Theoretical Analysis of The Thermodynamic Behaviour of Fe-Ni-Pd Melts at 1873K

44 | P a g e DOI: 10.5281/zenodo.17374650 Theoretical Analysis of The Thermodynamic Behaviour of Fe-Ni-Pd Melts at 1873K N.K. Roya, A.P. Singhb, N. Kumaric, R.P. Chaudharyd, R. K. Sahua, J. Mandale*, I. S. Jhad aUniversity Department of Chemistry, T.M. Bhagalpur University, Bhagalpur, 812007, India bMarwari College, T.M. Bhagalpur University, Bhagalpur, 812007, India cGovernment Polytechnic College , Purnea, Bihar, India dDepartment of Physics, M.M.A.M. Campus, Biratnagar, T.U., Nepal e*University Department of Physics, T.M. Bhagalpur University, Bhagalpur, 812007, India anabin[email protected], [email protected], [email protected], dr.p.cha[email protected], [email protected] e*jmanda[email protected]m, d[email protected]m Abstract The concentration dependent thermodynamic behaviour of Fe-Ni-Pd melts at 1873 K has been analysed in terms of the excess free energy of mixing, free energy of mixing, activity coefficient and activity of first element i.e. Fe on applying MIVM known as molecular interaction volume model. The computations of the mentioned thermodynamic functions require the input parameters which are ascertained from the parameters needed for the concerned binary system at 1873 K. The theoretical values for the excess free energy of mixing and activity computed from the analytical expressions deduced in the framework of MIVM, are in well agreement with the corresponding experimental results. The concentration dependence of the excess free energy of mixing, free energy of mixing, activity coefficient and activity of first element i.e. Fe for Fe-Ni-Pd system at 1873 K for three cross-sections namely : Ni Pd xx = 1:1, 1:2 and 1:3 indicate that Fe-Ni-Pd melts at 1873 K is more stable for cross-section 1:1 followed by 1:2 and 1:3 respectively. Keywords: MIVM model; excess free energy of mixing; activity coefficient; activity; free energy of mixing; FeNi-Pd melts 1. Introduction Fe-Ni-Pd liquid alloy is composed of three constituents like Fe, Ni and Pd. Obviously, three binaries namely FeNi, Fe-Pd and Ni-Pd are associated with Fe-Ni-Pd alloys. The Fe-Ni-Pd system has wide range of applications in material science as well as geoscience as high temperature alloys component of earth’s core, magnetic material etc. Therefore, Fe-Ni-Pd system become suitable for welding, casting, high temperature applications etc. Earlier, the morphological, crystallographical and martenstic characteristics of Fe-Ni-Pd system has been investigated experimentally [1]. The characteristics of Fe-Ni-Pd system relies upon the composition of phase relationship. 45 | P a g e DOI: 10.5281/zenodo.17374650 Therefore, the experimental studies of the phase diagram have been performed for Fe-Ni-Pd, and Fe-Ni-Pdbased multicomponent alloys namely Fe-Ni-Pd-S [2,3,4,5]. However, there is scarcity of theoretical an experimental data of the thermodynamics of Fe-Ni-Pd melts in the literature. Thus, Fe-Ni-Pd liquid alloy becomes a suitable candidate for the theoretical exploration. For the understanding of the thermodynamic behaviour of Fe-Ni-Pd alloys, it is necessary to have the knowledge of the mixing properties of the concerned binary alloys like Fe-Ni, Fe-Pd and Ni-Pd. The concentration dependent thermodynamic properties refer to the segregating nature of Fe-Pd melts at 1873 K and Ni-Pd melts at 1873 K as well as ordering nature of Fe-Ni melts at 1873 K [6]. It is interesting to mention that Fe-Pd system possess chemical complexes like FePd, FePd3, Fe2Pd and FePd2 [6, 7]. The interesting thermodynamic behaviour and potential applications i.e. in aerospace, telecommunications, electronics, gas turbine engines, magnetostrictive technologies, spintronic, biomedical applications, electrochemistry, as catalyst etc. of Fe-Ni melts, Fe-Pd melts and Ni-Pd melts have drawn the attentiveness of several researchers [8, 9, 10, 11, 12, 13, 14]. Therefore, the ternary system i.e. Fe-Ni-Pd melt is an interesting material which requires a theoretical exploration. In present research work, MIVM approach [15] is employed for the analysis of the thermodynamic properties namely excess free energy of mixing, E M G , free energy of mixing, M G , activity coefficient and activity of Fe-NiPd melts at 1873 K. Earlier, the thermodynamics of several binary liquid alloys such as Zn-Bi, Au-Ni, Au-Cu, Au-Pd, Ti-Al, Zn-Cd etc [17,18,19,20] and several ternary liquid alloys namely In-Bi-Sn, Al-Sn-Zn, Sn-Ag-Cu etc[21,22,23,24] by MIVM model. 2. Formalism The fluid based model, MIVM (i.e. molecular interaction volume model) has been described by Tao, 2000 [15] and obtained from statistical thermodynamics, fluid-phase equilibria and the fundamental concept of the nonrandom exchange of liquid molecules. The molecular excess free energy of mixing of the ternary molten alloys (1-2-3) is expressed as [15] 1 1 1 1 2 2 21 3 3 31 E m M m m m V Gx RT xV x V A x V A  = ++  2 2 2 2 1 1 12 3 3 32 ln m m m m V xx V xV A x V A  + ++  3 3 3 3 1 1 13 2 2 23 ln m m m m V xx V xV A x V A  + ++  2 21 21 3 31 31 11 1 2 21 3 31 1 12 12 3 32 32 22 1 12 2 3 32 1 13 13 3 23 23 33 1 13 2 23 3 ln ln ln ln 1 2 ln ln x A A x A A Zx x x A x A x A A x A A Zx x A x x A x A A x A A Zx x A x A x   +   ++     + −+   ++     +  + ++    (1) where, i Z (i = 1, 2, 3) stands for the first coordination numbers for the constituent molecules 1,2 and 3 respectively, i x ( i = 1,2,3) represents the concentration (or mole fraction) of the constituents 1,2 and 3 respectively 12 A , 21 A ; 23 A , 32 A ; and 31 A , 31 A denote the pair-potential energy interaction parameters between the constituent 46 | P a g e DOI: 10.5281/zenodo.17374650 elements 1-2, 2-3 and 1-3 for the concerned binary systems (1-2), (2-3) and (1-3) respectively, 1m V , 2m V and 3m V molar volume of elements 1, 2 and 3 respectively. Since, the ternary system (1-2-3) is composed of three binary systems (1-2), (2-3) and (1-3). Therefore, the potential energy interaction parameters are determined for each binary system. Let us consider a binary system ( ij− ) for which the expression molar excess free energy of mixing may be represented by [15] E GM RT = ln ln V Vmj mi xx ij x V x V A x V x V A i mi j mj ji j mj i mi ij         +     ++     - 2 xx Z A lnA Z A lnA ij ji ji j ij ij i x x A x x A i j ji j i ij   +  ++  (2) where, xi and xj are the compositions (or mole fractions) of i and j components respectively, i Z and j Z stands for the first coordination numbers for the constituent molecules i and j respectively. ij A and ji A denote the pairpotential energy interaction parameters and mi V and mj V are the molar volumes of elements i and j respectively. Now, ij A and ji A may be expressed as [15] exp ji ii Aji kT  −  =−   and exp ij jj Aij kT  −  =−   (3) with, ii  , jj  and ji  refer to pair potential energies for ii− , jj− and ij− respectively, so that ji ij  = , k and T represent Boltzmann constant and absolute temperature of the binary liquid alloy respectively. Again, the value of co-ordination number of an element of the binary system can be determined from the relation given below 33 () 42 exp 3 H T T rr mi mi mi oi Zr i i mi Z RTT rr c mi mi oi  −  − =   −  (4) with ∆Hmi and Tmi are enthalpy at melting temperature and melting temperature respectively; C Z is taken to be 12 for closed packed structure; 0.6022 NV i i i Vmi  == = molecular number density; mi r and oi r = initial and fist peak values of radial distribution function of the molten metal i near to melting temperature respectively, and R is the molar gas constant. The radial distances are expressed as 0.918 cov rd oi i = and rmi i  = (5) where, I and dcov i = atomic diameter and atomic covalent diameter respectively. The activity coefficients of the component µ (= ,ij ) of a binary melt in terms of E M G can be expressed as [24] 47 | P a g e DOI: 10.5281/zenodo.17374650 ln (1 ) E EM M G RT G x x     = + −  Therefore, (1 ) ln EE MM x GG RT RT x     − =+  (6) From equation (6), the activity coefficients for constituents i and j of the binary melt ij− can be represented respectively as ln EE j MM i i x GG RT RT x   =+  (7) ln EE i MM j j x GG RT RT x   =+  (8) where, (1 ) ji xx=− and (1 ) ij xx=− Employing equation (2) in (7) and (8), we can obtain respectively ( ) ( ) 22 22 ln ln ln ln 2 mj ji mi ji j i ji ji j ij ji mi ij i mi j mj ji i mi j mj ji j mj i mi ij i j ji j i ij V A V A x Z A A Z A A Vx xV x V A xV x V A x V xV A x x A x x A        = + − − +          + + + ++      (9) ( ) ( ) 2 2 22 ln ln ln ln 2 mj mj ji mi ij j ij ij i ji ji i ji j mj i mi ij i mi j mj ji j mj i mi ij j i ij i j ji V V A V A Z A A Z A A x x x V xV A xV x V A x V xV A x x A x x A        = − − − +          + + + ++      (10) The equations (9) and (10) are the analytical expressions from which i  and j  of the binary melt ij− can be computed. From definition, the activity coefficient and activity of the component  of a binary melt are related as ax     = (where, ,ij  = ) (11) From equation (11), the activities of the components i and j of the binary melt ij− can be expressed as i i i ax  = (12) j j j ax  = (13) The equations (12) and (13) can be utilized to determine the activities i a and j a respectively of the binary melt ij− . 48 | P a g e DOI: 10.5281/zenodo.17374650 The expressions for activity coefficients of the components of the alloy in infinite dilute solution range i.e. i x or i x → 0, are, respectively, given by ( ) 1 ln 1 ln ln ln 2 V A V A mj ji mi ji Z A Z A A i i ji j ij ij VV mi mj   = − − − +    (14) and ( ) 1 ln 1 ln ln ln 2 V A V A mi ij mj ji Z A Z A A j j ij i ji ji VV mj mi    = − − − +   (15) The pair-potential energy interaction parameters, Aij and Aji can be determined by solving equation (14) and (15) on applying Newton-Rapson method. Now, for a ternary melt (1-2-3), the activity coefficient of the component i is expressed as [15] 1 2 3 13 ln EE EMM M GG RT G x x xx     = + −      (16) Using equation (1) in (16), we can obtain 1 1 1 1 1 2 2 21 3 3 31 1 1 2 2 21 3 3 31 ln 1 ln mm i m m m m m m V xV xV x V A x V A xV x V A x V A   = + −  + + + +  2 1 12 3 1 13 1 1 12 2 2 3 3 32 1 1 13 2 2 23 3 3 mm m m m m m m x V A x V A xV A x V x V A xV A x V A x V −− + + + + 1 2 21 3 31 2 21 21 3 31 31 2 1 2 21 3 31 2 2 21 2 3 32 12 3 32 32 2 1 12 2 3 32 3 3 13 2 23 3 13 2 23 23 2 1 13 2 23 3 ( )( ln ln ) () [( )ln ln ] 1 2 ( ) [( )ln ln ] () Z x A x A x A A x A A x x A x A Z x A x x A A x A A x A x x A Z x A x A x A x A A x A x A x  ++  ++  +−  −+  ++  +−  +  ++  (17) which is the analytical expression for the activity coefficient of the component i of a ternary system (1-2-3). Now, the activity coefficient and activity of the component 1 may be related as 1 1 1 ax  = (18) The equation (17) and (18) are utilized to determine the theoretical values of the activity of component i of a ternary system (1-2-3). Now, the free energy of mixing, M G for a ternary system may be represented as E id M M M G G G RT RT RT =+ (19) 49 | P a g e DOI: 10.5281/zenodo.17374650 Again, for a ternary system, the ideal free energy of mixing, id M G at a given temperature T may be represented by   1 1 2 2 3 3 ln ln ln id M G RT x x x x x x= + + (20) Using equation (1) and (20), we get M G RT = 1 1 1 1 2 2 21 3 3 31 m m m m V xxV x V A x V A   ++  2 2 2 2 1 1 12 3 3 32 ln m m m m V xx V xV A x V A  + ++  3 3 3 3 1 1 13 2 2 23 ln m m m m V xx V xV A x V A  + ++  2 21 21 3 31 31 11 1 2 21 3 31 1 12 12 3 32 32 22 1 12 2 3 32 1 13 13 3 23 23 33 1 13 2 23 3 ln ln ln ln 1 2 ln ln x A A x A A Zx x x A x A x A A x A A Zx x A x x A x A A x A A Zx x A x A x   +   ++     + −+   ++     +  + ++      1 1 2 2 3 3 ln ln lnx x x x x x+ + + (21) which is the required analytical expression for M G of a ternary liquid alloys. 3. Result and Discussion The analytical expressions (1), (21), (17) and (18) are employed to compute / E M G RT , / M G RT , i  and i a respectively of Fe-Ni-Pd melts at 1873 K. For calculations, the input parameters are required which are evaluated from the concerned binary system i.e. Fe-Ni melts at 1873 K, Fe-Pd melts at 1873 K and Ni-Pd melts at 1873 K. The necessary parameters of pure Fe, Ni and Pd are presented in Table 1 [25]. For a binary system i-j the coordination number Zi and Zj of the constituents are determined from equation (4) which are presented in Table 2. The values of ji A and ij A are determined on solving equations (14) and (15) by Newton-Rapson method by using the experimental values of infinite dilute activity coefficients i.e. i   and j   [6]. The values of ji A and ij A slightly changed [15] in order to achieve a good agreement between the theory and experiment [6] for the excess free energy of mixing, / E M G RT for the concerned binary alloys. The best fit values ji A and ij A at required temperature are depicted in Table 2. Table 1. Input parameters for the pure metals [25] Metal,i Hmi  (KJ/mol) i  (×10-8cm) oi r (×10-8 cm) Vmi (cm3/mol) Fe 13.77 2.56 1.98 7.94[1+1.3×10-4(T-1808)] Ni 17.15 2.50 1.88 7.43[1+1.3×10-4(T-1728)] Pd 16.7 2.75 2.56 10.14[1+1.17×10-4(T-1828)] Table 2: Input Parameters for Binary Systems ij− T (K) mi V mj V ij A ji A i Z j Z i   j   50 | P a g e DOI: 10.5281/zenodo.17374650 (cm3/mole) (cm3/mole) Fe-Ni 1873 8.01 7.59 1.1037 1.0191 10.00 9.74 0.355 0.617 Ni-Pd 1873 7.59 10.19 0.9757 0.9395 9.74 11.53 2.306 2.805 Fe-Pd 1873 8.01 10.19 0.8761 0.9915 10.00 11.53 1.546 1.683 3.1 Results for E M G and a  ( , )ij  = of Fe-Ni, Fe-Pd and Ni-Pd at 1873 K The equation (2) is used to ascertain the values of / E M G RT for Fe-Ni, Fe-Pd and Ni-Pd at 1873 K as a function of concentration, p x (p = Fe, Fe and Ni). The theoretical results and correlated experimental results [6] presented in Fig. 1, exhibit well harmony in the complete range of composition, x = 0.1 to 0.9. For Fe-Ni system at 1873 K, E M G RT vs Fe x isotherm denotes the negative values of E M G RT in the complete composition range of Fe i.e. Fe x = 0.1 to 0.9 with minima at Fe x = 0.4 i.e. – 0.1853 (Theory) and -0.1777 (Experiment). Thus, the asymmetry in E M G RT as well as ordering nature [26] of Fe-Ni melts at 1873 K are well explained theoretically. It is required to point out that the theoretical results for E M G RT exhibit the marked departures from the experimental results i. e in the regions 0.2  Fe x  0.4 and 0.6  Fe x  0.8. For Fe-Pd melts at 1873 K, in whole composition region, Fe x = 0.1 to 0.9, E M G RT exhibits positive value at each concentration with maximum values at Fe x = 0.5 i.e. 0.2119 (Theory) and 0.2102 (Experiment). Thus, MIVM successfully represents the symmetry in E M G RT and segregating character [26] of Fe-Pd melts at 1873 K. For Ni-Pd liquid at 1873 K, E M G RT are positive in the complete composition range of Ni i.e. Ni x = 0.1 to 0.9 with maxima at Ni x = 0.5 i.e. 0.1209 (Theory) and 0.1194 (Experiment). Thus, the symmetry in E M G RT as well as segregating nature [26] of Ni-Pd melts at 1873 K are successfully explained by MIVM. 51 | P a g e DOI: 10.5281/zenodo.17374650 0.0 0.2 0.4 0.6 0.8 1.0 -0.2 -0.1 0.0 0.1 0.2 GE M/RT xp (GE M/RT)Theory Fe-Ni (GE M/RT)[6] Fe-Ni GE M/RT)Theory Fe-Pd (GE M/RT)[6] Fe-Pd GE M/RT)Theory Ni-Pd GE M/RT)[6] Ni-Pd Fig. 1 GE M/RT vs xp (p = Fe, Fe, Ni) for Fe-Ni melts at 1873 K, Fe-Pd melts at 1873 K and Ni-Pd melts at 1873 K For the computations of activity of both the components of each alloy namely Fe-Ni at 1873 K, Fe-Pd at 1873 K and Ni-Pd at 1873 K from the equation (12) and (13), the theoretical values of activity coefficients of both the components of the concerned binary alloys are required. For this, the equations (9) and (10) are utilized to determine the theoretical values of both the components of the alloys Fe-Ni at 1873 k, Fe-Pd at 1873 K and NiPd at 1873 K. The theoretical results and corresponding experimental results [6] for both the components of the alloys under consideration are depicted in Fig. 2 with respect to concentration. For Fe-Ni liquid alloys at 1873 K, the theoretical results for Fe a and Ni a relative to Fe x are compared with the corresponding experimental results [6] in Fig. 2. A reasonable agreement is observed between theory and experiment. The negative deviations of Fe a and Ni a from ideal mixing behaviour in the concentration region Fe x = 0.1 to 0.9, advocate the ordering character [26] of Fe-Ni melts at 1873 K. It is pointed out that the activity is a fortunate thermodynamic function which is measured by the experiment [26]. Thus, the harmony between theory and experiment authenticates the evaluated values of ji A and ij A . Fig. 2 indicates that the theoretical values and correlated experimental values [6] for Fe a and Pd a of Fe-Pd System at 1873 K are in well agreement in the complete composition region, Fe x = 0.1 to 0.9. Again, the positive departures of Fe a and Pd a from Raoults law confirm the segregation [26] in Fe-Pd melts at 1873 K. The agreement between theory and experiment validates the estimated values of ji A and ij A . For Ni-Pd melts at 1873 K, an excellent similarity is noticed between theory and experiment [6] in the region, Ni x = 0.1 to 0.9 for Ni a and Pd a in Fig. 2. Again, the small positive departures of Ni a and Pd a from linear law confirm the feebly segregating nature [26] of Ni-Pd melts at 1873 K. Therefore, the agreement between theory 52 | P a g e DOI: 10.5281/zenodo.17374650 and experiment provides the authenticity of the evaluated values of ji A and ij A . 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Activity xp (aFe for Fe-Ni)Theory (aFe for Fe-Ni)[6] (aFe for Fe-Pd)Theory (aFe for Fe-Pd)[6] (aNi for Ni-Pd)Theory (aNi for Ni-Pd)[6] Ideal (aNi for Fe-Ni)Theory (aNi for Fe-Ni)[6] (aPd for Fe-Pd)Theory (aPd for Fe-Pd)[6] (aPd for Ni-Pd)Theory (aPd for Ni-Pd)[6] Ideal Fig. 2 Activity vs xp (p = Fe, Fe, Ni) for Fe-Ni melts at 1873 K, Fe-Pd melts at 1873 K and Ni-Pd melts at 1873 K 3.2 Results for / E M G RT , / M G RT , Fe  and Fe a of Fe-Ni-Pd Liquid Alloys at 1873 K In present study, MIVM model has been employed to investigate the thermodynamic behaviour of Fe-Ni-Pd melts at 1873 K in terms of / E M G RT , / M G RT , Fe  and Fe a as a function of concentration of Fe. For the computations of the above thermodynamic properties of Fe-Ni-Pd melts at 1873 K, the required input parameters are presented below (taken from Table 2): Fe V = 1m V = 8.01 cm3/mole Ni V = 2m V = 12.13 cm3/mole Pd V = 3m V = 10.19 cm3/mole Fe Z = 1 Z = 10.00 Ni Z = 2 Z = 9.74 Pd Z = 3 Z = 11.53 ij A = 12 A = 1.1037 ji A = 21 A = 1.0191 (for Fe-Ni system at 1873 K) jk A = 23 A = 0.9757 kj A = 32 A = 0.9395 (for Fe-Pd system at 1873 K) ik A = 13 A = 0.8760 ki A = 31 A = 0.9915 (for Ni-Pd system at 1873 K) Now, expression (1) is utilized to determine the theoretical values of / E M G RT relative to the concentration of Fe for Fe-Ni-Pd liquid alloys at 1873 K for the cross-sections namely, : Ni Pd xx = 1:1, 1:2 and 1:3. The theoretical values of / E M G RT are tabulated in Table 3 and displayed in Fig. 3. It is evident that for cross-sections, : Ni Pd xx = 1:1, 1:2 and 1:3, / E M G RT display the positive values in the complete region of concentration i.e. Fe x = 0.1 to 0.9 for Fe-Ni-Pd at 1873 K. Again, the maximum values of / E M G RT are 0.1595 at Fe x = 0.3 for cross-section : Ni Pd xx = 1:3 followed by 0.1397 at Fe x = 0.1 for cross-section : Ni Pd xx = 1:2 and 0.1009 at Fe x = 0.1 for cross-section : Ni Pd xx = 1:1 respectively. This indicates that Fe-Pd melt dominates over Fe-Ni melt for the cross- 59 | P a g e DOI: 10.5281/zenodo.17374650 From present theoretical study, it may concluded that • The theoretical results for / E M G RT and activity of both the components of the binary alloys namely Fe-Ni melts at 1873 K, Fe-Pd melts at 1873 K and Ni-Pd melts at 1873 K, exhibit good harmony with the corresponding experimental results. • The observed symmetries in / E M G RT for Fe-Pd at 1873 K and Ni-Pd at 1873 K as well as asymmetry in / E M G RT for Fe-Ni at 1873 K are successfully reproduced. • The sign of / E M G RT in the complete concentration region confirms that Fe-Pd at 1873 K and Ni-Pd at 1873 K are segregating in nature whereas Fe-Ni at 1873 K is an ordering system. • The positive departures from linear law of the activity of both the components i.e. Fe a and Pd a for Fe-Pd system at 1873 K; Ni a and Pd a for Ni-Pd system at 1873 K, indicate that Fe-Pd melts at 1873 K and Ni-Pd melts at 1873 K are segregating alloys while Fe-Ni melt at 1873 K is an ordered alloy because Fe a and Ni a exhibit negative deviations from linear law. • The concentration dependence of / E M G RT for Fe-Ni-Pd system at 1873 K exhibits positive value at each concentration in the region, Fe x = 0.1 to 0.9 for each cross-section namely, : Ni Pd xx = 1:1, 1:2 and 1:3.. • In the entire composition range, Fe x = 0.1 to 0.9, / M G RT of Fe-Ni-Pd melts at 1873 K exhibit negative value at each composition for cross-section under consideration. The magnitude of negative values of / M G RT justifies that Fe-Ni-Pd system at 1873 K is more stable for cross-section, : Ni Pd xx = 1:1 followed by 1:2 and 1:3 respectively. • For each cross-section, the activity of Fe for Fe-Ni-Pd system at 1873 K, indicate positive deviation from linear law. Thus, MIVM model is a reliable and suitable model to explain the thermodynamic behaviour of binary as well as ternary alloys. References [1] Yildiz, G., Yildiz, Y.G. and Nezir, S. 2013. Bull. Mater. Sci. 36(1): 93-97 [2] Raghavan, V., 1992, Ind. Int. Metal., Culcutta, 1049-1052 [3] Makovicky, E. and Karup-Moller, S. 2016. Canadian Mineralogist 54: 377-400 [4] Raghavan, V., 2004, J. 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