A theoretical analysis of the electrochemical noise during the induction period of pitting corrosion in passive metals

361

J. Electroanal. Chem., 291 (1991) 361-375 Elsevier Sequoia S.A., Lausanne

A theoretical the induction

analysis of the electrochemical noise during period of pitting corrosion in passive metals

Part 2. The current noise associated formation in the passive film

with halide nucleus

T. Okada Industrial Products Research Institute, M. I. T I., Yatabe, Tsukuba, Ibaraki 305 (Japan) (Received

14 March

1990; in revised form 23 August

1990)

Abstract The formation of halide nuclei in the passive film during the induction period of pitting corrosion in passive metals is discussed using a stochastic model concerning the growth/diminishing processes of the hemispherical metal halides inside the metal oxide film. It was assumed that the electrochemical noise occurs owing to the formation of halide nuclei in the passive film. prior to pitting corrosion. From the stochastic analysis a distribution function was obtained for the size of the halide nuclei that form inward into the passive film. The survival probability of pit initiation, the specific parameter of pitting corrosion, was calculated as a function of time, and good correspondence was obtained with the reported experimental results. Also, the noise level was determined as the uncertainty of the distribution function for the radii of the halide nuclei. The calculated noise level was found to be appreciable and can then cause the electrochemical noise during the induction time.

INTRODUCTION

The “electrochemical noise” which is observed during the induction period of pitting corrosion in passive metals gives several kinds of information about the dynamic stability of the passive film [l]. In acidic media containing aggressive anions, a local dissolution current arises on the surface of the passive metals, which occurs around the imperfection point on the passive film or as a result of local fluctuations of the ionic concentration or of the electric field in the solution [2,3]. Okamoto et al. [4,5] observed two kinds of electrochemical noise in the course of the pit initiation stage, the first one at rather high frequencies around 10 Hz, and the second one at low frequencies around several Hz, when they studied stainless steels in chloride solutions. They attributed these types of noise to the process of 0022-0728/91/$03.50

0 1991 - Elsevier Sequoia

S.A.

362

destruction/repair of the passive film due to attack by chloride ions. Nachstedt and Heusler [6,7] measured the electrochemical noise of frequencies around 5 Hz during the induction period of pitting of passivated iron, and described it by a Lorentzian function, which they explained by the local thinning model of the oxide film. Shibata and Takeyama [8,9] attributed the statistical deviation of the observed pitting potentials and the induction time to the stochastic nature involved in the pit initiation processes, and proposed a stochastic model based on the birth and death processes of the pit embryo. Also, Williams et al. [lO,ll] used a stochastic model of the birth and death of pits, and explained the experimental results concerning the current noise and power spectra. These results indicate that the passive film destruction/repair process is potentially involved in the occurrence of pitting corrosion. I have proposed a two-step initiation hypothesis of pitting corrosion in which it is assumed that pitting occurs via a series of processes, each one including the stability/instability criteria caused by fluctuation processes [12,13]. It was also assumed that the occurrence of electrochemical noise at the induction period of pitting corrosion has certain correlations with the fluctuation processes in each proposed stability/instability criterion. The wide range of time constants of the electrochemical noise was attributed to the difference in the time constants of fluctuations in each step. In a previous paper [14], the process was considered related to aggressive anion adsorption/desorption on the surface of the passive film and successive transitional halide complex formation, which was thought to cause high frequency noise. In the present paper, the process associated with low frequency noise or spikes will be discussed in relation to the halide nucleus growth/diminishing processes, which is also a model of the passive film destruction/repair processes [2]. Other possible causes for the occurrence of electrochemical noise, such as gas evolution at the pit bottom or delayed diffusion processes in the solution, are outside the scope of this work and will not be considered here. THEORY

Halide nucleus formation

and growth inside the passive film

The process of pit initiation has been discussed for passive metals immersed in acidic media containing aggressive anions by use of a stability/instability criterion concerning halide nucleus formation in the passive film [2]. It was shown that halide nuclei can undergo stable growth when their size exceeds the critical radius r *, then they break through the passive film and bring about pitting. The model took it as a possible route for the occurrence of pitting that the coagulation of halide ions in the solution at the surface of the passive film triggers the formation of halide nuclei in the passive film. This could explain the feature of pit localization and also the experimentally observed current or potential fluctuation phenomena. It could be imagined that the step of halide nucleus formation appears as the final step in the

363

Fig. 1. Illustration

of the process

of halide nucleus

formation

in the passive

film.

induction period of pitting, so that the associated noise phenomena would be of practical interest. If the radius of the halide nuclei which grow inward into the passive film does not exceed the critical value r *, then the nuclei will finally diminish and no pitting will occur. The probability of the system staying in this condition can be considered as the survival probability, and has already been pursued experimentally by Shibata and Takeyama [8,9], Williams et al. [lO,ll] and Doelling and Heusler [15]. Stochastic

model

In Fig. 1 a halide nucleus of radius r which is formed from the surface of the passive film inward into the bulk is depicted schematically. The rate at which the radius of the nucleus grows from r to r + Ar is a function of r and E, the potential of the passive metal, and will be designated as Q+(r, E). The rate at which the nucleus diminishes from the radius r to r - Ar will be denoted as Q_(r, E). In this situation, the probability N(r, t) Ar, where the radius of a halide nucleus remains in the range r to r + Ar at time t, can be expressed by the following master equation:

aN(r, 4 at

Ar = Q+(r - Ar, E)N(r +Q_(r+Ar, +d(r,

- Ar, t) - Q+(r,

E)N(r+Ar,

E)N(r,

t)-Q_(r,

at

t)

E)Ar

where d(r, E) is the frequency at which halide nuclei potential E. Equation (1) can be rearranged as follows:

aN(r, t>

E)N(r,

t)

= - $[{Q+(r,

+d(r, E)

E) -

Q-(r,

of radius

r appear

(1) at a

E)}N(r, t>] (2)

where by definition /0

CQN(r, t) dr = 1

(3)

364

For simplicity it is assumed that each microscopic step of the growth or diminishing of halide nuclei proceeds very fast compared with the rate at which the halide nucleus exceeds the critical radius r * and goes into pitting. Then it can be assumed that the change in N(r, t) with time is correlated only to a minor extent with the pit initiation rate. The solution to eqn. (2) satisfying eqn. (3) is written as follows (see Appendix A): N(r,

t) =N(r,

0) exp{ -C(E)t}

D(r, E)

+

Q+(,.,

E)

Here

C(E)

D(r,

E) is the integral

D(r,

E) =/h(r,

_

Q_(,.,

[I - exp{ -C(E)t)l

E)

is a constant which does not depend of d(r, E):

r or t but

on either

E) dr

on E, and

(5)

0

In order to obtain C(E), eqn. (4) is differentiated with respect to t, and the critical point r = r * is considered, i.e. where the halide nuclei undergo stable growth: aN(r*,

t)

at

= -C(E)N(r*,

‘) +

C(E)D(r*, Q+(r*

E)

E) _

Q_(,.*

E)

(6)

The first term on the right-hand side of eqn. (6) represents the rate at which the so that C(E) can be expressed as halide nucleus of radius r = r * is consumed, follows: C(E)

= { Q+(r*,

=

E) - Q-(r*,

E)}/h*

Q*(E)

(7)

which will be equal to the net growth rate of halide nuclei of size r = r *, divided by the transition length h* for the nucleus to pass over at the critical radius. Next the physical meanings of Q+(r, E) and Q_(r, E) will be considered. It is apparent that

Q+(r,

E) -

Q_(r,

E) = =

$ (wwKl0 2vrr ‘zFd,,

(8)

where Z is the net current which flows through the halide nuclei, N is the number of halide nuclei per unit surface area of the passive metal, S is the area of the metal which contacts the solution, M,, is the “molar mass” of the passive oxide which is dissolved corresponding to 1 mol of electrons passed, and d,, is the density of the passive oxide. It can be expressed experimentally [2] that Z/NS

= iprX

(9)

365

where i is the current density which flows through constant which lies in the range 1 I x I 2 [2].

the halide

nuclei

and

x is a

i = k,c, (0) ’ exp( .$fE)

(10)

Here k, is the rate constant, c,(O) is the concentration of halide ions in the solution at the passive film/solution interface, q is the reaction order, 5 is a constant, and f= F/RT. From eqns. (8) and (9) it follows that Q+(r,

E) = qMoh*rx-‘i

E) - Q_(r,

4 MO = M,,/2zFd,,h D(r, D(r,

*

(11.2)

E) is not known,

but it will be searched

E) = a exp( -brW)fE(

Substituting N(r,

(11.1)

for in the form

E)

(12)

eqns. (11) and (12) into eqn. (A12) and using eqn. (3), it is found

co) =

f(2-x, F(2 -x>

w, r) w, m)

that

(13.1)

where f( 24, w, r) = r” exp( -br”)

(13.2)

F(u,

(13.3)

w, r) = i>(

Equation

N(r,

(13.1) can be rewritten

m> =*f(2

-x,

ql>

Here r(l)

l&x

u, w, r) dr

w, r)

is the gamma

function,

(14.1) and (14.2)

W

From

the above equations

N(r,

t) =N(r,

+ Bearing I0

as follows:

“rN(r,

we obtain

0) exp{ -Q*(E)t}

jc2-”

w7‘) [l

F(2 - xv w,

m>

-exp{

-Q*(E)t}]

(15)

in mind that at t = 0, N(r, 0) has a sharp peak at r = 0, so that 0) dr = 0

the average value (r) (r> =J, “rN(r, F(3-Xt = F(2-X,

(16) of the halide nuclei

at time t is obtained

as follows:

t) dr w9 O”) [I -exp{ w, m>

-Q*(E)t}]

(17)

366

Also, the dispersion

of r is obtained

0,’ ==Jm(r(r))‘N(r,

as follows:

t) dr

0

F(4 -x, = i F(2-X,

x[l

w, a) F(3 -x7 w, co) - [ F(2 -x,

w, m) ]‘[I-exp{-Q*(E)r}l} w, cc)

-ev{-Q*(E)t)l

Survival probability

The survival probability P(t) can be defined as the probability that a single pit embryo does not evolve into an observable pit. In the present model, this can be regarded as the fraction of halide nuclei whose radii r remain less than the critical value r *: Z’(t)

=l’*N(r,

Substitution

t) dr

09)

of eqn. (15) into eqn. (19) yields

P(t)=l-g(r*)[l-exp{-Q*(E)t}] g(r*)=l-

F(2-X,

w, r*)

F(2-X,

w, m>

Here it is assumed r = 0 and decreases

r* J0

N(r,

Shibata P(t)

(20.1) (20.2)

that N(r, 0), the initial value of N(r, very rapidly as r increases, so that

t), has a maximum

0) dr = 1

and Takeyama

= l-

and Williams

(21)

[8,9] obtained

the following

equation

experimentally:

(22)

&[I-exp{-(X+C)(I-lO)}l et al. [lO,ll]

introduced

the equation

P(t)=exp{-aA(l-TC)exp(-pT,.)}

(23)

where h and ~1are the birth and death parameters, respectively, of pitting Also, Doelling and Heusler [15] obtained the experimental formula P(t)

around

= exp{ -A(t)t}

where A(t) is a time-dependent correspond well with eqns. (20).

corrosion.

(24)

frequency

function.

These

experimental

results

361 TABLE

1

Functional

forms of the birth and death parameters for various values of the variables x and w a x=1 W=l

w=2

h

q,oi$(br*+l)exp(-br*)

q&F

P

q&${l-(br*+l)exp(-br*))

x

qMoi exp( - br * )

qMoi erf(&r*)

P

q&(1-exp(-br*))

qMoi erfc(fir

‘erf(r)=$J’exp(-q’)dn 710

1

exp(-

br*2)

*)

erfc(z)=l-erf(z)

A comparison will be made between the survival probability P(t) derived theoretically in eqns. (20) and the experimental results obtained by Shibata and Takeyama. Comparing eqns. (20) and (22), and using eqns. (7) and (ll), the birth and death parameters of pit initiation can be obtained as follows: h = &mi(r*)X-2g(r*)

(25.1)

~=q.Moi(r*)x-2{1-g(r”>}

(25.2)

X and 1-1are calculated for various values of the parameters x and w in Table For simplicity the case of x = 2 and w = 1 will be considered. A=

2z$$*

exp(

-br*)

1.

(26.1)

(26.2) where it is assumed that br * < 1, and the first term of the denominator expression of r* [2] is neglected compared with the second term:

for the

(27)

In eqn. (27), X, and h, are constants of the order of 1, L is the thickness of the passive film, A, is the specific conductivity of the halide nuclei, and a is the “jump distance” for ion transport in the passive film [2].

TABLE

2

The trend in the variation of the birth and death parameters ion concentration in the solution c,(O) and the passive J = decrease)

P

of the metal E, the halide L increase (f = increase,

c,(O)t

L?

T

t

1

_

_

?

Ef h

as the potential film thickness

The trends, whether A and ~1 increase or decrease when the potential of the passive metal E, the concentration of the halide ion in the solution ~~(0) and the passive film thickness L increase, are shown in Table 2. These trends hold for other selections of the parameters x and W. The results for the birth and death parameters in Table 2 agree well with those observed experimentally by Shibata and Takeyama in their stochastic analysis of pit initiation in stainless steels. * It could then be shown that the halide nucleus model which is applied to the stochastic analysis in the present work has a physical reality. The noise level The dissolution current which passes through the surface of the metal can be obtained from the radial distribution of the halide nuclei, which is given by eqn. (15). The fractional current Z(r, t) arising from halide nuclei of size r is 1( r, t) = i7rrXNSN( Then

Z(t)

r, t)

the average current

(28) expected

is

=JomZ(r, t) dr

F(2’ w’wak) [l -exp{ =i71NsF(2 ) -x,

The fluctuation

8Z(t) =

of the current

-Q*(E)t}]

can be expressed

(29) as follows:

w

-T&-q

l In the experiments by Shibata and Takeyama [8,9], failure by a pit was defined when the current rose above a certain threshold value. The time for this failure was used to evaluate the survival probability; hence the uncertainty should be taken into account concerning the value of the birth and death parameters [16]. The birth and death parameters derived in eqns. (26) are for the precursor states before growing pits, and would include overestimating the instances of pit initiation by neglecting the repassivation processes of actual pits. Then the present comparison should be regarded as that for general trends.

369 TABLE

3

Calculated

noise levels of the current

density

SI/I

for various

values of the parameters

x

w

61/I

1

1 2 1 2

0.707 0.503 1 0.962

2

It can also be written

as

H(t) -= ar = irnx NS Fe(!;l”‘wwi)

9 1

If eqns. (18) and following equation

[I -

exp{-Q*(E)t)]

(31) are substituted into is obtained at t + 00:

eqn.

(31) (30)

and

using 1 l/2

dI -2 I

p(1, ‘F(2,

F(3 -x, w, W> F(4 - x, w, cc> w, co) i F(2-X, w, cc) - 1 F(2-X,

w, a> W, cc)

I)

eqn.

(29)

the

(32)

Equation (32) gives the noise level of the current in the induction period of the pit initiation process. The calculated values of the noise level for various values of the parameters x and w are listed in Table 3. It can be seen from the table that the current noise expected as a result of halide nucleus growth/diminishing processes has an appreciable order of magnitude, so that it can easily be detected by the experiment. The power spectrum

of the current noise

It has been shown in the present work that the growth/diminishing process of halide nuclei in the passive film can cause a detectable level of electrochemical noise during the induction period of pitting corrosion. This model will be applied further to the power spectra of the current noise, i.e. the frequency distribution of the intensity of the current noise. For this purpose the current transient should be known along with time for each single occurrence of the halide nucleus growth/diminishing process. This can be obtained from i(t)

= iprX

(33)

and also by using eqns. (8) and (11): dr dt

x-2 =

qMolr

(34)

370

so that r = r0 exp( qMoit)

x=1

(35.1)

r = q&(

x=2

(35.2)

I + to)

Substituting

eqns. (35) into eqn. (33) we obtain

i(t)

= 9rr,i exp( qMoit)

x=1

(36.1)

i(t)

= nqLoi3( t + to)’

x=2

(36.2)

The power spectrum of the current noise will be calculated for each of these two cases. The power spectrum density GX,(w) of a variable x as a function of the frequency component w is expressed as follows [17]:

where X,(w) is the Fourier transform of the isolated in the measurable time range 0 < t < T as = JW xT( 1) exp( -jwt) -CC

&W

wave x,(t),

which is defined

dt

(38)

The current i(t) associated with the halide nucleus growth/diminishing process prior to pit initiation will now be considered in place of +(f). According to eqn. (36.1) or (36.2), this will increase along with the formation of a single halide nucleus, and becomes zero when the halide nucleus disappears at time t’. Then eqn. (38) becomes in this case

Z,<(w) = L”i(t)

exp(--jar)

dt

(39)

The power spectrum density G,,(w) for the current noise should then be the average for the entire lifetime t’ of the halide nuclei. t’ might range from 0 to the time t * needed for the nucleus to grow to the size r*. From eqns. (35) we obtain t* = -

1

.lnC

qM0’

t* = -

r*

qMOi

x=1

(40.1)

x=2

(40.2)

rO

- to

Also the time T is chosen as the time Tp in which one halide nucleus average, so that G,,(w)

=

$$/6*,Z&) P

I2 dt’

appears

on the

(41)

Then

the following

( 7Tro)2i

G,,(w)

=

L

(u)

=

fr2qhoi6

G,,

results can be obtained

Tp(4Moi)2

+

(see Appendix

ew(2choit* >

cd*

%4of *

B):

x=1

(42.1)

- 3w cos at*

P

8 + 8 + -t* L? w5

1 sin wt* + st*4

1

x=2

(42.2)

where it is assumed that r*/ro z+ 1 in the case x = 1, and t, = 0 in the case x = 2. G,,(w) takes the asymptotic value in the case x = 1:

GI,(~) a $-J

(43 .l)

2

For x = 2, it does not converge the limit w + co:

at the limit w + 0, but it takes the following

form at

(43.2)

DISCUSSION

The halide nuclei are supposed to appear as pit embryos, which occur in the precursor state before the pit initiation and propagation stages. It has sometimes been observed experimentally during the induction period of pitting corrosion that isolated current peaks or spikes appear which associate a charge of several 100 /.LC for one event, corresponding to a volume of lo3 pm3 [lo]. These are apparently due to the initiated and then repassivated pits, and other theoretical models [18] are required to analyse these propagating pits. Nachstedt and Heusler [6,7] reported on small current fluctuations, each event associated with a charge of several PC, and they explained these as due to the local thinning of passive films by non-uniform chemisorption of chlorides. The halide nucleus model utilized in this work would fall into events of the size as reported by Nachstedt and Heusler. However, the halide nuclei would grow in a stable manner if the critical condition was realized, and would initiate pitting [2]. This would lead to current spikes caused by those halide nuclei overcoming the critical size, although in some cases such initiated pits could be repassivated by other causes. Therefore, this model could be further extended to events of larger size where pit embryos go into initiating pits. Also, the model could be tested more appropriately for passive metals where the thickness of the passive film is large (more than 100 nm), so that halide nuclei of measurable dimensions would be plausible. The accumulation of experimental data in these passive metals is now required. The fluctuations of current accompanied by macroscopic events such as growing pits and their repassivation would include transport phenomena in solution within the pits, so they should be the subject of future study.

372 CONCLUSION

The following was shown for the electrochemical noise during the induction period of pitting corrosion in passive metals placed in aggressive media, by consideration of a stochastic model of the formation of halide nuclei. (1) The survival probability for the initiation of pitting was obtained based on the halide nucleus model, and good correlations were found with the reported experimental observations. (2) The physical meaning of the birth and death parameters of pit initiation was described, using microscopic parameters. (3) The noise level of the electrochemical fluctuation caused by growth/ diminishing of the halide nuclei was obtained, and it was shown to be of an appreciable order so that it can be measured. (4) The power spectrum density of the current noise was calculated, and it was shown to have the form l/o2 in the high frequency region, which was also in agreement with the experimental results. (5) The validity of the halide nucleus model in the pit initiation processes was shown to be acceptable. APPENDIX

A

If eqn. (2) is rewritten

using the method

N(rT‘1-

of the separation

D(r, E) Q+(,.,

E)

eqn. (Al) is written

R(r, E)

of variables

_

E)

=

C-42)

‘6.7 E)T(z, E)

as follows:

aT(t, E) at

Q_(r,

=

- &~+(6

E) - Q_(C

E)}%,

E)W,

E)]

(~3)

then 1

T(t,

aT(t, E)

at

E)

= qrl

E) ;[{Q+h

E) -

Q-try

E)}R(r,

E)]

(A4)

The left-hand side of eqn. (A4) is a function of t and E only, and the right-hand side is a function of r and E only, so in order that this equation is valid for the

313

change in t and r, both sides of the above equation It is put as C(E).

aT(t, E)

1

T(t,

E)

E) -

be functions

Q-(r,

(A5)

E)} + {Q+
Q_(r,

E)}$In R(r,

= C(E)

E) (A@

The solution T(t,

of E only.

= C(E)

at

${Q+(r,

should

to eqn. (A5) is

E) =K,(E)

exp{-C(E)t}

and the solution

(A7)

to eqn. (A6) is

R(r, E) = K,(E)

Q+(r,

E) 1 QP(r,

E)

exp’B(r7

(AS)

E)’

where

B(r’E,3 /

C(E) E) - Q_(r,

Q+(r,

E)

(A9)

dr

From

eqns. (A2), (A7) and (A@, we find that

N(r,

T) =N(r,

0) exp{ -C(E)t}

+N(r,

oo)[l -exp{

-C(E)}]

(AlO)

Here

N(rv ‘> =

~,(EME)

Q+(,.,

+

E)

Q+(r,

N(r7 03)=Q+(r,

_

Q_(,.,

D(r, E) E) - Q-(r, D(r, E) E) - Q_(r,

(A12)

E) N(r,

t) changes

exponentially

from

N(r, 0) to

B

First consider I,,(w)

(All)

E)

As can be seen from eqn. (AlO), N(r, co) in the course of time. APPENDIX

exP{B(ryE)t)

E)

= q,Il’!

the case x = 1. Substituting

jw

bd<

qMoi --jw)t’}

- l]

eqn. (36.1) into eqn. (39)

(Bl)

314

Take the complex conjugate of eqn. substitute the result into eqn. (41): G,,(o)

=

(rroi)2

f

P (qMoi)2+

a2

2

1

-

[

( qMoi)’ + u2 F

+o

(Bl),

exp(2q,,it t”

1 2q,,i

l+

multiply

{exp(q,,it*)(q,,i

it by eqn.

(Bl)

and

*) - 1

cos wt*

- qMoi}

sin at”)

then

(B2)

I Substitute eqn. (40.1) into the above equation eqn. (42.1) is obtained. Next consider the case x = 2. Substituting

and consider

the case r */r. B 1, then

eqn. (36.2) into eqn. (39)

exp(-jut’)-to}

I,,(b~)=~{(t’+t~)

1

(t’+to,‘--$

exp(-jut’)-ti+-$

(B3)

where C, = TqLoi3

(B4)

Take the complex conjugate result into eqn. (41)

of eqn. (B3), multiply

it by eqn. (B3) and substitute

the

- 3w cos wt*

+A CL@ [l-t,

coswt*-1

(

sin wt * +A t,Z t* -w cl_?[ (

+3 -- ,“,

[

[

+o

t*

cos at*

tot* cos ot* - 2to’ sin wt* t,4~ t*

+ tit*

i

,1

+ 2t* sin wt*

1

cos wt* - 1 + w sin ot* t* ( 11

sin wt*

+L (t*+toY4 a2 [ 5t*

sin&*

+t,4-F

I)

In the case to = 0, eqn. (42.2) is obtained.

1 (B5)

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