Wet Adhesion of Soap Bubble Bridge between Two Curved Surfaces

Xu, Peibao; Wang, Qi; Zhou, Lin; Yu, Yong; Zhao, Jun; Li, Kai

doi:https://doi.org/10.1155/2022/3340078

Mathematical Problems in Engineering

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Applied Mathematics for Engineering Problems in Biomechanics and Robotics 2022

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Research Article | Open Access

Volume 2022 | Article ID 3340078 | https://doi.org/10.1155/2022/3340078

Wet Adhesion of Soap Bubble Bridge between Two Curved Surfaces

Peibao Xu,¹Qi Wang,¹Lin Zhou,²Yong Yu,¹Jun Zhao,¹and Kai Li¹

Academic Editor: Carlos Llopis-Albert

Received25 Feb 2022

Revised26 Mar 2022

Accepted29 Mar 2022

Published16 Apr 2022

Abstract

Wet adhesion phenomenon of the soap bubble bridge is widespread in the climbing behaviors of geckos and insects, and the surface shapes generally have important impact on the adhesion behavior of soap bubble bridge. In order to focus on the effect of the shape on the wet adhesion, we study the adhesion behavior between two rigid surfaces containing different shapes such as concave, flat, and convex. For a given soap bubble bridge, the relationship between adhesion force and separation is numerically calculated, and the corresponding configuration of soap bubble bridge is also given. The results show that the adhesion force of soap bubble bridge decreases with the increase of separation. This trend is the same as the case of liquid bridge, although the volume of liquid bridge is constant while the volume of soap bubble bridge varies. The adhesion force with concave rigid surfaces is bigger than that of the other two kinds of rigid surfaces at the same separation. The wetting radius of the soap bubble bridge decreases with the increase of separation. The appearance of this phenomenon results from the equation of state for the gas in the soap bubble bridge. This research may improve the understanding of the mechanical mechanism of wet adhesion.

1. Introduction

Geckos and insects such as flies, ants, lizards, beetles, and spiders have evolved various structural feet suitable for climbing [1, 2]. Although the feet are often complex, they can still efficiently handle the mechanical loading through the adhesion [3, 4]. From the perspective of adhesion mechanism, the adhesion can be simply categorized into dry and wet adhesion. The dry adhesion based on van der Waals forces has been proved by experiments [5–7] and theories [8–11]. The wet adhesion between two adjacent surfaces is caused by the capillary force of the liquid bridge [12–14]. Understanding the adhesive characteristic has scientific value because it provides insights into how nature works [15, 16], in which a novel experimental method for adhesion measurement is proposed by using the centrifugal adhesion balance. This is helpful for guiding the artificial adhesion system.

Many experimental studies show that wet adhesion can be affected significantly by the morphology or the deformation of the layers [17–19]. The morphology or deformation of the layers generally changes the advancing and receding contact angles and in turn influences the profile of the droplet or liquid bridge, as well as their wetting properties and movement [15, 16, 20–27]. The key to study wet adhesion lies in the determination of the liquid bridge shapes [28]. A model of fiber substrate was proposed to study the effect of the fiber size on the strength of wet adhesion [29], and it is found that the strength increases with the decreasing radius of the fiber. The contact shapes, such as torus, horseshoe, and suction cup, have been often assumed to study adhesion strength in the adhesion system [7, 30–35]. As for this, the influence of different contact shapes, e.g., ring-like, convex, concave, and flat, on adhesion has been studied [36], and the contact shape of the ring-like has obvious advantages than the other ones. Therefore, it can be concluded that the contact shapes have an important effect on adhesion properties.

Compared with the wet adhesion of liquid bridge, very few research works have been carried out on the wet adhesion caused by the soap bubble, although the wet adhesion phenomenon of soap bubble bridge exists widely in nature and engineering. Usually, the basic principle of wet adhesion for soap bubble bridge was considered the same as that of the liquid bridge [37]. However, different from the liquid bridge with the unchanged volume, the volume of the soap bubble bridge varied during the wet adhesion. For the ideal gas in the soap bubble bridge, the product of volume and pressure is often considered to be unchanged. Therefore, it is possible to have different mechanical mechanism and behaviors [20]. The study of wet adhesion behavior of soap bubble bridge is important for the understanding of natural phenomena, including the inflation of lungs at birth [38] and adhesive locomotion of snails on microscale and nanoscale topographically structured surfaces in the biological field [39], as well as bioadhesive tapes in the medical field [40]. In addition, this research has potential applications in the engineering field, including the regulation of the volatilizing stage of aerogel production [41], the evaluation of the mechanical behavior of wet sand [42], and the treatment of waste water in industrial processes [43].

To understand the mechanical mechanism of wet adhesion caused by the soap bubble bridge, which is different from that of the liquid bridge, we will study the adhesion properties of soap bubble bridge between two rigid surfaces. Firstly, for a given soap bubble bridge between two rigid surfaces, we calculate the Laplace pressure [44] and corresponding volume by virtue of shooting method. Secondly, the adhesion force caused by the soap bubble bridge between different shapes of rigid surfaces will be calculated and discussed. Finally, we will explain the mechanical mechanism of different results through theoretical analysis. The research results may improve the understanding of the mechanical mechanism of wet adhesion between curved surfaces and have potential application value in the design of structures.

2. Theoretical Analysis Model

The wet adhesion properties provided by the soap bubble bridge between two rigid surfaces with different shapes, such as concave, flat, and convex, will be investigated in the present study. Hereon, we will take the concave rigid surfaces as an example to show the adhesion properties of soap bubble bridge, as shown in Figure 1. It shows that two identical rigid surfaces with the sphere center and curvature radius are connected by the soap bubble bridge with volume and height . The shape of soap bubble bridge is axisymmetric with the wetting radius of . A cylindrical coordinate system with the origin O is assumed to describe the inner and outer radii of the soap bubble bridge with and as well as the bridge profile .

Because of the surface tension of the liquid , there exists pressure difference inside and outside the soap bubble bridge. This pressure is called Laplace pressure and can be described by Young–Laplace equation [44]:where and are the inner and outer radii of the soap bubble bridge, respectively.

According to the equation of equilibrium, the contact angle of the soap bubble bridge, as shown in Figure 1(c), can be written aswhere and are the interfacial free energy densities of solid-gas and solid-liquid interfaces.

According to the static equilibrium conditions, the external force is equilibrium with the adhesion force provided by the soap bubble bridge, which is obtained from the surface tension and Laplace pressure , as shown in Figure 1(b). Thus, one obtainswhere can be explained as the apparent contact angle between the tangent line at the contact point and the horizontal line, as shown in Figure 1(c). For simplicity, the gas in the bubble is assumed to be the ideal gas and the Laplace pressure is determined by the Clapeyron equation [45]. Therefore, we can obtainwhere is the atmospheric pressure and the constant at a certain temperature depends on the mass of gas in the soap bubble bridge.

In order to obtain the volume of the soap bubble bridge, the profile of soap bubble bridge affected by the shape of rigid body should be determined. Therefore, we use and to describe the vertical displacement of the rigid body and the profile of soap bubble bridge in the current coordinate system, respectively. Hereon, we assume the axisymmetric rigid body is a spherical body. Therefore, the outline of the sphere in the plane is a circle. When the center of the rigid body is taken as the coordinate origin , the governing equation of the vertical displacement can be expressed aswhere is the horizontal coordinate in the rigid body. When we take O as the coordinate origin, the vertical displacement can be expressed as

When is positive, the contact surface, i.e., the shape of , is convex. When is negative, the contact surface is concave.

And the main radii of curvature can be expressed as and . Then, substituting these equations into equation (1), one obtains

According to the profile feature of soap bubble bridge, the boundary conditions for equation (7) can be expressed aswhere the angle depends on the bubble volume, the curvature of rigid surface, and the contact angle of the soap bubble bridge and is determined by

Due to the mirror symmetry of soap bubble bridge, one obtains

Based on equations (6) and (7), the volume of soap bubble bridge can be calculated by the integration:

Since both the upper and lower surfaces are rigid surfaces, the above equations are adequate to calculate the adhesion force and pressure. In Section 3, we will explain the numerical method for solving the above equations and discuss the calculation results.

3. Results and Discussion

For a given mass of gas in the soap bubble bridge, the Laplace pressure and the wetting radius are guessed to calculate the adhesion force and the shape of soap bubble bridge by virtue of the shooting method [20]. In the calculation, we first establish the shape of rigid surface for a given by equation (5), and then, for a given wetting radius , the shape of can be determined by equation (6). After that, for the different values of the Laplace pressure , we further calculate the profile of the soap bubble bridge by solving equation (7) with the corresponding boundary conditions’ equations (8) and (9). Based on the shape of the rigid surface and the profile of soap bubble bridge, the volume of soap bubble bridge can be calculated by equation (12). With the help of equation (4), we can get the different wetting radius and Laplace pressure for a given mass (namely, a given ) of gas in the soap bubble bridge. After the wetting radius and Laplace pressure have been calculated, the adhesion force between the two rigid surfaces can be obtained by using equation (3). And the corresponding deformation morphology can be obtained by equation (7).

In the following sections, we will discuss two main aspects: one is the effect of the contact surface morphology with different kinds of rigid surfaces on the adhesion force, and the other is to explore how the morphology of soap bubble bridge varies with the separation for the concave, flat, and convex rigid surfaces.

3.1. Adhesion Force

In order to obtain the adhesion force, we first plot the curves of Laplace pressure versus the product of pressure and volume for the different shapes of rigid surfaces such as concave, flat, and convex, as shown in Figure 2(a). Here, a characteristic length is given, and we define the following dimensionless parameters: , , , , , , , and . Generally, the interfacial tension depends on the type and concentration of surfactant. Hereon, from the typical value of the interfacial tension , as well as [20], , and , we can obtain the curvature radius of rigid surfaces and the dimensionless parameters . In the discussion, the typical radius of curvature of the soap bubbles is in the order of 1 mm [20], as shown in Figure 2(b). For a given and , the corresponding Laplace pressure can be obtained from Figure 2(b), and then, the adhesion force can be calculated by equation (3). For a given mass of gas, we can obtain the product of volume and pressure is constant by equation (4). However, the volume of soap bubble bridge can change, as shown in Figure 3, which is different from the case of liquid bridge with constant volume [37]. Therefore, it is interesting to study the behaviors of the adhesion property of the soap bubble bridge between two rigid surfaces.

(a)

(b)

(a)

(b)

It can be seen from Figure 2(a), for a given wetting radius , the Laplace pressure increases with the increase of mass of gas. For a given and , the Laplace pressure in the concave rigid surfaces is the smallest, and the Laplace pressure in the convex rigid surfaces is the biggest. The reason is that the volume between the two concave rigid surfaces is the biggest which results in the biggest at the same Laplace pressure. It can be seen from Figure 2(b), for a given , that the Laplace pressure decreases with the increase of wetting radius . This can be understood from equation (1). According to this figure, we can calculate the adhesion force and the configuration by using the different values of and (e.g., a combination of and for is shown by the red dot in Figure 2(b)).

Figure 3(a) shows the dependence of Laplace pressure in the soap bubble bridge on the surface separation. For a given mass of gas , the Laplace pressure increases with the increase of separation. This is because with the increase of separation, the wetting radius decreases, and in turn the Laplace pressure increases. And it also shows the dependence of volume of soap bubble bridge on the surface separation. For a given mass of gas , the volume of soap bubble bridge decreases with the increase of separation, which is different from liquid and solid and is counterintuitive. This can be understood by the constant product of volume and pressure. The mechanical mechanism for this phenomenon is different from that of liquid bridge in which the volume of liquid bridge was considered to be constant [20]. Hereon, the volume of soap bubble bridge varies with the Laplace pressure because of the characteristic of ideal gas. Figure 3(b) shows that the volume of soap bubble bridge decreases with the increase of Laplace pressure. For example, when the dimensionless Laplace pressure varies from -10000 to 0, the volume of soap bubble bridge will reduce by about 40%.

Then, we discuss the influence of shapes on the Laplace pressure and volume of soap bubble bridge. For a given mass of gas and separation , the wetting radius with convex rigid surfaces is the biggest and the wetting radius with concave rigid surfaces is the smallest. Therefore, the Laplace pressure in the soap bubble bridge for convex surfaces is the smallest and that for concave surfaces is the biggest by virtue of equation (1). Similarly, the changing trend of the volume of soap bubble bridge is opposite to that of Laplace pressure.

With the help of equation (3), we can get the curve of the adhesion force versus the separation, as shown in Figure 4(a). The results show that the adhesion force decreases with the increase of separation for the three kinds of contact surface. For the concave or convex surface with very small curvature, the adhesion force approaches that of the flat surface. For a given separation, the adhesion force between two concave rigid surfaces is the biggest, and the adhesion force between two convex rigid surfaces is the weakest. This can be explained as follows: for a given separation, compared with the case of flat contact surface, the wetting radius for concave contact surface is bigger, while the wetting radius for convex surface is smaller based on equations (4) and (12). The bigger wetting radii result in the bigger total pressure and total surface tension according to equation (3). Therefore, we can obtain that the order of the total pressure and surface tension from small to large is as follows: convex surface, flat surface, and concave surface.

(a)

(b)

Figure 4

The relationship between adhesion force and separation for the soap bubble bridge. (a) The comparison of adhesion force of the soap bubble bridge among different kinds of rigid surfaces and (b) the comparison of adhesion force for the different mass of soap bubble bridge. The adhesion force of the soap bubble bridge with concave rigid surfaces is stronger than that of the other two rigid surfaces, and the larger the soap bubble is, the greater the adhesion force is. The green dot in the figure denotes the contact between two convex rigid surfaces, and the magenta dot in the figure denotes the self-contact of the film of soap bubble.

To discuss the effect of the mass on the adhesion force, we take the soap bubble bridge with the concave rigid surfaces as an example. Figure 4(b) plots the relationship between the adhesion force and separation, for different mass of gas in the soap bubble bridge. For a given separation, the adhesion force increases with the increase of . This is because that the contact radius increases and in turn the total pressure and capillary force increase, which can be explained by equation (3).

3.2. Configuration of Soap Bubble Bridge

The trend of the adhesion force of soap bubble bridge is similar to the case of liquid bridge, but the mechanical mechanism for this phenomenon is different from that of liquid bridge in which the volume of liquid bridge was considered to be constant [20]. Hereon, the volume of soap bubble bridge varies with the separation because of the characteristics of ideal gas. Therefore, we have plotted the configuration of the soap bubble bridge for different separations, as shown in Figures 5–7. It can be seen from Figures 5 to 7 that the wetting radii of the soap bubble bridge with different kinds of rigid surfaces all decrease with the increase of separation. The volume of the soap bubble bridge decreases greatly with the increase of separation for concave rigid surfaces, as mentioned in Figure 3. Figure 5 also shows that the volume of soap bubble bridge varies greatly with the increase of separation. While the volume of soap bubble bridge decreases slightly with the increase of separation for flat and convex rigid surfaces, as shown in Figures 6 and 7. These results are caused by the characteristics of ideal gas, as described previously in Figure 3.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

4. Conclusions

In current study, the adhesion properties of the soap bubble bridge between two rigid surfaces with different shapes are investigated by considering the characteristics of ideal gas, which is different from that of liquid bridge. In order to better understand the mechanical mechanism of the soap bubble bridge, the surfaces containing different shapes such as concave, flat, and convex are assumed as rigid bodies. The studies show that the shapes of the rigid surfaces have important effect on the adhesion force and configuration. For a given mass of gas, the adhesion forces between different rigid surfaces decrease with the increase of separation. For a given separation, the adhesion force is the biggest for concave surfaces and the weakest for convex surfaces. For a given separation and shape of contact surfaces, the adhesion force increases with the increase of mass of gas. Accordingly, the volume of soap bubble bridge decreases with the increase of separation, which is counterintuitive. The volume is the biggest for concave surfaces and the smallest for convex surfaces. The volume of soap bubble bridge increases with the increase of mass of gas. The research results are beneficial to understand the mechanical mechanism of the climbing behaviors of geckos and insects and have potential application value in the design of elastic structures.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This research was funded by Natural Science Foundation of Anhui Province (Grant no. 2008085QA50), National Natural Science Foundation of China (Grant no. 12172001), Doctoral Startup Foundation from Anhui Jianzhu University (Grant no. 2020QDZ14), and the Key Project of Natural Science Research of Universities in Anhui (Grant nos. KJ2020A0449 and KJ2020A0452).

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