**1. Introduction**

Physical blending is an economical approach to achieve complementary material properties without developing a new polymer. Poly(butylene adipate-co-terephthalate) (PBAT) and polylactide (PLA) currently account for 13.5 and 18.8 percent of global bioplastic production capacity, respectively [1]. Blends made of commercially available PBAT and PLA tend to macrophase-separate and exhibit two glass transition temperatures (*Tg*) [2–4]. Therefore, unmodified PBAT/PLA blends have poor miscibility.

Experimentally, PBAT/PLA blends are often prepared by melt blending. At an elevated temperature for a sufficiently long time, transesterification can take place between the two polyesters, resulting in enhanced miscibility [5]. Two criteria often used to evaluate the blend miscibility [6–8] include (1) phase morphology: whether it is homogeneous down to the molecular level; (2) glass transition temperature: whether a binary blend exhibits a single *Tg*. The morphological investigation depends on the measuring points of the samples in scanning electron microscopy (SEM). A blend may not be truly miscible at the molecular level, even though it shows a homogeneous phase if examined on sufficiently large length scales [6]. Thermal analysis using differential scanning calorimetry (DSC) can only determine the miscibility of polymer blends with well-separated *T<sup>g</sup>* values [9]. Furthermore, different heating rates can lead to a variation in *T<sup>g</sup>* values to some extent. Therefore, both the blend preparation and the analysis can practically affect the actual blend miscibility.

To exclude the experimental influences, a theoretical study of the intrinsic miscibility is necessary. The concept of group contributions has been applied to calculate the solubility parameters and the blend miscibility [10]. According to the literature research, Dil et al. indicate that PBAT and PLA have the Hildebrand solubility parameters (HiSP) of 22.2 MPa1/2 and 21.9 MPa1/2, respectively [11]. Ding et al. report that the δ (PBAT) and δ(PLA) are 21.9 MPa1/2 and 20.7 MPa1/2, respectively [12]. As a result, the difference of solubility parameters *δ*(PBAT-PLA) is 0.3 or 1.2 MPa1/2. Materials with similar HiSP have a high affinity with each other [13]. However, the PBAT/PLA blend miscibility is still difficult to determine directly from the difference in solubility parameters. One reason for this is the irregular structure of PBAT molecules that consist of two randomly arranged monomers (BA and BT). Their molar ratio can differ due to the synthesis and degradation. If PBAT were degraded, the resulting lower molecular PBAT-chains might have different ratios of BA and BT. Another reason is that the intermolecular interaction depends on the molecular weights of both polymers, blend ratio, and temperature [14]. To the author's best knowledge, only Park et al. [15] have studied the miscibility of blends (PETG/PLA) with some similarity in molecular structure to PBAT/PLA blends from the theoretical aspect. Therefore, PBAT/PLA blends need more investigation regarding the solubility parameters and miscibility from the molecular level and the thermodynamic aspect. A good understanding of the intrinsic miscibility is not only of academic interest but also contributes to designing miscible PBAT/PLA blends without adding expensive additives. ity parameters and the blend miscibility [10]. According to the literature research, Dil et al. indicate that PBAT and PLA have the Hildebrand solubility parameters (HiSP) of 22.2 MPa1/2 and 21.9 MPa1/2, respectively [11]. Ding et al. report that the δ (PBAT) and δ(PLA) are 21.9 MPa1/2 and 20.7 MPa1/2, respectively [12]. As a result, the difference of solubility parameters *δ*(PBAT-PLA) is 0.3 or 1.2 MPa1/2. Materials with similar HiSP have a high affinity with each other [13]. However, the PBAT/PLA blend miscibility is still difficult to determine directly from the difference in solubility parameters. One reason for this is the irregular structure of PBAT molecules that consist of two randomly arranged monomers (BA and BT). Their molar ratio can differ due to the synthesis and degradation. If PBAT were degraded, the resulting lower molecular PBAT-chains might have different ratios of BA and BT. Another reason is that the intermolecular interaction depends on the molecular weights of both polymers, blend ratio, and temperature [14]. To the author's best knowledge, only Park et al. [15] have studied the miscibility of blends (PETG/PLA) with some similarity in molecular structure to PBAT/PLA blends from the theoretical aspect. Therefore, PBAT/PLA blends need more investigation regarding the solubility parameters and miscibility from the molecular level and the thermodynamic aspect. A good understanding of the intrinsic miscibility is not only of academic interest but also contributes to designing miscible PBAT/PLA blends without adding expensive additives. This study aimed to explore the dependence of the PBAT structure and the molecular weights of PBAT and PLA with different weight ratios on the blend miscibility. First,

To exclude the experimental influences, a theoretical study of the intrinsic miscibility is necessary. The concept of group contributions has been applied to calculate the solubil-

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This study aimed to explore the dependence of the PBAT structure and the molecular weights of PBAT and PLA with different weight ratios on the blend miscibility. First, group contributions theories were used to estimate the effect of the molar ratio of the two monomers in PBAT on the solubility parameter and the difference of solubility parameter between both polymers. Second, the Flory–Huggins model was used to establish phase diagrams and spinodal curves. Furthermore, the composition-dependent glass transition of miscible PBAT/PLA blends was estimated. The first novelty was to find a correlation between the molar ratios of the two monomers in PABT and the solubility parameter between PBAT and PLA. The second novelty was to simulate the phase diagrams and spinodal curves of PBAT/PLA blends, taking into account parameters including temperature, molecular weights, and the component ratios, as well as the calculated solubility parameters. group contributions theories were used to estimate the effect of the molar ratio of the two monomers in PBAT on the solubility parameter and the difference of solubility parameter between both polymers. Second, the Flory–Huggins model was used to establish phase diagrams and spinodal curves. Furthermore, the composition-dependent glass transition of miscible PBAT/PLA blends was estimated. The first novelty was to find a correlation between the molar ratios of the two monomers in PABT and the solubility parameter between PBAT and PLA. The second novelty was to simulate the phase diagrams and spinodal curves of PBAT/PLA blends, taking into account parameters including temperature, molecular weights, and the component ratios, as well as the calculated solubility parameters.

#### **2. Materials 2. Materials**

Poly(butylene adipate-co-terephthalate) (PBAT) (Figure 1) is a fully biodegradable polyester with two types of dimers: BT and BA. One dimer is the rigid section consisting of 1,4-butanediol and terephthalic acid monomers. The other dimer is the flexible section consisting of 1,4-butanediol and adipic acid monomers [16], resulting in high flexibility and high ductility [4]. The largest manufacturer, BASF (Ludwigshafen, Germany), produces PBAT under the brand name Ecoflex® (e.g., Ecoflex F Blend C1200). This polymer possesses a density of 1.26 g/mol, a number average molecular weight (*Mn*) of 52.1 kg/mol, and a polydispersity index of 2.0. Poly(butylene adipate-co-terephthalate) (PBAT) (Figure 1) is a fully biodegradable polyester with two types of dimers: BT and BA. One dimer is the rigid section consisting of 1,4-butanediol and terephthalic acid monomers. The other dimer is the flexible section consisting of 1,4-butanediol and adipic acid monomers [16], resulting in high flexibility and high ductility [4]. The largest manufacturer, BASF (Ludwigshafen, Germany), produces PBAT under the brand name Ecoflex® (e.g., Ecoflex F Blend C1200). This polymer possesses a density of 1.26 g/mol, a number average molecular weight (*Mn*) of 52.1 kg/mol, and a polydispersity index of 2.0.

**Figure 1. Figure 1.** Chemical structure of PBAT. Chemical structure of PBAT.

Poly(lactic acid) or polylactide (PLA) is a biodegradable and renewable aliphatic polyester [17]. Poly(lactic acid) is produced by direct polycondensation of lactic acid. Polylactide can be produced by ring-opening polymerization of cyclic lactide. PLA possesses two optical active and crystallizable isomeric forms: PDLA and PLLA (Figure 2). PLA exhibits many favorable features such as high modulus of elasticity, high strength and high transparency (in the amorphous state), and good processability [18]. The largest manufacturer, NatureWorks (Minnetonka, MN, USA), produces PLA under the trade name IngeoTM (for example, IngeoTM Biopolymer 2003D). This polymer has a density of 1.24 g/mol, a number average molecular weight (*Mn*) of 127.0 kg/mol, a polydispersity index of 1.6, and a D-isomer content of approximately 4.4%. Poly(lactic acid) or polylactide (PLA) is a biodegradable and renewable aliphatic polyester [17]. Poly(lactic acid) is produced by direct polycondensation of lactic acid. Polylactide can be produced by ring-opening polymerization of cyclic lactide. PLA possesses two optical active and crystallizable isomeric forms: PDLA and PLLA (Figure 2). PLA exhibits many favorable features such as high modulus of elasticity, high strength and high transparency (in the amorphous state), and good processability [18]. The largest manufacturer, NatureWorks (Minnetonka, MN, USA), produces PLA under the trade name IngeoTM (for example, IngeoTM Biopolymer 2003D). This polymer has a density of 1.24 g/mol, a number average molecular weight (*Mn*) of 127.0 kg/mol, a polydispersity index of 1.6, and a Disomer content of approximately 4.4%.

**Figure 2***.* Chemical structure of PLLA. **Figure 2.** Chemical structure of PLLA.

#### **3. Prediction of Solubility Parameters 3. Prediction of Solubility Parameters**

The basis of a molecular group contribution method is additivity. This assumes that the physical property of a polymer is calculable by the additive contributions from the individual structural and functional groups in its repeating unit. Using a group contribution method enables us to establish a correlation between the chemical structure of poly-The basis of a molecular group contribution method is additivity. This assumes that the physical property of a polymer is calculable by the additive contributions from the individual structural and functional groups in its repeating unit. Using a group contribution method enables us to establish a correlation between the chemical structure of polymers and their interaction. The term "solubility parameter" quantifies the intermolecular interaction.

mers and their interaction. The term "solubility parameter" quantifies the intermolecular interaction. Coleman et al. has reported a group contribution method to calculate the one-dimensional Hildebrand solubility parameter (HiSP) for polymers with *Tg* values below room temperature [19]. However, PLA possesses a glass transition temperature higher than room temperature (approx. 60 °C). Therefore, Coleman's method was inappropriate in this miscibility study of PBAT/PLA blends. Group contribution methods developed by van Krevelen [10] and Hoy [20] were used to estimate the three-dimensional HSP and onedimensional HiSP for polymers. Compared with the real system, the group contribution Coleman et al. has reported a group contribution method to calculate the onedimensional Hildebrand solubility parameter (HiSP) for polymers with *T<sup>g</sup>* values below room temperature [19]. However, PLA possesses a glass transition temperature higher than room temperature (approx. 60 ◦C). Therefore, Coleman's method was inappropriate in this miscibility study of PBAT/PLA blends. Group contribution methods developed by van Krevelen [10] and Hoy [20] were used to estimate the three-dimensional HSP and one-dimensional HiSP for polymers. Compared with the real system, the group contribution methods neglect the possible reactions between the blend components, such as transesterification while melt blending, which may change the molecular structures.

methods neglect the possible reactions between the blend components, such as transesterification while melt blending, which may change the molecular structures. In this section, these two different group contribution methods were applied to calculate the solubility parameters of PBAT and PLA. As mentioned in the Introduction, PBAT can consist of different ratios of its monomers (BA and BT). Furthermore, the se-In this section, these two different group contribution methods were applied to calculate the solubility parameters of PBAT and PLA. As mentioned in the Introduction, PBAT can consist of different ratios of its monomers (BA and BT). Furthermore, the sequence in the chain may change due to, e.g., polymer degradation. Considering these variations, the estimation of its solubility parameter was performed using three different assumptions:


#### (2) (PBT): only monomers of BT constitute the polymer. *3.1. Solubility Parameter Calculation According to van Krevelen's Method*

(3) (PBA): only monomers of BA constitute the polymer. *3.1. Solubility Parameter Calculation According to van Krevelen's Method*  The group contributions *Fdi*, *Fpi*, and *Ehi* [10] are applied to calculate HSP, including The group contributions *Fdi*, *Fpi*, and *Ehi* [10] are applied to calculate HSP, including dispersion interactions (*δ<sup>d</sup>* ), polar interactions (*δp*), and hydrogen bond interactions (*δ<sup>h</sup>* ), as well as the total Hildebrand solubility parameters (*δ<sup>t</sup>* or HiSP). The following equations are used for the calculation:

$$
\delta\_d = \frac{\sum F\_{di}}{V} \tag{1}
$$

$$
\delta\_p = \frac{\sqrt{\sum F\_{pi}^2}}{V} \tag{2}
$$

are used for the calculation:

$$
\delta\_{\rm li} = \sqrt{\frac{\sum E\_{\rm hi}}{V}} \tag{3}
$$

(2)

$$
\delta\_l = \sqrt{\delta\_d^2 + \delta\_P^2 + \delta\_h^2} \tag{4}
$$

where *Fdi* represents the group contributions of type *i* to the dispersion component *F<sup>d</sup>* of the molar attraction constant; *Fpi* represents the group contributions to the polar component *Fp*; *Ehi* is the hydrogen-bonding energy per structural group *i*; *V* is the molar volume. Details of the calculation are described in the Supplementary Material on the sheet "van Krevelen". The calculated HSP and HiSP using van Krevelen's method are listed (Table 1). the molar attraction constant; *Fpi* represents the group contributions to the polar component *Fp*; *Ehi* is the hydrogen-bonding energy per structural group *i*; *V* is the molar volume. Details of the calculation are described in the Supplementary Material on the sheet "van Krevelen". The calculated HSP and HiSP using van Krevelen's method are listed (Table 1).

ට∑ ଶ

**Polymer Type** *δ<sup>d</sup>* **[MPa1/2]** *δ<sup>p</sup>* **[MPa1/2]** *δ<sup>h</sup>* **[MPa1/2] HiSP [MPa1/2]** alternating PBAT 18.21 5.89 9.17 21.22 PBT 19.63 6.19 9.37 22.62 PBA 14.99 5.00 8.45 17.92 PLA 15.33 8.44 10.98 20.66 **Polymer Type** *δd* **[MPa1/2]** *δp* **[MPa1/2]** *δh* **[MPa1/2] HiSP [MPa1/2]**  alternating PBAT 18.21 5.89 9.17 21.22 PBT 19.63 6.19 9.37 22.62 PBA 14.99 5.00 8.45 17.92 PLA 15.33 8.44 10.98 20.66

**Table 1.** Calculated solubility parameters using van Krevelen's method. **Table 1.** Calculated solubility parameters using van Krevelen's method.

*Polymers* **2021**, *13*, 2339 4 of 11

ൌ

Due to the random arrangement of the copolymers BT and BA and possible different molar ratios, the solubility parameters for each segment is different. In the case of a BT-rich PBAT (molar ratio of BT ≥ 50%), the HiSP of PBAT was between 21.22 and 22.62, which are the values of alternating PBAT with BT/BA = 1/1 and PBT. In the case of a BA-rich PBAT (molar ratio of BA ≥ 50%), the HiSP of PBAT was between 17.92 and 21.22, the values of PBA and alternating PBAT with BA/BT =1/1. The calculated HiSP of PBAT (alternating) and PLA was 21.22 [MPa1/2] and 20.66 [MPa1/2], respectively, so the difference of total solubility parameter ∆δ between them was 0.56 [MPa1/2]. Since the HiSP of PLA was in the range of BA-rich PBAT (Figure 3), varying the molar ratio of the monomers in PBAT may lead to a smaller difference in solubility parameters and an increase in the intermolecular interaction between PBAT and PLA. Due to the random arrangement of the copolymers BT and BA and possible different molar ratios, the solubility parameters for each segment is different. In the case of a BTrich PBAT (molar ratio of BT ≥ 50%), the HiSP of PBAT was between 21.22 and 22.62, which are the values of alternating PBAT with BT/BA = 1/1 and PBT. In the case of a BArich PBAT (molar ratio of BA ≥ 50%), the HiSP of PBAT was between 17.92 and 21.22, the values of PBA and alternating PBAT with BA/BT =1/1. The calculated HiSP of PBAT (alternating) and PLA was 21.22 [MPa1/2] and 20.66 [MPa1/2], respectively, so the difference of total solubility parameter ∆δ between them was 0.56 [MPa1/2]. Since the HiSP of PLA was in the range of BA-rich PBAT (Figure 3), varying the molar ratio of the monomers in PBAT may lead to a smaller difference in solubility parameters and an increase in the intermolecular interaction between PBAT and PLA.

**Figure 3***.* Values and ranges of calculated solubility parameters using van Krevelen's method. **Figure 3.** Values and ranges of calculated solubility parameters using van Krevelen's method.

#### *3.2. Solubility Parameter Calculation According to Hoy's Method 3.2. Solubility Parameter Calculation According to Hoy's Method*

The methods of van Krevelen and Hoy, based on the same basic assumption of additivity, have the same order of accuracy. These two are of the same order of accuracy for predicting the solubility parameters [10]. However, van Krevelen and Hoy have applied different ways of calculation. Van Krevelen's method first predicts the partial solubility The methods of van Krevelen and Hoy, based on the same basic assumption of additivity, have the same order of accuracy. These two are of the same order of accuracy for predicting the solubility parameters [10]. However, van Krevelen and Hoy have applied different ways of calculation. Van Krevelen's method first predicts the partial solubility parameters, which are then used to calculate the total solubility parameter [10,21]. In contrast, Hoy's method first determines the total solubility parameter using the molar attraction functions and auxiliary equations. From the total, the three partial ones are calculated using additive molar functions and expressions for the components [20,22]. The equations used in Hoy's method are below.

$$\mathfrak{a}(P) = 777 \Delta\_T(P) / V \tag{5}$$

$$n = 0.5/\Delta\_T(P) \tag{6}$$

$$\delta\_l = (F\_l + \frac{\mathbf{B}}{\overline{\mathbf{T}}}) / V \tag{7}$$

$$\delta\_P = \delta\_l \left( \frac{1}{a(P)} \frac{F\_P}{F\_l + B/\overline{n}} \right)^{1/2} \tag{8}$$

$$\delta\_{\mathbb{H}} = \delta\_{\mathbb{I}} \left[ (\alpha(P) - \mathbf{1}/\alpha(P)) \right]^{1/2} \tag{9}$$

$$
\delta\_d = (\delta\_t^2 - \delta\_P^2 - \delta\_h^2)^{1/2} \tag{10}
$$

where *α(P)* is the molecular aggregation number of a polymer; ∆*T(P)* is the Lydersen correction for polymers derived by Hoy; *V* is the molar volume; B is a base value of 277; *Ft* is the molar attraction function; *F<sup>P</sup>* is the polar component of molar attraction function; *n* is the molecular aggregation number. For explanations of variables used before, see Section 3.1 "van Krevelen's method".

The three-dimensional HSP and the overall one-dimensional HiSP were calculated (Table 2). (For details of the calculation, see the Supplementary Material on the sheet "Hoy").

**Table 2.** Calculated solubility parameters using Hoy's method.


From the table above, it can be seen that BA-rich PBAT had a HiSP in the range of 20.8–21.73, while BT-rich PBAT possessed a HiSP in the range of 21.73 to 23.18 [MPa1/2]. The calculated HiSP for PBAT and PLA was 21.73 [MPa1/2] and 21.31 [MPa1/2], respectively, indicating a solubility parameter difference of 0.42 [MPa1/2]. Furthermore, Hoy's method shows the same tendency as van Krevelen's method: that the HiSP of PLA was in the range of the values of BA-rich PBAT (Figure 4). *Polymers* **2021**, *13*, 2339 6 of 11

**Figure 4***.* Values and ranges of calculated solubility parameters using Hoy's method. **Figure 4.** Values and ranges of calculated solubility parameters using Hoy's method.

According to the two group contribution methods, the mean value of the difference in calculated total solubility parameters between alternating PBAT and PLA was 0.49 According to the two group contribution methods, the mean value of the difference in calculated total solubility parameters between alternating PBAT and PLA was 0.49 MPa1/2 .

MPa1/2. The two different methods show consistency in the total solubility parameter and that PLA was in the range of the one of BA-rich PBAT. Furthermore, the same tendency

Based on the solubility parameters, the miscibility of PBAT/PLA blends can be further studied from the aspect of thermodynamics. The thermodynamic criterion of solubil-

where ∆*Gm* is the free energy of mixing; the ∆*HM* is the enthalpy of mixing (heat of mixing);

theory correlates the miscibility of a polymer blend with several parameters [24].

where ∆*GM* is the free enthalpy of mixing; R the gas constant of 8.31 kg٠m2/s2<sup>٠</sup>

ଵ <sup>+</sup> ଶ ଶ

A negative value of ∆*Gm* is generally required to obtain a miscible system. For low molecular weight materials, an increasing temperature generally results in an increase in miscibility as the *T*∆*SM* term increases, thus driving *GM* to be more negative [8,23]. However, both PBAT and PLA are high molecular weight molecules, implying that the negative contribution from the *T*∆*SM* term is small. An equation based on the Flory–Huggins

ଶ <sup>+</sup> ଵଶ

The phase diagram (Figure 5) offers a simulation of the PBAT/PLA blend miscibility with assumptions at room temperature (296 K, approximately 23 °C), which is important for the blend preparation by solution blending. PBAT structure was alternating; the HiSP difference was 0.49 MPa1/2; the density of both polymers was 1.25 kg/mol; the blend had the molecular weights: *Mn*52/127, *Mn*52/60, *Mn*52/30, and *Mn*30/30. As an example, *Mn*52/127 implies *Mn*(PBAT) is 52 kg/mol and *Mn*(PLA) is 127 kg/mol, respectively. Details of the calculation are in the Supplementary Material on the sheet "Flory–Huggins".

the absolute temperature; *V* the volume of the system; *ϕ*1 and *ϕ*2 the volume fraction of component, respectively; *δ1* and *δ2* are the solubility parameters; *V*1 and *V*2 are the molar volumes, respectively. Since the densities of PBAT and PLA are very close, for simplicity, we assumed that both polymers had the same density of 1.25 g/mol. Based on this assumption, for PBAT/PLA blends, the volume fraction is equal to the mass fraction.

∆ெ = ∆ெ − ∆ெ (11)

(ଵ − ଶ)ଶ (12)

mol٠K; *T*

ity of two dissimilar components is described by the equation:

*T* is the absolute temperature; ∆*SM* is the entropy of mixing.

**4. Simulation of the Blend Miscibility** 

∆ெ <sup>=</sup> ଵ ଵ

The two different methods show consistency in the total solubility parameter and that PLA was in the range of the one of BA-rich PBAT. Furthermore, the same tendency of the solubility parameter can be seen in both methods:

δ(PBA) < δ(PLA) < δ(alternating PBAT) < δ(PBT)
