Comment Cite This: Langmuir 2018, 34, 2581−2582
pubs.acs.org/Langmuir
Comment on “Millimeter-Sized Hole Damming” ABSTRACT: In a recent article, Ng and co-workers (Langmuir 2017, 33, 13892−13898; doi: 10.1021/acs.langmuir.7b03290) demonstrated the ability of containers with a small hole in their bottom to resist the drainage of water. They used Laplace’s capillary rise equation to explain their results, which grossly underestimated the onset of drainage. In this comment, an alternative analysis is offered. Equations are derived specifically for the drainage of liquid from a small hole in a horizontal surface. This approach yielded reasonably accurate predictions for Ng’s experimental data.
N
g and co-workers1 recently showed that small holes in horizontal surfaces can be used as passive valves to control flow. In their experiments, containers were constructed by drilling holes with diameters (D) of 1 to 4 mm in copper plates, treating the plates to make them relatively hydrophilic or superhydrophobic and then attaching the plates to plastic cylinders with silicone adhesive. Water was added through the open top of the containers. Subsequently, the height (h) of the water rose (Figure 1). Initially, capillary forces prevented
Consider the container depicted in Figure 1. The top of the container is open and thus exposed to atmospheric pressure. The bottom has a small round hole of diameter D. As liquid with surface tension γ and density ρ is added to the container, its height (h) rises. At the same time, liquid moves through the hole with an advancing contact angle of θa. Liquid reaches the bottom of the hole, where it is pinned.3−5 Capillary forces initially prevent flow or drainage. As liquid rises higher and hydrostatic pressure increases, the apparent contact angle (θ) of the bulging meniscus relative to the exterior horizontal wall also increases. The radius of curvature (R) of the protruding liquid interface creates a Laplace pressure (ΔpL), which is directed inward and upward,6
ΔpL =
2γ R
(1)
If the shape of the bulging liquid can be approximated as a section of a sphere, then its radius of curvature and the hole diameter can be related to the contact angle through the sine function,
sin θ =
D/2 R
(2)
The hydrostatic pressure (Δph) in the bulk liquid depends upon the height of the liquid above the bottom of the hole (h), the liquid density (ρ), and the acceleration due to gravity (g = 9.81 m/s2), Δph = ρgh
Figure 1. Depiction of a liquid in a container. (a) The top of the container is open. The bottom has a round hole of diameter D. The hole prevents drainage of liquid of height h. (b−d) Magnified views of the progression of liquid through the hole of a hydrophilic container, where the advancing contact angle (θa) is
