M.A.$Groeber1,$J.$Simmons1,$C.$Przybyla1,4,$J.$Tucker2,$A.D.$

Rolle@2,$M.$Comer3,$K.$Barmak2,$D.$Kinderlehrer2,$R.$

Schwab2,$S.$Ta’asan2,$D.$Rowenhorst5,$R.M.$Suter2,$G.S.$

Rohrer2,$S.F.$Li2,$C.M.$Hefferan2,$J.$Lind2,$X.$Tan2,$U.$Lienert6$

Upper$Tails$in$Grain$Size$

DistribuRons$

1$Air$Force$Research$Laboratory$

2$Carnegie$Mellon$University$

3$Purdue$University$$

4$Georgia$Tech$University$

5$Naval$Research$Laboratory$

6$Advanced$Photon$Source$
MoRvaRon$

MoRvaRon$to$Incorporate$Extreme$Values

NiZbase$superalloys

FaRgue$crack$iniRaRon$was$observed$

in$large$grains$oriented$for$slip$$

(Jha,$et$al.,$2007)

‘Forget(the(Representa/ve(Volume(Element,(show(me(the(Weakest'

Volume(Element’(–(paraphrased(from(Jim(Williams

?
Background$

•"3D"EBSD"data"from"FIB1SEM"

•"100μm"x"35μm"x"45μm"

•"0.25μm"Sec=on"thickness"

•"0.25μm"EBSD"point"spacing

•">"8500"Grains"in"total"

•">"5800"Grains"analyzed"

•"24"Voxel"minimum"size"(~"0.5μm)"

PM$IN100$NiZbase$Superalloy$Example
Combined$Probability$Plot$
High"Energy"Diffrac=on"Microscopy"(HEDM):"

Microstructure"Mapping"

•"Near1field"measurement""

•">"50keV"x1rays"(monochroma=c)"

•"1"–"2"μm"beam"height"

•"1.3"mm"beam"width"

•"1"–"4"μm"detector"pixels"

•"L"="5"–"10"mm"

•"Spa=al"resolu=on:""

2"–"4"μm

•"Orienta=on"resolu=on:"

<"0.5"degrees"

•"Analysis:"

15"–"50"layers"/"12hours"

Suter et(al,$Eng$Mat$&$

$$Tech$2007$

Suter et(al,$RSI$2006$

Lauridsen et(al.,$Appl

$$$Cryst$2001$

Carnegie

Mellon

MRSEC 5$

New:$100$layers$in$

less$than$1$day$
HP"Al:"Intra1granular"Misorienta=ons"

OrientaRons$

2$deg$boundaries$

•$Expanded$color$scale$$

•$2$deg$boundaries$

MisorientaRon$ Next:$DeviaRons$from$

average$within$2$degree$

boundaries:$4$degree$color$

scale$
Grain$Growth$Dynamics$in$High$Purity$Aluminum$

IniRalZFinal$MisorientaRon$

IniRal$state$boundaries$

Carnegie

Mellon

MRSEC 7$
8$

Geometrically$Necessary$DislocaRon$Content$

IniRal$State$ Final$State$

Calculated$as$the$ReadZShockley$scaled$energy$associated$with$each$point;$

max.$scaled$to$0.3.$$Trend$is$for$the$GND$content$to$decrease.$
9$

Analysis$by$LAGB$Content$

Matched$grains$between$the$

iniRal$and$final$states$according$

to$orientaRon.$

Excluded$edge$grains.$

Computed$the$change$in$volume$

between$snapshots.$

Computed$the$Low$Angle$

Boundary$Content$(as$a$rough$

measure$of$GND)$by$grain.$

Plo@ed$∆V$vs.$GND.$

Obtained$expected$trend$of$high$

stored$energy$leading$to$

decrease$in$size$(shrinks)$and$

vice(versa.$

-800

-600

-400

-200

0

200

400

600

800

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Al-final-trimmed-mi#2E6E802

!V

!

V

ave_misor
Three$Volumetric$Data$Sets$with$Well$Ordered$Grains$

1. Pure$Nickel:$42$layers,$4$micron$spacing,$0.16$mm3$

2. BiZdoped$Nickel:$62$layers,$4$micron$spacing,$0.24$mm3$

3. Pure$Copper:$177$layers,$4$micron$spacing,$0.56$mm3$

Next$slide:$movie$through$

volume$mesh$of$NiZBi$

3,496$grains;$~$23,598$GBs
StaRsRcs$extracRon$from$large$data$sets$

Neighbor$

misorientaRon$

angle$

distribuRon$
Σ27a

StaRsRcs$extracRon$from$large$data$sets$

Neighbor$

misorientaRon$

angle$

distribuRon$

ΣΣ2773 b
logrgroeber

Probability in %

0.5 1.0 1.5 2.0 2.5 3.0

0.1

1

5 20 50 80 95 99.9

Quantile: qnorm

IN100:$Probability$Plot$

log(radius) (μm)
Analysis:$IN100$

CDFs$of$Extreme$Values$as$a$FuncRon$of$Sample$Size
Analysis:$Po@s$model$

Histogram of R2050

R2050

Frequency

0 10 20 30

0 1000 2000 3000 4000

This$dataset$is$taken$from$a$snapshot$at$2050$MCS.$$Clearly,$logZnormal$is$a$

poor$descripRon$of$the$distribuRon$of$radii,$and$the$strong$upward$departure$

of$the$upper$tail$fits$with$the$apparent$cutZoff$in$size.$
0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.0 0.2 0.4 0.6 0.8 1.0

 from Seth, MC GG

CDF

5000

2500

1000

500

250

100

50

25

10

1

Sphere Equivalent Radius

2050$MCS$

Same$dataset$as$in$

previous$slide.$$44719$

grains.$

Analysis:$Po@s$model$
Analysis:$Ti$beta$21S$

Histogram of Rrowen

Rrowen

Frequency

0 10 20 30 40 50 60

0 200 400 600 800

log(Rrowen/20.16)

Probability in %

-3 -2 -1 0 1

0.1 1 5 20 50 80 95 99

Quantile: qnorm

Data$analyzed$from$a$dataset$with$4396$grains$for$betaZstabilized$Ti$alloy$(21S).$$Data$

courtesy$of$Dave$Rowenhorst,$NRL.$$3D$image$generated$by$serial$secRoning,$opRcal$+$

EBSD.$$

Rowenhorst$DJ,$Lewis$AC,$Spanos$G.$ThreeZdimensional$analysis$of$grain$topology$and$interface$

curvature$in$a$betaZRtanium$alloy.$Acta(mater.$2010;$58:$5511.$
Analysis:$BetaZTi$

2 4 6 8 10 12 14

0.0 0.2 0.4 0.6 0.8 1.0

BetaïTi from Dave Rowenhorst

CDF

5000

2500

1000

500

250

100

50

25

10

1

Radius

The$“bunching$up$of$the$CDFs$at$the$larger$sizes$suggests$that$there$is$an$upper$cutoff$

in$size.$$This$is$in$contrast$to$the$IN100$result.$$$
100nm 

Result is average grain size and grain size 

distribution. Reported data includes 21 

samples and 17,882 grains 

19 

Grain"Size"Measurements"Based"upon"TEM"

Observa=ons"
20 

 Stagnation of thin film grain growth

K. Barmak, et. al. Scripta. Mater., 40 (2006)

Strong {111} orientation texture

Al-film deposition

Growth experiments

MC&PDE simulations

Retarding"Forces"

Z$Grooving$

Z$Impurity$drag$$$

Z$Precipitate$pinning$$

Z$Triple$juncRon$drag$$

Z$Stress$on$boundary$

Network,$e.g.$residual$

stress
log(Rbarm)

Probability in %

-3 -2 -1 0

0.1 1 5 20 50 80 95 99

Quantile: qnorm

Analysis:$Thin$Films$

Histogram of Abarm

Abarm

Frequency

0 5 10 15

0 500 1000 1500 2000 2500

Data$provided$by$Katy$Barmak$for$areas$of$grains$in$thin$films$of$Al$analyzed$(mainly)$by$

Wayne$Archibald.$$Films$annealed$for$1,$2,$4$and$10$hours;$areas$normalized$by$average$

in$each$set.$$Areas$converted$to$circleZequivalent$radii.$Second$plot$has$addiRonal$data$

from$Derrick$Carpenter$and$Jihwan$Kim.$

The$distribuRon$is$remarkably$close$to$logZnormal,$with$rather$small$departures$at$the$

upper$and$lower$ends.$

lbarm$Reduced_Area_1_2_4_10_Derr_Jhiw

Probability in %

-3 -2 -1 0

0.1 1 5 20 40 70 90 99 99.9

Quantile: qnorm
In$keeping$with$the$upper$tail$seen$in$

the$nearly$logZnormal$distribuRon,$

there$is$li@le$sign$of$an$upper$cutZoff$

in$size$in$this$sampling$analysis.$

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.0 0.2 0.4 0.6 0.8 1.0

Barmak/ 1, 2, 4, 10, Derr, Jihw

CDF

5000

2500

1000

500

250

100

50

25

10

1

Area

Analysis:$Thin$Films:$Limits$
Combined$Probability$Plot$
Asymptote$for$Largest$Grain?$

The$median$of$

each$CDF$is$

extracted$and$

plo@ed$versus$the$

logarithm$of$the$

subZsample$size$

02468

0.15 0.20 0.25 0.30 0.35

log(Sample Size)

Median(Largest Member)

02468

1.5 2.0 2.5 3.0 3.5 4.0 4.5

log(Sample Size)

Median(Largest Volume)

02468

6 8 10 12 14

log(Sample Size)

Median(Largest Volume)

02468

0.38 0.40 0.42 0.44 0.46 0.48 0.50

log(Sample Size)

Median(Largest Volume)

(a)$Po@s$model$$ (b)$Ti$(betaZ21S)$

(c)$Ni$alloy$(IN100)$$ (d)$Al,$Cu$thin$film$
Summary$

• Damage$iniRaRon$is,$generically,$an$upper$tail$problem.$

• Large$datasets$from$simulaRon$and$3D$characterizaRon$enable$

upper$tails$to$be$invesRgated.$

• LogZnormal$can$be$applied,$as$suggested$in$the$literature,$but$only$

over$a$limited$range$of$the$data.$

• The$thin$film$data$most$nearly$approach$log$normal.$

• The$microstructures$from$the$Po@s$model$deviate$strongly$from$

lognormal;$the$subset$sampling$analysis$also$suggests$the$presence$of$

a$hard$cutoff$in$maximum$grain$size,$as$one$expects$from$coarsening$

theory$

• The$IN100$and$TiZ21S$microstructures$also$deviate$noRceably$from$

lognormal.$

• There$is$an$apparent$correlaRon$with$stagnaRon$or$pinning$of$grain$

growth:$stagnant$microstructures$approach$logZnormal.$

• Need:$more$(large)$datasets;$theoreRcal$distribuRons;$distribuRons$

of$other$microstructural$features;$apply$theorems$on$extremes$of$

samples.$
CriRcal$Issues$

•ExtrapolaRon$

•$QuanRficaRon$of$Error$in$ExtrapolaRng$CDFs$for$Large$Ns$

•$Extreme$Values$Not$Large$DeviaRons$

• Missing$Details$of$DistribuRon$Between$Breakdown$of$Lognormal$

and$Extreme$Values$(i.e.$2σ$to$4+σ)$

•$Neighborhoods$and$i.i.d.$AssumpRon$

•$How$to$Build$Neighborhoods$Around$Grains$Never$Before$Seen$

• i.i.d.$May$not$Apply$for$Abnormal$Grain$Growth$

•$ReZSampling$Discrete$Samples$

• $QuanRficaRon$of$Effect$of$Sampling$Discrete$Values$Instead$of$

True$DistribuRon$