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Massachusetts Institute of Technology

This document summarizes key concepts from a lecture on solutions to the diffusion equation when diffusivity is concentration-dependent or time-dependent. It discusses methods like the Boltzmann-Matano method for determining concentration-dependent diffusivity from diffusion couple experiments. It also covers anisotropic diffusion, describing how diffusivity is a tensor quantity and principal axes allow decoupling of diffusive fluxes. Related exercises in the textbook Kinetics of Materials are listed.

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48 views1 page

Massachusetts Institute of Technology

This document summarizes key concepts from a lecture on solutions to the diffusion equation when diffusivity is concentration-dependent or time-dependent. It discusses methods like the Boltzmann-Matano method for determining concentration-dependent diffusivity from diffusion couple experiments. It also covers anisotropic diffusion, describing how diffusivity is a tensor quantity and principal axes allow decoupling of diffusive fluxes. Related exercises in the textbook Kinetics of Materials are listed.

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© © All Rights Reserved
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Massachusetts Institute of Technology

Department of Materials Science and Engineering


77 Massachusetts Avenue, Cambridge MA 02139-4307

3.21 Kinetics of Materials—Spring 2006

February 27, 2006

Lecture 7: Solutions to the Diffusion Equation—I.

References
1. Balluffi, Allen, and Carter, Kinetics of Materials, Sections 4.3–4.5.
Key Concepts
• When the diffusivity D is concentration-dependent, the diffusion equation is nonlinear and closed-
form solutions to practical problems don’t exist. The “Boltzmann–Matano” method is a graphical one
for using a measured c(x)
� �profile from a diffusion-couple experiment to determine D(c), using the
� c1 � cL
� 1
relation D(c1 ) = − 2τ dcdx
c1 cR x(c) dc after setting the position x = 0 such that cR x dc = 0.

• Examination of asymmetry in an interdiffusion profile c(x) gives useful information about trends in
the concentration dependence of D(c): D will be larger on the side with the shallower c(x) profile,
and D will be smaller on the side with the steeper c(x) profile (see KoM Exercise 4.2).
• When D is time dependent (e.g., when temperature changes occur� during a diffusional process), a
simple approach using a time-weighted diffusivity defined by τD = 0t D(t ) dt allows Fick’s second
′ ′

law to be transformed into the alternate linear form ∂τ∂cD = ∇2 c. Familiar solution methods to solving
the diffusion equation such as error functions and point sources can be readily adapted to cases where
D is time dependent.
• In crystals and other anisotropic materials, D is generally anisotropic. Because D relates two vectors,
D is a second-rank tensor quantity. Note however that symmetry considerations dictate that for cubic
crystals, D is isotropic.
• The mathematical description of anisotropic diffusion depends on the choice of coordinate axes. Fre­
quently, the most convenient choice is parallel to high-symmetry crystal axes.
• When anisotropic diffusion is described in special coordinate axes termed principal axes, the diffu­
sivity tensor is diagonal, and diffusive fluxes along each principal axes are effectively uncoupled.
• Given a diffusivity tensor, finding its eigensystem (eigenvalues and eigenvectors) determines its prin­
cipal axes and the prinicipal values of the diffusivity tensor along the diagonal in the principal axis
coordinates.
• Crystal symmetry dictates the form of the diffusivity tensor in the crystal axis system, i.e., where the
non-zero terms will be, and which non-zero terms must be equal.
• A scaling transformation, KoM Eq. 4.64, permits solutions for isotropic D to be readily adapted to
cases in which D is anisotropic (see KoM Exercise 5.9).
Related Exercises in Kinetics of Materials
Review Exercises 4.1–4.8, pp. 91–97.

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