## 3 Results and Discussion
### Spherulitic Morphology and Crystal Growth of P3 HBHHx Samples
#### 3.4.5 Kinetic Analysis Based on the Lauritzen\(-\)Hoffman Model  
The crystal growth rate (\(G\)) of the neat P3 HBHHs and nucleated samples were simulated using the Lauritzen\(-\)Hoffman (LH) theory, which describes polymer crystallization kinetics by considering secondary nucleation and molecular transport as rate-determining steps [48; 49; 50]. This model provides insights into the activation energy and nucleation barrier that control the crystallization process. The Lauritzen\(-\)Hoffman equation expresses the growth rate \(G\) as follows:  
\[G=G_{0}\times\exp\left[-\frac{U^{*}}{R(T_{\rm c}-T_{\infty})}\right]\times\exp \left[-\frac{K_{\rm g}}{T_{\rm c}(\Delta T)f}\right] \tag{6}\]  
where \(G_{0}\) is the pre-exponential factor related to molecular mobility; \(U^{*}\) is the activation energy for segmental motion above the glass transition temperature; \(R\) is the universal gas constant; \(T_{\rm c}\) is the crystallization temperature; \(T_{\infty}\) is the reference temperature, typically approximated as \(T_{\rm g}-30\) K; \(K_{\rm g}\) is the nucleation constant, which accounts for the energy barrier of secondary nucleation; \(\Delta T=T_{\rm m}^{\circ}-T_{\rm c}\) is the supercooling (the difference between the equilibrium melting temperature \(T_{\rm m}^{\circ}\) and the crystallization temperature \(T_{\rm c}\)); \(f\) is a correction factor for temperature dependence, given by:  
\[f=\frac{2T_{\rm c}}{T_{\rm m}^{\circ}+T_{\rm c}} \tag{7}\]  
To determine \(K_{\rm g}\) and validate the applicability of the LH model, the experimental spherulitic growth rates obtained from POM observations were analyzed. The data were plotted according to the LH equation as follows:  
\[\ln G+\frac{U^{*}}{R(T_{\rm c}-T_{\infty})}\ {\rm versus}\ \frac{1}{T_{\rm c}( \Delta T)f} \tag{8}\]  
In such calculations, it is common to adopt the \"universal\" values reported by [PERSON] and [PERSON] [51], namely \(U^{*}\) = 1500 cal/mol and \(T_{\infty}=T_{\rm g}-30\) K. At low supercooling conditions (i.e., when \(\Delta T\) is small), crystallization kinetics are primarily governed by the nucleation term. As a result, the overall growth rate becomes relatively insensitive to variations in the \(U^{*}\) and \(T_{\infty}\), allowing for reliable fitting of the Lauritzen\(-\)Hoffman (LH) model using these standard values. The linearized form of the LH equation follows the relation \(y=mx+b\), where \(y=\ln G+\frac{U^{*}}{R(T_{\rm c}-T_{\infty})}\) and \(x=\frac{1}{T_{\rm c}(\Delta T)f}\). The slope \(m\) corresponds to the negative secondary nucleation constant \(-K_{\rm g}\), and \(b\) represents the intercept associated with the pre-exponential growth terms. The results are presented in Table 9.  
As shown in Figure 19, the LH plots for all samples exhibited strong linearity, with correlation coefficients (\(R^{2}\)) exceeding 0.99. This high degree of linear correlation supports the applicability of the LH model in describing the crystallization kinetics of both neat and nucleated P3 HBHHs. Two distinct slopes can be observed in Figure 19: one corresponding to neat P3 HBHHx and the other to the nucleated samples. This distinction suggests that crystallization proceeded in different kinetic regimes: regime III for the neat polymer and regime II for the nucleated samples due to the different temperature ranges used in each case. In regime III, which is typical of lower crystallization temperatures, crystal growth is governed by isolated nucleation events that result in higher \(K_{\rm g}\) values. Conversely, in regime II, which occurs at higher crystallization temperatures, crystal growth is characterized by prolific multiple nucleations with a niche separation similar to the stem width, leading to lower \(K_{\rm g}\) values. For neat P3 HBHHx, the highest \(K_{\rm g}\) value was obtained (4.10 \(\times\) 10\({}^{5}\) K\({}^{2}\)), consistent with its crystallization occurring at lower temperatures within regime III.