P Freton

J J Gonzalez

M Masquère

Frank Reichert
email: frank.reichert@siemens.com

Magnetic field approaches in DC thermal plasma modelling

Keywords:

The self-induced magnetic field has an important role in thermal plasma configurations generated by electric arcs as it generates velocity through Lorentz forces. In the models a good representation of the magnetic field is thus necessary. Several approaches exist to calculate the self-induced magnetic field such as the Maxwell -Ampere formulation, the vector potential approach combined with different kinds of boundary conditions or the Biot & Savart formulation. The calculation of the self-induced magnetic field is alone a difficult problem and only few articles of the thermal plasma community speak on this subject. In the present study different approaches with different boundary conditions are applied on two geometries to compare the methods and their limitations. The calculation time is also one of the criteria for the choice of the method and a compromise must be found between method precision and computation time. The study shows the importance of the current carrying path representation in the electrode on the deduced magnetic field. The best compromise consists of using the Biot & Savart formulation on the walls and/or edges of the calculation domain to determine the boundary conditions and to solve the vector potential in a 2D system. This approach provides results identical to those obtained using the Biot & Savart formulation over the entire domain but with a considerable decrease of calculation time.

I Introduction

The aim of this study is to asses various methods used to determine the contribution of the self-induced magnetic field created by the presence of the electric arc in thermal plasma modelling.

Numerous formulations have been proposed over the years to estimate the selfinduced magnetic field in DC or AC thermal plasma configurations. Chronologically, the first formulation, in the eighties, was the Maxwell -Ampere (M.A.) form which has been used in two-dimensional (2D) modelling in numerous articles such as Hsu [1] or Lowke [2]. It is very convenient in 2D on a structured grid but not really suited to an unstructured one. Moreover, it cannot be used in three dimensional (3D) modelling and due to the evolution of the models from 2D to 3D, alternatives formulations have been chosen in the nineties.

In order to estimate the self-induced magnetic field in 3D cases, the best way should be to use the Biot & Savart (B&S) equation. Some authors have chosen this formulation [3][4][START_REF] Daube | 3D simulation of a low voltage switching arc based on MHD equations[END_REF].

Nevertheless, for each point of the calculation domain it requires volume integration which can consume more than 90% of the computational time. Instead of using the B&S formulation, almost all authors use the potential vector (PV) approach ( [START_REF] Bernardi | [END_REF][7][8][9][10][11] for example).

The advantage of this formulation is that each vector potential component can be written as one equation having the same form as the generalized equation used as for the fluid description.

Nevertheless, the use of the PV formulation implies that care should be taken in the choice of boundary conditions. Indeed, the potential vector should be equal to zero at infinity. This specific condition cannot be reached in all models (problems in constricted parts and necessity to assume a large calculation domain). This point has been studied by some authors in RF plasmas configurations where people propose to calculate the magnetic field in an extended domain [12] [13] or by B&S equations [12]. In AC/DC, in order to overcome this problem, the authors very often consider a null flux as boundary condition for vector potential components at the modelled geometry edges [9][10] or a value equal to zero if the boundary is far from the arc [START_REF] Bernardi | [END_REF][7][8]. Very few authors, as He-Ping Li [11], propose, only for three dimensional DC configurations the use of Biot & Savart equation as a boundary condition for potential vector.

In this article, we want to point out that the MA formulation can be used only in the event of zero axial gradient of the current density. We would show that the use of the potential vectors without good boundary conditions can lead to wrong results and also to conclude on an alternative solution by studying a mixed method based on B&S and P.V.

formulations.

In order to present the study and the developments, a simple 2D configuration of a transferred arc burning in argon at atmospheric pressure has been chosen. A fluid model is used. For the electromagnetic part, the three "classical" formulations presented above (MA, PV and B&S) and the "mixed approach" are used to estimate the self-induced magnetic field.

A comparison is made on the results obtained with a current intensity value equal to 600A and we conclude on the importance of correctly taking into account the formulation for the selfinduced magnetic field description.

In the first section, we present the model and we detail various formulations to estimate the self-induced magnetic field and particularly the mixed solution. The first results obtained for an academic case corresponding to a cylinder crossed by a constant current intensity will be presented to validate our approach. Then the methods are applied to two geometries to test the method validity. The limits of the method are given in the conclusion.

II/ The model 1/ Fluid model

In order to describe the arc, we used a stationary fluid model. The Navier -Stokes equation is solved coupled with the mass and energy conservation equations. All these equations are written in the form of the generalised conservation equation (1) where φ is the variable solved, ρ the mass density, v the velocity vector, Γ φ the diffusion coefficient and S φ the source term. In our case, we consider this equation in a 2D axisymetrical stationary configuration. All the equations solved for the fluid model are summarized in table 1.

( ) ( ) φ φ φ φ S + = v ∇ ∇ ∇ (1)
In table1, v r and v z represent the radial and axial velocity components respectively, h is the enthalpy, κ and µ the thermal conductivity and the viscosity. P is the pressure, j r and j z the radial and axial current density components, σ and C p the electrical conductivity and the heat capacity, ε n the net emission coefficient.

2/ Electric and electromagnetic model a) Electric potential equation

In order to describe the electrical properties of the arc, we solved the potential scalar equations designated by the variable V presented in table 2.

From this potential, the axial and radial current density components j z and j r (equations (7-8))

are calculated.

z V = j z ∂ ∂ - (7) r V = j r ∂ ∂ - (8

-Maxwell Ampere equation

In the eighties and nineties some authors (for example [1][2]) proposed to calculate the self-induced magnetic field by using the equation:

R z dr rj R µ = B 0 0 (9)
In this equation (9), R is the integrated radius where the axial component of the current density exists, and µ 0 the vacuum permittivity. From this equation the azimuthal magnetic field induced by the arc current is calculated. This equation comes from the Maxwell-Ampere equation:

j µ = B 0 ∧ ∇ (10)
Nevertheless, equation ( 10) is a vectorial equation and must be verified for each component of the magnetic field vector. So, in case of an axisymetrical system, two equations must be satisfied for the azimuthal magnetic field:

( )

z j µ = rB r r 0 1 ∂ ∂ (11) r j µ = z B 0 ∂ ∂ - (12)
Assuming only equation ( 9) to deduce the azimuthal component ignores equation ( 12) and considers B θ invariant with the axial coordinate. Moreover, this formulation is only suitable in structured 2D cases.

-Biot & Savart formulation

It is possible to obtain the magnetic field using the B&S equation:

( ) | | - - ∧ ⋅ Volume dV ' r r ' r r j µ = B 3 0 4 ( 13
)
To use this equation integration over the whole volume must be done for each point. This formulation is valid in 2D as well as in 3D geometries. Some authors have proposed the selfinduced magnetic field be calculated by using this approach [3][4][START_REF] Daube | 3D simulation of a low voltage switching arc based on MHD equations[END_REF]. Nevertheless, the B&S formulation uses a large part of the computational time as integration over the volume must be done for each point of the mesh.

-Potential vector formulation with null flux boundary condition or null value boundary condtion

Nowadays, in order to estimate the magnetic field almost all authors use the potential vector formulation:

A = B ∧ ∇ ( 14
)
Where A is the potential vector which must verify equation (15), under the Poisson Gauge condition.

j µ = A 0 ⋅ (15)
In 2D axisymetrical configuration this leads to the three following equations:

z z j µ = z A 0 2 2 ∂ ∂ (16) r A + j µ = r A r r r r r r 0 1 - ∂ ∂ ∂ ∂ ( 17
) z A r A = B r z ∂ ∂ - ∂ ∂ ( 18
)
This formulation is very useful as equations ( 16) and ( 17) can be written in the general form of equation (1) giving a correct estimation of the self-induced magnetic field, taking into account all the components of the current density. Nevertheless, with this formulation a problem remains concerning the choice of the boundary conditions. Indeed, it is generally considered that the components of potential vector A tend to zero at an infinite distance. As infinity cannot be reached in the calculation domain, almost all authors consider a null flux for the boundary conditions of equations ( 16) and [START_REF] Conway | [END_REF] or a null value far from the arc [START_REF] Bernardi | [END_REF][7][8][9][10].

In this study, we will call this approach potential vector formulation with null flux boundary condition (PVNF) and the approach with a null value of the potential vector (PVNV)

-Potential vector formulation with B & S boundary condition

Another way to obtain the values of the potential vector components consists in using the B&S formulation:

| | dV ' r r j µ = A volume - 4 0 (19)
Obviously, the resolution of equation ( 19), like that of equation (13), is time consuming. Nevertheless, it can only be used in boundary conditions, whereas within the computational domain equations [START_REF] Abramovitz | Handbook of mathematical functions New York Dover Publications[END_REF][START_REF] Conway | [END_REF](18) should be used to obtain the self-induced magnetic field. We call this the mixed formulation (MF)

The development of equation (15) in axisymetrical cases requires the use of elliptical integrals. The mathematical approach used is presented in annex A.

III/ Validation of the electromagnetic models

To validate the magnetic field formulations presented above, we defined a theoretical case consisting of, a constant current density, crossing a cylinder as presented in figure 1. The cylinder has a radius of 2mm and is 40mm long. We assumed that it is crossed by a constant current density of 4.10 6 A/m 2 resulting to a current of 50A. The current is uniformly distributed in the central part (r<R). R is the radius of the part where the axial current density is not equal to zero.

The length of the cylinder is ten-fold its diameter. For this geometry, it is possible to assume analytical equations for A z , and B θ :

( ) ( ) R < r if r j µ = r B R > r if R j µ = r B z z 2 2r 0 2 0 (20) ( ) ( ) ( ) ( ) R < r if r j µ A = r A R > r if R r R j µ A = r A z z z z z z 4 0 ln 2 1 2 0 2 0 2 0 - + - (21)
These equations correspond to the case of an infinite cylinder but, in the present case, are valid in the center plane of the cylinder (central line in figure 1) Calculations were made for the PVNF, PVNV and the MF models. A comparison of the results obtained for the three approaches and the theoretical data given by equation (20) for the azimuthal value of the magnetic field are compared in figure 2.

In the cylinder (r<R=2mm), a good agreement was found between the theoretical result and the magnetic field calculated with the three approaches (MF, PVNV and PVNF).

Outside the cylinder, where there is no current, the MF and PVNV approaches give a better estimation of the magnetic field than PVNF. This is clearly due to the null flux boundary condition in the PVNF calculation. However, at the same time, this simple case does not consider a current density variation in axial or radial directions and only the axial component was considered. It is known that the distance of the boundary condition from the conducting zone can affect the accuracy of the results [12][13]. This point will be studied further in the geometry involving the electric arc.

IV/ Study of PVNF and MF approaches in thermal plasmas

1/ Geometries studied

In order to study the influence of various magnetic field formulations on the arc behaviour, we compared two academic configurations, presented in figure 3. In both cases we consider a transferred arc.

The first geometry (Geom1), consists of a cathode (AA'BB') (radius 2mm , length 5mm) and an anode (YY'ZZ') (3.5mm; 4mm). Near the cathode, an inlet (BC) is assumed.

Segment (CDW) is a wall located 15mm from the axis. Segment (WXY) is a pressure outlet enabling plasma flow out of the domain. This first geometry was chosen large enough for the arc to be able to expand freely within the volume. It is intended to be representative of a kind of free burning arc.

The second geometry Geom2 was constructed to be representative of cases where the plasma is more constricted (like in plasma torches or circuit breaker configurations).

Moreover, with this geometry we can observe the influence of the distance between the arc fringes and the radial boundary conditions. Also, we can study if ignoring the radial current density component in some formulations leads to a poor estimation of the self-induced magnetic field. Geom2 has the same electrode dimensions as Geom1. The inlet dimension (BC) is unchanged. Segment (CDEFG) is a wall that constitutes a convergent -divergent nozzle. Lines 1&2 are radial segments where quantities such as the temperature, the velocity or the magnetic field will be presented.

The choice of these two geometries was made to evaluate the influence of the magnetic field calculation for different thermal plasma configurations.

2/ Arc parameters and boundary conditions

In both Geom1 and Geom2, air plasma at atmospheric pressure is assumed. The current intensity is equal to 600A and the mass inlet flow (BC boundary) is 25Nl/min. The boundary conditions, except for the potential vectors, are given in table 3.

Concerning the electrical potential, a parabolic profile is imposed on the cathode tip. This is given by equation ( 22) where j max is the maximum value of the current density (chosen equal to 1.2 10 8 A/m 2 ) and R cat the radius verifying equation (24) for current intensity I. The radial component of the current density is assumed equal to zero.

2 max ) 1 ( ) ( cat z R r j r j - = if r < R cat (22) 0 ) ( = r j z if r > R cat ( 23
)
rdr j I cat R z = 0 2π (24)
3/ Boundary conditions for the magnetic field

In order to compare the various formulations, we considered several cases, described in table 4 for Geom1 and table 5 for Geom2. The generic number of the cases n°"X-Y" where X corresponds to the number of the geometry and Y the number of the case. All the cases and formulations are presented above and summarized in table 6.

-Cases 1-1, 1-2, 1-3: In some cases, we applied the mixed formulation by taking into account the current density distribution only within the plasma. Nevertheless, current densities also exist in the electrodes and particularly in the cathode. This current in the cathode plays a nonnegligible role on the potential vector amplitude and thus on the magnetic field. In order to study the influence of the cathode, we tested three cases. In case 1-2 only the current density distribution within the plasma is taken into account. In case 1-1 a constant distribution of the axial current density is imposed within the cathode. In the last case (case 1-3) a null flux of the potential vector components near the electrodes is assumed. For case 1-1, a constant distribution for j z within the cathode is used. Indeed, Gonzalez [14] have shown that the profile of the current density in the cathode is almost constant and that the change of this constant profile to a more complex one at the boundary between the electrode and the plasma occurs only in a very thin zone.

-Cases 1-4, 1-5: Case 1-4 is devoted to the results of the PVNF formulation applied to Geom1. This case corresponds to the way this method is usually used in the literature.

Nevertheless, to complete and compare the results with case 1-1, we have studied the PVNF formulation taking into account the current density within the cathode (like for case 1-1). This corresponds to case 1-5.

-Case 1-6 is devoted to the test of the PVNV formulation. Nervetheless, due to the results of cases 1-4 and 1-5, only the case taking into account the current density in the cathode is presented.

-Case 1-7 concerns the situation using the MA formulation in Geom1.

-Cases 2-1, 2-2, 2-3, and 2-4 are a repetition of cases 1-1, 1-4, 1-5 and 1-7 for the constricted geometry Geom2.

-Cases 2-5, 2-6, 2-7 and 2-8 : As Geom2 is constricted, we have also tested to impose the boundary conditions for the potential vector in a "radially" extended domain. Thus, for case 2-5 the domain is extended up to the radial value of segment (WX) (that is to say 15mm) and PVNF is applied. Case 2-6 corresponds to the same extended domain but with PVNV formulation. Finally, cases 2-7 and 2-8 are PVNF calculations for an extended radial domain up to respectively 9mm and 6mm.

4/ Results for Geom1.

a) General results for Geom1

First, we present "general" results obtained with geometry Geom1. As shown below, the more correct results are obtained by case 1-1. So temperature fields, velocity and current density components are proposed for this case in figures 4&5 in order to present the general properties of the arc created. In the right hand of figure 4, it is seen that a temperature maximum of 31kK is reached on the axis near the cathode. On the axis, the temperature decreases from this maximum to 12kK near the anode. Along the radial direction the arc expands as expected for this kind of geometry. Concerning the velocity a maximum of 1500m/s is reached under the cathode. In this area, the velocity is essentially created by the Lorentz force produced by the interaction of the self-induced magnetic field and the current densities of the plasma. The velocity decreases along the axis.

The components of the current density in the geometry are presented in figure 5. The axial component j z is maximal on the cathode where the parabolic condition ( 22) is imposed.

As it exists an expansion of the arc, there is an increase of the conductive section along the axis and thus a decrease of the magnitude of the axial current density. Within the cathode, the constant axial current density that we assumed can be observed. Concerning the radial component of the current density, it is maximal near the cathode with a value of around -7 10 7

A/m 2 . This maximum is out of the axis. The magnitude of the radial current density decreases away from the cathode.

b) Influence of the magnetic field formulation in Geom1

In this section, we compare the magnetic field obtained for the different cases (Table 4). The magnetic field obtained in case 1-1 is presented in figure 6. We can see that it is maximal on the side of the cathode with a value of around -70mT. This magnetic field is compared on the right part of figure 6 to the one obtained by a numerical calculation of equation ( 13) for all points of the geometry. We can conclude to a good agreement between the two cases. The magnetic field representation found for case 1-1 is similar to that deduced from the B&S formulation. Due to the good agreement, the other case will be compared with case 1-1. As the B&S formulation is the exact one, the comparison validates the MF method taking the current circulation in the cathode into account.

In figure 7, a comparison is given between case 1-1 (MF where the current in the cathode is considered) and case 1-2 (MF where the current in the cathode is not considered).

The aim now was to study if the current circulation on the cathode really needs to be considered as this point is never presented in the literature. Dashed lines correspond to levels A to K for case 1-1 (Left hand side). These lines are reported on the right hand side, related to case 1-2. The dashed lines must therefore be compared to the continuous lines. All figures involving a magnetic field use a similar representation. The magnitude of the magnetic field obtained in case 1-2 in the vicinity of the upper electrode is clearly smaller than in case 1-1, with a value near the cathode equal to -60mT. This will lead to differences on the estimated convective Lorentz forces. Axially, even if the behaviour of the magnetic field obtained in the two cases is similar, the values obtained without taking the current in the cathode into account remain smaller than those obtained in case 1-1. From this point, we can therefore conclude that the authors who consider the MF (like He Ping Li [11]) or B&S formulation without considering the current in the cathode, underestimate the magnetic field.

In figure 8, we compare case 1-1 and case 1-3, where null flux is considered for potential vector components on the arc attachment locations. In case 1-3, the magnetic field obtained near the cathode is more than 80mT and so higher than in case 1-1. Nevertheless, in other parts of the calculation domain, the differences between the two fields are weak.

In Figure 9 we plot the magnetic field obtained considering potential vectors with null flux conditions applied at all the boundaries (case 1-4). In this case, neither the radial nor the axial expansions seem to be well predicted. Furthermore, the null flux boundary condition on the potential vector leads to an overestimation of the magnetic field close to the cathode. This case corresponds to the most classical boundary conditions used in the literature when authors use PVNF formulation [9][10].

In order to compare with the MF case where current density is taken into the cathode we achieved a calculation of PVNF with this condition (case 1-5). The magnetic field obtained is plotted in figure 10. We can observe that the magnetic field obtained for PVNF taking into account the cathode is in quite good agreement with MF results on the centre of the plasma. Nevertheless, in the radial direction, the differences between the two formulations remain important.

In figure 11, comparison between case 1-1 and case 1-6 (PVNV taking into account the cathode) is presented. We can observe that a better agreement is obtained for this case even if PVNV condition seems to overestimate the radial values of the magnetic field.

Finally, in figure 12 we plot the magnetic field obtained by the MA equation (ignoring equation ( 12), case 1-7) and MF (case 1-1). The magnetic field obtained by the MA formulation is higher in the axial and radial directions than that obtained with the MF formulation. The results close to the cathode are suitable compared to those obtained in other parts of the domain such as in the vicinity of the anode.

In order to estimate the influence of these differences on arc properties, axial velocity and temperature profiles obtained in various cases are plotted in figures 13-14-15. In figure 13), we can observe that the behaviour of the velocity profiles following the different formulations is similar with a maximum value located at the position z=12mm. This maximum decreases from 12mm to the anode position. Cases 1-3, 1-4 and 1-7 lead to similar profile evolutions with a maximum value of 1500m/s at z=12 mm. For these three cases, the velocity is overestimated compared to case 1-1 which corresponds to the MF taking into account the cathode. In cases 1-1, 1-5 and 1-6 the maximum velocity value is 1350m/s. For these three cases, the current density is taken within the cathode and the formulation used seems not to play a role on the evolution of the velocity on the axis. This was not the case for magnetic field values.

Finally, due to the underestimation of the magnetic field in case 1-2, the velocity is clearly underestimated too.

The axial temperature variations following the choice of the different formulations used to represent the magnetic fields are reported in figure 13b. Except for case 1-2, the temperature profiles obtained for the various cases is almost the same. The most important difference of 700K between the temperature profiles is obtained between case 1-1 and the others at z=15mm.

We present (Figures 14a&b andFigures 15a&b) the radial evolutions for the axial velocity and the temperature for the two sections (line 1 and line 2) represented in the geometries of Figure 3.The velocity profiles are presented for the two lines. The conclusions are the same for the two lines: in cases 1-3, 1-4, 1-7 an overestimation of the axial velocity is found compared to the ones obtained in cases 1-1, 1-5 and 1-6. The velocity obtained in case 1-2 is underestimated.

The greatest difference occurs close to the axis for plasma radii lower than the electrode. This difference becomes lower going to the anode position.

For a temperature representation, the axial and radial temperature evolutions (Figure 13b and also Figures 15a&b) shows that MA, PVNF, MF, PVNV with or without current circulation in the cathode are sufficient to represent the temperature field correctly.

In figures 15a&b, we present the radial temperature profiles for line 1 and line 2. The differences in the approaches used to calculate the self-induced magnetic fields are very small for temperature variation.

5/ Results for geometry 2.

In this section, we studied if the conclusions obtained with Geom1 remain the same for a geometry where the arc is constricted (Geom2). All the cases studied are summarized in table 5. First, the general results obtained in case 2-1 (M.F. With cathode) are presented.

Secondly, the influence of the approach used to calculate the self-induced magnetic field is commented. For the second step, only PVNF, MA and MF are studied due to the fact that boundary conditions are too close to the conduction core to apply PVNV. Finally, a study on the position of an extended radial boundary condition for PVNF and PVNV is proposed.

a) General results for Geom2

The temperature and velocity fields obtained in case 2-1 are presented in figure 16.

For the velocity (left hand side in figure 16), the nozzle presence accelerates the fluid in order to maintain the mass flow rate. The maximum plasma velocity is 2000m/s and is reached on the axis at the nozzle exit. The maximum temperature (around 31kK) is obtained close to the cathode tip. In the nozzle, the temperature decreases to 16kK at the nozzle exit. ). These differences are present in the axial and radial directions. Furthermore, in the nozzle, the null flux boundary condition leads to a quasi null value of the magnetic field which is not physical. The trend obtained in the previous geometry with the comparison of formulations MA and MF with cathode is conserved (figure 20). Thus, MA leads to an overestimation of the magnetic field in the radial direction and to a lesser extension, in the axial direction.

As with Geom1 we present the influence of the magnetic field formulation on the velocity and temperature profiles which are plotted along the axis in figures 21a&b. The axial velocity profile on the axis is different for the four cases. In all geometries, MA and PVNF without taking into account the current within the cathode lead to an overestimation of the velocity compared with MF. The greatest difference (350m/s) is obtained at z=12.5mm. Note that differences also exist between MA and PVNF, particularly in the throat of the nozzle and after the exit (z=25mm). For case 2-3 which corresponds to the PVNF where current in the cathode has been taken into account, before the nozzle throat, the values of the velocity are close to the one of the MF case. Nevertheless, after the throat, the velocity is clearly underestimated, probably due to the bad estimation of the magnetic field.

Whereas velocities on the axis are different, the axial temperature profile does not seem to depend on the formulation used for the magnetic field. c) Influence of the domain dimension for the boundary condition for PVNF.. It seems that PVNF work "well" when null flux is applied to a boundary far from the conduction zone. In this section, we achieve a calculation, in Geom2 but with the potential vector equation solved on an extended domain. All cases calculated are presented in table 5.

-Case 2-5 calculates the potential vectors beyond the nozzle wall in an extended domain up to a radial boundary situated at r=15mm (in front of line WX (figure 3)).

-Case 2-6 corresponds to the same extended domain but applying PVNV.

-Cases 2-7 and 2-8 are calculations for respectively 9mm and 6mm extended domain.

The dimensions15mm, 9mm and 6mm have been chosen as they corresponds approximately to respectively 5, 3 and 2 times the characteristic arc core radius. The new calculation domain for Geom2 is presented in figure 22. Dashed domain corresponds to the extended domain for potential vector calculation.

The comparison of the magnetic field obtained with the mixed formulation and PVNF with full extended domain taking into account the cathode is presented in figure 23. In the plasma, the estimation of the magnetic field obtained with the extended domain is clearly closer to the reference than the one obtained without the extended. Nevertheless, as obtained for Geom1, the radial evolution far from the plasma (visible close to the anode) is badly represented.

In order to study the influence of the dimension of the extended domain, we propose to study the evolution of the magnetic field along the isotherm 10kK in figure 24. This isotherm can be seen as an image of the conduction zone and its position remains the same for all cases studied in Geom2. The evolution of magnetic field is plotted for a radial extended domain of 9mm (case 2-6) and 6mm (case 2-7). To complete the study, the magnetic field obtained with PVNV is also presented on the same figure. The results obtained with PVNV formulation are very close to those of the mixed formulation. For the PVNF formulation, results from 15mm and 9mm extended domains are also close to the reference case with slight differences. When the size of the extended domain is too reduced compared to the conduction zone (case 2-7 with 6mm extended domain), one can observe that the estimation of the magnetic field is clearly worse.

In order to conclude on the influence of the size of the extended domain on the arc characteristics, the evolution of the axial velocity for the various cases is represented in figure 25. For PVNF used with 15mm and 9mm extended domain and for PVNV case, the profile of the axial velocity obtained is close to the reference one but shifted by around 50m/s. As expected, due to differences observed on the magnetic fields calculated for the case 2-7 the velocity profile is further away from the reference one. Nevertheless, the maximum difference remains under 200m/s.

V/ Conclusion

The self-induced magnetic field conjugated with the current density leads to the Lorentz forces. These forces generate velocity in the arc core and, depending on the cathode geometry and the injected mass flow rate, can be totally responsible for the plasma characteristics. The self-induced magnetic field is due to current circulation not only in the plasma but also in the electrodes. Nevertheless current circulation in the electrode is never or rarely taken into account except in low-voltage circuit-breaker studies to represent arc displacement [15]. Several methods exist to represent the self-induced magnetic field such as the Biot & Savart formulation (which is the exact solution but is very time consuming), the Maxwell Ampere formulation (in two dimensional systems), the vector potential formulation (VP) with a null flux(PVNF) or an null value (PVNV) boundary condition on the walls. Due to the lack of studies on this point, this article was devoted to the comparison of several methods in two different geometries (Geom1 and Geom2). As the difficulty of the vector potential formulation is the choice of the boundary conditions, we propose a mixed formulation (MF) where the boundary values of the vector potential are determined by the resolution of the Biot & Savart formulation written for the vector potential. As the aim was also to obtain a satisfactory method in 2D, the elliptic integrals are used to assume the boundary conditions. Geom2 was a confined geometry compared to Geom1, with the applied boundary conditions close to the arc core. A study is also proposed on the interest of calculating the potential vector on an extended domain in such case. Table 6 summarize the results of the study which allow conclusions to be reached on several points:

-The self-induced magnetic field produced by current circulation in the cathode is essential to represent the velocity field. For PVNF and PVNV formulations, only this approach provides results comparable with the Biot & Savart formulation.

-For a representation of the temperature field, other methods such as PVNF (Vector potential with a null flux) or MA (Maxwell Ampere) give similar results.

-When the wall boundary condition of the geometry is close to the arc core, the PVNF formulation must be used with an extended domain on the radial direction for the potential vector calculation. The extension of this domain should be up to 3 times the characteristic arc core radius.

-The mixed formulation provides exactly the same results as the Biot & Savart formulation but requires much less calculation time. Indeed the triple integral is only calculated for points along the domain edges.

-The mixed formulation gives a good representation of the velocity field in the calculation domain. Indeed even though small differences exist in the temperature field, a wrong estimation of the self-induced magnetic field leads to large differences in the deduced velocities.

-The present study used a current intensity of 600A, in the stationary case. An increase of the current intensity leads to an increase of the self-induced magnetic field and can give erroneous results for the velocity field and the pressure waves in specific configurations such as for example high-voltage circuit breakers where the current intensity can reach 70 kA.

(

)

k E k K k rr' k = I 2 2 1 2 0,1,1 - - (A13)
As K(k) in equation (A9), E(k) can be approximated by an analytical expression [START_REF] Abramovitz | Handbook of mathematical functions New York Dover Publications[END_REF] ( ) ( )

( )(
)

4 1 4 3 1 3 2 1 2 1 1 0 1 4 1 4 3 1 3 2 1 2 1 1 0 / 1 ln m b + m b + m b + m b + b m + m a + m a + m a + m a + a = k E (A14)
Values for the coefficients for the calculation of E(k) and K(k) are given in table A1

B-B' 300 K - 0 m/s 0 = ∂ ∂ n V B-C 300 K 25Nl/min v z v r =0 0 = ∂ ∂ n V (Geom1) C- D (Geom2) C- D-E-F-G- H-W 300 K - 0 m/s 0 = ∂ ∂ n V W-X-Y 300 K 1 atm 0 = ∂ ∂ n v i 0 = ∂ ∂ n V Y-Y' 300 K - 0 m/s 0 = ∂ ∂ n V Y'-Z' 0 = ∂ ∂ n T - 0 m/s 0 V A-Z' 0 = ∂ ∂ n T 0 = ∂ ∂ n P 0 = ∂ ∂ n v i 0 = ∂ ∂ n V

Geometry

Magnetic field treatment

--

  ) b) Electromagnetic models In this study, we used four different formulations to estimate the self-induced magnetic field: -The Maxwell -Ampere equation (MA) -The Biot & Savart equation (B&S) The potential vector with null flux boundary conditions (PVNF) The potential vector with null value boundary conditions (PVNV) -The mixed formulation: Potential vector with B&S boundary conditions (MF)

Figure 17 ,

 17 Figure 17, the axial current density is maximum near the cathode where equation (22) is assumed as boundary condition. Along the axis, current density decreases as conduction radius increases. Concerning the radial current density, values up to 10 6 A/m 2 are obtained near the cathode and at the exit of the throat of the nozzle.

  field ( : Good, : Average , :Bad)

Case 1 - 1

 11 Geom1 : Freely expansion -M.F. -Biot & Savart on (A'-B'-B-C-D-W-X-Y-Y'-Z'), calculated from the current density within the domain and Savart on (A'-B'-B-C-D-W-X-Y-Y'-Z'), calculated from the current density within the domain Case 1 -3 G eom1 -M.F. -Biot & Savart on (B'-B-C-D-W-X-Y-Y'), calculated from the current density within the domain -Null flux of potential vector on (Y'-Z') and (A'-B') Case 1-4 Geom1 -P.V.N.F., on all boundaries, without taking into account the cathode Case 1-5 Geom1 -P.V.N.F., on all boundaries, calculated from the current density within the domain and the cathode. Case 1-6 Geom1 -P.V.N.V. on (DWX) and P.V.N.F. on (Y'-Z') and (A-B). Case calculated by taking into account the current density within the domain and the cathode.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Cylinder with a constant current density distribution

Figure 4 : 1 Figure 5 : 1 Figure 6 :

 41516 Figure 4: Fields of temperature (left) and velocity magnitude (right) for case 1-1

Figure 7 :Figure 8 :

 78 Figure 7: Comparison of azimuthal components of the magnetic field obtained in case 1-1 (M.F. with cathode) and case 1-2 (MF without cathode)

Figure 9 :Figure 11 :Figure 12 :

 91112 Figure 9: Comparison of azimuthal component of the magnetic field obtained in case 1-1 and case 1-4 (P.V.N.F.)

Figure 14 aFigure 16 :Figure 17 : 1 (Figure 19 :Figure 20 :

 14161711920 Figure 14 a & b: Velocity profiles along lines 1 and 2 (Respectively z=12.5mm and 20mm)

Figure 22 :Figure 24 :

 2224 Figure 22 : Geom2 with hatched extended domain

Figure 25 :

 25 Figure 25: Evolution of the axial velocity in geom. 2 depending on the size of the extended domain and the formulation.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Cylinder with a constant current density distribution

Figure 4 : 1 Figure 5 : 1 Figure 6 :

 41516 Figure 4: Fields of temperature (left) and velocity magnitude (right) for case 1-1

Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 (

 78910111213 Figure 7: Comparison of azimuthal component of the magnetic field obtained in case 1-1 (M.F. with cathode) and case 1-2 (MF without cathode)

Figure 14 a&b: Velocity profiles along lines 1 and line 2 ( 2 (Figure 16 :Figure 17 : 1 Figure 18 :

 14221617118 Figure 14 a&b: Velocity profiles along lines 1 and line 2 (Respectively z=12.5mm and 20mm)

Figure 22 :

 22 Figure 22 : Geom2 with hatched extended domain

Figure 25 :

 25 Figure 25: Evolution of the axial velocity in geom. 2 depending on the size of the extended domain and the formulation.

Table A1 :

 A1 Numerical constants for the calculation of the elliptic integrals.

					.
		a0	a1	a2	a3	a4
	K(k) 1.38629436112 0.09666344259 0.03590092383 0.03742563713 0.01451196212
	E(k)	0	0.44325141463 0.06260601220 0.04757383546 0.01736506451
		b0	b1	b2	b3	b4
	K(k)	0.5	0.12498593597 0.06880248576 0.03328355346 0.00441787012
	E(k)	0	0.24998368310 0.09200180037 0.03328355346 0.00526449639

Table captions Table 1 :

 captions1 Equations solved in the fluid model

Table 2 :

 2 Vector potential equation

Table 3 :

 3 Boundary conditions for geometries Geom1 & Geom2

Table 4 :

 4 Description of the magnetic field treatment for the cases studied for Geom1

Table 5 :

 5 Description of the magnetic field treatment for the cases studied for Geom2

Table 6 :

 6 Summary of the studied cases and of the results on the estimation of the magnetic

Table 1 :

 1 Equations solved in the fluid model

	Name	φ	Γ φ	S φ
	Electric potential	V	σ	0, No convective term	(6)
		Table 2: Vector potential equation

Table 3 :

 3 Boundary conditions for geometries 1 & 2.

Table 4 :

 4 Description of the magnetic field treatment for the studied cases for Geom1.

	Case 2-1	Geom2	-M.F.
			-Biot & Savart on (A'-B'-B-C-D-E-F-G-H-W-X-Y-
			Y'-Z'), calculated from the current density within the
			domain and the cathode
	Case 2-2	Geom2	-P.V.N.F., on all boundaries,without taking into
			account the current density within the cathode
	Case 2-3	Geom2	-P.V.N.F., on all boundaries, taking into account
			the current density within the cathode
	Case 2-4	G eom2	-M.A.
	Case 2-5	Geom2	-P.V.N.F., on all boundaries, taking into account
			the current density within the cathode and imposed
			at an extended radial distance r=15mm
	Case 2-6	Geom2	-P.V.N.V. taking into account the current density
			within the cathode and imposed at an extended
			radial distance r=15mm. P.V.N.F. on (ABC) and
			(ZYX)
	Case 2-7	Geom2	-P.V.N.F. taking into account the current density
			within the cathode and imposed at an extended
			radial distance r=9mm P.V.N.F. on (ABC) and
			(ZYX)
	Case 2-8	Geom2	-P.V.N.F., on all boundaries, taking into account
			the current density within the cathode and imposed
			at an extended radial distance r=6mm

Table 5 :

 5 Description of the magnetic field treatment for the studied cases for Geom2.

				Estimation	Estimation
			Boundary	of the	of the
		Number PVNF MA PVNV MF Cathode	condition	magnetic field in the	magnetic field far
			Distance	core	from the
					core
		1-1	15mm	good	good
		1-2	15mm	average average
	G	1-3	15mm	average	bad
	E
		1-4	15mm	average	bad
	O
	M	1-5	15mm	good	average
	1	1-6	15mm	good	bad
		1-7	-	average average
		2-1	Wall	good	good
		2-2	Wall	bad	bad
	G	2-3	Wall	bad	bad
	E	2-4	-	average average
	O	2-5	15mm	good	bad
	M	2-6	15mm	good	average
	2	2-7	9mm	good	bad
		2-8	6mm	bad	bad

Table 6 :

 6 Summary of the studied cases and of the results on the estimation of the magnetic field

Appendix A

Boundary condition for A z In order to calculate the Biot & Savart condition for A z we have to remember equation (A1).

Assuming that j z is constant for θ, we have

defined by equation (A4), the integral over θ can be written in function of ( ) ( ) a , r' r, I 0,0,0 written in equation (A5) in function of the complete first order elliptical integral K(k).

With :

More details can be found in [START_REF] Abramovitz | Handbook of mathematical functions New York Dover Publications[END_REF] and [START_REF] Conway | [END_REF]. From equation (A4) we can rewrite equation (A2) in term of elliptical integral :

The strong interest of equation (A7) is that I (0,0,0) can be expressed in terms of K(k) and that K(k) can be approximated by an analytical equation :

)

With m 1 =1-k 2 , a i and b i are constants given in [START_REF] Abramovitz | Handbook of mathematical functions New York Dover Publications[END_REF].

Boundary condition for A r

For calculating boundary condition for A r , we have to use equation (A10).

( )

This equation can be written in cylindrical system : The integral I (0,1,1) can be written in term of second and first order total elliptic integrals K(k) and E(k)


N Bibinov
email: nikita.bibinov@rub.de

N Knake

H Bahre

P Awakowicz

V Schulz-Von Der Gathen

Spectroscopic characterization of atmospheric pressure µ-jet plasma source

Keywords: 52.50.Dg, 52.80.Pi, 52.70.Kz

A new method for determination of plasma parameters at atmospheric pressure conditions is formulated and applied for characterization of a radio frequency µ-jet plasma source using He/O 2 mixture. Applying absolutely-calibrated optical emission spectroscopy and numerical simulation, the gas temperature in active plasma region and plasma parameters (electron density and electron distribution function) are determined. The steadystate concentrations of different species such as oxygen atom and ozone the plasma channel and in the effluent of the plasma source are calculated using measured plasma parameters and gas temperature. On the other hand, spatial distribution of steady-state densities of these species are measured using emission and absorption spectroscopy. A comparison of the results thus obtained and the validation of the new method against two-photon absorption laser-induced fluorescence spectroscopy measurements are discussed. In addition, the influence of the surface processes and gas flow regime on the loss of the active species in the plasma source are discussed.

Introduction

The micro-scaled atmospheric pressure plasma jet (µ-APPJ) [1] operated under a flow of He/O 2 gas mixture and at an excitation frequency of 13.56 MHz is a "cold" source of reactive species such as atomic oxygen, molecular metastable and ozone. Despite the wide field of possible applications such as in bio-medicine, the plasma-chemical processes of these kind of jet-like plasma sources are not yet clear and must be studied for the purpose of optimization of the sources. High collision rates of atoms and molecules, small diameters of the plasma channel and their influence on heterogenic processes, and different wall materials beside the plasma channel are examples of peculiarities of this plasma source which complicate understanding of plasma-chemical processes and thereby the optimization processes itself.

To gain knowledge on the processes occurring in the plasma, both measurements and modelling have to be applied while the synergetic combination of both may ideally cancel out the limitations of each individual method. Since probe measurements inside the µ-APPJ are difficult due to two reasons namely: i) the probe theory becomes complicated at high pressure; ii) the probe greatly perturbs the jet. Hence, most of the experimental data are derived from spectroscopic measurements. Phase-resolved optical emission spectroscopy (PROES) [2] provides information on excitation of species as well as on the plasma parameters in the plasma channel with temporal and spatial resolution. This technique highly relies on sophisticated collisional radiative models and is often limited to relative data of emission ratios such as in actinometry [3]. Two-photon absorption laser-induced fluorescence spectroscopy (TALIF) allows quite reliable determination of species' densities but is quite demanding in terms of equipment and usually limited to one distinct species [4,5].

Apart from the challenge of covering multiple time scales (electron dynamics in the range of ns and chemistry in the range of ms), numerical simulations, on the other hand, give a convenient insight into almost all plasma parameters and densities but are highly depending on the choice of assumptions, species, reaction rates, cross-sections, boundary conditions, etc. thus leaving a huge number of free parameters which must be experimentally determined or in other words be benchmarked [6].

To study the processes in the plasma source, we apply a combination of two methods namely optical emission spectroscopy (OES) and numerical simulation which supplement each other and allow to solve the problems which arise when these two methods are used separately. The gas temperature in plasma source is determined by OES using the rotational distribution of diatomic molecules, and numerical simulation of emission spectra. Electron velocity distribution functions are calculated for variable electric field applying numerical solution of Boltzmann equation in local approximation. Emission spectra are simulated and reliable plasma parameters (electron distribution function and electron density) are determined in comparison of simulated and measured emission spectra. At that absolute calibrated spectrometer is applied. The oxygen atoms and ozone densities in are measured applying emission and absorption spectroscopy. The measured species densities are compared with steady-state densities of oxygen atoms and ozone calculated by simulation of chemical kinetics based on plasma parameters determined for the studied experimental conditions.

Experimental setup

OES measurements

The µ-APPJ is a capacitively-coupled RF-plasma source (see figure 1) operated at an excitation frequency of 13.56 MHz. Two plane electrodes with thickness of 1 mm and length of 40 mm are separated by a gap of 1.3 mm (part A in figure 1). To exclude the influence of surrounding air on chemical kinetics in the effluent region and on the plasma conditions in the plasma channel, the µ-jet is provided with non-conductive plates (of PVC) with the same thickness and length as the electrodes (part B in figure 1) and quartz windows on both sides. This µ-jet plasma source is operated in a He/O 2 (1500 sccm/ 22.5 sccm) mixture and at a RF-sender power of 30 W. Gas velocity in plasma and effluent channels amounts to 19.5 m/s. We characterize the plasma conditions of µ-APPJ by applying OES and numerical simulation using the emission bands of nitrogen which have well-known mechanism of excitation, and high intensities because of high probability of spontaneous emission. To prevent influence of nitrogen admixture on the plasma conditions, the quantity of nitrogen admixture to He/O 2 gas mixture must be kept low. We use pure He (purity 5.0) and technical oxygen (purity 2.5) containing low admixture (about 0.5 %) of nitrogen. Concentration of nitrogen in total gas flow is not known and is determined. The emission spectrum of µ-jet plasma source is measured with a spatial resolution of about 1 mm using a broadband echelle-spectrometer (ESA-3000) and a grating spectrometer QE65000 (Ocean Optics). The former measures the emission spectrum simultaneously in about 100 spectral orders in the spectral range of λ = 200-800 nm with a spectral resolution of 0.015-0.06 nm while the latter measures spectrum in the spectral range of λ = 200-900 nm with a spectral resolution of about 1.3 nm. Both spectrometers are relatively and absolutely-calibrated using tungsten ribbon lamp and well known UV-emission spectra of N 2 and NO [7]. To increase the spatial resolution of OES characterization, the optical fiber of the spectrometers is provided with a diaphragm that limits the acceptance angle to 1.4°.

Absorption spectroscopy is used for determination of ozone concentration produced in the µ-APPJ. For this, the spectrum of a deuterium lamp (HAMAMATSU L625) transmitted through the gas gap is measured using the grating spectrometer QE65000 (see figure 1). The thickness of the absorbing layer of 1 mm is determined by the quartz panes. During absorption and OES measurements, the optical fibers are shifted parallel to the electrodes' surface along the gas gap in the x-direction with a step of 2 mm.

Determination of gas temperature

Gas temperature in active plasma volume is one of the most important parameter for simulation of plasma-chemical kinetics of plasma jet. The rate constants of chemical reactions and concentration of species depend on the gas temperature. We determine this parameter using nitrogen molecular emission in assumption that rotational and translation degrees of freedom of diatomic molecules have equal temperature at atmospheric pressure conditions. Rotational distribution in N 2 (C-B,0-0) vibrational band at λ=337.1 nm is used for determination of the gas temperature (T g ) in the active plasma volume. We simulate this vibrational band with the same spectral resolution of echelle λ/ nm T rot =370K Experiment spectrometer for varied rotational temperatures, and compare simulated and measured spectra for determination of gas temperature (see figure 3). Averaged rotational temperature with confidence interval in observed plasma volume is determined by fitting procedure. The gas temperature is approximately constant along plasma channel and amounts to 370 ± 20 K.

Simulations

Chemical kinetics model

Plasmas containing nitrogen and oxygen produce a large quantity of active species, and it is complicated to detect and quantify all these species. We approach this problem by simulation of chemical kinetics based on plasma parameters. In the simulation presented here, we include chemical reactions concerning the production and destruction of metastable molecules and atoms of oxygen and nitrogen, ozone and nitric oxide. Chemical reactions in He/O 2 /N 2 mixture included in the model are The rate constants for recombination reactions of oxygen atoms with the assistance of oxygen molecules (No. 14,16,18 in table 1) [8] multiplied by the factor of 0.54 [START_REF] Jeong | [END_REF] are used as the rate constants for recombination processes (No. 15,17,19 in table 1) with the assistance of helium atoms.

v N N N T k d N T D t N M x N M MN M M ⋅ ∇ + ⋅ ⋅ + ⋅ ⋅ = ∂ ∂ ) ( ) ( 2 .
Because of the small cross-section of the plasma channel -a rectangle with sides 1mm x 1.3 mm, the excited and chemical active species can be effectively lost on the surface of the walls despite high gas pressure, and therefore low diffusion velocity. Diffusion coefficients of 7.9•10 -5 m 2 /s, 1.1•10 -4 m 2 /s, 1.53•10 -4 m 2 /s and 8.3•10 -5 m 2 /s respectively for diffusion of O 2 , N 2 , O and O 3 in He [17,18] are used. For the simulation of diffusion of nitrogen atoms, the coefficient for diffusion of oxygen atoms is assumed because the difference of collision diameters of these atoms is smaller than 5% [19] .

Chemical kinetics in active plasma region (x = 0 -40 mm) and in the effluent (x > 40 mm) are different because of electron impact excitation and dissociation process in active plasma volume. In the effluent region, chemical active species are produced only in chemical reactions or reach that region with the gas flow.

If the plasma parameters (electron density and electron distribution function) are known, then the averaged excitation and dissociation rates of oxygen and nitrogen molecules (No.1-4 in table 1, No.26,27 in table 2) can be calculated with a spatial resolution of 1 mm. Determination of these plasma parameters by absolute calibrated OES and simulation is described below.

The temporal behavior of the chemically-active species produced in the discharge is described by the equation of continuity (1). Gain and loss of the active species by different reactions in the discharge channel, and their diffusion (according to Fick's law) are considered. (1) where N M and N N (in m -3 ) are the densities of species involved in the chemical reaction, D(T) (in m 2 ⋅s -1 ) is the diffusion coefficient, and k MN (in m 3 ⋅s -1 ) is the rate constant of the corresponding chemical reaction. This equation describes the temporal evolution of species' density due to diffusion (first term on the right hand side of equation ( 1)), due to different chemical reactions (second term) and motion of species with gas flow (third term) with velocity v (in m⋅s -1 ). ∇ x [M] is the density gradient of species M in axial direction. For simplification of calculation procedure, cylindrical symmetry of the plasma is assumed with diameter (d, in m) equal to averaged distance between the electrodes. Chemical kinetics is simulated using equation ( 1) which is solved numerically for O 2 (a 1 ∆),

O 2 (b 1 Σ), O, O( 1 D), N 2 (A 3 Σ u + )
, N, NO and O 3 concentrations simultaneously until steady-state density of the active species is attained.

Atomic species can be lost to the walls of the plasma source by adsorption, heterogenic recombination, production of molecules and desorption from the surface. The rate of the surface recombination of atoms depends on densities of physisorption and chemisorption centers on the surface [20], occupancy of chemisorptions centers, mobility of the atoms on the surface, etc.

Therefore, properties of materials of the plasma source [21], chemical active impurities in working gas and on the surface of the plasma source wall, the surface temperature [22] and treatment of the surface by fluxes of ions strongly influence efficiency of heterogenic recombination of atoms on the surface of the plasma source wall. On active centers that are occupied by some atomic species or impurities, ozone molecules can be dissociated. The loss of atomic species and ozone in reactions (2-4) to the walls of the plasma source is described using diffusion process to the wall of the plasma channel mentioned above, and probabilities of these reactions (γ) in equation ( 1) are varied from 0 up to 1 during simulation. These probabilities are unknown a priori because of influence of the uncertain considering conditions of the wall's surfaces.

O + wall → 1/2 O 2 (2)
O 3 + wall → O 2 + O (3) N + wall → 1/2 NO (4)
The chemisorption centers on the walls at our experimental conditions are occupied mainly by oxygen atoms because of low concentration of nitrogen and other impurities. Therefore, reactions for NO molecule production by adsorption of oxygen atoms, and N 2 molecules by adsorption of nitrogen atoms are not taken into account. Other possible processes like reactions with charged species, helium atoms, nitrogen atoms and different molecular radicals are not included in the model because of low density of these species.

Determination of plasma parameters

To simulate chemical kinetics in µ-APPJ, excitation and dissociation rates (No.

where N B -density of oxygen and nitrogen molecules in plasma volume (m -3 ); n e -electron density (m -3 ) ;

f v (E) -electron velocity distribution function (EVDF) (eV -3/2 ) normalized to fulfill the equation ; = or -cross section of electron impact excitation and dissociation of molecules correspondingly (m 2 ) [23,24]; m e , e -the electron mass (kg) and elementary charge (C);

E -the electron kinetic energy (eV). 1 ) ( 2 4 = ⋅ ⋅ E v dE E E f π ) (E B dis σ , ) ( 2 ) ( 2 4 dE E E m e E f n N R B E e v e B B σ π ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ) (E B exc σ B R ) (E B σ Table 1.
Chemical reaction scheme of ozone, oxygen molecules and oxygen atoms included in the model for simulation of chemical kinetics in the µ-APPJ and in the effluent. strongly on electron velocity distribution function and electron density. To determine these plasma parameters, we solve Boltzmann equation for electrons using variable electric field strength, simulate electron distribution functions, calculate emission spectra for variable electric field and determine plasma parameters by comparison of measured and calculated spectra. By solution of Boltzmann equation, only loss of electron kinetic energy must be taken into account and therefore only most important cross sections of electron collisions with heavy particles are needed. These cross sections are well known for various atomic and molecular gases, and hence the method can provide more reliable plasma parameters and excitation rates.

No. Reaction k No k No (T g =370K) Ref. O 2 (X) + e → O 2 (a 1 ∆) + e Rate constant is calculated from plasma parameters O 2 (X) + e → O 2 (b 1 Σ) + e Rate constant is calculated from plasma parameters O 2 (X) + e → 2O + e Rate constant is calculated from plasma parameters O 2 (X) + e → O( 1 D) + O + e Rate constant is calculated from plasma parameters O( 1 D) + O 3 → O 2 (a 1 ∆) + O 2 1.5⋅10 -17 m 3 s -1 1.7•10 -17 m 3 s -1 [8] O( 1 D) + O 3 → O 2 (b 1 Σ) + O 2 7.7⋅10 -18 m 3 s -1 8.6•10 -18 m 3 s -1 [8] O( 1 D) + O 3 → 2O 2 2.3•10 -17 m 3 s -1 2 . 6 •10 -17 m 3 s -1 [8] O( 1 D) + O 3 → O 2 + 2O 1.2•10 -16 m 3 s -1 1.3•10 -16 m 3 s -1 [8] O( 1 D) + O → O + O 7.5•10 -17 m 3 s -1 8.3•10 -17 m 3 s -1 [9] O( 1 D) + O 2 → O 2 + O 6.4 ⋅10 -18 exp(67/T) m 3 s -1 7.6•10 -18 m 3 s -1 [9] O + O 3 → O 2 (a 1 ∆) + O 2 6.3⋅10 -18 exp(-2300/T) m 3 s -1 1.3•10 -20 m 3 s -1 [8,10] O + O 3 → O 2 (b 1 Σ) + O 2 3.2⋅10 -18 exp(-2300/T) m 3 s -1 6.4•10 -21 m 3 s -1 [8,10] O + O 3 → 2O 2 9.5⋅10 -18 exp(-2300/T) m 3 s -1 2.0⋅10 -20 m 3 s -1 [8,10] 2O + O 2 → O 2 + O 2 1.9⋅10 -42 T -1 exp(-170/T) m 6 s -1 6.8⋅10 -46 m 6 s -1 [8] 2O + He → O 2 + He 1.0⋅10 -42 T -1 exp(-170/T) m 6 s -1 3.6⋅10 -46 m 6 s -1 [8,11] 2O + O 2 → O 2 (a 1 ∆) + O 2 1.3⋅10 -42 T -1 exp(-170/T) m 6 s -1 2.2⋅10 -45 m 6 s -1 [8] 2O + He → O 2 (a 1 ∆) + He 7.0⋅10 -43 T -1 exp(-170/T) m 6 s -1 1.2⋅10 -45 m 6 s -1 [8,11] 2O + O 2 → O 2 (b 1 Σ) + O 2 6.0⋅10 -43 T -1 exp(-170/T) m 6 s -1 1.0⋅10 -45 m 6 s -1 [8] 2O + He → O 2 (b 1 Σ) + He 3.2⋅10 -43 T -1 exp(-170/T) m 6 s -1 5.5⋅10 -46 m 6 s -1 [8,11] O 2 (a 1 ∆) + O 3 → O +2O 2 5.0⋅10 -17 exp(-2830/T) m 3 s -1 2.4⋅10 -20 m 3 s -1 [10] O + 2O 2 → O 3 + O 2 5.5•10 -43 T -1.20 m 6 s -1 4.6⋅10 -46 m 6 s -1 [12] O + O 2 + He → O 3 + He 3.3•10 -43 T -1.20 m 6 s -1 2.7⋅10 -46 m 6 s -1 [12] O 2 (a 1 ∆) + O → O + O 2 1.3⋅10 -22 m 3 s -1 1.4⋅10 -22 m 3 s -1 [10] O 2 (b 1 Σ) + O 3 → O + O 2 (a 1 ∆) 7.0⋅10 -18 m 3 s -1 7.8⋅10 -18 m 3 s -1 [9] O 2 (b 1 Σ) + O 3 → O +2O 2 1.5⋅10 -17 m 3 s -1 1.7⋅10 -17 m 3 s -1 [9]
33 O 3 + NO→ NO 2 + O 2 9•10 -19 exp(-1200/T) m 3 s -1 3.5•10 -20 m 3 s -1 [10] 34 2N 2 (A 3 Σ u + ) → N 2 (A 3 Σ u + ) + N 2 3.5•10 -16 m 3 s -1 3.9•10 -16 m 3 s -1 [14] 35 NO + N→ N 2 + O 3.1•10 -17 m 3 s -1 3.

Electron velocity distribution function

To determine the required plasma parameters, we compare measured and calculated emission spectra. The ratio of emission intensities of ionized molecular nitrogen and of neutral molecular nitrogen strongly depends on the RF-electric field ( ) and is used for determination of electron velocity distribution function. To calculate the emission spectrum, we simulate the electron velocity distribution function in He/O 2 mixture at atmospheric pressure by numerical solution of Boltzmann equation in "local" approximation (6) using the program code "EEDF" developed by Napartovich et al [25,26]. ,

where S coll -collision integral for electrons with heavy species.

"Local approximation" means that at every point in the plasma volume, EVDF can be determined from Boltzmann equation for homogeneous non-bounded plasma using local electric field strength. The "local approximation" is valid for highly collisional plasma such as the plasma at atmospheric pressure conditions [27]. Electric field strength in µ-APPJ is not known forehand. Hence, we simulate EVDF for different values of RF electric field. Using simulated EVDF, we calculate intensities of nitrogen molecular emissions and compare them with measured ones to determine the EVDF and also the electric field strength in active plasma volume. Intensities of the molecular vibrational bands N 2 (C-B,0-0, λ=337.1 nm) and N 2 + (B-X,0-0, λ=391.4 nm) are measured using calibrated echelle spectrometer. Emission of molecular nitrogen ions can be excited by electron impact (7) and by Penning ionization of nitrogen molecules (8) through collisions with helium metastable atoms He(2 1,3 S,2 3 P) excited by electron impact (9).

coll e v S f E m e t f = ∇ - ∂ ∂ ) ( r E r
where k is the rte constant for the numbered reaction listed above.

During collisions of helium metastables with nitrogen molecules, N (B) transition [START_REF] Siska | [END_REF].

Accordingly, population of ground vibrational level of N 2 + (B,0) amounts to 0.89 [31].

The steady state concentration of N 2 + (B,0) at our plasma conditions can be determined using balance equation ( 12).

k N N met ⋅ ⋅ ⋅ = ) 0 , ( ) 0 , ( 2 2 , X B N B C N A A - - + He B N He C N O B N O C N k k k k ) ( ) ( ) ( ) ( 2 2 2 2 2 2 , , , + + 0506 . 0 ) ( ) ( ) 0 , ( ) 0 0 , ( ) ( 2 2 2 2 2 2 2 = ⋅ + ⋅ + = + + + + + - - - He He B N O O B N X B N X B N B N N k N k A A Q 088 . 0 ) ( ) ( ) 0 , ( ) 0 0 , ( ) ( 2 2 2 2 2 2 2 = ⋅ + ⋅ + = - - - He He C N O O C N B C N B C N C N N k N k A A Q (12)
Emission intensity is proportional to the steady state concentration of N 2 + (B,0) molecules (13)

Intensity of neutral nitrogen emission N 2 (C-B,0-0) can be presented as ( 14)

The ratio of intensities of nitrogen molecular bands (15) depends strongly on the electron distribution function .

, (15) where and are quenching factors for N 2 (C) und N 2 + (B) excited states;

-the Einstein coefficients for spontaneous transitions (s -1 ) [32,33];

-the rate constants for quenching of respective nitrogen excited states by collisions with oxygen molecule and helium atoms (m 3 s -1 ) [32][33][34][35]; and -the rate constants for Penning-ionization of oxygen and nitrogen molecules by collisions with helium metastables [START_REF] Zhiglinskii | Handbook of Rate Constnts of Presses of Atoms, Ions, Electrons, and Photons[END_REF]; -the rate constant (m 3 s -1 ) for electron impact excitation of species M (He,N 2 ) into the respective states;

k N k k N b k Q n N k Q n N d I d I I I O He X B N exc B N e N C N exc C N e N X B N B C N λ λ λ λ ( ) 2 2 2 2 2 2 2 2 ) 0 , ( ) 0 , ( 10
8 9 ) 0 , ( O O q He He q X B N B N O He e N B N exc e N N k N k A N k N k k N n N b k n N + + = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ - + + + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ + = + + + + + + - - - - - 10
8 9 ) 0 , ( ) ( ) ( ) 0 , ( ) 0 0 , ( ) 0 0 , ( 2 2 2 2 2 2 2 2 2 2 2 k N k k N n N b k n N N k N k A A I O He e N B N exc e N O O B N He He B N X B N X B N X B N ( ) ) 0 , ( ) ( ) ( ) 0 , ( ) 0 0 , ( ) 0 0 , ( 2 2 2 2 2 2 2 2 2 C N exc e N O O C N He He C N B C N B C N B C N k n N N k N k A A I ⋅ ⋅ ⋅ ⋅ + ⋅ + = - - - - -
-the cross section of electron impact excitation for species M (He, N 2 ) (m 2 ) into the respective states [23,36];

Quenching of excited nitrogen molecules by collision with other atomic and molecular species is not taken into account because of low densities of the respective species in our experimental conditions.

Thus, we simulate electron velocity distribution function at variable RF-electric field, calculate intensity ratio of nitrogen molecular bands (15), compare the calculated ratio with a measured one to find the best fitted electric field and EVDF(x) at variable distance x from gas inlet of the plasma jet.

⋅ ⋅ ⋅ ⋅ = E M exc e v M exc dE E E m e E f k ) ( 2 ) ( 2 4 σ π ) (E M exc σ 3.

Determination of electron density near the gas inlet (n e 0 )

In the second step, we determine another starting parameter for simulation of chemical kinetics namely, electron density using the oxygen photoemission. The atomic lines O( 3 S o -3 P) at λ ∼ 844.6 nm can be excited by electron-impact dissociative excitation ( 16) of oxygen molecule or by electronimpact excitation of oxygen atom in ground state (O(2p 4 3 P) ) ( 17 Lifetime of the excited state O(2p 3 3p 3 P) concerning spontaneous transition at λ∼844.6 nm amounts to 33.3 ± 1.7 ns [31]. The rate constant of quenching of this excited state during collisions with oxygen molecules is equal to (7.8 ± 0.8)•10 -16 m 3 s -1 [37]. Excitation cross sections for processes (16) and (17) are presented in Ref. [32] and [39][40][41], correspondingly.

We determine a steady-state value of the electron density in the plasma channel using O( 3 S o -3 P) emission near the µ-APPJ gas inlet (x = 0). In contrast to the other points in the plasma jet where oxygen atom density increases also due to transport of the atoms with gas flow from other parts of the jet, mechanism of atom production near the gas inlet includes only dissociation of oxygen molecules.

We simulate plasma chemical kinetics based on the determined electron distribution function, and determine electron density at gas inlet, n e 0 =6.0⋅10 16 m -3 that corresponds to measured total intensity of O( 3 S o -3 P) emission excited in processes (16,17).

Determination of spatial distribution of electron density in the plasma channel (n e (x))

The density of nitrogen molecule ( ) in the plasma channel of 1.6•10 21 m -3 is determined (18) using measured intensity of neutral nitrogen (N 2 (C-B,0-0)) near the gas inlet of µ-APPJ ( ) and the plasma parameters ( n e 0 , ) determined in this point.

0 ) ( 2 B C N I - ) ( 0 E f v 0 2 N N (18)
Because of high gas flow and low probabilities of dissociation and ionization of nitrogen molecules in the active plasma volume, nitrogen concentration is considered constant along the plasma channel in x-

direction (

). This allows the determination of electron density along the plasma channel n e (x) from equation ( 19) using the measured nitrogen molecular emission intensity N 2 (C-B,0-0) and the electron distribution function ( ) determined by the procedure described in subsection 3.2.1.

(

) 19

Determination of oxygen-atom density

Now, the oxygen-atom density can be determined from equation (20) using the intensity of the O( 3 S o -3 P) transition and the plasma parameters (n e (x) and EVDF(x)) and the cross sections and for the excitation processes ( 16) and ( 17), respectively.

(

) 20
where are the rate constants for excitation of emission O( 3 S o -3 P) at λ ∼ 844.6 nm by electron impact of Y = O or O 2 ;

-is the quenching factor for the O(3p 3 P) excited state [37]. .

Results and discussion

The results presented here are shown for a helium flow of 1500 sccm with admixture of 1.5% oxygen and RF-power of 30 W. Plasma source is operated in stable α-mode [2]. The measured absorption spectrum (see figure 5) agrees with the well-known spectrum of ozone [42] in the spectral range 225 nm < λ < 300 nm. In the spectral range 200 nm < λ < 225 nm, the accuracy of absorption measurement decreases because of the reduced sensitivity of the applied spectrometer in this range. At steady-state conditions, plasma fills the whole gap between the electrodes (part A in figure 1). The RF-electric field, determined using the ratio of measured intensities of N 2 + (B-X,0-0) to N 2 (C-B,0-0) bands (figure 6 left) and numerically simulated ones (figure 6 right), increases slightly when moving along the gas gap from the gas inlet point but rapidly rises near the exit of the plasma channel (see figure 7). The determined electron density (using equation ( 19)) and measured absolute intensity of N 2 (C-B,0-0) emission (figure 8 left) is approximately constant along the plasma channel except in the region near the gas inlet (see figure 8 right). Probably, the negative ions produced in the He/O 2 discharge is the reason for variation of the plasma parameters. The negative ions are trapped within the plasma bulk because of positive plasma potential. But they can be transported with the gas-flow parallel to the walls and electrodes. Therefore, the density of negative ions may increase with distance from the gas inlet. When negative ion concentration becomes higher than electron density, the balance mainly between positive and negative ions concentrations provides quasi-neutrality of plasma. Positive ions are produced during electron-impact ionization. For this, electrons with kinetic energy higher than

) (E O exc σ ) ( 2 E O exc σ , 2 2 1 ) 844 ( ) 844 ( O exc e O O exc e O O O k n N k n Q I N ⋅ ⋅ ⋅ - ⋅ = - ⋅ ⋅ ⋅ ⋅ = E Y exc e v Y exc dE E E m e E f k ) ( 2 ) ( 2 4 σ π ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = - - - E C N exc e v e C N B C N N dE E E m e E f n Q I N ) ( 2 ) ( 2 4 ) 0 , ( 0 0 1 ) ( 0 ) 0 0 , ( 0 2 2 2 2 σ π ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = - - - E C N exc e x v N C N B C N e dE E E m e E f N Q x I x n ) ( 2 ) ( 2 4 ) ( ) ( ) 0 , ( 0 1 ) ( ) 0 0 , ( 2 2 2 2 σ
π 1 ) 844 ( - O Q ) (E f x v 0 2 2 ) ( N N N x N = 200
I N + 2 (B-X,0-0) / I N 2 (C-B,0-0) E/ V/mm
the ionization potentials of helium (24.59 eV) and oxygen (12.07 eV) are needed which in turn necessitates high electric field required for electron acceleration at plasma conditions. To obtain enough high ionization rate in electronegative plasma such as the plasma far away from the gas inlet, a small number of electrons at this point should have enough high energy which is provided by the evaluated electric field. By applying these plasma parameters and measured emission of O( 3 S o -3 P) at λ ∼ 844.6 nm, the steady-state oxygen atoms density is determined in plasma channel with spatial resolution (see figure 9).

The profile and absolute value of oxygen density presented in figure 9 correspond to data received in He/O 2 mixture applying TALIF measurements [4]. At all experimental conditions in TALIF measurements as well as in the experiments discussed here, the density of oxygen atom increases near the gas inlet and reaches some saturation level. The oxygen density measured in He/O 2 mixture applying TALIF at gas flow of 1500 sccm and 1.5% oxygen, and RF power of 25 W at 35 mm from the gas inlet amounts to 9.8⋅10 21 m -3 . This agrees well with oxygen density which is presented here in figure 9. In this work, plasma conditions like oxygen admixture and RF power were chosen to produce the maximum ozone concentration to have a chance to measure this concentration in the entire plasma channel applying absorption spectroscopy despite small optical length of 1 mm. These conditions are not usually applied for studying of oxygen atoms production in µ-jet because of probable of arcing near the end of the plasma channel. Comparison of oxygen densities measured using the method discussed here and using TALIF at variable plasma conditions and variable position in µ-jet's channel will be done in the future. . This behavior is in contradiction to measurements with molecular beam mass spectrometry (MBMS) [43] in the effluent of µ-APPJ (0.6 % O 2 in He) "unprotected" from the surrounding atmosphere. A significant increase of ozone density with distance from the nozzle was found and maximum value of 1.5•10 21 m -3 at a distance of 20 mm was measured. Probably, the observed differences can be explained as the influence of additional arrangement protected from surrounding atmosphere in our experiment and also the difference in oxygen admixture.

Simulation of chemical kinetics

To interpret the measured spatial profiles of species densities shown in figure 9 and figure 10, we simulate the chemical kinetics in the plasma channel and in the effluent of the plasma source. From the determined plasma parameters and known cross sections [23,24], the dissociation and excitation probabilities of nitrogen and oxygen molecules are calculated. Chemical kinetics in the plasma channel and in the effluent are simulated using the reaction scheme presented in table 1 and table 2.

Oxygen molecules are dissociated in the plasma channel with approximately constant efficiency, and a moderate increase near the end of the plasma channel due to increase in electric field. Therefore, as shown in figure 10, ozone production will be increased by extending the plasma channel and at constant RF voltage, which is a requirement for discharge stability and for prevention of "arcing mode".

The probabilities of heterogenic processes (2-4) can be varied from 0 up to 1 depending on the surface temperature, morphology, chemical composition of the particles adsorbed on the surface of plasma source walls, etc. but these are not known a priori. So to compare calculated densities of oxygen atoms and ozone molecule with measured ones (see figure 9 and figure 10), we simulate chemical kinetics in the plasma channel and in the effluent with probabilities of heterogenic processes of 0, 0.5 and 1. The simulated oxygen densities (figure 9) for all probability values increase with distance from the µ-APPJ gas inlet because of atoms transported by high gas flow. In the effluent region, the oxygen atoms density decreases gradually because of recombination processes (No.14-19 and 21,22 in the table 1).

The values of oxygen atom densities measured using OES and numerical simulation fall in between simulated ones (see figure 9). Recombination of the atoms on the µ-APPJ walls is important but not the main process by which the atoms are lost. By variation of the probabilities of heterogeneous processes from 0 to 1, the simulated steady-state concentration of oxygen atoms decreases by only 30 %.

The known values of recombination probability of oxygen atoms on the glass and metal surfaces amounts to about 0.01 and 0.2, respectively [20][21][22]44]. This is in accordance with the measured and simulated data in the region near the gas inlet (see figure 9). The oxygen atom density measured at distances for up to 30 mm from the gas inlet falls between curves simulated for probabilities of 0 and 0.5 of heterogenic processes.

One can see that, within the frame of the described model, the decrease of oxygen atoms density near the end of the plasma channel cannot be simulated. The simulated curves show increase of oxygen density near the end of the plasma channel because of increase of electric field. But the measured oxygen density in this region decreases, the reason of which is not clear now and must be investigated through variation of plasma conditions.

The ozone density in the plasma channel and in the effluent of the plasma source is also simulated for the same plasma conditions and by variation of the probabilities of heterogenic processes (see figure 10). The measured ozone density agrees with simulated one in the active plasma volume (x = 0-40 mm) in assumption of probabilities of 1 for all heterogenic processes (2-4) that are explicitly overstated. But also under this assumption, the ozone density in effluent region is about factor 2 lower than the simulated one. Probably some process (or processes), which is responsible for ozone destruction in the plasma channel and in the effluent, is not taken into account in the applied plasmachemical model.

In the frame of the applied plasma-chemical model, the active species (atoms and molecular metastables) are produced during electron impact (No.1-4 in table 1 and No.26,27 in table 2). Only reactions with neutral species are included in the reaction scheme concerning production and loss of oxygen, nitrogen atoms, oxygen molecular metastables, ozone and nitric oxide. Reactions between considered species and charged particles (electrons, positive and negative ions) possess high rate constant [45] but densities of these species are not high enough to affect chemical kinetics. According to our estimations, the density of charged particles must be higher than 10 18 m -3 for appreciable influence on simulated densities of active species. At our experimental conditions, electron density amounts to about 5⋅10 16 m -3 , and electronegativity (ratio of positive ion density to electron density) is usually not higher than 10 in plasmas with oxygen. This means the densities of charged species are lower than 10 18 m -3 and their influence on chemical kinetics can be neglected.

Molecular nitrogen, which is determined as impurity in plasma channel, can in general affect the production and loss of oxygen atoms and ozone molecules. We compare the loss of these species by participation of nitrogen atoms with total loss of species simulated in the frame of the applied model.

Simulated nitrogen atom density reaches a maxima of about 10 19 m -3 near the end of the plasma channel. This steady-state concentration corresponds to lifetime of oxygen atoms and ozone molecule, considering loss by reaction with nitrogen atoms (No.29,33 in table 2), which amounts to about 532 ms and 116 ms, correspondingly. These lifetimes are much longer than the residence time of the species (about 2 ms) in plasma channel considering the gas flow. Therefore, influence of nitrogen impurity on the loss of oxygen atoms and ozone molecule in plasma source can be neglected.

The lifetime of nitrogen molecules concerning dissociation amounts to about 170 ms. This value is much longer than the residence time of nitrogen molecules in the plasma channel and, as it was assumed in the introduction of plasma chemical model of µ-APPJ (in section 3.2.3), the density of nitrogen molecule in the entire plasma channel is constant.

Different molecular and atomic species are produced in µ-APPJ using He/O 2 /N 2 mixture. The steady state densities of these species in the plasma channel and in the effluent can be determined Plasma channel Effluent using experimental methods namely TALIF [4,5] and MBMS [43]. However, experimental detection of other species such as oxygen and nitrogen molecular metastables is complicated. This poblem can be solved by plasma chemical simulation using the determined plasma parameters and the known mechanisms of production and destruction of these species.

The metastables of oxygen molecules are excited in the µ-APPJ and flow along with the working gas. The steady state densities of these species are calculated within the frame of the model presented here. O 2 (a 1 ∆, E=0.98 eV) density reaches a maximum of about 7.5⋅10 21 m -3 near the end of the plasma channel (at x = 40 mm, see figure 11). In the effluent, the concentration of this metastable is high whereas the concentration of other oxygen molecular metastable namely O 2 (b 1 Σ, E=1.64 eV) is low because of effective quenching by collisions with ozone molecules.

Conclusion

By applying optical emission spectroscopy and numerical simulation, gas temperature and plasma parameters (electron density and electron velocity distribution function) in radio frequency atmospheric pressure µ-plasma jet operated in He/O 2 mixture are determined. The ozone and oxygen atom densities are measured using absorption and emission spectroscopy with spatial resolution of about 1 mm. Using determined plasma parameters, chemical kinetics in the plasma jet and in effluent are simulated, steady-state ozone and oxygen atom densities are calculated and compared with measured ones. Influence of heterogenic processes on chemical kinetics is discussed. According to our simulation, micro plasma jet operated in He/O 2 mixture can be used as source of atomic oxygen O( 3 P), oxygen molecular metastables O 2 (a 1 ∆) and ozone with active species concentration of about 500 ppm.

Figure 1 .

 1 Figure 1. Experimental setup for in situ absorption measurements and OES characterization of µ-APPJ (13.56 MHz) operated in He/O 2 mixture at atmospheric pressure. Plasma source consists of part A with RF electrodes, and part B with electrodes are replaced by plates of PVC. Gap between RF-electrodes (and PVC plates) amounts to 1.3 mm. One electrode is grounded. Active plasma volume and effluent are protected from the surrounding atmosphere by quartz panes. 1 -optical fiber of deuterium lamp, 2-optical fiber of spectrometer (ESA-3000 or QE65000) with diaphragm to confine acceptance angle.

Figure 2 .

 2 Figure 2. The measured emission spectrum of µ-APPJ operated with He/O 2 (1500 sccm/22.5 sccm) mixture and admixture of nitrogen. Emission spectrum is measured by applying grating spectrometer QE65000 in the middle of plasma channel (spectral resolution of ∆λ =1.3 nm). The molecular bands depicted in the spectrum (in different magnifications) are used for determination of gas temperature and plasma parameters (measured using echelle spectrometer ESA-3000 with spectral resolution of ∆λ =0.02-0.025 nm).

Figure 3 .

 3 Figure 3. Rotational structure of N 2 (C-B,0-0) vibrational band measured at the center of the plasma channel of µ-APPJ source operated with He/O 2 (1500 sccm/ 22.5 sccm) mixture. Simulated spectrum at rotational temperature of 370 K (bottom). Confidence interval amounts to ± 20 K. The measured spectrum (top) is shifted for clarity.

  b = 0.34 -the branching ratio for excitation of N 2 + (B,0) emission by Penning ionization (8) [29];

Figure 4

 4 Figure 4 shows electron distribution function at distance of 20 mm from the gas inlet of µ-APPJ operated with He/O 2 (0.985/0.015)) determined by this procedure. The RF electric field amounts to 310 V⋅mm -1 .

Figure 4 .

 4 Figure 4. Electron velocity distribution function in the plasma channel 20 mm away from the gas inlet of µ-APPJ operated with He/O 2 (0.985/0.015) mixture determined using OES and numerical simulation.

  ) O 2 + e → O(2p 3 3p 3 P) + O(2p 4 3 P) + e (16) O(2p 4 3 P) + e → O(2p 3 3p 3 P) + e (17)

Figure 5 .

 5 Figure 5. UV-absorption spectrum measured in the active plasma volume of the µ-APPJ (♦). Normalized absorption spectrum of ozone [42] (solid line). At λ = 253.7 nm, absorption coefficient of ozone is (3.072 ± 0.032)⋅10 4 m -1 .

Figure 6 .

 6 Figure 6. Spatial distribution of ratio of measured intensities of N 2 + (B-X,0-0) to N 2 (C-B,0-0) bands (left) in µ-APPJ (He/O 2 mixture (1500 sccm/22.5 sccm), RF-power of 30 W) and intensities ratio calculated applying eq.15 by variation of RF-field (right).

Figure 7 .

 7 Figure 7. Spatial distribution of averaged RF-electric field (right) determined using ratio of measured intensities of N 2 + (B-X,0-0) to N 2 (C-B,0-0) bands (figure 6 left) and numerical simulation (figure 6 right) in µ-APPJ (He/O 2 mixture (1500 sccm/22.5 sccm), RF-power of 30 W).

Figure 8 .

 8 Figure 8. Spatial distribution of the electron densities in active plasma region of RF-plasma source (right) determined using equation (19) and absolute intensity of N 2 (C-B,0-0) emission (left) measured in µ-APPJ (He/O 2 mixture (1500 sccm/22.5 sccm), RF-power of 30 W).

Figure 9 .

 9 Figure 9. Spatial distribution of the steady-state oxygen atom density measured using OES in µ-APPJ (♦) (He/O 2 mixture (1500 sccm/22.5 sccm), RF-power of 30 W). Atomic oxygen density measured using TALIF[4] at x = 35 mm ( ) (He/O 2 mixture (1500 sccm/22.5 sccm), RF-power of 25 W). Simulated oxygen atom density for different efficiency of oxygen atoms' heterogenic recombination (dashed line -without heterogenic recombination of the atoms, dashed dot line -recombination probability of the oxygen atoms on the walls of 0.5, solid line -recombination probability of 1). The length of the active plasma channel amounts to 40 mm.

Figure 10 .

 10 Figure 10. Ozone density measured using absorption spectroscopy (♦) in µ-APPJ (He/O 2 mixture (1500 sccm/22.5 sccm), RF-power of 30 W) and simulated applying the presented plasma-chemical model (dashed line -without heterogenic recombination of the oxygen atoms and destruction of ozone molecules; dashed dot line -probabilities of the oxygen atoms heterogenic recombination and ozone molecules destruction on the wall of 0.5; solid line -probabilities of the oxygen atoms heterogenic recombination and ozone molecules destruction on the wall of 1).

Figure 11 .

 11 Figure 11. Densities of oxygen metastables calculated applying presented plasma-chemical model without heterogenic loss of active species (solid line -density of O 2 (a 1 ∆), dashed line -O 2 (b 1 Σ) multiplied by 10).

23 2 γ

 2 listed in table 1 and table 2. Other reactions which are not included in this model have small rate constant or one (or both) of the reactants has too small concentration in our plasma conditions. We

	simulate chemical kinetics in µ-APPJ at steady-state conditions and determine densities of active
	species. The temperature dependencies of rate constants of some chemical reactions (No.10-20 in table
	1 and No.29-33 in table 2) are presented. For other reactions, the Arrhenius temperature dependence is
	assumed.

Table 2 .

 2 Chemical reaction scheme of nitrogen atoms and molecules and nitric oxide included in simulation of chemical kinetics in µ-APPJ and in the effluent.

	No.	Reaction	k No	k No (T g =370K)	Ref.
	26	N 2 (X) + e → N 2 (A 3 Σ u + ) + e	Rate constant is calculated from plasma parameters
	27	N 2 (X) + e → 2N + e	Rate constant is calculated from plasma parameters
	28	N + O + M → NO + M	1.8•10 -43 T -0.5 m 6 s -1	9.4•10 -45 m 6 s -1	[10]
	29	2N + M → N 2 (X 1 Σ g + ,A 3 Σ u + ) + M	8.3•10 -46 exp(500/T) m 6 s -1	3.2•10 -45 m 6 s -1	[10,13]
	30	N + O 2 → NO +O	1.1•10 -20 T exp(-3150/T) m 3 s -1	8.2•10 -22 m 3 s -1	[10]
	31	N 2 (A 3 Σ u + ) + O 2 → 2O + N 2	1.6•10 -18 (T/300) 0.55 m 3 s -1	1.8•10 -18 m 3 s -1	[14]
	32	N + O 3 → NO +O 2	5.0•10 -18 exp(-650/T) m 3 s -1	8.6•10 -19 m 3 s -1	[10]

where M = He, O 2 Plasma parameters (n e and f v (E)) are required for calculation of reaction rates

(5)

. In the presented work, plasma parameters are determined by applying OES and numerical simulation. This diagnostic method differs strongly from numerical simulation based on fluid dynamics applied in

[3,6]

. The latter is self-consistent solution of equations for mass and momentum conservation for all species, conservation of electron energy and Poisson's equation. This plasma-chemical model is based on big number of reactions. Many reactions involving positive and negative ions with not always well known cross sections must be taken into account and several assumptions have to be made. Therefore, excitation and dissociation rates of molecules calculated applying these plasma parameters can be unreliable. To enhance reliability of chemical kinetic simulation in µ-APPJ, we use a combination of two diagnostics namely numerical simulation and optical emission spectroscopy. Intensities of molecular bands and atomic lines in emission spectrum excited during electron impact depend

Table 3 .

 3 Elementary processes considered in simulations of EVDF in µ-APPJ operated in He/O 2 mixture at atmospheric pressure.

where O 2 (X,R), O 2 (X,v"=1-4) -rotational and vibrational excited oxygen molecules in ground electronic state respectively Elementary processes considered for simulation of the electron distribution function are presented in

table 3 .

 3 The plasma electrons are accelerated in electric field and lose kinetic energy in elastic and inelastic collisions with helium atoms and oxygen molecules. Possible influence of nitrogen impurity in He/O 2 mixture on calculated electron distribution function is determined. This influence is very low and is neglected in simulation.

  At our plasma conditions, mainly metastable atoms He(2 1,3 S,2 3 P) and resonance state He(2 1 P) are excited during electron impact. Rate constants for electron impact excitation of other excited states of helium atom are lower by more than two orders of magnitude. Lifetime of resonance state He(2 1 P) concerning spontaneous emission amounts to 0.56 ns. The resonance photons (λ = 584.3 nm) are "trapped" in helium and are absorbed by admixture molecules namely O 2 because of enough high concentration and high absorption coefficient[28]. Nitrogen and oxygen molecules are ionized by collisions with metastable helium atoms. Because of relative high concentration of O 2 and high rate constant[START_REF] Zhiglinskii | Handbook of Rate Constnts of Presses of Atoms, Ions, Electrons, and Photons[END_REF], the Penning ionization of oxygen molecules[START_REF] Mcewan | Chemistry of the atmosphere[END_REF] is the main quenching process of

			N 2 + e → N 2 + (B) + e			(7)
			He(2 1,3 S,2 3 P) + N 2 → N 2 + (B) + e		(8)
	No.	Reaction	No. He(1 1 S) + e → He(2 1,3 S,2 3 P) + e	Reaction	(9)
	37	O 2 (X) + e → O 2 (a 1 ∆) + e He(2 1,3 S,2 3 P) + O 2 → O 2 + + He + e 12	He + e → He(2 3 S, 19.83 eV) + e (10)
	38	O 2 (X) + e → O 2 (b 1 Σ) + e	13	He + e → He(2 1 S, 20.63 eV) + e
	39	O 2 (X) + e → 2O( 3 P) + e	14	He + e → He(2 3 P, 20.96 eV) + e
	40	O 2 (X) + e → O( 3 P) + O( 1 D) + e	15	He + e → He(2 1 P, 21.23 eV) + e
	41	O 2 (X) + e → O 2 (X,R) + e	16	He + e → He(3 3 S, 22.72 eV) + e
	42	O 2 (X,v"=0) + e → O 2 (X,v"=1-4) + e	17	He + e → He(3 1 P, 23.10 eV) + e
	43	O 2 (X) + e → O 2 (4.5eV) + e	18	He + e → He(4 1 P, 23.74 eV) + e
	44	O 2 (X) + e → O 2 + + 2e	19	He + e → He(5 1 P, 24.04 eV) + e
	45	O 2 (X) + e → O + O + + 2e	20	He + e → He + + 2e
	He(2 1,3 S,2 3 P) metastable. Therefore, steady state concentration of helium metastables (	met He N	)	is
	presented as:
			,

O 2 (X) + e → O 2 -

Acknowledgment

Financial support by German Research Foundation of the projects A1 and C2 of Research Group FOR1123 "Physics of Microplasmas" is gratefully acknowledged.