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Rheology studies of polyethylene/chemical blowing agent solutions within an injection molding machine.


Polymeric structural foams have been widely used in automotive, aerospace, construction, electronic, and packaging industries, because of their low density, high impact strength, and insulation properties. However, recent market pressures on processors to generate thinner, more complex part geometries have highlighted the lack of fundamental understanding and control inherent to foam processing. For the manufacture of thinner parts, high injection speeds and good flow characteristics are needed, requiring knowledge of the rheological properties of the thermoplastic/blowing agent mixtures at all stages throughout the injection molding process. Predictive rheological models of a gas/molten polymer solution are essential for the optimization of the processing conditions, development of numerical simulation regarding foam processing, and the development of new foam compounds and process technologies.

In recent years, more attention has been focused on the accurate measurement of the reduced viscosity of a polymer solution containing a dissolved gas [1-3] to better understand the foaming process. The viscosity of gas-polymer solutions cannot be measured by standard methods since the concentration of the blowing agent must be determined precisely and kept constant within the chosen rheology-measuring instrument. In off-line measurements, a gas-polymer solution can be charged into the rheometer from an equilibrium vessel while being maintained at a high pressure and temperature. Mendelson [4] designed a pressurized chamber at the exit of a plunger-type capillary viscometer in order to prevent bubble formation. Both Gerhardt et al. [5, 6] and Kwag et al. [7] measured the viscosity reduction of different polymer/C[O.sub.2] solutions by using a sealed high-pressure capillary rheometry with separate loading and back pressure assemblies. Sato et al. [8] devised a new viscometer for gas-polymer solutions that consisted of twin opposed piston-cylinders, a capillary, and a hydraulically controlled pressure device. The maximum shear rates achieved in off-line rheometers have been reported in the order of 1000 [s.sup.-1]. However, the equilibration of the melt with gas addition in these high-pressure rheometers can be quite complicated and time consuming. In addition, the amount of material charged into an off-line rheometer tends to be quite limited. Therefore, the results may not be applicable to large processing equipment.

In contrast to off-line devices, in-line rheometers are more capable of obtaining accurate rheological information of the polymer melt under conditions pertinent to the actual processing environment of an extruder or injection molding machine. Moreover, the formation of a gas-polymer solution is facilitated in an in-line rheometer by the mixing action of the screw, which leads to a more suitable condition for rheological measurements. Some research groups [1, 2, 9-12] have used an instrumented capillary die on their extruder for foam rheological measurements using C[O.sub.2] as a physical blowing agent. Other researchers [3, 13-15] have investigated the viscosity of gas-polymer solutions by a slit die rheometer on an extruder. With the flow rates of these extruders limited by their pumping capacity, the maximum shear rates quoted in those studies were in the order of 100 [s.sup.-1]. Lan and Tseng [16] studied the rheological behavior of a polypropylene and supercritical C[O.sub.2] mixture by a slit die on their injection molding machine. The shear rate in their work covered a larger range of shear rates (in the order of 10,000 [s.sup.-1]) than had been possible working with in-line extrusion rheometers; however, foam nucleation was observed in the slit die during their experiments. Throughout all these studies with in-line rheometers, the upper range of shear rate (approximately 10,000 [s.sup.-1]) was not sufficient to characterize the flow behavior found in the molding of thin parts.

In order to evaluate the rheological properties of a molten polymer with dissolved low molecular weight volatiles, researchers have attempted to quantify the effects of blowing agent on the viscosity of polymer melts. Lee et al. [1, 2, 9] proposed a generalized viscosity model containing eight adjustable parameters in which the free volume was described as a function of temperature, pressure, and gas concentration. The model was able to generate a master curve; however, the fact that eight fitted parameters were required from experimental data left the predictive capacity of the model in question. Areerat et al. [10] combined the Cross-Carreau model and the Doolittle equation (to describe the viscosity in terms of free volume) in order to predict the viscosity of a LDPE/C[O.sub.2] solution. With the Sanchez-Lacombe (S-L) equation of state and the solubility data, the free volume fraction of a solution was calculated to accommodate the effect of temperature, pressure, and C[O.sub.2] content. Gerhardt et al. [5] and Kwag et al. [7] used a viscoelastic scaling technique to shift experimental data to create a master curve, but the shift factors obtained in their work was empirically derived. Later, Gerhardt et al. [6] applied an equation of state model and free volume theory to predict the concentration scaling factor for the viscosity reduction. However, their approach did not consider the effect of pressure. Royer et al. [3, 15] employed two types of viscoelastic scaling models combined with predictions for [T.sub.g] depression (i.e., Chow model) to determine the effects of gas concentration and pressure on the viscosity of a gas-polymer solution. The first approach applied a set of equations analogous to the Williams-Landel-Ferry (WLF) equation for amorphous polymer with melt temperature between the glass transition temperature ([T.sub.g]) and [T.sub.g] + 100[degrees]C, whereas the second approach utilized equations of an Arrhenius form for semi-crystalline polymer with melt temperature over [T.sub.g] + 100[degrees]C. Overall, the inclusion of the concept of free volume and the use of viscoelastic scaling has been widely and effectively used in viscosity models for a variety of gas-polymer systems. This paper intends to follow a similar approach using a viscoelastic scaling method combined with the free volume concept to model the rheology of a gas-polymer system. The model presented improves upon the prior models by reducing the number of fitted parameters required to described the rheological properties of a gas-laden thermoplastic melt under high shear rates (as high as 200,000 [s.sup.-1]). The model is based on the rheological model of Areerat et al. [10] that is extended to include the concept of free volume.

Model Description

The shear thinning behavior of the LDPE/CBA system can be described by a generalized Cross-Carreau model for viscosity:

[eta]([dot.[gamma]]) = [[eta].sub.0]/[[1 + ([[[eta].sub.0][dot.[gamma]]]/[tau])[.sup.a]][.sup.(1-n/a)]] (1)

where [[eta].sub.0] is the zero-shear viscosity, and n, [tau], and a are constants in the model that are unique to the polymeric material. The four parameters are fit from the flow curve. According to Areerat et al. [10], at high shear rates where ([[eta].sub.0][dot.[gamma]]/[tau]) [much greater than] 1 holds true, Eq. 1 can be simplified to:

[eta]([dot.[gamma]]) [congruent to] [[eta].sub.0.sup.n]([dot.[gamma]]/[tau])[.sup.n-1]. (2)

This simplification assumes that when the temperature, pressure, and CBA content are varied, the viscosity of the polymer melt changes only through the zero-shear viscosity without affecting the material constants. Therefore, the shape of the viscosity curve is shifted according to the operating conditions. The zero-shear viscosity of a plastic can be explained as a function of the free volume fraction (f) [17]:

[[eta].sub.0] = A exp(B/f) (3)

where A and B are constants for the polymer and can be determined from the viscosity-shear rate curve of the pure polymer. For a gas-polymer solution, the free volume fraction can be expressed as follows:

f = [f.sub.r] + (1 - [f.sub.r])[[beta].sub.T](T - [T.sub.r]) - (1 - [f.sub.r])[[beta].sub.P](P - [P.sub.r]) + [phi]C. (4)

The coefficients [[beta].sub.T] and [[beta].sub.P] can be determined from pvT measurement of the pure polymer. The CBA expansion coefficient ([PHI]), which is a variable affected by CBA dissolution, is determined by fitting experimental data. It may be noted that Eq. 4 is similar to the model described by Areerat et al. [10]. In order to apply Eq. 4 to a semi-crystalline LDPE at normal processing conditions, an appropriate set of reference conditions must be chosen. The reference temperature should be at a temperature higher than the melting point of the semi-crystalline polymer where the free volume of the polymer has a linear relationship with temperature at constant pressure.

The similarity in shape of the flow curve for a gas-polymer solution to that of the pure polymer at the same temperature implies that the viscoelastic scaling method can be used to obtain master viscosity curves in which the effects of gas composition are represented by a viscosity scaling factor, [alpha]. The scaling factor is equal to the ratio of the zero-shear viscosity of the polymer/gas solution to the zero-shear viscosity of the pure polymer at the same temperature. During the viscosity measurements of the pure polymer and the gas-polymer solution for the same temperature, both blowing agent content and shear rate had an impact on the average pressure along the capillary die. Therefore, the scaling factor [alpha] is actually considered to be a product of scaling factors for both gas composition, [[alpha].sub.C], and pressure, [[alpha].sub.P] yielding: [alpha] = [[alpha].sub.C] * [[alpha].sub.P]. Thus, the master flow curves are constructed by plotting the scaled viscosity values [eta](c, [dot.[gamma]])/[alpha] vs. reduced shear rate [alpha][dot.[gamma]]. According to the definition of scaling factor, [alpha], we get,

ln [[alpha].sub.C] = ln([[[eta].sub.0](T, [P.sub.r], C)]/[[[eta].sub.0](T, [P.sub.r])]) = ([B/[f(T, [P.sub.r], C)]] - [B/[f(T, [P.sub.r])]]) (5a)

ln [[alpha].sub.P] = ln([[[eta].sub.0](T, P, C)]/[[[eta].sub.0](T, [P.sub.r], C)]) = ([B/[f(T, P, C)]] - [B/[f(T,[P.sub.r], C)]]). (5b)

Therefore, expanding Eq. 5 to include Eq. 4, we get:

ln([[alpha].sub.C][[alpha].sub.P]) = ([B/[[f.sub.r] + (1 - [f.sub.r])[[beta].sub.T](T - [T.sub.r]) - (1 -[f.sub.r])[[beta].sub.P](P - [P.sub.r]) + [phi]C]] - [B/[[f.sub.r] + (1 - [f.sub.r])[[beta].sub.T](T - [T.sub.r])]]). (6)

The value of B can be obtained from viscosity measurements for the pure polymer at the chosen reference temperature. Therefore, Eq. 6 provides predictive scaling for the effects of pressure and CBA concentration, and requires only the rheological properties of the pure polymer melt to be used.



A low density polyethylene (7.4 MI, Nova Chemicals) was used in this rheology study. The crystal melting temperature of the polyethylene was confirmed to be 108.5[degrees]C, as obtained by differential scanning calorimetry (DSC, TA Instruments) using a ramp rate of 10[degrees]C/min. Two chemical blowing agents (CBAs) were supplied by Clariant Co., namely an endothermic type masterbatch. Hydrocerol[R] HK40E (major constituents being sodium bicarbonate and citric acid in a polyethylene carrier), and an exothermic type masterbatch, Hydrocerol[R] 1191 (major constituent being azodicarbonamide in a polyethylene carrier). The decomposition behavior of the two CBAs were studied by DSC measurements. The endothermic CBA initiated gas evolution at 197.8[degrees]C with peak decomposition occurring at 210.7[degrees]C. The amount of gas yield was approximately 50 ml(STP)/g comprised of mostly carbon dioxide and water vapor. Similarly, the onset of decomposition was found for exothermic CBA to begin at 178.2[degrees]C with the maximum gas yielded at 199.3[degrees]C. The amount of gas yield was about 45 ml(STP)/g comprised of mostly nitrogen, carbon monoxide, and ammonia. Within this work, the CBA content was varied from 0 to 5 wt%.


In order to measure the viscosity of our gas-polymer mixture, an instrumented capillary tube die was used in place of the standard nozzle on an injection molding machine. A 55-ton Arburg Allrounder 320S injection molding machine (IMM) with a 30 mm 20 L/D plasticating unit was used for the experiments. The injection molding machine was operated up to its maximum injection speed, i.e., 16 cm/s to obtain the highest shear data possible. The designed in-line capillary rheometry nozzle for the injection molding machine is shown in Fig. 1. The capillary rheometry nozzle consists of an adaptor, a capillary die, and a nozzle, containing two pressure ports, and two temperature ports for flow measurements.

pvT Measurement

The pressure-specific volume-temperature (pvT) measurements pertinent to the processing conditions of an injection molding machine were required to determine the thermal expansion coefficient, [[beta].sub.T], and isothermal compressibility coefficient, [[beta].sub.P], used in the viscoelastic scaling method. Therefore, the pvT behavior of a pure LDPE and different LDPE/endothermic CBA mixtures was measured with an isothermal cooling procedure in the range of temperatures from 170 to 220[degrees]C and pressures from atmospheric to 62 MPa. The LDPE/endothermic CBA mixtures for the pvT measurements were prepared by blending the LDPE resin with the blowing agent in a melt mixer to achieve uniform dispersion. With DSC measurements indicating that the decomposition temperature for endothermic CBA began at 197.8[degrees]C, the blend was prepared between 140 and 150[degrees]C to avoid the risk of activating the blowing agent. Subsequently, the pvT measurements were performed in a modified capillary rheometer, equipped with a high-pressure sealing nut and a pressure transducer capable of measuring up to 200 MPa. The test was started by loading the chamber and heating it up to the maximum specified test temperature (220[degrees]C in our case). Pressure was then applied from atmospheric pressure to 62 MPa. The specific volume of the polymer sample was obtained by measuring the displacement of the piston as pressure and temperature changed. Figures 2 and 3 demonstrate pvT data measured for the pure LDPE and LDPE with 3 wt% endothermic CBA, respectively. From Fig. 3, a deviation between 170 and 190[degrees]C for pressures of 40 MPa and below indicates that the gas-laden LDPE experienced a transition for these conditions. The transition is related to the decomposition of the endothermic CBA and the subsequent expansion of the evolved gases.


Rheological Measurements

Viscosity measurements were performed with LDPE containing either an endothermic or exothermic CBA in the range of 0 to 5 wt%. The mixtures were processed at a nozzle temperature of 210[degrees]C and 230[degrees]C for the exothermic and endothermic CBA, respectively, to ensure complete decomposition of the additive. The complete set of experimental conditions for the two material systems are summarized in Table 1. The barrel and nozzle temperature settings were chosen to minimize premature decomposition and loss of gas through the hopper. In order to avoid the characteristic pressure variations attributed to mold filling, rheology properties were determined from repeated open air shots [18]. During the experiments, signals from each pressure transducer and thermocouple at the different locations in the nozzle were recorded by high-speed data acquisition. In all experiments, the viscosity of the gas-polymer solution was calculated using the Newtonian assumption and corrected using the Rabinowitsch equation [19]. The Bagley method [19] corrects the measured pressure drop for losses attributed to melt flow through the capillary die. In this work, it is assumed that entrance losses are significantly larger than exit effects, allowing the latter to be neglected in the calculations. To accomplish the Bagley correction, two capillary dies of constant diameter (2 mm), yet different lengths were used interchangeably in the in-line rheometry nozzle, namely 12 L/D and 1 L/D.




Validation of the In-Line Capillary Rheometer

The shear viscosity data were calculated from pressure measurements within the in-line rheometer at various injection speeds. During all rheological measurements, the temperature difference between the two thermocouples located before and after the capillary die showed only small variation between 0.4 and 2[degrees]C. The isothermal assumption made in the shear viscosity calculations was, therefore, considered to be valid for this work. Influences of pressure on the viscosity measurements were considered in terms of a shift factor, [[alpha].sub.P], as discussed in the model fitting section.

In order to guarantee that the measurements from the in-line capillary rheometer were consistent with off-line analysis, the shear viscosity data for our pure LDPE by the in-line capillary rheometer were compared to that obtained from a ROSAND (Bohlin Instruments) dual barrel capillary rheometer and ARES (TA Instruments) parallel plate rheometer at the same temperature. Figure 4 shows the viscosity data of LDPE melt at 210[degrees]C spanning seven decades of shear rate from measurements by the three different rheometers. The figure shows that the Cox-Merz rule was acceptable for our LDPE resin and that the in-line rheometer data was in good agreement with the other two off-line rheometers.

Validation of the Single-Phase Assumption

In all cases of the calculated shear viscosities, it was assumed that the gas and polymer melt remained as a single-phase solution across the capillary rheometer. However, being able to prove this assumption without the capacity to directly observe the flow is difficult since phase separation by the gas (i.e., foam nucleation) may be initiated by either mechanical or chemical superheat [20]. Under static conditions, initiation of nucleation requires a reduction of solubility of the gas in the melt which normally results from a significant drop in pressure or change in temperature. The resulting chemical superheat attributed to the supersaturated solution can be readily predicted by knowledge of solubility data. However, the screw rotation of an injection molding machine generates mechanical energy used to facilitate melting and enhance dispersion and dissolution of the evolved gas into the polymer melt. The mechanical superheat attributed to the shear and extensional stresses within the gas-laden melt has been observed by several researchers to cause nucleation at pressures considered to be well above the solubility of the gas in the polymer [21].


Based on the exit pressure of the capillary die and solubility data presented in the literature [10, 22], the evolved carbon dioxide and nitrogen from the chemical blowing agents were expected to remain in solution throughout the rheological analysis. However, considering the high shear rates employed within the present study, the possibility of dynamic nucleation was quite feasible. While the high pressures within the rheometer and nozzle for our process conditions do not allow for direct observation of nucleation, the occurrence of nucleation can be inferred from a discontinuity in the pressure drop across the die for all conditions above a critical shear rate as demonstrated in the work of Lan and Tseng [16]. Figures 5 and 6 show the pressure drop across the rheometer at its corresponding apparent shear rate for solutions containing the highest concentration of either endothermic or exothermic chemical blowing agent, respectively. The maximum standard error for the pressure drop across the die was 5.7% for results presented in the two figures, indicating that the data for 0 wt% and 5 wt% CBA were significantly different from one another. The pressure drop across the die of the gas-laden melts showed similar trends with apparent shear rate as that found for the pure polymer melt, with no discontinuities observed. Additionally, if foam nucleation were to occur within the capillary die, the viscosity should increase based on the decrease in free volume of the polymer melt [1, 15]. Such an observation has not been seen in the flow curves of our LDPE/CBA solutions, and so it can be inferred that the system was maintained as a single-phase within the capillary die during our experiments.



Viscosities of LDPE/CBA Solutions at Various Shear Rates and CBA Contents

The viscosity of the LDPE/CBA solutions was measured at various shear rates and CBA content. Shear viscosity curves are presented in Figs. 7 and 8 for the samples prepared at different levels of endothermic or exothermic CBA content and different processing temperatures. Figure 7 shows shear viscosity results of LDPE with different endothermic CBA concentration between 0 wt% and 5 wt%, at 230[degrees]C. Figure 8 shows shear viscosity results of LDPE with different exothermic CBA concentration, 0 wt% to 5 wt%, at 210[degrees]C. The maximum standard error in the determined viscosity values was 8.6% and 7.7% for LDPE with endothermic CBA and exothermic CBA solutions, respectively. The flow curves for the different LDPE/CBA solutions indicate that shear viscosity of the material showed a significant decrease at low shear rates with the increased addition of dissolved gas in the melt (evolved from the decomposition of the CBA), owing to the plasticizing effect of the gas on the melt. At high shear rates, the shear viscosity of the different LDPE/CBA solutions were similar, though still significantly different from the neat polymer. The shape of the flow curve for a gas-polymer solution is similar to that of the pure polymer at the same temperature, giving validation to our use of the simplified Cross-Carreau model (Eq. 2). In the literature, the viscosity reduction of polymer melt containing dissolved gas was reported up to 50% or more [3], while in our measurements, the viscosity reduction at the highest gas concentration was only 27% and 22% for the LDPE/endothermic CBA and LDPE/exothermic CBA solutions, respectively. The smaller reduction found in our experiments is attributed to the use of chemical blowing agents instead of physical blowing agents, which must be limited in their concentration due to residuals left over from the decomposition reaction that may influence the rheology (among other properties). In comparing the two chemical blowing agents, the reduction in shear viscosity was less for solutions containing the exothermic CBA as compared to the endothermic CBA despite a similar volumetric gas yield, especially at higher shear rates. The primary reason for this phenomenon is that the endothermic CBA has more of a plasticizing effect on the LDPE melt compared to the exothermic CBA. According to Bondi [23], the specific hard-core (incompressible) volume of C[O.sub.2] is larger than [N.sub.2], which results in higher free volume in solutions using an endothermic CBA.




Shear Viscosity Modeling

The similarity in the flow curves of different LDPE/CBA solutions to that of the pure LDPE melt allows for the use of proposed viscoelastic scaling method to model the viscosity. Master viscosity curves were constructed by combining a simplified Cross-Carreau model with viscoelastic scaling, which are shown in Figs. 9 and 10 for LDPE/endothermic CBA and LDPE/exothermic CBA solutions, respectively. The reference pressure chosen was 0.5 MPa, and the processing temperature for the LDPE/exothermic CBA solution (210[degrees]C) was selected as the reference temperature. The pvT data of the pure LDPE was used to isolate the effects of pressure and CBA concentration on the free volume fraction, f. From the isobaric line of 0.5 MPa in the pvT plot shown in Fig. 2, the slope of the specific volume at the reference pressure gave [[beta].sub.T] = 1.025 X [10.sup.-3] (1/[degrees]C). Similarly, the slope of the isothermal curve at 210[degrees]C in the pvT plot provided [[beta].sub.P] = 1.602 X [10.sup.-3] (1/MPa). The values of [[beta].sub.T] and [[beta].sub.P] obtained here were similar to those presented by Areerat et al. [10]. The parameters in Eq. 5 were estimated by non-linear regression using a least squares approach. The shift constants used for the viscoelastic scaling of LDPE/CBA solutions are given in Table 2. The strength of this approach to modeling the gas-polymer viscosity is that aside from the rheological properties of the pure polymer melt, only a small number of parameters for the equation of free volume fraction are required, i.e., three parameters for LDPE/endothermic CBA solutions and two parameters for LDPE/exdothermic CBA solutions.


The addition and dissolution of CBA influenced the thermal expansion coefficient and isothermal compressibility coefficient in the free volume fraction expression. It is shown in Figs. 9 and 10 that the viscoelastic scaling method that accommodated the effects of the CBA content, as well as pressure on the free volume fraction, quantitatively agreed with the experimental data. The resulting viscoelastic scaling factors for the LDPE/CBA solutions are examined in Fig. 11. The shift factor [alpha] represents the reduction of the shear viscosity relative to that of the pure LDPE melt. It was observed that the shift factor gradually decreased with increasing CBA content, more clearly demonstrating the plasticizing influence of the gas (C[O.sub.2] or [N.sub.2]) than shown in the respective flow curves (Figs. 7 and 8). It was also noted that the shift factors for endothermic CBA were always smaller than those for exothermic CBA, confirming our previous observations that the endothermic CBA exerted a greater plasticizing effect on the LDPE melt than exothermic CBA.


Comparison of Various Models

Since chemical blowing agents are used in our research instead of physical blowing agents, the residuals left over from the decomposition reaction are complex and the composition is difficult to be illustrated by equation of state. The models by Lee et al. [1, 2] and Royer et al. [15] are typical ones for describing the shear viscosity of gas-polymer solutions, which do not use equation of state to accommodate the effect of gas. Lee and Royer models have been reported in the literature to give a good description for the viscosity of gas-polymer solutions. Therefore, they are applied here to compare the performances with our present model.

In the model by Lee et al. [1, 2] for the viscosity of PS/supercritical C[O.sub.2] solutions, eight adjustable parameters in the generalized Cross-Carreau model and Arrhenius equation had to be independently determined for the material with and without the presence of gas. Lee's model was able to construct master curves for endothermic and exothermic CBA solutions, respectively. The concentration C in Lee model was the weight fraction of C[O.sub.2] and [N.sub.2] gas evolved from the decomposition of the endothermic and exothermic CBA, respectively. Master curves built by the Lee model were shown in Fig. 12 for the LDPE/endothermic and LDPE/exothermic CBA solutions, respectively. Royer et al. [15] employed equations of an Arrhenius form combined with a prediction of the glass transition temperature ([T.sub.g]) depression (i.e., Chow model) to model the viscosity of LDPE/C[O.sub.2] solutions. However, Chow's model for [T.sub.g] depression is not directly applicable for our gas-polymer solutions due to the complex composition of the blowing agents. The [T.sub.g] for the different CBA contents in this model was obtained by fitting the experiment data, not by using Chow's model. The Royer model has four parameters to be fitted including the glass transition temperature. With these considerations in mind, master curves were built by the Royer model for the LDPE/endothermic and LDPE/exothermic CBA solutions, as shown in Fig. 13.


It can be seen that all models satisfactorily described the shear viscosity of LDPE and LDPE/CBA solutions. All of the parameters in the models were determined using a least squares algorithm. In comparing the plots in Figs. 9 and 10 and Figs. 12 and 13 for LDPE/CBA solutions under high shear rate, it can be noted that Lee's model performs slightly better than the other two models. Royer's model and our model performed similarly for the viscosity behavior of LDPE and LDPE/CBA solutions. However, there are eight fitting parameters in Lee's model, whereas the Royer's model uses four parameters, and only three parameters are needed by the present model.



An in-line capillary rheometer was constructed for our injection molding machine that was found to give good agreement with standard off-line rheometers (i.e., parallel plate rheometer and dual-bore capillary rheometer). The in-line rheometer was used to measure the viscosity of a single-phase gas-polymer solution up to high shear rates in the order of [10.sup.5] [s.sup.-1]; a rheological analysis not easily accomplished using off-line rheometers due to their shear limitations, lack of correlation of flow conditions to processing machinery like injection molding machines, and the ready loss of gas from these off-line instruments. In addition, a model was proposed in this work that combined a simplified Cross-Carreau model and the free volume concept to describe the shear thinning behavior of various LDPE/CBA solutions over a large range of shear rates. The zero-shear viscosity was modeled by the Doolittle equation, with the free volume fraction used to accommodate the effects of temperature, pressure, and CBA content. The pvT data of a pure LDPE were used to determine the model parameters. The viscoelastic scaling scheme applied a concentration-dependent and pressure-dependent shift factor to scale both viscosity and shear rate, which successfully collapsed the viscosity data of LDPE/endothermic CBA and LDPE/exothermic CBA solutions to a single master curve at each temperature. The model developed in this paper has only three adjustable parameters and has been shown to provide acceptable predictions for LDPE/CBA solutions.


The authors wish to thank Moldflow Corporation for providing insight into the construction of the in-line rheometer. We thank Nova Chemical and Clariant on their donation of the resin and blowing agents, Edgar Gutierrez from INDESCA for his assistance with the pvT measurements, and Dr. Victor Bravo for his technical assistance.

A Constant, Pa-s
a Parameter in the generalized Cross-Carreau model
B Constant
C Weight fraction of the chemical blowing agent, %
f Free volume fraction
fr Reference free volume fraction of the pure polymer
n Parameter in the generalized Cross-Carreau model
P Pressure, MPa
[P.sub.r] Reference pressure, MPa
T Temperature, K
[T.sub.r] Reference temperature, K
[T.sub.g] Glass transition temperature, K
[[alpha].sub.C] Concentration-dependent shift factor
[[alpha].sub.P] Pressure-dependent shift factor
[[beta].sub.P] Isothermal compressibility coefficient
[[beta].sub.T] Thermal expansion coefficient
[phi] CBA expansion coefficient
[dot.[gamma]] Shear rate, [s.sup.-1]
[[eta].sup.0] Zero-shear viscosity, Pa-s
[eta] Melt shear viscosity, Pa-s
[tau] Parameter in the generalized Cross-Carreau model, Pa


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X. Qin, M.R. Thompson, A.N. Hrymak

MMRI/CAPPA-D, Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4L7, Canada

A. Torres

Application Group, INDESCA, Venezuela

Correspondence to: M.R. Thompson; e-mail:

Contract grant sponsor: Auto21 Network Centre of Excellence of Canada.
TABLE 1. Injection molding conditions for in-line rheology measurements
of LDPE.

 LDPE/endothermic LDPE/exothermic

Zone temp. (barrel) ([degrees]C)
 1 180 180
 2 190 180
 3 210 190
 4 230 210
Nozzle temp. ([degrees]C) 230 210
Injection speed (ccm/s) 0.1-110 0.1-110
Shot size (ccm) 8-50 8-50
wt% CBA 1-5 1-5

TABLE 2. Model parameters used in the viscoelastic scaling analysis.

Parameter LDPE CBA

A 2.428 2.428
B 2.386 2.386
[f.sub.r] 0.331 0.331
[T.sub.r] (K) 483.2 483.2
[P.sub.r] (MPa) 0.5 0.5
[[beta].sub.T] 1.025 X [10.sup.-3] 2.015 X [10.sup.-2]
[[beta].sub.P] 1.602 X [10.sup.-3] 3.47 X [10.sup.-3]
[PHI] 0.613
[tau] 4022.4 4022.4
n 0.398 0.398

Parameter CBA

A 2.428
B 2.386
[f.sub.r] 0.331
[T.sub.r] (K) 483.2
[P.sub.r] (MPa) 0.5
[[beta].sub.P] 3.315 X [10.sup.-3]
[PHI] 8.08 X [10.sup.-2]
[tau] 4022.4
n 0.398
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Author:Qin, X.; Thompson, M.R.; Hrymak, A.N.; Torres, A.
Publication:Polymer Engineering and Science
Date:Aug 1, 2005
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