## 3 Results and Discussion
### Kinetic Analysis Using the Avrami Equation
Figure 9.0 Degree of cure at different heating rates in the neat (**a**), 4 (**b**), 8 (**c**), and 10 wt% (**d**) systems. The following conclusions can be drawn from the above-described fitting straight line and data. First, \(k^{\prime}\) is a function of the temperature and is proportional to the temperature. Second, as can be seen from Tables 2 and 3, the \(n\) of all the systems shows a trend of increasing first and then decreasing. When the Avrami equation is used to study the curing of thermosetting resins, \(n\) is the scale for the formation and growth of microgel particles and will vary with the curing mechanism. An increase in the value of \(n\) represents the formation and initial growth of the microgel particles. However, in the late stage of curing, since the microgel particles are difficult to form and the growth of the gel particles is limited, the value of \(n\) is lowered. When the prepared BADCy resin nanoparticles were added into the matrix for curing, there were two processes. First, the matrix still underwent the formation and growth of microgel particles. Second, the added nanoparticles reacted with the matrix and appeared to undergo continuous growth in size. According to the results of the curing acceleration, it can be concluded that the reaction between the BADCy resin nanoparticles and the matrix is dominant. It is worth noting that the temperature corresponding to the change in \(n\) is markedly different. The turning points of the neat, 4, 8, and 10 wt% systems are 300, 240, 230, and 220 \({}^{\circ}\)C, respectively. This trend indicates that the mass content of the added BADCy resin nanoparticles is inversely proportional to the gelation time. Therefore, the added BADCy resin nanoparticles are capable of reconstructing the microscopic cross-linked network of the BADCy resin matrix. A difference in the trend of \(n\) means that the curing mechanism may change at the turning point. After all, curing and crystallization are not exactly the same. Although the formation and growth of the microgels during the curing process cannot give relatively clear qualitative information, like polymers' crystallization, the slope of the line is the scale of the Avrami index, \(n\), so it can be considered that this turning point explains the curing process with significant changes in the cross-linked mechanism. In the later stage of the curing process, the viscosity of the resin system shows an extreme increase, and the movement of the polymer segments is limited by the viscosity of the system. Therefore, the growth of microgels in the late curing process is probably affected by diffusion factors. To better illustrate this problem, a model of diffusion control was used for analysis. We assumed that the relationship between the rate constant, \(k(T)\), and temperature, (1/\(T\)), is consistent with the Arrhenius equation:  
Figure 9.1  
Equation (10) can be written as:  
Figure 9.2  
where \(E_{a}\) is the apparent activation energy, \(R\) is the molar gas constant, and \(A\) is a pre-exponential factor. Figure 11 is a plot of ln\(k\) versus 1/\(T\). It can be seen that the Arrhenius fitting plots of all the systems have an obvious turning point. The slope is a measure of the activation energy. It can be concluded that the activation energy in the low temperature stage is higher than that in the high-temperature stage. This indicates that the energy required for the formation and growth of the microgel particles has changed markedly. This phenomenon is common in heterogeneous reactions under diffusion-limited conditions [35; 36]. It further indicates that the diffusion factor is dominant in the late stage of the curing reaction. During the curing process, the quantity of microgels increases, and when a higher temperature is reached, a mass transfer restriction mechanism is established, and the microgels aggregate with each other, so that phase inversion occurs. This point of change is the gel point.