Thermodynamic and surface properties of liquid Mg–Zn alloys

Journal of Alloys and Compounds 329 (2001) 224–229

L

www.elsevier.com / locate / jallcom

Thermodynamic and surface properties of liquid Mg–Zn alloys a b, N. Jha , A.K. Mishra * a

b

Rajasthan Vidyapith, Purulia 723001, W.B., India Department of Physics, Sabour College, Sabour 813210, Bihar, India Received 2 February 2001; accepted 6 June 2001

Abstract A statistical thermodynamic theory, in conjunction with a complex formation model, was used to investigate the thermodynamic and surface properties of liquid Mg–Zn alloys. Our expressions reproduce the thermodynamic functions such as free energy of mixing and enthalpy of mixing. The study of concentration fluctuations in the long-wavelength limit (SCC (0)) provides structural information about liquid Mg–Zn alloys. SCC (0) was further used to obtain the chemical short-range order parameter (a1 ) to quantify the degree of order. The study shows that Mg atoms cause a decrease of surface tension in Mg–Zn liquid alloys.  2001 Elsevier Science B.V. All rights reserved. Keywords: Liquid alloys; Surfaces; Composition fluctuations; Thermodynamic properties

1. Introduction Complex formation models [1–7] are based on the fact [8] that the unlike atom interaction between component elements of alloys is an important factor responsible for complex formation. Various properties of Mg liquid alloys show anomalous behaviour as a function of concentration. The phase diagram of liquid Mg–Zn alloy [9] is complicated. Besides a congruently melting phase, MgZn 2 , there exists a number of incongruently melting compounds, Mg 2 Zn 11 , Mg 2 Zn 3 and MgZn, and the high-temperature phase Mg 51 Zn 23 (previously known as Mg 7 Zn 3 ). From extended X-ray absorption fine structure studies [10–13], a pre-maximum in the structure factor is observed in both liquid and amorphous Mg–Zn alloys. The occurrence of a pre-peak is attributed to chemical short-range ordering (CSRO) in liquid Mg–Zn alloys. Mg alloys are also particularly interesting due to their glass-forming ability (Mg–Zn, Ca–Mg). Mg–Zn is similar in structure to Ca– Mg and Ca–Mg is a good glass former. Mg–Zn glass was the first transition metal-free metallic glass [13]. The size effect (VMg /VZn 5 1.557, where V is the atomic volume at a specified temperature) in the Mg–Zn system is too small to cause the observed asymmetries. The anomalous behaviour of such alloys may be attributed

*Corresponding author.

to the strong interactions between unlike atoms [14–16], resulting in complexes in the melt. Here, we assume that the number m of A atoms and the number y of B atoms are energetically favoured to form chemical complexes Am By ( m and y are small integers):

m Mg 1 y Zn 5 Mg m Zny ( m 5 1, y 5 2) This concept has been used successfully to explain the mixing properties [17–20] of a variety of binary liquid alloys. In this study, liquid binary Mg–Zn alloys were chosen for investigation. The concentration fluctuations in the long-wavelength limit, SCC (0), are one of the most powerful microscopic functions for understanding the nature of interatomic interactions. Surface phenomena establish a link not only with the bulk properties but are also important from a metallurgical and catalytic point of view. Surface segregation is influenced by surface tension and the interaction among the atoms. A statistical thermodynamic model based on complex formation was used to calculate the interdependence of various thermodynamic properties, such as the free energy of mixing, the heat of mixing, the entropy of mixing, the activity coefficient, concentration fluctuations in the long-wavelength limit, the chemical short-range order parameter and surface tension of Mg–Zn liquid alloy. Section 2 deals with the thermodynamic and micro-

0925-8388 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 01 )01684-X

N. Jha, A.K. Mishra / Journal of Alloys and Compounds 329 (2001) 224 – 229

scopic functions, and the surface tension is included in Section 3. Section 4 is the conclusion.

225

2.2. Activity The activities, aA and a B , of the first and second species of the alloy are given by

2. Thermodynamic and microscopic functions

RT ln a i (i 5 A,B)

5 (≠GM / ≠Ni ) T,P,N 5 GM 1 (1 2 Ci )(≠GM / ≠Ci ) T,P,N

2.1. Excess thermodynamic properties of mixing The grand partition function, J, of an alloy consisting of NA A atoms and NB B atoms may be expressed in terms of the configurational energy, E:

O(q T )

J5

A

NA

(9)

The activity coefficients and their ratio are given by

gi 5 a i /Ci g 5 gA /gB

(10)

For m 5 1 and y 5 2:

(qB T )NB exp(( mA NA 1 mB NB 2 E) /k B T )

i

(1) where qi (T ) and mi are the atomic partition functions and the chemical potential of the ith (i 5 A,B) species, respectively, and k B and T refer, respectively, to Boltzmann’s constant and the absolute temperature. Eq. (1) can be solved [1] in the framework of the quasi-lattice theory to obtain various expressions for the thermodynamic and microscopic functions. After performing some algebra [21], one may obtain the free energy of mixing: XS

GM 5 G M 1 RT G

XS M

OC ln C i

(2)

i

5 CA CBW 1 FABWAB 1 FAAWAA 1 FBBWBB m 21 y 21 A B m 22 y A B m y 22 A B

(3)

m 21 y 21 A B m 22 y A B m y 22 A B

RT ln aA 5 Wf1B 1 WAB f2B 1 WBB f3B 1 (1 2 CB )K 1 GM (11a) RT ln a B 5 WF1B 1 WAB F2B 1 WBB F3B 1 (1 2 CB )K 1 GM (11b) where FB and fB are given as

F1B 5 (1 2 CB )2 3 4 16 3 ] F2B 5 ]61 1 2CB 2 6C 2B 1 ] 3 CB 2 2 CB F3B 5 2 ]41 1 CB 2 ]21 C 2B 1 C 3B 2 ]43 C 4B f1B 5 C 2B

(12) 2 B

f2B 5 2 C 1 1 ] 2

FAB 5 C C (2 2 C C ) FAA 5 C C (2 2 C C ), m $ 2 FBB 5 C C (2 2 C C ), y $ 2

(4)

Here, W is the order energy and WAB , WAA and WBB are the interaction energies. CA and CB are the concentrations of the first and second species of the alloy, G XS M is the excess free energy of mixing, and Fij denotes the probability of finding the i–j bond as a part of the complex. For m 5 1 and y 5 2:

(5)

3 B 4 B

3 ] 2

C 2 C

3 ] 4

4 B

f3B 5 2 C 1 C K 5 ln CA 2 ln CB

2.3. Concentration fluctuations in the long-wavelength limit ( SCC (0)) and short-range order parameter (a1 ) Concentration fluctuations in the long-wavelength limit (SCC (0)) have been widely used [22,23] to understand the alloying behaviour of liquid binary alloys. SCC (0) is related to GM and activities: SCC (0)

FAB 5 ]16 CB 1 C 2B 2 ]53 C 3B 1 ]12 C 4B FAA 5 0 FBB 5 ]41 CB 1 ]21 C B2 2 ]41 C B4

2 B

10 ] 3

21 5 RT(≠ 2 GM / ≠C A2 ) T,P,N 21 5 CB aA (≠aA / ≠CA ) T,P,N

5 CA a B (≠a B / ≠C )

The heat of formation, HM , can be obtained from Eq. (2) and the relation

Eq. (3) gives the value of SCC (0). For m 5 1 and y 5 2:

XS XS H XS M 5 G M 2 T(≠G M / ≠T ) P,C,N

SCC (0) 5 CA CB h1 1 CA CB (RT )21 [22W 1 2WAB (1

(6)

2 5CB 1 3C B2 ) 1 WBB (1 2 3C B2 )]j 21

Eqs. (2) and (6) yield A,B

H XS M 5 CA CB (W 2 T(≠W/ ≠T )) 1

OF (W 2 T(≠W / ≠T )) ij

ij

ij

i, j

(7) The entropy of mixing, SM , is given by SM 5 (HM 2 GM ) /T

(13)

21 B T,P,N

(8)

(14)

The Warren–Cowley [24,25] short-range order parameter (a1 ) is computed to quantify the degree of order. a1 can be determined experimentally from concentration–concentration and number density structure factors (SCC (q), SNN (q)). However, the determination of these two parameters is difficult for all types of binary alloys [22,26,27], but a1 can be evaluated theoretically [7,23]:

N. Jha, A.K. Mishra / Journal of Alloys and Compounds 329 (2001) 224 – 229

226

a1 5 (S 2 1) /(S(Z 2 1) 1 1)

(15)

S 5 SCC (0) /S id CC (0)

(16)

where Z is the coordination number, which is taken as 10 for our purposes.

2.4. Results and discussion It is clear from Section 1 that the likely complex to be formed in Mg–Zn liquid alloys is MgZn 2 . The system considered here is not a very strongly interacting system (GM (max) /RT 5 2 1.24). The expressions derived under weak interactions were used to compute GM (Eq. (3)), HM (Eq. (7)), SM (Eq. (8)), ln g (Eq. (10)), SCC (0) (Eq. (14)) and CSRO (Eq. (15)) as a function of concentration for Mg–Zn binary liquid alloys. The basic inputs for computation of the above functions are the interaction energy parameters. A reasonable choice of the interaction energy parameters is made from the experimental values for GM [28] at a couple of concentrations for Mg–Zn liquid alloys. To determine the variation of these energy parameters with temperature, we computed the enthalpy of mixing and entropy of mixing values for these alloys. The energy parameters (W,Wij ) for Mg–Zn liquid alloys are given in Table 1. The computed values of GM versus concentration are shown for Mg–Zn liquid alloy in Fig. 1 together with experimental values [28]. The computed values are in reasonable agreement with experiment. It can be seen from Fig. 1 that the values of GM are negative at all concentrations, being minimum around the stoichiometric concentration. It can be observed that the concentration dependence of HM can be explained if one considers the temperature dependence of the energy parameters for liquid Mg–Zn alloys. Experimental values of HM for Mg–Zn liquid alloys are not available for the whole concentration range. However, Agarwal et al. [29] compiled experimental data for the enthalpy of mixing of liquid Mg–Zn alloys at different temperatures. The temperature dependence of the energy parameters (DW, DWij ; D 5 ≠ / ≠T ) for liquid Mg–Zn alloys is given in Table 1. The computed values of HM versus concentration at 862, 893, 981 and 1073 K are given in Table 2 together with the

Table 1 Energy parameters for liquid Mg–Zn alloys Temp. (K) 923

862 893 981 1073

W/RT

WAB /RT

WAA /RT

WBB /RT

21.7

20.52

0

0.57

≠W/ ≠T 2 1.695R 2 1.895R 2 3.695R 2 4.095R

≠WAB / ≠T 0.08R 0.3R 1.9R 2.9R

≠WAA / ≠T 0 0 0 0

≠WBB / ≠T 0.04R 0.4R 0.8R 1.2R

Fig. 1. Energy of mixing (GM ) of liquid Mg–Zn alloy versus concentration of Zn (CZn ) at T 5 923 K. (———) Present value; (3) experimental value [28].

experimental values (the experimental values were taken from Fig. 2 of Agarwal et al. [29]). The computed and experimental values are in reasonable agreement. The last two equalities of Eq. (13) can be used directly to obtain SCC (0) from the observed activity data. Such SCC (0) are usually termed experimental values. The theoretical values of SCC (0) for Mg–Zn are computed from Eq. (14). These are shown in the upper part of Fig. 2 together with experimental and ideal (SCC (0)id 5 CA CB ) values. From SCC (0) , SCC (0)id , the existence of chemical ordering leading to complex formation is expected. Immediate insight into the local arrangements of atoms in the mixture is provided by a1 . a1 . 0 refers to like atoms pairing, a1 5 0 indicates a random distribution, and a1 , 0 corresponds to unlike atoms pairing as nearest neighbours. The computed values of a1 are shown in the lower part of Fig. 2. The values of a1 remain negative throughout the whole concentration range, which indicates that the liquid Mg–Zn alloy is an ordered alloy. The computed values of SM for Mg–Zn liquid alloy at 862, 893, 981 and 1073 K are given in Table 3. The experimental values for the entropy of mixing for liquid Mg–Zn alloy were estimated from experimental GM [28] and HM [29] values. These are also given in Table 3. The table shows fair agreement between calculated and experimental entropy of mixing values for liquid Mg–Zn alloys. The computed values of ln g for Mg–Zn liquid binary alloys versus concentration are given in Table 3, together

N. Jha, A.K. Mishra / Journal of Alloys and Compounds 329 (2001) 224 – 229

227

Table 2 Enthalpy of mixing of liquid Mg–Zn alloy 2 HM /RT 862 K

893 K

981 K

1073 K

CMg

Present

Expt.

CMg

Present

Expt.

CMg

Present

Expt.

CMg

Present

Expt.

0.066 0.1 0.106 0.166 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.269 0.360 0.377 0.507 0.566 0.699 0.774 0.796 0.767 0.688 0.553 0.351

0.251 – 0.353 0.530 – – – – – – – –

0.1 0.2 0.226 0.28 0.3 0.36 0.4 0.413 0.48 0.5 0.546 0.586 0.6 0.7 0.8 0.9

0.364 0.568 0.608 0.675 0.695 0.741 0.761 0.766 0.776 0.775 0.764 0.747 0.739 0.656 0.524 0.331

– 0.529 0.602 0.655 0.700 0.763 – 0.799 0.808 – 0.790 0.772 – – – –

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.733 0.8 0.846 0.893 0.9 0.947

0.325 0.487 0.569 0.592 0.568 0.509 0.424 0.391 0.321 0.268 0.210 0.201 0.131

– – – – – – 0.449 0.392 0.343 0.277 0.188 – 0.098

0.1 0.2 0.226 0.3 0.306 0.4 0.493 0.5 0.6 0.64 0.7 0.733 0.8 0.9

0.369 0.555 0.587 0.644 0.647 0.660 0.623 0.618 0.535 0.494 0.426 0.386 0.305 0.182

– – 0.547 – 0.642 0.702 0.635 – – 0.448 – 0.321 – –

with experimental observations [28]. Table 3 shows agreement between the theoretical and experimental values.

3. Expressions for surface tension By constructing a grand partition function for the surface and solving it in the framework of quasi-lattice theory, one obtains the equation for the surface tension of a binary solution given by Belton and Evans [30] and Angal and Roy [31], respectively, for ideal and non-ideal behaviour of the solution:

s

5 sA 1 (RT /A) ln(C AS /CA ) 1 (RT /A) ln(a AS /aA ) 5 sB 1 (RT /A) ln(C SB /CB ) 1 (RT /A) ln(a SB /a B )

(17)

and

s

5 sA 1 (RT /A) ln(C AS /CA ) 1 (RT /A) ln(g AS /gA ) 5 sB 1 (RT /A) ln(C SB /CB ) 1 (RT /A) ln(g SB /gB )

(18)

where si (i 5 A or B) is the surface tension of the pure component and A is the mean atomic surface area. C iS , a iS and g iS refer, respectively, to the concentration, activity and activity coefficient of the ith component at the surface. g Si is related to gi : ln g Si 5 n(ln gi containing C Si ) 1 q ln gi

Fig. 2. Top: SCC (0) for Mg–Zn liquid alloy versus concentration of Zn (CZn ) at T 5 923 K. (———) Present value; (3) experimental value calculated from activity data [28] using Eq. (13); (– ? –) ideal value (SCC (0)id ). Bottom: CSRO (a1 ) for Mg–Zn liquid alloy versus concentration of Zn (CZn ). (———) Calculated value at T 5 923 K.

where n and q, termed surface coordination fractions, are the fractions of the total numbers of nearest neighbours of an atom within the layer in which it lies and that in the adjoining layer, respectively, so that n 1 2q 5 1. The surface tension of binary liquid alloys, s, in the complex formation model is given by (for m 5 1 and y 5 2)

N. Jha, A.K. Mishra / Journal of Alloys and Compounds 329 (2001) 224 – 229

228

Table 3 Entropy of mixing and ratio of activity coefficients for liquid Mg–Zn alloy CMg

862 K

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ln(g ) 923 K

Entropy of mixing (SM /R)

0.146 0.260 0.343 0.394 0.412 0.397 0.349 0.267 0.151

893 K

981 K

1073 K

Present

Expt.

Present

Expt.

Present

Expt.

0.142 0.258 0.347 0.406 0.433 0.425 0.381 0.297 0.171

– 0.247 0.307 – – – – – –

0.181 0.340 0.473 0.575 0.639 0.655 0.613 0.500 0.301

– – – – – – 0.633 0.539 –

0.137 0.272 0.398 0.508 0.589 0.629 0.611 0.515 0.320

– – – 0.441 – – – – –

s 5 sA 1 (RT /A) ln[(1 2 C SB ) /(1 2 CB )] (19a)

s 5 sA 1 (RT /A) ln(C SB /CB )

9 2 F1B ) 2 qF1B ] 1 (W/A)[n(F 1B 9 2 F2B ) 2 qF2B ] 1 (WAB /A)[n(F 2B 1 (WBB /A)[n(F 93B 2 F3B ) 2 qF3B ]

Expt.

21.640 21.254 20.843 20.417 0.154 0.444 0.860 1.252 1.611

21.855 21.185 20.630 20.173 0.214 0.566 0.870 1.118 1.308

4. Conclusion

9 2 f2B ) 1 (W/A)[n( f 91B 2 f1B ) 2 qf1B ] 1 (WAB /A)[n( f 2B 2 qf2B ] 1 (WBB /A)[n( f 93B 2 f3B ) 2 qf3B ]

Present

(19b)

Here, primes denote C SB in place of CB in Eq. (12). F, f and F 9, f 9 are functions of the bulk and surface concentrations, respectively.

The phenomenon of complex formation in Mg–Zn liquid alloys and its effect on the concentration dependent mixing properties were investigated theoretically. Large negative values of GM and HM are sufficient to indicate that the intermetallic compounds MgZn 2 exist in the Mg– Zn system. The same energy parameters were used for the computation of the bulk as well as the surface properties. It was found that the energy parameters are temperature dependent. The negative values for the chemical shortrange order parameter indicate the presence of chemical order essential for complex formation. Study of SCC (0) reveals that there is a weak tendency for complex formation in liquid Mg–Zn alloy.

3.1. Results and discussion Eq. (19a,b) was solved numerically to compute the surface tension, s, of liquid Mg–Zn alloys at T 5 923 K. n and q were treated as parameters and taken as n 5 0.5 and q 5 0.25. The energy parameters required for the computation were the same as used for the bulk parameters. The mean atomic surface area, A, was calculated from the relation given by Laty et al. [32]: A 5 1.12(VM )2 / 3 where VM is the volume of the alloy. We used the linear volume based on the density data of Smithell’ s Metals Reference Book [33]. These volumes were used to calculate the mean atomic surface area, A. The surface tension of the component elements of the alloys at the specified temperature was also taken from the same book [33]. The computed values of s versus concentration for liquid Mg–Zn alloys are presented in Fig. 3. Because of a lack of experimental data we could not compare our theoretical results with experimental values. It can be seen from Fig. 3 that Mg atoms lower the surface tension of Mg–Zn.

Fig. 3. Surface tension of liquid Mg–Zn alloy versus concentration of Mg (CMg ) at 923 K. (———) Present value; (– ? –) ideal value.

N. Jha, A.K. Mishra / Journal of Alloys and Compounds 329 (2001) 224 – 229

Acknowledgements Financial support from C.S.I.R., New Delhi, is gratefully acknowledged by A.K. Mishra. The authors are grateful to Dr. R.N. Singh, Professor of Physics, S.Q. University, Oman, for encouragement and valuable discussions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

A.B. Bhatia, R.N. Singh, Phys. Chem. Liquids 11 (1982) 285, 343. A.B. Bhatia, R.N. Singh, Phys. Chem. Liquids 13 (1984) 177. F. Sommer, J. Non-Cryst. Solids 117 / 118 (1990) 505. R.N. Singh, F. Sommer, J. Phys. C 4 (1992) 5345. L.C. Prasad, R.N. Singh, Z. Naturforsch. 44a (1989) 529. A.K. Mishra, R.N. Singh, A.K. Rukhaiyar, B.B. Sahay, Phys. Status Solidi (a) 144 (1994) 335. N. Jha, S.M. Rafique, A.K. Mishra, R.N. Singh, V.N. Singh, Ind. J. Phys. 74A (2000) 461. H. Ruppersberg, H. Reiter, J. Phys. F 12 (1982) 1311. J.B. Clarck, L. Zabdyr, Z. Moser, Phase Diagrams of Binary Magnesium Alloys, ASM, Metals Park, OH, 1988. ¨ H. Rudin, S. Jost, H.J. Guntherodt, J. Non-Cryst. Solids 61 / 62 (1984) 291. P. Andonov, P. Chieux, J. Phys. Coll. 46 (C8) (1985) 81. ¨ E. Buhler, P. Lamparter, S. Steeb, Z. Naturforsch. A 42 (1987) 507. J. Hafner, J. Phys. F 18 (1988) 153.

229

[14] R.N. Singh, N.H. March, in: J.H. Westbrook, R.L. Fisher (Eds.), Principles, Intermetallic Compounds, Vol. 1, Wiley, New York, 1994. [15] L.C. Prasad, R.N. Singh, V.N. Singh, S.K. Chatterjee, Physica B 215 (1995) 225. [16] L.C. Prasad, A. Mikula, J. Alloys Comp. 299 (2000) 175. [17] R.N. Singh, I.S. Jha, D.K. Pandey, J. Phys. C 5 (1993) 2469. [18] S. Lele, P. Ramchandra Rao, Metall. Trans. 12B (1981) 659. [19] U. Gerling, M.J. Pool, B. Predel, Z. Metallkd. 74 (1983) 616. [20] K. Hoshino, W.H. Young, J. Phys. F 10 (1990) 1365. [21] A.B. Bhatia, D.E. Thornton, Phys. Rev. B 2 (1970) 3004. [22] C.N.J. Wagner, in: S. Steeb, H. Warlmont (Eds.), Rapidly Quenched Metals, North-Holland, Amsterdam, 1985, p. 405. [23] R.N. Singh, Can. J. Phys. 65 (1987) 309. [24] B.E. Warren, X-ray Diffraction, Addison-Wesley, Reading, MA, 1969. [25] J.M. Cowley, Phys. Rev. 77 (1950) 667. [26] H. Ruppersberg, H. Egger, J. Chem. Phys. 63 (1975) 4095. [27] S. Steeb, F. Falch, P. Lamparter, Z. Metallkd. 75 (1984) 599. [28] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys, ASM, Metals Park, OH, 1973. [29] R. Agarwal, S.G. Fries, H.L. Lukas, G. Petzow, F. Sommer, T.G. Chart, G. Effenberg, Z. Metallkd. 83 (1992) 216. [30] J.W. Belton, M.G. Evans, Trans. Faraday Soc. 41 (1945) 1. [31] R.D. Angal, D.L. Roy, Z. Metallkd. 73 (1982) 428. [32] P. Laty, J.C. Joud, P. Deste, Surf. Sci. 60 (1976) 109. [33] E.A. Brandes, G.B. Brook (Eds.), Smithell’s Metals Reference Book, 7th Edition, Butterworths–Heinemann, Oxford, 1992.