A non-gradient based algorithm for the determination of surface tension from a pendant drop: Application to low Bond number drop shapes

The pendant drop method is one of the most widely used techniques to measure the surface tension between gas–liquid and liquid–liquid interfaces. The method consists of fitting the Young–Laplace equation to the digitized shape of a drop suspended from the end of a capillary tube. The first use of digital computers to solve this problem utilized nonlinear least squares fitting and since then numerous subroutines and algorithms have been reported for improving efficiency and accuracy. However, current algorithms which rely on gradient based methods have difficulty converging for almost spherical drop shapes (i.e. low Bond numbers). We present a non-gradient based algorithm based on the Nelder–Mead simplex method to solve the least squares problem. The main advantage of using a non-gradient based fitting routine is that it is robust against poor initial guesses and works for almost spherical bubble shapes. We have tested the algorithm against theoretical and experimental drop shapes to demonstrate both the efficiency and the accuracy of the fitting routine for a wide range of Bond numbers. Our study shows that this algorithm allows for surface tension measurements corresponding to Bond numbers previously shown to be ill suited for pendant drop measurements.

Graphical abstractIn a pendant drop experiment, a non-gradient based algorithm can accurately fit small Bond number shapes. The images show decreasing Bond number for an air bubble in water. The lower limit for the new algorithm is Bond number of 0.01. This limit is a factor of ten smaller than the current limit.Figure optionsDownload full-size imageDownload as PowerPoint slide