
flows and the nonlinear response quantified with FT rheology. Oversampling of the stress
74
waveform is
n
eeded to achieve high S/N ratios. The oversampling number in eq. (1) mu
st be large
enough to reduce the electronic noise from the rheometer, but small enough as to not average out
the true nonlinear response of the polymer.
[11]
For neat systems, the deviation from electronic
noi
se to the true nonlinear response is typically marked by a quadratic dependence of
with
strain amplitude. However, power

law scaling less than 2 have been reported for various systems,
most notably in branched polymers
[8, 10, 12]
and highly loaded nanocomposites.
[6, 7]
In this
study, strain sweep tests at a fixed frequency of 1 rad/s were first tested for both composites and
matrix across two regime
s: SAOS (linear) and LAOS (nonlinear). The resulting storage modulus
and nonlinear parameters are presented in Fig
ure
4.
2.
Figure
4.2
Dynamic storage modulus for composites and matrix at 200
°
C for (a) 1 rad/s with
corresponding (b)
parameter (c
) Q parameter and (d) relative intensities acquired using FT
rheology at
= 0.8. Relative intensities shift factors are 1, 10, 100 for PP, PPNC3, and PPNC5,
respectively.
75
The onset of nonlinearity is observed from the storage modulus in Fig
ure
4.
2(a) f
or
composites at
= 0.04 and 0.02 for 3wt% and 5wt% composites, respectively, compared to the
matrix onset amplitude
at
= 0.4. However, these strains are less clear in the
nonlinear
parameter in Fig
ure
4.
2(b). At lower strain amplitudes in the SAOS regime the electronic noise
dominates the nonlinearity. For the matrix, a quadratic dependence with strain amplitude at
=
0.3 is clearly observed marking the true nonlinear response. It has been suggested
to consider
strains with
> 0.005 as the starting point for the nonlinear analysis
[5]
,
however this criterion
fails for the nanocomposites tested here
. In the nanocomposites, a region exists where the
is
nearly independent with strain due to the competing effects of the true nonlinear response and
electronic noise from the rheometer. For this study, this plateau in
is considered the noise
level criterion and only strains above this level are considered.
Nanocomposites do not show a quadratic scaling behavior in
at large strains above the
defined noise threshold. As consequence, no plateau region is observed for Q in Fig
ure
4.
2(c) for
nanocomposites.
Based on the results of Fig
ure
4.
2(b), it was determined that strain amplitudes
above
= 0.3 were less affected by noise and thus only these strains were considered for LAOS
testing. Validation of the oversampling method are represent
ed by shifted intensities from Fourier
transform of the stress waveform in Fig
ure
4.
2(d) resulting in large S/N ratios of 10,000:1.
The storage modulus for the composites and matrix under LAOS testing are presented in
Fig
ure
4.
3.
76
Figure
4.3
Dynamic storage modulus against strain amplitude at various frequencies for (a) matrix
(b) 3wt% clay nanocomposite and (c) 5wt% clay nanocomposite.
77
The dynamic storage modulus presented in Fig
ure
4.
1 showed the largest variation at low
frequencies where
particle

polymer interactions were dominant. This variation is also observed
with strain amplitude in Fig
ure
4.
3 for 0.1 rad/s in composites compared to the matrix. Here the
composites show an earlier onset of nonlinearity and an increase in the degree o
f strain softening
with increasing filler loadings as compared to the matrix. This behavior is similar to the Payne
effect, where an earlier onset of nonlinearity and increase in strain softening is due to the breakup
of a filler network. For dilute syst
ems, as in the nanocomposites presented in Fig
ure
4.
3(b,c), the
Payne effect is associated with the breakdown of entanglement networks formed by particle

attached chains.
[58]
The corresponding relative third harmonic
ratio for frequencies tested in Fig
ure
4.
3 are
presented in Fig
ure
4.
4.
Both composites in Fig
ure
4.
4(b,c) s
how a greater nonlinear response
over the matrix Fig
ure
4.
4(a) indicated by a larger
magnitude. The
magnitude for the
matrix increases with increasing frequency which is typically observed for unfilled polymer
systems.
[10]
However, this trend is not observed in nanocomposites, where instead an increase in
frequency leads to a decrease in the magnitude of
especially in the low frequenc
y regime
where particle

attached entanglements dominate the rheology. For all samples, the nonlinearity
collapses at frequencies above 5 rad/s with the PPNC3 composite in Fig
ure
4.
4(b) and matrix in
Fig
ure
4.
4(a) showing similar nonlinear behavior. This
suggests that entanglements with particle

attached chains are removed at higher frequencies and the nonlinear response is due to polymer

polymer interactions. This also corresponds with linear viscoelastic results in Fig
ure
4.
1, where
the moduli are equal
for the nanocomposites and matrix. A slightly higher nonlinearity is still
observed at higher frequencies for the PPNC5 case in Fig
ure
4.
4(c) which may be due to
hydrodynamic interactions caused by the increase in the number of particles present.
78
Figur
e
4.4
Relative third harmonic ratio against strain amplitude at various frequencies for (a)
matrix (b) 3wt% clay nanocomposite and (c) 5wt% clay nanocomposite.
79
The value of Q defined in
Equation 4.5
was determined as an intermediate step to acquire
and is presented in Fig
ure
4.
5. As consequence of the quadratic scaling in
with strain
amplitude for the matrix in Fig
ure
4.
4(a), a plateau in Q at strain amplitudes near
= 0.5 is
observed for all frequencies. The monotonic value of Q is direc
tly followed by a decrease in
magnitude with increasing strain amplitude. However, only a decrease in Q with increasing strain
amplitude is observed for the nanocomposites in Fig
ure
4.
5(b,c). Even over a larger strain
amplitude range in Fig
ure
4.
2(c), no
clear plateau in Q exists for the nanocomposites tested here
due to limitations in the rheometer. When Hyun and Wilhelm
[10]
originally defined
they
suggested that the true value was obtained by averaging the plateau region of Q over several strain
amplitudes. While this was possible for monodisperse unfilled polymer systems, the lack of a
plateau clearly presents a challenge for determining
in nanocomposites with strong particle

particle and particle

attached interactions. Hassanabadi et. al.
[7]
also did not observe a plateau in
Q over several strain amplitu
des for ethyl vinyl acetate nanocomposites filled with
montmorillonite clay.
Values of
were calculated in this work by setting it equal to the value of
Q at
= 0.5 for all samples across all frequencies. These effective
values are plotted agai
nst
frequency in Fig
ure
4.
6.
80
Figure
4.5
Nonlinear parameter Q against strain amplitude at various frequencies for (a) matrix
(b) 3wt% clay nanocomposite and (c) 5wt% clay nanocomposite.
81
Figure
4.6
Frequency dependence of
=0.5) for
nanocomposites and matrix.
While this may not be the true
value, it does provide a means for comparing the
nonlinear frequency dependence between composites and the matrix. As with the trends in
,
decreases with increasing frequency while the
opposite effect is observed for the matrix.
Ahirwal et. al.
[6]
tested the frequency dependence of
for polyethylene nanocomposites filled
with multiwal
led carbon nanotubes and found that above the percolation threshold,
decreased
with increasing frequency due to the breakup of the filler network. Without a filler network
present, the decrease in
is associated to the entanglement breakup of parti
cle

attached chains
suggesting that FT rheology is sensitive to the surface treatment of the clay and reaction with
compatibilizer. This is further supported by the monotonic decrease of PPNC3 towards the matrix
asymptote, where only polymer

polymer entan
glements exist. An increase in clay loading lead to
an overall increase in
, though the trends between both nanocomposites are nearly identical.
82
4.
4
Conclusion
s
Polypropylene

layered silicate nanocomposites with filler loadings below the percolation
t
hreshold were investigated under LAOS flows. Surface modification of the organically modified
montmorillonite clay filler with silane treatment created reactive sites for attachment with a maleic
anhydride grafted polypropylene with sufficient molecular w
eight to entangle with the
polypropylene matrix. The resulting particle

attached chain entanglements led to an increase in
the linear viscoelastic storage moduli at low frequencies suggesting a reduction in chain mobility
and slower relaxation dynamics.
The nonlinear response of the nanocomposites was investigated
using FT rheology and quantified by the relative third harmonic ratio
and
. In the unfilled
polypropylene matrix, the
followed a quadratic scaling with increasing strain amplitud
es and
increased in magnitude with increasing frequency. However, both nanocomposites displayed an
with a power law scaling less than 2 and a decrease in the magnitude with increasing
frequency. These trends resulted in
increasing with increasi
ng frequency for the polypropylene
matrix while a decrease with increasing frequency was found for both nanocomposites. This
suggests that
was sensitive to the entanglement network breakup formed by the slower relaxing
particle

attached chains. An in
crease in filler concentration also increased the magnitude of
while preserving the decrease with increasing frequency.
83
C
HAPTER
5
EFFECTS OF REACTIVE OLIGOMER ADDITIVES ON
MELT RHEOLOGY OF NYLONS: FOURIER TRANSFORM RHEOLOGY
5.1
Introduction
Successful rubber toughening of polyamide blends has been extensively studied in the
literature to improve the elastomeric properties of polyamide matrix through increases in impact
strength and increase in elongation at break
[80, 81]
. However, polyamides generally are generally
incompatible with natural rubber fillers leading to poor morphological structures and reduced
mechanical properties
[82]
. Reactive compatibilization has been frequent
ly used to promote
compatibility with the dispersed rubber phase using maleated elastomers which react with the
amine end groups and create a block copolymer with increased adhesion to the rubber phase
[83]
.
Large amplitude oscillatory shear flows combined with Fourier transform
rheology is
sensitive to polymer morphology and topology
[5]
. In the case of polymer blends, it has been used
to detect the miscibility of polymer blends c
ontaining nanocomposites
[19

21]
as well as detecting
the small morphological changes due to the addition of small droplets of a dispersed phase
[84]
.
In this stud
y polyamide blends reacted with an elastic oligomeric functionalized rubber are
investigated under large amplitude oscillatory shear. With the rheology of polyamides suggesting
a viscous dominant response, the aim of this study is to see what changes appe
ar in the nonlinear
behavior with the addition of oligomer. Fourier transform rheology and stress decomposition
techniques are identified as the best techniques for quantifying this effect.
84
5.2 Experimental
5.2.1 Materials
Polyamide copolymer with an 80:20 wt% ratio of nylon

6 to nylon

6,6 was obtained from
ExxonMobil. The copolymer was produced via a continuous polymerization process and received
in pelletized form. Blends consisting of the PA copolymer with an anhydride
functionalized
oligomer were produced by ExxonMobil using 5 and 10 wt% oligomer. Two oligomeric
polyisobutylene succinic anhydrides (PIBSA) with varying molecular weights (Mw = 750, 1000)
were obtained by Dover Chemicals.
5.2.2 Blend Preparation
Blends
consisting of 5 and 10 wt% oligomer were received from ExxonMobil after melt
mixed with polyamide copolymer via extrusion and pelletized. Blends produced in lab using
PIBSA were melt mixed in a Haake RheoDrive batch mixer at 80 rpm for 15 minutes with Ban
bury
blades. Polyamide copolymer and blends were then compression molded into square molds of
dimension 75x75x1 mm at 240
°
C in a Wabash compression molding machine. Samples were
vacuum dried for 3 days before testing, to ensure all moisture was removed.
5.2.3 Dynamic Shear Rheology
Rheological measurements were performed in an ARES (TA Instruments) strain

controlled
rheometer using a 25 mm parallel plate geometry. Linear viscoelastic frequency sweeps were
conducted at 230
°
C and
=0.03
from
=
0.1

1
00 rad/s at 7 frequencies per decade. Strain
sweep tests extending from small angle oscillatory shear (SAOS) flows to large amplitude
oscillatory shear (LAOS) flows were conducted at 230
°
C and 1 rad/s from
= 0.05
5 with 7
strains per decade and 5 cy
cles per strain amplitude.
85
5.2.4
Nonlinear Analysis
LAOS flows were interpreted using Fourier transform (FT) rheology
[11, 13, 14]
with a
home

written MATLAB code. Data acquisition from the ARES rheometer was performed using
a 16

bit resolution high

speed data acqu
isition (DAQ) card (PCIe

6341 X series, National
Instruments) with a 100 kS/sec/channel sampling rate.
The stress was Fourier transformed using
the fast Fourier transform algorithm to get the higher order harmonics,
(5.1)
Where
and
are the Fourier intensity and phase angles corresponding to the
harmonic. The
nonlinear parameter
=
was used to determine the degree of nonlinearity.
Stress decomposition methods
[16]
were implemented to decompose the total stress into
elastic and viscous stresses using the Fourier intensities and phase
angles in Equation 5.1,
(5.2)
(5.3)
Stress decomposition results were combined with Lissajous

Bowditch curves to qualitatively
describe the nonlinear behavior.
5.3
Resu
lts and Discussion
5.3.1 Linear Viscoelasticity
The linear viscoelastic frequency sweep for polyamide blends mixed with 5 wt% and 10
wt% PIBSA oligomer is shown in Figure 5.1.
86
Figure 5.1
Dynamic storage modulus comparing effect of oligomer addition
The linear viscoelastic data is sensitive to the morphological changes due to the addition
of oligomer. With 5 wt% oligomer addition, the polymer blend exhibits a greater solid

response
to deformation, particularly at lower frequencies. This response can
be attributed to the reactive
compatibilization between the nylon copolymer matrix and the PIBSA oligomer. Further addition
of oligomer to 10 wt % leads to an elastic modulus much less than the matrix response. While the
same extent of reaction is expec
ted, the excess PIBSA must phase separate to form a dispersed
region of lower viscosity due to the low molecular weight. Nonlinear rheology was used to further
classify the morphology of the system.
5.3.2 Nonlinear Viscoelasticity
LAOS flows were used t
o quantify the morphological changes through reactive
compatibilization leading to observed nonlinearities in the stress. One technique is to use Lissajous
plots for a qualitative inspection on the viscoelastic behavior of the stress. From linear viscoel
astic
experiments, the low frequency regime where
= 0.1 rad/s show the greatest variation between
blends and matrix. As frequency increased, the 5 wt% blend asymptotically approached the matrix
87
behavior, suggesting that the oligomer slows the relaxatio
n dynamics at low frequencies only. The
Lissajous and elastic stresses determined from Equation 5.2 are presented for
rad/s in
Figure 5.2(a) and
rad/s in Figure 5.2(b).
Figure 5.2
Lissajous curves with corresponding elastic stresses for varying oligomer
concentration and strain amplitude at (a)
= 0.1 rad/s, and (b)
= 1 rad/s.
For both frequencies, it is clear that the elastic behavior is increased by oligomer. This is
indica
ted by the transition to an ellipsoidal behavior in the stress waveform from a nearly viscous
response by the matrix. Inspection of the elastic stress also shows this increase in elastic behavior
through an increase in slope, where the matrix elastic stre
ss is nearly flat. Though a similar elastic
behavior is observed for the oligomer containing blends, the shear stress is significantly reduced
for the 10 wt% case.
A quantitative analysis may be made onto the magnitude of elastic stress buildup in each
system. First, the waveforms plotted in Figure 5.2 are instead mapped to the time

domain and
compared in Figure 5.3.
88
Figure 5.3
Elastic and viscous stress waveforms at
=1.14 and
=1 rad/s for (a) nylon copolymer
matrix (b) 5 wt% oligomer and (c) 10
wt% oligomer.
89
The viscous waveform in Figure 5.3 is similar to the overall shear stress waveform for all
samples indicating the response is viscous dominant. It is clear that upon addition of oligomer,
the elastic contribution increases over the matrix
response due to the increase in elastic stress
magnitude. The magnitude however, is similar between the oligomer cases though the waveforms
have completely different shapes. The maximum elastic stress was determined for each material
and plotted against
strain amplitude for
= 1 rad/s in Figure 5.4.
Figure 5.4
Normalized elastic stress against strain amplitude for
= 1 rad/s.
The contribution to the elastic stress is indeed greater for the oligomer containing blends.
Interestingly enough, though
the shear stress is significantly lower for the 10 wt% oligomer
containing blend, as shown in Figure 5.2(b), the relative increase in elasticity is independent of the
oligomer concentration, suggesting that it is dependent more so on the copolymer structur
e formed
between the amide linkages formed through reactive compatibilization. More evidence is needed
to determine this with certainty.
90
A final look into the effect that reactive compatibilization has on the nonlinear response is
the influence it has on
the
parameter. This is shown for
= 1 rad/s in Figure 5.5.
Figure 5.5
Effect of oligomer addition on the nonlinear
parameter for
= 1 rad/s.
The most pronounced effect the oligomer addition has on the nonlinearity is observed
through th
e
parameter. A slight increase is observed in the nonlinearity for 5 wt% oligomer
addition, though the dependence on strain is the same. At this point it has been speculated that the
10 wt% oligomer containing blends were fully reacted to the nylon
copolymer, leaving excess
oligomer available to phase separate. If this is indeed the case, then the
shows a remarkable
sensitivity to the dispersed oligomer phase, leading to an earlier onset of nonlinearity. The plateau
region in
extends t
hrough a wide strain amplitude range suggesting that the dispersed phase
is unable to contribute beyond
= 1. Reasons for this are still unexplained. Though it is clear
that the degree of reactive compatibilization can in some way be quantified using
LAOS.
91
5.4 Conclusion
s
and Recommendations
The effect of reactive compatibilization of nylon copolymer with an elastomeric PIBSA
oligomer was investigate using LAOS flows. Fourier transform rheology and stress decomposition
techniques showed that while an excess of oligomer led to the lowest shear
stress, the relative
elasticity was independent of the oligomer concentration. This suggests that the elastic stress was
dominated by the amide linkages formed in the matrix phase and had no dependence on the
potential dispersed phase. The suggestion th
at a dispersed phase exists was concluded by
remarkable differences in the nonlinear
parameter, where excess oligomer led to an earlier
onset of nonlinearity and a greater nonlinear over a wide range of strain amplitudes. Suggestions
for this resear
ch include the direct testing of amide linkages either through FT

IR methods or
titration techniques. This should answer if there is indeed an excess amount of oligomer present.
Further suggestions would include the imaging of these blends to determine i
f regions of phase
separation are observable.
92
C
HAPTER
6
CONCLUSIONS AND RECOMMENDATIONS
6
.1 Conclusions
This research focused on investigating the
nonlinear rheology of complex fluids using large
amplitude oscillatory shear (LAOS) flows. These flows
were analyzed using several techniques
including: Lissajous

Bowditch curves
to plot stress versus strain waveforms, Fourier transform
rheology to determine higher harmonic contributions to the stress, and
stress decomposition to
evaluate the elastic and vi
scous stress contributions in the nonlinear regime.
The
rheological properties associated with polymer nanocomposites can be directly related
to
both the microstructure as well as the type of interactions present in the system. Three types of
interaction
s are possible in nanocomposites: particle

particle, particle

polymer, and polymer

polymer. The particle

particle interactions are dependent on the size and concentration of the filler
phase.
For highly loaded nanocomposites, a critical concentration is
reached, known as the
percolation threshold, for which
a filler network
form
s
. The filler network is known to dominate
the rheology
through elastic (solid

like) responses. However, the processing range of these
systems is limited
since at large deformati
ons the filler network breaks down leading to a transition
into the nonlinear regime.
The breakdown of the filler network with increasing deformations is
know
n
as the Payne effect
.
When nanocomposites are
surface treated to promote compatibility with the
matrix chains,
stronger particle

polymer interactions
dominate the rheology. These interactions lead to hindered
chain mobility near the interface of the filler
, resulting is slower relaxation times and an increase
in the viscoelastic moduli at low frequ
encies.
Consequently, the
improved interactions between
93
the polymer matrix and the filler particles lead to increases in the nanocomposite properties such
as improved mechanical strength
, barrier properties and strain hardening behavior. These
improvemen
ts
also require less filler due to the strong entanglement networks which form between
particle

attached chains and the polymer matrix chains.
As with the breakdown of the filler
network,
the entanglement network formed by the slower relaxing particle

att
ached chains can
also break down
, leading to a similar Payne effect response.
The dynamics of these dilute nanocomposites containing strong entanglement networks
with particle attached chains was a major subject in this work.
Fourier transform rheology on
LAOS flows has been shown throughout the literature as a very sensitive method to determine
polymer
morphology, microstructure, and relaxation dynamics. However, relating
the trends
found
in nonlinear rheology to the
dynamics
of the tested material has always been a major
challenge. This motivated the development of the nonlinear viscoelastic
model for polymer
nanocomposites presented in this work.
Relating
linear relaxation mechanisms
(
double reptation
)
and nonlinear relaxa
tion mechanism
s
(
convective constraint release and chain retraction
)
to trends
observed in
FT rheology
resulted in a way to describe the nonlinearities in terms of
chain
dynamics.
Th
ree nonlinear parameters in FT rheology were used to quanti
fy the effect
of
entanglement networks formed by particle

attached chains:
the relative third harmonic
,
the
nonlinear parameter
and the zero

strain intrinsic nonlinearity
.
Through numerical
simulations and asymptotic expressions,
the
predicted nonlinearit
ies in the
viscoelastic
nanocomposite constitutive model w
ere
explored
.
The nonlinear behavior was found to be
dependent on three parameters: the strength of attachment (i.e.
slower particle

attached reptation
dynamics), the volume fraction of attached ch
ains
, and the
CCR parameter
.
94
Convective constraint release had been previously derived in an effort to
prevent
constitutive models from over predicting the shear thinning behavior at larger shear rates
. Since
this mechanism depends on the flow rate, it was uniformly applied to all
polymer chains present
in a system resulting in the same stress relaxation, regardless of the polymer architecture or
attachment to
nanoparticles. Thus, the idea of independ
ent CCR parameters was introduced here
so as to allow faster removal of entanglements made with particle

attached chains
.
Only when the
CCR rate of particle

attached chains is larger than the free matrix chains
, do the
trends in the
nonlinear parameters,
particular
, match with experiment.
Large amplitude oscillatory shear tests on polypropylene
layered

silicate nanocomposites
were
conducted to test the validity of the model presented in this work.
Particle

attached chains
were created using a silane t
reated montmorillonite nanoclay
reacted with a maleic anhydride
grafted polypropylene
. The PP

g

MA was of substantial molecular weight to
extend into the PP
matrix and form multiple entanglements.
The reduced chain mobility due to attachment with the
nan
oclay was confirmed through linear viscoelastic frequency sweeps. There, an increase in the
viscoelastic moduli at low frequencies
was reported for nanocomposites containing both 3 and 5
wt% in a 1:1 PP

g

MA : clay ratio.
Nanocomposites were subjected to
multiple frequencies ove
r
a range of strain amplitudes extending well into the nonlinear region. As predicted by the
viscoelastic model, the polypropylene nanocomposites
experienced a decrease in nonlinearity with
increasing frequency
, characterized by a
decrease in both
and
.
The unfilled matrix showed
the opposite trend
, having an increase in nonlinearity with increasing frequency.
This impli
ed
that
the
hindered chain mobility led to an earlier breakup of the particle

attached entanglement
network
resulting in a nonlinear response that was
governed by polymer

polymer interactions alone.
95
The final project was concerned with the reactive compatibilization of
an elastomeric
oligomer, PIBSA, with nylon copolymer.
The nylon copolymer was found t
o have a viscous
dominant response in its rheology. However,
due to the elastic nature of PIBSA, it was speculated
that the reaction with nylon would in some way impart some elastic behavior
to the resulting blend.
These blends were characterized using L
AOS flows combined with FT rheology and stress
decomposition methods. The stress decomposition
showed that an equivalent elastic response was
observed,
independent of the oligomer concentration.
However, the decrease in shear stress for
the largest oligo
mer concentration of 10 wt%
implied that excess oligomer was phase separated
from the blend.
Upon inspection of the
parameter, the 5 wt% and
nylon copolymer had similar
magnitudes in their nonlinearity; the
strain amplitude
onset of nonlinearity was
also comparable
.
However, in the presence of excess oligomer, the 10 wt% case
had the largest
magnitude
with
an onset of nonlinearity
an order of magnitude earlier than the other cases. This suggested that FT
rheology was sensitive
to
the morpholo
gy
of polymer blends
and
could be used to
determine at
what concentration phase separation would occur.
6.2 Recommendations
The present work has explored
the dynamics of polymer nanocomposites in both a
theoretical and experimental framework
for LAOS flows.
One of the most useful parameters in
describing the
relaxation and breakup of the particle

attached entanglement network is the zero

strain intrinsic nonlinearity,
. This parameter is a frequency dependent parameter
, though
acquiring
it experimentally is
tedious. Strain sweep tests from medium to large amplitudes must
be performed
for many frequencies if a high resolution
curve is desired.
Each test requires a
new sample
, which can be challenging if the material is only available
in low quantities. Therefore,
a recommendation is to investigate nonlinear frequency sweeps as a way to evaluate
the relaxation
96
phenomena. Most LAOS setups are

the

oversampling of the stress and strain
waveforms. Since oversampling depends on the excitation
frequency, m

the

However, by oversampling after the experiment is complete as is done in this work,
proper
oversampling can
be done on the waveforms acquired from a large amplitude frequency sweep
test.
97
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