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Polymer Testing 31 (2012) 417–424

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Polymer Testing

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Test equipment

A visualization system for observing plastic foaming processes undershear stress

A. Wong, C.B. Park*

Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8

a r t i c l e i n f o

Article history:Received 8 November 2011Accepted 22 December 2011

Keywords:Polymer foamNucleationVisualizationShear strainPolystyrene

* Corresponding author. Tel.: þ1 416 978 3053; faE-mail addresses: awong@mie.utoronto.ca (A.

utoronto.ca (C.B. Park).

0142-9418/$ – see front matter � 2012 Elsevier Ltddoi:10.1016/j.polymertesting.2011.12.012

a b s t r a c t

Previous studies offered theories to explain shear-induced bubble nucleation and growthphenomena in plastic foaming processes, but empirical verification was limited due todifficulty in observing these processes in situ under an easily adjustable and uniform shearflow. This study presents a novel visualization system that successfully achieved this goal.The system allows easy control of the critical experimental parameters: applied shearstrain, shear strain rate, temperature, pressure, pressure drop rate, plastic material andblowing agent. From a foaming visualization study of polystyrene, it was observed that cellnucleation rate and maximum cell density increased with the applied shear strain, whichwas due to the decreased local system pressure, detachment and growth of microvoids,and elongation of bubbles. This foaming visualization system provides a direct andeffective way to investigate the mechanisms of bubble nucleation and growth underdynamic conditions that simulates industrial plastic foaming processes.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Bubble nucleation and growth phenomena in plasticfoaming are key research subjects. This is because theydetermine the final structure of the foam, that is, bubblesize distribution and density, porosity and volume expan-sion ratio. These factors determine the plastic’s mechanical,thermal, acoustic and optical properties that relate toa wide range of its applications. Since bubble nucleationand growth phenomena are often encapsulated within thefoam processing equipment in extrusion foaming andinjection foam molding, early researchers conducted in-situ foaming observations through transparent slit dies[1–4] and transparent mold cavities [5,6]. Another in-linefoaming detection technique based on ultrasonicmeasurements through a transparent slit die was reportedby Tatibouët and Gendron [7]. In particular, Han and Yoo [6]reported that the level of stress in a plastic melt might have

x: þ1 416 978 0947.Wong), park@mie.

. All rights reserved.

a significant effect on bubble formation and growth ina structural foam molding process.

In a subsequent extrusion foaming study, Han and Han[2] pointed out that, in addition to nucleation by thermalfluctuations and cavitation, both shear stress near the diewall and flow around the die center could induce cellnucleation. Similar results were also reported by Tsujimuraet al. [3], Taki et al. [4], and Tatibouët and Gendron [7].However, bubble nucleation and growth phenomena ina continuous flow of plastic-gas solutions are highlycomplex, and their coupled thermodynamic, multi-phasefluid dynamic and rheological processes are difficult tothoroughly understand. Moreover, in these cases, the effectof shear or flow could not be examined in isolation. Otherresearchers, such as Otake et al. [8], Taki et al. [4] and Guoet al. [9], developed foaming systems to observe plasticfoaming in batch processes, but the plastic samples werefoamed under static conditions, so it was difficult toinvestigate the shear- or flow-induced bubble nucleationand growth processes that typify industrial foamingprocesses. Favelukis et al. [10] used a Couette apparatusdeveloped by Canedo et al. [11] to observe bubble

A. Wong, C.B. Park / Polymer Testing 31 (2012) 417–424418

nucleation and growth in a viscous liquid under simpleshear flow. However, the system was incapable of workingunder the high temperatures and pressures required inplastic foaming processes. Mackley et al. [12] and Mackleyand Spitteler [13] developed a capillary rheometer with anoptical section for viewing foaming processes understressed conditions. These instruments were capable ofsimulating a wide range of flow situations in extrusion andinjection molding. However, due to the nature of pressuredriven flows, the pressure and rate of deformation werenon-uniform and, hence, so was the induced shear flow.

To investigate the effect of shear stresses on plasticfoaming in isolation, some researchers developed batchfoaming systems that induced shear stresses [14,15] ora combination of shear stresses and vibrations [16,17] ina plastic-gas mixture at high temperatures and pressureswhich, when depressurized, generated foams. In particular,Chen et al. [14] developed a “cell stretch model” to explainshear-induced nucleation whereby bubble nuclei arestretched during shear flow. Due to their larger surface areathese nuclei expand more easily than spherical bubbles.Both Zhu et al. [16] and Gao et al. [17] demonstrated thatbubble densities were increased and bubble size uniformitywere improved by superimposing an oscillatory vibrationonto a shear flow in an orthogonal direction. Holl et al. [18],and Handa and Zhang [19] also investigated stress-inducedbubble nucleation, but their foamingexperimentswereonlyconducted in the solid state at relatively low temperatures.Also, in each of these studies [14–19] characterizationswerecarried out with scanning electron microscopy (SEM) afterthe foams had cooled and stabilized. Since cell coalescence,coarsening and collapse could occur to nucleated cellsbefore they were stabilized, the shear stress effect on cellnucleation could not be determined accurately.

Each of the early studies on bubble nucleation andgrowth has made a significant contribution to the under-standing of plastic foaming processes. At the same time, toverify and improve the existing theories of shear-inducedplastic foaming, it is crucial to obtain clear, empiricalbubble nucleation and growth phenomena data under theeffects of shear in an isolated manner. This has not beenachieved in the previously mentioned studies. In an earlierstudy, we developed a batch foaming visualization systemto observe the effect of extensional strain on bubblenucleation and growth [20]. Our current study presents thedevelopment of a novel visualization system for theobservation of plastic foaming processes under controllableshear strain and shear strain rate in batch process. Foamingexperiments of polystyrene (PS) blownwith carbon dioxide(CO2) were carried out to demonstrate the capability of thesystem, and to demonstrate the effect of shear strain on cellformation behavior based on in-situ foaming videos.

2. System development

The central objective of our studywas to develop a novelsystem by which to observe and capture plastic foamingprocesses under controllable shear strain and shear strainrate. To do so, the system employed the following threefunctions: i) application of a uniform simple shear flow toa plastic-gas melt under a high temperature and pressure;

ii) saturation of the plastic melt with gas and the subse-quent inducement of foaming by rapid depressurization;and iii) capture of the bubble formation and growthprocesses with fine temporal and spatial resolution. Thesefunctions had to be achieved while simultaneously main-taining easy control and adjustment of the following crit-ical parameters: applied shear strain, shear strain rate,system temperature, pressure, pressure drop rate, type ofplastic and type of blowing agent.

2.1. Function I: generating a uniform simple shear flow toa plastic melt at a high temperature and pressure

The first function was achieved with a sliding platemechanism, where a thin molten plastic film (e.g., 0.4 mmin thickness) was sandwiched between two flat sapphirelenses: a static sapphire lens on the top and a slidingsapphire lens on the bottom. As the bottom sapphire lensactuated linearly, a simple shear strain was applied to theplastic film. The design of this sliding plate mechanismwasinspired by the high-pressure sliding plate rheometer(HPSPR) developed by Koran and Dealy [21] that wasdesigned to measure the shear stresses of plastic melt. Forour system, sapphire lenses were used as sliding plates tomake the plastic sample visible during the foaming process.Instead of a sliding sapphire lens on the bottom, we hadalso considered using a rotating sapphire lens because of itscompact design, but rejected it due to the inherent non-uniformity of the shear field.

The sliding sapphire lenswas attached to ametal housingto forma sliding plate assembly. Themotion of this assemblywas controlled by a linear actuator via a drive shaft. Thelinear actuator (OrientalMotors EZA6) consisted of a steppermotor with accurate feedback control and was coupled toa lead screw to convert the rotary motion to linear. Theresolution and maximum speed of the linear actuator was0.01 mm and 300 mm/s, respectively, which permittedaccurate control of the shear strain and shear strain rate. Themotion of the motor was programmed and executed viaHyperTerminal software. A pair of photo-interrupters wereinstalled as limit sensors to prevent the moving plateassembly from colliding with the chamber body. Themaximum displacement of the sliding plate assembly was40mm. Based on themaximum actuator speed (300mm/s),and using a sample thickness of 0.4 mm, the upper limit onthe strain rate (dg/dt) is 750 s�1, but it can only be sustainedfor a maximum of 0.13 s due to the displacement constraint.Therefore, to achieve a steadier shear flow, the maximumdg/dt chosen was 100 s�1, which could be sustained fora periodof 1 s. Therefore, themaximumstrain (g) that canbegenerated is 100. By using a thinner plastic sample, a highermaximum g and dg/dt can also be achieved.

2.2. Function II: saturating the plastic melt with a highpressure gas and inducing foaming by rapid depressurization

To achieve the second function of foaming the plasticspecimen, it first needs to be saturated with a blowingagent at a high temperature and pressure. Foaming canthen be induced by the thermodynamic instability thatoccurs with rapid depressurization. To this end, the sliding

A. Wong, C.B. Park / Polymer Testing 31 (2012) 417–424 419

plate assembly was enclosed inside a stainless steelchamber. The chamber was positioned so that the plasticspecimen’s largest side was facing up while it was shearedhorizontally. The chamber was installed on a stand, onwhich the linear actuator and the gas inlet/release valveswere also mounted. The chamber temperature wascontrolled by two cartridge heaters with Proportional-Integral-Derivative (PID) feedback control and a resis-tance temperature detector (RTD) probe. The blowingagent was supplied by a gas cylinder via a syringe pump,which set and maintained the gas pressure inside thechamber. The top sapphire lens was backed by a metalcover with a slot for visualization.

Even at high temperatures, gas saturation into plasticvia diffusion processes can take a long time. Using theHPSPR [21], Park and Dealy [22] demonstrated that 99%saturation of CO2 into the center of a high-density poly-ethylene (HDPE) sample at 180 �C could take 190–230 min,while two days were required for a PS sample. An oscil-lating shear motion applied on the plastic decreased thesaturation time (td) required by half for the HDPE [22], butother plastics with lower gas diffusivity still had a muchlarger td. This could lead to degradation issues, especiallyfor heat-sensitive plastics. One main factor determining tdis the length of the diffusion path (l). If both the top andbottom surface of the plastic film is in contact with thesliding plates during the gas saturation phase, such as inthe HPSPR case, gas can only diffuse through the four sidesof the plastic film to the center of the sample, which resultsin a large l. Since td f l2 [23], a small increase of l cansignificantly increase td. Therefore, to decrease td for allmaterials, a mechanism was designed to lower the slidingsapphire window during the gas saturation phase underhigh pressure, so that the top surface of the plastic filmwasopen for gas diffusion. Once gas saturation was completed,the sliding sapphire window was moved upwards slightlyuntil the sample was in contact with the top sapphire lens(the static plate) so that the shear strain could be applied tothe plastic film. To achieve this, the rectangular metalhousing for the sliding sapphire lens was formed by twowedges that slid, with respect to each other, along a slantededge. An adjustment shaft that was inserted into thechamber on the opposite side of the motor was partiallythreaded with a fine pitch, which acted as a lead screw topush the upper wedge slightly along the shear direction.The adjustment and drive shafts were positioned collinearto each other by two brass bearings. A rectangular framesurrounding the chamber was designed to connect theadjustment shaft and the drive shaft/motor from outside ofthe chamber. This prevented the shafts from being forcedout of the chamber by the internal pressure and minimizedthe axial load on the motor. The adjustment shaft wasactuated by a threaded adjustment knob, which wasbacked by a thrust bearing on the rectangular frame. Whenthe upper wedge was pushed along the shear direction, itmoved slightly upwards. At the same time, the rectangularframe kept the lower wedge stationary. The upwardmotionstopped once the sample came into contact with the topsapphire lens. An excessive upward movement wouldcompress the sample, thus inducing a normal/compressivestress in it. This can affect the foaming behavior of the

plastic film and must be avoided. To prevent it, a microm-eter was also installed on the back side and collinear to theadjustment shaft to accurately monitor its position and,hence, the vertical position of the upper wedge. The plasticfilm must also have a uniform thickness to ensure itsproper adhesion to the sapphire lens and to prevent anylocal normal stresses.

Afterwards, the motor was actuated to pull the slidingplate assembly, the rectangular frame, the drive shaft andthe adjustment shaft together along in the shearing direc-tion, thus applying shear strain to the plastic film. As Fig. 1shows, this design allows the functions of gas saturationand shear strain application to be fulfilled independently,which is a good design practice according to the AxiomaticDesign principles [24,25]. Once shear strain was applied,a rapid pressure drop was induced in the chamber byopening a gas exit valve. At the same time, the third func-tion of the system came into play: a high-speed camerawastriggered to record the foaming process. In order tosynchronize the depressurization and video recordingprocesses, both the gas exit valve (a solenoid valve) and thehigh-speed camera were triggered simultaneously bya control panel programmed with the LabVIEW software.Meanwhile, the pressure data generated by a pressuretransducer was recorded during the depressurizationprocess, which could be used afterwards to determine thepressure drop rate. By adjusting the opening of a metervalve installed in series along the gas exit path, the pressuredrop rate could be adjusted. It is noted that the shear strain,and not the shear stress, is directly controlled in thissystem. Due to the viscoelastic nature of plastic, the stressapplied to the plastic specimen decreased over time.Therefore, the stress relaxation behavior of the plastic/gassolution should be considered.

In saturating and foaming a plastic sample, anothercritical design challenge is to prevent gas leaking from thechamber over a wide range of temperatures and pressures.To address this, the pressure seal between the chamberbody and the top sapphire lens is created by an O-ring thatfits into a groove on the chamber body. The dynamicpressure seals for the drive shaft and the adjustment shaftare each made with a spring-loaded cup seal that fits intoa brass bearing installed on the chamber body. The staticseal between each brass bearing and the chamber is ach-ieved with an O-ring that fits into a groove on the bearing.Based on the operating limits of the seals, the maximumoperating temperature and the pressure of the system are230 �C and 2900 psi (under the reciprocating movement ofthe drive and adjustment shafts at a speed of up to1500 mm/s), respectively.

2.3. Function III: capturing the bubble formation and growthprocesses with fine temporal and spatial resolution

To fulfill this function, a high-speed camera (PhotronUltima APX) with a magnifying lens was adapted from ourprevious studies [9,20]. The camera had a maximumrecording speed of 120,000 frames/s. Using the zoom lenswith high magnification, a field of view of w400 mm anda spatial resolving limit of 1–2 mm could be achieved. Dueto the high shutter speed and magnifying power, a high

Fig. 1. Operation of the moving plate assembly.

A. Wong, C.B. Park / Polymer Testing 31 (2012) 417–424420

intensity light source is needed. Ring lighting and coaxiallighting allow light to be provided from the top surface tothe plastic film and reflected to the zoom lens, so that thechamber does not need to be transparent from top tobottom along the optical axis (see Fig. 2a and b), which isnecessary if transmissive lighting is used (see Fig. 2c).Therefore, the chamber design could be simplified.However, it was observed that for both ring lighting andcoaxial lighting, the incident light was partially reflectedfrom the top sapphire lens surface, so that the cameraimages were blurred, and the contrast levels of the imagesdropped significantly. Consequently, transmissive lightingwas used. A center slot was incorporated into the upper andlower wedges to allow incident light from the bottom sideto pass through the plastic sample. An additional sapphirelens was installed on the bottom side of the chamber,where a fibre optic cable was attached as the transmissivelighting element. The pressure seal between the chamberbody and the additional sapphire lens was achieved witha sealing mechanism detailed in our previous work [20],which has been shown to work over repeated runs with noneed to replace the sealing O-ring. Themaximumoperatingtemperature and pressure also remained unchanged. Theincident light was provided to the fibre optic cable bya halogen lamp with controllable light intensity. Visuali-zation took place at the center region of the sample, wherethe edge effect that impacts the uniformity of the shearfield is minimized [21]. The optical systemwas mounted ona 3-way linear stage that allowed for accurate adjustmentof its position along three orthogonal axes: i) the opticalaxis; ii) in the shear strain direction; and iii) perpendicular

Fig. 2. a) Coaxial lighting; b) Ring lig

to the first two axes. Fig. 3 shows the cross-sectional viewof the foaming chamber. Fig. 4 shows a schematic of theoverall foaming visualization system.

3. Experimental materials and procedure

3.1. Experimental materials and sample preparation

To verify the system’s capability, plastic foamingexperiments on PS foamed with CO2 were conducted undervarious processing conditions. The plastic material used forthe foaming experiments was PS (Styron PS685D, DowChemical Ltd.). The PSmelt flow indexwas 1.5 g/10min andits density 1.04 g/cm3. The blowing agent used was CO2(99% pure, Linde Gas Canada). The PS material wascompression molded into thin films 0.4 mm thick usinga hot press at 180 �C. After release from the hot press thesamples were immediately quenched with water atapproximately 13 �C. Then, the samples were cut intorectangular shapes measuring 38 mm long, 24 mm wide,and 0.4 mm thick.

3.2. Experimental procedure

To carry out the foaming experiment, a rectangular-shaped plastic film was placed on top of the slidingsapphire window, which was positioned in the lowervertical position for gas saturation. The chamber was thenmaintained at the designated foaming temperature andpressure for 30 min to allow the blowing agent to dissolveinto the plastic film. In this study, the operating

hting; c) Transmissive lighting.

Fig. 3. Detailed design of the foaming chamber (cross-sectional view).

Fig. 4. Foaming visualization system with shear strain inducing ability.

A. Wong, C.B. Park / Polymer Testing 31 (2012) 417–424 421

A. Wong, C.B. Park / Polymer Testing 31 (2012) 417–424422

temperature and CO2 pressure were 180 �C and 500 psi,respectively. Afterwards, the adjustment shaft was movedmanually to raise the level of the sliding sapphire lensupwards until the plastic sample connected with the topsapphire window. A small-magnitude oscillatory shearstrain was then applied to the plastic sample along thelongest dimension, which had been shown to improve theadhesion of the plastic sample in a sliding plate rheometer[26]. The plastic sample was then held for another 10 minto allow the shear stress, induced by the oscillatory shearstrain, to diminish. A desired shear strain along the longestdimension was then induced in the plastic film. After thedesired strain at the desired strain rate had been reached,the motor was stopped, and the gas was released from thechamber. The rapid pressure drop inside the chambercaused foaming in the plastic film. At the same time, thefoaming process was captured in situ by the high-speedcamera, and the pressure data within the chamber wasrecorded.

4. Results and discussion

A set of three PS/CO2 foaming experiments were con-ducted to verify the capability of the foaming system. Inorder to study the shear strain effect in isolation, thesaturation pressure (P), average pressure drop rate (–dP/dtjavg), and system temperature (Tsys) were kept constant at3.45 MPa (500 psi), 12 MPa/s, and 180 �C, respectively.Based on Li et al.’s PS/CO2 solubility data, the dissolved CO2content in PS under these conditions was 1.27 wt% [27]. Thelevels of P and dP/dtjavg were selected at low levels, so thatthe gas content and pressure drop effect would not domi-nate over the shear strain (g) effect in the foamingprocesses. The PS samples were first foamed under staticconditions (i.e., g ¼ 0). Subsequent foaming experimentswere conducted with g ¼ 12.5 and 25 for both materialswhile keeping dg/dt constant at 25 s�1. Table 1 summarizesthe experimental conditions. We note that the top-to-bottom direction of the foaming videos was alignedwith the longest dimension of the plastic (38 mm), whichwas also the shear strain direction. The optical plane wasalong the 38 mm � 24 mm plane. Cell density with respectto the unfoamed volume (Nunfoamed) was characterizedbased on the foaming videos, and the method was detailedin our previous study [28].

Fig. 5 shows snapshots of the foaming videos of the PSfoaming experiments that used CO2. The cell density withrespect to the unfoamed volume (Nunfoamed) is shown inFig. 6, which demonstrates that the bubble nucleation rateandmaximum cell density increased compared to the staticcase (g ¼ 0) when a shear strain (g ¼ 12.5 and 25) wasapplied. Specifically, the maximum cell density increasedfrom 2.5 � 104 cells/cm3 (g ¼ 0) to 1.1 � 105 cells/cm3

Table 1PS/CO2 foaming experimental matrix.

Expt. # Tsys [�C] P [MPa] –dP/dtjavg [MPa/s] g dg/dt [s�1]

1 180 3.45 12 0 252 180 3.45 12 12.5 253 180 3.45 12 25 25

(g ¼ 12.5) and 1.6 � 105 cells/cm5 (g ¼ 25). Also, thefoaming process completion time decreased for thestrained cases. Pioneering researchers suggested that shearstress enhanced cell nucleation due to the conversion ofshear energy into the interfacial energy needed for cellnucleation [14–17,29]. However, since the cell nucleation,growth, and collapse processes were not observable, it wasdifficult to confirm if the increased final cell densities wereresulted from increased cell nucleation, decreased cellcoalescence and collapse, or a combination of both. Ourstudy visually confirmed that a higher number of cells werenucleated as the shear strain increased.

Several other factors might have increased the bubblenucleation rate in the strained cases. First, the local systempressure around microvoids or contaminants might havechanged due to the applied shear strain. If there wereextensional stress components in some local regions, thelevel of supersaturation, which is defined as the pressuredifference between the pressure within a bubble at thecritical size (Pbub,cr) and the local system pressure (Psys),would increase due to the decreased Psys. This concept issimilar to the foaming mechanism proposed by Albalaket al. [30] in their plastic devolatilization study and lateralso demonstrated by Leung et al. [31]. According to theclassical nucleation theory (CNT) [32,33], the increase ofsupersaturation level would effectively decrease the criticalradius (Rcr) and the free energy (W) required for cellnucleation. To be specific, the expressions of Rcr and W aregiven, respectively, as:

Rcr ¼ 2glg

Pbub;cr � Psys(1)

J ¼ Aexp��WkbT

�¼ Aexp

� 16pg3

lgF

3kbT�Pbub;cr � Psys

�2!

(2)

where glg is the surface tension at the liquid–gas interface,J is the nucleation rate, A is a pre-exponential factor, kb isBoltzmann’s Constant, Tsys is the local system temperaturearound the bubble, and F is the ratio of the volume of thenucleated bubble to the volume of a spherical bubble withthe same radius, which is related to the surface geometry ofthe nucleating agent and the contact angle (qc) between thepolymer-gas mixture and the nucleating agent. Due to thedecrease in Rcr, the bubble nucleation rate would increasethrough the growth of pre-existing microvoids when Rcrbecame less than the size of these microvoids. Alterna-tively, as W decreased, the required thermal instability tocause cell nucleationwas reduced, which also increased thecell nucleation rate. Due to the viscoelastic nature of the PS/CO2 mixture, these tensile stresses should be higher whena higher shear strainwas induced, which would explain theincreased cell density when the shear strain was increased.Second, the applied shear strain might have caused thedeformation of the existing microvoids into elongatedshapes, which, according to Chen’s “cell stretch model”,have greater potential to become nucleated cells owing totheir shape and increased surface area [14]. This conceptagrees with foaming videos, where a large number of theobserved bubbles in both strained cases were elongated.

Fig. 5. Snapshots of PS/CO2 foaming videos.

Fig. 6. Cell density vs. time of PS/CO2 foaming.

A. Wong, C.B. Park / Polymer Testing 31 (2012) 417–424 423

Third, additional gas cavities could have been generatedwhen the applied shear strain caused the detachment ofmicrovoids from contaminants or from the sapphiresurface [34]. Such gas cavities might be the seeds of bubblenucleation, as was observed for the g ¼ 12.5 case, wherea few elongated cavities were observed after the shearstrain was applied (See Fig. 5). Shortly after depressuriza-tion, these cavities started to grow.

5. Conclusion

A novel foaming visualization system has been devel-oped to observe and capture plastic foaming processesunder easily controllable shear strain and shear strain rate.The system allows easy control and adjustment of the

critical processing parameters: applied shear strain, shearstrain rate, system temperature, pressure, pressure droprate, type of plastic material and type of blowing agent. PSfoamingexperiments blownwithCO2 verified the capabilityof the system. This study confirmed the shear-induced cellnucleating phenomena that were suggested by the pio-neering research in this area. It was observed that the cellnucleation rate and maximum cell density increased withthe applied shear strain. These results could be attributed tothe decrease in local system pressure, detachment andgrowth of microvoids, and elongation of bubbles as shearstrain was applied. This foaming visualization systemprovides anewwayto investigate themechanismsofbubblenucleation and growth under the dynamic conditions thatsimulate industrial plastic foaming processes.

Acknowledgements

The authors are grateful to the Consortium of Cellularand Micro-Cellular Plastics (CCMCP), the Ontario ResearchFoundation (ORF), and the Natural Sciences and Engi-neering Research Council of Canada (NSERC) for theirfinancial support of this project.

Notations

A pre-exponent parameter for the cell nucleationrate equation, #/cm3-s

–dP/dtjavg average pressure drop rate, Pa/sF ratio of a nucleated bubble’s volume to a spherical

bubble’s volume with the same radius,dimensionless

kb Boltzmann’s Constant, m2-kg/s2-KJ cell nucleation rate, #/cm3-s

A. Wong, C.B. Park / Polymer Testing 31 (2012) 417–424424

Nunfoamed cell density with respect to the unfoamedvolume, #/cm3

Pbub,cr bubble pressure at the critical radius, PaPsys local system pressure around a bubble, PaRcr critical radius, cmTsys system temperature, �CW free energy barrier for cell nucleation, JWhom free energy barrier for homogeneous cell

nucleation, JWhet free energy barrier for homogeneous cell

nucleation, Jg shear strain, dimensionlessdg/dt shear strain rate, s�1

glg surface tension at the polymer/gas interface, N/m

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