## 3 Results and Discussion
### Isothermal Crystallization and Kinetic Analysis
#### 3.3.4 Analysis of Avrami Plots  
The linearity of the Avrami plots for neat P3 HBHHx and the nucleated samples across all tested isothermal temperatures confirms that the crystallization behavior follows the Avrami model with a good approximation (Table S3). The slope of each plot corresponds to the Avrami exponent (\(n\)), which provides insights into the nucleation mechanism and the dimensionality of crystal growth.  
For neat P3 HBHHx (Figure 13a), \(n\) ranged from 2.10 at 70 \({}^{\circ}\)C to 2.75 at 95 \({}^{\circ}\)C (Table 7), suggesting three-dimensional spherulitic growth with some degree of spatial confinement. These values are consistent with a combination of instantaneous and homogeneous nucleation, which is typically observed in systems with limited nucleation sites. The increase  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{\(T_{\text{c}}\) (\({}^{\circ}\)C)} & \multirow{2}{*}{\(n\)} & \multicolumn{2}{c}{\(Z\times 10^{6}\)} & \multirow{2}{*}{\(k\times 10^{3}\) (\(\text{s}^{-1}\))} & \multirow{2}{*}{\(\tau_{\text{(1/2)}}\) (s)} \\  & & & (\(\text{s}^{-n}\)) & & Avrami & Avrami \\ \hline \multirow{4}{*}{P3 HBHHx} & 70 & 2.10 & 16.11 & 5.26 & 160 \\  & 80 & 2.34 & 0.890 & 2.63 & 326 \\  & 85 & 2.54 & 0.046 & 1.29 & 670 \\  & 95 & 2.75 & 0.002 & 0.64 & 1366 \\ \hline \multirow{4}{*}{PHH\_05 BN} & 100 & 2.39 & 11.98 & 8.76 & 98 \\  & 105 & 2.20 & 8.38 & 4.98 & 170 \\  & 110 & 2.39 & 0.87 & 2.88 & 298 \\ \hline \multirow{4}{*}{PHH\_05 PHB} & 90 & 2.29 & 5.76 & 5.15 & 165 \\  & 95 & 2.10 & 11.23 & 4.43 & 189 \\  & 100 & 2.12 & 5.79 & 3.43 & 246 \\ \hline \multirow{4}{*}{PHH\_2 Talc} & 95 & 2.32 & 8.42 & 6.45 & 132 \\  & 100 & 2.68 & 0.27 & 3.54 & 246 \\ \cline{1-1}  & 105 & 2.88 & 0.02 & 2.04 & 431 \\ \hline \end{tabular}
\end{table}
Table 7: Isothermal crystallization kinetic parameters deduced from the Avrami equation for P3 HBHHx and the corresponding nucleating agents.  
Figure 13: Avrami plots from various isothermal crystallization temperatures for (**a**) P3 HBHHx, (**b**) PHH_05 BN, (**c**) PHH_05 PHB, and (**d**) PHH_2 Talc.  
in \(n\) with temperature suggests reduced spatial confinement, as fewer active nuclei are formed at higher temperatures, allowing larger spherulites to grow before impingement.  
In the case of PHH_05 BN (Figure 13b), \(n\) values remained relatively stable between 2.20 and 2.39 (Table 7) across the tested range (100-110 \({}^{\circ}\)C), suggesting a combination of instantaneous homogeneous nucleation and additional heterogeneous nucleation induced by the additive. The steeper slopes of these plots compared to those of the neat polymer reflect a higher rate constant (\(Z\)), confirming enhanced crystallization kinetics.  
PHB (PHH_05 PHB, Figure 13c) and talc (PHH_2 Talc, Figure 13d) also improved the crystallization behavior of P3 HBHHx. The Avrami exponent (\(n\)) ranged from 2.10 to 2.29 for PHH_05 PHB and 2.32 to 2.88 for PHH_2 talc (Table 7), indicating that the overall crystallization mechanism remained consistent with that of neat polymer. In both cases, the crystallization rate constant was significantly higher than that of neat P3 HBHHx, reflecting an improved nucleation efficiency.  
The crystallization rate constants followed the trend BN > PHB \(\approx\) Talc > neat P3 HBHHx. This observation is consistent with previous results from half-time and relative crystallinity analyses, reinforcing the conclusion that BN is the most effective nucleating agent for enhancing the crystallization of P3 HBHHx.  
Figure 14 illustrates the validation of the Avrami model by comparing the experimental crystallization rates and half-crystallization times obtained from DSC measurements with the theoretical values predicted by the Avrami equation. The strong correlation between the experimental and calculated data confirms that the crystallization behavior of both neat and nucleated P3 HBHHx closely follows the Avrami model, with minimal deviations.