Characteristics of spatiotemporal variations of primary and secondary streamers under pulsed-voltage in air at atmospheric pressure

A pulsed positive streamer discharge was simulated using a two-dimensional axisymmetric model to investigate the characteristics of primary and secondary streamers in air at atmospheric pressure and ambient temperature. The spatiotemporal variations of the reduced electric field and the electron density during propagation of the primary streamer were clarified, and their relationships with the applied voltage were discussed. The phenomenon of the secondary streamer was introduced according to the previously developed “attachment instability” theory, and the spatiotemporal variations of the net-attachment frequency were presented to validate the theory. The results indicated that variations in the reduced electric field and electron density can be approximately estimated by the theory even in conditions involving a pulsed voltage and non-uniform Laplacian field. Because the primary and secondary streamers have spatiotemporal characteristics related to the chemical reactivity in the streamer discharge, understanding these characteristics is valuable for the development of atmospheric-pressure plasma applications.


Introduction
A streamer discharge is the fundamental process of atmospheric-pressure-plasma, and it is widely used for various types of industrial applications such as removal of gaseous pollutants, [1][2][3][4][5][6] bio-medical applications 7,8) and applications involving plasma aerodynamics. 9,10) To ensure that the novel chemical reactivity of the atmospheric-pressure plasma can be exploited by several other industrial domains and to improve the associated efficiency, it is important to understand the physical and chemical characteristics occurring in a streamer discharge. However, because the streamer discharge is a temporally fast and spatially steep phenomenon, some parameters cannot be measured experimentally in a convenient manner. For example, the streamer discharge has a filament structure whose diameter is typically a few millimetres at most; in addition, its propagation velocity is of the order of a few millimetres per nanoseconds in atmospheric-pressure air in room temperature. [11][12][13][14] The electric field and the electron density inside and around the streamer vary in such spatiotemporal scales, which are difficult to obtain owing to the insufficient resolution capacity of current experimental devices. Under such circumstances, numerical simulations can be a powerful tool to investigate the chemical and physical characteristics of streamer discharges.
A streamer discharge develops primary and secondary streamers sequentially. [15][16][17] The primary streamer has a high electric field region at the wavefront, known as the streamer head; the streamer propagates rapidly between the electrodes insulated by the air and produces a considerable number of electrons. [18][19][20][21][22] In contrast, the secondary streamer propagates along the highly conductive channel where the primary streamer develops, with a relatively weak electric field. 23,24) The mean electron energy at the streamer head is reported to be more than 10 eV, whereas that in the secondary streamer is reported to be approximately 2-3 eV, which is not sufficient to lead to electron-impact ionization. 24,25) In terms of the chemical reactivity, the degree of reactions, which is estimated by the time-integration of the reaction rate, occurring in a primary streamer is less than that in the secondary streamer because the primary streamer propagates faster with a narrow streamer head whose thickness is reported to be typically 2-3 times smaller than the streamer radius; 26,27) consequently, the net duration of the high electric field is short. Although the intensity of the electric field in the secondary streamer is less than that of the electric field at the primary streamer head, the duration for which the secondary streamer holds its electric field is more than the corresponding value for the primary streamer, and therefore, the amount of production of some radicals is reported to be higher in the secondary streamer than the corresponding value in the primary streamer. 28,29) Thus, the primary and secondary streamers have different characteristics, which directly affect the amount of chemically active species produced. In the applications of atmospheric pressure plasma, various types of chemically active species such as N, O and OH are required to be used in various places such as on a solid wall, water surface, or in the gas phase. 30,31) It is thus important to know the characteristics of the spatiotemporal variations of primary and secondary streamers. Some previous studies investigated the characteristics of the primary and secondary streamers occurring under DC voltage. 18,19,23,24) However, because some applications employ a pulsed voltage waveform, the characteristics of the primary and secondary streamers occurring under pulsed voltage should be investigated.
In this study, the primary and secondary streamers occurring under a pulsed positive voltage waveform were simulated by using two-dimensional simulations, and their spatiotemporal characteristics were clarified. In particular, focusing on the spatiotemporal characteristics of the ionization rate at the primary streamer head and the phenomenon of the attachment instability in the secondary streamer, the influence of each phenomenon on the production of the oxygen radical was evaluated and discussed.

Simulation model
In this study, we applied a first-order electro-hydrodynamic model for electrons and positive and negative ions within a drift-diffusion approximation framework. The corresponding streamer model equations can be specified as follows: and D E N s ( ) / denote the charged particle density, charged particle velocity, particle chemical source term, and the mobility and diffusion coefficients, respectively, with the subscripts representing electrons (e) and positive (p) or negative (n) ions. In addition, e is the absolute value of the electronic charge, and e 0 is the permittivity of free space. The transport and source parameters related to electrons were calculated using BOLSIG+ 32) and published electron impact cross-sections. [33][34][35] Photoionization was taken into account through the application of three-exponential Helmholtz models 36,37) and solved using Poisson's equation. It is assumed that the photoionization of O 2 is caused by the radiation in the region of the spectrum where l < < ). 38,39) The vibrational kinetics for , where v and v 2 denote, respectively, the vibrational quantum number and vibrational quantum number of the bending level, were also taken into account. The rate coefficients of the vibrational kinetics were obtained as described in our previous work. 40) Figure 1(a) shows the electrode configuration and its computational domain. Equations (1)-(5) were solved under the assumption of 2D axisymmetric geometry. The actual (experimentally obtained) pulsed voltage, whose polarity is positive, shown in Fig. 1 was applied to the needle anode in the simulation. In the coordinate system used in the simulation, the z-axis was defined as extending from the anode tip to the plane cathode, and the r-axis was defined as extending outward from the central axis in the radial direction. The chemical reaction model assessed in this study included electron impact collisions (excitation, ionization, dissociation, recombination, attachment, and detachment), ion recombination, and the reactions of neutral molecules. The chemical reaction model is described in our previous paper. 41,42) Details regarding the model equations, numerical algorithms, and computational acceleration techniques employed in this study can be found in our previous paper. 25) 3. Results and discussion 3.1. Primary streamer In this section, first, the processes of streamer inception and propagation are introduced, and the simulation results are presented later. The inception process of the streamer head near the needle electrodes is shown in Fig. 2. Figure 2(a) shows the two-dimensional distributions of n e and E/N at t = 2 ns. By applying a high voltage to the electrode, the electrons accidentally existing in the space are accelerated, leading to the occurrence of the electron impact ionization reaction, which increases the electron density in the vicinity of the electrode. Owing to the electric field, the generated electrons drift toward the anode, whereas the ions generated simultaneously remain nearly stationary during the discharge pulse because they are heavier than the electrons. As a result, as shown in Fig. 2(c), the positive and negative charges are biased in the space, which, as shown in Fig. 2(d), causes a sharp charge distribution, resulting in the formation of the high E/N region, as shown in Fig. 2(b). This high E/N region is called a streamer head. Further ionization occurs in the vicinity of the streamer head, resulting in its movement toward the cathode. The streamer head advances from the anode to the cathode by repetition of this process. After the movement of the streamer head, the area, known as the streamer channel, reaches a quasi-neutral state; the presence of positive and negative charges results in a high conductivity, and the channel has a small electric field. Although the electron density is high in the streamer channel immediately after the streamer head has passed, the reactivity is generally relatively low because the electric field is small.
The photoionization reaction is an important phenomenon to explain the streamer discharge. [43][44][45] The photoionization reaction in air is a process in which oxygen molecules absorb light emitted from excited nitrogen, leading to ionization. Because the photoionization occurs in a wider range than the electron impact ionization does, the electrons may be generated in a region slightly farther from the electrode and the streamer head. Figure 3(a) shows the spatial distribution of the N 2 electron impact ionization rate and Fig. 3(b) shows the spatial distribution of the photoionization rate at t = 10 ns. Electron impact ionization occurs mainly near the high E/N region around the streamer head and barely occurs in the low E/N region in the streamer channel and in the area far away from the streamer head. Furthermore, the photoionization rate spreads spherically around the streamer head and in a wider range than the electron impact ionization rate does. Comparing Figs. 3(a) and 3(b), it can be noted that the electron impact ionization has a large local ionization rate but a narrow range, whereas photoionization has a small ionization rate but a wide range. Figure 4 shows the axial distributions of the N 2 ionization and photo-ionization rate at r = 0 mm indicated in Figs. 3(a) and 3(b). It can be seen that the electron impact ionization rate is approximately four orders of magnitude larger than the peak value of the photoionization rate. However, the electron impact ionization rate is dominant only in the range of approximately 0.17 mm of the streamer tip. Between the streamer head and the cathode, a pre-ionization region is formed as a result of the photoionization. Therefore, photoionization affects the electron density in front of the streamer head, and it could increase the propagation speed of the streamer head. Figure 5(a) shows the spatiotemporal variation of the electron density and E/N. Electrons are generated by the streamer head and distributed in the streamer channel behind the head. The values of E/N at the streamer head and in the channel are approximately 650 ∼ 700 Td and 20 Td, respectively. Figure 5(b) shows the spatiotemporal variations of the simulated discharge emission of light. The discharge emission intensity is assumed to be proportional to the N 2 (C 3 Π u ) density and calculated using line-of-sight integration of the N 2 (C 3 Π u ) density. In addition, the calculated emission intensity is integrated for 2 ns to compare the experimentally obtained ICCD images (e.g. in Refs. 46,47). The streamer head propagates towards the cathode while strongly emitting light. When the streamer head reaches the cathode, the high electric field area disappears, and the electric field redistribution starts; simultaneously the light emitting area develops again from the anode. At t = 40 ns, a region whose E/N is approximately 115 Td develops, extending from z = 0 to 5-6 mm, and the discharge light emission occurs strongly in this region. This area is known as the secondary streamer. At t = 100 ns, although E/N in the secondary streamer is still high, the emission is suppressed because the electron density   decreases in the secondary streamer. The characteristics of the secondary streamer are described in Sect. 3.2. Figure 6 shows the electron density and E/N distribution in the horizontal cross-section of the streamer channel at t = 22 ns. The cross-sections of the electron density and reduced electric field at z = 1, 4, 11, and 12 mm are illustrated. It can be seen that the streamer propagates with increase in its diameter. Although the electron density in the streamer channel is high, the chemical reactivity is relatively low because of the low E/N value in the streamer channel. However, although E/N around the edge of the streamer channel is higher than that in the streamer channel, the reactivity is not high because electrons are not present in abundance outside of the streamer channel. Figure 7(a) shows the temporal variations of the peak E/N and the propagation velocity of the primary streamer. The maximum value of E/N is approximately 1200 Td, attained at approximately t = 2 ns around the needle anode, which is estimated to be the required E/N for streamer inception to occur because the discharge current starts to increase at the same time, as shown in Fig. 7(b). After the inception of the streamer, the maximum value of E/N is approximately 700 Td at the streamer head during the propagation of the primary streamer. Although the applied voltage increases continuously during the propagation of the primary streamer, as shown in Fig. 7(b), it can be seen that the E/N at the streamer head does not change considerably under the simulation conditions. Furthermore, at the beginning of streamer inception, when the propagation velocity is 0.4 mm ns −1 , the streamer is temporally accelerated to approximately 1.0 mm ns −1 until it is absorbed into the cathode. This acceleration of the primary streamer propagation has been observed experimentally, 12,48) and acceleration due to increase in voltage has also been demonstrated. 17) That is, it can be understood that although the anode voltage is related to the acceleration of the streamer propagation speed, the E/N at the streamer head does not always directly affect it. Rather, as pointed out in Ref. 27, a correlation, as shown in Figs. 6 and 7, exists between the streamer diameter and velocity. It is considered that the diameter of the streamer increases as the anode voltage increases, resulting in the acceleration of the propagation velocity of the primary streamer. With increase  in the diameter of the streamer, the amount of pre-ionization generated near the tip of the streamer increases accordingly. Further simulations are needed to validate this estimation; however, this aspect is not the main objective of this study, and thus, it is not discussed further.

Secondary streamer
It is known that secondary streamers occur during streamer discharge in air at atmospheric pressure. The principles of secondary streamers were described by Bastien and Marode [49][50][51] and Sigmond 52) by using the "attachment instability" phenomenon. In this section, the principle of attachment instability is discussed using Fig. 8, and the results of the simulation are presented. Figure 8 shows the sum of the ionization frequencies of N 2 and O 2 , the two-body attachment frequency of O 2 , the three-body attachment frequency of O 2 , and the net attachment frequency, n , att.
calculated from the ionization and attachment reaction frequencies. Figure 8 is a double x plot, in which the lower axis is E/N and the upper axis is the voltage determined by multiplying the electric field E with the gap length d = 13 mm. As shown in Fig. 8, n att. changes dramatically in the region with E/N less than 120 Td. In particular, in the region with E/N ranging from 37 to 95 Td, n att. increases with increase in E/N, and this property is known as the negative resistance. This negative resistance region is a result of the fact that the two-and three-body reaction frequencies of O 2 have a different trend on E/N. The mechanism of the two-and three-body attachment reactions of O 2 are detailed in previous papers. [53][54][55][56] In this region, once E/N increases, n att. increases and the electron density decreases accordingly; consequently, the conductivity of the space decreases and E/ N increases further. In contrast, once E/N decreases, n att. decreases and the electron density increases accordingly; consequently, the space conductivity increases, leading to a further reduction in the E/N. As a result, the electric field cannot maintain the value of E/N in this negative resistance region, resulting in increase or decrease in the E/N until it deviates from the negative resistance region.
Immediately after the primary streamer reaches the cathode, many charged particles exist between the discharge gaps, and the conductivity of the gap is relatively low. That is, the contribution of the Poisson field generated by the charged particles is larger than that of the Laplace field generated by the electrode potential. Thus, it can be assumed that the almost homogeneous electric field is applied between the gaps immediately after the arrival of the primary streamer, and the secondary streamer phenomenon can be roughly explained in terms of the average electric field obtained by dividing the voltage V by the gap length d, as shown in the upper axis in Fig. 8. Under the conditions that we used in this simulation, the negative resistance region corresponds to V = 12-30 kV, as shown in Fig. 8. In a previous research that employed the same electrode structure as in this simulation, it was reported that the length of the secondary streamer increased linearly by 2-11 mm when the applied voltage was varied in the range of 16-28 kV 11) our simulations qualitatively reproduced this finding. 28,57) In particular, when a voltage having an average electric field in this negative resistance region is applied, the electric field is distributed in the region of E/N > 95 and E/N <37, that is, it is deviated from the negative resistance region. The region with E/ N > 95 is observed as the secondary streamer region. When E/N > 118, the attachment frequency becomes negative, and the streamer is easily transit to a spark discharge; 58) therefore, E/N = 118 Td is known as the breakdown electric field, (E/N) crit . In addition, it has been actually observed in experiments 59) that the secondary streamer region covers the entire discharge gap when a voltage larger than that in the negative resistance region and lower than (E/N) crit exists, such as 95 <E/N < 118. Although the extension length and electric field of the secondary streamer cannot be quantitatively determined by Fig. 8 alone, because these parameters are also related to other reactions such as detachment and recombination and variation in the gas density, it is possible to easily predict and explain the phenomena occurring in the secondary streamer by using the discussed attachment instability theory. Figure 9 shows the spatial distributions of (a) E/N, (b) electron density, (c) current density, and (d) n att at t = 75 ns. In the secondary streamer region, the E/N is almost uniformly distributed around 100 Td, and the electron density decreases accordingly. However, the current density, calculated as -e n v n v n v p p e e n n ( ) is almost homogeneous between the discharge gap. In other words, the electron density and E/N work in balance to maintain the continuity of the current  density along the gap. In the one-dimensional analysis by Sigmond,52) it was demonstrated that if a constant voltage is applied and the current exhibits continuity along the gap, the net attachment frequency is also homogeneous between the discharge gap. However, in the simulation carried out herein, the values of n att. in the secondary streamer and in the primary streamer were different, as shown in Fig. 9(d). Although the cause of this difference cannot be identified clearly, it is considered to be the difference in the considered dimension (one-dimensional or two-dimensional) or the difference in the voltage waveform used for the simulation (pulsed or DC).
The values of n att during the secondary streamer development can be observed in more detail using Fig. 10, which shows the spatiotemporal variations of n att . Immediately after the primary streamer arrives at the cathode at t = 25 ns, the values of E/N at positions z = 2.5 (labelled 1), 6.0 (labelled 2), and 9.5 mm (labelled 3) are noted to be similar, corresponding to the values in the negative resistance region. As time elapses, E/N changes to deviate from the negative resistance region. At t = 50 ns, the E/N at positions 1 and 3 is deviated from the negative resistance region, but E/N at position 2 is still in the negative resistance region because it is located at the tip of the secondary streamer. At time t = 75 ns, E/N at position 2 deviates to the lower side of the negative resistance region and forms a clear boundary between the secondary streamer region and the other region. Thus, the secondary streamer can be considered to change in space and time largely due to the relationship between the negative resistance region, E/N, and electron density.
Finally, the generation characteristics of the chemically active species in the secondary streamer region are shown in Fig. 11. Figure 11(a) shows the density distribution of O ( 3 P) at t = 100 ns, and Fig. 11(b) shows the temporal variations of the production amount of O ( 3 P) in the primary and secondary streamer regions. The amount of O ( 3 P) at the time when the primary streamer reaches the cathode at approximately t = 23 ns is nearly the same in both regions. Subsequently, a secondary streamer is formed, and electrons are accelerated by the electric field in the secondary streamer, resulting in a rapid increase in the amount of O( 3 P). Thus, the formation of the secondary streamer significantly affects the production characteristics of O( 3 P). Because it has been reported that the other discharge products including the chemically active species are mainly produced in the secondary streamer, 29) the chemical reactions that occur in the secondary streamer are critical in considering atmospheric pressure plasma processes.

Conclusions
In this paper, we discuss, on the basis of a conducted simulation, the characteristics of primary and secondary streamers generated by pulse discharge under an atmospheric-pressure-air environment. The formation and propagation of the primary streamer is influenced by electronimpact ionization and photoionization, which affects the preionization area formed on the front of the streamer head. In the pulse discharge with a pulse width of 100 ns or less, the anode voltage changes even during the development of the primary streamer, but the E/N at the streamer head does not change considerably during the development of the primary streamer. Furthermore, increase in the diameter of the streamer and the acceleration of the development rate are observed, and it is considered that there exists a correlation with the anode potential. Because the diameter of the streamer is related to the production volume of the chemically active species, and the propagation velocity of the streamer head is related to the electron density, it is possible to change the reactivity in the streamer discharge by changing the voltage waveforms in the same order as the propagation of the primary streamer.
The secondary streamer phenomenon is caused by the attachment instability phenomenon generated by the characteristics of the net attachment frequency in atmospheric pressure air. When the average electric field calculated from the voltage applied between the electrodes and the gap length is within the range of the negative resistance area, a deviation of the electric field distribution occurs between the gap, and the high electric field area becomes the secondary streamer area. In the secondary streamer, although the electric field intensity is relatively high, the net attachment frequency is also high, resulting in a rapid decrease in the electron density. As a result, after the formation of the secondary streamer, the electric field and electron density remain in balance to maintain the current density uniformly throughout the gap. The amount of O( 3 P) radicals generated in the secondary streamer is larger than that in the primary streamer. This is because the electrons generated in the primary streamer are accelerated by the electric field in the secondary streamer to cause chemical reactions. As described above, because the primary and secondary streamers have spatiotemporal  characteristics related to the chemical reactivity in the streamer discharge, it is important to understand these characteristics for the development of atmospheric-pressure plasma applications.