# Application of Mixing Rules for Adjusting the Flowability of Virgin and Post-Consumer Polypropylene as an Approach for Design from Recycling

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## Abstract

**:**

## 1. Introduction

## 2. Experimental

#### 2.1. Materials and Characterization

#### 2.2. Mixing Rules

## 3. Results

#### 3.1. Modeling with Virgin Blends

**Figure 1.**Melt flow rate depending on the weight fraction of vH4 of four virgin homopolymer mixtures vH8-vH4, vH12-vH4, vH20-vH4, and vH25-vH4.

#### 3.2. Modeling with Artificial Blends

#### 3.3. Modeling with Recyclate Blends

**Figure 9.**Parity plots for comparison of experimental and calculated values of commercial recyclates for (

**a**) model 3 and (

**b**) model 4.

**Table 8.**$MAE$ (left value, in g/10 min), $MRE$ (middle value, in %) and Pearson ${R}^{2}$ (right value) of recyclate mixtures r16-vH4 and r27-vH4 for model 3 and model 4.

Blend | Model 3 | Model 4 | ||||
---|---|---|---|---|---|---|

r16-vH4 | 0.47 | 5.45 | 0.995 | 0.16 | 1.82 | 0.999 |

r27-vH4 | 0.71 | 5.62 | 0.994 | 0.23 | 2.02 | 0.998 |

## 4. Modeling

#### 4.1. Symbolic Regression Analysis

#### 4.2. Symbolic Regression Results

**Figure 10.**Application of model 3, model 4 and MTF mixing rule on (

**a**) vH8-vH4, (

**b**) vH25-vH4, (

**c**) r16-vH4, and (

**d**) r27-vH4.

**Table 9.**$MAE$ (left value, in g/10 min), $MRE$ (middle value, in %), and Pearson ${R}^{2}$ (right value) of mixtures vH8-vH4, vH25-vH4, r16-vH4, and r27-vH4 for model 3, model 4, and MTF mixing rule.

Blend | Model 3 | Model 4 | MTF | ||||||
---|---|---|---|---|---|---|---|---|---|

vH8-vH4 | 0.13 | 2.04 | 0.997 | 0.18 | 2.94 | 0.996 | 0.08 | 1.35 | 0.998 |

vH25-vH4 | 0.17 | 1.43 | 0.999 | 0.59 | 5.70 | 0.997 | 0.13 | 1.04 | 1.000 |

r16-vH4 | 0.47 | 5.45 | 0.995 | 0.16 | 1.82 | 0.999 | 0.12 | 1.47 | 0.999 |

r27-vH4 | 0.71 | 5.62 | 0.994 | 0.23 | 2.02 | 0.998 | 0.13 | 1.07 | 1.000 |

**Table 10.**$MAE$ (left value, in g/10 min), $MRE$ (middle value, in %), and ${R}^{2}$ (right value) of mixtures vM3-vH4 and vB20-vH4 for the mixing rules model 3, model 4, and MTF.

Blend | Model 3 | Model 4 | MTF | ||||||
---|---|---|---|---|---|---|---|---|---|

vM3-vH4 | 0.29 | 4.44 | 0.991 | 0.40 | 5.84 | 0.986 | 0.38 | 5.52 | 0.987 |

vB20-vH4 | 0.74 | 7.51 | 0.985 | 0.39 | 3.89 | 0.995 | 0.44 | 4.15 | 0.994 |

## 5. Discussion

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Subfunctions of Equation (4) with $MF{I}_{2}=4g/10\mathrm{min}$

${A}_{1}=\frac{1}{{c}_{01}+{c}_{02}ln\left(MF{I}_{1}\right)}$ | (A1) |

${A}_{2}=\frac{{c}_{04}}{ln\left(MF{I}_{2}\right)+{c}_{05}{x}_{1}ln\left(MF{I}_{1}\right)}$ | (A2) |

${A}_{3}=\frac{{c}_{06}ln{\left(MF{I}_{1}\right)}^{2}\frac{{x}_{2}}{ln\left(MF{I}_{2}\right)}}{{c}_{07}+{c}_{08}ln\left(MF{I}_{1}\right)+{c}_{09}ln{\left(MF{I}_{1}\right)}^{2}-ln{\left(MF{I}_{1}\right)}^{3}}$ | (A3) |

${A}_{4}={c}_{10}{x}_{2}ln\left(MF{I}_{2}\right)$ | (A4) |

${A}_{5}={c}_{11}+{c}_{12}\frac{{x}_{2}}{ln\left(MF{I}_{2}\right)}$ | (A5) |

#### Appendix A.2. Model Coefficients

${c}_{00}$ | 0.0243327 | ${c}_{07}$ | 26.6035 |

${c}_{01}$ | 6292.96 | ${c}_{08}$ | −26.8233 |

${c}_{02}$ | −2934.41 | ${c}_{09}$ | 8.98627 |

${c}_{03}$ | −4.80228 | ${c}_{10}$ | −5.77251 |

${c}_{04}$ | 17.9609 | ${c}_{11}$ | −4.92978 |

${c}_{05}$ | −88.3534 | ${c}_{12}$ | −1.313 |

${c}_{06}$ | 0.0000276798 |

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**Figure 2.**Application of six mixing rules on (

**a**) vH8-vH4, (

**b**) vH12-vH4, (

**c**) vH20-vH4, and (

**d**) vH25-vH4.

**Figure 3.**Parity plots for comparison of experimental and calculated values of virgin mixtures for six different mixing rules: (

**a**) model 1, (

**b**) model 2, (

**c**) model 3, (

**d**) model 4, (

**e**) model 5, and (

**f**) model 6.

**Figure 4.**Melt flow rate depending on the weight fraction of vH4 of artificial mixtures vM1-vH4 and vM2-vH4.

**Figure 6.**Parity plots for comparison of experimental and calculated values of artificial recyclates for (

**a**) model 3 and (

**b**) model 4.

**Figure 7.**Melt flow rate depending on the weight fraction of vH4 of recyclate blends r16-vH4 and r27-vH4.

**Figure 12.**Scatter plots of MTF model: (

**a**) training and test set and (

**b**) validation set. The dashed lines indicate an absolute error of 1.06.

Designation | Material | MFR | Designation | Material | MFR |
---|---|---|---|---|---|

vH4 | HC205TF | 4 | vB8 | BD310MO | 8 |

vH8 | HD120MO | 8 | vB20 | BF970MO | 20 |

vH12 | HE125MO | 12 | r16 | Moplen QCP300P | 16 |

vH20 | HF955MO | 20 | r27 | Purpolen PP | 25 |

vH25 | HG385MO | 25 |

Set | Blend | MFR_{1} | MFR_{2} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

vH8 | vH12 | vH20 | vH25 | vB8 | vB20 | r16 | r27 | vH4 | |||

1 | vH8-vH4 | 100 | - | - | - | - | - | - | - | + | 100 |

vH12-vH4 | - | 100 | - | - | - | - | - | - | |||

vH20-vH4 | - | - | 100 | - | - | - | - | - | |||

vH25-vH4 | - | - | - | 100 | - | - | - | - | |||

2 | vM1-vH4 | 25 | 25 | 25 | 25 | - | - | - | - | ||

vM2-vH4 | 20 | 20 | 20 | 20 | 20 | - | - | - | |||

3 | r16-vH4 | - | - | - | - | - | - | 100 | - | ||

r27-vH4 | - | - | - | - | - | - | - | 100 | |||

4 | vM3-vH4 | 50 | 50 | - | - | - | - | - | - | ||

vB20-vH4 | - | - | - | - | - | 100 | - | - |

Mixture | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

${x}_{1}$ | 100 | 90 | 80 | 70 | 60 | 50 | 40 | 30 | 20 | 10 | 0 |

${x}_{2}$ | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

No. | Model | Equation | Source |
---|---|---|---|

1 | Linear | $MF{R}_{mix}={x}_{1}MF{R}_{1}+{x}_{2}MF{R}_{2}$ | [38] |

2 | K & M | $MF{R}_{mix}{}^{\frac{1}{n}}={x}_{1}MF{R}_{1}{}^{\frac{1}{n}}+{x}_{2}MF{R}_{2}{}^{\frac{1}{n}}$ | [39] |

3 | Arrhenius | $\mathrm{ln}\left(MF{R}_{mix}\right)={x}_{1}\mathrm{ln}\left(MF{R}_{1}\right)+{x}_{2}\mathrm{ln}\left(MF{R}_{2}\right)$ | [35] |

4 | Cragoe | $\frac{1}{\mathrm{ln}\left(LMF{R}_{mix}\right)}=\frac{{x}_{1}}{\mathrm{ln}\left(LMF{R}_{1}\right)}+\frac{{x}_{2}}{\mathrm{ln}\left(LMF{R}_{2}\right)}$ | [37] |

5 | Walther | $\mathrm{ln}\mathrm{ln}\left(MF{R}_{mix}+C\right)={x}_{1}\mathrm{ln}\mathrm{ln}\left(MF{R}_{1}+C\right)+{x}_{2}\mathrm{ln}\mathrm{ln}\left(MF{R}_{2}+C\right)$ | [40] |

6 | Bingham | $MF{R}_{mix}{}^{-1}={x}_{1}MF{R}_{1}{}^{-1}+{x}_{2}MF{R}_{2}{}^{-1}$ | [41] |

**Table 5.**$MAE$ (left value, in g/10 min) and $MRE$ (right value, in %) of virgin mixtures for all tested mixing rules.

Blend | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

vH8-vH4 | 0.14 | 2.37 | 0.06 | 0.94 | 0.13 | 2.04 | 0.18 | 2.94 | 0.27 | 4.41 | 0.38 | 6.09 |

vH12-vH4 | 1.16 | 15.46 | 0.52 | 6.52 | 0.20 | 2.28 | 0.11 | 1.28 | 0.32 | 4.24 | 0.71 | 9.05 |

vH20-vH4 | 2.09 | 24.31 | 0.87 | 9.53 | 0.25 | 2.62 | 0.12 | 1.35 | 0.58 | 6.26 | 1.42 | 14.25 |

vH25-vH4 | 3.43 | 32.17 | 1.20 | 9.93 | 0.17 | 1.43 | 0.59 | 5.70 | 1.34 | 11.84 | 2.86 | 22.73 |

Blend | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 |
---|---|---|---|---|---|---|

vH8-vH4 | 0.992 | 0.998 | 0.997 | 0.996 | 0.990 | 0.979 |

vH12-vH4 | 0.959 | 0.990 | 0.996 | 0.997 | 0.993 | 0.973 |

vH20-vH4 | 0.941 | 0.989 | 0.999 | 0.999 | 0.993 | 0.955 |

vH25-vH4 | 0.931 | 0.989 | 0.999 | 0.997 | 0.985 | 0.918 |

**Table 7.**$MAE$ (left value, in g/10 min), $MRE$ (middle value, in %), and Pearson ${R}^{2}$ (right value) of artificial recyclate mixtures for model 3 and model 4.

Blend | Model 3 | Model 4 | ||||
---|---|---|---|---|---|---|

vM1-vH4 | 0.19 | 2.09 | 0.998 | 0.12 | 1.61 | 1.000 |

vM2-vH4 | 0.18 | 2.01 | 0.998 | 0.27 | 3.63 | 0.996 |

**Table 11.**Errors for data points: $MAE$ (left, in g/10 min), $MRE$ (middle, in %), and Pearson ${R}^{2}$ (right) for Arrhenius, Cragoe, and MTF mixing rule.

Rule of Mixture | MAE | MRE | R^{2} |
---|---|---|---|

Arrhenius (model 3) | 0.29 | 3.29 | 0.995 |

Cragoe (model 4) | 0.22 | 2.67 | 0.997 |

MTF | 0.16 | 2.02 | 0.997 |

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**MDPI and ACS Style**

Traxler, I.; Marschik, C.; Farthofer, M.; Laske, S.; Fischer, J.
Application of Mixing Rules for Adjusting the Flowability of Virgin and Post-Consumer Polypropylene as an Approach for Design from Recycling. *Polymers* **2022**, *14*, 2699.
https://doi.org/10.3390/polym14132699

**AMA Style**

Traxler I, Marschik C, Farthofer M, Laske S, Fischer J.
Application of Mixing Rules for Adjusting the Flowability of Virgin and Post-Consumer Polypropylene as an Approach for Design from Recycling. *Polymers*. 2022; 14(13):2699.
https://doi.org/10.3390/polym14132699

**Chicago/Turabian Style**

Traxler, Ines, Christian Marschik, Manuel Farthofer, Stephan Laske, and Joerg Fischer.
2022. "Application of Mixing Rules for Adjusting the Flowability of Virgin and Post-Consumer Polypropylene as an Approach for Design from Recycling" *Polymers* 14, no. 13: 2699.
https://doi.org/10.3390/polym14132699