Debye counterion atmosphere

A first attempt to consider the role of the Debye counterion atmosphere on the transport of a surfactant ion through the DL was made by Mikhailovskij (1976, 1980) (cf. Kortilm 1966, Lyklema 1991). In contrast to a macro-kinetic model, Mikhailovskij derived kinetic equations for a multi-component system under the influence of an external electric field. The basis of this derivation was the set of Bogolubow equations for the partial distribution functions. As the result of the model derivation the following set of electro-diffusion equations is obtained. 

Some discussion is needed for the case of ionic surfactants. The charged micelles experience electrostatic interactions and they are not exactly hard spheres. They can be represented as such by taking into account the Debye counterion atmosphere. In this case the... 

As shown by Debye and Hiickel, due to the strong electrostatic interaction between the ions in a solution, the positions of the ions are correlated in such a way that a counterion atmosphere appears aronnd each ion, thns screening its Coulomb potential. The energy of formation of the counterion atmospheres contribntes to the free energy of the system called correlation energy. ... 

As expected, the D-H theory tells us that ions tend to cluster around the central ion. A fundamental property of the counterion distribution is the thickness of the ion atmosphere. This thickness is determined by the quantity Debye length or Debye radius (1/k). The magnitude of 1/k has dimension in centimeters, as follows ... 

FIG. 8 A calculated cross-section of the cylindrically symmetric distribution of condensed counterions held in common by a pair of identical parallel rodlike polyions when the partition function for the condensed layer is interpreted as a free volume. The numerical scale is in A, and the polyions pierce the page at 15 A. The Debye length equals 30 A, so the theoretically calculated condensed layer lies inside the Debye atmosphere, as required on physical grounds. Polymer charge spacing 1.7 A. 

Two kinds of counterions, condensed and those constituting a diffuse ion atmosphere that may be treated in the Debye-Htickel approximation, are clearly recognized in their distinct spatial distributions, and introduction of the partial polarizability tensor enables us to distinguish between contributions to the polarizability from these two kinds of ions. The contribution from condensed counterions to the radial components of the polarizability tensor is very small, as has hitherto often been postulated in various theories. That from the diffuse ion atmosphere is very large and cannot be neglected in the calculation of the anisotropy. 

As the electrolyte concentration is low and the Debye radius many times exceeds this distance an identification of the potential in this zone with the potential of the adsorbing ions is reasonable. At high electrolyte concentration the diffuse layer thickness can be comparable to this distance. Even if the counterions are indifferent their distribution in this layer cannot be neglected because it decreases the electrostatic component of surfactant ion adsorption., i.e. enhance its adsorption. In this case we have to consider a discreteness of charges must, the formation of a counter ion atmosphere around the adsorbed ions and their overlap with the neighbour adsorbed ions. 

The investigations of Mikhailovskij are significant for the discussion of the limits of the macro-kinetic approach and their extension, which can be established by the analysis of the transition from Eqs (7.74) to Eq. (7.75). To do so, Mikhailovskij concluded, that the influence of the Debye atmosphere of counterions on the transport of a higher valency ion through the diffuse layer can be neglected at sufficiently low background electrolyte concentrations. This limit, of course, has to be defined for different valences of surfactant ions, different surface activity etc. 

In the broad field of physical chemistry, the Boltzmann distribution law is fundamental to the derivation of the Debye-Htickel theory of electrolyte solutions. In the more narrow arena of interfacial and colloid science, it is applied to the determination of the ionic atmosphere around charged interfaces. In that context, the charge cloud is more commonly referred to as the electrical double layer (EDL). The concept is illustrated schematically (Fig. 5.2) for the situation in which a particle possesses an evenly distributed charge that is just balanced by the total opposite charge, the counterions in the electrical double layer. 

The concentration and nature of the electrolyte also has a significant impact on the stability of charged colloid dispersions. This was discussed in Section 3.3.2, where the concept of electric double layers was introduced. The electric double layer results from the atmosphere of counterions around a charged colloid particle. The decay of the potential in an electric double layer is governed by the Debye screening length, which is dependent on electrolyte concentration (Eq. 3.8). In the section that follows, the stability of charged colloids is analysed in terms of the balance between the electrostatic (repulsive) forces between double layers and the (predominantly attractive) van der Waals forces. 

A rigorous analysis of the model indicates that two modes of interaction must be considered the Debye-Hiickel ion atmosphere (adapted to cylindrical geometry), and the condensation of counterions on the line charge. 

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