MLB - Core Courses
-3.21Kinetic processes in materials
Jean-Philippe Péraud
Course 2
May 4th, 2012
1
Mass Diffusion in general
-Mathematical concepts
-Diffusion equation and
associated phenomena
-Solution methods
3 main parts
Diffusion processes
-relate microscopic effects
to macroscopic equations
and parameters
Phase Transformations
-continuous
-discontinuous
-kinetics of transformation
-stability problems 2
Forces and fluxes
• Force-flux relations
force=-grad(potential)
• Onsager
• Constraint on quantities
merges potentials.
network constraint,
electrochemical,
elastochemical potential…
3
Diffusivities and frames
*1
• Self-diffusivity *D
1
1
• Intrinsic diffusivity
• Interdiffusivity,
Darken equation
4
Interdiffusion & Kirkendall
JA
JV
C-Frame
JB
A
B
v
Vacancy sinks
Dislocation shrink
Vacancy sources
Dislocation climb
5
Solution of diffusion equation:
Toolbox
Point source
Step function
Fourier series
+ superposition principle
+method of images
6
Typical problems
C=0
J=0
7
Diffusivity and random walks
• Sequence of random
jumps
• Average displacement
=0
• Average squared
displacement
proportional to D
8
Diffusivity and random walks
• Simple models for
frequency of jumps
• More or less
complicated
depending on
diffusion mechanism
• Correlation factor
9
Diffusion in ionic crystals
D
Position depends on PO2
1/T
• Kröger-Vink notation,
Schottky, Frenkel
defects
• Be able to write the
equation of
incorporation of
impurities
• Use equation of
equilibrium
(Keq)+balance of
charges
• Identify diffusion
regimes
10
Other diffusion mechanisms. In brief.
• In grain boundaries
• In amorphous
materials
• Polymers (by
reptation)
11
Capillary phenomena: surface
smoothing
• By surface diffusion
• By vapor transport
+++
--12
Capillary phenomena: anisotropic
surface tension
13
Capillary phenomena: coarsening and
grain growth
• Diffusion limited
• Source-limited
• N-6 rule
Fluxes of atoms joining
or leaving the particle
shrinks
grows
14
Continuous transformations: spinodal
decomposition
• Due to concave free
energy profile in
miscibility gap
• Be able to explain CahnHilliard equation
• Kinetics: use perturbation
to derive critical and
thermodynamic
wavelength+amplification
factor
Credit: Balluffi, Allen, Carter, Kinetics of
Materials
15
Continuous transformations: orderdisorder transformation
• No energy barrier in
concave up regions
• Be able to explain AllenCahn equation
• Kinetics: use perturbation
to derive critical and
thermodynamic
wavelength+amplification
factor
Credit: Balluffi, Allen, Carter, Kinetics of
Materials
16
Nucleation
• Curve-to-curve and
tangent to curve
construction
• Calculate Rc and ΔGc
• Determine steady state
rate.
• Heterogeneous
nucleation: almost the
same thing
17
• Not covered: stability of moving interfaces
18
Advice
• Get some sleep
• Don’t panic
• Always try to answer (partial credit)
19
Questions
20