Accepted Manuscript

Nonequilibrium dissociation mechanisms in low temperature nitrogen and car-bon monoxide plasmas

M. Capitelli, G. Colonna, G. D’Ammando, V. Laporta, A. Laricchiuta

PII: S0301-0104(14)00101-3DOI: http://dx.doi.org/10.1016/j.chemphys.2014.04.003Reference: CHEMPH 9081

To appear in: Chemical Physics

Received Date: 7 November 2013Accepted Date: 6 April 2014

Please cite this article as: M. Capitelli, G. Colonna, G. D’Ammando, V. Laporta, A. Laricchiuta, Nonequilibriumdissociation mechanisms in low temperature nitrogen and carbon monoxide plasmas, Chemical Physics (2014), doi:http://dx.doi.org/10.1016/j.chemphys.2014.04.003

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Nonequilibrium dissociation mechanisms in low temperature nitrogen

and carbon monoxide plasmas

M. Capitelli1,2, G. Colonna?2, G. D’Ammando1 V. Laporta2,3, A. Laricchiuta2

1 Department of Chemistry, University of Bari (Italy)

2 CNR IMIP Bari (Italy)

3 Department of Physics and Astronomy, University College of London (UK)

Abstract

The role of vibrational excitation in affecting the dissociation under discharge conditions characterized

by reduced electric field E/N≤ 80 Td has been investigated in N2. The kinetic calculations have been

performed using a self-consistent approach, solving at the same time the master equation for the compo-

sition and the distribution of internal states (vibrational and electronic) and the Boltzmann equation for

the electron energy distribution function. The results show that vibrational mechanisms involving heavy

particle excited states dominate electron impact dissociation mechanisms involving the whole vibrational

ladder for E/N< 50 Td, the two mechanisms being competitive for E/N> 50 Td.

Keywords:

1. Introduction

A continuous interest is devoted to understand the role of vibrational energy in affecting chemical

processes involving molecules (N2, CO, CO2) under discharge and post-discharge conditions [1, 2, 3, 4].

Despite its apparent simplicity, the experimental determination of dissociation rates is very difficult,

while theoretical calculations considering only electron-molecule dissociation processes fail in repro-

ducing experimental dissociation rates even for simple molecules. This point was first emphasized by

the Polak group [5] in the 70’s, who showed that the experimental dissociation rate of nitrogen in glow

discharges at low E/N can not be predicted by the direct electron impact dissociation from the ground vi-

brational level of nitrogen (see also Ref. [6]). These experimental findings induced the plasma community

to develop sophisticated theoretical models to explain the dissociation rate of diatomic molecules under

electrical discharges beyond the direct impact dissociation. In particular a pure vibrational mechanism

(PVM) [7, 8, 9] was developed, based on a three step mechanism: (i) introduction of vibrational quanta by

resonant electron vibration excitation process (e-V); (ii) redistribution of the quanta by vibration-vibration

(V-V) and vibration-translation (V-T) energy exchange processes; (iii) overcoming by the same processes

of the last vibrational level linked to a pseudo-level located in the continuum, i.e. miming the dissociation

process. The corresponding PVM rates were found orders of magnitude higher than the corresponding

rates by direct electron-impact from the ground vibrational level υ = 0. Some of the hypotheses contained

in these results i.e. Maxwell distribution function for electrons and direct electron dissociation from υ = 0

were eliminated and a direct dissociation mechanism from all vibrational levels was considered [10]. In

any case the vibrational distribution of nitrogen was presenting a long plateau such as to promote disso-

ciation directly from PVM and indirectly by a direct electronic mechanism (DEM) including transitions

from all vibrational levels.

Preprint submitted to Elsevier April 9, 2014

These results however were based on the knowledge, poor at that time, of the relevant cross sections.

In particular the V-Ta rates involving nitrogen atoms were considered equal to the corresponding ones

induced by nitrogen molecules which neglect multi-quantum transitions. Moreover, the e-V rates were

estimated, by crude scaling law from the known e-V rates linking the first 8 levels. It should be noted

that these results have been obtained considering 45 vibrational levels in nitrogen, a problem solved in

the recently calculated rates and cross sections.

Inclusion of more realistic V-Ta rates from atomic nitrogen (considering also vibrational multi-quantum

transitions) was attempted many years ago by Armenise et al. [11]. The main result of this study was the

practical disappearance of PVM in dissociating nitrogen mainly due a new set of V-Ta. These rates are

continuously updated especially for the processes involving low-lying vibrational levels important for

the definition of a vibrational temperature [12, 13]. Insertion of these V-Ta rates in the kinetic model

confirmed the impossibility of PVM mechanisms based on the last vibrational level of the molecule [14].

The aim of the present paper is to reconsider the role of vibrational excitation in affecting the disso-

ciation rates of simple molecules in plasmas sustained by low reduced electric fields E/N. Under these

conditions the energy gained by electrons is mainly lost in pumping vibrational energy in the molecule

through resonant e-V processes promoting chemical processes activated by the vibrational energy. Under

these conditions e-V cross sections and rates take an important role which justifies the enormous effort

made by Laporta et al to compute complete sets of e-V cross sections for the N2 and CO systems [15, 16].

Moreover in the nitrogen case we are considering a resonant dissociation mechanism i.e.

e− +N2(X1Σ+

g ,υ) → N−2 (2Πg) → e− +2N(4S) , (1)

the cross sections of which have recently been calculated by extending the theory of e-V cross sections to

the continuum [16, 15] (Laporta et al [17]).

2. Model

A zero dimensional code coupling the vibrational kinetics of N2 with the electron energy distri-

bution function (eedf) and with plasma-chemistry (dissociation, ionization, electronic excitation) has

been developed in our laboratory to shed light on the different couplings existing in the non-equilibrium

plasma [14, 18]. In implicit form we write the vibrational kinetics as

(

dNv

dt

)

=

(

dNv

dt

)

e-V

+

(

dNv

dt

)

V-V

+

(

dNv

dt

)

V-Ta

+

(

dNv

dt

)

V-Tm

+

(

dNv

dt

)

e-D

+

(

dNv

dt

)

e-I

+

(

dNv

dt

)

e-E

+

(

dNv

dt

)

chem

+

(

dNv

dt

)

rec

,

(2)

where the different terms represent excitation and deexcitation processes involving the v-th vibrational

level due to e-V, V-V, V-Ta and V-Tm (vibration-translation energy exchange by molecule impact) pro-

cesses. In addition e-D, e-I and e-E represent the loss of vibrational energy due to electron impact disso-

ciation, ionization and excitation events starting from the v-th level, while the last two terms represent the

influence of plasmachemistry and recombination processes on the concentration of v-th level. The system

of vibrational master equations and the corresponding plasmachemistry equations are then coupled to the

Boltzmann equation for the eedf written in compact form as

∂n(ε, t)

∂t= −

∂JE

∂ε−

∂Jel

∂ε−

∂Jee

∂ε+Sin +Ssup , (3)

2

where n(ε, t) is the number of electrons in the energy range (ε,ε + dε) at time t. The different terms

on the right hand side of Eq. (3) describe the flux of electrons in the energy space due to (i) the electric

field E ( ∂JE

∂ε); (ii) the elastic collisions (

∂Jel

∂ε); (iii) the electron-electron collisions ( ∂Jee

∂ε); (iv) the inelastic

collisions (Sin); (v) the superelastic (second kind) collisions (Ssup). Details of the model can be found

in [18] where one can also find the detailed list of elementary processes inserted in the kinetics as well

as the corresponding sources of the different cross sections and rates. The present model considers 67

vibrational levels [15, 14] for nitrogen implying a rescaling of the old rates based on 45 vibrational levels.

Electron impact excitation, dissociation and ionization transitions involve all the vibrational ladder while

the same processes are promoted by bi-molecular collisions involving vibrationally and electronically

excited states.

3. Results

3.1. Nitrogen

Before examining the model results we want to compare old results obtained (i) by PVM [7, 8, 9];

(ii) by the direct electron-impact dissociation obtained by using the experimental Cosby cross sections

[19]; (iii) by multiplying the experimental electron-impact rates by a factor 70 to take into account the

role of vibrational excited molecules (Park’s model) [20] and (iv) by an upper limit of pure vibrational

mechanism (ulPVM) [3, 6]. In this last case we get the rate as a balance between the vibrational quanta

introduced by e-V processes and the loss of vibrational quanta by the dissociation process i.e.

K(ulPV M)d =

υmax

∑υ=1

υke-V(0 → υ)

υmax, (4)

where ke-V(0 → υ) is the rate coefficient of electron impact excitation process,

e− +N2(X1Σ+

g ,υ = 0) → N−2 (2Πg) → e− +N2(X

1Σ+g ,υ) , (5)

υmax = 67 is the number of vibrational states supported by the potential well of the N2 ground state.

Equation (4) is derived from the balance of the input of vibrational quanta by e-V processes and the

corresponding loss by dissociation (every dissociation event needs 67 vibrational quanta), i.e. no loss

of vibrational quanta occur through V-T relaxation. All the rates reported in Fig. 1 have been obtained

by considering a Maxwell distribution function for electrons and the complete set of e-V cross sections

recently calculated by Laporta et al. [15]. Inspection of Fig. 1 shows that K(ulPV M)d is orders of magnitude

higher than the experimental rate , KCosbyd [19], in the electron temperature range 7500 < Te < 30000 K,

while for higher temperatures the curves get closer, crossing at about Te ∼ 40000 K . In this temperature

range K(ulPV M)d is larger than the rate from the pure vibrational mechanism, which considers a non-

equilibrium vibrational kinetics including V-V, V-T and e-V processes and a ladder climbing model for

the dissociation process. K(ulPV M)d is also larger than the corrected rates estimated by Park , KPark

d [20],

by increasing the rate obtained from recommended cross section by Cosby contribution by a factor 70.

Let us now consider the results obtained by our time dependent model. Contrary to the results reported

in [21] we have run our time dependent equations at fixed electron density (molar fraction χe = 10−6),

pressure p = 5.6 torr and gas temperature T = 1000 K. These conditions roughly reproduce the experi-

ment of Polak et al. [5].

Figure 2 reports the time evolution of the theoretical impact dissociation rate from υ=0 and from all

vibrational levels, calculated according to Eq. (6)

Kkind (υ = all) =

υmax

∑υ=0

Kkind (υ) Kkin

d (υ) = kkind (υ)

Nυ

Ntot, (6)

3

10000 20000 30000 40000 5000010–13

10–12

10–11

10–10

10–9

Te [K]

Dis

soci

atio

n R

ate

Co

effi

cien

t [c

m3/s

]

N2

PVM

Kd(ulPVM)

KdCosby

KdPark

Figure 1: Nitrogen dissociation rate as a function of Te calculated according to different models. (solid line) K(ulPV M)d from accurate

e-V rates by Laporta et al. [15], (dashed line) dissociation rate from recommended cross section by Cosby [19], KCosbyd , (crosses)

global dissociation rate from Park [20], KParkd , (close circles) pure vibrational mechanism PVM [7, 8, 9].

Inspection of Fig. 2 shows that after 10−7 s the contribution from υ=0 and the global one are approxi-

mately the same (cold gas approximation, i.e. a condition where eedf depends only on E/N at fixed gas

temperature ). Then the two quantities start diverging, due to the contribution of vibrationally excited

molecules in affecting the dissociation process, as well as the eedf (hot gas approximation, i.e. eedf de-

pends not only on E/N but also on the population of vibrational and electronic states through the action

of the relevant superelastic collisions [18, 22]). Both the υ=0 contribution and the global one reach a

stationary condition in times of the order of 10−1-1 s. These stationary values are reported as a function

of E/N in Fig. 3 and compared with the experimental rates by Polak et al. [5].

10–9 10–7 10–5 10–3 10–110–13

10–12

10–11

t [s]

Dis

soci

atio

n R

ate

Coef

fici

ent

[cm

3/s

]

E/N = 60 Td

Kdkin(υ=all)

Kdkin (υ=0)

Figure 2: Temporal evolution of the dissociation rate from υ = 0 and the global one from all vibrational levels, at E/N=60 Td and

T =1000 K.

In this case the role of vibrational excited states is well evident bringing the theoretical global rates in

the same scale of experimental ones for E/N> 60 Td. Theoretical and experimental results however do

not agree for E/N< 50 Td. In this last case the theoretical results not only are lower than the experimental

ones but also do not reproduce the flat behavior of the experimental dissociation rate as a function of

E/N. A flat behavior is indeed shown by quantity K(ulPV M)d as a function of Te (Fig. 1) (i.e. to a first

approximation as a function of E/N) even though the corresponding theoretical values are more than

two orders of magnitude higher than the experimental values. Moreover the old PVM rates reported in

4

30 40 50 60 70 80

10–15

10–14

10–13

10–12

10–11

10–10

E/N [Td]

Dis

soci

atio

n R

ate

Co

effi

cien

t [c

m3/s

]

N2

Kdexp

Kdkin (υ=0)

Kdkin(υ=all)

Kh

Kdres

PVM

Figure 3: Nitrogen dissociation rates as a function of reduced electric field E/N, at p=5.6 torr and T = 1000 K , calculated according

to different models. (close squares) experimental dissociation rate with error bars [5], (open diamonds) present results from kinetics

with dissociation from υ = 0 and from all vibrational levels (t = 10−1 s in the time evolution), (close circles) pure vibrational

mechanism PVM [7, 8, 9], (open triangles) total rate coefficient of the heavy particle impact dissociation processes, Eq. (9), (open

squares) rate coefficient of the resonant dissociation process, Eq. (1).

Ref. [10] for E/N= 30 and 60 Td seem to fill the gap between theoretical and experimental values reported

in Fig. 3.

Going beyond the possibility of the dissociation from the last bound level of nitrogen we could assume

a vibrational mechanism [23, 24] involving vibrationally and electronically excited molecules, specifi-

cally

N2(X1Σ+

g ;10 < υ < 25)+N2(X1Σ+

g ;10 < υ < 25) → N2(X1Σ+

g )+2N , (7)

N2(X1Σ+

g ;14 ≤ υ ≤ 19)+N2(A3Σ) → N2(B

3Π;υ′ ≥ 13)+N2(X1Σ+

g ) → N2(X1Σ+

g )+2N. (8)

The global rate coefficients of these processes have been calculated considering the population of individ-

ual vibrational levels in the range reported in Eqs. (7) and (8) and the population of N2(A3Σ) and using

the values k7 = 3.510−15 cm3/s and k8 = 4.510−11 exp(−1765/T ) cm3/s reported in [24]

K7 =k7

χeN2

24

∑v=11

24

∑w=11

NvNw

K8 =k8NA

χeN2

19

∑v=14

Nυ

Kh = K7 +K8

(9)

where Nv, Nw and NA are the populations of the v,w vibrational levels and of the N2(A3Σ) electronic state

of the N2 molecule respectively, N is the total number density of N2 and χe = 10−6 is the electron molar

fraction. It should be noted that, in the present conditions, mechanism (7) prevails on (8). This mecha-

nism, first proposed by Guerra et al. [24] and later used by Dyatko et al. [25] and by Capitelli et al. [18],

involves the intermediate portion of the vibrational distribution and metastable nitrogen molecules in-

stead of the last vibrational level as in the original PVM model . The rate coefficient Kh at stationary

conditions (t ∼ 0.1− 1 s), divided by the electron molar fraction χe = 10−6 to obtain a pseudo-electron

impact rate coefficient, is reported in Fig. 3 and compared with the different models and with the ex-

perimental results. In this case the rates are in qualitative agreement with the experimental ones in the

5

0 10 20 30 4010–10

10–8

10–6

10–4

10–2

100

E/N = 30 Td

E/N = 40 Td

E/N = 50 Td

E/N = 60 Td

E/N = 70 Td

E/N = 80 Td

electron energy (eV)

eedf

(eV

-3/2

)

a)

0 2 4 6 8 1010–10

10–8

10–6

10–4

10–2

100

vibrational energy (eV)

vib

rati

onal

dis

trib

uti

on

b)

E/N = 80 Td

E/N = 30 Td

0 2 4 6 8 1010–10

10–8

10–6

10–4

10–2

100

vibrational energy (eV)

vib

rati

onal

dis

trib

uti

on

b)

E/N = 80 Td

E/N = 30 Td

Figure 4: Steady state (a) eedf and (b) vibrational distribution of N2 as a function of reduced electric field E/N, at p=5.6 torr and

T =1000 K.

30 40 50 60 70 801011

1012

1013

E/N [Td]

Num

ber

den

sity

[cm

-3]

N2(A3Σu)

Figure 5: Number density of the N2(A3Σ+u ) excited electronic state of N2 as a function of reduced electric field E/N, at p=5.6 torr

and T =1000 K.

6

0 10 20 30 40 50 60

10–21

10–19

10–17

10–15

10–13

10–11

vibrational level

Kdres [

cm3/s

]E/N = 30 Td

E/N = 50 Td

E/N = 80 Td

Figure 6: Vibrationally resolved rate coefficients for the resonant dissociation process described by Eq. (1), calculated according

to Eq. (11) using the eedf at steady state at different values of the reduced electric field E/N, at p=5.6 torr and T =1000 K.

whole E/N range. In particular they fill the gap between experimental values and theoretical ones based

on electron impact dissociation rates for E/N< 50 Td. Note that the flat behavior of Kh is due to the fact

that the quasistationary vibrational distributions (Fig. 4b), calculated with the complete vibrational and

electronically excited state kinetics, including the V-Ta energy transfer and neglecting surface processes,

and NA metastable concentration (Fig. 5) shows a much weaker dependence on the E/N compared to

Kkind (υ = all) and Kkin

d (υ = 0) as can be appreciated in Fig. 3. These last quantities, in fact depend on the

high-energy region of eedf, which is more affected by E/N values (see Fig. 4a).

In Fig. 3 we have also reported the global dissociation rate due to the resonant dissociation mechanism

described by Eq. (1) which is calculated using the following average over the vibrational distribution

Kresd =

vmax

∑υ=0

kresd (υ)

Nυ

Ntot, (10)

where kresd (υ) are the vibrationally resolved resonant dissociation rate coefficient, calculated using the

actual non-equilibrium eedf as

kresd (υ) =

√

2

me

∫ ∞

ε∗υσres

υ (ε)ε f (ε)dε , (11)

where ε∗υ is the threshold energy for the dissociation of N2(υ), σresυ are the resonant dissociation cross

sections calculated by Laporta et al [17] and f (ε) if the stationary eedf normalized according to

∫ ∞

0

√ε f (ε)dε = 1 . (12)

A sample of these rates have been reported in Fig. 6 at different E/N values as a function of vibrational

quantum number, showing that the resonant dissociation rates are important only for high lying vibrational

levels, the υ = 0 level being insignificant for the process. The global resonant dissociation rate is reported

in Fig. 3. The new mechanism, while presenting a flat dependence on E/N, shows however very low

values not competitive with the other mechanisms. It should be noted that the Kresd increases by an order

of magnitude if used is made of a Maxwell eedf with electron temperature Te obtained by the average

energy ε = 32kTe of the actual eedf.

7

3.2. Carbon monoxide

It is interesting to note that similar ideas apply to other important systems such as CO and CO2.

In the case of CO, Fig. 7 compares K(ulPV M)d obtained by the complete sets of e-V cross sections, re-

cently obtained by Laporta et al. [16], and the corresponding experimental electron-impact dissociation

rates[26].

10000 20000 30000 40000 5000010–13

10–12

10–11

10–10

10–9

Te [K]

Dis

soci

atio

n R

ate

Co

effi

cien

t [c

m3/s

]

CO

Kd(ulPVM)

KdCosby

Figure 7: Carbon monoxide dissociation rate as a function of electron temperature calculated according to different models. (solid

line) K(ulPV M)d from accurate e-V rates by Laporta et al. [16], (dashed line) dissociation rate from recommended cross section by

Cosby [26].

We note that, as in the nitrogen case, K(ulPV M)d � K

Cosbyd being also much larger than the few exist-

ing experimental values. As an example D’Amico and Smith [27] measured a characteristic dissociation

time of 30 s for pure CO under discharge conditions characterized by an average estimated electron en-

ergy of 0.5 eV and an electron density of 1010 cm−3 (p= 10 torr). Under these conditions one should

expect that the discharge power goes preferentially in the vibrational excitation of CO promoting a dis-

sociation process assisted by vibrational excitation while the electron impact dissociation time is several

orders of magnitude higher than the experimental value [28, 29]. The experimental vibrational distri-

butions of CO pumped either by IR laser [30] or electrical discharges [31] while showing well pumped

vibrational distributions up to υ = 40 rule out the possibility of a PVM based on the last vibrational

level of CO (υmax ≈ 80) because the opening of chemical and physical processes involving 20 < υ < 40

vibrational levels. In particular the chemical reaction activated by vibrationally excited states [4], i.e.

CO(υ)+CO(w) → C+CO2 , (13)

acts either as dissociation process or as relaxation one reducing in any case the gap between K(ulPV M)d and

experimental results. Old calculations [28, 29] for reaction (13) for pure CO for E/N= 310−16 V cm2, ne

= 1010 cm−3, p = 5 torr, T = 500 K give a characteristic time of 8.3 s not too far from the experimental

value. On the other hand DEM including the transitions from the whole set of vibrational levels should

increase up to a factor 100 the dissociation rate being in any case far from the experimental value. A

re-examination of vibrationally excited state assisted chemical reactions in CO urges to be reconsidered

and extended to the much more complex CO2 system. In this last case Legasov et al [32] were able to

reproduce the experimental results for the dissociation of CO2 under non equilibrium plasma conditions

(Te = 1− 2 eV, p = 1 atm, ne = 1012 cm−3) by using in Eq. (2) a ke-V(0 → 1) = 2 · 10−8 cm3/s and a

υmax ≈ 30 .

4. Conclusions and perspectives

The main conclusion of the present work for nitrogen dissociation is that vibrational excitation has a

twofold effect on the dissociation rate. The first is linked to the increase of the dissociation rate acting on

8

the electron energy distribution function while the second one is the increase of the dissociation rate due

to the inclusion of dissociation from all the vibrational levels. Apparently the dissociation rate from the

pure vibrational mechanism, i.e. by considering the pseudo-level above the last bound level of nitrogen

looses its importance because the small concentration of the last bound level (see Fig. 4(b)), at least under

the conditions studied in this paper. This observation requires a deeper insight of PVM values which

take into account better rates for the 67 vibrational level ladder of nitrogen. A mechanism involving

the intermediate portion of the vibrational distribution and metastable excited electronic states seem a

promising alternative to be used in explaining the dissociation rates of nitrogen at low E/N values. Finally

the average resonant dissociation rates, while presenting a flat dependence on E/N, show absolute values

orders of magnitude lower than the experimental values as well as the corresponding value for the other

mechanisms.

Similar considerations apply to the carbon monoxide dissociation rates. New theoretical models

should be developed to shed light on this process taking into account the enormous experimental data

base existing on the non-equilibrium vibrational distributions of CO as well as the recent new indirect

determination of the reaction rate of the elementary process (13)[30].

As a conclusion the new interest toward the understanding of chemical reactions under non-equilibrium

plasma conditions, in particular the role of direct and indirect vibrational mechanisms at low values of

E/N, opens new interesting perspectives. The state-to-state kinetics seems the best tool to describe the

complex phenomenology occurring in the plasma [33]. To this end a large effort for improving the nu-

merous necessary input data should be made taking into account old and new approaches. In particular

the rate coefficient of processes (7) and (8) should be improved, by further investigation on the dynamics

of the collisions involving highly vibrationally excited molecules. At the same time dedicated new exper-

iments should be welcome to be used for the validation of the present ideas. The final challenge should

be to apply these ideas to the dissociation of CO2 under non-equilibrium plasma conditions.

Acknowledgements

The research leading to these results has received funding from the European Community’s Seventh

Framework Programme (FP7/2007-2013) under grant agreement n. 242311.

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11

�

* The role of vibrational excitation in affecting N2 dissociation is put in

evidence.

* Calculations with new complete sets of e-N2 cross sections have been

performed.

* Vibrational mechanism dominates e-N2 dissociation for E/N<50 Td.

* Vibrational and electron dissociation mechanisms are competitive for

E/N>50 Td.