View source for Pearl drops

By Sung Hoon Kang

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Title: Pearl drops

Reference: J. Bico, C. Marzolin and D. Quere, Europhys. Lett. 47, 220-226 (1999).

==Soft matter keywords==

hydrophobic, surface roughness, contact angle, Young's equation, surface tension, Wenzel's equation

==Abstract from the original paper==

If deposited on a hydrophobic rough substrate, a small drop of water can look like a pearl, with a contact angle close to 180. We examine the conditions for observing such a phenomenon and show practical achievements where the contact angle can be predicted and thus quantitatively tuned by the design of the surface microstructure.

==Soft matter example==

(not finished)

For a flat uniform solid, the contact angle of a drop is given by Young's equation:

                cosθ = (f<sub>sv</sub>-f<sub>sl</sub>)/f<sub>lv</sub>               (1)

where f<sub>sv</sub>, f<sub>sl</sub>, f<sub>lv</sub> are surface tensions for solid/vapor, solid/liquid and liquid/vapor, respectively. 


The contact angle can be changed by introducing textured surface. In nature, the contact angles of water drops on some leaves such as lotus or lily-water are around 160° by having rough surfaces [1]. Synthetically, it was found that a hydrophobic surface with surface texture had a contact angle around 170° for water. In this paper, the authors analyzed parameters affecting the contact angle and discussed data obtained from model surfaces.


The effect of the surface roughness on contact angle was first formulated by Wenzel [2]. If we consider a small displacement dx of the contact line parallel to the surface as shown in Fig. 1, the new contact angle θ<sup>*</sup> can be calculated using the following approach. In this case, the change in the surface energy (per unit length of the contact line), 

                  dF = r(f<sub>sl</sub>-f<sub>sv</sub>)dx+f<sub>lv</sub>dx cosθ<sup>*</sup>     (2)

(r: the solid roughness, defined as the ratio between the real surface and the projected one)

[[Image:Bico et al Fig 1.jpg|thumb|center|600px| '''Fig. 1''' Infinitesimal spreading of a liquid wedge on a rough surface.]]

The equilibrium contact angle of the rough surface can be calculated by the minimum of F, which gives Wenzel's equation:

                 cosθ<sup>*</sup> = r cos θ                   (3)

(θ: Young's angle given from equation (1)) This model shows that the surface roughness amplifies the wetting or dewetting behavior because r > 1.


When θ > 90°, the surface energy of a dry solid is lower than the surface energy of the wet solid (f<sub>sv</sub><f<sub>sl</sub>). So, the contact line does not follow the solid surface and the drop styas on a composite surface with a patchwork of solid and air as shown by Barthlott and Neinhuis [1]. Then, the contact angle changes by the presence of the air pockets. We can consider this case by using the model surface shown in Fig. 2. For a crenellated surface as Fig. 2a, the change in surface energy by displacing the contact line dx parallel to the surface is equal to:

                 dF = φ<sub>s</sub>(f<sub>sl</sub>-f<sub>sv</sub>)dx+(1-φ<sub>s</sub>)f<sub>lv</sub>dx + f<sub>lv</sub>dx cosθ<sup>*</sup>                    (4)

where φ<sub>s</sub> is the solid fraction of the surface. The thin lines in Fig. 2 a) indicates liquid/vapor interfaces, which occupies a surface fraction of 1-φ<sub>s</sub> below the liquid drop. All these interfaces are pinned at the corners of the crenellations. From the equation (4), the equilibrium contact angle can be calculated by having minimum F:

                cosθ<sup>*</sup> = -1 + φ<sub>s</sub>(cos θ + 1)                (5)

The previous derivation assumed crenellation for modeling, but in general the fraction φ<sub>s</sub> is a function of the contact angle θ. If we assume that the solid surface has a hemispherical bump as shown in Fig. 2 b), the liquid/vapor interafaces are positioned where Young's condition is satisfied. Then, similar calculation gives the following relation:

               cosθ<sup>*</sup> = -1 + φ<sub>B</sub>(cos θ + 1)<sup>2</sup>                (6)

where φ<sub>B</sub> is the ratio of the surfaces of the spike bases over the total solid surface.

[[Image:Bico et al Fig 2.jpg|thumb|center|600px| '''Fig. 2''' Liquid deposited on a model surface with holes (a) crenellated surface; b) hemispherical bumps): for contact angles larger than 90°, air is trapped below the liquid, inducing a composite
interface between the solid and the drop.]]

Fig. 3 shows the plot of the equations (5) and (6) (dotted line). In both cases, the contact angle has a discontinuity, which is quite different from Wenzel equation. As sson as the contact angle is above 90°, air trapping occurs below the drop. These results showed good agreement with experimental data of the Kao group [3], which are shown in Fig. 4 for samples with different solid roughness.

[[Image:Bico et al Fig 3.jpg|thumb|center|600px| '''Fig. 3''' Cosine of the effective contact angle θ<sup>*</sup> of a drop on a rough model surface as a function of the cosine of the Young contact angle θ. r is the solid roughness and φ<sub>s</sub> is the surface solid fraction. On the hydrophobic side (cos θ < 0), the full line is eq. (5) and the dotted one is eq. (6).]]

[[Image:Bico et al Fig 4.jpg|thumb|center|600px| '''Fig. 4''' Experimental results of the Kao group (from [3]). The cosine of the effective contact angle θ<sup>*</sup> of a water drop was measured as a function of the cosine of the Young angle θ (determined on a flat surface of the same material), for different solid roughnesses (increasing from left to right). The
thin straight line is eq. (5) with φ<sub>s</sub> as an adjustable parameter.]]

==Reference==

1. C. Neinhuis and W. Barthlott, ''Planta'' 202, 1-8 (1997).

2. R. N. Wenzel, ''Ind. Eng. Chem.'' 28, 988-994 (1936); ''J. Phys. Colloid Chem.'' 53, 1466-1467 (1949).