The relation between macroscopic thermodynamic quantities and the thermodynamic arameters of defects formation in non-stoichiometric ZrO2

J. Phys.Chem. Solids, 1972,Vol. 33, pp. 1273-1278. PergamonPress. Printedin Great Britain

THE RELATION BETWEEN MACROSCOPIC THERMODYiNAMIC QUANTITIES A N D THE T H E R M O D Y N A M I C PARAMETERS OF DEFECTS FORMATION IN NON-STOICHIOMETRIC ZrO2 J. P. LOUP* and A. M. ANTHONYt (Received 30 June 1971 ; in revised f o r m 26 N o v e m b e r 1971 ) A b s t r a c t - T h e thermodynamic properties of nonstoichiometric zirconia were investigated on the basis of results of thermionic emission and conductivity experiments. These measurements established the pressure (Po~)s corresponding to stoichiometric zirconia in the 1270-2000~ temperature range, by using a microscopic model of an oxide exhibiting a slight deviation from stoichiometry log (P,n)s (atm) = ---~-~- + 7-8. The authors specify the relationship between the microscopic model involving structural elements. and the macroscopic model relating to chemical elements. Calculations are made of the variations in chemical potential of zirconium (A/xzr)s and monatomic oxygen (A~0s during the formation of the stoichiometric oxide. It was deduced that the slope of the tangent to the curve of the free enthalpy mixing of a grammeatom Agm(Xo) in the Z r - O system is positive for the stoichiometric composition:

(

dAgm]

dX0 ], -~ 5"2 eV.

The dissymmetry of this curve, revealing the tendency of zirconia to exhibit an excess of metal rather than an excess of oxygen, is in agreement with a theory, by Kroger, concerning transition metal oxide in a state of maximum oxidation. The determination of the oxygen pressure at stoichiometry leads to the derivation of various basic thermodynamic quantities pertaining to the nonstoichiometry of oxides. 1. INTRODUCTION

behaviour of a nonstoichiometric compound is essentially characterized by its free enthalpy of mixing, which varies with the deviation from stoichiometry, and with temperature. In the case of zirconia, the availability of experimental data concerning some samples of well defined compositions permits the satisfactory calculation, through interpolation, of this thermodynamic quantity. These compositions correspond either to the limit of

THE

THERMODYNAMIC

*Facult6 des Sciences de D A K A R , Senrgal. t C e n t r e de Recherches sur la Physique des Hautes Temprratures C.N.R.S.-45 Orleans 02, France.

the zirconia-zirconium phase, or to a region of near-stoichiometry. In the latter case, 'microscopic' analysis helps to interpret the experimental results. 2. MACROSCOPIC AND MICROSCOPIC DESCRIPTION OF AN OXIDE M O I + x A T HIGH TEMPERATURE

In the 'macroscopic' form of description, the oxide is defined by the ratio of oxygen and metal atoms, without specifying atomic defects. The free enthalpy of mixing Ag,,, per gramme-atom is given by the relation: A g , . = X M A /~ g -1- XoA/.L0

1273

(1)

1274

J . P . LOUP and A. M. ANTHONY

where XM, X0 = atomic fractions of elements PM, P0 = partial pressures AtzM = R T In PM

A/z0 = R T In P0The standard state, which provides the basis for defining Ag,,,, is formed, for each element, by isolated atoms[l]. Note that the expression (1) derives directly from the thermodynamic equilibrium of the solid and the gaseous phase. Following KrSger, the authors carried out a 'microscopic' analysis ,by introducing the concept of structural elements (groups consisting of an atom, or an atomic defect, and a lattice site)[2]. This is done assuming that the oxide MO exhibits a Frenkel disorder of 0. M,~(: element M in a normal site, with zero effective charge in relation to the periodic potential of the lattice. V02: oxygen vacancy doubly ionized, having the effective charge of 2 +. V~: intertitial site with zero effective charge. 02': intertitial oxygen that has trapped two electrons, having the effective charge of 2 --. The thermodynamic equilibrium of the solid with the gaseous phase is expressed by employing the concept of virtual thermodynamic potential: MM x + Oo x + otVI x = M(o) +

0~o);

~M.- + ~:oo-+ a ~ v . . = tZM.,)+ P-O.,) O0x = Oc,~+ V02"+ 2 e ; ~:0: = / ~

In ~-~

~:t: virtual thermodynamic potential of the element i. N*: number of sites in the sub-lattice M ( N * = N,11,,. = N ) or in the intertitial sub-lattice (N* = a N ) N,: number of elements of type i. In order to complete this description, the fotlowing relationships must be considered: - w i t h the conservation of sites (at high temperatures it is assumed that oxygen vacancies are mainly ionized): N = Noo. +Nvo,.; a N = Nv,z +No,,,

(6)

-- with electroneutrality: Nn + 2Nv~. = N e + 2No,2,

(7)

Note that we have ignored the coulomb interaction between atomic defects (Vo2-, Ot v) and electron charge carriers; this is permissible at high temperatures (T > 1300~ K), provided that the treatment is restricted to nearly stoichiometric compositions, where the concentrations of these charged species are low. It is possible to consider various approximations of the electroneutrality condition, depending on the pressure. This is derived (2)~ from relations (2-7), as follows:

+ ~:ro,.+2~e

(3) Oto) + Vt ~ = Oi 2' + 2h; IZo
where e: an electron in the conduction band h: a vacancy in the valence band ~tg~ = tz~ ~+ k T lnp~

/-~tg~: real chemical potential of the gaseous element i. [/zt~ standard chemical potential.

~ = ~i~ + k T

(4)

P02 l o w : 2Nv,,. = Ne; N e ~176

Po2 high: 2No~, = Nh; N h oo po+m The validity of these simple variation rules of electron concentration as a function of oxygen pressure has been clearly observed in conduction and emission[4,6,14]. The pressure (Po2)s corresponding to the stoichiometric oxide (Ne = Nh) can be deduced from laws of this type. For the interpretation of conduction measurements, electron mobilities

NON-STOICHIOMETRIC ZrO2 are assumed to be of the same order of magnitude. It is interesting to observe the relationships between this microscopic model and the classic macroscopic one. The free enthalpy G of the crystal is expressed i~n terms of the microscopic model as: G = N.U. scM,,.+ No.-~o,,- + N ~;.~:,,.

1275

thermionic emission measurements of zirconia as a function of oxygen pressure [3-6, 14], the microscopic analysis, outlined in Section 2, may be employed to interpret these results, the concentration of defects being low. This enables us to relate the pressure (Po,)s at which the oxide is stoichiometric to thermodynamic parameter of defect formation. As shown in Fig. I:

(8)

+ N,,,,'~:o,,'+ Nv.'-~v.'. + Ne~e + Nh~h.

3 10 +4

Log (Po2)s (atm) = - - ~ +

7-8;

By introducing the atomic fractions X0 and

XM, we can calculate the free enthalpy per gramme-atom: g,,, = (G..,1/'/N.~. + No,.+ No,,,) Noo. + No;. Xo = NM,,. + No.~ + No,,

(9)

1270~

< T < 2000~

We now derive (A/z0)s at stoichiometry may be calculated from table l by the relation:

Ala~=89189176 XM = 1--Xo

x

(15)

(10)

vV': Avogadro number. From relations (2-10), g,, is derived as:

g,,, = XMizM+ XoP.o+

(14)

W N,,,,. NM,,. + No,. + No,;

where AGo~ standard free enthaipy of dissociation of the oxygen molecule. The result as given in Table 1 Expression (1) enables us to calculate (A/Zzr)s since: Ag,,, - ] [ A G z0 r o - A G o0~ - A G s ]

(16)

[(~:,,,., + ~:v.,.) - (~:,,.- + ~:,,-)]

,A/'N~

-t N,v,,.+No,.+No,,. (~:e~-~h)

(11)

where: V-M= Xp.,~.u). The above relation leads to relation (1), taking into account the intrinsic equilibrium of the creation of defects: O0 x + VI x ~_. i7 2. _L. /'12'. 9 o - ,-.,. ~o..~+~v,.

=

~v..,

+~o,.. (12) (13)

for oxygen: Ah0 = -- 130 kcaig-l; As0 --~ -- 32 eu

to apply these

for zirconium: Ahzr = -- 265 kcalg-l; Aszr "=- -- 40 eu

O = e+h; O = ~e+ ~h. We can now proceed considerations to zirconia. 3. T H E R M O D Y N A M I C

where AG~ standard free enthalpy of formation of zirconia at temperature T, and AGs: free enthalpy of sublimation (tables of constants [7]) of zirconi urn. Knowing the variation in chemical potential Ap~ of the elements of ZrO2 with respect to temperature, we can calculate the variation in enthalpy Ahi and entropy As~ of these elements during the formation of the stoichiometric oxide.

STUDY OF ZIRCONIA

Having at our disposal conductivity and

(17)

The values obtained for variations in entropy are due mainly to the entropies of gaseous translation:

J. P. L O U P and A. M. A N T H O N Y

1276 , log {P02)S

i2

e Conductivity (5), (6)

10~ 1-o1<"

I

I

I

)

Fig. 1. Variation in the pressure (Po~)s corresponding to stoichiometric zirconium dioxide.

sotr(18OO~ = 43 eu st~(1800~

= 48 eu

(18)

Extrapolation of the tangent to the curve Agm(Xo) leads to the corresponding values of A/to and A/xzr (Fig. 2), whereas at each point

dAgm

dXo = A/zo-- A/xzr

(19)

Using the values from table 1, we obtain at stoichiometry,

(dAgm~ \ ~ 0 / ~ ' s = 5 eV.

(20)

Table 1. Free mixture enthalpy and chemical potentials of oxygen and zirconium of stoichiometric zirconium oxide T~

1273 1600 1700 1800 1900 2000

(Ag,,)s (Ap~0)s (Agzr)s (kcal/at.g) (kcal]at.g) (kcal/at.g) --129 --117 --114 --111 --107 --103

--88 --77 --74 --71 --68 --64,5

--211 --197 --193 --190 --185 --180

(dAg,,/dXo)s (kcal/at.g) +123 +120 +119 +119 +117 +115,5

It should be noted that Krtiger[1] made theoretical calculations of \(dAgm~ dXo/s for Fe~Oa and Co203, obtaining values of 4.7 and 8 e V respectively. These values are comparable to our experimentally obtained results. The slope is sharply positive because the oxygen potential must be much higher than that of zirconium to obtain a stoichiometric compound. In other words, zirconia shows a tendency to exhibit an excess of metal rather than an excess of oxygen. This conclusion has been confirmed by experimental data obtained through techniques other than electron emission[13,6] (Table 2), and is in agreement with Lorentz's rule[8] and Kroger's theory[I] concerning transition metal oxides in states of maximum oxidation. We have drawn the curve Agm (Xo) in Fig. 2 for the temperature of 1273~ the shape of the curve being corroborated by the experimental data mentioned above. The pressure of 10-35atm at which zirconia has the composition ZrOl.a3 is known because it corresponds to the limit Of the two-phase zirconia-zirconium region (Table 2). The flat portion of the curve corresponds to the wide

NON-STOICHIOMETRIC

/

~

h gm ( k.cal/at.g. ) ZrC"l'ZrOl,a3

ZrO2

1277

O,8

X0

!

/Z r01,83/ I

eO=l atm /

\ I/zr02

1 at m

-50. sS

s~

/

-100"S~

--

/

____

P02

= 10 - 3 5

Fig. 2. T h e change in free mixture enthalpy Ag,~ of the system Z r - O at 1273~

region of nonstoichiometric zirconia resulting from a lack of oxygen, while the steep incline of the curve reflects the great difficulty of introducing excess oxygen into the zirconia. It should be noted that as the temperature (dAg.,~ rises, ~ ] s gradually decreases (Table 1), corresponding to a widening of the curve Agm(Xo) and predictably, to an increase in the nonstoichiometric region of the oxide[9]. We will attempt to specify the reasons for which the emission and conductivity isotherms of zirconia shift towards higher pressures with increasing temperature. With the help of formula (15) and (17), it develops that: ASo~ Ah0~ (Po~)s = exp -- --R-- exp R---T-

(21 )

w ere Aho~= 2 A ho + 119 kcal - -- 140 kcal/ mol.g Since Aho2 is negative, the pressure (Po2)s rises with temperature. This displacement of the conductivity isotherm has been observed for many oxides: ZrO215], CaO, TiO2, Z n O [10], HfO2[11], Y_~O3[12]. In order to exemplify the structural disorder of zirconia, we previously suggested a

Table 2. Composition o f zirconium oxide as a function of oxygen pressure at 1273~ Composition Po~.(atm) Ref.

ZrOl .~ 10-35 13

Zr O,_, 10 -zn 6

ZrO._,..0.; 1 6

disorder of the anti-Frenkel[4] type (oxygen vacancy-interstitial oxygen). Thermodynamic data now enable us to describe the mechanism of the occurrence of these atomic defects in nearly stoichiometric compositions. This is most probably due to the passage of oxygen back and forth between solid and gaseous phases, on account of its chemical potential being much higher than that of zirconium. 4. CONCLUSION

We have shown that physical methods of measurement, such as thermionic emission and electrical conductivity, interpreted by means of microscopic analysis of nearly stoichiometric compositions, lead to the derivation of some important thermodynamic quantities pertaining to the nonstoichiometry of oxides. The complementary aspects of microscopic

I218

J . P . LOUP and A. M. ANTHONY

(structure elements) and macroscopic (chemical elements) description have been confirmed in the case of the oxide under study. It is shown that the macroscopic aspect is highly significant, as it enables us to break loose from reflections on the exact nature of atomic defects and their mutual interactions, to arrive at more fundamental thermodynamic criteria.

REFERENCES

1. KROGER F., J. Phys. Chem. Solids 29, p. 1889 (I 968). 2. KROGER F., The Chemistry o f lmperfect Crystals, North-Holland, Amsterdam (I 964). 3. LOUP J. P. and ANTHONY A. M., C.R. Acad. Sci. Paris, T 268, p. 772, Serie C (1969).

4. LOUP J. P. and ANTHONY A. M., Phys. Status Solidi38, 499 (1970). 5. VEST R. and T A L L A N N., J. Am. Ceram. Soc. 47, 472 (1965). 6. VEST R., T A L L A N N. and TRIPP W., J. Am. Ceram. Soc. 47, 635 (1964). 7. KUBASCHEWSKI O., EVANS E. and ALCOCK C., Metallurgical Thermochemistry, Pergamon Press, Oxford (1967). 8. LORENZ M.,Phys. Lett. 12, 161 (1964). 9. ANDERSON J., High Temperature Technology, London, Butterworths, p. 285 (1964). I0. RUDOLPH J.,Z. Naturforsch 14a, p. 727 (1959). 11. TALLAN N., TRIPP W. and VEST R., J. Am. Ceram. Soc. 50, 6, p. 279 (1966). 12. TALLAN N. and VEST R., J. Am. Ceram. Soc. 49, 8,401 (1966). 13. KOMAREK K. L. and SILVER M., Thermodynamics of Nuclear Materials, p. 749, International Atomic Energy Agency, Vienna (1962). 14. LOUP J. P. and ODIER P. H., Rev. Int. Hautes Temp. Refract. 7, 378 (1970).