Numerical modeling of a DBD in glow mode at atmospheric pressure
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Abstract
In this work, a fluid model of helium glow discharge at atmospheric pressure in a dielectric barrier discharge configuration has been developed and the discharge was numerically simulated. The transport equations for charged and excited species are selfconsistent coupled to the Poisson equation for the electrical field calculation. A finite difference method technique is adopted; a detailed numerical procedure modeling is given. The addition of some nitrogen impurities to the helium successfully reproduced the discharge evolution during the breakdown. The numerical results showed that the discharge has a structure similar to DC lowpressure glow discharges and confirms the establishment of the glow regime at atmospheric pressure under very adequate conditions. The profiles of physical and electrical discharge parameters knowing that, particle densities, electric field, drift velocity, voltages and discharge current are presented and analyzed. A detailed study was made of the effect of nitrogen impurities on the stability of the glow mode of the discharge, and on the evolution of its parameters.
Keywords
DBD Glow discharge Plasma Fluid model Helium Nitrogen impuritiesIntroduction
Dielectric barrier discharges (DBDs) can be used for generating of lowtemperature plasma under atmospheric pressure. As compared to the other types of gas discharge, the simplicity in the configuration and absence of the vacuum system of the DBD are obvious advantages for consideration of industrial applications. DBDs are used in many industrial applications, such as ozone generation [1, 2], decontamination [3, 4], lighting especially of UV light generation by means of DBD excimer lamps [5, 6], the manufacture of plasma display cells [7], modification and surface treatment [8, 9, 10, 11] and semiconductor manufacturing. This type of discharge can be produced when at least one of the electrodes is covered by a dielectric layer, such as glass, quartz or ceramics. An alternating voltage or a repetitively pulsed power source can be used to power the DBD cell. The function of the dielectric layer between the electrodes is to prevent the formation of arc discharge and to limit the current discharge.
A space gap of a few mm is commonly recommended in this case. In order to generate the plasma, the minimum peaktopeak applied voltage must be higher than the breakdown voltage of the gas.

The homogeneous discharge more precisely in glow mode, the physical parameters of this type of discharges can be controlled in space and in time which makes it possible to treat the surface more uniformly.

Otherwise, the parameters of the filamentary discharge are random and difficult to control. The filamentary and the homogeneous modes can often be readily distinguished with the human eye, because the homogeneous discharge forms are uniformly distributed over the electrode area, whereas the filamentary discharge consists of spatially divided filaments [12].
Due to small discharge geometries (the gap between electrodes generally does not exceed a few millimeters), the understanding of discharge physics is rather difficult, and numerical modeling is therefore an important tool for plasma discharge diagnostics.
DBD modeling can be classified into physical modeling and electrical modeling. Electrical modeling is the modeling by constructing an electrical circuit to characterize the overall discharge behavior of the DBD. The electrical modeling can be used to predict the external quantities such as external current and voltage [13, 14, 15, 16]. On the other hand, the physical modeling is based on theoretical model such as the ionization and fluid models. The physical modeling is commonly used to predict the interaction phenomena between the electrons, ions, molecules and metastables states in the discharge [17].
In the case of a glow discharge where the radial structure of the plasma parameters is homogeneous over the entire volume, the 1D model is rather advisable and has the great advantage of reducing the computation time unlike the twodimensional model.

The glow mode and the Townsend mode. Homogeneous DBD in glow mode has discrete discharge structures similar to DC glow discharges at low pressures. On the other hand, the DBD in Townsend mode has no quasineutral region but a positively charged space region, the electric field is not significantly altered in the discharge region, and the current density and electron density are relatively small compared to those of the glow mode [18, 19].
In this paper and in the order of 1 atm of pressure, we developed a 1D fluid model of a glow DBD in helium with a certain level of nitrogen impurities, taking into account the most significant chemical reactions. In fact, the amount of nitrogen added in helium is very small (~ 100 ppm), but its effect on the discharge behavior is very important, which has not been discussed in previous work in 2013 and 2016 for researchers in the same context in the laboratory group of electrical discharge and their applications in Oran. Algeria [20, 21]. The goal therefore is the study of the spatiotemporal evolution of the discharge parameters in a plane and parallel dielectric barrier configuration by means of numerical simulation. By the use of the finites differences method, this is adapted to the numerical processing of the model equations (already used in our preceding works [22, 23, 24]). The parametric behavior of the discharge is analyzed according to the optimization of the external parameters such as the width of the discharge gap, the thickness of the dielectrics, amplitude and frequency of the external voltage.
The paper is organized as follows: after a general introduction, Sect. 2 describes the DBD geometry. Fluid model descriptions, numerical procedure, simulation parameters and initial–boundary conditions are introduced in Sects. 3, 4, 5 and 6, respectively. Section 7 is devoted to interpretation of the results. The influence of the impurity rate on the discharge physical parameters is presented in Section 8.1. Section 8.2 is dedicated to the effect of impurity on the electrical properties of the discharge, and Sect. 9 summarizes the main conclusions and final remarks.
DBD geometry
Fluid model descriptions
A selfconsistent fluid model is used for the description of lowtemperature plasma generated by a DBD at atmospheric pressure, based on a set of balance equations derived from the Boltzmann transport equation. The onedimensional description considering the axial component x of the plasma, only, assumes in a first approach that the main characteristics of the discharge are not influenced by radial effects.
Reaction  Coeff  Value  Ref 

Direct ionization  
He + e → He^{+} + 2e  α _{ionis}  f(E)  [17] 
Direct excitation  
He + e → He^{*} + e  α _{ex}  f(E)  [17] 
Stepwise ionization  
He^{*} + He^{*} → He^{+} + He + e  k _{em}  f(E)  [17] 
Penning ionization  
He^{*} + N_{2} → He + N_{2}^{+} + e  k _{pm}  5 × 10^{−10} cm^{3}/s  [26] 
Radiative recombination  
He^{*} → He^{*} + hν  k _{rm}  7 × 10^{5} s^{−1}  [26] 
Conversion of metastables  
He^{*} + 2He → He_{2} + He  k _{2m}  2.5 × 10^{−34} cm^{6}/s  [26] 
 For electrons:$$ \frac{{\partial n_{\mathrm{e}} (x,t)}}{\partial t} + \frac{\partial }{\partial x}\varGamma_{\mathrm{e}} {(}x,t{) } = S_{\mathrm{e}} (x,t ) $$(1)
 For positives ions:$$ \frac{{\partial n_{\mathrm{p}} (x,t)}}{\partial t} + \frac{\partial }{\partial x}\varGamma_{\mathrm{p}} (x,t) = S_{\mathrm{p}} (x,t) $$(2)
 For the excited particles (helium metastables state):$$ \frac{{\partial n_{\mathrm{m}} (x,t)}}{\partial t} + \frac{\partial }{\partial x}\varGamma_{\mathrm{m}} (x,t) \, = \, G_{\mathrm{m}} (x,t) \,  \, L_{\mathrm{m}} (x,t) $$(3)n_{e}(x, t), n_{p}(x, t) and n_{m}(x, t) are electron, positive ion number and excited particles densities, respectively. Γ_{e}(x, t), Γ_{p}(x, t) and Γ_{m}(x, t) are their flux. The flux densities for electrons, ions and excited particles have the form:$$ \varGamma_{\mathrm{e}} (x,t) =  \mu_{\mathrm{e}} n_{\mathrm{e}} (x,t)E(x,t)  \frac{\partial }{\partial x}(D_{\mathrm{e}} n_{\mathrm{e}} (x,t)) $$(4)$$ \varGamma_{p} (x,t) = \mu_{\mathrm{p}} n_{\mathrm{p}} (x,t)E(x,t)  \frac{\partial }{\partial x}(D_{\mathrm{p}} n_{\mathrm{p}} (x,t)) $$(5)E(x, t) is the electric field, µ_{e} and µ_{p} are the mobility’s of particles, and D_{e}, D_{p}, D_{m} are diffusion coefficients for each type of particle.$$ \varGamma_{\mathrm{m}} (x,t) =  \frac{\partial }{\partial x}\left( {D_{\mathrm{m}} n_{\mathrm{m}} (x,t)} \right) $$(6)
q = 1602 × 10^{−19} C is the elementary charge.
ε_{0}= 8854 × 10^{−14} F/cm is the free space permittivity.
With V_{sd} (t) is the voltage of the solid dielectrics and C_{sd} its capacitance, S is the area of the electrode.
Numerical procedure
The interelectrode space is divided into N + 1 equal spatial intervals in x direction, where i = 0 is the boundary point at the left position and i = N + 1 is the boundary point at the right position, Δx the spacial steps along the x axis and Δt being the time step; superscripts k and k + 1 refer to time t^{k} and t^{k+1}, respectively (t^{k+1}= t^{k}+ Δt). Knowing that k =1 means the initial time and k = N is the final time.
The superscripts e, p, m in Eqs. (15), (16) and (17) represent the species of electrons, positive ions and metastables state.
After implementation of exponential form for fluxes of electron and ion densities (Scheme of Scharfetter and Gummel) [32], the relations (4) and (5) become:
Flux for ion density has the same form like flux for electron density.
The coefficients B_{1}, B_{2}, B_{3} for ions have the same form as A_{1}, A_{2}, A_{3} for electrons.
Equations (22), (23), (24) and (25) are the system of linear equations which are solved by Thomas’s algorithm. Details will be given in literature [33, 34].
Simulation parameters
All constants and coefficients necessary to carry out the calculations have been taken from the work on atmospheric pressure helium discharges presented by Radu and Bartnikas [17, 26]:
The electron diffusion coefficient is a function of electric field given by the relationships:
The ionization coefficient and the excitation coefficients are, respectively, represented in function of electric field by the relationships:
The stepwise ionization coefficient is defined as:
Definition of constants and coefficients for simulation
Parameter  Symbol  Value  Ref 

Pressure  P  760 torr  
Electron mobility  µ _{e}  987 cm^{2}/V s  [26] 
Ion mobility  µ _{p}  14 cm^{2}/V s  [26] 
Ion diffusion coefficient  D _{p}  0.354 cm^{2}/s  [26] 
Metastable diffusion coefficient  D _{m}  0.6 cm^{2}/s  [26] 
Electrode surface  S  12.56 cm^{2}  
Secondary emission coefficient  γ _{sec}  0.1 
Boundary and initial conditions
The boundary values of electric potential at the two electrodes are:
At the left electrode: V = V_{App}
At the right electrode: V =0 V.
Results and discussion
The DBD simulation conditions in He–N_{2} mixture with a rate of N_{2} is 100 ppm at atmospheric pressure are considered as follows: the discharge gap is d =0.5 cm, the amplitude V_{m} of the voltage source is 1.3 kV, while the frequency is fixed at 10 kHz. The dielectric thickness is a = 0.1 cm. The time step is 10^{−9} s.
The duration of the discharge is about 5 μs (time interval t_{2} − t_{1}) for the positive half cycle of the applied voltage, in this time interval, there is a sudden increase in the current from 1 to a maximum of 32.1 mA for the positive peak of the I_{d}, in the negative alternation of V_{app}, the discharge current keeps its behavior but in the sign (−) and the peak value of the negative peak reaches 33 mA, and the profile of the discharge current remains exactly the same from one cycle to another, i.e., it has the same periodicity as the external voltage. In our case, this behavior of the discharge current shows that there is only one breakdown per half cycle of the applied voltage, and this is the most important feature of the glow regime of DBD under atmospheric pressure.
For the temporal variation of the gas voltage V_{g} during one period of V_{app}: the voltage V_{g} increases from 705.3 V at time t = 0 s to 1.2 kV for t = 9 μs, which corresponds at the instant just before the first discharge appears. A value of 1.2 kV of voltage sufficient to maintain the breakdown of the gas, the fall of V_{g} after the moment of the peak of the current causes the extinction of the discharge and during the next half cycle of V_{app}, a new negative voltage V_{g} will trigger the ignition of the second discharge and so on for the cycles Following, the profile of the gas voltage during the time interval of the discharge follows the profile of the discharge current.
Therefore, at each half cycle of the discharge, the accumulation of charges in the inner layers of the dielectric barriers generates an opposite voltage called dielectric voltage V_{sd} which in turn reduces the voltage V_{g} and causes the extinction of the discharge avoiding the formation of the electric arc and consequently the generation of cold plasma.
When the calculation of the electric field is directly proportional to the electron and ion densities, we have chosen to represent the characteristics of the discharge for the helium–nitrogen impurities at atmospheric pressure in only one figure.
The discharge characteristics have similarities to those appeared in the dc glow discharge at low pressure, and they are distinguished by four discharge regions featuring the typical glow mode.
First, a cathode fall region exists where the electric field drops drastically from a maximum value of about 16.5 kV/cm to nearly zero because of large positive space charges near the cathode; in this region the density of ions reaches a maximum value of 4.7 × 10^{11} cm^{−3} and a maximum of 3.6 × 10^{11} cm^{−3} for the electrons. Thickness of the cathode fall is about 0.29 mm. At the cathode, the ions are the majority with respect to the electrons, because the electrons leave this region quickly under the effect of drift in the presence of a strong electric field and leaving behind them a quantity of the positive ions. Near the cathode, the electron density is small but never zeros (n_{e} ≈ 2 × 10^{7} cm^{−3}) because the cathode region is the source of electron generation by the secondary emission process, which is taken into account in our modeling.
The second region that corresponds to negative glow extends to 0.73 mm in thickness, where the electron and ion densities are equal and the electric field remains low. Then, the electric field increases and the electron and ion densities are very close: it is the Faraday dark space. Its thickness is of the order of 1.4 mm. In this zone, a small negative space charge is formed.
The 4th observed zone occupies the most space: it is the positive column, a region of electrically neutral plasma. Its width is 2.58 mm, and the electric field is constant. The value of this field is relatively low and close to 2 kV/cm. The electron and ion densities are equal; with a density is approximately 2 × 10^{10} cm^{−3}. In this region, the mobility of the electrons is reduced because of their interaction with the ions.
The distribution profile of the metastables density is very similar to the profiles of the electron and ion densities; with a maximum density calculated in the region of cathode fall is reached 6.3 × 10^{11} cm^{−3}, it is constant in the area of the positive column and its value is about 1.2 × 10^{11} cm^{−3}.
In terms of validation, our numerical simulation results of the discharge parameters were very similar to the results of the work in the same context, namely the result of literature [39, 40, 41, 42, 43, 44].
In this paragraph, we present a study on the kinetics of electrons and ions in helium in a glow discharge at atmospheric pressure.
The mobility of electrons and ions characterizes the speed with which an electron or ion can move through a medium, when driven by an electric field, while we find that the neutral and excited metastables particles tend to diffuse by the influence of the diffusion coefficient since their mobility is null.
When an electric field E is applied, electrons and ions respond by moving with a mean velocity called drift velocity.
The drift velocity is a parameter defined by the product of the mobility and the electric field W_{i}= sµ_{i}E, with µ_{i} is the mobility of the particles of type i and s is a parameter which is 1 for the positive ions and − 1 for the electrons, the drift velocity closely depends on the value of the electric field.
Influence of the impurity rate
Impurity effect on physical properties
In this part of work, we have varied the concentration rate of nitrogen as an impurity present in the gas with rates, respectively, 150 ppm, 100 ppm, 70 ppm, 50 ppm and 10 ppm; under the same previous conditions of the discharge, we will study the influence of the impurity rate on the evolution of charged and excited particles, as well as the spatial distribution of the electric field at the time when the discharge current is maximal.
For 150 ppm, 100 ppm, 70 ppm and 50 ppm, the discharge keeps its glow regime and at a rate less than or equal to 10 ppm, the densities of electron and ion are very low, and its profiles do not correspond to glow mode of the discharge at atmospheric pressure. Note that this difference in discharge behavior at ≤ 10 ppm is attributed to the fact that the contribution of the Penning effect to electron production is lower when the nitrogen concentration is decreased. The loss of electrons by recombination becomes predominant and is no longer compensated by the electronic production by the process of ionization and direct excitation.
At a rate = 10 ppm, the electric field is very weak and no distinct zone of the glow discharge appears.
Impurity effect on electrical properties
Let us discuss in this part the influence of the N_{2} concentration on the electrical properties of the discharge, in particular the gas voltage, i.e., the discharge voltage and the discharge current. We chose to plot the results for a three different levels of nitrogen impurities: 150 ppm, 100 ppm and without impurity (0 ppm).
Therefore, the effect of increase in the impurity concentration is directly proportional to the increase in the discharge voltages, which in turn accelerates the appearance of the discharge.
The influence of the impurities is manifested on the positive and negative peaks of the discharge current; for example, a value of (44.3 mA and − 49.1 mA), respectively, for + and − peaks recorded at 150 ppm of nitrogen and (32 mA and − 33.2 mA) recorded per 100 ppm of N_{2}, On the other hand, a zero current is observed for the whole of time in the case without impurities due to the impossibility to reach the minimum value of the breakdown voltage in this case.
The increase in current intensity is due to the abundance of electrons and positive ions (electric charge carriers), whenever the concentration of impurities is large (demonstrated results in Figs. 7 and 8). The increase in the concentration of impurities somewhat accelerates the onset of the first discharge and reduces its duration.
Conclusions and final remarks
A 1D fluid simulation of homogeneous dielectric barrier discharge at atmospheric pressure in He–N_{2} mixture has been developed in a parallelplate geometry, while adopting the assumption of the approximation of the local electric field.
Due to the rapid evolution of physical phenomena in the plasma and the strong coupling between the particle transport equations and the Poisson’s equation, we must adopt a very adequate computation time step. These calculations were made possible in a reasonable computation time using a very efficient implicit numerical scheme ‘unconditionally stable’ of the finite difference method for the momentum and momentum transfer equations.
The simulation results for the variation of the discharge current well reflect the behavior of a glow discharge (presence a single peak of current per half period) and a spatial structure of the electric field; electron and ion density are similar to that for an ordinary DC glow discharge a low pressure is formed during the breakdown. This confirms that the uniform DBD in atmospheric helium is a glow type discharge.
At atmospheric pressure condition, glow discharge can be stable and dependent on the external parameters of discharge, i.e., the amplitude and frequency of external voltage, width of the discharge gap and thickness of the dielectric barriers. The values of these external parameters have been mentioned in the “Results and discussion” section.
Our model indicates that Penning ionization of nitrogen impurities by helium metastables plays a major role; it is the dominant ionization mechanism in the discharge which strongly exceeds direct ionization, direct excitation and stepwise ionization. The effect of increasing the level of impurities is evident in increasing the velocity of the charged particles, in the discharge current peaks and the discharge voltage.
Without the addition of nitrogen impurities, no gas breakdown occurred and the simulation gave no results. As well, this type of ionization can occur at low electric field levels, which helps to maintain the discharge under these conditions and to produce more charged particles. We therefore conclude that Penning ionization is a necessary mechanism for obtaining a glow discharge at atmospheric pressure.
Notes
Author contributions
All numerical processing of the model equations, code programming set up in Fortran language, data preparation, figures and manuscript draft were carried out by AS. AWB provided guidance and advice to improve the quality of the manuscript. All authors read and approved the final manuscript.
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