Sintering of copper powder compacts

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Authors Imprescia, Richard John, 1932-

Publisher The University of Arizona.

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SINTERING OF COPPER POWDER COMPACTS

by

Richard John Im prescia

A T hesis Submitted to the Facu lty of the

DEPARTMENT OF METALLURGICAL ENGINEERING

In P artia l Fulfillm ent of the Requirements For the Degree of

MASTER OF SCIENCE

In the G raduate C o llege

THE UNIVERSITY OF ARIZONA

1 9 6 5

STATEMENT BY AUTHOR

This th e s is has been subm itted in partia l fu lfillm ent of re q u ire ­ments for an advanced degree a t The U niversity of Arizona and is d e p o s i ­ted in the U niversity Library to be made av a i la b le to borrowers under ru le s of the Library.

Brief quota tions from th is th e s is are a llow able w ithout sp ec ia l perm ission, provided tha t accu ra te acknow ledgm ent of source is made. R equests for perm ission for ex tended quota tion from or reproduction of th is m anuscrip t in whole or in part may be granted by the head of the major departm ent or the Dean of the G raduate C ollege when in his judgm ent the proposed u se of the m ateria l is in the in te re s ts of sch o la rsh ip . In a ll o ther in s ta n c e s , however, perm ission must be ob ta ined from the author.

SIGNED:

APPROVAL BY THESIS DIRECTOR

This th e s is has been approved on the da te shown below:

Professor of M eta llu rg ica l EngineeringD ate

ACKNOWLEDGMENTS

The author w ish es to ex p ress h is thanks to Dr. D. J. Murphy

for h is adv ice and encouragem ent during th is in v es tig a tio n . Thanks

are a lso ex tended to Dr. L. J. Demer for h is helpful comments during

the course of th is study, and to Mr. W. E. H orst for his help in

c larify ing se v e ra l important poin ts concerning the a n a ly s is of e r ro rs .

i l l

TABLE OF CONTENTS

Page

LIST OF TABLES. ............................... v

LIST OF ILLUSTRATIONS................................................................... v,i

ABSTRACT ................... v ii

I INTRODUCTION ................... 1

II SINTERING THEORY................... 4

, III SCOPE OF THE INVESTIGATION.................................................... 9

IV EXPERIMENTAL PROCEDURE ....................... 11

A. T est M ateria l and Specimen Preparation 11

B. Sintering ................................ 12

C. D ensity M easurem ents ..................... 14

V RESULTS AND DISCUSSION ............................................. 16

A. D en s if ica tio n D a t a ................................ 16

B. A ctivation Energies . . ................................................... 17

C. S intering M echanism s....... ......................... 19

D. S uggestions for Further Work . ......................................... 24

VI SUMMARY AND C O N C L U S IO N S ....................... 27

REFERENCES . . ................. 29

APPENDIX I .................................................................. 44

iv

Table

I

LIST OF TABLES

Page

D en s if ica tio n Parameter, a ............................................. 31

v

1

2

3

4

5

6

7

8

9

10

11

12

LIST OF ILLUSTRATIONS

Page

Sintering Furnace ...................... ; . . . ....................... 32

P ush-P u ll Rod and Sintering Boat Assembly ....................... 33

Sintering Boat and Thermocouple A s s e m b ly ............................. 34-

Plot of a v e rsu s t for C oarse S ize -F rac tio n ....................... 35

Plot of a v e rsu s t for Medium S i z e - F r a c t i o n ........................ 36

Plot of a v e rsu s t for Fine S ize -F rac tio n ........... 37

Plot of a v e rsu s T for C oarse S i z e - F r a c t i o n ........................ 38

Plot of a v e rsu s T for Medium S ize -F rac tio n . ................. 39

Plot of a v e rsu s T for Fine S ize -F rac tio n ........................... 40

Plot of Log t v e rsu s l / T ............................................... 41

Plot of t v e rsu s t . . . . . . . . . . . .......................................... 42c f

Schem atic R epresen ta tion of Sintering M e c h a n i s m s ............... 43

vi

ABSTRACT

The d en s if ica t io n of copper powder com pacts prepared from

th ree different powder s iz e - f ra c t io n s was in v e s t ig a te d . Sintering w as

carried out in the tem perature range 7 0 0 -8 5 0 “C, and over a time range

of 1-32 hours.

The ac t iv a t io n energy a s s o c ia te d w ith s in tering has been found

to be independent of powder p a r t ic le - s iz e , and equal to 76 k ca l/m o l.

Application of H erring 's a n a ly s is to the d e n s if ic a t io n data

g iv es the following re su l ts :

a . For low tem peratures the predom inant s in ter ing m echanism

is surface diffusion.

b. For higher tem peratures a p la s t ic flow m echanism is

operating during the ea r l ie r s ta g e s of s in tering , while

evapora tion and cond en sa tio n opera tes a t la te r s ta g e s .

c. The larger the p a r t ic le - s iz e the longer w ill be the

s in ter ing period during w hich a g iven m echanism is

opera ting .

v ii

I. INTRODUCTION

W hen a number of powder p a r t ic le s are brought into in tim ate

co n tac t and hea ted , adherence betw een th e p a r t ic le s occu rs . As the

time or the tem perature of heating is in c re a se d the ad h es io n in c re a se s

to the ex ten t th a t the powder m ass becom es a porous so lid having

some p h y s ic a l s trength . For su itab le tim es and tem peratures of

heating the porous so lid becom es more and more d ense , approaching

the d en s ity of the so lid m etal.

The driving force for the above d esc rib ed p ro c e ss is the

su rface free energy of the m ass of powder p a r t ic le s . A m easure of

the su rface free energy is the d ifference betw een the to ta l surface

a rea of the o v e r -a l l m ass of com pacted powder p a r t ic le s and the su r ­

face area of a so lid sphere of an eq u iv a len t volume. Therefore, when

powder p a r t ic le s are brought into in tim ate co n tac t the tendency is to

lower the free energy difference by d e c re a s in g the to ta l su rface area

of the com pacted p a r t ic le s . The sm aller the s iz e of the indiv idual

p a r t ic le s the g rea te r the free energy d ifference , and, hence, the

g rea te r the tendency toward the solid m etal. W hen th e se changes

are brought about by the add ition of h ea t the p ro cess is c a l le d

"s in te r in g . "

2

There are four b a s ic m ateria l transport m echanism s by w hich

s in ter ing is thought to occur. These are (a) evaporation and co n d en ­

sation , (b) su rface diffusion, (c) volume diffusion, and (d) p la s t ic

flow. It i s gen era lly thought (1) th a t w hile the f irs t two p ro c e sse s

can contribute to bond formation betw een p a r t ic le s and to the sphe-

ro id iz ing of angular vo ids, they cannot produce any app rec iab le

shrinkage and d en s if ica tio n . E ssen tia l ly , m ateria l is simply t r a n s ­

ported from one in te rna l surface to another by surface d iffus ion and

by evapora tion and condensa tion , leav ing the to ta l pore volume

unchanged. The major contribution to d e n s if ica t io n is u su a lly

a ttr ibu ted to one of the o ther two p ro c e sse s , volume d iffus ion or

p la s t ic flow. Volume diffusion is a p ro cess of m ateria l transport

orig inating in the random movement of atom s induced by thermal

v ib ra tion . When a concen tra tion g rad ien t e x is ts , such random

motion r e su l ts in a net transport of atom s down the g rad ien t. In the

s in ter ing of a pure su b s tan ce th e g rad ien t is thought to c o n s is t of

d iffe rences in the concen tra tion of la t t ic e d e fec ts under variously

curved su rfaces (2). P la s t ic flow is a p ro cess by w hich transport of

m atter occurs under the ac tion of s t r e s s . The origin of the s tre s s

lead ing to p la s t ic flow is cons ide red to be surface ten s io n .

Since sev era l rev iew s on s in te r ing are av a ila b le (1, 3-9),

i t is in tended in the following se c t io n to d is c u s s only the currently

3

important theo r ie s and some of the experim ental ev idence supporting

them.

II. SINTERING THEORY

The theory of s in tering in recen t years has been concerned

predom inantly w ith the m echanism of m ateria l transport during the

s in tering p ro c e ss . In sp ite of the effort which has been devoted to

the su b jec t there are s t i l l wide d ivergences of opinion regarding the

p rinc ipa l transport mechanism in c ry s ta l l in e su b s ta n c e s . These d if ­

fe rences are made m ost apparen t by the d ifferent schoo ls of thought

which e x is t in the U nited S ta tes and G reat Britain. In the U nited

S ta tes , a t ten tio n has been focused primarily on the d iffusion theory

of s in tering su g g es ted by Kuczynski (2). In G reat Britain, em phasis

has been on the p la s t ic flow th eo r ie s of M ackenzie and Shuttle worth

(10), and Clark and W hite (11).

Kuczynski examined th e o re t ic a l ly w hat the ra te of growth of

the region of co n tac t betw een a sphere and a plane surface on heating

should be in the c a s e s of the four b a s ic s in tering m echanism s. The

re su l ts of his a n a ly s is y ie lded the following re la t io n sh ip s :

2x CC t for p la stic flow;

3x cc t for evapora tion and condensa tion ;

5x OC t for volume diffusion;

4

x oC t for surface diffusion;

where x is the rad ius of the growing "neck" a t the co n tac t betw een

the sphere and the plane, and t is the s in tering tim e. By comparing

the re su l ts of s in tering experim ents conducted on spheres of s ilver

re s t in g on plane s ilv e r su rfaces , along with s im ilar experim ents

us ing copper, w ith the pred ic tions of h is theory, Kuczynski claim ed

to have e s ta b l is h e d th a t volume d iffus ion w as the predominant

mechanism for large p a r t ic le s a t high tem peratures , and th a t surface

d iffus ion w as involved a t lower tem pera tures and a t the beginning of

s in tering .

Lending support to the d iffus ion theo r ie s of s in te r ing is the

im portant work of A lexander and Balluffi (12). In the ir s tudy on the

s in tering of copper w ires wound in c lo se -p a c k e d arrangem ent on

spools of the same m ateria l, they observed th a t rap id shrinkage of

the pores betw een w ires took p lace only so long as the pores were in

co n tac t w ith a grain boundary along the wire surface . Pores were

shown to d im inish or d isap p e a r in the reg ions of grain boundaries but

not in reg ions betw een such boundaries . Their ex p lan a tio n for th is

e ffec t is th a t the grain boundaries a c t a s "p ipes" down w hich atoms

d iffuse to f il l the pores. It thus appears th a t th is e f fec t may not be

a ttr ibu ted to a p la s t ic flow m echanism, s in ce p la s t ic flow depends on

su rface tension , and s in ce the surface te n s io n on the w all of a pore

would not be expec ted to change app rec iab ly by the d isap p e a ran ce of

a grain boundary. There are, however, arguments in favor of the

p la s t ic flow theory.

The f ir s t of the p la s t ic flow theo rie s in order of time is tha t

of M ackenzie and Shuttleworth (10). They observed th a t the ra te of

c lo su re of pores in a porous com pact w as high compared to the ra te

of c lo su re of an iso la te d pore. To in terp re t th is o bserva tion they

advanced the argument th a t in the neighborhood of an iso la te d pore

the shear s t r e s s c a u se d by su rface te n s io n d e c re a se s w ith d is tan ce

from the pore, and therefore, beyond a c e r ta in d is tan ce , the s tre s s

w ill d im inish to a va lue below the y ie ld s t r e s s , no flow w ill be p o ss ib le ,

and consequen tly no pore c lo su re w ill occur. In continuing the ir a rg u ­

ment they s ta te th a t in a porous com pact the s t r e s se d reg ions overlap

and the y ie ld s t r e s s w ill be ex ceed ed and flow w ill occur, s low ly

filling up the p o re s . Their work p resen ted two equa tions for the

s in tering ra te , one for a purely v isc o u s m ateria l and the other for a

m ateria l exhib iting a y ie ld point (Bingham-type flow ).

The second theory is th a t of C lark and W hite (11) who

exam ined the p o s s ib i l i ty of p la s t ic flow of a layer of m ateria l to

form a neck be tw een two sp h e r ica l p a r t ic le s . It w as a ssum ed th a t

the p a r t ic le s rem ain sp h e r ica l and th a t flow or d iffus ion m ain ta ins

th is form w hile m ateria l is transfe rred to the neck cau s in g the volum es

of the p a r t ic le s to d e c re a se . The model is based on the m acroscopic

flow of m ateria l from the su rfaces of the spheres to the neck, the

driving force, here again , being su rface ten s io n . Equations were

derived for both purely v isc o u s and Bingham-type flow.

The th eo r ie s of M ackenzie and Shuttle worth and of Clark and

W hite are complementary, s ince one ap p lie s to the la te r s ta g e s of

s in ter ing during w hich the pores are iso la te d , w hile the other ap p lie s

to the early s ta g e s of s in tering w herein bonding betw een p a r t ic le s

tak es p lace . Attempts have been made (11, 13, 14) to fit e x p e r i­

m ental da ta to both of th e s e theo rie s , and in a ll c a s e s , i t was found

th a t to ob ta in good agreem ent w ith the theory i t w as n e c e s sa ry to

in troduce an em pirical correc tion factor. Because of th is correc tion

fac to r th e se theo r ie s are frequently d isco u n ted a s being invalid .

More recen tly , however, experim ents of the K uczynsk i-type (15)

have g iven new support to the p la s t ic flow theory. In th e s e e x p e r i ­

ments neck growth s tu d ies were made on spheres h ea ted on variously

t i l te d p lan es . Under th e se conditions i t w as concluded th a t p la s t ic

flow w as the predominant m echanism . This t i l t ing condition, although

s t i l l id ea lized , can be cons ide red to approximate more nearly the a c tu a l

s in tering conditions in a powder com pact.

In eva lua ting the experim enta l ev idence p re sen ted above on

the various th eo r ie s , i t becom es apparen t th a t i t probably would be an

overs im p lif ica tion to attem pt to d esc r ib e s in tering in terms of an

iso la te d m echanism . It is ce r ta in th a t d ifferent s in te r ing m echanism s

are predominant in d ifferent m ate ria ls , and th a t in a s ing le m ateria l

d ifferent m echanism s may be s ig n if ican t under various c ircu m stan ces .

It is not d iff icu lt to v isu a l iz e , for example, th a t p la s t ic flow might be

favored a t ea r l ie r s ta g e s of bond growth betw een p a r t ic le s when the

curvature a t the point of co n tac t is sharp and the s t r e s s e s in the neck

p o ss ib ly high. Also, i t may be re a so n a b le to assum e th a t a t lower

tem peratures , where su r fa c e - te n s io n e ffec ts are unim portant and vapor

p re ssu re s are too low for evapora tion and co ndensa tion to take p lace ,

surface or volume d iffus ion may predom inate . In the la te r s ta g e s of

long s in tering in te rv a ls the driving force for diffusion, i . e . , the defec t

concen tra tion gradient, may be no longer s ign if ican t, and su rface - te n s io n

e ffec ts may be neg lig ib le . Under th e s e conditions evapora tion and c o n ­

d en sa tio n may be the only important m echanism .

In conclusion , i t appears th a t the genera l course of s in tering

is s t i l l uncerta in . A g rea t many fac ts have been am assed , but as yet

no com plete theory b ased on f ir s t p r inc ip les is a v a i la b le .

III. SCOPE OF THE INVESTIGATION

Many stud ies have been made of the effect of various condi­

tions on sin tering. The effects of time, temperature, and p a rtic le -s ize

on sintering have been examined, to some extent, on most powdered

m aterials, but there appears to have been no system atic study of the

effect of these three variab les on the sintering of a single m aterial.

The present study wasr therefore undertaken as a step in acquiring

system atic knowledge of the sintering of powdered m aterials. For

p rac tical reasons, copper was se lec ted as the m aterial to be studied.

The specific aims of th is investigation have been to study

the sintering behavior of copper powder com pacts over a range of

sin tering tim es and sintering tem peratures, and to determine the

effect of powder p a rtic le -s ize on th is behavior. Throughout th is

study, the quantity used as a measure of the extent of sintering has

been "densification , " a s defined by a d im ensionless param eter in tro­

duced in a la te r section (Results and D iscussion).

In order to examine sintering behavior, two analyses have

been performed on the densification data . The f irs t of these is the

determ ination of the activa tion energies a sso c ia ted with sintering

of copper com pacts prepared from different powder p a rtic le -s iz e s .

The method used in th is determ ination is the same a s th a t employed

by Jordan and Duwez (16). The second a n a ly s is is an ev a lu a tio n of

the s in ter ing m echanism s w hich operate under the various conditions

of time, tem perature , and powder p a r t ic le - s iz e . The method used

here is the s c a l in g - la w c rite rion developed by Herring (17).

A d e ta i le d d is c u s s io n of the ap p lica t io n of the above

an a ly t ic a l tech n iq u es is g iven in Section V, Results and D isc u ss io n .

IV. EXPERIMENTAL PROCEDURE

A. Test M ate ria l and Specimen Preparation

The copper powder u sed w as th a t d es ig n a te d by th e m anufac- -

turer (M etals D is in teg ra ting Company, Elizabeth, New Jersey) as

M D -151 . The powder w as sc reen ed in to th ree s iz e - f ra c t io n s :

-150 +200, -200 +270, and -270 +325 mesh Tyler S e r ie s .

Throughout the rem ainder of th is paper th e se frac tions w ill be

referred to as coarse , medium, and fine f rac tions , re sp e c t iv e ly .

To e lim inate the e ffec t of s iz e seg reg a tio n during screen ing , each

frac tion w as blended by ro lling in a w ide-m outh bo ttle for two h o u rs .

Additional ag i ta t io n of the powder w as provided by includ ing in the

b o tt le s arrays of ben t aluminum w ires .

C y lindrica l spec im ens of approxim ate d im ensions 0 . 5 0 inches

in d iam eter x 0. 16 inches in th ic k n e ss w eighing 3. 00 grams each were

p re sse d in a non-lub rica ted , d o u b le -ac t in g h a rd e n e d -s te e l d ie . Com­

pacting p re ssu re for a l l p re ss in g s w ere 20, 000 psi; e ach specim en w as

m ain ta ined a t th is p re ssu re for approxim ately 10 seco n d s .

11

12

B. Sintering

All s in ter ing s tu d ie s were carried out in the tem perature range

7 00-850°C in a com m ercially - bu ilt e le c tr ic r e s i s ta n c e furnace (Hevi

Duty Company, Model 2012; see Fig. 1). The s in tering time range

covered w as 1 to 32 hours, and the a tm osphere was hydrogen. The

hydrogen w as dried by p ass in g the gas through a calcium su lfa te

d e s ic c a n t ; i t w as deox id ized by a DEOXO c a ta ly t ic purifier m anufac­

tured by Englehard Industr ie s , Newark, New Jersey .

The sin ter ing chamber c o n s is te d of a 2 - in ch d iam eter s ta in le s s

s te e l tube 24 in ch es long. The middle 12 in ch es re s te d w ith in the fur ­

nace and the two 6 - in ch ends protruded o u ts id e the furnace and were

coo led by w ate r cooling c o i ls fa s te n e d to them. One end of the furnace

w as arranged in a manner such th a t the s in te r ing boat w hich con ta ined

the spec im ens could be rapid ly in se r ted into the furnace hot zone or

re tr ieved into the w a te r -co o led end of the furnace tube by a ttachm ent

of a s te e l p u sh -p u l l rod (Fig. 2). In e ach s in tering run, five dup lica te

specim ens made from each powder s iz e - f ra c t io n w ere run s im u ltan eo u s ly .

F igures 2 and 3 i l lu s t ra te the s in ter ing boat u sed throughout

the experim ents. It c o n s is te d of four s te e l tu b es fa s te n e d in a v e r t ic a l

p o s it io n to a s te e l p la te . The tu b es were com pletely con ta ined w ith in

a d iam eter of 1 -3 /4 in ch es . The spec im ens to be s in te red were s tac k ed

13

w ith in the tu b es . To prevent the specim ens from s tick ing during

sin tering , a l igh t sprinkling of 100-m esh alumina powder w as p laced

betw een each spec im en and on the bottom p la te .

To help com pensate for the time tak en to h ea t the specim en,

the furnace w as in i t ia l ly h ea ted to a tem perature approxim ately 70° in

e x c e s s of the d es ired s in ter ing tem perature . Upon in se r t io n of the

specim ens into the hot zone, the furnace w as r e s e t for the correc t

tem perature .

By 'placing a therm ocouple in d ire c t co n tac t w ith a specim en in

the s in ter ing boat, m easurem ents were made of the hea ting and cooling

t im es a s s o c ia te d with in se r t io n and rem oval of the boat from the furnace

hot zone. This arrangem ent is i l lu s tra te d in Fig. 3. It w as determ ined

th a t a t 7 00°C approxim ately 1 -3 /4 min. were required to h ea t the s p e c i ­

men; a t 850°C 3 min. were required. The time required to cool the

specim ens to a tem perature of 450 °C from each of the above tem pera-\

tu res w as approxim ately 1 min. Sintering, then, w as e s s e n t ia l ly s topped

im m ediately upon removal of the spec im ens from the hot zone to the cold

zone.

The nature of th is s in ter ing system w as such th a t th e specim en

tem perature could be con tro lled no c lo se r than w ith in + 10°C of the

tem perature s e t on the con tro ller . This w as a regu lar and reproducib le

f luc tua tion due to the ON-OFF power c y c le of the contro lle r . Because

14

of the regu lar nature of th is f luc tua tion i t is fe l t th a t the effect, a t

l e a s t in part, w as com pensating (see Appendix I). C onsidering the

con tro lle r s e tt in g to have been the average tem perature, the tem pera­

ture of the hot zone w as held co n s ta n t to + 2° C over the 1 -3 /4 inch

leng th conta in ing the spec im ens.

C. D ensity M easurem ents

Before and af te r e ach s in ter ing run the d iam eter and th ic k n e ss

of each specim en were m easured . The volum es of the specim ens were

c a lc u la te d from th e s e d im ensions, and the d e n s i t ie s determ ined by

dividing the w eight of each spec im en by the c a lc u la te d volume.

W eights were determ ined on an a n a ly t ic a l b a lance to the n e a re s t

0 .5 mg, and d im ensional m easurem ents were made w ith a d ia l in d i ­

ca to r w hich read to the n e a re s t 0. 0001 inch . The va lue of th ic k n e ss

u sed for ca lc u la t io n of d en s ity w as the average of five m easurem ents

on each specim en. One m easurem ent w as tak en a t the c e n te r of the

specim en, and the other four were tak en a t the 3, 6, 9, and 12 o 'c lo c k

pos it io n s near the edge of the specim en. The va lue of d iam eter u sed

in the c a lc u la t io n s was determ ined by averaging four m easurem ents

tak en a t 90° to each o ther on the c ircum ference on the specim en.

W arpage during s in te r ing was s l ig h t. The w orst c a s e s of w arpage

occurred during the h igh-tem perature , long-tim e s in ter ing runs. Even

in th e se c a s e s the effec t of w arpage c au sed a maximum v ar ia tio n in

the d iam eter of only +0. 0005 inch, and in th ic k n e ss of +0 . 0003 inch.

The d en s ity v a lu e s used in the a n a ly s is of the da ta p re sen ted here are

the average v a lu e s of the d e n s i t ie s of five du p lica te specim ens s in te red

sim ultaneously .

V. RESULTS AND DISCUSSION

A. D en s if ica tio n Data

The m easure of d e n s if ica t io n employed throughout a ll a n a ly se s

of the experim enta l data has been a "d en s if ica t io n param eter" a, used

ea r l ie r by o ther in v es t ig a to rs (16), and defined a s

6 - 5 o

where 5 = s in te red density , 6 q = uns in te red density , and 5 .̂ =

th e o re t ic a l d en s ity (8. 96 g / c c for copper). This is a d im en s io n le ss

param eter, the va lue of which is in i t ia l ly zero before s in te r ing and

w hich becom es unity when the specim en a t ta in s th e o re t ic a l dens ity .

Table I g iv es the va lue of the d e n s if ic a t io n param eter for

the s in ter ing of the three powder s iz e - f ra c t io n s a t various tim es and

te m p e ra tu re s . The p rec is io n of the d en s ity v a lu e s from w hich the

v a lu e s of a were ca lc u la te d has been shown to be of the order of

+ 0 .03 g / c c (see Appendix I). The corresponding p re c is io n of va lu es

of a p resen ted has been determ ined to be ty p ica l ly +9% (see Appendix I)..

The da ta of Table I are p resen ted g raph ica lly in F igs. 4, 5,

and 6, and Figs. 7, 8, and 9. Figures 4, 5, and 6 are p lo ts of

16

17

d e n s if ic a t io n param eter v e rsu s s in ter ing time for c o n s ta n t tem pera tu res .

These p lo ts show d e n s if ica t io n ra te curves the shape of w hich is ty p ica l

of the s in tering of m ost powdered m eta ls . Figures 7, 8, and 9 are p lo ts

of d e n s if ic a t io n param eter v e rsu s tem perature for c o n s ta n t s in tering

tim es . Plots of th is type are u sefu l in the de term ination of the a c t iv a ­

tion energy a s s o c ia te d with s in tering d is c u s s e d below.

Two a n a ly s e s have been performed on the experim enta l data:

(a) the de term ination of the ac t iv a t io n energ ies a s s o c ia te d w ith d e n s i ­

f ica tio n of the three powder s iz e - f ra c t io n s , and (b) an a n a ly s is to

a ttem pt a d e l in ea t io n of the b a s ic s in tering m echanism s.

B. A ctivation Energies

If s in ter ing is cons ide red to be a p rocess d esc r ib ed by a

sim ple ra te equation, an ac t iv a t io n energy for the p ro cess can be

determ ined. The ra te equation (18) is g iven as

K - A exp(-

w here A is a cons tan t, Q is the ac t iv a t io n energy, R is the gas co n s ta n t

(1 .99 ca l/m o l) , and T is the ab so lu te tem perature .

The symbol K is the rate , e x p re sse d as the rec ip ro ca l of the

time for a frac tiona l change of some property to occur. The progress

of " reac tion" thus may be followed by o bserva tion of the changes in

18

such a s e le c te d property. In th is in v est ig a t io n the property se le c te d

to m easure the progress of s in tering w as the d e n s if ica t io n parameter,

a , as p rev iously defined. U sing logarithm ic notation the ra te equation

can be rew ritten as

In t ^ rt + C

where t is the sin tering time, and C is a co n s tan t. For d a ta obeying

th is equation , a p lot of the logarithm of s in tering time v e rsu s the

rec ip roca l ab so lu te tem perature w ill y ie ld a s tra igh t line, the s lope

of w hich is g iven by Q/R.

Figures 7, 8, and 9 permit the s e le c t io n of various s in tering

tim es and tem peratures for co n s ta n t v a lu es of the d e n s if ic a t io n p a ra ­

meter a . By plo tting th e s e se le c te d v a lu e s as log t v e rsu s l /T (Fig.

10), s tra ig h t l in e s are ob ta ined w ith s lo p es of Q / 2 . 3 R, ind ica ting

that, w ith in the lim its of time and tem perature a t w hich th e s e data

were co llec ted , the ra te equation is obeyed. It is to be noted tha t

for a g iven powder s iz e - f ra c t io n the s lo p es of the l in es in Fig. 10 are

approxim ately equal. It thus appears th a t the ac t iv a t io n energy is

independen t of the ex ten t of d en s if ica t io n . Furthermore, a com parison

of the curves of Fig. 10 (a), (b), and (c) rev ea ls th a t th e ac t iv a t io n

en e rg ies are approxim ately equal for e a c h of the th ree powder s iz e -

f rac tio n s . Thus it a lso appears th a t the a c t iv a t io n energy is independent

of p a r t ic le -s iz e . Evaluation of the s lopes of th e se l in es y ie ld s an

a c t iv a t io n energy of approxim ately 76, 000 ca l /m o l. This va lue of

a c t iv a t io n energy is in fair agreem ent w ith the va lue reported by Jordan

and Duwez (16) of 80, 000 ca l/m o l, and of Takasak i (19) of 87, 5.00

c a l /m o l. In both of the la t te r in v e s t ig a t io n s the m easure of the prog­

re s s of s in tering is id en tica l to th a t u sed in the p resen t study, i. e . ,

the d e n s if ica t io n param ater a . S intering conditions in a l l th ree in v e s t i ­

g a tions are com parable.

C. Sintering M echanism s

A method of d e l in ea t in g the tran sp o rt mechanism during s in ­

tering has been developed by Herring (17) in a th e o re t ic a l trea tm ent

of the e ffec t of change of s c a le on the four b a s ic s in ter ing m echanism s.

H erring 's a n a ly s is shows th a t for com pacts made from two different

powders of p a r t ic le s iz e and R^, but of id en tica l p roperties in a ll

o ther r e sp e c ts , the tim es required for each com pact to reach a given

d e n s if ic a t io n are re la ted by the equation t^ = 4’ nt^. Here, t^ and t^

are the tim es corresponding to com pacts of p a r t ic le s iz e R̂ and R^,

4> = R^/R^, and n is an in teger, the va lue of w hich is determ ined

by the s in tering m echanism as fo llow s:

n = 1, p la s t ic flow

n = 2, evapora tion and condensa tion

n = 3, volume d iffusion

n = 4, su rface d iffusion

From a plot of v e rsu s for equal amounts of d en s if ica tio n , a s tra igh t

l ine w ill be ob ta ined if the above sca lin g law s hold. Knowing the va lue

of 4> and the s lope of the line, the s in ter ing mechanism can be determ ined.

A range of p a r t ic le -s iz e s is a lw ays p re sen t in a powder, re g a rd ­

l e s s of how c lo se ly i t has been s ized . Therefore, to employ the s c a l in g -

law cr i te r ion it i s n e c e s sa ry to u se powders of w idely d iffe ren t average

s iz e s . In the p re sen t study the a n a ly s is has been app lied to the data

of the -150 +200 m esh and the -270 + 325 m esh powder s iz e - f ra c t io n s .

The p a r t ic le -s iz e s of the co a rse and fine f rac tions are d e s ig n a te d a s Rc

and R ,̂ re sp e c t iv e ly . These s iz e s are g iven as R^ = 74-104 microns,

and Rj, = 43-53 m icrons. The va lue of <j> = then, covers a range

of v a lu e s . Taking an average va lue of R̂ equal to 48 microns, 41 has

been determ ined from the above lim iting v a lu e s of R^ to l ie betw een

1. 54 and 2 .1 3 . Therefore, the following re la t io n sh ip s should hold

for the four s in tering m echanism s:

4> = 1 .54 - 2. 13, p la s t ic flow2

<(> = 2 .3 8 - 4 .5 5 , evapora tion and co n d en sa tio n3

4> = 3 .67 - 9 .7 2 , volume d iffus ion4

<j) = 5 .6 8 - 20 .80 , su rface d iffus ion

21

From the curves of d e n s lf ica t io n param eter v e rsu s s in tering

time (Figs. 4, 5, and 6), t im es have been se le c te d for sev era l s tag e s

of equal d e n s lf ic a t io n a t e ach tem perature , and the re su l ts have been

p lo tted in Fig. 11. The tim es corresponding to the co arse and fine

frac tions have been d es ig n a te d a s t and t^, re sp e c t iv e ly . An in s p e c ­

tion of th e se curves re v e a ls s ev e ra l in te re s t in g c h a ra c te r is t ic s . First,

the s lo p es of the l in es vary w ith tem perature , ind ica ting th a t the s in ­

tering m echanism is tem perature dependent. Second, in the c a s e of a l l

but the low est tem perature , the changes of s lope in Fig. 11 in d ica te

th a t changes occur in the mechanism operating during s in te r ing and th a t

th e s e changes are re la te d to p a r t ic le s iz e .

From a c a lc u la t io n of the slope, (|> of the l in e in Fig. 11

for a s in ter ing tem perature of 697 0C, a v a lu e of 20. 6 w as obtained,

which, co rresponds to a su rface d iffus ion m echanism for both coarse

and fine s iz e - f ra c t io n s over the range of tim es p lo tted . The ordinate

in d ic a te s th a t su rface d iffus ion w as operating w ithin the s in tering

in te rv a ls of 12-40 hours for the co a rse fraction; the a b s c i s s a in d ica te s

the same m echanism w ith in the in te rv a l of 1/2 to 2 hours for the fine

frac tion .

Similarly, the s in tering m echanism s for the o ther th ree

tem pera tu res and the in te rv a ls during w hich they were opera ting for

22

e ach s iz e - f ra c t io n have been determ ined. Following is a tab u la tio n

of th e se re su l ts :

, °c t , h rs. c tg, h r s . s lope m echanism

697 12-40 1 /2 -2 20. 6 surface d iffus ion

748748

5-1414-31

1 /2 -22-8

10. 82 .9

surface d iffus ion evaporation and condensa tion

807807

3-1919-34

1-99-15

1 .92 .5 •

p la s t ic flowevapora tion and condensa tion

850850

1-88-30

1 /2 -44-11

1.8 2. 9

p la s t ic flowevapora tion and condensa tion

A graph ica l p re sen ta t io n of th e se re su l ts is given in Fig. 12.

I t should be pointed out th a t the curves of Fig. 11 should

ex trapo la te to zero time. An in sp ec tio n of Fig. 11 show s tha t, with

the excep tion of the curve for 697°C, an add itiona l change in slope

would be n e c e s sa ry for th is ex trapola tion , ind ica ting th a t another

m echanism may have been operating a t short s in tering t im es . The

s lo p e s of th e s e curves obviously m ust be s tee p e r for th is s tag e of

s in ter ing than for the s tag e ju s t following. However, b e c a u se of the

overlapping of the ranges of p o s s ib le v a lu es of 4>n for the d ifferent

v a lu e s of n i t would be presum ptuous to hazard a g u e ss a t the

mechanism w ithout further supporting d a ta .

23

The above eva lua tion of s in tering m echanism s may be sum­

marized by the following four points:

a . For low tem peratures the predom inant s in tering m echanism

is su rface diffusion .

b. For higher tem peratures p la s t ic flow becom es important

some time af ter in i t ia l s in ter ing beg ins .

c. At la te r tim es evapora tion and con d en sa tio n is the

predominant sin tering m echanism .

d. The la rger the p a r t ic le -s iz e the longer w ill be the s in tering

period during w hich a g iven m echanism is operating .

The f ir s t of th e se po in ts is supported by the work of Kuczynski (2)

w herein he a lso concluded th a t surface d iffus ion w as the important

m echanism during low tem perature s in tering of copper. The second

point, concerning p la s t ic flow, has no such support. Kuczynski

claim ed th a t a t h igher tem peratures volume d iffus ion w as the p re ­

dominant mechanism ra ther than p la s t ic flow, a s is su g g es ted in the

p re sen t study. It w ill be reca lled , however, th a t K uczynsk i's e x p e r i ­

m ents w ere performed on id e a l iz e d sy s tem s, i. e . , p la te s and spheres ,

w hile the experim ents here have been on a c tu a l powder com pacts .

There is no re a so n to assum e th a t the s in ter ing m echanism should

be the same under both cond itions . Rather, in ligh t of the o b s e rv a ­

tion (15) th a t the mere ti l t ing of the p lane surface in a K uczynsk i-type

24

experim ent c a u s e s an apparen t change of mechanism from volume

d iffus ion to p la s t ic flow, i t would be ex p ec ted th a t an a c tu a l powder

com pact would s in te r d ifferently than an id e a l iz e d sp h e re -p la te system .

W ith regard to the third point of the above summary, i . e . ,

evapora tion and condensa tion predominating in the la te s ta g e s of

s in tering , there appears to be no other experim enta l support. However,

in a previous se c t io n (II. S intering Theory) i t w as poin ted out tha t th is

con c lu s io n may be re a so n a b le .

The l a s t point, concerning the ro le of p a r t ic le -s iz e on the

time in te rva l during which a g iven m echanism is operating, seem s

fairly re a so n ab le on the b a s is of energy co n s id e ra t io n s . The surface

free energy is l e s s for large p a r t ic le s than for small p a r t ic le s . C o n s e ­

quently, there is l e s s energy av a ilab le to drive a p a r t icu la r mechanism;

therefore, the mechanism could be ex p ec ted to operate over a longer

period of time.

D. S uggestions for Further Work

In the p resen t in v es t ig a t io n th e powder p a r t ic le - s iz e has been

shown to have no e ffec t on the ac t iv a t io n energy a s s o c ia te d with the

s in tering of copper com pacts . It is su g g es ted th a t an in te re s t in g

fo llow -up on th is work would be to determ ine the e ffec t of com pacting

p re ssu re on the a c t iv a t io n energy. It would not be surpris ing if the

ac t iv a t io n energy w as a lso independen t of th is va r iab le .

25

In c r i t ic iz in g his own work the w riter w ish es to su g g es t

sev e ra l improvements in techn ique th a t would benefit future s tu d ie s

s im ilar to the p resen t one. The f ir s t su g g es t io n s apply to in v e s t ig a ­

tions involving the determ ination of s in ter ing m echanism s by the

Herring a n a ly s is method. The p a r t ic le - s iz e ranges of the powders

m ust be minimized, and for a g iven experim ent the d ifference betw een

the average p a r t ic le -s iz e of the co arse and fine powders should be as

g rea t a s p o ss ib le , preferably a fac to r of ten or more. These two p re ­

cau tions would tend to minimize any overlapping of the ranges of v a lu es

of cj>n for the various v a lu e s of n in the Herring equation . In addition,

in H erring 's original, paper he developed the sca ling law s on the b a s is

of the s in ter ing of a "c lu s te r" of p a r t ic le s , though "c lu s te r" w as not

defined. Therefore, it is not c lea r th a t the a n a ly s is is ap p licab le to

specim ens prepared by com pacting a t high p re ssu re s (2 0, 000 psi in

the p resen t c a se ) . Experimental ev idence in th is area is a lm ost non­

ex is ten t , and no d is c u s s io n on th is su b je c t has been reported in the

l i te ra tu re . In the light of the p o ss ib i l i ty of a p re ssu re e ffec t on the

Herring a n a ly s is , i t may prove fruitful to perform experim ents in which

th is e ffec t could be eva lua ted .

Finally , i t is su g g es ted th a t in any work involving the s in te r ­

ing of copper powder, the powder f ir s t should be t re a ted in hydrogen

a t some in term ed ia te tem perature to remove any p o ss ib le su rface

oxide on the p a r t ic le s . Following th is trea tm ent the powder should

be com pacted im m ediately in to specim ens and s tored in an inert

a tm osphere until needed for s in ter ing . This s tep w as not tak en in

the p re sen t work. Although any su rface oxide w as ce r ta in ly removed

by the hydrogen gas during the f ir s t periods of heating, there is co n ­

s id e rab le u n cer ta in ty concerning the e ffec t th a t th is oxide may have

had on the course of s in tering during the in i t ia l s ta g e s .

VI. SUMMARY AND CONCLUSIONS

In th is in v es t ig a t io n the s in ter ing behavior of copper powder

com pacts has been s tud ied . C ylindrica l com pacts were prepared from

th ree d ifferent p a r t ic le - s iz e f rac tions of a commercial copper powder.

The powder was com pacted a t 20, 000 psi in a do u b le -ac tin g , non­

lub rica ted hardened - s te e l d ie . S intering experim ents w ere performed

a t four d ifferent tem peratures in the range 697-850°C . For each tem ­

perature s ix d ifferen t s in te r ing periods w ere u sed ranging from 1 to 32

h o u rs .

Two different a n a ly s e s were performed on the experim ental

data . In the f irs t a n a ly s is , the a c t iv a t io n energy a s s o c ia te d with

s in tering w as determ ined by the method of Jordan and Duwez. The

ac t iv a t io n energy has been shown to be independen t of powder p a r t ic le -

s iz e and ex ten t of dens if ica tion , and to be equal to 76 k ca l /m o l.

The second a n a ly s is employed H erring 's s c a l in g - la w crite rion

in the eva lu a tio n of the s in tering m echanism s operating under the v a r i ­

ous conditions of the experim ents . This a n a ly s is has led to the fo llow ­

ing conc lu s io n s :

a. At low tem peratures th e predominant s in tering m echanism

is su rface d iffusion .

27

At higher tem peratures p la s t ic flow is the important

m echanism for the early s ta g e s of s in tering , w hile

evapora tion and condensa tion predom inates in the la te

s ta g e s .

Powder p a r t ic le - s iz e a ffec ts the s in tering behavior

such th a t the la rger the p a r t ic le - s iz e the longer w ill

be the period during w hich a g iven mechanism is

operating .

REFERENCES

1. A. J. Shaler, Trans. AIME 185, 796 (1949).

2. C. G. Kuczynski, Trans. AIME 185, 169 (1949).

3. W. D. Jones, P rincip les of Powder M etallurgy (Edward Arnold,L td ., London, 1937).

4. P. E. W retblad and J. Wulff, in Powder M etallurgy ed ited by J. Wulff (ASM: C leveland , 1942), ch. 4.

5. F. N. Rhines, Trans. AIME 166, 47 4 (1946).

6. C. G. G oetzel, T rea tise on Powder M etallurgy (In te rsc ien ce P ublishers , Inc. , New York, 1950), ch. 35.

7. W. E. Kingston and G. F. Huettig, in The Physics of Powder M eta llu rgy , ed ited by W. E. Kingston (M cG raw -H ill Book C o . , 1951), ch. 1.

8. G. A. G each, in Progress in M etal P hysics , Vol. 4 ed ited byB. Chalm ers (Pergamon Press L td ., London, 1953), ch. 4.

9. W. D. Jones, Fundamental P rincip les of Powder M eta llu rgy , (Edward Arnold, L td . , London, I960), ch. 4.

10. J. K. M ackenzie and R. Shuttleworth, Proc. Phys. Soc. 62B, 833 (1949).

11. P. W. Clark and J. W hite, T ran s . Brit. Ceram. Soc. 49_, 305 (1949).

12. B. H. Alexander and R. W. Balluffi, Acta Met. 5, 666 (1957).

13. P. W. Clark, J. H. Cannon, and J. W hite, Trans. Brit. Ceram.Soc. 52, 1 (1953).

29

30

14. E. B. Allison and P. Murray, Acta Met. 2, 487 (1954).

15. P. Laurent and M. Eudier, Rev. Met. 43 , 271 (1951).

16. C. B. Jordan and P. Duwez, J. Met. L 96 (1949).

17. C. Herring, J. Appl. Phys. 2J_, 302 (1950).

18. W. D. Kingery, In troduction to C eram ics (John W iley and Sons,In c . , New York, i960), p. 288.

19. A. Takasaki, Sci. Repts. Tohoku Imp. Univ. A. 5, 366 (1953).

31

TABLE I

DENSIFICATION PARAMETER, a

Size Fraction ̂ (Tyler)

SinteringTime,hrs.

Sintering Temperature, °C

697° 748° 807° 850°

1 0. 003 0. 006 0. 057 0. 132

2 0. 006 0. 019 0. 085 0. 167

-150, 4 0. 013 0. 038 0. 120 0. 208

+ 200 8 0, 022 0. 057 0. 158 0. 255

16 0. 035 0. 083 0.199 0. 292

32 0. 047 0. I l l 0 .237 0. 336

1 0. 031 0. 045 0. 095 0. 156

2 0. 040 0. 066 0. 126 0. 193

-200, 4 0. 049 0. 082 0. 153 0.242

+ 270 8 0. 062 0. 100 0. 190 0. 288

16 0. 074 0. 118 0. 230 0. 337

32 0. 083 0. 145 0.267 0. 383

1 0. 042 0. 058 0 .112 0. 170

2 0. 051 0 .079 0. 140 0. 207

-270, 4 0. 063 0. 094 0. 167 0.257

+ 325 8 0. 069 0. 112 0. 204 0. 306

16 0, 085 0. 131 0. 243 0. 356

32 0. 094 0. 158 0. 280 0. 399

Figure 1. - -S in tering Furnace.

FIGURE 1

Figure 2. - -P ush -P u ll Rod and Sintering Boat Assembly.

33

FIGURE 2

Figure 3. - -S intering Boat and Thermocouple Assembly.

34

FIGURE 3

Figure 4. - -P lot of a v e rsu s t for C oarse S ize-F rac tion .

35

4 0

COARSE SIZE

TIME.hrs.

FIGURE 4

Figure 5. - -P lo t of a v e rsu s t for Medium S ize-F rac tion .

01*15

36

MEDIUM SIZE

807°C

748°C

00 10 20 30

TIME.hrs.

FIGURE 5

Figure 6. - - Plot of a v e rsu s t for Fine S ize-F rac tion .

01x1

0

37

03 020100

TIME.hrs.

FIGURE 6

Figure 7. - -P lo t of a v e rsu s T for C oarse S ize-F rac tion .

DU 10

4 0

COARSE SIZE

32 hrs.

3 0

CM

20

8 5 08 0 07 5 07 0 0T,°C

FIGURE 7

Figure 8. - -P lot of a v e rsu s T for Medium S ize-F rac tion .

OX

**

39

4 0

32 hrs.

MEDIUM SIZEr

3 0

20

7 0 0 7 5 0 8 0 0 8 5 0T,°C

FIGURE 8

Figure 9. - -P lo t of a v e rsu s T for Fine S ize-F rac tion .

Olx*X?

40

4 0 32 hrs.

FINE SIZE

3 0

CM

20

850800750700T,°C

FIGURE 9

Figure 10. - -P lot of Log t v e rsu s

TIM

E,

hrs.

41

a.COARSE

.10.1232

b.MEDIUM

0.20 .18 .16 .14 .12 .10

c. FINE

9.0 9.6 9.8 10.0 10.29.2 9.48.8l/TxlO 4, ^ ' 1

FIGURE 10

Figure 1 1 . - -P lo t of t v e rsu s t

42

40

30

20

O- 697°C

A- 807°C

o - 850^C

10

02010 150 5

t f ,hrs.

FIGURE 11

Figure 12. - -Schem atic R epresen ta tion of S intering M echanism s.

SINT

ERIN

G TE

MPE

RA

TU

RE

,°C

P L A . F L O . EVAR C O N D .850

R F. E. C.

I—CPLAS TI C FLOW EVAR COND.

807

M P L A . F L O . E. C.

r-C

748SURF. DIFF. EVAR COND.

H[ sjS.Q E. C.

c — coarse f — fine

r - C

697SURFACE DIFFUSION

J______ :____ L0 10 20

SINTERING TIME hrs.30 40

00

FIGURE 12

APPENDIX I

ERRORS

A. Sources of Error

There are four sources of error w hich a ffec t the re su l ts of the

experim ental work carried out in th is in v es t ig a t io n . These are:

a. Errors in s in tering tem perature

b. Errors in s in tering time

c . Errors in specim en d en s ity due to m easurem ent

d. Errors in specim en d en s ity due to such indeterm inate

fac to rs as tem perature g rad ien ts during s in ter ing and

o ther unknow ns.

An ev a lu a tio n of th e se errors is p re sen ted in the following

se c t io n s .

B. Errors in S intering Temperature

In the se c t io n on experim ental procedure i t w as pointed out

th a t the s in ter ing tem perature could vary a s much as + 1 0°C from th a t

s e t on the contro ller . This v a r ia tio n w as due to the ON-OFF power

c y c le of the tem perature contro ller . It w as a lso pointed out th a t th is

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var ia tion w as regu lar and reproducib le from cyc le to cy c le . The

s ta tem en t w as made th a t the e ffec t of th is var ia tion of tem perature

on d e n s if ica t io n may be com pensating . For th is e ffec t to be com pletely

com pensating the in c re a se in d e n s if ic a t io n of a spec im en due to heating

and cooling over a tem perature in terva l T to T + AT would have to be

e x ac tly ba lan ced by the d e c re a se in d e n s if ica t io n due to cooling and

heating over the in terva l T to T - AT. In examining th is p o ss ib i l i ty an

assum ption needs to be made as to the nature of the c y c l ic tem perature

var ia tion . It i s assum ed here th a t the v a r ia tio n is s inuso ida l , v i z . , i t

ta k e s as long for a specim en to h ea t up over a g iven tem perature in te rva l

during the ON cyc le of the contro ller a s i t does to cool over the same

in te rv a l during the OFF cy c le . The e ffec t of th is uniform heating and

cooling over a g iven tem perature in te rv a l may be exam ined by using the

d a ta of Fig. 8, which is a plot of d e n s if ica t io n param eter v e rsu s s in te r ­

ing tem perature for specim ens of the medium s iz e powder fraction .

C onsider, for example, a s in tering tem perature of 800°C for

the 8 hour s in tering run. The d e n s if ica t io n corresponding to th is

tem perature is 0. 174. For a tem perature v a r ia tio n of + 10°C there

is a v a r ia tio n in d e n s if ica t io n from 0. 157 to 0. 193. Therefore, the

va lue of d e n s if ic a t io n may be sp ec if ied a s a = 0. 174 + 0. 017,

0. 174 - 0. 019. In th is ca se , then, the errors are a lm ost com pletely

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com pensating . Similar co n c lu s io n s are reach ed for the o ther s in tering

conditions, as w ell as for the o ther powder s iz e - f ra c t io n s .

W hile the above i l lu s t ra t io n seem s p lau s ib le , i t m ust be

pointed out th a t the b a s is for the assum ption of a s in u so id a l tem pera ­

ture v a r ia tio n is not en tire ly sound inasm uch a s i t requ ires tha t the

h ea t t ra n sfe r ra te away from the s in te r ing chamber during the contro ller

OFF cy c le is ex ac tly equal to the hea t t ra n sfe r ra te to the chamber

during the ON cycle , a behavior w hich is q uan ti ta t ive ly unlikely .

G. Error in Sintering Time

In the sec t io n on experim enta l procedure i t w as pointed out

th a t the maximum error in sin tering time w as 3 m inutes. The e ffec t

of th is error on the d e n s if ica t io n param eter may be exam ined by reference ,

again , to the experim ental da ta . In Fig. 5, which is a plot of d e n s i f ic a ­

t ion param eter v e rsu s s in ter ing time for the medium s iz e powder fraction,

in the sho rt- tim e reg ions of the curves where th is error would be ex p ec ted

to have the g re a te s t effect, an error of 3 m inutes (0 .05 h r . ) is barely

d isce rn ib le and the shape of the curve is e s s e n t ia l ly unchanged.

D. Errors in D ensity Due to D im ensional M easurem ents

The con tribu tion of error in c a lc u la te d d en s ity due to d im en­

s iona l m easurem ents w as determ ined from the equation for the d en s ity

of a cy lind rica l specimen:

where W = specim en weight, D = spec im en diameter, and L = s p e c i ­

men th ic k n e ss . R eexpressed in logarithm ic form,

w hich g ives the re la t iv e error in d en s ity due to the errors in the th ree

m easured q u an ti t ie s W, D, and L. To determ ine the maximum p o ss ib le

error from the above exp ress ion , a l l term s w ere made pos it iv e , y ielding.

errors varied with the amount of spec im en warpage due to s in tering .

At high tem pera tu res and long s in ter ing tim es the w arpage c a u se d as

much as + 0 . 0005 in . v a r ia t io n in spec im en diam eter, and + 0 . 0003 in.

in th ic k n e ss . For the c a s e s of low tem perature and short s in tering

t im es the v a r ia t io n s rare ly ex ceed ed + 0 . 0002 in. in both diam eter

In 6 = In W - 2 In D - In L - In J- .

Upon d ifferen tia ting .

d6 _ dW „ AD AL

' A5 6

Experimental o bserva tion in d ica ted tha t the d im ensional

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and th ic k n e ss . For un s in te red specim ens the v a r ia t io n s were +0 . 0001

in . for d iam eter a n d +0 . 0002 in. for th ic k n e ss . W eight m easurem ents

were a lw ays w ith in +0 . 0005 g. U sing th e se v a lu es and ty p ica l v a lu es

of the m easured q u an ti t ie s (given in the following tab le) , the errors in

d en s ity were c a lc u la te d from the above formula a s summarized in the

following tab le for the th ree specim en conditions:

U nsin teredLow-Temperature, High-Tem perature,

Short-Time Long-Time

0 .1 6 0 + 0 .0002 in 0 .156 + 0.0002 in 0.147 + 0 .0003 in

D 0 .508 + 0.0001 in 0 .506 + 0 .0002 in 0 .476 + 0 .0005 in

W 3 .0 0 + 0 .0005 g 3 .0 0 + 0 .0005 g 3 .0 0 + 0.0005 g

6 5 .6 5 + 0. 01 g / c c 5 .8 3 + 0. 01 g / c c 7 .0 0 + 0. 03 g / c c

It is to be noted th a t the error in d en s ity due to m easurem ent in c reased

w ith time and tem perature of s in ter ing .

E. Errors in D ensity Due to Indeterm inate F actors

To ev a lu a te the to ta l in fluence of indeterm inate fac to rs an

experim ent w as performed in w hich tw enty specim ens w ere p ressed ,

s in tered , and handled in a ll o ther a s p e c ts by the same tech n iq u es as

w ere a ll spec im ens in the s in ter ing experim ents . The tw enty specim ens

w ere p re s se d from the m ed ium -s ize powder fraction, and w ere s in tered

a t 8 0 7 °C for 14 hours, conditions w hich w ere co n s id e red to be ty p ica l

of the s in ter ing experim ents performed in the o v e r -a l l s tudy. All

po s it io n s in the s in ter ing boat, which were normally occup ied by

specim ens prepared from the th ree powder s iz e - f ra c t io n s , w ere

occupied by one of the tw enty t e s t spec im ens. As a re su l t of th is

experim ent i t w as determ ined th a t spec im en pos it ion during s in tering

had no o bservab le e ffec t on s in te red d en s ity . Although n o ticeab le

v a r ia t io n s in d en s ity were observed from one pos it ion to another, there

w as no apparen t pa tte rn to the v a r ia t io n s . The average v a lu e s of

d en s ity for the uns in te red and s in te red spec im ens w ere 5 .7 4 g / c c

and 6 .4 5 g /c c , re sp ec tiv e ly , and the s tandard dev ia tion in both c a s e s

w as + 0 . 03 g / c c . Thus, the d e n s i t ie s for the uns in te red and s in te red

cond itions may be w ritten as :

un s in te red 6 = 5 .7 4 + 0 .03 g / c co —

sin te red 5 = 6 .4 5 + 0 .03 g /c c

This v a lu e of error, +0 . 03 g /c c , i s a to ta l error, v i z . , i t inc ludes the

error due to m easurem ent. If it is a ssum ed th a t th is error i s typ ica l of

the to ta l error in d en s ity to be ex p ec ted in th e se experim ents, the error

in d en s if ica t io n param eter may then be ca lc u la te d .

The d e n s if ic a t io n param eter is g iven by

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where 5 - s in te red density , 6 q = uns in te red density , and 6 = th e o re ­

t ic a l d en s ity . R eexpressed in logarithm ic form,

In a = In (6 - 6 ) - In (6 - 6 ).

Upon d ifferen tia ting ,

da d6 d 6 o d 6 t . d 6 0a 6 - 6 6 - 6 5 - 6 5 - do o t o t o

If i t is a ssum ed th a t the error in the th e o re t ic a l d en s ity 6 is zero, the

approxim ate form of the equation for maximum re la t iv e error in densifica*

tion may then be ex p re sse d as

a a r A6 A6

+ r ^ h +o o t o

Upon su b s ti tu tin g in to th is e x p ress io n the v a lu e s determ ined above for

6 + A6 the re la t iv e error in a w as determ ined to be

— = +o. 094 a —

Thus, the error in the d e n s if ica t lo n param eter due to errors in dens ity

w as determ ined to be approxim ately 9%.

In summarizing th e re s u l t s of the above error a n a ly s is , th ree

main po in ts have been covered:

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a . Errors in d en s if ica t io n a r is ing out of tem perature v a r ia ­

tions due to the contro lle r ON-OFF cycling are cons ide red

to be somewhat com pensating, though com plete com pensa­

tion is highly un likely .

b. Slight errors in time, such a s observed in th e s e ex p e r i­

m ents, have a sm all e ffec t on d en s if ica t io n .

c. The e ffec t of the errors in spec im en d e n s ity on the

d e n s if ic a t io n param eter, a, has been shown to be

approxim ately 9/4 for a ty p ic a l s in tering exam ple.

I