Reciprocal thickness of double layer

The first equation simply considers the effect of an applied field E on a point charge q balanced by the Stokes viscous drag and applies when particle velocity and a the particle diameter. The second equation, due to Smoluchowski is restricted in its applicability to cases where the radius of the particle is much greater than the thickness of the double layer (i.e., Ka>>l). The third incorporates cases 1 and 2 in an approach due to Henry( l). The reciprocal thickness of the double layer is given by... 

The double layer force, experienced by two approaching bodies, is determined by the potential of each body P0i and P02> the distance between them 6, the reciprocal thickness of the double layer x, and the dielectric constant of the liquid . 

When the ionic strength of the aqueous medium is increased, for instance by adding a salt, the electrical double layer is compressed. An approximation of the reciprocal thickness of the double layer, k, is given by the Debye-HUckel... 

By an accurate mathematical treatment, resembling Huckel s method, but now taking account fully of the deformation of the external field by the presence of the colloidal particle, Henry was able to show that the electrophoretic velocity is in all cases equal to/e C E/ti v), the numerical factor / depending upon the value of a, where a is the radius of the spherical or cylindrical particle (Henry considered only these two shapes of the particle) and represents the reciprocal thickness of the double layer, being equal to... 

One of the most important quantities to emerge from the Debye-Huckel approximation is the parameter k. This quantity appears throughout double-layer discussions and not merely at this level of approximation. Since the exponent kx in Equation (37) is dimensionless, k must have units of reciprocal length. This means that k has units of length. This last quantity is often (imprecisely) called the thickness of the double layer. All distances within the double layer are judged large or small relative to this length. Note that the exponent kx may be written x/k a form that emphasizes the notion that distances are measured relative to k in the double layer. 

Equation (1.9) is the linearized Poisson-Boltzmann equation and k in Eq. (1.10) is the Debye-Htickel parameter. This linearization is called the Debye-Hiickel approximation and Eq. (1.9) is called the Debye-Hiickel equation. The reciprocal of k (i.e., 1/k), which is called the Debye length, corresponds to the thickness of the double layer. Note that nf in Eqs. (1.5) and (1.10) is given in units of m . If one uses the units of M (mol/L), then must be replaced by IQQQNAn, Na being Avogadro s number. 

The quantity k is known to have the dimensions of a reciprocal length (p. 84), and l//c may be identified with the effective thickness d of the double layer if, at the same time, the electrical potential p in equation (9) is replaced by the elcctrokinetic potential f, this equation becomes... 

The last equation shows that the energy in PBS-dFFF is a function of the size and of the surface potential of the particle, of the Hamaker constant, and of the ionic strength of the carrier solution, as the reciprocal double-layer thickness is immediately related to the ionic strength of the suspending medium. Thus, selectivity in PBSdFFF results from differences in particle size or chemical composition of the particles and of the suspending medium, where the latter will affect the surface potential and the Hamaker constant of the particle, as well as the medium s ionic strength. 

A here is the potential at the surface (x = 0). The parameter k, can be thought of as the reciprocal of the double-layer thickness, 1/k, that is, the extent of counterion distribution away from the surface. This double-layer thickness is controlled, according to the theory, by certain variables in the clay-solution system as described by the equation... 

It will be recalled from Chapter 7 that has the dimensions of a reciprocal length and may thus be set equal to 1/8, 8 being the thickness of the Helmholtz double layer. Thus it may be seen that the expression utilized by Helmholtz for the potential of a parallel plate condenser... 

Debye Length A parameter in the Debye—Hiickel theory of electrolyte solutions, k-1. For aqueous solutions at 25 °C, k = 3.288y7 in reciprocal nanometers, where I is the ionic strength of the solution. The Debye length is also used in the DLVO theory, where it is referred to as the electric double-layer thickness. See also Electric Double-Layer Thickness. 

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