**1. Introduction**

Presently, plastics are used for every imaginable application. Because of the industry's focus on the benefits of the processing and application of plastics, important aspects of plastics have been neglected. In addition to the problematic use of finite resources such as petroleum, masses of plastic waste and the creation of plastic patches in oceans produce a major environmental threat [1]. In order to solve these problems without losing the positive properties of conventional plastics, the industry is in search of better alternatives. A very up-to-date and detailed overview of current research trends in biobased and biodegradable polymers can be found in [2]. A promising biopolymer that can make a small but important contribution along the way in the fight against the ocean garbage patches is poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV).

PHBV belongs to the group of polyhydroxyalkanoates (PHA) and is a copolymer of polyhydroxybutyrate (PHB) [3]. The linear molecular structure of PHBV consists of a linear carbonyl chain with randomly distributed methyl and ethyl side-chains [4]. PHBV is a biodegradable, biocompatible, hydrophobic, and non-toxic polymer [5,6]. It is produced, for instance, by the Gram-negative bacterium Ralstonia eutropha as an energy and carbon store. The production takes place under a lack of oxygen, nitrogen, and phosphor as well as a surplus of carbon [7]. To produce the characteristic side-chains of PHBV, propionic acid and valeric acid are added [8]. The brittle and stiff thermoplastic polymer shows viscoelastic properties [9]. In addition to the application of PHBV for films and packaging, there is also research on applications in medical technology. However, due to high production costs as well as its brittle and temperature-sensitive behavior, PHBV is not currently established in

**Citation:** Lajewski, S.; Mauch, A.; Geiger, K.; Bonten, C. Rheological Characterization and Modeling of Thermally Unstable Poly(3-hydroxybutyrate-co-3 hydroxyvalerate) (PHBV). *Polymers* **2021**, *13*, 2294. https://doi.org/ 10.3390/polym13142294

Academic Editors: José Miguel Ferri, Vicent Fombuena Borràs and Miguel Fernando Aldás Carrasco

Received: 28 June 2021 Accepted: 8 July 2021 Published: 13 July 2021

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**Copyright:** © 2021 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/).

industry and PHA only accounted for 1.7% of the global production capacities of bioplastics in 2020 [10]. Depending on the composition, the melting point of PHBV varies between 50 °C and 180 °C [11]. Temperatures above the melting point cause unstable behavior of PHBV.

and packaging, there is also research on applications in medical technology. However, due to high production costs as well as its brittle and temperature-sensitive behavior, PHBV is not currently established in industry and PHA only accounted for 1.7 % of the

*Polymers* **2021**, *13*, x FOR PEER REVIEW 2 of 10

global production capacities of bioplastics in 2020 [10].

Depending on the composition, the melting point of PHBV varies between 50 ◦C and 180 ◦C [11]. Temperatures above the melting point cause unstable behavior of PHBV. Thus, the influence of high temperatures leads to a strong degradation of the polymer chains. The thermal degradation processes with α-deprotonation, β-elimination, and random chain scission proceed via several intermediates. At the end of thermal degradation, propylene, acetaldehyde, ketene, acetone, and carbon dioxide occur [12,13]. Thus, the influence of high temperatures leads to a strong degradation of the polymer chains. The thermal degradation processes with α-deprotonation, β-elimination, and random chain scission proceed via several intermediates. At the end of thermal degradation, propylene, acetaldehyde, ketene, acetone, and carbon dioxide occur [12,13]. Viscosity curves are suitable tools for the analysis of the rheological properties of PHBV. These measurements are carried out in the molten state and can therefore

Viscosity curves are suitable tools for the analysis of the rheological properties of PHBV. These measurements are carried out in the molten state and can therefore accurately reflect the processing behavior of the plastics considered in the application. Even though PHBV has so far been used mainly in blends, the rheological characterization of the pure material is also important in order to obtain a statement of the expected material behavior. Only in this way can the pros and cons for the use of this biopolymer be assessed. accurately reflect the processing behavior of the plastics considered in the application. Even though PHBV has so far been used mainly in blends, the rheological characterization of the pure material is also important in order to obtain a statement of the expected material behavior. Only in this way can the pros and cons for the use of this biopolymer be assessed. The viscosity η is plotted double-logarithmically over the angular frequency ω or the

The viscosity η is plotted double-logarithmically over the angular frequency ω or the shear rate . *γ* (cf. Figure 1). A strong linear decrease at high frequencies is called the pseudo-plastic flow area or shear thinning behavior. At low frequencies, there is a constant viscosity called zero shear viscosity η0. Between the areas of zero shear viscosity and shear thinning, a transition occurs. The measured shear viscosity function of polymers with linear molecular structure (PS, PE-HD, PC, or even PHBV) is precisely approached with the Carreau model. The Carreau model, shown in Equation (1), calculates the viscosity η as a function of the shear rate . *γ* with three parameters: the zero shear viscosity A0, the reciprocal transient shear rate A1, and the gradient of the pseudo-plastic flow area A2. shear rate ሶ (cf. Figure 1). A strong linear decrease at high frequencies is called the pseudo-plastic flow area or shear thinning behavior. At low frequencies, there is a constant viscosity called zero shear viscosity η0. Between the areas of zero shear viscosity and shear thinning, a transition occurs. The measured shear viscosity function of polymers with linear molecular structure (PS, PE-HD, PC, or even PHBV) is precisely approached with the Carreau model. The Carreau model, shown in Equation (1), calculates the viscosity η as a function of the shear rate ሶ with three parameters: the zero shear viscosity A0, the reciprocal transient shear rate A1, and the gradient of the pseudo-plastic flow area A2.

**Figure 1.** Viscosity curve of thermoplastics. The blue curve is an approximated viscosity **Figure 1.** Viscosity curve of thermoplastics. The blue curve is an approximated viscosity curve derived from the Carreau model, and the orange curve is from the rheological measurement of PHBV.

curve derived from the Carreau model, and the orange curve is from the rheological measurement of PHBV. Because of the temperature sensitivity of thermoplastics, the viscosity curve shifts with the change of temperature. At higher temperatures, the viscosity curve shifts by −45◦ [14,15]. According to Equation (2), the temperature shift can be calculated in the Carreau model (3) with the temperature shift factor aT:

$$\eta\left(\dot{\gamma}\right) = \frac{a\_{\rm T} \cdot \mathbf{A}\_0}{\left(1 + a\_{\rm T} \cdot \mathbf{A}\_1 \dot{\gamma}\right)^{\mathbf{A}\_2}}\tag{2}$$

The nondimensional temperature shift factor a<sup>T</sup> results from the Arrhenius model as the best approach for the temperature-dependent viscosity function of semi-crystalline polymers like PHBV and can be calculated following Equation (3) [15–17]:

$$a\_T = \exp\left(\frac{E\_0}{\mathcal{R}} \left(\frac{1}{T} - \frac{1}{T\_0}\right)\right) \tag{3}$$

To plot viscosity curves, rheological measurements are necessary. Important values of a rheological measurement are the storage modulus G' and the loss modulus G". G' contains the reversible elastic energy, whereas G" represents the irreversible viscous energy component.

The complex viscosity can be used for an easier analysis of rheological measurements. As shown in Equation (4), the complex viscosity is calculated via the ratio of shear stress τ and the shear rate . *γ* or the angular frequency ω [18].

$$\eta^\* = \frac{\mathbf{r}(t)}{\dot{\gamma}(t)} = \frac{\tau\_0 \sin(\omega t + \delta)}{\gamma\_0 \omega \cos(\omega t)}\tag{4}$$

The empirical approach of Cox–Merz [19] in Equation (5) specifies the composition between the viscosity η and the amount of the complex viscosity η\*. At low frequencies, the viscosity corresponds to the amount of the complex viscosity if the shear rate takes the same value as the angular frequency [16].

$$\mathfrak{n}(\dot{\gamma}) = |\mathfrak{n}^\*(\omega)| \text{ if } \dot{\gamma} = \omega \tag{5}$$

Thus, the graph of the complex viscosity and angular frequency can be linked with the graph of the viscosity and the shear rate.

In a preliminary work, viscosity curves of PHBV were plotted from rheological measurements. A frequency sweep (FS) from 100 rad/s to 0.1 rad/s was performed to obtain the viscosity curves. The viscosity curves did not fulfill the expected and discussed model curves. As shown in Figure 1, the viscosity decreases at frequencies lower than 10 rad/s where a constant zero shear viscosity is expected. This decrease can be explained with thermal degradation.

Due to the extreme thermal degradation already occurring just above the melting temperature, it is difficult to accurately predict viscosity curves at temperatures below or above the measurement temperature. Moreover, the reasonable range of temperature and frequency as well as the zero shear viscosity at a certain temperature are unknown. However, for further rheological measurements, applications, and processing of PHBV, it is necessary to gain this knowledge. Therefore, the aim of this work is to determine important rheological parameters of PHBV. Additionally, we try to better understand the thermal degradation of PHBV by comparing and analyzing measured and calculated viscosity curves.
