Oscillations, Waves, and Interactions - GWDG

182 R. Mettin
concentration of species i in the liquid ci is proportional to the gas partial pressure
pi above the liquid: ci = pi/Hi with Henry’s constant Hi. 9
If we consider a static gas bubble in a liquid that is saturated with the gas, the
partial pressure inside the bubble will be higher than the static pressure outside
because of surface tension. Therefore the gas will go into solution and the bubble
will dissolve sooner or later. This process is even accelerated as the bubble shrinks,
since the surface tension grows with the inverse bubble size. For small bubbles in
the micrometer range, dissolution happens quite rapidly on the scale of milliseconds
to seconds [31]. The same holds for undersaturated liquids. Indeed, only for oversaturated
liquids a static diffusional equilibrium can be reached, which however turns
out to be unstable: bubbles either dissolve, or grow unlimited (and then rise to the
liquid surface due to gravity). This behaviour demands for a stabilization mechanism
of microbubbles in the context of cavitation nuclei [4,5].
If the bubble is not static, but oscillating in a sound pressure field, the bubble
size and the gas pressure change on the acoustic time scale which is typically fast
compared to the diffusional time scale. It turns out that the gas diffusion into the
bubble is favoured in such a case by three mechanisms [5]: (i) the surface effect,
i. e., a larger gas-liquid interface at low gas pressure in the expanded bubble cycle;
(ii) the shell effect, leading to a larger concentration gradient at the bubble wall
during the expanded bubble phase; and (iii) the nonlinear oscillation effect, causing
a longer time per period spent at large bubble sizes, i. e., at low inside gas pressures.
Mechanisms (i) and (ii) can be understood if one recalls that the diffusion is governed
by Fick’s law and thus the mass flow rate is proportional to the interface area and the
concentration gradient [5]. Mechanism (iii) follows from the unsymmetric potential
of the bubble oscillation, representing a hard spring oscillator for compression and a
soft spring for expansion [16].
The above effects can overbalance the dissolution tendency given by surface tension,
and oscillating bubbles can show a net growth for saturated and even undersaturated
liquids. This phenomenon has been named in the literature rectified diffusion [32,33].
In addition, the nonlinear bubble resonances and the dynamic Blake threshold can
give rise to diffusionally stable bubble sizes 10 This is used for example in bubble
traps, where single bubbles can be captured in an acoustic standing wave field close
to a pressure antinode. Bubble trap experiments have become famous for the singlebubble
experiments on sonoluminescence [30]. In water, usually significant degassing
was used to reach the diffusional equilibrium, but it has been demonstrated that also
in saturated liquid diffusion-stable single bubbles can be achieved [28].
For a spherically and periodically oscillating bubble, the gas diffusion is governed
9 This rule is only approximately valid, and Hi depends on both the solute and the
solvent.
10 The term equilibrium radius introduced above and its symbol R0 denotes the equilibrium
with respect to static pressure conditions, i. e., the resulting bubble radius if suddenly the
acoustic pressure would be turned off and the bubble came to rest (therefore also rest radius).
If we refer to the rest radius of a bubble with zero net gas diffusion, this would be denoted
as diffusional equilibrium radius.