The surface tension of planar interfaces has been widely studied, both from a theoretical and experimental perspective [1]. The surface tension of the planar surface depends on the same state-variables that characterise the coexistence state between vapor and liquid. For curved surfaces, such as those of droplets in equilibrium with a surrounding vapor, the surface tension is also a function of droplet-size. This dependence can be expressed in terms of the total and Gaussian curvatures, and can for a single-component system be formulated to second order by the Helfrich expansion.
The main quantities of interest in the Helfrich expansion is the Tolman length [2], which represents the first-order curvature correction compared to that of a plane surface, and the rigidity constants, representing the second order corrections. For water, the curvature corrections of the surface tension can be used to correct the erroneous temperature dependence of the nucleation rates predicted by classical nucleation theory [3]. These corrections may thus be of crucial relevance, as nucleation is often ubiquitous in physical, chemical and biological processes. Many important applications, such as the formation of oil-water emulsions, are however inherently mixture phenomena, yet the curvature-dependence of surface tension for mixtures has been given little attention in the literature.
This work is a step toward a deeper understanding of interfacial phenomena for mixtures. We discuss a generalized Helfrich expansion for multicomponent systems, which is model-independent and of general validity. We also present a route to obtain the coefficients in the expansion using a specific model for the components’ density profile, namely the square gradient theory [4-6]. Results from the square gradient theory are presented.
References
[1] J. S Rowlinson, B. Widom. Molecular Theory of Capillarity. Mineola, N.Y: Dover Publications, 2003.
[2] R. C. Tolman. “The Effect of Droplet Size on Surface Tension” The Journal of Chemical Physics 1949, 17: 333-337.
[3] Ø. Wilhelmsen, D. Bedeaux, D. Reguera. Communication: Tolman length and rigidity constants of water and their role in nucleation. The Journal of Chemical Physics 2015, 142: 171103.
[4] J. D. van der Waals, “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density” The Journal of Statistical Physics 1979, 20: 200-244.
[5] Ø. Wilhelmsen, D. Bedeaux, S. Kjelstrup, D. Reguera. Thermodynamic stability of nanosized multicomponent bubbles/droplets: The square gradient theory and the capillary approach. The Journal of Chemical Physics 2014, 140: 024704.
[6] E. M. Blokhuis, D. Bedeaux. Van der Waals theory of curved surfaces. Molecular Physics 1993, 80:705-720.
