Goniometer method

Sketch of the contact angle, as seen by a goniometer. In the top picture, the volume of the drop is being increased, and in the bottom it is being decreased. Each notated angle is an instance of a similar contact angle.

The simplest way of measuring the contact angle of a sessile drop is with a contact angle goniometer, which allows the user to measure the contact angle visually. A droplet is deposited by a syringe which is positioned above the sample surface, and a high resolution camera captures the image from the profile or side view. The image can then be analyzed either by eye (with a protractor) or more often is measured using image analysis software. This type of measurement is referred to as a static contact angle measurement.^[1]

The contact angle is affected not only by the surface chemistry but also by the surface roughness. The Young equation, which is the basis for the contact angle, assumes a homogeneous surface with no surface roughness. In case surface roughness is present, the droplet can be in Wenzel state (homogeneous wetting), Cassie-Baxter state (heterogeneous wetting) or in an intermediary state. The surface roughness amplifies the wetting behavior caused by the surface chemistry.

In order to measure the contact angle hysteresis, the sessile droplet can be increased gradually in volume. The maximum possible contact angle is referred to as the advancing contact angle. The receding contact angle can be measured by removing volume from the drop until dewetting occurs. The minimum possible contact angle is referred to as the receding contact angle.^[1] The contact angle hysteresis is the difference between the advancing and receding contact angle.^[1]

Wilhelmy method

An alternative method for measuring the contact angle is the Wilhelmy method, which employs a sensitive force meter of some sort to measure a force that can be translated into a value of the contact angle. In this method, a small plate-shaped sample of the solid in question, attached to the arm of a force meter, is vertically dipped into a pool of the probe liquid (in actuality, the design of a stationary force meter would have the liquid being brought up, rather than the sample being brought down), and the force exerted on the sample by the liquid is measured by the force meter. This force is related to the contact angle by the following equation:

${\displaystyle \cos \theta ={\frac {F-F_{\text{b}}}{I\sigma }},}$

where F is the total force measured by the force meter, F_b is the force of buoyancy due to the solid sample displacing the liquid, I is the wetted length, and σ is the known surface tension of the liquid.

The Zisman theory

The Zisman theory is the simplest commonly used theory, as it is a one-component theory, and is best used for non-polar surfaces. This means that polymer surfaces that have been subjected to heat treatment, corona treatment, plasma cleaning, or polymers that contain heteroatoms do not lend themselves to this particular theory, as they tend to be at least somewhat polar. The Zisman theory also tends to be more useful in practice for surfaces with lower energies.

The Zisman theory simply defines the surface energy as being equal to the surface energy of the highest surface energy liquid that wets the solid completely. That is to say, the droplet will disperse as much as possible, i.e. completely wetting the surface, for this liquid and any liquids with lower surface energies, but not for liquids with higher surface energies. Since this probe liquid could hypothetically be any liquid, including an imaginary liquid, the best way to determine the surface energy by the Zisman method is to acquire data points of contact angles for several probe liquids on the solid surface in question, and then plot the cosine of that angle against the known surface energy of the probe liquid. By constructing the Zisman plot, one can extrapolate the highest liquid surface energy, real or hypothetical, that would result in complete wetting of the sample with a contact angle of zero degrees.

The Owens/Wendt theory

The Owens/Wendt theory^[4] (after D. K. Owens and R. C. Wendt) divides the surface energy into two components: surface energy due to dispersive interactions and surface energy due to polar interactions. This theory is derived from the combination of Young's relation, which relates the contact angle to the surface energies of the solid and liquid and to the interface tension, and Good's equation (after R. J. Good), which relates the interface tension to the polar and dispersive components of the surface energy. The resulting principle equation is

${\displaystyle {\frac {\sigma _{\text{L}}(\cos \theta +1)}{2{\sqrt {\sigma _{\text{L}}^{\text{D}}}}}}={\frac {{\sqrt {\sigma _{\text{S}}^{\text{p}}}}{\sqrt {\sigma _{\text{L}}^{\text{p}}}}}{\sqrt {\sigma _{\text{L}}^{\text{D}}}}}+{\sqrt {\sigma _{\text{S}}^{\text{D}}}}.}$

Note that this equation has the form of y = mx + b, with

${\displaystyle {\begin{aligned}y&={\frac {\sigma _{\text{L}}(\cos \theta +1)}{2{\sqrt {\sigma _{\text{L}}^{\text{D}}}}}},\\m&={\sqrt {\sigma _{\text{S}}^{\text{p}}}},\\x&={\frac {\sqrt {\sigma _{\text{L}}^{\text{P}}}}{\sqrt {\sigma _{\text{L}}^{\text{D}}}}},\\b&={\sqrt {\sigma _{\text{S}}^{\text{D}}}}.\end{aligned}}}$

As such, the polar and dispersive components of the solid's surface energy are determined by the slope and intercept of the resulting graph. Of course, the problem at this point is that in order to make that graph, knowing the surface energy of the probe liquid is not enough, as it is necessary to know specifically how it breaks down into its polar and dispersive components as well.

To do this, one can simply reverse the procedure by testing the probe liquid against a standard reference solid that is not capable of polar interactions, such as PTFE. If the contact angle of a sessile drop of the probe liquid is measured on a PTFE surface with

${\displaystyle {\begin{aligned}\sigma _{\text{S}}^{\text{p}}&=0,\\\sigma _{\text{S}}^{\text{D}}&=18.0~{\text{mN}}/{\text{m}},\end{aligned}}}$

the principle equation reduces to

${\displaystyle \sigma _{\text{L}}^{\text{D}}={\frac {\sigma _{\text{L}}^{2}(\cos \theta +1)^{2}}{72}}.}$

Since the total surface tension of the liquid is already known, this equation determines the dispersive component, and the difference between the total and dispersive components gives the polar component.

The Fowkes theory

The Fowkes theory (after F. M. Fowkes) is derived in a slightly different way from the Owens/Wendt theory, although the Fowkes theory's principle equation is mathematically equivalent to that of Owens and Wendt:

${\displaystyle {\frac {1}{2}}\sigma _{\text{L}}(\cos \theta +1)={\sqrt {\sigma _{\text{S}}^{\text{p}}}}{\sqrt {\sigma _{\text{L}}^{\text{p}}}}+{\sqrt {\sigma _{\text{S}}^{\text{D}}}}{\sqrt {\sigma _{\text{L}}^{\text{D}}}}.}$

Note that by dividing both sides of the equation by ${\displaystyle {\sqrt {\sigma _{\text{L}}^{\text{D}}}}}$, the Owens/Wendt principle equation is recovered. As such, one of the options for the proper determination of the surface energy components is the same.

In addition to that method, it is also possible to simply do tests using liquids with no polar component to their surface energies, and then liquids that do have both polar and dispersive components, and then linearize the equations (see table 1). First, one performs the standard sessile drop contact angle measurement for the solid in question and a liquid with a polar components of zero (${\displaystyle \sigma _{\text{L}}^{\text{p}}=0}$; ${\displaystyle \sigma _{\text{L}}=\sigma _{\text{L}}^{\text{D}}}$) The second step is to use a second probe liquid that has both a dispersive and a polar component to its surface energy, and then solve for the unknowns algebraically. The Fowkes theory generally requires the use of only two probe liquids, as described above, and the recommended ones are diiodomethane, which should have no polar component due to its molecular symmetry, and water, which is commonly known to be a very polar liquid.

The Wu theory

The Wu theory (after Souheng Wu) is also essentially similar to the Owens/Wendt and Fowkes theories, in that it divides surface energy into a polar and a dispersive component. The primary difference is that Wu uses the harmonic means rather than the geometric means of the known surface tensions, and subsequently the use of more rigorous mathematics is employed.

The Schultz theory

The Schultz theory (after D. L. Schultz) is applicable only for very high energy solids. Again, it is similar to the theories of Owens, Wendt, Fowkes, and Wu, but is designed for a situation where conventional measurement required for those theories is impossible. In the class of solids with sufficiently high surface energy, most liquids wet the surface completely with a contact angle of zero degrees, and thus no useful data can be gathered. The Schultz theory and procedure calls to deposit a sessile drop of probe liquid on the solid surface in question, but this is all done while the system is submerged in yet another liquid, rather than being done in the open air. As a result, the higher “atmospheric” pressure due to the surrounding liquid causes the probe liquid droplet to compress so that there is a measurable contact angle.

The van Oss theory

The van Oss theory^[5] separates the surface energy of solids and liquids into three components. It includes the dispersive surface energy, as before, and subdivides the polar component as being the sum of two more specific components: the surface energy due to acidic interactions (${\displaystyle \sigma ^{+}}$) and due to basic interactions (${\displaystyle \sigma ^{-}}$). The acid component theoretically describes a surface's propensity to have polar interactions with a second surface that has the ability to act basic by donating electrons. Conversely, the base component of the surface energy describes the propensity of a surface to have polar interactions with another surface that acts acidic by accepting electrons. The principle equation for this theory is

${\displaystyle \sigma _{\text{L}}(\cos \theta +1)=2\left[{\sqrt {\sigma _{\text{L}}^{\text{D}}\sigma _{\text{S}}^{\text{D}}}}+{\sqrt {\sigma _{\text{L}}^{-}\sigma _{\text{S}}^{+}}}+{\sqrt {\sigma _{\text{L}}^{+}\sigma _{\text{S}}^{-}}}\right].}$

Again, the best way to deal with this theory, much like the two-component theories, is to use at least three liquids (more can be used to get more results for statistical purposes) – one with only a dispersive component in its surface energy (${\displaystyle \sigma _{\text{L}}=\sigma _{\text{L}}^{\text{D}}}$), one with only a dispersive and an acidic or basic component (${\displaystyle \sigma _{\text{L}}=\sigma _{L}^{\text{D}}+\sigma _{\text{L}}^{\pm }}$), and finally either a liquid with a dispersive and a basic or acidic component (whichever the second probe liquid did not have (${\displaystyle \sigma _{\text{L}}=\sigma _{\text{L}}^{\text{D}}+\sigma _{\text{L}}^{\mp }}$)), or a liquid with all three components (${\displaystyle \sigma _{\text{L}}=\sigma _{\text{L}}^{\text{D}}+\sigma _{\text{L}}^{+}+\sigma _{\text{L}}^{-}}$) – and linearizing the results.

It is naturally more robust than other theories, particularly in cases where there is a great imbalance between the acid and base components of the polar surface energy. The van Oss theory is most suitable for testing the surface energies of inorganics, organometallics, and surface containing ions.

The most significant difficulty of applying the van Oss theory is the fact that there is not much of an agreement in regards to a set of reference solids that can be used to characterize the acid and base components of potential probe liquids. There are however some liquids that are generally agreed to have known dispersive/acid/base components to their surface energies. Two of them are listed in table 1.