Coupled Effects of Electric Field and Moisture in Dielectric Breakdown: A Non-Equilibrium Thermodynamics Perspective

Introduction: This work investigates the interplay between electric field intensity and moisture in the dielectric breakdown of insulating materials. The objective is to develop a thermodynamic framework that captures the irreversible processes involved, highlighting the impact of moisture on the formation of conductive paths and system degradation. Methods: A phenomenological model is proposed, combining the Dielectric Breakdown Model (DBM) with principles from linear non-equilibrium thermodynamics. Coupled fluxes and forces are formulated using Onsager relations, and entropy production is evaluated under isothermal conditions. results: Figure~\ref{fig2} shows the dependence of the number of branches on the electric field intensity, illustrating the fractal-like growth behavior of the electric tree structure \cite{Niemeyer1984, Kudo1998}. This behavior is consistent with previous studies, which have established that higher electric fields reduce the number of active branches, leading to a more localized and intense dielectric breakdown. \begin{figure}[h] \centering \includegraphics[width=0.7\textwidth]{Figure 2 600dpi.jpg} \caption{Number of branches vs. electric field intensity, illustrating the fractal-like growth behavior.} \label{fig2} \end{figure} By integrating the linear phenomenological equations with the stochastic DBM framework, we obtain explicit expressions for the moisture and electric current fluxes: \begin{equation} J_{\mu} = T D \left(\frac{1}{T Nb}\right) + L_{\mu\phi} \left(\frac{E}{T}\right), \end{equation} \begin{equation} J_{\phi} = L_{\phi\mu} \left(\frac{1}{T Nb}\right) + T \kappa \left(\frac{E}{T}\right), \end{equation} where $Nb = C Br$ represents the effective number of moisture-conducting branches as predicted by the DBM model. The corresponding entropy production for the coupled system reads: \begin{equation} \sigma = T D \left(\frac{1}{T Nb}\right)^2 + T \kappa \left(\frac{E}{T}\right)^2 + 2 L_{\mu\phi} \left(\frac{E}{T}\right)\left(\frac{1}{T Nb}\right). \end{equation} Figure~\ref{fig3} compares the entropy production as a function of the electric field intensity in two scenarios: with and without coupling. \begin{figure}[h] \centering \includegraphics[width=0.7\textwidth]{Figure 3 600dpi.jpg} \caption{Normalized entropy production $\sigma$ vs. electric field intensity $E$: (a) no coupling ($L_{\mu\phi}=0$), (b) with coupling ($L_{\mu\phi} \neq 0$).} \label{fig3} \end{figure} As seen in Figure~\ref{fig3}, the entropy production increases significantly in the presence of moisture, indicating a higher degree of irreversibility in the system. This suggests that moisture not only facilitates conductive path formation, but actively enhances thermodynamic irreversibility through coupled flux mechanisms. Figure ~\ref{fig4} presents the dependency of entropy production on the electric field for various values of the coupling coefficient $L_{\mu\phi}$ . As $L_{\mu\phi}$ increases, the entropy production curve becomes significantly steeper. This behavior underscores the sensitivity of the system to even moderate moisture-electric coupling. Physically, it implies that materials with higher affinity between ionic/moisture fluxes and electric field exhibit accelerated degradation. It also highlights the potential of $L_{\mu\phi}$ as a critical design parameter in insulation systems. \begin{figure}[h] \centering \includegraphics[width=0.7\textwidth]{Figure 4 600dpi.jpg} \caption{Entropy production $\sigma$ vs. electric field intensity $E$ for different values of $L_{\mu\phi}$.} \label{fig4} \end{figure} These findings extend our earlier results, which connected the fractal analysis of tree growth with entropy production metrics in non-equilibrium systems \cite{Enciso2019, Razzitte2020}. Results: The model shows that moisture significantly enhances entropy production, especially when the coupling coefficient between ionic flux and electric field is non-zero. Numerical simulations demonstrate that entropy production increases with both electric field intensity and coupling strength, accelerating degradation pathways. discussion: Figure~\ref{fig2} shows the dependence of the number of branches on the electric field intensity, illustrating the fractal-like growth behavior of the electric tree structure \cite{Niemeyer1984, Kudo1998}. This behavior is consistent with previous studies, which have established that higher electric fields reduce the number of active branches, leading to a more localized and intense dielectric breakdown. \begin{figure}[h] \centering \includegraphics[width=0.7\textwidth]{Figure 2 600dpi.jpg} \caption{Number of branches vs. electric field intensity, illustrating the fractal-like growth behavior.} \label{fig2} \end{figure} By integrating the linear phenomenological equations with the stochastic DBM framework, we obtain explicit expressions for the moisture and electric current fluxes: \begin{equation} J_{\mu} = T D \left(\frac{1}{T Nb}\right) + L_{\mu\phi} \left(\frac{E}{T}\right), \end{equation} \begin{equation} J_{\phi} = L_{\phi\mu} \left(\frac{1}{T Nb}\right) + T \kappa \left(\frac{E}{T}\right), \end{equation} where $Nb = C Br$ represents the effective number of moisture-conducting branches as predicted by the DBM model. The corresponding entropy production for the coupled system reads: \begin{equation} \sigma = T D \left(\frac{1}{T Nb}\right)^2 + T \kappa \left(\frac{E}{T}\right)^2 + 2 L_{\mu\phi} \left(\frac{E}{T}\right)\left(\frac{1}{T Nb}\right). \end{equation} Figure~\ref{fig3} compares the entropy production as a function of the electric field intensity in two scenarios: with and without coupling. \begin{figure}[h] \centering \includegraphics[width=0.7\textwidth]{Figure 3 600dpi.jpg} \caption{Normalized entropy production $\sigma$ vs. electric field intensity $E$: (a) no coupling ($L_{\mu\phi}=0$), (b) with coupling ($L_{\mu\phi} \neq 0$).} \label{fig3} \end{figure} As seen in Figure~\ref{fig3}, the entropy production increases significantly in the presence of moisture, indicating a higher degree of irreversibility in the system. This suggests that moisture not only facilitates conductive path formation, but actively enhances thermodynamic irreversibility through coupled flux mechanisms. Figure ~\ref{fig4} presents the dependency of entropy production on the electric field for various values of the coupling coefficient $L_{\mu\phi}$ . As $L_{\mu\phi}$ increases, the entropy production curve becomes significantly steeper. This behavior underscores the sensitivity of the system to even moderate moisture-electric coupling. Physically, it implies that materials with higher affinity between ionic/moisture fluxes and electric field exhibit accelerated degradation. It also highlights the potential of $L_{\mu\phi}$ as a critical design parameter in insulation systems. \begin{figure}[h] \centering \includegraphics[width=0.7\textwidth]{Figure 4 600dpi.jpg} \caption{Entropy production $\sigma$ vs. electric field intensity $E$ for different values of $L_{\mu\phi}$.} \label{fig4} \end{figure} These findings extend our earlier results, which connected the fractal analysis of tree growth with entropy production metrics in non-equilibrium systems \cite{Enciso2019, Razzitte2020}. Discussion: The results confirm that moisture not only alters the local electric field distribution but also reinforces irreversible processes through coupling mechanisms. The coefficient Luø emerges as a critical parameter in assessing insulation performance under humid conditions. Conclusion: This study provides a theoretical framework for understanding moistureinduced dielectric breakdown from a thermodynamic perspective. The findings may assist in designing more robust insulating systems and motivate further exploration into nonlinear and non-extensive generalizations of irreversible processes.