Surface tension of droplets and Tolman lengths of real substances and mixtures from density functional theory.

The curvature dependence of interfacial properties has been discussed extensively over the last decades. After Tolman published his work on the effect of droplet size on surface tension, where he introduced the interfacial property now known as Tolman length, several studies were performed with varying results. In recent years, however, some consensus has been reached about the sign and magnitude of the Tolman length of simple model fluids. In this work, we re-examine Tolman's equation and how it relates the Tolman length to the surface tension and we apply non-local classical density functional theory (DFT) based on the perturbed chain statistical associating fluid theory (PC-SAFT) to characterize the curvature dependence of the surface tension of real fluids as well as mixtures. In order to obtain a simple expression for the surface tension, we use a first-order expansion of the Tolman length as a function of droplet radius R, as δ(R) = δ + δ/R, and subsequently expand Tolman's integral equation for the surface tension, whereby a second-order expansion is found to give excellent agreement with the DFT result. The radius-dependence of the surface tension of increasingly non-spherical substances is studied for n-alkanes, up to icosane. The infinite diameter Tolman length is approximately δ = -0.38 Å at low temperatures. For more strongly non-spherical substances and for temperatures approaching the critical point, however, the infinite diameter Tolman lengths δ turn positive. For mixtures, even if they contain similar molecules, the extrapolated Tolman length behaves strongly non-ideal, implying a qualitative change of the curvature behavior of the surface tension of the mixture.