1. Introduction to pattern formation in materials
Adrian Sutton
Department of Physics

3. Some microstructures in materials
Au-Cu martensite
Polycrystal
Ni3Al superalloy
Courtesy of David Grummon, Michigan State University

4. Defects in crystals

5. Edge dislocation

6. Screw dislocation

7. Screw dislocation

8. Mixed dislocation

9. Dislocation sources

10. The Frank-Read source

11. The most useful equation in dislocation theory
Resolving forces normal to dislocation line:
2T sin(d/2) = b dl
dl = Rd
T = b2
b/R

12. Applications of b/R
R = spacing of pinning centres in Frank-Read source
R = average spacing of particles in a dispersion hardened alloy
R = spacing of forest dislocations in work hardening (see later): ~bf
Setting  = th ~ /30 sets lower limit on length scale for dislocations to operate

13. Dislocation pileup in a cylinder

14. Forest interactions

15. Simulation of nanoindentation

16. Snapshots of dislocation network evolution obtained from a DD simulation
of [001] straining.
From: Vasily V. Bulatov et al., Nature 440, (27 April 2006)

17. The black lines correspond to the full [001] straining simulation,
the red lines are for the [011] straining, and the green lines are for
the 'doctored' [001] straining simulation in which two of the four
Burgers vectors are absent.
a, Flow stress as a function of strain.
b, Dislocation line density as a function of strain.
c, Flow stress versus total dislocation density.
d, The fraction of lines involved in multi-junction configurations as a function of the total line density.

18. Dislocation patterning in stage II in Cu at 77K
40 MPa
20 MPa
“Braids” on primary slip system comprising LC locks formed by reactions with inclined slip systems
From Prinz and Argon, Phys Stat Sol a 57, 741 (1980)

19. Dislocation pattern formation in stage II ctd
D = spacing of braids
B = Burgers vector
= shear modulus
= resolved shear stress
(D/b) = 7.80 (/)
Self-organization of dislocations
Into quasi-regular groupings called
‘patterning’

20. Grain growth

21. Large grains grow, small grains shrink

22. Driving force and kinetics
Driven by reduction in total grain boundary area per unit volume:
~D ( Jm-3 (Pa) or eV/Å3)
dD/dt = c/D  D2 = Do2 + 2ct
Ignores variation of  and mobility with 5 degrees of freedom of GBs, also ignores impurities: “abnormal” grain growth

23. Self-similarity
Remarkably uniform appearance: largest grain size / mean grain size is roughly
Steady-state distribution of grain sizes is log-normal: self-similar microstructure
Distribution reproduced by computer simulations
Log-normal distribution arises also in size distribution of precipitates during coarsening (Ostwald ripening) in some alloys, and in particle crushing

24. Log-normal distribution of grain sizes

25. Log-normal distribution
X = grain size
= average of log of grain size
 = standard deviation of log of grain size
A variable is log-normal distributed if it results from the product of many
small independent factors.
For example the long-term return rate on a stock investment can be
considered to be the product of the daily return rates.

26. Log normal distribution in grain growth
D/Di = (Df – Di)/Di = i = random fraction
Dm = (1+ m)(1+ l)(1+ k)…(1+ 1)D1
lnDm = ln(1+ m) + ln(1+ l) + ln(1+ k) + … + ln(1+ l) + lnD1
Central limit theorem then says lnD is normally distributed.
Same argument applies to particle coarsening and powder crushing