Electrical Resistivity and the Stability of Liquid Alloys J. L. T O M L I N S O N AND B. D. L I C H T E R

IN a

r e c e n t p a p e r , D a r k e n 1 p o i n t e d out that the s e c ond d e r i v a t i v e of the f r e e e n e r g y w i t h r e s p e c t to a t o m f r a c t i o n , w h i c h he c a l l e d the s t a b i l i t y , w a s a good i n d i c a t o r of the r e s i s t a n c e to c h e m i c a l c h a n g e in a b i n a r y m i x t u r e of l i q u i d m e t a l s . H e r e , w e w i l l show that m a x i m a in the e x c e s s s t a b i l i t y f u n c t i o n a r e a s s o c i a t e d w i t h m a x i m a in the e l e c t r i c a l r e s i s t i v i t y in t h r e e s y s t e m s c o m m e n t e d upon by D a r k e n and that the e x c e s s s t a b i l i t y c a n be thought of a s an i n d i c a t i o n of the e x t e n t to w h i c h e l e c t r o n s a r e t i e d up in b o n d i n g . T h e l a t t e r point w i l l be i l l u s t r a t e d by H a l l e f f e c t d a t a and the f l u c t u a t i o n s c a t t e r i n g m o d e l f o r the e l e c t r i c a l r e s i s t i v i t y of l i q u i d a l l o y s . 2'3 In the n o t a t i o n of T o m l i n s o n and L i c h t e r , s the s t a b i l i t y w o u l d be w r i t t e n a s

S-

d2 A G dZ~-

1 (1-Z~)

d-AG2 dZ~

d ~ xs dX2

10

Cd-Bi oE 1 0 0

8 -6

~>-" B 0 ~-

v

REDUCED RESISTIVITY

>.-

6

. -'1.'~

m 60 oo " 40 r'~ to 0

[2]

w tr

/ I

4

\~(

EXCESS STABILITY

20

/ /

/

0 0.0

io3 w (,,) x to

a

J. L. TOMLINSON is Research Physicist, Naval Weapons Center Corona Laboratories, Corona, Calif. B. D. LICHTER is Associate Professor of Materials Science, Vanderbilt University, Nashville, Tenfi. Manuscript submitted April 7, 1969.

I._1 m m

[1]

w h e r e X2 is the m o l e f r a c t i o n of c o m p o n e n t 2, AG is the r e l a t i v e i n t e g r a l m o l a r G i b b s f r e e e n e r g y , ~G2 is the r e l a t i v e p a r t i a l m o l a r f r e e e n e r g y of c o m p o n e n t 2, and the s u p e r s c r i p t x s i n d i c a t e s an e x c e s s q u a n t i t y , i . e . , the d i f f e r e n c e b e t w e e n the f r e e e n e r g y and the ideal free energy. F i g s . 1 to 3 show the e x c e s s stability ($xs) and the r e d u c e d r e s i s t i v i t y (Pexp - X~p~ - X2p2) f o r the s y s t e m s C d - B i , C d - S b , and M g - B i . E x p e r i m e n t a l r e s i s t i v i t y v a l u e s a r e t a k e n f r o m T o m l i n s o n and L i e h t e r , 3 M i l l e r et a l J and I l s c h n e r and W a g n e r . 5 In a l l c a s e s the e x c e s s s t a b i l i t i e s a r e t h o s e c a l c u l a t e d by D a r k e n ~ f r o m p u b l i s h e d data. In F i g . 3, the r e s i s t i v i t y and s t a b i l i t y s c a l e s d i f f e r f r o m t h o s e in F i g s . 1 and 2. T h i s i n f o r m a t i o n is s u m m a r i z e d and c o m p a r e d w i t h the e x c e s s f r e e e n e r g y t a b u l a t e d by H u l t g r e n et al. 8 in T a b l e I. One can s e e that the o c c u r r e n c e of a m a x i m u m in the r e d u c e d r e s i s t i v i t y - c o m p o s i t i o n c u r v e w i t h a m a x i m u m in the e x c e s s s t a b i l i t y is c o m m o n to t h e s e t h r e e s y s t e m s . In a d d i t i o n , the s i z e of the m a x i m u m in r e s i s t i v i t y is l a r g e s t f o r M g - B i , the s y s t e m w i t h the greatest excess stability. In M g - B i both m a x i m a o c c u r at the c o m p o s i t i o n MgsBi2. It w a s p o i n t e d out by I l s c h n e r and W a g n e r s t h a t t h i s is the c o m p o s i t i o n of a c o m p o u n d a p p e a r i n g in the s o l i d and that t h i s c o m p o s i t i o n c o i n c i d e s w i t h an i n f l e c t i o n in the c h e m i c a l p o t e n t i a l - m o l e f r a c t i o n plot. T h e y s u g g e s t that m o l t e n MgsBi2 i s e s s e n t i a l l y an i o n i c c o n s t i t u t i o n w i t h e q u a l but r e l a t i v e l y low c o n c e n t r a t i o n s of e x c e s s e l e c t r o n s and e l e c t r o n h o l e s . T h e v a r i a t i o n of r e s i s t i v i t y w i t h c o m p o s i t i o n a b o u t s t o i c h i o m e t r y w o u l d be a n a l a g o u s to the a d d i t i o n of

METALLURGICAL TRANSACTIONS

120

to

and the e x c e s s s t a b i l i t y w o u l d a p p e a r a s

S x s = d 2AGxs 1 dX~ - (l-X2)

i m p u r i t i e s to a s o l i d s e m i c o n d u c t o r c o m p o u n d . T h e v a r i a t i o n of e l e c t r i c a l r e s i s t i v i t y w i t h c o m p o s i t i o n in C d - S b is not n e a r l y so l a r g e a s in M g - B i ; h o w e v e r , a n e g a t i v e t e m p e r a t u r e d e p e n d e n c e of the r e s i s t i v i t y is o b s e r v e d by M i l l e r et a l J T h e y s u g g e s t that t h i s c a n b e e x p l a i n e d by d e s i g n a t i n g t h i s s y s t e m a s a l i q u i d s e m i c o n d u c t o r f o r m o l e f r a c t i o n s of c a d m i u m l e s s than 0.85. D a t a o b t a i n e d by E n d e r b y and W a l s h v f o r

0

D

I 0.2

I 0.4

I 0.6

I 0.8

"2 1.0

XCd Fig. 1--Reduced e l e c t r i c a l resistivity (measured resistivity minus a linear combination of the pure component r e s i s t i v i ties) and excess stability for the Cd-Bi system at 500~ vs composition. 120

12

l Cd-Sb

:.

~ 100

IO

I

-6 tD

~k >_" B 0

b.J m

~_ r 60 to re r~ 4 0 to o o w 20 re

6

~t

o3 tD __

STABILITY

II

~

4 , , 'u x w

/ 0

0.0

/

\ ,

w

i

,

0.2

0.4

0.6

0.8

\ -0

1.0

XCd Fig. 2--Reduced e l e c t r i c a l resistivity and excess stability for the Cd-Sb system at 500~ vs composition. VOLUME l , JANUARY 1970-305

CdSb show an i n c r e a s e in the Hall c o e f f i c i e n t , R, with t e m p e r a t u r e f r o m the m e l t i n g t e m p e r a t u r e to n e a r l y 600~ Th e r a t i o of the Hall c o e f f i c i e n t c a l c u l a t e d f r o m f r e e e l e c t r o n t h e o r y , a s s u m i n g a ll v a l e n c e e l e c t r o n s a r e in the conduction band, to the m e a s u r e d Hall c o e f f i c i e n t is 0.66 or e q u i v a l e n t to 2.3 e l e c t r o n s p e r a to m . If R is taken as a m e a s u r e of the e l e c t r o n d e n s i t y , this s u g g e s t s that 30 pct of the e l e c t r o n s a r e in bound o r l o c a l i z e d s t a t e s . The i n t e r p r e t a t i o n of the e l e c t r i c a l r e s i s t i v i t y of the C d - B i s y s t e m by the p r e s e n t a u t h o r s a a l s o supp o s e s that s o m e of the e l e c t r o n s a r e not f r e e to p a r t i c i p a t e in the conduction p r o c e s s , but the d e p a r t u r e f r o m the f r e e e l e c t r o n value is not so g r e a t as that m e a s u r e d f o r CdSb. The r e s u l t s of t h e s e o b s e r v a t i o n s on the t h r e e s y s t e m s shown a r e that when t h e r e is a m a x i m u m in the e x c e s s s t a b i l i t y - c o m p o s i t i o n c u r v e , t h e r e is a m a x i mu m in e l e c t r i c a l r e s i s t i v i t y and the l a r g e r the m a x i m u m in s t a b i l i t y , the l a r g e r the m a x i m u m and the n a r r o w e r the peak for e l e c t r i c a l r e s i s t i v i t y . This can be a t t r i b u t e d to an a b s e n c e of e l e c t r o n s a v a i l a b l e f o r conduction due to s t a b i l i z a t i o n of that p a r t i c u l a r c o m p o s i t i o n by e l e c t r o n bonds. T h i s is s u b s t a n t i a t e d by the Hall m e a s u r e m e n t s r e p o r t e d by E n d e r b y and Walsh. 7 We p r o p o s e to i l l u s t r a t e t h e s e o b s e r v a t i o n s with a m o d e l using the f l u c t u a t i o n s c a t t e r i n g a p p r o a c h . In a p r e v i o u s p a p e r the a u t h o r s 3 expanded and d i s c u s s e d the fluctuation s c a t t e r i n g m o d e l for the e l e c t r i c a l r e s i s t i v i t y of b i n a r y liquid alloys.Z In this m o d e l , the e l e c t r i c a l r e s i s t i v i t y is w r i t t e n as a s u m of t h r e e t e r m s , only one of which, the t e r m due to c o m p o s i t i o n f l u c t u a t i o n s , is s t r o n g l y n o n l i n e a r with c o m p o s i t i o n . That t e r m is w r i t t e n as A.T 1

Pc =

V m ' n a 1 - Xa

[z]

a~a

1 w h e r e B is a n o t h e r c o n s t a n t when the c o m p o s i t i o n and the t e m p e r a t u r e a r e not v a r i e d . T e n t a t i v e l y , we a s s u m e that the e x c e s s s t a b i l i t y depends i n v e r s e l y on the s q u a r e of the e l e c t r o n c o n c e n t r a t i o n . Thus, a

+ b

[51

w h e r e a and b a r e c o n s t a n t s . T h e n , one may w r i t e the s t ab i l i t y in t e r m s of the e x e e s s s t a b i l i t y as $ = $xs +RT/X1X2

= Sxs +D

[6]

w h e r e D is a c o n s t a n t at c o n s t a n t t e m p e r a t u r e and c o m p o s i t i o n . By r e a r r a n g i n g Eq. [5] and s u b s t i t u t i n g it and Eq. [6] into Eq. [4], one obtains p =A'

+B'$ xs +C' $XS-b Sxs + D

where A' = (-bB/a), 306

24 Mg-BI

x

E

REDUCED

I00

2o To

RESISTIVITY

r I

x

:~80

0

16

o

.,g

EXCESS STABILITY

)I---

/

60

N

I.-

12

"5 rn

I..-

w

q

40

I-.

8

m

if)

u

\

(:3 W =

0-~

0.0

I

0.2

0.6

0.4

U

4

20

0.8

•

W

0

1.0

XMg Fig. 3--Reduced electrical resistivity at 900~ and excess stability at 700~ for the Mg-Bi system vs composition,

Table 1. Maximum Excess Stability, Excess Free Energy, and Reduced Resistivity

Alloy

Excess Stability, cal

Excess Free Energy, cal

Reduced Resistivity, p ~2-cm

Cd-Bi Cd-Sb Mg-Bi

2100 9200 230,000

-85 -675 -5600

" 69 109 1136

aXe

w h e r e n is the e l e c t r o n c o n c e n t r a t i o n and V m is the m o l a r v o l u m e of the a ll o y (usually a slowly v a r y i n g function of c o m p o s i t i o n ) . Thus at constant t e m p e r a ture the r e s i s t i v i t y due to c o m p o s i t i o n fluctuations is a p p r o x i m a t e l y a c o n s t a n t d i v i d e d by the s t a b i l i t y and the e l e c t r o n c o n c e n t r a t i o n s q u a r e d , i . e . , Pc = C ~ n a g . Now the total r e s i s t i v i t y m a y be w r i t t e n as

sxs = ~

120

B' = B/a

V O L U M E 1, J A N U A R Y 1 9 7 0

[7] and C ' = C / a .

If IgXSl

>> Ibl and IDI, this r e d u c e s to p = B ' $ xs + const

[8]

i . e . , at constant t e m p e r a t u r e and c o m p o s i t i o n the r e -

s i s t i v i t y is a l i n e a r function of the e x c e s s st a bi l i t y if S x s is l a r g e . T h i s is p r o b a b l y the c a s e f o r M g - B i s i n c e S x s ~ 230 kcal and D - +9.6 kcal and b, which is the i n t e r c e p t on $ xs of 1 / n 2 = 0, is within 10 kcal

of the o r i g i n . Using the fluctuation s c a t t e r i n g mode l to find n, ~ we c a l c u l a t e a = 83 kcal p e r 1In 2 and b = - 4 kcal. If bSxsl is s m a l l c o m p a r e d to Ibl and IDI, a s i m i l a r r e s u l t is obtained with the r e s i s t i v i t y p r o p o r t i o n a l to the e x c e s s s t a b i l i t y , the only d i f f e r e n c e b ei n g an additive constant. In c o n c l u s i o n , it has been o b s e r v e d in t h r e e s y s t e m s that the magnitude of the r e d u c e d r e s i s t i v i t y tends to be l a r g e when the e x c e s s s t a b i l i t y is l a r g e . By a s s u m i n g that the e x c e s s s t a b i l i t y is l i n e a r with the r e c i p r o c a l of the e l e c t r o n c o n c e n t r a t i o n s q u a r e d and using the fluctuation s c a t t e r i n g m o d e l f o r e l e c t r i c a l r e s i s t i v i t y , we have shown that in the l i m i t s of l a r g e and s m a l l e x c e s s st ab i l i t y the r e s i s t i v i t y is a l i n e a r function of the e x c e s s s t a b i l i t y . It is s u g g e s t e d that this r e s u l t shows that l a r g e e x c e s s s t a b i l i t y is a s s o c i a t e d with a tendency t o w a r d l o c a l i z a t i o n of e l e c t r o n s f o r bonding p u r p o s e s . T h i s work was s u p p o r t e d by the National Science Foundation G r a n t s Nos. GK759 and GK2840. METALLURGICAL TRANSACTIONS

1. L.S. Darken: Trans. TMS-AIME, 1967, vol. 239, pp. 80-89. 2. S. Takeuchi and H. Endo: Tran~ Japan Inst. Metals, 1962, vol. 3, pp. 35-41. J. L. Tomlinsonand B. D. Lichter: Advan. Phys., 1967, vol. 16, pp. 501-12. 3. J. L. Tomlinson and B. D. Lichter: Trans. TMS-AIME, 1969, vol. 245, pp. 2261-67. 4. E. Miller,J. Paces, and K. L. Komarek: Tran~ TMS-AIME, 1964, vol. 230, pp. 1557-63. 5. B. R. Ilschner and C. Wagner:ActaMet., 1958, vol. 6, pp. 712-13. 6. R. Hultgren, R. L. Orr, P. D. Anderson, and K. K. Kelley: Selected Values of Thermodynamic Properties of Metals and Alloys, J. Wiley & Sons, New York, 1963. 7. J. E. Enderby and L. Walsh:Phil. Mag., 1966, vol. 14, pp. 991-1002. 8. J. L. Tomlinson: Ph.D. Thesis, Universityof Washington,Seattle, 1967.

An Indirect Method for the Determination of Intrinsic Microhardnesses of Microporous and Finely Divided Materials ERHARD KLAR

of t h e v o l u m e r a t i o of t h e p h a s e s of t h e c o m p o s i t e , t h e i n t r i n s i c h a r d n e s s e s of the i n d i v i d u a l p h a s e s , a n d t h e i r g e o m e t r i c s t r u c t u r e . T h e d i s p e r s i o n of s u c h a f r e q u e n c y d i s t r i b u t i o n , i . e . , t h e s c a t t e r i n g of the v a l u e s f r o m t h e i r a v e r a g e , i s a m e a s u r e of t h e h e t e r o g e n e i t y of t h e c o m p o s i t e a n d d e p e n d s on t h e s a m e v a r i a b l e s . It d e c r e a s e s a s t h e h a r d n e s s of t h e a u x i l i a r y p h a s e a p p r o a c h e s the i n t r i n s i c h a r d n e s s of t h e m a t e r i a l u n d e r s t u d y . T h e d i s p e r s i o n of a t w o - p h a s e composite has a minimum where both phases have equal hardness. The approximate intrinsic hardness of a m a t e r i a l c a n t h e r e f o r e b e o b t a i n e d f r o m e x t r a p o l a t i o n to z e r o o r m i n i m u m d i s p e r s i o n of e x p e r i m e n t a l hardness dispersions. F o r a s u c c e s s f u l a p p l i c a t i o n of t h i s m e t h o d , t h e load on the indentor must be chosen so that the areas c o v e r e d b y the i n d e n t a t i o n s a r e s m a l l e r t h a n w h a t c a n b e c o n s i d e r e d r e p r e s e n t a t i v e of t h e e n t i r e s t r u c t u r e . I T h e m e t h o d i s i l l u s t r a t e d in t h e f o l l o w i n g two e x a m p l e s . E x a m p l e No. 1. T h e e x p e r i m e n t a l d a t a s h o w n in F i g s . 1 to 3 w e r e o b t a i n e d o n p o r o u s c o p p e r c o m p a c t s m a d e b y i s o s t a t i c c o m p a c t i o n of a c o m m e r c i a l , recently reduced, -325 mesh copper powder. The c o m p a c t s w e r e t r e a t e d u n d e r h y d r o g e n a t 1000~ f o r

g i s w iN d e l y uE s e d a sS a n o nS destrucH A Rt e s t i n D t i v e m e t h o d f o r a s s e s s i n g m e c h a n i c a l p r o p e r t i e s of m e t a l s . In m u l t i p h a s e m a t e r i a l s a s p r o d u c e d b y p o w d e r m e t a l l u r g i c a l t e c h n i q u e s , o n e of t h e p h a s e s f r e q u e n t l y i s t h e p o r e p h a s e . In s u c h p o r o u s s y s t e m s t h e m e a s u r e d h a r d n e s s i s c a l l e d a p p a r e n t h a r d n e s s if the hardness indentation covers an area large enough to b e r e p r e s e n t a t i v e of t h e e n t i r e p a r t . It i s c a l l e d t r u e m a t e r i a l o r i n t r i n s i c h a r d n e s s if t h e i n d e n t a t i o n is taken well within the solid material. For a better u n d e r s t a n d i n g of c o r r e l a t i o n s b e t w e e n h a r d n e s s a n d m e c h a n i c a l p r o p e r t i e s of p o r o u s s y s t e m s , it i s d e s i r a b l e to know b o t h a p p a r e n t a n d i n t r i n s i c h a r d n e s s . W h i l e t h e m e a s u r e m e n t of t h e a p p a r e n t h a r d n e s s of a m a t e r i a l n o r m a l l y p r e s e n t s no p r o b l e m , the m e a s u r e m e n t of i t s i n t r i n s i c h a r d n e s s b e c o m e s d i f f i c u l t and eventually impossible as its particle size decreases or as its apparent density and pore size decrease. For such materials, even indentations taken w i t h m i c r o h a r d n e s s t e s t i n g e q u i p m e n t a r e too l a r g e to a v o i d i n t e r f e r e n c e f r o m n e i g h b o r i n g p o r e s . T h i s communication describes an indirect method for the a p p r o x i m a t e d e t e r m i n a t i o n of t h e i n t r i n s i c h a r d n e s s of s u c h f i n e l y d i v i d e d o r m i c r o p o r o u s m a t e r i a l s . P r e p a r a t i o n of t h e m a t e r i a l to b e t e s t e d c o n s i s t s i n t h e f a b r i c a t i o n of a s o l i d c o m p o s i t e . T h e a u x i l i a r y phase may be an inert and hardenable liquid such as s e l f - h a r d e n i n g r e s i n , o r a s u i t a b l e m o l t e n m e t a l . If the material under study is a porous part, it is imp r e g n a t e d w i t h t h e l i q u i d ; if it i s a p o w d e r , a u n i f o r m dispersion is prepared. The solid composite is caref u l l y p o l i s h e d b y a p p l y i n g t h e t e c h n i q u e of a l t e r n a t e mechanical polishing and etching. On the prepared surface, microhardness indentations are produced a t r a n d o m . Due to l o c a l v a r i a t i o n s of c o n c e n t r a t i o n of the p h a s e s , t h e i n d i v i d u a l h a r d n e s s v a l u e s f o l l o w a f r e q u e n c y d i s t r i b u t i o n t h e f o r m of w h i c h i s a f u n c t i o n ERHARD KLAR is Scientist, Metals Group, Glidden-Durkee Division of SCM Corporation, Baltimore, Md. Manuscript submitted March 20, 1969. METALLURGICAL TRANSACTIONS

,

,

~

,

I

I

I

,

,

,

so ~ 5o 0

~ 3o ~ 2o I

2

IlO

PERCENTAGE

210

I

I

50

L O W E R THAN

GIo

I 90

INDICATED

915

;S

HARDNESS

Fig. 1--Log p r o b a b i l i t y plots of r e s i n i m p r e g n a t e d copper compacts containing 52 vol pct Cu. C i r c l e s : h a r d n e s s of r e s i n is 7.0 kg p e r sq mm; d i s p e r s i o n of h a r d n e s s m e a s u r e m e n t s , cr = 1.196; T r i a n g l e s ; h a r d n e s s of r e s i n : 27.7 kg p e r sq mm; d i s p e r s i o n of h a r d n e s s m e a s u r e m e n t s , cr = 1.136.

9 8~I

I

I

I

I

o1 ~ 6 0

09

O9 nr

40

-Jr" (/1 rr h,

20

0

0

0

I

I

I

i

O. 2

0.4

0.6

0.8

VOLUME

FRACTION

1.0

CU

Fig. 2--Arithmetic a v e r a g e s of V i c k e r s h a r d n e s s e s of r e s i n i m p r e g n a t e d copper compacts. Upper curve: h a r d n e s s of r e s i n is 27.7 kg p e r sq mm; lower curve: h a r d n e s s of r e s i n is 7.0 kg p e r sq mrn. VOLUME l, JANUARY 1970 307