MECHANISMS OF CELL NUCLEATION, GROWTH, AND … · 2010-11-03 · MECHANISMS OF CELL NUCLEATION,...

MECHANISMS OF CELL NUCLEATION, GROWTH, AND COARSENING IN

PLASTIC FOAMING: THEORY, SIMULATION, AND EXPERIMENT

by

Siu Ning Sunny Leung

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

© Copyright by Siu Ning Sunny Leung 2009

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MECHANISMS OF CELL NUCLEATION, GROWTH, AND COARSENING IN

PLASTIC FOAMING: THEORY, SIMULATION, AND EXPERIMENT

Siu Ning Sunny Leung

Degree of Doctor of Philosophy, 2009

Department of Mechanical & Industrial Engineering

University of Toronto

ABSTRACT

This thesis highlights a comprehensive research for the cell nucleation, growth and

coarsening mechanisms during plastic foaming processes. Enforced environmental regulations

have forced the plastic foam industry to adopt alternative blowing agents (e.g., carbon dioxide,

nitrogen, argon and helium). Nevertheless, the low solubilities and high diffusivities of these

viable alternatives have made the production of foamed plastics to be non-trivial. Since the

controls of the cell nucleation, growth and coarsening phenomena, and ultimately the cellular

morphology, involve delicate thermodynamic, kinetic, and rheological mechanisms, the

production of plastics foams with customized cell morphology have been challenging. In light of

this, the aforementioned phenomena were investigated through a series of theoretical studies,

computer simulations, and experimental investigations. Firstly, the effects of processing

conditions on the cell nucleation phenomena were studied through the in-situ visualization of

various batch foaming experiments. Most importantly, these investigations have led to the

identification of a new heterogeneous nucleation mechanism to explain the inorganic fillers-

enhanced nucleation dynamics. Secondly, a simulation scheme to precisely simulate the bubble

growth behaviors, a modified heterogeneous nucleation theory to estimate the cell nucleation

rate, and an integrated model to simultaneously simulate cell nucleation and growth processes

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were developed. Consequently, through the simulations of the cell nucleation, growth, and

coarsening dynamics, this research has advanced the understanding of the underlying sciences

that govern these different physical phenomena during plastic foaming. Furthermore, the impacts

of various commonly adopted approximations or assumptions were studied. The end results have

provided useful guidelines to conduct computer simulation on plastic foaming processes. Finally,

an experimental research on foaming with blowing agent blends served as a case example to

demonstrate how the elucidation of the mechanisms of various foaming phenomena would aid in

the development of novel processing strategies to enhance the control of cellular structures in

plastic foams.

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To my beloved wife, Jody, and son, Ethan, for your endless love, strong support,

and inspiring encouragement during the long journey of my Ph.D. study. I

could not have done it without you. Your love is and will always be in my

heart.

To my parents, brother, and sister-in-law for your continuous care and love.

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ACKNOWLEDGMENTS

Throughout the course of my Ph.D. studies, there has been a multitude of people that

have made time at the University of Toronto a success. I would like to thank everyone that

supported, encouraged, and guided me to overcome a variety of challenges.

I am deeply indebted to my supervisor, Professor Chul B. Park, for his valued

supervision, guidance and encouragement throughout my research in the Microcellular Plastics

Manufacturing Laboratory. I would like to express my deep and sincere gratitude to him. His

understanding, encouragement and personal guidance have provided a good basis for my

research work and my future career.

I would also like to express my warm and sincere thanks to my Ph.D. thesis committee:

Professor Hani Naguib and Professor Charles Ward, both from the Department of Mechanical

and Industrial Engineering, for their invaluable advices throughout my Ph.D. thesis research. In

addition, I would like to thank Professor Markus Bussmann and Dr. Shau-Tarng Lee for their

feedback in my Ph.D. final oral examination.

My gratitude is extended to the School of Graduate Studies (SGS) at the University of

Toronto, to the Department of Mechanical and Industrial Engineering at the University of

Toronto, to the Ontario Centres of Excellence, and to the Ontario Graduate Scholarship Program

for providing academic scholarships. Also, I would like to thank the Consortium for Cellular and

Micro-Cellular Plastics (CCMCP) and the Natural Sciences and Engineering Research Council

(NSERC) of Canada for their funding and support in this research.

I would like to thank my colleagues and fellow researchers in the Microcellular Plastics

Manufacturing Laboratory for their help and friendship over the past years. Their advice and

support have been invaluable. Much of the work throughout this thesis research would not have

been possible without contributions from these fantastic people. My sincere gratitude goes to Dr.

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Guangjian Guo,Dr. Qingping Guo, Dr. Ryan Kim, Dr. John Lee, Dr. Kevin Lee, Dr. Kyungmin

Lee, Dr. Patrick Lee, Dr. Gary Li, Dr. Hongbo Li, Dr. Guangming Li, Dr. Takashi Kuboki, Dr.

Kumar, Dr. Moon, Dr. Mohammed Serry, Dr. Chunmin Wang, Dr. Jin Wang, Dr. Jing Wang, Dr.

Donglai Xu, Dr. Yoon, Dr. Wenge Zheng, Dr. Wenli Zhu, Dr. Zhenjin Zhu, Sue Chang, Nan

Chen, Raymond Chu, Mohammed Hasan, Peter Jung, Esther Lee, Richard Lee, Lilac Wang,

Anson Wong, and Hongtao Zhang. Their kind support and the stimulating discussions with these

intelligent people have made my graduate studies a pleasant journey.

I owe a big thanks to my family. I would like to thank my parents for bringing me to this

wonderful world and consistently supporting me under any circumstance; to my brother and

sister-in-law for their limitless cares and loves; to my wife, Jody, for her continuous supports

with her deep love, inspiring encouragement, and true understanding, especially during my most

difficult times; to my lovely son, Ethan, who motivates me with his big hugs and sweet smiles.

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Table of Contents

ABSTRACT .................................................................................................................................... ii

ACKNOWLEDGMENTS .............................................................................................................. v

Table of Contents .......................................................................................................................... vii

List of Tables ............................................................................................................................... xiv

List of Figures .............................................................................................................................. xvi

Chapter 1 INTRODUCTION ......................................................................................................... 1

1.1. Preamble ........................................................................................................................................1

1.2. Plastic Foams and Their Processing ..............................................................................................3

1.3. Challenges to Plastic Foams Production .......................................................................................4

1.4. Objectives of the Thesis ................................................................................................................5

1.5. Overview of the Thesis ..................................................................................................................7

Chapter 2 LITERATURE REVIEW & THEORETICAL BACKGROUND .............................. 11

2.1. Fundamentals of Blowing Agents ............................................................................................... 12

2.1.1. Physical Blowing Agents (PBAs)........................................................................................ 12

2.1.2. Chemical Blowing Agents (CBAs) ..................................................................................... 13

2.1.3. Formation of a Single-Phase Polymer-Gas Solution ........................................................... 13

2.2. Fundamentals of Cell Nucleation ................................................................................................ 16

2.2.1. Review of Nucleation .......................................................................................................... 16

2.2.1.1. Classical Homogeneous Nucleation ................................................................................ 17

2.2.1.2. Classical Heterogeneous Nucleation ............................................................................... 17

2.2.1.3. Pseudo-Classical Nucleation ........................................................................................... 18

2.2.2. The Classical Nucleation Theory (CNT) ............................................................................. 19

2.2.2.1. Free Energy Barrier for Homogeneous Nucleation ......................................................... 19

2.2.2.2. Free Energy Barrier for Heterogeneous Nucleation ........................................................ 21

2.2.3. Kinetics of Cell Nucleation ................................................................................................. 24

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2.3. Modeling of Cell Growth and Cell Coarsening ........................................................................... 26

2.3.1 The Single Bubble Growth Model ....................................................................................... 26

2.3.2 The Cell Model .................................................................................................................... 29

2.3.3 Cell Collapse, Cell Coarsening and Cell Coalescence during Plastic Foaming .................. 31

2.3.4 Mathematical Formulations to Describe Bubble Growth .................................................... 32

2.4. Experimental Studies on Plastic Foaming Mechanism ............................................................... 34

2.4.1. Heterogeneous Nucleation with Nucleating Agents ............................................................ 34

2.4.2. In-situ Visual Observation of Plastic Foaming .................................................................... 36

2.4.3. Stress-Induced Nucleation ................................................................................................... 37

2.5. Computer Simulation of Plastic Foaming ................................................................................... 38

2.5.1. Influence Volume Approach (IVA) ..................................................................................... 38

2.5.2. Modified Influence Volume Approach (MIVA) ................................................................. 40

2.5.3. Computer Simulation of a Continuous Foaming Process .................................................... 40

2.6. Summary of Literature Survey and Critical Analysis .................................................................. 41

Chapter 3 CELL NUCLEATION PHENOMENA IN PASTIC FOAMING .............................. 49

3.1. Introduction ................................................................................................................................. 49

3.2. Background and Research Methodology ..................................................................................... 50

3.2.1. Plastic Foaming Under Different Processing Conditions .................................................... 50

3.2.2. Plastic Foaming Using Nucleating Agents .......................................................................... 51

3.3. Theoretical Framework ............................................................................................................... 52

3.3.1. Classical View of Cell Nucleation ....................................................................................... 52

3.3.2. Dynamic Change of Rcr and Activation of Pre-existing Gas Cavities ................................. 54

3.3.3. Stress-Induced Nucleation ................................................................................................... 55

3.4. Experimental ............................................................................................................................... 56

3.4.1. Materials .............................................................................................................................. 56

3.4.2. Sample Preparation Materials .............................................................................................. 56

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3.4.3. In-situ Foaming Visualization ............................................................................................. 57

3.4.3.1. Experimental Procedures ................................................................................................. 57

3.4.3.2. Experiments to Study the Effects of -dPsys/dt, C, and Tsys on Cell Nucleation ................ 58

3.4.3.3. Experiments to Study the Effects of Talc on Cell Nucleation ......................................... 58

3.4.4. Characterization ................................................................................................................... 58

3.4.4.1. Effects of -dPsys/dt, C, and Tsys on Cell Nucleation ......................................................... 58

3.4.4.2. Effects of Talc on Cell Nucleation .................................................................................. 59

3.5. Results and Discussion ................................................................................................................ 60

3.5.1. Effect of Processing Conditions on Cell Nucleation ........................................................... 60

3.5.1.1. Effects of Pressure Drop Rate on Cell Nucleation .......................................................... 60

3.5.1.2. Effects of CO2 Content on Cell Nucleation ..................................................................... 60

3.5.1.3. Effects of Processing Temperature on Cell Nucleation ................................................... 61

3.5.2. Effect of Talc on Cell Nucleation ........................................................................................ 61

3.5.2.1. Effect of Talc Particles on Cell Nucleation Mechanism ................................................. 62

3.5.2.2. Effect of Talc Content on Cell Nucleation Mechanism in PS-Talc-CO2 Foaming ......... 63

3.5.2.3. Effect of Gas Content on Cell Nucleation Mechanism in PS-Talc-CO2 Foaming .......... 64

3.5.2.4. Effect of Surface Treatment of Talc on Cell Nucleation Mechanism in PS-Talc-CO2 Foaming……………. .......................................................................................................................... 65

3.5.2.5. Effect of Talc’s Particle Size on Cell Nucleation Mechanism in PS-Talc-CO2 Foaming….. ............................................................................................................ ………………….66

3.5.2.6. Effect of Processing Temperature on Cell Nucleation Mechanism in PS-Talc-CO2 Foaming….. ......................................................................................................................................... 67

3.6. Summary and Conclusions .......................................................................................................... 68

Chapter 4 BUBBLE GROWTH PHENOMENA IN PLASTIC FOAMING .............................. 85

4.1. Introduction ................................................................................................................................. 85

4.2. Modeling of Bubble Growth Dynamics ...................................................................................... 86

4.2.1. Simulation Model and Assumptions .................................................................................... 86

4.2.2. Mathematical Formulations ................................................................................................. 87

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4.2.3. Methodology of Computer Simulation ................................................................................ 89

4.2.4. Determination of Physical Parameters for Computer Simulation ....................................... 89

4.3. Experimental Verification ........................................................................................................... 90

4.3.1. Materials .............................................................................................................................. 90

4.3.2. Experimental Apparatus and Procedures ............................................................................. 90

4.4. Results and Discussion ................................................................................................................ 90

4.4.1. Experimental Results ........................................................................................................... 90

4.4.2. Determination of Physical Parameters for Computer Simulation ....................................... 91

4.4.3. Computer Simulation and Comparison with Experimental Results .................................... 91

4.5. Sensitivity Analyses .................................................................................................................... 92

4.5.1. Effect of Initial Bubble Radius Experimental Results ......................................................... 92

4.5.2. Effect of Initial Shell Radius (Rshell,t=t’) ................................................................................ 92

4.5.3. Effect of Diffusivity (D) ...................................................................................................... 93

4.5.4. Effect of Solubility (KH) ...................................................................................................... 93

4.5.5. Effect of Surface Tension (γlg) ............................................................................................. 93

4.5.6. Effect of Relaxation Time (λ) .............................................................................................. 94

4.5.7. Effect of Zero-Shear Viscosity (η0) ..................................................................................... 94

4.6. Summary and Conclusions .......................................................................................................... 95

Chapter 5 CELL STABILITY IN PLASITIC FOAMING ........................................................ 102

5.1. Introduction ............................................................................................................................... 102

5.2. Theoretical Framework ............................................................................................................. 104

5.2.1. Implementation of Cell Model to Model CBA-Based Bubble Growth and Collapse Processes ………………………………………………………………………………………….. 104

5.2.2. Mathematical Formulations ............................................................................................... 105

5.2.3. Determination of Critical Radius ....................................................................................... 106

5.3. Implementation of a Computer Simulation ............................................................................... 106

5.3.1 Numerical Simulation Algorithm ...................................................................................... 106

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5.3.2 Materials and Physical Parameters .................................................................................... 107

5.3.3 Initial Conditions ............................................................................................................... 108

5.4. Experimental Verification ......................................................................................................... 108

5.4.1 Sample Preparation ............................................................................................................ 109

5.4.2 Experimental Procedure .................................................................................................... 109

5.5. Results and Discussion .............................................................................................................. 109

5.5.1 Computer Simulation ......................................................................................................... 109

5.5.2 Computer Simulation vs. Experimental Simulation .......................................................... 111

5.5.3 Effect of Diffusivity on the Sustainability of a Bubble ..................................................... 112

5.5.4 Effect of Surface Tension on the Sustainability of a Bubble ............................................. 112

5.5.5 Effect of Solubility on the Sustainability of a Bubble ....................................................... 112

5.5.6 Effects of Viscosity and Elasticity on the Sustainability of a Bubble ............................... 113

5.6. Summary and Conclusions ........................................................................................................ 113

Chapter 6 SIMULTANEOUS COMPUTER SIMULATION OF CELL NUCLEATION & GROWTH ................................................................................................................................... 120

6.1. Introduction ............................................................................................................................... 120

6.2. Development of a Modified Heterogeneous Nucleation Theory ............................................... 121

6.3. Research Methodology .............................................................................................................. 123

6.3.1. Simultaneous Simulation of Cell Nucleation and Growth ................................................ 123

6.3.1.1. Overall Simulation Methodology .................................................................................. 124

6.3.1.2. Determination of Physical Parameters .......................................................................... 125

6.3.2. Experimental Verification ................................................................................................. 128

6.3.3. Impact of the Pbub,cr Approximation on Foaming Simulation ............................................ 129

6.3.4. Impact of the Psys Profile Approximation on Foaming Simulation ................................... 130


6.5.1. Simultaneous Simulation of Cell Nucleation and Cell Growth Phenomena ..................... 130

6.5.1.1. Computer Simulation and Experimental Verification of the Base Case ....................... 130

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6.5.1.2. Effects of the Rbub on γlg of a Critical Bubble ................................................................ 132

6.5.1.3. Sensitivity Analysis on the Effect of Contact Angle on the Computer Simulation ....... 132

6.5.1.4. Additional Experimental Verification under Various Processing Conditions ............... 133

6.5.2.1. Effects of Pressure Drop Rate and Dissolved Gas Content on Cell Size Distribution .. 134

6.5.2. Impact of Pbub,cr Approximation on Foaming Simulation ................................................. 134

6.5.3. Impact of Psys Profile Approximation on Foaming Simulation ......................................... 135

6.5.3.1. Validity of the Psys Profile Approximation on Calculated Cell Density ........................ 135

6.5.3.2. Validity of the Psys Profile Approximation on Calculated Cell Size .............................. 136


Chapter 7 PREDICTION OF PRESSURE DROP THRESHOLD FOR NUCLEATION ........ 152

7.1. Introduction ............................................................................................................................... 152

7.2. Methodology ............................................................................................................................. 153

7.2.1. Implementation of the Semi-Empirical Method ................................................................ 153

7.2.2. Implementation of the Theoretical Method ....................................................................... 154


7.3.1. Effect of –dPsys/dt on ΔPthreshold .......................................................................................... 155

7.3.2. Effect of Gas Content on ΔPthreshold .................................................................................... 156

7.3.3. Effect of Processing Temperature on ΔPthreshold ................................................................. 156

7.4. Sensitivity Analysis ................................................................................................................... 157

7.4.1. Effect of Surface Tension at the liquid vapor interface (γlg).............................................. 157

7.4.2. Effect of Relaxation Time (λ) ............................................................................................ 158

7.4.3. Effect of the Contant Angle (θc) ........................................................................................ 158

7.4.4. Justification of Termination Points of Simulations ........................................................... 159


Chapter 8 FUNDAMENTALS OF PLASTIC FOAMING USING CO2-ETHANOL BLEND BLOWING AGENT ................................................................................................................... 166

8.1. Introduction ............................................................................................................................... 166

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8.2. Experimental ............................................................................................................................. 169

8.2.1. Materials ............................................................................................................................ 169

8.2.2. Sample Preparation ............................................................................................................ 169

8.2.3. Rheology Measurement ..................................................................................................... 169

8.2.4. In-Situ Foaming Visualization........................................................................................... 170

8.2.5. Characterization ................................................................................................................. 170


8.3.1. Rheology ........................................................................................................................... 172

8.3.2. Effect of Ethanol Content on Foaming Behaviors ............................................................. 172

8.3.3. Hypotheses of Foaming Mechanism ................................................................................. 174


Chapter 9 SUMMARY, CONCLUDING REMARKS & FUTURE WORK ............................ 182

9.1. Summary ................................................................................................................................... 182

9.2. Key Contributions from this Thesis Research ........................................................................... 183

9.3. Recommendations and Future Work ......................................................................................... 188

References ................................................................................................................................... 191

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List of Tables

Table 3.1. Physical Properties of Polystyrene ........................................................................................... 70

Table 3.2. Physical Properties of Talc Particles ........................................................................................ 70

Table 3.3. Physical Properties of the Blowing Agent ................................................................................ 70

Table 3.4. Processing conditions to study the effect of pressure drop rate in PS-CO2 foaming (Tsys = 140˚C and C = 5.0 wt.%) ............................................................................................................................. 71

Table 3.5. Processing conditions to study the effect of dissolved CO2 content in PS-CO2 foaming (Tsys = 140˚C and –dP/dt|max = 22 MPa/s) ............................................................................................................... 71

Table 3.6. Processing conditions to study the effect of system temperature in PS-CO2 foaming (–dP/dt|max = 47 MPa/s and C0 = 5.0 wt%) ..................................................................................................... 71

Table 3.7. Processing conditions to study the effect of various processing conditions in PS-talc-CO2 foaming ........................................................................................................................................................ 72

Table 4.1. Thermo-physical and rheological parameters for PS/CO2 foaming system [T = 180˚C; Psat ~ 10 MPa] ....................................................................................................................................................... 96

Table 5.1. Properties of LDPE ................................................................................................................ 114

Table 5.2. Properties of Celogen® OT ..................................................................................................... 114

Table 5.3. Numerical values of physical properties of LDPE and N2 system at 160°C – 190°C ............ 114

Table 6.1. Comparison between different foaming simulation approaches ............................................ 138

Table 6.2. Processing conditions of PS-CO2 foaming for the base case of experimental verification .... 139

Table 6.3. Processing conditions to study the effects of pressure drop rate and dissolved CO2 content on PS-CO2 foaming ........................................................................................................................................ 139

Table. 6.4. Characteristic parameters of PS and CO2 for SL EOS .......................................................... 139

Table 6.5. Values of K12 for the SL EOS ................................................................................................. 139

Table 6.6. Summary of Psys drop rates considered in the simulations ..................................................... 140

Table 6.7. Parameters used in the simulations......................................................................................... 140

Table 7.1. Experimental conditions for foaming experiments and computer simulations ...................... 161

Table 7.2. One-way ANOVA results ...................................................................................................... 161

Table 8.1. Physical properties of polystyrene.......................................................................................... 177

Table 8.2. Physical properties of blowing agents .................................................................................... 177

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Table 8.3. A summary of experimental cases .......................................................................................... 177

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List of Figures Figure 1.1. A schematic of the basic steps during a plastic foaming process ............................................ 10

Figure 1.2. Schematic of the overall research strategy .............................................................................. 10

Figure 2.1. Homogeneous and heterogeneous nucleation in a polymer-gas solution ................................ 45

Figure 2.2. A schematic of a Harvey nucleus ............................................................................................ 45

Figure 2.3. Free energy change to nucleate a bubble homogeneously ...................................................... 45

Figure 2.4. A bubble nucleates on a smooth planar surface ...................................................................... 46

Figure 2.6. A bubble nucleates in a conical cavity with an apex angle of 2β ............................................ 47

Figure 2.7. A schematic of the cell model ................................................................................................. 47

Figure 2.8. A schematic of a cell (bubble and its influence volume) ........................................................ 48

Figure 2.9. Overall nucleation and bubble growth processes .................................................................... 48

Figure 3.1. The batch foaming visualization system ................................................................................. 73

Figure 3.2. A schematic of the dynamic change of Rcr and its relationship with Rbub ............................... 73

Figure 3.3. Micrographs of PS-CO2 foaming at different pressure drop rates (Tsys = 140˚C and C = 5.0 wt%) ............................................................................................................................................................ 74

Figure 3.4. Effect of pressure drop rate on PS-CO2 foaming: (a) pressure drop profiles & (b) cell density profiles ......................................................................................................................................................... 74

Figure 3.5. Micrographs of PS-CO2 foaming at different CO2 contents (Tsys = 140˚C & -dPsys/dt|max = 22 MPa/s) ......................................................................................................................................................... 75

Figure 3.6. Effect of dissolved gas content on PS-CO2 foaming: (a) pressure drop profiles & (b) cell density profiles ............................................................................................................................................ 75

Figure 3.7. Micrographs of PS-CO2 foaming at processing temperatures (–dP/dt|max = 47 MPa/s and C = 5.0 wt.%) ..................................................................................................................................................... 76

Figure 3.8. Effect of processing temperature on PS-CO2 foaming: (a) pressure drop profiles & (b) cell density profiles ............................................................................................................................................ 76

Figure 3.9. Micrographs of PS foaming with 2.1 wt% CO2 at 180°C: (a) pure PS and (b) PS + 5 wt% talc (CIMPACT 710) .......................................................................................................................................... 77

Figure 3.10. Micrographs of PS foaming with 2.1 wt% CO2 at 180°C: (a) pure PS at 2.20 s and (b) PS + 5 wt% talc (CIMPACT 710) at 1.56 s ......................................................................................................... 77

Figure 3.11. Schematics of the bubble formation phenomena .................................................................. 78

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Figure 3.12. A schematic of the extensional stress field around the talc agglomerate induced by the expanding bubble ........................................................................................................................................ 78

Figure 3.13. Micrographs of PS with 2.3 wt% CO2 at 180°C: (a) PS + 0.5 wt% talc (CIMPACT 710) and (b) PS + 5 wt% talc (CIMPACT 710) ......................................................................................................... 79

Figure 3.14. Micrographs of PS foaming with 2.3 wt% CO2 at 180°C: (a) PS + 0.5 wt% talc (CIMPACT 710) at 3.20 s and (b) PS + 5 wt% talc (CIMPACT 710) at 2.90 s ............................................................. 79

Figure 3.15. Micrographs of PS with 4.0 wt% CO2 at 180°C: (a) PS + 0.5 wt% talc (CIMPACT 710) and (b) PS + 5 wt% talc (CIMPACT 710) ......................................................................................................... 80

Figure 3.16. Micrographs of PS foaming with 4.0 wt% CO2 at 180°C: (a) PS + 0.5 wt% talc (CIMPACT 710) at 2.40 s and (b) PS + 5 wt% talc (CIMPACT 710) at 2.40 s ............................................................. 80

Figure 3.17. Micrographs of PS + 5.0 wt% talc with 2.3 wt% CO2 at 180°C: (a) CIMPACT 710 (untreated) and (b) CB7 (treated) ................................................................................................................ 81

Figure 3.18. Micrographs of PS + 5.0 wt% talc with 2.3 wt% CO2 at 180°C: (a) CIMPACT 710 at 2.90 s; (b) CB7 at 2.90 s ...................................................................................................................................... 81

Figure 3.19. A SEM micrograph of PS + 5 wt% talc (CB7) ..................................................................... 82

Figure 3.20. Distribution of talc particle sizes in PS-talc composites: (a) 0.5 wt% of untreated talc; (b) 5.0 wt% of untreated talc; (c) 0.5 wt% of surface-treated talc; and (d) 5.0 wt% of surface treated talc ..... 82

Figure 3.21. Micrographs of PS + 5.0 wt% talc (STELLAR 410) with 2.3 wt% CO2 at 180°C: (a) until 2.96 s; (b) at 2.82 s ...................................................................................................................................... 83

Figure 3.22. Micrographs of PS + 5.0 wt% talc (CIMPACT 710) with 2.1 wt% CO2 at 140°C: (a) until 2.10 s; (b) at 1.900 s .................................................................................................................................... 84

Figure 4.1. Numerical simulation algorithm of bubble growth dynamics ................................................. 97

Figure 4.2. In-situ visualization data of PS/CO2 batch foaming experiment [Tsys = 180˚C; Psat ~ 10 MPa] ..................................................................................................................................................................... 97

Figure 4.3. Measured bubble sizes at different time [Tsys = 180˚C; Psat ~ 10 MPa] .................................. 98

Figure 4.4. Pressure decay data [Tsys = 180˚C; Psat ~ 10 MPa] .................................................................. 98

Figure 4.5. Simulation results versus experimental observations.............................................................. 98

Figure 4.6. Effect of initial bubble radius (Rbub(t’,t’)) on predicted bubble growth behaviors .................. 99

Figure 4.7. Effect of initial shell radius (Rshell,t=t’) on predicted bubble growth behaviors ........................ 99

Figure 4.8. Effect of diffusivity (D) on predicted bubble growth behaviors ............................................. 99

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Figure 4.11. Effects of relaxation time (λ) on predicted bubble growth behaviors – (a) 0.0 s to 1.0 s and (b) 0.6 s to 1.0 s ......................................................................................................................................... 101

Figure 4.12. Effects of η0 on predicted bubble growth behaviors – (a) λ = 27.0 s and (b) λ = 0.1 s ....... 101

Figure 5.1. TGA curve of Celogen® OT at heating rates of 10°C/min and 20°C/min ............................ 115

Figure 5.2. A schematic of the experimental setup ................................................................................. 115

Figure 5.3. Simulated lifespan of a CBA-blown bubble at various degrees of saturation (x) ................. 116

Figure 5.4. Proposed mechanism of bubble growth and collapse in CBA-induced foaming: (a) heating, (b) bubble generation, (c) bubble expansion, (d) maximum bubble growth, (e) bubble collapse, and (f) bubble disappearance ................................................................................................................................ 116

Figure 5.5. Simulated bubble size (Rbub) and critical radius (Rcr) [x = 110%] ......................................... 117

Figure 5.6. Bubble growth and collapse phenomena with different CBA contents: (a) 0.25 wt% Celogen® OT and (b) 0.50 wt% Celogen® OT .......................................................................................... 117

Figure 5.7. Simulated vs. experimentally observed lifespan of bubbles ................................................. 118

Figure 5.8. Effect of diffusivity (D) on a bubble’s sustainability ............................................................ 118

Figure 5.9. Effect of surface tension (γlg) on a bubble’s sustainability .................................................... 118

Figure 5.10. Effect of solubility on a bubble’s sustainability .................................................................. 119

Figure 5.11. Effect of viscosity on a bubble’s sustainability ................................................................... 119

Figure 5.12. Effect of elasticity on a bubble’s sustainability .................................................................. 119

Figure 6.1. A bubble nucleated on a rough heterogeneous nucleating site – (a) a nucleating agent, and (b) the equipment wall .................................................................................................................................... 141

Figure 6.2. The overall computer simulation algorithm of plastic foaming ............................................ 142

Figure 6.3. Micrographs of a PS/CO2 batch foaming process ................................................................. 143

Figure 6.4. The smallest observable bubble being observed by the visualization system ....................... 143

Figure 6.5. Number density of the observable bubbles [θc = 85.7˚] ........................................................ 144

Figure 6.6. Rate of increase of the number density of observable bubbles [θc = 85.7˚] .......................... 144

Figure 6.7. Average CO2 concentration and the difference between Pbub and Psys .................................. 144

Figure 6.8. Volume expansion ratio of the PS foam ............................................................................... 145

Figure 6.9. Bubble sizes distribution at t = 0.6 second ............................................................................ 145

Figure 6.10. Deviation of Pbub from Psat at different Psys and wt% of CO2 [T = 180˚C] .......................... 145

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Figure 6.11. Curvature dependence of γlg of PS/CO2 system [Psat = 9.94 MPa; T = 180˚C] ................... 146

Figure 6.12. Effect of contact angle on the computer simulation result .................................................. 146

Figure 6.13. Simulation results versus experimental data of the PS/CO2 batch foaming processes ....... 146

Figure 6.14. Simulation results of average bubble radii (error bars = 3X standard deviations) .............. 147

Figure 6.15. Bubble radii distribution (C0 = 3.8 wt% & Tsys = 180˚C) .................................................... 147

Figure 6.16. Bubble radii distribution at various processing conditions (C0 = 5.9 wt.% & Tsys = 180˚C) ................................................................................................................................................................... 148

Figure 6.17. Effect of the Pbub,cr approximation on the predicted cell density ........................................ 148

Figure 6.18. Effect of the Pbub,cr approximation on the predicted cell nucleation rate ............................ 149

Figure 6.19. Effect of the Pbub,cr approximation on the predicted average gas concentration in the PS-CO2 solution ...................................................................................................................................................... 149

Figure 6.20. Deviation of Pbub,cr from Psat ............................................................................................... 149

Figure 6.21. Accumulated cell density versus time at different constant Psys drop rates ......................... 150

Figure 6.22. Maximum cell density versus -dPsys/dt (dash line: the step Psys drop) ................................ 150

Figure 6.23. Errors of simulated cell densities at different -dPsys/dt ....................................................... 150

Figure 6.24. Cell size distributions versus -dPsys/dt (dash line: the step Psys drop; error bar: 3X the standard deviation) .................................................................................................................................... 151

Figure 6.25. Errors of cell radii at different Psys drop rates relative to the step Psys ................................ 151

Figure 7.1. Overall research methodology to determine ΔPthreshold .......................................................... 162

Figure 7.2. Visualized batch foaming data taken from PS-CO2 foaming experiments ........................... 162

Figure 7.3. Effects of (a) –dPsys/dt, (b) CO2 gas content, and (c) Tsys on ΔPthreshold (error bars: 3X standard deviation) ................................................................................................................................................... 163

Figure 7.4. Effects of (a) –dPsys/dt, (b) CO2 gas content, and (c) Tsys on maximum cell density (error bars: 3X standard deviation) .............................................................................................................................. 164

Figure 7.5. Sensitivity analysis of surface tension’s effect on bubble growth ........................................ 165

Figure 7.6. Sensitivity analysis of relaxation time’s effect on bubble growth ........................................ 165

Figure 7.7. Sensitivity analysis of contact angle’s effect on simulated pressure drop threshold ............ 166

Figure 8.1. A schematic of the tandem foam extrusion system ............................................................... 178

Figure 8.2. Effects of blowing agent composition and melt temperature on shear viscosity of PS melt 178

xx

Figure 8.3. Snapshots of foaming visualization data of the experimental runs ....................................... 179

Figure 8.4. Effects of blowing agent composition on cell population density ........................................ 179

Figure 8.5. Effects of blowing agent composition on cell generation rate .............................................. 180

Figure 8.6. Effects of blowing agent composition on average cell radius ............................................... 180

Figure 8.7. SEM micrographs of PS foams obtained by (a) pure CO2, (b) CO2-EtOH blend (mCO2 : mEtOH = 60 : 40), and (c) pure EtOH .................................................................................................................... 181

Figure 8.8. The SEM micrograph (magnification = 1000X) of PS foams obtained by pure EtOH ........ 181

xxi

List of Symbols

A(Rcr) Surface area of a critical bubble, m2

Ahet(t) Area of unoccupied heterogeneous nucleation sites per unit volume of

polymer at time t, m2/m3

Ahet,0 Initial area of unoccupied heterogeneous nucleation sites per unit

volume of polymer, m2/m3

Alg Surface area of the liquid-gas interface, m2

Asg Surface area of the solid-gas interface, m2

Asl Surface area of the solid-liquid interface, m2

C(r,t,t’) Dissolved gas concentration at radial position r and time t for the

bubble nucleated at time t’, mol/m3

C0 Initial dissolved gas concentration in the polymer-gas solution,

mol/m3

Cavg(t) Average dissolved gas concentration in the polymer-gas solution at

time t, mol/m3

CR(t,t’) Dissolved gas concentration at the bubble surface at time t for the

bubble nucleated at time t’, mol/m3

Csat Saturated gas concentration, mol/m3

D Diffusivity, m2/s

D0 Diffusivity coefficient constant, m2/s

ΔED Activation energy for diffusion, J

F Ratio of the volume of the nucleated bubble at a heterogeneous

nucleating site to the volume of a spherical bubble with the same

xxii

radius, dimensionless

ΔFhet Free energy change for the heterogeneous nucleation of a bubble, J

ΔFhom Free energy change for the homogeneous nucleation of a bubble, J

H Henry’s law constant, dimensionless

Jhet Heterogeneous nucleation rate per unit surface area of heterogeneous

nucleating sites, #/m2-s

Jhom Homogeneous nucleation rate per unit volume of polymer, #/m3-s

Jtot Total nucleation rate per unit volume of polymer, #/m3-s

kB Boltzmann’s constant, m2-kg/s2-K

K12 Interaction parameter for the SL EOS, dimensionless

KH Ratio of the saturated gas concentration to the corresponding system

pressure, mol/N-m

m Mass of a gas molecule, g

n Number of bubbles, bubbles

n(Rcr) Number density of the critical bubbles, bubbles

ngen Number of moles of gas being generated as the CBA decomposes,

mol

N Number of gas molecules per unit volume of polymer, #/m3

NA Avogadro’s number, #/mol

Nb,foam Cell density with respect to the foam volume, #/m3

Nb,unfoam Cell density with respect to the unfoamed volume, #/m3

Pbub(t,t’) Bubble pressure at time t for the bubble nucleated at time t’, Pa

Pbub,cr Pressure inside a critical bubble, Pa

xxiii

PG* Characteristic pressure of the gas, Pa

PM* Characteristic pressure of the polymer, Pa

PP* Characteristic pressure of the polymer-gas solution, Pa

PR Reduced pressure of the polymer-gas solution, dimensionless

PRG Reduced pressure of the gas component, dimensionless

Psat Saturation pressure of the polymer-gas solution, Pa

Psys(t) System pressure at time t, Pa

ΔP Degree of supersaturation, Pa

ΔPthreshold Pressure drop threshold for cell nucleation, Pa

Q Ratio of the surface area of the liquid-gas interface of the bubble

nucleates on a heterogeneous nucleating site to the surface area of a

spherical bubble with the same radius, dimensionless

r Radial position from the centre of the nucleated bubble, m

rG Number of lattice sites occupied by a gas molecule in the polymer-

gas solution, lattice sites

rG0 Number of lattice sites occupied by a pure gas molecule, lattice sites

rP Number of lattice sites occupied by a mer in the polymer-gas

solution, lattice sites

rm Number of lattice sites occupied by a mer, lattice sites

Rbub(t’,t’) Initial bubble radius, m

Rbub(t,t’) Bubble radius at time t for the bubble nucleated at time t’, m

Rcr Critical radius, m

Rhet Radius of a spherical heterogeneous nucleating agent site, m

xxiv

Rg Universal gas constant, J/K-mol

Rshell, t=t’ Initial shell radius, m

Rshell(t,t’) Shell radius at time t for the bubble nucleated at time t’, m

bubR Fluid velocity at the bubble surface, m/s

t Time of simulation, s

t’ Nucleation time of a particular cell, s

tonset Onset time of cell nucleation, s

TR Reduced temperature of the polymer-gas solution, dimensionless

TRG Reduced temperature of the gas component, dimensionless

Tsys System temperature, K

u(r) Fluid velocity at radial position r, m/s

V Volume of the unfoamed polymer melt, m3

Vg Volume of a bubble, m3

VER Volume expansion ratio, dimensionless

Whet Free energy barrier for heterogeneous nucleation, J

Whom Free energy barrier for homogeneous nucleation, J

x Degree of gas saturation, dimensionless

Z Zeldovich factor, dimensionless

Greek letters

β Semi-conical angle, degrees

γa Surface tension of liquid a, N/m

γb Surface tension of liquid b, N/m

γexp Experimentally measured surface tension at the liquid-gas interface,

xxv

N/cm

γlg Surface tension at the liquid-gas interface, N/m

γsg Surface tension at the solid-gas interface, N/m

γsl Surface tension at the solid-liquid interface, N/m

η Shear viscosity, N/m2-s

η0 Zero-shear viscosity, N/m2-s

θa Contact angle of liquid a, degrees

θb Contact angle of liquid b, degrees

θc Contact angle, degrees

λ Relaxation time, s

μg Chemical potential of the gas inside the bubble, J/mol

μg,sol Chemical potential of the gas in the polymer-gas solution, J/mol

ρβ Probability density distribution of β, dimensionless

ρR Reduced density of the polymer-gas solution, dimensionless

ρRG Reduced density of the gas component, dimensionless

φG Close-packed volume fraction of the gas component, dimensionless

φP Close-packed volume fraction of the polymer component,

dimensionless

τrr Stress in the r direction, Pa

τθθ Stress in the θ direction, Pa

υ Rate at which molecules strike against an unit area of the bubble

surface, molecules/m2-s

1

Chapter 1 INTRODUCTION

1.1. Preamble

Plastics foaming is a polymer processing technology that involves the uses of blowing

agents, and sometimes other additives such as nucleating agents, to generate cellular structures in

a polymer matrix. Heightened needs for light weight materials with improved cushioning,

insulating, structural performances, and other characteristics are expected to push the worldwide

demands for plastic foams to increase continuously [1]. Among various foamed plastics,

thermoplastic foams remain as one of the most dominant classes. Due to a wide spectrum of

advantages such as good dielectric properties, strength and thermal resistances, their demand has

been projected to increase. Given the benefits being offered by new technology, the breadth of

plastic foam application is continuing to grow and the future potential has practically no limit.

Despite the significant success of the foaming industry, the extension of foamed

polymers into new markets, such as biomedical and pharmaceutical applications, hinges on the

2

ability to enhance control over the cellular morphology including cell density, void fraction, and

open- versus closed-cell structures. Continuous advancements in foaming technology over the

past couple decades have spurred increased interest in research and commercial applications. On

the one hand, the polymeric foaming process allows manufacturers to reduce their raw material

costs, which have risen dramatically in recent years due to ongoing increases in the price of

plastic resins. On the other hand, extensive research [2-9] has proven that plastic foams with high

cell densities, small cell sizes and narrow cell-size distributions can translate into notable

advantages in various applications.

In particular, microcellular foams (i.e., foamed plastic characterized by a cell density in

the range of 109 to 1015 cells/cm3 and an average cell size in the range of 0.1 to 10 μm) offer

superior mechanical properties, such as impact strength and fatigue life, over conventional foams

or their unfoamed counterparts. In this context, various investigations revealed that the notched

Izod impact strength of microcellular foams increases with their void fractions [2-6]. Seeler and

Kumar also demonstrated that the fatigue life of microcellular polycarbonate with a relative foam

density of 0.97 exceeded that of solid polycarbonate by over 400 percent [7]. In addition to the

improved mechanical properties, appropriate additives, blowing agents, and processing

conditions can all be chosen to alter or improve the thermal [8], acoustical [8], or optical [9]

properties of the plastic foams by tailoring the foam morphology.

The final foam morphology is governed by the cell nucleation, the cell growth, and the

cell coarsening during the foaming process. However, the controls of these phenomena are

challenging because they involves delicate thermodynamic, kinetic, and rheological mechanisms.

Although extensive experimental and theoretical investigations have been conducted in attempt

to elucidate the plastic foaming behaviors, the underlying mechanisms of the aforementioned

phenomena have not yet been clarified thoroughly.

3

1.2. Plastic Foams and Their Processing

Plastic foams possess cellular structures within the solid plastic matrices. The properties

of the final foams are derived from the properties of the polymer matrix and the retained gas, as

well as the foam morphology. Therefore, the choices of the base polymers, the blowing agents,

and the controls of the cell structures will influence the applications of the foamed plastics. In

general, foamed plastics can be classified in different ways: by nature as flexible, semi-flexible,

and rigid foams, by density as low- and high-density foams, by structure as open- or closed-cell

foams, and by cell density and pore size as fine-celled, microcellular, or nanocellular foams.

In the past few decades, plastic foams have been produced by processes such as batch

foaming, foam extrusion, and injection foam molding. The cellular structure in plastics may be

produced mechanically, chemically, or physically [10]. Regardless of the methods, the material

to be foamed is in a liquid or plastic state during the process. Mechanical foaming produces a

cellular structure by mechanically whipping or frothing of gases into a polymeric melt,

suspension, or solution. As the material hardens, it entraps gas bubbles in the polymer matrix,

and thereby yields the cellular structure. In chemical foaming processes, the decomposition of a

chemical blowing agent, either exothermic or endothermic, is used to produce gas and generate

the cellular structure. For example, an organic nitrogen compound decomposes and liberates

nitrogen gas to foam some types of PVC. The physical foaming process is another popular

method to produce plastic foams. Generating foams using this means consists of four major

steps: (i) dissolution of gas and homogenization of additives in a polymer matrix; (ii) cell

nucleation; (iii) cell growth; and (iv) stabilization of foam structures. The formation and

expansion of cells from the dissolved gas are achieved by reducing the pressure; or volatilization

of low-boiling liquid within the polymer mass either by application of external heat or under the

4

influence of the heat of reaction. A schematic of the basic steps during a typical plastic foaming

process using a physical blowing agent is illustrated in Figure 1.1.

1.3. Challenges to Plastic Foams Production

In recent years, the plastic foam industry (e.g., packaging, construction, and automotive

parts) has experienced serious regulatory, environmental, and economical pressures (i.e.,

alternative blowing agents, volatile organic compounds (VOC), and soaring oil and resin prices).

In plastic foams, bubbles are typically generated by the decomposing a chemical blowing agent

(CBA) that releases gases, or by injecting a physical blowing agent (PBA). An ideal physical

blowing agent should be environmentally acceptable, non-flammable, adequately soluble, stable

in the process, and should have an appropriate latent/specific heat, low toxicity, low volatility,

low vapour thermal conductivity, low diffusivity in the polymer, low molecular weight, and low

cost [11].

Prior to the 1990s, CFCs were widely used as blowing agents in manufacturing

polyurethane (PU), polystyrene (PS), and polyolefin thermal insulation foams, because they are

noncombustible, and have low toxicity, and low diffusivity in polymers. Furthermore, their low

thermal conductivity results in foams that also have excellent insulation properties. All these

properties make CFCs almost the ideal physical blowing agents. However, as early as 1974,

scientists recognized that rampant use of CFCs would have adversely affected the dynamic

equilibrium of stratospheric ozone, and thus these high-ODP substances were banned from

international use by the Montreal Protocol [12]. Finding a blowing agent to replace CFCs

subsequently became an urgent task for the foam industry.

Consequently, low-ODP hydrochlorofluorocarbon-based (HCFC-based) blowing agents,

such as HCFC-22, HCFC-141b, and HCFC-142b, have been used as alternatives. However, the

5

HCFC-based foams will be phased out in North America as of January 2010. New alternative

zero-ODP blowing agents are therefore urgently desired by the foam industry. The current

candidates for zero-ODP blowing agents include CO2, N2, hydrofluorocarbons (HFCs),

hydrocarbons (HCs), or their mixtures. Long-chain molecules such as butane have high

solubilities and low diffusivities and are favorable for producing low-density foams [13].

Nevertheless, the uses of HCs are limited due to their high flammability. The prolonged storage

time required to reduce the level of retained flammable blowing agents in HCs is also costly.

Inexpensive gaseous blowing agents, such as CO2 and N2, in contrast, have high diffusivities and

low solubilities. Although supercritical CO2 exhibits several advantages over traditional long-

chain blowing agents [14, 15], it remains challenging to use these gaseous blowing agents to

produce plastic foams [16-18]. Recently, gaseous blowing agents have been used as alternatives

to long-chain blowing agents to manufacture relatively high-density foams with volume

expansion ratios in the range of 1.2 to 15 folds (mainly less than 10 folds) [16-27]. The well-

known methods of inert gas-based foaming of relatively high-density foams are described in

various patents [18-24]. However, since the inert gas blowing agents have higher volatility and

higher diffusivity, than the long-chain blowing agents, gaseous blowing agents escape easily

during expansion [25-27]. Therefore, it is very difficult to obtain low-density foam with a large

expansion ratio of over thirty-fold. These blowing agents are more suitable to produce fine-celled

or microcellular foams [16-18, 28-31].

1.4. Objectives of the Thesis

In many cases, the uses of inert blowing agents (e.g., N2, CO2, Ar, … etc.) to produce

foamed plastics are non-trivial because of their low solubilities and high diffusivities. Therefore,

it remains challenging to achieve the spectrum of densities and structures desired for various

6

applications. Even with the relatively more soluble gases (e.g., CO2), the rheological properties

of the polymer melt are significantly affected and lead to difficulties in stabilizing foam

structures. Although active ongoing research is being conducted to extend the applications of

foamed plastics in new markets, including tissue engineering (e.g., bioscaffolds) and the

pharmaceutical industry (e.g., foam drug delivery vehicles), the ability to control and tailor the

cellular structures are crucial to succeed in these novel foam applications.

In order to enhance the control of plastic foaming processes to advance the current

technology and to emerge it into new markets, the ultimate goals of this research are to elucidate

plastic foaming processes, thereby aiding the industrial foaming companies to develop

innovative, industrially viable, cost-effective plastic foaming technologies, or to improve the

current technologies to produce plastic foam products with superior and controlled properties. To

achieve these goals, efforts will be made within three main objectives: (i) identify the underlying

mechanisms that control the cell nucleation, cell growth, and cell coarsening in plastic foaming;

(ii) estimate the onset point of bubble formation during polymeric foaming processes; (iii)

evaluate and improve the current theoretical models and simulation schemes to simulate overall

foaming processes.

Because cell nucleation, growth and coarsening phenomena are simultaneously affected

by many different processing parameters (e.g., processing temperature and pressure drop rate)

and material parameters (e.g., plastic type, blowing agent type and content and nucleating agent

type and content), an understanding of the underlying mechanisms cannot simply be yielded

from investigating real processing experiments or by employing theoretical approaches. As a

result, this research attempts to combine the theoretical studies and the experimental

investigation in order for them to complement each other. It is believed that this research will

serve as a bridge between industrial practice and theory as well as provide guidelines for the

7

plastic foaming industry to design the appropriate dies and processing systems, to optimize

processing conditions and to choose the appropriate materials. A schematic of the overall

research strategy is illustrated in Figure 1.2.

In summary, it is believed that, upon the achievements of (i), (ii), and (iii), it would help

to clarify the underlying physics of different phenomena involved in foaming plastics. The end

results of this research will provide some useful guidelines in developing processing strategies to

control the cell morphology of the plastic foams and thereby enhance the uses of alternative

blowing agents to produce foamed plastics.

1.5. Overview of the Thesis

Chapter 2 presents a literature survey on the theoretical and experimental studies on the

mechanisms of cell nucleation, cell growth, and cell coarsening in plastic foaming. It includes

the fundamentals of blowing agents, the fundamentals of cell nucleation, as well as the modeling

and computer simulation of the cell nucleation, growth, and coarsening in plastic foaming. Both

the theoretical studies and experimental investigations on polymeric foaming processes are

presented to demonstrate the current state in this research field.

Chapter 3 describes a comprehensive research on the cell nucleation phenomena in

plastic foaming. The experiments on the foaming behaviors of the polystyrene-carbon dioxide

system under different processing conditions, with or without the existence of inorganic fillers

(e.g., talc) were conducted to investigate the foaming mechanisms. Through the in-situ

visualization of the foaming behaviors, a new heterogeneous cell nucleation mechanism has been

proposed to explain the observed results. Furthermore, the effects of the sizes, contents and types

of talc particles, the blowing agent contents, as well as the processing temperatures on the stress-

induced nucleation mechanism were studied.

8

Chapter 4 discusses a research conducted to achieve an accurate bubble growth model

and simulation scheme to describe precisely the bubble growth phenomena that occur in

polymeric foaming. Using the accurately measured thermo-physical and rheological properties of

polymer/gas mixtures as the inputs for computer simulation, the growth profiles for bubbles

nucleated at different times were predicted and carefully compared to experimentally-observed

data. A polystyrene-carbon dioxide system was used herein as a case example.

In Chapter 5, a plastic foaming process using chemical blowing agents (CBAs) was

investigated to study the stability of nucleated cells during the later stage of plastic foaming. The

continuous change of Rcr was theoretically simulated and related to the sustainability of the

nucleated cells. The in-situ experimental results observed from a hot-stage system with optical

microscope were used to support the theoretically derived concept.

Chapter 6 describes the development of a modified nucleation theory and examines its

application, together with the bubble growth models presented in Chapter 4, to simultaneously

simulate the bubble nucleation and bubble growth phenomena. Using the developed computer

software, the effects of pressure drop rates and dissolved gas contents on the foaming behaviors

and the cell morphologies were studied. The simulation results were carefully compared with in-

situ visualization data. In addition, this chapter also clarifies the errors in predicting the cell

density and cell size being caused by two commonly adopted approximations to the system

pressure in computer simulation. The end results will provide useful guidelines to improve the

accuracy of simulating the cell nucleation phenomena during plastic foaming.

Chapter 7 presents a semi-empirical approach and a theoretical approach to determine the

onset time of cell nucleation during plastic foaming. The effects of the pressure drop rate, the gas

content, and the processing temperature on the pressure drop threshold for cell nucleation, which

is the amount of pressure drop below the solubility pressure to create a sufficient level of

9

supersaturation to initiate cell nucleation, were also explored. Finally, the pressure drop

thresholds being predicted from the two approaches were compared.

The research being discussed in Chapter 8 demonstrates how the elucidation of foaming

mechanism enhances the development of novel processing strategies to control foam

morphology. As a case example, it presents an experimental study on the foaming of polystyrene

using a blowing agent blend – carbon dioxide and ethanol. Through a series of in-situ

observations and the SEM analyses of polystyrene (PS) foaming using pure CO2, pure ethanol,

and CO2-ethanol blends as case studies, the fundamentals of plastic foaming using a blowing

agent blend were explored.

Chapter 9 provides a summary of contributions and concluding remarks for this thesis as

well as recommendations for future work, respectively.

10

-1Figure 1.1. A schematic of the basic steps during a plastic foaming process

2Figure 1.2. Schematic of the overall research strategy

11

Chapter 2 LITERATURE REVIEW &

THEORETICAL BACKGROUND

Plastic foams exhibits many useful properties to differentiate themselves from their solid

counterparts, which allow foamed plastic products to infiltrate into almost all aspects of our daily

lives and in many other novel applications. Improvements in the understanding of the underlying

sciences, the process technology and equipment, as well as the raw materials and their

availability have made it possible to produce useful foamed plastic articles. Although the foam

industry went through many difficulties in the past decades and is currently experiencing many

other challenges, extensive and continuous research efforts, by both the academia and the foam

industry, have offered a lot of insight into the advancement of the technology and expand the

applications of foams throughout new fields. These studies have also provided an invaluable

information base for researchers to explore the mechanisms and underlying sciences of various

phenomena that occur during plastic foaming. This chapter provides a comprehensive review of

12

previous literatures that contain studies of the fundamentals of plastic foaming. It also serves as

an overview of the current state of scientific research and how it complements the technology

advancement in the field.

2.1. Fundamentals of Blowing Agents

Plastic foaming usually consists of a gaseous phase, namely a blowing agent, which is

embedded in a polymer melt, to generate the cellular structure. Depending on the desired foam

morphologies or the applications of the foamed products, there are a great variety of suitable

blowing agents, which can be classified into physical blowing agents (PBAs) and chemical

blowing agents (CBAs). The former type is the gas being directly injected into the polymer melt

or polymer composite melt. The latter type evolves into gas when heat-induced chemical

decomposition occurs. In general, CBA is either dry-blended to the pelletized or powderized

polymer at a solid state, or mixed in a compounder at a temperature that is below the

decomposition temperature of the CBA.

2.1.1. Physical Blowing Agents (PBAs)

Traditionally, chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs) were

the most commonly-used PBAs for plastic foaming processes. Due to their ozone-depleting

potentials, the Montreal Protocol [12] and the related regulations have banned the uses of these

gases. Volatile organic compounds (VOCs) can also be used as blowing agents, but they are

flammable, detrimental to health, and react with ultraviolet light and nitrogen oxides to form

tropospheric ozone. Therefore, there is an increasing pressure to also regulate the uses of them.

Consequently, the plastic foam industry turned their attention to other potential replacements. In

particular, studies on hydrofluorocarbons (HFCs) have been conducted to investigate their

effectiveness as alternative blowing agents and have drawn a lot of interests from the industry

13

[32-33]. Various researchers have also investigated plastic foaming behaviors using carbon

dioxide (CO2), as well as inert gases such as nitrogen (N2), argon (Ar) and helium (He) [16, 19,

34-39]. However, HFCs, CO2, N2, Ar and He are less soluble and more diffusive in polymer

melts than their less environmental-friendly counterparts [32, 40-42]. These properties have

made achieving desired foam morphologies using these alternative blowing agents

technologically challenging because less gas is available for nucleating bubbles and their

subsequent growth.

2.1.2. Chemical Blowing Agents (CBAs)

CBAs are chemicals that generate gases upon their decomposition at high temperatures.

There are two major types of CBAs: exothermic and endothermic [10]. Most exothermic CBAs,

such as azodicarbonamide, generate N2 upon decomposition. In contrast, the primary gas

generated from endothermic CBAs, such as sodium bicarbonate and citric acid, is CO2.

Exothermic CBAs tend to decompose more readily than endothermic CBAs because the heat

generated upon their decomposition can trigger the decomposition of the neighbouring CBA

particles in a chain-like effect. The major advantages of using CBAs are that they do not require

any modification of the existing equipment, and it is easier to achieve an even distribution of gas

in the polymer matrix. However, they are more expensive than PBAs.

2.1.3. Formation of a Single-Phase Polymer-Gas Solution

The formation of a uniform polymer-gas mixture is critical to the production of high-

quality plastic foams. This is governed by the system pressure and the gas diffusion in the

polymer. Without achieving a uniform mixture, the resultant plastic foam will possess a non-

uniform cell structure and low cell density. For instance, during extrusion foaming or structural

foam molding processes, the system pressure prior to foaming must be higher than the solubility

pressure (i.e., also known as the saturation pressure) corresponding to the amount of injected

14

blowing agent. Otherwise, undissolved gas pockets can form and severely undermine the

uniformity of the resulting foam structures [42]. Typically, the system pressure is set to be much

higher than the solubility pressure to prevent any undissolved gas pockets and to accelerate the

gas dissolution process. In this context, accurate measurements of solubility data for various

blowing agents in different polymer melts are essential to plastic foaming processes [40- 41].

In the past, researchers have used different methods to measure solubilities of various

gases in polymers. In particular, the pressure decay method, which involves the measurement of

pressure changes inside a chamber as gas sorption by a polymer specimen takes place, was a

popular method due to its simplicity and low construction cost [43]. However, it was difficult to

use this method for molten polymers because it requires a high-resolution pressure sensor that

can be operated at elevated temperatures. Moreover, this method often requires a large polymer

sample, which translates into prolonged measurement time. Another method uses an

electrobalance to directly measure the mass uptake during sorption experiments [44-45]. This

method yields solubility measurements with high sensitivity in short measurement times.

However, due to the operating limits of the electrobalance, it works only at low temperatures. To

this end, researchers have designed systems to independently control the temperature of the

chamber and the electrobalance [46-47]. One drawback of this measuring technique is the effect

of convection-induced gas density variation on the measurement accuracy. This problem was

solved by another gravimetric method that utilizes a magnetic suspension balance (MSB)

developed by Kleinrahm and Wanger [48]. In this setup, the microbalance avoids the convection

effect by weighing the sample in a compartment that is isolated from the chamber containing it.

As a result, the apparatus can measure gas solubility and diffusivity in polymer at elevated

temperatures and pressures. Various researchers have adopted this method for gas solubility

measurement in polymer [40-41, 49-52]. However, due to the buoyancy effect of the swelled

15

polymer upon gas dissolution, the mass reading of the dissolved gas in the MSB, denoted as

apparent solubility, is lower than the actual solubility. In the absence of accurate pressure-

volume-temperature (PVT) data of the polymer-gas mixtures, various equations of state (EOS)

are typically used to estimate the extent of swelling and compensate for the buoyancy effect [40-

41, 49-53]. In particular, the Sanchez-Lacombe equation of state (SL EOS) [54] and Simha-

Somcynsky EOS (SS EOS) are two popular choices [55]. These EOSs assume molecules to be

arranged in lattices. On the other hand, some recent theories, including the Statistical Association

Fluid Theory (SAFT) [56-57], describe molecules to move freely in a continuous space.

Recently, a visualization system was developed to directly measure the PVT characteristics of

the polymer-gas mixtures [58], which would lead to more reliable solubility data. This would

also provide a means to verify the validity of the aforementioned EOSs in this context [58].

In addition to the system pressure requirement, it is also necessary to enhance the

diffusion of gas in a polymer to meet the processing time requirements in a continuous process

such as extrusion foaming. In general, the gas diffusivity in a polymer changes with temperature,

pressure, and gas concentration, and it can be approximated as [28, 59-65]:

0 exp D

g sys

ED DR T

⎛ ⎞Δ= −⎜ ⎟⎜ ⎟

⎝ ⎠ (2.1)

where D0 is the diffusivity coefficient constant; ΔED is the activation energy for diffusion; Rg is

the universal gas constant; and Tsys is the system temperature. Since the diffusivity increases with

temperature, the rate of gas diffusion is enhanced by an elevated temperature in typical extrusion

foaming processes. However, the diffusion process of gas in the polymer is still not fast enough.

Therefore, a technique for rapid solution formation is needed to achieve this task [28].

The aforementioned need is addressed by using a screw with high mixing and energy

transfer capability (e.g. barrier screw, energy transfer screw, and Barr screw). It homogenizes the

16

blowing agents or other additives in the polymer melt. Static mixers can also be installed to

improve the distributive and dispersive mixing of the materials. Moreover, a second stage

cooling screw is usually incorporated to enhance the uniformity of the temperature field [66]. In

summary, the aforementioned mixing strategies enhance the gas dissolution by redistributing the

local concentration of gas, increasing the area of the polymer-gas interface for mass transport,

and decreasing the striation distance for gas diffusion.

2.2. Fundamentals of Cell Nucleation

2.2.1. Review of Nucleation

In this thesis, the term, “cell nucleation,” is used generically to denote any process that

leads autogenously to the formation of a bubble in the polymer or polymer composite matrix.

Considering a polymer melt has been completely saturated with a blowing agent, the already

saturated system becomes supersaturated as the gas solubility reduces upon either temperature

increases [67-72] or pressure decreases [28, 31, 73-74]. Consequently, the polymer-gas solution

tends to form tiny bubbles in order to restore a low-energy stable state. The classical nucleation

theory (CNT) [75-78] classifies cell nucleation into two types – homogeneous nucleation and

heterogeneous nucleation. A schematic of these two types of nucleation is illustrated in Figure

2.1. CNT states that a bubble that has a radius greater than the critical radius (Rcr) tends to grow

spontaneously while the one that has a radius smaller than Rcr collapses. In addition to CNT,

another stream of thought postulated that pre-existing cavities or microvoids, which serve as

seeds for cell formation, exist in the supersaturated solution [79-83]. In light of these different

views, Jones et al. [84] proposed a classification system for cell nucleation. Under this scheme,

three types of cell nucleation can be defined. A similar classification system is adopted in this

thesis to discuss cell nucleation mechanisms in the context of plastic foaming.

17

2.2.1.1. Classical Homogeneous Nucleation

The classical homogeneous nucleation involves nucleation in the liquid bulk of a uniform

polymer-gas solution. There are no pre-existing gas cavitites present prior to the material system

becoming supersaturated. Han et al. [85], Lee [86], and Leung et al. [87] reported nucleation

rates during typical plastic foaming processes that were much higher than those calculated using

the CNT. Therefore, it is widely believed that this mechanism is not the route through which

cells form in plastic foams.

2.2.1.2. Classical Heterogeneous Nucleation

This type of nucleation suggests that supersaturation will result in the formation of a

bubble on a heterogeneous nucleating site (e.g., nucleating agents or impurities). Similar to the

homogeneous case, this form of nucleation suggests that the system initially contains no gas

cavity, either in the bulk or on the surface of heterogeneous nucleating sites. Wilt [88] showed

that heterogeneous nucleation at a smooth planar surface or a surface with conical or spherical

projections will not occur in an H2O-CO2 solution because the required level of supersaturation

is very high. A similar conclusion can be applied to plastic foaming because of the higher liquid-

gas interfacial tension. However, the study indicated that it is theoretically possible for classical

heterogeneous nucleation to occur in a conical cavity for an H2O-CO2 solution [88]. For plastic

foaming of a PS-CO2 system, Leung et al. [89, 90] also showed that this type of nucleation

activity could occur at a reasonably high rate theoretically, which qualitatively agreed with

experimental observations. Nevertheless, in all previous investigations of cell nucleation during

plastic foaming, the experimental data was not in good quantitative agreement with theoretical

predictions without the use of fitting parameters (e.g., pre-exponential factor [91], energy barrier

reduction factor [91], or θc [89]). Therefore, whether the classical nucleation theory can explain

the real mechanisms behind cell nucleation is still controversial.

18

2.2.1.3. Pseudo-Classical Nucleation

This form of nucleation includes homogeneous and heterogeneous nucleation from

metastable micro-bubbles or microvoids in the solution bulk, as well as pre-existing gas cavities

at the surface of the processing equipment or at the surface of suspended particles. When the

polymer-gas system is perturbed to a supersaturated state, the radius of curvature of each

meniscus is still less than Rcr as determined by CNT. Therefore, there exists a finite free energy

barrier to activate the expansion of the pre-existing gas cavities [86], which will finally be

overcome as the degree of supersaturation increases during the continuous pressure drop. As a

result, the pre-existing cavities will grow spontaneously.

Harvey et al. [79-82] were among the first who demonstrated that stable pre-existing

bubbles, also called the Harvey nuclei, might exist in conical pits on a heterogeneous nucleating

site. A schematic of a Harvey nucleus is shown in Figure 2.2. Lee [86] extended this concept and

proposed that shear flow during extrusion foaming would enhance the detachment of these pre-

existing gas cavities from the conical pits and form the cellular structures. However, Ward et al.

[83] suggested that the existence of Harvey nuclei required an interface with a contact angle (θc)

that is greater than 90°, which rarely exists. He then revealed that it is possible for tiny bubble

nuclei to exist in the bulk of liquid-gas solutions or in conical pits on the surfaces of suspended

solid particles when the gas concentration is slightly higher than the saturation level [83, 92].

Various literatures have also reported the presence of free volume in the polymer matrix [46].

Furthermore, during plastic foaming, supercritical fluid (e.g., supercritical CO2) diffusing into

the polymer matrix may dissolve the embedded low molecular weight impurities, including

entrapped solvents, residual catalysts, or low molecular oligomers [93]. The extraction of these

components from the polymer matrix will also create microvoids. In this context, Stafford et al.

reported that the extraction of a styrene oligomer from commercial polystyrene samples

19

increased the cell density and reduced the cell size when the samples were foamed with CO2

[94]. Ramesh et al. had found that thermally-induced microvoids and holes of sub-micron sizes

exist in high-impact polystyrene (HIPS) promoted the cell population densities of the foams [95-

97].

2.2.2. The Classical Nucleation Theory (CNT)

Gibbs [76] was the pioneer of the CNT studies, proclaiming the concept of critical radius

for bubble nucleation. Frenkel [98] and Cole [99] have undertaken extensive reviews of the

CNT. Tucker and Ward [100] performed an experimental study to verify the concept of the

critical radius, while Forest and Ward [101] have examined the nucleation of bubbles from

solutions containing dissolved gas. Using classical thermodynamics, various researchers have

derived the free energy barrier for homogeneous nucleation [83, 102-105] and those for

heterogeneous nucleation on different types of surfaces [88, 99, 106-111].

2.2.2.1. Free Energy Barrier for Homogeneous Nucleation

When the system is on the verge of bubble nucleation, the free energy change entailed by

the homogeneous formation of a bubble (ΔFhom) can be expressed as [83, 102-105]:

( )hom bub sys g lg lgF P P V AΔ γ= − − + (2.2)

where Pbub is the pressure inside the bubble; Psys is the surrounding system pressure; Vg is the

bubble volume; γlg and Alg are the interfacial energy and the surface area at the liquid-gas

interface, respectively. Using Equation (2.2), it is possible to plot the general relationship

between ΔFhom and the bubble radius (Rbub), which is depicted in Figure 2.3. In this figure, Rcr

represents the critical radius. The gas clusters with radii equal to Rcr are denoted as the critical

nuclei or the critical bubbles in this thesis.

20

Since the free energy change required to homogeneously form a critical bubble is

maximum, the corresponding state is an unstable equilibrium one. In other words, the nuclei

smaller than Rcr tend to collapse, while those larger than Rcr tend to grow. A bubble is thus

considered to have nucleated when its radius is larger than Rcr. The free energy change entailed

by forming a critical bubble is defined to be the free energy barrier for cell nucleation. By taking

the derivative of ΔFhom with respect to the Rbub and setting it to zero, it turned out that Rcr is

expressed as [83, 102-104].

lgcr

bub,cr sys

2R

P Pγ

=−

(2.3)

where Pbub,cr is the pressure inside a critical bubble. Combining Equations (2.2) and (2.3), the

free energy barrier for homogeneous nucleation (Whom) is [83, 102-104]:

( )

=−

3lg

hom 2

bub,cr sys

16W

3 P P

πγ (2.4)

Equations (2.3) and (2.4) indicate that both Rcr and Whom are functions of γlg and the degree of

supersaturation, which is defined as the difference between Pbub,cr and Psys in this thesis. Since a

critical bubble is at an unstable equilibrium state with its surrounding polymer-gas solution,

Pbub,cr can be determined by equating the chemical potential of the gas inside the bubble (i.e., μg)

and that of the gas in the polymer-gas solution (i.e., μg,sol). By further assuming the gas inside the

bubble obeys the ideal gas law and the polymer-gas solution is a weak solution, μg and μg,sol can

be expressed as Equations (2.5) and (2.6), respectively [112].

( ) ( ) bub,crg sys bub,cr g sys sys B sys

sys

PT ,P T ,P k T ln

Pμ μ

⎛ ⎞= + ⎜ ⎟⎜ ⎟

⎝ ⎠ (2.5)

( ) ( ) Rg ,sol sys sys R g sys sys B sys

sat

CT ,P ,C T ,P k T lnC

μ μ⎛ ⎞

= + ⎜ ⎟⎝ ⎠

(2.6)

21

where Tsys is the system’s absolute temperature; kB is the Boltzmann’s constant; CR is the gas

concentration at the liquid-gas interface; and Csat is the saturated gas concentration corresponds

to Psys. The chemical equilibrium condition (i.e., μg = μg,sol) can then be solved to determine

Pbub,cr:

R sysbub,cr

sat

C PP

C= (2.7)

Using Equation (2.7), Equation (2.3) can be rewritten as [112]:

lgcr

R syssys

sat

2R C P

PC

=−

γ (2.8)

Since both CR and Psys continuously changes during plastic foaming while the ratio between Psys

and Csat is approximately unchanged, the evolution of Rcr can be determined by Equation (2.8).

2.2.2.2. Free Energy Barrier for Heterogeneous Nucleation

Similarly, when the system is on the verge of bubble nucleation, the free energy change

entailed by nucleating a bubble heterogeneously (ΔFhet) can be expressed as [88, 99, 106-111]:

( ) ( )het bub sys bub sg sl sg lg lgF P P V A AΔ γ γ γ= − − + − + (2.9)

where Aij and γij are the surface area and the interfacial energy, respectively, of the interface

between phase i and phase j. The subscripts g, l, and s represent the gas, liquid, and solid phases,

respectively. Plotting the general relationship between ΔFhet and Rbub will result in a graph that is

similar to Figure 2.3 but with a lower maximum value. It also turned out that Rcr for

heterogeneous nucleation is identical to that of homogeneous case, which is expressed in

Equations (2.3) and (2.8). Equations (2.3) and (2.9) can be combined to deduce that Whet is:

( )

3lg

het hom2

bub,cr sys

16 FW W F

3 P P

πγ= =

− (2.10)

22

where F is the energy reduction factor for heterogeneous nucleation. Depending on the geometry

of the nucleating site, various expressions for F have been derived [88, 99, 106-111].

Considering the case where a bubble nucleates on a smooth planar surface (e.g., a piece

of unscratched glass), the system can be modeled as the situation in Figure 2.4. Hence, Fisher

[107] derived that F is a function of θc:

( )3

c cc

2 3cos cosF4θ θθ + −

= (2.11)

where θc is related to the interfacial energies by Young’s Equation [83, 105]:

sg sl lg ccosγ γ γ θ= + (2.12)

Using Equation (2.11), Wilt [88] determined that F and thereby Whet decrease as θc increases.

Equation (2.12) suggests that either a decrease in γsg or an increase in γsl would result in a larger

θc. Hence, the higher cell nucleation rate would actually be a result of the replacement of a high-

energy solid-liquid interface by a low-energy solid-gas interface and the generation of a smaller

liquid-gas interface when a bubble nucleates at the heterogeneous nucleating site.

For the foaming of polymer blends, the dispersed phase can also serve as a nucleating

agent. Depending on the relative compliances between the two phases, three scenarios, as

indicated in Figures 2.5 (a), (b), and (c), can happen when cell nucleation occurs. Firstly, Figure

2.5 (a) illustrates the case where the dispersed phase is spherical and much stiffer than the

matrix. This resembles the heterogeneous nucleation at a spherical projection. Fletcher [108] and

Wilt [88] investigated this case and derived an expression for F:

( )3

3c cc

32c c

c

1 a cos a cos1F ,a 1 a 2 32 g g

a cos a cos3a cos 1g g

θ θθ

θ θθ

⎡ ⎡⎛ ⎞ ⎛ ⎞+ += + + −⎢ ⎢⎜ ⎟ ⎜ ⎟

⎢ ⎝ ⎠ ⎝ ⎠⎣⎣⎤⎤⎛ ⎞ ⎛ ⎞+ + ⎥+ − −⎥⎜ ⎟ ⎜ ⎟⎥⎥⎝ ⎠ ⎝ ⎠⎦ ⎦

(2.13)

23

where a and g are defined in Equations (2.14) and (2.15), respectively. In Equation (2.14), Rhet

represents the radius of the spherical nucleating agent (i.e., the dispersed phase).

het

cr

RaR

= (2.14)

( )1

2 2cg 1 a 2a cosθ= + − (2.15)

Secondly, if the spherical dispersed phase is softer than the bulk phase, cell nucleation will occur

as indicated in Figure 2.5 (b). The situation is identical to the nucleation in a spherical cavity.

Cole [99] and Wilt [88] reviewed this case and deduced the following formulation for F:

( )3

2

3

3

11 1 3 12

2 3

c cc

c c

a cosθ a cosθF θ ,a mag g

a cosθ a cosθag g

⎡ ⎛ ⎞ ⎛ ⎞− −= − + −⎢ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎢⎣⎡ ⎤⎛ ⎞ ⎛ ⎞− −

− − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

(2.16)

Thirdly, if both the bulk phase and the dispersed phase have similar stiffness, cell

nucleation may deform both surfaces. Assuming the dispersed phase is much larger than the

critical radius, the situation could be simplified as if the cell nucleation had occurred at a flat

interface. Moore [109], Apfel [110] and Javis et al. [111] investigated the nucleation

phenomenon for the case depicted in Figure 2.5 (c) and derived the following formulation for F:

( ) ( ) ( )3 3 3 3a a a b b b

a b a b

3 3cos cos 3 3cos cosF , , ,

4γ θ θ γ θ θ

γ γ θ θ− + + − +

= (2.17)

In the plastic foaming industry, inorganic fillers (e.g., talc) are commonly added as cell

nucleating agents to enhance the foam quality. Due to the surface roughness, individual filler

particles or agglomerates of them usually contain conical pits. Cole [99] and Wilt [88] also

24

derived the F factor for this type of heterogeneous nucleating site, as shown in Figure 2.6, and

deduced the following expression:

( ) ( ) ( )2c c

c c

cos cos1F , 2 2 sin4 sin

θ θ βθ β θ β

β⎡ ⎤−

= − − +⎢ ⎥⎣ ⎦

(2.18)

For both homogeneous nucleation and heterogeneous nucleation, classical

thermodynamics suggests that the thermodynamic potential at which Rbub equals to Rcr is a

maximum. This state is thereby an unstable equilibrium state, and the polymer-gas solution is

metastable. The value of Rcr defines the required perturbation, provided by molecular motion or

any other external work, required to form a cell. However, the spontaneous formation of a cell of

size Rcr corresponds to a spontaneous decrease in the entropy of the corresponding

thermodynamic system [113]. Since such process contradicts the second law of thermodynamics,

classical thermodynamics cannot be used to predict when a gas bubble would form

spontaneously and take the system out of the metastable state. In this context, it is necessary to

consider kinetics when studying cell nucleation.

2.2.3. Kinetics of Cell Nucleation

As discussed in the previous section, kinetics is another fundamental principle that

governs cell nucleation in plastic foaming. Thus, in order to determine the cell nucleation rate

during foaming processes, it is necessary to combine the derived thermodynamic models with

kinetic theory.

According to CNT, a bubble is considered to be nucleated when its radius is larger than

Rcr [75-78, 83, 100, 102-105, 112], previous studies conducted by Blander and Katz [105]

determined that the nucleation rate, J, (bubbles/m3-s) is equivalent to the rate at which critical

bubbles gain molecules and grow, which can be expressed as:

25

( ) ( )cr crJ υA R n R Z= (2.19)

where υ is the rate per unit area at which molecules strike against the bubble surface; A(Rcr) is the

surface area of a critical bubble; n(Rcr) is the number density of the critical bubbles; and Z is the

Zeldovich factor. The value of υ can be approximated by the following formulation [105]:

2

bub ,cr

B sys

Pυ

πmk T= (2.20)

where m is the mass of a gas molecule. The quantity, n(Rc), is expressed in the form of an

Arrhenius equation [105]:

cB sys

Wn( R ) N expk T

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠ (2.21)

where the pre-exponential factor, N, is the number of gas molecules per unit volume for

homogeneous nucleation or the number of gas molecules per unit surface area of the nucleating

agents for heterogeneous nucleation; and W is the free energy barrier for cell nucleation. Blander

et al. indicated that there was no rigorous justification for the choice of N; however, they

believed that the possible errors in this pre-exponential factor should not be significant. Finally,

the Zeldovich factor, Z, is used to account for the thermodynamic fluctuations that affect the

number density of the critical bubbles determined by Equation (2.21) [78, 105, 114-115]. Hence,

the homogeneous nucleation rate (Jhom) and the heterogeneous nucleation rate (Jhet) can be

calculated by Equations (2.22) and (2.23), respectively.

( )

3

2

2 16

3lg lg

hom

B sys bub ,cr sys

γ πγJ N exp

πm k T P P

⎛ ⎞⎜ ⎟= −⎜ ⎟−⎝ ⎠

(2.22)

26

( )

32 3

2

2 16

3lg lg

het

B sys bub ,cr sys

γ πγ FJ N Q exp

πmF k T P P

⎛ ⎞⎜ ⎟= −⎜ ⎟−⎝ ⎠

(2.23)

where Q is the ratio of the surface area of the liquid-gas interface to that of a spherical bubble

with the same radius. The expressions of Q for different geometries of nucleating sites were

derived in various literatures [89, 99, 109-111]. For example, for a cell nucleated in a conical

cavity, Q can be expressed as a function of θc and β which is given by [89, 99, 105]:

( ) ( )cc

1 sin θ βQ θ ,β

2− −

= (2.24)

2.3. Modeling of Cell Growth and Cell Coarsening

Cell growth and collapse behaviors in polymer melts or other fluids have been active

research topics of many experimental and theoretical studies since 1917 [116-143]. Nearly all

bubble growth models can be classified into the single bubble growth model [116-136] and the

cell model (i.e., swarm of bubbles growing without interaction) [137-138, 127-130].

2.3.1 The Single Bubble Growth Model

The single bubble growth model studies the growth of a single bubble in an infinite sea of

liquid. Rayleigh [116], Epstein and Plesset [117], and Scriven [118] were among the earliest

researchers who used it as the base model to conduct theoretical or experimental investigations

on bubble growth or collapse behaviors. The analysis of diffusion-induced bubble growth and

collapse in viscous liquids with both mass and momentum transfer was pioneered by Barlow and

Langois [119]. They used the thin boundary layer approximation, which assumes that the

dissolved gas concentration gradient vanishes within a thin shell around the bubble. This is a

widely adopted approximation [139, 117, 120] that allows for a considerable degree of

simplification in the equations governing mass transfer in a viscous liquid. Furthermore, Barlow

27

and Langois, as well as various other researchers [117-123] also assumed that an unlimited

supply of gas was available, and thereby the uniform concentration far from the bubble surface

was always identical to its initial level. During actual plastic foaming, however, the availability

of dissolved gas is finite. Hence, the dissolved gas content will eventually be depleted and cannot

sustain the growth thereafter. Despite these shortcomings, their work is considered to be a great

improvement over previous studies [117], which had considered only the diffusion process in

bubble growth and had neglected the hydrodynamic effects.

Street et al. [124] considered the growth of a spherical stationary cavity in a viscoelastic

three-constant Oldroyd fluid. The related work of Fogler and Goddard [125], analyzed the

collapse of a large body of an incompressible viscoelastic liquid being modelled by a

viscoelastic, linear-integral model of the generalized Maxwell type. The latter found that fluid

elasticity retards the collapse of a void and produces prolonged, oscillatory motion. In both

cases, the driving mechanism for the volume change of the spherical cavity was assumed to be

the difference between the actual and equilibrium internal pressures. Since the mass transfer was

neglected, the cavity pressure was assumed to remain constant, which greatly simplified the

analysis. Unlike Street et al. [124] and Fogler and Goddard [125], Zana and Leal [121] coupled

the mass transfer process and the collapse-induced fluid flow through the internal pressure (i.e.,

concentration of dissolving gas), which does not remain constant. Considering both a Newtonian

liquid and a viscoelastic liquid of the modified Oldroyd “fluid B” type and an isothermal

condition, they showed that the collapse rate and the internal bubble pressure for a viscoelastic

liquid differ considerably from the values for a Newtonian liquid, especially during the early and

late stages of the process. However, they did not detected any bubble oscillation because of the

very short time span used in their computations (i.e., about one unit of dimensionless time versus

5 units of dimensionless time in Fogler and Goddard’s case).

28

Street et al. [124] extended the previous work [119] to formulate and numerically

simulate the bubble growth process for a liquid of an infinite medium under non-isothermal

conditions. Their theoretical model accounts for the heat, mass, and momentum transfer

processes governing the growth of a vapor bubble in a solution consisting of a viscous liquid and

a dissolved blowing agent. In their work, the viscous liquid was assumed to obey a non-

Newtonian (power law) fluid model. They identified that the most important parameters

controlling the growth rate are the diffusivity and the concentration of blowing agent, the

viscosity level of the melt, and the extent to which the liquid is shear thinning.

Han and Yoo [140] carried out an experimental investigation and a theoretical study to

elucidate the oscillatory behavior of a gas bubble in a viscoelastic liquid. The rheological

property of the fluid was modelled by the Zaremba-DeWitt model. They took into account both

the hydrodynamic and diffusion effects. Using a third order polynomial approximation for the

gas concentration profile around the bubble, the finite difference method was employed to solve

the governing equations. Their study showed that gas diffusivity has a profound influence on the

occurrence of oscillatory behavior. Furthermore, they indicated that while the melt elasticity

enhances the oscillatory behavior of bubble growth or collapse, the viscosity suppresses it.

Venerus et al. [126] formulated a rigorous model to describe bubble growth or collapse in

a non-linear viscoelastic fluid. The convective and diffusive mass transport as well as surface

tension and inertial effects had been taken into account in their study. They found that the

influence of non-linear fluid rheology on bubble growth dynamics is relatively minor when

compared to fluid elasticity. Using the developed model, they evaluated various approximations

being used by previous investigations and indicated that the thin boundary layer approximation

has a very limited range of applicability.

29

The aforementioned fundamental investigations on cell growth or cell collapse have led

to an increased understanding of the phenomena. However, during plastic foaming, a swarm of

bubbles grows simultaneously, and they are expanding in close proximity to one another with a

limited supply of gas. In this context, the practical application of the single bubble growth model

in the plastic foaming industry was very limited [137].

2.3.2 The Cell Model

In order to address the problematic approximations in the single bubble growth model,

Amon and Denson [137] introduced the well-known cell model, which suggested that a large

amount of gas bubbles grow in close proximity to each other in a polymer-gas solution. This

model represented a significant advancement in the field of bubble growth simulation. It

described the actual foaming situation more realistically by dividing the polymer-gas solution

into spherical unit cells consisting of equal and limited amounts of dissolved gas. A schematic of

the cell model is illustrated in Figure 2.7. In the Figure, Pbub(t,t’) is the bubble pressure at time t

for the bubble nucleated at time t’; Rbub(t,t’) and Rshell(t,t’) are the corresponding bubble radius

and polymer-gas solution shell radius, respectively; C(r,t,t’) is the dissolved gas concentration at

the radial position r; and CR(t,t’) is the dissolved gas concentration at the liquid-gas interface.

Assuming the polymer melt and the gas in the cell had behaved like Newtonian fluid and

an ideal gas, respectively, they applied the cell model to simulate the bubble growth during

plastic foaming. Unlike the single bubble growth model, their model yielded a finite radius for

the growing bubble. Furthermore, they concluded that the surface tension and the initial radius

have less effect on bubble growth dynamics than the thermodynamic driving force (i.e., the

degree of supersaturation) as well as the mass and momentum transfers. Later, they extended

their work to a low-pressure structural foam-molding process by considering the heat transfer,

solidification, and flow in the cavity [127-129]. The predicted bubble growth profiles were in

30

good qualitative agreement with the experimental measurements of the bulk density of

expanding thermoplastic polymeric foam. However, quantitative discrepancies existed between

the two. These differences were believed to be related to the omission of melt elasticity and

bubble coalescence in the model.

Arefmanesh and Advani also applied the cell model to a low-pressure structural foam-

molding process and studied the simultaneous growth of a given number of cells in a Newtonian

fluid [138]. In their work, they approximated the dissolved gas concentration gradient in the unit

cell as a polynomial profile. They further assumed that the cell growth is under an isothermal

condition, and that the gas inside each expanding cell behaves like an ideal gas. Later, they

extended their earlier researches by considering the viscoelastic properties (i.e., based on the

Upper-Convected Maxwell model) of the fluid [128] and the non-isothermal effects [129].

Effects of various parameters on the bubble growth dynamics were investigated in these studies.

The results showed that higher gas diffusivity enhances the bubble growth rate while higher

viscosity retards it, especially in the initial growing stage. Nevertheless, the effect of viscosity

becomes negligible as the growth proceeds.

Ramesh et al. proposed a modified viscoelastic cell model, which accounts for the effect

of dissolved gas content and the temperature on rheological and other physical properties [130].

Using the Upper-Convected Maxwell model to describe the viscoelastic property of the polymer-

gas solution, they simulated the growth of closely spaced spherical bubbles during the foaming

process and compared the results with experimental data. The simulation results indicated that

the predictions based on the modified cell model were in qualitative agreement with the

experimental data while the quantitative agreement was also satisfactory. The simulation results

demonstrated that gas loss to the surrounding, dissolved gas content, and transient cooling effects

are the most important factors that govern the bubble growth.

31

2.3.3 Cell Collapse, Cell Coarsening and Cell Coalescence during Plastic Foaming

Cell coarsening (also called cell ripening) and cell coalescence are the two mechanisms

through which the cellular structure degrades. The system will be more stable with fewer large

cells than with more small cells. Since the gas concentration in a small bubble is higher than that

in a large one, the gas concentration gradient will drive the gas from the smaller bubble to the

larger one. As a result, the smaller bubbles tend to get smaller, and eventually disappear. This

phenomenon is known as cell coarsening. Cell coalescence is a mechanism where two growing

contiguous cells in a polymer melt combined because of cell wall rupture. This usually occurs if

the stretched thin cell wall separating the two cells is not strong enough to sustain the tension

developed during cell growth.

Xu et al., using computer simulation and the empirically observed data in a batch

foaming process, investigated the bubble growth and collapse phenomenon in low-density

polyethylene (LDPE) foaming with CBA under atmospheric pressure [141]. A mathematical

model that accounts for the effects of diffusion, surface tension, viscosity, and elasticity has been

employed to investigate the fundamentals of the phenomena. Their study found that the

processing temperature, diffusivity, and dissolved gas concentration have dominant effects on

the lifespan of CBA-blown bubbles.

Zhu et al. studied the cell coarsening in plastic foaming through numerical simulation

[142]. A quadratic triangle-based finite element analysis with an implicit scheme for time

evolution is utilized to solve the governing equations for bubble growth and collapse dynamics.

Simulation results showed that larger bubbles grow while the smaller ones shrink due to the gas

diffusion from the smaller bubble to the larger bubble. It also reported that a shorter distance and

a larger size difference between the adjacent cells promote cell coarsening while a high bulk gas

concentration suppresses it. As an extension of this study, they explored the bubble growth and

32

collapse behaviors for a system with a central cell and eight surrounding cells [143]. The results

indicated that smaller nano-sized cells are doomed to collapse very quickly once they have

interacted with the larger cells, which makes them difficult to sustain.

Taki et al. investigated the effect of rheological behavior on the cell coalescence during

plastic foaming [131]. Using an in-situ foaming visualization system to observe the cell

coalescence phenomena, the cell coalescence dynamics for polymers with different rheological

behaviors were studied and four patterns of interface deformations were observed. For polymer

that exhibits Newtonian behavior, the planar interface that appears between the two spherical

bubbles gets thinner as the coalescence progresses. The phenomena can be characterized by a

combination of biaxial and planar elongation. For non-Newtonian polymers, the shape of the

interface changed from flat to parabolic and penetrates inside a bubble during the cell

coalescence. Their study also reported the effect of polymer’s strain-hardening on the magnitude

of the interface deformation and the coalescing time.

2.3.4 Mathematical Formulations to Describe Bubble Growth

The bubble growth and collapse can be analyzed by solving the governing equations for

both the mass transfer and the momentum transfer that occur between the nucleated bubbles and

the surrounding polymer-gas solution in the spherical coordinate system. Assuming the bubble is

spherically symmetric and the polymer-gas solution is incompressible, the continuity equation

for the flow around the growing bubble can be reduced to [132]:

( )22

1 r u( r ) 0r r

∂=

∂ (2.25)

where r is the radial position and u(r) is the fluid velocity at position r. Using Equation (2.25)

and the boundary condition of the radial velocity at the bubble surface, i.e.,

bub bubu( R ) R•

= (2.26)

33

The radial velocity of the polymer-gas solution can be expressed as [132]:

2

bub bub2

R Ru( r )r

•

= (2.27)

Using Equation (2.27), together with the assumption that the inertial forces are negligible and

that the pressure at the outer boundary of the shell equals to Psys(t), the momentum equation for

the polymer-gas solution that surrounds the bubble can be expressed as [132]:

R ( t ,t ')shell

lg rr θθbub sys

R ( t ,t ')bubbub

2γ τ τP ( t ,t ') P ( t ) 2 dr 0R r

−− − + =∫ (2.28)

where Pbub(t,t’) is the bubble pressure at time t of a bubble nucleated at time t’; τrr and τθθ are the

stress components in the r and θ directions, respectively. The two stress components can be

determined by any appropriate rheological model that can describe the particular material system

(e.g., polymer-gas solution) being considered in the simulation.

Assuming that the accumulation of the adsorbed gas molecules on the bubble surface is

negligible, the law of conservation of mass requires that the rate of change of the mass in the gas

bubble must be balanced by the mass of gas diffusing into or out of the bubble through its

surface. Together with the assumption that the gas inside the bubble obeys the ideal gas law,

Pbub(t,t’) can be related to the concentration gradient at the bubble surface by [132]:

( ) 3

bub bub 2bub

r Rg bub

P t ,t ' R ( t ,t ')d 4π C( r,t ,t ')4πR ( t ,t ') Ddt 3 R T r =

⎛ ⎞ ∂=⎜ ⎟⎜ ⎟ ∂⎝ ⎠

(2.29)

where Rg is the universal gas constant and D is the gas diffusivity in the polymer melt. With the

knowledge of the concentration gradient at the bubble surface, Equation (2.30) can be solved to

obtain the bubble pressure at a particular time. Therefore, it is necessary to determine the

concentration profile around the gas bubble. The diffusion equation can be written as [132]:

34

2

2bub bubbub2 2

R RC C D C Cr for r Rt r r r r r

•

∂ ∂ ∂ ∂⎛ ⎞+ = ≥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (2.30)

By choosing appropriate initial and boundary conditions that can closely describe the processing

conditions, Equations (2.28) through (2.30) constitute a complete set of governing equations to

simulate the bubble growth dynamics during plastic foaming.

2.4. Experimental Studies on Plastic Foaming Mechanism

Experimental investigations from plastic foaming revealed that foam appeared at low

supersaturation levels at which no nucleation should occur according to the theoretical

calculation based on the classical homogeneous nucleation theory [85-86]. Commercial resins

consist of numerous unknown additives as well as impurities, and the internal walls of the

processing equipment contain a lot of crevices. These facts evidence that cell nucleation happens

during typical plastic foaming is more likely to occur heterogeneously. Therefore, the elucidation

of the heterogeneous nucleation is essential to control the cellular structures of plastic foams.

2.4.1. Heterogeneous Nucleation with Nucleating Agents

During polymeric foaming processes, nucleating agents, such as inorganic fillers (e.g.

talc particles), organic phrases (e.g. elastomer phases), and nano-particles (e.g. nanoclay and

carbon nanotubes), are commonly used to facilitate the control of cellular structures and to

produce high-quality foam products. Various research groups have made tremendous efforts to

investigate the effects of different nucleating agents on polymeric foaming processes.

In the 1960s, Hansen et al. [133-134] was one of the pioneer groups to explore the effects

of nucleating agents on cell nucleation. As an extension of their work, a series of studies had

identified various effective nucleating agents for producing plastic foams. These nucleating

agents included azodicarbonamide [135], calcium carbonate [135-136], calcium stearate [135,

35

144-145], magnesium silicate [146], the mixture of citric acid and sodium bicarbonate [135, 146-

150], rubber particles [95-97], sodium benzoate [151], stearic acid [152], silica products [136,

153], talc [135, 144, 147-148, 86, 34, 154-155], and zinc stearate [135, 153]. Nowadays,

nanocomposite foams have received a lot of attention from both academic research and industrial

investigation due to the continuous evolution of nanotechnology in the past decade. During

nanocomposites foaming, nano-sized particles work as heterogeneous nucleating sites to promote

cell nucleation. Recent studies demonstrated that the nanoparticles (e.g., nanoclay [156-161],

carbon nanofibers [162], and single-walled carbon nanotubes [162]) could lead to a higher cell

density in plastic foams. These studies attributed to the promotion of heterogeneous cell

nucleation to the extremely fine dimensions and large total surface area of nucleating agents.

Furthermore, special properties such as better fire retardance, improved barrier resistance, and

higher thermal insulation can be achieved by selecting appropriate nanoparticles or a

combination of them [163-164].

Most previous studies accounted for the promotion of heterogeneous cell nucleation by

adding talc particles or other nucleating agents as a combined result of reducing the free energy

barrier to nucleate cells and increase the number of heterogeneous nucleating sites. Regarding

the effect of talc contents on cell nucleation, extrusion foaming experiments of various polymers

using different blowing agents [34, 154-155] showed that cell densities increased with talc

contents in most cases. These investigations also highlighted two additional facts. Firstly,

regardless of the types of blowing agents, further addition of talc particles beyond 10 wt% had

negligible effect on the cell density when compared with the cases with lower talc contents

[154]. Park et al. speculated that this was related to fact that the agglomeration of talc particles at

high loading did not create substantially more nucleating sites. Secondly, when CO2 was used as

the blowing agent, the correlation between the talc content and the final cell density was less

36

apparent [34, 154], especially at a high CO2 concentration (e.g. 5 wt%); however, the underlying

sciences of the phenomena has not yet been clearly identified.

2.4.2. In-situ Visual Observation of Plastic Foaming

Although intense research has been done in the past using small-scale industrial foam

processing equipment, such as extrusion foaming or structural foam molding systems, the bubble

nucleation and growth phenomena were not observed in these systems. Therefore, optimization

in processing strategies and conditions were still based on a trial-and-see approach in many

cases. In this context, experimental foaming simulation systems had been developed to capture

plastic foaming processes in-situ. In particular, Ohyabu et al. [165], Taki et al. [166], and Guo et

al. [167] developed batch foaming visualization systems to observe foaming under static

conditions using high-speed cameras with optical microscopes. Using these systems, various

studies were conducted to study the effects of various material parameters (e.g., the sizes,

contents and types of nucleating agents [133-136, 144-153], the types and contents of blowing

agent [32, 36-40, 42], and the types of polymers [16-18, 30] or polymer blends [3, 36]) and those

of processing conditions (e.g., pressure drop rates [18, 31, 34] and temperatures [18, 34]) on

foaming behaviors.

In the aforementioned experimental simulation systems, the stresses that were applied to

the plastic sample were minimal. However, in various foam processing technologies, plastics are

subjected to significant shear and extensional stresses, which can affect the final bubble density

and morphology [86, 168]. Therefore, while the simulation systems can offer valuable insight by

suppressing the shear to decouple the analysis of various experimental parameters, the systems

may not be ideal for understanding the foaming behavior in industrial plastic foaming processes.

Hence, various researchers have observed continuous foaming in-situ through transparent slit

dies [86, 168-170]. These studies have captured the dynamic foaming nature of industrial

37

processes. However, it is still challenging to establish a thorough understanding because both the

bubble nucleation and growth phenomena in a continuous flow of polymer-gas solutions involve

highly complicated thermodynamic, fluid mechanic and rheological concepts.

2.4.3. Stress-Induced Nucleation

Lee, one of the pioneers who studied the effect of shear field on cell nucleation, proposed

that the increase of cell density by adding nucleating agents might relate to the shear-induced

nucleation [86]. Nucleating agent particles tended to agglomerate together to form a cluster of

particles in the polymer melt. Due to the porous surfaces of the clumps, there may not be

complete wetting of the polymer melt on the nucleating agents. In these cases, gas molecules can

be trapped in the crevices on the nucleating agents to form pre-existing gas cavities. Lee

suggested that when the system pressure decreases, these cavities can expand and the shear force

being generated during extrusion foaming can pull them out of the crevices [86]. Eventually,

they can grow to form the final cellular structure.

Moreover, Lee suggested that the energy associated with the shear work in polymer flow

helps to overcome the energy barrier to promote cell nucleation. On the other hand, Han and Han

observed continuous foaming in-situ through transparent slit dies. They suggested that cell

nucleation can be induced by flow or shear stress due to the motion of gas clusters, even at

thermodynamically unsaturated conditions [85]. Later, Chen et al. [171] and Zhu et al. [172]

independently developed experimental foaming simulation systems to demonstrate effects of

shear stress on plastic foaming behavior. From the analyzed plastic foam samples, they both

concluded that shear could indeed promote cell nucleation. Using the concept of shear energy

suggested by Lee [86], they explained the shear-induced nucleation. However, these systems

were not equipped to capture foaming in-situ, and hence characterizations were still restricted to

the final foam structures. Therefore, while these studies illustrated the shear effects on foaming,

38

the validity of their theories in explaining shear-induced nucleation and growth remain unclear.

Despite the limitations in various experimental simulation systems, the aforementioned

studies had given invaluable insights in the investigation of heterogeneous nucleation and

provided a lot of important in-situ visualization data to verify various theoretical studies on the

simulation of the cell nucleation during plastic foaming. In most of these simulations, researchers

viewed the role of the nucleating agents to be providing more heterogeneous nucleating sites,

which promotes the more energetically favorable route (i.e., heterogeneous nucleation) for cell

nucleation. However, the underlying mechanisms of the nucleating agent-enhanced nucleation

have not yet been completely identified.

2.5. Computer Simulation of Plastic Foaming

2.5.1. Influence Volume Approach (IVA)

During foaming, both cell nucleation and cell growth are competing for the limited gas in

the polymer melt [31]. In light of this, Shafi et al. [173] and Joshi et al. [174] proposed an

outstanding model, called the Influence Volume Approach (IVA) that described the cell

nucleation and growth processes simultaneously to study the effects of various processing

conditions on the final cell size distribution. Shafi et al. [173], by assuming an instant pressure

drop, suggested that the nucleation rate is the highest at the beginning of the process because of

the high initial dissolved gas content (C0). As a nucleated bubble grows, both the bubble pressure

and the gas concentration at the bubble surface decrease. Through the gas diffusion into the

expanding bubbles, a concentration gradient is then generated in the polymer melt around the

bubble with time, as indicated in Figure 2.8. In this figure, the quantity S denotes the radial

position at which the dissolved gas concentration equals to the nucleation threshold (CS), which

39

is the dissolved gas concentration at which the nucleation rate is 2% or less of the nucleation rate

at the initial dissolved gas concentration. Thus, it is given by [173-174]:

hom,0 homS

0 B

W WC0.02 expC k T

⎛ ⎞−= ⎜ ⎟

⎝ ⎠ (2.31)

The volume of the polymer melt between the bubble surface and S is called the influence volume

(VS). Within Vs, the nucleation rate is negligibly small and is assumed to be zero. The volume of

the melt outside the influence volume and the bubble is called the non-influenced volume (VL).

In order to simplify the numerical simulation, the integral method is employed and it

approximates the gas concentration profile to be a polynomial function [173-174]:

( ) ( )4

0

1 1R

R

C r Cx

C C−

= − −−

(2.32)

3 3

3 3cb

r Rxr R−

=−

(2.33)

Using Equation (2.33) to define a new coordinate system (x), the problem has been transformed

from a moving boundary problem with variable boundary conditions to a fixed boundary

problem with constant boundary conditions. As a result, the mass balance equation (i.e.,

Equation (2.29)) and the diffusion equation (i.e., Equation (2.30)) were rewritten as:

bub 0 R bubR RH g sys

x 0cb bub

R ( C C ) dRdC 3CdC12πDK R Tdt V dx R dt=

−= − (2.34)

3 3

R bub 0 crcb 1

4H g sys R 0

0

4π( C R C R )V3K R T ( C C ) (1 x ) dx

−=

− −∫ (2.35)

Finally, since the nucleation rate (J) is very sensitive to the dissolved gas concentration, it

is important to evaluate J at the remaining gas content, which is approximated as the average gas

40

concentration (Cavg) in the non-influenced region. Through the overall gas mass balance, the

Cavg can be calculated by [173-174]:

3tbub ,cr bub

avg L 0 L0 L0 g sys

S ( t ,t ')t2

L0 R( t ,t ')

P ( t ,t ')R ( t ,t ')4πC V C V J( t')V ( t') dt'3 R T

J( t')V ( t') 4πr C( r,t ,t ')drdt'

= − −∫

∫ ∫

(2.36)

As more and more bubbles are nucleated and each bubble grows with time, the VL decreases with

time. Finally, nucleation ends when the entire VL becomes zero. The overall nucleation and

bubble growth process are indicated in Figure 2.9.

The IVA characterizes the interaction between cell nucleation and expansion behaviors

by considering the gas depletion around the nucleated bubbles. It also provides an in-depth

physical account of the bubble nucleation phenomena during plastic foaming. Various

researchers extended the IVA [174-175] or adopted a similar simulation scheme without using

the IVA [89, 176] to conduct computer simulations of cell nucleation and cell growth.

2.5.2. Modified Influence Volume Approach (MIVA)

Extending the IVA, Mao et al. [175] defined two distinct stages of bubble growth for

physical foaming. These two stages are termed as free and limited expansion and are controlled

by the bubble nucleation rate. The modified approach is called the modified influence volume

approach (MIVA). The MIVA assumed that cell nucleation would only occur in the free

expansion stage, during which the bubble pressure drops substantially while the dissolved gas

concentration only slightly changes. In the limited expansion stage, the continuous bubble

growth depletes the gas concentration significantly. They reported that the duration of the free

expansion stage is much shorter than the limited expansion stage. Furthermore, most of the

bubble volume expansion takes place in the second stage.

2.5.3. Computer Simulation of a Continuous Foaming Process

41

Combining the CNT and the bubble growth models with a non-Newtonian fluid model of

a flow, Shimoda et al. [176] conducted a series of computer simulations to study the

simultaneous cell nucleation and growth in a flow field, which also accounted for the effects of

surface tension, diffusivity, and viscosity on plastic foaming. Their study assumed that the cell

nucleation occurred heterogeneously on smooth planar surfaces. The computer simulation results

of polypropylene foaming, with and without using the concept of IVA, were compared with the

experimental data obtained by visual observations at a foaming extruder. They suggested that

polymers with lower surface tension and lower diffusivity are desired to produce foams with

finer cells. Furthermore, they also reported that temperature is a critical parameter to control the

gas diffusivity and viscosity in order to suppress cell growth and promote cell nucleation.

2.6. Summary of Literature Survey and Critical Analysis

In summary, experimental investigations yield limited information because the critical

nuclei are extremely small (i.e., in the scale of nanometers) and are difficult to be observed by

the existing technologies under experimental or actual processing conditions. Moreover, despite

the long-standing interest and research efforts regarding these processes, the simulation and

prediction of cell nucleation in polymer foaming remains challenging. As evidenced in various

studies, nucleation in typical polymeric foaming processes is more likely to happen

heterogeneously. In light of this, various researchers considered heterogeneous nucleation when

estimating the nucleation rate in their computer simulations [89, 176]. While most of these

researchers applied an energy reduction factor to account for the higher nucleation rate caused by

heterogeneous nucleation, they over-simplified the real situation by solely considering the

additives or the impurities enhance cell nucleation by reducing the free energy barrier to form

cells. Moreover, when theoretically predicting the cell nucleation rate, researchers have been

42

assuming the pressure inside the polymer-gas solution to be identical to the system pressure

inside the foaming equipment [89, 173-176]. Nonetheless, these simplifications and assumptions

seemed to be unrealistic, especially in the cases where nucleating agents existed. During the

pressure reduction, it is speculated that the discontinuity between the nucleating agents and the

surrounding polymer may lead to a local pressure field that differs from the bulk pressure.

Depending on the pressure field being established around the nucleating agents, the degree of

supersaturation may vary significantly within these local regions, and thereby affect the cell

nucleation behaviors. As a result, before trying to model the nucleation in plastic foaming, it is

first necessary to explore a new mechanism that can more realistically describe cell nucleation

phenomena.

As discussed in the previous sections, extensive research had been done to analyze

bubble growth [137-139, 173-175, 177-179] and bubble coarsening phenomena [141-143, 180]

in polymer-gas solutions. These studies have played significant roles in contributing to a more

complete understanding of cell growth and cell coarsening phenomena. However, in studying

bubble growth behaviors, almost all of these previous works have involved pure theoretical

studies without having verification experiments; only very limited experiments have addressed

the dynamic behavior of the phenomena in a polymer-gas solution [140, 170, 181-184].

Moreover, some of the physical parameters that were used to describe the material properties

adopted in these theoretical studies were unrealistic. Therefore, a more thorough investigation of

the bubble growth process, which bridges the theory and the actual plastic processing, is needed

to advance the understanding of the phenomena in this context. Recently, the in-situ observations

of both cell growth and cell coarsening have been made possible by the development of various

experimental foaming simulation systems being equipped with high speed CCD or CMOS

cameras [165-167, 185-186]. Therefore, once bubble nucleation has been initiated, it is capable

43

of obtaining in-situ visualization data of the subsequent cell growth and cell coarsening

phenomena during plastic foaming processes. These results, in conjunction with the accurately

measured thermo-physical and rheological properties of the polymer-gas mixtures, such as the

solubility, the diffusivity, the surface tension, the viscosity, and the relaxation time, have

provided a solid information base for this thesis research to enhance the development of a

theoretical model and a simulation scheme for bubble growth and bubble coarsening behaviors.

The advancement in foaming technology in the past decades has ignited the rapidly

growing use of polymer foams in various industrial domains. The vast majority of thermoplastic

polymer foams are prepared by extrusion foaming [187]. In this continuous process, cell

nucleation occurs inside the die after the pressure of the polymer-gas solution drops below the

solubility pressure. Upon cell nucleation, cells start to grow before the polymer-gas solution exits

the die. This cell growth phenomenon is termed “premature cell growth” [188]. An excess

amount of premature cell growth would lead to rapid cell growth upon die exit. This accelerated

cell growth will promote gas loss during the foam cooling process, so that the foam will shrink

before it stabilizes, and subsequently a low volume expansion ratio will result [7]. In order to

accurately determine the amount of premature cell growth, it is first necessary to identify the

onset point of cell nucleation. Many previous studies have assumed that cell nucleation occurs

right after the system pressure drops below the solubility pressures [8–9]. However, since cell

nucleation is a kinetic process, a certain amount of pressure drop beyond the solubility pressure

is needed to create a sufficient level of supersaturation to initiate cell nucleation. This pressure

drop is denoted as “pressure drop threshold (i.e., ΔPthreshold)” in this thesis. Fundamental

understanding of the mechanisms governing ΔPthreshold will assist the development of design

strategies in foaming systems to suppress premature cell growth and to better control cell

morphology, as well as the volume expansion ratio of foamed plastics. In particular, foamed

44

products with a high volume expansion ratio can be achieved while maintaining uniform cell

morphology. At the other end of the spectrum, by knowing the onset point of cell nucleation, it

will also aid in the development of innovative means to suppress cell growth to produce

nanocellular foamed products.

45

3Figure 2.1. Homogeneous and heterogeneous nucleation in a polymer-gas solution

4Figure 2.2. A schematic of a Harvey nucleus [79-82]

5Figure 2.3. Free energy change to nucleate a bubble homogeneously

46

6Figure 2.4. A bubble nucleates on a smooth planar surface [107]

(a) The dispersed phase is stiffer than the bulk phase [88, 108]

(b) The dispersed phase is softer than the bulk phase [88, 99]

(c) The dispersed phase and the bulk phase have similar compliances [109-111]

7Figure 2.5. A bubble nucleates at the interface between two polymer melts

47

8Figure 2.6. A bubble nucleates in a conical cavity with an apex angle of 2β

9Figure 2.7. A schematic of the cell model [137]

48

10Figure 2.8. A schematic of a cell (bubble and its influence volume) [173-174]

11Figure 2.9. Overall nucleation and bubble growth processes

t t + dt Nucleation ENDNucleated Bubble Influence Volume Residual Volume

49

Chapter 3 CELL NUCLEATION PHENOMENA

IN PASTIC FOAMING

3.1. Introduction

Extensive research has been conducted by both industrial foaming companies and

academia either to develop innovative, industrially viable, and cost-effective plastic foaming

technologies, or to improve the current technologies to produce plastic foam products with

superior and controlled properties. These studies have identified that various foaming strategies,

including the addition of nucleating agents, can enhance the control of plastic foam morphology.

However, the potential changes in the underlying mechanisms induced by these foaming

methodologies have yet to be elucidated completely.

This chapter discusses a comprehensive research, which elucidates the cell nucleation

mechanisms in various plastic foaming practices. The in-situ visualizations of polystyrene-

carbon dioxide (PS-CO2) foaming under different processing conditions (e.g., pressure drop rates

50

(-dPsys/dt), dissolved gas contents (C), and processing temperature (Tsys)) and talc-enhanced PS

foaming using CO2 are presented herein as case studies.

3.2. Background and Research Methodology

3.2.1. Plastic Foaming Under Different Processing Conditions

The effects of -dPsys/dt, C, and Tsys on the cellular structures of plastic foams were studied

carefully and systematically by a number of researchers using foam extrusion systems. Among

these processing parameters, -dPsys/dt and C have been identified to be two of the most

significant factors that govern cell nuclei density. When high -dPsys/dt or high C is employed, a

high cell density can be achieved because a large thermodynamic instability is instigated. While

many of such studies had also led to invaluable insights on the effect of Tsys on polymeric

foaming behaviors, the final conclusions seem to vary from one study to another. Therefore, a

general consensus on this subject seems to be lacking. For example, Park et al. [16] and Xu et al.

[34] conducted foaming experiments of high impact polystyrene (HIPS) and PS, respectively,

with CO2 using an extrusion foaming line. They had found that the effect of Tsys on cell density

was minimal. A similar trend was observed by Naguib et al. [155] in their study of

polypropylene (PP) being blown with butane. In contrast, in a study conducted by Lee et al. [18]

to investigate polycarbonate (PC) foaming with CO2, it was observed that cell density decreased

with increasing temperature.

The processing conditions are believed to affect polymeric foaming behaviors in many

different ways. For example, as C or Tsys varies, the viscosity of a polymer-gas solution changes,

which subsequently leads to a change in the shear force that is acted on the polymer-gas solution

during an extrusion foaming process. Lee et al. [86] and Chen et al. [171] showed that a shear

force could enhance cell nucleation significantly. On the other hand, both the surface tension

51

[189] and the melt strength of a polymer-gas solution also depend on C or Tsys. The change in

these material parameters will also affect cell nucleation and thereby cell density. Furthermore,

the gas diffusivity in a polymer melt is known to be a function of temperature [50], and it affects

cell growth and ultimately the final cell density. Due to the different and often competitive

effects of the processing conditions on various material parameters, it is difficult to thoroughly

understand the overall impact of them on foaming behaviors. In addition, since characterization

of foam is typically undertaken after the foam has been extruded from the die, some of the

phenomena during the foaming process are hidden.

In light of this, the research being presented in this chapter aims to re-examine the effects

of -dPsys/dt, C, and Tsys on PS foaming with CO2. Through the in-situ visualization of the

foaming processes using a batch foaming simulation system indicated in Figure 3.1 [167], it is

possible to probe the effects of these processing conditions at different stages of foaming.

Furthermore, the static nature in the batch foaming processes makes it possible to minimize the

shear effect to simplify the analysis. One might argue that the true mechanisms of foaming

would not be revealed because of the suppression of shear. Nevertheless, this study could serve

as the first step to investigate the individual effect of each processing parameter on the foaming.

3.2.2. Plastic Foaming Using Nucleating Agents

Addition of nucleating agents provides another route to control plastic foaming. Inorganic

fillers, such as talc particles, are commonly used as cell nucleating agents in polymeric foaming

processes. Previous experimental evidence has revealed that these particles promote cell

nucleation, which thereby increases the cell density of the foamed products. This improvement in

the foam morphology has been attributed to the lower free energy barrier for heterogeneous cell

nucleation and the increased number of heterogeneous nucleating sites. However, less is known

about the underlying mechanisms of the talc-enhanced nucleation and the role of talc particles.

52

Therefore, this chapter also presents visualization data of foaming experiments with PS-talc

samples using CO2 under various processing conditions. These in-situ observations demonstrated

that the expansion of nucleated cells triggers the formation of new cells around them despite the

lower gas concentrations in these regions. It is speculated that the growing cells are able to

generate tensile stress fields around the nearby filler particles, resulting in local pressure

fluctuations. The additional pressure drops lead to a further reduction of the critical radius for

cell nucleation in these local regions. As a result, the growth of pre-existing nuclei hidden in the

crevices of the inorganic fillers’ surfaces is promoted, causing the formation of new cells. A new

mechanism is proposed to elucidate the enhancement of cell nucleation during plastic foaming

with nucleating agents and to advance computer simulation technology of the process.

3.3. Theoretical Framework

3.3.1. Classical View of Cell Nucleation

The batch foaming experiments described herein investigated a polymeric foaming

process utilizing an isothermal decompression of a saturated polymer-gas system. Within the

pressure chamber, the plastic foam was allowed to expand freely. Based on the classical

nucleation theory (CNT) [75-78], a bubble with its radius equal to the critical radius (Rcr) (i.e., a

critical bubble) is at an unstable equilibrium state with its surroundings. Using classical

thermodynamics, the expressions to determine Rcr and the free energy barriers for homogeneous

nucleation and heterogeneous nucleation (i.e., Whom and Whet, respectively) were derived [83, 88,

99, 102-111]. These expressions are stated as Equations (2.4), (2.5), and (2.8) in Chapter 2 and

are restated herein as Equations (3.1) through (3.3):

lgcr

Rsys

H

2R C P

K

=−

γ (3.1)

53

( )

=−

3lg

hom 2

bub,cr sys

16W

3 P P

πγ (3.2)

( )

3lg

het hom2

bub,cr sys

16 FW W F

3 P P

πγ= =

− (3.3)

According to the CNT, the main driving force for cell nucleation is the degree of supersaturation,

ΔP, which can be expressed as:

= −bub ,cr sysP P PΔ (3.4)

Nucleating agents are commonly added to enhance the control of the foam structures. For

inorganic fillers (e.g., talc), the particles tend to aggregate together [86]. The surfaces of the

heterogeneous nucleating sites are rough and can be modeled as a series of conical cavities, as

indicated in Figure 2.6. Accordingly, Equation (2.18) and (2.24) are the appropriate expressions

for the energy reduction factor (F(θc, β)) and the geometric factor (Q(θc, β)) when calculating the

free energy barrier for heterogeneous cell nucleation. Combining the CNT and molecular

kinetics, the expressions to determine the homogeneous nucleation rate (Jhom) and the

heterogeneous nucleation rate (Jhet) were derived as:

2 ⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

lg homhom

B sys

γ WJ N expπm k T

(3.5)

( ) ( ) ( )23

2 ⎛ ⎞= −⎜ ⎟∫ ⎜ ⎟

⎝ ⎠

lg hethet β c

β c B sys

γ WJ ρ β N Q θ ,β exp dβπmF θ ,β k T

(3.6)

Equation (3.6) is a modified formulation to determine the heterogeneous nucleation rate (i.e.,

Equation (2.23)). Due to the surface roughness of the aggregates, the modified formulation uses

a probability density function (ρβ(β)) to account for the randomness of the semi-conical angle (β).

The derivation of Equation (3.6) is detailed in Chapter 6 of this thesis.

54

Due to the relatively high interfacial energy for typical polymer-gas system [189-191],

the calculated nucleation rates for homogeneous nucleation were reported to be negligible for

typical plastic foaming. This suggests that nucleation is heterogeneous in typical polymeric

foaming processes. The total number of nucleated cells per unit unfoamed volume of polymer,

Nb(t), within a time period, t, can then be estimated by:

( ) ( )= + ≈∫ ∫t t

b hom het het het het0 0N ( t ) J ( t') A t' J ( t')dt' A t' J ( t')dt' (3.7)

3.3.2. Dynamic Change of Rcr and Activation of Pre-existing Gas Cavities

Although continuum thermodynamics have provided numerous insights on the cell

nucleation process during plastic foaming, it does not adequately describe the real situation.

Some researchers suggest that free volumes as potential nucleating sites. Furthermore, a

nucleating agent or its agglomerates (i.e., talc particles) may have rough or porous surfaces, and

will not be totally wetted by the viscous polymer [192]. Therefore, the polymer-filler interfaces

provide extra crevices for gas molecules to accumulate to become pre-existing gas cavities.

During plastic foaming, both the free volumes within the polymer matrix and the pre-existing gas

cavities hidden at the polymer-filler interfaces can serve as seeds for bubble formation. However,

practically, the activation energy for homogeneous nucleation is much higher than that of

heterogeneous nucleation. Therefore, in most cases, heterogeneous nucleation will be dominant.

The expansion of these pre-existing gas cavities depends on the dynamic change of Rcr during a

foaming process. At the beginning of the process, a rapid system pressure drop dramatically

increases ΔP, leading to a significant reduction in Rcr according to Equation (3.1). Once Rcr

becomes smaller than the radii of curvatures of the pre-existing cavities, these seeds are activated

and start to grow into cells in the polymer matrix. Upon bubble expansion, Rcr starts to increase

due to the gas depletion around the growing bubble. If Rcr becomes larger than the bubbles’ radii

55

(Rbub) due to extensive gas depletion, these bubbles will shrink and collapse. A schematic of the

dynamic change of Rcr and its relationship with Rbub is depicted in Figure 3.2.

3.3.3. Stress-Induced Nucleation

During plastic foaming, the growth and collapse of bubbles as well as the local flow of

polymer-gas solutions will likely result in local pressure fluctuation. A promoted nucleation near

the surface of an expanding bubble caused by the surface stretching has been reported by

Albalak et al. [193]. To account for the phenomenon, the expression for the degree of

supersaturation (ΔP) (i.e., Equation (3.4)) can be rewritten as:

( )= − +bub,cr sys localP P P PΔ Δ (3.8)

where ΔPlocal is the difference between Psys in the bulk and the actual pressure at the nucleating

site. If the local region experiences a compressive stress, ΔPlocal will be positive. In contrast, if

the local region is under an extensional stress, ΔPlocal will be negative. As a result, Equations

(3.1) through (3.3) become:

( )=− +

lgcr

bub,cr sys local

2R

P P Pγ

Δ (3.9)

( )( )

=− +

3lg

hom 2

bub,cr sys local

16W

3 P P P

γ

Δ (3.10)

( )( )( )

( )= =− +

3lg c

het hom c2

bub ,cr sys local

16 F ,W W F ,

3 P P P

γ θ βθ β

Δ (3.11)

According to Equations (3.9) through (3.11), if there is an extensional stress at the local region

(i.e., ΔPlocal < 0), Rcr, Whom, and Whet will all decrease. In contrast, if there is a compressive stress

at the local region (i.e., ΔPlocal > 0), these thermodynamic parameters will all increase. The CNT

56

argue that any local extensive stress will reduce the free energy barriers for both homogeneous

nucleation and heterogeneous nucleation, and the cell nucleation rate will be higher because of

the promotion of heterogeneous nucleation. On the other hand, considering that cells grow from

the pre-existing gas cavities at the polymer-filler interfaces, the increase in ΔP leads to a more

rapid decrease in Rcr, and an earlier activation of the pre-existing gas cavities to form cells.

Therefore, it is critical for academic and industrial researchers to explore the factors affecting the

local pressure field and learn how to control them.

3.4. Experimental

3.4.1. Materials

The polystyrene being used in all experiments was Styron 685D polystyrene with a

weight-average molecular weight of 315 000 g/mol (The Dow Chemical Co.). The manufacturer

reports that PS 685D has no specific nucleating agents were added to PS 685D. The three types

of talc particles, Cimpact 710, CB7, and Stellar 410 (Luzenac) used for the experimental work

differ in either mean particle size or surface treatment. The physical blowing agent used for the

foaming experiments was 99% pure CO2 (Linde Gas). The physical properties of the polymer,

talc particles, and blowing agent are summarized in Tables 3.1 to 3.3.

3.4.2. Sample Preparation Materials

3.4.2.1. Preparation of Polystyrene-Talc Composites

For each of the three types of talc particles, a 20 wt% of talc masterbatch of PS was

prepared using a C.W. Brabender 3-piece mixer. Each masterbatch was then diluted with pure PS

using the same batch mixer to produce PS-talc compounds with talc contents of 0.5 and 5.0 wt%.

For the pure PS used in various experiments, the resins, without the addition of talc particles,

57

were processed by the batch mixer following the aforementioned procedures in order to ensure

the same processing history as that experienced by the PS-talc compounds.

3.4.2.2. Sample Preparation for In-situ Visualization Experiments

A compression molding machine (Carver Inc.) equipped with a digital temperature

controller was used to prepare film samples of the PS and the PS-talc compounds. PS resins or

PS-talc compounds were hot compression-molded into 200 μm thick films using the press, which

was pre-heated to a temperature above the glass transition temperature of PS. The PS and PS-talc

films were then punched into disc-shaped samples of about 6 mm in diameter for the foaming

visualization experiments.

3.4.3. In-situ Foaming Visualization

The setup of the batch foaming visualization system [167], as illustrated in Figure 3.1,

was used to observe the in-situ foaming behaviors of the aforementioned polymer-blowing agent

system. The system consists of a high-pressure, high-temperature chamber, a pressure-drop rate

control system, a data acquisition system for pressure measurement (i.e., a data acquisition board

and a computer), a gas supply system (i.e., a gas tank, a syringe pump, and valves), and an

optical system (i.e., objective lens, a light source, and a high-speed CMOS camera). With a

maximum frame rate of 120,000 frames per second, the CMOS camera is capable of capturing

any fast foaming processes.

3.4.3.1. Experimental Procedures

The foaming experiments were performed according to the following steps:

STEP 1: The chamber loaded with a PS or PS-talc sample was charged with CO2 at the pre-

determined saturation pressure, while the chamber temperature was controlled using a

thermostat.

STEP 2: The pressure and temperature of the chamber were maintained at the set points for 30

58

minutes to allow the sample to be saturated with CO2.

STEP 3: CO2 was released by opening the solenoid valve. The pressure transducer and the

CMOS camera captured the pressure decay data and in-situ foaming data, respectively.

The opening of the solenoid valve and the data capturing were synchronized by two

computers and the data acquisition system.

3.4.3.2. Experiments to Study the Effects of -dPsys/dt, C, and Tsys on Cell Nucleation

A series of batching foaming experiments were performed using the aforementioned

experimental simulation system. The pressure drop rates, the gas contents and the processing

temperature were independently controlled by adjusting the opening size of the solenoid valve,

the pressure of the syringe pump, and the set-point temperature of the temperature controller,

respectively. Consequently, the effects of each parameter on cell nucleation were studied

systematically. Various PS-CO2 foaming experiments serve as case examples in this thesis

research. The processing conditions being considered are summarized in Tables 3.4 through 3.6.

3.4.3.3. Experiments to Study the Effects of Talc on Cell Nucleation

A number of experiments were conducted to explore the effect of talc on the cell

nucleation mechanism of the PS-CO2 system. Processing conditions and material parameters

were altered to take into account the effects of sizes, contents, and types of talc particles, the

blowing agent contents, as well as the processing temperatures on the nucleation mechanism.

Each experimental case was conducted three times to test the repeatability of the experimental

results. Table 3.7 shows the processing conditions for these experimental simulations.

3.4.4. Characterization

3.4.4.1. Effects of -dPsys/dt, C, and Tsys on Cell Nucleation

To analyze the effect of –dPsys/dt, C, and Tsys on foaming behaviors, the continuous

changes in cell density data were obtained from the visualization data. Hence, N(t), the number

59

of cells within a superimposed circular boundary with an area of Ac at time t was counted at each

time frame. The radius of 10 randomly selected bubbles at time t (i.e., Ri(t), where i = 1…10)

were also measured. The cell density with respect to the foamed volume, Nfoam(t), and the cell

density with respect to the unfoamed volume, Nunfoam(t), were calculated using the following

equations:

32

foamc

N( t )N ( t )A

⎛ ⎞= ⎜ ⎟⎝ ⎠

(3.9)

unfoam foamN ( t ) N ( t ) VER( t )= × (3.10)

3n

ifoam

i

R ( t )4VER( t ) 1 N ( t )3 n

⎛ ⎞⎛ ⎞= + ×⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠∑π (3.11)

where VER(t) is the volume expansion ration of the plastic foam at time t.

The data was collected between the initial time and the time at which no new bubbles

were formed to extract the cell density profiles with respect to time. It should be noted that the

smallest observable bubbles by the system depends on the used magnification. Under the highest

magnification (i.e., 450X), the smallest observable bubble was approximately 2 – 3 μm in

diameter. Therefore, there could be a time delay between the moment of bubble nucleation and

the time at which the bubbles were observed, and this delay depends on the magnification power

being used in the experiments.

3.4.4.2. Effects of Talc on Cell Nucleation

To investigate the effects of talc on the cell nucleation mechanism under different

processing conditions, the extracted micrographs obtained from the in-situ visualization of the

foaming experiments were compared. To gain insight on the effect of the surface treatment of

60

talc particles on their dispersion in the polymer matrix, the cross-sections of the PS-talc

composites were analyzed using scanning electron microscopy (SEM) to investigate the

distribution of talc particles along the thickness direction. All PS-talc samples were fractured in

liquid nitrogen and their cross-sections were studied using SEM (JEOL, model JSM-6060).

3.5. Results and Discussion

3.5.1. Effect of Processing Conditions on Cell Nucleation

3.5.1.1. Effects of Pressure Drop Rate on Cell Nucleation

The dependence of nucleation behavior on the -dPsys/dt is demonstrated in Figure 3.3.

Figures 3.4 (a) and (b) depict the pressure profiles and the cell density profiles as functions of

time, respectively. By dropping Psys more rapidly (i.e., from 6 MPa/s to 47 MPa/s), the maximum

cell density increased from 2.37 × 107 cells/cm3 to 1.01 × 109 cells/cm3. It can be observed that

cells formed at earlier times and that cell densities increased at higher rates as -dPsys/dt increased.

According to Equations (3.1) through (3.3), Rcr, Whom, and Whet are functions of Psys. As Psys

drops at a higher rate, all three thermodynamic functions decrease more rapidly. As a result, the

growth of pre-existing gas cavities would be promoted, and eventually increase the cell density.

These results are consistent with parametric studies of PS-CO2 foaming in an extrusion foaming

line conducted by Park et al. [16] and Xu et al. [34].

3.5.1.2. Effects of CO2 Content on Cell Nucleation

The in-situ visualization data of the PS-CO2 foaming with different CO2 contents (C) are

illustrated in Figure 3.5. The pressure profiles and the cell density profiles as functions of time

are indicated, respectively, in Figures 3.6 (a) and (b). By increasing the C from 4 wt% to 7 wt%,

the maximum cell density increased from 2.4 x 107 cells/cm3 to 3.2 x 109 cells/cm3. Since the

interfacial energy at the liquid-gas interface (i.e., γlg) decreases as C increases, Equations (3.1)

61

through (3.3) suggest that Rcr, Whom, and Whet will all decrease. Furthermore, a more significant

swelling of the PS melt is expected when a larger amount of CO2 content is dissolved. This in

turn increases the sizes of the pre-existing gas cavities throughout the polymer matrix. These will

lead to a higher activation rate of the pre-existing gas cavities for expansion. As a result, the cell

density of the plastic foam will be promoted.

3.5.1.3. Effects of Processing Temperature on Cell Nucleation

Figure 3.7 depicts the micrographs of the in-situ foaming behaviors of PS-CO2 foaming

at different Tsys. The pressure profiles and the analyzed cell density profiles as functions of time

are plotted in Figures 3.8 (a) and (b), respectively. As Tsys increased from 140°C to 200°C, the

maximum cell density decreased slightly from 1.01 × 109 cells/cm3 to 2.10 × 108 cells/cm3.

These figures show that the cell density increased at earlier times as Tsys increased. In theory, as

Tsys increases, the mobility of gas molecules and that of polymer chains also increases, which

lead to a higher gas diffusivity. Furthermore, the thermal fluctuation will be higher at elevated

temperatures and increase the sizes of pre-existing gas cavities, and thereby it will promote the

initial nucleation rate. Meanwhile, γlg decreases with increasing temperature, which lowers Rcr,

Whom, and Whet and hence further increases the activation rate of pre-existing gas cavities.

Despite the higher initial nucleation or cell activation rates at higher temperatures, a

slightly lower maximum observable cell density was observed. One reason for this could be the

accelerated gas diffusion at a higher Tsys, leading to a faster rate of cell growth and gas loss to the

surrounding. Together with the higher initial nucleation rate in such cases, as time progresses,

the overall nucleation rate decreases at a higher rate. This implies cell nucleation is sustained for

a shorter period of time at a higher Tsys, which is suspected to cause the slight reduction in the

maximum cell density.

3.5.2. Effect of Talc on Cell Nucleation

62

3.5.2.1. Effect of Talc Particles on Cell Nucleation Mechanism

A pure PS disk sample and a PS-talc composite disk sample, which consisted of 5 wt% of

talc loading (i.e., CIMPACT 710), were pressurized to 5.5 MPa at 180°C. After complete

saturation, 2.1 wt% of CO2 [194] was dissolved in each sample. These samples were then

foamed by depressurizing at a rate of about 4.7 MPa/s; Figures 3.9 (a) and (b) show the

micrographs of the in-situ foaming phenomena. With the presence of 5 wt% of untreated talc

particles, the onset time of cell nucleation was earlier, and the cell density increased

dramatically. These results indicate that talc serves as an effective heterogeneous nucleating

agent during the foaming process, which is consistent with previous studies [34, 86, 135-136,

148-149, 154-155]. According to the CNT, the presence of talc will lower the free energy barrier

to nucleate cells [88, 99, 107-111]. Therefore, a lower degree of supersaturation was required to

nucleate new cells, which leads to earlier cell nucleation. During the rapid pressure decay, Rcr

will continuously reduce, ultimately becoming smaller than the radii of the pre-existing CO2

cavities that are ubiquitous on the rugged surfaces of talc agglomerates [86]. Subsequently, these

cavities will be activated and begin to grow. The omnipresence of CO2 cavities, together with the

lower free energy barrier for cell nucleation, enhanced the cell formation and resulted in higher

cell density.

In addition to the typical effects of talc on cell nucleation, a counter-intuitive

phenomenon was observed in the visualization experiments. Figures 3.10 (a) and (b) indicate that

during the foaming of the PS-talc sample, new cells were generated near the previously

nucleated and growing cells despite the excessive gas depletion in these regions. This

phenomenon was less pronounced when the pure PS sample was foamed. Figure 3.11 illustrates

the schematics of the observed bubble formation phenomenon in the PS-talc sample. The in-situ

observation offers insight into the underlying mechanism of talc-enhanced nucleation, thus

63

permitting a more complete explanation of the mechanism by which inorganic fillers promote

heterogeneous cell nucleation.

As nucleated bubbles expand, their growth will induce tangential stretching actions on

their surfaces and generate flow fields in the surrounding polymer-gas solution. The cell growth

will reduce the cell wall thickness, which will, in turn, bring the aggregates, together with the

polymer-gas solution, towards the growing bubble. As indicated in Figure 3.12, the extension

flow will generate a local pressure fluctuation around the side surfaces of the talc agglomerate.

This local pressure field is tensile and will result in a negative ΔPlocal. Using Equations (3.9)

through (3.11), it can be deduced that Rcr, Whom, and Whet will all be reduced, promoting the

nucleation of new cells as well as the growth of pre-existing nuclei, and eventually increasing the

cell density of the plastic foam.

By clarifying the roles of talc particles in promoting cell formation, it is possible to

generalize a set of criteria for ideal nucleating agents, which include:

(1) Ideal nucleating agents should be easily dispersible to increase the surface area for the

formation of pre-existing gas cavities and development of local stress field.

(2) Ideal nucleating agents should consist of a lot of crevices to entrap pre-existing gas

cavities.

(3) There should be an optimal compatibility between the nucleating agent and the polymer

melt in order to maximize both the dispersion of particles and the ability to entrap pre-

existing gas cavities.

3.5.2.2. Effect of Talc Content on Cell Nucleation Mechanism in PS-Talc-CO2 Foaming

By varying the talc (i.e., CIMPACT 710) loading from 0.5 to 5.0 wt%, a series of batch

foaming experiments were conducted to investigate the effect of talc content on the cell

nucleation phenomena. In the experiments, all PS-talc composite samples were foamed with CO2

64

at 180°C and a pressure drop rate of about 2.0 MPa/s. Since previous studies revealed that

increasing the talc content would enhance cell formation to different extents as the dissolved CO2

content varied [34, 154], the effect of talc content on the PS-talc foaming behaviors were

examined under two CO2 concentrations, 2.3 and 4.0 wt%, which correspond to Psat of 6.2 MPa

and 10.3 MPa, respectively [194]. Figures 3.13 (a) and (b) show that at a low CO2 concentration

(i.e., 2.3 wt%), increasing the talc content from 0.5 to 5.0 wt% advanced the onset time of cell

nucleation and increased the cell density. Most importantly, Figures 3.14 (a) and (b) provide

further evidence of the enhancement of bubble formation around the expanding bubbles,

especially when talc particles are present. With a higher talc content (Figure 3.14 (b)), the

number of tiny bubbles that formed around the expanding cells increased significantly. These

results suggest that the extensional stress-induced nucleation becomes more pronounced as talc

content increases.

As Figures 3.15 (a) and (b) illustrate, when the dissolved CO2 content was increased to

4.0 wt%, the onset time of cell formation only slightly advanced, while cell density was virtually

invariant when the talc content was increased from 0.5 to 5.0 wt%. These results, consistent with

the results of earlier studies using extrusion foaming [34, 154], reveal that the effect of higher

talc content on cell density of PS-CO2 foaming is negligible when the CO2 content is high. A

more detailed explanation will be provided in the next section in a discussion of the effect of gas

content on the phenomena.

3.5.2.3. Effect of Gas Content on Cell Nucleation Mechanism in PS-Talc-CO2 Foaming

Regardless of the talc content, the in-situ visualization data shown in Figures 3.13 (a) and

3.15 (a), as well as in Figures 3.13 (b) and 3.15 (b), show that the onset time of bubble formation

advanced and that the cell density increased significantly when the CO2 content increased from

2.3 wt% to 4.0 wt%. On the one hand, the higher CO2 content increased the abundance of CO2

65

molecules to form gas cavities. On the other hand, it also reduced γlg [189-190] and resulted in

decreases in Rcr, Whom, and Whet. These consequences, in turn, increased the nucleation rate of

new cells and enhanced the growth activation of the pre-existing cavities. Furthermore, as shown

in Figures 3.15 (a) and (b), there was virtually no trace of the extensional-stress induced

nucleation in either samples foamed at the higher CO2 concentration, suggesting that the pressure

fluctuation may not be significant when the CO2 content is high. Dissolving a larger amount of

CO2 in the PS promotes the plasticizing effect and reduces the viscosity and elasticity of the PS-

CO2 solution. In other words, the mobility of the fluid increases. It is speculated that the increase

in the fluidity of the PS-CO2 solution will reduce the potential extensional stress field that will be

established near the PS-talc interface. This will suppress the induction of the negative ΔPlocal

around the talc aggregates and lead to the lack of noticeable effects of increasing talc content on

cell density. Nonetheless, additional theoretical studies on the effect of viscosity and elasticity of

the polymer-gas mixture on stress distribution around the inorganic filler particles are needed to

verify this speculation.

3.5.2.4. Effect of Surface Treatment of Talc on Cell Nucleation Mechanism in PS-Talc-CO2

Foaming

The foaming behavior of PS-talc samples with surface-treated talc particles (i.e., CB7)

was studied and compared with that of PS-talc samples with talc particles without surface

treatment (i.e., CIMPACT 710) to examine the effect of surface treatment on the cell nucleation

mechanism. Both samples consisted of 5 wt% of talc and saturated with 2.3 wt% of CO2 at

180°C. The depressurization rates for both experiments were both about 2 MPa/s. Figures 3.17

(a) and (b) illustrate the micrographs of the in-situ foaming processes of the samples with the

untreated talc and the surface-treated talc. Comparing the visualization data, both the onset time

of cell nucleation and the cell density were virtually invariant in the two cases. Moreover, as

66

Figure 3.18 (a) and (b) indicate, in both cases the growth of nucleated cells promoted the

nucleation of new cells around them. However, the phenomenon was slightly less pronounced

than with the surface-treated talc. Since the surface treatment is believed to affect the

compatibility and thereby the bonding between PS and talc, it is surmised that the treatment has

an effect on the establishment of the stress field.

Unfortunately, the type of surface treatment was considered as confidential information

by the supplier, and was not available to this research. Therefore, a series of scanning electron

microscopy (SEM) pictures were taken of the unfoamed PS-talc samples to analyze the effects of

the surface treatment on the size distribution of talc particles. A sample SEM micrograph is

illustrated in Figure 3.19. The distributions of the talc particle sizes (i.e., the length of the largest

dimension) for varying loadings and surface treatments are illustrated in Figures 3.20 (a) through

(d). The analyzed data reveal that the surface treatment on the talc particles improved the

compatibility between PS and talc. The results were consistent with those of Alonso et al. [195].

As a result, it enhanced the dispersion of talc particles within the PS matrix and resulted in a

smaller mean particle size and lower standard deviation. It is speculated that the improved

compatibility between PS and talc particles will lead to a smaller amount of trapped CO2

cavities. Moreover, the increased compatibility between the polymer and the talc particles will

increase the wettability of the polymer on the talc surface, thereby reducing the size of θc and

increase Whet. In sum, although the improved dispersion of talc particles due to their surface

treatment increased the total surface area of the PS-talc interfaces, the stress-enhanced nucleation

of new cells by the expanding bubbles became less pronounced.

3.5.2.5. Effect of Talc’s Particle Size on Cell Nucleation Mechanism in PS-Talc-CO2 Foaming

The effect of talc particle size on the cell nucleation was investigated by observing the

foaming behavior of a PS-talc sample with larger talc particles (i.e., STELLAR 410) and

67

comparing the observed foaming phenomena to those obtained with smaller talc particles. Both

samples used 5 wt% of talc loadings, and they were saturated with 2.3 wt% of CO2 at 180°C

before depressurizing. The pressure drop rate for both was about 2 MPa/s. Figures 3.21 (a) and

(b) illustrate the micrographs of the in-situ foaming process of the sample with the larger talc

particles. A comparison of the extracted micrographs of both experiments (i.e., Figures 3.17 (a)

and 3.18 (a)) shows that the onset time of cell nucleation and the cell density were virtually

unaffected by the mean size of talc particles. Since the larger particles will create a smaller

surface area of the PS-talc interface as well as a smaller number of particles, cell nucleation

seems to be suppressed with the use of larger talc particles (see Equation (3.7)). But, the higher

degree of local pressure variation (i.e., ΔPlocal in Equation (3.11)) would enhance cell nucleation

with a larger particle size. Hence, the particle size would not influence the cell density. Further

studies will be needed to clarify the effects of talc particle size on local stress distribution and the

cell nucleation mechanism.

3.5.2.6. Effect of Processing Temperature on Cell Nucleation Mechanism in PS-Talc-CO2

Foaming

In order to explore the effect of the processing temperature on the heterogeneous

nucleation mechanism induced by the presence of inorganic fillers, a PS-talc sample with 5 wt%

of talc loading (i.e., CIMPACT 710) was saturated with 2.1 wt% of CO2 at a pressure of 5.0 MPa

[194] and set at a lower system temperature (i.e., 140°C). The experimental results were

compared with those obtained at a system temperature of 180°C (i.e., Figures 3.9 (b) and 3.10

(b)). Both experiments were conducted by depressurizing the chamber at a rate of about 4.7

MPa/s. Figures 3.22 (a) illustrates the in-situ foaming phenomena at the lower processing

temperature. Consistent with previous studies [196-197], the onset time of cell nucleation was

delayed and cell density was increased by lowering the foaming temperature. The later onset

68

time of cell nucleation at the lower processing temperature was a result of the higher free energy

barrier and the larger Rcr caused by the higher γlg. Meanwhile, the lower gas diffusivity

suppressed excessive cell growth. Results showed that a larger portion of gas contributed to the

nucleation of new cells and lead to a higher cell density. Furthermore, Figure 3.22 (b), when

compared to Figure 3.10 (b), clearly indicates that the number of smaller cells formed around the

expanding cells increased substantially when the system temperature was reduced from 180°C to

140°C. Reduction of the processing temperature will increase both the viscosity and the elasticity

of the polymer-gas solution. It is speculated that the restricted fluid mobility would enhance the

establishment of extensional stress (i.e., ΔPlocal < 0) on the side surfaces of the talc aggregates,

and thereby Rcr, Whom, and Whet would be reduced. If this is the case, the extensional stress-

induced nucleation around the expanding bubbles becomes more significant at lower

temperatures. However, a further investigation will be helpful to verify the relationships between

viscosity, elasticity, and the stress distribution around the polymer-filler interfaces during an

extensional flow.

3.6. Summary and Conclusions

The study being presented in this chapter illustrates the mechanisms under which the

pressure drop rate, the dissolved gas content, and the processing temperature affect polymeric

foaming behaviors. Through the effects of the processing conditions on various material

parameters (i.e., the interfacial energy, the diffusivity, the viscosity, and the relaxation time) and

the thermodynamic instability, a higher pressure drop rate and a higher dissolved gas content

promoted the cell density significantly. Meanwhile, the maximum cell density of the plastic foam

decreased slightly at higher temperatures, which could be due to a combined effect of the higher

gas depletion rate caused by the accelerated cell growth and the higher initial nucleation rate.

69

Through the elucidation of the effects of various processing conditions on cell nucleation, it will

enhance the development of processing strategies to optimize the foam morphology.

Besides the optimization of the processing conditions, the addition of cell nucleating

agents in polymer is a common way to enhance cell nucleation in foaming plastics. It is widely

believed that enhancement resulted from a lower free energy barrier through heterogeneous

nucleation. From these experimental results, it is deduced that the heterogeneous nucleation

mechanism, with the presence of nucleating agents, is closely related to local pressure

fluctuations around the polymer-nucleating agent interface. It is speculated that in the presence

of talc particles, the growing cells will generate extensional stress fields on the side surfaces of

the polymer-talc interfaces. An extensional stress field will result in a further pressure reduction

and a higher degree of supersaturation in the local regions. Consequently, Rcr, Whom, and Whet for

bubble nucleation will decrease, promoting the nucleation of new cells, or the growth of pre-

existing cavities. At a higher gas content, the reduction of viscosity and elasticity of the polymer-

gas solution may weaken the extensional stress field being generated and suppress additional

reduction of the local pressure. If this is the case, it may provide an explanation to the limited

impact of increasing talc content on the cell density when a higher CO2 content [34, 154].

Finally, the study observed that the lower processing temperature and higher talc content

promoted stress-induced nucleation, whereas improved PS-talc compatibility slightly reduced it.

However, the size of talc seems to have no significant effect on the phenomenon.

70

1Table 3.1. Physical Properties of Polystyrene

PS685D

MFI 1.5g/10 min Mn 120,000

Mw/Mn 2.6 Specific gravity 1.04 Glass transition temperature (Tg)

108°C

2Table 3.2. Physical Properties of Talc Particles

Name of Talc

Mean Size (μm)

Surface Treatment

Cimpact 710 1.7 No CB7 1.8 Yes

Stellar 410 10.0 No

3Table 3.3. Physical Properties of the Blowing Agent

Carbon Dioxide

Chemical formula CO2

Molecular weight 44.01 g/mol Boiling point -78.45 °C Critical temperature 31.05 °C Critical pressure 7.38 MPa

71

4Table 3.4. Processing conditions to study the effect of pressure drop rate in PS-CO2 foaming (Tsys = 140˚C and C = 5.0 wt.%)

Gas Content (C) [wt.%] / Psat [MPa]

Pressure Drop Rate (-dPsys/dt|max) [MPa/s]

System Temperature (Tsys) [˚C]

5.0 % / 12.1 6 140 5.0 % / 12.1 22 140 5.0 % / 12.1 32 140 5.0 % / 12.1 47 140

5Table 3.5. Processing conditions to study the effect of dissolved CO2 content in PS-CO2 foaming (Tsys = 140˚C and –dP/dt|max = 22 MPa/s)




4.0 % / 9.71 22 140 5.0 % / 12.1 22 140 6.0 % / 14.7 22 140 7.0% / 16.8 22 140

6Table 3.6. Processing conditions to study the effect of system temperature in PS-CO2 foaming (–dP/dt|max = 47 MPa/s and C0 = 5.0 wt%)




5.0 % / 12.1 47 140 5.0 % / 12.5 47 160 5.0 % / 12.9 47 180 5.0 % / 13.4 47 200

72

7Table 3.7. Processing conditions to study the effect of various processing conditions in PS-talc-CO2 foaming

Talc Size (μm) / Content (wt.%)

Surface-Treated (Y/N)

CO2 Content (wt %) / Psat (psi)

Tsys (°C)

0.0% N 2.1 / 800 180 1.7 / 0.5% N 2.1 / 800 180 1.7 /5.0% N 2.1 / 800 180

0.0% N 4.0 / 1500 180 1.7 / 0.5% N 4.0 / 1500 180 1.7 /5.0% N 4.0 / 1500 180 1.7 / 5.0% N 2.3 / 900 180 1.8 / 5.0% Y 2.3 / 900 180 10.0 / 5.0% N 2.3 / 900 180 1.7 /5.0% N 723 / 2.1 140

73

12Figure 3.1. The batch foaming visualization system [167]

13Figure 3.2. A schematic of the dynamic change of Rcr and its relationship with Rbub

74

14Figure 3.3. Micrographs of PS-CO2 foaming at different pressure drop rates (Tsys = 140˚C and C = 5.0 wt%)

(a) Pressure drop profiles (b) Cell density profiles

15Figure 3.4. Effect of pressure drop rate on PS-CO2 foaming: (a) pressure drop profiles & (b) cell density profiles

75

16Figure 3.5. Micrographs of PS-CO2 foaming at different CO2 contents (Tsys = 140˚C & -dPsys/dt|max = 22 MPa/s)


17Figure 3.6. Effect of dissolved gas content on PS-CO2 foaming: (a) pressure drop profiles & (b) cell density profiles

76

18Figure 3.7. Micrographs of PS-CO2 foaming at processing temperatures (–dP/dt|max = 47 MPa/s and C = 5.0 wt.%)


19Figure 3.8. Effect of processing temperature on PS-CO2 foaming: (a) pressure drop profiles & (b) cell density profiles

77

(a)

(b)

20Figure 3.9. Micrographs of PS foaming with 2.1 wt% CO2 at 180°C: (a) pure PS and (b) PS + 5 wt% talc (CIMPACT 710)

(a) (b) 21Figure 3.10. Micrographs of PS foaming with 2.1 wt% CO2 at 180°C: (a) pure PS at 2.20 s

and (b) PS + 5 wt% talc (CIMPACT 710) at 1.56 s

78

22Figure 3.11. Schematics of the bubble formation phenomena

23Figure 3.12. A schematic of the extensional stress field around the talc agglomerate induced by the expanding bubble

79

(a)

(b)

24Figure 3.13. Micrographs of PS with 2.3 wt% CO2 at 180°C: (a) PS + 0.5 wt% talc (CIMPACT 710) and (b) PS + 5 wt% talc (CIMPACT 710)

(a) (b) 25Figure 3.14. Micrographs of PS foaming with 2.3 wt% CO2 at 180°C: (a) PS + 0.5 wt% talc

(CIMPACT 710) at 3.20 s and (b) PS + 5 wt% talc (CIMPACT 710) at 2.90 s

80

(a)

(b)

26Figure 3.15. Micrographs of PS with 4.0 wt% CO2 at 180°C: (a) PS + 0.5 wt% talc (CIMPACT 710) and (b) PS + 5 wt% talc (CIMPACT 710)

(a) (b) 27Figure 3.16. Micrographs of PS foaming with 4.0 wt% CO2 at 180°C: (a) PS + 0.5 wt% talc

(CIMPACT 710) at 2.40 s and (b) PS + 5 wt% talc (CIMPACT 710) at 2.40 s

81

(a)

(b)

28Figure 3.17. Micrographs of PS + 5.0 wt% talc with 2.3 wt% CO2 at 180°C: (a) CIMPACT

710 (untreated) and (b) CB7 (treated)

(a) (b)

29Figure 3.18. Micrographs of PS + 5.0 wt% talc with 2.3 wt% CO2 at 180°C: (a) CIMPACT 710 at 2.90 s; (b) CB7 at 2.90 s

82

30Figure 3.19. A SEM micrograph of PS + 5 wt% talc (CB7)

(a) mean size = 3.10 μm & std. dev. = 1.55 μm (b) mean size = 1.76 μm & std. dev. = 1.00 μm

(c) mean size = 3.10 μm & std. dev. = 1.55 μm (d) mean size = 2.55 μm & std. dev. = 1.20 μm

31Figure 3.20. Distribution of talc particle sizes in PS-talc composites: (a) 0.5 wt% of untreated talc; (b) 5.0 wt% of untreated talc; (c) 0.5 wt% of surface-treated talc; and (d) 5.0 wt%

of surface treated talc

83

(a)

(b)

32Figure 3.21. Micrographs of PS + 5.0 wt% talc (STELLAR 410) with 2.3 wt% CO2 at 180°C: (a) until 2.96 s; (b) at 2.82 s

84

(a)

(b)

33Figure 3.22. Micrographs of PS + 5.0 wt% talc (CIMPACT 710) with 2.1 wt% CO2 at 140°C: (a) until 2.10 s; (b) at 1.900 s

85

Chapter 4 BUBBLE GROWTH PHENOMENA

IN PLASTIC FOAMING Reproduced in part with permission from “Leung, S.N., Park, C.B., Xu, D., Li, H. and Fenton, R.G., Computer

Simulation of Bubble-Growth Phenomena in Foaming, Industrial and Engineering Chemistry Research, Vol. 45, pp. 7823-7831, 2006.” Copyright 2006 American Chemical Society

4.1. Introduction

This chapter discusses a research conducted to achieve accurate bubble growth model and

simulation scheme to describe precisely the bubble growth phenomena that occur in polymeric

foaming. Using the accurately measured thermo-physical and rheological properties of polymer-

gas mixtures (i.e. the solubility, the diffusivity, the surface tension, the viscosity, and the

relaxation time) as the inputs for computer simulation, the growth profiles for bubbles nucleated

at different times were predicted and carefully compared to experimentally observed data

obtained from batch foaming simulation with online visualization (See Figure 3.1) [167].

Furthermore, a series of sensitivity analyses are presented to reveal the effects of the

aforementioned thermo-physical and rheological parameters on the cell growth dynamics. A

86

polystyrene-carbon dioxide (PS-CO2) system is used herein as a case example. The model being

established will allow us to thoroughly depict the growth behaviors of bubble nucleated at

varying processing conditions. The developed software will also serve as an important

component to simulate the overall foaming phenomena in chapter 6.

4.2. Modeling of Bubble Growth Dynamics

Bubble growth phenomena in polymer foaming involve a large number of bubbles

expanding in close proximity to each other in a polymer-gas solution. The well-known cell

model [137] is recognized as an appropriate model to describe such a situation. During plastic

foaming under isothermal conditions, bubble growth involves both mass transfer and momentum

transfer between the nucleated bubbles and their surrounding polymer-gas solution. Moreover,

polymer melts such as polystyrene (PS) melt, for example, are known to be viscoelastic.

Therefore, to determine the underlying physics that characterize the bubble growth dynamics, it

is necessary to simultaneously solve the continuity equation, momentum equation, constitutive

equations, and the diffusion equation subjecting to appropriate initial and boundary conditions.

4.2.1. Simulation Model and Assumptions

The cell model was used as the base model to simulate the bubble growth dynamics in

PS-CO2 foaming. The model assumes that a shell of a viscoelastic fluid with finite volume and a

limited amount of gas surrounds each bubble. A schematic of a nucleated bubble and its

corresponding polymer-gas solution shell is shown in Figure 2.7 in chapter 2. To implement the

cell model in the simulation algorithm to study the bubble expansion process, the following

assumptions are made:

(1) The bubble is spherically symmetric throughout the bubble growth process.

(2) The polymer-gas solution is incompressible.

87

(3) The initial bubble pressure can be determined by the thermodynamic equilibrium

condition (i.e., μg(Pbub, T) = μg,sol(Psys, T, CR)), where the chemical potential of the

gas in the gas bubble (i.e., μg) and that of the gas in the polymer-gas solution (i.e.,

μg,sol) can be determined by appropriate equations of state.

(4) The inertial forces and the effect of gravity on bubble growth are negligible.

(5) The pressure at the outer boundary of the shell at time t is equal to the applied

system pressure (Psys(t)) at that moment.

(6) The accumulation of the adsorbed gas molecules on the bubble surface is negligible.

(7) The gas inside the bubble obeys the ideal gas law.

(8) The diffusivity of the gas in the polymer-gas solution is constant.

(9) The bubble pressure can be related to the dissolved gas concentration at the

polymer-gas solution interface using Henry’s Law:

R sysbub

sat

C ( t ,t ')P ( t )P ( t ,t ')

C ( t )= (4.1)

where Csat is the saturated gas concentration at Psys.

(10) The bubble growth process is isothermal.

(11) The initial accumulated stress in the polymer-gas solution around the growing

bubble is zero.

4.2.2. Mathematical Formulations

The bubble growth dynamics can be analyzed by simultaneously solving the governing

equations for both the mass transfer and the momentum transfer that occur between the nucleated

bubbles and the surrounding polymer-gas solution in the spherical coordinate system. The

corresponding governing equations are stated as Equations (2.28) through (2.30) in chapter 2,

and they are restated here as Equations (4.2) through (4.4) [132]:

88

R ( t ,t ')shell

lg rr θθbub sys

R ( t ,t ')bubbub

2γ τ τP ( t ,t ') P ( t ) 2 dr 0R r

−− − + =∫ (4.2)

( ) 3

bub bub 2bub

r Rg sys bub

P t ,t ' R ( t ,t ')d 4π C( r,t ,t ')4πR ( t ,t ') Ddt 3 R T r =

⎛ ⎞ ∂=⎜ ⎟⎜ ⎟ ∂⎝ ⎠

(4.3)

2

2bub bubbub2 2

R RC C D C Cr for r Rt r r r r r

•

∂ ∂ ∂ ∂⎛ ⎞+ = ≥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (4.4)

The quasi-linear Upper-Convected Maxwell model [138] was employed to describe the

viscoelastic nature of the PS-CO2 solution. Using the Lagrangian coordinate transformation, the

constitutive equations that characterize the viscoelastic fluid can be reduced to the first-order

ordinary differential equations [138]:

2 2

bub bub bub bubrrrr3 3

bub bub

4R R R Rdτ 1 4ητdt λ y R λ y R

⎛ ⎞= − + −⎜ ⎟+ +⎝ ⎠

(4.5)

2 2

θθ bub bub bub bubθθ3 3

bub bub

dτ 2R R R R1 2ητdt λ y R λ y R

⎛ ⎞= − − +⎜ ⎟+ +⎝ ⎠

(4.6)

where λ is the relaxation time of the polymer-gas solution, η is the viscosity, and y is the

transformed Lagrangian coordinate, which is,

3 3buby r R ( t ,t ')= − (4.7)

Using the Lagrangian coordinate transformation, the momentum equation (i.e., Equation (4.2))

can be rewritten as:

( )

3 3R Rshell bublg rr θθ

bub sys 30 bub

2γ τ τP ( t ,t ') P ( t ) 2 dy 0R 3 y R

− −− − + =∫

+ (4.8)

Equation (4.4) is subjected to the initial and boundary conditions that are given by Equation (4.1)

and the following:

( ) 0C r,t ,t ' C for t t '= = (4.9)

89

shellC( R ,t ,t ') 0 for t t 'r

∂= ≥

∂ (4.10)

where C0 is the initial dissolved gas concentration in the polymer-gas solution. Equations (4.1)

through (4.10) constitute a complete set of equations that describe the bubble growth dynamics.

4.2.3. Methodology of Computer Simulation

The system of governing equations used to describe bubble growth dynamics are highly

nonlinear and coupled. This study used a numerical simulation algorithm that integrates the 4th

order Runge-Kutta method and the explicit finite difference scheme to solve them and thereby

simulate the cell growth dynamics. Figure 4.1 illustrate a flowchart of the simulation algorithm.

The finite difference scheme was found to converge when employing 100 or more mesh points.

Therefore, 100 mesh points were used to simulate the bubble growth phenomena.

4.2.4. Determination of Physical Parameters for Computer Simulation

The accurate measurements for the required thermo-physical and rheological properties,

such as the solubility, diffusivity, surface tension, viscosity, and the relaxation time of the

polymer-gas solution are critical to verify the validity of the computer simulation of the bubble

growth behaviors. For the PS-CO2 system considered in this study, the experimentally measured

values of the corresponding parameters are summarized in Table 4.1. Because the relaxation time

for a PS-CO2 system was not available, it was approximated by that of pure PS [198-199]. The

effect of this approximation on the simulation result was studied through a sensitivity analysis

and is discussed in the later section. The system pressure (Psys) and the temperature (Tsys) were

measured by a pressure transducer and a thermocouple, respectively. The initial bubble radius

(Rbub(t’,t’)) was assumed to be 1% larger than the critical radius:

( ) lgbub

bub ,cr sys

2γR t',t ' 1.01

P P= ×

− (4.11)

90

Finally, the initial shell radius, Rshell,t=t’, was estimated from the local cell density data around the

particular bubble obtained in the experimental foaming simulation system. The local cell

densities were determined from the micrographs using the following equation:

32

b ,unfoamed3

i

n 1N 4A 1 πR3

⎛ ⎞= ×⎜ ⎟⎝ ⎠ −∑

(4.12)

where Nb,unfoamed is the cell density with respect to the unfoamed polymer volume; n is the

number of bubbles within the local area, A, being considered; and Ri is the radius of the ith

bubble. Hence, the initial shell radii (Rshell,t=t’) can be determined by Equation (4.13).

13

3shell ,t t ' 0

b ,unfoamed

3 1 4R πR4π N 3=

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(4.13)

4.3. Experimental Verification

In order to verify the computer simulation, the predicted bubble growth profiles for

bubbles nucleated at different times were carefully compared with the in-situ visualization data.

4.3.1. Materials

The polymer used for the foaming experiments was PS (product name: PS101; NOVA

Chemical Inc.). The MFRs for this material is 2.2 g/10 min; and the specific gravity is 1.04. The

physical blowing agent employed was 99% pure carbon dioxide (CO2) (BOC Canada Ltd.).

4.3.2. Experimental Apparatus and Procedures

The setup of the batch foaming simulation system is illustrated as Figure 3.1 in the

previous chapter. The experimental procedures have been detailed in Section 3.4.3.1.


4.4.1. Experimental Results

91

Figure 4.2 shows a series of visualization images captured by the batch foaming

experimental system, which provided the cell density and the cell growth data at a given time.

Figure 4.3 graphed the experimentally-measured growth profiles of the four bubbles indicated in

Figure 4.2. The cell growth behaviors of these four bubbles were simulated and compared with

the in-situ visualization data.

4.4.2. Determination of Physical Parameters for Computer Simulation

The pressure profile recorded during the batch foaming experiment at 180˚C is illustrated

in Figure 4.4. This was fed into the simulation program to reflect the actual processing condition.

By estimating the nucleation time of bubbles 1 through 4 to be 0.038, 0.243, 0.321, and 0.440 s,

respectively, and using Equation (4.11), the values of Rbub(t’,t’) for these bubbles were

determined to be 0.181, 0.014, 0.014, and 0.013 µm, respectively. The local cell densities

bubbles 1 to 4 were measured to be 2.2 × 106, 3.5 × 106, 3.0 × 106 and 5.7 × 106 cells/cm3. Using

Equation (4.13), the corresponding values of Rshell,t=t’ for the four bubbles were determined to be

47.70, 40.86, 43.01, and 34.73 μm, respectively.

4.4.3. Computer Simulation and Comparison with Experimental Results

Figure 4.5 compares the simulated bubble growth profiles of the four bubbles with the

experimentally observed results up to 1.0 s. Since the bubble-to-bubble interactions become

significant after 0.6 s for bubble 2, the simulated profile starts to overestimate the actual size.

Nevertheless, it seems that the simulation program can precisely predict the bubble growth

behaviors before bubble-to-bubble interaction becomes significant. It is believed that the

simulation model precisely accounts for most of the underlying physics that describes the cell

growth dynamics. It can also be observed that the bubble growth profile is concave upward (i.e.,

bubble growth rate is increasing) at the very beginning moment of the process and becomes

concave downward (i.e., bubble growth rate is decreasing) thereafter. Since the retarding force

92

contributed by the surface tension is very large for a very small bubble and it continuously

reduces as the bubble grows, the bubble growth rate increases initially. This trend is more

pronounced for bubbles nucleated at an earlier time (e.g., bubble 1) because of the slower initial

growth rate due to the higher Psys. During the later stage, the reduced concentration gradient

around the bubble becomes significant and the retarding force due to the surface tension becomes

negligible, leading to the continuous reduction of the bubble growth rate.

4.5. Sensitivity Analyses

Finally, a series of sensitivity analyses was performed with bubble 1 to study the effects

of various simulation variables, as well as thermo-physical and rheological parameters on the

bubble growth phenomena. The results are illustrated in Figures 4.6 through 4.13.

4.5.1. Effect of Initial Bubble Radius Experimental Results

Figure 4.6 shows the effect of varying the initial bubble radii (Rbub(t’,t’)), from 0.179 μm

to 17.9 μm, on the simulation result. For the bubbles with smaller Rbub(t’,t’), the bubble pressures

during the initial growth process are higher, leading to a higher initial growth rate. Furthermore,

the simulated growth profiles for bubbles with Rbub(t’,t’) smaller than 1.79 μm are virtually

indistinguishable from each other. Since the critical radius is believed to be in the submicron

scale, the assumption that Rbub(t’,t’) is 1% larger than the critical radius (i.e., Rbub(t’,t’) = 0.181

μm) seems to be acceptable.

4.5.2. Effect of Initial Shell Radius (Rshell,t=t’)

Figure 4.7 shows the effect of varying the initial shell radius (Rshell,t=t’) between 28.8 μm

and 78.2 μm on the simulation. All four curves are overlapping at the early growing stage

because the gas concentrations around the cells are virtually the same until the later stage of

growth. A larger Rshell,t=t’ (i.e., a lower local cell density) will lead to a larger bubble because of

93

the higher gas content in the individual cell. In contrast, a smaller Rshell,t=t’ (i.e., a higher local cell

density) will result in a smaller bubble since the later growing stage is limited by the lower gas

content. The results also demonstrate that the average cell sizes of the foam products are smaller

when the cell density is higher.

4.5.3. Effect of Diffusivity (D)

The effect of diffusivity (D) on the bubble growth profiles was studied by varying its

value over a range of 5.0 × 10-10 m2/s to 6.0 × 10-9 m2/s; the simulation results are shown in

Figure 4.8. A higher D means faster diffusion rate of gas into the bubbles, resulting in higher cell

growth rate. For the cases of D = 4.0 × 10-9 m2/s and 6.0 × 10-9 m2/s, the bubble growth profiles

are virtually indistinguishable after 0.7 s because of the gas depletion within the shells.

4.5.4. Effect of Solubility (KH)

Figure 4.9 demonstrates the effect of solubility on the predicted bubble-growth behavior.

This was studied by varying the value of KH, which is the ratio of the dissolved gas content (i.e.,

in mol/m3) to the corresponding saturation pressure (i.e., in Pa), between 5.0 × 10-6 mol/N-m and

1.0 × 10-4 mol/N-m. A larger KH means that the polymer has higher gas solubility. Thus, the gas

content in the individual cell will be higher, increasing the bubble growth rate and the final cell

size. In contrast, a smaller KH (i.e., lower gas content) will lead to slower cell growth and a

smaller final bubble size.

4.5.5. Effect of Surface Tension (γlg)

The effect of surface tension (γlg) on the predicted bubble-growth profiles was

investigated by varying its value from 1.0 dynes/cm to 100 dynes/cm. Figures 4.10 (a) and (b)

show that a larger γlg suppress the initial cell growth rate because of the greater retarding force.

According to Equation (4.11), a larger γlg will also yield a larger Rbub(t’,t’) and hence a lower

94

initial bubble pressure; this ultimately leads to slower bubble growth. However, as the bubble

grows, the effect of γlg becomes less significant. Therefore, the overall bubble growth profiles are

insensitive to the changes in γlg.

4.5.6. Effect of Relaxation Time (λ)

The relaxation time (λ) is a characteristic parameter used to describe the viscoelastic

nature of a polymer melt. Physically, a longer λ (i.e., higher elasticity) means slower relaxation

and accumulation of stress around the growing bubble [126, 200]. Since our simulations were

focused on the initial bubble growth process, it is expected that a shorter λ will lead to a smaller

bubble because of the faster stress accumulation. Figures 4.11 (a) and (b) indicate that a shorter λ

(e.g., 0.1 s) results in a slightly slower bubble growth. Nevertheless, for a longer λ, the effect of

λ on the bubble growth was found to be negligible. It is noted that the bubble growth rate was not

affected significantly as the λ decreased to 0.1, whereas the value of λ for a pure PS melt is 27.0

s. However, further study would be required to determine the value of λ of a PS-CO2 solution.

4.5.7. Effect of Zero-Shear Viscosity (η0)

The zero-shear viscosity (η0), which relates to another source of retarding force that affect

cell growth dynamics, depends on the temperature, the pressure, and the gas content [201].

Figure 4.12 (a) illustrates the effects of η0 on the simulation when varying η0 from 100.0 N/m2-s

to 1.0 x 106 N/m2-s. At higher η0, the bubble grows more slowly as expected, but the effect of η0

is not pronounced. The low sensitivity of the simulation results on the changes of η0 is due to the

long relaxation time (i.e., λ = 27.0 sec.), which means a slower accumulation of retarding force,

used in the simulation. In order to illustrate the sensitivity of the predicted bubble growth

behavior for a less elastic fluid (i.e., a shorter λ), another set of simulations were performed by

setting λ to 0.1 second. Figure 4.12 (b) shows that a lower elasticity will increase the sensitivity

of the simulated bubble growth profile to the changes of η0. The overlapping of the initial growth

95

profiles (i.e., 0.0 to 0.1 s) shows that there is a delay of the stress accumulation due to the elastic

behavior of the polymer-gas solution. By reducing η0, the cell growth rate will increase.

Therefore, it is possible to tailor the cellular structures as well as the volume expansion ratios

through the control of η0 by adjusting the processing conditions or choosing the appropriate

materials.


Using the in-situ visualization data obtained from the experimental batch foaming

simulation system, the established mathematical model and simulation algorithm that describe

bubble growth dynamics have been verified. By carefully comparing the simulation results with

the experimentally observed data, it has been shown that the simulated growth profiles for

bubbles nucleated at different times can predict with precision the observed bubble growth

behaviors for different processing conditions. Therefore, it appears that the simulation program

accurately accounts for most of the physics that characterize bubble growth dynamics and can

thus serve as a powerful strategic tool for predicting bubble growth behavior during the early

stages of the polymeric foaming process (i.e., bubble-to-bubble interactions are negligible).

Finally, the established mathematical and simulation models have allowed for sensitivity

analyses to be performed to investigate the effect of each thermo-physical, rheological,

processing, and simulation parameters on bubble growth simulation. This developed software

will be integrated with the cell nucleation theory to simultaneously simulate both the cell

nucleation and the cell growth processes, which are presented in Chapter 6.

96

8Table 4.1. Thermo-physical and rheological parameters for PS/CO2 foaming system

[T = 180˚C; Psat ~ 10 MPa]

Parameters Methods Values/Equations Ref.

KH - Determined from solubility data, which was measured using a magnetic suspension balance (MSB).

8.422×10-5 mol/N-m[180˚C]

9.254×10-5 - 9.641×10-5 mol/N-m[150˚C]

7.862×10-5 - 8.361×10-5 mol/N-m[200˚C]

[49]

[194]

[194]

Diffusivity (D)

- Interpolated from the experimental data measured using an MSB.

1.0×10-9 m2/s - 2.5×10-9 m2/s[Remark: 150˚C - 200˚C]

[50]

Relaxation Time (λ)

- Determined using small-shear-strain oscillatory tests over a range of frequencies using a rheometer and the WLF equation.

27 s[Remark: this is the data for pure PS]

[198-199]

Surface Tension (γlg)

- Estimated from the measured data by pendant drop test at different saturation pressures and temperatures.

0.0256exp(-0.01922Psat×10-6) ×10-3 N/m [189]

Zero-Shear Viscosity (η0)

- Determined using a generalized Arrhenius equation that accommodates the effects of temperature, pressure, and dissolved gas concentration.

)

0sys

8sys

1272η 0.8298 expT 341.1

6.023 10 P 66.51C−

⎛= +⎜⎜ −⎝

× −

Pa-s

[Remark: T is in K; Psys is in Pa and C is in wt.%]

[201]

97

.34Figure 4.1. Numerical simulation algorithm of bubble growth dynamics

35Figure 4.2. In-situ visualization data of PS/CO2 batch foaming experiment [Tsys = 180˚C; Psat ~ 10 MPa]

0.916 s

0.416 s

0.833 s

0.333 s

1.000 s 0.750 s 0.666 s 0.583 s

0.500 s 0.250 s 0.166 s 0.083 s

100 μm

Bubble 1

Bubble 2

Bubble 4

Bubble 3

98

. 36Figure 4.3. Measured bubble sizes at different time [Tsys = 180˚C; Psat ~ 10 MPa]

37Figure 4.4. Pressure decay data [Tsys = 180˚C; Psat ~ 10 MPa]

38Figure 4.5. Simulation results versus experimental observations

99

39Figure 4.6. Effect of initial bubble radius (Rbub(t’,t’)) on predicted bubble growth behaviors

40Figure 4.7. Effect of initial shell radius (Rshell,t=t’) on predicted bubble growth behaviors

41Figure 4.8. Effect of diffusivity (D) on predicted bubble growth behaviors

100

42Figure 4.9. Effect of solubility (KH) on predicted bubble growth behaviors

(a) 0.0 s to 1.0 s (b) 0.03 s to 0.07 s

.43Figure 4.10. Effect of surface tension (γlg) on predicted bubble growth behaviors – (a) 0.0 s to 1.0 s and (b) 0.03 s to 0.07 s

101

(a) 0.0 s to 1.0 s (b) 0.6 s to 1.0 s .44Figure 4.11. Effects of relaxation time (λ) on predicted bubble growth behaviors – (a) 0.0 s

to 1.0 s and (b) 0.6 s to 1.0 s

(a) λ = 27.0 s (b) λ = 0.1 s

45Figure 4.12. Effects of η0 on predicted bubble growth behaviors – (a) λ = 27.0 s and (b) λ = 0.1 s

102

Chapter 5 CELL STABILITY IN PLASITIC

FOAMING

5.1. Introduction

Intense research efforts have been made in the past to develop foams with higher cell

density and smaller cell size. Although successful implementations of nanocellular plastics (i.e.,

foam with cell density that is higher than 1015 cells/cm3 and cell size that is less than 0.1 µm)

have been achieved using various batch foaming techniques, the high cost and slow production

rate associated with these techniques have limited their commercial applications. Therefore,

large-scale production of nanocellular plastics is still technologically challenging and

economically unviable. In this context, various researchers have attempted to understand the fate

of nano-bubbles [143, 202]. Despite the valuable insights offered by these researches, only a

limited number of publications have explored the fate and stability of bubbles in plastic foaming.

These studies mainly attributed the cell growth and collapse processes to the diffusion

103

phenomena. Actually, the fate of a cell in a polymer matrix can also be explained by the

Classical Nucleation Theory (CNT) [76, 98-99]. According to CNT, the critical radius (Rcr),

which is a state parameter, governs the growth and collapse of bubbles. In particular, bubbles

that are larger than Rcr grow spontaneously, whereas those that are smaller than Rcr collapse.

However, no study has been undertaken to apply the critical radius concept to explain the bubble

growth and collapse processes in polymeric foaming.

As discussed in chapter 3, a critical bubble is at a thermodynamic unstable equilibrium

with its surrounding. Therefore, the pressure inside it (i.e., Pbub,cr) is uniquely determined by the

state parameters, which include the system pressure (Psys), system temperature (Tsys), and

dissolved gas concentration (C). According to Equation (3.1), which is restated as Equation (5.1)

below, Rcr can be considered a function of the thermodynamic state.

lgcr

R syssys

sat

2γR C P

PC

=−

(5.1)

It was previously believed that Rcr is constant during a plastic foaming. However, as

discussed in chapter 3, during plastic foaming processes, Psys or CR or even both of them change

continuously, resulting in a continuous change in Rcr. Since Rcr decides the stability of a

nucleated cell, it is interesting to study its evolution during the process and examine the

sustainability of the nucleated bubbles under different conditions. In this research, the numerical

simulation system presented in Chapter 5 was modified to examine the bubble growth and

collapse processes in low density polyethylene (LDPE) foaming using a chemical blowing agent

(CBA). The objectives are to elucidate the mechanisms that govern the bubble growth and

collapse behaviors during plastic foaming, the relationship between the dynamic change of Rcr

and the fate of the generated bubble, as well as the dependences of bubble lifespan on various

104

thermo-physical parameters. The numerically simulated results were qualitatively compared with

results captured in-situ in batch foaming experiments.

5.2. Theoretical Framework

In general, a large number of bubbles are growing in close proximity to each other in the

polymer-gas solution during plastic foaming. To simulate the bubble expansion and collapse

behaviors of the CBA-blown bubbles, the cell model, which has be presented in details in

Chapter 4, together with appropriate adjustments in the mathematical models and simulation

algorithm, were adopted. Particularly, instead of simulating an isothermal process, the modified

simulation program considered the temperature increases during the heating process and the

changes of various thermo-physical parameters with the increasing temperature.

5.2.1. Implementation of Cell Model to Model CBA-Based Bubble Growth and

Collapse Processes

When simulating the CBA-based bubble growth and collapse processes, the following

assumptions have been made:

(1) The bubble is spherically symmetric throughout the cell growth/collapse processes.

(2) The polymer-gas solution is incompressible.

(3) The inertial forces and the effects of gravity on cell growth/collapse are negligible.

(4) The accumulation of the adsorbed gas molecules on the bubble surface is negligible.

(5) The gas inside the bubble obeys the ideal gas law throughout the bubble growth and

collapse processes.

(6) The polymer-gas solution is a weak solution.

(7) The dissolved gas concentration at the polymer-gas interface can be related to the

Pbub using Henry’s Law:

105

R sysbub

sat

C ( t ,t ')P ( t )P ( t ,t ')

C ( t )= (5.2)

(8) After the system has been heated up to the setpoint temperature, the bubble growth

and collapse processes are isothermal.

(9) The initial accumulated stress around the growing bubble is zero.

(10) The initial bubble volume is the same as the volume of the CBA particle(s).

5.2.2. Mathematical Formulations

The bubble expansion and shrinking mechanisms can be described by a standard group of

governing equations that include the following: (i) the momentum equation; (ii) the mass balance

equation over the bubble; and (iii) the gas diffusion equation in the surrounding polymer melt.

Using Assumption 3 and considering the surrounding pressure to be Patm, the dynamics of

the aforementioned system are governed by the conservation of momentum in the radial

direction. The corresponding momentum equation can be written as [132]:

−− − + =∫

Rshelllg rr θθ

bub atmRbubbub

2γ τ τP P 2 dr 0R r

(5.3)

The integration term in Equation (5.3) can be evaluated by considering the quasi-linear, upper-

convected Maxwell model with the Lagrangian coordinate transformation [132].

Using Assumption 4, the change rate of gas inside the expanding/collapsing bubble must

balance with both the rate at which gas diffuses in and out of the bubble and the rate of gas

generated through CBA decomposition. In this way, with the conditions of Assumption 5, the

mass conservation equation can be obtained:

=

⎛ ⎞ ∂= +⎜ ⎟⎜ ⎟ ∂⎝ ⎠

3gen2bub bub

r Rg sys bub

dnP R4π d C4πR D3 dt R T r dt

(5.4)

where ngen is the number of moles of gas being generated as the CBA decomposes.

106

With knowledge of the concentration gradient at the bubble surface, Equation (5.4) can

be solved with Equation (5.3) to obtain Pbub and Rbub at a particular instant. Therefore, it is

necessary to determine the concentration profile around the gas bubble. Based on Assumption 7,

the diffusion equation can be written as Equation (4.4), which is restated below [132]:

2

gen 2bub bubbub2 2

dn R RC C D c Cr for r Rdt t r r r r r

•

∂ ∂ ∂ ∂⎛ ⎞+ + = ≥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (5.5)

Equation (5.5) can be solved by imposing the following boundary and initial conditions:

For r > Rbub, ( ) = H atmC r,0 xK P (5.6)

For t > 0, ( ) =shell H atmC R ,t K P (5.7)

For t ≥ 0, ( ) =bub H bubC R ,t K P (5.8)

where KH is the ratio of the saturated gas concentration to the corresponding Psys and x is the

degree of gas saturation in the polymer melt. Equations (5.2) through (5.8) constitute a complete

set of equations that describes bubble growth and collapse behaviors in plastic foaming.

5.2.3. Determination of Critical Radius

Using Assumptions 7, Equation (5.1) can be rewritten as:

lgcr

R syssys

sat

2γR C P

PC

=−

(5.9)

This equation can be used to determine the continuously changing Rcr during plastic foaming. Its

values at different times can be compared with the Rbub to investigate the relationship between

Rcr and the sustainability of nucleated bubbles under different conditions.

5.3. Implementation of a Computer Simulation

5.3.1 Numerical Simulation Algorithm

107

Similar to Chapter 4, a numerical simulation algorithm, which integrates the explicit

finite difference scheme and the fourth-order Runge-Kutta method, was employed to solve

Equations (5.2) through (5.8), and thereby simulated the bubble expanding and shrinking

phenomena. The simulation involved two major numerical difficulties: (i) a moving boundary

and (ii) a steep concentration gradient at the bubble-polymer interface. The free moving

boundaries for the governing equations were immobilized by the Lagrangian coordinate

transformation [138]. A variable mesh, which had grid points clustered near the interface, was

used to overcome the challenges caused by the steep concentration gradient at the interface

during the later stage in the process.

In the experiments, bubble is generated by heating the system to a temperature that is

higher than the CBA’s decomposition temperature (Tdecomposition). The simulation program has

taken into account the ramping up of temperature in the experiment and the corresponding

changes in the thermophysical properties of the polymer-gas system. The overall numerical

simulation algorithm is illustrated in Figure 4.1 in Chapter 4.

5.3.2 Materials and Physical Parameters

The polymer and CBA considered in this study were low density polyethylene (LDPE)

supplied by Nova Chemicals (i.e., Novapol® LC0522A) and Celogen® OT supplied by Crompton

Chemicals, respectively. Tables 5.1 and 5.2 show a summary of their material properties.

A thermogravimetric analysis was done to study the decomposition behavior of Celogen®

OT using a thermogravimetric analyzer (TGA) (TA Instruments Q50). About 10 mg of pure

CBA sample was loaded into the TGA and heated from 30°C to 100°C at rates of 10°C/min, and

20°C/min. A stream of N2 was used for purging. Figure 5.1 illustrates the recorded sample

weight as a function of temperature. The result shows that the onset temperature of CBA

decomposition was about 155°C – 160°C. Moreover, the measured decomposition rate of

108

Celogen® OT was extremely high regardless of the heating rate. Therefore, the CBA particle was

assumed to decompose and generate gas instantaneously. Because the majority of gas released by

decomposing Celogen® OT was N2, all the physical constants were estimated based on the

system of LDPE and N2, which are summarized in Table 5.3.

5.3.3 Initial Conditions

It was further assumed that the amount of gas being released was in proportion to the

CBA mass. The average particle size of Celogen® OT was about 3 μm [141]. Assuming the CBA

particle had a cubic shape, for a single (i.e., 3 μm × 3 μm × 3 μm) Celogen® OT particle, the

number of moles of N2 being generated, being calculated from the specific gavity (i.e., 1.55) and

the gas yield (i.e., 125 cm3/g: 91% N2 and 9% H2O) [202], was found to be 2.335 × 10-13 mole.

According to Assumption 10, the initial Rbub generated by a 3 μm × 3 μm × 3 μm CBA particle

was estimated to be 1.86 μm. Hence, the initial Pbub was determined using the ideal gas law.

At the beginning of each experiment, the LDPE sample was equilibrated at 150 °C and

was saturated with N2 in air under Patm. During the rapid temperature increase, the solubility of

N2 in LDPE became higher, resulting in an under-saturation of N2. When the system temperature

was increased to 190°C, the degree of saturation was about 80%. Finally, when the CBA

decomposed, the generation of N2 might have led to saturation or over-saturation of N2 in the

LDPE melt. It was impossible, however, to determine precisely the degrees of saturation of N2 in

the LDPE melt. Consequently, various degrees of saturation (x), 80%, 100%, 105%, and 110%,

were considered in the computer simulation to cover different possible scenarios.

5.4. Experimental Verification

In order to verify the theory describing bubble growth and collapse phenomena, the

simulated results were compared with the experimental data of bubble growth and collapse as

109

observed using a hot-stage optical microscope-based image processing system illustrated in

Figure 5.2. The hot-stage (Linkam HFS 91) with a precision temperature controller (Linkam TP

93) was used to heat the system temperature to the desired level at a controlled rate. An optical

system, which consists of a high speed CMOS camera coupled with a high magnification zoom

lens and an optic fibre transmissive light source, was installed to allow for bright field

observation and video recording of the plastic sample during the foaming process.

5.4.1 Sample Preparation

Film samples of LDPE with 0.25 wt% and 0.50 wt% Celogen® OT were prepared using a

compression molding machine equipped with a digital temperature controller (Fred S. Carver

Inc.). LDPE powders were dry-blended with the specific amount of Celogen® OT powders. The

mixture was then molded into a 500 μm thick film by using a hot press, which was pre-heated to

a temperature above the LDPE’s melting point and below the CBA’s decomposition temperature.

5.4.2 Experimental Procedure

Experiments were conducted at two CBA contents (i.e., 0.25 wt% and 0.50 wt %). The

sample was first heated up and equilibrated at 150°C on the hot stage. Then, the system

temperature was rapidly ramped up to 190°C to initiate the bubble generation.


5.5.1 Computer Simulation

Figures 5.3 shows the simulated bubble growth and/or collapse behaviors under different

degrees of saturation. The results indicate that the bulk gas concentration, which was affected by

the CBA content and changed with time, influenced the maximum bubble size and the bubble

lifespan. When x was low (e.g. 80% and 100%), the simulated lifespan was extremely short (i.e.,

< 1 sec). The rapid dissolution of these small gas bubbles would quickly lead to the saturation or

110

over-saturation (i.e., x > 100%) of N2 in the LDPE melt, especially in the regions where the CBA

particles were dispersed densely. Once the degree of saturation was high enough (e.g., x =

110%), the CBA-blown bubbles would sustain and grow.

Figures 5.4 (a) through (f) illustrate the proposed mechanisms of bubble growth and

collapse in CBA-based foaming processes. Initially, during the heating process, the CBA

decomposes, and the gas generated increases the dissolved gas content in the polymer matrix.

Once the dissolved gas content is sufficiently high, the subsequent CBA decomposition will then

form a bubble that can sustain and grow. Due to its small size, this newly generated bubble has a

high internal pressure; therefore, a high concentration gradient will develop around it and cause a

rapid diffusion of gas from the bubble to its surroundings. As a result, a thin gas-rich layer forms

around the bubble. Meanwhile, the high pressure inside the bubble will lead to rapid

hydrodynamic-controlled bubble growth. When the bubble grows larger, its internal pressure

reduces, leading the gas concentration at the bubble surface to decrease dramatically. At that

moment, the gas-rich layer surrounding the bubble will become the source of gas sustaining the

continuous expansion of it. Eventually, due to the gas lost from the sample surface to its

surroundings, the bubble will shrink.

To elucidate the relationship between the Rcr and bubble growth and collapse behaviors,

the actual bubble radii (Rbub) and Rcr were compared at different times. Figure 5.5 shows the

evolution of Rcr during bubble growth and collapse processes when the LDPE melt was over-

saturated with N2 at a saturation level of 110%. The simulated result shows that Rcr was infinitely

large initially because the LDPE melt was fully saturated with N2. As T increased and the CBA

decomposed, a large amount of N2 was generated within a small volume, which dramatically

increased the N2 concentration at the polymer-gas interface. Hence, the Rcr reduced rapidly and

allowed for the generation of the bubble. After that, Rcr started to increase due to the gas

111

depletion around the bubble while it grew and the gas loss to its surroundings. Finally, the bubble

started to collapse when Rcr became larger than Rbub (i.e., 6 s). As a result, its internal pressure

increased and led to the reduction of Rcr. Nevertheless, the bubble continued to shrink and finally

disappeared as Rcr remained larger than Rbub.

5.5.2 Computer Simulation vs. Experimental Simulation

Figures 5.6 (a) and (b) show a series of micrographs taken in-situ during the CBA-based

foaming with 0.25 wt% and 0.50 wt% CBA, respectively. A bubble, which is indicated by an

arrow, has been chosen from each set of experiments to demonstrate its growth and collapse

phenomena. A typical curve describing the lifespan of a bubble was observed to have two stages

– bubble growth and bubble collapse. The bubble growth process was both hydrodynamic and

diffusion-controlled. The hydrodynamic-controlled bubble growth was predominant at the onset

of the process, while the diffusion-controlled process sustained the subsequent bubble growth.

As gas was continuously lost to the surroundings through the sample surface, the bubble

eventually shrank and completely dissolved into the polymer matrix.

Figure 5.7 shows that the computer-simulated bubble growth and collapse phenomena

agreed qualitatively with the experimental simulations. This confirms the validity of the

computer simulation model and the theory that supports it. On the other hand, a quantitative

discrepancy between the two sets of results was noted, and could be attributed to many possible

reasons. First, although the heating rate employed in the experiment was 1.5°C/sec, the actual

heating rate of the LDPE sample might have been lower due to the low thermo-conductivity of

polymer. To reflect the possible effect of the lower heating rate on bubble lifespan, a simulation

was run at a heating rate of 0.5°C/sec for comparison, which is illustrated in Figure 5.7. Second,

the bubble-to-bubble interactions in the experiments were not considered in the computer

simulations. Third, the initial thickness of the shell of LDPE melt around the bubble studied in

112

the computer simulation was estimated based on half of the sample thickness (i.e., 500 μm).

However, when a bubble was formed in the LDPE film sample during the experiment, only its

top and bottom surfaces experienced a similar situation while its side should have extended a

much longer distance from the surroundings (i.e., a longer diffusion path). Therefore, this would

lead to an overestimation of the gas depletion rate and thereby an underestimation of the bubble

lifespan. Nevertheless, the good qualitative agreement between the two sets of results gives a

good indication that the theory provides a realistic justification of the underlying mechanisms.

5.5.3 Effect of Diffusivity on the Sustainability of a Bubble

Figure 5.8 shows the effect of gas diffusivity on bubble growth and collapse behaviors. It

was observed that higher gas diffusivity led to increased maximum Rbub while shorter bubble.

With a higher diffusivity, more gas would be accumulated in the gas-rich region around the

bubble, increasing the initial bubble growth rate. This caused a higher gas depletion rate and

promoted the gas loss to the surroundings. The resultant increase in Rcr led to faster and earlier

bubble collapse.

5.5.4 Effect of Surface Tension on the Sustainability of a Bubble

Figure 5.9 illustrates the effects of surface tension on the sustainability of the CBA-

generated bubble. Both the maximum Rbub and the bubble lifespan increased with lower surface

tension. A lower surface tension reduces the retarding force on bubble growth, decreasing Pbub

according to Equation (5.3). This means that the gas content at the bubble-polymer interface

would be lower. Consequently, during the initial phase of bubble growth, the concentration

gradient between the gas-rich region and the bubble surface was higher, resulting in faster gas

diffusion into the bubble and a higher bubble expansion rate.

5.5.5 Effect of Solubility on the Sustainability of a Bubble

113

Figure 5.10 shows how the gas solubility in the polymer melt affects the fate of the CBA-

blown bubble. The simulation results indicated that higher gas solubility would lead to a larger

maximum Rbub and a longer bubble lifespan. With an increase in gas solubility, a larger amount

of gas would accumulate in the gas-rich region around the bubble. Therefore, this richer supply

of gas would fuel the expansion of the bubble and sustain it longer.

5.5.6 Effects of Viscosity and Elasticity on the Sustainability of a Bubble

The effects of melt viscosity and elasticity (i.e., measured by λ) on the sustainability of

the generated bubble are illustrated in Figures 5.11 and 5.12, respectively. According to Equation

(5.3), both parameters would influence bubble growth and collapse dynamics. However, the

simulation results suggested that the effects of these rheological parameters on the bubble’s

sustainability were negligible within the ranges of values considered in this study.


A series of computer simulations for bubble growth and collapse dynamics has

demonstrated the continuous change of the critical radius during plastic foaming and their

relationship to the bubble’s fate. The overall patterns of bubble growth and collapse phenomena

during various stages have been shown by both the theoretical and experimental results. It is

believed that when CBA decomposes, a gas-rich region around the newly formed bubble will

develop. This gas-rich region contributes to the bubble expansion during the initial phase of its

life cycle. Meanwhile, the continuous gas loss to the surroundings and the reduction of Pbub will

lead to the increase of Rcr. Finally, when Rcr becomes larger than Rbub, the bubble starts to

collapse. Furthermore, it has been found that diffusivity, solubility and surface tension are the

important parameters governing the fate of the generated bubble. It is believed that a lower

diffusivity, a higher solubility, and a lower surface tension will help to sustain the bubbles.

114

9Table 5.1. Properties of LDPE

Properties of LDPE LDPE (LC0522A)

Melt index (g/10min) 4.50

Density (g/cm3) 0.922

Melting temperature (°C) 110

.10Table 5.2. Properties of Celogen® OT [202]

Properties of CBA Celogen® OT

Gas, Yield (cm3/g) N2, 125

Specific gravity 1.55

Decomposition temperature (°C) 158 – 160

.11Table 5.3. Numerical values of physical properties of LDPE and N2 system at 160°C –

190°C [47, 203-205]

Physical Properties Values

D [m2/s] 3.10 × 10-9 – 6.04 × 10-9

KH [mol/m3Pa] 4.18 × 10-5 – 5.13 × 10-5

γlg [N/m] 0.026 – 0.028

η0 [N-s/m2)\] 1431.99 – 2450.0

λ [s] 0.00634 – 0.00909

115

46Figure 5.1. TGA curve of Celogen® OT at heating rates of 10°C/min and 20°C/min

47Figure 5.2. A schematic of the experimental setup

116

48Figure 5.3. Simulated lifespan of a CBA-blown bubble at various degrees of saturation (x)

(a)

(b)

(c)

(d)

(e)

(f)

49Figure 5.4. Proposed mechanism of bubble growth and collapse in CBA-induced foaming: (a) heating, (b) bubble generation, (c) bubble expansion, (d) maximum bubble growth, (e) bubble

collapse, and (f) bubble disappearance

117

50Figure 5.5. Simulated bubble size (Rbub) and critical radius (Rcr) [x = 110%]

(a) (b)

51Figure 5.6. Bubble growth and collapse phenomena with different CBA contents: (a) 0.25 wt% Celogen® OT and (b) 0.50 wt% Celogen® OT

118

52Figure 5.7. Simulated vs. experimentally observed lifespan of bubbles

53Figure 5.8. Effect of diffusivity (D) on a bubble’s sustainability

54Figure 5.9. Effect of surface tension (γlg) on a bubble’s sustainability

119

55Figure 5.10. Effect of solubility on a bubble’s sustainability

56Figure 5.11. Effect of viscosity on a bubble’s sustainability

57Figure 5.12. Effect of elasticity on a bubble’s sustainability

120

Chapter 6 SIMULTANEOUS COMPUTER

SIMULATION OF CELL NUCLEATION & GROWTH

6.1. Introduction

This chapter discusses the development of a modified nucleation theory and examines its

application to simulate the cell nucleation phenomena in plastic foaming. In reality, cells are

formed from pre-existing gas cavities during plastic foaming processes; however, it is extremely

difficult, if not impossible, to precisely determine the initial number of pre-existing gas cavities

and their corresponding sizes. Therefore, the modified nucleation theory discussed in this chapter

was developed on the basis of the classical nucleation theory, which predicts the free energy

barrier to form a bubble from no bubble. Although such an approach would not yield an accurate

quantitative description of the real cell formation phenomena, it would serve as a means to

qualitatively analyze the cause-and-effect relationships between various processing parameters

121

and the resultant cellular structures. Comparing to the classical theory, this modified theory

accounts for the random surface geometry due to the surface roughness of various heterogeneous

nucleating sites. During plastic foaming, once some cells have nucleated, subsequent growth of

these nucleated cells and nucleation of new cells occur simultaneously. The phenomena will

continue until the complete consumption of the dissolved gas in the polymer-gas solution or the

stabilization of the cellular structure upon cooling. Therefore, knowledge about the cell

nucleation and growth mechanisms and the interaction between them are indispensable for

controlling and optimizing the performance of various processing technologies utilized in the

foaming industry. An integrated model that combines the modified nucleation theory being

developed in this chapter and the bubble growth simulation model being presented in Chapter 4

was used to account for the simultaneous occurrence of both phenomena. The theoretical models

and the simulation scheme are verified by comparing the computer-simulated cell density with

the experimentally observed data of polystyrene-carbon dioxide (PS-CO2) foaming.

The developed program was used to verify the validities of two common approximations

about the system pressure (Psys) when simulating extrusion foaming processes. These include: (i)

the pressure of a critical bubble (Pbub,cr) equals to the gas saturation pressure (Psat); and (ii) Psys

drops from Psat to the atmospheric pressure (Patm) instantaneously. The end results will offer

guidelines to improve the accuracy of simulating the overall foaming behavior.

6.2. Development of a Modified Heterogeneous Nucleation Theory

Internally-added nucleating agents, impurities and unknown additives in the commercial

polymer, as well as the wall of the processing equipment can serve as heterogeneous nucleating

sites. Therefore, heterogeneous nucleation is believed to be the main mechanism through which

cells are formed in plastic foaming. Considering solid heterogeneous nucleating sites, the surface

122

roughness of these sites is likely to resemble a series of conical cavities, which are illustrated in

Figures 6.1 (a) and (b), respectively. Based on the classical nucleation theory (CNT) [76, 98-99],

various researchers derived the free energy barriers and the rates for both homogeneous

nucleation and heterogeneous nucleation. These formulations are stated as Equations (2.4),

(2.10), (2.22), and (2.23) in Chapter 2, and are restated as Equations (6.1) through (6.5) below,

where the free energy barriers for homogeneous nucleation (Whom) and heterogeneous nucleation

(Whet) are:

( )

=−

3lg

hom 2

bub,cr sys

16W

3 P P

πγ (6.1)

( )

3lg

het hom2

bub,cr sys

16 FW W F

3 P P

πγ= =

− (6.2)

In Equations (6.1) and (6.2), F is the ratio of the volume of a nucleated bubble to the volume of a

spherical bubble with the same radius, and it is a function of the contact angle (θc) and the semi-

conical angle (β) as indicated in Figure 6.1 (b). Its expression is stated in Equation (6.3) below:

( ) ( ) ( )2c c

c c

cos cos1F , 2 2 sin4 sin

θ θ βθ β θ β

β⎡ ⎤−

= − − +⎢ ⎥⎣ ⎦

(6.3)

Combining the above thermodynamically-derived formulations with kinetic theory, the

homogeneous nucleation rate (Jhom) and the heterogeneous nucleation rate (Jhet) were derived to

take the form of an Arrhenius equation, as indicated in Equations (6.4) and (6.5):

( )

3

2

2 16

3lg lg

hom

B sys bub ,cr sys

γ πγJ N exp

πm k T P P

⎛ ⎞⎜ ⎟= −⎜ ⎟−⎝ ⎠

(6.4)

( )

32 3

2

2 16

3lg lg

het

B sys bub ,cr sys

γ πγ FJ N Q exp

πmF k T P P

⎛ ⎞⎜ ⎟= −⎜ ⎟−⎝ ⎠

(6.5)

123

where Q is the ratio of the surface area of a nucleated bubble to the surface area of a spherical

bubble with the same radius. Similar to F, it is a function of θc and β, which is stated below:

( ) ( )cc

1 sin θ βQ θ ,β

2− −

= (6.6)

Simultaneous simulation of heterogeneous nucleation and growth had been reported

previously [176]. However, their assumption that all nucleating sites were smooth planar

surfaces was unrealistic. In reality, the shapes of different heterogeneous nucleating sites are

unlikely to be identical. Therefore, this study abandoned the assumption that all nucleating sites

are either smooth planar surfaces or a series of conical cavities with identical β. Instead, the

heterogeneous nucleating sites are modelled as a series of conical cavities with β randomly

distributed between 0° and 90°. As a result, Equation (6.5) was modified by incorporating a

probability density function (ρβ) to account for this, leading to the derivation of a modified

heterogeneous nucleation theory, which is stated in Equation (6.7) below:

( ) ( ) ( )( )

( )32

lg lg c3het β c 2

β c B sys bub ,cr sys

2γ 16πγ F θ ,βJ ρ β N Q θ ,β exp dβ

πmF θ ,β 3k T P P

⎛ ⎞⎜ ⎟= −∫⎜ ⎟−⎝ ⎠

(6.7)

Depending on the nature of the heterogeneous nucleating sites, different types of ρβ can

be applied to describe the surface characteristics. To account for the randomness of β at different

locations, a uniform probability density function (i.e., ρβ) from 0° to 90° was adopted.

6.3. Research Methodology

6.3.1. Simultaneous Simulation of Cell Nucleation and Growth

An integrate model that combines the modified heterogeneous nucleation theory and the

cell growth simulation model presented in chapter 4 was used to simulate simultaneously the cell

nucleation and growth. Assumptions (1) to (12) in Chapter 4 are maintained in the simulation.

124

6.3.1.1. Overall Simulation Methodology

Figure 6.2 shows a flowchart for the overall algorithm for the simultaneous simulation of

cell nucleation and cell growth. The simulation model and the subroutine for computing the

bubble growth profiles can be referred to Figures 2.7 and 4.1, respectively, in previous chapters.

It must be noted that cell nucleation rate varies continuously due to the continuous consumption

of gas and occupation of heterogeneous nucleating sites. Therefore, a time integration of the total

cell nucleation rate (Jtot) is needed to compute the cell density with respect to the unfoamed

volume of the polymer, Nb,unfoam(t). This is indicated as Equation (6.8) below:

tb ,unfoam tot0N ( t ) J ( t ')Vdt'= ∫ (6.8)

In the above formulation, V is the volume of the unfoamed polymer melt and Jtot is the sum of

the homogeneous nucleation rate and the heterogeneous nucleation rate per unit volume, which

can be computed by Equations (6.4) and (6.7), respectively, as below:

tot hom het hetJ ( t ) J ( t ) A ( t )J ( t )= + (6.9)

Since Jhet(t) represents the heterogeneous nucleation rate per unit area of heterogeneous

nucleating sites, the heterogeneous nucleation rate per unit volume is obtained by multiplying

Jhet(t) to the unoccupied area of the heterogeneous nucleating sites per unit volume, Ahet(t).

To account for the continuous reduction in both the gas content and unoccupied

heterogeneous nucleating sites, their values were updated in each time step. The average gas

concentration (Cavg(t)) that remains in the polymer-gas solution can be determined by:

( ) ( ) [ ]3

tbub bub

avg 0 tot0 g sys

4πR t,t ' P t ,t 'C ( t ) C J ( t') dt'

3R T= − ∫ (6.10)

125

where C0 is the initial dissolved gas concentration. The molar concentration of the dissolved

blowing agent obtained using Equation (6.10) can be multiplied to the Avogadro’s number (NA)

to compute the number of gas molecules per unit volume, as indicated in Equation (6.11) below:

avg AN( t ) C ( t )N= (6.11)

Ahet(t) is approximated by subtracting the projected area of nucleated bubbles on the nucleating

sites’ surfaces from the initial surface area of the nucleating agents per unit volume (Ahet,0):

( ) [ ]2,0

0

( ) , ' ( ') ( ') 't

het het bub het hetA t A R t t A t J t dt= − ∫π (6.12)

The remaining dissolved gas content as well as the area of the unoccupied heterogeneous

nucleating sites in the polymer-gas solution decreases continuously throughout the simulation

due to the gas consumption by the bubble nucleation and growth processes. Finally, when Cavg(t)

is sufficiently low and/or Ahet(t) is sufficiently small, the nucleation rate will be negligible and

this moment is considered to be the termination point of the simulation. Consequently,

information about the cell density and the bubble radii for each time step will be extracted from

the program for subsequent analyses. Table 6.1 summarizes the major differences between this

simulation approach and some other methodologies proposed in previous studies [173, 176].

6.3.1.2. Determination of Physical Parameters

The foaming system being investigated in this chapter is PS-CO2, which is the same

system being considered in the bubble growth investigation in Chapter 4. The values of various

material parameters summarized in Table 4.1 were adopted in the simulation. Nevertheless,

Equations (6.4) and (6.7) suggest that cell nucleation rates change exponentially as the interfacial

energy at the liquid-gas interface (γlg) varies. Therefore, the Scaling Functional Approach [206]

was used to account for the effect of the cluster size on γlg.

126

Additionally, an accurate prediction of cell nucleation rate also hinges on accurate

determination of the degree of supersaturation (ΔP), which depends on both the system pressure

(Psys) and the pressure inside a critical bubble (Pbub,cr). In this context, Psys can be directly

obtained from the pressure decay data being recorded during the experiments, while Pbub,cr is

determined from the thermodynamic equilibrium condition stated in Equation (6.13) below:

( ) ( )g ,bub bub ,cr g ,sol sysμ T ,P μ T ,P ,C= (6.13)

Both Psys and C decrease with time during the foaming process. Since μg,sol is known to be a

decreasing function of Psys and C, it is apparent that:

( ) ( )g ,sol sys g ,sol sat 0μ T ,P ,C μ T ,P ,C≤ (6.14)

The equality in Equation (6.14) only holds when Psys and C are equal to Psat and C0, respectively.

For a saturated polymer-gas solution with a given gas concentration, C0, the thermodynamic

equilibrium condition can be written as:

( ) ( )g ,sol sat 0 g ,gas satμ T ,P ,C μ T ,P= (6.15)

where μg,gas is the chemical potential of the gas surrounding the polymer. Using Equations (6.13)

through (6.15), it can be concluded that:

( ) ( )g bub ,cr g ,bub sat sys satμ T ,P μ T ,P for P P= = (6.16a)

( ) ( )g bub ,cr g ,bub sat sys satμ T ,P μ T ,P for P P< < (6.16b)

Despite the common approximation of Pbub,cr by Psat, Equations (6.16a) and (6.16b) indicate that

the equality is not valid unless Psys equals to Psat [206]. However, because cell nucleation does

not occur at Psat, it seems to be inappropriate to approximate Pbub,cr by Psat. Hence, it is of great

interest to evaluate the impact of using this approximation on the computer simulation.

127

Since the size of a critical bubble is in the sub-micron level, statistical thermodynamic

theories should be employed to determine the chemical potential of the gas in it. Following the

approach suggested by Li et. al. [206], the values of μg and μg,sol at specified values of T, Psys, and

C are determined based on the Sanchez Lacombe equation of state (SL EOS) [207]:

( )2R R R R R

m

1ρ P T ln 1 ρ 1 ρ 0r

⎡ ⎤⎛ ⎞+ + − + − =⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦ (6.17)

where PR, TR, and ρR are the reduced pressure, temperature, and density of the polymer-gas

solution, respectively, and rm is the number of lattice sites occupied by a mer. Using the SL EOS,

the values of μg can be determined by (6.17) [207]:

( )G G

0 G GR Rg G g R RG G G G 0

R R R R G

ρ P 1 1μ r R T 1 ln 1 ρ ln ρT ρ T ρ r

⎡ ⎤⎛ ⎞= − + + − − +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦ (6.18)

where rG0 is the number of lattice sites occupied by a pure gas molecule; and PR

G, TRG, and ρR

G

are the reduced pressure, temperature, and density for the gas component, respectively. On the

other hand, the value of μg,sol can be computed by [207]:

( )

0 2Gg ,sol g G P G R G P

P

G0 R R

G g R RG G 0R R R R G

rμ R T lnφ 1 φ r ρ X φr

ρ P 1 1r R T 1 ln 1 ρ ln ρT ρ T ρ r

⎡ ⎤⎛ ⎞= + − +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞

+ − + + − − +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

(6.19)

where φG and φP are the close-packed volume fractions of the gas and the polymer components;

rG and rP are the number of lattice sites occupied by a gas molecule and a mer in the polymer-gas

mixture; and XG is a function of the following:

( )* * *

G P MG * G

G R

P P 2PX

P T+ −

= (6.20)

where PG*, PP

* and PM* are the characteristic pressures of the gas, polymer, and the polymer-gas

128

mixture, respectively. PM* can be determined using Equation (6.21):

( ) ( )1

* * * 2M G P 12P P P 1 K= − (6.21)

where K12 is the interaction parameter for the SL EOS. Consequently, Pbub,cr can be estimated by

solving Equations (6.15) and (6.17) to (6.21) simultaneously.

The heterogeneous cell nucleation rate also depends on both F and Q, which are

functions of θc and β. As discussed earlier, the sizes of β were assumed to follow a uniform

distribution between 0˚ and 90˚. For θc, due to the difficulty in experimentally measuring it for

the processing condition under investigation herein, it was used as a free parameter in the

simulation. An iterative approach was employed to search for the value of θc that can best fit the

simulation result to the experimental data of a chosen foaming case. Then, the same value of θc

would be employed to simulate the foaming processes conducted under other processing

conditions to evaluate the choice of θc and to serve as additional verifications of the proposed

nucleation theory and simulation scheme.

6.3.2. Experimental Verification

The verification experiments were conducted by the batch foaming visualization system

illustrated in Figure 3.1 [167]. The PS-CO2 foaming experiment being presented in Chapter 4

was employed herein as the base case to verify the computer simulation. The corresponding

processing conditions are summarized in Table 6.3. Figure 6.3 shows the micrographs obtained

from the in-situ visualization of the PS-CO2 foaming, through which the cell density and cell size

data were extracted. This provided a useful information base for verifying the modified

nucleation theory. However, as illustrated in Figure 6.4, the smallest bubble that can be captured

by the equipment is about 3 to 5 μm in diameter. Since the critical bubble’s diameter is generally

in the scale of tens of nanometer, the equipment is unable to observe the bubble nucleation

129

phenomena in-situ. To ensure a fair comparison between the simulation and experimental results,

this study also simulated the number density of cells with diameters larger than 3 μm. Moreover,

the volume expansion ratio, VER, of the foam is difficult to be determined precisely due to the

non-uniform bubble size. Hence, the comparison was based on the cell density with respect to

the foam volume, Nb,foam(t), which can be related to Nb,unfoam(t) and VER(t), as below:

b ,unfoamb , foam

N ( t )N ( t )

VER= (6.13)

where

t 3tot bub0

4πVER( t ) 1 J ( t ')R ( t ,t ') dt'3⎡ ⎤= + ∫⎣ ⎦ (6.14)

After using the base case to determine an optimal value of θc, the six cases of PS-CO2 batch

foaming processes indicated in Table 6.3 were simulated and compared with the experimental

results. These comparisons serve as additional verifications of the modified nucleation theory as

well as case examples to theoretically investigate the effects of –dPsys/dt and C on plastic

foaming. The stress-induced nucleation demonstrated in Chapter 3 is believed to be negligible

due to the relatively high CO2 content and the absence of talc. Therefore, the computer

simulations did not consider the local pressure fluctuation in the polymer-gas solution.

6.3.3. Impact of the Pbub,cr Approximation on Foaming Simulation

In order to investigate the impact of substituting Psat for Pbub,cr in the computer simulation

of plastic foaming, this study simulated the cell densities of a PS-CO2 foaming process and

compared the results obtained from the thermodynamically determined Pbub,cr and also that

yielded with the aforementioned approximation. The processing conditions and the material

parameters being considered in this investigation are based on the base case presented in the

previous section (i.e., Tables 6.2 and 4.1). In order to solve for the value of Pbub,cr using

130

Equations (6.8) and (6.17) through (6.21), it is necessary to know the values of the characteristic

pressures, volumes, and temperatures of PS [194] and CO2 [49]. The values of these parameters

and the interaction parameter (K12) [194] are summarized in Tables 6.4 and 6.5, respectively.

6.3.4. Impact of the Psys Profile Approximation on Foaming Simulation

A series of simulations were conducted to study the foaming process at different -Psys/dt

drop rates (i.e., ranging between 107 and 1015 Pa/s), which are listed in Table 6.6. Linear Psys

profiles were adopted in these simulations. The other processing conditions, which are

summarized in Table 6.7, were kept constant in all simulation trials. The Psat that corresponded

to dissolving 5 wt% of CO2 in PS was about 12 MPa [194]. The cell density and cell size

distribution data were simulated in each trial (i.e., Trials 1 through 9). They were compared with

the simulation data in which the step Psys profile was adopted (i.e., Trial 10). The errors with

respect to the cell density and the cell size between each trial and trial 10 were evaluated.

Considering that order of magnitude analyses were adequate to study both the cell density and

the cell size in foaming research, the simulation results with errors within one order of magnitude

were deemed acceptable.


6.5.1. Simultaneous Simulation of Cell Nucleation and Cell Growth Phenomena

6.5.1.1. Computer Simulation and Experimental Verification of the Base Case

Following the aforementioned approach, the number density of the observable bubbles

(i.e., Rbub > 1.5 μm) during the PS-CO2 foaming process under various processing conditions

were simulated. It has been found that the best fit between the simulation result and the

experimentally-obtained number density of observable bubbles with respect to the foam volume,

which is illustrated in Figure 6.5, is achieved when θc was 85.7˚. The cell density with respect to

131

the foam volume (Nb,foam) and that with respect to the unformed volume, (Nb,unfoam) are also

shown in the figure. A delay can be observed between the number density of the nucleated

bubbles curve and the number density of the observable bubbles curve. It can also be observed

that significant cell nucleation only occurs after a finite amount of pressure drop.

Figure 6.6 illustrates the simulation and experimentally obtained rates of bubble

generation with respect to the foamed volume. The computer simulated nucleation rate with

respect to the unfoamed volume is also illustrated. The results show that the cell nucleation

process can be subdivided into three major stages: (i) the rapid increase of cell nucleation rate;

(ii) the achievement of maximum nucleation rate; and (iii) the rapid decrease of cell nucleation

rate. These three stages of the cell nucleation process in polymeric foaming can be explained by

the amount of the system pressure drop (i.e., Psat – Psys), the degree of supersaturation (i.e., ΔP =

Pbub,cr – Psys), and the dissolved gas concentration. The changes of these parameters are shown in

Figure 6.7. During the beginning stage of the process (i.e., 0.20 to 0.35 seconds), the rapid

increase of the nucleation rate was caused by the increase of Psat – Psys, which led to an increase

of Pbub,cr – Psys. During the intermediate stage of the process (i.e., 0.35 to 0.45 seconds), the

nucleation rate achieved its maximum level (~ 108 bubbles/cm3-s) as ΔP had reached its highest

level. In the final stage of the process (i.e., after 0.45 seconds), the rapid decrease of the cell

nucleation rate was due to the significant CO2 depletion. Therefore, despite the continuous

decrease of Psys, ΔP did not increase further. Furthermore, the continuous reduction of Ahet also

contributed to the decrease in the nucleation rate. The reduction of Nb,foam(t) after 0.5 second (see

Figure 6.5) was due to the negligible cell nucleation rate and the further increase of VER (see

Figure 6.8), which was caused by the continuous expansions of the nucleated bubbles in this later

stage of the process. Figure 6.9 illustrates the computer simulated bubble size distribution of the

PS foam at 0.6 second. The non-uniformity of the bubble sizes was caused by the cell nucleation

132

at different times. Figure 6.10 illustrates the calculated values of Pbub,cr for various Psys

conditions and CO2 contents in the PS-CO2 solution at 180˚C. The results indicate that Pbub,cr

equals to Psat only at the equilibrium conditions. As Psys decreases continuously, cell nucleation

and growth continue to occur and reduce the CO2 content. As a result, Pbub,cr starts to deviate

from Psat and begins to drop below it. The result, which is consistent with those achieved by Li et

al. [206], indicates that Pbub,cr deviates more significantly from Psat at lower Psys.

6.5.1.2. Effects of the Rbub on γlg of a Critical Bubble

Since γlg exhibits an exponential relationship (to the power of three) to the nucleation rate

(see Equations (6.4), (6.5) and (6.7)), it is critical to accurately estimate the value of this

parameter for the successful simulation of the plastic foaming process. In this study, γlg was

determined using Li’s approach [206]. The effect of the bubble radius, which is a function of

both the Psys and the dissolved CO2 content, on γlg is ploted in Figure 6.11. It can be observed

that γlg approaches the macroscopic surface tension (γexp) measured by Park et al. [189] when

Rbub is sufficiently large. However, its value decreases continuously as Rbub becomes smaller.

Furthermore, the effect of Rbub on γlg becomes more pronounced when Rbub is smaller than 5 nm.

6.5.1.3. Sensitivity Analysis on the Effect of Contact Angle on the Computer Simulation

The effect and sensitivity of the size of θc on the cell nucleation was studied by varying

its value. Figure 6.12 illustrates the simulation results when varying θc from 85.7˚. The results

indicate that a larger θc leads to an earlier nucleation onset time, higher final cell density, and

shorter nucleation duration. Equation (6.3) can be evaluated to explain the aforementioned

effects. It can be shown that F(θc,β) decreases when θc increases for all sizes of β [89, 209]. As

indicated in Equations (6.5) and (6.7), the heterogeneous nucleation rate increases as F(θc,β)

decreases, leading to earlier nucleation and higher cell density. However, because of the faster

gas consumption, the nucleation process occurs over a shorter duration.

133

6.5.1.4. Additional Experimental Verification under Various Processing Conditions

It should be emphasized that further experimental and theoretical investigations are

required to verify the validity of the current choice of θc. Hence, a series of computer simulations

of PS-CO2 foaming under different pressure drop rates or dissolved gas contents, which are

summarized in Table 6.3, were conducted and compared with experimental observations. The

simulated cell densities are plotted and compared with the experimental data in Figure 6.13.

Despite the discrepancies between the simulation and experimental results, it seems that the

modified nucleation theory explains certain quantitative facets of the experimentally nucleation

phenomena. As a result, the proposed theory and the simulation scheme offer a means to analyze

the cause-and-effect relationships between the material and processing parameters and the

foaming phenomena.

Figure 6.14 illustrates the average cell sizes obtained from the computer simulations. The

error bars represent three times the standard deviation for the simulated bubble radii. Figures

6.13 and 6.14 indicate that the increase in both -dPsys/dt and CO2 content leads to a higher final

cell density and smaller cell size. The driving force of bubble nucleation is ΔP. During plastic

foaming, ΔP increases initially as Psys decreases. Once ΔP becomes high enough to initiate cell

nucleation, the gas content in the polymer matrix starts to drop continuously due to the gas

consumption caused by both the nucleation of new cells and their subsequent growth. Finally, the

reduced gas content becomes the predominant factor governing ΔP, leading to a continuous

decrease of ΔP. As a result, a lower –dPsys/dt entailed a slower increase in ΔP. In other words, it

took a longer time for the polymer-gas system to achieve a sufficient ΔP to initiate a significant

amount of nucleation. For the PS-CO2 that was foamed under a higher –dPsys/dt, the faster

increase in ΔP resulted in a more rapid increase of the nucleation rate. Consequently, more cells

134

are nucleated within a shorter period of time. In other words, a larger amount of gas is consumed

to nucleate new cells rather than expand the nucleated cells, resulting in a higher cell density.

Furthermore, the simulation results indicate that a higher dissolved gas content will also

lead to a higher final cell density and a smaller average cell size. Since dissolving a larger

amount of gas into the polymer reduced γlg, an increase in the gas content significantly decreases

the free energy barrier for cell nucleation. As a result, a smaller amount of ΔP is needed to

initiate a significant amount of nucleation. Furthermore, because of the increased number of gas

molecules in the polymer-gas solution, the chance for a gas cluster or a pre-existing gas cavity to

be larger than Rcr is augmented. The nucleation rate increases significantly with the lower free

energy barrier and the higher gas content, leading to a higher cell density and smaller cell size.

6.5.2.1. Effects of Pressure Drop Rate and Dissolved Gas Content on Cell Size Distribution

Figures 6.15 and 6.16 illustrate the bubble size distribution when the nucleation process

has been completed for each experimental case. When the dissolved gas content is 3.8 wt%,

increasing –dPsys/dt slightly improves the bubble size uniformity and significantly reduces the

cell size. Moreover, the bubble size reduces significantly and becomes very uniform when the

dissolved gas content increases to 5.9 wt%. Because the final cell sizes depend on the nucleation

times of the bubbles, cell size would be more uniform if nucleation of all cells occurs within a

short period of time (i.e., high nucleation rate). Moreover, cells would be smaller because the

limited amount of gas is used to expand a larger number of cells. Both a higher –dPsys/dt and

higher gas content promote cell nucleation rate and shorten the entire nucleation process. Thus,

they would improve the bubble size uniformity and reduce the average bubble size.

6.5.2. Impact of Pbub,cr Approximation on Foaming Simulation

To study the effect of the Pbub,cr approximation on the foaming simulation, the PS-CO2

foaming under the processing conditions being summarized in Table 6.2 is used herein as case

135

examples. Figure 6.17 demonstrates that the Pbub,cr approximation leads to a significant

overestimation – by as much as three orders of magnitude – of the final cell density.

Furthermore, the computer simulation also shows that the approximation leads to earlier cell

nucleation. Both outcomes are caused by the higher predicted cell nucleation rate when Psat is

employed to approximate Pbub,cr. Figure 6.18 indicates that the highest nucleation rate computed

using the Pbub,cr approximation is about 1011 bubbles/cm3-s, which is about three orders of

magnitude higher than that calculated using the thermodynamically-determined Pbub,cr. The

elevated nucleation rate also leads to an overestimation of gas consumption, as shown in Figure

6.19, leading to a ore raid decrease in Cavg. To elucidate the effects of the Pbub,cr approximation

on predicting the cell density, cell nucleation rate, and the dissolved gas content, it is essential to

analyze the deviation of Pbub,cr from Psat during plastic foaming. Figure 6.20 shows that Pbub,cr

and Psat are equal only when Psys equals Psat. As Psys is decreasing rapidly during the process,

Pbub,cr also drops continuously and deviates from Psat. Moreover, a more rapid decrease in Pbub,cr

can be observed after ~0.4 s because of the significant gas depletion (see Figure 6.19). Figure

6.20 indicate that the approximation of Pbub,cr using Psat significantly exaggerates the magnitude

of ∆P, especially in the later stages of the foaming process. Using Equations (6.1) through (6.8),

it can be concluded that the Pbub,cr approximation significantly underestimates the free energy

barrier for cell nucleation and thereby overestimates the cell density, cell nucleation rate, and the

gas consumption rate. Therefore, this approximation should be abandoned in simulation.

6.5.3. Impact of Psys Profile Approximation on Foaming Simulation

6.5.3.1. Validity of the Psys Profile Approximation on Calculated Cell Density

Figures 6.21 through 6.23 illustrate that as -dPsys/dt increases beyond 1012 Pa/s, the final

cell density is within one order of magnitude of that of the step pressure drop profile case. This -

dPsys/dt is considered to be the Psys drop rate threshold (-dPsys/dt|threshold) for PS foaming using 5

136

wt% of CO2 above which the cell density does not change significantly. In most extrusion

foaming research, however, -dPsys/dt are in the order of 1010 Pa/s or lower [34], which are two

orders of magnitude lower than -dPsys/dt|threshold. In such cases, the step pressure profile

approximation will lead to an erroneous cell density calculation by several orders of magnitude.

Figure 6.23 shows the errors in predicted cell density caused by this approximation at different -

dPsys/dt. For example, when -dPsys/dt equals to 1010 Pa/s, the cell density is overestimated by

approximately three orders of magnitude. In summary, the step pressure profile approximation

will significantly overestimate the overall cell density in typical extrusion foaming processes.

6.5.3.2. Validity of the Psys Profile Approximation on Calculated Cell Size

As -dPsys/dt increases above 1011 Pa/s, which is one order of magnitude higher than the

typical -dPsys/dt in extrusion foaming research, the average cell size becomes within one order of

magnitude of the size calculated in the step pressure profile case. Figures 6.24 and 6.25 show the

errors in the simulated average cell sizes when using the approximation. For example,

whendPsys/dt is 1010 Pa/s, the average cell size is underestimated by approximately one order of

magnitude. Our analysis therefore shows that the step pressure profile approximation

significantly underestimates the average cell size.


A modified nucleation theory is proposed in this chapter. This theory accounts for the

irregular surface geometry on heterogeneous nucleating sites due to surface roughness. Software

that integrated this theory and the bubble growth simulation model presented in the Chapter 4

has been developed to simultaneously simulate both the cell nucleation and growth phenomena.

The simulation results were compared with batch foaming experimental results to evaluate the

validity of the modified nucleation theory. PS-CO2 foaming processes under various processing

137

conditions are presented as case examples. By making an appropriate choice for the contact

angle, a good agreement between the simulation results and the experimental results was

achieved. Using this contact angle, the modified nucleation theory also explains certain

quantitative facets of the nucleation phenomena under different processing conditions. It should

be emphasized that additional experimental and theoretical investigations are needed to verify

further the validity of the selected value. Nevertheless, the modified nucleation theory is believed

to provide an improved explanation of foaming process. It has also engendered qualitative

insights that will help to direct and expand an understanding of bubble nucleation.

The investigations that presented in the second part of this chapter had demonstrated the

impacts of approximating Pbub,cr to be Psat or assuming the Psys profile to be a step profile on

computer simulation of cell nucleation phenomena in plastic foaming processes. Firstly, with the

adoption of the Pbub,cr approximation, the computer simulation will predict an earlier onset time

for cell nucleation. Moreover, it also significantly overestimates the final cell density, bubble

nucleation rate, and the gas consumption rate. Therefore, the Pbub,cr approximation should be

avoided in the numerical simulation of any plastic foaming process. Secondly, the computer

simulations of the foaming behaviors under different -dPsys/dt demonstrate that the step Psys

profile approximation can lead to significant overestimation of cell density and underestimation

of cell size, relative to cases that use linear pressure drop profiles. It is clear that the Psys profile

significantly affect the predicted plastic foaming behavior, and it is inappropriate to adopt the

step profile approximation for typical processing conditions in most foaming research.

138

12Table 6.1. Comparison between different foaming simulation approaches Shaft’s

Approach [173] Shimoda’s

Approach [176] Approach used in this

thesis

Determination of Pbub,cr

Approximated by Psat Two cases were presented: Case 1 - approximated by Psat Case 2 - estimated by the average gas concentration and the Henry’s law constant

Determined by the thermodynamic equilibrium condition and Sanchez-Lancombe Equation of State (SL EOS) [206-207]

Determination of γlg

Considered the variation of surface tension with cluster size based on the long range intermolecular potential [204]

Approximated by the experimentally measured γlg without considering the cluster size effect

Considered the variation of surface tension with cluster size based on the Scaling Functional Approach [189, 206]

Determination of Cavg

Employed the influence volume approach

Two cases were presented: Case 1 - employed the influence volume approach [173] Case 2 - did not consider the influence volume

Did not consider the influence volume

Determination of the concentration profile around each nucleated bubble

Determined by solving the diffusion equation

Approximated by a 4th order polynomial

Determined by solving the diffusion equation

Determination of bubble growth profiles

Determined the growth profile for a bubble nucleated at t = 0 and assumed bubbles nucleated at different time follow the same growth profile

Determined the growth profiles for bubbles nucleated at different time using a set of governing equations

Determined the growth profiles for bubbles nucleated at different time using a set of governing equations

139

13Table 6.2. Processing conditions of PS-CO2 foaming for the base case of experimental verification




3.8 % / 10.0 50 180

14Table 6.3. Processing conditions to study the effects of pressure drop rate and dissolved CO2 content on PS-CO2 foaming




3.8 % / 10.0 7 180 3.8 % / 10.0 15 180 3.8 % / 10.0 23 180 5.9 % / 15.0 6 180 5.9 % / 15.0 16 180 5.9 % / 15.0 50 180

.15Table. 6.4. Characteristic parameters of PS and CO2 for SL EOS Substance P*

[MPa] V*

[cm3/g] T* [K]

PS 410.35 0.9093 746.1

CO2 720.30 0.6329 208.9 + 0.459T - 7.56 × 10-4T2

.16Table 6.5. Values of K12 for the SL EOS

Temperature [°C] 110 150 200

K12 -0.0767 -0.1240 -0.2015

140

.17Table 6.6. Summary of Psys drop rates considered in the simulations

Trial Number -Psys/dt (Pa/s)

1 107

2 108

3 109

4 1010

5 1011

6 1012

7 1013

8 1014

9 1015

10 ∞ (step profile)

.18Table 6.7. Parameters used in the simulations

Parameter Value

Tsys 140 °C

CO2 Content 5 wt%

Ahet,0* 10,000 cm2/cm3

θc 85.7° [89] *Estimated based on 5 wt% of Talc particles with radii of 1.7 µm and specific gravity of 2.8

141

(a) An aggregate of nucleating agent particles

(b) A conical cavity on the equipment wall

.0 58Figure 6.1. A bubble nucleated on a rough heterogeneous nucleating site – (a) a nucleating agent, and (b) the equipment wall

142

59Figure 6.2. The overall computer simulation algorithm of plastic foaming

143

-60Figure 6.3. Micrographs of a PS/CO2 batch foaming process

.61Figure 6.4. The smallest observable bubble being observed by the visualization system

100 μm

Smallest Observable

Bubble

0.416 sec.0.333 sec 0.500 sec.

0.750 sec.0.666 sec.0.583 sec.

0.916 sec.0.833 sec. 1.000 sec.

0.250 sec.0.166 sec.0.083 sec.

100 μm

144

.62Figure 6.5. Number density of the observable bubbles [θc = 85.7˚]

.63Figure 6.6. Rate of increase of the number density of observable bubbles [θc = 85.7˚]

.64Figure 6.7. Average CO2 concentration and the difference between Pbub and Psys

145

.65Figure 6.8. Volume expansion ratio of the PS foam

.66Figure 6.9. Bubble sizes distribution at t = 0.6 second

.67Figure 6.10. Deviation of Pbub from Psat at different Psys and wt% of CO2 [T = 180˚C]

146

68Figure 6.11. Curvature dependence of γlg of PS/CO2 system [Psat = 9.94 MPa; T = 180˚C]

69Figure 6.12. Effect of contact angle on the computer simulation result

70Figure 6.13. Simulation results versus experimental data of the PS/CO2 batch foaming processes

147

71Figure 6.14. Simulation results of average bubble radii (error bars = 3X standard deviations)

(a) –dPsys/dt|max = 7 MPa/s (b) –dPsys/dt|max = 15 MPa/s

(c) –dPsys/dt|max = 23 MPa/s

72Figure 6.15. Bubble radii distribution (C0 = 3.8 wt% & Tsys = 180˚C)

148

(a) –dPsys/dt|max = 6 MPa/s (b) –dPsys/dt|max = 16 MPa/s

(c) –dPsys/dt|max = 50 MPa/s

73Figure 6.16. Bubble radii distribution at various processing conditions (C0 = 5.9 wt.% & Tsys

= 180˚C)

74Figure 6.17. Effect of the Pbub,cr approximation on the predicted cell density

149

75Figure 6.18. Effect of the Pbub,cr approximation on the predicted cell nucleation rate

.76Figure 6.19. Effect of the Pbub,cr approximation on the predicted average gas concentration

in the PS-CO2 solution

.77Figure 6.20. Deviation of Pbub,cr from Psat

150

78Figure 6.21. Accumulated cell density versus time at different constant Psys drop rates

. 79Figure 6.22. Maximum cell density versus -dPsys/dt (dash line: the step Psys drop)

80Figure 6.23. Errors of simulated cell densities at different -dPsys/dt

151

.81Figure 6.24. Cell size distributions versus -dPsys/dt (dash line: the step Psys drop; error bar: 3X the standard deviation)

.82Figure 6.25. Errors of cell radii at different Psys drop rates relative to the step Psys

152

Chapter 7 PREDICTION OF PRESSURE DROP

THRESHOLD FOR NUCLEATION Reproduced in part with permission from “Leung, S.N., Wong, A., Park, C.B., and Guo, Q., Strategies to Estimate the Pressure Drop Threshold of Nucleation for Polystyrene Foam with Carbon Dioxide, Industrial & Engineering

Chemistry Research, Vol. 48, Issue 4, pp. 1921-1927, 2009.” Copyright 200 American Chemical Society

7.1. Introduction

In extrusion foaming processes, cell nucleation usually occurs inside the die after the

pressure of the polymer-gas solution drops below the solubility pressure. Upon cell nucleation,

cells start to grow before the polymer-gas solution exits the die. This cell growth phenomenon is

termed “premature cell growth”. An excess amount of premature cell growth would lead to rapid

cell growth upon die exit, promoting gas loss during the foam cooling process. This is because of

the thinner cell walls and the more direct gas loss path due to severe cell coalescence. As a result,

the foam shrinks before it stabilizes, resulting in a low volume expansion ratio [18, 34]. In order

to accurately determine the amount of premature cell growth, it is first necessary to identify the

onset point of cell nucleation. Many studies in the past assumed cell nucleation occurred right

153

after the system pressure (Psys) drop below the solubility pressures [18, 34]. However, since cell

nucleation is a kinetic process, Lee [208] suggests that a critical supersaturation is required to

take the system out of the metastable state and generate a cell. Considering physical foaming

process, a certain amount of Psys drop below the solubility pressure is needed to create a

sufficient level of supersaturation (ΔP) to initiate cell formation. This pressure drop is termed as

“pressure drop threshold” (ΔPthreshold) in this chapter. Fundamental understanding of the

mechanisms governing ΔPthreshold will assist the development of processing strategies to suppress

premature cell growth and to better control cell morphology as well as the volume expansion

ratio of foamed plastics. By knowing the onset point of cell nucleation, it is also possible to

develop innovative means to suppress cell growth to produce nano-cellular foams.

In the past, little effort has been made to study the mechanisms that govern ΔPthreshold due

to the difficulty in gathering such empirical data. The research being presented in this chapter

aims to fill this gap by investigating the effects of the pressure drop rate (-dPsys/dt), the dissolved

gas content, and the processing temperature (Tsys) on ΔPthreshold. To achieve this, a semi-empirical

and a theoretical approach were developed to determine the onset time of cell nucleation of PS-

CO2 foaming at various experimental conditions. ΔPthreshold results from the two approaches were

then compared. The effects of pressure drop rate, gas content, and temperature on ΔPthreshold were

studied.

7.2. Methodology

Figure 7.1 illustrates the overall research strategy, which includes a semi-empirical

approach and a theoretical approach.

7.2.1. Implementation of the Semi-Empirical Method

In the semi-empirical method, the batch foaming visualization system illustrated in Figure

154

3.1 [167] was employed to determine the time at which the first bubble occurred in each

experiment. As illustrated in Figure 6.4 in Chapter 6, the smallest bubble that can be captured by

the equipment is about 3 to 5 μm in diameter. Since the critical bubble’s diameter is believed to

be in the nanometer range in typical polymeric foaming processes, there is a time delay between

the onset moment of nucleation and the time at which the first bubble is captured. In this context,

the bubble growth simulation software being presented in Chapter 4 was utilized to estimate this

time delay to minimize the error when determining ΔPthreshold. Foaming experiments at different

processing conditions presented in Chapter 3, which are summarized in Table 7.1, were used to

elucidate the effects of –dPsys/dt, the dissolved gas content, and Tsys on ΔPthreshold.

The in-situ visualization data was analyzed to obtain the time at which the first bubble

occurred and its growth profile. Each experiment was carried out three times and the average

ΔPthreshold was determined. It should be noted that when studying the effect of Tsys on ΔPthreshold,

Psat was varied in order to maintain a constant CO2 content (i.e., 5.0 wt%).

Using the simulation algorithm being presented in Chapter 4, the growth profile of the

first bubble being observed in each experiment was simulated to depict the onset time (tonset) of

cell nucleation. This was achieved by finding the onset moment of cell nucleation that would

lead to the least squares fit between the simulated and the experimentally measured bubble

growth profiles. Consequently, the ΔPthreshold was determined by subtracting the system pressure

at tonset (i.e., Psys(tonset)) from the saturation pressure (Psat).

7.2.2. Implementation of the Theoretical Method

The theoretical method is based on the integrated model, which combines the modified

nucleation theory and the aforementioned bubble growth simulation model presented in Chapter

6 and Chapter 4, respectively. The onset time of nucleation was determined to be the time at

155

which the cell densities exceeded 10,000 bubbles/cm3 of unfoamed PS. One bubble observed in

the batch foaming visualization system (i.e., a circular viewing area of ~ 500 μm in diameter) is

equivalent to a cell density of ~10,000 bubbles/cm3. Since the onset times of nucleation

determined in the semi-empirical approach depended on the bubble growth profiles of the first

observable bubble, this metric on cell density was used to yield a meaningful comparison

between the onset times determined from the two approaches. It must be noted that the

aforementioned cell density (i.e., 10,000 bubbles/cm3) was adopted to define the onset time of

cell nucleation solely due to the limited optical resolution of the visualization system. In

industrial foaming processes, it is possible to define the initiation of bubble formation occurs

when the first cell has nucleated in the particular foam product.


Since the free energy barrier to initiate cell nucleation (i.e., Whom and Whet) and the

thermodynamic fluctuation (i.e., kBTsys) are inside the exponential term of Equations (6.5) and

(6.7), they would be the dominant factors that govern the nucleation rates and ΔPthreshold. Thus,

the discussion about the dependence of ΔPthreshold on –dPsys/dt, gas content, and Tsys focuses on

investigating the effects of these factors on Whom, Whet, and kBTsys.

7.3.1. Effect of –dPsys/dt on ΔPthreshold

Figure 7.2 shows a sample of foaming visualization data of PS-CO2 foaming at different

-dPsys/dt. The bubble growth profile of the first observable bubble was extracted in each

experimental case to estimate the onset time of nucleation by the semi-empirical approach.

Figures 7.3 (a) and 7.4 (a) show that in both approaches, ΔPthreshold remained approximately the

same while the maximum cell density was increased by raising –dPsys/dt from 22 MPa/s to 47

MPa/s. One-way Analysis of Variance (ANOVA) [17-18] was applied to confirm the lack of

156

effect of –dPsys/dt on ΔPthreshold with the semi-empirical results, and it was shown that the results

were indeed statistically insignificant (refer to Table 7.2). Figures 7.5 (a) and (b) indicate that the

cell nucleation rate increased more rapidly at higher –dPsys/dt and led to an earlier tonset. In the

beginning phase of the foaming processes (i.e., Nb,unfoam < 10,000 cells/cm3), it can be observed

that the nucleation rates were the same with the same amount of pressure drop. Meanwhile, the

CO2 content was virtually unchanged, leading to the same values of γlg and Pbub,cr in all cases.

Together with the constant Tsys, –dPsys/dt showed no effect on ΔPthreshold.

7.3.2. Effect of Gas Content on ΔPthreshold

The amount of CO2 content in polymer was varied from 4% to 7% in 1% increments by

adjusting the saturation pressure, which was determined from PS-CO2 solubility measurements

carried out using the gravimetric method with a magnetic suspension balance by Li et al. [194].

Figures 7.3 (b) and 7.4 (b) show that, in both approaches, higher CO2 content decreased

ΔPthreshold and increased the maximum cell density. Table 7.2 indicates the results of the one-way

ANOVA test. It indicates that the effect of CO2 content on were ΔPthreshold significant with higher

than 99% confidence. Figure 7.3 (b) shows that the semi-empirical results exhibited a slightly

steeper decrease of ΔPthreshold than that of the theoretical results, the trends agree well

qualitatively. The result can be explained by the effect of gas content on γlg. When there is a

higher gas concentration, γlg decreases [189] and thereby results in the reduction of Whom and Whet

as well as an increase in the nucleating rate. Hence, this explains the overall trend of ΔPthreshold

reduction with a higher CO2 content.

7.3.3. Effect of Processing Temperature on ΔPthreshold

Figures 7.3 (c) and 7.4 (c) indicate both approaches suggested that ΔPthreshold and the

maximum cell density decrease when increasing Tsys from 140 ºC to 200 ºC. However, a higher

Tsys would lead to a slight reduction in the maximum cell density. The one-way ANOVA test

157

about the significance of the temperature effect, shown in Table 7.2, shows that the results were

significant but with a lower confidence (i.e, 98%) than the previous case. This suggests that the

effect of Tsys is not as strong as that of the gas content in the ranges that were considered in this

study. This finding agrees with the theoretical results, which exhibit only a slight decreasing

trend in ΔPthreshold with increasing Tsys. In theory, an increase in Tsys increases the mobility of gas

molecules. The increased thermal fluctuation means that there is a higher chance of the gas

molecules forming clusters that are larger than the critical radius for cell nucleation. Therefore, a

higher Tsys would increase the nucleation rate. Furthermore, it would reduce γlg, but the changes

are very small in the considered pressure range, which directly related to the dissolved CO2

content range [189]. This implies that there would only be a slight decrease in Whom and Whet.

Hence, the decrease in ΔPthreshold with increasing Tsys is not as significant as the case with

increasing the gas content.

7.4. Sensitivity Analysis

Although γlg of PS-CO2 has been measured as a function of gas content and temperature,

the small radius of critical nucleus may not validate the use of this data because of the curvature

effect on surface tension [204, 206]. In addition, the data for relaxation times (λ) of PS-CO2

solutions is unavailable. Therefore, sensitivity analyses of these two parameters on bubble

growth profiles were undertaken to estimate the impact of such errors on the ΔPthreshold results

obtained in the semi-empirical approach. Furthermore, because of the unavailability of θc data

for the PS-CO-sapphire system, a constant value (i.e., 85.7°) was assumed at different system

temperatures. Hence, a sensitivity analysis was conducted to study the effect of the possible error

in θc on the theoretical predictions of ΔPthreshold of PS/CO2 foaming.

7.4.1. Effect of Surface Tension at the liquid vapor interface (γlg)

158

The effect of γlg on bubble growth was studied by varying its value from 1.923 mJ/m2 to

38.546 mJ/m2. The initial bubble radius in each case was assumed to be 1% larger than the

critical radius. It should be noted that the base case is γlg = 18.7 mJ/m2, which corresponds to

experimental case 6 in Table 7.1. The sensitivity analysis, illustrated in Figure 7.5, shows that the

effect of γlg on bubble growth was minimal. This result was consistent with the results being

presented in Chapter 4 [179]. Therefore, the value of γlg would have minimal effect on the fitting

of the simulated cell size data to the empirical results, and hence the estimation of tonset. This

means that the simulation results are valid despite the uncertainty of the validity of surface

tension data at the molecular level.

7.4.2. Effect of Relaxation Time (λ)

Figure 7.6 illustrates the effect of λ on bubble growth profiles was studied by varying its

value over a range of 0.1 s to 1000 s. It should be noted that the base case is λ = 27 s. Since the

simulations in this study focused on initial bubble growth, as discussed in Chapter 4, it was

expected that the higher rate of stress accumulation with a lower λ would lead to a smaller

bubble. But within the processing condition range being investigated in the simulation, the effect

of λ on the bubble growth profile was negligible, which was consistent with the finding in

Chapter 4 [179]. Since bubble growth processes are very insensitive even to a wide range of λ,

the validity of the simulation results carried out in this study should not be undermined by the

lack of available data on λ for PS-CO2 solutions.

7.4.3. Effect of the Contant Angle (θc)

θc is a parameter relates to the wettability of the polymer on the nucleating agent’s

surface (i.e., sapphire window). A larger θc means a worse wetting of the polymer on the

sapphire window, and thereby a better wetting of the gas on the sapphire window. This is

beneficial to cell nucleation. Therefore, it is expected that a larger θc will lead to a lower energy

159

barrier, a faster nucleation rate, and a lower ΔPthreshold. This is also reflected in the classical

nucleation theory and the modified nucleation theory as indicated in Equations (6.5) and (6.7),

respectively. In this study, because no data is available for the contact angle of the PS-CO-

sapphire system, similar to the computer simulation of the overall foaming process being

presented in Chapter 6, a constant value was assumed (i.e., 85.7°) for different system

temperatures in the computer simulation. Figure 7.7 shows the effects of the size of θc on

ΔPthreshold. It can be observed that the theoretically simulated ΔPthreshold were relatively sensitive

to the change of θc. This means that an accurate measurement of θc is critical to verify the

validity of the theoretical approach to predict ΔPthreshold. Therefore, the assumptions being made

on θc will need to be re-evaluated in the future once the data becomes available.

7.4.4. Justification of Termination Points of Simulations

The bubble growth simulation software used in this study assumed no interactions

between bubbles. Therefore, the simulations must be terminated before the bubbles have grown

to a point at which interaction between bubbles becomes significant. To this end, it is first noted

that the foamed cells matrix can be approximately represented by tetrahedral structures, in which

each bubble is centered at a vertex of a tetrahedron. Assuming that each of the sides of the

tetrahedrons is lo and each bubble has an identical radius ra, contact between adjacent bubbles

takes place when ra ≥ lo/2. Using a safety factor of two, the termination point of simulations was

chosen to be ra = lo/4 to ensure that interaction between bubbles is minimal.


Using a semi-empirical approach and a theoretical approach, analyses of the effect of

the –dPsys/dt, gas content, and Tsys on ΔPthreshold were conducted. The results from both

approaches have shown a reasonably good agreement qualitatively. Using One-Way ANOVA, it

160

was demonstrated that –dPsys/dt has no effect on ΔPthreshold, while ΔPthreshold decreases by

increasing the gas content and Tsys. The gas content showed a more significant effect than Tsys in

the ranges that were considered in this study. With the success in predicting ΔPthreshold in plastic

foaming, researchers and foam manufacturers would be able to identify the onset point of cell

nucleation during foam production processes. This additional piece of information will be an

invaluable input to enhance the development of strategies to suppress the pre-mature cell growth

or to produce plastic foams with nanocellular structures.

161

.19Table 7.1. Experimental conditions for foaming experiments and computer simulations

Cases Gas Content

[wt%] (Psat [MPa]) Max. –dPsys/dt

[MPa/s] Processing Temp. [˚C]

1 4.0% (9.71) 22 140

2 5.0% (12.1) 22 140

3 6.0% (14.7) 22 140

4 7.0% (16.8) 22 140

5 5.0% (12.5) 47 160

6 5.0% (12.9) 47 180

7 5.0% (13.4) 47 200

8 5.0% (12.1) 32 140

9 5.0% (12.1) 40 140

10 5.0% (12.1) 47 140

.20Table 7.2. One-way ANOVA results

Experimental Parameter P-Value Significance

[% Probability]

max –dPsys/dt [MPa/s] 0.886 < 12%

CO2 Gas Content [wt.%] 0.000 > 99%

Processing Temperature [°C] 0.012 > 98%

162

83Figure 7.1. Overall research methodology to determine ΔPthreshold

84Figure 7.2. Visualized batch foaming data taken from PS-CO2 foaming experiments

163

(a) (b)

(c)

.85Figure 7.3. Effects of (a) –dPsys/dt, (b) CO2 gas content, and (c) Tsys on ΔPthreshold (error bars: 3X standard deviation)

164

`

(a) (b)

(c)

. 86Figure 7.4. Effects of (a) –dPsys/dt, (b) CO2 gas content, and (c) Tsys on maximum cell density (error bars: 3X standard deviation)

165

87Figure 7.5. Sensitivity analysis of surface tension’s effect on bubble growth

88Figure 7.6. Sensitivity analysis of relaxation time’s effect on bubble growth

166

89Figure 7.7. Sensitivity analysis of contact angle’s effect on simulated pressure drop threshold

Chapter 8 FUNDAMENTALS OF PLASTIC

FOAMING USING CO2-ETHANOL BLEND BLOWING AGENT

8.1. Introduction

As one of the potential alternative blowing agents, carbon dioxide (CO2) has been

investigated extensively by various researchers as the blowing agent to foam thermoplastics.

167

However, the relatively low solubility and high diffusivity of CO2 in thermoplastic (e.g.

polystyrene (PS)) has resulted in various processing challenges such as open cells, blow holes,

and surface defects when producing thermoplastic foams. In order to circumvent these problems,

the uses of CO2, together with alcohol (e.g. ethanol (EtOH)), ketone (e.g. acetone), water, or

HFCs as co-blowing agents, have been suggested by various patent literatures [210-213].

Regarding the processing of PS foams using CO2, Gendron et al. [42] discovered that

blow holes occurred in the foam morphology when a high CO2 content (e.g. > 4 wt%) was used

due to a lack of solubility. With the addition of EtOH as the secondary blowing agent, the

problem of blow holes was resolved while the volume expansion ratios of the PS foams were

enhanced. Tsivintzelis et al. [214] identified that the addition of a small amount of EtOH in

Polycaprolactone (PCL)-CO2 foaming could improve the uniformity of cellular structure while

increasing the pore sizes. It was speculated that the promoted dispersion of the crystalline

structures in the PCL matrix or the enhanced plasticizing effects with the presence of EtOH

might explain the larger cell size and the improved cell size uniformity.

In order to elucidate the role of EtOH in thermoplastic foamimg (e.g. PS foaming) when

utilizing a blowing agent blend (e.g., CO2-EtOH), it will be useful to explore the interactions

between PS and EtOH, EtOH and CO2, as well as PS and CO2-EtOH blends. Although neither

pure CO2 nor pure EtOH can dissolve PS, it has been reported that CO2-EtOH supercritical

blends can serve as a solvent of low molecular weight PS [215]. Moreover, Simonsen et al. [216]

demonstrated that nano-sized bubbles were formed at the PS-EtOH interface when PS was

immersed into EtOH. Since EtOH molecules can form hydrogen bonds among themselves, the

formation of these nano-sized bubbles is believed to be related to the perturbation of hydrogen

bonding network among EtOH molecules by the hydrophobic surface. Recent research has

investigated the equilibrium solubility of various alcohols, including EtOH, in PS [217] and,

168

showed that 4.4 wt% to 7.1 wt% of EtOH was dissolved in PS in the temperature range between

55°C and 75°C under atmospheric pressures. Because the blowing agent’s solubility in plastics

typically increases with pressure, it is believed the solubility of EtOH in PS under typical

foaming conditions (i.e., high pressure and high temperature) may be substantially high.

Moreover, Bernardo et al. [217] showed that precipitates of alcohol had been observed after a PS

sample was immersed in hexadecanol at125°C for six days. In light of this, it is speculated that

if similar micro-droplets of EtOH form in the PS matrix during the plastic foaming process, the

micro-droplets may serve as seeds for bubble formation. In addition to solubility, the rheological

properties of the polymer-gas system are important factors that govern final foam structures.

Gendron et al. [42] found that the level of plasticization observed for CO2 and EtOH are

approximately the same (i.e., the reduction in the glass transition temperature (Tg) is -8°C/wt%).

However, the plasticization effect of CO2 is restricted at high CO2 content because of its limited

solubility. In this context, the presence of EtOH seems to be highly advantageous to PS-CO2

processing.

Moreover, little fundamental research on the mechanism of blowing agent mixtures has

been reported so far. To fill the knowledge gap, this research conducted in-situ observations and

rheological measurements of polystyrene (PS) foaming using pure CO2, pure EtOH, and CO2-

EtOH blends to improve the understanding on the fundamental mechanism of plastic foaming

using blowing agent mixtures. The study also serves as a case example how the elucidation of

the foaming mechanisms help to develop novel processing strategies to improve the quality of

plsatic foam. It is believed that, in the long-run, an improvement in the scientific understanding

of the foaming mechanism of using blowing agent blends in plastic foaming, is expected to

provide guidance to choose the optimal composition of blowing agent blends and offer insights

to develop new foaming technology.

169

8.2. Experimental

8.2.1. Materials

The polystyrene used in this study was obtained from the Dow Chemical Company (PS,

Styron PS685D). The blowing agents used in this study were carbon dioxide and EtOH, which

were obtained from BOC Gas Ltd. (99% purity) and Commercial Alcohols Inc. (Ethyl Alcohol

(Anhydrous)), respectively. Their physical properties are listed in Tables 8.1 and 8.2.

8.2.2. Sample Preparation

PS film samples were prepared using a compression molding machine equipped with a

digital temperature controller (Fred S. Carver Inc.). PS resins were hot pressed into a 200 μm

thick film at a temperature above the glass transition temperature of PS. The PS film was then

punched into small disc-shaped samples of about 6 mm in diameter.

8.2.3. Rheology Measurement

A tandem foam extrusion system, as indicated in Figure 8.1 [218], was employed to

investigate the shear viscosity of PS-CO2, PS-EtOH, and PS-CO2-EtOH solutions at a

temperature range of 140°C and 180°C. The first extruder was used to plasticate the polymer

resin and dissolve the blowing agent in the polymer melt, while the second extruder enhanced the

homogenization of the dissolved blowing agent in the polymer matrix. The flow rate of the melt

and the homogeneity of the melt temperature were controlled by the gear pump and the heat

exchanger, respectively. Phase separation was prevented by setting the average die pressure

between 3000 and 4000 psi. Finally, the influence of the composition of the blowing agents on

the viscosity was determined, and the shear thinning behavior could then be described over the

full range by the Cross equation, with a modified expression of the zero-shear viscosity that

accounts for the influence of temperature and pressure [201, 219]:

170

( )

01 n*

01−=

+

ηηη γ τ

(8.1)

0r

Aexp P CT T⎛ ⎞

= + +⎜ ⎟−⎝ ⎠

αη σ ϕ (8.2)

The fitting parameters, τ*, n, A, C, α, σ, φ and Tr can be determined using the least-square-fit

method.

8.2.4. In-Situ Foaming Visualization

The setup of the batch foaming visualization system [167], as illustrated in Figure 3.1

Chapter 3, was used to observe the in-situ foaming behavior of the aforementioned polymer-

blowing agent system. The foaming process was performed according to the following steps:

STEP 1: For pure blowing agent cases, the chamber loaded with the PS sample was charged with

CO2 or EtOH at the desired pressure, while the chamber temperature was controlled

using a thermostat. When using CO2-EtOH blend as the blowing agent, a weighted

amount of EtOH was preloaded in the chamber. After the chamber was heated up to the

desired temperature, it was immediately filled with CO2 at the desired pressure.

STEP 2: The pressure and temperature of the chamber were maintained at the set points for 30

minutes to allow the blowing agent to completely dissolve into the sample.

STEP 3: The blowing agent was released by opening the solenoid valve. The pressure transducer

and the CMOS camera captured the pressure data and foaming data, respectively.

All experiments were conducted at a saturation pressure (Psat) of 5.52 MPa (i.e., 800 psi)

and a pressure drop rate (-dPsys/dt) of about 8 MPa/s. Table 8.3 summarizes the studied blowing

agent compositions. Each experimental case was conducted three times to test for repeatability.

8.2.5. Characterization

171

To analyze the foaming behaviors, the cell density data was obtained from the foaming

visualization data. Hence, N(t), the number of cells within a superimposed circular boundary

with an area of Ac at time t was counted at each time frame. The radius of 10 randomly selected

bubbles at time t (i.e., Ri(t), where i = 1…10) were also measured. The cell density with respect

to the foamed volume, Nfoam(t), and the cell density with respect to the unfoamed volume, Nunfoam-

(t), were calculated using the following equations:

32

foamc

N( t )N ( t )A

⎛ ⎞= ⎜ ⎟⎝ ⎠

(8.3)

unfoam foamN ( t ) N ( t ) VER( t )= × (8.4)

3n

ifoam

i

R ( t )4VER( t ) 1 N ( t )3 n

⎛ ⎞⎛ ⎞= + ×⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠∑π (8.5)

The data was collected between t = 0 and the time at which no more new bubbles were

formed. The cell formation rates with respect to the unfoamed volume were computed by direct

differentiation of the cell density data. It should be noted that the smallest bubbles that could be

observed by the optical microscopic system depends on the magnification being used in the

experiments. Under the highest magnification (i.e., 450X), the smallest observable bubble was

approximately 3 – 5 μm in diameter. If the lowest magnification (i.e., 75X) was employed, the

smallest bubble that could be observed had a diameter of about 12 – 18 μm. Therefore, there

could be a time delay between the moment of cell nucleation and the time at which the bubbles

were observed, and this delay depended on the magnification power being used in the

experiments. In other words, the cell densities and bubble generation rates extracted from the

visualization data were based on the observable bubbles only. In addition to the cell population

172

densities and the bubble generation rates, the average cell growth profiles were also obtained.

In order to provide a more complete understanding on the differences in the foaming

behaviors when the blowing agent composition was varied, the cross-sections of the PS foam

samples were also analyzed using the scanning electron microscopy (SEM, JEOL, model JSM-

6060) to investigate the foam morphology along the thickness direction. PS foam samples were

fractured in liquid nitrogen.


8.3.1. Rheology

The plasticizing effect refers to the decrease of polymer melt viscosity when the polymer

melt was mixed with a low molecular weight substance. Figure 8.2 shows the plot of shear

viscosity versus the shear rate at 140oC and 180oC when different compositions of CO2 and

EtOH were added to the PS melt. The results show that the shear viscosity reduced as the

temperatures or the blowing agent contents increased. By comparing the viscosity data when 5

wt% of CO2 or 5 wt% of EtOH was injected into the PS matrix at 140oC and 180oC, it can be

found that the plasticizing effects of CO2 and EtOH on the PS melt were similar, which were

consistent with the previous study [42]. Furthermore, the viscosity of PS-CO2-EtOH system,

with 3 wt% or 5 wt% of each blowing agent, was even lower. It can be observed that the amount

of viscosity reductions, which were reflected by the distances between the two viscosity curves

under comparison, were roughly proportional to the total amount of blowing agent being added.

8.3.2. Effect of Ethanol Content on Foaming Behaviors

The effect of the initial EtOH content in the blowing agent blend (i.e., CO2 and EtOH) on

the foaming dynamics was studied at two EtOH contents (i.e., mCO2:mEtOH of 80:20 or 60:40).

The samples were also foamed using pure CO2 and pure EtOH. Figure 8.3 illustrates the

173

micrographs of the in-situ visualization of these experiments. The foaming behaviors of PS

blown by pure CO2 or CO2-EtOH blends were visualized under the minimum magnification (i.e.,

75X). However, the foaming behaviors of PS blown by pure EtOH, due to its high cell

population density and small cell size, were observed under the maximum magnification (i.e.,

450X). Therefore, the delays between the onset time of bubble formation and the time at which

the first bubble became observable were different, making it impossible to have a meaningful

comparison of the onset time of bubble formation between these two sets of data.

From these extracted frames, the observed cell densities at different times and the bubble

generation rates during the foaming processes were extracted and plotted in Figure 8.4 and

Figure 8.5, respectively. Each data point represents the average cell density obtained from the

three experimental runs and the error bars represent the standard deviations. The results indicate

that pure EtOH is a more powerful blowing agent than CO2, leading to a 5 orders of magnitude

increase in the cell density. For the samples blown with CO2-EtOH blends, the cell density and

the bubble generation rate for the samples blown with the mCO2:mEtOH ratio equals to 80:20 were

virtually indifferent from those for the sample being blown by pure CO2. However, the cell

density and the bubble generation rate for the samples blown with the mCO2:mEtOH equals to 60:40

were slightly higher than those for the sample being blown by pure CO2. Although it was

impossible to compare the onset time of PS-EtOH foaming with the other experimental cases, the

onset time of PS-CO2 foaming and PS-CO2-EtOH foaming could be compared as they were

observed under the same magnification power. The experimental results reveal that with the

presence of EtOH, the occurrence time of the first observable bubble was delayed when

comparing with the case where pure CO2 was used as the blowing agent.

In addition to the cell densities and the bubble generation rates, the effects of blowing

agent composition on the cell growth behaviors were investigated. The average bubble radii at

174

various times were analyzed, and the results were shown in Figure 8.6. It could be observed that

the bubble expansion rates decreased as EtOH content increased. Moreover, Figure 8.3 indicates

that the average bubble size was smaller when EtOH was added as a co-blowing agent.

Similar to the in-situ visualization data, the SEM micrographs (i.e., Figures 8.7 (a)

through (c)) of the PS foams indicated that the foaming behaviors of the PS-CO2 system and the

PS-CO2-EtOH systems were very different from that of the PS-EtOH system. On the one hand,

when pure CO2 or CO2-EtOH blends were used as the blowing agent, a single layer of large cells

(i.e., about 100 μm in size) were formed at the bottom of the samples (i.e., the polymer-sapphire

interface). In contrast, the cell morphology of the pure EtOH case resulted in a uniform

distribution of tiny cells (i.e., about 10 μm in size) throughout the entire foam thickness. A larger

foam expansion ratio was achieved in the pure EtOH case. Furthermore, by comparing the results

obtained by using pure CO2 and CO2-EtOH blend of mCO2:mEtOH equals to 60:40, it seems that

the presence of EtOH led to smaller cell sizes, which was consistent with the visualization data.

Figure 8.8 illustrated a 1000X SEM micrograph in a region near the top surface of the PS

foam being blown by pure EtOH. It can be observed that there existed some submicron-sized

bubbles in the unfoamed region around the large cells in the foam. However, the mechanism of

generating these nano-sized bubbles has yet to be identified.

8.3.3. Hypotheses of Foaming Mechanism

Based on the experimental results, a few hypotheses can be explored as potential

explanations of the different foaming behaviors in various cases. Firstly, based on the measured

solubility of EtOH in PS under atmospheric pressure [217], it is speculated that a substantial

amount of EtOH can dissolve in PS under typical conditions (i.e., high pressure and high

temperature) in foaming processes. As a result, a high degree of thermal instability (i.e.,

supersaturation) would be established upon the rapid pressure drop, leading to a faster nucleation

175

rate and higher cell population density. Secondly, the phase change of EtOH from a liquid state

to a gas state might have locally cooled down the polymer matrix and stabilized the cellular

structure before severe cell coalescence occurred. The pressure-volume-temperature

measurement conducted by Bazaev et al. [220] showed that the vaporization pressure of EtOH at

150°C and 200°C are about 1 MPa and 2.9 MPa, respectively. Therefore, when the system

pressure was dropped from the saturation pressure (i.e., 800 psi or 5.5 MPa) to the atmospheric

pressure during foaming, a large amount of heat would have been dissipated to vaporize the

EtOH (note: the latent heat of vaporization for EtOH is about 904 kJ/kg) and resulted in the

localized cooling. This cooling effect would increase the melt strength and contributed to the

stabilization of the foam structure. In other words, cell coalescence, which is a major factor that

leads to the non-uniform cellular structure, had been successfully avoided in the pure EtOH case

and thereby led to uniform cell morphology. Because of this and the promoted cell nucleation, a

large expansion ratio could be achieved. Furthermore, this localized cooling effect might lead to

a lower gas diffusion rate in the PS matrix, resulting in the slower bubble growth rate in the PS

foam blown by CO2 with the presence of EtOH. As a result, it caused the further delay of the

occurrence of the first observable bubble (i.e., cell size ~ 12 μm) as indicated in Figure 8.5.

Although the experimental results have provided new insights to construct various

interesting hypotheses in attempt to explain the roles of EtOH as the primary blowing agent or as

a co-blowing in PS foaming, further studies (e.g., solubility measurement) will be needed to

verify the validity of these hypotheses.


In this chapter, the possible roles of EtOH as the pure blowing agent or as a co-blowing

agent with CO2 in producing PS foam were investigated. The rheological measurement had

proven that EtOH has similar plasticization powers as CO2 in PS. Therefore, this fact provides

176

the foam industry another possible processing route to circumvent the processing challenges

caused by the limited solubility of CO2. Furthermore, the in-situ foaming visualization results

and the SEM analyses have guided us to the speculation of the potential roles of EtOH as the

primary blowing agent or co-blowing agent in manufacturing PS foam.

177

21Table 8.1. Physical properties of polystyrene

PS685D

MFI 1.5 g/10 min

Mn 120,000 Mw/Mn 2.6 Specific gravity 1.04 Glass transition temperature (Tg) 108°C

22Table 8.2. Physical properties of blowing agents

Carbon Dioxide Ethanol

Chemical formula CO2 C2H5OH Molecular weight 44.01 g/mol 46.069 g/mol Boiling point -78.45 °C 78.35 °C Critical temperature 31.05 °C 243.05 °C Critical pressure 7.38 MPa 6.38 MPa

23Table 8.3. A summary of experimental cases

Experiment Number Mass ratio of Blowing Agent (mCO2 : methanol )

1 100:0 2 80:20 3 60:40 4 0:100

178

90Figure 8.1. A schematic of the tandem foam extrusion system [215]

91Figure 8.2. Effects of blowing agent composition and melt temperature on shear viscosity of PS melt

179

92Figure 8.3. Snapshots of foaming visualization data of the experimental runs

93Figure 8.4. Effects of blowing agent composition on cell population density

180

94Figure 8.5. Effects of blowing agent composition on cell generation rate

95Figure 8.6. Effects of blowing agent composition on average cell radius

181

(a) Pure CO2

(b) CO2-EtOH blend (mCO2:methanol = 60:40)

(c) Pure EtOH

96Figure 8.7. SEM micrographs of PS foams obtained by (a) pure CO2, (b) CO2-EtOH blend (mCO2 : mEtOH = 60 : 40), and (c) pure EtOH

97Figure 8.8. The SEM micrograph (magnification = 1000X) of PS foams obtained by pure EtOH

182

Chapter 9 SUMMARY, CONCLUDING

REMARKS & FUTURE WORK

9.1. Summary

The cell nucleation, growth and coarsening mechanisms in plastic foaming were

investigated through a series of theoretical studies, computer simulations, and experimental

investigations in this thesis research. First, through the in-situ visualization of various batch

foaming experiments, the effects of processing conditions on cell nucleation phenomena were

studied. Second, a new heterogeneous nucleation mechanism was identified to explain the

foaming behavior with the existence of inorganic fillers (e.g., talc). Subsequently, an accurate

simulation scheme for the bubble growth behaviors, a modified heterogeneous nucleation theory,

and an integrated model for simulating the simultaneous cell nucleation and growth processes

were developed. Cell nucleation, growth, and coarsening dynamics were modelled and simulated

to enhance the understanding of the underlying sciences that govern these different physical

183

phenomena during plastic foaming. The impacts of various commonly adopted approximations

or assumptions were studied, resulting in useful guidelines for future work in the computer

simulation of plastic foaming processes. Furthermore, strategies were developed to predict the

onset point of cell formation in plastic foaming processes through the determination of the

minimum pressure drop required to initiate a reasonable cell nucleation rate, which is denoted as

ΔPThreshold. Finally, an experimental research was conducted to demonstrate how the elucidation

of the mechanisms of various foaming phenomena would aid in the development of novel

processing strategies (e.g., foaming with blowing agent blends) to enhance the control of cellular

structures in plastic foams.

9.2. Key Contributions from this Thesis Research

In summary, the theoretical, computer simulation, and experimental work conducted in

this study lead to the following contributions and conclusions:

1. Experimental simulations of plastic foaming were conducted to illustrate the mechanisms

under which the dissolved gas contents, the pressure drop rates, and the system

temperature affect polymeric foaming behaviors. When the initial gas content is higher,

the increased dissolved gas concentration and the reduced interfacial energy will lead to a

higher cell density. When a higher pressure drop rate is used, a more rapid increase in

thermodynamic instability will cause more bubbles to nucleate sooner. Consequently, a

larger portion of blowing agent will contribute to the formation of new cells, and thereby

the cell density will again be higher. Finally, even though a higher system temperature

will increase both the thermal fluctuation of the gas molecules and the initial cell

nucleation rate, the accelerated cell growth means that more gas will be consumed for the

cell growth and result in a slightly lower cell density.

184

2. A new heterogeneous nucleation mechanism has been discovered through the in-situ

visualization of talc-enhanced PS-CO2 foaming. Experimental evidence indicated that the

expansion of nucleated cells can trigger the formation of new cells around them despite

the lower gas concentrations in these regions. It is speculated that the growing cells are

able to generate extensional stress fields around the nearby filler particles, resulting in

local pressure fluctuations. The additional local pressure drops lead to a further reduction

of the critical radius and the free energy barrier for cell nucleation in these regions. As a

result, the heterogeneous nucleation of new cells and the expansion of the pre-existing

gas cavities are promoted. Together, these accelerate the cell formation and contribute to

the higher cell density. This proposed heterogeneous nucleation mechanism can be

extended to other heterogeneous systems (e.g., polymer blends, nanocomposites, and

semicrystalline polymers) to explain the enhanced cell nucleation phenomena.

3. The stress-induced nucleation is suppressed in the PS-talc-CO2 foaming by either

increasing the system temperature or the dissolved gas contents. At higher system

temperatures or higher blowing agent contents, the reduction in viscosity and elasticity of

the polymer-gas solution may weaken the extensional stress field being generated and

suppress the additional reduction of the local pressure. This provides a partial explanation

for the limited impact of increasing talc content on the cell density, when high carbon

dioxide content is used to foam polystyrene. It was also observed that higher talc content

promoted stress-induced nucleation, whereas improved PS-talc compatibility slightly

reduced it. The talc size seems to have no effect on the phenomenon.

4. Computer simulations of the bubble growth phenomena during plastic foaming were

conducted using experimentally recorded pressure decay data and the realistic values of

various physical parameters. The simulation results demonstrate good quantitative

185

agreements with the visualization data obtained from the in-situ physical foaming

processes of polystyrene and carbon dioxide.

5. Using the established mathematical model and simulation scheme for the bubble growth

phenomena, a series of sensitivity analyses were performed to investigate the effect of

various thermo-physical, rheological, processing, and simulation parameters on the

bubble growth dynamics. It has been shown that gas diffusivity and solubility are two of

the most important factors that govern cell growth dynamics. Although the expansion

behaviors of cells also depends on the initial bubble size, surface tension, and viscoelastic

properties, the effects are less prominent than those of gas diffusivity and solubility under

the processing range being considered.

6. Computer simulations of bubble growth and collapse dynamics during CBA-based

foaming processes were conducted to demonstrate the stability of cells in the later stage

of plastic foaming. The simulations reveal that upon the CBA decomposition, a gas-rich

region around the newly formed bubble will develop. This gas-rich region contributes to

the bubble expansion during the initial phase of its life cycle. Meanwhile, the continuous

gas loss to the surroundings and the reduction of Pbub will lead to the increase of Rcr.

Finally, when Rcr becomes larger than Rbub, the bubble starts to collapse.

7. The overall trend of the bubble growth and collapse phenomena during various stages has

been revealed. Both the theoretical and experimental results indicate that smaller bubbles

are less sustainable than larger ones. Therefore, it is always challenging to generate

polymer foams with sub-micron cell sizes. Furthermore, the computer simulation results

suggest that diffusivity, solubility and surface tension are important parameters that

govern the fate of the generated bubble. It is believed that a lower diffusivity, a higher

186

solubility, and a lower surface tension will enhance the sustainability of the bubbles

formed in CBA-based, pressure free foaming processes.

8. A modified heterogeneous nucleation theory has been developed to account for the

randomness of surface geometry of the heterogeneous nucleating sites. Using an

integrated model that combines the modified nucleation theory and the bubble growth

simulation model, computer simulations were conducted to study the PS-CO2 foaming

processes under various processing conditions (i.e., different pressure drop rates or

dissolved gas contents). The simulation results were compared with batch foaming

experimental results to evaluate the validity of the modified nucleation theory. Despite

some unavoidable discrepancies between the simulation results and experimental cell

density data, the research demonstrates that the modified nucleation theory provides an

improved explanation of the cell nucleation phenomena during plastic foaming and

explains certain quantitative facets of the experimentally observed data.

9. The impact of the commonly-used approximation to the pressure inside a critical bubble

(i.e., Pbub,cr = Psat) on the simulation of the nucleation phenomena in the polymeric

foaming processes was investigated. It was found that the simulation result based on the

approximation predicts an earlier onset time for bubble nucleation. Moreover, it also

significantly overestimated the final cell density, the cell nucleation rate, and the gas

consumption rate. In brief, it is recommended that the approximation be avoided in the

numerical simulation of foaming processes.

10. Through a series of simultaneous simulations of cell nucleation and growth phenomena,

it has been demonstrated that the commonly used step pressure drop profile

approximation in computer simulations can lead to significant overestimation of cell

187

density and underestimation of cell size. Thus, it is recommended that the approximation

be avoided in the numerical simulation of plastic foaming.

11. A semi-empirical approach and a theoretical approach have been developed to estimate

the pressure drop threshold for cell nucleation. This enhances the development of

processing strategies in foaming systems to suppress premature cell growth and to better

control cell morphology as well as the volume expansion ratio of foamed plastics.

Additionally, with the knowledge of the onset point of cell nucleation, it is possible to

develop innovative means to suppress cell growth in order to produce nano-cellular

foamed products.

12. The effect of pressure drop rate, gas content, and processing temperature on the pressure

drop threshold (ΔPthreshold) were studied. It has demonstrated that the pressure drop rate

has no effect on ΔPthreshold, while ΔPthreshold decreases with increasing gas content and

processing temperature. Moreover, the dissolved blowing agent content showed a more

significant effect than the processing temperature on ΔPthreshold.

13. The rheological measurements of PS-ethanol and PS-CO2 systems proved that ethanol

has similar plasticization powers as CO2 in PS. Therefore, this fact provides the foam

industry another possible processing route to circumvent the processing challenges

caused by the limited solubility of CO2.

14. The in-situ foaming visualization of PS-CO2-ethanol foaming processes and the SEM

analyses of the resulting foams reveal that the foaming behaviors of the PS-CO2 system

and the PS-CO2-EtOH systems were very different from that of the PS-ethanol system.

While the uses of pure CO2 or CO2-EtOH blends to foam PS resulted in a single layer of

large cells (i.e., about 100 μm in size), using ethanol as the lone blowing agent lead to a

uniform distribution of tiny cells (i.e., about 10 μm in size) throughout the thickness of

188

the PS foam. Furthermore, some submicron-sized bubbles were observed around the

large cells in the foam. Based on the measured solubility of ethanol in PS under

atmospheric pressure, it is speculated that the desirable foam morphology in the pure

ethanol case is related to the high solubility of ethanol in PS.

15. Although both pure CO2 and CO2-ethanol blends resulted in PS foams with similar cell

population densities, the foam obtained by using the blowing agent blend had smaller

cells. This is believed to be attributed to local cooling in the polymer matrix upon the

phase change of ethanol from a liquid state to a gas state. This cooling effect would not

only slower the cell growth rate but also increase the melt strength of the cell walls. As a

result, the cellular structure is stabilized before severe cell coalescence occurred.

9.3. Recommendations and Future Work

The following suggestions are made for directing the future research of the underlying

science behind the physical phenomena that occur during plastic foaming and for different novel

foaming technologies and strategies:

1. For research related to the cell nucleation phenomena, one limiting factor in analyzing the

effects of different processing parameters, material parameters or processing strategies is

the difficulty in capturing the nucleation event. The extremely high speed and small size

scale involved in cell nucleation makes it very challenging to elucidate the process. If

nucleation could be arrested early through a fast temperature quench, transmission

electron microscopy (TEM) images could be used to provide insight as to where and how

the cells nucleate. If possible, the development of an in-situ foaming visualization system

utilizing the TEM technology would significantly advance the scientific understanding in

this research field.

189

2. Cell nucleation in plastic foaming is controlled by both the thermodynamically defined

cell nucleation and the activation of pre-existing gas cavities. Therefore, a simulation

model that describes the two perspectives of cell formation would be beneficial to

thoroughly understand the cell nucleation phenomena.

3. Experimental measurements of the contact angles in various polymer-gas-nucleating

agent systems are also recommended. This interfacial property is critically important for

predicting the cell nucleation rate; however, it has not yet been clearly studied.

4. Fundamental studies on microvoids in polymer-gas solutions and the wetting behaviors

of polymer-gas solution on various heterogeneous nucleating agents would also be

beneficial to understand the cell nucleation phenomena. Possible directions of this

fundamental research include:

a. developing measurement techniques of the sizes and quantities of microvoids in

different polymers, and the sizes of trapped gas pockets at the polymer-nucleating

agent interfaces;

b. investigating the effects of different processing parameters (e.g., processing

temperature, system pressure, dissolved gas content) and material parameters (e.g.,

polymer blending) on the sizes of microvoids in the polymer matrix; and

c. exploring the effects of shear or extensional field on the microvoids or pre-existing

gas cavities.

5. Fundamental studies on the effects of polymer blends or blowing agent blends on cell

nucleation, growth, and coarsening phenomena. The end results of these investigations

would increase the flexibility to control and tailor the cellular structures and foam

properties.

190

6. Investigation of the heterogeneous cell nucleation mechanisms in various plastic or

plastic composite systems. It is believed that the cell nucleation mechanisms being

discovered in this thesis research could be extended to explain the heterogeneous cell

nucleation in other heterogeneous systems such as polymer blends, semicrystalline

polymers, and nanocomposites.

7. Development of experimental foaming simulation systems, with the abilities to conduct

in-situ visualization and to induce extensional and/or shear stress. Such a system would

provide a lot of valuable insights to elucidate and control the stress-induced nucleation.

8. Development of computer software to simulate the local stress fluctuation in the polymer

matrices during typical plastic foaming processes (e.g., extrusion foaming or structural

foam molding). These results could be utilized to develop strategies to control the stress

field and thereby the stress-induced nucleation.

9. Fundamental studies on the effects of elasticity and viscosity on the cell nucleation,

growth, and coarsening phenomena. Based on the in-situ visualization of the foaming

behaviors of talc-enhanced PS-CO2 foaming in this thesis, it is speculated that both

elasticity and viscosity of the polymer-gas solution affect the pressure fluctuation in the

polymer matrix, and thereby the cell nucleation mechanisms. Therefore, a more in-depth

study in this context would advance the plastic foaming technology.

191

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