Extreme conditions in a dissolving air nanobubble

Numerical simulations of the dissolution of an air nanobubble in water have been performed taking into account the effect of bubble dynamics (inertia of the surrounding liquid). The presence of stable bulk nanobubbles is not assumed in the present study because the bubble radius inevitably passes the nanoscale in the complete dissolution of a bubble. The bubble surface is assumed to be clean because attachment of hydrophobic materials on the bubble surface could considerably change the gas diffusion rate. The speed of the bubble collapse (the bubble wall speed) increases to about 90 m/s or less. The shape of a bubble is kept nearly spherical because the amplitude of the nonspherical component of the bubble shape is negligible compared to the instantaneous bubble radius. In other words, a bubble never disintegrates into daughter bubbles during the dissolution. At the final moment of the dissolution, the temperature inside a bubble increases to about 3000 K due to the quasiadiabatic compression. The bubble temperature is higher than 1000 K only for the final 19 ps. However, the Knudsen number is more than 0.2 for this moment, and the error associated with the continuum model should be considerable. In the final 2.3 ns, only nitrogen molecules are present inside a bubble as the solubility of nitrogen is the lowest among the gas species. The radical formation inside a bubble is negligible because the probability of nitrogen dissociation is only on the order of ${10}^{-15}$. The pressure inside a bubble, as well as the liquid pressure at the bubble wall, increases to about 5 GPa at the final moment of dissolution. The pressure is higher than 1 GPa for the final 0.7 ns inside a bubble and for the final 0.6 ns in the liquid at the bubble wall. The liquid temperature at the bubble wall increases to about 360 K from 293 K at the final stage of the complete dissolution.

Figure 1

The result of the numerical simulation for the dissolution of an air nanobubble into water saturated with air as a function of time for about 75.4 μs. The initial bubble radius is 100 nm. (a) The bubble radius $\left(R\right)$. (b) The number $\left(n_i\right)$ of molecules of each gas or vapor species inside a bubble with logarithmic vertical axis.

Figure 2

The result of the numerical simulation as a function of time for the last 255 ps of the complete dissolution of an air nanobubble into water. The condition is the same as that in Fig. 1 $(t=0$ in this figure corresponds to $t=75.361\mu \mathrm{s}$ in Fig. 1). (a) The bubble radius $\left(R\right)$ (dotted line) and the temperature $\left(T\right)$ inside a bubble (solid line). (b) Temporally integrated $pV$ work, energy loss due to thermal conduction and that by gas diffusion. (c) The Knudsen number $\left(K_n\right)$ and the de Broglie wavelength $\left({\lambda }_{\mathrm{dB}}\right)$ relative to the bubble radius $\left(R\right)$. (d) The pressure $\left(p_g\right)$ inside a bubble and the liquid pressure $\left(p_B\right)$ at the bubble wall. (e) The amplitude of the nonspherical component of degree $n=2$ relative to the instantaneous mean bubble radius. The initial $a_2$ is assumed as 1 nm with $d{a}_2/dt=0$. (f) The liquid temperature $\left(T_{L,i}\right)$ at the bubble wall. (g) The number $(n_{{\mathrm{N}}_2})$ of ${\mathrm{N}}_2$ molecules inside a bubble. (There is no other gas and vapor species inside a bubble at the final 2.3 ns.) (h) The rate of N atom production $\left(d{N}_{\mathrm{N}}/dt\right)$ and the total number $\left(N_{\mathrm{N}}\right)$ of N atoms production. (i) The mean free time $\left(t_{mf}\right)$ of gas molecules inside a bubble. (j) The density $\left({\rho }_g\right)$ inside a bubble. (k) The thermal conductivity (κ) of gas inside a bubble. (l) The diffusion coefficient $\left(D_i\right)$ of gas in liquid water. (m) The liquid density $\left({\rho }_{L,i}\right)$ at the bubble wall and that $\left({\rho }_{L,\infty }\right)$ far from a bubble. (n) The sound velocity $\left(c\right)$ in the liquid water at the bubble wall and that $\left(c_0\right)$ far from a bubble. (o) The surface tension at the curved bubble surface (σ) and that at a flat surface $\left({\sigma }_0\right)$ as a function of the liquid temperature $\left(T_{L,i}\right)$. (p) The viscosity $\left(\mu \right)$ of the liquid water at the bubble wall and that $\left({\mu }_0\right)$ far from a bubble.

Figure 3

The molar energy assumed in the present simulation for each gas species as a function of temperature.