A Sobota
email: a.sobota@tue.nl

J Van Dijk

M Haverlag

AC breakdown in near-atmospheric pressure noble gasses: II. Simulations

Keywords: numbers: 52.20.-j, 52.65.-y, 52.80.-s, 52.80.Dy, 52.80.Pi, 52.80.Tn, 52.80.+z

The effect of frequency on the characteristics of AC-driven breakdown processes in 0.7 bar argon is investigated by means of a two-dimensional fluid model. The geometry represents the HID lamp burner with a pin-pin electrode system forming a 7 mm electrode gap. The breakdown process is considered in the frequency range between 60 kHz and 1 MHz. The appearance of the discharge and the influence of the voltage frequency on its characteristics obtained in the simulations is in good agreement with the experimental data (accompanying paper -Sobota et al., submitted for publication in J.Phys.D: Appl. Phys., Special issue on LS12/WLED3 symposium). The role of the secondary electron emission from the electrode surfaces is demonstrated and linked to the lowering of the threshold voltage with the increase of frequency observed both in experiment and model.

Introduction

Breakdown processes driven by high-frequency AC voltage have been a topic investigated only for specific applications, for example for controlling and suppressing of the surface discharges on the printed circuit boards and similar semiconductor-based systems [START_REF] Pfeiffer | High-frequency voltage stress of insulation. Methods of testing[END_REF]. Related to this, the problem of surface and volume partial discharges occurring in partial vacuum have been of interest in aerospace insulating technology [START_REF] Koppisetty | Breakdown characteristics of helium and nitrogen at kHz frequency range in partial vacuum for point-to-point electrode configuration[END_REF]. On the other hand, AC discharges found their use in the science of material processing, where the glow discharges created in atmospheric pressure possess the uniformity that is not present in filamentary discharges obtained in pulsed mode [START_REF] Park | Gas breakdown in an atmospheric pressure radio-frequency capacitive plasma source[END_REF][START_REF] Shi | Large-volume and low-frequency atmospheric glow discharges without dielectric barrier[END_REF][START_REF] Walsh | Atmospheric-pressure gas breakdown from 2 to 100 MHz[END_REF]. Also, some research on ACdriven discharges has been done for the lighting industry when trying to design ignition concepts for high-frequency operated lamps [START_REF] Sießegger | Ignition concepts for high frequency operated HID lamps[END_REF] or lower their ignition voltage [START_REF] Beckers | Pulse, dc and ac breakdown in high pressure gas discharge lamps[END_REF].

When talking about the breakdown processes driven by high-frequency voltage, it is necessary to distinguish between the frequency ranges, as the discharge forms in different ways at different frequencies. The important parameter is actually a specific combination of the frequency, the electrode gap and the pressure of the gas, but if we imagine a system where the electrode gap and the gas pressure are kept constant, we can analyze the effect of the voltage frequency on the breakdown process. At low frequencies, provided that the voltage amplitude surpasses the threshold for breakdown, a breakdown event takes place in every half of every voltage cycle and it can be thought of as pulsed breakdown with very slow-rising voltage.

Increasing the frequency and keeping the voltage level just above the threshold for breakdown brings us to the range where breakdown events take more than one voltage cycle to complete. Even though the time necessary for the development of the charged channel to cross the gap can vary from just a fraction of a voltage cycle to few tens of voltage cycles [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF], the whole breakdown process still takes several cycles to complete, as the charge buildup at the electrode tips that precedes the formation of the ionized channel is necessary for its subsequent growth. The electron losses in this frequency range are drift-dominated. In a configuration with two pin electrodes the charged particle production volume is located around the electrode tips, as this is the area of highest electric field in the system. This production volume is also close to the electrode surface, bringing forward the potential significance of the secondary electron emission from the electrode material. A further increase of the driving frequency causes the transition to the diffusive mode, where the electron losses are diffusion-driven and the secondary electron emission from the electrode surface loses in relative significance as the ions in the system become insufficiently mobile. The description of the frequency ranges as presented here can be found in texts written on the breakdown phenomena in general [START_REF] Meek | Electrical breakdown of gases[END_REF]. In the high frequency range where the electron losses are drift-dominated not much research has been done.

The accompanying paper [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] presents the experimental results of the research done in the frequency range between 60 kHz and 1 MHz in argon and xenon at nearatmospheric pressure. The experiments were performed using 70W Philips high intensity discharge (HID) lamp burners. The electrode gaps in the experiments were 4 or 7 mm, and this particular combination of gas pressure, electrode gap and driving frequency positioned our experimental conditions in the high-frequency range where the electron losses were high and drift-dominated. We described the appearance of the discharge, which was found to develop over several voltage cycles. The ionized channel to cross the electrode gap formed in different ways for different frequencies -in the very last part of the last voltage cycle before breakdown in 60 kHz, to several cycles at 1 MHz. We showed that the threshold voltage needed for breakdown was decreasing with the increase in the driving frequency, which was a phenomenon that could not be explained in a satisfactory manner using the already proposed theories [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF][START_REF] Meek | Electrical breakdown of gases[END_REF]. We have also commented on the influence of the UV irradiation and 85 Kr used on occasion during the breakdown process.

The aim of this paper is to characterize high-frequency AC breakdown events in a noble gas by means of computer simulations, in the range where most other work on the subject has been experimental [1-5, 7, 8]. We show the results of the simulations done in 0.7 bar argon, in a pin-pin geometry, using high-frequency AC driving voltage. We vary the frequency between 60 kHz and 1 MHz, with an electrode gap of 7 mm, to reproduce as accurately as possible the experimental conditions in the work presented in the accompanying paper [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. The simulations show an agreement with the experiment regarding the value for the breakdown voltage, the appearance of the discharge and the behaviour of the threshold voltage for breakdown as a function of frequency. In addition, we show the relative significance of the secondary electron emission from the electrode surface for various heavy species.

System under consideration

The simulations presented in this paper were done in 0.7 bar pure argon atmosphere. Given the high pressure, the low ionization degree and the enclosed geometry, the temperature, pressure and density of the ground state argon were considered constant in the model. In the simulations, we have observed the evolution of the electrons, atomic metastables representing the two metastable (4s) states (Ar*) and atomic (Ar + ) and molecular (Ar + 2 ) ions. The initial densities were chosen to represent a state of low ionization degreea typical starting condition in cold HID lamps. The initial densities of electrons and atomic ions were 10 9 m -3 , while the densities of metastables and the molecular ions were presumed to be zero. An important parameter in these simulations was the secondary electron emission coefficient from electrode surface, which was defined separately for every species. We set these parameters to 0.07 for atomic ions and metastables and to 0.04 for the molecular ions. The secondary electron emission coefficients are not well known in general, and there is a big spread of values suggested in the literature [START_REF] Yu P Raizer | Gas discharge physics[END_REF][START_REF] Brown | Basic data of plasma physics[END_REF][START_REF] Ardenne | Tabellen zur Angewandten Physik I[END_REF][START_REF] Grigoriev | Hanbook of physical quantities[END_REF]. We chose the value 0.07 because it was chosen previously for breakdown simulations in similar conditions [START_REF] Brok | Modelling of Transient Phenomena in Gas Discharges[END_REF]. The secondary electron emission coefficient for molecular argon ions was estimated to 0.04 because the data [START_REF] Brown | Basic data of plasma physics[END_REF] suggests that the increase in mass of noble gas ions while keeping charge constant decreases the secondary electron emission coefficient in the assumed ratio (for reference, we compared the secondary electron emission coefficients of Ar + and Kr + , two ions with mass ratios of approximately 2).

Table 1 lists the reactions used in the simulations. In comparison with the fast development of the discharge in pulsed voltage mode, as described in [START_REF] Sobota | The role of metastables in the formation of an argon discharge in a two-pin geometry[END_REF], the development of the discharge we describe using these simulations is much slower (up to 100 µs compared to 150 ns in the pulsed mode). As a consequence of this prolonged simulation period, these simulations require a broader reaction list that includes the recombination processes as well, as given in table 1. The reaction list is divided into two parts, for better overview of the included processes. The first part features the excitation and ionization processes, most efficient being processes number 1 and 3, the excitation and direct ionization of the argon atom by electron collision. Reaction no. 4, the collisional process in which the molecular ions are created, is very important in near- atmospheric pressure chemistry because it involves a collision of an atomic ion with two ground state atoms, depleting the atomic ion supply. This means that the dominant ion species have much lower mobility than expected from atomic ions. Reactions involving the atomic metastables may not play the main role in the breakdown chemistry, but we have already shown that even for fast breakdown processes, they do give a contribution that cannot be neglected [START_REF] Sobota | The role of metastables in the formation of an argon discharge in a two-pin geometry[END_REF]. The rest of the process list consists of four recombination and one deexcitation process. The rates for excitation, deexcitation and ionization of argon atoms were obtained from the Boltzmann solver Bolsig+ [START_REF] Hagelaar | Solving the boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models[END_REF]. The geometry used in the simulations is shown in figure 1. It represents the real system used in the experiments performed on AC breakdown and described in the accompanying paper [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. The geometry represents a 70 W HID Philips burner made of YAG (Yttrium Aluminium Garnet) of relative permittivity r = 11.7, tungsten electrodes and the plasma region. The thickness of the burner wall is 0.6 mm, and the electrode radius is 0.3 mm. The background gas in the plasma region is argon at 300 K and 0.7 bar. The geometry of the burner in the model was constructed to resemble the one in the experiments as much as possible, regardless of the fact that the rectangular grid we used did not always allow for smooth surfaces. Imposing different geometry features that are not present in reality, such as 90 o corners in the dielectric burner, creates areas of unrealistically high electric fields at such places. We have therefore avoided approximating a sphere-like burner used in the experiments with a cylinder.

The electrodes used in the experiment were rod-shaped and 0.3 mm in diameter. In the model, we have made a slight adaptation, also shown in figure 1. The adaptation was made because in simulations using simple rod-shaped electrodes gave the highest electric field at the edges of the electrodes. As a consequence, a discharge would start forming at the edge to create a tunnel-shaped formation that would propagate through the gas volume. This never occurs in experiments because the electrode system is never perfectly symmetric, like in a 2D cylindrically-symmetric geometry of our fluid model. In addition, microstructures on the electrode surface create points of high local electric field. These two effects combined make it highly unlikely for the real discharge to incept and propagate in the same way it does in the model with simple rod-shaped electrodes.

To circumvent this problem, we introduced one extra cell in each electrode, in the middle of the electrode cross section, at the symmetry axis; we also took away one cell at the 90 o edge of the electrode, to lower the local electric field at that point. As a result, the discharge always initiates at the symmetry axis, making our model more similar to what happens in reality [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. We would like to stress that the changes we have made to the electrode geometry do not in any way challenge the validity of our results. We simply moved the spot of highest electric field from one place to another, otherwise not changing the electrode geometry.

There are two electrodes in the modelled system -one active and one grounded, like shown in figure 1. The potential of the grounded electrode is kept at zero during the simulations, while a simusoidal signal of amplitude A and frequency ν is applied to the active electrode,

V ActEl = A sin(2πνt). (1)
The simulations were done for four frequencies in the frequency range used in the experiments [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]: ν = 60 kHz, 100 kHz, 400 kHz and 1 MHz. The grid used in the simulations consists of rectangular elements of size (0.05×0.05) mm 2 . The present model does not yet support local grid refinement; this requires compromise in the chosen size of the grid cells, to accommodate for both the need for small cell size in the entire lamp volume and the prolonged calculation time in a breakdown process that can take up to 100 µs. Given the fact that our discharge does not appear filamentary in experiments and that its minimum thickness can be compared to the thickness of the electrodes [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF], cell dimension of 0.05 mm were found to be sufficiently small for the needs of this particular set of simulations.

The time step in the model was limited by highest allowed particle density change of 0.5 %, making the time step always smaller than 10 -11 s. The time needed to run a simulation scaled greatly with frequency of the driving voltage -from a few days for 60 kHz simulations to around three weeks for 1 MHz simulations, one simulation taking 100 % of one 2.8 GHz processor unit.

Model

The set of simulations presented in this paper characterizes the transition from the gaseous to the plasma state in 0.7 bar argon. The thermal equilibrium is not reached during the related experiments [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] nor is it assumed in the calculations. While the electron energy can reach above 10 eV, the energies of the ions in the process remain low. In addition, given the high gas pressure, the low ionization degree in the gas and the fact that the processes take place in an enclosed geometry, we assume the temperature, pressure and density of the background gas to be constant. The simulations describe the breakdown process through the interaction of five species: the electrons, the neutral gas atoms (Ar), the atomic metastables (Ar*) representing the two lowest-lying metastable states (4s) and the atomic (Ar + ) and molecular ions (Ar +

2 ). The model used for simulations is a 2D fluid model using a cylindrically-symmetric geometry. It is a part of the modelling toolkit Plasimo [START_REF] Van Dijk | The plasma modelling toolkit plasimo[END_REF] developed in Eindhoven University of Technology. The model was originally created and documented by Hagelaar and Kroesen [START_REF] Hagelaar | Modeling of microdischarges in plasma addressed liquid crystal displays[END_REF][START_REF] Hagelaar | Boundary conditions in fluid models of gas discharges[END_REF] for plasmas in display devices. It has been further developed for many other applications [START_REF] Van Dijk | The plasma modelling toolkit plasimo[END_REF]. As far as lighting purposes are concerned, the extensions done for modelling of the ignition process in fluorescent tubes [START_REF] Brok | A model study of propagation of the first ionization wave during breakdown in a straight tube containing argon[END_REF][START_REF] Brok | Numerical description of high frequency ignition of fluorescent tubes[END_REF] were done by van Dijk and Brok. It was later used for the simulations of the breakdown process with fast-rising voltage in near-atmospheric pressure gas [START_REF] Sobota | The role of metastables in the formation of an argon discharge in a two-pin geometry[END_REF]. For this set of simulations, we took into account the transport of all heavy species except the neutral gas atoms and secondary electron emission from the electrode surfaces.

The two-dimensional fluid model uses the control volume method for solving equations for the electrostatic potential and particle and electron energy density. The grid we used is composed of rectangular elements ∆x and ∆y in size that represent the plasma region, an electrode or a dielectric material. The geometry was cylindrically symmetric, making the grid elements ring-like structures in reality. The Poisson equation is solved in the plasma region and in the dielectric materials; the transport equations are solved only in the plasma region.

Fluid model

As mentioned, the model we used for the simulations presented in this paper was already used and described in [START_REF] Sobota | The role of metastables in the formation of an argon discharge in a two-pin geometry[END_REF] in a somewhat simpler form for pulsed discharges in 0.7 bar argon in a pin-pin geometry. We will shortly restate the main principles in the following section. f (E/N) Einstein relation [START_REF] Hagelaar | Solving the boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models[END_REF] A fluid model solves balance equations for particles and energy in order to describe a physical system. The general form of the balance equation for particle species p of density n p , flux Γ p and source term S p is as follows

∂n p ∂t + ∇ • Γ p = S p (2)
The density, flux and the net source term are functions of space and time. Net source term consists of contributions from the reactions that create or destroy the particles of the given type p. The source or the sink term for each reaction r is given as a product of the total number of created or destroyed particles c r,p in the given reaction, the densities of the reacting species n i and the rate coefficient k r . The net source term for the given species S p is given as the sum over all the relevant reactions.

S p = r c r,p k r i n i (3)
The flux Γ p is approximated by the drift-diffusion equation:

Γ p = µ p En p -D p ∇n p (4)
The first term in the drift-diffusion equation is the drift flux, characterized by the particle mobility µ p , proportional to the imposed electric field in the system E. The second term is the diffusive flux, proportional to the gradient of the particle density and the diffusion coefficient D p . The diffusion coefficients for the charged species are calculated from the Einstein relation (table 2).

D p = k B T p µ p q p (5)
There are two important values needed for the calculation of the diffusion coefficients -the particle mobility, and the ion temperature. Particle mobilities µ p are given as a simulation input parameter in a table, and the ion temperatures are calculated by the model. The ion temperatures are related to the temperature of the surrounding gas and given as [START_REF] Ellis | Transport properties of gaseous ions over a wide energy range[END_REF]

k B T i = k B T g + m i + m g 5m i + 3m g m g (µ i E) 2 (6)
T i are ion temperatures, T g is the temperature of the surrounding gas and m i and m g are the corresponding masses.

Because of the small electron mass which causes poor energy transfer in electronneutral collisions, the equation 6 does not hold for electrons. Instead, the local mean electron energy is given by the energy balance equation

∂(n ε ) ∂t + ∇ • Γ ε = -eΓ e • E + S ε (7)
The electron energy density is given as a product of the electron density and the mean electron energy n ε = εn e . The mean energy flux Γ ε can be expressed as [START_REF] Yu P Raizer | Gas discharge physics[END_REF][START_REF] Brok | A model study of propagation of the first ionization wave during breakdown in a straight tube containing argon[END_REF]]

Γ ε = 5 3 µ e En e ε - 5 3 n e D e ∇ε (8)
The second term in the equation above is the heat conduction flux, proportional to the mean electron energy gradient.

The first term on the right-hand side of equation 7 stands for the heating (energy gain) from the electric field. The second term S ε is an analog of the net source term for the particle density. The electron energy ε r can be gained or lost in reactions with the associated reaction rate R r .

S ε = - r c r ε r R r (9)
The local electron temperature necessary for the calculation of the electron diffusion coefficient as given in equation 5 can be calculated from the local mean electron energy as k B T e = (2/3)ε.

The electric field in the system is calculated by solving the Poisson equation.

∇ • ∇ϕ = -∇ • E = - p q p n p (10)

Treatment of charge at the interfaces of two media

The boundary conditions used in this model were described in detail in [START_REF] Hagelaar | Modeling of microdischarges in plasma addressed liquid crystal displays[END_REF]. Here we will describe them in short. The Poisson equation is solved in different regions in the computational domain, such as the discharge region and the dielectric region, with given permittivity . The electrode potentials are boundary conditions for the plasma-electrode boundary. For particle densities and electron energy density at open boundaries and the symmetry axis we use the homogenous Neumann boundary conditions. In the experiments [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] we took care not to place metallic surfaces in the proximity of the test lamp. Consequently, these boundary conditions regarding the Poisson equation should correctly reflect the reality in the experiments.

The boundary condition for the Poisson equation at the dielectric-plasma interface (the interface in the remainder of the paper) is determined by the surface charge density σ at the interface. The effect is given by Gauss's law

σ = dielectric E dielectric • e n -0 E plasma • e n ( 11
)
where the electric fields are given at the interface. The surface charge density is included in the Poisson equation, in an additional term on the right-hand side of equation [START_REF] Yu P Raizer | Gas discharge physics[END_REF].

The surface charge density σ resulting from the charged particles hitting the wall at the moment t has the following form [START_REF] Hagelaar | Boundary conditions in fluid models of gas discharges[END_REF]:

σ = t 0 p q p (Γ p • e n ) dt (12)
e n is the normal vector pointing towards the interface, and the flux of charge particles directed to the surface is given by [START_REF] Hagelaar | Modeling of microdischarges in plasma addressed liquid crystal displays[END_REF]

Γ p • e n = a p µ p (E • e n )n p + 1 4 v th,p n p - 1 2 D p (∇n p • e n ) ( 1 3 )
The parameter a p is equal to one if the drift velocity of the charged particles of type p is directed towards the interface, zero otherwise. The thermal velocity is determined by the particle temperature

T p v th,p = 8k B T p πm p (14)
The temperature T p is given by equation ( 6) for heavy particles and by the average energy value for electrons. Setting equation ( 13) as the boundary condition for the drift-diffusion equation ( 4), we get the expression that must be satisfied at the interface

µ p (E • e n )n p -D p (∇n p • e n ) = = a p µ p (E • e n )n p + 1 4 v th,p n p - 1 2 D p (∇n p • e n ). (15)
On the left-hand side of the equation above is the continuum expression for the flux, valid in the plasma volume, and on the right-hand side we find the analytical term for the particle flux toward the interface due to the drift motion in the electric field and the random motion of the particles. Ultimately, the expression for the heavy particle flux towards the interface is obtained in combination of equations ( 13) and ( 15)

Γ p • e n = (2a p -1)µ p (E • e n )n p + 1 2 v th,p n p (16)
Because of secondary electron emission, electrons require a more detailed treatment. We cannot simply use the total electron density n e in the equation ( 16), because it consists of the α electrons coming from the bulk of the plasma and the γ electrons due to secondary emission, n e = n α + n γ . Equation ( 16) can be rewritten for α electrons as [START_REF] Hagelaar | Modeling of microdischarges in plasma addressed liquid crystal displays[END_REF]

Γ α • e n = (2a e -1)µ e (E • e n )n α + 1 2 v th,e n α ( 17
)
The flux of γ electrons due to secondary emission has the following form [START_REF] Hagelaar | Modeling of microdischarges in plasma addressed liquid crystal displays[END_REF]:

Γ γ • e n = -(1 -a e ) p γ p (Γ p • e n ) ( 1 8 )
Γ p and γ p are the flux of the heavy particles of type p and the secondary electron emission coefficient for an impact of the same type of heavy particles on the interface, respectively. The term (1 -a e ) is included for the event that the electric field is directed away from the interface. The sum of the two fluxes ( 17) and ( 18) is used as the boundary condition for the total electron flux.

Results

The terms breakdown process, breakdown moment and breakdown voltage are extensively used in this paper. Their respective definitions from the experimental point of view still hold in case of simulations: the breakdown process is a phenomenon where a particular body of gas is turned into plasma when a significant amount of energy is introduced into the system, in our case through the electric field imposed by a system of electrodes.

The breakdown process ends at the breakdown moment, when conducting current starts flowing through the gas and the voltage across the electrode gap drops. The voltage associated with the breakdown moment, just before the sharp voltage drop, is called the breakdown voltage. In simulations, we define the breakdown moment as the moment when the current in the electrodes starts rising exponentially. A simulation is considered terminated as soon as the breakdown moment is reached.

Discharge evolution in time

It has been experimentally shown [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] that in our geometry the AC-driven discharges develop over several voltage cycles. The light emission in 0.7 bar argon discharges can be observed over an increasing number of voltage cycles as the frequency of the driving voltage increases. The time needed for the breakdown process is also not constant. The measurements we refer to and compare our simulation results with have been made with the aid of UV irradiation; we have argued extensively that the most important role of UV irradiation in this case was the minimization of the statistical lag time, and that there were no significant modifications of the breakdown process [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF].

In 60 kHz, the part of the breakdown process we were able to observe took less than one voltage cycle, and the development of a charged channel to cross the electrode gap took just a fraction of the last part of the last voltage cycle before breakdown. The accumulation of the charge at the electrode tips in the beginning of the breakdown process was observed as point-like light emission. The maximum frequency at which we have observed the breakdown process by means of an iCCD camera was 800 kHz, and at that frequency the visible part of the breakdown process took 33.5±5 voltage cycles. The discharge growth was not restricted to one last cycle, but spread over multiple voltage cycles, with a diffuse structure growing from the electrode tips, but not bridging the electrode gap in the time between two successive zero-crossings of the driving voltage.

Figure 2 shows moments in the simulated discharge development in 60 and 400 kHz discharges. The discharge development is represented by the argon metastables density. The light emission observed in the experiments [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] cannot be directly linked to any particle density, but the emitted light came mainly from radiative processes in the plasma, and the light intensity is proportional to the density of the species involved. Given the low ionization degree and the fact that the wavelengths observed in the experiment were in the visible and near-visible region, atomic metastables are good candidates for the task at hand. One could argue that the dynamics of the ion density would give different results because they are subject to drift, while the metastables only diffuse, but this would not be a correct assumption. First, as can be seen in figure 3, the atomic ion density is very low, because of the very efficient collisional process between argon ions and neutral atoms that associates an argon ion and an argon atom into a molecular ion. As the density is orders of magnitude below the levels of molecular ions or metastables, the contribution to the light emission made by the atomic ions would be negligible. The molecular ions, on the other hand, are twice as heavy as the atomic ions and do not have large drift distance, as will be shown later in the text. It is, therefore, safe to assume that the the atomic metastable density is a good representative of the simulated breakdown process.

Every frame shown in figure 2 except for the last pair of frames was taken at a zero-crossing of the driving voltage. The timing of each frame presented in figure 2 is shown in the graphs in figure 3. This choice is suitable to show the differences in the discharge growth in the two frequencies, because at zero-crossings the production rate of the charged particles is virtually zero, as the electric field across the gap is very small. Consequently, there is no processes that can quickly change the appearance of the discharge in the vicinity of the zero-crossings, which makes them a sort of stable points in which one can compare the results of two discharges of very different growth dynamics.

The left column of the figure shows the discharge evolution in time at 60 kHz. There was no propagation further than the electrode tip until the very last minimum, which is in agreement with our experimental results. The 400 kHz discharge shows different behaviour. At 95 µs after the start of the simulation, the discharge has already started propagating from both electrode tips. The subsequent two frames show further propagation and charge buildup in the channel at a zero-crossing and a minimum before the breakdown moment. The density was high in the entire volume between the electrodes already 1/2 cycle before the breakdown moment. This is also in agreement with the experimental results [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF], where we saw that the growth of the channel to cross the electrode gap took several voltage cycles. Both the 60 kHz and the 400 kHz discharges took more than one voltage cycle to form. The dynamics of the discharge growth when crossing the electrode gap was different in the two chosen frequencies, and it took only a small portion of the last voltage cycle in 60 kHz, but it was preceded by charge build-up at the electrode tips, which was necessary for the subsequent channel formation.

Figure 3 shows the evolution of particle densities averaged over the whole computational volume as a function of time, in 60 and 400 kHz discharges. One should notice from figure 2 that the main particle production sites and the places of highest particle densities are just in front of electrode tips. As the particle densities in these places are many orders of magnitude larger than in the volume away from the lamp axis, we can safely assume that the figures showing the mean particle density evolution in time actually reflects the dynamics at the electrode tips and subsequently in the area between them. The graphs in figure 3 show similar basic behaviour -the particle densities build up in every half of every voltage cycle for both frequencies, until the electrode gap is crossed and breakdown takes place, as shown in the previous figure. This is the key feature of AC discharges in this frequency range.

Figure 3 shows that electron and atomic ion densities follow the phase of the voltage across the active electrode, while the molecular ion and metastable densities rise in a step-like manner. The electron and atomic ion densities depend on electron temperature, as they are mostly created in reactions that involve electrons (see table 1). The electron temperature can follow the electric field profile in this frequency range, so it is understandable that the reaction coefficients for these processes will oscillate with the electric field. While the electron and atomic ion production rates are low, these particles can be lost in drift (electrons) or other reactions, creating other species, like the molecular ions. The densities of molecular ions and metastables depend among other on reactions that do not directly reflect the electron temperature, and therefore suffer no oscillations as a function of the voltage phase. There are no large losses as well because the metastables do not drift, the molecular ion drift is very limited (see further in the text) and they do not diffuse far. Thus the two different behaviour of particle densities as a function of time.

Breakdown voltage and formative time as function of frequency

The experiments have shown that breakdown voltage is a function of frequency in our experimental conditions [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. The increase of frequency caused the lowering of the breakdown voltage across the whole frequency range used in the experiments.

As finding the minimum voltage (the threshold voltage) needed for breakdown in simulations is an extremely long-lasting process, especially in high frequencies, we have decided to compare the characteristic voltages needed for breakdown in a different manner. We have devised two different ways of comparing the breakdown processes - first, using a unique voltage amplitude and comparing the formation time in several frequencies and second, finding the voltage amplitudes at which breakdown in different frequencies takes equally long.

The voltage amplitude and the formation time we took as a reference value was obtained from simulating breakdown in 60 kHz, using a slow voltage ramp, to find the true threshold voltage for the simulated breakdown process. We found that the threshold voltage was 1.75 kV and that the formation time was roughly 100 µs. One should note that choosing any value below 1.75 kV for simulating a 60 kHz discharge would not result in breakdown, as 1.75 kV was the lowest voltage amplitude at which the charged particle (mainly electron) production was bigger than the combined recombination, drift and diffusion losses.

Figure 4 shows the development of the electron and molecular ion densities at the same voltage amplitude (1.75 kV), for four different frequencies. We found that the time needed for breakdown decreases with the increase of frequency, which corresponds to the behaviour found when using an increasing amount of overvoltage [START_REF] Yu P Raizer | Gas discharge physics[END_REF]. The times needed for breakdown were 95.7 µs, 73 µs, 24.4 µs and 12.7 µs for 60 kHz, 100 kHz, 400 kHz and 1 MHz, respectively. This behaviour at the same voltage amplitude suggests that the threshold voltage needed for breakdown decreases with the increase of frequency.

The alternative way to compare breakdown events at different frequencies is to find the voltage amplitudes that had to be used in simulations to obtain the breakdown process that takes place in the same amount of time for every frequency. The voltage amplitudes for 100 µs breakdown processes were 1.75 kV, 1.72 kV, 1.6 kV and 1.55 kV for 60 kHz, 100 kHz, 400 kHz and 1 MHz, respectively. The results are plotted against the experimental data for breakdown voltage in the same conditions, in figure 5. The values we obtained in simulations show the same falling trend with the increase of frequency and similar values to the ones found in the experiments, even though the breakdown voltage seems to drop more slowly with the increase of frequency in the simulations. This, however, does not necessarily mean that the simulations are giving incorrect results. In the accompanying paper [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] we have shown that the breakdown process takes longer in higher frequencies. Therefore, if we would have allowed for more than 100 µs for the discharge development in the model, the voltage amplitude necessary would have been lower in higher frequencies, and the slope in figure 5 would have been more like the one obtained in the experiments. The low values coming out of simulations on the low frequency end of the graph are probably an effect of overestimating the secondary electron emission coefficient of the molecular argon ions. These ions have very low energies and it is possible that our value of 0.04 is an overestimate. However, we made this estimate according to the data we had, as already explained in section 2, and we did not want to tailor the value to make the results fit better with the experimental data.

The effect of secondary electron emission

We have been studying breakdown processes that are relatively long compared to the breakdown events using fast-rising pulsed voltage. Some additional parameters become important as the long breakdown times of 100 µs allow for drift and diffusions of ions and secondary electron emission from the electrode surfaces. Figure 6 depicts the influence of the secondary electron emission by metastable and atomic ion impact. For this set of simulations, we have changed the value of secondary electron emission coefficient for metastables and atomic ions (γ Ar * and γ Ar + , respectively) from 0.07 used in all the other simulations, to zero. The decision to make such a drastic change in the values of γ Ar * and γ Ar + , and to do it at the same time, was made to show how relatively unimportant secondary electron emission by these two particle types is. The figure shows that even disabling the secondary electron emission by metastables and atomic ions does not change the timing of the breakdown process significantly, and it certainly does not hinder it. The reason for this is very simple -the metastables are not charged and therefore do not drift back and forth around the electrode tip, they just drift away from the production site. The atomic ions, on the other hand, are depleted by the collisional process with neutral atoms, to produce molecular ions. Therefore, even if in theory the contribution to the secondary electron emission by atomic ions would be considered important, their low density (see figure 3) makes it negligible.

The influence of the secondary electron emission by molecular ions is much stronger. They are numerous because of the production process that creates them is very efficient in near-atmospheric pressures, and they are twice as heavy as the atomic ions and therefore do not drift as far. In a case when the secondary electron emission by the molecular ions is disabled (changed from γ Ar + 2 =0.04 to zero), the breakdown process cannot be sustained at the same voltage. Their role in an AC breakdown process in our geometry and frequency range is crucial.

Figure 7 shows the molecular argon ion density at the symmetry axis during one voltage cycle. We have chosen to show the data on the symmetry axis because this is also the shortest straight line that connects the tips of the two electrodes, and one might expect the drift of particles produced at an electrode tip to be directed along this path. The graph shows the behaviour of the Ar + 2 density in the first simulated voltage cycle at 60 kHz. The lines in the graph are 1/50 of the voltage cycle (3.33 × 10 -7 s) apart. The active electrode is situated on the left-hand side, and the grounded one is on the right. As the voltage on the active electrode starts rising in the first half of the voltage cycle, a sharp increase in particle density occurs at the tip of the active electrode. This charge build-up is followed by the drift of the molecular ions away from the electrode tip, as can be seen from the position of the maximum of the particle density moving towards the center of the electrode gap. The drift is caused by the high electric field at the positively charged active electrode. Subsequently, the polarity of the voltage on the active electrode changes, causing the molecular ions to drift back to the electrode surface, striking it and creating secondary electrons.

At the same time, the drift of the electrons is directed towards the grounded electrode, where they come into the high electric field area at the electrode tip and start the ionization and excitation processes that ultimately lead to the creation of new molecular argon ions near the tip of the grounded electrode. This can also be seen in the graph.

The losses that the molecular argon ions suffer during one voltage cycle, when they drift back and forth from the electrode tip is not a constant value, but is a function of frequency. We compared the losses in the second voltage cycle after the beginning of the simulation for the simulations done at the same voltage amplitude of 1.75 kV. The losses were 61%, 59%, 48% and 42% for 60 kHz, 100 kHz, 400 kHz and 1 MHz, respectively.

Figure 8 shows the molecular argon ion density at the symmetry axis in the whole 60 kHz breakdown process. One can see that in every voltage cycle there is efficient particle production, drift towards the center of the electrode gap, followed by the drift back to the electrode surface, at lower density. The arrow in the figure shows the direction of time. As the breakdown moment approaches, the central ion density grows as well, although the distribution is always symmetric with respect to the center of the electrode gap. This is another way to show that at 60 kHz there is no formation of the charged channel that would extend in one chosen direction from an electrode tip, before the very last part of the very last voltage cycle.

Figure 9 shows the same for 100 kHz discharges, but tells a different story about the discharge formation at higher frequencies. The asymmetric distributions of the molecular argon ions that occur after several voltage cycles at 400 kHz and 1 MHz show that there is partial discharge growth in the direction of the other electrode, and that it is initiated from both electrode tips, in several last voltage cycles. This is in very good agreement with the experimental results [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF], where we observed no formation of the charged channel in the gas volume before the very last voltage cycle before breakdown moment at lower frequencies (60 to 220 kHz), but such observations were made at higher frequencies (400 to 800 kHz).

The molecular ion densities are very low in the central part of the electrode gap for low frequencies -60 and 100 kHz, because the discharge does not partially propagate before the bridging of the electrode gap. When the gap is bridged, just before the breakdown moment, the central molecular ion densities rise to 10 21 m -3 , just like it can be seen in the profiles belonging to the 400 kHz and 1 MHz simulations. The reason why this happens earlier in the breakdown process in higher frequencies is partial bridging of the gap during the voltage cycles before the last one, which can be seen also in the experiments.

Discussion

In this paper we presented results of simulations in 0.7 pure argon atmosphere using AC driving voltage. The simulations were done using a two-dimensional fluid model and a cylindrically symmetric geometry. The aim of this paper was to make an overview of most important characteristics of the AC discharges in the frequency range where the discharge takes more than a single cycle to form and the electron losses are still drift dominated.

The model we used was adapted to describe the experimental conditions which we gave an account of in the accompanying paper [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. The discharges in the given experimental conditions were gas volume discharges that grew in the space between the electrode tips, which made it possible to model them in a 2D cylindrically symmetric geometry. The simulation results presented in this paper are in good agreement with the experimentally observed data [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] in several important respects -the appearance of the discharge, its behaviour as a function of frequency and the effect the change in frequency has on the minimum voltage needed for breakdown. The results presented in the previous section allowed us to explain two phenomena that were thus far explained in a way that was not satisfactory in our experimental conditions or not explained at all. These are presented in the following two subsections.

A lower threshold voltage for AC-driven discharges

The evolution of the particle densities takes multiple voltage cycles, which is essentially the reason why the AC-driven breakdown processes require lower threshold voltage than the pulsed breakdown in the same experimental conditions. First, the discharge is allowed a longer time to form. It has been shown many times that the formation time for the discharge is a steep function of the overvoltage used [START_REF] Yu P Raizer | Gas discharge physics[END_REF][START_REF] Moss | Avalanche process in an idealized lamp: I. Measurements of formative breakdown time[END_REF], as the first Townsend ionization coefficient is an exponential function of the applied electric field. From the experimental data [START_REF] Moss | Avalanche process in an idealized lamp: I. Measurements of formative breakdown time[END_REF] it can be seen that the discharge formation time is a bijective function of the overvoltage. If we make an additional assumption that the published data reflects a unique law that connects the formation time to the overvoltage, we are allowed to assume the reverse function, one that would connect the overvoltage to the discharge formation time. In this function, the overvoltage would decrease as the formation time increased. From this, we can draw a conclusion that the more time we allow our discharge to form, the lower the voltage that we have to supply. This has already been shown in pulsed-voltage experiments by changing the pulse slope [START_REF] Sobota | Speed of streamers in argon over a flat surface of a dielectric[END_REF]. Our AC discharges take as much as three orders of magnitude longer to form, and therefore can be expected to require lower voltage.

Another, more important reason why the AC breakdown processes require lower voltage than their pulsed equivalents is that the voltage changes polarity changes while the discharge forms, allowing for additional phenomena to help with the charged particle production. The heavy charged species in AC-driven breakdown processes drift back and forth around the electrode tips, making the secondary electron emission from electrode surfaces an important source of electrons. In our case, the molecular ions are the key species responsible for the secondary electron emission. This effect allows for an effectively higher production of charged particles than is the case in a unipolar fast discharge, even though the electron losses in this frequency range are substantial. In pulsed discharges, secondary electron emission is not likely to play any role, because the heavy particles are effectively immobile during the entire breakdown event. This has already been demonstrated [START_REF] Sobota | The role of metastables in the formation of an argon discharge in a two-pin geometry[END_REF].

The lowering of the breakdown voltage with the increase of frequency

The experiments [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] have shown that the threshold voltage needed for breakdown decreased with the increase of frequency. This phenomenon is also an effect replicated in the simulations and it has not been previously explained in a satisfactory way, at least as far as our experimental conditions are concerned. Two possible explanations were offered. First, there was a possibility that the electron drift losses were lowered with the increase of frequency [START_REF] Meek | Electrical breakdown of gases[END_REF] -we have already shown in the accompanying paper [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] that the electron drift losses were still very large in the whole frequency range we used, because the electron drift distance in one half of a voltage cycle was much larger than was the length of our electrode gap. There was certainly some lowering of the electron drift losses over the entire frequency range we used, since it spread over 1.5 orders of magnitude, but this effect alone cannot explain the effects we have observed on its own. We do expect the lowering of the electron drift losses to become significant with further increase of the voltage frequency.

Second, a conjecture was made that due to the high electron drift losses and low ion mobility, positive net charge would form in the electrode gap away from the electrodes, modify the electric field and allow the starting of an avalanche process at lower voltages as the frequency increased [START_REF] Meek | Electrical breakdown of gases[END_REF]. In our experimental and modelling conditions, we do not believe that this effect had a noteworthy influence on the discharge development. There are several reasons for this. One, we see the lowering of the threshold voltage in the frequency range between 60 kHz and 220 kHz, as well as in higher frequencies. The peculiarity of this lower frequency range is that the formation of the channel to cross the electrode gap happens in one voltage cycle or less, and that the charged channel starts growing from an electrode tip. There is, therefore, no reason to assume "islands" of high charge density in the electrode gap away from the electrode tips, which would start avalanches somewhere in the gas volume. The reason for the lowering of the breakdown voltage must be something else than charge accumulation in the electrode gap.

In addition, we have observed in the simulations that there really exists the threshold voltage under which the particle densities to not rise, the particles do not multiply and the breakdown can never occur, and that this voltage is a function of frequency. For example, a simulation of 400 kHz or 1 MHz discharge will result in breakdown at voltage amplitude of 1.6 kV. The same voltage amplitude at 100 kHz and 60 kHz will not be sufficient for charge build-up and eventual breakdown. We would like to focus on the beginning stages of the simulations, where there is still no high particle density of any species and the voltage has not gone through a phase cycle yet, making it impossible for the ions to drift back and forth in the electrode gap and create islands of high density. Even in these early stages in the simulations, there is a clear difference between the discharges at different frequencies, where some will subsequently grow and develop in at a given voltage amplitude, and some not. We do not believe the reason for this lies in the chemistry of the processes. At these frequencies, in a 0.7 bar argon atmosphere and a low ionization degree, the electron energy is given by and can follow the phase of the imposed electric field. Therefore, the electron energy is independent of frequency at the start of the simulation in our frequency range, and the starting reaction rates in the simulations are thus also independent of frequency.

The suggestion that the net charge build-up in the electrode gap is responsible for the lowering of the threshold voltage as the voltage frequency increases is, thus, not a satisfactory explanation for our observations. Besides, we were experimenting and modelling a pin-pin electrode system. The net charge needed to modify the electric field already given at the electrode tips is very high and can possibly be achieved very late in the breakdown process, after the particle densities have sufficiently increased. This effect would possibly be able to speed up the breakdown process long after it has already started, but not influence it from the beginning.

There is still an effect that causes the threshold voltage needed for breakdown to drop with the increase of frequency. We believe that the solution to this problem lies in the behaviour of the molecular ions in a single voltage cycle. As has been shown in figures 7, 8 and 9, the molecular ion density oscillates back and forth around the electrode tip as the phase of the voltage on the active electrode changes. In these oscillations, the molecular ions undergo collisions with other species in the plasma and their density decreases as a result. The decrease in a single voltage cycle is a function of frequency, simply because in higher frequencies there is less time available between the equivalent points in two subsequent voltage cycles for the collisionally-induced losses. Higher frequency, thus, means less molecular ion losses, and higher density to drift back to the electrode tip and induce secondary electron emission.

We have just explained how an increase of frequency can give an effective increase of electron production. The lowering of the electron losses that are drift-dominated with the increase of frequency is not considered a noteworthy contribution, but even if it was, this effect would add to the increase of the net electron production in a single voltage cycle. The total production rate of electrons can, therefore, be separated in two parts -the part that is caused by the chemistry in the breakdown process, and the part caused by the secondary electron emission. If we assume that at the threshold voltage for breakdown process one must have a certain constant minimum of the net electrons produced in a given time interval, and we know that the increase of frequency increases the part caused by the secondary electron emission, it is clear that the part caused by the chemistry can be lowered. The lowering of the contribution of the chemistry can be achieved by lowering of the mean electron energy via the imposed electric field. This is why the breakdown processes at higher frequencies are possible at lower voltages.

Outlook

We have shown that the simulations done on the HID lamp system and 0.7 bar argon using AC voltage gave results that are in very good agreement with the experiments presented in the accompanying paper [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. There are, however, several ways in which the model we used can be improved. The size of the computational cells in the model is a concern. The geometry should not be excessively coarse, because that would cause errors in the calculations of various key parameters, such as the electric fields in the system. On the other hand, very small size of computational cells would cause very long calculation times, especially at high frequencies. The implementation of the local grid refinement would greatly help this issue.

Additionally, the chemistry used in the model may not be sufficiently complex for the description of a breakdown process in near-atmospheric pressure. A possible improvement would be the addition of another effective atomic metastable state that would represent the (4p) and higher-lying excited states in the argon atom. Excimer molecules could be added as well. This would give a more correct effect of the excited species on the breakdown process.

Conclusion

This paper has shown the results of two-dimensional fluid model simulations of AC breakdown in 0.7 bar argon atmosphere. We have used a two-pin electrode system enclosed in an HID lamp geometry with fixed length (7 mm) of the electrode gap. The electrode gap and the gas pressure fixed, we were able to distill the effect of the voltage frequency (60 kHz to 1 MHz) in the overall breakdown process. We have shown its effect on the appearance of the discharge, the threshold voltage needed for breakdown and the timing of the breakdown process. The results of the simulations are in good agreement with the experimental data [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF], both in their qualitative behaviour as function of frequency, as in their quantitative behaviour in terms of breakdown voltages and formative times. We have also demonstrated the importance of the secondary electron emission from electrode surfaces, in particular by molecular ions, in the breakdown process and linked it to the lowering of the threshold voltage with the increase of frequency observed both in experiment and model.

Figure 1 .

 1 Figure 1. The schematic representations of the geometry used in the simulations. The geometry consists of the 400 × 95 square cell matrix. It is cylindrically symmetric, with the symmetry axis being the lower horizontal edge shown in the figure. The left electrode was always active, and the one on the right-hand side grounded. See the text for further details.

Figure 2 .

 2 Figure2. n Ar * at different points in time during the breakdown process in 60 kHz (left column) and 400 kHz (right column) discharges. Only a part of the geometry is shown -7 mm between the electrode tips, 2.25 around the symmetry axis. The symmetry axis is positioned at the bottom of every frame. The metastable densities are expressed on a logarithmic scale to better show the particle distribution. The timing of every frame is given in a column next to the frames.

Figure 3 .

 3 Figure 3.Particle densities averaged over the whole computational volume as a function of time. The graphs show the electron, metastable and atomic and molecular ion densities against the voltage form at the active electrode. The upper graph represents the breakdown process at 60 kHz and the lower one at 400 kHz voltage frequency. The graphs also show the timing of each frame in the figure2. Every frame except for the last one at each frequency was taken at the zero-crossing of the voltage.

Figure 4 .

 4 Figure 4. The time evolution of the electron and molecular ion densities averaged over the whole computational volume. The simulations shown in this graph were done at the same voltage amplitude of 1.75 kV.

Figure 5 .

 5 Figure 5. The voltage amplitudes at which the formation time in all four frequencies is 100 µs. The results are compared to the breakdown voltages measured in experiments.

Figure 6 .

 6 Figure 6. The influence of secondary electron emission by molecular ions and atomic metastables. The graph shows the time evolution of the particle densities averaged over the whole computational volume. The simulations were done at 100 kHz and 1.72 kV.

Figure 7 .

 7 Figure 7. Evolution of the axial density of molecular ions in the first cycle of a 60 kHz discharge, at 1.75 kV. The density is given on the symmetry axis between the electrode tips -the tip of the active electrode is at position zero, and the tip of the grounded electrode is at 7 mm. The inset shows the voltage amplitude on the active electrode during the collection of data. The lines in the graph are 1/50 of the voltage cycle apart. The red lines represent the evolution of the molecular ion density during the positive half-cycle, and the black ones during the subsequent negative half cycle.

Figure 8 .

 8 Figure 8. Evolution of the axial density of the molecular ions in the whole 60 kHz breakdown process, at 1.75 kV. The density is given on the symmetry axis between the electrode tips -the tip of the active electrode is at position zero, and the tip of the grounded electrode is at 7 mm. The lines in the graph are 1/50 of the voltage cycle apart.

Figure 9 .

 9 Figure 9. Evolution of the axial density of the molecular ions in the whole breakdown process, at 1.75 kV, at 100 kHz, 400 kHz and 1 MHz. The density is given on the symmetry axis between the electrode tips -the tip of the active electrode is at position zero, and the tip of the grounded electrode is at 7 mm. The lines in the graphs are 1/25 of voltage cycle apart: 0.4 µs at 100 kHz, 0.1 µs at 400 kHz and 0.04 µs at 1 MHz.

Table 1 .

 1 Reaction coefficients used in the simulations.

	No. Reaction	Reaction coefficients k r	Ref.
	1 e + Ar → e + Ar *	Bolsig +		[15]
	2 e + Ar * → 2 e + Ar +	Bolsig +		[15]
	3 e + Ar → 2 e + Ar +	Bolsig +		[15]
	4 Ar + + 2 Ar → Ar + 2 + Ar 2.5 × 10 -43 m 6 s -1	[16]
	5 2 Ar * → Ar + + Ar + e	1.2 × 10 -15 m 3 s -1	[16]
	6 2 Ar * → Ar + 2 + e	2.03 × 10 -15 m 3 s -1	[17]
	7 e + Ar * → e + Ar	Bolsig +		[15]
	8 2 e + Ar + → e + Ar	5.4 × 10 -39 ×T e	-9 2 m 6 s -1	[18]
	9 e + Ar + → Ar + hν	1 × 10 -17 m 3 s -1		[18]
	10 Ar + 2 + Ar → Ar + + 2 Ar 2.496 × 10 -36 m 3 s -1	[19]
	11 e + Ar + 2 → Ar

* + Ar 7 × 10 -13 (300K/T e ) 1/2 m 3 s -1

[START_REF] Balcon | Numerical model of an argon atmospheric pressure RF discharge[END_REF]

Table 2 .

 2 Transport coefficients used in the simulations. µ p is the particle mobility, and D p represents the diffusion coefficient.

	Species µ p	D p [Torr cm 2 s -1 ] Ref.
	e	f ( )	Einstein relation [15]
	Ar *	-	8 2 . 9 9 2	[ 1 3 ]
	Ar +	f (E/N) Einstein relation [15]
	Ar + 2

Acknowledgments

This work was supported by Philips Lighting.