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

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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

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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

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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)

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Questions

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