(551e) Modeling Surface Tension Using Gibbs'-Tolman Approach

The Gibbs-Tolman model of surface tension is the only classical thermodynamic model that predicts the surface tension from a bulk equation of state (EoS) [1]. The model relates surface tension (σ) to the vapor pressure (P_sat), and bounds of pressure for the metastable states i.e. (P_minand P_max); the two parameters λ_v and λ_l in this model are related to the characteristic lengths of the transition layers on the liquid and vapor sides of the interface. Typically, one expects this characteristic length parameter be positive. In some of the earlier literature, it was observed that modeling of experimental observations for several compounds using a suitable equation of state (such as van der Waals or Lee-Kesler) resulted in negative values for λ_v ; no reasons or justification for the same were presented.

To understand the reasons for this anomaly, the Gibbs – Tolman model was modified removing the underlying assumption of zero curvature of the Gibbs dividing surface. An expression for σ in terms of P_sat, P_min , P_max and the model parameters λ_v and λ_l , the curvature of the Gibbs dividing surface (c) and the distance between the surface of discontinuity and the surface of tension (y_t) is obtained. However, modeling the experimental surface tension values for various compounds, yielded parameters with poor confidence.

Later, a sensitivity analysis on the GT model showed that the length parameters turn positive for a certain range of P_minand P_max depending on the compound, with P_max being more sensitive than P_min . In addition, the original paper by Tolman also [2] emphasizes that the model requires reliable estimates of P_minand P_max . Thus, accurate experimental measurements or EoS estimates of P_minand P_max are necessary for validating the GT model. However, an extensive search in the literature did not provide us any methods to obtain reliable values for the same.

Hence, we propose a rearrangement of terms in the GT model to regress for the more sensitive term (i.e., P_max) along with the length parameters. The surface tension data for several compounds in the rearranged GT models using suitable EoS yields positive length parameters, as expected. We conclude that the origin of negative length parameters in the previous works was due to unreliable estimates of P_minand P_max.

References

[1] Tumram S, Kesava Rao K, Ananth MS, Augmented Gibbs-Tolman Model for Surface Tension, Langmuir, 33, 11687 (2017

[2] Tolman RC, The Superficial Density of Matter at a Liquid-Vapor Boundary, Chem. Phys., 17, 118 (1949)