Capillary adsorption of droplets into a funnel-like structure

The penetration of liquid from an infinite reservoir into a capillary tube has been studied for more than a century. However, if the size of the droplet is comparable to the pore size, a mathematical description of the penetration criterion, which is influenced by the droplet size, the intrinsic wettability, and the opening angle of the capillary tube, remains a knotty issue. In this contribution, we theoretically and numerically address the penetration criterion of the droplets with limited volume into a funnel-like pore structure. Our results show that there is a critical contact angle below which a certain-sized droplet can penetrate a hydrophobic pore. This critical contact angle is closely related to the opening angle and droplet size. Remarkably, our predictions extend the limit of the maximum contact angle for the complete penetration of droplets into capillary tubes in the literature. The critical contact angle becomes invariant for a given droplet size when the opening angle is larger than a certain threshold.

Furthermore, we find that for a constant opening angle, the critical contact angle decreases as the droplet size increases. As the droplet volume tends to become infinity, the opening angle has almost no effect on the penetration, and the critical contact angle asymptotically approaches 90°, which is consistent with previous works. Our observations shed light on a specific mechanism for a precise control of droplets in filtration systems and microfluidic platforms.