Buoyancy Force

Water-walking ability is one of the most intriguing applications for a superhydrophobic surface. The fundamental mechanism and calculation are addressed here and they will be pushed further for investigating and improving surface tension enhanced buoyancy in our work.

More than two thousand years ago, a Greek mathematician and physicist, Archimedes of Syracuse, discovered that weight of an object immersed in water is less than that in water. His findings are stated in his book On Floating Bodies [38]. The well-known Archimedes’

principles include:

Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced.

If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the difference between its weight and the weight of the fluid displaced.

A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced.

Usually, we can simply write these propositions by:

(eqn. 2-9)

where B is buoyancy and ρl, V, and g are density of object, volume of object in water, and acceleration of gravity respectively.

However, from our daily experiences, the Archimedes’ principle seems to be violated for a floating body denser than water. Therefore we must take surface tension into account. It keeps surfaces from wetting and helps water form meniscus shape around an object, resulting in increased buoyancy from curvature force, which is important for water-walking insects by keeping them totally afloat without piercing water surface.

Total buoyant force can be divided into two parts, “surface tension force (F_T)” and

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“pressure force (FP)”. J. B. Keller in 1998 showed that the vertical component of the surface tension force on a body partly submerged in a liquid equals the weight of liquid displaced by the meniscus; the vertical component of the pressure force on the body equals the weight of liquid which would fill the volume bounded by the wetted surface of the body [9]. The illustration is in Figure 2-17. The surface tension force originates from the pressure difference of curved fluid interface, known as Laplace pressure after P. S. Laplace [39]. When the vapor-liquid interface is curved, there exists a pressure difference due to surface tension and its value equals surface tension times mean curvature. For example, a soap bubble of radius r has a total surface free energy of 4πr²γ and, if the radius was to decrease by dr, then the change in surface free energy would be 8πrγdr. Since shrinking decreases the surface energy, the tendency to do so must be balanced by a pressure difference (∆P) across the bubble film such that the work against this pressure difference ∆P 4πr²dr is just equal to the reduction in surface free energy [13].

(eqn. 2-10)

(eqn. 2-11)

Judging from the above expression, one can reach a conclusion that the smaller the bubble, the greater the pressure of air inside relative to that outside. Nonetheless, it is necessary to invoke two radii of curvature to describe a curved surface. For a small section of curved surface shown in Figure 2-16, two radii of curvature, R₁ and R₂, are indicated. If the surface is displaced by a small distance outward, the change in area (A) will be

( )( ) (eqn. 2-12) The work done in forming this additional amount of surface is

( ) (eqn. 2-13)

There will be a pressure difference ∆P across the interface; it acts on the area xy and through a distance dz. The corresponding work is

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(eqn. 2-14)

From geometrical aspect, it is known that

(eqn. 2-15)

The two work terms must be equal, and together with substituting expression of dx and dy, it can be deduced that

( ) (eqn. 2-16)

It is apparent that equation 2-11 is just a special case of equation 2-16 for spherical shape where R1 equals R2. Moreover, for a plane surface, both radii are infinite and ∆P is therefore zero, i.e.

no pressure difference across the interface.

Consequently, as long as we know the two radii of curvature, we are able to calculate the Laplace pressure, and thus the surface tension force. This is valid only for a small surface. In order to calculate the total Laplace pressure over a larger area, we have to find a way to describe the average curvature analytically via classical Young-Laplace equation [13]:

( ) (eqn. 2-17)

The Laplace pressure at the meniscus is balanced with the hydraulic pressure, so

( ) (eqn. 2-18)

( ) (eqn. 2-19) where apostrophe stands for derivative with respect to the coordinate z, √ ( ) is the capillary length, ρl is the density of liquid, and γ is the surface tension of the liquid.

Now we can calculate the maximum buoyant force for a hydrophobic cylindrical rod floating on water as shown in Figure 2-18. θY is static contact angle and h is the depth of meniscus. It can be estimated by integrating the volume displaced by the meniscus and the volume displaced by the body itself. Before that, one needs to solve x in terms of z and θ, and

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this can be done with the help of Mathematica [40].

( ) √ (eqn. 2-20) ( ) √ (eqn. 2-21) √ √ ( ) (eqn. 2-22) The volume displaced by the wetted segment of the rod is

( ) (eqn. 2-23)

The volume of the dimple displaced by the meniscus is

∫ ( ) √ √ ( ) (eqn. 2-24)

Total buoyant force is the summation of V1 and V2. Subsequently, we should also determine when the rod will sink into water. There are two possible conditions of sinking. First, the total volume reaches its maximum, called volume criterion. Second, the nearest distance of the two menisci approaches zero, called distance criterion. It was shown by J. L. Liu [40] that volume should always be used to determine when a hydrophobic circular rod is expected to sink into the liquid.

These calculations will be modified to calculate the maximum buoyant force in our experiment later.

Figure 2-15: A sketch of meniscus forming around an object.

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Figure 2-16: An arbitrary curved surface expands by a small distance [13].

Figure 2-17: Vertical component of surface tension force (F_T) is indicated by the cross-hatched regions and is positive in (a) and negative in (b). The pressure force (FP) is the unhatched regions between body surface and the x axis [9].

Figure 2-18: A cylindrical rod floating on water subjected to a force F directed downward. h is the depth of meniscus, γ is the surface tension of liquid, θ_Y is the static contact angle on flat surface and R is the radius of the rod [40].

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