Acta Materialia 54 (2006) 119–130
www.actamat-journals.com
Composition evolution of nanoscale Al3Sc precipitates in an
Al–Mg–Sc alloy: Experiments and computations
Emmanuelle A. Marquis
a
a,b
, David N. Seidman
b,*
, Mark Asta b, Christopher Woodward
b,c
Materials Physics Department, Sandia National Laboratories, 7011 East Avenue, MS 9161, Livermore, CA 94550, United States
b
Materials Science and Engineering Department, Northwestern University, Evanston, IL 60208-3108, United States
c
Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright Patterson AFB, OH 45433, United States
Received 7 May 2005; received in revised form 18 August 2005; accepted 22 August 2005
Available online 25 October 2005
Abstract
Controlling the distribution of chemical constituents within complex, structurally heterogeneous systems represents one of the fundamental challenges of alloy design. We demonstrate how the combination of recent developments in sophisticated experimental high
resolution characterization techniques and ab initio theoretical methods provide the basis for a detailed level of understanding of the
microscopic factors governing compositional distributions in metallic alloys. In a study of the partitioning of Mg in two-phase ternary
Al–Sc–Mg alloys by atom-probe tomography, we identify a large Mg concentration enhancement at the coherent a-Al/Al3Sc heterophase
2
interface with a relative Gibbsian interfacial excess of Mg with respect to Al and Sc, Crel
Mg , equal to 1.9 ± 0.5 atom nm . The correspond2
rel
ing calculated value of CMg is 1.2 atom nm . Theoretical ab initio investigations establish an equilibrium driving force for Mg interfacial segregation that is primarily chemical in nature and reflects the strength of the Mg–Sc interactions in an Al-rich alloy.
2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Atom-probe tomography; Ab initio calculations; Al3Sc precipitates; Mg segregation; Coherent heterophase interface
1. Introduction
With continuing rapid increases in computing power,
computational modeling is increasingly augmenting
traditional empirical investigations in the design of technologically advanced structural materials. Most high-performance metallic alloys contain multiple alloying elements,
whose interactions govern the formation of strengthening
second-phases, partitioning behavior, and segregation at
internal interfaces such as grain boundaries or matrix/precipitate heterophase interfaces. Since such compositional
variations are a critical factor governing the mechanical
properties, theoretical understanding of interatomic interactions in multicomponent alloys is highly desirable from
the standpoint of designing such materials.
*
Corresponding author. Tel.: +1 8474914391/9252943287.
E-mail addresses: d-seidman@northwestern.edu, emarqui@sandia.gov
(D.N. Seidman).
In the case of Al alloys, scandium contributes significantly to improving strength by forming nanoscale coherent Al3Sc precipitates [1,2]. Additions of ternary elements
aim at improving mechanical properties and nanostructural stability. Transition metals, such as Ti and Zr tend
to decrease the coarsening kinetics of the L12 phase [3,4]
whereas Mg resides exclusively in the a-Al matrix in solid
solution [1]. This paper focuses on the effects of Mg additions on Al3Sc precipitation in Al–Sc alloys, which is
important for optimizing the mechanical properties of
these multicomponent alloys [5], and is part of a comprehensive study of the room temperature and elevated temperature (573 K) creep behavior of Al(Sc) based alloys
[3–10]. From a fundamental viewpoint, addition of Mg to
the a-Al/Al3Sc system, where Mg is an oversized atom
(12.08% by radius and 40.82 by volume [11]) in solid solution in the a-Al matrix, constitutes a simple and welldefined system for studying elemental partitioning and heterophase segregation. This research builds on our initial
1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2005.08.035
120
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
results obtained by high-resolution electron microscopy
(HREM), showing that Mg additions alter the morphology
of Al3Sc precipitates, that is, the {1 0 0} and {1 1 0} facets
disappear and the precipitates become spheroidal [6,7].
Asta et al. recently published first-principles calculations
demonstrating significant Mg segregation at coherent
{1 0 0} a-Al/Al3Sc coherent interfaces [12,13]. The present
study demonstrates how the behavior of Mg in a-Al/Al3Sc
alloys can be both measured at the subnanoscale level using
atom-probe tomography (APT) and predicted from ab initio calculations, thereby providing detailed insight into the
driving forces for phase partitioning and interfacial segregation in this model two-phase ternary aluminum alloy.
2. Experimental procedures
2.1. Specimen preparations
A cast Al alloy with nominal composition 2.2 Mg–0.12
at.% Sc was annealed at 618 C in air for 24 h (to ensure
uniformity of the Mg concentration throughout the material), quenched into cold water, and then aged in air at
300 C for times between 0.33 and 1040 h.
APT tips were obtained by a two-step electropolishing
procedure. The initial polishing solution of 30 vol.% nitric
acid in methanol was followed by a solution of 2 vol.% perchloric acid in butoxyethanol, used for final polishing to
produce a sharply pointed tip, with a radius of curvature
of less than 100 nm (Fig. 1). Field-ion microscopy (FIM)
analyses were performed at a background pressure of
105 Torr consisting of a mixture of 80% Ne and 20%
He; APT analyses were carried out under ultrahigh vacuum
conditions (1010 Torr) for pulsed field-evaporation, which
was performed with a pulse fraction (pulse voltage/steady
state dc voltage) of 20% and a pulse frequency of
1500 Hz. Specimens were maintained at temperatures below 30 K.
After aging for 0.5 h at 300 C, the precipitate number
density, 4 ± 2 · 1022 precipitate m3 [8], is sufficiently high
to perform APT random-area analysis. The error bars stated correspond to one standard deviation. After aging for
1040 h at 300 C, the number density of Al3Sc precipitates
decreases to about 3 ± 1 · 1021 precipitate m3, and random area APT analysis is no longer an efficient technique.
Hence, FIM imaging was first performed to locate precipitates that are just commencing to intersect the surface of a
tip before starting an analysis.
Fig. 1. Transmission electron micrograph of an APT tip, illustrating the
coherency strain contrast of the Al3Sc precipitates after aging at 300 C for
5 h.
2.2. Data analysis
A typical mass-to-charge state (m/n) spectrum is displayed in Fig. 2. The three isotopes of Mg are doubly
charged at 12, 12.5 and 13 a.m.u., with no hydride formation. The measured isotopic abundances are 78.4 ± 0.3%
for 24Mg2+, 10.5 ± 0.3% for 25Mg2+ and 11.1 ± 0.3% for
26
Mg2+, which agree favorably with the handbook values
of 79%, 10% and 11%. Scandium is also doubly charged
with multiple hydrides, exhibiting a possible overlap between Sc and singly charged Mg at m/n values of 24, 25
and 26 a.m.u. In some cases, peaks at 24, 25 or 26 a.m.u.
are detected, but the correct mass ratio of the isotope
24
Mg to the other two isotopes, 25Mg, and 26Mg, was not
found; therefore the corresponding ions were considered
to be Sc hydrides. Since one of the objectives of this study
is the measurement of Mg concentrations, this choice implies possibly under-estimating the Mg concentrations, in
particular in the proximity of the Al3Sc precipitates.
Fig. 2. Example of mass-to-charge state ratio (m/n) spectra from a tip
containing an Al3Sc precipitate.
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
Data visualization and analysis of data sets were performed using a software code, ADAM 1.5, which was
developed at Northwestern specifically for analyzing APT
data [14].
3. Results
In the as-quenched state, Mg and Sc appear homogeneously distributed although the standard statistical v2-test
used [15], which compares solute concentration distributions with binomial distributions of a perfectly random
solid solution, does not rule out whether or not Mg or Sc
atoms are homogeneously distributed in the a-Al matrix.
Precipitates are observed after aging for 0.33 h
(Fig. 3(a)). A cluster search algorithm with a maximum
separation distance of 0.7 nm is used to isolate the Sc
atoms constituting the precipitates. Taking into account
the spherical morphology of the precipitates [8], the composition was measured in spherical shells centered on the
center-of-mass of the selected Sc atoms. Eight precipitates
were analyzed yielding an average concentration of
22.4 ± 2.8 at.% Sc. The radius of these Al3Sc precipitates,
estimated using the radius of gyration of the Sc atoms, is
121
between 0.8 and 1.4 nm. Magnesium atoms are also present
inside the precipitates at a level, 4.3 at.%, which is approximately a factor of two greater than the average Mg concentration in the a-Al matrix.
After further aging, the precipitate radius increases
(Figs. 3(b) and (c)) and reaches about 4 nm at 1040 h
(Fig. 4). The spatial resolution of APT is illustrated in
Fig. 4, where an analysis performed near the 110 crystallographic pole reveals the {2 2 0} atomic planes perpendicular
to the analysis direction. The Sc concentration of the precipitates increases after 0.5 h and remains constant thereafter within experimental error, xSc = 27.4 ± 1.5 at.% Sc
(Table 1). The small discrepancy with the nominal composition, i.e. 25 at.%, is likely to be due to the different evaporation fields of Al and Sc that could not be
accommodated despite the low tip temperature <30 K.
On the other hand, the Mg concentration decreases with
increasing aging time to 0.9 at.% after 1040 h aging. The
Mg concentration within the Al3Sc precipitates is non-uniform with an enhancement at their centers as shown in the
example of Fig. 5.
For aging times longer than 2 h, a distinct Mg concentration enhancement at the a-Al/Al3Sc heterophase interface is also observed. The example displayed in Fig. 5 is
a proximity histogram [16] that calculates the average composition in shells of 0.4 nm thickness at a given distance
from the a-Al/Al3Sc heterophase interface. The interface
is defined by an isoconcentration surface at 18 at.% Sc.
The Mg concentration enhancement at the a-Al/Al3Sc heterophase interface is 200%, which is localized within 2 nm
at this heterophase interface. The maximum Mg concentration at this heterophase interface decreases slightly during
the early aging times and thereafter remains constant within experimental error. Fig. 6 displays the normalized average Mg concentration profile taken from the centers of the
precipitates for each aging time. The profiles are temporally invariant after 2 h.
4. Discussion
4.1. Mg segregation: experimental measurement
Fig. 3. Three-dimensional reconstruction of an analyzed volume, displaying only the Sc atoms, from a specimen aged at 300 C for: (a) 0.33 h; (b)
0.5 h; and (c) 5 h.
Mg segregation at the a-Al/Al3Sc interface is constant
with aging time, which suggests that the measured concentrations represent an equilibrium behavior. Moreover, the
root-mean-square
diffusion distance of Mg in Al is given
pffiffiffiffiffiffiffiffi
by 6Dt, where the factor 6 is for diffusion in 3-dimensions; D = 1.6 · 1016 m2 s1 is the diffusion coefficient
of Mg in Al [17] at 300 C and t is the aging time; the values are about 1.3 lm after 0.5 h aging at 300 C and
60 lm after 1040 h. The precipitate spacing is evaluated
using the square-lattice
pffiffiffiffiffiffiffiffiffiffiffiffiffi spacing approximation, which is
given by hRi ¼ 3 4p=3f , where ÆRæ is the mean precipitate
radius and f = 0.53 vol.% is the calculated volume fraction
of Al3Sc precipitates at 300 C. The precipitate spacing is
10 nm for ÆRæ equal to 1.1 nm (0.5 h aging), and 39 nm for
ÆRæ equal to 4.2 nm (1040 h aging). The root-mean-square
122
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
Fig. 4. Three-dimensional reconstruction of an Al3Sc precipitate after aging at 300 C for 1040 h; Al atoms are in blue, Mg atoms in green and Sc atoms in
red.
Table 1
Al3Sc precipitate composition as function of aging time at 300 C
Time (h)
Number of
precipitates
Al (at.%)
Mg (at.%)
Sc (at.%)
As quenched
0.33
0.5
2
5
30
1040
0
8
4
10
3
2
15
–
73.3 ± 3.5
68.8 ± 5.2
71.2 ± 1.4
68.9 ± 2.1
71.0 ± 2.8
69.4 ± 2.8
–
4.3 ± 2.6
4.09 ± 1.5
2.3 ± 0.6
3.1 ± 1.1
2.5 ± 0.8
0.9 ± 0.3
–
22.4 ± 2.8
28.5 ± 1.4
26.5 ± 1.4
28.1 ± 2.5
26.5 ± 2.3
29.2 ± 2.4
Fig. 6. Mg concentration normalized by the average Mg concentration in
the a-Al matrix for four different aging times (0.5, 2, 5, and 1040 h) at
300 C.
0
Crel
Mg ¼ CMg CSc
Fig. 5. Proximity histogram of an Al3Sc precipitate after aging at 300 C
for 1040 h; the colored areas correspond to the interfacial excesses of Al
(blue), Mg (green) and Sc (red).
diffusion distance of Mg is therefore always significantly
greater than the average center-to-center precipitate spacing and the system is at the very least in local thermodynamic equilibrium with respect to Mg segregation at the
a-Al/Al3Sc heterophase interface.
The relative Gibbsian interfacial excess concentration of
Mg with respect to Al and Sc provides a quantitative thermodynamic description of the observed equilibrium Mg
segregation, and it is given by [18]:
0
xaAl xaMg xaAl xaMg
0
0
xaAl xaSc xaAl xaSc
CAl
0
0
0
0
xaMg xaSc xaMg xaSc
xaAl xaSc xaAl xaSc
;
ð1Þ
where CMg, CSc and CAl are the Gibbsian interfacial ex0
cesses of Mg, Sc, and Al, respectively, and the xaj and xaj
are the concentrations of component j (j = Al, Sc or Mg)
in phase a (Al) and a 0 (Al3Sc). Fig. 5 displays the Gibbsian
excess quantities for Al (negative value), Mg and Sc (positive values) as areas under the concentration curves in the
proximity histogram [19]. The relative Gibbsian excesses of
Mg with respect to Al and Sc are listed in Table 2 for the
different aging times. The variations observed are within
the experimental errors. The similar values of the Gibbsian
excess of Mg relative to Al and Sc for all aging times implies that the system is at the very least in local thermodynamic equilibrium and the average value is Crel
Mg ¼
1:9 0:5 atom nm2 . For the Al–Sc–Mg system, the calculations of the solute concentrations at the interface of a
growing Al3Sc precipitate predict a depletion of Mg at
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E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
Table 2
Precipitate radius (from [7]), relative Gibbsian interfacial excess of Mg as function of aging time at 300 C, and maximum Mg enhancement factor
Aging time (h)
0.5
2
5
30
1040
Radius (nm)
a
Crel
Mg
matrix
cmax
Mg =cMg
1
1.5 ± 0.7
2.8 ± 0.3
–
1.7 ± 0.4
2.7 ± 0.2
2
1.7 ± 0.3
2.0 ± 0.2
–
2.22 ± 0.45
2.3 ± 0.5
4.2
1.85 ± 0.38
1.9 ± 0.5
a
Calculated using Eq. (1).
the a-Al/Al3Sc interface (see below and Table 3). Thus our
observation of a positive value of the relative Gibbsian
interfacial excess of Mg with respect to Al and Sc is further
evidence that we are observing a true thermodynamic equilibrium excess quantity.
The decrease in interfacial free energy associated with
Mg segregation can be estimated using [20]:
!
oc
rel
CMg ¼
;
ð2Þ
olMg
T;P
Crel
Mg
where
is the relative Gibbsian excess of Mg with respect to Al and Sc, and lMg is the chemical potential of
Mg. Assuming an ideal solid-solution, the chemical potential is lMg ¼ l0 þ k B T lnðxaMg Þ, and it yields the following
expression:
Crel
oc
Mg k B T
¼
.
ð3Þ
oxMg T;P
xaMg
At 300 C, the measured relative Gibbsian excess of Mg
2
with respect to Al and Sc is Crel
Mg ¼ 1:9 0:5 atom nm ,
and the average Mg concentration in the matrix is
xaMg ¼ 2:2 0:3 at.% Mg. Assuming Crel
Mg varies linearly
with concentration, the integration of Eq. (3) for xaMg varying from 0 to 0.022 yields a decrease in the interfacial free
energy of 15 mJ m2.
Several previous APT studies reported segregation
behavior of solute atoms at partially semi-coherent or
semi-coherent heterophase interfaces [22,23]. In the present
study, the coherency state of the a-Al/Al3Sc interface is
known from HREM observations, and was determined
to be perfectly coherent for all cases presented. Coherency
loss may occur when the precipitate diameter is sufficiently
large. For the lattice parameter misfit at 300 C, d 0.62%,
between the Al matrix containing 2.2 at.% Mg and the
Al3Sc phase, the spacing between the misfit dislocations
is a/d, where a 0.2 nm is the spacing between {2 0 0}
Table 3
Values of solute concentrations at the precipitate/matrix interface calculated from Eqs. (4) and (5) for spherical a 0 (Al3Sc) precipitates of radius R
growing in a supersaturated a (Al) matrix in a ternary Al–Sc–Mg alloy
0
planes; this yields a critical precipitate diameter for loss
of coherency of approximately 30 nm, which is much greater than the precipitate diameter measured after aging at
300 C for less than 1040 h.
The exact shape of the precipitates observed by APT
needs to be carefully considered. Firstly, the observations
may be subject to experimental artifacts, such as asymmetry of the tip, or misalignment of the analysis direction.
Also, the results depend strongly on the field evaporation
behaviors of the a-Al matrix and Al3Sc precipitates, which
may be significantly different. The measured local atomic
density of the a-Al matrix is indeed higher than that of
the precipitate phase, with an experimental density ratio
equal to about 1.4. It is consistent with the bright imaging
of the Al3Sc precipitates in FIM mode, indicating that the
precipitates exhibit a small protrusion effect, due to their
higher evaporation field. The width of the heterophase
interface may therefore be explained by the artificially
higher magnification of the Al3Sc precipitates and possibly
ion trajectory effects [24]. The approximately spheroidal
shape of the Al3Sc precipitates observed by HREM is,
however, reproduced and no significant distortions between the lateral dimension and the depth dimension are
observed.
4.2. Driving force for Mg segregation
In this section we employ first-principles calculations to
analyze the microscopic factors governing the pronounced
enhancement of Mg at the Al/Al3Sc interface. In this analysis we first make use of the theoretical framework provided by a model of diffusion-limited precipitate growth
kinetics, to ascertain whether the measured Mg enhancement may be due simply to capillary effects. The analysis
leads to the conclusion that the Mg enhancement measured
by APT cannot be interpreted simply as reflecting the effects of capillarity and solute-flux balance in a model of diffusion-limited growth. We subsequently use first-principles
calculations to establish that the interfacial enhancement
corresponds to an equilibrium segregation phenomenon
driven by a chemical driving force reflecting the attractive
interactions between Mg and Sc solute atoms.
0
R (nm)
^xaSc (at. %)
^xaMg (at.%)
^xaSc (at.%)
^xaMg (at.%)
2
4
1
2.5 · 104
1.8 · 104
9.0 · 105
2.2
2.2
2.4
25
25
25
3.4 · 107
4.0 · 107
4.8 · 107
These values were derived assuming values for the interfacial free energy, r
equal to 0.175 J m2 [21] that are independent of composition and radius,
as discussed in the text.
4.2.1. Interface solute concentrations in a model of
diffusion-limited growth
In this section we explore whether the solute–concentration profiles measured by APT can be interpreted as representing steady-state solutions to the solute diffusion
equation, subject to the boundary conditions imposed by
124
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
local thermodynamic equilibrium and flux balance at the
growing precipitate/matrix interface. In this analysis we
employ a model for diffusion-limited growth of precipitates
in a ternary alloy due to Kuehmann and Voorhees [25].
While in a binary alloy the condition of local thermodynamic equilibrium uniquely determines the interfacial solute compositions at the interface during diffusion-limited
growth, the situation is qualitatively different in a ternary
alloy. As described in [25], the conditions of local thermodynamic equilibrium provide only three equations (equality
of chemical potentials for each of the chemical species),
which do not determine uniquely the four independent values of the interface solute concentrations (two in each of
the two phases). Within a diffusion-limited-growth model
for a ternary alloy, the additional constraint required to
fix the interface compositions arises from the requirement
of mass conservation (flux balance) at the moving precipitate/matrix interface. As a consequence, interface compositions in a ternary alloy are dictated not only by bulk and
interfacial thermodynamic properties, but also by the ratio
of the diffusivities in the matrix phase, which affect the relative fluxes of each solute species.
For spherical precipitate geometries, employing a meanfield, quasi-steady-state solution to the diffusion equation
(neglecting off-diagonal terms in the diffusion matrix),
interface compositions can be derived from the following
equations [25]:
0
0
2c
V i ði ¼ Al; Sc; MgÞ;
R
ð^xaSc x1
Sc Þ
;
ð^xaMg x1
Mg Þ
0
lai ð^xaMg ; ^xaSc Þ lai ð^xaMg ; ^xaSc Þ ¼
a0
ð^xSc ^xaSc Þ
DSc
¼
a0
a
D
ð^xMg ^xMg Þ
Mg
0
0
ð4Þ
ð5Þ
where ^xaSc , ^xaMg , ^xaSc , and ^xaMg denote mole fractions of Sc and
Mg on the Al (a) and Al3Sc (a 0 ) sides of the interface for a
precipitate of radius R, c is the interfacial free energy, the
0
variables lai and lai correspond to bulk chemical potentials, and V i is the partial molar volume for species i in
1
the precipitate (a 0 ) phase. The variables x1
Sc and xMg correspond to far-field solute concentrations in the matrix
phase. The first three equations represented by Eq. (4) correspond to the well-known Gibbs–Thomson conditions
incorporating the effect of capillarity in the formulation
of the conditions for local thermodynamic equilibrium
[26], while Eq. (5) reflects the constraint imposed by solute
flux balance.
Since values for the chemical potentials in the ternary
Al3Sc intermetallic phase, required in the solution of Eqs.
(4) and (5), are unavailable from experimental measurements, we have employed first-principles free-energy models in calculations of the equilibrium phase compositions.
The bulk free energies were derived within a model of
non-interacting substitutional defects through a generalization of the approach outlined in [13,29–31]. Such a noninteracting defect model for the free energy is expected to
be highly accurate for the dilute solute concentrations considered in the present study. In this approach to modeling
the thermodynamics of the bulk a- and a 0 -phases, the freeenergy models take the following form:
F a ¼ F a0 þ k B TxAl ln xAl þ xSc ½DF aSc þ k B T ln xSc
þ xMg ½DF aMg þ k B T ln xMg ;
3n
0
0
a0 ;Al
Al
Al
k B TxAl
þ k B T ln xAl
F a ¼ F a0 þ
Al ln xAl þ xSc ½DF Sc
Sc
4
o
a0 ;Al
Al
þxAl
Mg ½DF Mg þ k B T ln xMg
1n
a0 ;Sc
Sc
Sc
Sc
k B TxSc
þ
Sc ln xSc þ xAl ½DF Al þ k B T ln xAl
4
o
a0 ;Sc
Sc
þxSc
½DF
þ
k
T
ln
x
;
B
Mg
Mg
Mg
ð6Þ
where in the a-Al phase xAl, xSc and xMg are the mole fractions of Al, Sc and Mg, while in the a 0 -Al3Sc phase xAl
Al and
xSc
Al denote Al mole fractions on the Al and Sc sublattices,
Sc
Al
Sc
respectively, and similarly for xAl
Sc , xSc , xMg and xMg . The
a
quantities DF j for the a-Al phase in Eq. (6) denote the free
energy to form a substitutional impurity of type j in pure Al;
at zero temperature these are simply the heats of solution
(DEaj ), while at finite temperature DF aj generally contains
entropic contributions of vibrational and
electronic origins.
0
Similarly, for the a 0 -Al3Sc phase DF aj ;k denotes the free energy to form substitutional impurity j on sublattice k.
The various defect energies entering in Eq. (6) were computed from first-principles using the ab initio total-energy
and molecular-dynamics program VASP (Vienna ab initio
simulation package) developed at the Institut für Materialphysik of the Universität Wien [32–34]. In these calculations, use was made of ultrasoft pseudopotentials [35],
the local-density approximation, and an expansion of the
electronic wave functions in plane waves. Further details
of the calculations can be found in [31]. Calculations of
the point-defect energies entering Eq. (6) were performed
employing 64-atom supercells, and were derived from the
relation DEj/;k ¼ Ej/;k E/ þ ðEk Ej Þ, where Ej/;k is the energy of the supercell of phase / containing an impurity of
type j on site k, E/ is the energy of the supercell for stoichiometric phase /, and Ej and Ek denote the energies per
atom of pure species j and k in their respective equilibrium
crystal structures, respectively.
The results of the supercell defect-energy calculations
are presented in Fig. 7. It is seen that Sc is calculated to
have a large and negative heat of solution in a-Al, while
the corresponding value for Mg is near zero. In the
a 0 -Al3Sc phase, all substitutional defects are seen to have
relatively large and positive formation energies. The DE
values given in Fig. 7 were used in Eq. (6) to derive bulk
chemical potentials, which formed the basis for the computation of the equilibrium concentrations given in Table 3.
In these calculations we also included vibrational-entropy
0
contributions to F a0 , F a0 and DF aSc , since these contributions
have been shown to be crucial for reproducing the measured binary Al–Al3Sc solvus boundary compositions
[29]. The details of the vibrational-entropy calculations
are given in [29,30].
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
125
Fig. 7. Calculated substitutional point-defect energies in Al and Al3Sc phases. Results for a-Al are plotted on the right and those for a 0 -Al3Sc on the left.
Solid and hatched bars denote defect energies on the Sc and Al sublattices in the a 0 -Al3Sc phase, while the white bars denote heats of solution for Mg and
Sc in a-Al.
Table 3 lists values of the interface concentrations calculated from Eqs. (4) and (5) for precipitates of radii 2 and
4 nm at an aging temperature of 300 C. For comparison
we also list compositions corresponding to two-phase equilibrium between bulk (R ! 1) a and a 0 -phases in a ternary
alloy with the composition Al–2.2 at.% Mg–0.12 at.% Sc
considered experimentally. For this alloy composition,
the bulk equilibrium compositions derived from first-principles (final row of Table 3) are in very reasonable agreement with the values of ^xaSc ¼ 7:2 104 at.% and
^xaMg ¼ 2:2 at.%, derived independently by Murray [36] from
empirical free-energy models.
In our calculations of the interface compositions listed
in Table 3, values for the far-field matrix compositions in
Eq. (5) were taken as x1
Sc ¼ 0:014 0:001 at.% and
x1
¼
2:35
0:02
at.%,
as
measured
by APT in alloys aged
Mg
for 1040 h [8]. We further made use of the following measured values for the solute diffusivities in Al:
DSc = 8.84 · 1020 m2 s1 and DMg = 1.62 · 1016 m2 s1
[17,32]. Additionally, we employed a value for c equal to
0.175 J m2, as derived from first-principles calculations
for planar interfaces between pure Al and stoichiometric
Al3Sc [21] (c is assumed to be constant and independent
of composition).
A comparison of the results in the first two rows of Table 3 with those corresponding to bulk phases (final row)
shows that the effects of capillarity and solute flux balance
are estimated to give rise to relatively small (10%)
changes in the matrix Mg concentration at the growing precipitate/matrix interface. Thus, the pronounced interfacial
enhancement of Mg measured by APT cannot be interpreted simply as reflecting the effects of capillarity and solute flux balance in a model of diffusion-limited precipitate
growth. Additional first-principles calculations, discussed
in the following section, suggest instead that the observed
interfacial segregation of Mg reflects an equilibrium segregation effect, arising from electronic interactions between
Sc and Mg atoms in Al.
4.2.2. Equilibrium segregation energies for Mg at a coherent
Al/Al3Sc interface
First-principles VASP calculations were conducted to
investigate the energetics of Mg solute atoms in the vicinity
of planar coherent Al/Al3Sc interfaces aligned parallel to
{2 0 0} crystallographic planes. The heats of solution for
Mg solute atoms were computed as a function of distance
in the vicinity of a planar Al/Al3Sc interface employing
supercell geometries with periodic boundary conditions.
Preliminary results from these calculations were reported
in [12] and further details were given in [13]. The supercells
used in these calculations contained periodic vectors in the
plane of the interface (a1 and a2) with lengths four times
the face-centered cubic (fcc) nearest-neighbor spacing:
a1 = a(2, 2, 0), a2 = a(2, 2, 0), where a is the fcc lattice constant. In the z-direction, normal to the interface, the periodic
length corresponded to 14 fcc unit cell dimensions. The total
number of atoms in these supercells was 412. The results presented below were derived using a value for the in-plane lattice constant (a) equal to that for the bulk Al3Sc phase; the
dimension of the supercell in the z-direction was then adjusted to give zero total zz-stress for a pure Al/Al3Sc interface (without Mg additions). Additional calculations were
performed using a value of the in-plane lattice constant
equal to that for bulk fcc Al, and only minor differences in
the calculated segregation energies were obtained. In this
section we report the solution energy of a single Mg atom
placed at various crystal sites in the vicinity of the Al/Al3Sc
(0 0 2) interface. For calculations in which Mg substitutes on
the Al side of the interface, the supercells contained a total of
nine unit cells of fcc Al and five unit cells of Al3Sc. Similarly,
for calculations of Mg impurity formation energies on the
Al3Sc side of the interface we employed supercells with five
fcc-Al unit cells and nine Al3Sc unit cells. Magnesium impurity formation energies were derived for substitution both on
Al and Sc sites in the supercell; in all cases the formation
energies on Sc lattice positions were sufficiently high to yield
negligible equilibrium Mg concentrations on these sites.
126
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
In Fig. 8 we plot the formation energies (DE) for substituting Mg for Al as a function of distance across the coherent (0 0 2) Al/Al3Sc planar interface. The values of DE
calculated on the Al3Sc side of the interface are roughly
0.6 eV larger than in pure Al, indicative of the strong energetic preference for the partitioning of Mg to the matrix
phase. In terms of the APT measurements, the most important feature of the results shown in Fig. 8 is the significantly
negative formation energy calculated on the Al side of the
interface at the crystallographic site corresponding to a second neighbor of the interface Sc atoms (the site labeled ‘‘1’’
in Fig. 8). The value of DE at this site is computed to be
0.1 eV lower than the heat of solution for Mg atoms in
bulk Al. This segregation energy (DE) thus provides a reasonably strong driving force for the equilibrium segregation of Mg atoms to the (0 0 2) coherent Al/Al3Sc interface.
To make explicit contact with the experimental results,
the energies derived from the first-principles calculations
have been used within a mean-field (Bragg–Williams) model
for the configurational free energy (similar to Eq. (6)) to
compute the equilibrium solute composition profiles across
a planar Al/Al3Sc {2 0 0} interface at 300 C. Within this
approximation, the equilibrium concentration of Mg atoms
at site j, xMg(j), is given as xMg ðjÞ ¼ x0Mg expf½DEðjÞ
DEð1Þ=k B T g, where x0Mg is the bulk Mg concentration in
a-Al, DE(j) is the formation energy of a substitutional
Mg impurity at site j near the interface, and DE(1) is the
heat of solution in bulk a-Al (far from the interface). The
calculated Mg concentration profile is shown in Fig. 9,
and was derived assuming a bulk Mg concentration in Al
of DE(1) = 2.4 at.%. The plot features a five-to sixfold
enhancement of the xMg in the plane positioned one lattice
constant from the interface Sc atoms. In comparison to the
…
APT results for Mg plotted in Fig. 5, the calculated concentration profile is considerably narrower with a larger value
for the Mg enhancement factor. These differences between
the experimental and theoretically calculated composition
profiles may be due to the effects discussed in Section 4.1
associated with the APT technique. Specifically, the measured width of the concentration profile at the Al/Al3Sc
interface is probably greater than the real width due to effects associated with the field-evaporation process [24],
which make a concentration profile appear broader than
it is (see Section 4.1).
Due to these differences in the width, a more meaningful
comparison between experiment and theory can be made in
terms of the integrated area under the Mg concentration
profiles. Specifically, the relative Gibbsian interfacial
adsorption segregation of Mg with respect to Al and Sc provides a quantitative thermodynamic description of the degree of equilibrium Mg segregation, If we interpret the
Mg segregation to reflect an equilibrium segregation effect,
we can estimate the values of the Gibbsian excess quantities
for Al (negative value), Mg and Sc (both positive values) as
the areas under the concentration curves in the proximity
histogram as displayed in Fig. 5 [29]. The value thus derived
2
from the experimental data is Crel
Mg = 1.9 ± 0.5 atom nm ,
which is found to be independent of aging time after 0.5 h
at 300 C. By comparison, Crel
Mg derived from the calculated
composition profiles is 1.2 atom nm2. Thus, experiments
and calculations yield values for the relative Gibbsian interfacial excess of Mg, which are approximately the same within the experimental uncertainty of ± 0.5 atom nm2. The
good level of agreement between experiment and theory
thus supports strongly the conclusion that the measured
interfacial enhancement of Mg reflects a pronounced
2
4
1
3
…
0.6
Al
∆ E (eV)
0.4
0.2
Al3Sc
0
-0.2
-5
-4
-3
-2
-1
z (Lattice Constants)
0
1
Fig. 8. Calculated formation energies for substitution of Mg for Al as a function of distance from the (0 0 2) a-Al/Al3Sc interface. A projection of the
interface atomic structure of the interfacial region is shown on top, with white and gray circles denote Al positions in planes parallel to the interface, and
black circles corresponding to Sc sites. In this projection the large circles denote atomic sites positioned 0.5 of a lattice constant below those indicated by
the smaller circles. The formation energies are plotted as a function of distance normal to the interface in the lower plot; white and gray circles denote
values of DE derived for the corresponding sites in each plane shown in the projection above.
127
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
Fig. 9. Calculated Mg concentration profile near the a-Al/Al3Sc interface. Compositions have been derived from the results in Fig. 8 assuming a bulk Mg
concentration in the Al phase of 2.4 atomic %.
Sc-Sc
200
Interaction Energy(meV)
equilibrium effect associated with the segregation of this
species to planar coherent Al/Al3Sc heterophase interfaces.
It is noteworthy that the calculated segregation energy
derived from the first-principles supercell calculations are
found to be highly insensitive to the imposed value of the
in-plane lattice parameter (the value of which was varied
in the calculations between the equilibrium lattice constants for pure Al and Al3Sc phases as described above).
Furthermore, the calculated segregation energy is found
to change by less than 10% if the interface positions are
fixed at ideal fcc positions. That is, the calculated segregation energy is found to be insensitive to the state of strain at
the interface. Thus the origin of the calculated segregation
energy appears not to originate from elastic strain energy.
Rather, we interpret the result as being a manifestation
of the nature of Mg–Sc electronic interactions in an Al-rich
alloy.
In support of this interpretation we have computed the
solute interaction energies up to fourth neighbor distances
for Mg–Mg, Sc–Sc and Mg–Sc pairs in pure Al employing
VASP and 216-atom supercells (6 · 6 · 6 primitive fcc unit
cells). The results are shown in Fig. 10. Strong and relatively long-ranged interactions are derived for both Mg–
Sc and Sc–Sc pairs. The magnitude and oscillating nature
of the Sc–Sc results are consistent with theoretical models
for transition-metal interactions in Al due to Carlsson
and Moriarty [38,39]. For both Sc–Sc and Mg–Sc we obtain repulsive interactions at first and third neighbors,
and attractive interactions at second and fourth. The magnitude and spatial variation of these interactions were
found to be primarily electronic in origin; very similar values to those plotted in Fig. 10 were obtained in supercell
calculations where the atoms were constrained to their bulk
fcc lattice sites (i.e., removing any elastically induced contribution to the interactions). In light of the results in
Fig. 10, the overall magnitude of the calculated segregation
energy, as well as the preferred binding site for Mg at the
Al/Al3Sc {2 0 0} heterophase interface, can be rationalized
as follows.
The supercell calculations yield a preferred binding site
for Mg, labeled ‘‘1’’ in Fig. 8, that contains the maximum
100
Mg-Mg
0
Sc-Mg
-100
-200
1.0
1.2
1.4
1.6
1.8
2.0
r/rnn
Fig. 10. Mg–Mg, Mg–Sc and Sc–Sc interaction energies in an Al host,
calculated as a function of separation r normalized by the nearestneighbor distance (rNN).
number of both second neighbors (one per Mg atom) and
fourth neighbors (four per Mg atom) to interface Sc atoms,
while featuring no nearest or third-nearest neighbor repulsive Mg–Sc pairs. Assuming that the alloy energetics are
dominated by pair interactions, the interaction energies
plotted in Fig. 10 can be used to derive a magnitude for
the Mg binding energy at site ‘‘1’’ equal to 0.14 eV. This result is in very good overall agreement with the value derived from direct supercell calculations (see Fig. 8).
Furthermore, the Mg impurity energy at the site labeled
‘‘2’’ in Fig. 8 is computed to be near zero in the direct
supercell calculations. This result is again consistent with
a bond-counting analysis considering that this site features
only third neighbor interactions with interface Sc atoms;
the magnitude of the third-neighbor Mg–Sc interactions
are shown to be relatively small in Fig. 10. Finally, both
sites ‘‘3’’ and ‘‘4’’ in Fig. 8 are calculated to have relatively
large positive Mg impurity formation energies on the
order of 0.2 eV. This result is again consistent with bond
128
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
counting, given the fact that these sites have two nearestneighbor bonds with interface-Sc atoms, each contributing
approximately 0.1 eV of interaction energy to DE according to Fig. 10. Overall, the very good agreement between
the directly calculated formation energies at the interface
sites, and the corresponding values derived from a relatively simple bond-counting analysis, suggests strongly that
Mg segregation at the Al/Al3Sc interface can be interpreted
as being a reflection of strong Mg–Sc electronic interactions in Al.
The type of bond-counting analysis described in the previous paragraph provides a convenient framework for analyzing the magnitude of the anisotropy associated with Mg
segregation to a coherent Al/Al3Sc interface. Such an analysis is interesting in light of the experimental observations
by Marquis et al. [8,14] showing that Mg additions to Al–
Sc lead to a pronounced change in the morphology of
Al3Sc precipitates from highly faceted (with well-developed
{1 0 0}, {1 1 0} and {1 1 1} facets) to a spheroidal morphology. Due to the higher areal densities of sites containing
attractive second and fourth neighbor interactions with
interface Sc atoms, Mg segregation is estimated to be
roughly a factor of three and 2.5 larger for {1 1 0} and
{1 1 1} interfaces, respectively, relative to the lower-energy
{1 0 0} orientation. We consider the analysis for {1 0 0} versus {1 1 0} orientations in detail in Fig. 11. This figure highlights with gray and hatched circles the atomic sites that are
predicted to have appreciable binding energies for Mg. In
the case of {1 0 0} discussed, there is one attractive site
per area a2, with a segregation energy of approximately
0.1 eV. By comparison, the
pffiffi{1
ffi 1 0} interface contains two
attractive sites per area, 2a2 . The site colored gray in
Fig. 11 contains two second-nearest-neighbor and five
fourth-neighbor Sc atoms. From the interaction energies
in Fig. 10, this site is thus predicted to have a binding energy of approximately 0.2 eV, leading to a site concentration at 300 C that is predicted to be nearly pure Mg
(assuming a bulk composition of 2.4 at.% and using a
Bragg–Williams model as above). The second binding site
for a {1 1 0} interface, indicated by the hatched circle in
Fig. 11, shares one fourth neighbor with interface-Sc atoms
and thus has a relatively small binding energy of roughly
0.025 eV, leading to a concentration enhancement of
about 1.7 times the bulk concentration at 300 C. The net
result is an enhancement of Crel
Mg by approximately a factor
of three for a {1 1 0} interface relative to {1 0 0}. A similar
analysis leads to the prediction of a value for Crel
Mg for
{1 1 1} orientations that is enhanced by approximately a
factor of two relative to {1 0 0}.
Using the Gibbs adsorption theorem, Mg segregation is
predicted to lower the free energy of {1 0 0}, {1 1 1} and
{1 1 0} interfaces by approximately 10, 20 and 30 mJ m2,
respectively. Magnesium segregation thus lowers the free
energy of {1 1 0} and {1 1 1} orientations by 10–20 mJ m2
relative to {1 0 0} orientations. In [13], calculated interfacial
free energies for Al/Al3Sc interfaces in binary Al–Sc showed
a {1 0 0} vs. {1 1 1} anisotropy of 25 mJ m2 at 300 C (with
the free energy of {1 1 0} estimated to be similar to {1 1 1} ).
The present results thus suggest that Mg segregation leads to
a substantial reduction in the anisotropy of the Al/Al3Sc
interfacial free energies, consistent with the observed reduction in precipitate faceting induced by the addition of Mg
[8,13] and the resulting spheroidal morphology.
Fig. 11. The projections compare the atomic geometries near {1 0 0} and
{1 1 0} interfaces. As in Fig. 8, small and large circles denote atomic sites
separated by 0.5 of a lattice spacing in the direction out of the page. Black
and white circles denote Sc and Al sites, respectively. The solid and
hatched gray circles denote Al sites near the interface displaying attractive
binding energies for Mg, as described in the text.
where xMg and xSc are the average concentrations of Mg
and Sc atoms, and gbMg–Sc is an average Gibbs binding free
energy between Mg and Sc atoms for first and second nearest-neighbors. In the case of a random solid solution, the
binding energy is zero therefore leading to an expected
4.3. Heterogeneous nucleation of Al3Sc precipitates?
As the aging time increases from 2 to 1040 h, the Mg segregation peak observed at early time splits into two peaks,
one at the interface separated from a peak at the center of
the precipitates (Fig. 6). In particular after 1040 h aging,
15–42 Mg atoms were detected at the center of the Al3Sc precipitates. It is speculated that Al3Sc precipitation occurs as a
result of interactions between Mg and Sc atoms with vacancies, which leads to a faster nucleation rate than in the binary Al–Sc alloy, as demonstrated by the microhardness
measurements [5]. Indeed, from an analysis of the early
stages of decomposition, the experimental number of Mg–
Sc dimers in the reconstruction is somewhat greater than
the expected number of pairs in the case of a perfectly random solid-solution. The experimental number of Mg–Sc
dimers defined by a maximum separation distance between
the Mg and Sc atoms of 0.5 nm, is 1082 for a total number
of ions, N, equal to 1.154 · 106; this yields a concentration
of Mg–Sc dimers equal to 0.0937 at.%. The theoretical concentration of dimers (first nearest and second nearest-neighbors), xMg–Sc, defined by the number of dimers divided by
the total number of atoms, is given by:
!
gbMg–Sc
xMg–Sc ¼ 18xMg xSc exp
;
ð7Þ
kB T
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
number of first and second nearest-neighbor dimers, given
by 18xMgxSc = 0.0693 at.%, The measured concentration
of dimers is 60% of the actual value, assuming an ion detection efficiency of 60%. An estimate of the average Gibbs
binding free energy in the as-quenched state is then calculated to be 0.040 eV. After 20 min aging, the measured concentration of dimers is 0.0617 at.%, compared to the
random solution concentration of 0.0484 at.%, which
yields an average binding energy of 0.037 eV. A positive value of gbMg–Sc indicates an attractive interaction between
atoms, in agreement with the first principles calculations
in Fig. 10. Al3Sc precipitation occurs as a result of interactions between Mg and Sc atoms with vacancies, which
leads to a faster nucleation rate than in the binary Al–Sc
alloy. The high concentration of quenched-in vacancies
and their high mobility, even at room-temperature, could
explain a fairly high number density of Mg–Sc dimers
and, during the formation of a stable nucleus involving
the Mg–Sc dimers, Mg atoms may get trapped within the
nanoscale growing Al3Sc precipitates, as Sc diffuses to the
precipitates.
We were unable to find diffusion data for any element in
Al3Sc in the literature. The presence of a Mg-rich Al3Sc
precipitate core after a long aging time, i.e. 1040 h,
(Fig. 7) indicates, however, a very small diffusivity of Mg
diffusivity
of Mgffi
in the Al3Sc phase. An estimate of the
qffiffiffiffiffiffiffiffi
ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Al3 Sc
in Al3Sc at 300 C is obtained using hd 2 i ¼ 6DMg
t,
where d is the precipitate diameter and the factor of six is
from three-dimensional diffusion. The estimated diffusivity
is therefore given by:
3 Sc
DAl
ffi
Mg
hd 2 i
.
6t
ð8Þ
For t = 1040 h, Eq. (8) yields a value of 2 ·
1023 m2 s1, which is approximately seven orders of magnitude smaller than the diffusivity of Mg in Al at 300 C
(1.62 · 1016 m2 s1) [37]. This indicates that it is difficult
for Mg to diffuse through the Al3Sc phase to the matrix,
which explains the shape of the Mg concentration profiles
observed in Figs. 6 and 7, even though the equilibrium ternary Al–Mg–Sc phase diagram [36] does not predict any
Mg solubility in the Al3Sc phase. And the first principles
calculations presented in Section 4.2 also indicate essentially no solubility of Mg in Al3Sc.
5. Conclusions
Magnesium segregation occurring at the perfectly coherent a-Al/Al3Sc heterophase interface was studied experimentally using atom-probe tomography (APT) for an
Al–2.2 Mg–0.12 at.% Sc alloy aged at 300 C.
The thermodynamic equilibrium Mg segregation
behavior corresponds to a measured relative Gibbsian
excess of Mg with respect to Al and Sc of 1.9 ±
0.5 atoms nm2.
129
In addition to the Mg segregation at perfectly coherent aAl/Al3Sc heterophase interfaces we also detected Mg at the
centers of Al3Sc precipitates, Fig. 7, which is kinetically
trapped since Mg is insoluble in Al3Sc. The diffusivity of
Mg in Al3Sc is estimated to be 2 · 1023 m2 s1, which is
approximately seven orders of magnitude smaller than
the diffusivity of Mg in Al at 300 C (1.62 · 1016 m2 s1).
The effects of capillarity and solute flux balance are theoretically estimated to give rise to relatively small
(10%) changes in the matrix Mg concentration at the
growing coherent a-Al/Al3Sc heterophase interface.
Therefore, the pronounced interfacial enhancement of
Mg measured by APT cannot be interpreted simply as
reflecting the effects of capillarity and solute flux balance
in a model of diffusion-limited precipitate growth.
The segregation of Mg at the coherent a-Al/Al3Sc heterophase interface was studied theoretically employing
ab initio calculations. These calculations demonstrate
that the driving force for segregation of Mg is due to
electronic interactions rather than elastic strain relaxation associated with highly over-sized Mg atoms. The
calculated value of the relative Gibbsian excess of Mg
with respect to Al and Sc is ca. 1.2 atoms nm2, which
is in good agreement with the experimental value.
The results of the supercell defect-energy calculations are
presented in Fig. 8. It is seen that Sc is calculated to have a
large and negative heat of solution in a-Al, while the corresponding value for Mg is near zero. In the a 0 -Al3Sc
phase, all substitutional point defects are seen to have relatively large and positive formation energies.
We have computed the solute interaction energies up to
fourth neighbor distances for Mg–Mg, Sc–Sc and Mg–
Sc pairs in pure Al employing VASP and 216-atom
supercells (6 · 6 · 6 primitive fcc unit cells), see results
in Fig. 11. Strong and relatively long-ranged interactions are derived for both Mg–Sc and Sc–Sc pairs. The
magnitude and oscillating nature of the Sc–Sc results
are consistent with theoretical models for transitionmetal interactions in Al.
Using the Gibbs adsorption theorem, Mg segregation is
predicted to lower the free energy of {1 0 0}, {1 1 1} and
{1 1 0} heterophase interfaces by roughly 10, 20 and
30 mJ m2, respectively. Magnesium segregation thus
lowers the free energy of {1 1 0} and {1 1 1} orientations
by 10–20 mJ m2 relative to {1 0 0} orientations.
These present theoretical results suggest that Mg segregation leads to a substantial reduction in the anisotropy of
the a-Al/Al3Sc interfacial free energies, which is consistent with our HREM observations of a reduction in precipitate faceting induced by the addition of Mg to Al–Sc
alloys [8].
Acknowledgments
This research is supported by the United States Department of Energy, Basic Sciences Division, under contracts
130
E.A. Marquis et al. / Acta Materialia 54 (2006) 119–130
DE-FG02-98ER45721 (EAM and DNS) and DE-FG0201ER45910 (MDA) and the Air Force Research Laboratory, the Air Force Office of Scientific Research under
contract F33615-01-C-5214 (CW). The authors thank
Professors P.W. Voorhees and D.C. Dunand for very interesting discussions, Dr. D. Isheim for help in using an atomprobe tomograph, and Alcoa Inc. and Ashurst Inc. for
supplying the Al–Sc master alloy. The calculations performed in this work made use of resources at the National
Energy Research Scientific Computing Center under
Contract No. DE-AC03-76SF00098, and the DOD High
Performance Computer Modernization Program at the
Aeronautical Systems Center-Major Shared Resource
Center on the IBM-SP3.
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