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~{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 {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. {figure}[h] [width=0.7]{Figure 2 600dpi.jpg} {Number of branches vs. electric field intensity, illustrating the fractal-like growth behavior.} {fig2} {figure} By integrating the linear phenomenological equations with the stochastic DBM framework, we obtain explicit expressions for the moisture and electric current fluxes: {equation} Jμ = T D (1/T Nb) + Lμφ (E/T), {equation} {equation} Jφ = Lφμ (1/T Nb) + T κ (E/T), {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: {equation} σ = T D (1/T Nb)^2 + T κ (E/T)^2 + 2 Lμφ (E/T)(1/T Nb). {equation} Figure~{fig3} compares the entropy production as a function of the electric field intensity in two scenarios: with and without coupling. {figure}[h] [width=0.7]{Figure 3 600dpi.jpg} {Normalized entropy production σ vs. electric field intensity E: (a) no coupling (Lμφ=0), (b) with coupling (Lμφ ≠ 0).} {fig3} {figure} As seen in Figure~{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 ~{fig4} presents the dependency of entropy production on the electric field for various values of the coupling coefficient Lμφ . As Lμφ 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 Lμφ as a critical design parameter in insulation systems. {figure}[h] [width=0.7]{Figure 4 600dpi.jpg} {Entropy production σ vs. electric field intensity E for different values of Lμφ.} {fig4} {figure} These findings extend our earlier results, which connected the fractal analysis of tree growth with entropy production metrics in non-equilibrium systems {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~{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 {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. {figure}[h] [width=0.7]{Figure 2 600dpi.jpg} {Number of branches vs. electric field intensity, illustrating the fractal-like growth behavior.} {fig2} {figure} By integrating the linear phenomenological equations with the stochastic DBM framework, we obtain explicit expressions for the moisture and electric current fluxes: {equation} Jμ = T D (1/T Nb) + Lμφ (E/T), {equation} {equation} Jφ = Lφμ (1/T Nb) + T κ (E/T), {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: {equation} σ = T D (1/T Nb)^2 + T κ (E/T)^2 + 2 Lμφ (E/T)(1/T Nb). {equation} Figure~{fig3} compares the entropy production as a function of the electric field intensity in two scenarios: with and without coupling. {figure}[h] [width=0.7]{Figure 3 600dpi.jpg} {Normalized entropy production σ vs. electric field intensity E: (a) no coupling (Lμφ=0), (b) with coupling (Lμφ ≠ 0).} {fig3} {figure} As seen in Figure~{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 ~{fig4} presents the dependency of entropy production on the electric field for various values of the coupling coefficient Lμφ . As Lμφ 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 Lμφ as a critical design parameter in insulation systems. {figure}[h] [width=0.7]{Figure 4 600dpi.jpg} {Entropy production σ vs. electric field intensity E for different values of Lμφ.} {fig4} {figure} These findings extend our earlier results, which connected the fractal analysis of tree growth with entropy production metrics in non-equilibrium systems {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.