Oxygen-ion diffusivity in hypostoichiometric zirconium dioxide

JOURNAL

OF NUCLEAR

27

MATERIALS

OXYGEN-ION

(1968)

DIFFUSIVITY

12-20.

Division,

Denver

Research

on

volume

calculations

diffusion

to evaluate

the diffusion

in zirconium knowledge and

of

metal. D=2.88 of

value

dioxide.

oxygen The

coefficients

These

diffusion

results

coefficients

conform

to

in

the

range

agreement

zirconium

zirconium of

with

600-850 those

x 1O-4 exp (-28,40O/RT)

donnant

les coefficients de zirconium

obtained

by

it la diffusion

dans des syst&mes bi-phas&,

il est possible

les coefficients

&cessitent

la connaissance

de la pellicule de l’oxygkne

1.

dans la zircone

d’oxyde

& la fois

dans le zirconium

avec celles obtenues

Es ist miiglich,

die Diffusionskoeffizienten Zirkondioxid

in

zienten

Ces calculs

von dioxid

de diffusion

bei einem

in Zirkonium in sich

und 850 ‘C: D=2.88

mhtallique.

Dieser

Introduction

Wert

miissen

als such

Sauerstoff ergibt

zu

von Berechnungen

Berechnungen

Oxidschicht

des Bpaisseurs

et des coefficients

dans

dans le

de 600 & 850 “C.

stoffionen

diese

de diffusion

ZrOz.

de l’oxyghne

oxide “C. This

en volume unidirectionnel d’kvaluer

de diffusion

cmz/sec

dans des Etudes anthrieures.

Verwendung

oxygene

USA

sous-stoechiom&rique

de temperature

in einer Richtung

pour l’ion

50210,

B la relation:

Cette valeur est en bon accord

studies.

En se basant sur des calculs applicables

Colorado

satisfont

l’oxyde domaine

relationship

cm2/sec for diffusion

hypostoichiometric

temperature

in

the

the scale

DIOXIDE

1967

D=2.88

ions

require

Denver,

Les r&ultats

it is possible

of the oxide

ZIRCONIUM

of Denver,

11 December

for oxygen

CO., AMSTERDAM

C. HAGEL

University

unidirectional

calculations

x 1O-4 exp (-28.400/RT)

is in good

earlier

to

systems

of both the thicknesses

oxygen

within

applicable

in two-phase

and W.

Institute,

Received

Based

PUBLISHING

IN HYPOSTOICHIOMETRIC

C. J. ROSA Metallurgy

0 NORTH-HOLLAND

stimmt

die

fiir Sauer-

bestimmen,

Zweiphasensystem. sowohl

die

Fiir

Dicke

der

Sauerstoff-Diffusionskoeffi-

bekannt

sein. Fiir die Diffusion

unterstiichiometrischem fiir

unter

der Volumendiffusion

Temperaturen

x 1O-4 exp (-28,40O/RT)

gut iiberein

Zirkon-

zwischen

600

cm2/sec.

mit Literaturdaten.

Methods commonly used to determine oxygen diffusion in metal oxides can be summarized as follows : (a) by IsO exchange and subsequent mass spectrographic analyses; (b) manometric

between the gas and the solid phases may not be sufficiently rapid to account for the constant concentration of 180 at the solid surface. Furthermore, there always exists an uncertainty of nonhomogeneity of the oxide particles used.

analysis of oxygen adsorption by oxygen deficient oxide particles; (c) electrical conductivity measurements; (d) interface migration involving two different phases (e.g. metal/oxide) or one phase but with a distinctive property change within that phase (e.g. color change within the oxide phase during oxidation) ; (e) proton activation analysis combined with autoradiography. Only methods (a) and (e) may be considered as direct experiments, all other methods are indirect. Each of these techniques has its limitations. For example, the theoretical interpretation of the exchange method is difficult. The exchange coefficient for oxygen

The purpose of this investigation is to evaluate the diffusion coefficients of oxygen ions in growing oxides. Accordingly, a mathematical analysis, based on Fick’s second law, is derived and its applicability is tested by the use of experimental results. During the past decade an enormous amount of results has been accumulated for the Zr-02 system and, therefore, oxidation of zirconium is an excellent test case for such calculations. In the present treatment, the rate of oxide growth and the knowledge about oxygen diffusion in the base metal are required in order to calculate the diffusivity of oxygen in the oxide. The tempera12

OXYGEN-ION

13

DIFFUSIVITY

ture range of 600 to 850 “C has been purposely

the

chosen because it is anticipated that the predominant factor controlling the oxidation

ments is still not very high and hence accurate

kinetics at these temperatures Before

the activation

the

applying

analyses

to

prior

and

mathematical

work

should

Lehr 1) applied

be

dominance

cited.

kinetics

energy

for oxygen

diffusion

in

of grain-boundary

diffusion at lower

treatment as reported by Crank a) 2.

in both ol-zirconium

and zirconium

Experimental Pure

oxide. They found a change in the activation energy for oxygen diffusion in the metal at about 650 “C, but no such change was observed for oxygen diffusion in zirconium dioxide for the temperature range of 400-850 “C. Their calculations were based on theoretical partitioning 3) of the total amount of oxygen, obtained from oxidation kinetics, between ol-zirconium and its oxide phase. Unfortunately,

99.93

wt

%,

zirconium

D

60

0

v x l

0

850 oc 0OOT

75ooc 7oooc 6OOOC

2

4

6

8

TIME, 1.

Zirconium

oxide

thickness

6;

samples

of

approximately 2 x 1 x 0.35 (cm) were exposed to oxygen at four temperatures: 700, 750, 800 and 850 “C. Before exposing the specimens to oxygen at a pressure of 400 Torr, they were subjected to wet abrasion on 200- through 600-grit Sic papers and subsequently polished with 8-, 6-, and l-,um diamond pastes. High purity, better than 99.9 vol %, oxygen was passed through a cold trap and columns

70

Fig.

measure-

temperatures.

Danckwerts’

and were able to estimate the oxygen-diffusion coefficients

oxidation

zirconium metal exhibits a moderate decrease below about 650 “C, and attribute this to the

of or-zirconium as a specific example,

following

Debuigne

the theoretical

of

determinations of oxygen diffusivities are difficult. Beranger and Lacombe 4) report that

is simple volume

diffusion. oxidation

precision

IO

12

14

16

hrs”‘-+

as function

of time at different temperatures.

14 containing

C.

anhydrous

J.

ROSA

CaSO4 and P205 in

order

thick-

followed the same pattern 'JLBLE

Zirconium

oxide

film

Time

thicknesses

1thickness

L

10.0 * 21.0 f

times

Hussey

and

Debuigne

2.0

23.0 5

20.0 & 0.6

36

24.0 & 2.0

(1) (P.I.)

30.4 *

(P.I.)

Present

96

40.0

(5) (P.I.)

1.6

Wallwork

117

44.5 _L 1.6

et al.

174

54.8

(6)

59.0

(6)

250

65.0

(6)

18

: 55

24

: 45

0.6

11.9 & 1.0

(1) (P.I.)

13.0 & 1.0

(P.I.)

17.0 + 0.4 1.0

(1) (P.I.)

21.0 -& 1.0

(P.I.)

16.0 i

48 78

(P.I.)

9.3 & 1.0 10.9 *

28.0 i

0.6

(1)

16

10.5 & 1.4

(P.I.)

36

14.3 & 1.4

(P.I.)

49

15.5 & 1.4

(P.I.)

81

19.4 & 1.4

(!?.I.)

144

25.0 & 1.2

(P.I.)

2

: 45

4.5

6 16 :25

8.4 *

36

125

0.8

10.2 _I 1.4 13.5

81 6 100

Gulbransen

7.81

, ,

2.76

(6)

(P.I.)

200

: 25

(P.I.)

(5)

1.6

34.8 *

(1)

(P.I.)

2.0

72

(5)

Lehr

(5) investigation

24

33.0 & 2.0

temperatures

Smeltzer and

24

48

and

Reference

20.6 & 2.0

22

-

oxidation

18

12

600 “C

1

(pm) 2.0

polishing on 0.05~,um

Measurements

of gray oxide

thicknesses were made by projecting the polished sections on a metallographic viewing screen.

19.9 & 1.0

6

700 “C

vibratory

16

6

750 “C

HAGEL

with additional

for various

15.0 & 0.6

1 48

800 “C

C.

Oxide

(hl

~ 6 I I 166

850°C

W.

y-Al203 solution.

to remove the residual water-vapor. Polishing of cross sections for oxide nesses determinations

AND

(1) and (1) (P.I.) (P.I.)

8.3 *

1.0

(7) (5)

9.5 *

1.0

(5)

150

10.1 & 1.0

(5)

186

10.7 & 1.0

(5)

209

11.9 & 1.0

(5)

259

12.7 & 1.0

(5)

Andrew

(7)

OXYGEN-ION

DIFFUSIVITY

No attempt was made to obtain independent data at 600 “C. For this temperature, reported

cation

the results

15

of volume

oxidation

analyses for longer

times may be justified.

by Hussey and Smeltzer 5) have been

utilized

because

material

these authors

used the same

and similar experimental

as those applied

techniques

in this investigation,

4.

Theory Consider the zirconium-oxygen

and

the

associated

during oxidation 3.

diffusion

Results Oxygen-deficient

zirconium

oxide

is

dark

diffusion

phase diagram couple

formed

of zirconium at temperature

T

below the a-/l transformation

as shown in fig. 2.

For a semi-infinite

and for the case

medium

gray, whereas ZrOz of virtually ideal composition

of concentration-independent diffusion coefficient D, the appropriate solution s) of Pick’s

is white.

second

Averages

of

ten

measurements

of

‘gray-oxide thicknesses formed on or-zirconium during oxidation for various times and temperatures are summarized in table 1 and plotted in fig. 1. The errors in thickness measurements are also based on ten measurements and represent the calculated average deviations from t#he mean values. The present results have been supplemented by data from other investigators. It is evident that, to a good approximation, the rate of oxide growth obeys a parabolic relationship after an initial period of more rapid growth; thus, the appli-

,

I

law is

c~= kl - (/cl -C:)

erf (x/Zvm)

at x> E,

(1)

for the metal phase (I), and CII = kz -

{(kz -

CII(CQ)}

erf

(x/2Vht)

at O
(2)

for the oxide phase (II). In these equations CI and cn represent the oxygen concentrations within the metal and the oxide as functions of distance x into the medium. kl and kg are values of the concentrations at the oxygen/oxide interface and DI

100% o2

I _____i___________~~~~~z___ I

PO,

_____-------

I____=1:-:I.:_ Q tzro2_x

-----+

I

I i i i i Fig.

2.

Schematic

Q T

TEMPERATURE

zirconium-oxygen zirconium

equilibrium oxidation

x/a; phase diagram at

temperature

x = DISTANCE

and associated

T

and

time

diffusion t>O.

couple formed during

16

C.

J.

ROSA

AND

and D~I indicate the oxygen diffusion coefficients in the metal Ci is the

and in the oxide,

initial

cr-zirconium

concentration

and cmco)

respectively. of oxygen

W.

C.

HAGEL

by eqs. (3) to (5) we obtain from relationships (1) and (2), at x=x1. that

in

is that concentration

of oxygen in the oxide which the error function

ki - C: = (CF - C:)/( 1 - erf yr)

(8)

and

curve would reach at infinity. If we take into account associated

with conversion

then

displacement

the

time dt represents oxide/metal

the volume

of metal to oxide,

dlri = Vrde/ Vn

the virtual

interface

due

the

These relationships

(YII~II/

VI).

(9)

can be substituted

into

of the

of eqs. (1) and (2) with elimination of the diffusion

diffusion

DII by means VII@:

Assuming that the concentration of oxygen in zirconium oxide at the surface corresponds to that in stoichiometric ZrOs, then. at time conditions,

CI= ki and CII = 6’:: for x = 0

these

(3)

and at x>O

- C:,)

_

of eq. (6) leads to (-

exp

1% VI~II

respect to x and coefficients DI and

erf

YII~II/~I)~

(yIr BdvI)

-

exp (-YJ

(c:‘-c:)

jh?y~( 1 -

=

c:,_clI.

(I())

erf yI)

Since the derivatives of the respective functions can be expressed as (erf

error

~1)’= d erf yi/d ye = (S/l%) exp ( - ii)

(11)

and

lim cr=Ci and lim cn=cn(oo) a!++a, x-t+CC

for t==O,

(4)

when considering the diffusion couple as a semiinfinite plate in the positive direction of x. The motion of the interface xi will depend on DII. It is also possible to express equivalently the movement of this boundary in terms of DI for the ix-solid solution, and accordingly xi = 2yr1/m

where yr and yn are dimensionless

y&G=

y&%

= yexpt. VI/2

[erf

(~II~II/

proportio-

VII,

where yexrt. is the slope of E versus It. The oxygen mass balance at interface

(6)

= d erf

relationship (C:$ -

C+I)

(10)

[erf

YII erf

exp

(YII~II/

VI)/dyrI

=

( -~IIVII/VI)“,

transforms

(12)

to

-- ’

@II VII/ VI)]

@II VII/ VI)

__ (P

-

(3) (erf

71 erfc

?I)’

= 2(Cl,

YI

_ CII) I

(‘3)

.

For the convenience of forthcoming calculations eq. (13) may be simplified by introducing the following arbitrary designations &II)=

[erf(yIIVII/~11’/{2y11

erf

(14)

(YIIVII/~I)}

and

&I) = (erf yr)‘/{$4

erfc: YI>

(15)

;

XI=

therefore,

yields

the resulting

cc:: -

- Dn(bc~~/bz),=,~-o

+ D1(h/b4,=,~+0 = = (C:, - CF)dxr/dt,

VI)]’

= (Zv1I/(p5?tI)f

(5)

or xi = 2yIIb’Dd

nality constants. Combining eq. (5) with the experimental determinations of ~=f(t), at a given temperature, we have that

= E VI/ VII

= CC:+ -C:,)/erf

the mass balance equation. Thus, differentiation

process. The ratio l’i/Vn is the inverse of the Pilling-Bedworth ratio and E is the experimentally measured thickness of the oxide layer.

0
CII(~)

within

position

to

k2 -

change

(7)

according to the notations used by Wagner and reported by Jost 9). For conditions implied

c:,)&II)

-

linear equation cc:’ - c:)t(yI)

==C&C:I.

is

= (16)

For a given temperature the value of E(m) can be evaluated by means of eq. (16) because

OXYGEN-ION

the oxygen

17

DIFFUSIVITY

concentrations

at the phase boun-

function

daries are known and QI)

follows directly from

oxidation

of time and temperature

the knowledge of Dr and yexpt. Consequently, for a given value of &n) a corresponding

is very

value

under oxygen

either

of yn may by

be obtained

numerical

or

from

eq.

graphical

wide

gravimetric

(14)

methods.

during the

process. The range of reported values and incomplete. measurements

From

at

thermo-

1100-1300

pressure of 1 atm, Kofstad

Ruzicka 10) estimated

that the value 0.001,

Because the analytical solution is rather involved

ZrOz-z

is less than

we shall attempt an indirect graphical solution,

periods

up to one day.

e.g., by plotting

into

account

for

and

of x in

equilibration

Taking

the presently

“C

their results

made

assumption

seems to be fair. The densities of or-zirconium and erf (yn Vn/ VI) versus YII. The unique values of yir thus obtained are utilized to determine the diffusion coefficient Dn from relationship (6). 5.

Evaluation

dioxide

TABLE

c;, G:

diffusion in zirconium

DI ; (cmz/sec) . yexpt. (cmt/sec) .

750

700

600

9.3 X 1O-4 0.439

9.3 X 10e4 0.439

9.3 X 10m4 0.439

9.3 X10m4 0.439

9.3 X10m4 0.439

1.476

1.479

1.483

1.487

1.511

1.511

1.511

1.511

2.23 x lo-lo 4.68 x 1O-s

7.45 x 10-l’ 3.02 X 10m6

1.89 x lo-l1 1.80 x 10-B

9.12 x lo-l3 1.22 x 10-B

6.79 x lo-10 6.45 x 10-B

of computed

3

values for oxygen-ion Temperature

850 (6)

E(w);

750

700

.

0.081

0.103

0.115

0.136

.

7.65

6.13

5.57

4.82

eq. eq.

(16).

104.1

106.6

oxide

(“C)

.

yn;fig.3.

DII;

800

diffusion in zirconium

eq. (15)

yI; eq. &I);

(“C)

800

TABLE Compilation

oxide

850

: : : : : 1.469 . * . . . 1.511

108.8

114.7

600 0.427 2.02 80.5

.

.

6.9 x 1O-2

6.8 x 1O-2

6.7 x 1O-2

6.6 x 10-Z

7.9 x 10-Z

(6) .

.

9.4 x 10-l”

5.1 x 10-10

2.2 x 10-10

8.1 x 10-11

2.6 x lo-l1

Note: All values calculated x
with aid of computer

re-

2

oxygen-ion

Temperature

.

and zirconium

5.82 12) g/cma,

oxygen/oxide and oxide/metal interfaces can be expressed in gOz/cma. The initial amount of oxygen in zirconium used in the present investigation was 9.3 x 10-4 gOz/cma, and this value has also been assumed for all calculations

It has already been mentioned that the concentration of oxygen at the surface was assumed to correspond to that in the stoichiometric form of ZrOa. At the present it is difficult to assess the degree of nonstoichiometry as a

Ci (gOz/cm3) c:1

6.49 11) and

spectively, and the contents of oxygen in the oxide and in the metal substrate are given by the appropriate equilibrium phase diagram la). Consequently, the concentrations of oxygen at

of results

Values used for calculating

are

expanded

erf(z) for x=0.001

interval

and exp( -z)

for

18 based

C.

on results

diffusion

coefficient

of other

J. ROSA

investigators.

for oxygen

AND

The

in a-zirconium

W.

C. HAGEL

Bedworth

ratio

a’ssumes the

Vrr/Vr has been

growth

of

oxide

used.

solely

This

in the

direction perpendicular to the interface and full is considered to be known with high accuracy and can be expressed by Dr= 5.4exp (- 50800/ constraint in the lateral directions. Unique solutions of eq. (lb), for different temperatures, RT) 14). These experimental results are summarized in table

2.

Knowledge

of

evaluation

are obtained these

values

of the remaining

permits

the

parameters:

yr:

exponential shown

from

the intersections

of both

curves for specific values of yrr as

in fig. 3. Fig. oxygen

1 shows

the

diffusion

E(~I), and yn. The numerical values thus obtained, together with t)he reference equations

coefficients

for

in hypost,oichiometric

zirconium

dioxide

from which they were obtained. are shown in table 3. The value, 1.52. for the Pilling-

from the data of this study? under conditions of oxidation of zirconium for times up to 260 h.

which have been calculated

0.30

0.28

0.26

0.24

0.22 t >.

0.20

>" c

0.18

Y 0, .C

0.16

N 0.14 >" . >*1 2

0.12

0.06

0.0 4

0.02 0.00 L

Fig. 3.

Plots

of

[VII exp {-

2

4

(~IIVII/VI)~}]/{~/~VIYIIS(YII)}

peratures.

6

6

and erf (~IIVII/VI)

12

IO

versus

yu for

different tem-

OXYGEN-ION

55O~C I I

800

I

9.2

I

I

9.6

19

DIFFUSIVITY

700 I

750 I 10.0

I 10.4

I

650 I IO 6

I

I 11.2

600 II 6'K-'x104

Fig. 4. Arrhenius plots for diffusion coefficients of oxygen in hypostoichiometric zirconium dioxide. Full line - this investigation. Dashed lines: A - Debuigne and Lehr I), B - Douglass’s results corrected by Smith 16).

Utilizing the least-squares analysis, the diffusion coefficient of oxygen anions may be expressed by &I=

2.88 x 1O-4 exp (-28

for the temperature 6.

400/RT) cm2/sec, (17)

range of 600 to 850 “C.

Discussion

The obtained value of 28.4 kcal/mole for the activation energy of diffusion is in good agreement with 29.3 kcal/mole given by Debuigne and Lehr, but it is considerably lower than that of 33.4 kcal/mole reported by Douglass 15) for oxygen diffusion in zirconia (ZrOl.ee4) sintered pellets. Smith 16) corrected Douglass’s results for the concentration of oxygen anion vacancies as a function of temperature and obtained the values of 1.1 x 10-3 (instead of 5.5 x 10-Z) for

the pre-exponential

factor and of 31.0 kcal/mole

for the activation energy. These corrected values are also shown in fig. 4, for comparison. It should be mentioned that Smith employed the “interruption kinetic technique” to measure the diffusivity of oxygen anion vacancies in zirconium dioxide films and obtained of Dn= 0.9 x 1O-3 exp (- 28 700/RT) temperature range of 334-470 “C.

a value for the

Acknowledgements The authors are indebted to Dr. C. B. Magee for valuable discussions through the course of this work and for critical review of the theoretical part. The writers wish to express their gratitude to the Denver Research Institute for technical and financial assistance in preparing this publication.

20

C.

J.

ROSA

AND

W.

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J.

Debuigne

Met.allurg.

and

P.

60 (1963)

Lehr,

Press,

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Mem.

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(Oxford,

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1957) p. 113 and B. E. Hopkins, alloys

Oxidation

(Butterwort,hs,

G. Beranger (1965)

5,

and I’. Lacombe,

London,

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Mat.

1~ -1

U. R. Wallwork, Acta

7, E.

Met.

A. Gulbransen

(1957)

W. W. Smeltzer

12 (1964)

9

8) J. Crank, ref. 2) p. 30 y, C!. Wagner, Diffusion in solids,

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R.

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New

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binary

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no. 6 (ed. J. 0. H. Donnay; and

alloys

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K.

Anderko,

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p. 1078 14

1 J. J. Kearns

15

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)

and J. N.

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D.

Douglass,

L.

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)

394

Academic

110 (1963)

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and C. J. Rosa, J. Metals

sot.

1 M. Hansen

409

and K. l?. Andrew,

P. Kofstad

13

J. Electrochem.

sot. 111 (1964) 1221

9

Jost;

monograph,

16

190

R. J. Hussey and W. W. Smeltzer,

W.

1960) p. 72

First

1962) p. 3G

4,

1.

)

911

J. Crank, The mathemat,ics

of

HACEL

(ed.

Clarendon

3,

C.

Corrosion

Atomic

1962) 1’. 224

T.

J. Electrochem.

Westinghouse

(1962)

of reactor

Energy

CN-13/l& Smith,

Chirigos,

WAPD-TM-306

Sot.

Agency,

materials Vienna,

112 (1965)

660