# Thermophysical Properties of Fe-Si and Cu-Pb Melts and Their Effects on Solidification Related Processes

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Thermodynamics and Surface Properties of Metallic Melts Representing Phase Separation and Strong Compound Forming Tendency

#### 2.1.1. Phase Separating Liquid Alloys and Self Aggregating Model (SAM)

#### 2.1.2. Compound Forming Liquid Alloys and Compound Formation Model (CFM)

#### 2.1.3. Quasi Chemical Approximation (QCA) for Regular Solution

#### 2.1.4. Perfect Solution Model

#### 2.2. Transport Properties: Viscosity

#### 2.3. Density/Molar Volume

#### 2.4. Structural Information: ${S}_{cc}\left(0\right)$ and ${\alpha}_{1}$ Microscopic Functions

## 3. Results and Discussion

#### 3.1. Thermodynamics of the Two Limiting Cases of Mixing: Cu-Pb and Fe-Si Liquid Alloys

#### 3.1.1. Cu-Pb

#### 3.1.2. Fe-Si

**Figure 1.**Concentration dependence of thermodynamic properties of liquid Cu-Pb alloys calculated for T = 1373 K together with the corresponding experimental data. The excess Gibbs free energy of mixing (${G}_{M}^{xs}$ curve 1); the Gibbs free energy of mixing ($\raisebox{1ex}{${G}_{M}$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.$ [29], curve 2); the enthalpy of mixing ($\raisebox{1ex}{${H}_{M}$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.$ [48], curve 3) and the activities of copper (${a}_{Cu}$ [50], curve 4a) and lead (${a}_{Pb}$, [55], curve 4b); (- - the ideal mixture).

#### 3.2. Surface Properties of Phase Separating and Compound Forming Liquid Alloys

#### 3.2.1. Self Aggregating Model (SAM) and Surface Properties of Cu-Pb Melts

#### 3.2.2. Compound Forming Model (CFM) and Surface Properties of Fe-Si Melts

#### 3.3. Molar Volume/Density of Cu-Pb Phase Separating and Fe-Si Compound Forming Liquid Alloys

#### 3.4. Viscosity of Cu-Pb Phase Separating and Fe-Si Compound Forming Liquid Alloys

#### 3.5. Microscopic Functions of Cu-Pb Phase Separating and Fe-Si Compound Forming Liquid Alloys

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$A,B$ | components of an $A-B$ alloy |

${a}_{i}$ ($i=A,B$) | activity of component $i$ |

${c}_{i}$ ($i=A,B$) | composition of component $i$ |

${c}_{i}^{s}$ ($i=A,B$) | surface composition of component $i$ |

$g$ | energetic term of CFM |

${G}_{M}$ | Gibbs free energy of mixing |

${G}_{M}^{xs}$ | excess Gibbs free energy of mixing |

${H}_{mix}$ | enthalpy of mixing |

${k}_{B}$ | Boltzmann’s constant |

${M}_{i}$ ($i=A,B$) | atomic mass of component $i$ |

${n}_{i}$ ($i=1,2,3$) | number of specie $i$ according to CFM in an $A-B$ alloy |

$N$ | Avogadro’s number |

$p,q$ | surface coordination fractions |

$R$ | gas constant |

$S$ | surface area of an alloy |

${S}_{cc}\left(0\right)$ | concentration fluctuations in the long wavelength limit |

${S}_{cc}\left(0,id\right)$ | concentration fluctuations for the ideal mixing |

$T$ | absolute temperature |

${V}_{i}$ ($i=A,B$) | atomic volume of the component $i$ |

${V}^{E}$ | excess volume |

${V}_{Alloy}$ | volume of a liquid $A-B$ alloy |

$Z$ | coordination number |

$W$ | interaction energy term of SAM |

${W}_{i}$($i=1,2,3$) | energetic terms of CFM |

$\alpha $ | mean surface area of an $A-B$ alloy |

${\alpha}_{i}$($i=A,B$) | surface area of atomic species $i$ |

${\alpha}_{1}$ | short-range order parameter |

$\beta $ | auxiliary function for the bulk phase |

${\beta}^{s}$ | auxiliary function for the surface phase |

${\gamma}_{i}$ ($i=A,B$) | activity coefficient of component $i$ |

$\varphi $, ${\varphi}^{s}$ | composition functions of the bulk and surface phases |

$\eta $ | viscosity of $A-B$ liquid alloys |

${\eta}_{i}$ ($i=A,B$) | viscosity of component $i$ |

$\mu ,\nu $ | stoichiometric coefficients of an intermetallic |

${\rho}_{i}$ $\left(i=A,B\right)$ | density of component |

${\rho}_{Alloy}$ | density of a liquid $A-B$ alloy |

$\sigma $ | surface tension of liquid $A-B$ alloys |

${\sigma}_{A}$ | surface tension of pure component $A$ |

${\sigma}_{B}$ | surface tension of pure component $B$ |

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**Figure 2.**Concentration dependence of: the excess Gibbs free energy of mixing (${G}_{M}^{xs}$ curve 1); Gibbs free energy of mixing ($\raisebox{1ex}{${G}_{M}$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.$, curve 2); the equilibrium number of complexes ${n}_{3}$ ($FeSi$ ) together with unassociated atoms ${n}_{1}$ (Fe) and ${n}_{2}$ (Si) for liquid Fe-Si alloys calculated by the CFM for T = 1823 K.

**Figure 3.**Concentration dependence of thermodynamic properties of liquid Fe-Si alloys calculated for T = 1823 K together with the corresponding experimental data: the enthalpy of mixing ($\raisebox{1ex}{${H}_{M}$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.$ [53,54,55], curve 1); the activities of iron (${a}_{Fe}$ [57], curve 2a) and silicon (${a}_{Si}$ [38,56], curve 2b). (- - the ideal mixture).

**Figure 4.**Surface composition (${C}_{Pb}^{s}$) vs. bulk composition (${C}_{Pb}$) for liquid Cu-Pb alloys calculated by the SAM (curve 1) and the QCA for regular solution (curve 2) for T = 1823 K.

**Figure 6.**Surface composition (${C}_{Si}^{s}$) vs. bulk composition (${C}_{Si}$) for liquid Fe-Si alloys calculated by the CFM (curve 1) and the QCA for regular solution (curve 2) for T = 1823 K.

**Figure 7.**Surface tension isotherms of liquid Fe-Si alloys calculated by: the CFM (curve 1), the QCA for regular solution (curve 2), and the perfect solution model (curve 3) for T = 1823 K. For a comparison, the available experimental data [18,64,65,66,67,68,69] obtained at the same temperature are shown.

**Figure 8.**Concentration dependence of the molar volume (curve 1) and the ideal mixture (curve 2) of liquid Cu-Pb alloys calculated for T = 1373 K. For a comparison, the molar volume values obtained from the density data [73] measured at the same temperature are shown.

**Figure 10.**Viscosity isotherms of liquid Cu-Pb alloys calculated by Moelwyn–Hughes’s (MH) model for T = 1373 K using different viscosity reference data of pure liquid metals: 1—the viscosity isotherms with the recommended data of Cu [74] and Pb [85] (curve 1a and curve 1b); 2—the viscosity isotherms with experimental data [82] (curve 2a and curve 2b). For a comparison, the experimental data [82,83,84] are shown (h—heating, c—cooling; curves 1b and 2b—the ideal mixture).

**Figure 11.**Viscosity isotherms of liquid Fe-Si alloys calculated by Moelwyn–Hughes’s (MH) model (curve 1) for T = 1823 K. The viscosity reference data of liquid Fe [76,85] and Si [77,85] together with available experimental datasets of Fe-Si melts [18,88,90,91] are shown; the ideal mixture (curve 2).

**Figure 12.**Composition dependent concentration fluctuations in the long-wavelength limit ${S}_{cc}\left(0\right)$ (curve 1), ${S}_{cc}\left(0,id\right))$ for the ideal mixing (curve 3) and chemical short-range order parameter ${\alpha}_{1}$ (curve 2) of liquid Cu-Pb alloys calculated for T = 1373 K. For a comparison with ${\alpha}_{1}$, the experimental data [22] are shown.

**Figure 13.**Composition dependent concentration fluctuations in the long-wavelength limit ${S}_{cc}\left(0\right)$ (curve 1), ${S}_{cc}\left(0,id\right))$ for the ideal mixing (curve 3), and chemical short-range order parameter ${\alpha}_{1}$ (curve 2) of liquid Fe-Si alloys calculated for T = 1823 K.

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**MDPI and ACS Style**

Novakovic, R.; Giuranno, D.; Lee, J.; Mohr, M.; Delsante, S.; Borzone, G.; Miani, F.; Fecht, H.-J. Thermophysical Properties of Fe-Si and Cu-Pb Melts and Their Effects on Solidification Related Processes. *Metals* **2022**, *12*, 336.
https://doi.org/10.3390/met12020336

**AMA Style**

Novakovic R, Giuranno D, Lee J, Mohr M, Delsante S, Borzone G, Miani F, Fecht H-J. Thermophysical Properties of Fe-Si and Cu-Pb Melts and Their Effects on Solidification Related Processes. *Metals*. 2022; 12(2):336.
https://doi.org/10.3390/met12020336

**Chicago/Turabian Style**

Novakovic, Rada, Donatella Giuranno, Joonho Lee, Markus Mohr, Simona Delsante, Gabriella Borzone, Fabio Miani, and Hans-Jörg Fecht. 2022. "Thermophysical Properties of Fe-Si and Cu-Pb Melts and Their Effects on Solidification Related Processes" *Metals* 12, no. 2: 336.
https://doi.org/10.3390/met12020336