I. Methodology supplementary material
0.025
0.015
2
2
<r >(Å )
0.020
0.010
0.005
0.000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
PAcc
FIG. 1. Mean square displacement
r 2 as a function of the acceptance ratio for a Cu64.5Zr35.5 liquid alloy at 1800K. The
circles are the NPT- MCS results obtained for 5x106 MC steps. The solid line is the cubic spline interpolation of the MCS results.
Probability density
0.12
0.10
A
0.08
0.06
0.04
0.02
0.00
0.0560
0.0565
0.0570
0.0575
0.0580
0.0585
0.0590
0.0595
0.0600
3
Probability density
Atomic density (atom/Å )
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-2160
B
-2150
-2140
-2130
-2120
-2110
Total internal energy (eV)
FIG. 2. Probability density functions for a Cu64.5Zr35.5 liquid alloy at 1800K of (A) the atomic density and (B) the total internal
energy. Circles represent MCS results (5x106 MC) steps and solid lines are Gaussian amplitude fit of these results.
1.4
LJ liquid
LJ gas
1.3
1.2
1.1
LJ gas + LJ liquid
*
T
1.0
0.9
0.8
0.7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0


FIG. 3: Phase diagram of the cut and shifted LJ interatomic model with
rc  4 . The solid line is the saturation density
calculated by the equation of state of Ref. 1 for a cut and shifted LJ interatomic potential with rc
 4 . The circles are the
equilibrium points used in this study to calculate the Gibbs energy of Cu-Zr melts at 1800K with
 =0.12eV and
  2.37Å .
To validate the acceptance ratio PAcc used in the Monte Carlo simulations presented in
our study, we have analyzed the impact of the acceptance ratio PAcc on the resulting mean square
displacement r 2 for a a Cu64.5Zr35.5 liquid alloy at 1800K. The results of these simulations are
presented in Fig. 1.
The validity and the precision of our Monte Carlo simulations were tested throughout our
work by analyzing atomic density (for NPT ensemble calculations) and energy distributions.
Example of Boltzmann like distributions fitted using Gaussian amplitude distribution functions
are presented in Fig. 2 for a Cu64.5Zr35.5 liquid alloy at 1800K.
The thermodynamic integration method has been implemented in our work in order to
evaluate the absolute Gibbs energy of Cu-Zr melts. An important condition to control when
performing this method is to avoid phase transitions when shifting from the reference interatomic
potential (in our case the Lennard-Jones fluid) to the interatomic potential (in our case the
modified embedded atom model formalism) that models the real thermo-physical behavior of the
system. The equilibrium state of each MCS used to evaluate the Gibbs energy of Cu-Zr melts is
presented in Fig. 3. In this figure, T* is the reduced temperature and  * is the reduced density
and are defined as T *  kT  and  *   N V  3 respectively.
II. Results and discussions supplementary material
105
100
BT (GPa)
95
90
85
80
75
0
200
400
600
800
1000
1200 1400
1600
1800
2000
T (K)
FIG. 4: Evolution of the isothermal bulk modulus of Zr-BCC as a function of temperature. The solid line represents the
assessment of Ref. 2. The open circles represent our MCS performed on a Zr-BCC supercell containing 432 atoms.
Since experimental data of BT for Cu-Zr melts are lacking in the literature, it was decided
to test the MC procedure on a solid BCC Zirconium supercell containing 432 atoms at high
temperature. The results of the MCS for pure Zr-BCC are presented in Fig. 4 as open circles. It is
important to mention that the value of BT at 0K presented in Fig. 4 was taken from the work of
Kim et al.3 The evolution of BT as a function of temperature proposed in the assessment of
Dinsdale2 is corroborating the results obtained by MCS. The relatively large error in the
evaluation of the isothermal bulk modulus by MCS shown in Fig. 4 is explained by the large
number of iterations needed to sample adequately supercell volumes in order to get good volume
fluctuations. The accurate evaluation of the bulk modulus was not a fundamental aspect of the
present work and longer MC simulations would be needed to improve the quality of the MC
results.
3600
C (m/s)
3400
3200
3000
2800
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
T (K)
FIG. 5: Calculated speed of sound using MCS in liquid Cu (solid circles) and liquid Zr (half-filled circles) as a function of
temperature. The solid line represents the linear regression of the experimental data of Ref. 4.
The longitudinal speed of sound C, which is related to the adiabatic bulk modulus Bs via
2
the well known equation Bs   C , with  the density of the melt, was measured in pure liquid
Cu by several authors.4-8 All the self-consistent physical and thermodynamic properties obtained
in the present study were used to convert BT into speed of sound values. The following equation
was used to convert BT into Bs:
Bs   BT  CP  V N  T  2  CP
(1)
The isobaric heat capacities CP used in Eq. (1) and calculated from our simulations are
35.0 J/mol/K for liquid Cu and 33.2 J/mol/K for liquid Zr. The resulting evolution of the speed of
sound as a function of temperature in pure liquid Cu and pure liquid Zr are presented in Fig. 5.
According to this figure, the speed of sound in Cu(liq.) and Zr(liq.) at their melting temperature
is 3240m/s and 3460m/s respectively. The experimental data for Cu(liq.) measured by Pronin
and Filippov4 are in good agreement with our results. No experimental data was found relatively
to the speed of sound in liquid Zr. Blairs9 proposed an empirical formulation of the speed of
sound at the melting point for many metallic elements and proposed a value of 3666 m/s for
Zr(liq.) which is 5% higher than the calculated MC value.
3.0
2.5
g(r)
2.0
1.5
1.0
0.5
0.0
2
3
4
5
6
7
8
r (Å)
FIG. 6: Total RDF of liquid copper. The solid line represents NPT-MCS results of the present study at 1600K. The circles are the
experimental data of Ref. 10 obtained at 1573K.
2.5
2.0
g(r)
1.5
1.0
0.5
0.0
2
3
4
5
6
7
8
r (Å)
FIG. 7: Total RDF of liquid zirconium. The solid line represents NPT-MCS results of the present study at 2200K. The circles are
the experimental data of Ref. 10 at 2173K and the stars are the experimental data of Ref. 11 at 2135K.
RDF data of pure Cu and Zr liquids have been presented in the important reference work of
Waseda.10 The total RDF calculated by MCS in this work for liquid Cu is in good agreement
with the data of Waseda10 (Fig. 6) while there are some discrepancies at medium-range-order for
liquid Zr as it can be observed in Fig. 7. The experimental results obtained from the recent study
of Schenk et al.11 presented in Fig.7 show a different internal structure for liquid Zr at 2135K
and corroborate our results. The differences obtained in the determination of RDF for pure liquid
Zr from diffraction measurements give a good idea of the precision of this type of measurements.
The conditions of the Monte Carlo / Molecular dynamics simulations to determine the impact of
the high densities used in the work of Jakse and Pasturel12 on the resulting thermal pressure are
presented in Table 1.
Table 1: Summary of the NPT/NVT MCS atomic densities and internal pressures of Cu-Zr melts at 1500K
Alloy
NPT-Atomic density
NVT-Density
Pth1* Pth2**
Atom/ Å3
Atom/ Å3
GPa
GPa
Cu28Zr72
0.0474
0.0523
9.9
--
Cu46Zr54
0.0527
0.0585
11.0
--
Cu64Zr36
0.0591
0.0661
12.2
11
Cu80Zr20
0.0651
0.0732
12.6
--
* Thermal pressure calculated using NVT-MD simulations
** Thermal pressure calculated from its physical definition: Pth2  
1500 K
0
1
 BT dT
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2
3
4
5
6
7
8
9
10
11
12
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