The Mathematical Model And Computer Simulation of Interdiffusion
Driven by The Reactions at Interfaces
M. Danielewski*, R. Filipek*, K. Holly**, M. Hetmañczyk*** and J. £askawiec***
*Fac. of Materials Sci. and Ceramics, Univ. of Mining and Metall., 30-059 Cracow, Poland
**Institute of Mathematics, Jagiellonian University, 30-059 Cracow, Poland
***Inst. of Materials Sci., Silesian Polytechnic, ul. Krasiñskiego 8, 40-019 Katowice, Poland.
Keywords: interdiffusion, computer simulation, partial differential equations, corrosion of metals.
Abstract
Computer simulation of interdiffusion driven by the reactions at interfaces base on the
generalized Darken model of interdiffusion which is applied for an open system. Its reformulated
variational enables to obtain an exact expression for the evolution of component distributions in
open systems. We show simulation of the evolution of the Fe-Cr-Mn-Si alloy composition
because of its sulphidation. An agreement between the computed and experimental results suggests
that postulated boundary conditions correctly describe the transport process.
Introduction
To show the computer simulation of inter-diffusion driven by the reactions at interfaces we base
on the model of the transient state oxidation of single-phase alloy and the generalized Darken’s
model of interdiffusion. The Figure 1 shows the simplified scheme of the scale growth process (a
selective reaction of a single element of a binary
alloy only). The mathematical model of
interdiffusion in the bounded alloy sheet showing
constant total concentration and variable
diffusivity of the components is a base of our
approach. Using the idea of generalized solution,
we will show the evolution of component
distributions in the reacting (sulphidized) metal
plate.
Theory
The particulars of this model for the closed
system [1] and the more general description of Figure 1 Schematic plot of the selective
interdiffusion that incorporates the equation of oxidation of multicomponent alloy.
motion can be found elsewhere [2].
Formation of external BO layer of thickness
Data: 1) molecular masses of the elements: M1, X(t) = 81(t) - X1(t).
. . . , Mr (gmol-1), where r - denotes the number
of components; 2) intrinsic diffusivities: D1, . . . , Dr (cm2s-1), which may depend on composition,
3) initial position of the right end of the system (its right border): 7, 4) time of the process
duration: t*, 5) initial distribution of the elements:
.
Physical laws: 1) the law of the mass conservation of an i-th element:
(1)
Following Darken drift flow idea we postulate that the flux of i-th elements is given by:
(2)
2) Postulate of the constant mixture concentration:
Initial conditions: 1) positions of the left and right ends of the system:
,
,
where indexes "1" and "2" denote the left and right end of the system. 2) initial density
distributions of the elements in the system:
Boundary conditions: 1) relation between the velocity of boundaries
and
mass flux at the boundaries:
i.e., we postulate that the velocity of
the boundary equals the velocity of the local mass center of the system, u (cms-1), at the boundary,
where:
2) We postulate that when the mass flow through the boundary occurs, i.e., system is open, then,
the flux of i-th component at the boundary equals its flow through the boundary [2]:
These functions can be calculated, e.g., from the known rate of reaction at the boundary.
Unknowns: 1) positions of the boundaries,
2) densities of all elements as a function
of time and position,
and 3) the drift velocity in the system,
.
Reformulation of the
problem and method of
computation has been
presented elsewhere [3,
4 ] . T he D Y FS YM
software used to compute
and its demo version is
available from the authors
[5].
Intrinsinc Diffusivities in
Fe-13Cr-18Mn-2Si
Alloy.
To show the
fea sibilit y o f t he
formulation of the problem
of interdiffusion we will
show the predicted and
measured distribution of
elements in the quaternary
diffusional couple. The
Figure 2 Interdiffusion in the Fe*Fe-13Cr-18Mn-2Si system,
main target of this
quaternary diffusional couple at 1273 K. The experimental and
experiment was to enable
calculated distributions of elements after 68 hours of annealing.
the calculations the intrinsic diffusivities in Fe-13Cr-18Mn-2Si steel. For calculations the following
data were used: thickness of the diffusional couple: 27 = 0.2 cm; global concentration in the alloy
c = 0.137 molcm-3; time of the annealing t* = 68 hours; the diffusional couple shown in Fig. 2
was formed by two metal pellets of thickness 7 = 0.1 cm each, having different composition.
The concentration profiles (initial and measured after diffusional annealing) allowed for the
computation of intrinsic diffusivieties [6], Table 1. Figure 2 shows the measured and calculated
density profiles of components in Fe-Cr-Mn-Si system (in the quaternary diffusional couple).
Table 3. Intrinsic diffusivieties in Fe-Cr-Mn-Si system at 1273 K.
No.
1
Intrinsic diffusivities, cm2 s-1
Diffusional couple
Fe*alloy;
alloy composition, %wt.
Comments
Fe-13Cr-18Mn-2Si
C - 0.3 %wt.
DFe
DCr
DM n
DSi
7.5@10-12
7.94@10-12
1.79@10-11
3.17@10-11
Interdiffusion in Sulphidized Fe-13Cr-18Mn-2Si Alloy In this section we will show the
usability of the generalized Darken model in modeling the complex diffusional processes and the
changes of density of elements at the alloy*scale interface (result of the reaction of its elements at
interfaces).
For computer simulations of the interdiffusion in sulphidized alloy the following data were used:
thickness of the sulphidized alloy sample: 27 = 1 mm; intrinsic diffusivities at 1273 K shown in
Table 1; c = 0.137 molcm-3; variable time of the reaction: t* 0 [0 ÷ 60 min); the sulphidized alloy
has constant initial composition: Fe-13Cr-18Mn-2Si in wt%.
The measured values of
sulphur uptake were used to
calculate the flux of sulphur as a
function of time (jS(t, 8i (t)))
and consequently the equivalent
fluxes of Fe and/or Mn through
the alloy*scale interfaces: jiL(t)
= ji(t, 81 (t)) and jiR(t), where i
= {Fe, Mn, Cr, Si}. In present
work we have neglected the
thermodynamic factor, which
affects the alloy transport
properties. This simplified
assumption (yi = ai) usually
markedly underrate the values
of intrinsic diffusivities. It does
not imply the composition
independent interdiffusion
coefficients. To show an impact
of the scale composition on the Figure 3. The computed distribution of Fe, Cr, Mn and Si in the
element’s distribution in the sulphidized Fe-13Cr-18Mn-2Si alloy at 1273 K and pS2 = 1@10-3
reacting alloy (near the Pa.
alloy*scale interface) the following sequence of the reactions was simulated (i.e., the following
mechanism of alloy sulphidation was analyzed):
1) the selective sulphidation of iron (formation of FeS) during the initial time period: 0 ÷ 20 min;
2) the simultaneous sulphidation of Fe and Mn (growth of (MnFe)S), the time period 20 ÷ 30 min;
3) the simultaneous sulphidation of Fe, Cr, Si (formation of (FeCr1.5Si0.5)S4): 30 ÷ 50 min (resulting
distributions of alloy components are shown in Fig. 3);
4) the simultaneous sulphidation of Mn and Fe ((MnFe)S): time period 50 ÷ 60 min (Fig. 3).
All the simulated reactions show that the distribution of all alloy components is affected by the
sulphidation process. Depending on the scale composition both depletion and enrichment are
observed. The interdiffusion process can generate the diffusional structure on initially uniform Fe
distribution, i.e., “uphill diffusion” of Fe, shown in Fig. 3.
The presented calculations and simulation of interdiffusion driven by the reactions at
interfaces show the qualitative agreement with the experimental results. They indicate the great
potential of the generalized Darken model of interdiffusion in the modeling of the oxidation
processes, the high temperature surface treatments and in general simulation of interdiffusion
driven by the reactions at interfaces.
Summary
The generalized Darken model describes the interdiffusion process in general case, i.e., when the
components’ intrinsic diffusivities vary with composition and the reactions of diffusing components
at the boundaries are allowed. The numerical computations of interdiffusion data in quaternary
diffusional couples show an agreement between the computed and experimental results and allow
to determine intrinsic diffusivities of all elements.
Contrary to the well known Onsager phenomenological approach, the Darken model of
interdiffusion does not directly takes into account that gradients of all components affect the flux
of each element. Instead Darken postulated generation of common drift velocity in the system, Eq.
(2). This postulate couples all fluxes and apparently allows for proper description of its behavior.
The modern mathematics now-days allow for the modeling, practical calculation and consequently
the better understanding of the real transport problems.
Acknowledgments
This work has been supported by the Polish State Committee for Scientific Research under
Grant No. 7 T08C 041 13.
References
1) M. Danielewski, K. Holly and W. Krzy¿añski, Polish J. Chem., 68, 2031 (1994).
2) K. Holly and M. Danielewski, Phys. Rev. B, 50, 13336 (1994).
3) M. Danielewski, S. Datta, G. Fisher, Y. Chan, R. Filipek and R. Bachorczyk, “Thermal
Stability and Interdiffusion in Multicomponent Coatings; Pt Modified β - NiAl Coating on
Superalloys”, this Conference.
4) M. Danielewski, R. Filipek, T. Walec and A. Milewska, Metallofiz. Noveishie Tekhnol.,
21, 97 (1999).
5) Demo version of DYFSYM:http://www.ceramika.agh.edu.pl/~rof/interdiffusion/demo.html
6) R. Filipek, “Interdiffusion in Multicomponent Systems”, Doctor Thesis, Kraków 1996.