DERIVATION OF STERN – GEARY EQUATION FOR CALCULATING CORROSION RATES FROM POLARIZATION DATA OBTAINED AT LOW CURRENT DENSITIES

29.2 DERIVATION OF STERN – GEARY EQUATION FOR CALCULATING CORROSION RATES FROM POLARIZATION DATA OBTAINED AT LOW CURRENT DENSITIES

Assume that the corrosion current, I corr , occurs at a value within the Tafel region for both anodic and cathodic reactions. Also assume that concentration polariza- tion and IR drop are negligible. Current relations are shown in Fig. 29.1 for the corroding metal polarized as anode by means of an external current to a potential

φ . Increasing anodic current, I a , is accompanied by decreasing cathodic current,

I c , because of the relation,

I appl = I a − I c (29.4) Similarly, for cathodic polarization, for which I appl changes sign, we have − I appl = I c − I a (29.5) For anodic polarization, we obtain

Figure 29.1. Polarization diagram for corroding metal polarized anodically from φ corr to φ.

DERIVATION OF STERN- GEARY EQUATION FOR C ALCUL ATING CORROSION R ATES

I a I φφ corr −

corr = Δ φβ = a log

− β a log

iA o a a iA o a a (29.6)

I log = a β a

I corr

where A a is the fraction of area that is anode, and i 0a is the exchange current density for the anodic reaction. Note that i 0a in the Tafel equation refers to current per unit local area (not total area); hence, I corr /A a is the current density to which i 0a applies. Likewise, if the metal is polarized an equal amount in the cathodic direction, we have

Δφ =− β c log

I corr

or

I c = I corr 10 − Δ φβ / c and I a = I corr 10 Δ φβ / a then

I appl = I corr ( 10 Δ φβ / a − 10 − Δ φβ / c ) (29.8) Expressed as a series, we have

If Δ φ / β c and Δ φ / β a are small, higher terms can be neglected and (29.8) can be approximated by

⎛ 1 1 I appl = 23 . I corr Δφ + ⎞ (29.9)

or

I appl

I a c 1 ββ a corr c = ⎛ ⎞ ⎛ ⎞ (29.10)

2.3 Δφ ⎝⎜ β a + β c ⎠⎟ = 23 . R ⎝⎜ β a + β c ⎠⎟ where R is the polarization resistance, determined experimentally by applying

both anodic and cathodic currents to the electrode and measuring the polariza- tion. The value of R used in Eq. (29.10) is the slope of the φ versus I graph at

the corrosion potential.

458 APPENDIX

Figure 29.2. Polarization diagram for metal corroding under control by oxygen depolarization.

Equation (29.10) is the Stern – Geary equation. Should the cathodic reaction

be controlled by concentration polarization, as occurs in corrosion reactions controlled by oxygen depolarization, the corrosion current equals the limiting diffusion current (Fig. 29.2 ). This situation is equivalent to a large or infi nite

value of β c in (29.10) . Under these conditions, (29.10) becomes

β a I appl

I corr a = = (29.11)

23 . Δ φ 23 . R

If a noncorroding metal (minimum local action) that polarizes only slightly (high value of i 0a ) is polarized anodically at moderate current densities, then i 0a can be substituted for I corr in Fig. 29.1 , and

I appl ββ

c Δφ a = (29.12)

23 . i o a β a + β c

indicating, as observed, that anodic polarization of many metals at low current densities is a linear function of applied current.

29.2.1 The General Equation

If φ corr falls outside the Tafel region, for example, when it approaches either the reversible anode or cathode potential, “ back - reactions ” become appreciable and (29.10) is subject to error. Mansfeld and Oldham [1] considered this case and

DERIVATION OF STERN- GEARY EQUATION FOR C ALCUL ATING CORROSION R ATES

formulated the required modifi cations. The derivation of the more exact form of (29.10) starts with modifying (29.4) . Because I appl changes sign at potentials above compared to below φ corr , choosing I appl and I a with the same sign — that is, anodic polarization — results in

I appl = I a − I ar − ( I c − I cr ) (29.13) where I ar and I cr are the back - reactions for the anodic and cathodic reactions,

respectively. Introducing the Tafel equation expressed in natural logarithms, η = b ln( I/I 0 ) and letting b a β = a /2.3, b c β = c /2.3, b ar β = ar /2.3, and b cr β = cr /2.3, we have

I I φφ − a φφ − a φφ − c φφ − c appl = o a exp − I o a exp − − I o c exp − + I o c e () xxp

b a ( b ar ) ( b c ) () b cr

(29.14) According to electrode kinetics, we obtain

1 1 nF a 1 1 nF c

b c b cr RT where n a and n c are constants related to the transfer coeffi cient. Using these rela-

and

b a b ar RT

tions to eliminate b ar and b cr , it follows that

φφ − a ⎡

() b ⎣⎢

RT ( ) ⎦⎥

nF a ( φ − φ )

I appl = I o a exp

1 − exp

( b c )) ⎣⎢ ( RT ) ⎦⎥

Differentiating (29.15) with respect to φ , we get

bnF a a nF a ( φ a − φ ) =

∂ I appl I o a φφ − a ⎡

b a () b a ⎣⎢ ( RT ) ( RT ) ⎦⎦⎥

b c () b c ⎣⎢ ( RT ) ( RT ) ⎦⎥

exp

− 1 exp

At the corrosion potential, I appl

= 0 and I a c = I corr = I . Letting Δ φ a φ corr = − φ a and Δ φ c φ = c − φ corr , (29.15) becomes

() b a ⎣⎢ ( RT ) ⎦⎥

− nF φ

I exp

corr = o a −

exp

() b c ⎣⎢ ( RT ) ⎦⎥

− nF Δ φ

= I c exp

c ⎡ 1 − exp

460 APPENDIX

and (29.16) becomes

bnF a a − n F Δ φ =

1 ⎛ ∂ I appl ⎞

R ⎝⎜ ∂ φ ⎠⎟ φφ = corr b a () b a ⎣⎢ ( RT ) ( RT ) ⎦⎥

exp

− 1 exp

b c () b c ⎣⎢ ( RT ) ( RT )) ⎦⎥

Substituting (29.17) into the fi rst part of (29.19) and (29.18) into the second part of (29.19) gives

nF a Δ φ a nF c nF c Δ φ cc ⎫ ⎨

− 1 − 1 1 ⎧ 1 1 nF

( RT ) ⎦⎥ RT ⎣⎢ ( RT ) ⎦⎥ ⎭

Equation (29.20) was developed by Mansfeld and Oldham [1] and is equivalent to the one derived originally by Wagner and Traud [2] . If the correction terms involving Δ φ a and Δ φ c can be neglected, (29.20) becomes the Stern – Geary equa- tion. On the other hand, if either Δ φ a or Δ φ c approaches zero, the corresponding correction term approaches infi nity. Small values of nF Δ φ / RT allow the exponential terms to be expanded, sim- plifying (29.20) to

If Δ φ c = 0.005 V, T = 298 ° K, and n c is assumed to be 0.5, then F / RT = 39 V −1 and n F Δ φ c / RT = 0.1. The corresponding correction term from (29.20) is 190 V −1 c , whereas the approximated correction term in (29.21) is 1/ Δ φ c = 200 V −1 . These correction terms are large compared to 1/ b a + 1/ b c = 46 V −1 (assuming β a β = c =

0.1 V). Should Δ φ c = 0.1 V, the corresponding correction term from (29.20) is

−1 3.2 V . If at the same time, Δ φ a = 0.5 V and n

a = 0.5, the correction term as given

by (29.20) equals 0.001 V −1 , which is negligible.

Additional approximations [1] that could be used if the corrosion potential is close to one of the reversible potentials are as follows. If n a Δ φ a < 2 RT / F (i.e., the corrosion potential is close to the reversible potential of the metal),

1 1 1 1 nF

= I corr a ⎛ + + − ⎞ (29.22)

⎝⎜ b a b c Δφ a 2 RT ⎠⎟

If n c Δ φ c < 2 RT / F (i.e., the corrosion potential is close to the reversible potential of the reduction reaction), we obtain

I corr ⎛

1 1 1 1 nF

= c + + − ⎞ (29.23)

⎝⎜ b a b c Δφ c 2 RT ⎠⎟

DERIVATION OF EQUATION EXPRESSING THE SATUR ATION INDEX

Figure 29.3. Relative errors in corrosion currents calculated by use of (29.10) , (29.21) , (29.22) , and (29.23) , instead of the exact equation (29.20) [1] ( with permission from Pergamon Press ).

The errors that result from each of these approximations to (29.20) are presented in Fig. 29.3 [1] , where n a = 2, n c = 1, b a = RT /F, b c = 2 RT / F , and φ c − φ a Δ φ = a + Δ φ c = 10 RT / F were assumed.