# Prediction of compatibility and chemical structure influencing plasticization of poly(3-hydroxybutyrate-co-4-hydroxybutyrate) (P3HB4HB)

Erwan Vasseur, Louis Leveuf, Vincent Le Saux, Stéphane Bruzaud

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Erwan Vasseur, Louis Leveuf, Vincent Le Saux, Stéphane Bruzaud. Prediction of compatibility and chemical structure influencing plasticization of poly(3-hydroxybutyrate-co-4-hydroxybutyrate) (P3HB4HB). Polymer, 2025, 336, pp.128872. 10.1016/j.polymer.2025.128872. hal-05319911

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# Prediction of compatibility and chemical structure influencing plasticization of poly (3-hydroxybutyrate-co-4-hydroxybutyrate) (P3HB4HB)

Erwan Vasseur<sup>a,b,c</sup>, Louis Leveuf<sup>b</sup>, Vincent Le Saux<sup>c</sup>, Stéphane Bruzaud<sup>a,* ID</sup>

## ARTICLE INFO

### Keywords:

Theoretical model plasticization

Compatibility

Entanglement molecular weight

Crystallinity

Polyhydroxyalkanoates

## ABSTRACT

This study aims to predict the plasticization of poly(3-hydroxybutyrate-co-4-hydroxybutyrate) (P3HB4HB) using two theoretical approaches: polymer-plasticizer compatibility via Hansen solubility parameters, and the chemical structure of polymers through the prediction of the entanglement molecular weight ($M_e$) based on Van Krevelen's method. Hansen solubility parameters were calculated for three P3HB4HB copolymers with varying 4HB molar contents (10%, 15% and 30%), allowing for the identification of potentially compatible plasticizers. In parallel, $M_e$ was theoretically predicted and compared to experimental rheological measurements. The discrepancies observed highlight the significant influence of crystallinity, particularly the size and distribution of spherulites, on the viscoelastic behavior of the materials. The transition to the rubbery plateau, analyzed through oscillatory rheology, confirmed the decisive role of the semi-crystalline structure in governing polymer chain mobility. This dual approach, combining chemical modeling and experimental validation, provides a deeper understanding of the plasticization mechanisms of P3HB4HB. It represents a valuable tool for guiding the formulation of more efficient bio-based materials tailored to industrial requirements.

## 1. Introduction

The polymer plasticization relies on two key concepts: compatibility and molecular structure. Compatibility determines the ability of a polymer and a plasticizer to exhibit thermodynamic compatibility, directly influencing the stability and mechanical material properties [1]. Molecular or chemical structure, on the other hand, plays a crucial role by governing the flexibility of polymer chains and their capacity to be modified by the plasticizer [2].

For both concepts, theoretical models enable reasonably reliable predictions. These models are based on analytical approaches, such as the evaluation of solubility parameters [3] for compatibility, or group contribution theories [4] to describe the molecular structure of the polymer. Using these tools, formulations can be optimized by anticipating material behavior.

The prediction of compatibility between polymer and plasticizer molecules is particularly relevant to the plasticization process. Indeed, good compatibility ensures uniform dispersion of the plasticizer within

the polymer matrix, thereby improving mechanical properties, flexibility, and crack resistance of the plasticized material. When polymer and plasticizer molecules are compatible, they exhibit chemical affinity, which promotes homogeneous mixing. In contrast, incompatibility may lead to phase separation, where the plasticizer migrates out of the polymer, resulting in a deterioration of material properties.

To predict this compatibility, Hansen solubility parameters are widely used in the literature [5,6]. These parameters account for three components: polarity, dispersion, and hydrogen bonding. By evaluating these components for both polymers and plasticizers, compatibility can be predicted based on the similarity or difference of these properties. Relying on these parameters, material designers can estimate polymer-plasticizer compatibility with reasonable accuracy. This helps guide material and formulation choices during the plasticization process, following the prediction method of Van Krevelen and Hoftyzer [7].

In addition, the chemical structure of both the polymer and the plasticizers plays a decisive role in the plasticization process. The flexibility of polymer chains directly affects their solvation by the

<footnote><sup>a</sup> Université Bretagne Sud, Institut de Recherche Dupuy de Lôme, UMR CNRS, 6027, 56 100, Lorient, France</footnote>

<footnote><sup>b</sup> Entreprise FIIISH, 8 Boulevard Gabriel Lippmann, 29 850, Gouesnou, France</footnote>

<footnote><sup>c</sup> ENSTA Institut Polytechnique de Paris, Institut de Recherche Dupuy de Lôme, UMR CNRS, 6027, 29 200, Brest, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: stephane.bruzaud@univ-ubs.fr (S. Bruzaud).</footnote>
plasticizer. This flexibility is closely linked to polymer chain entangle- ments in the amorphous state [8]. Entanglements are topological interaction points between polymer chains, where their movement is mutually constrained without any chemical bonding. They form a physical network that strongly influences the material’s mechanical and viscoelastic properties. There are three types of entanglements (Fig. 1) [9]:

The prediction of these entanglements can be based on a method derived from Van Krevelen’s group contribution theory. It uses the principle of additivity to analyze the chemical structure of polymers. Understanding how to plasticize poly(3-hydroxybutyrate-co-4- hydroxybutyrate) (P3HB4HB) is of great interest due to their potential as future-generation plastics [10,11]. These biodegradable polymers offer a sustainable alternative to conventional plastics while exhibiting promising mechanical properties [12]. Mastering their formulation and processing enables optimization of their performance for various ap- plications, thereby supporting the transition toward more environmen- tally friendly materials. Developing this expertise now offers a strategic advantage in addressing upcoming industrial and regulatory challenges.

However, the plasticization of P3HB4HB is a complex process due to the unique nature of this polymer [13–16]. P3HB4HB belongs to the polyhydroxyalkanoate (PHA) family, known for their semi-crystalline nature, mechanical stiffness, and biodegradability [17,18]. Moreover, hydrogen bonding is also a limiting factor in the plasticization of P3HB4HB. While it does not directly control chain mobility, it contrib- utes significantly to the cohesion of the material. During plasticization, these interactions are weakened, leading to a loss of overall cohesion. It is this loss of cohesion, rather than molecular mobility, that makes plasticization more challenging for certain PHA grades.

The main challenge lies in selecting a suitable plasticizer. Incom- patible plasticizers may migrate, leading to a loss of mechanical or thermal properties over time, as well as an overall degradation of ma- terial performance. Moreover, plasticizer concentration must be finely tuned: an insufficient amount will not achieve the desired effect, while excessive content may cause phase separation or alter the polymer crystallinity [19]. Beyond crystallinity, the density of entanglements within the amorphous phase of the polymer can also limit the plastici- zation of P3HB4HB. If entanglements are not sufficiently numerous, the increase in free volume induced by plasticization may lead to a loss of cohesion within the polymer matrix. This parameter has been scarcely investigated in the literature, and many assumptions are made in the way it is calculated [20]. Therefore, we aim here to compare theoretical predictions with an experimental method to highlight the entanglement density of three different P3HB4HB grades.

To better understand and anticipate the plasticization behavior of P3HB4HB, we adopted a dual approach combining theoretical

prediction and experimental validation. First, we evaluated polymer- –plasticizer compatibility based on Hansen solubility parameters, using group contribution methods to estimate the energetic profile of various P3HB4HB copolymers. Then, we focused on the molecular structure of the polymer by estimating the entanglement molecular weight (Me), both through Van Krevelen’s theoretical framework and through oscil- latory rheological measurements. The following section details the materials used and the methods implemented for this investigation.

## 2. Materials and methods

### 2.1. Materials

Poly(3-hydroxybutyrate-co-4-hydroxybutyrate) (P3HB4HB) with 10 %, 15 % and 30 % molar 4HB content were supplied by PHABUILDER Biotechnology Co., LTD (Beijing, China) and CJ BIO (Seoul, South Korea). For clarity and consistency, we will hereafter refer to these materials as P3HB4HB(10), P3HB4HB(15) and P3HB4HB(30), respectively.

### 2.2. Prediction method of energetic parameters using Hansen theory

In 1967 Hansen developed a method for evaluating the solubility parameter $\delta_H$ (MPa$^{1/2}$) of various solvents (Eq. (1)). Solubility refers to the ability of a substance (the solute) to dissolve into another material (the solvent) to form a homogeneous solution. The author Hansen [21] proposed dividing $\delta_H$ (MPa$^{1/2}$) into three components representing the main intermolecular forces: dispersive forces $\delta_d$ (MPa$^{1/2}$), polar in- teractions $\delta_p$ (MPa$^{1/2}$), and hydrogen bonding $\delta_h$ (MPa$^{1/2}$).

$$ \delta_H = \sqrt{\delta_d^2 + \delta_p^2 + \delta_h^2} \qquad (\text{Eq. 1}) $$

The dispersive component (Fig. 2a) in Hansen’s theory refers to the interaction between molecules involving dispersion or mutual repulsion. More specifically, it represents the attractive or repulsive forces result- ing from London dispersion forces, temporary intermolecular in- teractions due to fluctuations in electron distribution. This component is typically associated with the non-polar parts of molecules, where elec- trons are evenly distributed.

The polar component (Fig. 2b) in Hansen’s theory relates to in- teractions between molecules with polar characteristics, such as per- manent dipole-dipole bonds. In contrast to the dispersive component, which represents London forces, the polar component accounts for permanent electrostatic interactions and is associated with parts of molecules that have an uneven charge distribution.

The hydrogen bonding component (Fig. 2c) addresses specific

<figure><img src="image_3.png" /><figcaption>Fig. 1. Different types of entanglements.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 2. Energetic parameters and affinity sphere according to Hansen theory. (a). Diagram of the dispersive component (b). Diagram of the polar component. (c). Diagram of the hydrogen bonding component. (d). Diagram of the affinity sphere (c: theoretically compatible; ●: theoretically incompatible).</figcaption></figure>

interactions between hydrogen donor functional groups (e.g., hydroxyl groups -OH) and hydrogen acceptor groups (e.g., carbonyl groups -C=O). This component captures hydrogen bonding and similar interactions that can occur between molecules. In the case of P3HB4HB, hydrogen atoms appear to drive the specific interactions and tend to orient themselves toward the ester group (-COO-), which is the only polar functionality present in the structure. These are considered weak interactions.

Hansen's decomposition method allows the construction of a three-dimensional space where each molecule is represented by a point with coordinates ($\delta_d, \delta_p, \delta_h$). When a solute is tested with various solvents, those capable of dissolving it tend to cluster within a specific region in Hansen space (Fig. 2d). This region can be enclosed within a volume often approximated as a sphere for simplicity. Solvents inside the sphere can dissolve a given amount of solute, while those outside generally cannot. Hansen described each solute using four parameters: the center coordinates ($\delta_d, \delta_p, \delta_h$) and the sphere's radius $R_0$. For P3HB4HB, the radius $R_0$ is not experimentally known, but based on the polymer's structure, we estimate it at 8.5 MPa$^{1/2}$ [22].

The probability of a solute being solubilized by a solvent increase as the distance between the solute and solvent in Hansen space decreases [21] (Eq. (2)):

$$D = \sqrt{4(\delta_{d_{\text{plastifiant}}} - \delta_{d_{\text{poly}}})^2 + (\delta_{p_{\text{plastifiant}}} - \delta_{p_{\text{poly}}})^2 + (\delta_{h_{\text{plastifiant}}} - \delta_{h_{\text{poly}}})^2} \quad (\text{Eq. 2})$$

The Relative Energy Difference (RED) is defined by Hansen as the ratio of the solute-solvent distance D to the solubility sphere radius $R_0$ (Eq. (3)). This dimensionless value is a practical criterion for selecting suitable solvents. If RED ≤ 1, the solvent lies within the solubility sphere and should be able to dissolve the solute. If RED > 1, the solvent lies outside the sphere and is unlikely to be effective.

$$RED = \frac{D}{R_0} \quad (\text{Eq. 3})$$

To predict the center coordinates of a polymer's solubility sphere, we can apply the **Van Krevelen and Hoftyzer** method. In 1976, Van Krevelen and Hoftyzer demonstrated that solubility components can be calculated from group contribution values using the following equations, where V is the molar volume of the molecule (cm$^3$.mol$^{-1}$), i is the group index, F (kJ.mol$^{-1}$) and E (kJ.mol$^{-1}$) are the energy contribution terms (Eq. (4)):

$$\delta_d = \frac{\sum F_{d_i}}{V} \quad \delta_p = \frac{\sqrt{\sum F_{p_i}^2}}{V} \quad \delta_h = \sqrt{\frac{\sum E_{h_i}}{V}} \quad (\text{Eq. 4})$$

Van Krevelen and Hoftyzer also introduced correction factors to account for the nature of the group contributions. This method, widely cited in the literature [3], is the oldest and most used. It assumes that both the molar volume and group contributions remain constant for a given polymer or plasticizer. This approach is based on a statistical analysis of Hansen's experimental data and uses 28 functional and topological groups to represent molecules each contributing to the dispersive, polar, and hydrogen bonding components as defined by the equations above.

### 2.3. Prediction method of entanglement molecular weight using Van Krevelen theory

To theoretically predict the entanglement molecular weight (Fig. 3) based on chemical group contributions, the formula proposed by Souheng Wu [23] is used (Eq. (5) and Fig. 3).

$$M_e = \left( \frac{3 M_v}{\phi_0^{\frac{4}{3}}} \right) * \left( \frac{L_r}{M_r} \right)^{\frac{8}{3}} * \left( \frac{M_v}{(l_v)^2} \right)^2 \quad (\text{Eq. 5})$$

Thus, we need to determine the molecular weight per skeletal unit of the polymer, denoted as $M_v$ (g.mol$^{-1}$), the flexibility of the monomer units (3HB and 4HB), denoted as $L_r$ (g$^{1/4}$.cm$^{3/2}$.mol$^{-3/4}$), their skeletal bonds, denoted as $l_v$ (Å), and $\phi_0$ is a constant (2,51.10$^{23}$ cm$^3$ mol$^{1/2}$ g$^{-3/2}$).
<figure><img src="image_5.png" /><figcaption>Fig. 3. Schematic representation of the entanglement molecular weight (M~e~) (g·mol⁻¹). The small green spheres represent polymer chain segments, while the larger spheres indicate entanglement points. The entanglement molecular weight corresponds to the average molecular weight (distance) of a polymer chain between two entanglements (nodes). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

The average molecular weight per skeletal unit, commonly denoted as M~v~, is a measure of the average molecular mass of a polymer per repeating unit or skeletal unit (Eq. (6)). In other words, it represents the average mass of a single repeating motif within the polymer chain. This value is important for characterizing the molecular weight distribution in a polymer. It is calculated by dividing the molecular weight of a repeating unit (M~r~) (g·mol⁻¹) by the total number of skeletal units in the polymer (n~v~):

$$M_v = \frac{M_r}{n_v} \qquad (\text{Eq. 6})$$

Although the spatial arrangement of functional groups differs slightly between 3HB and 4HB repeating units, their calculated molecular weight (Mr = 86.09 g·mol⁻¹) and the number of skeletal atoms per unit (nv = 4) remain the same due to their very similar chemical structures. The flexibility of a polymer chain refers to its ability to rotate around its axis, thereby determining its spatial conformation. Accurate determination of the elementary rotational units is therefore crucial. To address this, Souheng Wu [23] proposed a formula originally based on the intrinsic viscosity of the polymer. This formula highlights the underlying mechanisms of polymer chain flexibility and employs the group contribution theory of Van Krevelen [4] (Table 2) to quantitatively assess this key characteristic.

Where L~r~ is the flexibility of a repeating unit, ΣK~i~ is the sum of the molar rigidity constants of the functional groups, B is a constant (equal to 4.2 for conventional polymers [23]), and n~r~ is the number of backbone atoms in a repeating unit (Eq. (7)).

$$L_r = \sum K_i + Bn_r \qquad (\text{Eq. 7})$$

For both the 3HB and 4HB monomers, the calculated flexibility of a repeating unit is identical: 66.45 g¹/⁴·cm³/²·mol⁻³/⁴.

In the case of P3HB4HB, a mixing rule is applied to define the skeletal unit of P3HB4HB according to its 4HB content. For 3HB, the average distance between the oxygen atom in the C=O group and the hydrogen atom in the CH₃ group is approximately 2.63 Å [24]. For 4HB, the average distance used is 2.32 Å [25].

## 2.4. Rheological method for measuring entanglement molecular weight (M~e~)

In rheology, it is possible to observe a rubbery plateau, which is a characteristic region in mechanical modulus curves as a function of

frequency or temperature. This plateau reflects a particular viscoelastic behavior of polymers and indicates the presence of entanglements between macromolecular chains. In this region, the elastic modulus G' (storage modulus) remains relatively constant, indicating that the material retains some mechanical integrity while still allowing for molecular mobility.

This phenomenon is explained by the existence of physical interactions between polymer chains, forming a temporary network of entanglements. These entanglements act as reversible linking points, restricting the movement of molecular segments on a macroscopic scale. The rubbery plateau is typically observed in amorphous polymers above their glass transition temperature (T~g~) and in crosslinked elastomers.

The appearance of a rubbery plateau in the viscoelastic spectrum indicates the presence of entanglements within the polymer matrix. These entanglements contribute to the material's mechanical properties and influence its deformation behavior as well as its processability (e.g., plasticization). However, this plateau does not appear below a certain critical molecular weight (M~c~). Its extent largely depends on the size of the polymer chains, i.e., the molecular weight of the polymer.

Graessley's equation [26] suggests that the modulus data of amorphous polymers in the rubbery plateau region can be related through a specific formula, where G~N~ = G' at G''~min~ for monodisperse polymers, or G' at δ~min~ for polydisperse polymers (Eq. (8) and Fig. 4).

$$G_N^0 = \frac{\rho RT}{M_e} \qquad (\text{Eq. 8})$$

Oscillatory rheological measurements were carried out using an Anton Paar MCR301 rheometer, operating in low-amplitude shear mode with a 25 mm parallel-plate geometry. The time-temperature superposition (TTS) method was implemented by applying a periodic dynamic strain of 0.1 %, with a frequency sweep ranging from 1 to 100 rad s⁻¹ and a temperature sweep specific to each material: from 130 °C to 90 °C for P3HB4HB(10), from 170 °C to 110 °C for P3HB4HB(15), and from 140 °C to 20 °C for P3HB4HB(30). These sweeps were performed in steps of -5 °C under air atmosphere.

The linear viscoelastic region was verified for all three polymers at their highest testing temperature, namely 130 °C for P3HB4HB(10), 170 °C for P3HB4HB(15), and 140 °C for P3HB4HB(30). A relatively low strain amplitude of 0.1 % was applied, ensuring that all measurements performed at each temperature were conducted within the linear viscoelastic regime.

To ensure the reliability of the obtained results, three sets of measurements were performed under identical experimental conditions. No significant variation was observed, either in the shape of the curves or in the calculated entanglement molecular weights. These observations confirm the robustness of the measurement method.

<figure><img src="image_6.png" /><figcaption>Fig. 4. Determination of G~N~⁰ for a monodisperse or polydisperse polymer.</figcaption></figure>
# 3. Results

## 3.1. Hansen theory energy parameters

To determine the energy parameters from Hansen theory, the molecular structure of the polymer under study must be decomposed. In the case of P3HB4HB, the polymer is a copolymer composed of 3HB and 4HB. To this end, we used the group contribution list proposed by Van Krevelen (Table 3 and Fig. 5).

For the 4HB repeating unit, we applied the same calculation method and obtained the following solubility parameters: $\delta_d = 17.5$, $\delta_p = 7.1$, $\delta_h = 10.1$ and $\delta_H = 21.4$.

Thus, to determine the Hansen solubility parameters (Table 4), we can apply a mixing rule that considers the molar proportions of the PHA samples studied. These proportions are 10%, 15%, and 30% of 4HB for P3HB4HB(10), P3HB4HB(15), and P3HB4HB(30), respectively.

To plasticize P3HB4HB materials, plastics companies must aim for compatibility with the corresponding Hansen solubility parameters. As demonstrated by Hildebrand and Scott [27]: “like dissolves like”. For example, for P3HB4HB(15), Table 5 presents an extensive list of potentially suitable plasticizers:

## 3.2. Prediction of entanglement molecular weight using Van Krevelen's theory

To predict the entanglement molecular weight of P3HB4HB using Van Krevelen's theory, we applied the equation developed by Souheng Wu [23]. In a similar way to the Hansen solubility parameter approach, we calculated the molar rigidity constants of the P3HB4HB molecules using a mixing rule. This method allows for the evaluation of the intrinsic flexibility of polymer chains based on their chemical structure and intra/intermolecular interactions. This theoretical estimation (Table 6) of $M_e$ serves as a useful tool for anticipating the formation of entanglement networks and optimizing the processing and plasticization properties of P3HB4HB materials.

## 3.3. Rheological measurement of the entanglement molecular weight

To determine $G_N^0$, oscillatory rheological measurements were performed at various temperatures within the viscoelastic region of the polymers (Fig. 7). The plateau modulus value was extracted from the rubbery plateau observed on the $G'$ curve at $\delta_{min}$, as our materials are polydisperse. Once $G_N^0$ is known, the entanglement molecular weight can be calculated using the Graessley relation (Table 7).

The analysis of the different regions of the graph provides insight into the discrepancies observed between the theoretical and experimental values of $M_e$. These aspects will be further developed in the discussion section.

<table><caption>**Table 1**

Physical properties of the studied polymers.</caption><thead><tr><th>Polymer</th><th>M<sub>n</sub> (g. mol<sup>-1</sup>)</th><th>I = M<sub>w</sub>/M<sub>n</sub></th><th>Density (at 25 °C)</th><th>T<sub>g</sub> (°C)</th><th>T<sub>m</sub> (°C)</th><th>ΔH<sub>m</sub> (J. g<sup>-1</sup>)</th></tr></thead><tbody><tr><td>P3HB4HB (10)</td><td>105000</td><td>4.6</td><td>1.24</td><td>-10</td><td>172</td><td>30</td></tr><tr><td>P3HB4HB (15)</td><td>191000</td><td>1.3</td><td>1.23</td><td>-20</td><td>172</td><td>34</td></tr><tr><td>P3HB4HB (30)</td><td>135000</td><td>2.6</td><td>1.22</td><td>-22</td><td>84</td><td>0.25</td></tr></tbody></table>

<table><caption>**Table 2**

List of group contributions according to Van Krevelen [4].</caption><thead><tr><th>Group</th><th>K<sub>f</sub> (g<sup>1/4</sup>.cm<sup>3/2</sup>. mole<sup>-3/4</sup>)</th><th>Group</th><th>K<sub>f</sub> (g<sup>1/4</sup>.cm<sup>3/2</sup>. mole<sup>-3/4</sup>)</th></tr></thead><tbody><tr><td>-CH<sub>3</sub></td><td>3.55</td><td>-CH</td><td>5</td></tr><tr><td>-CH<sub>2</sub>-</td><td>2.35</td><td>-OH</td><td>8</td></tr><tr><td>&gt;CH-</td><td>1.15</td><td>-O-</td><td>5.1</td></tr><tr><td>&gt;C&lt;</td><td>0</td><td>-COO-</td><td>9</td></tr><tr><td>-COOH</td><td>8</td><td>-OCOO-</td><td>27.5</td></tr><tr><td>-Cl</td><td>8.62</td><td>-CONH-</td><td>12.6</td></tr><tr><td>-C<sub>6</sub> aliphatic</td><td>18.25</td><td>-SO<sub>2</sub>-</td><td>12</td></tr><tr><td>-C<sub>6</sub> aromatic</td><td>16.3</td><td>&gt;Si&lt;</td><td>5</td></tr><tr><td>-C<sub>6</sub> aromatic o, m, p</td><td>40.8</td><td>Cycle structure</td><td>10</td></tr></tbody></table>

# 4. Discussion

## 4.1. Hansen solubility parameters

Experimentally evaluating Hansen solubility parameters is a complex task, requiring numerous manipulations to determine solubility in g·L⁻¹. However, it is possible to compare our results with values already reported in the literature. The study by Vermeer [22] is currently the key reference regarding PHBV solubility. It highlights exceptions among solvents capable of dissolving PHAs. For instance, according to Hansen theory, chloroform should not be an ideal solvent for PHBV, whereas in practice, it proves to be the most effective [22]. This illustrates the difficulty in accurately predicting the performance of a solvent or plasticizer. Experimental testing therefore remains essential for evaluating solubility or plasticization behavior.

To date, no scientific publication has provided precise solubility parameters for PHA copolymers based on 3HB and 4HB units. However, the authors Ong, Chen, and Don [28] calculated Hansen parameters for a P3HB4HB containing 16.5% 4HB. They reported the following values: $\delta_d = 16.7$ MPa¹/², $\delta_p = 7$ MPa¹/², $\delta_h = 10$ MPa¹/², and $\delta_H = 20.7$ MPa¹/², identical to those obtained for P3HB4HB(15), which contains 15 mol% of 4HB.

Our results are thus in perfect agreement with those in the literature for a P3HB4HB with a similar 4HB molar content. This consistency is partly due to the use of the same calculation method, thereby reinforcing the reliability of the values obtained. However, certain uncertainties should be taken into account.

Firstly, the molar volume used to calculate solubility parameters is based on the polymer density at a given temperature. Yet this density may vary depending on experimental conditions, introducing a margin of uncertainty into the calculations. Therefore, the results presented here are specific to a temperature of 25 °C and may differ under other thermal conditions.

Another point of discussion concerns the application of the mixing rule to calculate Hansen parameters. This approach may be challenged, since the physicochemical properties of P3HB4HB do not vary linearly with the 4HB molar content, as demonstrated by the study of Mitomo [29]. Given this limitation, an alternative method may be considered. For instance, Tomietto [30] used molar distribution to calculate the Hansen solubility parameters of a PHBV. This alternative approach could yield more accurate results by considering the actual polymer composition and its behavior in solution.

## 4.2. Entanglement molecular weight ($M_e$)

The entanglement molecular weights ($M_e$) of the studied polymers were determined using Van Krevelen's theoretical model and compared to the experimental values obtained from oscillatory rheology. The results are summarized in Table 8.

Indeed, the theoretical expression of the entanglement molecular
<table><caption>**Table 3**
Hansen solubility parameters of 3HB units.</caption><thead><tr><th colspan="5">Group contribution list proposed by Van Krevelen method</th><th colspan="4">Calculation using the Van Krevelen method</th></tr><tr><th>Group</th><th>Quantity</th><th>F<sub>di</sub></th><th>F<sub>pi</sub></th><th>E<sub>hi</sub></th><th>ΣF<sub>di</sub></th><th>ΣF<sub>pi</sub><sup>2</sup></th><th>ΣE<sub>hi</sub></th><th>Hansen solubility parameters</th></tr></thead><tbody><tr><td>-COO-</td><td>1</td><td>390</td><td>490</td><td>7000</td><td>1160</td><td>240100</td><td>7000</td><td></td></tr><tr><td>-CH<sub>2</sub>-</td><td>1</td><td>270</td><td>0</td><td>0</td><td>δ<sub>d</sub></td><td>δ<sub>p</sub></td><td>δ<sub>h</sub></td><td>δ<sub>H</sub></td></tr><tr><td>-CH&lt;</td><td>1</td><td>80</td><td>0</td><td>0</td><td>16.5</td><td>7.0</td><td>10.0</td><td>20.5</td></tr><tr><td>-CH<sub>3</sub></td><td>1</td><td>420</td><td>0</td><td>0</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Contributing groups of the 3HB unit according to Van Krevelen's group contribution list.</figcaption></figure>

<table><caption>**Table 4**
Hansen solubility parameters of the studied polymers.</caption><thead><tr><th>Polymer</th><th>Dispersive component δ<sub>d</sub></th><th>Polar component δ<sub>p</sub></th><th>Hydrogen bonding component δ<sub>h</sub></th><th>Hansen solubility parameter δ<sub>H</sub></th></tr></thead><tbody><tr><td>P3HB4HB (10)</td><td>16.6</td><td>7</td><td>10</td><td>20.6</td></tr><tr><td>P3HB4HB (15)</td><td>16.7</td><td>7</td><td>10</td><td>20.7</td></tr><tr><td>P3HB4HB (30)</td><td>16.8</td><td>7</td><td>10</td><td>20.8</td></tr></tbody></table>

weight is based on parameters subject to notable uncertainty. The statistical length of skeletal bonds (l<sub>v</sub>) used here was estimated from experimental measurements, which themselves carry inherent uncertainties. Moreover, this bond length may vary depending on experimental conditions such as temperature, atmosphere, or mechanical constraints.

Therefore, although the theoretical model provides a consistent order of magnitude, it comes with significant uncertainty. Let us consider an example to illustrate this variability: if the l<sub>v</sub> value is slightly lower for the 3HB molecule, then the ratio M<sub>v</sub>/l<sub>v</sub><sup>2</sup> increases, leading to a higher M<sub>e</sub> according to the Souheng Wu relation [23]. This reasoning shows that if l<sub>v</sub> varies, the ranking of the theoretical M<sub>e</sub> values could change. Consequently, it is preferable to rely on experimental values to validate trends and avoid misinterpretation regarding polymer entanglement.

Furthermore, the theoretical model is based on empirical relationships that do not account for certain influential parameters. For

example, the model does not account for crystallinity, which significantly affects the dynamics and mobility of polymer chains. This limitation is particularly relevant here, as the polymers under study are semi-crystalline rather than fully amorphous, making the theoretical assumptions less applicable. In particular, crystallization occurring during cooling reduces the length of the rubbery plateau (in frequency: 1.E+10 rad s<sup>-1</sup> for P3HB4HB(10), 1.E+12 rad s<sup>-1</sup> for P3HB4HB(15), and 1.E+13 rad s<sup>-1</sup> for P3HB4HB(30)), shifts this plateau toward higher apparent moduli, and requires the use of higher temperatures for M<sub>e</sub> estimation. In our study, these temperatures were 100 °C for P3HB4HB (10), 95 °C for P3HB4HB(15), and only 30 °C for P3HB4HB(30), which may contribute to an overestimation of the experimental entanglement molecular weights.

It is worth noting, however, that crystallinity appears to increase the variability of the measurements. Indeed, the experimental M<sub>e</sub> values for the semi-crystalline polymers P3HB4HB(10) and P3HB4HB(15) exhibit

<table><caption>**Table 6**
Theoretical entanglement molecular weights of the studied P3HB4HB samples.</caption><thead><tr><th>Polymer</th><th>Molar rigidity constant of one unit (ΣK<sub>i</sub>) (g<sup>1/4</sup>.cm<sup>3/2</sup>.mole<sup>-3/4</sup>)</th><th>Average skeletal bond length (Å)</th><th>Theoretical entanglement molecular weight (M<sub>e</sub> TH) (g.mol<sup>-1</sup>)</th></tr></thead><tbody><tr><td>P3HB4HB (10)</td><td>16.05</td><td>2.60</td><td>2075</td></tr><tr><td>P3HB4HB (15)</td><td>16.05</td><td>2.58</td><td>2126</td></tr><tr><td>P3HB4HB (30)</td><td>16.05</td><td>2.52</td><td>2331</td></tr></tbody></table>

<table><caption>**Table 7**
Experimental entanglement molecular weight of the studied P3HB4HB samples.</caption><thead><tr><th>Polymer</th><th>G<sup>0</sup><sub>N</sub> (MPa)</th><th>Density</th><th>Experimental entanglement molecular weight (M<sub>e</sub> exp) (g.mol<sup>-1</sup>)</th></tr></thead><tbody><tr><td>P3HB4HB (10)</td><td>2.12 ± 0.7</td><td>1.24</td><td>1819 ± 61</td></tr><tr><td>P3HB4HB (15)</td><td>2.07 ± 0.3</td><td>1.23</td><td>1817 ± 24</td></tr><tr><td>P3HB4HB (30)</td><td>1.75 ± 0.2</td><td>1.22</td><td>1754 ± 20</td></tr></tbody></table>

<table><caption>**Table 5**
Hansen solubility parameters of various plasticizers.</caption><thead><tr><th>Family</th><th>Plasticizer</th><th>Dispersive component δ<sub>d</sub></th><th>Polar component δ<sub>p</sub></th><th>Hydrogen bonding component δ<sub>h</sub></th><th>Hansen solubility parameter δ<sub>H</sub></th></tr></thead><tbody><tr><td rowspan="3">Polymeric plasticizers</td><td>Glycerol</td><td>17</td><td>11.8</td><td>28.6</td><td>35.2</td></tr><tr><td>PPG2000</td><td>15.7</td><td>9.4</td><td>23</td><td>29.4</td></tr><tr><td>PEG200</td><td>10.4</td><td>2.8</td><td>16.7</td><td>18.7</td></tr><tr><td rowspan="3">Terpenes</td><td>TEC</td><td>16.1</td><td>4.1</td><td>13</td><td>21.1</td></tr><tr><td>TBC</td><td>16</td><td>2.8</td><td>10.9</td><td>19.6</td></tr><tr><td>ATBC</td><td>16</td><td>2.6</td><td>8.6</td><td>18.4</td></tr><tr><td rowspan="2">Commercial plasticizers</td><td>Benzoflex®</td><td>17.7</td><td>2.9</td><td>7.9</td><td>19.6</td></tr><tr><td>Hexamoll®</td><td>16.3</td><td>1.57</td><td>5.6</td><td>17.3</td></tr><tr><td>Epoxides</td><td>ESBO</td><td>14.7</td><td>1.3</td><td>6.3</td><td>16</td></tr><tr><td>Phthalates</td><td>DOP</td><td>15.8</td><td>1.8</td><td>6</td><td>17</td></tr><tr><td>Solvent</td><td>Chloroform</td><td>17.9</td><td>11.9</td><td>3.9</td><td>21.9</td></tr></tbody></table>
<table><caption>**Table 8**

Theoretical and experimental entanglement molecular weights of the studied P3HB4HB samples.</caption><thead><tr><th>Polymer</th><th>Theoretical entanglement molecular weight (M<sub>e</sub> <i>t</i><sub>h</sub>) (g.mol<sup>-1</sup>)</th><th>Experimental entanglement molecular weight (M<sub>e</sub> exp) (g. mol<sup>-1</sup>)</th><th>Deviation (%)</th></tr></thead><tbody><tr><td>P3HB4HB (10)</td><td>2075</td><td>1819 ± 61</td><td>-12.3 ± 2.9</td></tr><tr><td>P3HB4HB (15)</td><td>2126</td><td>1817 ± 24</td><td>-14.5 ± 1.1</td></tr><tr><td>P3HB4HB (30)</td><td>2331</td><td>1754 ± 20</td><td>-24.7 ± 0.8</td></tr></tbody></table>

It is observed that the experimental $M_e$ values are consistently lower than the theoretical predictions, with the largest deviation for P3HB4HB(30) (-24.7 %), followed by P3HB4HB(15) (-14.5 %) and P3HB4HB(10) (-12.3 %). This difference between predicted and measured values can be explained by several structural and experimental factors.

greater dispersion and larger standard deviations compared to those obtained for the more amorphous P3HB4HB(30).

Other factors, such as molecular weight, polydispersity, the monomer unit distribution or the presence of branching, also influence $M_e$ values and may contribute to the discrepancies observed between theory and experiment [31].

### 4.2.1. Viscous regime: $G'$ and $G''$ at low frequency

At low frequencies ($1 < \omega < 100$ rad/s), a more pronounced separation between $G'$ and $G''$ is observed for P3HB4HB(10) and P3HB4HB (15) compared to P3HB4HB(30) (Fig. 6). This behavior is characteristic of semi-crystalline polymers, where the crystalline phase restricts polymer chain mobility, thereby increasing the material's stiffness. In contrast, P3HB4HB(30) exhibits a lower $G'$ modulus, indicating greater chain mobility.

Higher crystallinity generally promotes better structuring of entanglements, which should theoretically reduce the entanglement molecular weight ($M_e$) for polymers with similar physicochemical characteristics. However, in our study, the analyzed polymers also differ in other aspects, particularly in terms of density, which directly influences the $M_e$ calculation via the Graessley relation.

The analysis of moduli at low frequency shows that P3HB4HB(15) has the highest stiffness, followed by P3HB4HB(10), and finally P3HB4HB(30). This trend suggests that P3HB4HB(15) exhibits higher crystallinity. This hypothesis is further supported by the fusion enthalpy values (Table 1), as well as by the sharper transition between the viscous regime and the rubbery plateau observed for P3HB4HB(15) and P3HB4HB(10). In contrast, P3HB4HB(30) shows a more gradual transition, which is typical of amorphous materials.

### 4.2.2. Rubbery plateau: variation of $G_N^0$ at high frequency

At high frequencies ($10^6 < \omega < 10^{12}$ rad/s), a rubbery plateau is observed, corresponding to the regions where entanglements govern the mechanical behavior. P3HB4HB(10) exhibits the highest plateau, followed by P3HB4HB(15), and then P3HB4HB(30). A higher $G_N^0$ indicates either more strongly entangled polymer chains or a more organized crystalline structure. In our case, we assume that the increase in $G_N^0$ is primarily related to crystallinity. We suggest that P3HB4HB(10) forms larger spherulites than P3HB4HB(15), although the difference remains moderate.

While many studies have shown that crystallinity increases $G_N^0$ [32–35], few have established a direct correlation between spherulite size and $G_N^0$. To support this hypothesis, we rely on the analysis of the standard deviation and the temperature at which $G_N^0$ is determined.

### 4.2.3. Influence of standard deviation

A larger standard deviation indicates greater variability in the measurements, which may be linked to structural heterogeneity within the material. In semi-crystalline polymers, a broader distribution of spherulite size can lead to local variations in mechanical and rheological properties, affecting the repeatability of measurements. In our study, the higher standard deviation observed for P3HB4HB(10) could be explained by a wider size distribution of spherulites in its crystalline phase.

Indeed, spherulite formation during polymer crystallization depends on cooling conditions and chain mobility, sometimes resulting in broader size distributions, as reported in studies by Li and Lu [36,37]. We assume that such structural heterogeneity may influence local rheological properties, leading to greater variability in measurements. This hypothesis is further supported by the higher polydispersity of

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Evolution of the storage modulus ($G'$, solid lines) and loss modulus ($G''$, dashed lines) as a function of frequency for the studied P3HB4HB samples (P3HB4HB (30) in black, P3HB4HB(15) in red, and P3HB4HB(10) in blue). The reference temperatures used to construct the master curves are 140 °C for P3HB4HB(30), 170 °C for P3HB4HB(15), and 130 °C for P3HB4HB(10). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 7. Determination of the activation energy based on the variation of the plateau modulus G₀N with temperature, and interpretation of the probable spherulite size for the corresponding P3HB4HB.</figcaption></figure>

P3HB4HB(10) compared to P3HB4HB(15).

### 4.2.4. Influence of temperature

The temperature at which G₀N is determined also serves as an indicator supporting this hypothesis: the larger the spherulite size, the higher the G₀N. The transition to the rubbery plateau is directly influenced by the crystalline fraction of the polymer. A higher crystalline phase content or larger spherulites result in stronger intermolecular interactions within the material, requiring higher temperatures to soften the structure and reach the viscoelastic regime characteristic of the rubbery plateau.

In our case, we observe distinct transition temperatures among the studied materials: 100 °C for P3HB4HB(10), 90 °C for P3HB4HB(15), and 30 °C for P3HB4HB(30). This difference is supported by the data in Table 1, where P3HB4HB(30) exhibits a melting enthalpy close to zero, suggesting that it possesses an essentially amorphous structure. Conversely, P3HB4HB(10) shows a later transition at 100 °C, while P3HB4HB(15) reaches its plateau at an intermediate temperature of 90 °C. These observations further support the idea that spherulite size and organization directly influence the rheological behavior of the material and partially explain the variability observed in the measurements.

### 4.2.5. Approach using activation energy (Eₐ)

To further illustrate this analysis, we can apply the Arrhenius equation, which describes the relationship between the rate of a chemical process and temperature, assuming that the activation energy (Eₐ) remains independent of temperature (Eq. (9)):

$$k = A \cdot e^{-E_a/RT} \qquad (\text{Eq. 9})$$

In the case of a polymer, this equation can be adapted to determine the activation energy (Eₐ) required for chain relaxation or mobility. In other words, it corresponds to the energy required to initiate the disappearance of spherulites upon heating. The lower this activation energy, the less thermal energy is needed to process and plasticize the material.

In a semi-crystalline polymer, these chains are constrained not only by the crystalline phases but also by molecular entanglements (nodes). It is therefore relevant to quantify the energy threshold at which they can

reorganize within this network.

Since the plateau modulus G₀N follows the Arrhenius law [38], it can be expressed as (Eq. (10)):

$$G_{N}^{0} = G_{0}e^{-E_{a}/RT} \qquad (\text{Eq. 10})$$

Thus, increasing the temperature provides sufficient thermal energy to overcome the constraints imposed by the spherulites, thereby facilitating the relaxation of polymer chains. Fig. 7 illustrates the influence of spherulite morphology on Eₐ: larger spherulites may confine certain macromolecular chains more effectively within crystalline domains, limiting their mobility. This can lead to higher activation energies and elevated temperatures required to reach the rubbery plateau (G₀N). Indeed, by analogy with PVC, whose crystals are in the nano-to micro-scale range, a single macromolecular chain can be involved simultaneously in both the amorphous and crystalline phases. In the case of P3HB4HB, where spherulites are significantly larger, we suggest that a chain may be entirely confined within a crystalline domain, without the possibility of extending into a neighboring amorphous phase [39–41].

This analysis supports the hypothesis that spherulite size, regardless of the overall crystallinity, directly impacts the mechanical behavior of P3HB4HB, thereby complicating the experimental determination of the entanglement molecular weight.

The regression line equation applied to the Arrhenius law makes it possible to estimate the energy required for spherulite size to no longer restrict chain mobility within the entangled network. Although only 3 samples of P3HB4HB were tested due to their limited commercial availability, it appears that an energy of approximately 21.7 kJ mol⁻¹ is required to initiate this molecular mobility. Finally, visual interpretations are proposed for each polymer: P3HB4HB(30), being nearly amorphous, likely contains only a few small spherulites; P3HB4HB(15) has more numerous spherulites of intermediate size; and P3HB4HB(10) is characterized by large, well-structured spherulites. In conclusion, in our case, the larger the spherulites appear to be, the stiffer the material seems, making it more difficult to plasticize.

### 4.2.6. Impact of temperature on the determination of Mₑ

The temperature at which G₀N is measured has a direct influence on the evaluation of the entanglement molecular weight (Mₑ) when calculated using Graessley’s equation. If the measurement temperature
for $G_N^0$ varies between different polymers, the resulting $M_e$ values will not be directly comparable due to thermal effects. Moreover, crystallinity is not considered in Graessley's formula, which was developed for amorphous polymers. In the case of semi-crystalline polymers, this influence is even more pronounced, as the temperature required to reach the rubbery plateau depends on the crystalline fraction and the size of the spherulites, making comparative analysis more complex.

Altogether, these findings suggest that the crystalline structure and spherulite distribution play a fundamental role in determining $G_N^0$ and, by extension, in evaluating $M_e$. A combined approach involving both rheological and thermal analyses appears essential to better understand these phenomena and refine the interpretation of the results.

The theoretical determination of the entanglement molecular weight ($M_e$) provides an order of magnitude, but lacks precision due to uncertainties in input parameters, particularly the statistical bond length and density. For amorphous polymers, the experimental $M_e$ values obtained by oscillatory rheology are in good agreement with theory. However, for semi-crystalline polymers such as the P3HB4HB with 10% and 15% 4HB studied here, crystallinity plays a major role in altering chain dynamics and influencing the $G_N^0$ measurement temperature, which leads to a discrepancy between theoretical and experimental values. To account for these effects and obtain a representative value, we retain an average $M_e$ of 1750 g mol$^{-1}$ for P3HB4HB.

## 5. Conclusion

The effective P3HB4HB plasticization relies on a precise understanding of polymer-plasticizer compatibility and the chemical structure of the polymer. Through the combined application of Hansen solubility theory to predict compatibility and Van Krevelen's method to assess chain flexibility and entanglement density, relevant predictive tools were developed to guide plasticizer selection. These two approaches, although based on different theoretical principles, prove to be complementary for evaluating and predicting the plasticization potential of a polymer material.

The energetic parameters calculated using Van Krevelen's method were found to be consistent with values reported in the literature, confirming the robustness of this approach for P3HB4HB, particularly for copolymers containing 15% 4HB. The prediction of entanglement molecular weight ($M_e$) enabled us to anticipate the complexity of the polymer network and its influence on processing behavior.

However, the discrepancies observed between theoretical and experimental $M_e$ values highlight the major role of crystallinity particularly the spherulites size and their distribution in influencing the rheological properties of P3HB4HB. Oscillatory rheology proved to be an essential tool for quantifying the structuring of the polymer network. It clearly revealed the transition to the rubbery plateau and linked mechanical properties to morphological features, confirming the impact of spherulites on plasticization behavior.

Ultimately, this study demonstrates the value of a combined theoretical and experimental approach for evaluating the plasticization potential of P3HB4HB. It shows that while predictive models are useful for guiding plasticizer selection, experimental validation remains indispensable—especially for semi-crystalline polymers with high structural variability. This approach contributes to the optimization of formulations and the development of high-performance bio-based materials tailored to current industrial and environmental challenges.

## CRediT authorship contribution statement

**Erwan Vasseur:** Validation, Investigation, Formal analysis, Conceptualization.
**Louis Leveuf:** Supervision.
**Vincent Le Saux:** Supervision.
**Stéphane Bruzaud:** Validation, Supervision.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The authors are pleased to express their grateful acknowledgements to Mickael Castro for their assistance in the experimental work and their helpful discussions. I would also like to thank the ANRT for funding project number 2022/1175.

## Data availability

No data was used for the research described in the article.

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