Thermodynamic assessment of the K–Na and Cr–V system

ARTICLE IN PRESS

Physica B 403 (2008) 2877–2883 www.elsevier.com/locate/physb

Thermodynamic assessment of the K–Na and Cr–V system Y.A. Odusote Department of Physics, Olabisi Onabanjo University, P.M.B. 2002, Ago-Iwoye, Ogun State, Nigeria Received 11 January 2008; received in revised form 18 February 2008; accepted 21 February 2008

Abstract The assessment of the thermodynamic properties of K–Na and Cr–V molten alloys has been theoretically examined using a simple statistical mechanical model based on pairwise interaction to obtain higher-order conditional probabilities that describe the occupation of the neighbouring atoms in molten binary alloys. The optimised values of order energy o obtained are used to describe a number of thermodynamic quantities computed for different concentrations in the alloys at 384 and 1550 K, respectively. The study shows that there is a tendency for homocoordination (like atoms pairing as nearest neighbour) in K–Na and the existence of heterocoordination in Cr–V at all concentrations. Thus, the consistency between calculated and reported experimental thermodynamic values enforces the legitimacy of the findings. r 2008 Elsevier B.V. All rights reserved. Keywords: Order energy; Homocoordination; Conditional probabilities; Alkali metal

1. Introduction There has been significant increase in experimentalists and theoreticians interest in understanding the thermodynamic properties of liquid alkali metal alloys due to their use as coolants in nuclear reactors. Some efforts to create the coolants with controlled properties has led to increase in attention to multicomponent liquid alkali metal alloys. A number of theoretical studies have been performed on various properties of metals in the liquid state [1–10] which has greatly improved the understanding of the thermodynamics of molten alloys containing strongly interacting metals. Advances in these studies has been made possible due to the combination of the following theoretical approaches such as electron theory [11], pseudopotential formalism [12] with thermodynamic perturbation theories. In the present study, the assessment of the thermodynamic properties of K–Na an alkali metal alloy and Cr–V a transition molten alloy has been performed using a four atom cluster model (FACM) proposed by Singh [13] which is a simple statistical mechanical model based on pairwise interaction to obtain higher-order conditional Tel.: +234 8052203325.

E-mail address: [email protected] 0921-4526/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.02.029

probabilities that describe the occupation of the neighbouring atoms in molten binary alloys. The optimised values of order energy o obtained are used to calculate a number of thermodynamic quantities for different concentrations of the alloys at different temperatures. Apart from a high-academic interest, liquid K–Na and Cr–V alloys have been chosen for investigation because of their great commercial significance and the wide diversity of their physical and chemical properties. Besides, K and Na are alkali metals found in group 1A of the periodic table. Alloys of this group exhibit very fascinating properties in the liquid state. This group of metals are used primarily as a reducing agent in pharmaceutical, perfumery and general chemical industries. Due to excellent physical properties of alkali metals, mainly high-thermal conductivity coupled with low viscosity and low density, K–Na alloys are used in the heat-transfer systems as coolant in the valves of internal combustion engines and in nuclear reactors [14]. Chromium is alloyed (that is, mixed) with steel to make it corrosion resistant or harder. An example is its use in the production of stainless steel, a bright, shiny steel that is strong and resistant to oxidation (rust). It is also widely used in electroplating for appearance and wear in textile industries, in powder metallurgy, and to make X-ray

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targets and mirrors [15]. In addition, its component vanadium also has various industrial applications. It is used as an alloying element in different types of steel, where it increases strength and fatigue resistance. Aluminium is also alloyed with vanadium to produce master alloys, which are then added to titanium alloys used in jet engines, automotive industries and high-speed airframes [16,17]. Furthermore, vanadium is also used in speciality stainless steel for surgical instruments and high-speed tool steels. Vanadium foil is used as a bonding agent in binding titanium to steel and, because of its low fission neutron cross-section, vanadium also has nuclear applications. Vanadium pentoxide is used as a catalyst for the production of sulphuric acid by the contact process and for the production of maleic anhydride. All mentioned above confirm that for a proper understanding of the thermodynamic properties of these alloys reliable and comprehensive information regarding the energetics of mixing in binary alloys through the study of microscopic functions such as concentration–concentration fluctuations in the long-wavelength limit, S cc ð0Þ and Warren–Cowley chemical short-range order parameter, (CSRO), a1 , [2,4,7,18] are essential. In this study, a FACM based on pairwise interaction has been successfully used to compute the conditional probabilities enumerating the occupation of neighbouring sites by the atoms of the constituent elements in the liquid alloys. These are then utilised to compute the CSRO parameter, a1 as a function of concentration. It has been shown that conditional probabilities is closely related to the activity through the order energy, o. Thus, other thermodynamic quantities such S cc ð0Þ, free energy of mixing, chemical diffusion, heat of mixing and entropy of mixing are computed. The theoretical formalism of the FACM are discussed in Section 2, after which the results obtained are given in Section 3. In the final section, some concluding remarks are given.

The grand partition function X of a binary molten alloy AB, which consists of N A ¼ Nc and N B ¼ Nð1  cÞ atoms of elements A and B, respectively, where the total number of atoms, N, is equal to N A þ N B , can be expressed as X¼

E

NB bðmA N A þmB N B EÞ A qN ; A ðTÞqB ðTÞ e

Pij ¼ ebij ;

ði; j ¼ A; BÞ.

(2)

The complete discussion of FACM are given in Ref. [13]. After doing some algebra [13], one obtains an expression of the form s12  B1 s9  B2 s6  B3 s3  B4 ¼ 0,

(3)

where s¼

  fB PAA , fA PBB

wB1 ¼

(4)

1  3x , Z3

B2 ¼ 3x

(5)

ð1  xÞ , Z4

B3 ¼ 3x2

(6)

ð1  x=3Þ , Z3

(7)

B4 ¼ x3 , 1c x¼ c

(8) 

and

 bo Z ¼ exp , Z

(9)

w ¼ ZðAB  ðAA þ BB Þ=2Þ is the interatomic interaction energy, usually termed the order energy in regular solution theory and the parameter f in Eq. (4) is a constant which has to be eliminated in the final result. The activity ratio a defined as ða ¼ aB =aA Þ is related to s by the expression given as cf 1 ða; sÞ ¼ ð1  cÞf 2 ða; sÞ,

(10)

where f 1 ða; sÞ and f 2 ða; sÞ are defined as

2. Theoretical formalism

X

lattice sites are further subdivided into smaller cluster of just a few lattice sites in domain 1 and the remainder in domain 2. Arising from the above assumptions, one can define parameters Pij and the ij ’s as the bond energies for ij nearest neighbour bond such that

1 , b¼ kB T

(1)

where qi ðTÞ are the particle functions of atoms i (A or B) associated with inner and vibrational degrees of freedom, mA and mB are the chemical potentials and E is the configurational energy. Solving Eq. (1), requires two simplifying assumptions [13], the first assumption requires that the interactions between atoms should be of shortrange and effective only between nearest neighbours. The second assumption is that the atoms are located on lattice sites such that each site has Z-nearest neighbours. The

f 1 ða; sÞ ¼ a4 s4ZL þ

3a3 s3ZL 3a2 s2ZL asZL þ þ 3 Z3 Z4 Z

(11)

and f 2 ða; sÞ ¼

a3 s3ZL 3a2 s2ZL 3asZL þ þ þ1 Z3 Z4 Z3

(12)

here ZL ¼ Z  3 in Eqs. (11) and (12), Eq. (12) is solved numerically to determine the activity ratio for a given binary alloy from the knowledge of s obtained from the numerical solution of Eq. (3). The value of s required in the calculations is optimised in such a way that it gives a good overall representation of activity at all concentrations [19]. The main objective of the FACM is to express the degree of CSRO in terms of probabilities. To achieve this, one observes that in the framework of the model, the probability of finding an A atom or B atom on any lattice site depends on the nature of atoms already existing in the

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neighbouring sites [6]. One begins by stating the probability that all four lattice sites occupied by atoms A as ðA; A; A; AÞ and similar probabilities ði; j; k; lÞ can readily be reduced to higher-order conditional probabilities such that i=iji (the probability of finding i atom on a given lattice site while the other three sites in the cluster are occupied by i, j, and i atoms) and similar others [6]. Having this in mind, one can express the pairwise conditional probability PAB as PAB

ðA=BBÞ . ¼ ðA=BÞ ¼ ðB=ABÞ þ ðA=BBÞ

(13)

It is reasonable to express Eq. (13) in terms of higher-order conditional probabilities as ðA=BBÞ ¼

ðA=BBBÞ , ðB=ABBÞ þ ðA=BBBÞ

(14)

ðB=ABÞ ¼

ðB=AABÞ ðB=AABÞ þ ðA=ABBÞ

(15)

and it is possible to express the term ðB=ABBÞ in Eq. (14) as ðB=ABBÞ ¼ 1  ðA=ABBÞ. Eqs. (13)–(15) are only applicable in this model if we can express them in terms of a and s. The details of these derivations are also reported in [13,20]. Hence, the results are [6] ðA=BBBÞ ¼

ðA=ABBÞ ¼

1þ

asZL

1 , expð3bo=ZÞ

(16)

1 1 þ asZL expðbo=ZÞ

(17)

asZL expðbo=ZÞ . 1 þ asZL expðbo=ZÞ

(18)

and ðB=AABÞ ¼

By solving Eqs. (16)–(18), one can thus obtain the value of PAB defined in Eq. (13). A useful quantity which can be obtained from knowledge of PAB is the Warren-Cowley CSRO parameter, a1 , [2,4,7,18]. This parameter a1 is defined for nearest neighbour sites as PAB c and the limiting values of a1 lie in the range a1 ¼ 1 



c pa1 p1 ð1  cÞ



1c pa1 p1 ðcÞ

1 cp , 2 1 cX . 2

(19)

(20)

(21)

For equiatomic composition, the CSRO parameter, a1 , is found to be 1pa1 p1. Negative values of this parameter indicate ordering in the melt, and complete ordering is indicated by amin ¼ 1. In contrast, positive values of a1 1 indicate segregation, whereas phase separation takes place if amax ¼ 1. 1 One begins the thermodynamic assessment by using the optimised value of o obtained above to compute the excess

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Gibb’s free energy of mixing, G xs M and the concentration– concentration fluctuations in the long-wavelength limit, Scc ð0Þ. In achieving these, we have used xs GM ¼ G id M þ GM ,

(22)

Gid M ¼ RT fc þ ð1  cÞ lnð1  cÞg,

(23)

Gxs M ¼ RT fc ln gA þ ð1  cÞ ln gB g,

(24)

where the activity coefficients gA and gB can be easily obtained by the Fowler–Guggeheim method, as reported in Refs. [21,22] and related to the activity a by the standard relation [19]   b  1 þ 2c Z=2 gA ¼ , (25) cðb þ 1Þ  gB ¼

b þ 1  2c ð1  cÞðb þ 1Þ

aA ¼ cgA ;

Z=2 ,

(26)

aB ¼ ð1  cÞgB

(27)

and here b ¼ f1 þ 4cð1  cÞðZ2  1Þg1=2 .

(28)

With the aid of Eq. (22), S cc ð0Þ can be determined using thermodynamic relations:  2 1   q GM qaA 1 Scc ð0Þ ¼ RT ¼ ð1  cÞa A qc2 T;P;N qc T;P;N  1 qaB ¼ caB (29) qð1  cÞ T;P;N and if Eq. (22) is substituted in Eq. (29), the Scc ð0Þ becomes Scc ð0Þ ¼

cð1  cÞ  . Z 1 1 1þ 2 b

(30)

Once an appropriate optimised value of o is obtained, it can be used to compute G M =RT and S cc ð0Þ using Eqs. (22) and (30), respectively. The results can then be compared with experiments and deductions made therefrom. It is found that b in Eq. (28) depends on the interchange energy o through Eq. (9) and brings about the deviation from ideality. It should be pointed out that as o ! 0, b ! 1 and Scc ð0Þ ¼ cð1  cÞ ¼ Sid cc ð0Þ, the ideal values. The understanding gained through the study of S cc ð0Þ has been used by theoreticians as well as experimentalists to examine the nature of chemical diffusion in binary liquid alloys, which probably plays a significant role in many technological and corrosion phenomena. The expression that relates diffusion and Scc ð0Þ is given below, as reported in Refs. [7,23,24] DM S id ð0Þ ¼ cc Did S cc ð0Þ

(31)

here DM is defined as the mutual diffusion coefficient and Did is the intrinsic diffusion coefficient for an ideal mixture.

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For ideal mixing, Scc ð0Þ ! S id cc ð0Þ, i.e. DM ! Did ; for ordered alloys, S cc ð0ÞoS id ð0Þ, i.e. DM 4Did ; and similarly cc for segregation, DM oDid . The heat of mixing, H M , within the framework of the model used in the present study is obtained from the thermodynamic relations [5]:   qG M H M ¼ GM  T . (32) qT P On using Eq. (22) for G M in Eq. (32), one gets   2o 2 2   8RTc ð1  cÞ exp 1 do o ZkB T  HM ¼  ðb  1 þ 2cÞð1 þ bÞðb þ 1  2cÞ kB dT T (33) and the entropy of mixing is expressed as H M  GM . (34) T Based on the above formalism, all the essential equations required for the thermodynamic assessment of the binary alloys using FACM have been obtained.

SM ¼

3. Results and discussion The values of the fitted interaction parameters used for all calculations are tabulated in Table 1. While, Table 2(a) and (b) show the values of the conditional probabilities obtained using Eqs. (14) and (15) and the activity ratio for the alloys studied. The optimised values of order energy, o, obtained in the calculations of CSRO parameter, a1 , from Eq. (19) for the alloys were used to compute other thermodynamic quantities using Eqs. (22), (30), (31), (33) and (34). For both systems no experimental data were available to be used for calculations of the CSRO parameter, a1 , the coordination number, Z, in the liquid phase was taken as 10 [25]. When the values of the interaction energy parameters are well defined, the value of Z does not have any significant effect on the results obtained. The experimental values of Scc ð0Þ used for both systems were obtained from experimental free energy of mixing data taken from [26] using Eq. (29). 3.1. K–Na alloys From Table 2a, it is seen that the computed activity ratio for K–Na alloys are in very good agreement with the experimental values at all concentrations. The energetics of mixing have been assessed by using the optimised value of Table 1 Temperatures and fitted interaction parameters for the alloys System

T (K)

Z

o (eV)

do    1010

K–Na Cr–V

384 1550

10 10

0.0325 0.1415

252 35.6

dT

Table 2 Conditional probabilities ðA=BB; B=ABÞ and activity ratio for (a) K–Na alloys at 384 K ðA  K; B  NaÞ and (b) Cr–V alloys at 1550 K. ðA  Cr; B  VÞ (a) cNa

ðA=BBÞ

Activity ratio a ¼ aB =aA

ðB=ABÞ

Theory 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.096 0.186 0.274 0.360 0.448 0.540 0.637 0.743 0.862

0.884 0.780 0.683 0.590 0.500 0.409 0.316 0.219 0.115

(b) cV

ðA=BBÞ

ðB=ABÞ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.104 0.213 0.325 0.438 0.550 0.656 0.756 0.848 0.929

4.161 2.260 1.600 1.243 1.000 0.804 0.625 0.442 0.240

Expt. [26] 4.030 2.237 1.617 1.127 1.036 0.834 0.624 0.444 0.232

Activity ratio a ¼ aB =aA

0.905 0.819 0.717 0.610 0.500 0.389 0.282 0.180 0.085

Theory

Expt. [26]

21.336 7.716 3.633 1.874 1.000 0.500 0.275 0.129 0.046

9.137 7.716 3.633 1.889 1.314 0.829 0.453 0.208 0.069

0 -0.1 -0.2 -0.3 -0.4 GM RT

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K-Na

-0.5 -0.6 -0.7 -0.8 -0.9 Cr-V

-1 0

0.2

0.4

0.6

0.8

1

cNa,cV

Fig. 1. Free energy of mixing, GM =RT vs concentration for K–Na and Cr–V molten alloys at 384 and 1550 K, respectively. The solid lines represent theoretical values while the circles and the stars represent experimental values for K–Na and Cr–V alloys, respectively. cNa and cV are the Na and V concentrations in the alloys. The experimental data are from Ref. [26].

o obtained to compute the CSRO parameter, a1 , which were then utilised to compute both G M =RT and Scc ð0Þ for the alloys. The plot of G M =RT using Eqs. (22) and S cc ð0Þ using Eqs. (30) are presented in Figs. 1 and 2, respectively. It is seen that the computed values for both quantities are in very good agreement with the experimental values. It is interesting here to observe the deviation in calculated S cc ð0Þ values from the ideal mixture values S id cc ð0Þ ¼ cð1  cÞ. The

ARTICLE IN PRESS Y.A. Odusote / Physica B 403 (2008) 2877–2883

0.5

1

0.45

0.95 0.9

0.4

0.85

K-Na

0.35

0.8

0.3 DM Did

Scc(0)

2881

0.25

0.75

0.2

0.7

0.15

0.65

Cr-V

0.6

0.1

0.55

0.05

0.5

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

cNa,cV Fig. 2. Concentration–concentration fluctuations in the long-wavelength limit, Scc ð0Þ vs concentration for K–Na and Cr–V molten alloys at 384 and 1550 K, respectively. The solid lines represent theoretical values while the circles and the stars represent experimental values for K–Na and Cr–V alloys, respectively. The dots represent the ideal values. cNa and cV are the Na and V concentrations in the alloys.

0.6

0.8

1

cNa

Fig. 4. Concentration dependence of chemical diffusion, DM =Did using Eq. (31) for K–Na molten alloys at 384 K. cNa is the Na concentration in the alloys.

1.6 1.5

0.06 1.4

K-Na DM Did

0.04

1

0.02

1.3 1.2

0

1.1

-0.02

1 0

0.2

0.4

-0.04 -0.06 0

0.2

0.4

0.6

0.6

0.8

1

cV

Cr-V 0.8

1

cNa,cV

Fig. 5. Concentration dependence of chemical diffusion, DM =Did using Eq. (31) for Cr–V molten alloys at 1550 K. cV is the V concentration in the alloys.

Fig. 3. Calculated Warren–Cowley short-range order parameter, a1 vs concentration, using Eq. (19) for K–Na and Cr–V molten alloys at 384 and 1550 K, respectively. cNa and cV are the Na and V concentrations in the alloys.

0.25 0.2 0.15 HM RT

calculated S cc ð0Þ values are greater than the ideal values S id cc ð0Þ at all concentrations. The positive deviation from ideal values implies a tendency for homocoordination, i.e. like atoms K–K and Na–Na tend to pair as nearest neighbours. The S cc ð0Þ curve exhibits the maximum value of about 0.4697 at cNa ¼ 0:5. The position of maximum coincides with that reported by Ref. [27], obtained using a two atom cluster model. The positive values of CSRO parameter, a1 , over the concentration range for the K–Na (Fig. 3) further support a tendency towards segregation (like atoms pairing). The computed values of Scc ð0Þ have been used in Eq. (19) to obtain the concentration dependent chemical diffusion using Eq. (31), as shown in Fig. 4. It is observed that the ratio DM =Did is less than 1 at all concentrations. This also affirms the presence of self-coordination or homocoordination in the alloys. The temperature dependence of heat of mixing and entropy of mixing have been ascertained using Eqs. (33)

0.1 0.05 0 0

0.2

0.4

0.6

0.8

1

cNa Fig. 6. Heat of mixing, H M =RT vs concentration for K–Na alloys at 384 K. The solid lines represent theoretical values while the circles represent experimental values for K–Na alloys. cNa is the Na concentrations in the alloys. The experimental data are from Ref. [26].

and (34), respectively. One observes that the fits obtained for H M =RT (Fig. 6) shows a good agreement between the computed and experimental values except a slight deviation in magnitude in the concentration range 0:7pcNa p0:82,

ARTICLE IN PRESS Y.A. Odusote / Physica B 403 (2008) 2877–2883

2882 0

0.8 0.7

-0.05

0.6 0.5 SM R

HM RT

-0.1 -0.15

0.4 0.3

-0.2

0.2 -0.25

0.1

-0.3

0 0

0.2

0.4

0.6

0.8

1

0

0.2

cV

0.7 0.6 0.5 SM R

0.4 0.3 0.2 0.1 0 0.2

0.4

0.6

0.8

1

cV

Fig. 7. Heat of mixing, H M =RT vs concentration for Cr–V alloys at 1550 K. The solid lines represent theoretical values while the stars represent experimental values for Cr–V alloys. cV is the V concentration in the alloys. The experimental data are from Ref. [26].

0

0.4

0.6

0.8

1

cNa Fig. 8. Entropy of mixing, SM =R vs concentration for K–Na alloys at 384 K. The solid lines represent theoretical values while the circles represent experimental values for K–Na alloys. cNa is the Na concentrations in the alloys. The experimental data are from Ref. [26].

while SM =R (Fig. 8) shows a good agreement between calculated and observed values. The values of H M =RT and S M R are both positive (typical of segregating alloys) and symmetric around the equiatomic composition.

Fig. 9. Entropy of mixing, SM =R vs concentration for Cr–V alloys at 1550 K. The solid lines represent theoretical values while the stars represent experimental values for Cr–V alloys. cV is the V concentrations in the alloys. The experimental data are from Ref. [26].

in good agreement. It is interesting to observe for both systems studied that the GM =RT curves (Fig. 1) are symmetrical at equiatomic composition ðc ¼ 0:5Þ, with GM =RT values of about 0:4536 and 0:9650 for K–Na and Cr–V, respectively. This implies that Cr–V exhibits a higher tendency for compound formation than K–Na alloys. The S cc ð0Þ values for the Cr–V system at 1550 K clearly show that calculated S cc ð0ÞoS id cc ð0Þ in the whole concentration range. This negative deviation from Raoult’s law also indicates a tendency of complex formation and the presence of chemical order in the melt, that is substantiated by the negative values of CSRO parameter, a1 , as is shown in Fig. 3. The same mixing behaviour can be inferred from the values of chemical diffusion, DM =Did (Fig. 5). One notes that the ratio DM =Did is greater than 1 at all concentrations corresponding to compound formation and the presence of chemical order. The model, however, fails to reproduce the experimental heat of mixing, as shown in Fig. 7. Whereas, a comparison between calculated and experimental values of the entropy of mixing, SM =R, for Cr–V alloys (Fig. 9) shows a better agreement.

3.2. Cr–V alloys 4. Concluding remarks The computed values of the conditional probabilities and the activity ratio for Cr–V liquid alloys are given in Table 2b. The calculated and experimental values of the activity ratio are in reasonable agreements. Although, some minor discrepancies are observed at lower contents of V in the alloys. However, the agreement improves significantly with increasing concentration of V. Using the optimised value of o obtained here, the CSRO parameter, a1 , computed was used to assess the alloying behaviour expressed in terms of phase separation and compound formation by computing both G M =RT and Scc ð0Þ as functions of concentration. Figs. 1 and 2 show that the agreement between the calculated and experimental values of these quantities are

A simple statistical mechanical model based on pairwise interaction (FACM) has been used to evaluate the mixing behaviour of binary molten alloys and the results obtained are comparable to those from complex forming theories with more parameters as reported in Ref. [19]. The model is used to obtain the degree of chemical short-range order (CSRO) in terms of probabilities which was further used to compute order energy and the concentration dependent thermodynamic properties for K–Na and Cr–V alloys. The study shows that homocoordination among nearest neighbour exist in K–Na alloys whereas heterocoordination exists in Cr–V alloys. Comparison reveals that calculated values of the thermodynamic properties are generally in

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good agreement with the experimental results. The results obtained in the present study confirm the applicability of this approach for a complete description of binary systems which exhibit similar mixing properties. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

A.B. Bhatia, W.H. Hargrove, Phys. Rev. B 10 (1974) 3186. R.N. Singh, Canad. J. Phys. 65 (1987) 309. R.N. Singh, K.K. Singh, Modern Phys. Lett. B 9 (1995) 1729. R. Novakovic, M.L. Muolo, A. Passerone, Surf. Sci. 549 (2004) 281. R. Novakovic, D. Zivkovic, J. Mater. Sci. 40 (2005) 2251. O. Akinlade, Phys. Chem. Liq. 29 (1995) 9. Y.A. Odusote, L.A. Hussain, E.O. Awe, J. Non-Cryst. Solids 353 (2007) 1167. Y.A. Odusote, O.O. Fayose, P.J. Peace, J. Non-Cryst. Solids 353 (2007) 4666. M. Shimoji, Liquid Metals, Academic Press, New York, 1977. J.P. Hansen, Theory of Simple Liquids, Academic Press, London, 1986. N.W. Ashcroft, D. Stroud, Solid State Phys. 33 (1978) 1.

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[12] W.A. Harrison, Pseudopotentials in the Theory of Metals, Benjamin, New York, 1966. [13] R.N. Singh, J. Phys. Chem. 25 (1993) 251. [14] R.D. Kale, M. Rajan, Curr. Sci. 86 (5) (2004) 668. [15] Mineral Information Institute. hhttp://www.mii.org/i. [16] Metal Bulletin Monthly, MBM September, 2006, pp. 46–47. [17] A. Kostov, D. Zivkovic, B. Friedrich, J. Miner. Met. 42B (2006) 57. [18] R. Novakovic, E. Ricci, D. Giuranno, F. Gnecco, Surf. Sci. 515 (2002) 377. [19] O. Akinlade, Modern Phys. Lett. B 11 (1997) 93. [20] A. Cartier, J. Barriol, Phys. B81 (1976) 35. [21] R. Speiser, D.R. Poirier, K.S. Yeum, Scripta Metallurg. 21 (1987) 687. [22] K.S. Yeum, R. Speiser, D.R. Poirier, Metallurg. Trans. B3 (1989) 693. [23] R.N. Singh, F. Sommer, Z. Metallkd 83 (1992) 553. [24] L.C. Prasad, R.N. Singh, V.N. Singh, G.P. Singh, J. Phys. Chem. B102 (1998) 921. [25] B.C. Anusionwu, G.A. Adebayo, J. Alloys Compd. 329 (2001) 162. [26] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelly, Selected Values of the Thermodynamic Properties of Binary Alloys, American Society for Metals, Material Park, OH, 1973. [27] R.N. Singh, I.K. Mishra, Phys. Chem. Liq. 18 (1988) 303.