# Modeling of Impact Energy Release of PTFE/Al Reactive Material
## 2 Modeling of PTFE/Al Impact Energy Release
### Parameter Determination of JWL Equation for Unreacted PTFE/Al
\[E_{s}=-\int P_{s}d\overline{v}=\frac{A}{R_{1}}e^{-R_{1}\overline{v}}+\frac{B} {R_{2}}e^{-R_{2}\overline{v}}+\frac{C}{\omega}\overline{v} \tag{2}\]1  
In the formula, \(M_{i}\) is the material parameter and \(f_{i}\) is the mass fraction. Six unknown constant coefficients in the JWL equation of state need to be determined. \(c_{\varnothing}\) was determined by the Hill mixture law, and the other five parameters were determined according to the impact compression line of the PTFE/Al reactive material. Applying the JWL Equation of state (4) to the Hugoniot curve, Equation (12) is obtained:  
\[\left\{ \begin{array}{l}P_{H}=A\Big{(}1-\frac{\omega}{R_{1}^{2}\overline{b}} \Big{)}e^{-R_{1}\overline{v}}+B\Big{(}1-\frac{\omega}{R_{2}^{2}\overline{b}} \Big{)}e^{-R_{2}\overline{v}}+\frac{\omega}{S}E_{H}\\ E_{H}=E_{0}+\frac{1}{2}P_{H}(1-\overline{v})\end{array} \right. \tag{12}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Density (g/cm\({}^{3}\))** & **Sound Velocity (km/s)** & **Specific Heat Capacity (J\(\cdot\)kg\({}^{-1}\cdot\)k\({}^{-1}\))** & **S** \\ \hline
**Al** & 2.78 & 5.35 & \(8.9\times 10^{-4}\) & 1.49 \\
**PTFE** & 2.15 & 1.84 & \(1.05\times 10^{-3}\) & 1.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material density, sound velocity, specific heat capacity, and Hugoniot coefficient of PTFE and Al. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Density (g/cm\({}^{3}\))** & **Sound Velocity (km/s)** & **Specific Heat Capacity (J\(\cdot\)kg\({}^{-1}\cdot\)k\({}^{-1}\))** & **S** \\ \hline
**PTFE/Al** & 2.287 & 2.227 & \(1.002\times 10^{-3}\) & 1.646 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material density, sound velocity, specific heat capacity, and Hugoniot coefficient of PTFE and Al. When the reactive materials do not react, the initial internal energy \(E_{0}\) of the reactive material is 0 (regardless of chemical energy), and Equation (13) is obtained according to Equation (12):  
\[P_{H}=\frac{\left[A\left(1-\frac{\omega}{R_{1}5}\right)e^{-R_{1}5}+B\left(1- \frac{\omega}{R_{2}5}\right)e^{-R_{2}5}\right]}{\left(1+\frac{\omega}{2}-\frac {\omega}{2\omega}\right)} \tag{13}\]  
Parameters \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) in Equation (13) were fitted and determined by the impact compression line of PTFE/Al, and the JWL equation of state when the reactive material does not react was obtained. A genetic algorithm was used to determine five parameters in the JWL equation of state of unreacted PTFE/Al reactive material. The specific calculation method is as follows:  
1. In a specific range, a group of \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) were chosen. 2. In the range of a relative specific volume of 0.6-1, 500 values were taken at equal intervals to calculate the impact pressure of Equations (10) and (13). 3. Calculate the absolute value of the difference between the impact pressure of Equations (10) and (13), and then sum 500 sets of results as the objective function. 4. A genetic algorithm was used to optimize and fit five parameters continuously, and finally, the parameters of the unreacted JWL equation of state were obtained. Figure 1 is the Hugoniot curve and the JWL curve obtained by fitting. It is found that there are some differences between the two curves at low-pressure conditions, but at high-pressure situations, the two curves are in good agreement. The standard deviation of the difference between the two curves under 500 relative specific volumes is 0.0019 Mbar. The five parameters obtained by fitting are recorded in Table 3.