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Evidence for inhibition between pitting eventson carbon steel

In data published by Cheng and Luo ( Brit ish Corrosion Journal, 2000) , the variance toT. P. HUTCHINSONmean ratio of the number of pitting events on carbon steel is less than 1. It is here argued

that this is evidence for inhibition between events, not for promotion. T he branch of prob-ability known as renewal theory is useful for analysing counts of events occurring in time,

the intervals b etween t hem, and the intensity of events. BCJ /1913

T he author is in the Department of Psychology,M acquarie University, Sy dney, N SW 2109,A ustralia (phutchin@bunyip.bhs.mq.edu.au). M anuscript received 17 Sept ember 2001;

accepted 9 July 2002.

2002 IoM Communications L td. P ublished by Maney for the Institute of Materials, Minerals

and Mining.

There is no necessary connection between the mean andINTRODUCTIONthe coeYcient of variation of the intervals, in the sense thatCheng and Luo1 reported experiments on metastablethe same external conditions (e.g., as regards immersionpitting of A51670 carbon steel. In some conditions, pittingtime, potential, and chloride ion concentration) that lead(as indicated by the occurrence of a current transient) wasto a small mean (and thus a high rate of events) could alsorandom, in the sense of appearing to be a Poisson process:lead to a small coeYcient of variation (and thus thethe variance to mean ratio of the number of events in aappearance of inhibition between events).given time interval was close to 1. In other conditions (e.g.

Reference 4 is a standard work. The Encyclopedia ofdata for an immersion time of 5 h in Fig. 4 of Cheng andStatistical Sciences is an excellent starting point for manyLuo, a potential of 50 mV in Fig. 7, and a chloride iontopics within its scope; relevant articles include thatconcentration of 01M in Fig. 10), the Poisson model failed.by Smith.5Reading Ref. 1, the impression gained is that when the

Let r be the intensity of events, meaning that for anPoisson model was unsatisfactory, this was because ofinnitesimal length of time dt, r .dt is the probability of ancooperation (promotion) between the pitting events: thatevent. This intensity may be written r(t), to indicate possibleis, the occurrence of one event was associated with andependence on the time that has elapsed since the previousincreased probability of another event. It is argued in this

event. Assume the interevent intervals to be independent.paper that, on the contrary, the data indicate that when(If r(t) is constant, we have the Poisson process.)the Poisson model fails, this is because of inhibition

Example 1: suppose r(t) is an increasing power functionbetween the pitting events: that is, the occurrence of oneta (with (a>0). Example 2: suppose r(t) is 0 for tt, where t is some positive length of timeevent.(either a constant, or if a random variable, then with acoeYcient of variation less than 1).

VARIABILITY OF COUNTS AND OF TIME It seems fair to describe both examples as exhibitingINTERVALS inhibition: the intensity is 0 immediately following an event,

and is non-decreasing until the next event occurs. For bothConsidering the number of metastable pitting events perexamples, the coeYcient of variation of the intervals is lesstime interval, the ratio of variance to mean is seen to bethan 1, and thus the variance to mean ratio of the countseither close to 1 or less than 1 in Figs. 4, 7, and 10 ofis less than 1 also. (Notice that memory is unnecessaryCheng and Luo. The latter indicates an inhibitory process,for inhibition or cooperation to be observed, except in as

not a cooperative one. (In contrast, simulated data in Fig. 5 much as t being measured from when an event last occurredof Lunt et al.2 show a variance to mean ratio of greatercan be said to constitute memory. The model proposed bythan 1; the model that was used to generate these data wasWu et al.3 and simulated by Lunt et al.2 was not a renewala cooperative one.)3process, it had memory and there was an implied positiveIt is common to consider the variability of the timecorrelation between successive intervals.)intervals between successive events, rather than the varia-

bility of counts. Issues studied include the distribution andthe mutual dependence of these time intervals. When the DISCUSSIONintervals are independent and identically distributed and It is concluded that the data in Figs. 4, 7, and 10 of Chenghave an exponential distribution, the process is a Poisson and Luo are evidence for inhibition between metastableone. The intervals being independent and identically pitting events. (The author believes the autocorrelationsdistributed is a common nding in experimental work and shown in Figs. 5, 8, and 11 to arise from the length that ana common assumption in theoretical work. The term event lasts. The timescale is a matter of seconds, whereasrenewal process is then used. Relaxation of the assump- interevent intervals are tens or hundreds of seconds.) I have

tion of the exponential distribution is the usual way of no specialist knowledge of electrochemistry to enable megeneralising beyond the Poisson process. Probabilists are to say what the inhibitory process might be, but at pagecapable of stating precisely and proving theorems that 129, Cheng and Luo themselves describe one: The closeconnect the properties of the intervals between successive spatial proximity of two or more metastable pits formingevents with those of the numbers of events in a time period. within a short time interval can lower the chance ofFor example, asymptotically, the variance to mean ratio of occurrence for subsequent pits, because the former pitsthe count of events equals the square of the coe Ycient of serve as an anodic area with respect to the surroundingvariation of the lengths of the intervals. (The coeYcient region and inhibit the initiation of subsequent pits. Chengof variation is the ratio of standard deviation to mean.) and Luo give no reason why this process of inhibition

should require two or more pits; perhaps one is suYcient.These two ratios are both 1 for the Poisson process.

DOI 10.1179/000705902225004428 British Corrosion Journal 2002 Vol. 37 No. 3 239

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240 Hutchinson Inhibition between pitting events on carbon steel

3. b. wu, j. r. scully, j. l. hudson, and a. s. mikhailov:REFERENCESJ. Electrochem . Soc., 1997, 144, 16141620.

1. y. f. cheng and j. l. luo: Br. Corros . J., 2000, 35, 1254. d. r. cox: Renewal theory; 1962, London, Methuen.

130.5. w. l. smith: in Encyclopedia of statistical sciences, Vol. 8,

2. t. t. lunt, s. t. pride, j. r. scully, j. l. hudson, and a. s.3036: 1988, New York, NY, Wiley.

mikhailov: J. Electrochem . Soc., 1997, 144, 16201629.

British Corrosion Journal 2002 Vol. 37 No. 3

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