Categories: Fluid mechanics, Fluid statics, Physics, Surface tension.

In fluid statics, **wetting** is the ability of a given liquid to touch a given surface. When a droplet of the liquid is placed on the surface, the **wettability** determines the contact angle \(\theta\).

If \(\theta = 0\), we have **perfect** or **complete wetting**: the droplet spreads out over the entire surface. The other extreme is **dewetting** or **non-wetting**, where \(\theta = \pi\), such that the droplet “floats” on the surface, which in the specific case of water is called **hydrophobia**. Furthermore, \(\theta < \pi/2\) is **high wettability**, and \(\pi/2 < \theta < \pi\) is **low wettability**.

For a perfectly smooth homogeneous surface, \(\theta\) is determined by the Young-Dupré relation:

\[\begin{aligned} \alpha_{sg} - \alpha_{sl} = \alpha_{gl} \cos\theta \end{aligned}\]

In practice, however, surfaces can be rough and/or inhomogeneous. We start with the former.

A rough surface has some structure, which may contain “gaps”. There are two options: either the droplet fills those gaps (a **Wenzel state**), or it floats over them (a **Cassie-Baxter state**).

For a Wenzel state, we define the **roughness ratio** \(r\) as the relative increase of the surface’s area due to its rough structure, where \(A_{real}\) and \(A_{app}\) are the real and apparent areas:

\[\begin{aligned} r = \frac{A_{real}}{A_{app}} \end{aligned}\]

The net energy cost \(E\) of spreading the droplet over the surface is then given by:

\[\begin{aligned} E_{sl} &= (\alpha_{sg} - \alpha_{sl}) A_{real} = \alpha_{gl} A_{real} \cos\theta \\ &= \alpha_{gl} A_{app} r \cos\theta = \alpha_{gl} A_{app} \cos\theta^* \end{aligned}\]

Where we have defined the **apparent contact angle** \(\theta^*\) as the correction to \(\theta\) to account for the roughness. It is expressed as follows:

\[\begin{aligned} \boxed{ \cos\theta^* = r \cos\theta } \end{aligned}\]

For Cassie-Baxter states, where the gaps remain air-filled, we define \(f\) as the “non-gap” fraction of the apparent surface, such that:

\[\begin{aligned} E &= A_{app} \big( f (\alpha_{sg} - \alpha_{sl}) - (1 - f) \alpha_{gl} \big) \\ &= A_{app} \alpha_{gl} \big( f \cos\theta + f - 1 \big) \end{aligned}\]

Note the signs: for the solid-liquid interface, we “spend” \(\alpha_{sg}\) and “get back” \(\alpha_{sl}\), while for the gas-liquid interface, we spend nothing, but get \(\alpha_{gl}\). The apparent angle \(\theta^*\) is therefore:

\[\begin{aligned} \boxed{ \cos\theta^* = f (\cos\theta + 1) - 1 } \end{aligned}\]

We generalize this equation to inhomogeneous surfaces consisting of two materials with contact angles \(\theta_1\) and \(\theta_2\). The energy cost of the interface is then given by:

\[\begin{aligned} E &= A \big( f_1 (\alpha_{s1g} - \alpha_{s1l}) + (1 - f_1) (\alpha_{s2g} - \alpha_{s2l}) \big) \\ &= A \alpha_{gl} \big( f_1 \cos\theta_1 + (1 - f_1) \cos\theta_2 \big) \end{aligned}\]

Such that \(\theta^*\) for an inhomogeneous surface is given by this equation, called **Cassie’s law**:

\[\begin{aligned} \boxed{ \cos\theta^* = f_1 \cos\theta_1 + (1 - f_1) \cos\theta_2 } \end{aligned}\]

Note that the materials need not be solids, for example, if one is air, we recover the previous case for rough surfaces. Cassie’s law can also easily be generalized to three or more materials, and to include Wenzel-style roughness ratios \(r_1\), \(r_2\), etc.

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