A Model for the Sintering and Corsening of Rows of Spherical Particles

A model for the sintering and coarsening of rowsof spherical particles

F. Parhami a, R.M. McMeeking b,*, A.C.F. Cocks c, Z. Suo d

a Cypress Semiconductor, San Jose, CA 95134, USAb Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106, USA

c Engineering Department, Leicester University, Leicester, LE1 7RH, UKd Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

Received 11 November 1997; received in revised form 13 July 1998

Abstract

The formation of interparticle contacts and neck growth by grain boundary and surface diusion during the sintering

of rows of spherical particles is modeled. It is shown that rows of identical particles sinter into a metastable equilibrium

configuration whereas rows of particles with dierent sizes evolve continuously by sintering followed by relatively slow

coarsening. During coarsening, smaller particles disappear and larger particles grow. The model is based on a varia-

tional principle arising from the governing equations of mass transport on the free surface and grain boundaries.

Approximate solutions are found through the use of very simple shapes in which the particles are modeled as truncated

spheres and neck formation is represented by a circular disc between particles. Free and pressure assisted sintering of

rows of identical and dierent size particles are studied through this numerical treatment. The eect of initial particle

sizes, dihedral angles, diusivities, and applied compressive forces are investigated. 1999 Published by ElsevierScience Ltd. All rights reserved.

1. Introduction

Densification of a compact made of identicalspherical particles depends on the properties of thepowder material and the initial relative density ofthe compact (i.e. the ratio of current density to thefull theoretical density of the material). However,densification is limited by the evolution of con-tacting particles into equilibrium shapes. Thisshape is defined by the configuration in which thedecrease of surface area is balanced by the increase

of grain boundary area so that the total free energyis stationary. On the other hand, real powderparticles are not identical in size or shape and theequilibrium can easily be perturbed by coarsening.During coarsening, the total surface area is re-duced by the growth of larger particles at the ex-pense of the shrinkage of smaller particles andtheir eventual disappearance.

The modeling of the formation of interparticlecontacts and neck growth between powder parti-cles by grain boundary and surface diusion hasbeen the subject of study by several authors.Kuczynski (1949), Coble (1958), Ashby (1974) andSwinkels and Ashby (1981) have developed ana-lytical relationships for the size of the interparticlecontact radius and the center-to-center approach

Mechanics of Materials 31 (1999) 4361

* Corresponding author. E-mail: [email protected].

edu.

0167-6636/99/$ see front matter 1999 Published by Elsevier Science Ltd. All rights reserved.PII: S 0 1 6 7 - 6 6 3 6 ( 9 8 ) 0 0 0 4 9 - 0

velocity in terms of geometric variables and ma-terial properties. These relationships are based onsimplifying assumptions about the geometry of theneck and the particle shape. Furthermore, themodels are restricted to the very early stages ofsintering when the contact size and the radius ofcurvature at the neck are very small compared tothe particle radius.

Nichols and Mullins (1965), Bross and Exner(1979), Svoboda and Riedel (1995), Zhang andSchneibel (1995) and Bouvard and McMeeking(1996) relaxed the constraint of small contact ra-dius and neck radius of curvature and solved theresulting equations by numerical procedures forrealistic particle shapes. This approach providesmore accurate solutions for the evolution of par-ticle shape, interparticle contact size and particleshrinkage during sintering. Svoboda and Riedel(1995) also introduced a numerical procedurebased on the maximization of the rate of dissipa-tion of Gibbs free energy for a pair of particlesthrough grain boundary and surface diusion.They provided results for the evolution of inter-particle necks between particles of like size. Exceptfor Zhang and Schneibel (1995), none of the abovestudies predicted the equilibrium shape (Lange andKellett, 1989) for a pair or a row of identicalparticles which occurs when small perturbations ofgeometry produce no change in free energy of thesystem. Only Zhang and Schneibel (1995) com-puted the evolution of initially circular prisms intotheir equilibrium shape. Furthermore, little anal-ysis has been done for particles of unequal size,though recently Pan and Cocks (1995), through anumerical procedure, have introduced a two-di-mensional model for the coarsening of a row ofnonidentical circular prisms in the absence of ap-plied force.

The aim of the current paper is to use themethodology of Svoboda and Riedel (1995) ofenergy rate extremization to solve the problem of arow of spherical particles with dierent sizes sub-ject to free and pressure assisted sintering. Themodel provides insight into the processes by whichaggregates of powder of unequal size densify. Inaddition, such results will be useful as unit solu-tions for discrete element network studies (Parha-mi and McMeeking, 1998) of the macroscopic

behavior of powders at high temperature duringdensification.

In this paper, we develop our formulation forenergy rate extremization of a row of particlesthrough a variational principle. Then we use thisformulation to simulate the growth of interparticlecontact and shrinkage of a row of identical parti-cles. Thereafter, we extend the studies to simulatethe growth of interparticle contact between parti-cles of dierent sizes and thus the coarseningphenomenon.

2. Model formulation

2.1. Variational principle

Consider a row of spherical particles in contact,as shown in Fig. 1(a). At high temperature, atomstravel along the free surfaces and the interparticlecontacts to reduce the total free energy of surfacesand interfaces of the system. Energy is dissipatedin the process of mass transport, because atomsmust be driven over energy barriers during theprocess of diusion. For example, on being movedto another interstitial site, an atom dissolved in thelattice must first be raised to a higher energy beforedescending into the new position. The work doneto lift the atom over the barriers is not recoveredbut rather lost in heat, phonons and other ener-getic activities. An energy balance between sourcesand sinks occurs such that it extremizes the func-tional (Needleman and Rice, 1980; Svoboda andTurek, 1991; Svoboda and Riedel, 1995; Sun et al.,1996)

P _Gs Rs; 1where _Gs is the rate of change of the free energy ofthe system and Rs is one-half of the rate of energydissipation.

The dissipation is such that

Rs 12

ZAb

1

Dbjb jb dAb

1

2

ZAs

1

Dsjs js dAs; 2

where Ab and As are the grain boundary and sur-face areas respectively, jb and js are the fluxes ofmaterial on the grain boundary and on the free

44 F. Parhami et al. / Mechanics of Materials 31 (1999) 4361

surface respectively, Db and Ds are diusion pa-rameters given by

Db dbDbXkT ; 3

Ds dsDsXkT ; 4

where Db and Ds are the grain boundary andsurface atom diusivities respectively, db and ds arethe thicknesses within which diusion occurs onthe grain boundary and the surface respectively, Xis the atomic volume, k is Boltzmans constant andT is the absolute temperature. The fluxes j aredefined as the volume of material passing by dif-fusion through unit length in unit time.

The rate of change of the free energy of thesystem is the rate of change of the internal energyminus the external work rate. In the system underconsideration, internal energy is the sum of surfaceand grain boundary energies. Therefore

_Gs cs _As cb _Ab Fv; 5

where cs and cb are the surface and grain boundaryenergies per unit area respectively, F is the appliedforce and v is the velocity of one end of the row ofparticles relative to the other.

It can be demonstrated that the functional Phas a stationary minimum value with respect tocompatible variations of jb, js, _As, _Ab and v, andthat this leads to the appropriate governingequations for the mass transport problem on theactual free surface and grain boundary of the rowof particles (Needleman and Rice, 1980; Svobodaand Turek, 1991; Svoboda and Riedel, 1995; Sunet al., 1996). This is demonstrated in Appendix A.Consideration of the governing dierential equa-tions of the problem is found in Herring (1951),Asaro and Tiller (1972) and Chuang et al. (1979).

2.2. RayleighRitz method

Since a minimum principle is available, it can beused to obtain approximate solutions to theproblem using a RayleighRitz approach. Simplebut robust representations of the evolving shape ofthe row of particles can be used which depend ononly a few parameters of shape. In our calcula-tions, we have used the shape shown in Fig. 1(c) oftruncated spheres joined by discs. For a periodicrow of particles, this shape depends on only fourindependent parameters. The grain boundary is in

Fig. 1. (a) A row of spherical particles in contact. (b) Repre-

sentation of unit tangent vectors to the grain boundary and

particle surfaces at the neck and the velocity of the neck. (c) A

unit problem representing the geometry of a row of particles

during neck formation.

F. Parhami et al. / Mechanics of Materials 31 (1999) 4361 45

the middle of the disc. (A simpler shape of twotruncated barrels without the disc at the contactcould be used as invoked by Sun et al. (1996) intheir study of evolution of grain shape in cylin-drical lines. However, the initial behavior of verysmall contacts with undercutting is thought to bebetter represented by the presence of the disc.) Thecalculations are initiated with the disc absent (i.e.t 0) and the disc is allowed to appear or disap-pear according to the evolution of the numericalresults. The rates of change of the shape parame-ters become the degrees of freedom of the varia-tional functional P. However, the rates of changeof the shape parameters must preserve compati-bility and volume. For the periodic system shownin Fig. 1(a), this will eliminate one parameter sothat P depends finally on three independent de-grees of freedom. Minimizations of P with respectto these three parameters then provides threecoupled linear equations for the rates of change ofthe shape parameters which can be solved toprovide values for them. These rates of change ofshape parameters are then used for a finite timeincrement to update the shape of the system. Thisprocess is repeated so that the evolution of theshape in time can be computed incrementally. Inthis way, approximate solutions can be obtainedwith modest amounts of computation.

Note that the chosen shape in Fig. 1(c) doesnot conform to the dihedral angle at the neck. Inthe solution, the dihedral angle at the neck maynever be satisfied in the evolution of the simpleshapes used in the computation. The variationalmethod, however, produces the best overall shapewithin the class of shapes permitted by the de-grees of freedom employed. Thus, a given localcondition such as dihedral angle may not be sat-isfied but enforced only in the weak variationalsense. On the other hand, the vector fields for thefluxes js and jb on the particle surfaces and on thegrain boundary must be computed compatiblywith the rate of change of shape. The diusionfluxes on the surface and the contact obeyEqs. (A.4) and (A.5) for the chosen shape and theflux of material at the neck must be equal to thesums of the fluxes on the particle surfaces at theneck (Eq. (A.6)). Furthermore, the periodic sys-tem of Fig. 1(a) requires that no material enters

or leaves the unit geometry shown in Fig. 1(c); i.e.the surface fluxes at the top and bottom of theunit geometry are zero. This also means that thevolume of the unit cell shown in Fig. 1(c) staysconstant during the evolution of the shape. Theseconditions arising from the requirements ofcompatibility are essential conditions in the ter-minology of variational methods.

A further point is that the chosen shape is ca-pable of representing special configurations, themost notable being when the disc at the contact isabsent, i.e. t 0 (see Fig. 2(a) and (c)). The dihe-dral angle at the neck can then be representedexactly. The feature that in equilibrium, the grainboundary, which is taken to be flat, bisects thedihedral angle is not always satisfied. However,when the two particles are of equal size, all con-ditions can be met and this occurs in the equilib-rium condition depicted in Fig. 2(c). Anotherspecial shape which can be represented is a longcylindrical line when the neck disc absorbs thetruncated spheres as shown in Fig. 2(b).

We define a unit problem as shown in Fig. 1(c).In this figure, F is the applied force, a and b theradii of the larger and smaller particles respec-tively, t the thickness of a circular disc between thetruncated spheres, representing neck formation, Land h the distances between the circular disc andthe center of the larger and smaller particles re-spectively and x the radius of the circular disc. It isassumed that throughout the process, the initiallyspherical surfaces remain spherical but their radiimay grow or shrink. Similarly, the circular discmay thicken or get thinner and may increase orreduce its radius. The problem is therefore axi-symmetric.

The aim of the method is to find the rate ofchange of the independent degrees of freedom, a, band x in terms of geometric variables, the appliedforce, and material properties. The parameters ofthe model will be given in terms of the response ofthe unit cell of Fig. 1(c). From the geometry, therelative velocity vb across the grain boundary isvb _L _h _t. The compatibility condition be-tween the flux and relative velocity, see Eq. (A.5),can then be integrated to give

jq q2 _L _h _t; 6


where q is the radial distance in the grain boun-dary from the axis of symmetry and jq, which mustbe zero at q 0, is the radial flux in the grainboundary. Therefore, the virtual dissipation rate inthe grain boundary isZAb

1

Dbjb djb dAb 2p

Zx0

qDb

jqdjq dq

px4

8Db _L _h _td _L d _h d_t: 7

On the surface of the larger particle, the normalvelocity vn _a and Eq. (A.4) becomes

dj/d/ j/ tan / _aa; 8

where / is defined in Fig. 1(c) and j/ is the azi-muthal flux on the surface. Subject to j/ 0 at/ 0, integration of Eq. (8) givesj/ _aa tan /: 9Therefore the sum of the variations of the dissi-pation rate on the surfaces of the small and largeparticles isZAspheres

1

Dsjs djs dAs

2pa4

Ds

Zarctan L=x0

tan2 / cos / d/ _ad _a

2pb4

Ds

Zarctan h=x0

tan2 / cos / d/ _bd _b

2pDsa4ga _ad _a b4gb _bd _b; 10

where ga and gb are given by

ga Ln ax 11 x

a

2r !" # L

a; 11

gb Ln bx 11 x

b

2r !" # h

b: 12

On the free surface of the circular disc,Eq. (A.4) gives

djzdz _x; 13

where z is the direction perpendicular to the grainboundary. The grain boundary is taken to be po-sitioned at z 0. At z t/2, where the disk meetsthe large truncated sphere, the flux jz must be equalto j/. At this position / arctan L/x. Integra-tion of (13) for z > 0 therefore gives

jz _aa Lx _x zt2

z P 0: 14

The equivalent result for z6 0 is

jz _bb hx _x zt2

z6 0: 15

Fig. 2. (a) Configuration resulting from elimination of the cir-

cular plate. (b) Configuration resulting from elimination of

spherical surfaces. (c) Equilibrium configuration of a row

identical spherical particles.


The values of jz in Eqs. (14) and (15) generallydier at z 0. The dierence is equal to the fluxaway from the grain boundary. Enforcement ofthis condition will be taken care of below. Thevirtual dissipation rate on the disc surface isthereforeZAdisc

1

Dsjs djs dAs

2pxDs

Zt=20

_aaLx _xt

2

_xz

!

aLd _ax t

2d _x zd _x

!dz

2pxDs

Z0t=2

_bbhx _xt

2 _xz

!

bhd_b

x t

2d _x

zd _x

!dz

2pxDs

_aa2L2t2x2

_xaLt2

8x

!d _a

2pxDs

_bb2h2t2x2

_xbht2

8x

!d _b

2pxDs

_aaLt2

8x

_bbht2

8x _xt

3

12

!d _x: 16

It follows that dRs is the sum of Eqs. (7), (10) and(16).

To compute d _Gs we need to express the surfaceand grain boundary areas in terms of the geo-metric parameters identified above:

As 2paL 2pbh 2pxt 17and

Ab px2: 18Therefore

csd _As 2pcsad _L Ld _a bd _h hd _b xd_t td _x19

and

cbd _Ab 2pcbxd _x; 20whereas

F dv F d _L d _h d_t: 21

The result for d _Gs is computed from Eq. (5). TheRayleighRitz minimization is achieved by settingdP d _Gs dRs 0: Since dP must be zero forarbitrary variations of d _a; d _b; d _L; d _h; d_t and d _x, thecoecients of these six quantities in dP are set tozero. The resulting expressions are coupled linearequations for _a; _b; _L; _h; _t and _x having the form

k^h i

_^dn o

f^n o

; 22

where f _^dg are the rates of change of the six degreesof freedom. This vector plus the coecient matrix(termed the viscosity matrix by Sun et al., 1996)and the conjugate force vector are given in Ap-pendix B.

The set of variables used above are not inde-pendent. By imposing volume conservation andthe geometrical constraints given below, the setcan be reduced to three independent parameters.Geometric considerations require

L2 a2 x2 23and

h2 b2 x2: 24Volume conservation gives

t 1x2

V0p a2L 1

3L3

b2h 1

3h3

; 25

where V0 is the volume of two touching hemi-spheres representing the initial state as shown inFig. 1(a). The constraints in incremental form canbe written as

_^dn o

C _dn o

; 26

_dn o

_a_b

_x

8>:9>=>;; 27

where _a; _b and _x are the chosen independent de-grees of freedom. The constraint matrix [C] isgiven in Appendix C. With these constraints im-posed, the sum of the fluxes determined fromEqs. (14) and (15) at z 0 and Eq. (6) at q xequals zero, ensuring that continuity of flux wherethe grain boundary meets the free surface,Eq. (A.6), is satisfied.


The coupled linear equations for the indepen-dent variables are then

kf _dg ff g; 28where

k CTk^C 29and

ff g CTff^ g: 30The components of the coecient matrix k andthe generalized force ff g, which drive sinteringand coarsening, are given in Appendix D.

3. Numerical Procedures

3.1. Initial formation of interparticle contact

At each increment of time, _a; _b and _x are cal-culated from Eq. (28) and are used to update a, band x using Euler integration. The initial condi-tions are a a0, b b0 and x 0. It was found thatassigning very small values for the magnitude of xresults in instability of the numerical scheme. Thefollowing asymptotic approach is used to circum-vent this problem.

In the limit where x/a, x/b and t/x are muchsmaller than 1 and higher order terms in Eq. (28)are neglected, the following formulation results:

_x 4Dbgagb Dsga gb 2cs cbgagbxt2

4DbFpx3t

;

31

_a Ds2cs cbgata2

tDbF6px3a2ga

; 32

_b Ds2cs cbgbtb2

tDbF6px3b2gb

; 33

t 1x2

V0p 2

3a3 b3 x

4

4

1

a 1

b

: 34

It can be seen from Eqs. (11) and (12) that ga andgb are logarithmically divergent when x goes tozero. This could be used to simplify further theexpressions in Eqs. (31)(34). However, the loga-

rithmic singularities are so weak that it is prefer-able to retain the form as given, except for the firststep of the calculation. In this step 1/ga and 1/gbare considered negligible and therefore

_x 4Db2cs cbxt2

4DbFpx3t

: 35When _a= _x is computed using Eqs. (32) and (35), itis found that the ratio is always small unless Db isnegligible. We will always take Db to be compa-rable to Ds, so _a= _x can be taken to be zero for thefirst step. A similar result prevails for _b= _x, so in thefirst step of the calculation, _a _b 0 is used. Thismeans that throughout this step, a a0 and b b0and thus Eq. (34) simplifies to

t x2

4

1

a0 1

b0

: 36

Eq. (35) then integrates to give

x6 96Db 42cs cb1=a0 1=b02"

Fp1=a0 1=b0

#Te;

37where Te is the time elapsed since the particles firsttouched. When a0 b0, F 0, and the dihedralangle is 180 (cb 0), Eq. (37) is Cobles freesintering result (Coble, 1958). Therefore, we canregard Eq. (37) as a generalization of Coblesanalysis. Eq. (37) is used to compute x at the endof the first time increment and thereafterEqs. (31)(34) are used until x/a reaches a value of0.01. Eq. (28) is used thereafter with the expres-sions in Appendix D utilized for the matrix ofcoecients and the right-hand side array.

3.2. Elimination of the circular disc

At some point, the numerical simulationreaches a situation where the circular disc (i.e. t)disappears. This configuration usually arises latein the sintering process. The formulation must beadjusted for this case. After the disc disappears,there are two independent variables a and b asshown in Fig. 2(a). Furthermore, since t is zeroand remains zero, _t is also zero. The constraintequations are then


_^dn o

C _a_b

( )38

with C as given in Appendix E. The governingequations are thus

k _a_b

( ) f 1

f 2

( ); 39

where

k CT k^h iC 40

and

f 1f 2

( ) CTff^ g: 41

When k^ and ff^ g are evaluated in this case, t 0.The components of k and ff g are given in Ap-pendix E. After each increment, Eq. (28) is used tocheck whether the central disc will grow. If theresults show that _t6 0, the calculations are con-tinued with t 0 using Eq. (39). If _t > 0, the cal-culations are continued using Eq. (28).

3.3. Elimination of the spherical surfaces

In some cases, the configuration becomes acylinder due to the circular disc thickening to re-place the spherical surfaces which disappear. Thisconfiguration is depicted in Fig. 2(b) and results in

_x 12DsDbpcsxt pcbx2 Ft

Dbpx2t3 3Dspx3t2 42

with

t V0=px2; 43since only Eqs. (7), (16) and (19)(21) contributeto the result with x and t the only nonzero terms.This configuration is reached when the dihedralangle is large or when large compressive forces areapplied. It provides the metastable equilibriumshape for cb 0. When the original particles areidentical, this configuration, as shown in Fig. 2(b),can approximate the equilibrium configuration ofFig. 2(c). When the particles are not initiallyidentical, the cylindrical configuration approxi-mates the coarsened stage after the smaller particlehas completely disappeared.

4. Results and discussion

4.1. Free sintering of rows of identical particles

Fig. 3 shows the growth of the interparticlecontact radius x vs. time for various dihedral an-gles and Db=Ds 1: The shape of the unit geom-etry in the two main stages is shown. Note thatsomewhere between x=a0 101 and x=a0 1 thecircular disc disappears. The dips in our numericalresults are associated with this disappearance. Thedevelopment and disappearance of the circulardisc is similar to the development and disappear-ance of the undercutting at the neck in more exacttreatments. The numerical results are somewhaterratic at the beginning, corresponding to thetransition from the asymptotic formulation (i.e.Eqs. (31)(34)) to the full set of equations (i.e.Eq. (28)). This occurs when the neck is so smallthat the nonsmooth feature of the results here is ofno importance.

Also shown in the figure are Cobles analyticalsolution (Coble, 1958) and the numerical results ofSvoboda and Riedel (1995), Zhang and Schneibel(1995) and Bouvard and McMeeking (1996). TheCoble solution is as given in Eq. (37) with b0 a0,F 0 and cb 0. It is evident that early in theprocess our results agree very well with Coblesanalytical solution. Later, our results deviate fromCobles solution and reach an equilibrium config-

Fig. 3. Contact radius vs. time for the free sintering of a row of

identical particles for various dihedral angles.


uration. In this configuration, the driving force inEq. (39) is zero. By setting f 1 to zero with a b,hL and F 0, it is calculated that

x a sin W2; 44

where it should be recalled that cb 2cs cos w/2.The value of x in Eq. (44) is the exact value for theneck radius at equilibrium (Lange and Kellett,1989) as shown in Fig. 2(c), i.e. xeq. Volume con-servation can be used to show that, in terms of theinitial particle radius

xeqa0 sin

W2

32cos W

2 1

2cos3 W

2

1=3 : 45The relevant results for xeq/a0 for each dihedralangle are indicated in Fig. 3.

It can be seen that there is good agreement andthe simple model results reach the equilibriumconfiguration. Note that even though Cobles so-lution (Coble, 1958) is for the case of W 180, itshows good agreement with our results in the earlyand intermediate stage of sintering regardless ofthe dihedral angle. Presumably, this is due to aweak dependence of x on the dihedral angle, assuggested by the asymptotic result of Eq. (37).Only later in the process, and during the evolutionof the equilibrium configuration, does the dihedralangle W become important. Our results also showgood agreement with the numerical solutions ofSvoboda and Riedel (1995) and Bouvard andMcMeeking (1996) which are calculated forW 120 and Db=Ds 1 and W 130 andDb=Ds 2, respectively. When plotted on a loglog scale as in Fig. 3, it is dicult to discern anydierence between the results of Svoboda andRiedel (1995) and Bouvard and McMeeking (1996)despite the dierences in dihedral angles and sur-face diusion coecient. The dierence betweenour results and the results of Zhang and Schneibel(1995) calculated for W 150 and Db=Ds 1 ispresumably because their simulation is performedon a pair of prisms in contact and not a row ofparticles. The discrepancy is most notable in theequilibrium configuration. Thus the agreementwith more thorough numerical eorts is remark-able given that our calculations were done with a3-parameter shape approximation.

Fig. 4 shows the center-to-center distance be-tween neighboring particles for various dihedralangles vs. time. The results have been comparedwith Cobles prediction (Coble, 1958) and the re-sults of Zhang and Schneibel (1995) and Bouvardand McMeeking (1996). The agreement betweenour results for W 120 and those of Bouvard andMcMeeking (1996) for W 130 is very good. Asbefore, it is likely that the dierence between ourresults and those of Zhang and Schneibel (1995)calculated for W 150 is because their simulationrepresents the sintering of a pair of prisms and oursimulation represents sintering of a row of parti-cles. A result due to Coble (1958) is also shown inFig. 4. It can be obtained from Eq. (23) witha ao and x small compared to a0. When com-bined with t from Eq. (36), it gives, to first order,the center-to-center spacing

2L ta0

2 12

xa0

2: 46

This predicts the shrinkage poorly when it firstbecomes significant, but it later gives reasonableagreement with our results until equilibrium isapproached. The value predicted in our results forthe center-to-center distance at the equilibriumconfiguration is the exact result as calculated ac-cording to Fig. 2(c). This gives a prediction of

2Leqa0 2 cos

W2

32cos W

2 1

2cos3 W

2

1=3 ; 47

Fig. 4. Center-to-center distance vs. time for the free sintering

of a row of particles for various dihedral angles.


which is indicated for each dihedral angle inFig. 4. Our results compare well with the moreexact value shown in Fig. 4.

Fig. 5 shows how the radius of the particlevaries with time. This figure shows that initially theparticle radius does not change significantly, indi-cating that during the early stages of sintering,neck formation and shrinkage is mainly dominat-ed by grain boundary diusion. Surface diusion,which aects the radius, only becomes importantlater in the process. The results confirm Coblesassumption (Coble, 1958) that in the early stagethe particle radius does not change. This result isconsistent with the analysis that led to the result ofEq. (37). The reason for the dip in the plot ofparticle radius vs. time for W 150 is that in thiscase, the growth of the circular disc dominates thelater stages of sintering. As the disc grows, bothradially and axially, the radius of the sphericalregion gradually reduces until the radius of thedisc, x, equals the particle radius a. This occurs ata radius corresponding to the minimum of Fig. 5.At this point the spherical region disappears andthe shape of the particle is represented by a cyl-inder. In order to complete Fig. 5 we now interpretthe radius of the cylinder as the radius of theparticle. This radius gradually increases as theparticle spreads out radially and reduces in length.Close to the equilibrium configuration the spheri-cal region reforms at the center of the particle andspreads along its length. If we ignore the redevel-

opment of this spherical region a pseudo equilib-rium cylindrical profile is achieved when, a/a0 x/a0 1.37, which is very close to the true equilib-rium value of 1.38 indicated in Fig. 5 determinedfrom

aa0 1

32cos W

2 1

2cos3 W

2

1=3 : 48The results from the numerical simulation for theparticle (or disc) radius in the equilibrium config-uration for the other values of W are in exactagreement with the predictions of Eq. (48), whichare indicated on Fig. 5.

4.2. Free sintering of rows of dierent size particles

Fig. 6 shows the variation of the interparticleneck radius between two dierent size particleswith time for dierent ratios of initial particle ra-dii. These simulations are performed for W 120and Db=Ds 1. The shape of the unit geometry inthe two main stages is shown. The circular discdisappears somewhere between x/a0 101 and x/a0 1 and the dip in the plot is associated with thisdisappearance. It can be seen that the interparticleneck radius reaches a maximum level. Until thismaximum, the process is dominated by sintering,which causes neck growth. After the maximum,surface diusion dominates and the coarseningprocess results in the disappearance of the smallerparticle. The calculation was not continued there-after, but if this had been done, all three caseswould have evolved to the equilibrium configura-tion for the remaining equal sized particles. This isgiven by

xa0 1 b0=a0

3

3 cos W2 cos3 W

2

" #1=3 sin W2; 49

which indicates that further rearrangement wouldoccur in each case after the small particles disap-pear.

Also shown in Fig. 6 is Cobles analytical so-lution (Coble, 1958), as given in Eq. (37) witha0 b0, F 0 and W 180. Our results for b0/a0 0.9 are very close to Cobles predictionshowing that initially, neck growth does not de-pend strongly on the initial ratio of particle radii.

Fig. 5. Particle radius vs. time for the free sintering of a row of

identical particles for various dihedral angles.


Late in the process, however, the radius of thesmaller particle influences the neck growth. De-creasing the ratio b0=a0 results in a smaller maxi-mum value of x and the earlier disappearance ofthe smaller particle.

Fig. 7 shows the particle radii vs. time for dif-ferent initial ratios of particle sizes. Initially, theparticle sizes change gradually but later the smallerparticle shrinks rapidly and disappears while thelarger particle continues to grow gradually. Thesmaller particle has a finite radius when it van-ishes. It is eliminated by the grain boundaries oneither side fusing together, i.e. h 0. The smallerthe second particle the sooner it disappears. Fig. 8shows how the change in the particles center-to-center spacing varies with time. Cobles analyticalresult (Coble, 1958), as given in Eq. (46) for thecase a0 b0 is shown also. Our simulation forb0=a0 0.9 is similar to, but slower than, Coblesresult. Our results indicate that shrinkage is con-tinuous once it becomes significant and occursduring coarsening as well as earlier in sintering.However, the results indicate that diusional re-arrangement of material around the larger particleis the rate controlling mechanism for this process.

Eect of diusivity ratio: Fig. 9 shows thegrowth of the interparticle neck radius for an ini-tial size ratio of 0.7, W 120 and various diusi-vity ratios (Db=Ds). The figure shows that fastersurface diusivity results in the formation of larger

interparticle necks sooner. The results also indicatethat the maximum value of x=a0 is reached earlierand the neck disc is eliminated sooner when Db=Dsis low. This eect is clearer in Fig. 10 where theparticle radii are plotted vs. time for various ratiosof Db=Ds. It can be seen in this figure that thedisappearance of the smaller particle occurs soonerwhen grain boundary diusion is slow. Fig. 11 is aplot of the particles center-to-center distance vs.time for various values of Db=Ds. This figureshows that shrinkage occurs faster when grainboundary diusivity is low. In these plots the timehas been normalized in terms of the grain-boun-dary diusivity. Thus decreasing Db=Ds, results inan increase in the rate of surface diusion, allow-ing the material to redistribute around the parti-cles quicker. All the above observations areconsistent with a net increase in mobility of thematerial.

Eect of dihedral angle: Fig. 12 shows thegrowth of the neck radius for an initial size ratio of0.7 and Db=Ds 1 for various dihedral angles (W).This figure shows that a large dihedral angle re-sults in a high maximum value for the neck radius.When the particle sizes are unequal, a large dihe-dral angle causes early elimination of the smallparticles. As before, when W 150 the growth ofthe circular disc dominates the process and thespherical surfaces disappear. In this case theevolved shape (a cylinder) is an approximation of

Fig. 7. Particle radii vs. time for the free sintering of a row of

particles for various initial size ratios.


particles for various initial size ratios.


the stage where the particles have coarsened andequilibrium of the remaining equal sized particlesis reached. In this problem, the total number ofparticles remains constant, while, for smaller val-ues of W the total number of particles decreases asthe larger particles grow at the expense of thesmaller ones. This result is a direct consequence ofthe way in which we have idealized the micro-structure. In our formulation of the cylindricaldisc, we assume that the grain-boundary is locatedat the center of the disc. As soon as the second

spherical region disappears, the geometry is sym-metric and no further transfer of material canoccur between the two grains. In practice, coars-ening should continue in the same way as is ob-served for the other values of W considered inFig. 12. Thus, for W 150, our idealization over-constrains the evolution process. Additional de-grees of freedom are required to more accuratelymodel the system response. This can readily beachieved by replacing the disc by one or two

Fig. 10. Eect of grain boundary and surface diusivity ratio on

the free sintering of a row of particles with dierent sizes.

Particle radii vs. time.



Center-to-center distance vs. time.

Fig. 8. Center-to-center distance vs. time for the free sintering

of a row of particles for various initial size ratios.



Contact radius vs. time.


conical sections. Further consideration of this isbeyond the scope of this paper. It is important tonote, however, that a sucient number of degreesof freedom and idealization of the microstructureneeds to be adopted to allow the major features ofthe system response to be modeled. The fewer thedegrees of freedom the more thought that must begiven to the way in which the microstructure islikely to evolve.

Fig. 13 confirms the results of Fig. 12 byshowing that the smaller particle is eliminatedearlier as the dihedral angle increases. In fact, thesharp decrease of the neck radius is a consequenceof the disappearance of the smaller particle.Fig. 14 shows the influence of the dihedral angleon the way in which the center-to-center distancechanges with time. As before, only the early stagesof the result for W 150 can be considered to bereliable.

4.3. Pressure assisted sintering and coarsening

We now consider the influence of a superim-posed compressive force on the evolution process.Fig. 15 is a plot of neck radius vs. time for thesituation where all the particles are the same sizewith a dihedral angle of 120. This figure showsthat the application of a compressive force causesthe interparticle necks to grow quicker and to

eventually achieve a larger final size. For F/cs a0 0 and )10 the constrained equilibriumconfiguration consists of a row of truncatedspheres while for F=cs a0)100 the final config-uration is a row of cylinders. Perturbation of theseshapes resulted in no further evolution of micro-structure. These shapes therefore represent theminimum energy configurations within the limitedclass of profiles considered. The value for thecontact radius at equilibrium is marked on thefigure and can be found by setting the driving forceto zero (i.e. setting f 1 f 2 0 in Eqs. (E.12) and(E.13) for the spherical shapes and _x 0 inEq. (42) for the cylindrical shape). Fig. 16 shows aplot of the particles center-to-center distance(2L + t) vs. time. The final distance, labeled 2Leqin Fig. 16, is the equilibrium height of the unit cell.This can be computed from volume conservationand the final values for x/a0 are shown in Fig. 15.

Whereas the magnitudes of our results aresensible for the class of shapes considered, thetheoretical equilibrium shape of the particle sur-faces under a compressive force are curved, butnot spherical. Cannon and Carter (1989) haveprovided an analysis of the equilibrium shape, al-though they focus their attention on problems withtension applied rather than compression. Theformulae given by Cannon and Carter (1989) are,however, inconsistent with those in the paper by

Fig. 13. Particle radii vs. time for the free sintering of a row of

particles with an initial size ratio of 0.7 for various dihedral

angles.


particles with an initial size ratio of 0.7 for various dihedral

angles.


Carter (1988) serving as a source. After correctionto be consistent with Carter (1988) (i.e. in Cannonand Carter (1989) replace sin W/2 by )cos W/2 inEqs. 7(a) and 7(b)), Eq. 7(a) and (8) in Cannonand Carter (1989) can be used to compute pa-rameter values consistent with the compressiveforces used to obtain the results in our Figs. 15and 16. These parameters are then used in theirEq. 7(b) to evaluate L at equilibrium and in theirEq. (6) to give x at equilibrium. The results forW 120 are x/a0 1.534 and 2Leq/a0 0.533 forF/csa0)10 and x/a0 2.583 and 2Leq/a0 0.197for F/csa0)100. The final neck and particle

length shown in Figs. 15 and 16 are within 2% ofthese exact equilibrium values, thus demonstratingthat the simple particle shapes assumed here pro-vide a good approximation of the general featuresof the evolving microstructure.

In our simulation in which the particles areinitially of dierent radii we find that the centraldisc dominates the evolution process. As with thefree sintering situation considered in Section 4.2this configuration does not adequately reflect themajor features of the microstructure that is likelyto evolve. We therefore do not consider this situ-ation further here.

5. Closure

Evolution of identical spherical particles into anequilibrium configuration after sintering is pre-dicted using a simple numerical method. Resultsshow good agreement with Cobles analyticaltreatment (Coble, 1958) of the problem for earlyand intermediate configurations. The numericallypredicted parameters for the equilibrium shape arein good agreement with the analytical solutions.The simulation also predicts the sintering of dif-ferent size particles followed by the coarseningprocess as the large particle absorbs the smallerone. The ratio of initial particle sizes influences theprocess with a smaller value resulting in a smallermaximum size and an earlier disappearance of the

Fig. 16. Eect of compressive force on the sintering of a row of

particles with initial size ratio of 0.7. Center-to-center distance

vs. time.Fig. 14. Center-to-center distance vs. time for the free sintering

of a row of particles with an initial size ratio of 0.7 for various

dihedral angles.

Fig. 15. Eect of compressive force on the sintering of a row of

particles with initial size ratio of 0.7. Radius of contact vs. time.


neck. High grain boundary diusivity results in thelater disappearance of the smaller particle andslower shrinkage. A large dihedral angle leads tohigher maximum values for the neck radius andthe earlier elimination of the smaller particles. Theapplication of a compressive force results in fasterneck growth and shrinkage. In addition, a largerneck radius and a smaller center-to-center distanceat equilibrium occurs compared to free sintering.

Acknowledgements

This research was supported by NSF GrantDMR 9114560 to the University of California,Santa Barbara.

Appendix A. Weak vs. strong form of the governing

equations

The compatibility condition on the rate ofchange of surface area in Eq. (5) is

_As ZAs

jvndAs Xall b

Zneck

vneck s1 s2 dL; A:1where j is the sum of the principal curvatures of theparticle surface defined so that positive curvaturedenotes convexity. The velocity vn on As is the rateof motion of the particle surface in the outwardnormal direction. The second contribution to _As inEq. (A.1) comes from the motion of the locus ofpoints of connection between the grain boundaryand free surface of neighboring particles, which cancreate or destroy surface area. The rate of motionof each point is vneck and the unit tangent vectors tothe surfaces which attach to this point are s1 and s2,as shown in Fig. 1(b). The infinitesimal quantitydL is the length of a line element around the edge ofthe neck and the integral is taken around the neck.An integral is computed for each grain boundaryand summed as indicated where all b means sumover all grain boundaries.

Similarly, the rate of change of grain boundaryarea arises from the motion of the neck edge (as-suming that the grain boundary is flat or is sta-tionary within the system) and therefore

_Ab Xall b

Zneck

vneck sb dL; A:2

where sb is the tangent unit vector to the grainboundary at the neck, as shown in Fig. 1(b). Giventhat the particle interior is rigid, the relative ve-locity v of one end of the row compared to theother is composed of the sum of the relative ve-locities vb across all grain boundaries in the row ofparticles. Furthermore, the force F is equal to theintegral of the stress over each grain boundary.Therefore

Fv Xall b

ZAb

r dAbvb: A:3

Compatibility and volume conservation then re-quires that on As

vn $s js 0; A:4where $s is the gradient operator on the particlesurface. Similarly, on each grain boundary (whichare assumed to be flat)

vb $b jb 0; A:5where $b is the gradient operator in the grainboundary. Finally, flux continuity at the necksrequires

s1 js1 s2 js2 sb jb 0; A:6where js1 is the flux at the neck on the particle withtangent s1 as shown in Fig. 1(b) and js2 is the fluxat the neck on the particle with tangent s2.

Now let variation of js, jb, _As, _Ab and v occuralong with variations of vn, v

neck and vb, but suchthat compatibility (Eqs. (A.1)(A.6)) is obeyed forthe varied quantities. It follows that to first order(Needleman and Rice, 1980; Sun et al., 1996)

dP P js djs; jb djb; _As d _As; _Ab

d _Ab; v dv;

P js; jb; _As; _Ab; v

csd _As cbd _Ab

F dvZAb

1

Dbjb djb dAb

ZAs

1

Dsjs djs dAs:

A:7


Use of Eqs. (A.1) and (A.2) in terms of the vari-ations then gives

dP ZAs

csjdvn 1

Dsjs djs

dAs

Xall b

Zneck

cs s1 s2 cbsb dvneck dL

Xall b

ZAb

rdvb 1Db

jb djb

dAb: A:8

Eq. (A.4) is then used to replace dvn by $s : djsand Eq. (A.5) allows dvb to be replaced by$b : djb. This permits dP to be restated as

dP ZAs

cs$s jdjb cs$sj

1Ds

js

djs

dAs

Xall b

Zneck


Xall b

ZAb

$b rdjb 1

Dbjb

$br

: djb

dAb:

A:9

Use of the divergence theorem on As and on Abthen provides

dP Xall b

Zneck

cs j1s1 djs1 j2s2 djs2

rsb djb dLZAs

cs$sj 1

Dsjs

djs dAs

Xall b

Zneck


Xall b

ZAb

1

Dbjb $br

djb dAb; A:10

where the subscripts on j denote the dierentparticles meeting at L. Finally, use of Eq. (A.6)gives

dP Xall b

Zneck

csj1 r s1 djs1

csj2 rs2 djs2

dL

Xall b

Zneck


ZAs

cs$sj 1

Dsjs

djs dAs

Xall b

ZAb

1

Dbjb $br

djb dAb: A:11

This shows that for arbitrary compatible varia-tions from the problem solution, dP is zero. This istrue because the pointwise governing equations(Herring, 1951; Asaro and Tiller, 1972; Chuang etal., 1979) are: the flux conditions on the free sur-face and grain boundaries,

js Dscs$sj on As; A:12

jb Db$br on Ab A:13

the continuity of chemical potential at all junctionsbetween particle surface and grain boundary,

csj1 csj2 r on L A:14

and the dihedral angle is enforced at the neck,

cs s1 s2 cbsb 0 on L: A:15

(The latter can only be true if sb bisects the anglebetween s1 and s2 and this angle is the dihedralW 2 arcos cb=2cs). The conditions representedby Eqs. (A.14) and (A.15) are enforced by ex-tremely rapid local motion of material, so thateven if an initial shape does not satisfy theseequations instantaneously, mass transport en-forces them locally. Thereafter, the shape evolu-tions occurs with Eqs. (A.14) and (A.15)satisfied.

It is also readily shown that the functional P isminimized. Let the exact solution of the problem


be denoted by, js, jb, _As; _Ab and v. Since thefunctional P is stationary for these values,

P js djs; jb djb; _As d _As; _Ab d _Ab; v

dv

P js; jb; _As; _Ab; v

12

ZAb

1

Dbdjb djb dAb

12

ZAs

1

Dsdjs djs dAs A:16

to second order. Since Ds and Db are both posi-tive, the right-hand side is always greater than orequal to zero and therefore P is minimized by theexact solution.

Appendix B. Governing equations for a row of

bidiametrical spheres in terms of six degrees of

freedom

The governing equations for the six degrees offreedom _a; _b; _L; _h; _t and _x can be stated in matrixform as

k^h i

_^dn o

f^n o

; B:1where

_^dn o

_a_b_L_h_t

_x

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;B:2

and

_^fn o

2pcsL2pcsh2pcsa F2pcsb F2pcsx F2pcst 2pcbx

8>>>>>>>>>>>>>>>:

9>>>>>>>>=>>>>>>>>;: B:3

The nonzero entries in the coecient matrix are

k^11 pa2

xDs2xa2ga L2t

; B:4

k^16 paLt2

4Ds; B:5

k^22 pb2

xDs2xb2gb h2t

; B:6

k^26 pbht2

4Ds; B:7

k^33 k^34 k^35 k^43 k^44 k^45 k^53 k^54 k^55 px

4

8Db; B:8

k^61 paLt2

4Ds; B:9

k^62 pbht2

4Ds; B:10

k^66 pxt3

6Ds: B:11

Appendix C. Constraint equations to reduce six

degrees of freedom to three independent parameters

With_^dn o

defined as in Appendix A, the con-straint equations are

_^dn o

C_a_b

_x

8>:9>=>; C:1

with the nonzero terms of [C] as follows:

C11 C22 C66 1; C:2

C31 aL ; C:3

C33 C43 xL ; C:4

C42 bL ; C:5

C51 2aLx2 aL; C:6

C52 2bhx2 bh; C:7


C53 xLxh 2t

x: C:8

Appendix D. Governing equations in terms of three

independent degrees of freedom

The symmetric coecient matrix for Eq. (28) isgiven by

k11 a2L2 txDs 1

2Db

2Ds

a4ga; D:1

k12 k21 aLbh2Db

; D:2

k13 k31 aLxt 12Db t

4xDs

; D:3

k22 b2h2 12Db t

xDs

2Ds

b4gb; D:4

k23 k32 bhxt 12Db t

4xDs

; D:5

k33 x2t2

2

1

Db t

3xDs

D:6

with Eqs. (27) and (28) repeated for convenienceas

ga Ln ax 11 x

a

2r !" # L

a; D:7

gb Ln bx 11 x

b

2r !" # h

b; D:8

The generalized force vector is given by

f1 4csaLx 1x

2a xa

2L2 x

2

2L2

2aL

px2F ; D:9

f2 4csbhx 1x

2b xb

2h2 x

2

2h2

2bh

px2F ; D:10

f3 2csxaL b

h x

L x

h t

x

2cbx

2tpx

F :

D:11

Appendix E. Constraint matrix after the circular

disc has vanished reducing six degrees of freedom to

two independent parameters

With_^dn o

defined as in Appendix B, the con-straint equations are

_^dn o

C _a_b

( )E:1

with the nonzero terms of C as follows:C11 C22 1; E:2

C31 a x2 2hL

x2h L ; E:3

C32 b b2 h2

x2h L ; E:4

C41 a a2 L2

x2h L ; E:5

C42 b x2 2hL

x2h L ; E:6

C61 ah a2 L2

x3h L ; E:7

C62 bL b2 h2

x3h L : E:8

The symmetric matrix k for Eq. (39) is then

k11 2pa4gaDs

pa2L2

2Db; E:9

k12 k21 pabLh2Db

; E:10

k22 2pb4gbDs

pb2h2

2Db: E:11

and the generalized force vector for Eq. (39) is

f 1 2FaL

x2 2pcb

ah a2 L2 x2h L 2pcsL

2pcsax2h L a x

2 2hL b a2 L2 ;E:12


f 2 2Fbh

x2 2pcb

bL b2 h2 x2h L 2pcsh

2pcsbx2h L b x

2 2hL a b2 h2 :E:13

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Documents

A Model for the Sintering and Corsening of Rows of Spherical Particles