Gibbsian Thermodynamic Study of Capillary Meniscus Depth

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.