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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 surfaces wetability, 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;

# Cassie-Baxter

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 'incomplete coating' of surfaces by a liquid only. In the Cassie-Baxter state liquids sit upon asperities, resulting in air pockets that are bounded between the surface and liquid.

# Homogeneous surfaces

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]

# Equation origin

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.

The contact angle for the heterogeneous surface is given by,

The contact angle given by the Young equation is,

Thus by substituting the first expression into Young's equation, we arrive at Cassie's law for heterogeneous surfaces,

# History behind Cassie's law

Studies concerning the contact angle existing between a liquid and a solid surface began with Thomas Young in 1805.[7] The Young equation

reflects the relative strength of the interaction between surface tensions at the three phase contact, and is the geometric ratio between the energy gained in forming a unit area of the solid-liquid interface to that required to form a liquid air interface.[1] However Young's equation only works for ideal and real surfaces and in practice most surfaces are microscopically rough.

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.[5] They devised the Cassie-Baxter equation;

In 1948 Cassie refined this for two materials with different chemistries on both smooth and rough surfaces, resulting in the aforementioned Cassie's law

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.

# Examples

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]