## 3 Results
### Catalysis Process Characterization
As shown in this figure, the degradation depends strongly on the amount of oxygen vacancies and thickness of the films; however, the thickness is not a direct proportional parameter. In change, the inner diameter of the nanosheets or nanobowls has a direct relationship with degradation. This means that when the inner diameter of the nanosheets is larger, the degradation increases. This effect may be associated with the surface area of the nanomaterial because a larger surface area increases the adsorption of the molecules by bringing them closer to active sites on walls of the nanosheets [42]. In other words, a greater volume of dyed solution can be introduced into nanosheets. Therefore, the inner diameter is the main parameter to increase the oxidation and reduction processes to accelerate the catalysis process on the surface of TiO\({}_{2}\) films. Hence, the inner and outer diameter determine the porosity, in this way, a larger inner diameter and a thinner wall thickness increase the porosity, thus increasing the percentage of degradation of chemical and biological compounds [24; 42; 43]. In addition, other reports show that the length of the nanotubes influences catalytic activity because higher length increases the surface area and that increases the degradation [29; 37; 42]. In our case, TiO\({}_{2}\)-nc had a lower percentage of degradation than TiO\({}_{2}\)-nt because the nanocovities had a wider wall thickness, and a shorter inner diameter as well as their length which was 6.68 times shorter than nanotubes. Therefore, the degradation of methylene blue depends on the number of oxygen vacancies and the inner diameter and of the nanostructure. Figure 7: (**a**) percentage of degradation of methylene blue as a function of time for the four nanostructures; (**b**) comparison of defect sates, band gap energy, S\({}_{\text{u}}\) and final degradation percentage between the four films; and (**c**) comparison of geometric parameters, final degradation percentage and oxygen vacancies between the four nanomaterials. To understand the catalytic process occurring on TiO\({}_{2}\) nanostructures, in Figure 8 we show the mechanism of dye degradation. The process starts when the electron-hole pairs are generated by an excitation energy (hv) higher or equal than the band gap energy of the catalyst. After, the electrons travel from valence band to conduction band, so, the photogenerated charge carriers are separated which helps to develop the catalysis process. Thus, oxygen vacancies absorb O\({}_{2}\) on the surface of catalytic material, and these react with electrons to form the superoxide radical anions [23, 39, 44]. Posteriorly, the methylene blue (pollutant) reacts with superoxides starting the reduction process. This chemical reaction breaks down the dye into H\({}_{2}\)O and CO\({}_{2}\) and others chemicals [29, 44]. At the same instance, the holes generated on the valence band react with H\({}_{2}\)O and OH\({}^{-}\). This oxidation process leads to the formation of hydroxyl radicals (OH\({}^{-}\)). Consequently, the hydroxyl radicals react with organic and/or inorganic compounds, viruses, and bacteria leading to their decomposition. These chemical reactions are summarized in Equations (4)-(11) [29, 39, 44].