Date Published: March 01, 2019
Publisher: International Union of Crystallography
Author(s): Paul Benjamin Klar, Iñigo Etxebarria, Gotzon Madariaga.
Quantitative parameters of modulated crystal structures are determined from density functional theory calculations applied to superstructures obtained using the superspace formalism. The example of mullite is used to demonstrate that with this approach versatile aspects ranging from ordering phenomena to phase diagrams can be understood on a new level.
Computational methods have become a valuable tool in crystallography, partly triggered by the steadily improving computer power. Examples include relativistic Hartree–Fock calculations of atomic form factors as listed in the International Tables for X-ray Crystallography (Doyle & Turner, 1968 ▸), the estimation of anisotropic displacement parameters (ADPs) from phonon calculations (Madsen et al., 2013 ▸) or the investigation of phase diagrams (Wang et al., 2005 ▸). All computational methods applied to crystal structure models require invariably fully occupied sites because fractional atoms lack a physical basis. If there are partially occupied atomic sites in the average structure, large defect structures can be generated with random or systematic approaches (Proffen & Neder, 1997 ▸; Okhotnikov et al., 2016 ▸) for further investigation including the simulation of diffuse scattering (Proffen & Neder, 1997 ▸) or molecular dynamics (Matsui, 1996 ▸; Lacks et al., 2005 ▸; Adabifiroozjaei et al., 2018 ▸). For example, to study the structure of highly disordered meta-kaolin an 18-fold 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, although algorithms were developed to increase the number of atoms per unit cell to a few thousand (Goedecker, 1999 ▸; Mohr et al., 2018 ▸). For even larger systems, force-field (FF) methods are the only choice as these are significantly less demanding in terms of computational cost, but the accuracy is limited and depends strongly on the reliability of the FF parameters with which the atomic and molecular interactions are modelled (Leach, 2001 ▸). Nowadays, DFT methods have become the standard tool to study the structural and electronic properties of crystals when accuracy is essential (Sholl & Steckel, 2009 ▸). First-principles calculations have been applied successfully to a wide range of crystals: from systematic searches of potentially stable phases in multiferroic materials (Diéguez et al., 2011 ▸) to the validation of the structures of molecular crystals (van de Streek & Neumann, 2014 ▸).
In this section these results are compared with experiments. For that purpose, the DFT calculations of the energetically most stable superstructures of Section 3 were repeated with higher precision and applying dispersion correction (PBEsol-D), which is expected to improve the agreement between the calculated and experimental lattice parameters as pointed out in Section 2.2. Furthermore, the unified superspace model of the last section was used to predict additional superstructures with different compositions (M11, M14, M20, M43, M45), including compositions that are expected to be unstable according to the phase diagram (M11, M14). Selected characteristics of the superstructures are given in Table 1 ▸. These calculations with the ideal Al/Si ordering are labelled AS1.
The benefit of first-principles calculations for the investigation of modulated structures was demonstrated using the example of mullite. A systematic study based on FF and DFT calculations on supercells of different compositions and with different Si distributions allowed us to determine for the first time the details of the ideal Al/Si ordering pattern in mullite for vacancy concentrations 0 ≤ x ≤ 0.5. Quantitative modulation functions describing the displacive and occupational modulation were determined and compared, which indicated that the ordering mechanisms for all compositions are the same. On this basis a unified superspace model for the investigated solid solution range of mullite was established. A comparison with modulation functions based on X-ray diffraction experiments indicates that the applied method correctly determined the underlying Al/Si ordering pattern of the most ordered state of mullite. The sole energetical analysis of the solid solution range cannot explain the phase diagram of mullite. Nevertheless, a mechanism focusing on the structural flexibility due to the presence of oxygen vacancies was suggested to explain the observed stability range of mullite. Klar et al. (2018 ▸) showed that the superspace group Pbam(α0½)0ss allows one to derive the vacancy distribution pattern in mullite. As a consequence, the average structure and the superspace symmetry are sufficient to establish a superspace model from first principles.
For additional literature relating to the supporting information, see Burt et al. (2006 ▸) and Winter & Ghose (1979 ▸).