Theory for the equilibrium contact angle between a gas, a liquid and a solid

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Abstract
An alternative derivation is presented of the equation for the equilibrium contact angle between a gas, a liquid and a plane solid surface. The excess free energy of the system around the contact line is minimized with respect to the contact angle θ0 at constant surface area. Thus the surface tension (or surface free energy) does not appear explicitly in the derivation, although it can be calculated from the theory. The general theory is applied to a system in which the forces of attraction between atoms are of the unretarded London–van der Waals type. The equilibrium contact angle is found as a solution of A13/A11=½+¾ cos θ0–¼ cos3θ0 where A13 and A11 are the Hamaker constants of the gas–liquid and liquid–liquid interactions respectively. It is argued that Young's equation γSV=γLV cos θ0+γSL is incorrect in principle, because in deriving it, it is assumed that the excess free energy resides in planes such as the Gibbs dividing surfaces. Near the contact line the non-uniform interfacial zones overlap and the excess free energy at a point in a particular phase will depend on the properties of the two adjoining phases and their geometry. Thus it is essential in this instance to treat the free energy as a property of the bulk phases and not to idealize it as a surface property.

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