VISCOSITY OF Cd-In LIQUID ALLOYS

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VISCOSITY OF Cd-In LIQUID ALLOYS

B. Djemili, L. Martin-Garin, R. Martin-Garin, P. Hicter

To cite this version:

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JOURNAL DE PHYSIQUE Col Zoque C8, suppze'ment au n08, Tome 41, aoct 1980, page (38-363

V I S C O S I T Y O F C d - I n LIQUID A L L O Y S

B. Djemili, L. Martin-Garin, R. Martin-Garin and P. Hicter

Laboratoire de Thermodynamique e t Physico-Chimie Me'taZZurgiques, Associe' au C.N.R.S. ( L A 2 9 )

-

ENSEEG Domaine U n i v e r s i t a i r e , B. P . 44, 38401 Saint-Martin d 'Hzres, France.

Abstract.- Viscosity coefficient of 1 1 liquid alloys Cd-In has been measured by the oscillating cup method. The necessary mass specific values have been determined by an absolute method. The temperature range covered has allowed to evaluate with sufficient accuracy the evolution of the viscous flow energy with concentration.

An extension of the model of S. Takeuchi and all., to binary liquid alloys, using the Hicter's thermodynamic representation of simple liquid, is proposed and applied to the Cd;In system.

P. I n t r d u c t i o n

The Cd-In system presents several irnmriant

transformations (one eutectic ard tvm peritectics)

and tvm solid phases have been identified (1).

&I analysis of the roncentration deperdence of the enthalpies ard entropies of mixing, resulting £ran experimental data (2), shows t h a t there is a tendency towards ccmpourd formation i n melt near 25 a t % In, h e an intermetallic cunpound is occurring in the solid s t a t e (Cd31n)

.

The self

-

diffusion coefficients present also a peak i n the neigburhocd of xIn = 0.2 (3)

.

The present work has been unlertaken in view to complete the macroscopic information on Cd-In liquid alloys, by means of density a d viscosity measuranents.

where I is Inertia mentum of oscillating systan,

6 the lcgarithnic decrenent, R the radius of crucible, H the height of liquid i n crucible, Z an implicit f u ~ t i o n of

n,

p the density of liquid

ard T the period of oscillations.

3. Experimental r e s u l t s ard discussion Densities of liquid Cd, In an3 eight Cd-In alloys can be expressed by the follaring relations versus tenperature : p = A

-

ET; the values of constants A ard B a r e reported i n table 1.

Table 1

DENSITJ.JS OF THE LXXTEN Cd-In S Y m

-

Cd-In alloys Tanprature p= A

-

B T ( g x a-3 ) a t 8 In range cwere.3

TK A -B

0 594

-

683 8.584 9.24 x 10-4

2. Experimental Method

Densities of liquid Cd-In and Cd-In alloys were determined £ran their melting p i n t t o 700K, by the direct Archimedean method ( 4 ) , which consists to immerse in the liquid metal or'alloy a bob hung to a balance by a tungsten wire.

Viscosity coefficients were detemimd versus t e n - perature by

the

oscillating cup method, which is considered a s trre best metkd, to measure m e t a l l i c

liquid alloys viscosities (5). The determination

of logarithnic decrenent of the oscillations dam-

ping, due to the liquid placed in a c y l i r d r i c a l crucible, leads to t h e shear viscosity by the

follawing expression :

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JOURNAL DE PHYSIQUE

The accuracy of the density measurenents (0.5%) has

allow03 to evaluate significant molar volumes of mixing which a r e represented versus concentration

i n Fig. 1.

Cd 20 40 60 80 at / In Fig. 1

-

wlar

excess volume of molten Cd-In alloys

As expected the excess molar volume is d l

A-V

(T 1%) a s for

mst

of the liquid binary alloys. However, the change i n the sign of ?S for rich irdium alloys has t o be considered a s significant.

Wre of t h i s range of canpsition imreases

with tanperahre which leads t o

_-

< 0, where

av

v,

= (-1 aT

a N ~ d NCd = o is the limit partial volume of Cd for xCd + 0 (V is the volume of the alloy an3 Ncd the rnrmber of Cd atoms). This i s a n abnormal

behaviour which means that the volune occupied by Cd atan highly diluted in In s o l v e n t becanes

sMller when tanperature ircreases.

Table 2

CONSPANPS FOR EQJATION n = Ae (C. P )

- - -- - Cd-In alloys 10 x A at % In (C.P) E (K &/role-l) 0 5.93 1.85 9.81 5.35 1.85 19.70 5.19 1.60 29.6 5.05 1.44 39.50 5.24 1.26 49.50 4.77 1.27 59.50 5.33 1.09 69.50 ' 4.61 1.12 72.6 4.66 1.21 74.3 4.99 1.51 80 4.35 1.25

as.ao

4.73 1.17 100 4.19 1.31

Experimental results of viscosity coefficients can

be expressed by an Arrhenius l o w type : n = A exp (- E m ) , values of A am3 E constants are reported in table 2

1

20 40 60 8 0 a t % l n

Fig. 2 -Concentration deperdence of the activation

energy for viscaus flow of molten Cd-In alloys. The regxesentation of the viscous flow energy (E)

,

versus the In concentration is a w n in Fig. 2. The extraplation of this curve £ran x Cd = 0.9 t o xCd = 1 was m t given because of the

low

accuracy i n the measuranent of the w e Cd viscosity due

t o

the v o l a t i l i t y of this metal.

Fig. 3 shows the variation of viscosity with concentration a t t m tenperatures (600 and 673 K).

Fig. 3

-

Variation of the viscosity coefficient

of molten Cd-In alloys with alloy caqwsition

The isotherms of viscosity coefficients do not reflect the sharp maximum in the self-diffusion coefficients of In ard Cd, observed for

XIn = 0.2 (3)

.

This forbids an attempt in rgrte-

senting the correlation between self-diffusion

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The excess viscosity coefficient has been faund negative for the wf-role concentration range (Fig. 4 )

The maximum eff=t obtained for 40 a t % In is of about 40% referring to the linear law.

Fig. 4

-

mcess viscosity versus wncentration of molten Cd-In alloy

4. Model for the viscosity coefficient in liquid

binary alloys

S. Takeuchi

and

a l l (6) propsed a ph- -logical model for evaluating t h e v i s c o s i t y

coefficient

n

of pare liquid metals. This M e 1 is based on the calculation of the mmentutn transfer resulting £ran interatanic collisions due to the

oscillators, ard yields the following expression

for

n

:

4 a

n

= S .v.m p2 P(T)

1

g ( r ) r4 dr

0

where u is the frequency of vibration, m the mass

of an atan, p the atahic density, P(T) the

pp-

tion of vibrators i n the liquid, g (r) the radial distrikution function

and

a the value of rcorrespon-

ding to its f i r s t minimum.

Instead of ckosing the anpiric Gaussian function

of S. Takeuchi ard a l l (6), to evaluate P(T), we have taken Hicter

'

s ( 7 ) thermodynamic reprei.

sentation of a simple liquid where P(T) is given by : P(T) = q q v

+

qt*

q,

am3 qt are respectively the atanic partition functions of vibrators ard translators with :

--

where E is the transition energy above which an oscillator can be considered a s a translator, 6 the distarce of translation,

kg

the wristant of B o l t m , h the constant of Planck and T the

absolute temperature.

We pK0posed an extension of the W K E U C H I model t o

b h x y liquid alloys. In this ptrpose we express

the m t u m transferred between A-A, B-8 and A-B i n

the

alloy for a given comentration. The

s t r e q t h deduced £ran the total mamentun transferred has the following form :

Considerkg that the viscous flow is a macroscopic

concept we write :

T h i s tranq2osition fmn microscopic t o macroscopic analysis is self-consistent with Einstein representation of the oscillators i n the liquid alloy with a unique frequency v.

1t follows t h a t :

where P is the density of alloy, P (A)

a d

P (B) the proportions of vibrators A and B i n the liquid,

xA

ard ~g the concentrations of canpnents A a d B, and a, b and c are values of r correspording

respectively to the f i r s t minimum of t h e p a r t i a l p a i r

correlation functions g (r)

,

g ( r ) ard g(r)

.

AA BB AB

This exwession has been applied t o the system

Cd-In. The lack of exprimental data conc4rnix-g the partial pair correlation functions necessita- t e s their evaluation £ran a model. The Percus &

Yevick (8) hard sphere representation has been

used i n t h i s p l r p s e . The procedure of Frotopapas

and

a l l (9) has allowed the determination of

the

hard sphere diameters for different concentrations

and tenperatures

.

T-gh the Hard sphere Percus a d Yevick model

does not give a satisfactory representation of the

partial pair correlation functions gij (r) a f t e r Fourier transform of the pair partial structure factors, nevertheless it is sufficient to give a good evaluation of

the

rnrmber of f i r s t n e a r e s t

neigNxurs necessary for the viscosity calculation

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JOURNAL DE PHYSIQUE

C8-366

orderinq)

Fig. 5

-

Drperimental ~ o i n t s afd t h e o r e t i c a l curve (-

-

-) of v i s c o s i t y c o e f f i c i e n t molten Cd-In alloys.

I n order t o s i m p l i f y , w e have considered that

P (A) = P (B) ard evaluated this ~ o p r t i o n using l i n e a r laws versus concentration f o r E, v and 6.

Figure 5 represents the calculated v i s c o s i t y curve an3 the experimental r e s u l t s a t 600K. The agree- ment is f a i r l y goal. P a r t i c u l a r l y the linear re-

p e s e n t a t i o n versus concentration of the d i f f e r e n t parameters can be considered a s a f i r s t approxi- mation. However, these simple laws a r e s u f f i c i e n t

t o take account of the r e l a t i v e l y hi* negative excess v i s c o s i t y f o r Cd-In systan as it can be observed in Fig. 5.

Ackncwledgemmts: We are g r a t e f u l f o r useful dis- cussions w i L h Pr. P. DESRE and D r . J. BLEIXY.

BIBLI-HY

1. T. Heumann ard B. Predel, Z. Metallkde, Val. 50, pp 309-314 (1959), Vol. 53, pp 240-248 (1962)

2. B. P r e d e l a n d H. Berka, Z. Metallkde, Bd 67

H3, pp 198-204 (1976)

3. C. Berrard, C. Potard, P. H i c t e r , F. Durard, E. Bonnier, J. Chen. Phys. No. 10, pp 1525-1530

4. L. Martin-Garin, P. Bedon, P. D e s r B , J. Chen- Phys., NO 1, pp 112 (1973)

5. L. Martin-Garin, R. Martin-Garin, P. D e s r B ,

J. of the Less Carmon Metals, 59, pp 1-15 (1978)

7. P. H i c t e r , F. Ward, E. Bonnier, J. Chan. Phy. 68, 804 (1971)

8. J. E. W e r b y , D. M. N o r t h , Phys. Chan. Liq. Vol. 1, pp 1-11 (1968)

9. P. Protopapas, N. A. D. Parlee, Hight, Temp. Sciences, 7, pp 259-287 (1975).

Volume e f f e c t i n v o l v e d by t h e d i f f e r e n c e o f valence Z, between t h e A and B atoms i n an a l l o y d i l u t e d i n B, has been p r e v i o u s l y i n t e r p r e t e d by one o f us% u s i n g an e l e c t r o n i c model. I n t h i s model

dV

a = (xB)YB+o can be expressed by t h e f o l l o w i n g r e l a t i o n

aaoNe nzxZ cos 2kfb

a =

r3

s i n 2kfb

+

(1+2~a,,k~)~b'

where a, = @

,

n t h e number o f f i r s t neighbours,

2

z t h e m a t r i x valence, kf t h e Fermi ' s l e v e l wave v e c t o r and b t h e d i s t a n c e o f t h e f i r s t neighbours. For t h e a l l o y s o f t h e Cd-In system, r i c h i n I n , agreement between c a l c u l a t e d an experimental values o f a, r e s p e c t i v e l y -1.5 and -3.7, i s s a t i s f a c t o r y . However t h i s model i s n o t s u f f i c i e n t t o i n t e r p r e t t h e excess volume e v o l u t i o n versus temperature. x R. MARTIN-GARIN,

P.

BEDON e t

P.

DESRE, C.R. Acad.

Sc. Paris, t. 274, p. 676-679, (1972), S @ r i e C.

6. S. Takeuchi and T. Tida, c i t e d by H. Kihara zuYl