Discharge Enhancement Phenomenon and Streamer Control in Dielectric Barrier Discharge with Many Pores

: The surface and volume discharge enhancement phenomena and streamer propagation direction control in catalytic pores are signiﬁcant for the plasma catalytic degradation of pollutants. In this work, we use a two-dimensional particle-in-cell with Monte Carlo collisions model to explore the effect of lateral voltage on streamer enhancement and streamer propagation control for different driving voltages in pores of various shapes, sizes, and numbers. The driving voltage is applied to the top of the device, while the lateral voltages are applied at the left and right sides of the device. The surface and volume discharge enhancement phenomena become more signiﬁcant and streamer propagation is more restricted within a narrow channel as the lateral voltage (with the same values on the left and right sides) increases from − 5 kV to − 30 kV for a ﬁxed driving voltage of − 20 kV. In this case, both the volume and surface discharges are intensive, leading to highly concentrated plasma species in a narrow channel. Moreover, the streamer propagates in a straight direction, from top to the bottom plate, with the lateral voltage added on both sides. The streamer propagation, however, deviates from the center and is directed to the right side when the lateral voltage is applied to the left. Our calculations also indicate that increasing the number or size of the pores enhances both the volume and surface discharges.


Introduction
Due to the selectivity of catalytic materials and the high activity of plasma, plasma catalysis has been successfully applied to environmental protection activities. Plasma catalysis, an emerging branch of plasma processing, is thus attracting increasing attention [1][2][3][4][5][6][7][8]. In practice, two types of plasma catalysis exist according to the different positions of the catalyst: the single-stage type and the two-stage type [3]. In the single-stage type, the catalyst is placed inside the plasma discharging area, and two configurations exist: dielectric barrier discharge (DBD) [9][10][11] and corona discharge [1]. DBD is a kind of micro-discharge [12] in which the dielectric barriers are placed on the surface of the bare electrode. The filamentary mode [13] and Townsend mode [14] are two types of DBDs. Both can be operated at atmospheric pressure and generate non-equilibrium plasma [13]. The filamentary mode DBD can be treated as a streamer [15] and produces abundant plasma species without the need to heat the background gas remarkably.
However, the effect of lateral voltages on streamer enhancement and streamer propagation control for different driving voltages in many pores of various shapes, sizes, and numbers has not been studied so far. Therefore, in this work, we use a two-dimensional explicit PIC/MCC model to investigate the effect of the lateral voltage on the properties of streamer formation and propagation in a DBD reactor containing numerous pores with smooth boundaries. Section 2 describes the simulation model. Section 3 presents the numerical results and discussion about the shape, size, numbers of the catalytic pores, the effects of the magnitude and direction of the applied voltage, and the effects of lateral voltage on the streamer enhancement phenomenon and streamer propagation control. Section 4 concludes this paper.

Grid and Streamer
We divide the simulation space into 50,000 uniform grids, 1000 × 500. Each grid measures 2.0 × 3.3 µm. Under the applied voltage driving, seed electrons set under the top dielectric plate (x ∈ [0.9, 1.1] mm, y ∈ [1.29, 1.30] mm) will spread down to the bottom dielectric plate and form streamers. The seed electrons are supplied by a current density of 10 5 am −2 . Moreover, streamers formed from the seed electrons and propagating to the bottom dielectric plate give rise to electron and ion avalanches. Thus, we adopt a "particle merging algorithm" to merge the particles when their number in a grid exceeds 10. Rationally, the use of numerical parameters and consideration of numerical convergence are crucial to avoid incorrect results. To obtain the Debye length λ d and electron frequency ω pe , it is important to use the correct time step and space step. These values were set to 10 −14 s and 2.0 × 3.3 µm, respectively, in this work. As the results below show, we obtain the minimum Debye length and the maximum plasma frequency as approximately 10 µm and 3 × 10 13 rad/s, respectively, for a maximum electron density of 10 21 m −3 . Thus, to ensure numerical convergence, the selected space step and time step should be ∼2 µm and 10 −14 s, respectively. Otherwise, if the grid width exceeds 3λ D , the PIC algorithm will become self-heating, leading to "fine grid instability". In this work, self-heating is negligible as the grid width is not much larger than λ d [45,46].

VSim
VSim [47,48], a widely tested technical commercial software product, is used to solve the electrostatic (ES) and electromagnetic fields and study the movements of individual particles [49][50][51][52]. The following two basic processes must be conducted to study the movement of streamers: (1) We need to obtain the charge-current density distribution and electromagnetic field distribution in the streamers to compute the particle movement paths. (2) Then, by summing the mean of all the particle paths, we can track the overall movement of the streamers.
Two numerical methods could be used to deal with streamers: the fluid and dynamic approaches. The PIC/MCC model is the basic model of VSim, a very useful dynamic method. The PIC/MCC model is preferred for simulation of non-equilibrium state streamers whose electron velocity distribution is typically non-Maxwellian. Therefore, the VSim model, including the ES solver and particle moving algorithm, could be used to investigate low-temperature plasma at atmospheric pressure. This work used VSim version 9.1.

Particle-In-Cell Algorithm
Unlike the fluid method, whose electron velocity follows a Maxwellian distribution, the PIC algorithm is adopted to simulate individual dynamic behavior, and the speeds of the electrons and ions are self-consistent. As per the logic of this algorithm, "macro particles" are set to denote various real charged particles, and the space is divided into computational grids. The PIC algorithm defines the electromagnetic field on grid points, and we adopt a uniform Cartesian mesh in this work.

Particle Moving: The Newton Equations
Briefly, the calculation process for particle moving is called the "particle moving procedure." The time needed for one moving is known as the "time step" (δt), and the particle moving process is conducted using the Newton equations, as follows: x n+1 = x n + v n δ t .
n and n + 1 denote the nth and n + 1th time step, respectively. The time step defined in this work is 10 −14 s. The stability of the PIC algorithm and the Courant-Friedrichs-Lewy (CFL) condition limit the value range of δt. The CFL condition is as follows:

Electric Field Solution: The Poisson Equation
In this work, we adopt the Poisson equation to solve the electric field. The Poisson equation is as follows: − ·(ε φ) = ρ 0 .
To solve for electric potential in the Poisson equation, we adopt the Neumann boundary condition in the x direction and the Dirichlet boundary condition in the y direction. In this work, we define the PIC algorithm to solve the Poisson equation both in the discharge gap and the dielectric. However, we only need to solve the particle moving equation in the discharge gap given the dielectric barrier discharge properties. Moreover, should a charged particle arrive at the border or fall outside the simulation region, it will be absorbed, and a secondary electron will probably be emitted. The secondary electron emission coefficients for the dielectric surfaces of N + 2 and O + 2 are assumed to be the same as that of copper. In this work, we assume zero charge leak and zero electrical conductivity for the dielectric; specifically, we assume a perfect dielectric. The surface charge on the dielectric are calculated self-consistently by collecting all the charges deposited on it.

Reactions between Species: The Monte Carlo Collisions
This work considers several species: e, N + 2 , O + 2 , O − 2 , N 2 , O 2 , N * 2 and O * 2 . " * " denotes the excitation states of both N 2 and O 2 . We adopt the MCC method to deal with the reactions between the species. The reactions include electron-impact ionization, electron-impact excitation, elastic collision, and attachment, as seen in Table 1. The threshold energies data and cross sections are sourced from Ref. [53] and the LXCat database [54]. We consider molecules as the background and only trace the charged species. Table 1. Electron-impact ionization, electron-impact excitation, elastic collision, and attachment reactions between species. The threshold energies data and cross sections are sourced from Ref. [53] and the LXCat database [54].

Results and Discussion
In our previous works, we studied the filament discharge without pores [55] and with three pores [43]. We found that the surface discharge is enhanced on the surfaces of the catalyst pores compared with the micro-discharge inside the catalyst pores. However, the effect of a lateral voltage on the surface and volume discharge enhancement phenomena and the streamer propagation control for numerous pores of different shapes, sizes, and number were not considered. In this work, we investigate the effect law of these factors on the electron and ion densities, surface charge density, electric field, electron-impact ionization, and excitation rates in DBD with ε r = 4 for both the top and bottom dielectric plates with pores. The effects of different dielectric constants are important. We have studied the effect of different dielectric constants of the dielectric barrier and found that streamers are difficult to propagate to the pores as the dielectric constants increasing from 2 to 100, which is similar to a published work [8]. Moreover, we cannot get any valuable information from it, so we did not put the results here. is −20 kV, and the bottom electrode (y = 0 mm) is grounded. Before 0.3 ns, the streamer begins from the seed electron avalanches under the high driving voltage. The negative driving voltage drives the streamer to propagate from the upper plate to the lower plate, where a remarkable drift rate causes a new space-charge center at the streamer, thereby enlarging the streamer body [30,55]. Finally, the streamers propagate into the pores and produce plasma species inside the pores.

Different Shapes of Catalytic Pores
Some experiments have been operated to study porous dielectric discharge. Hensel et al. [24,25] reported stable discharge in porous dielectric material. As we know, it is difficult to experimentally measure discharge parameters such as electron and ion densities and reaction rates in detail. Thus, numerical simulations become indispensable in the study of porous dielectric discharge. It is worth mentioning that Zhang et al. [38] adopted a two-dimensional fluid model to study plasma properties for a steady glow discharge with one pore of different shapes. They found that the shape of the pore has a significant influence on electric field enhancement. Notably, tip-like structures could enhance the electric field significantly.
As shown in Figures 2 and 3, the surface discharge is stronger than the volume discharge, which is in line with our previous study [43]. Note in this model, no symmetrical or periodical condition is assumed, which will produce non-symmetrical results, similar to [56]. The reason is that in this case, the streamers start from random seed electrons. Applied voltages push seed electrons moving to the bottom dielectric plate. During the process, the velocity and direction of seed electrons movement are random. The maximum value of surface charge (N + 2 ) density occurs at the position where the streamer head arrives at in all cases. The maximum value is occurring in the vicinity of the pores or inside them, since streamers are propagating around the pores. The surface discharge and electric field are enhanced there, pushing the electrons and ions into the pores and generating more intensive plasma species along the surface of the pores. For example, the maximum value is approximately 2.8 × 10 −3 cm −2 occurring between the trapezoidal and triangle pores in f1 in Figure 3.
Note even with different shapes of pores, the electron density and surface charge density are similar from our simulations. The reason is that the pores are small enough to enhance the local field, and thus the shapes will not affect the results significantly. However, the numerical schemes adopted in VSim [47,48] will treat the elliptical pores more accurate, as the shape is smoother. In order to compare with our previous work [43] better and reduce the possible numerical errors, in the following works we will all adopt elliptical pores.

Different Numbers of Catalytic Pores
As the number of pores increases, the surface discharge enhancement phenomenon becomes more significant. Figure 4  As the pore number increases from 0 to 16, more intensive surface and volume discharges occur in the pores. Thus, more reactive species are generated in the pores (such as electrons, N + 2 , O + 2 , and O − 2 ), which are relevant for the active surface in plasma catalysis. To better understand this phenomenon, we present the density of N + 2 for different catalyst pore numbers, as seen in Figure 4. Highly concentrated electrons and ions are generated in the catalyst pores, enhancing plasma catalysis and providing the advantages of highly efficient reactivity and selectivity [5,6]. Achieving such advantages would be challenging in porous catalyst experiments.
The maximum value of the N + 2 density is 3.4 × 10 23 m −3 . As in our previous study [43], the maximum value appears on the surfaces of the pores. Abundant ions on the surface of the porous catalyst could improve the efficiency of plasma catalysis. One example is the elimination of 2-heptanone [57], a hazardous gaseous waste produced by anthropogenic activities. Ayrault et al. [57] selected 2-heptanone as the VOC [58,59] model molecule. Using a plasma catalytic device, which combines the highly efficient reactivity of plasma technology and the selectivity of catalysis, they improved the dissociation efficiency of 2-heptanone from less than 50% to 97%.  Both volume discharge and surface discharge occur in the pores, which agrees with the predictions in Refs. [8,44]. The pore size increases from 0 × 0 mm to 0.08 × 0.16 mm, surface discharge becomes dominant, and the maximum value of the O + 2 density is 1.3 × 10 23 m −3 , all of which agree with the predictions in Ref. [60]. These high-density ions could be used for improving the efficiency of the plasma catalysis process.  At the top, the applied voltage first increases from −8 kV to −32 kV, and the surface discharge is dominant. Then, both the surface and volume discharges are enhanced with sufficiently large driving voltage. Similar to the prediction in Ref. [38], the electric field is significantly enhanced near tip-like structures. As shown in Figure 7, the maximum value of the electric field, 1.0 × 10 8 Vm −1 , appears at the bottom of the catalytic pores, consistent with our previous study [43]. As seen in Figure 6, the maximum value of the electron-impact O 2 ionization rate is 7.5 × 10 22 m −3 s −1 . This value is as large as that in our previous work [43], 5.4 × 10 22 m −3 s −1 . Compared with the case without pores [43,55], the presence of pores likely enhances the ionization rate in a restricted region, which could effectively accelerate the chemical reaction rate and increase the surface discharge in the catalyst pores. The active surface of the catalyst could result in a stronger electric field along the pore sidewalls. In turn, the stronger electric field could further enhance the surface discharge. The electric field on the surfaces of the pores is much higher than that within their interiors. Thus, surface streamers could develop faster than volume streamers. However, when the external voltage is too large (over −20 kV), it is difficult to distinguish between surface discharge and volume discharge. Thus, this work adopted −20 kV as the appropriate external voltage to study plasma catalysis properties.   When lateral voltage is added to one side and its value increases from −10 to −60 kV, the direction of streamer propagation gradually biases to the other side. When the lateral voltage increases to −60 kV, the streamer does not even arrive at the bottom plate. However, irrespective of the change in the direction of the streamer, the surface discharge is always stronger than the volume discharge [43].

Effect of Lateral Voltage
When lateral voltage is added to two sides and its value increases from −5 to −30 kV, the direction of streamer propagation is almost vertical. Moreover, the surface discharge and volume discharge are both enhanced. As the volume discharge increases sharply, it is quite difficult to differentiate between them. Notably, when the left and right voltages are −30 kV, the volume discharge is stronger than the surface discharge. As the lateral voltage restricts the streamer propagation in the vertical direction, the streamers cannot spread along the pore surface. The streamers propagate directly into the pores. Moreover, the lateral voltage is so powerful that the streamers propagate within a narrow propagation channel only, thus realizing streamer propagation control. When streamers arrive at the surfaces of the pores, surface ionization waves exert a weak effect in the transverse direction. The electrons and ions are rebounded back to the pores, which enhances volume discharge. Hence, our results reveal a stronger transfer of volume discharge than surface discharge in the catalyst pores. The propagation direction of the streamer is controlled by intense lateral voltage and spread within a narrow propagation aisle. Volume discharge and surface discharge are concentrated and intensive.
Plasma catalysis is generating much interest, given the combined advantages of selectivity from catalysis and highly efficient reactivity from the plasma process [5,6]. This kind of configuration could generate abundant plasma species, enhance the efficiency of plasma catalysis, and improve the plasma catalysis process. There are varies potential applications for such a configuration, such as significantly improving the conversion and energy efficiencies of the synthesis of materials [4] and the treatment of greenhouse gas [29]. Furthermore, with lateral voltage, the prorogation of streamers can be controlled, and it may be used for gas switches [61] and other industrial applications such as oxidation of acetone [28].
In general, the size and numbers of catalytic pores, size of the applied top voltage, and magnitude and direction of the applied lateral voltage could affect the surface discharge and volume discharge enhancement phenomena as well as the propagation direction of streamers.
(1) As the number of pores increases, the discharge enhancement phenomenon becomes increasingly distinct. (2) As the pore size increases from 0 × 0 mm to 0.08 × 0.16 mm, the surface and volume discharges are more easily distinguished. (3) As the top applied voltage increases from −8 to −32 kV, the surface and volume discharge enhancement phenomena become apparent. (4) When the top voltage is −20 kV, as the left and right voltages (of the same size) increase from −5 to −30 kV, streamer propagation is restricted within a narrow channel.
As the volume discharge and surface discharge are both concentrated and intensive in catalyst pores, they could generate abundant plasma species, enhance the efficiency of plasma catalysis, and improve the plasma catalysis degradation process of pollutants.

Conclusions
Using a two-dimensional PIC/MCC model, we investigate the effect of lateral voltages on the surface and volume discharge enhancement phenomena and streamer propagation control in DBD with numerous micro-sized pores of different shapes, sizes, and numbers. We obtain the optimum discharge enhancement and propagation direction of streamers for different magnitudes of applied top and lateral voltage. The PIC model is a two-dimensional explicit and ES arithmetic model, while the MCC model is adopted to deal with the reactions of electrons and ions. The background gas is air (N 2 :O 2 = 80%:20%) under atmospheric pressure. The external voltages vary from −5 to −60 kV, and the electrodes are covered by SiO 2 with ε r = 4.
Our calculations indicate that the magnitude and direction of the applied lateral voltage could affect the streamer enhancement phenomenon and control the streamer propagation direction, and both surface discharge and volume discharge are enhanced. The streamer propagation is strongly restricted within a narrow channel with sufficiently large lateral voltages of the same value added on both sides. Notably, when the left and right voltages are applied, the volume discharge is stronger than the surface discharge. The streamer propagates directly into the pores. The volume discharge and surface discharge are both intensive, generate highly concentrated plasma species, enhance the efficiency of plasma catalysis, and improve the plasma catalysis process.
Moreover, the size and number of catalytic pores, as well as the magnitude of the top driving voltage could affect the surface and volume discharge enhancement phenomena: (1) As the number of pores increases from 0 to 16, more intensive surface and volume discharges occur on the pore surfaces and within the pores, and the surface and volume discharge enhancement phenomena become significant due to the increase in volume for ionization collisions. (2) As the pore size increases from 0 × 0 mm to 0.08 × 0.16 mm, the surface discharge enhancement phenomena become more obvious. (3) As the top applied voltage increases from −8 to −32 kV, many plasma species are generated on the pore surface and inside the pores, thus enhancing both surface discharge and volume discharge. Our results predict the mechanisms of the surface discharge and volume discharge enhancement phenomena, which are of great significance for deeper understanding of plasma catalysis and enhancing the efficiency of this process.

Acknowledgments:
The authors are very grateful to Ming-Xiang Gao for providing assistance with the figures.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: DBD dielectric barrier discharge PIC particle-in-cell MCC Monte Carlo collision VOC volatile organic compound AC alternating current ES electrostatic CFL Courant-Friedrichs-Lewy