Vertical force acting on partly submerged spindly cylinders

In nature, some aquatic insects, such as water striders, are able to stand on the water surface due to the lift force created at the curvature of the air-water interface.^1–3 Experiments have shown that a single leg of a water strider can provide a lift force on the water surface equivalent to approximately 15 times the total body weight of the insect.^2,4 This phenomenon, which is attributed to super-hydrophobicity and the effect of surface tension, has provided inspiration for several biologically-based miniature water-walking robots. The robots, whose legs consist of spindly cylinders decorated with hydrophobic materials, are based on the fundamental principles of the water strider's ability to float and move on the water surface.^1,5–11 It is of interest to consider how great a load these cylindrical legs can support on a water surface.

When an object comes into contact with a water surface, the air-water interface deforms and a meniscus develops. The air-water interface deformation due to surface tension, which is governed by Young-Laplace equation, plays an important role in the interaction between the water and object, especially for small objects. Keller¹² demonstrated that the vertical force acting on submerged objects, composed of the buoyant force and the vertical component of the surface tension force, is equal to the weight of water displaced. In the past decade since the advent of water strider robots, there has been substantial interest in better understanding the underlying mechanisms involved in interactions between water and partly submerged spindly cylinders.^{4–7,9,11,13–28} For instance, Sitti et al.⁵ performed a parametric study on the load capacity of a single cylindrical leg on water via a numerical method, and provided a set of rules regarding the design of the supporting legs of water strider robots. Vella et al.¹³ discussed how the density and radius of the cylinder affect the ability of cylinder to float on the water surface, and investigated the role of the contact angle. They also evaluated the effect of cylinder length on the maximum load on water.¹⁶ Another study by Jiang et al.⁴ modeled the water strider's leg as a spindly solid column and evaluated its flotation ability experimentally and theoretically. Feng et al.¹⁴ investigated the buoyant force and sinking conditions of a hydrophobic thin rod floating on water by solving Young-Laplace equation and Hwang et al.²² demonstrated that super-hydrophobicity plays a significant role in reducing the drag force and energy consumption of a cylindrical leg detaching from water. In the case of water strider robots, experiments have shown that the lift force decreases with increasing distance between neighboring legs.⁵ The leg distance, therefore, is set as large as possible in the design of water strider robots.^5–7,11,25 When the distance reaches a certain value, the problem reduces to a single-cylinder one. For this reason, previous studies have limited their analysis to only one leg, and few of them to date have investigated a multi-cylinder system by accounting for the cylinder center distance.^5,9,24,25

In the current article, we aim to present a quantitative study on the underlying mechanism of the vertical force acting on partly submerged spindly cylinders. First, the forces acting on the cylinders are analyzed using a simplified 2-D model, after which the profile of the curved air-water interface and the vertical force are numerically calculated using MATLAB. Finally, a parametric study is conducted to determine the effects of cylinder center distance, inclined angle, static contact angle, and radius on the vertical force. The findings of the parametric study are discussed and correlated with different design variables of the supporting legs of water strider robots.

By considering a group of long and thin horizontal cylinders in parallel contact with water and assuming that the section radius is much smaller than the length, the vertical force acting on these cylinders can be analyzed by a 2-D model as illustrated in Figure 1. Using this method, the complex air-water interface deformations at both ends of cylinders can be ignored. The resultant vertical force acting on the cylinders is equal to the weight of water displaced, and each cylinder possesses a force dependent not only on its own wettability, geometry, and position, but also on the properties of the nearest cylinders. In order to facilitate the following discussion, we view the two neighboring half cylinders as a single entity (the area enclosed by dotted lines in Figure 1). Note that only quasi-static processes are discussed here and the effect of dynamics falls outside the scope of this study.

As shown in Figure 1, the displaced water can be divided into two parts: purple areas (water displaced by the cylinders) and yellow areas (water displaced by the meniscus). Previous studies have demonstrated that the weight of water displaced by the meniscus is equal to the vertical component of the surface tension force, which is directed upward if the meniscus is depressed and downward if elevated.¹² Here we assume that all cylinders have the same radius (r) and static contact angel (θ_c). By using the geometric relations shown in Figure 1, the angles between the tangent lines at the three phase contact lines and the horizontal line (φ_T1, φ_T2), and the distances from the centers of cylinders to the free water surface (h_O1, h_O2), the vertical force can be calculated by:

where ρ is the density of water, g is the acceleration due to gravity, l is the length of cylinder, β₁ = φ_T1 − θ_c + 180° and β₂ = φ_T2 − θ_c + 180° are the submerged angles, and σ is the surface tension coefficient of air-water interface. The first term on the right-hand side of Eq. (1) represents the weight of water displaced by the cylinders and the second term denotes the vertical component of surface tension force. The curved air-water interface is described by Young-Laplace equation,

where h(x) denotes the profile of the air-water interface. The numerical solutions of Eq. (2) can be obtained with MATLAB assuming a set of boundary conditions at the three phase contact lines:

where d represents the cylinder center distance. Figure 2 illustrates the results of this numerical method which provide the changes in the curved air-water interface as the cylinders gradually press the water surface. In this case, r, d and α are set to 0.25 mm, 5 mm and 30°, respectively, and θ_c = 155.8° (the characteristic angle of super-hydrophobic copper wires we previously developed for fabricating a water strider robot¹¹). Here, the inclined angle, α, is defined as the angle between the line of the cylinder center and horizontal, as shown in Figure 1. The vertical forces obtained by these two half cylinders per unit length are displayed in Figure 2(f).

Although several research papers have proposed criteria for the sinking point where a single cylinder penetrates water surface,^11,14,19 it has been proven more complicated to identify the sinking point of a cylinder in multi-cylinder systems, especially for those composed of numerous cylinders with different inclined angles. In these more complex systems, the sinking points can only be solved if all the relevant details for each cylinder are available. The definitive sinking point is not discussed in this article.

When cylinder center distance decreases to a certain extent, individual meniscuses begin to merge and integrate with each other, which in turn induce a disturbance in the vertical force acting on the cylinders. For simplicity, we assume that the cylinders are deposited on the same level (h_O1 = h_O2 = h_O), (i.e., the inclined angle equals 0°). From Eq. (1)–(3), both the lift force as the cylinders press water surface and the drag force when they detach from water can be determined by considering a range of cylinder center distances (1 to 10 mm) and for the case of d = ∞, in which the cylinder center distance has no influence on the vertical force. That is, the case of d = ∞ equals a single-cylinder system. The results of this exercise are shown in Figure 3. The results corresponding to d = 10 mm are fairly similar to those corresponding to d = ∞, indicating that the effect of cylinder center distance becomes negligible at a certain point. Although the lift force increases with the cylinder center distance, the maximum lift force and relevant h_O both increase as cylinder center distance decreases, suggesting that cylinders with a smaller center distance possess a larger load capacity on water. As shown in Figure 3, the vertical force above free water surface (i.e., h_O < 0) is not a strong function of cylinder center distance, indicating that the cylinder center distance does not have a significant effect on the drag force during the detachment process.

In the case of water strider robots, when we consider that the value of surface tension force is proportional to the length of the three phase contact line, increasing the overall length of the supporting legs is an effective approach for improving load capacity. Given limited overall size, a multi-leg mechanism composed of several cylinders fixed horizontally and in line with an assigned center distance (Figure 4(a)), is widely used in the design of water strider robots, including such as STIDER,⁶ Water dancer II,¹⁰ and Screw-type water strider robot¹¹). Sitti et al.⁵ previously claimed the distance between the neighboring legs of water strider robots should be large enough to maximize the lift force, which is inconsistent with the above conclusion that cylinders with a larger center distance will have a smaller load capacity. One reasonable explanation for this contradiction is that the supporting legs of current water strider robots are usually fixed and deposited on the same level (Figure 4(a)), and when the two outermost legs gradually press on the water surface, they are likely to sink into the water before the maximum lift force is achieved.

As illustrated in Figure 1, the cylinders are often fixed or deposited on different levels intentionally or accidentally (i.e., the inclined angle α ≠ 0°). In order to correlate the inclined angle with the lift force, we explored three different angles (α = 0°, 15° and 30°). The variation in lift force as the cylinders gradually press on the water surface is depicted in Figure 5. Larger inclined angles exert an adverse effect on the lift force. For instance, at h_O2 = 3 mm, as the inclined angle α increases from 0° to 30°, the lift force F_V is reduced from 0.122 N to 0.064 N. We also observed that with a larger inclined angle, the lower cylinder is able to fall deeper below water surface without sinking.

The above analysis provides the following insights regarding the design of water strider robots: if the two outermost supporting legs shown in Figure 4(a) are arranged with an appropriate inclined angle upwards (Figure 4(b)), the inner legs can fall deeper without penetrating the water surface, thus providing improved load capability and stability on the water surface.

The superhydrophobicity, which is characterized by a high static contact angle and low roll-off angle originating from the hierarchical micro/nanoscale structures, plays an important role in the ability of water striders to float and walk on the water surface.^2,29 In order to evaluate how wettability, or static contact angle, affects the static interactions between water and cylinders with a limited center distance, we investigated both the maximum lift force and the maximum drag force, as well as relevant h_O, with the static contact angle ranging from 30° to 180°. The maximum lift force in the pressing process and the maximum drag force in the detaching process are shown in Figures 6(a) and 6(b), respectively.

As shown in Figure 6(a), both the maximum lift force and relevant h_O quickly increase as static contact angle increases for θ_c < 90°. There was little change when θ_c > 90°, indicating that increasing the static contact angle does not significantly improve the load capacity of hydrophobic surfaces. The results shown in Figure 6(b) suggest that larger static contact angles are associated with smaller maximum drag forces, and that the value of h_O which corresponds to the maximum drag force decreases with increasing static contact angle. This further suggests that a large static contact angle can effectively abate drag force and energy consumption (proportional to the integral of F_V times h_O over time) required for the cylinders to leave the water surface.

Although the role of hydrophobicity in the load capacity is limited, for some aquatic insects and water strider robots, a larger contact angle can improve their mobility on the water surface and reduce their energy consumption. It is clear that the hydrophobicity of the legs is an important design consideration in the fabrication of water strider robots.

It is well known that cylinders with a larger radius can displace a larger volume of water resulting in a larger lift force. In order to better understand the role of cylinder radius, we investigated the effect of radius on the maximum lift forces and the relevant proportion of surface tension force (f_σ), with the radius varying from 0.1 to 0.5 mm (Figure 7).

The maximum lift force increases nearly linearly with radius. However, increasing the radius will unavoidably result in an increase in the overall size and weight of cylinders, which may in turn cause the carrying capacity to decline, especially for cylinders with a greater density than that of water. In addition, increasing the radius can also increase the contact area between cylinders and water, resulting in increased resistance during movement over the water surface. One of the critical design requirements of water strider robots is that they should float on the water mainly by surface tension, providing an advantage over traditional aquatic devices and vehicles, such as boats, which float on water by displacing a large volume of water (i.e., the buoyant force). As shown in Figure 7, the proportion of surface tension force which corresponds to the maximum lift force varies inversely with the radius, indicating that cylinders with a larger radius may not be suitable for surface tension-dominated locomotion.⁵ 

A quantitative study was performed to investigate the underlying mechanism of the vertical force acting on spindly cylinders coming into contact with water surface by considering the effects of cylinder center distance, inclined angle, static contact angle, and radius. Results indicated that cylinder center distance does not have a significant effect on the drag force, but positively affects the maximum lift force. However, cylinders with a smaller center distance must fall deeper to achieve the maximum lift force. The effect of cylinder center distance becomes negligible when the distance reaches a certain point. While the inclined angle, considered for the first time in the current study, was shown to adversely affect the lift force, certain inclined angles may allow the lower cylinder to fall deeper below water surface without sinking. Cylinders with smaller center distances and larger inclined angles can therefore obtain a larger maximum lift force and load capacity if fixed and deposited properly. The static contact angle was shown to play a limited role in increasing the lift force and load capacity at angles greater than 90°, while hydrophobicity can effectively reduce the drag force and energy consumption when cylinders detach from water surface. Although a larger cylinder radius has a positive effect on the maximum lift force, increasing the cylinder radius can also increase the overall size and weight of aquatic devices and robots and decrease the proportion of surface tension force, which are undesirable for surface tension dominated locomotion. Findings from this study allow for a better understanding of the underlying mechanisms of static interactions between water and partly submerged spindly cylinders, and provide suggestions on how to improve the design of water strider robots. Considering that the legs of water strider robots are actually often not parallel to each other due to external factors such as water waves and winds, it is of essential to discuss cases in which cylinders are posited on a water surface with certain included angles. Our further studies will be focused on investigating the effects of the included angle on the interactions between water and partly submerged spindly cylinders and developing relevant numerical methods.

We thank Bernard Gerstman and an anonymous reviewer for their helpful comments on the manuscript. This work was financially supported by a self-planned task of The State Key Laboratory of Robotics and System of Harbin Institute of Technology (SKLRS200901C) and Natural Science Foundation of China (NSFC, Grant 50803013, and 51305098).