Isothermal Kinetics of Titanium-oxo-alkoxy Clusters Formation

In this article, the influences of titanium tetraisopropoxide Ti(OR)4 (R = Pr) concentration, molar ratio, h = [H2O]/[Ti(OR)4], and temperature, on the formation kinetics of the titanium-oxo-alkoxy clusters (TOAC), were studied. The TOAC formation isothermal kinetics was monitored by measuring absorbance changes versus time in the reaction mixture at predefined wavelength λ = 350 nm. It was determined that the isothermal rate of clusters formation is a power law function of titanium tetraisopropoxide concentration and the molar concentration of water (cw). The kinetic parameters α and β were calculated. The apparent activation energy Ea values in the clusters formation reaction has been calculated and correlated with the change of titanium tetraisopropoxide concentration and molar ratio. The model for mechanism of TOAC formation is proposed.


Introduction
Titania powders are widely used in industrial applications as pigments, opacifiers, catalysts, photocatalysts, fillers, cell separations and environmental remediation.The key application properties of TiO 2 are defined by the size and shape of particles and also by the particle size distribution [1][2][3][4][5].To create a TiO 2 with the predefined functional characteristics it is extremely important to know the mechanism and kinetics of TiO 2 particle formation in the initial phase of particle growth [6].
The kinetics of the sol-gel process based on titanium metal alkoxides has been studied in a number of papers [7][8][9][10][11].It is generally assumed, that in the first stage of the sol-gel process, hydrolyzed monomers are created and accumulated [9,[12][13].The induction time corresponds to the moment when the concentration of these monomers reaches the level of critical supersaturation.As a result of nucleation, primary particles are formed.They then grow by monomer addition and aggregation [10].
Barringer et al. [7] studied the production of spherical, monodispersed TiO 2 with a controlled hydrolysis of diluted ethanolic solution of titanium tetraethoxide -Ti(OC 2 H 5 ) 4 , at room temperature, at a molar ratio of water to tetraethoxide (h) always greater than 2.5.They found that the nucleation process is homogenous and that the rate of nucleation follows the stoichiometry of the reaction.
The kinetics of nucleation, when generating a monodisperse TiO 2 by the hydrolysis of Ti(OC 2 H 5 ) 4 in ethanolic solution, (with different titanium tetraethoxide concentrations c = 0.075─0.2M, molar ratios of water to tetraethoxide h = 5─13 and temperature T = 298 K) was investigated in the paper of Jean et al. [8].It was found that the induction time (I) can be described with the following equation: Golubko et al. [11] investigated the hydrolysis of Ti(OR) 4 (R = Et, i Pr, n Bu) in alcoholic solutions, with different titanium alkoxides concentrations (c = 0.19─0.67M), and different molar ratios h = 0.5─20, at room temperature.It was concluded that the induction time can be described with the same equation as in the paper of Jean et al. [8] and also suggested that the addition of water to perform hydrolysis of Ti(OBu) 4 , should be in two steps, which enables the decrease in rate of the solid precipitate formation, to regulate particles morphology in a wide range.
A spectroscopic study of the initial stage of the Ti(OPr i ) 4 sol-gel process in 2propanol (c = 0.1 M and h = 0.1─3) was presented out in the paper of Soloviev et al. [14].It was concluded that the hydrolysis-condensation reaction takes place during the mixing of reagents and that it is complete at low h ≤ 1.The molar ratio h > 1 is needed for particle growth.
The kinetics of hydrolysis and condensation of Ti(OPr i ) 4 in isopropanol, under neutral conditions (c = 0.08─0.28M, h = 1.5─7 and T = 298 K), was investigated in the paper of Soloviev et al. [6].It was found that the kinetics of the process is complex in nature and that it includes different processes: hydrolysis, condensation and aggregation of primary particles.The formation of primary particles (nucleation) takes place for a short time, after the initial hydrolysis and condensation.
The rate of individual particles mass growth (r) can be described with a power law equation: ( ) where k is the constant of the reaction rate, h is the molar ratio and α and β are kinetic parameters.
The effect of mixing on the Ti(OPr i ) 4 sol-gel process in isopropyl alcohol (c = 0.5─1 M, h = 2─4, T = 298 K) was investigated in the paper of Marchisio et al. [15].It was found that, processes of hydrolysis and condensation are much faster than mixing and that they have the decisive influence on kinetics and characteristics of the TiO 2 formation.
Azouani et al. [16] investigated nucleation and growth of titanium-oxo-alkoxy Ti x O y (O i Pr) z clusters which were created in the Ti(OPr i ) 4 sol-gel process in 2-propanol, at the hydrolysis ratio, h, between 1.0 and 2.6.Four different domains of the cluster/nanoparticle stability and growth kinetics were identificated: h < 1.45 (I), 1.45 ≤ h ≤ 1.75 (II), 1.75 < h ≤ 2.0 (III) and h > 2.0 (IV).In the fourth domain, the equation was derived, describing the induction kinetics: where c Ti is the initial Ti-atom concentration, h * = 1.45 and t ind is the induction time.
Azouani et al. [17] synthesized pure and doped TiO 2 nanoparticles in the sol-gel reactor with turbulent micromixing of two reactive fluids, containing titanium tetraisopropoxide and water, in T-mixer.It has been found that mixing, at the nucleation stage, has a strong impact on the nanoparticles polydispersity and mean size of the created TiO 2 .
Taking into account the complexity of all these processes, in this article, the influences of c, h, T parameters on TOAC formation kinetics have been investigated.

Experimental procedure
The kinetic study of the TOAC formation was carried out by measuring absorbance changes versus time in the reaction mixture, composed of titanium tetraisopropoxide, Ti(OPr i ) 4 , (98 + % Acros Organics), n-propanol provided by Lachner, 0.1% maximum water content and distilled water.
In each individual experiment, the molar concentration of titanium tetraisopropoxide (c -ranging from 0.04─0.07M) was fixed, as was the temperature at one of three different values (298 K, 308 K and 318 K), and the molar ratio h varied between 11 and 20 (h = 11, 14, 17, 20).
In the beaker-type reactor filled with 100 ml of the n-propanol solution, preliminarily heated to predeterminated temperatures, a precalculated amount of titanium tetraisopropoxide was added in order to obtain an appropriate value of c.The temperature of each individual experiment was kept constant, within ± 1 K.The reaction mixture was homogenized by stirring with a magnetic stirrer (450 rpm) and the required amount of distilled water was added drop-wise into the titanium alkoxide solutions in order to achieve a predeterminated molar ratio of water to titanium tetraisopropoxide.Immediately afterwards, an immersion quartz Suprasil probe (Hellma, 661.500-QX, optical path length 10 -2 m) was inserted in the beaker-type reactor for the measurement of absorbance changes versus time, at predefined time intervals.The optical probe is connected to the spectrometer through the standard fiberoptic cable (Hellma, 041.002-UV).Absorbance changes versus time were recorded on GBC Cintra 10e UV-Visible Spectometer at predetermined wavelength λ = 350 nm.
The time of clusters formation (t f ), was determined by the graphical method, as the section of extrapolation of the absorbance's linear time dependence with time axis.
The rate of clusters formation (υ f ) was calculated by using the equation: The apparent activation energy of the clusters formation reaction (E a ) was determined by the Arrhenius equation: where A is the pre-exponential factor, R is the gas constant and T is the temperature.

Tab. II The effects of c, h and T on the rate of titanium-oxo
According to Eq. 6, it is possible to determine the value of kinetic parameter α for the clusters formation process.Namely, the dependence ln υ f vs. ln c for h = const.and T = const.is linear and its plot is a straight line whose slope can be used for the kinetic parameter α calculation.The plot of ln υ f vs. ln c dependence, at temperature of 298 K and h ranging from 11─20, is presented in Fig. 2. In Tab.III the effects of experimental temperature T and h on the value of kinetic parameter α, for the clusters formation process, are displayed.We have established that at temperatures T = 298 K and T = 308 K α decreases, as h increases, while further increase of T leads to the α reduction if h ≥ 14, or parameter α remains constant if h ≥ 17.We have noticed that as the concentration of titanium tetraisopropoxide increases in the reaction mixture at T = 298 K, while the kinetic parameter β decreases from 7 to 4. Contrary to that, at temperatures T = 308 K and T = 318 K, for c ≥ 0.05, the kinetic parameter β is independent of c and equals 4 (β = 4).
Assuming that the initial hydrolysis reaction is very fast compared to the whole induction period, and that in order to describe correctly the kinetics of this process, instead of the amount of water put initially in the system, the amount of water remaining after the nucleation must be used, Soloviev obtained the following kinetic equation [6]: where α * and β * are Soloviev's kinetic parameters.In order to validate the Soloviev's kinetic model, the dependence ln υ f on ln (h -1.45) was examined within the tested ranges of c, h and T. On  In Tab.V, the values of kinetic parameter β * , obtained by applying Soloviev's model with different values of c and T, are shown.It can be noticed that kinetic parameters β * are nonintegers and vary complexly with c and T. Based on the values of α and β * , the value of Soloviev's model kinetic parameter α * was determined.In Tab.VI, as an example, the values of kinetic parameter α * are shown, for different values of c and h at T = 298 K.It should be noticed that kinetic parameters α * are also nonintegers.In a series of examined examples, values of α * were less than 1 or negative, which directly indicates that Soloviev's kinetic model isn't applicable for observing kinetics of TOAC formation.If we carefully analyze values of kinetic parameters α and β, shown in Tab.III and Tab.IV, it can be easily concluded that values of α are a) α = β, b) α > β or c) α < β.Bearing this in mind and accepting that the rate of hydrolysis is greater than the rate of cluster formation and also that the basic product of hydrolysis reaction is Ti(OR) 3 OH [19,20], it is possible to obtain the following kinetic models of TOAC formation: a) In case of α = β, ( ) ( ) ( ) c) In case of α < β, during the hydrolysis, not only Ti(OR) 3 OH is formed, but also some more complex, higher level products of hydrolysis, such as Ti(OR) 2 (OH) 2 or Ti(OR)(OH) 3 , which consequently interact and create [Ti α O β ](OR) 4α-2β .
In according to Day et al. [22], the titanium-oxo core structural formulas and degrees of condensation (DC = α/β) are shown in Tab.VII.
Tab. VII The titanium-oxo core structural formulas and degrees of condensation (DC), with different values of c and T, for h = 11 and h ranging from 14 -20.
[   Since the increase of υ f as T increases is exponential, at all investigated c (h = const.),the apparent value of energy E a in the clusters formation reaction was determined, by applying the Arrhenius equation.Tab.VIII shows the dependence of apparent activation energy E a on h and c.Based on the results given in Tab.VIII, it has been concluded that apparent activation energy E a decreases as the Ti(OPr i ) 4 concentration in the reaction mixture increases.The relatively low values of the apparent activation energy E a in the clusters formation reaction, which ranged from 16.2 kJ/mol to 87.7 kJ/mol, confirm the above given reaction mechanism.It also indicates that the energy barrier of the reaction is linked to the process of a transition complex formation.

Conclusions
In this article titanium-oxo-alkoxy clusters formation kinetics has been investigated.TOAC clusters formation is a reaction with complex kinetics whose elementary stages are hydrolysis Ti(OR) 4

Acknowledgement
This investigation was supported by the Ministry of Science and Technological Development of Serbia.

Fig. 1 .
Fig. 1.The TOAC formation isothermal kinetic curves, produced in the reaction between Ti(OR) 4 and H 2 O, in n-propanol solution, at c = 0.07 M and h = 17.

Fig. 2 .
Fig. 2. The plot of ln υ f vs. ln c dependence at temperature of 298 K.

Fig. 3 .
Fig. 3.The plot of ln υ f vs. ln c w dependence for titanium tetraisopropoxide concentration of 0.04 M.

Fig. 4 ,
as an example, the dependence ln υ f on ln (h -1.45) is shown, for c = 0.04 M. Tab.V The values of kinetic parameter β * , obtained by applying Soloviev's model, with different values of c and T.
-20, [Ti 5 O 7 ] clusters are formed.The increase of Ti(OR) 4 concentration leads to [Ti 7 O 4 ] or [Ti 5 O 4 ] clusters formation and therefore to decrease of DC values.Also, it has been observed that at 308 K and 318 K, for c = 0.04 M, [Ti 6 O 7 ] clusters are formed at h = 11, and [Ti 4 O 5 ] clusters are formed at h = 14 -20.Further more, the increase of Ti(OR) 4 concentration leads to the decrease of DC values and [Ti 4 O 4 ] clusters are formed.Established structural formulas [Ti 7 O 4 ], at c = 0.05 M and c = 0.07 M, for h = 11 and T = 298 K, are in agreement with the results of the X-ray diffraction and NMR spectroscopy Tab.VI The values of kinetic parameter α * for different values of c and h at T = 298 K.
to Ti(OR) 3 OH and the formation of titanium-oxo-alkoxy clusters [Ti α O β ](OR) 4α-2β through the alcoxolation reaction.New equation for isothermal rate of TOAC clusters formation has been established and correlated with Soloviev's kinetic equation.Processing parameters c, h, T were determined and apparent activation energy E a has been calculated and correlated with the change of c and h.Kinetic parameters α and β have been determined as well.It has been established that the values of kinetic parameters α and β change complexly as molar ratio h, titanium tetraisopropoxide concentration c and temperature T change.
[21]h were isolated from the reaction of Ti(OPr i ) 4 with water and methanol in the isopropanol solution[21].Therefore, TOAC formation in the reaction of Ti(OR) 4 with H 2 O in n-propanol solution is a complex kinetic reaction whose elementary stages are Ti(OR) 4 fast hydrolysis to Ti(OR) 3 OH or Ti(OR) n OH 4-2n and the formation of titanium-oxo-alkoxy clusters through the alcoxolation reaction.