Discussion on Electron Temperature of Gas-Discharge Plasma with Non-Maxwellian Electron Energy Distribution Function Based on Entropy and Statistical Physics

Abstract

Electron temperature is reconsidered for weakly-ionized oxygen and nitrogen plasmas with its discharge pressure of a few hundred Pa, with its electron density of the order of 1017m−3 and in a state of non-equilibrium, based on thermodynamics and statistical physics. The relationship between entropy and electron mean energy is focused on based on the electron energy distribution function (EEDF) calculated with the integro-differential Boltzmann equation for a given reduced electric field E/N. When the Boltzmann equation is solved, chemical kinetic equations are also simultaneously solved to determine essential excited species for the oxygen plasma, while vibrationally excited populations are solved for the nitrogen plasma, since the EEDF should be self-consistently found with the densities of collision counterparts of electrons. Next, the electron mean energy U and entropy S are calculated with the self-consistent EEDF obtained, where the entropy is calculated with the Gibbs’s formula. Then, the “statistical” electron temperature Test is calculated as Test=[∂S/∂U]−1. The difference between Test and the electron kinetic temperature Tekin is discussed, which is defined as [2/(3k)] times of the mean electron energy U=⟨ϵ⟩, as well as the temperature given as a slope of the EEDF for each value of E/N from the viewpoint of statistical physics as well as of elementary processes in the oxygen or nitrogen plasma.

Simulation and experimental study of a cold atmospheric pressure plasma and comparison of efficiency in boosting recombinant Endoglucanase II production in Pichia pastoris

“…The n e is an important contributor to ionization, plasma chemistry, and reactive species production [49,50]. According to simulation outputs, aggregations of electrons are formed around the inner electrode by voltage application to generate the avalanches and subsequent discharge.…”

Reconsideration of Temperature Determined by the Excited-State Population Distribution of Hydrogen Atoms Based on Tsallis Entropy and Its Statistics in Hydrogen Plasma in Non-Equilibrium State

“…Consequently, to determine the temperature of the non-equilibrium system, the distribution function must be approximated to a straight line, which is generally considered to correspond to the local derivative at each energy value in the Boltzmann plot. The definition of temperature in the non-equilibrium state, which does not obey the Maxwell-Boltzmann distribution, has received a great deal of attention in recent years [7,8]. For example, Álvarez et al attempted to describe the out-of-equilibrium free-electrons in cold plasmas and applied the developed theory assuming the electron entropy defined based on the Boltzmann H-theorem [7].…”

“…where v is the electron velocity and is the electron energy. However, in the realistic conditions of plasma, the relation T eff = T Gibbs does not always hold [8], and they solved the Boltzmann equation to obtain the EEDF of oxygen and nitrogen plasmas with inelastic collision cross-sections and calculated the Gibbs entropy. Eventually, these approaches made it possible to define some kinds of temperatures even in non-equilibrium states.…”

“…As Álvarez et al studied from the viewpoint of statistical physics, the temperature is given as a reciprocal value of the partial derivative of the entropy with respect to the internal energy [7]. On the other hand, Akatsuka et al examined another concept of the electron temperature of the non-equilibrium plasma on the relation between the entropy and the energy through the distribution function [8]. If the distribution function is known, the entropy of the system can be calculated, without concepts on equilibrium physics such as "free energy".…”