^{1,a)}, A. E. Gheribi

^{1}and P. Chartrand

^{1}

### Abstract

The design of multicomponent alloys used in different applications based on specific thermo-physical properties determined experimentally or predicted from theoretical calculations is of major importance in many engineering applications. A procedure based on Monte Carlo simulations(MCS) and the thermodynamic integration (TI) method to improve the quality of the predicted thermodynamic properties calculated from classical thermodynamic calculations is presented in this study. The Gibbs energy function of the liquid phase of the Cu–Zr system at 1800 K has been determined based on this approach. The internal structure of Cu–Zr melts and amorphous alloys at different temperatures, as well as other physical properties were also obtained from MCS in which the phase trajectory was modeled by the modified embedded atom model formalism. A rigorous comparison between available experimental data and simulated thermo-physical properties obtained from our MCS is presented in this work. The modified quasichemical model in the pair approximation was parameterized using the internal structure data obtained from our MCS and the precise Gibbs energy function calculated at 1800 K from the TI method. The predicted activity of copper in Cu–Zr melts at 1499 K obtained from our thermodynamic optimization was corroborated by experimental data found in the literature. The validity of the amplitude of the entropy of mixing obtained from the *in silico* procedure presented in this work was analyzed based on the thermodynamic description of hard sphere mixtures.

We would like to thank Professor Byeong-Joo Lee and Professor Arthur D. Pelton for the fruitful discussions that we had in the course of this work.

I. INTRODUCTION

II. METHODOLOGY

A. Interatomic potential

B. Monte Carlo simulations

C. Thermodynamic integration method

D. Thermodynamic model

III. RESULTS AND DISCUSSION

A. Atomic density and thermal expansion of Cu–Zr melts

B. Isothermal bulk modulus of Cu–Zr melts

C. Structural properties of Cu–Zr melts and amorphous alloys

D. Thermodynamic optimization of the liquid phase using the TI method

IV. CONCLUSION

### Key Topics

- Copper
- 72.0
- Optical microcavities
- 39.0
- Entropy
- 29.0
- Thermodynamic properties
- 29.0
- Gibbs free energy
- 28.0

## Figures

Partial RDF of Cu_{50}Zr_{50}(liq.) at 1800 K and 1500 K. The solid lines represent the MCS results of the present study; the circles (*T* = 1800 K) and the triangles (*T* = 1500 K) are the results obtained by Ref. 57.

Partial RDF of Cu_{50}Zr_{50}(liq.) at 1800 K and 1500 K. The solid lines represent the MCS results of the present study; the circles (*T* = 1800 K) and the triangles (*T* = 1500 K) are the results obtained by Ref. 57.

Atomic density as a function of temperature for different Cu_{1–x }Zr_{ x } melts. The solid lines represent the experimental results of Ref. 40 for Cu–Zr melts, the assessment of Ref. 90 for pure Cu(liq.), and the experimental results of Ref. 91 for pure Zr(liq.), respectively. The symbols represent: open circles = MCS of the present work, stars = experimental data of Ref. 92, filled diamonds = experimental data of Ref. 43. The dashed line represents the lowest eutectic temperature of the Cu–Zr system: Cu_{10}Zr_{7}(s) + CuZr(s) →Liquid.

Atomic density as a function of temperature for different Cu_{1–x }Zr_{ x } melts. The solid lines represent the experimental results of Ref. 40 for Cu–Zr melts, the assessment of Ref. 90 for pure Cu(liq.), and the experimental results of Ref. 91 for pure Zr(liq.), respectively. The symbols represent: open circles = MCS of the present work, stars = experimental data of Ref. 92, filled diamonds = experimental data of Ref. 43. The dashed line represents the lowest eutectic temperature of the Cu–Zr system: Cu_{10}Zr_{7}(s) + CuZr(s) →Liquid.

Atomic density as a function of the molar fraction of Zr for Cu–Zr: (a) liquid alloys. The solid line represents the exponential regression of the data of Ref. 40 (*T* = 1473 K). The open circles are the atomic densities obtained by MCS (*T* = 1500 K); (b) amorphous alloys (*T* = 300 K). The solid line represents the exponential regression of the data of Ref. 41. The symbols represent: open circles = MCS of the present study, stars = experimental data of Ref. 36, square = experimental data of Ref. 51.

Atomic density as a function of the molar fraction of Zr for Cu–Zr: (a) liquid alloys. The solid line represents the exponential regression of the data of Ref. 40 (*T* = 1473 K). The open circles are the atomic densities obtained by MCS (*T* = 1500 K); (b) amorphous alloys (*T* = 300 K). The solid line represents the exponential regression of the data of Ref. 41. The symbols represent: open circles = MCS of the present study, stars = experimental data of Ref. 36, square = experimental data of Ref. 51.

Evolution of the isothermal bulk modulus of Cu–Zr melts as a function of temperature. The solid line represents a linear regression of the MD simulation results of Ref. 43 for a Cu_{64.5}Zr_{35.5} liquid alloy. The open circles represent the MCS of the present work for the Cu_{64.5}Zr_{35.5} melt. The filled circles are the MCS results obtained for pure Cu(liq.); the half-filled circles are the MCS results for pure Zr(liq.).

Evolution of the isothermal bulk modulus of Cu–Zr melts as a function of temperature. The solid line represents a linear regression of the MD simulation results of Ref. 43 for a Cu_{64.5}Zr_{35.5} liquid alloy. The open circles represent the MCS of the present work for the Cu_{64.5}Zr_{35.5} melt. The filled circles are the MCS results obtained for pure Cu(liq.); the half-filled circles are the MCS results for pure Zr(liq.).

Total RDF of (a) Cu_{64}Zr_{36}, Cu_{57}Zr_{43}, Cu_{50}Zr_{50}, and Cu_{30}Zr_{70} liquid alloys. The filled circles represent the experimental data of Ref. 49 obtained at 1223 K, the hall-filled (1540 K) and open circles (1450 K) are the data of Ref. 45; The hall-filled (1400 K) and open triangles (1300 K) are the data of Ref. 50. Solid lines represent our NPT-MCS results. (b) Cu_{64.5}Zr_{35.5} amorphous alloy. The thick line represents the amorphous alloy obtained by NPT-MCS for an infinite quenching rate with *T* _{0} = 1200 K and *T* _{F} = 800 K followed by an equilibrium cooling to 300 K. The thin line represents the amorphous alloy obtained by equilibrating a bcc structure at room temperature using NPT-MCS. Experimental data of Ref. 44 (squares), Ref. 41 (triangles), and Ref. 39 (circles) are also presented in this figure.

Total RDF of (a) Cu_{64}Zr_{36}, Cu_{57}Zr_{43}, Cu_{50}Zr_{50}, and Cu_{30}Zr_{70} liquid alloys. The filled circles represent the experimental data of Ref. 49 obtained at 1223 K, the hall-filled (1540 K) and open circles (1450 K) are the data of Ref. 45; The hall-filled (1400 K) and open triangles (1300 K) are the data of Ref. 50. Solid lines represent our NPT-MCS results. (b) Cu_{64.5}Zr_{35.5} amorphous alloy. The thick line represents the amorphous alloy obtained by NPT-MCS for an infinite quenching rate with *T* _{0} = 1200 K and *T* _{F} = 800 K followed by an equilibrium cooling to 300 K. The thin line represents the amorphous alloy obtained by equilibrating a bcc structure at room temperature using NPT-MCS. Experimental data of Ref. 44 (squares), Ref. 41 (triangles), and Ref. 39 (circles) are also presented in this figure.

Comparison between partial RDF for NVT-MCS of the present work (solid lines) and NVT-AIMD (dashed lines) calculated by Ref. 56 for Cu_{28}Zr_{72}(a), Cu_{46}Zr_{54}(b), Cu_{64}Zr_{36}(c), and Cu_{80}Zr_{20}(d) melts at 1500 K.

Comparison between partial RDF for NVT-MCS of the present work (solid lines) and NVT-AIMD (dashed lines) calculated by Ref. 56 for Cu_{28}Zr_{72}(a), Cu_{46}Zr_{54}(b), Cu_{64}Zr_{36}(c), and Cu_{80}Zr_{20}(d) melts at 1500 K.

Simulation results for the integrand in the TI method for *X* _{Zr} equals to (a) 0, (b) 0.2, (c) 0.5, (d) 0.8, and (e) 1 at 1800 K. The solid lines represent the spline interpolation performed for each composition.

Simulation results for the integrand in the TI method for *X* _{Zr} equals to (a) 0, (b) 0.2, (c) 0.5, (d) 0.8, and (e) 1 at 1800 K. The solid lines represent the spline interpolation performed for each composition.

Comparison between the MCS (circles) and the calculated thermodynamic properties obtained using the MQMPA (solid lines) of Cu–Zr(liq.) at 1800 K (a) pair fractions, (b) enthalpy of mixing, and (c) entropy of mixing.

Comparison between the MCS (circles) and the calculated thermodynamic properties obtained using the MQMPA (solid lines) of Cu–Zr(liq.) at 1800 K (a) pair fractions, (b) enthalpy of mixing, and (c) entropy of mixing.

Comparison between predicted (solid line) activity of copper at 1499 K in Cu–Zr melts and experimental data of Ref. 8 (circles, *T* = 1499 K) and Ref. 13 (triangles, *T* = 1473 K).

Comparison between predicted (solid line) activity of copper at 1499 K in Cu–Zr melts and experimental data of Ref. 8 (circles, *T* = 1499 K) and Ref. 13 (triangles, *T* = 1473 K).

This figure presents the various entropy contributions (dashed lines) and the resulting total entropy of mixing (solid line) calculated from the hard sphere mixture theory. The circles represent the entropy of mixing obtained from the TI method. The effect of the electronic contribution using the Sommerfeld model is also clearly presented in this figure.

This figure presents the various entropy contributions (dashed lines) and the resulting total entropy of mixing (solid line) calculated from the hard sphere mixture theory. The circles represent the entropy of mixing obtained from the TI method. The effect of the electronic contribution using the Sommerfeld model is also clearly presented in this figure.

The solid line presents the fitted curve of as a function of *X* _{Zr} used in Eq. (20) to reproduce the entropy of mixing at 1800 K obtained from the TI method. The diamonds represent the experimental data of Ref. 100 obtained for Cu–Zr amorphous alloys, while the triangle symbol represents the experimental value for pure amorphous Zr determined by Ref. 101. The value of for pure Cu was estimated from the work of Ref. 100.

The solid line presents the fitted curve of as a function of *X* _{Zr} used in Eq. (20) to reproduce the entropy of mixing at 1800 K obtained from the TI method. The diamonds represent the experimental data of Ref. 100 obtained for Cu–Zr amorphous alloys, while the triangle symbol represents the experimental value for pure amorphous Zr determined by Ref. 101. The value of for pure Cu was estimated from the work of Ref. 100.

## Tables

Thermal expansion of Cu–Zr liquid and amorphous alloys with α_{ V } = *cte*.

Thermal expansion of Cu–Zr liquid and amorphous alloys with α_{ V } = *cte*.

Summary of the first-nearest-neighbor coordination numbers in Cu_{1–x }Zr_{ x } melts.

Summary of the first-nearest-neighbor coordination numbers in Cu_{1–x }Zr_{ x } melts.

Summary of the results obtained in NPT/NVT simulations and the thermodynamic functions calculated from the TI method at 1800 K.

Summary of the results obtained in NPT/NVT simulations and the thermodynamic functions calculated from the TI method at 1800 K.

Optimized thermodynamic parameters of the MQMPA model for the Cu–Zr liquid phase.

Optimized thermodynamic parameters of the MQMPA model for the Cu–Zr liquid phase.

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