Adsorption of pairwise interacting atoms: on the derivation of the interaction parameters between first and second neighbors from experimental data

Origin of 1-pK and 2-pK Models for Ionizable Water−Solid Interfaces

“…By considering a discrete charge Ising model, we have shown that simple mean-field models, which are commonly referred to as the 1-p K or the 2-p K model, usually provide an excellent description of the ionization process of water−solid interfaces. The success of such a simple mean-field description may seem surprising; for Ising models on various two-dimensional lattices it is well-known that mean-field models give poor results; particularly, if small clusters such as the ones we have employed here are considered. , There is, however, one fundamental difference between the Ising models discussed often in the literature and the present situation. The site−site interaction potentials of the Ising models studied in the literature are usually short range, mostly just of the nearest neighbor type. ,, This means that, only between sites that are nearest neighbors, the energy of interaction is nonzero.…”

“…The success of such a simple mean-field description may seem surprising; for Ising models on various two-dimensional lattices it is well-known that mean-field models give poor results; particularly, if small clusters such as the ones we have employed here are considered. , There is, however, one fundamental difference between the Ising models discussed often in the literature and the present situation. The site−site interaction potentials of the Ising models studied in the literature are usually short range, mostly just of the nearest neighbor type. ,, This means that, only between sites that are nearest neighbors, the energy of interaction is nonzero. In the context of ionization problems, such short-range potentials are typical for polyelectrolytes.…”

“…In the case of nearest neighbor pair interactions (short-range interactions), for a square lattice a mere four nearest neighboring sites account for all interactions. In such a situation, a mean-field model performs poorly. ,

4 Cumulative sum of the pair energies for the square lattice as a function of total number of nearest neighbors. Note the slow convergence to the plateau value which is also given in the first row of Table .

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Defining reactive sites on hydrated mineral surfaces: Rhombohedral carbonate minerals

“…Nevertheless, the quantitative characterization of the reactivity of individual primary surface sites is far from trivial because the proper calibration of these multi-site, multi-reaction models requires: (i) uniform and/or well-characterized mineral surfaces in terms of chemical composition and microtopography (Barrow et al, 1993;Piasecki et al, 2001), (ii) suitable and sufficient experimental data arising from various independent sources and carrying sufficient information to properly resolve the energetic contributions of individual surface sites (Rudziń ski et al, 1992(Rudziń ski et al, , 1998Piasecki et al, 2001), and (iii) the application of sophisticated mathematical treatments (Chandler, 1987;Jäger, 1991;Borkovec and Koper, 1994). For instance, it is well known that, because of compensating effects, the composite adsorption (or surface charge) isotherms obtained from titration experiments that are typically used for the calibration of adsorption chemical models, are largely insensitive to surface energetic heterogeneity, and therefore, additional data (e.g., potentiometric, electrokinetic, radiometric, calorimetric) are required to properly discriminate among potential heterogeneity models and prevent misleading over-interpretations of available data (van Riemsdijk et al, 1987;Blesa and Kallay, 1988;Cerník et al, 1995;Rudziń ski et al, 1992Rudziń ski et al, , 1998Lü tzenkirchen, 2005).…”