Discreteness-of-charge effect

Barlow, C. A., and J. R. MacDonald, Theory of discreteness of charge effects in the electrolyte compact double layer, AE, 6, 1 (1967). 

A. P. Winiski, A. C. McLaughlin, R. V. McDaniel, M. Eisenberg, and S. McLaughlin, An experimental test of the discreteness-of-charge effect in positive and negative lipid bilayers,... 

The influence of specific adsorption of nonreacting ions, chiefly anions, upon electrode kinetics at Hg electrodes is available from experiment and theory " . Equation (a) often proves to be approximately applicable, at least when the difference between the reaction plane and the outer Helmholtz plane are taken into account. However, marked discrepancies are noted that are attributed to local interactions between the reactant and the absorbing ions, including discreteness-of-charge effects - - . ... 

Additional alterations in the work terms with the electrode material for outer-sphere reactions may arise from discreteness-of-charge effects or from differences in the nature of the reactant-solvent interactions in the bulk solution and at the reaction plane. Thus metals that strongly chemisorb inner-layer solvent (e.g., HjO at Pt) also may alter the solvent structure in the vicinity of the outer plane, thereby influencing k bs variations in the stability of the outer-sphere precursor (and successor) states. Such an effect has been invoked to explain the substantial decreases (up to ca. 10 -fold) in the rate constants for some transition-metal aquo couples seen when changing the electrode materiaf from Hg to more hydrophilic metals such as Pt. Much milder substrate effects are observed for the electroreduction of more weakly solvated ammine complexes . 

A Model of the Electric Double Layer at a Completely Ionized Monolayer with Discreteness-of-Charge Effect... 

Tphe discreteness-of-charge effect (discrete-ion effect) is a general char-acteristic of electric double layers in aqueous media (I) and therefore should manifest itself in ionized monolayers. In a number of papers (2,3,4,5), one of the authors and co-workers investigated the role of this... 

In addition, the GCS model involves average potentials in the vicinity of the electrode and ignores the discrete nature of charges in solution. Such discreteness of charge effects have been treated and invoked to account for failures in the usual double-layer corrections (67). 

In the simple GCSG model already described, the charge densities are assumed uniformly smeared over each plane rather than in the form of discrete ions. Levine and co-workers [8] modified the GCSG model by introducing the discreteness of charge effect to account for various phenomena, mostly relating to adsorption at the mercury/water and silver iodide/water interfaces. Their theory shows that the electrostatic work of adsorption at the IHP is not zet /p but... 

The Frumkin correction in Eq. (24) reduces the EDL structure effect to its single characteristic, i/ i potential, whde real reactants represent a distribution of electric charges. A recent development has taken into account this distribution in the calculation of the work term, so that it depends on the whole profile of the potential across the EDL. Besides, the presence of the reactant perturbs this potential profile in the vicinity of the species (a kind of discreteness-of-charge effects [48] discussed in Sect. 2.1.11.3), due to both the field created by its charges and the cavity effects arising from the replacement of solvent molecules in the volume occupied by the solute species [63, 64]. 

This paradox (which remains unresolved even if the adsorption plane is shifted into the compact layer) represented a strong stimulus for the development of the ion adsorption theory. It was shown [249] that it originated from the calculation of the work term in Eq. (71) as an average potential in the adsorption plane. The qualitative explanation of experimental observations can be achieved, if one takes into account that the transfer of the ion into the adsorption site is accompanied by a redistribution of the other adsorbed ions that creates a hole for it with the diminished repulsive potential. This inapplicability of the approximation of the uniform distribution of the adsorbed ion charge along the plane is called discreteness-of-charge effects, which must be taken into account in the theory of this phenomenon. 

Another crucial factor is an adequate inclusion of these diffuse-layer and discreteness-of-charge effects into the model, which play a marked role if the overall electrolyte concentration is well below 1 M. It is why the adsorption parameters found by the interpretation of experimental data with the use of various isotherms derived mostly for uncharged species (Flory-Huggins, Frumldn, Temldn, etc) may strongly deviate from their proper values. 

The aforementioned diffuse-layer and discreteness-of-charge effects have been taken into consideration in the model proposed by Grahame and Parsons [26,250-252]. First, it was assumed (unlike in the Stern model) that the specifically adsorbed ions were located at the distance from the metal surface (in the inner Helmholtz plane ) ensuring their maximum bond strength, owing to the combination of forces of electrostatic and quantum-mechanical origins. It shows the need for the partial or even complete desolvation of the adsorbed species and its deep penetration into the compact layer. The position of this adsorption plane depends on all components of the system, metal, solvent, and adsorbed ion. 

Y.K. Leong, Yield stress and zeta potential of nanoparticulate silica dispersions under the influence of adsorbed hydrolysis products of metal ions—Cu(II), Al(III) and Th(IV). J. Colloid Interface Sci. 292(2), 557-566 (2005). doi 10.1016/j.jcis.2005.06.004 S. Levine, D. Calvert, G.M. Bell, Discreteness-of-charge effect in electric double layer theory. 

A more stringent approximation under certain conditions is the neglect of discreteness-of-charge effects including the finite size of charge carriers such as ions. This approximation may be expected to be weakest when the charges under consideration are completely blocked and when they may be specifically adsorbed at an electrode as well. This matter will be further considered in Section III. 

Following conventional practice, one may separate ea h AG into a chemical part independent of applied potential potential dependent part, ez X V or -ez X V. The X quantities introduced here are usually Bf t e order%f unity or less and are included to account at least ipproximately for discreteness of charge effects. Thus, X V are micr potentials rather than the macropote tial, V, Separation of AG yields AG. E ... 

For large surface charges (for example for ionized monolayers spread at an oil(0)/water(w) interface) discreteness of charge effect have to be considered [491 The counterions are then... 

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