22.2.1 du Noüy Ring

A very common tensiometric method is based on the technique first described by French physicist Pierre du Noüy in 1925 [1]. It uses a thin platinum ring that is slowly lifted off the surface of a liquid. The force required can be used to directly calculate the surface tension according to

$\begin{array}{ll}{F}_{\text{measured}}\hfill & ={\gamma }_{\text{liquid}}{l}_c\hfill \\ \hfill & ={\gamma }_{\text{liquid}}2\pi \left(r_{\text{inner}}+r_{\text{outer}}\right)\hfill \end{array}$

Remember that if the force has to be calculated, we always have to use the circumference l_c as the relevant length to be multiplied with the surface tension. The values measured using du Noüy rings usually have to be corrected to compensate for the shape of the liquid meniscus forming at the boundary of the ring that change the direction of action of the surface tension forces. These correction factors depend heavily on the shape of the rings used.

22.2.2 Wilhelmy Plate Method

Another method for measuring the surface tension is the Wilhelmy plate method, which was first used by the German chemist Ludwig Wilhelmy in 1863 [2]. In this setup a Wilhelmy plate is either immersed or retracted from a liquid with unknown surface tension (see Fig. 22.1a). In the immersion case, the surface will form an advancing contact angle with the liquid. In the retraction case, the surface will form a receding contact angle with the liquid. In both cases the surface tension will pull the plate into the liquid. The plate is held by an external force F_external that is adjusted such that the plate does not move. The balance of forces on the plate then is

$F_{\text{external}}+F_{\text{buoyancy}}-F_{\gamma \text{,}\text{gas/liquid}}-F_{\gamma \text{,}\text{liquid/solid}}-F_{\text{gravity}}=0$

with

$\begin{array}{ll}{F}_{\text{buoyancy}}\hfill & =\rho lw{h}_{1}g\hfill \\ F_{\gamma \text{,}\text{gas/liquid}}\hfill & =2\left(l+w\right)\gamma \cos \Theta \hfill \\ F_{\gamma \text{,}\text{liquid/solid}}\hfill & \approx 0\text{or}\text{cosnstant}\hfill \\ F_{\text{gravity}}\hfill & =\rho lw{h}_{0}g\hfill \end{array}$

Shortly before the plate detaches completely from the surface, F_external reaches its maximum values (see Fig. 22.1b). At this location no buoyancy forces are acting, and Θ = 0°. Therefore Eq. 22.1 simplifies to

$\begin{array}{ll}{F}_{\text{external}}-F_{\gamma ,\text{gas/liquid}}-F_{\text{gravity}}\hfill & =0\hfill \\ F_{\text{external}}-2\left(l+w\right)\gamma -\rho lw{h}_{0}g\hfill & =0\hfill \\ \gamma \hfill & =\frac{F_{external}-\rho l{h}_{0}g}{2\left(l+w\right)}\hfill \end{array}$

If F_external can be determined at sufficient precision, all terms of Eq. 22.1 can be calculated except F_γ,liquid/solid. For very small contact angles, this term is almost 0 (see Eq. 20.4). This condition is effected by using platinum as material for the Wilhelmy plate which will (when sufficiently clean) will form an almost zero contact angle with most liquids because of its high free surface energy (see Tab. 20.1).

Tab. 22.1

Liquids commonly used for determining free surface energies [7–9]. All values given are in mNm⁻¹

Substance	Surf. tens. γgas/liquid	Fowkes		Extended Fowkes			OCG
		${\text{γ}}_{\text{gas/liquid}}^{\text{d}}$	${\text{γ}}_{\text{gas/liquid}}^{\text{p}}$	${\text{γ}}_{\text{gas/liquid}}^{\text{d}}$	${\text{γ}}_{\text{gas/liquid}}^{\text{p}}$	${\text{γ}}_{\text{gas/liquid}}^{\text{h}}$	${\text{γ}}_{\text{gas/liquid}}^{\text{LW}}$	${\text{γ}}_{\text{gas/liquid}}^{+}$	${\text{γ}}_{\text{gas/liquid}}^{-}$
water	72.80	21.80	51	29.10	1.30	42.40	21.80	25.50	25.50
diiodomethane	50.80	0
glycerol	64	34	30	37.40	0.20	25.80	34	3.92	57.40
formamide	58.20	39	19	35.10	1.60	21.50	39	2.28	39.60
ethylene glycol	47.70	30.90	16.80	30.10	0	17.60	29	3	30.10
1-octanol	27.50	27.50	0				27.50	0	3.97
n-octane	21.62	21.62	0	21.62	0	0	21.62	0	0
chloroform	27.15	27.15	0				27.15	1.50	0

There are several modifications of the Wilhelmy plate method, with the most common one using a platinum rod instead of a plate. This allows measurement in smaller vessels and is thus suitable for applications where only minor amounts of the liquid can be provided. Compared to the du Noüy measurements, values obtained from the Wilhelmy plate method do not have to be compensated because there is shape-dependent meniscus formation at the plate surface.

22.2.3 Drop-Weight Method

The surface tension can also be derived by observing the characteristic deformation of a pendant drop at the end of a capillary (see Fig. 22.2). In these equations, surface tension balances gravity forces

$\begin{array}{ll}{F}_{\gamma }\hfill & =F_{\text{gravity}}\hfill \\ \gamma \pi d\sin \Theta \hfill & =mg\hfill \end{array}$

Eq. 22.2 is referred to as Tate’s law first described by Tate in 1864 [3]. If more and more liquid is pushed into the droplet via the capillary, the total mass of the droplet will increase. This increase is balanced by an increase in the contact angle, as all other parameters on the left-hand side of Eq. 22.2 are constant. The droplet will fall from the capillary once the contact angle reaches 90°, in which case

$\gamma =\frac{mg}{\pi d}$

Therefore the surface tension can be measured (in theory) by weighting the mass of droplets that fall off the capillary. If d cannot be determined, the surface tension can be measured in a differential manner by using a reference liquid with known surface tension (usually pure water or a suitable solvent). The same ratio $\frac{\eta }{m}$ will be constant for both liquids; therefore the surface tension to be determined can be obtained from the value of the reference liquid.

This method is referred to as the drop-weight method. The respective instrument for performing drop-weight-based surface tension determinations is referred to as a stalagmometer. In practical experiments, a larger portion of the droplet will remain attached to the capillary. Therefore a correction factor ξ is applied to Eq. 22.3 resulting in

$\gamma =\xi \frac{mg}{\pi d}$

The correction factor ξ is a function only of the capillary diameter d and the droplet mass m (actually of the droplet volume V to be precise) but not of the surface tension. This allows applying Eq. 22.4 once the droplet mass has been determined by looking up the correction factor in the reference literature.

22.2.5 Maximum Bubble Pressure Method

A very easy and convenient method for measuring the surface tension is referred to as the maximum bubble pressure method. In this method a capillary is immersed into the liquid to be measured, and a gas bubble is created inside the liquid using gas with controllable pressure (see Fig. 22.4). As the pressure increases, the size of the bubble increases until its diameter is identical to the diameter of the capillary (hemispherical bubble). In this case the Young-Laplace equation allows determining the surface tension using

$\begin{array}{ll}{p}_{\max }-p_0\hfill & =\gamma \left(\frac{1}{r_1}+\frac{1}{r_2}\right)\hfill \\ \hfill & =\gamma \frac{2}{r_{\text{capillary}}}\hfill \\ \gamma \hfill & =\frac{\left(p_{\max }-p_0\right)r_{\text{capillary}}}{2}\hfill \end{array}$

The maximum bubble pressure method is often used to measure the dynamic surface tension, as it allows measuring the development of the surface tension at a newly created interface. For example, in a surfactant solution, the molecules will (over time) assemble at the gas/liquid interface and change the surface tension. The changing surface tension can be measured conveniently at the changes in bubble size over time if the pressure is kept constant.

22.2.6 Spinning-Drop Method

Another method for measuring the surface tension is the spinning-drop method originally developed and described by Vonnegut [4]. This method uses a droplet of a liquid or a gas that is suspended in a second immersion fluid. If the whole setup is rotated at high speeds, the drop will deform under the action of the centrifugal force (see Fig. 22.5). The deformation of the droplet allows deriving the surface tension.

The method is very precise, and surface tensions down to several 10⁻⁶ mN m⁻¹ can be measured accurately. In addition, this is one of the few methods that allows measuring the interfacial surface tension between two liquids, one of which forms the droplet with the other one forming the surrounding liquid.

The elongated shape of the spinning drop can be well approximated as a cylinder with two hemispherical caps (see Fig. 22.6a). As the drop changes its surface area, two energy terms are involved: the change of energy because of the action of the surface tension and the change of energy because of the action of the centrifugal forces.

We will consider a drop being rotated at the angular velocity ω resulting in a drop of length L and diameter R.

Energy Contribution of the Surface Tension. The change of energy because of the surface tension is given by

$\begin{array}{ll}d{E}_{\text{surface}\text{tension}}\hfill & =\gamma dA\hfill \\ \hfill & =\gamma \left(d{A}_{\text{sphere}}+d{A}_{\text{cylinder}}\right)\hfill \\ \hfill & =\gamma \left(8\pi rdr+2\pi Ldr\right)\hfill \end{array}$

The area consists of the surface area of the cylinder and of the two hemispheres (a complete sphere in total). We have therefore used Eq. 3.105 and Eq. 3.107 for the infinitesimal surface change of a cylinder and a sphere, respectively. Integration of Eq. 22.7 yields

$\begin{array}{ll}\int d{E}_{\text{surface}\text{tension}}\hfill & ={\displaystyle {\int }_0^R\gamma \left(8\pi rdr+2\pi Ldr\right)}\hfill \\ E_{\text{surface}\text{tension}}\hfill & =\gamma \left(4\pi R^2+2\pi RL\right)\hfill \end{array}$

As L changes, it is better to express it as a function of the drop volume V , which obviously remains constant. Therefore

$\begin{array}{ll}{V}_{\text{drop}}\hfill & =V_{\text{sphere}}+V_{\text{cylinder}}\hfill \\ \hfill & =\frac{4}{3}\pi R^2+\pi R^{2}L\hfill \\ L\hfill & =\frac{V_{\text{drop}}-\frac{4}{3}\pi R^3}{\pi R^2}\hfill \\ \hfill & =\frac{V_{\text{drop}}}{\pi R^2}-\frac{4}{3}R\hfill \end{array}$

which yields after substitution in Eq. 22.8

$\begin{array}{ll}{E}_{\text{surface}\text{tension}}\hfill & =\gamma \left(4\pi R^2+2\pi R\left(\frac{V_{\text{drop}}}{\pi R^2}-\frac{4}{3}R\right)\right)\hfill \\ \hfill & =\gamma \pi \left(\frac{V_{\text{drop}}}{\pi R^2}-\frac{4}{3}{R}^2\right)\hfill \end{array}$

Here the surface tension value used is the interfacial surface tension between the liquid of the drop and the surrounding immersion liquid.

Energy Contribution of the Rotation. The energy that is due to the rotation is given by

$d{E}_{\text{rotation}}=\frac{1}{2}\Delta \rho {\omega }^2{r}^{2}dV$

where we used the differences in density of the two liquids ∆ρ. The change of volume dV can be expressed as

$\begin{array}{ll}dV\hfill & =d{V}_{\text{cylinder}}+d{V}_{\text{sphere}}\hfill \\ \hfill & =2\pi Lrdr+d{V}_{\text{sphere}}\hfill \end{array}$

where we used Eq. 3.106 for the infinitesimal change in volume for a cylinder (see Fig. 22.6b). Determining dVsphere is a bit more complicated because we need to consider the hemisphere as segments weighted with their respective distance to the center axis (see Fig. 22.6c). This effectively splits the hemispheres into individual rings with different heights h that can be determined geometrically as

$h=\sqrt{R^2-r^2}$

dV_sphere can then be expressed as

$\begin{array}{ll}d{V}_{\text{sphere}}\hfill & =hd{A}_{\text{circle}}\hfill \\ \hfill & =\sqrt{R^2-r^2}2\pi rdr\hfill \end{array}$

where we used Eq. 3.104 for expressing the infinitesimal change of the circle area. Note that we need to multiply this equation by two in order to account for both hemispheres. Inserting Eq. 22.12 and Eq. 22.13 into Eq. 22.11 yields

$\begin{array}{ll}d{E}_{\text{rotation}}\hfill & =\frac{1}{2}\Delta \rho {\omega }^2{r}^2\left(2\pi Lrdr+2\sqrt{R^2-r^2}2\pi rdr\right)\hfill \\ d{E}_{\text{rotation}}\hfill & =\Delta \rho {\omega }^2\pi \left(L{r}^3+2\sqrt{R^2-r^2}\right)dr\hfill \\ \int d{E}_{\text{rotation}}\hfill & =\Delta \rho {\omega }^2\pi {\displaystyle {\int }_0^R\left(L{r}^3+\sqrt{R^2-r^2}{r}^3\right)dr}\hfill \\ E_{\text{rotation}}\hfill & =\Delta \rho {\omega }^2\pi \left(\frac{L}{4}{R}^4+\frac{4}{15}{R}^5\right)\hfill \end{array}$

In the last step we replace L using Eq. 22.9 resulting in

$\begin{array}{ll}{E}_{\text{rotation}}\hfill & =\Delta \rho {\omega }^2\pi \left(\frac{1}{4}\left(\frac{V_{\text{drop}}}{\pi }{R}^2-\frac{4}{3}{R}^5\right)+\frac{4}{15}{R}^5\right)\hfill \\ \hfill & =\Delta \rho {\omega }^2\pi \left(\frac{1}{4}\frac{V_{\text{drop}}}{\pi }{R}^2-\frac{1}{15}{R}^5\right)\hfill \end{array}$

Energy Minimum. In order for the drop to achieve a stable form, the energy contributions upon change of the droplet’s shape must be minimized. Therefore

$\frac{d}{dR}\left(E_{\text{surface}\text{tension}}+E_{\text{rotation}}\right)\stackrel{!}{=}0$

Inserting Eq. 22.10 and Eq. 22.15 into Eq. 22.15 yields

$0\stackrel{!}{=}\frac{d}{dR}\left(\gamma \pi \left(\frac{2V_{\text{drop}}}{\pi R}+\frac{4}{3}{R}^2\right)+\Delta \rho {\omega }^2\pi \left(\frac{1}{4}\frac{V_{\text{drop}}}{\pi }{R}^2-\frac{1}{15}{R}^5\right)\right)$

$\begin{array}{l}0\stackrel{!}{=}-\gamma \pi \left(-\frac{2V_{\text{drop}}}{\pi R^2}+\frac{8}{3}R\right)+\Delta \rho {\omega }^2\pi \left(\frac{1}{2}\frac{V_{\text{drop}}}{\pi }R-\frac{1}{3}{R}^4\right)\hfill \\ 0\stackrel{!}{=}-\gamma \pi 2\left(-\frac{V_{\text{drop}}}{\pi R^2}-\frac{4}{3}R\right)+\Delta \rho {\omega }^2\pi R^3\left(\frac{1}{2}\frac{V_{\text{drop}}}{\pi R^2}-\frac{1}{3}R\right)\hfill \\ 0\stackrel{!}{=}-\gamma \pi 2L+\Delta \rho {\omega }^2\frac{1}{2}\pi R^3\left(\frac{V_{\text{drop}}}{\pi R^2}-\frac{4}{3}R+\frac{2}{3}R\right)\hfill \\ 0\stackrel{!}{=}-\gamma \pi 2L+\Delta \rho {\omega }^2\frac{1}{2}\pi R^3\left(L+\frac{2}{3}R\right)\hfill \\ \gamma =\frac{1}{2L}\Delta \rho {\omega }^2\frac{1}{2}{R}^3\left(L+\frac{2}{3}R\right)\hfill \\ \gamma =\frac{\Delta \rho {\omega }^2{R}^3}{2}\left(\frac{1}{2}+\frac{1}{3}\frac{R}{L}\right)\hfill \end{array}$

Eq. 22.18 allows determining the surface tension simply by measuring R and L of the rotating drop.

Very Long Drops. At higher rotation speeds, the droplet formed is merely a cylinder and $\frac{R}{L}\to \infty$, in which case Eq. 22.18 simplifies to

$\gamma =\frac{\Delta \rho {\omega }^2{R}^3}{4}$

Here it is only required to measure the radius of the droplet formed in order to determine the surface tension.

A Comment to the Usage of L. You may have noticed that we did not replace L when performing the integrations for Esurface tension (see Eq. 22.8) and Erotation (see Eq. 22.14). However, we inserted L prior to differentiating when determining the energy minimum (see Eq. 22.17). This is due to the fact that we developed the energy contributions such that they resulted in a constant L, as we were increasing only the diameter of the cylinder and the radius of the hemispheres. However, during the determination of the energy minimum, both L and R will change in order to keep the volume V_drop constant. This is why we cannot treat L as a constant when deriving Eq. 22.17.

22.2.7 Measurement by Capillarity

Surface tensions can also be measured by capillarity (see section 21) and can be derived directly from the capillary rise in a tube. For this the capillary height h_c must be determined as well as the contact angle Θ. By using Eq. 21.3, the surface tension can then be calculated.

22.3.1 Fowkes Model

American physicist Frederik Fowkes who first suggested the concept laid out in Eq. 22.20 in 1962 [6] assumed the interaction to be primarily dispersive and polar. Therefore Eq. 22.21 is rewritten as

$\left(\cos \Theta +1\right){\gamma }_{\text{gas/liquid}}=2\sqrt{{\gamma }_{\text{gas/liquid}}^{\text{d}}{\gamma }_{\text{gas/liquid}}^{\text{d}}}+2\sqrt{{\gamma }_{\text{gas/liquid}}^{\text{p}}{\gamma }_{\text{gas/solid}}^{\text{p}}}$

Experimentally, the dispersive component is first determined by a measurement using diiodomethane, which has no polar contribution. Therefore Eq. 22.22 can be used to solve for ${\text{γ}}_{\text{gas/liquid}}^{\text{d}}$. In the next step, water is used in order to solve Eq. 22.22 for ${\text{γ}}_{\text{gas/liquid}}^{\text{p}}$. As ${\text{γ}}_{\text{gas/liquid}}^{\text{d}}$ is known, the equation is easy to solve.

Tab. 22.1 lists the liquids most commonly used for measuring the free surface energy. It lists the contributions to the surface tension of the test liquids as the polar and the disperse contribution. In the following, we will provide an example using the Fowkes method for the determination of the free surface energy of polystyrene.

Determining the Free Surface Energy of Polystyrene Using the Fowkes Method. Contact angle measurements are carried out on a sheet of freshly cut polystyrene. Using diiodomethane, we measure the following contact angles: 21°, 16°, 15°, and 11°. From these values, the mean contact angle of 15.75° is calculated. The free surface energy, i.e., the surface tension of diiodomethane, is found to be 50.8 J m⁻² with ${\text{γ}}_{\text{gas/liquid}}^{\text{d}}=50.8$ mN m⁻¹ and ${\text{γ}}_{\text{gas/liquid}}^{\text{p}}=0$ mN m⁻¹ (purely disperse) as shown in Tab. 22.1. Using Eq. 22.22, we obtain

$\begin{array}{ll}\left(\cos \left(15.75°\right)+1\right)50.8\text{mN}{\text{m}}^{-1}\hfill & =2\sqrt{50.8\text{mN}{\text{m}}^{-1}{\gamma }_{\text{gas/solid}}^{\text{d}}}+2\sqrt{0\text{mN}{\text{m}}^{-1}{\gamma }_{\text{gas/solid}}^{\text{p}}}\hfill \\ \hfill & =2\sqrt{50.8\text{mN}{\text{m}}^{-1}{\gamma }_{\text{gas/solid}}^{\text{d}}}\hfill \\ {\gamma }_{\text{gas/solid}}^{\text{d}}\hfill & =9.91\times {10}^{-6}\text{mN}{\text{m}}^{-1}\hfill \end{array}$

In a similar experiment using water as test liquid, the following contact angles were measured: 71°, 68°, 72°, and 71°. From these values, the mean contact angle is 70.5°. As shown in Tab. 22.1, the following values for water are obtained: 72.80 mN m⁻¹ with ${\text{γ}}_{\text{gas/liquid}}^{\text{d}}=21.8$ mN m⁻¹ and ${\text{γ}}_{\text{gas/liquid}}^{\text{p}}=51$ mN m⁻¹ (mixed but predominantly polar). Using Eq. 22.22, we obtain

$\begin{array}{ll}(\cos (70.5°)+1)72.80\text{mN}{\text{m}}^{-1}\hfill & =2\sqrt{21.8\text{J}{\text{m}}^{-2}\times {10}^{-6}\text{mN}{\text{m}}^{-1}}+2\sqrt{51{\text{Jm}}^{-2}{\text{γ}}_{\text{gas/solid}}^{\text{p}}}\hfill \\ {\text{γ}}_{\text{gas/solid}}^{\text{d}}\hfill & =36.44\text{mN}{\text{m}}^{-1}\hfill \end{array}$

From these two experiments, the total free surface energy of polystyrene can be calculated to be γ_gas/solid = 36.44 mN m⁻¹, which is very close to the values reported (see Tab. 22.2).

We can already observe an interesting fact about polystyrene: the polymer contributes almost no dispersal forces to the free surface energy. The value is dominated by the contribution of the polar forces. This is one of the reasons why polystyrene is such an interesting material, e.g., for cell culture experiments for which polystyrene is the de facto gold standard.

22.3.2 Extended Fowkes Model

Fowkes later extended his original approach taking into account hydrogen interaction in addition to dispersive and polar interaction [10]. Because we have three unknown contributions, a minimum of three measurements must be performed. Tab. 22.1 lists the values for the hydrogen interaction for commonly used test liquids.

22.3.3 Owens, Wendt, Rabel, and Kaelble (OWRK) Model

The models proposed by Owens and Wendt [11] as well by Rabel [12] and Kaelble [13] use the original Fowkes model but change the way the experimental results are read from the data. Essentially, by dividing Eq. 22.22 by $2\sqrt{{\text{γ}}_{\text{gas/liquid}}^{\text{d}}}$, the following equation is obtained:

$\frac{\left(\cos \Theta +1\right){\gamma }_{\text{gas/liquid}}}{2\sqrt{{\gamma }_{\text{gas/liquid}}^{\text{d}}}}=\sqrt{{\gamma }_{\text{gas/solid}}^{\text{d}}}+\frac{\sqrt{{\gamma }_{\text{gas/liquid}}^{\text{p}}}}{\sqrt{{\gamma }_{\text{gas/liquid}}^{\text{d}}}}\sqrt{{\gamma }_{\text{gas/solid}}^{\text{p}}}$

Please note that this is a function of type y = mx + c with $x=\sqrt{{\text{γ}}_{\text{gas/liquid}}^{\text{p}}}$ and $c=\sqrt{{\text{γ}}_{\text{gas/liquid}}^{\text{d}}}$. For different liquids tested, for which we know both ${\text{γ}}_{\text{gas/liquid}}^{\text{p}}$ and ${\text{γ}}_{\text{gas/liquid}}^{\text{d}}$, we will obtain a linear function with the slope $\sqrt{{\text{γ}}_{\text{gas/liquid}}^{\text{p}}}$ from which we can directly obtain ${\text{γ}}_{\text{gas/liquid}}^{\text{p}}$. Extrapolating this line to the intersection with the vertical axis will give us ${\text{γ}}_{\text{gas/liquid}}^{\text{d}}$.

22.3.4 Wu Model

Wu suggested a different method for calculating the contribution of the polar and the disperse forces [14, 15]. According to Wu, Eq. 22.22 should be expressed as

$\left(\cos \Theta +1\right){\gamma }_{\text{gas/liquid}}=4\left(\frac{{\gamma }_{\text{gas/liquid}}^{\text{d}}{\gamma }_{\text{gas/solid}}^{\text{d}}}{{\gamma }_{\text{gas/liquid}}^{\text{d}}+{\gamma }_{\text{gas/solid}}^{\text{d}}}+\frac{{\gamma }_{\text{gas/liquid}}^{\text{p}}{\gamma }_{\text{gas/solid}}^{\text{p}}}{{\gamma }_{\text{gas/liquid}}^{\text{p}}+{\gamma }_{\text{gas/solid}}^{\text{p}}}\right)$

In this approximation the contributions are included by means of their geometric mean values. The experimental procedure is identical to the Fowkes method.

22.3.5 van Oss, Chaudhury, and Good (OCG) Model

Van Oss, Chaudhury, and Good reformulate the original Fowkes model and detail the nature of the polar interaction [16]. The polar interaction is associated with the action of donor acceptor pairs with two terms that reflect the concentration of donors in the liquid and acceptors in the surface and v.v. Furthermore, the dispersive contributions are corrected to account for Debye and Keesom interaction in addition to London dispersion. The resulting contribution is referred to as the Lifshitz/Van der Waals contribution ${\text{γ}}_{\text{gas/liquid}}^{\text{LW}}$. Eq. 22.22 is thus expanded to

$\left(\cos \Theta +1\right){\gamma }_{\text{gas/liquid}}=2\sqrt{{\gamma }_{\text{gas/liquid}}^{\text{LW}}{\gamma }_{\text{gas/liquid}}^{\text{LW}}}+2\sqrt{{\gamma }_{\text{gas/liquid}}^{\text{+}}{\gamma }_{\text{gas/solid}}^{\text{-}}}+2\sqrt{{\gamma }_{\text{gas/liquid}}^{\text{1}}{\gamma }_{\text{gas/solid}}^{\text{+}}}$

Occasionally, a reduced model is also used in which the dispersive contributions are unchanged compared to the Fowkes model. This model is then often referred to as van Oss and Good model. Again, Tab. 22.1 lists examples of the contribution of the Lifshitz/Van der Waals contribution for commonly used testing liquids.

22.3.6 Zisman Extrapolation

American chemist William Zisman was one of the most important scientists in surface physics. He did a number of important experiments in surface characterization and invented a number of important techniques. One of these is the concept of the critical surface tension γ_c. From this concept the Zisman extrapolation arises.

Zisman observed that when plotting the cosine of the static contact angle cos Θ of liquids with different surface tensions on the same solid, the values fall on a straight line [17]. This is true only for liquids that do not undergo strong hydrogen bonding, i.e., for liquids with lower surface tension. By extrapolating this line to cos Θ = 1, i.e., to a contact angle of Θ = 0, one can find the critical surface tension. It is defined as the surface tension a liquid would need to have in order to be made completely wet on this surface.

The critical surface tension is an important technique for characterizing materials because it is, in contrast to the free surface energies, a material constant and does not depend on a second fluid. Fig. 22.7 shows an example of such a Zisman extrapolation on a polymer surface made from a perfluorinated polyether acrylate. The liquids used are water, ethanol, dichloromethane, ethylene glycol, acetophenone, and isopropanol for which the surface tensions are shown in Tab. 20.2. The calculated critical surface tension is around γ_c ≈ 12 mN m⁻¹, which highlights one important property of polytetrafluoroethylene, of all fluorinated polymers: they have very low critical surface tensions (see Tab. 22.3). This means that in order for a liquid to spread on a fluorinated surface, the liquid must have a very low surface tension. Not many liquids fall into this category. The Zisman extrapolation is an exception to the techniques commonly used for characterizing the free energy of a surface. It does not account for the individual contributions of disperse, polar, or similar interactions to the overall surface energy. It also does not measure the free surface energy but, as stated, the critical surface tension, which in most cases amounts to almost the same value. The Zisman extrapolation should be applied only if one predominant interaction is expected, e.g., for the characterization of the wetting behavior of hydrocarbons on polymers which are both dominated by disperse forces. If this is not the case, the Zisman extrapolation tends to be inexact.

The Zisman extrapolation is the only relevant one-component method, as it does not measure the individual contributions to the free surface energy but assesses them as only one effect. As a result, all other techniques discussed so far are classified as two-component methods (if they account for two contributions, e.g., polar and disperse) or as multi-component methods if they include more than two effects.