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

4Þ

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

REFERENCES

1) W. Tompson: Fact, McGill University, Montreal, Canada.

2) G. Eriksson and K. Hack: Metall Trans. B. 21B (1990) 1013–1023.

3) B. Sundman: Thermo-calc, Royal Institute of Technology, Stockholm,

Sweden.

4) W. J. M. Van der Kemp, J. G. Blok, P. R. Van der Linde, H. A. J. Oonk

and A. Schuijﬀ: Calphad 18 (1994) 255–267.

5) H. Kojima and C. R. Masson: Canad. J. Chem. 47 (1969) 4221–4228.

6) K. Hirao and K. Kawamura: Material Design using Personal

Computer, (Shokabo, Tokyo, 1994) pp. 52–54.

7) D. K. Belashchenko: Russ. Chem. Rev. 66 (1997) 733–762.

8) D. K. Belashchenko, I. E. Gopengauz, A. B. Grytsenko and O. I.

Ostrovski: ISIJ Int. 32 (1992) 990–997.

9) D. K. Belashchenko, O. I. Ostrovski and Y. I. Utochkin: ISIJ Int. 38

(1998) 673–679.

10) T. Matsumiya, A. Nogami and Y. Fukuda: ISIJ Int. 33 (1993) 210–217.

11) M. W. Chase, Jr.: NIST-JANAF, Thermochemical Tables, 4th Ed., ed.

by American Chemical Society and American Institute of Physics,

(New York, 1998) p. 354, p. 718, p. 731.

12) C. E. Derrington, A. Lindner and M. O’Keeﬀe: J. Solid State Chem. 15

(1975) 171–174.

13) Y. Hiwatari and A. Ueda: J. Phys. Soc. Jpn. 49 (1980) 2129–2135.

14) R. A. Montain: J. Chem. Phys. 100 (1994) 8381–8384.

15) Y. Kaneko and A. Ueda: J. Phys. Soc. Jpn. 57 (1988) 3064–3073.

16) P. J. D. Lindan and M. J. Gillan: J. Phys. Condens. Matter 5 (1993)

1019–1030.

17) Y. Waseda and J. M. Toguri: The Structure and Properties of Oxide

Melts, World Scientiﬁc, (London, 1998).

18) E. T. Turkdogan: Physicochemical Properties of Molten Slags and

Glasses, The Metal Society, (London, 1983).

19) R. Ries and K. Schwerdtfeger: Arch. Eisenhuettenwes. 51 (1980) 123–

129.

20) A. K. Chatterjee and G. I. Zhmoidin: J. Mater. Sci. 7 (1972) 93–97.

21) S. C. Lee, J. J. Pak and S. H. Kim: Proc. 5th Int. Conf. on Molten Slags,

Fluxes and Salts, (The Iron and Steel Soc., 1997) pp. 823–829.

#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

i

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

4Þ

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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