# Reactive Molecular Dynamics Simulations of Polystyrene Pyrolysis
## 2 Results and Discussions
### Tissue Characterization and Carbon Ring Analysis of Pyrolysis Products
#### 2.3.3 Kinetic Analysis of PS Pyrolysis  
After establishing the carbon ring characterization of the pyrolysis organization, we can quantitatively analyze the PS pyrolysis process based on the kinetic theory of the reaction. According to [PERSON]'s work [33], the kinetic reaction model of PE pyrolysis can be derived from reduced time plots (RTPs) with the simulation results and then fit to Arrhenius parameters. Our work aims to simulate the preparation of activated carbon by PS pyrolysis, which is microstructured by irregularly cross-linked graphite layers, so the 6r can be used to characterize the conversion of the activated carbon. Therefore, using isothermal simulation data, we statistically analyzed the 6r evolution at various simulated temperatures with time to obtain a kinetic analysis of PS pyrolysis.  
The kinetic equation of a solid decomposition reaction can be expressed as follows:  
\[\frac{\mathrm{d}\alpha}{\mathrm{d}t}=k(T)f(\alpha)=Aexp(-\frac{E_{a}}{RT})f(\alpha) \tag{1}\]  
where \(\alpha\) is the conversion ratio of solid, \(f(\alpha)\) the kinetic model function, and \(t\), \(T\), \(k(T)\), \(A\), \(E_{a}\), and \(R\) are the simulation time, temperature, rate constant, pre-exponential factor, activation energy, and ideal gas constant, respectively.  
Figure 6: Variation in the carbon rings with time at different constant simulation temperatures. The subplot on the top left enlarges the shaded area.  
The RTP was constructed by plotting \(\alpha\) as a function of a reduced time, \(t=t_{\alpha}\), where \(t_{\alpha}\) is the time needed to attain a specific conversion (\(\alpha\) = 0.9) at an isothermal simulation temperature \(Ti\), and the kinetic equation can be changed to the following integral form:  
\[\frac{\mathrm{d}\alpha}{f(\alpha)}=k(T)dt \tag{2}\]  
\[\int\frac{\mathrm{d}\alpha}{f(\alpha)}=\int k(T)dt \tag{3}\]  
\[G(\alpha)=k(T)t \tag{4}\]  
where \(G(\alpha)\) is the integral form of \(f(\alpha)\). We know that PS pyrolysis at 2500 K, 2750 K, 3000 K, and 3250 K, will produce the same type of activated carbons. Therefore, the model functions listed in Table 1 are used to fit the 6r evolution data at these temperatures (Figure 7a), in which the Mample (first-order) model achieves the highest R-square value.  
Subsequently, a linear fit is applied to determine the values of \(k(T)\) on the \(G(\alpha)\)-t relationship at different simulated temperatures, as shown in Table 2. The activation energy is then calculated based on the ln\(k\)-\(1/T\) fitted curve, as Figure 7b depicts. The obtained R-square value is 0.94, and the simulated value of \(E_{a}\) is 159.74 kJ/mol, close to the experimental pyrolysis results [33].  
\begin{table}
\begin{tabular}{c c c c} \hline
**Reaction Model** & \(f(\alpha)\) & \(G(\alpha)\) & **R-Square** \\ \hline Power law (n = 3/2) & \(2/3\alpha^{-1/2}\) & \(\alpha^{3/2}\) & 0.8207 \\ One-dimensional diffusion & \(1/2\alpha^{-1}\) & \(\alpha^{2}\) & 0.9072 \\ Mample (first-order) & \(1-\alpha\) & \(-\ln(1-\alpha)\) & 0.9659 \\ Contracting sphere & \(3(1-\alpha)^{2/3}\) & \(1-(1-\alpha)^{1/3}\) & 0.8809 \\ Contracting cylinder & \(2(1-\alpha)^{1/2}\) & \(1-(1-\alpha)^{1/2}\) & 0.7959 \\ \hline \end{tabular}
\end{table}
Table 1: Differential and integration form of reaction models fitted for the ReaxFF-MD simulation of PS pyrolysis.  
Figure 7: (a) Theoretical solid decomposition models and RTP obtained from NVT ReaxFF-MD simulations at 2500 K-3250 K. (b) Fitted ln\(k\) vs. inverse temperature obtained from ReaxFF-MD simulations at 2500 K-3250 K.