On diffusion paths with “horns” and the formation of single phase layers in multiphase diffusion couples

doi:10.1016/j.actamat.2005.03.027

Acta Materialia

Volume 53, Issue 14, August 2005, Pages 3775-3781

https://doi.org/10.1016/j.actamat.2005.03.027 Get rights and content

Abstract

The current study using DICTRA finite difference software predicts sharp deviations from the linear zigzag diffusion paths predicted by an error function model for multiphase diffusion couples. The deviations appear as “horns” that protrude from the linear paths. It was found that the horn length is proportional to the composition vector component along the major eigenvector of the effective diffusivity matrix. Applying these results to a study on Ni–Cr–Al diffusion couples prepared from γ + β alloys, it was also found that the formation of single phase β layers could be attributed to the horns pointing away from each other, in which case the diffusion path could intersect the single phase β region of the phase diagram.

Introduction

Understanding multicomponent interdiffusion behavior is of considerable importance in a variety of applications, for example in predicting the useful life of high temperature coatings. As a result, extensive efforts have been made in recent years to develop numerical methods to solve interdiffusion problems effectively and efficiently. Among them, the error function model [1] is the simplest. It considers that precipitates can act as sources or sinks of solute, but assumes that the effective diffusivity of the composite is constant. It predicts that diffusion couples in the two-phase region will have a linear “zigzag” diffusion path.

DICTRA software provides a finite difference method of treating diffusion in multiphase systems, which can take into account a concentration-dependent diffusivity [2], [3]. As a result, DICTRA is not limited to small composition differences as is the error function model. A feature of the DICTRA simulations is deviations from the linear zigzag diffusion paths that are called “horns” [4]. In this work, it is shown how the DICTRA horns are related to the initial conditions. Then these results are used to explain the formation of single phase layers in certain multiphase diffusion couples.

Section snippets

Background

The error function model predicts that diffusion couples prepared from alloys in the two phase region of a phase diagram will have diffusion paths that consist of three straight line segments, following a “zigzag” course exemplified in Fig. 1. The line segments are parallel to eigenvectors of the effective diffusivity matrix. The effective diffusivity [D^eff] is defined as the product of the continuous phase diffusivity, D^γ, and the transformation matrix, C^TM [1]: $[D^{eff}] = [D^{γ}] \cdot [C^{TM}] .$ The elements

Characterization of “horns”

In ternary systems, the composition vector ΔC is defined as $[\begin{matrix} Δ C_{1} \\ Δ C_{2} \end{matrix}] = [\begin{matrix} C_{1}^{Right} \\ C_{2}^{Right} \end{matrix}] - [\begin{matrix} C_{1}^{Left} \\ C_{2}^{Left} \end{matrix}] .$ In this paper, a bold letter such as ΔC is used to represent a vector, while a normal one like ΔC is used to represent the magnitude of the corresponding vector. The composition vector is usually expressed in terms of coordinates that represent the solute elements, for example mole percent of Al and Cr for the Ni–Cr–Al system. However, for studying “horns”, it is better to express ΔC in terms of

Summary and conclusions

(1)
DICTRA simulations of Ni–Cr–Al, γ + β diffusion couples show sharp deviations from the linear zigzag paths, appearing as inward or outward pointing horns.
(2)
The horn length varies linearly with the component of the composition vector along the major eigenvector direction of the effective diffusivity matrix.
(3)
When horns form inward or outward can be predicted from the orientation of the composition vector.
(4)
The formation of a β layer in experimental γ + β diffusion couples could be associated with

Acknowledgments

The authors are grateful to the National Science Foundation for financial support under Grant No. DMR-0139705 and to Prof. E. Kvam at Purdue University who originally suggested that outward horns could lead to β layer formation.

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