Journal of Colloid and Interface Science 333 (2009) 557–562 Contents lists available at ScienceDirect Journal of Colloid and Interface Science www.elsevier.com/locate/jcis A non-gradient based algorithm for the determination of surface tension from a pendant drop: Application to low Bond number drop shapes Nicolas J. Alvarez a , Lynn M. Walker a , Shelley L. Anna a,b,∗ a Department of Chemical Engineering, Center for Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA b Department of Mechanical Engineering, Center for Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA article info abstract Article history: Received 20 December 2008 Accepted 24 January 2009 Available online 12 February 2009 Keywords: Surface tension Pendant drop Nelder–Mead algorithm Bond number Interfacial tension Numerical optimization Drop shape technique 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.  2009 Elsevier Inc. All rights reserved. 1. Introduction When a fluid is suspended from a capillary and surrounded by another fluid such that gravity acts along the axis of the capillary to distend the nominally spherical interface, the shape of the in- terface depends on the surface tension, γ , the characteristic size of the bubble or drop, R 0 , and the density difference between the two fluids, ρ . The Bond number, given by β = ρ gR 2 0 /γ , (1) is a dimensionless group describing the relative magnitude of forces due to gravity and surface tension. If the density differ- ence between the fluids is known and the size can be measured, then the surface tension can be determined from a measurement of the interface shape. This method of measuring surface tension, first realized by Andreas and coworkers [1], has come to be known as the pendant drop method. The method was suggested earlier by Worthington [2,3] and Ferguson [4], but measurements of drop coordinates proved difficult at that time. Andreas et al. overcame these issues by reformulating the Young–Laplace equation in a new coordinate system. * Corresponding author at: Department of Mechanical Engineering, Center for Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA. E-mail address: sanna@andrew.cmu.edu (S.L. Anna). Using the formulation of Andreas et al., the Young–Laplace equation is integrated to obtain a theoretical drop shape, which is then compared with an experimental drop shape to determine the surface tension between the two fluids. Before the availabil- ity of digital computers, drop shapes were analyzed by examining the ratio of radii of the drop at different axial positions, whose values were tabulated along with corresponding surface tension values [5]. This analysis, known as the selected plane method, is still carried out today when rough estimates (i.e. within 1 mN/m) of surface tension are of interest. However, when accuracy is re- quired it is necessary to solve a nonlinear least-squares problem to fit a calculated drop shape to a measured drop shape. In addition, the selected plane method only works for drops that fall within a selected range of Bond numbers. Although the particulars of the nonlinear least-squares fitting algorithms found in the literature might differ, the general proce- dure for each method remains the same. For instance, an image is first recorded by a CCD camera and digitized. An edge detec- tion method is used to extract the shape of the drop interface. The coordinates of the interface are then used to calculate the error be- tween computed theoretical shapes and the measured shape. The error is computed via an objective function, defined as the shortest distance between an experimental point and a point on the cal- culated interface. The procedure is repeated until the theoretical shape corresponding to the minimum error is found. The param- 0021-9797/$ – see front matter  2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2009.01.074