advances.sciencemag.org/cgi/content/full/4/11/eaao6051/DC1

Supplementary Materials for

The origin and stability of nanostructural hierarchy in crystalline solids

S. Meher*, L. K. Aagesen*, M. C. Carroll, T. M. Pollock, L. J. Carroll

*Corresponding author. Email: subhashish.meher@inl.gov (S.M.); larry.aagesen@inl.gov (L.K.A.)

Published 16 November 2018, Sci. Adv. 4, eaao6051 (2018)

DOI: 10.1126/sciadv.aao6051

This PDF file includes:

Fig. S1. A low-magnification SEM micrograph showing dendritic-interdendritic microstructure in F-11 alloy. Fig. S2. Coarsening behavior of secondary γ′ precipitates in F-11 alloy at 800°C. Fig. S3. Coarsening behavior of tertiary γ′ precipitates in F-11 alloy at 800°C. Fig. S4. Coarsening kinetics of secondary γ′ precipitates in F-11 alloy at 1000°C. Fig. S5. Air cooling rate curve of F-11 alloy after homogenization at 1285°C for 12 hours. Fig. S6. A dark-field TEM micrograph of homogenized samples of F-11 alloy shows nucleation of multiple generation of γ′ precipitates and no chemical splitting of secondary γ′ precipitates. Fig. S7. Coarsening kinetics of γ precipitates in the γ′ matrix in the interdendritic region in F-11 alloy during isothermal annealing at 800°C. Table S1. Compositions of the γ and γ′ phases in the interdendritic and dendritic regions after isothermal annealing at 800°C for 1500 hours. Table S2. APT estimated composition of the γ′ precipitates in the dendritic region after the homogenization heat treatment and after annealing at 800°C for various times. Table S3. Parameters used for phase-field modeling of the hierarchical microstructure. References (37–41)

Supplementary Figures

Fig. S1. A low-magnification SEM micrograph showing dendritic-interdendritic

microstructure in F-11 alloy.

Fig. S2. Coarsening behavior of secondary γ′ precipitates in F-11 alloy at 800°C. A plot of

log of radius (r) against log of time (t) of ’ precipitates in the dendritic region after 200 hours of

annealing at 800°C that indicates significant deviation and slower coarsening rate than predicted

by LSW theory.

Fig. S3. Coarsening behavior of tertiary γ′ precipitates in F-11 alloy at 800°C. (A) the

accelerated, and selective coarsening and dissolution of tertiary ’ precipitates was observed

during isothermal annealing at 800°C. (B) The mean radius evolution of secondary and tertiary ’

precipitates during isothermal annealing at 800°C.

Fig. S4. Coarsening kinetics of secondary γ′ precipitates in F-11 alloy at 1000°C. (A) a plot of

log(radius) of secondary ’ precipitates against log(time) for isothermal annealing at 1000°C for F-11 that

yields the inverse of temporal coarsening exponent suggesting classical coarsening as explained by LSW

model, (B) a plot of cube of mean radius (r3) of secondary ’ precipitates vs. time (t) for F-11 that yields

the LSW coarsening rate (K) of secondary ’ precipitates as 1.39 x 10-26 m3/s at 1000°C.

Fig. S5. Air cooling rate curve of F-11 alloy after homogenization at 1285°C for 12 hours.

Fig. S6. A dark-field TEM micrograph of homogenized samples of F-11 alloy shows

nucleation of multiple generation of γ′ precipitates and no chemical splitting of secondary

γ′ precipitates.

500 550 600 650 700

0

200

400

600

800

1000

1200

1400

Tem

per

atu

re (C

)

Time(minutes)

Coarsening kinetics of precipitates in ’ matrix in interdendritic region

The value of diffusivity of Re in ’ matrix as DReγ′

= 7.3 x 10-23

m2/s was obtained from the

coarsening kinetics of precipitates in ’ matrix in the interdendritic region.

This value of DReγ′

was obtained using the following equation,

K =8DRe

γ′(1−Ce)CeVmσ

9RT(Ceγ′−Ce)

2 (37)

Here, K= 3x 10-31

m3/s (experimentally calculated and shown in fig. S7), = 46.8 m

2/s (from

TC-PRISMA), and compositions of Re in and ’ phase (table S1).

Fig. S7. Coarsening kinetics of γ precipitates in the γ′ matrix in the interdendritic region in

F-11 alloy during isothermal annealing at 800°C. (A) temporal evolution of precipitates in ’

matrix shows no apparent change in spherical morphology of precipitates. (B) A plot of log of

radius (r) against log of time (t) shows the LSW coarsening exponent of 0.33. (C) A plot of cube

of radius (r) against time (s) gives the coarsening rate of precipitates.

0 1x106

2x106

3x106

4x106

5x106

6x106

0

500

1000

1500

2000

1.2 1.6 2.0 2.4 2.8 3.2

0.4

0.6

0.8

1.0

100 h

200 h

25 h 500 h

1500 h

100 nm

50 nm

50 nm

100 nm

100 nm

A B

C

time (s)

Log t

Log

rr3

(nm

)3

Slope =0.35

K = 3 x 10-31 m3/s

Table S1. Compositions of the γ and γ′ phases in the interdendritic and dendritic regions

after isothermal annealing at 800°C for 1500 hours.

Composition (at. %)

Ni Al Co Ta W Ru Re

F Bulk 67.1 13.6 7.6 3.0 1.5 5.9 1.3

Interdendritic

Region (800°C)

Overall 66.13 15.35 6.6 3.38 1.39 5.75 1.03

precipitate 53.5 5.8 19.5 0.5 1.8 17.2 1.6

’ matrix 68.0 17.5 4.5 3.9 1.7 3.8 0.9

Dendritic

Region (800°C)

Overall 66.02 13.57 7.8 2.6 1.77 6.4 1.62

matrix 61.2 6.5 15.0 0.6 1.5 12.1 3.3

’ precipitate 68.7 17.2 4.0 3.8 2.0 3.5 0.7

precipitate

(25h.)

59.0 6.2 15.3 0.7 1.5 13.5 3.8

Table S2. APT estimated composition of the γ′ precipitates in the dendritic region after the

homogenization heat treatment and after annealing at 800°C for various times.

Table S3. Parameters used for phase-field modeling of the hierarchical microstructure.

Quantity Value Reference

𝑐𝛾,0 0.13 (38)

𝑐𝛾′,0 0.235 (38)

k 2.62 × 1010 J/m3 Set as in (32)

𝐶1111𝛾

, 𝐶1122𝛾

, 𝐶4444𝛾

206.6, 148.5, 93.5 GPa (39)

𝐶1111𝛾′

, 𝐶1122𝛾′

, 𝐶4444𝛾′

201.4, 142.9, 100.2 GPa (40)

𝜖𝑖𝑗∗

0.00377 for 𝑖 = 𝑗 0 for 𝑖 ≠ 𝑗 (41)

Σ 46.8 mJ/m2

from TC-PRISMA® using TCNI8 and

MOBNI2 databases.

Λ 1 nm

W 3.09 × 108 J/m3

Κ 1.29 × 10-10 J/m

D 7.3 × 10-23 m2/s

from precipitate coarsening kinetics in

interdendritic region at 800°C (fig. S7)

M 2.8 × 10-33 m5/(J s)

L 2.8 × 10-14 m3/(J s)

’ precipitate Composition (at. %)

Ni Al Co Ta W Ru Re

Homogenized 66.8 16.5 5.0 3.4 1.9 5.0 1.0

800°C / 1 h 67.4 16.5 5.0 3.8 1.4 4.7 1.0

800°C / 25 h 68.0 17.2 4.0 3.9 1.9 3.9 0.9

800°C / 1500 h 68.7 17.2 4.0 3.8 2.0 3.5 0.7