(475f) Extension of Gibbs' Dividing Surface Analysis to Cavities That Intersect a Planar Surface: Calculation of the Line Tension of the Hard-Sphere Fluid

The behavior of a liquid drop resting on a solid surface is strongly influenced by the interfacial tensions that develop at the corresponding interfaces between the liquid, vapor and solid phases. The line tension, an additional contribution that arises at the linear interface or intersection of all three phases, also plays an important, but often neglected, role in the surface thermodynamics of sessile drops. The line tension is the one-dimensional analogue of the surface tension, and when properly included in a thermodynamic analysis is a necessary component of various expressions such as the Young Equation, the Neumann triangle relation and the one-dimensional version of the Gibbs Adsorption Isotherm. All of these relations show that the line tension is of little consequence for large droplets or bubbles, but may be quite influential at small sizes. In particular, the line tension has been shown to affect the stability of sessile droplets (Guzzardi et al., 2006, Phys. Rev. E, 73, 021602) and plays an important role in heterogeneous nucleation (Wang et al., 2001, Phys. Rev. E, 63, 031601; Auer and Frenkel, 2003, Phys. Rev. Lett., 91, 015703). Furthermore, a proper description of wetting phenomena may require information about the line tension, where it has been speculated that the line tension diverges upon the initial formation of a three-phase wetting line (Varea and Robledo, 1992, Phys. Rev. A, 45, 2645). Given that the line tension may be positive or negative, unlike the surface tension which has a well-defined sign, the generation of additional insights into the affect of the system's parameters on the line tension is crucial for developing an improved understanding and better control of various surface phenomena.

The line tension of sessile droplets has been studied previously using several approaches. The application of straightforward mechanical arguments yields a modification to the Young Equation. The use of the Gibbs' dividing surface analysis again leads to the same modified Young Equation, while additionally demonstrating that the line tension, like the surface tension, depends upon the location of the dividing surface (Navascués and Tarazona, 1981, Chem. Phys. Lett., 82, 586; Rusanov, 1987, Kolloid Zh., 49, 700; Rusanov et al., 2004, Coll. Surf. A, 250, 263). These and other analyses performed to date, however, are lacking in certain key respects. For one, they neither provide sufficient insight into the dependence of the line tension on the size of a droplet nor explicitly describe the (possible) asymptotic behavior of the line tension. (Auer and Frenkel (2003, Phys. Rev. Lett., 91, 015703) suggest that the line tension should exhibit a limiting value along with a simple dependence on the curvature of the linear interface, though their analysis was empirical in nature.) Also, while the thermodynamics of interfacial areas is well-developed, various and potentially useful linear analogues of some of these surface relations have yet to be derived. Once known, such linear relations could be used to predict the line tension and its resulting importance for various circumstances.

To obtain further understanding of the line tension of a three-phase interface, we revisit the Gibbs' dividing surface analysis and apply it specifically to the case of a spherical cavity overlapping a planar surface. In doing so, we introduce a temporary interfacial term that accounts for all interfacial free energy excesses that arise from the presence of the cavity. This interfacial term plays a role analogous to the surface tension in the classical Gibbs' analysis, which can be decomposed into terms related to the surface tension of the cavity and the planar surface and, most importantly, the line tension of the cavity-surface interface. Our analysis also leads to the Young Equation relevant to a cavity, a Gibbs line adsorption equation and the dependence of the line tension on the dividing surface. We obtain the dependence of the temporary interfacial term on the radius of the cavity that also identifies a limiting value of the line tension and an exact expression for the dependence of the line tension on the cavity radius (valid for the large cavity limit). While our method explicitly considers cavities, the thermodynamic analysis is still general, in that all of the obtained results can be straightforwardly extended to describe sessile drops or bubbles.

Since previous work has analyzed cavity growth within hard-sphere fluids, we apply the above thermodynamic relations to predict (for the first time) the line tension of cavities. The growth of cavities in contact with an impenetrable hard wall is described by the use of Scaled Particle Theory (Reiss et al., 1959, J. Chem. Phys., 31, 369) and, in particular, the more recent Inhomogeneous Scaled Particle Theory (I-SPT) (Siderius and Corti, 2005, Phys. Rev. E, 71, 036141; Siderius and Corti, 2007, Phys. Rev. E, 75, 011108). We find that I-SPT accurately computes the line tension for a number of cavity configurations, thereby revealing interesting geometric effects for certain system parameters. In particular, I-SPT shows that the line tension in the limit of infinite cavity radius may be either positive or negative depending on the fluid density and may invert from positive to negative at a fixed density as the cavity configuration changes. We also find that the line tension diverges as the line interface disappears (e.g., when a cavity detaches from the surface), an effect that closely resembles the divergence of the line tension at wetting and which may also play an important role in the behavior of depletion forces in colloidal systems near the Derjaguin limit (Herring and Henderson, 2007, Phys. Rev. E, 75, 011402). Finally, we compute the line adsorption at the linear interface, which may be both positive and negative (a result similar to the sign inversion of the line tension itself). Overall, the dividing surface analysis combined with I-SPT yields important insights into the behavior of the line tension of cavities within the hard-sphere fluid, results which are equally applicable to other fluids.