Diffusion in substitutional binary systems

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Diffusion in substitutional binary systems
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Objectives

• Darkens analysis of Kirkendalls experiment,
  continued
• The Boltzmann-Matano analysis

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From last class
X0
If D1ltD2 then J1ltJ2
Darken showed that
Darkens relation 2
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Darkens analysis of Kirkendalls experiment,
continued
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Goals of the analysis

• Quantify the marker movement based on the
  individual diffusion constants Done!
• Can we replace the two individual diffusion
  constant with a combined one that can be inserted
  into Ficks 2nd law? Done!
• How do we evaluate ?

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27-216 Transport in Materials The
Boltzmann-Matano analysis
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How do we choose the position of this?!
t t
CC
X
NB the next few slides provide details of the
Boltzmann-Matano analysis skip to find the next
topic
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Flipping the axes
C
x
R
C 0
In this analysis, it is useful to think of
flipping the axes such that one is integrating
distance (the independent variable) with respect
to concentration (the dependent variable)
(y-axis), instead of the other way around.
In other words, we are interested in the area
under the x x(C) curve.
x(C)
x
C
M
C
L
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Boltzmann-Matano analysis overview

• In what follows, an analysis is provided that
  shows you how to extract the interdiffusion
  coefficient from experimental data from a
  diffusion couple.
• The point with the greatest potential for
  confusion is the logic that is used to locate the
  Matano Interface based on the boundary
  conditions.
• So, think of it this way the boundary
  conditions allow one to relate the area under the
  concentration curve to a boundary condition this
  then means that the integro-differential equation
  (that is the solution to Ficks 2nd Law with
  variable diffusivity) only works for a particular
  choice of where to put the origin for the spatial
  coordinate (x).

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Boltzmanns Transformation
Ficks second law with variable diffusivity D (C
) shows
Slide from module 11, Inverse Methods, Glicksman
and Lupulescu
11
Boltzmanns Transformation
Writing in terms of ? and using the chain rule,
we obtain
Slide from module 11, Inverse Methods, Glicksman
and Lupulescu
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Boltzmanns Transformation
The Ficks second law, in its nonlinear form, may
be reduced (simplified) to a nonlinear ODE
Slide from module 11, Inverse Methods, Glicksman
and Lupulescu
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Matanos Geometry
Slide from module 11, Inverse Methods, Glicksman
and Lupulescu
The Boltzmanntransformed Ficks second law, and
the experimental configuration for finding D(C)
is called the BoltzmannMatano method.
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Matanos Geometry
Slide from module 11, Inverse Methods, Glicksman
and Lupulescu
The solute distribution provides a gradient,
dC/d?, that vanishes at C CR .
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Matanos Interface
Slide from module 11, Inverse Methods, Glicksman
and Lupulescu
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Matanos Interface
Slide from module 11, Inverse Methods, Glicksman
and Lupulescu
The Matano Interface is determined through a
massconservation condition
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Matanos Interface
Slide from module 11, Inverse Methods, Glicksman
and Lupulescu
Simplifying
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2. Identify the Matano Interface
Evaluating the interdiffusion coefficient,
D-tilde, according to the Boltzmann-Matano
analysis
x
3. Graphical evaluation at t
C
0
X0
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Summary

• Darkens analysis of Kirkendalls experiment
  continued
• The Boltzmann-Matano analysis Graphical way of
  determining

Darkens relation 2