A Molecular Expression for "Line Tension"

"Line tension", a concept that features in an additional term to the Young's equation, was introduced to describe the size dependence of contact angles of nanodroplets on surfaces. Although this concept describes the observations in a succinct, elegant manner, theorists have long had misgivings about the physical interpretation of the phenomenon. Papers have been published that attempt to nail down its value, which is reportedly very small (∼10 pN) and evidently even the sign has been uncertain. Attempts to interpret it in a mechanical manner analogous to interfacial tension, i.e., due to the curvature of the three-phase contact line, have run into conceptual problems that require invocations of ever more complex models. In this work, we have used molecular simulations to systematically relate "line tension" to the additional free energy per unit length of the three-phase line and found no direct relation. However, when we rederived the Young's equation without ignoring the interfacial molecules, we found a physically satisfying explanation for the size dependence of the contact angle of nanodroplets without invoking the curvature of the three-phase contact line. The new model does not have the elegant form of the modified Young's equation, but each parameter in it has an unambiguous physical interpretation. An approximate form of this model, linearized in the inverse droplet radius, yields a quantity that is mathematically analogous to what is conventionally called "line tension", but unpacked at the molecular level, showing that it is unrelated to a restoring force associated with the curvature of the macroscopic three-phase contact line.