Numerical and experimental study of a warming up effect of an underexpanded rarefied RF plasma jet outflowing into a flooded area
A Yu Shemakhin1, V S Zheltukhin2 and A A Khubatkhuzin2
1 Kazan Federal University, Kremlyovskaya 18, Kazan, Tatarstan 420008, Russia
2 Kazan National Research Technological University, Karl Marx Street 68, Kazan, Republic of Tatarstan 420015, Russia
A hybrid mathematical model is described of the low-pressure (13.3-133 Pa) RF plasma flow in the
transition regime at Knudsen 8 × 1 −3≤ Kn ≤ 7 × 1 −2 and carrier gas nozzle pressure ratio n = 10. The model is based on the statistical approach for the neutral component and the continuum model for electron and ion components of the RF plasma. Results of plasma flow calculations both in a free flow and in a stream with a sample at a prescribed electric field are described. It was found that the stream is warming up in the mixing zone which was confirmed by the correlation of numerical and experimental results.
RF plasma discharges at low pressures (P = 13.3 − 133 Pa) is used successfully for the modification of various materials: dielectric, conducting, semiconducting [1–5]. The plasma has the following properties: ionization degree is from 10−7 up to 10−5, electron density ne is from 1015 up to 1019 m−3, electron temperature Te is from 1 up to 4 eV, temperature of atoms and ions Ta is in range of (3 − 4) × 103 K in the plasma bunch and the ones is in range of Ta = (3.2 − 10) × 102 K in a plasma stream.
A seriously feature of the low pressure RF plasma stream is a transient flow mode between
the free-molecule flow and static continuum. When the Knudsen’s parameter is in range 8 × 10−3 ≤ Kn ≤ 7 × 10−2 as in under study case, using of Navier-Stokes equation is not correct for flow description [6,7]. At the same time, the electron and ion gases satisfy continuity, since their movement is determined not only to offset the gas flow, but also the action of the Coulomb force that prevents the separation of charges .
A direct statistical Monte-Carlo (DSMC) modeling [9–11] is widely used for the numerical solution of problems of rarefied gas dynamics. The method is based on the splitting of the Boltzmann’s equation on movie and collision processes that allows us to describe the gas- dynamic processes in the transition mode for neutral sparse . Heating of atoms and ions in a low-temperature plasma occurs mainly due to elastic collisions of ones with electrons, which is equivalent to the presence of an allocated heat source in the plasma stream. Therefore, a modification of Bird’s method for calculation of rarefied RF plasma flows and concordance of ones with the continuous model of charged particles is required.
A hybrid mathematical model, combining a kinetic model for the carrier gas flow at Knudsen 8 × 10−3 ≤ Kn ≤ 7 × 10−2 in [13–15] and the continuous model for charged particles is developed in . The aim of this work is studying of the effect of low pressure RF plasma jet overheating and comparing the numerical results with experimental data.
2. Mathematical model of low pressure RF plasmas stream
Mathematical model of the low pressure RF plasma stream [13–15] is constructed by neglecting the Hall effect, electron pressure gradient, the radiation energy loss , the electron attachment, the excitation of atoms, bulk recombination, formation of multiply charged ions and ions slipping. A direct electron impact as the basic mechanism of charged particles appearance  is assumed. We assume that the ion density is equal to the electron ones, the ion temperature coincides with the temperature of the neutral atoms.
Let us designate the radius of the cylindrical vacuum chamber as Rvk, the length of ones as Lvk, the radius of the plasma torch outlet as Rrk, subscripts inlet, outlet, body, walls will used for parameter value on inlet and outlet of the chamber, on the walls of the sample and the vacuum chamber, respectively.
The model includes:
1) the Boltzmann’s transport equation for neutral atoms:
∂f + c · ∂f
+ F- · ∂c
= S(f ), (1)
2) the equation of the electron continuity:
∂ne − div (D
— v n ) = ν n
e a e i e
3) the equation of the electron heating:
∂Te − div
( 5 \
λegrad Te − 2 kB neTeve +
+ kB δνcne (Te − Ta) = σE2 − νineEI . (3) 2
Here, c and r is vectors of the velocity and the coordinates of the atoms, respectively, f (c, r, t)
is the velocity distribution function of neutral atoms, S(f ) is the collision integral,
F- is the
reduced force which effects on neutral atoms at elastic collisions with electrons, ne is electron density, Da is ambipolar diffusion coefficient, νi is the ionization frequency, va is gas velocity, λe is thermal conductivity coefficient of electrons, cp is the heat capacity of the electron gas, νc is elastic collision frequency of electrons and atoms, σ is plasma conductivity, E is electric field strength, E = |E|, EI is ionization potential, kB is the Boltzmann’s constant, δ = me/2ma, me, ma are the electron and the atom masses. Coefficients Da, νi, λe are functions of the electron temperature [14–17],
F = − 1
grad WT , WT =
Ec dV dt, Ec = 2 kB δνcne(Te − Ta).
The system (1)–(3) is considered at the following initial conditions:
Here f0 is Maxwell velocity distribution function. Terms of impermeability on the body and walls boundary surfaces as well as sof t boundary conditions  on the inlet and outlet borders are defined for f (c, r, t).
Equations (1)–(3) is closed by following relations:
va(r, t) =
cf (c, r, t) dc, pa = nakB Ta,
ve = va − (Da/ne)grad ne, σ =
(ν2 + ω2)
ma ( 2
2 2 r 2
c − va
, c =
c f (c, r, t) dc,
Bird’s method is modified to take into consideration the distributed heat source density WT . A two-step iterative process is constructed to solve the problem (1)–(7). On the first step a solution of (1) is found by Bird’s DSMC method  to determine va and Ta. Then these values is used to solve problems (2), (4), (5) and (3), (4), (6). Further, solutions of these problems ne and Te are used to solve the equation 1 taking into account the distributed heat source power density WT . The process is repeated until the maximum of successive approximation ratios becomes less than the specified tolerance. The software package for calculating of low pressure RF plasma flow is developed by using OpenFOAM  environment on Linux OS.
3. Calculations of low pressure RF plasma flow overheating
The gas-dynamic characteristics of low pressure RF plasma undisturbed flow as well as stream with overflowing sample in the vacuum chamber of Rvk = 0.2 m, Lvk = 0.5 m and Rrk = 0.012 m on the center of the base plate is carried out. A cylindrical sample of radius Rb = 0.03 m and a height Lb = 0.02 m located in the plasma jet at a distance Ltb = 0.2 m from the inlet is assumed.
Flow input parameters are the following: the plasma forming gas is argon, gas flow rate G = 0.12 − 0.24 g/s, pressure Pinlet = 35 − 85 Pa, the temperature Tinlet = 400 − 600 K, the degree of ionization δi = 10−4. The initial pressure in the vacuum chamber P0 = 3.5 − 8.5 Pa.
As the results of calculations show, steady state flow is established for t 10−2 s. Distribution
graphs of velocity, temperature and pressure of the carrier gas, the concentration of electrons and the electron temperature generally given in [13–15]. However, that results do not describe the effects of gas overheating in the mixing zone of plasma flow and stationary gas in the vacuum chamber. Figures 1, 2 show the results of calculations of the characteristics of the plasma jet at G = 0.52 g/s, Tinlet = 500 K, gas velocity va,inlet = 1000 m/s, P = 60 Pa, P0 = 6 Pa, which corresponds to the flow pressure ratio n = 10. Curves on figure 1 shows that pressure decreases
at a distance of 0.05 − 0.15 m from the inlet (curves 2, 3) and increases near the sample(curve 4), which is associated with deceleration of the flow. Zone of a lowered pressure is created behind the sample, the pressure is leveled away from the sample, as in the model without sample .
As a result, numerical experiments revealed the heating effect of the jet on the periphery of the stream in the mixing zone of the plasma flow and the stationary gas in vacuum chamber. Curves on figure 2 shows that at a distance of 0.01 m from the inlet, the maximum temperature
Figure 1. Radial distribution of the pressure p in the cross-section of stream flowing around the sample.
Figure 3. Flange with thermocou- ples for the pilot study of the radial distribution of the temperature of the gas stream.
Figure 2. Ta radial temperature distribution in the cross section of the jet in the presence of the sample at different distances from the inlet.
Figure 4. Radial temperature dis- tribution of the carrier gas at the outlet of the plasmatron (experi- ment).
(∼ 650 K) is created on the periphery of the jet, and the minimum temperature (∼ 520 K) is observed at the center (curve 1).
The radial temperature profile is aligned with the distance from the jet. Thus, at a distance of
0.05 m from the inlet difference between temperature and the ones in the mixing zone δT ∼ 460 K, but on the flow axis δT ∼ 350 (curve 2) The cause of the gas overheating is most likely a sharp deceleration of the gas atoms at z = 0.15 m due to a collision with a stationary gas. Radial temperature profile is leveled at a distance of z = 0.15 m from the inlet closer to a bell-shaped profile; a maximum temperature Ta = 650 ◦C reaches at the center of flow (curve 3) . Gas is cooled to 570 ◦C at distance of 0.2 m when flow is collided with the sample, and at a distance from the axis ∼ 0.03 m a sharp temperature drop on ∼ 30 ◦C (curve 4). This effect is connected with the interaction of several factors. Firstly, the gas is cooled due to the flow expansion and interaction with the colder surface of constant temperature (300 K), which corresponds to the cooled sample. In the second, braking flow leads to gas heating, whereby the radial profile of the peak in the center of the jet becomes more acute as compared with the curve 3, and a gap at the distance of 0.3 m from the axis.
The pattern on the graph of the temperature distribution again appears pronounced dip (temperature difference between the axis and the edges of the stream is 100 K), probably caused by a result of cooling gas flow around the sample cold. The temperature distribution over the cross section is completely aligned with the jet at a distance of 0.3 m from the inlet, the profile becomes flattened. Maximum gas temperature T a = 420 K is achieved on the flow axis. The cooling gas flow caused by the expansion and uniformity of the profile of the cross section of the jet caused by heating due to elastic collisions of electrons with atoms, since the distribution of the electron temperature, as shown in [14, 15], in a section of the jet cross almost evenly.
4. Comparison of numerical results with experimental data
Experimental measurements of the plasma temperature at the exit of the plasma torch with thermocouples is made to confirm the revealed laws. The maximum allowable temperature was 1000 ◦C. The experiments were conducted at the facility for jet of RF inductive discharge, described in the works . Installation allows generating the RF plasma flows with the parameters specified in the introduction to this paper. Thermocouples were placed on the flange with a certain pitch along the radius (Figure 3). The flange is placed at the inlet to the vacuum chamber. Thermocouples measuring the temperature was allowed along the jet radius at a small distance from the inlet. The experiment revealed that temperature on the second thermocouple is higher than the ones in the jet center at a inductor current more than 1.8 A, a pressure in the vacuum chamber (for pumping) p = 60 Pa (figure 4), which indicates the presence of hot spots. Thus, experiments is confirmed and calculation results.
Peculiarities of the gas temperature distribution in the low pressure RF plasma jet into a vacuum chamber and the sample overflow are defined as a result of the numerical simulation at flow parameter pressure ratio 10. The overheating gas in the mixing zone of the plasma stream and stationary gas in the vacuum chamber is observed in the vicinity of the inlet. The difference between the maximum temperature and the temperature on the flow axis is 110 K. The radial temperature distribution is aligned with the distance from the inlet. In a zone of collision of a stream with the processed sample the jump of temperature caused by braking of a stream is observed. Temperature profile becomes uneven behind a sample that is caused as expansion of a stream, and due to interaction with colder sample. Temperature profile in the cross-cut section of a stream is leveled at distance of 0.3 m from an inlet.
The effect of an overheating up of a plasma stream on the periphery of a stream is consistent
with results of experimental studies.
The work was funded by RFBR, according to the research project No. 16-31-60081 mol a dk (theoretical part) and the Russian Ministry of Education (the basic part of the state task number 2196, the experimental part of the work).
 Abdullin I S, Zheltukhin V S, Sagbiyev I R and Shayekhov M F 2007 Modifikaciya nanosloev v vysokochastotnoj plazme ponizhennogo davleniya (Kazan: Izdatel’stvo Kazanskogo tekhnologicheskogo universiteta)
 Khubatkhuzin A A, Abdullin I S, Gatina E B, Zheltukhin V S and Shemakhin A Y 2012 Vestnik Kazanskogo tekhnologicheskogo universiteta 15 72
 Khubatkhuzin A A, Abdullin I S, Bashkirtsev A A and Gatina E B 2012 Vestnik Kazanskogo tekhnologicheskogo universiteta 15 71
 Shatayeva D R, Kulevtsov G N and Abdullin I S 2014 Vestnik Kazanskogo tekhnologicheskogo universiteta
 I S Miftakhov A O, Fadeyev and Shayekhov M F 2013 Vestnik Kazanskogo tekhnologicheskogo universiteta
 Belotserkovsky O M 1984 Chislennoe modelirovanie v mekhanike sploshnykh sred (Moscow: Nauka. Glavnaya redakcija physico-mathematicheskoj literatury)
 Dulov V G and Lukyanov G A 1984 Gazodinamika processov istecheniya (Moscow: Nauka)  Mitchner M and Kruger C H 1973 Partially Ionized Gases (NY: John Wiley and Sons)
 Bird G A 1976 Molecular gas dynamics and the direct simulation of gas flows (Clarendon Press: Oxford)  Pechatnikov Y M 2002 Vakuumnaya tehnika i tehnologiya 12 227
 Pechatnikov Y M Prikladnaya fizika 2 19
 Scheller G, Gottscho R A, Intrator T and Graves D B J. Appl. Phys. 64 4384
 Zheltukhin V S and Shemakhin A Y Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-
Matematicheskie Nauki 4 135
 Zheltukhin V S and Shemakhin A Y Mathematical models and computer simulations 6 101
 Abdullin I S, Zheltukhin V S, Khubatkhuzin A A and Shemakhin A Y Matematicheskoe modelirovanie gazodinamiki strujnykh techenij vysokochastotnoj plazmy ponizhennogo davleniya
 Raizer Y P Gas discharge physics (Berlin Heidelberg)
 Lymberopoulos D and Economou D J J. Appl. Phys. 73 3668
 OpenFOAM OpenFOAM Foundation. Free Open Source CFD 2011–2016 http://www.openfoam.org