Đề tài The influence of temperature and pressure on oxygen atom diffusion in ZrO2

Available at:  IC/2007/061 
United Nations Educational, Scientific and Cultural Organization 
and 
International Atomic Energy Agency 
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS 
THE INFLUENCE OF TEMPERATURE AND PRESSURE 
ON OXYGEN ATOM DIFFUSION IN ZrO2 
Nguyen Thanh Hai1 
Hanoi University of Technology, 01 Dai Co Viet Road, Hanoi, Vietnam 
and 
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy 
 and 
Do Cung Trang 
Hanoi National Pedagogic University, Km 8, Hanoi-Sontay Highway, Hanoi, Vietnam. 
Abstract 
By using the statistical moment method (SMM) to investigate the diffusion oxygen 
atoms in ZrO2 with fluorite crystal structure, we have obtained the activation energy (Q), 
diffusion coefficient (D), pre-exponential factor (D0) of oxy in ZrO2 as analytic functions, all 
these functions depended on temperature T and pressure P. The present analytical formulas 
include the anharmonic effects of the lattice vibrations and dipole polarization. The obtained 
results are applied to calculate by using Buckingham potential energy as Schelling and Zacate 
models. 
MIRAMARE – TRIESTE 
July 2007 
1 Junior Associate of ICTP: hai@mail.hut.edu.vn 
 2
1. Introduction 
ZrO2 is applicable to industry due to its combination of high temperature stability and high 
strength. 
In fact, the defect and the diffusion in crystal, as well as temperature and pressure 
effects, play an important role in many properties of materials [1]. Thus an investigation of the 
effect of those factors on the thermodynamic properties would be interesting. There were 
various methods used to investigate the above problem. Many theories and pratices focus on 
this issue [2, 3]. However, most of them do not give analytical expressions and they are too 
complicated in numerical calculation. In practice, it is also difficult to determine 
thermodynamic properties of materials in high temperature and pressure [4]. 
Recently, in [5] it was shown that by using the statistical moment method, we can 
investigate the self-difussion in silicon under high pressure. The obtained results are in 
agreement with experiment. By applying the similar moment method in [5], the present 
authors will study the oxygen atoms in ZrO2 with fluorite crystal structure in the range 
temperature 2550K - 2950K and pressure 0 GPa - 20 GPa . 
2. Theory approach 
The theory of atomic diffusion in solids was presented for the first time in 1885, in which it 
showed that the point defects, like the vacancies and interstitials, play an important role in 
determining the atomic diffusions coefficient in crystals [6,7]. The general expression of 
diffusion coefficient D can be written in the form [8]: 
2rngD vΓ= , (2.1) 
where r is the jump distance at temperature T; g is a coefficient which depends on the 
crystalline structure and the mechanism of diffusion , as follows: 
 1g n f= . (2.2) 
Here f is the correlation factor, and n1 denotes the number oxygen atoms of the first nearest 
neighbour from the central atom. In the first approximation, we get 
1
2
1
21
n
1-1f
n
−≈⎟⎟⎠
⎞
⎜⎜⎝
⎛≈
. (2.3) 
It is well known that vacancy-type lattice defects in solids (metals, alloys, 
semiconductors or oxides) play an important role in the process of both self- and impurity- 
 3
diffusions. In order to study the diffusion coefficient in solids as in eq. (2.1) we need to know 
the equilibrium concentration of the vacancies nv. From the minimization condition of the 
Gibbs free energy of the crystal with lattice vacancy, we obtain the concentration of the 
vacancy as: 
 ( ),exp fvv g T Pn θ
⎧ ⎫−⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭
, (2.4) 
with Bk Tθ = , and the change in the Gibbs free energy fvg due to the formation of an impurity 
vacancy can be given by [1]: 
*
0
f O O
vg U P V= − + ∆Ψ + ∆ (2.5) 
in which LTKT VVV −=∆ is the change of the atomic volume of the lattice when creating a 
vacancy, P is pressure. 
0
OU represent the sum of effective pair interaction energies between the zeroth oxygen 
atom (the central atom) and ith oxygen atoms in ZrO2 and can be written in the form: 
( )0 012O Oi iiU rϕ= ∑
r
, (2.6) 
ir
r is the equilibrium position of the ith oxygen atoms at temperature T, and Oi0ϕ the effective 
interaction energies between the zeroth and ith atoms. *O∆Ψ denotes the change in the 
Helmholtz free energy of the central oxygen atom by moving itself to a certain sink creating a 
vacancy in the oxide. *O∆Ψ is given by: 
 ( ) ( ),* * * *1O O O OB∆Ψ = Ψ −Ψ = − Ψ , (2.7) 
where *OΨ denotes the Helmholtz free energy per single atom in the perfect crystal Zr2O. 
*OΨ has been given by [9]: 
 ( ){ } { }2* 2 2 2 10 22 23 13 ln 1 3 2O O x xcthxU x e x cth xk γθθ γ− +⎛ ⎞⎡ ⎤Ψ = + + − + − ⎜ ⎟⎣ ⎦ ⎝ ⎠ + (2.8) 
 ( ) ( )3 2 22 1 1 243 4 1 1( 2 2 1 }3 2 2xcthx xcthxxcthx xcthxkθ γ γ γ γ+ +⎧ ⎫ ⎛ ⎞+ − + +⎨ ⎬ ⎜ ⎟⎩ ⎭ ⎝ ⎠ 
 ( )1/ 2 2 22 2 22 1 112 3 3 4 223 3 . 16 6 3 9 9 6
a kak a x cthx
K K K K K K K k
β ββ β β γ βθ θγ γ
⎡ ⎤⎡ ⎤ ⎛ ⎞+ − + − + + −⎢ ⎥⎜ ⎟⎢ ⎥ ⎝ ⎠⎢ ⎥⎣ ⎦ ⎣ ⎦
 4
in which θ
ω
2
O
x h= , ( )2 2*0212
O
Oi
i i eq
k m
u α
ϕ ω⎛ ⎞∂= ≡⎜ ⎟∂⎝ ⎠∑ , eqi i
O
i
u∑ ⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂= 4 0
4
1 48
1
α
ϕγ , 
eqi ii
O
i
uu∑ ⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂∂
∂=
βα
ϕγ 22 0
4
2 48
6
, ∑ ⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂∂∂
∂=
i eqiii
O
i
uuu γβα
ϕβ 0
3
2
1
, γ
β
3
2
−= kK , and 
Oω denotes the oxygen atomic vibrational frequency. 
( ),*OΨ denotes the free energy of the oxygen atom, after it moves to certain sink sites in the 
crystal Zr2O and is given by [1]: 
 ( ),* *O OBΨ = Ψ víi * * *1 2 1 1 2 2*(1 )12 2
O O O
O
n n n nB ψ ψ ψψ
− + + + +≈ + − . 
When pressures are zero, then (2.5) can be rewritten as: 
 ( ) *0 1f O Ovg U B= − + − Ψ , (2.9) 
because the change in the Gibbs free energy fvg >0, the Helmholtz free energy 
*OΨ <0 and 
*OBΨ < *OΨ , so we can get: 0*1 1 OUB< < + Ψ , in the averaging value approximation, we 
obtained: 
0
*1 2
O
O
UB ≈ + Ψ , (2.10) 
since the Gibbs free energy, G, can be written in the form: 
 G = H- TS (2.11) 
where 
PT
G ⎟⎠
⎞⎜⎝
⎛
∂
∂−= S is entropy and H represents the enthalpy of a system, then the change in 
the Gibbs free energy, fvg , due to the creation of a vacancy, is given by 
( , ) ( , ) ( , )f f fv v vg P T h P T TS P T= − (2.12) 
where fvh and 
f
vS are enthalpy and entropy required for the formation of a vacancy, 
respectively. 
The change in the Gibbs free energy associated with the exchange of the vacancy with 
the neighbouring atoms is equal to the inverse sign of ( *1
O∆Ψ ), and 
 5
*
1
Om
vg ∆Ψ−= (2.13) 
From the definition of the activation energy Q is the sum of the changes in the free energy for 
the formation fvg and migration
m
vg of a vacancy, we have: 
 *1
f m f O
v v vQ g g g= + = −∆Ψ , (2.14) 
In eq. (2.1), the attempt frequency Γ for atomic jumps is proportional to the transition 
probability of an atom, and has the form: 
⎭⎬
⎫
⎩⎨
⎧∆Ψ=Γ θπ
ω *1exp
2
OO
, (2.15) 
where ( ) *1*1 1' OO B Ψ−=∆Ψ , by using a way similar as obtained B, with a note that 
m f
v vg g< , we get the condition of B: 
 1 < B’ < 1 + ( ) *0 *
1
1O O
O
U B− − Ψ
Ψ . (2.16) 
 or 
( ) *0
*
1
1
' 1
2
O O
O
U B
B
− − Ψ≈ + Ψ . (2.17) 
Replacing eqs. (2.2), (2.4), (2.12), (2.15), into eq. (2.1), we obtained the expression of the 
diffusion coefficient D of oxygen atom in crystal ZrO2 as follows: 
**
2 1
1 0exp exp exp2
f f f OOO
v v vg TS gD n f r Dψωπ θ θ θ
⎧ ⎫ ⎧ ⎫⎧ ⎫ − −∆Ψ∆= − = −⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭ ⎩ ⎭ (2.18) 
with 
2
0 1D exp2
fO
v
B
Sn f r
K
ω
π
⎧ ⎫= ⎨ ⎬⎩ ⎭ (2.19) 
D0 pre-exponential factor of oxy in ZrO2 , combining (2.14) and (2.18) then (2.18) can be 
rewritten as follows: 
 ⎭⎬
⎫
⎩⎨
⎧ −=
TK
Q
B
expD D 0 , (2.20) 
replacing (2.5), (2.12) into (2.14) then the activation energy Q can be found in the form: 
* *
0 1
O O O f
vQ U TS P V= − + ∆Ψ −∆Ψ + + ∆ . (2.21) 
 6
From eqs. (2.20) and (2.21) we know that to determine the activation energy (Q) and 
diffusion coefficient (D) we also need to obtain the expression of the entropy fvS required for 
the formation of a vacancy. From the definition of the entropy nfvS required for the formation 
of n vacancies 
( )0fn
v
P
G G
S
T
∂ −⎛ ⎞= −⎜ ⎟∂⎝ ⎠ , in which Gibbs free energy G of the crystal 
consists of N atoms and n vacancies at temperature T and pressures P, has the form: 
n
c
f
vvo TSTPgNnTPGTPG −+= ),(),(),( (2.22) 
Here 0G is the Gibbs free energy of perfect crystal containing N atoms,
n
cS is the entropy of 
mixing: 
( )
!!
!ln
nN
NnKS B
n
c
+=
 , (2.23) 
then we found: 
f
fn f nv v c
v v v c
P P
g n SS N n N g T S
T T T
⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − − + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
 (2.24) 
From the nearest neighbour approximation, the entropy fvS , required for the formation of a 
vacancy can be obtained as 
1 2 1 2 1 2
1 1 ln( 1)
( ) ( ) ( )
f
f v
vf
f v P B
v
BP
gg
Tg kS N
n n T n n K T n n
⎛ ⎞∂− ⎜ ⎟∂⎛ ⎞∂ ⎝ ⎠= − − + +⎜ ⎟+ ∂ + +⎝ ⎠ (2.25) 
3. Results and discussions 
By applying the moment method in statistical dynamics, we calculated and found that in 
oxygen atoms of ZrO2 (with the cubic fluorite structure), the activation energy Q and the 
diffusion coefficient D depend a lot on temperatures and pressures. The atomic interactions are 
described by a potential function which divides the forces into long-range interactions 
(described by Coulomb’s Law and summated by the Ewald method) and short-range 
interactions treated by a pairwise function of the Buckingham form: 
 7
 6)exp()( r
C
B
rA
r
qq
r ij
ij
ij
ji
ij −−+=ϕ (3.1) 
where iq and jq are the charges of ions i and j respectively, r is the distance between them 
and ijij BA , and ijC are the parameters particular to each ion-ion interaction. In eq.(34), the 
exponential term corresponds to an electron cloud overlap and the 6/ rCij term any attractive 
dispersion or Van der Waal’s force. Potential parameters ijij BA , and ijC have most commonly 
been derived by the procedure of ‘empirical fitting’, i.e., parameters are adjusted, usually by a 
least-squares fitting routine, so as to achieve the best possible agreement between calculated 
and experimental crystal properties. The potential parameters used in the present study are 
listed in Table 1. 
 Using the potential parameters in Table 1 and our theory in Section 2, we obtain the 
values of the activation energies Q , and the diffusion coefficient D of oxygen in ZrO2 with the 
cubic fluorite structure. The SMM results for oxygen atom diffusion in ZrO2 in the 
temperature range 2550 - 2950K and the presures 0 – 20 Gpa using Schelling and Zacate 
potentials are summarized in Table 2. 
The calculated results for the activation energies Q , and the diffusion coefficient D of 
oxygen in ZrO2 show that they depend on the temperature and pressures. The temperature and 
pressure dependences of the diffusion coefficient D are strong. For example, at the same 
pressure P = 0, if the temperature T = 2550K then D ≈ 1.819×10-5 cm2.s-1 and when T=2950K 
then D≈6.569.10-5 cm2.s-1, at the same temperature T=2550K if the pressure P=0 then 
D≈18.19.10-6 cm2.s-1 but the pressure P=20 Gpa then D≈0.589.10-6 cm2.s-1. In general, at the 
same pressure, the diffusion coefficient D decreases when the temperature increases. 
However, at the same temperature the diffusion coefficient increases as the pressure becomes 
lower. The reason is that the increasing of the diffusion coefficient obeys the exponential 
function. This corresponds to Arrhenius’s law. 
For the activation energies Q , the temperature and pressure dependences are not much. 
At the temperature T= 2550K and P = 0 Gpa then Q ≈ 2.583 eV, and at the temperature T = 
2950K and pressure P = 20 Gpa then Q ≈ 3.415 eV. In general, the activation energies Q 
decrease as the temperature becomes lower. The reason for the temperature dependences of Q 
is given as follows: the temperature dependences of the free energies per single atom *OΨ , 
 8
*
1
OΨ have negative temperature dependences, because of the entropy term ( fvTS− ) in 
expression of fvg . On the other hand, at the same temperatures, the activation energies Q 
decrease as the pressure becomes higher. 
We found that if using difference potential, at the same pressure and temperature, the 
activation energies Q and the diffusion coefficient D are a little bit different. The main reason 
is the difference of the exponential term A.exp (-r/B) to the attractive dispersion term 6/ rCij 
in the Schelling and Zacate potentials. 
In conclusion, measuring the activation energies and the diffusion coefficient of 
oxygen in ZrO2 with the fluorite crystal structure at the high pressure and temperature is 
difficult [4]. By using SMM to investigate the process of oxygen atom diffusion in ZrO2 via 
vacancy mechanism, we obtained all those values for ZrO2. In our calculations, the 
anharmonicity effect of the lattice vibration is considered, so the applicable is not limited at 
the high temperature range (even near the melting temperature). The results also indicate that 
the dipole polarization effects play an important role in determining the activation energies 
Q (P,T) and the diffusion coefficient D (P,T). The obtained analytical formulas permit to apply 
not only for ZrO2 with fluorite crystal structure but are also very good for CeO2 and similar 
materials. 
Acknowledgments. This work was done within the framework of the Associateship Scheme 
of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. 
 9
References 
[1] Vu Van Hung, Nguyen Thanh Hai and Nguyen Quang Bau, Journal of the Physical 
Society of Japan, Vol. 66, No 11, pp.3494-3498 (1997). 
[2] L.A. Girifalco, Statistical Physics of Material, Interscience Publ., New York, London, 
Sydney, Toronto (1973) . 
[3] Science and Technology of Zirconia, edited by A. Heuer and L.W. Hobbs, Advances in 
Ceramics, Vol.3 (The American Ceramic Society, Westerville, OH, 1981); Science and 
Technology of Zirconia II, edited by A. Heuer and L.W. Hobbs, Advances in Ceramics, 
Vol.12 (The American Ceramic Society, Westerville, OH, 1984). 
[4] M. Wilson, U. Shonberger and M.W. Finnis, Phys. Rev. B 54, 9147 (1996). 
[5] Vu Van Hung, Jaichan Lee, K. Masuda Jindo and Phan Thi Thanh Hong, Journal of the 
Physical Society of Japan, Vol. 75, No 2 (2006) 
[6] G. Allan and M. Lannon, J. Phys. Chem. Solids, 37, 699 (1976). 
[7] K. Masuda –Jindo, J. de. Physique, 47, 2087 (1986). 
[8] Vu Van Hung, Hoang Van Tich and K. Masuda Jindo, Journal of the Physical Society of 
Japan, Vol. 69, No 8, pp.2691-2699 (2000). 
[9] Vu Van Hung, Nguyen Thanh Hai and Le Thi Mai Thanh, Proc. of the 6th National 
Conference on Physics, pp.48 -51 (2006). 
[10] O.P. Mamley and S.A. Rice, Phys. Rev., 117, 632 (1960). 
 10
Table 1. Short range potential parameters in ZrO2 
Interaction A (ev) B (A0) C [ev.(A0)6] 
02-- 02- 
Zr4+- 02- 
8547.96 
1502.11 
0.224 
0.345 
32.00 
5.1 Schelling [10] 
02-- 02- 
Zr4+- 02 
9547.96 
1502.11 
0.224 
0.3477 
32.00 
5.1 Zacate [2] 
Table 2. Calculated results using Schelling and Zacate potential 
Schelling potential Zacate potential 
P(Gpa) T(K) 2550 2650 2750 2850 2950 2550 2650 2750 2850 2950
Q(ev) 2.45 2.47 2.50 2.52 2.54 2.35 2.37 2.39 2.41 2.43
D(10-5cm2/s) 3.19 4.44 6.03 8.01 10.43 5.22 7.13 9.52 12.46 16.00
0 
Log D (m2/s) -8.49 -8.35 -8.21 -8.09 -7.98 -8.28 -8.14 -8.02 -7.90 -7.79
Q(ev) 2.3 2.40 2.42 2.44 2.46 2.38 2.40 2.42 2.44 2.46
D(10-5cm2/s) 4.61 6.33 8.49 11.15 14.38 4.61 6.33 8.49 11.15 14.38
1 
Log D (m2/s) -8.33 -8.19 -8.07 -7.95 -7.84 -8.33 -8.19 -8.07 -7.95 -7.84
Q (ev) 2.51 2.53 2.55 2.57 2.59 2.40 2.42 2.45 2.47 2.57
D(10-5cm2/s) 2.48 3.48 4.76 6.38 8.38 4.07 5.62 7.57 9.99 12.92
2 
Log D (m2/s) -8.60 -8.45 -8.32 -8.19 -8.07 -8.38 -8.24 -8.12 -8.00 -7.88
Q (ev) 2.59 2.61 2.63 2.66 2.68 2.48 2.51 2.53 2.55 2.57
D(10-5cm2/s) 1.69 2.41 3.35 4.54 6.03 2.81 3.93 5.37 7.17 9.38
5 
Log D (m2/s) -8.77 -8.61 -8.47 -8.34 -8.21 -8.55 -8.40 -8.26 -8.14 -8.02
Q (ev) 2.73 2.75 2.77 2.79 2.86 2.62 2.64 2.66 2.68 2.70
D(10-5cm2/s) 0.90 1.31 2.86 2.58 3.49 1.51 2.17 3.03 4.12 5.50
10 
Log D (m2/s) -9.04 -8.88 -8.72 -8.58 -8.45 -8.81 -8.66 -851 -8.38 -8.25
Q (ev) 3.01 3.03 3.05 3.07 3.09 2.89 2.91 2.93 2.95 2.98
D(10-5cm2/s) 2.54 3.89 5.77 8.32 11.70 4.42 6.63 9.65 13.68 18.93
20 
Log D (m2/s) -9.59 -9.40 -9.23 -9.07 -8.93 -9.35 -9.17 -9.01 -8.86 -8.72