Effects and optimization of hydrolysis conditions on the diameter distance ratio for metatitanic acid

The three factors (TiO2 concentration of TiOSO4 solution (X1), aging time (X2) and hydrolysis time (X3)) which had significant impacts on the diameter distance ratio were treated as independent variables, the diameter distance ratio (Y) was taken as the response value, and the factors and levels of BBD tests were shown in Table 1. The experimental accuracy and the impacts of experimental errors on the variance analysis was ensured through precise control of experimental conditions, consistency in operation and subsequent processing, consistency and stability in testing conditions, and 17 experiments were set up. Conducted the experiments according to the hydrolysis conditions as listed in Table 2, the experimental values, predicted values and particle size distribution for MA were also shown in Table 2.Table 1 Factors and levels of BBD experiments for the hydrolysis conditions.Table 2 Experimental design matrix, experimental diameter distance ratio results, predicted ones, and particle size distribution for metatitanic acid.Based on the results for the diameter distance ratio (Y) (Table 2), the coded regression relationship between diameter distance ratio and hydrolysis conditions (X1, X2, X3) could be written as the following coded Eq. (2).$$\begin{aligned} Y = & \, 1.10 \, – 0.0371X_{1} – 0.0039X_{2} – 0.0033X_{3} + 0.0022X_{1} X_{2} – 0.0040X_{1} X_{3} \\ & \quad + 0.0005X_{2} X_{3} + 0.0672X_{1}^{2} + \, 0.0162X_{2}^{2} + 0.0174X_{3}^{2} \\ \end{aligned}$$
(2)
The absolute values of the coefficients in Eq. (2) indicated the magnitude of the influence of each factor on the value of diameter distance ratio, and the positive or negative values of the coefficients indicated the positive or negative effects of the influence. It could be seen that the influence order was X1 > X2 > X3, that was, TiO2 concentration > aging time > hydrolysis time.The actual empirical relationship could also be written as the following Eq. (3) represented by actual values. This could not only predict the corresponding diameter distance ratio based on the equation resulting from the values of the hydrolysis conditions, but also could optimize the corresponding values of the hydrolysis conditions by presetting the diameter distance ratio value.$$\begin{aligned} Y = & \, 23.05943 \, – 0.23795X_{1} {-} \, 0.0415X_{2} {-}0.22X_{3} + 0.000045X_{1} X_{2} \\ & \quad – 0.0008X_{1} X_{3} + 0.0002X_{2} X_{3} + 0.000672X_{1}^{2} + 0.000647X_{2}^{2} + 0.0697X_{3}^{2} \\ \end{aligned}$$
(3)
The average particle size (DAV) for the MA samples in Table 2 ranged from 1.621 to 1.819 µm, D50 ranged from 1.604 to 1.784 µm, and the diameter distance ratio ranged from 1.101 to 1.235, indicating a relatively narrow particle size distribution for the MA particles.The effects of each hydrolysis condition were shown in Fig. 1a, b and c.When fixing the aging time at 25 min and hydrolysis time at 2.5 h, with the TiO2 concentration of TiOSO4 solution increasing, the diameter distance ratio showed increasing first and then decreasing, as showed in Fig. 1a. The TiO2 concentration mainly affected the initial nucleation quantity, hydrolysis rate, crystallizing growth and aggregating rate of MA. As the TiO2 concentration increased, the more MA crystals were generated was, the greater the aggregation situation was, leading to an increase in aggregation states of different size particles, thereby increasing the diameter distance ratio value of MA. When the TiO2 concentration exceeded 175 g/L, there were sufficient crystalline particles in the hydrolysis system to induce the growth and aggregation of MA particles, adjusting the uniformity of the aggregated particles and gradually reducing the diameter distance ratio value of MA. In addition, as the TiO2 concentration increased, the particle size of MA particles gradually decreased, which helped to reduce the diameter distance ratio value of particle size distribution. These combined effects showed that there was an appropriate TiO2 concentration for the narrower diameter distance ratio. The TiO2 concentration had the greatest impact on the diameter distance ratio value, indicating that the effects of this factor were obviously larger than the other two, and this could also be verified by the minimum P value (P < 0.0001) of this factor in the analysis of variance table.Figure 1Effects of hydrolysis conditions on the diameter distance ratio (a) Aging time at 25 min, hydrolysis time at 2.5 h; (b) TiO2 concentration at 175 g/L, hydrolysis time at 2.5 h; (c) TiO2 concentration at 175 g/L, aging time at 25 min.When fixing the TiO2 concentration at 175 g/L and hydrolysis time at 2.5 h, with the aging time increasing, the diameter distance ratio showed decreasing first and then increasing, as showed in Fig. 1b. The aging time mainly affected the number of hydrolysis induced crystal nucleus and the particle size distribution of MA. As the temperature of the hydrolysis system decreased, the reaction rate decreased, and the number of new crystal nuclei decreased, the particle size of MA aggregated particles could be adjusted. When the aging time was too short, the number of induced crystal seeds in the hydrolysis system was relatively small, and its size distribution was uneven, resulting in a wider particle size distribution and a larger diameter distance ratio of the formed primary agglomerate particles for MA. When the aging time was too long, as the growth and aggregation of induced crystal seeds were formed in the later stage, it would also lead to a wider particle size distribution and a larger diameter distance ratio for the MA particles. The effect of aging time on the diameter distance ratio was second only to the TiO2 concentration of TiOSO4 solution, as shown in Fig. 1b.When fixing the TiO2 concentration at 175 g/L and aging time at 25 min, with the hydrolysis time increasing, the diameter distance ratio showed decreasing first and then increasing, as showed in Fig. 1c. The hydrolysis time mainly affected the completeness degree of hydrolysis, the particle size and its distribution of MA. The hydrolysis time was relatively short, and the hydrolysis yield of the system was low, and there was still some un-hydrolyzed TiOSO4 solution. After the latter hydrolysis, due to the inconsistent secondary nucleation and incomplete adjustment of the particle size of precipitated MA particles, the size distribution of MA aggregated particles was wide, and the particle diameter distance ratio increased. When the hydrolysis time was too long, although the hydrolysis yield of the TiOSO4 solution increased, the corresponding hydrolysis temperature increased, causing changes in the nucleation and growth of MA, resulting in inconsistent grain growth, and ultimately widening the aggregated particle size distribution of MA and increasing the diameter distance ratio value. Adjusting the appropriate hydrolysis time would help the reaction to proceed more thoroughly and adjust the particle size distribution of MA.The experimental diameter distance ratio and predicted values were showed in Table 2 and Fig. 2. The predicted values were highly consistent with the experimental values, with the correlation coefficient R2 of 0.9989, as showed in Table 3. The adjusted correlation coefficient (R2adj = 0.9976) for the diameter distance ratio was very close to the correlation coefficient R2, which further proved the high consistency between predicted values and experimental data. As the predicted R2pred with the value of 0.9853 was also highly consistent with R2adj (0.9976), which also proved that the regression equation model was significant and reliable. The perturbation plot (Fig. 3) indicated that the influence factors of X1, X2, X3 on diameter distance ratio showed a curve effect, and from the variation amplitude and steepness of the perturbation plot, it could be seen that the influence of diameter distance ratio was greatest for factor X1, followed by factor X2, and the smallest for factor X3, consistent with the previous variance analysis results (Table 3).Figure 2Predicted diameter distance ratio vs actual values.Table 3 Variance analysis of response surface experiments results for the diameter distance ratio.Figure 3Perturbation plot for the diameter distance ratio.The impacts of these hydrolysis condition factors and their interactions on the diameter distance ratio could be determined through the variance analysis (Table 3). The F value was of 735.95, implying that the regression model was extremely remarkable, indicating that the differences in hydrolysis condition factors were very significant. The P value of the variance analysis (P < 0.0001) was much smaller than 0.01, showing that the significance level of the regression model was very high. The main hydrolysis condition factors (X1, X2 and X3) had very significant impacts on the diameter distance ratio (P < 0.01), and the interaction for X1X3 had very significant impacts (P < 0.01), while the other two interactions for X1X2 and X2X3 were not significant (P > 0.05). The P values of quadratic terms such as X12, X22 and X32 were all less than 0.0001, also had very significant impacts on the experimental diameter distance ratio (P < 0.01). The correlation coefficient R2 of the regression model between the predicted values and the experimental diameter distance ratios was very close to 1, indicating an excellent correlation between them. And the C. V. value was of 0.1971%, which was very low, indicating that the experimental values were reliable.The 3D diagrams and contours of the RSM were showed in Fig. 4, which could be clearly seen that under the optimal conditions, the synergistic interactions between the hydrolysis conditions and the diameter distance ratio, as well as the range of values for each factor. All the 3D diagrams were steep and concave, indicating they had synergistic impacts on the diameter distance ratio, with the smallest diameter distance ratio value. The closed ellipse of contour line for the hydrolysis factors indicated that they had synergetic interaction, and the effects of synergetic interaction between TiO2 concentration and hydrolysis time were the largest, consistent with the variance analysis results. The interaction between the three hydrolysis conditions determined the quality and quantity of hydrolysis induced crystal nucleus, hydrolysis rate, completeness degree of hydrolysis, simultaneously affected crystallizing growth and aggregation of MA, changed the aggregation state of metatitanic acid grains, primary agglomerates, and secondary aggregates, finally determined the particle size distribution and diameter distance ratio of MA. Controlling and optimizing hydrolysis conditions could help to obtain metatitanic acid with narrow diameter distance ratio. When TiO2 concentration was of 175 g/L, aging time was of 25 min, and hydrolysis time was of 2.5 h, the experimental minimum value of diameter distance ratio reached 1.101.Figure 4Response surface plots and contour lines of interaction factors on the diameter distance ratio.By stepwise regression of the obtained model equation, the predicted minimum diameter distance ratio value (1.100) and its corresponding optimal hydrolysis reaction conditions could be obtained as the following, the TiO2 concentration was of 177.181 g/L, aging time was of 23.911 min and hydrolysis time was of 2.691 h. For the convenience of experimental operations of hydrolysis conditions, TiO2 concentration of the validation experiment was adjusted to 177.2 g/L, aging time was adjusted to 23.9 min, and hydrolysis time was adjusted to 2.69 h. The verification tests results for the diameter distance ratio were listed in Table 4, and the average diameter distance ratio was of 1.100. The diameter distance ratio values in the verification test were very close, with a relative average standard deviation of only 0.14%, and the relative deviation between the verification value and the predicted value was also only 0.03%. It could be seen that the predicted values were highly consistent with the experimental data, which further proved the reliability of the predictive regression model.Table 4 Results of verification tests and particle size distribution for metatitanic acid.The XRD patterns for the obtained MA sample 18# and sample 19# were shown in Fig. 5. The patterns of the two samples showed little difference, clearly agreeing with the main diffraction peaks of the standard anatase phase (JCPDS 21-1272), indicating that the MA particles had a crystal structure of anatase TiO2. This was due to the presence of a large amount of SO42− ions in the hydrolysis system, which could easily promote the crystal structure of anatase TiO2. According to Scherrer formula, the calculated grain size of anatase face (101) for the MA samples was 22.4 nm for sample 18# and 22.8 nm for sample 19#, consistent with the slight difference in diffraction spectrum results.Figure 5XRD patterns for the obtained metatitanic acid.The SEM photographs for the obtained MA sample 18# and sample 19# were shown in Fig. 6. The MA samples mainly existed in the form of aggregated particles, which were clearly composed of smaller particles. The particle size distribution for the MA particles ranged from tens of nanometers to 1 μm. The size of dispersed smaller particles was less than 100 nm, and the smaller agglomerate particles were composed of the MA crystals. The difference in particle size between sample 18# and sample 19# was small, consistent with the results of particle size distribution and the value of diameter distance ratio. And the secondary nucleation promoted the formation of crystal clusters, then formed the primary agglomerates through surface nucleation, and lastly formed the micron aggregates by physical forces31.Figure 6SEM photographs for the obtained metatitanic acid.