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.

діоксид титану
sol–gel
nanocrystallized
anatase
рентгеноструктурний аналіз
Фур‘є-спектроскопія
fractional model
equilibrium point
fractional calculus
Похідні Капуто
  1. Ivanova T., Harizanova A., Koutzarova T., Vertruyen B.  Optical and structural characterization of TiO$_2$ films doped with silver nanoparticles obtained by sol–gel method.  Optical Materials. 36 (2), 207–213 (2013).
  2. Park H., Kim W.-R., Jeong H.-T., Lee J.-J., Kim H.-G., Choi W.-Y.  Fabrication of dye-sensitized solar cells by transplanting highly ordered TiO$_2$ nanotube arrays.  Solar Energy Materials and Solar Cells. 95 (1), 184–189 (2011).
  3. Ochiai T., Fujishima A.  Photoelectrochemical properties of TiO$_2$ photocatalyst and its applications for environmental purification.  Journal of Photochemistry and Photobiology C: Photochemistry Reviews. 13 (4), 247–262 (2012).
  4. Armstrong A. R., Armstrong G., Canales J., Bruce P. G.  TiO$_2$–B nanowires as negative electrodes for rechargeable lithium batteries.  Journal of Power Sources. 146 (1–2), 501–506 (2005).
  5. Macwan D. P., Dave P. N., Chaturvedi S.  A review on nano-TiO$_2$ sol–gel type syntheses and its applications.  Journal of materials science. 46 (11), 3669–3686 (2011).
  6. Wang S., Wu X., Qin W., Jiang Z.  TiO$_2$ films prepared by micro-plasma oxidation method for dye-sensitized solar cell.  Electrochimica Acta. 53 (4), 1883–1889 (2007).
  7. Wu C.-I., Huang J.-W., Wen Y.-L., Wen S. B., Shen Y.-H., Yeh M.-Y.  Preparation of TiO$_2$ nanoparticles by supercritical carbon dioxide.  Materials Letters. 62 (12–13), 1923–1926 (2008).
  8. Kim S. J., Park S. D., Jeong Y. H., Park S.  Homogeneous precipitation of TiO$_2$ ultrafine powders from aqueous TiOCl$_2$ solution.  Journal of the American Ceramic Society. 82 (4), 927–932 (1999).
  9. Neppolian B., Yamashita H., Okada Y., Nishijima H., Anpo M.  Preparation of unique TiO$_2$ nano-particle photocatalysts by a multi-gelation method for control of the physicochemical parameters and reactivity.  Catalysis Letters. 105 (1), 111–117 (2005).
  10. Ghorai T. K., Dhak D., Biswas S. K., Dalai S., Pramanik P.  Photocatalytic oxidation of organic dyes by nano-sized metal molybdate incorporated titanium dioxide  (M$_x$Mo$_x$Ti$_{1-x}$O$_6$) (M $=$ Ni, Cu, Zn) photocatalysts.  Journal of Molecular Catalysis A: Chemical. 273 (1–2), 224–229 (2007).
  11. Peng F., Cai L., Yu H., Wang H., Yang J.  Synthesis and characterization of substitutional and interstitial nitrogen-doped titanium dioxides with visible light photocatalytic activity.  Journal of Solid State Chemistry. 181 (1), 130–136 (2008).
  12. Bruns W., Ichim B., Söger C.  The power of pyramid decomposition in Normaliz.  Journal of Symbolic Computation. 74, 513–536 (2016).
  13. Nachit W., Touhtouh S., Ramzi Z., Zbair M., Eddiai A., Rguiti M., Bouchikhi A., Hajjaji A., Benkhouja K.  Synthesis of nanosized TiO$_2$ powder by sol gel method at low temperature.  Molecular Crystals and Liquid Crystals. 627 (1), 170–175 (2016).
  14. Crişan M., Brăileanu A., Răileanu M., Zaharescu M., Crişan D., Drăgan N., Anastasescu M., Ianculescu A., Niţoi I., Marinescu V. E., Hodorogea S. M.  Sol–gel S-doped TiO$_2$ materials for environmental protection.  Journal of Non-Crystalline Solids. 354 (2–9), 705–711 (2008).
  15. Sadek O., Touhtouh S., Hajjaji A.  The Rapid Identification of Solid Materials Using the ACP Method.  Environmental Sciences Proceedings. 16 (1), 22 (2022).
  16. Sadek O., Touhtouh S., Mahdi Bouabdalli E., Hajjaji A.  Development of a protocol for the rapid identification of solid materials using the principal component analysis (ACP) method: Case of phosphate fertilizers.  Materials Today: Proceedings (2022).
  17. Bouabdalli E. M., El Jouad M., Touhtouh S., Sadek O., Hajjaji A.  Structural studies on varied concentrations of europium doped strontium phosphate glasses.  Materials Today: Proceedings (2022).
  18. Horikawa T., Katoh M., Tomida T.  Preparation and characterization of nitrogen-doped mesoporous titania with high specific surface area.  Microporous and Mesoporous Materials. 110 (2–3), 397–404 (2008).
  19. Kim B.-H., Lee J.-Y., Choa Y.-H., Higuchi M., Mizutani N.  Preparation of TiO$_2$ thin film by liquid sprayed mist CVD method.  Materials Science and Engineering: B. 107 (3), 289–294 (2004).
  20. Muscat J., Swamy V., Harrison N. M.  First-principles calculations of the phase stability of TiO$_2$.  Physical Review B. 65 (22), 224112 (2002).
  21. Mo S.-D., Ching W. Y.  Electronic and optical properties of three phases of titanium dioxide: Rutile, anatase, and brookite.  Physical Review B. 51 (19), 13023 (1995).
  22. Ohno T., Akiyoshi M., Umebayashi T., Asai K., Mitsui T., Matsumura M.  Preparation of S-doped TiO$_2$ photocatalysts and their photocatalytic activities under visible light.  Applied Catalysis A: General. 265 (1), 115–121 (2004).
  23. Prasad K., Pinjari D. V., Pandit A. B., Mhaske S. T.  Phase transformation of nanostructured titanium dioxide from anatase-to-rutile via combined ultrasound assisted sol–gel technique.  Ultrasonics Sonochemistry. 17 (2), 409–415 (2010).
  24. Graaf G. H., Stamhuis E. J., Beenackers A. A. C. M.  Kinetics of low-pressure methanol synthesis.  Chemical Engineering Science. 43 (12), 3185–3195 (1988).
  25. Khajji B., Boujallal L., Elhia M., Balatif O., Rachik M.  A fractional-order model for drinking alcohol behaviour leading to road accidents and violence.  Mathematical Modeling and Computing. 9 (3), 501–518 (2022).
  26. Gouasnouane O., Moussaid N., Boujena S., Kabli K.  A nonlinear fractional partial differential equation for image inpainting.  Mathematical Modeling and Computing. 9 (3), 536–546 (2022).
  27. Ben-Loghfyry A., Hakim A.  Time-fractional diffusion equation for signal and image smoothing.  Mathematical Modeling and Computing. 9 (2), 351–364 (2022).
  28. Pawar D. D., Patil W. D., Raut D. K.  Fractional-order mathematical model for analysing impact of quarantine on transmission of COVID-19 in India.  Mathematical Modeling and Computing. 8 (2), 253–266 (2021).
  29. Fadugba S. E., Ali F., Abubakar A. B.  Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order.  Mathematical Modeling and Computing. 8 (3), 537–548 (2021).
  30. Kostrobij P. P., Markovych B. M., Ryzha I. A., Tokarchuk M. V.  Generalized kinetic equation with spatio-temporal nonlocality.  Mathematical Modeling and Computing. 6 (2), 289–296 (2019).
  31. Kostrobij P., Markovych B., Viznovych O., Zelinska I., Tokarchuk M.  Generalized Cattaneo–Maxwell diffusion equation with fractional derivatives. Dispersion relations.  Mathematical Modeling and Computing. 6 (1), 58–68 (2019).
  32. Odibat Z. M., Shawagfeh N. T.  Generalized Taylor’s formula.  Applied Mathematics and Computation. 186 (1), 286–293 (2007).
  33. Samko S. G., Kilbas A. A., Marichev O. I.  Fractional integrals and derivatives. Vol. 1.  Yverdon-les-Bains, Switzerland: Gordon and breach science publishers, Yverdon (1993).
  34. Lin W.  Global existence theory and chaos control of fractional differential equations.  Journal of Mathematical Analysis and Applications. 332 (1), 709–726 (2007).
  35. Kim W. B., Choi S. H., Lee J. S.  Quantitative Analysis of Ti-O-Si and Ti-O-Ti Bonds in Ti-Si Binary Oxides by the Linear Combination of XANES.  Journal of Physical Chemistry B. 104 (36), 8670–8678 (2000).
  36. Suppuraj P., Parthiban S., Swaminathan M., Muthuvel I.  Hydrothermal fabrication of ternary NrGO-TiO$_2$/ZnFe$_2$O$_4$ nanocomposites for effective photocatalytic and fuel cell applications.  Materials Today: Proceedings. 15 (3), 429–437 (2019).
  37. Himabindu B., Devi N. L., Kanth B. R.  Microstructural parameters from X-ray peak profile analysis by Williamson–Hall models; A review.  Materials Today: Proceedings. 47 (14), 4891–4896 (2021).
  38. Garrappa R.  On linear stability of predictor–corrector algorithms for fractional differential equations.  International Journal of Computer Mathematics. 87 (10), 2281–2290 (2010).