Wicking through a confined micropillar array

Abstract

This study considers the spreading of a Newtonian and perfectly wetting liquid in a square array of cylindric micropillars confined between two plates. We show experimentally that the dynamics of the contact line follows a Washburn-like law which depends on the characteristics of the micropillar array (height, diameter and pitch). The presence of pillars can either enhance or slow down the motion of the contact line. A theoretical model based on capillary and viscous forces has been developed in order to rationalize our observations. Finally, the impact of pillars on the volumic flow rate of liquid which is pumped in the microchannel is inspected.

Appendix

Equation (12) allows determining the critical pillar density $\phi _c$ in order to have $D=D_0$ with $\phi _c \ne 0$. The critical density verifies the relation

$$\begin{aligned} \phi _c \, \sqrt{\frac{\phi _m}{\phi _c}} \, \left( \sqrt{\frac{\phi _m}{\phi _c}}-1 \right) ^2 = \frac{ \overline{h}^2}{2 \overline{h} - 1} \end{aligned}$$

(17)

which has a solution only if $\overline{h}>0.5$. Equation (17) is solved numerically as a function of $\overline{h}$ and the solution is indicated in Fig. 6 by a white solid line.

For $\overline{h}>0.5$, the diffusivity ratio $D/D_0$ reaches a maximal value for a pillar density $\phi _{\mathrm{max}}$ which can be calculated by deriving Eq. (12) relatively to $\phi$ keeping $\overline{h}$ constant. Such a calculation is performed numerically, and the solution is indicated in Fig. 6 by a white dashed line. Finally, the pillar aspect ratio $\overline{h}_{\mathrm{max}}$ which maximizes the diffusivity ratio $D/D_0$ for a given pillar density verifies

$$\begin{aligned} \begin{array}{l} \phi \left( \sqrt{\frac{\phi _m}{\phi }} \, \left( \sqrt{\frac{\phi _m}{\phi }}-1 \right) ^2 + \overline{h}_{\mathrm{max}}^2 \right) = \overline{h}_{\mathrm{max}} \left( 1+ ( 2 \overline{h}_{\mathrm{max}} - 1) \phi \right) \end{array} \end{aligned}$$

(18)

The solution of Eq. (18) is presented by a white dotted line in Fig. 6.