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EXCESS VOLUME OF MIXING INDUCED BY SIZE
EFFECT IN LIQUID ALLOYS. A HARD SPHERE
SIMULATION
Cl. Lemaignan, P. Desre
To cite this version:
JOURNAL DE PHYSIQUE CoZZoque C8, suppZ&ent au n08, Tome 41, aoct 1980, page C8-305
EXCESS VOLUME OF M I X I N G INDUCED BY S I Z E EFFECT I N L I Q U I D ALLOYS, A HARD SPHERE ,SIrlULATION
C1. Lemaignan and P
.
~ e s r e *Laboratoire dlEtude de Za SoZidification, CENG
-
85 X , 38041 GrenobZe Ce'dex, France* ~ a b o r a t o i r e de Thermodynamique e t de Physico-Chimie Me'taZzurgiques, associe' au C.N.R.S., ENSEEG, B.P, 44, 38401 Saint Martin dlHeresJ France.
INTRODUCTION.-
To simulate t h e random s t r u c t u r e o f l i q u i d o r amorphous n e t a l s and a l l o y s , i t has been sugges- t e d t o b u i l d random packings o f h a r d spheres. This has been done e i t h e r w i t h packing o f s o l i d spheres ( l y 2 ) o r by computation ( 3 ' 4). The work already undertaken was mostly done or1 si.;cjle component packings
.
I n t h e case o f an a l l o y , t h e s i m u l a t i o n packing has t o be r e a l i z e d w i t h spheres o f d i f f e - r e n t diameters. Some s t u d i e s on packings o f spheres o r granules o f d i f f e r e n t diameters have been repor- t e d i n t h e 1 im i t e d case o f l a r g e d i f f e r e n c e s i n diameter(5y
6).
Those experiments show an increase i n packing f r a c t i o n f o r m i x t u r e s o f spheres. However, due t o l a r g e d i f f e r e n c e s i n diameter used i n those studies, these r e s u l t s are n o t a p p l i c a b l e t o m e t a l l i c alFoys f o r which d i f f e r e n c e s i n atomic r a d i i r a r e l y exceed 30%('I.
I n order t o f i n d , and p o s s i b l y t o measure a geometrical c o n t r i b u t i o n t o the volume o f m i x i n g found i n various a l l o y s , we have measured the packing f r a c t i o n o f m i x t u r e o f s l i g h t l y d i f f e r e n t spheres.EXPERIMENTAL PROCEDURE.
-
S t e e l b e a r i n g b a l l s o f d i f f e r e n t diameters
(P
= 3, 3 . 1 7 5 , 3 . 5 , 4 , 5 , 6 f 0.002 mm) were care- f u l l y cleaned and then b i n a r y mixed : s t a r t i n g from5
1.2 10 spheres, 3 mm i n diameter, small q u a n t i - t i e s o f spheres o f an o t h e r type were added, mixed
and t h e volume measured. The homogenity o f t h e mix- 2
t u r e was v e r i f i e d by
x
t e s t s on small samples o f-a\
t h e m i x t u r e . The procedure was repeated u n t i l t h e composition was 50 % (number % o f spheres). The two types o f spheres were then seperated, u s i n g a
r o l l i n g m i l 1 s o r t i n g device, and t h e experiment was repeated s t a r t i n g from t h e l a r g e diameter side.
The m i x t u r e was poured w i t h i n a p l e x i g l a s s c y l i n d e r : a t r a n s p a r e n t m a t e r i a l was used t o check t o randomness o f t h e packings along t h e w a l l s . The bottom o f the c y l i n d e r was grooved i n t h r e e d i f f e - r e n t d i r e c t i o n s w i t h d i f f e r e n t depths t o a v o i d r e - g u l a r p a c k i n g ~ o f t h e f i r s t l a y e r s a t t h e begin- n i n g o f t h e f i l l i n g . The whole c y l i n d e r was free t o r o t a t e around an h o r i z o n t a l a x i s ; t h i s degree o f freedom allowed us t o t o p p l e i t r a p i d l y over, i n o r d e r t o l e t the b a l l s f a l l f r e e l y down, t o l e a d t o a loose random packing. The f r e e surface was then l e v e l e d o f f and t h e h e i g h t o f t h e packing measured u s i n g a p i s t o n coated w i t h s o f t foam t o compensate i t s roughness.
The packing f r a c t i o n y, i s then simply de- f i n e d as t h e r a t i o o f t h e volume o f t h e s t e e l t o t h e i n n e r volume o f t h e c y l i n d e r . However two c o r r e c - t i o n s have t o be made : t h e bottom c o r r e c t i o n i s due t o the existence o f t h e grooves. T h e i r e f f e c t i s t h a t t h e p i s t o n does n o t l i f t b e f o r e a given number o f b a l l s has f i l l e d t h e volume o f t h e gro- oves. Those b a l l s (100 t o 600, depending on diame-
JOURNAL DE PHYSIQUE
I
t h e c y l i n d e r w i t h a l o n g p i p e g u i d i n g t h e b a l l sF i g u r e 1 : Loose packing f r a c t i o n o f t h e d i f f e r e n t m i x t u r e s o f spheres.
d/D = ( a )
.
75 ; (b).
6 ; (c).
5t e r ) have t o be substracted t o t h e mixture. The c o r r e c t i o n due t o t h e w a l l o f t h e c y l i n d e r was done according t o the work o f BENENATI(~), who showed t h a t t h e packing f r a c t i o n o f equal spheres o c i l l a - t e s around t h e mean value near t h e w a l l o f a c y l i n - der
.
The i n t e g r a l value o f those o s c i l l a t i o n s leads t o a " w a l l e r r o r " w h i c h , i n our case, i s an overestimate o f t h e volume o f t h e c y l i n d e r . Theinduced c o r r e c t i o n i s p r o p o r t i o n a l t o t h e w a l l sur- face area and l e t us increase t h e packing f r a c t i o n by 0.2 % f o r equal spheres, up t o 0.35 % a t i n t e r mediate concentration.
RESULTS.
-
For a1 1 the m i x t u r e s , two types o f packirgs have been r e a l i s e d , r e f e r e d usualy as "dense" and "1 oose"
.
Dense packi ngs were obtained by f i 11 i ngdown t o t h e f r e e s u r f a c e and t h e loose packings were obta,ined by t o p p l i n g t h e c y l i n d e r as explained above.
For equal spheres, dense packing f r a c t i o n s were found t o be :
yd = 0.639
5
0.002 and loose :yL = 0.614
+
0.002i n good agreement w i t h experiments p r e v i o u s l y repom ted('
'
F o r mixtures, we have p l o t t e d on f i g u r e 1
o n l y t h e loose packing f r a c t i o n s o f t h e mixtures, as they were more p r e c i s e l y d e f i n e d and e a s i e r t o d u p l i c a t e . W i t h i n t h e experimental p r e c i s i o n o f t h d experiment, i t i s c l e a r from those curves t h a t t h e packing f r a c t i o n does n o t change w i t h composition f o r diameter r a t i o i n t h e range 0. 7<DSma1 l/DLarge<l For h i g h e r d i f f e r e n c e s i n diameter, t h e packing f r a c t i o n increases s l i g h t l y , e s s e n t i a l l y on t h e small diameter s i d e ; however, t h e r e l a t i v e i n c r e a - se remains small ( < 5 % )
.
The maximum r e l a t i v e increase o f t h e 1 oose packing f r a c t i o n s versus diameter r a t i o i s p l o t t e d on f i g u r e 2, and i s com- pared w i t h o t h e r works. Our r e s u l t s a r e compatible w i t h them and t h e slope o f t h i s curve suggests t h e disappearance o f a geometrical e f f e c t on t h e pac- k i n g f r a c t i o n f o r diameter r a t i o above 0.7.DISCUSSION.-
F o r t h e l i m i t i n g case o f very l a r g e d i f - ferences i n diameters, i t i s obvious t h a t m i x i n g w i l l increase t h e packing f r a c t i o n : f o r l a r g e spheres, t h e i n t r o d u c t i o n o f small spheres i n t h e holes l e f t i n t h e network, w i t h o u t d i s t u r b i n g i t
Jeschar Yerazunis
A
This w o r k F i g u r e 2:
Maximum r e l a t i v e increase o f t h e l c o s e packing f r a c t i o n s as a f u n c t i o n of t h e diameter r a t i o .exchanged by a s i n g l e s o l i d sphere, t h e packing f r a c t i o n i s a l s o increased. For an " i n f i n i t e " d i f - ference i n diameter, t h e maximum increase i n pac- k i n g f r a c t i o n i s then given by :
Ymax = Yo + ( l - y 0 ) y o
,"
0.851However, f o r t h e i n t e r m e d i a t e s i t u a t i o n o f spheres o f s m a l l e r d i f f e r e n c e i n size, i t i s n o t p o s s i b l e t o i n t r o d u c e a sphere w i t h o u t d i s t u r b i n g t h e network and so f a r t h e r e i s no ways t o s o l v e a n a l y t i c a l l y t h i s problem o f pure s t a t i s t i c a l gee,- metry. A rough estimate o f t h e s i z e o f t h e i n t e r s -
b i t i a l s i t e i n s i d e t h e i r r e g u l a r t e t r a h e d r a o f t h e -.twork, leads us t o conclude t h a t t h e r e i s a very s..iall amount o f s i t e s l a r g e enough t o accept a sphere as an i n t e r s t i t i a l s e v e n w i t h a diameter o f o n l y one h a l f o f t h e diameter o f t h e o t h e r spheres o f t h e network(9).
C8-307 For l i q u i d a l l o y s , t h e experimental proper- t y usualy measured i s t h e d e n s i t y o r i t s r e c i p r o c a l
:
t h e molar volume. For i d e a l s o l u t i o n s , t h e molar volume i s a l i n e a r combination o f t h e molar volumes o f t h e l i q u i d components Videal = Vs
+
(1-C)VL, where C = atomic c o n c e n t r a t i o n o f t h e small specie and t h e subscripts S and L a r e f o r l a r g e and small. I n most cases a departure from i d e a l i t y e x i s t s t h a t leads t o an excess volumeVX,
= Vsol-
Videal& r s i m u l a t i o n g i v e s us t h e a b i l i t y t o express t h e cieometrical c o n t r i b u t i o n t o t h i s excess volume :
. .
where d i s t h e atomic diameter.
As an exemple t h e v a l u e o f t h i s geometrical excess volume i s p l o t t e d on f i g u r e 3 f o r t h e h y p o t h e t i c a l case o f a diameter r a t i o o f 1/2.
From t h e c o m p i l a t i o n a v a i l a b l e ( 7 ) , i t i s c l e a r t h a t more than 80 % o f a1 1 t h e metal1 i c atoms
0
have r a d i i l a y i n g between 1.25 and 1.80 A ( i . e . w i t h a diameter r a t i o between 0.7 and 1 ) . For a l l o y s o f those metals
,
t h e d i f f e r e n c e s i n atomic diameters aretoo small t o induce an i n c r e a s e i n d e n s i t y by geometrical e f f e c t . I f they show some excess v o l m on mixing, i t should be analysed i n term of chemi- c a l i n t e r a c t i o n s between u n l i k e atoms ( e l e c t r o n i c charge displacement on a: l o y i ng).
Of course t h i s s i m u l a t i o n cannot g i v e any e x p l a n a t i o n o f t h e p o s i - t i v e excess volume found i n some a l l o y s .CONCLUSION.-
C8-308 JOURNAL DE PHYSIQUE
d REFERENCES.
-
F i g u r e 3 : Geometrical m i x i n g volume f o r atoms o f diameter r a t i o 1/2. ( v e r t i c a l s c a l e u n i t = a o l a r volume o f t h e small c o n s t i - t u e n t ) .
found i n t h i s s i m u l a t i o n w i t h hard spheres can be used t o g i v e t h e value o f t h e s i z e e f f e c t on t h e molar volume o f m e t a l l i c a l l o y s . For most o f t h e common a l l o y s , t h i s e f f e t i s n e g l i g i b l e and a de- p a r t u r e from i d e a l i t y may suggest chemical i n t e - r a c t i o n s i n t h e 1 i q u i d . However s o f t e r i n t e r a t o m i c p o t e n t i a l should a1 low b e t t e r accomodation o f small i n t e r s t i t i a l s and hence h i g h e r s i z e e f f e c t s may be expected i n r e a l a l l o y s s p e c i a l l y when d i l u t e d .
(1)
J.D.
BERNAL, Proc. Roy. Soc. LondonA,
299 (1964).(2) J.L. FINNEY, Proc, Roy. Soc. London
A,
479 (1970).(3) J. BLETRY, Z. Naturforsch.
33,
327 (1978)( 4 )
T. ICHIKAWA, Phys. S t a t . Sol. (a)2,
293, (1975).( 5 ) S. YERAZUNIS and a l . , Nature
195,
33 (1962). ( 6 ) R. JESCHAR, Arch. Eisenhgttenwes.,X, 91 (1964) (7) R.P. ELLIOT, C o n s t i t u t i o n o f b i n a r y a l l o y s( F i r s t sup.) Mc. Grow H i l l , New York (1975)
p. 872.
(8) R.F. BENENATI, C.B. BROSILOW, A. I. Ch. E. Journal
8,
360 (1962).( 9 ) C1. LEMAIGNAN, Thesis Grenoble France 1980.
ACKNOWLEDGEMENTS.-
Special thanks a r e g i v e n t o Pr. H.
