**1. Introduction**

Considering the current situation of plastic consumption worldwide, the issue of endof-life of polymer materials has become a significant problem. Polyethylene (PE) accounts

**Citation:** Castéran, F.; Delage, K.; Hascoët, N.; Ammar, A.; Chinesta, F.; Cassagnau, P. Data-Driven Modelling of Polyethylene Recycling under High-Temperature Extrusion. *Polymers* **2022**, *14*, 800. https:// doi.org/10.3390/polym14040800

Academic Editor: Célio Bruno Pinto Fernandes

Received: 11 November 2021 Accepted: 15 February 2022 Published: 18 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

for most plastic packaging and, consequently, plastic waste [1]. As PE is a thermoplastic, the most common method for its recycling is mechanical recycling, which involves reprocessing the materials [2–4]. These processes can induce the formation of radicals by homolytic cleavage of the polymers, inducing degradation, branching or even crosslinking of the materials leading to different final properties [5–9]. Consequently, to these properties changes, the applications of the mechanically recycled polymers have to be adapted [10–12]. Whereas the majority of recycled high and low-density polyethylenes (HDPE and LDPE) are however produced that way, Ultra High Molecular Weight Polyethylenes (UHMWPE), mostly used for high-performance applications due to their superior mechanical properties, are more difficult or impossible to process due to their high viscosity [13]. The other principal way of recycling polymers is by chemical recycling, which consists of a chemical transformation leading to new raw materials. Whereas polyethylene terephthalate (PET) can be depolymerized into dimethyl terephthalate and ethylene glycol by mathanolysis [14], no such reactions are possible for PE. The main way of PE chemical recycling is then pyrolysis, leading to smaller carbonated molecules, which can, in theory, be reinjected into the chemical industry [2,6,15–21]. plastic, the most common method for its recycling is mechanical recycling, which involves reprocessing the materials [2–4]. These processes can induce the formation of radicals by homolytic cleavage of the polymers, inducing degradation, branching or even crosslinking of the materials leading to different final properties [5–9]. Consequently, to these properties changes, the applications of the mechanically recycled polymers have to be adapted [10–12]. Whereas the majority of recycled high and low-density polyethylenes (HDPE and LDPE) are however produced that way, Ultra High Molecular Weight Polyethylenes (UHMWPE), mostly used for high-performance applications due to their superior mechanical properties, are more difficult or impossible to process due to their high viscosity [13]. The other principal way of recycling polymers is by chemical recycling, which consists of a chemical transformation leading to new raw materials. Whereas polyethylene terephthalate (PET) can be depolymerized into dimethyl terephthalate and ethylene glycol by mathanolysis [14], no such reactions are possible for PE. The main way of PE chemical recycling is then pyrolysis, leading to smaller carbonated molecules, which can, in theory, be reinjected into the chemical industry [2,6,15–21].

Considering the current situation of plastic consumption worldwide, the issue of end-of-life of polymer materials has become a significant problem. Polyethylene (PE) accounts for most plastic packaging and, consequently, plastic waste [1]. As PE is a thermo-

*Polymers* **2022**, *14*, x FOR PEER REVIEW 2 of 23

**1. Introduction** 

Whilst thermal degradation of polymers that include heteroatoms in their structure (PMMA, for instance) leads to simple products and mechanisms, PE degradation is more complex [22]. A simplification of PE degradation mechanisms is presented in Figure 1. Whilst thermal degradation of polymers that include heteroatoms in their structure (PMMA, for instance) leads to simple products and mechanisms, PE degradation is more complex [22]. A simplification of PE degradation mechanisms is presented in Figure 1.

**Figure 1.** Simplified mechanisms of thermal degradation of polyethylene [23,24]. **Figure 1.** Simplified mechanisms of thermal degradation of polyethylene [23,24].

As shown in this figure, different degradation mechanisms are possible for PE thermal degradation. The process conditions then define their probability. Whereas for polymers with a more complex structure, the end-chain scission mechanism would be preponderant, in the case of polyolefins, random scissions are more significant [24]. Therefore, the higher the molecular weight of the polyolefin, the more random scissions occur, resulting in a narrowing of the molecular weight distribution [25]. As schematized in Figure 2, the molecular weight decreases with the increase of the temperature of the reaction, leading first to oligomers and then to smaller molecules whose nature depends on propagation and termination mechanisms. A short reaction time at high temperatures would favour β-scissions, leading to the formation of a certain yield of the ethylene monomer. However, longer reaction times tend to favour the production of cyclic compounds due to intramolecular and intermolecular transfers, which are less easy to valorize afterwards [25]. mal degradation. The process conditions then define their probability. Whereas for polymers with a more complex structure, the end-chain scission mechanism would be preponderant, in the case of polyolefins, random scissions are more significant [24]. Therefore, the higher the molecular weight of the polyolefin, the more random scissions occur, resulting in a narrowing of the molecular weight distribution [25]. As schematized in Figure 2, the molecular weight decreases with the increase of the temperature of the reaction, leading first to oligomers and then to smaller molecules whose nature depends on propagation and termination mechanisms. A short reaction time at high temperatures would favour β-scissions, leading to the formation of a certain yield of the ethylene monomer. However, longer reaction times tend to favour the production of cyclic compounds due to intramolecular and intermolecular transfers, which are less easy to valorize afterwards [25].

As shown in this figure, different degradation mechanisms are possible for PE ther-

*Polymers* **2022**, *14*, x FOR PEER REVIEW 3 of 23

**Figure 2.** Representation of the different products of polyethylene degradation depending on temperature and residence time, inspired from Vollmer et al. [25]. **Figure 2.** Representation of the different products of polyethylene degradation depending on temperature and residence time, inspired from Vollmer et al. [25].

The final products are then highly dependent on the processing conditions. Moreover, classic pyrolysis processes induce heat and mass transfer problems, leading to highly heterogeneous products without the possibility of controlling the degradation [25]. The final products are then highly dependent on the processing conditions. Moreover, classic pyrolysis processes induce heat and mass transfer problems, leading to highly heterogeneous products without the possibility of controlling the degradation [25].

Other innovative ways of PE recycling were recently studied. Manas et al. [26,27], for instance, studied the recycling of PE crosslinked by irradiations by using it as a filler for virgin LDPE, Elmanowich et al. [28] studied the use of supercritical fluids for PE recycling, and recent promising studies are about the enzymatic degradation of polymers [29,30]. Other innovative ways of PE recycling were recently studied. Manas et al. [26,27], for instance, studied the recycling of PE crosslinked by irradiations by using it as a filler for virgin LDPE, Elmanowich et al. [28] studied the use of supercritical fluids for PE recycling, and recent promising studies are about the enzymatic degradation of polymers [29,30].

The present work aims at controlling the thermo-mechanical degradation of PE by carrying out a twin-screw extrusion process at high temperatures (320 < T °C < 420). Whereas polyethylene extrusion usually leads to branching and crosslinking, the extrusion thermal conditions in this work favour degradation mechanisms closer to pyrolysis The present work aims at controlling the thermo-mechanical degradation of PE by carrying out a twin-screw extrusion process at high temperatures (320 < *T* ◦C < 420). Whereas polyethylene extrusion usually leads to branching and crosslinking, the extrusion thermal conditions in this work favour degradation mechanisms closer to pyrolysis conditions.

conditions. With the question of the process control comes the issue of its simulation. Extrusion simulation has been widely studied in the last decades. However, the complexity of the physical phenomena involved in the extrusion process involves either a long time and a significant computing power or a lot of hypotheses and simplifications. Furthermore, in this study, the viscoelastic properties of the materials evolve with their degradation. Due to the temperatures reached in the extruder, the materials are at the limit of pyrolysis. Moreover, the presence of oxygen involves additional chemical reactions due to high-tem-With the question of the process control comes the issue of its simulation. Extrusion simulation has been widely studied in the last decades. However, the complexity of the physical phenomena involved in the extrusion process involves either a long time and a significant computing power or a lot of hypotheses and simplifications. Furthermore, in this study, the viscoelastic properties of the materials evolve with their degradation. Due to the temperatures reached in the extruder, the materials are at the limit of pyrolysis. Moreover, the presence of oxygen involves additional chemical reactions due to high-temperature oxidation. Thus, the complexity of the degradation mechanisms increases the imprecision of the simulation.

perature oxidation. Thus, the complexity of the degradation mechanisms increases the imprecision of the simulation. On the other hand, machine learning does not need to understand these complex physics, only to have accurate experimental data to predict the results of new experiments.

This approach often leads to faster computing and sometimes to more precise results, the imprecision coming from unknown phenomena or simplifications unavoidable in classical simulations that are not necessary with machine learning. Such methodologies were successfully employed in previous works on various reactive extrusion systems [31,32]. imprecision coming from unknown phenomena or simplifications unavoidable in classical simulations that are not necessary with machine learning. Such methodologies were successfully employed in previous works on various reactive extrusion systems [31,32]. The following is a study of the degradation of HDPE and UHMWPE by twin-screw extrusion at high temperatures. Then, several methods will be applied and compared to

On the other hand, machine learning does not need to understand these complex physics, only to have accurate experimental data to predict the results of new experiments. This approach often leads to faster computing and sometimes to more precise results, the

*Polymers* **2022**, *14*, x FOR PEER REVIEW 4 of 23

The following is a study of the degradation of HDPE and UHMWPE by twin-screw extrusion at high temperatures. Then, several methods will be applied and compared to determine the extruded polyethylenes' final viscosities and molecular weights. Finally, the modelling of this process with the twin-screw extrusion Ludovic® simulation software (SC-Consultants, Saint-Etienne, France) will be compared to the results obtained with Machine-Learning methodologies. determine the extruded polyethylenes' final viscosities and molecular weights. Finally, the modelling of this process with the twin-screw extrusion Ludovic® simulation software (SC-Consultants, Saint-Etienne, France) will be compared to the results obtained with Machine-Learning methodologies. **2. Experimental Section** 

#### **2. Experimental Section** *2.1. Materials and Extrusion*

#### *2.1. Materials and Extrusion* HDPE (HDPE XRT70, TOTAL, Melt Flow Index (MFI: 190 °C/5 kg) = 0.7 g/10 min)

HDPE (HDPE XRT70, TOTAL, Melt Flow Index (MFI: 190 ◦C/5 kg) = 0.7 g/10 min) and UHMWPE (GUR 4130, Celanese, Melt Flow Index (MFI: 190 ◦C/21.6 kg) < 0.1 g/10 min) were extruded in an intermeshing corotating twin-screw extruder ZSE 18 MAXX/HPe (Leistritz, Nuremberg, Germany) for various temperatures, screw rotation speeds and exit flow rates, reaching 29 different configurations. These processing conditions are summarized in Table 1. and UHMWPE (GUR 4130, Celanese, Melt Flow Index (MFI: 190 °C/21.6 kg) < 0.1 g/10 min) were extruded in an intermeshing corotating twin-screw extruder ZSE 18 MAXX/HPe (Leistritz, Nuremberg, Germany) for various temperatures, screw rotation speeds and exit flow rates, reaching 29 different configurations. These processing conditions are summarized in Table 1**.** 

**Table 1.** Processing conditions used in this study.

**Table 1.** Processing conditions used in this study.


\* Tmax correspond to the maximal setpoint temperature cf. Figure 3.

As the Polyethylene samples are highly viscous, and in order to have a maximum of possibilities concerning flow rate and rotation speed combinations, the screw profile was designed with quite a low shear and few restrictive elements. The temperature was increased progressively along with the barrel blocks and progressively decreased before the die for security issues and to limit the degradation at the die exit. The polyethylenes were introduced at the extruder entrance with a gravimetric feeder. The die had a 3 mm diameter. At the exit, the materials were cooled by air and then pelletized. The extruder configuration is described in Figure 3. As the Polyethylene samples are highly viscous, and in order to have a maximum of possibilities concerning flow rate and rotation speed combinations, the screw profile was designed with quite a low shear and few restrictive elements. The temperature was increased progressively along with the barrel blocks and progressively decreased before the die for security issues and to limit the degradation at the die exit. The polyethylenes were introduced at the extruder entrance with a gravimetric feeder. The die had a 3 mm diameter. At the exit, the materials were cooled by air and then pelletized. The extruder configuration is described in Figure 3.

**Figure 3. Figure 3.**  Scheme of the screw and temperature profiles used. Scheme of the screw and temperature profiles used.

The details about the different extrusion configurations tested and the in-line measures are presented in the Appendix A of this article, in Table A1.

The torque, the Engine Power and the die pressure were measured for each experiment. Thermocouples placed around the centre of the extruder (*T<sup>c</sup>* at *L* = 38D) and in the die permitted to measure the melt temperature. However, it appears that the temperature measured is the one of the inner surface of the barrel and not precisely the bulk material temperature. Therefore, a manual thermocouple was used to measure it at the die exit of the extruder.

#### *2.2. Characterizations*

The rheological behaviour of the extruded and raw materials was studied using a DHR-2 (TA Instruments, New Castle, DE, USA), a stress-controlled rheometer. Frequency sweeps have been carried out at 190 ◦C from 100 to 0.01 rad·s <sup>−</sup><sup>1</sup> under nitrogen and with 1% deformation. The geometry used was 25 mm diameter parallel plates with a 1 mm gap.

The molecular weight distributions of five samples (two degraded HDPE, two degraded UHMWPE and raw HDPE XRT70) were measured using high-temperature steric exclusion chromatography (HT-SEC) using a Viscositek (Malvern Panalytical Ltd., Malvern, UK) device. The samples were previously dissolved in toluene at 100 ◦C for 30 min.

#### *2.3. Theoretical Methodologies*

#### 2.3.1. Determination of *Mw* from Viscoelastic Behaviour

The molecular weight distributions have been calculated using the TA instrument tool implemented in the Trios software (TA Instruments, New Castle, DE, USA). This tool, based on the double reptation theories [33,34], uses a model linking the molecular weight distribution to the relaxation modulus via the following relationship:

$$\mathbf{G}(t) = \mathbf{G}\_N^0 \cdot \left[ \int\_{\ln(M\varepsilon)}^{\infty} \mathbf{F}^{\frac{1}{\beta}} \, w(M) \mathbf{d} \ln(M) \right]^\beta \tag{1}$$

where *G*(*t*) is the linear viscoelastic relaxation modulus, *G* 0 *N* is the plateau modulus, *w*(*M*) is the molecular weight distribution, *F*(*M*, *t*) is the monodisperse relaxation function, and *β* takes the value 1 for simple reptation and 2 for double reptation. Several models exist to define the F function. In this study, as in most cases, a single exponential form described by Equation (2) is applied. Table 2 presents the definition of the constants used by the model and their values in this work, which correspond to classical values for polyethylene.

$$\begin{array}{l} F^{\frac{1}{2}}(M, t) = \exp\left(\frac{-t}{2\lambda(M)}\right) \\ \text{with } \lambda(M) = K\_{\lambda}(T)M^{\mathfrak{a}}; \\ K\_{\lambda}(T) = K\_{\lambda}(T\_{0})\exp\left(\frac{E\_{g}}{RT}\right) \end{array} \tag{2}$$

**Table 2.** Parameters used for the molecular weight distribution calculation from inverse rheology method.


#### 2.3.2. Determination of *Mw* from Measured Die Pressure 2.3.2. Determination of Mw from Measured Die Pressure

As the pressure measured at the extruder die depends on the viscosity of the viscous polymer, which can be related to the molecular weight, it is then possible to estimate the viscosity and subsequently the molecular weight without the need for post-process characterizations. Figure 4 summarises the different steps required to measure *Mw* from the die pressure measurement. As the pressure measured at the extruder die depends on the viscosity of the viscous polymer, which can be related to the molecular weight, it is then possible to estimate the viscosity and subsequently the molecular weight without the need for post-process characterizations. Figure 4 summarises the different steps required to measure *Mw* from the die pressure measurement.

**Figure 4.** Scheme of the Mw determination from die pressure measurement. **Figure 4.** Scheme of the *Mw* determination from die pressure measurement.

The first step is to determine the value of the viscosity and the shear rate in the die. For this purpose, the die is considered a capillary rheometer. The experimentally measured temperatures and pressures are then approximated to those of the narrowest section of the die. As in a capillary rheometer, an apparent shear rate ߛ ሶ and an apparent viscosity ߟ are calculated as a function of the flow rate *Q*, the die cross-section and length *r* and *L* and the difference between die and outside drop pressure *∆P* according to Equa-The first step is to determine the value of the viscosity and the shear rate in the die. For this purpose, the die is considered a capillary rheometer. The experimentally measured temperatures and pressures are then approximated to those of the narrowest section of the die. As in a capillary rheometer, an apparent shear rate . *γapp* and an apparent viscosity *ηapp* are calculated as a function of the flow rate *Q*, the die cross-section and length *r* and *L* and the difference between die and outside drop pressure ∆*P* according to Equation (3).

$$
\dot{\gamma}\_{app} = \frac{4 \text{ } \mathcal{Q}}{\pi \, r^3} \quad ; \quad \eta\_{app} = \frac{\Delta P \, r}{2 \, \text{L} \dot{\gamma}\_{app}} \tag{3}
$$

(3)

 ߟ ; ଷݎ ߨ ሶ ߛ ܮ 2 These values are valid at the temperature of the die ܶௗ, it is then necessary to calculate them at a reference temperature ܶ which is set at 190 °C in this study. This correction These values are valid at the temperature of the die *Tdie*, it is then necessary to calculate them at a reference temperature *T*<sup>0</sup> which is set at 190 ◦C in this study. This correction is then done assuming an Arrhenius dependence of the viscosity (Equation (4)).

$$\eta\_{app}(T\_{\text{diie}}) = a\_T \times \eta\_{app}(T\_0) \quad ; \quad a\_T = \exp\left(\frac{E\_a}{R} \times \left(\frac{1}{T\_0} - \frac{1}{T\_{\text{diie}}}\right)\right) \tag{4}$$

*Ea* = activation energy; *E<sup>a</sup>* = activation energy;

tion (3).

*R* = ideal gases constant. *R* = ideal gases constant.

Then, the Rabinowitsch correction (Equation (5)) has to be applied to calculate real shear rate and viscosity ߛሶ ௗ and ߟௗ. Then, the Rabinowitsch correction (Equation (5)) has to be applied to calculate real shear rate and viscosity . *γdie* and *ηdie*.

$$\begin{array}{rcl} \dot{\gamma}\_{di} = \frac{3m+1}{4m} \ \dot{\gamma}\_{app} \ \vdots \ \ \eta\_{di} = \frac{4m}{3m+1} \ \eta\_{app} \ \vdots \\\ \text{with } m-1 = \frac{\partial \log(\eta)}{\partial \log(\dot{\gamma})} \end{array} \tag{5}$$

డ ୪୭(ஓሶ) Usually, the parameter ݉ is obtained by measuring several values of ߛ ሶ and ߟ and calculating the slope of the resulting line by plotting ݈݃) ߟ = (݂)݈݃) ߛ ሶ *)).* However, in the present study, it is impossible to obtain more than one point of the curve because measuring another pressure value means changing the process and thus changing the material itself. A first regression is then performed considering a Carreau-Yasuda Usually, the parameter *<sup>m</sup>* is obtained by measuring several values of . *γapp* and *ηapp* and calculating the slope of the resulting line by plotting *log*(*ηapp*) = *f*(*log*( . *γapp*)). However, in the present study, it is impossible to obtain more than one point of the curve because measuring another pressure value means changing the process and thus changing the material itself. A first regression is then performed considering a Carreau-Yasuda model (Equation (6)) passing through this single point; it is thus considered that all degraded

PE still follow a Carreau-Yasuda law. *m* and *a* parameters are set as the ones of the raw HDPE XRT70 (*m* = 0.058 and *a* = 0.248). They have been determined by a frequency sweep with the rheometer and a fitting with the Carreau-Yasuda model. The Cox-Merz law was assumed to compare the actual shear viscosity and the complex viscosity measured by oscillatory rheometry tests. According to the literature [35], the viscosity at high shear rates does not depend on molecular weight. The viscosity curves of different molecular weight polyethylene samples converge then on an identical power-law behaviour. Making this assumption, *λ* and *η*<sup>0</sup> have been determined for each sample through classical regression methods, and the equation is thus completely described.

$$\eta\left(\dot{\gamma}\_{\prime}\ T\right) = \eta\_0 \cdot \left(1 + \left(\lambda \times \dot{\gamma}\right)^a\right)^{\frac{m-1}{a}}\tag{6}$$

*m* and *a* = dimensionless indexes;

*λ* [s] = characteristic time;

*η*<sup>0</sup> [Pa·s] = zero shear viscosity.

The *m* parameter can then be calculated with the resulting curve, which enables the calculation of real shear rate . *γdie* and viscosity *ηdie* in the die thanks to Rabinowitsch correction. Finally, the real viscosity curve passing through this point is estimated by a second regression with a Carreau-Yasuda model in the same way as the first regression.

It is then possible to link the average molecular weight *M<sup>w</sup>* to *η*<sup>0</sup> considering Equation (7). Often estimated at 3.4 for entangled polymers, the α exponent has been fixed at 3.6 in this study to be homogeneous with previous calculations and other HDPE studies [36–38] (the results being similar with the two values). The *K* constant can be determined from the values of *η*<sup>0</sup> and *M<sup>w</sup>* of the reference HDPE.

$$
\eta\_0 = \mathcal{K} \times M\_w^a \tag{7}
$$

#### **3. Modelling and Machine Learning**

*3.1. Simulation*

Ludovic® software (SC-Consultants, Saint-Etienne, France) is a well-known twin-screw extrusion simulation software. Due to its computing speed, ease of use and flexibility, it has proven its reliability in many different fields and applications. Initially developed for starch extrusion, it is now developed and used for plastics compounding and the pharmaceutical, cosmetics, food, and construction industries. This broad use makes it an interesting choice for this study and for comparison with new approaches such as machine learning.

Its efficiency is due to many simplifications and hypotheses and its adaptability to each situation. First, the melting process is considered instantaneous, but the user can also implement a melting model. Then, specific geometries are used depending on the element, allowing the flow calculation in only one dimension. Elements are divided into several sections in which the fluid is considered as Newtonian and isothermal. Specific viscosities depending on shear rate and temperature are however defined and chosen by the user. Dedicated articles were published for a complete description of the method [39].

Here, the software was configured to match screw profile, extruder, die and temperature profile with the experiments. Transfer coefficients were fixed at 50 W·m−<sup>2</sup> ·K−<sup>2</sup> for the die and 350 W·m−<sup>2</sup> ·K−<sup>2</sup> for the barrel, corresponding to similar and previous simulations [31,40].

Concerning the viscosity of the polyethylene, several options are available: Choosing a rheological model between Power Law and Carreau-Yasuda and indicating the corresponding parameters, implementing a new model, or entering a set of points (SoP) corresponding to rheological measurements. It is also possible to couple the viscosity with some simulation results, such as reaction rate (requires entering a description of the kinetics), total residence time, cumulated strain or total dissipated energy. Nevertheless, it requires knowing the relation between these parameters and viscosity. In our case, polyethylene viscosity evolves during the extrusion due to the degradation under high temperature. Berzin et al. [41] developed a method for coupling the starch viscosity variation with the SME on Ludovic® (SC-Consultants, Saint-Etienne, France) However, here, the dependence between degradation and the process parameters is unknown, and the objective is to use the simulation classically.

HDPE viscosity has been defined as following a Carreau-Yasuda model, in which parameters have been determined from rheological measurements on the raw polyethylenes.

Concerning UHMWPE, it appears that the available rheological measurement methods underestimate the viscosity. Several hypotheses and simplifications had then to be made in order to approximate it for our simulation.

As a first simplification, the elastic modulus (*G* 0 ) has been considered as constant and equal to its rubbery plateau value *G* 0 *N* , which can be calculated the following way:

$$G\_N^0 = \frac{\rho \mathcal{R} \ T}{M\_\varepsilon} \tag{8}$$

where *ρ* is the density of the polymer estimated at 930 kg/m<sup>3</sup> according to the supplier, *R* is the perfect gas constant, *T* is the reference temperature (*T* = 190 ◦C) and *M<sup>e</sup>* is the molecular weight between entanglements (1.25 kg/mol for polyethylene).

Then, as the loss modulus is significantly lower than the value of the elastic modulus, its contribution to viscosity calculation has been neglected.

Finally, the Cox-Merz hypothesis has been made, allowing to assimilate the actual viscosity *<sup>η</sup>* and shear rates . *γ* of the extruder to the complex viscosity *η* ∗ and angular frequency *ω*, thus obtaining the following approximation:

$$\eta(\dot{\gamma}) = \eta^\*(\omega) = \frac{G\prime(\omega)}{\omega} = \frac{G\_N^0}{\omega} \tag{9}$$

All the polymer characteristics used have been summarised in Table 3.

**Thermal Properties HDPE XRT70 UHMWPE GUR 4130** Heat Capacity [J kg−<sup>1</sup> K −1 ] 1550 1840 Density [kg m−<sup>3</sup> ] 947 930 Thermal Conductivity [W mK−<sup>1</sup> ] 0.35 0.41 Melting Temperature [◦C] 129 135 Melting enthalpy [kJ kg−<sup>1</sup> ] 190 122 Viscosity Law Carreau-Yasuda: Power Law: *η* . *γ* = *η*<sup>0</sup> ·(1 + *λ* × . *γ a <sup>m</sup>*−<sup>1</sup> *<sup>a</sup> η* . *γ* = *K* . *γ m*−1 *<sup>η</sup>*<sup>0</sup> = 2.5 <sup>×</sup> <sup>10</sup><sup>6</sup> Pa·<sup>s</sup> *λ* = 0.33 s *a* = 0.25 *m* = 0.058 Tref = 190 ◦C *<sup>E</sup><sup>a</sup>* = 30 kJ·mol−<sup>1</sup> *m* = 0 *<sup>K</sup>* = 2.86 <sup>×</sup> <sup>10</sup><sup>6</sup> Pa·<sup>s</sup> Tref = 190 ◦C *<sup>E</sup><sup>a</sup>* = 30 kJ·mol−<sup>1</sup>

**Table 3.** Thermal Properties and viscosity laws used in the simulation.

### *3.2. Machine-Learning*

Since a data-driven model is only fed by data, the more data there is, the more accurate the prediction will be. While today, in the era of "big data", one of the main concerns is to classify this huge amount of data successfully, this study is in the opposite situation dealing with the low data limit imposed by the number of extrusions and the available hardware. The dataset used in our study includes four types of inputs (HDPE or UHMWPE, flow rate, screw rotation speed and maximal imposed temperature), and only around 27 data, each corresponding to an extrusion configuration. To find a law linking the outputs to these inputs, it is consequently necessary to use algorithms able to perform with few data. As

regression methods are well adapted to this case, Support Vector Machine Regression (SVR) and Sparsed Proper Generalized Decomposition (sPGD), two regression methods, have been tested for those data. Their specific modes of operation are described hereafter. chine Regression (SVR) and Sparsed Proper Generalized Decomposition (sPGD), two regression methods, have been tested for those data. Their specific modes of operation are described hereafter.

UHMWPE, flow rate, screw rotation speed and maximal imposed temperature), and only around 27 data, each corresponding to an extrusion configuration. To find a law linking the outputs to these inputs, it is consequently necessary to use algorithms able to perform with few data. As regression methods are well adapted to this case, Support Vector Ma-

The process is as follows: the dataset is first randomly divided into training and test groups. Then, the training inputs and outputs are implemented in the algorithm, which, depending on the method, will "learn" from these data, creating a model linking the inputs to the outputs. The model is then tested with the remaining data, and the outputs predicted by it are compared to the real ones, allowing qualifying its accuracy. The process is as follows: the dataset is first randomly divided into training and test groups. Then, the training inputs and outputs are implemented in the algorithm, which, depending on the method, will "learn" from these data, creating a model linking the inputs to the outputs. The model is then tested with the remaining data, and the outputs predicted by it are compared to the real ones, allowing qualifying its accuracy.

#### 3.2.1. Support Vector Machine Regression—SVR 3.2.1. Support Vector Machine Regression—SVR

*Polymers* **2022**, *14*, x FOR PEER REVIEW 9 of 23

SVR is a derivate of the Support Machine Vector—SVM—classification method. It is a classical method known for its effectiveness in high dimensional spaces and is widely described in the literature, such as Smola et al. in 2004 [42]. As with all classification methods, SVM aims to find the equation of the limit between two classes. As it can be tricky or impossible to find this limit in the original space of the data, the strategy here is to represent the data in a higher-dimensional space where the equation of the limit would be more simple. This principle is represented in Figure 5. This transformation is carried out via a transformation function called "kernel" (noted *ϕ* in the figure). This kernel is defined by the user, depending on the system and its complexity. Here the "RBF" kernel was chosen as it is adapted to nonlinear systems. SVR is a derivate of the Support Machine Vector—SVM—classification method. It is a classical method known for its effectiveness in high dimensional spaces and is widely described in the literature, such as Smola et al. in 2004 [42]. As with all classification methods, SVM aims to find the equation of the limit between two classes. As it can be tricky or impossible to find this limit in the original space of the data, the strategy here is to represent the data in a higher-dimensional space where the equation of the limit would be more simple. This principle is represented in Figure 5. This transformation is carried out via a transformation function called "kernel" (noted *φ* in the figure). This kernel is defined by the user, depending on the system and its complexity. Here the "RBF" kernel was chosen as it is adapted to nonlinear systems.

**Figure 5.** Scheme of SVM principle. **Figure 5.** Scheme of SVM principle.

The decision surface defining the different class areas is defined according to two parameters *C* and *ε* defined by the user. ε represents the maximum error between the decision surface and the experimental points, and *C* characterizes the smoothness of this surface. To be more specific, a high value of *C* would make the decision surface fit exactly all the experimental points but can lead to overfitting. It would hardly represent reality and would not fit new points that are not part of the training points. On the contrary, a low value of *C* would smooth the decision surface, enabling more errors and bringing more realism. This principle can also be used for regression purposes, and this is how the SVR method used here works. Then, the decision surface does not represent a border between classes but a hyper-surface approximating the points, predicting numerical values for new testing points. The decision surface defining the different class areas is defined according to two parameters *C* and *ε* defined by the user. ε represents the maximum error between the decision surface and the experimental points, and *C* characterizes the smoothness of this surface. To be more specific, a high value of *C* would make the decision surface fit exactly all the experimental points but can lead to overfitting. It would hardly represent reality and would not fit new points that are not part of the training points. On the contrary, a low value of *C* would smooth the decision surface, enabling more errors and bringing more realism. This principle can also be used for regression purposes, and this is how the SVR method used here works. Then, the decision surface does not represent a border between classes but a hyper-surface approximating the points, predicting numerical values for new testing points.

#### 3.2.2. Sparsed Proper Generalized Decomposition—sPGD 3.2.2. Sparsed Proper Generalized Decomposition—sPGD

This regression method has been developed and fully described by Ibáñez et al. [43]. To give an idea of the principle of this method, let us consider an output ݕ that depends on two input parameters ݔଵ and ݔଶ. The simplest regression method, particularly in the case of few experimental data, is a linear regression such as described by Equation (10). This regression method has been developed and fully described by Ibáñez et al. [43]. To give an idea of the principle of this method, let us consider an output *y* that depends on two input parameters *x*<sup>1</sup> and *x*2. The simplest regression method, particularly in the case of few experimental data, is a linear regression such as described by Equation (10).

$$y(\mathbf{x}\_1, \mathbf{x}\_2) = a + b \times \mathbf{x}\_1 + c \times \mathbf{x}\_2 \tag{10}$$

where three experimental data, i.e., values of (*y*, *x*1, *x*2), are necessary to find *a*, *b* and *c* parameters of Equation (10).

However, the dependence between the inputs and outputs is often nonlinear, and calculating richer regressions would necessitate more data. The principle of sPGD is considering instead of (10) the following equation:

$$y(\mathbf{x}\_1, \mathbf{x}\_2) = X\_1(\mathbf{x}\_1) \times X\_2(\mathbf{x}\_2) \tag{11}$$

The dependence relation between the output *y* and the inputs *x*<sup>1</sup> and *x*<sup>2</sup> would thus be a product of functions depending each on *x*<sup>1</sup> and *x*2. To determine these functions without needing more data, the following method is applied. First, *X*1(*x*1) is fixed and *X*2(*x*2) is estimated thanks to the available data. Only two data are then necessary to have a linear function in each coordinate and then a bilinear regression with only three data (some regularisation being needed to avoid unphysical behaviours). However, a more significant amount of data would lead to richer approximations. Then, *X*1(*x*1) can be estimated by fixing *X*2(*x*2) to its just calculated value using the same data. With this method, three experimental data could lead to a quadratic dependence instead of a linear one with Equation (10).

#### 3.2.3. Stochastic Methods

Classically, the way of estimating the equation matching a maximum with data is to apply the classical least-squares procedure. However, another method called "stochastic" based on statistics is possible.

Let consider a set of two variables *x<sup>i</sup>* and *y<sup>i</sup>* that we attempt to describe by a linear function *y<sup>i</sup>* = *yi*(*xi*) = *ax<sup>i</sup>* + *b*. The classical lest squares procedure consists of minimizing *S*.

$$S = \frac{1}{2} \sum\_{i} (y\_i - f(\mathbf{x}\_i))^2 = \frac{1}{2} \sum\_{i} (y\_i - (a\mathbf{x}\_i + b))^2 \tag{12}$$

The number of samples is *n* (*i* = 1..*n*).

The linear regression applied at each data point reads:

$$\begin{aligned} b + a\mathbf{x}\_1 &= y\_1 \\ \dots & \dots \\ b + a\mathbf{x}\_{\mathbb{N}} &= y\_{\mathbb{N}} \end{aligned} \tag{13}$$

whose matric form reads

$$
\begin{pmatrix} \mathbf{x}\_1 & \mathbf{1} \\ \cdots & \cdots \\ \mathbf{x}\_n & \mathbf{1} \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} y\_1 \\ \cdots \\ y\_n \end{pmatrix} \tag{14}
$$

or using a more compact form

$$\mathbb{X}\left(\begin{array}{c} a\\b \end{array}\right) = \mathbb{Y} \tag{15}$$

The least-squares procedure used for solving the just overdetermined system consists of pre-multiplying the previous system by X*<sup>T</sup>* that results in

$$\mathbb{X}^T \mathbb{X} \begin{pmatrix} a \\ b \end{pmatrix} = \mathbb{X}^T \mathbb{Y} \tag{16}$$

or more explicitly

$$
\begin{pmatrix}
\sum \mathbf{x}\_i^2 & \sum \mathbf{x}\_i \\
\sum \mathbf{x}\_i & n
\end{pmatrix}
\begin{pmatrix}
a \\ b
\end{pmatrix} = 
\begin{pmatrix}
\sum \mathbf{x}\_i y\_i \\
\sum y\_i \\
\end{pmatrix}
\tag{17}
$$

It is then possible to prove that the solution of this system results

$$\begin{array}{c} a = \frac{\text{Cov}(\mathbf{x}, \mathbf{y})}{\text{Var}(\mathbf{x})}\\ b = \overline{y} - a\overline{x} \end{array} \tag{18}$$

with *y* and *x* the mean value in the sample of variables *y* and *x*, respectively.

We suppose now that for each input *x<sup>i</sup>* the response *y<sup>i</sup>* follow a Gaussian distribution with a standard deviation *σ<sup>i</sup>* . Let denotes by *Nσ*(.) the Gaussian distribution around the zero value with standard deviation *σ*. The least-squares procedure previously described is here modified. The idea consists in saying that each *y<sup>i</sup>* must follow a normal distribution centred at *ax<sup>i</sup>* + *b* and with a standard deviation *σ<sup>i</sup>* .

The optimization problem results then in the maximization of the following sum

$$S = \frac{1}{2} \sum\_{i} \mathcal{N}\_{\mathcal{O}\_i}(y\_i - f(\mathbf{x}\_i)) = \frac{1}{2} \sum\_{i} \mathcal{N}\_{\mathcal{O}\_i}(y\_i - (a\mathbf{x}\_i + b)) \tag{19}$$

To get back on a classical optimization problem expressed in terms of minimization, we just have to add a negative sign on the sum. The resolution cannot be made analytically, as was done in the case of the least-squares' procedure. We can however use a descent scheme.

Assuming that the standard deviation does not depend on *x*, one could look for the statistical regression with a minimum standard deviation. For that, the most straightforward procedure consists of, for a tentative *a* and *b* coefficients of the regression, compute the standard deviation of the sampling, and then apply the previous rationale for computing the regression, that is, for updating *a* and *b* coefficients and iterate until reaching convergence.

The procedure described above is easily generalized to handle multiparametric regressions with richer nonlinear regressions. This method was tried on 10 data corresponding only to HDPE extruded at 190 ◦C and coupled to SVR previously described methodology.

#### **4. Results and Discussion**

#### *4.1. Comparison of Estimated and Measured Viscosities*

The viscosities of the extruded materials deduced from the pressure measured at the die were compared to the rheometer measurements. The viscosity curves obtained from die pressure measurements are compared to data calculated from rheometer experiments in Figure 6A only for the samples resulting from HDPE 390 ◦C extrusion. Figure 6B compares the zero shear viscosities obtained for all samples with the two different methods. A Carreau-Yasuda model was thus used to fit the experimental viscosity curves, which validate the use of such a model for the degraded PE. With this type of representation, the closer the points on the *x* = *y* line are, the closer the values of the two methods for the same sample are. The samples from extrusions performed at different temperatures or with the UHMWPE give similar results. The results are fully detailed in Table A3 of the Appendix A.

Despite the assumptions made, the estimated viscosity curves are close to the rheometer experiments. However, the region of the Newtonian plateau at low frequency seems to be more pronounced for the rheometer data, leading to a higher value of the parameter *a* of the Carreau-Yasuda law. But the interest of this method is to avoid offline characterizations. Given the lack of information to model the curve correctly, it seems unlikely that better results can be obtained. Besides, the comparison of zero-shear viscosities gives satisfactory results and proves that this method can rapidly approximate the final viscosity of extruded materials.

#### *4.2. Molecular Weight Distribution*

Figure 7A shows the comparison of the values of weight average molecular mass (*Mw*) obtained with the three methods previously described, i.e., (i) viscoelastic behaviour measurements, (ii) from die pressure and (iii) from size exclusion chromatography (SEC). Only the five samples analyzed by SEC are in the 3D figure, the other ones being in the die pressure versus viscoelastic behaviour plane. Figure 7B compares *Mw* values obtained

by the three techniques and *Mn* values obtained by SEC and viscoelastic measurements. The results are fully detailed in Table A3 of the Appendix A. It appears that despite the simplifications they induce, the three methodologies give similar results for the determination of *Mw* for all samples, with a factor of less than 2 between values in most cases. Thus, it appears that, concerning the determination of *Mw*, the viscoelastic measurement method has few advantages over the die pressure method, which does not require offline characterizations. However, its advantages actually lie in the fact that this technique allows obtaining the complete distribution of molecular weights, unlike the die pressure method, which only allows obtaining the *Mw*. *Polymers* **2022**, *14*, x FOR PEER REVIEW 12 of 23

ments.

simplifications they induce, the three methodologies give similar results for the determination of Mw for all samples, with a factor of less than 2 between values in most cases. Thus, it appears that, concerning the determination of Mw, the viscoelastic measurement method has few advantages over the die pressure method, which does not require offline characterizations. However, its advantages actually lie in the fact that this technique al-**Figure 7.** (**A**) Comparison of weight average molecular weight (Mw) of degraded polyethylenes obtained from the estimated zero shear rate viscosity versus the one obtained from viscoelastic behaviour measurements. (**B**) Comparison of weight average (Mw) and number average (Mn) molecular weight values obtained with different specified methodologies. **Figure 7.** (**A**) Comparison of weight average molecular weight (*Mw*) of degraded polyethylenes obtained from the estimated zero shear rate viscosity versus the one obtained from viscoelastic behaviour measurements. (**B**) Comparison of weight average (*Mw*) and number average (*Mn*) molecular weight values obtained with different specified methodologies.

lows obtaining the complete distribution of molecular weights, unlike the die pressure

is the more precise methodology. It is consequently possible for this type of sample to determine the molecular weight distribution only from frequency sweep experiments, avoiding using SEC, which is a more complex and less accessible process involving the use of hot CMR solvents. However, for UHMWPE samples, Mn values are not as close to the SEC values. It can be due to structural differences with HDPE samples, inducing different parameters for the reptation model (relaxation time exponent, entanglement and

**Figure 8.** Molecular weight distribution obtained by SEC and viscoelastic measurements for (**A**) HDPE extruded at 390 °C, 3 kg/h and 1000 rpm and (**B**) UHMWPE extruded at 390 °C, 1 kg/h and

0

3456

**Log(M) (g/mol)**

SEC From viscoelastic behaviour

1

**P(M)**

The Mn values obtained are compared to SEC values in Figure 7B. Figure 8 compares complete molecular weight distributions obtained by SEC and calculated from viscoelastic measurements for an HDPE and a UHMWPE. These figures point out that concerning

**A: HDPE B: UHMWPE**

method, which only allows obtaining the Mw.

3456

**Log(M) (g/mol)**

reptation molecular weights, etc.).

1000 rpm.

0

1

**P(M)**

The *Mn* values obtained are compared to SEC values in Figure 7B. Figure 8 compares complete molecular weight distributions obtained by SEC and calculated from viscoelastic measurements for an HDPE and a UHMWPE. These figures point out that concerning HDPE samples, the distributions calculated are close to the ones obtained by SEC, which is the more precise methodology. It is consequently possible for this type of sample to determine the molecular weight distribution only from frequency sweep experiments, avoiding using SEC, which is a more complex and less accessible process involving the use of hot CMR solvents. However, for UHMWPE samples, *Mn* values are not as close to the SEC values. It can be due to structural differences with HDPE samples, inducing different parameters for the reptation model (relaxation time exponent, entanglement and reptation molecular weights, etc.). The Mn values obtained are compared to SEC values in Figure 7B. Figure 8 compares complete molecular weight distributions obtained by SEC and calculated from viscoelastic measurements for an HDPE and a UHMWPE. These figures point out that concerning HDPE samples, the distributions calculated are close to the ones obtained by SEC, which is the more precise methodology. It is consequently possible for this type of sample to determine the molecular weight distribution only from frequency sweep experiments, avoiding using SEC, which is a more complex and less accessible process involving the use of hot CMR solvents. However, for UHMWPE samples, Mn values are not as close to the SEC values. It can be due to structural differences with HDPE samples, inducing different parameters for the reptation model (relaxation time exponent, entanglement and reptation molecular weights, etc.).

**Figure 7.** (**A**) Comparison of weight average molecular weight (Mw) of degraded polyethylenes obtained from the estimated zero shear rate viscosity versus the one obtained from viscoelastic behaviour measurements. (**B**) Comparison of weight average (Mw) and number average (Mn) molec-

*Polymers* **2022**, *14*, x FOR PEER REVIEW 13 of 23

ular weight values obtained with different specified methodologies.

**Figure 8.** Molecular weight distribution obtained by SEC and viscoelastic measurements for (**A**) HDPE extruded at 390 °C, 3 kg/h and 1000 rpm and (**B**) UHMWPE extruded at 390 °C, 1 kg/h and 1000 rpm. **Figure 8.** Molecular weight distribution obtained by SEC and viscoelastic measurements for (**A**) HDPE extruded at 390 ◦C, 3 kg/h and 1000 rpm and (**B**) UHMWPE extruded at 390 ◦C, 1 kg/h and 1000 rpm.
