Contact angle

2.7 The SCHULTZ method

The method for calculating the surface energy according to SCHULTZ is only intended for use with high-energy solid surfaces. In the DSA1 program there are two methods implemented which are based on SCHULTZ: “SCHULTZ 1” and “SCHULTZ 2”. The theoretical requirements are the same for both methods; the difference lies in the test arrangement. As a result this section first described the theoretical principles and only then explains the differences between the two SCHULTZ methods.

2.7.1 Theoretical principles for the two SCHULTZ methods

High-energy solids are normally completely wetted by all liquids, so that their surface energy cannot be determined by using conventional contact angle measurements. In order to be able to investigate such systems at all the test arrangement must be altered: instead of being measured in air, the contact angle of a liquid drop (“drop phase”) on a solid is measured in a surrounding liquid phase (“bulk phase”).

The calculation of the surface energy assumes that the YOUNG equation also applies to a liquid/liquid/solid system:

$\sigma_s = \gamma_{sl\ drop} + \gamma_{ll  drop/bulk} \cdot \cos\theta$ (Equation 46)

In this case $\sigma_s$ represents the interfacial tension between the solid and the surrounding phase; $\gamma_{sl\ drop}$ the interfacial tension between the solid and the drop phase; and $\gamma_{ll drop\ /\ bulk}$ the interfacial tension between the two liquids.

With the equations of FOWKES and OWENS, WENDT, RABEL and KAELBLE (Equation 37, p.10) adapted for a liquid/liquid/solid system, the following equations are obtained for the drop phase and the surrounding phase:

$\gamma_{sl\ drop} = \sigma_s + \sigma_{l\ drop} -2 \sqrt {\sigma_s^D \cdot \sigma_{l\ drop}^D} - W_{sl\ drop}^P$ (Equation 47)

$\gamma_{sl\ bulk} = \sigma_s + \sigma_{l\ bulk} -2 \sqrt {\sigma_s^D \cdot \sigma_{l\ bulk}^D} - W_{sl\ bulk}^P$ (Equation 48)

$W^P_{sl}$ is the polar fraction of the work of adhesion, i.e. the interactions between the particular liquid and the solid.

If Equations 46, 47 und 48 are combined then the following relationship is obtained:

$\sigma_{l\ drop} - \sigma_{l\ bulk} + \gamma_{ll\ drop\ /\ bulk} \cdot \cos\theta = 2 \sqrt{\sigma_s^D} \cdot (\sqrt{\sigma_{l\ drop}^D} - \sqrt{\sigma_{l\ bulk}^D}) + W_{sl\ drop}^P - W_{sl\ bulk}^P$ (Equation 49)

2.7.2 SCHULTZ 1

In the “SCHULTZ 1” method only a single drop liquid is used and the surrounding phase is changed instead. The drop liquid used is normally water; the bulk phase is a liquid which is immiscible with water and with a lower density than water.

As in the FOWKES method (see Section 2.3) the calculation of the polar and disperse fractions of the surface energy is carried out in two steps.

At first the contact angle of water on the solid is measured in a range of purely disperse interacting liquids. Owing to the nonpolar character of the surrounding phase the term $W^P_{sl\ bulk}$ can be deleted from Equation 49. Equation 49 can then be adapted to conform with the general equation for a straight line:

$y = mx + b$ (Equation 50)

to give:

$\underbrace {\sigma_{l\ drop} - \sigma_{l\ bulk} + \gamma_{ll\ drop\ /\ bulk} \cdot \cos\theta}_{\gamma} = \underbrace {2\sqrt {\sigma_s^D}}_{m} \cdot \underbrace {(\sqrt {\sigma_{l\ drop}^D} - \sqrt {\sigma_{l\ bulk}^D})}_{x} + \underbrace {W_{sl\ drop}^P}_{b}$ (Equation 51)

If the term y is plotted against x then the disperse fraction of the surface energy of the solid $\sigma_s^D$ is can be calculated directly from the slope and $W^P_{sl\ drop}$ from the y-axis intercept

: Fig.6: Determining the surface energy according to SCHULTZ

In the second step the polar fraction of the surface energy of the solid is determined by using several surrounding phases which have polar fractions. As the term $W^P_{sl\ drop}$ from Equation 49 is now known, the polar fraction of the adhesion energy between the solid and the surrounding phase $W^P_{sl\ bulk}$ can be calculated for each individual surrounding phase.

According to FOWKES, this adhesion energy can be calculated from the geometric mean between the polar fractions of the surface tensions of the participating phases:

$W_{sl\ bulk}^P = 2 \cdot \sqrt {\sigma_s^P} \cdot \sqrt {\sigma_{l\ bulk}^P$ (Equation 52)

As a result, if $W_{sl\ bulk}^P$ is plotted against $\sigma_{sl\ bulk}^P$ then the required term $\sigma_s^P$ can be obtained from the slope of the regression curve.

2.7.3 SCHULTZ 2

In the SCHULTZ 2 method it is not the heavier liquid which is used as the drop liquid but the lighter one; the heavier liquid forms the surrounding phase. In order for this to be possible the test arrangement must be inverted: the drop is not present as a sessile drop on the solid but is suspended from it as a pendant drop:

: Fig.7: Test arrangements for the SCHULTZ method; left SCHULTZ 1, right SCHULTZ 2

In this arrangement the surrounding phase is retained and the contact angles of various drop liquids are measured. The advantage when compared with the SCHULTZ 1 method is that the contact angles of the drop phase to be measured are larger and can therefore be measured more accurately.

As in SCHULTZ 1 the disperse fraction of the surface energy of the solid $\sigma^D_s$ is measured first by using purely disperse interacting liquids. The difference from the calculation for SCHULTZ 1 is that the term $W^P_{sl\ bulk}$ is obtained from the intercept of the regression curve with the y-axis (see Fig.7) on the plot, whereas the term $W^P_{sl\ drop} $ is deleted from Equation 49.

In the second step the term $W^P_{sl\ drop}$ in Equation 49 is calculated from the contact angles of drop liquids with polar fractions for each test liquid. The polar of the surface energy of the solid $\sigma^P_s$ is obtained in a similar way to SCHULTZ 1 by using the equation

$W_{sl\ drop}^P = 2 \cdot \sqrt {\sigma_s^P} \cdot \sqrt {\sigma_{l\ drop}^P}$ (Gleichung 53)

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