How to Compute the Contact Angle inside an Opaque Capillary Tube: A Universal Equation

Abstract

Capillarity is a ubiquitous and important phenomenon in nature. The Jurin's law establishes a relationship between the contact angle and the liquid rising height inside a capillary tube and is widely employed to evaluate the wettability of transparent tubes since 1717. However, it fails once encounters the opaque counterparts because of the invisible rising height. Here, an idea that correlates the contact angle with an intrinsic property of a matter, i.e., the mass is proposed. A universal equation is thus established to compute the contact angle using the liquid mass inside the capillary tube. The equation is systematically verified against the height measurement method adopting a series of quartz capillary tubes, where a good agreement is attained. An in situ approach of high sensitivity is hence derived and demonstrated to evaluate the capillary tubes regardless of the wettability, transparency, or materials. A theoretical basis is provided to assess the surface wettability inside capillary tubes in a more general way and thus a new avenue is opened to investigate the heat and mass transfer under spatial limitation with well‐defined surfaces for portable devices, flexible electronics, etc.

Superwetting Capillary Tubes: Surface Science under Confined Space

“…A universal equation was recently established to compute the contact angle inside capillary tubes by Zhao and Sun. 33 It correlates the contact angle with the liquid mass instead of the conventional height, as displayed in Figure 2e. The contact angle can be computed by where Δm is the liquid mass inside a capillary tube and h 0 is the immersion depth.…”

“…An analogue manner is possible to monitor the liquid meniscus dynamics as the tube vertically moves up and down, in order to determine the θ rec and θ adv for capillary tubes, as shown in Figure 2l. 33 It is applicable for transparent capillary tubes, whereas it is impossible for opaque counterparts because of the invisible three-phase contact line. In this regard, further studies could be carried out to develop a novel and general approach to estimate the hysteresis inside capillary tubes, such as by measuring the variation of other parameters (e.g., mass or force) of the liquid column with tube movement.…”

“…Schematic diagrams showing the measurement of contact angle hysteresis on a flat surface (k) or inside a capillary tube (l). Panels a–g and j–l are adapted with permission from ref . Copyright 2022 Wiley.…”

“…A universal equation was recently established to compute the contact angle inside capillary tubes by Zhao and Sun . It correlates the contact angle with the liquid mass instead of the conventional height, as displayed in Figure e.…”

“…(a) Experimental and theoretical capillary rising height as a function of the tube diameter. Adapted with permission from ref . Copyright 2022 Wiley.…”