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Theory of Oxygen Tracer Diffusion Along Grain Boundaries and in the Bulk in Two-Stage Oxidation
Experiments. Part III: Monte-Carlo Simulations
Yuri Mishin, Jörg Schimmelpfennig, Günter Borchardt
To cite this version:
Yuri Mishin, Jörg Schimmelpfennig, Günter Borchardt. Theory of Oxygen Tracer Diffusion Along Grain Boundaries and in the Bulk in Two-Stage Oxidation Experiments. Part III: Monte-Carlo Sim- ulations. Journal de Physique III, EDP Sciences, 1997, 7 (9), pp.1797-1811. �10.1051/jp3:1997223�.
�jpa-00249681�
Theory of Oxygen Tracer Dilfusion Along Grain Boundaries and in the Bulk in Two-Stage Oxidation Experiments.
Part III: Monte-Carlo Simulations
Yuri Mishin (~), J6rg Schimmelpfennig (~) and Gfinter Borchardt (~>*) (~) Department of Materials Science and Engineering, Virginia Polytechnic
Institute and State University, Blacksburg, VA 24061-0237, USA
(~) Institut fur Allgemeine Metallurgie, SFB 180, Technische Universitht Clausthal, Robert-Koch-Strafle 42, 38678 Clausthal-Zellerfeld, Germany
(Received 23 October 1996, accepted 12 June 1997)
PACS.66.30.-h Diffusion in solids
Abstract. In Parts I and II of this work we developed a model of oxygen ~~O tracer diffusion
m a growing polycrystalline oxide film with parallel grain boundaries. In this paper we solve the basic equations of the model numerically using the Monte-Carlo approach. We introduce
a new simulation technique that takes into account the finite-size effect, the film growth, the effect of the oxygen chemical potential gradient across the film, and other factors We apply
this technique for the simulation of the most important cases encountered in two-stage oxidation experiments. The oxygen tracer profiles obtained demonstrate good agreement with the previous theoretical analysis, the finite-difference solution of the problem, and exact analytical solutions when available. We discuss possible extensions of the simulation method to provide a more
realistic description of the oxide growth.
Rdsumd. Dans la partie I et II de ces travaux, nous avons ddveloppd un modkle pour la diffusion du traceur ~~O dans
un film d'oxyde croissant avec des joints de grains parallkles. Dans cet article, nous donnons une solution numdrique des 4quations fondamentales par la m4thode de Monte-Carlo. Nous introduisons une nouvelle m4thode de simulation qui tient compte de la g40m4trie de couches minces, du gradient du potentiel chimique de l'oxygkne h travers la couche et d'autres paramktres. Nous utilisons cette technique afin de simuler les cas les plus frdquement rencontds dans des expdriences d'oxydation h deux dtapes Les profils de traceurs
obtenus sont en bon accord avec l'analyse thdorique antdrieur, avec les profits calculds par la mdthode des diffdrences finies et, s'il y en a, avec des solutions analytiques. Nous discutons
l'extension dventuelle de notre mdthode de simulation afin de fournir une description plus rdaliste de la croissance d'une couche d'oxyde
1. Introduction
In Parts I ill and II [2] of this work we have developed a model of ~~O oxygen tracer diffusion in two-stage oxidation experiments. In such experiments an alloy is first oxidized in oxygen-
bearing atmosphere with natural isotope composition (about 99.7995$l ~60) to form an oxide (*)Author for correspondence (e-mail: Guenter Borchardt©tu-clausthal.de)
@ Les #ditions de Physique 1997
layer, following which the oxidation continues in atmosphere enriihed with the stable isotope l~O After the two-stage oxidation the l~O isotope profile in the oxide scale 18 mea8ured by secondary ion mass spectrometry or some other experimental technique. The comparison of the experimental ~~O profile with theoretical models ha8 proved to be an effective tool for establishing the mechanism(s) of oxygen transport in the growing oxide film (see e.g. [3-6]).
It is therefore important to develop theoretical models of ~~O diffusion during the oxide film
growth. The principal objective of such models is to predict the isotope profiles depending on
the prevailing mechanism of oxygen transport (interstitial us. vacancy mechanism, lattice us.
grain boundary diffusion, etc.).
The earlier theoretical work in this area [6,7] was based on the spherical grain model that had been proposed for diffusion in fine-grained polycrystalline materials (see [8] and references
therein). In contrast, the model developed in Parts I and II of this work assumes that the grain
boundaries in the oxide film are parallel to one another and perjpendicular to the substrate
plane. The geometry in this model is shown in Figure I. After introducing some additional physical assumptions we have derived the equations describing the oxygen tracer transport in this model. Mathematically, these equations are two coupled driven diffusion equations with
appropriate boundary conditions. The drift terms arise from the chemical potential gradient
of oxygen across the oxide film.
Exact analytical solution of those equations is not possible. In reference 11,2] we considered
a few important limiting cases where the equations took a simplified form and their analytical
solution became feasible. Some of such limiting cases represent the conditions often met in
experiments. Furthermore, the analysis of the limiting cases made it possible to predict the tracer profiles in more general situations, at least on a qualitative level. However, it was pointed
out that in order to make the analysis more complete the basic equations of the model should be solved exactly using numerical computations.
In this paper we demonstrate a new numerical approach to simulation of oxygen tracer transport in a growing polycrystalline oxide film. The basic idea oi our approach is to simulate simultaneous diffusion of a large amount of tracer atoms using a hfonte-Carlo technique. We
apply this approach for the calculation of the tracer profiles in the framework of our model and compare the results with predictions based on the previous analysis.
Traditionally, diffusion problems are solved numerically using the finite-difference, finite- element or similar techniques. The Monte-Carlo approach has some advantages over such techniques, especially when applied to strongly non-uniform systems with variable sizes. A
Monte-Carlo algorithm is generally simpler and more flexible in modelling complicated bound- ary conditions and the boundary motion. Moreover, the hfonte-Carlo approach directly reflects the stochastic nature of the diffusion process, which makes the discussion of the simulation algorithm and the results more illustrative and easy-to-comprehend in comparison with the
traditional techniques. It should be emphasized that the algorithm we are going to introduce is not an atomistic simulation of oxygen transport. We deal with a phenomenological problem
and our way to solve it is to apply a random-walk algorithm on a macroscopic-size grid im- posed on the system. This approach is based on the equivalence of the Einsteinian and Fickian
definitions of the diffusion coefficient [9] as explained in more detail in Section 3.
Some comments should be made on the previous work where the Monte Carlo technique
similar to ours was used. Murch [10] was the first to apply this method to diffusion along dislocations. Later, Murch and Rothman iiIi successfully used this method to simulate coupled grain boundary and lattice diffusion in a polycrystal with parallel boundaries. The grain boundary diffusion was treated by Murch and Rothman ill] in terms of the discrete model of Benoist and Martin [12] and the simulation procedure was sligfttly different from that used
by Murch [10]. The work of Metsch et al. [13] was also based on Benoist and Martin's model
Atnlosphere
x
GBI
~~~~~
GB2
L
y Substrate
Fig. 1. Schematic geometry in the oxide growth model introduced m Parts I and II of this work.
GBI and GB2 are two grain boundaries in the oxide film, 6 is the grain boundary width, d is the distance between the grain boundaries, L is the oxide film thickness. We have strongly inflated the grain boundary width for the purpose of illustration; in reality 6 is orders of magnitude smaller than
both d and L.
and was generally similar to reference ill] except that only one isolated grain boundary was considered and the particles were diffused into the system simultaneously We wish to point out that these and other studies (e.g. [14-16] considered an instantaneous source of the particles at the surface, which means that the amount of particles diffused into the system was fifed (time- independent). In contrast, the two-stage oxidation process implies a constant concentration of the oxygen tracer at the surface, and thus a time-dependent amount of the tracer atoms in the system. Further, while the previous simulations were performed in semi-infinite systems, we
had to deal with a finite-size system as the tracer can often penetrate through the oxide film and spread over the oxide /substrate interface. Moreover, in contrast to the previous studies the thickness of the oxide film changes with time due to the continued oxidation. We therefore had to deal with diffusion in a system with a variable size. Finally, the oxygen transport occurs by the mechanism of driven diffusion, whereas the previous simulations only considered diffusion in the absence of driving forces. This explains why we could not simply adopt the existing
Monte-Carlo algorithms. Instead, we had to develop a new simulation technique that would be essentially more general and would meet the new important features of our problem.
In Section 2 we shall briefly describe our basic model and the tracer diffusion equations 11,2]
to the extent which is necessary for the understanding of the rest of the paper. In Section 3 we
shall introduce our simulation technique in detail. The simulation results and their comparison
with the analytical solutions are the subject of Section 4. In Section 5 we shall summarize and outline our future work
2. Theory
We consider a flat surface of metal M or an M-containing alloy. When exposed to an oxygen-
bearing atmosphere at elevated temperature the alloy oxidizes to form a continuous dense layer
of oxide Mmon. For example, we can think of an alumina (A1203) forming alloy oxidized in air. The oxide layer (or film) generally has a polycrystalline structure. In our model the
polycrystalline structure of the film is represented by an array of regularly spaced (distance d) grain boundaries that are parallel to one another and perpendicular to the substrate (Fig. I).
We assume that this structure forms during the oxide nucleation at early stages of growth,
which we do not consider in this work, and remains unchanged during the later stages of the oxidation process.
We are interested in the stage where the oxide growth occurs by diffusion transport of oxygen anions orland metallic cations across the film. In the framework of Wagner's theory ii?] the film will grow accordingly to the parabolic law with the effective rate constant
1 AD
Ke~ - w £~ De~d/l. ii)
Here, k is Boltzmann's constant, T is the temperature, ~to and ~tL are the oxygen chemical
potentials at the oxide latmosphere and oxide /substrate interfaces, respectively, and De~ is the effective diEusion coefficient in the film. The latter is given by ii,2j
Dea = u (D( + ~D[) + II u) (Do + ~DM)
,
(2)
where u m 6/d is the fraction of the cross-sectional area of grain boundaries, and the other
quantities are the respective self-diffusion coefficients in the grain boundaries (D[ and D[)
and in the lattice (Do and DM). In real conditions D[ » Do and D[ » DM.
At the second state of a two-stage oxidation experiment the ~~O concentration at the surface increases abruptly to a certain level co above the natural abundance of ~~O (cnat * 0.2005 at. $l)
and remains at this level thereafter. It was shown ii,2j that ~~O tracer transport at this stage is governed by the two coupled equation8, one for lattice diffusion and the other for grain
boundary diffusion:
~ ~ ~~~~~ ~~~ ~~~
l'
= V (D'VC') V'l'. (4)
t
In these equations c is the tracer concentration in the lattice (expressed as the atomic fraction
of ~~O in excess of cnat), c' is the tracer concentration in the grain boundaries, V and V' have
the meaning of the oxygen drift velocities in the lattice and in the boundaries, respectively. The coefficients D and D' in equations (3) and (4) are the respective tracer self-diffusion coefficients of ~~O These are related to the uncorrelated oxygen diffusion coefficients as D
= fDo and
D' = f'D[, where f and f' are the respective correlation factors. The drift velocities V and V' can be expressed as
V
= K/L and V'
= K'IL, (5)
where L is the film thickness and the partial rate constants K and K' are given by
K =
~ /~~ (Do + ~ (uD[ + (I u)DM)) d~t, (6)
~~ m
K' =
( /~~ (D[ + ~ [uD[ + (I u)DM)) d~t. (7)
~~ m
We note that K and K' are related to the effective rate constant Ke~ (Eq. (I)) by Ke~ =
UK'+ II u)K. (8)
The above equations demonstrate the important features that we discussed in Section I.
Namely, the tracer transport equations (Eqs. (3) and (4)) include both diffusion and drift terms, the latter being associated with the oxide growth. Furthermore, the drift velocities V
and V' are time-dependent: they are inversely proportional to the film thickness L (Eq. (5)),
the latter increases with time according to the parabolic law
L(t)
= 12KeR(t + to)l~~~ (9)
(to is the duration of the first (tracer-free) stage and t is the time of the second stage). Finally, equations (3) and (4) should be solved in a finite domain with a time-dependent extension L in the y direction. The exact boundary conditions for equations (3) and (4) were specified in
Parts I and II 11,2].
3. Simulation Procedure
In order to explain the basic idea of the simulation technique let us first consider diffusion in
a uniform isotropic medium with a diffusion coefficient D. Imagine that a regularly-spaced grid with a rectangular unit cell a~ x ay x az is superimposed on the system. The sides of the unit cell are parallel to the coordinate axes x, y and z. Suppose further that a particle
is placed in an arbitrary node of the grid and allowed to perform uncorrelated random walk through the grid by jumping between nearest-neighbouring nodes. The direction of every jump is decided by a random numbers generator according to the jump frequencies T~, Ty and Tz
in the corresponding directions. According to the Einstein equation, the diffusion coefficient of the particle can be expressed as
where (X~), (Y~) and (Z~) are the mean squared displacements of the particle in the corre-
sponding directions. These are obtained by averaging over many repeated walks having the
same duration t. For a truly random walk the calculation of the mean squared displacements
is straightforward and equation (10) becomes
D = a(T~
= a(Ty = a(Tz. (II)
Imagine now that we place a large number N of particles at the same node 0 and let them walk starting from that node, either simultaneously or one after another, for a certain time t. We
assume that the walks are independent from each other and that the particles do not avoid one
another. If the particles walk one after another, the current time receives an increment of I /W
after each jump, W
= 2(T~ + Ty + Tz) being the total jump frequency. If all N particles walk
simultaneously, each Monte-Carlo step involves two random numbers: one number chooses one of the particles for a jump and the other decides the jump direction. In this case the current
time receives an increment of I/(WN) after each jump. In either case, if the jump frequencies
are chosen according to equation ii1), the distribution of the terminal positions of the particles
will follow the Gaussian function centered at node 0 with the diffusion coefficient D [9,18,19].
On the other hand, the Gaussian function is the solution of Fick's equation
~~
= V (DVC) (12)
for diffusion from an instantaneous source located at node 0.
This important observation is actually very general: if the initial distribution of the particles
and their sources is set up properly, their new distribution after a random walk will be identical to the solution of the phenomenological Fick's equation (12) with the corresponding initial
a~ Su&ce
$ $ $ $ $ $
ay( $ $ $ $ $ $
(j jj~
]i z z z z i%
Ii z z z z ii
j z z z z m
$ $ $ $ $ 8
Subs~a~e
Fig 2. The grid imposed on the system for the simulation of oxygen tracer diffusion by the Monte- Carlo method. The circles mark the nodes of the grid and the numbers label the 8 different types of nodes The actual number of nodes in the simulations was greater than shown in this figure.
and boundary conditions. This is true equally for diffusion in non-uniform and/or isotropic systems of any dimensionality: the two approaches will always yield identical solutions to the diEusion problem. This identity reflects the fundamental fact of the equivalence of the Einsteinian (Eq. (10)) and Fickian (Eq. (12)) definitions of the diEusion coefficient and forms the theoretical ground for using the random walk on a grid as a mathematical tool for solving
phenomenologically-based diffusion problems.
The Monte-Carlo technique is also applicable for solving the driven diffusion equations such
as equations (3) and (4). Indeed, suppose that we slightly increase the jump frequency in the positive y direction at the expense of that in the negative y direction. Mathematically, this is
achieved by redefining Ty to
r+y = TVIi + E), (13)
where s is proportional to the applied driving force ((e( < I). Then, it can be easily shown
[9,18,19] that the particle distribution generated by an instantaneous source will displace as a
whole along the y axis with a velocity V given by equation
V =
~~~. (14)
ay
In a general case, the new distribution of the particles after the walk will represent the solution of equation (3) with the drift velocity V given by (14). We can therefore again apply the Monte Carlo approach to solve the diEusion problem. The drift parameter e should then be adjusted
to the given drift velocity V ma equation (14).
We shall now consider more specifically the procedure we applied for solving the ~~O diffusion equations (3) and (4). Due to the periodicity of the problem these equations were only solved in a two-dimensional rectangular domain 0 < z < d/2, 0 < y < L. In Figure 2 we show the
grid we used in the simulations. The grid contains 8 different types of nodes with different
jump frequencies between them. The jump frequencies between various groups of nodes are listed in Table I.
The grid periods in the x and y directions, a~ and ay, were generally different; they were
conveniently chosen depending on the ratio d/L, the lattice diffusion depth, and other factors.
The jump frequencies between the lattice nodes (type 4) were calculated as T~
= Dla( and
