Three-dimensional menisci in polygonal capillaries

Abstract

The shapes of gravity-free, three-dimensional menisci are computed from the augmented Young-Laplace equation. Incorporation of disjoining thin-film forces in the Young-Laplace relation eliminates the contact line, thereby eliminating the free boundary from the problem. To calculate a meniscus with finite contact angles, the conjoining/disjoining pressure isotherm must also contain an attractive, sharply varying, spike function. The width of this function, w, reflects the range of the thin-film forces. In the limit of w approaching zero, a solution of the Young-Laplace equation is recovered. The proposed calculation method is demonstrated for menisci in two different types of capillaries. In the first case, the capillary is regular-polygonal in cross section with either 3, 4, or 6 sides and with contact angles Φ ranging from 0 to 45°. In the second case, the capillary is rectangular in section with aspect ratios ranging from 1.2 to 5 and with Φ = 0°, 15°, or 30°. Gas-liquid menisci inside a square glass capillary of 0.5 mm inscribed radius are measured optically for air bubbles immersed in a solution of di- n-butyl phthalate and mineral oil. This liquid mixture exhibits a zero contact angle with the wall and matches the refractive index of the glass capillary, permitting precise visual location of the interface. Excellent agreement is found with the numerical results which further demonstrates that the limiting process of the proposed method is valid. Because it avoids the issue of locating the contact line, solution of the augmented Young-Laplace equation is a simple and powerful method for the calculation of three-dimensional menisci.