Internal Oxidation of Binary Alloys

SIAM J. APPL. MATH.

(C) 1985 Society for Industrial and Applied Mathematics

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

Vol. 45, No. 6, December 1985

006

INTERNAL OXIDATION OF BINARY ALLOYS* P. S. HAGANf, R. S. POLIZZOTTI"

AND

G. LUCKMAN’

Abstract. We derive and analyze the equations governing the oxidation of a binary alloy (composed of metals A and B) in an environment such that only the oxide of B, BOq, will form. Singular perturbation techniques are used to obtain the solutions which pertain after a rapid initial transient dies away. When Bo, the initial concentration of B in the alloy, exceeds a critical amount Bcr we find that there is only a single solution; it represents external oxidation of the alloy. However, when Bo < Bcr there are three solutions; two represent internal oxidation and one represents external oxidation. To determine which of these solutions actually governs the alloy’s oxidation, we numerically solve the full initial-boundary value problem to see which solution the initial transient settles down to. We find that when Bo < Bcr, the initial transient evolves to a solution representing internal oxidation. Thus we conclude that the alloy oxidizes internally when B < and it oxidizes externally if Bo> BCr.

nor

1. Introduction. Engineering alloys are able to withstand corrosive environments at high temperatures due to their ability to form a continuous layer of a protective metal oxide at the alloysurface (a protective external oxide scale) which separates the reactive metal from the aggressive environment. Corrosive agents from the environment, such as carbon and oxygen, and metal atoms from the alloy can only diffuse very slowly through a protective metal oxide. Thus if an alloy forms a continuous layer of such an oxide, it, will effectively protect the alloy from further corrosion. However, when an alloy is exposed to an oxidizing environment, it can oxidize in either of two modes. Under some conditions it will oxidize externally, and the metal oxide forms as the desired external oxide scale. Under other conditions it will oxidize internally, and the metal oxide forms as small particles dispersed within the alloy. If an alloy oxidizes internally, so no protective scale is formed, then it will continue to corrode and corrosive failure often ensues rapidly. In this paper we investigate quantitatively the internal oxidation of binary alloys and the conditions necessary for the alloy to oxidize externally. We derive and analyze the equations governing the oxidation of an alloy composed of metals A and B in solid solution. We assume that metal B has a higher affinity for oxygen than A has, and that the oxygen partial pressure of the alloy’s environment is low enough so that only the oxide of B, BOq, will form. Additionally, we assume that the oxide BOq is protective, so that the A, B, and O (oxygen) atoms in the alloy cannot diffuse through the oxide particles created by the oxidation reaction. Finally, we neglect all corrosive reactions (carburization, sulfidation,...) of the alloy except oxidation. Although the equations are derived only for the specific situation sketched here, they can be readily extended to treat more complicated metallurgical situations, such as the simultaneous oxidation of two competing metal species and the simultaneous oxidation and carburization of an alloy.

More specifically, the governing equations incorporate the following physical effects. Bulk reactions. The only significant bulk reaction is assumed to be

(1.1)

B+qO BOq,

* Received by the editors August 13, 1984, and in revised form December 20, 1984. Exxon Research and Engineering Company, Clinton Township, Annandale, New Jersey 08801. t Current mailing address, Olin Corporation, Metals Research Laboratories, New Haven, Connecticut 06511. 956

"

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

INTERNAL OXIDATION OF BINARY ALLOYS

957

which we shall treat as being very rapid and irreversible. In particular, reaction (1.1) is thus in equilibrium only when the concentration of B or O is zero; i.e., the B, O solubility product is effectively zero. Diffusive transport. We account for the diffusion of A, B, and O in the alloy, assuming that these species do not diffuse through the oxide particles. In particular, this diffusive blocking by the particles reduces the effective diffusion rate of O in the alloy. As we shall see, this reduction is the key cause of the transition from internal to external oxidation. Advective transport. Usually the oxide BOq occupies more volume than its constituent B atoms. Thus the expansion accompanying the oxidation of the B atoms in a region causes a bulk motion (advection) of the alloy away from the region, so that the extra volume can be accommodated. Similarly, the A and B atoms generally diffuse at unequal rates, so different regions in the alloy expand and contract as they suffer net gains and losses of metal atoms. This again causes bulk motion (the Kirkendall effect [7], [19]). The transition from internal to external oxidation results naturally from oxide blocking in our analysis. Oxide blocking is not accounted for in standard treatments of alloy oxidation (e.g. [2]-[4], but see [1]). Consequently, these treatments must be supplemented by experimentally found rules determining when external oxidation occurs. Additionally, we have incorporated diffusion of the inert metal A and advective transport so that our treatment is self-consistent; otherwise, the volume fractions of the alloy components would not sum to 100%. The physical effects we have incorporated are all necessary for a consistent treatment of internal oxidation and the transition to external oxidation, so we feel that our treatment is the simplest possible. However, other physical effects may prove to be quantitatively important in various alloys. See [5] for a detailed description of the metallurgical assumptions used. The equations governing alloy oxidation are derived in 2. In 3 we use singular perturbation techniques to find the solutions which pertain after a rapid initial transient dies away. When Bo, the initial concentration of B in the alloy, exceeds a critical amount Bcr, we shall find that there is only a single solution; it represents external oxidation of the alloy. However, when Bo < Bcr we shall find that there are three solutions; two represent internal oxidation and one represents external oxidation. Of these, one internal oxidation solution is always unstable, the other two solutions are

always stable. Clearly alloys with Bo> Bcr must form external oxide scales, since there are no solutions representing internal oxidation for these alloys. The situation is not clear-cut when Bo < Bcr, because there are two stable solutions. We resolve this situation in 4 by numerically solving the full initial-boundary value problem to see which solution the rapid initial transient evolves into. We shall find that when Bo < Bcr, the initial transient evolves to the stable internal oxidation solution. Thus we conclude that an alloy oxidizes internally if Bo < B and externally if Bo> Bcr. 2. The governing equations. Consider an arbitrary volume fl which is very small, yet is large enough to contain many oxide particles, as shown in Fig. 1. Divide into two parts, flz consisting of all the oxide particles and rIM consisting of the surrounding metal matrix. The concentrations of A, B, and O within any oxide particle will be extremely small, and can be safely neglected in our accounting. Thus we treat IIM as containing all the A, B, and O. Similarly, we regard all the oxide BOq as belonging to Ilz. We let e and 1-e be the respective volume fractions of fl and fz. We denote the concentrations of A, B, and 0 atoms within the metal matrix 1)4 as a, b, and c;

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

958

P. S. HAGAN, R. S. POLIZZOTTI AND G. LUCKMAN

FXG. 1.

we denote their concentrations in the total volume fl =rIM +l’z by A, B, and C. Similarly, we let the concentration of BOq within fz be z and its concentration in fl be Z. Clearly, the "apparent" concentrations A, B, C, Z are related to the "intrinsic" concentrations a, b, c, z by

A=ea, B=eb, C=ec, Z=(1-e)z.

(2.1)

We now consider the volumes of the four species. First, the O atoms diffuse interstitially and occupy no appreciable volume. Next, for binary alloys in solid solution, Vegard’s law 19] implies that the alloy’s volume per metal atom is A[ 1 + trb/(a + b) ]3 for some constants A and o-. The concentrations a and b represent the numbers of atoms per unit volume, so clearly in rim

(2.2) Now

Itrl

A(a + b)[1 + trb/(a + b)] 3-- 1. must be less than .15 for B to have any substantial solubility in A [19]. For

these values of

I 1, 2.2 is well approximated by

A(a + Ib) 1 with 1 + 3o’, even for atomic fractions b/(a + b) as large as 30%. Thus we shall use (2.3) instead of (2.2) as our volumetric law. Finally, the density of BOq within the (2.3)

oxide particles will be very nearly a fixed constant. So we shall assume that in fz,

z=l/Ak

(2.4)

for some constant k. We now nondimensionalize all concentrations by dividing them by l/A, the concentration of A atoms in pure A, a

(2.5)

A

A a ota, AA td,

A b ota AB ld,

b

B

In terms of these new concentrations, we have

(2.6) (2.7)

a + lb

1,

z=l/k,

and (2.1) still holds"

A=ea, B=eb, C=ec, Z=(1-e)/k. (2.8) In particular, (2.8) shows that the volume fraction 1-e occupied by the oxide BOq is simply kZ; the volume fraction occupied by the metals is just

(2.9)

e

1

kZ.

Clearly (2.6)-(2.9) must hold throughout the alloy since the volume ll is arbitrary.

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

INTERNAL OXIDATION OF BINARY ALLOYS

959

With these considerations out of the way, we turn to conservation laws. In what follows, we have chosen the intrinsic concentrations a, b, c and the apparent concentration Z as our dependent variables; (2.7)-(2.9) will be used to obtain A, B, C, z, and e as needed.

2.1. Conservation laws. Let the alloy’s bulk velocity be v(x) at point x. All species are swept along by this bulk motion, and the species A, B, and O also diffuse. Let r/1 denote the oxidation reaction rate. Clearly,

(ea)t =-V" J-V" (yea), (eb)t -V. J-V. (veb)- r/q, (ec), -V J-V. (vec)- qr/ q, z, -v. (vz) + n,

(2.10)

r

where the jo denote the diffusive parts of the fluxes. 2.2. Diffusion. If there were no oxide present, then

JA=-DA(b)Va, aa=-Da(b)Vb, a=-Do(b)Vc. (2.11) Here DA(b) and De(b) are the intrinsic [7], [11] diffusivities of A and B in the binary alloy, and Do(b) is the diffusion coefficient for oxygen. Suppose now that oxide particles are present. The diffusive fluxes of A, B, and O will clearly be reduced, since these species must diffuse around the impermeable oxide particles. Now diffusion is driven by gradients of the concentrations in the metal matrix IM, not by gradients of the apparent concentrations in the total volume 1". Indeed, if a were spatially constant, then no diffusion of A atoms would occur, regardless of the spatial distribution of e, and hence of A. Thus the fluxes are of the form: (2.12)

J=-EDA(b)Va, J=-ED(b)Vb, J=-EDo(b)Vc.

Here the factor E accounts for the reduction caused by the local volume fraction kZ of oxide particles. Although finding the effective properties of dispersed two-phase systems is a very active research area [8]-[14], no completely satisfactory method is available for determining E in terms of the statistics of the oxide particles. Many theories have been proposed; they all yield varying results for E. Until we obtain E from experimental measurements, we shall adopt the effective medium theory (EMT) [8]-[10], which is perhaps the most widely used. It yields 1-pkZ if kZ = 1/p. Here p- 1/(1-d), where d is the depolarization coefficient (see [15, Chap. 1]) in the direction of diffusion for a typically shaped oxide particle. In most alloys the oxide particles either are all needle-like or all plate-like particles oriented perpendicular to the alloy surface (parallel to the direction of diffusion); are all roughly spherical particles with all their length dimensions roughly equal; or are all needle-like particles oriented parallel to the alloy surface (perpendicular to the direction of diffusion). For these cases,

(2.13)

(2.14a) (2.14b) (2.14c)

E(kZ)=

(needles or plates perpendicular to the alloy surface), (roughly spherical particles), p p=2 (needles parallel to the alloy surface).

p-1

=

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

960

P. S. HAGAN, R. S. POLIZZOTTI AND G. LUCKMAN

Equation (2.13) exhibits a percolation threshold at Z 1/pk. This occurs because in EMT there remain no connected paths through the metal matrix when kZ exceeds lip. Such thresholds have been found experimentally in electrical conduction experiments

[14].

2.3. Advection. As the alloy oxidizes, bulk motions arise to keep the metal atoms and the oxide at their equilibrium densities; i.e., to keep the volumetric relations (2.6) and (2.9) satisfied. From (2.10) we find that

(2.15)

[ea+elb+kZ]t=-V.(J+IJ)-V’v[ea+elb+kZ]+(k-l)r/7.

Now e(a+lb)+kZ represents the sum of the volume fractions of the alloy’s components, which must always be unity. See (2.6) and (2.9). So,

(2.16a)

V. v=-V. (J+ lJ ) + k l) r/ 7.

Thus advection is caused by both diffusion (the Kirkendall effect [7], 19]) and reaction. We do not believe that either process should create vorticity, so we shall assume that

(2.16b)

V v=O.

2.4. Reaction. Experimentally, bulk oxidation reactions are extremely rapid [4]. Therefore we have written the reaction rate as r/7 =- r(e, b, c)/7, where 7 is the short time-scale of the reaction and r(e, b, c) is to be O(1) with r- 0 if e, b, or c is zero. As we shall see, the detailed dependence of r on e, b, and c will be of little importance. Gathering the preceding results together, we have

(2.17a) (2.17b) (2.17c) (2.17d) (2.17e)

(ca), =V" (EDAVa)-V" (yea), (eb), =V. (EDnb)- (veb)-r/7, (ec), =V. (EDoc)- (vec)-qr/7, z, -v. (vZ) + n, V.v= IV. [E(Dn-Da)Vb]+(k-l)r/7, V v=0,

r

where e=- 1-kZ. (2.17f) Here, DA---- DA(b), De =- De(b), Do =- Do(b), and E(kZ) is given by (2.13). These are

the key equations describing the alloy’s oxidation. They appear quite formidable, but the speed of the oxidation reaction will lead to great simplification. We now find the appropriate boundary and initial conditions to complete the theory. 2.5, Boundary and initial conditions. We assume that the alloy surface suffers no major distortions from the alloy’s expansion, so we approximate it as remaining planar as the oxidation proceeds. Since the choice of reference frames is arbitrary, we let the surface remain at x 0 for all time. Thus

(2.18)

V.ex-0

at x=0.

We neglect metal vaporization and vapor deposition, so there is no flux of metal atoms at the alloy surface

(2.19)

a bx 0

at x

O.

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

INTERNAL OXIDATION OF BINARY ALLOYS

961

Oxygen enters the alloy via reversible reactions occurring on the alloy surface. Typical is

H20 (gas) HE (gas)+ O (in alloy). It is believed [1]-[4] that these reactions usually occur fast enough to maintain the oxygen concentration in equilibrium at the alloy surface, so

(2.20)

c

c* at x 0.

Here c* is a function only of the oxygen activity (Po2) of the alloy’s environment. However, if the environment contains surface poisons (e.g., sulfur), then the surface reaction rates are dramatically reduced. In this case an alternative to (2.20) is required, and this alternative can lead to quite different results [16], [18]. When the alloy is first exposed to an oxidizing environment, the constant initial conditions

(2.21)

a =- l

lBo, b =- Bo, c--O, Z =- O at t=0

are appropriate. These initial conditions are also appropriate for determining whether an alloy will re-form its external oxide scale after spallation [4], [16], [18].

2.6. Consistency. Equation (2.17a)-(2.21) form a well posed mathematical problem describing alloy oxidation. From these equations, one easily finds that a + lb =- 1 and e 1- kZ for all and x. Thus these equations are consistent with the volumetric relations (2.6)-(2.9) used in their derivation. Since a only occurs in (2.17a), we shall ignore (2.17a) in our analysis and obtain a as 1-1b after solving the remaining equations. In the remaining sections, we shall analyze the equations in one space dimension. In one space dimension, we note that in any region where Z 0, the diffusion-induced part of the advection combines simply with the diffusion terms in (2.17a), (2.17b). Since e 1 and E- 1 when Z 0, we have

(2.22a)

(DAa,,)-(va), =-- (Da),-(vra), v rb )x veb =- Dwbx EDzbx

(when Z 0)

where the reaction-induced velocity is given by

(2.22b)

v

k l) r/ q,

and

(2.22c)

DM(b)=- lbDA(b)+(1-1b)DB(b).

The common diffusion coefficient DM is usually called the chemical or mutual diffusivity of A and B in the binary alloy [7], [11], [19]. 3. After the transient. Before analyzing (2.17)-(2.21), it is helpful to develop a physical picture of the alloy’s oxidation. To do this, recall that the oxidation reaction is in equilibrium only when either b 0 or c 0. So clearly in any region of the alloy where b---O(1), Q atoms could only exist for short times t O(7) and diffuse short distances x- O(x/Do) before reacting. Similarly, B atoms could only exist for short times and diffuse short distances in regions with substantial O concentrations. Therefore, at all times t//>> 1 the alloy must be divided into two regions as shown in Fig. 2: an oxidized region 0 :(t) where b is nonzero and thus

962

P. S. HAGAN, R. S. POLIZZOTTI AND G. LUCKMAN

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

Concentrations

t)

Surface

Reaction

x (Depth)

Zone FIG. 2. Qualitative concentration profiles for an internally oxidizing alloy at a fixed time t.

c 0. Clearly at.x so(t) there must be a thin boundary layer, the reaction zone, whose width ‘5 represents the distances O and B atoms interpenetrate before reacting. Moreover, virtually all of the oxidation must occur within the reaction zone since either b 0 or c 0 outside this boundary layer. So, within the reaction zone both the reaction and transport terms in (2.17) must be important; outside this layer the reaction terms can be neglected. For simplicity, let x (t) denote the intermediate limits

x st(t)

O,

[x st(t)] +o.

We are now led to the following picture of the alloy’s oxidation. There will be a short transient stage of oxidation when t---O(r/). This initial transient will die away as t/q becomes >> 1, and the physical situation will then be as shown in Fig. 2. The reaction zone :(t) will emerge from x =0 and advance continually into the alloy as the B atoms are depleted. In the region 0< x < :(t)- all the B has been consumed, so the O atoms will simply diffuse from the surface to the reaction zone. There they will rapidly encounter and react with B atoms, and thus the O atoms will not penetrate beyond x :(t) +. So c=0 in the region x> :(t) /. Also, Z =0 in this region since Z 0 initially and there has been no oxidation in this region. Finally, the consumption of B atoms within the reaction zone will cause additional B atoms to diffuse toward x :(t); consequently, A atoms must counter-diffuse away from x (t). Since only the region 0< x < :(t)- will contain oxide, the physical role of the oxide is to reduce the rate at which O atoms diffuse from the surface to the reaction zone. This reduction will give the B atoms more time to diffuse into the reaction zone, which will result in higher oxide concentrations, which will further reduce the diffusive We anticipate that if the initial concentration Bo is small enough, flux of O atoms, then the oxide blocking should simply lead to higher oxide concentrations than would have occurred otherwise; but if Bo is large enough, then the above nonlinear feedback process should "run away" and cause the formation of a layer of 100% oxide--an external oxide scale. We now establish the above physical picture mathematically by analyzing (2.17)(2.21) to find its solutions when t q >> 1. Instead of directly using t/q >> 1, we adopt the equivalent approach of using formal asymptotic methods based on r/ B. This is our criterion for external oxide scale formation. Examination of (2.13) and (3.14)-(3.16) shows that Ber depends on I, k, p, d(b)=Dt(b)/D(O), and 4 Vo (O) c*/ qV (O). Now D(b), Do(0), 1, k, and q depend only on which metals A and B are in the alloy; they can be obtained from the literature or experimentally. The parameter p can be found by examining an internally oxidized alloy to determine the prevalent oxide particle shape. Only the remaining parameter c* depends on the alloy’s environment; it can be obtained as a function of the environmental Po2 from published data or experimentally, and can be expected to be proportional to x/o. Given I, k, p, and DM(b) for any particular A and B, the turning point can be found at each value of b by solving (3.14)-(3.16). The resulting graph of BCr against b is a definite prediction of the minimum initial concentration Bo needed (to form an external oxide scale) as a function of the environmental Po2. We are currently engaged in verifying this prediction experimentally. In many alloys D(b) is approximately constant and is close to 1. Also, k is often around 2 for many oxides. For this case,

-

(4.5a)

trZo Bo- 1-Zo

(4.5b)

o

(1

e Erfc (tr),

Zo).,/d(1

2pZo)/ 2Zo.

kZo 1.00

" External Oxidation

.50

q=.5

.20 .10 Bo FIG. 4. Oxide volume fraction kZ versus the initial concentration Bo, for the case d(b) l, 1= 1, k 2 at .5. Shown are the internal oxidation solutions for needle-like or plate-like particles oriented perpendicular to the alloy (p ), for roughly spherical particles (p ), and for needle-like particles oriented parallel to the atto surface (p 2).

suace

969

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

INTERNAL OXIDATION OF BINARY ALLOYS

Bcr

.30

p=l .20

p=2 .10

.1

.01 FIG. 5. The turning point and p=2. p= 1,

p=,

Bcr

versus

b for the

case d b

1,

1, and k

2. Shown are the curves for

,

The bifurcation diagrams corresponding to (4.5a), (4.5b) at b =.5 are superimposed in Fig. 4 for the cases p 1, and 2. Figure 5 shows the resulting Bcr(b), again for p 1, and 2.

,

5. Conclusions. In the governing equations, we cannot be certain of the correctness of the factor E(kZ) given by (2.13). The different theories for E all yield results which vary to some extent. For example, we could have used differential effective medium theory [12], [13]. This would yield

E(kZ)=(1-kZ) p in place of (2.13). Figure 6 compares the Bcr(b) obtained using (5.1) with that obtained using (2.13). Since E (kZ) is uncertain, we plan to measure it experimentally for selected

(5.1)

alloys.

Scr

DE////__ p=l 5

,/ .01

.1

FZG. 6. The turning point Bcr versus qb for the case d (b) 1, 1, k 2, and p with E given by (5.1) (solid line) and with E given by (2.13) (dashed line).

.

Shown are the curves

Downloaded 03/16/15 to 38.98.219.157. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

970

P. S. HAGAN, R. S. POLIZZOTTI AND G. LUCKMAN

Mathematically, we have not shown that the bifurcation diagram always contains only one turning point, nor have we shown that the initial transient always selects the lower oxidation branch when Bo < Bcr. Additionally, in [17] we have only established the stability of the solutions to one-dimensional perturbations of the initial conditions, not to general three-dimensional perturbations. However, equations (3.14) and (3.19)-(3.21) provide quantitative results for the oxide volume fraction kZo, the oxidized zone’s width so(t), and the transition point Bcr(b), so our theoretical work can be tested experimentally. In particular, agreement between these predictions and experiment would show that the physical picture for internal oxidation and the formation of external oxide scales, developed in 3, must be essentially correct. We are currently carrying out experiments to test these predictions for selected alloys. In this paper we have only considered the simplest oxidation problem. Since alloys often form more than a single oxide, in the future we plan to theoretically investigate binary alloys in which both metals oxidize simultaneously. Since alloys are often attacked by corrosive agents besides oxygen, we also plan to investigate alloys which simultaneously oxidize and carburize. Appendix. Here we derive the inner equations which pertain within the reaction zone. For simpler exposition, we assume that the reaction rate is

(A.1)

r(e, b, c)= f(e)b"c"[l +. .] for b