# Imbibition by polygonal spreading on microdecorated surfaces

By Scott Tsai

## Contents

## Overview

Courbin et al use micropatterned surfaces to create different final shapes for spreading droplets. They show that they can control the final shape by changing the liquid. They describe a model for the velocity of the contact line, and they also show that the radii of the spreading drops scale with Washburn's scaling.

**Wetting On Patterened Surfaces**

By varying the liquid used and keeping the size of the drops the same, the authors were able to see seven different macroscopic scenarios. As Fig.2 shows, the seven scenarios are (i) a circle around a reservoir, (ii) an octagon around a reservoir, (iii) a square around a reservoir, (iv) a square, (v) an octagon, (vi) a rounded octagon, and (vii) a circle.

**Motion Of The Contact Line**

As will be shown, the contact line dynamics of the system are what determines the final shape of the drop.

To see the movement of the contact line, the authors used bright-field microscopy coupled with high-speed imaging. When the contact line reaches a row of posts, a single post is first wet, then a lateral propagation of wetting of the posts in the lateral direction occurs (Fig.3).

The spreading rate in the lateral direction is much higher than the spreading rate in the front of the contact line. The dynamics of spreading are largely controlled by the speed at with the front of the contact line wets the next row of posts. Since the inter-post distance is shorter along the diagonal and longer in the lattice axis direction, it is expected that there will be slower motion in the lattice axis direction.

By decreasing the wettability of the liquid, the authors were able to generate shapes from (i) to (vii), starting with the lowest wettability to the highest wettability. In (i), the low wettability causes the contact lines to pin, where in (ii) and (iii), the contacts lines have room to rearrange such that an octagon and a square can be formed while the reservoir of liquid remains circular. By increasing wettability more, unpinning starts to occur along the diagonals and not the lattice axes, so the square is formed (iv). Increasing the wettability some more unpins the contact line more and results in shapes (v), (vi), and (vii).

V, the velocity of the contact line, is controlled by a balance of capillary driving (<math>\gamma{\cos}{\theta}/{\mathrm d}</math>) and viscous resistance (<math>\eta{\mathrm LV}/{\mathrm d^2}</math>), where <math>{\mathrm L}</math> is the distance to the reservoir (Fig. 4). The velocity V is obtained and has the form <math>\mathrm V {\sim} (\mathrm d\gamma/\mathrm L\eta)\cos\theta</math>.

The authors also show in Fig.4 that they obtain the classical Washburn scaling <math>L \propto (\mathrm d\gamma/\mathrm L\eta)</math>

In Fig.5, the authors show how the seven resultant shapes corrospond to normalized values of diagonal or axis velocities and show that the shapes are deteremined by the ratio of the two velocities.

By Sung Hoon Kang

**Imbibition by polygonal spreading on microdecorated surfaces**

Reference: LAURENT COURBIN, ETIENNE DENIEUL, EMILIE DRESSAIRE, MARCUS ROPER, ARMAND AJDARI AND HOWARD A. STONE, Nature Mat. 6, 661-664 (2007).

## Soft matter keywords

wetting, pinning, hydrophilic, equilibrium contact angle, imbibition

## Abstract from the original paper

Micropatterned surfaces have been studied extensively as model systems to understand influences of topographic1 or chemical heterogeneities on wetting phenomena. Such surfaces yield specific wetting or hydrodynamic effects, for example, ultrahydrophobic surfaces, ‘fakir’ droplets, tunable electrowetting6, slip in the presence of surface heterogeneities and so on. In addition, chemical patterns allow control of the locus, size and shape of droplets by pinning the contact lines at predetermined locations. Applications include the design of ‘self-cleaning’ surfaces and hydrophilic spots to automate the deposition of probes on DNA chips. Here, we discuss wetting on topographically patterned but chemically homogeneous surfaces and demonstrate mechanisms of shape selection during imbibition of the texture. We obtain different deterministic final shapes of the spreading droplets, including octagons, squares, hexagons and circles. The shape selection depends on the topographic features and the liquid through its equilibrium contact angle. Considerations of the dynamics provide a ‘shape’ diagram that summarizes our observations and suggest rules for a designer’s tool box.

## Soft matter example

(not done yet)

When a surface structure is introduced in a material, the roughness of the material changes wetting behavior of the surface. Wenzel derived an expression for a contact angle with the microstructured surface as follows:

cosθ_{W}= rcosθ

(θ: contact angle of a surface without microstructures, θ_{W}: contact angle of a surface with microstructures, r is the ratio of the actual area to the projected area [1])

As a result, a hydrophobic surface (a surface with a contact angle larger than 90°) becomes more hydrophobic, whereas a hydrophilic surface (a surface with a contact angle smaller than 90°) becomes more hydrophilic [2].

In this paper, the authors enhanced the wettability by using a microtextured surface made of polydimethylsiloxane (PDMS) by soft-lithography technique based on the argument described above [3]. They studied the case when 0< θ_{eq} < θ_{c} < π/2 where θ_{eq} is the equilibrium contact angle and θ_{c} is the threshold which I couldn't understand the exact meaning. In this case, the liquid can penetrate the surface by partial imbibition[1] by the rough surface

## References

[1] R. N. Wenzel, "Resistance of Solid Surfaces to Wetting by Water". Ind. Eng. Chem. 28, 988-994, (1936).

[2] P.-G. de Gennes, F. Brochard-Wyart, D. Quéré, A. Reisinger, Capillarity and Wetting Phenomena (Springer, New York, NY, 2004).

[3] J. C. McDonald, G. M. Whitesides, Acc. Chem. Res. 35, 491-499 (2002).