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The interfacial nanobubbles (INBs) have been confirmed to exist, and have significant potential for applications in fields such as mineral flotation, aquaculture, and wastewater treatment. However, the microscopic nucleation process of INBs is still poorly understood. This study investigates the nucleation process and growth dynamics of INBs on smooth and rough surfaces under different levels of gas supersaturation. Molecular dynamics (MD) simulations using GROMACS software package are conducted to observe the microscopic nucleation process and the temporal evolution of the geometric characteristics of the INBs. Additionally, a growth dynamics model for INBs is derived based on the Epstein-Plesset gas diffusion theory, and the predictions from the model are compared with the MD simulation data. The results indicate that on smooth homogeneous surfaces, the curvature radius and width of INBs increase progressively with time after nucleation. This growth process is well captured by the theoretical model, indicating that the gas diffusion theory provides an accurate description of INB growth dynamics. In addition, the contact angle (measured on the gas side) during INB growth is not constant but increases initially before stabilizing. This phenomenon is caused by reducing solid-gas interfacial tension due to higher Laplace pressure, thus leading the contact angle to increase as the INB radius grows. Furthermore, on smooth homogeneous surfaces, INBs are observed to nucleate at 81 ns, 17 ns, 6 ns, and 1.3 ns under gas supersaturation levels of 100, 120, 150, and 200, respectively. This demonstrates that higher gas supersaturation significantly shortens the nucleation time. Additionally, as gas supersaturation increases, the growth rate of INBs after nucleation will also accelerate. However, at a gas supersaturation level of 50, no nucleation occurrs during the simulation period of 200 ns. Theoretical analysis reveals that the INBs can only nucleate and grow when the radius of gas aggregates exceeds the critical nucleation radius ($ {R}_{{\mathrm{critical}}} = \dfrac{\sigma }{\zeta {P}_{0}} $, where $ \sigma $ is the liquid-gas interfacial tension, $ \zeta $ is the gas supersaturation level, and $ {P}_{0} $ is the ambient pressure). As gas supersaturation decreases, $ {R}_{{\mathrm{c}}{\mathrm{r}}{\mathrm{i}}{\mathrm{t}}{\mathrm{i}}{\mathrm{c}}{\mathrm{a}}{\mathrm{l}}} $ increases, thus significantly increasing the difficulty of nucleation. On rough surfaces, pits with widths of 1 nm, 2 nm, 4 nm, and 10 nm are introduced. At a gas supersaturation of 50,where no INB nucleation occurrs on the smooth surfaces, gas nuclei rapidly form within the pits. However, only gas nuclei in pits with widths larger than 2 nm can grow into INBs. This is because in the growth process the pinning effect at the pit edges causes the curvature radius of the gas nucleus to initially decrease and then increase. Only when the minimum curvature radius exceeds the critical nucleation radius, can gas nuclei develop into INBs. The findings of this study provide more in-depth insights into the nucleation mechanism of INBs, and practical guidance for controlling their generation, and they also deliver theoretical support for relevant applications such as mineral flotation and other industrial processes. 
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Keywords:
- nanobubble nucleation /
- smooth surface /
- rough surface /
- gas supersaturation
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图 1 纳米气泡成核过程模拟模型 (a)光滑表面模型; (b)粗糙表面模型; (c)圆柱形纳米气泡示意图. 图中银色粒子代表固体原子, 绿色代表气体原子, 红色和白色分别代表水分子中的氧原子和氢原子, 1 atm = 1.013×105 Pa
Figure 1. Simulation model of the nanobubble nucleation process: (a) Smooth surface model; (b) rough surface model; (c) schematic diagram of cylindrical nanobubbles, the silver beads represent solid atoms, the green beads represent gas atoms, and the red and white beads represent oxygen and hydrogen atoms of the water molecule, respectively. 1 atm = 1.013×105 Pa.
图 2 气体溶解度与扩散系数模拟 (a)气体溶解度模拟模型; (b)水中气体浓度与气相气体浓度间线性关系拟合; (c)气体在水中均方根位移与时间t间关系
Figure 2. Simulation of gas solubility and diffusion coefficient: (a) Simulation model of gas solubility; (b) linear fitting of the relationship between gas concentration in water and gas concentration in the gas phase; (c) relationship between the mean square displacement (MSD) of gas in water and time t.
图 3 光滑均质表面上气体过饱和度为100时纳米气泡成核过程, 银色粒子代表固体原子, 绿色代表气体原子, 红色和白色分别代表水分子中的氧原子和氢原子, 分别展示了0, 81, 83, 85 , 100, 130 ns时的快照, 0 ns时气体分子均匀分布在体相液体中, 81 ns时固-液界面出现较小的气体聚集体, 83 ns时气体聚集体明显长大, 85 ns时气体聚集体呈现出纳米气泡形状, 100 ns和130 ns时气泡正在快速生长
Figure 3. Nucleation process of nanobubbles on a smooth homogeneous surface under a gas supersaturation of 100, in the figure, the silver beads represent solid atoms, the green beads represent gas atoms, and the red and white beads represent oxygen and hydrogen atoms of the water molecule, respectively, snapshots are shown at0, 81, 83, 85 , 100, and 130 ns. At 0 ns, gas molecules are uniformly distributed in the bulk liquid. By 81 ns, a small gas aggregate appears at the solid-liquid interface, at 83 ns, the gas aggregate has grown significantly, and by 85 ns, it has taken on a nanobubble shape, at 100 ns and 130 ns, the nanobubble is rapidly expanding.
图 4 光滑均质表面上气体过饱和度为100纳米气泡几何特征随时间的演化过程 (a)接触角随时间的变化; (b)曲率半径随时间的变化; (c)宽度随时间的变化. 其中蓝色曲线表示模拟数据, 红色曲线为拟合曲线, 插图为残差直方图
Figure 4. Evolution of the geometric characteristics of a nanobubble over time on a smooth, homogeneous surface with a gas supersaturation of 100: (a) Changes in contact angle over time; (b) changes in curvature radius over time; (c) changes in width over time, where the blue curve represents the simulation data, while the red curve represents the fitted curve, the inset shows the residual histogram.
图 5 气体过饱和度对光滑表面上纳米气泡成核动力学的影响 (a)纳米气泡成核所需时间与气体过饱和度的关系; (b)纳米气泡曲率半径随时间的演化; (c)纳米气泡宽度随时间的演化; (d)纳米气泡接触角(气体一侧)随时间的演化
Figure 5. Effect of gas supersaturation on the nucleation kinetics of nanobubbles on a smooth surface: (a) Relationship between the nucleation time of nanobubbles and gas supersaturation; (b) evolution of nanobubble curvature radius over time; (c) evolution of nanobubble width over time; (d) evolution of nanobubble contact angle (on the gas side) over time.
图 6 理论模型预测 (a)温差法和醇水替换法条件下纳米气泡曲率半径随时间的演化; (b)临界成核半径随气体过饱和的变化
Figure 6. The theoretical model predicts: (a) The evolution of the curvature radius of nanobubbles over time under the conditions of the temperature difference method and alcohol-water exchange method; (b) the variation of the critical nucleation radius with gas supersaturation.
图 7 粗糙表面气体过饱和度为50时纳米气泡成核微观过程, 在固体表面中部设置了一个凹坑, 其宽度4 nm, 深度1.34 nm, 分别展示了0, 7.48, 7.6, 50, 80, 130 ns时的快照, 0 ns时气体分子均匀分布在体相液体中, 7.48 ns时凹坑下边缘出现气体聚集体, 并在随后的0.12 ns内气体聚集体将凹坑完全填充, 从7.6 ns到50 ns凹坑中的纳米气泡的宽度被钉扎, 而高度显著增加, 到80 ns时纳米气泡已摆脱钉扎, 其高度和宽度均增加, 到130 ns时纳米气泡继续以无钉扎模式生长
Figure 7. Microscopic process of nanobubble nucleation on a rough surface at a gas supersaturation level of 50, a pit is created in the center of the solid surface, with a width of 4 nm and a depth of 1.34 nm, snapshots are shown at 0, 7.48, 7.6, 50, 80, and 130 ns. At 0 ns, gas molecules are uniformly distributed in the bulk liquid. At 7.48 ns, two gas clusters appear along the lower edges of the pit, and within the next 0.12 ns, the gas cluster fills the pit completely. From 7.6 ns to 50 ns, the width of the nanobubble within the pit is pinned, while its height increases significantly. By 80 ns, the nanobubble detaches from the pinning, with both its height and width increasing. At 130 ns, the nanobubble continues to grow in an unpinned mode.
图 8 凹坑宽度4 nm粗糙表面上气体过饱和度为50时纳米气泡几何特征随时间的演化 (a)宽度随时间的变化; (b)接触角随时间的变化; (c)曲率半径随时间的变化
Figure 8. Evolution of nanobubble geometric characteristics over time on a rough surface with a pit width of 4 nm at a gas supersaturation level of 50: (a) Variation of width over time; (b) variation of contact angle over time; (c) variation of curvature radius over time.
图 9 气体过饱和度为50时凹坑尺寸对纳米气泡成核的影响 (a)凹坑宽度1 nm, 0 ns时气体均匀分布在液相中, 0.14 ns时凹坑已被气相完全填充, 0.8 ns时气体分子逐渐进入凹坑气核中, 200 ns时凹坑气核中聚集了更多气体分子, 但气核未生长为纳米气泡; (b)凹坑宽度2 nm, 在5.12 ns时凹坑中形成气体聚集体, 在5.16 ns时凹坑被气相完全填充, 10 ns时凹坑气核中聚集了更多气体分子, 50 ns时凹坑气核高度增大, 形成凸起, 100 ns时凹坑气核挣脱凹坑, 生长为纳米气泡, 在150 ns和200 ns时纳米气泡以无钉扎模式快速生长; (c)凹坑宽度10 nm, 在50 ns时凹坑底部形成2个气体聚集体, 100 ns, 110 ns和120 ns时1个气体聚集体显著增大, 另一个气体聚集体在奥斯瓦尔德效应下逐渐缩小直至消失, 140 ns时纳米气泡完全占据凹坑, 180 ns时纳米气泡宽度被凹坑边缘钉扎, 高度逐渐增大, 240 ns时纳米气泡挣脱凹坑束缚, 以无钉扎模式增长
Figure 9. The effect of pit size on nanobubble nucleation at a gas supersaturation of 50: (a) Pit width of 1 nm, at 0 ns, the gas is uniformly distributed in the liquid phase, at 0.14 ns, the pit is fully filled with the gas phase, at 0.8 ns, gas molecules gradually enter the gas nucleus in the pit. At 200 ns, more gas molecules have accumulated in the pit's gas nucleus, but the gas nucleus has not yet grown into a nanobubble. (b) Pit width of 2 nm. At 5.12 ns, a gas aggregate forms in the pit, at 5.16 ns, the pit is fully filled with the gas phase, at 10 ns, more gas molecules accumulate in the gas nucleus in the pit, at 50 ns, the height of the gas nucleus increases, forming a protrusion. At 100 ns, the gas nucleus escapes from the pit and grows into a nanobubble, at 150 ns and 200 ns, the nanobubble grows rapidly in a non-pinned mode. (c) Pit width of 10 nm. At 50 ns, two gas aggregates form at the bottom of the pit, at 100 ns, 110 ns, and 120 ns, one of the gas aggregates significantly increases in size, while the other gradually shrinks and disappears due to the Ostwald effect, at 140 ns, the nanobubble fully occupies the pit, at 180 ns, the width of the nanobubble is pinned by the pit edge, and the height gradually increases, at 240 ns, the nanobubble escapes the pit's confinement and grows in a non-pinned mode.
表 1 分子动力学模拟中各原子类型参数
Table 1. LJ parameters of the different atoms used in molecular dynamics simulation.
i-j σ/nm ε/(kJ·mol–1) 电荷/e N-N 0.3698 0.7899 0 Ow-Ow 0.3166 0.6502 –0.8476 Hw-Hw 0 0 0.4238 Ow-N 0.3285 0.8050 0 S-N 0.3549 0.4661 0 S-Ow 0.3367 0.4247 0 ST-N 0.3549 0.4661 0 ST-Ow 0.3367 0.5946 0 
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