Shear Viscosity of CO2-Plasticized Polystyrene Under High Static

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Ind. Eng. Chem. Res. 2009, 48, 5460–5471

GENERAL RESEARCH Shear Viscosity of CO2-Plasticized Polystyrene Under High Static Pressures Maxwell J. Wingert, Shunahshep Shukla, Kurt W. Koelling, David L. Tomasko,* and L. James Lee* Department of Chemical and Biomolecular Engineering, The Ohio State UniVersity, Columbus, Ohio 43210

High-pressure shear viscosity measurements of polymer/supercritical fluid systems are numerous, but most involve pressure-gradient equipment such as a capillary or slit die with pressure drops exceeding 10%. Pressure change across the measurement region introduces some errors when quantifying the effect of blowing agents and other diluents on viscosity. This source of errors was removed by using a static-pressure, Couette rheometer. The viscosity of polystyrene (PS)/carbon dioxide (CO2) was measured from 140 to 180 °C and from 3 to 6 wt % CO2. The effect of pressure on diluent-free PS viscosity was required to calculate the CO2 concentration shift factor. Thus, viscosities at both atmospheric and elevated pressure (via helium, an insoluble gas) were measured using the same equipment and method. The Fillers-Moonan-Tschoegl (FMT) model was overdefined for obtaining free volume parameters, but it was a helpful correlation tool to obtain the pressure effect on viscosity. Pressure and concentration shift factors obtained from the experiments were compared with empirical free-volume plasticization models, one based on glass transition temperature, the other on P-V-T behavior. Unfortunately, neither model demonstrated quantitative agreement with the experimental data within the 3-6 wt % CO2 range examined. The WLF-Chow model, based on Tg data, fared better than the P-V-T-based model. Introduction Advancements in the science and technology of polymer processing with dissolved supercritical fluids (SCFs) are addressed in various reviews1–3 and pertain to applications such as thermoplastic foaming or impregnation of active pharmaceutical ingredients into polymers. A relatively difficult experiment within this topic is the viscosity measurement, because the diluents must be maintained in solution under pressurized conditions. Couette,4–7 parallel plates,8 falling ball,9 and sliding plate rheometers10,11 have been implemented when customized to accommodate pressure. Capillary12–18 and slit die19–29 rheometers are pressurized devices and hence are natural choices for measuring shear viscosity of these systems and for assisting in extrusion die design. Unfortunately, capillary and slit die rheometers possess two complications intrinsic to the presence of a pressure gradient. These complications may contribute to the scatter apparent in many literature master curves. First, polymer/diluent phase separation may occur within the measurement region of the die and two-phase flow is difficult to model.30 To avoid this, backpressure typically is raised to keep the fluid above the binodal; however, there are several studies in the literature with data points close to or below the saturation pressure predicted by other equilibrium solubility studies. Furthermore, phase separation during a capillary or slit die viscosity measurement is difficult to predict because it requires accounting for nucleation kinetics and for differences between solubility under static conditions and those under flowing conditions due to the presence of shear.31–37 Second, a change from atmospheric pressure, diluent-free polymer viscosity to high-pressure diluent-laden viscosity is quantitatively described by a ratio of viscosities at each condition at the same temperature. The viscosity of high-pressure diluent* To whom correspondence should be addressed. E-mail addresses: [email protected] (D.L.T.); [email protected] (L.J.L.).

laden viscosity is dependent on both the value of diluent concentration and on the elevated pressure. Common models that predict or correlate the effect of pressure on a viscosity measurement require a single pressure value to describe each measurement. A complication in assigning a pressure value arises for capillary and slit die rheometers because each viscosity measurement is calculated from a wide range of pressures, as large as 15-7 MPa from one measurement of a PS/CO2 slit die experiment.21 Unlike static pressure rheometers, varying shear rate requires several measurements at each temperature and concentration. Analysis of the results typically requires a new average or characteristic pressure for each measurement. It is possible to design die cavities to minimize the pressure drop, but in most literature studies, pressure drops exceed 10%. The two pressure-related complications are eliminated with a static-pressure rheometer.4–11 Of these rheometers, Oh et al.4,5 used a Couette rheometer. Their custom equipment relied upon a magnetic drive to transfer rotation across a high-pressure wall to a high-scale torque sensor (2.8 N · m). The authors successfully measured viscosity of PS/CO2 from 100 to 200 °C and shear rates ranging from 10-2 to 1 s-1. Unfortunately, they did not compute shift factors or other parameters to describe the pressure and diluent concentration effects on viscosity. The goal of this study is to investigate the effect of temperature, pressure, and concentration on the viscosity of the PS/CO2 system and compare to models using a static-pressure apparatus. A commercially available Couette rheometer from Anton Paar was used. The rheometer provides results on the pressure effect on neat PS viscosity and applies the Fillers-Moonan-Tschoegl (FMT) model38,39 to correlate them. Viscosity data of PS/CO2 was measured from 140 to 180 °C and from 3 to 6 wt % CO2. This paper critically examines the use of a glass transition temperature (Tg) based model, the Williams-Landel-Ferry-Chow (WLFChow) model21 as well as a P-V-T-based model adapted from Gerhardt et al.14 to predict the viscosity results. A computational

10.1021/ie800896r CCC: $40.75  2009 American Chemical Society Published on Web 04/29/2009

Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009

[ (

fluid dynamics (CFD) simulation was used to provide a measure of sensitivity to z, a WLF-Chow fitting parameter. Viscoelastic Scaling Equations Effect of Temperature. Free volume equations are commonly employed to describe time-temperature superposition and other shifts in viscosity data of certain fluids, including many polymers. The shift factor equations used in this study are all based on the Doolittle equation:40 η ) A exp(B/f)

(1)

where η is viscosity, f is the fractional free volume, and A and B are material-dependent constants. This study uses the most common definition of fractional free volume f ) (V - VΦ)/V

(2)

where V is the specific volume of the material and the subscript Φ refers to the occupied volume. A common adaptation of Doolittle’s equation is the Williams-Landel-Ferry (WLF) equation,41 which assumes a linear temperature dependence on fractional free volume from Tg, to about 100 °C + Tg. The WLF equation describes the ratio of zero-shear viscosity at T2 to that at T1: log(aT,P1,φ1) ) log

(

)

-c11(T2 - T1) η0(T2, P1, φ1) ) 1 η0(T1, P1, φ1) c2 + T2 - T1

c11

) B/f1

(4)

c12

) f1 /Rf

(5)

where B is the Doolittle constant from eq 1, f1 is the free volume at the temperature T1, and Rf is the thermal expansion coefficient of the free volume according to the following equations relating thermal expansion coefficients: Rf ) Rg - RΦ for T < Tg

(6)

Rf ) Rr - RΦ for T g Tg

(7)

where the subscripts g and r refer to the glassy and rubbery/ molten states, respectively. Effect of Temperature and Pressure. Penwell et al.42 sought a formula to build on WLF with the concept that pressure increases Tg in addition to increasing viscosity. Various measurements43–45 of glass transition temperature have reported a linear pressure dependence. The following equation for Tg,P was proposed: (8)

where A1 is the rate of change of Tg with pressure (0.29 K/MPa for PS43) and Tg,P0 is the glass transition temperature at atmospheric pressure. When this equation is inserted into eq 3, it becomes

A2 + A3P A4 - A1P

)]

(9)

where A2 ) -c1(T - Tg,P0), A3 ) c1A1, and A4 ) c2 + T Tg,P0. Penwell et al.’s equation is also useful when adapted to a shift factor equation. This is easily converted to a Tg-based pressure shift factor by dividing viscosity at one pressure by another: aT1,P,0 )

[ (

η0(T1, P2, 0) A2 + A3P2 ) exp 2.303 η0(T1, P1, 0) A4 - A1P2 A 2 + A 3 P1 (10) A4 - A1P1

)]

Often P1 will be atmospheric pressure and can be neglected when P2 is large enough (e.g., >20 MPa). A second model, the Fillers-Moonan-Tschoegl (FMT) model,38,39 is a semiempirical temperature and pressure scaling equation that estimates material parameters from rheological measurements along with polymer P-V-T information. The authors of the FMT theory found excellent correlation on temperature and pressure stress relaxation shift factors for PVC as well as Viton, EPDM, and Hypalon amorphous rubbers. The working form of the FMT model is the following shift factor from viscosity at the reference temperature and pressure (T1,P1) to a different temperature and pressure (T2,P2):

(3)

where c11 and c21 are constants and T1, P1, and φ1 are the temperature, pressure, and concentration of reference condition 1, respectively. The temperature shift factor, aT,P1,φ1, like all shift factors mentioned in this study, is a function of temperature, pressure, and diluent concentration. For brevity, the notation has been shortened from aT,P1,φ1(T,P,φ). The WLF equation contains two adjustable parameters, c11 and c21. The superscript 1 refers to the reference temperature T1 in eq 3. These constants are defined as

Tg,P ) Tg,P0 + A1P

η ) ηg exp 2.303

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log aT,P,φ1 )

-c1,1 1 [T2 - T1 - θ(P)]

(11)

c1,1 2 (P) + T2 - T1 - θ(P)

where the parameters are as follows

[

θ(P) ) c13(P)

1 + c14P2 1 + c14P1

]

[

- c15(P) ln

1 + c16P2 1 + c16P1

]

(12)

c1,1 1 ) B/2.303f1

(13)

c1,1 2 ) f1 /Rf (P)

(14)

c13(P) ) 1/krRf (P)

(15)

c14 ) kr /Kr*

(16)

c15(P) ) 1/kΦRf(P)

(17)

c16 ) kΦ /KΦ*

(18)

In the above equations, K is the isothermal bulk modulus, k (the Bridgman constant) describes the pressure dependence of the bulk modulus, and the calculation of both parameters should be done according to the work of Moonan et al.38 Also in the above equations, f1 is the fractional free volume at the reference temperature and pressure; * represents zero pressure (full vacuum); the first superscript 1 refers to the reference temperature; the second superscript 1 refers to the reference pressure; and the lack of a second superscript or asterisk refers to atmospheric pressure. Thus, c11, c21, Rf, and f1 are consistent terminology with the WLF model. Effect of Temperature, Pressure, and Diluent Concentration. There are a wide variety of models that have been used to predict viscosity depression due to high-pressure carbon dioxide and most are empirical with several adjustable parameters. Here, two semiempirical free-volume models are employed. The first model applies P-V-T data to predict viscosity reduction, while the other is based on glass-transition data.

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Figure 1. Glass transition temperature depression of polystyrene versus carbon dioxide headspace pressure from various experimental sources and the pressure-compensated Chow model prediction for 0.3209 J/g · K excess transition isobaric specific heat and various coordination numbers, z. The dashed line shows the Chow model with z ) 2 without the pressure effect on Tg.

Gerhardt et al.14 used a P-V-T model, based on the Doolittle equation (eq 1). When this model is applied to two conditions, it is possible to calculate a shift factor aT,P,φ )

[(

η0(T2, P2, φ2) 1 1 ) exp B η0(T1, P1, φ1) f2 f1

)]

(19)

where subscripts refer to conditions 1 or 2. This general free volume equation becomes an equation for the pressure shift factor aT1,P,φ1 when T1 ) T2 and φ1 ) φ2, but it can also calculate concentration shift factors, temperature shift factors, or combinations such as aT1,P,φ. Gerhardt et al.14 substituted eq 2 into eq 19 and added two assumptions. The first is that the occupied volume is independent of temperature, which is equivalent to RΦ ) 0. The second is that the Doolittle constant B is unity, yielding aT,P,φ )

(

)

η0(T2, P2, φ2) V1 - VΦ V2 - VΦ ) exp (20) η0(T1, P1, φ1) V2 V1

where V1 and V2 are the specific volumes at conditions 1 and 2, respectively. The specific volume is often estimated from an equation of state. Gerhardt et al.14 and Kwag et al.46 have applied this only to aT,P1,φ1 and aT1,P1,φ, but in this study, this equation is also evaluated on its effectiveness to predict aT1,P,φ1. When a diluent is included in the system, the occupied volume is expected to be different from diluent-free polymer and change with concentration. Gerhardt et al.14 suggest implementing a linear mixing rule: VΦ ) wdVΦ,d + (1 - wd)VΦ,p

(21)

where w is the weight fraction, the subscript d refers to the diluent component, and the subscript p refers to the polymer component. In their study, Gerhardt et al. measured carbon dioxide concentration shift factors of PDMS and found acceptable model agreement using a literature value VΦ,CO2 ) 0.589 cm3/g.47 This may suggest that both of the key assumptions in eq 20 are reasonable for PDMS/CO2. Kwag et al.46 measured carbon dioxide concentration shift factors for PS but found the Gerhardt et al. model limited in its ability to predict their experiment. A second common semiempirical model, the WLF-Chow model,21 combines eq 10 with the Chow model,48 an equation that describes the effect of diluent concentration on Tg depression:

( )

ln

Tg,mix,P0 Tg,P0

) β[(1 - θ) ln(1 - θ) + θ ln θ]

(22)

where Tg,mix,P0 is Tg,P0 of the polymer-diluent mixture and the

parameters θ and β are defined as θ)

Mp wd zMd 1 - wd

(23)

zR Mp∆Cp,Tg

(24)

β)

where Mp is the molecular weight of monomer (104.15 g/mol for PS), z is the coordination number, Md is the molecular weight of the diluent (44.01 g/mol for CO2), R is the gas constant (8.3145 J/mol · K), and ∆Cp,Tg is the excess transition isobaric specific heat of the polymer (0.3209 J/g · K for PS49). Chiou et al.50 were among the first to conduct an experimental study focused on which coordination number, z, is most appropriate for a particular polymer-CO2 pair. Chow suggested coordination numbers are independent of diluent identity and suggested values of two for PS48 and four for poly(methyl methacrylate)51 and poly(vinyl alcohol).51 Chiou et al. used this number as a fitting parameter and recommended values of either unity or two for the four polymers they studied. They concluded that z ) 1 is the best choice for PS/CO2, but this conclusion is unfortunately based on only one Tg measurement. Figure 1 compares the Chow model to a collection of five experimental Tg measurements52–56 versus CO2 headspace pressure. To generate the Chow prediction curve in Figure 1, two additional calculations are needed. First, the SanchezLacombe equation of state,57 fit to the data of Sato et al.,58 was employed to convert concentration to saturation pressure. Second, eq 8 was used to adjust the Tg prediction of the Chow model to Tg,P because the Chow model is based on ambient pressure, while all five measurements from Figure 1 were performed at pressured conditions. It has been assumed that the literature value,43 A1 ) 0.29 MPa-1, applies to both the PS and PS/CO2 systems. Figure 1 illustrates considerable y-axis variation among the experimental values. One notable reason for this is that all five sources use different techniques. Also, some variation can be attributed to the spread in CO2-free Tg. Indeed, the two most extreme curves52,53 were from the same laboratory using different techniques and different grades of PS, but found similar slopes of Tg,P depression versus pressure. Figure 1 clearly illustrates that the coordination number has a significant impact on the Chow model. When this number is two or greater, retrograde vitrification is predicted by the combination of Sanchez-Lacombe equation of state and Chow model. The same Tg,P data is plotted versus carbon dioxide concentration in Figure 2 and does not show retrograde vitrification. Retrograde vitrification has not been observed by experimental studies of bulk PS with CO2 dissolved in it.


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Table 1. Gel Permeation Chromatography Results of Molecular Weight, M, and Polydispersity Index, d

as received postrun 1 postrun 2

Mn, g/mol

Mw, g/mol

Mz, g/mol

d

87000 85800 86300

211700 211600 203100

436100 442100 415600

2.44 2.47 2.35

product of the two shift factors are equal for either path (i.e., aT1,P0,φ · aT1,P,φ1 ) aT1,P1,φ · aT1,P,0). Experimental Details Figure 2. Glass transition temperature predicted by the Chow model versus weight fraction CO2. Tg values were adjusted to compensate for the saturation pressure at its corresponding CO2 concentration.

Unfortunately Figure 1 seems to show mixed agreement of the Chow model to experiments based on carbon dioxide content. It is also challenging to describe the experimental data using a model given the variation of the experimental data. Given the CO2 content range investigated by this paper, reasonable agreement can be achieved by dividing the x-axis of Figure 1 into various sections. Below 2.4 MPa (which corresponds to 3 wt % CO2), the Chow model with any coordination number (1-4) appears to adequately describe the glass-transition profile. Between 2.4 and 5.3 MPa (which correspond to 6 wt % CO2), the coordination number three appears to best describe the four experimental data sets present in this range. Between 5.3 and 6.5 MPa, z ) 2 best describes the experiments; above 6.5 MPa, the model is far off the Tg behavior with only z ) 1 giving physically meaningful results. These conclusions about the optimal coordination number are dependent on using 0.3209 J/g · K49 for ∆Cp,Tg, which has also been selected by Chiou et al.50 and Shukla et al.59 However, Royer et al.21 instead applied a value of 0.2948 J/g · K,60 and Chow48 used 0.2593 J/g · K.61 Reassessments of the optimal integer z using these values are not pursued here. The WLF-Chow model21 invokes the work of Vinogradov and Malkin62 to allow removal of the zero-shear specification, avoiding extrapolating to the zero-shear viscosity. This assumes the shape of the viscosity-shear rate curve is unchanged when shifted as a function of temperature, pressure, and concentration. This assumption is followed in this study. The working form of the WLF-Chow model includes eqs 8 and 22-24 along with the following: log(aT1,P0,φ) ) log )

η(T1, P0, φ1) η(T1, P0, 0)

cg1(T1

- Tg,P0)

cg2 + T1 - Tg,P0

log(aT1,P,φ1) ) log )

(

(

η(T1, P1, φ1) η(T1, P0, φ1)

cg1(T1

) -

cg1(T1 - Tg,mix,P0) cg2 + T1 - Tg,mix,P0

)

- Tg,mix,P0)

cg2 + T1 - Tg,mix,P0

-

Materials. The polymer studied was polystyrene, CX-5197, manufactured by Total Petrochemicals (formerly Atofina) possessing a Tg of 100 °C. Gel permeation chromatography (GPC) analysis of the as-received polystyrene is found in Table 1. Zinc stearate, a lubricant/plasticizer commonly found in many commercial polystyrene grades, is not present in CX-5197. The carbon dioxide had a purity of >99.9% and was supplied by Praxair. Apparatus. The rheometer was an MCR 500 manufactured by Anton Paar. This controlled-stress rheometer relies upon an air bearing for its precision. In this study, the rheometer measured steady shear viscosities and shear rates by imposing a constant stress through controlled torque and measuring angular motion. The manufacturer’s software was used to calculate shear stress and shear rate. Per manufacturer specifications, the rheometer has a frequency range from 0.0001 to 100 Hz, a speed range from 10-7 to 1200 min-1, and a torque range is from 0.25 µNm to 200 mNm, with a resolution of 0.002 µNm and 0.5% torque accuracy. The rheometer is compatible with several geometry accessories; this study involved parallel plates operated in oscillatory mode and a modified Couette high pressure cell operated in constant stress mode. The high pressure measurements used a Couette geometry that includes an inset cup and bob placed inside a high pressure cell; a schematic is presented in Figure 3. A custom cell housing was used to achieve higher pressures at the desired temperature than the manufacturer’s limits. The cell holds pressures up to 20 MPa, and a burst disk is used for safety. High pressure carbon dioxide was delivered to the high pressure vessel via an ISCO 500D syringe pump. The pressure was monitored by a Honeywell Sensotec AG400 with an accuracy of 0.034 MPa. The

(25)

cg1(T1 - Tg,mix,P1) cg2 + T1 - Tg,mix,P1 (26)

The WLF-Chow model uses alternate definitions of the pressure and concentration shift factors to those used in many experimental studies, including this one. The main drawback of the WLF-Chow definitions is that η(T1,P0,φ) (and thus aT1,P0,φ) cannot be readily measured since the condition is thermodynamically unstable at desired concentrations. Nonetheless, the

Figure 3. Schematic of cup and bob in high pressure Couette rheometer.

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Table 2. Porous Cup Validation viscosity (Pa · s) at 40 °C test type

fluid 1

fluid 2

fluid 3

fluid 4

solid cup solid cup porous cup porous cup parallel plates parallel plates

9.44 9.42 9.65 9.77 10.1 10.9

45.3 45.0 46.6 46.5 49.4 50.2

76.6 76.5 79.0 79.3 86.4 82.8

16,600 16,500 17,000 16,900 16,800 17,600

rheometer rotation is guided by an air bearing, but the high pressure cell requires high-precision ball bearings which limits accurate measurements to signals above 100 µNm. This Couette cell has also been used by Flichy et al.7 for the rheology of poly(propylene glycol) with CO2, though due to customizing there are some differences between the two devices. Without customization or mixing, carbon dioxide equilibration times at desired operating conditions would be around one year due to axial diffusion58 down the long height of the bob (4.3 cm, excluding the neck). Thus, a porous cup was implemented to provide radial diffusion of CO2 through the walls to significantly reduce the diffusion distances. Made from sintered 316 stainless steel, this porous cup was able to reduce the equilibration time in the measurement region to a couple of hours because of the 1 mm one-dimensional diffusion distance. Under normal operation, the polymer below the bob does not influence the viscosity measurement. However, in diluent/ polymer systems, the region below the bob may not be sufficiently plasticized without sufficient equilibration time, and could affect the results. To ensure that this region (never greater than 1.0 cm diffusion distance) was sufficiently plasticized, more than 50 h of equilibration time was used. Secondary flows are possible errors in a Couette rheometer. In this study, both the Weissenberg and Taylor numbers are less than unity meaning neither Taylor-Couette instability nor purely elastic instability should occur. Additionally, it is not believed that the roughness of the porous Couette walls causes either slip (roughness has been shown to increase the critical stress for macroscopic slip63,64) or shear fracture (Mhetar and Archer63 observed that roughness did not create shear fracture). A series of 24 measurements of four grades of polydimethylsiloxane (PDMS) standard oils and melts were used to confirm that the porous cup functions equivalently to a standard, solid cup. These measurements were randomly blocked into three test devices: porous-cup Couette; solid-cup Couette; and oscillatory parallel plates operated on a separate Anton Paar MCR 300 rheometer. The three test methods show some differences, as shown in Table 2, but the three test methods are found to be statistically equivalent.65 Procedure. To measure the shear viscosity of CO2-free polystyrene, an oscillatory parallel-plates measurement was performed on compression-molded polystyrene discs using the MCR-500 from 150 to 240 °C. This test used a 1 mm gap, 5% strain, and a frequency, ω, of 0.1-10 s-1. The rheometer oven heated the sample using nitrogen gas in lieu of compressed air to suppress degradation. The average testing time was 13 min, and the pretesting thermal history (sample loading at high temperature) varied from 9 to 80 min (for 240 or 150 °C, respectively). The Couette rheometer was used to measure polystyrene viscosity at atmospheric pressure and at elevated pressures, with carbon dioxide or helium as pressurizing gas. Each polymer sample was tested at several temperatures and multiple con-

centrations. Loading void-free polystyrene into the Couette cavity requires a long procedure, details of which are located elsewhere.66 In this study, 3.0, 3.5, 4.0, 5.0, and 6.0 wt % CO2 were examined. Following the loading of polystyrene, time was allowed for CO2 saturation during which the temperature was maintained at 140 °C. The temperature was then slowly raised to the desired set point and pressure was continually adjusted to maintain the same CO2 concentration. Figure 4 shows a typical temperature profile. In runs 3 and 4, helium was used to study the pressure effect on viscosity. Helium is nearly insoluble in polystyrene,67 and the solid cup, which impedes dissolution of the gas into the measurement region, was used for these runs. In this study, degradation is a nuisance variable that directly affects the calculation of pressure and concentration shift factors, since the atmospheric pressure, diluent-free viscosity master curve was generated using samples with much shorter heating times. Two methods were used to test the influence of degradation. First, additional measurements were performed to measure viscosity at 180, 190, and 200 °C on the Couette rheometer following the high-temperature loading procedure without CO2/helium addition. The deviation between the Couette and the parallel-plates data ranged from 0.5% to 7.2%. Second, GPC was used to determine the molecular weight after the multiple-day experiment ended. These results are presented in Table 1 and indicate degradation was minor despite continuous heating for 8 or 11 days because oxygen was kept out of the sample and temperatures were kept below 200 °C. The GPC, performed in an industrial laboratory with precision of 5%, lacks sufficient resolution to state that the molecular weight is different between the three samples. Determination of CO2 Content. During operation of the rheometer, the concentration of carbon dioxide in polymer was not measured, but instead relied on sufficient dissolution time, valid solubility predictions, accuracy/uniformity of temperature,66 and accuracy of pressure sensors. Solubility is obtained from a correlation of a PS/CO2 magnetic suspension balance study by Sato et al.58 Sanchez-Lacombe parameters are found in Table 3 and the following expression is used to calculate the interaction parameter, kij: kij ) -0.000459T + 0.0859

(27)

where T is the temperature in degrees Kelvin. Figure 5 shows the agreement between the solubility predicted by the equation of state and experimental data.58,68,69 It is well-known how difficult it is to obtain solubility data for compressed gases in polymer melts primarily due to the fact that the polymer phase volume swells during the measurement. Most techniques rely on either an independent measurement of the swelling or a calculated correction to extract the solubility. The literature data we have correlated in this work properly accounts for swelling and we have recently published an analysis of the accuracy of the Sanchez-Lacombe equation of state (SL EOS) in addressing solubility, swelling, and Tg in polymer-gas systems.70 That paper includes details of the equation of state, mixing rules, and binary interaction parameters used in the present work to predict the concentration in our experiments from the temperature and pressure. While some other models may show better fits to different polymer-gas systems under some conditions, we have a high degree of confidence that our calculations of CO2 concentration in the melt is within (5%. Molecular weight varies between this study and each of the sources shown in Figure 5, but all are above the critical


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Figure 4. Operating diagram of temperatures and concentrations for the high-pressure rheometer during PS/CO2 run 2. Table 3. Sanchez-Lacombe Pure Component Characteristic Parameters material

T*, K

P*, MPa

F*, kg/m3

M, g/mol

CO2 PS

269.5 159.0

720.3 103.6

1580 803.4

44 156000

entanglement molecular weight and neither average molecular weight nor polydispersity index are expected to be a significant nuisance variable. Results and Discussion Shear Viscosity of Polystyrene-CO2. The viscosity of polystyrene with 3, 3.5, 4, 5, and 6 wt % CO2 was measured, from 140 to 180 °C. Figure 6 shows the specific locations of the PS/CO2 phase diagram measured in this study. At each temperature/concentration combination, steady-state viscosity was measured at six shear rates by imposing a 250, 400, 630, 1000, 1600, or 2500 Pa shear stress. Around one decade of shear

Figure 5. Experimental PS/CO2 solubility values from literature sources and agreement with the Sanchez-Lacombe equation of state.

Figure 6. PS/CO2 phase diagram with diamonds representing where viscosity was measured. The thick line is the average of the glass transition temperature from various literature sources.52–56 The contour lines represent a particular CO2 mass fraction, obtained using the SL EOS.57

Figure 7. Viscosity at 150 °C of PS/CO2 from the Couette rheometer (open shapes), time-temperature shift viscosity of PS from the parallel-plates rheometer (closed diamonds), and generalized Cross-Carreau model correlation (line). The closed diamonds are from a master curve formed with a reference temperature, T1, of 150 °C.

rates were measured at each condition; the friction in ball bearings was the primary reason for limiting this span. These stresses generally corresponded to low shear rates in the range 10-3-10-1 s-1. Viscosity results measured at 150 °C are shown in the open shapes of Figure 7. As expected, viscosity values are closer to the Newtonian plateau than to the power-law shearthinning regime. Figure 8 show viscosity results at 140, 160, 170, and 180 °C. Consistent with other publications, increasing temperature shifts the viscosity versus shear rate curve to lower viscosities and higher shear rates. All PS/CO2 measurements in this study are performed with a CO2 headspace, which results in equilibrium measurements (after sufficient saturation time). Thus, when at constant temperature, the higher CO2 content viscosity results were also performed at higher pressures than the lower CO2 concentration results. The results indicate the plasticizing effect of CO2 outweighs the increased pressure because the viscosity results at higher CO2 content also shift the curve to lower viscosities and higher shear rates. Shear Viscosity of Diluent-Free Polystyrene. Because it was desired to characterize the effect of shear rate on viscosity over several orders of magnitude, a master curve was created at atmospheric pressure without carbon dioxide. An oscillatory parallel-plates measurement was performed to produce this data; tables of viscosity results can be found elsewhere.66 The viscosity results were shifted to generate a master curve of η/aT,P0,0 versus ωaT,P0,0 at a reference temperature of 150 °C without any unique vertical shift factor. The closed shapes of Figure 7 show the master curve at a 150 °C reference temperature. The line in Figure 7 shows a correlation using the generalized Cross-Carreau model. Figure 9 illustrates the shift factors necessary to obtain this 150 °C master curve. These shift

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Figure 8. Shear viscosity of PS/CO2 from the Couette rheometer at 140 and 170 °C (a) and at 160 and 180 °C (b). Each shape represents a different combination of temperature and CO2 concentration.

Figure 9. Shift factors from polystyrene parallel-plates viscosity data, with best-fit WLF correlation and a literature WLF correlation. The reference temperature, T1, is 150 °C.

factors are commonly referred to as aT. However, in this paper, they are referred to as aT,P0,0 because the PS temperature shift factor very likely would be different at a different pressure and/ or diluent concentration (if present). A best-fit WLF correlation was based on shift factors from 150 to 200 °C since 200 °C (100 °C + Tg) is the approximate upper limit of the WLF equation. Shift factors at 220 and 240 °C follow a small, but noticeable deviation from the other data. It is also clear from this figure that the best-fit WLF parameters (c1g ) 13.2; c2g ) 74.1) match the monodisperse PS literature parameters of Pierson71 (c1g ) 13.9; c2g ) 50.0) reasonably well despite the noticeable difference in parameter values. The best-fit WLF parameters were changed from the reference temperature, 150 °C, to 100 °C, the glass transition temperature using the traditional equations found elsewhere.72 The 95% confidence intervals for c1g is from 13.2 to 13.8, and that for c2g is from 51.5 to 96.7. Combined Concentration and Pressure Shift Factor. The aT1,P,φ shift factor, η(T1,P1,φ1)/η(T1,P0,0), can be obtained from a master curve of viscosity-shear rate data with and without high-pressure carbon dioxide. As with CO2-free PS, viscosity was shifted an equal amount on both axes, which produces a graph of η/(aT1,P,φ) versus γ˙ aT1,P,φ at a given temperature. Figure 10 shows the master curve at 150 °C. While each high-pressure experiment only contains six viscosity/shear rate results, all five CO2 concentrations show a lack of scatter and the curvature of the transition from Newtonian to shear-thinning behavior matches both the Couette data and parallel-plates data. Master curves consisting of 140, 160, 170, and 180 °C Couette data showed similar behavior. Figure 11 shows aT1,P,φ values for all temperatures studied. These values represent the viscosity reduction from diluent-

free ambient-pressure viscosity to high-pressure CO2 and demonstrate a significant impact on polymer rheology. Because the conditions in this study are fully saturated, each data point in Figure 11 has been measured at a different pressure. Further calculations will isolate aT1,P1,φ, a key result in capillary and slit die studies, from aT1,P,φ. However, aT1,P,φ is still a useful single parameter to describe the effect of CO2 on viscosity. Assuming that extruders and foam injection molding machines operate above their saturation pressures, the results in Figure 11 represent the minimum aT1,P,φ possible for PS-CO2 in pressuredriven extrusion equipment. Both viscoelastic models previously introduced that can handle the effect of diluent concentration, P-V-T-based and Tg-based, can be directly compared on their prediction of viscosity reduction due to high-pressure diluent. Figure 12 compares this study’s experimental results with predictions from the two models. The WLF-Chow model is presented with four iterations based on choice of coordination number. It appears that the WLF-Chow model with z ) 3 is the best predictor among the two models when studying the region 3-6 wt % CO2. The results for 140, 160, 170, and 180 °C also suggest that WLF-Chow with z ) 3 is the preferred model to match the experimental results. Outside this concentration range, it is unknown which of these options represent the best prediction. Neither model shows excellent experimental agreement. Nonetheless, it is encouraging that, within this concentration range, the optimal integer value of z was three for both the WLF-Chow viscosity model and the Chow Tg depression model. Pressure Effect on Diluent-Free PS. The most convenient measure of CO2-induced viscosity reduction is aT1,P1,φ, but in order to calculate it from the Couette rheometer results, it is first necessary to obtain the aT1,P,0 shift factor, η(T1,P1,0)/ η(T1,P0,0), which is referred to as aP in many publications. The notation aT1,P,0 highlights that this shift factor is certainly a function of temperature and is very likely also a function of diluent concentration (if present). For neat polystyrene above the critical entanglement molecular weight, the available literature show significant disagreement as illustrated in Figure 13, which compares the pressure shift factor from atmospheric pressure to 7.0 MPa using best-fit results from four experimental studies.42,73–75 As previously mentioned, capillary and slit die rheometers have an ill-defined pressure term (P1) when calculating aT1,P,0. For these devices, two dominant approaches have been used to calculate the pressure effect of viscosity: analyze the nonlinearity of Bagley plots42,73,76–82 or operate with a backpressure,74,75,83–86 a technique commonly used on polymer/ diluent systems.


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Figure 10. Polystyrene master curve at 150 °C formed from PS parallel-plates data (diamonds) plus five sets of PS/CO2 Couette data (various shapes, identified by six arrows).

Figure 11. Viscosity shift factors from 140 to 180 °C representing the viscosity reduction from diluent-free atmospheric pressure PS to PS/CO2 at the saturation pressure.

Figure 12. Experimental viscosity shift factors at T1 ) 150 °C with Gerhardt et al. model prediction and WLF-Chow model prediction for various coordination numbers, z.

There are few studies in the open literature that document both atmospheric and high pressure measurements. The dearth of experimental data is likely because most devices that handle high pressure cannot also measure at atmospheric pressure, and it is undesirable to calculate the pressure effect using two separate devices. The Couette rheometer in this study is able to directly measure the pressure effect on PS viscosity. This is performed using helium as a mostly insoluble gas61 and by using a standard, nonporous inset cup to further prevent gas from diffusing into the measurement region. In this study, viscosity was measured at 180 and 200 °C at atmospheric and one elevated pressure (7.0 or 13.9 MPa, respectively), and at 190 °C at atmospheric pressure and five elevated pressures (2.4, 4.7, 7.0, 10.4, and 13.9 MPa) The two experimental shift factors with P1 ) 7.0 MPa are shown in Figure 13. The P-V-T-based model (eq 20) has been applied as a temperature shift factor to polydimethylsiloxane (PDMS)14 and

Figure 13. Effect of temperature, T1, on polystyrene pressure shift factor from the Couette rheometer, estimations from literature experimental studies, and three predictive models.14,38,42 The shift factor represents the viscosity ratio from atmospheric pressure to 7.0 MPa.

PS73 using VΦ as a single adjustable parameter. With VΦ set from the temperature shift factor, this model is a pure predictor of pressure shift factor, without any adjustable parameters. Figure 13 shows the equation does a poor job of predicting the PS pressure shift factor. This suggests that RΦ ) 0 and/or B is unity are not valid assumptions for diluent-free PS. The Tg-based (eq 10) model predictions are also included in Figure 13. While the Tg-based aT1,P,0 model (adapted from Penwell et al.42) described the capillary and slit die results fairly well, it failed to quantitatively describe the results from this study. Accurate aT1,P,0 measurements are needed to predict the pressure effect at the exact temperatures and pressures used by the PS/CO2 viscosity study. Hence, the FMT model38,39 was implemented to correlate and predict diluent-free viscosity. To evaluate the FMT model, Moonan and Tschoegl38 presented three options to estimate Rf(P), the pressure dependence of the free volume thermal expansion coefficient. In this study, the best correlation resulted from implementing their first option which assumes Rf is does not depend on pressure, Rf(P) ) Rf*. To implement the model, P-V-T data was first used to determine the material parameters Rr, the rubbery thermal expansion coefficient, and βr, kr, and Kr*, which are parameters that describe temperature and pressure effects on bulk modulus of the rubbery state. Then, some of the PS viscosity experimental results were correlated to obtain all other parameters. The parameters are summarized in Table 4. Figure 14 shows the agreement between experiment and correlated FMT model at 190 °C. Finally, the model is evaluated on its ability to predict the pressure effect at other temperatures. Figure 13 shows the predicted polystyrene pressure shift factor from atmospheric pressure to 7.0 MPa at various temperatures. The desired

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Table 4. Correlated FMT Parameters parameter

PS

parameter

PS

T1, K c11 c12, K Rr × 104, K-1 Kr*, MPa kr βr* × 103, K-1

463 6.12 167 5.84 984 11.2 6.20

B f1 KΦ*, MPa kΦ Rf × 104, K-1 RΦ × 104, K-1

1.00a 0.0709a 1290a 12.2a 4.24a 1.60a

a

These values represent one set of infinitely many solutions.

Figure 15. Carbon dioxide concentration shift factor at T1 ) 150 °C for polystyrene for this study, two experimental studies, and WLF-Chow model prediction when using the parameters from this study or from Royer et al.21

Figure 14. Polystyrene pressure shift factor at T1 ) 190 °C from Couette rheometer and correlation from FMT model.

agreement was observed for the measurement at 180 °C and 7.0 MPa (Figure 13) with the experimental aT1,P,0 ) 1.45 and a predicted aT1,P,0 ) 1.44. Desired agreement again occurred at 200 °C and 13.9 MPa, with the experimental aT1,P,0 ) 1.65 and a predicted aT1,P,0 ) 1.73. Pressure-effect data at lower temperatures were desired to provide further verification of the model since PS/CO2 measurements occurred as low as 140 °C. However, based on the large surface area of the bob used in this study coupled with torque and speed limits, 180 °C was the operational lower temperature limit for pressurized diluentfree PS. Fillers et al.39 mention that their model determines the Doolittle parameter, B, by regressing a three-parameter equation to raw aT1,P,0 data (e.g., Figure 14). Because this curve is overdefined by such an equation, we arbitrarily set B to unity and then evaluate the associated material properties f0, KΦ*, kΦ, Rf, and RΦ. This assumption has no affect on the correlation ability of the FMT model. Calculation of the Concentration Shift Factor. FMT model values of aT1,P,0 were used to calculate η(T1,P1,φ1)/η(T1,P1,0), the concentration shift factor aT1,P1,φ that is commonly known as ac in some publications. Much as with the temperature and pressure shift factors mentioned in previous sections, the terminology stresses that aT1,P1,φ is likely a function of both temperature and pressure. Concentration shift factors at 150 °C are found in Figure 15. Other experimental studies report the PS/CO2 concentration shift factor aT1,P1,φ and are also illustrated in Figure 15. Kwag et al.16 used an in-house capillary rheometer to directly measure viscosity with and without CO2 at 150 and 175 °C. Just as in this study, Kwag et al. calculated aT1,P,φ and aT1,P,0, independently, and then used a ratio to calculate aT1,P1,φ. Gendron and Correa29 and Royer et al.21 measured the viscosity of PS/CO2 in slit die rheometers. Gendron and Correa used a return-to-stream online rheometer to calculate aT1,P1,φ at 150 °C. They did not measure high-pressure diluent-free viscosity at 150 °C, but rather extrapolated from higher temperature measurements. Royer et al. measured PS/CO2 viscosity from 200 to 250 °C but did not report experimental aT1,P1,φ values. Thus, the WLF-Chow model is used to represent the findings of their study. The WLF-Chow parameters are changed to those implemented in their study: z ) 1, A1 ) 0.6 MPa-1, ∆Cp,Tg )

0.2948 J/g · K,60 c1g ) 13.7, and c2g ) 50. Because the A1 value used by Royer et al. is twice the values reported by experimental studies,43–45 it is possible that they were using it as a fitting parameter. The WLF-Chow model alone cannot predict aT1,P1,φ measurements or a model for aT1,P,0 are needed. Thus, aT1,P1,φ is calculated from (aT1,P0,φ · aT1,P,φ1)/aT1,P,0. Figure 15 illustrates that the WLF-Chow model using Royer et al.21 parameters did a poor job of predicting the experimental results from this study. It is possible that the measurements from Royer et al. poorly predict viscosity at 150 °C because they occurred at very high temperatures (some conditions exceeded 200 °C above the PS/ CO2 glass transition). Both Gendron and Correa29 and Kwag et al.16 report trends of higher aT1,P1,φ shift factors than this study. The difference in aT1,P,0 values are a key difference between Kwag et al. and this study. As previously noted, the aT1,P,0 values that Kwag et al. measured and then implemented into the aT1,P1,φ calculation are significantly lower than the predictions from this study. Indeed, if the experimental aT1,P,φ data of Kwag et al. were converted into ac data by the FMT model prediction used in this study, Kwag et al.’s aT1,P1,φ values would be even lower than those measured in this study. This would position the aT1,P1,φ results of this study between the experimental studies of Gendron and Correa (higher aT1,P1,φ values) and Kwag et al. (lower aT1,P1,φ values). Figure 15 is the best attempt at a direct comparison to other experimental studies, specifically Kwag et al.16 and Gendron and Correa.29 However, a nuisance variable may exist: aT1,P1,φ is likely a function of pressure. As previously mentioned, most other viscosity techniques require a pressure drop, making it problematic to define the pressure term that exists in the definitions of aT1,P,0 and aT1,P1,φ. For a given temperature and CO2 concentration, pressure should be lowest in this study. This assumes the measurements from Kwag et al. and Gendron and Correa were performed above the saturation pressures and the saturation pressures are the same as those from this study. On the basis of the WLF-Chow model combined with aT1,P,0 predictions, aT1,P1,φ decreases a noticeable amount with increasing pressure. Specifically, aT1,P1,φ at 3 wt % CO2 and 150 °C decreases 28% when the pressure, P1, increases from 7.4 to 14.4 MPa. This suggests additional pressure in an extruder is not the reason that the aT1,P1,φ is lower in this study than in the work of Gendron and Correa. Extrusion Die Simulation. A key motivation for this study is to assist in the die design of polystyrene foam extruders. When


Figure 16. Simulated pressure along the centerline of a capillary die and its entrance region for PS/CO2 viscosity predicted by two different values of z in the WLF-Chow model. The inlet temperature was 180 °C.

using the WLF-Chow model, it is desired to investigate whether the choice of Chow model coordination number has a significant effect on the pressure drop across an extrusion foaming die. A computational fluid dynamics (CFD) simulation using FLUENT was performed consistent with a recent simulation study.59 The capillary die from that study was simulated for 4.0 wt % CO2 dissolved in PS using a 180 °C inlet stream. The flow rate was 3.67 g/min and the capillary was 10 mm long with a 0.5 mm diameter. The initial part of the die system is the die insert over which little pressure drop occurs, and it comprises the first 23.4 mm of the length shown. The actual capillary die over which most of the pressure drop occurs comprises the remaining 10 mm of the die length. In this simulation, the WLF-Chow model21 is used to predict the plasticized polystyrene viscosity given diluent-free viscosity data.59 The parameters ∆Cp,Tg and A1 were again set to 0.3209 J/g · K49 and 0.29 MPa-1,43 respectively. Using a coordination number of three, the simulated pressure drop across the die was found to be 9.06 MPa. It was previously mentioned that three is the optimal integer for z assuming the other WLF-Chow parameters selected in this study. When alternatively selecting z ) 1 in the WLF-Chow model, the simulated pressure drop across the foaming die was found to be 9.74 MPa, an overprediction of 7%. An axial pressure plot comparison is shown in Figure 16. The nature of these pressure curves indicates that the pressure drop rate at the saturation surface would be lower for a simulation using z ) 3 than one using z ) 1, which suggests that the supersaturation downstream of the binodal for a simulation using z ) 3 would be lower when compared to a similar system using z ) 1. Thus, using the correct z value would affect not only the mechanical design of the die and of the extrusion system but also modeling of bubble nucleation and bubble growth processes for microcellular foams. Figure 17 shows the difference in temperatures along the die wall for z ) 1 as against z ) 3. The difference here is only 0.5 °C at the wall near the die exit (and is almost nothing at the center line). However, dissipative heating is far more pronounced under the lower temperature conditions at which plastic foams are usually manufactured in the industry, since the polymer melt is much more viscous at these lower temperatures. The impact of using the correct value of z should be more important at reduced temperatures and have a significant impact on the modeling of bubble nucleation and growth in polymer foams under these conditions. Because the only effect of a choice of coordination number in the WLF-Chow model is on Tg,mix,P0, a different coordination number has the same effect as varying CO2 content in the extrusion foaming die. Table 5 demonstrates that choosing z )

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Figure 17. Simulated melt temperature along a capillary die and its entrance region for PS/CO2 viscosity predicted by two different values of z in the WLF-Chow model. Table 5. WLF-Chow Model Results for Polystyrene-CO2 CO2 content 0.025 0.040 0.060

predicted Tg,mix,P0 (°C) z)1 z)3 79 71 62

72 62 49

3 for 4.0 wt % CO2 dissolved in PS has the same effect as choosing z ) 1 for 6.0 wt % CO2 dissolved in PS. Concluding Remarks The effect of pressure on PS shear viscosity was measured using a direct measurement of viscosity at atmospheric and elevated static pressure using the same equipment and method. This is a rare result. The FMT model was relied upon to predict the pressure shift factor at lower temperatures than measured. This model was demonstrated in the literature to have excellent predictive power all the way to the glass transition for two amorphous rubbers. The impact of temperature, pressure, and equilibriumsaturated concentration on the viscosity of PS/CO2 was measured and reported using shift factors. All three types of shift factors (temperature, pressure, and CO2 concentration) are likely functions of temperature, pressure, and CO2 content themselves. This study focused on two models derived from Doolittle’s free volume theory. The first model, adapted from the work of Gerhardt et al., is based on P-V-T data and it fails to quantitatively describe the aT1,P,φ shift factor. The Tg-based model, from Royer et al., shows better experimental agreement than the P-V-T-based model. For the range 0.03-0.06 weight fraction and the constants chosen in this study, the Chow model is best described by a coordination number, z, equal to three. The WLF-Chow model, too, is best described with this integer for PS/CO2. Outside of this range, agreement appears to worsen. Alternatively if z ) 1 behavior is assumed, the zero-shear viscosity increases 44%, the pressure drop across a 0.5 mm diameter capillary die is overpredicted by 7%, and the wall temperature rise due to viscous heat dissipation is overestimated by 0.5 °C. These results were obtained from a CFD simulation of a capillary die from a laboratory-scale extruder. The effect of z should become even more relevant for larger machines processing at commercial rates and much lower temperatures, where dissipative heating effects are more pronounced. The WLF-Chow model alone cannot predict aT1,P1,φ (also known as ac)sa model or measurements for aT1,P,0 (also known as aP) are needed. Head to head comparisons with the works of Kwag et al. and Gendron and Correa both report higher aT1,P1,φ

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values than this study. When Kwag et al.’s aT1,P,0 values are substituted with those of this study, this study reports aT1,P1,φ values intermediate to those studies. As mentioned above, pressure is expected to be a small nuisance variable in the headto-head comparison. Between the two models used in this study, it is recommended to use the WLF-Chow model to obtain PS viscosity depression due to high-pressure CO2 and to apply the FMT model for aT1,P,0 if aT1,P1,φ is desired. Acknowledgment This research was supported by the National Science Foundation under IGERT Grant No. 0221678, the Center for Advanced Polymer and Composite Engineering (CAPCE) at The Ohio State University, and the Ohio Third Frontier program. The authors thank Dehua Liu, Zhihua Guo, and Xiangmin Han for helpful discussions and Paula Slocum of Owens Corning for GPC analysis. Note Added after ASAP Publication: The version of this paper that was published on the Web April 29, 2009 had errors in eqs 3, 12, 25, and 26. The appearance of the variable Rf has been made consistent throughout the paper, and ref 54 has been revised. The correct version of this paper was reposted to the Web May 6, 2009. Supporting Information Available: Tabulated shift factors for the effect of pressure and concentration of CO2 on polystyrene viscosity at various temperatures. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Tomasko, D. L.; Li, H.; Liu, D.; Han, X.; Wingert, M. J.; Lee, L. J.; Koelling, K. W. Ind. Eng. Chem. Res. 2003, 42 (25), 6431–6456. (2) Kazarian, S. G. Polym. Sci., Ser. C 2000, 42, 78–101. (3) Han, C. D. Rheology and processing of polymeric materials; Oxford University Press: New York, 2007. (4) Oh, J.-H.; Lindt, J. T. 63rd Annu. Tech. Conf.-Soc. Plast. Eng. 2005, 2555-2557. (5) Oh, J.-H.; Lindt, J. T.; Ottoy, M. H. 60th Annu. Tech. Conf.-Soc. Plast. Eng., 2002, 2, 1920-1923. (6) Wingert, M. J.; Shen, J.; Davis, P. M.; Lee, L. J.; Tomasko, D. L.; Koelling, K. W. 63rd Annu. Tech. Conf.-Soc. Plast. Eng., 2005, 1143. (7) Flichy, N. M. B.; Lawrence, C. J.; Kazarian, S. G. Ind. Eng. Chem. Res. 2003, 42 (25), 6310–6319. (8) Ouchi, S.; Masubuchi, Y.; Shikuma, H. Intern. Polym. Process. 2008, 23 (2), 173–177. (9) Royer, J. R.; Gay, Y. J.; Adam, M.; DeSimone, J. M.; Khan, S. A. Polymer 2002, 43 (8), 2375–2383. (10) Park, H. E.; Dealy, J. M. Macromolecules 2006, 39 (16), 5438– 5452. (11) Park, H. E.; Dealy, J. M. 66th Annu. Tech. Conf.-Soc. Plast. Eng., 2008, 2534-2538. (12) Han, C. D.; Ma, C. Y. J. Appl. Polym. Sci. 1983, 28 (2), 851–60. (13) Han, C. D.; Ma, C. Y. J. Appl. Polym. Sci. 1983, 28 (2), 831–50. (14) Gerhardt, L. J.; Garg, A.; Manke, C. W.; Gulari, E. J. Polym. Sci., Part B: Polym. Phys. 1998, 36 (11), 1911–1918. (15) Gerhardt, L. J.; Manke, C. W.; Gulari, E. J. Polym. Sci., Part B: Polym. Phys. 1997, 35 (3), 523–534. (16) Kwag, C.; Manke, C. W.; Gulari, E. J. Polym. Sci., Part B: Polym. Phys. 1999, 37 (19), 2771–2781. (17) Lee, M.; Park, C. B.; Tzoganakis, C. Polym. Eng. Sci. 1999, 39 (1), 99. (18) Areerat, S.; Nagata, T.; Ohshima, M. Polym. Eng. Sci. 2002, 42 (11), 2234–2245. (19) Elkovitch, M. D.; Lee, L. J.; Tomasko, D. L. Polym. Eng. Sci. 2001, 41 (12), 2108–2125. (20) Royer, J. R.; DeSimone, J. M.; Khan, S. A. J. Polym. Sci., Part B: Polym. Phys. 2001, 39 (23), 3055–3066. (21) Royer, J. R.; Gay, Y. J.; Desimone, J. M.; Khan, S. A. J. Polym. Sci., Part B: Polym. Phys. 2000, 38 (23), 3168–3180.

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ReceiVed for reView June 6, 2008 ReVised manuscript receiVed March 29, 2009 Accepted April 6, 2009 IE800896R

Shear Viscosity of CO2-Plasticized Polystyrene Under High Static

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