Contact angle

2.3 The method according to FOWKES

By using the FOWKES method the polar and disperse fractions of the surface free energy of a solid can be obtained. Strictly speaking this method is based on a combination of the knowledge of FOWKES on the one hand and that of OWENS, WENDT, RABEL and KAELBLE on the other, as FOWKES initially determined only the disperse fraction and the latter were the first to determine both the components of the surface energy. The difference between the FOWKES method used by KRÜSS and the OWENS, WENDT, RABEL and KAELBLE method is that in the FOWKES method the disperse and the polar fractions are determined in succession, i.e. in two steps, while in the OWENS, WENDT, RABEL and KAELBLE method both components are calculated by using a single linear regression.

The calculation steps described below are only intended to explain the methods. When calculating the surface energy according to FOWKES you do not have to proceed in several steps; when the calculation is carried out these steps are processed internally by the program. The same applies for the “Extended FOWKES” method described in Section 2.4.


2.3.1 Step 1: Determining the disperse fraction

In this first step the disperse fraction of the surface energy of the solid is calculated by making contact angle measurements with at least one purely disperse liquid.

By combination of the surface tension equation of FOWKES for the disperse fraction of the interactions

\gamma_{sl} = \sigma_s + \sigma_l - 2 \sqrt {\sigma_s^D \cdot \sigma_l^D} (Equation 20)

with the YOUNG equation (Equation 16) the following equation for the contact angle is obtained after transposition:

\cos\theta = 2 \sqrt{\sigma_s^D} \cdot \frac {1}{\sqrt {\sigma_l^D}}- 1 (Equation 21)

and, based upon the general equation for a straight line,

y = mx + b (Equation 22)

\cos\theta is then plotted against the term 1 / \sqrt {\sigma_l^D} and can be determined from the slope m. The straight line must intercept the ordinate at the point defined as b=-1 (0/-1) As this point has been defined it is possible to determine the disperse fraction from a single contact angle: however, a linear regression with several purely disperse liquids is more accurate.

Fig. 3: Determining the surface energy according to FOWKES (1)

2.3.2 Step 2: Determining the polar fraction

For the 2nd step, the calculation of the polar fraction, Equation 20 is extended by the polar fraction:

\gamma_{sl} = \sigma_s + \sigma_l - 2 (\sqrt {\sigma_s^D \cdot \sigma_l^D} + \sqrt {\sigma_s^P \cdot \sigma_l^P}) (Equation 23)

It is also assumed that the work of adhesion is obtained by adding together the polar and disperse fractions:

W_{sl} = W_{sl}^D + W_{sl}^P (Equation 24)

and then as a third step YOUNG’s equation

\sigma_s = \gamma_{sl} + \sigma_l \cdot \cos\theta (Equation 25)

is added to the equation of DUPRÉ

W_{sl} = \sigma_s + \sigma_l - \gamma_{sl} (Equation 26)

to obtain the following relationship for the work of adhesion:

W_{sl} = \sigma_l (\cos\theta + 1) (Equation 27)

Now all the components required for the calculation of the polar fraction of the surface energy have been assembled. A combination of Equations 23, 24 and 27 produces  

W_{sl}^P = \sigma_l (\cos\theta + 1) - 2 \sqrt {\sigma _s^D \cdot \sigma_l^D} (Equation 28)

Based upon this relationship the contact angles of liquids with known polar and disperse fractions are measured and W_{sl}^P is calculated for each liquid. In this case a single liquid with polar and disperse fractions would be sufficient, although the results would again be less reliable.

As according to Equation 23 the polar fraction of the work of adhesion is defined by the geometric mean of the polar fractions of the particular surface tensions

W_{sl}^P = 2 \sqrt {\sigma_l^P \cdot \sigma_s^P} (Equation 29),

then, by plotting W_{sl}^P gegen 2 \sqrt {\sigma_l^P} and following this with a linear regression, the polar fraction of the surface energy of the solid can be determined from the slope. As in this case the ordinate intercept b is 0, the regression curve must pass through the origin (0;0).

Fig. 4: Determining the polar surface energy according to FOWKES (2)