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

22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
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:
••
3 •2 P
(1)
R R+ R = i
2
ρ0
•
where ρ 0 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. P i is the
dynamical pressure in the liquid at the bubble surface.
•
R 2s
Pi (t ) = Pgas (R , t ) − 4 µ −
− P0
R R
(2)
P 0 is the pressure in the liquid far from the bubble wall.
2σ
−
R
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.
•
Low frequency
signal generator
and − 4µ R and describe the influence of the
R
surface tension s and the dynamic viscosity µ
respectively. P gas 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:
3γ

2σ  R0  .
(3)
 
Pγaσ (R ,t ) =  P0 +
R0  R 

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
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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 nonspherical shapes.
High voltage
power supply
Micrometric
positioning
High voltage
nanosecond switch
Ballast
Resistor
5V-generator
silica
capillary
Pt wire
Cell
Si
HV probes
Resistor
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
1
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 1000KM3) 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.
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.
3. Results and discussion
The shape of a single bubble was recorded by the fastvideo camera. It was next digitalized and the contour of
the bubble wall was determined as shown in figure 2.
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.
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.
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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. 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. 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.
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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).
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