소결 중에 입자 내부 기공의 미세구조 해석

정용현, 이규혁, 김  헌, 편무실

명지대학교 화학공학과

Microstructural Analysis of Intra-Grain Pores during Sintering

Yong-Hyun Jung, Kyu-Heock Lee, Hern Kim, Moo-Sil Pyun

Department of Chemical Engineering, Myong Ji University

Introduction

One of the major structural characteristics of the specimen during the intermediate and final stages of sintering, namely, shrinkage and/or growth of the closed pores formed at the end of the initial stage of sintering.  A typical microstructure at the end of the initial stage of sintering consists of grains and pores.  The pores can be classified into two groups: (i) pores between or among the grains and (ii) pores inside the grains.  When gases which resist diffusion through the grains (such as argon or helium) are entrapped in these pores, the shrinkage or growth is strongly influenced by the gas pressure.  On the other hand, when the pores are empty or contain diffusible gases (such as hydrogen), the growth will be controlled significantly by lattice-diffusion of atoms or vacancies.  In this case material transport will occur between isolated pores and between the pores and the grain boundaries.  If many grain boundaries exist in the sintered material, the material transport between the pores and the grain boundaries will cause the shrinkage of pores and consequently the average size and total number of the pores will be reduced.

Very few systematic studies of the structure of pores and the variations in their geometric properties during the intermediate and final stages of sintering are available.  Most of the available literature consist of studies of the sintering behavior of inter-grain pores.  In contrast, the sintering behavior of intra-grain pores have received little attention because of the experimental difficulties involved in preparing samples.  We mention a few examples here so that the use of our computer simulations for this purpose can be illustrated.  The structural changes in pores lying within the grains have been studied optically [1-3] using a single large grain.  These authors observe that the porosities remain unchanged during sintering but that the shapes of the pores become more circular.  The literature also contains reports of experiments [4] which show that the intra-grain pores completely separated from grain boundaries cannot be removed easily during the intermediate and final stages of sintering.  These features and accompanying changes can be studied readily using our computer simulations [5].  In the present work we examine the sintering behavior of intra-grain pores which lie within a large grain.

Computer Experimental Procedure

Two concepts that are universally applicable to all sintering phenomena and are acceptable from a thermodynamic point of view may be stated as follows: (i) a reduction in the net interfacial free energy serves as the driving force for sintering, and (ii) this reduction occurs via several material transport mechanisms [6].  In the following development, we focus on the interfaces that act either as a source or as a sink for material transport.  The individual material transport mechanisms are not considered for this case.

We assume that the material is mapped onto a lattice of suitable topology, where each lattice site may be thought of as an individual atom, a 'material element', or a pore element.  The interaction between particles and pores can be considered as follows: initially, each lattice site is assigned an integer value S which is either 0 or 1.  The value S=0 identifies the site as a pore element and S=1 represents a solid element.  If a material element is adjacent to at least one pore site, then that element is defined as a surface element.

The energy associated with a site is specified by defining as interaction energy, E, between a site and its neighbors within a given distance.  The interaction energy is defined by the equation

Ei =[eqn: SUM from { {j}=0} to n] J(Si,Sj)	(1)

where J is positive and sets the scale of interfacial energy, J(0,0) is the pore-pore interaction energy, J(1,1) is the solid-solid interaction energy and J(1,0)=J(0,1) is the solid-pore interaction energy.  The pore-pore and solid-solid interaction energy is considered equal to zero.  The solid-pore interaction energy is considered equal to one.  The sum is taken over all neighbors within a circular neighborhood, with a prespecified radius, located around the site.

During each step, each lattice site is considered sequentially.  The energy (Eold) of a lattice site is calculated using Eq.(1), then state of the site is changed.  If the site is a pore it is changed to a solid and vice versa.  The energy of the new configuration (Enew) is then calculated, and the change is accepted or rejected based on the energy difference

□□ = Enew - Eold.	(2)

A reduction in energy leads to acceptance of the change, otherwise the change is rejected.  This process is repeated for every lattice site.  After all the lattice sites are visited the lattice structure is updated and the process is repeated.  The basic time unit is one Monte Carlo step (MCS) and corresponds to a number of change attempts equal to the number of sites in the lattice.  A square lattice of 100×100 has been used in the simulations reported here.

Results and Discussion

Fig. 1 shows the microstructural changes of a collection of pores within a grain.  One clearly sees that the larger pores become larger and smaller ones become smaller as sintering proceeds.  In addition, the pores become more circular.  This observation is very similar to that observed for the case of inter-grain pores during densification, but, in this case (i.e., for intra-grain pores), the grain does not shrink and therefore the total volume of the pores is conserved.  We also emphasize here an important advantage of computer simulations.  In the computer simulations results such as the above can be obtained for specifically identified pores (e.g., for pores lying in a single section of the sample).  Therefore, the changes in structural features and sintering kinetics can be studied in a rigorous manner.  In contrast, in physical experiments, one usually uses a number of samples, each sintered at a selected set of different conditions.  One then takes sections of such samples and studies the structural features.  This procedure, which generally cannot be avoided in real experiments, can introduce some uncertainties in the observations.

One quantitative observation can be obtained from the changes in pore-size distribution or average pore size as functions of time, which illustrates that the pore-size distribution becomes broader and the average pore size becomes larger.  This is in very good agreement with the experimental observations [1-3], who note the same behavior in their experiments.  We note that, in practice, one should control the intra-grain pores to be as small as possible in order to maintain a uniform size of pores in the sintered specimen or to prevent the formation of large pores, which is detrimental to densification.

Additional quantitative observation can be obtained from change in pore volume and also the changes in the number of pores as functions of time, which are also plotted as number-of-pores versus pore-volume in Fig. 2.  The growth of pores in this case may be treated by the theory of diffusion-controlled growth of particles precipitated in a saturated solution [7].  According to this theory the rate of growth of pores can be expressed by the following relations

Vp,avg  ∝  davg3  ∝   t   ∝  Nv-1	(3)

where Vp,avg is the average pore volume, davg is the pore diameter, and Nv is the total number of pores.  The correlations between the number of pores and the average pore volume shown in the above figures have been obtained using three simulation, and they agree well with the above relations.

Acknowledgement

This paper was supported in part by SPECIAL FUND for UNIVERSITY RESEARCH INSTITUTE, Korea Research Foundation.

References

1. Watanabe, R. and Masuda, Y., Modern Devel. in Powder Metall., 6, 377          (1974).

2. Watanabe, R., Tda, K., and Maduda, Y., Z. Metallkde., 67, 619 (1976).

3. Masuda, Y. and Watanabe, R., Mater. Res. Sci., 13, 3 (1980).

4. Hsueh, C. H. and Evans, A. G., Acta Metall., 30, 1269 (1982).

5. Philippou, G., Kim, H., and Rajagopalan, R., Computat. Mater. Sci., 4, 181       (1995).

6. Wong, B. and Pask, J. A., J. Am. Ceram. Soc., 62, 138 (1979).

7. Lifshitz, I. M. and Slyozov, V. W., J. Phys. Chem. Solids, 19, 35 (1961).

Fig. 1  Microstructures showing the growth of pores in a grain.

Fig. 2 The relationship between the number of pores and the average	pore volume