Laser assisted directional solidification of zirconia-based eutectics

Directionally solidified zirconia-based eutectic (DSE) fibres were obtained using the laser floating zone (LFZ) method. Two systems were investigated: zirconia-barium zirconate and zirconia-mullite. The purpose was to take advantage of zirconia properties, particularly as an ionic conductor and a mechanical rein-forcement phase. The influence of processing conditions in the structural and microstructural characteristics and their consequences on the electrical and mechanical behaviour were the focus of this thesis. The novel zirconia-barium zirconate eutectic materials were developed in order to combine oxygen ionic conduction through zirconia with protonic conduction from barium zirconate, promoting mixed ionic conduction behaviour. The mi-crostructure of the fibres comprises two alternated regions: bands having coarser zirconia-rich microstructure; and inter-band regions changing from a homogeneous coupled eutectic, at the lowest pulling rate, to columnar colony microstructure, for the faster grown fibres. The bands inter-distance increases with the growth rate and, at 300 mm/h, zirconia dendrites develop enclosed in a fine-interpenetrated network of 50 vol.% ZrO2-50 vol.% BaZrO3. Both phases display contiguity without interphase boundaries, according to impedance spec-troscopy data. Yttria-rich compositions were considered in order to promote the yttrium incorporation in both phases, as revealed by Raman spectroscopy and corroborated by the elemental chemical analysis in energy dispersive spectros-copy. This is a mandatory condition to attain simultaneous contribution to the mixed ionic conduction. Such results are supported by impedance spectrosco-py measurements, which clearly disclose an increase of total ionic conduction for lower temperatures in wet/reduction atmospheres (activation energies of 35 kJ/mol in N2+H2 and 48 kJ/mol in air, in the range of 320-500 ºC) compared to the dry/oxidizing conditions (attaining values close to 90 kJ/mol, above 500 ºC). At high temperatures, the proton incorporation into the barium zirconate is un-favourable, so oxygen ion conduction through zirconia prevails, in dry and oxi-dizing environments, reaching a maximum of 1.3x10-2 S/cm in dry air, at ~1000 ºC. The ionic conduction of zirconia was alternatively combined with another high temperature oxygen ion conductor, as mullite, in order to obtain a broad elec-trolytic domain. The growth rate has a huge influence in the amount of phases and microstructure of the directionally solidified zirconia-mullite fibres. Their microstructure changes from planar coupled eutectic to dendritic eutectic mor-phology, when the growth rate rises from 1 to 500 mm/h, along with an incre-ment of tetragonal zirconia content. Furthermore, high growth rates lead to the development of Al-Si-Y glassy phase, and thus less mullite amount, which is found to considerably reduce the total ionic conduction of as-grown fibres. The reduction of the glassy phase content after annealing (10h; 1400 ºC) promotes an increase of the total ionic conduction (≥0.01 S/cm at 1370 °C), raising the mullite and tetragonal zirconia contents and leading to microstructural differ-ences, namely the distribution and size of the zirconia constituent. This has important consequences in conductivity by improving the percolation pathways. A notable increase in hardness is observed from 11.3 GPa for the 10 mm/h pulled fibre to 21.2 GPa for the fibre grown at 500 mm/h. The ultra-fine eutectic morphology of the 500 mm/h fibres results in a maximum value of 534 MPa for room temperature bending strength, which decreases to about one-fourth of this value at high temperature testing (1400 ºC) due to the soft nature of the glassy-matrix.

Numerical and combinatorial applications of generalized Appell polynomials

This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.