九州大学大学院総合理工学府報告第 44 巻 第 2 号 20-31 頁 令和 5 年 2 月

Engineering Sciences Reports, Kyushu University
Vol. 44, No. 2, pp. 20-31, FEB, 2023

Thermodynamic properties of Cu–Pb–F ternary system
for the development of fluoride-ion batteries
Shangping ZHU*1 Hideyuki HARADA*2 Shoya KAWANO*2
Kumiko YAMAMOTO*2 Hidenori MIKI*3 Takeshi TOJIGAMORI*3
Kousuke NOI*3 and Satoshi IIKUBO*1, *2, †E-mail of corresponding author: iikubo.satoshi.472@m.kyushu-u.ac.jp†(Received November 30, 2022, accepted December 7, 2022)
A thermodynamic analysis of the Cu-Pb-F ternary system was performed by incorporating firstprinciples calculations into the CALPHAD method. To evaluate the Gibbs energy, the phonon
calculation was applied for some fluorides of the binary and ternary compounds. In addition, the
FCC interstitial model was considered for the fluorides of Cu and Pb. For the Cu-Pb binary system,
it is in good agreement with the experiment data. As for F-contained binary and ternary system,
Gibbs energies of the stoichiometric compounds obtained by thermodynamic analysis are in good
agreement with the phonon calculation, in which the existence of F makes it difficult to beinvestigated by experiment. Nevertheless, the data near CuPb2F6 shows significant discrepancy
between phonon calculation data and experiment data, which may be the reason for the inaccurate
reported CuPb2F6 crystal structure. Therefore, the crystal structure of reported CuPb2F6 was refinedby neutron diffraction (ND) and atomic pair distribution function (PDF), in which the location of F
atoms was obviously different after refinement. The formation energy of the refined structure basedon CuF2 and PbF2 is -9.494 kJ/mol of atoms, whereas the unrefined structure is 3.550 kJ/mol ofatoms, which shows the refined structure is more stable. The results of this study were helpful toclarify the phase equilibria of Cu-Pb-F system in the fluoride ion battery.
Key words: phase diagram, first-principles calculations, CALPHAD, fluoride

1. Introduction
Fluoride-ion batteries based on fluoride
anions,
which
have
the
highestelectronegativity, are promising candidates
as alternatives to lithium-ion batteries.1)
Relevant fluoride ion conductors have beenstudied in terms of their crystal structure,defects, ionic dynamics, etc.2-3) Among thereported fast fluoride-ion conductors, (1-x)
PbF2 ・ xMF2(Mg, Ca, Sr, Ba) exhibits
outstanding ionic conductivity of up to nearly
10-4 S∙cm-1.4-5) Copper, which exhibits highelectron conductivity, can also be treated as
an alternative candidate for a doping
material. In this field, knowledge regarding
the phase stability of compounds in the Cu*1 Department of Advanced Energy Engineering
Science*2 Graduate School of Life Science and Systems
Engineering, Kyushu Institute of Technology,
Kitakyushu 808-0196, Japan*3 Advanced Material Engineering Division, Higashifuji
Technical Center, Toyota Motor Corporation, 1200
Mishuku, Susono, Shizuoka 410-1193, Japan.

Pb-F system is an important factor in
fundamental studies, and basic information
is often obtained from phase diagrams.
Although it is difficult to collect systematic
experimental data on phase equilibria, thecalculation of phase diagrams (CALPHAD)
method6) is a useful approach for calculatingphase diagrams in multi-component andmulti-phase systems.
The CuF2–PbF2 experimental pseudo
binary phase diagram was obtained by M.
Samouël.7) The X-ray diffraction pattern andcrystal structure of CuPb2F6 were also
reported by Samouel et al.8) Furthermore,
Vaajamo et al. investigated the Cu–Pb binary
phase diagram.9) However, the other ternarycompound (CuPbF6) reported by B. G. Müller
et al.10) was not manifested in the reported
phase diagram, and Cu–F and Pb–F binary
phase diagrams have not been investigatedthus far, which makes it difficult to collectexperimental data. On the other hand, Johtoet al.11) noted that the thermodynamic
properties of the Cu–Pb–S ternary system令和 4 年度

九州大学大学院総合理工学報告 第 44 巻 第 2 号

21

were critically assessed and optimized using
the CALPHAD method and showed good
agreement with the experimental results.
To determine the difference between thecalculated and experimental data, atomic
pair distribution function (PDF) analysis of
the neutron powder diffraction data has beenused to compare the calculated PDF with theexperimental PDF and refine the crystal
structure parameters.12-13) PDF structure
modeling programs include small box
modeling methods,14) which use few
parameters,
and
big
box
modeling
programs,15) which vary the number of atoms
to maximize the agreement between thecalculated and experimental PDFs. There arethree widely used PDF structure modelingprograms: PDFgui,16) TOPAS,17) and diffpycmi.18)
In this study, first-principles calculations
were used to perform thermodynamic
assessments of stoichiometric compounds
using the CALPHAD method. The Gibbs
energy at finite temperatures was estimatedby the finite displacement method;19-20) then,the phase diagram was created using
PANDAT software,21) and the result was
optimized based on experimental data. The
difference between the realistic structure of
CuPb2F6 and the reported structure wasrevealed, and refinement of the reported
structure was carried out using neutron
diffraction and the PDFgui program. The
objective of this study was to clarify the phaseequilibria of the Cu–Pb–F ternary systemover the entire composition and phasestability of ternary compounds.

First-principles
calculations
wereperformed based on density functional theory(DFT) using the Vienna ab initio simulation
package (VASP) code.22, 23) The interaction
between the ion core and valence electrons
was described using the projector-augmented
wave method.24,25) The exchange andcorrelation functions were given by thegeneralized gradient approximation, as
proposed by Perdew et al.26, 27) For structural
optimization and enthalpy calculations, we
set the plane wave energy cutoff to 520 eV.
Brillouin zone sampling in a primitive cell
was performed using a Γ-point-centered kmesh that was limited to 0.15 Å-1. The
convergence criteria for electronic selfconsistency and the ionic relaxation loop were
set to 10-5 eV and 0.02 eV, respectively. The
Methfessel–Paxton smearing method with asmearing width of 0.2 eV and the tetrahedron
method with Blöchl corrections were used forstructural optimization and the enthalpy
calculation, respectively.
To examine the Helmholtz free energy, we
conductedafirst-principles
phononcalculation based on the finite displacement
method19, 20) using the VASP code. The
simplest approximation in lattice dynamics
theory is a harmonic approximation. Once thephonon frequencies, 𝜔, are solved under the
harmonic approximation, the Helmholtz freeenergy, 𝐹 , is calculated using the following
equation:1𝐹=ℏ𝜔 + 𝑘 𝑇
2

2. Experimental procedure

where ℏ , 𝑘 , 𝑇 , 𝑞 , and 𝑗 are the reduced
Planck constant, Boltzmann constant,temperature, wave vector, and phonon band
index, respectively. For this phonon
calculation, we used supercells to include aninteraction range of approximately 10.0 Å
with an atomic displacement distance of 0.02
Å. To estimate the force due to displacements
within the DFT calculation, we used a plane
wave energy cutoff of 400 eV, a convergencecriterion for the electronic self-consistency
loop of 10-6 eV, a Γ-point-centered k-meshlimited to 0.15 Å-1, and the Methfessel–
Paxton smearing method with a smearing
width of 0.2 eV.

2.1. Synthesis of CuPb 2F6 and X-ray
diffraction (XRD)
Pure PbF2 powder and pure CuF2 powder
with a ratio of 2:1 was synthesized using aplanetary ball milling machine at room
temperature. To ensure a single-phase solid
solution was obtained, milling was performedfor 72 h at 200 rpm.
X-ray diffraction (XRD) measurements
were used to analyze the sample. The XRD
measurements were carried out with
𝐶𝑢𝐾 (1.5406 Å) radiation, and the XRD data
were analyzed using the Rietveld method tomatch the reported crystal structure. 7-8)
2.2 First-principles calculations∑ ln 1 − exp −ℏ𝜔 ⁄𝑘 𝑇 ,2.3 CALPHAD
2.3.1 FCC interstitial

(1)

22

Thermodynamic properties of Cu–Pb–F ternary system for the development of fluoride-ion batteries

The crystal structures of pure Cu and Pb
are stable face-centered cubic (FCC) phases.
Furthermore, the binary systems of Cu or Pb
and F cannot be described by simply mixing
the two elements where the Gibbs free energyof the system increases monotonically with
the increase in F content, as this isinconsistent with reality. Therefore, we
considered the FCC interstitial model for the
fluorides of Cu and Pb. This model can
describe invasion of the Cu or Pb FCC lattices
by F atoms at vacancy sites. For instance, theGibbs energy of the FCC interstitial phase isexpressed using a two-sublattice model, in
which the Pb atoms are substituted with Cu.
For example, the molar Gibbs energy of the
FCC phase can be calculated using thefollowing equation:𝐺=
𝑥Cu ∘𝐺Cu + 𝑥 ∘𝐺
+ 𝑥F ∘𝐺F+ 𝑅𝑇(𝑥Cu 𝑙𝑛 𝑥Cu + 𝑥

𝑙𝑛 𝑥+ 𝑥F 𝑙𝑛 𝑥F )+ 𝑥Cu 𝑥 𝐿Cu,Pb + 𝑥 𝑥F 𝐿Pb,F + 𝑥Cu 𝑥F 𝐿Cu,F+ 𝑥Cu 𝑥 𝑥F 𝐿Cu,Pb,F(2)

where ∘𝐺𝑖 denotes the molar Gibbs energy ofelement 𝑖 in the solid state, R is the universal
gas constant, and the term 𝑥 is the mole
fraction of element 𝑖 in the ternary system.
The lattice stability parameter is describedby the following equation:∘𝐺− ∘𝐻= 𝑎 + 𝑏𝑇 +𝑐𝑇 𝑙𝑛 𝑇 + 𝑑𝑇 + 𝑒𝑇 + 𝑓𝑇 + 𝑖𝑇 +𝑗𝑇 ,(3)
where denotes the molar enthalpy of pureelement 𝑖 in its stable state at T = 25 ℃, andthe symbols a–j are coefficients. Thisparameter denotes the interaction energy
between 𝑖 and 𝑗 in the FCC phase, and it
exhibits
acompositional
dependency
following the Redlich–Kister polynomial:
𝐿 , = 𝐿 , + 𝐿 , (𝑥 − 𝑥 ) +𝐿 , (𝑥 − 𝑥 ) +⋅⋅⋅ + 𝐿 , (𝑥 −
𝑥) ,
(4)
where
𝐿,

= 𝐴 + 𝐵𝑇 + 𝐶𝑇ln𝑇+𝐷𝑇 +⋅⋅⋅,(5)in which the symbols A–D are coefficients.
The FCC solid solutions, which exhibit arange of non-stoichiometric alloys, aremodeled using the same FCC interstitialsolution approximation.

2.3.2 Liquid phase

The molar Gibbs energy of the liquid phase,, is calculated using the following
equation:
𝐺,
=𝑥 𝐺
+𝑥 𝐺
+
𝐺,

𝑅𝑇 ∑

,

(6)

(𝑥 − 𝑥 ) ⋯ ],(7)

𝑥 ln 𝑥 + 𝐺

where
𝐺

=𝑥 𝑥 [ 𝐿 ,(𝑥 − 𝑥 )

+ 𝐿 ,

+ 𝐿 ,

in which 𝐺
denotes the molar Gibbs
energy of element 𝑖 in the liquid state, R is
the universal gas constant, and 𝑥 is the mole
fraction of element 𝑖 in the A–B binary
system. 𝐺
is the excess Gibbs energy inthe system containing 𝐿 ,that has acompositional dependency following the
Redlich–Kister polynomial.
2.3.3 Gas phase

The gas phase of F was developed by
Scientific Group Thermo data Europe(SGTE)28) and provided by TCSAB.
2.3.4 Stoichiometric compounds

Binary compound phases with zero
homogeneity ranges (i.e., CuF2, PbF2, PbF3,
and PbF4) are treated as stoichiometriccompounds. The Gibbs energies can becalculated using the phonon calculation andthen fitted to equations.
2.3.5 Ternary compounds

In this study, two ternary phases wereconsidered. The crystal structures of CuPb2F6
and CuPbF6 were determined using thePearson’s crystal data. Then, the Gibbs
energies of these phases were calculatedusing phonon calculation and curve fitting toequations to obtain their coefficients.
2.4 Neutron diffraction (ND) and data
analysis
In this study, neutron diffraction (ND) wasused to modify the reported crystal structureof CuPb2F6. Time-of-light (TOF) ND
measurements were performed using thespecial-environment neutron diffractometer
SPICA at the Japan Proton Accelerator
Research
Complex(J-PARC).29)
This
instrument contains argon-filled positionsensitive detectors that are assembled in a

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九州大学大学院総合理工学報告 第 44 巻 第 2 号

single detector–component box. The detectorbanks consist of a back-scattering bank and amultipurpose bank that are arranged from175° to 5°, and the locations of these banks
are approximately 2 m from the sample. The
powder sample was placed in a cylindricalholder in which the atmosphere wascompletely replaced by argon gas, and theexperiment was carried out at room
temperature.
To visualize the crystal structure of the
CuPb2F6 phase and create modified atom
locations, full-profile fitting of the atomic
PDF from the obtained ND data was
performed using the PDFgui software. The
experimental PDF was obtained from theFourier transform of the structure factor.
S(Q) is the neutron structure factor, which
can be described by the following equation:
𝐼(𝑄)
S(Q) =
=1+
𝑁𝑏
( )
𝑑𝑟.
∫ 4𝜋𝑟 𝜌 [𝑔(𝑟) − 1](8)
We can then obtain the experimental PDF,
G (𝑟), according to the Fourier transform,
as follows:
G (r) =∫ 𝑄[𝑆(𝑄) − 1] sin(𝑄𝑟) 𝑑𝑄.
(9)When the crystal structure is previously
known, the PDF can also be calculated using
the PDFFIT program, and our equation forthe model PDF becomes the following:30)
𝐺 (𝑟) =∑ ∑𝛿(𝑟 − 𝑟 ) − 4𝜋𝑟𝜌 ,
(10)
〈 〉

where 𝐺 (𝑟) is the model PDF for a single
phase, 𝜌 is the average number density, N is
the number of atoms, and 〈𝑏〉 is the average
scattering length; the indices 𝑖 and 𝑗 sum over
all atom pairs in that phase, where 𝑏 and 𝑏
are the scattering lengths for atoms 𝑖 and 𝑗 ,respectively, and 𝑟 is the separation
distance between atoms 𝑖 and 𝑗 . Thus, thestructural parameters of the calculation are
refined to fit the experimental PDF to allow
the crystal structure to be obtained. The
calculation of the standard deviation is
described by the R-value, and the expected Rvalue is defined as follows:
𝑅

=

∑

( )

,

( )

(11)

where Gis the experimental PDF, and theweight for each data point is given by 𝑤(𝑟 ).

23

Therefore, the weighted R-value ( 𝑅 ) isdefined as follows:
∑

𝑅 =

( )[
∑

( )
( )

( )]
( )

(12)

.

Here, G
represents the calculation data.
In this study, we obtained the PDF data forthe experimental crystal structure using ND
and PDFgui in the first step, and then the
PDF data of the reported structure were also
calculated using Equation (12). Therefore,
the atoms of the reported structure could be
moved to fit the experimental data, and thus
the experimental crystal structure could be
predicted based on the goodness-of-fit of thetwo data sets with low 𝑅 .

3. Results and discussion
The space group and formation energy forthe stoichiometric compounds of Cu-Pb-F
system were list in Table A1, in whichenergy was calculated by First-principles
calculation.
3.1 Cu–Pb binary system
The lattice stability parameters for thepure elements obtained from the SGTE
databases28) are listed in Table A2. The
thermodynamic parameters for Cu–Pb are
listed in Table 3, which are included in the
Thermo-Calc
SSOL4.9
database.
The
calculated Cu–Pb phase diagram is shown in
Fig. 1 and is in good agreement with theexperimental data.31-33)
Liquid

1300
1100
900

This work
C.T. Heycock et al.[31]FCC

O.J. Kleppa et al.[32]W. Seith[33]

700
500
300

0
Cu

FCC
20

40

60

80

mol%Pb100
Pb

Fig.1 Calculated Cu–Pb phase diagram withexperimental results.

3.2 Cu–F and Pb–F binary systems (FCC
interstitial)
The Cu–F binary system consists of theliquid, gas, body-centered cubic (BCC), FCC,
hexagonal close-packed (HCP), and CuF2
phases The Pb–F binary system is composedof the liquid, gas, BCC, FCC, HCP, PbF2_FCC,
PbF2_pnma, PbF3, and PbF4 phases. In this24

Thermodynamic properties of Cu–Pb–F ternary system for the development of fluoride-ion batteries

Temperature(K)Temperature(K)study, the thermodynamic parameters were
calculated using a first-principles calculationin which non-stoichiometric compounds used
the FCC interstitial model. The database islisted in Table A3 and the calculated phase
diagrams are shown in Fig. 2(a) and Fig. 3.

Fig.3 Calculated Pb–F binary phase diagram.

Gibbs energy(CuF2)(J/mol)

As shown in Fig. 2(a), F is stable as F2, andthus the F2 gas phase is displayed on the Frich side. The CuF2 phase is stable below 760
K, while the FCC interstitial (FCC-int) is
stable above 760 K. We also matched theGibbs energy of CuF2 calculated by thethermodynamic and phonon calculation
analyses described in Fig. 2(b), which showsthat the results of the thermodynamic
analysis are in good agreement with theresults of the phonon calculation.
According to Fig. 3, the stoichiometriccompounds consist of PbF2_FCC, PbF3, andPbF4, in which PbF2_FCC is always morestable than PbF2_pnma. The FCC interstitial(FCC-int) in the Pb–F binary system is only
stable below 600 K. Fig. 4 shows the Gibbs
energy in the Pb–F system obtained by thephonon calculation and thermodynamic
analysis. The results are in good agreement
and indicate the stability of PbF2_FCC.

Fig.2 (a) Calculated Cu–F binary phase diagram,(b) Gibbs energy for CuF2.

Gibbs energy(PbF2_Pnma) (kJ/mol)
Gibbs energy(PbF4) (kJ/mol)

Gibbs energy(PbF3) (kJ/mol)

Gibbs energy(PbF2_FCC) (kJ/mol)In the Cu–F binary phase diagram, thestoichiometric compound is CuF2 with amelting point of 1109.15 K, which is the sameas the experimental data. The eutectic point
is located at 41% F and 992 K.

Fig.4 Gibbs energies of (a) PbF2_FCC, (b) PbF2_Pnma, (c) PbF3, and (d) PbF4.

令和 4 年度

九州大学大学院総合理工学報告 第 44 巻 第 2 号

25(a)1100
Temperature (K)Gas+Liquid
900

FCC-int

700

FCC
FCC

5

10

15
20
mol%(Pb)

FCC-int
25
30

PbF2

Gibbs energy(CuPbF6) (kJ/mol)0
CuF2

Gibbs energy(CuPb2F6)(kJ/mol)

500

Fig.5 (a)original pseudo CuF2-PbF2 phase diagram (b)pseudo CuF2-PbF2
phase diagram adjusted by experimental data (c) Gibbs energy for CuPbF6
in adjusted phase diagram(d) Gibbs energy for CuPb2F6 in original andadjusted phase diagram.

3.3 Cu–Pb–F ternary system
The Cu–Pb–F ternary phase diagram wascalculated using the three binary systems
above, and the ternary phases (CuPbF6 and
CuPb2F6) were calculated using phonon
calculationandadjusted
basedonexperimental data; the results are presented
in Table A3 and Fig. 5. In Fig. 5(a) and (b),the Fig. 5(a) is the calculated pseudo-binary
phase diagram of the CuF2–PbF2 system, inwhich the ternary phases just calculated byphonon calculation, and the Fig. 5(b) isadjusted by CuF2–PbF2 experimental data
based on the Fig. 5(a). This result in Fig. 5(a),has big different with experimental data,7)but the Fig. 5(b) is in good agreement withthe experimental results obtained by M.
Samouël.7) In particular, the Pb-rich side is
expected to cause the system to be in
equilibrium. Otherwise, one of the ternary
phases (CuPb2F6) appears on the pseudophase diagram (Fig. 5(b)), which has beenpreviously reported. In Fig. 5(c) and 5(d), theGibbs energies of the CuPbF6 and CuPb2F6
phases are shown, respectively, comparing
the thermodynamic analysis data with thephonon calculation results. We note that the
CuPbF6 phase has no diversity within the two

methods, but the Gibbs energy for the
CuPb2F6 phase obtained from the phonon
calculation is much higher than thethermodynamic analysis data which adjusted
by experimental data. This shows a clear
difference in CuPb2F6 before and after
adjustment, which may be a result of theinaccuracy of the reported structure.
As shown in Fig. 6, the isothermal sectionof the Cu-Pb-F phase diagram was calculated
at temperatures of 300 K, 500 K, 700 K, and1000 K. At 300 K, it consists of all of thebinary and ternary compounds reported, as
shown in Fig. 6(a). In particular, the CuPbF6
phase is present in the F-rich region, which
did not appear in the experimental phase
diagram. When temperature increases to 500
K, as shown in Fig. 6(b), the CuPbF6 phase
disappears and is disassembled into CuF2
and PbF4 phases, while the other phases
remain. Thus, the stable temperature rangeof the CuPbF6 phase is below 500 K.
Furthermore,asmall
regionof
PbF2_FCC+Liquid appears next to the Pb–F
binary side at 700 K, as shown in Fig. 6(c),because the melting point of Pb is 600 K. The
isothermal section at 1000 K is shown in Fig.
6(d). The CuF2 phase is unstable and

26

mo
l%
F

mo
l%
F

mo

l%
F

mo
l%
F

Thermodynamic properties of Cu–Pb–F ternary system for the development of fluoride-ion batteries

Fig.6 Isothermal sections of the Cu-Pb-F phase
diagram: (a) 300 K, (b) 500 K, (c) 700 K, (d) 1000 K.

transforms into the FCC-int phase, and the
CuPb2F6 is melted, as it has a melting pointof 891 ℃. In contrast, the PbF2_FCC, PbF3,
and PbF4 phases remain stable. The liquid
phase field extends continuously above therelevant eutectic points, in agreement withthe binary phase diagrams described above.
3.4 XRD and ND
The crystal structure of CuPb2F6 was
obtained from the experimental XRD data forthe pure material synthesized using CuF2
and PbF2, as shown in Fig. 7(a) and 7(b). The
phonon density of state (DOS) for CuPb2F6
wascalculatedusing
first-principles
calculations, as shown in Fig. 7(c). The
negative frequency of the phonon DOS
accounts for 12.2% of the overall calculation,
which indicates the dynamic instability ofthis crystal structure. Furthermore, theGibbs energies of CuPb2F6 for the adjusted
and unadjusted phase diagrams areinconsistent, which also indicates theinaccuracyofthecrystalstructure
ascertained by XRD. The low sensitivity forlight elements (e.g., Cu, F) in XRD may be thereason for the inaccurate CuPb2F6 structure.
Therefore, ND is employed to elucidate thestructure owing to its high sensitivity, evenfor light elements. In Fig. 8(a), theexperimental PDF is obtained using theFourier transform of the reduced structure
factor. We compared each peak of theexperimental data with the F–F, Cu–Cu, Pb–
Pb, Cu–F, Pb–F, and Cu–Pb pair calculation
data to confirm the existence of atom pairs foreach peak. The peaks are related to the F–F
pair, except those near 3 and 6 Å, which were
determined to be related to other pairs,
particularly Cu–F and Pb–F. The peak at 20
Å is related to all atomic pairs, and thus thecomparisons between the experiment andcalculation are conducted from 0 to 19 Å. A
comparison between the experimental data

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九州大学大学院総合理工学報告 第 42 巻 第 1 号

27

Fig.7 (a) XRD pattern for CuPb2F6, (b) reported crystal structure of CuPb2F6,(c) phonon density of state for the reported CuPb2F6 structure.

Fig.8 (a) Experimental neutron diffraction data transformed using PDFgui, (b) PDFs
for the experiment and unrefined structure, (c) PDFs for the experiment and refinedstructure, (d) phonon calculation for the refined crystal structure of CuPb2F6.

and the calculation for the unrefinedstructure with a weighted R-value ( 𝑅 ) of up
to 49.4% is shown in Fig. 8(b). In Fig. 8(c), thestandard deviation, 𝑅 , between theexperimental data and calculation data forthe refined structure, which was refined byfocusing on F–F, Cu–F, and Pb–F, decreases
to 13%. The results in Fig. 8(a) and 8(b) showthat the refined structure has good accuracy
with a realistic structure. The phonon
calculation for the refined structure showsthat the negative frequency decreases to 6.8%
compared to 12.2% for the unrefinedstructure, as shown in Fig. 8(d). There is stilla small amount of negative frequency
remaining, which may be because the firstprinciples phonon calculation with the finite
displacement method was calculated at 0 K,
and the ND was performed at room
temperature.

Fig.9 Structure of CuPb2F6:(a) unrefined, (b) refined.

The crystal structures of CuPb2F6 before
and after refinement are shown in Fig. 9(a)and 9(b), respectively. In terms of the locationof the atoms, there is no obvious difference for
Cu and Pb, but the F atoms are obviously
distorted. The structure of unrefined CuPb2F6
is an octahedron that rotates around the caxis, whereas the refined structure rotates

28

Thermodynamic properties of Cu–Pb–F ternary system for the development of fluoride-ion batteries

around the a-axis and b-axis. The results
show that the refined structure is locallydistorted compared to the unrefinedstructure, and the formation energy based on
CuF2 and PbF2 of the refined structure is 9.494 kJ/mol of atoms, which is more stable
than CuF2 and PbF2, whereas the formationenergy of the unrefined structure is 3.550
kJ/mol of atoms.

4. Conclusions
Thermodynamic analysis of the Cu–Pb–F
system and atomic pair distribution function(PDF) analysis of neutron powder diffraction(ND) data were performed, yielding thefollowing results.
(1) The Cu–F and Pb–F binary systems
were calculated, and the Gibbs energies ofthe stoichiometric compounds calculated bythermodynamic
analysisandphononcalculations were in good agreement. The
FCC interstitial model was used to calculate
the non-stoichiometric compounds, which
FCC-int phase shows stable in the
Isothermal sections of the Cu-Pb-F phase
diagram at 1000 K. Although the reliabilityof the model still need to be validated, it is
helpful to clarify the non-stoichiometriccompounds.
(2) For the Cu–Pb–F ternary system, thecalculated pseudo CuF2–PbF2 phase diagram
showedgood
agreementwiththeexperimental phase diagram, especially on
the Pb-rich side, and the phase stability of allcompounds was revealed in the isothermal
section of the Cu–Pb–F phase diagram.
Furthermore, a difference between thethermodynamic
analysisandphononcalculation for CuPb2F6 was observed, which
may be due to the accuracy of the reported
CuPb2F6 structure.
(3) PDF and ND were performed to refine
the crystal structure of CuPb2F6. The locationof the F atoms was obviously different in theunrefined and refined structures, whereas
the Cu and Pb atoms exhibited no obvious
change, which is the reason why the reported
structure obtained using XRD had low
accuracy. Finally, the negative frequency
remaining in the phonon calculation of therefined crystal structure may be caused by
the fact that the phonon calculation was
performed at 0 K and the ND was conductedat room temperature. It may be possible touse finite-temperature phonon calculations
to obtain results with no negative frequency.
Acknowledgment
This paper is based on results obtained
from a project, JPNP16001 and JPNP21006,
commissioned by the New Energy andIndustrial
Technology
Development
Organization (NEDO).
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30

Thermodynamic properties of Cu–Pb–F ternary system for the development of fluoride-ion batteries

Appendix

Table A1. Compounds of (Cu, Pb)F.
Formula
CuF2
CuF2
CuF
PbF2
PbF2
PbF3
PbF4
CuPb2F6
CuPbF6

Space group
Fm-3m
P2_1/c
F-43m
Fm-3m
Pnma
P-4c2
I4-mmm
C12-m1P-1

Formation Energy (kJ/mol of atom)-124.02-146.78-78.61-232.54-230.31-205.26-178.86-200.41-169.95

Table A2. Lattice stability parameters for Cu, Pb, and F.
Species

Phase

Cu

L

Lattice stability parameter, J/mol
°

− °𝐺

𝐺

=12964.736-9.511904T-5.849×10-21 T7= 13495.481 − 9.922344𝑇 − 3.642 × 10 𝑇

Pb

FCC

°

𝐺

− °𝐻

BCC

°

𝐺

− °𝐺

°

𝐺

− °𝐻

°

− °𝐺

HCP
FCC

L

𝐺

= −7770.458 + 130.485235𝑇 − 24.112392𝑇 ln 𝑇 −0.00265684𝑇 + 1.29223 × 10 𝑇 + 52478𝑇= −13542.026 + 183.803828𝑇 − 31.38𝑇 ln 𝑇 + 3.64167 ×
10 𝑇= 4017 − 1.255 × 𝑇= −7650.085 + 101.715188𝑇 − 24.5242231𝑇 ln 𝑇 −0.00365895𝑇 − 2.4395 × 10 𝑇
= −10531.115 + 154.258155𝑇 − 32.4913959𝑇 ln 𝑇 +0.00154613𝑇 + 8.05644 × 10 𝑇
= 4157.596 + 53.154045𝑇 − 18.9640637𝑇 ln 𝑇 +0.002882943𝑇 + 9.8144 × 10 𝑇 − 2696755𝑇 +8.05644 × 10 𝑇
= 4672.157 − 7.750257T − 6.0144 × 10 𝑇
= 4853.112 − 8.066587T − 8.05644 × 10 𝑇F

MOLE_GASGGASF2

GHSER= −9757.53536 − 3.03641497𝑇 − 29.08493𝑇 ln 𝑇 −0.007466065𝑇 + 1.16183617 × 10 𝑇 + 72117.3𝑇= −10225.5134 + 17.0719031𝑇 − 32.43806𝑇 ln 𝑇 −0.0028914815𝑇 + 2.13090167 × 10 𝑇 − 31206.07𝑇= −75010.7092 + 286.369351𝑇 − 66.19232𝑇 ln 𝑇 +0.004699418𝑇 − 9.36513 × 10 𝑇 + 22790040𝑇= 79866.236 − 46.25962𝑇 − 28.00029𝑇 ln 𝑇 + 4.8653845 ×
10 𝑇 − 6.238305 × 10 𝑇 − 96222900𝑇= 0.5 × GGASF2

Temperature,K
298.15 < 𝑇< 1357.77
1357.77 < 𝑇< 3200298.15 < 𝑇< 1357.77
1357.77 < 𝑇< 3200298.15 < 𝑇< 6000298.15 < 𝑇< 601
601 < 𝑇
< 12001200 < 𝑇< 5000298.15 < 𝑇< 601
601 < 𝑇
< 5000298.15 < 𝑇< 800
800 < 𝑇
< 25002500 < 𝑇< 68006800 < 𝑇< 10000

令和 4 年度

九州大学大学院総合理工学報告 第 42 巻 第 1 号

31

Table A3. Thermodynamic parameters for the binary and ternary systems.
System

Cu-Pb

Phase
and
model
L

FCC-int
BCC
Cu-F

L

Thermodynamic parameters, J/mol of model

𝐿
𝐿
𝐿
𝐿
𝐿
𝐿
°
𝐺
°
𝐺
°
𝐺

,
,
,
,
,
,
,

𝐺 − °𝐺
𝐿=0,

𝐿
FCC-int

Pb-F

CuF2
L

𝐿
𝐿
°
𝐺
°
𝐺

,
, :
, :
:

𝐺

PbF2

𝐿

PbF4
Cu-Pb-F,= 488000 + 30𝑇, :𝐿 , : = 55000 + 25𝑇
_
𝐺:
− °𝐺
− 2 °𝐺= −689907.6204 + 128𝑇 − 0.3𝑇 ln 𝑇

𝐺:
°
𝐺:

_

°

𝐺:

− °𝐺

𝐿

,

= −250000 − 20𝑇

𝐿,= 100𝑇

FCC-int

CuPb2F6=0°L

CuPbF6− 2 °𝐺 =− 689907.6204 + 128𝑇 − 0.3𝑇 ln 𝑇

=0

− °𝐺
− 2 °𝐺= −682907.6204 + 136.5𝑇 + 0.001𝑇 ln 𝑇
− °𝐺
− 3 °𝐺= −804786.388 + 208𝑇 − 0.02𝑇 ln 𝑇

°

PbF3

= 240000 + 10𝑇= −35000 + 25𝑇− °𝐺− 2 °𝐺= −429527.9934 + 135.55𝑇 − 0.1𝑇 ln 𝑇
− °𝐺 − 2 °𝐺 =− 540000 − 10.7𝑇,

𝐿
FCC-int

=0

− °𝐺

°

𝐿

− 4 °𝐺

𝐿,:= 32000 + 10𝑇

𝐿

,

:

=0

𝐺

: :

°

°

𝐺

298.15 < 𝑇< 6000=-4246.8+50T

°

: :

− °𝐺
− °𝐺

− °𝐺
− 2 °𝐺= −877665.1557 + 260𝑇 + 0.9𝑇 ln 𝑇

− 6 °𝐺− 6 °𝐺

Ref.

[30]= 31008 − 7.195𝑇= 15345 − 10.826𝑇= −6493 + 5.947𝑇= −18416 + 13.16𝑇= 28000 + 12𝑇:= 100000:= 45684 + 5.151𝑇:= 50000:− °𝐺 − 2 °𝐺 = −280000 − 30𝑇,

Temperature, K= −1312693.149 + 420𝑇 − 0.5𝑇 ln 𝑇
= −2060000 + 503.8𝑇 − 1𝑇 ln 𝑇

298.15 < 𝑇< 3000298.15 < 𝑇< 6000298.15 < 𝑇< 3000298.15 < 𝑇< 6000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000

This
work

This
work

298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000298.15 < 𝑇< 3000This
work