1

3D Cellular Automata modelling of Intergranular Corrosion

Simone Guiso1, Dung di Caprio2, Jacques de Lamare3, Benoît Gwinner4

1 Den-Service de la Corrosion et du Comportement des Matériaux dans leur Environnement

(SCCME), CEA, Université Paris-Saclay, F-91191, Gif-sur-Yvette, France,

simone.guiso@cea.fr

2 Chimie ParisTech, PSL Research University, CNRS, Institut de Recherche de Chimie Paris

(IRCP), F-75005 Paris, France, dung.di-caprio@chimieparistech.psl.eu

3 Den-Service de la Corrosion et du Comportement des Matériaux dans leur Environnement

(SCCME), CEA, Université Paris-Saclay, F-91191, Gif-sur-Yvette, France, jacques.de-

lamare@cea.fr

4 Den-Service de la Corrosion et du Comportement des Matériaux dans leur Environnement

(SCCME), CEA, Université Paris-Saclay, F-91191, Gif-sur-Yvette, France,

benoit.gwinner@cea.fr

Abstract

Intergranular corrosion (IGC) is one of the attacks that a material may suffer. It is characterized by a

preferential attack to grain boundaries, which induces an acceleration of the material degradation. As an

example, it can concern stainless steels (SS) when exposed to oxidizing media (e.g. 𝐻𝑁𝑂3) as in the

chemical industry [1] or in the nuclear fuel reprocessing industry [2]. Fully understand the degradation

mechanism is an important challenge.

IGC is characterized by the formation of triangular grooves at grain boundary level. Their progression

leads, with a certain periodicity, to grain dropping, which causes an acceleration of the corrosion

phenomenon [3]. From a quantitative point of view, the description of IGC kinetics is still incomplete:

in particular, the IGC evolution at bulk level, is quite uneasy to access by experiments. Numerical

simulations can, in this case, provide more information and improve the knowledge.

Cellular Automata (CA) can model any complex reality through simple rules and space-time

discretization, and are particularly suitable to model IGC [4-5]. The goal of this work is to simulate with

CA the kinetics of degradation by IGC, to match as accurately as possible the experimental results in

[3]. A hexagonal close-packed (HCP) grid is chosen for the simulations. The granular material structure

is modeled as a Voronoï diagram, while two corrosion probabilities (for grain and grain boundary

species) drive the time evolution. With appropriate time and space equivalences, CA simulations are

then able to reproduce accurately the experiments in terms of corrosion propagation velocity. The

surface morphology reflects the same pattern as in the experiments: after the first transitory phase, grains

start detaching from the material and accelerate the IGC process. The material-solution interface is then

studied through the time evolution of the solution-grain boundary distribution. Results show an

important dependence on the choice of the two corrosion probabilities.

Keywords: cellular automata, intergranular corrosion, modelling, PUREX process

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Introduction

Grain boundaries represent barriers to slip, and hence, they confer strength to metallic materials.

However, they are also sources of failure and weakness due to their relatively open structure,

compared to grains. This aspect allows grain boundaries to be subject to degradation and

important failures, even when « in service ». Grain boundary degradation may include

corrosion, cracking, embrittlement and fracture, often accelerated by precipitate segregations

[6].

Figure 1 – PUREX process

The intergranular corrosion (IGC) is a corrosion process of interest in nuclear engineering,

especially when dealing with the spent nuclear fuel reprocessing plants. The PUREX

(Plutonium and Uranium Refining by Extraction) process, schematized in Fig.1, which carries

out the treatment of the spent fuel, involves various interactions between nitric media and

stainless steels (SS), where IGC may occur in some rare cases [2]. In order to resist these nitric

media, SS containers are chosen in view of their corrosion resistance, and optimized and

qualified before their use: for instance, austenitic SS with a low carbon and precise silicon and

phosphorus contents are preferred. All these materials are protected by a stable passive layer.

In these conditions, SS corrosion is uniform and slow. However, too oxidizing conditions (e.g.

nitric media containing oxidizing ions at high temperature) may promote the deterioration of

this passive layer and generate intergranular corrosion phenomena. In these cases, the reduction

reaction is fast and can lead to a shift of the corrosion potential to the transpassive domain

(Fig.2), exposing materials to intergranular corrosion.

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Figure 2 - Schematic polarization curve for a SS

Experimentally, two cases of IGC may be distinguished, according to the chemical composition

and the thermal treatment of the involved SS [3]. The first case includes the so-called

“sensitized SS”, with a relatively high carbon content, which are subject to long treatments at

high temperatures (mainly between 500 and 800 °C). During this phase, Cr and C species

diffuse through SS and 𝐶𝑟23𝐶6 precipitates may be formed at grain boundary level. As

chromium diffuses slowly from the grain matrix to precipitate at grain boundaries, an adjacent

region to 𝐶𝑟23𝐶6 precipitates is generated, where the chromium content is low. In these regions,

the chromium content is not sufficient to form a passive film to protect the SS. Therefore, these

regions are preferentially dissolved, leading to IGC attacks [7]. The second case, instead,

involves the so-called “non-sensitized SS”. SS are here firstly treated to reduce the carbon

content (typically below 0.024 wt.%). In this way, during high temperature treatments,

chromium carbide precipitation does not take place. However, in highly oxidizing media (e.g.

𝐻𝑁𝑂3), IGC can still occur. Experimental tests showed that it is here characterized by the

formation of triangular grooves, as represented in Fig. 3.

Figure 3 – Cross section of a SS suffering IGC [3]

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During IGC attack, grooves progress inside the material. The progression may cause grain

detachment, with a certain periodicity, and accelerate the corrosion phenomenon, increasing the

SS degradation (Fig. 4).

Figure 4 – Grain detachment [8]

The complete comprehension of IGC degradation mechanisms is a real challenge that involves

both experimental tests and modelling. Recently, IGC modelling became more and more

important and useful, in particular for those cases in which obtaining results from experiments

are less convenient, not easily accessible and time-consuming.

A promising modelling approach to study these types of systems is cellular automata (CA). CA

models consist of a regular grid of finite automata (cells) that update their own state according

to the evolution of their neighboring cells, following precise transition rules [9]. CA can model

whatever dynamical and complex system by a simple space discretization and a list of rules that

allow the temporal transition from a time t to the following t+Δt. These characteristics let CA

be available for modelling biological, physical-chemical [10]–[12], and engineering systems

[13], [14].

IGC has been modelled with CA approach as well. Recently, Jahns et al. [15] developed a 2D

CA model to study intergranular oxidation obtaining a good agreement with experimental

results. Di Caprio et al. [4] developed a 3D stochastic CA model to study IGC, by improving

precedent 2D models [16]. On the other hand, Lishchuk et al. [17] improved his own brick-wall

model [18] by developing a 3D stochastic model to be compared with experimental results:

simulations presented, in this case, a good agreement.

The objective of the present paper is to apply CA methodology to study IGC attack. Firstly, a

3D stochastic model is presented in its main features; successively, IGC simulation results are

compared with experimental tests by Gwinner et al. [3]. The paper is organized as follows: the

model and the simulation setup are described in Chapter 2, while results and discussion are in

Chapter 3.

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Cellular automata approach

CA approach, firstly proposed and theorized by Von Neumann [19], is characterized by four

main elements:

1. A grid of identical cells

2. Each cell is defined by a state, which determines its main characteristics

3. The evolution of a cell is related to its neighboring cell states

4. The transition rules that defines when and how a cell may change its state

In our IGC model, a hexagonal close-packed (HCP) grid, represented in Fig. 5, was chosen for

simulations, due to its intrinsic isotropy on neighborhood [20].

Figure 5 – HCP grid representation

Three chemical states interact among each other and define the cell nature: GRN state defines

the grains, IGN defines the grain boundaries and SOL defines the corrosive agent in solution.

The neighboring pattern of a cell is clearly visible in Fig. 5. Each cell has 6 neighboring cells

in the horizontal plane (plane A) as well as respectively 3 neighboring cells in the plane below

and the plane above (plane B). The cells in the plane below and above have identical (𝑥, 𝑦)

coordinates. All neighboring cells are at equal distance from the central one. The system evolves

in time, depending on two corrosion probabilities, 𝑃𝑔𝑟𝑛 for the grain dissolution and 𝑃𝑖𝑔𝑛 for

grain boundary dissolution, according to the following rule: if an IGN cell is in contact with a

SOL cell, a random number (between 0 and 1) is associated to this GRN or IGN cell and

compared to the corresponding probability (𝑃𝑔𝑟𝑛 or 𝑃𝑖𝑔𝑛). If it is higher, the cell will keep its

state; in the other case, the cell is subject to corrosion and becomes a SOL cell.

In this model, we assumed to simulate a highly recirculating medium, so that no corrosion

products are accumulated in the solution. If a group of grain cells (GRN) is detached from the

main domain, conventionally placed at the bottom, its cells become SOL cells. Moreover, no

oxide layers are produced.

An example of the initial grid is presented in Fig. 6: the material microstructure was reproduced

with a Voronoï diagram. It consists of a partitioning of the 3D grid into regions that are defined

from some starting points, called seeds. Each seed progressively grows, along all directions:

when two spheres meet each other, a border is defined along their tangent plane. Nevertheless,

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sphere keep growing in the other directions, until all cells are covered. Once the Voronoï

structure is complete, states are assigned to cells: those who belong to spheres assume the GRN

state, while cells that belong to sphere borders assume the IGN state. Furthermore, the SOL

state is assigned to the two uppermost xy-layers: the solution path is so conducted from the top

to the bottom of the grid. Note that the top layer grain size distribution is different from the bulk

because these surface grains are cut.

Figure 6 – Representation of the 3D IGC model

Simulation setup

The goal of this study is to compare experimental and CA simulation results. Experimental tests

were performed by Gwinner et al. [3] on an AISI 310L stainless steel. This metal has a low

carbon content and was firstly thermally treated (homogenization treatment) to avoid the

precipitates formation that would have brought a type of intergranular corrosion proper of

sensitized SS (see Chapter 1). The SS was corroded by a solution of nitric acid containing an

oxidizing ion (𝑉𝑂2+) at boiling temperature, renewed after each corrosion period. SS specimens

were periodically removed and examined: after each cycle, the equivalent thickness loss (ETL)

was evaluated, as a function of the real mass loss, with the following equation,

𝐸𝑇𝐿𝑒𝑥𝑝(𝑡) =∆𝑚

𝜌∗𝑆∗ 104 (1)

where 𝐸𝑇𝐿𝑒𝑥𝑝 is the equivalent thickness loss from the beginning of the experiment (in µm),

∆𝑚 is the mass loss from the beginning of the experiment (in g), ρ the SS density (in g.cm-3)

and S is the specimen surface in contact with the solution (in cm2).

Once the ETL curve is obtained, the instantaneous corrosion rate 𝐶𝑅𝑒𝑥𝑝(𝑡) is calculated

through the following equation:

𝐶𝑅𝑒𝑥𝑝(𝑡) =∆𝐸𝑇𝐿𝑒𝑥𝑝(𝑡)

∆𝑡 (2)

where Crexp(t) is the corrosion rate calculated at the period of corrosion i (in µm.y-1),

∆𝐸𝑇𝐿𝑒𝑥𝑝(𝑡) is the equivalent thickness loss measured during the period of corrosion i (in µm)

and ∆𝑡 the duration of the period of corrosion i (in y).

CA simulations were here performed using a 1536*1536*512 cell grid. The Voronoï structure

was built to get an average ratio of around 80 1⁄ cells between grain and grain boundary

dimensions: this value is the result of an optimization process that takes into account the real

grain to grain boundary dimension ratio (around 10000 1⁄ ) and the available GPU memory,

physically limited.

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The choice of the two corrosion probabilities also required an optimization process. We

performed a study on the influence of a single corrosion probability on real corrosion velocity,

presented in Fig.7. As an example, we considered in this case a system that is subject to a

granular dissolution process. Results show that there is an exponential like relation between

the two.

Figure 7 – Relation between corrosion velocity and probability

Based on these results, we chose the two corrosion probabilities (𝑃𝑔𝑟𝑛 and 𝑃𝑖𝑔𝑛) in order to

simulate the same grain-boundary-to-grain velocity ratio that Gwinner et al. found in their

experiments [3], (between 5 and 10). Therefore, 𝑃𝑔𝑟𝑛 is set to 0.01, while 𝑃𝑖𝑔𝑛 is equal to 0.2.

Simulation results are expressed in terms of equivalent thickness loss (ETL in µm) and

corrosion rate (CR) with the following equations Eqs. (3) and (4):

𝐸𝑇𝐿𝑠𝑖𝑚(𝑖) =𝑁𝑚𝑎𝑡(𝑖=0)−𝑁𝑚𝑎𝑡(𝑖)

𝑁𝑥𝑦∗ 𝐴 (3)

where 𝑁𝑚𝑎𝑡 is the sum of the number of IGN and GRN cells, 𝑁𝑥𝑦 is the number of grid cells in

a xy-plane, and 𝐴 is a constant (in µm/pixel) that takes into account the geometrical scale

between two neighboring z-planes in a HCP lattice and the spatial equivalence between

experiments and simulations, 1 𝑝𝑖𝑥𝑒𝑙 ~ 1 𝜇𝑚. This spatial equivalence results from the real (80

µm) and the simulated (80 pixels) grain size.

𝐶𝑅𝑠𝑖𝑚(𝑖) =∆𝐸𝑇𝐿

∆𝑖∗ 𝐵 (4)

where 𝐵 takes into account the time equivalence to get the corrosion rate in 𝜇𝑚 𝑦⁄ : a single

simulation step is equivalent to 3 hours.

Results and discussion

A graphical example of the 3D cellular automata IGC model is presented in Fig.7 while

corrosion is in act. A 2D cut view is presented as well, where we can see the above-mentioned

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

0 0,2 0,4 0,6 0,8 1

Vel

oci

ty

𝑃_𝑔𝑟𝑛

8

triangular grooves, which penetrate into the material (see Fig. 3 for comparison with the

experimental tests in [3]).

Figure 8 – 3D representation of the 3D IGC model during the corrosion process (top) and 2D

cut plane view (bottom). The groove angles are here projected on a grid-orthogonal plane.

Figure 9 - Comparison between experimental results (left) and CA simulation results (right)

The comparison between experiments [3] and simulations is presented in Fig. 9. Simulations

results are here averaged on the simulation results of 10 different Voronoï diagrams. With these

equivalences, results overall show a good agreement. Both start with a transitory phase, where

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500

Co

rro

sio

n r

ate

(µm

/y)

Equ

ival

ent

thic

knes

s lo

ss (

µm

)

Time (h)

Equivalent thickness loss Corrosion rate

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grooves penetrate in the bulk and small grains gradually start detaching. The grooves go deeper,

more and bigger grains detach from the material, increasing the corrosion rate. After a while,

corrosion rate becomes constant and a stationary regime is reached: grains keep falling regularly

with a certain periodicity. The transitory phase is longer for CA simulations: grains take a while

before falling. Since it imposes a uniform grain distribution, it follows that there are only few

very small pieces of grains that are detached at the beginning. Experiments showed, instead,

that the corrosion rate grows immediately, thanks to small grains that leave soon the material.

A further study was conducted on the material-solution interface, through the grain boundary –

solution spatial distribution. The two probabilities are always chosen in order to respect the

velocity ratio that was obtained during the experimental tests. A graphical example of spatial

distribution is presented in Fig. 10.

Figure 10 – Representation of the grain boundaries-solution interface at i=350

The spatial distribution was calculated each 10 steps.

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Figure 11 – Evolution of the IGN-cell spatial distribution at grain boundary - solution

interface

As it can be seen on Fig. 11, the grain boundary – solution interface depth distribution is

Gaussian like and evolves with time. It is firstly peaked and narrow; then, the maximum

decreases and the peak enlarges. The mean depth, presented in Fig. 12, goes linearly in time.

The slope corresponds to the mean corrosion rate reached at steady state (Fig. 8). Its standard

deviation (σ) is defined in Eq. (6) and represented in Fig. 13.

𝜎(𝑖) =√∑ (ℎ𝑗−⟨ℎ(𝑖)⟩)

2𝑗

∑ 𝑤𝑗𝑗 (6)

Where h is the depth in µm, 𝑤𝑗 is the j-weight of a ℎ𝑗 depth and ⟨ℎ(𝑖)⟩ is the mean depth at i-

iteration.

Results show a “square root of time” regime, once a first initial layer of grains fell and grains

have their bulk size distribution.

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300 350 400 450

Wei

ght

[-]

Depth [µm]

i=50 i=150 i=250 i=350 i=450 i=550 i=650 i=750

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Figure 12 - Representation of the average interface depth as function of time

Figure 13 – Representation of the depth standard deviation as function of time with a linear

(in yellow) and a power regression (in red) for the two trends

Conclusion

In this paper, we developed a 3D cellular automata model to simulate the IGC phenomenon.

Chemical reactions kinetics were turned into corrosion probabilities, the model parameters that

rule the evolution of the system. A Voronoï diagram modelled grains and grain boundaries in

the material microstructure. We compared CA simulations with real IGC experiments: with

appropriate space-time equivalences, the two are comparable and show interesting results in

terms of mass loss and corrosion rate. A second study on the material-electrolyte interface

shows that CA model can also support experimental tests, by modelling the internal bulk

material, which is in general difficult to access experimentally. Results show a Gaussian

distribution, in a good agreement with expectations. The result is further confirmed by the

trends of the average depth and the standard deviation.

0

50

100

150

200

250

300

350

0 100 200 300 400 500 600 700

Dep

th [

µm

]

Time [-]

y = 1E-05x + 7E-19

y = 1,88E-04x5,03E-01

0,00001

0,0001

0,001

0,01

10 100 1000

Stan

dar

d D

evi

atio

n [

-]

Time [-]

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