Cassie's law

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;

are the contact angles for components 1 with fractional surface area

If there exist more than two materials then the equation is scaled to the general form of;

[4] Cassie's law takes on special meaning when the heterogeneous surface is a porous medium.

air gaps, such that the surface is no longer completely wet.

The Cassie-Baxter equation is more common in nature, and focuses on the 'incomplete coating' of surfaces by a liquid only.

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.

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.

as the free energy change per unit area caused by a liquid spreading,

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

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

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.

In 1936 Young's equation was modified by Robert Wenzel to account for rough homogeneous surfaces, and a parameter

was introduced, defined as the ratio of the true area of the solid compared to its nominal.

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.

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.

It is widely agreed that the water repellency of biological objects is due to the Cassie-Baxter equation.

, ultra low water adhesion due to minimal contact areas, and a self cleaning property which is characterised by the Cassie-Baxter equation.

[13] The microscopic architecture of the Lotus leaf means that water will not penetrate nanofolds on the surface, leaving air pockets below.

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.