Contact angle

2.5 The Owens, Wendt, Rabel and Kaelble method

According to OWENS, WENDT, RABEL and KAELBLE the surface tension of each phase can be split up into a polar and a disperse fraction:

\sigma_l = \sigma_l^P + \sigma_l^D (Equation 35)

\sigma_s = \sigma_s^P + \sigma_s^D (Equation 36)

The FOWKES method for calculating the surface energy has already been developed from this relationship. In contrast to the FOWKES method, in the OWENS, WENDT, RABEL and KAELBLE method the calculation of the surface energy of the solid takes place in a single step.

OWENS and WENDT took the equation for the surface tension

\gamma_{sl} = \sigma_s +\sigma_l - 2 (\sqrt {\sigma_s^D \cdot \sigma_l^D} + \sqrt {\sigma_s^P \cdot \sigma_l^P}) (Equation 37)

as their basis and combined it with the YOUNG equation  

\sigma_s = \gamma_{sl} + \sigma_l \cdot \cos\theta (Equation 38)

The two authors solved the equation system by using the contact angles of two liquids with known disperse and polar fractions of the surface tension. KAELBLE solved the equation for combinations of two liquids and calculated the mean values of the resulting values for the surface energy. RABEL made it possible to calculate the polar and disperse fractions of the surface energy with the aid of a single linear regression from the contact angle data of various liquids. He combined Equations 37 and 38 and adapted the resulting equation by transposition to the general equation for a straight line 

y = mx + b (Equation 39)

The transposed equation is shown below:

\underbrace {\frac {(1 + \cos\theta) \cdot \sigma_l}{2\sqrt {\sigma_l^D}}}_{\gamma} = \underbrace {\sqrt {\sigma_s^P}}_{m} \quad \underbrace {\sqrt {\frac {\sigma_l^P}{\sigma_l^D}}}_{x} + \underbrace {\sqrt {\sigma_sD}}_{b} (Equation 40)

In a linear regression of the plot of y against x, \sigma_s P is obtained from the square of the slope of the curve m and \sigma_s D from the square of the ordinate intercept b. 

Fig. 5: Determination of the disperse and polar fractions of the surface tension of a solid according to RABEL