Thermodynamically Consistent Vibrational-Electron Heating: Generalized Derivation for Excited State Populations

TL;DR

Addresses accurate prediction of electron temperature $T_e$ in non-equilibrium plasmas and extends a thermodynamically consistent vibrational-electron (V-e) heating model to include cooling via vibrationally excited states. Derives that the heating-to-cooling ratio remains $exp(theta_v/T_e - theta_v/T_v)$ even when electron cooling includes vibrationally excited populations, removing the previous ground-state dominance constraint. Introduces an effective activation energy $E_{eff}$ and an effective cooling rate $k_{e-v}$, and imposes detailed balance to set the closure coefficient $theta_e$, with thermodynamic consistency achieved by choosing $theta_e = theta_v$ anchored to the fundamental transition $v=0$ to $v=1$. Shows that the generalized derivation makes the macroscopic heating ratio invariant to excited-state populations under Boltzmann statistics, extending validity to flows where vibrational excitation significantly contributes to electron cooling, such as post-shock and relaxation regimes.

Abstract

Accurate prediction of electron temperature ($T_{\rm e}$) in non-equilibrium plasma flows is critical for applications ranging from hypersonic flight to plasma-assisted combustion. We recently proposed a thermodynamically consistent model for vibrational-electron (V-e) heating [Phys. Fluids 37, 096141 (2025)] which enforces convergence of $T_{\rm e}$ to the vibrational temperature ($T_{\rm v}$) at equilibrium. While the original derivation assumed electron energy loss was dominated by collisions with ground-state molecules, this Letter presents a rigorous generalization of the model. We demonstrate that the heating-to-cooling ratio $\exp(θ_{\rm v}/T_{\rm e}-θ_{\rm v}/T_{\rm v})$ with $θ_{\rm v}$ the characteristic vibrational temperature remains valid even when electron cooling interactions with vibrationally excited states are included. This derivation removes the previous constraint assuming ground-state dominance, thereby extending the model's validity to plasma flows where vibrationally excited populations contribute significantly to electron cooling.