# Applicability of the Cox-Merz Rule to High-Density Polyethylene Materials with Various Molecular Masses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

- Material 1 was a high viscosity hexene copolymer polyethylene compound (HDPE) for pipe applications (PE 100) with high density and an outstanding resistance to slow crack growth.
- Material 2 was a high-density polyethylene for injection and compression moulding.
- Material 3 was another high-density polyethylene for injection and compression moulding.

#### 2.2. Parallel-Plate Rheometer

#### 2.3. High-Pressure Capillary Rheometer (HPCR)

^{−1}. We calculated the wall shear stress ${\tau}_{w}$ according to [34]

#### 2.4. Slit-Die Extrusion Rheometer

^{3}/rev.), a bypass valve, and a slit die with a defined gap height of 0.8 mm and a width of 20 mm was used to determine the shear-rate-dependent viscosity through the slit. The temperature profile was adjusted for the measurement to always reach a melt temperature of 200 °C at the entrance of the die for every measurement point.

^{−1}.

## 3. Simulation

#### 3.1. Fitting of Experimental Data

#### 3.2. Simulation of Extrusion Equipment

^{−5}. Subsequently, we evaluated the pressure drop of the flow geometry.

#### 3.3. Extrusion Experiments with the Real Pipe Head

## 4. Results and Discussion

#### 4.1. Comparison of Plate-Plate Rheometry (PPR) to High-Pressure Capillary Rheometry (HPCR) and Extrusion Slit Rheometry

_{w}. As ${\eta}_{0}$ increases so too does the molecular weight of the polymer. The modified cross-law parameters for an HDPE melt are strongly related to the mass average molecular weight M

_{w}according to Equations 11 and 13 [39]. For the tested materials, the exponents of the equations are listed in Table 4. The values are in good accordance to the literature [39].

#### 4.2. Viscoelasticity of HDPE Materials

^{−1}and 400 s

^{−1}. For Material 2, the viscous part is more dominant up to 100 s

^{−1}, beyond which the elastic part becomes more dominant. For Material 3 the viscous part dominates between 10 s

^{−1}and 400 s

^{−1}.

#### 4.3. Comparison of Pipe-Head Simulations with Measured Rheology Curves

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Comparison of pressure-flow-based measurements and oscillatory parallel-plate measurement of HDPEs at 200 °C. (

**A**–

**C**) show the data for Materials 1, 2, and 3, respectively.

**Figure 3.**Comparison of the modified cross-law fits of Materials 1,2, and 3 with pressure-flow-based measurements (

**B**) and oscillatory parallel-plate measurement (

**A**) of HDPEs at 200 °C.

**Figure 4.**Storage and loss modules of three different HDPE materials. (

**A**–

**C**) show, respectively, the data of Materials 1, 2, and 3.

**Figure 6.**Comparison between the pressure drops according to two rheological models and experimental data at various mass flow rates for HDPE at 200 °C in a 32 mm pipe head. (

**A**–

**C**) show the simulation results for Materials 1, 2, and 3, respectively.

**Table 1.**Melt flow rate (MFR) and molecular weight distributions of the high-density polyethylene (HDPE) materials.

MFR (g/10 min) | M_{W} (g/mol) | M_{Z} (g/mol) | |
---|---|---|---|

Material 1 | 0.25 | 230,000 | 1,190,000 |

Material 2 | 1.5 | 110,000 | 550,000 |

Material 3 | 4.0 | 85,500 | 387,000 |

Parameter | Unit | Material 1 | Material 2 | Material 3 |
---|---|---|---|---|

${\eta}_{0}$ | Pa.s | 43,226 | 4559 | 1930 |

$\lambda $ | s | 0.362 | 0.125 | 0.0711 |

$m$ | - | 0.761 | 0.571 | 0.491 |

**Table 3.**High-pressure capillary rheometry (HPCR)-based modified cross-law parameters for HDPE melt at 200 °C.

Parameter | Unit | Material 1 | Material 2 | Material 3 |
---|---|---|---|---|

${\eta}_{0}$ | Pa.s | 56,796 | 3715 | 3175 |

$\lambda $ | s | 0.323 | 0.124 | 0.097 |

$m$ | - | 0.788 | 0.598 | 0.494 |

PPR | HPCR | |
---|---|---|

α | 3.09 | 3.12 |

β | 0.43 | 0.45 |

κ | 1.59 | 1.23 |

**Table 5.**Ratios of viscosities measured by PPR and HPCR at various shear rates $\dot{\gamma}$ for Materials 1–3.

Material 1 | Material 2 | Material 3 | |
---|---|---|---|

$\dot{\gamma}$ | $\phi $ | $\phi $ | $\phi $ |

5 | 2.12 | 1.27 | 1.05 |

50 | 2.22 | 1.29 | 1.13 |

150 | 2.33 | 1.32 | 1.20 |

400 | 2.41 | 1.36 | 1.23 |

**Table 6.**Comparison of simulated pressure drops with experimental data at various output rates for Materials 1–3.

Output | Material 1 | Material 2 | Material 3 | |||
---|---|---|---|---|---|---|

kg/h | $\mathit{\chi}$ | $\mathbf{\u0255}$ | $\mathit{\chi}$ | $\mathbf{\u0255}$ | $\mathit{\chi}$ | $\mathbf{\u0255}$ |

5 | 1.74 | 0.98 | 1.23 | 1 | 1.16 | 1 |

10 | 2.01 | 1.03 | 1.22 | 0.96 | 1.03 | 1 |

15 | 2.08 | 1.03 | 1.31 | 1.02 | 1.07 | 1.04 |

20 | 2.06 | 0.99 | 1.27 | 0.99 | 1 | 1.05 |

25 | 2.04 | 0.97 | 1.31 | 1.01 | 1 | 1.06 |

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**MDPI and ACS Style**

Rathner, R.; Roland, W.; Albrecht, H.; Ruemer, F.; Miethlinger, J.
Applicability of the Cox-Merz Rule to High-Density Polyethylene Materials with Various Molecular Masses. *Polymers* **2021**, *13*, 1218.
https://doi.org/10.3390/polym13081218

**AMA Style**

Rathner R, Roland W, Albrecht H, Ruemer F, Miethlinger J.
Applicability of the Cox-Merz Rule to High-Density Polyethylene Materials with Various Molecular Masses. *Polymers*. 2021; 13(8):1218.
https://doi.org/10.3390/polym13081218

**Chicago/Turabian Style**

Rathner, Raffael, Wolfgang Roland, Hanny Albrecht, Franz Ruemer, and Jürgen Miethlinger.
2021. "Applicability of the Cox-Merz Rule to High-Density Polyethylene Materials with Various Molecular Masses" *Polymers* 13, no. 8: 1218.
https://doi.org/10.3390/polym13081218