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 θ\theta.

If θ=0\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 θ=π\theta = \pi, such that the droplet “floats” on the surface, which in the specific case of water is called hydrophobia. Furthermore, θ<π/2\theta < \pi/2 is high wettability, and π/2<θ<π\pi/2 < \theta < \pi is low wettability.

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

αsgαsl=αglcosθ\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 rr as the relative increase of the surface’s area due to its rough structure, where ArealA_{real} and AappA_{app} are the real and apparent areas:

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

The net energy cost EE of spreading the droplet over the surface is then given by:

Esl=(αsgαsl)Areal=αglArealcosθ=αglAapprcosθ=αglAappcosθ\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 θ\theta^* as the correction to θ\theta to account for the roughness. It is expressed as follows:

cosθ=rcosθ\begin{aligned} \boxed{ \cos\theta^* = r \cos\theta } \end{aligned}

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

E=Aapp(f(αsgαsl)(1f)αgl)=Aappαgl(fcosθ+f1)\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” αsg\alpha_{sg} and “get back” αsl\alpha_{sl}, while for the gas-liquid interface, we spend nothing, but get αgl\alpha_{gl}. The apparent angle θ\theta^* is therefore:

cosθ=f(cosθ+1)1\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 θ1\theta_1 and θ2\theta_2. The energy cost of the interface is then given by:

E=A(f1(αs1gαs1l)+(1f1)(αs2gαs2l))=Aαgl(f1cosθ1+(1f1)cosθ2)\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 θ\theta^* for an inhomogeneous surface is given by this equation, called Cassie’s law:

cosθ=f1cosθ1+(1f1)cosθ2\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 r1r_1, r2r_2, etc.


  1. T. Bohr, Continuum physics: lecture notes, 2021, unpublished.