**2.4 Induced neutral gas perturbations**

Even if micro-discharges are non thermal plasmas, their propagation can affect the neutral background gas (Eichwald et al. 1997, Ono & Oda b, 2004, Batina et al, 2002). In all cases, the micro-discharges modify the chemical composition of the medium (Kossyi et al. 1992, Eichwald et al. 2002, Dorai & Kushner 2003). In fact, the streamer heads propagate high energetic electrons that create radicals, dissociated, excited and ionized species by collision with the main molecule of the gas. Indeed, we have to keep in mind the low proportion of electrons and more generally of charged particles present in non-thermal plasma. At atmospheric pressure, and in the case of corona micro-discharge, we have about one million of neutral particles surrounding every charged species. Therefore, the collisions chargedneutral particles are predominant. During the discharge phase (which is associated to the

Electro-Hydrodynamics of Micro-Discharges in Gases at Atmospheric Pressure 277

electric circuit model to calculate the micro-discharge impedance needed to follow the interelectrode voltage evolution. On the other hand, the calculated dissipated energy and momentum transfer are included as source terms in the background gas model in order to simulate the induced hydrodynamics phenomena like electric wind, pressure wave propagation, neutral gas temperature increase, etc. The neutral gas hydrodynamics influences both the discharge dynamics and the chemical kinetics results. For example, the charged transport coefficients depend on the neutral gas density and some main chemical reactions involving neutral species (like the three body reactions) are very dependant on both the gas temperature and density. Finally, the basic data models (Yousfi & Benabdessadok, 1996, Bekstein et al. 2008, Yousfi et al. 1998, Nelson et al. 2003) give the necessary parameters (such as the convective and diffusive charged and neutral transport coefficients, the charged-neutral and neutral-neutral chemical reaction coefficients, the fraction of the energy transferred to the gas from the elastic and inelastic processes, among

The electro-hydrodynamics model is an approximation of a more rigorous model. The kinetic description based on Boltzmann equations for the charged particles is probably the more rigorous theoretical approach. However the main drawback of the kinetics approach is linked to the treatment of the high number of electrons coming from ionization processes which involves huge computation times especially at atmospheric pressure. Therefore, the classical mathematical model used for solving the micro-discharge dynamics is the macroscopic fluid one also called the hydrodynamics electric model. Up to now, the most commonly used fluid model is the hydrodynamics first order model which involves the first two moments of Boltzmann equation (i.e the density and the momentum transfer conservation equation) for each charged specie coupled with Poisson equation for the space charged electric field calculation (Eichwald et al. 1996). In all cases, the momentum equation can be simplified into the classical drift-diffusion approximation. The obtained system of hydrodynamics equations is then closed by the local electric field approximation which assumes that the transport and reaction coefficients of charged particles depend only on the local reduced electric field E/N. The hydrodynamics approximation is valid as long as the relaxation time for achieving a steady state electron energy distribution function is short compared to the characteristic time of the discharge development. At atmospheric pressure and because of the high number of collisions, the momentum and energy equilibrium times are generally small compared to any macroscopic scale variations of the system. In the hydrodynamics approximation, the coupled set of equations that govern the micro-

> . *<sup>c</sup> cc c <sup>n</sup> nv S c*

<sup>∂</sup> +∇ = ∀

0 *c c c*

*E V* = −∇

(1)

*V qn* = − (3)

(4)

= −∇ ∀ (2)

*t*

ε Δ

∂

. *<sup>c</sup> nv µE D n c cc c c*

others) needed to close the total equation systems.

discharge evolution is the following:

current pulse), the radical and excited species are created inside the micro-discharge volume. But during the post-discharge phase (i.e. between two successive current pulses) these active species react with the other molecules and atoms and diffuse in the whole reactor volume. If a gas flow exists, they are also transported by the convective phenomena. However, convective transport can also be induced by the micro-discharges themselves. Indeed, the momentum transfers between heavy charged particles and background gas are able to induce the so called "electric wind". The random elastic collisions between charged and neutral particles directly increase the gas thermal energy. Furthermore, the inelastic processes modify the internal energy of some molecules thus leading to rotational, vibrational and electronic excitations, ionisation and also dissociation of molecular gases. After a certain time, the major part of these internal energy components relaxes into random thermal energy. However, during the lifetime of micro-discharges (some hundred of nanoseconds), only a fraction of this energy, which in fact corresponds mainly to the rotational energy and electronic energy of the radiative excited states, relaxes into thermal form. The other fraction of that energy, which is essentially energy of vibrational excitation, relaxes more slowly (after 10-5s up to 10-4s). The thermal shock during the discharge phase can induce pressure waves and a diminution of the gas density and the vibrational energy relaxation can increase the mean gas temperature (Eichwald et al. 1997). All these complex phenomena induce memory effects between each successive micro-discharge. In fact, the modification of the chemical composition of the gas can favour stepwise ionisation with the pre-excited molecule (like metastable and vibrational excited species), the gas density modification influences all the discharge parameters which are function of the reduced electric field E/N (E being the total electric field and N the background gas density) and the three body reaction that are also function of the gas density. Furthermore, the local temperature increase also modifies the gas reactivity because the efficiency of some reactions depends on the gas temperature following Arrhenius law. Therefore, the complete simulation of micro-discharges has to take into account all these complex phenomena of discharge and gas dynamics.

## **2.5 The complete micro-discharge model in the hydrodynamics approximation**

The complete simulation of the discharge reactor, in complement to experimental studies can lead to a better understanding of the physico-chemical activity triggered during microdischarge development and relaxation. Nowadays, in order to take into account the complex energetic, hydrodynamics and chemical phenomena that can influence the corona plasma process, the full simulation of the non thermal plasma reactor can be undertaken by coupling the following models:


Each model gives specific information to the others. For example, the electrohydrodynamics model gives the morphology of the micro-discharge, the electron density and energy as well as the energy dissipated in the ionized channel by the main chargedneutral elastic and inelastic collision processes. This information is coupled with the external

current pulse), the radical and excited species are created inside the micro-discharge volume. But during the post-discharge phase (i.e. between two successive current pulses) these active species react with the other molecules and atoms and diffuse in the whole reactor volume. If a gas flow exists, they are also transported by the convective phenomena. However, convective transport can also be induced by the micro-discharges themselves. Indeed, the momentum transfers between heavy charged particles and background gas are able to induce the so called "electric wind". The random elastic collisions between charged and neutral particles directly increase the gas thermal energy. Furthermore, the inelastic processes modify the internal energy of some molecules thus leading to rotational, vibrational and electronic excitations, ionisation and also dissociation of molecular gases. After a certain time, the major part of these internal energy components relaxes into random thermal energy. However, during the lifetime of micro-discharges (some hundred of nanoseconds), only a fraction of this energy, which in fact corresponds mainly to the rotational energy and electronic energy of the radiative excited states, relaxes into thermal form. The other fraction of that energy, which is essentially energy of vibrational excitation, relaxes more slowly (after 10-5s up to 10-4s). The thermal shock during the discharge phase can induce pressure waves and a diminution of the gas density and the vibrational energy relaxation can increase the mean gas temperature (Eichwald et al. 1997). All these complex phenomena induce memory effects between each successive micro-discharge. In fact, the modification of the chemical composition of the gas can favour stepwise ionisation with the pre-excited molecule (like metastable and vibrational excited species), the gas density modification influences all the discharge parameters which are function of the reduced electric field E/N (E being the total electric field and N the background gas density) and the three body reaction that are also function of the gas density. Furthermore, the local temperature increase also modifies the gas reactivity because the efficiency of some reactions depends on the gas temperature following Arrhenius law. Therefore, the complete simulation of micro-discharges has to take into account all these complex phenomena of

**2.5 The complete micro-discharge model in the hydrodynamics approximation** 

The complete simulation of the discharge reactor, in complement to experimental studies can lead to a better understanding of the physico-chemical activity triggered during microdischarge development and relaxation. Nowadays, in order to take into account the complex energetic, hydrodynamics and chemical phenomena that can influence the corona plasma process, the full simulation of the non thermal plasma reactor can be undertaken by



discharge and gas dynamics.

coupling the following models:


evolution,


The electro-hydrodynamics model is an approximation of a more rigorous model. The kinetic description based on Boltzmann equations for the charged particles is probably the more rigorous theoretical approach. However the main drawback of the kinetics approach is linked to the treatment of the high number of electrons coming from ionization processes which involves huge computation times especially at atmospheric pressure. Therefore, the classical mathematical model used for solving the micro-discharge dynamics is the macroscopic fluid one also called the hydrodynamics electric model. Up to now, the most commonly used fluid model is the hydrodynamics first order model which involves the first two moments of Boltzmann equation (i.e the density and the momentum transfer conservation equation) for each charged specie coupled with Poisson equation for the space charged electric field calculation (Eichwald et al. 1996). In all cases, the momentum equation can be simplified into the classical drift-diffusion approximation. The obtained system of hydrodynamics equations is then closed by the local electric field approximation which assumes that the transport and reaction coefficients of charged particles depend only on the local reduced electric field E/N. The hydrodynamics approximation is valid as long as the relaxation time for achieving a steady state electron energy distribution function is short compared to the characteristic time of the discharge development. At atmospheric pressure and because of the high number of collisions, the momentum and energy equilibrium times are generally small compared to any macroscopic scale variations of the system. In the hydrodynamics approximation, the coupled set of equations that govern the microdischarge evolution is the following:

$$\frac{\partial \mathfrak{m}\_c}{\partial t} + \vec{\nabla} \cdot \mathfrak{m}\_c \vec{\upsilon}\_c = \mathbf{S}\_c \qquad \forall c \tag{1}$$

$$
\mu\_{\rm c} \vec{v}\_{\rm c} = \mu\_{\rm c} \vec{E} - \stackrel{\rightleftharpoons}{D}\_{\rm c} \vec{\nabla} n\_{\rm c} \qquad \forall \sigma \tag{2}
$$

$$
\varepsilon\_0 \Delta V = -\sum\_c q\_c n\_c \tag{3}
$$

$$
\vec{E} = -\vec{\nabla} \, V \tag{4}
$$

τ

conductivity and *<sup>v</sup>*

power *j*.*E* 

variation.

ε

discharge phases relaxes after a mean delay time *<sup>v</sup>*

explained, all the discharge parameters ( *Sc* , *µc* , *Dc*

electron energy. The whole set of data includes:

electronic excitation coefficients,

approximation is not assumed valid)

diffusion coefficients,

processes,

Electro-Hydrodynamics of Micro-Discharges in Gases at Atmospheric Pressure 279

flux due to concentration and thermal gradients, Si the net rate of production per unit volume (due to chemical reactions between neutral species) and *Sic* simulates the creation of new neutral active species during the discharge phase by electron or ion impacts with the main molecules of the gas. h is the static enthalpy, T the temperature, k the thermal

generally assumed that the translational, rotational and electronic excitation energies relax quasi immediately into thermal form and that the vibrational energy stored during the

total momentum transferred from charged particles to the neutral ones. As already

dependent on the reduced electric field (E/N). Therefore the coupling of all the set of equations (1) to (9) for each charged and neutral chemical species will considerably enhance the complexity of the global hydrodynamics model. In fact, each gas density variation can directly affect the development of micro-discharges through the reduced electric field

Finally, the modelling of complex phenomena occurring inside non-thermal reactor filled with complex gas mixtures needs the knowledge of the electron, the ion and the neutral transport and reaction coefficients. The charged and neutral particles kinetics model is therefore one of the method in complement to the experimental one that can be used to calculate or complete the set of basic data. Concerning the charged particles, the more appropriate method to obtain the unknown swarm data is to use a microscopic approach (e.g. a Boltzmann's equation solution for the electron data and a Monte Carlo simulation for the ion data) based on collision cross sections (Yousfi & Benabdessadok, 1996, Bekstein et al. 2008, Yousfi et al. 1998, Nelson et al. 2003). On the other hand the most commonly used method to calculate the neutral swarm data in a gas mixture is the use of the classical kinetic theory of neutral gas mixture (Hirschfielder et al. 1954). The macroscopic charged particles swarm data are given over a large range of either the reduced electric field or the mean





The calculation of the scalar (e.g. ionization or attachment frequencies), vectorial (drift velocity), and tensorial (diffusion coefficients) hydrodynamics electron and ion swarm parameters in a gas mixture, needs the knowledge of the elastic and inelastic electronmolecule and ion-molecule set of cross sections for each pure gas composing the mixture. Each collision cross section set involves the most important collision processes that either

the vibrational energy. *Sh* and *Sv* are the fraction of the total electron

of some tens of micro-seconds. *Sqm*

transferred during the discharge phase into thermal and vibrational energy. It is

, *Sic* , *Sqm*

τ

, *Sh* and *Sv* ) are strongly

the diffusive

is the

the stress tensor. For each chemical species "i", mi is the mass fraction, *iJ*

These first four equations allow to simulate the behaviour of each charge particle "c" in the micro-discharge (like for example e, N2 +, O2 +, O4 +, O- , O2 - , among others). *nc* , *<sup>c</sup> v* , *Sc* ,

*µc* , *Dc* , *<sup>c</sup> q* are respectively the density, the velocity, the source term, the mobility, the diffusive tensor and the charge of each charge specie "c" involved in the micro-discharge. *V* and *E* are the potential and the total electric field. The source terms *Sc* represent for each charge specie the chemical processes (like ionization, recombination, attachment, dissociative attachment, among others) as well as the secondary emission processes (like photo-ionisation and photo-emission from the walls (Kulikovsky, 2000, Hallac et al. 2003, Segur et al. 2006)). The transport equations of charged particles are not only strongly coupled through the plasma reactivity but also through the potential and electric field equations. Indeed, in equation (3) the potential and therefore the electric field in equation (4) are directly dependant on the variation of the density of the charged species, obtained from solution of equations (1)-(2) requiring the knowledge of transport and reaction coefficients that in turn have a direct dependence on the local reduced electric field E/N. Therefore the simulation of micro-discharge dynamics needs fast and accurate numerical solver to calculate the electric field at each time step (especially in regions with high field gradients like near the streamer head and the electrode pin) and also to propagate high density shock wave.

Even if the solution of the first order hydrodynamics model allows a better understanding of the complex phenomena that govern the dynamics of charged particles in microdischarges, the experimental investigations clearly show that the micro-discharges have an influence on the gas dynamics that can in turn modify the micro-discharge characteristics. It is therefore necessary to couple the electro-hydrodynamics model with the classical Navier-Stockes equations of a compressible and reactive background neutral gas coupled with the conservation equation of excited vibrational energy (Byron et al. 1960, Eichwald et al. 1997).

$$\frac{\partial \rho m\_i}{\partial t} + \vec{\nabla} . \rho m\_i \vec{v} + \vec{\nabla} . \vec{J}\_i = \mathbf{S}\_i + \mathbf{S}\_{ic} \qquad \forall i \tag{5}$$

$$
\frac{
\partial \rho
}{
\partial t
} + \vec{\nabla} .\rho \vec{v} = 0 \tag{6}
$$

$$\frac{\partial \rho \vec{v}}{\partial t} + \vec{\nabla} \cdot \rho \vec{v} \vec{v} = -\vec{\nabla} P - \vec{\nabla} \cdot \vec{\tau} + \vec{S}\_{qm} \tag{7}$$

$$\frac{\partial \rho \hbar}{\partial t} + \vec{\nabla} \cdot \rho \hbar \vec{v} = \vec{\nabla} \cdot (k \vec{\nabla} T) + \frac{\partial P}{\partial t} + \vec{v} \cdot \vec{\nabla} P + \frac{\Rightarrow}{\tau} : \vec{\nabla} \vec{v} - \vec{\nabla} \cdot \sum\_{i} \vec{j}\_{i} h\_{i} + S\_{h} + \frac{\mathcal{E}\_{\upsilon}}{\tau\_{\upsilon}} \tag{8}$$

$$\frac{\partial \mathcal{E}\_v}{\partial t} + \vec{\nabla} \cdot \mathcal{E}\_v \vec{v} = \mathcal{S}\_v - \frac{\mathcal{E}\_v}{\mathbf{r}\_v} \tag{9}$$

The set of equations (5) to (9) are used to simulate the neutral gas behavior and to follow each neutral chemical species "i" (like N, O, O3, NO2, NO, N2 (A3∑u+), N2 (a'1∑u-), O2 (a1∆g), among others) that are created during the micro-discharge phase. In equations (5) to (9), ρ is the mass density of the background neutral gas, *v* the gas velocity, P the static pressure and

These first four equations allow to simulate the behaviour of each charge particle "c" in the

diffusive tensor and the charge of each charge specie "c" involved in the micro-discharge.

charge specie the chemical processes (like ionization, recombination, attachment, dissociative attachment, among others) as well as the secondary emission processes (like photo-ionisation and photo-emission from the walls (Kulikovsky, 2000, Hallac et al. 2003, Segur et al. 2006)). The transport equations of charged particles are not only strongly coupled through the plasma reactivity but also through the potential and electric field equations. Indeed, in equation (3) the potential and therefore the electric field in equation (4) are directly dependant on the variation of the density of the charged species, obtained from solution of equations (1)-(2) requiring the knowledge of transport and reaction coefficients that in turn have a direct dependence on the local reduced electric field E/N. Therefore the simulation of micro-discharge dynamics needs fast and accurate numerical solver to calculate the electric field at each time step (especially in regions with high field gradients like near the streamer head and the electrode pin) and also to propagate high density shock

Even if the solution of the first order hydrodynamics model allows a better understanding of the complex phenomena that govern the dynamics of charged particles in microdischarges, the experimental investigations clearly show that the micro-discharges have an influence on the gas dynamics that can in turn modify the micro-discharge characteristics. It is therefore necessary to couple the electro-hydrodynamics model with the classical Navier-Stockes equations of a compressible and reactive background neutral gas coupled with the conservation equation of excited vibrational energy (Byron et al. 1960, Eichwald et al. 1997).

> *i i i ic <sup>m</sup> mv J S S i <sup>t</sup>*

> > . 0 *v*

. . *qm <sup>v</sup> vv P S*

. .( ) . : . *<sup>v</sup> ii h*

 τ  τ

∂ ∂ (8)

*v*

 ε

τ

ρ

(5)

(7)

(6)

*i v*

the gas velocity, P the static pressure and

(9)

ε

τ

<sup>∂</sup> +∇ +∇ = + ∀

<sup>∂</sup> +∇ =

. . *<sup>i</sup>*

*t* ρ

∂ + ∇ = −∇ − ∇ +

*h P hv k T v P v J h S t t*

ε

<sup>∂</sup> +∇ = −

+∇ =∇ ∇ + + ∇ + ∇ −∇ + +

. *v v v v*

The set of equations (5) to (9) are used to simulate the neutral gas behavior and to follow each neutral chemical species "i" (like N, O, O3, NO2, NO, N2 (A3∑u+), N2 (a'1∑u-), O2 (a1∆g), among others) that are created during the micro-discharge phase. In equations (5) to (9), ρ is

*v S*

∂

ρ

*t* ε

∂

ρ

*t* ρ

∂ ∂

∂

ρ

∂

ρ

ρ

the mass density of the background neutral gas, *v*

+, O4

, *<sup>c</sup> q* are respectively the density, the velocity, the source term, the mobility, the

are the potential and the total electric field. The source terms *Sc* represent for each

+, O- , O2 -

, among others). *nc* , *<sup>c</sup> v*

, *Sc* ,

+, O2

micro-discharge (like for example e, N2

*µc* , *Dc* 

*V* and *E* 

wave.

τ the stress tensor. For each chemical species "i", mi is the mass fraction, *iJ* the diffusive flux due to concentration and thermal gradients, Si the net rate of production per unit volume (due to chemical reactions between neutral species) and *Sic* simulates the creation of new neutral active species during the discharge phase by electron or ion impacts with the main molecules of the gas. h is the static enthalpy, T the temperature, k the thermal conductivity and *<sup>v</sup>* εthe vibrational energy. *Sh* and *Sv* are the fraction of the total electron

power *j*.*E* transferred during the discharge phase into thermal and vibrational energy. It is generally assumed that the translational, rotational and electronic excitation energies relax quasi immediately into thermal form and that the vibrational energy stored during the discharge phases relaxes after a mean delay time *<sup>v</sup>* τ of some tens of micro-seconds. *Sqm* is the

total momentum transferred from charged particles to the neutral ones. As already 

explained, all the discharge parameters ( *Sc* , *µc* , *Dc* , *Sic* , *Sqm* , *Sh* and *Sv* ) are strongly

dependent on the reduced electric field (E/N). Therefore the coupling of all the set of equations (1) to (9) for each charged and neutral chemical species will considerably enhance the complexity of the global hydrodynamics model. In fact, each gas density variation can directly affect the development of micro-discharges through the reduced electric field variation.

Finally, the modelling of complex phenomena occurring inside non-thermal reactor filled with complex gas mixtures needs the knowledge of the electron, the ion and the neutral transport and reaction coefficients. The charged and neutral particles kinetics model is therefore one of the method in complement to the experimental one that can be used to calculate or complete the set of basic data. Concerning the charged particles, the more appropriate method to obtain the unknown swarm data is to use a microscopic approach (e.g. a Boltzmann's equation solution for the electron data and a Monte Carlo simulation for the ion data) based on collision cross sections (Yousfi & Benabdessadok, 1996, Bekstein et al. 2008, Yousfi et al. 1998, Nelson et al. 2003). On the other hand the most commonly used method to calculate the neutral swarm data in a gas mixture is the use of the classical kinetic theory of neutral gas mixture (Hirschfielder et al. 1954). The macroscopic charged particles swarm data are given over a large range of either the reduced electric field or the mean electron energy. The whole set of data includes:


The calculation of the scalar (e.g. ionization or attachment frequencies), vectorial (drift velocity), and tensorial (diffusion coefficients) hydrodynamics electron and ion swarm parameters in a gas mixture, needs the knowledge of the elastic and inelastic electronmolecule and ion-molecule set of cross sections for each pure gas composing the mixture. Each collision cross section set involves the most important collision processes that either

Electro-Hydrodynamics of Micro-Discharges in Gases at Atmospheric Pressure 281

radicals and excited species. In fact, and as already explained in the previous sections, in the non-thermal plasma reactor, the majority of the injected electrical energy goes into the generation of energetic electrons, rather than into gas heating. The energy in the microplasma is thus directed preferentially to electron-impact dissociation, excitation and ionization of the background gas to generate active species that, in turn, induce the chemical activation of the medium. As a consequence, the non-thermal plasma reactors at atmospheric pressure are used in many applications such as flue gas pollution control (Fridman et al., 2005, Urashima et Chang, 2010), ozone production (Ono & Oda b, 2004), surface decontamination (Clement et al., 2001, Foest et al., 2005) and biomedical field (Laroussi, 2002, Villeger et al., 2008, Sarrette et al., 2010). For many applications, particularly in the removal of air pollutants, decontamination or medicine field, the non-thermal plasma approach would be most appropriate because of its energy selectivity and its capability for

In micro-discharges the active species are created by energetic electrons during the primary and the secondary streamer propagation that last some hundred of nanoseconds. Despite these very fast phenomena, the energy transferred to the gas can initiate shock waves starting from the stressed high voltage electrodes. Furthermore, a part of the electronic energy is stored in the vibrational energy that relaxes in thermal form after some tens of microseconds. Anyway, it is worth to notice, that all the initial energy (chemical, thermal, among others) is transferred inside a very thin discharge filament i.e. in a very small volume compared with the volume of the plasma reactor. Therefore, the efficiency of the processes is correlated to the radical production efficiency during the discharge phase, the number of micro-discharges that cross the inter-electrode gap, the repetition frequency of the discharge and how the radicals are diffused and transported from the micro-discharge towards the

In the following sections, the discharge and the post-discharge phase are simulated using the hydrodynamics models presented in section 2.5 in the case of a DC positive pin-to-plan

The simulation conditions are described in detail in reference (Eichwald et al. 2008) as well as the used numerical methods and boundary conditions. To summarize, a DC high voltage of 7.2kV is applied on the pin of a pin-to-plane reactor filled with dry air at atmospheric pressure. The inter-electrode gap is of 7mm, the pin radius is equal to 25µm and photoionisation phenomenon is taken into account in the simulation. Results in Fig. 10 and 11 are

selected reactions. Because of the time scale of the discharge phase (some hundred of nanoseconds), the radical atoms and the main neutral molecules (N2 and O2) are supposed to remain static during the discharge phase simulation. Fig. 10 shows the reduced electric field (E/N) expressed in Td (1Td=10-21 Vm2 so that 500Td at atmospheric pressure is equivalent to an electric field of 12MVm-1). When the high voltage is applied to the pin, some seed electrons are accelerated in the high geometric electric field around the pin. A luminous spot is observed experimentally near the pin thus indicating the formation of excited species due to a high electronic energy. On can notice that the electrons move towards the pin. Furthermore, the electrons gain sufficient energy to perform electronic

+, N+ and O+) and two radical atoms (O, N) reacting following 10

and O2-), four

obtained by coupling equations (1) to (4) for electrons, two negative ions (O-

simultaneous treatment of pollutants, bacteria or cells for example.

whole reactor volume.

positive ions (N2+, O2

**3.2 Discharge phase simulation** 

corona reactor in dry air at atmospheric pressure.

affect the charged species transport coefficients or are needed to follow the charged species chemical kinetics and energy or momentum exchange. For example, in order to calculate the macroscopic electron swarm parameters in water vapor, 21 collision cross sections must be known involving the rotational, the vibrational and the electronic excitation processes as well as the ionization, the dissociative attachment and the superelastic processes.

One of the main difficulties is to validate for each pure gas that compose the mixture the chosen set of cross sections. To do that, a first reliable set of electron-molecule and ionmolecule cross section for each individual neutral molecule in the gas mixture must be known. Then, in order to obtain the complete and coherent set of cross sections, it is necessary to adjust this first set of cross sections so as to fit experimental macroscopic coefficients with the calculated ones estimated from either a Boltzmann's equation solution or a Monte Carlo simulation. The obtained solution is certainly not unique but as the comparisons concern several kinds of swarm macroscopic parameters having different dependencies on cross sections (ionization or attachment coefficient, drift velocity, transverse or longitudinal diffusion coefficient) over a wide range of reduced electric field or mean electron energy, most of the incoherent solutions are rejected. Finally, when the sets of cross section are selected for each pure gas, they can be used to calculate with a Bolzmann's equation solution or a Monte Carlo simulation the macroscopic charged species transport and reaction parameters whatever the proportion of the pure gas in the background gas mixture.

#### **2.6 Summary**

Micro-discharges are characterized by the development of primary and secondary streamers. As a function of the high voltage applied on the small curvature electrode (DC or pulse), the micro-discharges show either a mono-filament or a large branching structure. The passage from multi-filaments to mono-filament structure can be observed if a sufficiently large high voltage pulse is applied. The transition can be explained through the memory effects accumulated during the previous discharge. The primary streamers propagate fast ionization waves characterized by streamer heads in which the electric field is high enough to generate high energetic electrons like in an electron gun. The streamer head propagates a high charge quantity toward the inter-electrode gap. The micro-plasmas are generated behind the streamer heads. They are small conductive channels that connect the streamer head to the electrode stressed by the high voltage. The primary streamers are then followed by a secondary streamer which is characterized by an electric field extension that ensures the transition between the displacement current and the conductive one when the primary streamer arrives on the cathode. Both primary and secondary streamers create radicals and excited species by electron-molecule impacts. The elastic and inelastic energy transfers generate a chemical activity, a thermal energy increase of the gas and a neutral gas dynamics. To better understand all these complex phenomena, a hydrodynamics model can be used based on conservation equations of charged and neutral particles coupled to Poisson equation for the electric field calculation.
