EXPLORATION OF THE PHASE DIAGRAM OF Al-Li BINARY ALLOYS BY D.S.C. MEASUREMENTS AND MONTE CARLO METHODS

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EXPLORATION OF THE PHASE DIAGRAM OF Al-Li BINARY ALLOYS BY D.S.C.

MEASUREMENTS AND MONTE CARLO METHODS

F. Livet, Y. Brechet

To cite this version:

F. Livet, Y. Brechet. EXPLORATION OF THE PHASE DIAGRAM OF Al-Li BINARY ALLOYS BY D.S.C. MEASUREMENTS AND MONTE CARLO METHODS. Journal de Physique Colloques, 1987, 48 (C3), pp.C3-357-C3-363. �10.1051/jphyscol:1987341�. �jpa-00226572�

JOURNAL DE PHYSIQUE

C o l l o q u e C 3 , s u p p l & m e n t a u n 0 9 , Tome 48, s e p t e m b r e 1 9 8 7

EXPLORATION OF THE PHASE DIAGRAM OF A1-Li BINARY ALLOYS BY D.S.C.

MEASUREMENTS AND MONTE CARL0 METHODS

F. LIVET and Y. BRECHET

Laboratoire de Thermodynamique et Physico-Chimie Metallurgiques ENSEEG, Domaine Universitaire, B.P. 75,

F-38402 Saint-Martin-d'Heres. France

The AlLi FCC L m e t a s t a b l e phase diagram is modelized by an I s i n g harniltonian o f t h e a l l o r A nearehg neighbour i n t e r a t i o n V = 75 meV and a n e x t n e a r e s t neighbour i n t e r a c t i o n V = -16 meV c a n e x p l a i n t h e o%served Llp a n t i p h a s e energy and t h e observed FCC-I? s u r f a c e energy. The phase diagram o b t a l n e d from t h e s e v a l u e s is i n r e a s o n a b l e agr%ernent w i t h t h e experiment. The d i f f e r e n t i a l s c a n n i n g c a l o r i m e t r y anomalies observed i n t h i s system a r e e x p l a i n e d by dynamical e f f e c t s o f t h e FCC/L

unmixing p r o c e s s . 1 2

INTRODUCTION

The m e t a s t a b l e low t e m p e r a t u r e phase diagram o f A1-Li h a s been e x t e n s i v e l y s t u d i e d /1-4/. For an i n i t i a l composition C 4 1 4 a t LiX, i f a sample i n i t i a l l y i n s o l i d s o l u t i o n is quenched and aged a t T 5 5 8 K , a m e t a s t a b l e s i m p l e c u b i c L ₍ 6

o r d ' ) phase a p p e a r s w i t h t h e A 1 L i s t o e c h i o m e t r i c composition. X Ray and e i e c t r o n d i f f r a c t i o n experiments g i v e a p r 3 c i s e composition-temperature phase diagram. B e s t r e s u l t s a r e g i v e n on f i g . 4 ( r e f /I/, / 2 / , / 4 / ) .

Moreover, t h e k i n e t i c s o f t h e unmixing h a s shown t h a t t h i s system f o l l o w s t h e Lifchitz-Slyozov-Wagner behaviour (LSW) / 5 / , / 3 / , /4/. From such a s t u d y , t h e L /FCC

1 2 .

s u r f a c e energy /4/ h a s been o b t a i n e d a t 420 K : d ^{r. 14m~/m'.} From t h e d - s l o c a t l o n 5

c o n f i g u r a t i o n , t h e (111) a n t i p h a s e energy can b e e s h m a t e d t o d = ^140mJ/m / 6 / . R e c e n t l y , C l u s t e r V a r i a t i o n a l Method (CVM) calculatiox?s / 7 / p r e d i c t e d t h e e x i s t e n c e o f a new low t e m p e r a t u r e p h a s e , where Guinier-Preston (GP) zones s h o u l d a p p e a r w i t h t h e FCC symetry. T h i s p r e d i c t i o n is s u p p o r t e d by D i f f e r e n t i a l Scanning C a l o r i m e t r i g (DSC) anomalies observed i n A1-Li.

I n t h i s p a p e r , t h e microscopic m o d e l i z a t i o n o f t h e a l l o y w i t h t h e I s i n g Model i s d i s c u s s e d . Then f o l l o w Monte C a r l o c a l c u l a t i o n s o f t h e phase diagram. Reasonable v a l u e s o f t h e i n t e r a c t i o n e n e r g i e s a r e deduced. E v e n t u a l l y , DSC experiments a r e developped and t h e i r i n t e r p r e t a t i o n is d i s c u s s e d .

I- THE ISING MODEL I N A1-Li

The harniltonian of t h e system is w r i t t e n w i t h s f i r s t (V1) and a second (V2) nearest-neighbour i n t e r a c t i o n :

(1) H =

L

21

~ c j z , 1 1 J 2 i J

where C . = 1 i f s i t e i is occupied by a L i atom and Ci = 0 i f s i t e is occupied by an AI atom:

and : ( i j

>

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987341

C3-358 JOURNAL DE PHYSIQUE

I n t h e r e g i o n corresponding t o t h e m e t a s t a b l e g - L i phase diagram, (1) l e a d s t o an unmixing i f V > 0 and V ( 0 . It has been shown t h a t V = 0 l e a d s t o incomplete unmixing ( f i g . 2 ) / 9 / , / 9 / and2 t h e i n t r o d u c t i o n of a n e g a t i v e V 2 i s necessary here.

Rough e s t i m a t i o n of V1 and V can be obtained from da ^and

d s

One o b t a i n s h e r e : V ib -14 meV and V -8lmeV. This shows t h a t 13 = -V /V is small.

1 2 1

Formula ( 3 ) i s n e v e r t g e l e s s extremely u n p r e c i s e : d should be anisotropic, and d

should be c o r r e c t e d from non stoechiometry and from s g o r t range o r d e r e f f e c t s /lo/. 2

more p r e c i s e c a l c u l a t i o n can be o b t a i n e d from t h e Monte Carlo Method.

11- PHASE DIAGRAM AND SURFACE ENERGY

1. The Monte Carlo method is used h e r e (Metropolis a l g o r i t h m ) /11/ t o o b t a i n t h e most probable c o n f i g u r a t i o n s o f t h e system M.C. l a t t i c e has only 6,912 s i t e s and t h e i n i t i a l L i c o n c e n t r a t i o n C is f i x e d (canomical Ensemble). The c a l c u l a t i o n s a r e c a r r i e d a t v a r i o u s temperature2 with two ( s m a l l ) v a l u e s of R : ( I3 =.1 and R = .2).

The p e r i o d i c boundary c o n d i t i o n s a r e used.

The r e a c h of t h e e q u i l i b r i u m s t a t e is a c c e l e r a t e d i f t h e exchanges a r e c a r r i e d between any o f two s i t e s o f t h e l a t t i c e .

I n i t i a l l y , t h e system is d i s o r d e r e d . A t a low temperature two phases c o e x i s t i n t h e l a t t i c e :

( i ) a block of volume f r a c t i o n f v and of L i c o n c e n t r a t i o n C of t h e simple c u b i c LI2 phase and ( i i ) a block o f L i c o n c e n t r a t i o n C of t h e FCC s o l t d s o l u t i o n . One has :

Co = f C + (1-fv)Cm

v P

2. I n t e r p r e t a t i o n of t h e r e s u l t s

The r e s u l t o f each c a l c u l a t i o n is a n average of t h e "Warren Cowley parameters"

on about 20 c o n f i g u r a t i o n s . These parameters a r e a l s o a g r a g e d along t h e c u b i c e q u i v a l e n t o r i e n t a t i o n s and they a r e c a l c u l a t e d till t h e 28 neighbouring s i t e ( i e 444 i n t h e u s u a l 2h, 2k, 21 FCC n o t a t i o n ) . They can be w r i t t e n :

P i ( T . ) is t h e p r o b a b i l i t y t h a t two s i t e s i n t h e ith s h e l l of neighbours a t dkskancel r . a r e both occupied by L . l atoms. I n t h e range of c o n c e n t r a t i o n s and t e m p e r a t u r e s under s t u d y , a s Vl/kT > ^{1, it is} observed t h a t : P L . L i ( l ) = 0. This means t h a t no L i atom can be a f l r s t neighbour of a L i atom and t h a t t h e L non- sotechiometry only corresponds t o A 1 atoms occupying L i s i t e s . Fig.1 shows %$pica1 s e t of d ( r . ) o b t a i n e d w i t h : C = 12.5 a t L i % , R =0.1 and kT/V2 = -3.

F i g . 1 P l o t of t h e t h e WARREN-COWLEY parameters

a g a i n s t t h e d i s t a n c e

y.- Linear i n t e r - p o l a t i o n deduced

(formula 5)

I n a two p h a s e model, n e g l e c t i n g t h e s h o r t r a n g e o r d e r between d e f e c t s (which changes somewhat t h e f i r s t d ( T . ) ) , d ( T i ) c a n b e w r i t t e n :

5 C ~ ( - C ~ ; 2

4"

where : f (.f . ) = 1 i f r. c o r r e s p o n d s t o t h e d i s t a n c e between two s i m p l e c u b i c s i t e s ( 2 h , 2k and 2$ e v e n ) , +(%. ) = 0 e l s e where and ' I ~ ( T . ) i s t h e P a t t e r s o n f u n c t i o n o f t h e s i m p l e c u b i c phase. A& t h e r e s u l t s a r e a v e r a g e d l o v e r a l l t h e c u b i c e q u i v a l e n t d i r e c t i o n s , t h e a n i s o t r o p y o f

x(:r

T h i s is o f t e n c a l l e d P o r o d ' s f o r m u l a /12/ i n s m a l l a n g l e s c a t t e r i n g t e c h n i q u e s , V is t h e volume o f t h e L p h a s e and S i s t h e s u r f a c e o f t h e L /FCC i n t e r p h a s e .

1 2

Depending on 4 ( T . ) , two d i f f e r e n t s e t s ( & ^'&d d -1 of d ( r ^{. )} c a n b e d i s t i n g u i s h e d ( s e e f i g . ? ) , e a c h o f which e x h i b i t s a p p r o x i m a t i v e l y t h e l i n e a r b e h a v i o u r e x p e c t e d . From f i g . 1 , an T. 0 e x t r a p o l a t i o n o f d + and d - c a n b e o b t a i n e d . The two v a l u e s o b t a i n e d g i v e t h h p h a s e diagram by t h e f o l l o w i n g e q u a t i o n s :

( 5 b ) Co(l-Co)lim d - ( T ~ ) = -f C C -L(C -C,)

v p m o o ri -> 0

F i g . 2 Low t e m p e r a t u r e - low concen- t r a t i o n phase diagram deduced from f o r m u l a ( 5 ) :

* R = -V /V = -.2 6 R = - V / V =-.I

OR = 0 ?re* / 9 / )

Measured p h a s e diagram o f G L i ( / 2 / , / 3 / , / 5 / ) The t e m p e r a t u r e s i n d i c a t e d c o r r e s p o n d t o AlLi w i t h 0 = .2

Phase diagram T a b l e 1 shows t h e r e s u l t s o b t a i n e d w i t h t h e i n i t i a l c o m p o s i t i o n C = 1 2 . 5 a t % L i . A s i n a l l c a s e s h e r e , f is c l o s e t o 0 . 5 , t h e l i m i t i n g s u r f a c e s bgtween t h e two p h a s e s a r e assumed t o be p l a n a r and t h e i n t e r f a c e c u r v a t u r e do n o t modify t h e c a l c u l a t e d p h a s e diagram. These r e s u l t s a r e p l o t t e d on f i g . 2 . Though t h e e x t r a p o l a t i o n method is r a t h e r u n p r e c i s e , i t a p p e a r s t h a t i f I3 0.3

Y

t h e phase diagram is mainly dependent on kT/V i n t h e r e g i o n o f i n t e r e s t . T a b l e 1 shows t h a t C and C a r e e s s e n t i a l l y t h e same a t t h e same kT/V2, 2 though V v a r i e s

of a f a c t o r g f 2. ¹

\

9

5

JOURNAL D E PHYSIQUE

T a b l e 1 c a l c u l a t i o n o f t h e p h a s e diagram : C = 12.5 a t %

I n t e r f a c i a l e n e r g i e s I f t h e volume f r a c t i o n f i s o f t h e o r d e r of 0.3, t h e MC l a t t i c e c o n t a i n s a n a p p r o x i m a t i v e l y s p h e r i c a l p r e z i p i t a t e o f t h e L phase. I n s u c h a c a s e , it w i l l b e assumed t h a t Cm i s modified o f a s m a l l amour?? by t h e s u r f a c e t e n s i o n /13/ :

I n o r d e r t o o b t a i n V2 from ds , ( 3 ) i s now w r i t t e n :

t h e n : E (4 - RkT/2aV2 L O ~ ( C ~ + Ac/cm )

T h i s c a l c u l a t i o n h a s been c a r r i e d a t t h e c o m p o s i t i o n C = 9 a t % and R =0.2.

A s f o r m u l a ( 5 ) c a n o n l y c a l c u l a t e C and C s e p a r a t l y g i t h s t r o n g u n c e r t a n c i e s , i t i s now assumed t h a t C h a s h e r e thePsarne v z l u e a s t h a t c a l c u l a t e d on t a b l e 1.

The r a d i u s R is e s t i % a t e d from t h e l i n e a r b e h a v i o u r o f

(r

( P o r o d l s r a d i u s ) . T a b l e 2 shows t h e r e s u l t s o b t a i n e d . Frdm t h i s c a l c u l a t i o n

&NO. 9 and V \ry -15meV. I n s u c h a c a s e V15 75meV and R 9 . 2 . The c o r r e s p o n d i n g

A 1 - L i m e t a s t a b z e p h a s e diagram c a n t h e n b e g l v e n on f i g . 2 (R=0.2), t o g e t h e r w i t h t h e known e x p e r i m e n t a l r e s u l t s . The o b s e r v e d d i s c r e p a n c y i n C is e s s e n t i a l l y due t o t h e s i m p l i f i c a t i o n o f t h e FCC/L12 i n t e r f a c e i n t r o d u c e d i n t& two p h a s e model.

T a b l e 2 I n t e r f a c i a l e n e r g i e s and V from M.C. c a l c u l a t i o n s C = 9 a t % L i , R = -OV2/V1 = 0 . 2 ( s e e t e x g ) .

111- DYNAMICAL BEHAVIOUR OF THE ALLOY

I n t h e r e s u l t s o b t a i n e d h e r e , t h e p h a s e diagram is s i m p l e and no "GP" p h a s e i s observed. Due t o t h e s t r o n g r e p u l s i o n between L i atoms (Vl/kT > I ) , t h i s p h a s e diagram is c l o s e t o t h a t of a n unmixing p r o c e s s i n a s i m p l e c u b i c a l l o y : a t low t e m p e r a t u r e s , C and C a r e o n l y f u n c t i o n s o f kT/V

m P 2.

The r e p o r t e d e v i d e n c e of GP Zones 1 1 7 1 h a s now t o b e d i s c u s s e d . F o r t h i s r e a s o n , c a s e f u l DSC e x p e r i m e n t s h a v e b e e n c a r r i e d .

1. D.S.C. experiments

F i g . 3 shows a t y p i c a l DSC spectrum ( h e a t i n g r a t i o 40K/mn) o b t a i n e d h e r e from a sample of 40mg weight, C = 9.5 a t % Lithium maintained 3 minutes and 780 K , a t t h e n w a t e r quenched and aged 3 8 a y s a t room t e m p e r a t u r e .

F i g . 3 DSC S p e c t r a o b t a i n s from a sample quenched i n w a t e r and aged 3 days a t 300 K (

-

36-7mq

Two endothermic peaks a r e h e r e observed :

( i ) t h e f i r s t one, r e l a t i v e l y s o f t h a s i t s maximum a t N 415K and t h e c o r r e s p o n d i n g A H i s a p p r o x i m a t i v e l y 4 . 9 J / g ( i i ) t h e second one is r e l a t i v e l y s h a r p and s m a l l e r and i t s maximum i s a t 530K. The second c u r v e o f f i g . 3 c o r r e s p o n d s t o t h e same sample s l o w l y cooled from 780 K. The f i r s t peak has almost d i s a p p e a r e d and t h e second one is much s t r o n g e r and its maximum is o b t a i n e d a t a lower t e m p e r a t u r e

( T ~ 5 1 5 K ) .

The same behaviour is o b s e r v e d - o n f i g . 4 , where t h e sample h a s been aged 1 hour a t 373 K. I t a p p e a r s t h a t t h e f i r s t peak is observed a t a s l i g h t l y h i g h e r t e m p e r a t u r e ( d 4 2 5 K ) . Here H%5.6J/G. This l a t t e r v a l u e i n c r e a s e s somewhat w i t h l o n g e r a g i n g t i m e ( t ) a t t h e same t e m p e r a t u r e ( t i l l 7.2J/g i f t = 3 0 h o u r s ) .

Fig.4 DSC s p e c t r a o b t a i n e d f r o m a sample quenched i n water and aged l h a t 373 K

(

-

21-8 m

3

I n a l l c a s e s , t h e s i m p l e c u b i c A 1 Li phase i s observed a t room t e m p e r a t u r e by e l e c t r o n microscopy. The r a d i u s o f t h e p ? e c i p i t a t e s v a r i e s from .8nm a t 300°K t o 1 . 5 nm a f t e r l h a t 373 K. The r a d i u s of .8nm a t 300 K h a s been a l s o o b t a i n e d by s m a l l a n g l e s c a t t e r i n g /4/ i n a 11.5 a t % sample ( t h e i n n e r s t r u c t u r e o f t h e p r e c i p i t a t e is n o t known from t h e s e measurements).

C3-362 JOURNAL DE PHYSIQUE

2. I n t e r p r e t a t i o n o f t h e experiment : dynamical e f f e c t s

A s t h e L phase is o b s e r v e d , i n any sample o f t h e DSC experiment, t h e observed d i s s o l u i s o n is e x p l a i n e d w i t h o u t i n t r o d u c i n g a new phase. Q u a l i t a t i v e l y , it w i l l b e assumed t h a t t h e f i r s t DSC peak of f i g . 3 and 4 is due t o t h e r e d i s s o l u t i o n o f t h e p r e c i p i t a t e s c r e a t e d a t 293 K and 373K which have become u n d e r - c r i t i c a l when h e a t i n g . I n t h i s c a s e , t h e second peak c o r r e s p o n d s t o t h e d i s s o l u t i o n of a s m a l l q u a n t i t y of b i g p r e c i p i t a t e s c r e a t e d o r developped d u r i n g h e a t i n g . I n t h e s l o w l y c o o l e d sample, t h e L p r e c i p i t a t e s a r e c r e a t e d and developped a t h i g h e r t e m p e r a t u r e (no quenched-in v a c a g e s , t h e unmixing p r o c e s s is stopped under 450K). For t h i s r e a s o n , t h e two observed DSC peaks a r e confounded i n one.

2a. LSW model

I n t h i s system, t h e LSW k i n e t i c s of unmixing is observed / 3 / , / 4 / , / 5 / . T h i s behaviour r e s u l t s from a dynamical e q u i l i b r i u m of t h e p r e c i p i t a t e s of average r a d i u s

- R w i t h a s u p e r s a t u r a t i o n 6 c of t h e m a t r i x . The p r e c i p i t a t e s under a c r i t i c a l r a d i u s R d i s s o l v e , o v e r t h i s r a d i u s , t h e p r e c i p i t a t e s grow. I n t h e LSW d i s t r i b u t i o n , R a

%C The p r e c i p i t a t e s havg a s i z e between 0 and 3R/2 and t h e s i z e d i s t r i b u t i o n f u n c t t o n is : f ( R ) dR = 2/9 C /R P ( u ) w i t h P ( u ) d e f i n e d i n /14/, u =R/R.

-

An i n c r e a s e of T c a n 0 i n c r e a s e t h e c r i t i c a l r a d i u s R o v e r t h e maximum v a l u e 3R/2 o f t h e s i z e of p r e c i p i t a t e s and l e a d t o a d i s s o l u t i o n . 9 h e e n t h a l ~ v - ⁷ A H r e l e a s e d bv t h i s mechanism can b e simply e s t i m a t e d . The f r e e energy D G is deduced from :

A G = 9 _s ~Es {-

+

5

R 3 Re ~

-

and a S from %he entropy of a s o l i d s o l u t i o n o f - c o n c e n t r a t i o n C ( i f a l l t h e p r e c i p i t a t e s a r e ~ s s o l v e d )

9'

The tempergture T o f complete d i g s o l u t i o n is g i v e n by : R ( T * ) = 3:/2

From t h i s model t h e t e m p e r a t u r e T and t h e endothermic e f f g c t a H can be c a l c u l a t e d .

*The r e s u l t i s t h a t t h e e n t r o p y e f f e c t is far dominant. Numerical e s t i m a t i o n s g i v e T w 450 K i f a sample b a s been aged a t 300 K and H.u6.4J/gP i n good agreement w i t h t h e e x p e r i m e n t a l d a t a .

2b MC K i n e t i c a l c a l c u l a t i o n s

A s t h i s c a l c u l a t i o n can be d i s c u s s e d i n t h e c a s e of very s m a l l p r e c i p i t a t e s ( R s l n m ) , M.C. k i n e t i c a l s i m u l a t i o n s have been c a r r i e d i n such a system. The Kawasaki dynamics /15/ h a s been used w i t h 8.788 s i t e s and C z 9 . 5 at%.

F i g . 5 shows t h e energy e v o l u t i o n ( i n J / ( g ) of0 t h i s system when aged a t room t e m p e r a t u r e and suddenly h e a t e d a t 440 K. A f a s t i n c r e a s e o f energy is o b t a i n e d , followed by a r e l _ a t i v e l y slow d e c r e a s e .

The s m a l l L ( f? < lpnm) p r e c i p i t a t e s o b t a i n e d a t room t e m p e r a t u r e almost completly d i s a p p e a r b#ore t h a t a new p r e c i p i t a t i o n o c c u r s . From f i g . 5 , t h e endothermic energy is aE?!5J/g and t h i s . i s i n agreement w i t h t h e e x p e r i m e n t a l v a l u e s .

100 M C S

F i g . 5 Enthalpy e v o l u t i o n from MC s i m u l a t i o n a t room t e m p e r a t u r e

I * ) and t h e e f f e c t o f a sudden i n c r e a s e of t h e t e m p e r a t u r e a t 440 K ( m ) and(*)fron two d i f f e r e n t s t a t e s .

CONCLUSION

This model can q u a n t i t a t i v e l y e x p l a i n t h e thermodynamical p r o p e r t i e s of t h e m e t a s t a b l e g L i phase diagram. T h i s model i s v e r y rough. The atomic i n t e r a c t i o n s can be more extended, b u t t h i s s h o u l d n o t change t h e r e s u l t s o b t a i n e d . I n systems w i t h L low t e m p e r a t u r e phase diagram ( l i k e GLi o r . N i A l s e e / 1 6 / ) , V may be very

&gong, b u t a t l e a s t a s m a l l second n e a r e s t neighby& i n t e r a c t i o n is1 n e c e s s a r y t o e x p l a i n t h e thermodynamical p r o p e r t i e s o f t h e a l l o y .

ACKNOULEGMENTS The a u t h o r s want t o thank D r Brion (LTPCM) f o r having provided h i s e x p e r i e n c e and Knowledge o f DSC measurement.

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