Liquidus curves for eutectic systems

Volume 4 1A, number 5

PHYSICS LETTERS

LIQUIDUS

23 October 1972

CURVES FOR EUTECTIC

SYSTEMS

A.B. BHATIA Department

of Physics, University of Alberta, Edmonton,

Canada

and N.H. MARCH Department

of Physics, The University, Sheffield,

UK

Received 13 September 1972 The liquidus curves of the Na-K system are calculated from conformal solution theory, the input consisting of thermodynamic data for pure Na and K, plus an interchange energy characteristic of the solution.

It has been known for a long time that the liquidus curve of an ideal solution is approximately described by the equation [l] ln(1 - c2) = (L,,/R)

[T;l

- T-l]

,

L

T ACplO

Ll,

a_=_+ T

s

T,

Tl

--_

AT AC2

(2) form

-

(::;)n,T$$&,p

il-c2~l)cI,c2,p

(4) y1 being the activity. This is related to fil and to Q&O)

by

I-Q =~10@,T)+RTlnyl(l-c2)~~10(P,T)+X, this equation

?+?)pl

(5)

serving to define X in (4), and

1 =m--. 1 -c2

&lnrl 2

with the differential

($[RTln(yl

T

(1)

c2 being the concentration of element 2 and L,, the latent heat at freezing temperature T1 of pure liquid 1. To clarify the assumptions underlying (l), it is worth-while to briefly summarize the thermodynamic equations along the liquidus. When no mixed crystals are present, we have, in terms of the chemical potentials (the subscript zero referring to a pure substance and superscript S denoting solid) I&)(T) = P&T, 5)

--dT-

c2

&c(O)

Since S&O) -+ c2 as c2 -+ 0 and from (4) L + Llo in this case, we see from (3) that (dT/dc2)liquidus at c2 = 0 is -RTf/L,,, an exact result, already contained in and evidently F(T) is in(1). In (4) AC,,, = c;I&0 dependent of concentration. It is then readily shown that T

=(3)p,TjL/77

Z-p

= - c2(3J/0

s

RT2c2 S&L

(3)

F(T) dT = X(T, c2) + constant

and by letting T + T, the constant Hence

is found to be zero.

T

X(T,c2)=RTlnr1(1-c2)= where See(O) represents the concentration tions [2]. L is a generalized concentration latent heat defined by

(7)

j

fluctuadependent

F(T)dT

T, = jdT+ Tl

+ 7 T1

dT’ j

T’ AcplO(T”) T,,

T1

dT”.

(8) 397

Volume

4 1 A, number

5

PHYSICS

with y, = I and putting

Thus. for an ideal solution

T

exp [&

R7’In y, ‘i MY2

s dT’F(T’)j

We shall refer to this as the ideal liquidus curve. All the relevant properties of pure liquid I have been incorporated exactly, but the solution is taken as ideal. In the true solution. it then follows immediately that ‘ideal)/(’

_ “(T))

(IO)

where w is the interchange energy. and. following earlier work [I] we shall Taylor expand AC,,,,, around T, and neglect higher terms than the first. Then we find c.(t)

I 1 exp ( Wc2,1t)

’ = I”ideal

tA exp [(r

A)( I

t= T/T,. A=Ai ,dR*

I ‘,I ;

I

(13)

l=I,JRT,.

W=w/RT,. For Na-K, I = 0.86 at both ends of the phase diagram and ANa = 0.06, AK = 0. I6 [4]. From our work [Sj on partial structure factors, w/RTFa = 1 I and hence

No:

I

(11)

and Cideal(t) ~ ’

and from a knowledge of the properties of pure liquid I plus the experimentally determined liquidus curve the activity y, can be extracted along the liquidus curve. Knowledge of AC,,, from the melting point T, to the eutectic temperature would require measurements on super-cooled liquid and extrapolation will frequently be necessary. Therefore, to illustrate the

3807Y-

1972

(II)

(‘1)

i-1

Y] = (I

23 October

above remarks we shall adopt the model of conforma1 solutions [3] in which

C-Cc2

(1 ~~ tide,) =

LETTERS

I

I

I

I

I 04

O’S

I

I

I

K I

> 360

340

320

260

240

0

I

I

0’1

0’2

03

A~mw Fig. 1. Equilibrium

398

phase diagram

I

percent

I 0’6

I 0’7

1 0.8

I o-4

K

for Na-K system. Abscissa shows atomic concentration of K. Theoretical is shown as dotted line beyond the point where NazK compound forms.

liquidus

curve for Na

Volume 4 1A, number 5

PHYSICS LETTERS

WK = 1.2. In this way we find the liquidus curves in fig. 1, shown with measured data [4]. It should be cautioned that, in general, w is not necessarily independent of temperature. Further, the model of conformal solutions will not work when the forces between different species are markedly dissimilar. References [l] For a summary of this and refined theories, see, for example :

23 October 1972

A. Reisman, Phase equilibria (Academic Press, New York) or: I. Prigogine and R. Defay, Chemical tltermodynamics (Longmans, London 1965). [2] A.B. Bhatia and D.E. Thornton, Phys. Rev. B2 (1970) 3004. [3] H.C. Longuet-Higgins, Proc. Roy. Sot. A 205 (1951) 247. [4] Liquid metals handbook: Sodium (NaK) supplement, ed. C.B. Jackson (US Government Printing Office, Washington DC.). [ 51 A.B. Bhatia, W.H. Hargrove and N.H. March, to be published.

399