Kinetic Model for Nanocrystalline Anatase to Rutile Polymorphic Transformation

In this paper, kinetic models are presented for phase transformation from anatase to rutile in nanocrystalline samples at different temperature ranges. The models, in which the nucleation and growth are considered together, are able to predict the polymorphic transformation behavior of titania with better accuracy. In addition to researchers’ assumption that phase transformation of nanocrystalline anatase to rutile at lower temperatures occurs only with interface nucleation, the results reveal that the surface nucleation can also have a significant role in the kinetics of phase transformation in conditions of prolonged heating. In comparison to other published models, the advantage of these models is that no experimental data regarding the size of particles is required to study the kinetics of phase transformation of TiO2 nanoparticles.


Introduction
Titania (TiO 2 ) occurs in at least seven different polymorphic forms 1 .Anatase, rutile, and brookite are the most common polymorphous of titania.Titania nanoparticles are employed in a wide range of applications, such as photocatalysis, pigments, ceramics, cosmetics, gas sensors, and dye-sensitized solar cells.The material properties of TiO 2 nanoparticles are a function of the crystal structure, nanoparticle size, and morphology.For example, anatase crystals with regular crystal surfaces have a higher photocatalytic activity than those with irregular crystal surface 2 .Furthermore, the photocatalytic activity of nanocrystalline anatase is greater than that of rutile 2 , where the activity of nanocrystalline rutile increases with decrease in particle size 2,3 .Finally, the photocatalytic activity of amorphous titania is negligible 4,5 .On the other hand, using TiCl 4 , Zou et al. 6 have synthesized an amorphous TiO 2 sol which demonstrated a higher photocatalytic activity for the photodegradation of methylene blue, in comparison with anatase sol.
Understanding the mechanism of the phase transformation of titania is fundamental to the control of the microstructure and, thus, its properties.The transformation between polymorphs involves kinetics and thermodynamics of process.Different polymorphs have different stability due to surface energy effects, and the growth of nanoparticles changes the surface energy and can thus profoundly affect the rate and mechanism of nanoparticle transport 7,8 .
At standard conditions and macroscopic sizes, rutile is the most stable phase of titania thermodynamically, and upon heating, other phases are transformed to rutile 9,10 .Anatase-to-rutile is an irreversible transformation and there is no equilibrium transformation temperature, and that was observed at temperatures between 400 and 1200 ºC, depending on particle size, method of synthesis, impurity content, and the atmosphere [9][10][11] .It is observed that the transformation from anatase to rutile is initiated only when the particle size exceeds a certain size (~16-14 nm), which is attributed to the reversal of the two-phase stability in this particle size, which is also confirmed by thermodynamic analysis 13,14 .
The kinetics of the polymorphic transformation in titania from anatase to rutile has been studied intensively using anatase samples with different size at various temperatures [9][10][11]14,[15][16][17][18][19][20][21][22][23][24][25][26][27] . Genari ad Pasquevich 10 , and Shanon and Pask 11 , stated that the solid-phase transformation is a process of nucleation and growth, and they used conventional models for transformation kinetic interpretation from anatase (micrometer size) to rutile.It has been demonstrated that conventional models (including standard first-order reaction, standard second-order reaction, contracting spherical interface model, model of random nucleation and rapid growth, model of nucleation and growth of overlapping nu-clei, model of one-dimensional, linear, branching nuclei and a constant growth, and Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation) are not applicable in describing the kinetics of the phase transformation of nanocrystalline anatase 12,28 .Zhang and Banfield 27,28 , proposed kinetic models for TiO 2 polymorphic transformation from nanocrystalline anatase to rutile, where their models incorporate nucleation mechanisms alongside experimental data of the size of anatase nanoparticles.
In this paper, we propose new kinetic models to describe published experimental data for the anatase to rutile phase transformation in nanocrystalline samples.The results reveal that kinetic models presented in this paper are able to predict the polymorphic transformation behavior of titania with better accuracy.The results also show that in some cases, the transformation kinetics can be well described with a different interpretation than the previous interpretations.

Kinetic model
The fraction of transformation, α, for a dimorph phase transformation is defined 28 as: (1) where D, N and ρ represent the particle size, number of particles, and density of initial phase, respectively.Index 0 and 1 indicate, respectively, quantities at the initial reaction time (t = 0) and their values at any time t.According to Eq. ( 1), it can be written as: The number of particles of initial phase can be reduced, by interface nucleation (IN) and/or combined interface and surface nucleation (IN+SN), with time 27,28 (Fig. 1): where k 1 and k 2 represent the kinetic constants for the interface nucleation and the surface nucleation, respectively.These constants are a function of tem-perature and sample characteristics.Integrating Eqs.
(3) and ( 4) from time = 0 (when N 1 = N 0 ) to time t, we obtain: (Solve Eq. ( 4)) ( 6) The growth of nanoparticle in dry heat treatments is commonly described by the equation 29 : where k' is a constant, and m is the grain growth exponent representing the grain growth behavior.It is expected that the solid-state transformation behavior can be predicted using the kinetic models in which equations of the nucleation and growth are taken into consideration together.Accordingly, for the fraction of transformation, α, from one phase to another phase in the phase transformation processes in dry heating treatments, with inserting D 1 from Eq. ( 7) and N 1 /N 0 from Eq. ( 5) or (6) into Eq.( 1), we propose the following kinetic models: (Combining Eq. ( 1) with Eqs. ( 5) and ( 7) 1) with Eqs. ( 6) and ( 7)) .) ( 1 In Eqs. ( 8) and ( 9), k = k'/D 0 is an apparent kinetic constant relevant to the grain growth and k 1 N 0 can be viewed as one complex constant.
Following the predictions of these models, phase transformation behavior of titania nanoparticles were evaluated by the published experimental data in the literature.

Results and discussion
Phase transformation of nanocrystalline anatase to rutile at low temperature (< 600 °C) Fig. 2 shows the experimental data of formation of rutile in anatase (~6 nm) samples in the temperature range of 465-525 ºC 12 , alongside models curves ( 8) and ( 9).It may be observed that the experimental data at 465 and 480 ºC can be satisfactorily predicted by the kinetic model (8) (Fig. 2 (a)).The deviation of kinetic model (8) [combining interface nucleation correlation (5) with the growth correlation (7)], over longer reaction times at temperatures of 500 and 525 ºC, can be attributed to the presence of surface nucleation in the transformation process (an alternative interpretation other than that offered by Zhang and Banfield 28 ); that the experimental data related to total reaction time at these two temperatures can be very satisfactorily fitted using the kinetic model (9) [combined interface and surface nucleation correlation (6) together with the growth correlation ( 7)] (Fig. 2 (b)).Parameters of kinetic models of ( 8) and (9) were obtained for various temperatures by the fitting of experimental data and the method of non-linear leastsquares fitting, as listed in Table 1.
Ta b l e 1 -Parameters for the kinetic models (8) and ( 9) with m = 1, for transformation from anatase to rutile (Fig. 2) Kinetic model (8)  Kinetic model ( 9) Zhang and Banfield, 27,28 expressed that, at lower temperatures (less than 600 ºC), thermal fluctua-tion of Ti and O atoms in anatase is not strong enough to generate rutile nuclei on the surfaces of the anatase; and only considered the dominant interface nucleation at phase transformation kinetics.However, according to the ratio of rate of surface nucleation to that of interface nucleation, as where Eq. ( 1) has been substituted, we have: It is seen that the presence and dominance of a mechanism of nucleation is dependent on various parameters, like temperature, time, particle size, and also particle packing 30 .In Fig. 2 (b), at experimental points, the approximation beginning of the deviation of curves of models ( 8) and ( 9), anatase sample characteristics according to Ref. 12 , are as follows: F i g . 2 -Points represent transformation experimental data from anatase to rutile 12 , and curves are fitted with (a) kinetic model (8), and (b) kinetic model (8) {dashed lines} and kinetic model (9)

{solid lines}
Using kinetic constants in Table 1, alongside the above data in Eq. ( 10), the ratio of the rate of surface nucleation to that of interface nucleation (R), at 500 and 525 ºC, has been obtained to be greater than 1 (R T =500 ºC ≈ 17.4 and R T =525 ºC ≈ 14.4).Therefore, expectedly, the surface nucleation and interface nucleation can play an important role at phase transformation in such conditions of heating for a long time at temperatures below 600 ºC.
With data of k 1 N 0 in Table 1 at 465 and 480 ºC, activation energy of interface nucleation is obtained, E a (IN) = 216 kJ mol -1 and also using data of k 1 N 0 and k 2 listed in Table 1 for kinetic model ( 9), activation energies of interface nucleation and surface nucleation can be calculated: A comparison of these E a data suggests that, with the presence and increase in participation of surface nucleation at phase transformation, activation energy for interface nucleation increases, which is perhaps indicative that with coarsening of anatase nanoparticles and reducing their contact surfaces, the rate of interface nucleation is reduced.
Phase transformation of nanocrystalline anatase to rutile at high temperature (> 600 °C) Fig. 3 shows that the kinetic model ( 9) can also provide a good prediction of experimental data for the fraction of transformation of anatase (~8 nm) to rutile at temperatures ranging from 620 to 690 ºC.Model parameters were obtained for various temperatures by the fitting of experimental data and the method of non-linear least-squares fitting, as listed in Table 2. Arrhenius plot of the kinetic constants k 1 N 0 and k 2 listed in Table 2 are shown in Fig. 4, giving activation energies of interface nucleation and surface nucleation in phase transformation from anatase to rutile, respectively (Table 3).
Activation energy of surface nucleation obtained by the model presented by Zhang and Banfield 27 (listed in brackets in Table 3), in comparison with the value obtained with model ( 9), showed that, instead of using the experimental data of particle sizes of nanocrystalline anatase over the reaction times, a growth correlation is used and there is very little difference (< 3 %).This result points out that, even when the experimental data of particle sizes at different times are not available, this model can be a good replacement instead of their model.

F i g . 3 -Points represent transformation experimental data
from anatase to rutile 27 , and curves are fitted with Eq. ( 9) Ta b l e 2 -Parameters for the kinetic model ( 9) with m = 3, for transformation from anatase to rutile (Fig. 3) The greater difference in activation energy of interface nucleation (~26 %) may have been developed due to the use of only a series of experimental data (note the very large differences in values of kinetic constants for the interface nucleation, and their averages in Table 3 from Ref. 27 ) Kinetic model ( 9) also provides a very satisfactory fit to the experimental data for the fraction of transformation of anatase to rutile in the presence of alumina nanoparticles (Fig. 5) at 640 and 680 ºC.Using nonlinear least-square fitting, the data points in Fig. 5 were fitted with Eq. ( 9), and the parameters k 1 N 0 and k 2 were obtained (Table 4).With the data in Table 4, activation energies can be estimated: E a (IN) = 321 kJ mol -1 ; E a (SN) = 308 kJ mol -1 .It is observed that E a (SN) obtained from kinetic model ( 9), in comparison with E a (IN) is determined by the kinetic model employed by Zhang and Banfield 27 , again, the difference is very small (< 1.5 %).
Madras et al. 31 have stated that their kinetic model based on population balance equation is not able to interpret the experimental results of phase transformation of anatase to rutile in the presence of alumina nanoparticles at 680 ºC, due to scattering data and their anomalous behavior; and also their model did not provide a good fit of the experimental data at 640 ºC.

Conclusion
In this research, we have developed kinetic models in which consideration of the nucleation and growth resulted in successful simulation of the experimental data for the phase transformation of nanocrystalline anatase to rutile at various temperatures.Compared with similar models for the interpretation of behavior of phase transformation from anatase to rutile at the nanoscale, which are related to experimental data of particle sizes of anatase during the kinetic experiments, these models can be a very good substitute, and are only dependent on the experimental data of fractions of transformation.The results also indicate that at low temperatures (<600 ºC), where, according to studies, the transformation occurs only by interface nucleation (IN), SN addition to IN can play an important role in the kinetics of phase transformation due to heating for a long period of time (e.g.50 h at 500 ºC).We propose further evaluation of the predictability of kinetic models presented in this paper (Eqs.( 8) and ( 9)) concerning the behavior of solid state phase transformations using different kinetic data for other polymorphs of materials, such as γ-Al 2 O 3 → α -Al 2 O 3 .F i g .4 -Arrhenius plot of the kinetic constants k 1 N 0 and k 2 for transformation from anatase to rutile (at T > 600 ºC) F i g .5 -Points represent transformation experimental data from anatase to rutile 27 , and curves are fitted with Eq. ( 9 α -fraction of transformation, -D * -particle size, nm N * -number of particles,r -density, kg m -3 t -time, min, h k 1 -kinetic constants for the interface nucleation, h -1 k 2 -kinetic constants for the surface nucleation, h -1 k, k' -rate constant, nm h -1 m -grain growth exponent representing the grain growth behavior, nm h -0.33 T -temperature, °C, K R 2 -regression coefficient, -A 0 -frequency factors, h -1 E a -activation energies, kJ mol -1 *Index 0 and 1 indicate, respectively, quantities at the initial reaction time (t = 0) and their values at any time t

F i g . 1 -Fig. 1
Fig. 1 Diagram showing nucleation models in the solid-phase transformation