Superelastic Heating in Treanor-Gordiets Plasmas: A Unified Analytic Closure

TL;DR

Problem: In non-equilibrium plasmas with $T_v > T_g$, conventional harmonic models severely underestimate superelastic electron heating due to decoupled energy modes and neglect of overpopulated high-lying Treanor-Gordiets states. Approach: It derives a closed-form, thermodynamically consistent anharmonic gain function based on detailed balance and a second-order Dunham expansion, yielding a unified governing equation that captures the crossover between vibrational–vibrational (V-V) up-pumping and vibrational–translational (V-T) relaxation. Key contributions: The framework introduces the anharmonic energy gap ΔE_{n,m} ≈ m k_B θ_v [1 - δ(n,m)] with δ(n,m) = x_e(2n + m - 1) - y_e(3 n^2 + 3 n m + m^2 + 3 n + 1.5 m - 2.5) and a correction factor Φ_anh(n,m) = exp( m θ_v [ min( δ(n,m)/T_g, 1/T_v ) - δ(n,m)/T_e ] ), leading to the total heating rate Q_v-e = ∑_{n≥0} ∑_{m≥1} Q_{e-v}^{(n,m)} Φ_anh(n,m) exp( m θ_v / T_e - m θ_v / T_v ). The model enforces detailed balance, is thermodynamically consistent at equilibrium ($T_e=T_v=T_g$), and aligns with full state-to-state benchmarks at reduced cost, delivering a robust closure for predicting electron-temperature evolution across Treanor-Gordiets distributions. Significance: Enables accurate energy transfer predictions in hypersonic flows, PAC, and LIP by providing a thermodynamically consistent, computationally efficient closure for diatomic vibrational–electron coupling under strong non-equilibrium.

Abstract

In non-equilibrium plasmas where the vibrational temperature exceeds the gas temperature, conventional harmonic models underestimate superelastic electron heating rates by an order of magnitude or more. This failure stems from the artificial decoupling of energy modes, which ignores the exponential heating contributions from overpopulated high-lying states characteristic of Treanor-Gordiets distributions. We resolve this limitation by deriving a closed-form, thermodynamically consistent anharmonic gain function based on detailed balance and a second-order Dunham expansion. This formulation serves as a unified governing equation that naturally identifies the kinetic crossover between vibrational-vibrational (V-V) up-pumping and vibrational-translational (V-T) relaxation. This approach accurately predicts the Treanor minimum and recovers the accuracy of full state-to-state benchmarks at a fraction of the computational cost. The model provides a robust closure for predicting electron temperature evolution in applications ranging from hypersonic flows to plasma-assisted combustion.

Figures (1)

• Figure 1: Comparison of vibrational-electron heating rates for N$_2$ at $T_{\rm e}=2$ eV and $T_{\rm g}=1000$ K.