sinteing diagram.pdf

OVERVIEW 11

A SECOND REPORT ON SINTERING DIAGRAMS

F. B. SWINKELS and M. E ASHBY Cambridge Untverslty. Engineering Department. Trumpington Street.

Cambridge CB:! I PZ. England

(Receiwd 11 .4uyusr 1980)

Abstract-Sintering-mechanism diagrams are diagrams with axes of neck-size or density. and temperature. which identify rhtz jrlds qf domirmce of each of the several mechanisms which contribute 10 sintering. and show the rate or PY~VU~ qfsirurritly that all the mechanisms. acting together. produce. The present paper incorporates certain new ideas about sintering into the diagrams: the coupling of boundary diffusion and surface diffusions new criteria for the stages of sintering: and an approximate treatment of particle rearrangement. Diagrams showing hoc\ both the neck size and the dens&\ of compacts of wires and of spheres change with rime and temperature are developed. Their use is-illustrated b> an analysis of a large hod! of sintering data for both wires and spheres of Ag. Cu. Ni. Fe. W. NaCl and Stainless Steel.

Risum&-Les diagrammes de mecanismes de frittage sent rep&en& dans des axes taille des coliets (on dens&+--tem+rature: ils identifient les domaines dans lesquels prCdodomine chacun des mecanismes contribuant au frittage. et ils montrem la vitesse ou le taux de frittage que tous ces mecanismes produisem ensemble. Dans cet article. nous introduisons dans les diagrammes quelques idtes nouvellrs concernant le frittage: couplage des diffusions aux joints et en surface: nouveaux c&&es pour les stades de frittage et traitement approche du rearrangement des particules. Nous prtsentons des diagrammes montrant a la fois les variations de la taille des coliefs et de la densite d’agglom&ats de fils en fonction du temps et de la tempirature. Nous illustrons Milisation de ces diagrammes en analysant de nom- breuses donnees sur le frittage de fils et de spheres d’Ag. Cu. Ni. Fe. W, NaCl et acier inoxydable.

Zusamme&sau~--Diagramme der Sintermechanismen mit den Achsen Briickengr&e oder Dichte und Temperarur beschreiben die Be&he. in denen die verschiedenen Sintermechanismen dominieren; sie stellen die Geschwjndigkei~ oder das AusmaB des Sinterns dar. die siimtliche Mechanismen zusam- men erzeugen. Die vorhegende Arbeit beriicksichtigt im Diagramm einige neue Vorstellungen zum Sintern: die Verkoppelung von Korngrenz-und Obertliichendiffusion. neue Kriterien fiir die verschiedenen Stadien des S$erns. eine neue Behandlung der Umlagerung der Teilchen. Es werden Diapramme entwickelt. die die Anderungen von BriickengriSBe und Dichte von Proben aus Driihten und Kugeln mit der Zeit und der Temperatur beschreiben. Die Anwendung dieser Diagramme wird mit einer Anal?se einer gro&n Datenmenge Nr das Sintern von Drfhren und Kugeln aus Ag. Cu. Ni. Fe. W. NaCl und rostfreiem Stahl verdeutlicht.

1. INTRODUCTION

Six or more distinguishable mechanisms contribute to neck growth and densification when a powder aggregate is sintered (Fig. I and Table 1). It is helpful to have some way of displaying the range of dominance of each of these mechanisms; that is, the range of conditions over which a given mechanism contributes in an important way. One way of doing this was described in an earlier paper [t J. It described the construction of sinrering mechanism di~rarn~~ diagrams with axes of homologous temperature, T/T, (where TM is the melting temperature), and normalised neck radius, x/a (where Za is the particle diameter) showing thefield cf d~inanc~ of each mechanism, and (as contours superimposed on the fields) the net sj~feri#g rote or the sintering time.

In that paper, mechanisms of sintering were treated

as independent: no attempt was made to show how

sharply the transition from one field to the next took

Fig. 1. The mechanisms of sintering (the identifying numbers are defined in Table I). Alf lead to neck-growth.

Only mechanisms 4. 5 and 6 cause densification.

*.u. 29 t-4 259

260 SWINKELS AED ASWBY: SECOND REPORT ON SINTERING DIAGRAMS

Table 1. Mechanisms of sintering

Mechanism Transport path

1 Surface Diffusion

2 Lattice Diffusion

3 Vapour Transport

4 Boundary Diffusion

5 Lattice Diffusion

6 Lattice Diffusion

Source Sink Modification of equations

Non-Densifying Surface Neck Modified by the redistribution

associated with mechanism 4

Surface Neck Unchanged*

Surface Neck Unchanged*

Densifying Boundary Neck Major modification of driving force

to include redistribution of matter emerging from boundary by surface diffusion in neck

Boundary Neck Unchanged*

Dislocations Neck Unchanged*

* The physical assumptions of the sintering model are the same as those quoted or developed in the earlier paper [I], The treatment of the driving force, however. is modified somewhat (see Section 2).

place; and the treatment of the driving force for sintering resulted in an un~tisfactory positioning of the Stage I transition (discussed below in detail). The present paper incorporates a necessary coupling between certain mechanisms, illustrates the diffuse- ness of the transitions between mechanisms, and uses a more satisfactory treatment of the driving force. Many of the sintering-rate equations used here are identical with those used earlier; the reader will be referred to the First Report [l] and other reviews [2,3,4] for their derivation. But the modifications, which have an important effect on the diagrams, are dealt with in detail below. One further extension of the earlier paper is included here: diagrams showing the progress of densi~~tion (as well as that of neck-growth) are developed.

1.2 Stages of sintering and the stage transitions

It is convenient to think of sintering as occurring in four sequential stages. If loose powder particles are brought into contact, inter-atomic forces causes small circles-of-contact (necks) to form between them, and may also cause some particle-rearrangement. This Stage 0 of spontaneous adhesion occurs instanta- neously, and leads to a certain minimum initial neck size [S, 6.71. This is followed by Stage 1, a stage of neck-growth by diffusion during which the necks remain small and the indi~dual particles are still distinguishable. Stage 2 is an intermediate or transitional stage: the necks are now quite large and the pores are often assumed to be cylindrical (a valid assumption for wires, but not for spheres). When the pores become isolated and spherical. the final stage (Stage 3) of sintering has been reached.

The proper rate-equation for a mechanism depends on the stage of sintering. In treating the sintering of wires (Sections 2 and 3) we consider Stages 0, 1 and 2, as distinct, each with its own rate-equations; there is no Stage 3. In treating the sintering of spheres (Sections 4 and 51, on the other hand, we consider Stages 0, I and 3 as well-defined, each with its own

rate-equations; but we omit Stage 2. This is because pores in an aggregate of sintering spheres are never even approximately cylindrical; in the transition from Stage 1 to Stage 3, the pores form connected channels, but the channels vary in section and surface curvature, making the cylindrical approximation a very poor one. Instead, we link Stage 1 to Stage 3 with a transition region by interpolating between them (in a manner described below) to make them continuous.

1.3 Modijkations to the treatment of the mechanisms

The most important modification introduced in this paper is that of Mechanism 4: sintering by boundary- diffusion. The standard treatment of this mechanism (as used in the earlier paper) identi~es the driving force as the neck-curvature where the boundary inter- sects the neck, implicitly assuming that the atoms which flow into the neck out of the boundary are redistributed over the entire neck surface so quickly that the simple neck shape of Fig. 1 is maintained. But if more atoms arrive at the neck surface than can be redistributed by local surface-diffusion. a build-up will occur, reducing the curvature and thus the driving force for boundary diffusion. The curvatures will change until a baiance is reached. when the boundary flux matches the redistribution flux (Fig. 2).

The problem has been recognised for some time [4,8,9. lo] but there have only been two attempts to analyse it. The first one of these (the com- puter-simulation of Bross and Exner [ 11-J) included surface redistribution in calculating neck size and shape, but this simulation-method does not readily lead to explicit equations for the (modified) rate of sintering. The other (an approximate analytical treatment of Swinkels and Ashby [123) coupled boundary diffusion with surface-redistribution and partitioned the total driving force such that the rates of the two were matched. ft used the simplest geometry which could include such a coupling: the circular neck profile of earlier models was replaced by an ellipse. there- by creating a driving force for surface redistribution.

SWINKELS AND ASHBY: SECOND REPORT ON SINTERING DlAGRAMS 261 \ \ \ \ \ 1.l \ \ \ \ \ \ \ ’ I \ \

I_ x---d

\

Fig. 2. The matter which flows out of the boundary must ‘I redistributed over the surface. This requires that K, is ., .: negative than K , . and that the total driving force is ,;r.;!,aned so that the rates of boundar) diffusion and

surface redistribution match properly.

The calculation showed that redistribution could limit the rate of boundary diffusion; but the method of partitioning the driving force omitted an important aspect of the problem (the redistribution length. dl.

discussed below) which severely limits its usefulness. In this paper we develop and use a new and more complete model which incorporates surface-redistribution into all stages of sintering. for both wires and spheres.

This modification necessitates lesser changes in the treatment of Mechanism 1: the flux of matter from the surface no longer flows into the base of the neck. but is distributed over the nearby region (d2 in Fig. 2): and is driven by a greater curvature difference than before.

1.4 Precision of the diagrams

It should be clearly understood that the diagrams are based on approximate models, and on data, some of it uncertain, for the material properties which appear in them. The diagrams are no better (and no worse) than the equations and data used to construct them. We aim here at equations which predict the rates (and thus times) of sintering and densification to within a factor of 2, and neglect terms whose influence is much less than this. The chief source of error then

t Curvatures are taken to be positive when the centre-of- curvature lies within the material. negative when it lies outside it. Note that the symbol p is used for the component of neck curvature in the plane of figure for all stages of sintering. but that its definition depends on the stage of sintering and is given in the appropriate section.

lies in the material data (notably that for boundar! and for surface diffusion) used to evaluate the equations.

This lack of precision is not a deficiency of the sintering-diagram concept: it simply reflects the

present level of understanding of the sintering process and of material data relating to it. As better models and more accurate data become available. they can readily be incorporated into the diagrams. This present paper is just such an advance over the earlier one.

1.5 Definition of symbols

The symbols used m this paper have the following mean-

particle radius (m) modified particle radius (m) radius of disc of contact of two particles (m) one half of the interpenetration between two particles (m) radius of curvature of the neck (m)

curvatures (m-l) diffusion distances for surface redistribution and for surface diffusion (m) diffusive currents (m’s) rate of neck growth (m/s) rate of approach of particle centres (m s) densification rate rate of normal displacement of surface (m s) surface diffusion coefficient times efiectlve surface thickness: d,D. = d,D,.exp - Q,,‘RT(m3:s) lattice diffusion coefficient: D,, = Dol. exp - Q, ,RT(m’&) grain-boundar! diffusion coefficient times effective boundary thickness: C&D, = 6,D0,:.\p - Qb/R7(m3;s) vapour pressure: P, = PO exp - (Qvap ‘R T)(N fm’) surface free energy (J/m*) grain-boundary free energy (J,m*) atom or molecular volume (m3) Boltzmann’s constant (I.38 x 1O-‘3 J/K) Gas Constant (8.3 I J ,mol) absolute temperature (K) melting temperature (K) theoretical density (kg m’) initial densitv of powder compact (kg ‘m3) relative density shear modulus (N/m” Burgers’ Vector of dislocations. or the atomic or molecular diameter (m) dislocation density (me2) velocity of sound (taken as IO3 m/s) a smoothing function used to make non-densifying mechanisms go lo zero at the end of Stage 1

2 SINTERING OF WIRES

When two wires are joined by sintering only Stages 0 and 1 are involved.

2.1 Rate-equationsfor mechanisms 2. 3. 5 and 6,for two

wires

We consider first the Mechanisms 2. 3. 5 and 6 which are unaffected by the requirement of surface redistribution. Let the mean curvature+ of the neck

262 SWINKELS AND ASHBY: SECOND REPORT ON SINTERING DIAGRAMS

Table 2. Rate-equations for neck-growth and densitication

Wires Stage 0 Rate-equations

Spheres

Adhesion

L-Cl

[

ySa ’ ’ i=- for xc -

1 1% x

1=0 for x2 [ 5 1 12

2 L3 i=E for X< ?.-

X [ I 10/J 2 13 x = 0 for x 2 [ ‘& 1

Wires Sta8e 1 rate-equations

Spheres

Non-Densfying Mechanism

1. Surface diffusion from a surface source.

2. Lattia diffusion from a surface source.

3. Vapour transport from a surface source

Densijying Mechanism 4. Grain boundary diffusion

from a boundary source.

5. Lattice diffusion from a boundary source.

6. Lattice diffusion from dislocation sources.

Redisrribution Mechunism Surface diffusion

Neck 8rowth rate.

Linear shrinka8e rate.

Densification rate.

P _ 3D.;,Q(Ka - K,) z-

2kT

v, = Pep” Y$ & [ 1

1:2

w3 - KJ 0

P 4

= 3D,,&7.Q(l - Krx)

x2kT

c’ 5

= 6D,p@;,R(l - K,x)

.x’kT v, =

32npgD,;,R

xkT

p 6

= 2nx2ND,;,R

3kT (

4Px -- -Km n;,a >

p _ 6D&,R(K, - K,)(d, + 2dz) S, -

kTd,(d, + 3d2)

6

iq j.=’

X

A 1.814f,‘a -=- Ao ( 1 - !./a)3

v _ 3nxD,G,;,R(K, - K1) *-

d2kT

3nxD,y,R(K, K,) - 3 2

=

kT 1 2 r’, = 2nxpep, i’lR [ - R 1 kT 2nA,kT

(KJ - K,)

v _ 16nD&;‘,R 4-

xkT

v _ 8ns2p&VD,./$2 6-

_K _ *

9kT ( m

2;y

ti l,

= lZn.uD,d,y,R(K, - Kl)(d, + 2d2)

kTd,(d, + 3d2)

A 3 A(K)

L\O . jla -= A0 (1 - )Xl)4

Stage 2 wires

Rate equations Stage 3 spheres

7. Grain boundary diffusion from a boundary source.

8. Lattia diffusion from a boundary source.

Redisrriburion Mechanism Surface diffusion.

Neck growth rate.

Pd = 3D,&.xn(l - Kix) x2kT

. ZnpD,y,R(l - K,x) Ps =

x’kT

. v,, =

CD.&Y,R(K~ - KJ

;pkT

G = -kT(4;[D$&

u,, = 4nD&,R(K, - Kz)

In (2) kT

SWINKELS AND ASHBY: SECOND REPORT ON SINTERING DIAGRAMS

Table 2 continued

263

Rate equations

Linear shrinkage rate.

Stage 2 wires Stage 3 spheres

j = r; + i;

j= ri, + r’3

X x(x2 - p2)

Densification rate. is 1.814~.a _ _

A0 (1 - !.!a)’

1. The symbol p is used for the component of curvature of the pore in a plane normal to the neck. Its definition depends on the stage of sintering. and on whether wires or spheres are being sintered. See the text for the appropriate definition of p.

2. The constant in a number of rate-equations differs slightly from that used in Paper 1 [I] because we have used refinements ofdiffusion geometry as introduced by Rockiand [14]. The changes make very little difference to the diagrams.

3. The rate-equation for Mechanism 4. Stage I, for spheres, differs from that of Paper 1 both because of the coupling with surface redistribution and because of a change in boundary condition in integrating the rate-equation. We use

I 2x

f rpdr = Zx.x;.R

0 instead of

s

x 2n rpdr = 0

0 leading to the term [i - (Ktx,!2)3. The equivalent change. applied to wires. leads to the term (1 - Krx).

4. We have made changes. based on the calculations of Eadie er al. [15] in the way in which the lattice-diffusion mechanism (5) is related to that for boundary diffusion (4) for spheres. Stage 1. using the replacement

2nx6,DB = Znx*2%p.D,

instead of 2nxd,D, = 2nx2D,

with ICI replaced by K,. An equivalent relation is used for wires and Stage 3 for spheres. 5. The distances d, and d2 are related by:

(a) d, + d2 = p%

where

and

P= x2 + y* + a2 - ai - ZaJ,

Z(a, - x)

&Jrtan-’ “2 [ 1 X+-P

3 3 (Ks - K2) d* 1 (I<, - K2)

--=(I - 4 (.K, - K2);j; - 4 (K1 - K2)

6. The curvatures K,, K2 and K, are related by

K,K2 = K;

7. The curvature difference for non-densifying mechanisms are modified by a smoothing function S, given by

where

a = f (wires)

t 5 Q 5 ; (spheres).

8. The coupling of mechanism 4 with surface redistribution requires that

i; = t;,,


(Fig. 2) be

I(,= -’ P

where

becoming

X2

p=2m

(2.1)

(2.2)

if there is no densification. Then lattice diffusion from the boundary and from dislocations (Mechanisms 5 and 6) are driven by this curvature; and lattice diffusion from the surface and vapour (Mechanisms 2 and 3) are driven by the difference

K3 - K ,=!+A

a P

transport curvature

(2.3)

Matter flows into the neck region of each of these mechanisms at rates which are calculated by standard methods as used and referenced in the First Report [I]. The equations are listed in Table 2.

2.2 Rure-e9uafio~s,~r modified ~ecjla}lis~s I and lfir two wires

Consider first the flow of matter into the neck by boundary diffusion (Mechanism 4). It is driven by the curvature of the neck at A (Fig. 2) where the boundaq intersecrs if; call this K,. But this curvature will rapidly go to zero (removing the driving force) unless matter is redistribute from the region A to the regions B of more negative curvature, Kt. on either side (Fig. 2). This redistribution is driven by the difference (ICI-K& and occurs by surface diffusion. In addition to this redistribution-flux. there is a flux of matter into the regions B. by surface diffusion from more distant parts of the surface (regions C), driven by the curvature difference (1 /a-K J. This is Mechan- ism 1 of Table 1. modified because matter leaving C is deposited between B and C instead of flowing to A.

This new formulation of the problem involves two new curvatures (I< i and K 2) and two new distances (d, and d2) over which diffusion takes place (Fig. 2).

These four variables are determined by four equations. The first simply requires that the (geometric) mean curvature be equal to - 1 ‘p:

K,K, = K;t, f2.4)

The second requires that the total distance d, + d2 be equal to pB (Fig. 2):

dl -I= d2 = p0 (2.5)

The remaining two are continuity equations for the flux. We require that all the material flowing out of the boundary (I’,) be redistributed over the area d, [times unit depth) on either side of the boundary:

ti, = I’,, (2.6)

where r’,, is the total diffusion current (a volume per set) flowing away from A by surface diffusion.

We further require that the normal growth-rate of the neck surface between A and B. cussed by the deposition of the material from the boundary onto it. be equal to the normal growth-rate of the surface beyond B caused by the deposition of the surface flux from the surface (I’,) onto it:

c, r’l Zd,=;i; (2.7)

The individual fluxes i;. e, and psr. calculated by traditional methods. are listed in Table 2 under the heading Stage I’. These equations. sol‘ved by numerical methods, have been used in the construction of the maps shown later.

It is helpful to derive approximate analytical sol- utions for them also. To allow this, we simplify the equations. neglecting curvatures of order 1 a and. in Mechanism 4 of Table 2. the term 1 us. to give:

(2.8)

v,, = -4D,6,;;R

kTdl (Kz - Ki) (2.9)

These equations can be solved for the curvatures K t and K2 and the distances dl and dl. Substituting them

Table 3. Packing geometries for spheres

Geometry Density A!A,

Loose random packing -0.54 Dense random packing 0.636

f.c.c. or h.cx. packing 0.742 b.c.c. packing 0.68 I Simple cubic packing OS24

Plastic spheres. squeezed to full density 1.00

Typical powder packings prior to sintering OS-O.8

Coordination

-8 8-14 (mean: (11))

12 8(+6 = 14)

6

14

-

SWINKELS AND ASHBY: SECOND REPORT ON SINTERING DIAGRAMS 265

10

A:100

lo-” ' / I I 1 I 0.0 0.1 0.2 O-3 04

NORMALISCD NECK RADIUS X/a

Fig. 3. The curvature difference AK required to drive redistribution during Stage 1 of the sintering of wires. Note that, when A and the neck size are small, this difference is slight, and the neck has a circular profile; but when A and

x/n are both large. it deviates from a circle.

into equations (2.6) and (2.7) gives:

K2 - K, 3Adl -=-- K1 4x

K2--K, 3d: --;__ K2 4 d:

where

(2.11)

(2.12)

(2.13)

Making the further approximation (common to almost all sintering models) that d, + d2 = p, and

g 1000 1

0.1 0.2 03 04 NORMALISEQ NECK RAWJS x/a

Fig. 4. The driving forces involved in the sintering of two wires, when D,&/D,6, I: 100. The full line shows the mean curvature (I<,). the broken lines show the curvature K, at

the point A and the curvature K, at the point B.

0.0 0.1 0.2 0.3 04

NORnALlSED NECK RAOIUS x/a

Fig. 5. The redistribution length. d,, normalised by the average neck radius p. For small values of A and .~:a. the redistribution length is small; for large A and .x,ia is tends

to the value 0.5.

writing AK = KL - K, , we obtain:

dl_y{l-l+(~Af$) (2.14)

d2 =

and

AK 3Ap -I -+;-;[3(!$ +4(F)+ il”2 KI x

(2.16) which, with

K,K, = K:

completely determines K1, K2. d, and d2. This last equation is plotted in Fig. 3. It shows the

part of the driving force required to drive redistrihutiotl and the way in which this varies with A and ~/a. When A is small, redistribution is rapid. and requires almost no driving force (so AK is small). When the reverSe is true, most of the driving force is used to drive the redistribution mechanisms. The individual curvatures, K, and K, are plotted in Fig. 4, and the redistribution length dr [equation (2.14)] is shown in Fig. 5.

We now define the regime in which re-distribution is dom~~ont, limiting the rate of sintering, as that for which AK 3 K,. At the boundary of this regime the two lengths d, and d2 [equations (2.14) and (2.15)] are almost equal:

d, = 0.45 p I (2.17)

d2 = 0.55 p ..!

The boundary itself is defined by setting AK = K1 in

266 SWINKELS AND ASHBY: SECOND REPORT ON SINTERlNG DIAGRAMS

equation (2.16) to give

AP --23 X

from which

(2.18)

(2.19)

This result is plotted in Fig. 6 on the axes used for the sintering diagrams (using data for copper given later in Table 4). To the right of the full line. Mechan- ism 4 is controlled by boundary diffusion (that is. more than half of the total available driving force is used to drive the boundary flux); to the left. Mechan- ism 4 is controlled by the surface redistribution mech-

HC+lOLOGOUS TEHPERATURE T/T~ anism. The other cbntours show where AK/K1 has given values. When the activation energy for bound-

Fig. 6. The full line shows the transition from control of arv dj~us~on is less than that for surface diffusion. Mechanism 4 by boundary diffusion (right of full line) to control by surface redistribution (left to full line). using

redistribution controls neck growth on the left hand

data for copper (Table 4). The broken lines show the curva- side of the diagram (as in Fig. 6); but when the reverse

ture differences AL\KIK, used to drive the redistribution is true, it controls on the right. The diagram is inde- mechanism. pendent of a.

Table 4. Material data

Property Silver Copper Nickel

Atomic volume R Im3’ Burgers’ vector b (m) Melting point T,,, (K) Density A0 (kg,im3)

Shear modulus p (MN/m21 T-coefficient of p (K - ’ ) Dislocation density p (m-*) Surface energy ;‘s (J/m*)

Pre-exp. lattice diff. D,, (m’.‘s) Activ. energy, lattice diff. QIp (kJ/moie) Pre-exp, boundary diff. 6 f),, (m3/s) Activ. energy. boundary difX Qb (kJ/mole~

Pre-exp. surface diff. b, D,, (m3/s) Activ. energy. surface diff. Qs (kJ/mole) Pre-exp. vaporisation PO (MN/m’) Activ. energy. Vaporisation Q”,, (kJ/mole)

1.71 X 10-29 2.89 x IO-” 1234 1.05 x lo4

2.64 x lo4 4.36 x 1O-4 10’4 1.12 (i)

tg x 10T5 (m) (ml

6.94 x IO-” 89.8

I”; S

1.18 x lo-= 2.56 x 10-l* 1356 8.96 x IO3

4.21 x IO4 (b) 3.97 x 1o-4 (b) lOI 1.72 (i)

6.20 x 10-s (m) 207 5.12 x 1o-‘s ‘; 10s (t)

6.. x lO-‘O (aa) (aa)

1.23 x IO5 (gg) 324 (gg)

1.09 x lo-‘” 2.49 X lo-‘0 1730 8.90 x IO’

7.65 x IO4 Cc) 3.70 x 1o-4 (C) lOI 2.00 ci1

6.00 x lo--5 (n) 271 3.50 X 1o-‘5 1:; 115 (uf

4.40 x IO-‘2 (bb) 199 (bb)

7.4s x 105 401

(a) Neighbours J. R. and Alers G. A., Phys. Ret;. 111, 707 (1958). (b) Overton W. C. and Gaffney J., Phys. Reu. 98. 969 (1955). (c) Alers G. A.. Neighbours J. R. and Sato H., J. Phys. Chem. Solids 13.40 (1960). (d) Leese J. and Lord A. E., J. uppl. Phys. 39, 3986 (1968). (e) Using p = 0.375 E, and data from Kiister W., Z. Mefaffk. 39, 1 (1948). (f) Blackburn. L. D., ‘The Ge~erur~uf~ ~~isoch~}lo~s Stress-Sfrai~ &rues’, Paper presented at ASME Winter Annual

Meeting, New York. November (1972). (g) Lowrie R. and Gonas A. M.. J. appl. Phys. 38, 4505 (1967). (h) Durand M. A.. Phys. Rec. 50,449 (1963). (if Jones H.. Metal!. Sci. J. 5, 15 (1971). (j) Roth T. A., Mater. Sci. Engng. 18. 183 (1975). (k) Murr L. E., Wong G. 1. and Horylev R. J., Acca Metal/. 21, 595 (1973). (1) Gutshall P. L. and Gross G. E., J. appl, Phys. 36, 2459 (1965). (m) Peterson N. L., Solid St. Phys. 22, 429 (1968). (n) Kniepmeier M., Griindler M. and Helfmeier H., 2. Metallk. 67, 533 (1976). (0) Buffington F. S., Hirano K. and Cohen M., Acta Merail. 9,434 (1961). (p) Smith A. F. and Gibbs G. B.. Metals Sci. 3, 93 (1969). (qt Data of Anderlin R. L., Knight J. D. and Kahn M.. Trajzs. Merall. Sot. A.I.M.E. 233, 19 (1965). As modified by

Robinson S. L. and Sherby 0. D., Acra met& 17, 109 (1969).


3. SINTERING OF AN AGGREGATE OF WIRES

3.1 Stages 0 and 1 for an aggregate of wires

When a close-packed array of parallel wires is sintered (Fig. 7) neck growth and densification in Stage 0 and early Stage I are described by the rate-equations discussed in Section 2 and listed in Table 2. But as the necks grow, the pores become rounded, and the driving force for the non-densifying mechanisms (Tables 1 and 2) diminishes; it must obviously go to zero at the start of Stage II when the pore becomes cylindrical. If no densifying mechanisms operate in Stage 1, the pores become cylinders when the neck size, x/a, reaches the value xc&a = 0.35 [Fig. 7(a)]. But if only densifying mechanisms operate in Stage I, Stage II starts when xc&a reaches the value 0.44 [Fig. 7(b)].

In the earlier report [l], the driving force for non- densifying mechanisms was made to go to zero (for wires) by multiplying it by a smoothing function:

S=(l-‘> if x<xca,r (3.1)

S = 0 if x 2 xca,r

This method has two difficulties. First. by using a fixed value of yCRIT/a (of 0.42) it assumed a constant contribution from the densifying mechanisms. And second, it is entirely empirical, having no physical basis beyond the need to suppress these mechanism when x > .yCRIT.

We have replaced this by an equivalent, but physi- cally more acceptable scheme which incorporates variable densification. It is illustrated by Fig. 8. The non-densifying mechanisms draw material from the region P of length (or area) dWuRcE, and deposit it onto all or part of the region Q. of length (or area) dSINK, equal to the length (d, + d,) of Fig. 2. As the neck grows, the source-length diminishes and becomes zero when the pore becomes cylindrical. It is then that the driving force for non-densifying mechan-

isms becomes zero. We define the new smoothing function

S= d SOURCE

dsovm + &miK (3.2)

It varies from unity at the start of sintering to zero at the end of Stage 1. as the old one (equation

x-Iron y-Iron 304 L Steel Tungsten NaCl

1.18 x IO-r9 I.21 X lo-29 I.21 x lo-29 1.59 x 1o-*9 4.49 X lo-z’ 2.48 x IO-‘” 2.58 x 10-i’ 2.58 x 10-i’ 2.73 x 10-r’ 3.99 x 10-‘O 1810 1810 1680 3683 1074 7.62 x IO3 7.65 x 10’ 8.00 x IO3 1.93 x IO4 2.16 x IO3

6.4 x IO4 (d) 8.10 x IO4 8.1 x IO4 (f) I.55 x IO5 (g) I.51 X IO4 (h) 4.48 x lO-4 (d) 5.03 X lo-4 5.06 x lO-4 (f) I.04 X lo-4 (g) 6.82 x IO-“ (1~) 1or4 lOI lOI lOI lOI 2.10 (i) 2 (i) 2.15 (k) 2.65 (i) 0.28 (1)

1.90 x lo-4 (0) 1.80 x lO-4 (0) 3.70 x lo-5 239 (0) 270

7.50 x lo-‘* i:; 280 I!1

5.60 x 1o-4 2.50 x IO-’ (r) 585 I:1 217 (r)

I.12 x lo-” (v) 2.00 X lo-‘3 (w) 5.48 x lo-l3 (x) 6.20 x IO-‘” (y) 174 (v) I59 V 167 (w) 378 (x) 155 (Y)

2.50 X lo-’ (cc) I.10 x lo-i0 (cc) I.10 x IO-” (dd) 2.55 x 10-i) (ee) 1.00 x 10-i’ (8) 232 (cc) 220 (cc) 220 (dd) 326 (ee) 217 (8) 3.67 x IO’ 382 I%

3.67 x IO5 382 I%

3.67 x lo5 382 iz

3.23 x IO5 782 1z

2.97 x IO4 (gg) 182 (gg)

(r) Verrall R. A.. Fields R. J. and Ashby M. F.. J. Am. Ceram. Sot. 60. 211 (1977). (s) Hoffman R. E. and Turnbull D., J. appl. Phys. 22. 634 (1957). (t) Inferred by scaling data of materials of the same structure and of comparable melting points. The activation energy is

the same as that observed for Au in Cu grain boundaries. Austin A. E.. Richarchs M. A. and Wood E.. J. appl. Phrs. 37. 3650 (1966).

(u) Lange W., Haessner A. and Mischer G., Plrysico sratus Solidi 5, 63 (1957). (v) James D. W. and Leak G. M., Phil. Mag. 12.491 (1965). (w) Perkins R. A., Padgett R. A. and Tunali N. K.. MetaIl. Trans. 4. 2535 (1974). (x) Kreider K. G. and Bruggeman G., Trans. Metal/. Sot. A.I.M.E. 239, 1222 (1967). (y) Burke P. M., Ph.D. Thesis, Stanford University Metallurgy Dept. (1968). (z) Hough R. R.. Scripra metal/. 14, 559 (1970). (aa) Choi J. Y. and Shewmon P. G.. Trans. Metal/. Sot. A.I.M.E. 222. 589 (1962). (bb) Mills B.. Douglas P. and Leak G. M.. Trans. Metal/. Sot. A.I.M.E. 245. 1291 (1969). (cc) Blakely J. M. and Mykura H.. Acta merall. Il. 399 (1963). (dd) Assumed to be the same as for y-iron. Ref. (cc). (ee) Allen B. C.. Trans. Metal/. Sot. A.I.M.E. 236. 915 (1966). (ff 1 Approximated by lattice diffusivity times Burgers’ Vector for NaCI. (gg) Dushman S. and Lafferty J. M. (Eds), Scientific Foundarious oj’ Vacuum Techology Wiley, New York (1962).

268 SWINKELS AND ASHBY: SECOND REPORT ON SlNTERlNG DIAGRAMS

tbi

Fig. 7. The sintering of a close-packed agffregate of wires, illustrating the calculation of the density at the start of Stage 11, (a) when no densifying m~hani~ms operate in Stage I, and (b) when only densifying mechanisms operate

in Stage I.

3.1) did, and is used (as before) to multiply the curvatures driving the non-densifying mechanisms.

The geometry of Fig. 8 requires that, for close- packed wires, p is given by

P x2 + y2 + a2 - a: - 2ay

= %a, - 4

(3.3)

where 2y is the centre-to-centre approach of the wires due to densification and al is the current radius of each wire, which differs from a because material has been removed from its surface to fill the neck (a, is calculated by applying conservation of volume to the nondensifying mechanisms). In addition:

where

dsouscr = at@ - 7V3) (3.4)

d SINK = pe (3.5)

In constructing the maps, x, y and p are calculated in a step-like way, so that dsouRc, and dsfFac are known. and properly include the effect of densifi~tion. They are then used to calculate S (equation 3.2).

The end of Stage I is defined as the line S = 0; it appears on subsequent diagrams for wires as a broken line near .x/a = 0.40 or A/A, = 0.95. Its slope is caused by the varying contribution of densi~~ation across the diagram. The rates of the non-densifying, Stage 1, mechanisms (Mechanisms 1,2 and 3. Table 2) are modified by multiplying curvature difference which drives them by the factor S [equation 3.2): that for vapour transport, for example, becomes

W, - Kn)S.

3.2 Stage 2 for an aggregate of wires

Once the pore has become cylindrical, the driving forces for Mechanisms 1, 2 and 3 disappear, leaving only Mechanisms 4 and 5. The basic rate-equations for these two mechanisms are unchanged. There are, however, two modifications which enter Stage 2. The first is an obvious one: the mean curvature (which acts as the driving force for Mechanism 5) is now

Km= -’ P

(3.7)

where p is the mean radius of the pores, which, from the geometry of Fig. 8 is

By approximating the pore as a circle of radius p we

(3.6) Fig. 8. The pore between an aggregate of wires, showing . . . the dlmenslons tnvoived m the calculatmns.


STAGE: 2 WIRES

5fAGE 3 I

SPHERES

Fig. 9. The pore for late stage sintering, showing the dimensions involved in the calculations during (a) Stage II

for wires. and (b) Stage III for spheres.

obtain a second relationship, based on conversion of volume:

(a - y)(x + p) np2 ?ra2

2 =:--+-+.

6 12

Combining these two equations and solving for p gives

p = -2.53.x -t {3.87.x* i- 0.765~7”;” (3.8)

As before, Mechanism 4 can proceed only if the material it pumps into the neck is r~ist~b~ted across the surface of the pore, so a curvature-difference must exist to drive the redistribution (Fig. 9a). The matter entering the neck from the boundary does so at a rate

and the rate of redist~bution~ is

p _ c~,a,Y,w* - K2) SI -

kTdl (3.10)

t During Stage I material is plated onto the void surface as a wedge of maximum thickness at the point A (Fig. 2), diminishing to zero thickness at a distance (d, + d2) away. But during Stage 2, the pore remains geometrically identical, merely shrinkjng in size, so the redistributed material must be plated uniformly over the pore surface. In the numerical calculations we accomplish the transition by re+ placing the constant 4 in equations (2.9) by

c=4 +2 2P

(x + P - x,)

where x, is the neck size at the start of Stage 2.

For this stage (Fig. 9a) d2 has disappeared. and d, is simply

n dl = -

jp (3.11)

Equating Pk and r’,, (as before} and solving for K1 gives

Ki = ct - {c: + 4(1 + Crx)K:j”2

2(I + C,x) (3.12)

where

(3.13)

and X1 is given by KIKz = I(,,!,. The results readily simplify to

and

In the subsequent calculations the rates of neck growth and densification have been calculated from the rate-equations V, and 9, of Table 2, using these values of K, and K2. These resui?s are plotted in the same way as before in Figs 1 O- 13.

4. SINTERING OF TWO SPHERES

When two spheres are joined by sintering, only Stages 0 and 1 are involved. The rate-equations closely resemble those for a pair of wires. The differences arise because the new geometry involves slightly

ltf3 i u

0.0 01 0,2 o-3 04 0.5

NORHALISCD NECK R&MS x/a

Fig. 10. The curvature difference. AK/K,, required to drive redistribution during Stages I and II of the sintering of

wires (compare with Fig. 3).

270 SWINKELS AND ASHBY: SECOND

-0.0 0.1 0.2 0.3 04 05

NORMALISEO NECK RADIUS x/a

Fig. I I. The driving force for the sintering of wires during Stages I and II when D,S,/D,& = 100. The full line shows the mean curvature, K,. the broken lines show the curva-

tures K, and K, (compare with Fig. 4).

changed curvatures, and slight changes in the continuity equations. Both are discussed below.

4. I Rate-equations for mechanisms 2, 3, 5 and 6 for spheres

Mechanisms 2, 3, 5 and 6 are unaffected by the requirements of surface redistribution. The mean curvature at the neck (Fig. 2) is now

K,+i X

where p is given by equation 2.2. Lattice diffusion from the boundary and from dislocations (Mechanisms 5 and 6) are driven by this curvature; and lattice diffusion from the surface, and vapour transport

w 0.6 1 //I

-0.0 0.1 0.2 0.3 04 05

NORMALISED NECK RADIUS x/a

Fig. 12. The redistribution length, d,, normalised by the length n/3p for Stages I and II of the sintering of wires

(compare with Fig. 5).

REPORT ON SINTERING DIAGRAMS

0.0

full d&y,

COPPER

0.4 0.6 0.8 ;lO HOMOLOGOUS TEMPERATURE T/TM

Fig. 13. The transition from control of Mechanism 4 by boundary diffusion (right of full line) to control by surface diffusion (left of full line) for Stages I and II of the sintering of wires. The broken lines show the curvature differences AK;‘Ki used to drive the redistribution mechanism (com-

pare with Fig. 6).

(Mechanisms 2 and 3) are driven by the curvature difference

Matter flows into the each of these mechanisms at rates which are calculated by standard methods, as used and referenced in the First Report [I]. The equations are listed in Table 2.

4.2 Rate-equations for the modijed mechanisms I and 4 for spheres

As with wire, boundary diffusion can deliver matter steadily into the pore only if surface diffusion can redistribute it over the pore surface. There must then exist a curvature difference AK = K, - K, to drive the surface redistribution, leaving the lesser curvature (K,) to drive boundary diffusion (Mechanism 4). This in turn influences the rate of surface diffusion (Mech- anism 1) which is now driven by the curvature difference (2/a - K2). The need for surface redistribution introduces two new curvatures (K, and K2) and two new distances (d, and d2) over which diffusion takes place. They are related (as before) by equations (2.4) and (2.5), and by two continuity equations. As with wires (Section 2.2) we require that the redistribution flux be equal to the boundary flux where it enters the pore (equation 2.6) and that the rate of normal growth of the surface of the pore be continuous where the redistribution flux and the surface flux meet (equation 2.7).

If the equations for spheres given in Table 2 are now simplified as in Section 2 and substituted into these equations, they can be solved for K,. K,. d, and d,. The results are identical, but for a numerical constant of magnitude 3/4, with those of Section 2. Since

SWINKELS AND ASHBY: SECOND REPORT ON SINTERiNG DIAGRAMS ‘71

this approach is an approximate one. we may regard all results and figures for wires of Section 2 as apply ing to Stage 1 for spheres also.

In computing the maps of later sections we have used an iterative technique to solve the problem exactly. But from a practical point of view. the analytical results of Sectjon 2 given an adequate approximation to the true result.

5. SfNTERflVG OF Al; AGGREGATE OF SPHERES

When spherical particles of a single size are packed randomly [ 131. the density can vary between 0.56 and 0.64. and the coordination number (number of neighbours per particle) between 8 and 14. These results are compared with those for ordered packings of spheres in Table 3. If the particles are non-spherical. the densities are generally smaller; but if two or more sizes of particles are mixed. so that the smaller spheres fit between the larger ones. the density can h larger. It must be generally true that the packing geometry will change during sintering. so that the number of contact neighbours. and thus necks. per particle increases as sintering proceeds.

This introduces a new problem (one we did not have with wires). Not only is the initial packing a variable when aggregates of spheres are sintered. but the packing geometry may change during the sintering itself. A wire in a close-packed array of wires has 6 nearest neighbours throughout sintering: a sphere in a compact of spheres might start with 8 nearest neighbours and end up with 14. We shall first discuss sintering at fixed packing geometry and then introduce changes of geometry with time.

5.1 Stages 0 and I for a compact of spheres

When an aggregate of packed spheres of equal size is sintered, neck growth and densification in Stage 0 and early Stage I are described by the rate-equations discussed in Section 4 and listed in Table 2. But (as with wires) the pores become rounded as the necks grow and the driving force for the non-densifying mechanisms diminishes; it must go to zero at the start of Stage 3. when the pore becomes spherical. (Note that pores are never even roughly cylinidrical during the sintering of spheres, so no Stage 2 mechanisms are listed. We bridge the gap between Stage 1 and Stage 3 with a Transition stage described below.)

Throughout Stage I, we calculate the neck curvature, p. from equation 3.3. The coupling of boundary with surface diffusion is treated as follows. The rate- equations involved (Table 2) are

* v4 = (5.1)

u,, = 12axD,6,~,fJ(K, - KdW, + Id*) kTddt(d, + 3dz)

(5.2)

These vary with the quantity A and with x,/a. in a way

which closely resembles the equivalent equations for Stage 1 of’ wires plotted in Figs 3-6.

We assume that the packing geometry varies con- tinuously throughout Stage 1. Figure 14 shows plane sections through three possible geometries. The first is simple-cubic packing (A A0 = 0.52): the second is bee. packing (A ‘A0 = 0.68): and the last is fee. packing (A A0 = 0.74). The figure illustrates that the angle z can be used to characterise the structure. and that as z decreases from 90’ to 60’. the density rises from 0.52 to 0.74 in a roughly linear way.

We have used this as a basis of a device to include particle rearrangement into our calculations. The initial angle xi is chosen to match the initial densit! of the green compact. We have then kept track of the density throughout the computation of the maps and caused x to decrease from this initial value towards 60’. in a way which varies linearly with the density. and would cause r to become 60’ when A A0 reaches 1. This rearrangement contributes to the densification. over and above that caused by the diffusional removal of matter from the neck. (It incidentally causes the number n of contact neighbours. and thus necks-per- particle, to increase; the data in Table 3 is consistent with the approximate relationship

tr = 16A,A,, - 2

so that n increases towards the value 14 as sintering proceeds,)

In the calculations (described in more detail in Section 6) the current values of s. y and al are used to calculate the angle 8 [equation (3.6)] and p [equation (3.3)]. The angle z is calculated from the current value of the density. as described above. Then the new density is calculated from

_ _ tA&)iAo) A

Ao (1 - ?;‘aJ3

where Ai(Z)~AB is obtained from the equation of the line on Fig. 14. In this way, densi~cation both by neck growth and by particle rearrangement are included.

0 E 5 d

05 I

i I I

60 70 80 90

SMALLCST AwClE WWEEN PARTICLES ON A CLOSE-PACKED RANE, deg.

Fig. 14. The variation of density with packing peometr!. charactcrised by the angle 1.


Stage 1 ends when the section of the pore becomes circular. If the particles were close-packed this would simply be the neck size at which 0 = 60’. But we start with a non-close-packed geometry, characterized by the angle z (which decreases towards 60” as the density increases). It ‘is readily shown that, when z is a variable. equation 3.4 must be replaced by:

&,..,.=a~(~+~--~) -l-hen d,,,,, goes to zero, and the pore becomes

nrcular, when

This is the criterion we have used for the end of Stage 1.

5.2 Stage 3 and rhe transition stage. for a compact oj

spheres

The start of Stage 3 is calculated in the manner described in the next paragraph; then a transition zone is constructed by linearly interpolating between the values of A/A0 at the end of Stage 1 and the start of Stage 3, and using these interpolated values of density to calculate self-consistent values of x/a and y/a.

We assume that, at the start of Stage 3, all particles have 14 contact neighbours. packed so that the neck surfaces are coplanar with the faces of a regular tetrakaidecahedron (ICfaced solid, Fig. 15) and that no further changes in packing geometry occur. There is a pore at each corner (there are 24 comers each shared with 3 other particles). We assume further that the larger neck associated with each pore (Fig. 15) has the same size as that at the end of Stage I. (The density, of course. is larger because we have replaced cylindrical pores by spherical holes.) The new pore radius, p. which determined the driving force for

PI 1

Fig. 15. A particle at the start of Stage III. The spherical holes are located at the 24 coreners of the

tetrakaidecahedron.

Stage 3 sintering. is readily calculated by equating the volume V, of the regular tetrakaidecahedron of edge- length I

v =8 c \ i213

to the volume of matter per cell

4 v, r - za3

3

plus the volume of six pores

VP = 6 x ; np3 (5.6)

and noting (Fig. 15) that I = x + p. These considerations fix the start of Stage 3. As

explained, the gap between Stage 1 and 3 is bridged by a transition zone, based on a linear interpolation of A/A, between its value at the end of Stage 1 and that at the start of Stage 3. The further progress of Stage 3 is then calculated using the Stage 3 equations of Table 2 at fixed packing geometry. The numerical procedure is described briefly in Section 6.

Throughout Stage 3 the coupling of boundary and surface diffusion is included. The rate of this coupled process is calculated as follows. The pore, although almost spherical, cannot be a perfect sphere if there is to be any driving force for surface redistribution. If we think of a pore sitting on a planar grain boundary (the neck), then by symmetry, its shape is that of a surface of revolution about an axis normal to the plane of the neck. For such a surface, the rate of normal displacement of the surface due to a surface flux is

- -.- (5.7)

where y is the perpendicular distance from the axis of the surface of revolution (Fig. 9b) and s is the distance along the pore surface from the grain boundary. Assuming that the pore remains nearly spherical, y and s are related by Y = p cos s/p where p is the mean pore radius. For uniform distribution, roughly preserving the pore shape, we require that

1 - - = constant Y ds

(5.8)

The solution to the equation is

K - CIp’ln sec$ .(I II +C,ln sec?+tanz

(I P I> +C, (5.9)

P

using the boundary conditions

J, = 0 It

at SE- 2p


K=K,at s=O

K=K,at s=:p

we obtain, by in~egratjon. the flux of material leaving the boundary and flowing over the pore surface by surface diffusion. Expressed as a volume rate, the result is:

V SR

= 4~D,&$(K1 - Kz) In (2)kT

(5.10)

This we equate to the grain boundary flux, which is calculated by standard methods. To a sufficjent approximation, the boundary rate is

Vd = 2nD&@K 1

!

3 kT lnX?__ (5.11)

P 4

Equating the two rates we obtain

Using K,K2 = Ki we can solve explicitly for both K, and K2, and hence (by substitution in equation 5.11) obtain the rate df sintering by mechanism 4 when it is coupled to surface redistribution.

Once again, the way in which K, and K, vary with x/a and A are so similar to the results which we obtained for wires (Section 3 and Figs 10, 11 and 12) that the reader is referred to these for a description of the characteristics of the distribution mechanisms.

4 CONSTRUCI’ION OF SINTERlNG

DIAGRAMS

We construct two types of sintering diagrams, one showing neck-size the other density, as a function of temperature and time. Such diagrams for wires of two different radii are shown in Fig. 16, and for aggregates of spheres of two different radii in Fig. 17.

6. I Neck-size diagrams

The first type of diagram is the same as that shown in the earlier paper [ 11. The axes are normafised neck size .uJa and homologous temperature T/T,, where TM is the melting temperature of the material (Figs 16 and 17, upper diagrams). It shows the jelds over which each mechanism of neck-growth is dominant- that is, the range of neck size and temperature over which each mechanism contributes more to neck- growth than any other mechanism does. The boundaries of these fields can be found by equating pairs of the rate-equations (listed in Table 2) and solving for neck-size as a function of temperature. At a field boundary (shown as heavy lines on the Figures) two mechanisms contribute equally to the sintering rate. A

heavy broken line marks the transition from Stage I to Stage II or III: the mechanism does not change here, but the equation used to describe it does (see Table 2). On either side of some of the field boundaries is a shaded band. Within the non-shaded regions. a single mechanism contributes more than 5504 of the total neck-growth rate. Within the shaded regions, two or more mechanisms contribute in an important way to sintering, though none contributes more than 55%: the shading gives some idea of the ‘width’ of the field boundaries. FinaIly, the heavy dash-dot line shows where redistribution begins to control Mechanism 4 (boundary diffusion): on the side of this line Iabelled ‘SL’, surface-redistribution limits the rate; on the other side labelied ‘BL’, boundary diffusion does so. (Its construction was described in earlier sections and illustrated by Figs 6 and 13. It is the line along which the driving force is partitioned equally between the boundary flux r’, and the redistribution flux ri,a.) The part of the diagram labelled BL is aimost uninffuenced by the need for redistribution; the part labelled SL. however. reflects its pres- ence.

Superimposed on the fields are contours of constant time. The neck growth-rate is the sum of the contributions of the mechanisms listed in Table 2. according to the stage of sintering. The contours of constant time are computed by integrating this sum of the rate- equations with respect to time.

6.2 Density diagrams

The second type of diagram is new. The axes are relative density A/A, and homologous temperature T/TM (Figs. 16 and 17, lower diagrams). At first sight it would seem logical to show it on the fields of dominance of each densification mechanism, of which only two are ever important (M~hani~s 4 and 5). But where these dominate densi~cation they also dominate neck growth (so the field boundaries for both coincide); and where non-densifying mechanisms operate in Stage I. they influence densification (by changing the neck shape, and so the driving forces) even though they do not contribute directly to it. So more information is contained in the diagram if, as on the first type of diagram, we plot fields of d~inant neck-growrh mechanism, remem~ring that throughout Stages II and III, they are identical with those of dominant densification mechanisms. As before, heavy full lines show field boundaries; heavy broken lines show stage transitions; and the dash-dot line separ- ates the region of control of Mechanism 4 by redistribution (SL) from its control by boundary diffusion

(BL). Superimposed on the fields are co~rours of constant

time. The densifi~tion rate is the sum, appropriate to the stage of sintering, of the contributions of Mechan- isms 4, 5 and 6, listed in Table 2. The contours of constant time are computed by integrating this sum of the rate-equations.


Fig. 16. A set of sintering diagrams for a close-packed array of silver wires of two different radii: 10 and 100 pm. The upper diagrams show the neck growth, the lower show the density as a function of time and

temperature.

6.3 method of construction

The diagrams were constructed by numerical methods, using the data listed in Table 4. A practical method is as follows. Pick a starting value for T/TM (we generally used 0.4) and for neck size, x/a (the adhesive neck-size, see Table Z), and evaluate the rate equations for the volume flux ri of the mechanisms of a given stage of sintering. Sort and record the dominant mechanism. Form the sums given in Table 2 to give the net growth-rate J/a, linear shrinkage-rate j/u and densification-rate &Ao. Then increase x/a and T/TM in steps, evaluating these quantities at each step and calculating also the time interval At between steps in x/a, and sum the results to give the time t, the total shrinkage y/a and total density change A/A,. Finally, plot on axes of x/a or A/A, and T/TM the locus of points at which the dominant mechanism changes (the field boundaries), the locus of points at

which t has chosen values (the time contours) and the locus of points at which AK = K, (the boundary between boundary diffusion and surface redistribution-control of Mechanism 4). The 55% contours are added by the obvious modification of the same procedure.

7. APPLICATIONS

We conclude by illustrating the use of the maps in the interpretation of experiment and the design of in- tering procedures. The maps are constructed from the data listed in Table 4. Measurements (discussed below) are plotted onto each as bars or boxes, showing the observed values of neck size .x/a and temperature T, or of relative density A/A0 and T. The numbers on the bars or boxes are the observed times, in hours, to reach that neck size or density. They

allow a direct comparison of the maps with experi-


0-c lWOLO&S ~~~T~~

1.0 T/TM -~* ~~~~~~“~ T/T,

(11 (2)

Fig. 17. A set of sintering diagrams for an aggregate of spherical particles of silver of two different radii : 10 pm and 100 pm, The upper diagrams show the neck size, the lower show the density as a function of

time and tem~rature.

mental data.

7. I E.~peri~e~fs using wires

7.1.1 Alexander and Ballet [ 163. Sintered copper wires (a = 64~) wound in a close-packed array, onto a copper spindle. Their experiments compassed the range

T = 900-1075°C (0.86-0.99 TM) t-0-6OOh

x/a = 02-0.4 copper

A/A, = 0.92-0.97 i

The range of their experiments is shown as a box on Figs 18 and 19. Throughout these experiments, two mechanisms contribute in an important way to neck growth. Up to 10 h, neck growth is mainly by surface diffusion. Between about IO and 50 h, surface and lat-

tice diffusion contribute almost equally. Beyond that, lattice diffusion from the boundary (a densifyinp mechanism) becomes dominant. The predicted neck- sizes agree well with those measured on necks con- taining a grain boundary (typical discrepancy less than 12%). When boundaries migrated away from the pores, the necks ceased to grow. This can be incorporated into the diagram by suppressing the boundary dependent Mechanisms 4 and 5 when migration starts [I].

Alexander and Balluffi also measured the densificd- tion of their arrays of wires. Their observations are plotted onto Fig. 19, which is the density map corresponding to Fig. 18. The ex~rimenta~ data is not entirely self-consistent (due to presumably to the difficulty of measuring densification); the map agrees with it to within the experimental error.

7.1.2 Pranatis and Seigle [ 171. Studied neck growth between close packed wires of copper (a = 51 pm)


TfIlPCRATURf OC

300 450 600 750 900 1050 0.0

+ 8

HOtiOLOGUiJS TEMPERATURE T/TM

Fig. 18. A neck-size diagram for close-packed Copper wires: (n = 64 pm). Data of Alexander and BaIlu% [ 16) are

shown.

TEMPERATURE *C

HOMOLOGOUS TEMPfRAlURE “TM

Fig. 19. A density diagram for the wires of Fig. 18, again showing the data of Alexander and Ballufi [ 163.

and nickel (a = 64 p), under the following conditions:

T= 106O"C(O.98 TM) t = l-1OOh

I Copper

x/a = 0.2-0.43

T = 1400°C (0.97 TM) t=3-4OOh

1 Nickel

x/a = 0.21-0.43

Their data are shown on Figs 20 and 21. The agreement between measured and predicted neck-sizes is good (the maximum discrepancy in x/a for nickel is 5%; for copper it is 9%). In both cases surface diffusion (1) and lattice diffusion (5) contribute almost equally to neck growth and (as observed) there should be considerable shrinkage.

TEMPERATURE OC

300 WJ 600 750 900 1050

10-6

HOMOlOGOUS TEHPtRAlURf T/TM

Fig. 20. A neck-size diagram for close packed Copper wires: a = 51 pm. Data of Pranatis and Seigien [ 171 are

shown.

TfMPfRATURf 'C

0.6 0.8

HONOlOGOUS TEMPERATURE T/TM

Fig. 21. A neck size diagram for close-packed Nickel wires: LI = 64 pm. Data of Pranatis and Seigle [ 171 are shown.

The same authors sintered close-packed iron wires (a = 38 pm). The conditions were:

T = 875’C (0.63 TM) t=3-1OOOh

I Iron

XJU = 0.13-0.34

Figure 22 shows their results plotted onto an appropriate diagram for pure iron. Surface diffusion is the dominant mechanism for times up to 1 hour; beyond that, lattice diffusion and surface diffusion contribute about equally. The data given in Table 4 give a fair fit to Prantis and Seigle’s measurements, though (in this instance) the fit depends strongly on the choice of the surface diffusion coefficient. We favour the present choice because it gives a fair match also with the data of Matsumura [ 1 S] discussed below.

7.1.3 Matsumura [is]. Studied the sintering of iron, wound in close packed arrays. His experiments extend


0.6 04 14

HOnOLDtOUS TEMPfRATURf T/T6

Fig. 17. A neck size diagram for close packed Iron wires: P = 38 pm. Data of Pranatis and Seigle [ 171 are shown.

TfWERAflJRf *C

SW 150 1000 1250 1500 9.0,

I I

. 10-6

0.6 O-8 1.0 HOtiOLOGOUS TLHPERATURE l/T,

Fig. 23. A neck size diagram for close packed Iron wires: a = 75 Itm. Data of Matsumura 1187 are shown.

TfWfRATURf Y

300 650 600 750 900 1050

HOHa_OtOU5 TEHPERATURE l/Tpj 7.2.3 Kucz~~rski [21]. Measured the neck growth-

Fig. 24. A ?-sphere neck-size diagram for Copper: rate of copper spheres (a = 40~) on a copper plate. a = 57 pm, Data of Kingery and Berg [ 191 are shown. The ranges of temperature and time he used. and the

over both the x and the y-range:

T = 700-1350’c (0.X-0.9 TM) r = I-2000h

x:a = 0.075-0.24 Iron

The data are plotted on Fig 23. Sintering is predomi- nantly by surface diffusion. with a major contribution ( : 40’~ from lattice diffusion. The sudden decrease in sintering-rate at the x to ;’ phase boundary appears both on the map and in Matsumura’s data. But the map consistently predicts neck sizes which are too large (by a factor of about 2) or times to give a certain neck size which are too small (by a factor of about 5) implying that the surface diffusion coefficient used to construct the map is too rapid.

7.2.1 Kitt(tery und Berg [19]. Measured neck growth between two spheres of copper (N = 57 pm). Their measurements were made at very short times. cover- ing the range:

T = 9% 105O’C (0.9-5.97 7-M) t = 0.02-S h Copper

x;a = 0.124.26 I

Figure 24 shows that surface diffusion was the dominant neck-growth mechanism, with a major contribution from lattice diffusion (giving densification). The predicted neck sizes are in good agreement with the measurements {maximum discrepancy 4x1 except at the highest tem~ratur~ (where it jumps to 25”,). We have frequently noted that, very close to the melting point. theory underestimates neck-growth rates by surface diffusion; Kingery and Berg’s measurements are an example. We think that this is because surface diffusion (unlike most other diffusion processes) is rarely characterised by a single activation energy: and near TM its rate is faster than that expected by extra- polation from lower tem~ratures. The reader is referred to Mills PI al. [20] for diffusion data showing this effect.

7.2.2 Wilsm artd Shcwmar~ [3]. Measured neck growth in a row of copper spheres (a = 7.5 pm) under the following conditions

7 = 750-95O’C (0.75-0.90 T,) r = I-240h

I Copper

x/a = 0.1 l-0.32

Their data are plotted onto Fig. 25. The diagram supports their conclusion that surface diffusion is generally dominant. though it also indicates a large contribution from lattice diffusion. The agreement between predicted and measured neck sizes (or times) in this instance is extremely good: within the experimental error of the data ( 2 49;).


10-6

HOMOLOGOUS TEMPERATURE T/T"

Fig. 25. A ?-sphere neck-size diagram for Copper: a = 72 pm. Data of Wilson and Shewmon [3] is shown.

300

TEfiPtRATURE 'C

450 600 750 900 1050 0.0

3.10-s

-1.0 kz E

3x10-6 2

z Y

-1.5

10-6

-2.0 0.4 0.6 0.8 1.0

HOHOLOGOUS TEMPERATURE T/T,,

Fig. 26. A 2-sphere neck-size diagram for Copper: a = 40 pm. Data of Kuczynski [Zl] is shown.

TCMPERATURE 'C

300 I 450 600 750 900

__

HOKILCGOUS TIIlFfRATURE T/T,,,

Fig. 27. A 2-sphere neck-growth diagram for Silver: (I = 180 pm. Data of Kuczynski [Zl] are shown.

resulting range of neck size, was

T = 70&9OO’C (0.72-0.86 TM) t = 0.5-41 h

x/a = 0.1-0.33 Copper

Figure 26 shows these data, plotted onto the appropriate diagram. Agreement is fair: the maximum discrepancy in x/a is 359/,, although for most data points. the agreement is much closer: typically c lo”,/, discrepancy. The measurements all lie in the field for which surface diffusion from a surface source is dominant; but at the low temperature end, boundary diffusion contributes nearly half of the total. and at high temperatures lattice diffusion contributes in a major way ( - 40%).

The same author made similar measurements using silver spheres (a = 180 pm). They are reanalysed in Fig. 27. The data spans the ranges

T = 5OS8OO’C (0.63-0.87 TM) t=l-90h

I Silver

x/a = 0.033-O. 162

Agreement between theory and experiment in this case is very good over an exceptional range of temperature: the maximum error in predicting s/a is less than 12%. The data overlaps two fields (surface diffusion and boundary diffusion) with a major contribution from a third (lattice diffusion from the surface). The boundary mechanism is limited by boundary diffusion at the higher temperatures, but by surface redistribution at lower temperatures. The map (which of course includes the effects of redistribution) agrees with the data at low temperatures much more closely than a map in which redistribution is omitted [l]. This is the only data we have been able to find which permits a test of the redistribution model.

The diagram itself is interesting in showing both mechanisms 2 and 5: lattice diffusion around the pore and lattice diffusion from the boundary. Note, too, the steepened slope of the time contours in the surface-limited boundary diffusion regime: the redistributed mechanism is slowing down the sintering very considerably in this field.

Over the central shaded region, no one mechanisms is responsible for even half of the total neck growth- rate. Instead, four mechanisms all contribute about equally. This illustrates just how complicated the interpretation of sintering data can be, even in model experiments like this, and is a good example of how these diagrams can help in understanding what is going on.

7.2.4 Kingery and Berg [ 193. Also studied the sintering of sodium chloride spheres (a = 60 pm). The conditions were

T = 700-75O’C (0.91LO.96 TM) I = 0.025-0.25 h

1 NaCl

x/a = 0.074.22

In constructing a map (Fig. 28) we have had to guess parameters for surface diffusion (Table 4) for which

SWINKELS AND ASHBY: SECOND REPORT ON SINTELING DIAGRAMS 279

TEMPERAlURE *c no date are available. With this guess. the map shows 2OQ 300 LOO 500 600 700 two fields only: those of vapour transport and of

boundary diffusion. The data. which lies entirely in the vapour transport field. agree well with the predic- tions of the map. Our conclusion supports that of Kingery and Coble: vapour transport is dominant,

7.3 Esperinrerrts usirtg uggregates qf‘splreres

7.3.1 Eadie et al. [22] and Lee et al. [23]. Sintered aggregates of spherical silver particles (a = Z&22 pm) in air and in argon. and measured both neck growth and densifications. Their conditions covered the ranges

‘I- = 700-800-C (0.794.87 T,) -7.0 _ _ t = 2.6-4.7 h

04 0.6 0.8 1.0 xia = 0.18-0.37

1

Silver HOtlOLOGOUS TEMPERATURE T/T,,

Fig. 28. A 2-sphere neck-growth diagram for Sodium A/A0 = 0.62ZO.75

Chloride: a = 60 pm. Data of Kingery and Berg [19] are Some of the measurements of the neck growth in shown. air [23] are shown in Fig. 29. Agreement is good. and

the map confirms the authors’ conclusion that bound- TEMPERATURE Y ary diffusion is the dominant mechanism. with contri-

300 450 600 754 900 0.0 ' I butions from lattice diffusion and surface diffusion.

A more exacting comparison is shown in Figs 30 and 3 1. The first shows neck growth and the second shows the density of the same set of sintering silver spheres [22]. Both the neck sire and the density are less than that predicted by the diagrams-though a small increase in the rate of lattice diffusion (the dominant mechanism) would correct this. But we think the discrepancy may be real,(not merely a result of inaccurate diffusion data) and exist because the scheme of particle-rearrangement we have adopted (Section 5) may lead to rates of densification which are too rapid.

IOHESIPI

-2.0 7.3.2 Germatr [24]. Measured the densification of

04 0% 04 1.0 particles of a 304 L stainless steel (a = 2.5-20 pm: we HOMOLOGOUS TftlPfRATURf T/T"

Fig. 29. A neck-size diagram for a compact of Silver TEMPERATURE OC spheres: a = 20 pm. Data of Lee er al. [23] are shown. 300 450 600 150 900

TEMPERATURE 'C

300 L50 600 193 900 o-or ' I

XT5 z zl ---_---s&V4

r !z

>” F * a %

04 06 0% 10 i+UMWJGCUS TEMPERATURf T/TM HOHOLOGOUS TEnPERATURE T/T,,,

Fig. 30. A neck-size diagram for a compact of Silver Fig. 31. A density diagram for the same silver spheres as spheres: a = 22 pm. Data of Eadie er al. [22] are shown. Fig. 30. again showing the data of Eadie et al. [22].


TEMPERATURE 'C

750 1000 1250

, O-6 04 1.0 HOMOLOGOUS TEMPERATURE T/T,,

Fig. 32. A density diagram for a compact of 304 L Stain- less Steel spheres: a = 15 pm. Data of German [24] are

shown.

took a = 15 pm). The sintering conditions were

T = 99&l 153’C (0.7NI.85 TM) r=4h

I 304 L Steel

AlA0 = 0 66-0.75

Agreement between theory and experiment is remark- ably good (Fig. 32). Densification is by boundary diffusion, with major contributions from lattice diffusion. But it should be noted that surface diffusion has an important influence on neck growth and (because the neck size determines the driving force for all sintering mechanisms) surface diffusion influences the rate of densification even though it does not. by

1.0

TfMPfRATURf 'C

I ,

06 04 1.0 HOMOLOCOUS TEMPERATURE T/TM

Fig. 33. A density diagram for a compact of Copper spheres: a = 40 pm. Data of Rhines et al. [25] are shown.

1.0

0.9

2 vr t 0 i 0.8

;I 4 L

0.7

TEMPERATURE OC 1500 2000 2500 3000

0.6 04 0.6 04 1.0

HOMOLOGOUS TEMPERATURE T/TM

Fig. 34. A density diagram for a compact of tungsten spheres: 0 = 2 pm. Data of Kothari [27] are shown.

itself, cause densification. We think that this is the reason for the activation energy observed by German. which is close to that for surface and for volume diffusion.

7.3.3 Rhirles et al. [ZS]. Measured densification of spherical copper particles (a = 3744 pm: we used a = 40~m) which had been densely packed but not compressed. The conditions were

T = 915-104O’C (0.884.97 Thl) t=85h

I Copper

AlA0 = 0.660.75

The results are plotted on the appropriate map, shown as Fig. 33. The agreement is within the expected error in the models and data.

7.3.4 Kothari [26.27]. We have encountered one set of data, that of Kothari [26.27] describing the sintering of tungsten. which is inconsistent with the models described here: Data for particles of size a = 2 pm, corresponding to the conditions

T = 110&15OO’C (0.37-0.48 Th() t = 8.3 h

I Tungsten

A/A0 = 0.68-0.74

is shown on Fig. 34. The observed times (and thus rates) differ by a factor of more than 100 from those predicted by the models. Kothari studied other particle sizes also (a = 0.25 pm and a = 7.5 m): there is only a very weak dependence of densification on particle size (for the same T and t); according to the models, there should be a very strong one.

Two explanations are possible. One is that sintering is not diffusion-controlled. but controlled (for example) by an interface reaction. This seems unlike- ly-an interface reaction should slow down the rate of sintering, yet in Kothari’s experiments the tungsten sintered far faster than the models predict. The other

SWINKELS ASD ASHBY: SECOND REPORT ON SINTERING DIAGRAMS 281

is that the particles. being irregular in shape. sinter at a rate determined b! the radius of the average irregu- larity. or surface-bump. not at a rate determined b! the mean particle radius. Then the models should fit the data if evaluated for an ‘apparent‘ particle radius equal to that of the bumps. It is hard to test this idea with precision: but as far as we can tell. it IS capable of explaining the observations.

8. SL’MMAR\’ AND COSCLCSlOSS

(a) The paper describes two ways of assembling models for sintering into useful diagrams. one showing tterk size and the other showing drttsiry as a function of time and temperature. These diagrams are a development and extension of those reported earlier [ I]. The new diagrams include an important coupling between grain boundary and surface diffusion. and modifications to the treatment of particle arrangement and of the stages of sintering.

(b) The diagrams can be used to display the sintering behaviour of a specific material: as an aid to the design and interpretation of experiment; and as a way of visualizing commercial sintering schedules and the effects of changes in schedule.

(c) We tested the diagrams against I7 sets of experimental data for wires. pairs of spheres and aggregates of spheres of Cu. Ag. Ni. Fe. W. Stainless Steel and NaCl. The tests provide a check of almost all the equations which are used to construct the maps. We find that. for wires and pairs of spheres of Cu. Ni. Ag and Natl. agreement is excellent, and that the maps provide a way of identifying the mechanisms of sintering in each experiment. with no a priori assumptions. even when several mechanisms operate at once. For aggregates of particles of silver. copper and stainless steel the agreement is less good (probably because of the difficulty of describing geometry properly in the models), but the method still provides a practical tool for analysing both experiments and commercial sintering procedures. For iron wires the agreement is poor. almost certainly because the rate of surface diffusion differs from the values we have used. For tung sten aggregates the agreement is again poor. perhaps because of the irregular particle shape.

The coupling between surface and boundary diffusion, included here for the first time, generally slows the rate of sintering at low temperatures. Only one set

of data allows a real test of this inhibition and sup ports it.

The computational scheme used to construct such diagrams is a flexible one which readily allows of further extension: the inclusion of grain growth. pore dragging. and externally applied pressure or stress- once their effect on the individual mechanisms is known.

Ac~rlor~/rdgPnlPrlr-This work was supported by the Science Research Council.

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Ashb? M. F.. .4cru merall. 22. 275 (1974). Thummter F. and Thomma W.. Powder Murail. 115. 69 (1969) Wilson T. L. and Shewmon P. G.. Troos. Maral!. Sot. A.I.M.E. 2.X. 48 (1966). Coble R. L.. J. Am Ceram. Sot. 41. 55 (19581. Easterling K. E. and Tholen A. R.. Gorhenhurg Inst. Phys. Repr 73 (1971). Easterlmg K. E. and Tholen A. R.. Metall. Sci. J. 4. 13U ( 1970). Easterline K. E. and Tholen A. R.. Acra f~~~,r~~~. 20. iool 11972). Gessineer G. H.. Scripra metoll. 4. 673 11970). Johnson D. L.. Scripfa mt~o//. 4. 677 (1970). Nichols F. A.. Drl;‘crs atld Trorlsport in O.~jdc~. p. 459. Plenum Press. Neu York (1974). Brass P. and Exner H. E.. .4rta nrtml/. 27. 1013 (19791. Swinkels F. B. and Ashbl M. F.. Powder ;~era~~~rg~. 23. 1 (19801. Bernal J. D.. Proc. R. Sot. A 280. 799 (1964). Rockland J. G. R.. Acru metal/. 14. 1273 (1966). Eadie R. L.. Wilkinson D. S. and WeatherI! G. C.. Acre merall. 22. 1 I85 (1974). Alexander B. H. and Balluffi R. W.. Acru mefall. 5. 666 (1957). Pranatls A. L. and Seiple L.. Powder .~~e~ff~~~~g~. Pro- ceedings of an International Conf. held m Neu York. June 13- 17 1960 (Edited by W. Lesqnskj). Inter- science. New York (1961). Matsumura G.. Acfa merall. 19. 851 (1971). Kineery W. D. and Berg M.. J. appl. Php. 26. 1205 (1955). Mills B.. Douglas P. and Leak G. M.. Trans. Muruli. Sot. A.I.M.E. 245. 1291 (1969).

Kyczynski G. C.. Trans. Merafi. Sot. A.I.M.E. 185. 169 (1949). Eadie R. L.. WeatherI) G. C. and Aust K. T.. .4cra metal/ 26. 759 (1978). Lee W. L.. Eadie R. L.. Weatherly G. C. and Aust K. T.. Acta mefaff. 26. I837 (I 978).

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