Mixture segregation within sonoluminescing bubbles

Inside a sonoluminescing gas bubble in water, there will always be a mixture of gas and water vapor. To simplify the problem, Storey and Szeri (1999) considered the case of a mixture of the two monatomic inert gases, helium and argon. For this model, they assumed that no water vapor was present in the model. This avoids the complications caused by evaporation and condensation at the bubble interface. Due to thermal diffusion there will be a tendency for the gases to segregate due to the large temperature gradient. This phenom-enon was first a theoretical conjecture of the Chapman–Enskog theory and later confirmed by experimentation (Chapman and Cowling (1900)). In a rare gas mixture, molecules of the gas with the larger molecular mass will be driven to the cooler regions and molecules of the gas with the smaller molecular mass will be driven to the hotter regions. In sonolumines-cence, this thermal diffusion can be expected to play a part during the collapse of the bubble when the center is extremely hot and the bubble wall is relatively cool.

Figure 3.33 Figure taken from Barber et al. (1995). It shows the experimental result for the phase of light emission φs(t )^– for three different argon concentrations c∞/c0 = 0.00395, c∞/c0 = 0.0658, and c∞/c0 = 0.26, corresponding to a gas pressure overhead of p∞ = 3 mmHg, 50 mmHg, and 200 mmHg, respectively. Also shown is the relative phase of light emission for air bubbles: Stable SL is achieved for much higher concentration c^air∞/c0 = 0.2, corresponding to 150 mmHg. Hilgenfeldt et al. (1996) interpreted the drift in the phase of light emission as a result of the bubble growth through rectified diffusion, which is followed by a pinch-off of a microbubble when the bubble is running into the shape instability. (Barber et al. (1995).)

To analyze the problem, Storey and Szeri (1999) used a van der Waals type model for the equation of state (Vuong et al. (1999)) to avoid the shortcomings of less sophisticated equations of state. To solve the governing partial differential equations of the gas, Figure 3.34 Radial response of the bubble to one cycle of the applied acoustic field. The bubble is He–Ar (10% mass He) forced at a pressure amplitude of 1.3 (Storey and Szeri (1999) © Cambridge University Press.)

Figure 3.35 Radial response of the bubble to the applied acoustic field at the main collapse on two time scales. The marker particles were evenly spaced in the reference configuration. The outermost marker particle shows the evolving position of the interface. The bubble is He–Ar (10% mass He) forced at a pressure amplitude of 1.3. (Storey and Szeri (1999) © Cambridge University Press.)

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Chebyshev polynominals were used. They assumed that no transport occurs across the bubble surface. They also neglected heat transfer by radiation and made no attempt to model the light emission.

Storey and Szeri (1999) investigated a single bubble SL case with the following param-eters: liquid pressure P_l = 101 kPa, initial radius R₀ = 4.5 µm, initial temperature T₀ = 300 K, pressure of gas in bubble P₀ = P_l + 2σ/R₀, sound pressure amplitude ratio P_A = P_max/P₀ = 1.3 and sound frequency f = 26.5 kHz. The bubble contained a helium–argon mixture composed of 10% helium by mass (52.58% helium on a mole basis). The radial response of this bubble to one cycle of the applied acoustic field is shown on Fig. 3.34. Attention is then focused on the first main collapse of the bubble which occurs at approximately 20.6 ns from the beginning of the cycle. The next five figures show a beautiful sequence of events. Figure 3.35 shows the evolving radial position of selected marker particles in the bubble interior on two timescales which differ by two orders of magnitude. Time is shifted in this figure to put zero at the point of minimum radius. On the 100 ns timescale the collapse is quite sharp but on the 1 ns timescale the bubble smoothly reaches the minimum radius and expands.

Figure 3.36 shows the temperature history of the bubble center on the same two time scales as Fig. 3.35. The temperature rises an order of magnitude on a 1 ns time scale to a maximum of approximately 42,000 K. Figures 3.35 and 3.36 show the bubble motion at minimum radius and the temperature peak on a time scale of several hundred picoseconds.

Figure 3.37 shows the helium mole fraction at the bubble wall and center of the bubble as a function of time. The same two time scales from the previous two figures are used here.

Close examination reveals that compositional inhomogeneities develop much more slowly Figure 3.36 Temperature of the center of the bubble as a function of time at the main collapse. The bubble is He–Ar (10% mass He) forced with a pressure amplitude of 1.3. (Storey and Szeri (1999) © Cambridge University Press.)

than the peak dynamic and thermal fields. The species segregation is driven by the slow build up and release of heat throughout the collapse and not by the short burst of energy supplied to the bubble contents at the point of minimum radius. Note that at the bubble wall the mole fraction of helium decreases to a minimum slightly before t = 0 then increases. In contrast, at the center, helium continues to accumulate for some time after the collapse and then eventually the center becomes slightly argon rich during the expansion.

Another view of the composition field is provided in Fig. 3.38. In this figure are shown snapshots of the composition field within the bubble at intervals of 2 ns with the gray scale indicating the mole fraction of helium. Black indicates argon rich and white indicates helium rich. Upon close inspection it will be seen that the argon is heavily concentrated in a thin region near the wall up to the time of minimum radius. As the bubble expands, the helium continues to move to the center while the sharp argon “shell” relaxes by diffusion.

Storey and Szeri (1999) also considered the multibubble sonoluminescence of the same helium/argon mixture and obtained a maximum temperature at the bubble center of 22,000 K.

Yasui (2000) undertook computer simulations of bubble oscillations under multi-bubble sonoluminescence conditions taking into account the segregation of water vapor and noble gas inside a collapsing bubble. The MBSL was performed in 20°C water for helium, argon and xenon. The ambient bubble radius was 4 µm. The maximum bubble temperature Figure 3.37 Composition history of marker particles at the center and at the bubble wall as a function of time. The wall particle reaches the lowest mole fraction of helium while the center reaches the maximum. The bubble is He–Ar (10% mass He) forced with a pressure amplitude of 1.3. (Storey and Szeri (1999) © Cambridge University Press.)

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at the collapse occurred at the relatively low acoustic amplitude of about 1.5 bar. The maximum temperatures were 17,000 K for helium, and 24,000 K for argon and xenon.

Matula et al. (1996, 1997) describe experiments in microgravity in two abstracts.

3.6 INFLUENCE OF ARGON ON STABLE