5 cm. Flash 20 khz. Camera

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1 BUBBLE SIZE DISTRIBUTIONS AND STRUCTURES IN ACOUSTIC CAVITATION METTIN R., LUTHER S., LAUTERBORN W. Drittes Physikalisches Institut, Universitat Gottingen Burgerstr , D Gottingen (Germany) ABSTRACT The bubble size distribution in an acoustic cavitation streamer structure at khz is measured by photographic means. Recalculation of the bubble equilibrium radii leads to values in the range of a few micrometer. This is compared to previous results and a theoretical estimation based on surface stability of oscillating spherical bubbles. La distribution de grandeurs de bulle dans une structure lamenteuse de cavitation acoustique est mesuree par des moyens photographiques. Le recalcul des rayons d'equilibre de bulle mene aux valeurs de quelque micrometre. Ceci est compare aux resultats precedents et a une evaluation theorique basee sur la stabilite supercielle des bulles spheriques oscillantes. It is a well accepted fact meanwhile that most of the eects observed in sonochemistry are due to cavitation, i.e., the occurrence, motion, and violent collapse of bubbles in the irradiated liquid. Nevertheless, there exists remarkable few literature about sizes of the bubbles involved and about their locations in space. For a deeper understanding and enhancement of sonochemical reactions, however, such data seem to be quite valuable. Here we give a brief report on preliminary size measurement results from an acoustic cavitation experiment which has been constructed for investigation of bubble structure formation. I. EXPERIMENT The experimental setup is shown schematically in Fig. 1. The khz transducer excites a standing wave in the water with one pressure antinode near the geometric center of the transparent cuvette. For driving amplitudes above 100 kpa, bubble structures (\streamers" and \Acoustic Lichtenberg Figures") appear in the antinode region. See Leighton (1) for general remarks on such structures and Ref. (2) for a simulation approach. For the photographic investigations, a computer controlled video camera images the bubbles via a moderately magnifying optics. To provide a sharp standing image, a ash of duration 1 s is used. Double exposures, i.e., two subsequent ashes are possible to allow velocity measurements. 125

2 5 cm Flash khz Camera Figure 1: Schematic setup of the cavitation experiment: A piezoceramic disc transducer induces streamers in a rectangular PMMA cell lled with tap water. A microsecond ashlight illuminates the picture recorded by a computer controlled camera. A typical image obtained by the above technique is shown in Fig. 2. The bubble density in the streamers is not very high, which might be an unexpected feature when longer time exposures or the impression of the naked eye are recalled; e.g. compare some illustrations in Refs. (1) and (2). Therefore the bubbles can be counted and evaluated in size directly. To increase the spatial resolution, the ashes are set to a xed phase of the driving signal where the bubbles appear largest. Their size is close to the maximum at that moment, and one can recalculate equilibrium radii by means of a bubble model. II. RESULTS The evaluation of the picture in Fig. 2 results in rather small upper bounds of the bubble sizes. The central cluster is excluded from the evaluation as no individual bubbles can be resolved in that region. It has to be left open for future investigation if it consists of many small bubbles or of one or few large bubbles undergoing strong shape deformations. In the outer region, however, no bubble of maximum radius larger than about 70 m can be observed. Most imaged bubbles have maximum radii between 5 and 40 m. Objects of a diameter around 10 m, corresponding to a radius of 5 m, are at the limit of resolution in the present setup. To obtain the equilibrium radii, we assume spherical bubbles just caught at their maximum elongation (where we accept a small error by assumption of the maximum for all bubbles at the same driving phase). Such bubbles can be modelled by standard equations like the Keller-Miksis variant given in Ref. (3). A plot of 126

3 Figure 2: Double exposure of a streamer structure. Horizontal lenght: 6 mm. Time between ashes: 2.5 ms. Central pressure amplitude 150 kpa. maximum vs. equilibrium radii for realistic driving pressure amplitudes results in graphs like those given in the top of Fig. 3. Figure 2 was recorded with a central pressure amplitude of about 150 kpa. A recalculation with P ex = 130 kpa results in tha data shown in the bottom part of Fig. 3; the value has been chosen to give a more conservative upper bound and to account for the small phase errors and reduced excitation pressure for bubble positions a little o the antinode. The measured equilibrium radii clearly fall in the few micrometer range, accumulating above the Blake threshold (1). No bubble with R 0 larger than 10 m was detected. At the Blake threshold, the negative driving pressure peak approximately balances the surface tension of the bubble at rest, 2/R 0. At a pressure amplitude of 150 kpa and for water ( = N/m), this is the case for equilibrium radii of about 1 m. Thus, bubbles smaller than this value will not show signicant oscillation amplitudes (and remain undetected in our experiment), while larger bubbles can expand to many times their equilibrium size. The large expansion is followed by a violent collapse that is important for sonochemistry. From Fig. 2, also some bubble (projected) velocities could be determined. They ranged from about 3 cm/s up to cm/s. However, faster bubbles might be present in the double exposure, but with their correspnding second image too far apart for a unique identication. 127

4 R max [µm] kpa 130 kpa 110 kpa R 0 [µm] R 0 R max n R [µm] Figure 3: Top: Maximum radius R max vs. equilibrium radius R 0 of a spherical bubble driven at khz. The dierent lines correspond to the indicated excitation pressure amplitudes. Bottom: Bubble distribution of the counted number n vs. R for the picture shown in Fig. 2. The dashed data corresponds to measured R max, the solid boxes show the recalculated distribution of R 0 with P ex = 130 kpa. III. DISCUSSION The preliminary results presented here are in agreement with previous measurements of acoustic cavitation bubble sizes. In particular, the photographic data by Kuttru and Plass (4) and the data from Billo (5) obtained by means of holographic cinematography and recalculated in Ref. (6) is consistent with our ndings. Also, data recently given by Burdin et al. (7) for a dierent experimental setup using laser diraction techniques is in accordance with the quite small bubble sizes just above the Blake threshold. The experiments reveil a large discrepancy from the common, but obviously wrong assumption of a bubble size distribution near the linear resonance radii. The latter would result in equilibrium radii near 160 m at khz, which is not the case. A simple consideration of shape stabilty, similar to the calculations in Ref. (8), show 128

5 R 0 [µm] stable unstable P ex [kpa] Figure 4: Theoretical boundary of shape instabilities in the parameter plane of excitation pressure amplitude P ex and bubble equilibrium radius R 0. Driving frequency: khz. that at driving pressure amplitudes above 100 kpa pulsating bubbles become shape unstable (i.e., disintegrate) for equilibrium radii above about 10 m, see Fig. 4. Although the given stability border predicts destruction too early as additional stabilizing eects have been neglected, the main trend is clear: The larger the driving pressure amplitude, the smaller the surface-stable bubble sizes. Considering the additional fact that a \chemically active" bubble usually needs a strong collapse that is only possible beyond the Blake threshold, we end up at the conclusion that the important bubbles in sonochemistry range around a micrometer in size. Our experiment supports this conjecture. REFERENCES (1) Leighton, T.G., The Acoustic Bubble, Academic Press, London (1994). (2) Mettin, R. et al., Ultrasonics Sonochemistry, 6(1-2), 25 (1999). (3) Parlitz et al., J. Acoust. Soc. Am., 88, 1061 (1990). (4) Kuttru, H. and Plass, K., Acustica, 11, 225 (1961). (5) Billo, A., PhD thesis, TH Darmstadt (1997). (6) Mettin, R. et al., Phys. Rev. E, 56, 2924 (1997). (7) Burdin, F. et al., Ultrasonics Sonochemistry, 6(1-2), 43 (1999). (8) Brenner, M.P. et al., Phys. Rev. Lett., 75, 954 (1995). 129