**4. Simulation of the Blend Miscibility**

Based on the solubility parameters, the miscibility of PBAT/PLA blends can be further studied from the aspect of thermodynamics. The thermodynamic criterion of solubility of two dissimilar components is described by the equation:

$$
\Delta G\_M = \Delta H\_M - T\Delta S\_M \tag{11}
$$

where ∆*G<sup>m</sup>* is the free energy of mixing; the ∆*H<sup>M</sup>* is the enthalpy of mixing (heat of mixing); *T* is the absolute temperature; ∆*S<sup>M</sup>* is the entropy of mixing.

A negative value of ∆*G<sup>m</sup>* 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*∆*S<sup>M</sup>* term increases, thus driving *G<sup>M</sup>* to be more negative [8,23]. However, both PBAT and PLA are high molecular weight molecules, implying that the negative contribution from the *T*∆*S<sup>M</sup>* term is small. An equation based on the Flory– Huggins theory correlates the miscibility of a polymer blend with several parameters [24].

$$\frac{\Delta G\_M}{RTV} = \frac{\phi\_1}{V\_1} \ln \phi\_1 + \frac{\phi\_2}{V\_2} \ln \phi\_2 + \frac{\phi\_1 \phi\_2}{RT} (\delta\_1 - \delta\_2)^2 \tag{12}$$

where <sup>∆</sup>*G<sup>M</sup>* is the free enthalpy of mixing; R the gas constant of 8.31 kg·m2/s<sup>2</sup> ·mol·K; *T* the absolute temperature; *V* the volume of the system; *φ*<sup>1</sup> and *φ*<sup>2</sup> the volume fraction of component, respectively; *δ*<sup>1</sup> and *δ* <sup>2</sup> are the solubility parameters; *V*<sup>1</sup> and *V*<sup>2</sup> 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.

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". *Polymers* **2021**, *13*, 2339 7 of 11

would lead to a higher probability of miscibility.

miscibility at 463 K, approx. 190°C, was simulated (Figure 6).

**Figure 5.** Phase diagram of PBAT/PLA blends with various molecular weights at 296 K. **Commented [M11]:** Can you please change **Figure 5.** Phase diagram of PBAT/PLA blends with various molecular weights at 296 K.

From the phase diagram, the curve of the PBAT/PLA blends *Mn*52/127 with the respective molecular weight of 52 and 127 kg/mol is above the zero line, implying blend immiscibility made of the commercially available polymers. A similar trend is shown by the curve *Mn*52/60, representing the blend made of the original PBAT and PLA with molecular weights of 60 kg/mol. When the molecular weight of PLA was reduced to 30 kg/mol, *Mn*52/30 showed an interesting curve: in the region of low PBAT content (<15 wt.%), the value of free energy of mixing was slightly negative; this was followed by the Please if possible change comma in numbers on x axis into dot. **Commented [SS2R1]:** Thank you. I made the two changes. From the phase diagram, the curve of the PBAT/PLA blends *Mn*52/127 with the respective molecular weight of 52 and 127 kg/mol is above the zero line, implying blend immiscibility made of the commercially available polymers. A similar trend is shown by the curve *Mn*52/60, representing the blend made of the original PBAT and PLA with

hyphen (-) into minus sign (−)?

**Commented [M13]:** Can you please change

**Commented [SS4R3]:** Yes. I made this change.

hyphen (-) into minus sign (−)?

**Figure 6.** Phase diagram of PBAT/PLA blends with various molecular weights at 463 K.

Even at this higher temperature, *Mn*52/127 (PBAT/PLA blends consisting of original commercial polymers) showed a curve mostly close to or above the zero line, implying

curve with a PBAT mass fraction of 15–70 wt.%. Starting with a PBAT content of approximately 70 wt.%, the curve was in the negative region again. The PBAT/PLA blends with the molecular weight of 30 kg/mol (*Mn*30/30) showed negative values of the free energy of

PBAT), this curve exhibited a slight increase, while the curve shows even lower negative values at the PBAT/PLA ratio of about 20/80 or 80/20. Generally, a lower negative value

Since PBAT/PLA blends are often melt-blended at an elevated temperature, the blend

molecular weights of 60 kg/mol. When the molecular weight of PLA was reduced to 30 kg/mol, *Mn*52/30 showed an interesting curve: in the region of low PBAT content (<15 wt.%), the value of free energy of mixing was slightly negative; this was followed by the curve with a PBAT mass fraction of 15–70 wt.%. Starting with a PBAT content of approximately 70 wt.%, the curve was in the negative region again. The PBAT/PLA blends with the molecular weight of 30 kg/mol (*Mn*30/30) showed negative values of the free energy of mixing in the whole range. However, it can be seen that in the middle range (40–60 wt.% PBAT), this curve exhibited a slight increase, while the curve shows even lower negative values at the PBAT/PLA ratio of about 20/80 or 80/20. Generally, a lower negative value would lead to a higher probability of miscibility. From the phase diagram, the curve of the PBAT/PLA blends *Mn*52/127 with the respective molecular weight of 52 and 127 kg/mol is above the zero line, implying blend immiscibility made of the commercially available polymers. A similar trend is shown by the curve *Mn*52/60, representing the blend made of the original PBAT and PLA with molecular weights of 60 kg/mol. When the molecular weight of PLA was reduced to 30 kg/mol, *Mn*52/30 showed an interesting curve: in the region of low PBAT content (<15 wt.%), the value of free energy of mixing was slightly negative; this was followed by the curve with a PBAT mass fraction of 15–70 wt.%. Starting with a PBAT content of approximately 70 wt.%, the curve was in the negative region again. The PBAT/PLA blends with the molecular weight of 30 kg/mol (*Mn*30/30) showed negative values of the free energy of mixing in the whole range. However, it can be seen that in the middle range (40–60 wt.% PBAT), this curve exhibited a slight increase, while the curve shows even lower negative values at the PBAT/PLA ratio of about 20/80 or 80/20. Generally, a lower negative value would lead to a higher probability of miscibility. **Commented [M11]:** Can you please change hyphen (-) into minus sign (−)? Please if possible change comma in numbers on x axis into dot. **Commented [SS2R1]:** Thank you. I made the two changes.

Since PBAT/PLA blends are often melt-blended at an elevated temperature, the blend miscibility at 463 K, approx. 190 ◦C, was simulated (Figure 6). Since PBAT/PLA blends are often melt-blended at an elevated temperature, the blend miscibility at 463 K, approx. 190°C, was simulated (Figure 6).

*Polymers* **2021**, *13*, 2339 7 of 11

**Figure 5.** Phase diagram of PBAT/PLA blends with various molecular weights at 296 K.

**Figure 6.** Phase diagram of PBAT/PLA blends with various molecular weights at 463 K. hyphen (-) into minus sign (−)? **Figure 6.** Phase diagram of PBAT/PLA blends with various molecular weights at 463 K.

Even at this higher temperature, *Mn*52/127 (PBAT/PLA blends consisting of original commercial polymers) showed a curve mostly close to or above the zero line, implying **Commented [SS4R3]:** Yes. I made this change. Even at this higher temperature, *Mn*52/127 (PBAT/PLA blends consisting of original commercial polymers) showed a curve mostly close to or above the zero line, implying poor miscibility. With the decreased molecular weights of PLA, the curve of *Mn*52/60 displays a maximum slightly above the zero line (when PBAT content is approx. 50–60 wt.%) and two minima slightly below the zero line (when PBAT content is around 10 wt.% or 90 wt.%), indicating a relatively unstable state of miscibility. A small fluctuation of the temperature or composition may change the theoretical miscibility. When the molecular weights of PLA were 30 kg/mol, *Mn*52/30 blends showed negative values of the free energy of mixing in the whole range, indicating the blend should be miscible at 463 K. The blends with an *M<sup>n</sup>* of 30 kg/mol for each component demonstrated even lower values, indicating thermodynamically miscibility in the molten state at 463 K.

**Commented [M13]:** Can you please change

It is concluded that the mixing behavior depends strongly on the molecular weights of both components, their ratios, and the temperature. As shown in both phase diagrams, the miscibility of PBAT/PLA generally increases with decreasing molecular weights. At 296 K, the blends *Mn*52/127, *Mn*52/60, and *Mn*52/30 each showed a maximum positive value of ∆*G<sup>M</sup>* in the middle range (PBAT content: approx. 50 wt.%) in the curves, indicating poor miscibility of these blends, especially at the blend ratio of 50/50. Generally, in addition to the negative value of ∆*GM*, the second derivative of ∆*G<sup>M</sup>* with respect to the volume fraction of the second blend components was a necessary condition of the blend miscibility. At 296 K, the blends *Mn*30/30 showed negative values of ∆*G<sup>M</sup>* in the whole range of compositions while displaying two minima (at the ratio of about 10/90 and 90/10) and one maximum (at the ratio of 50/50). According to the second derivative of ∆*G<sup>M</sup>* (eq.12 = 0, at 30% volume fraction of PBAT), spinodal decomposition can occur at 296 K due to the negative value of free energy of mixing but beginning of the convex curve of the spinodal (Figure 7).

thermodynamically miscibility in the molten state at 463 K.

**Figure 7.** Spinodal curves for PBAT/PLA blends. **Figure 7.** Spinodal curves for PBAT/PLA blends.

(Figure 7).

At 463 K (e.g., for melt-blending), the curve of the blends *Mn*30/30 shows negative values in the entire range of compositions indicating good miscibility. Even the blends *Mn*52/30 would be miscible in the molten state while melt-blending at 463 K. After that, if the PBAT/PLA blends *Mn*52/30 were stored at a lower temperature between −28 °C and 61 °C (corresponding to the *Tg* values of both polymers) for a sufficiently long time, the PBAT chains would behave like rubber. For this reason, the blends *Mn*52/30 could change the miscibility from miscible in the molten state to partially miscible or immiscible at a lower temperature after a long enough time. However, the PBAT/PLA with an *Mn* of 30 kg/mol for each component should be miscible both at 296 K and 463 K, according to the simulation. At 463 K (e.g., for melt-blending), the curve of the blends *Mn*30/30 shows negative values in the entire range of compositions indicating good miscibility. Even the blends *Mn*52/30 would be miscible in the molten state while melt-blending at 463 K. After that, if the PBAT/PLA blends *Mn*52/30 were stored at a lower temperature between −28 ◦C and 61 ◦C (corresponding to the *T<sup>g</sup>* values of both polymers) for a sufficiently long time, the PBAT chains would behave like rubber. For this reason, the blends *Mn*52/30 could change the miscibility from miscible in the molten state to partially miscible or immiscible at a lower temperature after a long enough time. However, the PBAT/PLA with an *M<sup>n</sup>* of 30 kg/mol for each component should be miscible both at 296 K and 463 K, according to the simulation.

poor miscibility. With the decreased molecular weights of PLA, the curve of *Mn*52/60 displays a maximum slightly above the zero line (when PBAT content is approx. 50–60 wt.%) and two minima slightly below the zero line (when PBAT content is around 10 wt.% or 90 wt.%), indicating a relatively unstable state of miscibility. A small fluctuation of the temperature or composition may change the theoretical miscibility. When the molecular weights of PLA were 30 kg/mol, *Mn*52/30 blends showed negative values of the free energy of mixing in the whole range, indicating the blend should be miscible at 463 K. The blends with an *Mn* of 30 kg/mol for each component demonstrated even lower values, indicating

It is concluded that the mixing behavior depends strongly on the molecular weights of both components, their ratios, and the temperature. As shown in both phase diagrams, the miscibility of PBAT/PLA generally increases with decreasing molecular weights. At 296 K, the blends *Mn*52/127, *Mn*52/60, and *Mn*52/30 each showed a maximum positive value of ∆*GM* in the middle range (PBAT content: approx. 50 wt.%) in the curves, indicating poor miscibility of these blends, especially at the blend ratio of 50/50. Generally, in addition to the negative value of ∆*GM*, the second derivative of ∆*GM* with respect to the volume fraction of the second blend components was a necessary condition of the blend miscibility. At 296 K, the blends *Mn*30/30 showed negative values of ∆*GM* in the whole range of compositions while displaying two minima (at the ratio of about 10/90 and 90/10) and one maximum (at the ratio of 50/50). According to the second derivative of ∆*GM* (eq.12 = 0, at 30% volume fraction of PBAT), spinodal decomposition can occur at 296 K due to the negative value of free energy of mixing but beginning of the convex curve of the spinodal
