To model the gas heating effect without considering convective buoyancy effect, a heat conduction equation is added to couple with the fluid modeling equations. For
radial one-dimensional case, the heat conduction equation can be written as
where k is the thermal conductivity of background xenon gas, which is generally a function of gas temperature Tg [Hanley, 1973] and P(r) is a summation of ion power absorption through ion Ohmic heating and the elastic collision energy loss of electrons with background gas. Note the electron energy loss due to elastic collision is often ignored in a low-pressure discharge; however, it is important in a nearly atmospheric-pressure discharge like the xenon excimer discharge.
To obtain the spatial distribution of ionization rate we need to know the spatial distribution of background gas density, which is not uniform because of gas heating.
In the current study, we can obtain this important information through the coupling solution of the ideal gas law, uniform pressure distribution and conservation of mass in the radial direction. Firstly, uniform background gas pressure distribution is assumed since no convective flow along the radial direction, which leads to
( ) ( ) constant
g g
n r T r = regardless of the radial position. Secondly, conservation of mass
holds in the the radial position since the lamp is a close system without any flow in and out during operation, which leads to ( ) 2 0 ( )
(=8 mm) and r0 (=12.5 mm) is the inner and outer radii of the lamp, respectively, and Nb is the total number of molecules of the background gas that be directly be obtained
from the initial condition before turning on the power supply of the lamp.
6.2 Simulation Conditions
Employed xenon plasma chemistry and lamp configuration are the same as those in Chapter 3 except the discharge gap (=r0-ri) is changed to be 4.5 mm (ri=0.8 mm, r0=1.25 mm) for the purpose of comparison with experimental data as described later.
Radial one-dimensional computational domain extends from r=0 through r0. The
temperature at the outer electrode (r=r0) is fixed (=473 °K) based on the measurements [Lu, 2008] and the Neumann type boundary condition is employed at the axis (r=0) because of symmetry. The frequency f of the pulsed-voltage power supply is fixed at 60 kHz and the xenon gas pressure p is in the range of 100-700 torr throughout the study.
6.3 Results and Discussion
Figure 6-1 shows the simulation data of P172 (power emission of 172 nm line) and the measured illuminance under the near atmospheric-pressure condition in the range of 100-800 torr. Note that the direct comparison is difficult because of the fast decay of 172 nm VUV in space under the near atmospheric-pressure condition in the measurement and the simplified geometry employed in the current simulation.
Experimental results show that the discharge cannot even sustain if the gas pressure is larger than 440 torr. The simulations without consideration of gas heating show that
P172 levels off as the gas pressure exceeds 600 torr and the discharge extinguishes as the gas pressure is approximately 800 torr. The results also show that P172 are essentially the same as pressure is less than 500 torr for both the cases with and without consideration of gas heating in the simulations. However, the simulations show that, by including gas heating effect, P172 begins to decrease rapidly at p=500 torr and the discharge extinguishes as p³ 513 torr. These observations show that by
considering gas heating effect in the fluid modeling we can explain reasonably the extinguishment of the xenon discharge as the pressure exceeds some threshold. Two mechanisms about the extinguishment of the xenon discharge are explained in detail next.
Figure 6-2 shows spatial distributions of xenon gas temperature across the
discharge gap at various gas pressures. In the inner sheath region, the gas temperature is much higher than that in the bulk region and can reach up to 640°K at the inner
dielectric tube surface. Dramatic increase of gas temperature in this region is mainly caused by the large ionic Ohmic heating due to large electric field in the sheath, in which the Ohmic heating contributes most to the gas heating that is clearly shown in Figure 6-3. In addition, distributions of gas temperature in this sheath region are essentially the same at various background pressures. In the inner pre-sheath region (r
≈0.854-0.91 cm), the variation of gas temperature is slightly higher than that in the
bulk region. Although the influence of ionic Ohmic heating weakens in this region, the influence of energy loss of electron elastic collision with background natural gas becomes important in this region (see Figure 6-3). In the bulk region (r ≈0.86-1.22
cm), the gas temperature increases with increasing pressure mostly through absorbing the energy loss of electron elastic collision with background natural gas, as can be clearly shown in Figure 6-3. However, the increase rate of gas temperature with increasing background pressure jumps dramatically from 500 to 510 torr, which means that the gas heating is pronounced near this pressure.
Under the same gas heating source, Figure 6-4 shows the spatial distributions of xenon gas temperature with constant thermal conductivity across the discharge gap at various gas pressures. Interestingly, we have found opposite gas temperature distribution in the bulk region at various pressures. Tg is maximum in the bulk region at 100 torr if k=const due to the high variation of Tg in the sheath region. On the contrary, Tg is minimum in the bulk region at the case of 100 torr if k=f(Tg). Thus, thermal conductivity plays an important role in determining the temperature distribution. Figure 6-5 shows the variation of thermal conductivity in the region of Tg
= 400 – 700 K as k=f(Tg). The value of thermal conductivity increases from 0.007 to 0.012 (W/mK) with increasing gas temperature. When the value of thermal conductivity becomes larger, more heating source can be taken away from the outer
lamp tube to suppress the rise of gas temperature. Thus, at 100 torr as k=f(Tg), the increase of Tg in the outer sheath is lower than that as k=const. The effect of gas heating with k=f(Tg) in the bulk region increases with increasing background pressure.
Figure 6-6 shows the distribution of Tg at r =1 cm in the range of 100 – 513 torr.
Gas temperature rises from 496 to 531 (K) with increasing pressure. After reaching p
=510 torr, the bulk temperatures are almost the same. When the gas pressure exceeds 513 torr, discharge can not sustain, which the reason will be explained in the next few paragraphs.
Figure 6-7 shows spatial distributions of gas temperature and number density
across the discharge gap at 510 torr along with a horizontal line showing the gas number density that is the case without considering gas heating effect. It shows that the gas number density increases from the inner to the outer side in the gap because of the assumption of constant pressure inside the gap. The increase of background gas number density at the outer portion (> ~0.95 cm) enhances the three-body ion conversion from Xe+ to Xe2+ (No. 18 in Table 3-1), which further promotes the recombination reaction of Xe2+ and electrons that destroys electrons rapidly (No. 7 in Table 3-1). This can be further explained in more detail next.
Figure 6-8 shows the comparison of spatial averaged distributions of charged
species within the pre-breakdown and breakdown periods at 510 torr between the cases with gas heating and without gas heating. It clearly shows that both the number densities of electron and Xe2+ without gas heating are much larger than those with gas heating. Also there is almost no phase delay between the rising electron and Xe2+
number densities with gas heating during the breakdown period. Unlike the case without gas heating the abundant Xe2+ promotes the reaction of e- Xe2+ recombination, which eventually extinguishes the discharge. This can be demonstrated more clearly in Figure 6-9, which shows the comparison of cycle-averaged distributions of the source terms of e-Xe2+ recombination and Xe+-to-Xe2+ ion conversion, respectively. It shows that the source terms of the above two reaction channels are more pronounced in the region of r≈1-1.15 cm where the gas breakdown occurs in the first half cycle.
Figure 6-10 presents the spatial averaged distributions of electron temperature
(Te) and electron number density (ne) over a cycle in the range of 100-510 torr. It shows that the averaged electron temperature decreases with increasing xenon gas pressure because of electron energy loss due to elastic collision with background gas.
In the breakdown period where the electron number density rises up very rapidly, the averaged electron number density reach a maximum value at 500 torr and then becomes approximately the same with further increase of pressure. This leads to the existence of maximum electron power absorption at 500 torr as shown in Figure 6-11,
mainly because of the increasing electron energy loss due to elastic collision with background xenon atoms (Figure 6-12). From Figure 6-11 and 6-12, we can find the proportion of electron energy loss due to elastic collision in electron power absorption increases with increasing pressure, especially at 510 torr. As the background pressure exceeds 510 torr, fewer electrons as well as less electron energy can be utilized for the ionization which leads to the extinguishment of the xenon discharge as found in the current study.
6.4 Brief Summary of This Chapter
Major findings of this study in this chapter are summarized briefly as follows:
1. The simulations show that by including gas heating P172 begins to decrease rapidly at p= 500 torr and extinguishes as p> 513 torr. These observations show that by considering gas heating in the fluid modeling one can explain reasonably the observed extinguishment of the xenon discharge as the pressure exceeds some threshold as found in the experiments.
2. The major mechanisms of the above phenomena are described as follows: 1) Increasing pressure leads to higher gas heating because of increasing electron energy loss through the elastic collision with xenon atoms in the bulk region; and 2) The above leads to higher gas density at outer region of the gap (r≈ 1.1~1.16
cm), because of heat conduction and uniform pressure distribution, as compared
to the case without gas heating which promotes the three-body Xe+-to-Xe2+ ion conversion and e-Xe2+ recombination that greatly reduces the plasma density as pressure exceeds some threshold.
Chapter 7
Conclusion and Recommendations of Future Work
In this chapter, we first present the major findings in this thesis and then followed by the recommendations of future work in the research of xenon excimer discharges.