Critical Evaluation and Thermodynamic Optimization of the Cr–P and Cr–Fe–P Systems

Abstract

Existing thermodynamic descriptions of the whole Cr–Fe–P system are insufficiently accurate for understanding the thermodynamic behavior of the Cr–Fe–P materials during the manufacturing process. To construct a more precise and consistent thermodynamic database of the Cr–Fe–P system, thermodynamic modeling of the Cr–P and Cr–Fe–P systems was conducted using the CALculation of PHAse Diagrams (CALPHAD) approach based on critical evaluation of the experimental data. The modified quasichemical model and compound energy formalism were employed to describe the liquid and solid solutions, respectively. The Gibbs energies of stoichiometric compounds Cr₃P(s), Cr₂P(s), CrP(s), and CrP₂(s) were carefully determined based on reliable experimental data. The ternary (Cr,Fe)₃P, (Cr,Fe)₂P, and (Cr,Fe)P phosphides were modeled as solid solutions considering mutual substitution between Cr and Fe atoms. In addition, the phase equilibria of BCC_A2 and FCC_A1 solutions and the liquid phase of the ternary Cr–Fe–P system were also optimized for more accurate descriptions of existing phase equilibria and thermodynamic properties data. As an application of the present database, the experimentally unexplored thermodynamic properties and phase diagrams of the Cr–Fe–P system are predicted.

1. Introduction

In recent years, the Cr–Fe–P system has attracted increasingly wide focus among researchers since it is vital for the metallurgical process of amorphous alloys, stainless steels, and semiconductors [1,2,3,4,5]. Such materials always exhibit excellent corrosion resistance, strength, thermal sensitivity, and photoconductive properties, which primarily depend on their chemical composition and preparation process. In order to fabricate Cr–Fe–P materials with promising performance, fundamental investigations of the thermodynamic behavior of the Cr–Fe–P alloy are of great significance as they help the alloy design and process optimization in actual practice.

Until now, the Cr–Fe [6,7,8,9,10] and Fe–P [11,12,13,14,15,16,17] systems have been thermodynamically modeled by many investigators and were reoptimized recently by the present authors [18,19]. A thermodynamic assessment of the Cr–P system was performed by Miettinen [20], who proposed only a partial Cr–P phase diagram of the Cr-rich region. The extrapolated phase diagram for the remaining region shows much discrepancy. Furthermore, the assessment results of the Cr–P system exhibited less satisfactory agreement with the thermodynamic property data of intermediate Cr phosphides and inconsistency with the ternary Cr–Fe–P system. A thermodynamic assessment of the ternary Cr–Fe–P system was performed by Miettinen and Vassilev [21]. However, the assessed model parameters result in some deviations of phase equilibria and thermodynamic properties from existing data. Therefore, reoptimization of the Cr–Fe–P system is necessary to obtain a self–consistent and accurate thermodynamic description of this system.

The purpose of the present study is to conduct a critical thermodynamic optimization of the Cr–P and Cr–Fe–P systems to construct a more accurate thermodynamic database. The recently optimized Cr–Fe [18] and Fe–P [19] systems by the present authors were adopted by this work. The phase relation and thermodynamic properties of the Cr–P and Cr–Fe–P systems were optimized based on reliable experimental information. Particularly, the Gibbs energies of the liquid, CrP(s), CrP₂(s), (Cr,Fe)₃(P)₁, (Cr,Fe)₂(P)₁, (Cr,Fe)₁(P)₁, BCC_A2, and FCC_A1 phases were carefully determined to resolve the discrepancies left over in existing thermodynamic assessments [20,21]. The present Cr–Fe–P database was utilized to predict experimentally unexplored thermodynamic properties and phase diagrams. All the calculations were conducted using the FactSage 8.3 software [22].

2. Thermodynamic Models

2.1. Gas Phase

The gas phase of the Cr–Fe–P system is a mixture of Cr(g), Fe(g), P(g), P₂(g), and P₄(g) species. The molar Gibbs energy of gas phase ($G_T^{\mathrm{g}\mathrm{a}\mathrm{s}}$) was determined by Equation (1):

$$G_T^{\mathrm{gas}}={\displaystyle \sum x_{\mathrm{i}}\left(G_{\mathrm{i}}^{°}+RT\ln x_{\mathrm{i}}\right)}+RT\ln \left(f/P^{\theta }\right)$$

(1)

where $G_{\mathrm{i}}^{°}$ is the molar Gibbs energy (J/mol) of gas species i and was taken directly from the FactPS database of FactSage [22], R is the molar gas constant (=8.314 J/(mol·K)), $x_{\mathrm{i}}$ is the mole fraction of gas species i, f is the gas fugacity and identical to the gas pressure (in atm) at normal pressure, T is the temperature in Kelvin (K), $P^{\mathsf{\theta }}$ is the atmospheric pressure (=1 atm), and f is the gas fugacity that is identical to the gas pressure (in atm) at normal pressure.

2.2. Elementary Substance and Stoichiometric Compounds

The Gibbs energies of pure solid and liquid Cr, Fe, and P elements were taken from the SGTE [23] data compilation. No stoichiometric compound was considered for the binary Cr–Fe and ternary Cr–Fe–P systems. The Gibbs energies ($G_T^{°}$) of all intermediate compounds including Cr₃P, Cr₂P, CrP, and CrP₂ of the Cr–P system and Fe₃P, Fe₂P, FeP, and FeP₂ of the Fe–P system could be determined by their standard enthalpy of formation ${∆H}_{298.15 \mathrm{K}}^{°}$ (J/mol), heat capacity $C_{\mathrm{P}}$ (J/(mol·K)), and standard entropy $S_{298.15 \mathrm{K}}^{°}$ (J/(mol·K)), as expressed by Equation (2):

$$G_T^{°}=\left(\mathsf{\Delta }{H}_{298.15 \mathrm{K}}^{°}+{\displaystyle {\int }_{298.15 \mathrm{K}}^T{C}_P}dT\right)-T\left(S_{298.15 \mathrm{K}}^{°}+{\displaystyle {\int }_{298.15 \mathrm{K}}^T\frac{C_P}{T}}dT\right)$$

(2)

In particular, an additional Gibbs energy resulting from the magnetic contribution, $G^{\mathrm{m}\mathrm{g}}$ (J/mol), was applied to the Gibbs energies of Cr(BCC_A2, FCC_A1), Fe(BCC_A2, FCC_A1), and Fe₃P, which exhibits magnetic behavior, based on an empirical expression proposed by Inden [24] and modified by Hillert and Jarl [25]:

$$G^{\mathrm{mg}}=RT\ln \left(\beta +1\right)g(\tau )$$

(3)

where τ is expressed by $T/T^{\mathrm{*}}$, $T^{\mathrm{*}}$ is the Néel temperature $T_{\mathrm{N}}$ (K) for anti–ferromagnetic ordering or the Curie temperature $T_{\mathrm{C}}$ (K) for ferromagnetic ordering, β is the average magnetic moment of per mole of atoms expressed in Bohr magnetons (μ_B/mol), and $g\left(\tau \right)$ is a polynomial function that can be determined by Equation (4) [25]:

$$\begin{array}{l}g(\tau )=1-\left[{\displaystyle \frac{79{\tau }^{-1}}{140p}}+{\displaystyle \frac{474}{497}}\left({\displaystyle \frac{1}{p}}-1\right)\left({\displaystyle \frac{{\tau }^3}{6}}+{\displaystyle \frac{{\tau }^9}{135}}+{\displaystyle \frac{{\tau }^{15}}{600}}\right)\right]/D,\dots \tau \le 1\\ g(\tau )=-\left({\displaystyle \frac{{\tau }^{-5}}{10}}+{\displaystyle \frac{{\tau }^{-15}}{315}}+{\displaystyle \frac{{\tau }^{-25}}{1500}}\right)/D,\dots \tau >1\end{array}$$

(4)

where $D={\displaystyle \frac{518}{1125}}+{\displaystyle \frac{11692}{15975}}\left({\displaystyle \frac{1}{p}}-1\right)$, and the p value was considered as the fraction of the magnetic enthalpy absorbed above the critical temperature depending on the structure; for example, p is 0.40 for BCC_A2 phase and 0.28 for other common phases.

2.3. Solid Solutions

The Gibbs energies of solid solutions of the Cr–Fe–P system were described using the Compound Energy Formalism (CEF) [26] with consideration of their crystal structures. In the ternary Cr–Fe–P system, isomorphous Cr₃P and Fe₃P; Cr₂P and Fe₂P; and CrP and FeP can form solid solutions in the formulas of Me₃P, Me₂P, and MeP, respectively, through mutual substitution between Cr and Fe atoms. In the present study, the solid Me₃P, Me₂P, and MeP solutions were described using a two-sublattice model (Cr,Fe)_n(P) (n = 3 for Me₃P, n = 2 for Me₂P, and n = 1 for MeP). Their molar Gibbs energies can be calculated by Equation (5).

$$G_{{\mathrm{Me}}_{\mathrm{n}}\mathrm{P}}^{\mathrm{sol}.}=y_{\mathrm{Cr}}{G}_{{\mathrm{Cr}}_{\mathrm{n}}\mathrm{P}}^{°}+y_{\mathrm{Fe}}{G}_{{\mathrm{Fe}}_{\mathrm{n}}\mathrm{P}}^{°}+nRT\left(y_{\mathrm{Cr}}\ln y_{\mathrm{Cr}}+y_{\mathrm{Fe}}\ln y_{\mathrm{Fe}}\right)+{\displaystyle \sum _{\mathrm{i}=0,1,2\dots }{y}_{\mathrm{Cr}}{y}_{\mathrm{Fe}}{L^{\mathrm{i}}}_{\mathrm{Cr},\mathrm{Fe}:\mathrm{P}}}+G^{\mathrm{mg}}$$

(5)

where, $y_{\mathrm{C}\mathrm{r}}$ and $y_{\mathrm{F}\mathrm{e}}$ are site fractions of Cr and Fe in the substitutional lattice. $G_{{\mathrm{C}\mathrm{r}}_{\mathrm{n}}\mathrm{P}}^{°}$ and $G_{{\mathrm{F}\mathrm{e}}_{\mathrm{n}}\mathrm{P}}^{°}$ are Gibbs energies (J/mol) of Cr_nP and Fe_nP compounds, respectively. $L_{\mathrm{C}\mathrm{r},\mathrm{F}\mathrm{e}:\mathrm{P}}^{\mathrm{i}}$ is the adjustable interaction parameter. $G^{\mathrm{m}\mathrm{g}}$ is the magnetic contribution to the Gibbs energy. BCC_A2 and FCC_A1 solid solutions were described using the model (Cr,Fe,P)₁(Va)_j (j = 1 3 for BCC_A2 and 1 for FCC_A1), and their Gibbs energies per formula unit were calculated as follows:

$$\begin{array}{ll}{G}_{\mathrm{BCC}/\mathrm{FCC}}^{\mathrm{sol}.}=& \underset{\mathrm{i}=\mathrm{Fe},\mathrm{Cr},\mathrm{P}}{\Sigma }{x}_{\mathrm{i}}{G}_{\mathrm{i}}^{°}+RT\underset{\mathrm{i}=\mathrm{Fe},\mathrm{Cr},\mathrm{P}}{\Sigma }{x}_{\mathrm{i}}\ln x_{\mathrm{i}}+\underset{\mathrm{m}=0,1,2\dots }{\Sigma }{x}_{\mathrm{Cr}}{x}_{\mathrm{P}}{L}_{\mathrm{Cr},\mathrm{P}:\mathrm{Va}}^{\mathrm{m}}+\underset{\mathrm{k}=0,1,2\dots }{\Sigma }{x}_{\mathrm{Fe}}{x}_{\mathrm{P}}{L}_{\mathrm{Fe},\mathrm{P}:\mathrm{Va}}^{\mathrm{k}}\\ & +\underset{\mathrm{p}=0,1,2\dots }{\Sigma }{x}_{\mathrm{Cr}}{x}_{\mathrm{Fe}}{L}_{\mathrm{Cr},\mathrm{Fe}:\mathrm{Va}}^{\mathrm{P}}+\underset{\mathrm{q}=0,1,2\dots }{\Sigma }{x}_{\mathrm{Fe}}{x}_{\mathrm{Cr}}{x}_{\mathrm{P}}{}^{\mathrm{q}}L_{\mathrm{F}\mathrm{e},\mathrm{C}\mathrm{r},\mathrm{P}:\mathrm{Va}}+G^{\mathrm{mg}}\end{array}$$

(6)

where $x_{\mathrm{i}}$ is the mole fraction of component i; $G_i^{°}$ is the molar Gibbs energy (J/mol) of pure solid i (i = Cr, Fe, P); and $L_{\mathrm{C}\mathrm{r},\mathrm{P}}^{\mathrm{m}}$, $L_{\mathrm{F}\mathrm{e},\mathrm{P}}^{\mathrm{k}}$, $L_{\mathrm{F}\mathrm{e},\mathrm{P}}^{\mathrm{p}}$ and ${}^{\mathrm{q}}L_{\mathrm{F}\mathrm{e},\mathrm{C}\mathrm{r},\mathrm{P}}$ are adjustable interaction parameters of corresponding binary and ternary systems (J/mol). The sigma phase was described with a three-sublattice model (Fe)₈(Cr)₄(Fe,Cr)₁₈, and its molar Gibbs energy was calculated using Equation (7):

$$G_{\mathrm{Sigma}}^{\mathrm{sol}.}=y_{\mathrm{Fe}}{G}_{{\mathrm{Fe}}_8{\mathrm{Cr}}_4{\mathrm{Fe}}_{18}}+y_{\mathrm{Cr}}{G}_{{\mathrm{Fe}}_8{\mathrm{Cr}}_4{\mathrm{Cr}}_{18}}+18RT\left(y_{\mathrm{Fe}}\ln y_{\mathrm{Fe}}+y_{\mathrm{Cr}}\ln y_{\mathrm{Cr}}\right)+{\displaystyle \sum _{\mathrm{i}=0,1,2\dots }{y}_{\mathrm{Fe}}{y}_{\mathrm{Cr}}{L^{\mathrm{i}}}_{\mathrm{Fe}:\mathrm{Cr}:\mathrm{Fe},\mathrm{Cr}}}$$

(7)

where $G_{{\mathrm{F}\mathrm{e}}_8{\mathrm{C}\mathrm{r}}_4{\mathrm{F}\mathrm{e}}_{18}}$ and $G_{{\mathrm{F}\mathrm{e}}_8{\mathrm{C}\mathrm{r}}_4{\mathrm{C}\mathrm{r}}_{18}}$ are the Gibbs energies (J/mol) of Fe₈Cr₄Fe₁₈ and Fe₈Cr₄Cr₁₈ combinations, respectively. $L_{\mathrm{F}\mathrm{e}:\mathrm{C}\mathrm{r}:\mathrm{F}\mathrm{e},\mathrm{C}\mathrm{r}}^{\mathrm{i}}$ is the adjustable interaction model parameter (J/mol).

2.4. Liquid Solution

The modified quasichemical model (MQM) [27,28], accounting for the short-range ordering (SRO) of the nearest-neighbor atoms, was utilized to describe the liquid phases of all subsystems and the entire ternary system of Cr–Fe–P. Compared to the traditional Bragg–Williams random mixing model, the MQM gives a more realistic thermodynamic description of the liquid solution. In MQM, the atom pair formation Gibbs energy can be expressed as a polynomial of the pair fraction instead of the component fraction, and the coordination number of each component can be varied with composition to reproduce the SRO more easily.

For the binary A–B liquid solution, A and B atoms are distributed over the quasilattice sites, and the following atom pair exchanging reaction is considered in MQM:

$$(\mathrm{A}-\mathrm{A})+(\mathrm{B}-\mathrm{B})=2(\mathrm{A}-\mathrm{B});\mathsf{\Delta }{g}_{\mathrm{AB}}$$

(8)

where (A–A), (A–B), and (B–B) represent the first-nearest-neighbor pairs between components A and A, A and B, and B and B; and $\mathsf{\Delta }{g}_{\mathrm{A}\mathrm{B}}$ is the formation Gibbs energy of 2 moles of (A–B) pairs from 1 mole (A–A) pairs and 1 mole (B–B) pairs. The Gibbs energy of the A–B liquid phase was calculated as follows:

$$G_{\mathrm{AB}}^{\mathrm{Liq}.}=(n_{\mathrm{A}}{G}_{\mathrm{A}}^{°}+n_{\mathrm{B}}{G}_{\mathrm{B}}^{°})-T\mathsf{\Delta }{S}_{\mathrm{AB}}^{\mathrm{conf}.}+n_{\mathrm{AB}}(\mathsf{\Delta }{g}_{\mathrm{AB}}/2)$$

(9)

where $n_{\mathrm{A}}$ and $n_{\mathrm{B}}$ are the mole numbers of A and B atoms (mol), $G_{\mathrm{A}}^{°}$ and $G_{\mathrm{B}}^{°}$ are molar Gibbs energies of pure liquid A and B (J/mol), and ${\mathsf{\Delta }S}_{\mathrm{A}\mathrm{B}}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}.}$ is the configurational entropy of mixing (J/(mol·K)) given by random distribution of the (A–A), (A–B), and (B–B) pairs as follows:

$$\mathsf{\Delta }{S}_{\mathrm{AB}}^{\mathrm{conf}.}=-R(n_{\mathrm{A}}\ln X_{\mathrm{A}}+n_{\mathrm{B}}\ln X_{\mathrm{B}})-R\left[n_{\mathrm{AA}}\ln \left({\displaystyle \frac{X_{\mathrm{AA}}}{{Y_{\mathrm{A}}}^2}}\right)+n_{\mathrm{BB}}\ln \left({\displaystyle \frac{X_{\mathrm{BB}}}{{Y_{\mathrm{B}}}^2}}\right)+n_{\mathrm{AB}}\ln \left({\displaystyle \frac{X_{\mathrm{AB}}}{2Y_{\mathrm{A}}{Y}_{\mathrm{B}}}}\right)\right]$$

(10)

where $n_{\mathrm{A}\mathrm{A}}$, $n_{\mathrm{A}\mathrm{B}}$, and $n_{\mathrm{B}\mathrm{B}}$ represent the mole numbers of (A–A), (A–B), and (B–B) pairs (mol); $X_{\mathrm{A}\mathrm{A}}$, $X_{\mathrm{A}\mathrm{B}}$, and $X_{\mathrm{B}\mathrm{B}}$ are pair fractions of corresponding atom pairs; and $Y_{\mathrm{A}}$ and $Y_{\mathrm{B}}$ are coordination equivalent fractions of A and B atoms. The pair fractions $X_{\mathrm{A}\mathrm{A}}$, $X_{\mathrm{B}\mathrm{B}}$, and $X_{\mathrm{A}\mathrm{B}}$ and coordination equivalent fractions $Y_{\mathrm{A}}$ and $Y_{\mathrm{B}}$ were determined by Equations (11)–(15):

$$X_{\mathrm{AA}}=n_{\mathrm{AA}}/\left(n_{\mathrm{AA}}+n_{\mathrm{AB}}+n_{\mathrm{BB}}\right)$$

(11)

$$X_{\mathrm{AB}}=n_{\mathrm{AB}}/\left(n_{\mathrm{AA}}+n_{\mathrm{AB}}+n_{\mathrm{BB}}\right)$$

(12)

$$X_{\mathrm{BB}}=n_{\mathrm{BB}}/\left(n_{\mathrm{AA}}+n_{\mathrm{AB}}+n_{\mathrm{BB}}\right)$$

(13)

$$Y_{\mathrm{A}}=X_{\mathrm{AA}}+{\displaystyle \frac{1}{2}}{X}_{\mathrm{AB}}$$

(14)

$$Y_{\mathrm{B}}=X_{\mathrm{BB}}+{\displaystyle \frac{1}{2}}{X}_{\mathrm{AB}}$$

(15)

${\mathsf{\Delta }g}_{\mathrm{A}\mathrm{B}}$ in Equations (8) and (9) is the model parameter to reproduce the Gibbs energy of the A–B liquid solution (J/mol), and can be expanded as a polynomial in terms of the atomic pair fractions $X_{\mathrm{A}\mathrm{A}}$ and $X_{\mathrm{B}\mathrm{B}}$.

$$\mathsf{\Delta }{g}_{\mathrm{AB}}=\mathsf{\Delta }{g}_{\mathrm{AB}}^{°}+{\displaystyle \sum _{\mathrm{i}\ge 1}{g}_{\mathrm{AB}}^{\mathrm{i}0}}{X}_{\mathrm{AA}}^{\mathrm{i}}+{\displaystyle \sum _{\mathrm{j}\ge 1}{g}_{\mathrm{AB}}^{0\mathrm{j}}}{X}_{\mathrm{BB}}^{\mathrm{j}}$$

(16)

where ${\mathsf{\Delta }g}_{\mathrm{A}\mathrm{B}}^{°}$, $g_{\mathrm{A}\mathrm{B}}^{\mathrm{i}0}$, and $g_{\mathrm{A}\mathrm{B}}^{0\mathrm{j}}$ are adjustable model parameters (J/mol) that can be functions of the temperature. In MQM, the coordination numbers of A and B, and $Z_{\mathrm{A}}$ and $Z_{\mathrm{B}}$, can be varied with the composition to reproduce the SRO as follows:

$$\frac{1}{{\mathrm{Z}}_{\mathrm{A}}}=\frac{1}{{\mathrm{Z}}_{\mathrm{AA}}^{\mathrm{A}}}\left(\frac{2n_{\mathrm{AA}}}{2n_{\mathrm{AA}}+n_{\mathrm{AB}}}\right)+\frac{1}{{\mathrm{Z}}_{\mathrm{AB}}^{\mathrm{A}}}\left(\frac{n_{\mathrm{AB}}}{2n_{\mathrm{AA}}+n_{\mathrm{AB}}}\right)$$

(17)

$$\frac{1}{{\mathrm{Z}}_{\mathrm{B}}}=\frac{1}{{\mathrm{Z}}_{\mathrm{BB}}^{\mathrm{B}}}\left(\frac{2n_{\mathrm{BB}}}{2n_{\mathrm{BB}}+n_{\mathrm{AB}}}\right)+\frac{1}{{\mathrm{Z}}_{\mathrm{BA}}^{\mathrm{B}}}\left(\frac{n_{\mathrm{AB}}}{2n_{\mathrm{BB}}+n_{\mathrm{AB}}}\right)$$

(18)

where $Z_{\mathrm{A}\mathrm{A}}^{\mathrm{A}}$ is the value $Z_{\mathrm{A}}$ when all nearest neighbors of an A atom are A atoms, and $Z_{\mathrm{A}\mathrm{B}}^{\mathrm{A}}$ is the value of $Z_{\mathrm{A}}$ when all nearest neighbors of the A atom are B atoms. $Z_{\mathrm{B}\mathrm{B}}^{\mathrm{B}}$ and $Z_{\mathrm{B}\mathrm{A}}^{\mathrm{B}}$ are defined in an analogous manner. In this study, $Z_{\mathrm{C}\mathrm{r}\mathrm{C}\mathrm{r}}^{\mathrm{C}\mathrm{r}}=Z_{\mathrm{F}\mathrm{e}\mathrm{F}\mathrm{e}}^{\mathrm{F}\mathrm{e}}=Z_{\mathrm{P}\mathrm{P}}^{\mathrm{P}}=6$ [18,19], $Z_{\mathrm{P}\mathrm{F}\mathrm{e}}^{\mathrm{P}}=Z_{\mathrm{P}\mathrm{C}\mathrm{r}}^{\mathrm{P}}=Z_{\mathrm{F}\mathrm{e}\mathrm{C}\mathrm{r}}^{\mathrm{F}\mathrm{e}}=Z_{\mathrm{C}\mathrm{r}\mathrm{F}\mathrm{e}}^{\mathrm{C}\mathrm{r}}=6$ [18,19], and $Z_{\mathrm{F}\mathrm{e}\mathrm{P}}^{\mathrm{F}\mathrm{e}}=Z_{\mathrm{C}\mathrm{r}\mathrm{P}}^{\mathrm{C}\mathrm{r}}=3$ [19], as given in

Table 1. Optimized model parameters of the Cr–Fe–P system. Heat capacity, $C_P$ (J/(mol·K)); standard enthalpy of formation, ${∆H}_{298.15 \mathrm{K}}^{°}$ (J/mol); standard entropy, $S_{298.15 \mathrm{K}}^{°}$ (J/(mol·K)); adjustable interaction parameter, L (J/mol); Gibbs energy, G, g, $∆\mathrm{g}$ (J/mol); Curie temperature, $T_C$ (K); magnetic moment, $\beta$ (μ_B/mol).

Phase	Model Parameters
Liquid (Cr, Fe, P)	$Z_{\mathrm{CrCr}}^{\mathrm{Cr}}=Z_{\mathrm{FeFe}}^{\mathrm{Fe}}=Z_{\mathrm{PP}}^{\mathrm{P}}=6$ [18,19]
	$Z_{\mathrm{PCr}}^{\mathrm{P}}=Z_{\mathrm{PFe}}^{\mathrm{P}}=Z_{\mathrm{CrFe}}^{\mathrm{Cr}}=Z_{\mathrm{FeCr}}^{\mathrm{Fe}}=6$ [18,19] $Z_{\mathrm{CrP}}^{\mathrm{Cr}}=Z_{\mathrm{FeP}}^{\mathrm{Fe}}=3$ [19] $\mathsf{\Delta }{g}_{\mathrm{CrP}}=-\mathrm{49,078}-3.5564T+(\mathrm{21,548}+2.092T)X_{\mathrm{CrCr}}-\mathrm{29,957}{X}_{\mathrm{CrCr}}^2$ [*] $\mathsf{\Delta }{g}_{\mathrm{FeP}}=-\mathrm{56,902}+6.569T+(5481+3.033T)X_{\mathrm{FeFe}}-(\mathrm{11,966}-2.51T)X_{\mathrm{FeFe}}^2-9623X_{\mathrm{PP}}$ [19] $\mathsf{\Delta }{g}_{\mathrm{CrFe}}=-242-0.335T-(192.46+1.046T)X_{\mathrm{CrCr}}+83.68X_{\mathrm{FeFe}}$ [18] $g_{\mathrm{CrP}(\mathrm{Fe})}^{001}=8368$ [*], $g_{\mathrm{FeP}(\mathrm{Cr})}^{001}=-\mathrm{37,237.6}+7.5312T$ [*], $g_{\mathrm{FeCr}(\mathrm{P})}^{001}=4184+12.552T$ [*]
	“Toop–like” interpolation with P as an asymmetric component [*]
BCC_A2 (Cr,Fe,P)1(Va)3	$G_{\mathrm{Fe}:\mathrm{Va}}^{\mathrm{BCC}\_\mathrm{A}2}=G_{\mathrm{Fe}(\mathrm{BCC})}^{°}$, $G_{\mathrm{Cr}:\mathrm{Va}}^{{\mathrm{BCC}}_{-}\mathrm{A}2}=G_{\mathrm{Cr}(\mathrm{BCC})}^{°}$, $G_{\mathrm{P}:\mathrm{Va}}^{{\mathrm{BCC}}_{-}\mathrm{A}2}=G_{\mathrm{P}(\mathrm{BCC})}^{°}$ [*] $L_{\mathrm{Cr},\mathrm{P}:\mathrm{Va}}^{{\mathrm{BCC}}_{-}\mathrm{A}2}=-\mathrm{48,116}$ [*]
	$L_{\mathrm{Fe},\mathrm{P}:\mathrm{Va}}^{{\mathrm{BCC}}_{-}\mathrm{A}2}=-\mathrm{203,476}+15.48T+\mathrm{33,472}\left(y_{\mathrm{Fe}}-y_{\mathrm{P}}\right)$ [19]
	$L_{\mathrm{Fe},\mathrm{Cr}:\mathrm{Va}}^{{\mathrm{BCC}}_{-}\mathrm{A}2}=\mathrm{20,502}-9.68T$ [9] $T_{C\mathrm{Cr}:\mathrm{Va}}=-311$ [21], $T_{C\mathrm{Fe}:\mathrm{Va}}=1043$ [29] $T_{C\mathrm{Fe},\mathrm{P}:\mathrm{Va}}=-285$ [19], $T_{C\mathrm{Fe},\mathrm{Cr}:\mathrm{Va}}=1650-550(x_{\mathrm{Fe}}-x_{\mathrm{Cr}})$ [9] ${\beta }_{\mathrm{Fe}:\mathrm{Va}}=2.22$ [29], ${\beta }_{\mathrm{Cr}:\mathrm{Va}}=-0.008$ [6], ${\beta }_{\mathrm{Fe},\mathrm{Cr}:\mathrm{Va}}=-0.85$ [6]
FCC_A1 (Cr,Fe,P)1(Va)1	$G_{\mathrm{Fe}:\mathrm{Va}}^{\mathrm{FCC}}=G_{\mathrm{Fe}(\mathrm{FCC})}^{°}$, $G_{\mathrm{Cr}:\mathrm{Va}}^{\mathrm{FCC}}=G_{\mathrm{Cr}(\mathrm{FCC})}^{°}$, $G_{\mathrm{P}:\mathrm{Va}}^{\mathrm{FCC}}=G_{\mathrm{P}(\mathrm{FCC})}^{°}$ [*] $L_{\mathrm{Cr},\mathrm{P}:\mathrm{Va}}^{\mathrm{FCC}\_\mathrm{A}1}=L_{\mathrm{Cr},\mathrm{P}:\mathrm{Va}}^{\mathrm{BCC}\_\mathrm{A}2}=-\mathrm{48,116}$ [*] $L_{\mathrm{Fe},\mathrm{P}:\mathrm{Va}}^{\mathrm{FCC}}=-\mathrm{139,787}+6.49T$ [19] $L_{\mathrm{Fe},\mathrm{Cr}:\mathrm{Va}}^{\mathrm{FCC}\_\mathrm{A}1}=\mathrm{10,833}-7.477T-1410\left(x_{\mathrm{Fe}}-x_{\mathrm{Cr}}\right)$ [6] $T_{C\mathrm{Cr}:\mathrm{Va}}=-1109$, $T_{C\mathrm{Fe}:\mathrm{Va}}=-201$ [6] ${\beta }_{\mathrm{Cr}:\mathrm{Va}}=-2.46$ [6], ${\beta }_{\mathrm{Fe}:\mathrm{Va}}=-2.1$ [29]
Sigma (Fe)8(Cr)4(Fe,Cr)18	$G_{\mathrm{Fe}8\mathrm{Cr}4\mathrm{Fe}18}=8G_{\mathrm{Fe}(\mathrm{FCC}\_\mathrm{A}1)}^{°}+4G_{\mathrm{Cr}(\mathrm{BCC}\_\mathrm{A}2)}^{°}+18G_{\mathrm{Fe}(\mathrm{BCC}\_\mathrm{A}2)}^{°}+\mathrm{117,300}-95.96T$ [6] $G_{\mathrm{Fe}8\mathrm{Cr}4\mathrm{Cr}18}=8G_{\mathrm{Fe}(\mathrm{FCC}\_\mathrm{A}1)}^{°}+22G_{\mathrm{Cr}(\mathrm{BCC}\_\mathrm{A}2)}^{°}+\mathrm{92,300}-95.96T$ [6]
Me3P (Cr,Fe)3(P)1	$G_{\mathrm{Fe}:\mathrm{P}}^{{\mathrm{Me}}_3\mathrm{P}}=G_{{\mathrm{Fe}}_3\mathrm{P}}^{°}$ [19] $G_{\mathrm{Cr}:\mathrm{P}}^{{\mathrm{Me}}_3\mathrm{P}}=G_{{\mathrm{Cr}}_3\mathrm{P}}^{°}$ [*] $\mathsf{\Delta }{H}_{298.15 \mathrm{K}}^{°}({\mathrm{Cr}}_3\mathrm{P})=-\mathrm{184,880}$, $S_{298.15 \mathrm{K}}^{°}({\mathrm{Cr}}_3\mathrm{P})=93.3$ [*] $C_{\mathrm{P}}({\mathrm{Cr}}_3\mathrm{P})=99.258+0.0075T-\mathrm{760,000}{T}^{-2}+1.9\times {10}^{-5}{T}^2$ [*] $L_{\mathrm{Cr},\mathrm{Fe}:\mathrm{P}}^{{\mathrm{Me}}_3\mathrm{P}}=-\mathrm{52,718}+20.92T-\mathrm{18,828}(x_{\mathrm{Cr}}-x_{\mathrm{Fe}})$ [*]
Me2P (Cr,Fe)2(P)1	$G_{\mathrm{Fe}:\mathrm{P}}^{{\mathrm{Me}}_2\mathrm{P}}=G_{{\mathrm{Fe}}_2\mathrm{P}}^{°}$ [19] $G_{\mathrm{Cr}:\mathrm{P}}^{{\mathrm{Me}}_2\mathrm{P}}=G_{{\mathrm{Cr}}_2\mathrm{P}}^{°}$ [*] $\mathsf{\Delta }{H}_{298.15 \mathrm{K}}^{°}({\mathrm{Cr}}_2\mathrm{P})=-\mathrm{169,890}$, $S_{298.15 \mathrm{K}}^{°}({\mathrm{Cr}}_2\mathrm{P})=74$ [*] $C_{\mathrm{P}}({\mathrm{Cr}}_2\mathrm{P})=76.85+0.0058T-\mathrm{476,000}{T}^{-2}+9.8\times {10}^{-6}{T}^2$ [*] $L_{\mathrm{Cr},\mathrm{Fe}:\mathrm{P}}^{\mathrm{MeP}}=-\mathrm{45,187}+5.4392T$ [*]
MeP (Cr,Fe)1(P)1	$G_{\mathrm{Fe}:\mathrm{P}}^{\mathrm{MeP}}=G_{\mathrm{FeP}}^{°}$ [19] $G_{\mathrm{Cr}:\mathrm{P}}^{\mathrm{MeP}}=G_{\mathrm{CrP}}^{°}$ [*] $\mathsf{\Delta }{H}_{298.15 \mathrm{K}}^{°}(\mathrm{CrP})=-\mathrm{122,700}$, $S_{298.15 \mathrm{K}}^{°}(\mathrm{CrP})=46.4$ [*] $C_{\mathrm{P}}(\mathrm{CrP})=47.692+0.01233305T-\mathrm{200,250}{T}^{-2}+1.1051\times {10}^{-6}{T}^2$ [*] $L_{\mathrm{Cr},\mathrm{Fe}:\mathrm{P}}^{\mathrm{MeP}}=-\mathrm{20,083}$ [*]
CrP2 (Cr)1(P)2	$\mathsf{\Delta }{H}_{298.15 \mathrm{K}}^{°}({\mathrm{CrP}}_2)=-\mathrm{155,895}$, $S_{298.15 \mathrm{K}}^{°}({\mathrm{CrP}}_2)=63$ [*] $C_{\mathrm{P}}=68.476+0.0284548T-\mathrm{122,000}{T}^{-2}-6.253\times {10}^{-6}{T}^2$ [*]
FeP2 (Fe)1(P)2	$\mathsf{\Delta }{H}_{298.15 \mathrm{K}}^{°}=-\mathrm{191,100}$$, S_{298.15 \mathrm{K}}^{°}=51.05$ [19] $C_{\mathrm{P}}=77.52563+0.009348T-\mathrm{443,846}{T}^{-2}-1.1\times {10}^{-6}{T}^2$ [19]

* optimized in the present study.

.

The Gibbs energy of the ternary Cr–Fe–P liquid phase can be calculated by interpolating the Gibbs energies of its binary subsystems. In MQM, different geometric interpolation techniques [28] can be utilized depending on the nature of the involved binary liquid solutions to improve the predictive capability. In this work, an asymmetric “Toop–like” geometric interpolation with P as the “asymmetric component” was selected for the ternary Cr–Fe–P system, since the Cr–P and Fe–P liquid solutions deviate largely from the ideal mixing while the Cr–Fe liquid solution exhibits in almost ideal mixing. Based on this interpolation, the Gibbs energy and entropy of mixing of the ternary Cr–Fe–P liquid solution can be calculated by Equations (19) and (20):

$$G_{\mathrm{CrFeP}}^{\mathrm{Liq}.}={\displaystyle \sum _{\mathrm{i}=\mathrm{Cr},\mathrm{Fe},\mathrm{P}}{n}_{\mathrm{i}}{G}_{\mathrm{i}}^{°}}-T\mathsf{\Delta }{S}_{\mathrm{CrFeP}}^{\mathrm{conf}.}+{\displaystyle \sum _{\mathrm{j},\mathrm{k}=\mathrm{Cr},\mathrm{Fe},\mathrm{P}}^{\mathrm{j}\ne \mathrm{k}}\left(n_{\mathrm{jk}}/2\right)\mathsf{\Delta }{g}_{\mathrm{jk}}}$$

(19)

$$\mathsf{\Delta }{S}_{\mathrm{CrFeP}}^{\mathrm{conf}.}=-R{\displaystyle \sum _{\mathrm{i}=\mathrm{Cr},\mathrm{Fe},\mathrm{P}}{n}_{\mathrm{i}}\ln X_{\mathrm{i}}}-R\left[{\displaystyle \sum _{\mathrm{j}=\mathrm{Cr},\mathrm{Fe},\mathrm{P}}{n}_{\mathrm{jj}}}\left({\displaystyle \frac{X_{\mathrm{jj}}}{{Y_{\mathrm{j}}}^2}}\right)+{\displaystyle \sum _{\mathrm{k},\mathrm{m}=\mathrm{Cr},\mathrm{Fe},\mathrm{P}}^{\mathrm{k}\ne \mathrm{m}}{n}_{\mathrm{km}}\ln \left(\frac{X_{\mathrm{km}}}{2Y_{\mathrm{k}}{Y}_{\mathrm{m}}}\right)}\right]$$

(20)

where ${\mathsf{\Delta }g}_{\mathrm{j}\mathrm{k}}$ ($\mathrm{j},\mathrm{k}=\mathrm{C}\mathrm{r},\mathrm{F}\mathrm{e},\mathrm{P}$) is the pair formation Gibbs energy depending on the thermodynamic symmetry of the ternary system. ${\mathsf{\Delta }g}_{\mathrm{C}\mathrm{r}\mathrm{P}}$ and ${\mathsf{\Delta }g}_{\mathrm{F}\mathrm{e}\mathrm{P}}$ for the asymmetric Cr–P and Fe–P systems were calculated using Equations (21) and (22), respectively:

$$\mathsf{\Delta }{g}_{\mathrm{CrP}}=\mathsf{\Delta }{g}_{\mathrm{CrP}}^{°}+{\displaystyle \sum _{(\mathrm{i}+\mathrm{j})\ge 1}{g}_{\mathrm{CrP}}^{\mathrm{ij}}}{x}_{\mathrm{PP}}^{\mathrm{i}}{\left(x_{\mathrm{CrCr}}+x_{\mathrm{CrFe}}+x_{\mathrm{FeFe}}\right)}^{\mathrm{j}}+{\displaystyle \sum _{\mathrm{i}\ge 0,\mathrm{j}\ge 0,\mathrm{k}\ge 1}{g}_{\mathrm{CrP}(\mathrm{Fe})}^{\mathrm{ijk}}}{x}_{\mathrm{PP}}^{\mathrm{i}}{\left(x_{\mathrm{CrCr}}+x_{\mathrm{CrFe}}+x_{\mathrm{FeFe}}\right)}^{\mathrm{j}}{\left({\displaystyle \frac{Y_{\mathrm{Fe}}}{Y_{\mathrm{Cr}}+Y_{\mathrm{Fe}}}}\right)}^{\mathrm{k}}$$

(21)

$$\mathsf{\Delta }{g}_{\mathrm{FeP}}=\mathsf{\Delta }{g}_{\mathrm{FeP}}^{°}+{\displaystyle \sum _{(\mathrm{i}+\mathrm{j})\ge 1}{g}_{\mathrm{FeP}}^{\mathrm{ij}}}{x}_{\mathrm{PP}}^{\mathrm{i}}{\left(x_{\mathrm{CrCr}}+x_{\mathrm{CrFe}}+x_{\mathrm{FeFe}}\right)}^{\mathrm{j}}+{\displaystyle \sum _{\mathrm{i}\ge 0,\mathrm{j}\ge 0,\mathrm{k}\ge 1}{g}_{\mathrm{FeP}(\mathrm{Cr})}^{\mathrm{ijk}}}{x}_{\mathrm{PP}}^{\mathrm{i}}{\left(x_{\mathrm{CrCr}}+x_{\mathrm{CrFe}}+x_{\mathrm{FeFe}}\right)}^{\mathrm{j}}{\left({\displaystyle \frac{Y_{\mathrm{Cr}}}{Y_{\mathrm{Cr}}+Y_{\mathrm{Fe}}}}\right)}^{\mathrm{k}}$$

(22)

while ${\mathsf{\Delta }g}_{\mathrm{C}\mathrm{r}\mathrm{F}\mathrm{e}}$ for the symmetric Cr–Fe system was calculated by Equation (23):

$$\begin{array}{ll}\mathsf{\Delta }{g}_{\mathrm{CrFe}}=& \mathsf{\Delta }{g}_{\mathrm{CrFe}}^{°}+{\displaystyle \sum _{(\mathrm{i}+\mathrm{j})\ge 1}{g}_{\mathrm{CrFe}}^{\mathrm{ij}}}{\left(\frac{x_{\mathrm{CrCr}}}{x_{\mathrm{CrCr}}+x_{\mathrm{CrFe}}+x_{\mathrm{FeFe}}}\right)}^{\mathrm{i}}{\left(\frac{x_{\mathrm{FeFe}}}{x_{\mathrm{CrCr}}+x_{\mathrm{CrFe}}+x_{\mathrm{FeFe}}}\right)}^{\mathrm{j}}\\ & +{\displaystyle \sum _{\mathrm{i}\ge 0,\mathrm{j}\ge 0,\mathrm{k}\ge 1}{g}_{\mathrm{CrFe}(\mathrm{P})}^{\mathrm{ijk}}}{\left({\displaystyle \frac{x_{\mathrm{CrCr}}}{x_{\mathrm{CrCr}}+x_{\mathrm{CrFe}}+x_{\mathrm{FeFe}}}}\right)}^{\mathrm{i}}{\left({\displaystyle \frac{x_{\mathrm{FeFe}}}{x_{\mathrm{CrCr}}+x_{\mathrm{CrFe}}+x_{\mathrm{FeFe}}}}\right)}^{\mathrm{j}}{Y}_{\mathrm{P}}^{\mathrm{k}}\end{array}$$

(23)

where ${\mathrm{g}}_{\mathrm{C}\mathrm{r}\mathrm{P}}^{\mathrm{i}\mathrm{j}}$, ${\mathrm{g}}_{\mathrm{F}\mathrm{e}\mathrm{P}}^{\mathrm{i}\mathrm{j}}$, and ${\mathrm{g}}_{\mathrm{C}\mathrm{r}\mathrm{F}\mathrm{e}}^{\mathrm{i}\mathrm{j}}$ are binary model parameters (J/mol) and ${\mathrm{g}}_{\mathrm{C}\mathrm{r}\mathrm{P}\left(\mathrm{F}\mathrm{e}\right)}^{\mathrm{i}\mathrm{j}\mathrm{k}}$, ${\mathrm{g}}_{\mathrm{F}\mathrm{e}\mathrm{P}\left(\mathrm{C}\mathrm{r}\right)}^{\mathrm{i}\mathrm{j}\mathrm{k}}$, and ${\mathrm{g}}_{\mathrm{C}\mathrm{r}\mathrm{F}\mathrm{e}\left(\mathrm{P}\right)}^{\mathrm{i}\mathrm{j}\mathrm{k}}$ are ternary model parameters (J/mol) of the liquid solution.

3. Critical Evaluation and Thermodynamic Optimization

Thermodynamic optimization of the binary Cr–P and ternary Cr–Fe–P systems was conducted based on critical evaluation of all the experimental thermodynamic property and phase diagram data. The optimized model parameters of the whole Cr–Fe–P system are summarized in Table 1. The details of the present thermodynamic modeling are given in the following sections.

3.1. The Cr–P System

The experimental information of the partial phase diagram and various intermediate Cr phosphides, including Cr₃P(s) [30,31,32,33,34,35,36,37], Cr₂P(s) [32,33,38,39,40,41], Cr₁₂P₇(s)(Cr_1.7P(s)) [38,39,40,42,43,44,45], CrP(s) [31,33,34,46,47,48,49,50,51,52,53,54,55,56], Cr₂P₃(s) [49], CrP₂(s) [33,34,57], and CrP₄(s) [58,59], of the Cr–P system were reviewed by Venkatraman and Neumann [60]. Cr₃P(s), CrP(s), and CrP₂(s) are commonly confirmed as stable compounds of the Cr–P system. However, the existence of Cr₂P₃(s) was refuted by subsequent study [33] and the evidence for the presence of CrP₄(s) is insufficient. According to the literature [38,40], the stoichiometry of the compound between Cr₃P(s) and CrP(s) is ambiguous. So far, two primary phosphides, Cr₁₂P₇(s) and Cr₂P(s), have been proposed as intermediate compounds by various investigators [32,33,38,39,40,41,42,43,44,45]. Nevertheless, their thermodynamic stability is controversial in the literature. After careful examination, it was found that isomorphous Fe₂P(s) and Cr₂P(s) exhibit mutual solubility in each other to form a homogeneous ternary (Cr,Fe)₂P solid solution [31,32,61], and Cr₁₂P₇(s) has been reported to occur in a small homogeneity range [38]. Hence, Cr₁₂P₇(s) is more likely to be a metastable phase of the Cr–P system. It is well known that solid Cr and P show negligible solubility in each other. In the present study, stoichiometric Cr₃P(s), Cr₂P(s), CrP(s), and CrP₂(s) compounds, liquid and gas phases, are thus considered as stable phases of the Cr–P system.

3.1.1. The Cr–P Phase Diagram

The calculated Cr–P phase diagram calculated from the previous assessment and the present study are compared with the experimental data [32,62] in Figure 1. The early experimental data for the liquidus and solidus of the Cr–P system were reported by Vogel and Kasten [32] based on the thermal analysis and metallographic analysis. They suggested a eutectic reaction $\mathrm{l}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}(x_{\mathrm{P}}=0.134)=\mathrm{C}\mathrm{r}+{\mathrm{C}\mathrm{r}}_3\mathrm{P}\left(\mathrm{s}\right)$ at around 1376 °C. The other set of data by Zaitsev et al. [62] were integrated from the thermodynamic properties of Cr–P melts and Cr phosphides based on the Knudsen effusion experiments. Compared to the former experimental data, the data of Zaitsev et al. [62] are shifted towards the P-richer region and were less favored by both thermodynamic modelings. In the modeling of Miettinen [20], only a partial phase diagram of $x_{\mathrm{P}}=0~0.36$ was assessed with consideration of the liquid solution and two intermediate compounds (Cr₃P(s), Cr₂P(s)). The extrapolated diagram for $x_{\mathrm{P}}=0.36~1$ from Miettinen’s parameters shows a “concave shape” of liquidus lines, which are even below the “freezing temperature” in the range of $x_{\mathrm{P}}=0.69~0.86$, as shown in the figure. In the present study, the P-rich phase diagram was modified by introducing another two compounds, CrP(s) and CrP₂(s), which have been reported by many researchers [31,33,34,46,47,48,49,50,51,52,53,54,55,56,57]. The optimized invariant reactions of the Cr–P system from the present study are compared with those from the previous assessment and experimental data in

Table 2. Invariant reactions involved in the Cr–P system.

Type	Reactions	Temperature, °C	References
Eutectic	Liquid(xp = 0.134) → Cr(xp = 0) + Cr3P(xp = 0.25)	1371	[32]
	Liquid(xp = 0.134) → Cr(xp = 9.6 × 10−5) + Cr3P(xp = 0.25)	1376	[20]
	Liquid(xp = 0.134) → Cr(xp = 2.7 × 10−5) + Cr3P(xp = 0.25)	1376	Present study
Incongruent	Liquid(xp = 0.220) + Cr2P(xp = 0.333) → Cr3P(xp = 0.25)	1511	[20]
	Liquid(xp = 0.208) + Cr2P(xp = 0.333) → Cr3P(xp = 0.25)	1501	Present study
Congruent	Liquid(xp = 0.333) → Cr2P(xp = 0.333)	1642	[20]
	Liquid(xp = 0.333) → Cr2P(xp = 0.333)	1671	Present study
Eutectic	Liquid(xp = 0.428) → Cr2P(xp = 0.333) + CrP(xp = 0.5)	1469	Present study
Congruent	Liquid(xp = 0.5) → CrP(xp = 0.5)	1518	Present study
Incongruent	Liquid(xp = 0.926) + CrP(xp = 0.5) → CrP2(xp = 0.667)	1011	Present study
Eutectic	Liquid(xp = 0.9995) → CrP2(xp = 0.667) + P(xp = 1.0)	579	Present study

.

3.1.2. Thermodynamic Stability of Cr Phosphides

The thermodynamic stability of Cr phosphides has been widely investigated by many researchers [33,44,45,63,64,65,66,67,68]. Thermodynamic property data including the heat capacity (C_p) [45], standard enthalpy of formation (${∆H}_{298.15 \mathrm{K}}^{°}$) [44,45,63,64,65], standard entropy ($S_{298.15 \mathrm{K}}^{°}$) [44,45,63,64,65,66], formation Gibbs energy (${∆G}_f^{°}$) [44,45,67,68], and partial pressure of P₂(g) or P₄(g) over various Cr phosphides [33,44,45,68] were collected for critical optimization of their Gibb energies.

The C_p data of Cr₃P(s) and Cr₁₂P₇(s) were measured by Zaitsev et al. [45] using the differential scanning calorimetry method. As was discussed above, Cr₁₂P₇(s) should be a metastable intermediate phase of forming Cr₂P(s). Thus, the reported C_p values of Cr₁₂P₇(s) were converted to those of Cr₂P(s) in the present thermodynamic modeling. Figure 2 shows the calculated C_p curves of Cr₃P(s) and Cr₂P(s) from the previous assessment [20] and the present study compared with the experimental data [45]. However, the assessed C_p values by Miettinen [20] deviate somewhat from the experimental data. These discrepancies were resolved based on the present optimization, as shown in Figure 2a,b. The $C_P$ of CrP(s) and CrP₂(s), with no available experimental data in the literature, were determined based on those of Cr₃P(s) and Cr₂P(s): C_p(CrP) = 0.5{C_p(Cr₃P) + C_p(Cr₂P) − 3C_p(Cr, BCC_A2)}, C_p(CrP₂) = C_p(Cr₃P) + C_p(Cr₂P) − 4C_p(Cr, BCC_A2). The determined C_p(CrP) and C_p(CrP₂) functions depending on the temperature are given in Table 1.

The equilibrium P₂(g) partial pressure over various Cr–P alloys was measured by Myers et al. [44] using the mass–loss effusion method at 1194 K to 1768 K and Zaitsev et al. [45] using the Knudsen effusion method at 1341 K to 1704 K. The obtained high-temperature vapor pressure data were used to determine the ${∆H}_{298.15 \mathrm{K}}^{°}$ and $S_{298.15 \mathrm{K}}^{°}$ values of involved Cr phosphides based on formulated C_p values. Glaum and Gruehn [63] determined the ${∆H}_{298.15 \mathrm{K}}^{°}$ of CrP(s) to be −124.15 ± 8.4 kJ/mol using the transportation method. The ${∆H}_{298.15 \mathrm{K}}^{°}$ and $S_{298.15 \mathrm{K}}^{°}$ for different Cr phosphides were also assessed/estimated by Schlesinger [64], Miettinen [20], Pogorelyi [65], and Kubaschewski and Alcock [66]. The experimental data and thermodynamic assessment/evaluation of ${∆H}_{298.15 \mathrm{K}}^{°}$ and $S_{298.15 \mathrm{K}}^{°}$ for Cr₃P(s), Cr₂P(s), CrP(s), and CrP₂(s) are listed in

Table 3. Standard enthalpy of formation (from BCC_A2 Cr and white P) and standard entropy (J/(mol·K)) of Cr phosphides.

Species	${∆\mathit{H}}_{298.15 \mathbf{K}}^{°}$, kJ/mol	${\mathit{S}}_{298.15 \mathbf{K}}^{°}$, J/(mol·K)	Techniques	References
Cr3P	−184.26 ± 6.4	93.3 94.3	DSC KEM	[45]
	−123.23 ± 1.3	87.9	MLE	[44]
	−180.51	93.1	Estimation	[64]
	−177.58	108.4	Assessment	[20]
	−184.88	93.3	Assessment	Present study
Cr2P	−168.75 ± 5.4	74.0 74.4	DSC KEM	[45]
	−159.28	83.2	Assessment	[20]
	−169.89	74.0	Assessment	Present study
CrP	−112.07 ± 3.0	39.7	MLE	[44]
	−124.15 ± 8.4		TM	[63]
	−138.06	46.9	Estimation	[64]
	−117.04	46.4	Estimation	[65]
		46.0	Estimation	[66]
	−122.70	46.4	Assessment	Present study
CrP2	−232.94		Estimation	[65]
	−155.90	63.0	Assessment	Present study

.

It is noticeable that the ${∆H}_{298.15 \mathrm{K}}^{°}\left({\mathrm{C}\mathrm{r}}_3\mathrm{P}\right)=-123.23$ kJ/mol by Myers et al. [44] is much less negative than the −184.26 ± 6.4 kJ/mol by Zaitsev et al., −180.51 kJ/mol by Schlesinger [64], and −177.58 kJ/mol by Miettinen [20]. Such big differences could be caused by non-negligible errors of high-temperature vapor pressure measurements or C_p estimation. In the present study, the ${∆H}_{298.15 \mathrm{K}}^{°}$ for Cr₃P(s) was optimized to be −184.88 kJ/mol by favoring the experimental data of Zaitsev et al. [45]. The ${∆H}_{298.15 \mathrm{K}}^{°}$ for Cr₂P(s) from thermodynamic assessment [20] is about 9.5 kJ/mol less negative than the only experimental value [45], which was determined to be −169.89 kJ/mol with slight modification in the present study. The reported values of ${∆H}_{298.15 \mathrm{K}}^{°}$ for CrP(s) show some fluctuation. In the present thermodynamic modeling, more weight was given to the experimental data of Glaum and Gruehn [63] (−124.15 ± 8.4 kJ/mol), as shown in Table 3. No experimental data are available in the literature for ${∆H}_{298.15 \mathrm{K}}^{°}$ and $S_{298.15 \mathrm{K}}^{°}$ of CrP₂(s). The only estimated ${∆H}_{298.15 \mathrm{K}}^{°}$ value for CrP₂(s), −232.94 kJ/mol, was reported by Pogorelyi [65]. This value is apparently too negative and shows much inconsistency with the thermodynamic property data of other Cr phosphides. In the present study, the ${∆H}_{298.15 \mathrm{K}}^{°}$ for CrP₂(s) was optimized to be −155.90 kJ/mol to reproduce the equilibrium pressure data of P₄(g) over CrP and CrP₂(s) [33].

Based on the low-temperature C_p data and high-temperature vapor pressure data of Zaitsev et al. [45], $S_{298.15 \mathrm{K}}^{°}$ was determined to be 93.3 J/(mol·K) and 94.3 J/(mol·K) for Cr₃P(s) and 74.0 J/(mol·K) and 74.4 J/(mol·K) for Cr₂P(s), respectively. As is well known, there is always potential error in the derivation of $S_{298.15 \mathrm{K}}^{°}$ from high-temperature Gibbs energy data, while $S_{298.15 \mathrm{K}}^{°}$ determined from low-temperature C_p data is more reliable. Therefore, the $S_{298.15 \mathrm{K}}^{°}$ values measured by Myers et al. (87.9 J/(mol·K) for Cr₃P) [44] and assessed by Miettinen (108.4 J/(mol·K) for Cr₃P and 83.2 J/(mol·K) for Cr₂P) [20] are less reliable than those of Schlesinger (93.1 J/(mol·K) for Cr₃P) [64] and Zaitsev et al. (93.3 J/(mol·K) for Cr₃P and 74.0 J/(mol·K)) [45], which are adopted by the present study, as listed in Table 3. The $S_{298.15 \mathrm{K}}^{°}$ values of CrP(s) suggested by Myers et al. (39.7 J/(mol·K)) [44] is inconsistent with those estimation values by Schlesinger (46.9 J/mol) [64], Pogorelyi (46.4 J/(mol·K)) [65], and Kubaschewski and Alcock (46.0 J/(mol·K)) [66]. According to the present modeling, it was found that $S_{298.15 \mathrm{K}}^{°}=46.4$ J/(mol·K) for CrP(s) could result in an overall accurate reproduction of the available phase diagram and thermodynamic property data of the Cr–P system. The $S_{298.15 \mathrm{K}}^{°}$ for CrP₂(s) was determined to be 63.0 J/(mol·K) to reproduce the vapor pressure data [33].

Based on the optimized C_p, ${∆H}_{298.15 \mathrm{K}}^{°}$, and $S_{298.15 \mathrm{K}}^{°}$ of Cr₃P(s), Cr₂P(s), CrP(s), and CrP₂(s), the Gibbs energies of these compounds were determined from Equation (2).

Figure 3 shows the calculated formation Gibbs energies of Cr₃P(s) and Cr₂P(s) from the previous assessment [20] and the present study compared to the experimental data [44,45,67,68]. The experimental results of Zaitsev et al. [45] and Nagai et al. [67] are in reasonable agreement with each other but show much deviation from those of Myers et al. [44] and Pogorelyi [68]. The former two sets of experimental data are favored by the present modeling, which shows certain improvements compared to the previous assessment. The calculated equilibrium partial pressure of P₂(g) and P₄(g) over Cr₃P(s) and Cr(s), Cr₃P(s) and Cr₂P(s), Cr₂P(s) and CrP(s), and CrP(s) and CrP₂(s) are presented in Figure 4 along with the experimental data [33,44,45,68]. Likewise, the experimental results of Myers et al. [44] and Pogorelyi [68] that are in poor consistency are not taken into account, while those of Zaitsev et al. [45] and Faller and Biltz [33] are well reproduced in this work, as shown in Figure 4. It is therefore expected that the optimized Gibbs energies of intermediate Cr phosphides should be accurate.

3.1.3. Thermodynamic Properties of the Cr–P Liquid Solution

The thermodynamic properties of the Cr–P liquid solution are of primary importance for process optimization of the Cr–P alloys. The activities of Cr(l) and P(l) in high-P Cr–P melts (x_p = 0.08~0.368) at 1527 °C were also measured by Zaitsev et al. [62] using the mass spectrometry and Knudsen effusion techniques. The activity data were converted to corresponding partial chemical potentials and are presented in Figure 5 along with the previous and present modeling results. It is shown that the experimental data can be reproduced by both calculations within acceptable errors. So far, there is no available experimental information of the dilute Cr–P liquid solution due to the high melting point of Cr. According to the present optimization, the Henrian activity coefficient of P in Cr(l), ${\gamma }_{\mathrm{P} \mathrm{i}\mathrm{n} \mathrm{C}\mathrm{r}\left(\mathrm{l}\right)}^{°}$, which is to quantify the deviation of P behavior from the ideal behavior particularly for the dilute region, was determined as a function of temperature as follows:

$$\ln {\gamma }_{\mathrm{P} \mathrm{in} \mathrm{Cr}\left(\mathrm{l}\right)}^{\circ }=-\frac{\mathrm{20,742}}{T}-0.5285, 2180 \mathrm{K} < \mathrm{T} < 2573 \mathrm{K}$$

(24)

where T is the temperature in Kelvin (K). Based on the optimized ${\gamma }_{\mathrm{P} \mathrm{i}\mathrm{n} \mathrm{C}\mathrm{r}\left(\mathrm{l}\right)}^{°}$, the molar Gibbs energies for the dissolution of P(g) and P₂(g) and into liquid Cr(1 wt.% standard state) were also determined, as expressed by Equations (25) and (26):

$$\mathrm{P}\left(\mathrm{g}\right)={[\mathrm{P}]}_{\mathrm{in} \mathrm{Cr}\left(\mathrm{l}\right)}(1 \mathrm{wt}.\%), \mathsf{\Delta }{G}_T^{\circ }=-\mathrm{477,073}+70.4655\left(\mathrm{J}/\mathrm{mol}\right), 2180 \mathrm{K} < T < 2573 \mathrm{K}$$

(25)

$$0.5{\mathrm{P}}_2\left(\mathrm{g}\right)={[\mathrm{P}]}_{\mathrm{in} \mathrm{Cr}\left(\mathrm{l}\right)}(1 \mathrm{wt}.\%), \mathsf{\Delta }{G}_T^{\circ }=-\mathrm{227,128}+10.2395\left(\mathrm{J}/\mathrm{mol}\right), 2180 \mathrm{K} < T < 2573 \mathrm{K}$$

(26)

3.2. The Cr–Fe and Fe–P Systems

Thermodynamic reoptimization of the Cr–Fe and Fe–P systems was performed recently by the present authors [18,19]. The optimized model parameters of these two systems were adopted by this study. Figure 6 shows the phase diagrams of the Fe–Cr and Fe–P systems. As shown in the figure, the gas, liquid solution, sigma, FCC_A1, BCC_A2, and BCC_A2#2 were taken as stable phases of the Cr–Fe system and the gas, liquid solution, FCC_A1, BCC_A2, Fe₃P(s), Fe₂P(s), FeP(s), and FeP₂(s) as stable phases of the Fe–P system.

3.3. The Cr–Fe–P Systems

According to the literature [69,70,71,72,73], the mutual dissolution of isomorphous Cr₃P(s) and Fe₃P(s), Cr₂P(s) and Fe₂P(s), CrP(s) and FeP(s) to form Me₃P, Me₂P, and MeP in the formulas of (Cr,Fe)₃P, (Cr,Fe)₂P, and (Cr,Fe)P, respectively, occur in the ternary Cr–Fe–P system. Based on thermodynamic descriptions of the binary Cr–P, Cr–Fe, and Fe–P systems, the gas phase; red P; liquid solution; solid solutions including BCC_A2, FCC_A1, Sigma, Me₃P, Me₂P, and MeP; and stoichiometric compounds including CrP₂(s) and FeP₂(s) were considered stable in the ternary Cr–Fe–P system.

3.3.1. The Cr–Fe–P Phase Diagram

The phase equilibria of a variety of vertical sections of the Cr–Fe–P system were measured by Vogel and Kasten [32] by means of thermal analysis and microscopic analysis. Figure 7 shows the calculated vertical diagrams for the mass ratios of m(Fe):m(Cr) = 1:9, 2:8, 4:6, 5:5, 6:4, 7:3, 8:2, and 9:1 from previous assessment of the present optimization, compared to the experimental data. It can be seen that all experimental data are from the region of wt.%P < 28 and are basically reproduced by both calculations. It is noted that the melting points of Me₂P optimized by the present study are approximately 50 °C higher than those of the previous assessment, aiming at making more raised “Liquid + Me₂P” liquidus boundaries for a better match with the experimental data of the higher-P region, as shown in the figure. Nevertheless, these high-P liquidus data could not be perfectly reproduced. This is probably due to non-neglectable vaporization of P from the melts at such a high P concentration region. As a consequence of the present optimization, the eutectic reaction Liquid = Me₂P + MeP occurs in the lower-P composition, and the congruent melting points of MeP of the present study are 11~84 °C higher than those of the previous assessment, depending on the ratio of m(Fe):m(Cr).

Figure 8 shows the calculated solubility of P in FCC_A1 and BCC_A2 Fe–Cr alloys between 700 °C and 1140 °C depending on the Cr content, compared to the experimental data [32,72]. According to the experiments of Kaneko et al. [72], the solubility of P in BCC_A2 Fe–Cr solution decreases continuously with increasing Cr concentration and decreasing temperature. These data are well reproduced by adding one binary model parameter ($L_{\mathrm{C}\mathrm{r},\mathrm{P}:\mathrm{V}\mathrm{a}}^{\mathrm{B}\mathrm{C}\mathrm{C}\_\mathrm{A}2}=-\mathrm{48,116}$ J/mol) for BCC_A1 of the Cr–P system. However, the calculation results of Miettinen and Vassilev for 900 °C and 1000 °C are not matched with the experimental data, as shown in Figure 8a. This discrepancy is due to the overestimated stability of the FCC_A1 phase, as shown in Figure 8b. After careful examination, it was found that Miettinen [20,21] made a mistake by excluding the FCC_A1 phase in the calculation of P solubility in BCC_A2 solution. That is why the FCC_A1 was not appearing in their phase diagram. To resolve the inconsistency, the interaction model parameter ($L_{\mathrm{C}\mathrm{r},\mathrm{P}:\mathrm{V}\mathrm{a}}^{\mathrm{F}\mathrm{C}\mathrm{C}\_\mathrm{A}1}=L_{\mathrm{C}\mathrm{r},\mathrm{P}:\mathrm{V}\mathrm{a}}^{\mathrm{F}\mathrm{C}\mathrm{C}\_\mathrm{A}1}=-\mathrm{70,000}$ J/mol) assessed by Miettinen [20,21] was optimized to be a less negative value ($L_{\mathrm{C}\mathrm{r},\mathrm{P}:\mathrm{V}\mathrm{a}}^{\mathrm{F}\mathrm{C}\mathrm{C}\_\mathrm{A}1}=L_{\mathrm{C}\mathrm{r},\mathrm{P}:\mathrm{V}\mathrm{a}}^{\mathrm{B}\mathrm{C}\mathrm{C}\_\mathrm{A}1}=-\mathrm{48,116}$ J/mol) in this study, as listed in Table 1.

The homogeneity range of the Me₃P and Me₂P solid solutions at 800 °C were investigated by Kaneko et al. [73] using XRD and chemical analysis. They found a complete mutual dissolution between Cr₂P and Fe₂P into Me₂P solid solution and dissolution of 10.5 wt.%Cr in Fe₃P to form Me₃P solution at this temperature. These experimental results and reported P solubility data in BCC_A2 solution are compared with the present calculation in Figure 9. The existing data are well accounted for by the calculation results. Based on the present model parameters, Me₃P, Me₂P, and MeP were calculated to be complete solid solutions at 800 °C, as shown in the figure.

3.3.2. Thermodynamic Properties of the Cr–Fe–P Melts

The activity coefficient of P in molten Fe–Cr–P alloys at 1600 °C was studied by Frohberg et al. [74] using the distribution method and Yamada and Kato [75] using the Knudsen effusion method. Figure 10 shows the calculated activity coefficient of P (1 wt.% standard state) against the P and Cr contents of the Fe–Cr–P melts at 1600 °C, compared to the experimental data. As can be seen from the figure, the calculated activities by Miettinen and Vassilev [21] deviate distinctly from the experimental data. It should be noted that Figure 10b was plotted in the form of “${\mathrm{l}\mathrm{n}\gamma }_{\mathrm{P}}^{\mathrm{C}\mathrm{r}}$ against $x_{\mathrm{C}\mathrm{r}}$” by Miettinen and Vassilev [21]. That is why the data of Yamada and Kato [75] could still be reproduced despite a big discrepancy between the actual activity coefficient $f_{\mathrm{P}}(\mathrm{w}\mathrm{t}.\%)$ and experimental results. In this study, these discrepancies left in the previous assessment were successfully resolved, as shown in Figure 10a,b.

Based on the present thermodynamic database, the activities of P(l), Fe(l), and Cr(l) in the wide composition and temperature ranges of liquid Cr–Fe–P solution were calculated and compared with the Knudsen effusion experiment data of Zaitsev et al. [76] in Figure 11. It is shown that most activity data for a_Cr(l), a_Fe(l), and a_P(l) could be fitted except those of a_Cr at high Cr contents (x_Cr = 0.50~0.798). After careful examination, it was found that these experimental data are contradicted by the Gibbs–Duhem equation. From a thermodynamic point of view, such discrepancies between the modeling results and the experimental results could not be resolved.

3.3.3. Improvement of Present Optimization Compared to Previous Assessments

The present optimization shows some improvements compared to previous assessments of the Cr–P system [20] and Cr–Fe–P system [21]. For the Cr–P system, the phase diagram of x_p > 0.36 was determined based on consistent descriptions of the liquid phase and all intermediate compounds (Cr₃P(s), Cr₂P(s), CrP(s), and CrP₂(s)), and thermodynamic properties including heat capacity, standard enthalpy of formation, standard entropy, and Gibbs energy of Cr₃P and Cr₂P were optimized for more accurate reproduction of the experimental data. In the assessment of the Cr–Fe–P system by Miettinen and Vassilev [21], the phase equilibria of BCC_A2 and FCC_A1 solid solutions and the thermodynamic properties of the liquid solution were poorly determined. In the present optimization, more accurate descriptions were obtained by using fewer model parameters.

4. Predicted Phase Diagram of the Cr–Fe–P System

According to the optimized model parameters from the present study, the isothermal phase diagrams of 1000 °C and 1200 °C and liquid surface projection between 1000 °C and 2000 °C of the Cr–Fe–P system are predicted in Figure 12 and Figure 13. As shown in Figure 12, Fe and Cr atoms can also substitute each other completely to form Me₂P and MeP solid solutions at 800 °C and 1200 °C. However, Me₃P starts to melt at wt.%Fe > 67 when the temperature rises up to 1200 °C. The natural logarithmic Henrian activity coefficient of P (${\mathrm{l}\mathrm{n}\gamma }_{\mathrm{P}}^{°}$), which is of great importance to understand the thermodynamic behavior of P during production of the ferrochromium alloy, in the entire composition of Cr–Fe liquid solution at 1600 °C to 2100 °C is calculated in Figure 14.

5. Conclusions

Thermodynamic optimization of the binary Cr–P and ternary Cr–Fe–P systems in the entire composition range was performed based on the critical evaluation of the phase equilibria and thermodynamic properties data. The modified quasichemical model and compound energy formalism were used to describe the liquid and solid solutions, respectively. The CrP(s) and CrP₂(s) compounds were taken into account for the first time in the present thermodynamic modeling. The thermodynamic properties of Cr₃P(s), Cr₂P(s), CrP(s), and CrP₂(s) were carefully determined to reproduce reliable experimental data. In addition, the solubility of P in BCC_A2 and FCC_A1 phases and activity coefficient of P in Cr–Fe–P liquid solution were well optimized. Based on the present study, the discrepancies left in previous thermodynamic assessments of the Cr–P system and Cr–Fe–P system have been resolved with fewer model parameters. Any phase diagrams and thermodynamic properties within the Cr–Fe–P system can be predicted from the present thermodynamic database.