ic2006063 united nations educational scientific and cultural organization and international atomic energy agency the abdus salam international centre for theoretical physics

The calculated thermodynamic quantities of the cubic zirconia are in good agreement with the experimental results as well as those by ab initio calculations (in some cases, better res[r]

Available at: http://www.ictp.it/ pub_off IC/2006/063

United Nations Educational Scientific and Cultural Organization

and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

CALCULATION OF THERMODYNAMIC QUANTITIES OF CUBIC ZIRCONIA

BY STATISTICAL MOMENT METHOD

Vu Van Hung

Hanoi National Pedagogic University, Km8 Hanoi- Sontay Highway, Hanoi,Vietnam,

Hanoi University of Technology, 01 Dai Co Viet Road, Hanoi, Vietnam

and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

Le Thi Mai Thanh

Hanoi National Pedagogic University, Km8 Hanoi- Sontay Highway, Hanoi,Vietnam

Abstract

We have investigated the thermodynamic properties of the cubic zirconia ZrO2 using the statistical moment method in statistical physics The free energy, thermal lattice expansion coefficient, specific heats at the constant volume and those at the constant pressure are derived in closed analytic forms The present analytical formulas, including the anharmonic effects of the lattice vibrations, give accurate values of the thermodynamic quantities, which are comparable to

those of the ab initio calculations and experimental values The calculated results are in

agreement with experimental findings The thermodynamic quantities of cubic zirconia are predicted by using two different inter-atomic potential models The influence of dipole polarization effects on the thermodynamic properties for cubic zirconia is studied

MIRAMARE – TRIESTE

July 2006

1 Junior Associate of ICTP hai@mail.hut.edu.vn

1 Introduction

Zirconia (ZrO ) with a fluorite crystal structure is a typical oxygen ion conductor In order to 2

understand the ionic conduction inZrO , a careful study of the local behavior of oxygen ions 2

close to the vacancy and the thermodynamic properties of zirconia is necessary ZrO is an 2

important industrial ceramic combining high temperature stability and high strength [1] Zirconia

is also interesting as a structural material: It can form cubic, tetragonal and monoclinic or orthorhombic phases at high pressure Pure zirconia undergoes two crystallographic transformations between room temperature and its melting point: monoclinic to tetragonal at

T≈1443K and tetragonal to cubic at T ~2570K The wide range of applications (for use as an oxygen sensor, technical application and basic research), particularly those at high temperature, makes the derivation of an atomistic model especially important because experimental measurements of material properties at high temperatures are difficult to perform and are susceptible to errors caused by the extreme environment [2] In order to understand the properties of zirconia and predict them, there is a need for an atomic scale simulation Molecular dynamics (MD) has recently been applied to the study of oxide ion diffusion in zirconia systems [3-5] and the effect of grain boundaries on the oxide ion conductivity of zirconia ceramic [6] Such a model of atomic scale simulation requires a reliable model for the energy and interatomic forces First principles, or ab initio calculations give the most reliable information about properties, but they are only possible for very simple structures involving a few atoms per unit

experimental information is available at high temperatures (for example, in the case of zirconia,

> 12000C [7] ) In this respect, therefore, the ab initioand experimental data can be considered

as complementary Recently, it has been widely recognized that the thermal lattice vibrations play an important role in determining the properties of materials It is of great importance to take into account the anharmonic effects of lattice vibrations in the computations of the thermodynamic quantities of zirconia So far, most of the theoretical calculations of thermodynamic quantities of zirconia have been done on the basis of harmonic or quasi- harmonic (QH) theories of lattice vibrations, and anharmonic effects have been neglected

The purpose of the present study is to apply the statistical moment method (SMM) in the quantum statistical mechanics to calculate the thermodynamic properties and the Debye-Waller factor of cubic zirconia within the fourth-order moment approximation The thermodynamic quantities such as free energy, specific heats C and V C , and bulk modulus are calculated taking P

into account the anharmonic effects of the lattice vibrations We compare the calculated results

with the previous theoretical calculations as well as experimental results In the present study, the influence of dipole polarization effects on the thermodynamic properties, are studied We compare the dependence of the results on the choice of interatomic potential models

2 Method of calculations

2.1 Anharmonicity of lattice vibration

First, we derive the expression of the displacement of an atom Zr or O in zirconia, using the moment method in statistical dynamics

The basic equations for obtaining thermodynamic quantities of the crystalline materials are derived in the following manner We consider a quantum system, which is influenced by supplemental forces a i in the space of the generalized coordinates Q The Hamiltonian of the i

lattice system is given as

i

i

i Q a H

H = 0 −∑ (1) where H denotes the Hamiltonian of the crystal without forces0 a After the action of the i

supplemental forces a , the system passes into a new equilibrium state From the statistical i

Specifically, we use a recurrence formula [8-10]

a n

m n m

m m n

a n a

n a n a n

a

K i

m

B a

K Q

K K

1

) 2 ( 2

0 2 1

1 1

)!

2

∞

− +

+ +

∂

∂

⎟

⎠

⎞

⎜

⎝

⎛

−

∂

∂ +

where θ =k B T and K is the correlation operator of the nth order n

= − [[ , ]+ ]+ ]+ ]+

2

1

3 2 1

n

K (3)

In eq (2), the symbol expresses the thermal averaging over the equilibrium ensemble, a H

represents the Hamiltonian, and B2m denotes the Bernoulli numbers

The general formula (eq.(2)) enables us to get all the moments of the system and investigate the nonlinear thermodynamic properties of the materials, taking into account the anharmonicity effects of the thermal lattice vibration In the present study, we apply this formula to find the Helmholtz free energy of zirconia (ZrO ) 2

First, let consider the system zirconia composed of N atoms Zr and 1 N atoms O, we 2

assume the potential energy of system can be written as:

2 ) (

2

2

=

i

i i O io i

i i Zr

N

O Zr

Zr U C U

where U0 ,U0 represent the sum of effective pair interaction energies between the zeroth Zr

and ith atoms, and the zeroth O and ith atoms in zirconia, respectively In eq.(4), r is the i

equilibrium position of the ith atom, u its displacement, and i ϕio Zr,ϕio O,the effective interaction

energies between the zeroth Zr and ith atoms, and the zeroth O and ith atoms, respectively We

N

N C

N

N

, =

First of all let us consider the displacement of atoms Zr in zirconia In the fourth-order

approximation of the atomic displacements, the potential energy between the zeroth Zr and ith

atoms of the system is written as:

24

1

6

1 2

1 ) ( )

(

, , ,

4

, ,

3

, 2

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂

∂

∂ +

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂

∂ +

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂ +

= +

∑

∑

∑

η γ β α η

γ β

α α β γ η

γ γ

β

α α β γ α β β

α α β α β

ϕ

ϕ ϕ

ϕ ϕ

i i i i eq i i i i

Zr io

i i i eq i i i

Zr io i

i eq i i

Zr io i

Zr io i i

Zr

io

u u u u u u u u

u u u u u u u

u u u r

u r

(5)

In eq (5), the subscript eqmeans the quantities calculated at the equilibrium state

The atomic force acting on a central zeroth atom Zr can be evaluated by taking derivatives

of the interatomic potentials If the zeroth central atom Zr in the lattice is affected by a

supplementary force aβ, then the total force acting on it must be zero, and one can obtain the

relation

0 12

1

4

1 2

1

,

,

4

, ,

3

,

2

=

−

>

<

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂

∂

∂ +

>

<

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂

∂ +

>

<

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂

∑

∑

∑

β η γ α η

γ

α α β γ η

γ γ

α α β γ α

α α β α

ϕ

ϕ ϕ

a u

u u u

u u u

u u u

u u

u u

u

i i i i

eq i i i i

Zr io

i i

i eq i i i

Zr io i

i eq i i

Zr io

(6)

The thermal averages on the atomic displacements (called second- and third-order moments)

>

<u iαu iγ and <u iαu iγu iη > can be expressed in terms of <u iα > with the aid of eq.(2) Thus,

eq.(6) is transformed into the form:

2

k ky y da

dy y da

y

with β ≠γ =x,y,z and y≡<u i >

eq

Zr io

u

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

= 2 2 2

1

α

Zr

m ω

θ

ω

2

Zr

x= h (8)

∑ ⎥⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂ +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

=

eq i i

Zr io eq

i

Zr io

u u

4 4

4

6 12

1

γ β α

ϕ ϕ

In deriving eq (7), we have assumed the symmetry property for the atomic displacements in

the cubic lattice:

<u iα >= <u iγ >= <u iη > ≡ <u i > (10)

Equation (7) has the form of a nonlinear differential equations, since the external force a is

arbitrary and small, one can find the approximate solution in the form:

2 1

0 A a A a y

y= + + (11) here, y is the displacement in the case of no external force a Hence, one can get the solution 0

of y as: 0

A

k

2 2

0

3

2γθ

≈ (12)

In an analogical way, as for finding eq.(7), for the atoms O in zirconia ZrO , the equation for 2

the displacement of a central zeroth atom O has the form:

2

da

dy y

x x k

ky da

dy y da

y

with u i a ≡ ; y

θ

ω

2

O

x= h

2 2

1

O

i i eq

O io

m u

α

≡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

= ∑ (14)

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂ +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

=

i

eq i i

O io eq

i

O io

u u

4 4

4

6 12

1

γ β α

ϕ ϕ

i

eq i i i

O io

u u

( 2

1 3

γ β α

ϕ

β (16) Hence, one can get the solution of y of atom O in zirconia as 0

27

2 ) 1 coth ( 3 3

1 )[

6 1 ( 1 3 3

2 4

2 2 3

2 0

k x

x k K

K

A K

y

γ

β γθ

θ

γ γ

β

where the parameter K has the form:

γ

β

3

2

−

= k

K (18)

2.2 Helmholtz free energy of Zirconia

N

N C

N

N

, =

average atoms of m* =C Zr m Zr +C O m O

The free energy of zirconia are then obtained by taking into account the configurationally entropies S , via the Boltzmann relation, and are written as: c

ψ =C ZrψZr +C OψO −TS c (19) where ψZr and ψO denote the free energy of atoms Zr and O in zirconia , respectively Once the thermal expansion y of atoms Zr or O in the lattice zirconia is found, one can get the 0

Helmholtz free energy of the system in the following form:

ψZr =U0Zr +ψ0Zr +ψ1Zr (20) where ψ0Zr denotes the free energy in the harmonic approximation and ψ1Zr the anharmonicity contribution to the free energy [ 11-13] We calculate the anharmonicity contribution to the free energy ψ1Zr by applying the general formula

ψZr =U Zr +ψZr +λ∫<V >λdλ

0 0

0 ˆ (21)

where Vˆλ represents the Hamiltonian corresponding to the anharmonicity contribution It is straightforward to evaluate the following integrals analytically

=∫1< >

0

1 4 1

γ

γ

d u

0

2 0 2 2 2

γ

γ dγ

u

I i (22)

Then the free energy of the system is given by:

⎭

⎬

⎫

⎩

⎨

⎧

⎟

⎠

⎞

⎜

⎝

⎛ +

− +

− + +

≈

2

coth 1

3

2 coth

3 )]

1 ln(

[

2 2

2 2

0

x x x

x k

N e

x N

Zr

γ γ

θ

⎭

⎬

⎫

⎩

⎨

2

coth 1

)(

2 ( 2 ) 2

coth 1

( coth 3

4 3

2 1 2 1 2

2 4

3

x x x x x

x x x k

where U0Zr represents the sum of effective pair interaction energies between zeroth Zr and ith

atoms The first term of eq.(23) gives the harmonicity contribution of thermal lattice vibrations and the other terms give the anharmonicity contribution of thermal lattice vibrations The fourth-order vibrational constants γ1,γ2 are defined by:

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

=

i i eq

Zr io

u4

4 1

48

1

α

ϕ

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

∂

=

i

eq i i

Zr io

u

4 2

48

6

β α

ϕ

In an analogical way, as for finding eq.(23), the free energy of atoms O in the zirconia ZrO2

is given as:

⎭

⎬

⎫

⎩

⎨

⎧

⎟

⎠

⎞

⎜

⎝

⎛ +

− +

− + +

≈

2

coth 1

3

2 coth

3 )]

1 ln(

[

2 2

2 2

0

x x x

x k

N e

x N

O

γ γ

θ

⎭

⎬

⎫

⎩

⎨

2

coth 1

)(

2 ( 2 ) 2

coth 1

( coth 3

4 3

2 1 2 1 2

2 4

3

x x x x x

x x x k

6 9

9

) 3

2 ( [ 3 ] 6 6

[

2 4

1 2 3 1 2 2 / 1 1 3 2

2 2

2

2

− +

+

− +

−

k K K

ka K

a a

K K

N K

K

k

γ

β γ

β

Note that the parameters γ1,γ2 in eq.(25) have the form analogous to eq.(24), but we must to replace ϕio Zr,the effective interaction energies between the zeroth Zr and ith atoms, by ϕio O

quantities of zirconia The specific heats at constant volume C , V Zr C V O are directly derived from the free energy of system ψZr,ψO (23), (25), respectively, and then the specific heat at constant volume of the cubic zirconia is given as:

V O Zr V Zr

V C C C C

C = + (26)

We assume that the average nearest-neighbor distance of the cubic zirconia at temperature T can

be written as:

r1(T)=r1(0)+C Zr y0Zr +C O y0O (27)

in which y0Zr(T) and y0O(T)are the atomic displacements of Zr and O atoms from the equilibrium position in the fluorite lattice, and r1(0) is the distance r1 at zero temperature In eq.(27) above, y0Zr and y O0 are determined from Eqs (12) and (17), respectively The average

nearest-neighbor distance at T = 0K can be determined from experiment or the minimum condition of the potential energy of the system cubic zirconia composed of N1 atoms Zr and

2

1 0 1

0

U r

U r

∂

∂ +

∂

∂

=

∂

∂

0 ) ( 2

) (

2 1

⎠

⎞

⎜

⎝

⎛

∂

∂ +

⎟

⎠

⎞

⎜

⎝

⎛

∂

∂

i

i O io i

i Zr

r

N r

r

From the definition of the linear thermal expansion coefficient, it is easy to derive the result

αT =C ZrαT Zr +C OαT O (29) where

θ

α

∂

∂

Zr T

y r

∂

O T

y r

1(0) (30) The bulk modulus of the cubic zirconia is derived from the free energy of Eq (19) as:

T T

V

P V

⎠

⎞

⎜

⎝

⎛

∂

∂

−

T O Zr T Zr T

B C B C V

⎠

⎞

⎜⎜

⎝

⎛

∂

Ψ

∂

−

where P denotes the pressure, V0 is the lattice volume of the cubic zirconia crystal at zero

temperature, and the bulk moduli B T Ce and B T O are given by

=− α ⎜⎜⎝⎛∂∂VΨ∂θ ⎟⎟⎠⎞

k

Zr T

B Zr

T

2

3 , =− α ⎜⎜⎝⎛∂∂VΨ∂θ ⎟⎟⎠⎞

k

O T

B O

T

2

3 (32) Due to the anharmonicity, the heat capacity at a constant pressure,C P, is different from the heat

capacity at a constant volume,C V The relation between C P and C V of the cubic zirconia is

T P

V P

V

P T

V T C

⎠

⎞

⎜

⎝

⎛

∂

∂

⎟

⎠

⎞

⎜

⎝

⎛

∂

∂

−

=

2

VT B

C V +9αT2 T

= (33)

3 Results and discussions

3.1 Potential dependence of thermodynamic quantities

Using the moment method in statistical dynamics, we calculated the thermodynamic

properties of zirconia with the cubic fluorite structure 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

r

C B

r A

r

q q

ij ij

j i

where q i and q j are the charges of ions i and j respectively, ris the distance between them and

ij

ij B

exponential term corresponds to an electron cloud overlap and the C ij / r6 term any attractive

dispersion or Van der Waal’s force Potential parametersA ij,B ij and C ij 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 were taken

from Lewis and Catlow [14] and from Ref [28]

The potential parameters are listed in Table 1 Table 2 compares the zero K lattice

parameter predicted by ab initio calculations with previous calculations and two experimental

values The experimental values are derived from the high temperature neutron scattering data

[7] and to zero impurity in the cubic stabilized structure [19] We summarized here the results of

different ab initio calculations and compared them to experiment It is noted that the ab initio

calculations of lattice parameters were at zero K, but present results by SMM at temperatures T =

0 K and T = 2600 K, while experimental data at high temperatures (> 1500 K) [7]) The full-potential linearized augmented-plane-wave (FLAPW) ab inition calculation of Jansen [16], based

on the density functional theory in the local-density approximation (LDA), gives a0(A0)= 5.03, while Hartree-Fock calculations (the CRYSTAL code) give a0(A0)= 5.035 (both at zero K)

The linear muffin-tin orbital (LMTO) ab initio calculations of lattice parameter are larger than both experimental values and are in best agreement with the Hartree-Fock calculation [17] The potential-induced breathing model [18] (PIB) augments the effective pair potential (EPP) by allowing for the spherical relaxation (“breathing”) of the oxide anion charge density, gives

)

( 0

0 A

theory (DFT) within the plane-wave pseudopotential (PWP) [22] and RIP give a0(A o)=5.134,

and a0(A o)=5.162 These results and the CRYSTAL calculation [15] are larger than the experimental values Our SMM calculations give a lattice parameter a=5.0615(A0)and unit

values [7], FLAPW-DFT, LMTO and Hartree-Fock calculations

Table 3 lists the thermodynamic quantities of cubic fluorite zirconia calculated by the present SMM using potential 1 The experimental nearest-neighbor anion-anion separations r2O−O lie in the range 2.581−2.985A0[21], while the current SMM give 2.5931 A0 (without dipole polarization effects) and 2.6031A0 (with dipole polarization effects) at T = 2600 K, and in

agreement with the ab initio calculations [2] These calculations [2] used a potential fitted to ab

initio calculations using the oxide anion electron density appropriate to the equilibrium lattice parameter (2.581A0) as the fluorite analogue for all nearest-neighbor pairs The nearest-neighbor cation-anion separations r1Zr−O calculated by SMM lie in the range 2.2543-2.2669A0 (with

first-principles calculations give 2.236 0

A in cubic zirconia [23] We also calculated the bulk modulus

T

B of cubic zirconia as a function of the temperature T We have found that the bulk modulusB T

depends strongly on the temperature and is a decreasing function of T The decrease of B T with increasing temperature arises from the thermal lattice expansion and the effects of the vibration

O

neglected, gives the bulk modulus B T =201 GPa( ) The CIM calculations of the short-range

anion-cation interactions with and without full dipolar and quadrupolar polarization effects, gives

) (

194 GPa

polarization) for the fluorite structure are equal and greater than the experimental values while the SMM results of bulk modulus at high temperature (T = 2600 K ) are smaller than the experimental ones At lower temperatures the SMM calculations of bulk modulus give a much better agreement with experiment, because bulk modulus are the decreasing functions of the temperature Above about 2570 K (up to the melting point at 2980 K), zirconia assumes the

cubic fluorite structure In this phase the thermodynamic quantities (as the lattice parameter a ,

specific heats at constant volume and pressureC V,C P, and the bulk modulusB T, ) are

quantities of cubic fluorite zirconia calculated by the present SMM using potential 2 Tables 3, 4

show the thermodynamic quantities, a , C V,C P and B T, for the cubic phase of bulk zirconia as the functions of the temperature T Tables 3, 4 present the variations in temperature of the specific heats at constant volume and pressure C V,C P, in which specific heat C V depends slightly on the temperature, but the specific heat C P depends strongly on T and the linear thermal expansion coefficientα and bulk modulus B T depend strongly on the temperature The

i.e.α =10.5.10−6K−1 That experimental value is also close to the value calculated in the present study using potential 1 for the cubic phase of bulk zirconia at the temperature T = 2600 K For the specific heat capacity C P of the cubic zirconia, the reference data reported by Chase [26] gives C P ~ 640 J/ (kg.K) at T ~ 1400 K, while the current SMM using potential 1 gives C P = 9.4316 cal/(mol.K) (with dipole polarization effects) and C P = 8.8674 cal/(mol.K) (without dipole polarization effects) at T = 2600 K The lattice specific heats C V and C P at constant volume and at constant pressure are calculated using Eqs (26) and (33), respectively However, the evaluations by eqs (26) and (33) are the lattice contributions, and we do not include the contributions of lattice vacancies and electronic parts of the specific heats C V The calculated values of the lattice specific heats C V and C P, by the present SMM, may not be compared with the corresponding experimental values at high temperature region ( from T = 2600 K to the melting temperature) directly But the dependence of C P at temperature is in agreement with the experimental results