2.1. Theoretical framework and derivation of the criterion

Consider an underwater system with dissolvable gas bubbles, bulk water and a solid surface. Ignoring the influence of water vapour and temperature change, the evolution of the underwater liquid–gas interface consists of two major mechanisms: the dynamic process related to the mechanical force balance on the liquid–gas interface and the contact line, and the chemical balance related to the gas diffusion across the liquid–gas interface. This evolution can be analysed by minimising the total free energy ( $F_{t}$ ) of the underwater system, which is written as (Xiang et al. Reference Xiang, Huang, Lv, Xue, Su and Duan2017)

(2.1) \begin{equation} F_{{t}} = F_{{b}} + F_{{s}} + F_{{d}} + F_{{0}}. \end{equation}

Here, $F_{{b}}= (p_{{L}} - p_{{G}})V_{{G}}$ is the bulk energy for the gas bubble replacing the liquid, where $V_{{G}}$ is the volume of the air layer. Also, $F_{{s}} = \gamma _{\textit{LG}}$ ( $A_{\textit{LG}} + A_{\textit{SG}} \cos \theta _{{Y}}$ ) represents the energy of the liquid–gas interface, where $\theta _{{Y}}$ is Young’s contact angle, and $A_{\textit{LG}}$ and $A_{\textit{SG}}$ are the liquid–gas and solid–gas interface areas, respectively ( $A_{\textit{SG}}=0$ for suspended bubbles); and $F_{{d}}= n_{{G}}RT\ln [p_{{G}}/(sp_{{L}})]$ is the excess chemical potential of the free gas in the bubble relative to that of the gas dissolved in the surrounding bulk water, where $n_{G}$ is the gas mole number, $R$ is the universal gas constant, and $T$ is the ambient temperature. The last term, $F_{0}$ , is a constant that represents the free energy of the fully wetted state.

Consider an arbitrary morphology perturbation on the liquid–gas interface together with a perturbation on the gas mole number ( $\delta n_{{G}}$ ). Under such a perturbation, the changes of $V_{{G}}$ , $A_{\textit{LG}}$ and $A_{\textit{SG}}$ can be found by following geometric relations $\delta V_{{G}}=\int _{\varSigma _{\textit{LG}}}\delta x_{{N}}\,{\rm d}S,\ \delta A_{\textit{LG}}=\int _{\varSigma _{\textit{LG}}}\kappa\, \delta x_{{N}}\,{\rm d}S-\oint _{\varGamma }\delta x_{{tl}}\cos \theta\, {\rm d}l$ and $\delta A_{\textit{SG}}=\oint _{\varGamma }\delta x_{{tl}}\,{\rm d}l$ . Here, $\kappa$ is the curvature (twice the mean curvature) of the liquid–gas interface, with a positive value when it protrudes towards the liquid, $\theta$ is the contact angle, $\delta {{x}_{{N}}}$ represents the normal displacement of the liquid–gas interface $\varSigma _{\textit{LG}}$ , and $\delta x_{{tl}}$ is the displacement of the contact line $\varGamma =\partial \varSigma _{\textit{LG}}$ along the liquid–solid interface. Therefore, the first-order variation of the total free energy can be expressed as

(2.2) \begin{equation} \begin{aligned} \delta F_{{t}}={}&\int _{\varSigma _{\textit{LG}}}(p_{{L}}-p_{{G}}+\gamma _{\textit{LG}}\kappa )\,\delta x_{{N}}\,{\rm d}S\\ &{}+\gamma _{\textit{LG}}\oint _\varGamma \delta x_{{tl}}\,(\cos \theta _Y-\cos \theta )\,{\rm d}l+RT\ln \frac {p_{{G}}}{sp_{{L}}}\delta n_{{G}}. \end{aligned} \end{equation}

By setting (2.2) to zero for arbitrary perturbation, three equations can be obtained naturally: $p_{{G}}-p_{{L}}=\gamma _{\textit{LG}}\kappa ,\ \varTheta =\varTheta _{Y}$ and $p_{{G}}=sp_{{L}}$ (Xiang et al. Reference Xiang, Huang, Lv, Xue, Su and Duan2017). These equations describe the pressure balance across the liquid–gas interface (Laplace equation), the force balance of the contact line, and the diffusion (or partial pressure) balance between the gas bubble and dissolved gas, respectively. The diffusion equilibrium condition $p_{{G}}=sp_{{L}}$ can be also be explained by Henry’s law: when the gas concentration at the bubble surface equals the far-field gas concentration, there is no net diffusion across the interface (Lohse & Zhang Reference Lohse and Zhang2015a ; Zhang & Lohse Reference Zhang and Lohse2023). Furthermore, when the mechanical equilibrium and the diffusion balance are achieved simultaneously, the interface morphology should satisfy the relation $\gamma _{\textit{LG}}\kappa = (s-1 )p_{{L}}$ . This equilibrium condition can also be verified from the perspective of gas and mass transport dynamics. The gas and mass transport process follows the diffusion equation (Lv et al. Reference Lv, Xue, Shi, Lin and Duan2014; Lohse & Zhang Reference Lohse and Zhang2015b ), i.e. ${\rm d}[({{\gamma }_{\textit{LG}}}\kappa +{{p}_{{L}}}){{V}_{G}}]/{\rm d}t=-{{A}_{\textit{LG}}}{{D}_{G}}[{{\gamma }_{\textit{LG}}}\kappa +{{p}_{{L}}}-s{{p}_{{L}}}]/(l{{K}_{G}})$ , which governs the rate of gas exchange across the liquid–gas interface. Here, $l$ is the diffusion length, and ${D}_{G}$ and ${K}_{G}$ are the diffusive coefficient and Henry’s constant. When the system reaches equilibrium ${{\gamma }_{\textit{LG}}}\kappa = ( s-1 ){{p}_{{L}}}$ , the diffusion equation simplifies to ${\rm d}[({{\gamma }_{\textit{LG}}}\kappa +{{p}_{{L}}}){{V}_{G}}]/{\rm d}t=0$ , indicating that the net mass transfer between the bubble and the surrounding liquid ceases. This confirms that the material exchange has reached a balanced state, fully consistent with the derived equilibrium condition.

While setting the first-order variance of total free energy to zero ( $\delta F_{{t}}=0$ ) leads to the equilibrium state, it can be a minimum, a maximum or a saddle point of the free energy. A stable equilibrium state requires the second-order variance of the free energy to be positive for any perturbation, i.e. $\delta ^{2} F_{{t}}\gt 0$ . A general stability analysis for bubble morphology perturbation together with the perturbation $\delta {n_{G}}$ on an arbitrary surface is mathematically hard to discuss. However, since the gas diffusion is usually slow compared to liquid–gas interface dynamics in practice, we can decouple the evolution of the underwater gas bubble into two processes, i.e. a quick dynamic process with a sudden geometrical response of the gas bubble under hydrostatic pressures, followed by a slow chemical process with a gradual morphology evolution governed by the gas diffusion across the liquid–gas interface. In such a way, the stability analysis is also separated into two parts: the stability of the mechanical equilibrium and that of the chemical equilibrium.

First, we consider the mechanical process with no gas diffusion across the liquid–gas interface. For simplicity, we assume that the meniscus curvature $\kappa$ is uniformly distributed on the liquid–gas interface (but can vary with time), and the contact angle $\theta$ always equals Young’s contact angle ${\theta }_{{Y}}$ . Then for a given gas mole number $n_{{G}}$ , (2.2) can be simplified as $\delta {{F}_{{t}}}= ( {{p}_{{L}}}-{{p}_{{G}}}+{{\gamma }_{\textit{LG}}}\kappa )\,{\rm d}{{V}_{{G}}}.$ From this equation, we can easily find the mechanical stability criterion by setting the second-order derivative of total free energy regarding gas bubble volume to positive:

(2.3) \begin{equation} \left (\frac {\partial ^2F_{{t}}}{\partial V_{{G}}^2}\right )_{\!\! n_G}=-\frac {{\rm d}p_{{G}}}{{\rm d}V_{{G}}}+\gamma _{\textit{LG}}\frac {{\rm d}\kappa }{{\rm d}V_{{G}}}=\frac {p_{{G}}}{V_{{G}}}+\gamma _{\textit{LG}}\frac {{\rm d}\kappa }{{\rm d}V_{{G}}}\gt 0. \end{equation}

This criterion indicates that the rate of curvature change ( ${\rm d}\kappa /{\rm d}V_{G}$ ), which is determined by the surface structure, has a significant impact on the mechanical stability of the gas bubble. Specifically, if the rate of curvature change is positive ( ${\rm d}\kappa /{\rm d}V_{G} \gt 0$ ), then the mechanical equilibrium state is always stable. Moreover, an alternative criterion to determine the stability can be given out by examining the change of equilibrated volume at different ambient liquid pressures. Combining Laplace equation ${{p}_{{G}}}-{{p}_{{L}}}={{\gamma }_{\textit{LG}}}\kappa$ and (2.3), we can obtain a new criterion for the mechanically stable equilibrium state:

(2.4) \begin{equation} \frac {{\rm d}V_{{G}}}{{\rm d}p_{{L}}}=\left (\frac {{\rm d}p_{{G}}}{{\rm d}V_{{G}}}-\gamma _{\textit{LG}}\frac {{\rm d}\kappa }{{\rm d}V_{{G}}}\right )^{-1}=-\left (\frac {\partial ^2F_{{t}}}{\partial V_{{G}}^2}\right )_{\!\! n_G}^{-1}\lt 0. \end{equation}

That is, when increasing the liquid pressure, the volume of the gas bubble that is in mechanical equilibrium should shrink for the stability requirement.

Once the gas bubble reaches a stable mechanical equilibrium (notice that a static gas bubble that can be observed in the experiment is always mechanically stable), the gas diffusion dominates the bubble evolution. During this process, since the gas bubble is in mechanical equilibrium, one can determine the gas mole number $n_{{G}}$ by combining Laplace equation $p_{{G}}-p_{{L}}=\gamma _{\textit{LG}}\kappa$ with the ideal gas law $(p_{{G}}V_{{G}} = n_{{G}}RT)$ . By simplifying (2.2), the first-order derivative of free energy with respect to the gas mole number $n_{{G}}$ is written as ${\rm d}{{F}_{{t}}}/{\rm d}n_{{G}}=RT\ln [{{p}_{{G}}}/(s{{p}_{{L}}})]$ . Then the second-order derivative of free energy reads ${\rm d}^2{{F}_{{t}}}/{\rm d}n_{{G}}^{2}=RT\,{\rm d}{{p}_{{G}}}/({{p}_{{G}}}\,{\rm d}n_{{G}})={{\gamma }_{\textit{LG}}}\,RT\,{\rm d}\kappa /({{p}_{{G}}}{\rm d}n_{{G}})$ . Therefore, a diffusion stable gas bubble that can sustain for a long time requires

(2.5) \begin{equation} \frac {{\rm d}\kappa }{{\rm d}n_{{G}}}\gt 0. \end{equation}

Equation (2.5) is the necessary and sufficient condition for the stable state of underwater gas bubbles, which is independent of bubble-attached conditions, bubble size, wettability and morphology of the attached solid surface. Therefore, it is a universal criterion for evaluating underwater bubble stability. Equation (2.5) describes a competitive balance between gas diffusion and Laplace pressure, i.e. when gas dissolves from the bubble into the surrounding liquid, the morphological change of a stable bubble should result in an increasing curvature such that the reduced gas pressure in the bubble (due to increased Laplace pressure) will lead to gas diffusion from the surrounding liquid back into the bubble. The gas molecules diffusing between the bubbles and the surrounding water reach a balance, making the gas layer keep a stable state.

2.2. Verification of the criterion for underwater bubbles stability under various conditions

The stability of bubbles under various conditions is analysed using the criterion in (2.5). The results for suspended bubbles are shown in figure 1(b). When the gas mole number inside the bubble increases $({\rm d}n_{{G}} \gt 0)$ , the curvature of bubbles with different radii always decreases $({\rm d}\kappa \lt 0)$ , resulting in ${\rm d}\kappa / {\rm d}n_{{G}} \lt 0$ . This finding demonstrates that suspended bubbles cannot achieve a stable state, regardless of bubble size, surrounding pressure or dissolved gas saturation. To validate this prediction, the free energy of suspended bubbles is analysed (figure 1 c). Under oversaturated water conditions ( $s = 143.77$ ), an equilibrium state can be found ( $\delta F_{t} = 0$ ) at $r=10$ nm. Details of the conditions are provided in table 1. However, the energy curve at this equilibrium point (as the dashed line shows) is convex, demonstrating unstable conditions. Therefore, the conclusion of the free energy analysis aligns with the prediction of the criterion. The conclusion that suspended bubbles are unstable is consistent with findings reported in other literature, further supporting the criterion (Plesset & Sadhal Reference Plesset and Sadhal1982; Lohse & Zhang Reference Lohse and Zhang2015b ; Tan et al. Reference Tan, An and Ohl2021).

Table 1. Specific parameters used for calculating the free energy curves shown in figure 1.

The results for bubbles attaching to a flat surface are shown in figure 1(d–k). It is found that the stability criterion (2.5) is satisfied only on a hydrophobic surface with contact line pinning (figure 1 d). In this specific condition, bubbles can be stable. In contrast, for hydrophobic surfaces without pinning (figure 1 f), hydrophilic surfaces with pinning (figure 1 h), or hydrophilic surfaces without pinning (figure 1 j), bubbles do not achieve a stable state. (The specific case of gas molecule adsorption at the solid–liquid interface is not considered in this discussion; see Brenner & Lohse (Reference Brenner and Lohse2008).) These predictions are further validated by free energy analyses of nanobubbles, as shown in figures 1(e), 1(g), 1(i) and 1(k). The analyses confirm that stable states exist only for the pinned hydrophobic surfaces, consistent with the predictions of the criterion. This conclusion is also supported by other literature (Lohse & Zhang Reference Lohse and Zhang2015b ; Tan et al. Reference Tan, An and Ohl2021; Zhang et al. Reference Zhang, Zhu, Wood and Lohse2024), such as the Lohse–Zhang model (Lohse & Zhang Reference Lohse and Zhang2015b ).

The stability analysis of bubbles that are trapped in the surface structures is presented in figure 1(l–s). Similar to bubbles attached to flat surfaces, a stable state is found only on a hydrophobic surface with contact line pinning (figure 1 l) (Xu et al. Reference Xu, Sun and Kim2014; Xiang et al. Reference Xiang, Huang, Lv, Xue, Su and Duan2017). For hydrophilic surfaces with contact line pinning, bubbles do not achieve a stable state (figure 1 p). These predictions are validated through free energy analysis (figure 1 m,q) using micrometre-sized bubbles. The other two conditions, i.e. hydrophobic surfaces (figure 1 n) and hydrophilic surfaces (figure 1 r) without contact line pinning, also show that no stable state exists. The condition in figure 1(n) requires special attention, as the curvature of the liquid–gas interface remains constant regardless of variations in the number of gas molecules within the bubble, i.e. ${\rm d}\kappa / {\rm d}n_{G} = 0$ . Consequently, gas molecules will continuously dissolve out of or into the bubble, indicating that the system is in a unstable state, which satisfies mechanical equilibrium but does not achieve diffusion equilibrium. This prediction can be verified by the free energy graphs, which show a straight line along the horizontal axis (figure 1 o). Such states have also been observed experimentally, and have been discussed in previous studies (Poetes et al. Reference Poetes, Holtzmann, Franze and Steiner2010; Lv et al. Reference Lv, Xue, Shi, Lin and Duan2014).

From the above analysis, it can be concluded that this criterion applies to suspended, attached and trapped bubbles, covering a range of sizes including nanoscale, and extending to both hydrophilic and hydrophobic solid surfaces, including flat and structured geometries. As shown in figure 1, stable bubbles require the contact line to remain pinned, whether on flat (figure 1 d) or structured (figure 1 l) surfaces. Furthermore, all experimental observations of stable bubble states reported to date have occurred under contact line pinning conditions (Lohse & Zhang Reference Lohse and Zhang2015b ; Prakash, Xi & Patel Reference Prakash, Xi and Patel2016; Li et al. Reference Li, Quéré, Lv and Zheng2017; Xiang et al. Reference Xiang, Huang, Lv, Xue, Su and Duan2017; Zhang et al. Reference Zhang, Zhu, Wood and Lohse2024). If the contact line depins, then the bubble ultimately dissipates (Koishi et al. Reference Koishi, Yasuoka, Fujikawa, Ebisuzaki and Zeng2009; Papadopoulos et al. Reference Papadopoulos, Mammen, Deng, Vollmer and Butt2013; Domingues et al. Reference Domingues, Arunachalam, Nauruzbayeva and Mishra2018; Tan et al. Reference Tan, An and Ohl2021). Then we will explore the stable state that occurs after the contact line depins from the sharp edge.

3.1. Theoretical analysis of the new stable mode

We examine the stability of the bubble in a cavity with walls shaped as a solid of revolution of the sine function, $y_C = \sin (3\pi t_C / 20)$ (figures 1(t) and 2). The contact line is free to move along the cavity walls while maintaining constant contact angle $ {120}^\circ$ . Because the sidewall is curved, the liquid–gas interface adopts different curvatures in distinct regions: a convex curvature ( $\kappa \gt 0$ ) when $\alpha _{C} \gt {33}^\circ$ (figure 1 t) and a concave curvature ( $\kappa \lt 0$ ) when $\alpha _{C} \lt {33}^\circ$ (figure 1 $ v $ ). Focusing on the convex region at $r_{X} = {10.6}\,\unicode{x03BC}\textrm {m}$ (figure 1 t), an increase in the gas mole number ( ${\rm d}n_{G} \gt 0$ ) leads to an increased $r_{X}$ and a decrease in $\alpha _{C}$ (figure 2). The changes in both $r_{X}$ and $\alpha _{C}$ induce a reduction in curvature $\kappa$ . That yields ${\rm d}n_{G}/{\rm d}\kappa \lt 0$ , indicating the unstable state. This conclusion is further supported by the free energy analysis (figure 1 $ u $ ). This finding aligns with previous understanding that a stable state requires contact line pinning.

Figure 2. Design of the cavity with walls shaped as a solid of revolution of the sine function. (a) Original sine function: $y_{C} = \sin (3\pi t_{C}/20)$ , with $t_{C}$ ranging from 0 to 40 $\unicode{x03BC} \textrm {m}$ . (b) Rotation of the original sine function by ${60}^\circ$ anticlockwise around the origin $(0,0)$ , given by the transformation equations $r_{X} = \cos (-{60}^\circ )\, t_{C} + \sin (-{60}^\circ ) \sin (3\pi t_{C} / 20),\ y_{C} = -\sin (-{60}^\circ )\,t_{C} + \cos (-{60}^\circ ) \sin (3\pi t_{C} / 20)$ , with $t_{C}$ ranging from 0 to 40 $\unicode{x03BC} \textrm {m}$ . (c) The relationship between the rotated curve’s half corner angle $\alpha _{C}$ versus $r_{X}$ . The vertical dotted lines mark $r_{X} = {10.6}\,\unicode{x03BC}\textrm {m}$ and $r_{X} = {13.5}\,\unicode{x03BC}\textrm {m}$ , respectively.

In contrast, in the concave region at $r_{X} = {13.5}\,\unicode{x03BC}\textrm {m}$ (figure 1 $ v $ ), an increase in the gas mole number ( ${\rm d}n_{G} \gt 0$ ) results in increases in both $r_{X}$ and $\alpha _{C}$ (figure 2), which leads to ${\rm d}\kappa \gt 0$ , and yields ${\rm d}n_{G}/{\rm d}\kappa \gt 0$ . This indicates a stable state even as the contact line depins and moves, as confirmed by free energy analysis in figure 1, which demonstrates a new stable mode of surface bubble that has not been reported before. Moreover, our results show that bubble stability depends on surface geometry, with certain regions on a curved wall supporting a stable state, while others do not.

3.2. Geometric standard for solid surfaces supporting the new stable mode

Next, we will explore the specific surface geometries required for the bubble stable state. Equation (2.5) is applied to cavities with smooth sidewalls, ensuring the absence of contact line pinning. For simplicity, but without loss of generality, we select widely accepted topological models to represent cavity geometries: two-dimensional (2-D) grooves (a groove with uniform cross-section, figure 3 a,b) with an arbitrary but symmetric profile (figure 3 c) and three-dimensional (3-D) rotational symmetric pores (figure 3 d). The generatrix of the groove or the pore is characterised by $w(h)$ . The mechanical stability is hard to discuss analytically as it involves integrating the profile when calculating the gas volume. But in an experiment, as long as we can observe a non-dynamic gas bubble, it must be a mechanical stable equilibrium. For simplicity, we assume that the gas bubble is already mechanically stable. Under this assumption, the equilibrium position of gas dissolution satisfies ${{\gamma }_{\textit{LG}}}\cos ( {{\theta }_{{Y}}}-\alpha ( h ) )/w ( h )={{p}_{{L}}} ( s-1 ).$ The stability criteria (2.5) for gas diffusion can be simplified as

(3.1) \begin{equation} \frac {{\rm d}\kappa }{{\rm d}n_{{G}}}=\frac {{\rm d}\kappa }{{\rm d}h}\frac {{\rm d}h}{{\rm d}n_{{G}}}=\frac {w\left (h\right )\alpha ^{\prime }\left (h\right )\mathrm{sin}\left (\theta _{{Y}}-\alpha \left (h\right )\right )-w^{\prime }\left (h\right )\mathrm{cos}\left (\theta _{{Y}}-\alpha \left (h\right )\right )}{w\left (h\right )^2}\frac {{\rm d}h}{{\rm d}n_{{G}}}\gt 0. \end{equation}

where $\alpha '(h)=w''(h)\cos ^{2}\alpha$ . In order to calculate the sign of ${{\rm d}h}/{{\rm d}n_{{G}}}$ , we apply derivative regarding h to the ideal gas law and combine it with Laplace equation ${{p}_{{G}}}-{{p}_{{L}}}={{\gamma }_{\textit{LG}}}\kappa$ and (2.3),

(3.2) \begin{equation} RT\frac {{\rm d}n_{{G}}}{{\rm d}h}=\left (\gamma _{\textit{LG}}\frac {{\rm d}\kappa }{{\rm d}V_{{G}}}+\frac {p_{{L}}+\gamma _{\textit{LG}}\kappa }{V_{{G}}}\right )\frac {{\rm d}V_{{G}}}{{\rm d}h}V_{{G}}=\left (\frac {\partial ^2F_{{t}}}{\partial V_{{G}}^2}\right )_{\!\! n_{{G}}}\frac {{\rm d}V_{{G}}}{{\rm d}h}V_{{G}}. \end{equation}

Notice that the sign of ${ ( {{\partial }^{2}}{{F}_{{t}}}/\partial V_{{G}}^{2} )}_{{{n}_{{G}}}}$ denotes the mechanical stability, where positive and negative signs represent stable and unstable states, respectively. Then for a mechanically stable gas bubble, assuming that $V_{{G}}$ is a monotonic increasing function of $h$ ( ${\rm d}V_{{G}}/{\rm d}h \gt 0$ ), we can find that ${\rm d}h/{\rm d}n_{{G}} \gt 0$ . Therefore, the stability criteria (3.2) for gas diffusion can be simplified as

(3.3) \begin{equation} w(h)\,\alpha ^{\prime }(h)\sin (\theta _{{Y}}-\alpha (h))\gt \tan \left (\alpha (h)\right )\cos (\theta _{{Y}}-\alpha (h)). \end{equation}

Figure 3. Stable state of gas bubbles without contact line pinning. Schematics of symmetric cross-sections with (a) arbitrary and (b) wedge profiles, where $w(h) = \tan (\alpha )\,h$ , and $\alpha$ is a constant; (c) 2-D grooves with arbitrary profile, and (d) 3-D rotational symmetric pores. (e) Confocal images of the wedge-groove specimens with half corner angle $\alpha = 33 \pm {1.7 }^\circ$ , for top, 3-D images, and bottom, 2-D profile. (f) Experimental set-up: a sample fixed in a sealed resin glass box connected to a water tank. The submerged depth $H$ is adjusted by changing the vertical position of the water tank. Deionised water, saturated with atmospheric pressure over 2 days, is used. The evolution of the air layer is observed through a 40 $\times$ water immersion objective via an optical glass window. (g–j) The 3-D confocal images showing liquid–gas interface evolution as $H$ increases from 0 to 50 cm. Cyan indicates wedge groove wall; red indicates liquid–gas interface. (k) Cross-section images of the liquid–gas interface from (g), with $H$ increasing from 0 to 50 cm, where $h$ is the vertical distance from contact lines to the wedge groove tip. Due to light scattering near the curved liquid surface, the liquid–gas interface close to the wall is not clearly visible. To address this, a quadratic curve fitting method is applied to the interface contour to estimate the local profile and calculate the contact angle between the interface and the wall. (l) Cross-section images with $H$ increasing from 0 to 25 cm, then from 25 to 50 cm. (m,n) Quantified $h$ evolution from (k) and (l), with experimental results indicated by black squares ( $\alpha = {33}^\circ $ ) and red dots ( $\alpha = {38}^\circ$ ); see supplementary material figure S3 for details available at https://doi.org/10.1017/jfm.2025.10760. Dashed lines indicate theoretical prediction of $h$ using (3.8).

Notice that Young’s contact angle is in the range ${{\theta }_{{Y}}}\in [ 0,\pi ]$ . If we assume that $\alpha \in [ -{\pi }/{2}\;,{\pi }/{2}\; ]$ , which indicates that a certain value of $h$ only refers to one position of the interface, then we can simplify the stability criterion as

(3.4a) \begin{align}& \alpha ^{\prime }\left (h\right )\gt \dfrac {\tan \left (\alpha \left (h\right )\right )\cot \left (\theta _{{Y}}-\alpha \left (h\right )\right )}{w\left (h\right )}\quad \mathrm{if}\ \theta _{{Y}}-\alpha \left (h\right )\in \left (0,\pi \right )\!, \end{align}

(3.4b) \begin{align}& \alpha ^{\prime }\left (h\right )\lt \dfrac {\tan \left (\alpha \left (h\right )\right )\cot \left (\theta _{{Y}}-\alpha \left (h\right )\right )}{w\left (h\right )}\quad \mathrm{if}\ \theta _{{Y}}-\alpha \left (h\right )\lt 0\ \mathrm{or}\ \theta _{{Y}}-\alpha \left (h\right )\gt \pi. \end{align}

Although (3.4) defines the sufficient and necessary condition for the stability criterion, it is too complicated to be applied in practice. With some simplification, we can find two simpler, sufficient but not necessary conditions of the diffusion stable state:

(3.5a) \begin{align} \alpha \left (h\right )\gt 0,\ \alpha ^{\prime }(h)&\geq 0\mathrm{\quad and\quad }\theta _{{Y}}-\alpha \left (h\right )\gt \pi /2, \end{align}

(3.5b) \begin{align} \mathrm{or}\quad \alpha \left (h\right )\lt 0,\ \alpha ^{\prime }(h)&\geq 0\quad \mathrm{and}\quad 0\lt \theta _{{Y}}-\alpha \left (h\right )\lt \pi /2. \end{align}

Equation (3.5a ) describes a scenario on hydrophobic surfaces, where the cavity must exhibit an increasing width and a non-decreasing inclining angle from the bottom to the top, and the angle $\alpha (h)$ does not exceed the limit $\theta _Y-\pi /2$ . Equation (3.5b ) describes a scenario of hydrophilic cavities, where the width should decrease and the angle should not decrease from the bottom to the top, and the angle $\alpha (h)$ should be limited to $[\theta _Y-\pi /2, \theta _Y]$ . Specifically, for a wedge groove with constant angle $\alpha \gt 0$ , the meniscus curvature for a gas bubble is written as $\kappa =\cos (\theta _{{Y}}-\alpha )/(h\tan \alpha )$ . Then the stability criteria can be rigorously found by simplifying (2.5) together with ${\rm d}h/{\rm d}n_{{G}} \gt 0$ , which reads

(3.6) \begin{equation} \theta _{{Y}}-\alpha \left (h\right )\gt \pi /2, \end{equation}

which coincides with sufficient but not necessary conditions (3.5a ).

Then the equilibrium position of the contact line will be analysed as follows. In the wedge groove, the curvature of the liquid–gas interface can be expressed as $\kappa =\cos (\theta -\alpha )/(h\tan \alpha )$ , where $h$ is the vertical distance from the contact line to the tip of the wedge groove. Substituting $\kappa$ into the equilibrium equation, i.e. $\gamma _{\textit{LG}}\cos (\theta _{{Y}}- \alpha ( h ))/w( h ) = {p_{{L}}}(s-1)$ , we can obtain the equilibrium position of contact lines:

(3.7) \begin{equation} h=\frac {\gamma _{\textit{LG}}\cos (\theta -\alpha )}{\tan \alpha }\frac {1}{p_L\left (s-1\right )}. \end{equation}

It can be found from (3.7) that different positions of the liquid–gas interface $h$ correspond to different environmental conditions $(p_{L}, s)$ , indicating that the stable state has the potential to spontaneously adapt to the variations of the environmental conditions by changing its positions. Moreover, in the condition of the experiment in the main text, the gas saturation degree is $s = p_{{a}}/p_{{L}}$ , which renders (3.7) to be simplified as

(3.8) \begin{equation} h=\frac {-\gamma _{\textit{LG}}\cos (\theta -\alpha )}{\rho g\tan \alpha }\frac {1}{H}. \end{equation}

If the liquid–gas interface remains in this equilibrium position over an extended period, then we can conclude that the bubbles are in a stable state.

Similar to the above analysis on a 2-D symmetric groove, we can also discuss the diffusion stability for a 3-D pore with a rotational symmetric profile. Denote $r = w(h)$ as the radius of the pore, and $h$ as the vertical distance from the point of interest to the bottom of the pore. For a gas bubble at height $h$ , the meniscus curvature can be found by $\kappa = 2\cos (\varTheta _{Y} -\alpha (h))/r$ , where $\tan (\alpha ) = w'(h)$ . Compared to the 2-D groove, the interface curvature of the 3-D pore only has an additional factor 2, which does not have any impact when applying the stability criterion (2.5). Therefore, the above results for the diffusion stability as (3.3)–(3.8) also hold for the rotational symmetric 3-D pore.

3.3. Experimental evidence for the new stable state and geometric standard

Experiments are carried out to prove that stable states can be achieved without contact line pinning. The wedge grooves with different corner angles $2\alpha$ (figure 3(e) and supplementary material figure S1) are fabricated with the photoresist SU-8 (MicroChem Corp., USA) using a 3-D laser lithography system (Photonic Professional GT, Nanoscribe GmbH, Germany) (Xiang et al. Reference Xiang, Huang, Huang, Dong, Cao and Li2020), and then are hydrophobised with a commercial hydrophobic agent (Glaco, Japan) to achieve contact angle $\theta = 156 \pm 5.2 ^{\circ }$ . A laser scanning confocal microscope (A1R-PLUS, Nikon, Japan) is employed to observe the evolution of the liquid–gas interface in wedge grooves. The liquid pressure on the wedge grooves ( $p_{{L}}$ ) is composed of the atmospheric pressure ( $p_{{a}}$ ) and the hydrostatic pressure ( $\rho gH$ ), i.e. $p_{{L}}=p_{{a}}+\rho gH$ , where $\rho$ is the water density, $g$ is gravity, and $H$ is the submerged depth of the wedge grooves (figure 3 f).

Experimental images illustrating the submerged depth $H$ changing from 0 to 50 cm are as shown in figure 3(g–k), while the evolution of the meniscus position $h$ is presented in figure 3(m). Over time, the liquid–gas interfaces depin from the top edge of the wedge and freely move in the wedge. After 51 min of evolution (figure 3 k), their positions remained unchanged. From the perspective of gas and mass transport dynamics, the diffusion time scale is approximately 100 s (Lv et al. Reference Lv, Xue, Shi, Lin and Duan2014). Therefore, the 51 min observation period is sufficient for the bubble to reach a stable state. Moreover, by comparing the final value of $h$ with the theoretical prediction of (3.8) in figure 3(m), it is confirmed that the liquid–gas interface reaches a stable state. Subsequently, experiments are conducted with $H$ changing twice, and the results are shown in figures 3(l) and 3(n). First, when $H$ changes from 0 to 25 cm, the liquid–gas interface reaches a stable state for the first time (from 83 min to 145 min). Then as $H$ changes from 25 to 50 cm, the liquid–gas interface reaches a new stable state (from 190 min to 262 min). These experiments demonstrate the existence of a stable state without contact line pinning, and show that the freely moving contact line allows the system to adapt to changing environmental conditions.

The geometric standard for solid surfaces supporting the stable state is experimentally verified. Figures 4(a) and 4(b) show a stable gas column that remains unchanged for over 240 min when (3.5a ) is satisfied. In contrast, figures 4(c) and 4(d) show that when (3.5a ) is not satisfied, the gas column breaks into shrinking segments and disappears within 9 min. The cause of this segmentation is detailed in Concus & Finn (Reference Concus and Finn1969, Reference Concus and Finn1974), Langbein (Reference Langbein1990) and Xiang et al. (Reference Xiang, Huang, Huang, Dong, Cao and Li2020). The phase diagram for the stable state is shown in figure 4(e), with the threshold line defined by $\alpha =\theta -\pi /2$ . The region to the right of the curve, where (3.5a ) is satisfied, represents the stable state, while the left-hand region, where (3.5a ) is not satisfied, corresponds to the regime where bubbles gradually shrink and eventually disappear. To validate the phase diagram, experiments test the stability of gas bubbles in grooves with varying corner angles $2\alpha$ and contact angles $\theta$ . Solid points and hollow points in figure 4(e) represent the stable state and the fully wetted state, respectively. The experimental results align with the theoretical prediction of the phase diagram (additional results are available in supplementary material figure S3). Energy diagrams in figures 4(f) and 4(g) further confirm this. Figure 4(f) matches the energy profile of a fully wetted state, and figure 4(g) matches the stable gas state. These energy diagrams confirm, from a thermodynamic perspective, that gas bubbles can remain in a stable state when the solid surface adheres to the proposed geometric standard.

Figure 4. Geometry standard of solid surface supporting stable state. (a,b) Confocal images of stable gas bubbles, where $\theta ={156}^\circ ,\ \alpha ={33}^\circ$ and $H = {50}\,\textrm {cm}$ . (c) Gas bubbles evolution in the wedge groove, where $\theta ={116}^\circ ,\ \alpha ={33}^\circ$ and $H = {50}\,\textrm {cm}$ . (d) Enlarged image of (c). The insets of C–C and D–D show the cross-sections of the dotted boxes of the 3-D image. The menisci are convex. (e) Phase diagram of the stable states of gas bubbles, where the right- and left-hand regions indicate the stable state and fully wetted state, respectively. Energy diagrams of (f) the fully wetted state with $\theta ={116}^\circ ,\ \alpha ={33}^\circ$ and $H = {50}\,\textrm {cm}$ , and (g) the stable state with $\theta ={156}^\circ ,\ \alpha ={33}^\circ$ and $H = {50}\,\textrm {cm}$ . The red dot represents the energy local minima of the free energy curve. The blue ball indicates the minimum energy state that can be achieved in practice. (h) Schematics of the liquid–gas interface in the wedge groove corner. (i) Dependence of the contact line position $h$ on submerged depth $H$ . Lines show the theoretical prediction of (3.8), where the solid parts indicate the cases within the critical boundary, i.e. the stable region, and the dashed parts indicate the cases beyond the critical boundary, i.e. the unstable region. Solid and open dots indicate the experimental results of the stable state and fully wetted state, respectively. (j) Energy diagram of the fully wetted state with the wedge groove ( $\theta ={116}^\circ ,\ \alpha ={33}^\circ$ ) satisfies (3.5). Here, the submerged depth $H = {100}\,\textrm {cm}$ exceeds the critical value $H_{\textit{max}} = {88}\,\textrm {cm}$ . (k–m) Images of the liquid–gas interface evolution for (j).

Figure 5. Cross-section profiles and 3-D design drawing images of grooves and pores. For the hydrophobic sample with $\theta =156^\circ$ , the cross-section profiles are (a) $C_{1}$ , $\alpha =0.009375h^2+27$ , (b) $C_{1}$ , $\alpha= 0. 009375h^2+27$ , (c) $C_{2}$ , $\alpha=0.75h+30$ , (d) $\alpha=40^\circ$ . For the hydrophilic sample with $\theta =54^{\circ }$ , the cross-section profiles are (e) $C_{3}$ , $\alpha = - 0. 015625h^2+ 35$ , (f) $\alpha = - 25^{\circ }$ , (g) $\alpha=-15^{\circ }$ . The unit of $h$ here is microns.

The critical conditions for achieving a stable state are analysed based on (3.8), which shows that the submerged depth $H$ is inversely proportional to the interface position $h$ . Since the tip point of the groove cannot form a geometric singularity, $h$ has a minimum value $h_{\textit{min}}$ (figure 4 h). Correspondingly, $H$ has the maximum value $H_{\textit{max}} = -\gamma L_{{{G}}} \cos (\theta -\alpha )/\rho g\tan (\alpha )\,h_{\textit{min}}$ . Then the limitation $H_{\textit{max}}$ is experimentally verified. The lines in figure 4(i) correspond to the theoretical predictions from (3.8). The intersections of the solid and dashed lines denote the theoretical critical points $(H_{\textit{max}}, h_{\textit{min}})$ . The solid parts of the lines represent conditions within the critical boundary ( $H \lt H_{\textit{max}}$ ) where a stable state is maintained, while the dashed parts ( $H \gt H_{\textit{max}}$ ) indicate an unstable state. Experimental data, shown as solid and open dots, represent stable (lasting over 2 h) and unstable (vanishing within 30 min) liquid–gas interfaces, respectively. The experimental results (dots) are consistent with the theoretical predictions (lines) in figure 4(i), confirming that $H_{\textit{max}}$ represents the critical limit of the stable state. Figure 4(j) shows the energy diagram for $H \gt H_{\textit{max}}$ , confirming thermodynamically that no stable state exists. The evolution of gas bubbles under these conditions is shown in figure 4(k–m) (with additional data in supplementary material figures S4 and S5). These bubbles transition into a fully wetted state. In addition to grooves, stable gas bubbles can also form in pores, with a similar geometric criterion described by (3.5). Experimental validation for pores is provided in supplementary material figure S6.

3.4. Controlling gas bubble stability via surface geometry design

Figure 6. Control the stability of gas bubbles by designing the geometry of solid surfaces. Various structured solid surfaces are designed to stabilise the gas bubble. (a,b) A 3-D design drawing and confocal image of groove with the generatrix $C_{1}$ , $\alpha = 0.009375h^{2}+27$ . (c) Experimental result with $\theta ={156}^\circ$ and $H={30}\,\textrm {cm}$ . (d,e) Cross-section images with submersion times 120 min and 240 min, respectively. (f,g) Design drawing and confocal image of pore with the generatrix $C_{2}$ , $\alpha = 0.75h+30$ . (h–j) Experimental result with $\theta ={156}^\circ$ and $H={30}\,\textrm {cm}$ . (k,l) Design drawing and confocal image of pore with the generatrix $C_{3}$ , $\alpha = -0.015625h^{2}+35$ . (m–o) Experimental result with $\theta ={54}^\circ$ and $H={-30}\,\textrm {cm}$ . (p) Evolution of the liquid–gas interface in hydrophobic grooves and pores after submersion for over 120 min. For all samples, $\theta ={156}^\circ$ and $H={30}\,\textrm {cm}$ . (q) Experimental results on hydrophilic grooves and pores. For all samples, $\theta ={54}^\circ$ and $H={-30}\,\textrm {cm}$ .

We propose a method to control the stability of gas bubbles under varying environmental conditions by designing the geometry of solid surfaces. There are grooves or pores with generatrix $w(h)$ . For submerged depths $H_{1}$ and $H_{2}$ , the stable liquid–gas interface points are $w(h_{1})$ and $w(h_{2})$ , respectively. Detailed information is provided in figure 5. If all points on the piece of the sidewall between $w(h_{1})$ and $w(h_{2})$ satisfy the geometric standard, then the stable state can be achieved over the depth range from $H_{1}$ and $H_{2}$ . To validate this design method, experiments are conducted on hydrophobic and hydrophilic surfaces under oversaturated and unsaturated water conditions, using both straight and curved generatrix designs. The experimental results for the hydrophobic groove are shown in figure 6(a–e). The cross-section profile of the groove $C_{1}$ satisfies the criterion (3.5a ). Results for hydrophobic and hydrophilic pores are shown in figures 6(f–j) and 6(k–o). The generatrices $C_{2}$ and $C_{3}$ satisfy (3.5a ) and (3.5b ), respectively. Images in figures 6(d), 6(e), 6(i), 6(j), 6(n) and 6(o) represent the results of these three samples submerged underwater from 120 min to 240 min. These images show that liquid–gas interfaces remain stable over 120 min. Comparing the contact line location with the theoretically predicted stable value, we find that they are in a stable state. More results are shown in figure 6(p,q), which indicate that these cavities satisfying the geometric standard could make the gas bubbles achieve a stable state (see supplementary material figure S6 for more data).