Contact angle

2.4 The Extended FOWKES method

In the Extended FOWKES method the work of adhesion is not split up into just two fractions but into three: the disperse and polar fractions as well as the fraction $W_{sl}^H$ resulting from the hydrogen bridges:

$W_{sl} = W_{sl}^D + W_{sl}^P + W_{sl}^H$ (Equation 30)

The calculation of the surface energy accordingly is carried out in three steps instead of two. As in the first step of the FOWKES method the disperse fraction of the surface energy of a solid is determined from the contact angle data of a purely disperse liquid (see 2.3.1).

In the second step liquids with known polar and disperse surface tension fractions are selected ( $\sigma_l  P > 0$ ) , with a hydrogen bridge fraction $\sigma_l^H$ of O. In this way, as in the second step of the FOWKES method (see 2.3.2), the polar fraction of the surface free energy is first obtained with the aid of contact angle measurements (by subtracting the disperse fraction from the total work of adhesion).

$W_{sl}^P = \sigma_l (\cos\theta + 1) - 2 \sqrt {\sigma_s^D \quad \sigma _l^P}$ (Equation 31)

The determination of the polar fraction of the surface free energy of the solid is carried out as in the FOWKES method.

In a third step work is again carried out in a similar manner for the calculation of the hydrogen bridge fraction. Contact angles of liquids with known polar, disperse and hydrogen bridge fractions ( $(\sigma_l^H < 0)$ ) of the surface tension are measured. By extending Equation 23 by the hydrogen bridge fraction we obtain

$\gamma_{12} = \sigma_s + \sigma_l - 2 ( \sqrt {\sigma_s^D \cdot \sigma_l^D} + \sqrt {\sigma_s^P \cdot \sigma_l^P} + \sqrt {\sigma_s^H \cdot \sigma_l^H})$ (Equation 32)

As in Equation 28 the required fraction of the work of adhesion, i.e. the fraction $W_{sl}^H$ resulting from the hydrogen bridges, can be calculated for each contact angle by subtracting the known fractions (by including the YOUNG Equation (25)):

$W_{sl}^H = \sigma_l (\cos\theta + 1) - 2 (\sqrt {\sigma_s^D \cdot \sigma_l^D} + \sqrt {\sigma_s^P \cdot \sigma_l^P})$ (Equation 33)

Finally the hydrogen bridge fraction $\sigma_s^H$ of the surface energy of the solid can now be determined as described in Step 2 of the FOWKES method. According to Equation 32 the following relationship applies to $W_{sl}^H$ :

$W_{sl}^H = 2 \sqrt {\sigma_l^H} \quad \sqrt {\sigma_s^H}$ (Equation 34)

If $W_{sl}^H$ is plotted against $2 \sqrt {\sigma_l^H}$ then, $  \sqrt {\sigma_s^H}$ i.e. the hydrogen bridge fraction of the surface energy of the solid, is obtained from the slope of the regression curve.

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