Interpretation of Frenkel ’ s Theory of Sintering Considering Evolution of Activated Pores : I . Confirmation of the Time Constant

Frenkel’s theory of liquid-phase sintering was interpreted regarding pores as the activated component. The data of isothermal sintering shrinkage vs. time of the cordierite glass by Giess et al. are trained using the model by Nikolić et al. to obtain the relative density at varied temperatures and time. Then, the time constants are obtained as 1998.86mins, 388.21mins, 89.79mins and 26.11mins at 800°C, 820°C, 840°C and 860°C, respectively. The calculated time constants are close to that by the theoretical calculation, deviation of which arises from the fact that particle size is variable during the sintering process. The time constant determination is crucial to the research of the whole sintering process.


Introduction
Frenkel's liquid-phase sintering mechanism is most widely used in the liquid-phase sintering of materials.One reason is that the shrinkage is linear as a function of time, which provides accurate verification for the experimental results [1] .
Frenkel's liquid-phase sintering mechanism can only be applied successfully to explain the initial 10% of the isothermal sintering shrinkage for the mass transport governed by viscous flow.Many researchers attempted to use simpler linear shrinkage vs. time relation to explain the whole sintering process.Kingery attempted to add a constant y into the exponential part of time in order to make it deviate from the linear relation [2] .The constant y is definitely positive as the decrease in capillary size and porosity with proceeding of the sintering process [2,3] .Recently, taking account into the fact that the shrinkage is caused by decrease of activated volume, Nikolić et al. established the mathematic model of liner shrinkage vs. time during the isothermal sintering.However, in the model, time constant and other parameters are hard to confirm [4] .In this article, we use a practical equation to calculate the time constant, confirmation of which is pivotal in the research of the whole sintering process.

Experimental data
According to literature [5],

(
) ( ) Where ρ is the relative density at fixed time during the isothermal sintering, ρ 0 =0.617 is the initial relative bulk density, ΔR/R 0 is the relative diameter shrinkage, ΔH/H 0 is the relative height shrinkage, ΔR=R 0 -R, ΔH=H 0 -H, R 0 and H 0 are the initial radius and height, respectively, R and H are the instantaneous radius and length, respectively.The data of isothermal sintering shrinkage vs. time of the cordierite glass by Giess et al. is as shown in Fig. 1 and Fig. 2 [6] .The relative density at varied temperatures can be calculated by taking the data from Giess et   Fractional shrinkage vs. time of sintering for diameters [6] .

Model and data training 3.1. Relation between time constant and the relative density
To obtain an expression of time constant τ, Bordia and Raj resorted to the models for sintering of glass by Mackenzie and Shuttleworths [7] and by Scherer [8] which have been shown to fit well with the experimental data.Equating p 0 =2γ/r 0 , where γ is the surface tension and r 0 is the pore radius in the model by Mackenzie and Shuttleworth, Bordia and Raj obtained the following approximate expression for τ, the correction factor to account for porosity is not considered herein.
Where ρ g is the relative density of the bulk glass, η s is the viscosity of the glass at fixed temperature.For Equation (2), we will discuss it later that it is not suitable for the accurate determination of τ.

Data training and results
According to Fig. 1 and Fig. 2, the initial 5 data are in the range of Frenkel's liquidphase sintering.From Tab. 1, the relative density at 0min, 50mins, 120mins, 180mins and 240mins at 800°C are obtained, respectively.Then, using Equation ( 5) to train the obtained relative intensities in the software Origin version 8.0 [10] , and the Levenberg-Marquardt algorithm is adopted.The training results are shown in Fig. 3. Form Fig. 3, the time constant at 800°C is determined as 1998.89mins.
By the same means, the fitting results for that at 820°C, 840°C and 860°C are as shown in Figs.4-6.From Figs. 4-6, the time constants at 820°C 840°C and 860°C are determined to be 388.21mins,89.79mins and 26.11mins respectively, and are listed in Tab.II.It is obvious that the time constant in the stage of Frenkel's liquid-phase sintering decreases as the temperature increases.In order to confirm whether the fitting results are reasonable, we calculate the time constants at varied temperatures by Equation (2).As is shown in Table 2, the calculated time constants are close to that by practical fitting, deviation of which arises from the fact that pore radius r 0 is variable during the sintering process.

Conclusion
In this article, Frenkel's theory of liquid-phase sintering was interpreted regarding pores as the activated volume.And the mathematic model established by Nikolić et

Fig. 3 .
Fig. 3.The relative density vs. time at 800°C and the fitting results.

Fig. 4 .
Fig. 4. The relative density vs. time at 820°C and the fitting results.

Fig. 5 .
Fig. 5.The relative density vs. time at 840°C and the fitting results.

Fig. 6 .
Fig. 6.The relative density vs. time at 860°C and the fitting results.Tab.II Time constants at varied temperatures.
al. into equation(1).The shrinkage and relative density at varied temperatures are listed in Tab.I.