TSINGHUA SCIENCE AND TECHNOLOGY ISSN 1007-0214 06 /11 pp704-708 Volume 10, Number 6, December 2005

Predicting Interdiffusion in High Temperature Coatings*

J. E. Morral**

Department of Materials Science and Engineering, Ohio State University, Columbus, Ohio 43210, USA

Abstract: Interdiffusion can be a major cause of failure in coated parts that see service at elevated tempera-

tures. Ways to measure the extent of interdiffusion and mathematical equations for predicting these meas-

ures are given. The equations are based on the error function solution to the diffusion equation and do not

take into account variations of the diffusivity with composition. Also, when the substrate of the coating is

multiphase, the equations do not take into account the precipitate morphology, but do take into account that

precipitates can act as sinks or sources of solute as the average composition of the substrate varies. The

equations are meant to be alloy design tools that indicate how changing substrate or coating chemistry will

reduce the extent of interdiffusion.

Key words: interdiffusion; high temperature coatings; multicomponent diffusion; multiphase diffusion;

coating design

Introduction

When high temperature coatings interdiffuse with their underlying substrate, the changes in composition can reduce the life of a part either by reducing the oxida-tion resistance of the coating or by compromising the mechanical properties of the coating and/or substrate. In such cases, interdiffusion predictions can be equated to life-time predictions.

Interdiffusion can be predicted using various meth-ods depending on the accuracy needed and the data-bases available. Of these, the phase field method[1] is the most accurate. It can predict microstructure mor-phology and phase constitution with remarkable accu-racy, but requires the use of highly trained personnel and extensive computer time. A finite difference pro-gram, DICTRA[2], requires less training and runs on a PC, but cannot take precipitate morphology into account. A third option is to estimate interdiffusion

using mathematical equations. That is the option to be discussed herein. The equations are less precise be-cause they normally must assume that all kinetic pa-rameters are constant across the interdiffusion zone and that precipitates can only act as point sources or sinks of solute. However, their simplicity can make them a valuable tool for alloy and coating design.

In the following discussion, three measures of inter-diffusion will be defined: the diffusion distance, the amount of interdiffusion, and the composition of the coating surface. Regardless of the number of phases or components in the coating/substrate system, there are various approximations that can predict these measures. Therefore, even in the case of a two-phase, γ +β MCrAlY coating on a γ +γ ′ nickelbase superalloy that may contain five or more components, one could still use the single-phase equation to obtain a first approximation.

1 Diffusion Equation

The fundamental equation used to model diffusion is the well known differential equation[3] for the variation of concentration, C, with distance, x, and time, t:

﹡ Received: 2005-05-18 Supported by the NSF (No. DMR 0139705)

﹡﹡ E-mail: morral@matsceng.ohio-state.edu

J. E. Morral：Predicting Interdiffusion in High Temperature Coatings 705

xCD

xtC

∂∂

∂∂

=∂∂

(1)

When dealing with coating problems, it is helpful to normalize Eq. (1) by multiplying both sides by the ini-tial coating thickness squared to create normalized variables[4]. These are useful when plotting data that are obtained from samples having coatings with differ-ent initial thicknesses, 0.x However, in the present work, there is no advantage to normalizing the vari-ables. Therefore, solutions will be written explicitly in terms of ,x 0x , and t. Also Eq. (1) will be simplified to:

2

2

xCD

tC

∂∂

=∂∂

(2)

by assuming that the diffusivity, D, is constant. Equa-tion (2) is Fick’s Second Law[3].

For an n-component system, the same equation ap-plies except that the concentration, C, is an (n–1)-dimensional column vector and the diffusivity, D, is an (n–1)×(n–1) property matrix. In what follows the “rounded bracket notation” [5] will be used with col-umn vectors written as [C ) and square matrices as [D]. Rows and columns of square matrices will be written as (Di] and [Di), respectively. The subscript i is a num-ber which indicates which row or column is involved.

2 Measures of Interdiffusion

Three measures of interdiffusion can be calculated based on error function solutions to Eq. (2). One is the diffusion distance, Dx , which for tracer atoms is related to the root-mean-squared distance traveled by a tracer atom in time t. In terms of the error function solution,

Dx can be equated to the distance where the error

function argument, (2 )x Dt , is one, i.e., when erf (1)=0.84. Considering that the error function varies between 0 at the origin and 1 at infinity, it follows that the diffusion distance gives an estimate of where the majority of interdiffusion has occurred. The diffusion distance is illustrated in Fig. 1 on a schematic concen-tration profile. A second measure of interdiffusion is the amount of solute that has crossed the coating/substrate interface in the diffusion time, tD. It is illustrated in Fig. 1 by the cross-hatched area, S. A separate value is obtained for each solute and the values can be positive or negative.

A negative value corresponds to solute leaving the coating, as illustrated in Fig. 1. A positive value indi-cates that solute is entering the coating from the sub-strate. Related measures of the amount of interdiffu-sion are obtained by adding absolute values of S for each solute and the solvent[6] or by adding squares of S for each component in the alloy[7].

Fig. 1 Concentration profile of a same-phase, coat-ing/substrate system after interdiffusion illustrating three measures of interdiffusion: the diffusion dis-tance, , the amount of interdiffusion, S (cross-hatched area), and the concentration at the coating surface, .

Dx

SC

A third measure of interdiffusion is the concentra-tion of components at the surface, CS. This concentra-tion varies between the initial coating concentration, C0, and the final concentration reached, CF, once the coating has equilibrated with the substrate. There is a value of CS for each component in the alloy just as there was for S.

3 Same-Phase Approximation

It is possible for a high temperature coating and its substrate to have the same crystal structure, but it is unlikely that the diffusivity will be constant over the entire concentration range of the interdiffusion zone. However, the error introduced by not having a constant D can be minimized by selecting a diffusivity for the calculations that is representative of the concentration at 0x . Equations for the three measures are given in the following.

3.1 Binary systems

The equations for binary systems can be found in stan-dard textbooks on diffusion, for example Ref. [3].

D 2 Dx r t= (3)

Tsinghua Science and Technology, December 2005, 10(6): 704-708

706

0DtS r= − ∆π

C (4)

S 0

D

erf2

xC C C

r t∞

⎛ ⎞= + ∆⎜ ⎟⎜ ⎟

⎝ ⎠

0 (5)

in which 1

2r D= (6) and

0 0C C C∞∆ = − (7) The square root of the diffusivity, r, is used in these equations because it simplifies equations given later that apply to multicomponent systems. It can be seen above that not much can be done to retard interdiffu-sion in binary systems except to choose compositions for the coating and substrate that favor a reduced value for the diffusivity.

3.2 Ternary systems

For ternary systems, the equations are:

D 1 D2x r t= % (8)

( )0D1 1 2 2i i i

tS r C r= − ∆ + ∆π

0C (9)

(S 1 001 11 1 12

1 D

erf2i i i

xC C C C

r tα α α∞ −

⎛ ⎞= + ∆ + ∆⎜ ⎟⎜ ⎟

⎝ ⎠%)0

2 +

(1 002 21 1

2 D

erf2i

xC C

r tα α α−

⎛ ⎞∆ + ∆⎜ ⎟⎜ ⎟

⎝ ⎠%)0

22 2 (10)

where is the major eigenvalue and is the minor eigenvalue of the square root diffusivity matrix [r], de-fined by

1r% 2r%

[8]:

[ ] [ ]12r D= (11)

The [ ]α matrix and its inverse along with the ei-

genvalues of [r] are obtained from the equation:

[ ] [ ][ ][ ] 1r rα α −=% (12)

in which [ ]r% is a matrix with eigenvalues along the

diagonal and zeros as cross coefficients, while the in-verse of the alpha matrix, [ ] 1α − has columns that are

eigenvectors. The composition differences, 0iC∆ , are

given by: 0 0i i iC C C∞∆ = − (13)

Equation (8) follows from the error function solution

as described above, Eq. (9) is from Ref. [5], and Eq. (10) is adapted from Ref. [9]. The relationships be-tween a matrix and its eigenvalues and vectors can be found in any elementary book on matrix algebra and is contained in modern spread sheet software.

It can be seen in Eq. (8) that the diffusion distance is only a function of the diffusivity matrix as in the case of binaries. However, the amount of solute crossing the coating/substrate interface depends on the sum of two terms. These two terms can be adjusted so that the sum is zero because the concentration differences can be positive or negative as indicated by Eq. (13). Therefore, by alloy design one can prevent the loss of a key solute from the coating, e.g., Al, by setting SAl equal to zero and thereby making the coating/substrate interface a “zero-flux-plane[10].” In principle, an n-component sys-tem can have a zero flux plane at the interface for up to n–2 components[11]. Also, it is possible in principle to add the element to the coating by adjusting the concen-tration differences so that Si is positive. In practical ap-plications, such conditions may be impractical, but the equations can still be used to make modifications that will reduce interdiffusion and thereby increase coating stability.

With regard to the concentration at the surface, it is not possible to eliminate concentration changes with time, but one can slow the changes by reducing the

term. This will reduce the term in Eq. (10) that contains the major eigenvalue of [r]. The major eigenvector term is the most rapidly changing term in the equation at early times.

011 1 12 2( C Cα α∆ + ∆ 0 )

3.3 Multicomponent systems

The extension of the above treatment from ternary to higher order systems is simplified by writing the equa-tions in matrix form. The diffusion distance remains the same, just being a function of the major eigenvalue. However, the amount of interdiffusion and the concen-tration at the interface are given by:

[ ) [ ] )0DtS r ⎡= − ∆⎣πC (14)

) ) [ ] [ ][ ] )1S 0C C F Cα α−∞⎡ ⎡ ⎡= + ∆⎣ ⎣ ⎣ (15)

0

D

erf2ij ij

i

xF

r t

⎛ ⎞= δ ⎜⎜

⎝ ⎠%⎟⎟ (16)

where [S ) and [CS ) are n–1 element vectors, [∆C0 ) is

J. E. Morral：Predicting Interdiffusion in High Temperature Coatings 707

the composition vector with n–1 elements like Eq. (13), and δij is the Kronecker delta.

In this formulation, one can see that the amount of a particular solute that crosses the boundary, Si, is pro-portional to the dot product ( ] )0

ir C⎡∆⎣ between the

i-th row of [r] and the composition vector. Accordingly, when the two vectors are perpendicular to each other the amount of interdiffusion of that element is zero and there is a zero-flux plane at the initial interface.

With regard to the concentration at the surface, one can slow the changes by reducing the dot product ( ] )0

1 Cα ⎡∆⎣ . This will reduce the term that contains the

major eigenvalue of [r].

4 Different-Phase Approximation

When the coating and the substrate have different crystal structures, a layered structure forms. The structure may have the original phases only or have additional intermediate phases as shown in Fig. 2. In these cases, constant D solutions have been formulated that take into account the various layers, but they are beyond the scope of this short paper. However, the diffusion distance can be estimated by Eqs. (3) or (8) in which the major eigenvalue used is for the substrate phase. The amount of interdiffusion can be estimated, too, by assuming that most of the solute leaving the coating goes into the substrate phase. Then Eqs. (4), (9) or (14) can be applied using the diffusivity for the substrate phase and, instead of using the composition vector, using )BC⎡∆⎣ . The elements of the vector are

defined by Eq. (17) in which is the concentration BiC

Fig. 2 Concentration profile of a different-phase, coating/substrate system after interdiffusion illustrat-ing two measures of interdiffusion: the diffusion dis-tance, D , and the approximate amount of interdiffu-sion, S

xi (cross-hatched area).

at the boundary of the substrate phase as shown in Fig. 2:

B B2( )i i iC C C∞∆ = − (17)

For this case, one cannot estimate the surface concen-tration using linear equations.

5 Multiphase Approximation

When the substrate phase contains more than one phase, one can estimate an effective diffusivity,

effD⎡ ⎤⎣ ⎦ , which is related to the diffusivity in the matrix

phase, matrixD⎡ ⎤⎣ ⎦ , and the phase diagram by the follow-

ing equations[12,13]: eff matrix TMD D C⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (18)

matrixTM iij

j

CC

C∂

=∂

(19)

The transformation matrix, , changes the

variables in the diffusion equation from concentrations in the matrix, , to a local average concentrations of all phases present,

TMC⎡⎣ ⎤⎦

matrixiC

jC . The effective diffusivity

makes the simplifying assumption that precipitates act as point sources or sinks of solute, but do not otherwise contribute to the diffusion flux. The solution in this case is the same as the different-phase approximation, except that all the eigenvalues and vectors are obtained from the effective diffusivity. Accordingly, the diffu-sion distance and the amount of interdiffusion can be written as:

effD 1 D2x r t= % (20)

and

[ ) )eff BDtS r⎡ ⎤ ⎡= − ∆⎣ ⎦ ⎣πC (21)

The interpretation of these equations is the same as that given in the “different-phase approximation” sec-tion above.

6 Summary

Three measures of the extent of interdiffusion between a coating and its substrate have been defined: the diffu-sion distance, the amount of interdiffusion, and the concentration of solute at the coating surface. Equa-tions for predicting these quantities have been given

Tsinghua Science and Technology, December 2005, 10(6): 704-708

708

that are based on the error function solution to the dif-fusion equation. The equations assume that the diffu-sivity is a constant across the interdiffusion region. A number of other approximations are made as well de-pending on the constitution of the coatings, substrate, and interdiffusion zone. Equations for multicomponent systems are written in terms of matrices and matrix properties. These equations are readily computed using spread sheets when sufficient diffusivity and composition data are available.

Acknowledgements

The encouragement of University of Cincinnati Professor Ray Lin to write this paper is very much appreciated.

References

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[4] Morral J E, Barkalow R H. Analysis of coating/substrate

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[6] Morral J E, Thompson M S. Interdiffusion and coating design. Surface and Coating Technology, 1990, 43/44: 371-380.

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[8] Morral J E. Rate constants for interdiffusion. Scripta Met-all., 1984, 18: 1251-1256.

[9] Kirkaldy J S, Young D J. Diffusion in the Condensed State. London: Institute of Metals, 1987.

[10] Kim C W, Dayananda M A. Zero-flux planes and flux re-versals in Cu-Ni-Zn diffusion couples. Metall. Trans., 1979, 10A: 1333-1339.

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