Marcus P. Corrosion mechanisms in theory and practice
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4. Contribution of first-order and higher order nearest neighbors to the rate of
dissolution of A (soluble atoms) leading to a high-density percolation problem
Computer simulations with a Monte Carlo algorithm of the selective dissolution
of 2D square and 3D cubic lattices modeling binary alloys (A-B) are performed.
Dealloying Thresholds: Dealloying thresholds p* are defined by the value of p,
the fractional occupancy by A-type atoms, at which vacancy percolation occurs; i.e.,
continuous dissolution pathways start penetrating all the way through the lattice. Of
course, in the absence of any surface diffusion, vacancy percolation maps closely the
percolation of A atoms in the initial lattice. This is seen in Table 2, where no
diffusion or diffusion restricted to atoms B with zero or one first nearest neighbor
(lines 1 and 3) generates p* close to the theoretical percolation threshold on 2D
and 3D lattices (0.6 and 0.31, respectively). On the contrary, easy diffusion tends
to produce smooth front and layer-by-layer dealloying on 3D lattices as indicated
by p* close to the 2D percolation threshold (lines 5 and 6). This likely explanation
of p* larger than p
c
was questioned by in situ scanning tunneling microscopy
(STM) demonstrating that Ag atoms dissolve from terraces well below the 3D
percolation threshold [200].
Morphology of Dealloyed Structures on 2-D Lattices The roughness of the
dissolution front is characterized by its fractal dimension, d (estimated from the
mass theorem [190]). Variation of d as a function of p is shown in Figure 31,
displaying the classical transition toward the dimension of the infinite cluster at
the percolation threshold (d ≅ 1.60).
Easier diffusion of B produces features such as a smoother dissolution front,
internal vacancy clusters (polyatomic voids), and islands of A-type atoms hindered
from dissolution. Qualitatively similar conclusions are drawn on 3D lattices except
for the specific generation of pores with easier diffusion of B atoms as predicted [201]
by a nonstochastic approach. This tendency to generate a tunneling attack at the cost
of only surface diffusion could be considered as a likely explanation of pit
nucleation at the atomic level, with no need for the concept of passive film breakdown.
Kinetics of Dealloying and Structure of the Dealloyed Layer The dealloying
kinetics are represented by the time dependence of the flux of creation of vacancies,
available from the computer data. The results are compared with the prediction by
percolation theory, below and above the percolation threshold p*. Below p*, near-
exponential decay of the current with time is predicted in agreement with experiments.
Above p*, experiments show a decay much steeper than expected, attributed to
ohmic and mass transport effects in the porous interface. No mention is made of
the negative power law decay reported in Ref. 196 [Eq. (56)].
The Critical Potential E
c
The nature of the critical potential, (see Fig. 30)
was addressed in the framework of this model. No definite conclusions are
established. E
c
may be the balance point between the smoothing action of surface
diffusion and the roughening action of dissolution. Another explanation regards E
c
as a threshold potential above which percolation becomes possible within a
connected subset of A-type atoms with lowered reactivity due to their atomic
surrounding (high-density percolation problem).
156 Keddam
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Anodic Dissolution 157
Table 2 Summary of the Dissolution and Diffusion Rules Used in 2-D and 3-D Computer Simulations
Source: Ref. 32.
Copyright © 2002 Marcel Dekker, Inc.
The use of a percolation approach clarifies substantially the selective disso-
lution of single-phase binary alloys. Computer simulations support the existence of a
tight relationship between dealloying thresholds and site percolation thresholds.
However, a careful survey of the literature over the last 10 years leads to the striking
constatation that this important piece of work is totally ignored by other groups
active in the field. It should be emphasized that a significant dissolution of the noble
metal is repeatedly mentioned, e.g., Pd in Ag-Pd alloys [172], and is not allowed
in the percolation model of selective dissolution. Dissolution of both alloy components
at different rates is taken into account in the model of passivation of Fe-Cr alloys
[184,192], presented in the following.
Curvature Effects, Roughening Transition, and Critical Potential Following
the results of Monte Carlo simulations, the major contribution to the roughening
process and macroscopic dealloying comes from a kinetically determined
competition between dissolution and diffusion. The origin of roughening during the
dissolution of single-phase binary alloys has been revisited [179]. The role of
capillary phenomena, i.e., of surface curvature in the free surface energy, has been
considered in a quantitative approach to this problem. Dynamic balance between
dissolution and surface diffusion generates a modification of the surface profile
and is shown to give rise to unstable modes (roughening transition). Equality
between the wavelength of instabilities and the cluster size on the surface is
thought to occur at the critical potential E
c
. Reasonable consistency is claimed
with literature data on Ag-Au alloys [176].
158 Keddam
Figure 31. The fractal dimension d of the dissolution front as a function of p. For the
2-D rule (line 1 in Table 2). From ref [32].
Copyright © 2002 Marcel Dekker, Inc.
Percolation Model for Passivation of Binary Alloys, Case of Fe-Cr The
Fe-Cr alloys are well known to display easier passivation above a critical Cr
concentration close to 13 at % [180,186]. A computer approach was developed to
demontrate the main atomistic features of a percolation model for passivation of
binary alloys. In a way quite similar to that in Ref. 32, most aspects show up on
2D lattices. Computer simulations [191] were then extended to 3D [192]. Along
the same lines as sketched earlier [105,106], the Monte Carlo simulations make
use of dissolution probability rules of any atom that depend on its own atomic
neighborhood. The Cr atoms having at least two Cr atoms as first or next nearest
neighbors are considered as passivated. This allows Cr–O–Cr bridging bonds to
block the Fe atoms of the underlying atomic row. Application of these rules and
the resulting passivation at the surface of a random 2D lattice of Fe-Cr sites [184]
is sketched in (58). A wide set of p and relative dissolution rates was explored. In
all cases, larger Cr contents tend to passivate the alloy at the cost of an initial
selective dissolution confined to only a few monatomic layers.
Current-voltage trends in the dissolution-passivation range are restituted by
assuming Tafel kinetics for iron dissolution and a chemical rate-determining step
for chromium. Extension to 3D lattices [192] leads to similar conclusions but
highlights a critical concentration p
c
{1, 2} (the percolation threshold for interactions
up to the second nearest neighbor) on a 3D lattice in better agreement with the
transition of Fe-Cr to stainless steel behavior.
The atomistic approach to alloy dissolution shows a real possibility of
achieving a unified description, incorporating at the same time the kinetic and
geometric aspects of these intricate phenomena. Serious difficulties remain for
estimating the probabilities of reaction of individual atoms and their dependence
on local interaction. These models may involve as many parameters as in the
Anodic Dissolution 159
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macroscopic models of non-steady-state (impedance) responses of the anodic
dissolution and passivation processes. In that sense, they are subject to the same
kind of criticism. Estimation of the ability of the atomistic models at dealing with
these non-steady-state behaviors is certainly one of the next challenges in the field.
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