<figure><img src="image_35.png" /><figcaption>**Fig. 5.1:** Profile of the H atom axial velocity component along the jet centerline at 20 Pa (open square) and 100 Pa (solid triangle) background pressure. Also indicated is the local speed of sound for an Ar-H₂ mixture at $p_{\text{back}}$ = 20 Pa (circle), as it is determined from the measured parallel temperature $T_{\|}$.</figcaption></figure>

a background pressure ($p_{\text{back}}$) of 20 Pa, $M = 2.4$ at the arc nozzle exit and $M = 6$ in front of the stationary shock wave. A Mach number higher than 1 at the source exit ($M = 1$ correspond to a sonic orifice) is expected in view of the nozzle geometry. Then the velocity drops because of collisions between jet particles and background particles which results in the formation of a stationary shock front whose thickness is in the order of one local mean free path for H-Ar and Ar-Ar collisions. At $p_{\text{back}}$ = 20 Pa the Mach disk (end of the shock wave) is located around 100 mm from the arc nozzle in agreement with the result given by a theoretical expression for the Mach disk location [9, 15]. As can be seen from Fig. 5.1, the deceleration of the flow in the subsonic domain depends on the background pressure, because particles from the jet will lose their forward momentum more efficiently at higher pressure. The shape of the hydrogen drift velocity profile is somewhat quite similar to that of argon measured in the same conditions (see Fig. 3.8). It implies that the flow characteristics of the plasma jet are to a large extent determined by the argon atoms. However the fact that the H velocity is larger than the Ar velocity reveals that H and Ar are not perfectly coupled during the plasma expansion.
<figure><img src="image_36.png" /><figcaption>**Fig. 5.2:** Perpendicular temperature $T_⊥$ of H atoms as a function of the distance from the nozzle exit at 20 Pa (open square) and 100 Pa (solid triangle) background pressure. Also shown is the axial profile of the parallel temperature $T_‖$ at 20 Pa. The theoretical temperature profile in the supersonic domain is calculated from the Poisson adiabatic law with $\gamma = 1.3$ (solid line).</figcaption></figure>

## 5.4 Perpendicular and parallel temperature

The axial profiles of the perpendicular temperature $T_⊥$ associated with the velocity distribution perpendicular to a stream line of atomic hydrogen, depicted in Fig. 5.2, clearly reveal the presence of a stationary shock front. The shape of the axial temperature profiles complies with the theory which predicts a strong cooling effect in the supersonic domain and a dependence of both the position and the thickness of the shock front on $p_{back}$ [16]. Nevertheless, using Rankine-Hugoniot (R-H) relations [17] and taking a value of 5/3 for the adiabatic exponent $\gamma$, a Mach number ahead of the normal shock front of 2.6 is deduced from the temperature jump over the shock in the 20 Pa case, in disagreement with the value of 6 deduced from the axial velocity profile. This discrepancy might be explained by heat transfer from both the source and the hot background gas into the supersonic region which partly disturbs the effect of cooling in the core of the jet. This disturbance leads to an apparent smaller $\gamma$, meaning that the expansion is not adiabatic. By taking a value of 1.3 for $\gamma$, a Mach number ahead of the shock front of 5 is found and the calculated temperature development in the supersonic domain using the Poisson adiabatic law [16], agrees well with the measurements as can be seen in Fig. 5.2. In the background the gas temperature depends both on the thermal conductivity of the gas mixture and on the power input. The profile along the jet centerline of the parallel temperature $T_‖$ associated
with the velocity distribution parallel to a stream line measured at 20 Pa is also shown in Fig. 5.2. In the supersonic domain, the two temperature components match each other, i.e. the collision frequency is sufficiently high to maintain equilibrium and the continuum flow conditions hold. Just ahead of the stationary shock wave, we observe a departure from equilibrium: $T_{\parallel} > T_{\perp}$. In this region, the local mean free path becomes in the order of the jet radius, i.e. collisions become scarce, and equilibrium can not be preserved. The jet enters the so-called frozen regime: the parallel temperature “freezes” whereas the perpendicular temperature continues to decrease geometrically [9, 18]. Surprisingly, throughout the stationary shock front the deviation from equilibrium subsists whereas the collision frequency increases. This can be a direct consequence of an outward diffusion of H atoms. In order to explain the difference between $T_{\perp}$ and $T_{\parallel}$ within the shock wave, we here anticipate on the results discussed in the next section and in the following chapter. In the shock region, due to the numerous collisions with the background the temperature rises, which simply images an increase in disorder. A large H atom outflow however can be seen as an organized motion of particles in the radial direction which therefore leads to a lower spread of the radial velocity component in comparison with the spread in the axial velocity component. In other words, radial losses of H atoms act as a cooling mechanism and as a consequence $T_{\perp} < T_{\parallel}$. Behind the stationary shock wave, i.e. in the subsonic domain, the flow returns to equilibrium, $T_{\perp} \approx T_{\parallel}$, because of mixing with the background gas.

## 5.5 Anomalous H atom transport

### 5.5.1 Density profile along the jet axis

In contrast to the previously mentioned results, the measured H atom density profiles along the axis of the jet, depicted in Fig. 5.3, are in conflict with the theory. Throughout the stationary shock front no discontinuity (or jump) in density is observed in contrast to the Ar profile [10]. In the 20 Pa case the density even decreases over the shock front (Fig. 5.3) whereas the drop in drift velocity should be accompanied by a density jump with a factor of 3.7 according to the R-H relation (and as shown by the calculated profile). At 100 Pa the density is almost constant over the shock. The theoretical H atom density profile $n_H(z)$ along the jet centerline can be calculated from the measured H axial velocity profile $w_H(z)$, see Fig. 5.1, using a simple model for $n_H(z)$ derived from the conservation of the forward H flux. Under the assumption that the particles originates from a point source and flow along straight stream lines, the flux conservation in the supersonic domain reads in the case of a 45° expansion [16]

$$n_H w_H \pi (z + z_o)^2 = \phi_H^o, \qquad (5.1)$$

where $z_o$ is the position of the origin of the expansion ($z_o = 3$ mm, see chapter 6) and $\phi_H^o$ is the H flux in the plasma source. Throughout the stationary shock front, the diameter of the plasma jet is constant. In the subsonic domain the gas flows at a constant pressure and no recombination effects at the wall are taken into account. The expansion of a pure
<figure><img src="image_37.png" /><figcaption>**Fig. 5.3:** Axial density profile of H atom at 20 Pa (open square) and at 100 Pa (solid triangle) background pressure. The lines are drawn to guide the eye. Also shown is the model density profile at 20 Pa (circle) based on the axial velocity data of Fig. 5.1.</figcaption></figure>

Ar plasma can be satisfactorily described by this simple model [10]. The result of the calculation for $p_{\text{back}} = 20$ Pa is also depicted in Fig. 5.3. The observed density profile leads to a non-conservation of the forward flux of atomic hydrogen. Such effects can not be explained by volume recombination of atomic hydrogen because of the too low reaction rates [19]. Therefore the only explanation to this anomalous shock pattern is to consider the fact that H atoms radially escape the supersonic expansion.

### 5.5.2 H atom flux in the shock region

From measured radial profiles of H atom density, temperature and velocity components at $z = 8, 20, 50, 80, \text{ and } 100$ mm, several fluxes can be estimated in order to clarify the situation. From an integration of the product $n_H \times w_H$ over the shock front radius, we determined the forward flux of H atoms entering the shock front $\phi_H^{\text{in}}$ as well as the flux behind the shock front $\phi_H^{\text{out}}$. We found that $\phi_H^{\text{in}} = 3.3 \times 10^{19} \text{ s}^{-1}$ and $\phi_H^{\text{out}} = 3.9 \times 10^{18} \text{ s}^{-1}$ which means a one order of magnitude radial loss of H flux.

The mass focusing effect[9] can not explain this H outflow. The key of the problem is the following: there is a significant density gradient between the interior of the plasma jet and the surroundings, caused by the quasi absence of H in the ambient gas, that drives a radial diffusion of atomic hydrogen. In contrast, when the plasma jet turns from underexpanded into overexpanded, this gradient for Ar is opposite (non reactive species). From radial profiles, an averaged H atom radial flux $\phi_H^{\text{rad}}$ over the shock front can be estimated.
<figure><img src="image_38.png" /><figcaption>**Fig. 5.4:** Radial profile of the density (square), perpendicular temperature (circle), and radial velocity (triangle) of H atoms at $z = 8$ mm for a background pressure of 20 Pa (left) and for 100 Pa (right).</figcaption></figure>
We found $\phi_H^{\text{rad}} = 3.4 \times 10^{19} \text{ s}^{-1}$ which is in quite good agreement with the loss of forward flux inside the stationary shock front. In addition, an estimate of the same flux can be obtained using Fick's diffusion law. This gives $1.1 \times 10^{19} \text{ s}^{-1}$ which strongly encourages the idea of radial outflow induced by a density gradient. The determination of both fluxes is not straightforward and it is explained in detail in chapter 6.

### 5.5.3 Confinement in the jet core

Within the supersonic domain of the jet the density decays slightly faster than predicted by the model which means that it is not only geometrically determined [9, 16]. Moreover, the steepness of the density decay depends on $p_{\text{back}}$ in contrast with the prediction by the expansion theory. Again these effects can be explained by H atoms radially escaping the expansion by a diffusion process.

A study of the beam cross-section at a distance $z = 8$ mm from the arc outlet, of which the results are shown in Fig. 5.4, reveals that the hydrogen diffusion phenomenon is strong and that it occurs already at the beginning of the mixture expansion. The radial profile of both the radial velocity component and the perpendicular temperature reveal the barrel shock structure of the expansion and they show the influence of the background pressure on both the dimension of the jet and on the shock wave structure. The lower absolute H density in the 20 Pa case compared to the 100 Pa case, visible in both Fig. 5.3 and Fig. 5.4, can only mean that in between the nozzle and the measurement position already more H is lost from the plasma jet than in the 100 Pa case.

At low pressure the outward diffusion phenomenon is strong because of the large local mean free path for momentum exchange which creates a diffuse shock structure [20]. The higher density in the wings of the 100 Pa profile (Fig. 5.4) as well as the plateau in the axial density profile at 100 Pa (Fig. 5.3) can then be explained by a slower outward diffusion of H. It just takes more collisions for H to pass the barrel shock in the 100 Pa case than in the 20 Pa case. This means that H atoms are poorly confined within the supersonic domain of the plasma beam, the efficiency of the confinement depending on $p_{\text{back}}$, i.e. on the permeability of the shock structure.

### 5.5.4 Subsonic domain

In the subsonic domain, the H density is still decreasing. After diffusing out of the jet, H atoms travel towards the vessel wall where they can recombine to form molecular hydrogen. The large available wall area combined with the high recombination probability for H on a steel surface causes H to almost vanish from the residual gas. This is confirmed by the measurement of the dissociation degree at 300 mm from the arc outlet (i.e. in the background) which equals to 0,2 % for $p_{\text{back}} = 20$ Pa and to 1 % at 100 Pa. This creates very specific boundary conditions for the H density and results in the formation of large radial H density gradients all over the expansion which are responsible for the diffusion process of H out of the beam.
## 5.6 Conclusions

To conclude, we experimentally demonstrated that the expansion of atomic hydrogen when seeded in Ar can not be entirely described in terms of the adiabatic supersonic expansion theory. On the first hand, heat transfer leads to an apparent smaller adiabatic exponent. On the other hand, the striking absence of a H atom density jump across the shock front arises from a strong outward diffusion of H atoms. This process is driven by large radial density gradients induced by the loss of H atoms from the background gas by wall recombination. The Ar shock structure is partially transparent to H leading to a poor confinement of H atoms even at high pressure and therefore to a non efficient transport mechanism. This leads to a less than optimal use of the chemical potential of the plasma source. The loss of H atoms at the vessel wall should therefore be taken into account in the design of reactors using remote plasmas as an atomic hydrogen source.

## Acknowledgments

The authors gratefully acknowledge Prof. H. F. Döbele for his vital contribution in the initial phase of the two-photon LIF experiments in Eindhoven. This work is part of the research program of the Netherlands Foundation for Fundamental Research on Matter (FOM) and it is also financially supported by the Netherlands Technology Foundation (STW).
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<figure><img src="image_39.png" /></figure>
Chapter 6

Transport of ground-state hydrogen atoms in
an Ar-H₂ plasma expansion <fnref n="1" />

Abstract

The transport of ground state atomic hydrogen in the expansion of a thermal plasma gener-
ated from an Ar-H₂ mixture is studied by means of laser-based diagnostic techniques. The flow
of hydrogen atoms is investigated by two-photon excitation Laser Induced Fluorescence (LIF),
whereas Ar atoms are probed by LIF as well as by UV Rayleigh scattering. The transport of
Ar atoms can be fully understood in terms of a free jet flow; H atoms on the contrary exhibit
an anomalous behaviour. In the course of the plasma expansion, hydrogen atoms decouple from
the argon fluid by a diffusion process as a direct consequence of recombination of H atoms at the
vessel walls. It is shown, on the basis of experimental results, how plasma-surface interactions
can strongly influence the flow pattern of an atomic radical fluid.

<footnote n="1">¹ Adapted from:
S. Mazouffre, M.G.H. Boogaarts, I.S.J. Bakker, P. Vankan, R. Engeln, and D.C. Schram, *Phys. Rev E* **64**,
016401 (2001).</footnote>
## 6.1 Introduction

Understanding the transport of atomic radicals in a plasma expansion, i.e. the flow pattern and the interactions with the surrounding gas, is of interest from several aspects. From a fundamental point of view, a plasma expansion is a very general physical phenomenon which covers a broad range of dimensions, ranging from astrophysical objects [1] to small laser spots [2]. In comparison with the expansion of a neutral gas, plasma expansion is a more complex phenomenon. For instance, different kinds of particles are present (neutral, ion, electron, photon), current and electric fields can be generated, and the flow is often not isentropic. From a technological perspective, due to their high reactivity, atomic radicals play a major role in plasma chemistry and therefore are of relevance for industrial applications like deposition of thin films [3, 4, 5] and surface modification [6].

From a practical point of view it is advantageous to study the physics of expanding plasmas on the intermediate scale, as then the system is well suited for diagnostics. In this contribution, we focus on the transport of ground-state hydrogen atoms in the expansion of a thermal plasma generated from an Ar-H₂ mixture by a DC wall stabilized arc plasma source. Such a plasma jet can serve as an example, and the results obtained may be generalized to other kind of localized plasma expansion, like solar outbursts and laser spots. Atomic hydrogen is of particular interest. Such a light radical plays an important role in the field of nuclear fusion, e.g. Tokamak plasmas [7], as well as in numerous manufacturing techniques employing processing plasmas. Furthermore, H is one of the main components of both stellar and interstellar matter.

In order to clearly understand the dynamics of such a plasma jet, both neutral Ar atoms and H radicals have to be probed in such a way that the plasma flow is not disturbed. The local argon atom density is monitored by means of UV Rayleigh scattering, whereas the Ar velocity and temperature are measured by means of Laser Induced Fluorescence (LIF). The detection of ground-state hydrogen atoms is more cumbersome due to the large energy gap between the electronic ground state and the first excited states. Hydrogen atoms are spatially probed by means of Two-Photon Absorption Laser Induced Fluorescence (TALIF). By applying this method for different positions in the expansion, the H distribution in the plasma jet in terms of density, temperature, and velocity has been completely mapped.

In a foregoing paper [8], we have shown that the transport of H radicals, contrary to that of argon neutrals, can not be entirely described in terms of the well-established free jet flow picture. Hydrogen atoms exhibit a very specific behaviour. Whereas the occurrence of a stationary shock wave can clearly be identified on both the temperature profile and the velocity profile along the jet centerline, there is no density discontinuity throughout the shock front. This leads to a non-conservation of the H atom forward flux. It was stated that H radicals can escape the core of the plasma jet by a diffusion process induced by the presence of large density gradients between the jet and its periphery. The latter arise from the recombination of H at the vessel walls which leads to the formation of H₂ molecules. The H radical anomalous shock wave pattern is analyzed on the basis of detailed measurements, and it will be demonstrated that the hypothesis of outward diffusion controlled by density gradients can fully explain the measurements.
## 6.2 Plasma jet and experimental arrangement

The cascaded arc [9] is operated at a 40 A DC current, and with a cathode-anode voltage of 100 V. In normal condition a gas flow of 3.0 standard liters per minute (slm) Ar and 0.5 slm H₂ is used. The diameter of the arc channel is 3 mm, the stagnation pressure inside the arc is 0.6 atm, and the opening angle of the arc nozzle is 45°. The cascaded arc thermal plasma expands supersonically into a vacuum vessel (length 3 m, diameter 0.36 m) where the background pressure (p<sub>back</sub>) can be varied almost independently from the gas flow from 10 Pa to 1 atm. Because the plasma expands from the orifice of the arc into a region with a finite pressure, a well defined free jet shock wave structure [10, 11, 12] is produced due to the interaction of the supersonic flow with the ambient gas.

In the case of an Ar-H₂ mixture, a fast lowering in the electron density appears soon after the onset of the expansion [13]. At an electron temperature of 1 eV this ionization loss is due to the production of hydrogen atoms by the associative charge exchange reaction

$$Ar^+ + H_2 \rightarrow ArH^+ + H \qquad k = 1.2 \times 10^{-15} \text{ m}^3 \text{ s}^{-1}, \qquad (6.1)$$

followed by the dissociative recombination reaction [14, 15, 16]

$$ArH^+ + e^- \rightarrow Ar + H^* \qquad k \approx 10^{-13} \text{ m}^3 \text{ s}^{-1}. \qquad (6.2)$$

Thus, using a 6:1 Ar-H₂ mixture, at the arc outlet the plasma jet turns into almost a gas jet composed of Ar and H atoms and of H₂ molecules. Both the amounts of H and H₂ depend on the dissociation degree of the source which is very likely to be close to 1 [17]. Beyond the Mach disk, the gas expands subsonically and mixes with the residual background gas. Then the evolution of the flow is determined by the circulation pattern in the vessel, which is a direct consequence of the large residence time (τ<sub>res</sub> ≈ 1 s at 20 Pa). Furthermore, as the diffusion time is much shorter than the residence time (τ<sub>diff,H</sub> ≈ 1 ms at 20 Pa), there is ample time for plasma-wall interactions.

In this experiment ground-state hydrogen atoms are spatially probed by using two-photon absorption laser-induced fluorescence (TALIF) [18, 19, 20]. The experimental method is described elsewhere [20, 21], and only a short overview is presented here. A simplified scheme of the setup is depicted in Fig. 6.1. A tunable 20 Hz Nd:YAG pumped dye laser delivers radiation around 615 nm. The output of the dye laser is frequency-tripled using non-linear optical crystals resulting in 2 mJ of tunable UV light around 205 nm with a measured bandwidth of 0.18 cm<sup>-1</sup>. The UV laser beam is focused either perpendicular or parallel to the plasma expansion axis. Hydrogen atoms are excited with two 205 nm photons from the 1s<sup>2</sup>S ground state to the 3d<sup>2</sup>D and 3s<sup>2</sup>S states. The excitation is monitored by detection of the resulting fluorescence yield on the Balmer-α (H<sub>α</sub>) line at 656 nm using a gated photo-multiplier tube. A narrow bandwidth interference filter is used to isolate the H<sub>α</sub> line from the plasma emission. A slit mask is used to define the detection volume which dimensions are smaller than any gradient scale length. The dye laser frequency is calibrated by the simultaneous recording of the absorption spectrum of molecular iodine.
<figure><img src="image_40.png" /><figcaption>**Fig. 6.1:** Schematic view of the experimental arrangement. The UV laser beam is used to perform both TALIF and Rayleigh scattering measurements. It can be directed either perpendicular or parallel to the jet axis which allows for the measurement of ($T_\perp$, $w_r$) and ($T_\|$, $w_z$) respectively.</figcaption></figure>

From a spectral scan over the two-photon transition, the local H atom density, temperature, and velocity along the laser beam are measured. Note that the measurements are performed in succession. Absolute number densities are obtained by calibrating the TALIF setup by means of a titration reaction with NO₂ in a flow tube reactor [21, 22].

The UV laser beam at 205 nm is also employed to measure the argon heavy particle density by means of Rayleigh scattering [23, 24]. The polarized, scattered radiation at 205 nm is detected at an angle of 90° using a gated solar blind PMT, as shown in Fig. 6.1. A dielectric UV mirror centered on 205 nm ($\Delta\lambda = 10$ nm) is used to filter out the plasma background light. The amount of stray light is reduced by means of a set of diaphragms and by using windows at Brewster's angle. Although the amount of photons at 205 nm is much smaller than the amount of photons that can be generated at for instance 532 nm with our Nd:YAG laser, the number of scattered photons is similar. The enhanced scattering result from a much higher argon differential scattering cross-section at 205 nm; it is about 60 times larger than at 532 nm [25]: $\sigma = 3.6 \times 10^{-30}$ m² at 205 nm and $\sigma = 5.6 \times 10^{-32}$ m² at 532 nm. The current detection limit in an Ar-H₂ plasma is around $10^{20}$ m⁻³: it does not permit to accurately measure the Ar density in the shock region at a background pressure below 40 Pa.
## 6.3 Transport of atoms along the jet centerline

### 6.3.1 Argon atoms

In the case of an expanding argon plasma, it has been demonstrated experimentally [26] that the expansion of neutral atoms can be well understood in terms of an adiabatic supersonic expansion of an ideal gas [10]. Furthermore, the entire axial density profile can be described by a quasi-one dimensional model based on hydrodynamic equations [26], despite the fact that the Knudsen number $Kn$ inside the supersonic domain is rather high. For instance, at a 40 Pa background pressure, $Kn = 0.1$ ahead of the stationary shock wave. Later, this model has been extended to study the case of an expanding argon plasma seeded with a small amount of molecular hydrogen [13].

The development along the jet centerline of the Ar heavy particle density, measured by means of UV Rayleigh scattering for a 6:1 Ar-H₂ mixture, is shown in Fig. 6.2 for two different background pressures. Our data are in agreement with foregoing measurements obtained in the case of a pure Ar plasma [13]. It implies that the Ar flow is not disturbed by the presence of H and H₂. From the conservation laws in stationary state, and under the assumption that the particles originate from a point source and flow along straight stream lines, one can obtain the theoretical expression for the density development along the jet centerline

$$n(z) = n_o \frac{z_{\text{ref}}^2}{(z + z_o)^2} \qquad (6.3)$$

In Eq. 6.3, $n_o$ is the density in the source, $z_{\text{ref}}$ is a scaling length equal to the source nozzle radius in the case of a 45 degrees expansion, and $z_o$ is the position of the virtual point source. This formula is similar to the one used to describe the supersonic expansion of an inviscid gas through a sonic orifice in the field of gas dynamics [10, 27]. The axial density decay starts at some position $z_o$ depending on the source nozzle geometry, e.g. several nozzle diameters in the case of a straight orifice. This decrease in density describes a rarefaction effect due to the increase in the jet cross-section.

The measured Ar density profiles in Fig. 6.2, are fitted according to Eq. 6.3. As can be seen, the experimental data are well described by the theoretical expression. Up to the stationary shock front the density decrease is not affected by a change in $p_{\text{back}}$, meaning that the supersonic domain of the expansion can be considered as a closed domain surrounded by a shock wave structure [11, 26]. From the fit, both length parameters can be estimated. We find that $z_{\text{ref}} = 3.0 \pm 0.2$ mm and that $z_o = 3.0 \pm 0.2$ mm. This means that the virtual point source, i.e. the start of the expansion, is located 3 mm inside the nozzle. This is a direct consequence of the use of a diverging nozzle. The density jump across the normal shock wave, which results from the conservation of the forward flux, can clearly be seen in Fig. 6.2. From the Rankine-Hugoniot relation [28] the compression ratio is found to be 3 at 40 Pa and 4, i.e. the maximum compression ratio, at 100 Pa. In the subsonic region, a slow increase in the Ar density is observed which is due to a cooling of the gas by heat transfer, the static pressure being constant. The velocity of Ar atoms in an expanding
<figure><img src="image_41.png" /><figcaption>**Fig. 6.2:** Axial density profile of Ar neutrals at 42 Pa (open circles) and 100 Pa (solid triangles) in an Ar-H₂ mixture, and at 40 Pa in pure Ar (star).</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>**Fig. 6.3:** Axial velocity component profile along the jet centerline for Ar, H, and D atoms at a 20 Pa background pressure. The decoupling between H(D) and Ar inside the supersonic domain is clearly visible.</figcaption></figure>

Ar-H₂ thermal plasma jet created by a cascaded arc, has recently been determined by measuring the Doppler shift in a LIF experiment on metastable Ar [29]. The development of the Ar axial velocity component along the plasma jet axis at $p_{back}$ = 20 Pa is shown in Fig. 6.3 as well as a theoretical profile in the supersonic domain (see also chapter 3).

## 6.3.2 Hydrogen atoms

### Anomalous density profile

Contrary to the argon case, hydrogen atom axial density profiles do not exhibit any jump throughout the stationary shock front, as can be seen in Fig. 6.4, while the drift velocity drops, as can be observed in Fig. 6.3. This means that the forward flux of H atoms is not conserved in the shock region. This striking demonstration of an anomalous H atom shock pattern has been reported in a foregoing paper [8]. It is postulated that strong density gradients between the core of the jet and its surroundings, due to a low H concentration in the background gas, are responsible for an outward diffusion of H atoms. The efficient destruction of hydrogen ground-state atoms at the wall of the vessel [30], where they recombine to form molecular hydrogen [31, 32, 33], is responsible for the very low H background density. In neutral gas mixture expansions migration of the light species towards the edges of the jet has been observed within the first millimeters of the flow [12]. In this region this so-called mass focusing effect arises from a mass dependent kinetic energy.
<figure><img src="image_43.png" /><figcaption>**Fig. 6.4:** Axial H atom density profile at 16 Pa (square) and 42 Pa (triangle).</figcaption></figure>

<figure><img src="image_44.png" /><figcaption>**Fig. 6.5:** Axial H atom $T_{\perp}$ profile at 16 Pa (square) and 42 Pa (triangle). The lines represent a fit with $\gamma = 1.45$ (solid) and $\gamma = 5/3$ (dotted).</figcaption></figure>

One can exclude under our experimental conditions that H atoms recombine in volume during the expansion process to form molecular hydrogen, which would also lead to a non-conservation of the axial H flux. The chemical reactions that have to be considered are the three-body reactions [34, 35]

$$ \begin{array}{l@{\quad}l@{\quad}l@{\quad}l} \text{H} + \text{H} + \text{Ar} & \rightarrow & \text{H}_2 + \text{Ar} & k = 9.8 \times 10^{-42} \quad \text{m}^6 \text{s}^{-1}, \\ \text{H} + \text{H} + \text{H}_2 & \rightarrow & \text{H}_2 + \text{H}_2 & k \approx 2 \times 10^{-42} \quad \text{m}^6 \text{s}^{-1}, & (6.4) \\ \text{H} + \text{H} + \text{H} & \rightarrow & \text{H}_2 + \text{H} & k = 4.5 \times 10^{-44} \quad \text{m}^6 \text{s}^{-1}. \end{array} $$

Under our experimental conditions, these reaction rates are too small to lead to any significant recombination of hydrogen atoms during the residence time.

Furthermore, as already mentioned in [8], the steepness of the density decay within the supersonic domain depends on $p_{\text{back}}$, which means that H atoms which come from the source receive information about the standing ambient gas. This proves that a shock wave structure does not form a totally closed system. As shown in Fig. 6.4, the H axial density profile ahead of the shock wave can be well described by a modified point source expansion law of the form

$$ n(z) = n_o \frac{z_{\text{ref}}^{\beta}}{(z + z_o)^{\beta}}, \qquad (6.5) $$

where the exponent $\beta$ is larger than two, because an outward diffusion effect is superimposed on the normal rarefaction effect. The length parameters $z_{\text{ref}}$ and $z_o$ are taken to be the ones determined from the Ar measurements. The values of $n_o$ and $\beta$ depend on $p_{\text{back}}$ and are listed in Table 6.1. It can clearly be seen that $\beta$ gets closer to 2 as the pressure increases, meaning that H atoms get better confined inside the supersonic jet. In other
<table><caption>**Tab. 6.1:** Parameters governing the H atom expansion at different background pressures. The subscript *o* refers to the source region. The variable *β* is linked to the on-axis H atom density decrease, see Eq. 6.5, and *γ* is the quasi-adiabatic exponent.</caption><thead><tr><th>p<sub>back</sub> (Pa)</th><th>16</th><th>20</th><th>42</th><th>100</th></tr></thead><tbody><tr><td>n<sub>o</sub> (m<sup>-3</sup>)</td><td>(5.6 ± 1.0) × 10<sup>21</sup></td><td>(2.2 ± 0.2) × 10<sup>21</sup></td><td>(5.6 ± 1.0) × 10<sup>21</sup></td><td>(2.2 ± 0.2) × 10<sup>21</sup></td></tr><tr><td>β</td><td>2.65 ± 0.10</td><td>2.35 ± 0.05</td><td>2.20 ± 0.05</td><td>2.10 ± 0.05</td></tr><tr><td>γ</td><td>1.40 ± 0.10</td><td>1.40 ± 0.04</td><td>1.50 ± 0.15</td><td>1.45 ± 0.05</td></tr><tr><td>T<sub>o</sub> (K)</td><td>9000 ± 3000</td><td>4800 ± 600</td><td>8100 ± 3500</td><td>5600 ± 800</td></tr></tbody></table>

words, it becomes more difficult for H to cross the barrel shock wave at high $p_{back}$, since the local mean free path decreases. This also explains why the H density in the ambient gas depends on $p_{back}$, as can be seen in Fig. 6.4.

## Perpendicular temperature

The axial profile of the atomic hydrogen perpendicular temperature $T_{\perp}$ (associated with the velocity distribution perpendicular to a stream line) is depicted in Fig. 6.5 for two different background pressures<fnref n="1" />. The temperature profiles clearly reveal the presence of a stationary shock front across which $T_{\perp}$ rises due to the conversion of the jet particle kinetic energy into random thermal motion. As predicted by the theory, both the position and the width of the shock front depend on $p_{back}$ [12]. In the supersonic region of the expansion a strong cooling effect occurs because the thermal energy gained by the particles in the plasma source is converted into kinetic energy by means of collisions. For an adiabatic process, the energy equation can be replaced by the Poisson adiabatic law [28]

$$ \left(\frac{n}{n_o}\right)^{\gamma-1} = \frac{T}{T_o}, \qquad (6.6) $$

where $\gamma$ is the adiabatic exponent which is equal to $\frac{5}{3}$ for a monoatomic gas. By plotting the H atom temperature as a function of the density, the isentropic exponent can be evaluated. The value of $\gamma$ for various values of $p_{back}$ is listed in Table 6.1. As can be seen, $\gamma$ for H radicals has the almost constant value 1.4. The fact that the measured $\gamma$ is smaller than the theoretical $\gamma$ for a monoatomic gas means that, contrary to a neutral gas expansion, the supersonic expansion of a plasma is non-adiabatic [36]. As can be seen in Fig. 6.5, the measured temperature is well above the theoretical isentropic temperature. In the case of a plasma expansion the cooling of neutrals in the supersonic domain can be disturbed by heat transfer from both the source and the hot background gas behind the shock front. Re-entry of background gas into the supersonic domain through the barrel shock wave can also disturb the flow, especially at low pressure [29]. In Fig. 6.5, a fit to the measured data is shown using the adiabatic law with measured $\beta$ and $\gamma$. The source temperature $T_o$,

<footnote n="1">¹ The comparison between $T_{\perp}$ and $T_{\parallel}$ profiles is discussed in chapter 5.</footnote>
that corresponds to the temperature inside the arc nozzle, is found to be close to 0.5 eV,
independent of $p_{\text{back}}$ as it should.

## Axial velocity component

The measured axial profiles of the axial velocity components of H and D atoms are shown in Fig. 6.3 for a background pressure of 20 Pa. Also the Ar velocity profile is shown. The shape of the H and D velocity profile along the jet centerline is very similar to the Ar one. This means that the mechanisms responsible for the H and D axial velocity development are connected to those which determine the Ar profile. Throughout the supersonic domain, the H(D) velocity is higher than the Ar velocity. The decoupling between the Ar fluid and the radical fluid, which is already present at the source exit, is a direct consequence of losses of H atoms at the walls. A possible way of describing the hydrogen drift velocity is to consider this velocity as the sum of the argon drift velocity and a diffusion term<fnref n="2" />. The diffusion velocity, which contains implicitly the H density gradients, would control the H flow relative to the Ar flow. More theoretical and computational work is needed to validate this idea. Nonetheless, primary work using an hydrodynamic approach to model the Ar-H flow, seems to confirm this view. Behind the shock wave, all velocities are about the same, meaning that all particles are well coupled. This is a consequence of the much higher collision frequency in the subsonic domain. However, across the stationary shock front, the velocities differ substantially, the atomic hydrogen velocity being lower than the argon velocity. Because of collisions in the shock, we can expect the difference in velocity to slowly diminish. It can also be seen that the H(D) velocity starts to decrease earlier than the Ar velocity. This effect is not yet well understood.

# 6.4 Influence of the mass on the transport of radicals

## 6.4.1 Use of a light carrier gas

In order to study the influence of the mass on the radical transport mechanism, the carrier gas has been changed: argon ($m_{Ar} = 40$ amu) has been replaced by helium ($m_{He} = 4$ amu). The parameters are the following: the arc current is 50 A, the cathode-anode voltage is 140 V, the He flow is 2 slm, the H₂ flow is 0.5 slm and the background pressure is 20 Pa. The source geometry is identical to the one used in the Ar-H₂ mixture case. An admixture of less than 2 % Ar gas is used to prevent a too fast degradation of the cascaded arc cathode tips. Thus the average mass of the mixture is equal to 4.3 amu, using the fact that the plasma source delivers mainly H₂ in view of the low measured H density, see Fig. 6.6.

The results of the measurements are given in Fig. 6.6. The H density at the source exit is much less, by nearly 2 orders of magnitude at $p_{\text{back}} = 20$ Pa, than when Ar is used

<footnote n="2">² Starting from the relation $\vec{w}_H = \vec{w}_{Ar} + \vec{w}_H^{\text{diff}}$, a calculation based on two simplified momentum balances (H and Ar) coupled via a friction term and using the fact that $m_H \ll m_{Ar}$ shows that $\vec{w}_H^{\text{diff}}$ is the diffusion velocity as given by the Fick's diffusion law multiplied by the isentropic exponent $\gamma$.</footnote>
<figure><img src="image_45.png" /><figcaption>**Fig. 6.6:** Axial profiles of the H density and H perpendicular temperature $T_{\perp}$ when He is used as a carrier gas instead of Ar. The shock front structure is visible in the temperature profile. However, no discontinuity is observed in the H density profile.</figcaption></figure>

as a carrier gas [8]. The relatively low delivered amount of H arises directly from the way H atoms are created in a He-H₂ mixture [37, 38, 39]. The low H temperature at the arc outlet is due to the high thermal conductivity of the He-H₂ mixture. The sharp rise in the temperature, which occurs behind the arc exit as depicted in Fig. 6.6, reveals the occurrence of a stationary shock front. In comparison with the results obtained at the same p<sub>back</sub> for an Ar-H₂ mixture, two remarks can be made. First, the shock wave is located very close to the arc exit since the shock position varies with the stagnation pressure, which at constant flow decreases when the atomic mass decreases [12]. Second, the shock front, whose thickness is always in the order of one local mean free path for momentum exchange, is narrow despite a relatively small neutral-neutral momentum exchange cross section for He. This originates from the fact that the shock wave occurs close to the nozzle exit, where the He density is still high. According to the temperature jump, the Mach number *M* is found to be equal to 2.3 in front of the shock when using the Rankine-Hugoniot relation with a value of 1.5 for the isentropic exponent γ. With this value for *M*, the amplitude of the density jump should be 2.8. However, no significant density jump is observed across the shock wave. This again indicates an outward diffusion of H atoms. Despite the small mass ratio between He and H, H atoms leave the core of the plasma jet and recombine at the vessel wall. This result points out the importance of the density gradients in the H atom loss process.
<figure><img src="image_46.png" /><figcaption>**Fig. 6.7:** Axial H (square) and D (circle) density profile at 20 Pa with Ar as a carrier gas.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>**Fig. 6.8:** Axial H (square) and D (circle) $T_{\perp}$ profile at 20 Pa. Also shown is the axial D $T_{\|}$ profile (star) at 20 Pa.</figcaption></figure>

In the subsonic domain of the flow, the temperature decays fast because of the large thermal conductivity of the mixture, and the density remains relatively constant. Despite the low H density at the source outlet, it is slightly higher behind the stationary shock wave ($z > 100$ mm) than in the case where Ar is used as a carrier gas. This means that H is better transported when the mass of the carrier gas is small. The gain in confinement when the mass of the carrier gas is closer to that of the transported species is known for multicomponent neutral beam expansions [12]. But we demonstrated that despite a better confinement, the atomic hydrogen anomalous shock wave pattern does not vanish. A better transport is of relevance for the downstream chemistry since the amount of ground state radicals can be one of the limiting factors when an expanding plasma jet is used for thin film deposition [5] or surface modification. Then a good compromise has to be found between the quantity of radicals available in front of the surface and the momentum of the carrier gas which is also used to transport other particles like molecules of the precursor gas.

### 6.4.2 Isotopic effect

Another way to study the influence of the mass is to replace H by one of its isotopes, e.g. deuterium (D). The arc parameters are kept the same as in the case of H₂ seeding. To obtain absolute D density we assume that the two-photon absorption cross sections of H and D are identical. In Fig. 6.7, the axial H and D density profiles are shown for $p_{\text{back}} = 20$ Pa, and in Fig. 6.8 the axial H and D perpendicular temperature profiles are compared at 20 Pa. In the supersonic domain, after approximately one nozzle diameter, D is better confined inside the jet (higher density) because of its larger mass that results
in a lower diffusion velocity ($T_D \approx T_H$) as can be seen in Fig. 6.8.

As expected, no density jump is visible in the D density profile across the stationary shock wave, whereas the velocity decreases by a factor 3. The apparent difference between the H atom and the D atom perpendicular temperature may arise from the important radial losses of H atoms that act as a cooling mechanism (see chapter 3). Note that the two D atom temperature component remain in equilibrium over the course of the expansion, see Fig. 6.8, contrary to the H ones. In the subsonic domain, the higher H density is a consequence of the lower H temperature.

## 6.5 Confinement of radicals inside the jet core

In order to have a better insight in the lack of confinement of atomic hydrogen inside the core of the plasma jet generated from an Ar-H₂ mixture, it is of interest to study beam cross-sections at several axial positions. Also the permeability of the Ar barrel shock wave needs to be discussed. In Figs. 6.9, 6.10, and 6.11, radial profiles of the H atom density, perpendicular temperature, and radial velocity component are given for $p_{back} = 16$ Pa at $z = 2, 8, 20, 50, 80, \text{ and } 100$ mm behind the arc outlet.

### 6.5.1 Radial profiles of the H density

From the radial density profiles, we observe that the density of atomic hydrogen decreases fast as we move away from the source exit. This is a direct consequence of the rarefaction effect due to the increase of the beam diameter. However, this effect is enlarged by the outward diffusion phenomenon. Contrary to argon, the H jet is never overexpanded i.e. the H atom density inside the jet is never lower than outside the jet. Across the stationary shock front, from $z = 50$ mm to 100 mm, the radial H density profile is flat because the diffusion process levels out the density profile.

The shape of the density profile in the core of the plasma beam can be theoretically described. Such a shape corresponds to a situation in which the jet is underexpanded. We propose the following function

$$n(r, z) = n(z) \cos^a(\varphi), \qquad (6.7)$$

with the angle $\varphi$ given by

$$\varphi = \arctan \left( \frac{r}{z + z_o} \right), \qquad (6.8)$$

where $r$ is the radial distance and $z_o$ is the position of the virtual point source. In the field of gas dynamics, some more complex functions have been proposed based on experimental data [10, 40]. The results of fits using Eq. 6.7 are depicted in Fig. 6.9 at $z = 2$ and 8 mm. As can be seen in the graph, the function reasonably represents the experimental data inside the jet. However, to obtain such an agreement, the exponent $a$ in Eq. 6.7 has to be modified when changing the axial position $z$: $a = 2.4$ at $z = 2$ mm and $a = 5.5$ at $z = 8$ mm. In the classic supersonic expansion picture, the value of this exponent only
<figure><img src="image_48.png" /><figcaption>**Fig. 6.9:** Radial profiles of the density (top), perpendicular temperature $T_⊥$ (middle), and radial velocity $w_r$ (bottom) of H atoms for a background pressure of 16 Pa at $z = 2$ mm (left) and $z = 8$ mm (right). The H density profile in the core of the jet is modeled using Eq. 6.7 (solid line) with $a = 2.4$ and 5.5 at $z = 2$ mm and 8 mm respectively.</figcaption></figure>

depends on the source conditions and geometry and on $\gamma$, but it is a constant in the course of the expansion. Once more the increase in $a$ is connected to an outward loss of H radicals in particular in the outward regions of the jet.

### 6.5.2 Temperature and radial velocity

The radial profiles of the H atom perpendicular temperature $T_⊥$ clearly show evidence of the quasi-adiabatic cooling effect. They also reveal the structure of the barrel shock wave. It can be seen in Fig. 6.9 that at $z = 2$ mm in the core of the jet $T_⊥$ is not constant. The temperature is higher on axis than on the sides. This is purely a geometrical effect: the H atoms detected on the sides have travelled more in comparison with those on the jet axis. Then, since the former have undergone more collisions they converted a larger part of their thermal energy into kinetic energy which leads to a lower temperature. At $z = 8$ mm, this
<figure><img src="image_49.png" /><figcaption>**Fig. 6.10:** Radial profiles of the density (top), perpendicular temperature $T_\perp$ (middle), and radial velocity $w_r$ (bottom) of H atoms for a background pressure of 16 Pa at $z = 20$ mm (left) and $z = 50$ mm (right).</figcaption></figure>

geometrical effect is almost not noticeable anymore. The rise in $T_\perp$ when going off-axis is due to collisions with Ar atoms in the barrel shock wave [12]. This temperature jump defines the boundary of the plasma jet. Then, behind the barrel shock wave, $T_\perp$ decreases towards the vessel walls because of heat transfer to the background gas. As can be seen in Fig. 6.9, the background gas temperature is rather low, around 500 K. In the subsonic domain, the temperature of flowing H atoms is homogeneous over the jet radius, as can be seen in Fig 6.11, and relatively high. This shows that the heat gained in the source is transported mainly forward with some radial losses due to thermal conduction.

The radial profiles of the H atom radial velocity component also reveal the Ar shock wave structure. Because of the expansion process, the radial velocity first increases when moving away from the jet axis. Then when H atoms collide with Ar atoms in the barrel shock wave, the velocity decreases since it is converted into thermal energy, and the temperature increases accordingly. As expected, the position of the Ar barrel shock as deduced from the H radial velocity profile does coincide with the position deduced from the corresponding $T_\perp$ profile.
<figure><img src="image_50.png" /><figcaption>Fig. 6.11: Radial profiles of the density (top), perpendicular temperature $T_{\perp}$ (middle), and radial velocity $w_r$ (bottom) of H atoms for a background pressure of 16 Pa at $z = 80$ mm (left) and $z = 100$ mm (right).</figcaption></figure>

As can be seen in Figs. 6.9 and 6.10, the maximum value of the H radial velocity $w_r^{max}$ decreases as $z$ increases. This is not predicted in the classical expansion picture. Assuming that the total velocity does not vary with the radial position, the magnitude of the radial velocity component as a function of the radial position for a given axial position can be written in the classical picture as follows

$$w_r = w_{z,0} \sin(\varphi), \qquad (6.9)$$

where $w_{z,0}$ is the axial speed at $r = 0$ and $\varphi$ is the expansion angle given by Eq. 6.8. In the case of a free jet, the maximum radial velocity is always reached at roughly the same angle $\varphi$ for a given $p_{back}$, independent of $z$. Since up to the stationary shock wave the axial velocity increases slightly, as can be seen in Fig. 6.3, $w_r^{max}$ should then also increase a bit. However, it decreases by approximately a factor 3 between $z = 2$ and 20 mm. By using the measured argon axial drift velocity, shown in Fig. 6.3, it is possible to estimate with Eq. 6.9 the maximum value of the H radial velocity component in case of a perfect coupling between H radicals and Ar atoms. The velocity is calculated at the boundary of
<figure><img src="image_51.png" /><figcaption>**Fig. 6.12:** Profiles of the axial and radial component of the H velocity as well as the radial component of the Ar velocity as a function of the angle with the jet axis, recorded at z=8 mm behind the arc nozzle.</figcaption></figure>

the plasma jet of which the location is taken from the $T_⊥$ profile. The maximum speed is found to be equal to 1000, 900, and 900 m s⁻¹ at z = 2, 8, and 20 mm, respectively. At any z, the highest H radial velocity is always larger than the calculated value, which points out the importance of the radial escape of H radicals. This radial escape process is certainly most important at the beginning of the plasma expansion where the atomic hydrogen radial velocity component is highest. The discrepancy between the expected H radial velocity and the measured one not only confirms the idea of a decoupling between H and Ar atoms, but it also suggests that the H velocity is connected to the H density gradients which are the largest at the beginning of the expansion and which decrease with increasing z.

### 6.5.3 Permeable Ar barrel shock wave

The picture of a strong decoupling between H and Ar is substantiated by comparing the radial profile of the axial and radial velocity of the H atoms and the radial velocity components of the Ar atoms [29]. This is done in Fig. 6.12 for a position of 8 mm behind the arc outlet and at $p_{back}$ = 20 Pa. The data points are plotted as a function of the angle $\varphi$ with the expansion axis given by Eq. 6.8.

This figure confirms the picture of an outward diffusion of H. Moreover, it can be fully understood by considering that the H density gradient is the driving mechanism of the diffusion process. First, the graph clearly shows the decoupling between H and Ar atoms.
On the expansion axis, both radial velocity components are zero. As $r$ increases, i.e. further from the axis, the gap between the two velocities also increases, the maximum H radial speed being approximately 3 times higher than the maximum Ar speed. Second, H atoms penetrate and even cross the Ar barrel shock. The decay in Ar radial velocity, due to friction with the standing gas, indicates the position of the barrel shock. It follows from the picture that at $z = 8$ mm, the expansion angle is about 30°. But the H atom radial velocity only starts to decrease beyond the beginning of the Ar barrel shock wave. Moreover, when the Ar velocity is zero, the H velocity is still high, indicating that H atoms are still moving radially into the background gas, where Ar atoms are motionless. Finally, the H velocity becomes zero because of increasing friction with the neutral background gas. Then the directed motion is replaced by a random thermal motion. Third, even if H atoms penetrate the Ar barrel shock wave with still a significant axial velocity component, H mainly escapes the plasma jet radially. Indeed, the axial velocity component becomes zero whereas the radial component is still high. This is a direct consequence of the steep radial density gradients between the core of the plasma beam and its vicinity.

Another effect which has to be mentioned is the fact that beyond the plasma jet boundary the H atom radial velocity component is zero (see Figs 5.4 and 6.9 for clearness) whereas hydrodynamic calculations predict a negative velocity [42]. This proves that H atoms do not recirculate inside the vacuum chamber, i.e. H atoms are not trapped into recirculation vortices that surround the jet. In other words, H atoms diffuse out of the jet directly to the vessel walls where they stick or recombine.

## 6.6 Atomic hydrogen diffusion flux

To confirm the fact that hydrogen atoms escape the core of the plasma jet by a diffusion process we estimate, at a background pressure of 16 Pa, the losses of H radicals over the normal shock wave, i.e. from $z = 20$ mm to $z = 100$ mm, and compare them with a calculated diffusion flux<fnref n="3" />.

The loss of atomic radicals in the shock region can be estimated by determining the difference between the forward flux of H atoms entering the shock front, $\phi_H^{in}$, and the flux behind the shock, $\phi_H^{out}$.

<footnote n="3">³ Since H atom do not recombine in volume, the total H atom flux has to be conserved at any $z$ position throughout the plasma flow. The experimental verification of the flux conservation would however necessitate an extremely large dataset. Nevertheless, using our somewhat restricted dataset, the H flux conservation has been verified along the plasma jet axis. In cylindrical coordinates, and with cylindrical symmetry, the flux conservation reads $\frac{1}{r}\frac{\partial}{\partial r}(nrw_r) + \frac{\partial}{\partial z}(nw_z) = 0$. On the jet axis, i.e. at $r = 0$, and for every value of $z$, previous equation becomes $2n\frac{w_r}{r} = -\frac{\partial}{\partial z}(nw_z)$. The right hand side and the left hand side flux values are found to be in reasonable agreement meaning that the H atom flux evaluated on the jet centerline is conserved at a given $z$ position. Moreover we observe that both flux components decrease in magnitude in the course of the plasma expansion as a consequence of losses of H radicals on the jet axis.</footnote>
At a given $z$, the forward flux (in s⁻¹) is given by

$$ \phi^{\text{forward}} = \iint_S n w dS = \int_0^R n_z(r) w_z(r) 2\pi r dr, \qquad (6.10) $$

where $R$ represents the position of the boundary of the jet which is taken to be the radial position at which the H atom radial velocity is maximum. In order to calculate the forward flux, the radial profile of the H axial velocity component is needed. We will assume that the profile is identical to the one measured at $z = 8$ mm, which, in the core of the jet, can be well approximated by

$$ w_z(r) = w_z(0) \cos^2(\varphi), \qquad (6.11) $$

where the expansion angle $\varphi$ is given by Eq. 6.8. We find that $\phi_H^{\text{in}} = 3.3 \times 10^{19}$ s⁻¹ and $\phi_H^{\text{out}} = 3.9 \times 10^{18}$ s⁻¹, which means one order of magnitude loss of H flux.

Now this loss of H radicals has to be compared with the H radial flux across the shock front. At a specific axial position $z$, the differential radial flux of H atoms at the jet boundary $R$ is given by

$$ \frac{\partial}{\partial z} \phi_H^{\text{rad}} = 2\pi R n(z, R) w_r(z, R). \qquad (6.12) $$

With our data set, only an averaged H atom radial flux $\overline{\phi_H^{\text{rad}}}$ can be calculated over the shock front. The latter is obtained by multiplying the averaged differential flux<fnref n="4" /> by the shock thickness $L$, which is equal to 80 mm at 16 Pa. We find that $\overline{\phi_H^{\text{rad}}} = 3.4 \times 10^{19}$ s⁻¹, which is in good agreement with the decrease in forward flux. Thus we find that the total H atom flux is conserved throughout the shock front.

Finally, the radial flux can be compared with a diffusion flux. An estimate of the thermal flux can be obtained by using Fick's diffusion law. The radial diffusion flux of H atoms through the stationary shock wave is given by

$$ \phi_H^{\text{th}} = -\frac{1}{3} \lambda^{H-Ar} v_{th} \frac{\partial n_H}{\partial r} S = \frac{v_{th}}{n_{Ar} \sigma_{Ar-H}} \frac{\partial n_H}{\partial r} S, \qquad (6.13) $$

where $\sigma_{Ar-H}$ is the momentum transfer cross section [41], $v_{th}$ is the H atom thermal speed, and $S = 2\pi RL$ is the surface around the stationary shock wave. The H atom density gradient between the jet and the vessel wall can be estimated in first order from

$$ \frac{\partial n_H}{\partial r} = \frac{n_H(R) - n_H(\text{wall})}{R - r_{\text{vessel}}} \approx \frac{n_H(R)}{R - r_{\text{vessel}}}, \qquad (6.14) $$

where $r_{\text{vessel}} = 180$ mm is the radius of the vacuum chamber. From Rayleigh scattering measurements [13, 26], the average Ar density at the jet boundary is found to be equal to $6 \times 10^{20}$ m⁻³. The H diffusion flux is then equal to $1.1 \times 10^{19}$ s⁻¹. This value is lower than those calculated previously. The difference may arise from a too simple model for the diffusion coefficient of H atoms into an Ar-H₂ mixture. Moreover, the estimate of the H atom density gradient between the jet boundary and the vessel wall is also certainly too rough. A complete solution of the diffusion equation would therefore be necessary to confirm the picture of radial outflow induced by a density gradient.

<footnote n="4">⁴ The differential flux is calculated at $z = 20, 50, 80$ and $100$ mm and the mean value is used in the estimation of the diffusion flux. The corresponding value of $R$ are: $15, 20, 20$ and $20$ mm.</footnote>
## 6.7 Conclusions

Before giving the concluding remarks, it is worthwhile to provide the reader with a list of the most striking effects that support the idea of an outward diffusion of H atoms driven by plasma-wall interactions in the course of the expansion process

* pressure dependent H density decrease in the supersonic domain,

* no H density jump within the stationary shock wave,

* extremely low H concentration in the background gas,

* mass dependent decoupling between $\vec{w}_H$ and $\vec{w}_{Ar}$,

* H atom flow through the Ar barrel shock wave,

* zero H radial velocity component beyond the jet boundary.

In the expansion of a plasma generated from an Ar-H₂ mixture, hydrogen atoms are decoupled from argon atoms. As a consequence a peculiar situation is created where the plasma flow is composed of two fluids with their own macroscopic features. The Ar flow pattern can be entirely described using the well-established inert gas supersonic expansion theory. On the contrary the H flow pattern exhibits a significant departure from the classical expansion theory. The anomalous H atom flow characteristic is a direct consequence of the interaction between the plasma and the vessel walls where H atoms recombine to form molecular hydrogen.

The decoupling effect, of which the strength depends on the background pressure and on the reduced mass, is a general physical phenomenon which applies to most of atomic radicals as soon as the plasma expansion is surrounded by surfaces where the radicals can recombine with a high probability. In that case the surfaces, e.g. a reactor wall or dust particles, act as a sink for the radical. This in turn creates density gradients which are the driving force for this effect.

## Acknowledgments

The authors would like to acknowledge fruitful discussion with Prof. M.C.M. van de Sanden and Dr. J.A.M van der Mullen. The authors greatly appreciate the skillful technical assistance of M.J.F. van de Sande, A.B.M. Hüsken, and H.M.M. de Jong. This work is part of the research program of the Netherlands Foundation for Fundamental Research on Matter (FOM) and is partially founded by Euratom. It is also financially supported by the Netherlands Organization for Scientific Research (NWO).
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# Appendix 6A

## Dissociation degree

The dissociation degree $\alpha$ of the plasma source which characterized the strength of the source, i.e. the capability to generate radicals, is defined as

$$ \alpha = \frac{\phi_H}{2\phi_{H_2}^o} $$

where $\phi_H$ is the ground state atomic hydrogen flux and $\phi_{H_2}^o$ is the gas flow of molecular hydrogen at the arc inlet. In our experimental conditions we have: $\phi_{H_2}^o = 0.5$ slm = $2.2 \times 10^{20}$ s$^{-1}$. The knowledge of $\alpha$ in the source would necessitate the measurement of the H atom flux inside the arc channel, which is a difficult task. However, we will see that the knowledge of $\alpha$ in the core of the plasma jet can deliver information about $\alpha$ inside the source.

If there is no radial flux and no volume recombination, which is the case in the expansion of a neutral gas, then $\alpha$ can be estimated inside the plasma jet when knowing the forward flux. At a given $z$, the forward flux (in s$^{-1}$) is given by

$$ \phi^{\text{forward}} = \iint_S n w dS = \int_0^R n_z(r) w_z(r) 2\pi r dr, $$

where $R$ represents the position of the boundary of the jet, i.e. the beginning of the barrel shock wave. In this case, $\alpha$ is conserved throughout the supersonic domain (in the absence of mass-focusing effects). Therefore by measuring $\alpha$ in this part of the jet, one obtains its value at the source exit.

Since the radial dependence of the H axial velocity has been measured at $z = 8$ mm, see Fig. 6.12, it is possible to determine $\alpha$ at this location by using the density profile of Fig. 6.9. The boundary position, which is needed to calculate the forward flux, is taken from the Ar radial velocity profile shown in Fig. 6.12. We find that at 8 mm behind the arc outlet $\phi_H^{\text{forward}} = 8 \times 10^{19}$ s$^{-1}$ and thus $\alpha = 18\%$. It is likely that the dissociation degree inside the channel of the plasma source is close to unity when using such an Ar-H$_2$ mixture. Therefore, the low value of $\alpha$ measured at 8 mm behind the arc exit is an indication of the importance of the losses of H atoms in the arc nozzle as well as in the first few millimeters of the expansion.
# Appendix 6B

## Influence of the nozzle geometry

<figure><img src="image_52.png" /><figcaption>**Fig. 6.13:** Axial H density profile (left) and perpendicular temperature profile (right) measured at 20 Pa for 3 different arc nozzle opening angles: 0 degrees (circle), 12 degrees (star), and 45 degrees (square).</figcaption></figure>

The cascaded arc is often used as a H atom source for surface treatment or modification. In that case the amount of H atom transported towards the downstream region of the flow is of relevance. In this appendix the influence of the cascaded arc nozzle geometry, i.e. the opening angle, on the H atom density is investigated. The arc is operated in standard conditions: 40 A, 100 V, 3 slm Ar, and 0.5 slm $H_2$. Three different opening angles are compared: 0 degrees (straight), 12 degrees, and 45 degrees (without pre-expansion zone).

At the arc exit, the difference in density between the 3 nozzles arises from a geometrical effect, i.e. a difference in the jet diameter. The smaller the angle, the smaller the diameter over the first few millimeters; since the forward flux is approximately constant, the density increases. At large angle (45°), the plasma expands already inside the nozzle and therefore cools down already inside the nozzle that explains the lower temperature at the outlet. In view of the aforementioned applications, the difference in density has to be compared in the subsonic domain (in this zone the temperature is independent of the geometry). The H atom density appears to be higher when using a large opening angle. The confinement inside the jet is not enhanced; the higher H density results from lower H losses in the source nozzle. Indeed a large opening angle do not favor (nozzle)wall-recombination: the gas temperature is lower and the available area is less since the plasma starts to flow ahead of the source exit.
<figure><img src="image_53.png" /></figure>
# Chapter 7

## H atom fluid properties in a hydrogen plasma expansion<fnref n="1" />

### Abstract

The flow properties of the ground state hydrogen atom fluid in the course of the expansion of a weakly ionized hydrogen plasma created by a cascaded arc are studied by means of Two-photon Absorption Laser Induced Fluorescence (TALIF). The transport of H₂ molecules is investigated using Rayleigh scattering. In view of the low ionization degree (< 1 % as measured by Thomson scattering), the hydrogen plasma resembles a hot neutral gas. Furthermore the low dissociation degree measured at the source exit implies that H atoms flow in a H₂ environment. It is shown that the H atom expansion pattern is in disagreement with the neutral gas supersonic expansion theory. Indeed the transport of H atoms in the plasma jet is strongly influenced by surface-recombination processes. Because of the large density gradients between the core of the jet and its surroundings induced by the recombination of H atoms at the reactor walls, hydrogen atoms diffuse out of the plasma jet during the expansion process. When the surface loss probability is high, e.g. combination of a large wall-recombination probability with a long residence time, the losses of radicals by diffusion, and the subsequent decoupling between the inert gas fluid and the reactive gas fluid, can not be avoided even when the mass of the carrier gas is close to the mass of the radical.

<footnote n="1">¹ Adapted from:
S. Mazouffre, P. Vankan, R. Engeln, and D.C. Schram, *Phys. Plasmas* **8**, 3824 (2001).</footnote>
**7.1 Introduction**

The study of hydrogen plasma expansions is in the first place of particular interest since it concerns many different systems that cover a broad range of dimensions. Hydrogen being the most abundant form of matter in the universe, an object like solar outburst and a phenomenon like solar wind represent examples of astrophysical scale expanding hydrogen plasma [1]. At intermediate scale, hydrogen expansions are found in systems like the divertor region of a Tokamak plasma [2] and the remote plasma jets used for surface modifications.

In the second place, hydrogen plasma expansions contain light and highly reactive hy-
drogen atoms which play a key role in many chemical reactions encountered in plasma
processes. To clearly demonstrate the importance of H atom chemistry, two examples will
be given that are associated with energy and environmental problems humanity has to face
in the next century. Hydrogen atoms are involved in the plasma-aided fast deposition of
thin films of hydrogenated amorphous silicon (a-Si:H) needed for the next solar cell gener-
ation [3]. Atomic hydrogen radicals control the growth process (SiH$_4$ + H → SiH$_3$ + H$_2$,
SiH$_3$ being the main growth precursor) and thus the final film properties. In the field of
controlled nuclear fusion, atomic hydrogen plays a specific role. First, one wants to use
H atom sources for neutral beam heating of fusion devices (mainly via the production of
negative hydrogen ions H$^{-}$ [4]). Second, the interaction of neutral H atoms with the reactor
surfaces and the resulting reaction products, i.e. H$_2$ molecules, have drastic consequences
on the stability of the plasma and therefore on device performances [5, 6]. Numerous works
are nowadays devoted to this so-called hydrogen recycling process and they all require the
development of plasma sources capable of generating a high H atom flux.

All aforementioned processes are to a large extent governed by the transport mech-
anisms of H atoms in an expanding hydrogen plasma. In view of the applicability it
is therefore of importance to study the H atom flow pattern as well as the influence of
plasma-wall interactions on the transport properties. Moreover, the study of a plasma
expansion at a specific scale may serve as an example, and the results obtained can be
generalized to other kinds of localized plasma expansions with larger and smaller scales.
On a more fundamental point of view, this study serves as a test case to find new evidences
in order to confirm the fact that a radical expansion can differ strongly from a neutral gas
expansion. The anomalous transport of radicals in a supersonic plasma jet induced by
wall-recombination phenomena, i.e. the existence of a weak coupling between the inert gas
fluid and the reactive gas fluid, has been reported recently in the case of an Ar-H₂ and
He-H₂ mixture [7, 8]. In a hydrogen expansion, the mass of the carrier gas is close to that
of the radical, and the resulting enhancement of confinement especially is of interest.

In the experiments reported here, ground-state hydrogen atoms are spatially probed
using a two-photon excitation laser induced fluorescence technique in an expanding weakly
ionized hydrogen plasma produced by a cascaded arc. The determination of the atomic
hydrogen density, temperature, and velocity profiles enable to investigate in detail the
transport mechanisms of H atoms in the supersonic plasma jet. The molecular hydrogen
density profiles are measured by means of UV Rayleigh scattering that allows for discussions
about the coupling between the H fluid and the H₂ fluid over the expansion process.

## 7.2 Hydrogen plasma jet and optical system

### Plasma source and plasma jet

The hydrogen plasma is created by a cascaded arc [9]. The arc channel is composed of 4 insulated plates and has a diameter of 4 mm. The operating standard conditions are: a 55 A DC current, a cathode-anode voltage of 150 V and a H₂ gas flow of 3.5 standard liters per minute (slm). The stagnation pressure inside the arc is 0.14 bar. The thermal hydrogen plasma expands from a straight nozzle with a diameter of 4 mm into a roots-blower pumped vacuum vessel where the background pressure ($p_{back}$) can be varied, almost independently from the gas flow, from 10 Pa to 1 atm. The cascaded arc plasma source is mounted on a translation arm. Spatial scans through the expanding plasma can be performed by moving the arc relative to the intersection of laser beam and detection volume.

Since in this contribution we focus on the transport of H atoms, it is of interest to briefly present the ways H atoms can be produced in the plasma source where the electron temperature $T_e$ is around 1 eV. Ground-state atomic hydrogen can be generated by direct dissociation of H₂ molecule by electron impact

$$e + H_2 \rightarrow H + H + e. \qquad (7.1)$$

Yet the rate coefficient is relatively low at 1 eV [10, 11, 12]. A more favorable process at low temperature is the dissociative recombination of molecular ions

$$e + H_2^+ \rightarrow H^* + H \quad \text{and} \quad e + H_3^+ \rightarrow H + H + H. \qquad (7.2)$$

Both reactions are fast at low $T_e$ [12]. Note that $H_2^+$ can easily form $H_3^+$ when reacting with H₂ and this again leads to the formation of H atom

$$H_2^+ + H_2 \rightarrow H_3^+ + H. \qquad (7.3)$$

Another mechanism of interest for H atom production is the charge transfer between a proton and a rovibrationally excited hydrogen molecule [13, 14]

$$H_2^{v,J} + H^+ \rightarrow H_2^+ + H. \qquad (7.4)$$

The excited molecules are necessary to make the reaction exothermic. The energy deficit is around 2 eV, thus the charge exchange process is strongly favored for $H_2(v \ge 4)$. Note that the dissociation by impact with a proton of the H₂ molecule to form two H atoms is also possible at high v number [13]. The foregoing reactions have been proposed to explain the anomalous ion recombination, i.e. the fast ionization loss, in the expansion of an hydrogen plasma created by a cascaded arc [14]. Indeed without any influence of the arc parameters the plasma emerging from a cascaded arc with a straight nozzle is dark, which means that it contains mainly neutral particles, i.e. H atoms and H₂ molecules in our case. As measured by Thomson scattering, the ionization degree at the arc exit is less than 1 % [15].
# Laser-aided diagnostics

## H atom detection: TALIF

In the experiment reported in this contribution, ground-state hydrogen atoms are spatially probed by using a two-photon absorption laser-induced fluorescence (TALIF) technique [16, 17]. Only a short overview of the experimental method is presented here. A tunable 20 Hz Nd:YAG pumped dye laser delivers radiation around 615 nm. The output of the dye laser is frequency-tripled using non-linear optical crystals resulting in 2 mJ of tunable UV light around 205 nm with a measured bandwidth of 0.18 cm⁻¹. The UV laser beam is focused into the vacuum chamber either perpendicular or parallel to the plasma expansion axis. Hydrogen atoms are excited with two 205 nm photons from the 1s²S ground state to the 3d²D and 3s²S states. The excitation is monitored by detection of the resulting fluorescence yield on the Balmer-α line at 656 nm using a gated photo-multiplier tube. A narrow bandwidth interference filter is used to isolate the Hα line from the plasma emission. A slitmask is used to define the detection volume of which the dimensions are smaller than any gradient scale length. The dye laser frequency is calibrated by the simultaneous recording of the absorption spectrum of molecular iodine. From a spectral scan over the two-photon transition, the local H atom density, temperature (perpendicular and parallel to a stream line), and velocity (radial and axial component) are determined.

Absolute number densities are obtained from the ratio of the H LIF signal in the plasma to the LIF signal of a well defined amount of krypton [18, 19]. Krypton atom has a two-photon excitation scheme very similar to that of hydrogen atom. Furthermore, the ratio of the two-photon absorption cross section of Kr to H has been recently accurately measured (see chapter 4).

## H₂ molecule detection: Rayleigh scattering

The heavy particle density is measured by means of Rayleigh scattering using the UV laser beam. The polarized, scattered radiation at 205 nm is detected at an angle of 90° using a gated solar blind PMT. A dielectric UV mirror centered on 205 nm is used to filter out the plasma background light. The amount of stray light is reduced by a set of diaphragms and by using windows at Brewster's angle. The current detection limit in a H₂ plasma is 10²¹ m⁻³. The H and H₂ Rayleigh cross-sections are almost identical [20]. The local H₂ density is therefore determined by subtracting from the measured density the contribution of the H density measured by TALIF under the same conditions. A possible way to directly obtain the molecular species concentration by Rayleigh scattering is to use the depolarization effect [21].

## 7.3 H₂ density behavior along the jet axis

Because the hydrogen plasma created in the cascaded arc expands from a high pressure region into a low pressure region, a well defined free jet shock wave structure is produced [22]. Indeed, the H₂ density profile measured at $p_{back} = 100$ Pa along the jet axis, see Fig. 7.1,
<figure><img src="image_54.png" /><figcaption>**Fig. 7.1:** Axial $H_2$ density profile measured at 100 Pa by Rayleigh scattering (full square). The solid line is the result of a fit using Eq. 7.5 with $\beta = 2$ and $z_o = 1$ mm. The first point (open circle) is obtained under two assumptions: the injected $H_2$ flux is conserved and $H_2$ molecules flow at the local speed of sound.</figcaption></figure>

is characteristic for such a structure<fnref n="1" />. In the supersonic domain, the pronounced density decrease results from the rarefaction effect. Within the stationary shock wave, the $H_2$ density increases as a consequence of the forward flux conservation. In the shock region, two compression stages are visible (from 8 to 13 mm and from 20 to approximately 50 mm) which arise from the step by step adaptation of the jet static pressure to the background gas static pressure [24]. In between the $H_2$ fluid re-expands, and the density drops. A compression ratio, which gives indication about the strength of the shock, of 3, respectively 1.7, is found for the first, respectively the second, compression stage. Those numbers simply show that the pressure difference is higher before the first compression step. A Mach number in front of the shock wave $M = 3$ is deduced from the Rankine-Hugoniot relation [24] using the magnitude of the density jump over the entire shock wave. Behind the shock wave, the plasma flows at constant static pressure. Since the temperature drops because of heat transfer to the vessel walls, the $H_2$ density increases accordingly as can be seen in Fig. 7.1. Using the measured H atom temperature, the static pressure in the subsonic domain is around 100 Pa. The $H_2$ axial density behavior, especially in the shock region, is well predicted by a calculation using a hydrodynamic code based on the Navier-Stokes equation [25]. The $H_2$ density profile is also in good agreement with the H atom

<footnote n="1">¹ In previous works, the $H_2$ density has been measured by CARS [23]; however the laser beam configuration did not allow to accurately resolve the plasma jet structure in the first millimeters.</footnote>
temperature and velocity profile measured at 100 Pa as we will see.

In the supersonic domain, a theoretical expression for the density behavior is obtained from the flux conservation law in stationary state, and under the assumption that the particles originate from a point source and flow along straight stream lines (see chapter 2). The general expression for the particle density as a function of the distance to the source exit $n(z)$ reads

$$n(z) = n_o \frac{z_{\text{ref}}^{\beta}}{(z + z_o)^{\beta}}. \qquad (7.5)$$

In Eq. 7.5, $n_o$ is the density at the origin of the expansion, $z_{\text{ref}}$ is a scaling length and $z_o$ is the position of the virtual point source, i.e. the expansion origin. The exponent $\beta$ defines the steepness of the density decrease. In the case of a supersonic expansion of an inviscid gas through a sonic orifice, it is shown that the exponent $\beta$ is equal to 2, which corresponds to the classical rarefaction effect [22, 26]. The same value of 2 also applies in the case of the transport of neutral atoms in a plasma expansion [27]. From a fit of the supersonic H$_2$ density profile using Eq. 7.5 we find that $\beta = 2.0 \pm 0.1$ and $z_o = +(1.0 \pm 0.5)$ mm. The exponent $\beta$ correspond as expected to the case of an expansion of an inert gas; according to the value of $z_o$, the expansion starts close to the arc exit, as expected with a straight nozzle. According to the supersonic expansion theory, those two parameters are pressure independent. Setting $z_{\text{ref}}$ to the nozzle radius (2 mm), one obtains $n_o = (5 \pm 2) \times 10^{22} \text{ m}^{-3}$ which is in good agreement with the value of $4 \times 10^{22} \text{ m}^{-3}$ obtained at $z = 1$ mm under the assumptions that the injected H$_2$ flux is conserved and that H$_2$ molecules flow at the local speed of sound.

## 7.4 Axial H atom density and temperature profiles

### 7.4.1 Loss of H atoms by diffusion

As mentioned in the introduction, the flow pattern of H atoms is expected to be governed by wall-recombination processes. As a consequence, it should significantly deviate from the classical expansion behavior, i.e. from the free jet flow picture. The measured ground-state H atom density profile along the plasma jet centerline is shown in Fig. 7.2 for two different background pressures. The H atom density profile is indeed in disagreement with the H$_2$ molecule density profile shown in Fig. 7.1. At $p_{\text{back}} = 20$ Pa, there is no evidence of any density jump throughout the stationary shock wave. At $p_{\text{back}} = 100$ Pa, a density jump with an amplitude of 1.4 is visible but this is in disagreement with the magnitude of the H$_2$ density jump. Moreover, only the first compression stage is apparent: the H density solely drops behind $z = 10$ mm. The two H density profiles reveal a non-conservation of the H atom forward flux throughout the stationary shock wave in contradiction with the well established Rankine-Hugoniot relations. Behind the shock wave, the H density decreases as shown in Fig. 7.2, and since also the temperature decreases, hydrogen atoms do not flow at constant static pressure which is a sign for losses of H atoms.
<figure><img src="image_55.png" /><figcaption>**Fig. 7.2:** Ground-state H atom density profile along the jet centerline at $p_{back}$ = 20 Pa (open square) and $p_{back}$ = 100 Pa (solid triangle) in standard conditions. Also shown is the ground-state D atom axial density profile at 20 Pa background pressure (star) in the case of a H₂-D₂ mixture (3.1 slm H₂ and 0.4 slm D₂). The solid line represents a fit using equation 7.5 with $\beta$ = 2.4 and $z_o$ = 4 mm.</figcaption></figure>

From a fit of the H density profile ahead of the shock wave using Eq. 7.5, see Fig. 7.2, we find that $z_o$ = +(4 ± 1) mm and $\beta$ = 2.4 ± 0.2 at $p_{back}$ = 20 Pa. First, the expansion of H atoms starts already inside the plasma source nozzle, contrary to that of H₂ molecules. Second, the H density drop is faster than the expected one, meaning that a loss mechanism is superimposed onto the rarefaction effect. Note that in previous works devoted to the study of the transport of H atoms in an Ar-H₂ plasma, $\beta$ is also found to be larger than 2 as well as pressure dependent, and it approaches 2 at high background pressure [8]. Both effects support the idea of a decoupling between the H atom fluid and the H₂ molecule fluid which starts already in the source nozzle. Moreover, fixing $z_{ref}$ to the nozzle radius, that allows for comparison with the H₂ profiles, we find that $n_o$ = (2.5 ± 1.5) × 10²¹ m⁻³.

Even if ground-state hydrogen atoms can recombine in volume due to the following three-body reactions

$$H + H + H_2 \rightarrow H_2 + H_2 \quad \text{and} \quad H + H + H \rightarrow H_2 + H, \qquad (7.6)$$

the reaction rates [28] however are too low to lead to any significant loss of H atoms under our experimental conditions. Therefore the observed anomalous transport of H atoms arises from the fact that H escapes the plasma jet in the course of the expansion. As shown in previous papers [7, 8], the loss of H atoms is a direct consequence of the existence of large density gradients between the core of the plasma jet and its surroundings. The low H atom concentration in the background gas, which is responsible for a H radical
outward diffusion, results from the very efficient destruction of H atoms at the reactor walls when a high wall-recombination probability (Γ) is combined with a long residence time (τ). The wall-recombination probability on stainless steel for H atoms is known to be large: Γ<sub>ss</sub> = 0.1 - 0.2 [34, 35] at temperatures close to room temperature. The coefficient Γ depends on the surface temperature [30], on the state of the surface (roughness, presence of impurities, coverage by H) and on the energy of the incident H atom. That indicates how difficult it is to have an accurate estimate of Γ. However, the residence time inside our vacuum vessel is relatively long: τ ≈ 0.8 s at 20 Pa and τ ≈ 4 s at 100 Pa. That makes high losses of H radicals at the vessel walls, and thus a low H atom density in the background, very probable.

7.4.2 Perpendicular temperature development

The H atom perpendicular temperature (T⊥) profile along the jet centerline is plotted in Fig. 7.3 for two different background pressures. Due to conversion of thermal energy into kinetic energy, i.e. directed motion, the H temperature drops quickly in the supersonic domain. The temperature profiles clearly reveal the occurrence of a stationary shock wave across which the temperature rises because of collisions with the background gas particles. As can be seen in Fig. 7.3 both the position and the thickness of the shock wave depend on p<sub>back</sub>. In the subsonic part of the expansion, the temperature decreases due to heat transfer to the vessel walls. Behind the shock region, the measured H atom temperature is in good agreement with the rotational temperature of H<sub>2</sub> molecules measured by means of CARS [23]. The numerous collisions in the subsonic domain enable to maintain thermal equilibrium between H and H<sub>2</sub>.

Using the Poisson adiabatic law, the temperature at the origin of the H atom expansion T_o as well as the isentropic exponent γ can be determined from a plot of the H temperature as a function of the H density in the supersonic domain. It is found that γ = 1.55 ± 0.05 at p_back = 20 Pa and γ ≈ 1.7 at p_back = 100 Pa. At 20 Pa background pressure the plasma expansion is found to be non adiabatic since γ is smaller than its adiabatic value 5/3, as shown in Fig. 7.3. The fact that the supersonic expansion is not an adiabatic process may be explained by heat transfer from both the plasma source and the hot gas located behind the shock region. Another possibility is the entry in the jet of hot H<sub>2</sub> molecules, coming from the surrounding gas, in the last part of the supersonic domain where the mean free path for momentum exchange is large. The fact that the measured γ does not contain any molecular effects, especially at high pressure, whereas the jet is mainly composed of H<sub>2</sub> (γ<sub>mixture</sub> ≈ 1.3) is again an indication in favor of a decoupling between H atoms and H<sub>2</sub> molecules during the expansion process. The temperature T_o is equal to 2500 ± 500 K. This temperature correspond to the H temperature inside the arc nozzle, since z_o = +4 mm with this geometry. Therefore, the temperature inside the nozzle is different from the temperature inside the arc channel, which is about 1.2 eV [31]. In fact the plasma starts to cool down already in the arc nozzle and this drop in temperature may have a non negligible influence on plasma processes like ion-recombination for instance.

In the shock region, the H atom temperature profile measured at 100 Pa images the
<figure><img src="image_56.png" /><figcaption>**Fig. 7.3:** Ground-state H atom perpendicular temperature ($T_⊥$) profile along the jet centerline at $p_{back} = 20$ Pa (open square) and $p_{back} = 100$ Pa (solid triangle). Also shown is the atomic deuterium axial $T_⊥$ profile at 20 Pa background pressure (star) for a H$_2$-D$_2$ mixture (3.1 slm H$_2$ and 0.4 slm D$_2$). The theoretical H temperature profile in the supersonic domain is shown at 20 Pa using the Poisson adiabatic law with $\gamma = 1.55$ (solid line) and $\gamma = \frac{5}{3}$ (dashed line).</figcaption></figure>

H$_2$ density profile. For instance the second compression stage is noticeable: first the gas re-expands and then cools down (from $z \approx 13$ to $z \approx 20$ mm); secondly the expanding gas collide again with the background gas and the temperature rises or remains constant if the process is not too strong (from $z \approx 20$ to $z \approx 50$ mm). Note that a qualitative agreement between the H$_2$ density profile and the H atom temperature profile does not necessarily imply that $T_H = T_{H2}$ during the expansion.

### 7.4.3 Injection of molecular deuterium in the arc

In order to study the influence of the mass on the transport of radicals and particularly on the confinement of radicals inside the jet, deuterium is added to the hydrogen flow and the created D atoms are monitored in the H$_2$-D$_2$ expanding plasma. The flow mixture is composed of 3.1 slm H$_2$ and 0.4 slm D$_2$. The cascaded arc parameters stay unchanged.

The deuterium atom perpendicular temperature profile along the jet axis, measured at $p_{back} = 20$ Pa, is shown in Fig. 7.3, together with the H atom $T_⊥$ profile. Both profiles indicate the same shock wave location and width. In the supersonic domain as well as in the subsonic domain, H and D temperature profiles overlap reasonably. Throughout the stationary shock wave, the D atom $T_⊥$ is slightly lower than the H atom $T_⊥$ by approxi-
mately 20 %. This departure from thermal equilibrium in the shock region is not yet well understood. It may be caused by a difference in energy exchange in a D-H₂ collision as compared to an H-H₂ collision, H₂ molecules being the main collision partner.

The deuterium atom density profile for an H₂-D₂ plasma at a pressure of 20 Pa is compared to the hydrogen atom density profile for a pure H₂ plasma in Fig. 7.2. At the plasma source outlet, the ratio of the H density to the D density is approximately equal to the injected gas flow ratio. This supports the idea of a high dissociation degree and an efficient mixing inside the arc channel where the plasma is created. In the supersonic domain, we obtain $\beta = 2.0 \pm 0.1$ and $z_{source} = +(4 \pm 1)$ mm when using Eq. 7.5 to describe the density drop. The value of the exponent $\beta$ is equal to the value characteristic for a neutral gas supersonic expansion. Deuterium atoms are well coupled to H₂ molecules, as expected in view of the equal mass of the two particles, and thus they are relatively well confined inside the plasma jet, as indicated by the value of $\beta$. Nevertheless when the jet is underexpanded, losses of D atoms can occur in the barrel shock, i.e at the sides of the jet, where the mean free path is large and the radial velocity high. Across the stationary shock wave, the forward D atom flux is not conserved. In the shock region D atoms can escape the plasma jet because of the large mean free path. In the subsonic region, even if the confinement of D is better than that of H, D atoms are lost. The mass of the particle plays a role in the transport mechanism, since the loss of radicals arises from a diffusion process. It is shown however that when the mass of the carrier gas is close to that of the radicals the confinement inside the jet is improved but the losses by diffusion can not be avoided if the wall-recombination probability is high.

## 7.5 Recombination in the arc nozzle

From the measured H atom density and axial velocity it is possible to estimate the plasma source dissociation degree $\alpha$. This parameter characterizes the source strength, i.e. the capability of producing H radicals. The dissociation degree can be expressed as follows

$$ \alpha = \frac{\phi_H}{2\phi_{H_2}^0} = \frac{n_H}{n_H + 2n_{H_2}}, \qquad (7.7) $$

where $\phi_H$ is the ground-state atomic hydrogen flux and $\phi_{H_2}^0$ the gas flow of molecular hydrogen at the arc inlet. Under our experimental conditions, $\phi_{H_2}^0 = 1.56 \times 10^{21}$ s⁻¹. At the origin $z_o$ of the H atom expansion, i.e. inside the arc nozzle, $\alpha \approx 0.1$, if the plasma is assumed to flow with the local speed of sound $c_{s,o}$. Just behind the source exit ($z = 1$ mm), using the H and H₂ density profiles, one finds $\alpha \approx 0.02$. In our conditions, the low value of $\alpha$ does not arise from volume recombination of H atoms in view of the low three-body reaction rates [28]. For comparison, the dissociation degree in the background is found to be less than 1%. The low value of $\alpha$ in the downstream part of the flow is a direct consequence of the loss of H by diffusion during the expansion process. However, such a low dissociation degree in the source is not expected at an electron temperature of around 1 eV.
To a large extent, the low dissociation degree at the arc exit can be connected with the low electron density observed in the case of an expansion of a hydrogen plasma [14]. As already mentioned, the emerging plasma is dark. It thus appears that the expanding plasma jet is mainly composed of H₂ molecules. The low local value of α as well as the ionization loss need further clarification. A possible way to explain this experimental fact is to consider the production of vibrationally excited hydrogen molecules (in the electronic ground-state) on a surface. Neutral hydrogen atoms can indeed recombine on a surface to form excited H₂ molecules

$$H_{gas} + H_{ad} \rightarrow H_{2}^{v,J}_{gas}. \qquad (7.8)$$

This chemical reaction has been extensively studied both experimentally and theoretically (see for instance [32, 33] and [34]). Because the probability for recombinative desorption of H atoms on a Cu surface is close to unity [34, 35], H atoms can easily be lost via the previous reaction in the arc nozzle, which is made of Cu. Accordingly, a large amount of H₂<sup>v,J</sup> molecules can be generated. Atomic hydrogen ions H⁺, the most abundant ion, can easily exchange their charge with vibrationally excited H₂ molecule leading to H₂⁺ ions (see section 7.2). The dissociative recombination of H₂⁺ molecular ions is the main source of visible light at low electron density (production of H<sub>α</sub> radiation if H₂<sup>v</sup> (v > 5) or T<sub>e</sub> ≈ 1 eV).

Inside the arc nozzle, recombination processes are favored. In the standard cascaded arc configuration, the nozzle acts as the anode. The arc root however attaches at the nozzle inlet (nozzle length = 17 mm) and thus no energy is supplied in the largest part of the nozzle. Therefore surface-recombination of H atoms and the subsequent H₂<sup>v,J</sup> formation can explain the low value of both the dissociation degree and the ionization degree at the arc exit: H atoms are lost at the nozzle wall and the created H₂⁺ ions recombine before they escape the source.

CARS measurements performed on a pure hydrogen plasma [23] reveal a low density of excited H₂ molecules in the jet. At z = 20 mm, the density of H₂(v=0) is equal to 1.4 × 10<sup>21</sup> m<sup>-3</sup>, the density of H₂(v=2) is approximately equal to 10<sup>18</sup> m<sup>-3</sup> and the density of H₂(v=4) is less than 10<sup>17</sup> m<sup>-3</sup>. This may denote that all H₂<sup>v,J</sup> molecules produced in the arc nozzle by wall-recombination of H atom are quickly lost either via the formation of H₂⁺ ions (that explains the absence of visible light) or via the formation of negative hydrogen ions H⁻ by dissociative attachment of low energy (≈ 1 eV) electrons [36]

$$H_{2}^{v,J} + e \rightarrow H + H^{-}. \qquad (7.9)$$

There are indications for the production of H⁻ ions in a cascaded arc, e.g. the existence of blue radiation around 300 nm at the plasma jet periphery at high pressure [37] that may originate from the recombination of H⁻ [38]. The CARS measurements unfortunately could not be performed at the arc exit because of the crossed laser beam geometry and thus information concerning the energy distribution of H₂ molecules leaving the arc is still lacking. As it appears, the process of surface-recombination in the arc nozzle is complex, but it certainly needs further investigation since it plays a decisive role when the plasma jet is to be used as an efficient H ion or H radical source.
## 7.6 Axial velocity component and sound speed

Another important parameter to be determined in order to characterize the atomic radical transport mechanism is the flow velocity. The measured H atom axial velocity component profile along the jet centerline is shown in Fig. 7.4 and Fig. 7.5 at 20 Pa and 100 Pa background pressure respectively. Throughout the stationary shock wave, the distribution function is found to be non-Maxwellian. The departure from thermodynamic equilibrium in the shock region of an expanding plasma is studied in chapter 8. In this section we only consider the mean axial velocity in the shock region, i.e. the velocity calculated from the first moment of the distribution function. To clearly discern the three different flow regimes (supersonic, shock, subsonic), the flow velocity has to be compared with the local speed of sound in order to obtain the Mach number $M$. The speed of sound is given by

$$c_s = \sqrt{\frac{\gamma k_B T}{m}}, \qquad (7.10)$$

where $\gamma$ is the adiabatic exponent and $k_B$ the Boltzmann constant, and $m$ the mass. The on-axis development of the speed of sound is plotted in Fig. 7.4 and Fig. 7.5 for two different background pressures using the measured H atom parallel temperature ($T_‖$), the temperature associated with the velocity distribution parallel to a stream line, and using as mass number $m = 2$ amu and $\gamma = 1.3$ since H$_2$ is the major component of the jet. The sound speed at the expansion origin $z_o$ inside the arc nozzle can be calculated using $T_o$: with $\gamma = 1.3$, $c_{s,o} \approx 4000 \text{ m s}^{-1}$. At the plasma source exit the Mach number is already high, $M = 2$. A theoretical expression describing the evolution of the axial velocity along a stream line in the supersonic domain can be obtained from the Bernoulli equation and the Poisson adiabatic law [37]. If we consider that the plasma starts to expand with $c_{s,o}$, one finds for the velocity

$$w(z) = \frac{c_{s,o}}{\sqrt{\gamma - 1}} \sqrt{\gamma + 1 - 2 \left( \frac{z_{\text{ref}}}{z + z_o} \right)^{\beta(\gamma - 1)}}. \qquad (7.11)$$

Since the previous expression contains $\beta$ it accounts automatically for the outward diffusion effect when $\beta > 2$. In Fig. 7.4 the measured H atom axial velocity profile in the supersonic domain at 20 Pa is compared with a calculation using Eq. 7.11. The agreement is satisfactory with $\gamma = 1.55$. This again supports the idea of a H atom fluid partly decoupled from the H$_2$ molecule fluid in the course of the plasma expansion [8]. At this point however, we suffer from a lack of information about the transport properties of H$_2$ molecules. It is now clear that the problem concerning the coupling between H and H$_2$ in the jet can only be untangled by measuring the H$_2$ velocity and temperature profile along the jet axis.

As can be seen from Eq. 7.11 the maximum flow velocity depends on $T_o$ and accordingly on the source temperature. In our conditions, $w_{max} = 8600 \text{ m s}^{-1}$ for H atoms. However, before the hydrogen atoms can reach the maximum velocity they start colliding with the residual background gas meaning that the expansion never enters the so-called frozen regime [22]. The existence of a stationary shock wave is clearly revealed in Fig. 7.4 and
<figure><img src="image_57.png" /><figcaption>**Fig. 7.4:** Development of the atomic hydrogen axial velocity component $w_z$ along the jet axis at $p_{back}$ = 20 Pa (open square) and speed of sound $c_s$ calculated from the measured parallel temperature ($T_{||}$) with $m$ = 2 amu and $\gamma$ = 1.3 (circle). $M$ = 4.2 ahead of the stationary shock wave. The lines represent the outcome of a model describing the H axial velocity in the supersonic domain, see Eq. 7.11, with $\gamma$ = 1.55 (solid) and with $\gamma$ = 1.3 (dashed).</figcaption></figure>

Fig. 7.5. In the two figures, the effect of the background pressure on both the shock wave position and the shock wave thickness is visible. The Mach disk location $z_M$ is found to be equal to 50 mm at 20 Pa and approximately 40 mm at 100 Pa. At $p_{back}$ = 20 Pa, the measured shock thickness is equal to 40 mm. It can be compared with the H-H₂ collision mean free path ahead of the shock region since H atoms mainly collide with H₂ molecules. The mean free path is given by

$$ \lambda_{H-H_2}^{mfp} = \frac{1}{n_{H_2}\sigma_{H-H_2}}, \qquad (7.12) $$

where $n_{H_2}$ is the molecular hydrogen density and $\sigma_{H-H_2}$ the momentum exchange cross-section. The H₂ density has been measured by CARS [23]: $n_{H_2}$ = $4 \times 10^{20}$ m⁻³ at 20 Pa. The value for the cross-section can be found in literature [39]; at 500 K $\sigma_{H-H_2}$ ≈ $2 \times 10^{-19}$ m². At 20 Pa a mean free path for momentum exchange ahead of the shock wave of 12 mm is found. Thus the shock wave thickness is in the order of few $\lambda_{H-H_2}^{mfp}$. At this pressure the mean free path becomes in the order of the jet dimension (10 mm radius at $z$ = 10 mm if a 45° expansion is assumed). Therefore at this point the flow can not be satisfactorily described using the hydrodynamic theory and one must turn to the more complex gas kinetic theory.

The Mach number ahead of the stationary shock wave can be deduced from our mea-
<figure><img src="image_58.png" /><figcaption>**Fig. 7.5:** Development of the atomic hydrogen axial velocity component $w_z$ along the jet axis at $p_{back}$ = 100 Pa (solid triangle) and speed of sound $c_s$ calculated from the measured parallel temperature ($T_{||}$) with $m$ = 2 amu (circle). $M$ = 3.3 in front of the shock wave.</figcaption></figure>

surements: $M$ = 4.2 at 20 Pa and $M$ = 3.3 at 100 Pa. Using the Rankine-Hugoniot relations both the theoretical density and temperature jump across the shock wave can be calculated using these Mach numbers. With an adiabatic exponent of 1.55, the expected density jump is equal to 3.6 at 20 Pa and 3.2 at 100 Pa in complete disagreement with the measured values of 1.3 and 0.5 respectively (see Fig. 7.2). The observed discrepancy is a direct consequence of the losses of H atoms by diffusion combined with a weakening of the shock structure. The calculated temperature jump is equal to 5.9 at 20 Pa and 3.4 at 100 Pa, whereas the measured jump is equal to 3.3 and 2.7 at 20 Pa and 100 Pa respectively. A too small temperature jump across the shock wave confirms the idea of the invasion of the shock region by the surrounding background gas. The resulting energy transfer modifies the temperature. Behind the stationary shock wave, the velocity decreases because of momentum transfer to the background gas particles.

## 7.7 Plasma jet vicinity

In order to evaluate the difference in the H atom concentration and temperature between the core of the jet and the surrounding background gas, it is of interest to study jet cross-sections. The radial profile of the H atom density as well as the corresponding temperature profile have been measured at $p_{back}$ = 20 Pa at an axial position $z$ of 30 mm from the arc
<figure><img src="image_59.png" /><figcaption>**Fig. 7.6:** Radial profile of the H atom density (left) and perpendicular temperature (right) at 20 Pa background pressure measured at 30 mm behind the arc outlet. The H density outside the jet is relatively low due to wall-recombination processes. For comparison, the H₂ density around the plasma jet is approximately $3 \times 10^{21} \text{ m}^{-3}$ at 20 Pa. The H₂ rotational temperature in the background is found to be 600 K [23], in good agreement with the H temperature at large r position.</figcaption></figure>

outlet, i.e. in the shock region.

The density profile is shown in Fig. 7.6. In the core of the jet the H density is high, $n_H = 6 \times 10^{19} \text{ m}^{-3}$ whereas it is relatively low in the jet vicinity, $n_H \approx 10^{19} \text{ m}^{-3}$. Therefore, in the shock wave zone the H radical expansion is not overexpanded, contrary to that of non-reacting neutrals. As mentioned previously, the low H density in the background is a consequence of the recombination of H atoms at the walls of the vacuum chamber. For comparison, the background molecular hydrogen density at 20 Pa is about $3 \times 10^{21} \text{ m}^{-3}$ as measured by CARS [23]. The existence of strong density gradients at the jet boundary supports the idea of an outward diffusion process.

The radial profile of the H atom perpendicular temperature is also shown in Fig. 7.6. The temperature is on the average relatively high in the core of the jet, approximately 1100 K, however it is slightly lower around the jet axis. Indeed at $r \approx 0$, a dip is visible in the temperature profile in Fig. 7.6. The temperature is higher at the jet boundary because of collisions with the background gas. The temperature gradually decreases in the vicinity of the plasma jet to about 500 K due to heat transfer to the vessel wall. The measured H atom background gas temperature is in good agreement with the H₂ rotational temperature measured outside the jet by CARS [23].

## 7.8 Conclusions

The study of the transport of ground-state hydrogen atoms in the expansion of a thermal hydrogen plasma reveals that surfaces can have a great influence on the chemistry and
on the dynamics of a system containing reactive particles. The low dissociation degree as
well as the quasi-absence of charged particles at the plasma source exit, can be explained
by the recombination of H atoms on the arc nozzle wall and the subsequent formation
of vibrationally excited hydrogen molecules. As expected the transport of H atoms is
also strongly influenced by surface-recombination processes. The efficient loss of H atoms
at the vessel walls creates large density gradients between the jet and the surroundings
which in turn force H to diffuse out of the plasma jet. It is shown that when the surface
loss probability is high, e.g. combination of a large wall-recombination probability with a
long residence time, the losses of radicals by diffusion can not be avoided even when the
mass of the carrier gas is close to the mass of the radical. A full understanding of the
decoupling between the H fluid and the H₂ fluid, that results from the outward diffusion
of H atoms, during the expansion would however necessitate to also investigate the H₂
molecules transport properties. In other words, the “coupling problem” will not be solved
without the measurement of the H₂ velocity and temperature profiles under the same
experimental conditions.

This work stresses the importance of plasma-wall interactions in the manufacture of
a plasma source with specific requirements but also in the design of reactors devoted to
plasma-assisted chemistry. Nowadays the transport of radicals in a plasma expansion
starts to be better understood, and technological solutions can be proposed in order to
enhance the confinement inside the plasma jet (e.g. by using a vessel wall material with
a low surface-recombination probability). Nevertheless further investigations have to be
performed on plasma-wall interactions and the related generation of molecules. To better
understand the dynamics of the process and to investigate the possibility of using a surface
as a catalyst to favor the production of a desired molecule for instance. This may open in
the near future a new field in plasma-aided chemistry.

Acknowledgments

The authors would like to acknowledge fruitful discussion with Prof. M.C.M. van de
Sanden. The authors greatly appreciate the skillful technical assistance of M.J.F. van de
Sande, A.B.M. Hüsken, and H.M.M. de Jong. This work is part of the research program of
the Netherlands Foundation for Fundamental Research on Matter (FOM). It is financially
supported by the Netherlands Organization for Scientific Research (NWO) as well as the
Euratom foundation.
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# Chapter 8

## Behavior of the H atom velocity distribution function within the shock wave of a hydrogen plasma jet<fnref n="1" />

### Abstract

The evolution of the ground-state hydrogen atom velocity distribution function throughout the stationary shock wave of a supersonic hydrogen plasma jet ($3 < \text{Mach number} < 4$) is studied using laser induced fluorescence spectroscopy. A clear departure from thermodynamic equilibrium is observed. The H atom velocity distribution function can be decomposed into two Maxwellian distributions. The fast component of the distribution corresponds to the conditions upstream of the shock region. The slow component corresponds to the conditions in the shock region, i.e. within the shock front the mean velocity and the temperature of this atom group vary. Across the shock wave, the H atom population is gradually tranferred from the fast to the slow component by means of collisions. The development of the mean axial velocity is modeled using the Mott-Smith approach. Deviation from the theoretical shock profile is interpreted in terms of the non-conservation of both the H atom forward flux and momentum across the shock wave.

<footnote n="1">¹ Adapted from:
S. Mazouffre, P. Vankan, R. Engeln, and D.C. Schram, accepted for publication in *Phys. Rev. E* (2001).</footnote>
## 8.1 Introduction

The study of physical systems under non-equilibrium conditions has been the subject of numerous researches for a long period of time. An example of a system characterized by a departure from thermodynamic equilibrium is the interior of a shock wave [1]. Across a shock wave, a flowing medium exhibits a rapid change in its macroscopic properties, i.e. density, velocity, and temperature, over a short distance (merely a few mean free paths). Shock waves are nowadays relatively easy to produce and to control using several kinds of experimental facilities like wind tunnels, shock tubes, and free jets. Moreover laboratory shock waves offer the possibility to cover a vast ensemble of physical conditions: a broad range of Mach numbers (M) is accessible, the flow regime can be varied from continuum to rarefied, the shock wave can be created in an atomic gas, a molecular gas or a plasma, to only name a few.

As recently pointed out by Ramos *et al* [2], scientists are confronted with a paradox. Although the general properties of a shock wave are at present well established, there still exist numerous issues concerning the evolution of the distribution function within such a disturbed region. This peculiar situation is a direct consequence of the relative lack of detailed experimental studies in comparison with the large amount of theoretical works (see for instance [3, 4, 5] and references herein).

Most of experimental investigations of a gas(plasma) shock wave are devoted to the measurement of the development of macroscopic parameters. Several diagnostic techniques have been employed: electron-beam absorption [6] and electron beam induced fluorescence [7], Rayleigh scattering [8], laser-Doppler technique [9] and Thomson scattering in the case of charged particles [10]. Despite the fact that those studies furnish a large amount of data which can be compared with the prediction of different theories, i.e. hydrodynamic approach (Navier-Stokes equation) or kinetic theory (Boltzmann equation), they only provide indirect information concerning the distribution function. Up until now only a few works have been directly devoted to the study of the distribution function across a shock wave. In the work of Ramos *et al* [2] the rotational state distribution function of N₂ molecules is studied in a nitrogen free jet by means of Raman scattering, and in the work of Pham-Van-Diep *et al* [11] the velocity distribution function in a helium jet is measured by means of electron beam induced fluorescence (see also Ref. [12] for measurements in an argon jet). Those works confirm the main conclusions of Mott-Smith [13] and Glansdorff [14] about a bimodal distribution function in a shock region, even though the discovery of a third component, related with the invasion of the jet, is reported.

In the present contribution, the behavior of the velocity distribution function (VDF) of ground-state hydrogen atom is directly studied within the stationary shock wave of a weakly ionized expanding hydrogen plasma ($3 < M < 4$, depending on the background pressure) by means of two-photon absorption laser induced fluorescence (TALIF) spectroscopy. From the VDF, the local H atom density, mean velocity and temperature are also deduced. The large Doppler broadening of H atoms, combined with the relatively high temperature of the plasma medium, allows to obtain accurate data on the VDF. In this plasma environment, we are confronted with a very particular situation: neither the H atom forward flux nor
the H atom momentum is conserved within the shock wave [15]. Recombination of H atoms at the vessel walls induces large density gradients between the core of the plasma jet and its surrounding, which in turn force H to diffuse out of the jet [16]. This effect is very pronounced in the shock region where the mean free path is large. Because of recombination processes in the arc nozzle, the expanding plasma is mainly composed of H₂ molecules to which H atoms transfer most of their momentum. As a consequence the Rankine-Hugoniot relations are not valid for the H atom fluid, not even at high background pressure.

## 8.2 Experimental determination of the H atom VDF

Ground-state hydrogen atoms are spatially probed by using a two-photon absorption laser-induced fluorescence (TALIF) technique [17, 18, 19]. Only a short overview is presented here (see chapter 4 for more information). A tunable 20 Hz Nd:YAG pumped dye laser delivers radiation around 615 nm. The output of the dye laser is frequency-tripled resulting in 2 mJ of tunable UV light around 205 nm. The UV laser beam is focused parallel to the plasma expansion axis. Hydrogen atoms are excited with two 205 nm photons from the 1s²S ground state to the 3d²D and 3s²S states. The excitation is monitored by detection of the resulting fluorescence yield on the Balmer-α line at 656 nm using a gated photo-multiplier tube. An interference filter is used to isolate the Hα line from the plasma emission. A slitmask is used to define the detection volume which dimensions are much smaller than any gradient length. The dye-laser frequency is accurately calibrated by the simultaneous recording of the absorption spectrum of molecular iodine. A spectral scan over the two-photon transition provides a direct access to the local H atom velocity distribution function (VDF), which allows for discussion on the state of equilibrium of the plasma flow. From the measured line profile the H density and mean velocity are obtained. In the case of a Maxwellian distribution a H atom temperature can also be defined. The influence of the laser line profile (FWHM ≈ 0.18 cm⁻<fnref n="1" />) on the shape of measured spectral profile is neglected. This is legitimate in view of the low mass of H atoms and the relative high plasma temperature.

The experimental determination of the velocity distribution function of a particle group is based on the Doppler effect. The broadening of a spectral line results from the Doppler shift of the optical transition induced by the spread in velocity within a group of particles. A particle moving with velocity $v_z$ towards a probing light source absorbs a photon at a frequency $ν$ given by<fnref n="1" />

$$ν = ν_o \left( 1 - \frac{v_z}{c} \right), \qquad (8.1)$$

where $ν_o$ is the absorption frequency of the particle at rest, c is the speed of light in vacuum.

<footnote n="1">¹ In view of the low particle velocity in comparison with the speed of light, relativistic effects are not taken into account. For information, the longitudinal Doppler effect in relativity reads $ν = ν_o \sqrt{\frac{c+v}{c-v}}$ when the source and the observer move towards one another [20]. A transverse Doppler effect, that is purely a relativistic effect, is also predicted by the theory [20].</footnote>
If the Doppler effect is the main broadening mechanism, the measured absorption profile is essentially the distribution of the velocity component $v_z$. Indeed one can not directly probe the three-dimensional velocity distribution $F_v$, i.e. the density in velocity space, only the one-dimensional distribution $f_v(v_z)$ of the velocity component in the direction $z$ of propagation of the laser beam is measured. The 1D and 3D velocity distribution functions are related by

$$f_v(v_z) = \iint F_v dv_x dv_y, \qquad (8.2)$$

which corresponds to a projection in velocity space<fnref n="2" />. The energy distribution function $f_E(E)$ follows from a transformation of velocity into translational energy via $E = \frac{1}{2}mv^2$.

The hydrogen plasma is created by a cascaded arc [21]. The arc channel is composed of 4 insulated plates and has a diameter of 4 mm. The operating standard conditions are: a 55 A DC current, a cathode-anode voltage of 150 V and a H₂ gas flow of 3.5 standard liters per minute. The stagnation pressure inside the arc is 0.14 bar. The thermal hydrogen plasma expands from a straight nozzle with a diameter of 4 mm into a low background pressure ($p_{back}$) chamber. At the arc outlet the plasma turns into almost a hot gas, most electrons being consumed in the nozzle (ionization degree < 1%). Furthermore the generation of H₂ molecules by surface-recombination of H atoms is favored inside the arc nozzle which leads to a relatively low dissociation degree at the source exit. Thus H atoms flow in an environment mostly composed of H₂ molecules [15].

The flow pattern of H atoms in supersonic plasma jets generated from H₂ and from an Ar-H₂ mixture has been recently investigated by studying the H atom density, temperature and velocity development during the expansion process [15, 16, 22]. Here, we focus on the study of the plasma shock region with two main objectives: to examine the evolution of the H atom VDF in this region and to understand the related shock wave characteristics.

## 8.3 Non-Gaussian VDF within the stationary shock wave

The behavior of the ground-state H atom VDF is examined throughout the normal shock wave of the expanding hydrogen plasma. The system being in stationary state, the VDF only depends on spatial coordinates. The VDF is measured along the jet centerline, namely the velocity distribution is observed parallel to a stream line. Therefore only the axial component of the H atom velocity is determined. The departure from thermodynamic equilibrium is directly reflected in the shape of the velocity distribution function, which markedly deviates from the Gaussian form, i.e. the equilibrium form, within the shock wave, as we will see.

<footnote n="2">² If $F_v$ is isotropic then it can be reconstructed from $f_v(v_z)$.</footnote>
<figure><img src="image_60.png" /><figcaption>**Fig. 8.1:** Ground-state hydrogen atom velocity distribution function (VDF) measured at 20 Pa background pressure within the stationary shock wave of an expanding hydrogen plasma. In the shock region (10 to 50 mm), where departure from thermodynamic equilibrium is observed, the VDF consists of the sum of two Gaussian terms corresponding to the conditions upstream and downstream of the zone of silence.</figcaption></figure>
<figure><img src="image_61.png" /><figcaption>**Fig. 8.2:** Ground-state hydrogen atom velocity distribution function (VDF) measured at 100 Pa background pressure within the stationary shock wave of an expanding hydrogen plasma. In the shock region (6 to 30 mm), where departure from thermodynamic equilibrium is observed, the VDF consists of the sum of two Gaussian terms corresponding to the conditions upstream and downstream of the zone of silence. It can be seen that all H atoms are thermalized (Gaussian VDF) in front of the shock wave.</figcaption></figure>
Eight H atom velocity distribution functions recorded at different locations in the shock wave region are shown in Fig. 8.1 for a background pressure of 20 Pa. The local VDF is obtained from the measured spectral profile using the Doppler shift relation (see Eq. 8.1) to switch from frequency space to velocity space. At $p_{back} = 20$ Pa, the normal shock wave stretches from $z = 10$ mm to $z = 50$ mm. The Mach number prior to the shock equals 4. The end of the shock is defined as the position where $M = 1$.

Ahead of and behind the shock wave, the VDF is Gaussian. The corresponding energy distribution function is therefore Maxwellian, meaning that the H atom flow is in thermodynamic equilibrium. Throughout the stationary shock wave, there is a clear departure from thermodynamic equilibrium, since the VDF becomes non-Gaussian, as can be seen in Fig. 8.1. However, the H atom VDF can be decomposed into two Gaussian VDF, i.e. into two equilibrium components, corresponding to the conditions upstream of the zone of silence and in the shock wave. This so-called bimodal approximation of the distribution function is not arbitrary, but it arises directly from the definition of a shock wave which is a transition zone where a supersonic flow has to adapt to the ambient subsonic conditions. It was first proposed about half a century ago by Mott-Smith [13] and Glansdorff [14] in order to be able to solve the Boltzmann transport equation in a gas shock wave. The first component corresponds to the supersonically flowing particles that have not yet experienced any collisions with particles from the shock region: we name it the *fast* component. The second component reassembles particles which have experienced at least one collision and thus have become partly thermalized: we name it the *slow* component.

The bimodal approximation for the H atom VDF across the shock wave is found to be also valid at a background pressure of 100 Pa, as shown in Fig. 8.2. At this pressure the normal shock wave stretches from $z = 6$ mm to $z = 30$ mm, and $M = 3$ prior to the shock. Across the shock wave of the expanding plasma, the H atom population is gradually transferred from one group (the fast component of the VDF) to the other group (the slow component of the VDF) by means of collisions with the background particles, i.e. H₂ molecules under our experimental conditions. This transfer of population, and the subsequent emergence of a bimodal VDF, is clearly visible in Fig. 8.1 and Fig. 8.2. In Fig. 8.3 we plotted the contents of the two Gaussian components as well as the total H atom density across the shock wave for a background pressure of 20 Pa. The emergence of the slow component and the disappearance of the fast component can easily be seen. Note that, due to plasma-wall interactions, the total H atom density does not increase across the shock wave [15, 16, 22]. At the end of the shock wave, all H atoms have changed from an initial supersonic Gaussian VDF to a final subsonic Gaussian VDF and the system returns to equilibrium. Note that all H atoms are thermalized (Gaussian VDF) in front of the shock, i.e. they all have at least undergone one collision before the flow becomes subsonic. The fast component of the VDF contains only particles with a positive axial velocity whereas the slow component contains both particles with a positive and a negative axial velocity, and the dispersion in velocity (given by the width of the Gaussian VDF) is larger. This corresponds to an increase in disorder, i.e. an increase in the entropy of the H atom fluid throughout the shock wave.
<figure><img src="image_62.png" /><figcaption>**Fig. 8.3:** On-axis profile of the total H atom density, i.e. the contents of the VDF, across the stationary shock wave at 20 Pa background pressure (open triangle). The population of the fast component (solid square) decreases to finally vanish whereas the population of the slow component (solid circle) increases. At the end of the shock wave, only remain subsonically flowing H atoms.</figcaption></figure>

The H atom VDF within the shock region is well described by a bimodal approximation, as can be seen from the fit in Fig. 8.1 and Fig. 8.2. No evidence for the existence of a third component, the so-called scattered fraction, is experimentally found, contrary to what is reported in previous studies [2, 11, 12]. In References [11] and [12], one-dimensional VDF measured in He and Ar shock waves are compared with predictions from Monte Carlo Direct Simulation (MCDS) method based on realistic interatomic potential. An excellent agreement is found between experimental data and data calculated using MCDS. In particular, the MCDS method is able to predict the existence of a third group of particles. In other words, in those gas shock wave the Mott-Smith approach fails to represent the distribution function (Mott-Smith and MCDS predictions are compared in Ref. [11]).

In our case, the validity of a bimodal model may arise from the fact that, contrary to the Mott-Smith approach, the slow VDF does not correspond to the conditions dowstream of the zone of silence but it corresponds to the conditions in the shock wave. In other words, as we will see in the next section, the mean velocity and the temperature of the slow Gaussian component vary through the shock region, and therfore this component accounts for mixing effects [11, 12].

The scattered component may also originate from gas particles which penetrate the shock region from the jet boundary and from the Mach disk. This invasion can lead to the appearance of a third fraction since the initial VDF of these particles (as measured along
a stream line) is neither the fast one nor the slow one. Under our experimental conditions, due to wall-association processes H atom are almost absent in the background gas. As a consequence, the shock wave is invaded mainly by H₂ molecules, and no H atom scattered distribution is formed. The appearance of a third fraction in the VDF would in that case merely be visible in the H₂ molecule VDF. Note that this effect should be more pronounced at low background pressure where the invasion of the shock region by the background gas is favored [23, 24].

## 8.4 Velocity and temperature profiles along the jet axis

From the measured H atom velocity distribution function, a mean axial velocity as well as a parallel temperature, which represents the spread in velocity along a stream line, can be calculated for each Gaussian component. The velocity is determined from the Doppler shift of the peak. The temperature is determined from the Doppler broadening of the peak, which is the dominant broadening mechanism in our case, taking into account the bandwidth of the laser profile which is assumed to be Gaussian.

Throughout the stationary shock wave the average H atom axial velocity is obtained from the first moment of the VDF. In this region the H atom parallel temperature can not be strictly defined since the flowing medium is not in thermodynamic equilibrium. We define an average local parallel temperature $\hat{T}_{\parallel}$ by weighting the temperatures of the two Gaussian components with their corresponding densities

$$ \hat{T}_{\parallel} = \frac{n_{\text{fast}} T_{\parallel \text{fast}} + n_{\text{slow}} T_{\parallel \text{slow}}}{n_{\text{fast}} + n_{\text{slow}}}. \qquad (8.3) $$

The subscripts *fast* and *slow* refer to the two components of the distribution function. $\hat{T}_{\parallel}$ can be seen as the temperature which would be given by a hypothetical thermometer plunged into the shock region (ignoring the perpendicular temperature). The H atom velocity profile along the plasma jet centerline is shown in Fig. 8.4 for $p_{\text{back}} = 20$ Pa and for $p_{\text{back}} = 100$ Pa. On both graphs, the measured velocity is compared with the local speed of sound calculated from the measured H atom parallel temperature ($\hat{T}_{\parallel}$ across the shock wave) and using $m = 2$ amu and $\gamma = 1.3$ since the jet is mainly composed of H₂ molecules. The corresponding on-axis development of the H atom parallel temperature is shown in Fig. 8.5 for $p_{\text{back}} = 20$ Pa and for $p_{\text{back}} = 100$ Pa. The general shape of both the velocity and the temperature profile in the course of the plasma expansion has already been explained in preceding articles [15, 16]. In this contribution we only focus on the shock region.

As already mentioned, the *fast* component of the H atom VDF, i.e. the component which corresponds to the conditions upstream of the zone of silence, represents the fraction of hydrogen atoms that have not yet collided with the particles of the shock wave region, namely H₂ molecules. This component thus exhibits the flow properties of a supersonic beam [25]. The mean H atom axial velocity of the *fast* component remains high. At
<figure><img src="image_63.png" /><figcaption>**Fig. 8.4:** Profile of the H atom axial velocity (open triangle) along the jet centerline measured at 20 Pa (left) and 100 Pa (right). Throughout the shock wave the average H atom velocity is calculated from the first moment of the distribution function. The mean velocity of both the fast (solid square) and the slow (solid circle) component of the VDF are shown in the shock region. Also shown is the axial profile of the speed of sound (diamond) determined from the measured H atom parallel temperature with $m = 2$ and $\gamma = 1.3$.</figcaption></figure>

20 Pa background pressure the velocity is frozen at about 7500 $m s^{-1}$, whereas at 100 Pa the velocity still increases across the shock wave meaning that H atom can still convert part of the thermal energy they gained in the plasma source into kinetic energy. The corresponding parallel temperature decreases down to 300-200 K, meaning that the spread in velocity decreases. The fast component of the VDF is thus often referred to as the cold component.

The *slow* component of VDF, i.e. the component which corresponds to the conditions in the shock wave, contains H atoms that have experienced at least one collision with the particles located in the stationary shock wave. The mean axial velocity of the slow component is low at the beginning of the shock region, around $2000 m s^{-1}$. It increases across the shock wave due to collisions slow H atoms undergo with remaining supersonically flowing particles. The rise of the mean velocity of the VDF slow component is faster at the beginning of the shock region, as can be seen in Fig. 8.4. It arises from the fact that collisions of slow H atoms with fast $H_2$ molecules (the main collision partner) become quickly scarce because the population of the fast $H_2$ VDF component drops also fast across the shock wave to finally disappear. The temperature of the slow component is relatively high. It increases because kinetic energy is converted into thermal motion in a collision involving a fast H atom. In other words, directed motion is transformed into random motion within a shock wave. The slow component of the VDF is also often called the warm component.
<figure><img src="image_64.png" /><figcaption>Fig. 8.5: Axial profile of the H atom parallel temperature (open triangle) at 20 Pa (left) and at 100 Pa (right). Throughout the shock wave an average temperature is determined from the temperature of the two Gaussian components of the distribution function (see Eq. 8.3). The temperature of both the fast (solid square) and the slow (solid circle) component of the VDF are also shown in the shock region.</figcaption></figure>

## 8.5 Theoretical velocity profile in the shock region: Mott-Smith approach

The H atom average velocity measured throughout the stationary shock wave along the plasma jet centerline can be compared with a theoretical model based on a bimodal approximation of the velocity distribution function. Under the assumption that the VDF is the sum of a cold and a warm component, the density profile as well as the corresponding velocity profile of the flowing fluid can be calculated as shown by Mott-Smith [13] and Glansdorff [14]. The centerline velocity is given by

$$w(\tilde{z}) = \frac{w_2(1 + e^{4\tilde{z}/L})}{(w_2/w_1) + e^{4\tilde{z}/L}}, \qquad (8.4)$$

where $\tilde{z} = z - z_0$, $z_0$ being the center of the shock wave (point of inflection), $w_1$ and $w_2$ are the velocity ahead of and behind the shock respectively, and $L$ is the shock thickness defined as

$$L = (w_1 - w_2) \frac{dz}{dw} \Big|_{\max}, \qquad (8.5)$$

according to the Prandtl formalism [26]. We found that $L= 16.8$ mm and $9.7$ mm at 20 Pa and 100 Pa respectively. At 100 Pa, $L_{Hydro} = 2$ mm (see Eq. 2.28), $L_{MS} = 5$ mm [13], and $L_{Two-Fluids} \approx 6$ mm [14]. In both cases $L$ is underestimated, however, in comparison with a hydrodynamic calculation, models based on the solution of the Boltzmann equation provide a value closer to the real one. The velocity can also be expressed as a function
<figure><img src="image_65.png" /><figcaption>**Fig. 8.6:** Normalized H atom velocity (see Eq. 8.6) profile within the stationary shock wave of an expanding hydrogen plasma at 20 Pa (open triangle) and at 100 Pa (solid diamond) background pressure. The parameter $z_0$ corresponds to the center of the shock wave and L is the shock thickness. The solid line represents the calculated normalized velocity profile under the assumption of a bimodal distribution function based on the theory developed by Mott-Smith [13] and Glansdorff [14].</figcaption></figure>

of the Mach number M ahead of the shock front and the adiabatic exponent γ using the Rankine-Hugoniot relations. In order to make the results independent of the conditions, one can define a normalized velocity in the following way

$$ \tilde{w}(\tilde{z}) = \frac{w(\tilde{z}) - w_2}{w_1 - w_2}. \qquad (8.6) $$

When the measured (normalized) velocity is to be compared with the theoretical (normalized) velocity given by Eq. 8.4 (Eq. 8.6) a scaling length S needs to be introduced to take into account the shift in position between the inflection point of the density curve and the velocity curve. The parameter S is given by

$$ S = \frac{L}{4} \ln \frac{w_2}{w_1}. \qquad (8.7) $$

The comparison between experiments and calculations is shown in Fig. 8.6. As can be seen from the graph, the agreement is relatively good at the beginning of the shock wave even if the normalized velocity profile measured at 100 Pa deviates slightly from the predictions. However, in the final part of the shock region, a significant deviation is observed. This
effect has already been reported in literature in the case of on-axis density profiles measured in neutral gas shock waves [2, 6, 7]. It arises from the fact that the model developed by Mott-Smith is a one-dimensional (1D) normal shock wave model which does not account for scattering effects. Such effects, related to the invasion of the supersonic jet by the background gas, occurs for a 2D shock wave when the flow enters the rarefied regime [23], i.e. when the Knudsen number $Kn$ is large. It has indeed been observed that at high background pressure the agreement between experimental data and the Mott-Smith model is good and it deteriorates at low pressure. The Knudsen number ahead of a shock wave is the ratio of the mean free path for momentum exchange to the shock thickness. It reads

$$Kn = \frac{1}{n_{H_2}\sigma_{H-H_2}L} \qquad (8.8)$$

The temperature dependent momentum exchange cross-section $\sigma_{H-H_2}$ is taken from literature [27] and the $H_2$ density has been measured by means of CARS [28]. Under our conditions, $Kn = 0.7$ at 20 Pa and $Kn = 0.2$ at 100 Pa ahead of the shock wave meaning that the latter is formed in a transition flow regime. Surprisingly, the H atom velocity profile measured at $p_{back} = 20$ Pa and the one measured at $p_{back} = 100$ Pa exhibit a similar behavior in the final part of the shock wave, as can be seen in Fig. 8.6, in contradiction with measurements reported in literature. The two normalized velocity profiles deviate in approximately the same way from the theoretical calculation. In other words there is no clear influence of the background pressure. This remarkable fact is a consequence of the non-conservation of both the H atom forward flux and the H atom momentum throughout the stationary shock wave of the plasma jet [15, 16, 22]. In that case the Mott-Smith approach used to calculate the average velocity is not valid since it is based on the Rankine-Hugoniot relations. A theoretical calculation of the H atom velocity profile within the normal shock wave would therefore require a 2D model capable of accounting for a decoupling between the H atom fluid and the $H_2$ molecule fluid induced by H atom surface-recombination processes.

## 8.6 Conclusions

The specific ground-state hydrogen atom shock wave pattern formed in a weakly ionized hydrogen plasma jet can serve as a test case to validate predictions of kinetic models based on the Boltzmann equation used to study the behavior of the velocity distribution function and the fluid macroscopic properties within a shock wave. The two-photon absorption laser induced fluorescence spectroscopy is well adapted to extract local information about ground-state atoms at a microscopic scale, both in a gas and in a plasma environment. Using this diagnostic tool one can provide large datasets to support theoretical studies devoted to the shock wave properties. Such data are also of relevance in order to better understand the shock wave formation at low background pressure, i.e. in a rarefied flow regime, and the related invasion of the shock region by the residual gas particles, which is of interest in various fields like astrophysics and plasma-assisted chemistry.
Acknowledgments

The authors would like to acknowledge fruitful discussion with Prof. M.C.M. van de Sanden. The authors greatly appreciate the skillful technical assistance of M.J.F. van de Sande, A.B.M. Hüsken, and H.M.M. de Jong. This work is part of the research program of the Netherlands Foundation for Fundamental Research on Matter (FOM). It is financially supported by the Netherlands Organization for Scientific Research (NWO) as well as the Euratom foundation.
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# Chapter 9

## General Conclusion
Since all sections of this manuscript, which can be regarded as individual stories, have
their own conclusion, we will attempt in this general conclusion to give an overview of
the main outcomes of the last four years of fundamental research on expanding plasmas.
This chapter should therefore be seen as the counterpart of section 1.5 where the main
objectives of this work are presented in the form of a series of questions. The actual view
on the transport phenomena in intermediate scale plasma expansions will be presented
and discussed in the light of our findings. In doing so, we will also point to experimental
facts that still remain unclear or unexplained, and we will also present new problems that
emerged from that work. In order to open the way in the near future to experiments which,
as we hope, will contribute to confirm our present view.

Inert neutrals: validity of the gas expansion picture

The Rayleigh scattering measurements of the Ar and H₂ density profile along the jet axis in respectively Ar-H₂ and pure H₂ expanding plasmas confirm the idea that the expansion of inert neutrals can be described as a normal gas expansion [1,2]. The data also show that the inert particle properties such as the density, the velocity and the temperature can be well approximated using an hydrodynamic approach even in a rarefied flow regime. In the supersonic domain the set of simplified equations governing a steady isentropic flow can be used. Across the normal shock front the observed different compression stages are predicted by models based on the Navier-Stokes equation. Laser induced fluorescence measurements reveal that the supersonic plasma expansion is however not adiabatic, i.e. the adiabatic exponent γ is less than the expected value (1.3-1.4 instead of ⅝ for an atomic gas). It is well established that the presence of charged particles leads to a low γ [3]. However, with H₂ seeding, the ionization degree of the plasma is relatively low, and the explanation must be found elsewhere. The low value of γ may be caused by the relatively high heat conductivity at the higher plasma temperature. As a consequence energy can be supplied from the source and the downstream part of the zone of silence into the supersonic domain.

Atomic radicals: influence of surface chemistry

Contrary to neutrals, hydrogen radicals exhibit a deviation from the classical expansion
picture. As shown in this thesis, the transport properties of H atoms are very peculiar. For
instance, there is no density jump across the shock front whereas the axial velocity drops,
that corresponds to a non conservation of the H atom forward flux. Another remarkable
property is that in the core of the jet the H atom fluid is decoupled from the main gas
fluid, and the decoupling is already apparent at the plasma source exit. This is imaged for
instance in the fact that in an Ar-H₂ plasma the H atom velocity is higher than the Ar
atom velocity (see, for example, chapter 6). This anomalous transport of hydrogen radicals
is a direct consequence of the interaction between the plasma and adjacent surfaces. It is
clearly demonstrated that under certain circumstances, that are discussed below, surface
chemistry can strongly influence the plasma flow properties by way of recombination pro-
cesses.
The recombination of H atoms at the vessel walls to form molecular hydrogen is an efficient process under our conditions. A large surface is available, the loss (or recombination) probability Γ on stainless steel is high for H (Γ<sub>ss</sub> ≈ 0.2) and the residence time inside the vacuum chamber is long (τ<sub>res</sub> ≈ 1 s). This favorable combination of experimental conditions results in the creation of a large density gradient for H atoms between the jet and its surrounding (indeed H atom are almost absent in the background). In turn those gradients force H atoms to diffuse out of the plasma jet, even in the supersonic domain, and subsequently they are decoupled from the inert neutral fluid. The outward diffusion of H atoms makes possible an energy exchange between the interior of the jet and the outside, that explains the low value of γ. Moreover, it is shown that even if the mass of the radical is close to or identical to the mass of the carrier gas (H in H<sub>2</sub> and D in H<sub>2</sub> experiments), the losses of radicals can not be avoided. As a consequence only a small fraction of the H radicals produced in the source is available in the downstream region, creating a net loss of chemical potential.

One can wonder whether the anomalous radical transport is a very general phenomenon, or if it mainly concerns H atom in view of its small mass. In an unmagnetized plasma jet, we demonstrated that the outward losses solely depend on the thermal velocity, on Γ, on τ<sub>res</sub>, and on the momentum exchange cross-section. At the present moment, we have even more experimental evidences that support the idea of a general process. Some TALIF experiments have been performed on N<sub>2</sub> and Ar-N<sub>2</sub> plasma jets [4]. It is found that N atoms are better confined inside the jet (in comparison with H at identical pressure); nevertheless losses of N radicals are present [4]. Previous observations originate from a low Γ for N atoms [5] (Γ<sub>ss</sub> ≈ 0.001) combined with a relatively low thermal speed.

## Source strength and confinement

A point of concern is also the relatively low dissociation degree measured at the source exit: about 2 % and 7 % for a H<sub>2</sub> and Ar-H<sub>2</sub> plasma respectively (when calculated on axis). Indeed a high dissociation degree is expected in the arc channel in the case of hydrogen at a 1 eV electron temperature. This striking fact was already envisaged from earlier CARS and depolarization Rayleigh scattering experiments [6], however such a low fraction of H atoms was not expected. As we will see, the poor plasma source strength can be explained when involving plasma surface interactions. Once again, this is especially of interest for H atoms, because the loss probability on a Cu surface (the material the arc nozzle is made of) is close to unity. As described in chapter 7, since the arc root attached at the nozzle entrance, only a small amount of energy is supplied in the nozzle, and recombination mechanisms are favored. The H atoms are therefore also lost in the source nozzle via

$$H + H_{\text{wall}} \rightarrow H_2^{v,J}. \qquad (1)$$

The association of H atoms on metallic surfaces is known to generate vibrationally excited H<sub>2</sub> molecules [7]. Yet no significant amount of vibrationally excited H<sub>2</sub> molecules has been detected by CARS, neither in the jet nor in the background [6]. This is not in contradiction with aforementioned explanation; excited H<sub>2</sub> molecules can be deactivated
through collisions with particles (high density in the arc) and with the wall as well (very efficient process). They can also be involved in other reactions (see below). The situation needs further clarification, especially due to the fact that CARS measurements could not be performed at the arc outlet. In the case of a pure hydrogen plasma, the measured low fraction of H atom may be linked with the observed loss of ionization, i.e. the absence of visible radiation, through the reactions

$$H_2^{v,J} + H^+ \rightarrow H_2^+ + H \quad \text{and} \quad H_2^+ + e^- \rightarrow H^* + H. \qquad (2)$$

Subsequently the excited H atom emits light before leaving the arc. This process can also explain the presence of negative ions, formed by dissociative attachment of electrons on $H_2^{v,J}$ molecules, observed directly at high background pressure.

In the light of our observations we can propose a solution to increase the source strength and to improve the radical confinement inside the plasma jet, in order to obtain a high radical density in the downstream region. As regards the source, a good solution is to supply energy in the nozzle by driving the current outside the arc by means of an external anode. Another possibility is to vary the nozzle geometry in such way that H atoms only “see” a small surface (see appendix of chapter 6) and to play with the nozzle material (the requirements are a low $\Gamma$ and a high thermal conductivity). To confine the H radicals one may use a magnetic field! The external anode configuration can certainly provide a relatively high amount of ions; using an applied axial magnetic field B such as the Hall parameter $H_e \gg 1$, the electrons are magnetically confined. Because of charge neutrality restraint, the ions get also confined, and it may become more difficult for H atoms to leave the rotating plasma jet. The confinement may be enhanced because of the efficient charge exchange process between a proton and H: $k \approx 10^{-14} \text{ m}^3\text{s}^{-1}$ at 1 eV [8]. Such a confinement method is however not easy to achieve. Indeed, a strong magnetic field is required to magnetize the high electron(ion) density necessary to confine H by momentum exchange: $B \approx 1 \text{ T}$ for $n_e = 10^{21} \text{ m}^{-3}$.

## Charged particles as transient species

To a large extent, charged particles dominate the chemistry in a plasma environment. Charge exchange between an ion and a molecule may be an important step in the formation of a new molecule or a radical. For instance, the reaction

$$Ar^+ + H_2 \rightarrow ArH^+ + H \qquad (3)$$

is exothermic for ground-state $H_2$ by nature. Dissociative recombination of molecular ions is also an important process in chemistry (extremely fast reaction). Atoms and molecules can also be excited, ionized or even dissociated by electron impact. The transport of charged particles therefore deserves some attention. In retrospect, they can be considered as transient species in the sense that they are also efficiently consumed at walls. Plasma-surface interactions may accordingly influence the transport of ions in a (unmagnetized) plasma jet. Ions recombine at conducting surfaces to form neutrals; as a consequence
density gradients between the jet and its surrounding are created. The transport of ions
is however more complex than the transport of neutral radicals. It is affected by electric
fields and friction with electrons. Note that these two influences can be replaced by $\nabla p_e$ in
the electron momentum balance. It also depends on the Debye length<fnref n="1" /> and on the charge
exchange processes. Departure from the classical expansion behavior may be expected
especially in the case where the Debye length is large (low ionization degree) and where
the charge exchange process is non resonant.

Departure from TE within a shock wave

Another interesting characteristic of a plasma expansion is the departure from thermodynamic equilibrium (TE) within the stationary compression shock wave. The local velocity distribution function (VDF) of H and Ar atoms is measured by Doppler-shifted LIF. All VDF exhibit significant deviation from the Gaussian form, i.e. the equilibrium profile, both in the barrel shock wave and in the normal shock front. Such a behavior is however expected since a shock wave is a transition zone through which a supersonic flow has to adapt the local subsonic conditions. To work in a rarefied flow regime makes it easier to observe. The VDF can be well approximated by two Gaussian components, i.e. a bimodal approximation is valid. One component (fast and cold component) corresponds to the zone of silence conditions, i.e. it reassembles particles of the jet that have not yet collided with particles standing in the shock region. The other component (slow and warm component) corresponds to the conditions in the shock region, i.e. it represents particles that have already experienced at least one collision. Across a shock wave, population is gradually transferred from the fast to the slow component and the macroscopic properties of the two Gaussian components vary.

No third component, the so-called scattered component, has been observed [9]. The expla-
nation lies mainly in the fact that our bimodal approximation differs from the Mott-Smith
(MS) model [10]. In the MS approach, the fast component corresponds to the conditions
upstream of the normal shock wave and the slow component corresponds to the down-
stream conditions. In other words, in the MS model, the properties of the two Gaussian
components are linked to the supersonic and subsonic conditions. Only the content of
the two distributions vary through the shock wave. The existence of a third component
with intermediate properties then results from a blending between the supersonic and the
subsonic distributions.

Mixing with the background gas and recirculation pattern

The departure from TE results from collisions between particles of the jet and particles of the background. In other words, a shock wave is formed due to the invasion of the super-sonic domain by the background gas. A question that still remains concerns the formation

<footnote n="1">¹ The Debye length is the distance over which spontaneous charge separation takes place owing to thermal motion. Thus it defines the average distance over which an electric field may exist in the plasma.</footnote>
of the shock wave at low pressure: is the shock created by invasion from the Mach disk region (case in hydrodynamic regime) or is the invasion from the side also important or even dominant? As shown in the appendices, the invasion and the mixing with background gas does exist. The Ar atom VDF measured in the barrel shock of an Ar-H₂ plasma reveals that Ar atoms re-enter the jet. Seeding D₂ molecules in the background while burning a H₂ plasma, D atoms are found in the supersonic domain. The invasion process depends of course on the pressure, i.e. on the ratio of mean free path and shock dimension.

Mixing effects in the shock wave are very important for chemistry. In the supersonic zone, a high flux is available, combined with a high particle kinetic energy ($E_c \approx 2$ eV for Ar). Therefore the remaining question is the magnitude of the radial inward flux. At this moment it is still difficult to precisely quantify the invasion but some experiments could clarify the situation. For example, one can look at the invasion of an argon jet by Kr atoms using TALIF. Probing molecules by LIF in H₂-D₂ mixtures is also an interesting possibility (see [10] and appendix 2). The latter can also be used to establish at which location atoms are produced via

$$H(D) + D_2(H_2) \rightarrow HD + D(H). \quad (4)$$

Mixing between the plasma jet and the background gas is closely linked with the recirculation pattern which arises from the long residence time in the vessel: vortices in which particles are “trapped” are formed around the jet [11,12]. Recirculation plays an important role in transport phenomena. It offers several opportunities for particles present in the background (injected, created at the wall, or coming from the jet) to interact with the plasma jet.

## Towards plasma-surface interaction study

As pointed out in this work the interactions between plasma and surface have a large impact on both the transport properties and the source features with consequences on the chemistry. It thus appears to be of prime interest to study in more details those processes in the near future. To end this general conclusion, we here propose a series of investigations which would help in the understanding of plasma chemistry as a whole.

The study of the loss probability Γ as a function of the temperature is especially of interest since $T_{wall} \neq T_{source} \neq T_{substrate}$. Another important issue is the variation of Γ with the flux impinging onto the surface. The determination of the reaction product, including the state and the energy distribution, is important from a fundamental point of view to learn about the dynamics of recombination mechanisms in realistic conditions (so far experiments in UHV condition and well defined clean surfaces). For instance the vibrational distribution, in the electronic ground-state, of O₂ and H₂ molecules generated at a wall can be probed by means of LIF. It goes without saying that excited molecules play an important role in chemistry. Undoubtedly, the subject of formation of molecules on surfaces (CO, NH₃, H₂O to name a few) is very promising. On a fundamental perspective of course (e.g. production of molecules in interstellar medium), but also to investigate the possibility of using a surface as a catalyst to favor the production of a desired molecule.
References

[1] M.C.M. van de Sanden, R. van den Bercken, and D.C Schram, *Plasma Sources Sci. Technol.* **3**, 501 (1994).

[2] R.F.G. Meulenbroeks, R.A.H. Engeln, M.N.A. Beurskens, R.M.J. Paffen, M.C.M. van de Sanden, J.A.M. van der Mullen, and D.C. Schram, *Plasma Sources Sci. Technol.* **4**, 74 (1995).

[3] K.T.A.L. Burm, W.J. Goedheer, and D.C. Schram, *Phys. Plasmas* **6**, 2622 (1999).

[4] I. Bakker, *Properties of an Expanding Thermal Nitrogen Plasma*, internal report VDF/NT 01-06, Eindhoven University of Technology, The Netherlands (2001).

[5] S.F. Adams and T.A. Miller, *Plasma Sources Sci. Technol.* **9**, 248 (2000); H. Singh, J.W. Coburn, and D. Graves, *J. Appl. Phys.* **88**, 3748 (2000).

[6] R.F.G. Meulenbroeks, R.A.H. Engeln, J.A.M. van der Mullen, and D.C. Schram, *Phys. Rev. E* **53**, 5207 (1996).

[7] R.I. Hall, I. Cadez, M. Landau, F. Pichou, and C. Schermann, *Phys. Rev. Lett.* **60**, 337 (1988); C.T. Rettnwe and D.J. Auerbach, *J. Chem. Phys.* **104**, 2732 (1996).

[8] R.K. Janev, W.D. Langer, K. Evans, Jr., and D.E. Post, Jr, *Elementary Processes in Hydrogen-Helium Plasmas*, Springer-Verlag, Berlin (1987).

[9] A. Ramos, B. Maté, G. Tejeda, J.M. Fernández, and S. Montero, *Phys. Rev. E* **62**, 4940 (2000).

[10] S. Mazouffre, P. Vankan, I. Bakker, R. Engeln, and D.C. Schram, *Proceedings of Laser-Aided Plasma Diagnostics* **10**, Fukuoka, Japan (2001).

[11] R.F.G. Meulenbroeks, D.C. Schram, M.C.M. van de Sande, and J.A.M. van der Mullen, *Phys. Rev. Lett.* **76**, 1840 (1996).

[12] W.M.M. Kessels, A. Leroux, M.G.H. Boogaarts, J.P.M. Hoefnagels, M.C.M. van de Sanden, and D.C. Schram, *J. Vac. Sci. Technol. A* **19**, 467 (2001).
<figure><img src="image_66.png" /></figure>
# Appendix A

## Invasion of a supersonic jet by the background gas particles

The rarefaction process of a free jet flow and the subsequent invasion of the zone of silence by the background gas particles is a well-known problem. It has been already studied in the late sixties and in the early seventies, both theoretically [1, 2] and experimentally by mass spectrometric measurements [3]. It has been shown that the invasion is mass dependent in favor of the light species, and that this effect can be used for the separation of mixtures of gases and isotopes [3, 4]. On a technological point of view, it may have other interesting applications especially in the field of plasma assisted chemistry. Supersonic plasma jets are used for instance for the fast deposition of thin layers. In that case, an efficient mixing between the precursor gas injected at the jet periphery and the plasma jet particles is of crucial importance. Collisions in the shock region with supersonically flowing particles ($E \approx 4$ eV for Ar in a cascaded arc plasma) may enhance the reactions and thus speed up the deposition process. On a fundamental point of view, the invasion effect concerns the formation of a shock wave at low background pressure. Its influence on the velocity distribution function (VDF), and the related appearance of a scattered component, is also of particular interest.

In all studies, the zone of silence, or the stationary shock wave depending on $Kn$, is invaded by a species only present in the ambient gas surrounding the jet and not already in the jet. That naturally facilitates experiments like flux measurements since it is easy to distinguish between the different fluid components. The measurements performed in an expanding argon plasma jet (see chapter 3) on the Ar atom VDF by Doppler-shifted laser induced fluorescence reveal that in the rarefied regime Ar atoms from the ambient gas cross the barrel shock wave and penetrate the shock region.

In Fig. A.1 the Ar atom radial VDF (determined in a direction perpendicular to the jet axis) measured at $z = 50$ mm, i.e. in the shock region, is shown for $r = 0$ mm (on the jet centerline) and for $r = 22$ mm (in the barrel shock wave). In the two cases a strong deviation from thermodynamic equilibrium (TE) is observed. The VDF can nevertheless be decomposed into two Gaussian components, a high perpendicular temperature one, the *warm* component, and a low perpendicular temperature one, the *cold* component.

On the jet centerline, the cold component reassembles Ar atoms from the jet that have not
<figure><img src="image_67.png" /><figcaption>**Fig. A.1:** Radial Ar VDF recorded at 20 Pa at *z* = 50 mm and at *r* = 0 mm (left) and *r* = 22 mm (right). Also shown is the result of a fit using a bimodal approximation as well as the two individual Gaussian components.</figcaption></figure>

yet experience any collisions with the particles located in the shock wave. The temperature of that group is low due to the quasi-adiabatic cooling process during the supersonic expansion phase. The warm component corresponds to Ar atoms that originate from the plasma jet vicinity. Those atoms collide in the barrel shock wave with Ar atoms from the jet. The resulting energy exchange leads to an increase in the radial velocity spread and therefore to a rise in temperature. On the jet axis the average radial velocity is zero for both component as expected in view of the cylindrical symmetry of the expansion. Such a non-Gaussian Ar velocity profile, constituted of two equilibrium components of which both the origin and the properties can be perfectly defined, is a proof in favor of the invasion of the stationary shock wave by Ar atoms from the background gas. Indeed at 20 Pa the mean free path ahead of the shock front is in the order of the jet radius, the shock structure is diffuse and therefore the probability that particles enter the jet is relatively high.

In the barrel shock wave, the two velocity groups exhibit an opposite mean velocity. The velocity of the cold component is positive (for *r* < 0), i.e. the flow of Ar atom belonging to that group is directed outward, whereas the mean velocity of the warm component is negative, i.e. the Ar atoms move towards the jet core. The temperature of the jet particles increases, in comparison with the situation *r* = 0 mm, because of multiple collisions.

In Fig. A.2 the mean velocity as well as the temperature of the two Gaussian components of the Ar atom radial VDF are shown as a function of the radial position at *z* = 50 mm and at 20 Pa background pressure. For the cold component the maximum radial velocity is reached at *z* = 20 mm which defines the beginning of the barrel shock wave ($M$ = 1.7 at that location). Behind this limit the velocity decreases and the temperature increases as a consequence of collisions with the background gas. The barrel shock ends around *z* = 40 mm ($M$ = 1). Ar atoms situated in the surroundings of the jet enter the latter with a radial velocity of about 250 m s⁻¹. The temperature of this Ar atom group starts to change significantly solely for *r* < 20 mm, i.e. inside the jet, due to the numerous collisions,
<figure><img src="image_68.png" /><figcaption>**Fig. A.2:** Ar radial VDF Gaussian component mean velocity (left) and temperature (right) as a function of the distance from the jet axis measured at $z = 50$ mm and 20 Pa.</figcaption></figure>

and the maximum is reached at the jet center.

When moving towards the jet axis, the contents of the VDF decreases as can be seen in Fig. A.1, that means only a small fraction of the particles located in the jet vicinity can reach the interior of the jet. This effect is naturally pressure dependent: for instance it is observed that at 100 Pa hardly any particles from the outside reach the jet centerline in the shock wave region.

We demonstrated that the invasion of the supersonic domain of an expanding jet always exists at low pressure. This means that it does not only concern a category of particles different from the one that composes the flow [5]. Unfortunately the ground-state Ar flow<fnref n="1" /> entering the shock wave from the side can not be determined from our data set since we are probing metastable Ar atoms. Neither can be at the moment the Ar flux that originates from the downstream region of the shock wave. Such information would be valuable to study the formation of a shock wave at low pressure, therefore we may have to consider in the near future the local detection of Ar atoms.

[1] J.B. Fenn and J.B. Anderson, *Proceedings Rarefied Gas Dynamics* **2**, edited by J.H. de Leeuw, Academic Press, New York (1966).

[2] E.P. Muntz, B.B. Hamel, and B.L. Maguire, *AIAA J.* **8**, 1651 (1970).

[3] R. Campargue, *J. Chem. Phys.* **52**, 1795 (1970).

[4] J.W. Brook and V.S. Calia, *J. Energy* **4**, 199 (1980) and references herein.

[5] see also S. Mazouffre, M.G.H. Boogaarts, J.A.M. van der Mullen, and D.C. Schram, in *Heat and Mass Transfer under Plasma Conditions*, edited by P. Fauchais, J. van der Mullen, J. Heberlein, ANYAS, Vol. **891**, 348 (1999), for the invasion of a supersonic Ar jet by H and H₂.

<footnote n="1">¹ The ambient gas is mainly composed of ground-state Ar atoms, $n_{Ar}/n_{Ar}^* \approx 10^4$ in the supersonic jet vicinity at 20 Pa, and therefore the shock wave is mostly formed by collision with those atoms</footnote>
<figure><img src="image_69.png" /></figure>
# Appendix B

## Recirculation and downstream chemistry

Due to the large residence time inside the vacuum chamber ($\tau_{res} \approx 1$ s at 20 Pa) recirculation cells are formed around the plasma jet [1]. The created flow pattern allows for background particles to mix and react with the jet particles, mainly at the boundary of the supersonic domain and in the subsonic flow zone when $Kn < 0.2$. This recycling process is believed to play an important role in the plasma chemistry and kinetics downstream of the shock wave, e.g. in the formation of stable molecules or molecular fragments. In order to obtain a better insight on the recirculation process, $D_2$ is injected in the vessel while creating a hydrogen plasma in an arc, and one of the reaction products, D atoms in that case, is locally probed. This work is an extension of the work by Meulenbroeks et al in which HD molecules density and temperature are measured by CARS when burning the arc on a $H_2-D_2$ mixture [2]. The settings are: 0.4 slm $D_2$, 3.5 slm $H_2$, standard arc conditions. Deuterium atoms are produced mainly via the following reaction [2]

$$H + D_2 \rightarrow HD + D \quad k = 3 \times 10^{-19} \text{ m}^3 \text{ s}^{-1}. \qquad (B.1)$$

The local D density is measured by TALIF spectroscopy. In figure B.1 the D atom radial density profile measured at $z = 30$ mm at 20 Pa, i.e. shock region, is shown. The maximum D density is observed at the jet boundary (see Fig. 7.6). Atomic deuterium is also found inside the jet, meaning that D atoms or/and $D_2$ molecules invade the shock wave. In figure B.2 the D atom density profiles measured along the jet axis for two background pressures are shown. First, the D density depends on the background pressure. Second, the plasma jet supersonic domain is invaded (see Fig. 7.2). Third, the $D_2$ dissociation degree is extremely low: $\alpha \approx 10^{-4}$ in the subsonic domain at 20 Pa. Finally, at 100 Pa, the profile exhibits a maximum around $z = 200$ mm. These experimental data reveal that the ambient gas indeed mix with the plasma flow and that molecules injected in the background can actively participate in the chemistry.

A point of concern is the production of D atoms. Measurements support the idea of local production. In the jet vicinity, the D density decreases when the H density decreases while the $D_2$ density remains approximately constant. Moreover, the 100 Pa axial profile can be explained as follows: ahead of the Mach disk only few $D_2$ molecules have already entered the jet, whereas behind the shock wave $D_2$ fully mix with the jet; the $D_2$ density therefore
<figure><img src="image_70.png" /><figcaption>**Fig. B.1:** D atom radial density profile at $z = 30$ mm and 20 Pa, when $D_2$ is seeded in the background.</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>**Fig. B.2:** D atom density profile along the jet axis at 2 different pressures for $D_2$ vessel injection.</figcaption></figure>

increases at the beginning of the subsonic region and subsequently more and more D atoms are produced since the H atom density decreases only slowly; at some point however losses of D atoms get larger than production that leads to a decrease of the D density.

Assuming that the main loss mechanism for deuterium atoms is diffusion, the local D density can be estimated by

$$n_D = k n_H n_{D_2} \tau \quad \text{with} \quad \tau = \frac{\Lambda^2}{D} \approx \frac{3\Lambda_0^2}{\lambda v_{th}} + \frac{(2 - \gamma)R}{\gamma v_{th}}, \quad (B.2)$$

where $\tau$ is the diffusion time [3], $\Lambda_0$ the diffusion length, $\lambda$ the elastic collision mean free path, $v_{th}$ the thermal speed, $R$ the vessel radius, and $\gamma$ the wall-loss probability. As an example $n_D$ is calculated on axis at $z = 100$ mm and at 20 Pa. Using Eqs. B.1 and B.2 with $\gamma = 0.1$, and $n_{D_2} \approx 2 \times 10^{20} \text{ m}^{-3}$ from CARS measurements, one finds $\tau = 4$ ms and $n_D \approx 10^{19} \text{ m}^{-3}$ which is more than two orders of magnitude higher than the measured value. The discrepancy may arise from an erroneous $\gamma$ or from the fact that the reaction between D and $H_2$ is not taken into account. This simple estimation however reveals that the problem of D atom, or associated HD molecule, production as well as the issue of invasion of the supersonic domain by $D_2$, can only be solved by measuring the local $D_2$ density, using two-photon LIF spectroscopy for instance.

[1] W.M.M. Kessels, A. Leroux, M.G.H. Boogaarts, J.P.M. Hoefnagels, M.C.M. van de Sanden, and D.C. Schram, *J. Vac. Sci. Technol. A* **19**, 467 (2001).

[2] R.F.G. Meulenbroeks, D.C. Schram, M.C.M. van de Sanden, and J.A.M. van der Mullen, *Phys. Rev. Lett.* **76**, 1840 (1996).

[3] P.J. Chantry, *J. Appl. Phys.* **62**, 1141 (1987).
Summary

The expansion of a plasma from a hot and dense source to a low pressure environment
is a general physical phenomenon which concerns a large variety of objects such as super-
novae, solar flares, laser spots, and free jets produced from thermal plasma sources.

Over the last decade, the transport of charged particles as well as ground-state atoms and molecules has been extensively studied in laboratory scale plasma expansions. It has been shown that, to a large extent, the flow characteristics of ions and electrons and inert atoms and molecules could be interpreted in terms of a quasi-adiabatic supersonic expansion of a gas into a vacuum. In other words, the theory of the free jet flow developed in the field of gas dynamics could be well applied in the case of a plasma expansion, only minor corrections being necessary to account for field and current generation, and for the relatively high plasma temperature (departure from adiabatic flow).

However, at this point, questions remained about the transport properties of atomic
radicals, i.e. reactive neutral atoms such as H, N, C, in a supersonic plasma flow. This thesis
is mainly concerned with the study of transport mechanisms of ground-state hydrogen
atoms in the expansion of thermal plasmas generated by a cascaded arc from Ar-H₂ and
He-H₂ gas mixtures, and from H₂ gas. A clear understanding of transport phenomena in
free plasma jets containing hydrogen are of prime importance since H atoms play a key
role in plasma assisted surface modification processes (deposition of thin films, etching of
microstructures) as well as in the field of controlled nuclear fusion. The local macroscopic
properties of the H fluid (density, temperature, velocity) are determined by means of Two-
photon Absorption Laser Induced Fluorescence spectroscopy. The properties of Ar atoms
are measured by LIF (with a cw source) and UV Rayleigh scattering, and the behavior of
H₂ molecules is probed by UV Rayleigh scattering.

We found that the transport of atomic radicals can not be described using the classical
free jet flow picture. Contrary to inert particles, hydrogen radicals recombine on a surface
(the reactor walls) to form stable hydrogen molecules. This recombination process induces
huge density gradients between the plasma jet and its surrounding which in turn force the
radicals to diffuse out of the jet. It is demonstrated for the first time that, under specific
circumstances (high wall-loss probability and large residence time) surface-chemistry can
play a key role in the transport mechanisms of reactive particles.

The evolution of both the H and Ar atom velocity distribution function in the course
of the plasma flow, especially in the stationary compression shock wave where a strong
departure from thermodynamic equilibrium occurs, is also investigated, as well as the in-
vasion of the supersonic domain of the jet by the residual background gas. We found
that in a rarefied flow regime a bimodal approach can well describe the measured velocity
distribution function in the shock region. However, the two-component model of the ve-
locity distribution proposed in this thesis differs from the Mott-Smith approximation that
explains the absence of a third component, the so-called scattered component.

Stéphane Mazouffre, June 2001.
Acknowledgments

It will be obvious that the research undertaken over the last four years, which forms
the subject of this thesis, is the result of teamwork, and not the work of a single person,
however determined. I wish, then, to thank all those who have contributed, in whatever
degree, to the realization of this mamuscript.

I would like to thank first of all Daan Schram, my supervisor, for the confidence he has
shown in me and for the freedom of opinion to which he has accustomed me. His over-
whelming energy and his endless enthusiasm, although sometimes difficult to match even
for someone of my young age, have allowed me to make fast progress both professionally
and personally. I will never forget the hundreds of hours we spent talking about physics,
of course, but also just about life. And in 20 years' time, I will be telling my children for
the twentieth time how proud I am to have studied all those years ago under such a great
man....

Thanks to Joost van der Mullen for his ever cheerful but efficient supervision during the
first year of my thesis. I am sorry about the article that never was, and I sincerely hope
that someone will finish that study very soon. Et pourquoi pas ?

I must also thank the calmest person I have ever had the opportunity of meeting, a certain Maarten Boogaarts. I could not have wished for a better person to introduce me to experimental spectroscopy. And we did finally manage to write that article!

Many thanks to Richard Engeln, a.k.a. The Boss, my co-supervisor, for almost three years
spent working together with never a cross word. I am glad and proud to have been his
first Ph.D. student; I could not have wished for better. I will never forget the few days we
spent together in Limoges (you will have to give the snails another chance). As for those
170 pages, I suggest we buy two bottles And by the way, is it CARS or LIF?

Now is the time to thank Prof. Frieder Döbele and Prof. Nader Sadeghi for having shared
their great experience of laser spectroscopy with me. I also want to thank them, together
with Prof. M. E. H. van Dongen, for their constructive criticism after reading the first
draft of my thesis.

Of course, many thanks to everyone at the ETP group, and especially to Marco and Maikel
for having put up with me for four years in the same office (sorry for all the swearing, but is
it that surprising coming from a Frenchman?!); to Jeanne, our secretary, for having helped
me with all those wonderful forms; to Bertus, Herman, Ries and Jo, for having helped me
solve countless technical problems in computing, optics or mechanics; to Richard van de
Sanden for having taught me to stand up for my point of view come what may; and finally
to Edouard and Karine for the wonderful times in French.

Thank you to the residents of Cassandraplein 41, the most minimalist apartment I know
of. Thank you to Colin for these years of flat-sharing. It took us a while to get settled in
to each other, but it was worth it in the end. Sorry for having contributed to the ruining
of your accent. Zeez French! Wee weel never get zem to speek ze Eengleesh proper. Oh,
and thank you for the widescreen TV, which was much better for playing Doom... Thanks
Acknowledgments (continued)

to Tarik for the wild year we spent together (only my bank account thought it wasn't
that wild). Trix, Kix, Niks... Succes! Since you left we have got hold of a comfortable
couch. No more evenings slouched on a mattress wearing a pizza box as a sun visor. Car-
ole, even though your stay was relatively brief, it allowed us to get the apartment straight.
And nothing has changed since you left. We have kept up the good habits you got us into.
A big thank you to the IPM people for the months I spent in their company in that wonder-
ful house. I particularly want to thank one of them: my friend Peter. I know we will never
forget the first time we shook hands in that narrow corridor a few days after my arrival
in the Netherlands. Fate was smiling on us that day. Good luck with your thesis. I know
everything will work out. With a trainer like me, you should be prepared for anything!! I
also want to thank all the Thales members, and especially Ivo, as I have made him suffer
a bit recently, for having accepted me at some of their special parties. The wine was good
that night; we should have done it again.

Thank you to everyone at the Laser- und Plasmaphysik group at Essen University in Ger-
many for four very pleasant months I spent at their faculty, where I was able to have a
go at photon generation in the VUV, but also in the wonderful region of the Ruhrgebiet.
Thanks to Frieder Döbele, Uwe Czarnetzki, Thomas and Claudia, Norbert, Michael, Peter,
Timo, Kari and Anne.

Thanks to Philippe Supiot and Corinne Foissac of Lille University for having initiated me
into nitrogen plasmas. Sorry to Corinne for the lack of sleep and for the imposed diet. I
will always be in your debt. Thank you to Gerrit Kroesen and Gerjan for the few days we
spent in New York. It was a good idea to choose a hotel on Broadway after all. Thanks
to Armelle Vardelle, Jean Desmaison and Jean-Pierre Lecompte at the ENSI at Limoges
for having enabled me to teach a few classes at my alma mater, and for having advised me
on my future career. Thanks also to Lisette for having designed the cover for my thesis. I
think it is really stylish, and just what I wanted.

Finally, I want to thank with all my heart the people who mean more than anything else
to me, and who, despite being hundreds of kilometers away, have still had to suffer my
endless tales about plasma physics. Your help has been precious; never doubt that for
a second. A thousand thanks, then, to my Mum, to Raymond, Marie-Thérèse, Clément,
my godfather Serge, Giselle, my godmother Simone, Sylvian, my Laquaire grandparents,
Aimée and Christian. After the family come my friends, who are so dear to me. Thank
you Grégorie, Jean and Emmanuelle, David and Manuella, Alexandre and Valérie, Olivier,
Sophie and JB, Charles, Michel, Daniel, Christophe, Jeff and Sonia, Anne, Alexandrine,
Estelle, Gael and Fabien.

Bedankt, Thanks, Danke, Merci.
List of publications

* Transport of neutral atomic hydrogen in a supersonic plasma jet.
S. Mazouffre, M.G.H. Boogaarts, J.A.M. van der Mullen, and D.C. Schram, Heat and Mass transfer under Plasma Conditions, edited by P. Fauchais, J.A.M. van der Mullen, and J. Heberlein, ANYAS, Vol. 891, 348 (1999).

* Anomalous atomic hydrogen shock wave pattern in a supersonic plasma jet.
S. Mazouffre, M.G.H. Boogaarts, J.A.M van der Mullen, and D.C. Schram, Phys. Rev. Lett. **84**, 2622 (2000).

* A time-resolved experimental and computational study of two-photon LIF in a hydrogen plasma.
H.W.P. van der Heijden, M.G.H. Boogaarts, S. Mazouffre, J.A.M van der Mullen, and D.C. Schram, Phys. Rev. E **61**, 4402 (2000).

* The physics of plasma expansions.
D.C. Schram, S. Mazouffre, R. Engeln, and M.C.M. van de Sanden, Atomic and Molecular Beams, edited by R. Campargue, Springer, NY, 209 (2001).

* Density and temperature of N atoms in the afterglow of a microwave discharge measured by a two-photon laser induced fluorescence technique.
S. Mazouffre, C. Foissac, P. Supiot, P. Vankan, R. Engeln, D.C. Schram, and N. Sadeghi, Plasma Sources Sci. Technol. **10**, 168 (2001).

* Transport of ground-state hydrogen atoms in a plasma expansion.
S. Mazouffre, M.G.H. Boogaarts, I. Bakker, P. Vankan, R. Engeln, and D.C. Schram, Phys. Rev. E **64**, 016411 (2001).

* Influence of surface chemistry on the transport of H atoms in a supersonic hydrogen plasma jet.
S. Mazouffre, P. Vankan, R. Engeln, and D.C. Schram, Phys. Plasmas **8**, 3824 (2001).

* Flow dynamics and invasion by background gas of a supersonically expanding thermal plasma.
R. Engeln, S. Mazouffre, P. Vankan, D.C. Schram, and N. Sadeghi, accepted for publication in Plasma Sources Sci. Technol. (2001)
List of publications (continued)

* *Behavior of the H atom velocity distribution function within the shock wave of a hydrogen plasma jet.*

S. Mazouffre, P. Vankan, R. Engeln, and D.C. Schram,
accepted for publication in Phys. Rev. E. (2001)

* Plasma expansion: fundamentals and applications

R. Engeln, S. Mazouffre, P. Vankan, I. Bakker, and D.C. Schram,

Proceedings of the International Conference on Phenomena in Ionized Gases XXV,
to be published in Plasma Sources Sci. Technol.

* *Quantitative two-photon LIF measurements of atomic hydrogen densities, temperatures, and velocities in an expanding thermal plasma.*

M.G.H. Boogaarts, S. Mazouffre, H.W.P. van der Heijden, P. Vankan, J.A.M. van der Mullen, and D.C. Schram,

submitted for publication in Rev. Sci. Instrum.

* Wall-association processes in expanding thermal hydrogen plasmas.

S. Mazouffre, P. Vankan, W.M.M. Kessels, R. Engeln, M.C.M. van de Sanden, and D.C. Schram,

submitted for publication in IEEE Trans. Plasma Sci.

* TALIF performed on expanding nitrogen plasmas.

S. Mazouffre, I. Bakker, R. Engeln, P. Vankan, and D.C. Schram,
in preparation.
Stellingen

behorende bij het proefschrift

Transport Phenomena in
Plasma Expansions Containing Hydrogen
A Laser Spectroscopic Study

door

Stéphane Mazouffre

I

Wall-association processes can strongly influence the transport mechanisms of radicals in plasma expansions.

Dit proefschrift

II

Within the stationary shock wave of a plasma expansion the atom velocity distribution function can be decomposed into two Gaussians. One component corresponds to the un-hampered supersonic conditions. The other component originates in collision events.

Dit proefschrift

III

In a rarefied flow regime, the shock region and the supersonic domain of an expanding plasma are invaded by background gas particles.

Dit proefschrift

IV

Liberté. Égalité. Fraternité.

After more than four years spent in The Netherlands, I discovered that the country of individual freedom was not the one I thought of.
V

To love someone today is to take the risk of loving someone else tomorrow.

VI

Imagination creates dreams. Science destroys them by confronting us with reality.

VII

"Anticipatory plagiarism occurs when someone steals your original idea and publishes it a hundred years before you were born."

*Robert Merton*

VIII

The existence is ruled by free will, not by determinism.

"My life was running out, I knew not through what breach."

*Samuel Beckett, Molloy.*

IX

The only difference between Religion and Science is that Science does account for obser-
vational facts.

X

Controlled nuclear fusion could certainly be achieved if no reactor walls would be needed.

XI

Fortunately life is not always straightforward.