PLA: 4032D, molecular weight 130,000, Nature Works, United States of America.

Twin-screw extruder: CTE35 PLUS, Nanjing Kebelong Machinery Co., Ltd.

Vacuum drying oven: DHG101, Shanghai Rundai Automation Equipment Co., Ltd.

Extrusion cast machine: FDHU35, Guangzhou Putong Experimental Analytical Instruments Co., Ltd.

Biaxial tensile testing machine: KARO5, Bruckner, Germany.

The PLA casting films need to be prepared by the extrusion casting method before biaxial stretching. The casting temperature was set to 185–200°C and the screw speed was 20 r/min. The film thickness was controlled at 300–400 μm by adjusting the extrusion speed and the casting roller speed, then the film was cooled on the casting roll at 30°C. After the PLA casting films were obtained, the biaxial stretching experiment was carried out by using the biaxial stretching machine (KARO-5, Bruckner), which can collect the experimental data of load displacement during stretching. In this study, we mainly investigated the effect of the stretching ratio on the relaxation behavior of PLA during the biaxial stretching heat-setting stage. Thus, the stretching ratio was set as a variable in the experimental process, and the other experimental parameters were fixed. Based on this principle, BOPLA films with different ratios of 2 × 2, 3 × 3, 4 × 4, and 5 × 5 were prepared under the conditions of preheating for 30 s, stretching rate of 40 mm/s, and stretching temperature and heat setting temperature of 130°C. To ensure the repeatability of the experimental data, four repeated stretching were performed under each of the above conditions. The stretching diagram is shown in Figure 1, in which MD and TD represent the machine and transverse stretching directions, respectively.

The representative load-time curves with different stretching ratios are shown in Figure 2. It can be seen that the trend of load-time curves in MD and TD directions with different stretching ratios are basically the same. Upon reaching the yield point, the material begins to soften and the load drops briefly, and then, different degrees of rebound (increase in load) appear. The above trends become more apparent with the increase in the stretching ratio. It is worth noting that the load in the TD direction is higher than that in the MD direction with these four stretching ratios. This phenomenon may be related to the different crystal orientations in TD and MD directions. That is, the degree of crystal orientation in the TD direction of PLA films is higher than that in the MD direction during biaxial stretching. So, the mechanical properties of the TD direction are better than those of the MD direction. Meanwhile, it can also be seen that the difference in load between the TD direction and the MD direction shows an increasing trend with the increase in the stretching ratio, indicating that the effect of the stretching ratio on the crystal orientation in the TD direction is greater than that in the MD direction. In addition, when entering the heat setting stage, all four stretching curves show significant relaxation. Moreover, the relaxation amplitude exhibits an increasing trend with the increase in the stretching ratio.

To further study the rheological mechanism of the film during biaxial stretching, the load-displacement (time) relationship was transformed into the stress-strain (time) relationship. In the schematic diagram of biaxial stretching in Figure 1, we set L _M0, d _T0, and h ₀ to be the length, width, and height of the film at the beginning of stretching, respectively. F _M and F _T represent the machine and transverse stretching forces. V _M and V _T are the machine and transverse stretching rates. The volume of the film before stretching is expressed as: V 0 = L M 0 × d T 0 × h 0 (1)

The machine and transverse lengths in the stretching process can be expressed as: L M = L M 0 × 1 + V M × t (2) d T = d T 0 × 1 + V T × t (3)

So, the area of the film during stretching is expressed as: S = L M × d T = L M 0 × 1 + V M × t × d T 0 × 1 + V T × t (4)

For the convenience of calculation, we assume that the volume of the film remains unchanged during stretching, so the thickness during stretching can be expressed as: h = V 0 / S = L M 0 × d T 0 × h 0 / L M 0 × 1 + V M × t × d T 0 × 1 + V T × t = h 0 / 1 + V M × t × 1 + V T × t (5)

The machine cross-sectional area of the film during the stretching process can be written as: S M = d T × h = d T 0 × 1 + V T × t × h 0 / 1 + V M × t × 1 + V T × t = d T 0 × h 0 / 1 + V M × t (6)

Then, the machine stretching stress can be obtained: σ M = F M / S M = F M × 1 + V M × t / d T 0 × h 0 (7)

Similarly, the transverse stretching stress can be expressed as: σ T = F T / S T = F T × 1 + V T × t / L M 0 × h 0 (8)

Using Eqs 7, 8 to calculate the load-time data in Figure 2 and the obtained stress-strain curves during biaxial stretching as shown in Figure 3, it can be seen that the changing trend of the stress-time curve is basically consistent with the load-time curve. In the heat setting stage, with the increase in the stretching ratio, the difference between the stress curves of MD and TD directions gradually increases, and stress relaxation amplitude also increases. The reason for the above phenomenon may be that the large ratio of biaxial stretching accelerates the movement of molecular chains inside the film (Rhee and White, 2002), which leads to a significant change in macroscopic mechanical behavior, but the mechanism needs to be further investigated.

To further analyze the rheological behavior during the biaxial stretching process, the stress-time curve of the heat setting stage was extracted separately for analysis, as shown in Figure 4. It can be seen that the stress-time curves with different stretching ratios all decrease sharply at the initial stage of setting, and stress relaxation gradually tends to be gentle with the increase in time, indicating that the film has reached the setting effect within the preset setting time (Salem and Vasanthan, 2009). Meanwhile, the initial stress values of TD and MD both exhibit an increasing trend with the increase in the stretching ratio, and the difference between the two increases gradually. This indicates that the effect of large ratio stretching on the TD direction is more significant.

In order to further analyze the rheological mechanism of the film during the heat setting stage, the stress relaxation curves with different stretching ratios were fitted by the Kohlrausch-Willianms-Watts (KWW) equation (Wang et al., 2014). The expression of the KWW equation is as follows: the stress-time data of different relaxation curves is inputted and the corresponding σ ₀ is substituted for fitting analysis, so the corresponding β can be obtained. The only difference between the fitting of these four sets of curves is the inconsistency in the original stress-time data input under various stretching ratios. σ = σ 0 ⁡ exp − t τ c β (9)

Where σ ₀ is the initial stress, and τ _c is the characteristic relaxation time (Aval et al., 2020). The value of the extended exponential parameter β can reflect the dynamic inhomogeneity of the stress relaxation in the film, that is, the smaller β means the more significant dynamic inhomogeneity (Ediger, 2000; Zhao et al., 2015; Duan et al., 2022). The fitting curves for different stretching ratios are shown in Figure 5.

The obtained values of β by using the KWW formula to fit the stress relaxation curves at different stretching ratios are shown in Figure 6. It can be seen that the β in the MD and TD directions shows an increasing trend with the increase in the stretching ratio. Moreover, the values of β in the MD direction are persistently greater than in the TD direction. It indicates that the films become more homogeneous with the stretching ratios increase in the stretching process, and the deformation in the MD direction is more uniform than that in the TD direction, which may be ascribed to the large ratio of stretching that promotes the crystalline orientation of the molecular chains inside the film. As a result, the amorphous region is reduced, resulting in more uniform deformation.

In order to quantitatively analyze the rheological behavior of the PLA film, the flow unit model is introduced, in which the microstructure of the material is composed of hard and soft regions, and the soft is mixed with the hard (Wang et al., 2014). The hard region has a compact molecular structure, while the soft region consists of loosely arranged molecules. As the flow unit, the soft region undertakes the viscoelastic and viscoplastic deformation process of the material. Then, the activation rate of the rheological unit can be defined as n = 1-σ _r/σ0 (Wang et al., 2014), where σr is the equilibrium stress. The values of n under different stretching ratios are shown in Figure 7. It can be seen that n decreases with the increase in the stretching ratio, indicating that the crystallization orientation is promoted and the movement of the molecular chain is inhabited in the film at a larger stretching ratio, which corresponds to the more difficult condition of the transition from the hard region to the soft region in the microstructure of the film. As a result, the proportion of the hard region in which the energy is more stable is larger, and the internal structure of the film reaches a more stable state (Wang et al., 2014). Reference to the result of β is presented in Figure 6. It can be inferred that the activation rate decreasing will further result in a smaller dynamic inhomogeneity of the film.

In practice, the relaxation curves can exhibit a logarithmic variation of stress with time or a non-logarithmic one. The former case has been reported for a range of materials and deformation conditions and corresponds to a relaxation curve of the following equation (Wang et al., 2014): σ t = σ 0 + − k B T V a ln 1 + t C r (10)

The relaxation curves at different stretching ratios were fitted and analyzed by the above formula to further calculate the rheological unit volume. Where K _B is Boltzmann constant, τ 0 is the initial stress and C _r is a time-related constant. This is different from the free volume which was generated by the space gap between the crystal structure, grain shape, and grain size of the material. The apparent activation volume V _a represents the average volume of the rheological unit and is fully determined by knowing relaxation curves. The fitting curves under different stretching ratios are shown in Figure 8.

The volumes of the rheological unit which were obtained by fitting the stress relaxation curves under different stretching ratios are shown in Figure 9. As can be seen, the volume of the rheological unit decreases with the increase in the stretching ratio, which is consistent with the trend of activation rate. Combined with the dynamic inhomogeneity during the stress relaxation, it can be inferred that the rheological units of PLA films were mainly transformed from the amorphous region in the internal microstructure, meaning the higher crystallinity fraction with increasing stretching ratios leads to fewer rheological units This rule is consistent with the experimental result of characterizing the free volume by electron annihilation (Jia et al., 2015). Nevertheless, the internal physical mechanism needs to be further investigated.

The deformation of the film during biaxial stretching involves the local motion of the flow unit, which needs to cross the corresponding energy barrier to be activated (Gibbs et al., 1983). Due to the change, the activation energy will control the proportion of the rheological units. Thus, the volume of rheological units is also affected by the activation energy. In order to understand the connection between the activation energy and the volume of rheological units during the stress relaxation process of the film, the activation energy spectrum should be further analyzed. The flow units and activation energy in the film are widely distributed during biaxial stretching. The evolution of inelastic strain energy with time can be described by the activation energy spectrum (AES) (Gibbs et al., 1983). The variation of relaxation stress on time can be expressed by an integral equation: Δ σ t = ∫ 0 + ∞ P E θ E , T , t d E (11)

Where P(E) is the distribution intensity of the activation energy. The θ(E, T, t) is a characteristic function; it can be expressed as follow: θ E , T , t = 1 − exp − t τ = 1 − exp − v 0 t ⁡ exp E K B T (12)

Where v0 is an effective attack frequency of the order of the Debye frequency or less. During the heat setting process, the deformation with the characteristic relaxation time less than the experimental time is involved in stress relaxation. On the contrary, the deformation that the characteristic relaxation time is greater than the experimental time does not participate in the relaxation (Zhao et al., 2015). Correspondingly, the flow unit has a critical energy activation. When the activation energy of the flow unit is less than the critical activation energy, it will be activated to participate in the relaxation process; otherwise, it will not. In the frame of step-like approximation, the distribution intensity of the activation energy can be expressed as follows (Qiao et al., 2016): P E = − 1 k B T d σ t d ⁡ ln ⁡ t (13)

Where σ(t) is the time function of stress, and T is the temperature. The critical activation energy is calculated by the Arrhenius equation (Yang et al., 2021). E = K B T ⁡ ln v 0 t (14)

Based on Eqs. 13, 14, the activation energy spectrum during stress relaxation can be calculated. It should be pointed out that it is necessary to fit the relaxation stress-time curve with the power law formula before calculating the distribution intensity of the activation energy. The fitting curves are shown in Figure 10. Then, the activation energy spectrum is calculated based on the fitting curves.

According to Eqs. 13, 14, the obtained apparent activation energy spectra are presented in Figure 11. It can be seen that the shape of activation energy spectra is close to the Gaussian distribution, and the peak of the spectrum increases with the increase in the stretching ratio. In addition, the symmetry axis of the peak shifted toward the high energy value with the increase in the stretching ratio, indicating that more rheological units with higher energy barriers are activated during inelastic deformation at elevated stretching ratios. The movement of the rheological unit is limited which further results in a decrease in the activation rate. Meanwhile, the proportion of rheological units that can be activated is less, which means that the range of the soft region will gradually be smaller. Reference to the result of V _a is presented in Figure 9 and the value of β is presented in Figure 6. It can be inferred that the decrease in the rheological unit volume and the weaker dynamic inhomogeneity may be related to the high activation energy. This result further confirms that the high activation energy is related to the high crystallinity of the film.