# Imbibition by polygonal spreading on microdecorated surfaces

By Scott Tsai

## 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

Fig.1 Schematic of a micropatterened surface and a photograph showing the surface
Most studies on the wetting of patterened surfaces is concerned with increasing the hydrophobicity of the surface via increased roughness. In this study, the authors have focused on hydrophilic surfaces, where the increase in solid surface increases wettability. The paper presentated is concerened with the situation where the resultant contact angle is lower than the equilibrium contact angle of a drop sitting on a surface of the same material, but unpatterened. The authors use regularly patterened surfaces like the ones shown in Fig. 1.

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

Fig.2 Seven different macroscopic scenarios resulting from using different liquids

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).

Fig.3 Bright-field images of the motion of the contact line

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).

Fig.4 Log-log plot of the distance, L, versus time

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

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

Fig.5

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.