## 3 Results and Discussion
### Kinetic Analysis Using the Avrami Equation  
The prepared BADCy resin nanoparticles were added into the BADCy prepolymer at contents of neat, 4, 8, and 10 wt%. The nanoparticles were dispersed evenly in the matrix under high-sheer conditions. The DSC characterization was performed based on the following assumptions. First, the nanoparticles do not react with the matrix during the mixing. Second, the absolute area under the curve is proportional to the degree of cure, and \(\alpha\) is defined as the degree of cure at any time, \(t\):  
\[\alpha=\frac{\Delta H}{\Delta H_{0}}, \tag{4}\]  
where \(\Delta H\) is the enthalpy changes at any points, and \(\Delta H_{0}\) is the total enthalpy change during the heating process. Representative DSC curves are shown in Figure 8, in which (a), (b), (c), and (d) represent the neat, 4, 8, and 10 wt% system, respectively. It can be seen that the curing temperature is proportional to the rate of temperature increase for the same system. Also, the onset curing temperature decreases as the BADCy resin nanoparticle content increases. Figure 9 is obtained from the integration of Figure 8 and the curves of the degree of cure as a function of temperature can be obtained correspondingly in Figure 9. It shows that the degree of cure is inversely proportional to the rate of temperature increase for the same system. This is because a fast rate of temperature rise results in an incomplete curing reaction. At the same temperature, the degree of cure is proportional to the content of nanoparticles. This conclusion again demonstrates that the BADCy resin nanoparticles have a promoting effect on the matrix. Figure 7: Schematic illustration of free energy changes with size in the gel particle formation and growth stages. Under thermodynamic control, the final structures are in the state of minimum free energy. If kinetic control intervenes, the structure can be trapped in a metastable state (gel). The Avrami equation is:  
\[\alpha=1-\exp(-k^{n}), \tag{5}\]  
where \(\alpha\) is the relative crystallinity, \(n\) is the Avrami index reflecting the nucleation and growth mechanism, and \(k\) is the rate constant. [PERSON] improved the Avrami equation to enable it to be used for non-isothermal crystallization kinetics [34]. Assuming that non-isothermal crystallization consists of innumerable tiny isothermal crystallization steps, the Avrami equation can be modified to:  
\[\alpha=1-\exp(-k(T)/\phi^{m}), \tag{6}\]  
or  
\[\ln[-\ln(1-\alpha)]=\ln k(T)-m\ln\phi, \tag{7}\]  
where the rate constant, \(k\), is a function of \(T\), \(\phi\) is the heating rate, and \(m\) is the Ozawa index. Figure 8: Dynamic DSC curves at different heating rates of neat (**a**), 4 (**b**), 8 (**c**), and 10 wt% (**d**). When Equation (7) is used to describe the curing process of thermosetting resins, \(\alpha\) is the relative degree of cure, \(k\) is the curing rate constant, and \(n\) is the relative index describing the curing mechanism. At a given temperature, \(n\) is a constant. Equation (6) can be written as:  
\[1-\alpha=\exp(-k^{\prime}R^{n}), \tag{8}\]  
where \(\alpha\) is a function of temperature, \(R\) is the heating rate, and \(k^{\prime}\) is a function of temperature. Equation (8) also can be written as:  
\[\log[-\ln(1-\alpha)]=\log k^{\prime}-n\log R. \tag{9}\]  
Equation (9) shows that the curing kinetic parameters, \(k^{\prime}\) and \(n\), can be determined according to the degrees of curing at different heating rates. At a given temperature, a plot of \(\log[-\ln(1-\alpha)]\) on \(\log R\) should be a straight line with a slope of \(-n\) and an intercept of \(\log k^{\prime}\). Figure 10 is a graph made according to Equation (9) and Figure 9. The curves show an excellent linear relationship over the curing temperature range, which is consistent with the conclusions obtained from Equation (9). The result indicates that the modified Avrami equation can be used to describe the non-isothermal curing process of the BADCy resin systems. A series of rate constants and indices are listed in Tables 2 and 3.