22nd International Symposium on Plasma Chemistry July 5-10, 2015; Antwerp, Belgium

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Instability of cavitation bubbles formed by discharges in heptane in pin-to-plate configuration

A. Hamdan1, C. Noël2 and T. Belmonte2

1 Université de Lorraine, Institut Jean Lamour, UMR CNRS 7198, 54042 Nancy, France

2 CNRS, Institut Jean Lamour, UMR CNRS 7198, 54042 Nancy, France

Abstract: The way the wall of a cavitation bubble, obtained by nanosecond discharge in pin-to-plate configuration, is distorted is studied in heptane. The influence of the applied current is crucial to observe the ejection of gas droplets from the bubble to the liquid. Indeed, the maximum radius of the bubble correlates nicely with the discharge current but not so well with other discharge parameters like the deposited energy, the electric charge or the breakdown voltage. The way surface tension affects the shape of the bubble at the liquid/surface interface is also described.

Keywords: discharges in liquids, cavitation bubble, instabilities

1. General

The radius variation of a spherical cavitation bubble is very well described by the Rayleigh-Plesset equation for a non-compressive liquid [1]. The time evolution of the bubble radius R is given by:

0

2

23

ρiPRRR =+

••• (1)

where ρ0 is the liquid density at room temperature, and •R

and ••

R are the first and second time derivatives of R, i.e. the bubble wall velocity and acceleration. Pi is the dynamical pressure in the liquid at the bubble surface.

( ) ( ) 024 PRR

Rt,RPtP gasi −−−=

•sµ (2)

P0 is the pressure in the liquid far from the bubble wall.

Rs2

− and RR•

− µ4 and describe the influence of the

surface tension s and the dynamic viscosity µ respectively. Pgas is the pressure in the bubble. This pressure is a function of the bubble density, which is related to the bubble radius R, and then to the time t. Therefore, an equation of state is required for the bubble gas. By lack of data, a γ-law is commonly used to describe the gas equation of state:

( )γs 3

0

00

2

+=

RR

RPt,RPgas

. (3)

This approach is usually sufficient if the pressure in the bubble is not too high. Otherwise, the Gilmore equation must be preferred to account for the compressibility of the liquid [2]. Then, the thermodynamic properties of the liquid have to be introduced.

If the shape of the bubble is not spherical, no other model is available. When discharges are created in a dielectric liquid in a pin-to-plate configuration, generated bubbles are hemispherical. The importance of this

phenomenon is crucial but has never been studied in detail. In this preliminary work, we want to report some results showing the dynamics of bubbles with non-spherical shapes. 2. Experimental setup

The experimental setup used in this work is shown in figure 1 and described in detail in one of our former works [3]. Briefly, a DC high voltage power supply (Technix SR15-R-1200:15 kV–80 mA) fed a high-voltage solid-state switch (HTS-301-03-GSM) that delivered a maximum current of 2×30 A under a voltage up to 2×30 kV. A Pulsed High Voltage (PHV), up to +15 kV, was controlled by a low-frequency signal generator and applied to the power electrode, the other electrode being grounded. The on-time of one pulse was 200 ns.

Fig. 1. Experimental setup.

The setup was arranged in a pin-to-plate configuration.

The pin electrode is a platinum wire (50 µm in diameter). The plane electrode is a (100)-oriented silicon wafer. The two electrodes are immersed in the dielectric liquid (~ 10

High voltage power supply

Micrometricpositioning

Pt wire

CellSi

Ballast Resistor

Resistor

silicacapillary

HV probes

High voltage nanosecond switch

5V-generator

Low frequencysignal generator

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cm3). Anhydrous n-heptane (99%) was provided by Sigma-Aldrich. We have chosen to use a dielectric liquid with a relatively low purity because this choice is not of crucial importance in the present study. Indeed, highly reproducible breakdown voltage conditions are not needed.

For each impulse, we record by high-voltage probes the electrical signals that enable us to determine the intensity and voltage of the discharge. The resulting data are fitted with an RLC circuit. The energy deposited in the plasma is determined from the fitting parameters.

A fast-video camera (FASTCAM SA5 model 1000K-M3) has been used to record the bubble oscillations. The pictures have been acquired at a rate of 3.0×105 or 4.2×105 images-per-second for a corresponding window of 128×48 pixels.

3. Results and discussion

The shape of a single bubble was recorded by the fast-

video camera. It was next digitalized and the contour of the bubble wall was determined as shown in figure 2.

Fig. 2. Example of a digitalized image showing a bubble

at a certain stage of its evolution.

Then, the position of the bubble wall was determined as a function of time for different ballast resistances and applied voltage (figures 3 to 5 corresponding to ballast resistances of: 120 Ω, 350 Ω and 4000 Ω). The lower the ballast, the higher the current for a given voltage.

Other parameters like the deposition energy E, the deposited charge and the maximum current were determined in each situation. The time step between two successive images is determined from the acquisition frequency (it is either 3.3333 µs or 2.3809 µs). Images were separated according to whether the bubble was expanding or contracting during its first oscillation.

In figure 3, we notice that the bubble shape is spherical at its maximum radius but not at the beginning and the centre of the sphere is always located slightly under the surface of the silicon wafer. When the current is high, i.e.

when the ballast resistance is low or when the breakdown voltage is high, ejection of smaller bubbles can be observed. At lower current, wall instabilities are observed. If the current is weak, friction at the wall of the plate electrode due to surface tension induces a lift off at bubble edge. This effect is not visible on the wall of the pin electrode which is made of platinum and not silicon.

Fig. 3. Dynamics of cavitation bubbles for a ballast resistance of 120 Ω. Left: Expansion phase. Right:

Contraction phase. Blue arrows show the ejection of a small bubble. Red arrows show the formation of pinches.

During the contraction phase, the shrinkage of the

bubble is affected by the vicinity of the electrodes. The almost spherical shape of the bubble envelop is pinched near the walls, making at low or moderate current two times two local minima in radius (i.e. the four red arrows on the right column in figure 3). At high current, the development of strong instabilities distorts strongly the bubble shape and many pinches are observed.

If we can accurately reproduce in figure 4 the dynamics of a weakly-distorted bubble (the fifth one from top in figure 3), it does not mean much for the boundary conditions chosen to get this result (i.e. the initial and final bubble radii and the initial wall velocity) cannot be assessed independently [4]. In fact, the choice of boundary conditions determines the initial pressure at the inception of the bubble expansion phase, a parameter whose values are highly controversial and in all cases, poorly known.

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As theory was developed for spherical bubbles, the deposited energy being released at one spot and isotropically – which is not perfectly true even for laser cavitation – resorting to the Rayleigh-Plesset equation is highly questionable. Then, these results will be considered as non-relevant in the present situation.

Fig. 4. Experimental evolution of the bubble dynamics

together with a possible but non-relevant solution obtained with Rayleigh-Plesset’s equation.

When the ballast resistance increases for a given breakdown voltage, the corresponding current decreases and the development of instabilities at the bubble wall are limited (figures 5 and 6).

Fig. 5. Dynamics of cavitation bubbles for a ballast resistance of 350 Ω. Left: Expansion phase. Right:

Contraction phase. Red arrows show the formation of pinches.

Fig. 6. Dynamics of cavitation bubbles for a ballast resistance of 4000 Ω. Left: Expansion phase. Right:

Contraction phase. Red arrows show the formation of pinches.

The evolution of the bubble maximum radius versus the maximum current is depicted in figure 7. The higher the current, the bigger the bubble. However, a limitation in the maximum bubble radius seems to happen at high current, likely because of the aforementioned ejection phenomenon.

Fig. 7. Evolution of the bubble maximum radius versus

the maximum current for three different ballast resistance. Other evolutions of the maximum bubble radius versus the electric charge, the deposited energy or the breakdown voltage give less satisfying behaviours, which stress out

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the specific role of the discharge current on the bubble dynamics. 4. Conclusion

In this preliminary study, the dynamics of bubbles created by nanosecond pulse discharges have been studied in a pin-to-plate electrode configuration.

The applied current plays a key role in the development of bubble wall instabilities, leading to ejection of gas droplets in the most violent situation. The maximum radius of the bubble correlates nicely with the discharge current but not so well with other discharge parameters like the deposited energy, the electric charge or the breakdown voltage. We also noticed that the bubble lifts off from silicon but not from platinum.

The obtained raw data might be used next to propose a model accounting for the development of wall instabilities at high current. 5. References [1] M. Plesset, J. Appl. Mech., 16, 277 (1949) [2] F. R. Gilmore, “The Growth or Collapse of a Spherical Bubble in a Viscous Compressible Liquid,” Hydrodynamics Laboratory, California Institute of Technology, Pasadena, California, Report No. 26-4 (1952) 41 pp [3] A. Hamdan, C. Noël, J. Ghanbaja and T. Belmonte, Plasma Chem Plasma Process, 34, 1101 (2014). [4] A. Hamdan, C. Noel, F. Kosior, G. Henrion and T. Belmonte J. Acoust. Soc. Am. 134, 991 (2013).