## 1 Introduction  
Titanium dioxide (TiO\({}_{2}\)) photocatalyst has been extensively studied and widely applied in water and air treatment [1], paint [2], protective coatings [3], and cosmetics [4]. There are many advantages of TiO\({}_{2}\) such as chemical stability, self-cleaning, biocompatibility, and strong antibacterial qualities. TiO\({}_{2}\) with high photocatalytic and strong antibacterial activities has attracted much attention in decades [5]. TiO\({}_{2}\) photocatalyst has three polymorph structures, namely anatase, rutile, and brookite. High photocatalytic activity of TiO\({}_{2}\) under ultraviolet (UV) light is strongly depended on phase composition, crystallinity, crystallite size, and surface area [6, 7]. The relationship between TiO\({}_{2}\) phase and its photocatalytic activity has been an interest for many researchers. Although pure anatase phase of TiO\({}_{2}\) has been preferred, there is a limitation of fast recombination of photogenerated electrons (e\({}^{-}\)) and holes (h\({}^{+}\)). Some works have shown that the mixture of anatase-rutile [8, 9, 10, 11], anatase-brookite [12, 13] and brookite-rutile [14, 15] phases of TiO\({}_{2}\) powder in a certain ratio provided higher photocatalytic activity than the pure phase. [PERSON] et al. [11] found that the mixture of anatase and rutile phases with a ratio of 57:43 provides the highest hydroxyl radical (OH) formation and photocatalytic activity under UV irradiation. [PERSON] et al. [12] studied methanol photo-oxidation of TiO\({}_{2}\) powder and found that the mixture of anatase-brookite phases of TiO\({}_{2}\) showed higher photocatalytic activity than pure anatase phase. [PERSON] et al. [14] determined that the highest photodegradation of phenol was achieved by using TiO\({}_{2}\) composed of 72% of brookite and 28% of rutile. Although binary phases of TiO\({}_{2}\) provided higher photocatalytic activity than pure phase, there have been reports that ternary phases present higher photocatalytic activity than single [16] and binary phases [17]. [PERSON] et al. [16] investigated that the mixture of three phases including 65% anatase, 28% brookite, and 7% rutile phase showed higher photocatalytic activity degradation of 4-nitrophenol than pure rutile phase. [PERSON] et al. [17] found that the mixture of 30% anatase, 28% brookite, and 42% rutile showed higher photocatalytic activity over 90% degradation of methyl orange solution in 20 min than the mixture of anatase and rutile phase. The effect of synergetic phase junction and different band alignments of each phase provides e\({}^{-}\) and h\({}^{+}\) transfer at the interface of TiO\({}_{2}\). This can enhance e\({}^{-}\)-h\({}^{+}\) separation and reduce e\({}^{-}\)-h\({}^{+}\) recombination leading to an enhanced photocatalytic activity under UV light. However, a relationship between synthesis condition and mixture ratio of multiple phases of TiO\({}_{2}\) has still been a challenge for high photocatalytic activity. There are many synthesis methods such as hydrothermal [18], microwave [19], supercritical antisolvent [20], and sol-gel [21] methods for preparation of TiO\({}_{2}\) nanoparticles. Hydrothermal, microwave, and supercritical antisolvents provided high purity and high surface equipment for TiO\({}_{2}\) synthesis, whereas the sol-gel method can be undertaken at atmospheric pressure and room temperature. The sol-gel method is one of the promising methods for large-scale production [22]. However, this method suffers from controlling the shape and size distribution of TiO\({}_{2}\) nanoparticles. Generally, the smaller the particle size of TiO\({}_{2}\) leads to the larger surface area and higher photocatalytic activity under UV light irradiation. A grinding process is required for the high surface area of TiO\({}_{2}\) powder. Dry ball milling has been used widely to reduce the size of nanoparticles. However, wet ball milling provides smaller particle size and narrower size distribution [23]. [PERSON] et al.