Ionic clustering, theory

The remainder of Section I is devoted to a rather brief review of earlier work in the field in order to gain a little perspective. In Sections II to IV the basic results of the cluster method are derived. In Section V a very brief account of the application of the formal equations to some systems with short-range forces is given. Section VI is devoted to a review of the application to systems with Coulomb forces between defects, where the cluster formalism is particularly advantageous for bringing the discussion to the level of modern ionic-solution theory.86 Finally, in Section VII a brief account is given of Mayer s formalism for lattice defects69 since it is in certain respects complementary to that principally discussed here. We would like to emphasize that the material in Sections V and VI is illustrative of the method. This is not meant to be an exhaustive review of results obtainable. 

Friedman, H. L., Ionic Solution Theory Based on Cluster Expansion Methods, Interscience Publishers, New York, 1962. 

At a quantitative level, near criticality the FL theory overestimates dissociation largely, and WS theory deviates even more. The same is true for all versions of the PMSA. In WS theory the high ionicity is a consequence of the increase of the dielectric constant induced by dipolar pairs. The direct DD contribution of the free energy favors pair formation [221]. One can expect that an account for neutral (2,2) quadruples, as predicted by the MC studies, will improve the performance of DH-based theories, because the coupled mass action equilibria reduce dissociation. Moreover, quadrupolar ionic clusters yield no direct contribution to the dielectric constant, so that the increase of and the diminution of the association constant becomes less pronounced than estimated from the WS approach. Such an effect is suggested from dielectric constant data for electrolyte solutions at low T [138, 139], but these arguments may be subject to debate [215]. We note that according to all evidence from theory and MC simulations, charged triple ions [260], often assumed to explain conductance minima, do not seem to play a major role in the ion distribution. 

The abundant experimental data on ionic clusters reacting with neutral molecules has been used to test some of these collision theories. In the next subsection, we briefly review several papers where comparisons between measured and theoretical rate coefficients have been made, and we summarize some of the important conclusions concerning the reactivity of clusters. 

We will finish this paragraph by stating that the promising and very frequently used density functional theory (DFT) [6] is not generally applicable for molecular complexes. The reason for this is that it does not cover the intersystem correlation interaction energy, approximately equivalent to the classical dispersion energy. The DFT method yields reliable results for H-bonded and ionic clusters but fails completely in London-type clusters where the dispersion energy is dominant. 

Pickard, F.C., Dunn, M.E., Shields, G.C. Comparison of model chemistry and density functional theory thermochemical predictions with experiment for formation of ionic clusters of the ammonium cation complexed with water and ammonia atmospheric implications. J. Phys. Chem. A 2005,109(22), 4905-10. 

The only characterization technique that can directly view heterogeneities of the order of 1 nm, i.e., the size of the ionic clusters is electron microscopy. There have been very few electron microscopy studies of ionomer microstructure and these have failed to provide a satisfactory picture of ionic clusters (106). With the advancements that have been made in recent years in the optics and theories of electron microscopy, microscopists are in a better position to reexamine these materials (See the chapter by Williams). 

Ion clusters are commonly observed in the ionized forms of the perfluorinated membranes. The size of the clusters appears to be larger for sulfonate than for carboxylate membranes." " The size increases in the order Na, and Cs" and decreases with increasing number of functional groups per chain and with increasing temperature.As in the case of ethylene ionomers, the perfluorinated carboxylic acid membranes do not form ion clusters, at least in the dry state." The electrostatic interaction may be too weak to form ionic clusters. These observations are expected according to the Eisenberg theory (see Section II.2). 

Reference 7 reviews a number of electron microscopy studies of ionomer morphology in the period up to 1979. None of these studies makes a convincing case for the direct imaging of ionic clusters. This is because of the small size of the clusters (less than 5 nm based on scattering studies) and difficulties encountered in sample preparation. The entire problem was reexamined in 1980(21). In this study ionomers based on ethylene-methacrylic acid copolymers, sulfonated polypentenamer, sulfonated polystyrene and sulfonated ethylene-propylene-diene rubber (EPDM) were examined. The transfer theory of imaging was used to interpret the results. Solvent casting was found to produce no useful information about ionic clusters, and microtomed sections showed no distinct domain structure even in ionomers neutralized with cesium. Microtomed sections of sulfonated EPDM, however,... 

A model for ionic clustering in "Nafion" (registered trademark of E. I. du Pont de Nemours and Co.) perfluorinated membranes is proposed. This "cluster-network" model suggests that the solvent and ion exchange sites phase separate from the fluorocarbon matrix into inverted micellar structures which are connected by short narrow channels. This model is used to describe ion transport and hydroxyl rejection in "Nafion" membrane products. We also demonstrate that transport processes occurring in "Nafion" are well described by percolation theory. 

T.D. Gierke, G.E. Munn and EC. Wilson, Morphology of Nafion perfluorosulfonated membrane products, as determined by wide- and small-angle X-ray studies, J. Polym. Sci., Polym. Phys. Ed., 1981, 19, 1687-1704 W.Y. Hsu and T.D. Gierke, Elastic theory for ionic clustering in perfluorinated ionomers, Macromolecules, 1982, 15, 101-105. 

As the polarity difference between the two blocks increases, (xAB)cr the Krause (1969,1970) theory (see Section 4.7.1) will become smaller, or alternatively, phase separation will occur for shorter block lengths. A limiting case in this argument involves the carboxylic rubbers and ionomers, in which one block consists of isolated single salt (ionic) mers, incorporated in an otherwise hydrocarbon-type backbone. It should be emphasized that the ionic block length is one monomer unit long. Phase separation and formation of ionic clusters occurs because of the extreme difference in... 

Friedman HL (1962) Ionic solution theory based on cluster expansion methods, vol 3, Monographs in statistical physics. Wiley, New York... 

Hsu WY, Gierke TD (1982) Elastic theory for ionic clustering in perlluorinated ionomers. Macromolecules 15 101-105... 

Pisani [169] has used the density of states from periodic FIP (see B3.2.2.4) slab calculations to describe the host in which the cluster is embedded, where the applications have been primarily to ionic crystals such as LiE. The original calculation to derive the external Coulomb and exchange fields is usually done on a finite cluster and at a low level of ab initio theory (typically minimum basis set FIP, one electron only per atom treated explicitly). 

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