Introduction to pattern formation in materials Adrian Sutton Department of Physics

Some microstructures in materials Au-Cu martensite Polycrystal Ni3Al superalloy Courtesy of David Grummon, Michigan State University http://www.egr.msu.edu/classes/mse250/grummon/

Defects in crystals

Edge dislocation

Screw dislocation

Screw dislocation

Mixed dislocation

Dislocation sources http://www.gpm2.inpg.fr/axes/plast/MicroPlast/ddd/index.html

The Frank-Read source

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

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

Dislocation pileup in a cylinder http://www.gpm2.inpg.fr/axes/plast/MicroPlast/ddd/index.html

Forest interactions http://www.gpm2.inpg.fr/axes/plast/MicroPlast/ddd/index.html

Simulation of nanoindentation http://www.gpm2.inpg.fr/axes/plast/MicroPlast/ddd/index.html

Snapshots of dislocation network evolution obtained from a DD simulation of [001] straining. From: Vasily V. Bulatov et al., Nature 440, 1174-1178 (27 April 2006)

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.

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)

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’

Grain growth http://www.math.cmu.edu/people/fac/kinderlehrer.html

Large grains grow, small grains shrink

Driving force and kinetics Driven by reduction in total grain boundary area per unit volume: ~D (107-103 Jm-3 (Pa) or 10-4-10-8 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

Self-similarity Remarkably uniform appearance: largest grain size / mean grain size is roughly 2.5 - 3.0. 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

Log-normal distribution of grain sizes

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.

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