Behaviour of OH radicals in an atmospheric-pressure streamer discharge studied by two-dimensional numerical simulation

The simulations are performed in a 13 mm-gap point-to-plane electrode configuration according to our experimental setup. Streamer dynamics are described by the following system of equations:

$$\begin{eqnarray} &&\fl \frac{\partial n_s}{\partial t} + {\rm div} (n_s{\bit v}_s(E/N))= S_s(E/N), \tqs s={\rm e}, {\rm p}, {\rm n}, \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\fl n_{\rm e}{\bit v}_{\rme}(E/N) = n_{\rme}\mu_{\rme}(E/N) {\bit E} - D_{\rme}(E/N){\rm grad}(n_{\rme}), \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&\fl n_{\rm p}{\bit v}_{\rm p}(E/N) = n_{\rm p}\mu_{\rm p}(E/N){\bit E} , \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&\fl n_{\rm n}{\bit v}_{\rm n}(E/N) = n_{\rm n}\mu_{\rm n}(E/N){\bit E} , \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &&\fl {\rm div}{\bit E} = \frac{e}{\varepsilon_0}(n_{\rm p}-n_{\rme}-n_{\rm n}), \end{eqnarray} \tag{ 9 }$$

where n_s, v_s(E/N), S_s(E/N), μ_s(E/N) and D_s(E/N) are the charged particle density, charged particle velocity, particle chemical source term, mobility and diffusion coefficient, respectively. The subscript 's' denotes electrons (e) or positive (p) or negative (n) ions. E is the electric field, ε₀ is the permittivity of free space, and e is the absolute value of the electronic charge. The transport and source parameters involving electrons (such as v_e, D_e, and the reaction coefficients in S_s) are calculated using Bolsig+ software [20] with previously published e-V cross sections [21]. A detailed description of the chemical reaction model is given in the following section. In (7) and (8), all ion mobilities are assumed to be 2.2 × 10⁻⁴ m² V⁻¹ s⁻¹ [22]. This assumption is a rough estimation because ion mobilities are a function of reduced electric field E/N. However, since it is commonly accepted that the ion mobilities are significantly slower than the electron mobility due to a much larger mass, the ions hardly move (remain relatively static) within several nanoseconds. A similar value of ion mobility is used by Morrow and Lowke [23]. Ion diffusion is neglected for the same reason. Photoionization is taken into account through the three-exponential Helmholtz models [24, 25]. Equations (5) and (9) are solved using a general personal computer in an attempt to reduce the computational cost. Some numerical techniques such as the method of solving (5) and (9) are given in our previous paper [10].

Figure 2(a) shows the configuration of the electrodes and the computation domain. To simulate the development of a discharge, we chose a cylindrical domain of 18 mm height and 8 mm radius. The total number of grid points is N_z × N_r = 1792 × 256 with spatial steps from δz = 1 µm (near the point anode and plane cathode) to 10 µm (at the interelectrode gap) [26] in the z-direction and from δr = 2.5 to 200 µm in the r-direction. The spatial resolution is derived from Eichwald et al [27]. The point electrode has the form of a hyperboloid of revolution with a radius of 40 µm, corresponding to that used in our experiment. At the conducting anode, the positive ion fluxes are fixed to zero, while the negative ion and electron fluxes are estimated using the Neumann boundary condition [27]. At the conducting cathode, the Neumann boundary condition is applied to all positively and negatively charged fluxes [27]. All radial derivatives of the densities are set to zero at the open boundaries [28]. Figure 2(b) shows the waveform of the applied pulsed voltage and figure 2(c) shows an enlarged view of figure 2(b) and the discharge current calculated by an extension of Morrow and Sato's formula [29]. In this simulation, the actual pulsed voltage, which is measured from our experiments, is applied to the conduction anode tip. The discharge voltage V, which is set to 24 kV in figure 2(b), is defined by the peak voltage of the pulse. The rate of voltage increase is about 0.52 kV ns⁻¹.

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In our simulation, the neutral particles are assumed to be static. Much work remains to be done on the questions of gas heating and the decrease in gas density, which affect the electron distribution function through the variation of reduced electric field, E/N, and the Arrhenius constant through the gas temperature [30]. Actually, it has been suggested that the decrease in density due to gas heating is responsible for the formation of sparks, initiated by a discharge [31]. However, Popov [32] estimated that the gas heating is equal to 22–26 K at 3 µs under the same conditions as this paper, and the main heating occurred after the discharge (time t > 200 ns). Therefore, it is estimated that the gas heating does not affect the electron distribution function through E/N variations in the single discharge. At t > 3 µs, the V–T relaxation through H₂O molecules [18] can cause gas heating, which may affect Arrhenius constants through the gas temperature. Therefore, we compare the simulated data with the experimental data up to a time 3 µs after the discharge, which is considered to correspond to the pre-diffusion phase of neutral species in the discharge of a single-pulse streamer [33, 34].

We used a reduced reaction model including electron impact reactions (excitation, ionization, dissociation, recombination, attachment and detachment), ion recombination and the reactions of neutrals as shown in tables 1–3. For N₂ and O₂ molecules, the excited species correspond to effective states clustered around actual molecular levels as described in table 4, in accordance with the method proposed by Fresnet et al [35]. The rate constants of the electron impact reactions are calculated by an analysis of the Boltzmann equation. The effects of superelastic collisions induced by highly vibrationally excited molecules on the electron energy distributions (EEDF) are not considered in this simulation because the densities of vibrationally excited molecules are relatively small and have little effect in a single atmospheric-pressure pulse discharge [10, 36]. However, superelastic collisions are expected to have an effect under low-pressure discharge or repetitive discharge conditions [37, 38]. In the charged-particle kinetics, three positive ions: ${\rm N}_2^+$, ${\rm O}_2^+$ and H₂O⁺, four negative ions: ${\rm O}_2^-$, O⁻, OH⁻ and H⁻, and electrons are considered. Complex ions such as ${\rm O}_4^{+}$, ${\rm N}_4^{+}$ and ${\rm O}_2^+$ N₂, and the charge transfer reactions are not considered in this simulation. Reaction kinetics and rate coefficients of complex ions have been studied in many papers [39–42]. However, the effects of complex ions on radical production have only been investigated in simulated cases [43], and corroboration must be made through actual experiments. Even without such complex ions, the development of primary and secondary streamers was well reproduced in our reduced model, whose validity was confirmed by comparison between experimental and simulated streak photographs [10]. In addition, the simulated production of O and N radicals was also in good qualitative agreement with experimental results [10]. Thus, these effects are ignored in our simulation.

Table 1. Considered reactions involving electrons.

	Reaction	Rate constant (cm3 s−1)
(R1)	N2 + e → N2(v) + e	f1(E/N)
(R2)	N2 + e → N2(A1) + e	f2(E/N)
(R3)	N2 + e → N2(A2) + e	f3(E/N)
(R4)	N2 + e → N2(B) + e	f4(E/N)
(R5)	N2 + e → N2(a) + e	f5(E/N)
(R6)	N2 + e → N2(C) + e	f6(E/N)
(R7)	N2 + e → N2(E) + e	f7(E/N)
(R8)	N2 + e → N(4S) +N(2D) + e	f8(E/N)
(R9)	${\rm N}_2 + \rme \rightarrow{\rm N}_2^+ + 2\rme$	f9(E/N)
(R10)	${\rm N}_2 + \rme \rightarrow{\rm N}_2^+({\rm B}) + \rme$	f10(E/N)
(R11)	O2 + e → O2(v) + e	f11(E/N)
(R12)	O2 + e → O2(a) + e	f12(E/N)
(R13)	O2 + e → O2(b) + e	f13(E/N)
(R14)	O2 + e → O2(A) + e	f14(E/N)
(R15)	O2 + e → O(3P) +O(3P) + e	f15(E/N)
(R16)	O2 + e → O(1D) +O(3P) + e	f16(E/N)
(R17)	O2 + e → O(1S) +O(3P) + e	f17(E/N)
(R18)	${\rm O}_2 + \rme \rightarrow{\rm O}_2^+ + 2\rme$	f18(E/N)
(R19)	O2 + e → O−(2P) + O(3P)	f19(E/N)
(R20)	${\rm O}_2 + {\rm O}_2 + \rme \rightarrow{\rm O}_2^- + {\rm O}_2$	f20(E/N)
(R21)	H2O + e → H2O(ν2) + e	f21(E/N)
(R22)	H2O + e → H2O(ν1) + e	f22(E/N)
(R23)	H2O + e → H2O(ν3) + e	f23(E/N)
(R24)	${\rm H}_2{\rm O} + \rme \rightarrow{\rm H} + {\rm O}{\rm H} + \rme$	f24(E/N)
(R25)	H2O + e → O(P) + H2 + e	f25(E/N)
(R26)	H2O + e → H2O+ +2e	f26(E/N)
(R27)	H2O + e → OH− + H	f27(E/N)
(R28)	H2O + e → H− + OH	f28(E/N)
(R29)	H2O + e → H2 + O−	f29(E/N)

Table 2. Considered reactions involving charged particles.

	Reaction	Rate constant (cm3 s−1)	Ref
(R30)	${\rm N}_2^+ + \rme \rightarrow {\rm N}(^4{\rm S}) + {\rm N}(^4{\rm S})$	$1.80\times10^{-7}\bigg(\dsty\frac{300}{T_{\rme}}\bigg)^{0.39}$	[60]
(R31)	${\rm N}_2^+ + \rme \rightarrow{\rm N}_2$	4.00 × 10−12	[30]
(R32)	${\rm N}_2^+ + {\rm O}^- \rightarrow{\rm N}_2 + {\rm O}(^3{\rm P})$	4.00 × 10−7	[30]
(R33)	${\rm N}_2^+ + {\rm O}_2^- \rightarrow{\rm N}_2 + {\rm O}_2$	1.60 × 10−7	[30]
(R34)	${\rm O}_2^+ + \rme \rightarrow {\rm O}(^3{\rm P}) + {\rm O}(^3{\rm P})$	$1.95\times10^{-7}\bigg(\dsty\frac{300}{T_{\rme}}\bigg)^{0.7}$	[61]
(R35)	${\rm O}_2^+ + \rme \rightarrow{\rm O}_2$	4.00 × 10−12	[30]
(R36)	${\rm O}_2^+ + {\rm O}^- \rightarrow{\rm O}_2 + {\rm O}(^3{\rm P})$	9.60 × 10−8	[30]
(R37)	${\rm O}_2^+ +{\rm O}_2^- \rightarrow 2{\rm O}_2$	4.20 × 10−7	[30]
(R38)	${\rm H}_2{\rm O}^+ + \rme \rightarrow{\rm O}{\rm H} + {\rm H}$	3.80 × 10−7	[30]
(R39)	H2O+ + e → H2 + O(3P)	1.40 × 10−7	[30]
(R40)	${\rm H}_2{\rm O}^+ + \rme \rightarrow{\rm H} + {\rm H} + {\rm O}(^3{\rm P})$	1.73 × 10−7	[30]
(R41)	H2O+ + O− → H2O + O(3P)	4.00 × 10−7	[30]
(R42)	${\rm H}_2{\rm O}^+ + {\rm O}_2^- \rightarrow{\rm H}_2{\rm O} + {\rm O}_2$	4.00 × 10−7	[30]
(R43)	${\rm O}_2^- + {\rm H} \rightarrow{\rm H}{\rm O}_2 + \rme$	1.20 × 10−9	[62]
(R44)	${\rm O}_2^- + {\rm H} \rightarrow{\rm O}{\rm H}^- + {\rm O(P)}$	1.50 × 10−9	[62]
(R45)	${\rm O}_2^- + {\rm O}_2{\rm (a)} \rightarrow 2{\rm O}_2 + \rme$	2.00 × 10−10	[63]
(R46)	${\rm O}_2^- + {\rm O(P)} \rightarrow {\rm O}_3 + \rme$	3.00 × 10−10	[63]
(R47)	O− + O2(a) → O3 + e	3.00 × 10−10	[63]
(R48)	O− + O(P) → O2 + e	2.00 × 10−10	[63]
(R49)	H− + H2O → OH− + H2	3.80 × 10−9	[46]
(R50)	H− + H → H2 + e	2.00 × 10−9	[64]
(R51)	H− + O2 → HO2 + e	1.20 × 10−9	[65]
(R52)	OH− + H → H2O + e	1.40 × 10−9	[62]
(R53)	OH− + O(P) → HO2 + e	2.00 × 10−9	[65]

Table 3. Considered reactions for neutral species.

	Reaction	A (cm3 s−1)	β	Ea (K)	Ref
(R54)	N2(A1) + O2 → N2 + O(3P) + O(3P)	1.70 × 10−12	0	0	[32]
(R55)	N2(A1) + O2 → N2 + O2(b)	7.50 × 10−13	0	0	[32]
(R56)	N2(A1) + O2 → N2O + O2(P)	7.80 × 10−12	0	0	[66]
(R57)	N2(A1) + N2(A1) → N2 + N2(E)	1.00 × 10−11	0	0	[35]
(R58)	N2(A1) + N2(A1) → N2 + N2(B)	7.70 × 10−11	0	0	[32]
(R59)	N2(A1) + N2(A1) → N2 + N2(C)	1.60 × 10−10	0	0	[32]
(R60)	N2(A1) + O(3P) → N2 + O(1S)	3.00 × 10−11	0	0	[46]
(R61)	N2(A1) + O(3P) → NO + N(2D)	7.00 × 10−12	0	0	[35]
(R62)	N2(A1) + O(3P) → N2 + O(P)	2.00 × 10−11	0	0	[35]
(R63)	N2(A1) + H → N2 + H	2.10 × 10−10	0	0	[35]
(R64)	N2(A1) + OH → N2 + OH	1.00 × 10−10	0	0	[35]
(R65)	N2(A1) + H2O → N2 + H + OH	5.00 × 10−14	0	0	[35]
(R66)	N2(A1) + NO → N2 + NO(A)	6.90 × 10−11	0	0	[35]
(R67)	N2(A2) + N2 → N2(A1) + N2	1.00 × 10−11	0	0	[35]
(R68)	N2(A2) + O(P) → N2 + O(P)	2.00 × 10−11	0	0	[35]
(R69)	N2(A2) + H → N2 + H	2.10 × 10−10	0	0	[35]
(R70)	N2(A2) + OH → N2 + OH	1.00 × 10−10	0	0	[35]
(R71)	N2(A2) + H2O → N2 + H + OH	5.00 × 10−14	0	0	[35]
(R72)	N2(A2) + NO → N2 + NO(A)	6.90 × 10−11	0	0	[35]
(R73)	N2(A2) + O(P) → NO + N(S)	7.00 × 10−12	0	0	[32]
(R74)	N2(B) + O2 → N2 + O(3P) + O(3P)	3.00 × 10−10	0	0	[32]
(R75)	N2(B) + N2 → N2(A1) + N2	1.00 × 10−11	0	0	[35]
(R76)	N2(B) → N2(A1) + hν	1.50 × 105 (s−1)	0	0	[32]
(R77)	N2(a) + O2 → N2 + O(3P) + O(3D)	2.80 × 10−11	0	0	[32]
(R78)	N2(a) + N2 → N2(B) + N2	2.00 × 10−13	0	0	[66]
(R79)	N2(a) + N2 → N2 + N2	2.00 × 10−13	0	0	[35]
(R80)	N2(a) + H2 → N2 + H + H	2.60 × 10−10	0	0	[35]
(R81)	N2(a) + H2O → N2 + OH + H	3.00 × 10−10	0	0	[58]
(R82)	N2(a) + NO → N2 + N(4S) + O(3P)	3.60 × 10−10	0	0	[66]
(R83)	N2(C) + O2 → N2 + O(3P) + O(3P)	2.50 × 10−10	0	0	[32]
(R84)	N2(C) + N2 → N2(B) + N2	1.00 × 10−11	0	0	[32]
(R85)	N2(C) + N2 → N2(a) + N2	1.00 × 10−11	0	0	[35]
(R86)	N2(C) → N2(B) + hν	2.80 × 107 (s−1)	0	0	[35]
(R87)	N2(E) + N2 → N2(C) + N2	1.00 × 10−10	0	0	[35]
(R88)	${\rm N}_2^+{\rm (B)}\rightarrow{\rm N}_2^++h\nu$	1.40 × 107 (s−1)	0	0	[67]
(R89)	NO(A) → NO + hν	5.10 × 106 (s−1)	0	0	[35]
(R90)	N(2D) + O2 → NO + O(1D)	9.70 × 10−12	0	−185	[16]
(R91)	N(2D) + O2 → NO + O(3P)	1.50 × 10−12	0.5	0	[66]
(R92)	N(2D) + NO → N2 + O(3P)	1.80 × 10−10	0	0	[68]
(R93)	N(2D) + N2 → N(4S) + N2	1.70 × 10−14	0	0	[16]
(R94)	N(2D) + O(3P) → N(4S) + O(3P)	3.30 × 10−12	0	−260	[16]
(R95)	N(2D) + H2O → OH + NH	4.00 × 10−11	0	0	[16]
(R96)	N(4S) + HO2 → NO + OH	2.19 × 10−11	0	0	[69]
(R97)	N(4S) + NO → N2 + O(3P)	3.51 × 10−11	0	49.84	[46]
(R98)	N(4S) + N(4S) + N2 → N2 + N2	1.38 × 10−33	0	502.9	[70]
(R99)	N(4S) + NO2 → NO + NO	2.30 × 10−12	0	0	[46]
(R100)	N(4S) + NO2 → O(3P) + N2O	5.80 × 10−12	0	220	[71]
(R101)	N(4S) + O2 → NO + O(P)	4.47 × 10−12	1	3270.2	[72]
(R102)	O2(a) + N(S) → O(P) + NO	2.00 × 10−14	0	0	[71]
(R103)	O2(a) + O(P) → O(P) + O2	7.00 × 10−16	0	0	[66]
(R104)	O2(a) + O2 → O2 + O2	2.20 × 10−18	0.8	0	[73]
(R105)	O2(a) + N2 → O2 + N2	1.4 × 10−19	0	0	[74]
(R106)	O2(a) + NO → O(P) + NO2	3.49 × 10−17	0	0	[75]
(R107)	O2(a) + NO → NO + O2	2.50 × 10−11	0	0	[76]
(R108)	O2(b) + N2 → O2(a) + N2	2.10 × 10−15	0	0	[77]
(R109)	O2(b) + O2 → O2(a) + O2	4.10 × 10−17	0	0	[77]
(R110)	O2(b) + O(P) → O2 + O(P)	8.00 × 10−14	0	0	[77]
(R111)	O2(b) + H2O → O2 + H2O	4.60 × 10−12	0	0	[77]
(R112)	O2(b) + O3 → O2(a) + O2(a) + O(P)	1.80 × 10−11	0	0	[66]
(R113)	O2(b) + NO → O2(a) + NO	4.00 × 10−14	0	0	[66]
(R114)	O2(A) + O2 → O2(b) + O2(b)	2.90 × 10−13	0	0	[66]
(R115)	O2(A) + N2 → O2(b) + N2	3.00 × 10−13	0	0	[66]
(R116)	O2(A) + O(P) → O2(b) + O(D)	9.00 × 10−12	0	0	[66]
(R117)	O(3P) + O2 + N2 → O3 + N2	5.51 × 10−34	−2.6	0	[77]
(R118)	O(3P) + O2 + O2 → O3 + O2	6.01 × 10−34	−2.6	0	[77]
(R119)	O(3P) + O(3P) + N2 → O2 + N2	9.46 × 10−34	0	−484.7	[78]
(R120)	O(3P) + O(3P) + O2 → O2 + O2	3.81 × 10−33	−0.63	0	[79]
(R121)	O(3P) + N(4S) + N2 → NO + N2	6.89 × 10−33	0	−134.7	[78]
(R122)	O(3P) + NO + N2 → NO2 + N2	1.03 × 10−30	−2.87	−780.5	[80]
(R123)	O(3P) + HO2 → OH + O2	2.70 × 10−11	0	−224	[77]
(R124)	O(3P) + NO2 → O2 + NO	5.50 × 10−12	0	187.9	[77]
(R125)	O(1D) + O2 → O(3P) + O2	3.12 × 10−11	0	−70	[81]
(R126)	O(1D) + N2 → O(3P) + N2	2.10 × 10−11	0	−115	[81]
(R127)	O(1D) + H2 → OH + H	1.10 × 10−10	0	0	[77]
(R128)	O(1D) + H2O → OH + OH	2.2 × 10−10	0	0	[77]
(R129)	O(1D) + H2O → H2 + O2	3.57 × 10−10	0	0	[82]
(R130)	O(1D) + H2O2 → H2O + O2	5.20 × 10−10	0	0	[77]
(R131)	O(1S) + H2O → O(P) + H2O	3.00 × 10−10	0	0	[83]
(R132)	O(1S) + H2O → OH + OH	5.00 × 10−10	0	0	[83]
(R133)	O(1S) + H2O → H2 + O2	5.00 × 10−10	0	0	[83]
(R134)	O3 + H → OH + O2	1.40 × 10−10	0	480	[74]
(R135)	O3 + NO → NO2 + O2	3.16 × 10−12	0	−1563	[46]
(R136)	O3 + O(P) → O2 + O2	8.00 × 10−12	0	2060	[77]
(R137)	O3 + O(D) → O2 + O(P) + O(P)	1.20 × 10−10	0	0	[74]
(R138)	O3 + OH → HO2 + O2	1.70 × 10−12	0	940	[77]
(R139)	O3 + O3 → O2 + O(3P) + O3	7.16 × 10−10	0	11200	[84]
(R140)	OH + OH + N2 → H2O2 + N2	6.90 × 10−31	−0.8	0	[77]
(R141)	OH + OH + O2 → H2O2 + O2	6.05 × 10−31	−3	0	[74]
(R142)	OH + OH + H2O → H2O2 + H2O	1.54 × 10−31	−2	−183.6	[86]
(R143)	OH + OH → H2O2	2.6 × 10−11	0	0	[77]
(R144)	OH + OH → H2O + O(3P)	6.2 × 10−14	2.6	−945	[77]
(R145)	OH + H + N2 → H2O + N2	6.87 × 10−31	−2	0	[86]
(R146)	OH + H + H2O → H2O + H2O	4.38 × 10−31	−2	0	[86]
(R147)	OH + HO2 → H2O + O2	4.8 × 10−11	0	−250	[77]
(R148)	OH + N(4S) → NO + H	3.80 × 10−11	0	−85.39	[72]
(R149)	OH + O(3P) → O2 + H	2.40 × 10−11	0	−110	[77]
(R150)	OH + NO + N2 → HNO2 + N2	7.40 × 10−31	−2.4	0	[74]
(R151)	OH + NO + O2 → HNO2 + O2	7.40 × 10−31	−2.4	0	[74]
(R152)	OH + NO2 + N2 → HNO3 + N2	2.60 × 10−30	−2.9	0	[74]
(R153)	OH + NO2 + O2 → HNO3 + O2	2.20 × 10−30	−2.9	0	[74]
(R154)	HO2 + HO2 → H2O2 + O2	2.20 × 10−19	0	−600.2	[77]
(R155)	HO2 + NO → OH + NO2	3.60 × 10−12	0	−269.4	[77]
(R156)	HO2 + HO2 + N2 → H2O2 + O2 + N2	1.90 × 10−33	0	−980	[74]
(R157)	HO2 + HO2 + O2 → H2O2 + O2 + O2	1.90 × 10−33	0	−980	[74]
(R158)	HO2 + NO2 → HNO2 + O2	1.20 × 10−13	0	0	[85]
(R159)	H + HO2 → H2 + O2	1.75 × 10−10	0	1030	[86]
(R160)	H + HO2 → H2O + O(3P)	5.00 × 10−11	0	866	[87]
(R161)	H + HO2 → OH + OH	7.40 × 10−10	0	700.0	[86]
(R162)	H + O2 + O2 → HO2 + O2	5.94 × 10−32	−1.0	0	[46]
(R163)	H + O2 + N2 → HO2 + N2	5.94 × 10−32	−1.0	0	[46]

Table 4. Effective electronic states of N₂ and O₂ considered in this simulation.

Electronic state	Excitation energy (eV)	Effective state
N2(X, v = 0)	0	N2(X)
${\rm N}_2(A\,^3\!\Sigma_{\rm u}^+$, v = 0 ... 4)	6.17	N2(A1)
${\rm N}_2(A\,^3\!\Sigma_{\rm u}^+$, v = 5 ... 9)	7.00	N2(A2)
N2(B 3Πg)	7.35	N2(B)
N2(W 3Δu)	7.36	N2(B)
${\rm N}_2(A\,^3\!\Sigma_{\rm u}^+$, v > 10)	7.80	N2(B)
${\rm N}_2(B'\,^{3}\!\Sigma_{\rm u}^-)$	8.16	N2(B)
N2 $(a'\,^{1}\!\Sigma_{\rm u}^-)$	8.40	N2(a)
N2(a 1Πg)	8.55	N2(a)
N2(w 1Δu)	8.89	N2(a)
N2(C3Πu)	11.03	N2(C)
${\rm N}_2(E^{3}\!\Sigma_{\rm g}^+)$	11.88	N2(E)
${\rm N}_2(a''\,^{1}\!\Sigma_{\rm g}^+)$	12.25	N2(E)
O2(a 1Δg)	0.977	O2(a)
O2 $(b\,^{1}\!\Sigma_{\rm g}^+)$	1.627	O2(b)
O2(c 1Σu)	4.05	O2(A)

There are a large number of reaction pathways and rate constants that have been proposed to explain the relaxation kinetics from non-equilibrium conditions triggered by a discharge [45–44]; however, most of the rate constants are unknown or have only been estimated theoretically because of the difficulty of measuring such fast reactions. Therefore, we use only reliable reactions whose rate constants have been validated by direct measurements or indirect methods involving theoretical estimations and experimental results. For example, the quenching rate constant of N₂ $(A\,^3\!\Sigma_{\rm u}^+)$ by O₂ is reliable since it has been directly measured and its validity has been confirmed in other papers [47, 48]. The rates of the recombination reactions of N₂ $(A\,^3\!\Sigma_{\rm u}^+)$ are also reliable because the reaction kinetics of the N₂ discharge, including some impurities, have been well modelled with the theoretically estimated recombination rates even though the rates were not fully measured [35, 37]. On the other hand, some detachment processes from negative ions that play important roles in discharge relaxation phenomena [62, 63, 65] are not well defined. The detachments induced by excited molecules of nitrogen, such as N₂ $(A\,^3\!\Sigma_{\rm u}^+)$ and N₂(B ³Π_g), have not been clearly shown because the rate constants used in some studies on discharge modelling were taken from only one reference. Therefore, we use detachment reactions that have been investigated in more than one paper. For the other chemical reactions, we also attempted to find as many papers as possible with the help of the NIST database [49], and we chose the rate constants carefully. The vibration to vibration (VV) transitions and vibration to translation (VT) transitions are considered using our previously developed vibrational relaxation model [18]. However, they have no major effects in this simulation because the VV and VT transitions mainly occur several µs after the discharge [18].

Admittedly, the simplified reaction model does not describe all the kinetics triggered by the discharge. However, it is valid under limited conditions from the start of the discharge until 3 µs after the discharge pulse, as considered in this paper.

Figure 3 shows the spatial distribution of discharge light emission obtained from the optical emission of the second positive system (SPS) of N₂: C ³Π_u → B ³Π_g, whose intensity is proportional to the N₂(C ³Π_u) density. The numerically calculated SPS emission in the space r = (r, z) is spatially integrated along the line of sight. In addition, it is temporally integrated over 2 ns. Therefore, we calculate the time-integrated space-averaged optical emission [50, 51]. The primary streamer is formed near the anode tip and starts to propagate towards the plane cathode, as shown in figure 3. When the streamer head arrives at the plane cathode, the secondary streamer starts to extend. These characteristics of an atmospheric-pressure streamer discharge have been confirmed elsewhere in both experiments and simulations [10, 27, 52]. The light emission in humid air is similar to that in dry air. It is known that the changes in the spatial distribution and the time variation of the discharge emission depend on the gas composition and the shape of the applied pulsed voltage [52, 53]. Although the dependence of humidity on the discharge emission has not been clearly shown experimentally, the effect is estimated to be small from our simulated results. Figure 4(a) shows the axial distribution of the reduced electric field at several different times. The reduced electric field, E/N, is approximately 800 Td (1 Td = 10⁻¹⁷ V cm²) at the streamer head and it is almost constant when the streamer propagates to the plane cathode. However, the velocity of the streamer head and the reduced electric field in the streamer channel increase during its propagation through the gap. In our simulation, we confirmed that the propagation velocity of the streamer head and the reduced electric field in the streamer channel depend on the rate of increase in the applied pulsed voltage. The development of a reduced electric field in the primary streamer in humid air is not significantly different from that in dry air, obtained in our previous paper [10], because the ionization and photoionization rate coefficients are not changed for an admixture containing a few per cent of H₂O in atmospheric-pressure air. However, the axial distribution of the reduced electric field in the secondary streamer is slightly different in dry and humid air, as shown in figure 4(b). The reduced electric field in the secondary streamer, which is known to be determined by the balance between the rates of ionization and attachment [10, 54], slightly increases with humidity. This slight increase is caused by the electronegative character of H₂O. Figure 5 shows the total ionization and attachment rates in dry and humid air. The electron attachment reaction H₂O + e → H⁻ + OH causes the slight shift of the equilibrium point between the total ionization and attachment rates.

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Figure 6 shows the simulated axial distribution of the OH radical density at t = 3 µs after the discharge pulse has elapsed. This result indicates that a large number of OH radicals as well as O and N radicals are produced near the anode tip [10]. Compared with our previous experimental results [19, 55], the simulated axial distribution of OH in figure 6 is in good qualitative agreement. However, the simulated axial distribution of OH radicals has a stepwise change in density around the edge of the secondary streamer head, whereas the experimentally measured OH radical density decreases exponentially from the point anode to the plane cathode along the discharge axis. This is because the simulated axial distribution of the reduced electric field in the secondary streamer changes in a stepwise manner at the secondary streamer tip, as shown in figure 4(b). This stepwise distribution has also been reported in a previous paper [27] and is theoretically well defined [31]. However, our previous experimental results indicate different characteristics. We previously measured the axial distributions of O [56], N [57], and OH [19, 55] radical densities and showed that these radical densities decrease exponentially from the point anode to the plane cathode. For these experiments, the limitations of the laser-induced fluorescence method used make measuring a stepwise shape around the edge of the secondary streamer head, as shown in figure 6, difficult because of experimental inaccuracies. Therefore, it is difficult to distinguish whether the difference is merely an error caused by the experiment itself, or an error caused by a physical phenomenon not considered in our simulation.

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The number of OH radicals produced in a single discharge pulse is simulated as shown in figure 7. Then, the effect of humidity on the number of OH radicals generated is compared with the experimental results [19]. Figure 7 indicates that the number of OH radicals produced by the discharge is saturated with increasing water vapour, and the simulation result is qualitatively consistent with the experiments except for a difference in the total number of OH radicals within a factor of 2. The mechanism of OH radical production is discussed in detail in the following section.

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The temporal behaviour of OH density in the rectangular section indicated in figure 8(a) is plotted in figure 9. The observation volume is set at z = 3 mm below the point anode, where the reactivity is high in the secondary streamer channel. Figures 8(b) and (c), respectively, show the temporal variation of the reduced electric field and electron density at the observation volume for z = 3 mm. These figures show how electrons are produced in large numbers in the primary streamer and that the number of electrons decreases in the secondary streamer, concomitant with a significant decrease in the reduced electric field, E/N, as explained in our previous paper [10]. Figure 9 shows the time variation of the OH radical density with H₂O concentration at z = 3 mm, which is compared with the time variation in figures 8(b) and (c). From the comparison among figure 8(b) (E/N), figure 8(c) (electrons) and figure 9 (OH radicals), it is confirmed that the OH radicals are mainly produced in the secondary streamer and decrease in number after the discharge in two phases, labelled (i) and (ii). In particular, the rate of decrease in the second phase is accelerated with increasing humidity. This strong dependence of the decrease in OH radical density on humidity can explain the saturation of the number of OH radicals shown in figure 7. The detailed behaviour of OH radicals is explained in figure 10(a), which shows the time variation of the production and loss rates for each of the reactions leading to the formation of OH radicals as follows.

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Firstly, the mechanism of OH radical production is discussed. From the results in figure 10(a), OH radicals are mainly produced by the dissociation reactions of O(¹D) + H₂O and N₂(a) + H₂O in the secondary streamer. Electron dissociation reactions such as e + H₂O contribute less to OH radical production because the cross sections of these direct dissociation processes are low in the energy range of the secondary streamer compared with those of O(¹D) and N₂(a), as shown in figure 1. Although the dissociation of the reaction of O(¹D) + H₂O has been considered as one of the major pathways for OH radical production, the contribution of N₂(a) to OH-production has not been discussed before. Therefore, its validity should be discussed here. Fresnet et al [35] first suggested the importance of the reaction of N₂(a) + H₂O. They concluded that the decrease in the decomposition rate of NO with the increase in humidity in an atmospheric-pressure photo-triggered discharge is caused by the quenching reaction of N₂(a) by H₂O because N₂(a) plays the main part in the NO removal process. In addition, they measured the rate coefficient of the reaction of N₂(a) + H₂O indirectly from the temporal change in N₂(a) and NO densities under several humid-air conditions. However, branching ratio of the reaction of N₂(a) + H₂O is unknown. Magne et al [58] and Aleksandrov et al [59] assumed that the branching ratio of N₂(a) + H₂O → N₂ + O + OH is 1.0 in their plasma chemical reaction model and obtained good agreement between experimental and simulation results, although they did not pay sufficient attention to the branching reactions of N₂(a) + H₂O. In this work, we used the rate coefficients of N₂(a) + H₂O measured by Fresnet et al [35] and assumed the branching ratio to be 1.0, similarly to Magne et al [58] and Aleksandrov et al [59].

Next, the decrease in the number of OH radicals is discussed. The number of OH radicals produced in the secondary streamer rapidly decreases in two phases. The first decay (see (i) in figure 9) is caused by the reaction with O(³P), as shown in figure 10(a). The rate of OH-production by O(¹D) and the rate of OH loss by O(³P) are similar in the discharge phase since similar amounts of O(¹D) and O(³P) are produced by the direct dissociation process or dissociations by N₂(A, B, C). However, the produced O(¹D) are rapidly quenched by several species such as O₂, N₂ and H₂O. Therefore, OH radicals are destroyed just after the discharge by the reaction with O(³P), producing H + O₂. In figure 9, we can observe a temporal plateau of the OH radical density after the first decay and the subsequent second decay (see (ii) in figure 9). This phenomenon can be explained as follows using figure 10(b).

Figure 10(b) shows the time variation of the rates of reactions in three reaction processes: 'OH-production', 'OH-cycle' and 'OH-recombination', as shown in table 5. The rates of reactions in the OH-cycle change considerably from negative to positive at approximately 0.2–0.3 µs because the OH-cycle includes the OH-production reaction (R123) and (R161), and the OH loss reactions (R148) and (R149). The first negative rate in the OH-cycle is caused by the loss reactions (R148) and (R149) and the effect of the production reactions (R123) and (R161) emerges with increasing HO₂, which is mainly produced by the reactions (R162) and (R163) through H radicals formed by the reactions (R148) and (R149). This is why we call this process the OH-cycle. The OH-cycle results in the first decrease in OH density (i) and the subsequent plateau phase between (i) and (ii), as shown in figure 9.

The second decrease in OH density (ii) is explained as follows. From the results in figure 10(b), the decay of (ii) is caused by an OH-recombination process. The OH-recombination includes a number of OH loss processes involving the recombination reactions (R140)–(R144) or reactions with water-related radicals such as HO₂ and H (R145)–(R147). Therefore, the rate of OH-recombination increases with the square of the water concentration. The OH-recombination allows us to interpret the decay of (ii) in figure 10(b) and the saturation of the experimentally obtained number of OH radicals in figure 7.

Figure 11 shows the flow of OH-production by the atmospheric-pressure streamer discharge. OH radicals are mainly produced by the reactions of N₂(a) and O(¹D) with H₂O and are transformed into H₂O or H₂O₂ via several recombination reactions with water-related radicals.

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Under the present discharge conditions, we obtained OH-production pathways similar to that in figure 11. However, the pathways might be different under different conditions. For example, under the repetitive discharge conditions with a frequency of over 1 kHz, the gas temperature in the discharge volume might increase mainly by the vibrational relaxation phenomenon [18]. The recombination reactions in figure 11 have negative temperature coefficients, and therefore the rate of decrease of OH density after the discharge might be slower than that considered in this paper. To simulate the increase in the gas temperature accurately, the fast gas heating model [32] and the vibrational relaxation model [18] should be coupled with the Navier–Stokes equations to consider the effect of compressibility [30, 34]. In addition, the rate of current relaxation simulated by the extended Morrow and Sato equation [29], as shown in figure 2, is slightly faster than that observed experimentally [14]. The rates of the electron attachment and detachment reactions, which are considered to be important for current relaxation, need to be examined carefully. Moreover, complex ions may also affect the rate of current relaxation. Complex ions were ignored in this simulation because we were able to reproduce the primary and secondary streamers well without them. It may be necessary to evaluate the effect of complex ions on the production of radicals for a more precise simulation.