Limiting conditions for applying the spherical section assumption in contact angle estimation

The shape of liquid drops on solid surfaces deviates from the spherical as tension decreases and gravity effects start affecting the drop shape. This paper attempts to define this deviation and estimates the dimensionless Eotvos number limits above which the deviation becomes "significant." The use of these limiting values can facilitate estimation of contact angle in the following manner. It is well known that the equilibrium contact angle made by a liquid drop on a solid surface can be estimated from measurements of two drop parameters. These parameters can be any two chosen from the drop volume, height, and wetted radius. In case the effect of gravity on the drop shape is negligible, simple algebraic relations derived from the spherical section assumption exist, from which the contact angle can be estimated. In systems where the "spherical section" assumption is invalid, the Laplace equation for the drop shape has been solved numerically with any two of the above parameters as the constraints, to obtain the contact angle. In this paper, Eotvos numbers at which the deviation of the drop profile from the spherical is significant enough to result in contact angle deviation of 1 degrees are estimated. The limiting values of Eotvos number, expressed as a function of the original contact angle made by the spherical profile, are obtained by solving the Laplace equation for the drop shape with the drop volume and wetted radius constraints for decreasing values of Interfacial tension. These limiting values are also estimated for different drop sizes and for cases where the drop phase is heavier (sessile) and lighter (buoyant) than the surrounding fluid. The independence of the Eotvos number estimates from the sign of the density difference as well as the drop size is shown. These Eotvos number limits can be used to check if the spherical section assumption, with the resulting simple algebraic relations, can be used for contact angle estimation and other shape-related analysis for a system.