Numerical study on the gas heating mechanism in pulse-modulated radio-frequency glow discharge
School of Physics and Optoelectronic Engineering, Key Laboratory of Materials Modification by Laser, Ion and Electron Beams, Chinese Ministry of Education, Dalian University of Technology, Dalian 116023, China
† Corresponding author. E-mail:
wangdez@dlut.edu.cn
1. IntroductionRecently, there has been increased interest in nonthermal atmospheric pressure plasmas because of the capability of producing reactive species without expensive vacuum systems.[1–4] Among all kinds of power sources, the ratio-frequency (rf) power is almost the favorite one due to its advantage of lower breakdown voltage, higher density of reactive species, and easier generation of uniform glow plasmas.[5,6] However, its applications were restricted due to the relatively higher gas heating power compared to low frequency discharges.[7]
Pulse-modulated rf plasmas at high pressures have drawn intensive attention because they possess not only the trait of high density of reactive species of rf voltage power sources but also good control of plasma parameters.[8] Correspondingly, the discharge characteristics varying with modulated pulse frequencies and pulse widths[9] were studied experimentally and numerically in helium respectively. Some other works were also carried out, such as self-organized patterns[10] and the metastable particles in this discharge apparatus.[11] Recently, several groups claimed experimentally that pulse-modulation of the rf power could significantly change the gas temperature (
).[12–14]
Theoretical models simulating
for a self-consistent, non-equilibrium plasma discharge have been developed over the last 50 years. Pytte and Winsor[15] gave a simple two-temperature theoretical model for atmospheric helium arc plasma in a cylindrical duct. In the analysis, the radiation was ignored and plasma compositions were obtained from the Saha equation evaluated at the electron temperature
. In 1991, a non-equilibrium model was developed to predict the two-dimensional flow, electron and heavy particle temperatures, and number density distributions in cascaded arcs of monatomic gases.[16] Recently, Sakiyama used a one-dimensional non-equilibrium model to simulate the continuous wave rf atmosphere pressure glow discharge (APGD). He compared the results of considering
with and without the energy conservation equation for heavy particles, and found that there was little difference between them.[17] The
at the atmospheric pressure conditions was measured experimentally in 2008.[18] In 2012, a two-temperature non-equilibrium chemical model of the commercial code CFD-ACE+[19] was used to describe the tungsten inert gas arcs at atmospheric pressure. The energy conservation equations of electrons and heavy particles were treated separately, including the reaction heat terms and particle density gradients in this model, by which the heat transfer was described self-consistently. The system descriptions for the mechanism of gas heating can also be found elsewhere,[20,21] while the
in the discharge driven by pulse-modulated rf power at the atmospheric pressure has not been investigated quantitatively before.
In this paper, the
characteristics in the pulse-modulated rf glow discharge at atmospheric pressure conditions are investigated and the gas heating mechanisms are revealed with a one-dimensional two-temperature fluid non-equilibrium model. The spatiotemporal graphs of
in consistent rf discharge and pulse-modulated rf discharge are compared, then the
feature is also described. Additionally, the influences of the duty cycle on
are studied. Finally, the discharge mode changing from α to γ mode in the pulse-modulated rf discharge is investigated and the gas heating mechanisms are analyzed.
2. Numerical modelThe discharge is generated between two plane electrodes at atmospheric pressure. The pulse modulated rf voltage is applied to the upper electrode while the lower electrode is grounded. Helium is taken as the working gas, and five species are considered in the model: electrons (e), atomic ions (
), molecular ions (
), atomic metastable (
), and molecular metastable (
). The particle densities are governed by the continuity equations
 | (1) |
where
is the particle’s number density,
is the sum of source and loss terms of specific species resulting from series of reactions which are summarized in Table
1, and the subscripts e, i, and m represent electron, ion, and metastable, respectively.
The particle flow density
is computed via the diffusion-drift approximation
 | (2) |
 | (3) |
where
μ and
D are the mobility and diffusion coefficients, which have been listed distinctly in Table
1 in Ref [
28] as a function of gas density and gas temperature. The mobility of charged species is assumed to be constant, and the diffusivity is assumed to follow Einstein’s relation. The electrical field
is obtained by solving the current conservation equation instead of solving Poisson’s equation
 | (4) |
where
is the conductive current density calculated by
, with
being the unit charge of
is the total current density, and
is the permittivity, taking either
in the discharge gap or
in the dielectric barrier layer. The energy conservation equation for the electrons reads
 | (5) |
where
is Boltzmann’s constant,
is the electron temperature,
and
are the masses of electron and dominant background gas species, respectively,
is the electron momentum transfer collision frequency with the background gas,
is the energy lost per electron in an inelastic collision, and
is the rate-of-progress of reaction
. The second to the fifth terms on the left-hand side of Eq. (
5) are the electron convection and conduction term, the electron Joule heating term, the rate of electron energy loss due to elastic collisions with heavy particles, and the energy exchange term between the electrons and the heavy species through inelastic collision, respectively.
[29]
The
is solved by the equation
 | (6) |
where
is the heavy-particle number density (the sum of ions and neutral atoms),
is the thermal conductivity of the gas mixture, and
is the bond or ionization energy of inelastic collision with heavy particles. The third term at the left-hand side of Eq. (
5) represents the thermal flow of the gas. The three terms at the right-hand side of the equation are the contributions to the gas heating due to elastic collisions with electrons, the energy loss due to inelastic collision with heavy particles, and the effective ion Joule heating, respectively.
[30,31] The flow chart for coupling calculation of the equations is given in Fig.
1, and the equations above are solved numerically with explicit S–G scheme.
[32]
3. Results and discussionThe simulation parameters are chosen as follows: the discharge gap is fixed at 2 mm, and the relative permittivity constant
is set to 7.5. The gas pressure is 760 Torr, and the initial value of
is set to 300 K. The continuity boundary conditions are applied at the electrodes for the gas temperature. The secondary electron emission coefficient is assumed to be 0.01. The frequency of the rf power source is 13.56 MHz while the external voltage amplitude varies from 500 V to 600 V. The repetitive frequency of the pulse varies from 12.5 kHz to 100 kHz, and its width is set between
and
. The initial densities of electrons and atomic ions are both set to
, and the densities of the other charged particles and the metastable species are
in the whole discharge space. All the results presented in this paper are calculated after the discharge reaches fully periodically steady state.
3.2. Gas temperature of pulse-modulated rf dischargeIn order to assess the gas heating controlled by the modulated pulse source, we simulate the
with the same amplitude and frequency of the rf power as those (570 V, 13.56 MHz) in Subsection 3.1. The frequency of the pulse voltage is 12.5 kHz; the duty cycle of the power-on period is 30%, i.e., the width of one power-on time is set to
. After the discharge reaches its steady state, the waveforms of the driven voltage and discharge current density during about 1.5 pulse period are shown in Fig. 3. It can be seen from the figure that after the rf power is turned on, the discharge current density rises from almost zero to 65.9 mA/cm2, while the discharge current density drops gradually to zero after the pulse is switched off.
The spatiotemporal distribution of
in pulse-modulated rf discharge during one pulse cycle is given in Fig. 4(a). It can be seen that, with 30% duty cycle, the maximal of
is 328.4 K, which is 110.4 K lower than that in the corresponding continuous wave discharge shown in Fig. 2(a). It means that
during the discharge could be significantly lowered by the pulse-modulation rf power; the reason is further explained later in Fig. 5. The spatiotemporal distribution of
in this case is given in Fig. 4(b). The peak value of
is 3.50 eV, which is nearly equal to the value of the consistent rf discharge shown in Fig. 2(b). We must note from the figures that
reaches its maximum after the power is turned on and drops abruptly after the power is off, which differs with the characteristics of
. It can be explained by the fact that the electrons gain and lose energy easily and quickly due to the relatively small mass compared to the ions.
Figure 5(a) shows the time averaged contributions of three gas heating mechanisms for the consistent rf discharge, i.e., the ion Joule heating, heating due to elastic collisions, and heating due to heavy particle reactions. It can be seen that the elastic collision (33.44 W/cm3) is dominant in the central region (light cyan background) of the discharge; the ion Joule heating (23.28 W/cm3) has a peak value in the sheath region (light gray background) near the electrode; while the heating due to the heavy particle reactions (23.33 W/cm3) plays the most important role in the transition region between the sheath and the central regions. Figure 5(b) shows the same three mechanisms of
for the rf discharge with 30% duty cycle. We can see that the three components have similar spatial structures as those in the consistent rf discharge, while their magnitudes drop obviously. Being modulated by the pulse source, the maximum of heating due to elastic collisions in the central region is 4.94 W/cm3. The peak value of ion Joule heating in the sheath region is 2.48 W/cm3. The heating power due to heavy particles in the transition region reaches 3.55 W/cm3. That is to say, the peak values of the heating due to elastic collisions, ion Joule heating, and heavy particle collisions decrease about 28.50 W/cm3, 20.80 W/cm3, and 19.78 W/cm3, and drop to 14.8%, 10.7%, and 15.2%, respectively. It can be concluded that modulated by the pulse power, each of the components contributing to the gas heating changes considerably, while the elastic collision plays a more important role on the gas temperature adjustment.
Next, we keep the frequency and amplitude of the rf voltage fixed and change the pulse frequency. The peak values of
and electron density (
) for three pulse frequencies (12.5 kHz, 50 kHz, 100 kHz) are shown in Fig. 6. As for the f = 12.5 kHz case (triangle symbols),
when the duty cycle is 100%, then it decreases almost linearly with the reduction of the duty cycle,
when the duty cycle is 10%. The gas temperature decreases about 120 K, which is commonly beneficial to the material in the industrial applications. It must be noted in addition that
would decrease below
if the duty cycle of the pulse source is adjusted lower than a specific duty cycle (10.2%), which means that the discharge may be changed from the glow mode to the Townsend mode. There is one ideal value of the duty cycle by which both
can be adjusted and the glow mode can be achieved. It can also be seen that the ideal duty cycles for the cases of 50 kHz and 100 kHz are 26.5% and 37.2%, respectively.
3.3. Gas temperatures of different modes in the pulse-modulated rf dischargeIn order to investigate the
influenced by the external voltage, in this part, we keep all the parameters the same as those in Subsection 3.2 except the rf voltage amplitude. The parameters are as follows: the amplitude of the rf voltage varies from 510 V to 600 V while its frequency is fixed at 13.56 MHz. The frequency of the pulse voltage is 12.5 kHz, and the duty cycle of the power-on period is 50%, i.e., the width of the power-on is set to
. Figure 7(a) shows the discharge current density changes with the external voltage amplitude. The current density increases almost linearly with the rise of the voltage, while the value increases abruptly when the voltage amplitude exceeds 585 V, which is the same as the phenomenon of α–γ mode transition.[17] In the first mode, the
mode, the discharge current density is relatively low and the bulk plasma electrons obtain energy due to the sheath expansion. In the second mode, the
mode, the discharge current density is relatively high, the secondary electrons emitted by the cathode under ion bombardment in the sheath region play an important role in sustaining the discharge.
Figure 7(b) describes the peak value of
increasing with the current density. It is obvious that the higher current leads to higher gas temperature under the same input voltage. Lower temperature can be obtained in the pulse-modulated rf glow discharge than that in the rf plasma discharge. With the same current density (for example 90 mA/cm2 in the figure), the peak value of
in the pulse-modulated rf glow discharge (line with round symbols) is 390 K and the value in the consistent rf glow discharge (line with square symbols) is 608 K. Modulated by the pulse source with 12.5 kHz frequency and 50% duty cycle, the
is lowered by 218 K. This phenomenon has been observed in the experiment before.[2]
Figure 8(a) and 9(a) show the spatial distribution of
when the applied voltage amplitude is set to 570 V and 585 V, respectively. When
, one peak of
locates at the middle of the discharge gap; when
, two peaks of
appear near the electrodes. Moreover, the latter one (362 K) is higher than the former one (355 K).
In order to investigate the mechanism of the transition from
to
mode, we describe the contributions of the three mechanisms to the gas heating when the rf voltage amplitude is set to 570 V and 585 V in Figs. 8(b) and 9(b), respectively. It can be seen in the former case that the elastic collision (9.42 W/cm3) is dominant in the central region of the discharge. The ion Joule heating (5.28 W/cm3) has the peak value in the sheath region, while the heating due to the heavy particle reactions (6.94 W/cm3) plays the most important role in the transition region between the sheath and the central regions. In the latter case, the maximum of heating due to elastic collisions in the central region is 10.43 W/cm3, the ion Joule heating in the sheath region is 7.42 W/cm3, and the heating due to heavy particles in the transition region is 8.00 W/cm3. That is to say, the peak values of the heating due to elastic collisions, ion Joule heating, and heavy particle collisions increase about 1.01 W/cm3, 2.14 W/cm3, and 1.06 W/cm3 by quantity, and 10.7%, 40.5%, and 15.3% by proportion, respectively. It can be concluded that for the transition from α to γ mode in pulse modulated rf discharge, each of the three contributions to the gas heating increases, while the ion Joule heating is more important during the mode transition.
