MODELLING OF ELECTROCHEMICAL NOISE DUE TO THEACTIVATION·CONTROLLED DISSOLUTION OF METALS

R.A. COTIISCo"osion & Protection CentreUMIST, PO Box 88Manchester M60 1QD, UK

ABSTRACT. This paper considers the noise expected to be generated by the activation-controlledanodic dissolution ofa pure metal. An atomistic model of this process suggests that dissolutionwill occur in bursts, as ledges are 'unlocked' by the dissolution of a ledge atom, and then theexposed kink atoms dissolve rapidly. Examination of the statistics ofthis model suggests that theresultant electrochemical noise will have the characteristics ofshot noise, but with the magnitudeof the charge carrier being a function ofthe length ofledge that is dissolving. For this model thecu"ent noise power is proportional to the average number of atoms in each ledge and the totalcumnt flowing. For typical ledge lengths the charge in each pulse is such that the noise level willbe low relative to values that are nonnally measured for real co"oding systems, and this sourceof noise should be regarded as a lower bound for the possible noise characteristics.

1. Introduction

This paper is concerned with the electrochemical noise developed during the generalcorrosion of a pure metal. Only noise associated with the anodic reaction isconsidered. In addition to modelling the expected current noise spectrum, anobjective of the paper is also to demonstrate the value of visualization methods inproviding an appreciation of the characteristics of the underlying physical processes.At first sight the uniform general corrosion of a pure metal might be expected tobe a virtually noise-free process, with the only noise source being shot noise due tothe discrete nature of the charge carriers. Following the analysis of Shottky [1] thisshould give rise to a noise current given by

i;=2nElB

where:1;; = mean square noise currentn = number of electrons per charge carrier (typically 1, 2 or 3)E = charge on the electronI = current flowingB = bandwidth

(1)

However, if we examine the detailed atomistics of the dissolution process it becomes369

K. R. Trethewey and P. R. Roberge (eds.), Modelling Aqueous Corrosion, 369-379.© 1994 Kluwer Academic Publishers.

370

clear that it is actually likely to occur as a series of bursts with each burst deliveringmany electrons. A simple atomistic model is used to indicate the underlying statisticsof this process and this result is then used to derive model power spectruminformation.

2. Atomistic Model

The atomistic model used is a simplified version of a model used for the analysisof the corrosion of alloys [2, 3]. It is based on the premise that the probability ofdissolution of an individual atom is controlled by the physical environment of theatom. At its simplest the environment may be specified by the number of nearestneighbours, with fewer neighbours giving a higher probability of dissolution.

2-NelghbourMedium dissolution

to~~~~~:~~probability .'..,- .,..,,-....,

'"

l-NelghbourHigh dissolution

/ probability

Figure I - Types of atomic position for a 2-dimensional square crystal.

Ifwe consider a 2-dimensional slab of crystal with a square lattice (Figure 1), withone edge exposed to the solution, we can see that atoms with one neighbour willdissolve very quickly, atoms with 2 neighbours wiIJ dissolve at an intermediate rate,and atoms with 3 neighbours will dissolve very slowly. For a close-packed structurethe situation is similar, although the numbers of neighbours are different (Figure 2).As a result of these probabilities, atoms with only one neighbour wiIJ not beobserved (except very transiently). The bulk of the dissolution will be of atoms with2 neighbours, not least because such a dissolution will always leave a further atomwith 2 (or possibly 1) neighbours. However, this is not an indefinitely self-sustainingprocess, as eventually all of the row of atoms will be dissolved, leaving a situation

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with only 3-neighbour atoms exposed. In order for the dissolution to continue oneof these 3-neighbour atoms must dissolve. Once this has happened, two 2-neighbouratoms wiII be created and these wiII be dissolved rapidly. Thus the dissolutionprocess will consist of a delay until a 3-neighbour atom dissolves, followed by a burstof dissolution as the remaining row of atoms "unzips" with 2-neighbourdissolution.For a 3-dimensional crystal the process is similar, although slightly morecomplex as a result of the greater number of possible neighbours.

3-NelghbourMedium dissolution

~::~~~~:ba~probability

"

2-NelghbourHigh dissolution

/probability

Figure 2 - Types of atomic position for a 2-dimensional close-packed crystal.

In this situation we may describe several different types of atom:

Surface with one neighbour for a simple cubic crystal (3 for face-centred cubic, fcc),resulting in an atom sitting by itself on top of a plane of atoms. It wiII clearlydissolve very rapidly.

Kink with two neighbours for a simple cubic crystal (4 for fcc), resulting in an atomsituated at the end of a row of atoms. It will dissolve fast.

Ledge with three neighbours for a simple cubic crystal (5 for fcc), resulting in anatom situated in the middle of a row of atoms. It will dissolve relatively slowly.

Terrace with four neighbours for a simple cubic crystal (7 for fcc), resulting in anatom lying in a plane of atoms. It will dissolve very slowly, if at all.

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In 3-dimensional systems the situation is further complicated by the existence ofdislocations in the crystal. As a screw dislocation emerges from the free surface itgives rise to a ledge that cannot be removed by dissolution. Instead the ledgeeffectively "rotates" around the dislocation as the ledge atoms dissolve.As with the 2-dimensional case it can be seen that the dissolution will tend tooccur in bursts, triggered by the dissolution of a ledge atom, with the remainder ofthe ledge 'unzipping' by the dissolution of the exposed kink atoms. However, thisprocess will stop at the end of the row of ledge atoms, usually at an internal comerin the ledge (Figure 3).Consideration of the geometry of the situation also indicates that the probabilityof a burst of dissolution on a particular ledge will depend on the length of the ledge.Furthermore the length of the ledge will increase as a result of the dissolution of thetwo ledges either side of it. Thus as the ledge grows longer it dissolves more rapidlyand increases in length more rapidly, leading to a spiral. The size of the spiral willbe limited by interaction with ledges from neighbouring dislocations.

Figure 3 - Internal comers stopping ledge dissolution.

We can model this process by considering the growth of individual ledges using aMonte Carlo simulation (the atomistic approach is not feasible in practice becauseof the relatively large inter-dislocation spacing, even in heavily work-hardened

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crystal, which requires the treatment of impossibly large arrays of atoms). Figures4 and 5 show the resultant time record for a simulation considering 20 ledges ofmaximum length 500 atoms, while Figure 6 shows the corresponding powerspectrum, corrected to convert the average current flowing to 1 A m·2•

16

14

12

<' 10.s'E 8~~6(,)

4

2

00.0174 0.0176 0.0178 0.018 0.0182

Time (sec)0.0184 0.0186

Figure 4 - Noise due to ledge dissolution; high frequency time record from Monte Carlo simulation.

1.861.841.821.78 1.8Time (sec)

1.76

12..,.----------------------...,

11-1--+----------------f-------I

10+--+f----.-----+------J.--------19+---Ht------1H-c-+--.-+H---+-f-----+-.-,,-I--+---,-----1

<' 8-f1-t+--H1---II-1­

.s 7­c:~ 6~(,) 5

4

3-tt--fl+--'-------f'------'----'+J-----.:.-..:...,..-+-4--1--------I

2+--+-------.:..-------.:...-----------11-t----.---------r---.,.-----..------.----j1.74

Figure 5 - Noise due to ledge dissolution; intermediate frequency time record fromMonte Carlo simulation.

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However, in order to develop the noise spectrum expected from a real sample itis necessary to average over a very large number of dislocations, and a more efficientmethod is to develop the expected distribution of ledge lengths, and then derive theexpected noise spectrum from these, either by a Monte Carlo simulation, or bycalculation.

1E-19

N:x:N<~ 1E-20

;oQ.GlUl'0z 1E-21'EGl

~U

""-\~

\\

\

\"\. / •

\lUI

1E-221 10 100 1000 ooסס1 oo0סס1 1000ooo

Frequency (Hz)

Figure 6 - Power spectrum from Monte Carlo simulation.

For a simple cubic crystal the ledges will grow in length by one atom for everyrow dissolved from a neighbouring ledge. For a close-packed crystal the ledge willgrow in length by one for every row dissolved from a neighbour and it will alsoreduce in length by one for every row that it loses. In either case it can be seen thatthere should be a 'flat' distribution of ledge dissolution events from 1 up to themaximum length (while there will be a higher probability of a longer ledgedissolving, because it contains more ledge atoms to trigger the dissolution, there willalso be correspondingly fewer of a given length).The total number of atoms exposed will depend upon the geometry of the spiral.The precise analysis of this is complicated by the complex growth behaviour andboundary conditions. To a first approximation, the length of ledge exposed in aregion of crystal will be equal to the diameter of that region. Then the distributionsof times to the next ledge dissolution event will be that for a Poisson process, andthe times to the next event will have an exponential distribution.The dissolution of the ledge will give rise to a burst of charge. Each atomdissolving will generate 2 electrons (for a divalent metal), and the time betweenindividual atom dissolution events will also be exponentially distributed. Finallyaccount should also be taken of the position of the initial ledge atom dissolved

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within the ledge, which will result in one section of the ledge being longer than theother, and therefore taking longer to dissolve. However, the dissolution of the ledgewill be a rapid process in the timescale of typical electrochemical size measurements,and at low frequencies the ledge dissolution may be treated as an instantaneousburst of charge, the magnitude of which varies in accordance with the length of theledge as described above.Thus, at low frequencies, we model the dissolution of a pure metal as a Poissonprocess with each event consisting of a pulse of charge, with the magnitude of thecharge emitted being uniformly distributed in the range ZE to ZEm where Z is thenumber of electrons involved in the dissolution step, E is the charge on the electron,and m is the maximum number of atoms in the ledge (which will be related to thedislocation density). If we consider events for a given number of atoms, n, theresultant noise will be shot noise with a charge per event of ZEn. Hence

(i;r =2ZEnl*B

where r = the current due to this set of eventsand (i;) * = the noise due to this set of events

(2)

Given that the ledge dissolution events have a uniform distribution of charges in therange of ZE to ZEm, the total current flowing will be proportional to the sum of allof the relevant charges:

_m~(m~+l~)OCZE2

Then the current, 1*, due to events of n atoms will be given by

r 2zEII IZEm(m+l) toral

211 Im(m+1) toral

(3)

(4)

(5)

(6)

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Therefore,

(7)

The total noise will then be the sum of the events due to the U;l· terms:

4ZE 1- ltolill-m(m+1)(2m+1)

m(m+1) 6

(8)

(9)

(10)

This noise power will be observed at low frequencies, corresponding to a periodgreater than the longest ledge dissolution event. As the frequency increases the noisewill fall, until the high frequency limit is reached, corresponding to shot noise dueto the dissolution of individual atoms:

(11)

In order to put these results in perspective, we may consider the relevant values fora typical corrosion process. A typical corrosion current density for active corrosionwill be I A m-z (corresponding to a penetration rate of around 1 mm y-I). For a 1cmz electrode this implies a shot noise power of 2ZE x 10-4 A Z Hz, or 6.4 x 10-23 A Z

Hz-I (assuming z = 2).Typical dislocation densities range from around lOS to 10IZ cm-z, giving ledgelengths of around 10 nm to 30 ~m or 50 to 150,000 atoms. Hence the low frequencynoise power will be in the range 1.6 x 10-z1 to 4.8 X 10-18 AZ Hz-I. For comparison,Figure 7 shows the current noise for a 1 cmz steel electrode in aerated 1% NazS04

solution. It can be seen the observed current noise power is several orders ofmagnitude greater than that predicted by this model. This suggests that other noisesources are effective in this case, and careful studies on an ideal system (such ascopper dissolving in deaerated non-oxidising acid as a result of an applied potential)will be needed to determine whether or not the model proposed is correct.Note that the low frequency noise will greater than the high frequency shot noiseby a factor of approximately n, the characteristic number of atoms in a ledge.Looking at Figure 6, for which n is 500, it can be seen that this prediction is indeed

377

fulfilled.The above analysis has made one major simplifying assumption, namely that allgrains have a closest-packed plane lying parallel to the surface. This is clearly notgoing to be generally true, and we need to consider what is going to happen to otherplanes. When initially exposed such planes will offer a lot of ledges and kink atoms.However, it seems reasonable to assume that, after an initial transient, the ledgelengths will become comparable to those for the closest-packed planes, and on thisbasis we may expect that the noise spectrum will be similar to that for a surfaceconsisting entirely of closest-packed planes. This aspect of the process is currentlybeing examined through the atomistic model.

lE·l

lE·l1

N' lE·12

~(

~ lE·l

IQ. lE-l

lE·l

lE-l

0.001 0.1

1-1% -0.1% -0.01% -0.001% I

Figure 7 - Typical cu"ent noise spectrum for steel in 1% Na2S04 solution at 60°C.

A second major assumption has been that the dissolution of a row occurs beforethe next ledge dissolution event, so that at low frequencies the ledge dissolutionevent may be considered as a short pulse of current. Whether or not this is a validassumption will depend on the number of atoms in the ledge and the relativeprobabilities of dissolution of ledge and kink atoms. To a first approximation we areassuming that the probability of dissolution of a kink atom exceeds that of a ledgeatom by a factor that is greater than the number of atoms in the ledge. It is notclear whether or not this assumption is valid. If it is not, the ledge will tend to

378

dissolve as a number of shorter segments, with the typical length being of the orderof the ratio of the two dissolution probabilities, and this term should therefore beused in place of the maximum ledge length.We may estimate the frequencies below which the limiting noise power will beobserved by considering the number of ledge dissolution events occurring. If weconsider 1 cm2of material with an intermediate dislocation density of lOS cm,2, anda ledge length of about 5000 atoms, each ledge dissolution event will generate 10,000electrons, or 1.6 x 10-15 C. Hence there must be around 6 x 1010 events per secondat 1 A m2(10-4 A cm-2) . As there are around 108 dislocation cm,2, this implies thateach ledge generates a dissolution event around 100 times a second (this analysis isclearly complicated by the spiral of ledges of decreasing length around eachdislocation, but the above estimate seems reasonable for the longest ledges that willbe dissolving the most rapidly). Hence, provided our assumption that the ledgedissolution event is faster than the average period between such events is valid, thelow frequency limit for the noise spectrum will be at frequencies of the order of 100Hz. Thus in typical electrochemical noise measurements made at lower frequencieswe may expect activation controlled dissolution to be observed as white noise, witha noise current given by

3. Conclusions

:2 4 IBI ~-ZEm

" 3(12)

(1) An atomistic model of the activation controlled corrosion of a metal suggeststhat dissolution will occur in bursts, as ledges are 'unlocked' by the dissolution of aledge atom, and then the exposed kink atoms dissolve rapidly.

(2)Examination of the statistics of this model suggests that the resultantelectrochemical noise will have the characteristics of shot noise, but with themagnitude of the charge carrier being a function of the length of ledge that isdissolving.

(3)The current noise power is given by

:2 4 IBI ~-ZEm

" 3(13)

(4)For typical conditions of corrosion current density and dislocation density, the lowfrequency current noise power will be in the range 1.6 x 10,21 to 4.8 X 10,18N Hz,l.

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References

1. See King, RA. Electrical Noise, Chapman and Hall, London 1966.

2. Q. Song, RC. Newman, RA. Cottis and K. Sieradzki, Computer Simulation of AllovPassivation and Activation, Corrosion Science, Vol. 31, 1990,621-626.

3. RC. Newman, Q. Song, RA. Cottis and K. Sieradzki, ASTM STP 1154, 1992, 17-27.