Appendix 2 A Fortran code for determining a radial distribution function (RDF)

236 Pages·2004·6.49 MB·English

by  David Seidman

NORTHWESTERN UNIVERSITY

Decomposition Behavior in Model Ni-Al-Cr-X Superalloys: Temporal Evolution and
Compositional Pathways on a Nanoscale

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

For the degree

DOCTOR IN PHILOSOPHY

Field of Materials Science and Engineering

By

Chantal K. Sudbrack

EVANSTON, ILLINOIS

December 2004
© Copyright by Chantal K. Sudbrack 2004
All Right Reserved
ii
For my mother

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Abstract

Decomposition Behavior in Model Ni-Al-Cr-X Superalloys: Temporal Evolution and
Compositional Pathways on a Nanoscale

Chantal K. Sudbrack

In model Ni-Al-Cr-X superalloys, the compositional pathways and temporal
evolution of coherent γ' (L1 )-precipitation from an isothermally aged solid-solution,
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γ (FCC), is investigated at: (i) 600°C, to study nucleation, growth, and coarsening; and
(ii) 800°C, to study the influence of W on coarsening of a Ni-Al-Cr alloy.
In the quenched Ni-5.2 Al-14.2 Cr at.% alloy, radial distribution functions
establish Ni (Al,Cr)-type short-range ordering that extends 0.6 nm and is Cr depleted.
3
Phase separation at 600°C occurs by nucleation and growth, and the γ'-precipitates’
morphology is a mixture of isolated spheroids and spheroids in various stages of
coalescence. Sub-nanometer scale compositional profiles across the γ/γ' interfaces reveal:
(i) transient chemical gradients of Al depletion and Cr enrichment adjacent to the
precipitates; (ii) trapped Cr atoms in the growing precipitates; (iii) the interfacial width is
component dependent; and (iv) increased Al solubility in the γ'-precipitates resulting from
capillarity. For a quasi-steady state, the governing power-law time dependencies during
coarsening are compared to extant models and discussed in light of recent KMC
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simulations performed at Northwestern. Independent of the solute diffusivities, the γ/γ'
interfacial free-energy is determined from coarsening data to be 22 to 23 mJ m-2.
In Ni-9.8 Al-8.3 Cr at.% and Ni-9.7 Al-8.5 Cr at.%, spheroidal precipitates (5-15
nm diameter) form during quenching. Initially, chemical gradients exist in the γ'-
precipitates, however, they disappear after 1 h. After 16 h aging at 800°C, the
precipitates have a cuboidal morphology and align along the elastically soft <100>-type
directions. Particle size distributions and spatial pair correlation functions evolve
temporally, and are discussed in context of the morphological development of the γ'-
precipitates. The coarsening kinetics of the mean radius and interfacial area per unit
volume obey t1/3 and t–1/3 law, where the addition of W decreases the coarsening rate by a
third. The slower kinetics are attributed to W’s influence on elemental partitioning,
which leads to stronger partitioning of Al to the γ'-phase and Cr to the γ-phase, and to its
smaller diffusivity. Finally, an inflection-point method for determining reproducible
phase compositions from three-dimensional atom-probe data is described, which is
important for determining partitioning ratios.

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Acknowledgements

First and foremost, I wish to thank my advisor, Professor David N. Seidman, for
his guidance, encouragement, enthusiasm, and for providing a project, which molded my
scientific interests and challenged my analytical abilities and creativity. It has been a joy.
I am grateful to Dr. Ronald D. Noebe at NASA Glenn Research Center for his technical
guidance and his critical evaluation of the work as it progressed, as well as for providing
specimens and specimen machining. I would like to thank Professors Mark D. Asta and
Peter W. Voorhees for their keen interest and many fruitful discussions throughout my
Ph.D. Also, I extend a hearty “thank you” to Professor Dieter Isheim for his mentorship,
availability, and assistance on many aspects of this work.
A number of people have helped me experimentally and analytically, of which I
am thankful to:
• Dr. Thomas F. Kelly and the staff at Imago Scientific for use of the LEAP
microscope and generously offering their time, equipment, and instruction
• Dr. Olof C. Hellman for the software development and ongoing development of
ADAM, analytical and visualization software for 3DAP microscopy data
• Dr. Carelyn E. Campbell at the National Institute of Standards and Technology,
who graciously calculated diffusivities for a wide variety of compositions and a range
of temperatures
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• Professor Gautam Ghosh for assistance in calculating thermodynamic quantities
in ThermoCalc
• Dr. Zugang Mao for sharing his KMC simulation results and open discussion
• Various facility managers at Northwestern, in particular Dr. Kathleen Stair
(metallography), Ben Myers (SEM), and Dr. Gajendra Shekhawat (AFM)
• Undergraduate research assistants: Tiffany Ziebell, Jessica Weninger, Luis de la
Cruz, Nicholas Disabato, Gillian Hsieh, Brian Pasquini, Mark Murphey, and Alex
Vaynman
• Past and present members of Prof. Seidman’s group for instruction on countless
occasions, in particular Dr. Christian B. Fuller, Dr. Emmanuelle A. Marquis, Dr.
Jason T. Sebastian, and Mr. Richard A. Karnesky
• Past and present members of Prof. Voorhees’s research group for advice on
specimen preparation, image analysis, and scientific direction, in particular, Dr.
David Rowenhorst, Dr. Roberto Mendoza, and Dr. Alan C. Lund
• John Blatz du Rivage and Joshua Paul for providing code for several data analysis
modules
I am extremely fortunate and appreciative of the unwavering and constant support
from friends and family. In the interest of brevity, I’d like to single out only a few.
Dorianna, Sacha, Lou, and Luiz Sr., your belief in me is what made this possible. You
are in my heart always. Tracy, our chats over lunch, along a walk on the lake, and over a
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cup of coffee, have kept me laughing and balanced. Thanks so much; you are an amazing
friend.
This research is sponsored by the National Science Foundation under grants DMR-
9728986 and DMR-0241928. I am grateful to the NSF for partial support through a NSF
graduate student research fellowship. Furthermore, I would like to acknowledge The
Graduate School at Northwestern University for providing a Walter P. Murphy graduate
student fellowship during my first year of studies, as well as a dissertation year
fellowship in my last year.
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List of Symbols, Acronyms, and Abbreviations
aα lattice parameter of phase α
A' precipitated area fraction for a finite section
a
Co overall alloy concentration of component i
i
Cαor C concentration of component i in phase α
i i
<Cα> mean concentration (atomic fraction) of component i in phase α
i
Cα,eq equilibrium concentration of component i in phase α
i
<Cα,ff > mean far-field concentration of component i in matrix phase α
i
∆Cα supersaturation of component i in phase α, Cα,eq− <Cα(t)>
i i i
d flight-path distance of field-evaporated ion
Dα diffusion coefficient of component i in phase α
i
D diffusivity matrix
ii
D diffusion coefficient of vacancies
v
D diffusion coefficient of a monovacancy
1v
e elementary charge of an electron
E electric field
o
f fraction of precipitates that are coalesced
f geometric correlation factor
c
f(φ) ratio of the K(φ) to KKV
∆g strain energy per mole
s
Gk partial derivatives of the molar Gibbs free-energy of phase k
,ij
h etching height
hm migration enthalpy of a monovacancy
1v
<H'> mean caliper distance
J* steady-state nucleation rate
SS
k Boltzmann constant
B
KLSW coarsening rate constant for R according to the LSW theory
KKV coarsening rate constant for R according to the KV model
K(φ) coarsening rate constant for R for a system with a finite φ
<L> average cube length
m temporal exponent in the coarsening regime for ∆C
i
m/n mass-to-charge ratio
n temporal exponent in the coarsening regime for <R>
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N Avogadro’s number
A
N' number of precipitates per unit volume (areal density) for finite section
a
N number of dislocations per unit volume
d
N number of precipitates included in the analysis
ppt
N total number of atoms or events
TOT
N number of precipitates per unit volume (number density)
v
Ncor corrected number of atoms of component i
i
N3DAP measured number of atoms of component i (3DAP)
i
p temporal exponent in the coarsening regime for N
v
p difference in elemental partitioning in equilibrium, Cγ',eq −Cγ,eq
i i i
P' points per unit length for a finite section
l
r radial distance from precipitate center or from interface
r radius of dislocation core
c
R precipitate radius
<R(t)> time-dependent mean precipitate radius
<R(0)> mean precipitate radius at the onset of coarsening
R* radius of a critical nucleus
R average distance between dislocations
d
R ideal gas constant
g
R 2D precipitate radius
PS
s standard error of a mean
m
sm migration entropy of a monovacancy
1v
S interfacial area per unit volume
v
t aging time
t electronics delay offset
d
t time-of-flight
TOF
T temperature
T critical spinodal temperature
c
V volume
V applied voltage
o
V steady-state DC voltage
dc
V molar volume of the precipitate
m
V pulse voltage
p
Vβ molar volume of component i in the precipitate phase β
i
x2 root-mean squared diffusion distance
x