is the Debye length, κ−1. It is reciprocally proportional to the square root of the ion concentration C. In aqueous solutions it is typically on the scale of a few nanometers and the thickness decreases with increasing concentration of the electrolyte. The electric field strength inside the DL can be anywhere from zero to over 109 V/m. These steep electric potential gradients are the reason for the importance of the DLs. The theory for a flat surface and a symmetrical electrolyte is usually referred to as the Gouy-Chapman theory. It yields a simple relationship between electric charge in the diffuse layer σd and the Stern potential Ψd: There is no general analytical solution for mixed electrolytes, curved surfaces or even spherical particles. There is an asymptotic solution for spherical particles with low charged DLs. In the case when electric potential over DL is less than 25 mV, the so-called Debye-Huckel approximation holds. It yields the following expression for electric potential Ψ in the spherical DL as a function of the distance r from the particle center: There are several asymptotic models which play important roles in theoretical developments associated with the interfacial DL. The first one is "thin DL". This model assumes that DL is much thinner than the colloidal particle or capillary radius. This restricts the value of the Debye length and particle radius as following: This model offers tremendous simplifications for many subsequent applications. Theory of electrophoresis is just one example. The theory of electroacoustic phenomena is another example. The thin DL model is valid for most aqueous systems because the Debye length is only a few nanometers in such cases. It breaks down only for nano-colloids in solution with ionic strengths close to water. The opposing "thick DL" model assumes that the Debye length is larger than particle radius: This model can be useful for some nano-colloids and non-polar fluids, where the Debye length is much larger. The last model introduces "overlapped DLs". This is important in concentrated dispersions and emulsions when distances between particles become comparable with the Debye length. Electrical double layers The electrical double layer (EDL) is the result of the variation of electric potential near a surface, and has a significant influence on the behaviour of colloids and other surfaces in contact with solutions or solid-state fast ion conductors. The primary difference between a double layer on an electrode and one on an