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Critical Assessment of Activities of Structural Units in Fe–Al Binary Melts Based on the Atom and Molecule Coexistence Theory

  • Xue-min Yang EMAIL logo , Jin-yan Li , Fang-jia Yan , Dong-ping Duan and Jian Zhang
Published/Copyright: October 26, 2018
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High Temperature Materials and Processes
From the journal High Temperature Materials and Processes Volume 37 Issue 9-10

Abstract

A thermodynamic model for calculating the mass action concentrations Ni of structural units in Fe–Al binary melts based on the atom–molecule coexistence theory, i. e., AMCT–Ni model, has been developed and verified to be valid by comparing with reported activities aR,i of both Al and Fe relative to pure liquid as standard state in Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550  to 1700 °C). Furthermore, Raoultian activity coefficients γi0 of both Al and Fe in the Fe-rich corner or Al-rich corner of Fe–Al binary melts as well as the standard molar Gibbs-free energy changes ΔsolGm,i(l)→[i][%i]=1.0Θ,% of dissolved Al or Fe for forming [% Al] or [% Fe] as 1.0 in Fe–Al binary melts have also been determined by the developed AMCT–Ni model and verified to be accurate.

The reported activities aR,i of both Al and Fe from the literature can be well reproduced by calculated mass action concentrations Ni of free Al and free Fe in Fe–Al binary melts. A small effect of changing temperature from 1823 to 1973 K (1550  to 1700 °C) on reaction abilities of both Al and Fe from the available literature is also confirmed by calculated mass action concentrations Ni of free Al and free Fe in Fe–Al binary melts. The obtained activity coefficients γi of both Al and Fe in Fe–Al binary melts can be described by a quadratic polynomial function and a cubic polynomial function, respectively. Furthermore, accurate expressions of Raoultian activity coefficients γi0 of both Al and Fe in Fe-rich corner or Al-rich corner of Fe–Al binary melts are also obtained as lnγAl0=−9,646.5/T+2.196 and lnγFe0=−6,799.1/T−0.01367, respectively. In addition, expressions of the first-order activity interaction coefficients εii or eii or hii of both Al and Fe coupled with three activity coefficients γi or f%,i or fH,i relative to three standard states are also obtained from the developed AMCT–Ni model for Fe–Al binary melts.

Keywords: Assessment; Activity of aluminum; Activity of iron; Fe–Al binary melts; Mass action concentration; Reaction ability; Atom and molecule coexistence theory (AMCT); Structural units

Introduction

Thermodynamic properties of Fe–Al binary system, especially reaction ability of Al, i. e., activity aR,Al of Al relative to pure liquid Al(l) as standard state, have been attracting considerable attention of researchers and engineers since the 1930’s [1–20] after the pioneering study by Chipman [1] through Al partition LAl between liquid Fe and Ag at 1873 K (1600 °C) because pure Al and Al-containing alloys are indispensable deoxidizing agents for molten steel during steel refining processes. Furthermore, thermodynamic properties of Fe–Al binary system can also provide important information on refining the high temperature heat resistant alloys of Fe–Cr–Al ternary melts [7] and ceramic reinforcement materials for iron aluminide matrices [21]. The available studies on activity coefficient γAl of Al coupled with activity aR,Al of Al through aR,Al=γAlxAl in the literature prior to 1993 were reviewed by Jacobson and Mehrotra [17]. In addition, the available results of Raoultian activity coefficient γAl0 of Al in the Fe-rich corner of Fe–Al binary melts prior to 1977 and 2007 were also reviewed by Ichise et al. [15] and Kim et al. [19], respectively. The main experimental methods for measuring or determining activity aR,Al of Al in Fe–Al binary melts are summarized through a comprehensive literature survey by the present authors as follows: (1) measuring the Al partition LAl between liquid Fe and Ag melts through metal–metal equilibrium approach [1, 2, 3, 4, 5]; (2) measuring the electromotive force (EMF) by a galvanic cell of Al ion sensor in Fe–Al binary melts [6, 8, 14, 20]; (3) measuring the Al or Al-containing compounds vapor pressure in Fe–Al binary melts at the elevated temperatures by transportation, bubbling or isopiestic method [7, 9, 11, 12]; (4) determining the partial molar mixing enthalpy change ΔmixHm,Al of Al in Fe–Al binary melts by high temperature solution calorimeter [10]; (5) measuring the ion current ratio of Fe–Al binary melts through Knudsen cell–mass spectrometer [13, 15, 17, 18]; and (6) measuring nitrogen solubility through metal–nitride–gas equilibrium under various nitrogen partial pressures[19]. Numerous data sources [3, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20] on activity coefficient γAl of Al in Fe–Al binary melts are available in the literature; however, insufficient attention has been paid by researchers to develop a prediction model of activity coefficient γAl of Al in the full composition range of Fe–Al binary melts.

With the rapid development of computing science over the past several decades, the calculation of phase diagrams (CALPHAD) technique [22, 23, 24, 25] has become an important method to determine the accurate information of Fe–Al binary phase diagram [26, 27, 28] from 298 K (25 °C) to above the liquidus temperatures based on accumulated thermodynamic properties of Fe–Al binary melts, certainly including the accumulated data of activity aR,Al of Al in Fe–Al binary melts. Sundman et al. [22] simulated the integral molar Gibbs energy change ΔmixGm,Fe−Al of Fe–Al binary melts using a four-sublattice model based on the compound energy formalism (CEF). However, discrepancies between predicted results of the integral molar mixing enthalpy change ΔmixHm,Fe−Al of Fe–Al binary melts by Sundman et al. [22] and experimental results was too great, as pointed out by Paek et al. [24] In order to take into account the strong short-range ordering (SRO) exhibited in Fe–Al binary melts over the entire concentration range, the modified quasi-chemical model (MQM) [29, 30, 31] in the pair approximation was recently used by Phan et al. [23] and Paek et al. [24, 25] to model Fe–Al binary phase diagram. However, investigation on thermodynamic modelling of Fe–Al binary melts at about 1873 K (1600 °C) is not sufficient except for that by Akinlade et al. [32] through assuming the existence of a complex of the form FeAl3 to describe chemical short range order due to atomic level interactions in liquid Al–Fe alloys.

Five kinds of stoichiometrical intermetallic compounds as Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 can be formed through covalent bonds in solid Fe–Al binary alloys from the phase diagram of Fe–Al binary system [26, 27, 28]. Qin et al. [33] determined the liquid structure of Fe–Al binary melts through measuring the positions and heights of the pre-peaks of liquid Al–Fe alloys using a high temperature X–ray diffractometer. It was concluded by Qin et al. [33] that the atomic configuration with the Fe2Al5 type cluster persisted into the melts in the medium range and the Fe–Al distance in liquid Fe–Al alloys was nearly the same as that in the solid state. Zaitsev et al. [34]. confirmed by the integral effusion method and Knudsen mass spectrometry that three associates or clusters as FeAl, FeAl2, and Fe2Al5 existed in Fe–Al binary melts at elevated temperatures to form icosahedral quasicrystal (i-QC) precipitation. This means that atoms of both Fe and Al as well as at least three associates or clusters of FeAl, FeAl2, and Fe2Al5 can coexist in Fe–Al binary melts.

For the purpose of emphasizing the structural property of coexistence between atoms and associates or clusters or molecules in metallic melts, the conclusions from the literature [29, 32, 33, 34, 35, 36, 37, 38, 39], which were summarized by Yang et al. [40], are briefly introduced in this contribution. Hoffman et al. [35] used the diffuse intensity to specify the state of clustering or short–range order structures in metallic melts. Lee et al. [36] verified that the metallic melts could maintain a short-range order structures with the similar structures as those of the close–packed in solid. Bouteiller et al. [37]. suggested that the short–range order structures could exist in metallic melts. Sommer et al. [32, 38, 39] proposed that the short–range order structures in metallic melts could be treated as compounds. Pelton et al. [29] advised that the nearest–neighbor pairs in metallic melts should be considered as components for simplifying computation during developing the modified quasichemical model [29, 30, 31] for metallic melts. Evidently, the so-called short-range order structures [32, 33, 35, 36, 37, 38, 39] or nearest-neighbor pairs [29, 30, 31, 37] or associative complexes [34] in metallic melts have the same meaning like molecules or compounds or components as FexAly in Fe–Al binary melts.

Therefore, it has become a consistent viewpoint from results of Zaitsev et al. [34] that five kinds of stoichiometrical intermetallic compounds or associates or clusters of Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 can constitute the equilibrium phases in Fe–Al binary system at temperatures above liquidus lines. This means that two atoms of both Fe and Al as well as five molecules or associates or clusters or short-range ordering radicals of Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 can coexist in Fe–Al binary melts. It should be clarified that the covalent bonds of the applied molecules in this study is to some degree weaker than those of traditionally defined molecules.

Similar to the proposed ion and molecule coexistence theory (IMCT) by Zhang [41] for metallurgical slags [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55], the atom and molecule coexistence theory (AMCT) [40, 41, 56, 57, 58, 59, 60, 61] was also developed to represent reaction abilities of structural units by the mass action concentrations Ni of structural units as atoms/elements or molecules in metallic melts. It should be stressed that the defined [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] mass action concentrations Ni of structural units in both metallic melts and metallurgical slags are assigned pure liquid or solid matter i as standard state, like the traditionally applied activity aR,i of component i in slags in the classical metallurgical physicochemistry. Theoretically, the hypotheses of both the IMCT [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] for metallurgical slags and the AMCT [40, 41, 56, 57, 58, 59, 60, 61] for metallic melts are to some degree similar to those of the associated solution model [38, 39, 62, 63, 64, 65, 66], especially the ideal associated solution model [62, 63, 66], although some researchers [67, 68] argued about the limitations or weaknesses of the associated solution model [62, 63, 66].

It is well known from the classical metallurgical physicochemistry [69, 70] that three activities of elements in metallic melts have been widely applied to describe the reaction ability of element i in metallic melts as activity aR,i=γixi or a%,i=f%,i[%i] or aH,i=fH,ixi relative to three standard states. Activity aR,i of element i in metallic melts cannot be greater than 1.0, while it is not the case for activities a%,i and aH,i of element i in metallic melts. The sum of activities aR,i of all elements in a real metallic melt is not absolutely equal to 1.0, i. e., ΣaR,i=Σγixi≠1.0, inasmuch as activity coefficient γi of element i is usually not equal to 1.0. However, the sum of the mass action concentrations Ni of all structural units in a metallic melt or metallurgical slag is defined [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] as unity, i. e., ΣNi=1.0, in the developed AMCT–Ni model [40, 41, 56, 57, 58, 59, 60, 61] for metallic melts and the developed IMCT–Ni model [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] for metallurgical slags. Thus, values of the mass action concentrations Ni of structural units are not absolutely equal to the defined activities aR,i in metallic melts. Under this circumstance, the defined mass action concentrations Ni of structural units in metallic melts can be only matched with activities aR,i, rather than a%,i or aH,i, in metallic melts. Whether the defined mass action concentrations Ni of structural units and the measured or reported activities aR,i of elements has the same or similar variation tendency with mass content of structural units or elements is an ideal criterion for evaluating the accuracy and feasibility of the developed AMCT–Ni model for metallic melts.

According to the accumulated experiences of predicting reaction abilities of structural units in Fe–S [56], Fe–Si [56, 57], Fe–P [40], Fe–O [59] and Fe–C [60, 61] binary melts based on the AMCT, the thermodynamic model for calculating the mass action concentrations Ni of structural units, i. e., the AMCT–Ni model, in Fe–Al binary melts at elevated temperatures has been developed in this study. According to the critical assessment of nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] on activity coefficient γAl of Al as well as 14 collected data sources [1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 17, 19] on Raoultian activity coefficients γi0 of Al or Fe in Fe–Al binary melts, the data sources with ideal precision have been selected as criteria to verify the accuracy and feasibility of the developed AMCT–Ni thermodynamic model for Fe–Al binary melts. The ultimate aim of this study is to pave the way for developing a universal AMCT–Ni thermodynamic model for representing the reaction abilities of structural units or elements in any binary metallic melts by the defined mass action concentrations Ni of structural units.

Assessment of nine collected data sources on activity aR,i of Al or Fe in Fe–Al binary melts

Nine data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] on activity aR,i of Al or Fe in Fe–Al binary melts at various temperatures greater than 1373 K (1100 °C) were collected from an extensive literature survey by the present authors. Theoretically, measuring or determining activity aR,i of Al or Fe in nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] means obtaining activity coefficient γi of Al or Fe. Actually, nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] reported activity aR,Al of Al in Fe–Al binary melts from the standpoint of ferrous metallurgy, rather than the standpoint of aluminum refining. The expressions of activity coefficient γi of Al or Fe in Fe–Al binary melts at various temperatures from nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] have been chronologically summarized in Table 1. The available experimental methods in each data source are also described briefly in Table 1 for comparison. It should be pointed out that data source number of nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] in Table 1 is nominated as Di for easy distinction in the following text. In order to critically assess the accuracy, nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] on activities aR,i of Al or Fe in Fe–Al binary melts in Table 1 should be first classified.

Table 1:

Summary of reported activity coefficient γAl of Al in Fe–Al binary melts from various data sources and reported or calculated activity coefficient γFe of Fe at various temperatures greater than 1373 K (1100 °C).

No.Expressions of γγi of both Al and Fe (–)Experimental or calculation methodResearchersYearRef.Category(–)Note
D1(a)T=1873 K (1600 °C), xAl<0.2lnγAl=−3.48+5.99xAllnγFe=−1.55+7.78×10−5xAl−3.32xAl2Distribution of liquid Al between liquid Fe and Ag at 1873 K (1600 °C).Chipman and Floridis1955[3]I
D2(a)T=1873 K (1600 °C), 0<xAl<0.42lnγAl=−2.80+5.60xAllnγFe=−1.03×10−17−1.76×10−16xAl−3.12xAl2High-temperature solution calorimeter at 1873 K (1600 °C).Wooley and Elliott1967[10]I
D3(b)T=1588 K (1315 °C), 0.6<xAl<1.0lnγAl=−6.49+12.85xAl−6.37xAl2lnγFe=5.48−9.61xAl−0.36xAl2Transportation method by Al vapor pressure at 1588 K (1315 °C).Coskun and Elliott1968[11]II
D4(b)T=1573 K (1500 °C), 0.6<xAl<1.0lnγAl=2.76+5.35xAl−2.59xAl2lnγFe=5.48−9.61xAl−0.36xAl2Bubbling method by vapor pressure at 1473 K and 1373 K (1200 °C and 1100 °C).Mitani and Nagai1968[12]II
T=1473 K (1200 °C), 0.6<xAl<1.0lnγAl=0.53−1.65xAl+1.11xAl2lnγFe=−19.16+45.78xAl−27.51xAl2
D5T=1873 K (1600 °C), 0<xAl<1.0lnγAl=−3.77+11.89xAl−13.39xAl2+5.30xAl3lnγFe=0.158−2.02xAl+0.74xAl2−1.39xAl3Knudsen cell–mass spectrometer combination at 1873 K (1600 °C).Belton and Fruehan1969[13]IIISCd)
D6(a)T=1873 K (1600 °C), 0<xAl<0.03 lnγAl=−3.88+1.37xAl+325.13xAl2lnγFe=−4.438×10−7−1.316×10−4xAl−5.43xAl2EMF at 1873 K (1600 °C).Fruehan1970[14]I
D7(c)T=1673 K (1400 °C), 0.6<xAl<1.0lnγAl=−4.18+8.36xAl−4.18xAl2lnγFe=0.105+0.1987xAl−4.31xAl2Knudsen cell–mass spectrometer combination at 1673 K (1400 °C).Ichise et al.1977[15]IIISC
D8T=1873 K (1600 °C), 0<xAl<1.0 lnγAl=−2.88+4.87xAl−1.42xAl2−0.563xAl3lnγFe=0.0101−0.379xAl−0.789xAl2−2.448xAl3Critical literature review and evaluation at 1873 K (1600 °C).Desai1987[16]IIISC
D9(c)T=1573 K (1300 °C), 0.6<xAl<1.0lnγAl=−4.29+8.67xAl−4.38xAl2lnγFe=0.517−1.106xAl−3.347xAl2Knudsen cell–mass spectrometer combination at 1573 K (1300 °C).Jacobson and Mehrotra1993[17]IIISC
D101823 K (1550 °C)<T< 1973 K (1700 °C), 0<xAl<1.0lnγAl=−9,646.5T+2.196+(−6.753+22,741.8T)xAl+(−12,937.4T+4.497)xAl2lnγAl=−9,646.5T+2.196+−6.753+22,741.8TxAl+−12,937.4T+4.497xAl2AMCT–Ni model at 1823–1973 K (1550–1700 °C)Present authorsThis studyIII

  1. Note:

  2. (a) Expression of activity coefficient γFe of Fe was obtained from reported results of activity coefficient γAl of Al in Fe–Al binary melts through Darken’s quadratic formalism.

  3. (b) Expression of activity coefficient γFe of Fe was obtained from reported results of γAl through the No.1 Gibbs–Duhem integration method in eq. (2).

  4. (c) Expression of activity coefficient γFe of Fe was only regressed in Fe–Al binary melts with mole fraction of Al greater than 0.60 at temperatures lower than liquidus temperature.

  5. (d) SC means the original data of activity coefficient γFe of Al and γAl of Fe in Fe–Al binary melts are self-consistent, which confirmed by Gibbs–Duhem integration method in eq. (2) or eq. (4).

Classification of nine collected data sources on activity aR,i of Al or Fe in Fe–Al binary melts

Nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] on activity coefficient γi of Al or Fe in Fe–Al binary melts can be classified into three categories as I, II, and III in the seventh column of Table 1. Three data sources [3, 10, 14] of the first category by Chipman and Floridis [3] (D1), Wooley and Elliott [10] (D2), and Fruehan [14] (D6) reported activity coefficient γAl of Al in the Fe–rich corner of Fe–Al binary melts only at 1873 K (1600 °C). Two data sources [11, 12] of the second category by Coskun and Elliott [11] (D3) as well as Mitani and Nagai [12] (D4) provided activity coefficient γAl of Al in the Al–rich corner of Fe–Al binary melts at temperatures higher than 1373 K (1100 °C). Four data sources [13, 15, 16, 17] of the third category by Belton and Fruehan [13] (D5), Ichise et al. [15] (D7), Desai [16] (D8), as well as Jacobson and Mehrotra [17] gave activity coefficients γi of both Al and Fe in the full composition range of Fe–Al binary melts at temperatures higher than 1573 K (1300 °C).

Among the third category of four data sources [13, 15, 16, 17] in Table 1, Ichise et al. [15] (D7) reported activity coefficients γi of both Al and Fe in the full composition range of Fe–Al binary melts at 1673 K (1400 °C), while Jacobson and Mehrotra [17] (D9) provided activity coefficients γi of both Al and Fe at 1573 K (1300 °C). Certainly, temperatures at 1673 K (1400 °C) and 1573 K (1300 °C) were significantly smaller than melting point of pure Fe at 1811 K (1538 °C), but greater than melting point of pure Al at 933.45 K (660.45 °C) from Fe–Al binary phase diagram [26, 27, 28]. Therefore, some results of activity coefficients γi of both Al and Fe in the Fe–rich corner of Fe–Al binary melts by Ichise et al. [15] as well as Jacobson and Mehrotra [17] (D9) were only available in solid and liquid two-phases zone, rather than in the liquid Fe–Al binary melts.

Four data sources [13, 15, 16, 17] of the third category reported activity coefficients γi of both Al and Fe, other five data sources [3, 10, 11, 12, 14] of the first and second categories only provided activity coefficient γAl of Al, by which no results of activity coefficient γFe of Fe was provided. Under this circumstance, activity coefficient γFe of Fe in aforementioned five data sources [3, 10, 11, 12, 14] of the first and second categories should be calculated by Gibbs–Duhem integral equation [69, 70, 71, 72] or Darken’s quadratic formalism [73, 74, 75]. Meanwhile, the self-consistency of reported results between activity coefficient γAl of Al and γFe of Fe in four data sources [13, 15, 16, 17] of the third category should also be assessed by Gibbs–Duhem integral equation [69, 70, 71, 72].

Assessment of nine collected data sources on activity aR,i of Al or Fe in Fe–Al binary melts

Primary assessment of nine collected data sources on activity coefficient γAl of Al in Fe–Al binary melts

Comparison among reported results of activity coefficient lnγAl of Al for Fe–Al binary melts from nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] in Table 1 is shown in Figure 1. To display clearly, relationships of reported results of lnγAl from nine aforementioned data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] against mole fraction xAl of Al are illustrated in three sub-figures of Figure 1. Each sub-figure of Figure 1 only shows results of lnγAl in the same category, meanwhile the reported results of lnγAl at 1873 K (1600 °C) from critical assessment in the available literature prior to 1987 by Desai [16] (D8) are displayed in three sub-figures of Figure 1 as basis line. Obviously, the reported results of lnγAl from aforementioned nine data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] can be expressed by polynomial function as listed in Table 1, respectively.

Figure 1: Comparison among reported activity coefficient lnγAl\ln \gamma _{{\rm{Al}}} of Al relative to pure liquid Al(l) as standard state for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl{x_{{\rm{Al}}}} of Al at various temperatures greater than 1373 K (1100 °C), respectively.
Figure 1:

Comparison among reported activity coefficient lnγAl of Al relative to pure liquid Al(l) as standard state for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl of Al at various temperatures greater than 1373 K (1100 °C), respectively.

With respect to the reported [3, 10, 14] results of lnγAl in the Fe–rich corner of Fe–Al binary melts in Figure 1(a), i. e., the first category, the reported results of lnγAl by Wooley and Elliott [10] (D2) are in consistency with those by Desai [16] (D8) under condition of mole fraction xAl of Al less than 0.10, otherwise the reported results by Wooley and Elliott [10] (D2) are slightly greater than those by Desai [16] (D8). However, the reported results of lnγAl by Chipman and Floridis [3] (D1) as well as Fruehan [14] (D6) are smaller than those by Desai [16] (D8) under condition of mole fraction xAl of Al less than 0.20 or 0.03, respectively. The discrepancies between reported results by Desai [16] (D8) and those by Chipman and Floridis [3] (D1) or Fruehan [14] (D6) might be attributed to experimental uncertainties.

The reported [11, 12] results of lnγAl in the Al–rich corner of Fe–Al binary melts in Figure 1(b), i. e., the second category, by Coskun and Elliott [11] (D3) as well as Mitani and Nagai [12] (D4) are in good agreement with those by Desai [16] (D8) as criteria as shown in Figure 1(b) although there is a large temperature gap of 500 K (500 °C) from 1873 to 1373 K (1600 to 1100 °C) as listed in Table 1.

The reported results of lnγAl in the full composition range of Fe–Al binary melts in Figure 1(c), i. e., the third category, by Belton and Fruehan [13] (D5) are slightly greater than those by Desai [16] (D8) as criteria. The reported results of lnγAl by Ichise et al. [15] (D7) at 1673 K (1400 °C) as well as by Jacobson and Mehrotra [17] (D9) at 1573 K (1300 °C) are in good accord with those by Desai [16] (D8) at 1873 K (1600 °C) as criteria. It should be noted that results of lnγAl with mole fraction xAl of Al less than 0.6 by Ichise et al. [15] (D7) at 1673 K (1400 °C) and by Jacobson and Mehrotra [17] (D9) at 1573 K (1300 °C) are not displayed in Figure 1(c) due to the existence of two phases of solid and liquid in the melts and beyond the investigation interests in this study.

Methods of determining activity coefficient γFe of Fe in Fe–Al binary melts at fixed temperature

It is well known that accurate expression of activity coefficient γi of element i or component i should obey Gibbs–Duhem integral equation with the other element j or component j in an i–j binary system by [69, 70, 71, 72]

(1)xidlnaR,i+xjdlnaR,j=0;xidlnγi+xjdlnγj=0

According to the reported activity coefficient γAl of Al in the full composition range of Fe–Al binary melts at an interval of mole fraction xAl as 0.1 at a fixed temperature, activity coefficient γFe of Fe in Fe–Al binary melts can be calculated from eq. (1) through

(2)lnγFe=xAl=0xAl−xAlxFedlnγAl0<xAl<1.0

Using alpha function αj of element j as αj=lnγj/1−xj2=lnγj/xi2 in an i–j binary system with element j as solute, Gibbs–Duhem integral equation in eq. (1) can also be rewritten as [4, 72]

(3)lnγi=−αjxixj−xi=1.0xiαjdxi0<xi<1.0

With regard to four data sources [13, 15, 16, 17] of the third category in Figure 1(c) or Table 1, activity coefficient γFe of Fe in the full composition range of Fe–Al binary melts can be calculated through eq. (3) as

(4)lnγFe=−αAlxAlxFe−xFe=1.0xFeαAldxFe0<xFe<1.0

where the applied alpha function αFe of Fe in eq. (4) for Fe–Al binary melts is defined as [4, 72]

(5)αAl=lnγAl1−xAl2=lnγAlxFe2

It should be emphasized that the second term of Gibbs–Duhem integral equation in eq. (4) is equal to Raoultian activity coefficient γFe0 of Fe in the Al–rich corner of Fe–Al binary melts as [4, 72]

(6)lnγFe0=−xFe=1.0xFeαAldxFe0<xFe<1.0

In order to distinguish two kinds of Gibbs–Duhem integral equation in eqs (2) and (4), eq. (2) is named as the first method, while eq. (4) is assigned as the second method. Theoretically, the calculated results of activity coefficient γFe of Fe by eq. (2) should be in good consistency with those by eq. (4) based on the same results of activity coefficient γAl of Al. Considering the advantages of the second method in eq. (4) that Raoultian activity coefficient γFe0 of Fe in the Al–rich corner of Fe–Al binary melts can be accurately calculated by eq. (6), the second method in eq. (4) is applied to calculate activity coefficient γFe of Fe in the full composition range of Fe–Al binary melts.

With respect to the Fe-rich corner of Fe–Al binary melts, i. e., three data sources [3, 10, 14] of the first category in Figure 1(a) or Table 1, activity coefficient γFe of Fe as solvent can also be calculated by Darken’s quadratic formalism [73, 74, 75] as follows

(7)lnγFe=αFe−AlxAl20<xAl<0.2−0.4

The involved parameter αFe−Al in Darken’s quadratic formalism in eq. (7) can be obtained through the expression of lnγAl of Al as solute by [73, 74, 75]

(8)lnγAl=lnγAl0+αFe−AlxFe2−1=lnγAl0+αFe−Al−2xAl+xAl20<xAl<0.2−0.4

It can be deduced from Darken’s quadratic formalism in eqs (7) and (8) that it is not a difficult task to calculate activity coefficient γFe of Fe as solvent from the known activity coefficient γAl of Al as solute.

With regard to the Al-rich corner of Fe–Al binary melts, i. e., two data sources [11, 12] of the second category in Figure 1(b) or Table 1, Al is solvent relative to Fe as solute. Thus, activity coefficient γFe of Fe as solute in the Al–rich corner of Fe–Al binary melts cannot be calculated by Darken’s quadratic formalism from reported γAl of Al as solvent through [73, 74, 75]

(9)lnγAl=αAl−FexFe20<xFe<0.2−0.4
(10)lnγFe=lnγFe0+αAl−FexAl2−1=lnγFe0+αAl−Fe−2xFe+xFe20<xFe<0.2−0.4

Under this circumstance, activity coefficient γFe of Fe as solute in the Al–rich corner of Fe–Al binary melts was calculated by the first method in eq. (2) based on reported activity coefficient γAl of Al as solvent. After that, the involved parameter αAl−Fe in eq. (10) can be obtained according to the calculated activity coefficient γFe of Fe by the first method in eq. (2). Finally, the calculated activity coefficient γAl of Al by eq. (9) must be verified to be accurate enough with the reported ones. In one word, activity coefficient γFe of Fe as solute in the Al–rich corner of Fe–Al binary melts cannot be directly calculated through Darken’s quadratic formalism in eqs (9) and (10).

Thus, the absent results of activity coefficient γFe of Fe in aforementioned five data sources [3, 10, 11, 12, 14] in “Classification of nine collected data sources on activity aR,i of Al or Fe in Fe–Al binary melts” allotted to the first and second categories can be calculated by one or two of aforementioned three methods.

Assessment of self-consistency between activity aR,Al of Al and aR,Fe of Fe for three categories of collected data sources in Fe–Al binary melts

Taking the reported results of activity coefficient γAl of Al by Chipman and Floridis [3] (D1) as examples in the first category, comparison among calculated activity coefficient lnγFe of Fe by aforementioned two methods of Gibbs–Duhem integral equations in eq. (2) or eq. (4) and Darken’s quadratic formalism in eqs (7) and (8) for the Fe–rich corner of Fe–Al binary melts at 1873 K (1600 °C) is illustrated in Figure 2(a). Obviously, results of lnγFe by aforementioned two methods in eq. (2) or eq. (4) are in good consistency with those by Darken’s quadratic formalism in eqs (7) and (8) under the condition of mole fraction xAl of Al less than 0.20.

Figure 2: Comparison among calculated activity coefficient lnγFe\ln \gamma _{{\rm{Fe}}} of Fe based on activity coefficient lnγAl\ln \gamma _{{\rm{Al}}} of Al by No. 1 and No. 2 Gibbs–Duhem integral method or Darken’s quadratic formalism for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl{x_{{\rm{Al}}}} of Al at various temperatures greater than 1373 K (1100 °C), respectively.
Figure 2:

Comparison among calculated activity coefficient lnγFe of Fe based on activity coefficient lnγAl of Al by No. 1 and No. 2 Gibbs–Duhem integral method or Darken’s quadratic formalism for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl of Al at various temperatures greater than 1373 K (1100 °C), respectively.

With respect to the Al–rich corner of Fe–Al binary melts, taking the reported results of activity coefficient γAl of Al at 1588 K (1315 °C) by Coskun and Elliott [11] (D3) as examples in the second category, the calculated results of lnγFe by the first method in eq. (2) are more or less precise than those by the second method in eq. (4) as shown in Figure 2(b). However, no accurate results of lnγFe can be calculated by Darken’s quadratic formalism in eqs (9) and (10) from the reported results of activity coefficient γAl of Al. Therefore, direct application of Darken’s quadratic formalism in eqs (9) and (10) for calculating activity coefficient γFe of Fe as solute in the Al–rich corner of Fe–Al binary melts. i. e., the second category, is not a recommended method.

With regard to the full composition range of Fe–Al binary melts, taking the reported results of activity coefficient γAl of Al at 1873 K (1600 °C) by Belton and Fruehan [13] (D5) as examples in the third category, the calculated results of lnγFe by the first method in eq. (2) are almost identical with those by the second method in eq. (4) as shown in Figure 2(c). In addition, the calculated results of lnγFe by two methods in eqs (2) and (4) are in good consistency with those by Darken’s quadratic formalism in eqs (7) and (8) under the condition of mole fraction xAl of Al less than 0.50. However, Darken’s quadratic formalism in eqs (7) and (8) cannot be applied to obtain ideal results of lnγFe under the condition of mole fraction xAl of Al beyond 0.5 due to the intrinsic limitation of Darken’s quadratic formalism in eqs (7) and (8) as discussed in “Methods of determining activity coefficient γFe of Fe in Fe–Al binary melts at fixed temperature”. Thus, the second method in eq. (4) is applied to calculate activity coefficient γFe of Fe in the full composition range of Fe–Al binary melts because besides the required activity coefficient γFe of Fe, accurate results of Raoultian activity coefficient γFe0 of Fe in the Al-rich corner of Fe–Al binary melts can also be calculated.

Assessment of reported or obtained activity coefficient γFe of Fe in Fe–Al binary melts

Comparison among results of activity coefficient lnγFe of Fe for Fe–Al binary melts at different temperatures greater than 1373 K (1100 °C) from nine data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] in Table 1 is shown in Figure 3. It should be pointed out that the reported results of activity coefficient γFe of Fe from four data sources [13, 15, 16, 17] in the third category have been verified to be self-consistency with activity coefficient γAl of Al, while the results of activity coefficient γFe of Fe from other five data sources [3, 10, 11, 12, 14] in the first and second categories were calculated by one or two of three aforementioned methods described in details in “Methods of determining activity coefficient γFe of Fe in Fe–Al binary melts at fixed temperature”. To display clearly, relationships of results of lnγFe against mole fraction xAl of Al from nine data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] for Fe–Al binary melts are illustrated in three sub-figures of Figure 3. Similar to Figure 1, each sub-figure of Figure 3 only shows results of lnγFe in the same category, meanwhile results of lnγFe in Fe–Al binary melts by Desai [16] (D8) at 1873 K (1600 °C) are displayed as basis line in three sub-figures of Figure 3. Obviously, results of lnγFe from nine data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] for Fe–Al binary melts can be expressed by polynomial function as listed in Table 1, respectively.

Figure 3: Comparison among reported or calculated activity coefficient lnγFe\ln {\gamma _{{\rm{Fe}}}} of Fe relative to pure liquid Fe(l) as standard state for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl{x_{{\rm{Al}}}} of Al at various temperatures greater than 1373 K (1100 °C), respectively.
Figure 3:

Comparison among reported or calculated activity coefficient lnγFe of Fe relative to pure liquid Fe(l) as standard state for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl of Al at various temperatures greater than 1373 K (1100 °C), respectively.

It can be observed in Figure 3(a) that results of lnγFe after Chipman and Floridis [3] (D1), Wooley and Elliott [10] (D2), and Fruehan [14] (D6) in the first category are in good agreement with those by Desai [16] (D8) under the condition of mole fraction xAl of Al less than 0.40. However, it can be observed in Figure 3(b) that results of lnγFe after Coskun and Elliott [11] (D3) in the second category are slightly smaller than those by Desai [16] (D8) at 1873 K (1600 °C) under the condition of mole fraction xAl of Al greater than 0.50, while results of lnγFe after Mitani and Nagai [12] (D4) at 1573 K and 1473 K (1300 °C and 1200 °C) are greater than those by Desai [16] (D8) at 1873 K (1600 °C).

Figure 3(c) indicates that results of lnγFe by Belton and Fruehan [13] (D5) in the third category are slightly smaller than those by Desai [16] (D8) at 1873 K (1600 °C) under the condition of mole fraction xAl of Al less than 0.70, otherwise larger results of lnγFe can be obtained by Belton and Fruehan [13] (D5). The reported results of lnγFe by Ichise et al. [15] (D7) at 1673 K (1400 °C) are in good agreement with those by Jacobson and Mehrotra [17] (D9) at 1573 K (1300 °C) although there is a temperature gap of 100 K (100 °C). However, the reported results of lnγFe by Ichise et al. [15] (D7) at 1673 K (1400 °C) as well as by Jacobson and Mehrotra [17] (D9) at 1573 K (1300 °C) are smaller than those by Desai [16] (D8) at 1873 K (1600 °C). Considering the temperature gap of 200 K (200 °C) between data source by Desai [16] (D8) at 1873 K (1600 °C) and that by Ichise et al. [15] (D7) at 1673 K (1400 °C), the reported results of activity coefficients γi of both Al and Fe by Ichise et al. [15] (D7) at 1673 K (1400 °C) are thought to be more accurate than those by Jacobson and Mehrotra [17] (D9) at 1573 K (1300 °C). In addition, the discrepancies among results of lnγFe of Fe in Figure 3 are absolutely caused from differences among lnγAl of Al in Figure 1.

Combining activity coefficients γAl of Al in Figure 1(c) with activity coefficients γFe of Fe in Figure 3(c), the reported activity coefficients γi of both Al and Fe by Ichise et al. [15] (D7) at 1673 K (1400 °C) as well as by Desai [16] (D8) at 1873 K (1600 °C) can be treated as reliable ones in Fe–Al binary melts.

Assessment of reported Raoultian activity coefficients γi0 of Al and Fe in Fe–Al binary melts

Based on a thorough literature survey by the present authors, 14 data sources [1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 17, 19] on Raoultian activity coefficient γi0 of Al or Fe in Fe–Al binary melts at various temperatures were collected as listed chronologically in Table 2. Among 14 collected data sources [1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 17, 19], 11 data source [1, 2, 3, 4, 5, 6, 10, 13, 14, 15, 16, 19] reported results of γAl0 in the Fe-rich corner of Fe–Al binary melts, while six data sources [11, 12, 13, 15, 16, 17] provided results of γFe0 in the Al-rich corner of Fe–Al binary melts. In order to easily distinguish 11 data sources [1, 2, 3, 4, 5, 6, 10, 13, 14, 15, 16, 19] on γAl0 as well as six data sources [11, 12, 13, 15, 16, 17] on γFe0, 14 collected data sources [1,2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 17, 19] on γi0 of Al or Fe in Table 2 was nominated as Si in the following text. It can be found through comparing Table 1 with Table 2 that all nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] on activity coefficient γi of Al or Fe in Table 1 are embodied in 14 collected data sources [1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 17, 19] on γi0 of Al or Fe in Table 2. Thus, nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] assigned as Di in Table 1 were re-nominated as Si in Table 2, while the originally assigned numbers as Di in Table 1 are also listed in Si after equal symbol in Table 2.

Table 2:

Summary of expressions of Raoultian activity coefficients γi0 of both Al and Fe and corresponding standard molar Gibbs free energy change ΔsolGm,i(l)→[i][%i]=1.0Θ,% of dissolved liquid Al or Fe for forming [% Al] or [% Fe] as 1.0 referred to 1 mass percentage of Al or Fe as reference state in Fe–Al binary melt from various data sources.

No.T [K(°C)]ln γγAl0 (–)ln γFe0 (–)ΔsolGm,i(l)→[i][\bpercnti]=1.0Θ,\bpercnt (J/mol)εAlAl (–)εFeFe (–)eAlAl (–)Experimental or calculation methodResearchersYearRef.
i=Ali=Fe
S11873 (1600)−3.69NDa)−117,994.07NDNDNDNDRecalculated from experimental data of Al distribution between liquid Fe and Ag at 1873 K (1600 °C) by Chipman in 1934[1].Chipman1934,1948[1, 2]
S2=D11873 (1600)−3.48ND−114,690.61ND6.6ND0.049Same as that of D1 in Table 1.Chipman and Floridis1955[3]
S31823–1923 (1550–1650)NDND−48,976.2 − 32.32TNDNDNDNDRecalculation of Chipman’s data in 1934[1], in which Al distribution between liquid Fe and Ag at 1873 K (1600 °C).Chou and Elliott1956[4]
1873 (1600)−3.15ND−109,548.93NDNDNDND
S41873 (1600)−3.00ND−107,199.79NDNDNDNDRecalculation of experimental data by Chipman and Floridis in 1955[3], in which Al distribution between liquid Fe and Ag at 1873 K (1600 °C).Pehlke1958[5]
S51873 (1600)−2.76ND−103,601.39ND5.3ND0.043EMF at 1873 K (1600 °C).Wilder and Elliott1960[6]
S6=D21873 (1600)−2.80ND−10,3601.39ND5.6ND0.045Same as that of D2 in Table 1.Wooley and Elliott1967[10]
S7=D3b)1588 (1315)ND−3.42ND−115,425.01ND13.4NDSame as that of D3 in Table 1.Coskun and Elliott1968[11]
S8=D4b)1473 (1200)ND−4.48ND−120,094.48ND6.0NDSame as that of D4 in Table 1.Mitani and Nagai1968[12]
1373 (1100)ND−0.89ND−70,961.14ND3.8ND
S9=D5b)1873 (1600)−3.77−2.51−119,314.12−122,096.7310.24.40.087Same as that of D5 in Table 1.Belton and Fruehan1969[13]
S10=D61873 (1600)−3.88ND−120,902.71ND10.8NDNDSame as that of D6 in Table 1.Fruehan1972[14]
S11=D71673–1873 (1400–1600)−10,830T+3.01440T−4.30−90,040.62 − 7.30T3,658.1 − 80.04233T5,760T+3.92,280T+6.9390.9T+0.004Same as that of D7 in Table 1.Ichise et al.1977[15]
1873 (1600)−3.02−4.07−103,718.92−146,261.136.48.20.049
S12=D8b)1873 (1600)−2.90−3.60−105,709.54−139,018.805.284.4NDSame as that of D8 in Table 1.Desai1987[16]
S13=D9 b)1573 (1300)ND−3.96ND−121,460.09ND8.06NDSame as that of D9 in Table 1.Jacobson and Mehrotra1993[17]
S14c)d)1823–1923 (1550–1650)−279.83T−1.224ND–23,257.11–42.49TND2,224T+1.17ND111T−0.016 or 20.0T+5.87×10−3metal–nitride-gas equilibration under various nitrogen partial pressures.Kim et al.2007[19]
1873 (1600)−2.71–102,823.282.36ND0.043
S15=D101823–1973 (1550–1700)−7,266.2T+1.06−6,799.1T−1.367×10−2−80,199.53 − 14.07T−9,772.67 − 69.41T22,612.3T−6.708−29,523.3T+25.285203.3T−6.49×10−2AMCT–Ni model at 1823–1973 K (1550–1700 °C)Present authorsThis study
1873 (1600)−2.95−3.65−106,554.48−139,779.975.369.520.044

  1. Note:

  2. (a) ND means no datum or expression was reported by the original author(s).

  3. (b) Result of εFeFe was obtained by the present authors through eq. (20) based on the reported results of activity coefficient γAl of Al by the original researchers, in which the involved parameter αAl−Fe in eq. (20) was calculated by Darken’s quadratic formalism in eq. (10).

  4. (c) Expression of εAlAl as εAlAl=2,224T+1.17 was obtained by the present authors through relationship between εAlAl and αFe−Al in eq. (19) based on the reported αFe−Al.

  5. (d) Expression of eAlAl as eAlAl=20.0T+5.87×10−3 was obtained by the present authors through relationship between εAlAl and eAlAl in eq. (49) based on the reported αFe−Al.

Assessment of 11 collected data sources on Raoultian activity coefficient γAl0 of Al in Fe-rich corner of Fe–Al binary melts

The defined Raoultian activity coefficient γj0 of element j in the i-rich corner of i–j binary melts can be applied to link three activity coefficients as γj or f%,j or fH,j coupled with three activities as aR,j or a%,j or aH,j of element j relative to three different standard states by [57, 59, 60, 69, 72]

(11)γj0=γjf%,j×1/Mj+(100−1)/Mi[% j]/Mj+100−[% j]/Mi
(12)γj0=γjfH,j

In addition, Raoultian activity coefficient γj0 of element j in the i-rich corner of i–j binary melts plays a pivotal role in the standard molar Gibbs free energy change ΔsolGm,j(l)→[j][%j]=1.0Θ,% of dissolved liquid j(l) for forming [% j] as 1.0 in i–j binary melts referred to 1 mass percentage of j as reference state by [57, 59, 60, 69, 72]

(13)jl=[j (in liquidi,%j]=1.0),ΔsolGm,j(l)→[j][%j]=1.0Θ,%=RTln1/Mj1/Mj+99/Miγj0J/mol

To Fe–Al binary melts, the standard molar Gibbs-free energy change ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% and ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% in Fe–Al binary melts can be described as

(14)Al(l)=[Al](in liquid Fe, [%Al]=1.0),ΔsolGmΘ,Al%(l)→[Al][%Al]=1.0=RTln(1/MAl1/MAl+99/MFeγAl0)(J/mol)
(15)Fe(l)=[Fe](in liquid Al, [%Fe]=1.0),ΔsolGmΘ,Fe(l)→[Fe][%Fe]=1.0=RTln(1/MFe1/MFe+99/MAlγAl0)(J/mol)

Comparison among 11 collected data sources [1, 2, 3, 4, 5, 6, 10, 13, 14, 15, 16, 19] on γAl0 in the Fe-rich corner of Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C) is shown in Figure 4(a). Obviously, all 11 collected data source [1, 2, 3, 4, 5, 6, 13, 14, 15, 16, 19] on γAl0 were at 1873 K (1600 °C), which was higher than melting point of pure Fe at 1811 K (1538 °C) [26, 27, 28]. Only two data sources [15, 19] on γAl0 by Ichise et al. [15] (S11=D7) and Kim et al. [19] (S14) were expressed by formulae including temperature effect. The reported results of lnγAl0 at 1873 K (1600 °C) display a large variation range from −2.71 to −3.88. However, the reported values of lnγAl0 by four data sources [6, 10, 15, 19] such as by Wilder and Elliott [6] (S5) as well as Wooley and Elliott [10] (S6=D2) fluctuate in a narrow variation range from −2.70 to −2.78 at 1873 K (1600 °C), which are in good agreement with the reported ones from the proposed expressions of lnγAl0 by Ichise et al. [15] (S11=D7) and Kim et al. [19] (S14).

Figure 4: Comparison of reported results of Raoultian activity coefficient lnγAl0\ln \gamma _{{\rm{Al}}}^0 of Al by various researchers with obtained ones by developed AMCT–Ni{N_i} model (a) or comparison of reported results of standard molar Gibbs free energy change ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,%{\Delta _{{\rm{sol}}}}G_{{\rm{m, Al(l)}} \to {{{\rm{[Al]}}}_{{\rm{ [ \% Al] = 1}}{\rm{. 0}}}}}^{\Theta {\rm{, \% }}} by various researchers with obtained ones by developed AMCT–Ni{N_i} model (b) for Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.
Figure 4:

Comparison of reported results of Raoultian activity coefficient lnγAl0 of Al by various researchers with obtained ones by developed AMCT–Ni model (a) or comparison of reported results of standard molar Gibbs free energy change ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% by various researchers with obtained ones by developed AMCT–Ni model (b) for Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.

Similarly, comparison among the calculated ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% by 11 collected data sources [1, 2, 3, 4, 5, 6, 10, 13, 14, 15, 16, 19] on γAl0 is also displayed in Figure 4(b). The calculated results of ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% by 11 collected data sources [1, 2, 3, 4, 5, 6, 13, 14, 15, 16, 19] display a significant fluctuation range from −103 to −118 kJ/mol at 1873 K (1600 °C). However, the calculated results of ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% by eight collected data source [1, 2, 4, 5, 6, 10, 15, 16, 19] change in a narrow range from −103 to −109 kJ/mol at 1873 K (1600 °C), which are in good agreement with the reported ones by Ichise et al. [15] (S11=D7) and Kim et al. [19] (S14). Thus, the proposed expressions of γAl0 by Ichise et al. [15] (S11=D7) and Kim et al. [19] (S14) can be recommended to be accurate ones.

Assessment of six collected data sources on Raoultian activity coefficient γFe0 of Fe in Al-rich corner of Fe–Al binary melts

Comparison among six collected data sources [11, 12, 13, 15, 16, 17] on γFe0 in the Al-rich corner of Fe–Al binary melts over a temperature range from 1373 to 1973 K (1100 to 1700 °C) is shown in Figure 5(a). Because melting point of Al at 933.45 K (660.45 °C) is largely smaller than that of Fe at 1811 K (1538 °C), the temperature range of six collected data sources [11, 12, 13, 15, 16, 17] on γFe0 in the Al-rich corner of Fe–Al binary melts is larger than that of γAl0. Only one data source by Ichise et al. [15] (S11=D7) provided the effect of temperature on γFe0 in the Al-rich corner of Fe–Al binary melts over a temperature range from 1673 to 1873 K (1400 to 1600 °C). The reported results of γFe0 at 1873 K (1600 °C) in Figure 5(a) display a large variation range from −2.5 to −4.0. However, effect of temperature from 1373 to 1973 K (1100 to 1700 °C) on lnγFe0 is scatter enough by six collected data sources [11, 12, 13, 15, 16, 17].

Figure 5: Comparison of reported Raoultian activity coefficient lnγFe0\ln \gamma _{{\rm{Fe}}}^0 of Fe by various researchers with obtained ones by developed AMCT–Ni{N_i} model (a) or comparison of reported results of standard molar Gibbs free energy change ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,%{\Delta _{{\rm{sol}}}}G_{{\rm{m, Fe(l)}} \to {{{\rm{[Fe]}}}_{{\rm{ [ \% Fe] = 1}}{\rm{. 0}}}}}^{\Theta {\rm{, \% }}} by various researchers with obtained ones by developed AMCT–Ni{N_i} model (b) for Fe–Al binary melts over a temperature range from 1373 to 1973 K (1100 to 1700 °C), respectively.
Figure 5:

Comparison of reported Raoultian activity coefficient lnγFe0 of Fe by various researchers with obtained ones by developed AMCT–Ni model (a) or comparison of reported results of standard molar Gibbs free energy change ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% by various researchers with obtained ones by developed AMCT–Ni model (b) for Fe–Al binary melts over a temperature range from 1373 to 1973 K (1100 to 1700 °C), respectively.

Analogically, comparison among the calculated ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% by six collected data sources [11, 12, 13, 15, 16, 17] is also displayed in Figure 5(b). Evidently, changing temperature from 1373 to 1973 K (1100 to 1700 °C) can effectively affect the calculated values of ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% by six collected data sources [11, 12, 13, 15, 16, 17] as shown in Figure 5(b) although no visible effect of temperature on lnγFe0 can be observed in Figure 5(a). The effect of temperature from 1373 to 1973 K (1100 to 1700 °C) on the determined values of ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% by six collected data sources [11, 12, 13, 15, 16, 17] can be expressed by

(16)ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,%=42,899.17−99.29TJ/mol

Assessment of six collected data sources on first-order activity interaction coefficients εjj of Al and Fe in Fe–Al binary melts

According to Darken’s quadratic formalism [73, 74, 75], activity coefficient γj of element j or component j in an i–j binary system can be expressed as follows

(17)lnγj=lnγj0+αi−jxi2−1=lnγj0+αi−j−2xj+xj2 

Thus, the first-order activity interaction coefficient εjj of element j or component j related with activity coefficient γj in an i–j binary system can be derived from the definition of εjj as

(18)εjj=limxj→0lnγjxj=∂lnγj∂xjxj→0=−2αi−j

With respect to Fe–Al binary melts, εjj of Al and Fe can be expressed by

(19)εAlAl=limxAl→0lnγAlxAl=∂lnγAl∂xAlxAl→0=−2αFe−Al
(20)εFeFe=limxFe→0lnγFexFe=limxAl→1.0lnγFexFe=∂lnγFe∂xFexFe→0=−2αAl−Fe

As mentioned in “Methods of determining activity coefficient γFe of Fe in Fe–Al binary melts at fixed temperature”, the results of activity coefficient γFe of Fe in nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] were re-calculated to verify the self-consistency between activity coefficients γi of Al and Fe based on the reported activity coefficient γAl of Al in Fe–Al binary melts in Table 1. Therefore, the reported results of εAlAl by the original authors can be verified through the calculated values of αFe−Al by eq. (19), meanwhile the absent results of εFeFe can be obtained through the calculated results of αAl−Fe by eq. (20). The reported results of εAlAl by the original authors and the obtained results of εFeFe by the present authors via eq. (20) are listed in the seventh and eighth columns of Table 2, respectively. It should be pointed out that the collected results of εAlAl were also summarized by Kim et al. [19] (S14). Evidently, significant discrepancies of reported results of εAlAl from 2.26 to 10.8 at 1873 K (1600 °C) can be found in Table 2. Meanwhile, the obtained results of εFeFe also display a large fluctuation range from 4.4 to 13.4 at the same temperature in Table 2. Selecting accurate results of εAlAl and εFeFe is pivotal to determine precise values of activity coefficient γAl of Al in the Fe–rich corner of Fe–Al binary melts and activity coefficient γFe of Fe in the Al–rich corner of Fe–Al binary melts. However, it is difficult to recommend the accurate values of both εAlAl and εFeFe in Fe–Al binary melts from Table 2 without considering the accuracy of activity coefficients γi of both Al and Fe.

AMCT–Ni Thermodynamic model for Fe–Al binary melts

Hypotheses

It has been briefly introduced in “Introduction” that two atoms of both Fe and Al as well as five molecules of Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 are assumed to coexist in Fe–Al binary melts at elevated temperatures above liquidus lines. Therefore, the AMCT [40, 41, 56, 57, 58, 59, 60, 61] can be applied to describe structural properties of Fe–Al binary melts. The hypotheses of the developed AMCT–Ni model for Fe–Al binary melts can be summarized as (1) Fe–Al binary melts at elevated temperatures above liquidus lines are composed of seven structural units including two atoms of both Fe and Al as well as five molecules of Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 according to phase diagram of Fe–Al binary system [26, 27, 28] and viewpoints described in “Introduction”; (2) each structural unit occupies its independent position in Fe–Al binary melts; (3) 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; (4) reactions of forming aforementioned five molecules are under chemical dynamic equilibrium between simple atoms of both Fe and Al; (5) seven structural units in Fe–Al binary melts bear structural continuity in the investigated composition range; (6) chemical reactions of forming five molecules from Fe and Al atoms comply with the mass action law, i. e., KciΘ,R=aci/aR,FexaR,Aly=Nci/NFexNAly with subscript ci as FexAly [40, 41, 56, 57, 58, 59, 60, 61].

Establishment of AMCT–Ni Thermodynamic model for Fe–Al binary melts

Mole numbers of two atoms as Fe and Al before equilibrium or before forming five molecules in 100 g Fe–Al binary melts are assigned as b1=nFe0 and b2=nAl0 to represent chemical composition of Fe–Al binary melts. Two atoms of both Fe and Al as well as five molecules of Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Fe–Al binary melts at temperatures of interest above liquidus lines are summarized and assigned with exclusive numbers in Table 3. The defined [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] equilibrium mole numbers ni of seven aforementioned structural units in 100 g Fe–Al binary melts at the studied temperatures are also listed in Table 3. The total equilibrium mole number Σni of all seven structural units in 100 g Fe–Al binary melts can be expressed as

(21)Σni=n1+n2+nc1+nc2+nc3+nc4+nc5=nFe+nAl+nFe3Al+nFeAl+nFeAl2+nFe2Al5+nFeAl6mol
Table 3:

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

ItemStructural units as atoms or moleculesNo. of structural unitsMole numbers ni of structural units (mol)Mass action concentrations Ni of structural units (–)
Atom (2)Fe1n1=nFeN1=n1Σni=NFe
Al2n2=nAlN2=n2Σni=NAl
Molecules(5)Fe3Alc1nc1=nFe3AlNc1=nc1Σni=NFe3Al
FeAlc2nc2=nFeAlNc2=nc2Σni=NFeAl
FeAl2c3nc3=nFeAl2Nc3=nc3Σni=NFeAl2
Fe2Al5c4nc4=nFe2Al5Nc4=nc4Σni=NFe2Al5
FeAl6c5nc5=nFeAl6Nc5=nc5Σni=NFeAl6

According to the definition of the mass action concentrations Ni for structural units based on the AMCT [40, 41, 56, 57, 58, 59, 60, 61] for metallic melts, it can be calculated by

(22)Ni=niΣni

The chemical reaction formulae of forming five molecules, the related standard reaction equilibrium constants KciΘ,R relative to pure liquid matter ci(l) as standard state, and the representations of the mass action concentrations Nci of five molecules using KciΘ,R, N1=NFe and N2=NAl based on the mass action law are summarized in Table 4.

Table 4:

Chemical reaction formulae of possibly formed molecules or compounds as Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6, their standard equilibrium constants KciΘ,R, and mass action concentrations Nci relative to pure liquid matters as standard states, and regressed expressions of standard molar Gibbs free energy changes ΔrGm,ciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts over a temperature range from 1673 to 1873 K (1400 to 1600 °C) based on the AMC.

ReactionsExpressions of Kci\RTheta,R(–)Expressions of Nci(–)Expressions of ΔrGm,ci\RTheta,R(J/mol)Ref.
3[Fe]+[Al]=[Fe3Al]Kc1Θ,R=Nc1N13N2=NFe3AlNFe3NAlNc1=Kc1Θ,RN13N2=KFe3AlΘ,RNFe3NAl−120,586.85 + 48.61T[41]
−72,933.58 + 22.96TThis study
[Fe]+[Al]=[FeAl]Kc2Θ,R=Nc2N1N2=NFeAlNFeNAlNc2=Kc2Θ,RN1N2=KFeAlΘ,RNFeNAl−47,813.257 + 7.893T[41]
−58,063.73 + 12.62TThis study
[Fe]+2[Al]=[FeAl2]Kc3Θ,R=Nc2N1N22=NFeAl2NFeNAl2Nc3=Kc3Θ,RN1N22=KFeAl2Θ,RNFeNAl2130,186.64 − 84.582T[41]
−185,903.37 + 91.12TThis study
2[Fe]+5[Al]=[Fe2Al5]Kc4Θ,R=Nc4N12N25=NFe2Al5NFe2NAl5Nc4=Kc4Θ,RN12N25=KFe2Al5Θ,RNFe2NAl5−165,372.213 + 43.05T[41]
96,818.26 − 103.84TThis study
[Fe]+6[Al]=[FeAl6]Kc5Θ,R=Nc5N1N26=NFeAl6NAl6Nc5=Kc5Θ,RN1N26=KFeAl6Θ,RNFeNAl6−14,710.17 − 18.712T[41]
−26,299.94 − 12.81TThis study

The mass conservation equations of Fe and Al atoms in 100 g Fe–Al binary melts can be established based on aforementioned definitions [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] of ni, Ni, and Σni as

(23)b1=N1+3Nc1+Nc2+Nc3+2Nc4+Nc5Σni=(N1+3Kc1Θ,RN13N2+Kc2Θ,RN1N2+Kc3Θ,RN1N22+2Kc4Θ,RN12N25+Kc5Θ,RN1N26)=(NFe+3Kc1Θ,RNFe3NAl+Kc2Θ,RNFeNAl+Kc3Θ,RNFeNAl2+2Kc4Θ,RNFe2NAl5+Kc5Θ,RNFeNAl6)Σni=nFe0mol
(24)b2=(N2+Nc1+Nc2+2Nc3+5Nc4+6Nc5)Σni=(N2+Kc1Θ,RN13N2+Kc2Θ,RN1N2+2Kc3Θ,RN1N22+5Kc4Θ,RN12N25+6Kc5Θ,RN1N26)Σni=(NAl+Kc1Θ,RNFe3NAl+Kc2Θ,RNFeNAl+2Kc3Θ,RNFeNAl2+5Kc4Θ,RNFe2NAl5+6Kc5Θ,RNFeNAl6)Σni=nAl0mol

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, the following equation can be established

(25)N1+N2+Nc1+Nc2+Nc3+Nc4+Nc5=NFe+NAl+NFe3Al+NFeAl+NFeAl2+NFe2Al5+NFeAl6=1.0

The governing equations of the developed AMCT–Ni thermodynamic model for calculating the mass action concentrations Ni of structural units in Fe–Al binary melts are comprised of the equation group of eqs (23) and (25). Obviously, there are three unknown variables as N1=NFe, N2=NAl, and Σni with three independent equations in the established equation group. The unique solution of N1=NFe, N2=NAl, and Σni can be calculated by solving the algebraic equation group of eqs (23) and (25) through combining with the definition of Ni in eq. (22) in the case of knowing values of Kc1Θ,R=KFe3AlΘ,R, Kc2Θ,R=KFeAlΘ,R, Kc3Θ,R=KFeAl2Θ,R, Kc4Θ,R=KFe2Al5Θ,R, and Kc5Θ,R=KFeAl6Θ,R of reactions for forming five associated molecules or intermetallic compounds.

Determination of standard molar Gibbs free energy changes ΔrGm,ciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts

Unfortunately, no expressions of the standard molar Gibbs-free energy changes ΔrGm,ciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts at elevated temperatures above liquidus lines can be found in the literature. Hence, no data of the standard equilibrium constants KciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts can be determined at related temperatures. However, activity coefficients γi of both Al and Fe in nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] in Table 1 have been assessed in “Assessment of nine collected data sources on activity aR,i of Al or Fe in Fe–Al binary melts”. Thus, the involved KciΘ,R of reactions for forming five associated molecules can be mathematically regressed by substituting the reliable activities aR,i of Al and Fe, such as results by Ichise et al. [15] (D7) at 1673 K (1400 °C) and by Desai [16] (D8) at 1873 K (1600 °C), for the mass action concentrations Ni of Al and Fe in the developed AMCT–Ni thermodynamic model in eqs (23) and (25) for Fe–Al binary melts.

Determination method of standard equilibrium constants KciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts

With regard to a multiple linear regression equation Y=c1X1+c2X2+⋯+cnXn, when one coefficient, take coefficient c1 of the independent variable X1 as an example, is set as unity, other coefficients such as c2, c3, ⋯, and cn can be regressed by the sophisticated commercial software if there are enough groups of data for array of dependent variable Y as well as independent variables X1, X2, ⋯, and Xn.

One general equation in the form of Y=Kc1Θ,RXc1+Kc2Θ,RXc2+Kc3Θ,RXc3+Kc4Θ,RXc4+Kc5Θ,RXc5 can be derived by combining eqs (22) and (23). If one coefficient Xci of the standard equilibrium constant KciΘ,R is set as unity, five mathematical expressions can be derived to determine KciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts at a fixed temperature as

(26)1−b2+1N1−1−b1N23b2−b1+1N13N2=Kc1Θ,R+b2−b1+1N1N23b2−b1+1N13N2Kc2Θ,R+b2−2b1+1N1N223b2−b1+1N13N2Kc3Θ,R+2b2−5b1+1N12N253b2−b1+1N13N2Kc4Θ,R+b2−6b1+1N1N263b2−b1+1N13N2Kc5Θ,R−
(27)1−b2+1N1−1−b1N2b2−b1+1N1N2=3b2−b1+1N13N2b2−b1+1N1N2Kc1Θ,R+Kc2Θ,R+b2−2b1+1N1N22b2−b1+1N1N2Kc3Θ,R+2b2−5b1+1N12N25b2−b1+1N1N2Kc4Θ,R+b2−6b1+1N1N26b2−b1+1N1N2Kc5Θ,R−
(28)1−b2+1N1−1−b1N2b2−2b1+1N1N22=3b2−b1+1N13N2b2−2b1+1N1N22Kc1Θ,R+b2−b1+1N1N2b2−2b1+1N1N22Kc2Θ,R+Kc3Θ,R+2b2−5b1+1N12N25b2−2b1+1N1N22Kc4Θ,R+b2−6b1+1N1N26b2−2b1+1N1N22Kc5Θ,R−
(29)1−b2+1N1−1−b1N22b2−5b1+1N12N25=3b2−b1+1N13N22b2−5b1+1N12N25Kc1Θ,R+b2−b1+1N1N22b2−5b1+1N12N25Kc2Θ,R+b2−2b1+1N1N222b2−5b1+1N12N25Kc3Θ,R+Kc4Θ,R+b2−6b1+1N1N262b2−5b1+1N12N25Kc5Θ,R−
(30)1−b2+1N1−1+b1N2b2+6b1+1N1N26=3b2−b1+1N13N2b2−6b1+1N1N26Kc1Θ,R+b2−b1+1N1N2b2−6b1+1N1N26Kc2Θ,R+b2−2b1+1N1N22b2−6b1+1N1N26Kc3Θ,R+2b2−5b1+1N12N25b2−6b1+1N1N26Kc4Θ,R+Kc5Θ,R−

The derivation process of five sub-equations in eqs (26)–(30) is described in details in Appendix. Replacing N1=NFe and N2=NAl by reported activity aR,i of Al and Fe in Fe–Al binary melts as listed in Table 1 through critical assessment, the required KciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts at a fixed temperature in eqs (26)–(30) can be solved by the multiple linear regression approach because each formula in eqs (26)–(30) can be treated as an equation in the form of Y=Kc1Θ,RXc1+Kc2Θ,RXc2+Kc3Θ,RXc3+Kc4Θ,RXc4+Kc5Θ,RXc5 with Xc1 or Xc2 or Xc3 or Xc4 or Xc5 as 1.0.

Selection of determined standard equilibrium constants KciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts

In order to calculate KciΘ,R of reactions for forming the above-mentioned five associated molecules in Fe–Al binary melts at a fixed temperature, it is better to select activities aR,i of both Al and Fe in the full composition range of Fe–Al binary melts. Clearly, four data sources [13, 15, 16, 17] of the third category as described in “Classification of nine collected data sources on activity aR,i of Al or Fe in Fe–Al binary melts” and summarized in Table 1 should be selected. Among the four data sources [13, 15, 16, 17] of the third category, results by Ichise et al. [15] (D7) at 1673 K (1400 °C) and values by Desai [16] (D8) at 1873 K (1600 °C) were chosen based on the assessment conclusions in “Assessment of nine collected data sources on activity aR,i of Al or Fe in Fe–Al binary melts”.

The calculated KciΘ,R of reactions for forming the above-mentioned five associated molecules in Fe–Al binary melts at temperatures of 1673 and 1873 K (1400 and 1600 °C) by five methods in eqs (26)–(30) are summarized in Table 5. Five methods of calculating KciΘ,R of reactions for forming the above-mentioned five associated molecules in Fe–Al binary melts are assigned as method M1–M5, respectively. The obtained adjusted coefficients of determination Rˉ2 of reactions for forming five associated molecules at each temperature are also summarized in Table 5. The adjusted coefficient of determination Rˉ2 is closer to 1.0, the applied multiple linear regression method is more accurate.

Table 5:

Regressed values of standard equilibrium constants KciΘ,R of reactions for forming five associated molecules of Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 through multiple linear regression method by replacing mass action concentrations Ni of Fe and Al with reported activity aR,i of Fe and Al by Ichise et al. [15] (D7) at 1673 K (1400 °C) and by Desai [16] (D8) at 1873 K (1600 °C) through five mathematical equations as method M1–M5, adjusted coefficient Rˉ2 of determination, and corresponding values of standard molar Gibbs free energy change ΔrGm,ciΘ,R.

T [K(°C)]Method(–)Regressed value of KciΘ,R(a) (–)Adjusted coefficient Rˉ2 (–)Calculated value of ΔrGm,ciΘ,R(b) (J/mol)NoteOriginal data Ref.
Fe3AlFeAlFeAl2Fe2Al5FeAl6Fe3AlFeAlFeAl2Fe2Al5FeAl6
1673(1400)M117.1613.1615.88−181.0626.331.0−39,541.98−35,842.63−38,459.49ND(c)−45,494.20[15]
M211.9614.2411.09252.0330.920.9996−34,519.10−36,947.92−33,461.85−76,912.07−47,726.98Applied
M3−0.4415.377.07×10−40−31.860.478ND−38,004.2100,906.00NDND
M4016.6601.35×10−600.247ND−39,125.0ND188,036.50ND
M5010.9821.0507.61×10−60.333ND−33,323.9−42,378.60ND163,943.40
1873(1600)M16.839.122.66529.9625.261.0−29,926.81−34,423.61−15,238.11−97,680.79−50,288.49Applied[16]
M29.757.1012.97−166.6115.560.991−35,460.4−30,531.10−39,904.50ND−42,745.60
M34.189.79−7.90×10−5746.6728.661.0−22,258.6−35,519.40ND−10,3019.00−52,253.00
M408.238.79−7.30×10−800.666ND−32,823.50−33,852.60NDND
M504.9816.9603.10×10−60.666ND−25,009.80−44,084.40ND197,510.40

  1. Note:

  2. (a) The regressed standard equilibrium constant KciΘ,R of a reaction as negative has no physicochemical meaning because the real equilibrium constant KciΘ,R of a reaction must be positive.

  3. (b) The corresponding value of ΔrGm,ciΘ,R from KciΘ,R as negative is described as no datum, which is abbreviated as ND.

  4. (c) ND means no datum.

It should be emphasized that negative values of calculated KciΘ,R have no physicochemical meanings because the real standard equilibrium constant KciΘ,R of a reaction must be positive. It can be observed from Table 5 that only positive results of KciΘ,R can be obtained through method two (M2) based on data source by Ichise et al. [15] (D7) at 1673 K (1400 °C), while positive ones of KciΘ,R can be gotten through method one (M1) based on data source by Desai [16] (D8) at 1873 K (1600 °C).

The results of ΔrGm,ciΘ,R reactions for forming five associated molecules as Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Fe–Al binary melts at a fixed temperature can be calculated from the obtained KciΘ,R of the same reaction through the famous van’t Hoff isothermal equation as ΔrGm,ciΘ,R=−RTlnKciΘ,R [69, 70, 71]. The obtained expressions of ΔrGm,ciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts over a temperature range from 1673 to 1873 K (1400 to 1600 °C) are summarized in Table 4 for comparison with those by Zhang [41].

Comparison between calculated ΔrGm,ciΘ,R by present authors and reported ones by Zhang [41] for forming five associated molecules as Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Fe–Al binary melts over a temperature range from 1673 to 1873 K (1400 to 1600 °C) is illustrated in Figure 7. Basically, the calculated results of ΔrGm,ciΘ,R reactions for forming Fe3Al, FeAl, and FeAl6 in this study are in agreement with reported ones by Zhang [41] as shown in Figures 7(a), 7(b), and 7(e). However, the effect of changing temperature from 1673 to 1873 K (1400 to 1600 °C) on calculated values of ΔrGm,ciΘ,R of reactions for forming FeAl2 and Fe2Al5 in this study are not in consistency with those by Zhang [41] in Figures 7(c) and 7(d).

Figure 6: Comparison of reported first-order activity interaction coefficient εAlAl\varepsilon _{{\rm{Al}}}^{{\rm{Al}}} of Al (a) and εFeFe\varepsilon _{{\rm{Fe}}}^{{\rm{Fe}}} of Fe (b) by various researchers with obtained one by developed AMCT–Ni{N_i} model for Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.
Figure 6:

Comparison of reported first-order activity interaction coefficient εAlAl of Al (a) and εFeFe of Fe (b) by various researchers with obtained one by developed AMCT–Ni model for Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.

Figure 7: Comparison between regressed results of standard molar Gibbs free energy change ΔrGm,ciΘ,R{\Delta _{\rm{r}}}G_{{\rm{m, c}}i}^{\Theta {\rm{, R}}} in this study and reported ones by Zhang [41] for forming five complex compounds or molecules as Fe3Al, FeAl, FeAl2, Fe2Al5 and FeAl6 by Fe and Al atoms in Fe–Al binary melts over a temperature range from 1673 to 1873 K (1400 to 1600 °C), respectively.
Figure 7:

Comparison between regressed results of standard molar Gibbs free energy change ΔrGm,ciΘ,R in this study and reported ones by Zhang [41] for forming five complex compounds or molecules as Fe3Al, FeAl, FeAl2, Fe2Al5 and FeAl6 by Fe and Al atoms in Fe–Al binary melts over a temperature range from 1673 to 1873 K (1400 to 1600 °C), respectively.

As pointed out by Zhang [41], the reported activities aR,i of both Al and Fe by Jacobson and Mehrotra [17] (D9) at 1573 K (1300 °C) and those by Desai [16] (D8) at 1873 K (1600 °C) were applied to calculate values of ΔrGm,ciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts. Certainly, the reported activities aR,i of both Al and Fe under the condition of mole fraction xAl of Al less than 0.50 by Jacobson and Mehrotra [17] (D9) in the full composition range of Fe–Al binary melts at 1573 K (1300 °C) should not be applied because those results were obtained in two-phase zone of liquid and solid, rather than liquid melts. However, Zhang [41] applied all reported activities aR,i of both Al and Fe by Jacobson and Mehrotra [17] (D 9) at 1573 K (1300 °C). Thus, the obtained expressions of ΔrGm,ciΘ,R of reactions for forming five associated molecules in Fe–Al binary melts in Table 4 by the present authors are much reasonable than those by Zhang [41] and will be applied to the developed AMCT–Ni model in eqs (23) and (25).

Results and discussion for calculated mass action concentrations Ni of structural units by developed AMCT–Ni model for Fe–Al binary melts

The mass action concentrations Ni of seven structural units in the full composition range of Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C) have been calculated by the developed AMCT–Ni model in eqs (23) and (25) through the regressed expressions of ΔrGm,ciΘ,R of reactions for forming five associated molecules as Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Fe–Al binary melts in Table 4. Taking temperature at 1873 K (1600 °C) as an example, results of calculated Ni of seven structural units and the total equilibrium mole number Σni of all seven structural units in 100 g Fe–Al binary melts are tabulated in Table 6.

Table 6:

Chemical composition of Fe–Al binary melts with changing mole fraction xAl of Al from 0.0 to 1.0, calculated mass action concentrations Ni of seven structural units as Fe, Al, Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6, and total equilibrium mole number Σni of seven structural units in 100 g Fe–Al binary melts based on the AMCT at temperature of 1873 K (1600 °C).

No.Chemical composition of Fe–Al binary meltsCalculated mass action concentrations Ni of structural units (–)Σni (mol)
Mole fraction (–)Mass percentage (–)
xFexAl[% Fe][% Al]NFeNAlNFe3AlNFeAlNFeAl2NFe2Al5NFeAl6
11.000.00100.000.001.0000.0000.0000.0000.0000.0000.0001.79
20.950.0597.522.480.9450.0042.045×10−20.0313.156×10−52.641×10−104.716×10−141.68
30.900.1094.915.090.8810.0094.046×10−20.0701.756×10−41.998×10−89.353×10−121.58
40.850.1592.147.860.8060.0165.794×10−20.1195.610×10−43.809×10−73.641×10−101.50
50.800.2089.2210.780.7230.0276.992×10−20.1781.409×10−34.019×10−67.169×10−91.44
60.750.2586.1313.870.6350.0427.414×10−20.2463.044×10−32.945×10−59.392×10−81.39
70.700.3082.8517.150.5450.0637.018×10−20.3155.836×10−31.617×10−48.974×10−71.38
80.650.3579.3620.640.4590.0915.987×10−20.3801.007×10−26.889×10−46.504×10−61.38
90.600.4075.6424.360.3790.1254.638×10−20.4321.575×10−22.323×10−33.657×10−51.40
100.550.4571.6728.330.3070.1663.288×10−20.4652.251×10−26.296×10−31.621×10−41.42
110.500.5067.4232.580.2450.2132.139×10−20.4762.963×10−21.401×10−25.817×10−41.43
120.450.5562.8737.130.1910.2671.270×10−20.4653.623×10−22.625×10−21.750×10−31.43
130.400.6057.9842.020.1440.3296.700×10−30.4324.146×10−24.235×10−24.616×10−31.43
140.350.6552.7147.290.1030.4023.013×10−30.3784.428×10−25.897×10−21.095×10−21.42
150.300.7047.0152.990.0690.4861.112×10−30.3084.361×10−26.918×10−22.307×10−21.44
160.250.7540.8359.170.0440.5783.344×10−40.2313.903×10−26.591×10−24.136×10−21.50
170.200.8034.1065.900.0260.6738.089×10−50.1603.132×10−24.939×10−26.090×10−21.62
180.150.8526.7573.250.0140.7671.435×10−50.0982.189×10−22.751×10−27.188×10−21.84
190.100.9018.7081.300.0060.8581.478×10−60.0491.238×10−29.847×10−36.377×10−22.21
200.050.959.8290.180.0020.9395.300×10−80.0174.744×10−31.583×10−33.506×10−22.82
210.001.000.00100.000.0001.0000.0000.0000.0000.0000.0003.71

Comparison between calculated mass action concentration NAl of Al by developed AMCT–Ni model and reported activity aR,Al of Al from nine collected data sources for Fe–Al binary melts

Comparison between calculated NAl of Al at 1873 K (1600 °C) and reported activity aR,Al of Al from nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] in Table 1 at various temperatures for Fe–Al binary melts is illustrated in Figure 8. Similar to Figures 1, 2 and 8 is also separated into three sub-figures, in which each sub-figure of Figure 8 only shows comparison of reported activity aR,Al of Al in one category as summarized in Table 1 with calculated NAl of Al. Certainly, the reported activity aR,Al of Al by Desai [16] (D8) at 1873 K (1600 C) is displayed as basis line in three sub-figures of Figure 8 as criteria. The calculated results of NAl are almost identical with those of reported activity aR,Al of Al by Desai [16] (D8) at 1873 K (1600 C) in Figure 8(c) and are in good agreement with those of the reported activity aR,Al of Al in three categories as listed in Table 1 at various temperatures greater than 1373 K (1100 °C). The results in Figure 8 indicate that calculated NAl of Al by the developed AMCT–Ni model for Fe–Al binary melts can be applied to accurately replace the reported [3, 10, 11, 12, 13, 14, 15, 16, 17] activity aR,Al of Al in Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C).

Figure 8: Comparison of reported results of activity aR,Al{a_{{\rm{R, Al}}}} of Al by various researchers with obtained ones by developed AMCT–Ni{N_i} model for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl{x_{{\rm{Al}}}} of Al at various temperatures greater than 1373 K (1100 °C), respectively.
Figure 8:

Comparison of reported results of activity aR,Al of Al by various researchers with obtained ones by developed AMCT–Ni model for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl of Al at various temperatures greater than 1373 K (1100 °C), respectively.

The defined [40,41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] Ni of structural units are relative to pure liquid matter as standard state [40, 41, 56, 57, 58, 59, 60, 61], which are the same as those of activity aR,i of element i in metallic melts. The relationship of obtained activity coefficient lnγAl=lnNAl/xAl of Al against mole fraction xAl of Al in the full composition range of Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C) is displayed in Figure 9(a), respectively. Obviously, changing temperature from 1823 to 1973 K (1550 to 1700 °C) cannot result in a large variation tendency of obtained lnγAl=lnNAl/xAl. The obtained activity coefficient lnγAl of Al at four temperatures by the developed AMCT–Ni model can be accurately described by quadratic polynomial function as

(31)lnγAl=−3.09477+5.7031xAl −2.58117xAl2T=1823 K
(32)lnγAl=−2.95622+5.40893xAl−2.43139xAl2T=1873 K
(33)lnγAl=−2.82201+5.09042xAl−2.24907xAl2T=1923 K
(34)lnγAl=−2.69248+4.75429xAl−2.04118xAl2T=1973 K
Figure 9: Relationship between mole fraction xAl{x_{{\rm{Al}}}} of Al and calculated activity coefficient lnγAl=lnNAl/xAl\ln {\gamma _{{\rm{Al}}}} = \ln \left( {{N_{{\rm{Al}}}}/{x_{{\rm{Al}}}}} \right) of Al (a) or lnγFe=lnNFe/xFe\ln {\gamma _{{\rm{Fe}}}} = \ln \left( {{N_{{\rm{Fe}}}}/{x_{{\rm{Fe}}}}} \right) of Fe (b) by developed AMCT–Ni{N_i} model for Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.
Figure 9:

Relationship between mole fraction xAl of Al and calculated activity coefficient lnγAl=lnNAl/xAl of Al (a) or lnγFe=lnNFe/xFe of Fe (b) by developed AMCT–Ni model for Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.

The expression of obtained activity coefficient γAl of Al containing temperature effect from eqs (26)–(30) can be expressed by quadratic polynomial function as

(35)lnγAl=−9,646.5T+2.196+−6.753+22,741.8TxAl+−12,937.4T+4.497xAl21823 K<T<1973 K

The regressed expression of activity coefficient γAl of Al by eq. (35) is listed in Table 1 and nominated as the tenth data sources in D10 for comparison with others.

Comparison between calculated mass action concentration NFe of Fe by developed AMCT–Ni model and reported or obtained activity aR,Fe of Fe from nine collected data sources for Fe–Al binary melts

Comparison between calculated NFe of Fe at 1873 K (1600 °C) and reported or obtained activity aR,Fe of Fe from nine collected data sources [3, 10, 11, 12, 13, 14, 15, 16, 17] in Table 1 for Fe–Al binary melts at various temperatures is illustrated in Figure 10, respectively. Similar to Figures 1, 2, 8 and 10 is also separated into three sub-figures, in which each sub-figure of Figure 10 only shows comparison of reported or obtained activity aR,Fe of Fe in one category as summarized in Table 1 with calculated NFe of Fe. Certainly, the reported activity aR,Fe of Fe by Desai [16] (D8) at 1873 K (1600 C) is also displayed as basis line in three sub-figures of Figure 10 as criterion.

Figure 10: Comparison of reported or obtained results of activity aR,Fe{a_{{\rm{R, Fe}}}} Fe by various researchers with obtained ones by developed AMCT–Ni{N_i} model for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl{x_{{\rm{Al}}}} of Al at various temperatures greater than 1373 K (1100 °C), respectively.
Figure 10:

Comparison of reported or obtained results of activity aR,Fe Fe by various researchers with obtained ones by developed AMCT–Ni model for three categories of Fe–Al binary melts (a, b, c) with different ranges of mole fraction xAl of Al at various temperatures greater than 1373 K (1100 °C), respectively.

The calculated results of NFe of Fe are in good agreement with those of reported activity aR,Fe of Fe by Desai [16] (D8) at 1873 K (1600 C) in Figure 10(c) and are in good accord with those of activity aR,Fe of Fe in three categories as listed in Table 1 except for some results after Mitani and Nagai [12] (D4) at 1573 K (1300 C) and 1473 K (1200 C) in Figure 10(b). Thus, it can be deduced that the calculated NFe of Fe can be applied to precisely substitute the reported or obtained activity aR,Fe of Fe in Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C).

The relationships of calculated activity coefficient lnγFe=lnNFe/xFe of Fe against mole fraction xAl of Al in the full composition range of Fe–Al binary melts at four temperatures are displayed in Figure 9(b). Obviously, changing temperature from 1823 to 1973 K (1550 to 1700 °C) cannot lead to a large variation of obtained lnγFe=lnNFe/xFe. The obtained activity coefficient lnγFe of Fe at four temperatures by the developed AMCT–Ni model can be described by cubic polynomial function as

(36)lnγFe=0.02604−0.47217xAl−0.96978xAl2−2.26162xAl3T=1823 K
(37)lnγFe=0.03116−0.65002xAl−0.12923xAl2−2.88947xAl3T=1873 K
(38)lnγFe=0.03288−0.77165xAl+0.57861xAl2−3.46282xAl3T=1933 K
(39)lnγFe=0.03179−0.84479xrmAl+1.16592xAl2−3.98167xAl3T=1973 K

The expression of activity coefficient γFe of Fe including temperature effect from eqs (36)–(39) can also be described by cubic polynomial function as

(40)lnγFe=0.104−138.8T+ −5.405+8,952.3TxAl+27.190−51,256.3TxAl2+−24.911+41,268.4TxAl31823 K<T<1973 K

The regressed expression of activity coefficient γFe of Fe in eq. (40) is listed in Table 1 and allotted as the tenth data sources in D10 for comparison with others.

Assessment of obtained Raoultian activity coefficients γi0 of both Al and Fe by developed AMCT–Ni model for Fe–Al binary melts

Assessment of obtained Raoultian activity coefficient γAl0 of Al in Fe–rich corner of Fe–Al binary melts by developed AMCT–Ni model

Raoultian activity coefficient γAl0 of Al in the Fe-rich corner of Fe–Al binary melts can also be calculated from regressed expression of activity coefficient γAl of Al in eq. (35) under the condition of mole fraction xAl of Al at zero. The influence of changing temperature from 1823 to 1973 K (1550 to 1700 °C) on calculated γAl0 of Al in the Fe-rich corner of Fe–Al binary melts by the developed AMCT–Ni model can be directly obtained from eq. (35) as

(41)lnγAl0=−9,646.5T+2.196

The expression ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% can be obtained through eq. (13) or eq. (14) from expression of lnγAl0 of Al in eq. (41) as

(42)ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,%=−80,199.53−14.07TJ/mol

The obtained expressions of both lnγAl0 and ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% assigned as the fifteenth data source as S15 are also summarized in Table 2 for comparison with others. The obtained results of lnγAl0 of Al by eq. (41) are illustrated in Figure 4(a) in semi-solid pentagon for comparison with reported ones from 11 data source [1, 2, 3, 4, 5, 6, 10, 13, 14, 15, 16, 19], especially with reported ones by Ichise et al. [15] (S11=D7) and Kim et al. [19] (S14). Meanwhile, the obtained results of ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% by eq. (42) are also displayed in Figure 4(b) in semi-solid pentagon for comparison with reported ones from 11 data source [1, 2, 3, 4, 5, 6, 10, 13, 14, 15, 16, 19], especially with reported ones by Chou and Elliott [4] (S3), Ichise et al. [15] (S11=D7), and Kim et al. [19] (S14). The obtained values of lnγAl0 of Al by eq. (41) are in good accord with reported ones by Ichise et al. [15] (S11=D7) and Kim et al. [19] (S14) as shown in Figure 4(a). In addition, the obtained values of ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% by eq. (42) are in good consistency with reported ones by Chou and Elliott [4] (S3), Ichise et al. [15] (S11=D7), and Kim et al. [19] (S14) as illustrated in Figure 4(b). These findings imply that the developed AMCT–Ni model (S15=D10) can be applied to not only accurately predict reaction ability of Fe and Al but also precisely obtain γAl0 of Al and ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% in Fe–Al binary melts.

Assessment of obtained Raoultian activity coefficient γFe0 of Fe in Al–rich corner of Fe–Al binary melts by developed AMCT–Ni model

Similarly, Raoultian activity coefficient γFe0 of Fe in the Al-rich corner of Fe–Al binary melts can also be calculated from regressed expression of activity coefficient γFe of Fe in eq. (40) under the condition of mole fraction xAl of Al at unity, i. e., mole fraction xFe of Fe at zero. The influence of changing temperature from 1823 to 1973 K (1550 to 1700 °C) on obtained activity coefficient γFe0 of Fe in the Al-rich corner of Fe–Al binary melts by the developed AMCT–Ni model can be derived from eq. (40) as

(43)lnγFe0=−6,799.1T−0.01367

The expression ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% can be gotten through eq. (13) or eq. (14) from expression of lnγFe0 of Fe in eq. (43) as

(44)ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,%=−9,772.67−69.41TJ/mol

The obtained expressions of both lnγFe0 and ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% nominated as the fifteenth data source as S15 are listed in Table 2 for comparison with others. The obtained results of lnγFe0 of Fe by eq. (43) are illustrated in Figure 5(a) in semi-solid pentagon for comparison with the reported ones from six data sources [11, 12, 13, 15, 16, 17], especially with the reported ones by Ichise et al. [15] (S11=D7). Meanwhile, the obtained results of ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% by eq. (44) are also displayed in Figure 5(b) in semi-solid pentagon for comparison with the reported ones from six data sources [11, 12, 13, 15, 16, 17], especially with reported ones by Ichise et al. [15] (S11=D7). The obtained results of lnγFe0 of Fe by eq. (43) are in good accord with reported ones by Ichise et al. [15] (S11=D7) as shown in Figure 5(a). Meanwhile, the obtained values of ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% by eq. (44) are in good consistency with reported ones by Ichise et al. [15] (S11=D7) and by others in Figure 5(b). Therefore, besides accurate expressions of γAl0 and ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,%, the developed AMCT–Ni model (S15=D10) can also be applied to precisely determine γFe0 and ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% in Fe–Al binary melts.

Assessment of obtained first-order activity interaction coefficients εjj or ejjorhjj of Al and Fe by developed AMCT–Ni model for Fe–Al binary melts

Assessment of obtained first-order activity interaction coefficients εjj of Al and Fe in Fe–rich corner or Al–rich corner of Fe–Al binary melts by developed AMCT–Ni model

The relationship between mole fraction xAl of Al and obtained activity coefficient lnγAl=lnNAl/xAl of Al through eq. (35) for Fe–Al binary melts with mole fraction xAl of Al less than 0.01 at an interval of mole fraction xAl of Al as 0.005 at four temperatures is shown in Figure 11(a), respectively. Similar plots of mole fraction xFe of Fe against obtained activity coefficient lnγFe=lnNFe/xFe of Fe through eq. (40) at four temperatures are also illustrated in Figure 11(b). Certainly, intercept or slope of the regressed linear relationship in Figure 11(a) for Fe–Al binary melts with mole fraction xAl of Al less than 0.01 at four temperatures can be treated as lnγAl0 and εAlAl, respectively. Similarly, results of lnγFe0 or εFeFe can also be obtained from Figure 11(b). It should be pointed out that the obtained results of lnγAl0 from Figure 11(a) and lnγFe0 from Figure 11(b) are almost the same as lnγAl0 by eq. (41) and lnγFe0 by eq. (43), respectively. The obtained εAlAl by the developed AMCT–Ni model in Figure 12(a) can be described with reciprocal of temperature over a temperature range from 1823 to 1973 K (1550 to 1700 °C) by linear function as

(45)εAlAl=22,612.3T−6.708
Figure 11: Relationship between mole fraction xAl{x_{{\rm{Al}}}} of Al and obtained activity coefficient lnγAl\ln {\gamma _{{\rm{Al}}}} of Al (a) or relationship between mole fraction xFe{x_{{\rm{Fe}}}} of Fe and obtained activity coefficient lnγFe\ln {\gamma _{{\rm{Fe}}}} of Fe (b) by developed AMCT–Ni{N_i} model for Fe–Al binary melts with mole fraction xi{x_{i}} of Al or Fe less than 0.01 over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.
Figure 11:

Relationship between mole fraction xAl of Al and obtained activity coefficient lnγAl of Al (a) or relationship between mole fraction xFe of Fe and obtained activity coefficient lnγFe of Fe (b) by developed AMCT–Ni model for Fe–Al binary melts with mole fraction xi of Al or Fe less than 0.01 over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.

Figure 12: Relationship between reciprocal of temperature and obtained first-order activity interaction coefficient εAlAl\varepsilon _{{\rm{Al}}}^{{\rm{Al}}} coupled with activity coefficient γAl{\gamma _{{\rm{Al}}}} of Al (a) or εFeFe\varepsilon _{{\rm{Fe}}}^{{\rm{Fe}}} coupled with activity coefficient γFe{\gamma _{{\rm{Fe}}}} of Fe (b) by developed AMCT–Ni{N_i} model for Fe–Al binary melts with mole fraction xi{x_{i}} of Al or Fe less than 0.01 over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.
Figure 12:

Relationship between reciprocal of temperature and obtained first-order activity interaction coefficient εAlAl coupled with activity coefficient γAl of Al (a) or εFeFe coupled with activity coefficient γFe of Fe (b) by developed AMCT–Ni model for Fe–Al binary melts with mole fraction xi of Al or Fe less than 0.01 over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.

Meanwhile, the relationship between reciprocal of temperature T and obtained εFeFe by the developed AMCT–Ni model in Figure 12(b) can be expressed by linear function over a temperature range from 1823 to 1973 K (1550 to 1700 °C) as

(46)εFeFe=−29,523.3T+25.285

The obtained expression of εAlAl and εFeFe assigned as the fifteenth data source as S15 are also summarized in Table 2 for comparison with others. The obtained results of εAlAl by eq. (45) are illustrated in Figure 6(a) in semi-solid pentagon for comparison with reported ones from eight data source [3, 6, 10, 13, 14, 15, 16, 17], especially with reported ones by Ichise et al. [15] (S11=D7) and Kim et al. [19] (S14). Meanwhile, the obtained results of εFeFe by eq. (46) are also displayed in Figure 6(b) in semi-solid pentagon for comparison with reported ones from five data source [11, 13, 15, 16, 17], especially with reported ones by Ichise et al. [15] (S11=D7). The obtained values of εAlAl by eq. (45) are in good accord with reported ones by Wooley and Elliott [10] (S6=D2) as well as Jacobson and Mehrotra [17] (S13=D9), but greater than reported ones by Chipman and Floridis [3] (S2=D1) as well as Kim et al. [18] (S14) as shown in Figure 6(a). The obtained values of εFeFe by eq. (46) are close to reported results by Ichise et al. [15] (S11=D7), but greater than calculated results after Belton and Fruehan [13] (S9=D5) as well as Desai [16] (S12=D8), smaller than those after Coskun and Elliott [11] (S7=D3).

Assessment of obtained first-order activity interaction coefficients ejjorhjj of Al and Fe in Fe–rich corner or Al–rich corner of Fe–Al binary melts by developed AMCT–Ni model

The first-order activity interaction coefficient εjj has an explicit relationship with ejj or hjj in i–j binary melts as [58, 69, 71]

(47)ejj=1230.26εjj−1MiMj+1
(48)hjj=dlogfH,idxi=12.3026εjj

With regard to Fe–Al binary melts, εjj and ejj or hjj of Al and Fe can be correlated by

(49)eAlAl=1230.26εAlAl−1MFeMAl+1,eFeFe=1230.26εFeFe−1MAlMFe+1
(50)hAlAl=12.3026εAlAl,hFeFe=12.3026εFeFe

According to the obtained expressions of εjj of Al and Fe in eqs (45) and (46), the first-order activity interaction coefficient ejj and hjj of Al or Fe over a temperature range from 1823 to 1973 K (1550 to 1700 °C) can be deduced as

(51)eAlAl=203.3T−6.49×10−2,eFeFe=−61.9T+5.53×10−2
(52)hAlAl=9,820.3T−2.91,hFeFe=−12,821.7T+10.98

The obtained expression of eAlAl as eAlAl=80.5/T by Rohde et al. [76] was recommended by the JSPS (Japan Society for the Promotion of Science) [77] to be the accurate one [78, 79]. Thus, value of eAlAl as 0.043 at 1873 (1600 °C) through eAlAl=80.5/T by Rohde et al. [76] can be applied to be a criterion. About six data sources [3, 6, 10, 13, 15, 19] of eAlAl were collected by Kim et al. [19] (S14) as listed in the ninth column of Table 2. Comparison between obtained results of eAlAl by eq. (51) and reported ones from six collected data sources [3, 6, 10, 13, 15, 19] for Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C) is shown in Figure 13, respectively. Certainly, the recommended values of eAlAl as eAlAl=80.5/T by the JSPS [77] are also plotted in Figure 13 as criteria. The obtained results of eAlAl by eq. (51) are in good agreement with reported ones from five collected data sources [3, 6, 10, 15, 19] and recommended ones by the JSPS[77] only except for the greater value as eAlAl=0.087 after Belton and Fruehan [13] (D5).

Figure 13: Comparison of reported results of first-order activity interaction coefficient eAlAle_{{\rm{Al}}}^{{\rm{Al}}} of Al by various researchers with obtained ones by developed AMCT–Ni{N_i} model for Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.
Figure 13:

Comparison of reported results of first-order activity interaction coefficient eAlAl of Al by various researchers with obtained ones by developed AMCT–Ni model for Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C), respectively.

The expression of eAlAl as eAlAl=111/T−0.016 was reported by Kim et al. [18] (S14), however the expression of parameter αFe−Al involved in Darken’s quadratic formalism [73, 74, 75] as αFe−Al=−1,112T−0.585 was also obtained by Kim et al. [19] (S14). According to the relationship between εAlAl and αFe−Al in eq. (19), the expression of εAlAl after Kim et al. [19] (S14) should be derived by

(53)εAlAl=−2αFe−Al=2,224T+1.17

Thus, the recalculated expression of eAlAl after Kim et al. [19] (S14) by the present authors should be described through the relationship between eAlAl and εAlAl in eq. (49) by

(54)eAlAl=20.0T+5.87×10−3

Certainly, the originally reported results of eAlAl as eAlAl=111T−0.016 by Kim et al. [19] (S14) do not comply with the theoretical relationship between eAlAl and εAlAl in eq. (49). Under this circumstance, the recalculated results of eAlAl after Kim et al. [19] (S14) by the present authors in eq. (54) are also plotted in Figure 13 for comparison. Evidently, the recalculated values of eAlAl after Kim et al. [19] (S14) by the present authors in eq. (54) are smaller than obtained ones in eq. (51), reported ones from five collected data sources [3, 6, 10, 15, 19] and the recommended ones by the JSPS [77]. It can be concluded that results of αFe−Al and eAlAl by Kim et al. [19] (S14) are not self consistent due to unknown reasons.

To the knowledge of the present authors, no results of eFeFe as well as hjj of Al or Fe were reported in the literature, the obtained results of eFeFe and hjj of Al or Fe cannot be independently assessed up to now. Certainly, the obtained results of eFeFe and hjj of Al or Fe do not require further assessment based on the assessed results of εAlAl and εFeFe with enough accuracies in “Assessment of obtained first-order activity interaction coefficients εjj of Al and Fe in Fe–rich corner or Al–rich corner of Fe–Al binary melts by developed AMCT–Ni model”.

Relationships among calculated mass action concentrations Ni and equilibrium mole numbers ni for seven structural units in Fe–Al binary melts

Profiles of calculated mass action concentrations Ni of seven structural units in Fe–Al binary melts

The profiles of calculated Ni 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 (1600 °C) are illustrated in Figure 14. Clearly, results of NFe of Fe display a decrease tendency with an increase of mole fraction xAl of Al, while results of NAl of Al display an increase trend along with mole fraction xAl of Al. It is interesting to find in Figure 14 that NFeAl of FeAl exhibits a symmetrically inverse V–type profile along with mole fraction xAl of Al, in which the maximum value of NFeAl of FeAl at 0.48 reaches to mole fraction xAl of Al at 0.50. However, the calculated NFe3Al of Fe3Al displays an asymmetrically inverse V–type profile along with mole fraction xAl of Al, in which the maximum value of NFe3Al of Fe3Al at 0.074 meets mole fraction xAl of Al at 0.25. The similar effect of mole fraction xAl of Al on Nci of FeAl2 or Fe2Al5 or FeAl6 can also be observed in Figure 14, respectively. The maximum datum of Nci of FeAl2 or Fe2Al5 or FeAl6 corresponds mole fraction xAl of Al at 0.68, 0.73, and 0.86, respectively. In addition, the results of Nci of FeAl2 or Fe2Al5 or FeAl6 are very small compared with that of NFeAl of FeAl at xAl=0.48. It should be emphasized that the obtained profiles of Ni of seven structural units are in good agreement with those by Zaitsev et al. [34] under an assumption of existing three associates or clusters as FeAl, FeAl2, and Fe2Al5 in Fe–Al binary melts.

Figure 14: Relationship between mole fraction xAl{x_{{\rm{Al}}}} of Al and calculated mass action concentrations Ni{N_i} of seven structural units as Fe, Al, Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in full composition range of Fe–Al binary melts at 1873 K (1600 °C), respectively.
Figure 14:

Relationship between mole fraction xAl of Al and calculated mass action concentrations Ni of seven structural units as Fe, Al, Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in full composition range of Fe–Al binary melts at 1873 K (1600 °C), respectively.

Changing temperature from 1823 to 1973 K (1550 to 1700 °C) cannot cause an obvious variation tendency on profiles of calculated Ni of Al, Fe, Fe3A, FeAl, and FeAl6 as shown in Figures 15(a)–15(d) and 15(g). However, increasing temperature from 1823 to 1973 K (1550 to 1700 °C) can lead to a small decreasing trend of the maximum datum of NFeAl2 of FeAl2 from 0.057 to 0.027 in Figure 15(e), but can result in an increasing tendency of NFe2Al5 of Fe2Al5 from 0.050 to 0.11 in Figure 15(f).

Figure 15: Relationship between mole fraction xAl{x_{{\rm{Al}}}} of Al and calculated mass action concentrations Ni{N_i} of seven structural units as Fe (a), Al (b), Fe3Al (c), FeAl (d), FeAl2 (e), Fe2Al5 (f), and FeAl6 (g) in full composition range of Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.
Figure 15:

Relationship between mole fraction xAl of Al and calculated mass action concentrations Ni of seven structural units as Fe (a), Al (b), Fe3Al (c), FeAl (d), FeAl2 (e), Fe2Al5 (f), and FeAl6 (g) in full composition range of Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.

Relationship between calculated mass action concentrations Ni and calculated equilibrium mole numbers ni for seven structural units in Fe–Al binary melts

The relationship between calculated mass action concentrations Ni and equilibrium mole numbers ni for seven structural units in the full composition range of 100 g Fe–Al binary melts at four temperatures are illustrated in Figure 16, respectively. The nonlinear relationships between Ni and ni for Fe and Al can be observed in Figures 16(a) and 16(b). However, the spindle-shaped relationships of Ni against ni can be observed for Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Figures 16(c)–16(g). Furthermore, changing temperature from 1823 to 1973 K (1550 to 1700 °C) cannot result in a visible variation tendency of relationship between Ni and ni for both Fe and Al in Figures 16(a) and 16(b). Increasing temperature from 1823 to 1973 K (1550 to 1700 °C) cannot lead to an obvious variation trend of the spindle-shaped relationships for Fe3Al, FeAl, and FeAl6 in Figures 16(c), 16(d), 16(g). However, increasing temperature from 1823 to 1973 K (1550 to 1700 °C) can cause an obvious decreasing tendency of the spindle-shaped area surrounded by Ni against ni for FeAl2 in Figure 16(e). An opposite result can be observed in relationship of Ni against ni for Fe2Al5 in Figure 16(f). The spindle-shaped relationships between Ni and ni for Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Figures 16(c)–16(g) can be explained from the inverse V–type profiles of Ni along with mole fraction xAl of Al for Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Figures 15(c)–15(g).

Figure 16: Relationship between calculated mass action concentration NiN _{i} and calculated equilibrium mole number ni{n_i} for seven structural units as Fe (a), Al (b), Fe3Al (c), FeAl (d), FeAl2 (e), Fe2Al5 (f), and FeAl6 (g) in full composition range of Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.
Figure 16:

Relationship between calculated mass action concentration Ni and calculated equilibrium mole number ni for seven structural units as Fe (a), Al (b), Fe3Al (c), FeAl (d), FeAl2 (e), Fe2Al5 (f), and FeAl6 (g) in full composition range of Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.

To further explain the spindle-shaped relationships of Ni against ni for Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6, the profiles of ni of seven structural units along with mole fraction xAl of Al in the full composition range of 100 g Fe–Al binary at four temperatures are also displayed in Figure 17. Evidently, the profile of ni along with mole fraction xAl of Al in Figure 17 is similar with that of Ni in Figure 15 for the same structural unit.

Figure 17: Relationship between mole fraction xAl{x_{{\rm{Al}}}} of Al and calculated equilibrium mole number ni{n_i} for seven structural units as Fe (a), Al (b), Fe3Al (c), FeAl (d), FeAl2 (e), Fe2Al5 (f), and FeAl6 (g) in full composition range of Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.
Figure 17:

Relationship between mole fraction xAl of Al and calculated equilibrium mole number ni for seven structural units as Fe (a), Al (b), Fe3Al (c), FeAl (d), FeAl2 (e), Fe2Al5 (f), and FeAl6 (g) in full composition range of Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.

Changing temperature from 1823 to 1973 K (1550 to 1700 °C) cannot cause an obvious variation tendency of the maximum data of ni for Fe3Al, FeAl, and FeAl6 in Fe–Al binary melts in Figures 17(c), 17(d), and 17(g). The maximum value of nFe3Al of Fe3Al at 0.13 mol reaches to mole fraction xAl of Al at 0.25; the greatest result of nFeAl of FeAl at 0.68 mol corresponds to mole fraction xAl of Al at 0.50; the largest one of nFeAl6 of FeAl6 at 0.14 mol meets mole fraction xAl of Al at 0.90. However, increasing temperature from 1823 to 1973 K (1550 to 1700 °C) can result in a decreasing tendency of the maximum value of nFeAl2 of FeAl2 from 0.085 mol to 0.034 mol, and lead to an increasing trend of the greatest datum of nFe2Al5 of Fe2Al5 from 0.075 to 0.15 mol. It should be emphasized that the obtained spindle–shaped relationships of Ni against ni for Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Fe–Al binary melts are similar with those for structural units in Fe–Si binary melts reported by Yang et al. elsewhere [57].

It can be easily deduced from eq. (22) that slope of the relationships of Ni against ni for seven structural units in 100 g Fe–Al binary melts is equal to the reciprocal of Σni, i. e., 1/Σni. The nonlinear relationships of Ni against ni for seven structural units in 100 g Fe–Al binary melts in Figure 16 indicate that slope as 1/Σni is not constant. The total equilibrium mole number Σni of all seven structural units in 100 g Fe–Al binary melts in Figure 18 shows a asymmetric U–type relationship with mole fraction xAl of Al, i. e., a slowly decreasing tendency of Σni from 1.79 to 1.39 mol with an increase of mole fraction xAl of Al from 0.0 to 0.25, and a constant value of Σni at 1.40 mol with an increase of mole fraction xAl of Al from 0.25 to 0.60, and a rapid increasing trend of Σni with a further increase of mole fraction xAl of Al from 0.60 to 1.0.

Figure 18: Relationship between mole fraction xAl{x_{{\rm{Al}}}} of Al and calculated total equilibrium mole number Σni{\rm \Sigma} {n_i} in 100 g Fe–Al binary melts by developed AMCT–Ni{N_i} model for Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.
Figure 18:

Relationship between mole fraction xAl of Al and calculated total equilibrium mole number Σni in 100 g Fe–Al binary melts by developed AMCT–Ni model for Fe–Al binary melts at four temperatures of 1823, 1873, 1923, and 1973 K (1550, 1600, 1650, and 1700 °C), respectively.

Conclusions

A thermodynamic model for calculating the mass action concentrations Ni of structural units in Fe–Al binary melts based on the atom–molecule coexistence theory, i. e., AMCT–Ni model, has been developed and verified to be valid through comparing with the reported activities aR,i of both Al and Fe over a temperature range from 1823 to 1973 K (1550 to 1700 °C). Raoultian activity coefficients γi0 of both Al and Fe in the Fe-rich corner or Al-rich corner of Fe–Al binary melts as well as the standard molar Gibbs free energy changes ΔsolGm,i(l)→[i][%i]=1.0Θ,% of dissolved Al or Fe for forming [% Al] or [% Fe] as 1.0 in Fe–Al binary melts have been also obtained by the developed AMCT–Ni model and verified to be accurate. The main summary remarks can be obtained as follows:

1) The calculated mass action concentration NAl of free Al by the developed AMCT–Ni model is in good accord with reported activity aR,Al of Al relative to pure liquid Al (l) as standard state in Fe–Al binary melts. The obtained activity coefficient γAl of Al including temperature effect from 1823 to 1973 K (1550 to 1700 °C) by the developed AMCT–Ni model can be described by quadratic polynomial function as lnγAl=−9,646.5T+2.196+−6.753+22,741.8TxAl+−12,937.4T+4.497xAl2.

2) The calculated mass action concentration NFe of free Fe by the developed AMCT–Ni model is in good agreement with reported activity aR,Fe of Fe relative to pure liquid Fe(l) as standard state in Fe–Al binary melts. The obtained activity coefficient γFe of Fe including temperature effect from 1823 to 1973 K (1550 to 1700 °C) by the developed AMCT–Ni model can be expressed by cubic polynomial function as lnγFe=0.104−138.8T+−5.405+8,952.3TxAl+27.190−51,256.3TxAl2+−24.911+41,268.4TxAl3.

3) The obtained Raoultian activity coefficient γAl0 of Al in the Fe-rich corner of Fe–Al binary melts by the developed AMCT–Ni model as lnγAl0=−9,646.5T+2.196 is in good consistency with reported one from the literature. Meanwhile, the obtained standard molar Gibbs free energy change ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,% of dissolved liquid Al(l) for forming [% Al] as 1.0 in Fe–Al binary melts referred to 1 mass percentage of Al as reference state by the developed AMCT–Ni model as ΔsolGm,Al(l)→[Al][%Al]=1.0Θ,%=−80,199.53 − 14.07T J/mol is very close with reported ones from the literature.

4) The obtained Raoultian activity coefficient γFe0 of Fe in the Al-rich corner of Fe–Al binary melts by the developed AMCT–Ni model as lnγFe0=−6,799.1T−0.01367 is in good match with reported one from the literature. Meanwhile, the obtained standard molar Gibbs free energy change ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,% of dissolved liquid Fe(l) for forming [% Fe] as 1.0 in Fe–Al binary melts referred to 1 mass percentage of Fe as reference state by the developed AMCT–Ni model as ΔsolGm,Fe(l)→[Fe][%Fe]=1.0Θ,%=−9,772.67 − 69.41T J/mol is very close with reported ones from the literature.

5) The obtained first-order activity interaction coefficients εjj of Al and Fe coupled with activity coefficient γi by the developed AMCT–Ni model as εAlAl=22,612.3T−6.708 or εFeFe=−29,523.3T+25.285 is in good agreement with reported one from the literature. Meanwhile, the developed AMCT–Ni model also provides expressions of the first-order activity interaction coefficients ejj or hjj of Al and Fe coupled with activity coefficient f%,i or fH,i as eAlAl=203.3T−6.49×10−2 and eFeFe=−61.9T+5.53×10−2 or hAlAl=9,820.3T−2.91 and hFeFe=−12,821.7T+10.98, respectively.

6) The developed AMCT–Ni model for Fe–Al binary melts can be successfully applied to represent reaction ability of structural units in Fe–Al binary melts over a temperature range from 1823 to 1973 K (1550 to 1700 °C) based on the assumption of forming five molecules of Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Fe–Al binary melts. The reaction abilities of both Fe and Al show a competitive or coupling relationship each other in Fe–Al binary melts. Changing temperature from 1823 to 1973 K (1550 to 1700 °C) cannot result in an obvious variation of relationship between reaction abilities of both Fe and Al and mole fraction xAl of Al.

7) The calculated equilibrium mole numbers ni of seven structural units by the developed AMCT–Ni model can be recommended to represent mass contents of structural units in Fe–Al binary melts. The spindle-shaped relationships between calculated mass action concentrations Ni and equilibrium mole numbers ni for Fe3Al, FeAl, FeAl2, Fe2Al5, and FeAl6 in Fe–Al binary melts have been revealed to be the similar to those in Fe–Si binary melts. The calculated total equilibrium mole numbers Σni in 100 g Fe–Al binary melts display an U-type relationship with mole fraction xAl of Al from 0.0 to 1.0 over a temperature range from 1823 to 1973 K (1550 to 1700 °C).

Notes

The authors declare no competing financial interest. This work was presented in part at the 9th European Metallurgical Conference, June 25–29th, 2017, Leipzig, Germany, pp. 907–928.

Funding statement: This work is supported by the Beijing Natural Science Foundation (Grant No. 2182069) and the National Natural Science Foundation of China (Grant No. 51174186).

Appendix

Multiplied eq. (23) by mole number b2=nAl0 of Al in eq. (24) yields the following equation

(55)b1Σni×b2=b2N1+3b2Kc1Θ,RN13N2+b2Kc2Θ,RN1N2+b2Kc3Θ,RN1N22+2b2Kc4Θ,RN12N25+b2Kc5Θ,RN1N26=b2nFe0Σni

Similarly, multiplied eq. (24) by mole number b1=nFe0 of Fe in eq. (23) gets the following equation

(56)b2Σni×b1=b1N2+b1Kc1Θ,RN13N2+b1Kc2Θ,RN1N2+2b1Kc3Θ,RN1N22+5b1Kc4Θ,RN12N25+6b1Kc5Θ,RN1N26=b1nAl0Σni

Thus, the following equation can be obtained by subtraction eq. (56) from eq. (55) as

(57)3b2−b1+1Kc1Θ,RN13N2+b2−b1+1Kc2Θ,RN1N2+b2−2b1+1Kc3Θ,RN1N22+2b2−5b1+1Kc4Θ,RN12N25+b2−6b1+1Kc5Θ,RN1N26=1−b2+1N1−1−b1N2

This means that two equations of eqs (23) and (24) are reduced to one equation as eq. (57).

Inserting the expressions of KciΘ,R in Table 4 into eq. (25) yields

(58)Kc1Θ,RN13N2+Kc2Θ,RN1N2+Kc3Θ,RN1N22+Kc4Θ,RN12N25+Kc5Θ,RN1N26=1.0−N1−N2

Combining eqs (57) and (58) gives the following equation

(59)3b2−b1+1Kc1Θ,RN13N2+b2−b1+1Kc2Θ,RN1N2+b2−2b1+1Kc3Θ,RN1N22+2b2−5b1+1Kc4Θ,RN12N25+b2−6b1+1Kc5Θ,RN1N26=1−b2+1N1−1−b1N2

Dividing eq. (59) by coefficient of Kc1Θ,R or Kc2Θ,R or Kc3Θ,R or Kc4Θ,R or Kc5Θ,R yields five sub-equations as described in eqs (26)–(30), respectively.

Nomenclatures

ai

Activity of element i or compound i, (–);

aR,i

Activity of element i or compound i relative to pure matter i (l or s or g) as standard state with mole fraction xi as concentration unit and following Raoult’s law under the condition of taking ideal solution as reference state, i. e., aR,i=γixi, (–);

a%,i

Activity of element i referred to 1 mass percentage of element i as standard state with mass percentage [% i] as concentration unit and obeying Henry’s law under the condition of taking infinitely dilute ideal solution as reference state, i. e., a%,i=f%,i[%i], (–);

aH,i

Activity of element i relative to hypothetical pure matter i (l or s or g) as standard state with mole fraction xi as concentration unit and conforming to Henry’s law under the condition of taking infinitely dilute ideal solution as reference state, i. e., aH,i=fH,ixi, (–);

bi

Mole number of element i in 100 g metallic melts before reaction equilibrium for forming associated molecules or compounds, having the same meaning with ni0, (mol);

ci

Coefficient of independent variable Xi in multiple linear regression equation Y=c1X1+c2X2+⋯+cnXn, (–);

eii

First-order activity interaction coefficient of element i to i in metallic melts related with activity coefficient f%,i, (–);

f%,i

Activity coefficient of element i in metallic melts related with activity a%,i, (–);

fH,i

Activity coefficient of element i in metallic melts related with activity aH,i, (–);

ΔrGm,iΘ,R

Standard molar Gibbs free energy change of reaction for forming compound i based on activity aR,i for reactants and products, (J/mol);

ΔsolGm,j(l)→[j][%j=1\mdot0Θ,%

Standard molar Gibbs free energy change of dissolved liquid j(l) for forming [% j] as 1.0 in i–j binary melts referred to 1 mass percentage of j as reference state, (J/mol);

KiΘ,R

Standard equilibrium constant of chemical reaction for forming compound i based on activity aR,i for reactants or products, (–);

Mi

Relative atomic mass of element i, (–);

ni0

Mole number of element i in 100 g metallic melts before reaction equilibrium for forming associated molecule or compound, having the same meaning of bi, (mol);

ni

Equilibrium mole number of structural unit i in 100 g metallic melts based on the AMCT, (mol);

Σni

Total equilibrium mole number of all structural units in 100 g metallic melts based on the AMCT, (mol);

Ni

Mass action concentrations of structural unit i in metallic melts based on the AMCT, (–);

R

Gas constant, (8.314 J/(mol⋅K));

T

Absolute temperature, (K);

xi

Mole fraction of element i or compound i in metallic melts, (–);

X

Independent variable, (–);

Y

Dependent variable, (–);

[% i]

Mass percentage of element i or compound i in metallic melts, (×10−2,–).

Greek symbols
γi

Activity coefficient of element i related with activity aR,i, (–);

γi0

Raoultian activity coefficient of element i in infinitely dilute metallic melts relative to pure matter i (l or s or g) as standard state and taking infinitely dilute ideal solution as reference state, i. e., equal to value of γi,xi→0.0, (–);

εii

First-order activity interaction coefficient of element i in metallic melts related with activity coefficient γi, (–);

Subscripts
ci

Molecule i or compound i, (–).

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Received: 2017-02-10
Accepted: 2017-10-09
Published Online: 2018-10-26
Published in Print: 2018-10-25

© 2018 Walter de Gruyter GmbH, Berlin/Boston

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Numerical Simulation of the Electron Beam Welding and Post Welding Heat Treatment Coupling Process
  5. Effect of Ti and Ta on Oxidation Kinetic of Chromia Forming Ni-Base Superalloys in Ar-O2-Based Atmosphere
  6. Effects of Cerium on the Inclusions and Pitting Corrosion Behavior of 434 Ferritic Stainless Steel
  7. Critical Assessment of Activities of Structural Units in Fe–Al Binary Melts Based on the Atom and Molecule Coexistence Theory
  8. A Yield Stress Model for a Solution-Treated Ni-Based Superalloy during Plastic Deformation
  9. Stress Relaxation Behaviour and Creep Constitutive Equations of SA302Gr.C Low-Alloy Steel
  10. Effects of Inner Defects on Creep Damage and Crack Initiation for a Brazed Joint
  11. Experimental and Numerical Investigations on Hot Deformation Behavior and Processing Maps for ASS 304 and ASS 316
  12. Production of Iron Based Alloys from Mill Scale through Metallothermic Reduction
  13. Effect of Nb and V on Austenite Grain Growth Behavior of the Cr-Mo-V Steel for Brake Discs
  14. A Thermodynamic Study of the Reduction of a Limonitic Laterite Ore by Methane
  15. Electrochemical and Phase Analysis of Si(IV) on Fe Electrode in Molten NaCl-NaF-KCl-SiO2 System
  16. Characterization of Hot Deformation Behavior for Pure Aluminum Using 3D Processing Maps
  17. Effect of Chromium Addition on the Cyclic Oxidation Resistance of Pseudo-Binary (Mo,Cr)3 Si Silicide Alloy
  18. Equiaxed Solidification of 430 Ferritic Stainless Steel Nucleating on Core-Containing Ti
  19. FE Analysis of Dynamical Recrystallization during the Seamless Tube Extrusion of Semicontinuous Casting Magnesium Alloy and Experimental Verification
  20. Study on the Reblow Model for Medium-High Carbon Steel Melting by Converter
  21. Short Communication
  22. Effect of B2O3 on Slag-Metal Reaction between CaO-Al2O3-Based Mold Flux and High Aluminum Steel
  23. Review Article
  24. Computation of the Thermal Residual Stresses in SiC/SiC Composites with Multi-Layered Interphases by Using ANN with the Structure of Random Forest
  25. Research Articles
  26. Failure Analysis of the Corroded Water Wall Tube in a 50MW Thermal Power Plant
  28. CO2 Absorption of Powdered Ba2Fe2O5 with Different Particle Size
  29. Induced-Pitting Behaviors of MnS Inclusions in Steel
https://doi.org/10.1515/htmp-2017-0018
Keywords for this article
Assessment; Activity of aluminum; Activity of iron; Fe–Al binary melts; Mass action concentration; Reaction ability; Atom and molecule coexistence theory (AMCT); Structural units
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Numerical Simulation of the Electron Beam Welding and Post Welding Heat Treatment Coupling Process
  5. Effect of Ti and Ta on Oxidation Kinetic of Chromia Forming Ni-Base Superalloys in Ar-O2-Based Atmosphere
  6. Effects of Cerium on the Inclusions and Pitting Corrosion Behavior of 434 Ferritic Stainless Steel
  7. Critical Assessment of Activities of Structural Units in Fe–Al Binary Melts Based on the Atom and Molecule Coexistence Theory
  8. A Yield Stress Model for a Solution-Treated Ni-Based Superalloy during Plastic Deformation
  9. Stress Relaxation Behaviour and Creep Constitutive Equations of SA302Gr.C Low-Alloy Steel
  10. Effects of Inner Defects on Creep Damage and Crack Initiation for a Brazed Joint
  11. Experimental and Numerical Investigations on Hot Deformation Behavior and Processing Maps for ASS 304 and ASS 316
  12. Production of Iron Based Alloys from Mill Scale through Metallothermic Reduction
  13. Effect of Nb and V on Austenite Grain Growth Behavior of the Cr-Mo-V Steel for Brake Discs
  14. A Thermodynamic Study of the Reduction of a Limonitic Laterite Ore by Methane
  15. Electrochemical and Phase Analysis of Si(IV) on Fe Electrode in Molten NaCl-NaF-KCl-SiO2 System
  16. Characterization of Hot Deformation Behavior for Pure Aluminum Using 3D Processing Maps
  17. Effect of Chromium Addition on the Cyclic Oxidation Resistance of Pseudo-Binary (Mo,Cr)3 Si Silicide Alloy
  18. Equiaxed Solidification of 430 Ferritic Stainless Steel Nucleating on Core-Containing Ti
  19. FE Analysis of Dynamical Recrystallization during the Seamless Tube Extrusion of Semicontinuous Casting Magnesium Alloy and Experimental Verification
  20. Study on the Reblow Model for Medium-High Carbon Steel Melting by Converter
  21. Short Communication
  22. Effect of B2O3 on Slag-Metal Reaction between CaO-Al2O3-Based Mold Flux and High Aluminum Steel
  23. Review Article
  24. Computation of the Thermal Residual Stresses in SiC/SiC Composites with Multi-Layered Interphases by Using ANN with the Structure of Random Forest
  25. Research Articles
  26. Failure Analysis of the Corroded Water Wall Tube in a 50MW Thermal Power Plant
  28. CO2 Absorption of Powdered Ba2Fe2O5 with Different Particle Size
  29. Induced-Pitting Behaviors of MnS Inclusions in Steel
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