Abstract
In the presence of gravity or other external fields, liquid surface curvature deviates from a spherical shape and the surface configuration can be found by numerical integration of the Young–Laplace equation and the typical initial point for integration is the apex of the interface. The meniscus shape in large Bond number systems, which have the central portion of the interface flattened, cannot be determined with the apex as the initial point for integration. Here we find the depth of capillary menisci by considering an initial point for integration to be at the three-phase-contact-line (TPCL) and evaluate the curvature at the TPCL by free energy analysis and inspect the effect of different parameters on the interface shape. A new parameter—which is the deviation of equilibrium curvature at the TPCL from the spherical shape (SR)—is introduced and inspected and it was found that at a Bond number of 13 the maximum deviation, approximately 0.8 of spherical curvature, takes place while for large enough Bond numbers the curvature at the three-phase contact line is near the spherical shape (0.95 < SR < 1). A potential application of this approach is to measure the capillary rise at the TPCL to find the surface tension in high Bond number systems such as those with low surface/interfacial tensions.
Highlights
Finding the equilibrium shape of fluid interfaces has been frequently studied due to its importance in surface science1–17
In order to integrate the Young–Laplace equation numerically, we have changed the typical starting point for the integration from the apex of the interface as is used in typical interface shape calculations to be instead at the three-phase contact line, since the apex point cannot be used as the starting point for integration to predict the meniscus shape for systems with larger Bond numbers where the meniscus has some central portion flattened
It was found that the curvature at the three-phase contact line would be the highest acceptable curvature and this value correspond to the smallest system free energy
Summary
In the presence of gravity or other external fields, liquid surface curvature deviates from a spherical shape and the surface configuration can be found by numerical integration of the Young–Laplace equation and the typical initial point for integration is the apex of the interface. We find the depth of capillary menisci by considering an initial point for integration to be at the three-phase-contact-line (TPCL) and evaluate the curvature at the TPCL by free energy analysis and inspect the effect of different parameters on the interface shape. Lubarda and Talke assumed ellipsoidal shapes for sessile drops in the presence of gravity and did the minimization of free energy to get the equilibrium height and droplet spreading They compared their ellipsoidal results with the numerical solution and found that the ellipsoidal shape assumption is accurate for droplets with contact angles smaller than 120° and for droplets with sizes on the order of the capillary length. If we assume the liquid phase to be an incompressible liquid and the gas phase to be an ideal gas, we will have:
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