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Passivity and inhibition during the oxidation of metals at elevated temperatures
Corrosion Science, 1965, Vol. 5, pp. 751 to 764. Pergamon Press Ltd. Printed in Great Britain
PASSIVITY A N D INHIBITION D U R I N G T H E O X I D A T I O N OF METALS AT ELEVATED TEMPERATURES* CARL WAGNER Max-Planck-Institut ftir physikalische Chemie, G6ttingen, Gerraany. Abstract--Typical phenomena of passivity and inhibition during the oxidation of metals and alloys at elevated temperatures are considered in order to stimulate the development of appropriate methods for minimizing corrosion at elevated temperatures. R~sum6--On consid6re des ph6nom6nes typiques de passivit6 et d'inhibition pendant l'oxydation de m6taux et d'alliages aux temp6ratures 61evdes,et cela en rue de stimuler le d6veloppement de m6thodes appropri6es pour minimiser la corrosion gt ces temperatures. Zusanunenfassung--Clberlegungen iJber typische Erscheinungen von Passivit~it und Inhibition bei der Oxydation von Metallen und Legierungen bei h6heren Temperaturen werden angestellt, um die Entwicklung angemessener Methoden fiir die Einschrfinkung der Hochtemperaturkorrosion anzuregen.
Pe~epaT- C tte~bm CTHMy.rIIIpOBaHIIFIpaaBnv~ COOTBeTCTByIOIRHXMeTO~IOByMeHbmeHtifl ~¢oppoaH~ npa nOB~meHH~X TeMnepaTypax paccMaTp~IBalOTCa TnnnqH~e ~BJleHHff rlaCcHBnOCT~i II ~IHr~6~IpoBan~m eo npeMR OKHcJIeHHfl Mevaa~oB ~i CIIJIaBOB npa B~COHnX TeMnepaTypax. INTRODUCTION TWO well-known examples o f passivity are: (1) U p o n increasing the single electrode potential o f an iron electrode in l N H2SOo iron dissolves a n o d i c a l l y at a g r a d u a l l y increasing rate. A t a b o u t 0.47 V the rate decreases by a b o u t 4 o r d e r s o f m a g n i t u d e a n d becomes virtually i n d e p e n d e n t o f the electrode potential in a r a t h e r wide range. (2) I r o n dissolves readily in 0.1 M nitric acid but is virtually not a t t a c k e d by pure nitric acid. In b o t h cases the low rate o f dissolution under m o r e strongly oxidizing c o n d i t i o n s is ascribed to the f o r m a t i o n o f a dense film o f i r o n oxide which acts as a b a r r i e r for the transfer o f iron ions from the metal to the solution. U s i n g passivity as a purely p h e n o m e n o l o g i c a l term, one m a y state t h a t a metal is passive (a) if the steady-state rate o f a n o d i c dissolution o f a metal in a given e n v i r o n m e n t is lower at a m o r e noble single electrode potential t h a n at a less noble potential, o r (b) if u p o n increasing the c o n c e n t r a t i o n o f an oxidizing agent in an a m b i e n t solution the steady-state dissolution rate (without flow o f external current) is f o u n d to be less t h a n the dissolution rate at a lower c o n c e n t r a t i o n o f the oxidizing agent. These two definitions are equivalent u n d e r c o n d i t i o n s where the electrochemical m e c h a n i s m o f c o r r o s i o n applies. T h e decrease o f the dissolution rate o f a metal u p o n shifting the single electrode potential to a m o r e noble value a n d thus increasing the driving force represents a *Manuscript received 25 May 1965. 751
752
CARL WAGNER
special case of a negative differential resistance, which is frequently encountered in circuit elements of electrical engineering, e.g. amplifiers and wave generators. The foregoing phenomenological definition may be generalized. A metal may be called passive when the amount of metal consumed by a chemical or electrochemical reaction in a given time is significantly less under conditions corresponding to a higher affinity of the reaction (i.e. a greater de.crease in free energy) than under conditions corresponding to a lower affinity. The foregoing definition, however, does not cover all phenomena which are usually labelled as passivity, especially in the case of alloys. Thus a further generalization of the foregoing definition is suggested. A metal or alloy is called passive when the amount o f at least one o f the metallic components consumed by a chemical or electrochemical reaction in a given time is significantly lower at a higher affinity than at a lower affinity. In this definition the occurrence of a negative differential resistance is kept
as the most significant characteristic of the term passivity when it was first introduced by Sch6nbein. In spite of the foregoing rather general definition, however, it remains open to question whether this definition is applicable to all possible complex situations. A phenomenon closely related to passivity is inhibition. In contradistinction to passivity, the term inhibition is suggested for cases where at a constant affinity the rate of a reaction is significantly decreased upon changing the concentration or activity of a component which is not involved in the reaction under consideration. Using the foregoing definitions, one may ask whether phenomena of passivity and inhibition occur when metals or alloys are subject to oxidation at elevated temperatures. Typical examples are considered in what follows. The clarification of the mechanism is definitely the goal of research on corrosion. Nevertheless, it is helpful to use certain phenomenological terms before the mechanism has been ascertained. Detailed investigations show that many cases of both passivity and inhibition are due to the formation of well defined layers of metal compounds, especially metal oxides, or the formation of adsorption layers. Once the mechanism has been ascertained, use of terms such as passivity or inhibition becomes less relevant since these terms convey only a limited amount of information. THE OXIDATION OF COPPER, IRON, ZINC AND OTHER METALS AT TEMPERATURES WHERE VOLATILIZATION OF METALS IS SIGNIFICANT When copper is heated in a stream of air at 1000°C, a dense Cu~O scale is formed after a short induction period. Further oxidation takes place inasmuch as copper ions and electrons migrate across the Cu20 layer to the outer surface where they react with 02 molecules in order to form Cu20. The Cu20 layer confining the oxidation rate is analogous to an oxide layer on iron passivated by anodic polarization. In spite of this analogy, Cu covered by Cu20 at 1000°C is in general not classified as a passivated metal because the oxidation rate is not significantly lower than the rate observable under less strongly oxidizing conditions, e.g. at low oxygen partial pressures. Turkdogan, Grieveson, and Darken, 1 however, have recently shown that it is possible to realize conditions corresponding to an active and a passive state during the oxidation of Cu, Fe, Ni, and other metals at temperatures where the vapour pressure of the metal
Passivity and inhibition during the oxidation of metals at elevated temperatures
753
is appreciable and the vapour pressure of the metal oxide is insignificant. When under these conditions a metal is exposed to an oxygen-argon stream containing a low oxygen content, metal atoms vaporize from the surface and diffuse into the boundary layer up to a certain distance ~ where they encounter and react with O2 molecules which in turn diffuse into the boundary layer from the bulk stream. At the plane of encounter, metal atoms and O~ molecules form clusters of solid (or liquid) metal oxide which are carried away into the bulk stream by Brownian movement and convection and finally escape as fume. At the distance 8 from the metal surface at which the formation of solid (or liquid) metal oxide takes place, the partial pressures of metal atoms and oxygen molecules can be assumed to be vanishingly small. The fluxes of metal atoms, Jue, and O~ molecules,jo ,, must be equivalent. Thus JM~
--
DM~p~¢ 2 . -J o RT v
2 Do~p~~ v (A--5) R T
, - - - -
(1)
where v is the number of oxygen atoms reacting with one atom of metal for the formation of solid (or liquid) metal oxide, Dm~ and Dot are the diffusivities of metal atoms and Oz molecules, respectively, p ~ is the metal vapour pressure at the surface of the metal, p~, is the oxygen partial pressure in the bulk stream, and A is the effective thickness of the total diffusion boundary layer. From eq. (1) the value of S is found to be =
A 1 -t- (2Do, p~,/vDmcp~e)
(2)
In most cases DM, = Do, and p~, >> P~c and therefore ~ ,~ A, i.e. the fluxes of metal atoms and 02 molecules meet in the very vicinity of the metal surface. Substituting eq. (2) in eq. (I), one obtains for the rate of metal vaporization, which is equal to the rate of formation of metal oxide as fume, ° [ I ff V JM~=2D°'P°* 2DM~p~,] o vART Do, Po ,
(3a)
or
JMo -~ 9/)°---°'P°*if DMo = Do,, Po,° >> P~t, vART
(3b)
When the oxygen partial pressure p~, in the bulk stream is gradually increased, the value of ~ decreases according to eq. (2) and the rate JMe increases proportionally to p~, according to eq. (3b). The ratejMe determined by eq. (3b) can reach but not exceed the maximum vaporization rate of the metal into a vacuum according to the HertzLangmuir equation, JMc (max) = p~,/(2~MMo RT) 112 where MM© is the atomic weight of the metal.
(4)
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CARL WAGNER
Upon equating the right-hand members of eqs. (3b) and (4), the corresponding critical oxygen partial pressure p~, (crit) in the bulk stream is found to be
vApMe (RT) 112 p~, (crit)
~2-D~~ - M ~ 2
(5)
The value o f p ~ (crit) according to eq. (5) depends on the effective thickness A of the diffusion boundary layer and, therefore, on the flow rate and the dimensions of the sample. For example, for a copper sample 2 cm long at 1200°C (p~:, = 3.10 -6 atm), p°Oa (crit) was found to be 0.0035 atm at a flow rate of 40 cm/sec but only 0.0020 atm at a flow rate of 80 cm/sec. 1 When Po2 in the bulk stream is increased beyond P~2 (crit), the foregoing mechanism breaks down, i.e. in view of the supply of excess oxygen to the surface, the metal becomes saturated with respect to metal oxide, and a protective layer of metal oxide is formed. If a solid oxide layer is formed, its thickness first increases proportionally to the square root of time. After a sufficiently long time, the thickness is supposed to approach a steady-state value when the diffusion rate of ions and electrons across the solid oxide layer balances the rate of vaporization of metal oxide followed by diffusion across the gaseous boundary layer. In general, the vapour pressure of the metal oxide is much smaller than p~. (crit). Then, the steady-state corrosion rate after formation of a solid oxide layer is much lower than that of the bare metal. A stationary thickness of an oxide layer during the high temperature oxidation of a metal is analogous to the stationary thickness of an oxide film on passivated iron in HNO3 at room temperature when the flux of iron ions and electrons is equivalent to the rate of oxide dissolution. The decrease in the rate of metal oxidation after formation of a layer of solid metal oxide is especially large for zinc. At 400°C p~,, equals 1.0 × 10 -4 arm and, therefore, the maximum rate of vaporization ( = rate of formation of ZnO fume) is 2.5 × I0 -s mole Zn/cm 2 when the surface is free of oxide and other impurities which may impede vaporization. In contrast, when a layer of ZnO has been formed, the rate of oxidation is as low as 2.5 × 10 -1° mole Zn/cm 2 sec after one hour. 2 When after formation of a ZnO layer p~2 is decreased below p~. (crit), the ZnO layer remains stable since in view of the very low consumption of oxygen the oxygen partial pressure at the ZnO surface is virtually equal to that in the bulk stream. Thus upon first increasing p ~ and subsequently lowering p ~ , one has to expect a hysteresis loop as is schematically shown in Fig. 1. Small amounts of adsorbed molecules (less than required for the formation of a monolayer) may decrease the rate of vaporization of a solid metal considerably below the rate given by the Hertz-Langmuir equation (4) when the release of metal atoms from kinks is blocked. According to the definitions given above, this is a case of inhibition. By and large, presence of impurities in the gas phase or in the metal may diminish p~, (crit) considerably below the upper limit in eq. (5). THE OXIDATION
OF SILICON
AT E L E V A T E D
TEMPERATURES
Passing a stream of He and 02 of atmospheric pressure with a bulk oxygen partial pressure pOo, over liquid silicon at 1410°C, Kaiser and Breslin 3 observed that the
Passivity and inhibition during the oxidation of metals at elevated temperatures
PO~ FIG. 1.
755
PO~ (crit)
Corrosion rate of zinc in an argon-oxygen stream as a function of oxygen partial pressure.
steady-state oxygen content of the molten silicon is proportional to p°o.. ifPo2 ° < 0.01 atm but constant if p°o, > 0-01 atm. These observations may be interpreted in the following way. 4 At low oxygen partial pressures in the bulk stream, the supply of oxygen to the silicon surface is low. Then only gaseous SiO is formed by the reaction Si(l) + ½0z(g) = SiO(g).
f6)
As long as the silicon surface is bare, i.e. not covered by solid SiOz, oxygen is readily consumed and, therefore, the concentration of 02 at the silicon surface is vanishingly small. Thus the flux Jo2 across the hydrodynamic diffusion boundary layer with an effective thickness ~5 is
Jo, = P~,DoJ6RT For zero concentration of SiO in the bulk stream, the the silicon surface is Jsio =
PsioDsio/t3 RT
(7a)
~]UXjsioof SiO away from (7b)
where Psio is the partial pressure of SiO at the silicon surface, Dsi o is the diffusivity of SiO, and the effective thickness of the diffusion boundary layer is assumed to be the same as for diffusion of O2 in eq. (7a) for the sake of an approximation. Under steady-state conditions the net transport rate of oxygen in form of 02 or SiO must vanish. Thus
2jo, =Aio
(8)
Substituting eqs. (7a) and (7b) in eq. (8), solving for P~io, and letting Dslo = Do, as a fair approximation, one has
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CARL WAGNER
Psio = 2pg,
(9)
IfPsio according to eq. (9) is lower than the equilibrium partial pressure Psio(eq) for the reaction I Si(1) q- 1 S i O z f s ) = SiO(g)
(10)
the silicon surface remains bare. At 1410°C, Psio(eq) equals 0.015 atm. Thus, in view of eq. (9), the critical oxygen partial pressure p~, (crit) in the bulk stream for keeping the silicon surface bare is estimated to be o
1
Po, (crit) - ~Psio (eq) = 0.008 atm
(1 la)
In a more accurate calculation, different values of the diffusivities and the effective thickness of the boundary layer for diffusion of 02 and SiO have to be used. Then one obtains 4 o
1
x2
Po~ (crit) ----~ (Dsio/Dot) / Psio(eq) -----0.006 atm
(lib)
In contradistinction to eq. (5), the critical oxygen partial pressure is determined mainly by the equilibrium partial pressure Psio (eq) but does not depend on the effective thickness of the boundary layer, i.e. on the flow rate of the helium-oxygen mixture, or the dimensions of the sample. As long as the silicon surface is bare, the oxygen content of the melt is determined by the equilibrium Si(1) + O(in Si) ---- SiO(g), thus proportional to Psio, and in view of eq. (9) proportional to p ~ in accord with the observations by Kaiser and Bresling. 3 Moreover, the rate of oxidation, i.e., formation of SiO(g) is proportional to p~), according to eqs. (Ta) and (8), Jsio =
2poIDo,/8 RTlfPo~. < Po~ (cr i t) o
.
o
o
(12)
Upon increasing the oxygen partial pressure in the bulk stream above p~, (crit), the SiO partial pressure Psio at the surface becomes greater than Psio (eq). Then formation of solid SiO2 becomes possible. Hence, after formation of a compact SiO~ layer on the silicon surface, the melt becomes saturated with respect to SiO~(s) corresponding to a constant oxygen content. Moreover, the formation of gaseous SiO by a reaction between liquid Si and O2(g) will cease. Instead, solid SiOz is formed but at a much lower rate. This corresponds to a state of passivity. When after formation of a SiO2 layer at p~, > 0.01 atm the oxygen partial pressure is gradually decreased, theoretical calculations 4 indicate that the SiO2 layer should be stable down to an oxygen partial pressure as low as 3.10 -8 atm in the bulk gas. Thus a hysteresis loop for the rate of SiO volatilization on increasing and decreasing oxygen partial pressures in the bulk gas analogous to that shown in Fig. 1 is expected. 4
Passivity and inhibition during the oxidation of metals at elevated temperatures
757
By and large, a hysteresis loop is always supposed to occur if there are two steady states each of which is stable with respect to differential perturbations. T H E O X I D A T I O N OF A g - I n A L L O Y S When A g - I n alloys containing a few at.~o In are exposed to oxygen of atmospheric pressure at 550°C, internal oxidation takes place, 5, s i.e. oxygen dissolves in.the alloy, diffuses into the interior and reacts with In dissolved in Ag. Thus a dispersion of tiny In2Oa crystals in a matrix of silver is formed. The higher the indium content of the alloy, the greater is the volume fraction of precipitated In2Oa in the zone of internal oxidation which impedes diffusion of oxygen into the interior of the alloy. Thus at high In contents of the alloy, no internal oxidation takes place. Instead, indium diffuses to the surface of the alloy where a protective layer of In2Oa is formed. According to Dietrich and Koch, 5 transition from internal to external oxidation takes place when the In content exceeds 9 at. ~ . After careful cleaning of the alloy surface by electroetching in l N HNOa, Rapp 6 found that the transition takes place at about 15 at. 70 In. An increase in the In content corresponds to a higher affinity of the reaction = ~1 InoOafs) In(in Ag) + ~30o(g) . .
(13)
In view of the low rate of diffusion in solid IntO3, formation of a non-porous In2Oa layer causes a considerable drop in the rate of oxidation of In dissolved in Ag to In208. Thus transition from internal to external oxidation is another example of passivity according to the definition suggested above. The affinity of reaction (13) can also be increased by providing a higher oxygen partial pressure. An increase in the oxygen partial pressure, however, does not effect a transition from internal to external oxidation. On the contrary, lowering of the oxygen partial pressure causes transition from internal to external oxidation according to both experimental observations e and theoretical considerations. 6,7 For instance, an alloy with 5 at. 70 In shows internal oxidation at Po, ---- 1 atm, but transition to external oxidation occurs when Po, is reduced to 10 -5 atm. At high oxygen partial pressures, outward diffusion of In plays only a minor part. Accordingly the content of In in form of InzO3 in a zone of internal oxidation is virtually equal to the In content of the original alloy. At low oxygen partial pressures, e.g. 10-4 or 10 -e atm, the solubility of oxygen in silver is very low. Thus the diffusion rate of oxygen is small and outward diffusion of In becomes significant. Therefore the content of In in form of In2Oa in the zone of internal oxidation becomes greater than the In content of the original alloy and thus the volume fraction of precipitated In20 a can reach a sufficiently high value for the transition from internal to external oxidation. Figure 2 shows the In content required for the formation of a dense In2Oa layer as a function of the outer oxygen partial pressure according to experiments conducted by Rapp. 6 From Fig. 2 one recognizes that a lowering of the oxidation rate corresponding to a transition from internal to external oxidation is not directly related to the change in the affinity. Transition from internal to external oxidation corresponding to a significant drop in the oxidation rate of In takes place (1) upon increasing the In
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CARL WAGNER
Ag-In TnzO30z(g)
~
°''-c
o,, 4;
12-
'
8-
4(Inlernal oxidation)
o
-'6
-'4
-'2
o
log Poz (atrn)
FIG. 2. Transition from internal to external oxidation of Ag-In alloys at 550°C with experimental points x for specimens exhibiting local perturbations of the penetration of internal oxidation, i.e. start of the transition, and with experimental points o for specimens without internal oxidation, i.e. completion of transition according to Rapp: content associated with an increase in the affinity, and (2) upon lowering the oxygen partial pressure associated with a decrease in the affinity. Consequently, the definition of passivity suggested above holds only if the effect of changes in the affinity due to variations of all parameters at constant temperature are considered and there is at least one parameter whose variation yields a decrease in reaction rate associated with an increase in the affinity of the reaction. THE OXIDATION OF Cu-Be ALLOYS The oxidation of Cu-Be alloys has been studied by Fr~Shlich, 8 Dennison and Preece, ~ and most recently by Maak. 1° When alloys with less than 6"6 at. ~ Be are exposed to air of atmospheric pressure at 850°C, the oxidation rate is about the same as for pure Cu. The scale comprises an outer thin layer of CuO, a thicker dense layer of CuzO, a porous layer containing both Cu20 and BeO, and a zone of internal oxidation with tiny BeO crystals in a copper matrixY ° The oxygen concentration in copper coexisting with Cu20 at 850°C is about 0.01 at. ~o O, n i.e. of the same order of magnitude as the oxygen concentration in silver saturated with oxygen of atmospheric pressure at 550°C. ~2 Thus internal oxidation of Be in Cu-Be alloys packed in Cu20 takes place x3 in the same manner as internal oxidation of In takes place in A g - I n alloys heated in oxygen. U p o n oxidizing a Cu-Be alloy with less than 6.6 at. ~o Be, both Cu20 and BeO may nucleate at the surface. In view of the diffusion of oxygen into the alloy, however, further oxidation of Be to BeO occurs in the interior of the alloy rather than at the surface. Thus a complex scale consisting of Cu~O, CuO, and BeO as described above is formed, x°
Passivity and inhibition during the oxidation of metals at elevated temperatures
759
In contrast, a Cu-Be alloy containing 12.6 at. % Be is oxidized at a much lower rate during formation of a very thin, dense outer layer of BeO. 1° The general behaviour of Cu-Be alloys is analogous to that of Ag-In alloys considered in the last Section inasmuch as in both systems an increase in the content of the alloying component results in a drop of the over-all oxidation rate due t'o the formation of an outer dense layer of the oxide of the alloying component. Each of the "two oxidation reactions in the system Cu-Be may be considered separately. Upon alloying Cu with increasing amounts of Be, the activity of Cu decreases slightly in view of Raoult's law, whereas the activity of Be increases according to Henry's law. Consequently, the affinity of the reaction
2 Cu(alloy) + 1%(g) = Cuo.O(s)
(14)
is slightly decreased, whereas the affinity of the reaction I
(15)
Befalloy) + ~ Oz(g) -----BeO(s)
is increased. Upon alloying Cu with an amount of Be sufficient for formation of a dense scale of BeO, the oxidation rate of Cu becomes virtually nil, whereas the oxidation rate of Be remains finite but much lower than for alloys with a smaller Be content which form a complex scale, see Table 1. Thus the changes in the affinity and the rate have opposite signs for reaction (15) but not for reaction (14). The occurrence of opposite signs for the changes in affinity and rate of one reaction, however, is sufficient for using the term passivity according to the definition suggested above. TABLE I. OXIDATION OF C u - B e ALLOYS. Anp~/A = NUMBER OF MOLE B e OXIDIZED PER UNIT AREA WITHIN I H AT 8 5 0 ° C . VALUES FOR THE FIRST FOUR ALLOYS WERE CALCULATED FROM MICROSCOPICAL EXAMINATIONS OF SECTIONED SAMPLES. t° THE VALUE FOR THE LAST ALLOY HAS BEEN DEDUCED FROM WEIGHT GAIN MEASUREMENTS. 1°
at. ~o Be 0-77 1.5 2.7 6.6 12.6
Scale ") BeO in Cu, ~ Cu20 + BeO, CusO, CuO BeO
IOs An~/A mole Be/cm2 1"7 2.5 4"3 8-0 0.5
THE OXIDATION OF Cu-Zn-Al ALLOYS The general behaviour of Cu-Zn and Cu-AI alloys subject to oxidation is very similar to the behaviour of Cu-Be alloys discussed in the last Section. In particular, on Cu-Zn alloys with less than l0 at. % Zn, Rhines and Nelson ~4 have found the same type of complex scale which is observed on Cu-Be alloys with less than 6.6 at. Be. Moreover, in each of the systems Cu-Al, a large decrease in the oxidation rate
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CARL WAGNER
occurs at higher concentrations of the second component because of formation of ZnO and AlzOs, respectively, as the principal constituent of the scale. For exclusive formation of ZnO or Al~O3, concentrations of the alloying component as high as about 20 at. 7o are necessary, a,9,xs,~7 Although the oxidation rate of a C u - Z n alloy containing 29 at. 70 Zn is much lower than that of pure Cu, the oxidation rate can still be reduced significantly by small additions of AI as has been shown by Dunn 15 (see Fig. 3). The oxidation rate of an alloy consisting of 29 at. 70 Zn and 4.7 at. ~o AI is about as low as that o f a Cu-AI alloy with 19 at. 7o AI, or that of an alloy with 11 at. 70 A1, which has been preoxidized in H20 -t- H2 in order to obtain a scale consisting exclusively of Ai2Oa. x8 Seemingly a dense A1203 layer is formed at a considerably lower AI concentration in the presence of Zn than in the absence of Zn. This may be explained in the following way. Cu-AI alloys with 1 or 2 at. ~o A1 yield a complex scale similar to that observed on Cu-Be alloys. Internal oxidation of AI occurs 13 since the oxygen concentration in the alloy
I00-80-6o
o4-
0'2
0.0 0
0'5
I'0 AL,
FIG. 3.
1'~5
~0
%
Weight increase Am/A = oxygen take-up of Cu-Zn alloys with 30 wt. ~. Zn and 0 to 2 wt. ~ AI during 3 h at 850°C according to Dunn? 6
next to the scale determined by the two-phase equilibrium Cu 4 - C u 2 0 is about 0.01 at. ~o at 850°C. n In the case of a C u - Z n alloy with 29 at. 70 Zn, however, the oxygen concentration in the alloy next to the ZnO layer (determined by a two-phase equilibrium between Zn in the alloy with an activity azn = 0"03 ~8 and ZnO) is about I0 e times lower than in Cu coexisting with Cu=O. Thus, in the case of a C u - Z n alloy with 29 at. ~o Zn, hardly any internal oxidation of AI can be expected. When the surface of a C u - Z n - A I alloy is initially bare, Cu20, ZnO and Al2Oa may be nucleated at the beginning of oxidation. In view of the depletion of AI in the alloy next to the surface, AI atoms migrate towards the surface without being converted to AI~O3 on their way to the surface. Hence there is a sufficient supply of A1 atoms with which oxygen reacts preferentially in comparison to Cu and Zn in view of the highly negative standard free energy of formation of AI20 3. A content of 4.7 at. 70 A1 is ten times greater than the minimum content of 0.4 at. 70 A1 which is required for sufficiently rapid diffusion in order to cover the consumption of AI during the growth of an A12Oa, scale? 9 In addition, AI atoms diffusing toward the surface may also convert
Passivity and inhibition during the oxidation of metals at elevated temperatures
761
initially formed nuclei of Cu20 and ZnO to A12Os by virtue of the displacement reactions 2Al+3CuzO=6Cu
+AI20 s
(16)
2 AI q- 3 ZnO = 3 Zn -t- AI2Os
(17)
Thus the scale is supposed to consist exclusively of AIzOs. Although zinc atoms do not enter the scale, presence of zinc is important because Zn acts as a getter for oxygen during the initial stage of oxidation and prevents diffusion of oxygen atoms into the interior of the alloy followed by internal precipitation of AIzO3 which would preclude the formation of an outer A1203 scale. Basically the lowering of the oxygen solubility due to the presence of zinc has the same effect as the lowering of the oxygen solubility in the investigations by Rapp s on Ag-In alloys by reducing the outer oxygen partial pressure from 1 atm to 10-~ atm. An analogous effect may be expected when one replaces zinc by another metal whose affinity for oxygen is intermediate between the affinities of Cu and AI for oxygen (referred to 1 g-atom of oxygen). Further research seems highly desirable in view of technological implications. The excellent oxidation resistance of Fe-Cr-AI alloys 2°.a2 may be interpreted in the same way. Since the affinity of AI for oxygen is much greater than that of Fe or Cr, the scale will be virtually pure AI2Oa. Presence of Cr may be important only because of its tendency to act as an oxygen getter and to minimize internal oxidation of AI. According to Hagel 2s the degree of protection provided by an AI~Oa layer also depends on whether a or 7 Al2Os is formed. The transformation from 7 to ct AI20 s may be enhanced by certain alloying constituents. This is another possibility to explain the beneficial effect of alloying elements which do not enter an AlaOs scale. THE OXIDATION OF Mn-C ALLOYS Carbon dissolved in 3' iron may be removed by virtue of the reaction C02(g) q- C(in alloy) = 2 COfg)
(18)
To attain a low carbon content in the alloy, the carbon activity in the gas mixture must be low. Thus the COs/CO ratio must be high but below the equilibrium ratio for coexistence of iron with wiistite, and moreover, the total pressure of the COs-CO mixture must be low. In the same way, carbon dissolved in manganese may, in principle, be removed. Upon increasing the affinity of the reaction by increasing the C O J C O ratio, one may expect first an increase in the rate of decarburization. At higher C O J C O ratios, however, a scale of MnO is formed which prevents removal of carbon according to reaction (18) as is shown by experiments where Mn-C alloys were oxidized in air at 1000°C. ~4 According to the foregoing phenomenological definition, this is another case of passivity with a sharp decrease in the rate when the affinity of the reaction is increased. In contradistinction to the oxidation of Ag-In and Cu-Be alloys considered above, passivity is due to the formation of a layer of the oxide of the base metal Mn whose oxidation rate increases with an increasing C O J C O ratio while
762
CAgE WAGNER
the oxidation of carbon ceases. From a practical point of view, however, this case o f passivity is not interesting. THE I N H I B I T I O N OF THE OXIDATION OF N I C K E L IN THE PRESENCE OF Li20 AND RELATED PHENOMENA Pfeiffer and Hauffe 25 have found that the rate of oxidation of nickel at 1000°C is considerably reduced in the presence ol~ LifO vapour. Since presence of Li20 does not cause a significant change in the affinity of the oxidation of Ni to NiO, this is a typical example of inhibition according to the definition suggested above. The decrease in the oxidation rate is due to the dissolution of Li20 in solid NiO associated with a decrease in the concentration of cation vacancies which are essential for the migration of Ni ~+ ions across the NiO scale. Likewise, one may expect a decrease in the oxidation rate of Zn in the presence of oxides with a cation-to-anion ratio less than unity whereby cation vacancies are formed and interstitial ions are captured, which are essential for the migration of zinc ions across a ZnO scale. In accord herewith Gensch and Hauffe 26 have found that the oxidation rate of Zn at 400°C is lowered considerably by the presence of AI. In this system, however, the change in the oxidation rate may be due to the formation of a scale of AI2Oz rather than to a decrease in the concentration of interstitial zinc ions in ZnO, since ZnO on a Z n - A I alloy is thermodynamically not stable in view of the high free energy drop of the displacement reaction (17). Another example of inhibition is the decrease in the rate of chlorination or bromination of silver in the presence of small amounts of Cd as an alloying element. Since the silver halides exhibit predominantly ionic conduction, the migration of electrons via holes in AgC1 or AgBr is the rate determining step. 27 Dissolution of cadmium halide in AgCI or AgBr reduces the concentration of electron holes and accordingly the rate of the reaction between Ag and C12 or Br2. ~,~9
THE BEHAVIOUR OF A METAL IN A GAS PHASE I N V O L V I N G TWO O X I D I Z I N G COMPONENTS When nickel is heated in an Ar-I2 stream with a sufficiently low I2 content, the reaction Ni(s) + 12(g) = NiI2(g)
(19)
takes place. The ratejNixl is mainly determined by the diffusion of 12across the boundary layer to the Ni surface. In view of equations analogous to eqs. (7a) and (8), one has JNil~ = Jl,
=
DI 2
P~2/a R T
(20)
No solid NiI2 is formed unless the partial pressure PNa, at the metal surface exceeds the saturation pressure of NiI2. U p o n addition of a small amount of 02 to the A r - I 2 stream, a scale of solid NiO(s) is supposed to form,
Passivity and inhibition during the oxidation of metals at elevated temperatures 1
Ni(s) -4- : O2(g) = NiO(s) 2
763 (21)
and Nilz is formed by the consecutive reaction
I
NiO(s) + I2(g) = NiIz(g) + = Oz(g) 2
(22)
The rate of the reaction in eq. (22) is determined by the rate of diffusion of Nilz(g) from the NiO surface into the bulk stream according to an equation analogous to eq. (7b), JNilz = DNilzPNit2(eq)/t~ R T
(23)
where pNil,(eq) is the equilibrium partial pressure of Nil,, for reaction (22). The standard free energy change of reaction (22) is about 37 kcal at 700°C 3°,a~ corresponding to a ratio PNiI,(eq)/PI~. = 10-7atpo, = 0 . 0 1 atm. Hence the rate of Nilz formation at a surface covered by NiO according to eq, (22) is about 107 times lower than at a bare Ni surface according to (19). The lowering of the rate due to presence of oxygen is a typical case of inhibition rather than passivity since oxygen does not alter the affinity of the basic reaction (19). A more involved situation occurs when in the presence of two oxidizers two solid metal compounds can be formed. As an example consider silver heated in a mixture of sulphur and iodine vapour. At 200°C the reaction 2 Ag(s) + 21 S2 (g) = AgzS(s)
(24)
is much faster than the reaction 2 Ag(s) q- Iz(g) = 2 AgI(s)
(25)
Since the free energy drop for reaction (25) is much greater than that for reaction (24), AgI is much more stable than Ag2S. It is open to question, however, which partial pressure of I2 is actually required in order to enforce exclusive formation of Ag! and to prevent the rapid attack of silver by sulphur vapour. C O N C L U D I N G REMARKS The foregoing examples of observations on the oxidation of metals at elevated temperatures show that there is indeed a large variety of phenomena which may be classified either as passivity or as inhibition. This list of typical examples has not been drawn up because of satisfaction in classification but is intended to help other investigators in finding new ways for minimizing corrosion at elevated temperatures. Most remarkable are abrupt decreases in the rate of reactions such as the oxidation of Zn or Si in Ar-O2 mixtures, the oxidation of A g - I n and Cu-Be alloys, and the decarburization of M n - C alloys when the oxygen partial pressure or the concentration of an alloying metal is increased by a rather minor extent. In each of these cases, the
764
CARL WAGNER
transition from a bare metal surface to an oxide-covered surface or the f o r m a t i o n o f another oxide phase in the scale is decisive for the a b r u p t decrease of the rate. Theoretically, even a differential increase in the oxygen partial pressure may suffice in order to effect the a b r u p t decrease in the rate of oxidation of Si or Z n in A r - O z mixtures. Experimentally, however, one finds a certain range for the transition from a high to a low rate, where results are not well reproducible. O n the other hand, there are also cases with a c o n t i n u o u s decrease in rate. Such a behaviour occurs especially when the phase structure of a scale is not changed but the concentration of interstitial ions, ion vacancies, excess electrons, or electron holes is changed u p o n increasing the c o n c e n t r a t i o n of a " d o p e " . REFERENCES I. E. T. TURKDOGAN,P. GRIEVESONand L. S. DARKEN,J. Metals 14, 521 (1962); J. phys. Chem. 67, 1647 (1963). 2. C. GENSCHand K. HAUFFE,Z. phys. Chem. 196, 427 (1951). 3. W. KAISERand J. BRESLIN,J. appl. Phys. 29, 1292 (1958). 4. C. WAGNER,J. appl. Phys. 29, 1295 (1958). 5. I. DIETRICHand L. KocH, Z. Metallk. 50, 31 (1959). 6. R. A. RAPP, Acta Met. 9, 730 (1961). 7. C. WAGNER, Z. Elektrochem. 63, 772 (1959). 8. K. W. FROHLICH,Z. Metallk. 28, 368 (1936). 9. J. P. DEr,n~ISONand A. PREEeE,J. Inst. Metals 81,229 0952/53). 10. F. MAAK,Z. Metallk. 52, 538, 545 (1961). I I. M. HANSENand K. ANDERKO,Constitution of binary alloys, p. 604, McGraw-Hill Publishing Co,. New York (1958). 12. W. EIC,ENAUERand G. MOLLER,Z. Metallk. 53, 321,700 (1962). 13. F. N. RmN~, W. A. JOHNSONand W. A. ANDERSON,Trans. Am. Inst. Min. Metall. Engrs. 147, 205 (1942). 14. F. N. RHINESand B. J. NELSON,Trans. Am. Inst. Min. Metall. Engrs. 156, 171 (1944). 15. J. S. DtmN, J. Inst. Metals46, 25 (1931). 16. L. E. PRICEand G. J. THOMAS,J. Inst. Metals 63, 21 0938). 17. P. SPINEDI,Metallurgia ira/. 45,457 (1953). 18. W. SErrH and W. KRAUS,Z. Elektrochem. 44, 98 (1938). 19. C. WAGNER,J. Electrochem. Soc. 99, 369 (1952). 20. A. POR'rEvIN,E. PRaTEr and H. JOLIVET,Rev. Mdtall. 31, 101, 186, 219 (1934). 21. E. SCI-IEILand E. H. SCHULZ,Arch. EisenhiittWes. 6, 155 (1932/33). 22. E. SCHEILand K. KIWIT,Arch. EisenhiittWes. 9, 405 (1935/36). 23. W. C. HAGEL,Proc. Second lnternat. Congress on Metallic Corrosion, in press. 24. W. W. WEan, J. T. NORTONand C. WAG~ER,J. Electrochem. Soc. 103, 112 0956). 25. H. PrEIn:ERand K. HAUFFE,Z. Metallk. 43, 364 0952). 26. C. GENSCrland K. HAtrrEE, Z. phys. Chem. 196, 427 (1951). 27. C. WAGNER,Z. phys. Chem. B 32, 447 (1936). 28. W. HIMMLER,Z. phys. Chem. 195, 129 (1950). 29. K. HAUFFEand C. GENSCH,Z. phys. Chem. 195, 116 0950). 30. H. SCHXFER,H. JACOBand K. ETZEL,Z. anorg. Chem. 286, 42 (1956). 31. K. KIUKKOLAand C. WAGEreR,J. Electrochem. Soc. 104, 379 (1957).
PASSIVITY A N D INHIBITION D U R I N G T H E O X I D A T I O N OF METALS AT ELEVATED TEMPERATURES* CARL WAGNER Max-Planck-Institut ftir physikalische Chemie, G6ttingen, Gerraany. Abstract--Typical phenomena of passivity and inhibition during the oxidation of metals and alloys at elevated temperatures are considered in order to stimulate the development of appropriate methods for minimizing corrosion at elevated temperatures. R~sum6--On consid6re des ph6nom6nes typiques de passivit6 et d'inhibition pendant l'oxydation de m6taux et d'alliages aux temp6ratures 61evdes,et cela en rue de stimuler le d6veloppement de m6thodes appropri6es pour minimiser la corrosion gt ces temperatures. Zusanunenfassung--Clberlegungen iJber typische Erscheinungen von Passivit~it und Inhibition bei der Oxydation von Metallen und Legierungen bei h6heren Temperaturen werden angestellt, um die Entwicklung angemessener Methoden fiir die Einschrfinkung der Hochtemperaturkorrosion anzuregen.
Pe~epaT- C tte~bm CTHMy.rIIIpOBaHIIFIpaaBnv~ COOTBeTCTByIOIRHXMeTO~IOByMeHbmeHtifl ~¢oppoaH~ npa nOB~meHH~X TeMnepaTypax paccMaTp~IBalOTCa TnnnqH~e ~BJleHHff rlaCcHBnOCT~i II ~IHr~6~IpoBan~m eo npeMR OKHcJIeHHfl Mevaa~oB ~i CIIJIaBOB npa B~COHnX TeMnepaTypax. INTRODUCTION TWO well-known examples o f passivity are: (1) U p o n increasing the single electrode potential o f an iron electrode in l N H2SOo iron dissolves a n o d i c a l l y at a g r a d u a l l y increasing rate. A t a b o u t 0.47 V the rate decreases by a b o u t 4 o r d e r s o f m a g n i t u d e a n d becomes virtually i n d e p e n d e n t o f the electrode potential in a r a t h e r wide range. (2) I r o n dissolves readily in 0.1 M nitric acid but is virtually not a t t a c k e d by pure nitric acid. In b o t h cases the low rate o f dissolution under m o r e strongly oxidizing c o n d i t i o n s is ascribed to the f o r m a t i o n o f a dense film o f i r o n oxide which acts as a b a r r i e r for the transfer o f iron ions from the metal to the solution. U s i n g passivity as a purely p h e n o m e n o l o g i c a l term, one m a y state t h a t a metal is passive (a) if the steady-state rate o f a n o d i c dissolution o f a metal in a given e n v i r o n m e n t is lower at a m o r e noble single electrode potential t h a n at a less noble potential, o r (b) if u p o n increasing the c o n c e n t r a t i o n o f an oxidizing agent in an a m b i e n t solution the steady-state dissolution rate (without flow o f external current) is f o u n d to be less t h a n the dissolution rate at a lower c o n c e n t r a t i o n o f the oxidizing agent. These two definitions are equivalent u n d e r c o n d i t i o n s where the electrochemical m e c h a n i s m o f c o r r o s i o n applies. T h e decrease o f the dissolution rate o f a metal u p o n shifting the single electrode potential to a m o r e noble value a n d thus increasing the driving force represents a *Manuscript received 25 May 1965. 751
752
CARL WAGNER
special case of a negative differential resistance, which is frequently encountered in circuit elements of electrical engineering, e.g. amplifiers and wave generators. The foregoing phenomenological definition may be generalized. A metal may be called passive when the amount of metal consumed by a chemical or electrochemical reaction in a given time is significantly less under conditions corresponding to a higher affinity of the reaction (i.e. a greater de.crease in free energy) than under conditions corresponding to a lower affinity. The foregoing definition, however, does not cover all phenomena which are usually labelled as passivity, especially in the case of alloys. Thus a further generalization of the foregoing definition is suggested. A metal or alloy is called passive when the amount o f at least one o f the metallic components consumed by a chemical or electrochemical reaction in a given time is significantly lower at a higher affinity than at a lower affinity. In this definition the occurrence of a negative differential resistance is kept
as the most significant characteristic of the term passivity when it was first introduced by Sch6nbein. In spite of the foregoing rather general definition, however, it remains open to question whether this definition is applicable to all possible complex situations. A phenomenon closely related to passivity is inhibition. In contradistinction to passivity, the term inhibition is suggested for cases where at a constant affinity the rate of a reaction is significantly decreased upon changing the concentration or activity of a component which is not involved in the reaction under consideration. Using the foregoing definitions, one may ask whether phenomena of passivity and inhibition occur when metals or alloys are subject to oxidation at elevated temperatures. Typical examples are considered in what follows. The clarification of the mechanism is definitely the goal of research on corrosion. Nevertheless, it is helpful to use certain phenomenological terms before the mechanism has been ascertained. Detailed investigations show that many cases of both passivity and inhibition are due to the formation of well defined layers of metal compounds, especially metal oxides, or the formation of adsorption layers. Once the mechanism has been ascertained, use of terms such as passivity or inhibition becomes less relevant since these terms convey only a limited amount of information. THE OXIDATION OF COPPER, IRON, ZINC AND OTHER METALS AT TEMPERATURES WHERE VOLATILIZATION OF METALS IS SIGNIFICANT When copper is heated in a stream of air at 1000°C, a dense Cu~O scale is formed after a short induction period. Further oxidation takes place inasmuch as copper ions and electrons migrate across the Cu20 layer to the outer surface where they react with 02 molecules in order to form Cu20. The Cu20 layer confining the oxidation rate is analogous to an oxide layer on iron passivated by anodic polarization. In spite of this analogy, Cu covered by Cu20 at 1000°C is in general not classified as a passivated metal because the oxidation rate is not significantly lower than the rate observable under less strongly oxidizing conditions, e.g. at low oxygen partial pressures. Turkdogan, Grieveson, and Darken, 1 however, have recently shown that it is possible to realize conditions corresponding to an active and a passive state during the oxidation of Cu, Fe, Ni, and other metals at temperatures where the vapour pressure of the metal
Passivity and inhibition during the oxidation of metals at elevated temperatures
753
is appreciable and the vapour pressure of the metal oxide is insignificant. When under these conditions a metal is exposed to an oxygen-argon stream containing a low oxygen content, metal atoms vaporize from the surface and diffuse into the boundary layer up to a certain distance ~ where they encounter and react with O2 molecules which in turn diffuse into the boundary layer from the bulk stream. At the plane of encounter, metal atoms and O~ molecules form clusters of solid (or liquid) metal oxide which are carried away into the bulk stream by Brownian movement and convection and finally escape as fume. At the distance 8 from the metal surface at which the formation of solid (or liquid) metal oxide takes place, the partial pressures of metal atoms and oxygen molecules can be assumed to be vanishingly small. The fluxes of metal atoms, Jue, and O~ molecules,jo ,, must be equivalent. Thus JM~
--
DM~p~¢ 2 . -J o RT v
2 Do~p~~ v (A--5) R T
, - - - -
(1)
where v is the number of oxygen atoms reacting with one atom of metal for the formation of solid (or liquid) metal oxide, Dm~ and Dot are the diffusivities of metal atoms and Oz molecules, respectively, p ~ is the metal vapour pressure at the surface of the metal, p~, is the oxygen partial pressure in the bulk stream, and A is the effective thickness of the total diffusion boundary layer. From eq. (1) the value of S is found to be =
A 1 -t- (2Do, p~,/vDmcp~e)
(2)
In most cases DM, = Do, and p~, >> P~c and therefore ~ ,~ A, i.e. the fluxes of metal atoms and 02 molecules meet in the very vicinity of the metal surface. Substituting eq. (2) in eq. (I), one obtains for the rate of metal vaporization, which is equal to the rate of formation of metal oxide as fume, ° [ I ff V JM~=2D°'P°* 2DM~p~,] o vART Do, Po ,
(3a)
or
JMo -~ 9/)°---°'P°*if DMo = Do,, Po,° >> P~t, vART
(3b)
When the oxygen partial pressure p~, in the bulk stream is gradually increased, the value of ~ decreases according to eq. (2) and the rate JMe increases proportionally to p~, according to eq. (3b). The ratejMe determined by eq. (3b) can reach but not exceed the maximum vaporization rate of the metal into a vacuum according to the HertzLangmuir equation, JMc (max) = p~,/(2~MMo RT) 112 where MM© is the atomic weight of the metal.
(4)
754
CARL WAGNER
Upon equating the right-hand members of eqs. (3b) and (4), the corresponding critical oxygen partial pressure p~, (crit) in the bulk stream is found to be
vApMe (RT) 112 p~, (crit)
~2-D~~ - M ~ 2
(5)
The value o f p ~ (crit) according to eq. (5) depends on the effective thickness A of the diffusion boundary layer and, therefore, on the flow rate and the dimensions of the sample. For example, for a copper sample 2 cm long at 1200°C (p~:, = 3.10 -6 atm), p°Oa (crit) was found to be 0.0035 atm at a flow rate of 40 cm/sec but only 0.0020 atm at a flow rate of 80 cm/sec. 1 When Po2 in the bulk stream is increased beyond P~2 (crit), the foregoing mechanism breaks down, i.e. in view of the supply of excess oxygen to the surface, the metal becomes saturated with respect to metal oxide, and a protective layer of metal oxide is formed. If a solid oxide layer is formed, its thickness first increases proportionally to the square root of time. After a sufficiently long time, the thickness is supposed to approach a steady-state value when the diffusion rate of ions and electrons across the solid oxide layer balances the rate of vaporization of metal oxide followed by diffusion across the gaseous boundary layer. In general, the vapour pressure of the metal oxide is much smaller than p~. (crit). Then, the steady-state corrosion rate after formation of a solid oxide layer is much lower than that of the bare metal. A stationary thickness of an oxide layer during the high temperature oxidation of a metal is analogous to the stationary thickness of an oxide film on passivated iron in HNO3 at room temperature when the flux of iron ions and electrons is equivalent to the rate of oxide dissolution. The decrease in the rate of metal oxidation after formation of a layer of solid metal oxide is especially large for zinc. At 400°C p~,, equals 1.0 × 10 -4 arm and, therefore, the maximum rate of vaporization ( = rate of formation of ZnO fume) is 2.5 × I0 -s mole Zn/cm 2 when the surface is free of oxide and other impurities which may impede vaporization. In contrast, when a layer of ZnO has been formed, the rate of oxidation is as low as 2.5 × 10 -1° mole Zn/cm 2 sec after one hour. 2 When after formation of a ZnO layer p~2 is decreased below p~. (crit), the ZnO layer remains stable since in view of the very low consumption of oxygen the oxygen partial pressure at the ZnO surface is virtually equal to that in the bulk stream. Thus upon first increasing p ~ and subsequently lowering p ~ , one has to expect a hysteresis loop as is schematically shown in Fig. 1. Small amounts of adsorbed molecules (less than required for the formation of a monolayer) may decrease the rate of vaporization of a solid metal considerably below the rate given by the Hertz-Langmuir equation (4) when the release of metal atoms from kinks is blocked. According to the definitions given above, this is a case of inhibition. By and large, presence of impurities in the gas phase or in the metal may diminish p~, (crit) considerably below the upper limit in eq. (5). THE OXIDATION
OF SILICON
AT E L E V A T E D
TEMPERATURES
Passing a stream of He and 02 of atmospheric pressure with a bulk oxygen partial pressure pOo, over liquid silicon at 1410°C, Kaiser and Breslin 3 observed that the
Passivity and inhibition during the oxidation of metals at elevated temperatures
PO~ FIG. 1.
755
PO~ (crit)
Corrosion rate of zinc in an argon-oxygen stream as a function of oxygen partial pressure.
steady-state oxygen content of the molten silicon is proportional to p°o.. ifPo2 ° < 0.01 atm but constant if p°o, > 0-01 atm. These observations may be interpreted in the following way. 4 At low oxygen partial pressures in the bulk stream, the supply of oxygen to the silicon surface is low. Then only gaseous SiO is formed by the reaction Si(l) + ½0z(g) = SiO(g).
f6)
As long as the silicon surface is bare, i.e. not covered by solid SiOz, oxygen is readily consumed and, therefore, the concentration of 02 at the silicon surface is vanishingly small. Thus the flux Jo2 across the hydrodynamic diffusion boundary layer with an effective thickness ~5 is
Jo, = P~,DoJ6RT For zero concentration of SiO in the bulk stream, the the silicon surface is Jsio =
PsioDsio/t3 RT
(7a)
~]UXjsioof SiO away from (7b)
where Psio is the partial pressure of SiO at the silicon surface, Dsi o is the diffusivity of SiO, and the effective thickness of the diffusion boundary layer is assumed to be the same as for diffusion of O2 in eq. (7a) for the sake of an approximation. Under steady-state conditions the net transport rate of oxygen in form of 02 or SiO must vanish. Thus
2jo, =Aio
(8)
Substituting eqs. (7a) and (7b) in eq. (8), solving for P~io, and letting Dslo = Do, as a fair approximation, one has
756
CARL WAGNER
Psio = 2pg,
(9)
IfPsio according to eq. (9) is lower than the equilibrium partial pressure Psio(eq) for the reaction I Si(1) q- 1 S i O z f s ) = SiO(g)
(10)
the silicon surface remains bare. At 1410°C, Psio(eq) equals 0.015 atm. Thus, in view of eq. (9), the critical oxygen partial pressure p~, (crit) in the bulk stream for keeping the silicon surface bare is estimated to be o
1
Po, (crit) - ~Psio (eq) = 0.008 atm
(1 la)
In a more accurate calculation, different values of the diffusivities and the effective thickness of the boundary layer for diffusion of 02 and SiO have to be used. Then one obtains 4 o
1
x2
Po~ (crit) ----~ (Dsio/Dot) / Psio(eq) -----0.006 atm
(lib)
In contradistinction to eq. (5), the critical oxygen partial pressure is determined mainly by the equilibrium partial pressure Psio (eq) but does not depend on the effective thickness of the boundary layer, i.e. on the flow rate of the helium-oxygen mixture, or the dimensions of the sample. As long as the silicon surface is bare, the oxygen content of the melt is determined by the equilibrium Si(1) + O(in Si) ---- SiO(g), thus proportional to Psio, and in view of eq. (9) proportional to p ~ in accord with the observations by Kaiser and Bresling. 3 Moreover, the rate of oxidation, i.e., formation of SiO(g) is proportional to p~), according to eqs. (Ta) and (8), Jsio =
2poIDo,/8 RTlfPo~. < Po~ (cr i t) o
.
o
o
(12)
Upon increasing the oxygen partial pressure in the bulk stream above p~, (crit), the SiO partial pressure Psio at the surface becomes greater than Psio (eq). Then formation of solid SiO2 becomes possible. Hence, after formation of a compact SiO~ layer on the silicon surface, the melt becomes saturated with respect to SiO~(s) corresponding to a constant oxygen content. Moreover, the formation of gaseous SiO by a reaction between liquid Si and O2(g) will cease. Instead, solid SiOz is formed but at a much lower rate. This corresponds to a state of passivity. When after formation of a SiO2 layer at p~, > 0.01 atm the oxygen partial pressure is gradually decreased, theoretical calculations 4 indicate that the SiO2 layer should be stable down to an oxygen partial pressure as low as 3.10 -8 atm in the bulk gas. Thus a hysteresis loop for the rate of SiO volatilization on increasing and decreasing oxygen partial pressures in the bulk gas analogous to that shown in Fig. 1 is expected. 4
Passivity and inhibition during the oxidation of metals at elevated temperatures
757
By and large, a hysteresis loop is always supposed to occur if there are two steady states each of which is stable with respect to differential perturbations. T H E O X I D A T I O N OF A g - I n A L L O Y S When A g - I n alloys containing a few at.~o In are exposed to oxygen of atmospheric pressure at 550°C, internal oxidation takes place, 5, s i.e. oxygen dissolves in.the alloy, diffuses into the interior and reacts with In dissolved in Ag. Thus a dispersion of tiny In2Oa crystals in a matrix of silver is formed. The higher the indium content of the alloy, the greater is the volume fraction of precipitated In2Oa in the zone of internal oxidation which impedes diffusion of oxygen into the interior of the alloy. Thus at high In contents of the alloy, no internal oxidation takes place. Instead, indium diffuses to the surface of the alloy where a protective layer of In2Oa is formed. According to Dietrich and Koch, 5 transition from internal to external oxidation takes place when the In content exceeds 9 at. ~ . After careful cleaning of the alloy surface by electroetching in l N HNOa, Rapp 6 found that the transition takes place at about 15 at. 70 In. An increase in the In content corresponds to a higher affinity of the reaction = ~1 InoOafs) In(in Ag) + ~30o(g) . .
(13)
In view of the low rate of diffusion in solid IntO3, formation of a non-porous In2Oa layer causes a considerable drop in the rate of oxidation of In dissolved in Ag to In208. Thus transition from internal to external oxidation is another example of passivity according to the definition suggested above. The affinity of reaction (13) can also be increased by providing a higher oxygen partial pressure. An increase in the oxygen partial pressure, however, does not effect a transition from internal to external oxidation. On the contrary, lowering of the oxygen partial pressure causes transition from internal to external oxidation according to both experimental observations e and theoretical considerations. 6,7 For instance, an alloy with 5 at. 70 In shows internal oxidation at Po, ---- 1 atm, but transition to external oxidation occurs when Po, is reduced to 10 -5 atm. At high oxygen partial pressures, outward diffusion of In plays only a minor part. Accordingly the content of In in form of InzO3 in a zone of internal oxidation is virtually equal to the In content of the original alloy. At low oxygen partial pressures, e.g. 10-4 or 10 -e atm, the solubility of oxygen in silver is very low. Thus the diffusion rate of oxygen is small and outward diffusion of In becomes significant. Therefore the content of In in form of In2Oa in the zone of internal oxidation becomes greater than the In content of the original alloy and thus the volume fraction of precipitated In20 a can reach a sufficiently high value for the transition from internal to external oxidation. Figure 2 shows the In content required for the formation of a dense In2Oa layer as a function of the outer oxygen partial pressure according to experiments conducted by Rapp. 6 From Fig. 2 one recognizes that a lowering of the oxidation rate corresponding to a transition from internal to external oxidation is not directly related to the change in the affinity. Transition from internal to external oxidation corresponding to a significant drop in the oxidation rate of In takes place (1) upon increasing the In
758
CARL WAGNER
Ag-In TnzO30z(g)
~
°''-c
o,, 4;
12-
'
8-
4(Inlernal oxidation)
o
-'6
-'4
-'2
o
log Poz (atrn)
FIG. 2. Transition from internal to external oxidation of Ag-In alloys at 550°C with experimental points x for specimens exhibiting local perturbations of the penetration of internal oxidation, i.e. start of the transition, and with experimental points o for specimens without internal oxidation, i.e. completion of transition according to Rapp: content associated with an increase in the affinity, and (2) upon lowering the oxygen partial pressure associated with a decrease in the affinity. Consequently, the definition of passivity suggested above holds only if the effect of changes in the affinity due to variations of all parameters at constant temperature are considered and there is at least one parameter whose variation yields a decrease in reaction rate associated with an increase in the affinity of the reaction. THE OXIDATION OF Cu-Be ALLOYS The oxidation of Cu-Be alloys has been studied by Fr~Shlich, 8 Dennison and Preece, ~ and most recently by Maak. 1° When alloys with less than 6"6 at. ~ Be are exposed to air of atmospheric pressure at 850°C, the oxidation rate is about the same as for pure Cu. The scale comprises an outer thin layer of CuO, a thicker dense layer of CuzO, a porous layer containing both Cu20 and BeO, and a zone of internal oxidation with tiny BeO crystals in a copper matrixY ° The oxygen concentration in copper coexisting with Cu20 at 850°C is about 0.01 at. ~o O, n i.e. of the same order of magnitude as the oxygen concentration in silver saturated with oxygen of atmospheric pressure at 550°C. ~2 Thus internal oxidation of Be in Cu-Be alloys packed in Cu20 takes place x3 in the same manner as internal oxidation of In takes place in A g - I n alloys heated in oxygen. U p o n oxidizing a Cu-Be alloy with less than 6.6 at. ~o Be, both Cu20 and BeO may nucleate at the surface. In view of the diffusion of oxygen into the alloy, however, further oxidation of Be to BeO occurs in the interior of the alloy rather than at the surface. Thus a complex scale consisting of Cu~O, CuO, and BeO as described above is formed, x°
Passivity and inhibition during the oxidation of metals at elevated temperatures
759
In contrast, a Cu-Be alloy containing 12.6 at. % Be is oxidized at a much lower rate during formation of a very thin, dense outer layer of BeO. 1° The general behaviour of Cu-Be alloys is analogous to that of Ag-In alloys considered in the last Section inasmuch as in both systems an increase in the content of the alloying component results in a drop of the over-all oxidation rate due t'o the formation of an outer dense layer of the oxide of the alloying component. Each of the "two oxidation reactions in the system Cu-Be may be considered separately. Upon alloying Cu with increasing amounts of Be, the activity of Cu decreases slightly in view of Raoult's law, whereas the activity of Be increases according to Henry's law. Consequently, the affinity of the reaction
2 Cu(alloy) + 1%(g) = Cuo.O(s)
(14)
is slightly decreased, whereas the affinity of the reaction I
(15)
Befalloy) + ~ Oz(g) -----BeO(s)
is increased. Upon alloying Cu with an amount of Be sufficient for formation of a dense scale of BeO, the oxidation rate of Cu becomes virtually nil, whereas the oxidation rate of Be remains finite but much lower than for alloys with a smaller Be content which form a complex scale, see Table 1. Thus the changes in the affinity and the rate have opposite signs for reaction (15) but not for reaction (14). The occurrence of opposite signs for the changes in affinity and rate of one reaction, however, is sufficient for using the term passivity according to the definition suggested above. TABLE I. OXIDATION OF C u - B e ALLOYS. Anp~/A = NUMBER OF MOLE B e OXIDIZED PER UNIT AREA WITHIN I H AT 8 5 0 ° C . VALUES FOR THE FIRST FOUR ALLOYS WERE CALCULATED FROM MICROSCOPICAL EXAMINATIONS OF SECTIONED SAMPLES. t° THE VALUE FOR THE LAST ALLOY HAS BEEN DEDUCED FROM WEIGHT GAIN MEASUREMENTS. 1°
at. ~o Be 0-77 1.5 2.7 6.6 12.6
Scale ") BeO in Cu, ~ Cu20 + BeO, CusO, CuO BeO
IOs An~/A mole Be/cm2 1"7 2.5 4"3 8-0 0.5
THE OXIDATION OF Cu-Zn-Al ALLOYS The general behaviour of Cu-Zn and Cu-AI alloys subject to oxidation is very similar to the behaviour of Cu-Be alloys discussed in the last Section. In particular, on Cu-Zn alloys with less than l0 at. % Zn, Rhines and Nelson ~4 have found the same type of complex scale which is observed on Cu-Be alloys with less than 6.6 at. Be. Moreover, in each of the systems Cu-Al, a large decrease in the oxidation rate
760
CARL WAGNER
occurs at higher concentrations of the second component because of formation of ZnO and AlzOs, respectively, as the principal constituent of the scale. For exclusive formation of ZnO or Al~O3, concentrations of the alloying component as high as about 20 at. 7o are necessary, a,9,xs,~7 Although the oxidation rate of a C u - Z n alloy containing 29 at. 70 Zn is much lower than that of pure Cu, the oxidation rate can still be reduced significantly by small additions of AI as has been shown by Dunn 15 (see Fig. 3). The oxidation rate of an alloy consisting of 29 at. 70 Zn and 4.7 at. ~o AI is about as low as that o f a Cu-AI alloy with 19 at. 7o AI, or that of an alloy with 11 at. 70 A1, which has been preoxidized in H20 -t- H2 in order to obtain a scale consisting exclusively of Ai2Oa. x8 Seemingly a dense A1203 layer is formed at a considerably lower AI concentration in the presence of Zn than in the absence of Zn. This may be explained in the following way. Cu-AI alloys with 1 or 2 at. ~o A1 yield a complex scale similar to that observed on Cu-Be alloys. Internal oxidation of AI occurs 13 since the oxygen concentration in the alloy
I00-80-6o
o4-
0'2
0.0 0
0'5
I'0 AL,
FIG. 3.
1'~5
~0
%
Weight increase Am/A = oxygen take-up of Cu-Zn alloys with 30 wt. ~. Zn and 0 to 2 wt. ~ AI during 3 h at 850°C according to Dunn? 6
next to the scale determined by the two-phase equilibrium Cu 4 - C u 2 0 is about 0.01 at. ~o at 850°C. n In the case of a C u - Z n alloy with 29 at. 70 Zn, however, the oxygen concentration in the alloy next to the ZnO layer (determined by a two-phase equilibrium between Zn in the alloy with an activity azn = 0"03 ~8 and ZnO) is about I0 e times lower than in Cu coexisting with Cu=O. Thus, in the case of a C u - Z n alloy with 29 at. ~o Zn, hardly any internal oxidation of AI can be expected. When the surface of a C u - Z n - A I alloy is initially bare, Cu20, ZnO and Al2Oa may be nucleated at the beginning of oxidation. In view of the depletion of AI in the alloy next to the surface, AI atoms migrate towards the surface without being converted to AI~O3 on their way to the surface. Hence there is a sufficient supply of A1 atoms with which oxygen reacts preferentially in comparison to Cu and Zn in view of the highly negative standard free energy of formation of AI20 3. A content of 4.7 at. 70 A1 is ten times greater than the minimum content of 0.4 at. 70 A1 which is required for sufficiently rapid diffusion in order to cover the consumption of AI during the growth of an A12Oa, scale? 9 In addition, AI atoms diffusing toward the surface may also convert
Passivity and inhibition during the oxidation of metals at elevated temperatures
761
initially formed nuclei of Cu20 and ZnO to A12Os by virtue of the displacement reactions 2Al+3CuzO=6Cu
+AI20 s
(16)
2 AI q- 3 ZnO = 3 Zn -t- AI2Os
(17)
Thus the scale is supposed to consist exclusively of AIzOs. Although zinc atoms do not enter the scale, presence of zinc is important because Zn acts as a getter for oxygen during the initial stage of oxidation and prevents diffusion of oxygen atoms into the interior of the alloy followed by internal precipitation of AIzO3 which would preclude the formation of an outer A1203 scale. Basically the lowering of the oxygen solubility due to the presence of zinc has the same effect as the lowering of the oxygen solubility in the investigations by Rapp s on Ag-In alloys by reducing the outer oxygen partial pressure from 1 atm to 10-~ atm. An analogous effect may be expected when one replaces zinc by another metal whose affinity for oxygen is intermediate between the affinities of Cu and AI for oxygen (referred to 1 g-atom of oxygen). Further research seems highly desirable in view of technological implications. The excellent oxidation resistance of Fe-Cr-AI alloys 2°.a2 may be interpreted in the same way. Since the affinity of AI for oxygen is much greater than that of Fe or Cr, the scale will be virtually pure AI2Oa. Presence of Cr may be important only because of its tendency to act as an oxygen getter and to minimize internal oxidation of AI. According to Hagel 2s the degree of protection provided by an AI~Oa layer also depends on whether a or 7 Al2Os is formed. The transformation from 7 to ct AI20 s may be enhanced by certain alloying constituents. This is another possibility to explain the beneficial effect of alloying elements which do not enter an AlaOs scale. THE OXIDATION OF Mn-C ALLOYS Carbon dissolved in 3' iron may be removed by virtue of the reaction C02(g) q- C(in alloy) = 2 COfg)
(18)
To attain a low carbon content in the alloy, the carbon activity in the gas mixture must be low. Thus the COs/CO ratio must be high but below the equilibrium ratio for coexistence of iron with wiistite, and moreover, the total pressure of the COs-CO mixture must be low. In the same way, carbon dissolved in manganese may, in principle, be removed. Upon increasing the affinity of the reaction by increasing the C O J C O ratio, one may expect first an increase in the rate of decarburization. At higher C O J C O ratios, however, a scale of MnO is formed which prevents removal of carbon according to reaction (18) as is shown by experiments where Mn-C alloys were oxidized in air at 1000°C. ~4 According to the foregoing phenomenological definition, this is another case of passivity with a sharp decrease in the rate when the affinity of the reaction is increased. In contradistinction to the oxidation of Ag-In and Cu-Be alloys considered above, passivity is due to the formation of a layer of the oxide of the base metal Mn whose oxidation rate increases with an increasing C O J C O ratio while
762
CAgE WAGNER
the oxidation of carbon ceases. From a practical point of view, however, this case o f passivity is not interesting. THE I N H I B I T I O N OF THE OXIDATION OF N I C K E L IN THE PRESENCE OF Li20 AND RELATED PHENOMENA Pfeiffer and Hauffe 25 have found that the rate of oxidation of nickel at 1000°C is considerably reduced in the presence ol~ LifO vapour. Since presence of Li20 does not cause a significant change in the affinity of the oxidation of Ni to NiO, this is a typical example of inhibition according to the definition suggested above. The decrease in the oxidation rate is due to the dissolution of Li20 in solid NiO associated with a decrease in the concentration of cation vacancies which are essential for the migration of Ni ~+ ions across the NiO scale. Likewise, one may expect a decrease in the oxidation rate of Zn in the presence of oxides with a cation-to-anion ratio less than unity whereby cation vacancies are formed and interstitial ions are captured, which are essential for the migration of zinc ions across a ZnO scale. In accord herewith Gensch and Hauffe 26 have found that the oxidation rate of Zn at 400°C is lowered considerably by the presence of AI. In this system, however, the change in the oxidation rate may be due to the formation of a scale of AI2Oz rather than to a decrease in the concentration of interstitial zinc ions in ZnO, since ZnO on a Z n - A I alloy is thermodynamically not stable in view of the high free energy drop of the displacement reaction (17). Another example of inhibition is the decrease in the rate of chlorination or bromination of silver in the presence of small amounts of Cd as an alloying element. Since the silver halides exhibit predominantly ionic conduction, the migration of electrons via holes in AgC1 or AgBr is the rate determining step. 27 Dissolution of cadmium halide in AgCI or AgBr reduces the concentration of electron holes and accordingly the rate of the reaction between Ag and C12 or Br2. ~,~9
THE BEHAVIOUR OF A METAL IN A GAS PHASE I N V O L V I N G TWO O X I D I Z I N G COMPONENTS When nickel is heated in an Ar-I2 stream with a sufficiently low I2 content, the reaction Ni(s) + 12(g) = NiI2(g)
(19)
takes place. The ratejNixl is mainly determined by the diffusion of 12across the boundary layer to the Ni surface. In view of equations analogous to eqs. (7a) and (8), one has JNil~ = Jl,
=
DI 2
P~2/a R T
(20)
No solid NiI2 is formed unless the partial pressure PNa, at the metal surface exceeds the saturation pressure of NiI2. U p o n addition of a small amount of 02 to the A r - I 2 stream, a scale of solid NiO(s) is supposed to form,
Passivity and inhibition during the oxidation of metals at elevated temperatures 1
Ni(s) -4- : O2(g) = NiO(s) 2
763 (21)
and Nilz is formed by the consecutive reaction
I
NiO(s) + I2(g) = NiIz(g) + = Oz(g) 2
(22)
The rate of the reaction in eq. (22) is determined by the rate of diffusion of Nilz(g) from the NiO surface into the bulk stream according to an equation analogous to eq. (7b), JNilz = DNilzPNit2(eq)/t~ R T
(23)
where pNil,(eq) is the equilibrium partial pressure of Nil,, for reaction (22). The standard free energy change of reaction (22) is about 37 kcal at 700°C 3°,a~ corresponding to a ratio PNiI,(eq)/PI~. = 10-7atpo, = 0 . 0 1 atm. Hence the rate of Nilz formation at a surface covered by NiO according to eq, (22) is about 107 times lower than at a bare Ni surface according to (19). The lowering of the rate due to presence of oxygen is a typical case of inhibition rather than passivity since oxygen does not alter the affinity of the basic reaction (19). A more involved situation occurs when in the presence of two oxidizers two solid metal compounds can be formed. As an example consider silver heated in a mixture of sulphur and iodine vapour. At 200°C the reaction 2 Ag(s) + 21 S2 (g) = AgzS(s)
(24)
is much faster than the reaction 2 Ag(s) q- Iz(g) = 2 AgI(s)
(25)
Since the free energy drop for reaction (25) is much greater than that for reaction (24), AgI is much more stable than Ag2S. It is open to question, however, which partial pressure of I2 is actually required in order to enforce exclusive formation of Ag! and to prevent the rapid attack of silver by sulphur vapour. C O N C L U D I N G REMARKS The foregoing examples of observations on the oxidation of metals at elevated temperatures show that there is indeed a large variety of phenomena which may be classified either as passivity or as inhibition. This list of typical examples has not been drawn up because of satisfaction in classification but is intended to help other investigators in finding new ways for minimizing corrosion at elevated temperatures. Most remarkable are abrupt decreases in the rate of reactions such as the oxidation of Zn or Si in Ar-O2 mixtures, the oxidation of A g - I n and Cu-Be alloys, and the decarburization of M n - C alloys when the oxygen partial pressure or the concentration of an alloying metal is increased by a rather minor extent. In each of these cases, the
764
CARL WAGNER
transition from a bare metal surface to an oxide-covered surface or the f o r m a t i o n o f another oxide phase in the scale is decisive for the a b r u p t decrease of the rate. Theoretically, even a differential increase in the oxygen partial pressure may suffice in order to effect the a b r u p t decrease in the rate of oxidation of Si or Z n in A r - O z mixtures. Experimentally, however, one finds a certain range for the transition from a high to a low rate, where results are not well reproducible. O n the other hand, there are also cases with a c o n t i n u o u s decrease in rate. Such a behaviour occurs especially when the phase structure of a scale is not changed but the concentration of interstitial ions, ion vacancies, excess electrons, or electron holes is changed u p o n increasing the c o n c e n t r a t i o n of a " d o p e " . REFERENCES I. E. T. TURKDOGAN,P. GRIEVESONand L. S. DARKEN,J. Metals 14, 521 (1962); J. phys. Chem. 67, 1647 (1963). 2. C. GENSCHand K. HAUFFE,Z. phys. Chem. 196, 427 (1951). 3. W. KAISERand J. BRESLIN,J. appl. Phys. 29, 1292 (1958). 4. C. WAGNER,J. appl. Phys. 29, 1295 (1958). 5. I. DIETRICHand L. KocH, Z. Metallk. 50, 31 (1959). 6. R. A. RAPP, Acta Met. 9, 730 (1961). 7. C. WAGNER, Z. Elektrochem. 63, 772 (1959). 8. K. W. FROHLICH,Z. Metallk. 28, 368 (1936). 9. J. P. DEr,n~ISONand A. PREEeE,J. Inst. Metals 81,229 0952/53). 10. F. MAAK,Z. Metallk. 52, 538, 545 (1961). I I. M. HANSENand K. ANDERKO,Constitution of binary alloys, p. 604, McGraw-Hill Publishing Co,. New York (1958). 12. W. EIC,ENAUERand G. MOLLER,Z. Metallk. 53, 321,700 (1962). 13. F. N. RmN~, W. A. JOHNSONand W. A. ANDERSON,Trans. Am. Inst. Min. Metall. Engrs. 147, 205 (1942). 14. F. N. RHINESand B. J. NELSON,Trans. Am. Inst. Min. Metall. Engrs. 156, 171 (1944). 15. J. S. DtmN, J. Inst. Metals46, 25 (1931). 16. L. E. PRICEand G. J. THOMAS,J. Inst. Metals 63, 21 0938). 17. P. SPINEDI,Metallurgia ira/. 45,457 (1953). 18. W. SErrH and W. KRAUS,Z. Elektrochem. 44, 98 (1938). 19. C. WAGNER,J. Electrochem. Soc. 99, 369 (1952). 20. A. POR'rEvIN,E. PRaTEr and H. JOLIVET,Rev. Mdtall. 31, 101, 186, 219 (1934). 21. E. SCI-IEILand E. H. SCHULZ,Arch. EisenhiittWes. 6, 155 (1932/33). 22. E. SCHEILand K. KIWIT,Arch. EisenhiittWes. 9, 405 (1935/36). 23. W. C. HAGEL,Proc. Second lnternat. Congress on Metallic Corrosion, in press. 24. W. W. WEan, J. T. NORTONand C. WAG~ER,J. Electrochem. Soc. 103, 112 0956). 25. H. PrEIn:ERand K. HAUFFE,Z. Metallk. 43, 364 0952). 26. C. GENSCrland K. HAtrrEE, Z. phys. Chem. 196, 427 (1951). 27. C. WAGNER,Z. phys. Chem. B 32, 447 (1936). 28. W. HIMMLER,Z. phys. Chem. 195, 129 (1950). 29. K. HAUFFEand C. GENSCH,Z. phys. Chem. 195, 116 0950). 30. H. SCHXFER,H. JACOBand K. ETZEL,Z. anorg. Chem. 286, 42 (1956). 31. K. KIUKKOLAand C. WAGEreR,J. Electrochem. Soc. 104, 379 (1957).