Acta Materialia 55 (2007) 2183–2199
www.actamat-journals.com
Phase stability and structural features of matrix-embedded
hardening precipitates in Al–Mg–Si alloys in the early
stages of evolution
M.A. van Huis
a,b,*
, J.H. Chen
a,b
, M.H.F. Sluiter
b,c
, H.W. Zandbergen
a
a
c
National Center for HREM, Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, NL-2628 CJ Delft, The Netherlands
b
Netherlands Institute for Metals Research, Delft University of Technology, Mekelweg 2, NL-2628 CD Delft, The Netherlands
Virtual Materials Laboratory, Department of Materials Science, Delft University of Technology, Mekelweg 2, NL-2628 CD Delft, The Netherlands
Received 6 July 2006; received in revised form 25 September 2006; accepted 16 November 2006
Available online 30 January 2007
Abstract
The strength of Al–Mg–Si aluminium alloys depends critically on nanometre-size MgxSiyAlz-type precipitates that have a face-centered cubic-based structure. In this work, a large number of early structures are investigated by means of first-principles calculations.
Both platelet-type and needle-type precipitates are considered. Calculations show that for alloys with an Mg:Si ratio smaller than
one, needle-type precipitates with Si pillars extending in the needle direction are energetically favoured. The formation of Si pillars
and the low density cylinder is described. For alloys with an Mg:Si ratio larger than one, platelet-type precipitates consisting of stacked
layers of Mg, Si and Al atoms are energetically favoured. Using both the information on the formation enthalpies and the calculated
lattice mismatch with the Al matrix, it is discussed which structures are likely to be formed. The earliest, most favourable structures with
high Al content are the needle-type initial-b00 Mg2Si3Al6 structure and the platelet-type structures {MgSi}2Al10, {MgAl}1Al10, Mg3Si2Al5
and Mg2Si1Al3.
2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Aluminium alloys; Nanostructure; Ordering; Precipitation
1. Introduction
Aluminium is increasingly used in the car industry
because of its combination of strength and low weight.
The low weight improves the car fuel efficiency, and it is
also used to adjust the weight balance of the car. After
the thermal treatments applied in the aluminium production factory, further processing takes place at the car factories. First, the aluminium sheets are shaped into car parts
by stamping. At this stage the aluminium should be easily
*
Corresponding author. Address: National Center for HREM, Kavli
Institute of Nanoscience, Delft University of Technology, Lorentzweg 1,
NL-2628 CJ Delft, The Netherlands. Tel.: +31 152782272; fax: +31
152783251.
E-mail address: [email protected] (M.A. van Huis).
deformable. After it has been painted, it undergoes the final
heat treatment, which is the bake hardening process at a
temperature of 180 C. Here, a high density of MgxSiyAlz
precipitates are formed that are responsible for the large
increase in strength. After this last thermal treatment the
material should be hard and stiff for passenger protection.
In order to ensure that the material is easily deformable
during stamping, hardly deformable after paint bake hardening and that the material properties do not degenerate
during storage, the composition of the alloy and the
sequence of thermal treatments need to be fine-tuned very
carefully, and detailed knowledge of the evolution of precipitation during all industrial processing steps is of great
importance.
There are two main reasons why knowledge of the
early precipitation stages is important: First, in the bake
1359-6454/$30.00 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2006.11.019
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M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
hardening process that is currently applied by the aluminium industry, the annealing temperatures are too low and
the annealing times too short to reach the b00 Mg5Si6 phase
with maximum hardness. Therefore, in the cars currently
being fabricated with these 6xxx Al alloys, the hardness
actually depends on precipitates in an earlier stage, which
makes the study of these early stages so important. Second,
knowledge of the early precipitation stages is required to
develop pre-ageing treatments in order to prevent natural
ageing. There is a storage time of typically a few months
between the production of Al sheets and further processing
at car factories. This natural ageing at room temperature
leads to lower hardness after the quick-bake hardening.
The hypothesis is that small clusters or precipitates are
formed at room temperature and that these clusters should
dissolve or transform before the large monoclinic hardening precipitates can be formed. It is necessary to know
the structure of these unwanted small clusters in order to
develop better pre-ageing treatments.
The smallness of the clusters at the early stages makes
investigation by means of electron microscopy very
demanding. Atom probe field ion microscopy (APFIM)
has been applied to the early precipitates stages in AA6xxx
alloys [1] and provides information on the size and composition of the precipitates, but the spatial resolution is very
limited and, therefore, no detailed structural information
is obtained from these experiments. There is great potential
for improving the properties through computer simulations
to optimize the microstructure by fine-tuning the thermal
treatments and compositions.
In a previous work [2], the authors performed first-principles calculations on the phase stability of the late phases
(b00 to b) and discussed their mutual structural relationships. In this work, the early stages that still have a facecentered cubic (fcc)-based structure were investigated by
means of first-principles calculations. Ravi and Wolverton
[3] and Chakrabarti and Laughlin [4] reviewed the litera-
ture on AA6xxx alloys. The contents of Si and Mg are in
the range 0.5–1.0 wt.%, usually with an Mg:Si ratio smaller
than one. A large number of structures were identified or
proposed, based on experimental and theoretical results
[5–15]. The generic precipitation sequence for the AlMgSi
alloys is as follows [2]:
SSSS → clusters → initial- β”
β” (Mg5Si 6)
→ pre-β”
β’ (Mg9Si5) → β (stable).
ð1Þ
U2, U1
where SSSS is the supersaturated solid solution. The various phases are listed in Table 1. The phase transformation
from pre-b00 to b00 (or directly to U2) marks the transition
from fcc-based structures to non-fcc-based structures.
Early phases means all the fcc-based phases (SSSS up to
pre-b00 ), while the other phases (up to b) are the late phases.
In this work, the general term ‘‘GP zone’’ will not be used,
as this word is not specific: it has been used for all early
structures and for b00 . The MgxSiyAlz precipitate structures
that were reported in the literature will now be briefly discussed, starting with the smallest clusters.
Much is still unclear about the structure of the earliest
clusters (size < 5 nm), and reports in the literature are
partly inconsistent. In alloys with ratio Mg:Si = 2, platelet-like structures were observed by means of high resolution transmission electron microscopy (HRTEM) after
artificial natural ageing at a temperature of 343 K. In order
to explain their observations, Matsuda et al. [8] suggest
that the platelets are {0 0 1}fcc composite {MgSi} slices
which consist of alternating Æ1 0 0æ columns of Mg and Si
atoms. Additional Al layers can be inserted in-between
the {MgSi} layers. The phases {MgSi}1Al2 and
{MgSi}1Al6 of this type were calculated by Ravi and Wolverton [3]. This work considers many more possibilities, for
different stoichiometries, arrangements and spacings of Al
Table 1
Overview of the complex precipitation sequence in MgSiAl 6xxx alloys
Stage
Solute concentration
(at.%)
Shape
Description, features
SSSS
0–5%a
Point defects
Atomic clusters
1–15%a
Clusters (1–5 nm)
Initial-b00
Pre-b00
b00
Late phases: B 0 , b 0 ,
U1, U2
b
10–25%
15–45%
35–60%
50–90%
60–100%
Dimers, trimers,
clusters
3d, 2d
Needle
Needle
Needle
Rod/lath
Supersaturated solid solution: substitutional Mg and Si atoms, large
concentration of vacancies
Early clustering of solute atoms, large degeneracy in formation enthalpy of
structures, entropy effects very strong
Spherical, APFIM [1] and platelets, HRTEM [8]
Monoclinic cell, single Si pillars [15]
Monoclinic cell [9], double Si pillars, low density cylinder appears
monoclinic cell [5] double Si pillars, LDC underwent 0.5b shift
Collection of hexagonal, trigonal and orthorhombic phases [2]
95–100%
Cube
Stable Mg2Si bulk phase with anti-fluorite structure [16], excess Si forms the
diamond structure
For each stage, a tentative Al concentration is given based on results found in the literature (EDS measurements, experimental and theoretical structural
refinements).
a
No real distinction can be made between the precipitate and the matrix. The particles are too immature and contain too much Al to determine where
the interface is.
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
2185
Fig. 1. Platelet-type, fcc-based precipitates. The Mg1Si1 structure (IV, Table 3) is one of the energetically most favourable fcc composite phases when the
cell dimensions are confined to Al matrix values. Al layers can be inserted as (0 0 1) planes, separating {MgSi} composite planes. Here, the Matsuda
platelets (II) are formed that were reported experimentally [13]. {MgAl} platelets are also considered. Alternatively, Al layers can be inserted as (0 1 0)
planes, thereby forming platelets with Mg/Si < 1 (I) or Mg/Si > 1 (III).
layers, as displayed in Fig. 1. The thermodynamically
metastable Mg1Al3 phase (L12-type) is also included, this
phase can be considered as a {MgAl} platelet with a spacing consisting of a single Al layer: Mg1Al3 = {MgAl}1Al2.
Murayama et al. [1] have applied three-dimensional atom
probe field ion microscopy (3D-APFIM) to alloys with
Mg:Si ratios of 2 and 1, after artificial natural ageing at
343 K. They did not find any platelets. Instead, they found
spherical clusters 2 nm in size, with a high atomic Al concentration of 80%. It is not clear whether the Mg and Si
atoms are arranged in any particular way.
Recently, a very early monoclinic stage has been identified experimentally with a composition close to
Mg2Si3Al6 [15]. This phase will be called initial-b00 , as it
can be considered an early precursor of the pre-b00 phases.
The Al concentration is relatively high at 60%. It is
observed experimentally that the cell dimensions are commensurate with the Al lattice and that the atomic positions are very close to that of the Al matrix. Hence, this
early structure is very coherent with the embedding Al
matrix. This structure and a few variations of this structure are shown in Figs. 2(a)–(d) and 3. The precipitates
are needle-shaped, and their most important feature are
the Si pillars that extend along the b-axis. It was determined experimentally that the Si pillars form a stable skeleton, while the composition of the atoms in-between the
columns is variable. This is a new and conceptually useful
model, because it allows for fluctuations in stoichiometry
within the precipitate, and therefore it can accommodate
much configurational entropy, which is very advantageous
at this stage of the precipitation sequence. These early precipitates are possibly needle-shaped, because growth by
extension of the stable Si columns (along the b-axis) is
energetically favoured. Growth in the lateral directions is
not favourable, as this requires nucleation of new Si
columns.
The b00 phase is important because it gives the best hardness [9,11]. It has a monoclinic structure with composition
Mg5Si6 that was found by means of quantitative electron
diffraction [5]. The cell dimensions and atomic coordinates
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Fig. 2. Evolution of the needle-type precipitates. All structures are periodic in all dimensions, and the needle stretches along the b-axis, because this
direction has the smallest lattice mismatch with the Al matrix. First, solute atoms agglomerate into parallelogram-type structures ((a)–(c)). Two such
clusters together form the initial-b00 structure [15] with single Si2 columns, on positions 6 and 8 ((d) and (e)). As more Al atoms are replaced by solute
atoms, double Si2 columns (positions 6 and 8 and 9 and 10) are formed ((f)–(j)), and the LDC is also formed (positions 2–5 become Mg atoms). In the more
evolved stages, slices of Mg hexagons are formed (j) that are the main structural unit of the late phases [2].
of the Mg5Si6 structure are shown schematically in Fig. 2i,
where two conventional monoclinic unit cells are displayed.
The cluster of five magnesium atoms in the Mg5Si6 structure is often called the ‘‘eye’’, as they resemble eyes in the
HREM images [5]. In this work, the ‘‘eye’’ will be called
the ‘‘low density cylinder’’ (LDC) as this column, with a
basis of five atoms, extends along the b-axis and has a
low atomic concentration. The pre-b00 structures were studied by Marioara et al. [9,11] and are a variation of the b00
Mg5Si6 structure shown in Fig. 2i. In the Marioara model
[9], the Mg atoms in the LDC are partly replaced by Al
atoms so that the composition becomes (Mg + Al)5Si6.
Furthermore, the Mg/Al atom in the center of the LDC
(Mg atom #1 in Fig. 2i) is shifted by 0.5b, which brings this
atom into an fcc-type configuration. Consequently, the
whole precipitate structure becomes an fcc-based structure.
Pre-b00 structures are displayed in Fig. 2f and h. The structures are relatively commensurate with the Al lattice that
can be easily recognized in Fig. 2a–c. The late phases (b 0 ,
B 0 , U1, U2) consist of trigonal, hexagonal and orthorhombic structures and were investigated in a previous work [2].
Finally, the stable b phase is the cubic anti-fluorite Mg2Si
phase that is present in over-aged specimens.
In order to investigate the earliest stages of precipitation, when precipitates are still too small to be characterized by means of electron microscopy, electronic density
functional total energy calculations were performed on
selected precipitate structures. This has provided insight
in the energetics of the precipitation process. We did not
use the embedded atom method or related semi-empirical
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
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2. Methodology and computational details
The formation enthalpies of the phases mentioned in
Tables 1–3 and displayed in Figs. 2–4 were calculated using
the first-principles VASP code [17–19], with the efficient
ultrasoft (US) pseudopotentials [20] and employing both
the local density approximation (LDA) and the generalized
gradient approximation (GGA). Energy and k-point convergence was ascertained for all systems in order to limit
the uncertainty to 1 meV atom1 or 0.1 kJ mole1. All calculations were performed with an energy cut-off of 200 eV.
For all structures, the density of the Monkhorst–Pack kpoint grid was increased until the above-mentioned accuracy was reached. For all grids thus obtained, the linear
k-point spacing was <0.017 Å1. For the b00 Mg5Si6 structure, for example, a k-point grid of 4 · 16 · 10 (pertaining
to the reciprocals of the a-, b- and c-axes, respectively) was
required.
The calculated enthalpies of the structures cannot be
directly compared with the experimental data as they
depend on the choice of pseudopotentials. Formation
enthalpies, formulated as differences of enthalpies, however, are in principle independent of the potentials and
can be compared with experimental calorimetric data,
where available. The formation enthalpy with respect to
the solid solution (SS) is defined as
a
a
sub
DH form
Þ
SS ðMgx Siy Alz Þ ¼ H ðMgx Siy Alz Þ xH ðMg
yH ðSisub Þ zH ðAlfcc Þ
ð2Þ
Table 2
Most favourable interatomic distances calculated using VASP-GGA,
derived from the thermodynamically most stable binary phases; the L10
structure for Mg1Si1 is displayed in Fig. 4(a)
All structures
Al–Al
Mg–Mg
Si–Si
Mg–Si
Mg–Al
Si–Al
a
Fig. 3. Structural refinement of the initial-b00 Mg2Si3Al6 structure: (a)
experimental coordinates; the arrows indicate which atoms can be
swapped [15]; (b) refined atomic positions (VASP-GGA) with the cell
dimensions confined to be commensurate with the Al matrix, using a
primitive C-centered cell; (c) as (b), but now using a conventional
monoclinic cell, i.e., without symmetry operations caused by the Ccentering; (d) refined structure (VASP-GGA) after full relaxation, both
cell dimensions and atomic positions. Structures (b) and (c) both
transform into the C-centered (d) structure when the cell dimensions are
relaxed. Structural details are listed in Tables 6 and 7.
schemes, because they are generally not as accurate and
reliable as density functional theory. In particular, the preferred tetrahedral orientation of the Si atoms (with four
nearest neighbours making an angle of 109) is best
approached with DFT codes.
fcc-type structures
d (Å)
Structure
d (Å)
Structure
2.86
3.19
2.36
2.75
3.01
2.76
fcc Al
hcp Mg
diamond Si
Mg2Si (b)
Mg1Al1 (L10)
Si1Al1 (L10)a
2.86
3.20
2.75
2.89
3.01
2.76
fcc Al
fcc Mga
fcc Sia
Mg1Si1 (L10)
Mg1Al1(L10)
Si1Al1 (L10)a
Not observed experimentally.
Table 3
Calculated structure for the Mg1Si1 phase, Al matrix coherent (atomic
positions relaxed) and fully relaxed (both cell and ions relaxed), calculated
with VASP-GGA – the structure is displayed in Fig. 4(a)
Property
Al matrix constrained
Full relaxation
Space group
a (Å)
b (Å)
c (Å)
b
(x, y, z)Mg1
(x, y, z)Si1
P2/m
4.05
4.05
4.05
90.0
(0.744, 0.000, 0.256)
(0.231, 0.500, 0.231)
P2/m
4.29
3.91
4.29
93.6
(0.735, 0.000, 0.265)
(0.287, 0.500, 0.287)
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M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
Fig. 4. Correspondence between the Mg1Si1 Matsuda phase displayed in (a) and (b) and the pre-b00 Mg5Si6 phase (d). The pre-b00 structure contains two
slices of the Matsuda phase separated by a planar defect. Si2 pillars stretching along the b-axis are formed across this planar defect. The 0.5b shift, which is
the phase transformation from pre-b00 to b00 , is shown in (d) and (e).
where the EMg,sub and ESi,sub are the enthalpies of Mg and
Si atoms on substitutional sites in the Al matrix. When the
formation enthalpy is negative, there is an energy gain with
respect to the solid solution. The values of H(Mgsub) and
H(Sisub) and other details can be found in Ref. [2].
An important aspect that should be considered is the
interaction with the aluminium matrix. Because of interface
strains caused by the interaction with the surrounding Al
lattice, the unit cell of the precipitate material may have
a different shape and volume from that without the influence of the Al lattice (at standard pressure, and considering
the precipitate material as bulk). Therefore, the precipitate
structures were calculated in two modes:
relaxing only the atomic positions while constraining the
cell dimensions to be commensurate with the Al matrix;
relaxing both the atomic positions and the lattice
parameters (a, b, c, a, b, c) in order to find the lowestenthalpy structure.
The precipitate will be called commensurate with the Al lattice when the corner points of the unit cell of the precipitate
structure coincide with lattice points of the Al fcc matrix.
This is analogous to the definition used by Sutton and Balluffi [21] for a commensurate interface.
The first mode has the advantage that it is closer to the
structural properties of the precipitates as embedded in Al,
the latter mode has the advantage that phases also can be
studied that are less commensurate or incommensurate
with the Al matrix. Finally, in order to compare structures
with different Al content, the formation enthalpy can also
be expressed in kilojoules per mole of solute atoms, instead
of kilojoules per mole. This transformation is achieved as
follows: DHSS [kJ mole solute1] = DHSS [kJ mole1]/
(xSi + xMg), where xSi and xMg are the atomic fractions
of Si and Mg in the precipitate MgxSiyAlz (x = xMg,
y = xSi). The use of this transformation is that the threedimensional common tangent hull is projected on a twodimensional plane, as explained in Ref. [2].
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
3. Results
A very large number of early structures were calculated,
and the phases displayed in Figs. 1–4 are only a small selection. There is an infinite number of possibilities for arranging the three different atoms in an fcc-type arrangement, so
no attempt is made to list or calculate them all. It is also
not very useful to try and find the one ‘‘most favourable’’
early structure, as the entropy effects are very strong in
the early stages, and there will be a large variation in stoichiometry and arrangements, i.e., there is not a single
‘‘most favourable’’ structure. In general, however, there
will be an alternation of Mg and Si atoms, as they compensate for their different size relative to Al atoms. While it is
relatively easy to determine which phases are stable at thermal equilibrium, the determination of the ‘‘most metastable’’ phases is complicated as the relative stability of
metastable phases changes during the evolution of precipitation. The Al content in the precipitate is indicative of the
progress of the precipitation. So when comparing the stability of early stages, it is useful to compare structures with
similar Al content. From the very large collection of calculated data, we present here the structures that are energetically favourable (given a certain fraction of Al), and we
discuss common trends and typical features (such as the
Si2 pillars in the needle structures). First, the main structural features of the phases will be discussed, followed by
a discussion of the thermodynamic stability of the phases.
Finally, the lattice mismatch with the Al lattice is discussed
for the phases with the most favourable formation
enthalpy.
3.1. Structural features and relations
The thermal treatment of the industrial process starts
with quenching to room temperature after hot rolling,
giving a supersaturated solid solution with substitutional
Mg and Si atoms, and a certain concentration of vacancies
( 0.01–0.10 at.%). In this work, the influence of vacancies
on the precipitation process is not considered, as those questions are better addressed by other methods (such as molecular dynamics and the kinetic Monte-Carlo method). The
substitutional atoms will start to form small clusters, such
as dimers and trimers. This is often advantageous, because
the formation energy of the strain field of such a cluster is, in
general, smaller than the sum of the energy of the strain
fields of the isolated point defects. The strain fields are
caused by the different size of the Mg and Si atoms in comparison with Al. When Mg and Si are combined, they compensate for their large and small atomic size, causing less
strain in the surrounding Al lattice. A second aspect is the
chemical bonding. Considering the binary phase diagrams
Mg–Si, Mg–Al and Si–Al, the Si–Al bonding is the weakest
bond, actually there is no stable composite AlSi bulk phase
at thermal equilibrium. Mg–Al and Mg–Si both have stable
bulk phases, of which the Mg–Si bond (Mg2Si structure) is
the strongest. The atomic size effect translates into ideal
2189
interatomic distances, shown in Table 2. So, the ideal Si–
Si interatomic distance, for example, is relatively small
(2.36 Å in the diamond structure), while the ideal Mg–Al
and Mg–Si interatomic distances are larger than the interatomic distance in the Al matrix: dAl–Al = 2.86 Å.
The structures whose formation enthalpies and relaxed
lattice parameters were calculated in this work will now
be described: first, the platelet-type structures (Fig. 1) that
have the most favourable formation enthalpy for Mg:Si
ratios larger than one, followed by the needle-type structures (Fig. 2) that have the most favourable formation
enthalpy for Mg:Si ratios smaller than one. The quantum
mechanical refinement of the Mg2Si3Al6 initial-b00 structure, which is the earliest precipitate structure that has been
fully described experimentally (Fig. 3), will also be shown.
Finally, the correspondence between the platelet-type
phases and the needle-type phases (Fig. 4), will be shown,
notably the relationship between Mg1Si1 and Mg5Si6.
3.1.1. Platelet-type phases
The simplest fcc-based structures that the solute atoms
can form are the Mg1Si1-type L10 structures, displayed in
Fig. 1. They consist of stacking (0 1 0)fcc planes consisting
of Mg, Si and Al atoms. Please note that the Mg1Si1 structure in Fig. 1.IV is orthorhombic and that the structure is
not precisely L10-type, because the atomic positions, in
particular those of the Si atoms, deviate from the perfect
fcc lattice. Structural details are given in Table 3 and displayed in Figs. 1.IV and 4(a). When the cell shape is
allowed to relax, the cell adopts a diamond shape, as shown
in Fig. 4(a). The Mg1Si1 phase has never been observed
experimentally by any technique. This is probably caused
by the large lattice mismatch of this structure, as will be
discussed in Section 3.2.2. To reduce the strain partially,
Al atoms can be added in several ways. Al layers can be
inserted in-between the Mg(0 1 0) and Si(0 1 0) layers, as
shown in Fig. 1 (platelet type I for Mg/Si < 1 and platelet
type III for Mg/Si > 1). Alternatively, (0 0 1) layers of Al
can be inserted parallel to the monoclinic axis, so that Al
slices are inserted in-between {MgSi} composite (0 0 1)
planes (platelet type II in Fig. 1). From an {1 0 0} view,
the {MgSi} composite planes consist of alternating rows
of Mg and Si atoms, which result in alternating bright
and dark spots in high-resolution transmission electron
microscopy (HREM) images [8]. The current calculations
show that, in the case of a precipitate with Mg:Si = 1, the
(1 0 0) stacking parallel to the monoclinic axis is favoured
over (0 1 0) stacking (similar to platelet types I or III in
Fig. 1, but then with one Mg ending layer and one Si ending layer at the top/bottom).
From the discussion of the formation enthalpies discussed in Section 3.2.1, it will become clear that the
needle-type structures are energetically more favourable
in the case of Mg:Si < 1. This is probably due to the fact
that, in platelet type I, there are Si//Al interfaces which
are energetically less favourable from the point of view of
chemical bonding (see the discussion above). The Matsuda
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M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
structures seem relatively favourable, however, for the case
of structures with Mg:Si > 1 (platelet type III) where there
are no Si–Al bonds at all. It will also be shown below that
the Mg2Si1Al3 phase displayed in Fig. 1.III(a) has a favourable formation enthalpy, especially when considering that
the Al content is 50 at.% so that it can be formed at early
precipitation stages. This phase consists of three-layer
Mg/Si/Mg platelets, separated by three Al layers. Structural details are given in Table 4. Other separations were
also calculated, such as separation by one Al layer
(Mg2Si1Al1) or separation by five Al layers (Mg2Si1Al5),
but these turned out slightly less favourable when the structures were constrained to Al lattice dimensions. Apparently, the three Al layers separating the Mg2Si platelets
are just sufficient to accommodate the strain induced by
the platelet. Furthermore, the {MgAl}1Al10 structure, a
{MgAl} monolayer platelet separated by five Al layers
(similar to Fig. 1.II(d)), is energetically very favourable.
3.1.2. Needle-type precipitates
The phases displayed in Fig. 1 all have a platelet-like
morphology. However, the earliest experimentally detected
precipitates in AA6xxx alloys with Mg:Si < 1 have a spherical or needle-like morphology with a monoclinic unit cell
[15]. As will become clear from the calculation of the formation enthalpies, these needle-type structures are energetically favoured for compositions with Mg:Si < 1. Several of
these phases are shown in Fig. 2. All these structures are
commensurate or nearly commensurate with the Al fcc lattice, which is easily recognized in Fig. 2(a)–(c). The
Mg2Si3Al6 structure by Chen et al., displayed in
Fig. 2(d), is the earliest structure that is experimentally well
described [15]. Details of this structure, such as experimental and relaxed coordinates, are given in Fig. 4 and in
Tables 6 and 7. The most important feature of this structure is the Si pillar (consisting of atoms #6 and #8 in
Fig. 2(d)), which extends along the b-axis. Comparing this
structure with the more evolved pre-b00 structures, such as
Mg2Si6Al3 (Fig. 2(f)) and Mg4Si7 (Fig. 2(h)), the main difference is that the latter structures have four Si pillars in the
monoclinic cell, while the initial-b00 structure has only two
such pillars.
Table 4
Calculated structure for the Mg2Si1Al3 phase, coherent with the Al matrix
(only atomic positions relaxed) and fully relaxed (both cell and ions
relaxed), calculated with VASP-GGA – the structure is shown schematically in Fig. 1.III(a)
Property
Al matrix constrained
Full relaxation
Space group
a (Å)
b (Å)
c (Å)
b
(x, y, z)Mg1
(x, y, z)Si1
(x, y, z)Al1
(x, y, z)Al2
P2/m
4.05
12.15
4.05
90.0
(0.747,
(0.231,
(0.250,
(0.249,
P2/m
4.18
12.23
4.18
90.3
(0.746,
(0.223,
(0.250,
(0.250,
0.337,
0.500,
0.000,
0.160,
0.253)
0.231)
0.750)
0.249)
0.336,
0.500,
0.000,
0.159,
0.254)
0.223)
0.750)
0.250)
Starting from the Mg2Si3Al6 structure by Chen et al.,
Mg and Si atoms were substituted by Al atoms in order
to find the earliest structure which has a stable Si pillar.
This structure, which we call the cluster structure, is displayed in Fig. 2(c). It consists of a parallelogram Mg2Si2
with an additional Si atom at the corner of the parallelogram, whereby a Si pillar is formed. When one more atom
is replaced by Al, the structures displayed in Fig. 2(a) or (b)
are obtained. While these phases are identical in composition, the structure in Fig. 2(a) (parallelogram structure) is
the energetically most favourable of the two. In the structure in Fig. 2(b), the Si pillar is disrupted by the Al for Si
substitution and, consequently, the Si–Si interatomic distance is larger than the interatomic distances in the fcc Al
lattice. It is the small Si–Si interatomic distance that makes
the Si pillars in the later structures so favourable. As dSi–Si
decreases, it becomes closer to the ideal Si–Si interatomic
distance of 2.35 Å in diamond Si, as shown in Table 2.
As the precipitation progresses, the Si–Si distance in the
pillar is decreasing, as shown in Table 5. dSi–Si in the pillars
decreases from 2.68 Å in the cluster structure to 2.50 Å for
the most evolved pre-b00 structures. The cluster structure
shown in Fig. 2(c) is the earliest structure where a Si pillar
(with dSi–Si < dAl matrix) is found by means of calculations.
The initial-b00 structure (Fig. 3(d)) consists of two such
‘‘clusters’’ in the monoclinic cell, which are positioned with
respect to each other by a translation of (1/2, 1/2, 0) using
monoclinic lattice vectors. However, the two clusters do
not have the same shape. Apparently, the strain caused
by the presence of the clusters is too large to accommodate
two Si pillars in the monoclinic cell when it is constrained
by the Al lattice.
There are two main differences between the initial-b00
structures and the pre-b00 structures. In the pre-b00 structures, there are four Si pillars per monoclinic cell instead
of two, and the low density cylinder (LDC) consisting of
a ring of Mg atoms is present (see also Table 1), whereas
it is absent in the initial-b00 structures. Considering the initial-b00 structure in Fig. 2(d), the position of the Mg atom at
position 7 is incompatible with the pre-b00 structures which
have a Si atom at that position. Therefore, we propose the
transition structure shown in Fig. 4(e), where this Mg atom
has shifted to position 2 or 4, thereby partly building the
LDC (consisting of atoms 1–5 in Fig. 2(g) and (i)) that is
so typical of the pre-b00 structures. The Mg4Si7 structure
shown in Fig. 2(g) is the energetically most favourable needle-type precipitate when the cell dimensions are fully
relaxed, as will be discussed below.
However, when the cell dimensions are confined to be
commensurate with the Al matrix, the Mg4Si8Al10 structure
(Fig. 2(j)) is the energetically most favourable structure for
Mg/Si < 1. The double Si pillars are enclosed by hexagons
consisting of Mg atoms, and the other half of the monoclinic cell consists of Al atoms. Considering the periodicity
in three dimensions, this structure consists of slices of ‘‘Mg
hexagons’’ oriented in {3 0 1}Al planes, separated by slices
of Al with approximately equal thickness. Comparing
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
2191
Table 5
Overview of Si–Si interatomic distance in the Si pillar for various structures
Structure
Composition
dSi–Si (Å)
Reference structures
Al matrix
Si diamond
Al
Si
2.86
2.35
Si pillars without Mg
Si dimer in Al
Si pillar in Al
Si2Al106
Si1Al10
2.89
2.96
Si2 in 3 · 3 · 3 fcc Al supercell
Fig. 2(b) with Al replacing Mg
Si pillars with Mg
Cluster structure
Initial-b00
Initial-b00
Pre-b00
Pre-b00
Pre-b00
Pre-b00
b00
Mg2Si3Al17
Mg2Si3Al6
Mg3Si3Al4
Mg2Si6Al3
Mg4Si6Al1
Mg4Si7
Mg5Si6
Mg5Si6
2.68
2.66
2.63
2.47,
2.51,
2.49,
2.52,
2.43,
Fig. 2(c)
Fig. 2(d) [15]
Fig. 2(e), transition
Fig. 2(f)
Fig. 2(h)
Fig. 2(g)
Fig. 3(d)
Figs. 2(i), 3(e)
2.73
2.57
2.50
2.61
2.47
Details, references
The phases are listed in the order of decreasing Al concentration. The cell dimensions of all structures were confined to Al lattice dimensions. In the case of
the Si columns (pre-b00 structures in Fig. 2), there are two typical Si–Si distances: within the Si pillar (atoms #6 and #8) and an interpillar distance (atoms
#6 and #9), which is even smaller (note that atoms #6 and #8 have different y-coordinate, while #6 and #9 have the same y-coordinate).
Fig. 2(j) and (g), the structure can be considered Mg4Si7type, but including an interface with the Al matrix. As
the Mg4Si8Al10 structure is energetically very favourable,
clearly this interface is energetically very favourable. A
few variations to the hexagon structure were calculated.
The Si atom at the origin of the unit cell can be replaced
by an Al atom, so that the composition becomes
Mg4Si7Al11. Also, the other Si atoms outside the Mg hexagon can be replaced by Al, giving compositions Mg4Si6Al12
and Mg4Si4Al14. All These phases will be referred to as
‘‘hexagon’’ structures and are listed in the formation
enthalpy tables. The ‘‘hexagon slices’’ are the main building
blocks of the late phases [2].
3.1.3. Refinement of the initial-b00 structure
The first-principles refinement of the initial-b00 structure
by Chen et al. is shown in Fig. 3 and listed in Tables 6 and
7. The experimental coordinates [15] are shown in Fig. 3a,
while Fig. 3b shows the first-principles refined parameters
when a primitive cell is used for the C-centered monoclinic
structure (Table 7). The monoclinic cell drawn in Fig. 3b
consists of two such primitive cells, and the result of using
a primitive cell is that the two clusters are forced to have
the same shape. Fig. 3c shows the refined parameters when
the conventional monoclinic cell is used, and is equal to
the structure drawn in Fig. 2d. This structure is energetically
slightly more advantageous than the primitive cell in Fig. 3b.
Here, one cluster evolves into the cluster structure including
a Si pillar, while the second cluster (on the left-hand side in
Fig. 3b) assumes approximate fcc lattice positions. When
the lattice parameters of the configurations shown in
Fig. 3b and c are relaxed, they both evolve into the same
structure, shown in Fig. 3d. Here the two clusters have the
same shape. Thus, when full relaxation is applied, the primitive cell and the monoclinic cell give the same result.
3.1.4. Relationship between platelet- and needle-type
structures
Comparing the platelet-type Matsuda structures in
Fig. 1 with the needle-type monoclinic structures in
Fig. 2, the difference between the structures appears large.
However, the two types have common features. Fig. 4
shows the correspondence between the Mg1Si1 phase and
the pre-b00 Mg5Si6 phase. In Fig. 4(b), bulk Mg1Si1 is shown
with a monoclinic unit cell and with the atoms at fcc positions. The middle section of this cell (0.25 < x < 0.75) corresponds relatively well with the pre-b00 structures shown in
Fig. 4(d). Similarly, also the left-hand and right-hand sides
of the pre-b00 cell (x < 0.25 and x > 0.75) are relatively similar to the Mg1Si1 phase. In other words, the Mg5Si6 phase
consists of two slices of Mg1Si1, separated by a planar
defect. This planar defect can be described by two structural defects: an Mg vacancy dislocation in a {3 0 1}Al plane
(indicated with red lines shown in Fig. 4) and a glide of one
Mg1Si1 slice with shift vector s = (1/2, 1/2, 0)Al with
respect to the other Mg1Si1 slice at the other side of the planar defect. The introduction of this ‘‘defect’’ leads to the
formation of the very favourable Si pillars, which are the
structural basis of the initial-b00 , pre-b00 and b00 structures.
The Si pillars are indicated by ovals in Fig. 4. Here, it
should be remarked that it is unlikely that the Matsuda
phase transforms into the Mg5Si6 phase, as the removal
of an entire plane of atoms (the {3 0 1}fcc Mg vacancy
plane) requires massive diffusion. It is more likely that
the Mg5Si6 phase grows with this ‘‘defect’’ already included
in the nucleus. The Mg5Si6 phase may be more favourable
than the Mg1Si1 phase from the point of view of alloy stoichiometry (Mg/Si < 1), but it may also be possible that the
lattice mismatch of the Mg1Si1 phase with the Al matrix
forces misfit dislocations (in order to keep the precipitate
small and commensurate). The lattice mismatch of a
2192
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
Table 6
Experimental and quantum mechanically refined positions for the initial-b00 phase Mg2Si3Al6 shown in Fig. 3
Space group
a (Å)
b (Å)
c (Å)
a ()
b ()
c ()
x
Mg1
Mg2
Mg3
Mg4
Si1
Si2
Si3
Si4
Si5
Si6
Al1
Al2
Al3
Al4
Al5
Al6
Al7
Al8
Al9
Al10
Al11
Al12
0.9776
0.4867
0.1958
0.7145
0.0000
0.5000
0.1534
0.6534
0.2587
0.7587
0.8197
0.3197
0.0985
0.5924
0.9076
0.4136
0.7533
0.2533
0.5470
0.0379
0.8788
0.3758
Experimental coordinates [15]
Commensurate with Al matrix,
relaxation of atomic positions
Full relaxation of cell dimensions
and atomic positions
Pm
14.60
4.05
6.40
90.0
105.3
90.0
y
Pm
14.60
4.05
6.40
90.0
105.3
90.0
Cm
14.97
4.14
6.21
90.0
94.2
90.0
0.0000
0.5000
0.5000
0.0000
0.5000
0.0000
0.0000
0.5000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.0000
0.5000
z
x
y
z
x
y
z
0.3424
0.3555
0.9667
0.9394
0.0000
0.0000
0.2268
0.2268
0.3671
0.3671
0.1152
0.1152
0.5091
0.4788
0.5879
0.5515
0.6879
0.6606
0.7515
0.7394
0.8606
0.8121
0.9537
0.4724
0.1783
0.6628
0.0000
0.4693
0.1270
0.6657
0.2845
0.7858
0.8162
0.3176
0.0907
0.6013
0.9117
0.4051
0.7262
0.2160
0.5352
0.0415
0.8572
0.3480
0.0000
0.5000
0.5000
0.0000
0.5000
0.0000
0.0000
0.5000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.0000
0.5000
0.2663
0.2037
0.9040
0.9558
0.0000
0.8882
0.1607
0.2695
0.3336
0.3888
0.1085
0.0437
0.4405
0.4625
0.5622
0.4745
0.6810
0.5899
0.6935
0.7223
0.8178
0.7544
0.0222
0.5222
0.1905
0.6905
0.0000
0.5000
0.2056
0.7056
0.3519
0.8519
0.8513
0.3513
0.1442
0.6442
0.9551
0.4551
0.7585
0.2585
0.5720
0.0720
0.8767
0.3767
0.0000
0.5000
0.5000
0.0000
0.5000
0.0000
0.0000
0.5000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.5000
0.0000
0.0000
0.5000
0.3000
0.3000
0.0608
0.0608
0.0000
0.0000
0.3625
0.3625
0.4770
0.4770
0.1839
0.1839
0.5540
0.5540
0.5974
0.5974
0.7570
0.7570
0.8053
0.8053
0.8887
0.8887
Note that the origin of the unit cell has been redefined with respect to the coordinates given in Ref. [15] in order to make the Si pillar coincide with the Si
pillars of the pre-b00 phases (Fig. 2).
number of important precipitate phases is discussed at the
end of this section. Also the 0.5b shift is shown schematically in Fig. 4. When the central atom in the LDC (atom
1) undergoes a shift along 0.5b, the b00 Mg5Si6 structure is
formed, as shown in Fig. 4(e).
3.2. Phase stability and commensurability with the Al matrix
3.2.1. Thermodynamic stability of the early phases
The formation enthalpies of all considered MgxSiyAlz
precipitates are listed in Table 8, calculated using GGA
and LDA. The listing order corresponds approximately
to the order of the evolution of the precipitates, where Si
and Mg atoms replace more and more Al atoms during
the annealing treatment. Considering that the MgSiAl system is a ternary alloy, the composition-dependent phase
stability graph is a three-dimensional convex tangent hull.
Figs. 5 and 6 show a projection that allows comparison
of structures with different Al content, as explained in
Ref. [2]. There are three atomic concentrations, xMg, xSi
and xAl, but because the sum of the concentrations is unity,
there are only two independent variables that we choose as
xMg and xSi. The aluminium concentration xAl = 1 Table 7
Quantum mechanically refined (VASP-GGA) positions for the initial-b00
phase Mg2Si3Al6 shown in Fig. 3(c) and (d)
Space group
a (Å)
b (Å)
c (Å)
a ()
b ()
c ()
x
Mg1
Mg2
Si1
Si2
Si3
Al1
Al2
Al3
Al4
Al5
Al6
0.4669
0.1854
0.0000
0.6517
0.2867
0.8257
0.1050
0.9163
0.7339
0.5501
0.3669
Commensurate with Al matrix
Full relaxation
P1
7.58
6.40
7.58
104.7
31.0
104.7
y
P1
7.77
6.21
7.77
94.1
30.9
94.1
0.2860
0.9544
0.0000
0.2350
0.3981
0.1177
0.4741
0.5643
0.6667
0.7562
0.8323
z
x
y
z
0.4669
0.1854
0.0000
0.6517
0.2867
0.8257
0.1050
0.9163
0.7339
0.5501
0.3669
0.5222
0.1905
0.0000
0.7056
0.3519
0.8513
0.1442
0.9551
0.7585
0.5720
0.3767
0.3000
0.0608
0.0000
0.3625
0.4770
0.1839
0.5540
0.5974
0.7570
0.8053
0.8887
0.5222
0.1905
0.0000
0.7056
0.3519
0.8513
0.1442
0.9551
0.7585
0.5720
0.3767
Note that here a primitive cell is used instead of the C-centered monoclinic
cell defined in Table 6. The lattice parameters of the primitive (p) and
monoclinic (m) cell can be converted as follows: ap = 0.5 Æ (am + bm),
cp = 0.5 Æ (am bm) and bp = cm.
Table 8
Formation enthalpies calculated using Eq. (2) using GGA and LDA
Structure
Description, references
Solute ratio xMg/
(xMg + xSi)
DHSS
Relax ions
Relax all
Relax ions
Relax all
1.00
0.00
0.00
0.00
0.00
0.00
0.00
0.0
NA
NA
33.1
64.1
4.7
46.6
0.0
0.0
0.0
1.2
59.7
4.7
46.6
0.0
NA
NA
28.4
17.5
0.0
0.0
0.0
4.7
46.6
3.5
13.1
0.0
0.0
0.0
NA
NA
22.7
55.6
6.2
34.5
0.0
0.0
0.0
1.3
45.8
6.2
34.5
0.0
NA
NA
16.5
21.1
0.0
0.0
0.0
6.2
34.5
4.9
11.2
0.0
0.0
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.50
0.75
0.83
0.88
0.90
0.92
0.93
0.94
6.7
1.0
1.2
0.6
0.5
0.3
0.3
0.2
1.3
0.9
0.2
0.1
0.1
0.1
0.1
0.1
4.4
0.2
0.4
0.0
0.0
0.1
0.1
0.1
3.6
2.0
0.6
0.5
0.3
0.3
0.2
0.2
2.7
0.4
0.5
0.3
0.4
0.4
0.5
0.5
0.7
0.5
0.5
0.3
0.2
0.2
0.2
0.1
0.4
1.9
0.6
0.4
0.2
0.1
0.0
0.1
3.8
2.1
0.6
0.5
0.4
0.3
0.3
0.3
0.80
0.67
0.57
0.69
0.63
0.56
2.9
4.7
6.8
1.5
3.7
6.0
1.2
2.3
3.1
2.6
3.3
3.9
1.1
2.6
4.3
1.1
2.5
4.1
2.6
3.3
3.7
2.6
3.3
3.9
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.00
0.50
0.67
0.75
0.80
0.83
0.86
0.88
0.33
0.50
0.60
0.67
0.71
0.75
0.78
11.8
8.8
6.1
4.6
3.7
3.0
2.6
2.3
9.5
7.5
5.8
4.5
3.9
3.2
2.9
9.5
8.0
5.7
4.4
3.4
2.9
2.5
2.2
7.1
4.6
3.8
3.2
2.8
2.4
2.3
13.9
4.0
2.4
1.8
1.5
1.3
1.0
0.9
7.6
5.4
4.5
4.0
3.5
3.2
2.8
16.2
4.8
2.9
2.0
1.7
1.4
1.1
1.0
10.0
8.2
6.5
5.4
4.5
4.0
3.4
5.8
5.9
4.4
3.7
3.0
2.6
2.4
2.2
5.5
4.6
3.7
2.9
2.6
2.2
2.1
5.0
5.4
3.8
3.0
2.3
2.0
1.8
1.5
4.1
2.7
2.2
1.9
1.7
1.4
1.4
14.6
4.3
2.3
1.4
1.1
0.8
0.5
0.4
8.1
5.6
4.5
3.9
3.2
2.9
2.5
15.3
4.8
2.9
2.1
1.7
1.4
1.1
1.1
9.4
7.5
5.9
4.9
4.1
3.7
3.1
0.33
0.33
0.33
0.33
0.00
0.25
0.50
0.63
32.8
20.1
13.6
10.0
25.6
18.9
12.6
10.1
0.2
4.4
2.8
2.2
7.0
5.5
3.7
2.2
25.5
15.1
10.7
8.0
18.4
12.9
8.8
6.5
0.4
3.7
1.8
1.4
6.7
5.9
3.7
2.9
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
Relax all
Transition platelets
Mg4Si1Al11
Platelet
Mg4Si2Al10
Fig. 1.IIf
Mg4Si3Al9
Platelet
Fig. 1.IIc
D Hbulk
DHSS
Relax ions
MgAl binary phases
Mg2Al2
Mg1Al3
Fig. 1.IIe
{MgAl}Al4
{MgAl}Al6
Fig. 1.IId
{MgAl}Al8
{MgAl}Al10
{MgAl}Al12
{MgAl}Al14
Fig. 1.IIa
LDA
DHbulk
Relax all
NA
1.00
0.00
1.00
0.00
1.00
0.00
Fig. 1.II
Fig. 1.IV
Fig. 1.IIb
GGA
Relax ions
Elementary phases
Al fcc
Bulk
Mg hcp
Bulk
Si diamond
Bulk
Mg fcc
fcc imposed
Si fcc
fcc imposed
Mg sub
Substitutional
Si sub
Substitutional
Platelets
Mg:Si = 1
Mg1Si1
{MgSi}1Al2
{MgSi}1Al4
{MgSi}1Al6
{MgSi}1Al8
{MgSi}1Al10
{MgSi}1Al12
{MgSi}1Al14
{MgSi}2Al2
{MgSi}2Al4
{MgSi}2Al6
{MgSi}2Al8
{MgSi}2Al10
{MgSi}2Al12
{MgSi}2Al14
Mg:Si < 1
Mg1Si2 fcc
Mg1Si2Al1
Mg1Si2Al3
Mg1Si2Al5
Mg:Si > 1
Al concentration xAl
Fig. 1.I
Fig. 1.Ia
Fig. 1.III
2193
(continued on next page)
2194
Table 8 (continued)
Structure
Description, references
Solute ratio xMg/
(xMg + xSi)
Al concentration xAl
GGA
LDA
Relax ions
Relax all
0.00
0.25
0.50
0.63
0.17
0.38
0.50
0.00
15.4
8.0
5.2
3.8
9.2
7.0
5.4
15.8
7.8
4.1
3.4
2.8
6.0
5.1
4.2
8.1
3.3
6.0
4.1
3.2
8.7
6.4
5.3
2.9
10.9
9.9
5.9
4.2
11.9
8.4
6.5
10.6
7.6
2.9
2.1
1.7
3.7
3.1
2.5
8.2
4.7
2.0
2.0
1.6
3.0
2.8
2.3
4.9
8.0
8.8
5.7
4.2
10.9
7.9
6.3
7.4
10.9
9.8
5.8
4.3
11.6
8.2
6.4
10.7
DHbulk
Relax ions
D Hbulk
DHSS
Relax all
Relax ions
Relax all
Relax ions
DHSS
Relax all
Thomas [3]
0.67
0.67
0.67
0.67
0.60
0.60
0.60
0.67
Si1Al10
Mg1Al10
Mg1Si1Al20
Mg1Si2Al19
Mg1Si2Al19
Mg2Si1Al19
Si2Al9
Mg2Si2Al18
Mg2Si3Al17
Mg2Si4Al16
Si2 column
Mg2 column
MgSi column
Mg1Si2 column
Mg1Si2 column
Mg2Si1 column
2 Si column
Parallelogram Fig. 2a
Cluster, Fig. 2c
Big cluster
0.00
1.00
0.50
0.33
0.33
0.67
0.00
0.50
0.40
0.33
0.91
0.91
0.91
0.86
0.86
0.86
0.82
0.82
0.77
0.73
4.5
0.6
2.1
4.0
4.0
2.4
9.0
3.2
5.3
7.5
4.4
0.4
2.1
3.7
3.9
2.2
8.6
2.5
4.7
6.9
0.3
0.2
0.3
0.4
0.5
0.2
0.5
1.5
1.5
1.4
0.1
0.0
0.3
0.7
0.5
0.3
0.1
2.2
2.1
2.0
4.7
0.7
2.3
3.9
3.9
2.3
8.5
2.7
4.4
6.2
3.1
0.5
1.5
2.9
2.8
1.9
6.2
1.9
3.5
5.2
1.6
0.2
0.4
0.4
0.5
0.2
2.2
1.0
0.8
0.6
0.0
0.0
0.4
0.5
0.6
0.2
0.1
1.8
1.7
1.6
Initial-b00
Mg2Si2Al7
Mg2Si3Al6
Mg2Si2Al7
Mg4Si5Al13
Fig. 2d
[15]
[15]
0.50
0.40
0.50
0.44
0.64
0.55
0.64
0.59
8.2
11.2
7.8
10.2
7.8
6.8
4.9
7.4
1.1
2.4
1.6
1.3
1.5
6.8
4.4
4.0
6.2
8.5
5.8
7.8
5.8
5.2
3.9
5.8
1.2
2.0
1.6
1.2
1.6
5.3
3.5
3.2
Hexagons
Mg4Si6Al12
Mg4Si7Al11
Mg4Si8Al10
Fig. 2j
Hexagon
Hexagon
Hexagon
0.40
0.36
0.33
0.55
0.50
0.45
10.6
10.8
11.0
8.0
8.5
9.0
3.0
4.9
6.8
5.5
7.1
8.8
8.0
7.7
7.6
8.6
6.6
7.0
18.5
4.4
6.1
19.1
5.5
6.7
Transition
Mg2Si4Al5
Mg3Si3Al5
Mg3Si4Al4
Fig. 2e
Transition
Transition
Transition
0.33
0.50
0.43
0.45
0.45
0.36
15.3
10.7
13.1
13.4
7.2
9.2
2.5
3.3
5.1
4.4
6.8
9.0
11.8
7.3
8.9
10.3
5.1
7.7
1.9
3.8
5.3
3.4
6.0
6.5
Pre-b00
Mg0Si6Al5
Mg1Si6Al4
Mg2Si6Al3
Mg3Si6Al2
Mg4Si6Al1
Mg5Si6
Mg4Si7
Fig. 2f–h
[3]
[3]
[2]
[3]
[3]
[3]
[2]
0.00
0.14
0.25
0.33
0.40
0.45
0.36
0.45
0.36
0.27
0.18
0.09
0.00
0.00
28.8
25.8
23.6
21.7
18.9
17.6
21.6
24.2
21.2
17.9
12.3
10.8
13.4
11.1
3.4
0.0
2.6
5.0
8.2
10.0
9.7
1.2
4.6
8.4
14.4
16.3
14.2
20.3
25.0
21.4
18.7
16.1
12.8
10.6
14.9
19.8
16.9
13.9
8.3
7.4
8.7
6.4
6.1
2.0
1.3
4.5
8.3
11.0
9.3
0.9
2.5
6.0
12.3
13.7
12.9
17.8
b00 (non-fcc)
Mg5Si6
Fig. 2i
[5]
0.45
0.00
8.5
4.0
19.0
23.5
1.7
0.3
20.0
21.9
Fig. 1.IIIa
Fig. 1.IIIb
See Ref. [2] for a definition of DHbulk. The structures were calculated in two modes: (a) keeping the cell dimensions fixed to the Al matrix values (i.e., the precipitate structure is commensurate with the Al
matrix) while allowing the atoms to relax (‘‘relax ions’’ mode) and (b) allowing both the cell dimensions and the atomic positions to relax (‘‘relax all’’). All enthalpies are in kJ mole1 of solute.
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
Mg2Si1
Mg2Si1Al1
Mg2Si1Al3
Mg2Si1Al5
Mg3Si2Al1
Mg3Si2Al3
Mg3Si2Al5
Mg2Si1Th
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
2195
(xMg + xSi). It is customary to look at the enthalpy results
in terms of the relative solute ratio xMg/(xMg + xSi).
There are many experimentally observed metastable
phases that would never appear at thermal equilibrium.
The entropy of the supersaturated solid solution can be
estimated at 12 kJ mole1 at a temperature of 500 K [2],
and therefore most structures exist as transitory states only.
Here, it should be mentioned that the structures themselves
also have configurational entropy, but it is beyond the
scope of this work (entropy calculations are extremely time
consuming) to calculate the configurational entropy of all
the structures.
As this work is in the early stages, only fcc-type structures are considered. The b00 structures are not selected,
because the atom at the center of the LDC (position #1)
is displaced by a 2 Å (shift of Dy = 0.5b = 2.03 Å) with
respect to the corresponding Al matrix position, see also
Fig. 6. Formation enthalpy DHSS (kJ mole1 of solute) for early fcc-type
structures with an Al content of at least 50 at.%, calculated using VASPGGA. The grey scale indicates the Al content (white is 100 at.% Al, black
is 0 at.% Al). (a) Relaxing atomic positions with the cell dimensions
constrained to be commensurate with the Al matrix; (b) relaxing both the
cell dimensions and the atomic positions.
Fig. 5. Formation enthalpy DHSS (kJ mole1 of solute) for all considered
fcc-type structures calculated using VASP-GGA. The grey scale indicates
the Al content (white is 100 at.% Al, black is 0 at.% Al). (a) Relaxing
atomic positions with the cell dimensions constrained to be commensurate
with the Al matrix; (b) relaxing both the cell dimensions and the atomic
positions.
Fig. 4(d) and (e). The structural relations and phase stability of the non-fcc structures of the late phases were discussed in a previous work [2]. Fig. 5(a) shows the
formation enthalpy of all fcc-type structures calculated
using VASP-GGA, for the case where the cell dimensions
are confined to Al matrix values, while Fig. 5(b) displays
the formation enthalpy for the same fcc-type structures
for the case where the cell dimensions are fully relaxed.
In both cases, the atomic positions were fully relaxed.
VASP-LDA produced qualitatively similar results, which
are not displayed here. It is clear that the phases Mg1Si1
(Fig. 4(a), structural details in Table 3) and pre-b00 phases
Mg4Si7 (Fig. 2(g), structural details in Ref. [2]) and hexagon Mg4Si8Al10 (Fig. 4(j)) dominate the ground state hull.
The pre-b00 structures Mg4Si6Al1 and Mg3Si6Al2 (structural
details in [3]) fall near the ground state hull. For alloys with
Mg:Si > 1, the Mg2Si1Al3 phase (displayed in Fig. 2.III(a),
2196
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
structural details in Table 4) is a rather favourable phase.
These observations are combined in Fig. 7(a), where a ternary Gibbs’ triangle is drawn. In comparison with the work
performed by Ravi and Wolverton [3], the phases Mg4Si7,
Mg4Si8Al10, Mg2Si1Al3pl and Mg3Si2Al3pl appear as new
phases in the Gibbs’ triangle (‘‘pl’’ indicates platelet). It
can be concluded that, for compositions with Mg/Si > 1,
the platelet-type structures are the most favourable fcc-type
phases, while for compositions with Mg/Si < 1, the monoclinic needle-type structures are the most favourable fcctype phases.
The formation of precipitate structures is most likely
limited by the diffusivity of the solute Mg and Si atoms
in the solid solution. The earliest phases will therefore contain a large fraction of Al [1,15]. In order to obtain phase
stability information on these earliest stages, an additional
constraint was applied. When only those structures are
selected that have an Al concentration of at least 50 at.%,
the intermediate common tangent hulls shown in Fig. 6
Fig. 7. Gibbs’ triangle for the ternary alloy MgSiAl, showing which fcctype structures are stable or energetically favourable. Near the tangent
hull has been defined as dDH < 0.015 kJ mole1 of solute (dDH is the
vertical distance in energy to the tangent line in Figs. 5 and 6). (a) Most
stable structures when all fcc-type structures are selected and (b) earliest
metastable structures with an Al content of at least 50 at.%.
are obtained. When the cell dimensions are fully commensurate with the Al matrix, the needle-type Mg4Si7Al11
structure (variation to Fig. 2(j)) is on the tangent hull,
and so are the Matsuda platelet {MgSi}2Al10 (similar to
Fig. 1.II(c)) and the platelets Mg2Si1Al3 (Fig. 1.III(a),
Table 4) Mg3Si2Al5 (similar to Fig. 1.III(b)). When the cell
dimensions are fully relaxed, the initial-b00 phase Mg2Si3Al6
and the Matsuda {MgSi}2Al4 phase (Fig. 1.II(c)) appear on
the tangent hull. It should be mentioned here that configurational entropy has been ignored. In reality, the initial-b00
phase Mg2Si3Al6 is probably more favourable, as this precipitate model allows for multiple arrangements, implying
a favourable entropy. The Si pillars form the fixed skeleton
of this structure, while the composition of the atoms inbetween the pillars is variable [15]. In Fig. 3(a), arrows
indicate how Mg and Al atoms can be swapped. The energetically most favourable structures with high Al content
(Fig. 5) were plotted separately in a Gibbs triangle, displayed in Fig. 7(b). The procedure of applying selection criteria described above finally gives a metastable tangent hull
in which several experimentally determined early precipitate phases appear: the initial-b00 phase Mg2Si3Al6 and
the Matsuda platelets of Fig. 1.II.
It is remarkable that the Mg4Si7 phase has not been
observed experimentally (yet), despite the very low formation enthalpy after full relaxation, as shown in Fig. 5(b).
One possible reason might be an early phase transition
from pre-b00 to b00 when the precipitate still contains Al
[2], so that the pre-b00 Mg4Si7 phase is never reached.
Another aspect is the large lattice mismatch of this phase
with the Al matrix.
3.2.2. Misfit of precipitate structures with the Al lattice
Structures with a large lattice mismatch have energetically less favourable interfaces, and therefore have a lower
probability of being formed. This aspect is particularly
important for early structures, because the interface/volume fraction is high, and therefore the contribution of
the interface formation enthalpy to the overall formation
enthalpy (Htotal = DHSS + Hinterface) is high.
Furthermore, the calculated misfit can provide insight
into the possibility of coherency strengthening, whereby
dislocations in the matrix interact with the strain fields that
are caused by the presence of coherent misfitting precipitates [22]. For example, the initial-b00 Mg2Si3Al6 precipitate
described by Chen et al. is experimentally observed to be
fully coherent [15], while the fully relaxed structure (see
Fig. 3d) is not commensurate with the Al matrix. Therefore, the precipitate as embedded in the Al matrix is only
coherent because strain fields force it to be coherent. This
precipitate structure is thus likely to induce coherency
strengthening. However, coherency strengthening is only
one of the mechanisms causing precipitation hardening.
Also mechanisms such as chemical strengthening, stacking-fault strengthening, modulus hardening and order
strengthening contribute to the precipitation hardening
[22,23]. As the contribution of other strengthening mecha-
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
nisms cannot be extracted from the current calculations, we
cannot predict which precipitates are the best hardening
precipitates. Nonetheless, as already mentioned, the lattice
misfit is also important from the point of view of interface
energy and therefore the probability that the precipitate
will be formed.
Table 9 lists the calculated lattice parameters after full
relaxation and the corresponding lattice mismatch with
the Al lattice for a number of important structures that
are also present in the Gibbs triangles in Fig. 7. The lattice
mismatch is defined as g = (aprec aAl)/aAl, where aprec is
the lattice parameter of the precipitate, and aAl is the corresponding distance in the Al matrix when the precipitate
is fully commensurate. Both aprec and aAl are tabulated in
Table 9. The volumetric expansion ratio (free expansion
volume divided by the volume of the corresponding Al lattice) is listed as well.
Although the b00 Mg5Si6 structure is not an fcc-type
structure, it has been included in the table for comparison,
because it has an experimentally well-defined structure and
is a well-known effective hardening precipitate [1,5,9,11].
The structure has a considerable lattice mismatch along
the a- and c-axes (3.6% and 5.3%, respectively), but a very
small lattice mismatch along the b-axis (calculated lattice
parameter 4.07 Å vs 4.05 Å for pure Al). Therefore, growth
along the b-axis (the direction corresponding to the long
dimension of the needle) is energetically favoured. The volumetric expansion ratio is 1.098, which is the largest expansion ratio in the list.
Two aspects are important in order to predict which
early precipitate phases have a high probability of being
formed: (i) they should have a favourable formation
enthalpy, i.e., on or near the tangent hull in Fig. 7(a) or
(b), and (ii) the lattice mismatch with the surrounding Al
matrix should be small. In the scientific community, a lattice mismatch of 5–6% or more is considered large (in those
cases, misfit dislocations will certainly be necessary to
relieve coherency stresses), and a lattice mismatch of 0–
2% is considered small. As this work deals with early precipitates where the influence of lattice mismatch (affecting
the interface energy) is important, it is considered that a
lattice mismatch must be <2% in at least one direction
for a reasonable likelihood of formation of the precipitate.
The interface energy does not depend only on the lattice
mismatch, and the interface energies cannot be determined
quantitatively. It will now be discussed qualitatively which
structures, among the phases that have a favourable formation enthalpy, are also favourable from the point of view of
lattice mismatch.
First, the phases with Mg:Si < 1 are discussed. Considering the lattice parameters of the fully relaxed pre-b00 Mg4Si7
phase in Table 9, it is clear that the lattice mismatch with
the Al lattice is 10% in all three directions in which the
change in the monoclinic angle is 15. So despite the
favourable bulk formation enthalpy in of this structure
(see Fig. 5(b)), it is a very unfavourable phase from the
point of view of lattice mismatch, which explains why this
2197
phase has never been observed experimentally. Also the
pre-b00 Mg3Si6Al2 phase has a very large lattice mismatch.
The parallelogram Mg2Si2Al18 and cluster Mg2Si3Al17
structure (Fig. 2(a) and (c)) are examples of the earliest
structures and have a very small lattice mismatch but, considering Fig. 6, they do not have a favourable bulk formation enthalpy. However, the initial-b00 Mg2Si3Al6 phase
found experimentally by Chen et al. (Fig. 2(d), Ref. [15])
combines a favourable formation enthalpy with a small lattice mismatch of 1–3% in two directions. So considering the
precipitates which contain more Si than Mg atoms, the
Mg2Si3Al6 phase is a phase that is likely to be formed.
Considering the phases with Mg:Si = 1, the Matsuda
phase Mg1Si1 is not likely to be formed, because the calculated lattice mismatch is relatively large at 3–6%. This
might be the reason why this phase is never observed. However, the {MgSi}2Al10 phase (similar to Fig. 1.II(c)) is both
energetically favourable and has a small lattice mismatch
of 1% along the a-axis. As an example, the {MgSi}1Al2
platelet depicted in Fig. 1.II(b) is also included in Table
9. Although this very early phase is energetically not very
favourable, it possibly describes the platelets observed in
Fig. 6 in Ref. [8]. These platelets consist of composite
{MgSi} or {MgAl} layers separated by one Al layer. The
lattice mismatch in the lateral directions is small, 1.1%,
while the lattice mismatch perpendicular to the layers is
6.0%. The formation of a platelet is then advantageous as
the lattice mismatch is minimal in two orthogonal
directions.
Considering the phases with Mg:Si > 1, the calculations
predict that the Mg2Si1Al3 (Fig. 1.I(a)) and Mg3Si2Al5
platelets are favourable precipitate structures, with a formation enthalpy near the tangent hull (Fig. 5) and a lattice
mismatch along the b-axis of 0.8% and 0.0%, respectively
(Table 9). For precipitates containing mainly Mg solute
atoms, the {MgAl}1Al10 platelet (similar to Fig. 1.II(d))
is favourable with a very small lattice mismatch (<1%) in
all directions.
Summarising the results on the earliest phases, the calculations predict that the initial-b00 phase Mg2Si3Al6, and the
platelets {MgSi}2Al10, {MgAl}1Al10, Mg2Si1Al3 and
Mg3Si2Al5 are energetically favourable precipitates. It also
becomes clear that total energy calculations that consider
the precipitate as bulk, have limitations. A few phases that
appear to be energetically favourable in the formation
enthalpy diagrams have too large a lattice mismatch with
the Al matrix, so that it is unlikely that they will be formed.
The smaller the precipitate, the larger the contribution of
the interface enthalpy to the total formation enthalpy,
and therefore this systematic error in the calculations is larger for early precipitates. Furthermore, entropy effects
become more important at the earliest stages because the
number of defects and the variation in stoichiometry
(between precipitates and within the precipitate) is large
at these stages. Although entropy and dynamic effects are
not included in the total energy calculations, by combining
the information on formation enthalpy and lattice mismatch,
2198
Table 9
Lattice parameters and lattice mismatch with the Al matrix for various energetically favourable structures, calculated using VASP-GGA
Relaxation
Lattice dimensions
Lattice mismatch (%)
Volumetric
expansion V/VAl (–)
a (Å)
b (Å)
c^ (Å)
Mg1Al3
Constrained
Relaxed
4.05
4.14
4.05
4.14
4.05
4.14
90.0
90.0
2.4
2.4
2.4
0.0
1.074
{MgAl}1Al10pl
Constrained
Relaxed
4.05
4.08
4.05
4.08
12.14
12.24
90.0
90.0
0.8
0.8
0.9
0.0
1.026
Constrained
Relaxed
4.05
4.29
4.05
3.91
4.05
4.28
90.0
93.6
6.1
3.3
5.8
3.6
1.085
Constrained
Relaxed
4.05
4.00
4.05
4.00
4.05
4.29
90.0
90.0
1.1
1.1
6.0
0.0
1.037
{MgSi}2Al10pl
Constrained
Relaxed
4.05
4.09
4.05
3.92
28.32
29.80
90.0
90.0
1.0
3.0
5.2
0.0
1.031
{MgSi}2Al4pl
Constrained
Relaxed
4.05
4.07
4.05
3.86
8.09
8.90
90.0
90.0
0.6
4.5
10.0
0.0
1.057
Constrained
Relaxed
4.05
4.18
12.14
12.23
4.05
4.18
90.0
90.3
3.3
0.8
3.3
0.3
1.076
Constrained
Relaxed
4.05
4.16
20.23
20.23
4.05
4.16
90.0
90.0
2.8
0.0
2.8
0.0
1.058
Mg2Si2Al18 parallellogram
Constrained
Relaxed
14.60
14.35
4.05
4.02
6.18
6.15
105.3
101.8
1.8
0.7
0.5
3.5
0.971
Mg2Si3Al6 initial-b00
Constrained
Relaxed
14.60
15.10
4.05
4.12
6.18
6.21
105.3
94.1
3.4
1.8
0.5
11.2
1.058
Constrained
Relaxed
14.60
15.84
4.05
3.88
6.18
6.33
105.3
103.8
8.5
4.2
2.5
1.4
1.065
Constrained
Relaxed
14.60
15.77
4.05
3.93
6.18
6.33
105.3
106.4
8.0
2.8
2.5
1.1
1.075
Constrained
Relaxed
14.60
14.90
4.05
3.76
6.18
6.96
105.3
119.9
2.1
7.1
12.7
14.6
1.068
Constrained
Relaxed
14.60
16.00
4.05
3.62
6.18
6.84
105.3
90.4
9.6
10.5
10.7
14.9
1.086
Constrained
Relaxed
14.60
15.87
4.05
3.97
6.18
6.36
105.3
102.5
8.7
1.9
3.0
2.8
1.098
Constrained
Relaxed
14.60
15.13
4.05
4.07
6.18
6.51
105.3
110.5
3.6
0.6
5.3
5.2
1.098
Mg1Si1 (Table 3)
{MgSi}1Al2pl
Mg3Si2Al5pl
Mg4Si7Al11 hexagon
Mg4Si8Al10 hexagon
Mg3Si6Al2 pre-b00
Mg4Si7 pre-b
00
Mg5Si6 pre-b
00
00
Mg5Si6 b
ga
gb
gc?
The a- and b-axes correspond to the corresponding lattice parameters, while the c^ dimension has been chosen perpendicular to a and b (in order to obtain the lattice mismatch in three orthogonal
directions). For the Matsuda (mat) structures shown in Fig. 1.II, the c-axis is chosen perpendicular to the {MgSi} plane. The b00 Mg5Si6 structure is the only non-fcc-based structure and has been
included for comparison because it is a well-known and well-described precipitate structure. Precipitates that combine a favourable formation enthalpy (Figs. 5 and 6) with a small lattice mismatch
(g < 2%) in at least one direction are in bold. Both the fully relaxed lattice parameters are the lattice parameters when commensurate with the Al lattice (constrained) are given.
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
Mg2Si1Al3pl (Table 4)
b ()
Angle difference
Db ()
M.A. van Huis et al. / Acta Materialia 55 (2007) 2183–2199
the energetically most favourable structures are easily identified and very unfavourable structures are ruled out.
Hopefully, in the future, enhanced computational power
will allow dynamic atomistic simulation of the formation
and
properties
of
precipitates
as
embedded
in aluminium using, for example, molecular dynamics
employing the modified embedded atom method (MEAM).
This would allow a more detailed investigation of various
aspects, such as nucleation energies, interfacial energies,
the strain fields within and surrounding the precipitate,
and investigation of the changes in formation enthalpy
as a function of deformation (this work considers only
two extreme situations: fully commensurate and free
relaxation).
4. Conclusions
The formation enthalpies of commensurate and fully
relaxed precipitate structures, and the lattice parameters
after full relaxation were calculated for a large number of
early fcc-based structures. The calculations show that, for
phases with a Mg:Si ratio larger than one, platelet-type
structures are energetically favoured, while for phases with
a Mg:Si ratio smaller than one, the needle-type structures
with Si columns are favoured. More specifically, the calculations predict that the monoclinic Mg1Si1 and pre-b00
Mg4Si7 structures are the most favourable phases among
the fcc-based structures. However, the calculated lattice
mismatch of these structures with the Al matrix is high
(6–10%), which makes it unlikely that these phases will
be observed experimentally. Either they remain very small
(not visible) or do not exist. Energetically favourable early
structures (with a high fraction of Al atoms) that also have
a favourable calculated lattice mismatch are the initial-b00
Mg2Si3Al6 phase with single Si pillars, and the platelets
{MgSi}2Al10, {MgAl}1Al10, Mg2Si1Al3 and Mg3Si2Al5.
During the precipitation sequence of the needle-type precipitates, the Si–Si interatomic distance in the Si pillars
decreases as more Al atoms in the precipitate are replaced
by solute Mg and Si atoms.
2199
Acknowledgement
This research was carried out under project number
02EMM023 in the framework of the Strategic Research
programme of the Netherlands Institute for Metals Research in the Netherlands (www.nimr.nl) in collaboration
with the Foundation for Fundamental Research on Matter
(FOM).
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