WAGNER’S THEORY WAGNER’S THEORY OF

METAL OXIDATION

http://home.agh.edu.pl/~grzesik

OxidantOxidant

Scale

Schematic illustration of scale formation according to Tamann (1920)

Metal Metal

Scale12 2X Me MeX+ →

Schematic illustration of the classic Pfeil experiment (1929)

Air

ScaleSiO

Air

Fe

Scale

Fe

SiO2

SiO2

Literature on Wagner’s theory

1. C. Wagner, Z. phys. Chem., 21, 25 (1933)

2. C. Wagner, Z. phys. Chem., 32, 447 (1936)

3. C. Wagner, Atom Movements ASM, Cleveland, 1951

1. F. Gesmundo, Solid State Phenomena, 21&22, 57 (1992)

2. T. Norby, „Wagner-type theory for oxidation of metals with co-

transport of hydrogen ions”, 196th Electrochemical Society Meeting,

Honolulu, Hawaii, October 1999

3. P. Kofstad, „High Temperature Corrosion”, Elsevier Applied Science

Publications of other authors describingWagner’s theory

Publishers Ltd., London-New York, 1988

4. P. Kofstad, „High-Temperature Oxidation of Metals”, John Wiley &

Sons, Inc, New York-London-Sydney, 1978

5. A.S. Khanna, „Introduction to High Temperature Oxidation and

Corrosion”, ASM International, Materials Park, 2002

6. S. Mrowec, „An Introduction to the Theory of Metal Oxidation”,

National Bureau of Standards and the National Science Foundation,

Washington, D.C., 1982

Wagner’s theory of metal oxidation: assumptions

1. The scale growing on the metal surface is compact in its entirety.2. Diffusion of reagents in the scale takes place in the form of ions and

electrons through point defects in the crystalline lattice of the reaction product.

3. The scale formation process is determined by diffusion, the rate of which is lower than that of chemical reactions at the interface.

4. Electronic and ionic defects in the crystalline lattice of the scale travel 4. Electronic and ionic defects in the crystalline lattice of the scale travel due to ambipolar diffusion.

5. Diffusion of reagents in the scale proceeds under the influence of a lattice defect gradient caused by the oxidant chemical potential gradient.

6. Point defect concentrations are established at the scale interfaces, thank to which the oxidation process proceeds according to the parabolic rate law.

7. There is a state close to thermodynamic equilibrium at the scale interfaces.

fd

dx N

d

dxz F

d

dxii

a

ii= = − +

η µ ϕ1

j c B fi i i i=General equation expressing the flux of component “i”:

j i – flux of component “i”Bi – mobility of component “i” f i – driving force of component “i”

dx N dx dxa

j c BN

d

dxz F

d

dxi i ia

ii= − +

1 µ ϕ

ηi – electrochemical potential of component “i” µi – chemical potential of component “i”Na – Avogadro’s numberF –Faraday’s constant, C/molϕ – electrical potential zi – valence of component “i”

Nernst-Einstein equation:

BD

kTii=

Di – diffusion coefficient of component “i”

Taking the equation above into account, the flux of component “i” can be expressed as:

j cD

RT

d

dxz F

d

dxi ii i

i= − +

µ ϕcan be expressed as:

j cD

RT

d

dxz F

d

dxc cc c

c= − +

µ ϕ

Me MeX X2

jc

ja

je

j cD

RT

d

dxz F

d

dxa aa a

a= − +

µ ϕ

j cD

RT

d

dxz F

d

dxe ee e

e= − +

µ ϕ

ze = -1, therefore:

j cD

RT

d

dxF

d

dxe ee e= − −

µ ϕ

By inserting the fluxes into the equation above, the electrical potential gradient can be determined:

z j z j jc c a a e+ − = 0

Because the migration of point defects in the scale takes place due to ambipolar diffusion, the electroneutrality condition must be fulfilled:

d

dx F

c D zddx

c D zddx

c Dddx

z c D z c D c D

c c cc

a a aa

e ee

c c c a a a e e

ϕµ µ µ

= ⋅− − +

+ +1

2 2

The chemical potential of the metal and oxidant is associated by Gibbs-Duhem’s equation (for an isothermal-isobaric process):

N d N dMe Me X Xµ µ+ = 0

NMe and NX – denote molar fractions of the metal and oxidant, respectively

For compounds exhibiting minor deviation from stoichiometry, the ratio of the metal and oxidant molar fractions is inversely proportional to their valences:

N zN

N

z

zX

Me

c

a

=

Thus, the Gibbs-Duhem equation assumes the following form:

dN

Nd

z

zdMe

X

MeX

c

aXµ µ µ= − = −

Ionization reactions of the metal and oxidant:

Me Me z e

X X z e

zc

za

c

a

⇔ +

⇔ −

+

−

In thermodynamic equilibrium conditions:

µ µ µ

µ µ µMe c c e

X a a e

z

z

= +

= −µ µ µX a a ez= −

After inserting the correlations above, cation and anion fluxes can be expressed by the following equations:

j cD

RT

c D

z c D z c D c D

d

dxc cc e e

c c c a a a e e

Me= − ⋅+ +2 2

µ

j cD

RT

c D

z c D z c D c D

d

dxa aa e e

c c c a a a e e

X= − ⋅+ +2 2

µ

If electron mobility is significantly higher than ionic defect mobility, i.e. Da << De >> Dc, then:

j cD

RT

d

dxc cc Me= −

µ

j cD

RT

d

dxa aa X= −

µ

Scale growth, dX, is the result of reactions at both interfaces:

dX dX dXc a= −

Increase in scale thickness as a function of time at both its interfaces is described by Stefan’s condition, which assumes the following form:

dX

dt

j

c Me Xa a

X a b

=′

′ ( )

dX

dt

j

c Me Xc c

Me a b

=′′

′′ ( )

Therefore:

dX

dt

j

c Me X

j

c Me Xc

Me a b

a

X a b

=′′

′′−

′′( ) ( )

After inserting the expression describing the fluxes into the previous equation, scale thickness increase can be expressed as:

( ) ( )dX

dt

c D

c Me X RT

d

dx

c D

c Me X RT

z

z

d

dxc c

Me a b

Me a a

X a b

a

c

Me= −′′

−′

1 1µ µ

( ) ( )dX

dt RT

c D

c Me X

c D

c Me X

z

z

d

dxc c

Me a b

a a

X a b

a

c

Me= −′′

+′

1 µ

Taking into account that:

dX

dt RTD

z

zD

d

dxMea

cX

Me= − +

1 * * µ

c D c Dd d i i= *

Meanwhile:

- self-diffusion coefficient of component “i”Di*

Because the correlation between self-diffusion coefficients of a given substance and its chemical potential gradient is unknown, the average value must be calculated:

dX

dt RTXD

z

zD dMe

a

cX Me

Me

Me

= − +

′

′′

∫1 * * µ

µ

µ

If it is assumed that the scale formation process proceeds

′ = +

′′

′

∫kRT

Dz

zD dp Me

a

cX Me

Me

Me1 * * µµ

µ

If it is assumed that the scale formation process proceeds according to the parabolic rate law, then:

From the Gibbs-Duhem equation and the correlation between chemical potential and oxidant pressure, the following equation can be derived:

dz

zd

z

zRT d pMe

c

aX

c

aXµ µ= − = −

1

2 2ln

The parabolic oxidation rate constant is thus:

′′p

′ = +

′

′′

∫kz

zD D d pp

c

aMe X X

p

p

X

X1

2 2

2

2

* * ln

Application of Wagner’s metal oxidation theory

[ ] [ ]K V h pMn S= ′′ ⋅ ⋅• −2 1 2

2

/

′ = +

′

′′

∫kz

zD D d pp

c

aMe X X

p

p

X

X1

2 2

2

2

* * ln

′ =′

′′

∫k D d pp Me Xp

p

X

X1

2 2

2

* ln

Example I: Mn1-yS

xSMn22

1 Sh2VS ++′′⇔ •

[ ] [ ]2 ′′ = •V hMn

[ ]′′ = ⋅ ⋅V K pMn S0 63 1 3 1 6

2. / /

′pX2

2

[ ]D D V K D pMe Me Mn Me S* / /.= ⋅ ′′ = ⋅ ⋅ ⋅0 63 1 3 1 6

2

′ = ⋅ ⋅ ⋅ ⋅′

′′

∫k K D p d pp Me S Sp

p

S

S1

20 63 1 3 1 6

2 2

2

2

. ln/ /

′ = ⋅ ⋅ ⋅ ⋅ =k K D p Dp Me S Me3 0 63 31 3 1 6

2. / / *

( ) [ ]′ = + ⋅ = ⋅k p D def Dp Me1 * ~

Example II: calculating the self-diffusion coefficient (Fueki-Wagner method)

′ = +

′

′′

∫kz

zD D d pp

c

aMe X X

p

p

X

X1

2 2

2

2

* * ln

′ =′′

∫k D d p

pX1 2

* ln

Application of Wagner’s metal oxidation theory

′ =′∫k D d pp Me X

pX2 2

2

* ln

d k

d pD

p

Xp

Me

X

′=

′′ln

*

22

1

2

Dd k

d pMe

p

XpX

*

ln= ⋅

′

′′

22

2

Example II: calculating the self-diffusion coefficient (Fueki-Wagner method), cont.nickel oxidation

Fueki-Wagner:

Lindner:

DRTNi

* , exp= ⋅ ⋅ −

−11 10

503003

Application of Wagner’s metal oxidation theory

Lindner:

Moore:

DRTNi t( )

* , exp= ⋅ ⋅ −

−1 7 10

560002

DRTNi t( )

* , exp= ⋅ ⋅ −

−3 9 10

442004

Note: the results above were obtained for the same oxygen pressure value

THE ENDTHE END