Capillary as a liquid diode

We present a theoretical investigation of liquid imbibition into capillaries with axially varying geometry. We show that using appropriate geometry and wettability, converging capillaries, in principle, can act as a liquid diode. We explain the underlying mechanics and conditions that make a converging capillary a liquid diode. Further, we describe a methodology to enhance the rate of liquid imbibition into the proposed diode. We mathematically show that creating a wettability gradient along the axial length of the capillary diode enhances the rate of liquid imbibition compared to a capillary diode with uniform wettability. Our mathematical model predicts that using a wettability gradient along the axial length of the capillary diode decreases the imbibition time by more than 30% compared to a capillary diode with uniform wettability.

Figure 1

Schematic illustrating the physical mechanism behind the workings of a converging capillary as a liquid diode. (a) A straight capillary with uniform contact angle of $\theta ={90}^{\circ }$ forming a fluid-liquid interface of zero curvature. (b) A converging capillary with uniform contact angle of $\theta ={90}^{\circ }$ and inclination angle $\varphi >0$. Due to the converging profile, the liquid imbibing from the left end of the capillary forms a concave interface with air, whereas the liquid imbibing from the right end of the capillary forms a convex interface with air.

Figure 2

Schematic showing a imbibition into a converging capillary with a wall inclination angle $\varphi$ and whose local solid-liquid contact angle $\theta \left(l\right)$ and radius of the capillary $h\left(l\right)$ is a function of the axial distance $l$ from the inlet.

Figure 3

Schematic illustrating the effect of wettability on the radius of curvature of a gas-liquid interface. For a capillary with a uniform cross section, the radius of curvature of the gas-liquid interface is smaller for a lower value of the contact angle, that is, $R_2<R_1$ for ${\theta }_2<{\theta }_1$. The small radius of curvature, in turn, leads to higher capillary suction.

Figure 4

Liquid imbibition into a diode capillary (converging capillary) with uniform wettability and a diode capillary with an axial contact angle variation. The contact angle of the inner surface of the diode capillary varies such that the cosine of the sum of the contact angle $\theta$ and inclination angle $\varphi$ vary linearly along the axial length of the capillary. These calculations have been performed for $\varphi =5^{\circ }, h_o=1\mathrm{mm}$, and water as the working fluid. For a diode capillary with uniform wettability, $\theta ={90}^{\circ }$. The inset shows a comparison of predictions of our mathematical model with the experimental results of Gorce et al. [6], wherein the radius $h_o=330 \mu \mathrm{m}, L=6$ mm, $\theta ={75}^{\circ }$, and glycerol is used as the working fluid. Using an axial wettability gradient in a capillary diode reduces the time of liquid imbibition by more than 30% compared to a capillary diode with uniform wettability.

Figure 5

Effect of the contact angle on liquid imbibition into converging capillaries with uniform wettability. These calculations have been performed for converging capillaries with $\varphi =5^{\circ }$ and water as the imbibing liquid. A converging capillary with $\theta ={90}^{\circ }$ is a diode, as $\delta >{90}^{\circ }$ for liquid entering from the narrow end. However, a converging capillary with $\theta ={70}^{\circ }$ is not a diode, as $\delta <{90}^{\circ }$ for liquid entering from the narrow end.