Colloids and Surfaces A: Physicochem. Eng. Aspects 333 (2009) 12–18 Contents lists available at ScienceDirect Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa Investigation of the Neumann triangle for dodecane liquid lenses on water Robert David a,∗ , Stephanie M. Dobson a , Zahra Tavassoli a , M. Guadalupe Cabezas b , A. Wilhelm Neumann a a Department of Mechanical & Industrial Engineering, University of Toronto, 5 King’s College Road, Room MB56, Toronto, ON, Canada, M5S 3G8 b Department of Mechanical, Energetic and Material Engineering, University of Extremadura, Avda De Elvas s/n, E-06071 Badajoz, Spain article info Article history: Received 25 July 2008 Accepted 5 September 2008 Available online 16 September 2008 Keywords: Contact angle Neumann triangle Liquid lens Line tension Interfacial tension abstract Liquid lenses of dodecane resting on a water surface are imaged and analyzed computationally. The equi- librium condition for the three-phase line is a vector balance of three interfacial tensions known as the Neumann triangle. Measurement of the interfacial tensions, and two angles, allows the relation to be verified. Despite use of purified dodecane and the highest accuracy methods currently available at the millimetre length scale, a discrepancy with the Neumann triangle is found. The results, as well as literature data, are not consistent with interpretation as a thermodynamic line tension or as a pseudo-line tension. Alternative possibilities are briefly discussed. © 2008 Elsevier B.V. All rights reserved. 1. Introduction The balance of interfacial tensions at the three-phase line of a sessile liquid drop on a solid substrate is given by Young’s equation. Owing to the difficulty of measuring solid surface tensions, verifica- tion of Young’s equation is not straightforward [1]. This difficulty is removed if the solid substrate is replaced by a liquid one. The sessile drop now becomes a liquid lens, and Young’s equation is general- ized to a vector balance known as the Neumann triangle [2]. The experimental verification of the Neumann triangle is the subject of this contribution. As a thermodynamic equilibrium condition for a simple capillary system, the Neumann triangle is of fundamental interest; it is also of applied interest, for example in foams [3]. The Neumann triangle is derived by minimization of the free energy of the liquid lens system. The free energy consists of the interfacial energies (i.e., works of formation) of the three liquid–fluid interfaces. However, a complete accounting of the free energy must also include energy associated with the line phase [4]. If this energy (which we denote by ) is included, the triangle becomes a quadrilateral [5], as depicted in Fig. 1(a) for the system studied here. Recent experimental studies of the Neumann triangle have been focused on inferred measurement of the line tension vector. Two strategies have been used. In the first (used here), the three inter- ∗ Corresponding author. E-mail address: robertd@mie.utoronto.ca (R. David). facial tension magnitudes and the orientations of at least two of them are measured; if they do not close the Neumann triangle, the remaining distance can be interpreted as a missing fourth vector, the line tension (Fig. 1(b)). The second strategy is based on the fact that the length of the line tension vector scales inversely with the lens radius, while the lengths of the other three vectors are inde- pendent of the lens radius (Fig. 1(a)). Thus, a significant line tension would cause one or more angles of the quadrilateral to change with lens size; no such change would be observed for a negligible line tension. It is possible to quantify measurements of size-dependent lens angles as measurements of line tension. Individual experiments have focused on measuring line tension for lenses within a narrow range of sizes at some typical length scale. Studying lenses with diameters of a few millimetres, Chen et al. reported line tension of about -10 -6 J/m for lenses of dodecane [6,7] and higher alkanes [8] on water. At the sub-millimetre level, three studies using interferometry have been published. Dussaud and Vignes-Adler [9] examined lenses of octane on saline, reporting line tension of +10 -10 to 10 -9 J/m. Aveyard et al. [10] studied dode- cane lenses on water and reported  of about +10 -11 J/m. Takata et al. [11] examined hexadecane lenses on an aqueous solution of sur- factant, and calculated line tension in the 10 -11 J/m range, changing sign as a function of the surfactant concentration. At the microme- tre level, Stöckelhuber et al. [12] found , for a drop of water resting on a dodecane surface, to be of order 10 -8 J/m. Finally, Wallace and Schürch [13,14] studied drops of dibutylphthalate in a wide range of radii from 1 to 200 m, and reported line tension of about +10 -8 J/m. The lower liquid in their experiments was so dense that 0927-7757/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2008.09.018