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A MODEL FOR THE SINTERING OF SPHERICAL

PARTICLES OF DIFFERENT SIZES BY SOLID STATE

DIFFUSION

J. PAN{, H. LE{, S. KUCHERENKO and J. A. YEOMANS

School of Mechanical and Materials Engineering, University of Surrey, Guildford GU2 5XH, U.K.

(Received 30 September 1997; accepted 29 March 1998)

Abstract ÐIn this paper the numerical scheme developed by Pan and Cocks (Acta metall . 43, 1395±1406,1995) is used to simulate the co-sintering process of two spherical particles of dierent sizes by coupled

grain-boundary and surface diusion. The numerical analysis reveals many interesting features of the co-sintering process. For example, it is found that the shrinkage between the two particles is not aected sig-ni®cantly by the size dierence of the two particles as long as the dierence is less than 50%. Based on thenumerical results, empirical formulae for the characteristic time of the co-sintering process and for theshrinkage rate between the two particles are established. The empirical formulae can be used to developconstitutive laws for early-stage sintering of powder compacts which take into account the eect of particlesize distribution. To demonstrate this, a densi®cation rate equation for compacts with bimodal particle sizedistributions is derived.# 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Sintering is a crucial step for the fabrication of a

ceramic component from a powder compact. Two

things occur during sintering: grain growth and

densi®cation. The driving force for the microstruc-

tural changes is the excess free energy associated

with the large free-surface area of the ®ne powders.

The actual mechanism of matter redistribution

depends on the material system, the particle size,

the sintering temperature and the level of external

pressure, if such pressure is applied. Coupled grain-

boundary and surface diusion is often the domi-

nant mechanism for the sintering of a ®ne particle

compact which is subjected to a moderate

pressure [1,2]. This is a mechanism where grain-

boundary diusion transports matter to the junc-

tions between grain-boundaries and pore surfaces,

and surface diusion redistributes that matter ontothe pore surfaces. A typical example of sintering by

this mechanism is the pressureless sintering of

alumina powder with a particle size of about 3±

8 mm at 14008C. In this paper we concentrate on

this coupled diusion mechanism only.

Considerable eorts have been made to under-

stand the sintering process. Constitutive laws have

been developed that enable ®nite element analysis

to be performed for the sintering process [3]. The

analysis can predict the history of stress, strain,

relative density and grain size at any location of a

component during the sintering process [4].

However, the current generation of densi®cation

laws assume uniform particle size and ignore the

fact that most of the commercial powders consist of

particles with a wide range of sizes. There havebeen relatively few studies examining the eect of

particle size distribution on sintering. Patterson and

Benson [5] did an experimental study on the eect

of powder size distribution on sintering. Ting and

Lin [6] derived a shrinkage rate equation for pow-

der compacts taking into account the eect of par-

ticle size distributions. Grain growth was also

considered in their model. Ting and Lin did not in-

vestigate the detailed kinetics of the co-sintering

process between particles of dierent sizes. It was

simply assumed that the sintering rate equation

between two particles of dierent sizes takes exactly

the same form as that between two particles of single size. For bimodal powder mixtures,

German [7] proposed a simple model for prediction

of the sintered density vs mixture composition from

the knowledge of the densi®cation of the large and

small powders. Probably the ®rst theoretical study

of the co-sintering process between a pair of spheri-

cal particles of dierent sizes was made by Coble [8]

who used the approximiate geometric relationships

for identical particles to derive an approach rate

equation for the non-identical particles. Recently

Tanaka [9] re-investigated this problem. Assuming

that the two particles maintain their truncated

spherical shape during the co-sintering process,Tanaka obtained the rate equations for the sintering

and coarsening of the two-particle system. More

Acta mater. Vol. 46, No. 13, pp. 4671±4690, 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(98)00144-X

{To whom all correspondence should be addressed.{Current address: Department of Engineering, University

of Cambridge, Cambridge CB2 1PZ, U.K.

4671



recently Parhami et al . [10] investigated a similar

problem using a variational approach. Three

degrees of freedom were used to de®ne the geome-

try of a representative unit of a row of particles of two dierent sizes. Using the classical Rayleigh±

Ritz method, a numerical solution was obtained for

the co-sintering process. It is important to realize

that the selected degrees of freedom in these models

limit the co-sintering process to a speci®c kinetic

route which can be very dierent from the actual

one. For example, it was assumed that the main

parts of the two particles remain spherical during

the sintering process. This is not quite correct since

surface diusion is often not fast enough for the

two particles to maintain the near-equilibrium

shape. The accuracy of these models can only be

found when they are compared with a full solution.

The main purpose of this paper is to investigate

the sintering kinetics of two spherical particles of

dierent sizes by means of computer simulation.

Such a computer simulation was made possible by

a numerical scheme developed recently by Pan and

Cocks [11]. Ignoring the interaction between thepair of particles and the particles surrounding

them, the two particles can be considered as a

representative unit of a powder compact. The nu-

merical analysis is shown to reveal many interest-

ing features of the co-sintering process. Based on

the numerical results, empirical formulae are estab-

lished in order to describe the various aspects of

the co-sintering process analytically. These empiri-

cal formulae can be used to develop constitutive

laws for early-stage sintering taking into account

the eects of particle size distributions. To demon-

strate this, a densi®cation rate equation is derived

for powder compacts with bimodal particle size

distributions based on the empirical formula.

Fig. 1. The co-sintering process of two particles in contact with each other (schematic drawings).

PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES4672



Experimental work is underway to verify the major

®ndings of this numerical study.

2. DRIVING FORCE FOR SINTERING

Figure 1(a) shows the representative unit that is

considered in this work. The total free energy of the

system is

E

s

gsdAs

gb

ggbdAgb 1

where gs and ggb are the speci®c energies of the free

surface and the inter-particle boundary (the grain-

boundary), respectively. The tendency to reduce E

is the driving force for a shape change of the sys-

tem. For the two-particle system, it is obvious that

E reaches its minimum value when the two particlesbecome one perfect sphere.

There are several mechanisms by which matter

redistribution can be achieved. These include vis-

cous ¯ow, lattice diusion, evaporation and con-

densation, and coupled grain-boundary and surface

diusion. As mentioned in the Introduction

(Section 1), only the last mechanism is considered

here. The shape evolution is controlled by the

kinetic law as well as by the driving force. A

comprehensive discussion about the roles played

by driving forces and kinetic laws, respectively, in

the microstructral evolution has been given by Sun

et al . [12].

Where the grain-boundary meets the free surface,the equilibrium between the surface tensions of the

two particles and the grain-boundary tension has to

be maintained. As shown in Fig. 1(b), this require-

ment of equilibrium tends to ``bend'' the grain-

boundary towards the small particle with ends

pinned at the junction. The grain-boundary then

moves towards the smaller particle to ¯atten itself.

The combination of these two mechanisms results

in migration of the boundary towards the smaller

particle as long as the junction itself moves. The

¯attening of the grain-boundary is assumed to be

much faster than the movement of the junction.

3. THE KINETIC LAW

The usual linear kinetic law (Fick's law) is

assumed for grain-boundary and free-surface diu-

sion. The diusive ¯ux j , de®ned as volume of mat-

ter ¯owing across unit area perpendicular to the

¯ux direction per unit time, is assumed to depend

linearly on the gradient of the chemical potential m

of the diusing species:

j ÀDd

kT r m 2

where D is diusivity, d is the thickness of the

layer through which the material diuses, k isBoltzmann's constant and T is the absolute tem-

perature.

Along the grain-boundary, the gradient of the

chemical potential is directly related to the gradi-

ent of stress, s, acting normal to the grain-

boundary [13], i.e.m ÀOs 3

where O is the atomic volume. Along the free

surface, the gradient of the atomic chemical poten-

tial is related to the gradient of the free-surface

curvature [13], i.e

m ÀOgsk 4

Here gs is the surface tension and k is the principal

curvature of the surface.

In equation (2), Dd should be replaced by

Dgbdgb for grain-boundary diusion and by Dsdsfor free-surface diusion. The subscripts ``gb'' and

``s'' represent grain-boundary and free surface,respectively.

4. NUMERICAL SCHEME AND NON-DIMENSIONALIZATION

The coupled grain-boundary and surface diu-

sion problem is generally too dicult to solve ana-

lytically. Cavity growth and sintering are two

opposite phenomena. Under many practical circum-

stances both processes can be controlled by the

coupled diusion mechanism. For cavity growth, a

steady state solution was obtained by Chuang and

Rice [14] and later a self-similar solution wasobtained by Chuang et al . [15]. For the sintering

problem of uniform particles, Svoboda and Riedel

obtained an analytical solution for the so-called

``small scale'' diusion problem [16]. For more gen-

eral situations, numerical methods have to be used.

Pharr and Nix [17] studied the cavity growth pro-

blem while Bross and Exner [18] studied the sinter-

ing problem using similar numerical methods at

almost the same time. More recent studies of sinter-

ing using numerical analysis include the work by

Bouvard and McMeeking [19] and Zhang and

Schneibel [20]. These eorts have considerably

improved our understanding of sintering kinetics.

However, there is a common problem to all these

previous studies: surface diusion was assumed to

be symmetric about the grain-boundary and conse-

quently the sintering of particles of dierent sizes

cannot be studied using these numerical schemes

since matter diuses from the smaller particle to the

larger one when the two particles in contact are of

dierent sizes.

Recently, a general numerical method has been

developed by Pan and coworkers [11, 21], which can

be used to simulate microstructural evolution con-

trolled by solid state diusion and grain-boundary

migration. Using this numerical method, the evol-

ution history of a prescribed network of grain-boundaries with internal and external free surfaces

can be followed. The grain-boundaries and the free

PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES 4673



surfaces can take any arbitrary shape. As a demon-

strating example of the numerical scheme, the

co-sintering process of two cylinders of dierent

radii was simulated [11].In this work, the numerical scheme described in

Ref. [11] is used with some straightforward modi®-

cations to consider the axisymmetric problem of the

two spherical particles instead of the two cylinders

considered previously. The underlying principle

used in this study is exactly the same as that

described in Ref. [11], i.e. the chemical potential at

the junction between the grain-boundary and the

free surface is taken as an unknown. Matter conser-

vation in the vicinity of the junction is used to

determine the chemical potential. As soon as this

chemical potential is known, the central ®nite dier-

ence scheme is used to determine the migration vel-

ocity of the free surface. The approach velocity

between the two particles is obtained analytically

for the diusion problem in the circular disk of

contact between the two particles. The pro®le of the

system is then updated using the direct Euler inte-

gration scheme and the entire procedure is repeated

for a required number of time steps. Rather than

repeat the details of the numerical scheme here,

emphasis is placed on the physical merits of the

numerical analysis.

It proves convenient to discuss the numerical

results in terms of non-dimensionalized groups of

material properties that control the sintering kin-

etics. A reference ``strain rate'' is de®ned as:

egb DgbdgbO

kT

gs

r42

5

where r2 is the initial radius of the large particle. In

this paper, all the lengths are scaled by r2 and the

various physical variables are non-dimesionalized in

the following way:

"k kr2,

"s sr2

gs

"

j

j

egbr22

"W W

egbr2

"t egbt 6

where W is the approaching velocity between the

two particles and t is the time. The free energy E

can be non-dimensionalized as

"E E

r22gs

s

d "As

gb

"ggbd "Agb 7

in which

"ggb ggb

gs

8

and

d "A dA

r2

2

Then equation (2) becomes

" j gb r "s 9

for the grain-boundary diusion and

" j s "Dsr "k 10

for the free-surface diusion, in which

"Ds Dsds

Dgbdgb

11

Two non-dimensionalized groups of the material

properties have emerged from the above analysis:

"Ds, which represents the relative importance of the

two diusion processes, and "ggb, which determines

the dihedral angle, C, through the equation

cosC "ggb

212

The numerical results are presented using "Ds and C

as the input material properties. To apply the

results to any real material system, data required by

equation (5) as well as "Ds and "ggb need to be avail-

able so that equation (6) can be used to transform

the non-dimensionalized results into those for the

material system concerned.

5. OVERVIEW OF THE COMPUTER SIMULATION

The computer simulations presented cover a wide

range of dierent combinations of r1/r2, "Ds, C, and

"sI, which represent the ratio of particle radii, the

ratio of surface diusivity over grain-boundary dif-

fusivity, the dihedral angle and the normalized

average stress applied on the grain-boundary, re-

spectively. The ratio of particle radii, r1/r2, is varied

between 0.1 and 1.0 and "Ds is varied between 0.01

and 100. Three dierent values of the dihedral

angle, i.e. C = 458, C = 608 and C = 758, andthree levels of the applied stress are used. Table 1

summarizes all the dierent cases. In total about

150 simulations were performed covering the

various dierent cases.

Ideally, the numerical analysis should start from

zero contact area between the two particles.

Numerically, however, the analysis has to start

from a small initial contact area. In the numerical

Table 1. An overview of the computer simulation

r1/r2 "Ds C "sI

0.1, 0.2, 0.3, 0.01, 0.1, 458, 608, 758 0.0, 5.0, 50.00.5, 0.7, 0.9, 1.0, 100.01.0




simulations, the radius of the initial contact area

has been taken as 0.1r1.

The total volume of the two particles should

remain constant throughout the co-sintering processbut numerical errors cause volume ¯uctuations

during the simulation. In all the simulations per-

formed, the maximum ¯uctuation of the total

volume was within 4% of the initial volume of the

small particle. This indicates the high accuracy of

the numerical scheme.

The co-sintering process can be divided into two

distinct stages, i.e. the stages before and after the

grain-boundary disappears. Once the grain-bound-

ary disappears, the system becomes one particle

which evolves eventually into a perfect sphere as

shown in Fig. 1. The second stage, which is comple-

tely controlled by surface diusion, is of little prac-

tical interest since at this stage the interactionbetween the system and the surrounding particles

becomes signi®cant. All the simulations were there-

fore terminated as soon as the grain-boundary

migrates out of the system.

6. CHARACTERISTICS OF SHAPE EVOLUTION OFTHE TWO-PARTICLE SYSTEM

One of the purposes of the numerical study is to

understand how the two-particle system evolves.

Fig. 2. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5,"Ds=1.0 and C = 608. (a) "t 0, (b) "t 1X153Â 10À2, (c) "t 21X95 Â 10À2, (d) "t 28X19 Â 10À2.




Based on such an understanding, a simpler model

of the system can be proposed and approximate sol-

utions can be obtained using the variational

approach described by Parhami et al . [10].It is obvious that the closer the sizes of the two

particles, the greater is the in¯uence of the smaller

particle on the shape evolution of the entire system.

This can be seen by comparing the numerical result

from the case of r1/r2=0.5 with that of r1/r2=0.1 as

shown in Figs 2 and 3, respectively. In these

examples C = 608 and "Ds=1. For r1/r2=0.5 the

large particle changes its shape rapidly from a

sphere into a bulb. For r1/r2 =0.1, however, the

large particle manages to maintain its original

shape.

Varying "Ds (the ratio of surface diusivity to

grain-boundary diusivity) changes whichever of

the two diusion processes dominates the co-sinter-

ing process. It is the slower process that controlsthe overall rate, for example, when "Ds=100 grain-

boundary diusion dominates and when "Ds=0.01

surface diusion dominates. Figures 4 and 5 show

the cases for "Ds=0.01 and 100, respectively, which

can be compared with Fig. 2 where "Ds=1. In these

examples, C = 608 and r1/r2=0.5. It can be

observed that although "Ds varies over a range of

®ve orders of magnitude, the pattern of the shape

evolution remains similar. The grain-boundary dif-

fusion-controlled case shows a slightly smaller

grain-boundary during the entire process. These ob-

Fig. 3. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.1,"Ds=1.0 and C = 608. (a) "t 0, (b) "t 1X227Â 10À4 and (c) "t 2X9 Â 10À4.




servations are also true for other ratios of particle

radii. The numerical results suggest that "Ds mainly

in¯uences the rate of the process not the pattern of

evolution.

The dihedral angle has a dramatic eect on the

pattern of the shape evolution. This is clearly

demonstrated by comparing Fig. 6, where C = 458,with Fig. 2, where C = 608, for r1/r2=0.5 and by

comparing Fig. 7, where C = 458, with Fig. 8,

where C = 608, for r1/r2=0.9. In these examples,"Ds=1.0. It can be seen that a smaller dihedral

angle helps the small particle to maintain a more

rounded shape throughout the co-sintering process.

As mentioned in the Introduction (Section 1), in

eort to model the co-sintering process, Tanaka [9]

assumed that the two particles maintain their trun-cated spherical shape as matter is transferred from

the small particle to the large one while Parhami et

Fig. 4. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5,"Ds=0.01 and C = 608. (a) "t 0, (b) "t 0X8078, (c) "t 10X56 and (d) "t 20X05.




al . [10] modi®ed Tanaka's model by introducing a

cylindrical disk between the two spheres. From the

computer simulations presented above it can be

seen that the actual shape evolution of the two-par-

ticle system is very dierent from those assumed by

Tanaka and Parhami et al . In fact the two-particle

system can be better approximated by a system thatconsists of two truncated spheres connected by a

truncated cone, as shown in Fig. 9(a). The approxi-

mate system is completely determined by ®ve geo-

metric parameters: the radii of the large and small

truncated spheres, r1 and r2, the radii of the top

and bottom sections of the truncated cone, r1 and

r2, and the height of the truncated cone, h. Matter

conservation requires that only four of the ®ve par-

ameters are independent; the system therefore hasfour degrees of freedom. The system starts from

h = 0 and r1=r2=r0, where r0 is the initial neck

Fig. 5. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5,"Ds=100 and C = 608. (a) "t 0, (b) "t 0X6 Â 10À3, (c) "t 7X171Â 10À3 and (d) "t 9X687Â 10À3.




size. By comparing the numerical results for all the

dierent cases, it is found that for r1/r2 larger than

0.5, the junction between the truncated cone and

the large truncated sphere can be regarded as a

smooth one. As a consequence, the degrees of free-

dom of the approximate system can be reduced to

three as shown by Fig. 9(b). For r1/r2 less than 0.5,the approximate model with four degrees of free-

dom, shown by Fig. 9(a), is more appropriate.

7. TIME TO DISAPPEARANCE OF THE GRAIN-BOUNDARY

A characteristic time describing the co-sintering

process is the time taken for the inter-particle

boundary (i.e. the grain-boundary) to migrate out

of the system. In the following discussions, this

characteristic time is referred as to"

td (which is non-dimesionalized by egb). Once the grain-boundary

has disappeared, grain growth is complete as far as

Fig. 6. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5,"Ds=1.0 and C = 458. (a) "t 0, (b) "t 7X948Â 10À2, (c) "t 65X49 Â 10À2 and (d) "t 80X77 Â 10À2.




the two particles are concerned and it is experimen-

tally dicult to distinguish the system from its sur-

rounding particles in a powder compact.

The dependence of "td on relative diusivity "Ds

and the ratio of particle radii r1/r2 is shown in

Figs 10±12, for C = 458, C = 608 and C = 758, re-spectively. The interesting feature of the numerical

results is that "td depends linearly on r1/r2 on the

log±log plots. The numerical results can be best

®tted using the following empirical formula

"td 0X15 Á "DÀ0X85s

ggb

gs

Á

r1

r2

4X63

13

which is plotted in Figs 10±12 using solid anddashed lines to compare with the numerical results.

Using equations (5) and (6) we obtain

Fig. 7. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.9,"Ds=1.0 and C = 458. (a) "t 0, (b) "t 12X68 (c) "t 15X25 and (d) "t 16X39.




td 0X15 ÁkT

OgsDgbdgb

Á

Dgbdgb

Dsds

0X85 ggb

gs

Á

r1

r2

4X63

Á r42 14

Equation (14) suggests that the in¯uence of grain-

boundary diusivity on the characteristic time is

much weaker than that of surface diusivity. This is

expected since it is the local surface diusion of

matter from the small particle to the large one that

controls the grain-boundary migration. The grain-boundary is pinned at the junction between the

grain-boundary and the free surface, and can only

Fig. 8. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.9,"Ds=1.0 and C = 608. (a) "t 0, (b) "t 0X7869, (c) "t 5X242 and (d) "t 5X84.




migrate when the junction moves by surface diu-

sion. Grain-boundary diusion has little in¯uence

on this process. It does, however, control the neck

growth and shrinkage between the two particles

which will be discussed in the Section 8. Equation(14) also suggests that the in¯uence of the size of

the large particle on the characteristic time is much

weaker than that of the small one. This is simply

because the grain-boundary always migrates

through the small particle.

Equation (14) breaks down when r1/r2approaches unity for which "td should be in®nity in

Fig. 9. An approximate model of the two-particle system (a) using four independent degrees of freedomand (b) using three independent degrees of freedom.

Fig. 10. The time to disappearance of the grain-boundary at various values of the relative diusivityand ratio of particle radii. The symbols represent the numerical results; the solid and dashed lines rep-

resent the empirical formula of equation (13). C = 458.




theory. In reality the symmetric con®guration of

two identical particles is unstable and a small per-

turbation of the particle geometry or material prop-

erty can destroy the symmetry. This was observed

in the numerical simulation for identical particles

during which numerical errors destroy the sym-

Fig. 12. The time to disappearance of the grain-boundary at various values of the relative diusivityand ratio of particle radii. The symbols represent the numerical results. The solid and dashed lines rep-


Fig. 11. The time to disappearance of the grain-boundary at various values of the relative diusivityand ratio of particle radii. The symbols represent the numerical results; the solid and dashed lines rep-





metry slightly and cause the grain-boundary to

migrate towards one of the particles.

8. SHRINKAGE AND CONTACT SIZE

Based on their ®nite dierence studies, Bouvard

and McMeeking [19] suggested the following

empirical formula to describe the relationship

between the approach velocity, W , and the radius

of the contact area, x, between two identical par-

ticles

W

regb

a

xar4

b

xar2

sIr

gs

15

where r is the radius of the particles, sI is the aver-age stress transmitted onto the contact area, a is a

pure number which depends on "Ds and b is a pure

number which is insensitive to "Ds and sI

.

Figures 13±15 present the numerical results from

this study relating"W to "x for various values of r1/

r2 and for "sI=0 (Fig. 13), "sI=5 (Fig. 14), and

"sI=50 (Fig. 15), respectively, where

"W W

r2egb

"x x

r2

and

"sI sIr2

gs

In these examples,"

Ds=1 and C = 608. The nu-merical results can be ®tted using a modi®ed ver-

sion of equation (15)

W

r2egb

0X5

1

r1

r2

&a

xar24

b

xar22

sIr2

gs

'

16

which is plotted in the ®gures using solid and

dashed lines. Here a and b have been taken as 9

and 8, respectively, which is consistent with the

values used by Bouvard and McMeeking [19]. It

can be seen that the numerical results con®rm

equation (15) for the case of uniform particles. Asthe size of the small particle decreases, the approach

rate decreases. This is incorporated in the empirical

formula given by equation (16) by introducing a

factor 0.5(1 + r1/r2). The interesting feature of the

numerical results is that the particle ratio does not

have a signi®cant eect on the approach rate pro-

vided that r1/r2 is larger than 0.5. This is especially

the case when stress is applied as shown by Figs 14

and 15. As the matter redistribution from the grain-

boundary to the neck is a local event in the early

stages of the co-sintering process, the relative size

of the two particles only comes into play at a later

stage. The numerical results indeed indicate that W

depends on r1/r2 in the later stages. When stress is

Fig. 13. Relationship between the shrinkage rate and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (16). C = 608,

"Ds=1 and "sI=0.




applied, it tends to dominate the driving force for

the co-sintering process and further weakens the in-¯uence of r1/r2. From Figs 13±15 it can be seen that

equation (16) breaks down for r1/r2 less than 0.5.

The numerical results can be better ®tted by intro-ducing a dierent factor, 0.5(1 + (r1/r2))x for


"Ds=1 and "sI=5.


"Ds=1 and "sI=50.




example, to equation (16) where x is an empirical

exponent. No attempt to do so has been made here

since such ®tting does not give any further insight

into the process.Figure 16 shows the numerical results relating the

shrinkage W to the radius of the contact area x

between the two particles. Coble used a simple re-

lationship between the two variables [8]

w

r

1

2

x

r

2

17

in which r is the initial radius of the particles. For

two identical particles, the numerical results con®rm

equation (17) when the neck radius is larger than

0.2r. For non-identical particles, the numerical re-

lationship can be ®tted using a modi®ed version of

equation (17)

W

r2

0X5

1

r1

r2

Àz1

d

x

r2

2

18

in which d and z are empirical parameters.

Equation (18) is plotted on Fig. 16 using z = 1.5

and d = 2.4 for comparision with the full numerical

results. The empirical ®tting breaks down when the

neck size is very small or when r1/r2 is less than 0.5.

Figure 17 presents the numerical results relating

the normalized shrinkage, W /r2, to the normalized

time, tegb, for three dierent values of r1/r2 and for"Ds=1 and C = 608. Again it can be seen that the

size ratio of the two particles has almost no eecton the shrinkage. This observation is true for the

entire range of values of "Ds and C covered in

Table 1. From Fig. 17, it can also be observed that

W /r2 is linearly dependent on tegb on the log±log

scale which suggests the following empirical re-lationship

W

r2

Àltegb

Á1an19

where l and n are empirical parameters. It is found

that n is insensitive to "Ds and C and is within the

range between 3 and 4. The empirical parameter l

is found to be dependent on "Ds and C. For

example, l = 0.006316, 0.4453 and 1.1874 for"Ds=0.01, 0.1, and 1.0, respectively when C = 608.

At any ®xed normalized shrinkage, the shrinkage

rate is linearly dependent on l. The numerical

results suggest that as surface diusivity increasesover three orders of magnitude relative to grain-

boundary diusivity, the shrinkage rate increases

over the same orders of magnitude.

Figure 18 presents the numerical relationship

between W /r2 and tegb covering "Ds=0.01±10,

C = 458 ±758 and r1/r2=0.5±1.0. Equation (19) is

plotted on top of the numerical results using n = 4

and l = 1.1874. Equation (19) is invalid for "Ds lar-

ger than 10, i.e. the linear relationship between W /

r2 and tegb on the log±log scale breaks down when"Ds is larger than 10. This is shown in Fig. 19 for an

extreme case of "Ds=100.

Probably the most interesting numerical resultpresented in this section is that for a wide range of

Fig. 16. Relationship between the shrinkage and the contact radius at various values of the ratio of par-ticle radii. The solid and dashed lines represent the empirical formula of equation (18). C = 608, "Ds=l

and "sI=0.




material parameters, the shrinkage and shrinkage

rate between the two particles are not signi®cantly

aected by the size dierence between the two par-

ticles. This numerical ®nding simpli®es the task of

constructing densi®cation laws for powder compacts

when taking into account the eect of size distri-

butions. It means that the size distribution only in-¯uences densi®cation (in the early stages) by

in¯uencing the initial density and the number of

contacts of a powder compact. The diusion kin-

etics between the particles are not aected at least

in the early stages of the sintering process. This

conclusion is not valid if the size dierence between

the particles is larger than 50%.

9. A DENSIFICATION RATE EQUATION FORPOWDER COMPACTS WITH A BIMODAL

PARTICLE SIZE DISTRIBUTION

For a powder compact which consists of particles

of only two dierent radii, Rl and Rs, for large and

small particles, respectively, nl and ns are the num-

ber fractions of the large and small particles. There

are three dierent types of contacts between the

particles; nss, nll and nls represent the number frac-

Fig. 17. The shrinkage as a function of time for various values of the ratio of particle radii. C = 608and "Ds=1.

Fig. 18. The shrinkage as a function of time. The shaded band represents the numerical results covering"Ds=0.01±10, C = 458 ±758 and r1/r2=0.5±1.0. The solid line represents the empirical formula of

equation (19) using n = 4.




tions of contacts between small and small, large

and large, and large and small particles, respect-

ively. Turner [22] demonstrated that the number

fractions of the dierent contacts can be expressed

as

nll

nlRl

nlRl nsRs

2

ls 2nlRlnsRsÀ

nlRl nsRs

Á2

nss

nsRs

nlRl nsRs

2

20

Furthermore Turner found that the number frac-

tions of the contacts in a string of two sizes of par-

ticle can be given as

nsll 2nllRl

2nllRl nlsRl Rs 2nssRs21

nsls

nlsRl Rs

2nllRl nlsRl Rs 2nssRs

22

nsss

2nssRs

2nllRl nlsRl Rs 2nssRs

23

where the superscript ``s'' denotes a string.

In order to obtain the densi®cation rate equation,

either equation (16) together with equation (18) or

equation (19) can be used. For simplicity, the latter

is used here, i.e. it is assumed that the normalized

shrinkage rates for the three dierent types of con-tacts are the same. The line strain rate of the pow-

der compact is simply

e Àns

ll nsls ns

ss

2nsllRl ns

lsRl Rs 2nsssRs

W 24

in which W is the shrinkage rate of a contact.

Using equation (20) and equations (21)±(23), it canbe shown that

nsls

nsll

"V s

1 À "V s

1 "Rs

"R2s

25

and

nsss

nsll

"V s

1 À "V s

21

"R3s

26

where "Rs=Rs/Rl and "V s is the volume fraction of

the small particles in the compact. Using

equations (25 and 26), equation (24) can be rewrit-

ten as

e À1

2wÀ

"Rs, "V sÁ W

Rl

27

in which

wÀ

"Rs, "V sÁ

"R3s

À1 À "V s

Á2 "V s

À1 À "V s

ÁÀ1 "Rs

Á"Rs "V 2s

"R3s

À1 À "V s

Á2 0X5 "V s

À1 À "V s

ÁÀ1 "Rs

Á2 "Rs "V 2s"Rs

28

Identifying r2 in equation (19) as Rl and substitut-

ing equation (19) into equation (27) gives

e À1

2nw "Rs, "V slegb

1ant1Ànan 29

Fig. 19. The shrinkage as a function of time for various values of the ratio of particle radii. C = 608and "Ds=100.




Referring to the relative density of the compact as

D and the densi®cation rate as D then

D

D À3e

D

D0

1

330

in which D0 is the initial density of the compact.

Combining equation (29) with equation (30) gives

D

D

3

2nwÀ

"Rs, "V sÁ D

D0

1

3Àlegb

Á1ant1Ànan 31

which can be integrated with respect to time giving

1 À

D0

D

1

3

1

2wÀ

"Rs, "V sÁÀlegbÁ

1ant1an 32

Eliminating t from equations (31) and (32) and

noticing equation (5) results in

D

D Awn

À"Rs, "V s

Á 1

R4l

D

D0

1

31 À

D0

D

1

3

Hfd

Ige

1Àn

33

in which A is the only parameter that depends on

material properties and temperature:

A

321Àn

2n

!Á

DgbdgbOgs

kT

!Á l 34

As discussed in Section 8, l is an empirical par-ameter which depends on Dsds/Dgbdgb and gs/ggb,

and increases as Dsds/Dgbdgb increases. In practice

A can be determined experimentally as a single par-

ameter of the material which is advantageous as the

constituent properties, especially the thermal diu-

sivity, are dicult to measure accurately.

It must be pointed out that equation (33) is onlyvalid for the early stage of sintering during which

the interaction between dierent contacts is insignif-

icant and the contact number is not aected by

density changes of the compact although it would

be possible to release the second limitation. The

conclusion that size dierence does not aect the

shrinkage between two particles can be used to

develop densi®cation rate equations for powder

compact with a known size distribution. This has

not been pursued here but rather equation (33) has

been constructed to establish a macroscopic conse-

quence of the numerical ®ndings so that exper-

iments can be performed to verify the numerical

results.

Function wÀ

"Rs, "V sÁ

represents the dierence in

densi®cation rate between a uniform powder com-

pact and a bimodal powder compact. wÀ

"Rs, "V sÁ

is

plotted in Fig. 20 where the eects of volume frac-

tion and size of the small particles can be seen

clearly.

10. CONCLUSIONS

Particle size distribution can in¯uence the sinter-

ing process in two respects, i.e. grain growth and

densi®cation. The two processes become strongly

coupled in the intermediate stage of the sinteringprocess. In the early stage, however, they can be

treated separately. For early stage sintering,

equation (14) can be used to formulate a rate

Fig. 20. Eect of the volume fraction "V s and relative size "Rs of the small particles on the initial densi®-cation rate of a powder compact with a bimodal particle size distribution.




equation for grain growth, and equations (16) and

(19) can be used to formulate a rate equation for

densi®cation. The most interesting ®nding of this

numerical study is that the shrinkage and shrinkagerate between two particles is not signi®cantly

aected by the size dierence of the two particles as

long as the dierence is less than 50%. The densi®-

cation rate equation, equation (33), for powder

compacts with bimodal particle size distribution, is

directly based on this numerical conclusion and

therefore can be used to verify the numerical ®nd-

ings experimentally.

Acknowledgements ÐThe authors wish to thank A. C. F.Cocks for his invaluable suggestions throughout thiswork. Parts of the numerical simulations were performedby ®nal year project students including A. Sem, H. D.

Luong and C. Poth in the Department of MechanicalEngineering of the University of Surrey. This research is®nancially supported by the EPSRC (grant GR/K78102),which is gratefully acknowledged.

REFERENCES

1. Ashby, M. F., Acta. metall., 1974, 22, 275.2. Swinkels, F. B. and Ashby, M. F., Acta metall., 1981,

29, 259.3. Cocks, A. C. F., Acta. metall., 1994, 42, 2191±2210.4. Du, Z. Z. and Cocks, A. C. F., Acta. metall., 1992,

40, 1981.

5. Patterson, B. R. and Benson, L. A., Prog. PowderMetall., 1984, 39, 215.

6. Ting, J.-M. and Lin, R. Y., J. Mat. Sci., 1994, 29,1867.

7. German, R. M., Metall. Trans., 1992, 23A, 1445.8. Coble, R. L., J. Am. Ceram. Soc., 1973, 56, 461.9. Tanaka, H., J. Ceram. Soc. Japan, 1995, 103, 138.

10. Parhami, F., Cocks, A. C. F., McMeeking, R. M. andSuo, Z., private communication.

11. Pan, J. and Cocks, A. C. F., Acta. metall., 1995, 43,1395±1406.

12. Sun, B., Suo, Z. and Cocks, A. C. F., J. Mech. Phys.Solids, 1996, 44, 559.

13. Herring, C., in The Physics of Powder Metallurgy, ed.W. E. Kingston. McGraw-Hill, New York, 1955.

14. Chuang, T. and Rice, J. R., Acta. metall., 1973, 21,1625.

15. Chuang, T., Kagawa, K. I., Rice, J. R. and Sills, L.B., Acta. metall., 1979, 27, 265.

16. Svoboda, J. and Riedel, H., Acta. metall., 1995, 43, 1.

17. Pharr, G. M. and Nix, W. D., Acta. metall., 1979, 27,1615.

18. Bross, P. and Exner, H. E., Acta. metall., 1979, 27,1013.

19. Bouvard, D. and McMeeking, R. M., J. Am. Ceram.Soc., 1996, 79, 666.

20. Zhang, W. and Schneibel, J. H., Acta metall. mater.,1995, 43, 4377.

21. Pan, J., Cocks, A. C. F. and Kucherenko, S., Proc. R.Soc. A, 1997, 453, 2161.

22. Turner, C. D., Fundamental Investigations of theIsostatic Pressing of Composite Powders, Ph.D. thesis,Engineering Department, Cambridge University, 1994.


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