Thermodynamically Consistent Vibrational-Electron Heating: Generalized Model for Multi-Quantum Transitions

TL;DR

The paper addresses the need for a thermodynamically consistent vibrational-electron heating closure in non-equilibrium plasmas and generalizes previous single-quantum models to include multi-quantum overtone transitions. It derives a channelwise decomposition of the cooling flux $Q_{\\rm e-v}^{(m)}$ and, using detailed balance and Boltzmann vibrational populations, obtains $Q_{\\rm v-e}=\\sum_{m=1}^{\\infty} Q_{\\rm e-v}^{(m)} \\exp\\left(\\frac{m\\theta_{\\rm v}}{T_{\\rm e}}-\\frac{m\\theta_{\\rm v}}{T_{\\rm v}}\\right)$, guaranteeing zero net energy transfer at equilibrium $T_{\\rm e}=T_{\\rm v}$. A key finding is that neglecting hot-band transitions in prior models yields a heating error $\\\\varepsilon=\\exp(-\\theta_{\\rm v}/T_{\\rm v})$, which can exceed 40% when $T_{\\rm v} \\gtrsim \\theta_{\\rm v}$, thereby impeding thermal relaxation. The framework, based on a harmonic oscillator assumption and Boltzmann vibrational distributions, extends applicability to higher $T_{\\rm e}$ regimes relevant to hypersonic flow, PAC, and LIP, while acknowledging possible non-Boltzmann distributions in strongly non-equilibrium states.

Abstract

Accurate prediction of electron temperature ($T_{\rm e}$) is critical for non-equilibrium plasma applications ranging from hypersonic flight to plasma-assisted combustion. We recently proposed a thermodynamically consistent model for vibrational-electron heating [Phys. Fluids 37, 096141 (2025)] that enforces the convergence of $T_{\rm e}$ to the vibrational temperature ($T_{\rm v}$) at equilibrium. However, the original derivation was restricted to single-quantum transitions, limiting its validity to low-temperature regimes ($T_{\rm e} \lesssim 1.5$ eV). In this Letter, we generalize the model to include multi-quantum overtone transitions, extending its applicability to high-energy regimes. We demonstrate that previous models neglecting hot-band transitions incur a systematic heating error of $\exp(-θ_{\rm v}/T_{\rm v})$, where $θ_{\rm v}$ is the characteristic vibrational temperature. This error exceeds 40\% when $T_{\rm v}$ is greater than $θ_{\rm v}$, effectively preventing thermal relaxation. To correct this, we derive a formulation where the total heating rate is a summation of channel-specific cooling rates $Q_{\rm e-v}^{(m)}$, each associated with a quantum jump $m$, scaled by a thermodynamic factor $\exp(mθ_{\rm v}/T_{\rm e}-mθ_{\rm v}/T_{\rm v})$. This generalized model preserves thermodynamic consistency by ensuring zero net energy transfer at equilibrium.

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