Interfaces, Capillarity, and Microdrops - Microfluidics for Biotechnology - page 123

Figure 3.71 (a) Droplet penetrating between the pillars. (b) Droplet sitting on top of the pillars (fakir
effect). q W is the Wenzel contact angle and q C is the Cassie contact angle.

tween a Wenzel droplet and a Cassie droplet? All these questions are discussed next.
Let us first investigate the case of a hydrophobic substrate.

Hydrophobic Substrate
The contact angle of a sessile drop sitting on microfabricated pillars has been the
subject of many investigations recently. As we have seen previously, Young's law
defines the contact angle on the substrate material

γ γ -

SG SL

(3.89)

cos θ =

γ

LG

If the drop penetrates between the pillars, one can write the Wenzel angle as

θ = (3.90)
where q W is the Wenzel contact angle and r is the roughness of the surface. If the
drop stays on top of the pillars, one can write the Cassie law under the form

cos

W r

cos

cos

θ = f

cos

θ +

(1

- f

) cos

θ (3.91)

C 0

where q C is the Cassie contact angle, q 0 is the contact angle with the layer of air,
and f is the ratio of the contact surface (top of the pillars) to the total horizontal
surface. If the pillars are not too far from each other, the value of q 0 is roughly q 0
= π (Figure 3.72).
Equation (3.91) then simplifies to

= - + + ( ) (3.92)
The two relations (3.90) and (3.91) can be plotted in a [cos q Young cos q real ] dia-
gram (Figure 3.73) [39-44]. In such a representation, the two equations correspond

cos 1 1 cos

θ f θ

C

Figure 3.72 Sketch of a Cassie drop (fakir effect). The interface between the pillars is roughly
horizontal.