## 3 Results and Discussion
### Adsorption and Photocatalytic Processes  
Heterogeneous photocatalytic processes comprise several steps, such as physical [51] or chemical [68] adsorption, chemical reactions occurring between the species formed at the catalyst surface followed by degradation and reaction products formation (such as halides, metals, inorganic acids, organic compounds), and adsorption/desorption of these products. Therefore, the description of heterogeneous photocatalysis with specific adsorption modelling tools also involves the consideration of secondary chemical processes (pseudo-adsorption) [69], to which is added the influence of substrate morphology, especially in the inter-particle diffusion stage. Figure 5: pH changes in the photocatalyst films vs. total HNO\({}_{3}\) 0.1M volume (black line) and the derivative curve (dpH/dV, blue line), (**a**). sample A10, (**b**). sample B10, (**c**). sample C10. Since the adsorption step determines the rate in heterogeneous photocatalytic processes, its study becomes necessary for understanding the photodegradation mechanism. The adsorption mechanism involved four steps: (a) surface pollutant diffusion, (b) the mass transfer between the substrate and liquid phase, (c) catalyst surface pollutant adsorption, and (d) adsorbed molecules diffusion through catalyst pores [70; 71; 72]. Due to the strong adsorption of dye molecules on the ternary heterostructure surface, the photocatalysis process takes place at a considerable rate higher than Fly ash-TiO\({}_{2}\) materials (see in the Supplementary material Figure S2 for adsorption process and Figure S3 for adsorption/photocatalysis process). MB adsorption was firstly conducted in a dark condition (see Figure 6), followed by its photocatalytic degradation with and without the addition of H\({}_{2}\)O\({}_{2}\) under UV irradiation (see Figure 7). The adsorption behaviour can serve as an essential role in the photocatalytic degradation reaction. The need for kinetic investigations in heterogeneous photocatalysis is closely related to the photoreactor design as well as the choice of that reaction mechanism sequence with a determining effect on the photoreactor performance. The highest adsorption efficiency (in the dark) was 75% (without H\({}_{2}\)O\({}_{2}\)) and 90% (with H\({}_{2}\)O\({}_{2}\)), both corresponding to sample C10 due to its morphology, characterised by a high number of small pores. Under UV irradiation, the sample A10 becomes more efficient (96% without H\({}_{2}\)O\({}_{2}\) and 99% with H\({}_{2}\)O\({}_{2}\)) due to the higher TiO\({}_{2}\) content and the presence of larger pores allowing a multi-scattering effect [73]. Figure 6: The adsorption process on ternary TiO\({}_{2}\)–Cu\({}_{x}\)S–FA materials: (**a**) without H\({}_{2}\)O\({}_{2}\) and (**b**) with H\({}_{2}\)O\({}_{2}\). Figure 7: The adsorption/photocatalysis process on ternary TiO\({}_{2}\)–Cu\({}_{x}\)S–FA materials: (**a**) without H\({}_{2}\)O\({}_{2}\) and (**b**) with H\({}_{2}\)O\({}_{2}\). Sample A10 have a better suppression of the electron-hole recombination, which allows the generation of oxidative species (h\({}^{+}\), HO-) [74]. It is known that fast electron-hole pairs recombination correlated with lacking efficient export channels induce a significant reduction of the photocatalytic efficiency [75; 76]. The superoxide radicals (HO-) generated in the presence of surface holes (h\({}^{+}\)) present a strong ability of oxidative degradation and are used as active species in MB degradation reaction. Based on the present experimental results which consider satisfactory photocatalytic degradation activity and good charge separation, the ternary heterostructure can be further tested in dynamic system applications. The superoxide radicals (HO) generated in the presence of surface holes (h\({}^{+}\)) present a strong oxidative degradation ability and are used as active species in MB degradation reaction.