Interpretation of Frenkel’s theory of sintering considering evolution of activated pores: III. Determination of equilibrium sintering time

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*)_{Corresponding author: johnyucl@aliyun.com}

_____________________________

doi: 10.2298/SOS1502215Y

UDK

549.632; 622.785

Interpretation of Frenkel’s Theory of Sintering Considering

Evolution of Activated Pores: III. Determination of Equilibrium

Sintering Time

C.-L. Yu

*)

, D.-P. Gao, F. Wang, R.-J. Huo, X.-M. Hao, X.-L. Xie

School of Materials Science and Engineering, Shaanxi University of Science &

Technology, Xi’an 710021, P. R. China; 2. School of Materials Science and

Engineering, Chongqing University of Technology, Chongqing 400054, P. R. China

Abstract:

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.

Keywords: Equilibrium time determination; Activated pores; Frenkel’s theory of liquid-phase sintering; Cordierite glass

1. 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.

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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.

2. 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].

2.2. 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 v0, 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,

0

1 (

1) exp(

)

v

t

τ

+

= +

+

−

(1)

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],

0

n

R t

K

R

τ

Δ ⎛ ⎞

= ⎜ ⎟_{⎝ ⎠} (2) 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

E

t

K

RT

ϕ

τ

⎛

⎞ ⎛

=

_⎜

−

_{⎟ ⎜}

⋅

⎝

⎠ ⎝

⎞

⎟

⎠

(3)

Where K0 is a constant, R - the gas constant, E - the activation energy and T -

sintering temperature. The parameter

ϕ

t

τ

⎛ ⎞

⎜ ⎟

⎝ ⎠

represents a measure of the degree of sintering.

Provided that sintering is determined by the transport of activated pores volume, then

ϕ

t

τ

⎛ ⎞ ⎜ ⎟ ⎝ ⎠ can be defined as the ratio between the equilibrium activated volume and the effective activated volume given by,

t v v

ϕ

τ

+ ⎛ ⎞ = ⎜ ⎟

(3)

0

1 1 (

1) exp(

)

t

v

t

v

ϕ

τ

+

⎛ ⎞ =

⎜ ⎟

⎝ ⎠

₊

₋

(5)

Thus, the following equation can be derived to describe the sintering kinetics as,

0

_{1 (}

_{1) exp(}

₎

n

R

K

t

v

t

R

v

τ

+

Δ

₌

⎛ ⎞

⋅⎜ ⎟

_{⎝ ⎠}

+

−

(6)

Where

K

₀

exp

E

RT

⎛

⎞

=

_⎜

−

⎝

⎠

⎟

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 τ800ºC=1998.86mins, τ820ºC=388.21mins,

τ840ºC=89.79mins and τ860ºC=26.11mins, respectively.

3. Data Training and Discussion

For the research, according to reference [12], we obtain:

max

(

1 exp

)

tτ

ρ

−

ρ

Δ = Δ

−

(7) Where Δ =

ρ

(

ρ ρ

− ₀

)

ρ

₀ , Δ

ρ

_max= −

(

1

ρ

₀

)

ρ

₀,

ρ

is the relative density of cordierite glass at instant sintering time,

ρ

₀ - the initial relative density,

Δ

ρ

- the variation of relative density. Thus eq. (7) can be rewritten as:

ρ

=1- 1-

(

ρ

0

)

exp -

(

t

τ

)

(8)

and

v

= −

1 ρ

, then

v

₀

= −

1 ρ

₀

=

0.383

,

v

+

= −

1 ρ

+,

ρ

+ - the relative density in

the equilibrium state. Then taking

v

+

= −

1 ρ

+ into eq. (8) we obtain:

(

ln

0.383 t

= −

τ

V

+

)

(9)

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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].

4. Conclusion

In this article, the mathematic model established by Nikolić et al. was used to obtain the equilibrium sintering time at varied sintering temperatures for the sintering of cordierite glass by Giess et al. 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. In short, the model established by Nikolić et al. can be applied successfully to predict isothermal sintering equilibrium time of the cordierite glass at varied temperatures.

Acknowledgments

This work is co-supported by the Research Fund for Ph.D in SUST, P. R. China under Grant No. BJ09-11, the Special Scientific Research Project for the Education Department of Shaanxi Province, P. R. China under Grant No. 12JK0459, the Innovation Fund for Industrial Technologies of Xi’an City under Grant No. CX1256(5), the Ministry of Science and Technology Innovation Fund for Small and Medium Enterprises, P. R. China under Grant No. 12C26216106760, the Innovation Team Fund of SUST, P. R. China under Grant No. TD12-05, and the “6333” Scientific Innovation Project of Baoji City, P. R. China under Grant No. 2009FWPT-9.

5. References

1. M. M. Ristic, S. Dj. Milosevic. Frenkel’s theory of sintering. Sci Sintering, 2006, 38(1): 7-11.

2. D. Bernard, D. Gendron, J. M. Heintz, et al. First direct 3D visualisation of microstructural evolutions during sintering through X-ray computed microtomography. Acta Mater, 2005, 53(1): 121-128.

3. Ya. I. Frenkel. Viscous flow of crystalline bodies under action of surface tension. J Phys (U. S. S. R.), 1945, 9: 385-39. (in English)

4. W. D. Kingery, M. Berg. Study of initial stages of sintering solids by viscous flow, evaporation-condensation, and self-diffusion. J Appl Phys, 1955, 26(10): 1205-1212. 5. G. C. Kuczynski, I. Zaplatynskyj. Sintering of glass. J Am Ceram Soc, 1955, 39(10):

349-350.

6. E. A. Giess, C. F. Guerci, G. F. Walker, et al. Isothermal sintering of spheroidized cordierite-type glass powders. J Am Ceram Soc, 1985, 68(12): C-328-C-329.

7. E. A. Giess, J. P. Fletcher, L. W. Herron. Isothermal sintering of cordierite-type glass powders. J Am Ceram Soc, 1984, 67(8): 549-552.

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9. N. S. Nikolić, M. V. Nikolić, S. M. Radić, et al. Sintering as a process of transport of activated volume. Sci Sintering, 2002, 34(1): 53-56.

10. C. P. Flynn. Point Defects and Diffusion. Oxford: Clarendon Press, 1972.

11. M. V. Nikolić, N. Labus, M. M. Ristić. A phenomenological analysis of sintering kinetics from the viewpoint of activated volume. Sci Sintering, 2005, 37 (1): 19-25. 12. R. K. Bordia, R. Raj. Analysis of sintering of a composite with a glass or ceramic

matrix. J Am Ceram Soc, 1986, 69(3): C-55-C-57.

Са р а: ,

. Р

,

Н . К , 800ºC, 820ºC, 840ºC

860ºC 7014,42, 1569,65, 368,92 114,61 , . Р

,

. , Н ,

.

К учн р чи: ђ , ,