Pressure in underwater spark discharge initiated with the help of bubble injection and its evaluation based on H-alpha line broadening

A general description of the underwater discharge generated between plane-to-plane electrodes with a nitrogen bubble injection was given in [8]. After voltage application, a streamer discharge develops in the injected gas bubble as a result of an avalanche process, and propagates along the inner side of the bubble surface [8, 14]. When the streamers reach the anode side of the bubble, they continue to propagate in the water. As soon as one of these streamers bridges the inter-electrode gap, the electrical breakdown takes place and the underwater spark is generated. Its visible spectrum is shown in figure 2.

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The electrical and optical emission properties of such underwater discharges depend on the delay τ in the application of the voltage behind the bubble injection. The first column of figure 3 shows examples of current and voltage waveforms for three typical discharges, which differ in terms of the size of the injected nitrogen bubble just before initiation of the spark. The processes in three types of discharge can be described as follows. (a) Large τ {∼27 ms}: the injected bubble almost interconnects the electrodes, and the pre-breakdown streamers propagate easily along the inner side of the bubble wall up to the bubble front facing the high-voltage electrode. They finally have to travel only a small distance in undisturbed water. Breakdown (spark initiation) occurs almost immediately after voltage application. Since the spark channel in the water is very short, a small volume of water is evaporated, the pressure increase in the plasma column is relatively low, and the H_α line is clearly visible in the spectrum. (b) Medium τ {∼24 ms}: the size of the bubble is around 50% of the inter-electrode gap, and the pre-breakdown streamers have to travel about one half of their path in the water. Due to this, breakdown is delayed {∼20 μs} following the voltage application. The development of a longer discharge channel in water accompanies evaporation of larger amounts of water and increases pressure in the discharge channel and in the bubble, meaning that the H_α line broadens and its peak value is reduced. (c) Small τ {∼20 ms}: the bubble occupies approximately 30% of the inter-electrode gap and the pre-breakdown streamers have to bridge a large distance through the water. The onset of the spark is even further delayed {nearly 40 μs} after voltage application, and a larger amount of water is evaporated, with still higher pressure. The H_α line further broadens and starts to be hardly noticeable above the background. Unfortunately, the video frames corresponding to the time period of spectroscopy measurements were overexposed, and did not contain any information. Instead, images of the bubble one frame before breakdown are presented in the second column of figure 3. The third column of the figure depicts the visible spectra of the spark discharge immediately after breakdown; a change in the amplitude and width of the H_α line (at a wavelength of 656.3 nm) are clearly visible (the backgrounds {dashed lines} of this spectral line were obtained by extrapolation from their neighbourhoods).

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The results presented above (and in [9]) encouraged us to use the H_α line profile (which is clearly discernible in the red part of the visible spectra) as a diagnostic tool for the spark channel. An analysis of spectral lines is usually done in circular frequency ω scale and with subtraction of the background (see figure 4). An error in background determination causes the line asymmetry and hence uncertainty in the fitting of the theoretical profiles to line wings. Negative Δω (= ω−ω₀, ω₀ is the central circular frequency of the spectral line) correspond to the red wings of the spectral lines, where the background is lower, and hence the preference for fitting is on the negative side of Δω. The first two graphs (discharges (a) and (b)) suggest that the H_α line profiles are better approximated by Lorentzian curves (impact approximation of pressure broadening) than by Gaussian ones (Doppler broadening due to thermal motion of radiating atoms), while the third graph (discharge (c)) shows a part of the spectrum that is too narrow (in comparison with the line width) to decide. A Voigt function (folding of Lorentzian and Gaussian functions) can be distinguished in our environment from a purely Lorentzian one only if the Gaussian width is at least two thirds of the Lorentzian one. This gives Δω_{FWHM−Doppler} ∼4 × 10¹³ s⁻¹ for discharge type (a), which corresponds to a temperature as high as ∼3 × 10⁸ K ∼26 keV, far above the highest value given for underwater discharges (5.2 × 10⁴ K ∼4.5 eV [15]). This disqualifies the spectral line profiles from being used for temperature determination in the underwater spark channel. The term 'temperature' implies a local thermodynamic equilibrium, which is a prerequisite for its use. In this case, the prerequisite is evidently fulfilled, because the mean free time between collisions of water vapour molecules at 100 °C and atmospheric pressure {∼2.25 × 10⁻¹⁰ s} is much smaller than the investigated time interval of our discharge {∼1 × 10⁻⁶ s}. Knowledge of temperature is essential for: (i) estimating the plasma composition (degree of dissociation, excitation, etc), and the associated types of collisions; (ii) assessing the applicability of various approximations; and (iii) estimating the pressure in the discharge channel. For discharge type (a), we calculated the temperature from the ratio of the intensities I/I' of the two Balmer lines with wavelengths λ, λ', oscillator strengths f_mn, f'_mn and excitation energy eU_m, eU'_m [16]:

$$\begin{align}\frac{I}{{I^{^{\prime} }}} = \frac{{{{\lambda ^{^{\prime} }}^3}}}{{{\lambda ^3}}}\frac{{{f_{mn}}}}{{{{f^{^{\prime} }}_{mn}}}}{e^{ - \frac{{e{U_m} - e{{U^{^{\prime} }}_m}}}{{kT}}}},\end{align} \tag{ 1 }$$

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giving

$$\begin{align}T = \frac{{hc}}{k}\frac{{\frac{1}{{\lambda ^{^{\prime} }}} - \frac{1}{\lambda }}}{{{\text{Ln}}\left[ {\frac{I}{{I^{^{\prime} }}}} \right] - {\text{Ln}}\left[ {\frac{{{{\lambda ^{^{\prime} }}^3}}}{{{\lambda ^3}}}\frac{{{f_{mn}}}}{{{{f^{^{\prime} }}_{mn}}}}} \right]}}.\end{align} \tag{ 2 }$$

The line intensity ${I_{{\text{line}}}} = \int\nolimits_{{\text{end of blue wing}}}^{{\text{end of red wing}}} {{I_{{\text{s line}}}}} \left( \lambda \right){\text{d}}\lambda$, where the line spectral intensity I_{s line}(λ) is the measured spectral intensity I_s(λ) minus background spectral intensity I_{s background}(λ) {I_{s line}(λ) = I_s(λ)—I_{s background}(λ)}, we see that the largest error comes from uncertainty of background determination, and this is namely at H_α line (due to its largest width).

For I_Hβ /I_Hγ , we have

$$\begin{align}T\left[ {\text{K}} \right] = \frac{{3.551 \times {{10}^3}}}{{{\text{Ln}}\frac{{{I_{H\beta }}}}{{{I_{H\gamma }}}} - 0.6424}},\end{align} \tag{ 3 }$$

which for discharge (a) gives a value of ∼8000 K. For discharges (b) and (c), the higher members of the Balmer series lines are very small, and are therefore unsuitable for temperature determination. The temperature value given above is higher than indicated in the model [1], but smaller than in all other experiments (for example, Martin [17] measured a value of ∼30 000 K; Robinson et al [15] measured temperatures up to 52 000 K; and Vokurka and Plocek [18] found temperatures between 11 000 and 18 000 K).

On the other hand, since discharges type (b) and (c) need to heat more particles than type (a) for the same driving circuit, we expect that they will have even lower temperatures. To conclude this part, we note that our temperature determination is not extremely convincing. For simplicity, a temperature of 8000 K will be used for the plasma channel in the following numerical calculations, but where possible, the whole assumed temperature range of 2000–25 000 K will be considered.

Pressure broadening is usually classified according to the strength of the interaction causing a frequency change Δω for a radiating particle interacting with a perturbing particle at a radius vector r,

$$\begin{align}\Delta \omega \left( r \right) = {C_q}/{r^q},\end{align} \tag{ 4 }$$

where C_q is the interaction constant, and q is an integer [19]. The most important cases of interaction are: (i) the linear Stark effect {q = 2}; (ii) resonance interaction between identical particles, or interaction between a charged perturber and a quadrupole {q= 3}; (iii) the quadratic Stark effect {q = 4}; and (iv) the Van der Waals interaction {q = 6}. Since our spark discharge burns in a mixture of water vapour (evaporated when streamers/spark pass through a water slab) and nitrogen (the gas used for the injected bubble) {the ratio H₂O:N₂ differs among the types of discharge, being smallest in type (a) with the largest bubble, and largest in type (c) with the smallest bubble}, the first relevant broadening mechanism is the Van der Waals one {q = 6}. Here, the radiating atom is hydrogen and the perturbers are molecules of water vapour, some products (oxygen) of their dissociation, and molecules and atoms of nitrogen. In this case [19, 20],

$$\begin{align}{C_6}\sim \left( {{e^2}{\alpha _{\text{p}}}{a_0}^2/\hbar } \right) \times \left( {5/2} \right){n^*}^4\sim 1.78641 \times {10^{ - 30}}\,{\text{c}}{{\text{m}}^6}\,{{\text{s}}^{ - 1}}\end{align} \tag{ 5 }$$

with elementary charge e, polarisability α_{p H2O} ∼ 1.44 × 10⁻²⁴ cm³ [21], α_{p O} ∼ 1.57 × 10⁻²⁴ cm³ [21], α_{p N} ∼ 3 × 10⁻²⁴ cm³ [22], the Bohr radius a₀, the reduced Planck constant ħ, and the effective principal quantum number n^* . Strictly speaking, C₆ has a specific value for each radiating atom/perturber pair; fortunately, however, these specific values do not differ greatly, and a medium value seems to be a good compromise. Polarisability has in cgs units dimension cm³, while in SI units dimension kg⁻¹ s⁴ A²; this is why we use the cgs system in the majority of our calculations.

However, when we look at the temperature dependence of water dissociation at atmospheric pressure and thermal equilibrium (see figure 5; the calculation is similar to that in [23]; at higher pressure, the curves are shifted down and stretched to higher temperatures), it becomes clear that even resonance broadening {q = 3} must be considered. In this case [19],

$$\begin{align}{C_3}\sim \frac{{{e^2}{f_{12}}}}{{2m{\omega _0}}}\sim 4.41 \times {10^{ - 8}}{\text{ c}}{{\text{m}}^3}{\text{ }}{{\text{s}}^{ - 1}}\end{align} \tag{ 6 }$$

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where e is the elementary charge, f₁₂ is the oscillator strength of the given transition [24] (a full list of energy transitions contributing to the hydrogen Balmer-alpha line can be found in [25]), m is the mass of the electron, and ω₀ is the line central frequency on the ω-scale. Hence, C₃ has a specific value for each spectral line.

Perturbers passing along a straight line with velocity v by the radiating atom (of a different sort than the perturbers) change the phase of this radiation and broaden the radiating spectrum. In the case of non-overlapping collisions, the broadened line spectrum leads to a Lorentz (dispersion) profile [19, 20]:

$$\begin{align}I\left( {\Delta \omega } \right) = \frac{1}{\pi }\frac{{\left( {\gamma /2} \right)}}{{{{\left( {\Delta \omega - \Delta {\omega _{\text{s}}}} \right)}^2} + {{\left( {\gamma /2} \right)}^2}}},\end{align} \tag{ 7 }$$

where I(Δω) is the line intensity as a function of the circular frequency deflection Δω from the central point ω₀,

$$\begin{align}{\gamma _{{\text{VdW}}}} = 8.08{\text{ }}{C_6}^{2/5}{\text{ }}{v^{3/5}}{\text{ }}N\end{align} \tag{ 8 }$$

is the full width at half magnitude (FWHM) of the given line (on the ω-scale), and

$$\begin{align}\Delta {\omega _{{\text{s VdW}}}} = 2.94{\text{ }}{C_6}^{2/5}{\text{ }}{v^{3/5}}{\text{ }}N\end{align} \tag{ 9 }$$

is the shift of the centre of the line ω₀ due to collisions (where v is the thermal velocity, and N is the density of perturbers). Note that the ratio γ/Δω_s has a constant value of 2.75, and independent determination of these two quantities cannot yield any additional information. Starting with an experimentally determined value of γ for discharges (a), (b), and (c), we can calculate the appropriate particle densities N and pressures p(= N k_B T with the Boltzmann constant k_B, temperature T) in Pa as well as in atm (physical atmospheres). The results are summarised in table 1.

The applicability of this approximation is usually examined with the help of the Weisskopf radius ρ_W, where

$$\begin{align}{\rho _{{\text{W VdW}}}} = {\left( {1.178{\text{ }}{C_6}/v} \right)^{1/5}}\end{align} \tag{ 10 }$$

(in our case ∼7 × 10⁻⁸ cm) is the impact parameter of a perturber colliding with a radiating atom which causes a phase shift of radiation 1. The impact approximation is valid if ρ_W ≪ N^−1/3 (mean distance between the particles d; see figure 6). At the same time, the region of applicability is restricted to the interval Δω < v/ρ_w (inverse duration of one collision) ∼1.9 × 10¹³ s⁻¹ i.e. for the central part of the spectral line profile. Due to this, the pressures determined above are not relevant.

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Perturbers passing along a straight line with velocity v by a radiating atom (of the same sort as the perturbers) change the phase of this radiation and broaden the radiating spectrum. In the case of non-overlapping collisions, the broadened line spectrum also leads to the Lorentz (dispersion) profile in (7), in which the damping constant γ_res is

$$\begin{align}{\gamma _{{\text{res}}}} = 2{\text{ }}{\pi ^2}{\text{ }}{C_3}{\text{ }}N,\end{align} \tag{ 11 }$$

which does not depend on the thermal velocity of perturbers, and hence does not depend on the temperature of the spark channel, and the shift of the line centre Δω_{s res} is

$$\begin{align}\Delta {\omega _{{\text{s}}{\text{ }}{\text{res}}}}\sim 0.\end{align} \tag{ 12 }$$

If the experimentally found broadening of the H_α line is caused by the impacts of hydrogen atoms/molecules, their appropriate densities and pressures can be calculated as summarised in table 2.

Even here, it is essential to test the applicability of this approximation: The Weisskopf radius ρ_W

$$\begin{align}{\rho _{{\text{W res}}}} = {\left( {2{\text{ }}{C_3}/v} \right)^{1/2}}\end{align} \tag{ 13 }$$

must be much smaller than the mean distance between the perturbers (ρ_W ≪ N^−1/3). The reality is shown in figure 7, and it is obvious that even in this case, this approximation is not applicable.

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In the case of higher densities, the simultaneous action of more perturbers must be considered [19, 20], although their motion is neglected. The simplest case is the one, where the phase shift of the emission of the radiating atom is caused by the nearest perturber. From the probability that this nearest perturber has distance r follows fitting curve to line profile in the form

$$\begin{align}I\left( {\Delta \omega } \right) = {k_1}\frac{3}{q}{\left( {\frac{{\Delta {\omega _x}}}{{\Delta \omega }}} \right)^{\left( {q + 3} \right)/q}}{\text{exp}}\left[ { - {{\left( {\frac{{\Delta {\omega _x}}}{{\Delta \omega }}} \right)}^{3/q}}} \right],\end{align} \tag{ 14 }$$

where (similarly to (4))

$$\begin{align}\Delta {\omega _x} = {C_q}{\text{ }}{r_x}^{ - q} = {C_q}{\text{ }}{\left( {\left( {4/3} \right){\text{ }}\pi {\text{ }}N} \right)^{\left( {1/3} \right)q}}\end{align} \tag{ 15 }$$

is a normal shift corresponding to a mean distance r_x of the perturbers. The fitting curve in (14) has two maxima located at

$$\begin{align}\Delta {\omega _{{\text{max}}}} = \pm {\left\{ {3/\left( {q + 3} \right)} \right\}^{q/3}}{\text{ }}\Delta {\omega _x}\end{align} \tag{ 16 }$$

with size

$$\begin{align}I\left( {\Delta {\omega _{{\text{max}}}}} \right) = {k_1}{\text{ }}\left( {3/q} \right){\text{ }}{\left\{ {\left( {q + 3} \right)/3} \right\}^{\left( {q + 3} \right)/3}}{\text{ exp}}\left[ { - \left( {q + 3} \right)/3} \right].\end{align} \tag{ 17 }$$

To fit this curve to the experimental profile, we have two degrees of freedom: the amplitude (coefficient k₁), and the fitting curve parameter Δω_x . During the fitting procedure (see figure 8), we usually try to (i) tie the approximating function in (14) to the end points of the line segment marked 'FWHM a' (the FWHM of the experimental line profile); and (ii) achieve a reasonably good fit to the wings of this line (the approximating function (14) is obscured around the line centre). In practice, we usually search directly for Δω_x where I(Δω_x ) = (3 k₁/q) exp[−1]; care should be taken since this equation has four roots, and only the two outer ones are correct. For the NN approximation, we compared three different fittings (amplitude preference, Δω_x preference, no preference), and the standard deviation of density determination was smaller than 15%. Simultaneously, we can find the half width at half magnitude Δω_HWHM and its relation to Δω_x :

$$\begin{align}\Delta {\omega _x} = f \times \Delta {\omega _{{\text{HWHM}}}},{\text{ }}\end{align} \tag{ 18 }$$

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where f is a specific number for each discharge.

The NN approximation for the Van der Waals interaction (q = 6) has a fitting curve in the form

$$\begin{align}{I_{{\text{VdW}}}}\left( {\Delta \omega } \right) = {k_{1{\text{ VdW}}}}\frac{1}{2}{\left( {\frac{{\Delta {\omega _{x{\text{ VdW}}}}}}{{\Delta \omega }}} \right)^{3/2}}{\text{exp}}\left[ { - {{\left( {\frac{{\Delta {\omega _{x{\text{ VdW}}}}}}{{\Delta \omega }}} \right)}^{1/2}}} \right],\end{align} \tag{ 19 }$$

where

$$\begin{align}\Delta {\omega _{x{\text{ VdW}}}} = {C_{6{\text{ }}}}{r_x}^{ - 6} = {C_6}{\text{ }}{\left( {\left( {4/3} \right){\text{ }}\pi {\text{ }}N} \right)^2}.\end{align} \tag{ 20 }$$

Hence,

$$\begin{align}N = \left( {3/\left( {4\pi } \right)} \right) \times {\left\{ {\Delta {\omega _{x{\text{ }}{\text{VdW}}}}/{C_6}} \right\}^{1/2}},\end{align} \tag{ 21 }$$

and the pressure p is then

$$\begin{align}p = N{\text{ }}{k_{\text{B}}}{\text{ }}T.\end{align} \tag{ 22 }$$

Our calculations of N and p (where pressure p is evaluated for a temperature T = 8000 K) based on the fitting procedure described above are given in table 3.

The NN approximation for resonance interaction (q = 3) has a fitting curve in the form

$$\begin{align}{I_{{\text{res}}}}\left( {\Delta \omega } \right) = {k_{1{\text{ res}}}}{\text{ }}{\left( {\frac{{\Delta {\omega _{x{\text{ res}}}}}}{{\Delta \omega }}} \right)^2}{\text{ exp}}\left[ {{\text{ }} - \frac{{\Delta {\omega _{x{\text{ res}}}}}}{{\Delta \omega }}} \right],\end{align} \tag{ 23 }$$

where

$$\begin{align}\Delta {\omega _{x{\text{ res}}}} = {C_3}{\text{ }}{r_x}^{ - 3} = {C_3}{\text{ }}\left( {\left( {4/3} \right){\text{ }}\pi {\text{ }}N} \right).\end{align} \tag{ 24 }$$

Hence,

$$\begin{align}N = \left( {3/\left( {4\pi} \right)} \right) \times \left\{ {\Delta {\omega _{x{\text{ res}}}}/{C_3}} \right\}\end{align} \tag{ 25 }$$

and the pressure p is then

$$\begin{align}p = N{\text{ }}{k_{\text{B}}}{\text{ }}T.\end{align} \tag{ 26 }$$

Our calculations of N and p (where the pressure p is evaluated for a temperature T = 8000 K) based on the fitting procedure described above are given in table 4.

Even in this case we have quasi-static approximation, at which for example quasi-static electric field of stationary ions splits spectral lines of atoms with one valence electron into equidistant components. These are then broadened by an impact effect of much faster electrons. The resulting line profile is a superposition of these broadened components. Fortunately, we needn´t follow this rather complicated theory, in which deflections from Holtzmark field, contribution of 'slow' electrons to quasi-static field, Debye screening, and so on are respected, because H_α line profiles were tabulated in a relatively large range of temperatures and densities in [26]. From these tables we learnt that (1) the temperature dependence is very weak (esp. at low densities), (2) deflections from the Lorentz profiles are relatively small, (3) the half line-width (Δω_HWHM—half width at half magnitude) can be approximated at 10 000 K by the power function

$$\begin{align}{N_{\text{e}}} = 23.957 \times \Delta {\omega _{{\text{HWHM}}}}^{1.2731},{\text{ }}\end{align} \tag{ 27 }$$

where N_e is density of electrons in cm³. With that we can easily find results summarized in table 5.

As mentioned in section 2, the amplitude of the pressure wave in water (at a distance of 70 mm from the axis of the electrodes) was measured using the Müller-Platte needle probe. The results are shown in table 6. It should be borne in mind that this is only an indirect measurement (the amplitude of the pressure wave falls with distance from the discharge channel).

To estimate the partial pressures of the water vapour, and the nitrogen in the discharge channel, we consider (in the ideal gas approximation) the following processes: (i) during the first 3 μs (the exposure time of the spectrograph camera), the spark discharge (a) evaporates a water column of the radius R ({we selected R = 0.1 cm—the first free parameter}; the spark discharges (b) and (c) evaporate radii that are proportional to the square root of the energy input in discharges (b) and (c) normalized to energy input to discharge (a) during this time interval) and the length z1 (determined from shadow pictures {see the second column of figure 3}; z1 is a distance between the bubble front and the high-voltage electrode). Hence, the volume V1 of water evaporated in z1 thick water slab (V1 = π R² z1) is proportional to the energy input during the first 3 μs. From these data, an evaporated water volume per unit length V1u (V1u = π R²) can be calculated. The evaporated water in V1 has partial pressure p_{V1 H2O}. (ii) The bubble volume V2 and the bubble half-contour length z2 are also determined from shadow pictures (second column of figure 3). The spark discharge propagating along the inner wall of the bubble [14] evaporates a volume of water proportional to V1u × z2 with proportionality coefficient η, the second free parameter (we used η∼ 0.5). This evaporated water creates a partial pressure p_{V2 H2O} in V2. The pressures p_{V1 H2O} and p_{V2 H2O} adiabatically balance out (an irreversible process), and also partly penetrate into volume V3 (see below) creating a resulting partial pressure p_H2O. (iii) The bubble feeding tube with volume V3 has a constant pressure of nitrogen p_{V3 N2}, and only a small part s (third free parameter, we used s∼ 0.03) of this V3s (V3s = s× V3) takes part in fast pressure changes. (iv) The nitrogen pressure in the bubble p_{V2 N2} and in the part of feeding volume V3s adiabatically expands to V1, yielding the same value of p_N2 for V1, V2, and V3s. (v) The total pressure p_tot in V1, V2, and V3s is p_tot = p_H2O+ p_N2. These results are shown in table 7.