Study of some thermodynamic properties of ZrxCe1-xO2/CeO2 systems by statistical moment method

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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41 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