## 2 Cure Modeling and Heat Generation
### The Compensation Effect to Determine the Pre-Factor  
Determining the pre-factor \(A\) in the Arrhenius equation is independent of the isoconversional method decribed above. If Equation (1) is rearranged the following relationship is obtained:  
\[\ln\left(\frac{1}{f_{k}(\alpha)}\frac{\mathrm{d}\alpha}{\mathrm{d}t}\right)= \ln A_{k}-\frac{E_{k}}{RT} \tag{6}\]  
Here the index \(k\) denotes different kinetic models as for example given in Table 1.  
The time dependency of the conversion follows directly from the DSC data (cf. Equation (2)) and thus is known. For each \(\alpha\), the inverse of the different kinetic functions can simply be calculated. Hence, the left hand side of Equation (6) is known for each time \(t\). Since constant heating rates are assumed, time and temperature correlate linearly with each other. The left hand side of Equation (6) over the inverse temperature needs then to be linearly fitted for each model \(k\). This will yield different sets of \(\ln A_{k}\) and \(E_{k}\). It has to be mentioned that here, and only here, the pre-factor and activation energy are not considered to be dependent on the conversion.  
After sets of \(\ln A_{k}\) and \(E_{k}\) are determined for different models \(k\) (e.g., all the models in Table 1) it was shown [62; 63; 64; 65; 66; 67] that these lie all on a line and the following linear relationship holds true:  
\[\ln A_{k}=a+b\cdot E_{k} \tag{7}\]  
This is the so called compensation effect with the compensation parameters \(a\) and \(b\). The interested reader is refered to the cited sources for a justification of these equations.  
Once the compensation parameters are known the conversion dependent pre-factor can easily be determined by:  
\[\ln A_{\alpha}=a+b\cdot E_{\alpha} \tag{8}\]  
A summary of the above is the following: The baseline corrected heat flow data of all dynamic experiments is taken to calculate the left hand side of Equation (6) for all models given in Table 1. Then these values are fitted over the inverse temperature for \(20\%<\alpha<80\%\) to obtain the sets of \(\ln A_{K}\) and \(E_{k}\). The interval of \(\alpha\) was chosen as proposed by [PERSON] et al. [24]. However, smaller or larger intervals have no significant influence on the values of the compensation parameters.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Reaction Models** & \(\mathbf{f(\alpha)}\) \\ \hline
1 & Power law & \(4\alpha^{3/4}\) \\
2 & Power law & \(2\alpha^{2/3}\) \\
3 & Power law & \(2\alpha^{1/2}\) \\
4 & Power law & \(2/3\alpha^{-1/2}\) \\
5 & 1D diffusion & \(1/2\alpha^{-1}\) \\
6 & [PERSON] (1 st order) & \(1-\alpha\) \\
7 & [PERSON]–[PERSON] & \(4(1-\alpha)[-\ln(1-\alpha)]^{3/4}\) \\
8 & [PERSON]–[PERSON] & \(3(1-\alpha)[-\ln(1-\alpha)]^{2/3}\) \\
9 & [PERSON]–[PERSON] & \(2(1-\alpha)[-\ln(1-\alpha)]^{1/2}\) \\
10 & 3D diffusion & \(3/2(1-\alpha)^{2/3}[1-(1-\alpha)^{1/3}]^{-1}\) \\
11 & Contracting sphere & \(3(1-\alpha)^{2/3}\) \\
12 & Contracting cylinder & \(2(1-\alpha)^{1/2}\) \\
13 & 2D diffusion & \([-\ln(1-\alpha)]^{-1}\) \\
14 & n-th order & \((1-\alpha)^{n}\) \\
15 & Sestak-Berggren & \(\alpha^{m}(1-\alpha)^{n}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Some reaction models (information taken from Table 1 in [27]).  
Finally, as described above, a linear fit over all these sets yields \(a\) and \(b\).