Bulk and surface properties of Co–Fe and Fe–Pd liquid alloys

Journal of Non-Crystalline Solids 394–395 (2014) 9–15

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Bulk and surface properties of Co–Fe and Fe–Pd liquid alloys R.P. Koirala a, J. Kumar b, B.P. Singh a, D. Adhikari c,⁎ a b c

University Department of Physics, T.M. Bhagalpur University, Bhagalpur, India Metals and Ceramics Division, Research Institute, University of Dayton, Dayton, USA Department of Physics, M.M.A.M. Campus, Biratnagar, Tribhuvan University, Nepal

a r t i c l e

i n f o

Article history: Received 13 February 2014 Received in revised form 26 March 2014 Available online 24 April 2014 Keywords: Thermodynamic properties; Short range order; Quasi-chemical approach; Surface segregation

a b s t r a c t We have employed quasi-chemical approximation to study the concentration dependence of bulk thermodynamic properties of two liquid alloys, Co–Fe and Fe–Pd at temperatures of 1863 K and 1873 K, respectively, at fixed atmospheric pressure. In order to investigate the surface structure in the alloys we have also performed a comparative study of surface tension using two statistical models. The study of the alloys indicates a positive deviation from the Raoultian behaviour with a higher degree of segregation in Fe–Pd system than that in the Co–Fe system. The surface tension is found to increase, both in Co–Fe and Fe–Pd alloys, with the addition of Fe-component. Furthermore, the surface segregation of the atoms of the metal with lower surface tension (Pd atoms in Fe–Pd alloy and Co atoms in Co–Fe alloy) is pronounced. Maximum surface segregation is expected to occur in the Co–Fe and Fe–Pd alloys around the compositions xCo = 0.5 and xFe = 0.8, respectively. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Iron is one of the most commonly used metals which can be alloyed with many elements to produce materials with desirable characteristics. The alloys of iron with transition metals like cobalt and palladium in particular are of special importance to researchers and have been studied by many authors [1–5]. The Co–Fe alloys in suitable composition possess improved properties such as high mechanical strength and soft magnetic character, with high permeability and high Curie temperature, which are required for advanced power applications [5]. Fe–Pd system is the basis for ferromagnetic shape memory alloys that work as magnetically switchable smart materials [6]. Palladium resists corrosion and it can be used as electrodes in low-temperature fuel cells [7]. Noble alloys containing palladium are used in dentistry as a high quality stiffener in dental inlays and bridgework. Such interesting applications of the alloys in solid state led us to give attention in these alloys. The understanding of thermo-physical properties helps in producing new reliable materials required particularly for high temperature applications. The thermodynamic functions such as the free energy of mixing (GM), heat of mixing (HM), entropy of mixing (SM) and chemical activities (ai) provide information about the bonding strength and the maximum stable composition of the alloys. The microscopic properties such as the concentration fluctuation in long wavelength limit (Scc(0)) and the Warren–Cowley short-range order parameter (α1) give idea about the local ordering of atoms. The surface tension effects play a vital role in the formation of solid alloys by solidification process of the melts. Several metallurgical phenomena such as, crystal growth, ⁎ Corresponding author. E-mail address: [email protected] (D. Adhikari).

http://dx.doi.org/10.1016/j.jnoncrysol.2014.04.001 0022-3093/© 2014 Elsevier B.V. All rights reserved.

welding, gas absorption, and nucleation of gas bubbles are mainly guided by the surface tension phenomena [8]. Indeed processes like kinetics of phase transformation, catalytic activity etc. in metal solutions can be comprehended from the knowledge of surface properties. Thus in order to understand the physics and chemistry of structure development during solidification in the binary system, the study of surface properties is very essential. For theoretical study of the thermo-physical properties of binary liquid alloys several models have been employed [9–15]. In the current work we intend to study the mixing behaviour of two liquid alloys, Co–Fe at 1863 K and Fe–Pd at 1873 K, at fixed pressure, most likely, at the atmospheric pressure by computing their thermodynamic and surface properties using statistical models. To study the thermodynamic properties we make use of quasi-chemical approximation (QCA) [9]. Due to unavailability of experimental data on surface properties of the alloys at hand we cannot compare the results which can be obtained from a model. Because of this we intend to perform a comparative study of the surface properties using Butler's model [13] and quasichemical approach [14]. In the following section, the basic formalism for calculation is presented. Results and discussion of the work are outlined in Sections 3 and 4 respectively. The conclusions are provided in Section 5. 2. Formalism The QCA is a statistical formulation which is appropriate for studying segregating binary liquid alloys. According to this model, a binary liquid alloy is assumed to consist of NA atoms of component A and NB atoms of component B situated at equivalent sites. There is short ranged atomic interaction between the nearest neighbours that forms a polyatomic

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R.P. Koirala et al. / Journal of Non-Crystalline Solids 394–395 (2014) 9–15

matrix leading to the formation of like atom clusters or self-associates of the type iA and jB, where i and j are the number of atoms in the clusters of elements A and B, respectively [8]. Under this assumption of QCA, thermodynamic and microscopic functions of binary liquid alloys can be computed. 2.1. Thermodynamic functions Consider one mole of a binary liquid alloy A–B consisting of NA (=xAN) atoms of A-component and NB (=xBN) atoms of B-component in the bulk phase, N being the total number of atoms, and xA and xB are the concentrations expressed in mole fractions such that xA + xB = 1. The free energy of mixing, GM for binary liquid alloys in QCA can be obtained using the expression [8]: GM ¼ RT ½xA ln xA þ xB ln xB þ xA ln ð1−ϕÞ þ lnφ þ xA xB φ W

φ¼

1 ; 1−xA ϕ

j γ¼ : i




∂GM ∂NA




;

RT ln aB ¼

T;P;N B

∂GM ∂N B

Scc ð0Þ ¼

ð3Þ

T;P;N A

Using Eqs. (1) and (3), the working expressions for the activities are obtained as: 2

lnaA ¼ ln ðxA φð1−ϕÞÞ þ xB ϕφ þ ðφxB Þ

W RT

ð9Þ

T;P

ð2Þ

 :

  2 2 −1 Scc ð0Þ ¼ RT ∂ GM =∂xA :

ð1Þ

Here, R is the universal gas constant, T is the temperature of alloy, and W is the ordering energy parameter. The term γ and W are the model parameters in QCA that are independent of alloy composition but may depend on temperature T and pressure P. The chemical activity is another basic thermodynamic quantity. The activities can be measured as a function of composition and temperature by several experimental methods, such as measurements of electro-motive force, vapour pressure, etc. Theoretically the activity, ak (k = A,B) of constituent elements A and B in binary liquid alloys are obtained from the following standard thermodynamic relations: RT ln aA ¼

The concentration fluctuations in long wavelength limit (Scc(0)) and Warren–Cowley [16,17] short-range order parameter (α1) are important microscopic functions which have emerged as powerful tool to extract important information on the nature and degree of local ordering of atoms in the binary liquid alloys. The Scc(0) in particular is useful in ascertaining the segregation in binary liquid alloys. The (Scc(0)) is obtained from the second composition derivative of the Gibbs function as [18]:

  2 It may be pointed out that the second derivative ∂ GM =∂xA 2 is a measure of stability for the binary solution, which was originally introduced by Darken in 1967. On using Eq. (1) in Eq. (9), the expression for Scc(0) can be put in the form:

where 1 ϕ ¼ 1− ; γ

2.2. Structural functions

ð4Þ

xA xB 1−xA xB f ði; j; ωÞ

ð10Þ

with

f ði; j; ωÞ ¼

2

W 2 2 γ −ðγ−1Þ ðxA þ γ xB Þ RT : ðxA þ γ xB Þ3

ð11Þ

The value of (Scc(0)) can also be obtained by taking the first composition derivative of the activity as given below: −1

−1

Scc ð0Þ ¼ xB aA ð∂aA =∂xA ÞT;P ¼ xA aB ð∂aB =∂xB ÞT;P :

ð12Þ

In the literature, the Scc(0) determined by using the measured activity data in Eq. (12) is usually referred to as an experimental Scc(0) [18]. In order to ascertain the degree of atomic order in the alloy, Warren– Cowley short range order parameter (α1) can be estimated from the knowledge of the interaction energy parameter, W. For this parameter, the following expression is available [15]: α1 ¼

λ−1 ; λþ1

ð13Þ

where λ is an auxiliary function defined by: W : lnaB ¼ ln ðxA φÞ þ γxA ϕðϕ−1Þφ þ ðφxA Þ RT 2

ð5Þ

The entropy of mixing (SM) is calculated from the standard thermodynamic relation given by:
 ∂GM : SM ¼ − ∂T P

Scc ð0Þ ¼

ð7Þ

HM

  ∂W ¼ xA xB φ W−T : ∂T

ð8Þ

c ð1−cÞ : 1 þ 1=2Z ð1=λ−1Þ

ð15Þ

From Eqs. (13) and (15), the short range order parameter, α1 immediately follows in the form: α1 ¼

The heat of mixing is another important thermodynamic quantity which can be measured directly from calorimetric experiments. Theoretically it can be calculated from the following expression, derived by solving Eqs. (1) and (7):

ð14Þ

Here Z represents the co-ordination number in the bulk. In terms of the function λ, the concentration fluctuation in long wavelength, Scc(0) can be expressed as:

ð6Þ

Using Eq. (1), expression for SM can be worked out as: ∂W SM ¼ −R ½ xA lnxA þ xB ln xB þ xA ln ð1−ϕÞ þ ln φ−xA xB φ : ∂T

h i1=2 2 : λ ¼ ðxB −xA Þ þ 4xA xB expð2W=ZkB T Þ

ðS−1Þ S ðZ−1Þ þ 1

with

S¼

Scc ð0Þ : xA xB

ð16Þ

2.3. Surface properties As discussed before, the experimental determination of surface tension involves several difficulties in many cases. As alternative to the experimental methods, several theoretical models [11,13,14] have

R.P. Koirala et al. / Journal of Non-Crystalline Solids 394–395 (2014) 9–15

been developed to obtain data on the surface tension of metals and alloys. In the work, we plan to study the composition dependence of surface properties of Co–Fe and Fe–Pd systems in molten state using the following two statistical approaches. (i) The first approach we employ for the theoretical estimation of the surface properties of the alloys is Butler's model [13]. This model is based on the assumption of hypothetical monatomic layer on the surface of the liquid mixture. The monatomic layer is considered as a separate phase that is in thermodynamic equilibrium with the bulk. The surface tension, σ of a binary liquid solution in this model can be expressed as [19]: σ¼

μ sA −μ bA αA

¼

μ ks

μ sB −μ bB αB

¼ ::::::::::: ¼

μ sk −μ bk αk

ð17Þ

where and denote, respectively, the chemical potentials in the hypothetical surface and that in the bulk, and αk the molar surface area of pure component k (k = A, B). From Eq. (17) the expressions for the surface tension has been derived in the following forms involving the partial excess free energy of mixing in the bulk, GE,b k and that at the surface, GE,s k ; and surface concentrations in the bulk, xbk and that at the surface, xsk (k = A,B) [20]: σ ¼ σA þ

1 αA

σ ¼ σB þ

1 αB

" E;s

E;b

E;s

E;b

GA −GA þ RT ln

" GB −GB þ RT ln

xsA xbA

s

xB xbB

# ð18aÞ

# ð18bÞ

¼β

E;b Gk :

ð19Þ

The value of β is assumed a constant number in the calculations. The area of the monatomic surface layer for the component k is commonly calculated from the following relation [19]: 1=3

α k ¼ 1:091 NA Ωk

2=3

ð20Þ

where NA is Avogadro's constant and Ωk is the molar volume of the component k. The molar volume of a component can be calculated from its molar mass and density. Eqs. (18a) and (18b) are solved together for the surface concentration and the surface tension of a binary liquid solution is estimated. (ii) For the comparison of the results, the second approach from which we compute the surface properties is the quasi-chemical method. In this method the surface tension of binary liquid alloys can be computed by simultaneously solving the following pair of equations [14]: " σ ¼ σ A þ Γ1 þ Y

p ln

#  ð1 þ λÞ λs þ xsA −xsB ðλ þ xA −xB Þ −q ln ð1 þ λÞxA ð1 þ λs Þ ðλ þ xA −xB Þ ð21aÞ

" σ ¼ σ B þ Γ2 þ Y

Γ1 ¼

kB T ð2−pZ Þ xS ln A ; 2α xA

Γ2 ¼

kB T ð2−pZ Þ xS ln B ; 2α xB

#  ð1 þ λÞ λs þ xsB −xsA ðλ þ xB −xA Þ −q ln p ln ð1 þ λÞxB ð1 þ λs Þ ðλ þ xB −xA Þ ð21bÞ

Y¼

Z kB T : 2α ð21cÞ

Here Z is co-ordination number in the bulk and α is the mean atomic surface area as given by α¼

X

xk α k

ðk ¼ A; BÞ

ð22Þ

where the atomic area of hypothetical surface for each component is calculated as [18]: −2=3

Vk

2=3

:

ð23Þ

Here Vk is the atomic volume of species k and NA stands for Avogadro's number. p and q are surface coordination fractions which are defined as the fractions of the total number of nearest neighbours made by an atom within its own layer and that in the adjoining layer. These are related to each other through the relation, p + 2q = 1. The ordering energy term W is a temperature dependent quantity and for an alloy at a given temperature, it can be estimated from thermodynamic considerations. The function λ is concentration dependent auxiliary variable defined in Eq. (14). The function λs for surface is obtained from Eq. (14) by replacing the bulk concentrations by respective surface concentrations and the coordination number in the bulk Z by coordination number of the surface atoms, Zs. Here Zs is calculated from the relation [8]: s

Z ¼ ðp þ qÞZ:

where σA and σB are surface tension of pure metals A and B, respectively. The partial excess free energy of mixing of component k at the surface is related to that in the bulk through a parameter β as: E;s Gk

where,

α k ¼ 1:102 NA

μ bk

11

ð24Þ

In order to understand the nature of atomic ordering on the surface of the liquid alloys, the concentration fluctuations at the surface Sscc(0) can be computed using the following expression [8]:  s s s s

s  s −1 : Scc ð0Þ ¼ xA xB 1 þ z =2λ 1−λ

ð25Þ

3. Results In general the mixing properties of binary metal alloys vary as a function of composition, temperature and pressure. As the study of the liquid alloys is usually carried out at the fixed pressure, most likely, the atmospheric pressure, the thermo-physical properties of a binary liquid alloy at a given temperature vary with the composition of the alloy. In the present work, we have studied the composition dependence of the thermodynamic and surface properties of Co–Fe and Fe–Pd liquid alloys at constant temperature, the results of which are discussed in the following section. 3.1. Thermodynamic properties The fundamentals of mixing of liquid metals forming binary alloys are generally discussed in terms of the enthalpic and entropic functions. To determine these functions in the framework of QCA, the model fit parameters γ and W are needed. These parameters were estimated for Co–Fe alloy at 1863 K and Fe–Pd alloy at 1873 K by method of successive approximation using their experimental data of free energy of mixing (GM) and chemical activities (ai) [21] in Eqs. (1), (4) and (5) over the whole range of concentration. The calculated values of the free energy of mixing of the alloys are compared with the experimental values [21] in Fig. 1. The plot shows good agreement between the two sets of the values. The theory predicts the minimum values of free energy of mixing, GM (=−0.629RT for Co–

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R.P. Koirala et al. / Journal of Non-Crystalline Solids 394–395 (2014) 9–15

Fig. 1. Free energy of mixing (GM), heat of mixing (HM) and entropy of mixing (SM) versus bulk concentration (xA): (a) Co–Fe liquid alloy at 1863 K (b) Fe–Pd liquid alloy at 1873 K [Theory — solid curve; Experiment — circles].

Fig. 2. Activities (aA and aB) versus bulk concentration (xA): (a) Co–Fe liquid alloy at 1863 K (b) Fe–Pd liquid alloy at 1873 K [Theory — solid curve; Experiment — circles].

Fe alloy and −0.478 RT for Fe–Pd alloy) at the equi-atomic composition which are close to the experimental results (−0.605 RT for Co–Fe alloy and −0.483 for Fe–Pd alloy). The plot for the chemical activities of the alloy components shows that there is a good agreement between the sets of computed and experiment results [Fig. 2]. As in the free energy of mixing for the liquid alloys, the data for the chemical activity are found better matching for the Fe–Pd alloy than Co–Fe alloy. For the computation of the entropy of mixing (SM) and the enthalpy of mixing (HM) of the alloys using QCA, the temperature derivative of order energy term (∂W/∂T) is required. It was estimated by the iterative procedure using the experimental data [21], and the entropy of mixing and the heat of mixing were computed respectively from Eqs. (7) and (8). W The values of the QCA parameters γ, RT and R1 ∂W estimated for the two ∂T alloys are presented in the following Table.

at a whole range of concentrations from Eq. (10). To maintain consistency in the calculations we have used the same values of the ratio of self associates and the order energy term as those estimated for the computation of thermodynamic properties. The experimental Scc(0) were computed from Eq. (12). The theoretical and experimental values along with the ideal values of Scc(0) are plotted in Fig. 3. The peak value of Scc(0) occurs around the equi-atomic composition. The calculated and observed values of Scc(0) are close to the ideal values towards either end of compositions and at intermediate composition the Scc(0) values are all greater than ideal values. The structure regarding the atomic arrangements in the two alloys was further accessed by calculating the Warren–Cowely short range ordering parameter, α1 using Eq. (16). This parameter was found positive at all concentrations.

Alloy

T

γ

W/RT

1 ∂W R ∂T

Co–Fe Fe–Pd

1863 K 1873 K

2.75 0.92

+0.510 +0.899

+0.937 +0.240

The heat of mixing of the Co–Fe alloy is found negative whereas in the Fe–Pd alloy it is positive at all compositions. Moreover, both the heat of mixing and entropy of mixing in the former are observed asymmetrical about the equi-atomic composition but almost symmetrical in the latter. 3.2. Microscopic functions In order to ascertain the nature of atomic ordering in the alloys the values of the microscopic function Scc(0) of the alloys were computed

3.3. Surface properties To examine the surface properties of a binary liquid alloy at a temperature, the surface tensions and densities of the constitute metals are required at that temperature. The density (ρ) and surface tension (σ) of metals Co, Fe and Pd were separately calculated at the temperature of investigation (T) for both the alloys from the following relations [22]: Metal

Density

Surface Tension

Co Fe Pd

ρ = 7760 − αo (T − To) kg/m3 ρ = 7015 − αo (T − To) kg/m3 ρ = 10490 − αo (T − To) kg/m3

σ = 1873 − γo (T − To) mN/m σ = 1872 − γo (T − To) mN/m σ = 1500 − γo (T − To) mN/m

R.P. Koirala et al. / Journal of Non-Crystalline Solids 394–395 (2014) 9–15

13

Fig. 3. Concentration fluctuations in long wavelength limit, Scc(0) and short range order parameter, α1 versus bulk concentration (xA): (a) Co–Fe liquid alloy at 1863 K (b) Fe–Pd liquid alloy at 1873 K [Theory — solid curve; Experiment — circles; Ideal — broken curve; Thin broken curve — α1].

The values of the melting temperature (To), the temperature coefficients of density (αo) and temperature coefficient of surface tension (γo) of the metals were taken from Smithells metals reference book [22]. In Butler's model, the area of the hypothetical mono-atomic surface layer was calculated for each of the two alloys from Eq. (20) using the densities of the metals at the temperature T. Taking the partial excess free energy of each component in the bulk [21], Eqs. (18a) and (18b) were simultaneously solved for the surface compositions for both Co– Fe and Fe–Pd liquid alloys. The plot of surface composition as a function of bulk composition is depicted in Fig. 4. The surface tensions of the liquid alloys were then computed as function of bulk concentration from Eqs. (18a) and (18b). In order to ascertain the reliability of the results from theoretical calculations, we need experimental data of the surface tension of the alloys. Due to unavailability of experimental data of the surface tension of the alloys at hand, we could not compare our result obtained from the previous model. An alternative to this is to use some other theoretical models. Here we used Eqs. (21a) and (21b), of quasi-chemical approach and solved them together for the surface composition by considering the effects of coordination number in the bulk and that at the surface. The introduction of Zs in the formulation is expected to yield better results. In the calculations the surface coordination fractions of the component metals were taken as p = 0.5 and q = 0.25, the surface coordination number for either alloy was calculated from Eq. (24). The value of energy parameter, W needed in the calculations for each alloy was taken from thermodynamic calculations of QCA. Next the surface tension values of the alloys were calculated from either of the Eqs. (21a) and (21b). The surface tensions of the alloys computed from the two approaches are plotted in Fig. 5. The nature of atomic arrangements on the surface of the liquid alloys were investigated by estimating surface Scc(0) from Eq. (25), using

Fig. 4. Surface concentration (xsA) versus bulk concentration (xA): (a) Co–Fe liquid alloy at 1863 K (b) Fe–Pd liquid alloy at 1873 K [Butler's model — solid curve; Quasi-chemical approach — circles].

the same energy parameter used in previous calculations. The plot of surface Scc(0) of the alloys as function of bulk concentration is shown in Fig. 6.

4. Discussion 4.1. Thermodynamic properties The free energy of mixing is an important thermodynamic property which gives an idea about atomic bonding and stability of the alloys. The computed values of free energy of mixing of the Co–Fe and Fe–Pd liquid alloys are in good agreement with the observed values [21]. The uncertainty in the experimental data in integral free energy of mixing is ± 418 J mol− 1 at xCo = 0.5 for Co–Fe alloy and ± 209 J mol− 1 at xFe = 0.5 for Fe–Pd alloy [21]. The other experimental values of free energy of mixing were calculated from the selected values using Gibbs–Duhem relation [21]. The positive values of W in both the alloys indicate that there is a tendency of homo-coordination of atoms in the alloys. Furthermore the Fe–Pd system is found to be more interacting than the Co–Fe system. This can be attributed to larger difference in the electro-negativity values of the component metals in the Fe–Pd system as compared to the Co–Fe system. Further the values of γ for the alloys indicate stronger associations among Fe atoms resulting in greater number of atoms in the cluster of Fe than that in the cluster of Co atoms in Co–Fe alloy and of Pd atoms in Fe–Pd alloy. The heat of mixing contains information about the nature of binding in the liquid alloys. Similarly, entropy of mixing helps to draw important

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R.P. Koirala et al. / Journal of Non-Crystalline Solids 394–395 (2014) 9–15

In order to examine how well the QCA model explains the alloying properties, the method of regression analysis, a data generating process, was used to fit fourth order polynomials to the sets of calculated and experimental data for the thermodynamic functions GM, SM and HM of the alloys. The peak values of GM, SM and HM of the alloys predicted by the method of interpolation are compared in the following table. GM (J mol−1)

Alloy Co–Fe

Fe–Pd

−9680.30 at XA = 0.47 Experimental −9368.01 at XA = 0.49 Calculated −7386.50 at XA = 0.52 Experimental −7499.82 at XA = 0.49

Calculated

SM (J mol−1 K−1)

HM (J mol−1)

4.006 XA = 3.762 XA = 5.237 XA = 5.186 XA =

−2594.56 at XA = 0.65 −2594.64 at XA = 0.59 2458.44 at XA = 0.47 2212.20 at XA = 0.49

at 0.38 at 0.41 at 0.49 at 0.49

This list shows some asymmetry in the thermodynamic properties of the Co–Fe alloys and very small asymmetry in the Fe–Pd alloy, about the equi-atomic composition. The asymmetry may arise due to the interaction between atoms having unequal number of valence electrons (e.g., vCo − vFe = 1) or due to interaction between atoms of different sizes (e.g., VFe/VPd ≈ 0.79) in the alloys. From the polynomial fits the coefficients of determination (r2) were obtained for the thermodynamic properties GM, SM and HM of the alloys which are displayed in the following table. The coefficients of determination can be examined to see how well the regression curve represents the data. The range for the coefficient of determination is 0 b r2 b 1.

Fig. 5. Surface tension (σ) Vs bulk concentration (xA): (a) Co–Fe liquid alloy at 1863 K (b) Fe–Pd liquid alloy at 1873 K [Butler's model — solid curve; Quasi-chemical approach — circles].

structural information of binary liquid alloys and to determine the equilibrium state of the mixture. Our study reveals that the theoretical data for the entropy of mixing and the heat of mixing are found in better agreement with the experimental results, obtained from direct reaction calorimetry methods, for the Fe–Pd system as compared to that for the Co–Fe system. The uncertainty in the experimental data in integral heat of mixing and entropy of mixing are, respectively, ±418 J mol−1 and ±0.251 J mol−1 K−1at xCo = 0.5 for Co–Fe alloy, and ±1463 J mol−1 and ±0.084 J mol−1 K−1 at xFe = 0.5 for Fe–Pd alloy [21]. The remaining experimental values of heat of mixing were calculated from the selected values using Gibbs–Duhem relation [21]. The computed values of heat of mixing of the liquid alloys indicate that the alloys are weakly interacting systems. The behaviour of entropy is often complicated for many of the binary liquid alloys and plays important role to determine the magnitude and sign of the enthalpic effects. Calculations of activity show that the theoretical and experimental data are in good agreement. Activity represents the measure of tendency of a component to leave the solution. The deviations in the nature of a solution from ideal behaviour can be incorporated into activity. The magnitudes of activities are assumed to depend on the interactions among the constituent species of the system i.e. on the bond energies. The knowledge of activity for a class of similar binary systems is a basis to understand correlation in their behaviour and therefore can also be used for the prediction of the behaviour of more complex systems.

Fig. 6. Surface Scc(0) Vs bulk concentration (xA): (a) Co–Fe liquid alloy at 1863 K (b) Fe–Pd liquid alloy at 1873 K.

R.P. Koirala et al. / Journal of Non-Crystalline Solids 394–395 (2014) 9–15 Coefficient of determination (r2)

Alloy

SM

GM

Co–Fe Fe–Pd

15

HM

Calculated

Experimental

Calculated

Experimental

Calculated

Experimental

0.9993 0.99873

0.99941 0.99861

0.9988 0.99928

0.99902 0.99879

0.9998 0.9991

0.9985 1

The values of r2 for the properties suggest that about 99.9% of the total variation in the properties can be explained by the regression equations. The other about 0.1% of the total variation in the properties remains unexplained. This means the QCA well explains the thermodynamic properties of the alloys. 4.2. Structural properties The mixing properties of binary alloys in liquid state are considered to be due to energetic and structural readjustment of the constituent atoms. The structural information about the liquid alloys can be derived from the values of the microscopic functions, Scc(0) and α1. Our calculations of Scc(0) and α1 suggest that both the alloys are segregating in nature. It may be worthwhile to note that for larger γ, tendency of ordering is significant even if the order energy term W is positive [18]. ΩFe Here for the Co–Fe alloy, γ ¼ Ω N1. For lower concentration of Fe, the Co role of γ is not significant and hence the Co–Fe alloy behaves slightly ordering in nature below the composition xA = 0.4. On the other hand γ plays a dominant role at higher concentration of Fe, where theoretical values of Scc(0) are found greater than the ideal values [Fig. 3a] and the alloy behaves as segregating. In Fe–Pd alloy at 1873K, the Scc(0) values indicate relatively larger segregation at all compositions [Fig. 3a,b]. The short range order parameters computed for the alloys also indicate that both Co–Fe and Fe–Pd systems are segregating, with comparatively higher degree of segregating tendency in Fe–Pd system. In calculations of the short range order parameter the value of the coordination number Z was taken 10. It is to be noted that on varying the value of Z, the depth of α1 changes but the overall nature, along with the position of minima, is retained. 4.3. Surface properties The structural analysis shows that in Co–Fe system at 1863 K, the surface layer is slightly enriched with Co-atoms at lower bulk concentrations of Co-atoms. In Fe–Pd system at 1873 K surface segregation of Pd-atoms is pronounced over the entire range of bulk composition. The result shows that surface tension of the alloy decreases with increase in the bulk concentration of the metal with smaller surface tension. The calculation of surface Scc(0) indicates that maximum surface segregation occurs around the compositions Co0.5Fe0.5 and Fe0.8Pd0.2 of the alloys. The tendency of surface segregation is noticed higher in the Fe–Pd system as compared to the Co–Fe system. This indicates that the surface segregation is pronounced when the electro-negativity difference between the metals in the alloy is large. It may be proper to point out that when there is a larger difference in surface tensions of metals, the mixing shows strong segregation of the component with smaller surface tension at the surface and the other in the bulk. On the other hand, with very comparable values of the surface tension of the metals surface compositions do not differ much from the bulk compositions. 5. Conclusion The theoretical study shows positive deviation in the mixing properties from the Raoultian behaviour indicating segregation in Co–Fe and

Fe–Pd alloys. There is higher degree of segregation in Fe–Pd system at 1873 K than that in the Co–Fe system at 1863 K. When there is a larger difference in surface tension between the solvent and solute, the segregating behaviour of the component is more significant. The surface tension study under different assumptions shows that with the addition of Fe-component in the alloys, the surface tension increases both in Co–Fe and Fe–Pd alloys with reasonably good consistency. Furthermore, the surface segregation of the atoms of the metal with lower surface tension (Pd atoms in Fe–Pd alloy and Co atoms in Co–Fe alloy) is pronounced. Maximum surface segregation is expected to occur in the Co–Fe and Fe–Pd alloys around the compositions xCo = 0.5 and xFe = 0.8, respectively. Acknowledgements We are grateful to Prof. L.N. Jha (Former Head, Central Department of Physics, T.U., Kathamandu), Prof. Pradeep Raj Pradhan (Post-Graduate Department of Physics, M.M.A.M. Campus, T.U., Biratnagar) and Dr. I.S. Jha (Post-Graduate Department of Physics, M.M.A.M. Campus, T.U., Biratnagar) for fruitful suggestions and inspiring discussions. References [1] A.A. Couto, P.I. Ferreira, J. Mater. Eng. 11 (1989) 31, http://dx.doi.org/10.1007/ BF02833751. [2] M.A. Vasiliev, J. Phys. D. Appl. Phys. 30 (1997) 3037, http://dx.doi.org/10.1088/00223727/30/22/002. [3] Y.A. Sonvane, J.J. Patel, Pankajsinh B. Thakor, P.N. Gajjar, A.R. Jani, Adv. Mater. Res. 665 (2013) 143, http://dx.doi.org/10.4028/www.scientific.net/AMR.665.143. [4] T. Nishizawa, K. Ishida, J. Phase Equilib. 5 (1984) 250, http://dx.doi.org/10.1007/ BF02868548. [5] W. Gasior, P. Fima, Z. Moser, Arch. Metall. Mater. 56 (2011) 13, http://dx.doi.org/10. 2478/v10172-011-0002-3. [6] S.G. Mayr, A. Arabi-Hashemi, New J. Phys. 14 (2012) 1, http://dx.doi.org/10.1088/ 1367-2630/14/10/103006. [7] Ermete Antolini, Energy Environ. Sci. 2 (2009) 915, http://dx.doi.org/10.1039/ b820837a. [8] B.C. Anusionwu, C.A. Madhu, C.E. Orji, PRAMANA J. Phys. 72 (2009) 951, http://dx. doi.org/10.1007/s12043-009-0088-6. [9] R.N. Singh, Can. J. Phys. 65 (1987) 309, http://dx.doi.org/10.1139/p87-038. [10] F. Sommer, J. Non-Cryst. Solids 117 (1990) 505, http://dx.doi.org/10.1016/00223093(90)90580-F. [11] L.C. Prasad, R.N. Singh, G.P. Singh, 27 (1994) 179, http://dx.doi.org/10.1080/ 00319109408029523. [12] D. Adhikari, B.P. Singh, I.S. Jha, J. Non-Cryst. Solids 358 (2012) 1362, http://dx.doi. org/10.1016/j.jnoncrysol.2012.03.008. [13] J.A.V. Butler, Proc. R. Soc. A 135 (1932) 348, http://dx.doi.org/10.1098/rspa.1932. 0040. [14] R. Novakovic, J. Non-Cryst. Solids 356 (2010) 1593, http://dx.doi.org/10.1016/j. jnoncrysol2010.05.055. [15] A.B. Bhatia, R.N. Singh, Phys. Chem. Liq. 11 (1982) 285, http://dx.doi.org/10.1080/ 00319108208080752. [16] B.E. Warren, X-ray Diffraction, Addison-Wesley, Reading, MA, 1969. 227. [17] J.M. Cowley, Phys. Rev. 77 (1950) 669, http://dx.doi.org/10.1103/PhysRev.77.669. [18] R.N. Singh, F. Sommer, Rep. Prog. Phys. 60 (1997) 57, http://dx.doi.org/10.1088/ 0034-4885/60/1/003. [19] D. Adhikari, I.S. Jha, B.P. Singh, Adv. Mater. Lett. 3 (2012) 226, http://dx.doi.org/10. 5185/amlett.2012.3324. [20] J. Lee, W. Shimoda, T. Tanaka, Mater. Trans. 45 (2004) 2864, http://dx.doi.org/10. 2320/matertrans.45.2864. [21] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelly, Selected Values of the Thermodynamic Properties of Binary Alloys, ASM, Metal Park, 1973. [22] Brandes, Smithells, Metals Reference Book, Sixth ed., 1983. (Sec. 14-6.)