A constitutive theory for polyolefins in large amplitude oscillatory shear.
An interesting way of characterizing the nonlinear viscoelastic behavior of a molten plastic is in large amplitude oscillatory shear. This test is interesting because the strain amplitude and the frequency of the test can be varied independently. In other words, the Weissenberg number (We = [Lambda][[Gamma].sub.0] [Omega]) and the Deborah number (De = [Lambda][Omega]) can be varied independently. Until recently, this test has not attracted the attention of practitioners because of long standing difficulties in both measurement and interpretation.
The development of a sliding plate rheometer incorporating a shear stress transducer for molten plastics has recently removed some important experimental limitations in realizing large amplitude oscillatory shear flow (1). Versions of this rheometer have been commercialized first by the MTS Systems Corporation (2), and now by the Interlaken Technology Corporation (3).
Until recently, no constitutive theory had been found which could accurately predict the shear stress response for any molten plastic in large amplitude oscillatory shear (4). Since an accurate constitutive equation is the rheologist's main tool for interpreting rheological measurements, considerable time has recently been invested in evaluating classical theories in large amplitude oscillatory shear (5).
Very recently, Jeyaseelan et al. (6) used a simple constitutive theory to interpret the large amplitude oscillatory shear behavior of molten low-density polyethylene (LDPE). Specifically, the nonlinear oscillatory shear behavior was interpreted in terms of a constitutive theory based on entanglement kinetics. This theory differs from the kinetic theory originally proposed by Acierno et al. (7) in one important way - it employs a kinetic rate expression originally proposed by Liu et al. (8). In this work, the large amplitude oscillatory shear response was used to measure the kinetic rate constants for shear induced disentanglement, and for reentanglement due to thermal motion. In the present paper, this theory is extended to other polyolefin melts by measuring the kinetic rate constants for an atactic polystyrene, a linear low-density polyethylene, two high-density polyethylenes, and a polyisobutylene. A robust constitutive theory is used to interpret the large amplitude oscillatory shear behavior of seven molten polyolefins. The model contains just three nonlinear parameters ([[Kappa].sub.1] [[Kappa].sub.2] and "m") whose meanings are well defined in terms of entanglement kinetics.
Jeyaseelan et al. (6) have found the following simple constitutive theory to accurately model the large amplitude oscillatory shear behavior of one low density polyethylene film resin:
[Tau] = [Sigma][[Tau].sub.i], (1)
[[Tau].sub.i]/[G.sub.i] + [[Lambda].sub.i] [Delta]/[Delta]t [[[Tau].sub.i]/[G.sub.i]] = 2 [[Lambda].sub.i]D (2)
[G.sub.i] = [G.sub.oi][x.sub.i], (3)
[Mathematical Expression Omitted]
where [x.sub.i] are a set of scalar structural variables each ranging from 0 to 1, [G.sub.i] are the structure-dependent relaxation moduli for the structure-dependent relaxation times [[Lambda].sub.i], and where [Delta]/[Delta]t is the upper convected or contravariant derivative:
[Delta][[Tau].sub.i]/[Delta]t = d[[Tau].sub.i]/dt - [Nabla]v[[Tau].sub.i] - [[Tau].sub.i][Nabla][v.sup.T] (5)
Equations 1 to 5 were originally proposed by Acierno et al. (7). To model the evolution of the entanglement structure a suitable kinetic rate equation is required. Jeyaseelan et al. (6) used the kinetic rate expression proposed by Liu et al. (8):
[Mathematical Expression Omitted].
where the second invariant of the rate of deformation tensor, [II.sub.D] = 2 tr([D.sup.2]), reduces to [Mathematical Expression Omitted] for simple shear, [[Kappa].sub.1] is the kinetic rate constant for thermal regeneration of entanglements, [[Kappa].sub.2] is the rate constant for shear induced disentanglement, and "m" is a dimensionless parameter associated with the elasticity in the melt. When [x.sub.i] = 1, we recover the well known structure independent model, the upper convected Maxwell model or its equivalent integral form called the Lodge rubberlike liquid.
In this paper we have examined seven polyolefins. We have combined all previously published large amplitude oscillatory shear measurements on melts (four materials) with new measurements on three other materials in our analysis. In this discriminating flow, we will determine if the model can be used as a general tool for probing entanglement kinetics in polyolefin melts.
Compared are two LDPE film resins whose large amplitude oscillatory shear behavior have been previously published (1, 4). The first is the IUPAC X LDPE [TABULAR DATA FOR TABLE 1 OMITTED] melt characterized using a sliding plate rheometer incorporating a shear stress transducer developed at McGill university (1). The second is a LDPE (DFDQ 4400) characterized on a Rheometrics mechanical spectrometer with the cone-plate geometry by Tsang and Dealy (4). A LLDPE (Dowlex 2037) film resin is examined which was previously characterized in large amplitude oscillatory shear with an MTS Direct Shear Rheometer (6). This rheometer is a commercial version of the sliding plate rheometer developed at McGill University (1). An injection molding grade of atactic polystyrene whose large amplitude oscillatory shear behavior was previously measured with the sliding plate rheometer developed at McGill University (9, 10) is then examined. New experimental data are ported and evaluated for two HDPE blow molding resins characterized on an MTS Direct Shear Rheometer. New experimental data are also reported for a food grade polyisobutylene melt which was characterized in large amplitude oscillatory shear using a sliding plate rheometer originally developed by Dealy and Soong (11, 12) for studying materials at room temperature and later adapted by Giacomin (9).
Linear relaxation spectra for the LLDPE (Dowlex 2037), both HDPEs, and the atactic polystyrene resins were calculated for the other materials from G[prime]([Omega]) and G[double prime]([Omega]) measurements using the technique of linear regression with regularization outlined by Orbey and Dealy (13). To determine the kinetic rate constants [k.sub.1] and [k.sub.2] a technique recently reported by Jeyaseelan and Giacomin was used (14). To see how well the model predicts the observed behavior, shear stress vs. shear rate plots are used, a technique suggested by Tee and Dealy (15). For each material, comparisons with two different large amplitude oscillatory shear experiments are reported.
To evaluate the constitutive theories, Eqs 1 to 6 were solved subject to the initial condition:
[Mathematical Expression Omitted]
using a sixth order Runge-Kutta-Verner technique. Roughly 20 s of CPU time are required on a CRAY YMP2/216 supercomputer for each computation. Following a fitting procedure reported by Jeyaseelan and Giacomin (14), about 200 such model evaluations are required to determine the optimum values of the model parameters [k.sub.1], [k.sub.2], and "m".
The molecular weights were determined by size exclusion chromatography in various laboratories and these are summarized in Table 1.
The linear relaxation spectra for LDPE IUPAC X, LDPE (DFDQ 4400), and LLDPE (Dowlex 2037) are previously published (16, 4, 6). The linear relaxation spectra for PS (STYRON 683) and HDPE (Experimental Grade) are given in Tables 3 and 4 respectively. The linear spectrum for HDPE (SCLAIR 56B/3830) at 180 [degrees] C calculated by Hatzikiriakos (17) is reproduced in Table 5. The linear spectrum for atactic polyisobutylene (Vistanex LM-MS) proposed by Soong (11) is reproduced in Table 6. The best fit values for [[Kappa].sub.1] and [[Kappa].sub.2] are recorded in Table 2 for each system studied. For each material, two conditions of strain amplitude and frequency were examined.
[TABULAR DATA FOR TABLE 2 OMITTED]
Table 3. Linear Relaxation Spectrum for Atactic Polystyrene, STYRON 683 at 190 [degrees] C.
[[Lambda].sub.i] (s) [G.sub.i] (Pa)
0.0001 1108 0.001 10,830 0.01 66,820 0.1 41,390 1. 12,390 10. 810.1 100. 22.21 1000. 1.692 Table 4. Linear Relaxation Spectrum for HDPE (Experimental Grade) at 190 [degrees] C.
[[Lambda].sub.i] (s) [G.sub.i] (Pa)
0.0001 65,140 0.001 243,900 0.01 117,500 0.1 40,530 1. 10,530 10. 2396 100. 521.5 1000. 158.2 Table 5. Linear Relaxation Spectrum for HDPE, SCLAIR 56B/3830 at 180 [degrees] C.
[[Lambda].sub.i] (s) [G.sub.i](Pa)
6.816 x [10.sup.-5] 1.027 x [10.sup.7] 0.0410 93,460 0.5051 20,570 3.171 4545 7.496 932.2 26.88 1001 278.7 192.2
Figures 1 and 2 show the good agreement for the LDPE IUPAC X at 150 [degrees] C. Figure 3 shows equally good agreement for another low density polyethylene, LDPE DFDQ 4400 at 170 [degrees] C. Figure 4 shows that the overall shear stress amplitude is slightly overpredicted at the higher strain amplitude of 9.55. The nonlinear elastic parameter, m, is unity for both systems, but their kinetic rate constants differ considerably.
Figures 5 and 6 also show good agreement for a linear low density polyethylene at 190 [degrees] C. In Fig. 5, the loop area is underpredicted indicating that the lost work is slightly over predicted. Under the more severe conditions (higher [[Gamma].sub.o] and higher [Omega]) in Fig. 6, the overall shear stress amplitude is also slightly overpredicted. The nonlinear elastic parameter, m, is unity for this system.
Figures 7 and 8 show good agreement for an experimental grade of high density polyethylene at 190 [degrees] C. Equally good agreement is found for another high density polyethylene, SCLAIR 56B, at 180 [degrees] C in Fig. 9. Figure 10 shows that the overall shear stress amplitude is slightly overpredicted for more severe test conditions (higher [[Gamma].sub.o], same [Omega]). The kinetic rate constants differ considerably. The nonlinear elastic parameter, m, is unity for both the high density polyethylenes studied here.
Table 6. Linear Relaxation Spectrum for Polyisobutylene, Vistanex LM-MS at 23 [degrees] C.
[[Lambda].sub.i] (s) [G.sub.i] (Pa)
0.001 365,530 0.01 213,050 0.1 106,390 1. 39,325 10. 5646.5
Comparing the two low density polyethylenes with the three linear polyethylenes, we may conclude that long-chain branching does not affect the nonlinear elastic parameter, m, in the case of polyethylene.
Figure 11 shows reasonable agreement for the atactic polystyrene at 190 [degrees] C. In this case, both the overall shear stress amplitude and the loop area are slightly overpredicted. Better agreement is found at the different conditions shown in Fig. 12. Here the nonlinear elastic parameter, m = 0.28, is considerably less than unity. Polystyrene has relatively large side chains compared to the hydrogen atoms on polyethylenes. Since this is the main difference between polystyrene and polyethylene, we suspect that the nonlinear elastic parameter, m, is sensitive to the type of side chain.
To test this hypothesis, we considered the atactic polyisobutylene melt at 23 [degrees] C shown in Fig. 13. The loop area is underpredicted, indicating that the lost work is slightly overpredicted. Under the more severe conditions of Fig. 14, the overall shear stress amplitude is also slightly overpredicted. More importantly, for the polyisobutylene the nonlinear elastic parameter m = 0.20 is considerably below unity as was found for polystyrene. The nonlinear elastic parameter, m, does indeed seem to be sensitive to side chain type.
Since m = 1 for all the polyethylenes and since these span ranges of molecular weights and polydispersities, clearly "m" is independent of molecular weight and polydispersity. Since both polyisobutylene and polystyrene are substituted olefins, we know their chain stiffnesses are higher than that of polyethylene. But we also know that polymers of aromatic olefins give stiffer chains than those of aliphatic olefins. So the values of "m" measured for polyethylene, polyisobutylene, and polystyrene do not rank with chain stiffness. Clearly additional work will be required to uncover the subtle role of molecular structure on the nonlinear elastic parameter "m".
For all materials studied, and under all conditions studied, the model discrepancies are small. The model discrepancies observed for the polystyrene are qualitatively different from those observed for polyisobutylene. The one with the straight side chain causes an overprediction of the lost work, whereas the one with the aromatic side chain causes the lost work to be underpredicted.
To give some idea of the ranges of Weissenberg and Deborah numbers over which the theory has been tested, we constructed Table 7. We define the Weissenberg and Deborah numbers for each material in terms of the longest relaxation time in its discrete spectrum, [[Lambda].sub.1]:
We = [[Lambda].sub.1] [[Gamma].sub.[Omega]][Omega] (8)
De = [[Lambda].sub.1] [Omega] (9)
Table 7 shows that, in this paper, the theory has been variously tested over 5 decades of Deborah number, and over 4 decades of Weissenberg number. But Table 7 records just two tests on each material. What if we systematically vary the Deborah and Weissenberg numbers for a given material?
Figures 15 through 18 summarize a set of 16 measurements on the IUPAC X LDPE at 150 [degrees] C taken over a 4 by 4 grid of strain amplitudes and frequencies. These Figures compare the measured values with the corresponding model predictions. For the shear stress measurements, 128 points are gathered over 4 cycles. To avoid transient effects following start-up of oscillatory shear we begin gathering data with the seventh cycle. By this time, the shear stress normally becomes [TABULAR DATA FOR TABLE 7 OMITTED] a standing wave that can be represented as a Fourier series of odd harmonics:
[Sigma](t) = [summation of] [[Sigma].sub.m] sin(m[Omega]t + [[Delta].sub.m]) where m = 1, odd to [infinity] (10)
where the amplitudes, [[Sigma].sub.m]([Omega], [[Gamma].sub.o]), and phase angles, [[Delta].sub.m]([Omega], [[Gamma].sub.o]) of the harmonics depend on both strain amplitude and frequency. Following Giacomin and Dealy (10), we perform a discrete Fourier transform on both the measured and predicted shear stress waves to determine [[Sigma].sub.m] and [[Delta].sub.m]. The lower limit of unity for the strain amplitude range used for Figs. 15 through 18 is where nonlinear effects are already clearly in evidence. The upper limits of strain amplitude and frequency ranges used for Figs. 15 through 18 corresponds to the onset of melt fracture.
Figure 15 shows reasonable agreement between the measured and predicted surface for the behavior of the amplitude of the first harmonic, [[Sigma].sub.1]. Figure 16 shows similarly reasonable agreement for the phase angle of the first harmonic, except that the frequency dependence predicted by transient network theory for [[Delta].sub.1] is slightly weaker than the measured behavior.
Figures 17 and 18 show reasonable agreement for the amplitude and phase angle of the third harmonic. The predicted [[Sigma].sub.3] are significantly higher for large strain amplitudes and frequencies. Figure 18 shows that the frequency dependence of the predicted [[Delta].sub.3] is weaker than the measured behavior. So for a given material, reasonable model validity obtains over the ranges of strain amplitude and frequency which can be accessed experimentally.
Many years ago several integral models were proposed with discrete relaxation moduli depending on the second invariant of the rate of deformation tensor. These models included the Bird-Carreau (18), the Carreau-B (19), and the MacDonald-Bird-Carreau (20) models. Subsequently, MacDonald et al. (21) found that in large amplitude oscillatory shear the Bird-Carreau model introduced far too much frequency dependence. MacDonald (22, 23) later reported the same problem existed with the Carreau-B and the MacDonald-Bird-Carreau models. Making the discrete spectrum depend on the second invariant of the rate of deformation tensor has since been considered a bad idea (24). On this basis, Mewis and Denn (25) criticize Eq 6, suggesting that it might not reduce to linear viscoelasticity. Figures 15 through 18 show no pathological behavior in the limit of low strains at high frequency. We can also report that no such pathological behavior has been observed outside the region of experimentally accessible frequencies and strain amplitudes. Hence, the way the dependency on the second invariant of the rate of deformation tensor is introduced matters. The kinetic rate expression in Eq 6 introduces this dependency in an acceptable way.
A general purpose structural network theory has been used as a framework for the interpretation of large amplitude oscillatory behavior of polyolefin melts in terms of their entanglement kinetics. A qualitative difference in the large amplitude oscillatory shear behavior has been identified that distinguishes the five polyethylenes examined from two other polyolefins, polystyrene, and polyisobutylene.
The authors acknowledge the financial support of the Advanced Research Program of the State of Texas. The support of Academic Computer Services and the Supercomputer Center of Texas A & M University was invaluable. This work was inspired by the preliminary calculations of Mr. Jason G. Oakley (now with the Department of Energy, Washington, DC).
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A. J. GIACOMIN Polymer Processing Research Group Department of Mechanical Engineering University of Wisconsin Madison, Wisconsin 53706-1572
R. S. JEYASEELAN Rheology Research Laboratory Mechanical Engineering Department Texas A&M University College Station, Texas 77843-3123
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|Author:||Giacomin, A.J.; Jeyaseelan, R.S.|
|Publication:||Polymer Engineering and Science|
|Date:||May 15, 1995|
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