# Difference between revisions of "Pearl drops"

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

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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), | 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> | + | 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) | ||

+ | 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) | ||

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

## Revision as of 01:53, 9 March 2009

By Sung Hoon Kang

Title: Pearl drops

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

## Contents

## 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

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

cosθ = (f_{sv}-f_{sl})/f_{lv}(1)

where f_{sv}, f_{sl}, f_{lv} 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^{*} 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_{sl}-f_{sv})dx+f_{lv}dx cosθ^{*}(2)

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

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

cosθ^{*}= r cos θ (3)

## 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).