Hard-core ionic model

Figure 2. Two morphological models used to describe the origin of the ionic SAXS maximum observed for Nafion (a) the modified hard-sphere model depicting interparticle scattering and (b) the depleted-zone core—shell model depicting intraparticle scattering. (Adapted with permission from ref 36. Copyright 1981 American Chemical Society.)...

In practice, this hard-core model is too simple to predict reliably the ground-state structure of ionic compounds such as the alkali halides that are located in the upper left-hand corner of the AB structure map in Fig. 1.9. Nevertheless, it provides a simple introduction to the importance of the radius ratio in determining structural stability. 

In many physically important cases of localized adsorption, each adatom of the compact monolayer covers effectively n > 1 adsorption sites [3.87-3.89, 3.98, 3.122, 3.191, 3.214, 3.261]. Such a multisite or 1/n adsorption can be caused by a crystallographic Me-S misfit, i.e., the adatom diameter exceeds the distance between two neighboring adsorption sites, and/or by a partial charge of adatoms (A < 1 in eq. (3.2)), i.e., a partly ionic character of the Meads-S bond. The theoretical treatment of a /n adsorption differs from the description of the 1/1 adsorption by a simple Ising model. It implies the so-called hard-core lattice gas models with different approximations [3.214, 3.262-3.266]. Generally, these theoretical approaches can only be applied far away from the critical conditions for a first order phase transition. In addition, Monte Carlo simulations are a reliable tool for obtaining valuable information on both the shape of isotherms and the critical conditions of a 1/n adsorption [3.214, 3.265-3.267]. 

Short-range repulsion A first improvement is related to the repulsive forces which become effective at short inter-ionic distances. In the original model, steric - or hard-core - repulsive forces prevent two ions i and J from coming closer than the sum of their ionic radii r,- and rj. The short-range repulsion energy is infinite if Rij < r, + ry, and zero otherwise, which may be written in the form ... 

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