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BY-NC-ND 3.0 license Open Access Published by De Gruyter February 5, 2015

Numerical Prediction of the Thermodynamic Properties of Ternary Al-Ni-Pd Alloys

  • Maryana Zagula-Yavorska EMAIL logo , Jolanta Romanowska , Sławomir Kotowski and Jan Sieniawski

Abstract

Thermodynamic properties of ternary Al-Ni-Pd system, such as exGAlNPd, µAl(AlNiPd),µNi(AlNiPd) and µPd(AlNiPd) at 1,373 K, were predicted on the basis of thermodynamic properties of binary systems included in the investigated ternary system. The idea of predicting exGAlNiPd values was regarded as calculation of values of the exG function inside a certain area (a Gibbs triangle) unless all boundary conditions, that is values of exG on all legs of the triangle are known (exGAlNi, exGAlPd, exGNiPd). This approach is contrary to finding a function value outside a certain area, if the function value inside this area is known. exG and LAl,Ni,Pd ternary interaction parameters in the Muggianu extension of the Redlich–Kister formalism were calculated numerically using the Excel program and Solver. The accepted values of the third component xx differed from 0.01 to 0.1 mole fraction. Values of LAlNiPd parameters in the Redlich–Kister formula are different for different xx values, but values of thermodynamic functions: exGAlNiPd, µAl(AlNiPd), µNi(AlNiPd) and µPd(AlNiPd) do not differ significantly for different xx values. The choice of xx value does not influence the accuracy of calculations.

PACS.: 02.60.C6

Introduction

The knowledge of phase diagrams and thermodynamic properties of alloys is necessary for many metallurgical and diffusion controlled processes. The possibility of reactions, the stability of phases, the reaction paths, the diffusion paths and many others physical properties depend on it. The aim of thermodynamic investigation is determination of partial and integral quantities of the system depending on concentrations of elements, pressure and temperature. Various methods are available and can be chosen depending on the system and on the properties that shall be determined [1]. Phase diagrams are elaborated by experimental methods such as: thermal analysis, microstructure examination, pressure measurements and others. However, the experimental determination of phase diagrams is a time-consuming and costly task, because the number of possible systems increases drastically with the number of elements. Experimental information for entire phase diagrams is available for most of the binary systems, but experimental information becomes increasingly sparse as the number of constituent elements increases (for ternary, quaternary and higher order systems) [2]. In this context, it is useful to estimate thermodynamic data of multicomponent systems from the constituent binary systems. There are many methods of modeling thermodynamic properties and calculations of phase diagrams in complex systems on the basis of thermodynamic properties of binary alloys constituting the complex alloy. Geometrical models may be used for prediction of excess Gibbs energies of a ternary homogenous solution from the corresponding binary data.

Methods of extrapolating thermodynamic properties of alloys into multicomponent systems are based on the summation of the binary and ternary excess parameters. The formula for performing this is based on various geometrical weightings of the mole fractions. Binary compositions are chosen by using geometric correlations in an isothermal Gibbs triangle [3]. The ternary excess Gibbs energy is calculated as a sum of binary excess energies multiplied by weighting factors ωij.

(1)exGijk=ωij·exGij+ωik·exGik+ωjk·exGjk

where the weighting factor ωij is:

(2)ωij=NiNjxi(j)xj(i)

where Ni is the ternary mole fraction component i and xi(j) is the binary mole fraction of component i in the i–j. Compositions of binary alloys are chosen using different models [3].

The other approach is a semi-empirical Calphad method [46]. It is a combination of experimental observation and theoretical modeling and depends on the quality of available experimental data. This approach is based on the modeling of multicomponent systems starting from pure components followed by the more complex ones (binary and ternary). The basic mathematical method is a minimization of the excess Gibbs energy for a given temperature, pressure and overall composition. This approach is common to all currently available software packages for the modeling of thermodynamic properties and phase diagrams of multicomponent systems [6].

The excess Gibbs energy exG describes the influence of non-ideal mixing behavior on the thermodynamic properties of a solution phase. The Muggianu [7] extension of the Redlich–Kister formalism [8] is a widely accepted description of the excess Gibbs energy:

(3)exG=xixjz=0mzLij(xixj)z+xixkz=0mzLik(xixk)z+xjxkz=0mzLjk(xjxk)z+ijknxixjxkLijk,z=0,1,,m,

where zLij are binary and Lijk are ternary temperature-dependent interaction parameters optimized on the basis of the available thermodynamic and phase diagram data.

(4)Lijk=0Lijk·xi+1Lijk·xj+2Lijk·xk

The knowledge of the values of excess Gibbs energy and chemical potentials of components in multicomponent systems is essential for many purposes. One of them is the simulation of the diffusion process. The chemical potential is the driving force of the diffusion process. Very often there is no experimental thermodynamic data of the multicomponent system in which the diffusion process takes place, therefore it is impossible to calculate the excess Gibbs energy and chemical potentials of components in the multicomponent system. In such a case, thermodynamic properties of a ternary system are predicted on the basis of thermodynamic properties of binary systems included in the investigation ternary system.

Diffusion takes place in many technological processes. One of them is formation of protective coatings. Aluminide diffusion coatings are deposited on the turbine blades in the engine hot section. The blades are made of nickel superalloys. Aluminide coatings protect nickel superalloys against high temperature oxidation and hot corrosion [9, 10] Unfortunately, theirs long-term oxidation resistance and adherence to the substrate are not sufficient. Therefore aluminide coatings are modified by platinum or palladium [11, 12]. Platinum improves stability of the coating microstructure by eliminating chromium-rich precipitation from the outer coating layer and prevents diffusion of refractory elements, such as molybdenum, vanadium and tungsten into the outer layer. Palladium-modified aluminide coatings have higher oxidation and hot corrosion resistance than the conventional aluminide coatings [12]. Palladium-modified aluminide coatings are used as bond coatings for thermal barrier coatings (TBC) [12].

The technological process of the coatings’ deposition will be better elaborated if the diffusion of elements forming the coating is known. As to simulate the diffusion of a certain element, the value of the chemical potential of this element in the system in which diffusion takes place should be known. In a case of the Al-Ni-Pd system there is no experimental data about values of thermodynamic functions. This paper presents a numerical approach to predicting thermodynamic properties of ternary Al-Ni-Pd systems on the basis of thermodynamic properties of binary systems included in the investigated ternary system.

The idea of predicting exGijk values is regarded as calculation of values of exG function inside a certain area (a Gibbs triangle) unless all boundary conditions, that is values of exG on all sides of the triangle, are known (exGij, exGik, exGjk). This approach is contrary to finding a function value outside a certain area, if the function value inside this area is known (this issue is well known in mathematics). In this approach, weighting of each mole fraction is the same. This model was successfully applied to Cu-Sn-Zn, Bi-Cu-Ni, Ag-Au-Bi, In-Sn-Zn and Cu-Fe-Sn alloys [13, 14] and results agree very well with the values obtained by the Calphad method.

Calculations

Values of the excess Gibbs energy of the Al-Ni-Pd system were predicted on the basis of the values of the excess Gibbs energy of the Al-Ni, Al-Pd and Ni-Pd systems. The applied approach is as follows:

Unless the formula (3) and zLij parameters are known, the idea of calculating Lijk parameters can be regarded as solving an eq. (3), when all boundary conditions (values of exGij, exGik, exGjk) are known. The approach proposed in this paper is as follows: if all boundary conditions that is exGij values on all sides of the Gibbs triangle (Figure 1) are known, a function value inside the triangle, exGijk can be found.

Figure 1 The Gibbs triangle.
Figure 1

The Gibbs triangle.

Calculations were performed using the Excel program and Solver. First, exGij values on all legs of the triangle were calculated and the concentration of the third component xx was accepted 0. Next, an assumption was made that exGij value on the leg of the triangle and for the small concentration of the third component xx was the same. Lijk were calculated on the basis of this assumption.

The concentration of the third component was assumed to be 0.1 mole fraction. In other words, exGij value for each point of the Gibbs triangle (bold line) and for the corresponding point of the inner triangle exG (dotted line) (see Figure 1) was assumed to be the same.

(5)exGij=exGijkforxk=0.1
(6)exGik=exGijkforxj=0.1
(7)exGjk=exGijkforxk=0.1
Lijk parameters were calculated numerically on the basis of this assumption using the Excel program and Solver.

Thermodynamic parameters for binary Al-Pd alloys were accepted from Li et al. [15], for binary Al-Ni alloy were accepted from Huang [16] and for Ni-Pd from Ghosh et al. [17].

Li et al [15] assessed the Al-Pd system by means of the Calphad technique. Phases of the system were thermodynamically modeled and the parameters of the excess Gibbs energy formula (3) Lij were optimized. The calculated phase equilibria and thermodynamic properties, including the phase diagram (Figure 2), agree well with the experimental data [15].

Figure 2 The Al-Pd phase diagram [15].
Figure 2

The Al-Pd phase diagram [15].

The diffusion process and formation of the palladium-modified aluminide coatings takes place at the temperature between 1,273 and 1,473 K, therefore calculations were performed for T = 1,373 K. At this temperature aluminum-rich alloys are liquid, whereas nickel-rich and palladium-rich alloys are solid (see Figures 2 and 3). As the diffusion process takes place in the solid state, calculations were performed for nickel-rich (fcc (γ)) and palladium-rich phases (fcc (Pd)). For the simplification and as an example of illustrating and idea of predicting, the calculations were performed for the whole concentration range, but for the further use only the range of high nickel or palladium concentration, in which phases accepted for calculations are stable, may be used. Thermodynamic parameters Lij for binary Al-Pd alloys, for the fcc phase, (Pd) were accepted from Li et al [15]:

(8)LAlPd=164947.99+23.32T+112770.3929.38TxAlxPd

Values of the LAlPd parameter for T = 1,373 K were inserted to eq. (3) as the Lij parameter.

Huang and Chang [16] elaborated the Al-Ni phase diagram (Figure 3) on the basis of experimental results and calculations. They modeled the fcc (γ) phase as a disordered solution and the substitutional solution model was applied.

Figure 3 The Al-Ni phase diagram [16].
Figure 3

The Al-Ni phase diagram [16].

Thermodynamic parameters Lik for binary Al-Ni alloys were accepted for the fcc (γ) phase [16]:

(9)LAlNi=168292+16T+32712xAlxNi+7998+35TxAlxNi2

Values of the LAlNi parameter for T = 1,373 K were inserted to eq. (3) as the Lik parameter.

The Ni-Pd phase diagram was optimized by Ghosh et al. [17] (Figure 4). Both liquid and solid phases were modeled as substitutional solutions.

Figure 4 The Ni-Pd phase diagram [17].
Figure 4

The Ni-Pd phase diagram [17].

Thermodynamic parameters Ljk for binary Ni-Pd solid phase were accepted:

(10)LNiPd=15866.3675.6399T+5276.78+2.0358TxNixPd+12038.2371+8.322TxAlxNi2

Values of the LNiPd parameter for T = 1,373 K were inserted to eq. (3) as the Ljk parameter.

As a result of calculations the following values of ternary L parameters in eq. 4 for T = 1,373 K were obtained:

(11)0LAlNiPd=66523=671041LAlNiPd=552074=5.5·1052LAlNiPd=699675=70105

Finally the following LAlNiPd parameter was obtained:

(12)LAlNiPd=66523xAl+552074xNi+699675xPd

Values of excess Gibbs energy of ternary alloys calculated on the basis of eq. 3 with L parameter presented in eqs (8)–(10) and (12) and excess Gibbs energies of binary alloys are presented in Table 1 and in Figure 5.

Table 1

Values of binary (exGij) and ternary (exG) excess Gibbs energy.

XAlXNiXPdexGijexG
0.060.930.01−6.8 × 103−8.0 × 103
0.110.880.01−1.3 × 104−1.4 × 104
0.160.830.01−1.9 × 104−1.9 × 104
0.200.780.02−2.4 × 104−2.4 × 104
0.250.740.01−2.9 × 104−2.8 × 104
0.300.690.01−3.2 × 104−3.1 × 104
0.350.640.01−3.5 × 104−3.4 × 104
0.400.590.01−3.6 × 104−3.5 × 104
0.450.540.01−3.7 × 104−3.6 × 104
0.490.490.02−3.7 × 104−3.6 × 104
0.540.450.01−3.5 × 104−3.5 × 104
0.590.400.01−3.3 × 104−3.3 × 104
0.640.350.01−3.0 × 104−3.0 × 104
0.690.300.01−2.7 × 104−2.7 × 104
0.740.250.01−2.2 × 104−2.3 × 104
0.780.200.02−1.8 × 104−1.9 × 104
0.830.160.01−1.3 × 104−1.4 × 104
0.880.110.01−8.5 × 103−1.0 × 104
0.930.060.01−4.0 × 103−5.6 × 103
0.980.010.010−1.6 × 103
0.930.010.06−3.2 × 103−4.8 × 103
0.880.010.11−6.7 × 103−8.2 × 103
0.830.010.16−1.0 × 104−1.2 × 104
0.780.020.20−1.4 × 104−1.5 × 104
0.740.010.25−1.8 × 104−1.9 × 104
0.690.010.30−2.2 × 104−2.2 × 104
0.640.010.35−2.5 × 104−2.5 × 104
0.590.010.40−2.8 × 104−2.8 × 104
0.540.010.45−3.1 × 104−3.1 × 104
0.490.020.49−3.3 × 104−3.2 × 104
0.450.010.54−3.5 × 104−3.4 × 104
0.400.010.59−3.5 × 104−3.4 × 104
0.350.010.64−3.5 × 104−3.4 × 104
0.300.010.69−3.4 × 104−3.3 × 104
0.250.010.74−3.2 × 104−3.1 × 104
0.200.020.78−2.8 × 104−2.8 × 104
0.160.010.83−2.3 × 104−2.3 × 104
0.110.010.88−1.7 × 104−1.8 × 104
0.060.010.93−9.4 × 103−1.1 × 104
0.010.010.980−2.6 × 103
0.010.060.93−1.5 × 103−3.5 × 103
0.010.110.88−2.7 × 103−4.3 × 103
0.010.160.83−3.8 × 103−4.9 × 103
0.020.200.78−4.6 × 103−5.4 × 103
0.010.250.74−5.2 × 103−5.8 × 103
0.010.300.69−5.7 × 103−6.0 × 103
0.010.350.64−5.9 × 103−6.1 × 103
0.010.400.59−6.1 × 103−6.1 × 103
0.010.450.54−6.0 × 103−6.0 × 103
0.020.490.49−5.9 × 103−5.9 × 103
0.010.540.45−5.6 × 103−5.6 × 103
0.010.590.40−5.3 × 103−5.3 × 103
0.010.640.35−4.8 × 103−5.0 × 103
0.010.690.30−4.3 × 103−4.6 × 103
0.010.740.25−3.7 × 103−4.1 × 103
0.020.780.20−3.0 × 103−3.7 × 103
0.010.830.16−2.3 × 103−3.2 × 103
0.010.880.11−1.6 × 103−2.7 × 103
0.010.930.06−8.0 × 102−2.2 × 103
0.010.980.010−1.7 × 103
Figure 5 Values of binary exGij$\left({{}^{ex}{G_{\rm ij}}} \right)$ and ternary exG$\left({{}^{ex}G} \right)$ excess Gibbs energy.
Figure 5

Values of binary exGij and ternary exG excess Gibbs energy.

In order to check how much the choice of the xx value influences values of LAlNiPd parameters and excess Gibbs energy calculated on the basis on these parameters, analogues calculations were performed for the range of xx values from 0.01 to 0.1. Results of calculations are presented in Table 2 and Figures 611. Although values of L parameters change with the xx value, the final results, that is the excess Gibbs energy, do not chance significantly (only about 10%).

Table 2

The calculated values of L parameters and the excess Gibbs energy.

xx0LAlNiPd1LAlNiPd2LAlNiPdexGxAl=xNi=xPd= 1/3 [kJ/mol]exGxAl=xNi=2/5,xPd= 1/5 [kJ/mol]exGxAl=xPd=1/10,xNi= 4/5 [kJ/mol]
0.01−6.7 × 1045.5 × 1057.0 × 105−1.9 × 104−2.4 × 104−1.1 × 104
0.02−5.8 × 1045.6 × 1057.1 × 105−1.9 × 104−2.4 × 104−1.1 × 104
0.03−4.9 × 1045.7 × 1057.2 × 105−1.8 × 104−2.3 × 104−1.1 × 104
0.04−4.1 × 1045.8 × 1057.3 × 105−1.8 × 104−2.3 × 104−1.1 × 104
0.05−3.3 × 1045.9 × 1057.3 × 105−1.8 × 104−2.3 × 104−1.0 × 104
0.06−2.5 × 1045.9 × 1057.4 × 105−1.8 × 104−2.3 × 104−1.0 × 104
0.07−1.8 × 1046.0 × 1057.4 × 105−1.7 × 104−2.2 × 104−1.0 × 104
0.08−1.3 × 1046.1 × 1057.4 × 105−1.7 × 104−2.2 × 104−1.0 × 104
0.09−8.3 × 1036.2 × 1057.4 × 105−1.7 × 104−2.2 × 104−1.0 × 104
0.01−6.7 × 1045.5 × 1057.0 × 105−1.9 × 104−2.4 × 104−1.1 × 104
Figure 6 exGAlNiPd for xNi:xAl = 1:1 for different xPd values.
Figure 6

exGAlNiPd for xNi:xAl = 1:1 for different xPd values.

Figure 7 exGAlNiPd for xAl:xPd = 1:1 for different xNi values.
Figure 7

exGAlNiPd for xAl:xPd = 1:1 for different xNi values.

Figure 8 exGAlNiPd for xPd:xNi = 1:1 for different xAl values.
Figure 8

exGAlNiPd for xPd:xNi = 1:1 for different xAl values.

Figure 9 Values of palladium chemical potential in Al-Ni-Pd alloys.
Figure 9

Values of palladium chemical potential in Al-Ni-Pd alloys.

Figure 10 Values of nickel chemical potential in Al-Ni-Pd alloys.
Figure 10

Values of nickel chemical potential in Al-Ni-Pd alloys.

Figure 11 Values of aluminum chemical potential in Al-Ni-Pd alloys.
Figure 11

Values of aluminum chemical potential in Al-Ni-Pd alloys.

Values of chemical potentials of palladium, nickel and aluminum were derived from the excess Gibbs energy according to the following formulas:

(13)μpd=exGXAlexGXAlXNiexGXNi=120171XAl3XNi+XAl2(2XNi2+XNi0.541.11Xpd+0.60Xpd0.01XNi3Xpd+XNi20.07+0.01XpdXpd++XAlXNi1.22+XNi2+5.82Xpd2+XNi0.54+9.19)Xpd
(14)μNI=exGXAlexGXAlXPdexGXPd=80114XAl3XNi1224.26XNi2XPd2+XNiXPd2(807193+122426XPd)+XAlXPd(1329296355207425XNi2+144863.30XPd139935151XNiXPd+XAl2(80114XNi2144863.30XPd+XNi(32712+133046.02XPd)
(15)μAl=exGXNiexGXNiXPdexGXPd=1836.39XNi3XPd+XNi2(16143867367279XPd)XPd++XAl2XNi(80114XNi+66523.01XPd2)++XNiXPd(2360995+1614387XPd+1836.39XPd2)++XAl(80114XNi3+XNi2(327121104148.5XPd)+7243165XPd2139935151XNiXPd2

Results of calculations are presented in Figures 611.

Discussion

The Al-Ni-Pd system has not been optimized yet and values of thermodynamic functions have not been determined. Nevertheless, binary systems Al-Ni, Al-Pd and Ni-Pd are well known and described. Therefore in this paper, values of thermodynamic functions, excess Gibbs energy and chemical potentials of aluminum, nickel and palladium were predicted numerically based on the values of thermodynamic functions of binary alloys constituting the investigated ternary alloys. As there is no other data about the Al-Ni-Pd system, the results obtained in this work cannot be compared with the experimental data or any other calculations. The applied numerical method was validated on other systems, for which there are other thermodynamic data, for instance on Cu-Sn-Zn, Bi-Cu-Ni, Ag-Au-Bi, In-Sn-Zn and Cu-Fe-Sn alloys [1314, 1820] and results obtained by the numerical method agree very well with values obtained by the Calphad method (Figures 12 and 13).

Moreover, the applied numerical method is better than the geometric models [3], for instance for the Cu-Sn-Zn alloys (Table 3).

Table 3

Values of exGCuSnZn calculated by different methods.

xCuxSnxZnexGCuSnZn Calphad [J/mol]exGCuSnZn numerical method [J/mol]exGCuSnZn Kohler model [21] [J/mol]exGCuSnZn Muggianu model [7] [J/mol]exGCuSnZn Chou model [22] [J/mol]
0.60.10.3−5,132−5,204−5,721−5,612−5,596
0.50.250.25−4,484−4,521−5,759−5,189−4,905
0.60.30.1−4,907−4,906−5,051−5,178−5,696
Figure 12 Comparison between the excess Gibbs energy of the Cu-Sn-Zn system calculated by the numerical approach and by other authors.
Figure 12

Comparison between the excess Gibbs energy of the Cu-Sn-Zn system calculated by the numerical approach and by other authors.

Figure 13 Comparison between the excess Gibbs energy of the Bi-Cu-Ni system calculated by the numerical approach and by other authors.
Figure 13

Comparison between the excess Gibbs energy of the Bi-Cu-Ni system calculated by the numerical approach and by other authors.

Therefore, it seems that the presented results may be regarded as valuable and reliable ones.

Conclusions

Values of excess Gibbs energy and chemical potentials of aluminum, nickel and palladium in ternary Al-Ni-Pd alloys at 1,373 K were predicted numerically for three cross-sections (xAl:xNi=1:1, xPd:xNi=1:1, xPd:xAl=1:1 and for alloys of the following contents: (xAl=1/3, xNi=1/3, xPd=1/3, xAl=2/5, xNi=2/5, xPd=1/5, xAl=1/10, xNi=4/5, xPd=1/10). The obtained values of the excess Gibbs energy do not change significantly with the change of the predicting procedure, namely the accepted xx value. Predicted values of nickel and aluminum chemical potentials do change significantly with the change of the xx value, along the examined cross-sections. Values of palladium chemical potential change significantly with the change of the xx value for xPd>0.8.

The results obtained in this work will be used to model the diffusion process in the palladium-modified aluminide coatings. Therefore, the obtained results will be applied to alloys of high nickel content and low palladium and aluminum content and for these alloys, there are the smallest deviations of the predicted thermodynamic functions.

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Received: 2014-6-26
Accepted: 2014-12-28
Published Online: 2015-2-5
Published in Print: 2016-1-1

©2016 by De Gruyter

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