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Calculation of thermodynamic properties

Materials Transactions, Vol. 46, No. 3 (2005) pp. 643 to 650
#2005 The Mining and Materials Processing Institute of Japan

Calculation of Thermodynamic Properties and Phase Diagrams
for the CaO-CaF2 , BaO-CaO and BaO-CaF2 Systems
by Molecular Dynamics Simulation
Won-Gap Seo*1 , Donghong Zhou*2 and Fumitaka Tsukihashi
Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo,
Kashiwa 277-8561, Japan
The thermodynamic properties for the CaO-CaF2 , BaO-CaO and BaO-CaF2 systems were calculated by molecular dynamics (MD)
simulation using the simple Born-Mayer-Huggins type potential model. The interatomic potential parameters were determined by fitting the
thermodynamic properties of pure CaO, BaO and CaF2 . The calculated thermodynamic properties for CaO, BaO and CaF2 were in good
agreement with measured results, and the superionic conductivity on the solid-solid phase transition of CaF2 has also been successfully assessed
by MD simulation. The ÁH M , ÁSM and ÁGM for each binary system were calculated based on the thermodynamic parameters obtained by MD
simulation and thermodynamic solution model. The calculated enthalpy interaction parameters for the BaO-CaF2 system represented the
possibility of formation of the compounds such as BaOÁCaF2 in the BaO-CaF2 system. The calculated phase diagrams for the CaO-CaF2 and
BaO-CaO systems were in good agreement with experimentally measured and CALPHAD method results. The calculated eutectic points for the
CaO-CaF2 and BaO-CaO systems were about 20 mol% CaO at 1650 K and about 20 mol% CaO at 2050 K, respectively. The BaO-CaF2 system
has also been estimated the liquidus lines in the CaF2 -rich and BaO-rich region by MD simulation.
(Received May 31, 2004; Accepted December 1, 2004)
Keywords: molecular dynamics simulation, thermodynamics, phase diagram, calcium oxide, barium oxide, calcium fluoride


1.

Introduction

Molecular dynamics (MD) simulation is widely used as the
powerful tool for the calculation of structural, dynamical and
thermodynamic properties of the molten slags and fluxes at
high temperature. Recently, the thermodynamic properties
and phase diagrams for the multiphase molten slags and
fluxes are generally calculated using computer-based software packages such as FactSage1,2) and Thermo-Calc.3)
These programs calculate the themochemical equilibria and
phase diagrams in various systems by thermodynamic
modeling based on the thermodynamic databases. However,
the application of these calculation methods is limited
because the experimentally measured thermodynamic databases are required for the calculation of thermodynamic
properties of multiphase molten slags and fluxes. On the
other hand, MD simulation is to calculate the thermodynamic
properties based on the dynamic quantities of individual
particles in the solid and fluid simulation cells without any
basic database. Therefore, the thermodynamics properties of
various systems which are difficult to be measured by
experimental methods can be effectively estimated.
The CaO-based slag systems such as the CaO-CaF2 , CaOCaF2 -SiO2 and BaO-CaO-CaF2 systems are generally used in
steelmaking process. Especially, the CaO-based slag systems
containing barium oxide are attractive with the possibility of
application in hot metal pretreatment on their high basicity
and low melting temperature. However, in spite of the
importance of these slag systems, the thermodynamic
properties and phase diagrams of barium oxide systems have
*1Graduate

Student, The University of Tokyo.

*2Formerly Graduate Student, Department of Advanced Materials Science,

Graduate School of Frontier Sciences, The University of Tokyo. Now at
Mitsubishi Electric Corporation, Wakayama 640-8686, Japan

many obscure respects. Kemp et al.4) recently reported the
phase diagram for the BaO-CaO system calculated by
CALPHAD (CALculation of PHAse Diagram) method,
which shows the eutectic point of 14 mol% CaO at 2180 K.
The phase diagram for the BaO-CaF2 system measured by
Kojima et al.5) partially represents the phase equilibrium up
to about 15 mol% BaO in CaF2 -rich region. The availability
of phase diagrams for barium oxide ternary systems such as
BaO-CaO-CaF2 system are also limited.
Therefore, the purpose of present research is to determine
the optimum potential model for the calculation of thermodynamic properties of the CaO-CaF2 , BaO-CaO and BaOCaF2 systems and calculate the thermodynamic properties for
each binary system by MD simulation. Finally, the phase
diagrams for the CaO-CaF2 , BaO-CaO and BaO-CaF2
systems are estimated from the thermodynamic parameters
obtained by MD calculation.
2.

Molecular Dynamics Calculation

2.1 Interatomic potential
The interatomic potential models of MD simulation for the
oxide and halide systems have been proposed by Hirao et
al.,6) Belashchenko et al.7–9) and many other researchers.
These interatomic potential models show good agreement
with structural properties of solid, glass and liquid phases
measured by experiments. However, these models have a
limitation for the calculation of thermodynamic properties
such as fusion data of the CaO, BaO and CaF2 system.
In this study, the potential energy for MD simulation was
calculated by the summation of pairwise interactions between ions i and j that was the Busing approximation of BornMayer-Huggins form of eq. (1).


644

W.-G. Seo, D. Zhou and F. Tsukihashi

ij ðrÞ ¼



Zi Á Z j e2
i þ  j À rij
þ f0 ðbi þ bj Þ exp
rij
bi þ bj

ð1Þ

where rij is the interatomic distance between ions i and j, Zi is
the valence of the ion i, e is the electron charge, f0 is the
standard force of 6:9478 Â 10À11 N (units constant), i and bi
are the repulsive radius and softness parameter of the ion i,
respectively. The interatomic pairwise potential terms of
eq. (1) represent the Coulomb and short-range repulsion
interactions without the dispersion terms. In this study, for
the calculation of thermodynamic properties in the molten
binary CaO-CaF2 , BaO-CaO and BaO-CaF2 systems, the
interatomic potential parameters were calculated based on
the thermodynamic properties, especially fusion properties
such as melting temperature and enthalpy of fusion of CaO,
BaO and CaF2 . The interatomic potential parameters for CaO
were taken from Matsumiya et al.10) that was successfully
reproduced the thermodynamic properties of CaO as shown
in Fig. 1. The optimum interatomic potential parameters for
BaO and CaF2 were calculated by fitting the thermodynamic
properties of BaO and CaF2 with measured results by fixing
the interatomic potential parameters of Ca-Ca, Ca-O and O-O
pairs for CaO. The interatomic potential parameters used in
this study are listed in Table 1.
2.2 Methods for calculation
The MD simulations were carried out using the isobaric
and isothermal (N-p-T) ensemble. Temperature is controlled
by velocity scaling method. Pressure is controlled by
Parrinello and Rahmann method at atmospheric pressure.

300
Present work

Enthalpy, HT-H1000K, kJ/mol

250

(Heating from solid phase)

Present work
(Cooling from liquid phase)

200

Observed11)

150
100
50

CaO

0

3.
1000 1500 2000 2500 3000 3500 4000

Temperature, K
Fig. 1 Calculated and observed enthalpies of solid and liquid CaO as a
function of temperature.

Table 1

Parameters of interatomic potential used for simulation.
Zi

Ca
Ba

The atomic configurations of initial cells for solid phases
were taken from the respective unit cell structures. The CaO
and BaO crystal structures were composed of 1000 (Ca 500
and O 500) and 1000 (Ba 500 and O 500) atoms according to
an array of 5 Â 5 Â 5 unit cells of rocksalt structure. The
CaF2 crystal structure was composed of 1500 (Ca 500 and F
1000) atoms according to an array of 5 Â 5 Â 5 unit cells of
CaF2 structure. The atomic configurations of initial cells for
liquid phases were set to be random in the cubic cell. The
total number of atoms was taken from 1000 to 1500. The
densities of initial cells for CaO, BaO and CaF2 liquid phases
were adopted to be 3340 kg/m3 , 5720 kg/m3 and 3180 kg/
m3 , respectively based on the densities of solid CaO, BaO
and CaF2 at room temperature and the densities of CaOCaF2 , BaO-CaO and BaO-CaF2 systems were determined to
be 3180–3340 kg/m3 , 3340–5720 kg/m3 and 3180–5720 kg/
m3 , respectively. All simulations have been verified using
systems of 3000 atoms and there have not noticed relevant
differences.
The periodic boundary conditions were employed for each
simulation system. The long-range Coulomb interactions
have been summated by Ewald method. The equations of
motion were integrated by fifth-order Gear’s predictorcorrector algorithms using a time step Át ¼ 1 Â 10À15 s.
The run durations of all simulations were carried out for
30000 time steps. At the region around the critical points
such as phase transition temperatures, the simulations were
carried out using long runs up to 100000 time steps. The
simulations for solid phases were started from the room
temperature structures of each solid crystal and then heated
up to the required temperatures. The simulations for liquid
phases were heated to the initial temperature of 4000 K and
thermally equilibrated during the 30000 time steps in order to
stabilize the highly energetic atomic configurations of initial
cells, and then were cooled stepwise from 4000 to 1400 K. In
this study, the effect of cooling rate on the MD calculation
results of all simulation systems has been verified using
cooling rate of 0.1 K per step and relevant differences were
not observed. Therefore, in this study, the effect of cooling
rate was assumed to be negligible. The various properties for
the each system were calculated by statistical analyses of
velocities and positions data after reaching the thermal
equilibrium of each stimulation system. All MD calculations
were carried out using WinMASPHYC program (Fujitsu).

2

i (nm)

bi (nm)

0.19995

0.02101

2

0.25500

0.02685

O

À2

0.18400

0.01300

F

À1

0.14848

0.01160

Results and Discussion

3.1 Pure CaO, BaO and CaF2
The enthalpies for solid and liquid phases of CaO, BaO and
CaF2 were calculated as a function of temperature. The
enthalpies of simulated system can be directly calculated
from the internal energy, pressure and volume values
obtained by MD calculation. The calculated enthalpies are
compared with observed values at the sufficiently high
reference temperatures above the Debye temperature to
neglect the quantum correction terms in this study. The
enthalpy of simulated system (HT ) can be calculated by
eq. (2). The internal energy (UT ), which is given by eq. (3) is
obtained as the sum of potential and kinetic energy calculated
by MD simulation. The heat capacity at constant pressure


Calculation of Thermodynamic Properties and Phase Diagrams for the CaO-CaF2 , BaO-CaO and BaO-CaF2 Systems

645

240

250

Present work

Present work
200

Present work

200

Enthalpy, HT-H1000K, kJ/mol

Enthalpy, HT-H1000K, kJ/mol

(Heating from solid phase)
(Cooling from liquid phase)

Observed 11)
150
100
50

BaO

0
1000

1500

2000

2500

3000

Present work
160

(Cooling from liquid phase)

Observed11)

120
80
40

CaF2

0
800

3500

1200

1600

2000

2400

2800

Temperature, K

Temperature, K
Fig. 2 Calculated and observed enthalpies of solid and liquid BaO as a
function of temperature.

Table 2

(Heating from solid phase)

Fig. 3 Calculated and observed enthalpies of solid and liquid CaF2 as a
function of temperature.

Calculated and observed thermodynamic properties for CaO, BaO and CaF2 .
CaO

BaO

CaF2

Observed

Calculated

Observed

Calculated

Observed

Calculated

Melting
temperature (K)

3200 Æ 50

3210 Æ 10

2285 Æ 5

2290 Æ 10

1691 Æ 5

1700 Æ 10

Áfus H  (kJ/mol)

79.5

55.2

58.6

27.5

29.7

20.0

4.8
(1424 K Æ 20)

2.1
(1265 K Æ 10)

Átrs H  (kJ/mol)

(Cp ), eq. (4), can be calculated from the temperature
dependence of enthalpy calculated by eq. (2).
HT ¼ UT þ PVT
XX
3
UT ¼
ij ðrÞ þ NkT
2
iCp ¼ ð@H=@TÞp

ð2Þ
ð3Þ
ð4Þ

where N is the number of ions of system, k is the Boltzmann’s
constant and T is the absolute temperature.
Figures 1, 2 and 3 show the calculated and observed11)
enthalpies as a function of temperature for CaO, BaO and
CaF2 at reference temperature of 1000 K. The calculated
enthalpies of fusion and melting temperatures of CaO and
BaO are to be 55.2 kJ/mol at 3210 Æ 10 K and 27.5 kJ/mol at
2290 Æ 10 K, respectively. In Fig. 3, the temperature dependence of calculated enthalpies for CaF2 shows the solidsolid phase transition ( -
phase) at 1265 K, and the melting
temperature and enthalpy of fusion are to be 1700 Æ 10 K and
20.0 kJ/mol. The melting temperatures for CaO, BaO and
CaF2 calculated by potential model in this work are in good
agreement with measured results11) of 3200 Æ 50 K, 2285 Æ
5 K and 1691 Æ 5 K, respectively. The calculated enthalpies
of fusion for CaO, BaO and CaF2 show lower than observed
values.11) These differences are considered due to the
overestimation of Coulomb energy by assuming that the
CaO, BaO and CaF2 in this study are perfect ionic crystal.
However, the variations of enthalpy with temperature of
CaO, BaO and CaF2 calculated by MD simulation, in spite of

the perfect crystal cells without defects such as vacancy and
dislocation, are in good agreement with observed results.11)
Therefore, the potential model used in this study is
reasonable to the calculation of thermodynamic properties
of CaO, BaO and CaF2 systems. The calculated thermodynamic properties for CaO, BaO and CaF2 are summarized in
Table 2.
The superionic conductivity on the solid-solid phase
transition ( -
phase) of CaF2 has been assessed by MD
calculation such as pair distribution functions, mean square
displacements and self-diffusion coefficients of Ca and F
ions. Figure 4 shows the pair distribution functions of Ca-Ca,
Ca-F and F-F in low-(a-phase, 800 K), high-temperature
(
-phase, 1500 K) solid CaF2 and liquid CaF2 (2000 K). The
pair distribution functions for the simulated system can be
calculated by eq. (5).
V X hnij ðr À Ár=2; r þ Ár=2Þi
gij ðrÞ ¼
ð5Þ
Ni Nj j
4r 2 Ár
where hnij ðr À Ár=2; r þ Ár=2Þi is the average number of
ion j surrounding ion i in a spherical shell within r Æ Ár=2,
Ni and Nj are the total number of ions i and j, V is the volume
of the system. The calculated pair distribution functions for
the cation and anion, gCa-F ðrÞ and the anion and anion, gF-F ðrÞ
in
-CaF2 show that the F ions are strongly disordered
distribution in the regular site of solid CaF2 like liquid phase.
The ionic diffusivity of solid and liquid CaF2 can be
calculated by mean square displacements of ions as a
function of time. The self-diffusion coefficients of Ca and F


646

W.-G. Seo, D. Zhou and F. Tsukihashi

6

Ca-Ca
Ca-F
F-F

0
β -CaF2 (1500K)

4
2
0

α -CaF2 (800K)

4
2
0.2

0.6

0.4

0.8

1.2

1.0

1.4

Distance, nm

MSD ¼ hjrðtÞ À rð0Þj2 i
ð6Þ
1
ð7Þ
D ¼ ðhjrðtÞ À rð0Þj2 iÞ
6t
where rðtÞ and rð0Þ are the position of the ions at time t and
initial position of the ions at zero time, respectively. h i means
the ensemble average, D is the self-diffusion coefficient.
Figure 5 shows the mean square displacements of Ca and F
ions calculated as a function of time for -CaF2 (800 K),
CaF2 (1500 K) and liquid CaF2 (2000 K). The mean square
displacements of Ca and F ions in -CaF2 show constant
values with time. However, the F ions in
-CaF2 show drastic

0.15

Ca ions
F ions

2

Mean square displacements, nm

-8

-9

-10

DF Present work
DCa Present work
12)

DF Derrington et al.

-11

0.5

0.6

0.7
3

ions can be estimated by the slopes of mean square
displacements calculated as a function of time. The mean
square displacements (MSD) and the self-diffusion coefficients of ions can be calculated by eqs. (6) and (7),
respectively.

1200

CaF2

10 / T, K

Fig. 4 Calculated pair distribution functions for -CaF2 ,
-CaF2 and liquid
CaF2 .

0.8

2000K
0.05

800K

1500K
1

2

3

4

5

Time, ps
Fig. 5 Mean square displacements of Ca and F ions as a function of time
for -CaF2 (800 K),
-CaF2 (1500 K) and liquid CaF2 (2000 K).

0.9

-1

Fig. 6 Calculated self-diffusion coefficients of Ca and F ions for
-CaF2
and liquid CaF2 at various temperatures with measured results.

increase with increasing time. These results show that the Ca
ions do not diffuse in solid CaF2 , on the other hand the F ions
in
-CaF2 diffused from the regular site of CaF2 lattice.
Figure 6 shows the self-diffusion coefficients of Ca and F
ions in
-CaF2 and liquid CaF2 . The self-diffusion coefficients of F ions in
-CaF2 calculated by MD simulation are
in good agreement with measured results by Derrington et
al.12) The pair distribution functions, mean square displacements and self-diffusion coefficients of CaF2 assessed in this
work are also in good agreement with previous researchers’
investigations calculated by Monte Carlo calculation13,14) and
MD simulation by using soft-core potential model15) and
shell model.16)
3.2 CaO-CaF2 , BaO-CaO and BaO-CaF2 systems
3.2.1 Calculation of enthalpy of mixing, entropy of
mixing and Gibbs energy of mixing
The enthalpies of mixing for the CaO-CaF2 , BaO-CaO and
BaO-CaF2 systems can be calculated by MD simulation at
various compositions and temperatures. The enthalpy of
mixing was calculated as a difference between the enthalpy
of solution at certain composition and the sum of the
enthalpies of pure components according to eq. (8).
ÁH M ¼ HAÀB À ðXA HA þ XB HB Þ

0.10

0.00
0

1400

2000 1800 1600

2 -1

2

-7

Self diffusion coefficients, log D, m s

Pair distribution functions, gij(r)

4

0
0.0

Temperature, K

Liquid-CaF2 (2000K)

ð8Þ

where HAÀB is the molar enthalpy of A and B binary solution,
HA and HB are the standard molar enthalpies of component A
and B, XA and XB are the mole fractions of component A and
B, respectively.
Figures 7(a), (b) and (c) show the enthalpies of mixing for
the CaO-CaF2 , BaO-CaO and BaO-CaF2 systems calculated
as a function of composition at various temperatures. The
enthalpies of mixing of each binary system show the negative
values in a whole composition, and they do not show the
large temperature dependence. Especially, the enthalpy of
mixing of the BaO-CaF2 system shows the exothermic
behavior larger than those of the CaO-CaF2 and BaO-CaO
systems, due to the effect of interactions between Ba and F


Calculation of Thermodynamic Properties and Phase Diagrams for the CaO-CaF2 , BaO-CaO and BaO-CaF2 Systems

-100

(a) CaO-CaF2

BaO-CaF2
∆H /(XBaO· XCaF2 ), kJ/mol

0

M

Enthalpy of mixing, ∆H , kJ/mol

2

-2

1600K
2000K
2400K
2800K

-6
0.2

0.4

0.6

1800K
2200K
2600K
3000K

0.8

1.0

Mole fraction CaO

2

-140

1400K
1600K
1800K
2000K
2200K

-160

-180
0.0

0.2

0.4

0.6

0.8

1.0

Mole fraction BaO
Fig. 8 Calculated enthalpy interaction parameters (ÁH M =ðXBaO Á XCaF2 Þ)
as a function of composition for the BaO-CaF2 system at various
temperatures. (standard state: liquid).

(b) BaO-CaO
0

M

Enthalpy of mixing, ∆H , kJ/mol

-120

M

-4

0.0

647

-2

-4
2200K
2600K
3000K

-6
0.0

0.2

0.4

0.6

2400K
2800K

0.8

1.0

(c) BaO-CaF2

0

M

Enthalpy of mixing, ∆H , kJ/mol

Mole fraction CaO

-20

parameters, and the mixture become stable state at 50 mol%
BaO. This result suggests the possibility of formation of the
compounds such as BaOÁCaF2 in the BaO-CaF2 system.
The thermal properties such as internal energy, volume and
pressure of the systems can be calculated by MD simulation.
However, the entropy of mixing cannot be directly calculated
by MD simulation. Therefore, in this study, the entropy of
mixing was calculated by the fractions of ions in the binary
melts, assuming that the CaO-CaF2 , BaO-CaO and BaOCaF2 melts are completely ionic solution and all ions in the
melts have random configurations. These assumptions are
supported by calculated pair distribution functions, gij ðrÞ and
running coordination numbers, Nij ðRÞ of each binary system.
The running coordination numbers for the simulated system
can be calculated by eq. (9).
ZR
Nij ðRÞ ¼ 4i
r 2 gij ðrÞdr
ð9Þ
0

-40
1400K
1800K
2200K

-60
0.0

0.2

0.4

0.6

1600K
2000K

0.8

1.0

Mole fraction BaO
Fig. 7 Calculated enthalpies of mixing as a function of composition for the
(a) CaO-CaF2 , (b) BaO-CaO and (c) BaO-CaF2 systems at various
temperatures. (standard state: liquid).

ions in the BaO-CaF2 melts. Figure 8 shows the enthalpy
interaction parameters (ÁH M =ðXBaO Á XCaF2 Þ) calculated as a
function of composition at various temperatures of the BaOCaF2 system. The calculated enthalpy interaction parameters
show the minimum values at each temperature when the XBaO
equals 0.5. It represents that the BaO-CaF2 system shows the
strong composition dependence of the enthalpy interaction

where i is the partial number density of ion i and R is the
distance of the first minimum of gij ðrÞ. The calculated pair
distribution functions and running coordination numbers of
Ca-Ca, Ca-O, Ca-F, O-O, O-F and F-F in 50 mol% CaO50 mol% CaF2 melt at 2400 K shown in Fig. 9 represent that
all ions in the simulation cell are randomly distributed, which
do not have specific ionic bonding such as network structure.
Typically, the molten slags and fluxes containing BaO and
CaO show the high basicity, and BaO and CaO in these melts
have the role of network modifier.17,18) Therefore, these
oxides in melts are characterized by the ionic nature, and do
not have covalent bonding structure. The molten slags and
fluxes containing CaF2 show the decrease of viscosity and
melting temperature with the addition of CaF2 in melts.18) It
also represents that the Ca and F ions in melts do not have any
structure, and all ions are randomly distributed. These
previously measured results are in good agreement with the
results of structural properties in the melts calculated by MD
simulation. Therefore, these assumptions of random configuration applied for the calculation of entropy of mixing of
each binary system in this study are reasonable.


W.-G. Seo, D. Zhou and F. Tsukihashi

Pair distribution
functions, gij(r)

Running coordination
numbers, Nij(r)

648

6
Ca-Ca
Ca-O
Ca-F
O-O
O-F
F-F

4
2
0

50mol%CaO-50mol%CaF2
2400K

4
2
0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

Distance, nm
Fig. 9 Calculated pair distribution functions and running coordination
numbers of Ca, O and F ions in the 50 mol% CaO-50 mol% CaF2 melt at
2400 K.

The configuration entropy makes a great contribution to
the entropy of mixing in the ionic melts, and the thermal
entropy is numerically much less than the configuration
entropy. In this study, the thermal entropy is assumed to be
negligible. The entropy of mixing is expressed by eq. (10).
A
B
ÁSM ¼ SA+B
Conf À ðXA SConf þ XB SConf Þ
!1
0 n
X
Ni ! C
B
B i¼1
C
B
C
SConf ¼ k lnB Q
n
C
@
ðNi !Þ A

ð10Þ

i¼1

A
B
where SA+B
Conf , SConf and SConf are the configuration entropies of
the A and B binary, pure A and pure B solutions, k is the
Boltzmann’s constant and Ni is the number of ion i per mole
of system. Figure 10 shows the calculated entropies of
mixing for the CaO-CaF2 , BaO-CaO and BaO-CaF2 systems.
The Gibbs energies of mixing for the CaO-CaF2 , BaOCaO and BaO-CaF2 systems were calculated as a function of
composition at various temperatures. The Gibbs energy of
mixing was calculated from the enthalpy and entropy of
mixing based on the thermodynamic parameters obtained
from MD simulation and ionic solution model. Figures 11(a),
(b) and (c) show the calculated Gibbs energies of mixing for
the CaO-CaF2 , BaO-CaO and BaO-CaF2 systems.
3.2.2 Calculation of phase diagrams for the CaO-CaF2 ,
BaO-CaO and BaO-CaF2 systems
The phase diagrams for the CaO-CaF2 , BaO-CaO and
BaO-CaF2 systems were estimated by Gibbs energies of
mixing calculated as a function of composition at various
temperatures. The Gibbs energies of fusion of pure BaO, CaO
and CaF2 for the calculation of phase diagram were evaluated
from the heat capacities at constant pressure based on the
temperature dependence of enthalpies calculated by MD
simulation, eq. (4). Figure 12 shows the Gibbs energies of
fusion of pure BaO, CaO and CaF2 calculated as a function of
temperature with observed results.11) These calculation

Fig. 10 Calculated entropies of mixing for the CaO-CaF2 , BaO-CaO and
BaO-CaF2 systems as a function of composition.

results are lower than observed values with decreasing
temperature. As stated above, these differences are considered due to the underestimation of enthalpies of fusion for
pure BaO, CaO and CaF2 based on the overestimation of
Coulomb energy by assuming that the BaO, CaO and CaF2 in
this study are perfect ionic crystal.
Figure 13 shows the calculated phase diagram for the
CaO-CaF2 system compared with measured results by Ries et
al.19) and Chatterjee et al.20) The calculated eutectic composition and temperature for the CaO-CaF2 system are about
20 mol% CaO and 1650 K, respectively. The calculated phase
diagram is in good agreement with measured results of the
eutectic point of 20 mol% CaO at 1630 K.
Figure 14 shows the calculated phase diagram for the
BaO-CaO system. The phase diagram for the BaO-CaO
system has not been measured experimentally. Recently,
Kemp et al.4) reported the phase diagram with the eutectic
point about 14 mol% CaO at 2180 K obtained by CALPHAD
method. They calculated the phase diagram of BaO-CaO
system from estimated excess thermodynamic properties.
The excess enthalpies and entropies were obtained by the
relationship of Redlich-Kister coefficients with empirically
fitted parameters based on previously measured thermodynamic properties of various oxide and halide mixtures. In
Fig. 14, the phase diagram for the BaO-CaO system
calculated by MD simulation shows the eutectic point about
20 mol% CaO at 2050 K. This result has a difference about
6 mol% CaO and 130 K with the eutectic point reported by
Kemp et al.4) However, the calculated phase diagram for the
BaO-CaO system shows similar shape as phase equilibrium
obtained by CALPHAD method. The calculated and observed eutectic points for the CaO-CaF2 and BaO-CaO
systems are summarized in Table 3.
Figure 15 shows the calculated and measured phase
diagrams for the BaO-CaF2 system. The phase diagram for
the BaO-CaF2 system has been measured by Kojima et al.5)
Only CaF2 -rich region up to about 15 mol% for the BaOCaF2 system was measured. In the present work, the phase
diagram for the BaO-CaF2 system cannot be also calculated


Calculation of Thermodynamic Properties and Phase Diagrams for the CaO-CaF2 , BaO-CaO and BaO-CaF2 Systems

60

Gibbs energy of fusion, ∆G fusion , kJ/mol

(a) CaO-CaF2

0

M

Gibbs energy of mixing, ∆G , kJ/mol

4

-8
-12
1600K
2000K
2400K
2800K

-16
-20
0.2

0.4

0.6

1800K
2200K
2600K
3000K

0.8

1.0

Present work
11)
Observed

40
CaO

o

-4

0.0

649

20
BaO

0
CaF2
-20
1200

Mole fraction CaO

1600

2000

2400

2800

3200

(b) BaO-CaO

0

Fig. 12 Calculated and observed Gibbs energies of fusion of pure BaO,
CaO and CaF2 as a function of temperature.

M

Gibbs energy of mixing, ∆G , kJ/mol

Temperature, K

-4

-8
3200

-12

-16
0.0

0.2

0.4

0.6

2800

2400K
2800K

0.8

1.0

(c) BaO-CaF2

0

2400

2000
Present work
19)
Ries et al.
20)
Chatterjee et al.

1600

M

Gibbs energy of mixing, ∆G , kJ/mol

Mole fraction CaO

Temperature, K

2200K
2600K
3000K

CaO-CaF2

1200
0.0

-20

0.2

0.4

0.6

0.8

1.0

Mole fraction CaO
Fig. 13 Calculated phase diagram for the CaO-CaF2 system.

-40

1400K
1800K
2200K

-60

1600K
2000K

3500

0.0

0.2

0.4

0.6

0.8

1.0

Mole fraction BaO

in a whole composition range because of the possibility of
formation of the compounds such as BaOÁCaF2 based on the
results of calculated enthalpy interaction parameters of the
BaO-CaF2 system. However, the liquidus lines in CaF2 -rich
and BaO-rich region of the BaO-CaF2 system have been
estimated by MD simulation. In Fig. 15, the calculated
liquidus line of the BaO-rich region in the BaO-CaF2 system
shows drastic decrease with the addition of CaF2 . Lee et al.21)

3000

Temperature, K

Fig. 11 Calculated Gibbs energies of mixing as a function of composition
for the (a) CaO-CaF2 , (b) BaO-CaO and (c) BaO-CaF2 systems at various
temperatures. (standard state: liquid).

BaO-CaO

2500

2000

1500

1000
0.0

Present work
4)
W.J.M. van der Kemp et al.
0.2

0.4

0.6

0.8

1.0

Mole fraction CaO
Fig. 14 Calculated phase diagram for the BaO-CaO system.


650

W.-G. Seo, D. Zhou and F. Tsukihashi
Table 3

Calculated and observed eutectic points for the CaO-CaF2 and BaO-CaO systems.
BaO-CaO

CaO-CaF2


Observed19;20Þ

Calculated

Calculated
(CALPHAD)

Calculated

1633

1650

2180

2050

20

20

14

20

Temperature (K)
XCaO (mol%)

2500

model. The calculated phase diagrams for the CaO-CaF2 and
BaO-CaO systems were in good agreement with experimentally determined ones and with obtained ones by CALPHAD
method. The possibility of formation of the compounds such
as BaOÁCaF2 in the BaO-CaF2 system was suggested by the
calculated enthalpy interaction parameters for the BaO-CaF2
system. The liquidus lines in CaF2 -rich and BaO-rich region
of the BaO-CaF2 system have also been estimated by MD
simulation. From these results, we have successfully demonstrated that MD simulation can be used for the calculation
of thermodynamic properties and the estimation of phase
diagrams for the oxide and halide systems at high temperature.

Temperature, K

BaO-CaF2
Present work
5)
H. Kojima et al.

2000

1500

1000
0.0

0.1

0.2

0.3

0.7

0.8

0.9

1.0

Mole fraction BaO
Fig. 15 Calculated phase diagram for the BaO-CaF2 system.

reported the rapid decrease of melting temperature with the
addition of BaO in the CaO-CaF2 system. These results may
be due to the enthalpy of mixing of the BaO-CaF2 system
smaller than those of the CaO-CaF2 and BaO-CaO systems
from MD calculation results.
These calculation results are concluded that the MD
simulation with optimized potential model is a useful method
for the calculation of thermodynamic properties and the
estimation of phase diagrams for the oxide and halide
systems at high temperature.
4.

Conclusions

The thermodynamic properties for the CaO-CaF2 , BaOCaO and BaO-CaF2 systems were calculated by MD
simulation using simple Born-Mayer-Huggins type potential
model with the optimized potential parameters. The calculated thermodynamic properties of pure CaO, BaO and CaF2
were in good agreement with experimental results. The
superionic conductivity on the solid-solid phase transition of
CaF2 has also been successfully assessed from the pair
distribution functions, mean square displacements and selfdiffusion coefficients calculated by potential model in this
study. The phase diagrams for the CaO-CaF2 , BaO-CaO and
BaO-CaF2 systems were calculated by thermodynamic
parameters obtained from MD simulation and ionic solution

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