MECHANISMS OF CELL NUCLEATION, GROWTH, AND … · 2010-11-03 · MECHANISMS OF CELL NUCLEATION, GROWTH, AND COARSENING IN PLASTIC FOAMING: THEORY, SIMULATION, AND EXPERIMENT Siu Ning
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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
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
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 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 4.5. 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 5.6. 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 6.16. 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
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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
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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
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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
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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.
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. Results and Discussion
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.
4.6. Summary and Conclusions
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.
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.
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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
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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
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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.
6.4. Results and Discussion
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
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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
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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.
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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.
6.5. Summary and Conclusions
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
Gas Content (C) [wt.%] / Psat [MPa]
Pressure Drop Rate (-dPsys/dt|max) [MPa/s]
System Temperature (Tsys) [˚C]
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
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.
7.3. Results and Discussion
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
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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.
7.5. Summary and Conclusions
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%
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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. Results and Discussion
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.
8.4. Summary and Conclusions
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
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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.
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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