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Applications of deterministic chaos theory to corrosion
Corrosion Science, Vol. 35, Nos 1-4, pp. 751-760, 1993
0010-938X/93 $6.(X/+ 0.00 Pergamon Press Ltd
Printed in Great Britain.
APPLICATIONS
OF
DETERMINISTIC CORROSION
CHAOS
THEORY
TO
J. STRINGER* and A. J. MARKWORTHf * Research Materials Support, Electric Power Research Institute, 3212 Hillview Ave, Palo Alto, C A 94303, U.S.A. +Battelle Columbus Laboratories, Columbus. Ohio, U.S.A.
A b s t r a c t - - I n recent years there has been a considerable increase in interest in the behavior of non-linear dynamical systems. U n d e r some circumstances, these systems exhibit a behavior which has been termed deterministic chaos. This m e a n s that the system is always behaving in a deterministic way, that is, nonrandom; but the detailed behavior is extremely sensitive to the initial conditions, and in the long range is completely unpredictable. O n e effect of this is that if a numerical model is being analyzed, then the roundoff errors and methods of approximation can also contribute to the long-range predictability. U n d e r some circumstances, this lack of predictability is undesirable, and the aim in that case is to ensure that the system does not move into a chaotic regime. However, the chaotic regime corresponds to very good mixing, and in systems such as chemical reactors this may be the desirable condition. The lack of predictability does, however, present control problems. In this paper, the possibility that some corrosion processes may exhibit chaotic behavior is examined, and the implications of such behavior are discussed. INTRODUCTION
IF a dynamical system is fully specified by a set of n independent variables P l, P2, p,,, its state at any time is described by a set of specific values of these variables. This can be thought of as a point in an n-dimensional state space, and the changes in the system which occur as a result of a perturbation form a p a t h - - a ' t r a j e c t o r y ' - - i n this state space. In the absence of an external force, an initially displaced system will tend to move towards an equilibrium state: thus, a damped pendulum will eventually come to rest. This is called the 'attractor' for the system, and in the case of a d a m p e d pendulum the attractor is a point. In other c a s e s - - f o r example, an undamped p e n d u l u m - - t h e attractor may be a closed path in the state space, called a limit cycle or orbit. The rate at which an initially displaced system approaches its attractor is typically described in terms of an exponential decay, with a different exponent for each of the n dynamical variables. These exponents are called the Lyapunov exponents, and for a normal decay process they will be negative. An important result is that if a system is allowed to relax from two different positions close together in state space, the trajectories will converge as they approach the attractor. Generally, there is a limit to how far a system can be displaced and still relax towards a given attractor. The region of state space surrounding an attractor within which the system will relax towards it is called the basin of attraction, and the boundaries of the basin of attraction are important in understanding how to control a dynamical system. If a system behaves in a wholly random fashion, there is no attractor, and consequently the Lyapunov exponents are zero. The trajectories of systems initially at two different but neighboring locations in state space will not correlate at all: they may converge, or diverge, or remain parallel. There may still be a boundary in state space which defines the range over which the behavior is random, however. It is also •
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conceivable that the behavior of one or more variables is random while others are not: for example, a biatomic molecule may follow a random path in space, while its internal vibrational modes are completely deterministic. In this case, the Lyapunov exponents for the deterministic components arc non-zero and negative; and an attractor exists in a subspace of the n-dimensional state space. A special case is where one or more of the Lyapunov exponents are positive. In this case, the trajectories from two initially adjacent positions diverge, but the system remains wholly deterministic: its behavior is not random. It cannot be absolutely predicted, however, and it is important to recognize that this lack of predictability is intrinsic in the existence of the positive Lyapunov exponent; it cannot be removed by improved accuracy of observation. In practical terms, limited predictability may well be possible; this depends on the magnitude of the positive Lyapunov exponent or exponents, which measure how rapidly adjacent trajectories diverge. In spite of this behavior, an attractor does exist for the system, but the existence of divergent trajectories means that the attractor is not simple. It is called a 'strange attractor', and generally it has non-integer dimensions--it is a 'fractal'. There is also a basin of attraction, and in at least some cases the boundaries of the basin are also fractal. Proofs for the generality of some of these statements do not appear to exist at present, and are the subjects of discussion: for the purposes of this paper this is not important. A dynamical system which behaves in this way is said to exhibit 'deterministic chaos', although recently it has been more commonly called simply chaos. This is unfortunate, because the common meaning of the word chaos is more appropriately reserved for random behavior. The divergence of the trajectories from two closely adjacent starting positions, which is the essential characteristic of dynamical chaos, can be readily visualized in several everyday situations. The easiest to imagine is the kneading of dough containing currants: the process involves rolling the dough, folding it, rolling again, and so on. The objective is to separate the currants, although they may have been adjacent initially. The two rotating hooks in a kitchen mixer which are used for mixing dough in the home produce chaotic motion in the dough. Although the essential mathematics of the behavior are of considerable antiquity, the emergence of deterministic chaos as a topic of practical interest dates from the work of Lorenz, 1 who attempted to model the behavior of the atmosphere in terms of a simple set of equations derived from a Rayleigh-B6nard model. He was able to show that the system described by these equations would exhibit deterministic chaos, and made the point about the consequent fundamental long-range unpredictability of the system and its sensitivity to very small variations in initial conditions. Since then, there have been many studies reported. The popular book by Gleick 2 describes some of the history of the developments after Lorenz. It is relatively easy to determine whether a given set of equations does or does not exhibit deterministic chaos. A necessary condition is that the equations are nonlinear: linear equations will not exhibit a transition to chaos. In the past, there has been relatively little study of non-linear processes in engineering because of the problems in solving the equations: the tendency has been to linearize them, and this will preclude the detection of a chaotic instability. Much of the recent work related to the applications of deterministic chaos to practical systems has concentrated on developing a set of equations which appears to
Applications of deterministic chaos theory to corrosion
753
describe the system and then inspecting the behavior of these equations. A second approach has been to study the experimental behavior of systems and attempt to determine whether the behavior is indeed chaotic. Much of this second approach has used a technique first developed by Packard et al. 3 and later justified theoretically by Takens. 4 This showed that a 'portrait' of the system trajectory in the ndimensional state space could be derived from the behavior of a single variable p~ by constructing another n-space from either time delay quantities pt(t), p t ( t + At), Pl (t + 2At) . . . . . p l ( t + (n--1)At), where At is a time interval sufficiently great that the self-correlation of the variable p ~is small; or by using derivatives of the variable: p l, dp~/dt, &"~p l / d 2t . . . . . d n- t p J d n - ~t. This method has been widely used to try to get an idea of the form of the attractor for the system. It is far from easy. In the first place, it is very important that the time interval At is properly chosen, and the conditions for this are often not examined by the investigators. In the second place, the experimental values of the variable Pl are always contaminated by noise, which one hopes is random: removal of the noise from the signal is not easy, and sometimes this can give the impression in the portrait that a strange attractor is present when perhaps it is not. A further problem is that often the portrait is presented as a two-dimensional projection of a three-dimensional space. The projection can be rotated by varying At, but some difficult inferences must be made if the dimension of the attractor is greater than 3. There are other, more quantitative methods of dealing with the experimental information which are beyond the scope of this paper. Methods exist for determining the values of the Lyapunov exponents; the existence of at least one positive exponent is a proof that the system is chaotic. The dimensionality of the attractor can be determined by an embedding method. This can be easily understood in principle. Suppose there is an attractor with dimension m. If the dimension is measured when the attractor is represented in a space of dimension w, if w < m, the apparent dimension of the attractor will be w; but if w > m, the apparent dimension will be m. Note that for a strange attractor m will not be an integer, although w is always an integer; this does not affect the method. By embedding the attractor in increasingly higher dimension spaces and plotting the apparent dimension versus the dimension of the embedding space, the dimension of the attractor can be determined. This seems easy and unequivocal, but in fact the information required to determine the dimension increases roughly as the power of the dimension; for low-dimension attractors the technique works well, but for high-dimension attractors the information requirements will exceed that available. This concept of 'information' is very important. It may seem that if an analog recorder measures the time dependence of a variable, the linear trace can be divided without limit, and thus the amount of information contained is infinite. This is not so. The information is limited by a number of factors: the characteristic time of the total measuring system is an obvious limitation, but the self-correlation of the signal mentioned above is also a factor. Methods exist for testing the information content of the signal: it is important that these should be applied. If the first approach--that of using a set of trial equations--is used, the test will be if the transitions to chaos predicted are consistent with the behavior of the dynamical system. If the second approach--that of using dynamical information to test for the existence of chaos--is used, the ultimate aim must be to use that knowledge to identify the important variables, and hopefully use this information to synthesize a
754
J. STRINGERand A. J. MARKWORTH
set of equations to describe the behavior of the system. However, this is far from easy. The dimension of the attractor does not give an indication of the number of significant physical variables. For example, a limit cycle is a line, and thus has dimension one, irrespective of the dimension of the space that is required to draw the cycle in terms of the dynamical variables. The method relies more on comparing the form of the attractor with an 'atlas' of attractors from known sets of equations that lead to chaotic behavior. So far, there appears to have been no example of this step having been completed. This is an issue of major importance which merits greatly increased study. The last point concerns the 'route to chaos'. In many, but not all, cases the single trajectory typical of a well-behaved deterministic system first splits into two. This is called a bifurcation. Next, each of these bifurcates, and further bifurcations, follow until the situation becomes wholly chaotic. In those systems where chaos is to be avoided, the appearance of a birfurcation is the first sign of trouble, and thus the bifurcation is the principal item of interest, rather than the eventual chaotic behavior itself. POSSIBLE EFFECTS OF DYNAMICAL CHAOS IN CHEMICAL PROCESSES Dynamical chaos has been observed in some chemical reactions, in particular the Belousov-Zhabotinsky reactions. One effect is to develop patterns in the reaction products, and it might be expected that these would move towards a self-similar pattern. A second possibility is the appearance of two different behaviors on a surface, corresponding to a bifurcation: this might result in pitting, but not related to specific microstructural features in the underlying alloy. The observation that in corrosion reactions the system is so far from equilibrium that non-linear effects might be expected appears to have been made first by Bertrand, 5 and later papers, for example by Bertrand et al. ,6 have discussed 'instability and bistability' in the growth of corrosion scales. Bistability appears to be essentially the same as bifurcation, but the equivalence with formal chaos theory is not altogether clear. The problem that these investigators have been addressing is the formation of layered scales, in particular on titanium oxidized in the range 630-1000°C, which is referred to as 'mechanical' layering, and on Fe-22wt% Cr-5.Swt% Al sulfidized at temperatures below 740°C, which is termed 'chemical' layering. A model is presented for chemical layering in a binary alloy A - B , where CA is the mole fraction of A 7. The scale is considered to be in three zones: an inner zone I, which is a thin substoichiometric layer; zone II, which consists of layers of two different oxidation products; and zone III, which is a homogeneous layer of one of the oxidation products. The alloy composition at the interface with zone i is AsBs; the two oxidation product phases are labeled X and Y; Cx and Cy are the compositions of the two product phases at the interface between zone I and zone II, and Cx + Cy = 1. The non-linearity is introduced in the kinetics of the interface process at the interface between zone I and zone II, as a result of cooperativity. A kinetic constant is then written as: k = ko exp (vCx),
(1)
where v, which is positive, represents the cooperativity strength. The following equation is then generated:
Applications of deterministic chaos theory to corrosion
755
CY t0.9 branch 0 0
i
07
I
06 05 04 03 O~ I
o
~el ~Bs
FIG. I.
Cy versus
o2 os
o~,-2o~
o~ o~ o~
"-B s
Cu~calculated from equation (2), showing 'bistability'. From Bertrand] Reproduced by permission of Sci-Tecb Publications.
Cy exp ( - v C y ) / ( 1 - Y) = (1//3) {(1 -
CBs)/(CBs},
(2)
where fi is a non-dimensional parameter which takes into account the kinetic constants of the chemical processes. Plotting Cy versus CBs generates an S-shaped curve (Figure 1), which is three-valued for Cy between C1B~,and C2B~. This results in a cyclic oxidation behavior generating alternate layers of X and Y, as the As in the 'reservoir', which is zone I near the interface, moves from one of these extremes to the other as the different phases are produced. (Note: the papers referred to here are only a few of several by this group on this topic.) Oscillatory composition fluctuations have sometimes been observed in scales, for example in the oxidation of Fe-18% Cr at 1200°C s, which are difficult to rationalize without a similar 'bistability' argument. (We are grateful to Prof. Wood for bringing this result to our attention.) It seems probable that the generation of layered scales, or the evolution of quasiperiodic composition fluctuations within a scale, may well be examples of an instability which is closely related to a bifurcation. If this is indeed the case, it would be worth inspecting equations similar to that derived by Bertrand and co-workers to see if further bifurcations, or a transition to deterministic chaos, are possible, and if the conditions are physically attainable. A recent paper by Kirkaldy 9 is perhaps relevant. This discusses possible deterministic chaos effects in eutectoid transformations, and specifically considers the pearlite reaction. The conventional description of this reaction to produce a uniform laminar structure is essentially linear and does not show bifurcations or chaos. However, Kirkaldy points out that the equations are derived assuming a (physically unreasonable) planar initial interface, and also define a physically unattainable velocity state under some conditions. Kirkaldy introduces a difference equation, shows that successive bifurcations and eventual chaos in the laminar spacing result as the relative supersaturation increases, and argues that the behavior implied by this is consistent with the formation of upper bainite. Other work by Unger and Klein m has shown that spinodal decompositions result in structures which exhibit fractal geometry in the early stages, and this may be a signal of dynamical chaos. Akuezue and Stringer 11 have used a difference equation substitution in multicomponent diffusion theory, and also show the possibility of chaotic instabilities.
756
J. STRINGERand A. J. MARKWORTH
The point of their analysis is not the form of the steady-state diffusion profiles, but the approach to this steady-state from the initial non-equilibrium state. It is easy to show that
dYi(2)/dt = k 2 + Y/(2)(1 - Yi().)),
(3)
where I1,.(2) is the concentration of the ith component at the dimensionless position 2 and k2 is a parameter dependent on position but independent of time; this equation is essentially similar to the well-known logistic equation used to describe population changes. This can be recast as a difference equation: Yi(2)n+l = k2 + k, Yi(2)n(1 - Yi(2)n),
(4)
where kl = Aci, the difference in the concentration of the ith species across the diffusion couple. The evolution of this function is well known, and it undergoes successive bifurcations leading eventually to chaos, depending on the values of k 1 and k2. The question is whether values of these two parameters which will lead to instabilities are physically reasonable. Akuezue and Stringer show that for two experimental situations, the carburization of iron and the aluminizing of nickel, the values of k 1 and k 2 correspond to situations where bifurcation does not take place. This is not the same as proving that bifurcation cannot occur in binary diffusion systems, but it does suggest that it is unlikely. The situation for ternary diffusion couples has also been examined for the Ni-Cr-A1 system, and these suggest that conditions may arise where instabilities may appear in the evolution of the system from the initial condition towards the steady state. It seems likely that, if behavior of this sort does arise, it will most likely be close to phase boundaries. More analysis is required. The use of difference equation substitution to achieve these results is a cause for question, since it is necessary to show that the instabilities are intrinsic to the process and not an artifact of the substitution. This step in the discussion is difficult and is often neglected. Kirkaldy also mentions the possibility of 'theorectical illegality', but argues that in fact the discrete representation is probably more physically reasonable than the conventional continuous form for the problem he considered. In all these situations, the key point is that when a system is far from equilibrium, non-linear effects are to be expected; these may (but not necessarily have to) result in instabilities such as bifurcations and dynamical chaos. Systems which are close to equilibrium might be expected to behave in an essentially linear fashion. Careful analysis is required to identify experimental situations where effects of this sort might be expected from quantitative theoretical arguments, but the appearance of complex and fine-scale microstructures such as are found in upper bainite or spinodal decompositions may also serve as an empirical clue to the existence of this kind of behavior. There have been several recent papers concerned with possible dynamical chaos in corrosion and electrochemical dissolution. These include papers by Kawczynski et al.,12 Diem and Hudson, 13Bassett and Hudson,14 Li et al.,15 Sharland et al.,16 Koper and Gaspard,17 Hudson and Bassett,ls and Corcoran and Sieradzki. 19 METAL PASSIVATION AND OSCILLATIONS IN ANODIC DISSOLUTION The phenomenon of passivation is one in which bifurcation seems consistent with the physical behavior, and generally the systems of equations which have been
Applications of deterministicchaos theory to corrosion
757
developed are non-linear, so that unstable behaviors might be anticipated. These possibilities have been studied by Markworth et al. 2° The reaction schemes of Sato 21 and Talbot and Oriani 22 were analyzed, and a reaction scheme incorporating aspects of both was postulated. A metal M is dissolving in a solution. Any given point on the surface of the metal may be bare, covered with adsorbed M O H , or covered with adsorbed MO. The adsorbed M O H and MO passivate the underlying metal. There are three variables: the concentration of the metal in solution, C; the fractional surface coverage by MO, 00; and the fractional coverage by M O H , 0OH. The fraction of the surface which is bare is then 0M, where 0M = 1 -- 00 -- 0OH. The following reactions may occur: M---~ M + + e
(kl)
M + H 2 0 = MOH~,q + H ÷
(k2, k_2)
nM + + a n ---*M,,A MOHaq = MOHad
(5) (6)
(k3)
(7)
(k4, k_4).
(8)
These reactions are essentially from Talbot and Oriani. MOH~d ~
MOad
+
H+ + e
(ks)
M + MOad + H 2 0 ~ 2MOHad
(9)
(k6).
These equations are from Sato. The following reduced variables are introduced: r
=
(10)
k_41, Y = K C , K =- k4/k_4,
p = k l K / k _ 4 , q = kB/k_4 K n - 1 , r = ks~k_4, s = k f f k _ 4.
From the above reactions, in terms of the reduced variables, the following three differential equations were deduced: d Y / d r = pOM -- q Y "
(l 1)
d0oH/dr = YOMf (OoH ) -- [r + g (0OH]0OH q- 2sOoO M
(12)
d0o/dr
=
rOoH
(13)
-- s O 0 0 M.
Markworth et al. then assumed that f(0OH ) = 1,
g(0OH ) = eo~°,
n = 1;
and chose the parameters p=2xl0
-4,
q=lxl0
-3,
r=2xl0
5.
The value of s was varied. Trajectories were then calculated for 601 different values of s. To reduce transient effects, the results from the first 300 000 time units (equivalent to about 250-300 oscillations) were discarded, and the results from the next 300 000 time intervals were reported. Figure 2 shows a bifurcation diagram, plotting Y versus s. Figure 3 shows a projection of the attractor on the OoH--Y plane, calculated for s = 9.7 x 10 -5. The Lyapunov exponents associated with each of the three dimensions of the equations were determined using the method described by Wolf et al.;23 the
758
J. STRINGERand A. J. MARKWORTH 0.114 0.112 iil",~7':~:i'ii:i ~' ;::i~:'!!i; j,
0.11 O. 108
[, , ; ,
0 . 106 O. 1 0 4
O. 1 0 2 I 0.1 _ 9600
9650
9700
9750
9800
s x 108
FIG. 2. A bifurcation diagram for 9600 < s x 108 < 9800 from equations (11)-(13). From Markworth et al. 2°
v a l u e s a r e 21 = 0.553, 22 = 0.000,23 = - 1 5 . 7 3 3 ( b i t s / p e r i o d ) . T h e e x i s t e n c e of a p o s i t i v e L y a p u n o v e x p o n e n t is a q u a n t i t a t i v e v e r i f i c a t i o n t h a t t h e a t t r a c t o r is c h a o t i c . F r o m t h e s e e x p o n e n t s , a L y a p u n o v d i m e n s i o n c a n b e c a l c u l a t e d for t h e a t t r a c t o r : t h e r e s u l t is d~ = 2.035. T h e r e a r e a n u m b e r o f w a y s of c a l c u l a t i n g t h e d i m e n s i o n o f a n
0.116
0.114
0.112
Y
0.11
O. 108
O. 1 0 6
O. 1 0 4 0.1
z 0.2
i 0.3
i 0.4
I 0.5
0.6
~oH FIG. 3. A projection of the attractor on the O o H - Y plane, calculated for s = 9.7 x 10-5 from equations (11)-(13). From Markworth et al.2°
Applications of deterministic chaos theory to corrosion
759
attractor, and a discussion of these is outside the scope of the present paper, but the fact that this dimension is a non-integer is a further indication that the attractor is chaotic. T h e existence of a great deal of c o m p l e x structure in this system of equations is of considerable interest, but of course it does not m e a n a great deal if it does not c o r r e s p o n d to some changes in b e h a v i o r in the practical system. To this end, M a r k w o r t h and Rollins and their co-workers have been designing simple experiments in which the p a r a m e t e r s can be varied within the range where the system of equations would predict bifurcation and chaos. Recently, D e w a l d et al. 24 have reported on current oscillations during the anodic dissolution of c o p p e r in acetate buffer using a rotating disc electrode. These experiments have recently been extended into the regime where chaos would be predicted, and chaotic b e h a v i o r has indeed been observed and an attractor has been determined. These results will be presented elsewhere. SUMMARY Non-linear dynamics exhibit the possibility of different kinds of instability, including bifurcation and deterministic chaos. It is not clear what the implications of this are for oxidation and corrosion, or indeed w h e t h e r instabilities of these kinds are possible. It has been p r o p o s e d that scale laminations in high t e m p e r a t u r e oxidation and the passive-active transition in a q u e o u s corrosion may be examples, and investigations of these possibilities are continuing. T h e very large deviation from equilibrium in the initial stages of an oxidation reaction suggests that linear approximations m a y well be unsuitable here, and instabilities may well be possible in the a p p r o a c h to a quasi-equilibrium configuration. It has been claimed that this m a y be possible in m u l t i c o m p o n e n t diffusion couples. In solid-state transformations, other situations which m a y be chaotic in origin have been p r o p o s e d : the pearlite to u p p e r bainite transition and spinodai decompositions are examples. Further w o r k is necessary, but care must be taken in the interpretation of results, and in particular the c o m m o n m e t h o d of time-series phase portraits must take account of the restrictions imposed by information theory.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10, 11. 12. 13, 14, 15, 16,
REFERENCES E. N. LORENZ,J. atmos. Sci. 20, 130 (1963). J. GLEICK,Chaos: Making a New Science. Penguin Books, London (1987). N. H. PACKARD,J. P. CRUTCHFIELD,J. D. FAR~ERand R. S. SHAW,Phys. Rev. Lett. 45,712 (1980). F. TAKENS,Lecture Notes in Mathematics (eds D. A. RANDand L. S. YOUNG),p. 366. SpringerVcrlag, Berlin (1981). G. BERTRAND,in Synergetics, Far From Equilibrium (eds A. PACAI!LTand C. VIDAL), p. 147. Springer-Verlag, Berlin (1979). G. BERTRAND,A. SANFELD,G. WATELLEand J.-P. LARPIN,J. Chim. Phys. 83,695 (1986). G. BERTRAND,Solid State Phenomena 3/4, 257 (1988). D. P. WHIrrLEand G. C. WOOD,J. electrochem, Soc. 114,986 (1967). J. S. KIRKALDV,Scripta Met. Mater. 24,179 (1990). See for example C. UNOERand W. KLEIN,Phys. Rev. B 29, 2698 (1984). H. C. AKUEZUEand J. STRINGER,to be published. A. L. KAWCZYNSKI,W. RACZYINSKIand B. BARANOWSLI,Z. Phys. Chem., Leipzig 269, 596 (1988). C. B. DIEMand J. L. HUDSON,A. I. chem. E. ,1.33,218 (1987). M. R. BASSErrand J. L. HUDSON,J. phys. Chem. 93, 2731 (1989). W. LI, K. NOBEand A. J. PEARL,STEIN,Corros. Sci. 31,615 (1990). S. M. SHARLAND,C. M. BISHOP,P. H. BALKWILLand J. STEWART,in Advances in Localized Corrosion
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17. 18. 19. 20. 21. 22. 23. 24.
J. STRINGERand A. J. MARKWORTH (eds H. S. ISAACS, U. BERTOCCI, J. KRUGERand S. SMIALOWSKA),p. 109. National Association of Corrosion Engineers, Houston, TX (1990). M. T. M. KOPER and P. GASPARD,J. phys. Chem. 95, 4945 (1991). J. L. HUDSON and M. R. BASSETT,Rev. Chem. Engng 7, 109 (1991). S. G. CORCORANand K. SIERADZKI,J. electrochem. Soc. 139, 1568 (1992) A. J. MARKWORTH,J. K. McCoY, R. W. ROLLINS and P. PARMANANDA,Final Report to EPRI on Research Project RP2426-25 (April 1990). N. SATO, in Passivity of Metals (eds R. P. FRANKENTHALand J. KRUGER), p. 29. The Electrochemical Society, Princeton, NJ (1978). J. B. TALBOTand R. A. ORIAN1,Electrochim. Acta 30, 1277 (1985). A. WOLF, J. B. SWIFT, H. L. SWINNEYand J. A. VASTANO,Physica 16D, 215 (1985). H. D. DEWALD, P. PARMANANDAand R. W. ROLLINS, J. electroanal. Chem. 306, 297 (1991).
0010-938X/93 $6.(X/+ 0.00 Pergamon Press Ltd
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APPLICATIONS
OF
DETERMINISTIC CORROSION
CHAOS
THEORY
TO
J. STRINGER* and A. J. MARKWORTHf * Research Materials Support, Electric Power Research Institute, 3212 Hillview Ave, Palo Alto, C A 94303, U.S.A. +Battelle Columbus Laboratories, Columbus. Ohio, U.S.A.
A b s t r a c t - - I n recent years there has been a considerable increase in interest in the behavior of non-linear dynamical systems. U n d e r some circumstances, these systems exhibit a behavior which has been termed deterministic chaos. This m e a n s that the system is always behaving in a deterministic way, that is, nonrandom; but the detailed behavior is extremely sensitive to the initial conditions, and in the long range is completely unpredictable. O n e effect of this is that if a numerical model is being analyzed, then the roundoff errors and methods of approximation can also contribute to the long-range predictability. U n d e r some circumstances, this lack of predictability is undesirable, and the aim in that case is to ensure that the system does not move into a chaotic regime. However, the chaotic regime corresponds to very good mixing, and in systems such as chemical reactors this may be the desirable condition. The lack of predictability does, however, present control problems. In this paper, the possibility that some corrosion processes may exhibit chaotic behavior is examined, and the implications of such behavior are discussed. INTRODUCTION
IF a dynamical system is fully specified by a set of n independent variables P l, P2, p,,, its state at any time is described by a set of specific values of these variables. This can be thought of as a point in an n-dimensional state space, and the changes in the system which occur as a result of a perturbation form a p a t h - - a ' t r a j e c t o r y ' - - i n this state space. In the absence of an external force, an initially displaced system will tend to move towards an equilibrium state: thus, a damped pendulum will eventually come to rest. This is called the 'attractor' for the system, and in the case of a d a m p e d pendulum the attractor is a point. In other c a s e s - - f o r example, an undamped p e n d u l u m - - t h e attractor may be a closed path in the state space, called a limit cycle or orbit. The rate at which an initially displaced system approaches its attractor is typically described in terms of an exponential decay, with a different exponent for each of the n dynamical variables. These exponents are called the Lyapunov exponents, and for a normal decay process they will be negative. An important result is that if a system is allowed to relax from two different positions close together in state space, the trajectories will converge as they approach the attractor. Generally, there is a limit to how far a system can be displaced and still relax towards a given attractor. The region of state space surrounding an attractor within which the system will relax towards it is called the basin of attraction, and the boundaries of the basin of attraction are important in understanding how to control a dynamical system. If a system behaves in a wholly random fashion, there is no attractor, and consequently the Lyapunov exponents are zero. The trajectories of systems initially at two different but neighboring locations in state space will not correlate at all: they may converge, or diverge, or remain parallel. There may still be a boundary in state space which defines the range over which the behavior is random, however. It is also •
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752
J. STRINGERand A. J. MARKWORTH
conceivable that the behavior of one or more variables is random while others are not: for example, a biatomic molecule may follow a random path in space, while its internal vibrational modes are completely deterministic. In this case, the Lyapunov exponents for the deterministic components arc non-zero and negative; and an attractor exists in a subspace of the n-dimensional state space. A special case is where one or more of the Lyapunov exponents are positive. In this case, the trajectories from two initially adjacent positions diverge, but the system remains wholly deterministic: its behavior is not random. It cannot be absolutely predicted, however, and it is important to recognize that this lack of predictability is intrinsic in the existence of the positive Lyapunov exponent; it cannot be removed by improved accuracy of observation. In practical terms, limited predictability may well be possible; this depends on the magnitude of the positive Lyapunov exponent or exponents, which measure how rapidly adjacent trajectories diverge. In spite of this behavior, an attractor does exist for the system, but the existence of divergent trajectories means that the attractor is not simple. It is called a 'strange attractor', and generally it has non-integer dimensions--it is a 'fractal'. There is also a basin of attraction, and in at least some cases the boundaries of the basin are also fractal. Proofs for the generality of some of these statements do not appear to exist at present, and are the subjects of discussion: for the purposes of this paper this is not important. A dynamical system which behaves in this way is said to exhibit 'deterministic chaos', although recently it has been more commonly called simply chaos. This is unfortunate, because the common meaning of the word chaos is more appropriately reserved for random behavior. The divergence of the trajectories from two closely adjacent starting positions, which is the essential characteristic of dynamical chaos, can be readily visualized in several everyday situations. The easiest to imagine is the kneading of dough containing currants: the process involves rolling the dough, folding it, rolling again, and so on. The objective is to separate the currants, although they may have been adjacent initially. The two rotating hooks in a kitchen mixer which are used for mixing dough in the home produce chaotic motion in the dough. Although the essential mathematics of the behavior are of considerable antiquity, the emergence of deterministic chaos as a topic of practical interest dates from the work of Lorenz, 1 who attempted to model the behavior of the atmosphere in terms of a simple set of equations derived from a Rayleigh-B6nard model. He was able to show that the system described by these equations would exhibit deterministic chaos, and made the point about the consequent fundamental long-range unpredictability of the system and its sensitivity to very small variations in initial conditions. Since then, there have been many studies reported. The popular book by Gleick 2 describes some of the history of the developments after Lorenz. It is relatively easy to determine whether a given set of equations does or does not exhibit deterministic chaos. A necessary condition is that the equations are nonlinear: linear equations will not exhibit a transition to chaos. In the past, there has been relatively little study of non-linear processes in engineering because of the problems in solving the equations: the tendency has been to linearize them, and this will preclude the detection of a chaotic instability. Much of the recent work related to the applications of deterministic chaos to practical systems has concentrated on developing a set of equations which appears to
Applications of deterministic chaos theory to corrosion
753
describe the system and then inspecting the behavior of these equations. A second approach has been to study the experimental behavior of systems and attempt to determine whether the behavior is indeed chaotic. Much of this second approach has used a technique first developed by Packard et al. 3 and later justified theoretically by Takens. 4 This showed that a 'portrait' of the system trajectory in the ndimensional state space could be derived from the behavior of a single variable p~ by constructing another n-space from either time delay quantities pt(t), p t ( t + At), Pl (t + 2At) . . . . . p l ( t + (n--1)At), where At is a time interval sufficiently great that the self-correlation of the variable p ~is small; or by using derivatives of the variable: p l, dp~/dt, &"~p l / d 2t . . . . . d n- t p J d n - ~t. This method has been widely used to try to get an idea of the form of the attractor for the system. It is far from easy. In the first place, it is very important that the time interval At is properly chosen, and the conditions for this are often not examined by the investigators. In the second place, the experimental values of the variable Pl are always contaminated by noise, which one hopes is random: removal of the noise from the signal is not easy, and sometimes this can give the impression in the portrait that a strange attractor is present when perhaps it is not. A further problem is that often the portrait is presented as a two-dimensional projection of a three-dimensional space. The projection can be rotated by varying At, but some difficult inferences must be made if the dimension of the attractor is greater than 3. There are other, more quantitative methods of dealing with the experimental information which are beyond the scope of this paper. Methods exist for determining the values of the Lyapunov exponents; the existence of at least one positive exponent is a proof that the system is chaotic. The dimensionality of the attractor can be determined by an embedding method. This can be easily understood in principle. Suppose there is an attractor with dimension m. If the dimension is measured when the attractor is represented in a space of dimension w, if w < m, the apparent dimension of the attractor will be w; but if w > m, the apparent dimension will be m. Note that for a strange attractor m will not be an integer, although w is always an integer; this does not affect the method. By embedding the attractor in increasingly higher dimension spaces and plotting the apparent dimension versus the dimension of the embedding space, the dimension of the attractor can be determined. This seems easy and unequivocal, but in fact the information required to determine the dimension increases roughly as the power of the dimension; for low-dimension attractors the technique works well, but for high-dimension attractors the information requirements will exceed that available. This concept of 'information' is very important. It may seem that if an analog recorder measures the time dependence of a variable, the linear trace can be divided without limit, and thus the amount of information contained is infinite. This is not so. The information is limited by a number of factors: the characteristic time of the total measuring system is an obvious limitation, but the self-correlation of the signal mentioned above is also a factor. Methods exist for testing the information content of the signal: it is important that these should be applied. If the first approach--that of using a set of trial equations--is used, the test will be if the transitions to chaos predicted are consistent with the behavior of the dynamical system. If the second approach--that of using dynamical information to test for the existence of chaos--is used, the ultimate aim must be to use that knowledge to identify the important variables, and hopefully use this information to synthesize a
754
J. STRINGERand A. J. MARKWORTH
set of equations to describe the behavior of the system. However, this is far from easy. The dimension of the attractor does not give an indication of the number of significant physical variables. For example, a limit cycle is a line, and thus has dimension one, irrespective of the dimension of the space that is required to draw the cycle in terms of the dynamical variables. The method relies more on comparing the form of the attractor with an 'atlas' of attractors from known sets of equations that lead to chaotic behavior. So far, there appears to have been no example of this step having been completed. This is an issue of major importance which merits greatly increased study. The last point concerns the 'route to chaos'. In many, but not all, cases the single trajectory typical of a well-behaved deterministic system first splits into two. This is called a bifurcation. Next, each of these bifurcates, and further bifurcations, follow until the situation becomes wholly chaotic. In those systems where chaos is to be avoided, the appearance of a birfurcation is the first sign of trouble, and thus the bifurcation is the principal item of interest, rather than the eventual chaotic behavior itself. POSSIBLE EFFECTS OF DYNAMICAL CHAOS IN CHEMICAL PROCESSES Dynamical chaos has been observed in some chemical reactions, in particular the Belousov-Zhabotinsky reactions. One effect is to develop patterns in the reaction products, and it might be expected that these would move towards a self-similar pattern. A second possibility is the appearance of two different behaviors on a surface, corresponding to a bifurcation: this might result in pitting, but not related to specific microstructural features in the underlying alloy. The observation that in corrosion reactions the system is so far from equilibrium that non-linear effects might be expected appears to have been made first by Bertrand, 5 and later papers, for example by Bertrand et al. ,6 have discussed 'instability and bistability' in the growth of corrosion scales. Bistability appears to be essentially the same as bifurcation, but the equivalence with formal chaos theory is not altogether clear. The problem that these investigators have been addressing is the formation of layered scales, in particular on titanium oxidized in the range 630-1000°C, which is referred to as 'mechanical' layering, and on Fe-22wt% Cr-5.Swt% Al sulfidized at temperatures below 740°C, which is termed 'chemical' layering. A model is presented for chemical layering in a binary alloy A - B , where CA is the mole fraction of A 7. The scale is considered to be in three zones: an inner zone I, which is a thin substoichiometric layer; zone II, which consists of layers of two different oxidation products; and zone III, which is a homogeneous layer of one of the oxidation products. The alloy composition at the interface with zone i is AsBs; the two oxidation product phases are labeled X and Y; Cx and Cy are the compositions of the two product phases at the interface between zone I and zone II, and Cx + Cy = 1. The non-linearity is introduced in the kinetics of the interface process at the interface between zone I and zone II, as a result of cooperativity. A kinetic constant is then written as: k = ko exp (vCx),
(1)
where v, which is positive, represents the cooperativity strength. The following equation is then generated:
Applications of deterministic chaos theory to corrosion
755
CY t0.9 branch 0 0
i
07
I
06 05 04 03 O~ I
o
~el ~Bs
FIG. I.
Cy versus
o2 os
o~,-2o~
o~ o~ o~
"-B s
Cu~calculated from equation (2), showing 'bistability'. From Bertrand] Reproduced by permission of Sci-Tecb Publications.
Cy exp ( - v C y ) / ( 1 - Y) = (1//3) {(1 -
CBs)/(CBs},
(2)
where fi is a non-dimensional parameter which takes into account the kinetic constants of the chemical processes. Plotting Cy versus CBs generates an S-shaped curve (Figure 1), which is three-valued for Cy between C1B~,and C2B~. This results in a cyclic oxidation behavior generating alternate layers of X and Y, as the As in the 'reservoir', which is zone I near the interface, moves from one of these extremes to the other as the different phases are produced. (Note: the papers referred to here are only a few of several by this group on this topic.) Oscillatory composition fluctuations have sometimes been observed in scales, for example in the oxidation of Fe-18% Cr at 1200°C s, which are difficult to rationalize without a similar 'bistability' argument. (We are grateful to Prof. Wood for bringing this result to our attention.) It seems probable that the generation of layered scales, or the evolution of quasiperiodic composition fluctuations within a scale, may well be examples of an instability which is closely related to a bifurcation. If this is indeed the case, it would be worth inspecting equations similar to that derived by Bertrand and co-workers to see if further bifurcations, or a transition to deterministic chaos, are possible, and if the conditions are physically attainable. A recent paper by Kirkaldy 9 is perhaps relevant. This discusses possible deterministic chaos effects in eutectoid transformations, and specifically considers the pearlite reaction. The conventional description of this reaction to produce a uniform laminar structure is essentially linear and does not show bifurcations or chaos. However, Kirkaldy points out that the equations are derived assuming a (physically unreasonable) planar initial interface, and also define a physically unattainable velocity state under some conditions. Kirkaldy introduces a difference equation, shows that successive bifurcations and eventual chaos in the laminar spacing result as the relative supersaturation increases, and argues that the behavior implied by this is consistent with the formation of upper bainite. Other work by Unger and Klein m has shown that spinodal decompositions result in structures which exhibit fractal geometry in the early stages, and this may be a signal of dynamical chaos. Akuezue and Stringer 11 have used a difference equation substitution in multicomponent diffusion theory, and also show the possibility of chaotic instabilities.
756
J. STRINGERand A. J. MARKWORTH
The point of their analysis is not the form of the steady-state diffusion profiles, but the approach to this steady-state from the initial non-equilibrium state. It is easy to show that
dYi(2)/dt = k 2 + Y/(2)(1 - Yi().)),
(3)
where I1,.(2) is the concentration of the ith component at the dimensionless position 2 and k2 is a parameter dependent on position but independent of time; this equation is essentially similar to the well-known logistic equation used to describe population changes. This can be recast as a difference equation: Yi(2)n+l = k2 + k, Yi(2)n(1 - Yi(2)n),
(4)
where kl = Aci, the difference in the concentration of the ith species across the diffusion couple. The evolution of this function is well known, and it undergoes successive bifurcations leading eventually to chaos, depending on the values of k 1 and k2. The question is whether values of these two parameters which will lead to instabilities are physically reasonable. Akuezue and Stringer show that for two experimental situations, the carburization of iron and the aluminizing of nickel, the values of k 1 and k 2 correspond to situations where bifurcation does not take place. This is not the same as proving that bifurcation cannot occur in binary diffusion systems, but it does suggest that it is unlikely. The situation for ternary diffusion couples has also been examined for the Ni-Cr-A1 system, and these suggest that conditions may arise where instabilities may appear in the evolution of the system from the initial condition towards the steady state. It seems likely that, if behavior of this sort does arise, it will most likely be close to phase boundaries. More analysis is required. The use of difference equation substitution to achieve these results is a cause for question, since it is necessary to show that the instabilities are intrinsic to the process and not an artifact of the substitution. This step in the discussion is difficult and is often neglected. Kirkaldy also mentions the possibility of 'theorectical illegality', but argues that in fact the discrete representation is probably more physically reasonable than the conventional continuous form for the problem he considered. In all these situations, the key point is that when a system is far from equilibrium, non-linear effects are to be expected; these may (but not necessarily have to) result in instabilities such as bifurcations and dynamical chaos. Systems which are close to equilibrium might be expected to behave in an essentially linear fashion. Careful analysis is required to identify experimental situations where effects of this sort might be expected from quantitative theoretical arguments, but the appearance of complex and fine-scale microstructures such as are found in upper bainite or spinodal decompositions may also serve as an empirical clue to the existence of this kind of behavior. There have been several recent papers concerned with possible dynamical chaos in corrosion and electrochemical dissolution. These include papers by Kawczynski et al.,12 Diem and Hudson, 13Bassett and Hudson,14 Li et al.,15 Sharland et al.,16 Koper and Gaspard,17 Hudson and Bassett,ls and Corcoran and Sieradzki. 19 METAL PASSIVATION AND OSCILLATIONS IN ANODIC DISSOLUTION The phenomenon of passivation is one in which bifurcation seems consistent with the physical behavior, and generally the systems of equations which have been
Applications of deterministicchaos theory to corrosion
757
developed are non-linear, so that unstable behaviors might be anticipated. These possibilities have been studied by Markworth et al. 2° The reaction schemes of Sato 21 and Talbot and Oriani 22 were analyzed, and a reaction scheme incorporating aspects of both was postulated. A metal M is dissolving in a solution. Any given point on the surface of the metal may be bare, covered with adsorbed M O H , or covered with adsorbed MO. The adsorbed M O H and MO passivate the underlying metal. There are three variables: the concentration of the metal in solution, C; the fractional surface coverage by MO, 00; and the fractional coverage by M O H , 0OH. The fraction of the surface which is bare is then 0M, where 0M = 1 -- 00 -- 0OH. The following reactions may occur: M---~ M + + e
(kl)
M + H 2 0 = MOH~,q + H ÷
(k2, k_2)
nM + + a n ---*M,,A MOHaq = MOHad
(5) (6)
(k3)
(7)
(k4, k_4).
(8)
These reactions are essentially from Talbot and Oriani. MOH~d ~
MOad
+
H+ + e
(ks)
M + MOad + H 2 0 ~ 2MOHad
(9)
(k6).
These equations are from Sato. The following reduced variables are introduced: r
=
(10)
k_41, Y = K C , K =- k4/k_4,
p = k l K / k _ 4 , q = kB/k_4 K n - 1 , r = ks~k_4, s = k f f k _ 4.
From the above reactions, in terms of the reduced variables, the following three differential equations were deduced: d Y / d r = pOM -- q Y "
(l 1)
d0oH/dr = YOMf (OoH ) -- [r + g (0OH]0OH q- 2sOoO M
(12)
d0o/dr
=
rOoH
(13)
-- s O 0 0 M.
Markworth et al. then assumed that f(0OH ) = 1,
g(0OH ) = eo~°,
n = 1;
and chose the parameters p=2xl0
-4,
q=lxl0
-3,
r=2xl0
5.
The value of s was varied. Trajectories were then calculated for 601 different values of s. To reduce transient effects, the results from the first 300 000 time units (equivalent to about 250-300 oscillations) were discarded, and the results from the next 300 000 time intervals were reported. Figure 2 shows a bifurcation diagram, plotting Y versus s. Figure 3 shows a projection of the attractor on the OoH--Y plane, calculated for s = 9.7 x 10 -5. The Lyapunov exponents associated with each of the three dimensions of the equations were determined using the method described by Wolf et al.;23 the
758
J. STRINGERand A. J. MARKWORTH 0.114 0.112 iil",~7':~:i'ii:i ~' ;::i~:'!!i; j,
0.11 O. 108
[, , ; ,
0 . 106 O. 1 0 4
O. 1 0 2 I 0.1 _ 9600
9650
9700
9750
9800
s x 108
FIG. 2. A bifurcation diagram for 9600 < s x 108 < 9800 from equations (11)-(13). From Markworth et al. 2°
v a l u e s a r e 21 = 0.553, 22 = 0.000,23 = - 1 5 . 7 3 3 ( b i t s / p e r i o d ) . T h e e x i s t e n c e of a p o s i t i v e L y a p u n o v e x p o n e n t is a q u a n t i t a t i v e v e r i f i c a t i o n t h a t t h e a t t r a c t o r is c h a o t i c . F r o m t h e s e e x p o n e n t s , a L y a p u n o v d i m e n s i o n c a n b e c a l c u l a t e d for t h e a t t r a c t o r : t h e r e s u l t is d~ = 2.035. T h e r e a r e a n u m b e r o f w a y s of c a l c u l a t i n g t h e d i m e n s i o n o f a n
0.116
0.114
0.112
Y
0.11
O. 108
O. 1 0 6
O. 1 0 4 0.1
z 0.2
i 0.3
i 0.4
I 0.5
0.6
~oH FIG. 3. A projection of the attractor on the O o H - Y plane, calculated for s = 9.7 x 10-5 from equations (11)-(13). From Markworth et al.2°
Applications of deterministic chaos theory to corrosion
759
attractor, and a discussion of these is outside the scope of the present paper, but the fact that this dimension is a non-integer is a further indication that the attractor is chaotic. T h e existence of a great deal of c o m p l e x structure in this system of equations is of considerable interest, but of course it does not m e a n a great deal if it does not c o r r e s p o n d to some changes in b e h a v i o r in the practical system. To this end, M a r k w o r t h and Rollins and their co-workers have been designing simple experiments in which the p a r a m e t e r s can be varied within the range where the system of equations would predict bifurcation and chaos. Recently, D e w a l d et al. 24 have reported on current oscillations during the anodic dissolution of c o p p e r in acetate buffer using a rotating disc electrode. These experiments have recently been extended into the regime where chaos would be predicted, and chaotic b e h a v i o r has indeed been observed and an attractor has been determined. These results will be presented elsewhere. SUMMARY Non-linear dynamics exhibit the possibility of different kinds of instability, including bifurcation and deterministic chaos. It is not clear what the implications of this are for oxidation and corrosion, or indeed w h e t h e r instabilities of these kinds are possible. It has been p r o p o s e d that scale laminations in high t e m p e r a t u r e oxidation and the passive-active transition in a q u e o u s corrosion may be examples, and investigations of these possibilities are continuing. T h e very large deviation from equilibrium in the initial stages of an oxidation reaction suggests that linear approximations m a y well be unsuitable here, and instabilities may well be possible in the a p p r o a c h to a quasi-equilibrium configuration. It has been claimed that this m a y be possible in m u l t i c o m p o n e n t diffusion couples. In solid-state transformations, other situations which m a y be chaotic in origin have been p r o p o s e d : the pearlite to u p p e r bainite transition and spinodai decompositions are examples. Further w o r k is necessary, but care must be taken in the interpretation of results, and in particular the c o m m o n m e t h o d of time-series phase portraits must take account of the restrictions imposed by information theory.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10, 11. 12. 13, 14, 15, 16,
REFERENCES E. N. LORENZ,J. atmos. Sci. 20, 130 (1963). J. GLEICK,Chaos: Making a New Science. Penguin Books, London (1987). N. H. PACKARD,J. P. CRUTCHFIELD,J. D. FAR~ERand R. S. SHAW,Phys. Rev. Lett. 45,712 (1980). F. TAKENS,Lecture Notes in Mathematics (eds D. A. RANDand L. S. YOUNG),p. 366. SpringerVcrlag, Berlin (1981). G. BERTRAND,in Synergetics, Far From Equilibrium (eds A. PACAI!LTand C. VIDAL), p. 147. Springer-Verlag, Berlin (1979). G. BERTRAND,A. SANFELD,G. WATELLEand J.-P. LARPIN,J. Chim. Phys. 83,695 (1986). G. BERTRAND,Solid State Phenomena 3/4, 257 (1988). D. P. WHIrrLEand G. C. WOOD,J. electrochem, Soc. 114,986 (1967). J. S. KIRKALDV,Scripta Met. Mater. 24,179 (1990). See for example C. UNOERand W. KLEIN,Phys. Rev. B 29, 2698 (1984). H. C. AKUEZUEand J. STRINGER,to be published. A. L. KAWCZYNSKI,W. RACZYINSKIand B. BARANOWSLI,Z. Phys. Chem., Leipzig 269, 596 (1988). C. B. DIEMand J. L. HUDSON,A. I. chem. E. ,1.33,218 (1987). M. R. BASSErrand J. L. HUDSON,J. phys. Chem. 93, 2731 (1989). W. LI, K. NOBEand A. J. PEARL,STEIN,Corros. Sci. 31,615 (1990). S. M. SHARLAND,C. M. BISHOP,P. H. BALKWILLand J. STEWART,in Advances in Localized Corrosion
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J. STRINGERand A. J. MARKWORTH (eds H. S. ISAACS, U. BERTOCCI, J. KRUGERand S. SMIALOWSKA),p. 109. National Association of Corrosion Engineers, Houston, TX (1990). M. T. M. KOPER and P. GASPARD,J. phys. Chem. 95, 4945 (1991). J. L. HUDSON and M. R. BASSETT,Rev. Chem. Engng 7, 109 (1991). S. G. CORCORANand K. SIERADZKI,J. electrochem. Soc. 139, 1568 (1992) A. J. MARKWORTH,J. K. McCoY, R. W. ROLLINS and P. PARMANANDA,Final Report to EPRI on Research Project RP2426-25 (April 1990). N. SATO, in Passivity of Metals (eds R. P. FRANKENTHALand J. KRUGER), p. 29. The Electrochemical Society, Princeton, NJ (1978). J. B. TALBOTand R. A. ORIAN1,Electrochim. Acta 30, 1277 (1985). A. WOLF, J. B. SWIFT, H. L. SWINNEYand J. A. VASTANO,Physica 16D, 215 (1985). H. D. DEWALD, P. PARMANANDAand R. W. ROLLINS, J. electroanal. Chem. 306, 297 (1991).