POSESIÓN EFECTIVA

There have been numerous interatomic potential based studies performed on ceria and zirconia in recent years (44-48) while the number of ab initio quantum mechanical studies is more limited. The reason for this is clear: until recently, calculations o f this nature were far too expensive to be cost effective, meaning that substandard basis sets generally had to be used. Because o f this, results from interatomic potential (IP) studies were often more reliable and accurate than the ab initio studies. Ab initio calculations

* These concentrations were chosen based on the earlier work by Glushkova et al. (82) where they found that a minimum o f 8 mol% yttria was needed to stabilise c-ZrO: at room temperature.

Chapter 4: Fluorite

are usually preferable to IP calculations since they make no assumptions about the nature of the interactions between the ions in the material. As these interactions define the overall chemistry of the material, the detailed analysis permitted by QM studies can be of great importance in better understanding the chemical (catalytic) properties o f the material.

The work discussed in this thesis forms the first joint (periodic) Hartree-Fock study o f both ceria and zirconia. Although similar work has been performed on either ceria or zirconia independently, such as the papers by Hill and Catlow (49); Stefanovich et al. (50), and Orlando et a l (51) for example, these studies are not directly comparable to one another in the same way as the calculations reported here.

It is therefore important to reinvestigate these two oxides with a more accurate procedure, and in a consistent manner to enable a direct comparison o f the calculated properties o f ceria and zirconia. In this section, I shall provide a brief summary o f the important conclusions from papers in the literature most relevant to the current study.

4.2.1 Ceria

Few studies have been performed on either the bulk or surface structure o f ceria employing a Hartree-Fock approach, and little more with DFT methods. The only HF study of note is that by Hill and Catlow in 1993 (49). Even as recently as eight years ago, it was only possible to perform such calculations using minimal basis sets on the ions. In actual fact, this particular work employed a minimal (ST0-3G) basis set on the oxygen ions along with a minimal basis set for a cerium atom; such a combination is “unbalanced”. It is our belief that such basis functions would not permit a sufficiently large enough variational freedom of the orbitals on the anions: the only unoccupied orbitals present in this description o f ceria are the outermost valence orbitals on the cerium ions, therefore these orbitals must be used to reproduce the chemical properties of the material as a whole.

Hill and Catlow concluded that ceria is a partially covalent insulator, with a cubic lattice parameter of 5.385 Â and a bulk modulus o f 357 GPa; the (Mulliken) charge assigned to the cations was +2.35 |e| (and therefore -1.17 |e| to the anions), and the calculated HF band gap reported as 11.25 eV. They quote an experimental value, determined by Wuilloud et al. using XPS measurements (52), o f between 4-5 eV. More

Chapter 4: Fluorite

specifically, the metal c/-band dispersion was 5.80 eV and that of the oxygen p-bands, 8 . 8 6 eV.

The literature contains several well developed (and much tested) potential sets which have been used to successfully model the bulk and surface properties o f ceria: a comprehensive IP study of all {hkl} surfaces in which + P' < 20 was perfomied by Vyas, etal. (46,48) using the two potential sets he derived (and are listed for reference on page 124, in tables 6.1 and 6.2). In this work, Vyas concluded that potential set 1 best described the properties o f simple surfaces and o f the bulk material - including defect structures and simulated bulk annealing studies; potential set 2 was created to improve upon potential set 1 in studying high index faces where the original potential set failed. Vyas et al. used slabs having a (region I) thickness of approximately 25 Â, much larger than is possible in any QM study at present. This potential set was also used more recently by Baudin et al. (53), who performed a Molecular Dynamics (MD) simulation of the {011} and {111} surfaces of ceria.

The Vyas study was extended in the course of the current calculations to provide additional data not reported in the literature; chapter 6 details our extended IP study of the {0 1 1} and {1 1 1} surfaces examined here, while various structural properties of the bulk are reported later in this discussion for reference.

4.2.2 Zirconia

Zirconia currently has many more practical applications than ceria, due in part to its abundance on Earth which makes zirconia the cheaper of the two. This has created a greater driving force towards performing ab initio calculations on zirconia-based systems, despite their extreme computational cost.

Orlando et al. examined the cubic and tetragonal phases o f bulk zirconia, and also the {011} surface. Although ab initio results were desirable, the low symmetry of the non-cubic phases and the thickness of the surface slabs studied, meant that lower quality basis functions had to be used. Their basis set was not minimal, however it did make use o f a large-core ECP on the Zr ions: a choice which leaves only the outermost four metal ion electrons to be studied explicitly in the calculations. Their choice substantially reduces the cost o f the calculations, but may significantly reduce the confidence we can

Chapter 4: Fluorite

In the work by Stefanovich, et al. (50), both periodic HF and IP calculations were performed on pure and doped zirconia systems; the aim of this work being to analyse the energetics of the phase transitions occurring between the three ambient pressure phases. The periodic HF calculations performed in this paper made use of the same small-core Hay-Wadt pseudopotential used here, with the valence basis functions reported in their earlier paper (54); for cost reasons a Durand-Barthelat pseudopotential (55) was also applied to the oxygen ions.

Their analysis o f the non-cubic phases vWll be studied in greater detail later in this thesis; the study o f the cubic phase attributed a (Mulliken) charge of +3.038 |e| to the Zr ions (and therefore -1.519 |e| to the oxygen ions) in a cubic unit cell with a lattice parameter o f 5.154 Â.

In the work by Fabris et al. (56) an empirical tight-binding (TB) approach was used, the implementation of which is discussed in Reference (56), and the connected work by Finnis et al. (57) used a Linear Muffin Tin Orbital scheme (LMTO) to study the properties o f the cubic and tetragonal phases of zirconia, and in particular the energetics associated with the phase transformation between the two.

Of the remaining key QM studies on ZrO], such as the work by Stapper et al. (58) and that by Christensen and Carter (59), the basis functions used are plane-waves; these do not require the laborious and careful optimisation o f the atom-centred Gaussian basis sets used in CRYSTAL, however, the non-localised nature o f the plane wave basis functions makes it difficult to derive detailed chemical interpretations from the results obtained - which were a key requirement of the calculations reported here.

The work by Christensen and Carter used the LDA Hamiltonian with a plane wave cut-off energy o f 800 eV, and applied a A-point separation o f 0.05 A'^; for comparison the CASTEP calculations performed here using ultrasoA pseudopotentials with an energy cut-off o f 430 eV, and the same ^-space grid density.

Interatomic potential calculations on stoichiometric cubic zirconia are few and far between, as the phase has such restricted stability. Instead, many of the IP studies employ potential sets optimised for use in doped systems; several of these sets were tested on the pure phase, but their performance was less than adequate. The best

Chapter 4: Fluorite

potential set we have found in the literature to date is that derived by Balducci, et a l (44,47,60) shown in table 6.3.