Microscopic Drop Profiles and the Origins of Line Tension

Abstract

The line tension associated with the three-phase contact line of a liquid drop on a rigid substrate is calculated as the excess free energy per unit length of the contact line relative to the free energy of the drop/substrate system when the interaction between the free liquid interface and the substrate is turned off. To perform this calculation, the interaction energy is treated in the Derjaguin approximation and the microscopic drop profile determined accordingly. An explicit expression for the line tension as a function of the macroscopic contact angle for the case of an infinite drop is obtained as a quadrature over the interaction energy per unit area between planar half-spaces. The magnitude of the line tension is shown to be the product of the free liquid surface tension and the length scale associated with the interaction energy. For attractive interaction energies the line tension is negative but when the interaction has both an attractive component (which must dominate at small separations) and a repulsive component which dominates at larger separations, the line tension can be positive. Some simple physical models of the interaction are considered to illustrate these points. For the case of a finite drop, we show that the concept of line tension as a macroscopic quantity which may be added to the free energy balance, giving rise to deviation from Young's equation for the contact angle, is a valid one in the sense that a full treatment of the microscopic drop shape yields the same macroscopic picture of the drop near contact. The line tension for a finite drop is demonstrated to be the value calculated for the infinite drop to within terms of the order of the ratio of interaction energy range to drop radius. Hence, the treatment of line tension as a macroscopic concept is valid down to drop sizes of the order of the range of the interaction energy. Finally, by applying the Derjaguin approximation in a different (but equally valid) way and deriving an alternate expression for the line tension, we are able to elucidate the range of validity of the approximation itself. Line tensions calculated by these methods would appear to be accurate for contact angles up to ∼20°.