Coalescence characteristics of bulk nanobubbles in water: A molecular dynamics study coupled with theoretical analysis

The coalescence of two nanobubbles (NBs) in water is a process of great importance to many industrial applications. In this work, we study the coalescence of two equal-sized nitrogen NBs in water using molecular dynamics (MD) simulations and continuum-based theoretical analysis. We vary the NB diameter from 30 to 50 nm and study the coalescence characteristics including the expansion speed of the capillary bridge between two coalescing NBs, the dynamic regime of NB coalescence, the diameter of fully merged NBs, and the temperature variation of NBs during the coalescence process. For all cases, we show the MD simulation results can be well understood by the theoretical models developed in this work. Due to the large Laplace pressure in the model NBs, the diameter ratio of fully merged NBs to their daughter NBs is $\sqrt{2}$, which explains the recent experimental result showing that the size of NBs in water is distributed discretely with a uniform increment factor of $\sqrt{2}$ [Ma et al., J. Phys. Chem. B 124, 5067 (2020)]. The expansion of gas inside the coalescing NBs and the heat transfer between the gas NB and surrounding liquid leads to fluctuations of gas temperature during coalescence. From the theoretical analysis, we find the coalescence dynamics of NBs is in the crossover regime where neither viscous stress nor inertial stress in the surrounding liquid dominates even when the viscous stress is more than ten times higher than inertial stress. In the range of Ohnesorge number from 0.33 to 0.82, we show the scaling exponent for the capillary bridge radius vs time at late times of NB coalescence is around 0.75 ± 0.05, which is considerably higher than 0.5 in the viscous-dominated regime.

Figure 1

(a) Schematic of coalescing NB system and (b–d) snapshots from MD simulation model. The snapshots from the MD model show 40-nm ${\mathrm{N}}_2$ NBs coalescing in liquid water at 0, 100, and 200 ps. Green, blue, and white dots represent nitrogen, oxygen, and hydrogen atoms, respectively. The liquid water phase is hidden to clearly display the coalescing ${\mathrm{N}}_2$ NBs and the water liquid-gas interface. In (d), $r_L$ is the principal radius at the bridge minimum on the liquid side.

Figure 2

(a) Snapshot from MD simulation of a liquid water slab coexisting with a nitrogen gas and water vapor mixture. Green, blue, and white dots represent nitrogen, oxygen, and hydrogen atoms, respectively. (b) Density profiles of water (${\rho }_{{\mathrm{H}}_2\mathrm{O}})$ and nitrogen (${\rho }_{{\mathrm{N}}_2})$. (c) Pressure profiles in the normal ($P_N$) and tangential ($P_T$) directions.

Figure 3

The pressure-tensor autocorrelation function (PACF) and the running integral of the PACF of the model liquid water at a temperature of 300 K and density of 981 $\mathrm{kg}/{\mathrm{m}}^3$.

Figure 4

(a) A snapshot of the NEMD simulation used to obtain the liquid-gas interfacial thermal conductance of the model system. Yellow, green, blue, and white dots represent gold, nitrogen, oxygen, and hydrogen atoms, respectively. (b) Temperature profile at steady state from MD data. The dashed line in (b) indicates the position of the liquid-gas interface.

Figure 5

(a) MD data (measuring snapshots) and theoretical prediction [Eq. (4) with $c=0.85]$ of the minimum bridge radius $r_b\left(t\right)$ during the first 200 ps of coalescence of two 40-nm ${\mathrm{N}}_2$ NBs submerged in the model water ($\mathrm{Oh}=0.71)$. The inset shows the same MD data on logarithmic scale with early and late time slope fits. (b) Viscous, inertial, and acceleration terms in Eq. (4) over time $t$. The inset shows the ratio of viscous to inertial terms over time $t$.

Figure 6

(a) MD data (measuring snapshots) of the dimensionless bridge radius $r_b/R$ for two 40-nm diameter coalescing ${\mathrm{N}}_2$ NBs in water. Measurement uncertainties are smaller than the circle symbols showing the MD data. The horizontal dashed line at $r_b/R$ $=\sqrt{2}$ is the theoretical radius of the fully merged NBs. The four insets are snapshots of the model system at 0.8, 2.0, 4.0, and 6.5 ns. (b) Temperature variation of $N_2$ inside the 40-nm diameter coalescing NBs and ${\mathrm{H}}_2\mathrm{O}$ in the surrounding liquid from MD simulations, and the temperature prediction [Eq. (10)] of the gas inside the coalescing NBs. The two insets are snapshots of the model system at 0.5 and 1.3 ns.

Figure 7

(a–d) MD data (measuring snapshots) and theoretical prediction [Eq. (4)] of the dimensionless bridge radius $r_b/R$ over time $t$ for 30-, 35-, 40-, and 50-nm diameter NBs. The horizontal dashed line at $r_b/R=0.45$ marks the upper limit of late coalescence times. The radii, $\mathrm{Oh}$ numbers, and $c$ values are shown for all simulation cases. (e–h) MD data and slope fits at early and late coalescence times for the bridge radius $r_b\left(t\right)$ on logarithmic scales.

Figure 8

Ratios of viscous to inertial terms for ${\mathrm{N}}_2\text{−}{\mathrm{H}}_2\mathrm{O}$ and LJ fluid systems [13]. The error bars show the minimum and maximum ratios calculated during late coalescence times. The number below each symbol shows the NB diameter.

Figure 9

Theoretical scaling exponents during late coalescence times for 50-nm NBs as the viscosity ${\eta }_L$ of water is varied from $0.3$ to 10 mPa s in Eq. (4). The dashed lines represent the range of $\mathrm{Oh}$ ($0.330.82)$ studied.