Effects of porosity in a model of corrosion and passive layer growth
We introduce a stochastic lattice model to investigate the effects of pore formation in a passive layer grown with products of metal corrosion. It considers that an anionic species diffuses across that layer and reacts at the corrosion front (metal-oxide interface), producing a random distribution...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Effects of porosity in a model of corrosion and passive layer growth / F.D.A. Aarão Reis // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33803: 1–8. — Бібліогр.: 38 назв. — англ. |
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F.D.A. Aarão Reis 2019-06-19T12:43:20Z 2019-06-19T12:43:20Z 2017 Effects of porosity in a model of corrosion and passive layer growth / F.D.A. Aarão Reis // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33803: 1–8. — Бібліогр.: 38 назв. — англ. 1607-324X PACS: 82.45.Bb, 05.40.-a, 68.35.Ct DOI:10.5488/CMP.20.33803 arXiv:1710.01548 https://nasplib.isofts.kiev.ua/handle/123456789/157017 We introduce a stochastic lattice model to investigate the effects of pore formation in a passive layer grown with products of metal corrosion. It considers that an anionic species diffuses across that layer and reacts at the corrosion front (metal-oxide interface), producing a random distribution of compact regions and large pores, respectively represented by O (oxide) and P (pore) sites. O sites are assumed to have very small pores, so that the fraction Φ of P sites is an estimate of the porosity, and the ratio between anion diffusion coefficients in those regions is Dr < 1. Simulation results without the large pores (Φ = 0) are similar to those of a formerly studied model of corrosion and passivation and are explained by a scaling approach. If Φ > 0 and Dr 1, significant changes are observed in passive layer growth and corrosion front roughness. For small Φ, a slowdown of the growth rate is observed, which is interpreted as a consequence of the confinement of anions in isolated pores for long times. However, the presence of large pores near the corrosion front increases the frequency of reactions at those regions, which leads to an increase in the roughness of that front. This model may be a first step to represent defects in a passive layer which favor pitting corrosion. Представлено стохастичну ґраткову модель для вивчення ефектiв формування пор у пасивному шарi, який вирощується з продуктами корозiї металу. Вважається, що анiони дифундують через цей шар i реагують на корозiйнiй межi (межi роздiлу металу та оксиду), утворюючи випадковий розподiл компактних областей i великих пор, якi вiдповiдають вузлам видiв О (оксид) i Р (пора). Припускається, що вузли виду О мають дуже малi пори, тобто частка Φ вузлiв виду P приблизно вiдповiдає пористостi, а спiввiдношення коефiцiєнтiв дифузiї анiонiв у цих областях Dr < 1. Результати моделювання для випадку вiдсутностi великих пор (Φ = 0) подiбнi до результатiв для попередньої моделi корозiї та пасивацiї i пояснюються масштабним пiдходом. При Φ > 0 i Dr 1 спостерiгаються суттєвi змiни в ростi пасивного шару i жорсткостi корозiйної межi. Для малих Φ зауважено сповiльнення швидкостi росту, яке можна пояснити тривалим обмеженням руху анiонiв в iзольованих порах. Однак присутнiсть великих пор поблизу корозiйної межi збiльшує частоту реакцiй в цих областях, приводячи до збiльшення жорсткостi межi. Дану модель можна розглядати як перший крок при описi дефектiв у пасивному шарi, якi сприяють герметичнiй корозiї. This work is dedicated to the memory of my friend and scientific collaborator Jean-Pierre Badiali, who introduced me to the area of corrosion modelling. This work was supported by CNPq and Faperj (Brazilian agencies). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Effects of porosity in a model of corrosion and passive layer growth Ефекти пористостi в моделi корозiї та росту пасивного шару Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Effects of porosity in a model of corrosion and passive layer growth |
| spellingShingle |
Effects of porosity in a model of corrosion and passive layer growth F.D.A. Aarão Reis |
| title_short |
Effects of porosity in a model of corrosion and passive layer growth |
| title_full |
Effects of porosity in a model of corrosion and passive layer growth |
| title_fullStr |
Effects of porosity in a model of corrosion and passive layer growth |
| title_full_unstemmed |
Effects of porosity in a model of corrosion and passive layer growth |
| title_sort |
effects of porosity in a model of corrosion and passive layer growth |
| author |
F.D.A. Aarão Reis |
| author_facet |
F.D.A. Aarão Reis |
| publishDate |
2017 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Ефекти пористостi в моделi корозiї та росту пасивного шару |
| description |
We introduce a stochastic lattice model to investigate the effects of pore formation in a passive layer grown
with products of metal corrosion. It considers that an anionic species diffuses across that layer and reacts
at the corrosion front (metal-oxide interface), producing a random distribution of compact regions and large
pores, respectively represented by O (oxide) and P (pore) sites. O sites are assumed to have very small pores,
so that the fraction Φ of P sites is an estimate of the porosity, and the ratio between anion diffusion coefficients
in those regions is Dr < 1. Simulation results without the large pores (Φ = 0) are similar to those of a
formerly studied model of corrosion and passivation and are explained by a scaling approach. If Φ > 0 and
Dr 1, significant changes are observed in passive layer growth and corrosion front roughness. For small Φ,
a slowdown of the growth rate is observed, which is interpreted as a consequence of the confinement of anions
in isolated pores for long times. However, the presence of large pores near the corrosion front increases the
frequency of reactions at those regions, which leads to an increase in the roughness of that front. This model
may be a first step to represent defects in a passive layer which favor pitting corrosion.
Представлено стохастичну ґраткову модель для вивчення ефектiв формування пор у пасивному шарi,
який вирощується з продуктами корозiї металу. Вважається, що анiони дифундують через цей шар i реагують на корозiйнiй межi (межi роздiлу металу та оксиду), утворюючи випадковий розподiл компактних
областей i великих пор, якi вiдповiдають вузлам видiв О (оксид) i Р (пора). Припускається, що вузли виду
О мають дуже малi пори, тобто частка Φ вузлiв виду P приблизно вiдповiдає пористостi, а спiввiдношення коефiцiєнтiв дифузiї анiонiв у цих областях Dr < 1. Результати моделювання для випадку вiдсутностi
великих пор (Φ = 0) подiбнi до результатiв для попередньої моделi корозiї та пасивацiї i пояснюються
масштабним пiдходом. При Φ > 0 i Dr 1 спостерiгаються суттєвi змiни в ростi пасивного шару i
жорсткостi корозiйної межi. Для малих Φ зауважено сповiльнення швидкостi росту, яке можна пояснити
тривалим обмеженням руху анiонiв в iзольованих порах. Однак присутнiсть великих пор поблизу корозiйної межi збiльшує частоту реакцiй в цих областях, приводячи до збiльшення жорсткостi межi. Дану
модель можна розглядати як перший крок при описi дефектiв у пасивному шарi, якi сприяють герметичнiй корозiї.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/157017 |
| citation_txt |
Effects of porosity in a model of corrosion and passive layer growth / F.D.A. Aarão Reis // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33803: 1–8. — Бібліогр.: 38 назв. — англ. |
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AT fdaaaraoreis effectsofporosityinamodelofcorrosionandpassivelayergrowth AT fdaaaraoreis efektiporistostivmodelikoroziítarostupasivnogošaru |
| first_indexed |
2025-11-24T18:50:51Z |
| last_indexed |
2025-11-24T18:50:51Z |
| _version_ |
1850486981832736768 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 3, 33803: 1–8
DOI: 10.5488/CMP.20.33803
http://www.icmp.lviv.ua/journal
Effects of porosity in a model of corrosion and
passive layer growth
F.D.A. Aarão Reis∗
Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói RJ, Brazil
Received April 24, 2017, in final form June 23, 2017
We introduce a stochastic lattice model to investigate the effects of pore formation in a passive layer grown
with products of metal corrosion. It considers that an anionic species diffuses across that layer and reacts
at the corrosion front (metal-oxide interface), producing a random distribution of compact regions and large
pores, respectively represented by O (oxide) and P (pore) sites. O sites are assumed to have very small pores,
so that the fractionΦ of P sites is an estimate of the porosity, and the ratio between anion diffusion coefficients
in those regions is Dr < 1. Simulation results without the large pores (Φ = 0) are similar to those of a
formerly studied model of corrosion and passivation and are explained by a scaling approach. If Φ > 0 and
Dr � 1, significant changes are observed in passive layer growth and corrosion front roughness. For small Φ,
a slowdown of the growth rate is observed, which is interpreted as a consequence of the confinement of anions
in isolated pores for long times. However, the presence of large pores near the corrosion front increases the
frequency of reactions at those regions, which leads to an increase in the roughness of that front. This model
may be a first step to represent defects in a passive layer which favor pitting corrosion.
Key words: stochastic model, corrosion, passivation, diffusion, porosity, roughness
PACS: 82.45.Bb, 05.40.-a, 68.35.Ct
1. Introduction
If a metal is in contact with an aggressive solution, a corrosion process begins and leads to the
formation of a layer of oxide or hydroxide which protects the metal. However, the corrosion process
continues due to the transport of ions across that passive layer. Figure 1 illustrates this process. In
conditions where dissolution of the passive layer is very slow, its thickness increases in time and the
corrosion rate decreases. Several models were already proposed for this process and were successfully
applied to describe experiments with various materials [1–4]. Stochastic models in lattices (also called
cellular automata or kinetic Monte Carlo) were also studied by several authors to represent coarse grained
features of passive layers, such as temporal and spatial scaling of the thickness and of the roughness of
the interfaces with the metal (corrosion front; see figure 1) and the solution [5–12].
Reference [10] proposed a model in which diffusion of anions was the main transport mechanism
and in which reactions occurred at the corrosion front. This model implicitly assumed that the passive
layer had pores where ion transport is possible, but that layer was considered homogeneous. A scaling
approach predicted a crossover between an initial regime with constant velocity of the corrosion front
and rapid roughening (Kardar-Parisi-Zhang (KPZ) scaling [13]) and a long time regime with parabolic
(t1/2) displacement and slow roughening (diffusion-limited erosion (DLE) scaling [14]). Recently, refer-
ence [15] extended that model to represent the dissolution-reprecipitation mechanism [16] that leads to
the formation of an altered layer during the weathering of a mineral.
In this work, we introduce a corrosion-passivation model similar to that of [10] but considering an
inhomogeneous passive layer, with random distribution of compact regions and large pores. Such pores
∗E-mail: reis@if.uff.br
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33803-1
https://doi.org/10.5488/CMP.20.33803
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
F.D.A. Aarão Reis
O−−
OH−
O2H
O
−
−−
H
H+
+OH
M+n (film)
M+n(film) M+m(soln)
Metal Passive layer Solution
Corrosion front
M
Figure 1. (Color online) Scheme of the phases in the corrosion process and transport of ions in the passive
layer. The advance of the corrosion front occurs where those ions react, leading to a growth of the passive
layer at the regions indicated with dashed curves.
may be filled with solution, act as traps for the moving anions, and eventually form preferential paths for
diffusion. On the other hand, diffusion in the remaining parts of the layer (called the compact region) is
assumed to be much slower. Simulations of the model in two dimensions (square lattice) show that the
thickening of the passive layer and the roughning of the corrosion front depend on the porosity and on
the ratio of diffusion coefficients in the compact regions and in the large pores. For small values of that
ratio, a remarkable decrease in the growth rate may be observed for small porosity, while the roughness
of the corrosion front increases.
The defects of the oxide or hydroxide layer formed during metal corrosion play an important role in
pitting corrosion [17]; for instance, they facilitate the transport of aggressive anions from the electrolyte
to the metal surface and increase the frequency of metastable pitting events [18]. This model of an
inhomogeneous passive layer is a first step to represent such defects and determine their possible effects.
The rest of this paper is organized as follows. In section 2, we present the model and briefly discuss
the relations with other models of passive layer growth. In section 3, we analyze the time evolution of
the thickness of the passive layer. In section 4, we analyze the time evolution of the roughness of the
corrosion front. In section 5, we summarize our results and present our conclusions.
2. The model of corrosion and passive film growth
The model is defined in a square lattice and the lattice constant is taken as the length unit. Each site is
expected to represent a homogeneous region with several atoms or molecules. Thus, the lattice constant
corresponds to some nanometers or larger sizes in possible applications. The sites may be of four types,
as shown in figure 2 (a): metal (M), oxide (O), solution (S) at z 6 0, and pore (P) at z > 0.
The configuration at time t = 0 has solution at all points with z 6 0 and metal at all points at z > 0,
as shown in figure 2 (b). The passive layer is produced at z > 0 by continuous transformation of M sites
into O or P sites. O sites represent regions with very small pores, which permit slow ion transport but
which have a negligible contribution to the total porosity; the set of O sites is called the compact region
of the layer. P sites represent larger pores and their volume fraction is assumed to be equal to the total
porosity.
We assume that an anionic species is produced in electrochemical water decomposition at the solid-
solution interface z = 0. This species is represented by a mobile particle A that can occupy any lattice
site; one particle A is shown in the initial configuration of figure 2 (b). At each step of the corrosion-
passivation process, one particle A is released at a randomly chosen S site with z = 0, executes a random
walk on sites O and P, and eventually reacts with a site M to produce a new O or P site. A new particle A
is released only after the previous one has reacted; thus, a single particle A is present in the lattice at each
time.
The random walk of particle A considers a Moore neighbourhood [19], in which nearest neighbor
(NN) and next nearest neighbor (NNN) sites are randomly chosen for each step, as illustrated in figure 2 (c)
(yellow sites are used to represent any type of lattice site in that figure). The chosen neighbor is called
the target site. Possible outcomes of the step attempt are illustrated in figures 2 (d)–(f). If the target is an
S site, then A does not move because the model does not describe transport in solution (for this reason,
33803-2
Porosity of a passive layer
Large pore
Solution (b)
0
z
x
(c)
prob. 1(d)
prob.
prob. Dr
1−Dr
(f)
prob. p prob.
prob.
Φ
1−Φprob.
1−p
(a) Metal
Oxide
(e)
Figure 2. (Color online) (a) Four types of the lattice sites. (b) Initial configuration of the lattice with a
particle A (red circle) at its initial position in the line z = 0. (c) Possible directions of step attempts of the
particle A. Yellow color is used here for sites that may be of any type. (d)–(f) Step attempts of a particle
A with target P, O, and M sites, respectively. The step probability does not depend on the current site
(yellow) of the particle A.
such process is not shown in figure 2). If the target site is P [figure 2 (d)], then the step is executed with
probability 1. If the target site is O [figure 2 (e)], then the step is executed with probability Dr; we consider
Dr < 1, which represents a slower diffusion in the compact regions in comparison with the diffusion in
the large pores. If the target site is M [figure 2 (f)], then a reaction occurs with probability p; the overall
consequence of the reaction is represented by the annihilation of particle A and the transformation of the
M site into an O site (with probability 1 − Φ) or into a P site (with probability Φ).
The stochastic rules of the model lead to the formation of a passive layer with a fraction Φ of P sites.
This is the value of porosity because we assumed that pores of regions represented by the O sites are
very small. The mechanism of pore formation considered here is an oversimplified description of a much
more complex process. In a real corrosion-passivation process, we expect an increase in the volume of
the oxide or hydroxide film after reactions, which generates stress that may lead to rearrangement of
molecules. However, the aim of our model is not to represent such details, but to investigate the effect of
inhomogeneity in the ion transport on the layer growth and on the morphology of the corrosion front.
The parameter Dr is a simple form of representing that inhomogeneity and may be interpreted as a
ratio of diffusion coefficients. However, it is certainly a difficult quantity to be measured experimentally
because it would be necessary to measure two diffusion coefficients in very small homogeneous regions
of a passive layer.
We performed simulations of the model for several values of the parameters p, Dr, andΦ using lattices
of lateral size L = 512 sites. Average quantities were obtained considering 100 different realizations of the
corrosion-passivation process for each parameter set; this is necessary to reduce the noise effects, since a
single realization does not represent all the relevant microscopic environments. The maximal thickness
of the passive layer studied here is Hmax = 400, although for some parameter sets the simulations were
restricted to Hmax = 100. The simulations were run in a dual Xeon E5520 computer with Linux for
approximately six days.
In the implementation of our model, a new particle A is released at z = 0 only after the previous
particle A has reacted. This implies that the calculation of the time of layer growth is not as direct as in
33803-3
F.D.A. Aarão Reis
the model of [10], in which the time scale was set by the rates of the diffusion and reaction processes.
Here, we separate particles A in sets of L particles that are consecutively released (an average of one
particle per column x) and assume that all particles of the i-th set left their initial positions at time iδt,
with i = 0, . . . ,Hmax − 1. The time interval δt is taken as 1/Dr, which is the average time for a particle
A to jump to a neighboring O site; this mimics a sequential release of particles A at each column. For
each particle A, the transport time is computed as the total number of step trials until reacting with an
M site. The average time for the thickness to increase from H = i to H = i + 1 is then calculated as the
average of release time plus transport time in the i-th set of particles A. As the passive layer thickens, the
transport times become much larger than the release times, and consequently the former gives the main
contribution to the total growth time.
3. Growth of the passive layer
The thickness of the passive layer is shown in figures 3 (a) and 3 (b) as a function of time for Dr = 0.1
and Dr = 0.01, respectively, and for several values of the porosity Φ, with reaction probability p = 1. At
long times, the results for smaller values of p are similar, as shown in the inset of figure 3(a).
In all cases, a diffusive displacement of the corrosion front is observed at long times:
H ≈ DCFt1/2. (1)
Here, DCF has a dimension of a diffusion coefficient and measures how fast the corrosion front moves.
Alternatively, a corrosion rate may be defined as the ratio H/t, but this is a time decreasing quantity. For
φ = 0, we obtain DCF ∼ Dr; this is the case of a compact layer, in which our model is equivalent to the
long time limit of the model introduced in [10]; indeed, that model predicts the proportionality between
DCF and Dr. As Φ increases, different trends are observed for Dr = 0.1 and Dr = 0.01: in the former,
the corrosion rate at a given time increases with Φ, although for small Φ this increase is very small; in
the latter, a nonmonotonous variation of that rate is observed, with a decrease for low Φ and subsequent
increase.
The estimates of DCF/Dr are suitable to characterize those trends. Figure 4 shows DCF/Dr as a
function of the porosity Φ, for two values of Dr and p = 1. For Dr = 0.1, DCF has negligible changes for
Φ 6 0.1, but increases for large Φ. The sets of neighboring P sites may form paths where the particles A
can move faster, but isolated P sites have an opposite effect and confine them for long times; when the
particle A reaches an isolated P site, it takes a long time to move to a neighboring O site and there is
Figure 3. (Color online) Thickness of the passive layer as a function of time for the parameters indicated
in the plots. In (a), the inset compares data for different values of p.
33803-4
Porosity of a passive layer
Figure 4. (Color online) Estimates of DCF/Dr as a function ofΦ for Dr = 0.1 (red squares) and Dr = 0.01
(green triangles). In both cases p = 1.0.
a large probability that it returns to the previous P site; see step probabilities in figure 2. These effects
are balanced for small Φ, but an increase in the porosity eventually leads to an increase of DCF. The
relation between DCF and Dr is consequently nonlinear due to the complex pore connectivity, which is
well known from percolation theory [20]. For Dr = 0.01, a small porosity has a much more drastic effect:
DCF decreases by a factor ≈ 4 in comparison with the compact oxide layer for Φ = 5% and Φ = 10%.
This occurs due to the trapping of the particles A in isolated pores for long times. Dr begins to increase
for larger porosity because longer channels of P sites appear and compensate that effect.
A diffusive displacement of the corrosion front [equation (1)] was already observed in the oxidation
of iron or iron nitride in different conditions [21–24]; however, in some cases it was shown that the rate
limiting process was a diffusion of iron cations [23, 24] and was not a diffusion of anionic species, as
proposed in our model. The diffusive law was also obtained in several models, e.g., those of [25–27].
However, other models and experimental observations suggest a slower growth of passive layers as a
consequence of additional energy barriers for ion diffusion or dissolution of that layer [1].
The results presented here show that porosity of the passive layer has a nontrivial effect on ion
diffusion. Such effects may also appear in systems with additional energy barriers if the transport in large
pores is much faster than that in the more compact regions of the passive layer.
4. Roughness of the corrosion front
The corrosion front may have overhangs because all exposed M sites are subject to react with
particles A, but to calculate the roughness it is useful to define a single-valued interface {h}. At each
position x (column x), h is the topmost position z of an M site in that column; the orientation of the z
axis is shown in figure 2 (b). The height h is the minimum value of z for an M site in that column.
The roughness W (t) of the corrosion front is defined as the root mean square fluctuation of the
interface {h} at time t. Here, we analyze the variation of W as a function of the thickness H instead of
the time t because this approach helps the interpretation of the results in the light of kinetic roughening
theory [28, 29], which is mostly based on models of interfaces moving with constant velocity.
Figure 5 shows the square roughness as a function of H for compact films (Φ = 0), two values of Dr,
and two values of p. The semilogarithmic plot was useful to highlight the scaling as
W2 ∼ ln H. (2)
This slow roughening appears because the peaks of the corrosion front (i.e., the most prominent M sites)
33803-5
F.D.A. Aarão Reis
Figure 5. (Color online) Roughness of the corrosion
front as a function of its average displacement, for
compact layers grown with the values of Dr and p
indicated in the plot.
Figure 6. (Color online) Roughness of the corrosion
front as a function of the passive layer thickness for
the values of the parameters indicated in the plot.
The dashed line has slope 0.15.
have a larger probability of reacting with the incident particles A than the valleys of that interface. In
these conditions, overhangs in the corrosion front are very rare.
The scaling in equation (2) is explained by the equivalence between our model with p = 1 and Φ = 0
and the DLE, in which equation (2) is predicted in two dimensions [14]. Such scaling was also observed
in the models of passive layer growth of [9, 10]. Figure 5 also shows the saturation of the roughness at
short times, which is expected due to the small value z = 1 of the dynamical exponent of DLE [14]. In
three-dimensions, the DLE theory predicts a finite and small interface roughness at all times, which is
consistent with observations of altered layers in mineral weathering [15].
Figure 5 shows that the roughness obtained for Dr = 0.1 and p = 0.1 is much larger than the roughness
obtained with other model parameters (although the absolute value of W is still very small). This is a
case in which the reaction rate is small compared to the rate of diffusion, so that the particle A scans a
large region of the metal-oxide interface before reacting. In this case, [10] predicts that the roughening
at short times is similar to that of the Eden model [30, 31], in which all exposed M sites have the same
probability to react. The roughening of the Eden model is in the class of the KPZ equation [13], which
gives W ∼ H1/3 in two dimensions. Indeed, for Dr = 0.1 and p = 0.1, an initial rapid roughening is
observed in figure 5 and, for H ≈ 3, a crossover to the logarithmic scaling of equation (2) is observed.
Reference [10] considered much smaller values of p for compact layers, which showed crossovers at
much larger thicknesses.
In figure 6 we show W as a function of H for the porous films and for one of the compact deposits.
For Dr = 0.1, the presence of porosity leads to an increase in the roughness. Again, the reduction of the
reaction probability p also contributes to the increase of the roughness. However, for Dr = 0.01, a much
larger change in the roughness of porous layers is observed. For small Dr, the passive layer is highly
inhomogeneous to the diffusion of particles A, which leads to their confinement at the P sites for long
times, as discussed above. This facilitates reactions at the M sites localized in regions with larger local
concentrations of P sites. The more rapid growth of the front at those regions leads to an increase in the
roughness.
Figure 6 shows that the roughness increases approximately as W ∼ H0.15 in the thickness range
60 6 H 6 400 for Dr = 0.01. This scaling relation differs from those obtained in the most frequently
studied models of interface growth [28, 29]. Possibly this occurs because the simulated thicknesses are
relatively small and a crossover to another scaling will appear at longer times. However, such investigation
is beyond the scope of the present work due to the much longer simulation times that would be required.
33803-6
Porosity of a passive layer
5. Conclusion
We studied a model of metal corrosion and growth of a passive layer in which an anionic species
diffuses across that layer and reacts at the metal-oxide interface. The model for a homogeneous layer is
similar to that of [10]. Here, we advanced over that work by considering that large pores may be formed
after the reactions and that the diffusion coefficient of anions in those pores is larger than that in the
compact regions. The total porosity of such inhomogeneous layers is assumed to be dominated by the
volume fraction of the large pores.
When the ratio between diffusion coefficients in compact regions and in large pores is small, significant
effects of the porosity are observed. For small porosity, a slowdown of the passive layer growth is observed
because anions are frequently trapped in isolated pores; for the ratio 10−2 considered in our simulations,
this occurs up to 20% of porosity. On the other hand, the roughness of the corrosion front increases
with the porosity because the random distribution of pores near that front increases the effective noise
amplitude of the roughening process.
The inhomogeneity in our model of passive layer is completely random, but even in this case it leads
to an increase in the roughness of the corrosion front. This result is consistent with the hypothesis that the
defects of an oxide/hydroxide film formed on a metal are preferential points for nucleation of metastable
pits [18], i.e., for the initiation of a localized rapid corrosion which is followed by repassivation. It is
important to recall that other types of random disorder or surface inhomogeneity in corrosion models
are also associated with an increase of roughness of corrosion fronts [32–35] or with the changes in the
morphology of corrosion pits [36–38]. For these reasons, we believe that an improvement of the present
model may be useful for a quantitative description of the phenomena related to corrosion and passivation.
Acknowledgements
This work is dedicated to the memory of my friend and scientific collaborator Jean-Pierre Badiali,
who introduced me to the area of corrosion modelling.
This work was supported by CNPq and Faperj (Brazilian agencies).
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Ефекти пористостi в моделi корозiї та росту пасивного шару
Ф.Д.А. Аарао Райс
Iнститут фiзики, Федеральний унiверситет Флумiненс, Авеню Лiторанеа, 24210-340 Нiтерой
Рiо-де-Жанейро, Бразилiя
Представлено стохастичну ґраткову модель для вивчення ефектiв формування пор у пасивному шарi,
який вирощується з продуктами корозiї металу. Вважається, що анiони дифундують через цей шар i ре-
агують на корозiйнiй межi (межi роздiлу металу та оксиду), утворюючи випадковий розподiл компактних
областей i великих пор, якi вiдповiдають вузлам видiв О (оксид) i Р (пора). Припускається, що вузли виду
О мають дуже малi пори, тобто частка Φ вузлiв виду P приблизно вiдповiдає пористостi, а спiввiдношен-
ня коефiцiєнтiв дифузiї анiонiв у цих областях Dr < 1. Результати моделювання для випадку вiдсутностi
великих пор (Φ = 0) подiбнi до результатiв для попередньої моделi корозiї та пасивацiї i пояснюються
масштабним пiдходом. При Φ > 0 i Dr � 1 спостерiгаються суттєвi змiни в ростi пасивного шару i
жорсткостi корозiйної межi. Для малих Φ зауважено сповiльнення швидкостi росту, яке можна пояснити
тривалим обмеженням руху анiонiв в iзольованих порах. Однак присутнiсть великих пор поблизу коро-
зiйної межi збiльшує частоту реакцiй в цих областях, приводячи до збiльшення жорсткостi межi. Дану
модель можна розглядати як перший крок при описi дефектiв у пасивному шарi, якi сприяють гермети-
чнiй корозiї.
Ключовi слова: стохастична модель, корозiя, пасивацiя, дифузiя, пористiсть, жорсткiсть
33803-8
https://doi.org/10.1007/s11581-013-0953-7
http://www.ams.org/notices/200302/fea-gray.pdf
https://doi.org/10.1149/1.2428232
https://doi.org/10.1080/18811248.1989.9734313
https://doi.org/10.1007/BF01675263
https://doi.org/10.1016/S0040-6090(99)00377-6
https://doi.org/10.4028/www.scientific.net/MSF.461-464.481
https://doi.org/10.1016/j.corsci.2008.06.004
https://doi.org/10.1007/s10008-012-1935-9
https://doi.org/10.1080/00018739700101498
http://projecteuclid.org/euclid.bsmsp/1200512888
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https://doi.org/10.1103/PhysRevB.75.115405
https://doi.org/10.1016/j.electacta.2008.01.075
https://doi.org/10.1103/PhysRevE.79.041604
https://doi.org/10.1016/j.corsci.2016.07.028
https://doi.org/10.1103/PhysRevE.91.022403
https://doi.org/10.1149/2.0521602jes
https://doi.org/10.1016/j.corsci.2015.11.034
Introduction
The model of corrosion and passive film growth
Growth of the passive layer
Roughness of the corrosion front
Conclusion
|