Microscopic strain, macroscopic strain and the thermodynamics of phase transitions in minerals
Almost any change in the local structure of a crystal, such as replacement of one cation by another of different size, reordering of cations between neighbouring crystallographic sites, ordering of magnetic dipoles, Jahn–Teller distortions etc., will give rise to two types of strain. Firstly, there will be a unit cell scale strain which will decay in magnitude as it extends away from the point where the structural change has occurred. Secondly there will be a net strain at a much longer length scale such as might be detected in the average lattice parameters of hundreds to thousands of unit cells. Both will be associated with a change in elastic energy. If the microscopic strain fields remain isolated or overlap in an uncorrelated way, the increase in elastic energy would be associated predominantly with local elastic strain heterogeneity. If the local strain fields interact in a correlated or collective manner, the change in elastic energy could be understood in terms of macroscopic distortions of a homogeneous material. The former might provide an appropriate model for solid solution formation, while the latter describes more or less what happens in a phase transition.
For a complete description of solid state processes in minerals, it is necessary to understand the length scales over which such strain fields operate for a given type of structural change. In the limit of local strain fields only extending to first and second nearest neighbours, say, a model based on only a few unit cells might be sufficient to account
Figures & Tables
The present book shows the arguments which have been considered in the EMU School (No. 4), dealing with Energy Modelling in Minerals; these arguments have been selected in order to provide examples of application of the most advanced theories to several cases. It should be pointed out that although the ultimate solution of our problems should involve “ab initio” quantum-mechanical calculations, at present such sophisticated procedures are far from being routine. Therefore, although “ab initio” approaches will play an ever-increasing role in the future and some important and most recent examples of such approaches are illustrated here, the greatest part of the contributions is dealing with empirical atom-atom calculations. Remarkably enough, such “semi-empirical” applications are often quite successful, providing excellent (or comparatively excellent) results in spite of their more or less approximate nature. It often happens that the methods here illustrated are some steps ahead of the current level of empirical treatment, thereby indicating a possible way of improvement by figuring out routines to be adopted in practice. If some methods seem to be too speculative to be actually usable, here they also are shown, in view of their possible discussion, or just to indicate a way to obtain promising developments. Among the descriptions of practical methods and results, some purely theoretical arguments have been inserted; these arguments — although abstract — according to our opinion are fundamental for earth scientists. Owing to the present status of the art, in a number of arguments there is no unique opinion with respect to their theoretical treatment as it is explained by different authors. Instead of having all of them discarded except the one which looks to be the most appropriate to the Editor (who might sometimes be personally involved in the question), most of such controversial points have been left just as they are, in the original draft of their advocates. Accordingly, the reader might find some discrepancies between some articles and others, which may lead to some obscurity; there are, however, several good reasons in favour of our behaviour. First of all, with a few exceptions we apologize about, our attention in inviting the contributors has been extended to all the principal authors in the world, with no limitation to a group of particular friends; moreover, the presence of different opinions in the context might give rise to interesting debates and critical objections; a further point is that the validity of the different treatments is shown per se by either the level of the theory and most of all by the agreement with the corresponding experimental data. Since we have to do with an advanced school, and in line with what should be a scientific procedure, it is important to provide the user with the possibility of choosing what seems to be the most appropriate method among a number of selected possibilities, rather than yielding to the assertion that something is indeed the ultimate and unquestionable “truth”.