Abstract. Some thermodynamic quantities of ZrxCe1-xO2/CeO2 systems are investigated by
using the moment method in statistical dynamics taking into account the anharmonicity
effects of lattice vibrations. The analytic expression of the Gibbs free energy, Helmholtz free
energy, and specific heats at the constant volume of ZrxCe1-xO2/CeO2 systems are obtained.
The lattice parameter and specific heats at the constant volume of the ZrxCe1-xO2/CeO2
systems are calculated as functions of the temperature, pressure, concentration of Zr and the
thickness ratio d2/d1 of the ZrxCe1-xO2/CeO2 systems by using the Buckingham potential. The
effects of temperature (for the temperature range T = 0K ÷ 2900K), pressure (P = 0; 5; 10; 20;
30; and 40GPa), concentration of Zr (x = 2%; 4%; 6%; 8%), and the thickness ratio d2/d1 (d2/d1
= 1÷ 20) on the lattice parameter and specific heats at the constant volume of the ZrxCe1-
xO2/CeO2 systems will be discussed in details. The dopant dependence of the lattice constants
of Ce1−xZrxO2 at T = 300K and zero pressure, and the pressure dependence of the lattice
constants of CeO2 at T = 300K are compared with the available experimental and other
theoretical results.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0068
Natural Sciences 2018, Volume 63, Issue 11, pp. 41-51
This paper is available online at
STUDY OF SOME THERMODYNAMIC PROPERTIES OF ZrxCe1-xO2/CeO2 SYSTEMS
BY STATISTICAL MOMENT METHOD
Vu Van Hung
1
,
Le Thi Thanh Huong
2
and Dang Thanh Hai
3
1
University of Education, Vietnam National University, Hanoi
2
Hai Phong University,
3
Vietnam Education Publishing House
Abstract. Some thermodynamic quantities of ZrxCe1-xO2/CeO2 systems are investigated by
using the moment method in statistical dynamics taking into account the anharmonicity
effects of lattice vibrations. The analytic expression of the Gibbs free energy, Helmholtz free
energy, and specific heats at the constant volume of ZrxCe1-xO2/CeO2 systems are obtained.
The lattice parameter and specific heats at the constant volume of the ZrxCe1-xO2/CeO2
systems are calculated as functions of the temperature, pressure, concentration of Zr and the
thickness ratio d2/d1 of the ZrxCe1-xO2/CeO2 systems by using the Buckingham potential. The
effects of temperature (for the temperature range T = 0K ÷ 2900K), pressure (P = 0; 5; 10; 20;
30; and 40GPa), concentration of Zr (x = 2%; 4%; 6%; 8%), and the thickness ratio d2/d1 (d2/d1
= 1÷ 20) on the lattice parameter and specific heats at the constant volume of the ZrxCe1-
xO2/CeO2 systems will be discussed in details. The dopant dependence of the lattice constants
of Ce1−xZrxO2 at T = 300K and zero pressure, and the pressure dependence of the lattice
constants of CeO2 at T = 300K are compared with the available experimental and other
theoretical results.
Keywords: Statistical moment method, high pressure, oxide superlattice.
1. Introduction
Study of the thermodynamic properties of semiconductor and oxide superlattices has become
quite interesting in recent years since semiconductor and oxide super lattices play an important
role in technological applications, especially in the electronic industry. The structural and thermo-
mechanical properties of semiconductor superlattices were studied by using the lattice dynamics
approach [1] and ab initio molecular dynamic simulations [2]. The structural characteristics of
undoped CeO2/ZrO2 superlattices were investigated by Wang et al. [3]. This study revealed
several interesting properties of the layered film structure beyond their improved ionic conduction.
When the individual layer thickness in superlattices consisting of alternating layers of ceria and
zirconia doped with gadolinia (Gd2O3) was reduced below approximately 50 nanometers, the ionic
conduction of this superlattice began to increase greatly with respect to either bulk material [4].
The increase in conduction observed in the ceria/zirconia superlattice could prove to be a
breakthrough in this regard.
Received September 7, 2018. Revised October 15, 2018. Accepted October 22, 2018.
Contact Dang Thanh Hai, e-mail address: dthai@nxbgd.vn
Vu Van Hung,
Le Thi Thanh Huong and Dang Thanh Hai
42
It is remarkable that despite the widespread use of CeO2/ZrO2 and doped ceria/zirconia
superlattices like as ZrxCe1-xO2/CeO2 in oxygen sensors, and other related applications for several
decades, the studies of thermodynamic quantities of doped ceria/zirconia superlattice are not fully
understood in these materials and continue to be an important focus of experimental and
theoretical studies.
The purpose of present article is to study effects of the temperature, pressure and dopant
concentration on the thermodynamic properties of the ZrxCe1-xO2/CeO2 superlattice using the
statistical moment method (SMM) [5-6]. The lattice parameter and specific heats at the constant
volume of ZrxCe1-xO2/CeO2 superlattice are calculated as functions of the temperature, pressure
and doped concentration by using the Buckingham potential.
2. Content
2.1. Theory
The ZrxCe1-xO2/CeO2 oxide superlattice consists of ZrxCe1-xO2 and CeO2 layers, where both the
ceria (CeO2) and doped ceria (ZrxCe1-xO2) layers are assumed to be in the cubic fluorite structure,
consistent with experimental observations [3]. In the fluorite structure, cations are located in the
face-centered cubic positions, with eight oxygen ions in the positions.
Figure 1. Structure of ZrxCe1-xO2/CeO2 superlattices
In the ZrxCe1-xO2/CeO2 present oxide superlattice with very thin layers and small lattice
mismatch, the difference in lattice parameter between the layers can be completely compensated
for by strain. We assume that the thickness of ZrxCe1-xO2 layer is d1 and this oxide layer consists
of N1 atoms with NZr atoms of Zr, NCe atoms of Ce and NO atoms of O (as shown in Fig.1) then
1 ,Zr Ce ON N N N (1)
1
,
/ 3
Zr Zr
Zr Ce
N N
x
N N N
1 1 1 12, (1 ), .
3 3 3 3
Zr Ce Zr O
xN N N N
N N N x N
(2)
Similarly, CeO2 ceria layer is supposed to have the thickness d2 and consist of N2 atoms with
*
CeN atoms of Ce and
*
ON atoms of O then
2CeO1 2x xZr Ce O
1d 2d
Study of some thermodynamic properties of ZrxCe1-xO2/CeO2 systems by statistical moment method
43
* *
2 .Ce ON N N (3)
So
1 2( ) ,N N N n (4)
where N is the number of atoms and n is the period of the ZrxCe1-xO2/CeO2 oxide superlattice.
In order to investigate the thermodynamic properties of the ZrxCe1-xO2/CeO2 oxide
superlattice, we firstly consider the change of Gibbs free energy of ZrxCe1-xO2 system when NZr
atoms of Ce are replaced by Zr atoms in CeO2 crystal. The substitution of an atom Ce by an atom
Zr causes the change of the free Gibbs energy fZrg
as
0 ,
f Ce
Zr Zrg u (5)
where Zr is the free energy of an atom Zr in the ZrxCe1-xO2 system, and 0
Ceu
is the cohensive
energy associated with atom Ce of the cubic-fluorite CeO2.
Because of the ZrxCe1-xO2 system is supposed to be built by substituting NZr atoms Zr into the
positions of Ce atoms of cubic-fluorite CeO2 system then the Gibbs free energy of system has an
approximate form as
1
2
,
d f
Zr Zr cCeOG N g TS
1
2
1 ,
3
d f
Zr cCeO
xN
G G g TS
(6)
where cS is the configuration entropy of ZrxCe1-xO2 system, and
1
2
d
CeOG is the Gibbs free energy of
CeO2 system with 1N of Ce and O atoms 1( ).Ce ON N N
From Eqs. (5), and (6), we obtain the Gibbs free energy of ZrxCe1-xO2 system
1
2 0
( ) ,
d Ce
Zr Zr cCeOG G N u TS
1
2
1 1
0 ,
3 3
d Ce
Zr cCeO
xN xN
G G u TS
(7)
where 1
2
d
CeOG is the Gibbs free energy of the CeO2 system with 1N of Ce and O atoms
1 1
2 2 1
,
d d
CeO CeOG PV (8)
and P denotes the hydrostatic pressure, V1 is the volume of the ZrxCe1-xO2 system.
Because of CeO2 ceria layer of oxide superlattice is supposed to have the thickness d2 and
consist of N2 atoms with
*
CeN atoms of Ce and
*
ON atoms of O, then the Gibbs free energy is
given by
2 2
2 2 2
.
d d
CeO CeOG PV (9)
From Eqs. (7), (8) and (9), it is easy obtain the Gibbs free energy of the ZrxCe1-xO2/CeO2
oxide superlattice
2
2
sup ( ),
d
CeOG n G G
Vu Van Hung,
Le Thi Thanh Huong and Dang Thanh Hai
44
1 2
2 2
sup 1 1
1 0 2 .
3 3
d dCe
Zr cCeO CeO
xN xN
G n PV u TS PV
(10)
The Helmholtz free energy 1
2
,
d
CeO
2
2
,
d
CeO of ZrxCe1-xO2 and CeO2 layers is then written by
taking into account the configurational entropies 1 ,
d
cS and
2d
cS via the Boltzmann relation as [5,6]
1 1
2 1 1
,
d d
Ce Ce O O cCeO C N C N TS
1 1
2
1 12 ,
3 3
d d
Ce O cCeO
N N
TS
(11)
2 2
2 2 2
,
d d
Ce Ce O O cCeO C N C N TS
2 2
2
2 22 ,
3 3
d d
Ce O cCeO
N N
TS
(12)
where ,Ce O are the total Helmholtz free energies of an Ce and O atoms, CeC , OC denote
concentrations of Ce, O atoms ( CeC = , ), respectively. Using SMM the quasi-harmonic
contributions to the free energies of Ce, and O
atoms in the CeO2 crystal are treated as [5]
2
0 0 0
1
(| |) 3 ln(1 ) ,
2
CexCe Ce Ce
Ce i i Cei
u r x e
(13)
2
0 0 0
1
(| |) 3 ln(1 ) .
2
OxO O O
O i i Oi
u r x e
(14)
In Eqs. (13), (14) ,Rx Ox are given by
/
,
2 2
CeCe
Ce
k m
x
2
0
2
1
,
2
Ce
i
Ce i
i eq
k
u
(15)
2
20
2
1
,
2
O
i
O Oi
i eq
k m
u
/
.
2 2
OO
O
k m
x
(16)
where x, y, or z, and 0
Ce
i (or 0 )
O
i is the interaction potential between the 0-th and the
i-th Ce (or O) atoms, and ,iu iu are , -Cartesian components of the displacement of i-th
ion,
and Ce (or )O is the vibration frequency of Ce (or O) atoms, and Bk T (kB - the Boltzmann
constant), and m is the average atomic mass of the system, .Ce Ce Zr Zr O Om C m C m C m
Using Eqs. (1), (2), (3), (4), (10), (11), (12), (13) and (14) we obtain the Gibbs free energy of
ZrxCe1-xO2/CeO2 oxide superlattice
sup *0
2
,
3 3
Ce
Ce O Zr Zr c
N N
G N u PV nTS
(17)
2 2sup 0 023 ln 1 3 ln 1
3 3
Ce Ox xCe O
Ce O
N N
G u x e u x e
Study of some thermodynamic properties of ZrxCe1-xO2/CeO2 systems by statistical moment method
45
2 *0 0
2 2
1 1
3 ln 1 .
3 3
1 1
ZrxCe Zr
Zr c
N x N x
u u x e PV nTS
d d
d d
2sup 0 0 0
2 2
1 1
2
1 ln 1
3 3 3
1 1
CexCe O Zr
Ce
N x N N x
G u u u N x e
d d
d d
2 *
2
1
ln 1 ,
1
Zrx
Zr c
N x
x e PV nTS
d
d
(18)
where 1 2 ,V n V V
1 2* .
d d
c c c cS S s S
(19)
In order to calculate the average nearest-neighbor distance (NND) between two intermediate
atoms a(P,T) in ZrxCe1-xO2/CeO2 oxide superlattice at temperature T and various pressure P, we
can use the minimum condition of the Gibbs free energy of ZrxCe1-xO2/CeO2.
sup
0.
G
a
(20)
From Eqs. (18) and (20), we obtain the following equation
0 0 01 2
3 3(1 ) 3 3(1 )
Ce O ZrSM u u uG x x
N
a X a a X a
coth coth coth
2
2 2 (1 ) 2
Ce Ce Ce O O O Zr Zr Zr
Ce O Zr
x x k x x k x x kx
N
k a k a X k a
,
V
p
a
(21)
2 1with / .X d d
By numerically solving Eq.(21), one can determine the average nearest-neighbor distance
(NND) between two intermediate atoms a(P,T) at temperature T and pressure P of ZrxCe1-xO2/CeO2.
In the case of zero pressure (P = 0), using Eq. (17) we can find the Helmholtz free energy of
ZrxCe1-xO2/CeO2
sup *
0
2
( ) ,
3 3
Ce
Ce O Zr Zr c
N N
N u nTS
*
0
2
1
2
( ) ,
3 3 3
1
Ce
Ce O Zr c
N N N x
u nTS
d
d
(22)
Vu Van Hung,
Le Thi Thanh Huong and Dang Thanh Hai
46
2
*
0
2
1
( ) ,
3
1
Ce
CeO Zr c
N x
N u nTS
d
d
(23)
where
1 2
2
2
( ).
3 3
d d
CeO Ce O c c
N N
nT S S
(24)
Applying the Gibbs-Helmholtz relation, E
, and using Eq.(23), we obtain the
expression for the energy of ZrxCe1-xO2/CeO2 oxide superlattice
2
sup 0
0
2 2
1 1
1
,
3 3
1 1
Ce
Ce
CeO Zr
ux N x
E E E u
d d
d d
(25)
2
2
2 ,
CeO
CeO
T
E NT
T
(26)
2 .
Zr
Zr
T
E NT
T
(27)
Furthermore, the specific heat at constant volume CV of ZrxCe1-xO2/CeO2 oxide superlattice is
determined as
sup sup
,V B
E E
C k
2
0
0
2 2
1 1
1
.
2 3
1 1
Ce
Ce
CeO Zr B
V V V
u
u
k Nx x
C C C
d d
d d
(28)
2.2. Results and discussion
To calculate the thermodynamic quantities of ZrxCe1-xO2/CeO2, we will use the Buckingham
potential [7]
6
exp( / ) ,
i j ij
ij ij ij
q q C
A r B
r r
(29)
where qi and qj are the charges of the i-th and the j-th ions, r is the distance between them and the
parameters Aij, Bij and Cij are empirically determined by Ref.[8-11]. The potential parameters of
ZrxCe1-xO2/CeO2 systems are given in Table 1.
Study of some thermodynamic properties of ZrxCe1-xO2/CeO2 systems by statistical moment method
47
Table 1. The potential parameters Aij, Bij and Cij of ZrxCe1-xO2/CeO2 systems
Interaction A(eV) B(Å) C(eV.Å6)
O
2-
- O
2-
Ce
4+
- O
2-
Zr
4+
- O
2-
9547.92
1809.68
1502.11
0.2192
0.3547
0.3477
32.00
20.40
5.10
Potential 1
O
2-
- O
2-
Ce
4+
- O
2-
Zr
4+
- O
2-
9547.92
2531.50
1502.11
0.2192
0.3350
0.345
32.00
20.40
5.10
Potential 2
O
2-
- O
2-
Ce
4+
- O
2-
Zr
4+
- O
2-
22764.3
1986.83
985.87
0.1490
0.3511
0.3760
27.89
20.40
0.00
Butler
In the case of the zero thickness d2 2( 0),d using Eq.(21) one can determine the average
nearest-neighbor distance (NND) between two intermediate atoms a(P,T) of ZrxCe1-xO2 layer. The
lattice constants of the ZrxCe1-xO2 system with the different dopant concentrations at temperature T =
300K and zero pressure are presented in Fig.2. One can see that the lattice constant of ZrxCe1-xO2
decreases with the increasing dopant concentration. The variation of the lattice parameter of ZrxCe1-
xO2 as a function of Zr concentration in the present work by SMM (empty square), and our SMM
results are in good agreement with the results of empirical equations [12], MD simulation [13], and
experiments [14 -17].
Figure 2. The dopant dependence of the lattice constants of the cubic ﬂuorite Ce1−xZrxO2 system
at T = 300K and zero pressure. The dark yellow circles are the results from Emprirical equation
[12], the pink triangles with line and symbol are the results from the molecular dynamics (MD)
simulations [13], red triangle, purple dimonds, blue stars and olive pentagons are the
experimental results [14-17].
In Figures 3 and 4, we present the lattice parameter of the ZrxCe1-xO2/CeO2
systems as
functions of thickness ratio d2/d1 at the different composition Zr (x = 2%; 4%; 6%; 8%) at room
temperature, pressure P = 5GPa, and T = 900K, and P = 15GPa, respectively. As it can be seen
from these two figures, the lattice parameters of the ZrxCe1-xO2/CeO2
systems are the increasing
functions of thickness ratio d2/d1. Furthermore, when the thickness ratio d2/d1 of the ZrxCe1-xO2/CeO2
systems increases to 15, the superlattice constants have the same value as ceria's lattice constant.
Vu Van Hung,
Le Thi Thanh Huong and Dang Thanh Hai
48
Figure 3. Thickness dependence of lattice
constants of ZrxCe1-xO2/CeO2
with Zr different
concentrations (x = 2%; 4%; 6%, and 8%) at
room temperature and pressure P = 5GPa
using potential 2.
Figure 4. Thickness dependence of lattice
constants of ZrxCe1-xO2/CeO2
with Zr different
concentrations (x = 2%; 4%; 6%, and 8%) at T
= 900K, and P = 15GPa using potential 2.
Figure 5. Pressure dependence of lattice
constants of ZrxCe1-xO2/CeO2
system with
thickness d2 = d1, and Zr concentration x =
2% at various temperature (T = 0K; 300K;
600K; 900K; 1200K; 1500K; 1800K; 2300K;
2600K and 2900K) using potential 2
Figure 6. Pressure dependence of lattice
constants of ZrxCe1-xO2/CeO2
system with
thickness d2 = 20d1, and Zr concentration x =
2% at room temperature using potentials 1,2
and Butler potential and experimental results
of CeO2 [18]
The pressure dependence of the lattice constants of the ZrxCe1-xO2/CeO2
system with Zr
concentration x = 2% and thickness d2 = d1, and d2 = 20d1 at various temperature and room
temperature are presented in Figs.5 and 6, respectively. One can see that the lattice constant of
ZrxCe1-xO2/CeO2
system decreases with the increasing pressure. In the case of ZrxCe1-xO2/CeO2
Study of some thermodynamic properties of ZrxCe1-xO2/CeO2 systems by statistical moment method
49
system with a thickness ratio d2/d1 of 20 and a small dopant concentration (x = 2%), the
thermodynamic properties of the ZrxCe1-xO2/CeO2 system are similar to those of the ceria (CeO2
layers with thickness d2). The variation of the lattice parameter of ZrxCe1-xO2/CeO2 as a function of
pressure in the present work by SMM calculations using potentials 1, 2 and Butler potential (as
presented in Figs. 6 and 7), and our SMM results are in good agreement with the experiments [18]
and the ab initio calculations [19]. One can see in Fig. 6 that the lattice parameters calculated by
using potentials 1 and 2 are very similar. The small difference between the two calculations simply
comes from the difference in cerium-oxygen interaction potentials, since the ionic Coulomb
contribution and the oxygen-oxygen potential are the same for potentials 1 and 2.
In Fig.8 we show the temperature dependence of the lattice constants of ZrxCe1-xO2/CeO2
system with thickness d2 = d1, and Zr concentration x = 2% at various pressures calculated by using
potential 2 for the temperature range T = 0K - 2900K. The lattice constant of ZrxCe1-xO2/CeO2
system increased smoothly with an increase of temperature due to the thermal expansion.
Figure 7. Pressure dependence of lattice
constants of ZrxCe1-xO2/CeO2
system with
thickness d2 = 20d1, and Zr concentration x =
2% at room temperature using potential 2 and
experimental results [18] and ab initio
calculations of CeO2 [19]
Figure 8. Temperature dependence of
lattice constants of ZrxCe1-xO2/CeO2
system with thickness d2 = d1, and Zr
concentration x = 2% at various pressures
(P = 0; 5; 10; 20; 30, and 40GPa) using
potential 2
The thickness ratio dependence of the specific heats at constant volume CV of the ZrxCe1-xO2/CeO2
system with Zr concentration x = 2% at zero pressure and room temperature are presented in Fig.9.
As it can be seen from this figure, the specific heats at constant volume CV of the ZrxCe1-xO2/CeO2
systems are the increasing functions of thickness ratio d2/d1. But the value of the specific heats at
constant volume CV of the ZrxCe1-xO2/CeO2 system with different thickness ratios of d2/d1 at the
same temperature does not differ much.
In Fig.10, we show the temperature dependence of the specific heats at constant volume CV
of ZrxCe1-xO2/CeO2 system with thickness d2 = d1, and composition of Zr (x = 0.2) at zero pressure.
The calculated specific heats CV increase briskly with temperature in the temperature range below
750K. As also shown in Fig.10, the specific heat CV depends weakly on the temperature for the
temperature region from 750K up to the melting temperature. These properties of ZrxCe1-xO2/CeO2
system with thickness ratio d2/d1 = 1, and a small dopant concentration, are similar to those of
ceria (CeO2 layers with thickness d2).
Vu Van Hung,
Le Thi Thanh Huong and Dang Thanh Hai
50
Figure 9. Thickness ratio dependence of the
specific heats at constant volume CV of ZrxCe1-
xO2/CeO2 system with composition of Zr (x =
0.2) at zero pressure and room temperature
calculated using potential 2 (P2)
Figure 10. Temperature dependence of the
specific heats at constant volume CV of
ZrxCe1-xO2/CeO2
system with thickness d2 =
d1, and composition of Zr (x = 0.2) at zero
pressure calculated using potentials 1, 2 (P1,
P2) and Butler potential (B)
3. Conclusions
In conclusion, the SMM calculations have been performed to investi