^{1,a)}, Leon Rice

^{1}and William Thurston

^{1}

### Abstract

*Non-linear* correlations are found between the melt strength and fundamental shear flow properties such as low frequency loss tangent, crossover frequency, and zero-shear-rate viscosity for a series of polypropylene (PP) melts. The very good non-linear correlations suggest *rheotens* as a reliable *rheological test* to characterize extensional rheology relevant to fabrication, despite its non-homogeneous and non-isothermal flowkinematics. The rheotens model of Doufas [J. Rheol. 50, 749–769 (2006)] with a *modified* Giesekus (MG) viscoelastic constitutive equation is expanded to the case of PP melts. A single set of molecular parameters per material predicts the rheotens force curves very well over a wide range of processing conditions. The rheotens model is proposed as a tool for determination of rheological parameters of constitutive equations applicable to the simulation of complex polymer processes. Molecular considerations and predictions of the rheotens model are extensively discussed. A multi-mode MG model predicts the *non-linear steady shear data* (shear and first normal stresses) very favorably satisfying linear viscoelasticity. The oscillatory shear data and model predictions satisfy both the Cox–Merz [J. Polym. Sci.28, 619–621 (1958)] and Laun [J. Rheol.30, 459–501 (1986)] rules. The model exhibits stable numerical behavior without singularities or turning points in the prediction of steady shear viscosity even at quite high shear rates (e.g., on the order of ), problems that have been reported for other constitutive equations.

We would like to thank Sunoco Chemicals for supporting this research and permitting its publication. A preliminary version of this work was presented at the International Congress of Rheology, Monterey, CA, August 2008.

I. INTRODUCTION

II. EXPERIMENT

A. Materials

B. Dynamic and steady shear flow experiments

C. Capillary shear flow experiments

D. Rheotens experimental set-up and process conditions

III. MATHEMATICAL MODELS

A. Rheotens constitutive based model

B. Rheotens model input parameters

C. Multi-mode MG constitutive formalism

IV. RESULTS AND DISCUSSION

A. Experimental results and shear/extensional rheology correlations

B. Modeling results of rheotens experiment

1. Rheotens model predictive capability

2. Temperature and strain rate profile predictions

3. Apparent elongational viscosity predictions

4. Effective extrudate swell

5. Molecular considerations of rheotens model

C. Modeling results of steady shear flow experiments

D. Consistency of modeling approach

1. Number of viscoelastic modes

2. Non-linear shear viscoelasticity modeling

V. CONCLUSIONS

### Key Topics

- Extensional flows
- 44.0
- Amorphous metals
- 40.0
- Viscosity
- 35.0
- Constitutive relations
- 34.0
- Shear rate dependent viscosity
- 34.0

## Figures

Typical rheotens force curves for the PP resins studied with temperature dependence. Testing conditions: 2.58 g/min, . (a) die temperature. (b) die temperature.

Typical rheotens force curves for the PP resins studied with temperature dependence. Testing conditions: 2.58 g/min, . (a) die temperature. (b) die temperature.

Non-linear correlations of shear and extensional rheology of PP melts at and . (a) Melt strength vs loss tangent at 0.1 rad/s. (b) Melt strength vs crossover frequency. For melt strength, the temperature refers to the die temperature. Testing conditions for the rheotens experiments: 2.58 g/min, .

Non-linear correlations of shear and extensional rheology of PP melts at and . (a) Melt strength vs loss tangent at 0.1 rad/s. (b) Melt strength vs crossover frequency. For melt strength, the temperature refers to the die temperature. Testing conditions for the rheotens experiments: 2.58 g/min, .

Melt strength vs zero-shear-rate viscosity at 200 and . For the melt strength, temperature refers to the die temperature. Testing conditions for the rheotens experiments: 2.58 g/min, . The solid line is shown only to “guide the eye” with respect to the trend between the two rheological parameters.

Melt strength vs zero-shear-rate viscosity at 200 and . For the melt strength, temperature refers to the die temperature. Testing conditions for the rheotens experiments: 2.58 g/min, . The solid line is shown only to “guide the eye” with respect to the trend between the two rheological parameters.

Comparison of experimental rheotens force curves with MG rheotens model predictions at various die temperatures for PP F. Testing conditions: 2.58 g/min, . The same set of molecular parameters , , and was used for the rheotens model predictions for PP F as shown in Table III.

Comparison of experimental rheotens force curves with MG rheotens model predictions at various die temperatures for PP F. Testing conditions: 2.58 g/min, . The same set of molecular parameters , , and was used for the rheotens model predictions for PP F as shown in Table III.

Comparison of experimental rheotens force curves with MG rheotens model predictions at various throughputs for PP F. (a) Rheotens force curves. (b) Melt strength vs throughput. Testing conditions: die temperature, . The same set of molecular parameters , , and was used for the rheotens model predictions for PP F, as shown in Table III.

Comparison of experimental rheotens force curves with MG rheotens model predictions at various throughputs for PP F. (a) Rheotens force curves. (b) Melt strength vs throughput. Testing conditions: die temperature, . The same set of molecular parameters , , and was used for the rheotens model predictions for PP F, as shown in Table III.

Prediction of non-isothermal and non-homogeneous flow kinematics of the rheotens experiments. (a) Filament temperature near the take-up wheels vs take-up speed at various throughputs. (b) Strain rate profile along the spinline at 2.58 g/min at various take-up speeds. PP F, die temperature, . In (a), the horizontal dotted line represents the die temperature.

Prediction of non-isothermal and non-homogeneous flow kinematics of the rheotens experiments. (a) Filament temperature near the take-up wheels vs take-up speed at various throughputs. (b) Strain rate profile along the spinline at 2.58 g/min at various take-up speeds. PP F, die temperature, . In (a), the horizontal dotted line represents the die temperature.

Predicted profiles of apparent elongational viscosity vs strain rate at a position near the take-up wheels for PP F at various throughputs and die temperatures. The temperature in the legend refers to the predicted filament temperature near the take-up wheels. Molecular parameters used in the simulations are included in Table III.

Predicted profiles of apparent elongational viscosity vs strain rate at a position near the take-up wheels for PP F at various throughputs and die temperatures. The temperature in the legend refers to the predicted filament temperature near the take-up wheels. Molecular parameters used in the simulations are included in Table III.

(a) Comparison of predictive capability of rheotens force curve for viscoelastic (MG) and Newtonian constitutive models. (b) Comparison of predicted tensile stress difference vs strain rate. PP F, die temperature, 2.58 g/min, . For the MG rheotens model, , , and listed in Table III were used. For the Newtonian rheotens model, , , and . The model predictions refer to a filament position near the take-up wheels. The stress difference is calculated as the difference of the tensile minus the radial extra stress at a given strain rate.

(a) Comparison of predictive capability of rheotens force curve for viscoelastic (MG) and Newtonian constitutive models. (b) Comparison of predicted tensile stress difference vs strain rate. PP F, die temperature, 2.58 g/min, . For the MG rheotens model, , , and listed in Table III were used. For the Newtonian rheotens model, , , and . The model predictions refer to a filament position near the take-up wheels. The stress difference is calculated as the difference of the tensile minus the radial extra stress at a given strain rate.

(a) Predicted molecular chain extension [Eq. (13)] in the (drawing) direction. (b) Percent fractional chain extension [Eq. (4)] as a function of strain rate at a position near the take-up wheels for PP A, PP E, and PP G. Rheotens testing conditions: die temperature, 2.58 g/min, . The arrows represent the experimental filament break point for each material.

(a) Predicted molecular chain extension [Eq. (13)] in the (drawing) direction. (b) Percent fractional chain extension [Eq. (4)] as a function of strain rate at a position near the take-up wheels for PP A, PP E, and PP G. Rheotens testing conditions: die temperature, 2.58 g/min, . The arrows represent the experimental filament break point for each material.

(a) Percent fractional chain extension [Eq. (4)] as a function of strain rate at a position near the take-up wheels for PP E with various combinations of molecular parameters for the OG rheotens model. (b) Comparison of predictive capability of rheotens force curve of OG vs MG constitutive models. OG (1) , , ; OG (2) , , ; OG (3) , , ; OG (4) , , ; OG (5) , , . Material: PP E. Rheotens testing conditions: die temperature, 2.58 g/min, .

(a) Percent fractional chain extension [Eq. (4)] as a function of strain rate at a position near the take-up wheels for PP E with various combinations of molecular parameters for the OG rheotens model. (b) Comparison of predictive capability of rheotens force curve of OG vs MG constitutive models. OG (1) , , ; OG (2) , , ; OG (3) , , ; OG (4) , , ; OG (5) , , . Material: PP E. Rheotens testing conditions: die temperature, 2.58 g/min, .

(a) Predictive capability of multi-mode MG model for steady shear viscosity flow curves at various temperatures. (b) First normal stress difference as a function of shear rate at . The experimental viscosity flow curves derive from (i) steady shear (rotational) experiments and (ii) superposition of oscillatory shear data with application of the Cox–Merz rule and capillary shear data. data derive from (i) steady shear experiments and (ii) oscillatory shear experiments employing the Laun (1986) approximation represented by Eq. (7). For each material, the same set of non-linear molecular parameters , (Table IV) were used for each relaxation mode coinciding with those used in the rheotens model (Table III). Model predictions for are extended in the vicinity of the maximum shear rate where capillary data are available.

(a) Predictive capability of multi-mode MG model for steady shear viscosity flow curves at various temperatures. (b) First normal stress difference as a function of shear rate at . The experimental viscosity flow curves derive from (i) steady shear (rotational) experiments and (ii) superposition of oscillatory shear data with application of the Cox–Merz rule and capillary shear data. data derive from (i) steady shear experiments and (ii) oscillatory shear experiments employing the Laun (1986) approximation represented by Eq. (7). For each material, the same set of non-linear molecular parameters , (Table IV) were used for each relaxation mode coinciding with those used in the rheotens model (Table III). Model predictions for are extended in the vicinity of the maximum shear rate where capillary data are available.

Sensitivity of , non-linear molecular parameters of multi-mode MG model on predictions of steady shear viscosity flow curves. Predictions are for PP G at using the discrete relaxation spectrum of Table IV. Model Set 1: , (MG, rheotens parameters; best prediction). Model set 2: , (upper-convected Maxwell). Model set 3: , (FENE-like). Model set 4: , . Model set 5: , . Model set 6: , .

Sensitivity of , non-linear molecular parameters of multi-mode MG model on predictions of steady shear viscosity flow curves. Predictions are for PP G at using the discrete relaxation spectrum of Table IV. Model Set 1: , (MG, rheotens parameters; best prediction). Model set 2: , (upper-convected Maxwell). Model set 3: , (FENE-like). Model set 4: , . Model set 5: , . Model set 6: , .

## Tables

List of PP resins studied in this work with material/rheological properties. The melt shear rheological parameters (zero-shear-rate viscosity and activation energy for flow) are inputs to the rheotens model of Doufas (2006). The shear data are based on frequency sweep dynamic experiments as described in Sec. II. Zero-shear-viscosity is calculated from Eq. (1) using the discrete relaxation spectrum. The zero-shear-rate viscosity determined from the dynamic experiments is identical with that determined from the steady shear flow experiments (see Fig. 11).

List of PP resins studied in this work with material/rheological properties. The melt shear rheological parameters (zero-shear-rate viscosity and activation energy for flow) are inputs to the rheotens model of Doufas (2006). The shear data are based on frequency sweep dynamic experiments as described in Sec. II. Zero-shear-viscosity is calculated from Eq. (1) using the discrete relaxation spectrum. The zero-shear-rate viscosity determined from the dynamic experiments is identical with that determined from the steady shear flow experiments (see Fig. 11).

Processing conditions of rheotens experiments employed in this work. The values of the testing parameters in parentheses are the default values used in this work, unless otherwise indicated.

Processing conditions of rheotens experiments employed in this work. The values of the testing parameters in parentheses are the default values used in this work, unless otherwise indicated.

Molecular/rheological parameters of studied PP resins determined by fitting a single experimental force rheotens curve to the Doufas (2006) rheotens model.

Molecular/rheological parameters of studied PP resins determined by fitting a single experimental force rheotens curve to the Doufas (2006) rheotens model.

Examples of discrete relaxation spectra with non-linear molecular/rheological parameters , of multi-mode MG model used for prediction of steady shear flow data. The relaxation spectra correspond to . The parameters , correspond to those used in the rheotens model for each material (Table III).

Examples of discrete relaxation spectra with non-linear molecular/rheological parameters , of multi-mode MG model used for prediction of steady shear flow data. The relaxation spectra correspond to . The parameters , correspond to those used in the rheotens model for each material (Table III).

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