Small bubble can exist everywhere in the nature and industrial liquid systems, which is a spherical void under or in the liquid. Nanobubbles include bulk nanobubbles and surface nanobubbles (SNBs) [1]. In all kinds of environments, SNBs survive with smaller contact angles of the gaseous phase and higher stability than the common-sized ones. Figure 1 shows a SNB of spherical cap on the solid substrate. Its contact angle is usually between 10 and 30° and lateral size is several or more than ten times of the height.

Nanobubbles are usually generated by electrochemical reactions [2] or solvent exchange process [3]. They have been widely used in a number of technical fields and have recently been reported to have wide prospect in cancer treatment and medical imaging. The nanobubbles have obvious application effect in bio-medicine [4] and water oxygenation [1], and they are effective in creating slip boundary conditions in fluid transport [5, 6] and can increase the efficiency in mineral particle flotation [7]. However, a small bubble on the surface would also influence the electrochemical reaction processes [8], and decrease the efficiency of photocatalytic reactions in solar conversion [9] and cause decompression illness [10]. It is essential to understand the principle of nanobubbles for their utilization and to avoid their harmfulness.

The concept of SNBs and the relevant studies date back about 40 and 20 years, respectively. There has been a lot of investigations focus on the generation, the properties, and the applications of SNBs. The SNBs were firstly realized in the force measurement experiments between two surfaces [11]. When two hydrophobic surfaces are close to each other, and the relation between the force and the distance presents a ladder shape, which is different from the long-range gravity and electrostatic interaction. During a long time, the study progress of SNBs was limited by experimental conditions, and high-tech microscopes or other technical methods were needed to observe bubbles of nano scale. In 2000, Lou [12] and Ishida [13] got the first images of SNBs on different substrates using the atomic force microscope (AFM). These earliest experimental results verified the assumption of SNBs proposed by Parker [14] and laid a solid foundation for the later study of SNBs.

Researchers are interested in the extraordinary properties especially for the small gas-side contact angle θ [15–17] and the long-term stability of SNB [18–20]. On one hand, its gas-side contact angle is much smaller than the angle determined by Young’s equation [21], which is the criterion for the equilibrium of the three-phase contact point [22, 23]. On the other hand, the long surviving time is far more than the predicted by Epstein and Plesset’s theory [24]. For a nanobubble with curvature radius of 100 nm, the pressure inside (mainly Laplace pressure) is about 14 times of the standard atmosphere. It means that the bubble will shrink and disappear in ∼μs as soon as its appearance. However, experiments show that SNBs can exist for hours or even days [18–20]. It is several orders of magnitude greater than the above-mentioned theory.

In terms of theoretical research, a considerable amount of literature has been published discussing about SNBs. Most of these studies focus on their long-term life and other features and try to explain the amazing stability. Ducker argued that some contaminants are adsorbed on the interface of SNBs, which may lead to the decrease of surface tension and make the SNBs stable [16]. Das’ simulation showed that the surfactant will cut down the surface tension by half, but this is not enough for the stability [25]. However, all the theory was denied by the experiments which show that there is gas exchange across the interface [26]. Brenner firstly put forward the dynamic equilibrium model including the gas absorption on hydrophobic surface that the gas gets into the bubble near the substrate is the supplement for that gets out across the interface [27]. Yasui took the potential of Van Der Waals to improve the dynamic model, and showed that the density inside the bubble is not uniform [28]. Inspired by the coffee ring effect, several models involving contact line pinning were proposed by Liu [29, 30], Weijs [31] and Lohse [32]. Using the Popov’s solution for the evaporation of liquid droplet [33], Lohse and Zhang demonstrated the dynamic evolution to equilibrium of the SNBs with gas oversaturation in water [32]. It is a pity that this model cannot explain the stable bubbles in the under-saturated environment, which was found in some experiments [34, 35]. Tan combined the hydrophobic adsorption with contact line pinning-oversaturation theory to explain that bubbles can survive in under-saturated water, and also presented that how hydrophobicity and oversaturation work together to stabilize SNBs [36]. Furthermore, Tan got the timescale ∼1,000 s of bubble evolution with the dynamic gas concentration in water with height of 1 mm [37]. Recently, Petsev argued that the adsorption of gas molecules at the substrate modifies the energy of solid-gas interface, and thus reduces the gas-side contact angles and the pressure inside [38]. However, none of those models can account for all the outstanding properties of SNBs, especially their stability on the hydrophilic surface, such as mica or glass.

With the contact line pinning, as a surface nanobubble dissolves, its volume decreases, the curvature radius increases and the Laplace pressure reduces. The surface tension determines the internal pressure of the nanobubble and thus influences the dynamic evolution of nanobubble. Ducker and Das have shown that the interface containing some impurity can reduce the surface tension [16, 25]. In the calculations of the stabilization process, surface tension is a crucial parameter, determining the pressure inside and concentration around the bubble. In this work, we investigate the effects of variable surface tension on the evolution and stability of SNBs, which the surface tension follows the Tolman or state dependence. With the smaller surface tension in the gas-liquid interface, the results show that the SNBs can exist with larger contact angle in the same environment and even survive on a highly hydrophilic surface.

The solubility of gas in liquid is a constant at certain pressure and temperature. Usually, the solubility increases with increasing pressure, but there is an inverse relation with temperature. The gas saturation ratio s and gas oversaturation ζ ₀ are defined as fallow respectively s = c ∞ c s , (1) ζ 0 = c ∞ c s − 1 = s − 1 , (2) where c _s is the gas solubility at standard atmospheric pressure, and normal temperature of T = 293K; c _∞ is the far-field dissolved gas concentration (environment concentration) which is a given condition and determines ζ ₀ . If the concentration c(z) along the z axis (height direction) is the same as c _∞, the gas oversaturation ζ(z) is also the same. Brenner [27] firstly used the hydrophobic attraction in the dynamic equilibrium mechanism of SNBs base on the results of some molecular dynamics (MD) simulations [39–41] and experiments [42, 43]. Those study demonstrate that there is a gas enrichment layer near solid hydrophobic surfaces when gas dissolves in the liquid, leading to a spatially distributed c(z). When the hydrophobic adsorption is taken into account, the gas concentration, as the solution of the diffusion equation, exists with the form as follow c ( z ) = c ∞ ⁡ exp ( − ϕ 0 e − z / λ k B T ) , (3) In Equation 3, the attractive potential ϕ(z) = ϕ ₀ e ^−z/λ is a function of the distance z, and ϕ ₀ is a representation of the surface’s wettability (for hydrophilic, ϕ ₀ > 0; hydrophobic, ϕ ₀ < 0; neutral, ϕ ₀ = 0), λ = 1 nm is the characteristic distance of the interaction, k _B is the Boltzmann constant and T is the temperature. And the gas oversaturation can be expressed as ζ ( z ) = s × exp ( − ϕ 0 e − z / λ k B T ) − 1. (4)

In the SNBs’ evolution theory, the gas diffuses from the interface out to the infinity following the diffusion equation (Eq. 5) [31]. The well-known Young-Laplace’s law (Eq. 6) determines the pressure inside the bubble and Henry’s law (Eq. 7) defines the gas concentration around the gas-liquid interface. ∂ c ∂ t = D × ∇ 2 c , (5) P b = P 0 + 2 γ R , (6) c ( R , t ) = P b P 0 c s , (7) where D is the diffusion constant, P ₀ is the atmospheric pressure, γ is the surface tension, R is the curvature radius. R can be calculated by the bubble bottom radius L and the real-time contact angle using R = L / sin(θ).

With the concentration difference between in and out of the interface, the SNBs will dissolve or grow, and the concentration gradient determines the diffusion rate. The evolution problem of a dissolving pinned SNB under liquid is analogous to that of an evaporating pinned droplet. For this problem, Popov has derived the exact solution [33], which is consistent with the experiments. Lohse and Zhang extended this theory to calculate the evolution of pinned SNBs [32]. Tan embedded spatially varying oversaturation ζ (z) (Eq. 4) into the pinning-oversaturation stabilization theory about SNBs’ evolution and showed that the stability of SNBs does not necessarily require strict gas oversaturation [36]. With the above definitions, the hydrophobic attraction model (HA model) is used to calculate the evolution of the contact angle as follows. The mass changing rate of the gas in a spherical nanobubble with a fixed footprint radius L and any contact angle θ obeys d m d t = − π L D c s h f ( θ ) ∫ 0 h ( 2 γ L P 0 sin ⁡ θ − ζ ( z ) ) d z , (8) where f ( θ ) = sin ⁡ θ 1 + cos ⁡ θ + 4 ∫ 0 ∞ 1 + cosh ⁡ 2 ⁡ θ ξ sinh ⁡ 2 ⁡ π ξ tanh [ ( π − θ ) ξ ] d ξ , (9) is a geometric term of Popov’s theory [33]. The total mass of bubble is calculated by the gas density ρ _g and the volume of spherical cap m = π L 3 ρ g ( cos 3 ⁡ θ − 3 ⁡ cos ⁡ θ + 2 ) 3 ⁡ sin 3 ⁡ θ . (10)

Then the changing rate of contact angle at any real-time θ is obtained by Eq. 8 and Eq. 10 d θ d t = − D c s ρ g L 2 h ( 1 + cos ⁡ θ ) 2 f ( θ ) ∫ 0 h ( 2 γ L P 0 sin ⁡ θ − ζ ( z ) ) d z . (11)

Given an initial contact angle θ ₀ and L, the dynamic evolution of SNB can be calculated and plotted by Eq. 11. Note that the first part of the above expression before the integral part is negative, whether the contact angle increases or decreases entirely depends on the integral term, which is the difference between the gas getting in at the bottom and those getting out at the upside of the bubble. The integral term turning into zero means that the SNBs reach equilibrium. In this paper, the parameters are set as follows, D = 2 × 10^–9 m²/s, c _s = 0.017 kg/m³, ρ _g = 1.165 kg/m³, P ₀ = 101,325 Pa, which are the same as the HA model.

Equations 8–11 show that the dynamic evolution of SNBs is entirely determined by the geometric parameters (L and θ), gas oversaturation ζ(z) along the z axis and some constants including surface tension γ. Surface tension is the tendency of liquid surface at rest to shrink into the minimum surface area possible. At liquid–air interface, surface tension results from the greater attraction of liquid molecules to each other (due to cohesion) than to the molecules in the air (due to adhesion), and for ambient water, it is 72 mN/m. Actually, surface tension is equal to surface energy per unit surface area and adhesion is the inevitable result of minimizing the surface free energy.

Both the recent experiments and molecular dynamics (MD) simulations demonstrate that the density of SNBs is much higher than that of gas at standard atmosphere [44, 45]. In addition, Luo puts forward that the interface between the gas and the liquid is a special layer, which is called coupled layer of gas and liquid, and its density is between the pure water and the gas [46]. With the curving interface between gas and liquid, the pressure inside the bubble is larger than outside P ₀. The following points indicate that the surface tension decreases with the increasing internal pressure and density. The larger pressure, the little difference is in density and molecular attraction between the two phases, and the adhesion is getting larger while the cohesion remains unchanged; the larger pressure, the more gas is trapped on the surface of the liquid and the surface energy decreases; the larger pressure, the more gas diffuses into the water and the surface energy decreases. Whether the reduction of the difference between cohesion and adhesion, or of the surface energy would lead to a smaller surface tension.

In the above analysis, the surface tension of the bubble’s interface will decrease in some way and has effects on the dynamic model. In this paper, we introduce two variable forms of surface tension into the HA model, and conduct a study about the effects of surface tension on the stability of SNBs. One of the two forms is Tolman-dependent surface tension, the other is state-dependent, which are already proved and used in other study.

Many researches about SNBs focus on their long life and small contact angles. The HA model reveals that only contact line pinning is required and the attractive hydrophobic potential can make a supplementary for the gas oversaturation, which is not a necessary condition for the stability of SNBs. In this work, we take the Tolman-dependent and state-dependent surface tension into the HA model. The surface tension changes with the curvature radius R or the saturation ratio s. It slows down the changing rate of evolution process, and enlarges the contact angle at the equilibrium. The SNBs are more stable on the hydrophobic or neutral surface with larger contact angle and even can survive on the highly hydrophilic surface under the same oversaturation degree.

The surface tension of Tolman-dependence related to the curvature radius R has little effect on the evolution of SNBs in large scale (such as L = 50 or 100 nm), while it has a great influence in small SNBs (such as L = 20 nm), in which the initial curvature radius R is close to the Tolman length. When the SNBs reach equilibrium, the surface tension maintains a constant and the SNBs hold larger contact angle. It is similar in the state-dependence, surface tension is beneficial to the stability of SNBs. With the saturation ratio s = 3, the bubbles on the highly hydrophilic surface (ϕ ₀ = 1∼5 k _B T) can also be stable, and those on the hydrophobic surface grow up. In conclusion, the two variable surface tensions, which are verified providing SNBs’ mechanical equilibrium and thermodynamic stability, can make the SNBs holding larger contact angle and more stable. The present study enriches our understanding of the mechanism for the stability of SNBs. In the future, much more research is urgently needed on the phase interface’s surface tension, which is a new perspective on the SNB and closely related to its thermodynamic stability. At the same time, more experiments and simulation studies should be carried out to provide some guidance for theoretical analysis of the stability mechanism.