Contact angle

2.2 Equation of State

The equation of state was obtained during the search for a method of determining the surface energy of a solid from a single contact angle measurement by using a liquid with known surface tension.

Starting with the equation of Young

\sigma_s = \gamma_{sl} + \sigma_l \cdot \cos\theta (Equation 16)

it can be seen that a second equation is required which also describes the surface energy of the solid as a function of the interfacial tension solid/liquid and the surface tension of the liquid:

\sigma_s = f  (\gamma_{sl}, \sigma_l) (Equation 17)

From thermodynamic considerations it was first demonstrated that such an equation valid for all systems must exist. By using an enormous volume of contact angle data the required equation of state was determined empirically:

\gamma_{sl} = \sigma_l + \sigma_s - 2\sqrt{\sigma_l \cdot \sigma_s} \cdot e{^{-\beta (\sigma_l - \sigma_s)}}^2 (Equation 18)

The value 0.0001247 was determined for the constant \beta in the exponent. If the equation of state is inserted in Young’s equation then a new equation is obtained which allows the calculation of the surface tension of the solid \sigma_s from a single contact angle if the surface tension \sigma_l is known: 

\cos\theta = -1+2 \sqrt {\frac{\sigma_s}{\sigma_l} \cdot e}{^{-\beta (\sigma_l - \sigma_s)}}^2 (Equation 19)

In the calculation of the surface energy with the help of the equation of state the type of interactions which lead to the formation of the interfacial tensions (polar or disperse interactions) are not taken into account. However, the assumption that the knowledge of the surface tension of the liquid alone is sufficient has been disproved by experiments in which the contact angles of liquids with similar high surface tensions and differing fractions of polar interactions were measured. It appears that the disperse and polar fractions of the surface tensions must be taken into account; this means that the equation of state only provides useful results when only disperse interactions are present or when these are in the majority.