![Structure parameters. Weighted mean and standard deviation of the fitted structure parameters, as a function of time delay, depicted for one oscillation period. R is the bubble radius, σ the FWHM of the interface between cavitation bubble interior and water, and ρ the mass density of the bubble core. The fit parameters correspond to the model outlined above, with the applied correction scheme discussed in the supplementary material. The right column shows an enlarged section of the collapse and bubble rebound. The radial dynamics, i.e. the trajectory R(t), is fitted to the Gilmore model [41], with fixed driving frequency νa=87.6kHz . The fit parameters pa=1.72(1)bar , Rn=1.72(1)μm , γ=1.65(3) and Δt0=4.32(4)μs , corresponding to the acoustic driving amplitude, the equilibrium radius of the oscillating bubble, the polytropic exponent of the gas mixture, and the relative timing of collapse ( Δt0 was set to zero at the first collapse).](https://www.researchgate.net/publication/378217418/figure/fig4/AS:11431281227685548@1709696830398/Structure-parameters-Weighted-mean-and-standard-deviation-of-the-fitted-structure_Q320.jpg)
March 2024
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Single bubble sonoluminescence (SBSL) is the phenomenon of synchronous light emission due to the violent collapse of a single spherical bubble in a liquid, driven by an ultrasonic field. During the bubble collapse, matter inside the bubble reaches extreme conditions of several gigapascals and temperatures on the order of 10 000 K, leading to picosecond flashes of visible light. To this day, details regarding the energy focusing mechanism rely on simulations due to the fast dynamics of the bubble collapse and spatial scales below the optical resolution limit. In this work we present phase-contrast holographic imaging with single XFEL pulses of a SBSL cavitation bubble in water. X-rays probe the electron density structure and by that provide a uniquely new view on the bubble interior and its collapse dynamics. The involved fast time-scales are accessed by sub-100 fs XFEL pulses and a custom synchronization scheme for the bubble oscillator. We find that during the whole oscillation cycle the bubble’s density profile can be well described by a simple step-like structure, with the radius R following the dynamics of the Gilmore model. The quantitatively measured internal density and width of the boundary layer exhibit a large variance. Smallest reconstructed bubble sizes reach down to R ≃0.8 μm, and are consistent with spherical symmetry. While we here achieved a spatial resolution of a few 100 nm, the visibility of the bubble and its internal structure is limited by the total X-ray phase shift which can be scaled with experimental parameters.
![Experimental setup: Cavitation bubbles are seeded by an infrared pump laser close to a glass disc held inside an open cuvette filled with water. The jetting bubbles are probed by XFEL pulses while an optical camera observes from the side. a Schematic of the x-ray beam path in quasi-parallel beam illumination (not to scale). b The cuvette made from 3 mm acrylic plates has interior dimensions: a=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=$$\end{document}12 mm, b=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=$$\end{document}15 mm and c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=$$\end{document}36 mm. The glass plate (i.e. the blue disc) is placed in the horizontal plane, so both the x-ray and optical imaging feature a side view of the jetting bubble (perpendicular view). c Here, the glass surface is normal to the x-ray beam (parallel view)](https://www.researchgate.net/publication/377875696/figure/fig1/AS:11431281224396305@1708225340090/Experimental-setup-Cavitation-bubbles-are-seeded-by-an-infrared-pump-laser-close-to-a_Q320.jpg)
![Optical and x-ray holographic imaging of a bubble jetting towards a rigid (glass) surface. a Optical image sequence of the bubble dynamics taken with 75 kfps (i.e. 13.3μs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$13.3\,\upmu {s}$$\end{document} between frames). Here, the stand-off distance is γ=0.57±0.03\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0.57\pm 0.03$$\end{document}. The lower panels present x-ray holographic images captured at the corresponding instant marked with the legend “x-ray” in (a). Panel b shows a side view of the jetting cavity as an X-ray hologram, while in (c) the jetting of a second bubble with similar dynamics is observed from a parallel view (normal to the glass disc). The x-ray light reveals the bubble interior, invisible in the optical shadowgraphs of (a). The arrows indicate some of the features commonly present in the holograms: i x-ray light blocked by the border of the hole of the laser mirror; ii thick interfacial line; iii bright arc; iv double shadow at the jet liquid column; v bubble pedestal stripes; and vi Blake splash](https://www.researchgate.net/publication/377875696/figure/fig3/AS:11431281224396307@1708225340710/Optical-and-x-ray-holographic-imaging-of-a-bubble-jetting-towards-a-rigid-glass_Q320.jpg)
![Bubble jetting observed from a perpendicular view, as explained in Fig. 1c, and contrasted with the full 3D numerical simulations, computed for a measured stand-off distance γ=0.57±0.03\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0.57\pm 0.03$$\end{document}. a x-ray holographic images of the jet dynamics and its impact on the solid surface. The images are compared with frames from the numerical simulations with time indicated in μs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu {s}$$\end{document} after the laser pulse. In those, the gas–liquid interface is highlighted by a black contour drawn along a plane crossing the centre of the bubble. The Blake splash can be clearly seen right after the jet impact, i.e. at t≃100μs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\,\simeq \,100\,\upmu {s}$$\end{document} (as pointed out by the blue arrows. b Optical images acquired simultaneously to the x-ray images from a direction orthogonal to the x-ray propagation. As the jetting points predominantly away from the camera, the bubble appears nearly symmetrical when viewed from this perspective. The two types of images are paired by the capital letters in the lower right corner of the frames. The red contours show the numerical simulations from the perspective of the optical camera, cut through the origin. Therefore, the moment of impact appears slightly retarded as compared to the according frames in (a). Multimedia view: (Video Fig. 3.avi)](https://www.researchgate.net/publication/377875696/figure/fig4/AS:11431281224396308@1708225341131/Bubble-jetting-observed-from-a-perpendicular-view-as-explained-in-Fig1c-and-contrasted_Q320.jpg)
![Parallel view of the bubble collapse as the jet hits the glass plate for a case with γ=0.56±0.03\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0.56\pm 0.03$$\end{document}. The scale bars represent 250μm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$250\,{\upmu \text {m}}$$\end{document}. a X-ray images of the jet impact observed from a parallel view (see Fig. 1c). The asymmetry of the splash and the cavity collapse is clearly visible from this perspective. For example, the splash is predominantly directed towards the lower right corner of the panels in (a) and (b). Here the time in μs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu {s}$$\end{document} was obtained by matching the experimental frames with the simulations presented in panel (b). The gas–liquid interface in the simulation is rendered in a semi-transparent turquoise colour, while the liquid is left white. c Synchronous optical images taken perpendicular to the glass disc and orthogonal to the x-ray beam direction](https://www.researchgate.net/publication/377875696/figure/fig5/AS:11431281224422513@1708225341708/Parallel-view-of-the-bubble-collapse-as-the-jet-hits-the-glass-plate-for-a-case-with_Q320.jpg)
![Holographic phase retrieval and cavitation bubble density
a X-ray hologram (normalized intensity I/I0) of a cavitation bubble at Δt = 10 ns, exhibiting strong contrast at the inner interface (gas/shockwave) and outer interface (shockwave/equilibrium water). For phase retrieval, the hologram is averaged along the polar angle to obtain the radial intensity distribution. b Radial intensity distribution of (a) and intensity obtained from numerical propagation of the RFP retrieved phase (see (c)). c In a forward model approach the projected phase ϕ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\phi }$$\end{document} of the bubble is retrieved by minimizing the difference to the radial intensity distribution (Radially Fitted Phase, RFP). d Retrieved phase of (a) using the AP algorithm, for comparison. The phase distribution reflects the deficit density in the core and excess density in the shockwave. e The average along the polar angle of the AP reconstruction is compared to the result obtained from RFP (c). f Radial three dimensional phase ϕ reconstructed from the RFP projected phase (c). The right ordinate shows the calculated density distribution of the cavitation bubble for an ellipticity factor ϵ ≈ 0.8. Scale bars: 10 μm (a, d).](https://www.researchgate.net/publication/352220891/figure/fig2/AS:1032665025228800@1623217936009/Holographic-phase-retrieval-and-cavitation-bubble-density-a-X-ray-hologram-normalized_Q320.jpg)
![Cavitation dynamics
a Radius of bubble and shockwave boundary RB and RSW. Each scatter dot represents one processed cavitation event. The color scales with the bubble’s energy (shared colorbar with (b), logarithmic scale). b Radial 3d phase profiles ϕ(R) of cavitation events with 2–3 μm bubble boundary radius (dashed box in (a)). The radial phase was reconstructed from the RFP phases ϕ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\phi }$$\end{document}. The color represents EB. The median of all phase distributions is shown in black. The phase shift of vacuum to water ϕvac is shown for comparison. c Median of phase profiles for different ranges of RB, showing how the median phase evolves with time. Here, only cavitation events with EB between 66 and 130 μJ were used. The color represents the median of the time delay Δt. The (smoothed) envelope of the shockwave’s phase shift (black) is used to calculate the shockwave’s peak pressure ppeak as a function of the distance to the bubble center R. dppeak(R) obtained from the envelope of the shockwave’s phase shift for energy ranges EB between 7 and 66 μJ (low EB), 66 and 130 μJ (med. EB) and 130–250 μJ (high EB).](https://www.researchgate.net/publication/352220891/figure/fig4/AS:1032665025228810@1623217936228/Cavitation-dynamics-a-Radius-of-bubble-and-shockwave-boundary-RB-and-RSW-Each-scatter_Q320.jpg)
