Minimization of free energy is used to calculate the equilibrium verticalrise and mensicus shape of a liquid column between two closelyspaced, parallel planar surfaces that are inert and immobile. States of minimum free energy are found using standard variationalprinciples, which lead not only to an Euler-Lagrangedifferential equation for the drop shape, but also to the boundary conditionsat the three-phase junction where the liquid meniscus intersects thesolid walls. The analysis shows that the classical Young-DuprÈ equation for thethermodynamic contact angle is valid at the three-phase junction, asalready shown for sessile drops with or without the influence of agravitational field. When both solid walls are composed of the same material, the meniscus shapeis symmetric about the midpoint and the vertical rise of the liquid isin excellent agreement with the predictions of the classical Laplace-Youngequation, provided that the spacing between the walls is very narrow. When the walls have dissimilar surface properties, the meniscusgenerally assumes an asymmetric shape. The height of capillary rise depends on spacing between the walls andalso on the difference in contact anglesat the two surfaces. The capillary rise between dissimilar walls can be closelyapproximated by an equation for the height of the liquid column proposedby O'Brien, Craig, and Peyton , provided that the heightis taken to be the average elevation of the meniscus. When the contact angle at one wall isgreater than 90_, the meniscus can have an inflection point separatinga region of positive curvature from a region of negative curvature, theinflection point being pinned at zero height. However, this conditiononly arises when the spacing between the walls exceeds athreshold value that depends on the difference in contact angles.
Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
and Garboczi, E.
Capillary Rise between Planar Surfaces, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=861608
(Accessed December 6, 2023)