Theoretical analysis of liquid-phase sintering: Pore filling theory

PII:

Acta mater. Vol. 46, No. 9, pp. 3191±3202, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(97)00489-8 1359-6454/98 $19.00 + 0.00

THEORETICAL ANALYSIS OF LIQUID-PHASE SINTERING: PORE FILLING THEORY SUNG-MIN LEE and SUK-JOONG L. KANG Department of Materials Science and Engineering and Center for Interface Science and Engineering of Materials, Korea Advanced Institute of Science and Technology, Yusong-gu, 373-1 Kusong-dong, Taejon 305-701, South Korea (Received 6 January 1997; accepted in revised form 4 November 1997; accepted 5 December 1997) AbstractÐBased on the pore ®lling model and previous calculations, a new liquid-phase sintering theory has been developed for compacts containing isolated pores with size distribution. The relevance of the model has been critically examined in consideration of previous experimental and theoretical results. The developed pore ®lling theory considers both densi®cation and grain growth during sintering, in contrast to the classical Kingery theory which does not take grain growth into account. The e€ects of such various parameters as pore size distribution, pore and liquid volume fraction, dihedral and wetting angle, particle size (scale), entrapped gas, etc., can be predicted. The present theory appears to describe well the microstructural development during liquid-phase sintering in re¯ecting real phenomena. # 1998 Acta Metallurgica Inc.

1. INTRODUCTION

When a powder compact is sintered in the presence of a liquid phase, fast densi®cation and grain growth take place in the early stage of sintering, even on heating to the sintering temperature [1]. In the very early stage of liquid-phase sintering, particle rearrangement may also occur when the amount of liquid is suciently high [2±7]. In this period, the grains are well-packed, their skeleton may form and the interconnected pores may become isolated. Sometimes, the grain skeleton forms before reaching liquid-phase sintering temperature [1]. Further densi®cation then occurs with the elimination of isolated pores, which is much slower and determines the overall densi®cation kinetics of the compact [1, 8]. The time needed for the elimination of isolated pores is, at least, one order of magnitude longer than that needed for particle rearrangement and pore isolation. For the elimination of isolated pores, two mechanisms were proposed: contact ¯attening [9±12] and pore ®lling [13±15]. The contact ¯attening mechanism proposed by Kingery [9] describes that a center-to-center approach of particles, i.e. a shrinkage of the compact, occurs by continuous material transport from the contact area between the grains to the o€-contact area through a thin liquid ®lm present in the area. Pore densi®cation then occurs continuously during the material transport. The validity of the liquid-phase sintering model based on this mechanism, however, has been questioned by several researchers because of some assumptions which appear to never apply to the real powder

compact [16±20]. First, it is thought that the driving force for the contact ¯attening, in other words, the driving force for the grain shape accommodation has been improperly described. Though Kingery considers the capillary pressure of the liquid in the compact to be the driving force, it may not be the force increasing the chemical potential of the atoms in the contact area. Even though the compact contains isolated pores, the grains are under hydrostatic pressure with the same liquid meniscus at the specimen surface as at the pore surface. Therefore, the continuous accommodation of the grain shape cannot be expected [20]. According to the contact ¯attening theory, the grain shape becomes more and more anhedral until the complete elimination of the pores, in contrast to the real microstructure development during liquid-phase sintering [1]. Second, the Kingery theory assumes no grain growth. Observation and theoretical calculation of the shape change of growing grains, however, reveal that, if the shape of the grains changes, the shape change does not occur by contact ¯attening but mainly by grain growth [16]. Subsequent model experiments and theoretical calculations demonstrate again that the shape change occurs by grain growth [7, 17±19]. Finally, the contact ¯attening theory assumes the presence of a liquid ®lm at the contact area between the grains, i.e. the dihedral angle of zero degree. If the dihedral angle is greater than zero degree, densi®cation is expected to occur very slowly by grain boundary di€usion or volume di€usion, as in solid-state sintering [10]Ðan incredible consequence.

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The instantaneous ®lling of liquid into the isolated pores was observed in real powder compacts [1, 21] as well as in some model systems [7, 13, 14, 22, 23]. Figure 1 shows an example of pore ®lling and the subsequent microstructural change around a liquid-®lled pore [7]. Figure 1(a) shows the successive shape changes in the grains around a large pore during the sintering of Mo±Ni. The specimen was cyclically sintered three times by cooling and reheating between isothermal holdings at 14608C. The etch boundaries formed within the growing grains, such as grain A and grain B, reveal the shape of the grains after

Fig. 1. Microstructures of 96Mo±4Ni specimen around pores: (a) before liquid ®lling, (b) just after liquid ®lling and (c) being homogenized. Specimen sintered at 14608C in a cycle of 60±30±30 min [7].

each sintering cycle. On examining the etch boundaries, it can be seen that the grains grow laterally along the pore shape, indicating that the pore does not continuously shrink by material deposition at the pore surface but remains intact for the long sintering time. Such an isolated pore was eliminated by the instantaneous ®lling with liquid, as shown in Fig. 1(b), forming a liquid pocket at its site. The shape of the grains around the liquid pocket demonstrates that the pocket was formed just before quenching the specimen. Upon prolonged sintering, the liquid pocket was also eliminated by microstructural homogenization. The etch boundaries in grain C in Fig. 1(c) reveal that microstructural homogenization occurred by a preferential material deposition of the grains at the concave surfaces (indicated by an arrow) and growth toward the liquid pocket center. Therefore, as long as the pore is not ®lled with liquid, the grains grow around the pore; once the pore ®lling occurs, the grains grow into the liquid pocket, resulting in microstructural homogenization. The pore ®lling is thought to be a result of grain growth [15, 24]. Figure 2 depicts schematically the microstructures at the specimen surface and pore surface during grain growth [23]. Because of the hydrostatic pressure in the liquid, the liquid pressures at the specimen surface and pore surface are the same with the same liquid menisci radius, if the gas pressure in the pore is the same as that outside the specimen. With grain growth, the liquid meniscus radius of the compact increases linearly [25] and can become equal to the pore radius, resulting in complete wetting of the pore surface (the critical moment for the pore ®lling), as schematically shown in Fig. 2(b). Then, an imbalance of liquid pressures at the specimen surface and pore surface arises with further increase in grain size, because

Fig. 2. Schematic showing the liquid ®lling of pore during grain growth: (a) before pore ®lling, (b) critical moment of ®lling and (c) liquid ¯ow right after critical moment. No e€ect of gases entrapped in pore is assumed. P is the pore and r is the radius of curvature of liquid meniscus (r1
LEE and KANG: PORE FILLING THEORY OF LIQUID-PHASE SINTERING

the liquid meniscus radius at the pore surface is limited by the pore size while that at the specimen surface is not [Fig. 2(c)]. Although the analysis was made for spherical pores, such expected pore ®lling behavior was also observed in real powder compacts with irregular pores as well as spherical ones [1]. One of the important consequences of the pore ®lling mechanism is that the pore ®lling must occur in temporal sequence depending on size: smaller pores earlier and larger ones later. This prediction was also con®rmed in real powder compacts [1]. The contributions of pore ®lling and contact ¯attening relative to densi®cation can be estimated as in a previous calculation [16]. The solid volume transported by contact ¯attening was inconsiderable, while the increase in average grain size was relatively fast. The time needed for densi®cation by pore ®lling was estimated to be, in general, a few orders of magnitude shorter than that by contact ¯attening. Furthermore, in this estimation, the contribution of contact ¯attening was overestimated, because the grain shape was assumed to change during the process from a sphere to one of anhedral equilibrium. But, from the early stage of liquidphase sintering, the grain shape is one observably

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close to anhedral equilibrium for a given liquid pressure in the compact: so the driving force for shape change must be inconsiderable. Based on these previous observations and analyses, a model of liquid-phase sintering (pore ®lling model) has already been proposed [20]. Figure 3 shows the proposed liquid-phase sintering model [20]. Figure 3(a) depicts schematically the microstructure of a compact containing pores of di€erent sizes. As long as the pores are stable, the volume fraction of liquid surrounding the grains does not change with grain growth. Once the surface of smaller pores is completely wetted as a result of grain growth to the critical size, the liquid spontaneously ®lls the pores, as illustrated in Fig. 3(b). With the pore ®lling, the compact density measured by the water-immersion method must increase. In terms of microstructure, however, the result of the pore ®lling is that the liquid menisci recede at specimen and intact pore surfaces, and thus there is a sudden decrease in liquid pressure. The pressure decrease can also be understood as a reduction of the e€ective liquid volume surrounding the grains in the bulk away from the liquid pockets formed. The situation is similar to the suction of liquid from a dense compact by pores, resulting in

Fig. 3. Illustrations of pore ®lling and shape accommodation during liquid-phase sintering: (a) just before liquid ®lling of small pores (Ps) (at a critical condition), (b) right after ®lling of small pores, (c) grain shape accommodation by grain growth and homogenization of microstructure around liquid pockets formed at pore sites during prolonged sintering and (d) just before liquid ®lling of large pore (Pl) [20].

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a substantially lower fraction of liquid for each grain, as in a previous model experiment [17]. Because of the liquid pressure decrease, i.e. the capillary pressure increase, by pore ®lling, the shape of the grains tends to become more anhedral during their growth in order to meet the sudden change in liquid pressure. Meanwhile, the homogenization of the microstructure around the liquid pockets formed will proceed, as shown in Fig. 3(c), resulting again in a homogeneous microstructure containing large stable pores. Specimen shrinkage is expected to occur during the microstructural homogenization. The microstructural homogenization must contribute to the increase in the e€ective liquid volume fraction in the bulk except in the liquid pockets. Even though the pore ®lling model was initially developed for powder compacts containing isolated pores, the pore ®lling mechanism is believed to be the major densi®cation mechanism from the beginning of liquid-phase sintering. When the particles are rearranged with liquid ¯ow and a skeleton of grains forms, the pores, either interconnected or isolated, are randomly distributed in a dense grain±liquid matrix, as typically shown in Fig. 4(a) [1]. At this stage, already, some pores are partially or completely ®lled with liquid as indicated by circles. The shape of the grains in Fig. 4(a) does not observably change during extended sintering, as shown in Fig. 4(b), implying that a near equilibrium shape for the initial volume fraction of liquid has already been attained at this stage, regardless of the shape and connectivity of the pores. The question may be just one of microstructure scale and grain size to pore size ratio. As explained thus far, the densi®cation kinetics of liquid-phase sintering appears to be controlled by the liquid ®lling of pores from its beginning, except for the particle rearrangement stage by liquid ¯ow. Particle rearrangement, however, may be limited to systems with a very large volume fraction of liquid and a low dihedral angle of almost zero degree [3±7]. For systems with a dihedral angle greater than zero degree, for example W±Ni±Fe, a skeleton of grains can form even during heating to the temperature of liquid-phase sintering. Therefore, the overall kinetics of liquid-phase sintering is believed to be governed by the pore ®lling which occurs on the complete wetting of the pore surface by grain growth. The grain growth thus appears to control the pore ®lling and densi®cation from the beginning of liquid-phase sintering. In this study, based on the present review and discussion of previous investigations, the liquidphase sintering model [20] and theoretical calculations [26] have been extended to the analysis of the sintering kinetics of powder compacts under various experimental conditions. The developed theory, namely the pore ®lling theory, can predict the e€ects of such various parameters as pore size dis-

Fig. 4. Microstructures of W (5 mm)±1Ni±1Fe specimens sintered at 14608C for (a) 10 min and (b) 1 h [1].

tribution, liquid and pore volume fraction, wetting and dihedral angle, scale (grain size), entrapped gas, etc. 2. THEORETICAL MODEL AND CALCULATION

During liquid-phase sintering, the measured pore intercept length over time exhibited size distribution and sequential pore elimination relative to size [1]. In the present analysis, as an example, the initial pore size distribution is assumed to show a lognormal distribution curve, as shown in Fig. 5, in accordance with a previously measured interceptlength distribution [1]. The distribution curve in Fig. 5 has an interval of 0.01 in initial pore size/ grain size ratio, starting at the ratio of 10 and ending at the ratio of 40. Any size distribution of pores may be taken for the analysis; however, the procedure for the calculation is the same. The e€ect of pore size distribution will be discussed by taking three types of distributions into account, lognormal,

LEE and KANG: PORE FILLING THEORY OF LIQUID-PHASE SINTERING

Fig. 5. Various pore size distributions used in calculation for a given compact porosity.

normal and Weibull distributions, as shown in Fig. 5. For a given pore size distribution, the evaluation of the liquid meniscus radius during pore ®lling and microstructure homogenization is critical for the quantitative calculation of densi®cation. The liquid meniscus radius is determined by the size, shape and packing geometry of the grains, the e€ective liquid volume fraction, the wetting and dihedral angle, and the ratio of liquid/vapor interfacial energy to solid/liquid interfacial energy [25]. For the calculation, closely packed mono-size grains (cubic-close-packing) were assumed. The grains immersed in liquid were also assumed to easily attain local equilibrium con®gurations during their growth for any e€ective volume fraction of liquid. Figure 6 shows the calculated variation of the mean liquid meniscus radius, r, with liquid volume fraction and wetting angle, y, for 08 (a) and 308 (b) dihedral angle [25]. For the calculation, the ratio of liquid/vapor interfacial energy glv to solid/liquid interfacial energy gsl was taken as 7. For low liquid volume fractions, less than about 5 vol.%, the e€ect of the wetting angle on the liquid meniscus radius is negligible; but for high liquid volume fractions, the e€ect is signi®cant. This result implies that for high liquid volume fractions, the wetting angle becomes a major parameter for densi®cation. From Fig. 6, it can also be seen that for a given liquid volume fraction, the liquid meniscus radius increases with increasing dihedral angle. In order to calculate the liquid meniscus radius during densi®cation, it is critical to estimate the e€ective volume fraction of liquid, f e€ l , because the liquid meniscus radius depends strongly on f e€ l . The e€ective volume fraction of liquid in the specimen is determined by the initial liquid volume fraction f il, the volume of pores ®lled with liquid (liquid pockets) and the homogenization of liquid pockets, as already explained in Fig. 3. Figure 7 shows a schematic for the calculation of f e€ l . After the pore ®lling, a microstructural homogenization at the liquid pocket site occurs with the growth of the grains towards the pocket center,

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leading to the squeezing-out of liquid from the pocket to the neighboring grain±liquid bulk. The increase in e€ective liquid volume by the squeezed liquid is then equivalent to the volume of the homogenized region at the liquid pocket site. The contribution of the microstructural homogenization to f e€ l should proceed until the grains surrounding the pocket grow up to the pocket center. Microstructure homogenization is also related to the specimen shrinkage, because the liquid meniscus and the grain shape should change as long as overall microstructure homogenization proceeds. When the pore ®lling occurs, the grain shape tends to accommodate itself to grain growth, which results in the specimen shrinkage. The microstructure homogenization around the liquid-®lled pores occurs concurrently with the shape accommodation and redistributes the liquid and solid. Since the volume shrinkage, which occurs after the pore ®lling, corresponds to the volume of the liquid-®lled pores, it is assumed in the calculation that the homogenized volume in all the liquid pockets is equivalent to the volume shrinkage. The calculated rate of specimen shrinkage may then be the minimum rate obtainable during liquid-phase sintering. As explained so far, after the grain rearrangement in the very early stage of sintering, grain growth is believed to determine the densi®cation and shrink-

Fig. 6. Variation of the mean liquid meniscus radius with liquid volume fraction and wetting angle y for (a) 08 and (b) 308 dihedral angle [25]. The mean liquid meniscus radius r is normalized to grain radius R.

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Fig. 7. Schematic for the calculation of the homogenization of liquid-®lled pores on e€ective liquid volume in the compact at a certain instant of liquid-phase sintering. (a) small pores already ®lled with liquid and completely homogenized, (b) medium pores already ®lled with liquid and partially homogenized in the outer shell of the former pore and (c) large pores not yet ®lled with liquid.

age of the compact. Grain growth follows a kinetic law as n

n 0

G ÿ G ˆ Kt,

…1†

where G is the grain size at the time of observation at t = t, G0 the grain size at t = 0 and K the kinetic constant. The value of n depends on the growth mechanism and the system [27, 28]. In general, the exponent n is known to be 3 for di€usion-controlled growth, while the exponent may be indeterminable for reaction-controlled growth [29±31]. In case of di€usion control, the proportional constant K in equation (1) must depend on the volume fraction of liquid, fl, in the compact, in contrast to no dependence of K on fl for interface-reaction control. The dependence of K on fl for di€usion-control has long been studied theoretically as well as experimentally [32±36]. In our investigation, we assumed a di€usion-controlled grain growth; however, the calculation procedure for densi®cation is the same and the physical meaning of the results must also be similar for the other growth mechanism. For the dependence of K on fl, we used   0:05 0:8 , …2† K ˆ K0 f eff l where K0 is a constant independent of liquid volume fraction. Equation (2) is obtained from previous experimental data on the Co±Cu system [36]. The ®gure 0.05 in equation (2) is introduced in order to take 5 vol.% as the standard. Then, as in Fig. 7, the homogenized volume in a liquid pocket j during microstructure homogenization, V jhomo , can be expressed as   …t drt
dt, …3† V jhomo ˆ ÿ 4pr2t
dt 0 where rt is the radius of the liquid pocket being homogenized at time t = t and

drt 1 dG : ˆÿ 2 dt dt

…4†

The e€ective liquid volume fraction, f e€ l , relative density, d, and shrinkage, (1 ÿ l/l0), are then calculated by V il ÿ f

eff l

ˆ

m X

…V jp ÿ V

jˆk‡1

V is ‡ V il ÿ

m X

…V jp ÿ V jhomo †

,

…5†

jˆk‡1

X d ˆ1ÿ

j homo †

jˆm‡1

V is ‡ V il ‡

V

j p

X

V

jˆm‡1

j p

,

…6†

and 2 1ÿ

m X

V

6 6 l jˆk‡1 ˆ1ÿ6 61 ÿ l0 l03 4

j homo

31=3 V jp 7 7 jˆ1 7 : ÿ 3 l0 7 5 k X

…7†

Here, V il is the initial volume of liquid, V is the initial volume of solid, Vpj the volume of pore j ®lled with liquid for j Rm or the volume of un®lled pore j for jr m + 1, l0 the initial dimension of the specimen, l the dimension of the specimen at time t and k the maximum size of the completely homogenized liquid-®lled pores (liquid pockets). For the parameters included in equations (1)±(7), the typical values measured in real systems [35, 36] were taken as: K0/G30=0.5 sÿ1 (for example, G0=1 mm, K0=5  10ÿ19 m3/s). 3. RESULTS AND DISCUSSION

As described in Section 2, our sintering theory takes various systems and sintering parameters into

LEE and KANG: PORE FILLING THEORY OF LIQUID-PHASE SINTERING

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account for the prediction of sintering kinetics. We will ®rst describe the general behavior of liquidphase sintering by pore ®lling and then discuss the e€ects of various parameters. 3.1. General behavior of sintering and e€ect of pore size distribution To understand the general behavior, we ®rst take a system with a dihedral angle of 08, a wetting angle of 08, a liquid volume of 5%, a pore volume of 10% and no entrapped gas. Other parameters, such as the packing geometry of grains and interfacial energies, are ®xed as noted in Section 2. On varying these parameters, the calculated sintering kinetics changes; however, the overall trend is unchanged. Figure 8 shows the calculated densi®cation, shrinkage, e€ective liquid volume fraction and the maximum size of the liquid-®lled pores with sintering time or the normalized grain size for two kinds of pore size distributions, lognormal and Weibull. As the sintering time increases, the relative density continuously increases; its overall shape is similar to that measured in real powder compacts [1, 2, 8]. The relative density in Fig. 8(a) indicates the density measured by the water-immersion technique because the densi®cation occurs through the instantaneous liquid-®lling of pores. On the other hand, shrinkage occurs by grain-shape accommodation and change after the pore ®lling; hence, it is slower than densi®cation. The shrinkage curves in Fig. 8 also imply that the microstructural homogenization, i.e. the attainment of an equilibrium microstructure, takes a long time even after the ®nal densi®cation by pore ®lling. During densi®cation and shrinkage, the e€ective liquid volume fraction, f e€ l , which determines the liquid meniscus radius at the pore and specimen surfaces, changes as shown in Fig. 8(b). In general, the fraction decreases sharply at the beginning of pore ®lling toward a minimum value, but increases again when the relative density becomes almost to the original liquid 100%. The increase in f e€ l fraction, however, is slow because of the slow microstructure homogenization with grain growth. with sintering Such a calculated variation of f e€ l time is also expected from the schematic microstructures in Fig. 3 and is in good agreement with the measured variation in a previous investigation [14]. re¯ects, of course, the comThe variation of f e€ l bined e€ects of liquid suction by pores during pore ®lling and the supply of liquid from the liquid-®lled pores to the grain±liquid bulk through microstructural homogenization. During most of the period of densi®cation, the reduction in liquid volume due to pore ®lling appears to be dominant. The maximum size of liquid-®lled pores is directly related to the e€ective liquid volume. As long as the reduction of f e€ l is inconsiderable at the initial stage of the liquid ®lling of small pores, the maximum

Fig. 8. Calculated curves of (a) relative density and shrinkage, (b) e€ective liquid volume fraction with sintering time and (c) maximum size of liquid-®lled pores and e€ective liquid volume fraction with normalized grain size for lognormal and Weibull distributions of pores.

size is almost linearly proportional to grain size, as shown in Fig. 8(c). When the reduction becomes considerable with successive pore ®llings, the maximum size is no more linearly proportional to grain size; extended grain growth is needed for further pore ®lling. These sintering kinetics and behavior are much a€ected by pore size distribution, as shown in Fig. 8, even though the total porosity is unchanged. Since the pore ®lling occurs sequentially as the grains grow (smaller earlier, larger later), the densi®cation must be much faster in a powder compact with ®ne pores (lognormal distribution) than in a compact with coarse pores (Weibull distribution).

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For 99.5% theoretical density, the densi®cation time for pores with the Weibull distribution is about 3.8 times that for pores with the lognormal distribution. The slower densi®cation for the Weibull distribution results in a slower reduction of e€ f e€ l and also in a slower recovery of f l to the original value [Fig. 8(b)]. Because of the slower reat the early stage of densi®cation, duction of f e€ l however, the medium size pores are ®lled earlier than they are for the lognormal distribution, as shown in Fig. 8(c). The linearity between the maximum size of the liquid-®lled pores and the grain size maintains longer. But, as the densi®cation proceeds with the liquid ®lling of large pores, the deviation from linearity becomes much more considerable because of the longer reduction time of f e€ l . A similar retardation e€ect with large pores has also been predicted even for compacts containing pores of an average size of 25 mm with normal size distribution in Fig. 5. The sintering time to get 99.5% theoretical density for a compact containing pores with a standard deviation s of 5 mm is calculated to be 1.33 times that for a compact with a standard deviation s of 1 mm. The calculated sintering time for the normal distributions, however, is longer than that for the lognormal distribution but shorter than that for the Weibull distribution. The relative sintering times for lognormal distribution, normal distribution with 1 mm deviation, normal distribution with 5 mm deviation and Weibull distribution are 1, 2.26, 3.02 and 3.80, respectively. The calculated results on the e€ect of pore size distribution clearly show that densi®cation is much retarded in a compact with a high fraction of large pores, although the porosity is unchanged. In this respect, it is essential to reduce the size of the particles with a low melting point, because the pores are usually formed at their sites. The results may also demonstrate the fundamentals of liquid-phase sintering: pore ®lling, e€ective liquid volume fraction, microstructural homogenization and their interrelation.

sintering time for the pore ®lling is proportional to f ÿ3 because the grain size increases with t1/3. l Otherwise, the exponent of the liquid volume fraction must be larger than ÿ3 (for example, ÿ2.8). For a powder compact containing pores of various sizes, however, the situation becomes somewhat complicated, because f e€ l decreases with pore ®lling. The reduction of f e€ means a reduction in the l liquid meniscus radius and, at the same time, an increase in the grain growth rate. The former delays pore ®lling, while the latter promotes it. These two opposite e€ects are introduced in a real powder compact. In our case, the exponent was found to be ÿ2.9 instead of ÿ3. When the dependence of the grain growth rate on f e€ l is much reduced, the exponent can become smaller than ÿ3. In any case, however, the exponent is around ÿ3, because the exponent of the grain growth equation [equation (1)] is assumed to be 3. The reduction in the densi®cation rate with the increase in porosity can be easily understood, because the reduction of f e€ l depends on the volume of pores ®lled with liquid. The e€ect of porosity, however, is less considerable than that of the liquid volume fraction. This result arises from the fact that the liquid volume fraction directly a€ects f e€ l over the whole specimen while only a fraction of the pores does, i.e. the unhomogenized liquid-®lled ones. 3.2.2. Dihedral and wetting angle. The e€ects of the dihedral and wetting angle can also be estimated when we calculate the equilibrium microstructure and the liquid meniscus radius, as in a previous investigation [25]. Figure 10 shows the densi®cation and shrinkage curves with dihedral (a) and wetting angle (b) for a powder compact containing pores of 10 vol.% and liquid of 5 vol.%. In

3.2. E€ects of various parameters 3.2.1. Pore and liquid volume fraction. Figure 9 delineates the calculated sintering time to 99.5% relative density of the compacts with various liquid and pore volume fractions ( f il of 2±8% and V ip of 5±15%). When we take a compact containing 5 vol.% liquid and 10 vol.% pores as a standard sample, in this particular case, the sintering time appears to be proportional to ( f il)ÿ2.9 and (V ip)2.2. As a ®rst approximation, the liquid meniscus radius r is linearly proportional to the liquid volume fraction fl, if fl is not very high, less than about 8 vol.%, as shown in Fig. 6(a). Then, the critical grain size for the liquid ®lling of a pore of the same size is inversely proportional to fl. If the dependence of grain growth on fl is neglected, the

Fig. 9. Calculated sintering time to 99.5% relative density with various liquid and pore volume fractions.

LEE and KANG: PORE FILLING THEORY OF LIQUID-PHASE SINTERING

Fig. 10. Calculated curves of relative density and shrinkage with sintering time for (a) 08 and 308 dihedral angle with 08 wetting angle and (b) 08, 208 and 408 wetting angle with 08 dihedral angle.

terms of the dihedral angle, densi®cation enhances as the angle increases. This result is due to the increase in the liquid meniscus radius with dihedral angle for a given grain size and liquid volume. For most liquid-phase sintered materials, however, the variation in dihedral angle is not considerable: between zero and a few tens of degrees. Then, as shown in Fig. 10(a), the enhancement is less than double. The e€ect of the wetting angle is more pronounced than that of the dihedral angle, as shown in Fig. 10(b). But the e€ect is opposite: densi®cation retards as wetting angle increases. When the wetting angle increases from 08 to 208 and 408, the densi®cation time increases to 2.3 and 7 times for the present case. The liquid meniscus radius increases also with increases in wetting angle, as in the case of the dihedral angle. For the pore ®lling, however, the pore surface must be completely wetted by liquid. Because of the considerable retardation in pore wetting with increases in wetting angle [24], this e€ect is a much more important factor in the densi®cation than is the radius increase. The retardation calculated in Fig. 10(b) may show the maximum e€ect. In a real powder compact, the size of grains surrounding a pore is not constant but has a distribution and the shape of the pore is, in general, irregular. Under this condition,

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the pore ®lling can occur earlier, before the estimated period, because the pore ®lling condition can be easily ful®lled when a partial liquid-®lling of the pore occurs [24]. Therefore, the e€ect of the wetting angle in a real powder compact can be less than the calculated one. The e€ect of the wetting angle on densi®cation, however, may also appear during the very initial stage of liquid-phase sintering, the particle rearrangement stage. The initial pore size and porosity may increase as the wetting angle increases [3±6] so that the overall densi®cation may be further retarded by the retardation resulting from the pore ®lling condition. 3.2.3. Scale. Since densi®cation occurs by pore ®lling which is in turn determined by grain growth, the e€ect of scale on densi®cation is expected to follow the scale e€ect on grain growth. This expectation was already con®rmed by calculating the densi®cation of compacts with similar microstructure but of di€erent scale [26]. The exponent in a scaling law for densi®cation is therefore equal to that for grain growth, 3 in our di€usion-controlled growth. The dependence of shrinkage on scale is also the same as that of densi®cation. This result derives from the dependence of microstructural homogenization on grain growth. The e€ect of average pore size on densi®cation can also be estimated, based on the scaling law. When the size of all pores in a powder compact is doubled, the densi®cation time increases a little bit more than eight times. The di€erence between this result and that predicted by the scaling law, eight times, comes from the fact that an additional time is needed for grains to become also doubled in their size for the scaling law application. 3.2.4. Entrapped gas. When slowly di€using gases (inert gas) are entrapped within pores, densi®cation is much retarded and full densi®cation is never reached unless very high external pressure is applied [22, 37±41]. Under an inert sintering atmosphere, as a simple case, the pore ®lling condition and the gas bubble size after densi®cation are determined by: 2glv 2g ˆ Pin ÿ lv rs rp

…8†

 3 2glv rp 2g ˆ Pin ÿ lv , rs rb rb

…9†

Pout ÿ

Pout ÿ

where Pin is the pressure of the entrapped inert gas just before the isolation of a pore, Pout is the external inert gas pressure, rp is the pore radius, rs is the radius of the liquid meniscus at specimen surface and rb is the radius of the gas bubble. The size of the gas bubble after pore ®lling is determined by and grain size. On the other hand, the homf e€ l ogenization of the liquid pocket is limited by the bubble size.

3200

LEE and KANG: PORE FILLING THEORY OF LIQUID-PHASE SINTERING

extended sintering in Fig. 11(a) re¯ects this e€ect. On the other hand, when pore coalescence occurs during extended sintering, the density is reduced because of pore expansion [38±40]. This e€ect was not taken into account for the calculation: the calculation shows the maximum attainable density during pressureless sintering. A similar analysis can also be made for the e€ect of external gas pressure on densi®cation. For a powder compact containing no gas within the pores and under 1 atm external gas pressure, i.e. positive external pressure of 1 atm, the densi®cation time was calculated to be about 2/3 that of normal sintering with fast di€using gas. In sintering practice, powder compacts may be sintered in a gas mixture, slow di€using and fast di€using. The external gas species and pressure may also be changed during sintering. Even for these complicated conditions, the e€ect of the entrapped inert gas can also be estimated with modi®cation of equations (8) and (9), as in previous investigations [22, 37, 41]. The sintering kinetics should be modi®ed; however, the overall shapes of curves must be similar to densi®cation and f e€ l those shown in Fig. 11. 4. CONCLUSIONS Fig. 11. Calculated curves of (a) relative density and shrinkage with sintering time and (b) maximum size of liquid-®lled pores and e€ective liquid volume fraction with normalized grain size for fast and slow di€using (inert) gas atmosphere of 1 atm.

Figure 11 plots the calculated densi®cation curves, e€ective liquid volume fractions and maximum sizes of the liquid-®lled pores during a powder compact sintering in a slowly di€using inert gas of 1 atm (solid lines) or in a fast di€using gas of 1 atm (dashed lines, as comparison). For the calculation, all other parameters were taken as the same as in the above calculations and glv as 1.7 J/m2. Note that the initial densi®cation of a compact containing inert gas is faster than that of a compact without entrapped inert gas. This result can be understood when we think about the changes in the e€ective liquid volume fraction and in the maximum size of the liquid-®lled pores during densi®cation, as shown in Fig. 11(b). At the beginning, the reduction in the e€ective liquid volume is less with inert gas because of the gas bubbles formed in the liquid-®lled pores. Therefore, the large pores can be ®lled earlier, resulting in faster densi®cation. When almost all the pores are ®lled with liquid, however, additional densi®cation occurs only by a reduction in bubble size. The densi®cation rate is then much more reduced and essentially a limiting density is reached. At this stage, the reduction in bubble size derives from the increase in liquid pressure resulting from the increase in the liquid meniscus radius due to grain growth. The very slow densi®cation during

A new theory of liquid-phase sintering, namely the pore ®lling theory, has been developed for powder compacts containing pores of various sizes. The theory is based on the microstructural development observed in previous investigations [1, 7, 13, 14, 21± 23] and on our previously developed liquid-phase sintering model [20]. By calculating the changes in and the the e€ective liquid volume fraction f e€ l liquid meniscus radius during densi®cation, it was possible to critically evaluate the e€ects of such various practical sintering parameters as pore size distribution, porosity, liquid volume fraction, dihedral and wetting angle, particle size and sintering atmosphere. It has been found that pore size distribution has a considerable e€ect on sintering kinetics of powder compacts with same porosity. The densi®cation of a powder compact with a Weibull distribution is retarded a few times more than that with a lognormal distribution. For compacts with same average pore size, the densi®cation of pores with a broad size distribution is slower than that with a narrow size distribution. The liquid volume fraction f il, which directly a€ects the e€ective liquid volume and the liquid meniscus radius, has a fraction f e€ l strong e€ect on densi®cation. The sintering time appears to be proportional to about ( f il)ÿ3 in the case of di€usion-controlled grain growth. This dependence results from two things: the linear proportionality of the liquid meniscus radius to the liquid volume fraction, and the grain growth kinetics with annealing time. The e€ect of porosity is

LEE and KANG: PORE FILLING THEORY OF LIQUID-PHASE SINTERING

also found to be considerable but less than the e€ect of liquid volume fraction, because only unhomogenized liquid-®lled pores a€ect f e€ l while liquid over the whole volume fraction directly a€ects f e€ l specimen. Dihedral angle increase enhances densi®cation but not by much. In contrast, wetting angle increase considerably retards densi®cation, several times over with angle increases of from zero degrees to a few tens of degrees. The e€ect of scale is determined by the kinetics of grain growth, because densi®cation occurs by pore ®lling with grain growth. The exponent in the scaling law for densi®cation is therefore the same as that for grain growth. The calculated e€ect of the entrapped gas is somewhat di€erent from that conventionally expected: retardation of densi®cation right from the beginning of the elimination of isolated pores. At the beginning of the elimination of isolated pores, the densi®cation rate of a compact with inert gas is faster than that without it. This result is related to less reduction in e€ective liquid volume fraction during pore ®lling because of the presence of gas bubbles in the liquid-®lled pores. During extended sintering, however, the densi®cation essentially stops because of the entrapped inert gas. In the development of this theory, in contrast to the classical Kingery theory which neglects grain growth, both densi®cation and grain growth, two essential processes occurring during sintering, were considered. The classical theory is, in fact, a twoparticle model, similar to that of solid-state sintering. For solid-state sintering, the description of microstructural development was possible through the consideration of grain growth as well as densi®cation and their interaction [42, 43]. In this respect, the present theory has a similar implication in terms of the development of sintering theory and provides a better understanding of the real phenomena. Densi®cation in liquid-phase sintering, however, is basically di€erent from that in solid-state sintering. In solid-state sintering, the densi®cation occurs by material transport from the grain boundary, as in two-particle type sintering. In liquid-phase sintering, the densi®cation appears to occur by pore ®lling with grain growth; the densi®cation by contact ¯attening of the two-particle type is negligible. The sintering kinetics is therefore determined by grain coarsening, in contrast to the case of solid-state sintering. In our pore ®lling theory, the grains are assumed to be spherical. We can well de®ne and describe the sintering processes: pore ®lling, grain shape accommodation and microstructure homogenization. If the grains are faceted, it may be dicult to quantitatively describe the sintering kinetics. Nevertheless, the basic concepts of this pore ®lling theory are believed to apply, because the liquid meniscus must also increase with grain size and the complete wetting of the pore surface must result in pore ®lling.

3201

The overall behavior of sintering and the e€ect of various sintering parameters should be similar to those evaluated for compacts with spherical grains. The diculties observed in densifying compacts with faceted grains can then be understood as a result of the low rate of grain growth. AcknowledgementsÐThis work was supported by the Korea Advanced Institute of Science and Technology and also by the Korea Research Foundation. REFERENCES 1. Park, J.-K., Kang, S.-J. L., Eun, K. Y. and Yoon, D. N., Metall. Trans., 1989, 20A, 837. 2. Cannon, H. S. and Lenel, F. V., in Proc. Plansee Seminar 1952, ed. F. Benesovsky. Metallwerk Plansee, 1953, p. 106. 3. Whalen, T. J. and Humenik, M., Jr, in Sintering and Related Phenomena, ed. G. C. Kuczynski, N. A. Hooton and C. F. Gibbon. Gorden and Breach, New York, 1967, p. 715. 4. White, J., in Sintering and Related Phenomena, ed. G. C. Kuczynski. Plenum Press, New York, 1973, p. 81. 5. Huppmann, W. J., in Sintering and Catalysis, ed. G. C. Kuczynski. Plenum Press, New York, 1975, p. 359. 6. Huppmann, W. J. and Riegger, H., Acta metall., 1975, 23, 965. 7. Kang, S.-J. L., Kaysser, W. A., Petzow, G. and Yoon, D. N., Powder Metall., 1984, 27, 97. 8. Kingery, W. D. and Narasimhan, M. D., J. Appl. Phys., 1959, 30, 307. 9. Kingery, W. D., J. Appl. Phys., 1959, 30, 301. 10. Gessinger, G. H., Fischmeister, H. F. and Lukas, H. L., Acta metall., 1973, 21, 715. 11. Svoboda, J., Riedel, H. and Gaebel, R., Acta mater., 1996, 44, 3215. 12. Mortensen, A., Acta Mater., 1997, 45, 749. 13. Kwon, O.-J. and Yoon, D. N., in Sintering Process, ed. G. C. Kuczynski. Plenum Press, New York, 1980, p. 203. 14. Kwon, O.-J. and Yoon, D. N., Int. J. Powder Metall. Powder Tech., 1981, 17, 127. 15. Park, H.-H., Cho, S.-J. and Yoon, D. N., Metall. Trans., 1984, 15A, 1075. 16. Kang, S.-J. L., Kaysser, W. A., Petzow, G. and Yoon, D. N., Acta Metall., 1985, 33, 1919. 17. Lee, D.-D., Kang, S.-J. L. and Yoon, D. N., Scripta metall., 1990, 24, 927. 18. Kaysser, W. A. and Petzow, G., Z. Metallkd., 1985, 76, 687. 19. Kaysser, W. A., Zivkovic, M. and Petzow, G., J. Mater. Sci., 1985, 20, 578. 20. Kang, S.-J. L., Kim, K.-H. and Yoon, D.N., J. Am. Ceram. Soc., 1991, 74, 425. 21. Kang, S.-J. L. and Azou, P., Powder Metall., 1985, 28, 90. 22. Cho, S.-J., Kang, S.-J. L. and Yoon, D. N., Metall. Trans., 1986, 17A, 2175. 23. Kang, S.-J. L., Greil, P., Mitomo, M. and Moon, J.H., J. Am. Ceram. Soc., 1989, 72, 1166. 24. Park, H.-H., Kwon, O.-J. and Yoon, D. N., Metall. Trans., 1986, 17A, 1915. 25. Park, H.-H., Kang, S.-J. L. and Yoon, D. N., Metall. Trans., 1986, 17A, 325. 26. Kang, S.-J. L., Kim, K.-H. and Lee, S.-M., in Sintering Technology, ed. R. M. German, G. L. Messing and R. G. Cornwall. Marcel Dekker, New York, 1996, p. 221. 27. Lifshitz, I. M. and Slyozov, V. V., J. Phys. Chem. Solids, 1961, 19, 35.

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37. Kang, S.-J. L. and Yoon, D. N., in Horizons of Powder Metallurgy, Part II, ed. W. A. Kaysser and W. J. Huppmann. Verlag Schmid, 1986, p. 1214. 38. German, R. M. and Churn, K. S., Metall. Trans., 1984, 15A, 747. 39. Kim, J.-J., Kim, B.-K., Song, B.-M., Kim, D.-Y. and Yoon, D. N., J. Am. Ceram. Soc., 1987, 70, 734. 40. Kang, S.-J. L. and Yoon, K. J., J. Eur. Ceram. Soc., 1989, 5, 135. 41. Yoon, K. J. and Kang, S.-J. L., J. Eur. Ceram. Soc., 1990, 6, 201. 42. Brook, R. J., J. Am. Ceram. Soc., 1969, 52, 56. 43. Spears, M. A. and Evans, A. G., Acta metall., 1982, 30, 1281.