In this study, the nanostructures of pure ZnO and ZnO:xCe³⁺ were prepared using statistic design—factorial design 3^(3-1)—3-level and mixed-level factorials and fractional with replicates in the central point, totaling 11 experiments. The experiments were performed using the microwave-assisted hydrothermal (MAH) method with Ce³⁺ concentration of 2 and 4 mol% at 60, 110, and 160 ℃ for 10, 20, and 30 min. X-ray diffraction (XRD), scanning electron microscopy (SEM), and optical diffuse reflectance were used to characterize the products. The fractional factorial design indicated the optimal design area, and the studies were continued by ridge analysis. The analysis of variance (ANOVA), the Pareto, and the model adjusted to the conditions proposed in this study due to the determination coefficient of 99.9%, variance (R²), and response surface generated were satisfactory, thus having an optimization in the process of obtaining ZnO doped with Ce.

Synthetic inorganic pigments are most widely used in ceramic applications due to their excellent chemical and thermal stability and their lower toxicity to both human and environment as well. In the present work, black ceramic pigment CoFe₂O₄ has been synthesized by the complex polymerization method (CPM) with good chemical homogeneity. In order to study the influence of variables on the process of obtaining pigment through CPM, 2^(5-2) fractional factorial design with resolution III was used. The variables studied in the mathematical modeling were: citric acid/metal concentration, pre-calcination time, calcination temperature, calcination time, and calcination rate. Powder pigments were characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), and UV–visible (UV–Vis) spectroscopy. Based on the results, the formation of cobalt ferrite phase (CoFe₂O₄) with spinel structure was verified. The color of pigments obtained showed dark shades, from black to gray. The model adjusted to the conditions proposed in this study due to the determination coefficient of 99.9% and variance (R²) showed that all factors are significant at the confidence level of 95%.