Quadratic elongation and the variance of bond angles are linearly correlated for distorted octahedral and tetrachedral corrdination complexes, both of which show variations in bond length and bond angle. The quadratic elongation is dimensionless, giving a quantitative measure of polyhedral disortion which is independent of the effective size of the polyhedron.

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Expressions are developed for the components of the linear Lagrangian, linear Eulerian, finite Lagrangian (Green's), and finite Eulerian (Almansi's) strain tensors in terms of a crystal's lattice parameters before and after a deformation. The development has been undertaken with the concepts and notations of linear algebra.

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Previous studies show the demineralization of biogenic, amorphous, and crystalline forms of silica is enhanced in the presence of alkali and alkaline earth cations. The origins of this effect are difficult to explain in light of work suggesting predominantly weak outer-sphere type interactions between these ions and silica. Here we investigate the ability of M(II) aqua ions to promote the hydrolysis of Si-O bonded interactions relative to ion-free water using electronic structure methods. Reaction pathways for Si-O hydrolysis are calculated with the B3LYP and PBE1PBE density functionals at the 6-31G(d) and 6-311+G(d,p) levels in the presence of water, and both inner- and outer-sphere adsorption complexes of Mg(2+)(6H(2)O) and Ca(2+)(6H(2)O). All reaction trajectories involving hydrated ions are characterized by one or more surmountable barriers associated with the rearrangement of ion-associated water molecules, and a single formidable barrier corresponding to hydrolysis of the Si-O bonded interaction. The hydrolysis step for outer-sphere adsorption is slightly less favorable than the water-induced reaction. In contrast, the barrier opposing Si-O hydrolysis in the presence of inner-sphere species is generally reduced relative to the water-induced pathway, indicating that the formation of inner-sphere complexes may be prerequisite to the detachment of Si species from highly coordinated surface sites. The results suggest a two-part physical model for ion-promoted Si-O hydrolysis that is consistent with experimental rate measurements. First, a bond path is formed between the cation and a bridging oxygen site on the silica surface that weakens the surrounding Si-O interactions, making them more susceptible to attack by water. Second, Si-O hydrolysis occurs adjacent to these inner-sphere species in proportion to the frequency of ion-associated solvent reorganization events. Both processes are dependent upon the particular ion hydration environment, which suggests measured cation-specific demineralization rates arise from differential barriers opposing reorganization of ion-associated solvent molecules at the silica-water interface.