The influence of oxide grain boundaries on diffusion-controlled oxidation

Corrosion

Science,

1967, Vol. 7. pp. 165 to 169. Pergamon

SHORT

Press Ltd. Printed

COMMUNI

THE INFLUENCE OF OXIDE DIFFUSION-CONTROLLED

in Great

Britain

CATION

GRAIN

BOUNDARIES OXIDATION*

G. B. GIBBS Central Electricity Generating Board, Berkeley Nuclear Laboratories,

ON

Berkeley, Glos., England

Abstract-When metal oxidation proceeds by cation diffusion along gram boundaries as well as through the volume of the oxide layer, this layer will grow uniformly with parabolic kinetics if cations have a high surface diffusivity on the oxide. The influence of gram boundaries may be accounted for by using an effective diffusion coefficient which is the sum of the volume diffusion coefficient and the grain boundary diffusivity weighted according to the grain size. Growth by anion diffusion should be relatively insensitive to the presence of grain boundaries if internal stresses produced by volume changes at their junctions with the parent metal limit the boundary diffusion flux. 1. INTRODUCTION

THEREis considerable experimental evidence that the rate-controlling process in the growth of oxide films on the surfaces of many metals and alloys is diffusion through the oxide of one particular ionic species.This may be an anion or a cation. Several theoretical treatments of the processhave been published and are reviewed by Hatie.]They generally supposethat: (a) Concentrations of mobile ions at both interfaces are essentially independent of time. This is becauseinterfacial reactions necessaryto establishlocal thermodynamic equilibrium are sufficiently rapid to compensate addition or removal of ions by diffusion. (b) The varying concentrations of charged speciesacross a semi-conducting oxide film give rise to an electric field which enhancesdiffusion transport. (c) A quasi-steady-state model, in which the ion flux at any time is the equilibrium one for the flrn thickness at that time, is adequate to describe the growth of oxide. (d) Ionic transport is predominantly by a volume diffusion mechanism. With the above assumptions, the rate of increase of film thickness X with time t becomes dX/dt = ZD

Q (C,

-

C,)/X

(1)

leading to a parabolic growth law X2 = 220

Q (Co -

Cl) t

(2)

Here C,, and C, are the numbers of mobile ions per unit volume of oxide at metaloxide or oxide-gas interfaces defined so that Co > C,, D is the volume diffusivity and L2is the volume of oxide per ion. Z is an “enhancement factor” which measures *Manuscriptreceived20 September 1966. 165

G. B. Gm~s

166

-r

Ji

y co Cl

OXYGEN

1 x D

METAL

D'

7r-

(a)

W

OXYGEN SURFACE I I

Cl

OXYGEN

I I

l/IPXIDEIIcn ” ,,1,1,,:: --__

-6-7---d--

*. S-8/ _--_-

METAL

METAL

Cd)

(Cl

Fro. 1. Idealized models for grain boundary diffusion and oxidation.

the effect of the electric field in increasing the oxidation rate above that given by the steady-state diffusion solution in the absence of a field (equation (1) with 2 = unity); see Fromhold.B The form of equation (1) suggests that the influence of changes in diffusion geometry may be investigated by considering a hypothetical model in which the migrating ions are uncharged. This approximation will be made here in an attempt to determine the influence of grain boundaries on the kinetics of oxide layer growth. 2. CRITIQUE

OF IRVING’S

MODEL

The diffusivity D’ of any species in a grain boundary, width 6, is often orders of magnitude higher than its dBu.sivity D in the adjacent lattice. Whipple3 has considered the time-dependent diffusion problem for a grain boundary in a semi-infinite medium with the geometry of Fig. I(a) and the boundary conditions C = C,, at x = 0 for all y and t C=Oatx>Oforallyandt

$0

He gives equations for the shape and position of any constant concentration contour such as C, as a function of time. Irving4 supposed that if C,, and C, are chosen to be the appropriate concentrations at the oxide-gas and oxide-metal interfaces when anion migration dominates, the C, contour in Fig. l(a) will also represent the profile of the growing oxide film, Fig. l(b), and its change of shape and position with time will give the average growth

Short communication

167

rate. On this basishe derived a number of different rate laws, depending on the grain size. The analysis may be criticized on the following grounds: (a) An oxide profile of the form illustrated in Fig. l(b) would develop only if the volume per metal ion were essentially the same in the pure metal as in the oxide. Elementswhich develop a protective oxide film generally exhibit an increasein volume per metal ion on oxidation. This will generate internal stresseswhich constrain the film to grow uniformly, as indicated in Section 3.2. (b) Even for an idealized system in which internal stressesmay be neglected, Whipple’s analysis is not appropriate for Fig. l(b) becausethe boundary conditions of the diffusion problem are different. Whipple’s equation for the movement of the C, contour assumesthe existence from t = 0 of a semi-infinite grain boundary with diffusivity D’ in a semi-infinite matrix with constant diffusivity D. There are no discontinuous changes in concentration within the diffusion zone. However, the hypothetical arrangement of Fig. I(b) has only a finite grain boundary in the oxide film and volume diffusivity in the film differs from anion diffusivity in the “semiinfinite” metal base. Further, there is a sharp discontinuity in anion concentration at the oxide-metal interface; concentration in the metal at this interface is C,,, = C, exp - (AG/kT)

< C1

where A G is the free energy change for a single anion crossing the boundary. In principle a new solution of the diffusion equation satisfying these special boundary conditions could be constructed but this will not be attempted here in view of the first objection (a). 3. QUASI-STEADY-STATE

MODELS

On account of the large discontinuities in concentration at the interfaces, the rates of interface movement are small compared with the rate of ion migration across a growing oxide film. Therefore the diffusion flux at any time is approximately equal to the steady-state flux for the particular film geometry at that timem 6This is true whether growth occurs by anion or cation migration. The steady-state approximation is applied here to both types of diffusion control. A further simplification arisesbecause reasonablephysical assumptionsabout the metal-oxide systemspredict uniform oxide layer growth. 3.1 Cation diffusion control An appropriate model for growth by cation migration is Fig. l(c). It is assumed that interface reactions are sufficiently rapid to maintain a constant concentration C,, at normal sites of the oxide-metal interface and at grain boundary siteswhere the diffusion flux is greater.* It is further supposed that rapid surface diffusion can redistribute the larger amounts of material flowing along grain boundaries to give a uniform film growth rate. Rapid surface diffusion appears to be a general characteristic of solid systems.6 For idealized square grains of side d, the total diffusion flux through the film is * Vacancies created at the metal surface diffuse away rapidly through the volume of metal at oxidation temperatures.

168

G. B.

GIBW

This suggeststhat the parabolic growth law, equation (2), should be written with an “effective diffusivity” D* defined by D* = D + 2D%/d

(4)

The same equation applies for a very fine-grained oxide having d < X, cf. Hart.’ Thus, when grain size is sufficiently small, an oxide film will grow appreciably faster than if only volume diffusion were allowed, but the parabolic growth law will be unaltered. Writing Q and Q’ for the activation energiesassociatedwith D and D’, respectively, it is expected that Q’ < Q. An Arrhenius plot of the parabolic growth constant (corrected for temperature variations of (C, - C,)) should therefore be linear at high temperatures with a slope proportional to Q. At lower temperatures where D’S z Dd the slope should decreasetowards proportionality with Q’. Then greater scatter of experimental points is expected owing to oxide grain sizevariations between nominally identical specimens.Also the apparent low temperature activation energy may be lessthan Q’ if grain size tends to increase with increasing temperature of oxidation. A possible example of this type of behaviour has been reported8 for oxidation of the intermetallic compound UCu, in CO,. 3.2 Anion diffusion The problem of anion diffusion control may be considered by reference to Fig. l(d) where a hypothetical very thin oxide film is assumedto exist at t = 0 with constant concentrations C, and C, at the interfaces. Initial diffusion flow will be greatest down the grain boundaries producing larger amounts of new oxide near the oxide grain boundary-metal interface (broken line in Fig. l(d)). The volume change accompanying the formation of oxide will produce internal stressesat the tip of the grain boundary which increase the local equilibrium concentration of anions and correspondingly reduce the boundary diffusion flux. If individual grain boundaries are sufficiently widely spacedso that the stressfields at their ends do not interact, internal stressesare expected to build up until boundary and volume diffusion fluxes are approximately equal and the oxide-metal interface advances uniformly. In this case equation (2) may be used without modification. The magnitude of internal stressesproduced by grain boundary diffusion may be estimated by noting that when D’ $ D, the final concentration C, at the tip of a boundary must be close to C,,. Writing C, = C1 exp s Q/k T z C, where Qis some average local stressand fi is the volume change when an anion com*Vacancies created at the metalsurface diffuse away rapidly through the volumeof the metalat oxidation temperatures.

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169

bines to produce new oxide at the metal interface, an upper limit for the internal stress is o = (kT/Q)

in C,/C,

The oxide layer will be non-protective if (I exceeds the fracture strength of and gives rise to spalling. Taking typical values of kT - IO-l3 ergs, R N lO-23 cm3 and =f - IO9 dynes cm-2, of > Q for (C,/C,) 2 1.1 The elastic strains associatedwith internal stressesare N o/E and since Young’s moduli E have values typically - 1012dynes cm-” they are small and are established at the beginning of an oxidation anneal. 4. CONCLUSIONS

When oxide layer growth is controlled by migration of cations through the volume and along grain boundaries of the oxide, a uniform film thickness may be maintained by rapid surface diffusion. A quasi-steady-state model for this situation predicts a parabolic growth law, X2 = 2ZD* r;2(Co - C&, where D* is the sum of the volume diffusion coefficient and the grain boundary diffusivity weighted according to the grain size. When anion migration determines the oxidation rate, grain boundary diffusion cannot appreciably enhance film growth if internal stressesare developed at the junctions of oxide grain boundaries and parent metal and limit the grain boundary flux. In this casethe usual rate law, X2 = 220 !A (C, - CJt, with D the volume diffusivity, should be a good approximation. If the internal stressesexceed the fracture strength of the oxide, spalling may occur after which a simple diffusion analysis is invalid. Acknowledgemenr-This

paper is published by permission of the Central Electricity Generating Board.

REFERENCES HALJFFE,Metol Oxidation. Plenum, New York (1965). T. FROMHOLD, J. phys. Chem. Solids 24, 1081 (1963). T. P. WHIPPLE, Phil. Mag. 45, 1225 (1954). A. IRVING, Corros. Sci. 5,471 (1965). 5. G. B. GIBBS, J. nucl. Mater. 20, 303 (1966). 6. P. G. SHEWMON, Diffusion in Solids. McGraw Hill, New York (1963). I. E. W. HART, Acta metall. 5, 597 (1957). 8. J. J. S~oses, R. J. PEARCEand I. WHIITLE, Trans. Am. Inst. Min. metall. Engrs 233, 1676 (1965).

1. K. 2. A. 3. R. 4. B.