Cassie's law, or the Cassie equation, describes the effective contact angle θc for a liquid on a chemically heterogeneous surface, i.e. the surface of a composite material consisting of different chemistries, that is, non-uniform throughout.[1] Contact angles are important as they quantify a surface's wettability, the nature of solid-fluid intermolecular interactions.[2] Cassie's law is reserved for when a liquid completely covers both smooth and rough heterogeneous surfaces.[3]
More of a rule than a law, the formula found in literature for two materials is;
\cos\thetac=\sigma1\cos\theta1+\sigma2\cos\theta2
where
\theta1
\theta2
\sigma1
\sigma2
\cos\thetac
N | |
=\sum | |
k=1 |
\sigmak\cos\thetak
N | |
\sum | |
k=1 |
\sigmak=1
Cassie's law takes on special meaning when the heterogeneous surface is a porous medium.
\sigma1
\sigma2
180\circ
\cos(180)
-1
\cos\thetacb=\sigma1\cos\theta1-\sigma2
Unfortunately the terms Cassie and Cassie-Baxter are often used interchangeably but they should not be confused. The Cassie-Baxter equation is more common in nature, and focuses on the '
The Cassie-Baxter equation is not restricted to only chemically heterogeneous surfaces, as air within porous homogeneous surfaces will make the system heterogeneous. However, if the liquid penetrates the grooves, the surface returns to homogeneity and neither of the previous equations can be used. In this case the liquid is in the Wenzel state, governed by a separate equation. Transitions between the Cassie-Baxter state and the Wenzel state can take place when external stimuli such as pressure or vibration are applied to the liquid on the surface.[6]
When a liquid droplet interacts with a solid surface, its behaviour is governed by surface tension and energy. The liquid droplet could spread indefinitely or it could sit on the surface like a spherical cap at which point there exists a contact angle.
Defining
E
E=\sigma1(\gamma
s1a |
-\gamma | |
s1l |
)+\sigma2(\gamma
s2a |
-\gamma | |
s2l |
)
where
\sigma1
\sigma2
\gammasa
\gammasl
The contact angle for the heterogeneous surface is given by,
\cos\thetac=
E | |
\gammala |
\gammala
The contact angle given by the Young equation is,
cos\theta
|
Thus by substituting the first expression into Young's equation, we arrive at Cassie's law for heterogeneous surfaces,
\cos\thetac=\sigma1\cos\theta1+\sigma2\cos\theta2
Studies concerning the contact angle existing between a liquid and a solid surface began with Thomas Young in 1805.[7] The Young equation
cos
\theta | ||||
|
In 1936 Young's equation was modified by Robert Wenzel to account for rough homogeneous surfaces, and a parameter
r
\cos\thetaw=r\cos\thetay
shows that the apparent contact angle, the angle measured at casual inspection, will increase if the surface is roughened. Liquids with contact angle
\thetaw
The notion of roughness effecting the contact angle was extended by Cassie and Baxter in 1944 when they focused on porous mediums, where liquid does not penetrate the grooves on rough surface and leaves air gaps. They devised the Cassie-Baxter equation;
\cos\thetac=\sigma1\cos\theta1-\sigma2
\cos\thetac=\sigma1(\cos\theta1+1)-1
\sigma2
(1-\sigma1)
In 1948 Cassie refined this for two materials with different chemistries on both smooth and rough surfaces, resulting in the aforementioned Cassie's law
\cos\thetac=\sigma1\cos\theta1+\sigma2\cos\theta2
Following the discovery of superhydrophobic surfaces in nature and the growth of their application in industry, the study of contact angles and wetting has been widely reexamined. Some claim that Cassie's equations are more fortuitous than fact with it being argued that emphasis should not be placed on fractional contact areas but actually the behaviour of the liquid at the three phase contact line.[10] They do not argue never using the Wenzel and Cassie-Baxter's equations but that “they should be used with knowledge of their faults”. However the debate continues, as this argument was evaluated and criticised with the conclusion being drawn that contact angles on surfaces can be described by the Cassie and Cassie-Baxter equations provided the surface fraction and roughness parameters are reinterpreted to take local values appropriate to the droplet.[11] This is why Cassie's law is actually more of a rule.
It is widely agreed that the water repellency of biological objects is due to the Cassie-Baxter equation. If water has a contact angle between
0<\theta<90\circ
90\circ<\theta<180\circ
150\circ<\theta
One example of a superhydrophobic surface in nature is the Lotus leaf.[12] Lotus leaves have a typical contact angle of
\theta\sim160\circ
The Cassie–Baxter wetting regime also explains the water repellent features of the pennae (feathers) of a bird. The feather consists of a topography network of 'barbs and barbules' and a droplet that is deposited on a these resides in a solid-liquid-air non-wetting composite state, where tiny air pockets are trapped within.[14]