Interpretation of Frenkel ’ s Theory of Sintering Considering Evolution of Activated Pores : III . Determination of Equilibrium Sintering

In this article, the Frenkel’s theory of liquid-phase sintering was interpreted regarding pores as the activated volume. The mathematical model established by Nikolić et al. was used to infer the equilibrium sintering time at varied sintering temperatures during the isothermal sintering of codierite glass by Giess et al. Through the calculation, the equilibrium time at 800oC, 820oC, 840oC and 860oC is inferred to be 7014.42mins, 1569.65mins, 368.92mins and 114.61mins, respectively. The equilibrium time decreases as the temperature increases. And the theoretical value is in good accordance with the experimental results. Thus, the model established by Nikolić et al. can be applied successfully to predict the equilibrium sintering time of the cordierite glass at varied temperatures during isothermal sintering.


Introduction
Frenkel's theory of liquid-sintering is a significant liquid-phase sintering mechanism, in the model of which the shrinkage is linear as a function of time [1] .Now it is well accepted that the continuous open-pores formed by voids, produced by the isolated particles in the initial stage of sintering, will gradually close and decrease with the process of sintering.And it has been proved by X-ray computed microtomography [2] .
In the Frenkel's theory of liquid-sintering, closed pores with radius r are formed by the merged particles with radius r and thus independent pores are formed in the initial stage of sintering, hence the relation of linear shrinkage to time is determined [3] .For the ideal glass system, the interfacial radius, formed by two spheric glass particles, as a function of sintering time is linear [4~5] .However, in the past decades, Frenkel's theory is proved to only fit well with the initial 10% of the whole sintering shrinkage for the general viscous flow mass transport system.
For the spherical glass particles, Giess et al. indicated that determination of the relation of shrinkage to time should take into account the final equilibrium shrinkage, which can be described by the Avrami equation [6] .Thus the reasonable relation of shrinkage to time should take into account the final equilibrium state.However, some equilibrium parameters, such as equilibrium time, are difficult to determine.In this article, we attempt to use the mathematical model established by Nikolić et al. to train the data from Giess et al., and try to get the equilibrium sintering time at varied sintering temperatures for cordierite glass.

Data and Mathematical Model 2.1. Data
The data of fractional shrinkage vs. time used in this article are taken from the research of isothermal sintering of cordierite glass by Giess et al.The detail preparation parameters and other parameters are given in reference [7].The data are as shown in Fig. 1 and listed in Tab.I in reference [8].

Mathematical Model
From the isothermal sintering process of the cordierite glass, the sintering can be described as a process from unsTab.state to equilibrium state.At the beginning of sintering, the volume of effective activated pores is defined as v 0 , and in the equilibrium system the effective activated pores is defined as v + [9] .At any time, the degree of sintering can be characterized by the volume of effective activated pores -v.During the sintering process, volume of effective activated pores -v gravitates towards the equilibrium state -v + .In this process, the reduction of the free energy follows the thermodynamic law.
According to the reference [10], reduction in effective activated volume can be defined as, Where τ in min is a time constant.In general, kinetics of the sintering process can be expressed in the relation of linear shrinkage for diameters to time as [11] , Where K -the sintering rate constant which depends on the sintering temperature and time, n -a constant which depends on the mechanism and the sintering process, t -sintering time.The sintering rate constant is then defined as, 0 exp Where K 0 is a constant, R -the gas constant, E -the activation energy and Tsintering temperature.The parameter represents a measure of the degree of sintering.
Provided that sintering is determined by the transport of activated pores volume, then can be defined as the ratio between the equilibrium activated volume and the effective activated volume given by, Then taking into account eq. ( 1) we obtain: Thus, the following equation can be derived to describe the sintering kinetics as, Where ⎠ ⎟ only depends on E and T. The data of fractional shrinkage vs. time from Tab.I in reference [8] are trained by eq. ( 6).According to our previous analysis [8] , the time constants are τ 800ºC =1998.86mins,τ 820ºC =388.21mins,τ 840ºC =89.79mins and τ 860ºC =26.11mins, respectively.

Data Training and Discussion
For the research, according to reference [12], we obtain: ( ) ) , ρ is the relative density of cordierite glass at instant sintering time, 0 ρ -the initial relative density, ρ Δ -the variation of relative density.Thus eq. ( 7) can be rewritten as:  8) we obtain: Fig. 1.Effect of sintering temperature on the equilibrium sintering time.
The sintering equilibrium time of cordierite glass can be calculated by eq. ( 9).The equilibrium time at 800ºC, 820ºC, 840ºC and 860ºC is 7014.42mins,1569.65mins,368.92mins and 114.61mins, respectively.Fig. 1 shows the variation of equilibrium time with sintering temperature.The equilibrium time decreases as the temperature increases.And the theoretical value is in accordance with the experimental results [7] .