Role of grain boundaries in sintering

ROLE

OF GRAIN H.

BOUNDARIES and

ICHINOSEt

G.

IN SINTERING”

C. KUCZYNSKIf

The effect of the presence of grain boundaries on tha sintering rate was investigated by measuring the growth of necks between three twisted wires as a function of time and temperature. It wss found that, in the presence as well as absence of the grain boundary in the neck, volume diffusion controls the process. The measured rate of neck growth was in good agreement with that calculated from the theory. The rate of approach of centers of wires was also studied. In the absence of grain boundaries no approach of centers wss observed. In the presence of grain boundaries, the centers did approach et a rate which was measured. This study allowed distinction between grain boundary and surface sinks of vacancies during sintering; in the absence of grain boundaries, the diffusion flux of vacancies is from the region just beneath the neck to the adjacent free surface; in the presence of grain boundaries, vacancies migrate from the region just, beneath the neck to the grain boundary. ROLE

DES

JOINTS

DE

GRAINS

LORS

DU

FRITTAGE

L’effet de la presence de joints de grains SUPla vitesse de frittage a 6th Btudie par la mesure de croissance des colliers, entre trois fils tordus en fonction du temps et de la temperctture. On a trouve que, en presence aussi bien qu’en absence de joints de grains dans le collier, la diffusion en volume controle le processus. La vitesse de croissance des colliers mesuree, Btait en bon accord avec celle calculee suivant la theorie. La vitesse de rapprochement des centres des fils a et6 aussi etudiee. En absence de joints de En presence de joints de grains, les centres grains, aucun rapprochement de centres n’rt Bte observe. n’approchaient avec une vitesse qui a et& mesuree. L’etude a permis de distinguer entre les joints de grainset les chutes en surfaces des lacunes pendant le frittage, en absence de joints de grains, le flux de diffusion des lacunes se dirige de la region juste en dessous du collier vers la surface libre adjacente; en presence de joints de grains les laounes migrent de la region immediatement sous adjacente au collier vers les joints de grains. DIE

BEDEUTUNG

DER

KORNGRENZEN

FUR

DIE

SINTERUNG

Urn zu untsrsuchen, welchen EinfluD die Anwesenheit vorn Kongrenzen auf die Sinterungsgeschwin digkeit hat, wurde das Wachstum der “H&e” zwischen drei verdrillten Driihten in Abhiingigkeit von Zeit und Temperatur gemessen. Es zeigte sich, dal3 sowohl mit als such ohne Korngrenzen in den H&en die Volumdiffusion maogebend fiir diesen ProzeD ist. Die gemessene Wachstumsgeschwindigkeit der Hiilse stimmt gut mit der nach der Theorie berechneten iiberein. AuDerdem wurde die Geschwindigkeit Ohne Korngrenzen wurde keine untersucht, mit der sioh die Drahtachsen aufeinander zu bewegen. derartige Bewegung der Drahtachsen beobachtet, wiihrend bei Anwesenheit von Korngrenzen eine solche estgestellt und die Gesohwindigkeit gemessen wurde. Auf Grund dieser Untersuchung konnte entschieden werden, ob sich Leerstellen beim Sintern an Korngrenzen oder an der Oberfliiche anlagern; die Leerstellen kommen in jedem Fall aus der Gegend unmittelbar unter dem Hals; sie wandern dann an die n&chste freie Oberfliiche, wenn keine Korngrenzen da sind. dagegen an die Korngrenze, wenn eine solehe vorhanden ist.

INTRODUCTION

The phenomenological of its first stage adjacent lished.

particles

during form,

The motivating

each other or with a flat plate a rather simple relation-

theory of sintering, especially which

bonds

between

has been fairly

ship between the neck radius x, time t and temperature

the

of sintering 7’ has been found:(l)

well estab2” -

stresses and gradients are due

am

to the unbalanced surface forces perpendicular to the sharply curved surface in the neck formed between two

adjacent

mechanisms plastic

flow,

condensation volume

particles.

Consequently

of extending

this neck are:

evaporation

from

convex

where a is the radius

the possible viscous parts

function

or

flow.

and

diffusion.

METALLURGICA,

VOL.

19, MARCH

1962

of the particle

and A(T)

only appropriate

n and m allow us to identify

during the process.

a

for a given the

The particles

viscous or plastic flow if 12 = 2 and m = 1 evaporation and condensation if n = 3 and m = 1 volume diffusion if n = 5 and m = 2

with

* This paper is based on a part of the thesis submitted by H. Ichinose in partial fulfillment of the requirements for the degree of Doctor of Philosophy to the Graduate School of the University of Notre Dame. Received September 1, 1961. 7 Department of Metallurgical Engineering, University of Notre Dame, Notre Dame, Indiana. ACTA

A(T)t

sinter by

In the case of simple

systems such as two spheres or wires in contact

The exponents

flow predominant

in the cavities (neck) of the system and

or surface

of temperature

=

surface diffusion if n = 7 and m = 3 These relations were verified in a series of experiments involving spherical particles and wires.(r-6) The 209

210

ACTA

METALLURGICA,

VOL.

10,

1962

where D’ is the coefficient of self-diffusion

of vacancies

and p the smaller radius of curvature of the neck. AC,is yV,C,,/RTp

excessvacancyconcentration, Cc is the vacancy

concentration

As COD’ = D, the alp -

IO2 or In

The where

under a flat surface.

coefficient

of

self-diffusion

(u/p) N 5 the expression

and

for the flux

can be written J

TVOD

=

5 RTp2 This flux must be equal to the rate of increase of the

V per its area A, d V/A&.

neck volume

If two wires

of radius a are sintered to each other and there is no grain boundary in the neck, the volume per unit axial length

B

A

experiment nance

of the volume

case equation

diffusion

In this

(1) can be written in the form X5

we obtain X5

the predomi-

mechanism.

KYVODt

(2)

s=j+-

of the neck as in Fig. l(a) and

p E x2/2a. Therefore after integration of equation (3),

on sintering of wires and spheres of metals

such as copper and silver demonstrated

V g xp and the area A = 2p,

of the neck

where x is half-width

Fig. 1. Schematic representation of a section through two sintered wires: (A) without grain boundary, (B) with grain boundary.

8V4Dt

3

iTp

If the vacancy

sinks are in grain boundary

p z x2/4a V g

2xp/d2

integration

X5

R the gas constant and D the self-diffusion coefficient of the metal; K is a constant depending on

volume,

the geometry path.

this paper. Equation the

of the sample and also on the diffusion

The exploration

of the latter is the subject

(2) can be derived from consideration

vacancy

concentration

underneath

the

of of

highly

curved surface of the neck, represented schematically in Fig. 1. It can be shown’132) that in that area there exists an excess

concentration

tional to the curvature. visualized

of vacancies

The growth

as the replacement

of these vacancies

by

cies diffuse in the opposite

direction

The vacanand have to be

either on the surface of the system

away

from the neck (Fig. la) or in a grain boundary(3)

(Fig.

lb).

The former

mechanism

does not change appreciably

of vacancy

elimination

the distance between the

grain boundary

(5)

RT

3

K corresponding

In the case of sintering

of spherical

to

is about particles

this ratio should be close to 5.3. It is possible to verify this theory by measuring

directly

the radius x of the

neck formed between two wires or spherical as a function

particles

of time, in the presence and absence of

the grain boundary

in the neck area.

EXPERIMENTAL

PROCEDURES

The principal experimental was to eliminate

problem in this research

the grain

boundary

in the neck

between two particles.

Because of the small radius of of the neck, the grain boundary has a

curvature tendency

to stay in the neck during sintering.

ever Zaplatynsky(7)

observed

by recrystallization

point.

This

method

boundary-free

necks.

latter,

thick

was

was adopted on

into very

near the melting to obtain

A layer of copper

electroplated

How-

that a layer of copper

plated on the copper wire can be converted large crystals

centers of the spheres or wires because the replacement the material

162/2rr y V, D t

and surface sinks, respectively,

atoms are essentially supplied from the surface.

In the filling the neck comes from the

and A = 2p in which case the

Thus the ratio of the constant

propor-

of the neck is

atoms diffusing from inside of the system. eliminated

-. =p a2

2.83.

(Fig. lb),

yields

V, its atomic

where y is surface tension of the metal,

(4)

RT

99.999%

about

Cu

grain 14 p

wires

of

lenticular area in Fig. l(b). In this case the center to center distance should decrease with time. As the symmetry of the neck can be considered cylindrical, the flux of vacancies J through its area is approxi-

0.005 in. diameter from the standard CuSO, solution with a little phenol added to insure smoothness of the In some deposit ; current density was 33 mA/cm2. specimens a very thin layer of molybdenum was

mately

introduced

J=

TD’AC P In (alp)

between

the

core

of the wire and the

plated layer of copper, as an inert marker. Molybdenum was plated from aqueous solution of molybdic

ICHINORE

acid in ammonium prepared

acetate.(s)

by uniformly

on a jeweler’s

AND KUCZYNSKI:

lathe.

wires were removed

GRAIN

The specimens

twisting

together

BOUNDARIES

IN

SINTERING

211

were

three wires

To prevent untwisting

when the

from the lathe, the ends of the

wires were bound with copper wire to two copper pins mounted on a copper block. Plain as well as plated wires were thus twisted and sintered in vacuum at 950, 1000 and 1050°C.

At certain intervals

of time

ranging from a few hours to 250 hr, the specimens were withdrawn from the furnace and mounted in lucite

by polymerizing

monomer

liquid

methyl

in a small amount of benzoyl

methacrylate peroxide.

FIG. 3. Photomicrograph of a section through wires plated with copper, sintered at 1050°C for 50 hr. Etched with potassium dichromate solution. / 240

FIG. 2. Photomicrograph of a section through plain wires sintered at 1050°C for 50 hr. Etched with potassium dichromate solution. x 240

The

sintering

represented

of

plain

wires

resulted

by the photomicrographs

grain boundaries

visible

in necks

in Fig. 2.

in the necks between

The them

persisted almost to the end, that is, to the final closure of the internal cylindrical pore. In contrast, the copper plated wires and especially the copper-molybdenum plated wires sintered, developing large single crystals in the necks, as can be ascertained

from Figs.

3 and 4. All these specimens were etched in potassium dichromate solution. RESULTS

AND

DISCUSSION

The widths of the necks formed between the wires, as those represented in Figs. 2,3 and 4, were measured under the microscope. 3

The results were plotted

as

FIG. 4. Photomicrograph of a section through wires plated with molybdenum and copper, sintered at 1050°C for 50 hr. Etched with potassium dichromate solution. x 240

METALLURGICA,

ACT.4

212

VOL.

10,

1961

depicted in the conventional TEMPERATURE

1050-C

wire

plot given in Fig. 6. All

the points, those corresponding without grain boundaries

0

plain

A

Cu ploted

(g)

(g)

straight

line

.

CUplated

($1

following

equation :

This

which

is in fair

coefficient

can

be

agreement

of copper

to the necks with and

fall on essentially

the same

approximated

with

the

as determined

by the

self-diffusion

by radioactive

tracers.(s) Additional

verification

from measurements 10

0

20

30 TIME

40

50

60

in hrs.

time.

of the centers of the wires with single crystal necks between them did not change with time. However the centers of wires separated by grain boundaries in the neck did approach each other. A plot of the logarithm

Fm. 5. Plot of x5/a” versus time for the specimens sintered at 1050°C.

x6/a2 against

wires as a function

of this model was obtained

of the center to center approach of of time. As expected the distance

One of these graphs is given in

Fig. 5. The proportionality that the predominant

of x5/a” to time indicates

mechanism

of sintering

in this

case is that of volume diffusion. Moreover the slopes of the lines corresponding to the necks with grain boundaries

are higher

than

those

obtained

for the

single crystal necks. The ratio of these slopes is about 3, in excellent agreement with the predicted value of the ratio of constants

K discussed in the Introduction.

It should be noted that the layer of molybdenum

does

not seem to obstruct the sintering, for the points obtained from the necks formed between molybdenum plated

33

’

1

4.0

4.5

1 Log

wires fall on the same line as those obtained

for copper-plated were essentially

wires.

From equations diffusion de~rmined

were

Of course

all these necks

single crystals. (4) and (5) the coefficients

calculated

slopes.

from

the

Their ~mperature

I

I

1

6.0

5.0 5.5 TIMEfinsec)

FIG. 7. Rate of approach of centers of plain copper wires.

of selfof the ratio of the center-to-center

experimentally dependence

IO-*1

is

against the logarithm

I

Since AL s (5) we obtain

approach,

AL/L

of time, is given in Fig. 7.

2p, AL/L E x2/&9 and from equation AL -= L

_____. ---_

(6)

The slopes of the straight lines in Fig. 7 are very close to Q verifying the above equation. of self-diffusion

calculated

from

results with the help of equation

The coefficient

these experimental (6) yields

( - 55,400) D = 2.4 exp -%?cm2/sec in good agreement

with that obtained

growth measurements. The comparison of Fm. 6. Self-diffusion coefficients of copper obtained from neck’s growth, as the function of time.

coefficients

the

of copper obtained

experiments(l,3,7,10-13)

with

values

from the neck

of

self-diffusion

from various sintering those

obtained

from

ICHINOHE

*ND

KUCZYNSKI:

GRBIN

BOUNDARIES

IN

SINTERING

213

values seem to fall fairly well on a straight line which can be best approximated

STEIGMAN,etol,TRACER ROLLIN,TRACER MAIER,etal,TRACER RAYNOR,et al,TRACER KUCZYNSKI, SPHERE PLATE COHEN,etal;ALEXANOEF et al, WIRE-PLATE v SHALER,SPHERESPHERE n OEORICK,et al, SPHERE-SPHERE

D =

by the equation

( - 53,000)

2.4 exp ~-RT--

This dependence

of self-diffusion

temperature

is in fairly good

found

the results

from

described

cm2/sec coefficient

agreement

of the sintering

upon with those experments

in this paper. ACKNOWLEDGMENT

. The authors

are indebted

Research for sponsoring

to the Office of Naval

this research.

REFERENCES

i

0 0

SPHERE-SPHERE APPROACH OF

0

ZAPLATYNSKY, WIRE- CYLINDER

0 v A

KUCZYNSKI 8 ICHINOSE WIRE-WIRE APPROACH OF CENTERS SHRINKAGE OF PORE

0.7

0.8

0.9 1.0 1000 T°K

0

I.1

1.2

I.3

Fro. 8. Cumulative plot of self-diffusion coefficients of copper obtained from sintering data and measured with radioactive tracers techniques.

measurements

with radioactive

tracers(Q.14~15)is given

in Fig. 8 in the form of conventional

plot.

All these

1. G. C. KUCZYNSKI, Trans. Amer. Inst. Min. (Metnll.) Engrs. 185, 169 (1949). 2. G. C. KUCZYNSRI, J. AppZ. Phw. 21. 632 (1950). 3. W. D. KINGERY and M:BERc, J. AppZ. 2hy.y: 26, 120,5 (1955). 4. R. I. COBLE, J. Amer. Ceram. Sot. 41, 55 (1958). 5. G. C. KUCZYNSKI, L. ABERNATHY and J. ALLAN, Kinetics of High Temperature Processes p. 163. Technology, Cambridge, Mass. Press. 6. G. C. KUCZYNSKI, J. &pZ. Phys. 20, 1160 (1949). 7. I. ZAPLATYNSKY, Ph.D. Thesis, Metallurgical Engineering Department, University of Notre Dame (1957). 8. M. J. KSYCKI and L. F. YNTEMA, J. EZectr. Chem. Sot. 96, 48 (1949). 9. J. STEICMAN, W. SCHOCKLEY and F. C. NIX, Phys. Rev. 56, 13 (1939). 10. G. COHEN and G. C. KUCZYNSKI, J. AppZ. Phys. 21, 1339 (1950). 11. G. C. KUCZYNSKI and H. ICHINOSE, to be published. 12. A. J. SHALER, Trans. Amer. Inet. Min. (Metall.) Engrs. I

185,796 (1949). 13. J. H. DEDRICK and A. GERDS, J. AppZ. Phys.

20, 1042

(1949). 14. D. V. ROLLIN, Phys. Rev. 55, 231 (193!1). 15. M. S. MAIER and H. R. NELSON, Trans. Amer. Inet. Min. (MetnZZ.) E’ngrx. 147, 39 (1942).