Phys. kondens. Materie 7, 383--389 (1968)

Heat of Mixing in Liquid Alloys SHIGERU TAMAKI and IcRI~O SHIOTA Research Institute for Iron, Steel and Other Metals, Tohoku University, Sendai, Japan Received April 11, 1967; in revised form March 13, 1968 A calculation of the heat of mixing in liquid alkali alloys is presented. The two body correlation function is used in the calculation. The theoretical value compares satisfactorily with the experimental one in the Na--K system. Les auteurs d6crivent un calcul de la chaleur de m61ange dans un alliage liquide des m6taux alealins. Ils utilisent pour cela, la fonction de correlation entre deux corps et la valeur th6orique ealcul6e correspond de fagon satisfaisante s la valeur exp6rimentale dans le syst~me Na--K. Die Mischungswarme yon fliissigen Alkalilegierungen wird bereehnet. Dabei wird die Paarkorrelationsfunktion verwendet. Der theoretische Wert zeigt eine befriedigende ~bereinstimmung mit den Mel]werten im Na--K-System. 1. Introduction The two body correlation function a(q) or the radial distribution function g (r) is an important factor for the interpretation of physical properties in the study of liquid metals. ZIMA~ [1, 2] and his co-workers have developed the theory of electrical resistivity and thermo-power of liquid metals using a (q). WATAB]~ et al. [3] have explained the Knight shift of liquid sodium at constant volume using a(q). I n the field of molecular theory of liquids, JOHNSON et al. [4] have found the effective ion-ion interaction potential of liquid metals from the BornGreen and Percus-Yevick equation in which the measured radial distribution function g (r) was used. I n this paper, an a t t e m p t of the calculation of the heat of mixing lin liquid N a - K alloys is made using a (q). CHRISTMAN and HUNTINGTON [5] calculated the heat of mixing in the liquid Na-K(50--50) alloy. The advantage of their treatment is the absence of the structure factor in diagonal matrix elements of the A. P. W. Hamiltonian. I n contrast, our calculation using a (q) is rather phenomenological, but most practical. By thermodynamical measurements, it is easy to observe the heat of mixing in liquid alloys, since the mixing is very rapid. There exist systematic investigations on the heat of mixing in the liquid state by KLErrA and his co-workers [6] and TA~EUCI~I and UE~UaA [7]. In liquid metals, we need not consider the strain energy which is a very important quantity in the solid state. Therefore, it is possible to estimate easily the heat of mixing by a quantum mechanical calculation. The heat of mixing in a liquid alloy is equal to the change of the total energy with alloying. In section 2 we show the method of calculation of the total energy and explain the role of the effective ion-ion interaction, and in section 3 and 4 the calculated value is discussed.

384

S. TA~Ax: and I. SE:OTA: 2. Evaluation of the total Energy

We proceed to calculate the total energy of the system of ions and electrons in liquid metals or alloys assuming that the nearly free electron is a good approximation. After HA~R~SON [8], the total energy Etot of a system divided by the number of ions is written as follows: Etot : Ed -~- Eel, 1 Ed :

~

(1) (2)

~. Vd (r~ - - r]) ,

i,j 1

Ee~ = ~- ~ E(R) = [2~0/(2~) 8] ~ E ( R ) d k ,

(3)

k
where Vd (r~ -- rj) means the direct Coulomb interaction between the ions i and ], Eel is the total energy of the electrons assumed to be independent and ~0 is the atomic volume. From the second order perturbation, E (k) is written as follows; k 2 4- § ~ S*(q)S(q)
(4)

where Z(q) -

Z*(q) = s(-

q)

J and

= Qo 1 f e -i(k+ q)~ W (r) ei~ tiT.

(5)

Eq. (5) is the Fourier component of the bare potential of an ion with respect to the wave vector q. The total energy contributed from the free electron energy h 2k2/2m is written by the following expression:

Z

f 0

3 h 2k~ 2.21 4z~k2dk----5 -Z 2m - r~ r y ,

4zke(h2k2/2m)dk

(6)

/

where rs means the radius of the sphere occupied by an electron (in Bohr radius units). The electron energy is altered by the introduction of electron-elektron interactions. In the Hartree-Fock approximation and the treatment of NozI]~I~ESPINES [9], the corrected free electron energy Ere is given as Ere : 2.21/r~ -- 0.916/r s + Ec, Ec ~-- (-- 0.115 ~- 0.0311nrs).

(7) (8)

The first order perturbation with alloying is mainly due to the electrons at the Fermi surface. Therefore, the value of will be important. This value is equal to the limiting value for q = 0 of the matrix element
r(q) =- = ( - - 4 ~ Z e e / q ~ + fl)/t~oe(q),

(9)

Heat of Mixing in Liquid Alloys

385

where Z means the number of valence electrons, fl is the repulsive potential due to the core electron and e (q) is the dielectric constant. In the limit of q -+ 0, Eq. (9) is reduced to 2 rim V(q) = = -- ~ E r . (10) q-->0

After HAI~RISON, the third term of Eq. (4) (hereafter to be called Ebs, the band structure energy) is equal to

Ebs = ~ S*(q) S(q). F(q), q,o

F(q)=

(11)

[i2" e(q)--i

4~ ~~

(12)

where W0 = e(q). W. Let us give a physical meaning to Ebs from the self-consistent field method. Suppose the charge fluctuations surrounding the ions i and j in a metal are e(~nq and e(~nJq, respectively, in a q space. The indirect interaction between the ions i and j through conduction electrons, Vqi - j , is expressed with the Poisson equation as follows :

(13)

~ ] W~ (~nq- 4~e2 i-e d n~ -

4 ~q2 +

[,

(14) (15)

Iw~-~

where W~- " and W~-e are the bare potentials of i and j ion, respectively and I [ means the form factor of the pseudo-potential. Then, V~- j is

v~-~ - I w':~ I w~-~

(16)

4~te2 q~

We should consider the contribution of screening by conduction electrons. The effective indirect interaction potential (Vq-J) s is given by the self-consistent field method as V~- j - (V~-J) s = e (q). V~- j . (17) From Eqs. (16) and (17), (V~_i) s _ -

-

1--e(Cl) .]W~-~llW~-~ e(q) 4zte2 q~

(18)

Therefore, total effective ion-ion interaction Vq-J(tot) composed of the direct Coulomb interaction 47~Z2e2/q2 and the indirect ion-electron-ion interaction (V~-J) s is

Vq-q~ot) - 4"~z2e2~ + 1 -~(q)~(q) I w~-~ wT~I

(19)

q2 To make a full theory, we should consider the effects of electrontransfer and the core repulsion between the ions i and j. A detailed discussion of the former effect is given by CHRISTMAN and ItU~TI~GTO~ and S~IMOaI [10]. In the alloys of the Na-K system, this effect is not so large and we will not take it into account. The

386

S. TAMAKIand I. SalOTA:

latter effect is due to interaction from the overlapping of the closed shells of the ions i and ?'. This repulsion potential is well k n o w n as Born-Meyer potential [11] V(I~) (r) -~ A e(grs_r)/e

(20)

= 0,345 X 10-s c m , where A is a constant. The Fourier component of Vlr~) is written as 8 z~A . e2rS/e " q~ -- (1/02) V(~)(q) -- ~v (~2 + 1/02)3 9

(21)

As the value of A is 1.25 x 10 -12 for both sodium and potassium metal, we can choose the same value of A for all N a - K alloys. The final form of the effective ion-ion potential V (q) is

v(q) = V~-J(tot) + v(~)(q).

(22)

Therefore, the radial effective ion-ion potential is V (r) = ~ V (q) " eiqr .

(23)

q

The energy Eeff contributed from the effective potential between ions is

Ee.

= ~

1

~ V (rl - - rj), i.j 1 ~ ei"q(r,- r~) 1 = 2 N ~ V(q) 9 = ~-~ q

i.j

V(q)'S*(q)S(q).

(24)

q

The last expression of Eq. (24) is equal to the sum of Ed and Ebs, as far as we neglect the term of core repulsion V (1~)(q). Therefore, the second order perturbation Ebs is related to the effective ion-ion potential. I n hquid metals, this s u m m a t i o n in Eq. (24) is changed to the integration over the pair distribution function g (r) as follows; Eeff-- 2

V

f V(r)'4~r2"g(r)'dr'

(25)

0 co

1 ~V(q)" ~---2V

I V f e -iqr "47~r 2" g(r) 9dr.

q~0

(26)

0

The radial distribution function g(r) is transformed into the pair correlation function a ( q ) as J~

f e--iqr" 47~ r 2" g (r)" dr = a (q) -- 1,

(27)

0

where 1

a (q) : ~ - S* (q). S (q). Thus, Eefr is

(28)

co

1 f 4 x e q 2 " V ( q ) . {a(q) - - 1 } - d q . Eeft ~ ~16n

(29)

0

Therefore, the total energy Etot is Etot = Z f e -{- + Eef f.

(30)

H e a t of Mixing in Liquid Alloys

387

3. Heat of Mixing I n a binary alloy of elements A and B, the binding energy E b i n d is given b y the total energy and the ionization energy I, Ebind :

- - (Etot + I ) .

(31)

Suppose the form of the conduction band is not altered appreciably with alloying, then the change of b y alloying m a y mainly result from . Thus, the following expression

~ d

I W[

= --

2

(32)

AEF

is premissible as a first approximation (A: variation by alloying). The heat of mixing A E AB is AEAB = Ebind AB _ (1 Z) Ebind A - - Z E ~ nd, (33) where Ebind A~ , Ebind A B a n d Ebind are t h e binding energy of the AB alloy, of the A metal and of the B metal, respectively. The sum of the second and third t e r m on the right hand side of Eq. (33) is shown b y the linear line L in Fig. 1. I f the heat of mixing is negative, the curve o f Ebind versus concentration is above L (curve U ) a n d below in the other case (D). The heat of mixing E AB is equal to L - - U (or L - - D ) .

A

atomic fraction of B B

1 9

U

EB

Fig. 1. Schematic figure of binding energies of alloys; U a n d D curves are Eblnd of N. F. E. alloys of A a n d B, L is the algehraical m e a n value of pure states

The numbers of pairs A - - A , A - - B and B - - B in AB are, respectively, number of A - - A pairs -- (1 -- Z)gN~ 2 2 Z(1 -- Z)N2 number of A - - B pairs -2 '

(34)

number of B - - B pairs -- z2N2 2 '

(36)

(35)

N = (1 -- z) A + z B .

(37)

I f we apply the same pair correlation function a (q) to all pairs in the AB system, for simplicity, the heat of mixing AE AB is co

zjEX s

1

-

16 z~s

~{(1

-

Z) 2 V ~ ( q ) + 2Z(1

--

--AB (q) Z) V ~ ( q ) + ~..2VBB

-- (1 -- Z) V ~ ( q ) -- Z V ~ ( q ) ) " 4~q2{a(q) -- 1}dq +

+A

92 . 2 1

(r~)2

(r~)2

(r~)~

(38)

~s + A ( l n r s ) + A V (-~- ) ( q ) ,

where the lower suffixes of V ~ mean the A - - B components and the upper suffixes are the correlated pair. The last three terms on the right hand side of

388

S. TA~.Ir and I. SI~IOTA:

Eq. (39) are contributions to the heat of mixing from the parts of 1/rs, In rs and V(1~). These terms are the second order contribution. If T7AA~. AA BB AEAB is --AB-~- VAA and -T7~B... -AB ~ V B B ' then -

-

.

. 16~a . . Z( 1

4~q 2 {2 vA~(q) x~

Z)

i { l + -9-" 2.21 9 (r~)2

1--7.

v~(~)

Z }

(r:).z

(r

-

v ~~B B (q)} d~ {~(q) - - 1

(39)

9

4. Result of Na-K Alloys and Discussion We applied the pseudo-potential of HAm~iso~ hi the effective ion-ion potential of Na-Na, Na-K and K - K in the Na-K system. Then, for example,

V-~a-~:(q) -~ T

+

4z~e~

e(q)

+ V(m(q),

(40)

q2

V~-~'~(q) =

4~e 2

~ +

e(q) -- 1

~(q)

(--

49ze2 + 27) 2 q2

4~ q2

+ V(~)(~),

where we used the following relation [12], 4~e 2 n {1 4k~--q 2 2kr+q e ( q ) _ ~ l + q~ 2 + 8kF.q "In ~ g " EF

}

.

(41)

(42)

On evaluation, the first term on the right hand side of Eq. (39) has a negative sign and the second term has a positive one. For example, in 50--50 alloy, the

200

o

E m o o

~100

K

Fig. 2.

Heat

.I

.3 .5 .7 atomic f r a c t i o n ( k - x ) of

.9

Na

Na of mixing in liquid N a - - K alloys; ------ experimental (after YOKOK&WAand KLEPPA), theoretical

Heat of Mixing in Liquid Alloys

389

first t e r m is -- 255 eal/mol and the second one is -~ 17 cal/mol. Thus the heat of mixing in the 5 0 - - 5 0 alloy is a b o u t - - 2 1 8 cal/mol. The calculation in all N a - K systems was done as shown in Fig. 2. The calculated curve agrees satisfactorily with the experimental observation b y YOKOKAWA and KLEPPA. Since the relative error between the case using the a (q) observed b y Gr~GRICH and H~ATO~ and t h a t using the observed values for various compositions b y O~To~ et al. is a b o u t 10~o, we m a y use the same pair correlation function a(q) in a n y composition for simplicity [13, 14]. As calculated b y CHRISTMAN and HUNTINGTON, the charge transfer from N a to K is a b o u t 0.04 e. Therefore, the contribution to the effective ion-ion interaction from charge transfer is a higher order correction. I n the case of an alloy whose components have different negativities, the consideration of charge transfer m a y be important. I n N a - K systems, the correction from the last three terms of Eq. (38) is 5 . . . 10 cal/mol.

Acknowledgements. Authors express their cordial thanks to Prof. M. WATABE for his suggestion to introduce the effective ion-ion interaction. Thanks are due to Prof. S. TAK]~VCHIfor his encouragement and also due to Prof. G. Busch for his noteworthy comment and for a critical reading of the manuscript. References 1. ZI~AN, J. M.: Phil. Mag. 6, 1013 (1961). 2. ]~IIA.DLEY, C. C., T. E. F~_BER, E. G. WILSO:N, and J. M. ZIMAN:Phil. Mag. 7, 865 (1962).

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

WATABE,M., 2VI.TANAKA,H. ENDO, and B. K. JONES: Phil. Mag. 12, 347 (1965). Jom~soN, M. D., P. HUTCHINSON,and N. H. MARC~: Proc. Roy. Soc. A 282, 283 (1964). Cm~ISTMAN,J. R., and H. B. ItCNTINGTON:Phys. Rev. 189 A, 83 (1965). YOXOKAWA,T., and 0. J. KL]~PPA:Private communication to Prof. H. EN])Oand J. chem. Phys. 40, 46 (1964). TAKEUCHI, S., and O. U ~ u ~ A : Private communication. HARRISON,W. A. : Pseudo-potentiM in the theory of metals. New York: Benjamin, W. A., Inc. 1966. NOZIkRES,P., and D. PINEs: Phys. Rev. l l l , 442 (1958). S~IMOJI, M.: J. Phys. Soe. Japan 14, 1525 (1959). MoTT, N. F., and H. JONES: The theory of metals and alloys. Oxford 1936. Z~A~, J. M. : Principles of the theory of solids. Cambridge 1965. GISrGRIeE,N. S., and LE RoY HEATON:J. chem. Phys. 34, 873 (1961). ORTON,B. ]~., B. A. S]tAw, and G. I. W n m i ~ s : Aeta Metallurgica 8, 177 (1960). Prof. SHIGERUTAMAKI Dept. of Physics, Faculty of Science Niigata University Niigata/Japan

26 Phys. kondens. Materie, Bd. 7