Retraction: Representation of Reaction Ability for Structural Units in Fe–Al Binary Melts

Abstract

A thermodynamic model for calculating the mass action concentrations of structural units in Fe–Al binary melts based on the atom–molecule coexistence theory, i.e., AMCT–N_i model, has been developed and verified through comparing with the reported activities of both Al and Fe in Fe–Al binary melts in a temperature from 1573 K to 1873 K by different researchers. The calculated mass action concentration N_Al of Al or N_Fe of Fe can be applied to ideally substitute the measured activity a_{R, Al} of Al or a_{R, Fe} of Fe relative to pure liquid Al(l) or Fe(l) as standard state in Fe–Al binary melts. The following equations were derived for the activity coefficient of Al in natural logarithmic form ln γ_Al and the calculated activity coefficient of Fe in natural logarithmic form ln γ_Fe in the temperature range from 1573 K to 1873 K.   

Temperature dependences of the activity coefficient of Al in natural logarithmic form and the calculated activity coefficient of Fe in natural logarithmic form were given   

The obtained values were compared with the results of the previous investigations.

1. Introduction

The knowledge of the thermodynamics of Fe–Al binary melts is important both from a process metallurgy point of view and for an understanding of the thermochemical behavior of fiber–reinforced iron aluminized composites.¹⁾ For these reasons, the reaction abilities of elements, especially activity coefficient γ_Al of Al and γ_Fe of Fe in Fe–Al binary melts have attracted tremendous attention in the past several decades.^{2,3,4,5,6,7,8,9,10)} Chipman and Floridis²⁾ measured the distribution of aluminum between liquid iron and silver, and combined these data to extrapolate activate coefficient γ_Al of Al in Fe–Al binary melts up to 9.66 mass% Al at 1873 K as γ_Al = –3.477+6.0X_Al. Wilder and Elliott³⁾ determined the activity of aluminum in liquid Al–Ag alloys between 973 K and 1173 K by a galvanic cell technique, and their results were extrapolated to 1873 K to reexamine Chipman et al.’s reported data. They reported ${\gamma }_{\text{Al}}^0$ , ${\epsilon }_{\text{Al}}^{\text{Al}}$ and $e_{\text{Al}}^{\text{Al}}$ values in liquid iron as 0.063, 5.3 and 0.043 at 1873 K, respectively. Wooley and Elliott⁴⁾ measured the partial molar heats of solution of aluminum at 1873 K in the high–temperature solution calorimeter in the range of 0<x_Al<0.56 for the Fe–Al system, They reported ${\gamma }_{\text{Al}}^0$ , ${\epsilon }_{\text{Al}}^{\text{Al}}$ and $e_{\text{Al}}^{\text{Al}}$ values as 0.061, 5.6 and 0.045 at 1873 K, respectively. Coskun and Elliott⁵⁾ measured the activity of aluminum in liquid Fe–Al system by an improved version of the transportation method in which the metallic vapor was collected by its solution in a metallic condenser, and proposed the equation log γ_Al = –3.0(1–x_Al)². Using Knudsen cell–mass spectrometry, Belton and Fruhan⁶⁾ directly measured the activity of aluminum in liquid Fe–Al alloys up to 20.29 mass% Al at 1873 K. Ichise et al.⁷⁾ also used the same experimental technique to determine the activity of aluminum in liquid Fe–Al alloys up to 8.21 mass% Al in the temperature range from 1673 K to 1873 K. They reported the values of ${\gamma }_{\text{Al}}^0$ , ${\epsilon }_{\text{Al}}^{\text{Al}}$ , ${\gamma }_{\text{Fe}}^0$ and ${\epsilon }_{\text{Fe}}^{\text{Fe}}$ in liquid iron as 0.049, 6.4, 0.017 and 8.2 at 1873 K, respectively. Jacobson and Mehrotra⁸⁾ determined the activities of Fe and Al in Fe–Al alloys at 1573 K using the ion–current–ratio technique in a high–temperature Knudsen cell–mass spectrometry, and reported the values of ${\gamma }_{\text{Al}}^0$ and ${\gamma }_{\text{Fe}}^0$ at 1573 K as 0.016 and 0.019. In 2007, Kim et al.⁹⁾ investigated the thermodynamics of aluminum in liquid iron in the temperature range of 1873 K–1973 K and reported the values of ${\gamma }_{\text{Al}}^0$ as 0.066, 0.069 and 0.071 at 1873 K, 1923 K and 1973 K. Recently, Zaitsev et al.¹⁰⁾ reported that three associative molecules as FeAl, FeAl₂, and Fe₂Al₅ could exist in Fe–Al binary melts based on the experimental results using Knudsen cell–mass spectrometer and an integral variant of the effusion method under the condition of super high oil–free vacuum. But after checking previous literatures and phase diagram of Fe–Al binary system, Zhang¹¹⁾ proposed that five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ could exist in Fe–Al binary melts. Other researchers^{12,13,14,15,16,17,24,25,26)} have reached the similar viewpoint for metallic melts that molecules can exist in metallic melts besides atoms. Several attempts^24,25,26) can well describe the thermodynamic properties of metallic melts on the concept of ionic or molecular constituents associated with artificial parameters, and most of the regressed artificial parameters were obtained by experimental results.²⁰⁾

Although various experimental technique were adopted to measure the activity of Al and Fe in liquid Fe–Al alloys, to the knowledge of the present authors, no prediction model was established to predict or evaluate the activity of Al and Fe in liquid Fe–Al alloys. In this study, the AMCT–N_i thermodynamic model for calculating the mass action concentrations N_i of structural units in Fe–Al binary melts in the temperature range from 1573 K to 1873 K has been developed without any regressed parameters. The reported activities of Al and Fe by different researchers in a temperature range from 1573 K to 1873 K were chosen as evaluation criteria to verify the accuracy of the developed AMCT–N_i thermodynamic model for Fe–Al binary melts. It should be emphasized that the reported activities of Al and Fe are relative to pure liquid Al(l) and Fe(l) as standard state, i.e., a_{R, Al} of Al and a_{R, Fe} of Fe. The ultimate aim of this study is to pave the way for developing a universal AMCT–N_i thermodynamic model for representing the reaction abilities of structural units in any binary metallic melts by the calculated mass action concentrations N_i of structural units in the binary melts.

2. AMCT–N_i Thermodynamic Model for Fe–Al Binary Melts

2.1. Hypotheses

It has been briefly demonstrated in Section 1 that atoms of Fe and Al, and molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ can coexist in Fe–Al binary melts. Therefore, the atom–molecule coexistence theory, i.e., AMCT, can be applied to describe the structural characteristics of Fe–Al binary melts. The hypotheses of the developed AMCT–N_i model for Fe–Al binary melts can be briefly summarized as follows: 1) the Fe–Al binary melts at elevated temperature are composed of seven structural units including two atoms as Fe and Al as well as five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ according to the phase diagram of Fe–Al binary melts^18,21) and viewpoints described in Section 1 from the literature;¹¹⁾ 2) each structural unit occupies its independent position in Fe–Al binary melts; 3) the elements of both Fe and Al in Fe–Al binary melts will take part in reactions of forming five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in the form of atoms; 4) the reactions of forming five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ are under the chemical dynamic equilibrium between the simple atoms of both Fe and Al; 5) five structural units in Fe–Al binary melts as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ bear the structural continuity in the investigated composition range; 6) the chemical reactions of forming five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ from Fe and Al obey the mass action law.

2.2. Establishment of AMCT–N_i Thermodynamic Model for Fe–Al Binary Melts

The mole numbers of two atoms as Fe and Al before equilibrium or before forming five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in 100–g Fe–Al binary melts were assigned as b₁ = $n_{\text{Fe}}^0$ and b₂ = $n_{\text{Al}}^{\text{0}}$ to represent the chemical composition of Fe–Al binary melts. Two atoms of Fe and Al as well as five molecules of Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in Fe–Al binary melts at the studied temperatures were summarized and assigned with exclusive numbers in Table 1. The defined equilibrium mole numbers n_i of all above–mentioned five structural units as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in 100–g Fe–Al binary melts at the studied temperatures were also listed in Table 1. The total equilibrium mole number Σn_i of all seven structural units in 100–g Fe–Al binary melts can be expressed as   

$$\begin{array}{l}\mathrm{\Sigma }{n}_i=n_1+n_2+n_{\text{c1}}+n_{\text{c2}}+n_{\text{c3}}+n_{\text{c4}}+n_{\text{c5}}\\ \hspace{1em}\mathrm{\hspace{0.17em} } \mathrm{\hspace{0.17em} }=n_{\text{Fe}}+n_{\text{Al}}+n_{{\text{Fe}}_{\text{3}}^{}\text{Al}}+n_{\text{FeAl}}+n_{{\text{FeAl}}_{\text{2}}}+n_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}+n_{{\text{FeAl}}_{\text{6}}}& \left(\text{mol}\right)\end{array}$$(1)

Table 1. Expressions of structural units as atoms or molecules, their equilibrium mole numbers n_i and mass action concentrations N_i in 100–g Fe–Al binary melts based on the AMCT.

Item	Structural units as atoms or molecules	No. of structural units	Mole numbers of structural units (mol)	Mass action concentrations of structural units Ni (–)
Atom(2)	Fe	1	n1 = nFe	$N_1=\frac{n_1}{\mathrm{\Sigma }{n}_i}=N_{\text{Fe}}$
	Al	2	n2 = nAl	$N_2=\frac{n_2}{\mathrm{\Sigma }{n}_i}=N_{\text{Al}}$
Molecules(5)	Fe3Al	c1	nc1 = $n_{{\text{Fe}}_{\text{3}}\text{Al}}$	$N_{\text{c1}}=\frac{n_{\text{c1}}}{\mathrm{\Sigma }{n}_i}=N_{{\text{Fe}}_{\text{3}}\text{Al}}$
	FeAl	c2	nc2 = nFeAl	$N_{\text{c2}}=\frac{n_{\text{c2}}}{\mathrm{\Sigma }{n}_i}=N_{\text{FeAl}}$
	FeAl2	c3	nc3 = $n_{{\text{FeAl}}_{\text{2}}}$	$N_{\text{c3}}=\frac{n_{\text{c3}}}{\mathrm{\Sigma }{n}_i}=N_{{\text{FeAl}}_{\text{2}}}$
	Fe2Al5	c4	nc4 = $n_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}$	$N_{\text{c4}}=\frac{n_{\text{c4}}}{\mathrm{\Sigma }{n}_i}=N_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}$
	FeAl6	c5	nc5 = $n_{{\text{FeAl}}_{\text{6}}}$	$N_{\text{c5}}=\frac{n_{\text{c5}}}{\mathrm{\Sigma }{n}_i}=N_{{\text{FeAl}}_{\text{6}}}$

According to the definition of mass action concentrations N_i for structural units based on the AMCT^11,18,19,20) for metallic melts or the IMCT^22,23) for metallurgical slags, the mass action concentrations N_i of structural units i in metallic melts can be calculated by   

$$N_i=\frac{n_i}{\mathrm{\Sigma }{n}_i}\hspace{1em}(-)$$(2)

The chemical reaction formulas of forming five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆, the related standard reaction equilibrium constants $K_{\text{c}i}^{\mathrm{\Theta }\text{, R}}$ relative to pure liquid matter as standard state, and the representations of mass action concentrations N_ci of five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ using $K_{\text{c}i}^{\mathrm{\Theta }\text{, R}}$ , N₁ (N_Fe) and N₂ (N_Al) based on the mass action law were summarized in Table 2.

Table 2. Chemical reaction formulas of formed molecules, their standard equilibrium constants $K_{\text{c}i}^{\mathrm{\Theta }\text{, R}}$ , and mass action concentrations N_ci of formed molecules in Fe–Al binary melts based on the AMCT.

Reactions	$K_{\text{c}i}^{\mathrm{\Theta }\text{, R}}$ (–)	Nci (–)
3[Fe]+[Al]=[Fe3Al]	$K_{\text{c1}}^{\mathrm{\Theta }\text{, R}}=\frac{N_{\text{c1}}}{N_1^3{N}_2^{}}=\frac{N_{{\text{Fe}}_{\text{3}}\text{Al}}}{N_{\text{Fe}}^{\text{3}}{N}_{\text{Al}}^{}}$	$N_{\text{c1}}=K_{\text{c1}}^{\mathrm{\Theta }\text{, R}}{N}_1^3{N}_2^{}=K_{[\mathrm{F}{\mathrm{e}}_3\mathrm{A}\mathrm{l}]}^{\mathrm{\Theta }\text{, R }} N_{\text{Fe}}^{\text{3}}{N}_{\text{Al}}^{}$
[Fe]+[Al]=[FeAl]	$K_{\text{c2}}^{\mathrm{\Theta }\text{, R}}=\frac{N_{\text{c2}}}{N_1^{}{N}_2^{}}=\frac{N_{\text{FeAl}}}{N_{\text{Fe}}^{}{N}_{\text{Al}}^{}}$	$N_{\text{c2}}=K_{\text{c2}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^{}=K_{\text{[FeAl]}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^{}$
[Fe]+2[Al]=[FeAl2]	$K_{\text{c3}}^{\mathrm{\Theta }\text{, R}}=\frac{N_{\text{c2}}^{}}{N_1^{}{N}_2^2}=\frac{N_{{\text{FeAl}}_{\text{2}}}}{N_{\text{Fe}}^{}{N}_{\text{Al}}^2}$	$N_{\text{c3}}=K_{\text{c3}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^{\text{2}}=K_{{\text{[FeAl}}_{\text{2}}\text{]}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^{\text{2}}$
[Fe]+[Al]=[Fe2Al5]	$K_{\text{c4}}^{\mathrm{\Theta }\text{, R}}=\frac{N_{\text{c4}}}{N_1^{\text{2}}{N}_2^5}=\frac{N_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}}{N_{\text{Fe}}^2{N}_{\text{Al}}^5}$	$N_{\text{c4}}=K_{\text{c4}}^{\mathrm{\Theta }\text{, R}}{N}_1^2{N}_2^5=K_{{\text{[Fe}}_{\text{2}}{\text{Al}}_{\text{5}}\text{]}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{\text{2}}{N}_{\text{Al}}^{\text{5}}$
[Fe]+6[Al]=[FeAl6]	$K_{\text{c5}}^{\mathrm{\Theta }\text{, R}}=\frac{N_{\text{c5}}}{N_1^{}{N}_2^6}=\frac{N_{{\text{FeAl}}_{\text{6}}^{}}}{N_{\text{Fe}}^{}{N}_{\text{Al}}^{\text{6}}}$	$N_{\text{c5}}=K_{\text{c5}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^6=K_{{\text{[FeAl}}_{\text{6}}\text{]}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^6$

The mass conservation equations of two atoms as Fe and Al in 100–g Fe–Al binary melts can be established based on the above–mentioned definitions^{11,18,19,20,22,23)} of n_i, N_i, and Σn_i as follows   

$$\begin{array}{l}{b}_1=\left(N_1+3N_{\text{c1}}+N_{\text{c2}}+N_{\text{c3}}+2N_{\text{c4}}+N_{\text{c5}}\right)\mathrm{\Sigma }{n}_i\\ \hspace{1em}=(N_1+3K_{\text{c1}}^{\mathrm{\Theta }\text{, R}}{N}_1^3{N}_2^{}+K_{\text{c2}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^{}\\ \hspace{1em}+K_{\text{c3}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^2+2K_{\text{c4}}^{\mathrm{\Theta }\text{, R}}{N}_1^{\text{2}}{N}_2^5+K_{\text{c5}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^6)\\ \hspace{1em}=(N_{\text{Fe}}+3K_{\text{c1}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^3{N}_{\text{Al}}^{}+K_{\text{c2}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^{}+K_{\text{c3}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^2\\ \hspace{1em}+2K_{\text{c4}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{\text{2}}{N}_{\text{Al}}^5+K_{\text{c5}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^6)\mathrm{\Sigma }{n}_i=n_{\text{Fe}}^0\hspace{1em}\hspace{1em}\text{(mol)}\end{array}$$(3a)

  

$$\begin{array}{l}{b}_2=(N_2+N_{\text{c1}}+N_{\text{c2}}+2N_{\text{c3}}+5N_{\text{c4}}+6N_{\text{c5}})\mathrm{\Sigma }{n}_i\\ \hspace{1em}\hspace{0.17em}=(N_2+K_{\text{c1}}^{\mathrm{\Theta }\text{, R}}{N}_1^3{N}_2^{}+K_{\text{c2}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^{}+2K_{\text{c3}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^2\\ \hspace{1em}\hspace{0.17em}+5K_{\text{c4}}^{\mathrm{\Theta }\text{, R}}{N}_1^{\text{2}}{N}_2^5+6K_{\text{c5}}^{\mathrm{\Theta }\text{, R}}{N}_1^{}{N}_2^6)\mathrm{\Sigma }{n}_i\\ \hspace{1em}\hspace{0.17em}=(N_{\text{Al}}+K_{\text{c1}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^3{N}_{\text{Al}}^{}+K_{\text{c2}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^{}+2K_{\text{c3}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^2\\ \hspace{1em}\hspace{0.17em}+5K_{\text{c4}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{\text{2}}{N}_{\text{Al}}^5+6K_{\text{c5}}^{\mathrm{\Theta }\text{, R}}{N}_{\text{Fe}}^{}{N}_{\text{Al}}^6)\mathrm{\Sigma }{n}_i=n_{\text{Al}}^0\hspace{1em}\hspace{1em}\text{(mol)}\end{array}$$(3b)

According to the principle that the sum of mole fraction of all structural units in a fixed amount of metallic melts under equilibrium condition is equal to unity as 1, the following equation can be established   

$$\begin{array}{l}{N}_1+N_2+N_{\text{c1}}+N_{\text{c2}}+N_{\text{c3}}+N_{\text{c4}}+N_{\text{c5}}\\ =N_{\text{Fe}}+N_{\text{Al}}+N_{{\text{Fe}}_{\text{3}}\text{Al}}+N_{\text{FeAl}}+N_{{\text{FeAl}}_{\text{2}}}+N_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}+N_{{\text{FeAl}}_{\text{6}}}=1.0& (-)\end{array}$$(4)

The governing equations of the developed AMCT–N_i thermodynamic model for calculating the mass action concentrations N_i of structural units in Fe–Al binary melts are comprised by the equation group of Eqs. (3) and (4). Obviously, there are three unknown parameters as N₁ (N_Fe), N₂ (N_Al), and Σn_i with three independent equations in the established equation group of Eqs. (3) and (4). The unique solution of N_i, Σn_i, and n_i can be calculated by solving the algebraic equation group of Eqs. (3) and (4) through combining with the definition of N_i in Eq. (2) after knowing the values of $K_{\text{c1}}^{\mathrm{\Theta }\text{, R}}$ ( $K_{{\text{Fe}}_{\text{3}}\text{Al}}^{\mathrm{\Theta }\text{, R}}$ ), $K_{\text{c2}}^{\mathrm{\Theta }\text{, R}}$ ( $K_{\text{FeAl}}^{\mathrm{\Theta }\text{, R}}$ ), $K_{\text{c3}}^{\mathrm{\Theta }\text{, R}}$ ( $K_{{\text{FeAl}}_{\text{2}}}^{\mathrm{\Theta }\text{, R}}$ ), $K_{\text{c4}}^{\mathrm{\Theta }\text{, R}}$ ( $K_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}^{\mathrm{\Theta }\text{, R}}$ ), and $K_{\text{c5}}^{\mathrm{\Theta }\text{, R}}$ ( $K_{{\text{FeAl}}_{\text{6}}}^{\mathrm{\Theta }\text{, R}}$ ).

The standard equilibrium constant $K_{\text{c}i}^{\mathrm{\Theta }\text{, R}}$ of reactions for five molecules as Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in Fe–Al binary melts at a fixed temperature can be calculated based on the obtained standard molar Gibbs free energy change ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, c}i}^{\mathrm{\Theta }\text{, R}}$ of the same reaction from the mass action law as ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, c}i}^{\mathrm{\Theta }\text{, R}}$ = –RT ln $K_{\text{c}i}^{\mathrm{\Theta }\text{, R}}$ based on $K_{\text{c}i}^{\mathrm{\Theta }\text{, R}}$ = $a_{{\text{Fe}}_x{\text{Al}}_y}/\left(a_{\text{Fe}}^x{a}_{\text{Al}}^y\right)$ . Meanwhile, the standard molar Gibbs free energy changes ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, c}i}^{\mathrm{\Theta }\text{, R}}$ of reactions for forming Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ from Fe and Al have been reported by Zhang^11,18) relative to pure liquid matter as standard states as follows   

$$\begin{array}{l}3\left[\mathrm{F}\mathrm{e}\right]+\left[\mathrm{A}\mathrm{l}\right]=\left[\mathrm{F}{\mathrm{e}}_3\mathrm{A}\mathrm{l}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{{\text{m, Fe}}_{\text{3}}\text{Al}}^{\mathrm{\Theta }\text{, R}}=-RTln\left(\frac{a_{{\text{R, Fe}}_{\text{3}}\text{Al}}}{a_{\text{R, Fe}}^3{a}_{\text{R, Al}}}\right)=-120\mathrm{\hspace{0.17em} }586.85+48.61T& \left(\text{J/mol}\right)\end{array}$$(5)

  

$$\begin{array}{l}\left[\mathrm{F}\mathrm{e}\right]+\left[\mathrm{A}\mathrm{l}\right]=\left[\mathrm{F}\mathrm{e}\mathrm{A}\mathrm{l}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, FeAl}}^{\mathrm{\Theta }\text{, R}}=-RTln\left(\frac{a_{\text{R, FeAl}}}{a_{\text{R, Fe}}^{}{a}_{\text{R, Al}}}\right)=-47\mathrm{\hspace{0.17em} }813.257+7.893T& \left(\text{J/mol}\right)\end{array}$$(6)

  

$$\begin{array}{l}\left[\mathrm{F}\mathrm{e}\right]+2\left[\mathrm{A}\mathrm{l}\right]=\left[\mathrm{F}\mathrm{e}\mathrm{A}{\mathrm{l}}_2\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{{\text{m, FeAl}}_{\text{2}}}^{\mathrm{\Theta }\text{, R}}=-RTln\left(\frac{a_{{\text{R, FeAl}}_{\text{2}}}}{a_{\text{R, Fe}}^{}{a}_{\text{R, Al}}^{\text{2}}}\right)=130\mathrm{\hspace{0.17em} }186.64-84.582T& \left(\text{J/mol}\right)\end{array}$$(7)

  

$$\begin{array}{l}2\left[\mathrm{F}\mathrm{e}\right]+5\left[\mathrm{A}\mathrm{l}\right]=\left[\mathrm{F}{\mathrm{e}}_2\mathrm{A}{\mathrm{l}}_5\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{{\text{m, Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}^{\mathrm{\Theta }\text{, R}}=-RTln\left(\frac{a_{{\text{R, Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}}{a_{\text{R, Fe}}^{\text{2}}{a}_{\text{R, Al}}^{\text{5}}}\right)=-165\mathrm{\hspace{0.17em} }372.213+43.05T& \left(\text{J/mol}\right)\end{array}$$(8)

  

$$\begin{array}{l}\left[\mathrm{F}\mathrm{e}\right]+6\left[\mathrm{A}\mathrm{l}\right]=\left[\mathrm{F}\mathrm{e}\mathrm{A}{\mathrm{l}}_6\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{{\text{m, FeAl}}_{\text{6}}}^{\mathrm{\Theta }\text{, R}}=-RTln\left(\frac{a_{{\text{R, FeAl}}_{\text{6}}}}{a_{\text{R, Fe}}^{}{a}_{\text{R, Al}}^{\text{6}}}\right)=-14\mathrm{\hspace{0.17em} }710.17-18.712T& \left(\text{J/mol}\right)\end{array}$$(9)

3. Results and Discussion for Calculated Mass Action Concentrations N_i of Structural Units in Fe–Al Binary Melts

3.1. Comparison between Calculated Mass Action Concentration N_Al of Al or N_Fe of Fe and Reported Activity a_{R, Al} of Al or a_{R, Fe} of Fe by Different Researchers

To improve the readability of this article, chemical composition of Fe–Al binary melts, reported activity a_{R, Fe} of Fe or a_{R, Al} of Al in a temperature range from 1573 K to 1873 K by different investigators,^5,6,7,8,21) calculated mass action concentrations N_i of seven structural units including Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆, and calculated total equilibrium mole number Σn_i of structural units in 100–g Fe–Al binary melts were summarized in Table 3.

Table 3. Chemical composition of Fe–Al binary melts, reported activity a_{R, Fe} of Fe or a_{R, Al} of Al in a temperature range from 1573 K to 1873 K by different investigators, calculated mass action concentrations N_i of seven structural units including Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆, and calculated total equilibrium mole number Σn_i of structural units in 100–g Fe–Al binary melts based on the developed AMCT–N_i thermodynamic model for Fe–Al binary melts.

The comparison between the calculated mass action concentration N_Al of Al and the reported activity a_{R, Al} of Al by different researchers^5,6,7,8,21) relative to pure liquid Al(l) as standard state in Fe–Al binary melts with changing mole fraction x_Al of Al from 0 to 1 in a temperature range from 1573 K to 1873 K was illustrated in Fig. 1(a). The comparison between the calculated mass action concentration N_Fe of Fe and the reported activity a_{R, Fe} of Fe by different researchers^6,7,8,21) relative to pure liquid Fe(l) as standard state in Fe–Al binary melts with changing mole fraction x_Fe of Fe from 0 to 1 in the same temperature range was also shown in Fig. 1(b). The calculated mass action concentration N_Al of Al has a very excellent agreement with the reported activity a_{R, Al} of Al by different researchers^5,6,7,8,21) in Fe–Al binary melts as shown in Fig. 1(a). Meanwhile, the calculated mass action concentration N_Fe of Fe in the full composition has a very good 1:1 corresponding relationship with the reported activity a_{R, Fe} of Fe by different researchers^6,7,8,21) in Fe–Al binary melts as shown in Fig. 1(b). Obviously, the calculated mass action concentration N_Al of Al or N_Fe of Fe can be successfully applied to substitute the reported activity a_{R, Al} of Al or a_{R, Fe} of Fe by different researchers^5,6,7,8,21) in Fe–Al binary melts in the full composition range in a temperature range from 1573 K to 1873 K.

Fig. 1.

Relationship between calculated mass action concentration N_Al of Al and reported activity a_{R, Al} of Al by different researchers relative to pure liquid Al(l) as standard state (a), and relationship between calculated mass action concentration N_Fe of Fe and reported activity a_{R, Fe} of Fe by different researchers relative to pure liquid Fe(l) as standard state (b) in Fe–Al binary melts in the temperature range from 1573 K to 1873 K, respectively.

3.2. Determination of Raoultian Activity Coefficient ${\gamma }_{\mathrm{A}\mathrm{l}}^0$ of Al and ${\gamma }_{\mathrm{F}\mathrm{e}}^0$ of Fe from Calculated Mass Action Concentration N_Al of Al and N_Fe of Fe in Fe–Al Binary Melts

The good 1:1 corresponding relationship between the calculated mass action concentration N_Al of Al and the reported activity a_{R, Al} of Al by previous researchers^5,6,7,8,21) in Fe–Al binary melts as described in Fig. 1(a) shows that the calculated mass action concentration N_Al of Al, like the measured activity a_{R, Al} of Al, can be applied to determine the activity coefficient γ_Al of Al in Fe–Al binary melts as γ_Al = N_Al/x_Al.

The mass action concentrations N_i of seven structural units in Fe–Al binary melts with changing mole fraction x_Al of Al from 0 to 0.005 at an interval of x_Al as 0.00005 at temperature of 1573 K, 1673 K, 1773 K, and 1873 K have been calculated based on the developed AMCT–N_i model for Fe–Al binary melts.

The relationship between mole fraction x_Al of Al and the calculated activity coefficient of Al in natural logarithmic form ln γ_Al in Fe–Al binary melts with changing mole fraction x_Al of Al from 0 to 0.005 at an interval of x_Al as 0.00005 at temperatures of 1573 K, 1673 K, 1773 K, and 1873 K was illustrated in Fig. 2(a), respectively. The corresponding relationships at the above–mentioned four temperatures can be expressed by the linear equations as follows   

$$\begin{array}{l}ln{\gamma }_{\text{Al}}=-\text{3}\text{.810}+\text{4}\text{.476}{x}_{\text{Al}}\\ 0<x_{\text{Al}}\le 0.005,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T=1\mathrm{\hspace{0.17em} }573\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(10a)

  

$$\begin{array}{l}ln{\gamma }_{\text{Al}}=-\text{3}\text{.397}+\text{4}\text{.248}{x}_{\text{Al}}\\ 0<x_{\text{Al}}\le 0.005,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T=1\mathrm{\hspace{0.17em} }673\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(10b)

  

$$\begin{array}{l}ln{\gamma }_{\text{Al}}=-\text{3}\text{.056}+\text{4}\text{.011}{x}_{\text{Al}}\\ 0<x_{\text{Al}}\le 0.005,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T=1\mathrm{\hspace{0.17em} }773\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(10c)

  

$$\begin{array}{l}ln{\gamma }_{\text{Al}}=-\text{2}\text{.537}+\text{3}\text{.599}{x}_{\text{Al}}\\ 0<x_{\text{Al}}\le 0.005,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T=1\mathrm{\hspace{0.17em} }873\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(10d)

Fig. 2.

Relationship between mole fraction x_Al of Al and calculated activity coefficient of Al in logarithmic form ln γ_Al (a), and relationship between mole fraction x_Fe of Fe and calculated activity coefficient of Fe in logarithmic form ln γ_Fe in Fe–Al binary melt at temperatures of 1573 K, 1673 K, 1773 K, and 1873 K, respectively.

The relationship between mole fraction x_Fe of Fe and the calculated activity coefficient of Fe in natural logarithmic form ln γ_Fe in Fe–Al binary melts with changing mole fraction x_Fe of Fe from 0 to 0.005 at an interval of x_Fe as 0.00005 at temperatures of 1573 K, 1673 K, 1773 K, and 1873 K was illustrated in Fig. 2(b), respectively. The corresponding relationship at the above–mentioned four temperatures can be expressed by the linear equations as follows   

$$\begin{array}{l}ln{\gamma }_{\text{Fe}}=-\text{3}\text{.837}+\text{6}\text{.701}{x}_{\text{Fe}}\\ 0<x_{\text{Fe}}\le 0.005,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T=1\mathrm{\hspace{0.17em} }573\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(11a)

  

$$\begin{array}{l}ln{\gamma }_{\text{Fe}}=-\text{3}\text{.753}+\text{7}\text{.627}{x}_{\text{Fe}}\\ 0<x_{\text{Fe}}\le 0.005,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T=1\mathrm{\hspace{0.17em} }673\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(11b)

  

$$\begin{array}{l}ln{\gamma }_{\text{Fe}}=-\text{3}\text{.701}+\text{8}\text{.066}{x}_{\text{Fe}}\\ 0<x_{\text{Fe}}\le 0.005,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T=1\mathrm{\hspace{0.17em} }773\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(11c)

  

$$\begin{array}{l}ln{\gamma }_{\text{Fe}}=-\text{3}\text{.686}+\text{8}\text{.176}{x}_{\text{Fe}}\\ 0<x_{\text{Fe}}\le 0.005,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T=1\mathrm{\hspace{0.17em} }873\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(11d)

The Raoultian activity coefficient ${\gamma }_{\text{Al}}^0$ of Al in the infinitely dilute Fe–Al binary melts related with activity a_{R, Al} of Al can be defined as the value of ${\gamma }_{\text{Al, }{x}_{\text{Al}}\to 0.0}$ . Therefore, the intercepts of the linear functions in Eq. (10) and in Fig. 2(a) can be treated as the values of the Raoultian activity coefficient ln ${\gamma }_{\text{Al}}^0$ of Al in the infinitely dilute Fe–Al binary melts at the above–mentioned four temperatures. The effect of temperature from 1573 K to 1873 K on the determined Raoultian activity coefficient in natural logarithmic form ln ${\gamma }_{\text{Al}}^0$ of Al in the infinitely dilute Fe–Al binary melts was illustrated in Fig. 3 and can be regressed as   

$$\begin{array}{l}ln{\gamma }_{\text{Al}}^0=\text{2}\text{.6917}-\text{10}\mathrm{\hspace{0.17em} }\text{222}\text{.7/}T\\ 1\mathrm{\hspace{0.17em} }573\mathrm{K}\le T\le 1\mathrm{\hspace{0.17em} }873\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(12)

Fig. 3.

Relationship between reciprocal of temperature 1/T and calculated Raoultian activity coefficient of Al in logarithmic form ln ${\gamma }_{\text{Al}}^0$ (a), and relationship between reciprocal of temperature 1/T and calculated Raoultian activity coefficient of Fe in logarithmic form ln ${\gamma }_{\text{Fe}}^0$ (b) in the infinitely dilute Fe–Al binary melts in a temperature range from 1573 K to 1873 K.

Similarly, the Raoultian activity coefficient ${\gamma }_{\text{Fe}}^0$ of Fe in Fe–Al binary melts related with activity a_{R, Fe} of Fe can be defined as the value of ${\gamma }_{\text{Fe, }{x}_{\text{Fe}}\to 0.0}$ . Therefore, the intercepts of the linear functions in Eq. (11) and in Fig. 2(b) can be considered as the values of the Raoultian activity coefficient ln ${\gamma }_{\text{Fe}}^0$ of Fe in Fe–Al binary melts at the above–mentioned four temperatures. The effect of temperature from 1573 K to 1873 K on the determined Raoultian activity coefficient in natural logarithmic form ln ${\gamma }_{\text{Fe}}^0$ of Fe in Fe–Al binary melts was illustrated in Fig. 3(b) and can be regressed as   

$$\begin{array}{l}ln{\gamma }_{\text{Fe}}^0=-\text{2}\text{.8573}-\text{1}\mathrm{\hspace{0.17em} }\text{521}\text{.6816/}T\\ 1\mathrm{\hspace{0.17em} }573\mathrm{K}\le T\le 1\mathrm{\hspace{0.17em} }873\mathrm{K}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(-)\end{array}$$(13)

To evaluate the reliability of Eqs. (12) and (13) in this study, several reported Raoultian activity coefficient ln ${\gamma }_{\text{Al}}^0$ of Al and ln ${\gamma }_{\text{Fe}}^0$ of Fe by previous researchers^3,4,7,8,9) were summarized in Table 4 and displayed in Fig. 3 for comparison. Obviously, the reported ln ${\gamma }_{\text{Al}}^0$ of Al by previous researchers have good agreement with the determined Raoultian activity coefficient ln ${\gamma }_{\text{Al}}^0$ of Al in the infinitely dilute Fe–Al binary melts. This suggests that the determined Raoultian activity coefficient ln ${\gamma }_{\text{Al}}^0$ of Al in the infinitely dilute Fe–Al binary melts in a temperature range from 1573 K to 1873 K has an ideal accuracy. Meanwhile, the accuracy of the predicted Raoultian activity coefficient ln ${\gamma }_{\text{Fe}}^0$ of Fe in Fe–Al binary melts in a temperature range from 1573 K to 1873 K can be verified from Fig. 3(b).

Table 4. Comparison of reported activity coefficient ${\gamma }_{\text{Al}}^0$ and ${\gamma }_{\text{Fe}}^0$ in Fe–Al binary melts from different researchers.

No.	Investigator	T(K)	${\gamma }_{\text{Al}}^0$ (–)	${\gamma }_{\text{Fe}}^0$ (–)	Method	Ref
1	Jacobson	1573	0.016	0.019	Ion–current–ratio technique	8)
2	Ichise	1673	0.027	0.02	Knudsen Cell Mass Spectrometry	7)
4	Wilder	1873	0.063	–	Fe–Al/Ag–Al(estimated from Chipman’s data)	3)
5	Woolley	1873	0.061	–	Heats of solution in Fe–Al	4)
7	Ichise	1873	0.049	0.017	Knudsen Cell Mass Spectrometry	7)
8	KIM	1873	0.066	–	Meta–gas equilibration	9)
9	Present study	1573	0.022	0.021	AMCT–Ni model	–
10	Present study	1673	0.033	0.023	AMCT–Ni model	–
11	Present study	1773	0.047	0.0247	AMCT–Ni model	–
12	Present study	1873	0.062	0.025	AMCT–Ni model	–

3.3. Prediction of Mass Action Concentrations N_i of Structural Units in Full Composition Range of Fe–Al Binary Melts

As a representative, the relationship between mole fraction x_Al of Al and the calculated mass action concentrations N_i of seven structural units as Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in the full composition range of Fe–Al binary melts at 1873 K was illustrated in Fig. 4. The effect of changing temperature from 1573 K to 1873 K on the relationship of the calculated mass action concentrations N_i of seven structural units as Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ against mole fraction x_Al of Al in the full composition range of Fe–Al binary melts was shown in Fig. 5, respectively.

Fig. 4.

Relationship between mole fraction x_Al of Al and calculated mass action concentration N_i of seven structural units as Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in the full composition range of Fe–Al binary melts at an interval of mole fraction x_Al of Al as 0.05, respectively.

Fig. 5.

Relationship between mole fraction x_Al of Al and calculated mass action concentration N_i of seven structural units as Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in the full composition range of Fe–Al binary melts at temperatures of 1573 K, 1588 K, 1673 K, and 1873 K, respectively.

It can be observed from Figs. 4 and 5(a) that the calculated mass action concentration N_Al of Al shows a slow increase tendency with an increase of mole fraction x_Al of Al from 0 to 0.40, and then displays a sharp increase trend with an increase of mole fraction x_Al of Al from 0.40 to 1.0 at the above–mentioned four temperatures. Meanwhile, an opposite variation trend of the calculated mass action concentration N_Fe of Fe against mole fraction x_Fe of Fe can be found at the above–mentioned four temperatures. Changing temperature from 1573 K to 1873 K cannot cause an obvious variation on the calculated mass action concentration N_Al of Al as well as N_Fe of Fe as shown in Figs. 4 and 5(a), respectively.

The reverse V–type relationship between mole fraction x_Al of Al and the calculated mass action concentration Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in the full composition range of Fe–Al binary melts at temperatures of 1573 K, 1673 K, 1773 K, and 1873 K can be found in Figs. 5(b)–5(f). As shown in Figs. 5(c) and 5(f), changing temperature from 1573 K to 1873 K cannot cause an obvious variation on the calculated mass action concentration N_FeAl of FeAl or $N_{{\text{FeAl}}_{\text{6}}}$ of FeAl₆ in Fe–Al binary melts. The greatest value of the calculated mass action concentration N_FeAl of FeAl is 0.75 corresponds to mole fraction x_Al of Al as 0.5; the largest value of the calculated mass action concentration $N_{{\text{FeAl}}_{\text{6}}}$ of FeAl₆ is 0.064 corresponds to mole fraction x_Al of Al as 0.9. This implies that the magnitude of the calculated mass action concentration N_ci of molecule ci as Fe_xAl_y has a close relationship with the ratio of atom Fe to atom Al.

As shown in Fig. 5(b), the maximum value of the calculated mass action concentration $N_{{\text{Fe}}_{\text{3}}\text{Al}}$ of Fe₃Al is 0.152 corresponds to mole fraction x_Al of Al as 0.2 at 1573 K; the greatest value of the calculated mass action concentration $N_{{\text{Fe}}_{\text{3}}\text{Al}}$ of Fe₃Al as 0.08 meets mole fraction x_Al of Al as 0.3 at 1873 K. Increasing temperature from 1573 K to 1873 K can lead to a decrease of the maximum value of the calculated mass action concentration $N_{{\text{Fe}}_{\text{3}}\text{Al}}$ of Fe₃Al from 0.152 to 0.08. As illustrated in Fig. 5(d), the largest value of the calculated mass action concentration $N_{{\text{FeAl}}_{\text{2}}}$ of FeAl₂ is 0.102 corresponds to mole fraction x_Al of Al as 0.6 at 1873 K; the greatest value of the calculated mass action concentration $N_{{\text{FeAl}}_{\text{2}}}$ of FeAl₂ is 0.0136 corresponds to mole fraction x_Al of Al as 0.6 at 1573 K. Increasing temperature from 1573 K to 1873 K can cause an increase of the largest value of the calculated mass action concentration $N_{{\text{FeAl}}_{\text{2}}}$ of FeAl₂ from 0.102 to 0.0136. As described in Fig. 5(e), the largest value of the calculated mass action concentration $N_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}$ of Fe₂Al₅ as 0.1 corresponds to mole fraction x_Al of Al as 0.7 at 1573 K; the greatest value of the calculated mass action concentration $N_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}$ of Fe₂Al₅ is 0.0312 corresponds to mole fraction x_Al of Al as 0.037 at 1873 K. Increasing temperature from 1573 K to 1873 K can cause a decrease of the largest value of the calculated mass action concentration $N_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}$ of Fe₂Al₅ from 0.1 to 0.0312.

It should be specially emphasized that the sum of the calculated mass action concentrations N_i of seven structural units as Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in Fe–Al binary melts should be unity as 1 as expressed in Eq. (4) in Section 2.1. Therefore, the calculated mass action concentration N_i of seven structural units in Fe–Al binary melts should be competitive or coupled each other. The competitive or coupled effect of the calculated mass action concentrations N_i of seven structural units in Fe–Al binary melts is also valid in the cases changing temperature from 1573 K to 1873 K.

3.4. Relationship between Calculated Equilibrium Mole Number n_i and Mole Fraction of x_i of Structural Units in Fe–Al Binary Melts

It has been described in Section 2.2 that the calculated equilibrium mole numbers n_i is an important parameter to represent mass content of structural units in Fe–Al binary melts under equilibrium, like mole fraction x_Al of Al or mole fraction x_Fe of Fe, according the AMCT^11,18,19,20) or the IMCT.^22,23)

The relationship of the calculated equilibrium mole number n_i of each structural unit against mole fraction x_Al of Al shown in Fig. 6 is similar with that of the calculated mass action concentration N_i of each structural unit against mole fraction x_Al of Al in Fe–Al binary melts at the above–mentioned four temperatures as shown in Fig. 5. Changing temperature from 1573 K to 1873 K cannot bring an obvious variation on the calculated equilibrium mole number n_Fe of Fe as well as n_Al of Al, respectively. It can be observed from Fig. 6(a) that the calculated equilibrium mole number n_Al of Al shows a slow increase tendency with an increase of mole fraction x_Al of Al from 0.0 to 0.4, and then displays a sharp increase trend with an increase of mole fraction x_Al of Al from 0.4 to 1.0. An opposite variation trend of the calculated equilibrium mole number n_Fe of Fe against mole fraction x_Al of Al can also be found from Fig. 6(a).

Fig. 6.

Relationship between mole fraction x_Al of Al and calculated equilibrium mole numbers n_i of seven structural units as Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ in the full composition range of Fe–Al binary melts at temperatures of 1573 K, 1588 K, 1673 K, and 1873 K, respectively.

As shown in Figs. 6(c) and 6(f), changing temperature from 1573 K to 1873 K cannot cause an obvious variation of the calculated equilibrium mole number n_FeAl of FeAl and $n_{{\text{FeAl}}_{\text{6}}}$ of FeAl₆ in Fe–Al binary melts. The greatest value of the calculated equilibrium mole number n_FeAl is 0.75 corresponds to mole fraction x_Al of Al as 0.5; the largest value of the calculated equilibrium mole number $n_{{\text{FeAl}}_{\text{6}}}$ of FeAl₆ is 0.136 corresponds to mole fraction x_Al of Al as 0.9.

As described in Fig. 6(b), increasing temperature from 1573 K to 1873 K can lead to a decrease tendency of the maximum value of the calculated equilibrium mole number $n_{{\text{Fe}}_3\text{Al}}$ of Fe₃Al from 0.186 to 0.106. The maximum value of the calculated equilibrium mole number $n_{{\text{Fe}}_3\text{Al}}$ of Fe₃Al is 0.186 corresponds to mole fraction x_Al of Al as 0.2 at 1573 K; the greatest value of the calculated equilibrium mole number $n_{{\text{Fe}}_3\text{Al}}$ of Fe₃Al as 0.106 meets mole fraction x_Al of Al as 0.2 at 1873 K. As illustrated in Fig. 6(d), increasing temperature from 1573 K to 1873 K can lead to an increase tendency of the maximum value of the calculated equilibrium mole number $n_{{\text{FeAl}}_{\text{2}}}$ of FeAl₂ from 0.0184 to 0.158 corresponds to mole fraction x_Al of Al as 0.7. As shown in Fig. 6(e), increasing temperature from 1573 K to 1873 K can cause a decrease tendency of the maximum value of the calculated equilibrium mole number $n_{{\text{Fe}}_{\text{2}}{\text{Al}}_{\text{5}}}$ of Fe₂Al₅ from 0.135 to 0.482 meets mole fraction x_Al of Al as 0.7.

4. Conclusions

A thermodynamic model for calculating the mass action concentrations of structural units in Fe–Al binary melts based on the atom–molecule coexistence theory, i.e., AMCT–N_i model, has been developed and verified through comparing with the reported activities of both Al and Fe in Fe–Al binary melts in a temperature from 1573 K to 1873 K from the literature. The main summary remarks can be obtained as follows.

(1) The calculated mass action concentration N_Al of Al or N_Fe of Fe in Fe–Al binary melts has a good 1:1 corresponding relationship with the reported activity a_{R, Al} of Al or a_{R, Fe} of Fe relative to pure liquid Al(l) or Fe(l) as standard state in a temperature range from 1573 K to 1873 K from different researchers. The calculated mass action concentration N_Al of Al or N_Fe of Fe can be applied to ideally substitute the measured activity a_{R, Al} of Al or a_{R, Fe} of Fe relative to pure liquid Al(l) or Fe(l) as standard state in Fe–Al binary melts. Therefore, the developed AMCT–N_i model for Fe–Al binary melts can be successfully applied to represent the reaction ability of structural units in Fe–Al binary melts in a temperature from 1573 K to 1873 K.

(2) The values of the Raoultian activity coefficient ${\gamma }_{\text{Al}}^0$ of Al and ${\gamma }_{\text{Fe}}^0$ of Fe in the infinitely dilute solution of Fe–Al binary melts in a temperature range from 1573 K to 1873 K can be expressed as ln ${\gamma }_{\text{Al}}^0$ =2.6917–10222.7/T and ln ${\gamma }_{\text{Fe}}^0$ =–2.8573–1521.6816/T, respectively. The calculated values of the Raoultian activity coefficient ${\gamma }_{\text{Al}}^0$ of Al or ${\gamma }_{\text{Fe}}^0$ of Fe has good agreement with the reported ${\gamma }_{\text{Al}}^0$ of Al or ${\gamma }_{\text{Fe}}^0$ of Fe by previous researchers.

(3) The reaction abilities of structure units Fe, Al, Fe₃Al, FeAl, FeAl₂, Fe₂Al₅ and FeAl₆ show a competitive or coupling relationship in the investigated temperature range.

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