Forecasting interface roughness from kinetic parameters of corrosion mechanisms

Forecasting interface roughness from kinetic parameters of corrosionmechanisms

P. Cordoba-Torres a, R.P. Nogueira b,1, V. Fairen a,*a Departamento de Fısica Matematica y Fluidos, Universidad Nacional de Education (UNED), Apdo. 60141, 28080 Madrid, Spain

b Laboratorio de Corrosao, PEMM/COPPE/UFRJ, C.P. 68505, CEP21945-970 Rio de Janeiro, Brazil

Received 24 September 2001; received in revised form 4 April 2002; accepted 2 May 2002

Abstract

This paper is aimed at investigating the possibility of predicting interface roughness behavior from reaction kinetic parameters of

a simple open-circuit potential corrosion mechanism. A cellular automaton algorithm simulates the evolution of a dissolving

metallic front according to specific rules that govern transitions between the different states associated with surface reactants. A first

mathematical model presents a rigorous mesoscopic treatment incorporating kinetic parameters and purely morphologic descriptors

and yields an analytical expression accounting for the interplay between reaction kinetics and morphology. Comparing the mean

cell-lifetimes of active species taking part in the model does this. A new morphological descriptor built on reaction kinetic

parameters is introduced and shown to represent the interface roughness very well. Finally, a simplified version of the model, which

resorts to the exclusive use of measurable electrochemical quantities, is proposed. Results are promising, showing the approach to be

an interesting tool for the understanding and even forecasting of the complex interaction between reaction kinetic parameters and

morphological features of metal j electrolyte interfaces. # 2002 Elsevier Science B.V. All rights reserved.

Keywords: Corrosion; Surface structure; Pattern formation; Kinetic model; Reaction mechanism; Theory

1. Introduction

The dynamic evolution of dissolution fronts leads to

different surface profiles related to a wide variety of

intrinsic properties of the system. As a consequence,

roughness appears to be a ubiquitous characteristic of

electrochemical interfaces, which is known to exert a

feedback effect in the evolution of the process itself, as is

the case with porous electrodes. It is also clear that the

influence of morphology on kinetics must be present far

beyond the simple overall reaction rate increase asso-

ciated with an enhanced electrode-electrolyte area*/for

example, the connection of certain surface topographic

features with a higher reactivity. In fact, the electrode

morphology and reaction kinetics are highly interde-

pendent and a close investigation of the behavior and

evolution of electrochemical interfaces should not

neglect this issue.

Some recent papers have pointed out experimentally

the influence of morphology on kinetic parameters. As

an example, Parkhutik [1] has found evidence of the

dependence of Si electrochemical kinetics on the mor-

phology of surface passive films, whilst Bock and Birss

[2] related kinetic differences observed in the Ir3�/4�

reaction that were ascribed, at least partially, to the

structure of Ir oxide films. Nevertheless, this interplay

between morphology and reaction kinetics remains

poorly understood since it arises from some generic

evidence and no consistent theory has been proposed

because of the great complexity of the matter and the

great difficulty in conceiving and performing specific

feasible experiments.

In this sense, we have presented in a previous paper

[3], a first approach to this issue by using a cellular

automaton algorithm that models the evolution of a

corrosion front. The impact of mesoscopic interfacial

heterogeneities on the classical electrochemical descrip-

tion, which is based on a macroscopic homogeneous

* Corresponding author. Tel.: �/34-913-987124; fax: �/34-913-

986697

E-mail address: [email protected] (V. Fairen).1 Present address: UPR15 du CNRS, Physique des Liquides et

Electrochimie, 75252 Paris Cedex 05, France.

Journal of Electroanalytical Chemistry 529 (2002) 109�/123

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treatment, was also investigated. This was achieved by

means of the development of a set of mesoscopic charge

and mass balance equations that included effects of

morphological mechanisms intervening in the dissolu-tion process. Results have shown that variations in

reaction kinetic parameters of the model lead to changes

in the preferential location of active dissolution sites.

This was found to be closely related to surface rough-

ness. It was also seen that deviations from the expected

standard macroscopic predictions appeared, both qua-

litatively and quantitatively, as a consequence of the

interplay between reaction kinetics and morphology.Cross-effects led to the segregation of reactants and to

the detachment of non-dissolved clusters, designated

ndc, which are related to a sort of non-electrochemical

mass loss and have already been referred to in the

literature [4�/6]. Nevertheless, although clearly disclosed,

the morphology-kinetics relationship on the mesoscopic

scale seems to be a complex problem deserving specific

treatment, which is the aim of the present paper.This paper is then a first attempt at formalizing a

mathematical model that conveys a comprehensive

description of the coupling between the morphology

and reaction kinetics at a corroding surface. The

mathematical development presented in the following

sections aims at providing two new parameters, defined

as Da and Dak, in terms of reaction kinetic parameters

of the model, which we will find to be closely correlatedto interface morphological descriptors [3]. In a first

approach, taking advantage of the mesoscopic balance

equations that include morphological effects [3], the

mathematical modeling makes explicit the influence of

reaction kinetics on surface roughness by means of the

construction of an expression, the differential aging

factor, Da, in terms of the lifetimes of surface reactants

present on the simulated metallic surface. It varieswithin the interval 0.755/DaB/2 and conveys informa-

tion on the differentiation in the range of times of

residence of cells on the interface. For example, if the

time of residence of cells freshly incorporated into the

surface from the metal bulk is much longer than that

corresponding to ‘old’ surface cells, we shall tend the

lower limit for Da. When all cells have, instead, similar

times of residence, Da:/1 Finally, whenever most cellsget trapped almost permanently at some early stage of

the dissolution route, then Da approaches from below

its upper limit. It will not then be surprising to obtain a

correlation between this parameter and surface rough-

ness. In the first case, the process will not depart very

much from a layer by layer dissolution (no roughness),

while in the second the dissolution proceeds in a purely

random way, generating a surface with a very char-acteristic degree of roughness, and in the third the metal

surface becomes extremely corrugated. In Section 6, it

will be shown that this parameter can be evaluated fairly

well solely from the knowledge of kinetic constants, and

that the resulting approximation, Dak, will permit us to

predict the degree of roughness of the interface.

Neither Ref. [3] nor this paper deals with atomic scale

phenomena, such as the well known influence on ratesdue to specific arrangements in the lattice, as at terraces

and kinks, where the reactivity has been found to be

different by many orders of magnitude [7,8]. The range

of scales in which this formulation is valid has been

discussed in Ref. [3]. There, we estimated a lower limit to

be somewhere between 1 and 10 mm, well above the level

of atomic resolution.

It is also important to note that the aim of this paperis to establish the influence of the different steps of an

electrochemical mechanism on the resulting morphology

of the metal surface, regardless of mass transport

limitations. For that purpose we shall simplify the

electrolyte and work in the reaction-limited domain.

Thus, all the transport processes from and into the

electrolyte will be supposed very fast and will be

neglected as in Ref. [3].Hence, this study is in some way the continuation of

Ref. [3]. It investigates the same simple model of a

corroding bivalent metal, which is simulated by a

cellular automaton algorithm that we summarize in

the next section. Readers are then advised that a more

exhaustive discussion about scaling phenomena, inter-

facial mesoscopic heterogeneity and also about the

validity of the usual macroscopic treatment of electro-chemical interfaces has already been proposed there.

2. Electrochemical mechanism and simulation

Both the electrochemical model and the cellular

automaton algorithm that constitute the basis of the

present study have been presented and discussed ex-

haustively in Ref. [3]. For the sake of completeness, we

present in this section a very brief summary with theminimal information deemed necessary for a self-suffi-

cient reading of the text.

2.1. Electrochemical mechanism

The dissolution mechanism considers a bivalent non-

identified metal and a generic cation, both leading to the

presence of two intermediate adsorbed species on the

metallic surface. The reaction scheme is given by:

M ?k1

k�1

M(I)ads�e� (1)

M(I)ads 0k2

M(II)sol�e� (2)

C��e� 0k�3

Cads (3)

C��Cads�e� 0k�4

C2;sol (4)

This is one of the simplest mechanisms that can be

P. Cordoba-Torres et al. / Journal of Electroanalytical Chemistry 529 (2002) 109�/123110

related to metal dissolution processes. Under certain

conditions, it can stand, for example, for the extended

corrosion of pure iron, for which different models have

been proposed*/with one or several intermediate spe-cies [9�/11]. Models involving a single intermediate

species (cf. reactions (1) and (2)) can reproduce the

steady state behavior of the interface at corrosion

potential, but cannot explain, for instance, results

from electrochemical impedance. Keddam et al. [10,11]

established the need to consider four intermediate

species in order to account for non-stationary behavior

in sulfuric acid at large anodic overpotentials. It isinteresting to note, however, that they used three-

electrode electrochemical cells so that anodic and

cathodic reactions took place on well-defined, separated

iron working-electrode and platinum counter-electrode,

respectively. In the case of the present study, we actually

intend to simulate with our model the spontaneous

generalized corrosion of a metal*/i.e. at a free corro-

sion potential*/in the presence of adsorbed species: oneintermediate species from the dissolving metal and a

second one, coming from the cathodic counterpart.

Under these conditions, cathodic and anodic reactions

take part simultaneously at the same electrode, without

any restriction of their spatial distribution whatsoever.

We cannot pretend to recover the results of Keddam et

al. and we, therefore, choose the simplest model includ-

ing a single adsorbate intermediate form that can satisfyopen-circuit behavior as is the case of reactions (1) and

(2). On the other hand, the unidentified cathodic

reactions (3) and (4) can be seen as a generic emulation

of the hydrogen adsorption/desorption following the

Volmer�/Heyrovsky reaction. The Tafel reaction is not

considered here since we suppose that the interface

follows the Langmuir isotherm and no lateral interac-

tion between adsorbed species is allowed. Nevertheless,since the central idea of this paper is to evaluate the

close interplay between kinetics and morphology re-

gardless of the specific electrochemical reaction model,

we do not keep this illustrative identification and work

with unidentified reactive species.

As in ref. [3], we have chosen a purely reaction-limited

model. Therefore, diffusion is considered to be a fast

process compared with electron transfer: reactionproducts*/M(II)sol and C2,sol*/are immediately re-

moved from the interface during the simulation and

reactions (2) and (4) are taken as irreversible. Also, in

order to preclude a C� concentration gradient, we

assume that the interface is constantly supplied with a

surplus of cations. Consequently, the forward anodic

reaction by k3 can be taken as negligible.

2.2. Stochastic cellular automaton computation model

Cellular automata [12,13] are discrete dynamical

systems, i.e. space, time and the system states are

discrete. A space grid is defined in which each regular

point, called a cell, can have a finite number of states.

Transitions between these states are made according to a

well prescribed set of rules. All cells are synchronouslyupdated, the state of any cell at time t depending on the

state of that cell and that of its neighbors at time t�/1.

In ref. [3], we designed a cellular automaton model

consisting of a rectangular, two-dimensional spatial

array. According to the proposed dissolution mechan-

ism*/reactions (1)�/(4)*/, each cell (or site) of the

cellular automaton can assume four possible occupation

states: empty [8 ], metal [M], intermediate adsorbate[M(I)ads], cation adsorbate [Cads]. Transitions between

states obey a set of probabilistic rules that mimic the

above electrochemical mechanism*/a complete list of

transition rules can be found in ref. [3]. Each significant

allowed transition is given a probability of success

R1�probability of [M] 0 [M(I)ads] (5)

corresponding to reaction (1), forward;

R2�probability of [M(I)ads] 0 [8 ] (6)

corresponding to reaction (2);

R3�probability of [M(I)ads] 0 [M] (7)

corresponding to reaction (1), backward;

R4�probability of [M] 0 [Cads] (8)

corresponding to reaction (3). And, finally:

R5�probability of [Cads] 0 [M] (9)

corresponding to reaction (4).

Rules (8) and (9) deserve a particular treatment due to

their specific characteristics. In both cases, the existence

of C� cations in contact with M site is implicitly

assumed. C� are then assumed to be in surplus at the

interface. Computationally, this means that a C� is athand whenever a metal surface cell either in state [M] or

in state [Cads] is liable to undergo a transition according

to (8) or (9), respectively. Otherwise, C� cations are

irrelevant and are not explicitly considered in the

automaton, as are also reaction products, M(II)sol and

C2,sol. Furthermore, the presence of a Cads bounded to a

metal site is considered to block the reactivity of the

latter as long as it remains attached, taking thus intoaccount the inhibiting effect of hydrogen intermediates

on the dissolution of Fe [14]. We proceed to an

algorithmic simplification of this process, which consists

in actually replacing the state metal site by the state

cation adsorbate . The reverse process happens when

reaction (4) occurs: C2,sol is instantly released and the

metal site returns to an active form.

Given a set of reaction probabilities,{Ri}, as definedin (5)�/(9), the corresponding simulation starts from an

initial grid in which all cells, except those at the top row,

are metal*/in state [M]. Those at the top row are

P. Cordoba-Torres et al. / Journal of Electroanalytical Chemistry 529 (2002) 109�/123 111

initially in the ‘empty’ state*/[8 ] and model the initial

electrolyte, which is always considered structureless*/

empty cells*/for our purpose. The simulation proceeds

and metal cells dissolve progressively from top tobottom of the grid. The stochastic nature of the

elementary transition processes roughens the initially

smooth metal surface until, after some relaxation time

[3], the whole process progresses in a stationary regime,

characterized by fluctuating coverage fractions of sur-

face reactants around well defined mean values we call

steady-state values. From time to time, the junction of

dissolving inlets isolates clusters of non-dissolved cellsfrom the metal bulk. They were designated ndc in ref. [3]

and must consistently be accounted for in the rate

equations governing the process.

As is usual in classical electrochemistry we can write

mass and charge balance equations at steady state,

which for the cellular automata are [3]:

R1U3�(R2�R3)U1� j1�0 (10)

R4U3�R3U2� j2�0 (11)

xR2U1�R5U2�R3U1�(R1�R4)U3� j3�0 (12)

(x�1)�

X3

i�1

ji

R2U1

(13)

J�(R1�R4)U3�(R2�R3)U1�R5U2 (14)

where U1; U2; U/3�/1�//U/1�//U/2, stand for the steady-

state coverage ratios of M(I)ads, Cads and M, respec-

tively; R1 through R5 denote the reaction probabilities

defined above; J is the dimensionless net current density,

which is considered null at open circuit conditions; j1; j2/

and j3 stand for the averaged ndc loss rate of M(I)ads,Cads and M, respectively. Finally, x is the mean number

of newly exposed M sites that, as a result of a given cell

dissolution, become part of the interface*/see ref. [3] for

a thorough interpretation of this descriptor. Beside the

standard electrochemical contributions, Eqs. (10)�/(12)

contain terms originating in the rough topography of

the metal surface: j1/�//j3 and xR2U1; which would not be

present were the metal surface totally smooth.In ref. [3] it was shown that x; besides giving

information about the preferential locations of active

dissolution sites, is a very good descriptor of surface

roughness. Moreover, Eq. (13) describes the close

relationship between the net ndc production and inter-

face roughness*/represented by x: It can be seen [3] that

in the limit of very low roughness no ndc are formed and

x 0 1:/Expressions for the cellular automaton predicted

steady-state values of coverage fractions for M(I)ads

and Cads are then easily obtained from (10)�/(13):

U1�R1R5

(R4 � R5)(R2 � R3) � R1R5

�j2R1 � j1(R4 � R5)

(R4 � R5)(R2 � R3) � R1R5

(15)

U2�R4(R2 � R3)

(R4 � R5)(R2 � R3) � R1R5

�j1R4 � j2(R1 � R2 � R3)

(R4 � R5)(R2 � R3) � R1R5

(16)

It is interesting to note that the first terms on the rhs

of Eqs. (15)�/(16) are precisely those expressions for the

steady-state values derived from the standard macro-

scopic rate equations for reactions (1�/4), provided we

make a one to one correspondence between R1, R2, R3,R4, R5 and k1, k2, k�1, k�3 and k�4, respectively. The

j/-dependent contributions to (15)�/(16) are purely

mesoscopic�/having no macroscopic counterpart-and

are related to ndc production.

3. Preliminary ideas on reaction kinetics vs. interface

roughness

Morphology is the result of random individual events

driven by the mechanism. As a consequence of this, the

metal j electrolyte interface evolves to a rather involved

boundary with inlets and overhangs on several scales,with a degree of roughness dependent, as we shall see

later, on the specific values of the kinetic parameters.

Fractal geometry [15] provides a language for a quali-

tative and a quantitative characterization of surface

roughness. Fractals objects are self-similar: they are

deterministically*/deterministic fractals*/or statis-

tically*/random fractals*/invariant under scale trans-

formations. Quantitative characterization of a fractalappears when trying to get a finite measure of its volume

V (l) independently of the unit of measurement l . This

can be done by covering the object with df-dimensional

balls of linear size l and volume ldf (df is not an integer

number). The volume of the object is estimated by the

expression N (l)ldf, where N (l) is the smallest number of

balls needed to cover it completely. In order to obtain

scale invariance of the measure we have N (l)�/l�df. Thedimension df is called the Hausdorff dimension or,

commonly, the fractal dimension , and is defined as

liml00 lnN(l)=ln(1=l): For fractal objects we obtain

dTB/dfB/dE, where dT is the topological dimension of

the object and dE is the embedding dimension*/the

smallest Euclidean dimension of the space in which the

object can be embedded.

Surfaces belong generally to a broad class of fractalscalled self-affine fractals [16], which are statistically

invariant under non-isotropic transformations*/a self-

affine function h (x ) must be rescaled in a different way


horizontally (x 0/bx) and vertically (h 0/bah , with an

exponent a"/1) in order to obtain invariance. The

morphology of a rough interface is usually characterized

by this value of a , which is called the roughness exponent

[16]. In addition, it is possible to associate with a self-

affine surface a local fractal dimension, df. On short

length scales a and df are related through the expression

df�/2�/a [16].The high level of irregularity displayed by the major

part of the profiles obtained from the simulations, with

many overhangs in all scales, precludes an efficient

estimate of the roughness exponent a . In contrast, on

the working scales of the simulations, df is well-defined

and can be estimated easily. Consequently, we shall

resort to it as a roughness descriptor when quantifying

disorderly surfaces. There are many methods for mea-

suring the fractal dimension, the best known being the

box counting method [17]. This consists in dividing theregion of the dE-dimensional space where the object is

located in a hypercubic lattice of cell spacing l . The

number of boxes of volume ldE which overlap with the

structure can be used as a definition for N (l) in the

above definition of fractal dimension.

The set of control parameters is given by {R1, R2, R3,

R4, R5} but the presentation of results may sometimes

demand the use of the alternative set {/U/1, U/2, J; R2, R3}.The reasons for choosing U1; U2 and J are obvious: All

calculations were performed by simulating corrosion

potential conditions, so that the electronic charge

balance always delivered a mean zero net current, and

both U1 and U2 are physically relevant quantities. For

each simulation, transition probabilities were selected

such as to satisfy specifically chosen target values for

U1; U2; and a dimensionless zero net current density,J�0 (corrosion potential). This left two degrees of

freedom from the five transition probabilities Ri . The

choice fell on R2 and R3 and it was not arbitrary

inasmuch as for any M(I)ads site at the interface they

control, respectively, the final transition to dissolution

for that site (M(I)ads0/M(II)sol) or its re-entry in the

‘game’ recovering the M state (M(I)ads0/M). For given

values of U1 and U2; we shall work between two limitingcases: The limit R2/R30/�, in which straight dissolution

of metal is favored*/transition (6) is much faster than

transition (7)*/, and the case R2/R30/0 where the rate-

determining step is the dissolution of M(I)ads*/transi-

tion (7) much faster than transition (6). As in ref. [3],

simulations were carried out for different lattice sizes*/

minimal size: 10002.

3.1. Dependence of surface roughness on M(I)ads

coverage fraction

We start by summarizing in Fig. 1, how interface

roughness*/represented by the box-counting dimension

dbc*/depends on the fraction U/1 of active dissolution

sites, M(I)ads, for three different values of the Cads

coverage fraction, U/2. Each one of these is conveniently

displayed in a separate window. In all cases the U/1 rangehas a natural upper limit at 1�U2:Curves corresponding

to different values of the ratio R2/R3 are represented in

each window. In the limit R2/R30/0 (dissolution is not

favored), the results are parallel to the U/1 axis*/the

roughness does not depend on M(I)ads surface con-

centration*/while curves of ever greater convexity

appear as the ratio R2/R3 increases (increasingly favored

dissolution). The limiting lower curve holds for R2/R30/

�.

Before presenting the complete formal treatment

accounting for the overall behavior depicted in Fig. 1,

Fig. 1. Box-counting dimension (dbc) vs. M (I )ads mean interface

coverage fraction (/U1) according to the R2/R3 ratio: R2/R30/0 (j);

0.25 (m); 1.5 (m); and R2/R30/� (%). Each window stands for a

different value of Cads mean coverage: U2/�/0.001 (a); 0.25 (b) and 0.55

(c). The dotted line corresponds to the pure random dissolution dbc

value.


we proceed to a preliminary analysis that physically

justifies that theoretical approach, in terms of the

electrochemical mechanism. We do it with the help of

Fig. 2 for the simplest case of Fig. 1a, with negligible U/2,

because this case emphasizes the predominant role of the

dissolution route. As indicated in the left column of Fig.

2, variations in the value of U/1 are associated with shifts

in the relative magnitude of the transition probabilities

of the elementary electrochemical steps along the

dissolution route.

3.2. Roughness versus M(I)ads coverage fraction: the

limiting cases U1 0 0 and U1 0 1/

All curves in Fig. 1a merge at both end points, U1 00; U1 0 1: At the left end of the U/1 range, compara-

tively small probabilities of successful transitions out of

the intact metal state lead to high stationary fractional

values for M and, correspondingly, U3 0 1 (case (a) in

Fig. 2). Interface cells accumulate in that state and it

takes a long time for any cell to leave for an M(I)ads

Fig. 2. Electrochemical model mechanism with indication of steps with low probability transitions (left column) and corresponding schematic

representation of local configuration of the interface (right column). Four representative cases are displayed that illustrate four different situations on

Fig. 1a. Cases (a), (b) and (d) are on the line dbc�/1.36, defined by the limit R2/R30/0, while case (c) holds for the minimum of the curve generated in

the limit R2/R30/�. Legend for the identification of the nature of each cell is shown at the top of the right column. Each case has its box-counting

dimension displayed on its corresponding right column window.


state. Randomly, from time to time, one M site makes a

successful transition to M(I)ads. It does not stay long in

this state (transitions out of it are fast) and it either exits

the automaton by dissolving or jumps back again tostate M. The roughness development process has then

become the result of a purely random site-ejection

mechanism. Something similar happens at the opposite

end of the U1 range. Comparatively small probability

values inhibit transitions out of state M(I)ads and its

occupation fraction approaches the limit U/0/1 (case (d)

in Fig. 2). All interface sites have equal probability of

dissolving and once again the corrosion front movesforward by randomly removing interface cells. Bound-

ary sites in both cases (a) and (d) of Fig. 2 are then

probabilistically equivalent as far as their ‘time-till-

dissolution’ is concerned, this entailing an equivalent

degree of roughness*/corresponding synoptic diagrams

of how the interface may look locally are depicted side

by side on the right column in Fig. 2. In both cases, (a)

and (d) of Fig. 2, we recover the value dbc�/1.36. Thisparticular value of the box-counting dimension agrees

exactly with what would be obtained in parallel simula-

tions of a purely random dissolution process: [M]0R

[8 ];that is, the simplest imaginable dissolution mechanism

in which all the metal cells are in the same single state

and sites belonging to the surface are randomly ejected

with a given probability R.

3.3. Roughness at intermediate values of M(I)ads

coverage fraction: dependence on R2/R3

Besides the limiting surface profile illustrated in Fig.

2a and d, the noticeable fact in Fig. 1a is the behavior of

roughness in a bi-component surface, with both M and

M(I)ads present in non-negligible amounts. Here the

ratio R2/R3 is a discriminating factor. The simplest case

is offered by the limit R2/R30/0 (inhibited dissolution),in which roughness is again independent of U/1 (upper-

most curve in Fig. 1a, corresponding to the case of Fig.

2b). Here, after a transition M0/M(I)ads has occurred,

the reverse transition is favored with respect to straight-

forward dissolution. Interface sites switch back and

forth between these states M and M(I)ads until some

randomly chosen unit hits a successful dissolution,

M(I)ads0/M(II)sol. The dissolution process, depicted inFig. 2b, is then similar to cases (a) and (d) of Fig. 2 and

this is why we also find dbc�/1.36.

Something different happens in the opposite limit, R2/

R30/� (straightforward dissolution unimpeded)*/low-

ermost curve in Fig. 1a, corresponding to the case of Fig.

2c. Here, an M(I)ads has very little chance of returning to

M (R3 much lower than R2). Once an M site belonging to

the bulk has become part of the interface, it is just a matterof time for it to attain the M(I)ads state and ‘wait there’ in

order to complete the dissolution process. Dissolution is

consequently favored at ‘older’ interface sites, which are

left at the rear of the advancing corrosion front (the front

in the automaton advances in the downward direction).

The emergence of a spontaneous segregation of species

(M(I)ads sites, with shorter lifetimes, at the back edge of theinterface and M sites, with longer lifetimes, at its

forefront) smooths out the interface and lowers the

degree of roughness. This is illustrated in the synoptic

representation of the cellular automaton corresponding to

Fig. 2c (right panel).

3.4. Dependence of roughness on Cads coverage fraction

The effect of a varying fraction of adsorbed cations,Cads, on roughness is somewhat different. We show in

Fig. 3 results of simulations for three significant values

of the M(I)ads fraction. As was done in Fig. 1, they have

been displayed separately for convenience. The general

trend here is a monotonic dependence of roughness on

the Cads fraction. There appears to be a dependence on

the ratio R2/R3 that clearly unfolds in Fig. 3b and c:

uppermost curves stand for the limit R2/R30/0 (dissolu-tion inhibited) while lowermost curves stand for R2/

R30/� (dissolution favored). In Fig. 3a all these curves

merge in one, which means that roughness does not

depend on R2 and R3 for very small values of U/1. We

shall clarify all this in the mathematical treatment of

later sections.

However, as was done in the case of Fig. 1, some

preliminary ideas on the relationship between mechan-ism and morphology can be outlined in advance for the

simplest case of Fig. 3a (/U1�10�3); for example. Here,

the limit U/20/0 just about attains the case of Fig. 2a and

will not be examined again. Three representative situa-

tions are represented in Fig. 4 for non-trivial values of

U2: In Fig. 4a, U/2 is small and U/3, the coverage fraction

for M, is large. This is because both possible transitions

out of state M are slow (recall that reaction C��/e�0/

Cads actually ‘consumes’ an M site in the cellular

automaton): sites in state M remain for a long time in

that state. U2 is small, so there are only a few Cads sites

and their lifetime is short (R5 is larger than R4). In Fig.

4b (/U/2�/0.5) the fraction of Cads has increased simply

because their release through R5*/reaction (4)*/has

been slowed down while the forward reaction (3) has

simultaneously been accelerated. The residence time forthese Cads sites has lengthened, a fact that conveys a

freeze of those same sites as the front keeps moving

downward. Roughness increases, and we witness the

emergence of dendrite-like Cads extrusions while active

dissolution sites are preferably located deep into elec-

trolyte inlets. This segregation process intensifies in the

case of Fig. 4c (/U2�0:9) since a Cads release is even

more severely restricted (R5 becomes very low). As aconsequence of this trapping effect of metallic sites by

Cads, roughness attains high values and electrochemical

dissolution happens preferably at sites newly incorpo-


rated into the interface while those older Cads sites are

harvested essentially through j2; that is, by means of

ndc detachment [3].

3.5. Summary

One conclusion starts to emerge from the previous

discussion on the results of Figs. 1 and 3. If dissolution

of an intact metal cell occurs easily and unimpeded, as

happens in the case of Fig. 2c, roughness development is

not stimulated. On average, older interface sites dissolveearlier than new ones and corrosion front propagation is

accompanied by a razing of old sites trailing behind. On

the other hand, roughness is actually favored by the

existence of some hindrance on the dissolution route of

intact metal sites. This can be in the form of a very slow

step in the dissolution route as in cases (a) and (d) of

Fig. 2, which can eventually be coupled or not to a

significant reversal of this dissolution route as in case (b)

in Fig. 2, or even in the form of some temporary

trapping in a long lived blocking state (Cads, for

example, as detailed in Fig. 4). Whatever this ‘impedi-ment’ might be, a given fraction of interface cells see

their dissolution eventually delayed, while newly dis-

closed metal sites might occasionally have a better

opportunity of dissolving faster, simply because they

still have a chance of completing the dissolution route

straightforwardly. This is strikingly apparent upon

examining Fig. 4.

In summary, some sort of relationship seems to emergebetween the degree of roughness and the evolution of ‘old’

interface sites as compared with that of ‘new’ ones. For

example, we have just mentioned what happens in the

cases of Fig. 4, in which new M sites are, on average, short

Fig. 3. Box-counting dimension (dbc) vs. Cads mean interface coverage

fraction (/U2) according to the R2/R3 ratio: R2/R30/0 (j); 0.25 (m); 1.5

(m); and R2/R30/� (%). Each window stands for a different value of

M(I)ads mean coverage: U2/�/0.001 (a); 0.1 (b); and 0.25 (c). The dotted

line corresponds to the pure random dissolution dbc value.

Fig. 4. Electrochemical model mechanism with indication of steps

with low probability transitions (left column) and corresponding

schematic representation of local configuration of the interface (right

column). Three representative cases are displayed that illustrate three

different situations on Fig. 3a: (a) low Cads coverage; (b) medium Cads

coverage; (c) high Cads coverage. Legend for the identification of the

nature of each cell is shown at the top of the right column. Each case

has its box-counting dimension displayed on its corresponding right

column window.


lived in comparison to older sites trapped in state Cads: the

roughness increases as this difference becomes more

apparent. We have seen the opposite behavior for case (c)

of Fig. 2: older sites dissolve faster than new sites, and theroughness is relatively low. An intermediate situation is

offered by cases (a) and (d) of Fig. 2, in which there is no

distinction between old and new sites as far as dissolution

is concerned.

4. The differential aging factor

4.1. Electrochemical mechanism and surface reactants

lifetimes

A formalization of what we discussed in the preceding

sections can be built from ground by starting from the

concept of cell-lifetime, which we can define for any

accessible state as the average time elapsing for one site

placed in that state until it is no longer part of the

interface. Of course, the concept of cell-lifetime makes

sense only for cells belonging to the interface, not forthose in the metal bulk. In their definition these mean

times must take into account that a given site may leave

the interface by either following the dissolution route

and end up in state M(II)sol or by being collected by an

ndc. In the present model they are associated with states

M(I)ads, Cads and M, and denoted t1, t2 and t3,

respectively. t2, for example, is the average time it takes

a site in state Cads to disappear into the electrolyte. It isimportant to note that these cell-lifetimes are averaged

values evaluated after the cellular automaton had

reached stationary state conditions.

From a strictly electrochemical point of view, that is,

first disregarding morphological features (ndc),ti can be

easily calculated in terms of the transition probabilities

{Ri}, according to reactions (1)�/(4), and by taking into

account that the mean time corresponding to a transi-tion of probability R is given by 1/R .

For the Cads state only one transition is possible.

Reaction (4) stands for the cathodic release of Cads*/

liberating the blocked M cell*/with probability R5, with

a mean time of 1/R5. The resulting free M site has in its

future evolution, an average time t3 till it disappears.

Thus, we have:

t2�1

R5

�t3 (17)

M(I)ads cells have, in contrast, two possible transi-

tions: one anodic, governed by reaction (2), which leads

to the dissolution of the cell with probability R2, and a

cathodic one*/reaction (1, backwards)*/in which thecell recovers the M state with probability R3. The overall

mean time associated with the transition out of this state

is then given by 1/(R2�/R3). Besides this, one has to

consider two further possibilities. If the M(I)ads cell

dissolves, no supplementary time has to be taken into

account. On the contrary, if the transition is cathodic

and the cell goes back to M, one must, as well as in thecase of a transition out of Cads, add t3, i.e. the mean time

corresponding to the future evolution of the recovered

M site. Finally, considering the relative probabilities of

undergoing an anodic (no extra time) or cathodic (t3

before disappearance) transition, we can write

t1��

1

R2 � R3

�(1�R3t3) (18)

Finally, an M cell is liable to undergo transitions to

two distinct states according to reactions (1, forward)

and (3), leading to M(I)ads and Cads with probabilities R1

and R4, respectively. This gives an average reaction time

for this cell of 1/(R1�/R4). Taking into account the

relative probabilities of transitions and the futureevolution of the resulting M(I)ads or Cads cells, as

presented before, we obtain:

t3��

1

R1 � R4

�(1�R1t1�R4t2) (19)

By solving Eqs. (17)�/(19), we obtain analyticalexpressions of ti as functions of the set of transition

probabilities {Ri}. This leads to the following relation:

t2�/t3�/t1, confirming the intuitively expected result

from our electrochemical scheme without incorporating

the harvesting action of ndc: a cell in state M(I)ads is

‘closer’ to dissolution than a cell in state M, which in its

turn is closer than a Cads site, independently of the

reaction probabilities.

4.2. The effect of non-dissolved clusters on lifetimes

Not surprisingly, the actual cell-lifetimes obtainedfrom simulations are smaller than those from Eqs. (17)�/

(19). This is because the electrochemical dissolution

process considered so far is not the unique mechanism of

cell extraction from the metal surface. We recall from [3]

that the coalescence of independent active dissolution

tunnels may disconnect clusters of non-dissolved cells

(ndc) from the metal bulk. This constitutes an additional

mechanism of mass loss concerning, at different levels, allcells belonging to the surface, and certainly alters the cell-

lifetimes computed in Section 4.1. In fact, this mechanism

is highly heterogeneous inasmuch as it is more likely to

take place at surface peaks than at valleys, but in Eqs.

(10)�/(13), as we did in [3], the corresponding contribution

has been averaged over the surface. At each time step, n ,

the number of surface cells collected by ndc belonging to a

given reactant, i , are counted, and the correspondingnumber divided by the actual surface length. This

procedure yields ji (n ), and ji (i�/1, 2, 3) in Eqs.

(10)�/(13) is defined as the time average (over n ) of ji(n ).


In order to obtain more reliable expressions for the ti

than those given by Eqs. (17)�/(19), extra probabilities

accounting for ndc cell extraction have to be incorporated

into the previous argument. We have not, at present, an

ab initio procedure for evaluating the statistical distribu-

tion of ndc and the ji contributions can be evaluated only

from the computer output. We need, however, to

complete our argument on lifetime information in terms

of the probability for a given surface cell of being

collected by an ndc. In [3] this problem was solved byassuming that each term ji was proportional to the

coverage fraction of the corresponding reactant, that is,

there exist three numbers Ri , such that j1�R6U1; j2�R7U2; j3�R8U3: These relationships constitute the

empirical definition for these three numbers and,

accordingly, define the procedure for their evaluation:

R6�j1

U1

; R7�j2

U2

and R8�j3

U3

(20)

where, R6, R7 and R8 are now interpreted as probabil-

ities of transitions between the state ‘belonging to the

surface’ and a new one ‘belonging to an ndc’ for M(I)ads,

Cads and M, respectively.

Eq. (20) is an approximate modeling of whatreally occurs on the surface because it supposes that

all cells in a given state have the same probability

of being collected by ndc. We have already pointed

out that this is not strictly true because for each cell

this probability increases with the age (or residence

time) of the site. The longer a cell has been attached

to the surface the higher is its chance of being

harvested by an ndc, because this cell is preferablylocated at surface peaks. In spite of this, the results

obtained prove that this effect is not very significant as

far as the general validity of what follows is concerned

and we shall accept Eq. (20) as a reasonable

approximation.

In the interface evolution, R6, R7 and R8 are

associated with irreversible transitions leading directly

to the disappearance of sites and, as such, must beincluded in the cell-lifetimes calculations. An argument

similar to that leading to Eqs. (17)�/(19) determines the

correction to their corresponding expressions. Skipping

details, we find:

t1��

1

R1 � R3 � R6

�(1�R3t3) (21a)

t2��

1

R5 � R7

�(1�R5t3) (21b)

t3��

1

R1 � R4 � R8

�(1�R1t1�R4t2) (21c)

The solution to Eq. (21) gives the final expressions ofti as a function of the transition probabilities:

from which expressions for t1 and t2 follow immediately

with Eq. (21).

The analysis of Figs. 1�/4 has brought to the forefront

of the discussion the idea of a connection between the

degree of roughness and the comparison of cell-lifetimes

of what we loosely termed old and new interface sites.As a matter of fact, these terms can be given a more

precise meaning if we associate the name ‘old’ to a site

already attached to the surface at any time once the

steady state has been attained, while the term ‘new’

identifies a site that is just being recruited at that

moment. It is forcibly an M cell and has not yet had

the opportunity of undergoing any transition, regardless

of how fast the transition might be. The lifetimes we areassigning to each type of site can be denoted as told and

tnew, respectively. told is the result of averaging the cell-

lifetime over all interface sites:

told�X3

i�1

Uiti (23)

while, tnew is logically given by t3.

4.3. Definition of the differential aging factor, Da

The relation between these two times, told and tnew,

can be an indicator of the way in which the interface

propagates into the metal bulk. As discussed before, if

tnew is larger than told, old cells are more prone to

disappear earlier. This has a smoothing effect on thesurface since protuberances are mainly formed by the

accumulation of old cells. If told is larger than tnew the

opposite happens, meaning that new M cells tend to be

very active sites that make notches in the metal bulk,

enhancing roughness. This is why we define the ratio

between these two times as the differential aging factor,

Da, and obtain an expression of it as function of ti :

Da�told

tnew

�

X3

i�1

Uiti

t3

(24)

Eq. (24) can be further expanded by introducing the

t3��

(R5 � R7)R1 � (R2 � R3 � R6)(R4 � R5 � R7)

(R5 � R7)(R1(R2 � R6)) � (R2 � R3 � R6)(R8(R5 � R7) � R4R7)

��

Y

W(22)


definitions made in (20) into the mesoscopic balance

equations Eqs. (10)�/(14) and obtaining expressions for

Ui :

U1�R1(R5 � R7)

(R1 � R2 � R3 � R6)(R4 � R5 � R7) � R1R4

(25)

U2�R4(R2 � R3 � R6)

(R1 � R2 � R3 � R6)(R4 � R5 � R7) � R1R4

(26)

Eq. (24) together with Eqs. (21)�/(22), (25) and (26)

deliver the final expression of Da as a function of the

whole set of transition probabilities:

where Y and W are the numerator and denominator of

the expression of t3 as defined in Eq. (22).

It is interesting to note that, although it can be seen as

a mostly kinetic parameter, Da incorporates informa-

tion about morphological features indirectly through

R6, R7 and R8.

5. Retrieving interface roughness from Da

Our aim now is to check the connection between the

degree of roughness of the surface and the recently

defined parameter Da. For that purpose, we start by

showing in Figs. 5 and 6 the behavior of Da with

varying concentrations of the two adsorbed species.

They are to be compared with Figs. 1 and 3, respec-tively. Da presents qualitatively the same features shown

by the box counting dimension for the whole range of

adsorbate coverage. This means that Da efficiently

describes, as well as dbc, the interface behavior according

to surface reactant occupation. Results in Figs. 5 and 6

seem then to indicate that Da can be, at least qualita-

tively, a reaction kinetics-based descriptor of interface

morphology.In order to confirm this suspicion we proceed to the

straightforward comparison of Da and different rough-

ness indicators. Fig. 7 shows its relationship with the

box-counting dimension dbc, the normalized mean

length L//L0 of the coastline and also x: The box-

counting dimension is considered for obvious reasons.

The so-called roughness parameter, L//L0, is still con-

sidered by many authors as a measure of roughness and,consequently, also included. For completion, x; which

was shown to be closely related to dbc in Ref. [3], is

displayed in a third panel. Values for Da were obtained

from Eq. (27) for a complete set of simulations (includ-

ing those represented in Figs. 5 and 6). The three

morphology descriptors increase monotonically with

Da. For reasons of consistency three significant cases

deserve to be highlighted.

A first important remark about results in Fig. 7

concerns the limit-case of purely random dissolution, as

introduced in Section 3.2. Values of dbc, L and x

obtained under these conditions have been indicated in

each plot (dotted line)*/it is worth recalling here the

value 1.36 for dbc, as obtained for cases (a), (b) and (d)

of Fig. 2. As expected, in all three cases, Da�/(told/

tnew)�/1, meaning that all surface cells, whether old or

new, have equal probability of undergoing dissolution.

We can also observe in Fig. 7 that the lower limit

attained in simulations is Da:/0.75. This is the result of a

process in which R30/0 or, equivalently, k�10/0, in (1),

while surface coverage fractions for Cads, M(I)ads and M,

tend to 0.0, 0.5 and 0.5, respectively. More will be said on

this value in the following section. What is interesting to

notice here is that a hypothetical prolongation of the

curves in Fig. 7 to Da values lower than 0.75 leads, in the

limit Da0/0, to dbc0/1 and L 0 L0; with L0 as the initial

surface length. It would be approached whenever tnew/

told, that is, if old cells disappeared much faster rate than

new cells did, and would display an evolving surface that

remained as smooth as it was initially, i.e. a layer by layer

dissolution. This makes the picture consistent, although

the limit itself, of course, cannot be reached in the present

model.

The third case that we addressed specifically is given

by the upper limit of the Da range. Values for the box-

counting dimension are close to dbc:/2, revealing the

existence of a convoluted surface profile that ‘fills’

densely the whole two-dimensional array (/L//L00/�).

We deal with it in Section 6.

The slight dispersions observed for increasing rough-

ness*/high values of dbc, L; x and Da*/do not

invalidate the assumption made on Da and can prob-

ably be attributed to the average approximation made in

Eq. (20), since the influence of ndc production on the

morphology is certainly more complex, mainly at high

levels of roughness.

Results presented in Figs. 5�/7 clearly confirm that,

not only qualitatively but also quantitatively, Da can be

used as a good kinetics-based descriptor of the mor-

phology of the interface. Moreover, Da provides a

better understanding of how the electrochemical steps

Da�W [R1(R5 � R7)2 � R4(R2 � R3 � R6)2]

Y 2(R5 � R7)(R2 � R3 � R6)�

R4R5(R2 � R3 � R6)2 � (R5 � R7)2[(R2 � R3 � R6)2 � R1R3]

Y (R5 � R7)(R2 � R3 � R6)(27)


of a model have an influence on the roughness of the

metallic surface.

6. The reaction-restricted differential aging factor

However, in spite of the importance of this compre-

hension, the mesoscopic treatment seems to be of

theoretical use only since it requires the knowledge of

some parameters (R6, R7 and R8) that are neither a

priori available nor for the time being measurable

quantities in experimental electrochemistry. In order toovercome this practical limitation we must look for a

simplified version of Eq. (27) in terms of experimentally

measurable reaction kinetics quantities exclusively.

The simplest way to do so is to proceed to a crude

approximation, which consists in disregarding contribu-

tions from ndc collection in the evaluation of cell-

lifetimes and verify if it fulfils our purpose. We start by

neglecting ndc contributions in balance Eqs. (10)�/(14)

and by replacing transition probabilities Ri by rate

constants ki of reactions (1)�/(4), which are in principle

obtainable from electrochemical experiments. Eqs.(14)�/(16), at steady-state corrosion potential, reduce to:

(k1�k�3)Uk

3 �(k2�k�1)Uk

1�k�4Uk

2 �0 (28)

Uk

1 �k1k�4

(k�3 � k�4)(k2 � k�1) � k1k�4

(29)

Uk

2 �k�3(k2 � k�1)

(k�3 � k�4)(k2 � k�1) � k1k�4

(30)

Fig. 6. Differential-aging factor (Da) vs. Cads mean interface coverage

fraction (/U2): Da values were calculated using Eq. (27) for all

simulations displayed in Fig. 3. The dotted line corresponds to the

pure random dissolution (Da�/1).

Fig. 5. Differential-aging factor (Da) vs. M(I)ads mean interface

coverage fraction (/U1): Da values were calculated using Eq. (27) for

all simulations displayed in Fig. 1. The dotted line corresponds to the

pure random dissolution (Da�/1).


Also, from Uk

3/�/1�//Uk

1/�//Uk

2 we get:

Uk

3 �k�4(k2 � k�1)

(k�3 � k�4)(k2 � k�1) � k1k�4

(31)

where the superscript k stands for this simplified

reaction-restricted treatment.

Eqs. (28)�/(31) are identical to the expressions that rise

from the standard macroscopic electrochemical ap-

proach (charge and mass balance equations) for theelectrochemical mechanism represented by reactions

(1)�/(4). Nevertheless, they are not to be taken here as

a ‘conceptual’ extension toward macroscopic systems of

the previous mesoscopic analysis, but instead as a

convenient result of a simplifying assumption on the

effects of a rough interface. Within this context, expres-

sions accounting for the electrochemical cell-lifetime, tik,

of surface reactants are obtained from Eqs. (17)�/(19)

tk1 �

�1

k2 � k�1

�(1�k�1t

k3)

tk2 �

1

k�4

�tk3

tk3 �

�1

k1 � k�3

�(1�k1t

k1�k�3t

k2) (32)

Once solved, they yield:

tk1 �

k�1k�3 � k�1k�4 � k1k�4

k1k2k�4

tk2 �

k�1k�3 � k�1k�4 � k1k�4 � k2k�3 � k2k�4 � k1k2

k1k2k�4

tk3 �

k�1k�3 � k�1k�4 � k1k�4 � k2k�3 � k2k�4

k1k2k�4

(33)

The corresponding cell-lifetime of an unidentified

surface cell toldk , is written:

tkold�

X3

i�1

Uk

i tki (34)

and, accordingly:

Dak�tk

old

tknew

�1�Uk

2

�tk

2 � tk3

tk3

��Uk

1

�tk

3 � tk1

tk3

�(35)

where, we have made use of Uk

3 �1�Uk

1 �Uk

2 :/We can rewrite Eq. (35) in a more convenient form by

involving the following relationship:

k�3(k2�k�1)�k1k2 (36)

easily deducible from Eqs. (28)�/(31). At the same time,

we also obtain from Eqs. (30), (33) and (36)�tk

2 � tk3

tk3

��Uk

2 (37)

and�tk

3 � tk1

tk3

��

k2

k2 � k�1

(1�Uk

1) (38)

from Eqs. (28) and (33).

Finally, by substituting Eqs. (37) and (38) in (35), we

obtain

Dak�1�(Uk

2)2�Uk

1(1�Uk

1)

�k2

k2 � k�1

�(39)

Fig. 7. Interface roughness descriptors (dbc, L//L0, x) vs. differential-

aging factor (Da). L is the time-average boundary length in the

stationary regime and L0 its initial value given by the width of the

automaton grid. The pure random dissolution case is shown as the

reference in each curve (dotted lines). Solid line in the dbc vs. Da and L//

L0 vs. Da panels is a hypothetical prolongation of data in order to

highlight the limit Da0/0, for which the profile is perfectly flat.

Representative points cover the (/U1; U2)/-parameter space in the range:

0.0015//U1; U2/5/0.998.


Eq. (39) shows in an analytical way that roughness,

represented by Dak, has a second order dependence on

Cads concentration that is in good agreement with the

general behavior of Figs. 3 and 6. It also reproduces thedependence on M(I)ads coverage presented in Figs. 1 and

5. Curves generated by Eq. (39) are plotted in Fig. 8a

and b for, as typical examples, the same range of values

of Figs. 1a and 3b, respectively. This emulates very well

the actual roughness behavior even at the limiting

conditions discussed in Fig. 2a and d, for which Eq.

(39) confirms that roughness is independent of the ratio

between transition probabilities*/here identified forconvenience to potentially measurable kinetic constants.

It is easily seen from Eq. (39) that, for Uk

1/�/0.5, Uk

2/�/

0 and k�1�/0, we reach the lowest possible value:

Dak�/0.75, which presumably corresponds to the

‘smoother’ profile*/lowest roughness*/obtainable

within the requirements of the present model. The

maximum values for Dak is 2 and is obtained when Uk

1/

�/0 and Uk

2/�/1. In simulations narrowly closing thisregime*/for R50/0, or k�40/0, in reaction (4)*/an

increasing number of cells become permanently trapped

in the Cads state. An ever decreasing population of active

dissolution sites is confined to an ever denser network of

burrows until the corrosion process finally stops. The

result displays the two-dimensional cellular automaton

grid densely covered by Cads.

All this is consistent with what has been said in former

sections and paves the way to a direct comparison between

Da and Dak, which we do in Fig. 9. In general, there is a

good quantitative agreement between both parameters.

Below the purely random dissolution characteristic values(Da�/Dak�/1), there is a fully matching equivalence

between the two descriptors. Beyond this crossover

deviations of at most 10% are to be associated to the

effects of ndc which are neglected in the Dak evaluation.

An interesting feature disclosed in Fig. 9 is that at very

high levels of roughness, there is a net tendency of those

deviations to vanish again. This complex behavior must be

investigated from an exhaustive description of ndcproduction and composition, which will be a matter

for a future publication concerning also segregation of

surface reactants. For the moment, and according to the

aims of this paper, it is worth stressing that results

presented in Figs. 8 and 9 clearly indicate that our

treatment is able to forecast the morphological features of

the interface from the knowledge of standard experi-

mental parameters in electrochemistry.

7. Concluding remarks

In the present paper, we have characterized the onset

of spatial statistical patterns in terms of the underlying

electrochemical kinetics. No diffusion is involved in the

process. This may be somewhat surprising inasmuch as

spatial pattern formation in a chemical context, whetherit be deterministic or statistical, is mostly linked to the

presence of diffusion. Sooner or later diffusion-con-

trolled processes appear to dominate most of the work

in the development of the theory of self-affine interfaces,

for example [16�/18].

We have studied in the present paper the formation of

interface patterns in a reaction-dominated system. The

term pattern applies here not in a regular but in astatistical sense, inasmuch as they have well-defined

statistical properties arising from a collection of inde-

pendent, random individual events. We have concluded

Fig. 8. Reaction-restricted differential-aging factor (Dak) calculated

from Eq. (39) vs.: (a) M(I)ads mean interface coverage fraction (/U1); (b)

Cads mean interface coverage fraction (/U2): Curves have been obtained

for the same parameter values as in Fig. 1a and Fig. 3b, respectively.

Fig. 9. Reaction-restricted differential-aging factor (Dak) vs. complete

differential-aging factor (Da) for the same set of simulations of Fig. 7.


that the generation of rough interface patterns is strongly

linked to time-dependent features of the independent

interface constituents, and it can be analyzed in terms of

the state transitions of an individual cell and its relativeposition with respect to the forward parts of the

advancing corrosion front. In an abuse of language,

we can say that a ‘differentiation in time’, when produced

in a proper arrangement, induces a ‘differentiation in

space’. In this paper, we have done nothing more than to

give a mathematical form to this simple idea.

This ‘differentiation in time’ is obviously the sole

responsibility of the working of the electrochemicalmechanism. Here, the additional mechanism of ndc

collection does not substantially alter the reasoning. The

direction of progress of the corrosion front (downwards

in our automaton) breaks the grid symmetry by

distinguishing a ‘rear’ from a ‘front’ in the corroding

surface. We have seen that an unimpeded passage of the

metal into dissolution favors the average distribution of

sites that are soon to dissolve at the ‘rear’ while sites atthe ‘front’ take longer to do it. This particular arrange-

ment of sites tends to produce relatively smooth

interfaces. The converse happens if there is some

obstruction in the mechanisms that traps sites for a

long time. Long-lived, trapped sites remain trailing at

the ‘rear’ forming long metal protrusions while dissolu-

tion is preferably taking place at the ‘front’. Consistently

with the automaton rules, were it not for the ndcchopping action these protrusions would attain huge

proportions in some cases. We believe that the conclu-

sion, we have just reached does not depend on the

specific mechanism at hand in the present paper and we

thus infer that our mathematical modeling could be

generalized to different dissolution mechanisms. Never-

theless, the statement remains to be proved and this is

left for a future publication.

8. List of symbols

Cads cation adsorbate site (or cell)

Da differential aging factor

dbc box-counting dimension

/J/ dimensionless mean current density

K superscript standing for the simplified reaction-

restricted treatment

KI dimensionless rate constants of electrochemical

reactions/L/ steady-state length of the corrosion front

L0 initial length of the corrosion front

M metal site (or cell)

M(I)ads intermediate adsorbate site (or cell)

Ndc non dissolved clusters

RI probabilities of state transitions

/Ui/ averaged steady-state coverage fraction of sur-

face reactants

/ji/ averaged ndc loss rate of each surface reactants

tI average time taken by a site in state i todisappear into the electrolyte

told mean lifetime of sites already belonging to the

surface at step n

tnew mean lifetime of sites reaching the surface at

step n

/x/ mean number of newly exposed M sites after

one cell dissolution

Acknowledgements

P.C.-T. acknowledges a predoctoral fellowship from

The Ministerio de Educacion y Cultura, Spain. R.P.N.

has enjoyed a postdoctoral fellowship from The Vice-

rrectorado de Investigacion, UNED, Spain, undercontract 2001V/PRYT/19�/I�/D. Finally, the authors

wish to express their thanks to K. Bar-Eli for his fruitful

discussions.

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Forecasting interface roughness from kinetic parameters of corrosion mechanisms

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