A. Öchsner 1,2 , W. Miszuris 3 , J. Grácio
1 Centre for Mechanical Technology and Automation
2
Department of Mechanical Engineering
University of Aveiro, Portugal
3
Faculty of Mathematics, Informatics and Mechanics
Warsaw University, Poland
ABSTRACT: When oxygen dissolves from atmosphere and diffuses into an alloy phase during oxidation, less noble alloy components may react to form oxide particles within the metal. This process is termed internal oxidation. Classical approaches to describe this phenomenon were derived under many strong simplifications, such as: constant diffusion coefficients, certain boundary conditions and semi-infinite sample.
The presented approach is based on the finite difference solution of general diffusion equations coupled through the stoichiometry of reaction between oxygen and the considered element. The main enhancement is the consideration of concentration dependent diffusion coefficients, concentration dependent source terms and arbitrary time-dependent boundary conditions formulated as a concentration, a flux or mixed conditions. Furthermore, the finite dimension of the specimen is incorporated. This general treatment also allows the incorporation of the energy balance.
1. INTRODUCTION
Internal oxidation is the process, by which oxygen diffuses into an alloy and causes sub-surface precipitation of oxides of one or more alloying elements. The growth of the oxide layer with time is an important parameter which can give information about the internal damage of the alloy.
Corresponding processes also take place when elements such as sulphur, carbon, nitrogen etc. diffuse into alloys and cause precipitation of sulphides, carbides and nitrides, respectively. Several treatments are available in literature and the classical approaches, e.g. Wagner’s Theory and its modifications, can be found in the books of
Birks and Meier (1), Kofstad (2) and Khanna (3).
The mathematical and numerical treatment is demonstrated for a binary alloy B-M.
Metal B is assumed to be a noble metal under the oxidation conditions, i.e. no oxide of B can be formed. Metal M may form an oxide MO 

 The reaction of dissolved oxygen (O) with M atoms in the alloy may be written as:
M
 
O

MO

. (1)
The presented approach is based on the solution of the general diffusion equations for
M and O coupled through the stoichiometry of reaction between oxygen, O, and the considered element. The main enhancement is the consideration of concentration dependent diffusion coefficients, concentration dependent source terms, differing kinetics of the reaction and arbitrary time-dependent boundary conditions formulated as a concentration, a flux or mixed conditions. Furthermore, the finite dimension of the specimen is incorporated. The numerical solution of the coupled diffusion reaction equations is based on a finite difference scheme which enables the incorporation of the mentioned non-linear effects.
1 - 46

2. CLASSICAL APPROACH
The simplest approach for the mathematical description of the internal oxidation of alloys is based on the assumption that the M atoms are immobile compared with the oxygen atoms (1-3). The classical approach for the mathematical treatment of the internal oxidation was proposed by Wagner (4) and is still applied by engineers for the evaluation of experiments. He considered not only inward diffusion of oxygen, but also the outward diffusion of less noble alloy elements M. In this approach, a sharp reaction front is assumed and no component can cross this front. Fick’s second law for both moving components can be expressed for the concentrations c under the assumption of constant diffusion coefficients D
O
and D
M
as:
 c
O
 t

D
O
 2 c
O
 x
2
;
 c
 t
M

D
M

For the special case of initial conditions ( t = 0) of type
2 c
M
 x
2
. (2) c
O

0 and c
M
 c
0
M for x

0 , and boundary conditions ( t > 0) of type (x =

: reaction front), c
O
 c
0
O for x

0, c
O

0 for x
  and c
M
 c
0
M for x
 
,
(3) c
M

0 for x
 
, (4) the general solution of Eqs. (2) is given for a semi-infinite sample as: c
O
 c
0
O

1
 x D t )
1/ 2

 ; c
M
 c
0
M



1
 erf[( / 2 D t )
1/ 2
] erfc(

)
1/ 2 
, (5) where

is the so-called rate constant and

the diffusion coefficient ratio D
O
/ D
M
.
Oxide MO  is formed at the reaction front and the assumption of a sharp reaction front implies that the flux of oxygen atoms is

times the outward flux of the M atoms at the front:

D
O 
 c
O
 x
 x
 
 
D
M 

 c
M x
 x
 
. (6)
Combining of Eqs. (2) and (6) yield an expression for the determination of the rate constant

which is used in a parabolic expression for the thickness of the internally oxidized zone:
  
D t )
1/ 2
. (7)
A more general solution was proposed by Böhm and Kahlweit (5). They assumed that the concentration will not reduce to zero at the reaction front but are equal to c
O
(

) = c
O m and c
M
(

) = c
M m , respectively, where the product c
O m c
M m is the critical concentration necessary for nucleation of oxide particles. To determine the parameter
 in the solution of type (5), a materials balance for a volume element is performed.
This approach allows a penetration of the components O and M. However, since the solution is based on equations of type (5), the same assumptions (i.e. constant diffusion coefficients and semi-infinite sample) still hold and material non-linearity can not be incorporated. The most general approach is to consider a system of coupled diffusion reaction equations and to solve this system based on numerical
1 - 47

approximation methods, which will be presented in detail in the next section for the finite difference method (FD).
3. GENERAL FORMULATION OF THE PROBLEM
Let us consider a general 2D system of partial differential equations of the parabolic type represented by the system of diffusion-reaction equations (6-8):

 c t
O

 x

D
O
 c
O
 x

  f ch

(8)
 c
M
 t


  x 
D
M
 c
M
 x 
 f ch

(9)


T


 t x

D
T

T
 x

 f th

(10) within the domain: ( )
D D
O

M
and D
T
[ a b ] [0 T

] . We assume that diffusion coefficients
and the chemical and thermal sources f ch
and f th
are generally speaking depending on the variables ( ) and as well as on the problem solutions: c ,
O c and T . The initial conditions to the system are:
M c ( 0)
O
  c 0
O
( )
 c
M x c 0
M
( )

( 0) T 0 ( ).
(11)
Boundary conditions at both sides of the interval [ ] are given in a general form to carry out all possible combinations, which may appear in practice:
  c
O O
 
O

 c
O
 x
 g

O
 x
 a b


2 2
  c
M M
  
M
 c
M
 x
 g

M
 x
 a b


2 2

T

T
 
T


T
 x
 g

T
 x
 a b


2 2



(12)
(13)
(14) where we also assume that all functions in the boundary conditions may not only depend on the space variable x but also on the problem solutions. As a result, any nonlinear boundary conditions are considered in general scheme.
     
O M T O M T
  g

O
 g

M
 g

T
      
O M T O M
 
T
 g

O
 g

M
 
T
(
 
0
 c
M T
)
4. FD FORMULATION OF THE PROBLEM
Let

N

T N
1
,
and
(
M

)
1 be numbers of nodes in t and x directions respectively, and h
 b a M - the lengths of the corresponding increments. Below the following notation will be consequently used: x i
 
(

1)
 t j

M

( j

1)
 i
 
M
  j
   
N
(15)
1 - 48

Problem Eqn. (8) - (13), which is written in matrix representation, has a diagonal form. Later on, we discuss for simplicity without any loss of generality only the scalar case of one equation with respective initial and boundary conditions. In the general case, the problem solution is sought in the same way. Thus, the modeling equation is:

 w


 t x 
D
 w
 x 
   
(16)
Let us introduce the standard notations:
( i j
)
 w


 w
 x

(
 j
)
 w
 w i
  j  h
 w
 t

(
 j
)
 w i j 1

 w


 w
 x

(
 j
)
 w i
  j
 w h
.
We will use for the second derivative the following representation:
  w
 x
D
 x

(
 j
)

1
2




 x
D


 x w

(
 j
)



 x
D


 x w

(
 j
)
1 h
2
( k i
  j w i j
 k w

 k i
  j w i j
)

 where coefficients k are calculated from:
(17)
(18)
(19) k i
  j

1
2
( D i
  j

D )
 k i
  j

1
2
( D

D i
  j
)
 k

1
2
( D i
  j

2 D

D i
  j
)

(20)
This representation is symmetrical with respect to the number i . In order to guarantee the convergence of the FD solution, one can use an indirect scheme replacing Eq. (19) and introducing the new parameter

(0 1) in the representation:
  w
 x
D
 x
(1

)
  w
 x
D
 x
(

 j
)
 
  w
 x
D
 x

( i j

1
)

(21)
1
 h
2

( k i
  j w i j
 k w i j

 k i
  j w i j
)

 h
2
( k i
  
1 w i
  
1
 k i j 1 w i j 1
 k i
  
1 w i
  
1
)

Finally, the FD representation of the equation (16) takes the form:

( w i j 1
 w )
  
)( k i
  j w i j
 k w i j

 k i
  j w i j
)

(22)

( k i
  
1 w i
  
1
 k i j 1 w i j 1
 k i
  
1 w i
  
1
)
 h
2
((1
 
) f
  f i j 1
)
 where
 h

. Let us note that the sources f can be a function depending drastically in some region with respect to the solution of the problem, while all other material parameters are rather changing slowly. For this reason, in the right-hand side of equation (22) also mixed representation is applied. Finally, this equation can be rewritten in the following form:
 h
2
((1
 
) f
  l i
  
1 w f i j 1 i
  
1

 l l i j 1 i
  j w i
  j w i j 1

 l w
 l i
  
1 w i
  
1 l
 i
  j w i
  j
)

(23) with new notations:
1 - 49

l i
  j l i
  
1
(1

) k i
  j

  k i
  
1
 l l i j 1

(1

) k

  k i j 1
)
 l i
  j l i
  
1

(1

) k i
  j

 k i
  
1

(24)
This equation is valid for any 1 i M

1 . In the case
 
0 one receives the direct scheme which can be unstable depending on the chosen value of
 
0 . Moreover, because coefficients in the equation are functions of its solution, it is impossible to guarantee convergence and stability of such FD scheme a priori. For this reason one can decide to use an indirect scheme with
 
0 . Moreover, in the case of
   one can expect the best convergence rate.
The boundary conditions at the ends of the interval [ ] should be also written by its
FD representations:
  j

1 w
1
 
1

 j


1 h
( w
2
 
1
 w
1
 
1
)
 g
 j

1

 j


1 w
M
  
1

 j


1
( w
M
  
1
 w
M j 1
)
 h g
 j

1
.
(25)
(26)
One can rewrite the boundary condition in the following forms: l
1

 
1 w
1
 j
 l

2
 
1 w
2 1
 hg j


1
 l

M
  
1 w
M
  
1
 l

M j 1 w
M j 1
 hg
 j

1
 l
1
 j 1

( h
  j

1
  j


1
)
 l

2
 
1
   j


1
 l

M
  
1

( h
  j

1
  j


1
)

(27) l

M j 1
   j


1

(28)
If the solution at any time t
 t j
is known, then the solution for the next moment t
 t j

1
has to be calculated from the system of equation (23) with boundary conditions (27). For this reason, let us introduce the following notation: w j

1






 w w
1
 
1
2
 
1
 w
M
  
1







L j

1













 l l
1

1
 
1
0
 j 1
0
0
0 l

2
 
1 l
2
 
1 l
2
 
1

0
0
0
0
0
0
0 l
3
 
1 l
3
 
1

0
0 l
4
 
1







0
 l
M
  
1
0
0
0
0
0
 l
M
  
1 l

M j 1
0
0




0
0
 l
M
  
1 l

M
  
1









(29)
Corresponding problem (23), (27) takes the form:
L j

1
( w j

1
) w j

1
 b
(0) j

1
( w j
)
 b
(1) j

1
( w j

1
)
 j 1 2 N (30) where components of the vector b j

1
are defined in (22) and (27): b
1
(0) b
1
(1)  hg
 j

1
 b
(0)
M

1
(0) b m b
(1)
M

1
 hg j


1

  h
2
(1
 
)
(1) b m f
 m

1  p m 1 l w
  h
2
 f m j 1
 m 2 3

M
(31)
The solution for any fixed j

1 will be sought with an iteration method. Namely, we will construct a sequence u q
, q 1 2 3 by the following scheme:
1 - 50

u
1
 w j

L j

1
( u u q
) q

1
 b
(0) j

1
( w j
)
 b
(1) j

1
( u q
)
 q 1 2 3 w j

1
 q lim
 u q

It is important to note that L j

1
is the diagonally dominant matrix. In fact,
 l i
  
1
   l i
  
1
 
( k i
  j
 k i
  j
)
  k k
 l i j 1
 
(32)
(33)
(34)
Here, we have taken into account that components k in (20) are positive. It remains to check only the first and last lines in matrix L j

1
using definitions (28):
 l
1
 j 1
 l

2
 
1
  l

M
  
1
 l

M j 1
 
(35)
This means that the solution u q
exists for any q . The process should be stopped in the case that any of the following conditions is valid: u q

1
 u q

 u q

1
 u
(

)

(36)
This finishes the FD solution to one dimensional problem. In the case of the reactiondiffusion equations system, the only difference is that the scalar values have to be replaced by the respective vectors or diagonal matrices, while all estimates are still valid.
5. NUMERICAL EXAMPLES
The above described numerical procedure has been used for the simulation of the internal oxidation of the system Ag-Cd with the respective data given in (5) for the constant temperature T = 850°C: D
O
= 2.6 10-5 cm
2
/s
-1
; D
Cd
= 3.4 10
-9
cm
2
/s
-1
; c
O
0
=
1.48 10
-5
mol O/cm
3
; c
Cd
0
= 9.3 10
-4
mol Cd/cm
3
; time t max
= 10 h and sample thickness l max
= 0.25 cm. For our reference simulation we assumed the second order reaction rate of the form r = kc
O c
Cd
. Results in Figs. 1 and 2 are presented in a normalised way where the following transformations were used: u
1
= c
O
/ c
O
0
; u
2
= c
Cd
/ c
Cd
0
;

= t / t max
; x = l / l max and D
1
= D
O t max
/ l max
2
. In comparison with our reference state, also non-linear modifications of the diffusion coefficients and higher-order reaction rates were investigated.
Figure 1: Influence of the diffusion coefficient formula on the normalised concentration profiles for different normalised times

.
1 - 51

Figure 2: Influence of the source formula on the normalised concentration profiles for different normalised times

.
We assumed that the oxygen boundary condition is of Dirichlet type on the left and of
Neumann type on the right side of the sample. Whereas, Neumann conditions were assigned on both sides of the sample for cadmium. Figure 1 illustrates the influence of the non-linearity of the diffusion coefficient while the second figure shows the influence of the reaction rate on the concentration profiles for different times. As it can be seen, both effects influence significantly the concentration distributions and should be incorporated in the evaluation of such experiments.
Figure 3: Influence of boundary conditions on the right and side of the specimen and the specimen thickness on the normalised concentration profiles for different normalised times

.
The influence of the boundary condition on the right hand side of the sample is shown in Fig. 3 (left side) and the influence of the specimen thickness in Fig. 3 (right side).
As it could be expected, there is a significant influence only near the right-hand side boundary for the cases under consideration. In the authors’ opinion, this result follows from the dominating influence of the source. If someone uses the classical approach, the influence of boundary conditions might be more drastic. However, it should be emphasized here that even in the general case of modeling with sources, correct usage of the boundary conditions guaranties the accurate simulation of the process.
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It turned out that the proposed numerical procedure is stabile for various non-linear effects and enables us to investigate further configurations of diffusion-reaction problems. In order to enable the easy usage of our code, we will provide in the future a user-friendly interface.
ACKNOWLEDGMENTS
A. Öchsner is grateful to Deutsche Forschungsgemeinschaft (DFG) for financial support. The authors are thankful to Prof. G. Mishuris for fruitful discussions.
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(4) Wagner W.: Reaktionstypen bei der Oxydation von Legerungen. Z. Elektochem.
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(5) Böhm G. and Kahlweit M.: Ueber die innere Oxydation von Metallegierungen.
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(6) Aris R.: Elementary Chemical Reactor Analysis. Prentice-Hall, UK 1969.
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