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Tem perature Depen dence of the Young s

M odulus and Shear M odulus of Pu re Nickel,

P latinum , and M olybdenum

R. F ARRARO AND REX. B. McL ELLA N

A pulse-echo t echnique u t i l i z ing a wave gu ide opera t ing a t h igh t emp erat ures has been

used to measu re the variat io n in the Young's modulus and shear modulus of pure pla t i -

num, nickel , and molybdenum in the tem per atu re r ange from 25 to 1000~ With the

except ion of Ni in the temperature range below the Curie point , l inear behavior was found

for a l l th ree meta l s . The s ign i f i cance of the l inear re la t ion between t empe ratu re and the

elast ic c onstants for Mo to the obser ved non l ine ari ty of the Arr he niu s plot of the diffusiv-

i ty of carbon in Mo is discussed.

I~ EC ENT work 1 has pointed out the im port ance of

cons ider ing the vo lume changes in so l id so lu t ion for -

mat ion in unders tand ing the fo rm of the in terac t ions

between the component atoms. The effects of such

volume changes may be part ial ly elucidated by a

knowledge of the variat ion of the elast ic propert ies

of the solut ion with solute conc entrat ion. This has

been the mot ivat ion for me asur emen ts o f the var ia -

t ion of the elast ic propert ies of austeni te at high tem-

pera ture s as a funct ion of carbon concent ra t ion .2 How-

ever , the t empera ture va r ia t ion of the e las t i c p rope r-

t i es o f pure meta l s , especia l ly a t h igh t empe ratur es , i s

also of impo rtan ce in elucidat ing the cause of depart-

u res f rom Arrhen ius behavior observed in cer ta in

case s of int er s t i t ia l diffusion. In the case of the de-

par tu r e f rom Arrhe n ius behavior observed in the d i f -

fusivi ty of car bon and ni t roge n in bcc i ron, Borg 3 has

re la ted the acceler a t ion in d i f fus iv i ty found on approach-

ing the Curi e tem per atu re with the change found in the

elas t i c cons tan t s in th i s t empe ra tu re range. The non-

l inear i ty in the e las t i c moduli of i ron has been meas-

ured by Dever 4 us ing bcc i ron s ing le crys ta l s . Dever

has a l so d i scus sed the anomalous d i f fus ion behavior

of N and C in i ron in ter ms of the effect of the non-

l inear var ia t ion in the e las t i c modul i on the ac t iva t ion

entropy AS for int ers t i t i al diffusion. Such models are

closely re late d to the proposal of Wutt ig5 who con-

side red dire ct ly the effect on AS due to the change in

the magnet ic s tate. Since the elast ic moduli are the

second der ivat ive s o f the to ta l Helmhol tz f ree energy

with respect to s t rain, i t i s clear that the degree of

sp in order wi l l be ref l ec ted in the e las t i c cons tan t s .

Despi te diff erenc es in formul at ion, the models of

Borg, 3 Deve r, 4 and Wuttig5 ascr i be the dev ia t ions

f rom norm al d i f fusion behavior to t empe ratur e

var ia t ions in the thermodyna mic functions charac-

ter i z in g a

s i n g l e s t a g e

diffusion mechanism. Such a

s ing le-s tage m echan ism has a l so been invoked by Ki -

mur a and Yoshioka 6 to account for the diffusion

anomaly in the Mo-C sys te m. Thei r analys i s i s based

on the nonl i nea r var iat io n in the Young's modulus of

Mo observ ed by K6ster . 7 However, this model has

R. FARRARO is Graduate Student and REX B. MCLELLAN

is Professor of Materials Science at Rice University, Houston,

Texas, 77001.

Manuscript submitted January 25, 1977.

been recen t ly cr i t i c i zed by McLel lans who has pointed

out that a so ftening of the par ent m etal lat t ice wil l lead

to var ia t ions in the a t t empt f requency of the in te rs t i t i a l

atom giving ris e to a curva ture in the opposi te sens e

to that ari s in g from the effect on AS. A more l ikely

explanat ion for the Mo-C sys tem is the two-stage

mechan ism involv ing C- jump s between bo th the octa-

hedra l and tet r ahe dra l s i tes in the bcc lat t ice. Such

a me chan ism has bee n proposed ear l ie r . 9'1~

In order to reso lve the quest i on posed above, i t was

decided to meas ure the e las t i c p ro per t i es o f pure

Mo in the temp era tur e r ange in which the diffusion

anomal y is observe d. In addi t ion to the relat iv ely

sl ight c urva ture found by KSster 7 in the temp era tur e

variation of the Young's modulus of Mo, he found a

much more pronounced effect for Pt and a very lar ge

effect in Ni which s t rongly depends on the magnet ic

s ta te . Thus , i t was decided to re inves t ig a te Ni and Pt

also.

E X P E R I M E N T A L M E T H O D

A p u l s e - e c h o t e c h n i q u e u t i l i z i n g a m a g n e t o s t r i c t i v e

w a v e g u i d e w a s e m p l o y e d t o d e t e r m i n e t h e e l a s t i c

c o n s t a n t s o f t h e m a t e r i a l s . T h e s a m p l e s , i n t h e f o r m

o f w i r e s c o n f o r m i n g t o t h e t h i n - l i n e l i m i t , 2 w e r e

b u t t w e l d e d t o t h e e n d o f t h e w a v e g u i d e . T h e s a m p l e s

w e r e h e l d a t t e m p e r a t u r e i n a h o r i z o n t a l f u r n a c e h a v -

i n g a c e n t r a l c o n s t a n t t e m p e r a t u r e z o n e o f : ~ 3 ~ a t

1 0 0 0 ~ u n d e r a p r o t e c t i v e f l o w o f h i g h p u r i t y a r g o n

g a s . T h e t e m p e r a t u r e w a s m e a s u r e d u s i n g c h r o m e l -

a l u m e l t h e r m o c o u p l e s h e l d w i t h i n 1 m m o f t h e s a m p l e

w i r e s . T h e t h e r m o c o u p l e s c o u l d b e t r a v e r s e d a l o n g

t h e l e n g t h o f t h e s a m p l e . T h e d e t a i l s o f t h e e x p e r i -

m e n t a l p r o c e d u r e h a v e b e e n d i s c u s s e d p r e v i o u s l y . 2

T h e Y o u n g s m o d u l u s E a n d t h e s h e a r m o d u l u s G

a r e r e l a t e d t o t h e l o n g i t u d i n a l ( 1) a n d t r a n s v e r s e ( t)

s o u n d v e l o c i t i e s i n t h e m a t e r i a l b y t h e r e l a t i o n s , 2

E = V ~ p

G = V~p

where p is the densi t y of the medium . Cor rec t io ns

were made for both length and densi ty changes with

tempe ratu re us ing values o f the thermal expans ion

coefficients of the me tals n

The Ni and Pt samples were MARZ Grade mater ia l

METALLURGICAL TRANSACTIONSA VOLUME 8A, OCTOBER 1977-1563

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obta ined f rom Ma ter ia l s R esear ch Corpora t ion in the

form of 0.049 cm diam wires . The Mo was made into

a wire of the same d iam ete r by rotat in g a rod of zone

refined Mo in an etching solut ion.

RESULTS

The res ul t s obtained can be seen in Figs. 1, 2, and

3, showing plots of E and G as funct ions of tem per a-

ture . The valu es of E found for Ni are shown in Fig. 1.

The t r ia ngul ar data points (4) are due to KSster. In

the t emperature range below the Cur ie t emperature ,

the measu reme nts were made in a sa tura t in g magnet ic

field. l~ The depth of pronounced mi nim um seen in the

fer ro magnet i c reg ion in the demagnet ized s ta te i s de-

pendent on the thermal his tory and grain s ize of the

sample.12 T his effect has b een ascr ibe d to the effect

of res idual s t re sse s on the magnet ic o rder ing . 12 The

data t aken in the curr en t inves t iga t ion are i l lus t ra te d

by fi l led circl es (o taken on heat ing) and open circle s

(o taken on cool ing). The sampl e was anne aled for 20

h a t 1000~ pr io r to making the mea sure ment s . Af ter

terminating the experiment a metallographic exami-

nation showed the grains to be equiaxed and to have a

mean diameter about one-third of the wire diameter.

2 6

9 H e a t i n g N i

o C o o h n g

K o s | e r

z

t 6

1 4

i J ] I ..... I, J I ' + 'l

0 ( ID O 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 S O 0 9 0 0 I 0 0 0

~ C

Fig . 1 - - Tempe ratu re var iat ion of the Young s modulus o f

nickel.

~8

1 6 8 4

14

1 2 .

x t O

0 B "

0 6 "

O4

0

9 Heat i nq

~ . . ~ o C o o l in g P t

A

K os t er

I

- ---

I - - f - ' - - l I ' - - I

I 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0

~

- ~- i

8 0 0 9 0 0 I O 0 0

Fig. 2--Temperature variation of the Young s modulus E and

shear modulus G of platinum.

3 4

3

~ - ~ ~ ~ ~9 Heot l nq

Cool i ng

M o

z~ KOsfer

~o o + ~ - - , ~ o so*el a do K lerk

K )O 2 0 0 5 0 0 6 0 0 9 0 0 I 0 0 0

0 0 8 0 00 0 7 0 0

~ C

3 0

2 8

~ 6

z

. l l

O 9

Fig . 3 - - Tempe ratur e var iat ion of the Young s modulus E and

the shea r modulus G of molybdenum.

The pos i t ions of the m aximum and mi n imum cor-

respo nd closely to those found by KSster 12 The data

do not show any nonl inear i ty above the Cur ie t empe ra-

tu re and a l ea s t -s quar es regr ess i on of the data po in t s

g ives the represen ta t ion ,

E = (2.42 - 9.3 3 10 -4 T 0 .09 2) 1011

N / r n 2

Figu re 2 shows the data found for Pt . The sample had

been anneale d in pure ar gon for 20 h at 1000~ prio r

to making the mea su re men ts . Both the E and G values

found were repr oduci ble on heat ing ( 9 ) and cool ing ( o )

and the t emperature was cycled severa l t imes . In con-

t ras t to the E values determine d by Kbs ter , a l inear

var ia t ion wi th the t em perat ure was found . A leas t -

squares reg res s ion to the data y ie lds the repres en ta -

t ions .

E = (1 .6 8 - 3.38 x I0 -~ T 0.046) 1011

N / m 2

G = (0.6 24 - 1.15 10 4 T 0.0 15) 1011

N / m L

The data obtained for Mo are shown in Fig. 3. Again,

no depar tur e f rom l inear i ty was detec ted. The data

are in good accord wi th the low temperature measure-

ment s of Bolef and de Klerk. 13 A least squ are s analy-

s i s y ie lds :

E = (3 . 12 - 4.18 10 -4 T 0.63) 1011

N / m 2

G = (1.27 - 2.23 10 -4 T 0.0 33) 1011

N / m 2

DISCUSSION

With the exception of Ni below the Curie tempera-

ture none of the measured elastic properties ex-

hibited a nonlinear variation with temperature. The

reason for the discrepancy between the current meas-

urements and those of KSster is not obvious.

The present measurements on Mo do not support the

model of Kimura and Yoshioka in which the anomalous

diffusion behavior in the Mo-C system is based upon a

nonlinear variation of the elastic properties of the

parent metal with temperature. It has been pointed

outs that if the diffusion anomaly is due to a dual site

occupancy mechanism the effect may be observable

in the thermodynamic properties of the Mo-C solution

at high temperatures. Thermodynamic measurements

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o n M o C s o l u t io n s a t h ig h t e m p e r a t u r e s a r e c u r r e n t ly

b e i n g m a d e .

A C K N O W L E D G M E N T

T h e a u t h o r s a r e g r a t e f u l f o r t h e f i n a n c i a l s u p p o r t

p r o v i d e d b y th e R o b e r t A . W e l c h F o u n d a t i o n .

REFEREN ES

1. C. Wagner:Ac ta Me t . , 1971, vol. 19, pp. 843--49.

2. W. J Arnoult and R. B. McLellan: Acta Met. , 1975, vol. 23, pp. 51-56.

3. R. V. Borg: m Diffusion m Body.Centered Cubtc Metals, p. 225, Am. Soc.

Metals, 1965.

4. D. J. Dever: J Appt . Phy s , t972, vol. 43, pp. 3293-3305.

5 M, Wuttlg: Scr. Met. , 1975. vol. 5, pp 33-36.

6. H. Kimura and K. Yoshioka. Mats . Sc t Eng , 1976, vol. 24, pp. 171-177.

7. W. Koster: Z Me ta t lk , 1948, vol. 39, pp. 1-9

8. R. B. McLellan M a t s . S c i E n g , 1977, vol 28, pp, 161-163.

9~ R. B. McLellan, M. L. Rudee, and T. Ishlbachi. Trans. TMS-AIME, 1965,

vol. 233, pp. 1938-1943.

10 R H. Cond~t and D. N. Beshers: Trans. TMS-AIME, 1967, vol. 239, pp. 680-

687.

11. C J. Smithells: 'Metals Referen ce B ook, Third Edition, 1962, Butterworth's,

Wasbangton.

I2. W. Koster Z Metallk, 1943, vol 3, pp. 57-67.

13. D. I Bolef and J. de Klerk J. A p p l . P h y s , 1962, vol. 33, pp. 2311-2314.

METALLURGICAL TRANSACTIONs A VOLUME 8A, OCTOBER 197 7-15 65