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+---
+title: "Wetting"
+firstLetter: "W"
+publishDate: 2021-03-29
+categories:
+- Physics
+- Fluid mechanics
+
+date: 2021-03-29T16:20:44+02:00
+draft: false
+markup: pandoc
+---
+
+# Wetting
+
+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](/know/concept/young-dupre-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.
+