Statistical Mechanics of Surface Tension of Curved Interface, I

Abstract

From the microscopic point of view, surface tension is defined in two different ways: the statistical~ thermodynamic and so to speak, the kinetic. In this paper these two definitions are examined with regard to a cylindrical interface between liquid· and. vapor, and it is shown that under certain ~onditions-these two definitions become equivalent with each other and also with the usual thermodynamic definition. The methods used are those due to Harasima, Maclellan, and Kirkwood and Buff. The possibility of extending these methods to a spherical interface are brieRy mentioned, leaving detailed calculations to be covered by future papers. § 1. Introduction Surface tension measured as a macroscopic quantity may be defined in two different and distinct ways from the microscopic point of view. The one, statisticahhermodynamic, is defined as: r=aF/as.~ (1) where F is the free energy of the two-phase system considered and SiJ~ the area of the interface. Calculations of surface tension based on this concept were made by Fowler, Harasima and MacLellan with regard to plane interfaces. The other definition is, ,so to speak, kinetic, and is based on calculations of the force across a plane strip perpendicular to the interface. Calculations according to this definition were carried out by Kirkwood and Buff, and MacLellan for the case of a plane interface. These two definitions· of surface tension are, of course, mutually independent. But it was shown by Kirkwood and Buff, and MacLellan that in the case of a plane interface these two definitions are equivalent and lead to the same result. The case of a curved interface presents a number of difficulties. For example, surface tension has two principal values corresponding to the two principal curvatures. Moreover, in the case of a closed surface, it is not clear in the statistical thermodynamic definition which quantities should be kept constant in performing the partial differentiation with respect to the interfacial area. The kinetic definition itself does not present any conceptual difficulty but calculations of stress dyadic for a curved interface become very involved, which makes the method of approximation a subject for discussion.