OF POLYMERIC SUSPENSIONS: POLYMER NANOCOMPOSITES AND WATERBORNE COATINGS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Jianhua Xu, B.Eng.

♦ ♦ ♦ ♦ ♦

The Ohio State University 2005

Dissertation Committee: Approved by: Professor Kurt W. Koelling, Advisor

Professor Stephen E. Bechtel ______Advisor Professor David L. Tomasko Chemical Engineering Graduate Program

ABSTRACT

The complexity inherited in the polymeric suspensions makes prediction and control of their rheological properties difficult. Much effort has been devoted to characterize these properties and considerable improvements have been made. However, much more work is desired in order to develop new products and understand the underlying knowledge about the flow behavior of the suspensions. In this work, the rheology of two types of polymeric suspensions, namely waterborne coatings and polymer/nanoclay and polymer carbon nanofibers has been studied.

As the waterborne coatings are non-Newtonian, the depends on the shear rate and the strain history, which makes the flow in circular pipes complicated. The rheology of a type of metallic automotive basecoat is measured using a highly accurate and the data is fitted to several constitutive models. The drop of the paint flow in a straight circular pipe has been measured in a pilot plant, and the steady state laminar flow pressure drop over the pipe length has been calculated using numerical methods. Sisko model and Carreau model has been found to be able to predict the pressure drop accurately using the material parameters obtained from rheological measurements. The power law model is applicable for flows where the Reynolds number is less than 100. The effects of thixotropy can be neglected during normal continuous

ii operation. The relatively higher initial pressure drop caused by the thixotropy of the paint occurs during start up is found to be 1.25 times the pressure drop at steady state. The paint studied shows abnormal temperature effects on the rheology. Steady shear viscosity

decreases from 10 to 35 °C, and then increases at higher temperatures up to 45 °C.

In the study of the rheology of carbon nanofiber (CNF) suspensions in Newtonian

, it is found that the dispersion of the nanofibers determines the suspension

rheology. The suspensions contain up to 5wt% CNFs. Steady shear of

untreated sonicated suspensions are higher than that of better dispersed treated sonicated

suspensions, due to the existence of large clumps and strong inter-particle interactions

that block the flow. The measurements of small amplitude oscillatory shear and yield

reveal that the both untreated sonicated and treated sonicated suspensions exhibit

solid-like behavior when the nanofiber concentration is higher than 2wt%. Morphology

investigations show that the acid treatment of the treated nanofibers disintegrates the

agglomerates exists in the as-received CNFs and therefore improves dispersion, it also

weaken them, as the subsequent sonication break most of the treated CNFs.

Two methods have been used to prepare the composites, such that the effect of the

preparation methods on the dispersion and possible damage to the CNFs can be studied.

Our results show that the melt blending method using twin screw extruder can produce

composites with homogeneous dispersion of the CNFs, but at the cost of the length of the

CNFs. By careful analysis, we found that the CNFs in the melt blended composites have

been greatly reduced. On the other hand, a solvent cast methods, in which a solvent was

used as buffer and a high powerful ultrasound probe was used, can prevent the CNFs

from breaking while achieve satisfactory dispersion.

iii Extensive shear and extensional rheological studies have been conducted on the

polystyrene/CNF composites. Our results show that the elastic and viscous moduli (G’ and G”) are greatly improved with higher CNF concentrations. Higher CNFs in the SC composites have more impact on the increase. Also, the elastic modulus of the SC5 and

SC10 composites show plateau, such that the modulus become less dependent of the frequency. This is an indication of a network structure. Network structure is also found by measuring the yield stress of the composites. Transient behavior, including transient shear viscosities and normal forces, can also be related to the evolution of the CNF orientation and structure. Orientation of the CNFs induced by shear flow is also measured, both as a function of the shear strain and of the shear rate. Two 2-D pictures in mutually orthogonal planes provide vital information of the 3-D fiber orientation.

Mathematical models have been used to interpret the effect of the length of the CNFs and

the interactions between them. Second order orientation tensor is used to capture the fiber

orientation. Our model results show that the rheology of the composites can be captured

and the orientation of the CNFs can be predicted and coupled with the rheology.

iv

Dedicated to my wife and parents.

v

ACKNOWLEDGEMENT

I thank my advisor, Dr. Kurt W. Koelling, for his guidance, inspiration, and support that made this work possible. I wish to express my gratitude to Dr. Stephen E.

Bechtel for his knowledge and insight in the mathematical modeling. I thank Yingru

Wang for performing most of the mathematical modeling of the rheology of carbon nanofiber systems. Special thanks to Dr. David Tomasko for serving on my committee and for providing constructive comments.

I thank Paul Green, Carl Scott and Leigh Evrard in the Department of Chemical and Biomolecular Engineering for their generous help in the experimental setup and other details relating to mechanical machining and electrical testing.

I would also like to thank Shubho Bhattacharya of Honda of America

Manufacturing for providing the paint samples, equipment for the pilot plant, and more importantly, his effort to push the paint project forward.

Financial support comes from National Science Foundation, Honda of America

Manufacturing, and Center for Advanced Polymer and Composite Engineering.

Finally, I thank my parents for their unfaltering love, encourage and support through out the years. I also thank my wife, Chun Zhang, for her endless love and tender care.

vi

VITA

March 24, 1972······································· Born – Shaanxi, P.R. China

September 1989 – June 1993·················· B.S. Chemical Engineering, Zhengzhou Institute of Technology Zhengzhou, Henan, P.R. China

September 1993 – June 2000·················· Chemical Process Engineer, Process Lead Engineer, Process Discipline Group Leader. China Hualu Engineering Corporation Xian, Shaanxi, P.R. China

September 2000 – August 2001·············· University Fellowship The Ohio State University Columbus, Ohio, U.S.A.

September 2001 – December 2004········· Graduate Research Associate, Department of Chemical Engineering, The Ohio State University, Columbus, Ohio, U.S.A.

January 2005 – Present··························· CAPCE Fellowship Center for Advanced Polymer and Composite Engineering, The Ohio State University, Columbus, Ohio, U.S.A.

vii

PUBLICATIONS

1. J. Xu, S. Chatterjee, K.W. Koelling, Y. Wang and S.E. Bechtel. “Shear and extensional rheology of carbon nanofiber suspensions.” Rheologica Acta (2005), 44(6), 537-562. ISSN:0035-4511.

2. J. Xu, K.W. Koelling, Y. Wang, and S.E. Bechtel. “Rheology of polystyrene/carbon nanofiber composites,” Annual Technical Conference - Society of Plastics Engineers (2005), 63rd, 1950-1954. ISSN:0272-5223.

3. J. Xu, K.W. Koelling, Y. Wang, S.E. Bechtel, and M.G. Forrest. “Rheology of Molten Polystyrene/Carbon Nanofiber Composites,” PMSE Preprints (2005), 92 170- 172. ISSN:1550-6703.

4. J. Xu and K.W. Koelling, “Temperature-Dependence of Rheological Behavior of a Metallic Automotive Waterborne Basecoat.” Progress in Organic Coatings (2005), 53(3), 169-176. ISSN:0300-9440.

5. S.E. Bechtel, K.W. Koelling, Y. Wang and J. Xu. “Characterization and modeling of carbon nanofiber suspensions.“ AIP Conference Proceedings Vol. 712(1) pp. 311- 315. June 10, 2004. ISSN:0094-243X.

6. K.W. Koelling, S.E. Bechtel, J. Xu and Y. Wang. “Characterization of Nanoparticle/Polymer Melt Composites“, AIP Conference Proceedings Vol. 712(1) pp. 316-320. June 10, 2004. ISSN:0094-243X.

7. J. Xu. “Safety Analysis in Piping & Instrument Diagram”, Hualu Technology Forum, Vol. 1, pp. 1-7, 1999.

viii

FIELDS OF STUDY

Major field: Chemical Engineering

Concentration: Rheology of complex fluids

ix

TABLE OF CONTENTS

ABSTRACT...... ii

ACKNOWLEDGEMENT ...... vi

VITA...... vii

LIST OF TABLES...... xv

LIST OF FIGURES ...... xvi

Chapters

1. INTRODUCTION ...... 1

1.1 Polymer / Carbon Nanofiber Composites...... 2 1.2 Waterborne coatings...... 3 1.3 Significance of Research...... 4 1.4 Scope of study...... 6 1.5 Outline...... 7 2. LITERATURE REVIEW ...... 9

2.1 Experimental studies...... 9 2.1.1 Introduction...... 9 2.1.2 Rheology of fiber suspensions ...... 10

x 2.1.2.1 Newtonian solvent...... 10 2.1.2.2 Non-Newtonian solvent...... 13 2.1.2.3 Extensional rheology...... 16 2.1.2.4 Effects of fiber aspect ratio...... 16 2.1.2.5 Fiber orientation...... 17 2.1.2.6 Carbon nanotube/ nanofiber suspensions...... 20 2.1.3 Rheology of waterborne coatings ...... 22 2.2 Dynamic theories of fiber suspensions ...... 29 2.2.1 Introduction...... 29 2.2.2 Fiber orientation...... 30 2.2.3 Rheology of fiber suspensions ...... 31 3. RHEOLOGY OF CARBON NANOFIBER SUSPENSIONS ...... 36

3.1 Introduction...... 36 3.2 Materials and Preparation of Suspensions ...... 38 3.3 Morphological Characterization...... 40 3.4 Rheological Characterization...... 45 3.4.1 Steady state shear measurements ...... 45 3.4.2 Small amplitude oscillatory shear measurements ...... 50 3.4.3 Measurements of temperature effects ...... 53 3.4.4 Extensional rheology...... 54 3.5 Modeling...... 58 3.5.1 Kinetic Theory Models ...... 58 3.5.2 Modeling predictions for treated sonicated nanofiber suspensions ...... 65 3.5.2.1 Elastic dumbbell models...... 66 3.5.2.2 Rigid dumbbell models...... 71 3.5.2.3 Modeling predictions for treated unsonicated nanofiber suspensions .. 74 3.5.2.4 Modeling predictions for untreated sonicated nanofiber suspensions .. 75 3.6 Electrical Conductivity Measurement...... 78 3.7 Conclusions...... 78 4. DYNAMIC AND STEADY SHEAR OF POLYSTYRENE/CARBON NANOFIBER

COMPOSITES...... 98

xi 4.1 Introduction...... 98 4.2 Experimental procedures...... 100 4.2.1 Materials ...... 100 4.2.2 Sample preparation...... 101 4.3 Experimental Results...... 103 4.3.1 Morphology and Dispersion of CNFs in the Polymer Matrix ...... 103 4.3.2 Linear ...... 106 4.3.3 Start-up of steady shear...... 111 4.3.4 Steady State Rheology ...... 112 4.3.5 Cox-Merz rule...... 115 4.4 Nanoparticle/Polymer Melt Composite Constitutive Model ...... 116 4.5 Application of the model to CNF/PS melt composites...... 121 4.5.1 Procedures to deduce material constants in the model from experimental measurements...... 121 4.5.2 Models for the Melt Blended composites ...... 128 4.5.3 Models for the Solvent Cast composites...... 129 4.5.4 Investigation of polymer-fiber interaction...... 130 4.6 Conclusions...... 131 5. TRANSIENT RHEOLOGY OF PS/CNF COMPOSITES AND CNF ORIENTATION

INDUCED BY FLOW...... 160

5.1 Introduction...... 160 5.2 Experimental Procedures...... 161 5.3 Experimental Results...... 163 5.3.1 Transient shear rheology...... 163 5.3.2 Orientation Characterization...... 170 5.3.3 Effect of pre-orientation on transient shear ...... 174 5.3.4 Shear induced orientation at steady state ...... 176 5.3.5 Shear induced orientation at different shear strain...... 177 5.3.6 Transient extensional rheology...... 178 5.4. Conclusions...... 181 6. RHEOLOGY OF WATERBORNE AUTOMOTIVE PAINT ...... 209

xii 6.1 Shear rheology and pressure drop prediction...... 209 6.1.1 Introduction...... 209 6.1.2 Experimental ...... 210 6.1.3 Results and Discussion...... 212 6.1.3.1 Thixotropy of the paint ...... 212 6.1.3.2 Rheology of the paint and model fitting ...... 216 6.1.3.3 Reynolds number...... 217 6.1.3.4 Pressure drop measurement in the pilot plant and model prediction.. 218 6.1.4 Conclusion ...... 218 6.2 Temperature effects on the rheology of waterborne automotive paints ...... 228 6.2.1 Introduction...... 228 6.2.2 Experimental ...... 230 6.2.3 Results and Discussion...... 232 6.2.3.1 Linear viscoelasticity...... 232 6.2.3.2 Thixotropy...... 232 6.2.3.3 Pseudo Steady state viscosity...... 234 6.2.3.4 Apparent yield stress...... 236 6.2.3.5 Ford cup #4 ...... 237 6.2.3.6 Identification of the network structure...... 238 6.2.4 Conclusions...... 239 6.3 Color Degradation and Rheological Changes Caused by Prolonged Circulation in Paint Delivery System ...... 246 6.3.1 Introduction...... 246 6.3.2 Experiment setup...... 246 6.3.3 Rheological Measurements Procedures...... 249 6.3.4 Color Degradation Test Procedures ...... 250 6.3.5 Results and Discussions...... 250 6.3.5.1 Viscosity variation...... 250 6.3.5.2 Use of Inline Viscometer ...... 254 6.3.5.3 Thixotropy...... 256 6.3.5.4 Dynamic measurement...... 260 6.3.5.5 Color Degradation...... 261

xiii 6.3.6 Conclusion ...... 263 7. CONCLUSIONS AND RECOMMENDATIONS ...... 265

7.1 Conclusions...... 265 7.1.1 CNF dispersion...... 265 7.1.2 Effect of CNFs on suspension rheology ...... 267 7.1.3 Relationship between CNF orientation and rheology...... 268 7.1.4 Rheology of waterborne coatings ...... 268 7.2 Recommendations and future work...... 270

Appendix

A. SOME MISCELLANEOUS AND UNSUCCESSFUL EXPERIMENTAL RESULTS

...... 272

A.1 Results from PS/CNF Composites Using Another Type of Polystyrene...... 272 A.1.1 Thermal conductivity of the Nova PS/CNF composites...... 273 A.1.2 Measurement of glass transition temperature (Tg) ...... 274 A.2 Electrical Conductivity Measurement of PS/CNF Composites...... 274 A.3 Electrical Conductivity Measurement of CNF Aqueous Suspensions ...... 274

BIBLIOGRAPHY...... 280

xiv

LIST OF TABLES

Table Page

Table 6.1. Measured properties of the melt blended and solvent cast composites, and their nanofiber and polymer constituents...... 134

Table 6.2. Slopes of G'(ω) and G"(ω) at low frequencies on log-log plot...... 135

Table 6.3. Characteristic relaxation times of MB and SC composites ...... 136

Table 6.4 Values of CI and corresponding second order orientation tensor components of MB composites at different shear rate ...... 137

Table 6.5 Values of CI and corresponding second order orientation tensor components of SC composites at different shear rate...... 138

Table 6.6. Model predictions of melt blended and solvent cast composites...... 139

Table 6.7 Model predictions of modified Giesekus model...... 140

Table 8.1. Experimental parameters...... 248

xv

LIST OF FIGURES

Figure Page

Figure 1.1 A typical SEM micrograph of as-grown carbon nanofibers (From Applied Science, Inc.)...... 8

Figure 2.1 Flow reversal of glass fiber in Newtonian solutions (fiber concentration = 8 vol%). The shear rate in both directions was 43.7 s-1. [Ganani and Powell10] ...... 35

Figure 3.1. SEM micrographs of (a) as-received, (b) untreated sonicated, (c) treated unsonicated, and (d) treated sonicated carbon nanofibers (scale bar: 50 μm)...... 82

Figure 3.2. SEM micrographs of (a) as-received, (b) untreated sonicated, (c) treated unsonicated, and (d) treated sonicated carbon nanofibers (scale bar: 10 μm)...... 83

Figure 3.3. Optical microscopy photos of (a) 0.5wt% untreated sonicated and (b) 0.5wt% treated sonicated carbon nanofiber suspensions (scale bar: 100 μm)...... 84

Figure 3.4. Steady shear viscosity of untreated sonicated suspensions: (a) experimental measurements, (b) predictions of elastic dumbbell models with isotropic hydrodynamic drag with or without hydrodynamic interaction, (c) predictions of elastic dumbbell models with anisotropic hydrodynamic drag with or without hydrodynamic interaction, (d) predictions of rigid dumbbell models with isotropic hydrodynamic drag with or without hydrodynamic interaction...... 85

Figure 3.5. Shear viscosity as a function of shear stress of the untreated sonicated suspensions: (a) experimental measurements, (b) predictions of elastic dumbbell models with anisotropic hydrodynamic drag with or without hydrodynamic interaction, (c) predictions of rigid dumbbell models with isotropic hydrodynamic drag with or without hydrodynamic interaction...... 86

xvi Figure 3.6. Steady shear viscosity of treated sonicated suspensions: (a) experimental measurements, (b) elastic dumbbell model predictions, (c) rigid dumbbell model predictions...... 87

Figure 3.7. Steady shear viscosity of treated unsonicated suspensions: (a) experimental measurements, (b) elastic dumbbell model prediction, (c) rigid dumbbell model predictions...... 88

Figure 3.8. Relative viscosity of treated sonicated and untreated sonicated suspensions as a function of fiber volume fraction...... 89

Figure 3.9. Linear viscoelasticity of untreated sonicated nanofiber suspensions...... 90

Figure 3.10. Elastic modulus G′ of treated sonicated suspensions: (a) experimental measurements, (b) elastic dumbbell model predictions, (c) rigid dumbbell predictions...... 91

Figure 3.11. Viscous modulus G" of treated sonicated suspensions: (a) experimental measurements, (b) elastic dumbbell model predictions, (c) rigid dumbbell model predictions...... 92

Figure 3.12. Effects of temperature on 2wt% untreated sonicated suspensions: (a) experimental measurements of the steady shear viscosity, (b) elastic dumbbell model predictions of the steady shear viscosity, (c) experimental measurements of the relative viscosity, (d) elastic dumbbell model predictions of the relative viscosity...... 93

Figure 3.13. Effects of temperature on 2wt% untreated sonicated suspensions: (a) experimental measurements of the apparent yield stress, (b) elastic dumbbell model predictions of the apparent yield stress, (c) Arrhenius plot, (d) linear viscoelasticity...... 94

Figure 3.14. Extensional and shear viscosity of 1wt% untreated sonicated suspension. Fresh sample was measured using 2mm jets, and then the sample was stretched at 600 s-1 for 13 hours. After that, the sample was measured again using 2mm jets followed by measured by 1mm jets. Then the sample was stretched at 2000 s-1 for another 2 hours. It was measured again using 1mm jets. Temperature 28 °C...... 95

Tr =η (ε) /η(γ) Figure 3.15. Trouton ratio, e , of 2wt% untreated sonicated suspension calculated by the standard method ( γ = ε )and Jones method (γ = 3ε )...... 96

xvii Figure 3.16. Comparison of complex viscosities (η*) and steady shear viscosities (η) of 1wt% and 5wt% untreated sonicated nanofiber suspensions. At 1wt% the Cox-Merz rule holds...... 97

Figure 4.1. Optical microscopy images of a 5wt% (a) melt blended (MB) composite and (b) solvent cast (SC) composite. Scale bar: 40 microns...... 141

Figure 4.2. TEM micrographs of (a) a 5wt% MB composite and (b) a 5wt% SC composite. Scale bar: 5 microns...... 142

Figure 4.3. Elastic modulus (G’) of (a) MB and (b) SC composites with various CNF concentrations at 200°C...... 143

Figure 4.4. Shifting factor of MB and SC composites...... 144

Figure 4.5. Shift of crossover point of G’ (solid symbols) and G” (open symbols) of (a) MB composites and (b) SC composites...... 145

Figure 4.6. Transient shear viscosity η+ of the pure PS at shear rates 0.0001-10s-1. (Curves for 0.01 s-1 and below fall on top of each other.)...... 146

Figure 4.7. Start up of steady shear at shear rates 0.0001 – 10 s-1. (a) Transient shear viscosity, (b) primary normal stress difference. SC10 composite. T=200°C...... 147

Figure 4.8. Steady shear viscosities of (a) MB composites and (b) SC composites...... 148

Figure 4.9. Relative viscosity of MB and SC composites...... 149

Figure 4.10. First normal stress difference N1 under steady shear of (a) MB and (b) SC composites...... 150

Figure 4.11. Applicability of Cox-Merz rule of (a) MB and (b) SC composites...... 151

Figure 4.12. (a) Fit of model with optimum value λ = 0.329 s to oscillatory shear measurements of melt blended pure polymer melt, (b) fit of model with optimum values λ = 0.329 s and α = 0.392 to steady shear measurements of melt blended pure polymer melt...... 152

Figure 4.13 Model predictions of (a) the viscosity and (b) first normal stress difference compared to the experimental steady shear measurements for melt blended composite systems based on the fitting results of Table 4.4...... 153

Figure 4.14 Values of CI at each shear rate and the corresponding trendlines of MB and SC composites...... 154

xviii Figure 4.15 Model predictions of (a) the viscosity and (b) first normal stress difference compared to the experimental steady shear measurements for melt −0.507 blended composite systems based on CI = 0.0066γ ...... 155

Figure 4.16. (a) Fit of model to oscillatory shear measurements of solvent cast pure polymer melt, (b) Fit of model to steady shear measurements of solvent cast pure polymer melt...... 156

Figure 4.17 Model predictions of (a) the viscosity and (b) first normal stress difference compared to the experimental steady shear measurements for melt blended composite systems based on the fitting results of Table 4.5...... 157

Figure 4.18 Model predictions of (a) the viscosity and (b) first normal stress difference compared to the experimental steady shear measurements for melt −0.729 blended composite systems based on CI = 0.0001γ ...... 158

Figure 4.19 Comparisons of Giesekus model and Modified Giesekus model at mass concentration c=2wt% for (a) MB composite and (b) SC composite...... 159

Figure 5.1.Photos of samples during RME tests. Sample was made of polystyrene. Extension rate = 0.1 s-1...... 183

Figure 5.2. Widths of the stretch sample measured at left, middle and right of the air table. Black line is the best fit of the average width...... 184

Figure 5.3. Transient shear viscosity η+ at shear rates 0.0001-10s-1. of (a) SC0 (Curves for 0.01 s-1 and below fall on top of each other.) and (b) SC10 composite. T=200°C...... 185

Figure 5.4. Start up of steady shear at different shear rates. (a) pure PS (SC0) and (b) SC10 composite. T=200°C...... 186

Figure 5.5. Start up of steady shear as a function of strain at shear rate of 1s-1 of (a) MB composites and (b) SC composites. T=200°C...... 187

Figure 5.6. Reduced stress of flow reversal tests at shear rate of 1s-1 of (a) pure PS (SC0) and (b) SC10 composites. T=200°C...... 188

Figure 5.7. First normal stress difference (N1) of flow reversal tests at shear rate of 1s-1 of (a) pure PS (SC0) and (b) SC10 composites. T=200°C...... 189

Figure 5.8. Effect of pre-shear on the transient shear stress at shear rate of 1s-1 of (a) pure PS (SC0) and (b) SC10 composites. T=200°C...... 190

xix Figure 5.9. Effect of pre-shear on the transient N1 at shear rate of 1s-1 of (a) pure PS (SC0) and (b) SC10 composites. T=200°C...... 191

Figure 5.10. Schematic drawings of the evolution of network structure in flow reversal tests...... 192

Figure 5.11. CNF orientation induced by extensional flow. (a) plane parallel to the flow, (b) plane perpendicular to the flow, (c) histogram of orientation with respect to flow direction in the plane parallel to the flow, and (d) histogram of orientation with respect to cutting direction in the plane perpendicular to the flow direction...... 193

Figure 5.12. Schematic patterns of nanostructure in a disk-shaped shear specimen made by injection molding method. The size of the nanoparticles is greatly exaggerated in these schematics...... 194

Figure 5.13. Schematic patterns of nanostructure in a disk-shaped shear rheometry specimen made by compression molding method. The size of the nanoparticles is greatly exaggerated in these schematics...... 194

Figure 5.14. Effects of pre-orientation of the CNFs on the transient stress. (a) SC5 composite, (b) MB5 composites. Shear rate = 0.1 s-1. T=200°C...... 195

Figure 5.15. Coordination system of the shear sample...... 196

Figure 5.16. Special cases of the 2-D orientation tensor...... 197

Figure 5.17. Effect of shear rate on the shear induced CNF orientation at steady state condition T=200°C. MB5 composite. Injection molded disk samples using center gated mold...... 198

Figure 5.18. TEM micrograph of the un-sheared sample in 1-3 plane and 2-3 plane. MB5 composite. Injection molded disk samples using center gated mold...... 199

Figure 5.19. TEM micrograph of the sample sheared at 0.1 s-1 for 600s (60 strain units). MB5 composite. Injection molded disk samples using center gated mold...... 200

Figure 5.20. TEM micrograph of the sample sheared at 10 s-1 for 6s (60 strain units). MB5 composite. Injection molded disk samples using center gated mold...... 201

Figure 5.21. Transient CNF orientation induced by simple shear at 0, 0.4, 3 and 6 strain units. T=200°C. MB5 composite. Injection molded disk samples using center gated mold...... 202

xx Figure 5.22. Transient CNF orientation induced by simple shear at 0.4 strain units. T=200°C. MB5 composite. Injection molded disk samples using center gated mold...... 203

Figure 5.23. Transient CNF orientation induced by simple shear at 3 strain units. T=200°C. MB5 composite. Injection molded disk samples using center gated mold...... 204

Figure 5.24. Transient reduced stress of MB0 and MB5 composites at 10 s-1. T=200°C. MB5 composites. Injection molded disk samples using center gated mold...... 205

Figure 5.25. Transient extensional viscosity of (a) MB composites and (b) SC composites. Extensional rate = 0.1 s-1. T=170°C...... 206

Figure 5.26. (a) Transient extensional viscosity of pure PS (MB0). Solid line is 3 times of transient shear viscosity at 0.01 s-1. (b) Transient Trouton ratio of pure PS. T=170°C...... 207

Figure 5.27. (a) Transient extensional viscosity of MB5 composite. Solid line is 3 times of transient shear viscosity at 0.01 s-1. (b) Transient Trouton ratio of MB5 composite. T=170°C...... 208

Figure 6.1 Schematic drawing of the pilot plant...... 220

Figure 6.2 Thixotropy loop. (a) Shear history in a thixotropical loop. (b) Typical response of a thixotropic ...... 221

Figure 6.3 Viscosity decay during start up of steady shear at shear rate of 200 s-1...... 222

Figure 6.4 Pressure drop observed in the start up of constant flowrate...... 223

Figure 6.5 Pressure drop in response to a step change of the flowrate from 2.5 gpm to 0.7 gpm...... 224

Figure 6.6 Pressure drop of steady state flow...... 225

Figure 6.7 Steady state shear viscosity and model fitting. (a) Carreau model. (b) Sisko and power law model...... 226

Figure 6.8 Comparison of measured pressure drop with model predictions...... 227

Figure 6.9. Thixotropic behavior of the metallic basecoat. (a) Shear history in a thixotropical loop. (b) stress response of a thixotropic fluid. (c) Steady shear at constant shear rate of 200 s-1. The measurements were performed at room temperature...... 240

xxi Figure 6.10. Degree of thixotropy of the metallic basecoat at various temperatures...... 241

Figure 6.11. Linear viscoelasticity of the metallic basecoat. (a) G’ vs. frequency. (b) tan δ vs. frequency...... 242

Figure 6.12. Pseudo steady state viscosity of the metallic and non-metallic basecoats. (a) Steady shear viscosity vs. shear rate. (b) viscosities at shear rates of 0.03 s-1 and 600 s-1...... 243

Figure 6.13. Apparent yield stress at various temperatures of the metallic basecoat...... 244

Figure 6.14. Ford cup #4 measurement of the metallic basecoat...... 245

Figure 6.15. Viscosity profiles change with circulation time...... 251

Figure 6.16. Decrease Percentage of power law parameter n...... 254

Figure 6.17. Thixotropy loop...... 257

Figure 6.18. Steady shear at constant shear ...... 258

Figure 6.19. Change of thixotropy of the paint during circulation with various length of circulation times...... 259

Figure 6.20. Typical change of elastic and viscous moduli (G’, G”) during circulation ...... 261

Figure 6.21. A typical result of the color measurement. Results shows that for sample 2 and 3 (longer circulation time), there is larger color travel than uncirculated sample (sample 0) ...... 262

Figure 6.22. Color degradation results show that almost all the samples are lighter than standard...... 263

Figure A.1. Thermal conductivities of Nova PS/CNF composites...... 276

Figure A.2. Relative increase of the thermal conductivities of MB and SC composites...... 277

Figure A.3. Glass transition temperature (Tg) of MB composites...... 278

Figure A.4. Electrical conductivity of CNF suspensions...... 279

xxii CHAPTER 1

INTRODUCTION

Polymeric suspensions are mixtures of solid particles and polymeric matrix. At room temperature, these suspensions can be either solid or . Common solid polymeric suspensions includes polymer composites, especially fiber reinforced thermoplastics or thermosets. These polymer composites usually have a unique combination of physical properties, including mechanical, thermal and electrical properties. Most liquid polymer suspensions are inks, coatings and foods. In both solid and liquid polymer suspensions, polymer usually serves a contributor to mechanical properties of the final product. Due to the useful properties of these polymeric suspensions, they are being produced in large quantities. The mixing process in the production plays an important role, as dispersion of the fillers greatly affects the properties of the suspensions. All mixing process happens when the matrix is liquid. In the case of solid polymer composites at room temperature, the mixing process usually happens at elevated temperatures. The energy used to mix and shape the composites is dominated by the rheology of the composites. Unfortunately, the rheology of the polymeric suspensions cannot simply be characterized by viscosity. Almost all polymeric

1 suspensions are non-Newtonian, where the shear viscosity is a function of shear rate, time, and filler orientation. Only Newtonian fluids can be described by a single constant, namely viscosity. As a result, a comprehensive characterization of the rheology of the polymeric suspensions is needed in order to build a process that is energy efficient. This study is mainly concentrated on the rheological properties of and polymer / carbon nanofiber nanocomposites and waterborne coatings.

1.1 Polymer / Carbon Nanofiber Composites

With discovery of carbon nanotubes,1 properties of polymer/carbon nanotube

(CNT) composites have been extensively studied. Numerous papers have been published on the mechanical, thermal and electrical properties of composites of CNT in both thermoset and thermoplastic polymers, and promising results have been found. However, low volume of production and high cost of the CNT greatly limit development and application.

Carbon nanofibers (CNF), which have diameters of 100-200 nm and lengths of 1-

100 microns, may serve as a substitute for the carbon nanotubes. Recent researchers have found that CNF/polymer composites have similar properties as CNT/polymer composites.2-4 Carbon nanofibers are mainly produced by catalytic vapor grown method in which hydrocarbon such as benzene and methane is decomposed on the surface of metal catalyst at the temperature of about 700-1200 °C.5, 6 By varying the type of catalyst and other operating conditions, length and diameter of the CNFs can be controlled. These

CNFs usually have a structure of multiple layers of concentric cylinders of carbon layers

2 with a hollow core filament. As grown CNFs are highly entangled, forming “balls” of nanofibers as large as millimeters (see Figure 1.1). Therefore, dispersion of them in polymer matrix is a challenging problem.

CNFs have been melt compounded using twin screw extruder into polymers including polypropylene, polystyrene, nylon, polycarbonate, polyester and poly(ether ether ketone) (PEEK).2-4 These composites exhibit reduced uniform surface electrical conductivity, while maintaining mechanical properties of matrix polymers. They can be used to make conductive paints, coatings, films, tubes, sheets, and parts for electrostatic painting, electro-magnetic interference (EMI) and electro-static discharge (ESD) applications. In addition, these composites also provide improved strength, stiffness, dimensional stability and thermal conductivity, etc. It makes the polymer/CNF composites a very promising material for a wide range of applications in automotive, aerospace, electronic and chemical industries.

1.2 Waterborne coatings

Generally speaking, there are four basic components in any liquid coatings: resins, solvents, pigments and additives. In waterborne coatings, water is used as primary solvent, whereas volatile organic compounds (VOC) are used in conventional solvent borne coatings. Comparing to solvent-borne coatings, the most significant advantage of waterborne coatings is low VOC levels, and therefore beneficial to the environment and personal health. The final film can be formulated to have about the same qualities and for metallic coatings, the waterborne coatings usually have better appearances because more shrinkage occurs during drying process which causes better flake orientation. For some applications, the waterborne coatings have wider operating conditions.

3 In the 1970s more than 90% of the paints worldwide are solvent-borne.7 With the oil embargo and the more stringent environment requirements in the 1970s, coatings containing less solvent are desired. Several solutions have been developed and waterborne coatings are gaining acceptance due to their advantages. In the automotive industry, General Motors’ Oshawa Truck Plant is known as a world first to use waterborne basecoat in 1986 in normal production scale.8 Since that time, waterborne coatings have been used more and more widely. Up to 1994, over 95% of coatings for automotive manufacturing have been changed to waterborne technique. In the architectural market, which occupies about 45-55% of the whole paint and coatings market, interior coatings for commercial and residential applications are nearly 100% waterborne coatings.7

1.3 Significance of Research

The superior properties of polymer/CNF composites have gained acceptance and the composites have significant potential for wide application. However, problems occur when the nanoparticles are incorporated into the polymer matrix. Greatly increased viscosities will make extrusion and injection molding difficult, and enhanced viscoelasticity may cause more molded stress which in turn can cause of molded parts. The desired properties are determined by the morphology of the particles, such as the the electrical and thermal conductivity is maximized if the CNFs are aligned in one direction. Therefore the control of CNF orientation during processing is critical.

We have only a very limited knowledge about its rheological behavior of the polymer/CNF composites. One paper has been published 9 on CNF composites and another two papers on multi-walled nanotubes. Although the mechanical properties of

4 polymer/CNF composites have been widely studied, and promising application have been proposed, the processing of these new materials are still not known to us. The above mentioned studies show that the shear rheology is not too much difference from their polymer matrices at high shear rates, but this is far from enough to give a full picture of the rheology of the composites in order to make an effective design of the desired product and establish appropriate processing conditions. In addition, study of CNF orientation has not been seen in this area.

Although waterborne coatings show many advantages, change over from solvent- borne require a lot of works. As far as the rheology is concerned, unlike the solvent-borne coatings that are generally Newtonian, waterborne coatings are extremely shear thinning.

In addition, the existence of thixotropy makes the flow behavior more complex. The design equations for solvent-borne coatings are no longer valid for waterborne coatings.

However, some companies, such as Honda, are stilling using those old equations for revamping of their current paint delivery systems. This greatly hinders design and improvement of the production process.

As a result, a comprehensive rheological characterization of polymer/CNF composites and waterborne coatings including linear and non-linear rheology and particle orientation in shear and extensional flows is very important, and currently this information is lacking. This research aims at obtaining a comprehensive rheology of the nanocomposites and waterborne coatings through performing a series carefully designed experiments and analyzing the experimental data. Together with the modeling, a deeper understanding of the fundamental mechanism is to be gained. With this knowledge, we

5 can predict more precisely about the pressure drop, particle orientation, dimensional instability etc, which are problems encountered in the processing, so that we can design and optimize relevant processes better.

1.4 Scope of study

To address the problems in the design issues related with the polymer/CNF composites and properties of waterborne coatings, the scope of this studied is listed below:

• Perform steady and dynamic shear and extensional flow experiments on polymer solution/CNF and polymer melt/CNF systems to measure their shear and extensional response.

• Measure particle orientation and structure development by freezing polymer melt nanocomposites at different shear and extensional strains, and using OM, TEM, SEM.

• Correlate nanoparticle orientation and structure development with the rheological response.

• Determine the effect of the addition of nanoparticles on the shear and extensional viscosity of polymer solutions and melts, as a function of nanoparticle size and concentration.

• Determine how nanoparticle loading and surface treatment of nanoparticles will affect the of viscoelastic polymers.

• Develop models for determining the coupled orientation and rheology of these CNF polymer suspensions, and employ them to gain insight into the experimental data base.

• Perform steady and dynamic shear rheological measurements on a waterborne paint using high precision rheometer.

• Conduct steady state flow and start up flow of the paint in a pilot plant to determine the relationship of pressure drop and flow rate.

• Develop a mathematical model such that the pressure drop in steady state flow can be predicted at a given flow rate.

6 • Quantify the effects of the paint thixotropy on the pressure drop prediction during start up.

• Use the models in concert with the experimental measurements to develop an understanding of the fundamental mechanisms

1.5 Outline

General description of this research and its significance as well as the scope of study is presented in Chapter 1. Chapter 2 gives a review about rheology and some morphology of polymer/CNF composites and waterborne coatings. In addition, rheology of fibers suspended in low viscosity solutions is also reviewed, due to their similarities to the studied systems and large amount of available literature. Chapter 3 presents the results from experimental investigations about the rheology of CNFs in low viscosity

Newtonian solution. Chapter 4 shows the results of rheological characterization of polymer melt/CNF composites. Thermo-mechanical modeling is also presented in this chapter. In Chapter 5, the CNF orientation as a function of shear rate and type of flow, i.e. shear and extensional, as well as transient shear and extensional rheology of the composites, are displayed. The molecular modeling employs a Jeffery-type model to simulate the experimental results from the polymer nanocomposites. The emphasis of the modeling is on the coupling orientation of the particles with the rheology of the composites. The experimental measurements are discussed and detailed discussion on the theory on fiber/polymer composites is provided. The rheology of waterborne coatings is presented in Chapter 6. Measurement of the rheological properties, pressure drop over circulation systems, as well as temperature dependence of the rheology is discussed.

7

Figure 1.1 A typical SEM micrograph of as-grown carbon nanofibers (From Applied Science, Inc.).

8 CHAPTER 2

LITERATURE REVIEW

2.1 Experimental studies

2.1.1 Introduction

There have been numerous experimental investigations on polymeric suspensions.

For the waterborne coatings, the rheology depends largely upon their formulations. As the formulations vary in a wide range of combination of pigments and chemicals in order to meet various requirements for the coatings, the rheological properties also changes from product to product. Similar variations exist in polymer nanocomposites due to the usage of different fillers and polymer matrices. However, some phenomenological rheological properties are similar, such as shear thinning, viscoelasticity, yield, thixotropy and other time dependent behaviors. The literature review below describes the rheological phenomena and possible underlying mechanism in the categories of fiber suspensions and waterborne coatings.

9 2.1.2 Rheology of fiber suspensions

The study of the rheology of fiber suspensions began in the late 1960’s when glass fiber products had been widely produced in large quantities. For interested readers, good reviews can be found in the works of Ganani and Powell10 and Maschmeyer and

Hill11.

2.1.2.1 Newtonian solvent

There are some good reasons why rheological studies of suspensions should be started from Newtonian solvents. From practical application point of view, there are some industrial suspensions whose solvents are Newtonian, such as in paper pulp and drilling mud where water is the solvent. Theoretically, selecting a Newtonian solvent will greatly reduce the complexity of stress analysis. Besides, when the solvent is Newtonian, any non-Newtonian behavior of the suspensions can be attributed to the particles.

Yield stress is understood as a stress below which the fluid does not flow continuously. It is usually explained that structures are formed within the fluid that resist the flow. In the light of this statement, adding particles into solutions will usually generate yield stress, which is observed experimentally. In the case of fibers suspensions, yield stress appears when the fiber concentration is high. In the work of Maschmeyer et al

12, the tube flow of concentrated glass fiber suspensions in Newtonian silicon oils was investigated. Yield stresses were observed and were influenced strongly by the fiber length.

For pure Newtonian fluids, there is no elastic behavior. When fibers are added to the solution, however, the suspension may develop some elasticity depending on the nature of the fibers. In order to have elasticity, the energy must be able to be stored in the

10 suspension in some way. For flexible fibers, the energy can be stored through of the fibers, or through the possible structures that are formed by the fibers. In the case of non-interacting non-Brownian rigid fibers, the suspensions should not develop elasticity.

Ganani and Powell10 found that in non-colloidal suspensions of rigid rods (glass fiber), the elastic modulus G’ was independent on the strain, showing no elastic effects.

However, Kotsilkova13 showed that suspensions of long glass fibers in Newtonian silicon oils exhibited viscoelastic behavior. Elastic and viscous moduli (G’ and G”) were sensitive to the aspect ratio of the glass fibers. It is suggested that the viscoelasticity was caused by the possible structure formed by the long glass fibers.

Fiber suspensions may show time dependent behaviors when the flow starts from rest. When the flow starts, the stress does not increase instantly. Rather, it takes some time for the stress to become steady. Sometimes stress overshoot is observed, indicating the shear stress undergoes a maximum with time. Ganani and Powell10 studied nearly monodisperse glass fiber suspensions in Newtonian solutions. The suspensions still exhibited Newtonian steady shear viscosity but showed stress overshoot during the start up of constant flow. This stress overshoot was also observed by Chaouche and Koch14.

Possible mechanism for this phenomenon is that it takes some time for the fibers to orient with the flow during start up. Before start up, the fibers are orientated randomly. When the flow starts, the fiber rotates, or even tumbles, causing the stress to grow. At a certain strain, when considerable amount of fibers are in the process of tumbling, the stress overshoot happens. After the strain, most of the fibers align to the flow, and the stress decreases and reaches steady state.

11 Apart from start up flow, flow reversal experiments are also used to probe transient properties. For fiber suspensions, the time dependent of the stress is often used to deduce the evolution of the fiber orientation during flow. Ganani and Powell10 showed that after the stress reaches steady in the start up of the flow, reverse the direction of the flow will cause the fibers to tumble and reorient (Figure 2.1).

At low fiber concentrations, the suspensions still exhibit Newtonian shear viscosity. The viscosity increases with the fiber concentration. Ganani and Powell10 showed that the glass fiber suspensions are Newtonian in dilute and semi-dilute regime.

Chaouche and Koch14 found the same results with their nylon fibers in Newtonian fluids at semi-dilute regime.

At higher fiber concentrations, the viscosity of the suspensions becomes shear rate dependent. This dependence becomes more obvious with increasing fiber loadings.12

Chaouche and Koch14 studied nylon fibers in several Newtonian fluids and found that the suspensions were generally Newtonian but showed shear thinning at low shear rates. The shear thinning was stronger at higher fiber concentrations and weaker at lower solvent viscosity. They attributed the shear thinning to the formation of flocs under flow which was hold by the adhesion force. Under flow, the flocs were broken by the shear stress, therefore showing shear thinning behavior.

12 2.1.2.2 Non-Newtonian solvent

As stated before, study rheology of suspensions with Newtonian solvents has merits in the aspect of both industrial application and fundamental understanding of the rheology as a functions of fiber properties. However, as one of the most common fiber suspensions is fiber reinforced plastics, the study with non-Newtonian solvent is also very important.

The linear viscoelasticity of nylon filled with glass fibers was studied by Park and

Kim15. It was found that Relatively simple mathematical equations in the form of η' = a/(1 + b ω) c were used to correlate the dynamic viscosity (η’) with fiber concentration adequately in the range of 0.04-0.3. At lower concentration (<0.04), the modeling was difficult because the fibers were randomly oriented such that the fiber interaction might dominate. Above the threshold concentration (0.04), the fibers tended aligned and therefore the interactions decreased. Silane coupling agent was found to greatly increase the dynamic viscosity and yield stress. Ganani and Powell10 showed that adding fibers into a non-Newtonian solvent increased the elastic modulus (G’) of the suspensions at all frequency ranges. At very low fiber concentrations (φ = 0.02 and 0.05), the G’ was found to be a function of second power of frequency.

Non-Newtonian fluids often shows shear thinning behavior. In fiber suspensions with non-Newtonian solvents, the shear thinning behavior usually becomes stronger, especially at low shear rates. Nicodemo and Nicolais16 studied the rheology of short glass fiber in poly(ethyl oxide) aqueous solutions with the fiber loading of 0.01-0.07 by volume. The length of the glass fibers were about 530 microns and the average aspect ratio of the glass fiber was 37. A Weissenberg rheogoniometer with cone and plate 13 geometry (5cm diameter and 2° angle) was employed in the experiments. It was found that the addition of the fibers increased the degree of shear thinning. Low shear viscosity in the range where plateau Newtonian viscosity occurs for the pure solvent increased with the decrement of shear rate. A superposition of the viscosities at different volume fractions of the fibers was made and it was found that the relaxation time was increased with the addition of the fibers. The authors suggested that the fiber aggregation would be the main cause to these rheological changes. Ganani and Powell10 studied nearly monodisperse glass fiber suspensions in non-Newtonian solutions and found that the addition of glass fiber increased shear viscosity, more pronouncedly in the low shear

Newtonian region than in the shear thinning region.

The steady shear viscosity of fibers suspensions based on polymer melts is greatly affected by the addition of fibers. Han and Lem17 measured shear viscosity of glass fiber filled unsaturated polyester with a cone and plate rheometer and found shear thinning behavior at very low shear rate instead of the Newtonian plateau for the pure matrix.

Becraft and Metzner18 studied the flow of glass fiber filled polypropylene experimentally with the glass fiber concentrations ranging from 10 to 40% by weight. The length glass fibers were about 500-1200 microns and the aspect ratio was about 30-60. Instron capillary rheometer was used to measure the viscosity. The results showed that the viscosity of the suspensions increased markedly with increasing fiber concentration at low shear rate regions but the viscosity didn’t change much at high shear regions.

14 In order to quantify the effects of the fibers, a large number of equations have been developed, both empirically and theoretically. The empirical equation proposed by

Maron and Pierce19 is given here because of its simplicity and the ability to characterize the suspensions.

1 ηr = (2.1) [1 − (φ / A)]2

where φ is the volume fraction of the filler, ηr is relative viscosity of the suspension (ratio of suspension viscosity to solvent viscosity), and A is an empirical parameter. Although the equation was proposed for spherical fillers, it has been expanded to fiber suspensions by relating parameter A with the aspect ratio of fibers. Kitano et al20 found that A decreases linearly with increasing average fiber aspect ratio from the results of polymer melts filled with various fibers. Relationship between relative viscosity and fiber content have also been proposed in similar form of the above equation.21-25

The normal stress difference of Ramazani et al26 studied glass fiber suspensions in different fluids with the purpose to evaluate the effects of fiber concentration and suspending fluid on the rheology of the suspensions. An ARES strain-controlled rheometer with parallel plate geometry (diameter 25mm, gap ~1.5mm) was used to measure the rheological properties. The glass fibers had diameters of 16 microns and average aspect ratio of about 15. The results showed that larger aspect ratio had about the same effects as higher fiber content on the rheological properties. With the increase of fiber concentration, the steady shear viscosity and first normal stress difference increased

15 markedly at low shear rates and but they remained about same as the matrix at high shear rates. The transient behavior of fiber suspensions in Boger fluids showed that the orientation of the fiber was greatly affected by the preshear before the tests. The stress overshoot were showed to be functions of shear rate. A modified Folgar-Tucker equation was used to predict rheological properties of fiber suspensions and quantitative agreements were obtained for steady shear viscosities in dilute and slightly above semi- dilute regimes. Qualitative predictions could be obtained for transient properties.

2.1.2.3 Extensional rheology

Extensional rheology is much less studied, partly due to the difficulties in measuring extensional viscosities. However, the extensional rheology is very important and it dominates in some productions such as fiber spinning, blow molding and planner stretching. Shuler et al27 studied the effects of fiber concentration on shear and extensional viscosities of polyamide 6,6 filled with glass fibers. In the range of 30-40 vol% of fiber loading, the extensional viscosities depend moderately on the fiber content, but not on fiber aspect ratio. Comparing with the shear viscosities of the matrix, the filled polymers showed 4 orders of magnitudes higher of extensional viscosity.

2.1.2.4 Effects of fiber aspect ratio

Aspect ratio of the fibers determines the fiber movement and interactions during flow. At the same flow condition, longer fibers tend to move slower than shorter ones, and therefore it need longer time to reach steady state. At steady state, Ganani and

Powell10 found that increasing aspect ratio increased the dependence of relative viscosity on fiber volume concentration. Ramazani et al26 showed for low aspect ratio (~15) glass fibers, the variation in the aspect ratio had about the same effects as the fiber

16 concentration. However, other studies showed that fiber aspect ratio did not affect much on steady state viscosity, even for very long fibers. Shuler et al27 investigates the glass filled nylon composites (aspect ratio ~550) and showed that the steady shear and extensional viscosities did not depend on the fiber aspect ratio. Miles et al28 studied flow behavior of dilute long fiber suspensions. The fiber concentrations were less than 3 vol%.

Fibers with two different aspect ratios (1120 and 500) were used to investigate the effects of aspect ratio on the flow. They found that in long smooth tubes when the fibers were completely aligned to the flow, the viscosity of the suspensions did not depend on the aspect ratio of the fibers. At the entrance of the tube, however, when the contraction of the flow disturbed the orientation of the fibers, the entrance pressure drop was affected dramatically by fiber concentrations and aspect ratio.

The fibers may form structures under quiescent conditions. In Maschmeyer et al’s investigation12, the yield stresses of glass fibers in Newtonian silicon oils were observed and were influenced strongly by the fiber length.

2.1.2.5 Fiber orientation

The fiber orientation is, of course, very important for polymer composites. The fiber interferes the flow of the suspensions and the flow change the orientation of the fibers. Many researches have been conducted in determining the fiber orientation under flow. Petrich et al29 investigated the relationship between fiber orientation and stress in the fiber suspensions in the semi-dilute and semi-concentrated regime. Glass fibers with diameters of 9-12 microns and aspect ratio of 50 were used as fillers. Opaque tracer fibers were used and light refraction technique was employed to capture the orientation. High viscous Newtonian fluids were used as suspending media. Measured orientation

17 information was incorporated in to a model based on Jeffery’s orbit was used to predict the rheological properties. Agreements between model predictions and experimental measurements of shear viscosity were found, but the observed shear thinning could not be predicted.

In the study conducted by Folgar and Tucker30, fiber orientation was studied in the suspensions of nylon and polyester fibers in silicone oil using Couette geometry.

Photographs were taken during steady state shear and the fiber orientation and position was digitized using computer. Angles between the fibers and flow direction were measured and orientation distribution was calculated. A mathematical model based on

Jeffery equation was used and the simulation results agreed satisfactorily with the measured steady state fiber orientation.

Fiber orientation during extrusion was studied by several researchers. Wu31 investigated the extrudate surface morphology at different extrusion rate. The composites contained 25 vol% glass fiber (length: 2.54 cm, diameter: 0.0381 cm) and 75 vol% poly

(ethylene terephthalate). An order-disorder transition was found. The surface of the extrudate was smooth at low extrusion rates, and it became very rough and the glass fibers protruded out when the rate increased. At very high rate, the extrudate become smooth again. The PET matrix could produce smooth extrudates at all extrusion rates (1-

10,000 s-1), so it was the glass fibers that cause the order-disorder transition. By studying the fiber distribution from the cross sections, the author claimed that the fiber migration caused the transition. Stronger shear thinning of the suspension could decrease the fiber disorder as the more uniform profile could be produced.

18 Becraft and Metzner18 studied the flow of glass fiber filled polymer experimentally and the morphology and orientation of the glass fibers were investigated.

The glass fiber concentrations ranged from 10 to 40% by weight. The length glass fibers were about 500-1200 microns and the aspect ratio was about 30-60. Instron capillary rheometer was used to measure the viscosity. The fiber concentration profiles were determined from the cross-section photos of the extruded samples through a circular die.

The fibers were aligned with flow even the flow was slow. The surface smoothness and the shape of the cross-section of the extrudates were investigated under different extruding speed. It was found that for the 40% suspensions, the extrudates had more uniform diameter and smoother surface. The radial fiber migration reported by Wu31 was not found in this study. The results suggested that the rigidity of the fibers and the uniformity velocity profile at the die exit might be two most important factors.

Knutsson and White32 explored the fiber orientation in the extrudate of glass fiber polycarbonate composites. By observing the surface of the extrudate using scanning electron microscopy, the fiber orientation angle to the flow direction was determined. It was found that there existed a limit extrusion rate around 100 s-1 above which the orientation suddenly increased.

Barbosa and Bibbo33 tried to correlate fiber orientation with suspension rheology.

Rigid fibers were suspended in Newtonian solvents. The orientation of the fiber was deduced from the measurements on the transient shear viscosity and the first normal stress difference.

19 2.1.2.6 Carbon nanotube/ nanofiber suspensions

With the emergence of new materials such as carbon nanotubes and nanofibers, some rheological characterization of their suspensions have been conducted. In the work of Kinloch et al34, The rheology of aqueous multi-walled carbon nanotube suspensions was studied using a Rheometrics Dynamics Stress SR200 rheometer with parallel plates geometry. The MWNT was chemically treated using a concentrated acid mixture (3:1 of sulphuric acid/nitric acid) for an hour. Aqueous suspensions were prepared by making a dilute suspension first and then evaporating part of the water to obtain suspensions with desired fiber concentrations. Up to 11 vol% fiber content could be prepared. The small amplitude oscillatory shear experiments in linear range showed that both G’ and G” were independent of frequency, indicating a strong gel-like behavior. Both moduli increased with MWNT concentration and the relationship could be modeled by a power law type correlation. Shear viscosity indicated strong shear thinning with apparent yield stress.

Bingham model was found to fit the experimental data better than the Herschel-Bulkley model. The suspensions were thixothopic and the structure recovery time was about 60 minutes. The authors suggested that microstructure similar to flocculated networks was formed in the concentrated suspensions.

Lozano et al9 conducted a comprehensive study of mechanical, electrical and rheological properties carbon nanofibers/polypropylene composites. Vapor grown carbon nanofibers were purified and functionalized and then compounded into PP using a

Banbury-type mixer. Complex viscosity from the dynamic shear experiments showed that the viscosity of the composites was barely affected by less than 10wt% of nanofibers.

20 Higher nanofiber concentration led to higher viscosity especially at low shear rates. High shear rate viscosity was less increased and this ensured same easy processibility of the composites as the pure polymer matrix.

Pötschke et al35 conducted experiments with MWNT/polycarbonate composites.

A master batch of 15wt% was purchased from Hyperion and it was diluted to different

MWNT concentrations by adding polycarbonate using a twin screw extruder. Samples for rheological measurements were made from compression injected bars and a Rheometrics

ARES rheometer was used. Monotonic increase in the dynamic viscosity with increasing

MWNT concentration was observed at all frequencies, with much more increase at low frequencies than at high frequencies. For composites having more than 2wt% of MWNT low frequency plateau was developed in G’. In addition, G’ was greater than G” and both became less dependent on the frequency. This indicated formation of interconnected network structure at high MWNT concentrations. A percolation threshold of about 2wt% was observed through rheological data and this was supported by the measurements from electrical conductivity.

Summary: Shear viscosity of suspensions increases with the addition of fibers.

The increase of viscosity has strong relationship with the fiber concentration and aspect ratio. For Newtonian solvents, the suspension exhibits shear thinning behavior, especially at low shear rates. High shear viscosities tend to be about the same as the suspending solvent. For non-Newtonian solvents, the addition of fibers usually enhances the non-

Newtonian behavior. G’ and G” generally increase with increasing fiber loadings.

Sometimes thixotropy develops in some systems. The suspensions rheology depends on fiber orientation and some work has been done in this respect. Stress overshoot is usually

21 observed in the start up of shear flow due to the change of fiber orientation. Fibers can be aligned to the flow even the shear strain rate is small. When the fibers are completely aligned to the direction of the flow, fiber length does not affect the suspension viscosity.

2.1.3 Rheology of waterborne coatings

Generally the waterborne paint contain about 60wt% water as solvent, 20-30 wt% resin, thickener, pigments and other components. For metallic effects, aluminum metal flakes are added. Based on the design of resin systems, generally there are three types of waterborne coatings: water-emulsion (latex), water-soluble and water-reducible. The first two systems are two-phase systems (three-phase if solid pigments and flakes are considered), consisting of spherical polymer particles dispersed in continuous water solutions. The rheology relies more on the size of the particle sizes than the molecular weight of the polymers. In the third system, polymers do not form particles. The system is just polymer solutions. Therefore, the molecular weight has more effects on the viscosity. However, the viscosity of the coatings is dominated by the thickeners. Without them, the mixture of water and spherical deformable polymer particles will have a too low viscosity such that sufficient film thickness during application cannot be formed.

Therefore, thickeners have to be used to increase the shear viscosity. In order to obtain the best final appearance of the paint film, and to facilitate the processing procedures like storage and delivering and during application steps, the rheology of the paint should be carefully designed. During storage, the paint should have relatively high yield stress so that the solid pigments won’t settle. When the paint is delivered through pipe, the viscosity should be low so save the pumping energy. At the spray gun, the paint should be thin enough so that good atomization can be obtained. However, as soon as the paint

22 droplets meet the substrate, the paint should have higher viscosity to form film with sufficient thickness. When the drying starts, viscosity should be low to ensure good leveling, but still be high enough to prevent running and sagging. If the paint is to be applied using a roller, the spattering should be minimized. As a result, through years of development, the rheology of the waterborne paint is designed to be shear thinning, thixotropic and viscoelastic.

Lu36 investigated the viscoelastic properties of PVC latex paints. High PVC paints were observed to have high cross over frequency of G’ and G” than that of low PVC paints. Both G’ and G” showed power law dependence on the frequency. The paint leveling was found to be correlated well with G’ and low frequency (0.01 rad/s).

Osterhold37 studied a waterborne basecoat containing polyurethane microgels and polyacrylic thickener. A controlled-stress rheometer (VOR, Bohlin) with Couette geometry was used. Small amplitude oscillatory shear was used the probe the viscoelastic properties. The elastic modulus (G’) showed power law dependence on the frequency. At a constant frequency, the G’ decreased with increasing temperature from 10 to 40 °C due to the effect of microgel structure and not of the matrix.

Hester and Squire38 gave a general guideline for the design and control of the rheology of waterborne coatings. Functions of three commonly used associative thickeners were analyzed and their advantages and disadvantages were discussed. The waterborne coatings should have relatively low shear viscosities to ensure good flow and leveling, but the viscosities should be high enough to build up film thickness. Low extensional viscosity was desirable in order to minimize spattering during application.

Mechanisms of thickening effects were proposed but it was a subject of debate. Casson

23 equation was proposed to model the shear viscosity. Range of shear rates under different process steps (storage, spraying and drying) along with required shear viscosity and yield stress was tabled so that desired films could be produced. The technique of dynamic mechanical analysis was used to correlate spatter resistance with elasticity of the coatings. It was found that relatively low elastic modulus (G’) decreased spatting and lower complex viscosity would result in better flow and leveling properties.39 The rheology during curing of the latex coatings was often described using Mooney equation which incorporated the shape, packing factor and volume concentration of latex particles.40 An empirical equation relating leveling with coating rheological properties was proposed by Jackman.41

Harakawa et al42 formulated an aqueous acrylic dispersion with metal flakes in order to simulate the rheology of waterborne metallic basecoats. A Haake RS-100 stress- controlled rheometer with cone-and-plate geometry was used to measure the rheological properties. They found that a ratio of G’/G” of around 1 gave the best final metallic appearance, comparing 0.5 found by other researchers.43, 44 They also analyzed the required shear viscosities during spraying, setting and baking stage.

In the works of Fernando et al45, effects of waterborne rheology on spray atomization and roll misting were reviewed. Although both atomization and misting involved droplet formation, the former was desired during spray but the latter was not desired during roll application. Shear rates in roll application were about 104 - 105 s-1, while about 105 - 106 s-1 in spray application. However, no successful correlation between spattering and shear rheology had been found since 1978. On the other hand, extensional viscosity was found to be related with the misting tendency.

24 Due to the fact that the waterborne usually show inferior leveling and open time comparing to solvent-borne paints, the effects of associative thickeners on the paint rheology were further studied by Reuvers46. Rheological measurements were performed on a Bohlin controlled-stress rheometer with cone-and-plate geometry. Through the analysis of experimental results, the authors argued that the mechanisms of the HEUR type of associative thickeners were two fold. The thickeners formed network-like structures with latex particles and hence increased viscosity, and the thickeners increased the effective volume concentration of the latex particles. The network structures were found to be desirable since it improved the performance of the waterborne paint.

Weiss et al7 studied the rheological changes of waterborne automotive basecoats caused by circulation. A nonmetallic waterborne basecoat was tested in a pilot scale circulation system. Different pumps and back-pressure regulators were used to find which device would cause the most rheological change. A stress-controlled rheometer (SR-200,

Rheometrics) with Couette geometry was used to perform the rheological measurements.

The basecoat was found to be shear thinning and thixotropic. The results showed that both rotary circumferential piston and centrifugal pumps caused severe rheological changes to the waterborne basecoat. Low shear BPR had fewer effects than the conventional BPR, but the effects were very small comparing to those caused by the pumps. Despite the rheological degradations, there were no significant changes in the final appearance of the paint film. The authors also calculated that the Ford #4 cup would generate a shear rate of about 480 s-1. The shear rates in the circulation system were estimated to be 0-10 s-1 in the mixing tank, 100-1000 s-1 in the pipeline and 1000-10000 s-1 in the pumps, valves and other in-line instruments.

25 The ability to reflect light is a very important factor for the final film of automotive basecoat. Usually metal flakes are added in order to increase the light reflection effects. Ideally the metal flakes are aligned with the contour of the substrate after the paint dries, so that the light can be reflected like a mirror. When the light beam is fixed in a certain angle, changing viewing angle will make the paint film look light or dull. Standard methods have been established to measure the lightness at different viewing angles. As expected the alignment of the flake during paint drying is controlled by the rheology of the paint.

Osterhold37 studied the light reflection of a waterborne basecoat containing polyurethane microgels and polyacrylic thickener. He defined a flop-index based on the lightness value at 20, 45 and 70 degrees. He found that for the sprayed materials the flop- index decreased with the phase angle measured in small amplitude oscillatory shear experiments, and increased as G’ increases or yield stress increases. However no such relationship had been found for unsprayed materials.

Boggs et al47 studied several fully formulated waterborne paints in order to correlate rheological properties with flake orientation. A Bohlin CS-10 rheometer with concentric cylinder geometry was used to measure the rheology. Samples were capped to avoid evaporation of the solvent during the measurement. Steady shear viscosity was measured before and after the spray application, and dynamic shear experiments were performed to evaluate the time dependency and thixotropy of the paints. A Zeiss multiangle goniospectrophotometer “MMK-III” with 45° illumination angle was used to measure the reflected lightness at different view angles. A metallic flop index was defined to characterize the light travel based on the measurement at 25 and 70 degrees.

26 The shear viscosity showed shear thinning behavior and there was no low shear

Newtonian plateau. In the time dependence experiments, the paint sample was sheared at

50 Pa for 30 seconds to completely remove the structures in the paint. Immediately after that, dynamic shear started at 1Hz (strain not stated) and the increase of elastic modulus was monitored with time. It was found that for some paints, the G’ became close to steady state after 30 seconds while one paint required up to 500 seconds. Effects of three different types of thickeners on paint rheology was evaluated and it was found that purely associative thickeners did not provide much increase of the low shear viscosity and alkali swellable thickeners did. Synthetic and natural clay were found to be able increase low shear viscosity, possibly due to the bridging structures formed by the charged platelets.

Results from the orientation study showed that the flake orientation was independent of shear viscosities or elastic modulus. The phase angle from the dynamic shear experiments was found to have a closer correlation to the flake orientation. The lower the phase angle, the better the flakes would orient along the surface of the substrate.

The possibility of the metal flakes in the paint being damaged by the circulation device was studied by Bankert48. Pumps and back pressure regulator were the devices that could produce the highest shear rates. When the paint was circulated in the systems, the flakes might be crumpled by the high shear rates and hence caused paint degradation.

Since the paint was usually circulated for a long period of time, the degradation would become so severe that the final appearance of the paint film was unacceptable. The author cited experimental results from Graco’s laboratory and showed that the degradation tended to level off after the paint had been circulated for about 150 turns. A maximum theory was proposed that the degradation caused by the whole circulation system was

27 equivalent to that caused by the single device which generated the highest shear rates. For a normal system, this device was identified to be the back pressure regulator (BPR). A new design of BPR was proposed with increased contact surface comparing with a conventional BPR. Bankert49 later constructed a pilot plant at Graco to study further in this subject. By comparing the effects of different pumps and BPRs, he found that the rheology and appearance (gloss, depth of image and orange peel) of the final film had no correlation with the number of paint being circulated through pumps. However, the light travel was found to be related with different pumps and BPRs. The piston pumps caused the minimum degradation and the rotary lobe pump caused the highest degradation. The specially designed low shear BPR caused almost no degradation while the conventional

BPR did.

Summary: Shear thinning, thixotropy and viscoelastic are desired properties for waterborne paints. As the requirements for paints are so diversities, paints are formulated with very complex components. Detailed specification, such as low shear or high shear viscosity, degree of thixotropy and required viscoelasticity, should be specified for each paint that has different ingredients. Rheological and/or color degradation can be caused by prolonged circulation in the paint delivery system, and high shear generating devices such as pumps and back-pressure regulators have been identified to be the main cause.

Differently formulated paints may have different sensitiveness to the degradation.

28 2.2 Dynamic theories of fiber suspensions

2.2.1 Introduction

With the success of wide application of filled polymer composites, especially the fiber reinforced polymer composites, there has been a need that the processibility and product quality of the composites can be predicted. As a result, many phenomenological models that include particle concentration and aspect ratio have been developed from experimental results. These models usually have relatively simple mathematical forms the computation is fast. However, these models cannot predict particle orientations. On the other hand, theories regarding the particle interaction and orientation have also been developed. Some have been used to predict the experimental data adequately. Most theories assume that the suspended particles are spherical or rodlike due to their simple geometrical shape. In the following review of dynamic theories, only those with rodlike assumptions and are capable of predicting particle orientations are included.

Fiber suspensions are broadly divided into three regimes: dilute, semi-dilute and concentrated.50 In dilute suspension a fiber can freely rotate in any direction without touching other fibers. In this case, nL3 〈〈 1, where n is the number of fibers in a unit volume and L is the length of the fibers. A suspensions is called semi-dilute when 1 〈〈 nL3 〈〈 L/D, where D is the diameter of fibers. The distance between two neighboring fibers is greater than the diameter but less than the length of the fiber. As a result, a fiber can only move in two freedoms. Finally, in the concentrated regime when nL3 >> L/D, the distance between two neighboring fibers is less than the diameter, such that the fiber can only rotate along its axis.

29 2.2.2 Fiber orientation

The central topic of this issue is to develop a set of constitutive equations that the macroscopic rheological properties can be predicted based on the microscopy knowledge of the suspension system including suspending fluid, fiber size, volume concentration and orientation. In order to achieve this goal, the fiber orientation has to be expressed in a certain way. The first one to study fiber orientation in suspension is Jeffery.51 He analyzed the motion of a single ellipsoidal particle in a Newtonian solvent and developed an evolution equation which is referred to as Jeffery Orbit. His work is so fundamental that it became the starting point of almost all modeling of fiber orientation.

Theatrically, the fiber orientation can represented by an orientation distribution function (ODF) based on the force analysis on an individual fiber. The ODF describes the orientation of all fibers with respect to a specified direction. This direct simulation method requires to construct an ensemble consisting a lot of fibers in order to obtain a statistical distribution.52-56 By doing this, the computation time becomes incredibly long that makes this type of calculation unacceptable in some cases. For the purpose of reducing computation time, methods that uses a second order tensor to represent the fiber orientation have been proposed.57-59 They are usually close approximations using pre- averaging concept of the higher order tensor in the orientation distribution functions which are difficult to solve.

Another way to derive the orientation distribution function is through Doi theory, which assumes rigid rods suspended in Newtonian fluids. The Doi theory is based on

Onsager’s60 free energy expression which includes pair-wise interactions between the rigid-rods due to excluded volume effects. Doi type model has been widely used in the

30 area of liquid crystalline polymers and different variations have been developed to account for different situations.61, 62 Just as the theories based on Jeffery’s model, different closure approximations have been developed to reduce computation time.63, 64

Excellent review of the development of Doi type model can be found in Bhave’s work.65

2.2.3 Rheology of fiber suspensions

In the dilute regime, Jeffery equation could give satisfactory results. However, beyond the dilute regime, the fibers cannot rotate freely without touching other fibers and therefore the fiber-fiber interactions come into play. The neighboring fibers confine the movement of a fiber. Batchelor66 studied rod-like particles in non-dilute regime under extension flow. The fiber interactions were assumed to be hydrodynamic and Brownian effects were neglected. An equation relating elongational viscosity to the size of particle and shear viscosity was derived. Batchelor’s equation was shown to have good agreements with experimental results.67 Folger and Tucker30 claimed that the interactions between fibers tend to randomize the fiber orientation. By assuming that these interactions will cause fibers to re-orientate independently, as analogous to the concept of

Brownian motion, the authors set up a phenomenological equation to account for the interactions which contains a coefficient that has to be determined through experiments.

Dinh and Armstrong68 developed a rheological model for semi-concentrated rigid-rod suspensions. They assumed that physical contacts between fibers were rare and therefore most of the interactions were hydrodynamic. Starting from analyzing the stress on a single fiber, they modeled the fiber interactions as relative “slippage” between each other along the axis and estimated the drag coefficient based on Batchelor’s cell model.66

Effects of hydrodynamic interactions have also been studied by Mackaplow and

31 Shaqfeh.54 A set of integral equations was set up to solve for the hydrodynamic interaction of a large number of fibers. These equations were applicable to suspensions up to simi-dilute regime and were shown to be able to reproduce published shear and extensional viscosity data. Phan-Thien and Graham69 developed a phenomenological constitutive equation based on experimental results that effective specific viscosity increases with increasing fiber volume fraction at high fiber loadings. Although the

Jeffery’s model is based on dilute regime where fibers can rotates without contacting other fibers, Koch and Shaqfeh70 calculated the distance between the fibers and showed that the model can be applied to semi-dilute regime with good approximation.

However, the limitation of all these models above is that they neglect physical contacts of fibers. As one can imagine, at high fiber concentration, the fibers are going to physically touch each other during flow. The existence of apparent yield stress in fiber suspensions implies that mechanical contacts exists at rest.20, 71 Due to the difficulties in modeling the physical contact, few papers have been published in this area.

Sundararajakumar and Koch55 proposed a model to account for the mechanical contact between the fibers while neglecting the hydrodynamic interactions. A method which was developed for molecular simulation of hard sphere systems72 was used to find the points of contact and frictionless contact was assumed. The results showed that in dilute and semi-dilute regime hydrodynamics dominated the fiber orientation. At higher fiber concentrations, mechanical contacts contributed to the abrupt increase in shear viscosity and normal stress. Servais et al73 further analyzed the nature of the contacts and pointed out that there were three type of forces at the points of contact: normal force, friction force and lubrication force. Toll’s results were used to calculate the number of contact

32 points74 and a model was constructed with the assumption that all interactions happened at the contact points and the hydrodynamic effects were neglected. The model prediction was in reasonable agreement with measured shear viscosity.

The Jeffery equation was developed assuming the Newtonian solvent. Efforts has bee put on extension the equation to non-Newtonian solvents. Grmela et al75 modeled the viscoelastic matrix as dumbbells and extended the Jeffery’s equation to the application of suspensions with viscoelastic solvents. Following Grmela’s idea, Ramazani et al26 modified Jeffery equation for the fiber using a Hookean energy model for the non-

Newtonian matrix. Quantitative agreement between model prediction and experimental results was obtained. The Doi type model is also based on Newtonian solvents. Becraft and Metzner18 added a viscous dissipation term with which the shear thinning effects of the solvents were taken into consideration. The modified Doi model was found to agree well with steady shear viscosity of the glass fiber filled polypropylene.

All the models we have discussed up to now are based on straight rigid-rod. In reality the fibers are sometimes not strictly straight. One can expect that the behavior of a suspension containing straight fibers should be different from that containing curved ones. Experimental results suggest that curved/flexible fibers may cause higher suspensions viscosity than straight fibers.76-78 The effects of fiber curvature have been studied by Joung et al79, 80 through direct simulation. The curved fiber was modeled on the concept of chain and beads described by Bird et al81. Short range interactions between fibers and solvents was assumed to be lubrication forces as proposed by Yamane et al53, and long range interactions was assumed to be hydrodynamic effects. The simulation

33 results showed that curved fiber would cause market increase in the suspension viscosity.

Goto et al’s claim was confirmed that suspension viscosity was inversely dependent on the fiber stiffness.78

Almost all the models assume that the rigid-rods have the same diameter and length. The influence of polydispersity of the fiber length has been investigated by

Marrucci and Grizzuti.82 The polydispersity was treated as a simple extension of the theory for monodispersed fibers. Basically it is in the form of summation of equations for a spectrum of fiber lengths. Models based on Doi and Edward’s work was developed and the simulation results showed that the polydispersity played a more important role than the interaction potential in concentrated solutions below nematic phase.

34

Figure 2.1 Flow reversal of glass fiber in Newtonian solutions (fiber concentration = 8 vol%). The shear rate in both directions was 43.7 s-1. [Ganani and Powell10]

35 CHAPTER 3

RHEOLOGY OF CARBON NANOFIBER SUSPENSIONS

3.1 Introduction

Carbon nanotubes have been of great interest since their discovery in 1991 by

Iijima 1, due to their excellent electrical, mechanical, and thermal properties. Intensive studies have been conducted on polymer/carbon nanotube composites and some exciting results have been found 83-92. However, due to their limited availability and hence the prohibitive price, it seems unlikely that carbon nanotubes will be widely used in the near future. As an alternative, vapor-grown carbon nanofibers, which have average diameters of about 100 to 200 nm, can be used to produce polymer reinforced nanocomposites.

Since nanofibers and nanotubes have similar structures and physical properties, polymer composites filled with carbon nanofibers may have similar properties as those filled with carbon nanotubes. Therefore, studies on the nanofibers will directly facilitate their applications. Moreover, these nanofibers can be produced in large quantity at an affordable price, making them amenable for industrial scale applications.

36 With their extraordinary physical properties, the carbon nanofibers are promising candidates for reinforcing or conducting fillers. Kennel et al. 93 reported that although the structure of carbon nanofibers is not as regular as that of nanotubes, and hence nanofibers have relatively inferior mechanical properties, yet they still possess impressive mechanical characteristics 88-92. The nanofibers have graphene edge planes, which improve physical bonding with other materials. This property may lead to successful applications in the development of polymer/carbon nanofiber composites. Carbon nanofibers used as reinforcement and conducting agents in polymer matrices have been studied by Lozano and Barrera 2 and Lozano et al. 9. They compounded up to 15wt% carbon nanofiber into polypropylene using a twin-screw extruder and discovered that agglomerates are eliminated by the high shear in the extruder. The tensile strength of the composite remained unaltered from polystyrene, but the dynamic modulus increased

350%. In addition, a percolation threshold for electrical conduction of 9-18% was found.

Thermal behavior was not significantly changed due to the chemical treatment of the nanofibers. Ma et al. 3 studied Poly (ethylene terephthalate) resin compounded with 5wt% carbon nanofiber (Pyrograf III) using different methods, including ball milling, high shear mixing in the melt, and twin screw extruding. They found no significant reinforcement in the tensile strength but considerable increase in compressive strength and torsional modulus.

In contrast to the abundant literature on polymer/carbon nanotube composites, far less work has been published for polymer/carbon nanofiber composites 88-92. Further, most of the published works concentrate on the mechanical and electrical properties; none concentrate on the rheology of carbon nanofiber suspensions in low viscosity

37 solvents. Our objective in this paper is to study the morphology and rheology of nanofibers suspended in a Newtonian solvent. We select a Newtonian solvent so that any non-Newtonian properties of the suspensions can be attributed solely to the addition of the carbon nanofibers (CNFs). The CNFs as delivered have a strong tendency to clump into mm-sized agglomerates. Different methods are employed to disperse the large agglomerates, and the resulting dispersion effects and the rheology of the suspensions are carefully studied.

3.2 Materials and Preparation of Suspensions

The CNFs used in this study was Pyrograf® III (type PR-24-PS) made by Applied

Sciences, Inc. It is produced by decomposing organic vapors at elevated temperature in the presence of metal catalysts. These nanofibers have a tree-ring structure with a hollow core. The nanofibers we obtained were not pelletized and were in powdered form. Large agglomerates of a centimeter in diameter were observed, but most of the agglomerates were less than 1 mm in diameter. The as-received powders were sieved through #60 standard sieve (mesh size 0.25mm) before preparing suspensions.

In order to improve dispersion, the surfaces of some of the nanofibers were treated using a technique similar to that described by Esumi et al. 94. Acidic functional groups were added to the nanofiber surface so that the nanofibers would be more chemically compatible with hydrophilic solvents. 10 g of the nanofiber was refluxed with 200 ml of a concentrated sulfuric acid-nitric acid mixture (1:3 by volume of 96.1% sulfuric acid and 70.4% nitric acid). The reaction mixture was heated at a constant temperature of 140 °C for one hour. A combination of two water condensers in series, additionally cooled by ice, was used to cool the acid vapors. After cooling the reaction

38 mixture, it was diluted with water. Exploiting the difference of densities between the acid mixture and the aqueous fiber solution, centrifugation of the mixture was used to remove a greater part of the acid as supernatant. The fibers were neutralized further by repeated dilution with water and centrifugation. However, centrifugation became ineffective once the fiber-water solution density was comparable to the acidic mixture.

The nanofibers were washed with more water to neutralize the any remaining acid and vacuum filtered (polycarbonate membrane, 5 μm pore diameter). Finally, after removing the residue from the filter membrane, the resultant treated nanofibers were dried in vacuum at 80°C. About 55-60% yield in the treatment was obtained, with the only losses occurring during the filtration process, wherein the polycarbonate filter was unable to retain some of the smaller fibers.

The solvent used to prepare nanofiber suspensions was a 90wt% glycerol in water solution made by mixing 99.5wt% pure glycerol with distilled water. Both sieved untreated and treated carbon nanofibers were used to prepare suspensions, so that the effects of treatment on the dispersion and rheology could be studied.

The sieved untreated nanofibers were weighed and added into the solvent and then the suspensions were sonicated using an ultrasonic bath (100W) for 3 hours. The sonication time was selected based on previous experience that longer sonication would not provide better dispersion. Six untreated sonicated nanofiber suspension samples were prepared with concentration varying from 0.5wt% to 5wt%. Six sonicated suspensions of treated nanofibers were prepared using the same procedures. A 1wt% suspension sample was made by mixing treated nanofibers with the solvent with mechanical stirring for 36 hours without sonication. In total, 13 suspensions were prepared, six untreated sonicated

39 suspensions of different concentrations from 0.5wt% to 5wt%, six treated sonicated suspensions in the same concentrations, and one treated unsonicated suspension of 1 wt%. The size of all the samples for shear rheology was about 40 ml. The sample size for extensional rheology was about 300 ml.

3.3 Morphological Characterization

The suspension morphology was characterized using a Philips XL 30 scanning electron microscope equipped with secondary and back-scatter detectors operating at 30 kV. Magnifications were in the range of 10,000 to 12,000 times. For the as-received nanofibers, conductive glue was used to fix the powder to the SEM stub. For the suspensions, the SEM samples were prepared using the following procedure: One drop of the suspension from the 1wt% suspensions was diluted with about 2ml distilled water.

Care was taken so that this drop contained fibers that were representative of the whole sample. Then several drops of the diluted suspensions were dropped onto aluminum SEM stubs and were left in open air for overnight. As the liquid on the stubs contained glycerol, the stubs were put in vacuum oven at 60 °C for an hour to completely dry. A thin film of dried nanofibers was formed on each stub surface. The dilution of the suspensions and the drying speed were carefully controlled such that an optimum film thickness of nanofibers was formed. If the suspensions were not sufficiently diluted then the dried film contained too many nanofibers and no individual nanofibers could be identified. Too high a drying rate caused the liquid to contract and eventually form a small dried spot on the stub in which the nanofiber were too tightly compacted to be clearly observed. The stubs were then sputter coated with platinum at 18mA for 60 seconds using a Pelco Model 3 sputter coater. Although carbon nanofibers are

40

conductive, the sputtered SEM samples gave sharper images. The dispersion of nanofibers in the suspensions was also examined using optical microscopes, with the magnification in the range of 4 to 400 times.

The as-received nanofibers, purified by the supplier, were found using scanning electron microscopy to contain highly entangled aggregates but no visible impurities

(Figure 3.1a and Figure 3.2a). Almost all the individual nanofibers were not straight, forming clumps with a characteristic size of 20 by 50 μm. These clumps were often clustered to form larger weakly bound agglomerates with diameters up to millimeters or even centimeters. The diameter of the nanofibers was rather uniform, in the range of 100-

200 nm. Nanofiber length, however, was in a wide range from several μm up to 100 μm.

As the nanofibers were so entangled, it was usually difficult to find both ends of a single nanofiber, so that an accurate distribution of the length was impossible to obtain. After analyzing many SEM micrographs, it was estimated that more than half of the as- received nanofibers were about 5-20 μm in length.

The dispersion of carbon nanofibers and nanotubes in any solvent is known to be difficult (Esumi et al. 1996, Glasgaw et al. 2002, Lake et al. 2002, Shi et al. 2003,

Enomoto et al. 2004, McKenzie et al. 2003). Therefore it is important to explore the effect of dispersion of the suspension on rheological properties. Figure 3.1 and Figure 3.2 show the difference in dispersion between the as-received nanofibers and the suspensions prepared by the three different methods discussed in section II. The untreated sonicated sample (Figure 3.1b) contains many nanofiber clumps and some partially dispersed individual nanofibers, but no large agglomerates. The representative size of the clumps is

41 20 by 40 μm, similar to those in the as-received nanofibers. These clumps either directly touch each other, or more often are connected by the dispersed nanofibers. We conclude that the sonication disintegrates most of the large agglomerates and some of the clumps.

Both treated unsonicated suspensions (Figure 3.1c) and treated sonicated suspensions

(Figure 3.1d) contain no nanofiber clumps. Some individual fibers and small clusters can be seen in the treated unsonicated suspension while there are few clusters in the treated sonicated suspension. This clearly verifies that the acid treatment greatly improves the dispersion of the nanofibers.

More details can be seen in the higher magnification of Figure 3.2. In the untreated sonicated suspension (Figure 3.2b), the length of the individual nanofibers is observed to be about 5-15 μm. Comparing with the as-received nanofibers (Figure 3.2a), no obvious fiber shortening due to sonication has been found. The treated unsonicated suspension (Figure 3.2c) contains more uniformly dispersed fibers than that in the untreated sonicated suspension (Figure 3.2b). The long fibers still entangle with each other but they are not as tightly compacted. In addition, there are some short fibers with the length of less than 1 μm. It is deduced that the acid treatment not only separates the existing less-than-1-μm fibers from long fibers, but also shortens some long fibers and thus creates more short fibers. When the treated fibers are sonicated using an ultrasonic bath, it is apparent that the suspension contains almost exclusively less-than-5-μm short fibers (Figure 3.2d). Long fibers with length of 5-10 μm exist but they are much less in number. In this case, the sonication greatly shortens the nanofibers.

42 The strong acid treatment of a carbon surface is not new. Herrick 95 and Herrick et al 96 oxidized the surface of carbon fibers using nitric acid and observed increased surface area and surface functionality. Recently, Esumi et al. 94 used a mixture of concentrated nitric acid and sulfuric acid to treat carbon nanotubes in order to obtain a better dispersion. Increased dispersion has been attributed to the fact that acidic groups were added to the nanotubes and they readily disperse in polar solvents and form well- dispersed suspensions. Esumi et al did not report any damage to the nanotubes.

Mechanical damage of carbon nanotubes caused by ultrasound was first reported by Lu et al. 97. They observed that the ultrasound caused bending and buckling, and with longer time of sonication, the outer layer of the nanotubes were stripped off to form amorphous graphite. More powerful ultrasound and long sonication time cause more damage to the nanotubes. However, the damage to the carbon nanofibers caused by the ultrasonication bath in this study is different. The nanofibers are shortened while the diameters remain unchanged. The nanofiber surface is still smooth with no amorphous graphite observed in the sample. Further analysis of the as-received nanofibers at higher magnification shows that many contain bends and other curvatures, and some have bamboo-like structures with the distance between joints of about 500 nm. Most of the nanofibers are not perfectly straight, leading to structurally weak points. These weak points were attacked by the strong acid and thus become weaker and subsequent sonication caused the nanofibers to break. Liu et al. 98 sonicated single-walled carbon nanotubes in the presence of a mixture of concentrated sulfuric and nitric acid (3:1) and reduced the tubes from “endless” to several hundred nanometers.

43 The analysis clearly shows that sonication of the as-received nanofibers does not lead to fiber damage. After being treated with strong acid, these nanofibers become much weaker and the subsequent sonication breaks most of the nanofibers. The acid treatment followed by sonication not only opens the nanofiber clumps and disperses individual fibers, but also breaks most of them.

It should be noted that in preparing the SEM samples of the suspensions, it is difficult to guarantee that in each micrograph there are the same amount of nanofibers.

Even if same amount of fibers were put on the SEM stubs, it was difficult to ensure the formation of a homogeneous film of uniform thickness. The SEM micrographs are employed in this study to reveal the important features of nanofiber length, radius, and the conclusions discussed above based not only on the SEM micrographs shown in

Figure 3.1 and Figure 3.2, but also on hundreds of additional SEM micrographs.

Optical microscopy was used to study the dispersion on a larger scale. Figure 3.3 shows the micrographs for the untreated sonicated suspension and treated sonicated suspension. In the treated sonicated suspension (Figure 3.3b), the nanofibers are uniformly dispersed with only a few small clumps visible. These observations reflect the same trends as the SEM micrographs. Although the individual nanofibers are too small to be seen with optical microscopy, at this magnification (100x), in Figure 3.3a (untreated sonicated suspensions) we observe unbroken nanofiber clumps and clouds with partially dispersed nanofibers.

44 3.4 Rheological Characterization

Shear rheological measurements were made using a Rheometrics Fluid

Spectrometer (RFS II) from Rheometrics, Inc. A Couette geometry (bob diameter 32mm, bob length 33.3mm, cup diameter 34mm) with water bath was used for all measurements.

For steady shear viscosity, the shear rate range was from 0.02 to 500 s-1. For small amplitude oscillatory shear, the strain sweep was done first to determine the linear viscoelastic regime. The frequency was from 0.1 to 15 Hz. The first normal stress difference was measured using a Rheometrics Mechanical Spectrometer (RMS) 800.

Cone and plate (50mm diameter, 0.04 radian cone angle) geometry was used. Unless otherwise specified, all the measurements were done at 25°C.

The extensional rheology was characterized using the Rheometrics RFX extensional rheometer. A 250ml suspension was put in the test beaker and jets with different diameters were used to cover the extensional rate from 2 to 2000s-1; 1mm diameter jets were used for extensional rate from 100 to 2000s-1, and 2mm jets were for 2 to 200s-1. The separation distance is the same as the diameter of the jets.

3.4.1 Steady state shear measurements

The steady state shear viscosity measurements of the six untreated sonicated samples are plotted in Figure 3.4a. The viscosity of the suspensions increases monotonically with the nanofiber concentration by 3 orders of magnitude at low shear rates when concentration changes from 0.5wt% to 5wt%. The suspensions exhibit strong shear thinning behavior. For example, the low shear viscosity of 5wt% untreated

45 sonicated suspensions increases 3 orders of magnitude and the high shear viscosity increases by a factor of 8. Beyond 5wt% we found that the suspensions are too viscous to be effectively mixed by sonication.

Figure 3.6a displays the steady state shear viscosity for four treated sonicated samples. These samples show almost constant viscosity over a wide range of shear rates, increasing with the nanofiber concentration. The viscosity of the 5wt% treated sonicated suspensions is less than twice as much as that of solvent. The amount of the viscosity increases was much less than that of the untreated sonicated suspensions.

As the solvent is Newtonian, all non-Newtonian behavior of the suspensions is due to the addition of the nanofibers. We conjecture that in the untreated sonicated suspensions, the nanofiber clumps and the dispersed individual fibers entangle with each other, causing the viscosity to increase. The entanglement is relatively strong at small shear rates and becomes weaker at high shear rates due to alignment of some fibers to the flow. Hence the untreated sonicated suspension exhibits shear thinning. Ganani and

Powell 10 proposed a mechanism that the shear exerted on some parts of the suspensions was higher than other parts, and this might lead to a drift of low shear Newtonian plateau to lower shear rates. This non-uniform shear theory works as well for the untreated nanofibers suspensions. The shear stress is transported through both the continuous solvent phase as well as the nanofiber networks, and both mechanisms may have comparable strengths. However, the networks may break and become aligned to the flow under stress, so the total stress cannot be transported uniformly, leading to shear thinning in the suspensions.

46 By plotting the shear viscosity versus stress, the apparent yield stress of the suspensions can be examined. Flocculated suspensions are known to have yield stresses at high particle loadings when interactions between the particles are high enough to form continuous 3-D network structures. The untreated sonicated nanofiber suspensions in this study are of this fluid type. They show no zero-shear viscosity but an apparent yield stress. Figure 3.5 shows that the apparent yield stress starts to become obvious when the nanofiber loading is 1wt%. At 2, 3, 4, and 5wt% the suspensions have clearly defined yield stresses of 1, 3, 6, and 9 Pa respectively. Therefore it is reasonable to conclude here that the interactions between the nanofibers are large enough in most of the untreated sonicated suspensions such that continuous 3-D network structures exist and generate yield stresses.

We attempted to measure the first normal stress coefficient for all of the suspensions, but the normal force was too small for the rheometer to detect. Start up of steady shear tests were performed and the shear stress reached steady state almost immediately after the onset of the shear, showing little time dependent behavior. Results from thixotropic loop experiments revealed no obvious thixotropic behavior.

For the treated sonicated suspensions, the situations are different than with the untreated sonicated suspensions. As the treated sonicated suspensions contain no nanofiber clumps, they have more individual particles in a unit volume. One may expect that more individual particles will have more surface area, so that the viscosity would be higher than the untreated sonicated suspensions if there were no interactions between particles. However, this is not the case. As analyzed in the morphology section, the major differences between the treated sonicated and untreated sonicated suspensions are that the

47 latter contain nanofiber clumps and partially dispersed nanofibers which form network interactions. The interactions are so strong that they overrule the effects from the increase of the surface area. Further proof of the interaction is described in the linear viscoelasticity section. The treated sonicated suspensions are mainly composed of sub- micron nanofibers and the average aspect ratio is less than 10. As these short fibers are also well dispersed due to the effect of acid treatment, the interactions between them are small even at high concentrations.

As seen from the SEM micrographs (Figure 3.1 and Figure 3.2), although the treated unsonicated suspensions contain many more long (5-20μm) nanofibers and therefore more entanglements between the nanofibers than do the treated sonicated suspensions, the effects of the entanglements in treated unsonicated suspensions to the viscosity can be neglected at low nanofiber concentrations. Experimentally, we observe that the treated sonicated and treated unsonicated suspensions at 1wt% exhibit about the same viscosity to within experimental error (Figure 3.7a). Figure 3.7a also shows that the viscosity of both suspensions have little strain rate dependence. The nearly Newtonian viscosity may be due to the small aspect ratio which reduces shear rate dependence of viscosity. Brenner 99 showed that for suspensions of oblate and prolate spheroids the shear thinning behavior diminished when the aspect ratio decreased to unity. As a result, the traditional theory for the dilute suspensions might be a good approximation.

Many researchers have found that particle size does not necessarily affect the viscosity of suspensions. For spherical filler systems, Krieger’s 100 investigation of monodisperse suspensions shows that the complex viscosity is not a function of the size of the filler. This result holds even for concentrated systems 101. Choi and Krieger 102

48 studied sterically stabilized monodisperse systems and found that the relative viscosity is independent of the particle size. Therefore introducing more surface area does not necessarily increase the viscosity. Miles et al. 28 studied the apparent viscosities of glass fiber/sucrose suspensions using a capillary viscometer and found that the viscosity is dependent on the fiber volume fraction but is independent of the fiber length and shear rate. Some semi-empirical models predict higher viscosity with larger fiber aspect ratio and it might be the result of increased fiber interactions.

For dilute suspensions, the Einstein equation can be used to predict the viscosity.

Bachelor 103 modified the equation by adding a second order correction to account for hydrodynamic interactions,

η /η = 1 + 2.5φ + 6.2φ 2 s . (3.1)

In Eq.(3.1), η and ηs are the viscosities of suspension and the solvent, respectively, and φ is volume fraction of the filler. Krieger and Dougherty 23 put forward a semi-empirical equation that can be used for concentrated suspensions:

φ [η ] ⎛ 1 ⎞ m η /η = ⎜ ⎟ s ⎜1−φ φ ⎟ ⎝ m ⎠ , (3.2)

where φm is the maximum packing fraction and [η] is the intrinsic viscosity. Other

104 24 equations ( , ) also utilize φm , which takes values between 0.5 and 0.75 for monodispersed hard spheres. The relative viscosities of treated and untreated suspensions at high shear range (100 s-1) are plotted as a function of nanofiber volume fractions in

Figure 3.8. It is clear that the Einstein and Bachelor equations are not able to accurately predict the viscosities. This may be due to interactions between some long fibers in the

49 suspensions neglected in Eq.(3.1). When the Krieger-Dougherty model is used to fit the data, satisfactory fitting is obtained with the φm=0.2, [η]=12 for treated sonicated suspensions, and φm=0.05, [η]=30 for untreated sonicated suspensions. These φm values are low as compared to 0.5-0.75 for spheres, and hence lose the original physical

105 meanings as a maximum packing factor. Giesekus found that φm ranges from 0.2 to

106 0.4 for glass fibers. Harzallah and Dupuis studied the rheology of TiO2 particles in polymer solutions and obtained low values of φm ranging from 0.07 to 0.22 from fitting the experimental data to the Krieger-Dougherty equation (Eq.(3.2)). Although the TiO2 particles formed irregular shape of clusters, the authors didn’t relate the morphology to the low φm nor gave other explanations. Here we propose that this low φm is the result of particle interactions and aspect ratio.

3.4.2 Small amplitude oscillatory shear measurements

The linear viscoelasticity of each suspension was studied to investigate the effect of microstructures formed by the carbon nanofibers. Strain sweep measurements were performed to determine the extent of the linear viscoelastic regime. For the untreated sonicated suspensions, it was found that the maximum strain limit for the linear regime decreases with the increase of nanofiber loading, from 10% strain at 1wt% loading to

0.1% strain at 5wt% loading. This phenomenon was also observed by Aral and Kalyon 107 in 40 vol% spherical suspensions. From results of flow visualization, they suggested that this is due to internal slip of the particles. This idea is similar to the non-uniform shear theory proposed by Ganani and Powell 10 to explain shear thinning. Figure 3.9a,b show the elastic modulus G′ and the viscous modulus G′′ versus the dynamic frequency. Both moduli increase with more nanofiber loading, and G′ becomes higher than G′′ at all 50 frequencies studied when the suspension contains 3wt% or more. The fact that G′ is greater than G′′ is an indication of solidlike behavior, as defined in rheology textbooks

(see, for example, Macosko 108). Further, the low-frequency plateau of the moduli in G′ is also a sign of solidlike behavior. A possible explanation for this solidlike behavior is that the nanofibers form a network microstructure. This microstructure can be easily broken down by deformation and rebuilt quickly when the deformation is removed. Therefore the suspensions are shear thinning but not thixotropic. At low concentrations, the microstructure is weak so the solvent dominates the rheology and the suspensions still show liquidlike behavior. Beyond a critical concentration, which is 3 wt% for the system studied here, the network structure becomes stronger and hence elastic solidlike behavior is observed. This type of microstructure is verified from the morphology study, which shows that the dispersed nanofibers are entangled with each other and also with the nanofiber clumps.

Non-terminal low-frequency rheological behavior has been widely observed in polymer/nanoclay systems (109-111), ordered block copolymers 112 and smectic liquid crystalline small molecules 113. Conventional filled polymer composites also exhibit this behavior and the accepted reason is that the filler and the polymer strongly interact such that large domains are formed. Larson 113 studied the smectic liquid crystals and suggested that the non-terminal low-frequency rheological behavior is due to the long- range domain structure and the presence of defects. Krishnamoorti and Giannelis 111 argued that in the end-tethered polymer layered silicate nanocomposites, the presence of silicate layers and the lack of complete relaxation of the confined polymer chains cause

51

the solidlike response at low frequencies. Solomon’s 110 results from the polypropylene/clay hybrid materials suggested that the low frequency plateau was the result of network structure and not the orientational relaxation of individual platelets.

Interestingly, the low-frequency plateau in G′ presented in the untreated sonicated suspensions, also appears in the treated sonicated suspensions (Figure 3.10a). There is a wide spread of elastic modulus at low frequencies while the elastic modulus G′ becomes less dependent on the fiber suspension at high frequencies. The elastic modulus generally increases with fiber loading. The viscous modulus G′′ monotonically increases with the fiber concentration at all frequencies (Figure 3.11a). As there are no nanofiber clumps and many of the fibers have been shortened to less than 1μm, the above-mentioned network structure should not exist. From the size of the particles, it is suggested the

Brownian motion and colloidal forces come into play. The Brownian effects become important when the particle size is less than 500nm 114. A measure of the Brownian effect is the rotary diffusivity (Dr0), which is the rate at which a particle reorients through

Brownian motion. Using the formula in Larson 115, the rotary diffusivity of the treated suspensions is calculated to be about 0.04 s-1. When the external deformation rate is small, the Brownian motion tends to make the suspensions liquid-like and the colloidal forces tend to make the suspensions solid-like. Since acid groups are added to the nanofibers during the surface treatment, electrostatic forces exist in the suspensions. That act like springs and cause elastic effects 108. The Brownian and colloidal forces compete with each other, and at low frequencies the latter dominates, causing the plateau in the elastic modulus. 52 The untreated sonicated nanofiber suspensions were found to roughly obey the

Cox-Merz rule, especially at high nanofiber concentrations. The empirical Cox-Merz rule states that the complex viscosity (η*) versus the dynamic frequency obtained from small amplitude oscillatory shear is numerically equal to the steady shear viscosity (η) versus the shear rate. Although there is no general explanation for such a relationship, it is widely accepted and used for isotropic polymeric solutions and polymer melts. Figure

3.16 shows that at 1wt%, the Cox-Merz rule holds well except for the low shear region

(less than 5 s-1). However, for 5wt% suspensions, the complex viscosity is about an order of magnitude higher than the steady shear viscosity but the degree of shear thinning is about the same. Therefore, the Cox-Merz rule holds only for low nanofiber concentration suspensions. Kinloch et al. 34 showed that for concentrated aqueous nanotube dispersions, complex viscosity was about 5 orders of magnitude greater than the steady shear viscosity.

3.4.3 Measurements of temperature effects

Temperature effects on the steady shear and relative viscosities of 2wt% untreated sonicated suspensions are plotted in Figure 3.12a-d and Figure 3.13a-d. The steady shear viscosity decreases as the temperature increases over the entire range of shear rates

(Figure 3.12a). The difference between viscosities at different temperatures is smaller at low shear rates than that at high shear rates. Figure 3.12c shows the relative viscosity

(ratio of suspension viscosity to solvent viscosity) as a function of shear rate at different temperatures. There is a clear trend that the relative viscosity is higher at lower temperatures except for the first few low shear rate points at 35°C and 45°C, implying that the contribution of the nanofibers to the viscosity of the suspension is relatively

53 larger than that of solvent at higher temperatures. The solvent viscosity decreases rapidly while the interactions between the nanofibers might remain the same, making the latter more important. At high shear rates, the relative viscosities at all temperatures plateau at the same value. This suggests that the microstructures are rather weak and can be broken down to approximately the same extent at high shear rates. High shear rate viscosities (at

200 s-1) of the suspension and the solvent as a function of temperature are plotted in the

Arrhenius plot in Figure 3.13c. The fact that the two curves are almost parallel indicates that the solvent dominates the high shear viscosity. This is reasonable, as the microstructures have been broken down by the shear.

The apparent yield stress σay decreases as temperature increases (Figure 3.13a), implying that the strength of network microstructures comes from both particle-particle contact/entanglement and particle-solvent interaction. When temperature increases, while the particle-particle contact presumably stays the same, the particle-solvent interaction becomes weaker due to the decreased solvent viscosity, causing σay to decrease. The loss G′′/G′ decreases from above 1 to about 0.2 with increasing temperature (Figure

3.13d) and this is another indication that the suspension becomes more solidlike at higher temperatures.

3.4.4 Extensional rheology

The results of the extensional rheology of 1wt% untreated sonicated suspension are shown in Figure 3.14. The suspension exhibits extension rate thinning behavior over the range of extension rate examined (the apparent extensional viscosity decreases with increasing extension rate). It is clear that the untreated nanofibers cause a dramatic change in the extensional rheology. The extension rate thinning is not surprising because

54 the breakup of the network structure decreases the viscosity, a mechanism similar to the shear thinning. The usual extension thickening (also called strain hardening) of polymer melts is a phenomenon that at a certain constant extension rate, the transient extensional viscosity increases with strain. If steady state can be reached, even some polymer melts exhibit extension rate thinning 116. Extension rate thickening is also seen in the literature.

Eastman et al 117 found that the aqueous solutions of polyvinylpyrrolidone (PVP) containing sodium dodecylsulphate (SDS) were extension rate thickening and they suggested that it was due to the increase in the dimensions of PVP/SDS complexes because of the repulsion between SDS aggregates.

The comparison of extensional and shear viscosity is through the calculation of the Trouton ratio, Tr, defined as

Tr = ηe (ε) /η(γ), (3.3)

where ηe is the extensional viscosity at the extensional rate of ε and η is the steady shear viscosity at the shear rate of γ . Jones et al. 118 proposed that γ = 3ε should be used to calculate the Tr. This method has been adopted by Meadows 119 and Viebke et al. 120. The Tr has been theoretically predicted to be 3 for Newtonian fluids and some researchers observed this value experimentally 121, 122. However, Eastman 117 reported that the Tr for was about 4.3 using the Rheometrics RFX. Our observation for the 90wt% glycerol water solution is about 4.0. The deviation from the theoretical value of 3 may be due to the inertia of the fluid, when being pulled into or pushed from the opposing jets.

55 The Tr of 1wt% untreated sonicated suspension calculated by both standard method (γ =ε ) and Jones method (γ = 3ε ) is plotted in Figure 3.15. The Tr decreases with extension rate from 80 using Jones method or 50 using standard method to 10, which is the same using either method. For Newtonian fluids, the Trouton ratio is 3, which has been observed experimentally and proved theoretically. The large Tr as seen here indicates strong non-Newtonian behavior. Bachelor 66 studied the stress in a non- dilute suspension of elongated particles subjected to extensional flow. He found that when the average lateral distance of particles was between the diameter and length of the elongated particle, the extensional viscosity was sensitive to the length, due to the hydrodynamic effects between the aligned particles. The Tr is calculated using

Bachelor’s equation 123 to vary from 5 to 35 for different fiber lengths in the untreated sonicated suspension, which falls in the range of our experimental results. This implies that the hydrodynamic effects play an important role in the extensional flow. However,

Bachelor’s equation cannot predict the extension rate thinning behavior. We propose that the extension rate thinning behavior may be the results of physical contacts of the nanofibers and the formation of network microstructure. Under extensional stress, the network breaks down and the nanofibers orient to the flow and thus cause the viscosity to decrease. In other words, it is due to the yield stress of the suspension. Cohu and Magnin

124 studied fully formulated coil-coating paints which are dispersions of mineral or organic pigments in organic solvents with polymeric binders. They found that these paints were extension rate thinning with decreasing Tr from about 20-30 to the

Newtonian range 3-7. They attributed this observation to the existence of yield stress of the paints. Anklam et al. 125 made a similar observation with water-in-oil emulsions. They

56 argued that the yield stress of the emulsions dominates the force on the nozzle arm and concluded that fluids with high yield stress should not be measured using opposed nozzle .

Extension rate thinning has also been found by Viebke et al. 120 in their concentrated latex dispersions. They also found that the Tr was independent of the extension rate. Meadows et al. 119 found that hydroxyethylcellulose solutions exhibits slight strain thinning and that it maybe attributed to the “stiffer” conformation of the cellulose polymer and thus hindered the stretch of the coils. Another possible mechanism proposed by the authors is that the breakup of overlapped coils offsets the strain- hardening coil stretch of individual polymer molecules. Kapoor et al. 126 found that the waxy corn starch solution was also extension rate thinning and they cited Dontula et al.’s argument 127 that the instrument effects on the measured extensional viscosities using opposed nozzle rheometers are significant, hence only limited conclusion could be drawn from the data.

In order to further probe the particle interactions, the stability of the suspension after long time stretching at high extension rate was examined for 1wt% untreated sonicated suspension. After the extensional viscosity measurement was done for the fresh suspension using 2mm jets, the sample was then stretched at 600 s-1 extension rate for 13 hours. The “mix-cycle” function provided by the rheometer was utilized to pull the liquid into the jets and then push them out. In each mix-cycle, 40ml suspension was stretched.

In the 13 hours, the total amount of suspensions that had been stretched was about 81250 ml, which was equivalent to 325 times of the amount of the sample. The extensional viscosity was measured again and was found to be identical to the fresh sample data.

57 Then 1mm jets were used and the viscosity was measured at higher extensional rate.

After that, the sample was further stretched at 4000 s-1 extensional rate for 2 hours, which is equivalent to 20 times of the amount of the sample, and the measured extensional viscosity did not change. Therefore it is concluded that the untreated sonicated sample is very stable under extension. The morphology of the fresh and stretched suspension was studied using optical microscopy, and the results showed no difference. The high extension rate did not change the length or interactions between the nanofibers.

3.5 Modeling

3.5.1 Kinetic Theory Models

In this paper we develop models for the carbon nanofiber suspensions from SEM measurements of their morphology, and through these models deduce the bulk rheological properties of the composite systems from the microstructural measurements.

We chose to model the nanofiber suspensions with kinetic theory-based elastic and rigid dumbbell models 81, 115, 128, 129.

Dumbbell models are of appropriate complexity for a first attempt at microstructurally based modeling of the disperse treated sonicated nanofiber suspensions.

In the dumbbell models the carbon nanofiber suspended in a Newtonian solvent is idealized as a pair of beads joined by either a massless elastic spring connector (elastic dumbbell) or massless rigid connector (rigid dumbbell) with orientation along the unit vector u . The dumbbell models assume the solution is dilute, i.e. the dumbbells are assumed to move independently. Such models are customarily applied to polymer solutions, where the dumbbells idealize the polymer molecules; here we investigate their ability to describe nanofiber suspensions. Importantly, in our model all material

58 coefficients in the governing equations are specified directly in terms of separate, independent primitive measurements of the constituents (the characteristic length and radius of the nanofibers measured in our morphology study, density of the nanofibers, and density and viscosity of the solvent), as well as the mass concentration of the fibers in the suspension and absolute temperature.

For the isolated nanofiber in the glycerol/water solvent, the two salient physical features of the particle/solvent interaction are the force on the fiber from the solvent, distributed over the surface of the fiber, and the possible elasticity of the fiber (its ability to stretch or bend under load and then return to its original configuration when the load is removed). The dumbbell as applied here is an idealization of the fiber in which the distributed load on the fiber and mass of the fiber are localized in the two beads, and the elasticity of the fiber both in stretching or bending, if present, is modeled as a linear spring. We consider the spring in the elastic dumbbell to be Hookean, i.e. the tension in the spring is proportional to the bead separation. The relation between the length and radius of the nanofibers and connector length and bead radius of their dumbbell idealization must be calibrated.

The forces assumed acting on the beads are Brownian forces, intramolecular force, and hydrodynamic drag force; external forces such as gravitational and electrical forces are neglected (Bird et al. 1987). For the elastic dumbbell model, the intramolecular force is modeled as spring force acting on the bead. For the rigid dumbbell model, the intramolecular force is the constraint force in the rigid rod. Hydrodynamic drag force

F (h) is the force of resistance experienced by the bead as it moves through the solution 81:

F (h) =⋅ζ ⎡ rvv − +' ⎤ , (3.4) ⎣ v ( )⎦ 59 where rv is the average bead velocity, v the imposed homogeneous flow field of the solvent at the bead, v' the perturbation of the flow field due to the motion of the bead at the other end of the dumbbell, and ζ the friction tensor.

We pursue six different dumbbell models of the nanofiber suspensions:

• the elastic dumbbell with isotropic hydrodynamic drag (friction tensor

ζ = ζ I , as given by Stokes’s law with friction coefficient ζ = 6πηsr ,

where ηs is the solvent viscosity and r the bead radius) without

hydrodynamic interaction ( v' zero in Eq.(3.4)) 81, 115, 128, 129;

• the elastic dumbbell with isotropic hydrodynamic drag and with

hydrodynamic interaction ( v' nonzero in Eq.(3.4)) 81, 130;

• the elastic dumbbell without hydrodynamic interaction but with

−−11⎛⎞α p p anisotropic hydrodynamic drag (ζ =−ζ ⎜⎟IT, where T is the ⎝⎠nkT

particle contribution to the stress, n the number of fibers per unit volume,

k =1.3807 ×10−23 J / K the Boltzmann constant, T absolute temperature,

and α the mobility factor) (81, 128;

• the elastic dumbbell with anisotropic hydrodynamic drag with

hydrodynamic interaction 81, 128;

• the rigid dumbbell with isotropic hydrodynamic drag without

hydrodynamic interaction 81, 115, 128, 129;

• the rigid dumbbell with isotropic hydrodynamic drag with hydrodynamic

interaction 81, 115, 128, 129.

60 The questions we answer in this study are:

• How effective is the dumbbell idealization in modeling the nanofiber

suspensions?

• Are the carbon nanofibers better modeled as rigid or elastic dumbbells?

• Should anisotropic hydrodynamic drag be included in the elastic dumbbell

models?

• Should hydrodynamic interaction be included in the elastic or rigid

dumbbell models?

The primitive microstructural parameters in the dumbbell models of our nanofiber/Newtonian solvent suspensions are:

a = length of the spring link (for elastic dumbbells) or length of rigid rod (for rigid dumbbells) (m),

r = radius of the beads (m),

ρ f = density of the nanofibers (kg/m3),

ρ 3 s = density of the solvent (kg/m ), (3.5)

c = mass concentration of the suspension (dimensionless),

T = absolute temperature (K),

η s (T ) = Newtonian solvent viscosity (Pa·s),

α = mobility factor, only for the elastic dumbbells with anisotropic hydrodynamic drag (dimensionless).

61 In the kinetic theory the constitutive equation for the suspension is produced by integrating over a representative volume of dumbbells and solvent yielding that the

Cauchy stress T is a superposition of a solvent contribution 2ηs D and a particle contribution T p :

1 TDT=+2η p , D = (∇v + (∇v)T ) . (3.6) s 2

For the rigid dumbbell models, the particle contribution to the stress T p is (Bird et al. 1987):

without hydrodynamic interaction:

D Tuup = 3nkTλ , (3.7) Dt

with hydrodynamic interaction:

(2) p (2) D ⎡⎤λ2 TuuuuI=−−−31(3)nkTλ2 ⎢⎥(1) nkT nkT , (3.8) Dt ⎣⎦λ2

Where uu=∫∫ uuf () u,tdθ dφ is the phase-space average of the second order orientation tensor uu , with f (u,t ) the orientation distribution function, and

D ⎛ ∂ ⎞ (•) = ⎜ + v ⋅∇⎟(•) + (•)W −W (•) − (•)D − D(•) , Dt ⎝ ∂t ⎠

11 WvvDvv=∇−∇()(),TT =∇+∇() (). 22

For the elastic dumbbell models, the orientation tensor uu can be mathematically eliminated, producing the following closed-form constitutive equations

81:

62 with isotropic hydrodynamic drag:

D TTDpp+=λη2 , (3.9) Dt p

with anisotropic hydrodynamic drag:

ppD λ pp TT++λ αη() TTD ⋅=2 p . (3.10) Dt η p

The Hookean elastic dumbbell model with isotropic hydrodynamic drag, Eq.

(3.9), is referred to as the Oldroyd fluid-B model, and the Hookean elastic dumbbell model with anisotropic hydrodynamic drag, Eq (3.10), is referred to as the Giesekus model.

(2) The material constants η , λ ,η p , λ2 etc. in the constitutive equations (3.6)-(3.10) are expressed in terms of the primitive parameters in list (3.5) 81, 131:

ρ c φ = volume fraction= s , ρ f +−c()ρρsf

φ ρsc 1 n = number of fibers per unit volume = 22=⋅; ππraρρρfsf+−c() ra

for the elastic dumbbell models with hydrodynamic interaction:

r h = hydrodynamic interaction parameter = , ( h =0 if neglect hydrodynamic a interaction),

πη ra2 λ = relaxation time = s , ⎛⎞6 21kT⎜⎟− h ⎝⎠π

η p =particle contribution to the viscosity

63 ρηca1 ==nkTλ ss ⋅⋅ ; ρρρfsf+−c()2r ⎛⎞6 ⎜⎟1− h ⎝⎠π

for the rigid dumbbell models with hydrodynamic interaction:

πη ra2 λ = relaxation time = s , (3.11) 2kT

3r h = hydrodynamic interaction parameter = , ( h =0 if neglect hydrodynamic 4a interaction),

λ 1 πη ra2 λ == s , h 112−−hhkT

−11− ⎡ 22⎤⎡2 ⎤ 1 ⎛⎞12⎛⎞rrπη ra ⎛⎞ 12 ⎛⎞ λλ()=−+⎢11hh⎜⎟⎥⎢ ==s 11 −+ ⎜⎟ ⎥ , 2 ⎜⎟626⎜⎟akTa ⎜⎟ ⎜⎟ ⎣⎢ ⎝⎠⎝⎠⎦⎣⎥⎢ ⎝⎠ ⎝⎠ ⎦⎥

−11− ⎡ 22⎤⎡2 ⎤ 2 ⎛⎞12⎛⎞rrπη ra ⎛⎞ 12 ⎛⎞ λλ()=−⎢12hh⎜⎟ 1 −⎥⎢ =s 12 − ⎜⎟ 1 − ⎥. 2 ⎜⎟62⎜⎟akTa ⎜⎟ 6 ⎜⎟ ⎣⎢ ⎝⎠⎝⎠⎦⎣⎥⎢ ⎝⎠ ⎝⎠ ⎦⎥

In this paper we retain the microstructural basis of the dumbbell models: the characteristic lengths a and radii r of the model dumbbell are inferred from our morphological SEM measurements, independent of rheological measurements apart from a single calibration. The relaxation time λ and particle viscosity ηp are then deduced from a and r, and measurements of densities ρs and ρf, mass concentration c, temperature T, and solvent viscosity ηs, also independent of rheological measurements; see Eqs. (3.11).

Only the mobility factor α of elastic dumbbell models with anisotropic hydrodynamic drag is deduced from a fit to rheological measurements.

64 3.5.2 Modeling predictions for treated sonicated nanofiber suspensions

Recall from the morphological characterization given in section 3.3 that the treated sonicated suspensions contain well-dispersed fibers with radii in the range 100 to

200nm and lengths mostly in the range of 1 to 5μm, with few fibers of length 5 to 10μm and essentially no clumps. Because of the absence of clumps, it is physically reasonable to model the treated sonicated suspensions with the kinetic theory model of dumbbells in

a viscous solvent. The representative fiber length a f and radius rf are measured from the

SEM images (Figure 3.2d and hundreds of others analyzed) to be a f =2μm and the radius

50, 115 rf =100nm. According to Doi-Edwards theory , fiber suspensions are divided into

3 dilute, semidilute and concentrated regimes as follows: when n a f < 1, the suspension is

3 a f 3 a f said to be dilute, when 1< , 2rf 2rf the solution is concentrated. The number of fibers per unit volume n

a ρsc 1 f = ⋅ 2 . For the treated sonicated suspensions, the transition =10, ρρρfsf+−c()π raf 2rf

3 and n a f =0.5, 0.9, 1.8, and 4.5 for c=0.5, 1, 2, and 5wt%, respectively. Hence, according to Doi-Edwards classification, the 0.5wt% and 1wt% treated sonicated suspensions are dilute, and the suspensions are semidilute for weight percent 2 to 5wt%.

The relation between the length a f and radius rf of the physical nanofiber and the connector length a and bead radius r in its dumbbell idealization must be calibrated. We

assume that the relation between ( a , r ) and ( a f , rf ) for the treated sonicated fibers is the same at all concentrations in all flows, and therefore our calibration utilizes only one

65 experiment. This protocol is consistent with dilute suspensions where entanglements do not alter the effective spring constant. We choose to calibrate with the 2wt% concentration small amplitude oscillatory shear experiment: a small amplitude oscillatory shear experiment so as not to involve the parameter α , and 2wt% so as to avoid the possible inertia effects at lower concentrations and possible particle interaction at higher concentrations. We perform the calibration as follows.

3.5.2.1 Elastic dumbbell models

For the elastic dumbbell models, the linear viscoelastic material functions G' ,

G'' and η' , η'' are 81:

⎛⎞nkTλω2 G'()ωωηωω = '' () = ⎜⎟, ⎜⎟2 ⎝⎠1+()λω (3.12) ⎛⎞nkTλ G''='=()ωωηωωη () ⎜⎟ + . ⎜⎟s 2 ⎝⎠1+()λω

In terms of the primitive microstructural parameters of list (3.5) these become

⎛⎞ ⎜⎟ ⎜⎟ρηπωckT23 a G'ωωηωω = '' = ⎜⎟ss , () () 2 ⎜⎟⎡⎤⎛⎞6 ⎜⎟⎡⎤⎢⎥22 2 224 2 ρρρfsf+−cTk-h+ra()41⎜⎟ πηω s ⎜⎟⎣⎦⎢⎥⎜⎟π (3.13) ⎝⎠⎣⎦⎝⎠ ⎛⎞ ⎜⎟22 ⎛⎞6 21ρηck T a⎜⎟ - h ⎜⎟ss⎜⎟π G''='=ωωηωωη⎜⎟ + ⎝⎠ , () () s 2 ⎜⎟⎡ ⎛⎞6 ⎤ ⎜⎟⎡⎤⎢ 22 2 2342⎥ ρρρfsf+−ckTr-h+ra()41⎜⎟ πηω s ⎜⎟⎣⎦⎢ ⎜⎟π ⎥ ⎝⎠⎣ ⎝⎠ ⎦

66 r where h = 0 without hydrodynamic interaction and h = with hydrodynamic interaction. a

Notice that these forms do not contain the primitive parameter α , so that α will not interfere with our calibration relating a and r to the microstructure measurements

a f and rf .

The parameters of list (3.5) that appear in Eq. (3.13) besides a and r are measured independently of the small amplitude oscillatory shear experiments:

temperature T =25°C, solvent viscosity at 25°C η s = 0. 154 Pa·s, fiber density ρ f =1750

3 3 kg/m , solvent density ρs =1236 kg/m , and fiber concentration c = 0.5, 1, 2, 3, 4, and

5wt%. The characterization relating a and r to the measured fiber length a f and radius rf are obtained by selecting a and r to minimize the error

22 N ⎧⎡⎤⎡⎤⎛⎞ ⎛⎞2 ⎫ ⎪ '''nkTλ nkTλωi ⎪ (3.14) δηωη=−+⎨⎢⎥⎢⎥log() log⎜⎟ + log ηω() − log ⎜⎟⎬ ∑ 10 expis 10⎜⎟22 10 exp i 10 ⎜⎟ i=1 ⎪⎢⎥⎢⎥11++()λωii() λω ⎪ ⎩⎣⎦⎣⎦⎝⎠ ⎝⎠⎭

' '' and comparing to a f and radius rf . In Eq. (3.14) η exp (ωi )and η exp ()ωi are the

experimental measured values of η' and η'' at the N discrete frequencies ωi for a particular concentration c . The fitting results using the experimental data at c = 2wt%

(for which N = 22) are a =0.23μm, r =10.5nm for the elastic dumbbell models without hydrodynamic interaction and a =0.22μm, r =10.7nm for the elastic dumbbell models with hydrodynamic interaction.

67 For the elastic dumbbell models with or without hydrodynamic interaction, the

r ratio of the fiber radius to the fiber length is small ( h =≈004. ). Therefore for the a treated sonicated suspensions, the predictions of the elastic dumbbell models without hydrodynamic interaction are essentially the same as the elastic dumbbell models with hydrodynamic interaction.

Compared with the representative fiber length a f =2 μm and the radius rf =100 nm measured from the SEM images of the treated sonicated suspensions (Figure 3.2d), we see that the model length a of the spring link and the model radius r of the beads are

proportional to the measured a f and rf by essentially the same factor 0.11. This factor, obtained from oscillatory shear data for the 2wt% concentration, is employed at all concentrations and in both oscillatory and steady shear with no refitting, since the characteristic fiber length and radius are the same at all concentrations, in either type of shear flow.

Applying the fiber length a =0.23 μm, radius r =10.5 nm and different mass concentration c to small amplitude oscillatory shear Eq. (3.13), the elastic dumbbell models predict the elastic modulus G′ and viscous modulus G′′ of the treated sonicated suspensions shown in Figure 3.10b and Figure 3.11b. Comparing these predicted moduli to the corresponding measured moduli displayed in Figure 3.10a and Figure 3.11a, we see that the elastic dumbbell models are successful in capturing the trends exhibited in the measured response: The elastic dumbbell models predict that the addition of the particles into the Newtonian solvent creates an elasticity in the composite suspension, as verified in the experiments; note from the right hand side of Eq. (3.13) that the predicted elastic 68 modulus G′ is zero when the concentration c is zero, but nonzero for nonzero c. In agreement with the small amplitude oscillatory shear measurements, both elastic modulus

G′ and viscous modulus G′′ in the elastic dumbbell predictions monotonically increase with nanofiber concentration. The slopes and orders of magnitude of G′ and G′′ predicted by the elastic dumbbell models are also consistent with the experimental data; the models overpredict the separation of G′ at high frequencies. As the elastic dumbbell models do not take colloidal forces into account, the plateau in the elastic modulus G′ observed in the suspensions at low frequencies could not be captured by the models.

In the simulation of steady shear, the elastic dumbbell model with isotropic hydrodynamic drag (i.e. the Oldroyd fluid-B model) predicts a constant viscosity

2 nraπηs ηγ() =+ ηsps η =+ η , (3.15) ⎛⎞6 21⎜⎟− h ⎝⎠π

independent of the shear rate γ , and the elastic dumbbell model with anisotropic hydrodynamic drag (the Giesekus model) predicts shear-rate dependent non-constant viscosity 81, 128:

222 (11−−ψψ) nraπηs ( ) ηγ() =+ ηsp η =+ η s , (3.16) 112+−()α ψαψ⎛⎞6 112 +−() 21⎜⎟− h ⎝⎠π

69 where

1− χ ψ = , 1+ ()1− 2α χ

1/ 2 1/ 2 2 2 . 2 ⎡ ⎛ . ⎞ ⎤ ⎡ ⎛ ⎞ ⎤ πηs ra 1+16α()1− 2α γλ −1 ⎢1+16α()1− 2α ⎜γ ⎟ ⎥ −1 ⎢ ⎜ ⎟ ⎥ ⎜ 2kT ⎟ ⎣⎢ ⎝ ⎠ ⎦⎥ ⎣⎢ ⎝ ⎠ ⎦⎥ χ = 2 = 2 . ⎛ . ⎞ ⎛ . πη ra 2 ⎞ 8α()1− 2α γλ s ⎜ ⎟ 8α()1− 2α ⎜γ ⎟ ⎝ ⎠ ⎝ 2kT ⎠

The same microstructural parameters a =0.23 μm and r =10.5 nm inferred from the SEM morphological study that were found to be successful in predicting the small amplitude oscillatory shear behavior of the treated sonicated suspensions at all concentrations also are successful in predicting their steady shear behavior: when we insert a =0.23 μm and r =10.5 nm into Eq. (3.16) and fit to the experimental data of

Figure 3.6a, we deduce α = 2.50×10-11, 6.40×10-11, 6.40×10-11, 2.40×10-11 for c=5wt%,

2wt%, 1wt%, and 0.5wt%, respectively, reflecting the slight shear thinning behavior of the treated sonicated suspensions. Because these values of α are very close to zero, the predictions of the Giesekus model are essentially the same as those of the Oldroyd fluid-

B model (Figure 3.6b). Comparing Figure 3.6a and Figure 3.6b, we note that the elastic dumbbell models with a and r deduced from microstructural morphology correctly predict the magnitude of the steady shear viscosity of the treated sonicated suspensions, as well as its dependence on fiber concentration.

70 3.5.2.2 Rigid dumbbell models

The linear viscoelastic material functions G' , G'' and η' , η'' predicted from the

3r rigid dumbbell models are ( h = 0 without hydrodynamic interaction, h = with 4a hydrodynamic interaction) 129:

⎛⎞ 31nkTλ 2ω 31 nkT G'ωωηωω = '' = ⎜⎟⋅=⋅h , () () 2 ⎜⎟51−− 2hh1+ λ ω 2 51 2 ⎛⎞ ⎜⎟()()h 2 ⎛⎞2kT ⎝⎠⎜⎟11+−()h ⎜⎟ ⎜⎟πηra2 ω ⎝⎠⎝⎠s

2 ⎛⎞⎡ 13− hh⎛⎞()λω 3 1⎤ ⎜⎟⎢ h ⎥ G''='=()ωωηωωη () sh +nkTλ ⋅−⋅⎜⎟1 22 −⋅ ⋅ ⎜⎟⎢12−−hh⎜⎟ 511++λω 512 λω ⎥ ⎝⎠⎣ ⎝⎠()hh()⎦ ⎡⎤⎛⎞ ⎢⎥⎜⎟ ⎢ 13⎜⎟ 1 ⎥ (3.17) ⋅−⋅1 ⎢ ⎜⎟2 ⎥ 12− h 5 2 ⎛⎞2kT 2 ⎢ ⎜⎟11+−h ⎥ nraπηs ω ⎢ ⎜⎟()⎜⎟2 ⎥ =ηωs + ⎝⎠πηsra ω . 2 ⎢ ⎝⎠⎥ ⎢ 31hh()1− ⎥ ⎢−⋅ ⋅ ⎥ 512− h 2 2 ⎢ 1 ⎛⎞πηsra ω ⎥ 1+ 2 ⎜⎟ ⎢ 2kT ⎥ ⎣⎦⎢ ()1− h ⎝⎠⎥

As with the elastic dumbbell models, the constants of proportionality of a and r

to a f and rf are also obtained from a fit to 2wt% small amplitude oscillatory shear

experiment by selecting a and r to minimize the error:

⎛⎞' 2 ⎧⎫log10ηω exp ()i − ⎜⎟⎪⎪ ⎜⎟⎪⎪2 ⎨⎬⎡⎤⎛⎞13− hh⎛⎞λω 3 1 ⎜⎟⎢⎥⎜⎟()h ⎪⎪log10η s +nkTλh ⋅−⋅⎜⎟ 1 22 −⋅ ⋅ N ⎜⎟⎢⎥⎜⎟12−−hh⎜⎟ 511++()λω 512 () λω . (3.18) δ = ∑⎜⎟⎩⎭⎪⎪⎣⎦⎝⎠⎝⎠hh i=1 ⎜⎟2 ⎜⎟⎧⎫⎛⎞2 ⎪⎪'' ⎜⎟31 nkTλωh ⎜⎟+log⎨⎬ηω()−⋅ log ⎜⎟10 expi 10 ⎜⎟51− 2h 2 ⎜⎟⎪⎪⎜⎟1+()λωh ⎝⎠⎩⎭⎝⎠()

71 The best fit to the small amplitude oscillatory experimental data for the 2wt% suspension is obtained with a = 0.50 μm and r = 45.8 nm for rigid dumbbell model without hydrodynamic interaction and a = 0.49 μm and r = 48.2 nm for rigid dumbbell model with hydrodynamic interaction. Because the results of these two models are so close, the hydrodynamic interaction can be ignored. Compared with the representative fiber length a f = 2 μm and the radius rf = 100 nm measured from the SEM images of the treated sonicated suspensions (Figure 3.2d), we see that the model length of rigid rod

a = 0.25 a f ; the model radius of the beads r = 0.46 rf .

Applying the fiber length a = 0.50 μm, radius r = 45.8 nm and different mass concentration c to small amplitude oscillatory shear Eqs. (3.17), the rigid dumbbell models predict the elastic modulus G′ and viscous modulus G′′ of the treated sonicated suspensions shown in Figure 3.10c and Figure 3.11c. A comparison with the experimental measurements of Figure 3.10a and Figure 3.11a indicates that the rigid dumbbell models correctly predict the creation of elasticity with the addition of the nanofibers, but fail to capture the magnitude and increasing trends of elastic modulus G′ at high frequencies, and accurately predict the magnitude of viscous modulus G′′, but underpredict its dependence on nanofiber concentration.

In the simulation of steady shear flow, the rigid dumbbell models predict the following shear-rate dependent viscosities, without hydrodynamic interaction 81:

. ⎡⎤1824 1326 ηη=+s nkT λ⎢⎥1(046) −() λγ +() λγ − λγ ≤. ⎣⎦35 1925 . (3.19) -1/3 ηη≈+s 0.. 678nkT λλγ() ( λγ > 0 46)

72 with hydrodynamic interaction 81, 129:

⎧⎛⎞⎡1181326− h 24 ⎤⎫ ⎪⎜⎟⎢⎥1−+()λγhh() λγ −⎪ ⎪⎝1− 2h ⎠ ⎣ 35 1925 ⎦⎪ . ηη=+shnkT λ⎨⎬(046) λγ ≤. ⎛⎞⎛338h ⎞⎡2 ⎤ ⎪⎪ (3.20) −−+⎜⎟⎜ ⎟⎢⎥1 ()λγh ⎩⎪⎪⎝⎠⎝512− h ⎠⎣ 35 ⎦ ⎭

1− h -1/3 ⎛⎞ ηη≈+s 0.. 678nkT λhh⎜⎟() λγ ( λγ > 0 46) ⎝⎠12− h

We insert the microstructural parameters a =0.50 μm and r =45.8 nm inferred from small amplitude oscillatory shear experiment of 2wt% suspension into Eq. (3.19),

Eq. (3.20) and evaluate at the different concentrations c. The two rigid dumbbell models give essentially the same predictions which are shown in Figure 3.6c. A comparison with the experimental measurements shown in Figure 3.6a reveals that the rigid dumbbell models with and without hydrodynamic interaction are in agreement with experiment in their prediction of a nearly constant viscosity, but both of them underpredict the magnitude of viscosity at high nanofiber concentration.

Comparing Figure 3.6, Figure 3.10, and Figure 3.11, we conclude that the single- mode elastic dumbbell models based on microstructural measurements successfully model the behavior of treated sonicated suspensions over the range of nanofiber concentrations except at 5wt%, which is in the semidilute regime, whereas the rigid dumbbell models based on microstructural measurements are less successful. We surmise that the success of the dumbbell models is due to the similarity of the physical system to assumptions of the models, namely a dilute suspension of particles in a Newtonian solvent, and that the elastic dumbbell models are more successful than the rigid dumbbell models since, although the actual nanofibers are rigid in extension, they exhibit a recoverable tendency to bend or unbend that must be incorporated in the model. 73 3.5.2.3 Modeling predictions for treated unsonicated nanofiber suspensions

Unlike the treated sonicated suspensions, on which we performed small amplitude oscillatory shear and steady shear experiments at concentrations of 0.5, 1, 2, 3, 4, and

5wt%, for the treated unsonicated suspensions we performed these shear experiments only at the single concentration 1wt%. We recall that comparing the SEM images of the treated sonicated suspensions (Figure 3.2d) and the treated unsonicated suspensions

(Figure 3.2c), the treated unsonicated suspension has longer fibers and a number small clusters. The characteristic length of the treated unsonicated fibers required for a single- mode model is five times longer than that of the treated sonicated fibers, while the fiber radius is unchanged by the sonication. Based on this observation, we insert

aaunsonicated= 5 sonicated , rrunsonicated= sonicated into the response functions G′ ()ω , G′′ ()ω , and

η()γ predicted by the six dumbbell models (Eqs. (3.12)-(3.13), (3.15)-(3.17), and (3.19)-

(3.20)) at 1wt% fiber concentration. We ignore the presence of the small clusters. Figure

3.7b and Figure 3.7c display the predictions of steady shear viscosity given by the elastic and rigid dumbbell models, respectively, for both the treated sonicated and treated unsonicated suspensions. The mobility factor α of the Giesekus model is close to zero (α

= 9.00×10-12 for c = 1wt%) so that the effect of the anisotropic hydrodynamic drag is small. Therefore the predictions of elastic dumbbell models with isotropic hydrodynamic drag (Oldroyd fluid-B model) are the same as those of the elastic dumbbell models with anisotropic hydrodynamic drag (Giesekus model). The ratio of particle radius to length

r ( h ==0. 008 ) is small, so that the effect of hydrodynamic interaction could be ignored. a

Therefore, the predictions of the four elastic dumbbell models with or without

74 hydrodynamic interaction are same and the predictions of the two rigid dumbbell models with or without hydrodynamic interaction are also essentially the same. Figure 3.7b shows that the elastic dumbbell models for the treated unsonicated suspension overpredict the difference of viscosity between the two suspensions. Figure 3.7c shows the prediction of the rigid dumbbell models with or without hydrodynamic interaction is in agreement with the experiment. We caution that this behavior is for one experimental point at relatively low nanofiber concentration.

3.5.2.4 Modeling predictions for untreated sonicated nanofiber suspensions

Recall our morphological study of the untreated sonicated suspensions revealed the presence of clumps of entangled fibers. A theory of aggregated structures which accounts for both disperse fibers and fiber clumps in a Newtonian solvent would be desirable to model the untreated sonicated nanofiber suspensions, but at present, a theory taking into account the different length scales of the disperse fibers and aggregates does not exist. In the absence of such a theory, we instead introduce the concept of “effective length” into the dumbbell models, combing the disperse fibers and fiber clumps into a single length scale.

The effective spring length auntreated = 21. 7 μm of the Oldroyd fluid-B model for the untreated sonicated suspensions is obtained by keeping r =10. 5 nm unchanged and

selecting auntreated to minimize the error

N 2 ⎡ ⎤ δ =−+∑ ⎣log10ηγ exp() log 10 ( ηs nkT λ )⎦ . (3.21) i=1

We insert this effective length into Eq. (3.15) to obtain the steady shear viscosity predicted by the Oldroyd fluid-B model for the untreated sonicated suspensions, shown in

75 Figure 3.4b; the Oldroyd fluid-B model cannot capture the shear thinning behavior of the

suspensions. The effective spring length auntreated = 345 μm of the Giesekus model for the untreated sonicated suspensions is obtained by keeping r =10. 5 nm unchanged and minimizing the error

2 2 N ⎡⎤⎛⎞()1−ψ δηγηλ=−+⎢⎥log() log ⎜⎟nkT . (3.22) ∑ 10 exp 10 ⎜⎟s 112+−αψ i=1 ⎣⎦⎢⎥⎝⎠()

Inserting this effective length into Eq. (3.16) produces the prediction of steady shear viscosity of the untreated sonicated suspensions shown in Figure 3.4c. Comparing with the experimental measurements (Figure 3.4a), we see that despite its debatable physical basis, the effective length modeling approach in the Giesekus model (elastic dumbbells with anisotropic drag) captures the steady shear behavior of the untreated sonicated suspensions: the orders of magnitude of steady viscosity predicted by the

Giesekus model are consistent with the experimental data, the Giesekus model correctly predicts that the viscosity of the suspensions increases with the nanofiber concentration and it captures the significant shear thinning behavior (α = 1.82×10-9, 4.89×10-9, 2.13×10-

8, 4.51×10-8, 4.23×10-7, and 9.00×10-7 for c=5wt%, 4wt%, 3wt%, 2wt%, 1wt%, and

0.5wt%, respectively). Comparing Figure 3.5a, Figure 3.5b, and Figure 3.5c, we note that the Giesekus model incorporating effective length predicts the correct dependence of viscosity on shear stress, except at low shear rates. In the above elastic dumbbell models,

r the ratio of particle radius to length is so small that the influence of the hydrodynamic a interaction on the relaxation time is miniscule.

76 The effective rigid rod length auntreated = 33. 49 m for the rigid dumbbell model of

untreated sonicated suspensions is obtained by keeping runsonicated = 45. 8 nm unchanged

and selecting auntreated to minimize the error:

2 ⎡⎤log10ηγ exp () − ⎢⎥ ⎢⎥⎛⎞⎧⎛⎞⎡1181326− h 24 ⎤⎫ N 1−+λγ λγ − ⎢⎥⎜⎟⎪⎜⎟⎢⎥()hh() ⎪ δλγ=≤∑ ⎜⎟⎪⎝1− 2h ⎠ ⎣ 35 1925 ⎦⎪ ( 0. 46), i=1 ⎢⎥log ηλ+ nkT (3.23) 10 ⎜⎟sh⎨⎬ ⎢⎥⎛⎞⎛338h ⎞⎡2 ⎤ ⎜⎟⎪⎪−−+1 λγ ⎢⎥⎜⎟⎪⎪⎜⎟⎜ ⎟⎢⎥()h ⎣⎦⎢⎥⎝⎠⎩ ⎝⎠⎝512− h ⎠⎣ 35 ⎦ ⎭ N 2 δηγη=−+⎡⎤log log 0. 678nkTλλγ ( )-1/3 ( λγ> 0.. 46) ∑ ⎣⎦10 exp() 10 ()s i=1

We apply this effective length into Eq. (3.19) or Eq. (3.20) to obtain the rigid dumbbell model prediction of the steady shear viscosity of the untreated sonicated suspensions, shown in Figure 3.4d. We note that the model captures the steady shear behavior of the untreated sonicated suspensions qualitatively, predicting the shear thinning behavior of the suspensions and the dependence of viscosity on the mass concentration but failing to predict the magnitude of the steady shear viscosity at high mass concentration.

Recall from Eqs. (3.11) that in the microstructurally based forms of the dumbbell

models the relaxation time λ and the particle contribution to the viscosity η p are explicit functions of temperature; no calibration or fitting is allowed aside from experimentally measuring solvent viscosity as a function of temperature. We insert T =5°C, 15°C, 25°C,

35°C, and 45°C and c =2wt% into Eq. (3.16), and note that the model accurately captures the measured temperature dependence of the untreated sonicated suspensions in shear

(Figure 3.12).

77 All six dumbbell models with the effective length concept fail to capture the observed behavior of the untreated sonicated suspensions in small amplitude oscillatory shear.

3.6 Electrical Conductivity Measurement

It would be ideal to measure the electrical conductivity of the suspensions, so that the network structure of the CNFs can be further detected. The measurement was attempted but no reliable results were obtained. More details about the measurements can be found in Appendix A.

3.7 Conclusions

The morphology and rheology of carbon nanofiber suspensions were studied. The as-received nanofibers contained millimeter and centimeter size agglomerates, with most nanofibers entangled with each other. The diameter was rather uniform, in the range 100-

200 nm, and the length was estimated to be from 5 to 20 μm. Dispersion of the as- received nanofibers was difficult. Sonication could only disperse part of the nanofiber agglomerates, and many nanofiber clumps of the size of 20 by 50 μm remained in the suspensions. Surface treatment with a concentrated acid mixture improved overall dispersion greatly by opening the nanofiber clumps and possibly shortening some fibers.

The nanofibers, weakened by the treatment, were shortened by subsequent sonication to less than 1 μm in length.

The untreated sonicated suspensions exhibited extreme shear thinning behavior and the viscosities increased monotonically with fiber loading. The untreated suspensions possessed apparent yield stress at the nanofiber loading above 2wt%. We suggest that this is caused by the network structures formed from the partially dispersed nanofibers and

78 nanofiber clumps. The treated sonicated and treated unsonicated suspensions exhibited a nearly constant shear viscosity as a function of shear rate and the viscosities gradually increased with fiber concentration. At 1wt% fiber concentration, the treated sonicated and treated unsonicated suspensions had about the same viscosities, although the average length of the nanofibers was different. When the Krieger-Dougherty model was applied to the suspensions, good fits were obtained for viscosities at high shear rates. However, the maximum packing coefficient, φm, was small compared to that for spherical particles.

This is believed to be the effects of particle aspect ratio and of interactions between particles. Small amplitude shear studies showed that for the untreated sonicated suspensions there was a critical concentration of 3wt% above which the suspensions showed solidlike behavior. G′ was greater than G′′ at all frequencies studied and a low- frequency plateau was observed. This could be attributed to the continuous 3-D network structures between the nanofibers. The appearance of low-frequency plateau in G′ in the treated sonicated suspensions, which do not have network structures, is the result of the colloidal forces between the treated nanofibers.

Study of temperature effects on the rheology of the 2wt% untreated sonicated suspension showed that the suspension became more solidlike (G′/ G′′ increased) at higher temperatures although the overall shear viscosity decreased. This is because as temperature rises, the viscosity of the solvent decreases greatly while the interaction between the nanofibers remains about the same, inducing a solid-like behavior. At high shear rates, the shear viscosity was dominated by the solvent that follows the Arrhenius

79 law. The extensional viscosity of 1wt% untreated sonicated suspensions was observed to be extension rate thinning, with a decreasing Trouton ratio due to the breakup of the network structure and nanofiber alignment to the flow.

We investigated the ability of elastic and rigid dumbbell models to describe the rheological behavior of the aqueous nanofiber suspensions. We retained the microstructural basis of the elastic dumbbell models and rigid dumbbell models by inferring the connector length and bead radius in the models from SEM measurements of the suspensions, and from these and measurements of densities, concentration, and solvent viscosity, computed the relaxation time and particle viscosity in the models. We found that the elastic dumbbell models successfully capture both the small amplitude oscillatory shear and steady shear behavior of the treated sonicated suspensions, while the rigid dumbbell models fail to capture the increasing trends of the elastic modulus G′ at high frequency. We conjecture that the success of the dumbbell models is due to the similarity of the physical system to assumptions of the models, and that the elastic dumbbell models are more successful than the rigid dumbbell models since, although the actual nanofibers are rigid in extension, they exhibit a recoverable tendency to bend or unbend that must be incorporated in the model. Because of the small ratio of radius to

r length of the carbon nanofibers, the predictions of the elastic and rigid dumbbell a models with or without hydrodynamic interaction are indistinguishable.

80 The elastic dumbbell models and the rigid dumbbell models based on SEM measurements of microstructure for treated unsonicated suspensions which ignore the small clumps present in these suspensions overpredict the difference of the treated unsonicated suspension’s viscosity and that of the treated sonicated suspension at the one concentration for which we have an experimental comparison.

The elastic dumbbell models with anisotropic hydrodynamic drag with or without hydrodynamic interaction correctly predict the observed steady shear behavior of the untreated sonicated suspensions if we introduce the concept of an effective dumbbell length to account for the presence of large agglomerates. The rigid dumbbell models predict the shear thinning behavior of the untreated sonicated suspensions but fail to predict the magnitude of the steady shear viscosity. The elastic dumbbell model with isotropic hydrodynamic drag incorporating effective length fails to predict the strongly shear thinning behavior of the untreated sonicated suspensions. All six dumbbell models fail to capture the observed small amplitude oscillatory shear behavior of the untreated sonicated suspensions.

81

(a) (b)

(c) (d)

Figure 3.1. SEM micrographs of (a) as-received, (b) untreated sonicated, (c) treated unsonicated, and (d) treated sonicated carbon nanofibers (scale bar: 50 μm). 82

(a) (b)

(c) (d)

Figure 3.2. SEM micrographs of (a) as-received, (b) untreated sonicated, (c) treated unsonicated, and (d) treated sonicated carbon nanofibers (scale bar: 10 μm). 83

(a) (b)

Figure 3.3. Optical microscopy photos of (a) 0.5wt% untreated sonicated and (b) 0.5wt% treated sonicated carbon nanofiber suspensions (scale bar: 100 μm).

84 103 103

5wt% 4wt% 5wt% 3wt% 2wt% 2wt% 102 102 1wt% 1wt% 0.5wt% 0.5wt% 0 wt% 0wt%

101 101 Viscosity (Pa.s) Viscosity (Pa.s) Viscosity

100 100

10-1 10-1 10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103 Shear rate (s-1) Shear rate (s-1)

103 103 5wt% 5wt% 4wt% 4wt% 3wt% 3wt% 2wt% 2 2wt% 102 10 1wt% 1wt% 0.5wt% 0.5wt% 0wt% 0wt%

101 101 Viscosity (Pa.s) Viscosity

Viscosity (Pa.s)Viscosity 100 100

10-1 10-1 10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103 Shear rate (s-1) Shear rate (s-1)

Figure 3.4. Steady shear viscosity of untreated sonicated suspensions: (a) experimental measurements, (b) predictions of elastic dumbbell models with isotropic hydrodynamic drag with or without hydrodynamic interaction, (c) predictions of elastic dumbbell models with anisotropic hydrodynamic drag with or without hydrodynamic interaction, (d) predictions of rigid dumbbell models with isotropic hydrodynamic drag with or without hydrodynamic interaction. 85

103 103 5wt% 5 wt% 4wt% 4 wt% 3wt% 3 wt% 2wt% 102 102 2 wt% 1wt% 1 wt% 0.5wt% 0.5 wt% solvent 0 wt%

101 101 Viscosity (Pa.s) Viscosity Viscosity (Pa.s) Viscosity 100 100

10-1 10-1 10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103 Shear stress (Pa) Shear stress (Pa)

103 5 wt% 4 wt% 3 wt% 102 2 wt% 1 wt% 0.5 wt% 0 wt%

101

Viscosity (Pa.s) Viscosity

100

10-1 10-2 10-1 100 101 102 103 Shear stress (Pa)

Figure 3.5. Shear viscosity as a function of shear stress of the untreated sonicated suspensions: (a) experimental measurements, (b) predictions of elastic dumbbell models with anisotropic hydrodynamic drag with or without hydrodynamic interaction, (c) predictions of rigid dumbbell models with isotropic hydrodynamic drag with or without hydrodynamic interaction.

86 100 100

5wt% 5wt% 2wt% 2wt% 1wt% 1wt% 0.5wt% 0.5wt% 0wt% 0wt%

Viscosity (Pa.s) Viscosity (Pa.s) Viscosity

10-1 10-1 10-1 100 101 102 103 10-1 100 101 102 103 Shear rate (s-1) Shear rate (s-1)

100 5wt% 2wt% 1wt% 0.5wt% 0 wt%

Viscosity (Pa.s) Viscosity

10-1 10-1 100 101 102 103

-1 Shear rate (s )

Figure 3.6. Steady shear viscosity of treated sonicated suspensions: (a) experimental measurements, (b) elastic dumbbell model predictions, (c) rigid dumbbell model predictions.

87

100 100

Viscosity (Pa.s) Viscosity (Pa.s) Viscosity

Unsonicated 1wt% treated unsonicated Sonicated 1wt% treated sonicated 10-1 10-1 -1 0 1 2 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 Shear rate (s-1) Shear rate (s-1)

10 0

1 wt% treated sonicated 1wt% treated unsonicated

Viscosity (Pa.s) Viscosity

10-1 100 101 102 103 Shear rate (s-1)

Figure 3.7. Steady shear viscosity of treated unsonicated suspensions: (a) experimental measurements, (b) elastic dumbbell model prediction, (c) rigid dumbbell model predictions.

88

8 treated sonicated 7 untreated sonicated

) modelstreatedmodel K.D. s 6 η Bachelor / η untreated KD 5

4 Krieger- Dougherty (1959) 3

2 Relative viscosity ( viscosity Relative

1 Einstein & Bachelor (1977)

0 0 0.010.020.030.040.05 Volume fraction (φ)

Figure 3.8. Relative viscosity of treated sonicated and untreated sonicated suspensions as a function of fiber volume fraction.

89 10 3 103

10 2 102

10 1

101 10 0 G' (Pa.s)

G" (Pa.s) 100 10-1 5wt% 5wt% 4wt% 4wt% 3wt% 3wt% -2 10 2wt% 10-1 2wt% 1wt% 1wt% 0.5wt% 0.5wt% 10 -3 10-1 100 101 102 103 10-2 10-1 100 101 102 103 Frequency (rad/s) Frequency (rad/s)

Figure 3.9. Linear viscoelasticity of untreated sonicated nanofiber suspensions.

90

1 101 10

0 100 10

-1 10-1 10 G' (Pa) G' (Pa) 10-2 10-2 5wt% 5wt% 4wt% 4wt% 3wt% -3 -3 10 3wt% 10 2wt% 2wt% 1wt% 1wt% 0.5wt% 0.5wt% 10-4 10-4 0 1 2 100 101 102 10 10 10 Frequency (rad/s) Frequency (rad/s)

10 1 5wt% 4wt% 3wt% 0 10 2wt% 1wt% 0.5 wt% 10-1

G' (Pa) 10-2

10 -3

10 -4 10-1 100 101 102 Frequency (rad/s)

Figure 3.10. Elastic modulus G′ of treated sonicated suspensions: (a) experimental measurements, (b) elastic dumbbell model predictions, (c) rigid dumbbell predictions.

91

2 2 10 10

101 101

100 100

G" (Pa) G" (Pa.s)

5wt% 5wt% 4wt% 4wt% 10-1 3wt% 10-1 3wt% 2wt% 2wt% 1wt% 1wt% 0.5wt% 0.5wt%

10-2 10-2 100 101 102 100 101 102 Frequency (rad/s) Frequency (rad/s)

102

101

100

G" (Pa)

5wt% 4wt% 10-1 3wt% 2wt% 1wt% 0.5wt%

10-2 10 -1 100 101 102 103 Frequency (rad/s)

Figure 3.11. Viscous modulus G" of treated sonicated suspensions: (a) experimental measurements, (b) elastic dumbbell model predictions, (c) rigid dumbbell model predictions. 92

103 103 5 oC 5 oC 15 oC 15 oC 2 2 o 10 o 10 25 C 25 C 35 oC 35 oC o o 45 C 45 C 101 101

100 100 Viscosity (Pa.s) Viscosity (Pa.s) Viscosity

10-1 10-1

10-2 10-2 10 -2 10-1 100 101 102 103 10-2 10-1 100 101 102 103 Shear rate (s-1) Shear rate (s-1)

103 103 o 5 C 5 oC o 15 C 15 oC o 25 C 25 oC o o )

) 35 C s 35 C s

o η o 2 45 C / 2 45 C

η 10 η/η 10 ty ( ty i os

1 101 10 Relative viscosity ( viscosity Relative Relative visc Relative

0 100 10 -2 -1 0 1 2 3 10-2 10-1 100 101 102 103 10 10 10 10 10 10 -1 Shear rate (s-1) Shear rate (s )

Figure 3.12. Effects of temperature on 2wt% untreated sonicated suspensions: (a) experimental measurements of the steady shear viscosity, (b) elastic dumbbell model predictions of the steady shear viscosity, (c) experimental measurements of the relative viscosity, (d) elastic dumbbell model predictions of the relative viscosity.

93

103 103

5 oC 5 oC 15 oC 15 oC 2 2 10 25 oC 10 25 oC 35 oC 35 oC 45 oC 45 oC 101 101

100 100

Viscosity (Pa.s) Viscosity (Pa.s) Viscosity

10-1 10-1

10-2 10-2 10-1 100 101 102 103 10-1 100 101 102 103 Shear stress (Pa) Shear stress (Pa)

101

10

1

100

G"/G' solvent 0.1 suspension 8 oC o 16 C Viscosity (at 200 1/s) (Pa.s) 24 oC 33 oC 42 oC 0.01 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037 10-1 1/Temp (1/K) 10-1 100 101 102 103 Frequency (rad/s)

Figure 3.13. Effects of temperature on 2wt% untreated sonicated suspensions: (a) experimental measurements of the apparent yield stress, (b) elastic dumbbell model predictions of the apparent yield stress, (c) Arrhenius plot, (d) linear viscoelasticity. 94

103 fresh (2mm) 13hr stretch (2mm) 13hr stretch (1mm) 2hr stretch (1mm) 102 shear viscosity

101

Viscosity (Pa.s) Viscosity

100

10-1 10-2 10-1 100 101 102 103 104 105

Extension/shear rate (s-1)

Figure 3.14. Extensional and shear viscosity of 1wt% untreated sonicated suspension. Fresh sample was measured using 2mm jets, and then the sample was stretched at 600 s-1 for 13 hours. After that, the sample was measured again using 2mm jets followed by measured by 1mm jets. Then the sample was stretched at 2000 s-1 for another 2 hours. It was measured again using 1mm jets. Temperature 28 °C.

95

80

70 Standard 60 Jones (1987)

50

40

Trouton ratio 30

20

10

0

100 101 102 103 Shear rate (s-1)

Tr =η (ε) /η(γ) Figure 3.15. Trouton ratio, e , of 2wt% untreated sonicated suspension calculated by the standard method (γ =ε )and Jones method (γ = 3ε ).

96

103

η* (5wt%) η (5wt%) η* (1wt%) 102 η (1wt%)

(Pa.s) 1 10

η∗

η,

100

10-1 10-2 10-1 100 101 102 103 -1 Shear rate (s ), frequency (rad/s)

Figure 3.16. Comparison of complex viscosities (η*) and steady shear viscosities (η) of 1wt% and 5wt% untreated sonicated nanofiber suspensions. At 1wt% the Cox-Merz rule holds.

97 CHAPTER 4

DYNAMIC AND STEADY SHEAR OF POLYSTYRENE/CARBON NANOFIBER

COMPOSITES

4.1 Introduction

Graphitic carbon nanofilaments have been of great research interest since the discovery of the single wall carbon nanotube (SWNT) in 1991 1. Due to their extraordinary mechanical, electrical, and thermal properties, the SWNTs seem to have promising future serving as reinforcement fillers in composite systems, or improving the thermal and electrical conductivities of the composite. Experimental results so far have confirmed enhanced performance in these applications. However, due to the high cost of production, the price of SWNTs is often prohibitive.

Carbon nanofiber (CNF) is now available as a low cost alternative to SWNTs.

CNFs are roughly 100 times larger than SWNTs in both diameter and length, and are available at 1/500 the cost. These CNFs are typically made by decomposing organic vapor in the presence of metal catalyst (e.g. iron) at high temperature 132-137. Due to their excellent thermal138, electrical139, and mechanical properties140, CNFs have been incorporated into a variety of thermoplastic and thermoset polymers to make polymer

98 nanocomposites. These thermoplastic polymers include polycarbonate141-144, polyethylene

145, polypropylene2, 9, 142, 146-151, poly(ethylene terephthalate) 152, 153, nylon 6 147, nylon 66

148, 151, and polyphenylene sulfide 147, 154. For thermoset matrix, epoxy has been mostly used 147, 154-158. Electrical and thermal conductivities of the polymer/CNF composites increase markedly with only a small amount of CNFs (less than 10wt%) 147. Applications of polymer/CNF composites include electrostatic dissipation (ESD) and electromagnetic interference (EMI) shielding. Mechanical strength also increases with the addition of

CNFs into the polymer, which could lead to smaller and lighter parts in a variety of applications 148.

However, the enhancement of performance properties by the addition of CNFs to polymers is accompanied by changes, often negative, in processing properties. Because of these changes, equipment used to process pure polymers may not be capable of processing the same polymers filled with CNFs. Rheological studies of the polymer/CNF composites must be performed in order to evaluate the effect of carbon nanofibers on processing. Compared to the above abundant literature on the properties of the solidified polymer/CNF composites, far less can be found on the rheological properties of these composite systems in their molten processing phase 9, 145, 149, 159. Linear viscoelasticity and steady shear viscosity of composites with different polymer matrices have been measured, but none explicitly relate rheological measurements to morphology of the composites.

This paper addresses these deficiencies: The results of this work include the preparation of polymer/CNF composites with uniform dispersion by two different techniques (melt blending and solvent casting), comparing the damage done to the CNFs

99 by these processing techniques to the as-received CNFs, investigation of the effect of dispersion and size of CNFs on the composite rheology, construction of mathematical models that deduce the rheology of the melt composites, and connect this rheology and process conditions to the development of CNF nanostructure, providing a tool for predicting enhanced performance properties of parts with process flow induced CNF orientation.

4.2 Experimental procedures

4.2.1 Materials

The polystyrene used in this study is Fina (CX5197) produced by Atofina. It has a

3 density of 1000g/m , Mw of approximately 200,000g/mol, and polydispersity index of

2.4. Polystyrene was chosen as our thermoplastic polymer matrix because its rheology has been well characterized, so that any deviation caused by the addition of the CNFs can be readily identified. Further, the molecular structure of polystyrene is simple and no crystalline structure is present in solid phase, so that the nanostructure of the CNFs developed during flow in its molten phase can be preserved when cooled to its solid phase.

The carbon nanofibers used in this study are Pyrograf® III (type PR-24-PS) made by Applied Sciences, Inc. They are produced by decomposing organic vapors at elevated temperature in the presence of metal catalysts. These nanofibers have a tree-ring structure with a hollow core. The nanofibers we obtained were in powdered form consisting largely of fiber agglomerates. Large agglomerates of a centimeter in diameter were

100 observed, but most of the agglomerates were less than 1 mm in diameter. The as-received

CNFs in these agglomerates were characterized using scanning electron microscopy to have diameters of 100 to 200 nm and lengths of 1 to 50 microns.

4.2.2 Sample preparation

In this study, two methods have been used to prepare CNF/PS composites: melt blending and solvent casting. The composites made by melt blending are referred to here as MB composites, and composites made by solvent casting as SC composites.

In the melt blending process, CNF powder and PS pellets are fed into a DACA microcompounder, a small twin-screw extruder in which the compounding time, temperature and screw speed can be controlled. Typically five minutes compounding time at 180°C and 150 rpm achieves the best CNF dispersion with the least polymer degradation (i.e. breaking of polymer chains). Longer compounding time does not generate better CNF dispersion, but can lead to more polymer degradation. The composite is extruded through a 1mm die and then cut into pellets with approximate length of 2mm. Composites containing 2, 5, and 10wt% CNF were prepared using the melt blending process for this study.

In the solvent casting process, PS pellets are dissolved in tetrohydrofuran (THF), and the CNFs are added. The resulting 90% THF suspension is then sonicated for 0.5 hour using a Sonic Dismembrator (Fisher Scientific) at 20 kHz and power level of 760 watts per liter. After sonication, the suspensions are heated to drive off most of the solvent. The residual material, still containing about 5wt% THF solvent, is broken into powder using a blender and then further dried in a vacuum oven at 80°C for 7 days. The dried powder is heated to 200°C and compressed through a 1mm diameter die, and the 101 resulting rod is cut into pellets with approximate length of 2mm. This process prevents the bubbles that would otherwise develop in the compression molding process to make disks, described below. For this study, composites containing 2, 5, and 10wt% CNF were prepared using the solvent casting method.

Both the MB and SC composite pellets were compression molded into disks of

25mm diameter and 0.6-1mm thickness using a hot press at 190°C. These disks served as specimens for the rheometers of section C employed to characterize the composite’s melt behavior. Although the CNFs in the pellets were oriented in the flow direction through the 1mm die, the orientation of the pellets in the compressed disks was random. The disks were stored in a vacuum oven at 70°C prior to the measurement, to prevent the absorption of moisture or air; if stored in open air, bubbles are generated during the re- melting in the rheometer.

The thermal processing in both melt blending and solvent casting causes thermal degradation to the polymer matrix. In order to maintain repeatability and allow for comparison, special care has been taken to minimize the degradation for both MB and SC composite disks. Pure PS was processed using both melt blending and solvent casting methods to serve as a control sample.

Shear rheology of the melt composites was measured using a strain controlled rheometer from TA Instruments (ARES LS2) with both torque transducer (0.02g.cm to

2000g.cm) and normal force transducer (2g to 2000g). Parallel plates with 25mm in diameter were used for all measurements. The gap distance was kept in the range 0.5-

102 0.9mm for all tests. The molded disks were allowed to rest at the measure temperature for

15 min to relax the residue stress introduced by the compression molding process. Unless otherwise stated, all tests were conducted at 200°C.

A transmission electron microscope (Philips CM 12 at 120 kV) was used to observe the nanometer-scale CNF dispersion and structure in the composites, and a scanning electron microscope (Philips XL 30 equipped with secondary and back-scatter detectors, operated at 30 kV) was used to characterize the as-received CNFs. An optical microscope (Olympus BH-2) was used to study the CNF dispersion at a larger scale. The glass transition temperatures (Tg) of all the composites were measured using a

Differential Scanning Calorimetry (TA Instrument 2920).

4.3 Experimental Results

In this section, we present and relate our measurements of bulk (i.e. macroscale) small amplitude oscillatory shear, steady state shear, and transient shear rheology of the

CNF/PS composites, and our nanoscale measurements of CNF morphology and dispersion in the polymer matrix. In section IV these measurements are used to benchmark and validate a nanostructurally based thermo-mechanical model for the melt composites.

4.3.1 Morphology and Dispersion of CNFs in the Polymer Matrix

The morphology and dispersion of CNF in composites prepared by the two methods has been characterized using both optical microscopy and transmission electron microscopy (TEM). The combination of complementary length scales of these two studies results in comprehensive measurements, with some features accessible through

TEM, and others through the optical microscope.

103 The CNFs used in this study have diameters ranging from 100 to 200 nanometers.

Since the smallest object that can be observed by optical microscopes is about 200nm, determined by the wavelength of the visible light, only those CNFs in the largest range of diameters can be observed with an optical microscope. In contrast, TEM can capture all of the CNFs, from largest to smallest, and hence TEM images are used for the measurement of the nanoscale dispersion and structure in the composites. However, TEM requires thin slices of sample, with typical thicknesses of the order of 100nm, depending on the capability of the TEM. With our system the maximum thickness was found be

800nm, beyond which the images are not clear. With even these thickest slices, most of the CNFs in the TEM samples will be cut during the sectioning, especially the longer fibers. Thus, TEM cannot capture the true length of the CNFs. Therefore measurement of the length of the CNFs after being processed through either melt blending or solvent casting is accomplished with optical microcopy; the CNF/PS composite is dissolved in the solvent THF, with composite to solvent ratio of about 8:100 by weight, and a thin film of the resulting THF/CNF/PS suspension is viewed through an optical microscope.

Figure 4.1 shows photographs through an optical microscope of 5wt% MB and SC composites suspended in THF. We observe in these images, and many others not shown of MB and SC composites in all CNF concentrations, that the CNFs in MB composites have lengths in the range 1-20 micron, while in SC composites the range is 4-60 microns.

The diameters of the CNFs in both MB and SC composites were measured in TEM images to be in the same 100 – 200 nm range measured in the as-received fibers.

Combining the length and diameter measurements, we find that a representative aspect ratio for the MB composites is 40, and for the SC composites is 160.

104 From the TEM images of sections of solidified MB and SC composites shown in

Figure 4.2 and many others in all CNF concentrations, we learn that in the micron scale the CNFs are uniformly dispersed in MB composites, while there are micron-scale regions almost devoid of CNFs in SC composites, mostly due to CNF settling during the drying process; on a millimeter scale, the SC composites exhibit good uniformity.

Summarizing, our optical and TEM images of the CNF/PS composites reveal that the CNFs are well dispersed in the composites created by either melt blending or solvent casting, at least to millimeter scale, compared with the large, tightly packed CNF agglomerates in the as-received form. However, it is noticeable from both TEM and optical micrographs that the CNFs in MB composites are shorter than that in SC composites. It is likely that the CNFs in melt blended composites were severely sheared during compounding, achieving uniform dispersion in the micron scale at the cost of damage to the CNFs. In contrast, the sonication in the solvent casting process successfully dispersed, at least to the millimeter scale, most of the agglomerates in the as- received CNFs, while largely preserving the original length of the CNFs.

Fiber suspensions can be divided into three concentration regimes 160 based on the number of fibers per unit volume n, fiber length L, and fiber diameter D. The suspension is said to be in dilute when the nL3 <<.1. In this regime, the fibers can rotate freely without touching the neighboring fibers. When 1 << nL3 << L/D, the suspension is in the semi-dilute regime, in which the fibers collide other fibers during rotation. For nL3>>

L/D the suspension is in the concentrated regime, with frequent physical contact between fibers. Table 4.1 summarizes the measurements of our TEM and optical microscopy studies, which include all of the properties necessary to determine the concentration

105 regime for each composite. In our labeling scheme, MB2 is the 2wt% melt blended composite, SC5 is the 5wt% solvent cast composite, etc. Using our measurements of composite systems and their CNF and PS constituents, we compute that the melt blended composite MB2 with CNF concentration of 2wt% is in the semi-dilute regime, and the

MB composites with higher CNF concentrations, and all SC composites, are in the concentrated regime. In particular, the SC5 and SC10 composites have much higher nL3 values, leading us to expect that interparticle forces will play a significant role on their rheological behavior.

4.3.2 Linear Viscoelasticity.

In our linear viscoelastic characterization of the response of the CNF/PS melt composites to small amplitude oscillatory shear, we first determined the boundaries of the linear regime over which elastic modulus G’ and viscous modulus G” are independent of amplitude by running strain amplitude sweeps from 0.01% to 10% at different frequencies on each sample. We found that the boundary of the linear regime is 1% strain for the MB2, MB5, and SC2 composites, and 0.1% strain for the MB10, SC5, and SC10 composites, so that higher CNF concentration and/or longer CNF lengths reduces the linear domain of viscoelasticity. This may be evidence that the structure formed by the

CNFs is weak: even a small strain may alter the structure and make the response non- linear.

Figure 4.3 shows the master curves of elastic modulus G’ of the MB and SC composites, as well as samples of pure polystyrene which have undergone the same mixing process as the melt blended composites (denoted MB0) and the solvent casting process (denoted SC0). The small amplitude oscillatory shear experiments were

106 performed over the temperature range of 140 – 220°C. The master curves were generated by shifting the measured data to 200°C following the principles of time-temperature superposition; the shift factors (aT) of all the MB and SC composites as well as the pure polymer control samples are plotted in Figure 4.4. Our measurements show marked effects of the CNFs on the linear viscoelastic moduli, especially at the low frequencies

(see Figure 4.3 and Figure 4.4). For both melt blended and solvent cast composites, the elastic modulus G’ increases monotonically with increasing CNF concentration. The amount of increase in the SC composites is larger than that in the MB composites for a given CNF concentration, an effect of aspect ratio of the CNFs.

At low frequencies, homogeneous polymers typically exhibit the terminal behavior of G’ ∝ω2 and G” ∝ω , so that the slopes of log G’ vs. log ω and log G” vs. log

ω at low frequencies are near 2 and 1, respectively. Table 4.2 lists the low-frequency slopes of d(logG’)/d(logω) and d(logG”)/d(logω) deduced from our small amplitude oscillatory shear measurements on the MB and SC composites and pure PS samples. Both the pure polymer sample that underwent the melt blending process and the pure polymer sample that underwent solvent casting process exhibit the terminal behavior typical of homogeneous polymers, with slopes near 2 and 1, but both the melt blended and solvent cast CNF/PS melt composites do not, with deviation of the slopes from 2 and 1 increasing with CNF concentration. Further, at any CNF concentration the slopes of G’ and G” for

SC composites are smaller than those of MB composites, indicating that the longer fibers in the SC composites have a greater influence on the viscoelasticity of the composites.

107 Non-terminal behavior is an indication of solid-like, elastic response, with a long relaxation time, and is also typical of physical gelation as described by other researchers:

Larson et al 113 studied smectic liquid crystals and suggested that non-terminal low- frequency rheological behavior is due to long-range domain structure and the presence of defects. This shows that the addition of rod-like particles is enough to produce non- terminal behavior, without the requirement of entanglement. However, Krishnamoorti 161 showed that typical terminal behavior is observed in poly(dimethylsiloxane) clay composites in which the ends of polymer chains are not chemically bonded to the clay surface,. In later work, Krishnamoorti and Giannelis 111 argued that in end-tethered polymer layered silicate nanocomposites, the inclusion of silicate layers and the lack of complete relaxation of the confined polymer chains cause the solid-like response at low frequencies. Solomon’s results 110 from the polypropylene/clay hybrid materials suggested that the low frequency plateau was the result of non-Brownian network structure formed by the individual clay platelets. Pötschke et al 35 studied multi-walled carbon nanotube (MWNT)/polycarbonate systems and found that a low-frequency plateau in G’ occurs when the MWNT loading is 2wt% or higher. They suggested that the

MWNTs form interconnected structures and dominate the rheological behavior at low frequencies. Liu et al 162 further analyzed Pötschke’s data 35 and showed a gellation point of about 1.6wt% for their composites.

It is likely that during flow of our CNF/PS melt composites (especially those in the concentrated regime; see Table 4.1) the CNFs interact with each other. We conjecture, based on the above observations from our linear viscoelastic characterizations of the melt composites and control samples of PS that the shorter CNFs in MB

108 composites interact with each other is less than the longer CNFs in the SC composites.

Continuing this conjecture, the shortness of the CNFs in the MB melt composites renders them essentially rigid rods that can be confined by their neighboring CNFs such that they cannot rotate freely, but for which fiber entanglement is almost impossible. In contrast, the much longer CNFs in the SC composites can more readily entangle with each other to form network structure based on physical contact. At the higher CNF concentrations and lower frequencies these structures respond at longer time scales than the composites without CNF structure, producing solid-like behavior. As a result, decrease in the slope of

G’ and G” of SC composites is more dramatic than that of MB composites. We discovered similar non-terminal behavior is for CNFs suspended in glycerol/water solvents 163. In those systems the Newtonian solvent does not contribute any viscoelastic behavior, and the non-terminal behavior is generated entirely by interactions between the

CNFs.

The linear viscoelastic response of the CNF/PS melts was measured at the temperature range of 140°C to 220°C, and time-temperature superposition was used to generate master curves in Figure 4.3. It is found that the temperature shift factors of

Figure 4.4are independent of the CNF concentration and length, and the same as those of the pure polystyrene. The theoretical foundation of time-temperature superposition is that the relaxation times of the polymer chains have the same temperature dependence 164. In the light of this statement, and noting that the addition of CNFs to the polystyrene does not alter the shift factors, it seems that the mobility of the polymer chains are not constrained by the CNFs. (Polymer nanoclay composites 110, 111 are similar systems in which inorganic particles are observed to have marked effects on G’ and G”, but not on

109 the shift factors of time-temperature superposition.) The mobility of the polymer chains was further probed by measuring the glass transition temperature (Tg) as a function of mass concentration. We did not detect apparent change of Tg from pure PS to composites with up to 10wt% CNFs (data not shown here). By definition, Tg is the temperature above which the polymer chains are mobile. Thus, the lack of change in Tg of all composites also suggests that presence of CNFs does not effect the mobility of the polymer chains.

The relaxation times of the composites are further investigated by examining the shift of the cross over point of G’ and G” curves vs. frequency. The characteristic relaxation time of the composite is the inverse of the frequency at the cross over point 131.

Figure 4.5a and Figure 4.5b exhibit the cross over points of G’ and G” for the melt composites and control samples, and Table 4.3 list the relaxation times deduced from these points. In Figure 4.5a, the crossover point for MB composites shifts slightly to the left with increasing CNF concentration, corresponding in the slight increase of the characteristic relaxation time with CNF concentration displayed in Table 4.3. In SC composites (Figure 4.5b), however, the crossover point shifts slightly to the left when the

CNF concentration is less than 2wt%, but shifts more than a decade to the left in 10wt% composite. This indicates that at low CNF concentrations the characteristic relaxation times in SC composites behave like those in MB composites, i.e., they increase slightly with CNF concentration. However at high CNF concentrations the relaxation behavior of the SC composite departs from that of the corresponding MB composite. For instance, in the 10wt% SC composite the characteristic relaxation time is about nine times the relaxation time of pure polystyrene matrix, compared to one and half times for the 10wt%

MB composite. Summarizing, the MB composites gradually become more elastic as the

110 CNF concentration increases up to 10wt%; the SC composites behave similarly at very low CNF concentrations up to 2 wt%, but become significantly more elastic at 10wt%.

The gradual increase of relaxation time, as well as the G’ and G”, in the MB composites and the low-concentration SC composites is likely the result of fiber-matrix interactions, while the dramatic change in the SC10 composite is most likely caused by the entanglements of the CNFs, since the CNFs in SC composites are much longer that that in MB composites. These structures have longer relaxation time and make the composite more elastic.

4.3.3 Start-up of steady shear

Before displaying the transient shear viscosity η+ of the MB and SC melt composites, we show in Figure 4.6 the transient shear viscosity of the pure PS for shear rates from 0.0001 to 10s-1 . The η+ curves at shear rates 0.01s-1 and below fall on top of each other in the figure. It can be seen that for the as-received PS, at small shear rates the transient viscosity η+ gradually approaches steady state from below, but at large shear rates overshoots the steady state value and then approaches steady state from above.

Figure 4.7 shows the transient shear viscosity η+ and primary normal force

+ difference N1 at various shear rates of SC10 composite. Comparing to Figure 4.6, the η of SC10 is larger than that of PS at the corresponding shear rate, but the shape of the curves are similar for shear rates between 0.1 and 10s-1. However, at very small shear rates, e.g. 0.0001-0.01s-1, the shapes of viscosity curves are very different. Instead of having identical η+ over this range like the pure polymer, SC10 composites possess markedly increased viscosities over the range 0.0001-0.01s-1. Compared to the pure

111 polymer, it takes longer times for the shear viscosity of the composite to reach steady state, if it ever does. Due to the lower torque limit of the torque transducer on the rheometer, tests at shear rates smaller than 0.0001s-1 cannot produce reliable data and therefore the results are not shown. The normal stress differences N1, shown in Figure

4.7b, generally experience larger overshoots than the η+, and the overshoots occur at a longer time. Normal stress differences at shear rates lower than 0.01s-1 are too small to accurately measure.

4.3.4 Steady State Rheology

For CNF concentrations 2wt% and below, and shear rates 0.01s-1 or greater, we observed that the viscosity reaches a true steady state within 5 minutes (data not shown for brevity). However for higher concentrations and lower shear rates the identification of steady shear viscosity presents some challenges, since the viscosity never reaches a steady state in times typical in polymer processing (see for instance Figure 4.7a, in which the viscosity of the SC 10wt% composite at shear rate 0.0001 s-1is still evolving after 12 hours). For these cases, the value assigned as the steady viscosity for the purpose of comparison to model predictions of steady shear is the viscosity at 2-4 strain units.

Given this rule, the steady state viscosities of MB and SC composites are shown in Figure 4.8. Each data point was obtained from a start up of steady shear test with a fresh sample. Both MB and SC composites show increased viscosities and more shear thinning with higher CNF loadings, especially at low shear rates, with more pronounced increases in viscosity for SC composites. Furthermore, the plateau of the viscosity at low shear rates gradually disappears with increasing CNF concentrations, especially for the

SC10 composite. The slope of the viscosity curve for the SC10 composite on the log-log

112 plot is -1, indicating that this composite possesses a yield stress. The occurrence of yield stress further supports our conjecture of network structure in the SC composites with higher CNF concentration.

To further reveal the effects of CNF concentration and length on the rheology of the melt composites, relative viscosities ηr of the MB and SC composites are plotted against shear rate in Figure 4.9. Any deviation from the line ηr =1 can be attributed to the

CNFs in the composites. The relative viscosity ηr for both types of composites decreases with increasing shear rate. Further, at any CNF concentration ηr of the SC composite at low shear rates is much higher than that of the corresponding MB composite, an indication that the longer CNFs in the SC composites increase the viscosity of the composite much more than the short CNFs in the MB composites. This difference of relative viscosity between SC and MB composites increases with CNF concentration, and

-1 decreases as shear rates increases, until at shear rate of 10 s the ηr of both composites is essentially the same.

Primary normal stress differences N1 of the composites are displayed in Figure

4.10. In MB composites, the measured N1 for all CNF concentrations are about the same as that of pure PS polymer, with MB10 having slightly higher N1. In contrast, the N1 shows much more dependence on CNF concentration in SC composites. Higher CNF

-1 concentration leads to higher N1. At shear rate of 0.1s , the N1 of SC10 is more than 10 times higher than that of pure PS polymer. However, the effects of CNFs on N1 quickly

-1 diminish at higher shear rates. At 10s , the CNF does not contribute to N1 at all.

113 It is likely that the aforementioned network structure in the composites determine the viscosity and N1 at steady state as well. The nanostructure increases the resistance of the flow, while the applied shear rate flow may be powerful enough to destroy part or most of the nanostructure so that the composite’s viscosity is reduced. Both CNF concentration and length affect the strength of the nanostructure so that the MB and SC composites exhibit different steady state viscosity and N1 at same shear rates. MB composites contain shorter CNFs so that the strength of the nanostructure, which is mostly likely determined by the entanglement of the CNFs, is weaker. Therefore the enhancement of the viscosity in MB composites is much smaller than that in SC composites. This is the same as the trend in the linear viscoelasticity measurement. This is especially true at lower shear rates, as only part of the nanostructure is destroyed by the shear flow. It is also likely that due the hindrance of its neighboring CNFs, the CNFs are not aligned to the flow direction at low shear rates as well as that at higher shear rates. As a result, the remaining nanostructure and CNF orientation generate more resistance to the flow, leading to higher viscosity at low shear rates. This resistance also results in enhanced N1. At even smaller shear rates, the shear flow is too weak and the nanostructure is relatively strong enough to withstand the flow, resulting to the appearance of yield stress. This is seen in the case of SC5 and SC10 composites. In the same manner, high shear rate flow destroys more nanostructure and creates better CNF orientation to the flow direction, so the viscosity of the composite decreases and the effect of the CNFs on N1 diminishes.

114 The Peclet number (Pe) is the ratio of hydrodynamic force to the Brownian force.

For fibers in a polymer melt the Peclet number in steady shear flow can be defined as

3 Pe=ηm γ L /kBT(ln(L/D)-0.8), where ηm is the viscosity of the polymer melt, γ is the shear rate, L is the fiber length, D is the fiber diameter, kB is the Bolzmann constant, and T is the temperature 115. From the steady shear viscosity measurements of this section for pure polymer (see Figure 4.8), and our measurements of CNF geometry summarized in Table

4.1, Pe is calculated to be of order of 105-107 for our systems. Therefore in the modeling to follow in section IV the Brownian forces can be neglected and the CNFs can be treated as non-Brownian particles.

4.3.5 Cox-Merz rule.

The empirical Cox-Merz rule 165 states that the complex viscosity (η*) versus the dynamic frequency obtained from small amplitude oscillatory shear is numerically equal to the steady shear viscosity (η) versus the shear rate. Although there is no general explanation for such a relationship, it is widely successful in describing the observed behavior of isotropic polymeric solutions and polymer melts. As η* is more readily measurable experimentally than η, this rule provides a convenient way to estimate the η over a relatively wide range of shear rates. Figure 4.11 shows that the Cox-Merz rule holds for the MB2, MB5, and SC2 composites, but not the MB10, SC5, and SC10 composites. Steady state viscosities of the MB10 and SC5 composites are less than half an order of magnitude lower than the corresponding dynamic viscosity, while that of

SC10 is more than an order of magnitude lower. As MB10, SC5, and SC10 show evidence of a yield stress, they are expected not to obey the Cox-Merz rule, just like other polymer composites with yield stress 166, 167. 115 4.4 Nanoparticle/Polymer Melt Composite Constitutive Model

In this section we develop, benchmark, and validate models for the CNF/PS melt composite systems studied through experiments in sections 4.2 and 4.3. These models are motivated by the behavior we observed in the experiments, and have been found to accurately capture the relationships between the rheological properties, processing conditions, and fiber nanostructure measured in the experiments.

In a previous paper 163 we study composite systems of CNFs suspended in glycerol-water. Model systems of either elastic or rigid dumbbells in a Newtonian solvent with isotropic or anisotropic hydrodynamic drag, with or without hydrodynamic interaction are investigated; it is found that the most successful model (the elastic model with anisotropic hydrodynamic drag and negligible hydrodynamic interaction) successfully captures the rheological behavior we observe in our experiments on the

CNF/ glycerol-water systems. The behavior of CNFs in a polymer melt we observed in the experiments described in section 4.2 and 4.3 is fundamentally different from the behavior of CNFs in a glycerol-water solvent we measured and reported in the previous paper 163, in that the polymer melt without nanoparticles exhibits shear thinning, extensional, and viscoelastic properties that are not observed in the Newtonian solvent, and our experiments gave strong evidence of significant interaction of the CNFs with the polymer matrix and/or other CNFs. The constitutive equations which we found to best capture the observed features of our CNF/PS melt composites are (in indicial notation):

c p CNF τ ij=−pDδη ij +2 s ij + ττ ij + ij , (4.24) p pppDτ ij αλ τλij++() ττη ik kj =2 pD ij , (4.25) Dt η p

116 τηηφCNF =+22⎡⎤AD a + B D a + a D ++ CD Fa D . (4.26) ij[ s] ⎣⎦ kl ijkl( ik kj ik kj) ij ij r with da ij =−+−−+∏−Wa aWχδ Da aD24 Da C12/ ma , dt ()(ik kj ik kj ik kj ik kj kl ijkl )()I D ij ij (4.27) These coupled equations model the polymer melt matrix as viscoelastic and strain rate dependent, and model fiber – fiber and fiber – matrix interaction. Equation (4.24),

168 c proposed by Azaiez , expresses the total stress τ ij in the polymer/nanocomposite

system as the sum of the stress contribution 2ηs Dij from a Newtonian solvent (if present),

p the stress contribution τ ij from polymer molecules with fiber inclusions, the stress

CNF contribution τ ij in the carbon nanofibers, and the pressure p maintaining

incompressibility. In the Newtonian contribution 2ηs Dij in equation (4.24), Dij the

∂vi symmetric part of the velocity and ηs the solvent viscosity. ∂x j

Equation (4.25) is Giesekus model which predicts the strain rate dependent viscoelastic behavior of the polymer matrix. In eq. (4.25),

Dτ p d ij =−+−−τττττpppppWWDD is the upper convected derivative of τ p , with Dt dt ij ik kj ik kj ik kj ik kj ij

Wij the skew part of the Eulerian velocity gradient; the constants η p , α , λ in eq. (4.25) are the polymer viscosity, mobility factor, and relaxation time, respectively, of the melt phase of the polymer matrix. In our study we investigated a generalization of equation

(4.25) presented in Azaiez168,

117 p Dτ ij pppαλ m(1−σ ) pp λστττ++ij() ik kj +()aaD ik ττη kj + ik kj =2 p ij , (4.28 Dt η p 2 ) which includes the effect of polymer-fiber interaction through the parameter σ.

For the composite systems we study in this paper, we found that σ is effectively 1 (see section 4.5.4 to follow), so that there is negligible polymer-fiber interaction and the modified Giesekus equation (4.28) reduces to the Giesekus equation (4.25).

In equation (4.26), derived by Tucker169, η is the viscosity contribution from the

p τ12 polymer matrix; in simple shear in the 1-2 plane η = withγ the shear rate. Dr is the γ rotary diffusivity due to Brownian motion, and φ is volume fraction of fibers. Parameters

A, B, C, and F are in general functions of the particle aspect ratio hLD= / , with L the particle length and D the particle diameter. In dilute regimes 169:

h2 6ln 2h − 11 3h2 ， ( ) ， ， A = B = 2 C = 2 F = , 2ln2⎣⎦⎡⎤()h − 15. h ln() 2h − 0. 5 (4.29) and in semi-dilute regimes 68:

h2 A = ， B = 0 ， C = 0 ， F = 0 , (4.30) 3ln() 2hDf / with hf a characteristic distance between a fiber and its nearest neighbors:

2h π f = for aligned fibers, (4.31) D φ 2h π f = for random fibers. (4.32) Dh2φ Since the melt composites we study in this paper are in either the semi-dilute or concentrated regimes (see Table 4.1), we employ choices (4.30) of parameters A, B, C, and F , rather than (4.29) (there have been no values of A, B, C, and F proposed in the 118 literature for the concentrated regime). Note that the term multiplied by F in equation

(4.26) embodies the effects of Brownian forces, so that the selection in (4.30) of F = 0 is consistent with our assessment in section 4.3.4 that the order of magnitude of the Peclet number in the flows we investigate of our melt composites is such that the effects of the

Brownian forces are negligible.

The properties of a nanoparticle/polymer composite, both during processing (such as viscosity and relaxation time) and in its end state (such as modulus and conductivities) are mesoscale properties, the result of combining the contributions of many nanoscale particles. The measure of nanostructure that combines the contributions of a collection of particles to dictate the mesoscale properties of the nanocomposite is the second order

orientation tensor aij that occurs in Eq. (4.26), defined as the dyadic product of fiber orientation p with itself averaged over the orientation space containing a sufficient number of particles 57:

appd= ψ pp, (4.33) ij∫ i j ( ) where ψ(p) is the orientation distribution function, i.e. the probability that the fiber is oriented along the unit vector p. The evolution equation (4.27) for the mesoscale quantity

aij averaging the composite response over an ensemble of nanoparticles is obtained from the nanoscale equations governing the behavior of the individual particles as follows: The orientation distribution function ψ(p) evolves according to the evolution equation 57:

∂ψ ∂ =− ()piψ , (4.34) ∂∂tpi and the rate of change of orientation p with time for a single nanoparticle is 57

11∂ψ pWpDpDpppDiijjijjklklir=+−()(χ ) −, (4.35) 2 ψ ∂pi

119 where χ is a shape parameter related to the particle aspect ratio,

h2 −1 χ = . (4.36) h2 +1 Following Advani 57 as suggested by Folger and Tucker 30 for concentrated

12/ suspensions of large fibers, in equation (4.35) we employ DCrID= 2 ∏ , where ∏D is

the second invariant of the symmetric part Dij of the velocity gradient and CI is the interparticle interaction parameter which measures the intensity of fiber interaction in the

composite. Multiplying the kinetic equation (4.34) by pijp , integrating over orientation space for a representative number of particles, and inserting eq. (4.35) gives the evolution equation (4.27) of the second order orientation tensor aij, where m is the dimension of the space (m=2 for planar flows and m=3 for 3-dimensional flows).

Note that in addition to the second order orientation tensor aij , Eq. (4.26) also

involves the fourth order orientation tensor aijkl defined as

appppd= ψ pp. (4.37) ijkl∫ i j k k ( ) System (4.24)-(4.27) as presented is underdetermined. To restore closure one must adopt an approximation that relates the fourth-order orientation tensor aijkl in Equations (4.26) and (4.27) to the second order orientation tensor aij. Popular closure approximations are:

1. The linear closure approximation,

11 aˆ =−δ δδδδδ + + + aaaaa δ + δ + δ + δ + δ, ijkl35() ijklikjliljk 7 () ijklikjlklijjlikjkil (4.38) works well when the fibers remain nearly random, but introduces an artificial instability into the equations for highly aligned suspensions 170;

120 2. the quadratic closure approximation

aaaijkl= ij kl , (4.39) performs well for highly aligned states, but introduces steady-state errors for more random states 170

3. the hybrid closure approximation,

afafaijkl=−()1 ˆ ijkl + ijkl with f =127det()− aij , (4.40) mixes linear and quadratic forms according to scalar measure of orientation and performs well over the entire range of orientation 170;

4. the orthotropic closure approximation58 , developed based on the assumptions that any approximate fourth-order tensor must be orthotropic, that its principal axes must match those of the second-order tensor, and that each principal fourth-order component is

58 a function of just two principal values of the second-order tensor . For CI ≥ 001. and a wide variety of flows, the orthotropic closure approximation provides more accurate solutions than any previous closure approximations58.

4.5 Application of the model to CNF/PS melt composites

4.5.1 Procedures to deduce material constants in the model from experimental

measurements

To characterize the CNF/PS melt composites in the context of constitutive model

(4.24)-(4.27), we deduce the particle aspect ratio h from our measurements of the

morphology of the nanofibers, and the constitutive properties λ , α , η p , ηs from flow measurements in our steady shear and small amplitude oscillatory shear experiments, reported in section 4.3.

121 Recall from Table 4.1 that the representative aspect ratio for the MB composites was measured to be 40, and the representative aspect ratio for the SC composites was measured to be 160.

The constitutive parameters λ , α , η p , ηs in the characterization (4.25) of the

CNF/polymer melt composite are constants and properties of the polymer melt alone.

Hence they are determined from experiments on the pure polymer. In the characterization

of the MB composites, λ , α , η p , ηs are deduced from steady and oscillatory shear measurements of polystyrene that has gone through the melt blending process. In the

characterization of the SC composites, λ , α , η p , ηs are deduced from steady and oscillatory shear measurements of polystyrene that has gone through the solvent casting

process. As will be seen in section 4.5.4.5.2 and 4.5.4.5.3, the values of λ , α , η p , ηs for these two polystyrenes are different.

In detail, the material constants characterizing the pure polystyrenes are deduced as follows:

In simple shear flows, with the appropriate choice of coordinate system the

velocity is of the form vx== γx2 e1 ; shear rate γ is constant in steady shear, and

harmonic in small amplitude oscillatory shear. Zero shear viscosity η0 =+ηηs p is the

τ limit of 12 as γ becomes small in steady shear; we identify η as the measured γ 0 viscosity at shear rate γ = 0.001s-1. Since there is no solvent in the polymer melt, its

solvent viscosity ηs is zero. Therefore ηs = 0 and η p is given by the measured η0 .

122 For small amplitude oscillatory shear experiments, Eq. (4.25) gives the functional dependence of the storage modulus G ' and loss modulus G" for the polymer melt on the

material constants λ , α , η p , ηs and frequency ω as

⎛⎞η η λω2 G" ()ω = ⎜⎟ηω+ p , G' ()ω = p . (4.41) ⎜⎟s 2 2 ⎝⎠1+()λω 1+()λω The relaxation time λ of the polymer melts is then obtained by minimizing the error

⎧⎫2 ⎡⎤⎛⎞⎛⎞η ⎪⎪⎢⎥logG" ωη−+ log ⎜⎟⎜⎟p ω ⎪⎪10 exp()ii 10 s 2 n ⎢⎥⎜⎟⎜⎟1+ λω ⎪⎝⎠⎪⎣⎦⎝⎠()i δ = 1 ∑ ⎨⎬2 , (4.42) i=1 ⎪⎪⎡⎤⎛⎞ηλω2 ⎪⎪⎢⎥p i +log10G' exp()ωi − log 10 ⎜⎟2 ⎪⎪⎢⎥⎜⎟1+ λω ⎩⎭⎣⎦⎝⎠()i where G'exp ()ωi and G"exp (ωi ) are the measured storage and loss moduli at

frequency ωi , η p =η0 is the measured zero shear viscosity, and n is the number of measurements; in our study for each of the MB and SC melts there are 24 experiments with frequency from ω = 0. 001 to 15s-1 (refer to the points for MB0 and SC0 in Figure

4.3). We search over the physically reasonable domain 010< λ < s, first using a coarse

mesh, then restricting λ to a smaller domain with a refined mesh to minimize δ1 .

Finally, we verify that the best fit is not near the edge of the domain.

For steady shear at shear rate γ , Equation (4.25) implies that shear viscosity η()γ and first normal stress difference N1 of the pure PS are of the forms

()1−ψ 2 ψ (1−αψ ) ηγ() =+ η η N ()γλη = 2 sp112+−α ψ 1 p λ 2αψ1− (), () , (4.43)

123 with

2 1/2 ⎡⎤⎛⎞. ⎢⎥11612+ ααγλ()−−⎜⎟ 1 ⎣⎦⎝⎠ χ = 2 1− χ ⎛⎞. ψ = 812ααγλ()− ⎜⎟ 1+ ()1− 2α χ , ⎝⎠ .

With ηs = 0 , η p =η0 , and λ deduced from small amplitude oscillatory shear measurements as described above, the mobility factors α of the polymer melts are obtained by minimizing the error

2 2 n1 ⎡⎤⎛⎞()1−ψ δηγη=−⎢⎥log() log ⎜⎟ 2∑ 10 expip 10 ⎜⎟112+−αψ i=1 ⎢⎥⎝⎠() ⎣⎦, (4.44) 2 n2 ⎡⎤⎛⎞ψαψ()1− +−⎢⎥logN ()γλη log⎜⎟ 2 ∑ 10 1exp ip 10 ⎜⎟2 i=1 ⎢⎥⎣⎦⎝⎠λα()1− ψ -1 where ηγexp ()i is shear viscosity measured at shear rate γ = 0. 0001∼ 10 s (n1=9).

N γ is first normal stress difference measured at a smaller range of shear 1exp ()i

-1 rateγ = 01. ∼ 10 s (n2=5) Due to the lower torque limit of the torque transducer. The

search to minimize δ2 is over the physically reasonable domain 01<<α , following the same procedure as with λ . This exhaustive search method we have adopted to sequentially minimize the errors (4.42) and (4.44) is feasible since there are only two decoupled 1-parameter searches over well-defined domains.

124 With the constitutive parameters λ , α , η p , ηs known, for a specified steady

shear flow vx== γx2 e1 we can solve Equation (4.25) to obtain the stress contribution

p from the pure polymer τ ij in Equation (4.24) and strain rate dependent polymer viscosity

τ p η = ij in Equation (4.26). γ

The final parameter remaining to be determined is the interparticle interaction parameter CI. With symmetric and skew parts D and W of the velocity gradient specified

by the steady shear flow vx= = γx2 e1 and inserted the orientation evolution Equation

(4.27), using closure approximation (4.39), and assuming homogeneous steady behavior

d ( a = 0 ) produces the following system of equations for the components of the dt ij

symmetric tensor aij :

02213,=+γχγaaaaCa12() 12 − 12 11 +I γ ( − 11 ) 11 11 ⎛⎞2 026,=−+γγχγaa22 11⎜⎟ aaaCa 22 +−− 11 12I γ 12 22⎝⎠ 22 11 ⎛⎞ 026,=+γχγaaaaCa23⎜⎟ 23 − 12 13 −I γ 13 22⎝⎠ (4.45)

02213,=−γχγaaaaCa12 +()() 12 − 12 22 +I γ − 22 11 ⎛⎞ 02=−γχγaaaa13 +⎜⎟ 13 − 12 23 − 6,CaIγ 23 22⎝⎠

02=−χγaa12 33 + 2 CI γ () 13. − a 33

In our study of the processing of CNF/PS melt composites we have measured the effect of shear processing conditions on the development of nanofiber structure with

TEM images of sectioned samples. One of our findings is that the end-state fiber orientation in a polymer nanocomposite depends strongly on the strain rate the

125 nanocomposite undergoes during processing. If interparticle interaction parameter CI is considered to be independent of strain rate, the steady shear rate γ cancels from all six

Equations (4.45), so that their solution is a second order orientation tensor aij that is a function only of the aspect ratio h and CI , independent of strain rate. This model prediction is unacceptable, given its inconsistency with the physically observed dependence of CNF orientation on strain rate. We conclude that in our model the interaction parameter CI must be considered as a function of strain rate. Note that when

CI is a function of strain rate, strain rate does not cancel from Eqs. (4.25), so that

orientation tensor aij is a function of strain rate.

With fixed representative aspect ratio h for melt blended and solvent cast composites, separately at each shear rate γ = 0.01, 0.03, 0.1, 0.3, 1, 3, 10s-1 we search over CI by solving Equations (4.24)-(4.27) specialized to steady shear flow, and minimizing error

N 2 δηγηγ=−⎡⎤log log , (4.46) 3∑ ⎣⎦ 10 exp()ii 10() model () i=1

summed over the three viscosity measurements (2wt%, 5wt% and 10wt%) at that shear rate. (Note we only fit to the experimental measurements of steady shear viscosity

ηexp ()γi . According to the fitting results obtained by fitting to the shear viscosity, we predict the first normal stress difference N1.) With D and W specified by the steady shear

flow vx== γx2 e1 and aijkl expressed in terms of aij through a closure approximation,

126 aij can be solved as a function of aspect ratio h (through χ ) and interaction parameter CI from equation (4.27). Equations (4.24), (4.25), (4.26) then are a closed set of coupled

p CNF c equations for τ ij , τ ij , and τ ij .

As we will display in the next section, for both types of composites the above seven fitting results of CI at discrete values of strain rate are found to lie on a curve that

q can be closely approximated by CpI = γ . Accordingly, in addition to the above pointwise determination of CI as a discrete function of γ , we also obtain CI as the

q continuous power law function CpI = γ of γ by searching through parameter space ( p , q ) in the solutions of Equations (4.24)-(4.27) and minimizing error (4.46) summed over the 21 measurements of steady shear viscosity at all seven shear rates γ = 0.01 to 10s-1.

(Also, we only fit to the measurements of steady shear viscosity and then predict the first normal stress difference N1.) We omit the data with shear rates lower than 0.01 s-1 mainly because we are aiming at possible industrial applications and these extremely low shear rates are below the typical processing range. Another reason we omit the small shear rate data in modeling is that the interparticle forces dominate the rhoelogy, as can be seen from Figure 4.8 that the viscosity of SC composites increases significantly due to the relatively large interparticle interactions. Although the constitutive equations we selected, i.e. Eqs. (4.24)-(4.27), contain a interparticle interaction parameter CI, the whole structure of the equations are based on dilute fiber suepensions30, where interparticle interactions are not present. As a result, this set of equations is not applicable for fiber suspensions/composites where interparticle interactions are large.

127 4.5.2 Models for the Melt Blended composites

The measured zero shear rate viscosity for pure polymer subjected to the melt blended process is 9864 Nm/s2, so that in the melt blended CNF/PS melt composite

2 model η p ==η0 9864 Nm/s . Searching over the domain 010< λ < s and minimizing error (4.42) for the 24 small amplitude oscillatory shear experimental measurements

G'exp ()ωi and G"exp ()ωi of the MB pure polymer melt with frequency from ω = 0. 001 to 15s-1 produces λ = 0.329 s. The corresponding fits to the measured storage modulus

G ' and loss modulus G" are shown in Figure 4.12a. Searching over 01<<α and minimizing error (4.44) for 9 steady shear viscosities measurements from γ = 0.0001 to

-1 -1 10s and 5 first normal stress difference N1 from γ = 0.1 to 10s of the MB pure polymer produces α = 0.392 . The fits to measured hear viscosity and first normal stress difference

(N1) are shown in Figure 4.12b.

Recall from Table 4.1 the representative aspect ratio of MB composites is 40. If

CI is assumed to be a function of strain rate, we search CI by solving Equations (4.24)-

(4.27) with the minimum error (4.46) summed over N=3 experimental measurements of shear viscosities at each shear rate and obtain CI =6.83E-02, 6.83E-02, 2.65E-02, 9.70E-

03, 2.70E-03, 1.90E-03 and 9.90E-03 corresponding to shear rate γ = 0.01, 0.03, 0.1, 0.3,

-1 1, 3, and 10s (Table 4.4 shows the values of CI and corresponding components of

second order orientation tensor aij ). Figure 4.13 shows the model predictions of the viscosity and first normal stress difference of the steady shear experiments. Based on the

q trend of CI versus shear rate (Figure 4.14), we propose CI has form CpI = γ . Searching through parameter space ( p , q ) by minimizing error (4.46) summed over the 21

128 experimental measurements of steady shear viscosity, we finally obtained p =×6.600 10-3 and q =− 0.507 . (Table 4.6). Figure 4.15 shows the model predictions

−0.507 with CI = 0.0066γ .

Comparing the model prediction and the experimental measurements shown in

Figure 4.15, we conclude for the MB composite system, the model with assumption of

q −0.507 interparticle interaction parameter CpI ==γγ0.0066 successfully capture shear thinning behavior at low shear rate and gives an accurate prediction of the composite viscosity magnitude. The model captures the correct trend of the first normal stress difference while it over-predict the stress magnitude for all mass concentrations.

4.5.3 Models for the Solvent Cast composites

The measured zero shear rate viscosity for the pure polymer subjected to solvent

2 2 casting is 11160 Nm/s , so that η p =η0 =11160 Nm/s for the solvent cast CNF/PS melt composite models. Similarly to the procedure employed for pure polymer of MB composites, searching over the domain 010< λ < s by minimizing error (4.42) for 24

-1 oscillatory shear measurements of G'exp (ωi ) and G"exp (ωi ) (ω = 0001. to 15s ) on the

SC pure polymer produces λ = 0.321 s (Figure 4.16a). Searching over 01<<α and minimizing error (4.44) for 9 steady shear viscosity measurements (γ = 0.0001 to 10s-1)

-1 and 5 first normal stress difference N1 measurements (γ = 0.1 to 10s ) on the SC pure polymer produces α = 0.479 (Figure 4.16b).

129 Based on the optical micrograph in Figure 4.1.b and many others like it, we set the particle aspect ratio h to be 160. We search parameter CI to minimize error (4.46) summed over N=3 experimental measurements at each shear rate and finally obtain CI

=2.82E-03, 1.39E-03, 5.81E-04, 2.21E-04, 5.90E-05, 2.70E-05 and 9.80E-05 corresponding to shear rate γ = 0.01, 0.03, 0.1, 0.3, 1, 3, and 10s-1 (Table 4.5). Figure

4.17 shows the model predictions of the steady shear viscosity and first normal stress difference of the steady shear experiments. Based on the trend of CI versus shear rate

q (Figure 4.14), we propose CI has form CpI = γ . Searching through parameter space

( p , q ) by minimizing error (4.46) summed over the 21 experimental measurements of steady shear viscosity, we finally obtained p =×1.00 10-4 a and q =−0.729 (Table 4.6).

−0.729 Figure 4.18 shows the model predictions with CI = 0.0001γ .

Comparing the model prediction and the experimental measurements shown in

Figure 4.15, we conclude for the SC composite system, the model with assumption

q −0.729 CpI ==γγ0.0001 successfully capture shear thinning behavior at low shear rate and gives an accurate prediction of the composite viscosity magnitude. The model captures the correct trend of the first normal stress difference while it over-predict the stress magnitude for all mass concentrations.

4.5.4 Investigation of polymer-fiber interaction

In addition to modeling the polymer matrix with the Giesekus model, Equation

(4.25), we also investigated the modeling of the matrix with the generalization (4.28) of the Giesekus model proposed by Azaiez168, which includes the effect of polymer-fiber interaction through a fiber/polymer matrix interaction parameter σ. This parameter σ , in

130 general a function of mass concentration of the particles, characterizes the effect of particle orientation on the hydrodynamic drag acting on the polymer molecules 168. In the special caseσ =1, the drag force on the polymer molecules is independent of the fiber orientation, and the usual Giesekus model (4.25)81 is obtained. Utilizing the modified

Giesekus model, we search through parameter space ( p , q , σ2 , σ5 , σ10 ) by solving equations (4.24), (4.28), (4.26) and (4.27) with the minimum error (4.46) summed over the 21 measurements of steady shear viscosity ηγ( ) at shear rate γ = 0.01-10s-1.

Figure 4.7 shows the fitting results of CI and σ using the modified Giesekus

model. We notice the most different cases are σ 2 = 0.89 for MB composite and σ 2 = 0.78 for SC composites. Figure 4.19 shows the comparison of model prediction (steady shear viscosity and first normal stress difference) between the Giesekus model and the modified Giesekus model. It is obvious that for MB composite the model predictions of the modified Giesekus model and Giesekus model are almost same and for SC composite the modified Giesekus model are only marginally better than the Giesekus model.

Therefore, we conclude that the interaction between the nanofibers and the polymer are negligible in our experiments and the modified Giesekus equation (4.28) reduces to the

Giesekus equation (4.25), which is capable of capturing the rheological behavior of our composite systems.

4.6 Conclusions

In this paper we first prepared two PS/CNF composites with different lengths of

CNFs. The MB composites ware produced by using a microcompounder and the CNFs were uniformly dispersed in the PS matrix. However, the lengths of the CNFs were only

131 about 1/5 of the as-received CNFs. The SC composites were prepared using a solvent casting method. The lengths of the CNFs were maintained to be about the same as those in the as-received CNFs. The dispersion of CNFs in SC composites was not as uniformly dispersed as those in MB composites, but on a millimeter scale, the homogeneous dispersion was achieved.

Rheological studies showed that both the storage modulus (G’) and loss modulus

(G”) of both type of composites increased monotonously with the CNF concentration.

Moreover, at low frequencies, plateau of G’ occurred for SC composites at high CNF concentrations, showing solid-like behavior. This behavior indicated that network nanostructure existed in these composites. The shifting factors derived from time- temperature superposition of the linear viscoelasticity measured at 140-220C are almost identical for all composites. This indicates that the CNFs and the nanostructures formed by the CNFs in composites are non-Brownian.

It has been found that the steady state viscosities increased with higher CNF loadings and the composites exhibited more shear thinning. SC composites showed higher relative viscosity than MB composites because the CNFs in the former are much longer. At higher CNF concentrations, both MB and SC composites showed yield stress.

All these rheological behavior could be attributed to the interactions between the

CNFs in the composites. At high concentrations, the CNFs form continuous network of nanostructures, which possess longer relaxation times, causing the G’ to plateau at small frequencies. When subjecting to flow at constant shear rates, part of the nanostructure is

132 destroyed. Higher shear rates destroy the nanostructure to a larger degree and thus the shear thinning behavior. At very low shear rates, the nanostructure remain intact and caused yield stress at higher CNF concentrations.

The experimentally measured rheological behavior is modeled by a rigid rod in a viscoelastic solvent, in which the carbon nanofiber is described by rigid rod model and polymer matrix is described by Giesekus model. The fiber-fiber interaction parameter CI which is assumed to be a power law function of shear rate is investigated.

For both MB and SC composite systems, the model successfully predicts the increasing trend of shear viscosity with the mass concentration. The model can also capture shear thinning behavior and predict the viscosity magnitude of the composite system. The model captures the correct trend of the normal stress while it cannot quantitatively predict the normal stress magnitude accurately.

133

MB composites SC composites CNF length (L), nm 1,000 - 20,000 4,000 - 60,000 CNF diameter (D), nm 100 - 200 100 - 200 CNF aspect ratio (L/D) 10 - 100 20 - 500 CNF representative aspect ratio 40 160 CNF density, kg/l 1.8 1.8 PS melt density, kg/l 1.0 1.0 MB2 MB5 MB10 SC2 SC5 SC10 CNF mass fraction (wt%) 2 5 10 2 5 10 CNF volume fraction 1.12 2.84 5.81 1.12 2.84 5.81 CNF number density (n), m-3 4.5e16 1.1e17 2.3e17 1.1e16 2.8e16 5.8e16 nL3 23 58 118 370 930 1900 semi- concen- concen- concen- concen- concen- regime dilute trated trated trated trated trated

Table 4.1. Measured properties of the melt blended and solvent cast composites, and their nanofiber and polymer constituents

134

MB Composites SC Composites wt% d(logG’)/d(logω) d(logG”)/d(logω) d(logG’)/d(logω) d(logG”)/d(logω) 10 1.14 0.85 0.44 0.40 5 1.53 0.97 1.14 0.62 2 1.68 0.97 1.30 0.88 0 1.93 0.99 1.91 0.96

Table 4.2. Slopes of G'(ω) and G"(ω) at low frequencies on log-log plot.

135

Characteristic relaxation times (s) wt% MB Composites SC Composites 10 0.091 0.50 5 0.078 0.13 2 0.071 0.071 0 0.063 0.065

Table 4.3. Characteristic relaxation times of MB and SC composites

136

-1 shear rate (s ) CI a11 a12 a22 a33 0.01 6.83E-02 0.6658 0.2037 0.1669 0.1672 0.03 6.83E-02 0.6658 0.2037 0.1669 0.1672 0.1 2.65E-02 0.8008 0.1862 0.0994 0.0998 0.3 9.70E-03 0.8917 0.1494 0.0539 0.0544 1 2.70E-03 0.9521 0.1034 0.0237 0.0242 3 1.90E-03 0.9619 0.0925 0.0187 0.0194 10 9.90E-03 0.8903 0.1502 0.0547 0.0550

Table 4.4 Values of CI and corresponding second order orientation tensor components of MB composites at different shear rate

137

-1 shear rate (s ) CI a11 a12 a22 a33 0.01 2.82E-03 0.9510 0.1064 0.0245 0.0246 0.03 1.39E-03 0.9692 0.0856 0.0154 0.0155 0.1 5.81E-04 0.9826 0.0649 0.0087 0.0087 0.3 2.21E-04 0.9908 0.0473 0.0046 0.0046 1 5.90E-05 0.9962 0.0303 0.0019 0.0018 3 2.70E-05 0.9977 0.0232 0.0011 0.0012 10 9.80E-05 0.9946 0.0361 0.0027 0.0027

Table 4.5 Values of CI and corresponding second order orientation tensor components of SC composites at different shear rate

138

Melt Blended (MB) Solvent Cast (SC) λ 0.329 s * 0.321 s + α 0.392 * 0.479 + * + η p 9864 Pa.s 11160 Pa.s * + ηs 0 Pa.s 0 Pa.s h 40** 160++ q q CI fit to pointswise fit to CpI = ()γ : fit to pointswise fit to CpI = ()γ : values of CI : values of CI: −0.472 −0.627 −0.507 −0.729 CI = 0.0074γ CI = 0.0001γ CI = 0.0066γ CI = 0.0001γ

* measured from pure polymer that had undergone the melt blending process

(MB0).

**from morphological study of MB composites

+ measured from pure polymer that had undergone the solvent casting process

(SC0).

++ from morphological study of SC composites

Table 4.6. Model predictions of melt blended and solvent cast composites.

139

Melt Blended (MB) Solvent Cast (SC) h 40** 160++ q −0.49 q −0.71 CI CpI ==()γγ0.0070 CpI ==()γγ0.0001

σ σ 2 = 0.89 (MB2) σ 2 = 0.78 (SC2)

σ 5 = 1.00 (MB5) σ 5 = 0.83 (SC5)

σ10 = 1.00 (MB10) σ10 = 1.00 (SC10)

Table 4.7 Model predictions of modified Giesekus model

140

(a) (b) Figure 4.1. Optical microscopy images of a 5wt% (a) melt blended (MB) composite and (b) solvent cast (SC) composite. Scale bar: 40 microns.

141

(a) (b) Figure 4.2. TEM micrographs of (a) a 5wt% MB composite and (b) a 5wt% SC composite. Scale bar: 5 microns.

142

1.E+7 1.E+7

1.E+6 1.E+6

1.E+5 1.E+5

1.E+4 1.E+4 G' (Pa) G' (Pa) 1.E+3 1.E+3 MB10 SC10 1.E+2 MB5 1.E+2 SC5 MB2 SC2 1.E+1 1.E+1 MB0 SC0 1.E+0 1.E+0 1.E-2 1.E+0 1.E+2 1.E+4 1.E+6 1.E-2 1.E+0 1.E+2 1.E+4 1.E+6

Frequency (rad/s) Frequency (rad/s) (a) (b) Figure 4.3. Elastic modulus (G’) of (a) MB and (b) SC composites with various CNF concentrations at 200°C.

143

10000 MB0 MB2 1000 MB5 MB10 100 SC0 SC2

aT SC5 10 SC10 WLF

1

0.1 0.002 0.0021 0.0022 0.0023 0.0024 0.0025 1/T (1/K)

Figure 4.4. Shifting factor of MB and SC composites.

144

1.E+10 1.E+10

1.E+09 1.E+09 SC10 (x1000) MB10 (x1000) 1.E+08 1.E+08

SC5 (x100) 1.E+07 1.E+07 MB5 (x100)

1.E+06 1.E+06 MB2 (x10) SC2 (x10)

1.E+05 1.E+05 SC0

MB0 G' and G" (Pa) G' and G" (Pa)

1.E+04 1.E+04

1.E+03 1.E+03

1.E+02 1.E+02

1.E+01 1.E+01 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 Frequency (rad/s) Frequency (rad/s) (a) (b) Figure 4.5. Shift of crossover point of G’ (solid symbols) and G” (open symbols) of (a) MB composites and (b) SC composites.

145

1E+5

0.01 0.1 1E+4 0.3 1

3 Viscosity (Pa.s)

10

1E+3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 Time (s)

Figure 4.6. Transient shear viscosity η+ of the pure PS at shear rates 0.0001-10s-1. (Curves for 0.01 s-1 and below fall on top of each other.)

146

1E+07 1E+06 0.0001 10 1E+05 3 1E+06 1 0.3 0.001 1E+04 0.1 1E+05 0.01 1E+03

0.1 N1 (Pa) 0.3 1E+02

Viscosity (Pa.s) 0.01 1 1E+04 3 1E+01 10

1E+03 1E+00 0.01 1 100 10000 0.01 0.1 1 10 100 1000 10000 Time (s) Time (s) (a) (b)

Figure 4.7. Start up of steady shear at shear rates 0.0001 – 10 s-1. (a) Transient shear viscosity, (b) primary normal stress difference. SC10 composite. T=200°C.

147

1.E+07 1.E+07

10 10 5 1.E+06 5 1.E+06 2 2 0 0

1.E+05 1.E+05 Viscosity (Pa.s) Viscosity (Pa.s) 1.E+04 1.E+04

1.E+03 1.E+03 0.0001 0.001 0.01 0.1 1 10 0.0001 0.001 0.01 0.1 1 10 Shear rate (1/s) Shear rate (1/s) (a) (b) Figure 4.8. Steady shear viscosities of (a) MB composites and (b) SC composites.

148

1000

MB10 MB5 100 MB2 SC10 SC5 SC2 10 Relative Viscosity

1 0.0001 0.001 0.01 0.1 1 10 Shear rate (1/s)

Figure 4.9. Relative viscosity of MB and SC composites.

149

1.E+06 1.E+06

1.E+05 1.E+05

1.E+04 1.E+04

1.E+03 1.E+03 N1 (Pa) 10 N1 (Pa) 1.E+02 5 1.E+02 10 2 5 1.E+01 0 1.E+01 2 0 1.E+00 1.E+00 0.01 0.1 1 10 0.01 0.1 1 10 Shear rate (1/s) Shear rate (1/s)

(a) (b)

Figure 4.10. First normal stress difference N1 under steady shear of (a) MB and (b) SC composites.

150

1.E+5 1.E+6

1.E+5 1.E+4 ) Pa.s

(Pa.s) 1.E+4 η∗, η η∗, η ( 1.E+3 SC10 SC5 MB10 MB5 1.E+3 MB2 MB0 SC2 SC0 MB10 SS MB5 SS SC10 SS SC0 SS MB2 SS MB0 SS SC5 SS SC2 SS 1.E+2 1.E+2 1.E-2 1.E-1 1.E+0 1.E+1 1.E+2 1.E+3 1.E-2 1.E-1 1.E+0 1.E+1 1.E+2 1.E+3 Shear rate (1/s), Frequency (rad/s) Shear rate (1/s), Frequency (rad/s) (a) (b) Figure 4.11. Applicability of Cox-Merz rule of (a) MB and (b) SC composites.

151

1.E+05 1.E+05

1.E+04 1.E+04 1.E+03

G'(Pa) 1.E+02 1.E+03 Viscosity (Pa.s) Viscosity model model 1.E+01 experiment experiment 1.E+02 1.E+00 1.E-04 1.E-02 1.E+00 1.E+02 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 Frequency (rad/s) Shear rate (1/s)

1.E+05 1.E+06

1.E+05 1.E+04 1.E+04

1.E+03 1.E+03 N1(Pa) G"(Pa) 1.E+02 model 1.E+02 model 1.E+01 experiment experiment 1.E+00 1.E+01 1.E-01 1.E+00 1.E+01 1.E+02 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 Frequency (rad/s) Shear rate (1/s)

(a) (b) Figure 4.12. (a) Fit of model with optimum value λ = 0.329 s to oscillatory shear measurements of melt blended pure polymer melt, (b) fit of model with optimum values λ = 0.329 s and α = 0.392 to steady shear measurements of melt blended pure polymer melt.

152

MB MB model2 1.00E+06 model5 1.00E+05 model10 exp2 1.00E+05 exp5 exp10 1.00E+04 model2 N1 1.00E+04 model5 model10 exp2 Viscosity (Pa.s) Viscosity 1.00E+03 exp5 exp10

1.00E+03 1.00E+02 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E-01 1.00E+00 1.00E+01 Shear rate (1/s) Shear rate (1/s)

(a) (b) Figure 4.13 Model predictions of (a) the viscosity and (b) first normal stress difference compared to the experimental steady shear measurements for melt blended composite systems based on the fitting results of Table 4.4.

153

CI vs. Shear rate SC MB 1.00E+00 Power (MB) y = 0.0074x-0.472 Power (SC) 1.00E-01

1.00E-02 CI 1.00E-03

1.00E-04 y = 0.0001x-0.627 1.00E-05 0.01 0.1 1 10 Shear rate

Figure 4.14 Values of CI at each shear rate and the corresponding trendlines of MB and SC composites.

154

2% 1.00E+05 1.00E+06 5% 10% 1.00E+05 exp2% exp5% 1.00E+04 exp10% 2% 1.00E+04 5% 1.00E+03 10% exp2% exp5% 1.00E+02 exp10%

1.00E+03 1.00E+01 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E-02 1.00E-01 1.00E+00 1.00E+01

(a) (b) Figure 4.15 Model predictions of (a) the viscosity and (b) first normal stress difference compared to the experimental steady shear measurements for melt blended composite −0.507 systems based on CI = 0.0066γ .

155

1.E+05 1.E+05

1.E+04

1.E+03 1.E+04

G'(Pa) 1.E+02 model experiment model 1.E+01 (Pa.s) Viscosity experiment 1.E+00 1.E+03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E-04 1.E-02 1.E+00 1.E+02 Frequency (rad/s) Shear rate (1/s)

1.E+05 1.E+06

1.E+05 1.E+04 1.E+04

1.E+03 1.E+03 model G"(Pa) N1(Pa) model 1.E+02 experiment 1.E+02 experiment 1.E+01

1.E+01 1.E+00 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E-01 1.E+00 1.E+01 1.E+02 Frequency (rad/s) Shear rate (1/s)

(a) (b) Figure 4.16. (a) Fit of model to oscillatory shear measurements of solvent cast pure polymer melt, (b) Fit of model to steady shear measurements of solvent cast pure polymer melt.

156

model2 SC 1.00E+05 SC 1.00E+06 model5 model10 exp2 1.00E+05 exp5 exp10 1.00E+04 1.00E+04 model2 N1 1.00E+03 model5 model10 Viscosity (Pa.s) Viscosity exp2 1.00E+02 exp5 exp10 1.00E+03 1.00E+01 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E-02 1.00E-01 1.00E+00 1.00E+01 Shear rate (1/s) Shear rate (1/s)

(a) (b) Figure 4.17 Model predictions of (a) the viscosity and (b) first normal stress difference compared to the experimental steady shear measurements for melt blended composite systems based on the fitting results of Table 4.5.

157

2% 1.00E+06 1.00E+06 5% 10% 1.00E+05 exp2% 1.00E+05 exp5% 1.00E+04 exp10% 2% 5% 1.00E+03 10% 1.00E+04 exp2% exp5% 1.00E+02 exp10%

1.00E+03 1.00E+01 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E-02 1.00E-01 1.00E+00 1.00E+01

(a) (b) Figure 4.18 Model predictions of (a) the viscosity and (b) first normal stress difference compared to the experimental steady shear measurements for melt blended composite −0.729 systems based on CI = 0.0001γ .

158

MB2 1.00E+05 MB2 1.00E+06

Giesekus 1.00E+05 experiment Modified Giesekus 1.00E+04 1.00E+04 N1 1.00E+03 Viscosity (Pa.s) Viscosity Giesekus 1.00E+02 exp2 Modified Giesekus 1.00E+03 1.00E+01 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 Shear rate (1/s) Shear rate (1/s)

(a)

SC2 1.00E+05 SC2 1.00E+06

Giesekus 1.00E+05 experiment Modified Giesekus 1.00E+04 1.00E+04 N1 1.00E+03 Viscosity (Pa.s) Viscosity Giesekus 1.00E+02 exp2 Modified Giesekus 1.00E+03 1.00E+01 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 Shear rate (1/s) Shear rate (1/s)

(b) Figure 4.19 Comparisons of Giesekus model and Modified Giesekus model at mass concentration c=2wt% for (a) MB composite and (b) SC composite.

159 CHAPTER 5

TRANSIENT RHEOLOGY OF PS/CNF COMPOSITES AND CNF

ORIENTATION INDUCED BY FLOW

5.1 Introduction

In the previous chapter, we discussed linear viscoelasticity and steady state shear rheology of the PS/CNF composites. However, most the production processes are transient and involve large strain and strain rates. Also, in addition to the shear rheology, extensional rheology plays a large role in many processes, especially injection molding, blow molding and fiber spinning. As a result, a better understanding of the transient shear and extensional rheology of the composites are necessary.

The CNF orientation can be induced by the flow, but no literature has been published to experimentally observe and quantify the CNF orientation. However, the performance properties of the final parts are largely dependent on the CNF orientation.

Therefore, knowledge of the CNF orientation measurement, characterization of the CNF orientation induced by the flow, and relationships of the CNFs and rheology of the composite are needed.

160 This chapter is devoted to the investigation of the lacking knowledge on the transient shear and extensional rheology of the PS/CNF composites and the orientation of the CNFs induced by flow. Additionally, the effect of the CNF loadings and lengths will be examined. The results of this work can be used directly for the design of the processing equipment, such as extruders and injection molding machines, as well as in the fundamental research in the area of the particle filled viscoelastic media.

5.2 Experimental Procedures

The materials used in this chapter are the same as were used in the previous chapter. In summary, two types of the polystyrene (PS) /carbon nanofiber (CNF) composites will be prepared and used to study the effects of the CNF loadings and lengths. Melt blended (MB) composites and solvent cast (SC) composites with 2, 5, 10 wt% CNFs were used. The CNF lengths in SC composites are about 160 microns, and those in MB composites are about 40 microns.

The shear experiments will be conducted on a strain controlled rheometer (ARES

LS2 by TA Instruments) with 25 mm parallel plates. Two types of samples were made for shear rheology. One was made using an injection molding method with a center-gated mold. A micro-injection molding machine (DACA) was used. The barrel temperature was 200°C and the mold temperature was 60°C. The injection pressure was 50 psi. The other type of sample for shear rheology was made using a compression molding method.

Although the CNFs in the pellets were oriented in the flow direction through the 1mm die, the orientation of the pellets in the compressed disks was random. The sample disks were stored in a vacuum oven at 70°C prior to the measurement to prevent the absorption of

161 moisture or air; if stored in open air, bubbles are generated during the re-melting in the rheometer. Unless otherwise mentioned, compression molded disks were used in all shear tests.

Extensional rheology was measured using a Rheometrics Melt Extensiometer

(RME), which is basically a heating chamber and four rotating belts. A high resolution force measurement system is used to measure the force generated by the flow. The belts have rough surfaces with tongues so that good gripping of the sample can be achieved.

Pins were also used to prevent the molten sample from being excessively compressed before tests. Details of the structure and the usage of the RME can be found in its user manual. All tests using the RME were conducted at 170°C. Unless otherwise noted, all samples for RME tests were made using the compression molding method.

Due to the possible error in the stretching speed, a video camera was used to calibrate the extension rate. Pure polystyrene was used and the width of the sample is measured from the pictures extracted from the video. One picture was taken every second.

Photos at every 10 seconds from start of stretching till the sample breaks are shown in

Figure 5.1. The width of the stretched sample was measured at three points: left, middle and right of the air table (the copper rectangular shaped object as seen in the photos).

Average values of the three measurements from each pictures were calculated and a best- fit (trend) line was created using Microsoft Excel. Note that the average is taken here because the measured widths of the sample at different points are not the same due to the unevenness of the illuminating light. In the middle of the sample where the light was brighter, the sample appeared wider. Thus average values of the widths were used. The results of the width measurements are shown in Figure 5.2. The speed of change in the

162 width is one half of the true extensional rate 131. The slope of the fit multiplied by -2 gives the true extensional rate, which is 0.077 s-1 for a nominal extensional rate of 0.1 s-1.

Similar result was also found by other researchers 171. The calibration was done at 145°C for shear rate of 0.1 s-1. This result is assumed applicable to tests at 170°C for all extensional rates.

5.3 Experimental Results

5.3.1 Transient shear rheology

The stress response upon start up of steady shear was measured with the SC0 and

SC10 composites at different shear rates to investigate the evaluation of the CNF orientation and structure. Figure 5.3a,b display the results of the measured transient shear viscosity of these composites. The pure polymer exhibits typical behavior of a homogeneous linear polymer. At small shear rates, e.g. less than and equal to 0.01 s-1, the

SC0 shows linear viscoelasticity. The transient shear viscosity is independent of shear rate and the final steady state viscosities are identical. At higher shear rates, shear thinning and stress overshoot appear. The transient shear viscosities of the SC10 composite at relatively large shear rates, e.g. larger than 0.1 s-1, are very similar to those of the SC0 composite. However, at smaller shear rates, the transient viscosity as a function of time is quite different from that of pure polymer. Most significantly, the linear viscoelastic behavior is not observed. Instead of falling on top on each other, the transient shear viscosity at small shear rates, e.g. less than 0.01 s-1, increases markedly with decreasing shear rates. In other words, more shear-thinning behavior is observed.

163 The differences in the transient shear behavior between SC0 and SC10 can be seen more clearly when the reduced stress is plotted with strain, as seen in Figure 5.4.

+ The reduced stress is defined as the ratio of transient stress to steady state stress τ /τ∞.

Shear rates between 0.1 and 10 s-1 were probed. The magnitude of the overshoot is a strong function of shear rate for both pure PS and SC10 composite. Figure 5.4a displays the reduced stress for pure PS. At shear rate of 0.1 s-1 and higher, measurable stress overshoot can be detected. Higher shear rates give rise to larger overshoot. In contrast with the reduced stress of SC10 composites (Figure 5.4a), three distinct features can be found. First, the height of the overshoot for SC10 composite is always larger than that of pure PS at the same shear rate. Second, the positions of the peek of the overshoot of the pure PS are almost identical for all shear rates, while they move towards smaller strain units for SC10 composites at higher shear rates. Thirdly, the widths of the overshoot are about the same for pure PS for all shear rates, and they decrease with higher shear rates for SC10 composites. The width of the overshoot is defined as the distance between the two points where the reduced stress intersects with the horizontal line where the reduced stress equals one.

All three features are due to the CNFs in the SC10 composite. More pronounced overshoot in SC10 composite is due to the change of the CNF nanostructure. The CNFs change their orientation during the flow. Before reaching steady state, they introduce extra stress to the composite, leading to stress overshoot. The shift of the peek position is the combination of the effects of the CNFs and polymer in the SC10 composites. For comparison, the positions of the peeks are the same, independent of the shear rate for the pure polymer. This is typical for most homogeneous polymers and composites in which

164 the fillers are non-Brownian 110, 115. However, although the CNFs in the SC10 composites are also non-Brownian, the positions of the overshoot become dependent on the shear rates. This behavior is mostly likely caused by the interactions of the CNFs. The shear flow tends to cause the fibers to tumble, while the inter-particle interactions may prevent the CNFs from tumbling. The flow competes with the interactions, such that the stress response is the result of these two actions. In the un-sheared sample, the CNFs are randomly oriented. When the sample is sheared, some CNFs are aligned to the flow direction induced by shear flow. However, if the shear rate is small, the inter-particle interactions dominate. This prevents the CNFs from being aligned to the flow direction, which means that the CNFs are still close to random even when steady state is reached. If the shear rate is large, the flow destroys the inter-particle interactions and aligns the

CNFs to the flow direction better. The result of the competition between the flow and inter-particle interactions is that the peek of the overshoot occurs at a smaller strain units for smaller shear rate flows than the larger ones. Moreover, if we observe Figure 5.4 carefully, we can see that the positions of the overshoot of SC0 and SC10 are about the same at 10 s-1 shear rates, and at smaller shear rates the overshoot occurs at a smaller strain for SC10 composites than for SC0 (pure polymer). This is also an indication that the strength of the inter-particle interactions of the CNFs is relatively large at smaller shear rates, and it can be neglected at higher shear rates. The width of the overshoot might also be the effect of the inter-particle interactions. As they are relatively stronger at smaller shear rates, the competition between the interactions and the flow lasts longer, and so the overshoot becomes wider. At higher shear rates, flow dominates so that the widths of the overshoot of SC10 and SC0 are almost the same.

165 The effects of inter-particle interactions of CNFs are also seen in the start up of steady shear experiments for all MB and SC composites as a function of CNF loading.

Figure 5.5 shows the reduced stress of both MB and SC composites in the start up of steady shear at 1 s-1 shear rate. For MB composites (Figure 5.5a), the reduced stress is about the same with CNF loadings of 5 wt% and lower. At 10wt% CNF loading, the reduced stress is markedly higher. In contrast, The SC composites show much higher reduced stress than MB composites. At the same CNF loading, the SC composites

(Figure 5.5b) show significantly higher overshoot than MB composites (Figure 5.5a).

This is the effect of the length of the CNFs. As the CNFs in the SC composites are much longer (aspect ratio of 160 in SC composites and 40 in MB composites), both the extra stress caused by pure length and the stress resulting from the more inter-particle interactions between CNFs are larger. Longer CNFs have more chances to contact each other and possibly entangle, therefore the inter-particle interactions of SC composites are larger than MB composites. That’s why larger stress overshoots are seen in the SC composites. In particular, the effects of the inter-particle interactions of CNFs can be seen in both MB composites and SC composites. The peeks of the overshoots shift to the left with higher CNF loadings, which can be contributed to higher interactions. If there were no inter-particle interactions, positions of the overshoots would be the same, as shown by other researchers26, 110, 115. We can safely claim that this is the effect of increasing inter- particle interactions at higher CNF loadings since the overshoots shift to the left. Larger interactions prevent CNFs from tumbling, such that the peek of the overshoot

166 occurs at smaller strain units. This is similar to the case when shear rate becomes smaller, and interactions becomes relatively larger, and the overshoots shift to the left, as seen in

Figure 5.4b.

The evolution of the CNF structure was further probed with the flow reversal experiments. In this experiment, the sample was sheared at 1 s-1 for 40 seconds, then the shear stopped and the sample was allowed to rest for a certain period of time. After that, the shear was re-started at the same shear rate but in the opposite direction, and the stress response was recorded with time. Figure 5.6a,b shows the reduced stress of pure PS and

SC10 composite as a function of time. For pure PS, the stress overshoot is small if the reversed flow is started without rest. When the sample is allowed to rest for 120s or longer, the pure PS is fully relaxed. The stress overshoot in the reversed flow is exactly the same as the forward flow (Figure 5.6a). In the case of SC10 composites, however, the overshoot in the reversed flow is a much stronger function of the rest time (Figure 5.6b).

When the rest time is zero, the stress gradually increases to reach steady state, showing no overshoot at all. If more rest time is allowed, an overshoot starts to appear. At 1200s rest time, pronounced overshoot appears and if the rest time is 2400s, larger overshoot is seen. Surprisingly, even for a rest time of 2400s, the nanostructure still did not relax to its un-sheared state, as the overshoot of the initial forward startup is still much higher. The dependence of the stress response in the reversed flow on the length of the rest time is a clear indication that the CNF nanostructure is able to evolve even under quiescent conditions.

167 First normal stress difference (N1) of the same tests are shown in Figure 5.7. For pure polymer matrix (SC0), the N1 is about 5000 Pa for all the flows (Figure 5.7a) at steady state. However, for SC10 composites, N1 is found to be not only to be a function of time during the reverse flow but also a strong function of the rest time before the reverse flow. N1 of the initial forward startup increases rapidly and then decreases from zero slowly to reach steady of about 20000 to 30000 Pa. When there is no rest time, an immediate reverse flow generates N1 that is quite different from the forward startup flow.

The N1 decreases rapidly from 28000 Pa to nearly zero, and then increases slowly approaching the same steady state value (20000 to 30000 Pa). When the sample has 120 seconds rest time or more, the N1 relaxes to near zero at the start of the reverse flow and then increases gradually until steady state is reached. Also, longer rest time leads to larger transient N1 in the reverse flow before steady state is reached. This result further shows that the CNF structure evolutes even after the flow has stopped.

Another flow reversal experiment was done to study the effect of pre-shear on the transient shear stress. In this test, the sample was first sheared at 1 s-1 for 40 seconds, and then the shear stops and the sample was allowed to rest for 120 seconds. After that, the shear was re-started, either in the same direction as the pre-shear or in the reversed direction. The results are shown in Figure 5.8. For pure PS, the reduced stress for all the three flows are identical (Figure 5.8a). The rest time of 120s is long enough to relax the polymer chains to their original un-sheared condition. In the SC10 composite, the second flow in the same direction with the first flow gives a smaller stress overshoot, since the

CNFs have been aligned by the first flow and they have not relaxed to the un-sheared condition after 120s of rest time. However, the more interesting phenomenon is that if the

168 second flow is in the reversed direction, there is no overshoot at all. This is in contradiction to what we had expected, which was that a reverse flow will generate more fiber tumbling so that the overshoot should be larger than a same direction flow. One possible explanation could be that the inter-particle interactions in the SC10 composite prevent CNFs from tumbling in the reverse flow once the fibers are aligned to one direction.

Figure 5.9 depicts the change of N1 of pure PS and SC10. The results show that after 120 seconds of rest time, a same direction flow generates larger N1 than a reverse direction flow before reaching the same steady state. The result here and that in Figure

5.7 seem to suggest that when the network structure of the CNFs is stretched, larger N1 will arise. This is similar to the concept that for polymer melts normal stress appears when the polymer chains are deformed away from their equilibrium configurations by the flow.131 If the network structure is allowed to return to its un-stretched equilibrium state,

N1 decreases. This theory can explain all the measured N1 both in transient tests (Figure

5.7 and Figure 5.9 ) and in steady states (Figure 4.10 in the previous chapter). Before shear, the CNFs are randomly oriented, so that the network structure is in an un-stretched state, hence the N1 is zero. Upon shear, the structure is stretched and N1 increases. If there is an overshoot in N1, then it means that the stretching speed is larger than the relaxation speed. Steady state is reached when the stretching and relaxation reaches equilibrium. Higher shear rate flows stretch the network structure more and so the N1 increases. That is why N1 at steady state increases with shear rate (Figure 4.10 in the previous chapter). If steady state is reached and the shear is re-started in the reversed direction with various length of rest time (Figure 5.7), the network will relax to a

169 different state during the rest time. If there is no rest, the reverse flow deforms the stretched network structure in the opposite direction, which is essentially equivalent to relaxing the structure. Thus, the N1 decreases to zero from the previously established steady state value. As the reversed flow continues, the network structure is stretched further in the opposite direction and so the N1 increases again. When steady state is reached again in the reversed flow, the morphology of the network structure is no different than the forward flow, such that the same N1 at steady state results. If the sample is allowed to rest for a period of time, the network structure tends to relax to its un-stretched state. Longer rest time will lead to a state closer to the fully relaxed state. As a result, the N1 for reversed flow with rest time of 120s or more starts at around zero and gradually increases. Longer rest time leads to a faster arrival of the steady state in the reverse flow, and generates higher N1. In the case of the second flow being same direction flow or reversed direction flow as in Figure 5.9, the network structure in the same direction flow is still largely preserved due to the slow relaxation process, so that its stretched condition is attained soon once the same direction flow starts. A schematic drawing of this network structure evolution during flow reversal test is shown in Figure

5.10.

5.3.2 Orientation Characterization

As one can imagine, the rheology of the polymer/CNF composites is affected by the CNF orientation. In order to find out the relationship between the rheology and CNF orientation, we need to be able to measure both and then use models to couple them. In this study, we have developed a novel way to prepare the samples and measure the CNF orientation. The sample is deformed at elevated temperature and then quenched using dry

170 ice to solidify the sample, which fixes the CNF orientation. The sample is then cut in two perpendicular planes and TEM is used to examine the angle of the CNF with respect to the flow direction. The thickness of the TEM section is 800 nm. Challenges in this analysis include inference of 3-D orientation from 2-D micrographs, and the interplay between the nano-length-scales of particle geometry, width of a TEM section of the nanocomposite, and the transparency of the section. Figure 5.11 are TEM micrographs showing the orientation induced during extensional flow in two perpendicular planes and histograms of the particle orientation and length distribution. These micrographs serve as benchmarks of the alignment of the CNFs in a plane parallel to the flow direction and a plane perpendicular to the flow direction in a nearly perfectly aligned sample. It is clear that the TEM micrograph of the plane parallel to the flow direction shows strong CNF alignment to the flow direction. The micrograph of the plane perpendicular to the flow direction shows a random orientation, as we expected. Another significant difference is that the lengths of the CNFs in the parallel plane are much longer than those in the perpendicular plane. This is understandable, since the CNFs are aligned in the parallel plane, so that the full length can be seen in the picture. While in the perpendicular plane, most of the CNFs were cut, so that we can only see the cross section of the CNFs. One of the most significant marks to identify a cross section view is the existence of a circular shape, as seen in Figure 5.11b. This is the key to differentiate a short CNF from a cross section of a long CNF.

The two 2-D pictures, such as those in Figure 5.11, can be viewed as projections of 3-D fibers on two orthogonal planes. We have attempted to construct 3-D orientation distribution function from these two 2-D TEM micrographs, but we met insurmountable

171 obstacles. The key missing information is that one fiber seen in one plane cannot be paired with its counterpart in another plane. In another word, we don’t know which two projections come from the same fiber. Due to our imaging technique (TEM), there is no way to build this pair-wise relationship. The microtoming process used to prepare TEM sections destroys the sample. As a result, there is no way to reconstruct 3-D orientation from two 2-D projections.

Fortunately, the injection molded disks used for shear rheology have a clearly defined planner orientation, such that most of the CNFs are oriented in the thickness plane. TEM micrographs of as molded samples and samples subjected to known shear history show that the CNFs are long in the thickness plane and short in the cross section plane. With this information, we can quantify 2-D CNF orientation using an orientation tensor. The computation of the 2-D orientation tensor is exactly the same for the 3-D orientation tensor as seen in the previous tensor, only with setting the out-of-the-plane angle θ to 90°. The 2-D orientation tensor for the shear sample (Figure 5.15) is

⎡⎤aa11 13 aij = ⎢⎥ (5.1) ⎣⎦aa13 33

a11 = (5.2)

a13 = (5.3)

a33 = (5.4)

where φ is the angle of the CNF with respect to the axis 1 (see Figure 5.15). Angle brackets < > is a length weighing operator using the length of the kth fiber

172 k ∑ Lakij = k . (5.5) ∑ Lk k

Special cases of the 2-D orientation tensor aij are shown in Figure 5.16. When the fibers are perfectly aligned to direction 1, then the a11=1, and all other elements of the tensor are zero. Similarly, when the fibers are perfectly aligned to direction 2, a22=0 and all others are zero. If the fibers are aligned with a 45° angle to direction 1 and 2, then all the elements of the tensor are 0.5.

The endpoints of a CNF were connected by a straight line. In a typical photo, 600 to 2000 fibers were digitized in this way. The lines were drawn manually in AutoCAD with the TEM photos as the background. Thus, the positions of endpoints of the fibers are determined. The lengths of the lines and the angles with respect to a baseline were then calculated using the functions provided by the AutoCAD. A program coded in AutoLISP was made to automate the calculation. The results, containing measured angles and lengths, were exported and used for the calculation of the 2-D orientation tensor aij.

Part of the reasons why the CNFs are only oriented in the thickness plane rather than tumbling in the plane of shear direction and gradient of the shear is due to the normal force of the polymer matrix. As pointed out by Laun 172, the possible tumbling of fibers distorts the stream lines and introduces normal forces. The resulting torque can prevent the fiber from tumbling and keep it within limited angles from the flow direction.

In fact, that is also the reason why an injection molded disk has a planner CNF orientation in the first place.

173 Another reason we used injection molded sample disks for the orientation study is that due to the fact that the CNF orientation can be induced by the processing, it is very difficult to make samples for rheological tests in which the CNFs are randomly oriented.

In order to study the evolution of the orientation, we need to know the orientation of the

CNFs before and after a known deformation. As a result, we used the injection molding method to create samples with repeatable CNF orientation. This will serve as the initial state and will be used to compare with CNF orientation after the sample has been subjected to strain, so that the whole history of CNF orientation can be deduced. For the shear rheology, which requires disk shaped samples, a center-gated mold with diameter of 25mm and thickness of 1 mm was used. For the extensional rheology where a bar shaped sample is used, we used a tensile bar mold. The middle of tensile bar was cut out and further machined to the required dimensions. The CNF orientation in the disks was characterized using the technique mentioned above and the results are shown in Figure

5.12. In this specimen, the CNFs orient circumferentially in the mid-plane, while radially near the surface. The compression molded samples contains macroscopically randomly oriented CNF. A schematic drawing of the CNF orientation is shown in Figure 5.13.

5.3.3 Effect of pre-orientation on transient shear

With the ability to create and measure the CNF orientation in the samples for shear rheology, the effect of CNF pre-orientation on the rheology of the PS/CNF composites was studied. Shear samples made by the injection molding method and compression molding method were made and they were used to measure transient shear rheology. Figure 5.14a shows the different stress response of the start up of steady shear test at 0.1 s-1 shear rate of SC5 composite. The reduced stress of the injection molded

174 sample with pre-orientation shows a large overshoot before reaching steady state. On the other hand, the reduced stress of the compression molded sample with random CNF orientation shows a small overshoot followed by a large undershoot, before reaching a steady state.

However, the effect of CNF pre-orientation on the transient stress is negligible for the MB5 composites. As seen in Figure 5.14b, the reduced stresses for sample of MB5 composite with random orientation and pre-oriented orientation are identical. Contrasting

Figure 5.14a to b, the CNFs in SC5 composites play a larger role than those in MB5 composites. As the average length of CNFs in SC composites are much longer than that of MB composites, this clearly identifies that longer CNFs have larger impact on the effect of CNF pre-orientation on transient shear.

We conjecture that there are two mechanisms involved. One is that longer CNFs need more time to rotate, producing larger extra stress, so that different pre-orientation leads to different stress response. The other mechanism comes from the inter-particle interactions of the CNFs. Longer CNFs generate larger interactions, and different initial

CNF orientation leads to different degree of interactions. During shear, these interactions may be broken down by the flow and the CNFs may be oriented to the flow, so that the amount of excess stress produced by these interactions reduces. The evolution of the interactions and CNF orientations varies with initial orientations, resulting in different shear response.

175 5.3.4 Shear induced orientation at steady state

To facilitate discussion in the orientation, the coordination system of the shear sample is shown in Figure 5.15. The CNF orientation subject to simple shear at steady state has been studied. The results are shown in Figure 5.17. The pictures shown in the figure are obtained from sections on the surface and parallel to the surface and 1mm from the edge. The position of the section is shown as red box in the figure. Figure 5.17a shows the CNF orientation of an as-molded sample using the injection molding method, indicating that the CNFs are oriented to radially. From the values of the orientation tensor, we see that a11 = 0.33 and it is much smaller than a33 which equals 0.67. This means that more fibers are aligned in direction 3 (radius direction) than in direction 1

(shear flow direction). After the sample has been sheared at 0.1 s-1 for 60 strain units, the

CNFs are re-oriented to the flow direction, as seen in Figure 5.17b. a11 increased to 0.46.

However, if the sample is sheared at 10 s-1 for the same amount of strain units, the CNFs can be aligned even more to the flow direction Figure 5.17c. a11 is 0.64, larger than a33.

At 60 strain units, steady state is reached for both 0.1 s-1 and 10 s-1 shear rates. This result clearly shows that the CNF orientation can be oriented to the flow direction, and more interestingly, the stronger alignment to the flow direction is obtained with larger shear rates. More detailed results showing TEM micrographs in both 1-3 plane and 2-3 plane and histograms of orientation angles and lengths are shown in Figure 5.18 to Figure 5.20.

Many constitutive equations that have been developed so far can only predict fiber orientation as a function of strain, for example, Jeffory’s equation 51 and Folger and

30 Tucker’s modification with CI as a constant . According to their results, different strain rate will produce the same orientation at steady state. Fiber orientation as a function of

176 shear rate has also been found by several other researchers. Ramazani et al. 26 studied transient shear rheology of 15 vol% fiber-filled highly elastic fluid (fiber length 1/8” mesh size) and found that pre-shear at different shear rates generated different subsequent transient shear responses. From this observation, they claimed that the fiber orientation induced by the pre-shear depends on the shear rates of the pre-shear, and increasing fiber alignment to the flow direction is found with increasing shear rate. Similar phenomena is also found by Vaxman et al. 173 in the extrudates of their glass-fiber filled polyphenylene ether/polystyrene blend (Nyrol) and also by Mutal for glass fiber filled polypropylene 174.

5.3.5 Shear induced orientation at different shear strain

The evolution of the CNF orientation is further studied by measuring the CNF orientation at different strain units at the shear rate of 10 s-1. This relatively large shear rate (10 s-1) was chosen because we have found that flows with larger shear rate align

CNFs better to the flow direction. Therefore, larger change in orientation could be seen at

10 s-1. The TEM micrographs at the strain unit of 0, 0.4, 3 and 6 superimposed with transient viscosity are shown in Figure 5.21. Before the shear starts, the CNFs are mainly oriented radially, and a11 = 0.33. After 0.4 strain unit of shear at 10s-1, there is a significant change in the CNF orientation, as more CNFs are oriented to the shear flow direction, a11 = 0.58. At 3 strain units, slightly more CNFs are aligned to the flow direction, but the change is much smaller than the first 0.4 strain units. At this time, a11 =

0.63. When the shear comes to 6 strain units, there is almost no difference in the orientation with a11=0.64. The transient viscosity reaches the peak of the overshoot at

177 0.4 strain unit, and at 3 strain units steady state is reached. Therefore, most of the change in the CNF re-orientation has completed by the peak of the viscosity overshoot. Detailed results are shown in Figure 5.18, Figure 5.20, Figure 5.22 and Figure 5.23.

Just as we have discussed above that different initial CNF orientation in the shear samples does not have significant effect on the shear response, now we study if measurable change in CNF orientation in MB5 composites introduces any difference in stress response. In other words, if the CNF re-orientation induced by shear flow generates different shear response than the PS matrix. Figure 5.24 shows the result of reduced stress of MB0 and MB5 composites sheared at 10 s-1. The reduced transient stresses are identical for these two composites. This further confirms that although the addition of

CNFs to polymer increases the viscosity of the composites, the effect of CNF orientation is negligible for MB5 composites at relatively high shear rates (10 s-1). Here we don’t claim that the effect of CNF orientation is negligible for MB composites with any CNF concentrations, but we are more likely to believe that this is the case for 5wt% and lower.

5.3.6 Transient extensional rheology

Transient extensional rheology of both MB and SC composites were measured at

170°C. The temperature is different than that tested for shear rheology (200°C) because at 170°C the sample can generate enough stress for the transducer to measure. At higher temperature, the sample is too soft and the stress is too small. At lower temperatures, the sample is too hard such that the gripping of the belt to the sample is a problem. Our preliminary experiments performed at 150°C showed a considerable amount of slippage between sample and belts (results not shown here).

178 Figure 5.25a,b show transient viscosity of MB composites and SC composites as a function of time at the extensional rate of 0.1 s-1. The transient viscosities of both MB and

SC composites increase with higher CNF loadings. More increase is found in the SC composites. The effect of the length of the CNFs on the extensional viscosity is similar to that on the shear viscosity.

Now let’s see the effect of the CNFs on the extensional viscosity at different extensional rates. Figure 5.26a shows the transient viscosity of the pure PS matrix (MB0).

It is seen that at relatively low extensional rate, the transient extensional viscosity is very close to the 3 times of the transient shear viscosity. As the extensional rate increases, strain hardening appears. The strain hardening is more clearly seen in Figure 5.26b, where the transient Trouton ratio is plotted against time. The Trouton ratio is defined as the ratio of the transient extensional viscosity to the transient shear viscosity. For

Newtonian and linear viscoelastic fluids, the Trouton ratio is 3. Now we see that for pure

PS, the Trouton ratio for extensional rate of 0.077 and 0.23 s-1 is very close to 3, indicating that at these extensional rates, the PS is very close to the linear viscoelasticity regime. At 0.77 s-1 extensional rate, apparent strain hardening occurs, as the maximum

Trouton ratio increases to 3.8.

The presence of the CNFs greatly changes the transient viscosity of the composite. With 5wt% CNF, the transient extensional viscosity is higher than that of the pure PS (Figure 5.27a). More importantly, instead of exhibiting strain hardening, the

MB5 shows strain softening behavior. The extensional viscosity at larger extensional rates is enclosed in the extensional viscosity at lower extensional rates. The Trouton ratio as a function of time, shown in Figure 5.27b, describes this behavior more clearly. The

179 maximum Trouton ratio decreases from more than 5 at the extensional rate of 0.023 and

0.077 s-1 to a little higher than 4 when the extensional rates increase to 0.77 s-1. On the other hand, at low extensional rates, say 0.077 s-1, the maximum Trouton ratio of MB5 is about 5, much higher than that for pure PS, which is about 3.

Many molten polymers exhibit strain hardening behavior, especially ones with linear molecular structure 131. According to the Doi and Edward theory 50, strain hardening occurs when the extension rate is faster than the relaxation of the polymer chain, so that the contour of the chain is stretched beyond its equilibrium length. The tendency of the polymer chain to retract back to equilibrium creates additional stress, causing higher extensional viscosity, and so strain hardening appears. Higher extensional rates give rise to more strain hardening. That is why we see that the pure PS shows strain hardening behavior at the extensional rate of 0.77 s-1. However, the presence of CNFs in the composites may prevent the polymer chains from stretching. At low extensional rates, the polymer chains are not stretched anyway, but the interactions of the CNFs increases stress more in the extensional flow so that the extensional viscosity increases. These interactions are more important in extensional flow than shear flow, so that the Trouton ratio increases. When the extensional rate is higher, the polymer chains are confined by the CNFs, so that the possible strain hardening coming from the stretching of the polymer chain does not occur. At the same time, the higher extensional rate aligns the CNFs better so that the extensional viscosity drops, just like the case of shear thinning behavior for the shear flows. Further, the Trouton ratio drops too, as the effect of alignment of the CNFs

180 is larger on the extensional viscosity than on the shear viscosity. Other researchers have also reported that solid particles in polymer melts cause weakened strain hardening behavior 175-177.

The CNF orientation was measured for the extensional flows. Bar shaped samples were used. Due to the fact that the bars were machined from injection molded tensile bars, they shrank in the RME chamber when heated. Therefore, a metal mold with rectangular cavities which have the same dimensions of the bars was used. The machined bars were inserted into the cavities and sandwiched by another two metal plates on both sides of the mold. The sandwich was then pressed tightly and heated to 180°C under pressure to anneal the bars. Very small amount of flow occurred during this annealing process. The annealed bars were then used to measure the extensional viscosities.

Orientation of the CNFs was measured at the center of the bar in two perpendicular planes before the stretching and during the stretching with different strain units. Samples were cooled with dry ice to fix the CNF orientation. The results show that the CNFs were aligned to the stretching direction before stretching. Same CNF orientation was also found for samples after the stretching. These results show that the CNFs are aligned to the stretching direction. However, the results are not so interesting and TEM pictures are not shown here.

5.4 Conclusions

Transient shear rheology of PS/CNF composites has been studied. Inter-particle interactions in the composites play a large role in determining the rheological properties, especially at low shear rates. The peaks of the stress overshoot in the start up of steady shear tests shifts to smaller strain units for SC10 composites with increasing shear rate,

181 and the widths of the overshoot become narrower. At the same CNF concentration, the height of stress overshoot is higher for SC composites, as the lengths of the CNFs in SC composites are much longer than those in MB composites. Longer CNFs also increases

N1 in the SC composites.

Flow reversal experiments shows that the CNF structure in SC10 composites evolutes under quiescent conditions, as the stress response in the reverse flow shows strong dependence on the length of the rest time. The first normal stress difference N1 also changes with different amounts of rest time. Possible reasons for these experimental observations are that the CNF network structure evolutes under shear, and during rest the structure rebuild to a certain degree. Higher N1 results from the stretched network structure. Different amounts of rest time generate different structure states and give rise to the observed stress and N1 response.

CNF orientation has been experimentally quantified. 2-D TEM photographs have been taken for shear samples at different flow conditions. The results show that flow with higher shear rates induces better alignment of CNFs to the shear direction than low shear rate flows. Also, major change of the CNFs occurs at the peak of the stress overshoot.

The pre-orientation of the CNFs have a significant effect on the transient shear rheology at small shear rates.

Transient extensional viscosities of the composites were measured and we found that higher CNF loading and longer CNFs will increase the transient extensional viscosities. The presence of the CNFs suppresses the strain hardening behavior of the composites.

182

0 s 10s

20s 30s

40s 50s

60s 62s

Figure 5.1.Photos of samples during RME tests. Sample was made of polystyrene.

Extension rate = 0.1 s-1.

183 10

y = 6.829e-0.0384x

Left 1 Middle

Width (mm) Right Avg Expon. (Avg)

0.1 020406080 Time (s)

Figure 5.2. Widths of the stretch sample measured at left, middle and right of the air table. Black line is the best fit of the average width.

184 1E+5 1E+07 0.0001

1E+06 0.001 0.01 0.1 1E+4 0.3 1E+05 0.01 1 0.1 3 0.3 Viscosity (Pa.s) Viscosity Viscosity (Pa.s) 1 1E+04 10 3 10

1E+3 1E+03 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 0.01 1 100 10000 Time (s) Time (s)

Figure 5.3. Transient shear viscosity η+ at shear rates 0.0001-10s-1. of (a) SC0 (Curves for 0.01 s-1 and below fall on top of each other.) and (b) SC10 composite. T=200°C.

185 1.5 1.5

1.4 10 1.4 10 3 3 1.3 1.3 1 1 1.2 1.2 0.3 0.3 0.1 1.1 1.1 0.1 1 1

0.9 0.9 Reduced stress Reduced stress 0.8 0.8

0.7 0.7 0.6 0.6 0.01 0.1 1 10 100 0.01 0.1 1 10 100 strain Strain

Figure 5.4. Start up of steady shear at different shear rates. (a) pure PS (SC0) and (b) SC10 composite. T=200°C.

186 1.2 1.2 -1 startup 1 s-1 Start up 1 s 1.15 1.15 MB10 SC10 MB5 SC5 1.1 1.1 MB2 SC2 MB0 SC0 1.05 1.05

1 1 Reduced Stress Reduced Reduced Stress Reduced

0.95 0.95

0.9 0.9 0.1 1 10 100 0.1 1 10 100 Time (s) Time (s) Figure 5.5. Start up of steady shear as a function of strain at shear rate of 1s-1 of (a) MB composites and (b) SC composites. T=200°C.

187 1.2 1200s rev 1.15 120s rev 1.1 0s rev 1.05 Forw ard startup 1

0.95

Reduced stress 0.9

0.85

0.8 010203040 Time (s)

1.2 2400s 1.15 1200s 120s 1.1 0s Forward startup 1.05

1

0.95

Reduced stress Reduced 0.9 0.85

0.8 0 10203040

Time (s)

Figure 5.6. Reduced stress of flow reversal tests at shear rate of 1s-1 of (a) pure PS (SC0) and (b) SC10 composites. T=200°C.

188 60000

50000 1200s rev 40000 120s rev 0s rev 30000 startup

N1 (Pa) N1 20000

10000

0

-10000 0 10203040 Time (s)

(a) 60000 2400s 50000 1200s 120s 0s 40000 Forward startup

30000

20000 N1 (Pa)

10000

0

-10000 010203040

Time (s)

(b) Figure 5.7. First normal stress difference (N1) of flow reversal tests at shear rate of 1s-1 of (a) pure PS (SC0) and (b) SC10 composites. T=200°C.

189

1.2 120s rev 1.15 120s same dir 1.1 Forw ard startup

1.05

1 Reduced stress

0.95

0.9 010203040 Time (s)

1.2 120s rev 1.15 120s same Forward startup 1.1

1.05

1 Reduced stress Reduced

0.95

0.9 0 10203040

Time (s)

Figure 5.8. Effect of pre-shear on the transient shear stress at shear rate of 1s-1 of (a) pure PS (SC0) and (b) SC10 composites. T=200°C.

190 60000

50000 Forw ard startup 40000 120s same dir

30000 120s rev

N1 (Pa) N1 20000

10000

0

-10000 0 10203040 Time (s)

(a)

60000 Forward startup 50000 120s same dir 120s opposite dir 40000

30000

20000 N1 (Pa)

10000

0

-10000 010203040

Time (s)

(b)

Figure 5.9. Effect of pre-shear on the transient N1 at shear rate of 1s-1 of (a) pure PS (SC0) and (b) SC10 composites. T=200°C.

191

Figure 5.10. Schematic drawings of the evolution of network structure in flow reversal tests.

192

(a) (b)

300 100 90 250 80

200 70 60 150 50 40 Frequency Frequency 100 30 20 50 10 0 0 -90 -60 -30 0 30 60 90 -90 -70 -50 -30 -10 10 30 50 70 90 Angle Angles with repect to flow direction (degree) (c) (d)

Figure 5.11. CNF orientation induced by extensional flow. (a) plane parallel to the flow, (b) plane perpendicular to the flow, (c) histogram of orientation with respect to flow direction in the plane parallel to the flow, and (d) histogram of orientation with respect to cutting direction in the plane perpendicular to the flow direction.

193

(a) (b)

Figure 5.12. Schematic patterns of nanostructure in a disk-shaped shear rheometry specimen made by injection molding method. The size of the nanoparticles is greatly exaggerated in these schematics.

Figure 5.13. Schematic patterns of nanostructure in a disk-shaped shear rheometry specimen made by compression molding method. The size of the nanoparticles is greatly exaggerated in these schematics.

194 1.1

1.05

1 Reduced stress 0.95 pre-oriented random

0.9 0 200 400 600 800 Time (s)

1.1

1.05

1 Reduced stress 0.95 random

pre-oriented

0.9 0 50 100 150 200 250 300 Time (s)

Figure 5.14. Effects of pre-orientation of the CNFs on the transient stress. (a) SC5 composite, (b) MB5 composites. Shear rate = 0.1 s-1. T=200°C.

195

Figure 5.15. Coordination system of the shear sample. Shaded areas are the positions of the slices used for TEM.

196 ⎡10⎤ aij = ⎢ ⎥ ⎣00⎦

⎡00⎤ aij = ⎢ ⎥ ⎣01⎦

⎡0.5 0.5⎤ aij = ⎢ ⎥ ⎣0.5 0.5⎦

Figure 5.16. Special cases of the 2-D orientation tensor.

197

⎡ 0.33 − 0.01⎤ ⎡ 0.46 − 0.08⎤ ⎡0.64 0.06⎤ aij = ⎢ ⎥ aij = ⎢ ⎥ aij = ⎢ ⎥ ⎣− 0.01 0.67 ⎦ ⎣− 0.08 0.54 ⎦ ⎣0.06 0.36⎦

Figure 5.17. Effect of shear rate on the shear induced CNF orientation at steady state condition T=200°C. MB5 composite. Injection molded disk samples using center gated mold.

198 2-3 plane (cross section) 2-3 plane (surface)

300 1000

200

500 Frequency Frequency 100

0 0

0 1 2 3 4 5 6 0 1 2 3 4 Length Length

200 200

100 100 Frequency Frequency

0 0

-100 0 100 -100 0 100 Angle Angle

Figure 5.18. TEM micrograph of the un-sheared sample in 1-3 plane and 2-3 plane. MB5 composite. Injection molded disk samples using center gated mold.

199 2-3 plane (cross section) 1-3 plane (surface)

800 400 700 600 300 500 400 200 300 Frequency Frequency 200 100 100 0 0

0 1 2 3 0 1 2 3 4 5 Length Length

90

150 80 70 60 100 50 40 30 Frequency 50 Frequency 20 10 0 0

-100 0 100 -100 0 100 Angle Angle

Figure 5.19. TEM micrograph of the sample sheared at 0.1 s-1 for 600s (60 strain units). MB5 composite. Injection molded disk samples using center gated mold.

200 2-3 plane (cross section) 1-3 plane (surface)

200 150

100

100 Frequency Frequency 50

0 0

0 1 2 0 1 2 3 Length

150 70

60

50 100 40

30 50 Frequency Frequency 20

10

0 0

-100 0 100 -100 0 100 Angle

Figure 5.20. TEM micrograph of the sample sheared at 10 s-1 for 6s (60 strain units). MB5 composite. Injection molded disk samples using center gated mold.

201 8000

7000

6000

5000 ⎡ 0.58 − 0.07⎤ aij = ⎡ 0.63 − 0.03⎤ ⎢− 0.07 0.42 ⎥ aij = ⎢ ⎥ ⎣ ⎦ ⎣− 0.03 0.37 ⎦ 4000 ⎡ 0.33 − 0.01⎤ ⎡0.64 0.06⎤ aij = ⎢ ⎥ aij = ⎢ ⎥ 3000 ⎣− 0.01 0.67 ⎦ ⎣0.06 0.36⎦ Viscosity (Pa.s) Viscosity

2000

1000

0 0123456 Time (s)

Figure 5.21. Transient CNF orientation induced by simple shear at 0, 0.4, 3 and 6 strain units. T=200°C. MB5 composite. Injection molded disk samples using center gated mold.

202 2-3 plane (cross section) 1-3 plane (surface)

400 400

300 300

200 200 Frequency Frequency 100 100

0 0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 1 2 3 4 5 length Length

100 100

50 50 Frequency Frequency

0 0

-100 0 100 -100 0 100 angle Angle

Figure 5.22. Transient CNF orientation induced by simple shear at 0.4 strain units. T=200°C. MB5 composite. Injection molded disk samples using center gated mold.

203 2-3 plane (cross section) 1-3 plane (surface)

700 500 600 400 500

400 300

300 200 Frequency Frequency 200 100 100

0 0

01234 0 1 2 3 4 5 6 length Length

200

100

100 50 Frequency Frequency

0 0

-100 0 100 -100 0 100 angle Angle

Figure 5.23. Transient CNF orientation induced by simple shear at 3 strain units. T=200°C. MB5 composite. Injection molded disk samples using center gated mold.

204

1.4

1.2

1

0.8

0.6 MB5 Reduced stress 0.4 MB0

0.2

0 0123456 Time (s)

Figure 5.24. Transient reduced stress of MB0 and MB5 composites at 10 s-1. T=200°C. MB5 composites. Injection molded disk samples using center gated mold.

205 1.E+07

1.E+06

1.E+05

MB0 Viscosity (Pa.s) Viscosity MB2 1.E+04 MB5 MB10

1.E+03 1.00E-01 1.00E+00 1.00E+01 1.00E+02 Time (s)

1.E+07

1.E+06

1.E+05

SC0 Viscosity (Pa.s) Viscosity SC2 1.E+04 SC5 SC10

1.E+03 1.00E-01 1.00E+00 1.00E+01 1.00E+02 Time (s)

Figure 5.25. Transient extensional viscosity of (a) MB composites and (b) SC composites. Extensional rate = 0.1 s-1. T=170°C.

206 1.E+06

1.E+05 0.77 0.23

Viscosity (Pa.s) Viscosity 0.077 3*shear

1.E+04 0.1 1 10 100 Time (s)

6

5

4 0.77 0.23 Trouton Ratio 0.077 3

2 0.1 1 10 100 Time (s)

Figure 5.26. (a) Transient extensional viscosity of pure PS (MB0). Solid line is 3 times of transient shear viscosity at 0.01 s-1. (b) Transient Trouton ratio of pure PS. T=170°C.

207 1.E+06

1.E+05 0.77 0.23 Viscosity (Pa.s) Viscosity 0.077 0.023 3*shear

1.E+04 0.1 1 10 100 Time (s)

6 0.77 0.23 5 0.077 0.023

4 Trouton Ratio

3

2 0.1 1 10 100 Time (s)

Figure 5.27. (a) Transient extensional viscosity of MB5 composite. Solid line is 3 times of transient shear viscosity at 0.01 s-1. (b) Transient Trouton ratio of MB5 composite. T=170°C.

208 CHAPTER 6

RHEOLOGY OF WATERBORNE AUTOMOTIVE PAINT

6.1 Shear rheology and pressure drop prediction

6.1.1 Introduction

As the waterborne coatings are environmentally benign, energy saving and exhibiting superior performance, they are replacing the conventional solvent-borne coatings in the past two decades 7. In an automotive manufacturing plant, the paint is kept circulating in the delivery system during normal operation to prevent solid particles from settling. The circulation speed is one of the major control parameters, as too slow a speed will cause solid settlement and too high a speed will waste pumping energy. For the solvent-borne coatings, the flow speed in the main pipe is accepted by the industry to be about 60 feet per minute. However, as the waterborne coatings don’t have much settling troubles, some waterborne paint shops are running at lower flow velocity such as 35 fpm.

Lower circulation speed means less paint circulation and reduced pumping energy, which leads to less operation cost. Unlike the Newtonian solvent-borne coatings that have constant shear viscosities, waterborne coatings are extremely shear thinning and usually

209 thixotropic, complicating the design and operation of the painting facilities. Therefore it is ideal if we can calculate the pressure drop accurately over a wide range of operation conditions so that an optimum circulation system design can be sought. Although the rheology of waterborne paint has been extensively studied 36, 38, 46, 178, 179, the pipe flow properties of the paint have been less investigated. Some people use power law model but it does not cover the viscosity behavior over a wide range of shear rates, and the thixotropy is neglected. Some designers are even still using the Ford cups to measure the viscosity, which is designed for Newtonian coatings. Although the Ford cups are still widely used for waterborne coatings, they are only used for quick one-point measurements and quality control purposes by shop floor operators. Thus, without further study, it is almost impossible to accurately predict pressure drops, flow rates and energy consumption in a commercial paint circulation system.

The purpose of this study is to experimentally measure the pressure drop of a waterborne metallic basecoat flow in a straight pipe both under steady state laminar flow and during start up of constant flowrate, and to determine a constitutive model or models that are good to predict the pressure drop for steady state flow as well as to find a method to account for the effect of thixotropy for engineering calculation.

6.1.2 Experimental

The paint used in this study is a metallic waterborne basecoat produced by Du

Pont (Type YR525). The paint contains about 60-70 wt% of water, 30-40 wt% polymers and 2wt% aluminum flakes.

210 The paint was circulated in a pilot plant that consists of a 7-gallon tank equipped with low shear agitator, a circumferential double-action piston type rotary pump (APV

DW2/006/10), a bag filter (mesh size 75 μm), a volumetric flow meter (GPI Model

A108GMA025NA1), an in-line viscometer (Visco Pro 1000, Cambridge Applied

Systems, Inc.), a back-pressure regulator (Model 208997, Series E, Graco) and 1” stainless steel pipe connects the above devices from the tank down to the P2 pressure gauge, and after that, there is a section of 60 feet ID 0.5 inch plastic tube. This plastic tube was used to simulate the main pipe of a real paint circulation loop. The tube was arranged in a large nearly horizontal loop so that the elevation change can be neglected and, due to its large aspect ratio, the tube can be treated as a straight pipe. The rotary pump was controlled by an invertor so that the speed of the driving-motor could be adjusted and the flowrate could be controlled. The tank was covered with a lid equipped with an agitator to prevent the paint from drying. However, it was vented directly to the atmosphere through a small hole so that the pressure in the tank was taken as 0 psig and the pressure drop over the tube is the reading of the P2 pressure gauge which is located at the beginning of the tube. The schematic drawing of the pilot plant setup is shown in

Figure 6.1.

Paint samples were collected at the exit of the tube. Rheological characterizations were made using Rheometrics RSF II rheometer with Couette tool at the circulation temperature. The paint sample in the rheometer was capped with a moisturized liner so that the drying of the sample during the measurement was minimized. For each sample, several types of characterizations were made. In the thixotropical loop, the sample was sheared from 0 to 600s-1 linearly in 30 seconds and then immediately sheared from 600s-1

211 back to 0 (Figure 6.2a). The area enclosed between the up curve and down curve is an indication of thixotropy (Figure 6.2b). Larger enclosed area denotes higher degree of thixotropy. The start up of constant shear from rest was also done to illustrate the thixotropy. The sample was allowed to rest long enough and then a constant shear rate was imposed to the sample. In the steady shear rate sweep, the shear rate first increased from 0.02 s-1 to 600 s-1 and then decreased back to 0.02 s-1.

Ford cup #4 measurements were performed as per the ASTM D1200 standard.

The Ford cup has been widely used in the automobile industry to monitor the viscosity. It is designed for Newtonian or near Newtonian fluids, which is the case for the solvent- based paint. Although the waterborne paint is strongly shear thinning, which seems to renders it improper to use the Ford cup any longer, the automobile industry still keeps the practice for quick viscosity measurements. The viscosity from the Ford cup measurement was used here as if the paint were Newtonian.

6.1.3 Results and Discussion

6.1.3.1 Thixotropy of the paint

As the paint shows thixotropy, the viscosity changes with time under shear, so the pressure drop over a pipe is also changing with time. Therefore, a detailed study of the thixotropy must be conducted first. Thixotropy loop and start up of steady shear from rest are standard methods to probe the thixotropy. The enclosed area in Figure 6.2b and the viscosity decay in Figure 6.3 show that the paint exhibits thixotropy. As the paint is formulated to have this behavior, the thixotropy is expected. One explanation of the

212 thixotropy is that the paint contains microstructures that are broken under shear, so the viscosity decreases. These microstructures rebuild when the shear is removed and the fluid regains its original viscosity 180.

For engineering design and the practical operation purpose, more emphasis is put on the maximum pressure drop which occurs at the beginning of start up and the steady state pressure drop when the plant operates continuously, as the change of pressure drop only happens for a relatively short period of time after start up. For most of the time, the circulation system runs continuously. Therefore the first challenge is how to obtain experimental data of pressure drop at steady state flow versus the flowrate over a wide range. It is common to think that one can circulate the paint at a certain flowrate to reach steady state and record the pressure drop, and then go to another flowrate. However the problem is that in order to obtain many data points, many circulations should be done. As each circulation requires at least half an hour to reach steady state and another several hours to regain the original viscosity, the total experiments would take too long such that the drying of the paint would be a problem. Using fresh paint each time will require too much paint and may induce constancy problems as the formulation of the paint varies from batch to batch. Therefore, a relatively fast, but still accurate method is needed.

As a result, three types of circulation were performed: start up of constant flowrate, step change of flowrate and steady state flow. These circulations were designed to get the maximum and minimum pressure drop that may occur, and find out the effects of thixotropy during the circulation. In the case of start up of constant flowrate, the loop is filled with paint and allowed to rest for at least 2 hours. Therefore the structures of the paint are assumed to have been fully rebuilt before start up. This is to avoid the effects of

213 the filling procedure during which the paint is sheared so that the true maximum pressure drop can be obtained. Then the circulation is started and the flowrate is maintained at a certain constant value. The instantaneous pressure drop over the tube is recorded with time for about an hour. Several runs at different flow rates are done to illustrate the thixotropy effects. Figure 6.4 shows the results and for all the runs the pressure drop decreases very quickly during the first 5 minutes, and then decreases slowly until almost reach steady state after 30 minutes. This is similar to the viscosity decay during start up of constant shear measurement as shown in Figure 6.3. The pressure drop decreases more rapidly for large flowrate runs than that for small flowrate runs. This is because faster flow imposed greater shear to the paint so that the microstructures in the paint were broken in a shorter period of time than that in the slower flow. However, the ratio of the initial pressure drop to the steady state pressure drop remains to be about the same value of 1.25. This ratio is the same as the initial viscosity to the steady state viscosity during the viscosity decay (Figure 6.3). The ratio can be used as a safety factor to account for the thixotropy for engineering calculation purpose. Once the steady state pressure drop is calculated, the maximum pressure drop which occurs during start up can be easily calculated through multiplying by the safety factor. This also implies that although the flow rates are different, the final degrees of the microstructure breakdown of all the runs are about the same. Faster flow will caused the microstructures to break down faster, but not to a larger degree.

As the microstructures are broken down under shear, it is desirable to explore if when the flowrate, hence shear rate, suddenly drops, some of the microstructures will rebuild. Figure 6.5 shows the results of the experiments in which the paint was circulated

214 at 2.5 gpm for 2 hours to reach steady state and then suddenly the flowrate dropped to 0.7 gpm. It is clear that after the flowrate drops, the pressure drop over the tube remains the same for another 2 hours. The slight increase at the very beginning of the low flowrate flow is considered to be due to the stabilization of the flow, since if it were due to the rebuild of the microstructures, the process would have happen sometime later than the onset of flow as the microstructures need some time to rebuild. This indicates the microstructures in the paint doesn’t rebuild under slow flow. One possible reason is that even at low flowrate, the shear in the pump is still high enough to prevent the microstructures from rebuilding. Therefore, if the paint microstructures have been broken up at a relatively large flowrate and the pump is still running, the microstructures won’t rebuild, even at lower flowrate. In this case, the flow will be steady state once a new flowrate is set.

Therefore, a series of steady state pressure drop versus flowrate were readily obtained by first circulating the paint at 2.5 gpm for half an hour and then decreasing the pump speed stepwisely. At each step, the pressure drop was recorded with the flowrate and then decreased the pump speed again until lower limit of the flow meter was reached.

The process took about 30 minutes. Then the pump speed was increased stepwisely and the pressure drop was recorded until to the higher limit of the flow meter. The purpose of the hysteresis loop is to verify that there is no paint microstructure rebuild during the experiments.

Figure 6.6 shows the results together with the initial and steady state pressure drops from the start up of constant flowrate from rest. It can be seen that the up curve and the down curve are almost exactly the same, showing that there was no microstructures

215 rebuild during the experiment. This verified that the rebuild of the microstructures requires a complete quiescent environment; even slow flow will stop the rebuilding process. The fact that the steady state pressure drops are almost the same as the pressure drops after 30 minutes of the start up of constant flowrate experiment runs indicates that as long as steady state is obtained, the degree of the breakdown of the pant microstructures are the same no matter of what flowrate it is running at.

Up to now, we have obtained the pressure drop of paint flow through the tube over a wide range of flowrate. These are the data upon which our modeling work will be based.

6.1.3.2 Rheology of the paint and model fitting

In the rheological measurement of the steady state shear viscosity, both rate sweep up and sweep down were done. Since during the sweep up measurement the microstructures in the paint are breaking down gradually, the afterwards sweep down measurement recorded the “true” steady state viscosity as the microstructures have been fully broken down. Therefore the sweep down data is used for the model fitting. Three constitutive equations are investigated in this study: power law model (Equation 1),

Carreau model (Equation 2) and Sisko model (Equation). Power law model is mathematically the simplest of the three but it cannot represent the viscosity in the full range of shear rates. Carreau model and Sisko model are capable of predicting the viscosity accurately and they are still simple enough for engineering design purpose.

216 The fitting results of the models are shown in Figure 6.7a,b. It is clear that both

Carreau model and Sisko model work very well that the fitted lines go through almost every data point. The power law model is also acceptable but it lacks the accuracy than the other two models. This difference can also be seen in the pressure drop prediction in a pipeline discussed later.

Power law model:

η = mγ n−1 (6.1)

Carreau model:

η − η ∞ = [1+ (λγ)2 ](1−n) / 2 (6.2) η0 − η∞

Sisko model (1958)

n−1 η = η∞ + mγ (6.3)

6.1.3.3 Reynolds number

Although the Carreau model and Sisko model give a better fitting of the viscosity data, the overall viscosity profile doesn’t deviate much from a power law fluid. For the ease of discussion, the generalized Reynolds number for power law fluids that is defined by Metzner and Reed 181 is used in this paper to characterize the fluid flow. The expression can be reduced to the traditional definition when m=μ and n=1, which is the case of Newtonian fluids. The definition of Reynolds number for Sisko model is given by

Bird 81 and no definition for Carreau model is available. 217 n 8ρv 2−n Dn ⎛ n ⎞ Re = ⎜ ⎟ (6.4) m ⎝ 6n + 2 ⎠

6.1.3.4 Pressure drop measurement in the pilot plant and model prediction

The relationship between the pressure drop and the flowrate for laminar flow in the straight tabular pipe can be found in standard textbooks as the solutions to the Navier-

Stokes equations. However, analytical solutions are only available for Newtonian fluids and some other simple fluids such as power law fluids. In the case of Carreau model and

Sisko models, numerical methods have to be used to obtain the results. A simple computer code was written for this purpose. The experimental data and the predictions of different models are shown in Figure 6.8. Both Sisko model and Carreau model can accurately predict the pressure drop. The power law model works fine at low flowrate region (Re < 100), but underpredicts the pressure drop when the flowrate is large (Re >

100). Therefore, the use of power model to predict pressure drop is only limited to small flow rates. Carreau model and Sisko model are suitable for wider flow rate range.

6.1.4 Conclusion

The pressure drop in straight pipe of laminar flow of a metallic automotive basecoat has been experimentally studied. The rheology of the paint has been measured and fitted using several rheological models. Numerical calculation shows that the Sisko model and Carreau model can accurately predict the pressure drop of the steady state laminar flow in a round pipe, with an average error of less than 10%. Power law model is only good at low flow rates (Re < 100) and it underpredicts the pressure drops at high flow rates. The thixotropy of the paint is observed in the pilot plant circulation and is characterized by the rheometer. The maximum pressure drop, which occurs at the

218 beginning of the startup flow, is fount to be 1.25 times the steady state pressure drop.

This relationship can be used in the engineering design to account for the effects of thixotropy.

219

60' ID 0.5" tube

M

Figure 6.1 Schematic drawing of the pilot plant.

220

700 25

600 20 500 ) -1 15 400

300 10 Shear rate (s 200 Shear stress (Pa) 5 100

0 0 0 102030405060 0 200 400 600 800 Time (s) Shear rate (s-1)

Figure 6.2 Thixotropy loop. (a) Shear history in a thixotropical loop. (b) Typical response of a thixotropic fluid.

221

0.16

0.14

0.12

0.1

0.08

0.06 Viscosity (Pa.s) 0.04

0.02

0 0 50 100 150 200 250 300 350 Time (s)

Figure 6.3 Viscosity decay during start up of steady shear at shear rate of 200 s-1.

222

Thixotropy in circulation

30.0 0.45 gpm 0.87 gpm 1.28 gpm 25.0 1.70 gpm 2.52 gpm

20.0

15.0

Pressure drop (psi) 10.0

5.0

0.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 Time (min)

Figure 6.4 Pressure drop observed in the start up of constant flowrate.

223

25.0

20.0

15.0

10.0 pressure drop (psi)

5.0

0.0 0 40 80 120 160 200 240 280 time (min)

Figure 6.5 Pressure drop in response to a step change of the flowrate from 2.5 gpm to 0.7 gpm.

224

30

25

20

15

Init dP Pressure drop (Psi) drop Pressure 10 SS dP

Down curve

5 Up curve

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Flowrate (gpm)

Figure 6.6 Pressure drop of steady state flow.

225

Carreau model fitting

10

Exp data Carreau

1 0.01 0.1 1 10 100 1000 Viscosity (Pa.s) 0.1

0.01 Shear rate (s-1)

a.

Sisko mode fitting 100 exp data Sisko power law 10

1 0.001 0.01 0.1 1 10 100 1000 Viscosity (Pa.s)

0.1

0.01 -1 Shear rate (s )

b

Figure 6.7 Steady state shear viscosity and model fitting. (a) Carreau model. (b) Sisko and power law model.

226

50.0 exp data Newtonian 45.0 power law 40.0 Sisko Carreau 35.0

30.0

25.0

20.0 pressure drop (psi) drop pressure 15.0

10.0

5.0

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 flowrate (gpm)

Figure 6.8 Comparison of measured pressure drop with model predictions.

227 6.2 Temperature effects on the rheology of waterborne automotive paints

6.2.1 Introduction

With the requirements from the environmental protection agencies, more and more of the traditional solvent-borne coatings have been replaced by low-VOC-emitting coatings7. Since the first commercial-scale application of the waterborne basecoat in the automotive industry started in mid-1980’s, the environmentally benign waterborne paint technology has been replacing the traditional solvent-borne technology in the automotive industry. Among others, waterborne coatings take up more than 50% of the total amount of coatings in Europe the United States and up to 90% in Germany 8, 182, 183. Other impetus of the technological changeover includes better performance and lower cost provided by the waterborne coatings 7, 8.

Although the waterborne coatings possess some intriguing advantages, there are difficulties that have to be overcome before successful applications can be made. One of the most important problems is the control of the rheology, which determines the quality and appeal of the final film. For a Newtonian fluid, its viscosity can describe its rheology and it is constant at a certain temperature, and independent of whether it is being sheared and/or how long it has been sheared. However, most waterborne coatings are highly non-

Newtonian. Their viscosities are not only functions shear rate, but in most cases are functions of time as well. Studies in the past 8, 184 have shown that for a waterborne basecoat, the rheology should be carefully designed such that the performance of the paint is superior during the whole handling cycle, from storage to spray application and drying. The required rheology should enable the basecoat to have a high viscosity during storage that leads to minimal settling. Under moderate agitation and during circulation,

228 the viscosity should be low that the basecoat can be easily pumped. At the spraying gun or bell, the viscosity should be very low so that the paint can be properly atomized. As soon as the paint is sprayed on to the work piece, the viscosity should build up quickly to prevent the paint from running or sagging. However, the viscosity should still be low enough to enable the paint to level off, providing a smooth film and good flake orientation.

The rheology of waterborne coatings has been studied for many years. As the rheology of the coating is critical to the final appearance, people have developed many ways to control the rheology. Different types of thickeners or modifiers have been developed suitable for a wide variety of waterborne coatings 185-188. Konishi studied the time-dependent behavior and showed that the paint had high limiting sagging thickness and pinholing thickness 189. Steady-state viscosity profiles, creep flow and thixotropic measurements were carried out by Boggs et al 190 in order to determine the connection between these and aluminum flake orientation. Osterhold 179 reviewed the rheological behavior of waterborne paint and described some important measuring methods, such as oscillation tests and determination of yield points. Other researchers have studied the role of the rheology in spraying, leveling, coating structure and other relevant factors 36, 45, 191,

192.

Much less effort has been spent on the temperature dependence of the rheology of a waterborne paint. Kim studied the rheology of a latex paint at different temperatures and found that the viscosity followed the Arrhenius equation very well 193. Osterhold

229 found that the G’ of two types of polyurethane microgels decreases as the temperature increases 37. Churella et al. observed the increased G’ with temperature and they attributed this to the formation of a thin film on the latex paint 194.

As there is almost no published literature about the temperature dependence of the rheological behavior of a metallic automotive basecoat, the purpose of this study is to investigate the subject. In this paper a comprehensive shear rheological characteristics of a metallic automotive basecoat at a range of temperatures is studied.

6.2.2 Experimental

A commercially available silver metallic waterborne basecoat was used in this experiment. It contains about 50-70wt% water, 2-3wt% aluminum flakes, 5-15wt% acrylic latex, 5wt% thickeners and other ingredients. The viscosity had been adjusted to be appropriate for spray application. A non-metallic basecoat with similar composition but with no aluminum flakes was also used for comparison purpose. The rheological characterization was performed on a Rheometrics RSF II rheometer, with a transducer that is capable of measuring torques in a range of 0.02g⋅cm to 20g⋅cm. A Couette tool with temperature control with accuracy of +/- 0.5°C was used. The paint sample bottle was soaked in a water bath before loading so that the sample had the same temperature as the tool. For each load of sample, a series of rheological characterizations were performed with the sequence of small amplitude oscillatory shear, thixotropical loop and the steady state shear. The drying of the paint during measurements was minimized by covering the Couette tool with a cap lined with a layer of water-saturated material.

230 In small amplitude sinusoidal oscillation tests dynamic strain sweep experiments had been done beforehand to determine the linear region, where the elastic modulus G’ and viscous modulus G” are not functions of the strain, and 1% strain was determined.

The oscillation was performed with frequencies ranging from 0.1 to 15 Hz. In the thixotropic loop test, the sample was sheared from 0 to 600 s-1 linearly in 30 seconds and then immediately sheared from 600 s-1 back to 0 (Figure 6.9a) in 30 seconds.

Pseudo steady state viscosity was measured using the shear rate sweep down procedures. The word pseudo is used here because the data shown here is obtained at a stage very close to steady state, but not at the true steady state. As seen in Figure 6.9c, it takes 300 seconds for the viscosity to reach steady state when sheared at 200 s-1. Longer time is needed if sheared at smaller shear rates. To get true steady state viscosity at various shear rates, the experiment will last so long such that drying of the paint sample becomes an issue. On the other hand, use fresh paint for each shear rate will be too time consuming. For the simplicity of the experiments, a pre-shear (delay) time of 30 seconds and measure time of 10 seconds at each shear rate was chosen for all the tests. The sweep test started from high shear rate (600 s-1) to low shear rate (0.02 s-1), so that the structure in the sample can be broken at high shear rates and close to steady state conditions can be obtained at lower shear rates after the 30 seconds pre-shear. This procedure was kept the same for all steady shear viscosity measurements so that a direct comparison can be made between different temperatures.

Ford cup #4 measurement was performed as per the ASTM D1200 standard.

231 6.2.3 Results and Discussion

6.2.3.1 Linear viscoelasticity

Small amplitude oscillatory shear has been used to characterize the linear viscoelastic behavior of the metallic basecoat, and the results are shown in Figure

6.11a,b. Plot (a) shows that the elastic modulus G’ decreases a little with temperature rising from 10°C to 20°C, and then increases monotonically with temperature from 20 to

45°C for all frequencies. There is an acceleration of the rate of increase with increasing temperature. From 20 to 30°C, the G’ increases gradually; when the temperature is higher, G’ increases much faster. The G’ become less dependent on the frequency and greater than the G” at higher temperatures, which implies that the basecoat behaves more like an elastic solid than a liquid. The tangent of the phase angle, tan δ = G”/G’, is the ratio of elastic modulus to viscous modulus, and it is an indicator of the fluid’s viscoelasticity. When tan δ is greater than 1, then the fluid is more liquid-like; lower than

1, solid-like. The tan δ vs. frequency, shown in plot (b), further reveals this trend as with higher temperature. The average of tan δ at 40 and 45 °C is about 0.15, much less than 1, indicating that the viscous modulus is less than 1/6 of the elastic modulus. This means that at these temperatures, the basecoat exhibits much stronger solid-like behavior.

6.2.3.2 Thixotropy

Figure 6.9a,b,c show that the metallic basecoat exhibits a thixotropic behavior.

Plot (a) shows the shear pattern and plot (b) shows the corresponding shear stress vs. time. It can be seen that the magnitude of the shear stress when the shear rate increases

(up curve) is greater than that when the shear rate decreases (down curve). As the

232 definition of viscosity is the ratio of shear stress to shear rate, the viscosity of the up curve is greater than that in the down curve. The size of the area enclosed in the two curves is an indication of the degree of thixotropy. Larger enclosed area indicates higher degree of thixotropy. Plot (c) shows when the paint is subjected to a constant shear rate.

The viscosity decreases as the shear time increases. An accepted theory to explain thixotropy is that certain network structure forms in the fluid. When subject to shear the structure is broken down, hence the viscosity decreases. After the fluid is allowed to rest, the structure rebuilds and thus the fluid gains its original viscosity 180.

The existence of the thixotropy complicates the measurement of the rheology, as the loading process shears the paint sample and therefore its viscosity may change. In order to obtain repeatable measurements, the loading procedure has to be maintained to be the same for each sample, so that the shear history of the sample before measurements is the same. In this study, before loading the paint samples to the rheometer, the paint is gently shaken for 2 minutes. The loading procedure and the rheological measurement sequences are all kept the same. Therefore, same the shear history was applied for each sample. Figure 6.10a-h shows the change of degree of thixotropy with the increase of temperature. At 10°C, the up curve collapses onto the down curve so there is almost no thixotropy. When the temperature increases, more thixotropy is evident. From the enclosed area between the curves, the degree of thixotropy gradually increases when the temperature increases from 10°C to 45°C. Furthermore, an overshoot in the up curve appears at 40°C. It may indicate that the structure was not broken down gradually, instead a large amount of structure was broken down at the same time. This is similar to the overshoot that appeared in the start-up of constant shear of some relatively strong 233 viscoelastic materials 131. At 45°C, the overshoot is bigger and thus the paint is more elastic at this temperature. The appearance of the overshoot suggests that the paint shows strong elastic properties at high temperatures. Further, the magnitude of the shear stress is much higher than lower temperatures. This means the overall shear viscosity at 45°C is much higher than that at lower temperatures. This is quite different from our common sense that the viscosity is lower at higher temperature. The increase of viscosity is further discussed in the next section of this study. The overshoot in the up curve at low shear rate is also interpreted as “static yield stress”, which usually appears only when the sample is sheared for the first time after a prolonged storage time and disappears afterwards 195.

The increase of the degree of thixotropy as the temperature increases implies that the paint develops a 3-dimensional network structure at high temperatures, and more structure is formed at higher temperatures.

6.2.3.3 Pseudo Steady state viscosity

Figure 6.12a,b shows the temperature dependence of the pseudo steady state shear viscosity profile. Generally the viscosity increases as temperature increases (Figure

6.12a) for the metallic basecoat. At very low shear rate, the viscosity increases up to 500 times. Figure 6.12b shows in detail that the viscosity at low shear rate region decreases slowly with temperature from 10°C to 20°C, and then increases dramatically with the temperature; while the high shear viscosity decreases slowly when the temperature increases up to 35°C and then increases slowly up to 45°C. This viscosity change with temperature is reversible, such that the viscosity increases with increasing temperature

234 and decreases back to its original viscosity when temperature drops. However, the viscosity of the non-metallic basecoat decreases slightly with temperature. In this semi- log plot it is almost independent of the temperature.

This type of reversible viscosity increase with temperature in the metallic basecoat is very interesting. Most often the temperature dependence of viscosity will follow the WLF equation for polymer melts and the Arrhenius relationship for solutions, by which the viscosity decreases as temperature increases 131. Nonetheless, reverse temperature dependence of viscosity has been reported by researchers in colloidal systems 102 and biodegradable polymer aqueous solutions 196-199. Chio and Krieger 102 argued that in their gelled dispersion of PMMA colloids sterically stabilized in silicone oils, a “steric-elastic” force is present, which increases at elevated temperatures, causing the viscosity to increase. The biodegradable poly(ethylene oxide)-poly(propylene oxide)- poly(ethylene oxide) triblocks (PEO–PPO–PEO) aqueous solution has been found by many researchers to have reverse thermo gelation phenomena (low viscosity at low temperature and gels at higher temperature). Various reasons have been proposed, including intrinsic changes in micellar properties 196, entropic gain of the system 197, hard sphere crystallization process 198, and the formation of a 3-dimensional network 199. In the coating systems, such phenomenon has never been reported. Other researchers have found non-reversible viscosity increase with increasing temperature in coating systems and they attributed it to curing 200 or drying 201 at high temperatures. In our experiments the evaporation of the water is well controlled such that no obvious drying was observed.

On the other hand, the coalescing procedure during curing is not readily reversible and the cross-linking procedure is not reversible at all, so the reversible viscosity increase

235 observed in this paint is not caused by drying or curing. Thus, these results suggest that the viscosity increase is not caused by the curing, but by the formation of structure in the paint at higher temperatures. This structure grows as the temperature increases and grows faster at higher temperatures, forming a gel. The paint can therefore be viewed as a two- phase suspension with one phase being the structure and the other being the remaining liquid. When temperature increases, the structure grows with a trend to increased suspension viscosity and, on the other hand, the viscosity of the remaining liquid decreases with a trend to decreased suspension viscosity. These two effects compete with each other and result in the first decrease and then increase in the viscosity of the paint. In contrast, the non-metallic basecoat does not have the capability to form structure and therefore its viscosity decreases at high temperature following the Arrhenius rule.

6.2.3.4 Apparent yield stress

To further probe the network structure formed in the paint, the viscosity if plotted against the stress as shown in Figure 6.13. At 10-20°C, the viscosity decreases gradually with the applied stress, showing the typical shear thinning behavior. As the temperature increases, the viscosity precipitates dramatically within a narrow stress range and then decreases gradually with stress. This sudden decrease of viscosity can be attributed to the breaking down of the network structure at the apparent yield stress. This apparent yield stress is often observed in flocculated suspensions at high particle loadings when interactions between the particles are high enough to form continuous 3-D network structures 202.

236 6.2.3.5 Ford cup #4

It is common that in paint shops the Ford cups (or ISO cups) are used for quality control. The Ford cup is made up of a metal cup with an orifice at the bottom. The size of the cup and the orifice is standardized in the ASTM standards. The time for the fluid to flow through the orifice under gravity is an index of the viscosity. The Ford cup is a single-point rheometer and is designed for Newtonian or near Newtonian fluids.

However, due to its simple geometry and ease of use, it is still used for non-Newtonian waterborne coatings for quality control. The Ford cup #4 measurements for the metallic basecoat at different temperatures are showed in Figure 6.14. The measured time in seconds decreases when the temperature increases from 10°C to 30°C, and then increases from 35 to 45°C. Comparing the viscosities corresponding to the time in seconds with the measured steady shear viscosities, the Ford cup #4 is estimated to be equivalent to a shear rheometer running at constant shear rate of about 200 s-1. As also can be seen that the trend of the Ford cup seconds is almost the same trend as the steady state viscosity at high shear rate curve in Figure 6.12b.

It is worthwhile to mention that during the Ford cup measurement the phenomena of thixotropy and viscoelasticity was very obvious. When the temperature was low, the paint was just like a Newtonian liquid. When the sample temperature was 45°C, a marked viscosity increase could be visually detected, as flow was considerable slower. The paint that flowed out from the Ford cup was collected and used to redo the measurement immediately without rest. It was found that the time measured dropped dramatically until

237 stable after 7 to 8 times. The thixotropy must have played an important role here. As a result, the seconds shown in Figure 6.14 was the average from the first several measurements of a sample.

6.2.3.6 Identification of the network structure

In order to further investigate the nature of the structure and the fundamental reason for the increased viscosity at high temperature in the metallic basecoat, we need to look into the composition of both basecoats. The thickeners are long chain surfactants and they associate with the latex to increase the viscosity of the paint. Although the thickeners and acrylic latex may form microgels and generate non-Newtonian behavior 42,

191, 203, the viscosities of suspensions containing microgels decrease with increasing temperature 37, 193, 194, 203. The major difference in the composition between the metallic and non-metallic basecoats is that the former contains aluminum flakes and the latter does not. Since the non-metallic basecoat does not show this increased viscosity at high temperature behavior, it is most likely that the aluminum flakes played an important role in forming the structure in the metallic basecoat. These disk-shaped flakes have an average diameter of about 20 μm and thickness of less than 1 μm. At high temperatures the size of the latex migrogels becomes smaller 203, 204, so that the thickeners associates more with the aluminum flakes. With the aid of the association, these flakes interact with each other and form a 3-dimensional continuous network, just like the case when plate- like clay suspended in water 202. The occurrence of this structure greatly increases the elasticity and viscosity of the paint, and it is also the source of the yield stress. Under shear stress the structure breaks and when the stress is relieved the structure rebuilds, giving the basecoat a thixotropic behavior. When temperature drops, the latex microgels

238 swell 203, 204 and interact more with the thickeners, such that aluminum flakes lose the support from the thickeners and therefore the structure collapses. As a result, the paint becomes more fluid-like, less viscous, and the yield stress and thixotropy disappear. This whole process is determined by the chemical properties of the latex, thickener and aluminum flakes.

6.2.4 Conclusions

The effect of temperature on the rheology of a metallic waterborne automotive basecoat is studied. It is found that the degree of thixotropy increases as the temperature increases. The results of the linear viscoelastic characterization show that the paint possesses network structure at high temperatures, which leads to solid-like behavior. The steady shear viscosity vs. shear rate reveals that the paint is more shear thinning at high temperatures. The viscosity at low shear rate increases dramatically with temperature.

The high shear viscosity decreases gradually from 10°C to 30°C, and then increases gradually up to 45°C. It is surmised that the aluminum flakes in the basecoat interact with themselves as well as other ingredients at high temperatures to form a continuous 3- dimentional network structure, which leads to increased elasticity, viscosity, thixotropy and the appearance of yield stress. The Ford cup is estimated to be equivalent to a single point rheometer with a shear rate of about 200 s-1 for the paint studied.

239

700 35

600 30

500 25 ) -1 400 20

300 15 Shear rate (s 200 Shear stress (Pa) 10

100 5

0 0 0 15304560 0 200 400 600 800 Time (s) Shear rate (s-1)

0.07

0.06

0.05

0.04

0.03

Viscosity (Pa.s) 0.02

0.01

0 0 50 100 150 200 250 300 350 Time (s)

Figure 6.9. Thixotropic behavior of the metallic basecoat. (a) Shear history in a thixotropical loop. (b) stress response of a thixotropic fluid. (c) Steady shear at constant shear rate of 200 s-1. The measurements were performed at room temperature.

240 35 35 10 C 15 C 30 30 25 25 20 20 15 15 10 10 5 5 0 0 0 200 400 600 800 0 200 400 600 800

35 35 20 C 24 C 30 30 25 25 20 20 15 15 10 10 5 5 0 0 0 200 400 600 800 0 200 400 600 800

35 35 30 C 30 30 35 C 25 25 20 20 15 15

Shear stress (Pa) 10 10 5 5 0 0 0 200 400 600 800 0 200 400 600 800

35 100 40 C 30 45 C 80 25 20 60 15 40 10 20 5

0 0 0 200 400 600 800 0 200 400 600 800

Shear rate (s-1)

Figure 6.10. Degree of thixotropy of the metallic basecoat at various temperatures.

241 1000

Temp (oC) 100 10 15 20 25 10 30

G' (Pa) 35 40 45 1

0.1 0.1 1 10 100 Freqency (Hz)

10 Temp (oC)

10 15 20 25 1 30 tan 35 40 45

0.1 0.1 1 10 100 Frequency (Hz)

Figure 6.11. Linear viscoelasticity of the metallic basecoat. (a) G’ vs. frequency. (b) tan δ vs. frequency.

242

10000 Temp (oC)

10 1000 15 20 100 25 30 35 10 40 45

1 Viscosity (Pa.s)

0.1

0.01 0.01 0.1 1 10 100 1000 Shear rate (s-1)

100000 metallic 0.03 1/s

10000 metallic 600 1/s

1000 non-metallic 0.03 1/s non-metallic 600 1/s 100

10

Viscosity (Pa.s) Viscosity 1

0.1

0.01 0 1020304050 Temperature (°C)

Figure 6.12. Pseudo steady state viscosity of the metallic and non-metallic basecoats. (a) Steady shear viscosity vs. shear rate. (b) viscosities at shear rates of 0.03 s-1 and 600 s-1.

243

10000

10 Temp (oC) 1000 15 20 25 100 30 35 40 10 45

1 Viscosity (Pa.s)

0.1

0.01 0.01 0.1 1 10 100 Shear stress (Pa)

Figure 6.13. Apparent yield stress at various temperatures of the metallic basecoat.

244

25

23

21

Seconds 19

17

15 10 15 20 25 30 35 40 45 Temperature (oC)

Figure 6.14. Ford cup #4 measurement of the metallic basecoat.

245 6.3 Color Degradation and Rheological Changes Caused by Prolonged Circulation

in Paint Delivery System

6.3.1 Introduction

In an automotive manufacturing plant, paints are delivered to the spraying points through circulation systems. To keep the ability to spray at any time and to prevent the solids in the paints from settling, the paints have to be circulated continuously at high pressure. The pumps and backpressure regulators impose a great amount of shear to the paint during long period of circulation, and the metal flakes may be crumpled or some components of the paint changes. As a result, the final appearance of the paint may not be as good as the fresh paint. This type of color degradation occurs especially after long vacations when the circulation systems have run for several days without replenishment of fresh paint.

In this report we intend to investigate the color degradation and the rheological changes by doing a series of experiments under different flowrates and backpressures.

We hope to find out when and how much the degradation occurs and then an optimum operating condition may be sought. Most of the experiments were done using the rotary pumps and an additional experiment was done using piston pump in order to make direct comparison between the pumps.

6.3.2 Experiment setup

The pilot plant consists of a 7-gallon tank equipped with low shear agitator, circumferential double-action piston type rotary pump (APV DW2/006/10), a bag filter

(mesh size 75 μm), a volumetric flow meter (GPI Model A108GMA025NA1), an in-line

246 viscometer (Visco Pro 1000, Cambridge Applied Systems, Inc.), a back-pressure regulator (Model 208997, Series E, Graco) and 1” SS pipe connects the above devices from the tank down to the P2 pressure gauge, and after that, there is a section of 60 feet

ID 0.5 inch plastic tube. This plastic tube was used to simulate the main pipe of a real paint circulation loop. The rotary pump is controlled by an invertor so that the driving- motor speed can be adjusted. The schematic drawing of the pilot plant setup is shown in

Figure 6.1. An air-driven double-acting piston pump (Model 223-954, Graco) was also used to compare how different pumps would affect the rheology of the paint.

A silver waterborne metallic automotive basecoat (YR-525M, DuPont) was used for the degradation test. In order to illustrate the effect of the circulation on the rheology of the paint, a 2-factor 3-level full factorial experiment was designed. The flowrates were

0.3, 0.6 and 0.9 gallon per minute (gpm) and the backpressures of the backpressure regulator (BPR) were 0, 20 and 40 psi. All combinations of these levels lead to a 9-run experiment matrix. These factors were selected in order to simulate a wide operating condition including the real operating conditions at Honda’s East Liberty plant, and were based on previous experience of the paint properties. These experiments are designed to study when the rheological change appears and how much it will be under different flow conditions for the circulation using rotary pump. An additional run was done using a piston pump, so that the degradation of the paint caused by different pump can be compared. The experimental parameters are tabled in Table 6.1. However, the design of this matrix assumes that when the setting of one factor changes, the effects caused by another factor remains the same so that the change in the results will be attribute to the changing factor. Unfortunately this is not the case in this work. If the backpressure is

247 changed from 0 psi to 20 psi while flowrate is maintained at 0.6 gpm, the pump speed has to be increased to generate pressure and the opening of the regulator has to be reduced to maintain the flowrate. As a result, both the pump and the regulator are changed and the final results will reflect the effects of both the pump and the regulator, not just one factor as we expected. Ultimately, the matrix of the experiments will not disclose clearly which factor, the pump or the backpressure regulator, causes the degradation. Better design should include runs that the paints are circulated by pressurized tanks, not the pumps, such that the effects of the backpressure regulator can be investigated. Then use pumps to circulate the paint with a backpressure regulator whose properties are known. This way, the effects caused by pumps and BPR can be isolated. Other researchers have done it this way7, 49.

Run Type of Flowrate Backpressure number pump (gpm) (psi) 1 Rotary 0.3 0 2 Rotary 0.9 40 3 Rotary 0.6 20 4 Rotary 0.3 40 5 Rotary 0.9 0 6 Rotary 0.3 20 7 Rotary 0.6 40 8 Rotary 0.6 0 9 Rotary 0.9 20 10 Piston 0.6 20

Table 6.1. Experimental parameters.

248 Paint samples were collected at the beginning and periodically during each circulation run. The first sample was taken after an initial circulation for 5 minutes running at 3.0 gpm with the BPR fully open. This initial circulation was designed to mix the paint with a very small amount of remaining water in the system from the previous cleaning procedure of the circulation system. Another 3 samples were taken after 24, 48,

72 hours after the start of the designed circulation. All the samples were taken at the exit of the return tube. Each sample contains a 32oz sample and a 6oz sample. The 32oz samples sent to the paint supplier for the color degradation test. The 6oz samples were used for the rheological measurement. The circulation time was converted into “turns” which means the number of turnovers of the paint that has been circulated in the system.

6.3.3 Rheological Measurements Procedures

All rheological measurements were performed on Rheometrics RSF II. The

Couette tool with water bath temperature control was used. Unless otherwise stated, all the rheological measurements were performed at 24°C.

For each load of sample, a series of rheological measurements were designed.

They are frequency sweep, thixotropy loop, then let the paint rest for 5 minutes, and followed by shear rate sweep up and sweep down. Before loading the sample, the sample bottle was gently shaken for 10 minutes to ensure the paint was homogeneous and did not contain air bubbles. The same loading and test procedure were kept for all samples to ensure each rheological measurement had about the same immediate shear history. The measurement began immediately after the sample was loaded.

249 The frequency range was 0.1 to 15 Hz. The strain was determined to be 1% from previous strain sweep tests to make sure that it was in the linear viscoelastic regime. The thixotropy loop test involves the shear rate increase linearly in 30 seconds from 0 to 600 s-1 and then decrease back to 0 in another 30 seconds. The 5 minutes’ rest was designed to let the paint structure rebuild, as part of it has been broken during the thixotropy loop test. The shear rate sweep test increases the shear rate from 0.02 s-1 to 600 s-1, and then immediately decreases back to 0.02s-1.

In order to further detect the thixotropy effect, the steady shear at a constant shear rheological measurement was performed on a new loading of the sample. Before loading, the sample bottle was also gently shaken for about 10 minutes. The measurement began immediately after the sample was loaded. The paint was shear at 200 s-1 for 300 seconds.

6.3.4 Color Degradation Test Procedures

The color degradation tests were performed by Du Pont as per the PPG stands for color measurement.

6.3.5 Results and Discussions

6.3.5.1 Viscosity variation

The steady state shear viscosity versus the shear rate is obtained from the rate sweep rheology tests. In following discussions, the sweep down curves are used. A typical example of the viscosity profile variations during the circulation (Run 3) is shown in Figure 6.15. It clearly shows that the viscosity increases dramatically as the circulation time increases, especially in the low shear regions. The circulation time is converted in to

“turns” which is the number of turnovers of the paint in the system. The paint exhibits

250 greater shear thinning as the circulation time increases. As there is not much curvature in the viscosity profiles in a log-log plot, power law model would be appropriate to fit the data.

Power law model has been widely used due to its simplicity and the capability to capture a wide range of materials. It is a two-parameter model in the form of Equation

(6.1). The parameter m indicates the viscosity at the shear rate of 1 s-1. The parameter n indicates the slope. If n=1 then the line is horizontal, and it means the material has constant viscosity, like water. If n < 1 then there is a slope and the material exhibits shear thinning behavior, like most polymeric materials. From the Figure 6.15 it can be seen that the viscosity at the high shear end doesn’t change much. Therefore, it is appropriate that the power law model parameter n is used to quantify the degree of variation in the paint viscosity caused by the circulation.

Power law model: η = mγ n (6.5)

0.6 gpm, 20 psi

100 0 turn 150 turns 314 turns 10 494 turns

1 0.01 0.1 1 10 100 1000 Viscosity (Pa.s) 0.1

0.01 Shear rate (s-1)

Figure 6.15. Viscosity profiles change with circulation time.

251

As the paint is a complicated two-phase mixture of many ingredients, its physical properties are subject to change. Even the same batch of paint was used in all the 10 runs, the initial samples of each run are still different. Therefore, it is better to compare the percentage of property change within one run. Figure 6.16 shows the percent of decrease in n values of the paint samples taken during circulation under different backpressures.

For all the backpressures, the curves of different flowrates roughly fall on each other and the solid lines are the trend lines. The phenomenon indicates that flowrate is not a significant factor. At different flowrates, the total shear the circulation system imposes to the paint is about the same. At the same backpressure, in order to increase flowrate, the pump has to run faster and the opening of the BPR is slightly larger. The pump will impose more shear while the BPR will impose less. The two effects counteract each other so that the overall shear condition is about the same. Slightly different can still be seen, however, that the change of n for small flowrates is greater than for large flowrates. This may be explained that although the shear condition is about the same for different flowrates, in small flowrates each unit volume of flowrate will bear slightly larger shear than that in large flowrates. However, this cannot be verified without further investigation.

On the other hand, backpressure is obviously a significant factor, in that the curves are significantly different at different backpressures. Higher backpressure settings are obtained by increasing pump speed and decreasing the opening of the BPR. Both actions will increase the shear to the paint. However, in our experiments, we cannot isolate the two effects.

252 Figure 6.16 shows that at zero backpressure, there is a slight trend that the n value linearly decreases to about 10% of its initial value after about 800 turns. At 20 psi backpressure, the n values decrease more dramatically as the circulation turns increases, and then level off after about 200 turns. The maximum degree of decrease is about 45% after 800 turns. At 40 psi backpressure, there is a even more clear trend that the n decreases about 45% within only 100 turns. The maximum degree of decrease is about the same as that of the 20 psi backpressure condition. From these it can been see clearly that the settings of the backpressure regulator (BPR) greatly influence the rheological changes of the paint.

The 20 psi backpressure plot in Figure 6.16 also shows a curve of the piston pump. As both rotary pump and piston pump were experimented at 20 psi backpressure and 0.6 gpm flowrate, the difference should be attribute to the pump only. For rotary pump, the n decreases about 35% after 200 runs, while for piston pump it decreases only less than 10%. After 400 turns the curve for rotary pump almost reaches the maximum

45% decrease, while the piston pump curve shows about 20%. This is clear that rotary pump caused much more viscosity change than the piston pump.

Another indication of the viscosity increase is the pump speed. As the paint viscosity increases, the pressure drop over the pipes increases, and the pumping capacity at that speed decreases as per the pump performance curve. During the circulation, the pump speed was increased to maintain the settings of the BPR. In all the 10 runs, the amount of pump speed increase is in the same trend as the pressure drop increase using power law model and the parameters fitted from corresponding rheological data.

However, pressure drop increase was not observed. In some cases, pressure drop

253 decreases slightly was observed. The possible reason may be the large variation of the pressure gauge readings. Two types of error may be resulted. One is that the reading is not repeatable. In this part the maximum error is estimated to be about +/- 40%. Average error is +/- 20%. The other is that the gauge will not respond to very slow pressure change, as in all 10 runs the readings are almost constant.

0 psi backpressure 20 psi backpressure 40 psi backpressure 80.0 80.0 80.0 0.3 gpm 0.3 gpm 0.3 gpm 70.0 70.0 70.0 0.6 gpm 0.6 gpm 0.6 gpm 60.0 0.9 gpm 60.0 0.9 gpm 60.0 0.9 gpm piston 0.6 gpm 50.0 50.0 50.0

40.0 40.0 40.0

30.0 30.0 30.0

20.0 20.0 20.0

Percent of decrease in n 10.0 Percent of decrease in n 10.0 Percent of decrease in n 10.0

0.0 0.0 0.0 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 turns turns turns

Figure 6.16. Decrease Percentage of power law parameter n.

6.3.5.2 Use of Inline Viscometer

The inline viscometer used in this work is able to measure temperature and viscosity continuously. The effective part contains a chamber, a piston, a flow deflector, and some coils. The deflector deflects the flow into the chamber, and two coils move the piston back and forth magnetically in the chamber at a constant force. The piston travel time is tracked and converted into viscosity. Therefore, the viscometer is a constant stress rheometer.

The readings of the inline viscometers can be correlated satisfactorily with the rheological data obtained using RSF II. The shear stress backed out is quite constant,

254 indicating the viscometer worked very well. The shear rate is calculated to be from 100 to

300 s-1. The deflector is efficient so that the fluid in the chamber is the same as that in the main flow. Generally speaking, this inline viscometer captures the general trend of the viscosity change during the circulation. The percentage increase of the inline viscometer readings is about 1.5 to 3 times the percentage of decrease in n. However, this variation may be too large for the quality control purpose. This variation may be caused by several factors. The most important factor is that the viscometer can only impose one constant stress, which is designed for Newtonian fluids. The working principles and calibration procedures determine that the viscometer would misinterpret the viscosity profile for a non-Newtonian fluid. A better design might be that the viscometer can impose two levels of stresses. The viscosities at the two stresses would give power law model parameters.

This is a very simplified stress sweep idea. The stress sweep is realized in a typical stress- controlled rheometer that the stress is changed (either increased or decreased) progressively over a wide range so that the viscosity corresponding to each stress level is measured. This way, the viscometer would work in a much wider range of fluid types.

The second factor is the thixotropy, which causes viscosities to change with time. The current design of the inline viscometer cannot capture this type of time effect. A better way to monitor the change of thixotropy may be the thixotropic index discussed below.

The Ford cup measurement was performed on some samples. As expected, as the circulation time increases, the Ford cup seconds increases, indicating the viscosity increase. The percentage of increase of the Ford cup seconds turns out to be the about the same percentage of decrease in n values. As the Ford cup is designed for Newtonian or near Newtonian fluids, whilst the paint shows extreme shear thinning behavior, this

255 similarity can only be attributed to coincidence. However, this coincidence may be used, at least for the paint studied, for the quality control purpose. The equivalent shear rate of the Ford cup #4 was evaluated to be about 100 s-1, using the power law model and the corresponding rheological data obtained on RSF II.

The percentage of increase of Ford cup seconds does not mean that the pressure drop over the pipe increase to the same amount. If the paint were a Newtonian, this would be true. Some constitutive models should be used to calculate the pressure. For the paint studied, power law model is used with the parameters fitted from the corresponding viscosity profile to predict the pressure drop. However, the pressure gauge didn’t show the predict pressure drop change.

6.3.5.3 Thixotropy

The classic definition of thixotropy is that the when a fluid is subject to external shear, the structure in the fluid is broken, rendering the viscosity to decrease. When the external shear is removed, the structure is allowed to rebuild, so that the fluid regain its original viscosity. Although other definitions exist, the key idea is the structure-break- down-and-rebuild.

Experimentally there are several ways to characterize the thixotropy. As thixotropy is a time dependent behavior in nature, transient viscosities should be measured. The most widely used methods includes thixotropy loop, in which the shear rate increase linearly from a very low level to a high level, and then decrease to the same very low level. As the time scale for the fluid structure to rebuild is usually much longer than the time need to break down, up and down curve will not be identical, see Figure

6.17. The area between the two lines is a measure of thixotropy. The larger the area is,

256 the greater the thixotropy. Another widely used method is steady shear at a constant shear rate. The viscosity is measured as a function of time. At the beginning the fluid has its structure, so the viscosity is high. As the fluid is being sheared, the structure is gradually broken down, so that the viscosity gradually decreases. A steady state will be reached after a long time shearing when the structure has been fully broken down at that shear rate, see Figure 6.18. Therefore, the amount of decrease of the viscosity with time is a characteristic of the thixotropy. In this article, both methods were used to capture the thixotropy. The two methods show exactly the same trend of the change in the thixotropy.

For the simplicity of quantification, the steady shear at a constant shear was used to define a thixotropy index.

Thixotropy loop 620-3

700 35

600 30

) 500 25 -1

400 20

300 15

Shear rate (s 200 10 Shear stress (Pa)

100 5

0 0 015304560 0 200 400 600 800 Time (s) Shear rate (s-1)

Figure 6.17. Thixotropy loop

257 0.16

0.14

0.12

0.1

0.08

0.06 Viscosity (Pa.s) 0.04

0.02

0 0 50 100 150 200 250 300 350 Time (s)

Figure 6.18. Steady shear at constant shear

In this paper a thixotropy index (Ti) is defined in the steady shear at a constant shear rate experiments to quantify the degree of thixotropy. It is defined as the difference of initial viscosity and steady state viscosity divided by the steady shear viscosity, see

Equation (6.6). The larger the Ti, the greater the thixotropy.

(η − η ) Ti = init SS (6.6) ηSS

Figure 6.19 shows the change of the Ti with circulation turns at different backpressure settings. In the 0 backpressure plot, data of all 3 flow rate runs fall on the same trend line, and this is roughly true for other two plots. The Ti is about 0.11 for the fresh paint, and gradually increases to abut 0.16 after 800 turns of circulation. In the 20 psi backpressure plot, the Ti increases quickly from about 0.11 to nearly 0.28 in 200 turns, and then gradually increase to about 0.34 after 800 turns. In the 40 psi backpressure

258 plot, the Ti increases very rapidly to 0.32 in about 100 turns and then increases very slowly until reach the maximum of 0.34. The 20 psi backpressure data 40 psi backpressure data suggests that there is a maximum limit of Ti. When the Ti reaches the limit, further shear will not affect the thixotropy.

In the 20 psi backpressure plot, the piston pump run data is also showed. The Ti increases linearly from 0.11 to 0.27 after about 550 turns. Comparing the piston run data with the trend line of the rotary pump data, it is clear that the trend line has larger value than the piston pump data. In lower turns, the difference may be as large as 100%. This indicates that the rotary pump causes much more shear to the paint than the piston pump.

Compared with Figure 6.16, the thixotropy index almost has the same trend as the change of viscosity. This similarity suggests that the shear induced by the circulation changed certain aspect of the paint that causes both the increase of viscosity and thixotropy. Further analysis of this factor is probed using the dynamic method.

0 backpressure 20 psi backpressure 40 psi backpressure

0.4 0.4 0.4

0.35 0.35 0.35

0.3 0.3 0.3

0.25 0.25 0.25

0.2 0.2 0.2 thix index thix index 0.15 0.15 thix index 0.15 0.3 gpm 0.1 0.1 0.1 0.6 gpm 0.3 gpm 0.3 gpm 0.9 gpm 0.05 0.05 0.6 gpm 0.05 0.6 gpm piston 0.6 gpm 0.9 gpm 0.9 gpm 0 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 Turns Turns Turns

Figure 6.19. Change of thixotropy of the paint during circulation with various length of circulation times.

259 6.3.5.4 Dynamic measurement

Dynamic oscillation method has long been used to study the viscoelasticity of a fluid. In this article, small amplitude sinusoidal oscillation was used to study the linear viscoelasticity of the paint. As in the linear region, the internal microstructure of the fluid is not destroyed, it is a strong tool to probe the microstructure of a liquid. The elastic modulus G’ and viscous modulus G” are measured at different frequencies. A typical change in G’ during the circulation (Run 620) is shown in Figure 6.20. It clearly shows that the circulation greatly affect the G’ and G”. G’ and G” increase very fast within 150 turns, then slow down and finally reach a ultimate limit after 494 turns. It is noticed that the G’ and G” curves are rather straight and have about the same slope in the log-log plot.

This phenomenon implies the “onset of a critical elastic network” 205, such as the gels near their gel points. This explains that the thixotropy increases during the circulation.

Before the circulation, the paint does not have much structure, so it exhibits almost no thixotropy. With the increase of the circulation time, the paint is sheared by the pump so that the it develops a certain network structure. This structure greatly increases the viscosity of the paint, as seen in the viscosity profile versus shear rate. However, this kind of structure is weak and can be easily broken down by the shear. Therefore, as the shear continues, the structure kept being broken down, so the viscosity decreases. This is what happens during the steady shear at constant shear rate. Therefore, the thixotropy increases.

The empirical “Cox-Merz” rule works very well with the paint. The complex viscosity is almost identical to the steady state viscosity, except for a few data points in the low shear region.

260 0.6 gpm, 20 psi 0.6 gpm, 20 psi 0.6 gpm, 20 psi 100 100 10

10 10 δ 1 tan G' (Pa)

G" (Pa) 0.01 0.1 1 10 100 1 1 0.01 0.1 1 10 100 0.01 0.1 1 10 100

0 turn 0 turn 0 turn 150 turns 150 turns 150 turns 0.1 314 turns 0.1 314 turns 0.1 314 turns freq (s-1) 494 turns freq (s-1) 494 turns freq (s-1) 494 turns

Figure 6.20. Typical change of elastic and viscous moduli (G’, G”) during circulation

6.3.5.5 Color Degradation

A typical result of the color measurement is shown in Figure 6.21.

The color travel is defined as:

DL = L15 – L75.

The percent change of color travel over uncirculated sample is

% = DL/DL0 *100, where DL0 is the color travel of uncirculated sample.

The results of color degradation are plotted in Figure 6.22. According to the PPG standards for color measurement, a positive change indicates that the sample is lighter than standard. Therefore, it seems that the paint quality is enhanced after circulation. The degradation doesn’t occur. However, this is contradicting with what we expected.

According to our past experience, the color travel of circulated paint should be less than uncirculated paint. This contraction cannot be explained yet. It is possible that there were some measurement inconsistence, as the paint samples had been put aside for five months

261 before measurements. The measurements of all the samples took about a month, during which the operating condition of the booth might be different. All these could affect the measured data.

BPR=20psi, Q=0.3gpm

140.00

120.00

100.00

0 80.00 1 2

L Value 60.00 3

40.00

20.00

0.00 0 20406080100120 Angle (Deg)

Figure 6.21. A typical result of the color measurement. Results shows that for sample 2 and 3 (longer circulation time), there is larger color travel than uncirculated sample (sample 0)

262 35.00

30.00

25.00 0.3 gpm, 0 psi 0.6 gpm, 0 psi 20.00 0.9 gpm, 0 psi 0.3 gpm, 20 psi 0.6 gpm, 20 psi 15.00 0.9 gpm, 20 psi 0.3 gpm, 40 psi 10.00 0.6 gpm, 40 psi 0.9 gpm, 40 psi 5.00 piston 0.6 gpm, 20 psi

0.00 % change of color travel over uncirculated 0 200 400 600 800 1000 -5.00 Turns

Figure 6.22. Color degradation results show that almost all the samples are lighter than standard.

6.3.6 Conclusion

The circulation system causes permanent rheological changes to the waterborne metallic paint. It was identified that the rotary pump with piston rotors causes much more rheological change than the double acting piston pump. The rotary pump did not cause much rheological change at low operating pressure while at high operating pressure it did. Therefore the settings of the backpressure regulator should be as low as possible.

Different flow speed was observed to cause about the same effects when the flow speed was converted to the circulation times. This indicates that the flowrate is not a significant factor as far as the rheological change is concerned.

The change of viscosity profiles and the degree of thixotropy showed that when the BPR is fully open, the paint is not affected much by the circulation. However, when 263 the BPR is 20 psi the viscosity and the degree of thixotropy increased progressively, until they leveled off after about 400 circulation turns. At 40 psi BPR setting, the viscosity and the degree of thixotropy increased dramatically within about 200 turns, reaching about the same maximum as in 20 psi backpressure setting.

Dynamic viscosity analysis shows that the paint developed a kind of elastic structure during the circulation. This elastic structure increased the viscosity and degree of thixotropy of the paint. Compared the complex viscosity with steady shear viscosity profile, the Cox-Merz rule works very well for this waterborne paint.

For the quality control purpose the functions of an in-line viscometer and the Ford cup were evaluated. The in-line viscometer is a constant shear stress device and the shear rate varies with the fluid viscosity. As the in-line viscometer used in this article is designed for Newtonian fluid, it is not recommended for the quality control purpose for the pseudoplastic paint, although it showed the general trend of the viscometer change.

The Ford cup is designed to measure the viscosity for Newtonian or near Newtonian fluid, however, the Ford cup seconds happened to have the same trend as the viscosity change. This coincidence can be used to monitor the quality of the paint for the studied paint only. Cares should be taken for other types of paints, because just like the in-line viscometer, the amount of measured viscosity change may not reflect the true value.

The results of color degradation measurements show that the paints are even lighter (brighter) after being circulated, which is contradicting to our past experience.

Possible reasons is that there were inconstancies during the measurement.

264

CHAPTER 7

CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

7.1.1 CNF dispersion

The as-received nanofibers are entangled and contain millimeter and centimeter size agglomerates. The extent of entanglement and agglomeration are determined using the CNF production process. As such, dispersion of the CNFs without breaking them could be a problem. This problem has been studied using various techniques and in two types of matrices. In the case of a Newtonian fluid matrix with a viscosity of about 0.1

Pa-s at room temperature the uniform dispersion of CNFs was difficult to achieve.

Regular mechanical mixing using a magnetic stirring bar did not work. Ultrasonication using a sonication bath proved to be less efficient. The sonication could break loose some of the CNFs from the original agglomerates, but it does not eliminate aggregations completely. Most likely, the sonication is only strong enough to break the not-so-tightly bundled CNFs.

265 Surface treatment proved to work well. In this work, the CNFs were treated with a mixture of sulfuric acid and nitric acid. Presumably, carbolic acid or hydroxyl groups attached to the surface after chemical treatment. We did not characterize the exact composition and amount of these groups, but the effect of the acid treatment is obvious.

The dispersion of the treated CNFs is much better than untreated CNFs. For example, treated CNFs disperse well using just a magnetic stirring bar. However, it seems that the strength of the CNFs is weakened by the treatment, most likely because the sonication process broke most of the CNFs into smaller pieces. Nonetheless, evidence of physical damage was not found from surface smoothness of the treated CNFs.

Dispersion of the CNFs in polymer melts seems to be relatively easy, with about

10wt% CNFs in polystyrene dispersing with no problem. However, further examination showed that the CNFs had been greatly broken due to the high stress induced by the extruder. In order to determine the best possible dispersion in polymer melt and achieve a homogeneous dispersion of CNFs in polymer without breakage, we employed a solvent casting method. A solvent was used as a buffer and a high power sonicator was used to disperse the un-treated (i.e. as received) CNFs in polymer/solvent solution, after which the solvent was driven out to leave the desired composite. It was found that the dispersion was satisfactory, although during evaporation of the solvent CNFs start to settle.

Regardless, this slight in-homogeneity was assumed to have no effect on the composite rheology.

266 7.1.2 Effect of CNFs on suspension rheology

The presence of the CNFs greatly impacts the suspension rheology whether the matrix was a low viscosity Newtonian solution or highly viscous polymer melt.

Additionally, the CNFs introduce extra stress to the suspensions. As a result, the modulus, viscosity, and normal force increase with higher CNF concentrations. More importantly, due to the small size of the CNFs the interactions between them are large, especially physical contacts or entanglements if the CNFs are long enough. These interactions generate some special phenomena, such as plateau of G’ at low frequencies, yield stress, and shifts of the cross-over point of G’ and G”. These phenomena are interesting from the standpoint of academic research, but they may not be important in terms of industrial scale production. In industrial scale production, the shear rate is usually large, up to 100 to 1000 s-1 in extruders and injection molding machines. Most of the special phenomena disappear at these shear rates.

Shear and elongational viscosity of the suspensions have been captured by the models we proposed. It seems that these CNF suspensions are not so different from traditional fiber suspensions, where the fibers are mm scale or even larger. The most significant difference is that certain phenomena, which occur for large fibers such as glass and conventional carbon fibers, occur for CNFs at a much lower particle concentration. Therefore, models for large fibers can be applied to the CNFs, with some parameters changed. The underling principles of large fibers and CNFs are the same. The interactions between the CNFs are relatively large at low concentration, where larger fibers have the same interactions, but at a higher fiber concentration.

267 7.1.3 Relationship between CNF orientation and rheology

Although the orientation of CNFs has detectable effects on the rheology of the suspensions, their effects seem to be small. In the case of PS/CNF composites, the effect of orientation can only be detected at 0.1 s-1 (and possibly lower). At higher shear rates, the orientation effect disappears. As we know that generally the rheology of the composites depends on the rheology of the matrix, loading of the particles, interactions between the particles and the orientation of the particles. It is very likely that the first three items play a larger role at higher shear rates, such that the effect of the particle orientation is almost negligible.

7.1.4 Rheology of waterborne coatings

Generally speaking, the waterborne coatings, whether metallic or non-metallic, are extremely shear thinning. Due to the use of water as the solvent, the viscosity of the coatings is usually too low if thickeners are not used. Although thickeners are also used in solvent-borne coatings, their contributions to the rheology are not as large as the thickeners for waterborne coatings. Needless to say, the chemical structure and the mechanism of thickening are very different for the thickeners in solvent borne and waterborne coatings. Regardless, shear thinning behavior is, usually, a desired property for any coating and waterborne coatings happen to be more so. This makes some of the waterborne coatings have better performance properties than the solvent borne ones.

Thixotropy is relatively mild, but easily detectable, for the metallic coatings we studied. Compared to the non-metallic coatings, the thixotropy seems to come from the presence of the metal flakes. However, since we don’t have the exact composition of the coatings, we don’t have 100% confidence in this assessment. We found that for the

268 purpose of pressure drop calculation in the paint circulation systems the thixotropy can be safely neglected. In fact, this is also the coating manufacturer’s suggestion. Previous studies using more complicated models to take the thixotropy into account are theoretically better and academically more interesting, but I found them less useful in practice. Many experimental difficulties have to be overcome and many times bold assumptions have to be made. For interested readers, please refer to Susan Porter’s

Master’s degree thesis.

Temperature dependence of the viscosity is simple for the solvent borne coatings.

The viscosity drops with higher temperature. However, we found a reverse phenomenon for our metallic waterborne coatings. In the temperature range of 10 to 50°C, the viscosity increases with temperature and even gels at about 50°C. Likewise, the viscosity decreases with decreasing temperature. Since this process is reversible, there should be no chemical reaction or curing involved. Comparing to the results of a similar formulated non-metallic waterborne coating, we contribute the reverse phenomenon to the presence of the metal flakes and their interactions with the thickeners. From the standpoint of production there is nothing to worry about, as the reverse temperature dependence is only obvious at very small shear rates. At high shear rates, the change in viscosity is very small. For the design of paint circulation system, where the typical shear rate is about 100 to 1000 s-1, the reverse temperature dependence of the viscosity can be neglected.

For details, please refer the conclusion section in each chapter.

269 7.2 Recommendations and future work.

In the study of CNF in Newtonian solutions, we used a sonication bath to dispersion the CNFs but found it less effective. At later time of my PhD research, I gained access to a high powerful ultrasound probe, which was used in our studies of

CNF/PS composites. The ultrasound probe was found to be powerful enough to disperse

CNFs in relatively low viscosity solutions. Thus, in addition to the two types of suspensions that we prepared in Chapter 3, we could have prepared a third one, a fully dispersed CNF suspension without damage to the CNFs. Thus, our discussion could be more interesting. Thus my first recommendation is to prepare new samples using the powerful ultrasound probe and see what will happen.

In the characterization of the lengths of the CNFs, we could have used statistical methods to gain a more accurate measure of the length of the CNFs. We only did that in the case of the TEM photos of the CNF orientations, and used the information to calculate the orientation tensor. In the case of the CNF in Newtonian solutions, we didn’t use this method, as we found that direct comparison of the SEM photos of the CNFs were enough to illustrate the same idea. Also, to measure the length of the CNFs, better SEM or other pictures are needed, which requires more time and energy. Nevertheless, having statistics of the CNF lengths would be the best method to make comparisons.

270 Future work on the PS/CNF composites should include the measurement of the performance properties, mainly mechanical and electrical properties. We did try to measure the electrical properties, but due to the limitation of the equipment, we couldn’t get any meaningful measurement of the conductivities on our composites. It would be much more interesting if a more sensitive conductivity meter is available. This would allow the CNF network structure to be inferred from the conductivity measurement.

We didn’t perform mechanical tests on our composites, but we know that the composites become more brittle with higher CNF loadings. The Young’s modulus may have increased, but we didn’t measure it. The Young’s modulus of almost any type of composite is usually higher than that of the pure polymer matrix. But at the same time, the presence of the solid particles makes the composite brittle. We have noticed the brittleness for our PS/CNF composites, especially at higher CNF concentrations. On the other hand, higher CNF concentration is needed for the electrical and/or thermal conductivities. So for product development purposes, there might be a sweet point, that the gain in the modulus and conductivity can justify the loss in the flexibility.

271 APPENDIX A

SOME MISCELLANEOUS AND UNSUCCESSFUL EXPERIMENTAL

RESULTS

A.1 Results from PS/CNF Composites Using Another Type of Polystyrene

In the beginning of the research on the PS/CNF composites, the polystyrene made by Nova Chemicals were used. At that time, we had two types of PS to choose from. One was made my Nova Chemicals, and the other by Atofina Chemicals. The PS made by

Nova came in very small beads. Average diameter of the beads was about 200-300 microns. The PS made by Atofina was in the form of 3mm pellets. We figured that the small size PS made by Nova would have a better mix between the PS and CNF, so it was chosen. However, until very late of the research when a lot of rheology had been done, it was found that the repeatability of the rheological tests was not very good. Most often there was 20-50% error between tests. Sometimes the error was in the order of 10 times, especially for transient tests. A series of tests were done to determine the causes for the bad repeatability in the rheology tests. The results showed that most likely it was either heat or oxidation or the combination of both that degraded the polymer.

272 After that, we switched to the PS made by Atofina (Fina) and obtained excellent repeatability. On average, the error between tests is about 5-10%, much higher than those tests made on Nova PS. As a result, all materials made from Nova PS were discarded and the composites were prepared again using Fina PS.

The dispersion of the CNFs in both type of PS is indistinguishable. The rheology of the composites made with different PS matrices is slightly different. But in principle, all the results we found in Chapter 4, which is based on Fina PS, applies to the composites made from Nova PS. Therefore, the rheology of composites made from Nova

PS is not shown here.

Two important experiments that were performed using composites made from

Nova PS and not repeated with those made from Fina PS were (1) the measurement of the thermal conductivities of the composites, and (2) Tg (glass transition temperature) of the composites. Since it is not appropriate to put it in any chapter, I decided to put them in this appendix.

A.1.1 Thermal conductivity of the Nova PS/CNF composites.

The thermal conductivity of the Nova PS/CNF composites were measured using a modulated differential scanning calorimeter (MDSC) made by TA Instruments (model

2920). The test samples were made and the test procedures were taken following the manual of the device. The results are shown in Figure A.1. The conductivities of MB composites remain the same with up to 2wt% of the CNFs. At 5wt% and 10wt%, the conductivity increases in a linear fashion. For SC composites, there is no change with

2wt% CNF and less. But with 5wt%, the thermal conductivity increases much more than the MB composite with same CNF loading. In fact, the conductivity of 5wt% SC

273 composites is higher than 10wt% MB composites. Since the lengths of the CNFs are much longer, the difference in the conductivity is not so surprising. The long CNFs form a possibly continuous network structure that can conduct heat much better. More contrast between the MB and SC composites can be seen in Figure A.2, where the relative increase of the thermal conductivity is plotted against the CNF loading.

A.1.2 Measurement of glass transition temperature (Tg)

The Tg of MB composites was measured using the DCS 2920. We didn’t find too much change in the composites due to the presence of the CNFs. The measurement is plotted in Figure A.3. We see that there might be a slight trend that the Tg increases with

CNF loadings, but the increase is too small comparing to the experimental error.

Therefore, the conclusion from this figure is that the addition of the CNF does not affect the Tg of the composite.

A.2 Electrical Conductivity Measurement of PS/CNF Composites

Electrical conductivity measurement of both MB and SC composites based on the

Fina type of PS has been attempted. However, no successful results has been obtained.

The so-called four probe measurement using the equipment in the Electrical Engineering at OSU produced no data at all. It seems that the conductivity of the compostes, including the SC10, is too small for the equipment to measure.

A.3 Electrical Conductivity Measurement of CNF Aqueous Suspensions

Attempts have been made in measuring electrical conductivity measurement of untreated and treated CNF suspensions. The preparation and characterization of the suspensions are described in Chapter 3. A digital conductivity meter (Orion 1230) borrowed from Civil Engineering at OSU was used. The meter consists of two square

274 metal (platinum) electrodes with dimension of about 5mm x 5mm facing each other with about 5mm gap size. The meter was able to give conductivity directly. It was calibrated with NaCl aqueous solutions with various salt concentrations and the conductivity measurements agreed very well with the published literature.

However, during the measurement of the suspensions, the readings were very unstable, especially for suspensions with higher CNF concentrations (greater than 3wt%).

The readings tend to increase dramatically in minutes, and usually reached a maximum after a period of time (on the order of several minutes). This maximum was not quite repeatable. Each time a different maximum was obtained. Because of this unreliable reading, the data that we chose to present was somewhat subjective (see Figure A.4). This figure is for reference only.

275

0.25

0.2

0.15

0.1 SC composites

0.05 MB composites thermal conductivity (W/mK) 0 0 5 10 15 wt% CNF

Figure A.1. Thermal conductivities of Nova PS/CNF composites.

276 40

35 Melt blended (short) 30 Solvent cast (long) 25 20 15 conductivity 10

% increase in thermal 5 0 125 carbon nanofiber concentration (wt%)

Figure A.2. Relative increase of the thermal conductivities of MB and SC composites.

277 110

108

106

Tg (C) 104

102 MB composite

100 0 5 10 15 wt% CNF

Figure A.3. Glass transition temperature (Tg) of MB composites.

278 100 Untreated max treated

10

1 0123456 Conductivity (uS/cm)

0.1 fiber content (wt%)

Figure A.4. Electrical conductivity of CNF suspensions.

279

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