THE RHEOLOGY AND THERMODYNAMICS OF POLYMER NANOCOMPOSITES
By
Jonathan E. Seppala

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Chemical Engineering
2010

ABSTRACT
THE RHEOLOGY AND THERMODYNAMICS OF POLYMER
NANOCOMPOSITES
By
Jonathan E. Seppala
Recent advancements in the quality and quantity of nanoparticles have increased their use
in polymer nanocomposites. Their small size creates the opportunity for interesting size
ratios and limits. One of these, the nanoparticle limit, where the particle is smaller than the
suspending polymer radius of gyration, has resulted in surprising effects to the mechanical
properties among them a decrease in the melt viscosity and an increased Young’s modulus in
the solid state. Dispersing nanoparticles within polymers plays a critical role in nanocomposite properties, and the dispersion stability is very sensitive to the confinement limit (h < Rg )
and solubility limit (a < Rg ). Bimodal blends of linear polystyrene with tightly crosslinked
polystyrene nanoparticle having a small volume fraction of unconfined chains or a small
volume fraction of chains larger than the nanoparticle fail to show any unique features.
Knowing the sensitivity of these limits allows for proper selection of polydisperse nanoparticles and polydisperse linear chains producing a fully polydisperse system with reduced
viscosity. Rheology gives a rough estimate of dispersion in a system of tightly crosslinked
polystyrene nanoparticles in linear perdeuterated polystyrene via either a viscosity reduction
or adherence to Einstein’s predicted viscosity increase. Dispersion or agglomeration in these
systems is confirmed though small angle neutron scattering and solubility and mixing energies are calculated via Guinier and Virial analysis. These results follow the scaling penalty
from chain stretching as nanoparticle radius approaches linear polymer radius of gyration.
However, even in the agglomerated case rapid precipitation produced a well dispersed sys-

tem that phase separated during thermal annealing. Refinements to the rapid precipitation
blending technique yielded an ideal polymer concentration for proper blending and subtle
difficulties are discussed. Finally the melt viscosity of poly(methyl methacrylate) C60 and
poly(methyl methacrylate) F e3 O4 nanocomposites show a reduced viscosity, results which
are contrary to recently reported experiments. The method of dispersion is the key difference
between this work and what has been reported in the literature. Young’s Modulus was also
measured with only the poly(methyl methacrylate) F e3 O4 nanocomposite resulting in an
increase, indicating a dependence of unusual reinforcement on nanoparticle size and spacing.
These details and other phenomena are discussed in this work.

To my wife Clare.

iv

ACKNOWLEDGMENT

First and foremost I would like to thank my adviser Michael Mackay for his guidance as an
adviser and rheology expert. It is difficult to remember how little I knew about rheology
and research in general when I started. There are an unknown number of details regarding
each that I leaned from Dr. Mackay throughout the years and I would not hesitate to point
any new graduate student in his direction.
Special thanks to BASF for their funding of my work; none of it would have been possible
without their generous contribution.
Though moving universities is a difficult and disruptive event, it brought me into contact
with far more diverse group of people and expertise. My wife, Clare on the other hand
moved away from friends and family without benefiting quite as much. I owe her a great
deal of gratitude for the sacrifices she made while I pursued my PhD.
From the Mackay group, thank you to Jon Kiel and Tzuchia Tseng who were my best
friends, constant companions, and competent colleagues. Additional thanks to Dr. David
Bohnsack for teaching me how to use the instruments in our lab and for being available to
discuss my project even after his graduation.
At the University of Delaware, I would like to thank the MSEG Department for making
us feel welcome following our move and helping us to get adjusted to the UDel community.
Special thanks to Professor Norman Wagner in CHEG for use of his rheology equipment,
the neutron scattering expertise of his group, and for allowing me to join in their group
meetings.
Neutron scattering can be very complex experiment to setup and to analyze. Fortunately
v

I was aided by Dr. Yun Liu, Dr. Aaron Eberle, and Dr. Armin Optiz. Thank you to the
staff in CHEMS at MSU: Jennifer Somerville, Jennifer Peterman, Nikki Shook, and Joanne
Peterson; as well as the staff in MSEG atUDel: Charles Garbini, Christine Williams, Dionne
Putney, and Steve Beard.
Moving also afforded me two sets of graduate school friends, at MSU: Katie Shipley,
Ming Wei Lau, Dan Olds, Megan Ramanowich, Luke Granlund, and Shannon Nicely; and
atUDel:Conan Wieland, Laura Polvich, Katie Feldman, Carl Giller, and Bakathyr Ali - each
of whom contributed in their own ways whether it was working together on course projects
or discussing research problems.
I also would like to thank my parents, Fay and Dennis, and sisters, Jean and Josie for
their support throughout the years and my Aunts, Laurie and Sharon, who were always
interested in the technical details of my work and who provided a much needed outside
perspective.

vi

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures

ix

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

1 Introduction
1.1 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
5

2 Bimodal Blends
2.1 Experimental Methods .
2.2 Results & Discussion . .
2.2.1 Bidisperse Blends
2.2.2 Bidisperse Blends
2.3 Polydisperse Systems . .
2.4 Summary . . . . . . . .

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- 40NP
- 230NP
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3 Dispersion and Mixing Energy of Ideal
3.1 Introduction . . . . . . . . . . . . . . .
3.2 Experimental Methods . . . . . . . . .
3.3 Results & Discussion . . . . . . . . . .
3.3.1 120dPS-40NP . . . . . . . . . .
3.3.2 120dPS-230NP . . . . . . . . .
3.4 Summary . . . . . . . . . . . . . . . .
4 PMMA Nanocomposites
4.1 Introduction . . . . . . .
4.2 Experimental Methods .
4.3 Results & Discussion . .
4.3.1 Rheology . . . .
4.3.2 DMA . . . . . . .
4.3.3 WAXS . . . . . .
4.4 PMMA-Fe3 O4 . . . . .
4.4.1 Rheology . . . .
4.4.2 SAXS . . . . . .
4.4.3 Young’s Modulus
4.5 Summary . . . . . . . .

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76
76

5 Rapid Precipitation
5.1 Introduction . . . .
5.2 Experimental Setup
5.3 Results . . . . . . .
5.4 X-Ray Scattering of
5.5 Summary . . . . .

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Polystyrene
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CdSe Nanocomposites
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6 Experimental Techniques
6.1 Rheology . . . . . . . . . . . . . . . .
6.1.1 Simple Shear . . . . . . . . .
6.1.2 Oscillatory Shear . . . . . . .
6.1.3 Plate Alignment . . . . . . . .
6.1.4 Wall Slip . . . . . . . . . . . .
6.1.5 Sample Stability . . . . . . .
6.2 SAXS . . . . . . . . . . . . . . . . .
6.2.1 Absolute Intensity Calibration
6.2.2 Experimental Setup . . . . . .
6.2.3 Results . . . . . . . . . . . . .
6.2.3.1 Absorbance . . . . .
6.2.3.2 SAXS . . . . . . . .
6.2.3.3 Correction Factor . .
6.2.4 Summary . . . . . . . . . . .
7 Conclusions

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80
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94
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105
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109
111

A Master Curves of Bimodal Poly(styrene) Blends With Cross Linked Polystyrene
Nanoparticles
115
B Master Curves of Poly(styrene-co-acrylonitrile) With Cross Linked Poly(styreneco-acrylonitrile) Nanoparticles
126
C Master Curves of Poly(methyl methacrylate) C60 and F e3 O4 Nanocomposites
128
D Rheology of Poly(3-hexylthiophene)

132

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

viii

LIST OF TABLES

2.1

2.2

2.3

2.4

3.1

3.2

3.3

Mn , Mw and PDI of neat bidisperse PS blends from GPC. PDIs are similar
to those of commercially available polystyrenes, 1.5. . . . . . . . . . . . . . .

13

Mn , Mw , PDI and radius of PS nanoparticles. Radius based on bulk density
of polystyrene (1.04 g/cm3 ). † Rh DLS in toluene at 30 ◦ C. ‡ Rg SANS in 120
kD perdeuterated polystyrene (120dPS) at 170 ◦ C. †† 230NP particles were
agglomerated in SANS study, Rh in d-cyclohexane is reported instead.[1] . .

14

0
Terminal viscosity ratios (ηr ) and change in terminal viscosities (∆η 0 ) of
bimodal linear polystyrene and 2 v% 40 kD tightly crosslinked polystyrene
nanoparticles[2] (40NP). As volume fraction of confined chains increases the
viscosity reduction increases. . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

∗
Complex viscosity ratios (ηr (ω)) for 2.0 v% 40NP blends with reduced viscosity at selected frequencies. Again the highest viscosity material has the
largest reduced viscosity, however even the 75 v% 393 bimodal blend has a
reduced viscosity at industrially relevant frequencies. . . . . . . . . . . . . .

19

Peak position from Gaussian fit to Kratky Plots (Figure 3.3) of four volume
fractions of 40 kD tightly crosslinked polystyrene nanoparticles (40NP) in 120
kD perdeuterated linear polystyrene measured at 170 ◦ C. Peak positions are
constant regardless of volume fractions and the characteristic length scale (d)is
approximately the same size as the Rh for these nanoparticles which is a good
indication of dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Mn , Mw , PDI (determined by GPC) and hydrodynamic radius (determined
by DLS) of crosslinked polystyrene nanoparticles used in this work. Hydrodynamic radius determine by DLS in toluene at 30 ◦ C. . . . . . . . . . . . .

35

Results of Guinier fits to scattering profiles of 40 kD tightly crosslinked polystyrene
nanoparticles[2] (40NP) in 120 kD perdeuterated linear polystyrene (120dPS)
at 170 ◦ C. The radius of gyration is in excellent agreement with Rh and literature values[1]. The intensity at zero wavevector (I0 ) will be used to determine
solubility and mixing energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
ix

3.4

Kratky peak position from four volume fractions of 230 kD tightly crosslinked
polystyrene nanoparticles[2] (40NP) in 120 kD perdeuterated linear polystyrene
(120dPS) measured at 170 ◦ C. The large characteristic length (d) indicates
the 230NPs are phase separating and not surprisingly the high volume fractions have larger agglomerates since there is more opportunity for individual
nanoparticles to come into contact. . . . . . . . . . . . . . . . . . . . . . . .

42

3.5

Results of Guinier fits to scattering profiles of 230 kD tightly crosslinked
polystyrene nanoparticles[2] (230NP) in 120 kD perdeuterated linear polystyrene
(120dPS) at 170 ◦ C. The large Rg is a clear sign of phase separation which
increases with volume fraction of 230NP. Aggregate size vs volume fraction is
plotted in Figure 3.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6

Peak positions from Gaussian fit to peak in Kratky Plots (Figure 3.13) of 230
kD tightly crosslinked polystyrene nanoparticles[2] in 120 kD perdeuterated
linear polystyrene (120dPS) at two different annealing times 15 minutes and
3 hours at 170 ◦ C, measured at 127 ◦ C. Characteristic length scale calculated
from q = π/d, longer annealing time has a significantly larger 230NP aggregate
size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Material properties of PMMA-C60 nanocomposites from master curves shifted
to a reference temperature of 170 ◦ C. The high viscosity 388 kD PMMA and
low thermal degradation point prevented measurement down to the terminal region, the complex viscosity at a frequency of 0.1 rad is reported ins
b The plateau modulus is determined using the Min method[3], G0 =
stead.
N
G (ωtan(δ)=min ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Glass transition temperature of 388 kD PMMA C60 nanocomposite at four
volume fractions of C60 . The glass transition temperature for the composite
is unchanged from the neat PMMA. . . . . . . . . . . . . . . . . . . . . . . .

66

Peak positions determined from Gaussian peak fits to Kratky Plots, Figure
4.12, of 388 kD PMMA with two volume fractions of F e3 O4 . Characteristic
length scale is reported using the following conversion, q = 2.6/d based on a
hard sphere scatterer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.1

4.2

4.3

5.1

Preparation specifications for monodisperse and polydisperse polystyrene nanocomposite blended by rapid precipitation. . . . . . . . . . . . . . . . . . . . . . . 82
x

5.2

5.3

6.1

Peak positions determined from Gaussian peak fits to Kratky Plots, Figure
5.6, of 393 kD linear polystyrene (393PS) with three volume fractions of oleic
acid stabilized CdSe Nanocrystals (oa-QNC). Characteristic length scale is
reported using the following conversion, q = 2.6/d based on a hard sphere
scatterer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Results of linear fit to low q-region in Guinier Plots, Figure 5.7, of 393 kD
linear polystyrene (393PS) with three volume fractions of oleic acid stabilized
CdSe Nanocrystals (oa-QNC). . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Complex viscosities from PDMS standard at a single point oscillatory test at
0.1 rad/s and ever decreasing plate gaps. . . . . . . . . . . . . . . . . . . . .

99

6.2

Instrument configuration for transmission measurements. . . . . . . . . . . . 105

6.3

Instrument configuration for SAXS measurements. . . . . . . . . . . . . . . . 106

6.4

Intensity at 0 angle for an empty cell (EC) and L18 glassy carbon (GC) sample.107

6.5

Intensity at q=0.06˚ (interpolated) for an empty cell (EC) and L18 glassy
A
carbon (GC) sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.6

Differential scattering cross section (APS) and intensity (Rigaku) of L18 Glassy
Carbon Stand at q=0.06˚. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A

xi

LIST OF FIGURES

2.1

2.2

2.3

2.4

Molecular weight distribution of bidisperse linear polystyrene blends as determined by GPC. The dashed vertical lines indicate the smallest chain confined by a given volume fraction of 40 kD tightly crosslinked polystyrene
nanoparticles[2]. The critical molecular weight for entanglement Mc is also
noted, solid vertical line. For interpretation of the references to color in this
and all other figures, the reader is referred to the electronic version of this
dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Molecular weight distribution of bidisperse linear polystyrene blends as determined by GPC. The dashed vertical lines indicate the smallest chain confined by a given volume fraction of 230 kD tightly crosslinked polystyrene
nanoparticles[2]. The critical molecular weight for entanglement Mc is also
noted, solid vertical line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

0
Terminal viscosity ratios (ηr ) of linear bimodal blends with 40 kD tightly
crosslinked polystyrene nanoparticles[2] (40NP). Theory line indicates the Einstein prediction[4] with the Batchelor correction[5] (ηr = 1 + 2.5φ + 6.2φ2 ).
As the fraction of high molecular weight linear chains (393PS) is increased the
melt viscosity moves to a reduced value. Terminal viscosities are reported as
the η 0 parameter from a fit of the Carreau-Yasuda Model[6, 7] following conversion of the complex viscosity to steady shear using the Cox-Merz Rule[7, 8].
Complex viscosity and moduli master curves for these systems are available
in Appendix A. All data shifted to a reference temperature of 170 ◦ C. . . . .

17

Terminal viscosity ratios of bimodal linear polystyrene with 230 kD tightly
crosslinked polystyrene nanoparticles[2] (230NP). Theory line indicates the
Einstein prediction[4] with the Batchelor correction[5] (ηr = 1+2.5φ+6.2φ2 ).
All nanocomposites with 75PS and 230NP essentially behave as Einstein predict which indicates the nanoparticles are phase separated. Terminal viscosities are reported as the η 0 parameter from a fit of the Carreau-Yasuda
Model[6, 7] following conversion of the complex viscosity to steady shear using the Cox-Merz Rule[7, 8]. Terminal viscosity ratios with 8.5 v% 230NP
blends with any v% of 75PS are extrapolated. Complex viscosity and moduli
master curves for these systems are available in Appendix A. All data shifted
to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . .

20

xii

2.5

Terminal viscosity ratios and plateau modulus ratios of poly(styrene-co-acrylonitrile)
with crosslinked poly(styrene-co-acrylonitrile) nanoparticles. Theory line indicates the Einstein prediction[4] with the Batchelor correction[5] (ηr = 1 +
2.5φ + 6.2φ2 ). Simple Equation (2.3) and Tsenoglou (Equation 2.4) indicate
results based on blending with a volume of material with essentially zero viscosity. The ratio of G0 is also included while it captures most of the decrease
N
at high frequency it does not account for the large decrease a low frequency,
which is typical of these nanocomposites. Terminal viscosities are reported as
the η 0 parameter from a fit of the Carreau-Yasuda Model[6, 7] following conversion of the complex viscosity to steady shear using the Cox-Merz Rule[7, 8].
Terminal viscosity ratios with 8.5 v% 230NP blends with any v% of 75PS are
extrapolated. Complex viscosity and moduli master curves for these systems
are available in Appendix B. All data shifted to a reference temperature of
170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1

0
Terminal viscosity ratios (ηr ) of 120 kD perdeuterated linear polystyrene
(120dPS) with 4 volume fractions of 40 kD tightly crosslinked polystyrene
nanoparticles[2] (40NP). Theory line indicates Einstein’s relation[4] with Batchelor’s correction[5] (ηr = 1 + 2.5φ + 6.2φ2 ). The reduced viscosity typical of
a well dispersed nanoparticle in a polymer melt is present. Terminal viscosities are reported as the η 0 parameter from a fit of the Carreau-Yasuda
Model[6, 7] following conversion of the complex viscosity to steady shear using the Cox-Merz Rule[7, 8]. Complex viscosity and moduli master curves for
these systems are available at the end of this chapter, Figure 3.14. All data
shifted to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . .

33

One dimensional neutron scattering profiles (I(q) vs q) from the azimuthally
averaged two dimensional scattering profiles of four volume fractions of 40kD
tightly crosslinked polystyrene nanoparticles[2] (40NP) in 120 kD perdeuterated linear polystyrene (120dPS) at 170 ◦ C. As expected the scattering intensity increases with increasing 40NP content and has the shape of a particle-like
scatterer with incoherent background. . . . . . . . . . . . . . . . . . . . . . .

34

Kratky plots (q 2 I(q) vs q) constructed directly from the one dimensional scattering profiles of four volume fractions of 40 kD tightly crosslinked polystyrene
nanoparticles (40NP) in 120 kD deuterated linear polystyrene (120PS) with
the incoherent background subtracted. A peak in the Kratky plot indicates
a particle like structure, which is expected for this system. Furthermore the
peak positions are constant for all volume fractions. . . . . . . . . . . . . . .

35

3.2

3.3

xiii

3.4

Guinier plots (Ln(I(q)) vs q) of the azimuthally averaged scattering profile
from 4 volume fractions of 40 kD tightly crosslinked polystyrene nanoparticles[2]
(40NP) in 120 kD perdeuterated linear polystyrene (120dPS). Solid lines indicate fits to Equation (3.1), fits ranged from 0.0009 < q 2 < 0.0025˚−2 and the
A
radius of gyration and intensity at zero wavevector were determined. From
the fits the average radius of gyration is 2.8 ± 0.1 nm indicating excellent
dispersion. The results of the fits to the Equation (3.1) are listed in Table 3.3. 36

3.5

Radius from Guinier fit of 4 volume fractions of 40 kD tightly crosslinked
polystyrene nanoparticles[2] (40NP) in 120 kD perdeuterated linear polystyrene
(120dPS). The dashed line marks the hydrodynamic radius (Rh ) determined
by DLS, Table 3.2. The two radii are in excellent agreement and the SANS
results indicate excellent dispersion. . . . . . . . . . . . . . . . . . . . . . . .

38

Virial Plot, concentration of scatterer over intensity at zero wavevector against
concentration of scatterer (c/I0 vs c) for 40 kD tightly crosslinked polystyrene
nanoparticles[2] in 120 kD perdeuterated linear polystyrene at 170 ◦ C. Solid
line indicates linear regression fit to data set. Second Virial Coefficient, A2 , is
proportional to the slope and Mw is the inverse of the intercept according to
Equation (3.3.1). A2 = (7.66±0.34)x×10−5 cm3 mol/g 2 a value which proves
the nanoparticles are soluble in the 120 kD perdeuterated linear polystyrene
at 170 ◦ C. Also Mw = 40.0 ± 0.3 kD suggesting a slight dimerization during
nanoparticle synthesis when compared to the Mw of the linear precursor at
37.1 kD but reinforcing the dispersion claim. . . . . . . . . . . . . . . . . . .

40

Neutron scattering profiles (I(q) vs q) of the azimuthally averaged 2D scattering profiles from 4 volume fractions of 230 kD tightly crosslinked polystyrene
nanoparticles[2] (230NP) in 120 kD perdeuterated linear polystyrene (120dPS).
Like the scattering profiles of the 120dPS-40NP system, Figure 3.2, the scattering intensity increases with increasing 230NP content and has the shape of
a particle-like scatterer with incoherent background. . . . . . . . . . . . . . .

43

Kratky plots (q 2 I(q) vs q) of the azimuthally averaged scattering profile from
4 volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2]
(230NP) in 120 kD perdeuterated linear polystyrene (120dPS) at 170 ◦ C. A
peak in the Kratky plot indicates a particle like structure, which is expected
for this system. However the peak positions decrease with volume fraction
of 230NP which corresponds to an increase in the characteristic length scale
(d) of the scatter (230NP). Peak positions and characteristic length scales are
listed in Table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.6

3.7

3.8

xiv

3.9

Guinier plots (Ln(I(q)) vs q) of the azimuthally averaged scattering profile
from 4 volume fractions of 230 kD tightly crosslinked polystyrene[2] (230NP)
in 120 kD perdeuterated linear polystyrene (120dPS) at 170 ◦ C. Solid lines
indicate fits to Guinier Equation, Equation (3.1) and the radius of gyration
and intensity at zero wavevector were determined. Similar to the characteristic
length scale from the Kratky plots (Figure 3.8, Table 3.4 the radius of gyration
increases with volume fraction of 230NP and is significantly larger than the
Rh and literatures values[1] for these nanoparticles, a clear sign of phase
separation. The intensity at zero wavevector (I0 ) will be used to determine
solubility and mixing energy. The results of the fits to the Equation (3.1) are
listed in Table 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.10 Aggregate size from 4 volume fractions of 230 kD tightly crosslinked polystyrene
nanoparticles[2] (230NP) in 120 kD perdeuterated linear polystyrene (120dPS)
at 170 ◦ C. The large 230NP aggregate size, relative to Rh of individual
nanoparticle is a clear sign of phase separation. . . . . . . . . . . . . . . . .

46

3.11 Virial plot from Guinier fits of 230 kD tightly crosslinked polystyrene nanoparticles[2]
(230NP) in 120 kD perdeuterated linear polystyrene (120dPS). Negative slope
indicates insolubility of the 230NP in 120dPS. . . . . . . . . . . . . . . . . . 47

0
3.12 Terminal viscosity ratios (ηr ) of 120 kD perdeuterated polystyrene (120dPS)
with four volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2].
Theory line indicates Einstein’s relation[4] with Batchelor’s correction[5] (ηr =
1 + 2.5φ + 6.2φ2 ) which this system follows indicating colloidal behavior[9, 10,
11, 12] instead of well dispersed nanoparticles as seen in the 120dPS-40NP
terminal viscosity ratios, Figure 3.1. Terminal viscosities are reported as the
η 0 parameter from a fit of the Carreau-Yasuda Model[6, 7] following conversion of the complex viscosity to steady shear using the Cox-Merz Rule[7, 8].
Terminal viscosity ratios with 8.5 v% 230NP blends with any v% of 75PS are
extrapolated. Complex viscosity and moduli master curves for these composites are included at the end of the chapter, Figure 3.15. All data shifted to a
reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . 48

3.13 Kratky plots (q 2 I(q) vs q) of the azimuthally averaged scattering profile from
4 volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2]
(230NP) in 120 kD perdeuterated linear polystyrene (120dPS), measured at
127 ◦ C. Two annealing times were used, longer annealing time results in larger
230NP aggregates regardless of 230NP volume, indicated by shift of peak to
lower q (larger characteristic length scale of scatterer). . . . . . . . . . . . .
xv

50

3.14 Master curves of 120 kD perdeuterated linear polystyrene (120dPS) with four
volume fractions of 40 kD tightly crosslinked polystyrene nanoparticles[2]
(40NP). Results shifted to a reference temperature of 170 ◦ C. . . . . . . . .

53

3.15 Master curves of 120 kD perdeuterated linear polystyrene (120dPS) with four
volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2]
(230NP). Results shifted to a reference temperature of 170 ◦ C. . . . . . . . .

54

4.1

4.2

4.3

4.4

4.5

4.6

Images of 388 kD PMMA C60 nanocomposites at 3 different volume fractions
of C60 . C60 is a strong absorber in the UV range[13] resulting in the dark
purple-brown color at low loading. . . . . . . . . . . . . . . . . . . . . . . . .

60

Complex viscosities from master curves of 77 kD poly(methyl methacrylate)
three volume fractions of C60 . The viscosity of the nanocomposites are decreased with respect to the neat PMMA at all viscosities. Complete master
curves for these systems are available in Appendix C: Figure C.1. All data
shifted to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . .

61

Complex viscosities from master curves of 388 kD poly(methyl methacrylate)
three volume fractions of C60 . The viscosity of the nanocomposites are decreased with respect to the neat PMMA at all viscosities. Complete master
curves for these systems are available in Appendix C: Figures C.2. All data
shifted to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . .

62

Terminal viscosity ratios of 77 kD poly(methyl methacrylate) with three volume fractions of C60 . Terminal viscosities are reported as the η 0 parameter
from a fit of the Carreau-Yasuda Model[6, 7] following conversion of the complex viscosity to steady shear using the Cox-Merz Rule)[7, 8]. All data shifted
to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . .

64

Viscosity ratios from complex viscosity at a frequency of 0.01 rad/s from
master curves of 388 kD poly(methyl methacrylate) three volume fractions of
C60 . Complete master curves for these systems are available in Appendix C:
Figure C.2. All data shifted to a reference temperature of 170 ◦ C. . . . . . .

65

Temperature ramps of 388 kD PMMA and PMMA-C60 nanocomposites. The
glass transition temperature is the same for the neat PMMA and all C60
nanocomposite, indicated by the constant peak position. Tg s are listed in
Table 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

xvi

4.7

Wide angle x-ray scattering profile of 388 kD PMMA C60 nanocomposite at
four volume fractions of C60 results are shifted vertically for clarity. Neat
C60 is also shown for comparison. The lack of matching peaks indicates no
C60 crystallites are present. . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Relative Young’s modulus of 388 kD poly(methyl methacrylate) with two volume fractions of C60 . Mori-Tanaka relationship, Yr = 1 + 5.3φ, is included
for comparison. The Young’s Modulus scales well with the Mori-Tanaka relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Relative zero-shear viscosity (extrapolated) of 388 kD PMMA Fe3 O4 nanocomposites with two different volume fractions of Fe3 O4 . The high viscosity of
388 kD PMMA and low degradation temperature prohibited measurement in
the terminal region. The 2.0 v% F e3 O4 nanocomposite has a significantly
lower viscosity than predicted by Einstein’s hydrodynamic theory, by 5.0 v%
F e3 O4 the system adheres to the Einstein’s prediction indicating phase separation. Terminal viscosities are reported as the η 0 parameter from a fit of
the Carreau-Yasuda Model[6, 7] following conversion of the complex viscosity
to steady shear using the Cox-Merz Rule[7, 8]. Complex viscosity and moduli
master curves for these systems are available in Appendix C. All data shifted
to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . .

71

4.10 Small angle x-ray scattering profile of 388 kD PMMA Fe3 O4 nanocomposites
with two different volume fractions of Fe3 O4 , at ambient temperature. . . .

72

4.11 Guinier plots (Ln(I(q)) vs q) constructed from x-ray scattering profiles of
388 kD PMMA Fe3 O4 nanocomposites with two different volume fractions
of Fe3 O4 , at ambient temperature. . . . . . . . . . . . . . . . . . . . . . . .

74

4.12 Kratky plots (q 4 I(q) vs q) constructed from x-ray scattering profiles of 388 kD
PMMA Fe3 O4 nanocomposites with two different volume fractions of Fe3 O4 ,
at ambient temperature. A peak in the Kratky plot indicates a particle like
structure, which is expected for this system. . . . . . . . . . . . . . . . . . .

75

4.13 Relative Young’s Modulus of 388kD PMMA Fe3 O4 nanocomposites with two
different volume fractions of Fe3 O4 . Mori-Tanaka relationship, Yr = 1 + 5.3φ,
is also plotted for comparison. The 2.0 v% F e3 O4 nanocomposite greatly out
performs the Mori-Tanaka model, by 5.0 v% F e3 O4 the system adheres to
the Mori-Tanaka model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.8

4.9

5.1

Solutions of polymer and polymer-nanoparticle blends in toluene at three
concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii

83

5.2

5.3

5.4

5.5

Precipitates of 393 kD polystyrene and 393 kD polystyrene with 3 v%25NP
crosslinked polystyrene nanoparticles. Notice the cloudy supernatant in samples with nanoparticles precipitated from 5 and 25 mg/mL solutions. . . . .

84

Precipitates of Polystyrol 168N and Polystyrol 168N with 25NP crosslinked
polystyrene nanoparticles. Again notice the cloudy supernatant in samples
with nanoparticles precipitated from 5 and 25 mg/mL solutions (375, 376). .

85

Scattering profiles of 393 kD linear polystyrene (393PS) with 3 different volume fractions of oleic acid stabilized CdSe nanocrystals (oa-QNC). . . . . . .

88

Absolute scattering profiles of 393 kD linear polystyrene (393PS) with 3 different volume fractions of oleic acid stabilized CdSe nanocrystals (oa-QNC).

89

5.6

Kratky plots created from absolute scattering profiles of 393 kD linear polystyrene
(393PS) with 3 different volume fractions of oleic acid stabilized CdSe nanocrystals (oa-QNC). Single peak is indicative of particle like structure. . . . . . . 90

5.7

Guinier plots created from absolute scattering profiles of 393 kD linear polystyrene
(393PS) with 3 different volume fractions of oleic acid stabilized CdSe nanocrystals (oa-QNC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.1

Strain sweep of GK1604/23 hyperbranched polystyrene-anhydride nanoparticle run at 130 ◦ C and 100 rad/s. Linear region extends from 0.01 to 10%
strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

6.2

Complex viscosity of PDMS standard at different plate gaps. The drop off in
viscosity denotes the lower usable gap with the current alignment. . . . . . . 100

6.3

Terminal viscosity of 393 kD polystyrene standard at different plate gaps. The
stable value indicates no wall-slip is present. . . . . . . . . . . . . . . . . . . 101

6.4

Frequency sweeps at 170 ◦ C poly(styrene-co-acrylonitrile) with cross linked
poly(styrene-co-acrylonitrile) nanoparticles. Overlap of material properties
indicated stability of system during the time frame of the typical experiment. 103

6.5

Differential scattering cross section and scattering intensity for glassy carbon
standard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.6

Differential scattering cross section of glassy carbon from APS and Rigaku
(corrected). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
xviii

A.1 Master curves of 75 kD polystyrene with two volume fractions of 25NP cross
linked polystyrene nanoparticles. Results shifted to a reference temperature
of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.2 Master curves of 75 kD polystyrene with three volume fractions of 140NP cross
linked polystyrene nanoparticles. Results shifted to a reference temperature
of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.3 Master curves of 75/25% 75/393 kD polystyrene with two volume fractions
of 25NP cross linked polystyrene nanoparticles. Results shifted to a reference
temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.4 Master curves of 75/25% 75/393 kD polystyrene with three volume fractions
of 140NP cross linked polystyrene nanoparticles. Results shifted to a reference
temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.5 Master curves of 50/50% 75/393 kD polystyrene with two volume fractions
of 25NP cross linked polystyrene nanoparticles. Results shifted to a reference
temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.6 Master curves of 50/50% 75/393 kD polystyrene with three volume fractions
of 140NP cross linked polystyrene nanoparticles. Results shifted to a reference
temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.7 Master curves of 25/75% 75/393 kD polystyrene with two volume fractions
of 25NP cross linked polystyrene nanoparticles. Results shifted to a reference
temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.8 Master curves of 25/75% 75/393 kD polystyrene with three volume fractions
of 140NP cross linked polystyrene nanoparticles. Results shifted to a reference
temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.9 Master curves of 393 kD polystyrene with two volume fractions of 25NP cross
linked polystyrene nanoparticles. Results shifted to a reference temperature
of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.10 Master curves of 393 kD polystyrene with three volume fractions of 140NP
cross linked polystyrene nanoparticles. Results shifted to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
xix

B.1 Master curves of poly(styrene-co-acrylonitrile) (SAN) with cross linked poly(styreneco-acrylonitrile) nanoparticles (SAN-NP). Results shifted to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
C.1 Master curves of 77 kD poly(methyl methacrylate) with three volume fractions
of C60 . Results shifted to a reference temperature of 170 ◦ C. . . . . . . . . . 129
C.2 Master curves of 388 kD poly(methyl methacrylate) with three volume fractions of C60 . Results shifted to a reference temperature of 170 ◦ C. . . . . . 130
C.3 Master curves of 388 kD with two volume fractions of F e3 O4 . Results shifted
to a reference temperature of 170 ◦ C. . . . . . . . . . . . . . . . . . . . . . . 131
D.1 Material properties of P3HT for a temperature ramp from 30-250 ◦ (forward
cycle) and from 250-30 ◦ C (reverse cycle) each at 3 ◦ C/minute. A peak
in the loss tangent (Tg ) is noted as well as approximate crystalline melting
temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
D.2 Material properties of P3HT as a function of time at 235 ◦ C. Note the sudden
increase in stiffness after 1200 seconds. The vertical lines indicate times when
strain sweeps (dashed lines) Figure D.3 or frequency sweeps (solid lines) Figure
D.4 were conducted. The dot-dash line indicates a final frequency sweep
followed by a final strain sweep, Figure D.5. All subsequent figures share this
legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
D.3 Two strain sweeps of P3HT before and after increase in stiffness at 235 ◦ C.
Legend is the same as Figure D.2 . . . . . . . . . . . . . . . . . . . . . . . . 135
D.4 Four frequency sweeps of P3HT following increase in stiffness at 235 ◦ C.
Legend is the same as Figure D.2 . . . . . . . . . . . . . . . . . . . . . . . . 136
D.5 Final frequency sweep and strain sweep. The sample begins to show non linear
behavior in at high strain after several hours at 235 ◦ C. Legend is the same
as Figure D.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xx

Chapter 1
Introduction

1

One of the primary draw backs in polymer-composite production is the increase in viscosity of the polymer-composite melt.[10, 11] It is generally accepted that addition of particles
to a fluid will increase the viscosity of the composite fluid. In a simple hydrodynamic theory
proposed by Einstein the viscosity of a suspension scales proportionally to the volume of
the particles suspended via the following equation, ηr = 1 + 2.5φ, where ηr is the viscosity
ratio (suspension/neat) and φ is the volume fraction of particle.[4] Many subsequent models, theoretical and empirical have been developed which accurately cover a wide variety of
materials.[10] These models share one common feature a viscosity increase that scales with
the volume fraction of the particle added. For all of these models the particle is assumed
to be rigid and for a wide variety of systems the particle is larger than the individual solvent molecules. This is also true for most polymer nanocomposites, where, even though the
polymer chains are quite large, the particle introduced is still larger.

As interest in nanoscience and nanoparticles has grown the uniformness and quantity of
nanoparticles has improved. Gram quantities of highly uniform very small, less than 10nm,
nanoparticles made from a variety of materials, organic and inorganic, are now available.[2]
Many of the common inorganic nanoparticles, F e2 O4 , CdSe[14], etc, are available with
surfactants and stabilizing agents allowing inorganic materials to be suspended in organic
solvent, which is crucial for suspending in most polymers. These nanoparticle are being
blended with a variety of polymer in a effort to produce new properties in otherwise well studied polymers. Changes is absorbance, dielectric strength[15], tailored refractive index[16] and
even anti-bacterial[17] properties are being incorporated. However, many of these nanocomposites have large scale agglomeration due to nanoparticle size, dispersion method, and/or
insolubility.
2

Recent work has been published detailing the effect of well dispersed nanoparticles which
have a radius smaller than the radius of gyration of the suspending polymer, the primary
effect being the decrease in melt viscosity of the composite. This is contrary to the general
understanding of suspensions in polymeric materials. In a effort to understand the decrease
in viscosity several other effects and some requirements were discovered. These are listed
below.

1. Non-Einstein-Like viscosity decrease upon addition of nanoparticles.[18, 19, 20]

2. Faster-than-expected diffusion of nanoparticle in a polymer matrix.[21]

3. Chain swelling upon addition of nanoparticles.[22, 23, 24, 25, 26, 27, 28]

4. Confinement requirement for Non-Einstein-Like decrease.[29]

5. Entanglement[30] requirement for Non-Einstein-Like decrease.[29]

6. Appropriate method for dispersing nanoparticles in a polymer matrix.[20]

7. Indirect reinforcement of polymers by nanoparticle addition.[20]

The first and most surprising effect is the afore mentioned decreased terminal viscosity. It
is generally accepted that addition of nanoparticles to an fluid increases the viscosity.[9,
10, 11, 12] Einstein first described this with a simple hydrodynamic argument where the
fluid around a spherical non interacting particle takes a longer more torturous path.[4] This
change in the flow field results in an increased viscosity. There are currently many different
models and theoretical descriptions available for a wide variety of particles. However to date
none describe a decreased viscosity upon particle addition.
3

In Chapter 2 of this thesis, the subtleties of nanoparticle confinement of polymer chains
and the solubility of nanoparticles in linear chains are addressed with respect to melt viscosity reduction. Bimodal linear polystyrene blends composed of a monodisperse high and
low molecular weight standards blended with two sizes of tightly crosslinked polystyrene
nanoparticles[2]. The nanoparticle volume fraction chosen leaves a portion of the linear low
molecular molecular weight content unconfined and the viscosity reduction scales indirectly
with the unconfined volume. The viscosity reduction is surprisingly sensitive to volume of
unconfined chains as only a small fraction completely suppresses the effect. The same is true
for solubility, using a larger nanoparticles which are insoluble in the linear low molecular
weight content the viscosity scale with Einstein’s relation.[4] With a better understanding
of these limits a polymer-nanocomposite with polydisperse linear chains and polydisperse
crosslinked nanoparticles with reduced viscosity was created.

In Chapter 3 of this thesis, the dispersion characteristics and miscibility of ideal nanoparticles in a polymer melt are extended into new particle/polymer size ratios which is the
key to nanoparticle phase stability[31, 32, 33]. Two sizes of tightly crosslinked polystyrene
nanoparticles[2] blended with perdeuterated monodisperse linear polystyrene were studied
via small angle neutron scattering (SANS) and rheology. SANS results of a properly confined (h < Rg ) and size matched (a < Rg ) system show well dispersed nanoparticles, with a
negative mixing energy. When compared with previous work at other nanoparticles to polymer size rations the mixing energy scales appropriately with the mixing theory developed
by Mackay and coworkers[20]. Rheology of this system shows a reduced viscosity. SANS
results on the unconfined (h > Rg ) and/or sized mismatched system (a ≈ Rg ) shows large
agglomerates (40X the individual nanoparticle) that start from a well dispersed state and
4

grow during thermal annealing. The melt viscosity of this system follows Einstein’s relation.
In Chapter 4 of this thesis, the methods of dispersion are used to make polymer-nanocomposites
with reduced melt viscosity and increased glassy toughness. Polymer-nanocomposites with
monodisperse poly(methyl methacrylate) (PMMA) standards, commercially available Fullerenes
(C60 ) and oleic acid stabilized magnetite (F e3 O4 ) are prepared by rapid precipitation. Two
molecular weights of linear PMMA were used each producing a C60 composite with reduced
melt viscosity but no change to the Young’s Modulus. A combination of high molecular
weight PMMA and F e3 O4 produced a composite with both reduced viscosity and increased
Young’s Modulus (unusual reinforcement). Small and wide angle x-ray scattering (SAXS
and WAXS) confirmed dispersion quality.
In Chapter 5 of this thesis, some details of rapid precipitation are reported. Blends of
tightly crosslinked polystyrene nanoparticles[2] with monodisperse and polydisperse polystyrene
and prepared by rapid precipitation using a range of polymer concentrations solution. A maximum concentration was determined by examining the precipitate, at 25 mg/mL or above
the nanoparticles remained suspended in the supernatant. This concentration happens to
be just above the overlap concentration for the molecular weight used. Using this limit a
polymer-nanocomposite of oleic acid stabilized cadmium selenide quantum nanocrystals[14]
in monodisperse linear polystyrene is produced. A well dispersed polymer-nanocomposite
was created and confirmed though small angle x-ray scattering (SAXS).

1.1

Research Objectives

The primary objective of this work is to expand the understand of the confinement and solubility limits for polymer-nanocomposites with reduced viscosity. Using controlled bimodal
5

blends these two limits are investigated and the results are used to create a polydispersepolydisperse polymer-nanocomposite. Dispersion and mixing energy of ideal polymer-nanocomposites
are determined for new size ratios further confirming the mixing theory proposed in the
literature.[20] PMMA multifunctional polymer-nanocomposites are created and characterized. Finally rapid precipitation is studied to created better nanoparticle dispersions.

6

Chapter 2
Bimodal Blends

7

Well dispersed nanoparticles that are smaller than the radius of gyration of the neat
polymer form a stable suspension which have unusual viscosity properties. It was previously
shown that decreases in melt viscosity were seen for systems where a large enough volume of
nanoparticles were added to confine all of the chains in a given polymer matrix.[19] Here we
study bidisperse linear blends with ideal nanoparticles to see if the viscosity decrease occurs
in these system. Confinement is achieved when the interparticle half-gap (h) is less than the
radius of gyration. The interparticle half gap is calculated using Equation 2.1, where a is
the radius of the particle, φm is the maximum packing fraction (0.638 for random packing)
and φ is the volume fraction of the nanoparticle.

h
φm
= 3
−1
a
φ

(2.1)

The radius of gyration (Rg ) for polystyrene (PS), the polymer used in this work, is
calculated using Equation 2.2 based on a fit to other work[34], where Rg is in nanometers
and M W is the molecular weight of a given chain in kiloDaltons. For monodisperse systems
the chain size can be calculated from the number average molecular weight (Mn ) or weight
average molecular weight (Mw ) since the difference in chain size between the two will be
very small. Obviously this will not be the case for bidisperse or polydisperse systems.

Rg[nm] = 0.83 M W [kD]

(2.2)

For the purpose of this experimental set we wished to selectively confine specific portions of a bidisperse blend. Blends of low and high molecular weight monodisperse linear
polystyrene standards with small and large tightly crosslinked polystyrene nanoparticles[2]
are prepared by the rapid precipitation technique.[20]
8

There are several limits that can be addressed just based on chain length, nanoparticle size
and nanoparticle volume. Consider that the nanoparticles are displacing entanglements in a
time (or frequency) dependent manner. If the material properties are measured at frequency
or shear rate faster than the particle diffusion time the nanoparticle-polymer interactions
may appear as entanglements. The chains themselves are still limited by their remaining
entanglements[30] and are restricted to motion by reptation[35, 36, 37, 38] along their contour
length[39, 40]. However, moving to lower frequencies or shear rates the nanoparticles can
diffuse away during the deformation resulting in a decrease in entanglement density per chain
and a decreased viscosity. In this scenario the lowest theoretical viscosity will be a system
where enough nanoparticles are present to disrupt all of the chain entanglements. Since
the nanoparticles are diffusing uninhibited by the entanglements seen in previous work[21]
the viscosity reduction is most pronounced in the terminal region. This system would have
a baseline viscosity at the Rouse limit (η ∝ M 1 )[36, 38] plus the increase predicted by
Einstein’s hydrodynamic theory with Batchelor correction where applicable (ηr = 1 + 2.5φ +
6.2φ2 )[4, 5] assuming the volume fraction of nanoparticles required was still within the proper
range. The upper bound on viscosity is that of a neat system which typically scales to the
well known 3.4-3.8 power (η ∝ M 3.4 )[41, 42, 43] for most linear polymers. This means
for a larger molecular weight a larger decrease can be achieved since there is a wider range
between the Rouse limit and the entangled limit.

While some may argue the viscosity decrease is simply that of a blended system with
a low viscosity additive it should be pointed out that the viscosity decrease is less than
that predicted by addition of a volume having zero viscosity and the tightly crosslinked
polystyrene nanoparticles[2] themselves have no terminal viscosity since in the pure state
9

they are a jammed system.[1] A thorough understanding of the limits of the viscosity decrease
will allow it’s predictable use in future systems.

2.1

Experimental Methods

75 kD and 393 kD monodisperse linear polystyrene standards (Scientific Polymer Products),
here labeled 75PS and 393PS, were used as received.
40 kD and 230 kD tightly crosslinked polystyrene nanoparticles[2], here labeled 40NP
and 230NP, were synthesized from a 20% benzocyclobutene functionalized linear polystyrene
prepared by Brooke Van Horn according to a method present here [2]. Tightly crosslinked
nanoparticles were prepared in the following manner a dilute suspension (1 mg/mL) of the
linear precursor in benzyl ether is dripped into a nitrogen padded bath of vigorously stirring
benzyl ether at 250 ◦ C. Under these conditions the linear precursor will intramolecularly
crosslink. Benzyl ether is distilled off from the suspension until approximately 10 mL remains. This suspension is precipitated into a 10X volume of stirring methanol. After settling
overnight the solids are separated from the suspensions via a polyamide membrane (0.2µm
pore diameter) and resuspended in THF and precipitated again in methanol. This process is repeated 3 times in total before the purified solids are transferred to a vacuum oven
at 40 ◦ C for at least 1 week to drive off all of the solvent and non-solvent. The size of
synthesized nanoparticles were characterized using light scattering on a Protein Solution
Dynapro dynamic light scattering unit (now a division of Wyatt). Intensity fluctuations in
the light scattered from the diffusing particles are related to the particle diffusion coefficient. A hydrodynamic radius is reported, calculated from the Stokes-Einstein relationship
k T
D = 6πη B R [44, 45].
solv h
10

Blends of 75 kD and 393 kD linear polystyrene were prepared in 25 v% increments from 0
to 100 with and without 40 kD and 230 kD tightly cross linked polystyrene nanoparticles[2].
All blends were prepared by suspending solids in tetrahydrofuran (THF), a good solvent,
at 10 mg/mL overnight and precipitating into 10X volume of vigorously stirred methanol,
a nonsolvent. The solutions were added drop-wise via a separation flask into the methanol.
White precipitates formed immediately. Slurries were left to settle overnight before excess
methanol was removed from the top via a vacuum apparatus. Remaining solvent-nonsolvent
was evaporated off under ambient conditions overnight before placing the solids in a vacuum
oven at 40 ◦ C for more than one week.

Dried powder was placed into a 8-mm vacuum press under 0.4 kN at 130 ◦ C for 15min
then 0+ kN at 170 ◦ C for 1.75 hours (> 10τd [46]). Disks were then placed in the vacuum
oven overnight. Rheological experiments were conducted on a strain controlled ARES-G2
rheometer. 8mm parallel plates were heated to 150 ◦ C for 30 minutes then zeroed. Samples
were loaded on hot plates. The plates are closed to 1.0 N and heated to 180 ◦ C to adhere the
disk to the plates. The temperature was then dropped to 150 ◦ C under auto compression
to maintain good contact. The system was then equilibrated for 20 minutes before starting
a test. A strain sweep was conducted to determine the linear viscoelastic region which
ends around 8% strain. Frequency-temperature sweeps were run from 150 to 210 ◦ C in 20
◦ C increments. For temperatures below 210 ◦ C, the frequency sweep ran from 100 to 1
rad/s, at 210 ◦ C the frequency range was extended down to 0.1 rad/s. A constant gap was
maintained by the thermal compensation control with a thermal coefficient of expansion of
2.6µm/◦ C. Boltzmann Time-temperature superposition[30] is used to shift frequency sweeps
from different temperature into a single master curve with a reference temperature of 170
11

◦ C.
The terminal viscosity is reported from a fit of the Carreau-Yasuda Model (η(γ) =
˙
n−1
η0 [1 + (λγ)a ] a )[6, 7] after converting the oscillatory shear data into steady shear data
˙
using the Cox-Merz Rule (η(γ = ω) = |η ∗ |(ω))[7, 8]. The plateau modulus is determined
˙
using the MIN method[47] where the value of the storage modulus at the minimum of Tan(δ)
is reported,
(G0 exp = G (ω)tan(δ)→minimum ).
N

2.2

Results & Discussion

As noted earlier it is generally understood the viscosity ratio (ηr , blend viscosity/neat viscosity) increases linearly with the volume fraction of filler added, as shown in Einstein’s
relation[4] ηr = 1 + 1.5φ + 6.2φ2 with Batchelor correction[5] for φ > 2%. Consequently the
magnitude of change in viscosity will scale with the viscosity of suspending fluid. At constant
volume fraction filler (constant increase in viscosity ratio) the magnitude of viscosity increase
will be larger for a high viscosity fluid compared to a low viscosity fluid. The same should
hold true for polymer nanocomposites with reduced viscosity found in the literature.[19, 29]
By blending 75PS and 393PS we were able to create a controlled bidisperse blend that
captures the range of chain sizes and polydispersity present in commercial polymers. From
Table 2.1, we see the polydispersity index (PDI) of the bidisperse blends range from 1.24 to
1.68, which compares well with the 1.5 PDI of commercial polystyrenes. From the molecular weight distribution (MWD), Figures 2.1 & 2.2, we can also determine what chains are
confined and therefore are capable of exhibiting a reduced viscosity. In order to achieve
confinement the interparticle half-gap, h, must be less than the Rg of the suspending chains.
12

Table 2.1: Mn , Mw and PDI of neat bidisperse PS blends from GPC. PDIs are similar to
those of commercially available polystyrenes, 1.5.
Sample
Mn (kD)
75PS
79.7 (2.6Mc )
25v% 393PS
140
50v% 393PS
205
75v% 393PS
280
393PS
341 (11.6Mc )

Mw (kD)
85.2
235
305
347
361

PDI
1.07
1.68
1.49
1.24
1.06

In Figures 2.1 & 2.2 we show the molecular weight distribution (MWD) for all of the
bidisperse blends as determined by GPC. The vertical lines indicate the smallest chain confined for a given volume fraction and size of nanoparticle, see Table 2.2, equal to those used
in this study. This means any chains higher in molecular weight than the vertical line will
be confined by that nanoparticle size at that volume fraction. From Figure 2.1 one can see
2.0 v% of the 230 kD tightly crosslinked polystyrene nanoparticles (230NP)[2] will confine
almost all of the 393PS, leaving the 75PS unconfined. 5.0 v% 230NPs will confine all of the
393PS and 1/4 of the 75PS. Moving to 8.5 v% 230NP will again confine all of the 393PS
and all of the 75PS. Due to the smaller size of the 40 kD tightly crosslinked polystyrene
nanoparticles[2] (40NP) they will confine significantly smaller chains at the same volume
fraction of 230NP. This equates to needing fewer nanoparticles to confine a particular chain
length. Again looking at Figure 2.2 the 2.0 v% 40NP line suggests all of the 393PS leaving
1/4 of the 75PS unconfined. As the volume fraction of 40NP is increased to 4.7 v% all the
chains should be confined. In Tables 2.1 and 2.2 we show the number and weight average
molecular weights and polydispersity for the bidisperse blends and polystyrene nanoparticles,
respectively. Included in Table 2.2 are the radii of the nanoparticles used in this study.
13

Figure 2.1: Molecular weight distribution of bidisperse linear polystyrene blends as determined by GPC. The dashed vertical lines indicate the smallest chain confined by a given
volume fraction of 40 kD tightly crosslinked polystyrene nanoparticles[2]. The critical molecular weight for entanglement Mc is also noted, solid vertical line. For interpretation of the
references to color in this and all other figures, the reader is referred to the electronic version
of this dissertation.

Table 2.2: Mn , Mw , PDI and radius of PS nanoparticles. Radius based on bulk density of
polystyrene (1.04 g/cm3 ). † Rh DLS in toluene at 30 ◦ C. ‡ Rg SANS in 120 kD perdeuterated
polystyrene (120dPS) at 170 ◦ C. †† 230NP particles were agglomerated in SANS study, Rh
in d-cyclohexane is reported instead.[1]
Sample Mn (kD) Mw (kD)
kD
kD
40NP
36.5
37.1
230NP
231
248

PDI
1.02
1.08

14

a (nm) † Rh (nm) ‡ Rg (nm)
nm
nm
nm
2.4
3.0
2.8
†† 5.9
4.4
7.2

Figure 2.2: Molecular weight distribution of bidisperse linear polystyrene blends as determined by GPC. The dashed vertical lines indicate the smallest chain confined by a given
volume fraction of 230 kD tightly crosslinked polystyrene nanoparticles[2]. The critical
molecular weight for entanglement Mc is also noted, solid vertical line.

15

2.2.1

Bidisperse Blends - 40NP

In Figure 2.3, we show the terminal viscosity ratios (calculated by fitting the Carreau-Yasuda
Model) for bimodal blends with two concentrations of 40NP, 2.0 v% and 4.7 v%. Focusing on
blends with 2.0 v% 40NP, we see that the viscosity increases for 75PS which is not unexpected
since a portion of the chains are unconfined (h > Rg ). 2.0 v% 40NP in 393PS has a large
viscosity decrease, also not unexpected since all of the chains are confined in this system
(h < Rg ). If we look at the bidisperse blends, we see that as 393PS content is added, the
reduced viscosity ratio becomes larger. From these five points, it is clear that high molecular
weight content receives the most benefit from reduced viscosity by nanoparticle addition. 50
v% 393PS has a PDI of 1.5, similar to commercial blends, and the nanocomposite created
with this bidisperse blend has a slight reduced viscosity. We are confident that the reduced
viscosity effect of nanoparticles will be useful in commercial polymers and the effect will be
more pronounced with higher molecular weight materials. The data point for 75PS +4.7 v%
40NP is not included since pressed disks continually cracked before rheological measurements
could be preformed. Although all chains would be confined the viscosity ratios for the 4.7
v% 40NP blends are not as impressive as the 2.0 v% 40NP blends. We suspect that the
nanoparticles at this high loading are on the verge of phase separation.
The viscosity ratios for 2.0 v% 40NP bidisperse blends that exhibit a reduced viscosity in
Figure 2.3 are listed in Table 2.3. The change in terminal viscosity is also listed to highlight
the magnitude of the reduced viscosity in high molecular weight blends. Not only is the
magnitude higher due to the higher viscosity of the suspending matrix but the decrease in
the ratio of the terminal viscosity is also larger. Since a higher molecular weight material has
a larger difference between the Rouse regime and the entangled polymer limit the viscosity
16

0
Figure 2.3: Terminal viscosity ratios (ηr ) of linear bimodal blends with 40 kD
tightly crosslinked polystyrene nanoparticles[2] (40NP). Theory line indicates the Einstein
prediction[4] with the Batchelor correction[5] (ηr = 1 + 2.5φ + 6.2φ2 ). As the fraction of high
molecular weight linear chains (393PS) is increased the melt viscosity moves to a reduced
value. Terminal viscosities are reported as the η 0 parameter from a fit of the Carreau-Yasuda
Model[6, 7] following conversion of the complex viscosity to steady shear using the Cox-Merz
Rule[7, 8]. Complex viscosity and moduli master curves for these systems are available in
Appendix A. All data shifted to a reference temperature of 170 ◦ C.

17

0
Table 2.3: Terminal viscosity ratios (ηr ) and change in terminal viscosities (∆η 0 ) of bimodal linear polystyrene and 2 v% 40 kD tightly crosslinked polystyrene nanoparticles[2]
(40NP). As volume fraction of confined chains increases the viscosity reduction increases.
0
Sample
ηr ∆η 0 (Pa-s)
75PS
1.33
+1300
25 v% 393PS 1.12
+4600
50 v% 393PS 0.98
-4000
75 v% 393PS 0.92
-38000
393PS
0.78
-250000

drop can be and is significantly larger.
Industrial equipment operates at shear rates in the 102 − 103 s−1 so it is important to
demonstrate the viscosity reduction occurs at all frequencies. By employing the Cox-Merz
Rule[8, 7] η(γ = ω) = |η ∗ |(ω), a valid assumption for polystyrene, we can assume the
˙
oscillatory response is the same as the steady shear response. Table 2.4 lists the viscosity
ratios for the 2.0 v% 40NP blends with a reduced viscosity at selected frequencies. The
reduced viscosity is present in significant magnitude at all frequencies although the effect is
lessened further from the terminal region. Where the frequency/shear rate is approaching
the diffusion rate of the nanoparticles. These reductions are more substantial considering
standard fillers at this volume fraction would produce an increase of at least 5 %, a ratio of
1.05. By using well dispersed nanoparticles instead of traditional fillers new properties can
be incorporated into polymers and processibility can be improved through reduced viscosity.

2.2.2

Bidisperse Blends - 230NP

The 230NP nanoparticles are very close in size to the 75PS chains, as such they approach the
solubility limit for nanoparticles in a polymer matrix.[20] As high molecular weight content
18

∗
Table 2.4: Complex viscosity ratios (ηr (ω)) for 2.0 v% 40NP blends with reduced viscosity
at selected frequencies. Again the highest viscosity material has the largest reduced viscosity,
however even the 75 v% 393 bimodal blend has a reduced viscosity at industrially relevant
frequencies.
∗
Sample
ηr
ω=
101 rads
s
50 v% 393PS
0.99
75 v% 393PS
0.92
393PS
0.91

∗
ηr
102 rads
s
0.99
0.89
0.93

∗
ηr
103 rads
s
1.00
0.91
0.95

is added, these nanoparticles should become more soluble. In Figure 2.4 we again show
terminal viscosity ratios, this time for three volume fractions of 230NP; 2.0 v%, 5.0 v% and
8.5 v%. In general, most of the blends that contained 75PS and 230NP have an increase in
viscosity close to that of Einstein’s relation. The 393PS blended with 230NP resulted in a
steady decrease in viscosity as more nanoparticles were added, Figure 2.4. For the 393PS
sample the 230NP are significantly smaller than the Rg of the chains making them more
soluble. The increase that was present in all bidisperse blends with 75PS and pure 75PS
with 230NP shows the solubility of the nanoparticles is independent of the number of small
chains present, at least at the volume fractions of 75PS studied. From this experiment we
can conclude that in order to make a nanocomposite with reduced viscosity the nanoparticles
should be significantly smaller than any chain present in the suspending polymer.

2.3

Polydisperse Systems

Part of the purpose of this experimental set was the determine the industrial feasibly of
nanocomposite melts with reduced viscosity. BASF was kind enough to provide us with a
polydisperse linear chain (SAN) (Mw = 197 kD, PDI = 1.5) and a polydisperse crosslinked
19

Figure 2.4: Terminal viscosity ratios of bimodal linear polystyrene with 230 kD
tightly crosslinked polystyrene nanoparticles[2] (230NP). Theory line indicates the Einstein
prediction[4] with the Batchelor correction[5] (ηr = 1 + 2.5φ + 6.2φ2 ). All nanocomposites
with 75PS and 230NP essentially behave as Einstein predict which indicates the nanoparticles are phase separated. Terminal viscosities are reported as the η 0 parameter from a fit
of the Carreau-Yasuda Model[6, 7] following conversion of the complex viscosity to steady
shear using the Cox-Merz Rule[7, 8]. Terminal viscosity ratios with 8.5 v% 230NP blends
with any v% of 75PS are extrapolated. Complex viscosity and moduli master curves for
these systems are available in Appendix A. All data shifted to a reference temperature of
170 ◦ C.

20

nanoparticle (Rh = 1 − 3 nm) (SAN-NP). These systems were prepared and characterized
in the same manner as the previous systems, with the exception of methyl ethyl ketone (2butanone) being used as the solvent. In Figure 2.5 we show the viscosity ratios in the terminal
region and at high frequency. Also included in the figure are some comparisons to blending
rules assuming the volume added has a viscosity of 0, for the sake of calculating the actual
value 1 × 10−36 Pa-s was used. For the simple blending case, labeled simple, the viscosity
was calculated by a volumetric average Equation (2.3). Tsenoglou’s blending rule for linear
polymer was also used, equation 2.4[48] where ηB is the blend viscosity, υi,j is the volume of
the component, and ηi,j is the viscosity of the component. Both cases are presented to show
how the nanoparticle reduction cannot be described by a simple blending even assuming the
viscosity of the nanoparticle is zero which would never be the case. Interestingly enough the
decrease at high frequency is fairly well captured by the decrease in plateau modulus.

ηblend = ηpolymer φpolymer + ηnanoparticle φnanoparticle

n

ηB = 2

2.4

(2.3)

n

ηi ηj
υi υj
ηi + ηj
i=1 j=1

(2.4)

Summary

In this work we clearly see two different failures of the nanocomposite melts with reduced viscosity. We will begin our discussion by addressing the results with 230 kD tightly crosslinked
polystyrene nanoparticles[2] (230NP). Tuteja and Mackay[20] have already postulated the favorable and unfavorable energies that drive or in this case prevent the dispersion of nanopar21

Figure 2.5: Terminal viscosity ratios and plateau modulus ratios of poly(styrene-coacrylonitrile) with crosslinked poly(styrene-co-acrylonitrile) nanoparticles. Theory line indicates the Einstein prediction[4] with the Batchelor correction[5] (ηr = 1 + 2.5φ + 6.2φ2 ).
Simple Equation (2.3) and Tsenoglou (Equation 2.4) indicate results based on blending with
a volume of material with essentially zero viscosity. The ratio of G0 is also included while
N
it captures most of the decrease at high frequency it does not account for the large decrease
a low frequency, which is typical of these nanocomposites. Terminal viscosities are reported
as the η 0 parameter from a fit of the Carreau-Yasuda Model[6, 7] following conversion of the
complex viscosity to steady shear using the Cox-Merz Rule[7, 8]. Terminal viscosity ratios
with 8.5 v% 230NP blends with any v% of 75PS are extrapolated. Complex viscosity and
moduli master curves for these systems are available in Appendix B. All data shifted to a
reference temperature of 170 ◦ C.

22

ticles in linear polymer. Keeping the nanoparticles dispersed results in a favorable energy
gain by increasing molecular contacts[49] that would otherwise be unavailable to the system in an agglomerated case. However, there is a penalty associated with the dispersion of
nanoparticles, physically this is due to the chains stretching[22, 23, 24, 25, 26, 27, 28] to accept the nanoparticles resulting in an entropic loss to the chains[50, 51]. This penalty scales
with (a/Rg )3 [20] based on stretching of polymer brushes[33, 51], which obviously becomes
very large as the nanoparticle radius (a) approaches the polymer chain length (Rg ). This explains the phase separation we see in the bimodal blends with 230NP. Since the 75 kD linear
polystyrene (75PS) and the 230 kD tightly crosslinked polystyrene nanoparticles[2] (230NP)
are approximately the same size the benefit of dispersion is outweighed by the penalty incurred by stretching[22, 23, 24, 25, 26, 27, 28] around such a large (relative to the chain)
nanoparticle. One would expect that as the average molecular weight of the suspending
chains is increased a nanoparticle at a fixed radius would become more soluble and therefor
more likely to produce a viscosity decrease. At the two extremes (monodisperse chains) only
75 kD linear polystyrene (75PS) and only 393 kD linear polystyrene (393PS) blended with
230NP this holds quite true. The monodisperse 393 kD linear polystyrene (393PS) has a
reduced viscosity at all volume fractions of 230NP and mondisperse 75 kD linear polystyrene
(75PS) has an increase a all volume fractions of 230NP. Which is not much of a surprise, the
230NPs are soluble in the high molecular weight content (393PS) and insoluble in the low
molecular weight content (75PS). However, the viscosity of the 75% 393PS bimodal blend
with 230NP shows a very large increase in viscosity, higher than that off all other systems.
And unlike the bimodal blends with 40 kD tightly crosslinked polystyrene nanoparticle[2]
(40NP) the viscosity decrease does not correspond to the volume fraction of low molecular

23

weight chains. Specifically the 25% and 75% 393PS polystyrene systems appear to be flipped
in this respect.
Moving to the smaller nanoparticle system we see the sensitivity to the confinement limit
even with a small number of unconfined chains. Looking at the results for the bimodal
blends with 40 kD tightly crosslinked polystyrene nanoparticles[2] (40NP) there is only a
small volume fraction of unconfined chains yet they have a strong influence on the viscosity.
There is little theoretical reasoning for why confinement is so important or why it should scale
with the Rg (Mw ). However, there is a significant amount of experimental data[19, 20, 29]
in the literature and in this work showing the limit of h < Rg being quite important to the
viscosity reduction. The results suggest an equilibrium state where only the larger chains
are swollen and the smaller chains are unaffected by the nanoparticles resulting in the classic
increase in viscosity[9, 10, 11, 12]. We can again turn to Tsenoglou’s blending rule[48] and we
find that while it over predicts the blend viscosity for the intermediate bimodal blends with
2.0v% nanoparticles it does so no worse than what would be calculated for the neat bimodal
systems. Since Tsenoglou’s relationship is based on the relaxation time of the blended chains
it should be a valid approximation for the viscosity as it would for a linear-linear bimodal
blend. In either case, the neat system or the nanocomposite, polydispersity of the chains
will effect the prediction leading to the discrepancy between the prediction and the actual
blended viscosities. Still only a small volume fraction of the 75PS chains are unconfined,
even in the pure 75PS case.
Finally looking at the polydisperse systems we see the results are quite promising for
industrial applications, this particular size of nanoparticle is above the confinement limit
and small enough to be soluble in the suspending chains. The viscosity reduction is still
24

present even at high frequency though that decrease correlates to the decrease in plateau
modulus and can almost be described by using Tsenoglou’s blending rule for a material with
zero viscosity.
From this work we conclude that the minimum amount of nanoparticles required for a
viscosity drop does not need to be calculated solely on the smallest chain present. However
there is no appropriate limit that can be determined form this experimental set. It was also
observed that blends made with nanoparticle which are of similar size to the polymer matrix
will not give favorable results. In general the best results can be obtained with long chains
and small nanoparticles. Under these conditions only a small volume of nanoparticles would
be required to achieve a decrease in the terminal viscosity limiting the likelihood of phase
separation.

25

Chapter 3
Dispersion and Mixing Energy of
Ideal Systems

26

3.1

Introduction

As nanoparticles are becoming available on a larger and more uniform scale, they are finding
their way into composite materials: sometimes replacing contemporary colloidal sized particles since the nanoscale particles produce the same benefits, increased tensile strength and
modulus, at lower loading in polyimide-clay composites.[52] Other composite materials rely
on new functionality intrinsic to the nanoparticle such as conductivity.[53] In either case, the
performance of the composite material is dependent on the dispersion of the nanoparticles.
This is especially true for optical nanocomposite materials which rely on the particle being
small enough to prevent light scattering[54]. If the nanoparticles cannot be dispersed or
do not remain dispersed, the composite material will lose performance or functionality altogether. Understanding the thermodynamics governing nanoparticle suspension in polymeric
systems is critical for these nanocomposites.
We begin our study of nanoparticle dispersion by looking at the size relationship between
the nanoparticles and suspending polymer. It has been shown that when the suspended
particle becomes smaller than the suspending chain the melt viscosity of the composite
decreases[19, 18, 55] contrary to well established results and theory from larger particle
suspensions.[10] Suspensions of well dispersed nanoparticles in a polymer matrix have been
shown to produce other unusual material properties; faster than expected diffusion of the
nanoparticle [21] and good-solvent-like chain swelling[27].
The dispersant phase of many polymer-nanocomposites are comprised of inorganic materials with a surface coating to make them compatible with polymers and/or organic
solvent.[56] Inorganic nanoparticles can be synthesized in large quantities in uniform sizes[57,
58, 59] and the chemistry for attaching organic solubilizing groups is well understood.[14, 60]
27

Unfortunately, inorganic nanoparticles are chemically different from the suspending polymer
complicating the forces present by introducing attractions or repulsions based on the chemistry at the surface and the polymer matrix. This chemical disparity can lead to phase
separation[61, 62, 63] and/or agglomeration of nanoparticles into larger clusters. By using organic nanoparticles, specifically tightly crosslinked polystyrene nanoparticles[2], in a
polymer matrix, again polystyrene, an refractive index match ideal system[11, 64] is created
where size and shape dominate the polymer and nanoparticle arrangement (packing).
Scattering has long been used to determine interparticle forces by measuring the Second Virial Coefficient of dilute concentrations of suspended particles.[65, 66, 67] We can
also measure the particle size and molecular weight using the appropriate analysis. While
the crosslinked polystyrene nanoparticle polystyrene polymer system is ideal, measuring the
morphology is complicated by a lack of contrast between the two components. With comparable chemical structures and density the optical, X-ray, and electron contrast are minimal.
However, perdeuterated monodisperse linear polymer has a large neutron contrast with respect to the hydrogen rich nanoparticles. Making neutron scattering an excellent method to
study this ideal system.

3.2

Experimental Methods

120 kD perdeuterated linear monodisperse polystyrene (Polymer Source, Inc; Montreal, QC,
Canada) was used as received, here referred to as 120dPS. The as received material was
run through the GPC to confirm molecular weight and distribution, compared against a 5
point polystyrene standard, Mn = 122 kD and Mw = 130 kD. NMR was used to confirm
deuteration, compared against a protonated chain of similar Mw .
28

40 kD and 230 kD tightly crosslinked polystyrene nanoparticles[2] were synthesized from
a 20% benzocyclobutene functionalized linear polystyrene prepared by Brooke Van Horn
according to a method present here[2]. Tightly crosslinked nanoparticles[2] were prepared
in the following manner a dilute suspension (1 mg/mL) of the linear precursor in benzyl
ether is dripped into a nitrogen padded bath of vigorously stirring benzyl ether at 250 ◦ C.
Under these conditions the linear precursor will intramolecularly crosslink with minimal
intermolecular side reactions. Benzyl ether is distilled off from the suspension until 10 mL
remains. This suspension is precipitated into a 10X volume of stirring methanol. After
settling overnight the solids are separated from the suspensions via a polyamide membrane
(0.2µm pore diameter) and resuspended in THF and precipitated again in methanol. This
process is repeated 3 times in total before the purified solids are transferred to a vacuum
oven at 40 ◦ C for at least 1 week to drive off all of the solvent and non-solvent. The
size of synthesized nanoparticles were characterized using dynamic light scattering (DLS)
on a Protein Solution Dynapro dynamic light scattering unit (now a division of Wyatt).
Intensity fluctuations in the light scattered from the diffusing particles is related to the
particle diffusion coefficient. A hydrodynamic radius is reported, calculated from the Stokesk T
Einstein relationship D = 6πη B R [44, 45].
solv h

Blends of 40 kD and 230 kD tightly crosslinked polystyrene nanoparticles[2] (40NP and
230NP) and 120 kD linear perdeuterated polystyrene standard (120dPS) are prepared by
rapid precipitation. An appropriate mass of nanoparticle and deuterated polymer are dispersed in a good solvent tetrahydrofuran (THF) at a concentration of 10 mg/mL. The
suspension is left on a rocking mixer for several days to ensure the polymer is suspended
properly. Each suspension is sonicated for 60 minutes to ensure the nanoparticles are well
29

dispersed. Following sonication each suspension is precipitated into a 10X volume of stirring
methanol. Solids are immediately separated via a polyamide membrane (0.2µm pore diameter) in a B¨chner funnel - vacuum flask apparatus with a water aspirator. The semi-dried
u
solids are then rinsed with methanol and transfered to a vacuum oven at 40 ◦ C at for at
least one week to ensure all solvent and non-solvent is removed.
Disks for rheology are prepared by placing dried powder into an 8-mm vacuum press
under slight pressure at 170 ◦ C for 60 minutes (> 10τd [46]).
Disks for SANS are prepared by placing dried powder into an 8-mm vacuum press under
slight pressure at 170 ◦ C for 15 minutes. Disks were then loaded into Al washers and into
scattering cells where they were annealed in a vacuum oven for 45min at 170 ◦ C before being
loaded into the neutron beam line sample chamber.
Rheological experiments were conducted on a strain controlled ARES-G2 rheometer. 8mm parallel plates were heated to 130 ◦ C for 60 minutes then zeroed. Samples were loaded
on hot plates. Plates are closed to 1 N and heated to 170 ◦ C to adhere the disk to the
plates. The temperature was then dropped to 130 ◦ C under auto compression to maintain
good contact. The system was then equilibrated for 30 minutes before starting a test. A
strain sweep was conducted to determine the linear viscoelastic region which ends around
8% strain. Frequency-temperature sweeps were run from 100 to 0.1 rad/s and 130 to 190
◦ C in 20 ◦ C increments. A constant gap was maintained by the thermal compensation
control with a thermal coefficient of expansion of 2.6 µm/◦ C. Boltzmann Time-temperature
superposition[30] is used to shift frequency sweeps from different temperature into a single
master curve with a reference temperature of 170 ◦ C.
Small angle neutron scattering (SANS) measurements were carried out using the 3030

m NG-3 SANS instrument at the National Institute of Standards and Technology (NIST),
Center for Neutron Research (NCNR) in Gaithersburg, MD. Two instrument configurations
were used for nanocomposites with 40 kD tightly crosslinked polystyrene nanoparticles[2]
(40NP). The first configuration had a sample to detector distance of 133 cm with a 20 cm
detector offset, 8 beam guides, and 6 ˚ wavelength, giving an expected scattering vector (q)
A
range of 0.0304-0.4472 ˚−1 . The second configuration had a sample to detector distance of
A
400 cm with a 0 cm detector offset, 5 beam guides, and 6 ˚ wavelength, giving an expected
A
q range of 0.0086-0.1179 ˚−1 . All the measurements were performed at 170 ◦ C with a
A
1/4 inch aperture. Four instrument configurations were used for nanocomposites with 140
kD tightly crosslinked polystyrene nanoparticles[2] (230NP). The first configuration had a
sample to detector distance of 133 cm with a 20 cm detector offset, 8 beam guides, and 6
˚ wavelength, giving an expected scattering vector (q) range of 0.0304-0.4472 ˚−1 . The
A
A
second configuration had a sample to detector distance of 400 cm with a 0 cm detector offset,
5 beam guides, and 6 ˚ wavelength, giving an expected q range of 0.0086-0.1179 ˚−1 . The
A
A
third configuration had a sample to detector distance of 1317 cm with 0 cm offset, 2 beam
guides, and 6 ˚ wavelength, giving an expected q range of 0.0046-0.0360 ˚−1 . The fourth
A
A
configuration had a sample to detector distance of 1317 cm with 0 cm offset, 0 beam guides,
and 8 ˚ wavelength, giving an expected q range of 0.0010-0.0257 ˚−1 . Experiments were
A
A
performed at 170 ◦ C and 127 ◦ C with a 1/4 inch aperture. Data reduction and analysis
of the data was done using the SANS and USANS Data Reduction and Analysis software
provided freely by the NCNR at NIST.[68]

31

3.3
3.3.1

Results & Discussion
120dPS-40NP

In Figure 3.1 we show the terminal viscosity ratios for 120 kD perdeuterated linear polystyrene
(120dPS) with four volume fractions of 40 kD tightly crosslinked polystyrene nanoparticles[2]
(40NP). The addition of crosslinked polystyrene nanoparticles reduces the melt viscosity,
contrary to Einstein’s[4] and others predictions and a large body of experiments with larger
particles.[10] This reduction in viscosity is consistent with prior studies[20, 29, 55] of well
dispersed nanoparticles with radii smaller than the radius of gyration (a < Rg ) of the suspending polymer and enough nanoparticles are present to achieve confinement (h < Rg ),
1v%, produces this effect.[20] Complete master curves for these composites are included at
the end of the chapter, Figure 3.14.
In Figure 3.2 we show the one dimensional scattering profiles of four volume fractions
of 40 kD tightly crosslinked polystyrene nanoparticles[2] (40NP) in 120 kD perdeuterated
linear polystyrene (120dPS). The scattering intensity increases regularly with nanoparticle
content, as expected, and the shape of scattering profile is consistent with particle-based
systems with an incoherent background.
Kratky plots (q 2 I(q) vs q) made from the scattering profiles, Figure 3.3, exhibit a single
peak centered at 0.07-0.08 ˚ that becomes more pronounced at higher volume fraction, which
A
is indicative of particle like structure. In Table 3.1 the peak positions and characteristic
length scales are reported.
Guinier plots, Figure 3.4, were constructed directly from the reduced SANS data taken
at 170 ◦ C and using Equation (3.1)[69, 70] the intensity at zero wavevector, I(0), and the
32

0
Figure 3.1: Terminal viscosity ratios (ηr ) of 120 kD perdeuterated linear polystyrene
(120dPS) with 4 volume fractions of 40 kD tightly crosslinked polystyrene nanoparticles[2]
(40NP). Theory line indicates Einstein’s relation[4] with Batchelor’s correction[5] (ηr = 1 +
2.5φ+6.2φ2 ). The reduced viscosity typical of a well dispersed nanoparticle in a polymer melt
is present. Terminal viscosities are reported as the η 0 parameter from a fit of the CarreauYasuda Model[6, 7] following conversion of the complex viscosity to steady shear using the
Cox-Merz Rule[7, 8]. Complex viscosity and moduli master curves for these systems are
available at the end of this chapter, Figure 3.14. All data shifted to a reference temperature
of 170 ◦ C.

33

Figure 3.2: One dimensional neutron scattering profiles (I(q) vs q) from the azimuthally averaged two dimensional scattering profiles of four volume fractions of 40kD tightly crosslinked
polystyrene nanoparticles[2] (40NP) in 120 kD perdeuterated linear polystyrene (120dPS)
at 170 ◦ C. As expected the scattering intensity increases with increasing 40NP content and
has the shape of a particle-like scatterer with incoherent background.

Table 3.1: Peak position from Gaussian fit to Kratky Plots (Figure 3.3) of four volume
fractions of 40 kD tightly crosslinked polystyrene nanoparticles (40NP) in 120 kD perdeuterated linear polystyrene measured at 170 ◦ C. Peak positions are constant regardless of volume
fractions and the characteristic length scale (d)is approximately the same size as the Rh for
these nanoparticles which is a good indication of dispersion.
Sample
0.5v% 40NP
1.0v% 40NP
2.0v% 40NP
4.0v% 40NP

Peak Position (˚−1 )
A
0.08127
0.08068
0.08004
0.07646

34

Size d = 2.6 (nm)
q
3.2
3.2
3.2
3.4

Figure 3.3: Kratky plots (q 2 I(q) vs q) constructed directly from the one dimensional scattering profiles of four volume fractions of 40 kD tightly crosslinked polystyrene nanoparticles
(40NP) in 120 kD deuterated linear polystyrene (120PS) with the incoherent background
subtracted. A peak in the Kratky plot indicates a particle like structure, which is expected
for this system. Furthermore the peak positions are constant for all volume fractions.

Table 3.2: Mn , Mw , PDI (determined by GPC) and hydrodynamic radius (determined
by DLS) of crosslinked polystyrene nanoparticles used in this work. Hydrodynamic radius
determine by DLS in toluene at 30 ◦ C.
Sample Mn (kD) Mw (kD)
40NP
36.5
37.1
230NP
231
248

35

PDI Rh (nm)
1.02
3.0
1.08
7.2

Figure 3.4: Guinier plots (Ln(I(q)) vs q) of the azimuthally averaged scattering profile
from 4 volume fractions of 40 kD tightly crosslinked polystyrene nanoparticles[2] (40NP)
in 120 kD perdeuterated linear polystyrene (120dPS). Solid lines indicate fits to Equation
(3.1), fits ranged from 0.0009 < q 2 < 0.0025˚−2 and the radius of gyration and intensity at
A
zero wavevector were determined. From the fits the average radius of gyration is 2.8 ± 0.1
nm indicating excellent dispersion. The results of the fits to the Equation (3.1) are listed in
Table 3.3.
z-average radius of gyration , Rg z , of the nanoparticles were determined and reported in
Table 3.3.

−[qRq ]2
3
I(q) = I(0)e

(3.1)

Using I(0) and the concentration of nanoparticles in the polymer matrix Virial plots were
constructed, Figure 3.6. Using Equation (3.3.1)[65, 66, 69], the Second Virial Coefficient,
A2 , was calculated from the slope and the weight average molecular weight, Mw , was determined from the intercept. The results are in excellent agreement with the literature[1] and
36

Table 3.3: Results of Guinier fits to scattering profiles of 40 kD tightly crosslinked
polystyrene nanoparticles[2] (40NP) in 120 kD perdeuterated linear polystyrene (120dPS) at
170 ◦ C. The radius of gyration is in excellent agreement with Rh and literature values[1].
The intensity at zero wavevector (I0 ) will be used to determine solubility and mixing energy.
Sample
0.5v% 40NP
1.0v% 40NP
2.0v% 40NP
4.0v% 40NP

I0 (cm−1 )
0.64
1.25
2.32
4.27

±
0.01
0.02
0.03
0.05

Rg z (˚)
A
28.7
28.0
29.3
28.9

±
0.5
0.4
0.4
0.3

complimentary experiments, Table 3.2.

c
= K[1 + 2A2 cMw ]
I(0)
NA
K=
Mw υ 2 (∆ρ)2 )
=

Mw
NA V 2 (∆ρ)2 )

(3.2)
(3.3)
(3.4)

From the Guinier fits, Figure 3.4 & Table 3.3, the Rg was calculated to be 2.8 ± 0.1
nm suggesting the nanoparticles are well dispersed since it would be much larger were the
particles agglomerated. Also the Rg determined from SANS is very close to the hydrodynamic radius (Rh ) of 3.0 nm determined by DLS in toluene (a good solvent), Table 3.2.
Interestingly, using the Mn to calculate the particle radius, and assuming a density of 1
g/mL (a reasonable assumption), we find it to be 2.4 nm. This is smaller than the measured
radius (a) assuming hard sphere behavior; a =

5 R = 3.6 nm and results in a Buchard’s
3 g

2
P-ratio (P 2 = Rg /R2 ) of 1.16 placing the density distribution between that of a uniform
37

Figure 3.5: Radius from Guinier fit of 4 volume fractions of 40 kD tightly crosslinked
polystyrene nanoparticles[2] (40NP) in 120 kD perdeuterated linear polystyrene (120dPS).
The dashed line marks the hydrodynamic radius (Rh ) determined by DLS, Table 3.2. The
two radii are in excellent agreement and the SANS results indicate excellent dispersion.

38

sphere and a Gaussian chain.[71] So, the particles may be swollen in the melt state and do
not behave as hard spheres. Note the particles could be agglomerated to increase the radius
of gyration; however, the molecular weight should accordingly increase to a value in excess
of 100 kD which is certainly not the case.
From the Virial plot, Figure 3.6, the Mw calculated from the intercept is 40.0 ± 0.3 kD.
Although a little higher, this value agrees with the weight average molecular weight 37.1
kD, determined for the linear precursor by GPC, as some dimerization may occur during the
synthesis of the nanoparticles. A2 calculated from the slope of the Virial Plot, Figure 3.6,
is (7.66 ± 0.34)x × 10−5 cm3 mol/g 2 . A comparison of A2 for a hard-sphere with the same
radius and molecular weight calculated using the following equation, A2 = 16Na a3 /3M 2 ,
A
was found to be 2.9 × 10−4 cm3 mol/g 2 , A 2
= 0.26. Using the square well potential,
2−HS
Equation (3.5), one can find A2 = 1 − (λ3 − 1)[exp( /kT ) − 1] assuming a typical value for
λ, 1.5, a well depth of about -0.35kT is calculated. Although the square-well potential is
a crude approximation for this system the energy is consistent with van der Waal’s force,
0.3-0.4kT.




∞






U (r) = −







0


r < 2a
2a < r < λ2a

(3.5)

r > λ2a

The Huggins interaction parameter χ can be calculated from A2 using Equation (3.6).[72]

χ=

2
1 1 V1 M2 A2
−
2
P1 2
V2
39

(3.6)

Figure 3.6: Virial Plot, concentration of scatterer over intensity at zero wavevector
against concentration of scatterer (c/I0 vs c) for 40 kD tightly crosslinked polystyrene
nanoparticles[2] in 120 kD perdeuterated linear polystyrene at 170 ◦ C. Solid line indicates linear regression fit to data set. Second Virial Coefficient, A2 , is proportional to
the slope and Mw is the inverse of the intercept according to Equation (3.3.1). A2 =
(7.66 ± 0.34)x × 10−5 cm3 mol/g 2 a value which proves the nanoparticles are soluble in the
120 kD perdeuterated linear polystyrene at 170 ◦ C. Also Mw = 40.0 ± 0.3 kD suggesting
a slight dimerization during nanoparticle synthesis when compared to the Mw of the linear
precursor at 37.1 kD but reinforcing the dispersion claim.

40

Where P is the ratio of the reduced volume to a reduced reference volume, in this work
we will use the volumes in their unreduced form and the Kuhn segment length for PS is
used as the reference volume. The subscript 1 denotes the polymer chain or solute and 2
denotes the nanoparticle. This relationship between χ and A2 is limited to systems with
uniform segment density and no dependence for χ on concentration.[72] These assumptions
are valid for our system since it is comprised of linear chains and the concentrations of the
nanoparticles are small and cover a short range. From this expression χ is calculated to be
−0.047 ± 0.002, a favorable mixing energy. This is quite surprising given blends of linear
protonated and deuterated polystyrene have a positive χ and phase separate[73, 74]. If this
system was composed of to linear chains of the same molecular weights χ would be 1.6 ± 0.6
which is below 0.5 so the system would be stable. However, mixing is more favorable for the
linear-nanoparticle system than the linear-linear system especially as molecular weights are
increase.
Comparing to a larger chain larger nanoparticle system in the literature, 473 kD perdeuterated polystyrene with 210 kD crosslinked polystyrene nanoparticles (χ = −1.2 × 10−3 )[20],
we see the smaller system has more favorable mixing energy. Initially this may seem counterintuitive as larger chains have a smaller swelling penalty, however, that penalty scales as
(a/Rg )3 )[20] and the combination of larger chains and larger nanoparticles results in a
higher swelling penalty, (5.9/21)3 > (2.8/11)3 .

3.3.2

120dPS-230NP

Moving to larger nanoparticles, 230 kD tightly crosslinked polystyrene nanoparticles[2] (230NP)
were blended with 120 kD perdeuterated linear polystyrene (120dPS). In Figure 3.7 we show
41

Table 3.4: Kratky peak position from four volume fractions of 230 kD tightly crosslinked
polystyrene nanoparticles[2] (40NP) in 120 kD perdeuterated linear polystyrene (120dPS)
measured at 170 ◦ C. The large characteristic length (d) indicates the 230NPs are phase
separating and not surprisingly the high volume fractions have larger agglomerates since
there is more opportunity for individual nanoparticles to come into contact.
Sample
Peak Position (˚−1 )
A
0.5v% 230NP
0.00914
1.0v% 230NP
0.00745
2.0v% 230NP
0.00211
4.0v% 230NP
0.00192

Size d = 2.6 (nm)
q
34.4
42.2
148.9
163.6

the one dimensional scattering profile of four volume fractions of 230NP in 120dPS. Problems with the instrument control software prevent acquisition of the low q-range for the
lowest two volume fractions of nanoparticles. The scattering profiles of the nanocomposites with 230NP look similar to the profiles of the 120dPS-40NP nanocomposites, though
the scattering intensity is significantly higher, indicating a larger scatterer volume in the
q-range measured. This increase is somewhat expected since the 230 kD tightly crosslinked
polystyrene nanoparticles[2] (230NP) are approximately 2 times larger than the 40 kD tightly
crosslinked polystyrene nanoparticles[2] (40NP), however, that size does not account for such
a large increase in scattering intensity.
Kratky plots (q 2 I(q) vs q) made from the scattering profiles of the 140NP composites,
Figure 3.8, exhibit a single peaks that become more pronounced at higher volume fraction,
again indicative of particle like structure. However, the the peaks do not occur at the same
position. Peak locations are listed in Table 3.4, the peak position decrease with increasing
volume fraction of 230NP. This corresponds to an increased scatterer size as volume factions
increase, which is not unexpected if the system 230NP nanoparticles are phase separating.
Fitting to the Guinier region, Figure 3.9 to determine Rg and I0 for the 120dPS-230NP
42

Figure 3.7: Neutron scattering profiles (I(q) vs q) of the azimuthally averaged 2D scattering profiles from 4 volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2]
(230NP) in 120 kD perdeuterated linear polystyrene (120dPS). Like the scattering profiles
of the 120dPS-40NP system, Figure 3.2, the scattering intensity increases with increasing
230NP content and has the shape of a particle-like scatterer with incoherent background.

43

Figure 3.8: Kratky plots (q 2 I(q) vs q) of the azimuthally averaged scattering profile from
4 volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2] (230NP) in
120 kD perdeuterated linear polystyrene (120dPS) at 170 ◦ C. A peak in the Kratky plot
indicates a particle like structure, which is expected for this system. However the peak
positions decrease with volume fraction of 230NP which corresponds to an increase in the
characteristic length scale (d) of the scatter (230NP). Peak positions and characteristic length
scales are listed in Table 3.4.

44

Figure 3.9: Guinier plots (Ln(I(q)) vs q) of the azimuthally averaged scattering profile
from 4 volume fractions of 230 kD tightly crosslinked polystyrene[2] (230NP) in 120 kD
perdeuterated linear polystyrene (120dPS) at 170 ◦ C. Solid lines indicate fits to Guinier
Equation, Equation (3.1) and the radius of gyration and intensity at zero wavevector were
determined. Similar to the characteristic length scale from the Kratky plots (Figure 3.8,
Table 3.4 the radius of gyration increases with volume fraction of 230NP and is significantly
larger than the Rh and literatures values[1] for these nanoparticles, a clear sign of phase
separation. The intensity at zero wavevector (I0 ) will be used to determine solubility and
mixing energy. The results of the fits to the Equation (3.1) are listed in Table 3.5.

systems we see a significantly larger than expected radius of gyration listed in Table 3.5.
Again this is a clear sign of the 230NP nanoparticles phase separating.
A Virial plot (c/I0 vs c, constructed form the Guinier fit data, is not particularly relevant
for this phase separated system. The calculated Second Virial coefficient is negative, though
not directly related to the individual nanoparticles, and the molecular weight is quite large,
2.11 × 108 g/mol. However this molecular weight is consistent with average agglomerate
45

Table 3.5: Results of Guinier fits to scattering profiles of 230 kD tightly crosslinked
polystyrene nanoparticles[2] (230NP) in 120 kD perdeuterated linear polystyrene (120dPS)
at 170 ◦ C. The large Rg is a clear sign of phase separation which increases with volume
fraction of 230NP. Aggregate size vs volume fraction is plotted in Figure 3.10.
Sample
0.5v% 140NP
1.0v% 140NP
2.0v% 140NP
4.0v% 140NP

I0 (cm−1 )
±
2692
24
16174
807
77188
445
244018
3541

Rg z (˚) ±
A
270.8
2.2
361.5
12.7
469.4
3.6
605.0
7.0

Figure 3.10: Aggregate size from 4 volume fractions of 230 kD tightly crosslinked
polystyrene nanoparticles[2] (230NP) in 120 kD perdeuterated linear polystyrene (120dPS)
at 170 ◦ C. The large 230NP aggregate size, relative to Rh of individual nanoparticle is a
clear sign of phase separation.

46

Figure 3.11: Virial plot from Guinier fits of 230 kD tightly crosslinked polystyrene
nanoparticles[2] (230NP) in 120 kD perdeuterated linear polystyrene (120dPS). Negative
slope indicates insolubility of the 230NP in 120dPS.
molecular weight of (8.64 ± 7.88) × 107 g/mol.
We can again see the phase separation through rheology. In Figure 3.12 we show the
terminal viscosity ratios for the same 120dPS-230NP “nanocomposite” systems, again at four
different volume fractions. Einstein’s relation is also included and you can see this systems
follow it quite well. In our experience adherence to Einstein’s relation simply means the
nanoparticle have agglomerated to the point where they cannot produce a reduced viscosity
and instead are acting more like a classic colloidal system[9, 10, 11, 12]. Complete master
curves for these composites are included at the end of the chapter, Figure 3.15.
Finally enough beam time was given to allow to look at the effect of annealing time
on the 120dPS-230NP “nanocomposite” systems. Remember each disk was annealed a for
47

0
Figure 3.12: Terminal viscosity ratios (ηr ) of 120 kD perdeuterated polystyrene (120dPS)
with four volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2]. Theory
line indicates Einstein’s relation[4] with Batchelor’s correction[5] (ηr = 1 + 2.5φ + 6.2φ2 )
which this system follows indicating colloidal behavior[9, 10, 11, 12] instead of well dispersed
nanoparticles as seen in the 120dPS-40NP terminal viscosity ratios, Figure 3.1. Terminal
viscosities are reported as the η 0 parameter from a fit of the Carreau-Yasuda Model[6, 7]
following conversion of the complex viscosity to steady shear using the Cox-Merz Rule[7, 8].
Terminal viscosity ratios with 8.5 v% 230NP blends with any v% of 75PS are extrapolated.
Complex viscosity and moduli master curves for these composites are included at the end of
the chapter, Figure 3.15. All data shifted to a reference temperature of 170 ◦ C.

48

15 minutes before being loaded into the sample cells. In this portion of the experiment
two of the disks were further annealed at 170 ◦ C for 3 hours and the remaining two were
loaded at 127 ◦ C. Unfortunately a lack of samples prevented a proper experimental setup
i.e., comparison at constant volume fraction. Instead the samples that previously showed the
largest agglomerates (2 and 4 v% 140NP), were not annealed any further and the samples
with the smallest agglomerates (0.5 and 1 v% 140NP) were annealed for the extra 3 hours.
All samples were measured at 127 ◦ C.
In Figure 3.13 we show the Kratky plots for the two different annealing conditions. The
two highest concentration (short annealing time) composites have a peak at a much higher q
value than the two lowest concentrations (long annealing time) indicating larger agglomerates
as the sample worked toward equilibrium. In Table 3.6 the peak positions and characteristic
length scales are reported. Although this experimental set was not ideal, it does show the
increase in in agglomerate size with volume fraction and annealing time. Furthermore it
points to the fact that the 230NP nanoparticles started off in a well dispersed state. In
the 15 minute annealing time cases the characteristic size is only about 2-3 that expected
for a single particle, even at the high volume fraction where agglomerates have a better
opportunity to form.
The driving force for the agglomeration is probably two fold. First the 140NP particles
are approaching the size of the 120 kD dPS and the penalty for chain stretching[22, 23, 24,
25, 26, 27, 28] is too high, remember it scales with (a/Rg )3 [20] which is based on stretching
of polymer brushes[51, 33] and results in an entropic loss to the chains[50, 51] The scaling
for the penalty for 230NP - 120 kD dPS system (5.9/11)3 is an order of magnitude higher
than the 25NP - 120 kD dPS system (2.8/11)3 . Secondly the volume fractions (chosen for
49

Figure 3.13: Kratky plots (q 2 I(q) vs q) of the azimuthally averaged scattering profile from
4 volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2] (230NP) in 120
kD perdeuterated linear polystyrene (120dPS), measured at 127 ◦ C. Two annealing times
were used, longer annealing time results in larger 230NP aggregates regardless of 230NP
volume, indicated by shift of peak to lower q (larger characteristic length scale of scatterer).

Table 3.6: Peak positions from Gaussian fit to peak in Kratky Plots (Figure 3.13) of 230 kD
tightly crosslinked polystyrene nanoparticles[2] in 120 kD perdeuterated linear polystyrene
(120dPS) at two different annealing times 15 minutes and 3 hours at 170 ◦ C, measured at
127 ◦ C. Characteristic length scale calculated from q = π/d, longer annealing time has a
significantly larger 230NP aggregate size.
Annealing Time
3 hours
3 hours
15 minutes
15 minutes

v% 230NP
0.5v%
1.0v%
2.0v%
4.0v%

Peak Position (˚−1 ) d =
A
0.00858
0.00673
0.0232
0.0180

50

2.6 (nm)
q
36.6
46.6
13.5
17.5

an accurate Virial analysis) are below that of the confinement limit (h < Rg ). Again when
a chain is in the presence of a nanoparticle is swells or stretches to a non-equilibrium state.
Below the confinement condition some segments of the chain are not swollen which causes
a contraction of the swollen segment near the sparsely spaced nanoparticles. This process
excludes the nanoparticles driving a phase separated condition.

3.4

Summary

The relationship between the nanoparticle size and suspending chain size is quite clear from
this work. In the system where the nanoparticle is significantly smaller than the suspending chain the mixing is very favorable as that ratio approaches unity the penalty due to
chain swelling (or stretching)[22, 23, 24, 25, 26, 27, 28] and subsequent entropic loss to the
chains[50, 51] over takes the entropic gain of nanoparticle dispersion. In the system with the
smallest nanoparticle to chain size ratio 120dPS-40NP we see favorable mixing, (negative
χ), and a well dispersed system. The 40 kD tightly crosslinked polystyrene nanoparticles[2]
are in a slightly swollen state and although a slight attractive force is present between
the nanoparticles it is not enough to overcome the entropic loss associated with agglomeration. Furthermore this system exhibits a reduced viscosity in the melt state a result
that is often seen in composited with well dispersed spherical nanoparticles.[19, 20, 55] As
the ratio of nanoparticle size to polymer length approaches unity the penalty from chain
stretching[22, 23, 24, 25, 26, 27, 28] becomes to high and the nanoparticles phase separate as
seen in the 120dPS-230NP system. Although a contribution due to lack of confinement can
not be ruled out as a driving force for the agglomeration. Looking at agglomerate size with
respect to annealing time we can conclude that blending by rapid precipitation produced a
51

well dispersed system and thermally annealing allowed the composite to move toward it’s
equilibrium state with the nanoparticles agglomerated, a process requiring several hours.
Melt rheology of these systems followed Einstein’s prediction, as expected. We also see that
rheology, which is quite sensitive to structure, worked as an excellent tool for determining
dispersion. Though it is only useful for determining whether or not the nanoparticles were
dispersed. Subtle details such as particle size can only be determined with respect to the suspending chain size, however, it is an excellent method for a quick determination of dispersion
if other methods such as scattering or TEM are unavailable.

52

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure 3.14: Master curves of 120 kD perdeuterated linear polystyrene (120dPS) with four
volume fractions of 40 kD tightly crosslinked polystyrene nanoparticles[2] (40NP). Results
shifted to a reference temperature of 170 ◦ C.
53

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure 3.15: Master curves of 120 kD perdeuterated linear polystyrene (120dPS) with four
volume fractions of 230 kD tightly crosslinked polystyrene nanoparticles[2] (230NP). Results
shifted to a reference temperature of 170 ◦ C.
54

Chapter 4
PMMA Nanocomposites

55

4.1

Introduction

Fullerenes (C60 ) has been added as a dispersed phase into many polymer composite systems
resulting in a variety of effects on the composite properties and new properties intrinsic to
C60 . Polymer composite of PMMA and C60 offer protection from ionizing radiation[75] and
reduced flammability[76]. The small size of the nanoparticle also changes the morphology.
Inclusion of C60 distorts or completely disrupts the regular network formation in block
copolymers.[77] Nanoparticles including C60 prevent de-wetting of polymer thin films.[78, 79]
Derivatives of C60 like phenyl-C61-butyric acid methyl ester (PCBM) are used heavily in
organic electronic devices such as polymer solar cells.[80, 81, 82] However, changes to the
material properties are very dependent on dispersion which in turn is dependent on blending
method. Using an evaporative blending method, either film casting or spin coating, results in
a phase separated system especially in composite systems with chemically dissimilar systems.
This phase separation leads to aggregation of the nanoparticles resulting in typical colloidal
dispersion results such as an increase in melt viscosity[10, 11] for PMMA-C60 [83] and PSC60 [55]. Blending by rapid precipitation resulting in a decrease of melt viscosity in PS-C60
composites[55] as well as increased chain dynamics[84]. C60 is well known to have a low
solubility in many organic solvents[85] and most polymer melts[86]. In this work we blend
C60 and F e3 O4 nanocrystals with PMMA using rapid precipitation where we expect a good
dispersion and a decrease in melt viscosity.
56

4.2

Experimental Methods

77 kD PMMA and 388 kD PMMA were purchased from Scientific Polymer Products, Inc
and used as received. C60 was purchased from Term-YSA (Moscow, Russia) and used as
received. F e3 O4 nanoparticles, Rh = 6 nm, were purchased from Ferrotec (San Jose, CA,
USA) with an oleic acid solubility agent and used as received. Composite blends were prepared by rapid precipitation. Due to the low solubility of C60 in organic solvents it was
dissolved separately in toluene at 5 mg/mL, sonicated for 1 hour and filtered though a 0.2µm Teflon syringe filter to removed aggregates. Solid PMMA was added along with the
appropriate volume of C60 solution and pure toluene until a polymer concentration of 10
mg/mL was achieved. PMMA and F e3 O4 were prepared by adding the appropriate volume
of each to toluene again holding the concentration of polymer at 10 mg/mL. These solutions
were left to equilibrate for several days. The color varied from clear (pure PMMA) to dark
purple increasing on the C60 content and clear to dark brown with increasing on F e3 O4
content. Prior to precipitation the solutions were sonicated for 60 minutes. The polymerC60 and polymer-F e3 O4 solutions are added drop-wise into a 10X volume of vigorously
stirring methanol. Precipitates formed immediately, colored similarly to their solution eventually settling our leaving a clear supernatant. The supernatant was decanted off and the
remaining solvent/nonsolvent is separated from the precipitates by vacuum filtration using
a polyamide membrane (0.2µm pore diameter). Solids are placed in a vacuum oven at 40
◦ C, approximately 1 week.
Rheological disks are prepared by pressing the dried solids in an 8-mm barrel die with a
vacuum attachment. 77 kD PMMA samples are annealed at 160 ◦ C for 2 hours (> 10τd [46])
and 388kD PMMA samples are annealed at 180 ◦ C for 8 hours (> 10τd [46]) both under
57

slight pressure to shape the disk, examples are shown in Figure 4.1.
Rheological properties are measured on an ARES-G2 strain controlled rheometer with 8mm parallel plates running TRIOS v1.7 and the associated firmware. PMMA is significantly
stiffer than PS, with a friction coefficient 6X larger and a very low thermal degradation
temperature 204 ◦ C limiting the experimental range to conduct rheological experiments.
For 77kD PMMA samples the plates were preheated to 150 ◦ C at least 1 hour before zeroing
the gap, samples were loaded on hot plates and 0.1 ± 0.05 N axial force is applied using the
automatic compression feature. A time sweep at 0.1% strain and 0.5 Hz is run for 1 hour
under auto compression to ensure contact with plates. The 388 kD PMMA is run under
similar condition however due to the stiffness of the material 170 ◦ C was chosen as the
starting temperature. Strain sweeps at the lowest temperature and highest frequency were
run to determine the linear region, 1.5% strain, and subsequent experiments were run at this
stain value or below. Frequency sweeps were run in 20 ◦ C increments up to 190 ◦ C. Timetemperature superposition[30] is used to shift frequency sweeps from different temperature
into a single master curve with a reference temperature of 170 ◦ C.
Pressed disks were also used to look for C60 diffraction peaks, an indication of dispersion, using wide angle x-ray scattering. Small angle x-ray scattering was used to measure
dispersion of F e3 O4 . X-ray scattering was done on a Rigaku Ultima IV with a copper
anode, wavelength 1.54 ˚, and equipped with a single point scintillation crystal detector.
A
Wide angle scans were run with the following settings: Scanning Mode = 2Theta, Scanning
Type = FT, X-Ray Power = 40kV/44mA, Divergence Slit = 1.0 mm, Divergence Height
Limiting Slit = 5 mm, Secondary Slit = 0.2 mm, Receiving Slit = 0.2 mm, Start Angle =
5◦ , Stop Angle = 65◦ , Step = 0.01◦ , Fixed Time = 5 s, and Theta = 10◦ . Small angle
58

scans were run with the settings except these: Receiving Slit = 0.1 mm, Start Angle = 0.1◦ ,
Stop Angle = 5◦ , Step = 0.005◦ , Fixed Time = 3 s, and Theta = 0◦ . In order to properly
scale the intensity for the SAXS profiles transmissions were also collected using Start Angle
= -0.1◦ , Stop Angle = 0.1◦ , Step = 0.005◦ , Fixed Time = 0.5 s, Theta = 0◦ and 0.4 mm
of Al absorber was added to attenuate the beam, preventing damage to the detector. The
peak intensity of each transmission was compared to an empty cell peak to determine the
absorption, see section 6.2. Samples were run at room temperature.

Dynamical mechanical analysis (DMA) and tensile test were performed on a RSA3 (TA
Instruments, New Castle, DE). Films were pressed from unused rheology disks using a Carver
Model C press with hot plates. Rectangular bars were cut from each using a simple jig and
a razor, samples were 0.09-0.12 mm thick by 5 mm wide by 2-3 cm long. Samples for DMA
and tensile were loaded in a similar manner on rectangular tension compression geometries
with the clamps tightened to 50 cN-m and the samples were pretensioned to 50 grams force.
Tensile test were run at room temperature (22◦ C), with a Hencky strain of 0.01. If a low
strain ”toe” region was present the strain was corrected by extrapolation from the linear
region to determine the proper zero strain point. Young’s modulus was determined from
the slope of a fit to the linear region at low strain. DMA test were run in at a frequency of
10 rad/s from 60◦ C to 175 ◦ C at a ramp rate of 3 ◦ C/minute and a strain of 0.05. Static
force tracking dynamic force at 125% was enabled to ensure a good signal to noise ratio as
the sample softened. A Gaussian peak was fit to the loss tangent to determine the glass
transition temperature (Tg ). The RSA3 was controlled using TA Orchestrator V7.2.1.1.
59

Figure 4.1: Images of 388 kD PMMA C60 nanocomposites at 3 different volume fractions
of C60 . C60 is a strong absorber in the UV range[13] resulting in the dark purple-brown
color at low loading.

4.3

Results & Discussion

Not surprisingly as C60 is added to the PMMA the color of the composite moves from
clear to a deep purple, (the color of bulk C60 ). The color change is obvious in Figure
4.1. The precipitates, disks, and films produced were all uniform in color and texture. The
low volume fraction of C60 was chosen based on the small volume fractions required to
achieve confinement[20], limited solubility of C60 [77], and for comparison to other PMMAC60 composites in the literature[83].

4.3.1

Rheology

Overall, addition of C60 to PMMA decreases the complex viscosity over the full range of
frequencies when compared to the neat material, Figures 4.2 & 4.3. This effect is more
pronounced in the high molecular weight PMMA system, Figure 4.3, which is consistent
with the bimodal blends, Chapter 2, and the literature, higher molecular weight chains
typically result in greater viscosity reduction.[29]
The material properties for the two sets of PMMA-C60 composites are listed in Table
60

Figure 4.2: Complex viscosities from master curves of 77 kD poly(methyl methacrylate)
three volume fractions of C60 . The viscosity of the nanocomposites are decreased with
respect to the neat PMMA at all viscosities. Complete master curves for these systems are
available in Appendix C: Figure C.1. All data shifted to a reference temperature of 170 ◦ C.

61

Figure 4.3: Complex viscosities from master curves of 388 kD poly(methyl methacrylate)
three volume fractions of C60 . The viscosity of the nanocomposites are decreased with
respect to the neat PMMA at all viscosities. Complete master curves for these systems are
available in Appendix C: Figures C.2. All data shifted to a reference temperature of 170
◦ C.

62

Table 4.1: Material properties of PMMA-C60 nanocomposites from master curves shifted
to a reference temperature of 170 ◦ C. The high viscosity 388 kD PMMA and low thermal
degradation point prevented measurement down to the terminal region, the complex viscosity
at a frequency of 0.1 rad is reported instead. b The plateau modulus is determined using the
s
Min method[3], G0 = G (ωtan(δ)=min ).
N
PMMA Mw (kD) C60 (v%) η 0 or a η ∗ (ω = 0.1 rad ) (Pa-s)
s
6
77
4.65 × 10
77
0.07
3.30 × 106
77
0.70
4.33 × 106
77
1.38
4.58 × 106
a 5.85 × 106
388
a 4.04 × 106
388
0.07
a 3.34 × 106
388
0.70
a 2.88 × 106
388
1.32

ηr
1
0.71
0.93
0.98
1
0.69
0.57
0.49

b G0 (MPa)
N
0.462
0.258
0.361
0.272
0.634
0.442
0.347
0.311

4.1. The terminal viscosity was determined by fitting the Carreau-Yasuda model[6] to the
complex viscosity. The plateau modulus is determined using the MIN method[47] where the
value of the storage modulus at the minimum of Tan(δ) is reported,
(G0 exp = G (ω)tan(δ)→minimum ).
N
Looking at the terminal viscosity ratios for 77 kD PMMA composites, in Figure 4.2 we
see the largest decrease in viscosity is from the smallest C60 volume fraction. Subsequent
addition of C60 increases the viscosity ratio though it remains below the value predicted by
Einstein’s theory and below 1. Phase separation is a likely culprit, as nanoparticles are less
soluble in short chains and it is reasonable to believe C60 has less favorable interactions with
PMMA than the more chemically similar PS.
Unfortunately the experimental setup used to measure the 388 kD PMMA-C60 composite
did not capture the terminal region. Still relevant deductions can be made from the complex
viscosity at low frequency, for these system η ∗ (ω = 0.1rad/s) will be used as a substitute.
63

Figure 4.4: Terminal viscosity ratios of 77 kD poly(methyl methacrylate) with three volume
fractions of C60 . Terminal viscosities are reported as the η 0 parameter from a fit of the
Carreau-Yasuda Model[6, 7] following conversion of the complex viscosity to steady shear
using the Cox-Merz Rule)[7, 8]. All data shifted to a reference temperature of 170 ◦ C.

64

Figure 4.5: Viscosity ratios from complex viscosity at a frequency of 0.01 rad/s from master
curves of 388 kD poly(methyl methacrylate) three volume fractions of C60 . Complete master
curves for these systems are available in Appendix C: Figure C.2. All data shifted to a
reference temperature of 170 ◦ C.
In Figure 4.3 we show the complex viscosity ratios (at ω = 0.1rad/s) for 388 kD PMMA and
three volume fractions of C60 . Unlike the lower molecular weight case as the C60 volume
fraction is increased the viscosity reduction continues to improve. In this system we expect
the nanoparticles to be more soluble as the ratio of nanoparticle radius to chain length is
smaller resulting in a lower amount of chain swell and thus a smaller penalty for dispersion.

4.3.2

DMA

To rule out changes in material properties due to glass transition temperature, seen in thin
polymer films[87, 88, 25, 89, 90], DMA was preformed on one group of the composites. In
65

Table 4.2: Glass transition temperature of 388 kD PMMA C60 nanocomposite at four
volume fractions of C60 . The glass transition temperature for the composite is unchanged
from the neat PMMA.
PMMA Mw (kD) C60 (v%) Tg (◦ C) ±
388
144.1
0.1
388
0.07
145.7
0.2
388
0.70
144.8
1.6
388
1.32
146.5
0.1

Figure 4.6 we show the loss tangent as a function of temperature, all of the peaks overlap
around 145 ◦ C, indicating no change in the glass transition temperature. The actual peaks
and associated error with the Gaussian peak are listed in Table 4.2. This values a bit
high for Tg values of PMMA reported in the literature (110-120 ◦ C). This was initially
attributed to the method in which the RSA3 determines temperature. Unlike the ARES-G2
this instrument cannot measure temperature at the sample instead a set of thermocouples
are placed near the hot gas inlet resulting in high temperature. However, similar values
have been reported for PMMA on other DMA type instruments.[83] These results are not
intended to state the precise Tg merely the similarity between the neat material and the C60
composites to show that it cannot account for changes to the melt state properties.

4.3.3

WAXS

Wide angle x-ray scattering (WAXS) was used to determine if any C60 crystallites were
present in the PMMA-C60 composites. This method has been used as a gauge for C60
phase separation.[77, 91] In our composites no sign of C60 is present in the WAXS profiles,
Figure 4.7. The scattering profiles for the PMMA samples clearly show the amorphous halo.
Pure C60 is included to indicate peak positions.
66

Figure 4.6: Temperature ramps of 388 kD PMMA and PMMA-C60 nanocomposites. The
glass transition temperature is the same for the neat PMMA and all C60 nanocomposite,
indicated by the constant peak position. Tg s are listed in Table 4.2.

67

Figure 4.7: Wide angle x-ray scattering profile of 388 kD PMMA C60 nanocomposite at
four volume fractions of C60 results are shifted vertically for clarity. Neat C60 is also shown
for comparison. The lack of matching peaks indicates no C60 crystallites are present.

68

Figure 4.8: Relative Young’s modulus of 388 kD poly(methyl methacrylate) with two
volume fractions of C60 . Mori-Tanaka relationship, Yr = 1+5.3φ, is included for comparison.
The Young’s Modulus scales well with the Mori-Tanaka relationship.
The glassy state mechanical properties of the PMMA-C60 are rather uninteresting but
are included for completeness and comparison to other composites. The results show no
change in solid state mechanical properties, Figure 4.8, at least within the sensitivity of the
instrument used.

4.4

PMMA-Fe3O4

Along with the C60 PMMA nanocomposites were also made with F e3 O4 which has been previously shown to produce a substantial viscosity reduction in poly(styrene) composites.[55]
Much like the C60 the F 33 O4 imparted it’s color characteristics in to the composite moving
69

from a clear to a dark brown with increasing concentration. Again the precipitates and the
disks pressed from them were uniform in color and texture. It is important to note the that
volume fractions of F e3 O4 are much higher than those of C60 and only the high molecular
weight system was studied (388 kD PMMA). Both the molecular weight of the suspending
chain and the volume fraction of the nanoparticle were chosen to ensure solubility of the
nanoparticles[20] and confinement of the chains[29]. In the case of the F e3 O4 which has a
radius of 5-6 nm a larger volume fraction was required to ensure the half-gap was less than
the Rg of the suspending polymer. Also the nanoparticle size is approaching the Rg of the
lower molecular weight polymer (77 kD PMMA) so in the interest of solubility only the 388
kD PMMA was used.

4.4.1

Rheology

In Figure 4.9 we show the terminal viscosity ratio of Neat PMMA and PMMA with two
volume fractions of F e3 O4 (2 and 5v%). An improved experimental procedure allowed
data collection to a lower frequency compared to the PMMA-C60 systems. The terminal
viscosity was again determined by fitting to the Carreau-Yasuda model. At 2 v% loading
we see a decrease in the melt viscosity, similar to poly(styrene) F e3 O4 systems found in the
literature[55], though not to the same magnitude.

4.4.2

SAXS

The high electron contrast between the F e3 O4 and PMMA allows the use of X-Rays to
quickly measure dispersion. Instead of looking for crystallites as was the case for C60 we
can use small angle scattering to determine the size of the nanoparticles and/or aggre70

Figure 4.9: Relative zero-shear viscosity (extrapolated) of 388 kD PMMA Fe3 O4 nanocomposites with two different volume fractions of Fe3 O4 . The high viscosity of 388 kD PMMA
and low degradation temperature prohibited measurement in the terminal region. The 2.0
v% F e3 O4 nanocomposite has a significantly lower viscosity than predicted by Einstein’s
hydrodynamic theory, by 5.0 v% F e3 O4 the system adheres to the Einstein’s prediction
indicating phase separation. Terminal viscosities are reported as the η 0 parameter from a fit
of the Carreau-Yasuda Model[6, 7] following conversion of the complex viscosity to steady
shear using the Cox-Merz Rule[7, 8]. Complex viscosity and moduli master curves for these
systems are available in Appendix C. All data shifted to a reference temperature of 170 ◦ C.

71

Figure 4.10: Small angle x-ray scattering profile of 388 kD PMMA Fe3 O4 nanocomposites
with two different volume fractions of Fe3 O4 , at ambient temperature.

gates. In Figure 4.10 we show the one dimensional scattering x-ray scattering profile for two
volume fractions of F e3 O4 . Absolute intensity is reported as our instrument has been calibrated against a glassy carbon standard supplied and characterized at APS by Jan Ilavsky’s
group[92], see Section 6.2. The scattering intensity increases with increasing volume fraction
of nanoparticles as expected.
Using the same technique as the SANS work in Chapter 3 the one dimensional scattering
profile are plotted as Ln(I(q)) vs q 2 also know as a Guinier plot, Figure 4.11. In this form
fitting the low q region to the Guinier equation (Ln(I) = −(qRg )/3 + Ln(I0 )) gives us the
intensity at zero wavevector (I0 ) and the Rg of the scatterer. This technique is limited to
the region of 0.5 < qRg < 1.0 though qRg up to 1.5 acceptable for polydisperse scatterers.
72

Table 4.3: Peak positions determined from Gaussian peak fits to Kratky Plots, Figure
4.12, of 388 kD PMMA with two volume fractions of F e3 O4 . Characteristic length scale is
reported using the following conversion, q = 2.6/d based on a hard sphere scatterer.
˚
PMMA Mw (kD) F e3 O4 (v%) qP eak (A−1 )
388
2.0
0.0450
388
5.0
0.0475

d (nm)
5.8
5.5

The particles should be 5-6 nm in radius resulting in an upper bound of q 2 = 1/502 =
4 × 10−4 ˚ which is within the experimentally measured range. Unfortunately linear fits in
A
the low-q region did not satisfy these requirements, qRg > 1.5, most likely due to proximity
to the primary x-ray flux. Although the slope of the intensity at low-q is slightly higher
for the 5 v% system which indicates a larger scatterer. This is consistent with viscosity
results as the 5 v% system is approaching the Einstein’s predicted value. Regardless we will
turn to another method of determine size scales of scatterers. Here we assume hard sphere
scatterers and covert a the peak position from a Gaussian fit using q = 2.6/d, Table 4.3. The
characteristic length scales determined from the peaks are consistent with the expected size
of the nanoparticles, 5-6 nm. Thought this analysis suggest the high volume fraction has a
smaller scatterer, contrary to the incomplete Guinier analysis. Regardless the characteristic
length scale is consistent with the size of the magnetite for both volume fractions. However
the we suspect there to be some agglomeration, not captured in the Kratky plot but in the
incomplete Guinier analysis.
Turning instead to Kratky plots (q 4 I(q) vs q) shown in Figure 4.12 we see peaks characteristic of individual spherical scatterers. The peak position can be used to determine a
characteristic scatterer size.
73

Figure 4.11: Guinier plots (Ln(I(q)) vs q) constructed from x-ray scattering profiles of 388
kD PMMA Fe3 O4 nanocomposites with two different volume fractions of Fe3 O4 , at ambient
temperature.

74

Figure 4.12: Kratky plots (q 4 I(q) vs q) constructed from x-ray scattering profiles of 388
kD PMMA Fe3 O4 nanocomposites with two different volume fractions of Fe3 O4 , at ambient
temperature. A peak in the Kratky plot indicates a particle like structure, which is expected
for this system.

75

4.4.3

Young’s Modulus

Finally we look at the Young’s Modulus for the PMMA-F e3 O4 composites. In Figure 4.13
we show the relative Young’s Modulus for the neat PMMA and PMMA with two volume
fractions of F e3 O4 . The results here echo those found when measuring the rheological
properties. In this case we see a large increase in the Young’s Modulus for the 2 v% case
that returns to the predicted value for the 5 v% case. Again the relative Young’s Modulus
is compared to the Mori-Tanaka model for infinitely hard spheres, a reasonable model for
F e3 O4 in PMMA. The unusual increase at 2 v% is attributed to unusual reinforcement
by the pseudo-fiber effect where regions of higher density overlap forming a high density
network. The formation of these high density regions are attributed to an increase in the
density fluctuations around the nanoparticles, partially described in simulations of polymer
near spherical surfaces[93] and polymer-particle systems[94, 95, 96, 97, 98, 99] though the
equilibrium state is not captured due to computational challenges[100, 101, 102, 103], though
excess free-volume has been measured by experiment[104]. The return to the classic response
is taken along with the viscosity and weak Guinier analysis to be a clear sign of phase
separation in the 5 v% sample.

4.5

Summary

In this work we see a rich difference in the results generated by the incredibly small C60 vs
the larger F e3 O4 . In the case of the C60 we see a large decrease in viscosity, larger than that
see for the polystyrene-polystyrene systems in Chapters 2 and 3, and larger than the decrease
seen with the F e3 O4 from this work. There are a few things to consider when describing
these results. First, and foremost, if we assume the diffusion[21, 84] of the nanoparticles to be
76

Figure 4.13: Relative Young’s Modulus of 388kD PMMA Fe3 O4 nanocomposites with two
different volume fractions of Fe3 O4 . Mori-Tanaka relationship, Yr = 1 + 5.3φ, is also plotted
for comparison. The 2.0 v% F e3 O4 nanocomposite greatly out performs the Mori-Tanaka
model, by 5.0 v% F e3 O4 the system adheres to the Mori-Tanaka model.

77

the driving force for the viscosity reduction than it is not surprising that the incredibly small
C60 molecules diffuse faster resulting in a faster terminal relaxation (the exact mechanism
remains unknown). The small size of the C60 will also effect is solubility[19, 20, 29] and
chain confinement limits[19, 20, 29]. Remember that nanoparticle must be smaller than the
suspending chain (a < Rg ) and the interparticle half-gap must also be smaller than the chain
(h < Rg ). Each of these limits is easily achieved at low volume fraction due to the small
radius of C60 making it an ideal nanoparticle for this work. Chemical incompatibility may
also play a role, C60 has a limited solubility is many solvents, including polymers (<2 v%),
being below that limit the C60 remains dispersed but unfavorable interactions may drive the
chains to swell a bit larger than what is seen in the PS-PS systems[27]. The larger swelling
may also allow for faster diffusion, though the penalty due to swelling[20] must remain small
compared to favorable separation of the C60 . Unfortunately this system did not result in any
beneficial changes to the glassy state mechanical properties, instead resulting in essentially
no change. It is clear that the results presented in this work, blended by rapid precipitation,
are significantly different than systems prepared by other methods.[83]

With the larger F e3 O4 nanoparticles we see somewhat different results. Though higher
volume fractions were required to satisfy the confinement requirement. While this appears
to make a direct comparisons difficult if the systems were made below the confinement condition would be useless as they would phase separate resulting in classic colloidal conditions
with an increased viscosity[9, 10, 11, 12]. The PMMA-F e3 O4 composites clearly show the
importance of dispersion. In both the melt viscosity and solid state Young’s Modulus we
see the unusual effects of nanoparticles for only the 2 v% case. In the 2 v% F e3 O4 case
we clearly see the reduced viscosity and indirect reinforcement effect nanoparticles have on
78

polymer chains. However the 5 v% case results in the expected increases for both viscosity
and Young’s Modulus from well understood theories, Einstein-Batchelor and Mori-Tanaka.
It is not clear from this single data set if 5 v% is actually a solubility limit or if the dispersion
technique for this system could be improved. Although best-practices were employed a few
improvements, details to the rapid precipitation technique will be discussed in Chapter 5.
Overall it is clear that nanoparticles greatly effect the material properties of PMMA and
proper dispersion through rapid precipitation is key to creating these interesting systems.

79

Chapter 5
Rapid Precipitation

80

5.1

Introduction

All of the composites presented in this dissertation were prepared by rapid precipitation as it
is the preferred method for producing a composite will well dispersed nanoparticles. During
the course of making the various systems examined and may others not listed some practical limits arose that should be discussed. The rapid precipitation method uses a solvent
and nonsolvent common to both the polymer and the nanoparticle. However, during the
many system precipitated the precipitant quality and uniformness of the material properties
depended on the concentration of the polymer (or nanoparticle in solution). Initial work
with somewhat concentrated systems (25-50 mg/mL) tended to result in a single precipitation having two distinctly different solid types. In those cases the a fibrous precipitate
would be in a cloudy (particulate like) supernatant that would sometimes settle out after an
extended period (weeks). Nanocomposites made with system like this would have wildly different material properties. Before continuing to make new samples the appropriate starting
concentration needed to be determined.

5.2

Experimental Setup

Two different linear polymer, monodisperse 393 kD polystyrene (Scientific Polymer Products) and polydisperse Polystyrol 168N (BASF) where blended with 25NP tightly crosslinked
polystyrene nanoparticles and 393 kD polystyrene was blended with oleic acid stabilized cadmium selenide quantum nanocrystals both using rapid precipitation. Neat and composite
solutions were prepared at three polymer concentrations (5, 25, 50 mg/mL) in toluene and
precipitated into a 10X volume of stirring methanol. Image were taken before precipitation
81

Table 5.1: Preparation specifications for monodisperse and polydisperse polystyrene
nanocomposite blended by rapid precipitation.
Sample #
365
366
367
368
369
370
371
372
373
374
375
376

Polymer
393PS
393PS
393PS
393PS
393PS
393PS
168N
168N
168N
168N
168N
168N

v% 25NP
3.0
3.0
3.0
3.0
3.0
3.0

Conc. polymer in Toluene (mg/mL)
5
25
50
5
25
50
5
25
50
5
25
50

and several weeks after precipitation.

5.3

Results

As seen in Figure 5.1 each of the solutions is clear as expected. Each is labeled with a sample
number, the preparation specifications for which are listed in Table 5.1.
Moving to the look of the precipitates several weeks after precipitation, Figure 5.2 and
5.3. In the samples with the highest concentration of polymer in solution 50 mg/mL (367,
370, 373, and 376) both the neat and composite precipitate have a fibrous almost nest
looking collection of solids. This collection was typically found spun around the magnetic
stir bar used to vigorously stir the methanol. Also in the composite blends, those with 25NP
nanoparticles (370, 376), there is a cloudy supernatant.
Clearly the concentration of the polymer-nanoparticle solution prior to precipitation is
a crucial parameter. Form this experimental set we see that polymer concentrations above
82

(a) 393 kD polystyrene in toluene at 5 mg/mL (365), 25 mg/mL (366), and 50 mg/mL (367)
and 393kD polystyrene with 3 v% 25NP crosslinked polystyrene nanoparticles (relative to
the polymer) in toluene at 5 mg/mL (368), 25 mg/mL (369), and 50 mg/mL (370).

(b) Polystyrol 168N in toluene at 5 mg/mL (371), 25 mg/mL (372), and 50 mg/mL (373)
and Polystyrol 168N with 3 v% 25NP crosslinked polystyrene nanoparticles in toluene at 5
mg/mL (374), 25 mg/mL (375), and 50 mg/mL (376).
Figure 5.1: Solutions of polymer and polymer-nanoparticle blends in toluene at three
concentrations.

83

(a) Precipitates from 5 and 25 mg/mL 393 kD polystyrene solutions with and without
crosslinked polystyrene nanoparticles. Supernatants in samples with nanoparticles are cloudy
(370, 376).

(b) Precipitates from 50 mg/mL 393 kD polystyrene solutions with and without crosslinked
polystyrene nanoparticles. All supernatants are clear.
Figure 5.2: Precipitates of 393 kD polystyrene and 393 kD polystyrene with 3 v%25NP
crosslinked polystyrene nanoparticles. Notice the cloudy supernatant in samples with
nanoparticles precipitated from 5 and 25 mg/mL solutions.

84

Figure 5.3: Precipitates of Polystyrol 168N and Polystyrol 168N with 25NP crosslinked
polystyrene nanoparticles. Again notice the cloudy supernatant in samples with nanoparticles precipitated from 5 and 25 mg/mL solutions (375, 376).
25 mg/mL result in a phase separated system. While this does not prove conclusively that
the polymer concentration is the primary issue, the limit falls very close to the overlap
concentration for polystyrene at this molecular weight. In the next section we put this
knowledge to use, dispersing oleic acid stabilized CdSe nanocrystals in polystyrene.

5.4

X-Ray Scattering of Polystyrene CdSe Nanocomposites

Directly measuring dispersion of polystyrene nanoparticles in linear polystyrene is quite
difficult due to the similarity of intrinsic properties i.e., electron density, refractive index,
etc. Even indirect methods such as scattering sufferer from the same limitations unless
85

neutron scattering is employed. To overcome this issue a nanoparticle with higher electron
contrast, for this work oleic acid CdSe quantum nanocrystals (oa-QNC) dispersed in 393 kD
linear monodisperse polystyrene standard was used. The heavy metal content of the CdSe
provides a large amount of electron contrast relative to polystyrene. This allows the use of
small angle x-ray scattering to determine the size of the nanocrystals, giving a measure of
the dispersion in the polymer matrix.
X-ray scattering was done on a Rigaku Ultima IV with a copper anode, wavelength 1.54 ˚,
A
and single point scintillation crystal detector. Wide angle scans were run with the following
settings: Scanning Mode = 2Theta, Scanning Type = FT, X-Ray Power = 40kV/44mA,
Divergence Slit = 1.0 mm, Divergence Height Limiting Slit = 5 mm, Secondary Slit = 0.2
mm, Receiving Slit = 0.2 mm, Start Angle = 5◦ , Stop Angle = 65◦ , Step = 0.01◦ , Fixed
Time = 5 s, and Theta = 10◦ . Small angle scans were run with the settings except these:
Receiving Slit = 0.1 mm, Start Angle = 0.1◦ , Stop Angle = 5◦ , Step = 0.005◦ , Fixed Time
= 3 s, and Theta = 0◦ . In order to properly scale the intensity for the SAXS profiles
transmissions were also collected using Start Angle = -0.1◦ , Stop Angle = 0.1◦ , Step =
0.005◦ , Fixed Time = 0.5 s, Theta = 0◦ and 0.4 mm of Al absorber was added to attenuate
the beam, preventing damage to the detector. The peak intensity of each transmission was
compared to an empty cell peak to determine the absorption, see section 6.2. Experiments
were run at room temperature.
The analysis technique employed will be the same as that used for SANS since the
theory of the two experimental techniques are essentially the same. The primary difference
between the two techniques being the nature of the electron and it’s indefinite position,
otherwise known as the electron cloud or orbital. A small correction is used to account
86

for this, Equation (5.1), where re = 0.282 × 10−5 nm, electron or Thomson radius and Z =
Atomic Number though it does not effect the outcome significantly at low angle.

(

2
δΣ
2 1 + [cos(Θ)]
)(Θ) = Z 2 re
δΩ
1

(5.1)

In Figure 5.4 we see the one dimensional scattering profile, unlike the SANS work in
Chapter 3 this instrument has a point detector so this profile is not from a two dimensional
average. This is the raw data, uncorrected for thickness, sample transmission, and absolute
intensity. We see the expected an increase in scattering intensity with nanoparticle concentration, however, the scaling of the intensity is definitely incorrect for the highest volume
fraction since it has almost the same value as the next lowest. At the very least we expect
the intensity to be proportional to the volume of the scatterer, I(q) ∝ I0 V δΣ (q). This
δΩ
exemplifies the importance of properly correcting for sample thickness and transmission,
furthermore to get the absolute intensity the instrument must be calibrated against a known
standard. Though it is immediately obvious that sphere like particle with low polydispersity
is present, indicated by the local peak.
In Figure 5.5 we see the thickness, transmission and absolute intensity corrected scattering
profiles. It is now clear that the intensity is scaling correctly with the volume fraction of the
scatterer and as in the Figure 5.4 we see a local peak indicating a spherical scatterer.
To further deduce the size of the scatter we turn to the Kratky plot (q 4 I(q)vsq), Figure
5.6, we see a more pronounced peak occurring at approximately the same q position for each
volume fraction. Fitting a Gaussian peak to the profile we can determine the exact peak
position in q-space. Then assuming a spherical scatterer we can calculate a characteristic
scatterer size d, q = 2.6/d. The results of this analysis are list in Table 5.2. From this analysis
87

Figure 5.4: Scattering profiles of 393 kD linear polystyrene (393PS) with 3 different volume
fractions of oleic acid stabilized CdSe nanocrystals (oa-QNC).

88

Figure 5.5: Absolute scattering profiles of 393 kD linear polystyrene (393PS) with 3 different volume fractions of oleic acid stabilized CdSe nanocrystals (oa-QNC).

89

Figure 5.6: Kratky plots created from absolute scattering profiles of 393 kD linear
polystyrene (393PS) with 3 different volume fractions of oleic acid stabilized CdSe nanocrystals (oa-QNC). Single peak is indicative of particle like structure.
we see the scatterers are all the same size, 2.3 nm, an excellent indication of dispersion. This
value also corresponds well to the size determined by TEM, 2.2 nm.[63]
The final method for determination of dispersion will be though Guinier analysis. In
Figure 5.7 we show Guinier plots (Ln(I)vsq 2 ) and by fitting the low q-region to Ln(I(q)) =
−(qRg )2 /3 + Ln(I0 ) we can determine the Rg of the scatterer as well as the intensity at
zero wavevector, I0 , which could be used for subsequent analysis not relevant to this work.
Again we are limited by qRg < 1 especially for these nanoparticles which have a very low
polydispersity. Using this analysis we see the same radius 2.3 nm however the variance
increases significantly at the volume fraction of the nanoparticles decreases. At the lowest
volume fraction the Guinier analysis was unable to converge on linear fit that satisfied the
90

Table 5.2: Peak positions determined from Gaussian peak fits to Kratky Plots, Figure 5.6,
of 393 kD linear polystyrene (393PS) with three volume fractions of oleic acid stabilized
CdSe Nanocrystals (oa-QNC). Characteristic length scale is reported using the following
conversion, q = 2.6/d based on a hard sphere scatterer.
PS Mw (kD)
393
393
393

oa-QNC (v%)
0.7
2.4
7.7

qP eak (˚−1 )
A
0.1111
0.1119
0.1091

d (nm)
2.3
2.3
2.3

Table 5.3: Results of linear fit to low q-region in Guinier Plots, Figure 5.7, of 393 kD linear
polystyrene (393PS) with three volume fractions of oleic acid stabilized CdSe Nanocrystals
(oa-QNC).
PS Mw (kD)
393
393
393

oa-QNC (v%)
0.7
2.4
7.7

Rg (nm) ±
2.3
1.0
2.3
0.7

I0 (cm−1 ) ±
6.8
1.3
20.2
1.0

qRg < 1 limit. Again this is most likely do to the low volume of scatterers and proximity
to the primary x-ray flux. The results of the Guinier analysis are listed in Table 5.3. The
intensity at zero wavevector (I0 )is also reported but no subsequent analysis was preformed.

5.5

Summary

It is clear that choosing the proper concentration is crucial to the success of blending polymer
chains and nanoparticles via rapid precipitation. It appears that using polymer concentrations below the overlap concentration are key, though this was not proven unequivocally.
Regardless is it important to see the precipitates settle out in a short amount of time and
that they are uniform in texture and color. Permanent suspension of some solids is a definite
91

Figure 5.7: Guinier plots created from absolute scattering profiles of 393 kD linear
polystyrene (393PS) with 3 different volume fractions of oleic acid stabilized CdSe nanocrystals (oa-QNC).

92

indication of phase separation and separation via filtering will lead to a very heterogeneous
solid.
Using this knowledge we were able to produce a set of composites with oleic acid stabilized quantum nanocrystals dispersed in polystyrene. The dispersion was characterized
by small angle x-ray scattering. Even at several different volume fractions proper use of
rapid precipitation created a composite with well dispersed nanoparticles even though the
solubilizing layer as a bulk liquid is incompatible with polystyrene.

93

Chapter 6
Experimental Techniques

94

6.1

Rheology

The field of rheology encompasses the flow and deformation of matter. This wide ranging
definition covers a variety of materials and methods. From macroscopic to microscopic,
experiments and theory the field of rheology covers a wide variety of materials and methods.
At its core it is the study of deformation by two interrelated mechanisms; force or stress
and displacement or strain. In this section I will cover the basics related to the rheological
studies done as part of this work.

6.1.1

Simple Shear

One of the most basic deformations, simple shear, is described by the flow field in Equation
(6.1) and (6.2). If material was sandwiched between two plates of infinite area and one of
the plates were subjected to a constant velocity in x relative to the other a simple shear flow
would be produced.

Vx = f (x, y)

(6.1)

Vy = Vz = 0

(6.2)

In a stress-controlled rheometer a constant force is supplied to move on of the plates the
stress (σ) on the system will be known and the resulting displacement ∆x can be measured
and given a known sample thickness (h) the strain (γ) can be determined. The opposite is
true for a strain-controlled rheometer, the stain is supplied and the force is measured. In
either method a time dependent modulus G(t) can be calculated via Equation (6.3)[7]. If
95

the strain rate is also known then the viscosity under those measurement conditions can be
determined.

G(t) ≡

σ(t)
γ0

(6.3)

Unfortunately no instrument has infinitely large plates to work with so other geometries
were developed to mimic this flow and various limitations an errors for each have been
well studied.[105] The geometry most often utilized in this work is the 8-mm parallel plate.
Parallel plates work by turning one plate relative to the other along a central axis. Although
the stress field is inhomogeneous they can replicate the infinite plate since they can be
continuously turned without losing contact with the sample. They are well adapted for use
with polymer since it is rather easy (compared to other geometries) to make samples that
fill them appropriately.

6.1.2

Oscillatory Shear

Steady shear poses a problem to polymeric systems, the constant shearing will pull chains
into a non-equilibrium state. Typically this will permanently damage the sample but even
a subtle over straining will pull chains out of their entanglement (tubes) without allowing
them to re-entangle (re-enter a new tube) rendering the sample unsuitable for study. To over
come this oscillatory shear is used. Instead of a constant shear direction the strain (or stress)
follows a sinusoidal amplitude, Equations (6.4) and (6.5), passing back through zero. This
technique, if used properly i.e., in the linear region, the polymer will remain in equilibrium
through out the test (baring thermal degradation).
96

γ = γ0 sin ωt

(6.4)

σ(t) = σ0 sin ωt + δ

(6.5)

In order to determine the linear region the material must be subjected to a strain sweep,
typically under the conditions where the sample is most stiffest, low temperature and high
frequency. In Figure 6.1 we show a typical strain sweep, in this example a hyperbranched
polystyrene-anhydride nanoparticle supplied by BASF. The strain sweep was run from 0.01
to 100% strain, at 100 rad/s (typical upper end of a frequency sweeps) and 130 ◦ C (typical
starting temperature for polystyrene, Tg + 25 ◦ C). Referring again to Figure 6.1 a large
linear region (plateau in G’, G” and η∗) exists from approximately 0.01% strain up to 10%
strain above 10% strain both the storage and loss modulus (and there for the viscosity)
begin decreasing slightly with a sharp drop off (non-linear response) above 20% strain also
called the non-linear region. The usable non-linear region ends around 60% strain where you
can see the torque begin to decease, this is a classic sign of sample failure most likely do to
edge fracture. Beyond 60% strain the sample is no longer in contact with the plates and the
data should be ignored. Subsequent experiments (such as a frequency temperature sweep)
with this material should be run at a maximum strain of 10% to ensure a linear response.
The measured linear region will remain valid for all temperatures higher than 130 ◦ C and
frequencies lower than 100 rad/s.
97

Figure 6.1: Strain sweep of GK1604/23 hyperbranched polystyrene-anhydride nanoparticle
run at 130 ◦ C and 100 rad/s. Linear region extends from 0.01 to 10% strain.

98

Table 6.1: Complex viscosities from PDMS standard at a single point oscillatory test at
0.1 rad/s and ever decreasing plate gaps.
Gap (mm)
0.986
0.897
0.796
0.695
0.595
0.497
0.377
0.286
0.231
0.185
0.144
0.096
0.048

6.1.3

1/Gap (mm−1 )
1.01
1.11
1.26
1.44
1.68
2.01
2.65
3.50
4.33
5.41
6.94
10.4
20.8

η ∗ (ω = 0.1rad/s) (KPa-s)
0.709
0.541
1.943
0.805
1.256
1.163
1.024
1.282
0.907
1.106
1.019
0.257
0.158

Plate Alignment

Ensuring proper plate alignment is crucial to the accuracy of experimental results in parallel
plate and cone-and-plate geometries. The simplest method for determining alignment is a
series of measurements while decreasing the gap. Eventually the scale of any misalignment
(axial center, normal vector alignment) will become on the order of the plate gap. Simply
put the miss alignment will become a large factor in the measurement. All shear experiments
in this dissertation use 8-mm parallel plates for which the alignment was checked with high
viscosity PDMS standard provided by TA Instruments. A series of single point oscillatory
test at 0.1 rad/s and ever decreasing plate gap were run at room temperature, the results of
which are listed in Table 6.1 below.
In Figure 6.2 the measured viscosity at 1 rad/s for several the plate gaps (h) are plotted.
It is immediately obvious there is a sharp drop off in the terminal viscosity around 0.1 mm,
99

indicating the lower usable limit. Above 0.1 mm the terminal viscosity is independent of the
plate gap, though the values start to vary wildly above 0.6 indicating the upper limit. The
gap used for all of the system studied in this work was between 0.18 mm and 0.36 mm well
within the usable range.

Figure 6.2: Complex viscosity of PDMS standard at different plate gaps. The drop off in
viscosity denotes the lower usable gap with the current alignment.

6.1.4

Wall Slip

6.1.5

Sample Stability

Ensuring samples are stable during the course of an experiment is very important to gain the
full understanding of the system being studied. If the polymer degrades or if a suspension
100

Figure 6.3: Terminal viscosity of 393 kD polystyrene standard at different plate gaps. The
stable value indicates no wall-slip is present.

collapses over the course of a frequency-temperature sweep Boltzmann-Time-Temperature
superposition will sometimes fail or worse it will work but the results are the combination
of a non static system. One quick check that can be done is the classic hysteresis loop. For
the case of a frequency temperature sweep returning to one of the lower temperature at the
end of an experiment will give a good indication of the stability of the system being studied.
In Figure 6.4 we show sweeps of poly(styrene-co-acrylonitrile) with cross linked poly(styreneco-acrylonitrile) nanoparticles. The 1st cycle frequency sweep was measured during the typical frequency-temperature sweep (130, 150, 170, 190, 210 ◦ C) used to make a master curve,
the second frequency sweep was measured following the last frequency sweep (210 ◦ C) during
the frequency-temperature sweep experiment. As you can see the two frequency sweep are
101

in excellent agreement, which is typical for most of the system report in this dissertation,
indicating the stability of the nanoparticle dispersion during the course of the frequency
temperature sweep.

6.2
6.2.1

SAXS
Absolute Intensity Calibration

Bench top x-ray instruments are readily available from several manufacturers. Modern machining and control techniques allow their use as a small angle x-ray scattering (SAXS)
instruments, over a limited yet useful range. Typically bench top setups are employed to
measure size scales with SAXS. However, subsequent analysis requires absolute intensity,
which is unknown. In order to determine the absolute intensity a correction factor is employed. The correction factor is determined by scattering from a well characterized standard,
in the this case glassy carbon supplied and characterized at the Advanced Photo Source (Argonne National Lab).[92] By measuring the standard on the bench top instrument a simple
scaling factor can be calculated. The procedure and equations are taken directly from Dreiss’
paper on the subject.[106] The characteristic momentum scattering Equation (6.6) is shown
below.

Is (q) =

δΣ
Ns (q)
= I0 (λ)A∆Ωη(λ)Ts (λ)ds (q) + BGs
ts
δΩ

(6.6)

The scattering intensity, Is (q), is the measured quantity of photons, Ns (q), detected
per unit time, ts , where q = 4π/λ sin( Θ ). I0 (λ) is the incident x-ray flux which is itself
2
a function of wavelength, λ. A is the illuminated area, ∆Ω, is the solid angle element
102

(a) Complex Viscosity

(b) Storage & Loss Moduli
Figure 6.4: Frequency sweeps at 170 ◦ C poly(styrene-co-acrylonitrile) with cross linked
poly(styrene-co-acrylonitrile) nanoparticles. Overlap of material properties indicated stability of system during the time frame of the typical experiment.
103

(pixel detector size), η(λ) is the detector efficiency (a function of wavelength), Ts (λ) is the
sample transmission (a function of wavelength), ds is the sample thickness and BGs is the
background. The calibration for our instrument will use ( δΣ )s , the differential scattering
δΩ
cross section of a well characterized standard (Glassy Carbon). The differential scattering
cross section of a sample can be determined by Equation (6.7), where ’s’ indicates the sample
and ”st” indicates the Glassy Carbon Standard. Background scattering typically very weak
and can be ignored.

(

δΣ
[Is (q) − BGs ]
dst Tst
δΣ
)s (q) = ( )st (q)
δΩ
δΩ
ds Ts
[Ist (q) − BGst ]

(6.7)

If the standard and sample are measured under identical conditions the remaining variables from Equation (6.6) cancel out. The calibration factor can be derived from all of
the ”st” labeled variables, Equation (6.8) below. Accurate measurements of the calibration
standard can then be used to correct unknown samples.

CF = (

δΣ
dst Tst
)st (q)
δΩ
[Ist (q) − BGst ]

(6.8)

Once the correction factor has been determined the differential scattering cross section
of an unknown sample can be calculated from Equation (6.9), below.

(

δΣ
[Is (q) − BGs ]
)s (q) = CF ×
δΩ
ds Ts
104

(6.9)

Table 6.2: Instrument configuration for transmission measurements.
Scanning Mode
=
Scanning Type
=
Source Voltage
=
Source Current
=
Divergence Slit
=
Divergence Height Limiting Slit =
Secondary Slit
=
Receiving Slit
=
Start
=
Stop
=
Step
=
FT
=
Fixed Angle (Theta)
=
Al absorber thickness
=

6.2.2

2Theta
FT
40kV
44 mA
1.0 mm
5.0 mm
0.2 mm
0.1 mm
-0.100◦
0.100◦
0.005◦
0.5 s
0◦
0.4 mm

Experimental Setup

Glassy carbon sample was provided by Jan Ilavsky at APS, labeled as Glassy Carbon L18.
The sample was characterized at APS on 32ID-B USAXS line. Differential scattering cross
section as a function of wavevector was supplied for calibration of local instruments. The
glassy carbon sample thickness is 0.100 cm which was confirmed locally. Rigaku UltimaIV
was warmed up to 40kV at 44mA for 30 minutes. This instrument has a copper source
providing a x-ray wavelength of 1.54˚(Cu Kα ). Standard automatic alignment procedure for
A
SAXS was followed. Tables 6.2 and 6.3 detail the instrument configuration for transmission
and SAXS measurements. Al absorbers were used during the transmission to prevent damage
to the detector. 5 empty cell measurements and 5 sample measurements were conducted for
each configuration. Intensities are reported in counts per second (cps). In x-ray nomenclature
the scattering angles are reported as 2Θ (this is not 2 × Θ) for this work the scattering angles
are converted to q-space using q = 4π/λ sin( Θ ).
2
105

Table 6.3: Instrument configuration for SAXS measurements.
Scanning Mode
=
Scanning Type
=
Source Voltage
=
Source Current
=
Divergence Slit
=
Divergence Height Limiting Slit =
Secondary Slit
=
Receiving Slit
=
Start
=
Stop
=
Step
=
FT
=
Fixed Angle (Theta)
=

2Theta
FT
40kV
44 mA
1.0 mm
5.0 mm
0.2 mm
0.1 mm
0.070◦
5.000◦
0.005◦
0.5 s
0◦

It is important to note that this calibration is only valid for the slit configurations listed
in Table 6.3, changes to the source flux or detector flux will result in a different correction
factor, see Equation (6.6). However using a different q-range does not change the correction
factor. Changes to the transmission setups will not effect the result.

6.2.3

Results

6.2.3.1

Absorbance

The results for the peak intensities for the empty cell runs and glassy carbon standard are
list in Table 6.4. The calculated transmission of the glassy carbon, Tst = 0.538 ± 0.011, is in
excellent agreement with Dreiss’ reported value, (0.527 ± 0.003), with a similar sample[106]
and within the acceptable transmission range of 1/3 to 2/3.
106

Table 6.4: Intensity at 0 angle for an empty cell (EC) and L18 glassy carbon (GC) sample.

Run
1
2
3
4
5
Average
Standard Deviation

EC Peak Intensity
cps
21311
20822
21365
21712
21092
21260
296

GC Peak Intensity
cps
11507
11122
11559
11600
11432
11444
170

Table 6.5: Intensity at q=0.06˚ (interpolated) for an empty cell (EC) and L18 glassy
A
carbon (GC) sample.
Run
EC I(q = 0.06) (cps)
1
3
2
3
3
5
4
4
5
2
Average
3
Standard Deviation
1

6.2.3.2

GC I(q = 0.06) (cps)
2138
2240
2261
2207
2153
2200
48

SAXS

In Figure 6.5 I show the differential scattering cross section vs wavevector q (provided by
APS) and the scattering intensity vs wavevector q (measured locally) of the L18 Glassy Carbon standard provided by APS. Overall the two curves are in excellent agreement. However
there are slight differences in the plateau region, similar to those experienced by Dreiss.[106]
To avoid including unnecessary error in the calibration factor the data from our instrument
will be corrected at q=0.06 ˚. The intensities at q=0.06 ˚are listed in Table 6.5.
A
A
107

Figure 6.5: Differential scattering cross section and scattering intensity for glassy carbon
standard.

108

Table 6.6: Differential scattering cross section (APS) and intensity (Rigaku) of L18 Glassy
Carbon Stand at q=0.06˚.
A
Instrument
Rigaku UltimaIV
32ID-B (APS)

6.2.3.3

I(q = 0.06)
Units
2196
cps
−1 sr−1
30.7087
cm

Correction Factor

Using Equation (6.8) the correction factor was calculated to be
7.51 × 10−4

30.71cm−1 (0.1cm×0.537)
=
2196phs−1

s . This value was then applied to Equation (6.9), treating the standard
sr−ph

as an unknown sample. The resulting corrected value is shown in Figure 6.6 along with the
original differential scattering cross section from APS.

6.2.4

Summary

The correction factor for the Rigaku UltimaIV was determined to be 7.51 × 10−4

s .
sr−ph

This value can be used with the scattering intensity, transmission and thickness to calculate
the differential scattering cross section of an unknown sample. In the future the correction
factor will be calculated as a function of wavevector, CF (q), and error will be included to
achieve a more robust correction.

109

Figure 6.6: Differential scattering cross section of glassy carbon from APS and Rigaku
(corrected).

110

Chapter 7
Conclusions

111

In this work polymer-nanocomposites with well dispersed spherical nanoparticles were
studied to learn more about the reduced viscosity phenomena. The solubility of nanoparticles in polymer melts and confinement aspect nanoparticles on polymer chains with respect
to reduced viscosity were studied using bimodal blends. The sensitivity of viscosity reduction to confinement limits was investigated by leaving a portion of the bimodal linear chains
unconfined. Determined the partial confinement does not produce a complete reduction,
indicating the nanoparticle’s effect on the polymer chains must be uniform throughout the
sample. The sensitivity to nanoparticle solubility was also studied, again by choosing a
low molecular weight linear component of the bimodal blend that the nanoparticles would
be insoluble in. The same technique was used to measure viscosity reductions with partial
solubility. Similar to the confinement limit if the nanoparticles are insoluble in some of the
chains the reduction is suppressed. In the insoluble case the viscosity follows that of Einstein’s relation. Then with a more through grasp of these limits a polydisperse-polydisperse
polymer-nanocomposite with reduced viscosity was created.

Next the dispersion of ideal nanoparticles in a polymer melt was characterized via neutron scattering and compared to separate rheological experiments. This provided a direct
relationship between reduced melt viscosity and nanoparticle dispersion. Systems with well
dispersed nanoparticles exhibit a reduced viscosity. Agglomerated system exhibit an increased viscosity following Einstein’s relation[4]. Mixing energy scaling with (a/Rg ) follows
mixing theory of Mackay and co-authors[20]. Also rapid precipitation produces well dispersed system regardless of solubility and lack of confinement or solubility drive a phase
separation upon annealing.
112

Finally attempts to make a PMMA-nanocomposite with reduced viscosity and increased
toughness gave mixed results. Using C60 and F e3 O4 nanoparticles nanocomposites with
reduced viscosity were made but only one volume fraction of F e3 O4 resulted in an increased
Young’s Modulus. The remaining systems followed the Mori-Tanaka theory for infinitely
stiff additives. Concentration limits for blending polystyrene and crosslinked polystyrene
nanoparticles were identified, these limits were then used for preparing subsequent samples.
Moving forward a clear understanding of the phase separation and viscosity of nanoparticles below the confinement limit should be investigated. The diffusion characteristic of
nanoparticles with should be studied with well defined chemically similar (to the suspending
polymer) at a few nanoparticle radii to compare reduced viscosity to nanoparticle size and
hence diffusion. With the diffusion-viscosity relationship being compared by volume fraction,
confinement ratio (h/Rg < 1), and normalized by a viscosity increase to the Rouse limit by
the nanoparticles.

113

APPENDIX

114

Appendix A
Master Curves of Bimodal
Poly(styrene) Blends With Cross
Linked Polystyrene Nanoparticles

115

(a) Complex Viscosity

(b) Storage & Loss Moduli
Figure A.1: Master curves of 75 kD polystyrene with two volume fractions of 25NP cross
linked polystyrene nanoparticles. Results shifted to a reference temperature of 170 ◦ C.

116

(a) Complex Viscosity

(b) Storage & Loss Moduli
Figure A.2: Master curves of 75 kD polystyrene with three volume fractions of 140NP
cross linked polystyrene nanoparticles. Results shifted to a reference temperature of 170 ◦ C.

117

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure A.3: Master curves of 75/25% 75/393 kD polystyrene with two volume fractions of
25NP cross linked polystyrene nanoparticles. Results shifted to a reference temperature of
170 ◦ C.
118

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure A.4: Master curves of 75/25% 75/393 kD polystyrene with three volume fractions
of 140NP cross linked polystyrene nanoparticles. Results shifted to a reference temperature
of 170 ◦ C.
119

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure A.5: Master curves of 50/50% 75/393 kD polystyrene with two volume fractions of
25NP cross linked polystyrene nanoparticles. Results shifted to a reference temperature of
170 ◦ C.
120

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure A.6: Master curves of 50/50% 75/393 kD polystyrene with three volume fractions
of 140NP cross linked polystyrene nanoparticles. Results shifted to a reference temperature
of 170 ◦ C.
121

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure A.7: Master curves of 25/75% 75/393 kD polystyrene with two volume fractions of
25NP cross linked polystyrene nanoparticles. Results shifted to a reference temperature of
170 ◦ C.
122

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure A.8: Master curves of 25/75% 75/393 kD polystyrene with three volume fractions
of 140NP cross linked polystyrene nanoparticles. Results shifted to a reference temperature
of 170 ◦ C.
123

(a) Complex Viscosity

(b) Storage & Loss Moduli
Figure A.9: Master curves of 393 kD polystyrene with two volume fractions of 25NP cross
linked polystyrene nanoparticles. Results shifted to a reference temperature of 170 ◦ C.

124

(a) Complex Viscosity

(b) Storage & Loss Moduli
Figure A.10: Master curves of 393 kD polystyrene with three volume fractions of 140NP
cross linked polystyrene nanoparticles. Results shifted to a reference temperature of 170 ◦ C.

125

Appendix B
Master Curves of
Poly(styrene-co-acrylonitrile) With
Cross Linked
Poly(styrene-co-acrylonitrile)
Nanoparticles

126

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure B.1: Master curves of poly(styrene-co-acrylonitrile) (SAN) with cross linked
poly(styrene-co-acrylonitrile) nanoparticles (SAN-NP). Results shifted to a reference temperature of 170 ◦ C.
127

Appendix C
Master Curves of Poly(methyl
methacrylate) C60 and F e3O4
Nanocomposites

128

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure C.1: Master curves of 77 kD poly(methyl methacrylate) with three volume fractions
of C60 . Results shifted to a reference temperature of 170 ◦ C.

129

(a) Complex Viscosity

(b) Storage & Loss Moduli

Figure C.2: Master curves of 388 kD poly(methyl methacrylate) with three volume fractions
of C60 . Results shifted to a reference temperature of 170 ◦ C.
130

(a) Complex Viscosity

(b) Storage & Loss Moduli
Figure C.3: Master curves of 388 kD with two volume fractions of F e3 O4 . Results shifted
to a reference temperature of 170 ◦ C.

131

Appendix D
Rheology of Poly(3-hexylthiophene)
Poly(3-hexylthiophene) is the currently the work horse polymer of organic solar cell researchers. The rheological properties were briefly studied in an attempt to provide additional
information, such as melt viscosity to other group members. Sepiloid P-100 P3HT (BASF
& Reike Metals) was pressed under vacuum into 8-mm rheological disk at 235 ◦ C for 15
minutes.
For oscillatory shear temperature ramps the disks were loaded on to an ARES-G2 at room
temperature and a 3 ◦ C/minute ramp was run from 30-250 ◦ C and back down with a strain
of 0.1%, frequency of 10 rad/s. Axial force was set to 0.03±0.01 N with force tracking (Axial
Force > Dynamic Force at 1.0%) with a minum force of 1.0 N to manintain contact. To
compensate for the decrease in visocsity though the melting temperature automatic strain
adjustment was employed to keep the oscillartory torque between 30 and 10000 µN − m
bounded to a minimum strain of 0.1% and 8.0%. All in a nitrogen atmosphere.
For oscillatory shear time sweeps, strain sweeps, and frequency sweeps the disks were
loaded at 235 ◦ C after allowing the instrument to thermally equilibrate for 1 hour. Time
132

sweeps were run at 10% strain and 10 rad/s.
From the temperature ramp, Figure D.1, we determine the Tg for this type of P3HT
to be 35.1 ◦ C and the crystalline melting temperature to be between 197.9 ◦ C and 227.4
◦ C. A time sweep was run to determine thermal stability in nitrogen above the melting
temperature.
In Figure D.2 we show a composite time sweep covering 3.5 hours with intermittent
frequency and strain sweeps. The P3HT starts off as a viscous liquid that steadily increases
in viscosity until approximately 20 minutes (Figure D.2) where we see a G’ G” crossover.
From the strain sweeps (Figure D.3 taken before (D.3(a)) and after (D.3(b)) the cross over
we can again see the change from liquid like to solid like. The set of intermittent frequency
sweeps (Figure D.4), taken about ever 30 minutes remain fairly constant. In the final strain
sweep, shown in Figure D.5(b) a non-linear region appears at high strain.
There are two likely reasons for the increase in stiffness of the material over the duration
of the time sweep. The P3HT known for is instability in air could still be degrading via
crosslinking, however, the used disks readily re-disperse in toluene ruling out permanent
network formation. However P3HT can readily crystallize and it seems reasonable to suggest
the extra thermal mobility is allowing the polymer to for a crystalline network. If this is
the case it would begin excluding nanoparticles at that temperature after 20 minutes. This
would effect processing conditions if nanoparticle dispersion is desired. Though the ideal
organic solar cell is a interdigitated network so using the high temperature crystallization to
drive a phase separation could be quite beneficial.

133

(a) Storage and Loss Modulus

(b) Complex Viscosity and Loss Tangent
Figure D.1: Material properties of P3HT for a temperature ramp from 30-250 ◦ (forward
cycle) and from 250-30 ◦ C (reverse cycle) each at 3 ◦ C/minute. A peak in the loss tangent
(Tg ) is noted as well as approximate crystalline melting temperatures.
134

Figure D.2: Material properties of P3HT as a function of time at 235 ◦ C. Note the sudden
increase in stiffness after 1200 seconds. The vertical lines indicate times when strain sweeps
(dashed lines) Figure D.3 or frequency sweeps (solid lines) Figure D.4 were conducted. The
dot-dash line indicates a final frequency sweep followed by a final strain sweep, Figure D.5.
All subsequent figures share this legend.

(a) Initial Strain Sweep

(b) Strain Sweep at 1200s

Figure D.3: Two strain sweeps of P3HT before and after increase in stiffness at 235 ◦ C.
Legend is the same as Figure D.2
135

(a) Frequency Sweep at 4300s

(b) Frequency Sweep at 6300s

(c) Frequency Sweep at 8300s

(d) Frequency Sweep at 10300s

Figure D.4: Four frequency sweeps of P3HT following increase in stiffness at 235 ◦ C.
Legend is the same as Figure D.2

136

(a) Final Frequency Sweep

(b) Final Strain Sweep

Figure D.5: Final frequency sweep and strain sweep. The sample begins to show non linear
behavior in at high strain after several hours at 235 ◦ C. Legend is the same as Figure D.2

137

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