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A theoretical analysis of the electrochemical noise during the induction period of pitting corrosion in passive metals
349
J. Electroanal. Chem., 297 (1991) 349-359 Elsevier Sequoia S.A., Lausanne
A theoretical the induction
analysis of the electrochemical noise during period of pitting corrosion in passive metals
Part 1. The current noise associated with the adsorption/desorption processes of halide ions on the passive film surface T. Okada Industrial Products Research Institute, M. I. T. I., Yatabe, Tsukuba, Ibaraki 30.5 (Japan) (Received
2 March
1990; in revised form 23 August
1990)
Abstract
It has been demonstrated theoretically that the adsorption/desorption process of halide ions on the surface of a passive metal can cause electrochemical noise during the induction period of pitting corrosion on passive metals. The process was considered by using a mode1 in which the lattice cation of a passive film makes a transitional complex with the halide ions in the solution, followed by local dissolution into the solution. A stochastic analysis was applied to the system, and the distribution function was obtained for the coordination number around the surface cation of the lattice of the passive film. The noise level was determined as the uncertainty of the distribution function for the coordination number. The noise level was rather low when the probability for the formation of the transitional halide complex was high. The noise level increased as the coordination number of halide ions necessary for the formation of the transitional complex increased, or as the activity of the halide ions in the solution decreased. When the van der Waals condition was established concerning the adsorption of halide ions on the passive film, the noise level reached the maximum value.
INTRODUCTION
Passive metals are of practical significance because they reveal a high corrosion resistance in corrosive environments owing to the protective power of the passive film which forms on the surface of these metals [1,2]. The protective power of a passive film may come from microscopic self-repairing processes which occur instantaneously at the local sites of dissolution at the film/solution interface. This protective power might be a common feature for passivity, although the thickness of the passive film L changes along with the material and also the environment; e.g. for stainless steels L ranges from 3 to 5 nm, for nickel-based alloys from 0.5 to 1 0022-0728/91/$03.50
0 1991 - Elsevier Sequoia
S.A.
350
nm, for chrome alloys from 1 to 2 nm, for titanium and zirconium from 100 to 200 nm, for tantalum about 200 nm, and barrier and porous layers of aluminum make L more than tens of pm [3,4]. These metals are intrinsically very active in the sense of thermodynamic stability, and belong to “less noble metals” in the electrochemical series. However, if they are polarized to positive potentials in a wet environment, they form oxide films on the surface and become passive because oxide films are the cause of the protective nature of such metals. In acidic environments containing aggressive ions such as halide ions (e.g. Cll or Br- ions), however, the passive films are sometimes destroyed locally and a very specific phenomenon, called pitting corrosion, occurs. Pitting corrosion causes hazardous effects to materials, and attracts practical interest. It also gives several kinds of information about the dynamic stability of the destruction/repair processes on the passive film surface [5-81. Consequently, recent research work has been putting much emphasis on the analysis of the dynamic stability of the passive film accompanying the occurrence of pitting corrosion by use of non-stationary measurements [9,10]. In particular, the “electrochemical noise” which is observed during the induction period of pitting corrosion has been recognized to be closely related to the metal dissolution/redeposition processes or to the passive film destruction/repair processes which proceed on the passive film surface or inside it, and much theoretical and experimental research work has been carried out [ll-151. Shibata and Takeyama [16] introduced a stochastic analysis to the problem of the statistical deviation of the pit initiation potential and the induction time observed experimentally for stainless steels, and evaluated the pitting corrosion resistance of the material. Williams et al. [17] related the electrochemical noise during the induction time of pitting corrosion to the birth and death stochastic processes and proposed a model in which local acidification of the solution in contact with the passive film brings about the pit initiation. Also, Heusler and co-workers [18-201 measured the spontaneous fluctuations of the passive current in iron electrodes accompanying the addition of chloride ions, and proposed a model based on local fluctuations of the passive film thickness due to non-uniform chemisorption of the chloride. If the probabilistic features of pitting corrosion are noted, then the dynamic balance between the appearance and diminishing of pit embryos will be an important factor in considering the stability of pitting processes. This will be the same point of view as that for studying the passivity phenomena as the dynamic balance of breakdown and repair processes of the passive film. This point of view takes into account the kinetic stability of pitting corrosion, unlike the earlier considerations concerning the static stabilities for the pitting corrosion phenomena. Then it will be possible to discuss the mechanism of pitting reactions from the analysis of, for example, the electrochemical noise, the frequency decomposition of signals and the power spectra maps. I have been concerned mainly with the kinetic consideration of pitting phenomena, and have proposed a model in which pitting is assumed to occur via the probabilistic fluctuation processes. The occurrence of pitting was studied by consid-
351
eration of the stability and instability criteria applied to the possible steps, first the perturbation in the solution and then the appearance of halide nuclei in the passive film [21-231. In this paper, it is assumed that the occurrence of electrochemical noise at the induction period of pitting corrosion has certain correlations with the fluctuation processes proposed before. The phenomena which bring about such noise will be studied and some physical meaning of the stochastic parameters concerning the pit initiation will be clarified based on the proposed model. The observed electrochemical noise is usually composed of waves with several frequency bands, and it appears that the phenomena which cause those of high frequency (> 10 Hz) and those of moderate or low frequency (< 10 Hz) are different. So, the phenomena with high frequency will be considered in this paper, and those with low frequency will be discussed in subsequent papers. THEORY
The adsorption of halide ions on the passive film surface Okamoto et al. [ll] observed current noise of relatively high frequencies in the induction period of pitting corrosion of stainless steels immersed in chloride solutions. They regarded such noise to be caused by the passivity breakdown/repair processes which might sustain the passive state. This kind of noise appeared normally after the metals had been placed into aggressive environments, so it was assumed that aggressive anions (chloride ions) took part in its occurrence. The process of passive metal dissolution was treated using the transitional halide complex theory proposed earlier by Hoar et al., and an equation for the dissolution rate was obtained [24]. In this equation it was assumed that the metal cation makes a transitional complex with the halide ions and dissolves into the solution if the number of halide ions (coordination number) adsorbed around a lattice cation at the surface of the passive film exceeds a fixed value. A critical state was discovered, where the halide ion adsorption/desorption process suffers a drastic change.
-----.
I’ OH- H20'\ h
\,‘--
xz__o$
\“,,,‘~L
Fig. 1. Illustration of the elementary steps of adsorption/desorption cations on the surface of the passive film.
of halide
ions around
the lattice
352
In this paper the adsorption/desorption process of halide ions on the passive film surface will be considered from a stochastic point of view, and the appearance of current noise which is assumed to result from the fluctuation in the system will be discussed together with its properties. It will be important to consider to what degree each elementary step of halide ion adsorption/desorption exerts an effect on the current noise of passive metal dissolution into the solution. In Fig. 1, the elementary processes of adsorption/desorption of halide ions which occur on the surface of the passive metal are depicted schematically. When the number of coordinated halide ions around a metal cation exceeds a certain value, the metal cation passes over an energy barrier, drawn by ligand halide ions, and dissolves into the solution. The smallest number of coordination of halide ions, which is required to form this transitional complex, will be denoted as nM. Stochastic
model
For simplicity it is assumed that each elementary step of the adsorption of halide ions around a metal cation on the surface of the passive film does not interfere with each other. Each adsorption step can be considered as Markovian. The fraction of metal ions on the surface of the passive film which is coordinated by n halide ions at time t is designated as P,,(t). The following master equations [25] can be written for its time variation: &0(t)
= h-,P,(t)
UP,,
= L(n
-G%(t) + lP,+,(t)
(1.1) - P,(N,-
+4{N,-(n-lHP,-,(t) $P,(r)
= Lb
n) +~-I+&) l
(1.2)
n,
(1.3)
+ l)P,+,(t)
-{X1(N,-n)+X-,n+K~}P,(t) +~,{Ns-(~-~))PnAt) ;P.\(t)=
-{A-,N,+ + hP,,-1(t)
hA~Piv,(t) (1.4)
where N, is the number of adsorption sites around the metal cation at the surface of the passive film, and K~ is the dissolution rate constant of the lattice metal cations, which are coordinated by n ( nM < n < N,) halide ions, forming the transitional complex. It is assumed that the adsorption/desorption of halide ions onto the lattice cations occur very fast so that a quasi-equilibrium condition is attained: hr, h-1 ~KM
(2)
A, and A_, are the adsorption and desorption frequencies of the halide ions per unit time, respectively, and can be related to the rate constants of the adsorption
353
and desorption
processes:
A, = +X(O)
(3.1)
h-r
(3.2)
= k-r
Here c,(O) is the concentration of halide ions in the solution at the passive film/solution interface. Equations (1) are microscopic expressions for the elementary steps, but they can be converted into a macroscopic expression by averaging operations. The average coordination number of halide ions around the surface metal cation, (n), can be expressed as
l%(t>
(n> =c
(4)
If K~ can be neglected eqns. (1) and (4) &n,
= -X-,(n)
+ WY
in eqn. (1) then the following
equation
is obtained
from
(5)
- W)
which is the rate equation of the adsorption/desorption steps of the halide ions. The function P,(t) will now be obtained for the initial conditions: P,(O) = 1
(6.1) nzo
P,(O) = 0 The solution
(6.2)
to eqn. (1) is as follows (see Appendix):
(7) From
eqns. (4) and (7) we obtain
(n> =&A, ::,
(see Appendix)
[l-ex~{-(A,+~-~)~)l -1
Then it can be seen from eqn. (7) that the average number of halide ions which are coordinated around the surface metal cation increases asymptotically from 0 to N,W(% + A-,). The dispersion
of n is expressed
x[h_,+h,exp{-(h,+X_,)t}]
l
23 means
I&
unless otherwise specified.
as follows (see Appendix):
(9)
354
The noise level The dissolution current of the metal ions can be obtained from the fractional distribution function of the coordination number of halide ions around the surface metal cations given by eqn. (7). Suppose that the number of metal cations per unit surface of the passive film (i.e. the surface density of the lattice metal cations) is uM and the charge of the metal cation is z, then the average current density (I) is given by
(10.1)
(1) =c4 0
z, = i
n
(10.2)
zefJMKMPn(t) nM
where e is a unit charge. On the other hand, the dispersion equation:
of the current
density
is given by the following
0: = c (1, - (W2CW
= ~z;P,(t) =
- (z)2
(ze%fKM)2(XM -XL)
(11 .l)
where
(11.2)
TABLE Calculated
1 noise levels of the current
density
61/l
for various
parameter
values
A.
N,
X,+h-,
6
5/6
4/6
5
3/6 4/5
4
3/5 3/4
4 3 2 3 2 2 3 2 2 2
0.938 0.991 0.999 0.900 0.982 0.891 0.942 0.993 0.913 0.949
0.258 0.094 0.026 0.334 0.135 0.350 0.248 0.082 0.309 0.231
355
If u, can be equated (10) and (11)
--1
-J -
to the fluctuations
of the current
density
I, then from eqns.
(12)
1
XM
In Table 1, the values of SZ/I, i.e. the noise level for the current density, the time calculated for various values of the parameters are listed. In the calculation for P,z(t) in eqn. (7) was set at t + cc, because each adsorption/desorption process of the halide ions proceeds very rapidly compared with the variation of the fluctuation of the current density. From the table it can be seen that the noise level of the current density for passive film dissolution via the transitional halide complex ranges between 0.03 and 0.35. The noise level at the critical state Equation
h A, + A-,
(8) can be rearranged
(n> =@+c9) =
cc
which is the coverage n,/N,
into
(13) of adsorbed
halide ions at the equilibrium
state. If
= 19,
is defined, this means the minimum required coverage of the adsorbed above which lattice cations on the surface of the passive film form a halide complex with the halide ions, and dissolve into the solution. The where the in Table 1 were chosen so that dM < Q, i.e. the condition activity in the solution is so high that the coverage of halide ions around lattice cations becomes high. It the following condition is chosen:
e, dM
(14) halide ions transitional parameters halide ion the surface
(15)
then the probability of the surface metal cations transferring into the transitional state becomes very low. The noise level for these conditions are shown in Table 2. Comparison of Tables 1 and 2 shows that the system changes abruptly from a state where passive film dissolution proceeds smoothly to a state where current fluctuation starts, when the coverage of halide ions becomes small, reaching a critical state. It should be noticed, however, that in such cases the current densities are also small and become comparable to the passive current density of the metal, which was not considered here, so this does not necessarily mean an increased current noise.
356 TABLE Calculated
2 noise levels of the current
6
5
4
5/6 4/6 3/6 2/6 4/5 3/5 215 3/4 214
density
5 4 3 2 4 3 2 3 2
SI/I
at the critical
state
0.733 0.680 0.656 0.649 0.737 0.683 0.663 0.738 0.668
0.598 0.685 0.724 0.736 0.597 0.682 0.713 0.595 0.674
Next another type of critical state will be considered, where the halide ion adsorption equilibrium becomes very unstable. It has been reported that current noise can be expected when the condition of a van der Waals loop is realized concerning the competitive adsorption between the halide and the hydroxide ions, or when the adsorption isotherm shows a very steep rise due to the shift of the electric potential of the Helmholtz plane by adsorbed halide ions [24]. In these cases, 19, will fluctuate between nearly 0 and nearly 1, so X~ = OS. Then from eqn. (12) the current noise level tiecomes SI/Z = 1. This corresponds to a very large current noise.
DISCUSSION
Electrochemical noise originates from statistical fluctuations in the rate of electrochemical processes in equilibrium or non-equilibrium conditions, especially when there are uneven distributions of reactive species or catalytic centres in space. More generally, shot noise of a very low level has been demonstrated to arise owing to electron-transfer processes across the electrode/solution interface [26-291, but special precautions should be taken in measuring such noise. In the case where large deterministic processes are induced by microscopic (molecular) events, relatively large fluctuations accompanying phase growth can be expected to appear as electrochemical noise, like two-dimensional or three-dimensional nucleation of crystal growth, ion transport through biological membranes, electrocatalysis, etc. [30-321. The noise observed during the induction period of pitting corrosion can be categorized into two parts: one is low noise arising from elementary surface reactions, and the other is high noise, which can be described by discrete phase transition models. Both of these are induced by molecular fluctuations and also should have non-equilibrium characters.
351
In this paper, the noise accompanying the local dissolution of passive films was considered using a stochastic model concerning the transitional metal-halide complexes, and the following was shown: (1) The current noise level ranged around 0.03-0.35, depending on the situation, if the activity of halide ions in the solution was so high that the probability for transitional halide complex formation was also high. (2) This noise level increased when nM, the smallest value of the coordination number of halide ions around the surface metal cations required for making the transitional complex, increased. (3) For the same value of n M, the noise level increased when the activity of the halide ions in the solution decreased and their surface coverage at the equilibrium state decreased. (4) When the adsorption equilibrium was very unstable, so that the van der Waals condition appeared [24], the noise level attained maximum values. Because the effects could have become minute if the events were averaged in time or in space, careful measurements are required to observe intrinsic electrochemical noise. Okamoto et al. [ll] and Heusler and co-workers [18,19] observed electrochemical noise at a high frequency band in stainless steels or passivated irons, after the addition of chloride to the electrolyte. Although there was not much experimental evidence for the occurrence of adsorption/desorption processes in these stages, such high frequency noise was suggestive of the surface reaction kinetics in accordance with the present consideration. Further experimental studies are required to confirm the predictions of the present study. APPENDIX
Define G(x,
the generating
function
G(x,
t) as follows:
t) =xx”P,,(t)
(AI)
Here x is an arbitrary (Al), we obtain
variable
(0 < x < 1). Ignoring
K~
in eqn. (1) and using eqn.
x, t) =CX_,(n + l)x”~,+&) &G(
-C{ X,(N, - n) + Ll”}xnPn(r)
+D,{ x - (n - l)b”e-*(d Again
substituting
j&x, r) = (1 Putting that G(x,
eqn. (Al) into the above equation, x)( (X-i
t = 0 in eqn. (Al) 0) = 1
(AZ)
+ Q)&G(x, and using
we obtain
t) -X,&C+, the initial
conditions
t))
(A3)
(6.1) and (6.2), we find
(A4)
358
The solution
to eqn. (A3) satisfying
Gb, t) =
l
h,
[A-,
+A_,
+A,
eqn. (A4) is
exp{ -(A, +A-,)t}]
i
+
h,
:1,, [l
+h-,)fllxjNs
-exp(-(A,
(A51
Expanding eqn. (A5) and comparing it with eqn. (Al), eqn. (7) can be obtained. Also, differentiating eqn. (Al) with respect to x and putting x = 1, the following equation is obtained:
(A6) The right-hand side of eqn. (A6) is equal to (n), according of eqn. (A5), the left-hand side of eqn. (A6) becomes
&(I,
t) =N+
1
;1,_1 [I- ev{-(A, +X-l)fll
Then eqn. (8) is obtained. The dispersion of n is written a2n = (2)
to eqn. (4). Also, by use
(A-3
as
- (L+
(A8)
Also,
(n’) =Cn’P,(t) = &(x&G(l,
t)
(A9) Substitution
of eqns. (A9) and (8) into eqn. (A8) gives eqn. (9).
REFERENCES R.P. Frankenthal and J. Kruger (Eds.), Passivity of Metals, The Electrochem. Sot., New Jersey, 1978. M. Froment (Ed.), Passivity of Metals and Semiconductors, Elsevier, Amsterdam, 1983. N. Sato, Denki Kagaku, 46 (1978) 584. J.W. D&e (Ed.), Oxides and Oxide Films, Vol. 2, Marcel Dekker, New York, 1973. 2. Szklarska-Smialowska, Corrosion, 27 (1971) 223. M. Janik-Czachor, J. Electrochem. Sot., 128 (1981) 513C. H.-H. Strehblow, Werkst. Korros., 35 (1984) 437. T. Okada, Boshoku Gijutsu, 36 (1987) 383. M. Duprat (Ed.), Electrochemical Methods in Corrosion Research, Material Science Forum, Vol. 8, Tram Tech Publications, Switzerland, 1986.
359 10 R.A. Oriani and H.J. Engell (Eds.), The German-American Colloquium on Electrochemical Passivation, Corros. Sci., 29 (1989) No. 2/3. 11 G. Okamoto, T. Shibata, S. Nishiyama and K. Tachibana, Boshoku Gijutsu, 23 (1974) 439. 12 U. Bertocci, J. Electrochem. Sot., 127 (1980) 1931. 13 U. Bertocci, J. Electrochem. Sot., 128 (1981) 520. 14 S. Magaino, M. Yasuda and H. Yamada, Boshoku Gijutsu, 34 (1985) 157. 15 M. Keddam, M. Kuraiti and Pallota, Corrosion, 43 (1987) 454. 16 T. Shibata and T. Takeyama, in Metallic Corrosion, Proc. 8th Int. Congress on Metallic Corrosion, DECHEMA, Frankfurt/Main, 1981, p. 146. 17 D.E. Williams, C. Westcott and M. Fleischmann, J. Electrochem. Sot., 132 (1985) 1796, 1804. 18 R. Doelling and K.E. Heusler, Z. Phys. Chem., N.F., 139 (1984) 39. 19 K. Nachstedt and K.E. Heusler, Z. Phys. Chem., 160 (1988) 131. 20 K. Nachstedt and K.E. Heusler, Electrochim. Acta, 33 (1988) 311. 21 T. Okada, J. Electrochem. Sot., 131 (1984) 241. 22 T. Okada, J. Electrochem. Sot., 132 (1985) 537. 23 T. Okada, Corros. Sci., 31 (1990) 453. 24 T. Okada, Corros. Sci., 26 (1986) 839. 25 K.E. Shuler (Ed.), Advances in Chemical Physics, Vol. 15: Stochastic Processes in Chemical Physics, Interscience, New York, 1969. 26 G.C. Barker, J. Electroanal. Chem., 21 (1969) 127. 27 V.A. Tyagai and N.B. Luk’yanchikova, Elektrokhimiya, 3 (1967) 316. 28 V.A. Tyagai, Elektrokhimiya, 3 (1967) 1331. 29 M. Seralathan and S.K. Rangarajan, J. Electroanal. Chem., 208 (1986) 13, 29. 30 P. Bindra, M. Fleischmann, J.W. Oldfield and D. Singleton, Faraday Discuss. Chem Sot., 56 (1974) 180. 31 M. Fleischmann, M. Labram, C. Gabrielli and A. Sattar, Surf. Sci., 101 (1980) 583. 32 M. Fleischmann, L.J. Li and L.M. Peter, Electrochim. Acta, 34 (1989) 475.
J. Electroanal. Chem., 297 (1991) 349-359 Elsevier Sequoia S.A., Lausanne
A theoretical the induction
analysis of the electrochemical noise during period of pitting corrosion in passive metals
Part 1. The current noise associated with the adsorption/desorption processes of halide ions on the passive film surface T. Okada Industrial Products Research Institute, M. I. T. I., Yatabe, Tsukuba, Ibaraki 30.5 (Japan) (Received
2 March
1990; in revised form 23 August
1990)
Abstract
It has been demonstrated theoretically that the adsorption/desorption process of halide ions on the surface of a passive metal can cause electrochemical noise during the induction period of pitting corrosion on passive metals. The process was considered by using a mode1 in which the lattice cation of a passive film makes a transitional complex with the halide ions in the solution, followed by local dissolution into the solution. A stochastic analysis was applied to the system, and the distribution function was obtained for the coordination number around the surface cation of the lattice of the passive film. The noise level was determined as the uncertainty of the distribution function for the coordination number. The noise level was rather low when the probability for the formation of the transitional halide complex was high. The noise level increased as the coordination number of halide ions necessary for the formation of the transitional complex increased, or as the activity of the halide ions in the solution decreased. When the van der Waals condition was established concerning the adsorption of halide ions on the passive film, the noise level reached the maximum value.
INTRODUCTION
Passive metals are of practical significance because they reveal a high corrosion resistance in corrosive environments owing to the protective power of the passive film which forms on the surface of these metals [1,2]. The protective power of a passive film may come from microscopic self-repairing processes which occur instantaneously at the local sites of dissolution at the film/solution interface. This protective power might be a common feature for passivity, although the thickness of the passive film L changes along with the material and also the environment; e.g. for stainless steels L ranges from 3 to 5 nm, for nickel-based alloys from 0.5 to 1 0022-0728/91/$03.50
0 1991 - Elsevier Sequoia
S.A.
350
nm, for chrome alloys from 1 to 2 nm, for titanium and zirconium from 100 to 200 nm, for tantalum about 200 nm, and barrier and porous layers of aluminum make L more than tens of pm [3,4]. These metals are intrinsically very active in the sense of thermodynamic stability, and belong to “less noble metals” in the electrochemical series. However, if they are polarized to positive potentials in a wet environment, they form oxide films on the surface and become passive because oxide films are the cause of the protective nature of such metals. In acidic environments containing aggressive ions such as halide ions (e.g. Cll or Br- ions), however, the passive films are sometimes destroyed locally and a very specific phenomenon, called pitting corrosion, occurs. Pitting corrosion causes hazardous effects to materials, and attracts practical interest. It also gives several kinds of information about the dynamic stability of the destruction/repair processes on the passive film surface [5-81. Consequently, recent research work has been putting much emphasis on the analysis of the dynamic stability of the passive film accompanying the occurrence of pitting corrosion by use of non-stationary measurements [9,10]. In particular, the “electrochemical noise” which is observed during the induction period of pitting corrosion has been recognized to be closely related to the metal dissolution/redeposition processes or to the passive film destruction/repair processes which proceed on the passive film surface or inside it, and much theoretical and experimental research work has been carried out [ll-151. Shibata and Takeyama [16] introduced a stochastic analysis to the problem of the statistical deviation of the pit initiation potential and the induction time observed experimentally for stainless steels, and evaluated the pitting corrosion resistance of the material. Williams et al. [17] related the electrochemical noise during the induction time of pitting corrosion to the birth and death stochastic processes and proposed a model in which local acidification of the solution in contact with the passive film brings about the pit initiation. Also, Heusler and co-workers [18-201 measured the spontaneous fluctuations of the passive current in iron electrodes accompanying the addition of chloride ions, and proposed a model based on local fluctuations of the passive film thickness due to non-uniform chemisorption of the chloride. If the probabilistic features of pitting corrosion are noted, then the dynamic balance between the appearance and diminishing of pit embryos will be an important factor in considering the stability of pitting processes. This will be the same point of view as that for studying the passivity phenomena as the dynamic balance of breakdown and repair processes of the passive film. This point of view takes into account the kinetic stability of pitting corrosion, unlike the earlier considerations concerning the static stabilities for the pitting corrosion phenomena. Then it will be possible to discuss the mechanism of pitting reactions from the analysis of, for example, the electrochemical noise, the frequency decomposition of signals and the power spectra maps. I have been concerned mainly with the kinetic consideration of pitting phenomena, and have proposed a model in which pitting is assumed to occur via the probabilistic fluctuation processes. The occurrence of pitting was studied by consid-
351
eration of the stability and instability criteria applied to the possible steps, first the perturbation in the solution and then the appearance of halide nuclei in the passive film [21-231. In this paper, it is assumed that the occurrence of electrochemical noise at the induction period of pitting corrosion has certain correlations with the fluctuation processes proposed before. The phenomena which bring about such noise will be studied and some physical meaning of the stochastic parameters concerning the pit initiation will be clarified based on the proposed model. The observed electrochemical noise is usually composed of waves with several frequency bands, and it appears that the phenomena which cause those of high frequency (> 10 Hz) and those of moderate or low frequency (< 10 Hz) are different. So, the phenomena with high frequency will be considered in this paper, and those with low frequency will be discussed in subsequent papers. THEORY
The adsorption of halide ions on the passive film surface Okamoto et al. [ll] observed current noise of relatively high frequencies in the induction period of pitting corrosion of stainless steels immersed in chloride solutions. They regarded such noise to be caused by the passivity breakdown/repair processes which might sustain the passive state. This kind of noise appeared normally after the metals had been placed into aggressive environments, so it was assumed that aggressive anions (chloride ions) took part in its occurrence. The process of passive metal dissolution was treated using the transitional halide complex theory proposed earlier by Hoar et al., and an equation for the dissolution rate was obtained [24]. In this equation it was assumed that the metal cation makes a transitional complex with the halide ions and dissolves into the solution if the number of halide ions (coordination number) adsorbed around a lattice cation at the surface of the passive film exceeds a fixed value. A critical state was discovered, where the halide ion adsorption/desorption process suffers a drastic change.
-----.
I’ OH- H20'\ h
\,‘--
xz__o$
\“,,,‘~L
Fig. 1. Illustration of the elementary steps of adsorption/desorption cations on the surface of the passive film.
of halide
ions around
the lattice
352
In this paper the adsorption/desorption process of halide ions on the passive film surface will be considered from a stochastic point of view, and the appearance of current noise which is assumed to result from the fluctuation in the system will be discussed together with its properties. It will be important to consider to what degree each elementary step of halide ion adsorption/desorption exerts an effect on the current noise of passive metal dissolution into the solution. In Fig. 1, the elementary processes of adsorption/desorption of halide ions which occur on the surface of the passive metal are depicted schematically. When the number of coordinated halide ions around a metal cation exceeds a certain value, the metal cation passes over an energy barrier, drawn by ligand halide ions, and dissolves into the solution. The smallest number of coordination of halide ions, which is required to form this transitional complex, will be denoted as nM. Stochastic
model
For simplicity it is assumed that each elementary step of the adsorption of halide ions around a metal cation on the surface of the passive film does not interfere with each other. Each adsorption step can be considered as Markovian. The fraction of metal ions on the surface of the passive film which is coordinated by n halide ions at time t is designated as P,,(t). The following master equations [25] can be written for its time variation: &0(t)
= h-,P,(t)
UP,,
= L(n
-G%(t) + lP,+,(t)
(1.1) - P,(N,-
+4{N,-(n-lHP,-,(t) $P,(r)
= Lb
n) +~-I+&) l
(1.2)
n,
(1.3)
+ l)P,+,(t)
-{X1(N,-n)+X-,n+K~}P,(t) +~,{Ns-(~-~))PnAt) ;P.\(t)=
-{A-,N,+ + hP,,-1(t)
hA~Piv,(t) (1.4)
where N, is the number of adsorption sites around the metal cation at the surface of the passive film, and K~ is the dissolution rate constant of the lattice metal cations, which are coordinated by n ( nM < n < N,) halide ions, forming the transitional complex. It is assumed that the adsorption/desorption of halide ions onto the lattice cations occur very fast so that a quasi-equilibrium condition is attained: hr, h-1 ~KM
(2)
A, and A_, are the adsorption and desorption frequencies of the halide ions per unit time, respectively, and can be related to the rate constants of the adsorption
353
and desorption
processes:
A, = +X(O)
(3.1)
h-r
(3.2)
= k-r
Here c,(O) is the concentration of halide ions in the solution at the passive film/solution interface. Equations (1) are microscopic expressions for the elementary steps, but they can be converted into a macroscopic expression by averaging operations. The average coordination number of halide ions around the surface metal cation, (n), can be expressed as
l%(t>
(n> =c
(4)
If K~ can be neglected eqns. (1) and (4) &n,
= -X-,(n)
+ WY
in eqn. (1) then the following
equation
is obtained
from
(5)
- W)
which is the rate equation of the adsorption/desorption steps of the halide ions. The function P,(t) will now be obtained for the initial conditions: P,(O) = 1
(6.1) nzo
P,(O) = 0 The solution
(6.2)
to eqn. (1) is as follows (see Appendix):
(7) From
eqns. (4) and (7) we obtain
(n> =&A, ::,
(see Appendix)
[l-ex~{-(A,+~-~)~)l -1
Then it can be seen from eqn. (7) that the average number of halide ions which are coordinated around the surface metal cation increases asymptotically from 0 to N,W(% + A-,). The dispersion
of n is expressed
x[h_,+h,exp{-(h,+X_,)t}]
l
23 means
I&
unless otherwise specified.
as follows (see Appendix):
(9)
354
The noise level The dissolution current of the metal ions can be obtained from the fractional distribution function of the coordination number of halide ions around the surface metal cations given by eqn. (7). Suppose that the number of metal cations per unit surface of the passive film (i.e. the surface density of the lattice metal cations) is uM and the charge of the metal cation is z, then the average current density (I) is given by
(10.1)
(1) =c4 0
z, = i
n
(10.2)
zefJMKMPn(t) nM
where e is a unit charge. On the other hand, the dispersion equation:
of the current
density
is given by the following
0: = c (1, - (W2CW
= ~z;P,(t) =
- (z)2
(ze%fKM)2(XM -XL)
(11 .l)
where
(11.2)
TABLE Calculated
1 noise levels of the current
density
61/l
for various
parameter
values
A.
N,
X,+h-,
6
5/6
4/6
5
3/6 4/5
4
3/5 3/4
4 3 2 3 2 2 3 2 2 2
0.938 0.991 0.999 0.900 0.982 0.891 0.942 0.993 0.913 0.949
0.258 0.094 0.026 0.334 0.135 0.350 0.248 0.082 0.309 0.231
355
If u, can be equated (10) and (11)
--1
-J -
to the fluctuations
of the current
density
I, then from eqns.
(12)
1
XM
In Table 1, the values of SZ/I, i.e. the noise level for the current density, the time calculated for various values of the parameters are listed. In the calculation for P,z(t) in eqn. (7) was set at t + cc, because each adsorption/desorption process of the halide ions proceeds very rapidly compared with the variation of the fluctuation of the current density. From the table it can be seen that the noise level of the current density for passive film dissolution via the transitional halide complex ranges between 0.03 and 0.35. The noise level at the critical state Equation
h A, + A-,
(8) can be rearranged
(n> =@+c9) =
cc
which is the coverage n,/N,
into
(13) of adsorbed
halide ions at the equilibrium
state. If
= 19,
is defined, this means the minimum required coverage of the adsorbed above which lattice cations on the surface of the passive film form a halide complex with the halide ions, and dissolve into the solution. The where the in Table 1 were chosen so that dM < Q, i.e. the condition activity in the solution is so high that the coverage of halide ions around lattice cations becomes high. It the following condition is chosen:
e, dM
(14) halide ions transitional parameters halide ion the surface
(15)
then the probability of the surface metal cations transferring into the transitional state becomes very low. The noise level for these conditions are shown in Table 2. Comparison of Tables 1 and 2 shows that the system changes abruptly from a state where passive film dissolution proceeds smoothly to a state where current fluctuation starts, when the coverage of halide ions becomes small, reaching a critical state. It should be noticed, however, that in such cases the current densities are also small and become comparable to the passive current density of the metal, which was not considered here, so this does not necessarily mean an increased current noise.
356 TABLE Calculated
2 noise levels of the current
6
5
4
5/6 4/6 3/6 2/6 4/5 3/5 215 3/4 214
density
5 4 3 2 4 3 2 3 2
SI/I
at the critical
state
0.733 0.680 0.656 0.649 0.737 0.683 0.663 0.738 0.668
0.598 0.685 0.724 0.736 0.597 0.682 0.713 0.595 0.674
Next another type of critical state will be considered, where the halide ion adsorption equilibrium becomes very unstable. It has been reported that current noise can be expected when the condition of a van der Waals loop is realized concerning the competitive adsorption between the halide and the hydroxide ions, or when the adsorption isotherm shows a very steep rise due to the shift of the electric potential of the Helmholtz plane by adsorbed halide ions [24]. In these cases, 19, will fluctuate between nearly 0 and nearly 1, so X~ = OS. Then from eqn. (12) the current noise level tiecomes SI/Z = 1. This corresponds to a very large current noise.
DISCUSSION
Electrochemical noise originates from statistical fluctuations in the rate of electrochemical processes in equilibrium or non-equilibrium conditions, especially when there are uneven distributions of reactive species or catalytic centres in space. More generally, shot noise of a very low level has been demonstrated to arise owing to electron-transfer processes across the electrode/solution interface [26-291, but special precautions should be taken in measuring such noise. In the case where large deterministic processes are induced by microscopic (molecular) events, relatively large fluctuations accompanying phase growth can be expected to appear as electrochemical noise, like two-dimensional or three-dimensional nucleation of crystal growth, ion transport through biological membranes, electrocatalysis, etc. [30-321. The noise observed during the induction period of pitting corrosion can be categorized into two parts: one is low noise arising from elementary surface reactions, and the other is high noise, which can be described by discrete phase transition models. Both of these are induced by molecular fluctuations and also should have non-equilibrium characters.
351
In this paper, the noise accompanying the local dissolution of passive films was considered using a stochastic model concerning the transitional metal-halide complexes, and the following was shown: (1) The current noise level ranged around 0.03-0.35, depending on the situation, if the activity of halide ions in the solution was so high that the probability for transitional halide complex formation was also high. (2) This noise level increased when nM, the smallest value of the coordination number of halide ions around the surface metal cations required for making the transitional complex, increased. (3) For the same value of n M, the noise level increased when the activity of the halide ions in the solution decreased and their surface coverage at the equilibrium state decreased. (4) When the adsorption equilibrium was very unstable, so that the van der Waals condition appeared [24], the noise level attained maximum values. Because the effects could have become minute if the events were averaged in time or in space, careful measurements are required to observe intrinsic electrochemical noise. Okamoto et al. [ll] and Heusler and co-workers [18,19] observed electrochemical noise at a high frequency band in stainless steels or passivated irons, after the addition of chloride to the electrolyte. Although there was not much experimental evidence for the occurrence of adsorption/desorption processes in these stages, such high frequency noise was suggestive of the surface reaction kinetics in accordance with the present consideration. Further experimental studies are required to confirm the predictions of the present study. APPENDIX
Define G(x,
the generating
function
G(x,
t) as follows:
t) =xx”P,,(t)
(AI)
Here x is an arbitrary (Al), we obtain
variable
(0 < x < 1). Ignoring
K~
in eqn. (1) and using eqn.
x, t) =CX_,(n + l)x”~,+&) &G(
-C{ X,(N, - n) + Ll”}xnPn(r)
+D,{ x - (n - l)b”e-*(d Again
substituting
j&x, r) = (1 Putting that G(x,
eqn. (Al) into the above equation, x)( (X-i
t = 0 in eqn. (Al) 0) = 1
(AZ)
+ Q)&G(x, and using
we obtain
t) -X,&C+, the initial
conditions
t))
(A3)
(6.1) and (6.2), we find
(A4)
358
The solution
to eqn. (A3) satisfying
Gb, t) =
l
h,
[A-,
+A_,
+A,
eqn. (A4) is
exp{ -(A, +A-,)t}]
i
+
h,
:1,, [l
+h-,)fllxjNs
-exp(-(A,
(A51
Expanding eqn. (A5) and comparing it with eqn. (Al), eqn. (7) can be obtained. Also, differentiating eqn. (Al) with respect to x and putting x = 1, the following equation is obtained:
(A6) The right-hand side of eqn. (A6) is equal to (n), according of eqn. (A5), the left-hand side of eqn. (A6) becomes
&(I,
t) =N+
1
;1,_1 [I- ev{-(A, +X-l)fll
Then eqn. (8) is obtained. The dispersion of n is written a2n = (2)
to eqn. (4). Also, by use
(A-3
as
- (L+
(A8)
Also,
(n’) =Cn’P,(t) = &(x&G(l,
t)
(A9) Substitution
of eqns. (A9) and (8) into eqn. (A8) gives eqn. (9).
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