Abstract
The benefit of computational methods applying density functional theory for the description and understanding of modulated crystal structures is investigated. A method is presented which allows one to establish, improve and test superspace models including displacive and occupational modulation functions from first-principles calculations on commensurate structures. The total energies of different configurations allow one to distinguish stable and less stable structure models. The study is based on a series of geometrically optimized superstructures of mullite (Al4+2xSi2-2xO10-x) derived from the superspace group Pbam(α0½)0ss. Despite the disordered and structurally complex nature of mullite, the calculations on ordered superstructures are very useful for determining the ideal Al/Si ordering in mullite, extracting atomic modulation functions as well as understanding the SiO2-Al2O3 phase diagram. The results are compared with experimentally established models which confirm the validity and utility of the presented method.
Highlights
Computational methods have become a valuable tool in crystallography, partly triggered by the steadily improving computer power
The coordinates of the most stable structures are only slightly displaced from the expected coordinates. These results are taken as an indication that a qualitative assessment of the relative stability of a certain supercell is possible with the FF calculations, but for the determination of the ideal Al/Si ordering more accurate density functional theory (DFT) calculations must be carried out
In combination with the known occupational modulation parameters of O3 and O4 (Klar et al, 2017b) the exact parameters to describe the specific Al/Si ordering scheme can be determined because in the present example Si22 is always bonded to O3b (ÁSbiw22 = ÁObw3b, tbSwi22 = tbOw3b) and Al22 only occurs in triclusters together with Al3 (ÁAbwl22 = ÁAbwl3, Figure 6 x2 of T site of M40 #1 (PBE)
Summary
Computational methods have become a valuable tool in crystallography, partly triggered by the steadily improving computer power. To study the structure of highly disordered meta-kaolin an 18fold supercell with 282 atoms was used for calculations applying density functional theory (DFT) to support the refinement of the pair distribution function (White et al, 2010). The size of those systems quickly exceeds the possibilities of current ab initio calculations on modern clusters, algorithms were developed to increase the number of atoms per unit cell to a few thousand (Goedecker, 1999; Mohr et al, 2018). As there are several fundamental questions concerning the crystal structure and the phase diagram of mullite, it is an ideal playground to study the capabilities and benefits of firstprinciples calculations on modulated structures
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