Theoretical analysis of liquid-phase sintering: Pore filling theory

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 ofMaterials, 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 theoryhas been developed for compacts containing isolated pores with size distribution. The relevance of themodel has been critically examined in consideration of previous experimental and theoretical results. Thedeveloped pore ®lling theory considers both densi®cation and grain growth during sintering, in contrast tothe classical Kingery theory which does not take grain growth into account. The e�ects of such variousparameters as pore size distribution, pore and liquid volume fraction, dihedral and wetting angle, particlesize (scale), entrapped gas, etc., can be predicted. The present theory appears to describe well the micro-structural development during liquid-phase sintering in re¯ecting real phenomena. # 1998 Acta Metallur-gica 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, par-

ticle 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®ca-

tion 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 par-

ticle rearrangement and pore isolation.

For the elimination of isolated pores, two mech-

anisms 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 com-

pact, occurs by continuous material transport from

the contact area between the grains to the o�-con-

tact 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 mechan-

ism, 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 con-

tains isolated pores, the grains are under hydro-

static 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 ex-

periments and theoretical calculations demonstrate

again that the shape change occurs by grain

growth [7, 17±19]. Finally, the contact ¯attening the-

ory assumes the presence of a liquid ®lm at the con-

tact 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�u-

sion, as in solid-state sintering [10]Ðan incredible

consequence.

Acta mater. Vol. 46, No. 9, pp. 3191±3202, 1998# 1998 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/98 $19.00+0.00PII: S1359-6454(97)00489-8

3191

The instantaneous ®lling of liquid into the iso-

lated 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 micro-

structural 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 iso-

thermal 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

each sintering cycle. On examining the etch bound-

aries, it can be seen that the grains grow laterallyalong the pore shape, indicating that the pore doesnot continuously shrink by material deposition at

the pore surface but remains intact for the long sin-tering time. Such an isolated pore was eliminated

by the instantaneous ®lling with liquid, as shown inFig. 1(b), forming a liquid pocket at its site. The

shape of the grains around the liquid pocketdemonstrates that the pocket was formed justbefore quenching the specimen. Upon prolonged

sintering, the liquid pocket was also eliminated bymicrostructural homogenization. The etch bound-

aries in grain C in Fig. 1(c) reveal that microstruc-tural homogenization occurred by a preferentialmaterial deposition of the grains at the concave sur-

faces (indicated by an arrow) and growth towardthe liquid pocket center. Therefore, as long as the

pore is not ®lled with liquid, the grains growaround the pore; once the pore ®lling occurs, the

grains grow into the liquid pocket, resulting inmicrostructural homogenization.

The pore ®lling is thought to be a result of graingrowth [15, 24]. Figure 2 depicts schematically themicrostructures at the specimen surface and pore

surface during grain growth [23]. Because of thehydrostatic pressure in the liquid, the liquid press-

ures at the specimen surface and pore surface arethe same with the same liquid menisci radius, if thegas pressure in the pore is the same as that outside

the specimen. With grain growth, the liquid menis-cus radius of the compact increases linearly [25] and

can become equal to the pore radius, resulting incomplete wetting of the pore surface (the critical

moment for the pore ®lling), as schematicallyshown in Fig. 2(b). Then, an imbalance of liquidpressures at the specimen surface and pore surface

arises with further increase in grain size, because

Fig. 1. Microstructures of 96Mo±4Ni specimen aroundpores: (a) before liquid ®lling, (b) just after liquid ®llingand (c) being homogenized. Specimen sintered at 14608C

in a cycle of 60±30±30 min [7].

Fig. 2. Schematic showing the liquid ®lling of pore duringgrain growth: (a) before pore ®lling, (b) critical moment of®lling and (c) liquid ¯ow right after critical moment. Noe�ect of gases entrapped in pore is assumed. P is the poreand r is the radius of curvature of liquid meniscus

(r1<r2, r2rr2') [23].

LEE and KANG: PORE FILLING THEORY OF LIQUID-PHASE SINTERING3192

the liquid meniscus radius at the pore surface is lim-

ited by the pore size while that at the specimen sur-face is not [Fig. 2(c)]. Although the analysis wasmade for spherical pores, such expected pore ®lling

behavior was also observed in real powder com-pacts with irregular pores as well as sphericalones [1]. One of the important consequences of the

pore ®lling mechanism is that the pore ®lling mustoccur in temporal sequence depending on size:smaller pores earlier and larger ones later. This pre-

diction was also con®rmed in real powdercompacts [1].The contributions of pore ®lling and contact ¯at-

tening relative to densi®cation can be estimated asin a previous calculation [16]. The solid volumetransported by contact ¯attening was inconsider-

able, while the increase in average grain size wasrelatively fast. The time needed for densi®cation bypore ®lling was estimated to be, in general, a few

orders of magnitude shorter than that by contact¯attening. Furthermore, in this estimation, the con-tribution of contact ¯attening was overestimated,

because the grain shape was assumed to changeduring the process from a sphere to one of anhedralequilibrium. But, from the early stage of liquid-

phase sintering, the grain shape is one observably

close to anhedral equilibrium for a given liquid

pressure in the compact: so the driving force forshape change must be inconsiderable.Based on these previous observations and ana-

lyses, a model of liquid-phase sintering (pore ®llingmodel) has already been proposed [20]. Figure 3shows the proposed liquid-phase sintering

model [20]. Figure 3(a) depicts schematically themicrostructure of a compact containing pores ofdi�erent sizes. As long as the pores are stable, the

volume fraction of liquid surrounding the grainsdoes not change with grain growth. Once the sur-face of smaller pores is completely wetted as a

result of grain growth to the critical size, the liquidspontaneously ®lls the pores, as illustrated inFig. 3(b). With the pore ®lling, the compact density

measured by the water-immersion method mustincrease. In terms of microstructure, however, theresult of the pore ®lling is that the liquid menisci

recede at specimen and intact pore surfaces, andthus there is a sudden decrease in liquid pressure.The pressure decrease can also be understood as a

reduction of the e�ective liquid volume surroundingthe grains in the bulk away from the liquid pocketsformed. 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) justbefore 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 liquidpockets formed at pore sites during prolonged sintering and (d) just before liquid ®lling of large pore

(Pl) [20].

LEE and KANG: PORE FILLING THEORY OF LIQUID-PHASE SINTERING 3193

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 duringtheir growth in order to meet the sudden change inliquid pressure. Meanwhile, the homogenization of

the microstructure around the liquid pocketsformed will proceed, as shown in Fig. 3(c), resulting

again in a homogeneous microstructure containinglarge stable pores. Specimen shrinkage is expectedto occur during the microstructural homogeniz-

ation. The microstructural homogenization mustcontribute to the increase in the e�ective liquidvolume fraction in the bulk except in the liquid

pockets.Even though the pore ®lling model was initially

developed for powder compacts containing isolatedpores, the pore ®lling mechanism is believed to bethe major densi®cation mechanism from the begin-

ning of liquid-phase sintering. When the particlesare rearranged with liquid ¯ow and a skeleton ofgrains forms, the pores, either interconnected or iso-

lated, are randomly distributed in a dense grain±li-quid matrix, as typically shown in Fig. 4(a) [1]. At

this stage, already, some pores are partially or com-pletely ®lled with liquid as indicated by circles. Theshape of the grains in Fig. 4(a) does not observably

change during extended sintering, as shown inFig. 4(b), implying that a near equilibrium shapefor the initial volume fraction of liquid has already

been attained at this stage, regardless of the shapeand connectivity of the pores. The question may be

just one of microstructure scale and grain size topore size ratio.As explained thus far, the densi®cation kinetics

of liquid-phase sintering appears to be controlledby the liquid ®lling of pores from its beginning,except for the particle rearrangement stage by liquid

¯ow. Particle rearrangement, however, may be lim-ited to systems with a very large volume fraction of

liquid and a low dihedral angle of almost zerodegree [3±7]. For systems with a dihedral anglegreater than zero degree, for example W±Ni±Fe, a

skeleton of grains can form even during heating tothe temperature of liquid-phase sintering.Therefore, the overall kinetics of liquid-phase sinter-

ing is believed to be governed by the pore ®llingwhich occurs on the complete wetting of the pore

surface by grain growth. The grain growth thusappears to control the pore ®lling and densi®cationfrom the beginning of liquid-phase sintering.

In this study, based on the present review anddiscussion of previous investigations, the liquid-phase sintering model [20] and theoretical

calculations [26] have been extended to the analysisof the sintering kinetics of powder compacts under

various experimental conditions. The developed the-ory, namely the pore ®lling theory, can predict thee�ects of such various parameters as pore size dis-

tribution, liquid and pore volume fraction, wettingand dihedral angle, scale (grain size), entrapped gas,

etc.

2. THEORETICAL MODEL AND CALCULATION

During liquid-phase sintering, the measured poreintercept length over time exhibited size distribution

and sequential pore elimination relative to size [1].In the present analysis, as an example, the initialpore size distribution is assumed to show a lognor-

mal distribution curve, as shown in Fig. 5, in ac-cordance with a previously measured intercept-length 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 end-ing at the ratio of 40. Any size distribution of pores

may be taken for the analysis; however, the pro-cedure for the calculation is the same. The e�ect ofpore size distribution will be discussed by takingthree types of distributions into account, lognormal,

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


normal and Weibull distributions, as shown inFig. 5.

For a given pore size distribution, the evaluationof the liquid meniscus radius during pore ®lling and

microstructure homogenization is critical for thequantitative calculation of densi®cation. The liquid

meniscus radius is determined by the size, shapeand packing geometry of the grains, the e�ectiveliquid volume fraction, the wetting and dihedral

angle, and the ratio of liquid/vapor interfacialenergy to solid/liquid interfacial energy [25]. For

the calculation, closely packed mono-size grains(cubic-close-packing) were assumed. The grainsimmersed in liquid were also assumed to easily

attain local equilibrium con®gurations during theirgrowth for any e�ective volume fraction of liquid.

Figure 6 shows the calculated variation of themean 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 ofliquid/vapor interfacial energy glv to solid/liquid

interfacial energy gsl was taken as 7. For low liquidvolume fractions, less than about 5 vol.%, the e�ect

of the wetting angle on the liquid meniscus radius isnegligible; but for high liquid volume fractions, the

e�ect is signi®cant. This result implies that for highliquid volume fractions, the wetting angle becomesa major parameter for densi®cation. From Fig. 6, it

can also be seen that for a given liquid volume frac-tion, 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 thee�ective volume fraction of liquid, f l

e�, because theliquid meniscus radius depends strongly on f l

e�. The

e�ective volume fraction of liquid in the specimen isdetermined by the initial liquid volume fraction f l

i,

the volume of pores ®lled with liquid (liquid pock-ets) and the homogenization of liquid pockets, asalready explained in Fig. 3.

Figure 7 shows a schematic for the calculation off le�. After the pore ®lling, a microstructural hom-

ogenization at the liquid pocket site occurs with thegrowth of the grains towards the pocket center,

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 squeezedliquid is then equivalent to the volume of the hom-

ogenized region at the liquid pocket site. The con-

tribution of the microstructural homogenization tof le� 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 meniscusand the grain shape should change as long as over-

all microstructure homogenization proceeds. When

the pore ®lling occurs, the grain shape tends to ac-

commodate itself to grain growth, which results inthe specimen shrinkage. The microstructure hom-

ogenization around the liquid-®lled pores occurs

concurrently with the shape accommodation andredistributes the liquid and solid. Since the volume

shrinkage, which occurs after the pore ®lling, corre-

sponds to the volume of the liquid-®lled pores, it isassumed in the calculation that the homogenized

volume in all the liquid pockets is equivalent to the

volume shrinkage. The calculated rate of specimenshrinkage may then be the minimum rate obtain-

able 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. 5. Various pore size distributions used in calculationfor a given compact porosity.

Fig. 6. Variation of the mean liquid meniscus radius withliquid 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.


age of the compact. Grain growth follows a kinetic

law as

G n ÿ G n0 � 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 kineticconstant. The value of n depends on the growthmechanism and the system [27, 28]. In general, the

exponent n is known to be 3 for di�usion-controlledgrowth, while the exponent may be indeterminable

for reaction-controlled growth [29±31]. In case ofdi�usion control, the proportional constant K inequation (1) must depend on the volume fraction of

liquid, fl, in the compact, in contrast to no depen-dence of K on fl for interface-reaction control. The

dependence of K on fl for di�usion-control has longbeen studied theoretically as well asexperimentally [32±36].

In our investigation, we assumed a di�usion-con-trolled grain growth; however, the calculation pro-

cedure for densi®cation is the same and the physicalmeaning of the results must also be similar for theother growth mechanism. For the dependence of K

on fl, we used

K � K0

�0:05

f effl

�0:8

, �2�

where K0 is a constant independent of liquidvolume fraction. Equation (2) is obtained from pre-vious experimental data on the Co±Cu system [36].

The ®gure 0.05 in equation (2) is introduced inorder to take 5 vol.% as the standard.

Then, as in Fig. 7, the homogenized volume in aliquid pocket j during microstructure homogeniz-ation, V j

homo, can be expressed as

V jhomo � ÿ

�t0

4pr2t ��drtdt

�� dt, �3�

where rt is the radius of the liquid pocket beinghomogenized at time t = t and

drtdt� ÿ 1

2

dG

dt: �4�

The e�ective liquid volume fraction, f le�, relative

density, d, and shrinkage, (1ÿ l/l0), are then calcu-lated by

f effl �

V il ÿ

Xmj�k�1�V j

p ÿ V jhomo�

V is � V i

l ÿXmj�k�1�V j

p ÿ V jhomo�

, �5�

d � 1ÿ

Xj�m�1

V jp

V is � V i

l �X

j�m�1V j

p

, �6�

and

1ÿ l

l0� 1ÿ 1ÿ

Xmj�k�1

V jhomo

l30ÿ

Xkj�1

V jp

l30

266664377775

1=3

: �7�

Here, V li is the initial volume of liquid, V s

i the in-itial volume of solid, Vp

j the volume of pore j ®lledwith liquid for jRm or the volume of un®lled pore

j for jrm + 1, l0 the initial dimension of the speci-men, l the dimension of the specimen at time t andk the maximum size of the completely homogenized

liquid-®lled pores (liquid pockets). For the par-ameters included in equations (1)±(7), the typicalvalues measured in real systems [35, 36] were taken

as: K0/G03=0.5 sÿ1 (for example, G0=1 mm,

K0=5�10ÿ19 m3/s).

3. RESULTS AND DISCUSSION

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

Fig. 7. Schematic for the calculation of the homogenization of liquid-®lled pores on e�ective liquidvolume in the compact at a certain instant of liquid-phase sintering. (a) small pores already ®lled withliquid and completely homogenized, (b) medium pores already ®lled with liquid and partially homogen-

ized in the outer shell of the former pore and (c) large pores not yet ®lled with liquid.


account for the prediction of sintering kinetics. Wewill ®rst describe the general behavior of liquid-

phase sintering by pore ®lling and then discuss thee�ects of various parameters.

3.1. General behavior of sintering and e�ect of poresize distribution

To understand the general behavior, we ®rst takea system with a dihedral angle of 08, a wettingangle 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 inter-facial energies, are ®xed as noted in Section 2. Onvarying these parameters, the calculated sintering

kinetics changes; however, the overall trend isunchanged.Figure 8 shows the calculated densi®cation,

shrinkage, e�ective liquid volume fraction and themaximum size of the liquid-®lled pores with sinter-ing time or the normalized grain size for two kinds

of pore size distributions, lognormal and Weibull.As the sintering time increases, the relative densitycontinuously increases; its overall shape is similar tothat measured in real powder compacts [1, 2, 8]. The

relative density in Fig. 8(a) indicates the densitymeasured by the water-immersion techniquebecause the densi®cation occurs through the instan-

taneous liquid-®lling of pores. On the other hand,shrinkage occurs by grain-shape accommodationand change after the pore ®lling; hence, it is slower

than densi®cation. The shrinkage curves in Fig. 8also imply that the microstructural homogenization,i.e. the attainment of an equilibrium microstructure,

takes a long time even after the ®nal densi®cationby pore ®lling.During densi®cation and shrinkage, the e�ective

liquid volume fraction, f le�, which determines the

liquid meniscus radius at the pore and specimensurfaces, changes as shown in Fig. 8(b). In general,the fraction decreases sharply at the beginning of

pore ®lling toward a minimum value, but increasesagain when the relative density becomes almost100%. The increase in f l

e� to the original liquid

fraction, however, is slow because of the slowmicrostructure homogenization with grain growth.Such a calculated variation of f l

e� with sinteringtime is also expected from the schematic microstruc-

tures in Fig. 3 and is in good agreement with themeasured variation in a previous investigation [14].The variation of f l

e� re¯ects, of course, the com-

bined e�ects of liquid suction by pores during pore®lling and the supply of liquid from the liquid-®lledpores to the grain±liquid bulk through microstruc-

tural homogenization. During most of the period ofdensi®cation, the reduction in liquid volume due topore ®lling appears to be dominant.

The maximum size of liquid-®lled pores is directlyrelated to the e�ective liquid volume. As long as thereduction of f l

e� is inconsiderable at the initial stageof the liquid ®lling of small pores, the maximum

size is almost linearly proportional to grain size, as

shown in Fig. 8(c). When the reduction becomesconsiderable with successive pore ®llings, the maxi-

mum size is no more linearly proportional to grain

size; extended grain growth is needed for furtherpore ®lling.

These sintering kinetics and behavior are much

a�ected by pore size distribution, as shown inFig. 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 acompact with coarse pores (Weibull distribution).

Fig. 8. Calculated curves of (a) relative density and shrink-age, (b) e�ective liquid volume fraction with sintering timeand (c) maximum size of liquid-®lled pores and e�ectiveliquid volume fraction with normalized grain size for log-

normal and Weibull distributions of pores.


For 99.5% theoretical density, the densi®cationtime for pores with the Weibull distribution is

about 3.8 times that for pores with the lognormaldistribution. The slower densi®cation for theWeibull distribution results in a slower reduction of

f le� and also in a slower recovery of f l

e� to the orig-inal value [Fig. 8(b)]. Because of the slower re-duction of f l

e� at the early stage of densi®cation,

however, the medium size pores are ®lled earlierthan they are for the lognormal distribution, asshown in Fig. 8(c). The linearity between the maxi-

mum size of the liquid-®lled pores and the grainsize maintains longer. But, as the densi®cation pro-ceeds with the liquid ®lling of large pores, the devi-ation from linearity becomes much more

considerable because of the longer reduction timeof f l

e�.A similar retardation e�ect with large pores has

also been predicted even for compacts containingpores of an average size of 25 mm with normal sizedistribution in Fig. 5. The sintering time to get

99.5% theoretical density for a compact containingpores with a standard deviation s of 5 mm is calcu-lated to be 1.33 times that for a compact with a

standard deviation s of 1 mm. The calculated sinter-ing time for the normal distributions, however, islonger than that for the lognormal distribution butshorter than that for the Weibull distribution. The

relative sintering times for lognormal distribution,normal distribution with 1 mm deviation, normaldistribution with 5 mm deviation and Weibull distri-

bution 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 largepores, although the porosity is unchanged. In thisrespect, it is essential to reduce the size of the par-ticles with a low melting point, because the pores

are usually formed at their sites. The results mayalso demonstrate the fundamentals of liquid-phasesintering: pore ®lling, e�ective liquid volume frac-

tion, microstructural homogenization and theirinterrelation.

3.2. E�ects of various parameters

3.2.1. Pore and liquid volume fraction. Figure 9delineates the calculated sintering time to 99.5%

relative density of the compacts with various liquidand pore volume fractions ( f l

i of 2±8% and V pi of

5±15%). When we take a compact containing

5 vol.% liquid and 10 vol.% pores as a standardsample, in this particular case, the sintering timeappears to be proportional to ( f l

i)ÿ2.9 and (V pi )2.2.

As a ®rst approximation, the liquid meniscusradius r is linearly proportional to the liquidvolume fraction fl, if fl is not very high, less than

about 8 vol.%, as shown in Fig. 6(a). Then, thecritical grain size for the liquid ®lling of a pore ofthe same size is inversely proportional to fl. If thedependence of grain growth on fl is neglected, the

sintering time for the pore ®lling is proportional to

f lÿ3 because the grain size increases with t1/3.

Otherwise, the exponent of the liquid volume frac-tion must be larger than ÿ3 (for example, ÿ2.8).For a powder compact containing pores of various

sizes, however, the situation becomes somewhatcomplicated, because f l

e� decreases with pore ®lling.

The reduction of f le� means a reduction in the

liquid meniscus radius and, at the same time, an

increase in the grain growth rate. The former delayspore ®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 beÿ2.9 instead of ÿ3. When the dependence of the

grain growth rate on f le� is much reduced, the expo-

nent can become smaller than ÿ3. In any case, how-

ever, the exponent is around ÿ3, because theexponent 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 le� 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 factthat the liquid volume fraction directly a�ects f l

e�

over the whole specimen while only a fraction of

the pores does, i.e. the unhomogenized liquid-®lledones.

3.2.2. Dihedral and wetting angle. The e�ects of

the dihedral and wetting angle can also be esti-

mated when we calculate the equilibrium micro-structure 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 con-

taining pores of 10 vol.% and liquid of 5 vol.%. In

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


terms of the dihedral angle, densi®cation enhances

as the angle increases. This result is due to theincrease 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 pro-

nounced than that of the dihedral angle, as shown

in Fig. 10(b). But the e�ect is opposite: densi®cationretards 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 pre-

sent 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 wet-

ting with increases in wetting angle [24], this e�ectis 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 com-

pact, the size of grains surrounding a pore is not

constant but has a distribution and the shape of thepore is, in general, irregular. Under this condition,

the pore ®lling can occur earlier, before the esti-

mated 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 re-

arrangement stage. The initial pore size and poros-

ity 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 fol-

low the scale e�ect on grain growth. This expec-

tation was already con®rmed by calculating the

densi®cation of compacts with similar microstruc-

ture 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 atmos-

phere, as a simple case, the pore ®lling condition

and the gas bubble size after densi®cation are deter-

mined by:

Pout ÿ 2glvrs� Pin ÿ 2glv

rp�8�

Pout ÿ 2glvrs� Pin

�rprb

�3

ÿ 2glvrb

, �9�

where Pin is the pressure of the entrapped inert gas

just before the isolation of a pore, Pout is the exter-

nal inert gas pressure, rp is the pore radius, rs is theradius 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

f le� and grain size. On the other hand, the hom-

ogenization of the liquid pocket is limited by the

bubble size.

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

with 08 dihedral angle.


Figure 11 plots the calculated densi®cationcurves, e�ective liquid volume fractions and maxi-

mum sizes of the liquid-®lled pores during a powder

compact sintering in a slowly di�using inert gas of1 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 contain-ing inert gas is faster than that of a compact with-

out entrapped inert gas. This result can beunderstood when we think about the changes in the

e�ective liquid volume fraction and in the maxi-

mum size of the liquid-®lled pores during densi®ca-tion, as shown in Fig. 11(b). At the beginning, the

reduction in the e�ective liquid volume is less withinert 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. Whenalmost all the pores are ®lled with liquid, however,

additional densi®cation occurs only by a reductionin 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 sizederives from the increase in liquid pressure resulting

from the increase in the liquid meniscus radius dueto grain growth. The very slow densi®cation during

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 reducedbecause of pore expansion [38±40]. This e�ect wasnot taken into account for the calculation: the cal-

culation shows the maximum attainable densityduring pressureless sintering.A similar analysis can also be made for the e�ect

of external gas pressure on densi®cation. For apowder compact containing no gas within the poresand under 1 atm external gas pressure, i.e. positive

external pressure of 1 atm, the densi®cation timewas calculated to be about 2/3 that of normal sin-tering with fast di�using gas.In sintering practice, powder compacts may be

sintered in a gas mixture, slow di�using and fastdi�using. The external gas species and pressure mayalso be changed during sintering. Even for these

complicated conditions, the e�ect of the entrappedinert gas can also be estimated with modi®cation ofequations (8) and (9), as in previous

investigations [22, 37, 41]. The sintering kineticsshould be modi®ed; however, the overall shapes ofdensi®cation and f l

e� curves must be similar to

those shown in Fig. 11.

4. CONCLUSIONS

A new theory of liquid-phase sintering, namelythe pore ®lling theory, has been developed for pow-

der compacts containing pores of various sizes. Thetheory is based on the microstructural developmentobserved in previous investigations [1, 7, 13, 14, 21±

23] and on our previously developed liquid-phasesintering model [20]. By calculating the changes inthe e�ective liquid volume fraction f l

e� and theliquid meniscus radius during densi®cation, it was

possible to critically evaluate the e�ects of such var-ious practical sintering parameters as pore size dis-tribution, porosity, liquid volume fraction, dihedral

and wetting angle, particle size and sintering atmos-phere.It has been found that pore size distribution has

a considerable e�ect on sintering kinetics of powdercompacts with same porosity. The densi®cation of apowder compact with a Weibull distribution isretarded a few times more than that with a lognor-

mal distribution. For compacts with same averagepore size, the densi®cation of pores with a broadsize distribution is slower than that with a narrow

size distribution. The liquid volume fraction f li,

which directly a�ects the e�ective liquid volumefraction f l

e� and the liquid meniscus radius, has a

strong e�ect on densi®cation. The sintering timeappears to be proportional to about ( f l

i)ÿ3 in thecase of di�usion-controlled grain growth. This

dependence results from two things: the linear pro-portionality of the liquid meniscus radius to theliquid volume fraction, and the grain growth kin-etics with annealing time. The e�ect of porosity is

Fig. 11. Calculated curves of (a) relative density andshrinkage with sintering time and (b) maximum size ofliquid-®lled pores and e�ective liquid volume fraction withnormalized grain size for fast and slow di�using (inert) gas

atmosphere of 1 atm.


also found to be considerable but less than the

e�ect of liquid volume fraction, because only unho-mogenized liquid-®lled pores a�ect f l

e� while liquidvolume fraction directly a�ects f l

e� over the whole

specimen. Dihedral angle increase enhances densi®-cation but not by much. In contrast, wetting angleincrease considerably retards densi®cation, several

times over with angle increases of from zero degreesto a few tens of degrees. The e�ect of scale is deter-

mined by the kinetics of grain growth, because den-si®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. Thecalculated e�ect of the entrapped gas is somewhatdi�erent from that conventionally expected: retar-

dation of densi®cation right from the beginning ofthe elimination of isolated pores. At the beginningof the elimination of isolated pores, the densi®ca-

tion rate of a compact with inert gas is faster thanthat without it. This result is related to less re-

duction in e�ective liquid volume fraction duringpore ®lling because of the presence of gas bubblesin the liquid-®lled pores. During extended sintering,

however, the densi®cation essentially stops becauseof the entrapped inert gas.

In the development of this theory, in contrast tothe classical Kingery theory which neglects graingrowth, both densi®cation and grain growth, two

essential processes occurring during sintering, wereconsidered. The classical theory is, in fact, a two-particle model, similar to that of solid-state sinter-

ing. For solid-state sintering, the description ofmicrostructural development was possible throughthe 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 andprovides a better understanding of the real phenom-ena.

Densi®cation in liquid-phase sintering, however,is basically di�erent from that in solid-state sinter-ing. In solid-state sintering, the densi®cation occurs

by material transport from the grain boundary, asin two-particle type sintering. In liquid-phase sinter-

ing, the densi®cation appears to occur by pore ®ll-ing with grain growth; the densi®cation by contact¯attening of the two-particle type is negligible. The

sintering kinetics is therefore determined by graincoarsening, in contrast to the case of solid-state sin-tering.

In our pore ®lling theory, the grains are assumedto be spherical. We can well de®ne and describe the

sintering processes: pore ®lling, grain shape accom-modation and microstructure homogenization. Ifthe grains are faceted, it may be di�cult to quanti-

tatively describe the sintering kinetics. Nevertheless,the basic concepts of this pore ®lling theory arebelieved to apply, because the liquid meniscus must

also increase with grain size and the complete wet-ting of the pore surface must result in pore ®lling.

The overall behavior of sintering and the e�ect ofvarious sintering parameters should be similar to

those evaluated for compacts with spherical grains.The di�culties observed in densifying compactswith faceted grains can then be understood as a

result of the low rate of grain growth.

AcknowledgementsÐThis work was supported by theKorea Advanced Institute of Science and Technology andalso by the Korea Research Foundation.

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Documents

Theoretical analysis of liquid-phase sintering: Pore filling theory