The Master Sintering Curve for Pressureless Sintering of TiO 2

A Master Sintering Curve (MSC) for rutile TiO2 was constructed for pressureless sintering using constant heating rate dilatometry data based on the combined-stage sintering model. Construction of the master sintering curve is described and the validation is proved with rutile TiO2 under different thermal histories. The concept of master sintering can be used to predict the sintering shrinkage and final density and calculate the activation energy, and a value of 105 KJ/mol for TiO2 was obtained. With one temperature dependent parameter determined experimentally, it became possible to describe accurately the densification behavior of TiO2 from the initial to final stages of sintering.


Introduction
There has been an increasing interest in nanocrystalline TiO 2 in powder and ceramics forms.A nanocrystalline TiO 2 powder is widely used for its catalytic, photo-catalytic and gassensing properties.Moreover, nanocrystalline TiO 2 ceramics are also of great importance due to the low-creep and superplastic property at room temperature.However, preparing a fullydense fine-grained TiO 2 is difficult because the grains grow rapidly at the later stages of sintering [1].
Sintering has long been recognized as a very complicated process, involving the evolution of microstructure through the action of several different transport mechanisms.Early sintering studies were focused on rigorously defined ideal geometric models that represented only one of the three stages in the process [2][3].Several attempts at more-general simulation of sintering have been reported.Hassold, Chen and Srolovit [4] used Monte-Carlo methods in the final stages of sintering in two dimensions and showed microstructure evolution features.
One of the ultimate objectives for sintering studies is to be able to predict densification results under different thermal histories for a given processing method.It has been reported that the geometric parameters related to sintering often are functions only of density for a given powder and green-body process, provided that one diffusion mechanism dominates in the sintering process.Based on this report, the concept of a master sintering curve has been developed that characterized the sintering behavior for a given powder and green-body process regardless of the heating profiles.The formulation and construction of the master sintering curve are given in this paper.When this new method is used, densification behavior can be predicted under arbitrary temperature-time excursions following a minimal set of preliminary experiments, and these predictions can be used in planning sintering strategies.Moreover, deviations from the assumption of a single mechanism can be observed readily.
Three polymorphs of titania occur in nature: rutile (tetragonal), anatase (tetragonal) and brookite (orthorhombic).Rutile is the only stable phase, whereas anatase and brookite are metastable at all temperatures, transforming to rutile when they are heated.Titania normally undergoes an anatase /rutile phase transformation in the temperature range from 600 to 700 ℃ [5].To avoid the anatase/rutile phase transformation and simplify the sintering process, we chose rutile TiO 2 as the subject.
The aim of the present work is to construct and validate a master sintering curve for TiO 2 for pressureless sintering using constant heating rate dilatometry data based on the combined-stage sintering model.With the help of MSC of TiO 2 , we can predict the sintering shrinkage and final density, calculate the activation energy and describe accurately the densification behavior of TiO 2 from the initial to final stages of sintering.

Construction of the Master Sintering Curve
The master sintering curve can be derived from the densification rate equation of the combined-stage sintering model [6].For the development of master sintering curves, the parameters in the sintering rate equations are separated into (a) those related to the microstructure and (b) those related to time and temperature terms, on the opposite sides of the equation.If there exists only one dominant diffusion mechanisms (either volume or grain boundary diffusion), the terms that define the microstructural evolution and are independent of the thermal history can be represented by [7,8].
where k is the Boltzmann constant, γ is the surface energy, Ω the atomic volume, D 0 is the preexponential factor, ρ 0 is the green density of the powder compact, Γ is the collection of microstructure scaling parameter.The time and temperature dependent side of the equation can be represented as theta parameter, Θ, where t is the instantaneous time, which is usually a function of temperature.Q is the activation energy, R is the gas constant, T the absolute temperature.For isothermal portion of the sintering runs, Eq. ( 2) can be simplified to: ( , ( )) exp( ) where t i is the duration of the isothermal portion of the run.For a constant heating rate, Eq.
(2) can be written as: where c is the heating rate used and T 0 is the temperature below which no sintering takes place.

( ) ( , ( )) t T t ρ Φ =Θ
(5) The relationship between the density (ρ) and Θ (Eq.( 5)) is defined as the master sintering curve.For the construction of MSC, a series of runs at different temperatures (isothermal) or constant heating rates over a range of heating rates is needed.If the activation energy of sintering is unknown, it has to be estimated in order to obtain the master sintering curve.
For the construction of MSC, the integral of Eq. ( 2) and the experimental density should be known.Dilatometry can be conveniently used to determine the density since the instantaneous density at all times can be obtained from the dilatometric data.For the calculation of Θ, the activation energy for the sintering process must be known.If the activation energy is unknown, it can be estimated with good precision from Θ versus density (ρ) data.For this purpose, a particular value of activation energy is chosen and ρ-Θ curves are constructed for each heating rate.If the curves fail to converge, a new value of activation energy is chosen and the calculations are repeated.This procedure should be continued until all the curves converge showing that the activation energy is the acceptable one for sintering.A curve can be then fitted through all the data points, and then convergence of data to the fitted line can be quantified through the sum of residual squares of the points with respect to the fitted line.The best estimate of Q will be the value of the minimum in the plot of activation energy versus mean residual squares [7].

Experimental procedure 3.1 Material
Nanocrystalline rutile TiO 2 was purchased from Zhejiang Zhoushan nanomaterials Corporation, China.The main characteristics of the as-received rutile TiO 2 powders are given in Tab.I.

Green body preparation
The as-received powder was first pressed in a mold at room temperature at a relatively low press to impart handling strength uniformly to the green body.Then, the green body was pressed by cold isostatic pressing at 200 MPa with a dwelling time of 8 minutes.The green body was 50×5×5 mm in size.The green density of these samples was about 42% of the theoretical density (TD).

Dilatometry
The shrinkage of TiO 2 was measured with a push rod type dilatometer in the axial direction.The length change measurements were made by a linear voltage differential transducer (LVDT) which was maintained at a constant temperature by means of water circulation from a constant temperature bath.The temperature was measured using a calibrated thermocouple placed directly above the sample.Sintering was performed in air using a dilatometer.For non-isothermal sintering (constant heating rate), two heating rates of 2, 5C/min were used to reach the desired temperature without holding.Up to 200 0 C, all the samples used for dilatometric experiments were heated using a heating rate of 10 0 C /min to drive out moisture and volatile impurities.As the sample was heated, its temperature and length values were measured continuously with the help of a thermocouple and LVDT, respectively.
The green density was measured by a geometric method, and the sintered density was 0 measured by the Archimedes method with deionizer water.The densities as well as open and closed porosity levels of the specimens were determined by the Archimedes method.The value of 4.25 g/cm 3 was used as the theoretical density.

Results and discussion
Fig. 1 shows the relative density versus temperatures for TiO 2 under different heating rates.It can be seen from the figure that the relative density at a low heating rate is higher than that at a high heating rate for the same temperature.The relative density was converted from the shrinkage curve.The shrinkage values were converted into relative density using the following relation [9]: where ρ and ρ 0 are the densities of the sintered and green pellets, respectively.The curves have the familiar shape and generally shifted to higher temperatures with increasing heating rate.It can be noted that the sintered densities obtained at any temperature showed a modest but systematic dependence on the heating rate.The density was found to increase at 1100 ℃ approaching a maximum value greater than 90% of the TD.The maximum density was found to depend upon the heating rate, the higher the heating rate the lower the sintered density.As mentioned earlier, one of the essential data for obtaining the master sintering curve is the activation energy.For this, the density data for TiO 2 obtained from the dilatometric measurements, and Θ values obtained from Eq. (2) are employed.A ρ-Θ curve is then constructed for all the heating profiles for a chosen value of activation energy (50 kJ/mol) as shown in Fig. 2(a).It can be seen that the curves for different heating rates are not converging.Now a new value of activation energy is chosen and the calculation is repeated.The curves at 100, 150 and 200 kJ/mol are shown in Fig. 2(b)-(d), respectively.The best convergence occurs at around 100 kJ/mol.Fig. 3 gives the mean residual squares for the various values of activation energy and the minimum has been found to be for 105 kJ/mol.The activation energy thus obtained was found to be in reasonable agreement with the values reported in the literature (96.2 kJ/mol) [10].The activation energy for densification is a characteristic quantity that elucidates the fundamental diffusion mechanisms during the sintering process.ionally, it has been obtained from the shrinkage rate from either isothermal he ing Tradit at or constant heating rate experiments.The present result indicates that an alternate method exists on the basis of MSC for the determination of the activation energy of sintering if it is unknown.

ig. 4 of
From the knowledge of the activation energy of sintering, a MSC for TiO 2 was co tru

. Validation of Master Sintering Curve
Before applying MSC, it is necessary to validate it with a sufficient number of experim It can be seen that the value of Θ changed dramatically from 10 -40 at the beginning to10 -28 at the higher density end.Despite a 2.5-fold rise in the heating rate, the individual sintering curves have merged reasonably close to a single curve.This result suggests that there must be a general curve, regardless of the sintering path, which is what we have defined as the MSC [11][12][13][14][15].

5
ents.The experiments were carried out as the following.Three pellets of rutile TiO 2 were compacted under the same conditions as the ones used in section 3.2.The experiments were carried out in a dilatometer.The first sample pellet was sintered by heating up to 900 0 C and holding at that temperature for 4 h.The second and third pellet was sintered at 1000 0 C for 3 h and 1100 0 C for 2 h, respectively.The sintered density of the three pellets was measured by the Archimedes method.It was found to be 61%, 80% and 92% of theoretical density for the pellets sintered at 900 0 C /4 h, 1000 0 C / 3 h and 1100 0 C 2 h, respectively.From the dilatometric data, the Θ values were calculated for isothermal holding using Eq.( 3).These values were shown on the master sintering curve in Fig. 5.It can be seen that the values for all the three temperatures with different periods of time are on the MSC, validating the concept of MSC.sinte Acknowledgments ring using constant heating rate dilatometry data based on the combined-stage sintering model.Construction of the master sintering curve is described and the validation is proved with rutile TiO 2 under different thermal histories.The MSC for TiO 2 provides a useful tool to predict the densification behavior from the initial to final stages of sintering.The concept of MSC can be used to calculate the activation energy for sintering.The activation energy for sintering for TiO 2 is 105kJ/mol.The density continuously determined by the dilatometer during pressureless sintering was plotted against the integral of a temperature function over time, Θ (t, T (t)).It agreed well with the densities determined by the Archimedes method for different heating histories.It is proved that the density profile versus time-temperature integral values, Θ, can be used to predict the final density regardless of the heating history.Master sintering curves enable prediction of the final density for a given heating history, or heating schedule in order to achieve a certain relative density level at a given temperature.

Fig. 1
Fig.1 Relative density (%TD) versus temperature for TiO 2 for different heating rates

A
master sintering curve (MSC) for rutile TiO 2 was constructed for pressureless Master sintering curve for TiO 2 .The curve is constructed using an activation energy is shown in Fig.4.
Characteristics of the as-received rutile TiO 2 powders