Recently, non-thermal atmospheric pressure plasma (APP) has received considerable attention due to their practical applications in the field of plasma medicine, material processing, and pollution control.^{[1– 5]} To address the specific requirements of various applications, a range of different types of non-thermal APP sources have been reported.^{[6– 9]} Plasma needle as a typical non-thermal APP source, plasma discharge usually generates in a gap (mm range) between the powered needle tip (∼ 100  μ m in diameter) and the cathode surface.^{[8, 9]} The sharpened tip is driven by different power sources, including kHz ac, RF, microwave, pulsed dc, and He or Ar is typically used as the source gas to sustain the stable plasma.

With deepening research into the non-thermal APP, researchers pay more and more attention to the detailed generation mechanism and exact features of this peculiar plasma. For instance, several research groups demonstrated that plasma jets are not continuous but discrete bullet-like plasma luminous volumes travelling at speeds of 10⁴  m/s– 10⁶  m/s.^{[10– 14]} Based on experimental and modeling studies, ^{[15– 18]} it is commonly accepted that the “ bullets” are ionization waves, which are quite similar to cathode-directed streamers. And, these plasma “ bullets” are hollow taken by using an intensified charge coupled device (ICCD) once it emerges as a propagating streamer in the open air. Over the hollow “ bullet” in the plasma jet of He thereare various opinions but up to now no consensus has been reached. Sakiyama et al.^[15] compared the light emission measurements with a one-way coupled model with local field approximation, concluding that the Penning effect must play an important role in the formation of the hollow bullets. However, by turning the Penning reactions on and off in their modes, Naidis, ^{[19, 20]} and Naja et al.^[21] have shown that the Penning ionization is not essential for the bullets propagation and their hollow shape formation. Furthermore, Chang et al.^[22] reported that the annular structure of the plasma plume in ambient, in the case of He, is not due to the Penning effect on the He/air interface. To date, the mechanisms of the acceleration behavior of the plasma bullets have achieved considerable progress. It was concluded that there are two main reasons for the acceleration of the plasma bullets in open air.^{[23, 24]} One is due to the diffusion of N₂, which leads to Penning ionization and finally leading to the acceleration of the bullets. Another one is due to the change of the permittivity of the medium surrounding the plasma plume.

As mentioned above, the backdiffusion of N₂, in other words, the gas flow rate of source gas exerts significant influence on streamer propagation. Several experimental studies have been carried out to investigate the effect of gas flow rate on APP discharge characteristics, such as jet length, ^{[25, 26]} distributions of long-lifetime active species in the discharge tube, ^[13] and gas temperature.^{[27, 28]} Besides, it is important to note that the streamer is driven by the high electric field in our APP needle discharge, unlike the bullets in the open air (afterglow region) discussed above. Consequently, the metallic needle tip radius should play a role in the streamer propagation. Therefore, the dependence of APP needle discharge dynamics on the gas flow rate and curvature radius of the needle tip is discussed in this paper. The purpose of this paper is to take a step in this direction.

In this paper, we propose a multi-species, self-consistent, two-dimensional axially symmetric plasma model which incorporates detailed helium– air finite-rate plasma chemistry. In this model, air is taken to mean a mixture of nitrogen (79%) and oxygen (21%). The effects of the helium gas flow rate and the curvature radius of the needle tip on the dynamics of the streamers are performed. Several important results are obtained and briefly summarized. Penning ionization and charge exchange reactions seem to play no significant role in streamer propagation. With increasing He gas flow rate from 0.2  SLM to 0.8  SLM, the solid circle emission shapes from oxygen and nitrogen can be made into hollow profiles and the average streamer propagation velocity decreases. Besides, with decreasing the needle tip curvature radius from 200  μ m to 100  μ m, the spatially resolved plasma radical distributions remain almost the same, except that around the near-tip region their peak values are getting to be twice or more.

Our model consists of a coupled set of models: a neutral gas fluid model and a plasma dynamics model in two-dimensional cylindrical coordinates. More specially, steady state equations are solved with the neutral gas flow, whereas we solve time-dependent equations with the plasma discharge dynamics. The schematic diagram of the plasma needle and the simulation domain is shown in Fig.  1. The simulation domain in this study is similar to that previously reported by Sakiyama et al.^[29] The simulation domain (5  mm× 5  mm) corresponds to the neutral gas model and the smaller region ABCHIJM (2  mm× 2.5  mm) to the plasma dynamics model, which is a part of the neutral gas simulation domain. The needle tip (identified by the points AMLin Fig.  1) is a semi-sphere with a radius 0.2  mm or 0.1  mm (0.2  mm in Fig.  1). The total length of needle is 2.5  mm. The needle is concentric with an insulator tube, which has an inner radius of 1.5  mm and an outer radius of 2.5  mm. The dielectric constant of the insulator is 5ε ₀ (ε ₀ is vacuum permittivity). The edges of the insulator tube (points I and G in Fig.  1) are rounded with a curvature radius of 50  μ m. The needle tip is flushed with the end of tube. The cathode dielectric is simplified as a planar quartz plate (BDPN in Fig.  1). The gap distance between the needle tip and the cathode surface is fixed at 2.5  mm. The thickness of the quartz plate is 0.2  mm with relative permittivity 5 unless otherwise stated. The electrical potential is zero at the backside of the quartz plate (NP in Fig.  1), where a grounded metal plate is assumed to be attached. A 4.0-kV positive dc voltage is applied on the needle tip. Besides, an external RC (R = 2  kΩ , C = 1  pF) circuit is assumed to be added to the plasma discharge, which is used to prevent the discharge from arcing. The voltage is assumed to be a square wave ‘ pulse’ , i.e., the peak voltage is applied instantaneously at the beginning of the simulation and remains constant for the simulation duration. The pure He gas flows into the domain from the boundary between the needle and the insulator tube (LK in Fig.  1).

Fig.  1.

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	Fig.  1. Side view schematics of the plasma needle and the simulation domain.

The governing equations for the neutral gas model consist of the total mass continuity equation, Navier– Stokes equation, and convection– diffusion equation as follows:

where u = (u, w) is the velocity field, ρ is the gas density at the standard state (helium 0.1664  kg/m³, air 1.293  kg/m³), p is the pressure, η is the dynamic viscosity (helium 1.94× 10^{− 5}  Pa· s and air 1.82× 10^{− 5}  Pa· s), D is the diffusion coefficient (helium in air 1.82× 10^{− 5}  m²/s), c_i is the molar concentration of air or helium (mol/m³), and F = (0, 0, − ρ · g) is the body force field (buoyancy, g the gravity constant). The boundary conditions are summarized in Table  1. The velocity distribution of the helium at the inlet (LK in Fig.  1) is given by a fully developed laminar flow (i.e. Poiseuille flow) with 0.2  SLM or 0.8  SLM. The pressure at the exit (DE and EF) is fixed at 1  atm (1  atm = 1.01325× 10⁵  Pa) and the gas temperature is 300  K. No-slip velocity is applied on the solid surfaces of the needle (AML in Fig.  1), the dielectric tube (FGIK in Fig.  1) and the treated surface (BCD in Fig.  1). The hydrodynamic flow/mixing by plasma and photo-ionization mechanisms are not considered in this model, and background pre-ionization is assumed instead. The initial electron and ion densities in the gas are considered to be uniform to 10⁷  cm^{− 3}.

Table 1.

Table 1.

Table 1. Boundary conditions for the neutral gas model. The symbols (A– M) correspond to the vertexes in Fig.  1. ur and uz are r and z components of the velocity field, respectively. n is the normal vector to the boundary. ur uz ci AB ur = 0 AML ur = 0 uz = 0 D∇ ci · n = 0 KL ur = 0 uz = u0a ci = ci0, i = He FGIK ur = 0 uz = 0 D∇ ci · n = 0 DE uz = 0 ci = ci0, i = air EF ur = 0 ci = ci0, i = air BD ur = 0 uz = 0 a) Deduced from the helium gas flow rate. b) Corresponds to zero stress tensor.

Table 1. Boundary conditions for the neutral gas model. The symbols (A– M) correspond to the vertexes in Fig.  1. ur and uz are r and z components of the velocity field, respectively. n is the normal vector to the boundary.

The plasma dynamics model is based on the classical fluid model, which is governed by the continuity equations with the drift– diffusion approximation, the electron energy conservation and Poisson’ s equation are as follows:

where subscript i, e, and He indicate the i-th species, electrons, and helium gas, respectively. Γ is the flux in the drift– diffusion approximate, R_{i j} is the j-th reaction rate corresponding to Table  3, q is the charge, μ is the mobility, E is the electric field, ϕ is the potential and ε is the mean electron energy. Δ E and K_inel are the energy loss to inelastic collisions and the corresponding reaction rate, respectively. ε ₀ is vacuum permittivity.

The boundary conditions for the particle flux at the needle, the insulator and the treated surface are given as follows:

where e, i, m, and ε indicate the electrons, ions, metastables, and electron energy, respectively. T_i and T_g are the ion and background gas temperature. In this paper, we take T_i = T_g = 300  K. n is the normal vector pointing toward the surface and γ is the second electron emission probability. The secondary electron coefficient is 0.05 in this model. The surface charge accumulation is considered on the dielectric surface. α _s and are switching functions depending on the inner product of E and n, defined as:

The self-bias potential at the surface of the quartz plate is calculated by Gauss’ s law:

where D₁ and D₂ are the electric displacement at the two sides of the dielectric surface, J_i · n and J_e · n are the normal component of the total ion current density and total electron current density on the cathode surface, respectively. σ _s is the net surface charge accumulated on the dielectric surface. The boundary conditions are summarized in Table  2.

Table 2.

Table 2.

Table 2. Boundary conditions for the plasma dynamics model. The symbols (A– M) correspond to the vertexes in Fig.  1. Subscripts e, i, and m indicate electron, ions, and metastables, respectively. ne ni nm nε ϕ AML Eq.  (8) Eq.  (9) Eq.  (10) Eq.  (11) 4.0  kV AB KL FGIK Eq.  (8) Eq.  (9) Eq.  (10) Eq.  (11) ε 0 ∇ ϕ = σ s DE EF BCD Eq.  (8) Eq.  (9) Eq.  (10) Eq.  (11) Eq.  (13)

Table 2. Boundary conditions for the plasma dynamics model. The symbols (A– M) correspond to the vertexes in Fig.  1. Subscripts e, i, and m indicate electron, ions, and metastables, respectively.

The plasma chemistry considered in this model consists of 17 different species (e, He, He⁺ , , He* , , O₂, O₂ (V), O₂ (R), , , , O, O⁻ , N₂, , N₂ (V)) and 51 elementary reactions as shown in Table  3. The helium and helium– air chemistry reaction sub-mechanisms are taken from Refs.  [21], [30], and [31], and the air chemistry sub-mechanisms are taken from Refs.  [21], [30], and [32]. For reactions which produced O₂ and N₂ indifferent rotational and vibrational states, their reaction rates are calculated by cross section data from Bolsig+ and fitted to analytic functions of the mean electron energy.^[33] Penning ionization of O₂ and N₂ by helium metastables and charge transfer reaction are also taken into account.

Table 3.

Table 3.

Table 3. Helium– air chemistry reactions used in this model. Helium– air chemistry (molecules-meters-electron volt) Rxn Reaction A B C Activation energy Refs. Helium chemistry R1 e + He→ e + He BOLSIG+ 0 [33] R2 e + He→ e + He* BOLSIG + 19.8 [33] R3 e + He→ 2e + He+ BOLSIG+ 24.6 [33] R4 e + He* → 2e + He+ 4.66 × 10− 16 0.6 4.78 4.78 [31] R5 1.268 × 10− 18 0.71 3.4 3.4 [31] R6 2He* → e + He+ + He 4.5 × 10− 16 0 0 – 15.0 [31] R7 e + 2He+ → He + He* 5.386 × 10− 13 – 0.5 0 0 [31] R8 1.3 × 10− 45 0 0 0 [31] R9 1.0 × 10− 43 0 0 0 [31] R10 e + He+ → He* 6.76 × 10− 19 – 0.5 0 0 [31] R11 e + He* → e + He 2.0 × 10− 16 0 0 0 [31] R12 2e + He+ → e + He* 6.186 × 10− 39 – 4.4 0 0 [31] R13 e + He+ + He→ He + He* 6.66 × 10− 42 – 2 0 0 [31] R14 2.8e × 10− 32 0 0 0 [31] R15 3.5e × 10− 39 0 0 0 [31] R16 1.5e × 10− 39 0 0 0 [31] R17 2.8e × 10− 32 0 0 0 [31] Air – hemistry R18 e + O2 → e + O2 (VIB3v1) BOLSIG + 0.579 [33] R19 e + O2 → e + O2 (VIB4v1) BOLSIG + 0.772 [33] R20 e + O2 → e + O2 (A1) BOLSIG + 0.977 [33] R21 e + O2 (A1) → e + O2 BOLSIG + – 0.977 [33] R22 e + O2 → e + O2 (B1) BOLSIG + 1.627 [33] R23 e + O2 (B1)→ e + O2 BOLSIG + – 1.627 [33] R24 e + O2 → e + O2 (EXC) BOLSIG + 4.5 [33] R25 e + O2 (EXC)→ e + O2 BOLSIG + – 4.5 [33] R26 e + O2 → O + O− BOLSIG + 3.6 [33] R27 e + O2 → e + 2O BOLSIG + 5.58 [33] R28 e + O2 → e + O + O (1D) BOLSIG + 8.4 [33] R29 BOLSIG + 12.06 [33] R30 e + N2 → e + N2 BOLSIG + 0 [33] R31 e + N2 → e + N2 (VIBv1) BOLSIG + 0.2889 [33] R32 e + N2 → e + N2 (VIB3v1) BOLSIG + 0.8559 [33] R33 e + N2 → e + N2 (VIB4v1) BOLSIG + 1.134 [33] R34 e + N2 → e + N2 (VIB5v1) BOLSIG + 1.4088 [33] R35 e + N2 → e + N2 (A) BOLSIG + 6.17 [33] R36 BOLSIG + 15.6 [33] R37 3.464 × 10− 12 – 0.5 0 0 [31] R38 5.17 × 10− 43 – 1 0 0 [31] R39 2 × 10− 13 0 0 0 [31] R40 2 × 10− 37 0 0 0 [31] R41 O− + O → O2 + e 5 × 10− 13 0 0 0 [31] R42 1 × 10− 13 0 0 0 [31] R43 3.165 × 10− 30 – 0.8 0 0 [31] R44 4.184 × 10− 44 – 2.5 0 0 [31] Helium– air interactions R45 1.0 × 10− 13 0 0 0 [30] R46 2.6 × 10− 16 0 0 0 [30] R47 2.6 × 10− 16 0 0 0 [30] R48 7.0 × 10− 17 0 0 0 [30] R49 7.0 × 10− 17 0 0 0 [30] R50 5.0 × 10− 16 0 0 0 [30] R51 5.0 × 10− 16 0 0 0 [30] Note: (i) Species ‘ M’ in reactions R40 represents third-body species (He, O2, and N2). (ii) The tabulated reaction rates are given in the Arrhenius form. (iii) Units: Two-body reaction rate coefficient (m3· s− 1). Three-body reaction rate coefficient (m6· s− 1). Electron temperature Te (eV). Gas (heavy particle) temperature Tg (K).

Table 3. Helium– air chemistry reactions used in this model.

The electron transport coefficients (mobility and diffusivity) are evaluated using the Boltzmann solver in advance, ^[33] while the transport coefficients applied for the ions and radical species are taken from Refs.  [34]–   [36]. The generalized Einstein relation is used to include the effect of high electric field for ion diffusivity. The radical species diffusion coefficients follow the assumption of binary diffusion in helium. It is important to note that the ionization and transports parameters for electrons in helium– air mixtures are governed by the values of reduced electric field and air mole fraction. According to previous studies, ^{[19, 20]} the electron ionization reaction coefficient is nearly independent of the air mole fraction (< 1%). Therefore, in our case, as the gas flow rate is high enough (i.e. > 0.5  SLM), the influence of air on the electron impact reaction rate and electron transports parameters is negligibly small. On the other hand, as the gas flow rate is low (at 0.2  SLM), the effect of the large air concentration on electron impact reaction rate cannot be neglected anymore.

All the governing equations given above are solved by COMSOL with the finite element method. First, we solve Eqs.  (1)– (3) in order to obtain a steady state fluid flow field of the neutral gas. Then the flow field results are coupled to the time-dependent solver, which is applied to solve the plasma dynamic Eqs.  (4)– (7). The free triangle mesh method is used to discretize the governing equations. The plasma domain requires a more refined, but more localized, mesh than neutral simulation. Close to the needle tip and the treated surface, the maximum mesh size is 1  μ m and the mesh size increases following a geometric progression to 10  μ m and is kept constant in the discharge gap (ABCHIJM in Fig.  1). The mesh constructed to resolve the neutral gas fluid model consists of 13221 cells. On the other hand, the mesh constructed to describe the plasma dynamic model consists of 298530 cells and the total number of degree of freedom is 2361412.

A single streamer discharge mechanism of the helium atmospheric pressure plasma needle has been studied numerically in a two-dimensional axisymmetric model. It is found that the needle tip curvature radius has little effect on the streamer dynamics, except that around the near-tip region the peak densities of plasma parameters are getting to be twice the value or more. It is also concluded that gas flow rate plays an important role in streamer propagation. Air impurities stimulate the Penning ionization and cause a significant increase in the streamer propagation speed. Besides, Penning ionization reactions also have an impact on streamer propagation but not to as big as electron-impact ionizations. With increasing the gas flow rate from 0.2  SLM to 0.8  SLM, the emissions resulting from reactive oxygen and nitrogen species change from solid circle to hollow shapes, which are contributed to the large radial concentration gradient of air impurities concentration.