Density Functional Study of the b¢¢ Phase in Al-Mg-Si Alloys
PER HARALD NINIVE, OLE MARTIN LØVVIK, and ARE STRANDLIE
The b¢¢ phase is the major hardening precipitate in Al-Mg-Si alloys. It was studied by atomistic
calculations based on density functional theory (DFT), using an atomistic model where the
precipitate was embedded in an Al matrix. This allowed quantifying and visualizing the
coherency strain in the matrix and within the precipitates. The elastic strain was found to
decrease exponentially in the matrix as a function of distance from the precipitate interface. The
formation enthalpy of several different chemical compositions of b¢¢ was calculated, and the
most stable composition was found to be Mg5Al2Si4. A study of the calculated valence charge
density and electron localization function showed that the covalency network between Si-atoms
in the precipitate structure is broken when the precipitate contains Al.
DOI: 10.1007/s11661-014-2214-4
The Minerals, Metals & Materials Society and ASM International 2014
I.
INTRODUCTION
Al-Mg-Si alloys are among the most widely used
lightweight materials in commercial industry today due
to their favorable chemical and mechanical properties,
as well as being relatively low priced. Precipitation is the
most important hardening mechanism of Al-Mg-Si
alloys. The precipitate phases are formed either during
heat treatment followed by storage at room temperature
(natural aging) or during artificial aging. After extrusion
at typically 673 K to 773 K (400 to 500 C) followed
by a rapid cooling, a supersaturated solid solution
(SSSS) is formed in the alloy. During aging, the solute
atoms start clustering in the fcc Al lattice, and precipitates are formed. The precipitation sequence in Al-MgSi alloys is agreed to be[1,2]
SSSS ! atomic clusters ! GP zones ! b00
! b0 ; U1; U2; B0 ! b:
When peak hardness is reached, the alloy has been
found to contain a high concentration of the metastable,
needle-shaped b¢¢ phase.[3–5] These are coherent in the
needle direction, while in the plane orthogonal to the
needle direction, elastic strains arise due to lattice mismatch between the precipitate and the Al matrix. These
strain fields hinder propagation of dislocations throughout the matrix, and thus the material is hardened. The
crystal structure of b¢¢ has previously been found via use
of high-resolution transmission electron microscopy
(HRTEM) and quantitative electron diffraction refinements.[6–7] The unit cell is monoclinic, and the orientational relationships to the fcc Al matrix are given by
PER HARALD NINIVE, Ph.D. Student, and ARE STRANDLIE,
Professor, are with the Gjøvik University College, Teknologiveien 22,
2815 Gjøvik, Norway. OLE MARTIN LØVVIK, Professor, is with
the SINTEF Materials and Chemistry, Blindern P. O. Box 124, 0314
Oslo Norway and also with the Department of Physics, University
of Oslo, Gaustadalleen 21, 0349 Oslo, Norway. Contact e-mail:
o.m.lovvik@fys.uio.no
Manuscript submitted August 7, 2013.
Article published online February 13, 2014
2916—VOLUME 45A, JUNE 2014
½100b00 k ½230Al ; ½010b00 k ½001Al ; ½001b00 k ½310Al : ½1
The monoclinic angle is defined by the angle between
[230]Al and ½310Al in bulk Al, which is 105.3 deg.
The chemical composition of the b¢¢ unit cell was for a
long time believed to be Mg5Si6, since this was indicated
by energy dispersive X-ray spectroscopy,[7] and also
supported by DFT studies.[8] However, a recent combined atom-probe tomography (APT) and DFT study
showed that the structure most likely contains an
amount of Al.[9]
Due to the difficulty of determining the chemical
composition via observational techniques, accurate
theoretical studies of the structure are needed. In
Reference 8, several compounds of b¢¢ were studied by
use of DFT calculations. Mg5Si6 was found to be the
favored composition, but only cases where Al replaced
Mg were considered. In the DFT study of Reference 9, Al
was allowed to occupy any of the sites in the asymmetric
unit cell (see Figure 1(b)), and Mg5Al2Si4 was found to be
the energetically most stable configuration. In this structure, the Al atoms are situated at the Si3-sites in the b¢¢
unit cell. Another configuration, Mg4Al3Si4, was also
found to be more stable than Mg5Si6. Here, the Mg1-site
is also occupied by Al, in addition to the Si3-sites. A
recent study of b¢¢ precipitates by use of high-angle
annular dark-field scanning transmission electron microscopy (HAADF-STEM)[10] indeed showed Mg4Al3Si4 to
be the most likely composition.
In order to make a correct and accurate analytical
study of b¢¢, we employ a model where the precipitates
are modeled as semi-infinite needles embedded in an Al
matrix, that is, infinite along the needle (longitudinal)
direction and of finite size in the transversal plane
orthogonal to the needle direction. In the DFT study of
potential b¢¢ compositions performed in Reference 9, b¢¢
was modeled as a separate bulk phase. However, since
b¢¢ only exists as needle-shaped precipitates in an
aluminum matrix, the model presented in the current
work should give a more correct description than the
bulk model does. We will thus be able to predict the
energetically favored b¢¢ compound in an appropriate
METALLURGICAL AND MATERIALS TRANSACTIONS A
way, by calculating the formation energies of different
theoretical compositions of b¢¢ with varying contents of
Mg, Si, and Al.
Since our model includes the corners and edges of the
precipitate–matrix interface, a detailed and realistic
strain field in the Al matrix as well as inside the
precipitates can also be simulated. Furthermore, our
calculations provide new insight into the electronic
structure and the nature of bonding within the precipitates and at the interfaces.
II.
METHODOLOGY
A large number of supercells corresponding to a
variety of precipitate sizes and spacings were created.
The Diamond program[11] was used to visualize and
construct the precipitate models, using published experimental crystal structures for Al and b¢¢.[7,12] The
precipitate models were based on an N 9 N 9 1 Al
supercell (with N an integer between 6 and 12), replacing
Al atoms in the center of the cell with an integer number
of precipitate formula units, NFU (see Figure 1(a) for an
example). The precipitates were constructed using the
published crystal structure of b¢¢[7] and placed into the
matrix according to the interfacial relations given in Eq.
[1]. Precipitate models were made for each of the three
previously reported compositions Mg5Si6, Mg5Al2Si4,
and Mg4Al3Si4.
The modeled precipitate sizes ranged from a single
formula unit up to 16 (4 9 4) formula units. The
periodic spacing between precipitates, dn, varied from
8 Å up to 40 Å. The DFT calculations were carried out
using VASP (Vienna ab initio simulation package)[13–15]
with the projector-augmented wave (PAW) method,[16]
and within the generalized gradient approximation
(GGA) by Perdew, Burke, and Ernzerhof (PBE).[17] In
precipitate models containing less than 400 atoms, a
2 2 16 k-point grid was used. In the case of
precipitate models containing more than 400 atoms, a
1 9 1 9 8 k-point grid was used. The quasi-Newton
Fig. 1—(a) Atomistic model of a NFU = 16 (4 9 4) b¢¢ precipitate
embedded in an Al matrix. The monoclinic unit cell is marked with
yellow lines, along with the corresponding directions in the Al
matrix. The white frame designates the periodic supercell. (b) One
formula unit of b¢¢ showing the different atomic sites, based on the
original Mg5Si6 structure (Color figure online).
METALLURGICAL AND MATERIALS TRANSACTIONS A
method was used for the relaxations, with remaining
forces below 0.01 eV/Å.
Cubic 4 9 4 9 4 bulk Al supercells containing 256
atoms were used for reference energy calculations, with
one Al atom substituted by a single Mg or Si atom.
Here, a C-centered 5 9 5 9 5 k-point grid was used.
Data for the valence charge density plots, electron
localization function (ELF) plots, and density of states
(DOS) plots were taken from the output of the VASP
calculations.
III.
RESULTS AND DISCUSSION
A. Elastic Properties
One of the biggest advantages of our model is that it
makes us able to study the elastic strain that arises due
to the lattice mismatch between the precipitates and the
Al matrix. This is very difficult to measure experimentally. There are many possible ways one could quantify
and visualize the elastic strain based on the DFT relaxed
structures. The simplest and most straightforward way
to describe the strain in the Al matrix is to calculate the
deviation of the local lattice constant from that of bulk
aluminum along a given direction, and display how it
varies throughout the matrix using a contour plot, as
shown in Figures 2(a) and (c). These plots show the
strain around a single formula unit precipitate, measured along the [100]Al and [010] Al axes, respectively.
The compressions in the Al matrix are due to the
precipitate occupying more space than would a pure Al
matrix. The Si1-Si1-distance in the bulk model is 5.24 Å,
while the corresponding distance in bulk Al is 4.04 Å, so
one formula unit occupies almost 30 pct more space
than Al along the [100] axis. The Si3-Si3-distance in the
bulk model is 8.91 Å, while the corresponding distance
in the Al lattice is 8.08 Å (two periods in the Al lattice).
So along the [010] axis one precipitate formula unit
occupies 10 pct more space than Al. We see from the
contour plots that the maximum compression is clearly
larger along the [100] axis than along [010]. At the same
time, there is tension along the [010] axis in the areas
with compression along [100], and vice versa.
Figures 2(b) and (d) show 1D plots of the deviation
along with an exponential fit. Similar measurements
from supercells with larger precipitates than one single
formula unit have also been included. Here, the strains
were measured along lines starting from the interface by
the Si1-site (for strain along [100]) and by the Si3-site (for
strain along [010]) of the formula unit at the upper right
corner of the precipitate.
The strain falls off nearly exponentially as a function
of distance from the precipitate interface both along the
[100] and the [010] axis. The maximal compression is less
along the [010] axis than along the [100] axis, but the
strain field is clearly falling off at a slower rate along
[010].
We also see local maxima in strain near the Si1- and
Si3-site atoms for larger precipitates. However, the most
important cause of strain around b¢¢ precipitates consisting of more than one formula unit is the periodic
VOLUME 45A, JUNE 2014—2917
Fig. 2—(a) The strain in the x-direction of the Al matrix surrounding a single formula unit of Mg5Al2Si4 b¢¢, quantified as the deviation from
the bulk Al lattice constant. The atom positions in the matrix are indicated by black dots. (b) The distortion DaAl in the x-direction of the Al
matrix as a function of distance dn from the precipitate interface along the dashed line shown in (a). Similar calculations for other precipitate
sizes are also included. The dashed line is a linear regression fit in the log plot, fitted with equal weight to all the calculated points; it displays a
quite good fit to an exponential behavior. In all calculations included, supercells of 48.5 Å (corresponding to 12 periods in the Al lattice) in the
x- and y-direction were used. (c) A similar plot to that shown in (a), but here with the strain in the y-direction. (d) The distortion DaAl in the
y-direction of the Al matrix as a function of distance dn from the precipitate interface along the dashed line shown in (c) (Color figure online).
mismatch between the b¢¢ structure and the Al matrix.
The experimental mismatch between Al and b¢¢ is 3.8 pct
along ½230Al =½100b00 and 5.3 pct along ½001b00 =½310Al .[7]
The calculated strain along these two directions is
shown in Figure 3, in which we have also included the
strain within the precipitate. Since the matrix and the
precipitate have different reference values, we have
plotted the strain as the deviation in percent from the
corresponding experimental periodic distances. The
maximum tension in the Al matrix is pretty similar in
both directions, while the maximum compression is
clearly larger along [310] than along [230] due to larger
mismatch. The strain will generally be higher near the
interface of large precipitates than in the small ones,
since we will have more periodic distances and thus a
larger total mismatch. The relative difference in strain
between the three compositions was found to be small,
even though their relaxed bulk volumes are clearly
different.
In the precipitate models consisting of 9 or 16 formula
units, we were not able to run simulations on supercells
large enough to contain the entire strain field. This will
2918—VOLUME 45A, JUNE 2014
Fig. 3—Contour plots of the strain in the ½230Al =½100b00 direction
(a) and in the ½310Al =½001b00 direction (b) in a supercell containing a
NFU = 16 (4 9 4) Mg5Al2Si4 b¢¢ precipitate. The reference values in
the Al matrix are 7.30 Å in the [230]Al direction and 6.40 Å along
[310Al : In the precipitate, the reference values are 7.58 Å (=a/2)
along ½100b00 and 6.74 Å along ½001b00 . The blue dots represent the
Al positions, and the red crosses designate the atoms of the precipitate. The dashed lines give a rough indication of the interface
between the precipitate and the matrix. The black arrows show the
directions in which the strain was measured. The strain fields of the
other two compositions (Mg5Si6 and Mg4Al3Si4) were found to be
qualitatively very similar to those shown here (Color figure online).
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 4—Contour plots of the calculated anisotropy field in the Al matrix around four b¢¢ precipitates with different sizes. The anisotropy was
quantified as the local lattice parameter in the x-direction ([100]) divided by that in the y-direction ([010]). Thus, there is positive strain in the
x-direction (x/y > 1) in the red areas, while the strain is positive in the y-direction in the blue areas (x/y < 1) (Color figure online).
hence lead to overlapping strain fields near the boundaries of the supercell. An expected consequence of this is
that the calculated fall-off rate of the strain field will be
too slow. This is shown in Figure 2(d), where the fall-off
in supercells containing precipitates of 9 or 16 formula
units clearly lie above the exponential fit as we go
further away from the precipitate interface.
Another way to visualize the distortion is to compare
the local strain in the [001] direction to that in the [010]
direction (see Figure 4); this quantifies the anisotropy of
the strain. The anisotropy is clearly largest close to the
precipitate interface, especially near the Si1- and Si3sites, but decreases nearly exponentially outwards along
the [100]Al and [010]Al directions. The anisotropy is
caused by what was seen in the strain plots in Figure 2,
where we have compression in the [100] direction at the
same time there is tension in the [010] direction, and vice
versa. In addition, anisotropy is to some extent also
caused by the volume change in the relaxed matrixprecipitate supercell. In the largest precipitate models,
the total extent of the supercell in the [100] and in the
[010] direction is about 1 pct larger than that of a
corresponding bulk aluminum cell.
METALLURGICAL AND MATERIALS TRANSACTIONS A
The deformation of the precipitates from their bulk
structure can furthermore be quantified by the change in
unit cell volume and monoclinic angle when embedded
in the Al matrix. Figure 5 shows the calculated b¢¢
conventional unit cell volume and the monoclinic angle
(averaged across the precipitate) plotted as a function of
model precipitate size. Expectedly, we see that the mean
unit cell volume tends to go toward the bulk values as
the precipitate sizes increase. We expect the precipitate
structure to be more bulk-like in the inner regions, far
away from the interface. It should then follow that large
precipitates will have a more bulk-like structure on
average than small ones, in which the entire crosssection will be strongly affected by the surrounding Al
matrix.
The average monoclinic angles, on the other hand,
seem to be rather unaffected by the precipitate size. For
Mg5Al2Si4 and Mg4Al3Si4, the values are already close
to those of the respective bulk models, while there is a
clear deviation for Mg5Si6. This indicates that the
Mg5Si6 bulk structure will not be recovered for any
precipitate size, which again is an indication that Mg5Si6
is not the favored structure of b¢¢.
VOLUME 45A, JUNE 2014—2919
Fig. 5—(a) The b¢¢ cell volume in the precipitate model vs precipitate
size (NFU) calculated for the three previously reported compositions,
in addition to the volume of b¢¢ based on the experimentally measured cell parameters. (b) A similar plot of the average monoclinic
angle vs precipitate size. The error bars show the variation of the
angles across the precipitates. The bulk values for each of the three
previously reported compositions are shown as data points at
NFU ! 1.
B. Electronic Structure and Bonding
An investigation of the electronic structure and the
bonding nature of the b¢¢ precipitates can provide insight
in their relative stabilities. To this end, we have studied
the valence charge density and the ELF[18] of the
different b¢¢ compositions in the precipitate model.
Together, these functions provide a comprehensive view
of the spatial distribution of chemical bonds; ionic
bonds will be seen as accumulation of valence charge on
certain nuclei, while covalent bonds are displayed as
regions of high localization. The ELF can be considered
to be a measure of the inverse kinetic energy of the
electrons—thus, a high degree of localization along with
a noteworthy valence charge density designates a
covalent bond. A metallic bond will be shown as a
distributed charge density and low values of the ELF.
Contour plots of the valence charge density for a Mg5Si6
and a Mg5Al2Si4 precipitate model are shown in
Figure 6, and corresponding plots of the ELF are
shown in Figure 7. Mg4Al3Si4 was not included since
2920—VOLUME 45A, JUNE 2014
the charge density and ELF are qualitatively identical to
those of Mg5Al2Si4. As was also seen in Reference 19,
the concentration of charge between Si nearest neighbor
atoms is a prominent feature in the Mg5Si6 structure.
We see that there is also a relatively high ELF around
and between the Si atoms, supporting the previous
report on a covalent network between the Si atoms in
the Mg5Si6 structure.[20] This network is broken, however, if the Si3-site is substituted by Al (Figures 6(b) and
(d)). The only remaining covalent bonds are those
between the Si1-site atoms. Thus, in the Mg5Al2Si4 and
Mg4Al3Si4 structures, there are more metallic and to
some extent ionic bonds.
The charge transfer between the precipitate atoms and
the matrix near the interface appears to be rather small,
indicating little or no covalent bonding between the
atoms in the matrix and the atoms in the precipitates.
The DOS provides further information about the
bonding nature and the atomic interactions within the
precipitates.
Figure 8 shows plots of the DOS for a formula unit of
b¢¢ in the bulk model and in the precipitate model for the
three most stable compositions. The width of the bands
is very similar for each of the compositions, indicating
that the exchange of Al at the Si3- and the Mg1-sites has
a rather small effect on the bonding energies.
Figure 9 displays the local DOS (LDOS) for three
single Al atoms, one located in the matrix (far away
from the precipitate), one at the interface between
matrix and precipitate, and one in the middle of the
precipitate. Expectedly, the LDOS characteristics for the
atom in the matrix differ from those of the other two,
since its environment is purely metallic. In the precipitate, Al is influenced by Mg and Si. However, the DOS
is very similar at the interface and in the interior. The
peak at 6 eV originates from the b¢¢s states. This peak
is also present for Al atoms at the interface, indicating
that there is an overlap in the LDOS between the
precipitate and the matrix, which means that there is
bonding at the interface.
C. Energy Calculations
In order to find the energetically most stable composition of b¢¢, we compared the total energy of an Al
supercell with an embedded precipitate to that of a solid
solution containing the corresponding number of (noninteracting) solute atoms spread throughout the matrix.
We quantified this by the precipitate energy, Eprec,
defined by
Eprec ¼ ðEtot Ntot EAl NMg EMg NSi ESi Þ=NFU : ½2
Etot is the total electronic energy of the supercell containing the precipitate, Nx is the number of atoms of
element x, Ex is the solute energy per atom of element
x, and NFU is the number of formula units in the
precipitate cross-section. The solute energies were
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 6—2D plot of the valence charge density (electrons/Å3) of a NFU = 9 (3 9 3) Mg5Si6 (a, c) and a NFU = 9 (3 9 3) Mg5Al2Si4 (b, d) precipitate in an Al matrix in the (100) plane at z = 0 and in the (001) plane at x = 0. Plots are shown for cross-sections (a, b) and length sections
(c, d). The dashed lines give a rough indication of the interface between the precipitate and the Al matrix (Color figure online).
calculated using a cubic 256 atom Al supercell. For
Al, it was then given by
EAl ¼ EAl256 =256:
½3
The solute energies EMg and ESi correspond to the
energy penalty we must pay for replacing an Al atom
with a Mg or Si atom in the supercell:
EMg ¼ EAl255 Mg EAl256 ;
½4
ESi ¼ EAl255 Si EAl256 :
½5
In Figure 10, the precipitate energies of several different
chemical configurations of b¢¢ are plotted as a function
of the Mg content relative to the Si content in the
precipitate. In these calculations, the precipitate crosssection size was 9 (3 9 3) formula units. The lines define
the convex hull which represents the minimum precipitate energy for a given Mg to (Mg+Si) ratio in the
alloy. According to these calculations, Mg5Al2Si4 is
favored in Mg-rich alloys, while the Mg5Si6 configuration
METALLURGICAL AND MATERIALS TRANSACTIONS A
is favored in Si-rich alloys. The Mg4Al3Si4 composition
lies above the convex hull, which is somewhat surprising
considering the results from the HAADF-STEM studies
in Reference 10.
Figure 11 shows the precipitate energies of the three
most stable compositions plotted as a function of crosssection size. From our results it is clear that Mg5Al2Si4 is
the thermodynamically most stable composition of
b¢¢, given that we have free access to solute elements.
This result confirms that the Si3-sites are occupied by Al.
However, it does not seem energetically favorable for
the Mg1-site to be occupied by Al. The Mg4Al3Si4
composition is also less stable than Mg5Si6, though the
energy difference between these two compositions is
smaller for larger precipitate sizes. This might suggest
that Mg4Al3Si4 is more energetically stable than Mg5Si6
in large precipitates. There are, however, no indications
that Mg4Al3Si4 at any point will be more stable than
Mg5Al2Si4. As was discussed in Reference 10, a possible
explanation of the inconsistency with the HAADFSTEM observations could be that the composition of b¢¢
VOLUME 45A, JUNE 2014—2921
Fig. 7—Contour plots of the ELF of b¢¢ precipitates. In the red areas (ELF greater than 0.5), the electrons are more localized than in a uniform
electron gas, while in the blue areas (ELF less than 0.5) they are less localized. Plots are shown for cross-sections (a, b) and length sections (c, d)
of the Mg5Si6 (a, c) and Mg5Al2Si4 (b, d) models (Color figure online).
is inhomogeneous, that is, the composition near the
interface may differ from that in the interior. This is due
to the contraction of the cell parameters of b¢¢ near the
interfaces, favoring Al instead of Mg at the Mg1-site,
since Al has a smaller atomic radius than Mg. This was
indeed indicated in the HAADF-STEM analysis. It is
also likely that the precipitate composition is determined
by the composition near the interface. Since the growth
direction is likely to be inside-outwards, the interface
composition will be frozen when it is covered by a new
interface layer. However, we should then expect
Mg4Al3Si4 to be the preferred composition for very
small precipitates (e.g., NFU = 1), which is clearly not
the case. It is also conceivable that temperature or
entropy effects can affect the stability of the precipitates.
However, it is difficult to argue that this could affect the
relative stability of these different compositions enough
to explain the observed discrepancy.
Thus, the only explanation we have left, given that the
HAADF-STEM observations are correct, is that the
configuration of b¢¢ is governed by kinetic effects during
precipitate growth. One possibility is that the growth is
asymmetric, favoring filling of Mg on Mg1 sites before
the surrounding precipitate is formed. Also, it is possible
2922—VOLUME 45A, JUNE 2014
that due to local strains, the barrier for diffusive jumps
between matrix and precipitate is higher for Al than for
Mg, so that Al is not able to jump back into the matrix
when a vacancy is present next to the interface of the
growing precipitate. However, studies of such kinetic
effects are beyond the scope of this work.
Finally, we cannot exclude the possibility that the
HAADF-STEM observations might be incorrect, as
there are several potential error sources in the analysis
of the experimental data.
IV.
SUMMARY AND CONCLUSIONS
The b¢¢ phase has been investigated by use of DFT
calculations at the GGA level, employing a periodic
atomistic model without vacuum where the precipitate
was embedded in an Al matrix. Our model allowed us to
study directly the coherency strain in the Al matrix and
in the precipitate. From this, we were able to make a
prediction of the range of the strain field in the Al matrix,
and to study how the structure of the precipitate itself is
affected by the matrix. The strain field was found to
decrease nearly exponentially along the [100] and [010]
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 8—DOS for a formula unit b¢¢ in each of the three most stable compositions in the bulk (a, b, and c) and in the precipitate model (d, e, and
f), including partial DOS for the s states (blue) and the p states (red). The plots were smoothed using a Fast Fourier Transform (FFT) filter with
5 points (Color figure online).
Fig. 9—LDOS projected on s states of Al. The Al atom is placed in
a b¢¢ Mg4Al3Si4 precipitate in the bulk model (red), in the precipitate
model (green), at the precipitate–matrix interface (blue), and in the
Al matrix (purple) (Color figure online).
directions as a function of distance from the precipitate–
matrix interface. Close to the interface, the Al lattice was
distorted by up to 5 pct along the [230] and [310]
directions due to the lattice mismatch between the matrix
and the precipitate. An investigation of the valence
charge density and the ELF for different compositions of
b¢¢ showed that the covalent network found in the Mg5Si6
structure is partially broken if the Si3-site is occupied by
Al. Thus, the Mg5Al2Si4 and Mg4Al3Si4 structures have
more ionic and metallic bonds. The DOS showed similar
characteristics for each of the three most stable compositions, though Mg5Si6 had the largest number of states
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 10—The precipitate energy as a function of NMg/N(Mg+Si). The
three most stable compositions are indicated. Energies of most possible configurations where Al is substituted at Mg- and/or Si-sites are
included. However, we have not allowed Mg at Si-sites or vice versa.
below the Fermi level. Furthermore, the energetics of the
b¢¢ precipitates were studied in order to find their most
likely chemical composition. Mg5Al2Si4 was found to be
the thermodynamically most stable composition, which
is in agreement with recent APT and DFT studies.[9]
On the other hand, the structure observed in recent
HAADF-STEM studies, Mg4Al3Si4,[10] was found to be
less stable than both Mg5Al2Si4 and Mg5Si6. The most
likely explanation of this discrepancy is that the resulting
composition of b¢¢ is governed by kinetic effects during
the precipitate growth, which are not accounted for in
our calculations.
VOLUME 45A, JUNE 2014—2923
Fig. 11—The precipitate energy vs precipitate size (the number of
formula units in the cross-section) for the three most stable compositions of b¢¢.
This work has demonstrated that atomistic modeling
and DFT can be used to perform realistic calculations on
precipitates in their real environment, predicting their
interactions with the surrounding matrix. This method can
thus be very useful for future studies of similar systems.
ACKNOWLEDGMENTS
Computation time from the NOTUR consortium is
gratefully acknowledged.
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METALLURGICAL AND MATERIALS TRANSACTIONS A