The mathematical fractional modeling of TiO_2 nanopowder synthesis by sol–gel method at low temperature

Titanium dioxide is a compound of oxygen and titanium with the formula TiO$_2$ present in nature and manufactured on an industrial scale.  It is used in several fields and applications such as cosmetics, paint, food, photocatalyst, electrodes in lithium batteries, dye solar cells (DSSC), biosensors, etc., given its importance and its various fields of application, there are several methods of synthesis of TiO$_2$ such as the sol–gel method widely used to obtain nanoparticles.  In our study, on the one hand we synthesized titanium dioxide nanopowders crystallized in the anatase phase at a crystal size of $49.25$ nm with success using titanium tetraisopropoxide (TTIP) as precursor by the sol–gel method.  The powders obtained were analyzed by X-ray diffraction (XRD) with CuK$_\alpha$ radiation ($\lambda=0.15406$ nm) and Fourier transform infrared spectroscopy (FTIR) in the wave number range $4000-400$ cm$^{-1}$, and on the other hand we present a mathematical model for the prediction of the TiO$_2$ concentration as a function of time and the concentration of reactants by using the fractional order derivative more precise than the whole order derivative, we study the existence and the uniqueness of the solutions.  In addition, we determine the points of equilibrium.  Numerical simulations and their graphical representations are made to visualize the efficiency of this model.

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Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 616–626 (2022)