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Phase Transformations 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
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MLB - Core Courses-3.21-
Kinetic processes in materials
Jean-Philippe PéraudCourse 2
May 4th, 2012
1
3 main partsMass Diffusion in general-Mathematical concepts-Diffusion equation and associated phenomena
-Solution methods
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
• Self-diffusivity *D
• Intrinsic diffusivity
• Interdiffusivity,Darken equation
1 1
*1
4
Interdiffusion & Kirkendall
A B
JA
JBJV
v
Vacancy sinksDislocation shrink
Vacancy sourcesDislocation climb
C-Frame
5
Solution of diffusion equation:Toolbox
Step function Point source
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 factor9
Diffusion in ionic crystals• 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
1/T
D
10
Position depends on PO2
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
14Fluxes of atoms joining or leaving the particle
shrinksgrows
Continuous transformations: spinodal decomposition
• Due to concave free energy profile in miscibility gap
• Be able to explain Cahn-Hilliard equation
• Kinetics: use perturbation to derive critical and thermodynamic wavelength+amplification factor
Credit: Balluffi, Allen, Carter, Kinetics of Materials
15
Continuous transformations: order-disorder transformation
• No energy barrier in concave up regions
• Be able to explain Allen-Cahn 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