Han Xiao-Yu, Wang Jun-Hong, Chen Mei-E, Zhang Zhan, Li Zheng, Li Yu-Jian. Similarity principle of microwave argon plasma at low pressure. Chinese Physics B, 2018, 27(8): 085206
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Similarity principle of microwave argon plasma at low pressure
Han Xiao-Yu, Wang Jun-Hong †, Chen Mei-E, Zhang Zhan, Li Zheng, Li Yu-Jian
Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China
Project supported by the National Natural Science Foundation of China (Grant No. 61331002), the National Key Basic Research Program of China (Grant No. 2013CB328903), and the Fundamental Research Funds for the Central Universities, China (Grant No. W15JB00510).
Abstract
In order to validate the similarity principle of microwave breakdown, a two-dimensional (2D) fluid model of low-pressure microwave argon plasma is established and solved by the finite-element method. Proportional conditions are used in this model to build three different breakdown processes that meet the premise of a similarity principle, and these breakdown processes are called “similar cases” in this paper. Similar cases have proportionately sized breakdown regions, where the ratio of frequency of incident microwave f to gas pressure p (f/p), and the reduced field E/p in them are kept the same. All the important physical parameters such as electron density, electron temperature, and reduced electric field can be obtained from the simulation of this model. The results show that the parameters between similar cases are in constant ratio without changing with time, which means that the similarity principle is also valid in microwave breakdown.
Microwave plasmas are widely used in industry for depositing, heating, and coating. Compared with the plasmas generated by classical direct current (DC) or low frequency technologies, the plasmas generated by microwave technology have larger electron density and higher power utilization efficiency.[1,2] However, in the traditional theoretical study of microwave plasma, the time for simulating the breakdown models by fluid-based methods is quite long and the requirement for computation resource is very high, especially for the extreme cases with very high pressure or with large breakdown area. In order to improve the computation efficiency, the methods based on the similarity principle for DC or low frequency plasmas have been put forward.[3,4] With the similarity principle, extreme case problems can be effectively solved by approximating them to general and computable models. However, for microwave breakdown, the premises and effectiveness of the similarity principle are not clear yet.
The similarity principle is a statement of conditions under which similar discharges take place in the determined area. In the past few decades, theories of the similarity principle have been studied by many researchers. Paschen first revealed that the breakdown voltage depends on a constant product pd, where p is the gas pressure and d is the length of the breakdown gap or the distance between anode and cathode.[5] By this theory, a problem with very high pressure can be equivalent to another problem with lower pressure but larger length of breakdown gap. This original general similarity theory was extended by Holm to another problem, in which the potentials were determined by space charges as well as by charges on the electrodes.[6] For high-frequency breakdown, the definition of a similarity law was first given by Margenau in 1948,[7] and then Jones and Morgan studied the high-frequency (3.5 MHz∼70 MHz) breakdown between wires and coaxial cylinders in air and in hydrogen respectively at low pressure (lower than 2.66 kPa). The results proved that Margenau’s theory of the similarity principle in high-frequency breakdown is correct, and the parameters of breakdowns with different air pressures and breakdown sizes are kept in certain ratios.[8] Recently, Fu et al. carried out numerical simulations to investigate the validity of the similarity principle of argon glow discharges at low pressure, showing that the proportional relation can be given by the similarity law between two discharge gaps, and also extends the traditional Paschen law for the gas breakdowns to non-uniform electric field conditions.[9]
However, very few studies or verifications of a similarity principle for high-frequency problems can be found after the work of Jones and Morgan. Furthermore, it is hard to find the studies of the similarity principle for breakdown problems in microwave-frequency band (300 MHz–300 GHz), although many difficult breakdown problems occur in current high power microwave applications. An equivalent method based on the similarity principle in the microwave band could offer a way to simplify the study of breakdown in a large area into a similar case with a computable proportional small size. From the results of the small sized case, the breakdown parameters of large size problem can be extrapolated.
In this paper, microwave (with frequency higher than 2 GHz) breakdowns in similar processes are studied numerically using a two-dimensional (2D) fluid model for verifying the validation of the similarity principle in the microwave band. Parallel plate waveguides are used to build the breakdown region with argon in it as the working gas. By this study, the correctness of the similarity principle in microwave breakdown can be verified comprehensively, and the proportional relations between the important physical parameters, including the electron densities, electron temperature, and the distribution of electric field, of microwave breakdowns in similar cases can be found.
2. Theories of similarity principle in microwave breakdown
2.1. Brief review of similarity principle in DC-discharge
Relations between parameters of two similar systems for DC-discharge, derived under the similarity principle, are listed in Table 1, where k is the scale-down ratio. The dominant processes in the breakdown are the collisions between electrons and argon atoms, the diffusion and drift of the electrons and ions. The microcosmic physical parameters used to describe this process are the mean free path of electron λ and the reduced electric field E/p. Under the condition of constant temperature, λ is inversely proportional to the number density of gas n, and n depends linearly on air pressure p, so the macroscopic parameter p, which is easy to measure, can be used to represent the microscopic parameter λ, because of λ ∝ 1/p. Hence, pd is proportional to the number of collisions Nc, where d is the distance between the electrodes and E/p is proportional to the energy gained by each electron from every collision. Consequently, the influences of other parameters can be ignored, and as long as pd and E/p are kept the same in different breakdown processes, the number of collisions and the energy gained by each electron from one collision are nearly the same. Then the whole breakdown processes between similar cases are considered to be the same macroscopically, so the premises of the similarity principle in DC-discharges can be written as[10,11]
Table 1.
Table 1.
Table 1.
Relations between parameters of two systems with similar DC discharges.[12,13]
.
Physical parameters
Similar relations
Number density of particles
n1 = n2/k2
Electric field at corresponding position
E1(x1) = E2(x2)/k
Mean free path of particles
λ1 = kλ2
Energy gained by each collision
E1λ1 = E2λ2
Surface charge density of electrodes
σ1 = σ2/k
Space charge density
ρ1 = ρ2/k2
Current density
j1 = j2/k2
Time interval of each process
dt1 = kdt2
Table 1.
Relations between parameters of two systems with similar DC discharges.[12,13]
.
2.2. Premises of similarity principle in microwave breakdown
For the case of microwave breakdown with low pressure, there are two phenomena which are different from those of static discharges. First, the direction of the electric field in microwave band changes so fast that the ions and even the electrons could not reach the cathode within one period, so they just oscillate in the space with the frequency of microwave. Second, the secondary ionization does not play an important role as in the case of static discharge, the breakdown process is determined only by the primary ionization and the loss of electrons caused only by the diffusion to the walls.[7] Hence, the concept of ‘electrode’ is unimportant in the microwave breakdown, and the actual path traversed by an electron is not determined by the concentration of gas atoms (gas pressure p) nor by the distance between electrodes d, like that in the case of an electron driven by a uni-directional field, but is determined by the microwave frequency f. Thus, as long as f/p and E/p are kept the same, the whole breakdown process is considered to be the same. The relations of the similarity principle in microwave breakdown are[7,8]
In addition, in microwave plasma, there is no need to calculate the surface charge density of an electrode nor current density, which are vital parameters in DC-discharge as shown in Table 1.
3. Modeling
3.1. Breakdown regions and boundary conditions
Figure 1 shows the diagram of the high power microwave (HPM) system for argon breakdown, the microwave with power higher than 1000 W is generated by HPM source, and is transmitted into the breakdown region through a rectangular waveguide. The waterloaded circulator is used to absorb the reflected HPM and guarantee the safety of HPM source.
Fig. 1. Diagram of HPM system for argon breakdown.
Figure 2 shows the geometrical structures of the breakdown regions of three similar cases. Here, we refer to the largest one as the prototype case and the smaller two as the scale-down cases, their scale-down factor k values are set to be 2 and 5 respectively. In order to obtain similar breakdowns, f/p and E/p are ensured to be the same. Hence, for the prototype case, a1 = 0.25 m, b1 = 0.05 m, the air pressure p1 = 200 Pa, the power of incident microwave Power1 = 1000 W, and the frequency of incident microwave f1 = 2.45 GHz. For the scale-down case with k = 2, a2 = 0.125 m, b2 = 0.025 m, p2 = 400 Pa, Power2 = 2000 W, and f2 = 4.9 GHz. For the scale-down case with k = 5, a3 = 0.5 m, b3 = 0.01 m, p3 = 1000 Pa, Power3 = 5000 W, and f3 = 12.25 GHz. Two observation lines l1 and l2 are set to be in the regions. For each case, l1 is set to be on the central axis, l2 is set to be on the line with one fifth of the distance between two plates of waveguide, which is in close proximity to the lower plate.
Fig. 2. Schematic diagrams of the similar cases: (a) prototype case, (b) scale-down case (k = 2), and (c) scale-down case (k = 5).
The computational domain of each case is shown in Fig. 3, where o–a and c–b represent the two plates of the parallel plate waveguide. The fluid equations of microwave plasma are computed in the region of o–a–b–c, and o–c and a–b are set as the waveguide ports for the excitation of incident wave working at TEM mode. Assuming that no reflection occurs on the ‘wall’ boundary b–c and o–a, which means that for b–c and o–a, zero gradient boundary condition for electrons is used, and the metastable atoms Ar* and ions Ar+ are absorbed due to the surface reactions on the wall boundary: Ar* ⇒ Ar and Ar+ ⇒ Ar. The specific proportionate values for similar cases are shown in Table 2. The gas temperature is set to be 300 K, which approaches the room temperature.
where ne is the electron density, Re is the source of electrons consumed by ionization or recombination, Γe is the electron density flux, ue is the electron mobility, and De is the electron diffusion coefficient.
The electron energy transport equations are
where nε is the electron energy density, Rε is the source of energy loss caused by the inelastic collision, e is the unit charge, Γε is the electron energy flux, uε is the electron energy mobility, and Dε is the electron energy diffusion coefficient.
where Nn is the number density of target species in the j-th reaction, kj is the reaction coefficient of the j-th reaction, which can be obtained from the cross sectional data as follows:
where is a constant, σj(ε) is the cross section of the j-th reaction, and f(ε) is the electron energy distribution function used in this model.
The electron energy source is given by
where Δεj is the energy loss of the j-th reaction, and qe is the elementary charge.
The equations of heavy particles are given by
where ρ is the density of gas, p is the heavy particle type, Rp is the production or destruction of particle p in reactions, wp is the mass fraction of particle p, and dp is the diffusive mass flux of particle p.
All the reactions calculated by Eqs. (5)–(15) are shown in Table 3. The rates of reactions (1)–(5) in Table 3 can be calculated by the cross section data, and the rate coefficients for reactions (6)–(8) are set to be constant, and their values are 6.4 × 10−10 cm3⋅s−1, 2.3 × 10−15 cm3⋅s−1, and 1.8 × 10−6 cm3⋅s−1 respectively. The microwave breakdown model is computed by the finite-element method.
Table 3.
Table 3.
Table 3.
Main reactions and particles for argon plasma.[17–22]
.
No.
Reactions
Description
Δε/eV
1
e+Ar ⇒ e+Ar
Elastic collision
2
e+Ar ⇒ e+Ar*
Excitation
11.83
3
e+Ar* ⇒ e+Ar
Superelastic
−11.5
4
e+Ar ⇒ 2e+Ar+
One step ionization
15.76
5
e+Ar* ⇒ 2e+Ar+
Step-wise ionization
4.24
6
Ar*+Ar* ⇒ e+ Ar+Ar+
Penning ionization
–
7
Ar*+Ar ⇒Ar+Ar
Metastable quenching
–
8
Ar* ⇒ hν+Ar
Radiation
–
Table 3.
Main reactions and particles for argon plasma.[17–22]
.
4. Results and discussion
The distributions of reduced electric field E/p in similar cases are kept the same by using specific combinations of incident powers and waveguide cross section areas to meet the premise of similarity principle, and their values along the observation line l1 are shown in Fig. 4. In order to make results more clear, we use px, the product of air pressure p, and position x, as the horizontal ordinate in this figure. From Fig. 4 we can see that the reduced electric fields of the three similar cases are nearly the same in the beginning (<1 × 10−10 s) and meet the premise of the similarity principle of microwave discharge perfectly. The peak values of E/p in similar cases are 3.13 V⋅cm−1⋅Pa−1, 3.11 V⋅cm−1⋅Pa−1, and 3.11 V⋅cm−1⋅Pa−1 respectively. We could obtain that the peak values of electric field are 3.13 V⋅cm−1⋅Pa−1 at a pressure of 200 Pa for the prototype case, 6.22 V⋅cm−1⋅Pa−1 at 400 Pa for the scale-down case of k = 2, and 15.55 V⋅cm−1⋅Pa−1 at 1000 Pa for the scale-down case of k = 5. The continued proportion of the three maximum values of electric field is equal to 1:1.99:5. Similarly, we can obtain that at the time of 10−6 s, the peak values of electric fields are 3.03 V⋅cm−1⋅Pa−1 at the pressure of 200 Pa for the prototype case, 4.46 V⋅cm−1⋅Pa−1 at the pressure of 400 Pa for the scale-down case of k = 2, and 7.55 V⋅cm−1⋅Pa−1 at the pressure of 1000 Pa for the scale-down case of k = 5. The continued proportion of the maximum values of electric field is equal to 1:1.47:2.49. The results show that as time goes on, the breakdown occurs and the deviations begin to show up. At the time of 10−2 s, the plasma becomes steady. The peak values of electric field are 0.096 V⋅cm−1⋅Pa−1 at the pressure of 200 Pa for the prototype case, 0.210 V⋅cm−1⋅Pa−1 at the pressure of 400 Pa for the scale-down case of k = 2, and 0.589 V⋅cm−1⋅Pa−1 at the pressure of 1000 Pa for the scale-down case of k = 5, respectively. The continued proportion of the three maximum values of electric field is equal to 1:2.188:6.13. These proportions of peak values still have a big deviation from the theoretical proportion of 1:2:5, but the reduced electric fields E/p are nearly the same almost everywhere in similar cases at this time as shown in Fig. 4(c).
Fig. 4. (color online) Distributions of reduced electrical field along l1 of three waveguides at different times: (a) 1 × 10−10 s, (b) 1 × 10−6 s, and (c) 1 × 10−2 s.
Figure 5 gives the 2D distributions of electron density in three similar cases. From Figs. 4 and 5, we can see that in the very beginning, free electrons are generated just in the area where the electric field is strong, so the distribution of electrons is similar to that of an electric field. As time goes on, the electron density near the incident terminal keeps growing, and the electric field becomes weaker. This is because when the electron density reaches a certain value, the frequency of electron plasma will be greater than that of incident wave, which prevents the electromagnetic wave from propagating within the plasma.[23,24] The profiles of the 2D distributions of electron density in Fig. 5 are almost the same in three similar cases at any time, which can be a preliminary verification of the similar principle.
Fig. 5. (color online) Spatial distributions of the electron density in similar cases: (a) b = 0.05 m, time = 1 × 10−6 s; (b) b = 0.05 m, time = 1 × 10−2 s; (c) b = 0.025 m, time = 1 × 10−6 s; (d) b = 0.025 m, time = 1 × 10−2 s; (e) b = 0.01 m, time = 1 × 10−6 s; and (f) b = 0.01 m, time = 1 × 10−2 s.
Comparisons of electron densities along the observation line l1 at 1 × 10−6 s and 1 × 10−2 s are given in Fig. 6. It can be observed that the discharge regions are quite coincident with the distributions of electrical field shown in Fig. 4. The peak values of electron density are all located near the peaks of electrical field, which means that the excitations and ionizations keep taking place in these regions. According to the relations shown in Table 1, the continued proportion of the values of electric field should be 1:4:25. Taking peak values of electron density along l1 for example, at 1 × 10−6 s, the electron densities are 1.57 × 1015 m−3, 7.00 × 1015 m−3, and 4.78 × 1016 m−3 respectively, the continued proportion of the peak values of electron density is equal to 1:2.112 :5.522. At 1 × 10−2 s, the peak values of electron density are 9.96 × 1017 m−3, 4.21 × 1018 m−3, and 2.66 × 1019 m−3 respectively, the continued proportion of the peak values of electron density is equal to 1:2.052 :5.172. The ratios of electron density between similar cases satisfy the proportional relations shown in Table 1 very well.
Fig. 6. (color online) Distributions of electron density along observation line1. at (a) 1 × 10−6 s and (b) 1 × 10−2 s.
The electron densities of the three similar cases as a function of time, along the observation lines l1 and l2 in the waveguide, are shown in Fig. 7. It is observed that the ratios of electron density to the squared scale-down factor, i.e., ne/k2, at the corresponding positions of similar cases for every time moment, are almost the same, which indicates that the ionization processes at every corresponding position of the three cases are similar, and the distribution curves are coincident with each other.
Fig. 7. (color online) Electron densities of similar cases at different times at (a) observation line 1, and (b) observation line 2.
The mean electron density of the whole breakdown region and the ratios of mean electron density to squared scale-down factor of the three similar cases at different times (from 1 × 10−6 s∼1 × 10−2 s) are listed in Table 4. The results show that the continued ratio of the similar cases are approximate to 1::. The maximum deviation is 10% less than the accurate relation of similarity principle, and in most of time period, it is about 5% With the time goes on, the deviation turns smaller.
The mean electron temperatures along the central axis of the three similar cases are shown in Fig. 8. As shown in this figure, the trends of the mean electron temperature in different cases are basically the same at any time, which are consistent with the theoretical relations listed in Table 1.
The electron temperature reaches a peak value of about 5 eV in the prototype case and scale-down case (k = 2), and reaches about 5.9 eV in scale-down case (k = 5) after the electrons have been accelerated by high power microwave, these accelerated electrons gain enough energy for collisions and reactions listed in Table 3. Then electron temperature decreases because the energy is transferred to other particles rapidly due to frequent inelastic collisions as time goes on. Significant low mean electron temperature at about 0.9 eV is observed in the bulk plasma for all three cases, because the electrons move toward the incident waveguide port as time goes on and reflect most of the incident power. Hence, nearly all the incident energy is blocked in the front part of waveguide. So most of electrons in the remaining part of the waveguide can only gain very limited energy from incident microwave. For the low-pressure microwave breakdown, the generation of the charged particles is mainly determined by the electron ionization, which depends mainly on the electron temperature. Therefore, if the same trend of electron temperature between similar cases is observed, the same energy Eλ will be obtained and the discharges in similar cases will be comparable to each other. Thus, the similarity principle relation is also valid here in the microwave breakdown at low pressure.
Table 4.
Table 4.
Table 4.
Densities of electrons in similar cases and ratios.
.
Time
Prototype (k = 1)
Scale-down case (k = 2)
Scale-down case (k = 5)
Ratio
1 × 10−6
4.09 × 1014
1.51 × 1015
8.37 × 1015
1:1.922:4.522
1 × 10−5
3.90 × 1015
1.45 × 1016
7.37 × 1016
1:1.932:4.352
1 × 10−4
2.89 × 1016
1.38 × 1017
5.81 × 1017
1:1.932:4.352
1 × 10−3
1.85 × 1017
8.21 × 1017
4.91 × 1018
1:2.112:5.152
1 × 10−2
1.92 × 1017
7.88 × 1017
4.90 × 1018
1:2.022:5.052
Table 4.
Densities of electrons in similar cases and ratios.
Fig. 8. (color online) Time-dependent mean electron temperatures along the central axis in similar cases.
The similarity principle of the microwave plasma at the steady-state is the focus of this study. The relations of plasma parameters at steady-state among similar cases are checked and confirmed with the similarity theory.
The results discussed above are in good consistence with similarity relations at the steady state of plasma, even though there are clear deviations in transient state. This phenomenon may be caused by the nonlinear processes such as stepwise ionizations in the breakdown process, of which the creation rate of corresponding particles progressively deviates from the prediction of similarity principle, and the deviation turns bigger as the scale-up factor k increases.[25,26] The detailed precondition and application scope of the similarity principle in microwave band will be conducted in in our future work.
5. Conclusions
In this work, the correctness of similarity principle in microwave breakdown is verified, and the premises of similarity principle in microwave breakdown are given. A 2D fluid model of argon plasma is established and three similar cases with a continued proportion of 1:2:5 in size are studied. By comparing the breakdown parameters in these similar cases, the proportional relations in microwave breakdown are found. The electron density distributions and mean electron temperatures of the three cases are approximately the same, the electric field is proportional to the scale-down factor k, and the electron density and ion density are proportional to the squared scale-down factor k2. The results show that in microwave breakdown, the proportional relations of the similarity principle are similar to those in DC-discharges, but with new premises.
