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. Contact angles are important as they quantify a surface's wettability, the nature of solid-fluid intermolecular interactions. Cassie's law is reserved for when a liquid completely covers both smooth and rough heterogeneous surfaces.
More of a rule than a law, the formula found in literature for two materials is;
where and are the contact angles for components 1 with fractional surface area, and 2 with fractional surface area in the composite material respectively. If there exist more than two materials then the equation is scaled to the general form of;
, with.
Cassie-Baxter
Cassie's law takes on special meaning when the heterogeneous surface is a porous medium. now represents the solid surface area and air gaps, such that the surface is no longer completely wet. Air creates a contact angle of and because =, the equation reduces to:, which is the Cassie-Baxter equation.
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 '
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.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.Defining as the free energy change per unit area caused by a liquid spreading,
where, are the fractional areas of the two materials on the heterogeneous surface, and and the interfacial tensions between solid, air and liquid.
The contact angle for the heterogeneous surface is given by,
, with the interfacial tension between liquid and air.
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
Young's law
Studies concerning the contact angle existing between a liquid and a solid surface began with Thomas Young in 1805. The Young equationreflects 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. However Young's equation only works for ideal and real surfaces and in practice most surfaces are microscopically rough.
Wenzel state
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. Known as the Wenzel equation,shows that the apparent contact angle, the angle measured at casual inspection, will increase if the surface is roughened. Liquids with contact angle are known to be in the Wenzel state.
Cassie-Baxter state
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;, sometimes written as where the has become.