Thermodynamic analysis of nanocrystal formation in the TiO$_2$–H$_2$O (NaOH, HCl) system

A thermodynamic analysis of the crystallization of titanium dioxide in the anatase, brookite, and rutile modifications from aqueous salt solutions was performed, taking into account the influence of medium pH, temperature, reagent concentration, and the specific surface energy $(\sigma)$ of the phases. It was shown that the choice of the $\sigma$ value for the thermodynamic analysis of anatase crystallization is decisive: at $\sigma_A$ = 0.3 J/m$^2$, the minimum particle size is determined by the crystallochemical criterion ($l_{min}$ $\sim$5–7 nm), while at $\sigma_A$ = 1.3 J/m$^2$, it is determined by thermodynamic criteria ($d_{crit}$ $\sim$8 nm, $d_{eq}$ $\sim$12 nm). Using $\sigma$ values most closely approximating the conditions of a hydrated TiO$_2$ surface ($\sigma_R$ = 1.79, $\sigma_B$ = 1.0, $\sigma_A$ = 1.13 J/m$^2$), the regions of possible crystallization for each modification were determined. Rutile can crystallize in a relatively wide pH range of 0.8–14 (25$^\circ$C) and 1.1–10.2 (200$^\circ$C), and the minimum particle sizes of rutile under these conditions are determined by thermodynamic criteria – $d_{crit}$ and $d_{eq}$. For brookite and anatase in acidic and alkaline conditions (pH $\sim$1–3 and 9–14), the minimum particle sizes, as for rutile, are also determined by thermodynamic criteria, whereas in the neutral region, they are determined by the crystallochemical criterion $l_{min}$. Based on the analysis of structural transitions, it was established that anatase can transform into rutile or brookite at particle sizes larger than $\sim$16 nm. The calculated size for the brookite $\to$ rutile transition is $\sim$712 nm.