MLB - Core Courses -3.21 Kinetic 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

*1 1

Diffusivities and frames

• Self-diffusivity *D 1 • Intrinsic diffusivity • Interdiffusivity, Darken equation 4

Interdiffusion & Kirkendall

J A J V J B A B

C-Frame

v Vacancy sinks Dislocation shrink Vacancy sources Dislocation climb 5

Solution of diffusion equation: Toolbox Step function Point source Fourier series + superposition principle +method of images 6

C=0

Typical problems

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

D

Diffusion in ionic crystals

Position depends on P O2 1/T • • • • Kröger-Vink notation, Schottky, Frenkel defects Be able to write the equation of incorporation of impurities Use equation of equilibrium (K eq )+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 Fluxes of atoms joining or leaving the particle • Source-limited • N-6 rule shrinks grows 14

Continuous transformations: spinodal decomposition • Due to concave free energy profile in miscibility gap • Be able to explain Cahn Hilliard equation Credit: Balluffi, Allen, Carter, Kinetics of

Materials

• Kinetics: use perturbation to derive critical and thermodynamic wavelength+amplification factor 15

Continuous transformations: order disorder transformation • No energy barrier in concave up regions Credit: Balluffi, Allen, Carter, Kinetics of

Materials

• Be able to explain Allen Cahn equation • Kinetics: use perturbation to derive critical and thermodynamic wavelength+amplification factor 16

Nucleation

• Curve-to-curve and tangent to curve construction • • Calculate R c and ΔG c 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