Vibrational Kinetics

Abstract

In this chapter we want to describe the vibrational kinetics of anharmonic diatomic molecule submitted to the action of different elementary processes including electron, molecule and heavy-particle collisions. A rich literature does exist on the subject including previous books written by different authors (Capitelli and Molinari 1980; Capitelli 1986; Gordiets et al. 1988; Capitelli et al. 2000; Fridman and Kennedy 2004; Fridman 2012). The chapter contains different experimental and theoretical examples which clarify the numerous applications of the vibrational kinetics under non equilibrium conditions.

Appendices

Appendix 1: Non-equilibrium Vibrational Distributions: General Considerations

A well developed vibrational distribution under non-equilibrium plasma conditions can be thought formed by three parts. The first part can be considered a Boltzmann one at $\vartheta _{1}$, followed by a plateau ending in another Boltzmann distribution at the gas temperature. They reflect the dominance of elementary processes acting in the plasma. As an example the plateau can be formed due to the interplay of VV quasi-resonant processes

$$\displaystyle{ \text{A}_{2}(v) + \text{A}_{2}(v) \rightarrow \text{A}_{2}(v + 1) + \text{A}_{2}(v - 1) }$$

(7.61)

and VT_M transitions.

$$\displaystyle{ \text{A}_{2}(v) + \text{A}_{2} \rightarrow \text{A}_{2}(v - 1) + \text{A}_{2} }$$

(7.62)

Limiting our analysis to these two processes and considering quasi-stationary conditions we have the following relation

$$\displaystyle{ -N_{v}N_{\text{A}_{ 2}}k_{v,v-1} - N_{v}N_{v}k_{v,v-1}^{v,v+1} + N_{ v+1}N_{v-1}k_{v+1,v}^{v-1,v} = 0 }$$

(7.63)

Considering the detailed balance principle VV resonant rates (see Eq. (7.23)) we can write

$$\displaystyle{ -N_{v}N_{\text{A}_{ 2}}k_{v,v-1} - N_{v}N_{v}k_{v+1,v}^{v-1,v}\exp \left [-\frac{\varDelta _{v,v}^{\text{VV}}} {k_{B}T} \right ] + N_{v+1}N_{v-1}k_{v+1,v}^{v-1,v} = 0 }$$

(7.64)

In the plateau we can assume, as a first approximation

$$\displaystyle{ N_{v} \approx N_{v+1} \approx N_{v-1} }$$

(7.65)

so that Eq. (7.65) can be simplified as

$$\displaystyle{ -N_{\text{A}_{ 2}}k_{v,v-1} - N_{v}k_{v+1,v}^{v-1,v}\exp \left [-\frac{\varDelta _{v,v}^{\text{VV}}} {k_{B}T} \right ] + N_{v}k_{v+1,v}^{v-1,v} = 0 }$$

(7.66)

giving

$$\displaystyle{ N_{v} = N_{\text{A}_{ 2}} \frac{k_{v,v-1}} {k_{v+1,v}^{v-1,v}}\left \{1 -\exp \left [-\frac{\varDelta _{v,v}^{\text{VV}}} {k_{B}T} \right ]\right \}^{-1} }$$

(7.67)

Assuming harmonic oscillator rates (crude approximation)

$$\displaystyle\begin{array}{rcl} k_{v,v-1} = vk_{1,0}& &{}\end{array}$$

(7.68)

$$\displaystyle\begin{array}{rcl} k_{v+1,v}^{v-1,v} = v^{2}k_{ 1,0}^{0,1}& &{}\end{array}$$

(7.69)

we get a v ⁻¹ dependence of the plateau

$$\displaystyle{ \frac{N_{v}} {N_{\text{A}_{ 2}}} = \frac{1} {v} \frac{k_{1,0}} {k_{1,0}^{0,1}}\left \{1 -\exp \left [-\frac{\varDelta _{v,v}^{\text{VV}}} {k_{B}T} \right ]\right \}^{-1} }$$

(7.70)

i.e. a plateau slightly declining with v. It is interesting to develop the energy difference in the exponential factor assuming a simple anharmonic oscillator (see Eqs. (7.16)), (7.22), and expanding the exponential in power series, the equation at the plateau becomes

$$\displaystyle{ \frac{N_{v}} {N_{\text{A}_{ 2}}} = \frac{1} {v} \frac{k_{1,0}} {k_{1,0}^{0,1}}\left \{1 -\exp \left [- \frac{2\omega _{e}\chi _{e}} {k_{B}T}\right ]\right \}^{-1} \approx \frac{1} {v} \frac{k_{1,0}} {k_{1,0}^{0,1}}\left [ \frac{2\omega _{e}\chi _{e}} {k_{B}T}\right ]^{-1}. }$$

(7.71)

The VV kinetics in the presence of a source of vibrational quanta is such to create the so called Treanor distribution (Treanor et al. 1968) i.e.

$$\displaystyle{ \left ( \frac{N_{v}} {N_{v+1}}\right ) =\exp \left [ \frac{\varepsilon _{1}} {k_{B}\vartheta _{1}} - 2 \frac{\varepsilon _{1}\chi _{e}v} {k_{B}T_{g}}\right ] }$$

(7.72)

χ _e is the anharmonicity constant and $\vartheta _{1}$ and T _g are respectively the non-equilibrium vibrational temperature and the gas temperature. This distribution presents a minimum when the exponential becomes null, i.e.

$$\displaystyle{ \left (v\right )_{\text{min}} = \frac{T_{g}} {2\chi _{e}\vartheta _{1}} + \tfrac{1} {2} }$$

(7.73)

Appendix 2

Let us consider the VT terms restricting them to the mono-quantum transitions . The sequence of the following reactions can be considered

$$\displaystyle\begin{array}{rcl} \text{A}_{2}(0) + \text{A}_{2}& \rightarrow & \text{A}_{2}(1) + \text{A}_{2} {}\\ \text{A}_{2}(1) + \text{A}_{2}& \rightarrow & \text{A}_{2}(2) + \text{A}_{2} {}\\ \text{A}_{2}(2) + \text{A}_{2}& \rightarrow & \text{A}_{2}(3) + \text{A}_{2} {}\\ {\ldots }& & {\ldots } {}\\ \text{A}_{2}(v) + \text{A}_{2}& \rightarrow & \text{A}_{2}(v + 1) + \text{A}_{2} {}\\ \end{array}$$

The different equations are interconnected so that we can write for the levels

$$\displaystyle\begin{array}{rcl} \left (\frac{dN_{0}} {dt} \right )& =& -N_{0}N_{\text{A}_{ 2}}k_{0,1}^{\text{A}_{2} } + N_{1}N_{\text{A}_{ 2}}k_{1,0}^{\text{A}_{2} } {}\\ \left (\frac{dN_{1}} {dt} \right )& =& N_{0}N_{\text{A}_{ 2}}k_{0,1}^{\text{A}_{2} } - N_{1}N_{\text{A}_{ 2}}k_{1,0}^{\text{A}_{2} } - N_{1}N_{\text{A}_{ 2}}k_{1,2}^{\text{A}_{2} } + N_{2}N_{\text{A}_{ 2}}k_{2,1}^{\text{A}_{2} } {}\\ \left (\frac{dN_{2}} {dt} \right )& =& N_{1}N_{\text{A}_{ 2}}k_{1,2}^{\text{A}_{2} } - N_{2}N_{\text{A}_{ 2}}k_{2,1}^{\text{A}_{2} } - N_{2}N_{\text{A}_{ 2}}k_{2,3}^{\text{A}_{2} } + N_{3}N_{\text{A}_{ 2}}k_{3,2}^{\text{A}_{2} } {}\\ \end{array}$$

At the stationary conditions we can write

$$\displaystyle\begin{array}{rcl} \left (\frac{dN_{0}} {dt} \right )& =& -N_{0}N_{\text{A}_{ 2}}k_{0,1}^{\text{A}_{2} } + N_{1}N_{\text{A}_{ 2}}k_{1,0}^{\text{A}_{2} } = 0 {}\\ \left (\frac{dN_{1}} {dt} \right )& =& N_{0}N_{\text{A}_{ 2}}k_{0,1}^{\text{A}_{2} } - N_{1}N_{\text{A}_{ 2}}k_{1,0}^{\text{A}_{2} } - N_{1}N_{\text{A}_{ 2}}k_{1,2}^{\text{A}_{2} } + N_{2}N_{\text{A}_{ 2}}k_{2,1}^{\text{A}_{2} } = 0 {}\\ \left (\frac{dN_{2}} {dt} \right )& =& N_{1}N_{\text{A}_{ 2}}k_{1,2}^{\text{A}_{2} } - N_{2}N_{\text{A}_{ 2}}k_{2,1}^{\text{A}_{2} } - N_{2}N_{\text{A}_{ 2}}k_{2,3}^{\text{A}_{2} } + N_{3}N_{\text{A}_{ 2}}k_{3,2}^{\text{A}_{2} } = 0{}\\ \end{array}$$

Summing the first two equations we get

$$\displaystyle{ -N_{1}N_{\text{A}_{ 2}}k_{1,2}^{\text{A}_{2} } + N_{2}N_{\text{A}_{ 2}}k_{2,1}^{\text{A}_{2} } = 0 }$$

while summing the first three equations we get

$$\displaystyle{ -N_{2}N_{\text{A}_{ 2}}k_{2,3}^{\text{A}_{2} } + N_{3}N_{\text{A}_{ 2}}k_{3,2}^{\text{A}_{2} } = 0 }$$

i.e.

$$\displaystyle\begin{array}{rcl} N_{1}k_{1,2}^{\text{A}_{2} }& =& N_{2}k_{2,1}^{\text{A}_{2} } {}\\ N_{2}k_{2,3}^{\text{A}_{2} }& =& N_{3}k_{3,2}^{\text{A}_{2} } {}\\ \end{array}$$

and after applying the detailed balance principle on the rates we get

$$\displaystyle\begin{array}{rcl} \frac{N_{2}} {N_{1}}& =& \frac{k_{1,2}^{\text{A}_{2}}} {k_{2,1}^{\text{A}_{2}}} =\exp \left [\frac{\varepsilon _{2} -\varepsilon _{1}} {k_{B}T} \right ] {}\\ \frac{N_{1}} {N_{2}}& =& \frac{k_{2,3}^{\text{A}_{2}}} {k_{3,2}^{\text{A}_{2}}} =\exp \left [\frac{\varepsilon _{3} -\varepsilon _{2}} {k_{B}T} \right ] {}\\ \end{array}$$

By making the same treatment on the first equation, we get

$$\displaystyle{ \frac{N_{1}} {N_{0}} = \frac{k_{0,1}^{\text{A}_{2}}} {k_{1,0}^{\text{A}_{2}}} =\exp \left [\frac{\varepsilon _{1} -\varepsilon _{0}} {k_{B}T} \right ] }$$

The three levels submitted at the action of VT processes present at the stationary conditions a Boltzmann distribution at the gas temperature. This conclusion can be extended to the whole ladder of vibrational levels at the stationary conditions. It can be shown that also the time evolution of vibrational distributions under the action of VT mono-quantum transitions keeps its Boltzmann character if the initial condition is characterized by a Boltzmann distribution. More complex is the situation when multi-quantum VT transitions are inserted in the master equation .