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Acta Materialia 69 (2014) 126–134
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Detailed atomistic insight into the b00 phase in Al–Mg–Si alloys
Per Harald Ninive a, Are Strandlie a, Sverre Gulbrandsen-Dahl b, Williams Lefebvre c,
Calin D. Marioara d, Sigmund J. Andersen d, Jesper Friis d, Randi Holmestad e,
Ole Martin Løvvik f,g,⇑
a
Gjøvik University College, PO Box 191, NO-2802 Gjøvik, Norway
SINTEF Raufoss Manufacturing, PO Box 163, NO-2831 Raufoss, Norway
c
Groupe de Physique des Matériaux, UMR CNRS 6634, Université de Rouen, BP 12, Avenue de l’Université, 76801 Saint Etienne de Rouvray, France
d
SINTEF Materials and Chemistry, PO Box 4760 Sluppen, NO-7465 Trondheim, Norway
e
NTNU, Department of Physics, NO-7491 Trondheim, Norway
f
SINTEF Materials and Chemistry, PO Box 124 Blindern, NO-0134 Oslo, Norway
g
University of Oslo, Department of Physics, Gaustadalleen 21, NO-0349 Oslo, Norway
b
Received 18 February 2013; received in revised form 22 January 2014; accepted 25 January 2014
Abstract
The b00 phase is the major hardening precipitate in Al–Mg–Si alloys. The present study aims to improve understanding of the industrially important hardening process by systematically combining advanced microscopy, ab initio atomistic calculations and strength measurements of Al–Mg–Si alloys containing b00 precipitates. The microscopy identified Mg4Al3Si4 as the most likely precipitate
composition, with possibilities for compositional variation within a single precipitate. The atomistic calculations enabled quantification
of precipitate formation energies and strain fields inside and around the precipitates. Associated measurements of precipitate size, hardness and yield strength in samples only containing b00 precipitates gave new, empirical relations between these parameters. This demonstrated that the particle sizes were effectively larger than directly observed, owing to coherency strain fields. The study demonstrates that
atomistic insight now can be directly linked to the bulk elastic properties of advanced structural materials.
Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Transmission electron microscopy; Density functional theory; Aluminium alloys
1. Introduction
Lightweight materials play an increasingly important
role in the quest to improve quality of life and reduce
energy consumption. Alloys of the Al–Mg–Si system are
heat-treatable and extrudable, exhibiting properties such
as high strength-to-weight ratio, good formability and weldability, superb corrosion resistance, high electrical and
thermal conductivity and attractive surface appearance.
Being relatively low-priced and with good recycling proper⇑ Corresponding author at: University of Oslo, Department of Physics,
Gaustadalleen 21, NO-0349 Oslo, Norway.
E-mail address: o.m.lovvik@fys.uio.no (O.M. Løvvik).
ties, they will continue to be very important materials for a
broad variety of products.
These alloys are extruded between 400 and 500 °C.
Rapid cooling ensures a supersaturated solid solution
(SSSS), after which the solutes start clustering in the facecentred cubic Al lattice. Optimal strength is attained after
a few hours’ final ageing treatment between 150 and
200 °C. The strength is caused by a high number density
of nano-sized, needle-shaped b00 precipitates, usually
together with Guinier–Preston (GP) zones [1,2]. With
increasing temperature and aging time, the number density
decreases, while the precipitates coarsen and are replaced
by more stable types. The precipitation sequence in Al–
Mg–Si alloys is [3,4].
http://dx.doi.org/10.1016/j.actamat.2014.01.052
1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
P.H. Ninive et al. / Acta Materialia 69 (2014) 126–134
SSSS ! atomic clusters ! GP-zones ! b00
! b0 ; U1 ðType AÞ; U2 ðType BÞ; B0 ðType CÞ ! b
All metastable precipitates are elongated and coherent along
h001iAl. Normal to this, coherency stresses arise around the
precipitates in {001} Al planes. The surrounding elastic strain
fields slow down propagation of dislocations, which means that
higher applied stress is required to obtain deformation.
The typical cross section of a b00 precipitate is between 1
and 15 nm2, while the length is between 30 and 100 nm [3].
It has a monoclinic crystal structure closely related to the
Al matrix; this was previously determined by high resolution transmission electron microscopy and quantitative
electron diffraction refinements [5]. The orientation relationship with Al is (see Fig. 1a)
½100b00 P½230Al ; ½010b00 P½001Al ; ½001b00 P½310Al :
ð1Þ
127
The measured monoclinic angle of b00 is close to 105.3°,
the angle between ½2 3 0Al and ½3 1 0Al . For brevity, the
interfaces defined by ½1 0 0b00 P ½2 3 0Al and ½0 0 1b00 P ½3 1 0Al
will in the following be designated by the general Al planes
{3 2 0} and {1 3 0}.
For several years, the accepted chemical composition of
b00 was Mg5Si6, suggested by energy dispersive X-ray spectroscopy [5] and density functional theory (DFT) calculations [6]. However, a recent atom probe tomography
study of the b00 structure [7] showed an Al content of
20–30% in all investigated precipitates. The observations
were supported by DFT calculations, which showed that
Mg5Al2Si4 and Mg4Al3Si4 were the most energetically
favourable bulk compositions [7]. In the former composition, the Si3 sites in Mg5Si6 were substituted by Al, while
the Mg1 sites were also replaced by Al in the latter (see
Fig. 1b and c).
Fig. 1. Atomistic view of b00 precipitates. (a and b) HAADF-STEM image of a large b00 precipitate. Interfacial relations between the precipitate and the Al
matrix are shown. The area delimited by the white square is shown enlarged in (b), where noise was reduced by filtering all distances shorter than 0.17 nm.
One b00 unit cell corresponding to the original Mg10Si12 structure [5] is shown by double white lines, together with an atomic overlay. The connection to the
Al matrix along both {3 2 0} and {1 3 0} interfaces is also shown. One asymmetric unit cell, used for the calculations in this paper, is drawn with white lines.
(c) Its atomic overlay is expanded, and the different atomic sites as defined in Refs. [5,7] are indicated. (d) Atomistic model of a b00 precipitate embedded in
an Al matrix. The Mg and Si sites are shown as blue and red balls, respectively, while the Al matrix atoms are shown as grey balls. The Mg10Si12 unit cell is
indicated, along with the corresponding orientations in the Al matrix. The white square indicates the supercell used in the DFT calculations. (For
interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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P.H. Ninive et al. / Acta Materialia 69 (2014) 126–134
2. Microscopy investigations
Cs-probe corrected high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM)
enables extraction of structural and chemical information
on atomic columns from a thin specimen [8–10]. A very fine
probe (0.1 nm in diameter) was scanned across a specimen oriented along a zone axis, and the intensity of atomic
columns could be related to their mean atomic number.
This justifies the usual expression of “Z-contrast” associated with this technique.
A HAADF-STEM image of a b00 precipitate is presented
in Fig. 1a and b. The structure is very well defined along
the [001]Al zone axis. An atomic overlay based on the original model [5] is shown in (b) and the atomic sites are
defined in (c). Based on a well-established quantitative
approach [11], the intensity distribution of the six different
atomic columns was estimated.
Fig. 2 demonstrates that a clear distinction between the
different elements is achievable, despite their small difference in atomic numbers. It is also clear that Mg2 and
Mg3 are equivalent, as are Si1 and Si2. The most striking
result is that the Mg1 columns are more intense than those
of Mg2 and Mg3, whereas Si3 columns are less intense than
those of Si1 and Si2. This supports the prediction of Ref.
[7], that both Mg1 and Si3 contain significant fractions of
aluminium. A comparison of Fig. 2a, b and c even suggests
that the Mg1 and Si3 sites probably consist of pure Al. This
is confirmed by HAADF-STEM image simulations provided in the supplementary material. Nevertheless, the
present authors investigated whether the relatively broad
distribution of intensity in Fig. 2a and b was only attributable to statistical fluctuations or whether a correlation
existed between atomic column composition and location
in the precipitate. For Si3 columns, this could not be found.
However, a relationship between the composition and relative position to the interface could be seen for Mg1 columns (Fig. 2d). The intensity of Mg1 columns along the
{1 3 0} interfaces was significantly higher than that of other
Mg1 columns, suggesting that the Mg1 sites contain more
aluminium in this region of the precipitate. In contrast,
the intensity of Mg1 columns along the {3 2 0} interfaces
suggests that they are richer in magnesium.
Could the systematic intensity deviations in Fig. 2d be
due to experimental artefacts or assumptions? One supposition was that the depth of atomic columns is relatively
constant across the image; this is clearly reasonable, given
the very small size of the image. A second assumption was
that cross-talk artefacts [12,13] can be neglected. This could
stem from intensity of the electron probe channelling in an
atomic column appearing in neighbouring columns as the
thickness increases [12]. However, a simulation study demonstrated that the cross-talk artefact occurs only for very
thick specimens along the h1 0 0i zone axis of aluminium
[14]. Since the inter-column distances in b00 are of the same
order as in Al along [0 0 1]Al and since the density of atomic
columns along [0 0 1]Al and [0 1 0]b00 are identical [7], no
cross-talk should occur in the precipitate. Finally, the
HAADF intensity could be reduced as a result of lattice
strain [15,16]. The distortion field around the precipitate
was not extracted from the HAADF-STEM observations
here [17], but such work is in progress by some of the
Fig. 2. Intensity distributions of the symmetry related atomic columns of the b00 precipitate visible in the HAADF-STEM image of Fig. 1a and identified in
Fig. 1c. (a–c) Intensity for (a) Mg sites and (b) Si sites in the precipitate and for (c) Al columns in the matrix. (d) The lower part of the b00 precipitate of
Fig. 1a. The image has been smoothed, Wiener filtered and deconvoluted of the electron probe intensity distribution following a conventional protocol for
column intensity extraction in HAADF-STEM [14]. Blue circles and red squares indicate locations of Mg1 sites of particularly high (>7 a.u) and low (<6.8
a.u) intensity according to (a). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
P.H. Ninive et al. / Acta Materialia 69 (2014) 126–134
authors. Nevertheless, the calculated lattice strain in the
precipitate in the next section demonstrated that the strain
amplitude was large where the higher HAADF intensity
was recorded for Mg1 columns. An effect of lattice strain
would rather be to reduce this intensity [15]. HAADFSTEM image simulations provided in the supplementary
material also fail to identify any artefacts induced by lattice
strain. In conclusion, the local variations of column intensities can be directly attributed to relative variations in column composition.
3. Ab initio calculations
Previous DFT studies on the precipitates of Al alloys,
including b00 , have been restricted to bulk precipitate structures [6,7,18] or two-dimensional interfaces [19,20]. The latter can be helpful: e.g. to generate Wulff constructions of
precipitates. Finkenstadt and Johnson [20] also used such
interfaces to study wetting of such interfaces and the
non-equilibrium thermodynamics governing the growth
of precipitates. In the present work, realistic atomistic
models were constructed, with precipitates embedded in
Al; see Fig. 1d for an example. This implies a number of
benefits: corners and edges of the interface are included;
the strain fields in Al and inside the precipitate can be estimated; and the formation energy of the precipitate can be
appropriately calculated. This is the only way to obtain a
correct description, since b00 is metastable and only exists
in the Al matrix.
A number of models with varying size and precipitate
composition were investigated (see Appendix A). They
contained whole numbers of formula units NFU; each formula unit consists of 11 atoms and is defined in Fig. 1c. A
model with NFU = 16 is shown in Fig. 1d.
The elastic strain was calculated by comparing the interatomic distances in the DFT relaxed Al matrix with and
without precipitates. Fig. 3a and b show contour plots of
the calculated distortion around a precipitate with
NFU = 1, demonstrating that strain is generated by Al
atoms being pressed outwards from the precipitate. This
is because of larger average interatomic spacing in the precipitate than in Al. The strain was largest close to the Si1
and Si3 sites (see Fig. 1c), but the plot in Fig. 3c shows that
it decreased exponentially when moving away from the precipitate along [1 0 0]Al (horizontal) or [0 1 0]Al (vertical). The
approximate relationship is DaAl e0.415x, where x is the
distance from the interface. At 1 nm away from the interface, the strain is reduced to <1%.
The distortion of the precipitates was also studied.
Fig. 3d and e show contour plots of the calculated distances between Mg1 atoms in the ½1 0 0b00 and ½0 0 1b00 directions for a Mg5Al2Si4 precipitate. (Similar results were
achieved with Mg5Si6 and Mg4Al3Si4.) Distortion was
defined relative to the experimentally measured lattice constants of 7.58 Å (a/2) and 6.74 Å (c) [5]. The precipitate was
contracted by 3.7% and 5.0% in the ½1 0 0b00 and ½0 0 1b00
directions, owing to the lattice mismatch (Fig. 3f). The
129
calculated monoclinic angle of b00 approached the bulk
relaxed angle for the largest precipitates (Fig. 3g). This
may be partially due to shorter inter-precipitate distances.
The precipitate stability was quantified by the precipitate energy Eprec
SS
Eprec ¼ ðEtot N Al EAl N Mg ESS
Mg N Si E Si Þ=N FU ;
ð2Þ
where Etot is the total electronic energy of the supercell
containing the precipitate, Nx is the number of atoms of
element x, and ESS
x is the solid solution energy per atom
of element x. ESS
was
calculated using a 256 atom supercell
x
of bulk aluminium
EAl ¼ EAl256 =256
255
EAl ;
256 256
255
¼ EAl255 Si EAl :
256 256
ð3Þ
ESS
Mg ¼ E Al255 Mg ð4Þ
ESS
Si
ð5Þ
Using the solid solution energies as reference is consistent with previous studies on this system [7], and is most
relevant when precipitates are formed from impurities present as a SSSS. An alternative is to use energies of the standard states as in Ref. [6], but this is most appropriate when
bulk quantities of the impurities are present when the precipitates are formed.
The calculated precipitate energies using the definition in
Eq. (2) are shown in Fig. 4. They were 2 eV, monotonically decreasing with precipitate size. They were apparently
converging towards an asymptotic value, but complete convergence was not obtained with the largest precipitates
(NFU = 16). Mg5Al2Si4 was clearly the most stable composition among the three candidates, giving further evidence that
Al is present in the b00 structure. Nevertheless, the values of
Eprec of Mg5Si6 and Mg5Al2Si4 were almost the same for
NFU = 1, indicating that a larger amount of Si is favourable
in the early stages of the precipitation process; this is consistent with previous experimental results [21].
It is surprising that Mg4Al3Si4 was the least stable of the
three compositions, in apparent contradiction to the
HAADF-STEM results above. The energy penalty of Mg4Al3Si4 compared with Mg5Si6 decreases with increasing
precipitate size, but not rapidly enough to make Mg4Al3Si4
the most stable composition.
A possible explanation to this could be an inhomogeneous composition of b00 , as suggested in Fig. 2d. Inhomogeneities between the interface and interior of the
precipitate could result from strain fields close to the interface. Since Mg is larger than Al, the Mg1 site may contain
more Al in areas with contraction. This was indeed found
along the {1 3 0} interface (Fig. 3e), in the same region
where the HAADF-STEM Z-contrast indicated high Al
content of Mg1 sites (Fig. 2d). A similar correspondence
between tension and low intensity of Mg1 was found in
the [0 0 1]b00 direction along the {3 2 0} interface. It is thus
concluded that the most important stabilization mechanism for Al occupying the Mg1 sites is at the interface.
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P.H. Ninive et al. / Acta Materialia 69 (2014) 126–134
Fig. 3. DFT calculated strain and precipitate cell parameters. (a and b) The calculated strain along (a) [1 0 0]Al and (b) [0 1 0]Al around a Mg5Al2Si4
precipitate with NFU = 1. The strain is represented as the local deviation (in Å) from the bulk Al lattice constant, measured along the horizontal axis. The
dots designate the atomic positions in the matrix. (c) Exponential fit of the strain (shown as a dashed line) as a function of distance dn from the b00 -Al
interface along the horizontal axis indicated by the red dashed line in (a). Data points are included for all the precipitate sizes. (d and e) Contour plot of the
calculated distance between neighbouring Mg1 site atoms along the (d) [1 0 0]b00 and (e) [0 0 1]b00 directions across a Mg5Al2Si4 precipitate with NFU = 16
placed in an Al matrix. Contraction and tension are defined relative to the experimental distances (7.58 and 6.74 Å), marked in red. The strongest
contraction is seen along the {1 3 0} interface. (f) The calculated cell volume and (g) the monoclinic angle of b00 as a function of the number of precipitate
formula units. The bulk values (for a pure b00 crystal structure) are included at NFU ! 1. (For interpretation of the references to colour in this figure
legend, the reader is referred to the web version of this article.)
Fig. 4. The calculated precipitation energy vs. number of precipitate
formula units. Results are given for Mg5Si6 (blue diamonds), Mg5Al2Si4
(red circles) and Mg4Al3Si4 (green triangles). Lines are drawn as a guide to
the eye. (For interpretation of the references to colour in this figure legend,
the reader is referred to the web version of this article.)
It is not unlikely that the precipitate composition is
determined from the composition at the interface layer.
The cross-sectional growth direction of the precipitate is
probably inside-outwards, and the interface composition
should be frozen when covered by a new interface layer
(the diffusivity inside the precipitate should be low at the
relevant temperatures). This will give rise to an experimental precipitate composition (Mg4Al3Si4) with higher energy
than the most stable composition (Mg5Al2Si4).
In order to investigate the stability of b00 precipitates with
inhomogeneous compositions, the precipitate energies of
three different cases of inhomogeneous precipitates were
calculated. Using a nine-formula unit Mg5Al2Si4 precipitate as a starting point, some of the Mg1 site atoms were
replaced by Al. The three cases considered were substitution of Mg1 by Al along the {1 3 0} and {3 2 0} interfaces,
and in the middle of the precipitate. The resulting precipitate energies are shown in Table 1. As the calculated values
clearly show, none of the inhomogeneous compositions is
more stable than Mg5Al2Si4, even though it is slightly more
preferable to replace Mg1 atoms along the interfaces than
in the middle of the precipitate. This points to a weak tendency towards enrichment of Al at the interface sites of b00 ,
but not strong enough to explain the overall composition
seen by microscopy.
P.H. Ninive et al. / Acta Materialia 69 (2014) 126–134
Table 1
Calculated precipitate energies for various inhomogeneous compositions
of b00 ; the first entry is the precipitate energy of a homogeneous Mg5Al2Si4
b00 ; the entries named IF {1 3 0} and IF {3 2 0} represent energies calculated
where one or two Mg1 site atoms were substituted by Al along the {1 3 0}
and along the {3 2 0} interfaces; in the entry named “Middle of prec.”, two
Mg1 site atoms in the middle of the precipitate were substituted by Al;
Nsubst. is the total number of Mg1 site atoms substituted by Al in the
precipitate.
Mg5Al2Si4
IF {1 3 0}
IF {3 2 0}
Middle of prec.
Nsubst.
Eprec (eV)
0
2
4
2
4
2
2.312
2.259
2.190
2.252
2.182
2.250
The calculations were carried out at 0 K, without zeropoint corrections, temperature or entropy effects. Even if
this level of theory has been very successful for a number
of similar solid-state systems (e.g. see Ref. [22]), it is clear
that these effects influence the calculated stability of precipitates. As an example, the configurational entropy of the
impurities in solid solution serves to stabilize them as the
temperature increases. This means that the present results
are valid for a certain temperature range only; a more thorough investigation of the phase diagram of the precipitates
would require calculations far beyond the scope of this
paper. Also, one cannot exclude that the Mg4Al3Si4 phase
was stabilized relative to Mg5Al2Si4 as a result of temperature or other mechanisms not included in the calculations.
It was shown in earlier studies (e.g. see Ref. [20]) that the
precipitation thermodynamics can be strongly affected by
solute effects (e.g. the Suzuki effect [23]) at interphase
boundaries. This can lead to the formation of compositions
other than that which is thermodynamically favoured at
zero temperature DFT. Thus, the equilibrium calculations
should, in the future, be complemented by calculations
allowing for non-equilibrium growth.
To exclude the possibility of effects from the exchange–
correlation functional, a representative set of the calculations were repeated within the local density approximation
(LDA). Though the LDA generally gave somewhat lower
values of Eprec than GGA-PBE, the differences were found
to be less than 60 meV or 4%. The present results are thus
numerically robust within DFT with respect to all approximations and numerical parameters.
131
has so far not been taken into account. The DFT calculations above quantify these strain fields, and can be used as
additional input in the models. This was employed using
effective hardening radii rather than the measured ones.
More sophisticated models could involve the exponential
decrease in the strain field, but that is beyond the scope
of this paper.
Bright-field transmission electron microscopy (TEM)
was used to measure the precipitate size in Al–Mg–Si alloys
containing only b00 type precipitates. Three alloys and three
heat treatments were used to provide a large range of hardening and precipitate sizes. Other microstructural features
such as crystallographic grain size, morphology and texture
were characterised and found to be homogeneous for the
samples analysed. Measurements of Vicker’s hardness
and yield strengths were performed on the same samples,
providing a unique correspondence between b00 precipitate
size and mechanical strength.
The yield strength ry can be represented as the sum of
the intrinsic strength ri = 10 MPa [22], the solid solution
hardening rss and the precipitate hardening rp: ry = ri +
rss + rp. The precipitate hardening in a multicrystalline
sample rp can be found from the strengthening stress in a
single crystal sp from rp = Msp, where M = 3.1 is the Taylor factor. Fig. 5a shows the correspondence between the
calculated sp and the measured ry. sp was calculated from
Eq. (8) in Appendix A. The calculations were based on
both the original as-measured cross-section radii of the precipitates and of revised radii, where 1 nm was added
because of the coherency strain field around the precipitates. Restricting the slope to M = 3.1 gave a reasonable
fit, and extrapolating this fit to sp = 0 gave an initial yield
strength of ri + rss = 120 and 100 MPa for the original and
revised radii. An estimation of rss from rss = RkjCj2/3,
where kSi, kMg and kMn are 66.3, 29.0 and
29.0 MPa wt.%2/3, respectively [27] and calculating the
solid concentrations Cj using the Alstruc model [28] gave
(a)
(b)
4. Hardening mechanisms
It is well established that the precipitate volume fraction
and size are the dominating parameters determining the
hardening in Al–Mg–Si alloys [24]. For non-shearable precipitates, this follows the Orowan equation [25], which
includes the effective inter-particle spacing as a parameter.
Several empirical models of the relationship between precipitate morphology and hardening have been published
[26], but the coherency strain field surrounding the particles
Fig. 5. Correspondence between experimental and calculated hardness:
(a) the increased strengthening stress Dsp calculated from the precipitate
sizes and compared with the measured yield strength ry. The calculation
was based on the original (as-measured) precipitate radii as well as revised
ones, where the radii were increased by 1 nm, owing to the coherency
strain field. The straight line is the best fit to the revised data using a slope
of M = 3.1 (the Taylor factor for Al), and crosses the ry axis at 100 MPa.
(b) The same sets of radii were used to calculate Vicker’s hardness
calculated from the precipitate sizes and to compare with that directly
measured in the experiments.
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P.H. Ninive et al. / Acta Materialia 69 (2014) 126–134
ri + rss 6 100 MPa, in clear favour of the revised radii. A
least-squares fit without restricting the slope would give
M = 1.7 and ri + rss > 190 MPa, which is clearly
unreasonable.
Vicker’s hardness measurements are widely used to
characterise the strength of alloys, and it is convenient to
have empirical connections between such measurements
and the yield strength. Such relations have previously been
based on alloys containing more than one type of
precipitate [28], and the present data set including only
b00 precipitates gives an important update. A linear fit of
the correspondence between ry (in MPa) and Vicker’s
hardness HV (in VPN) gave
HV ¼ 0:14ry þ 64
ð6Þ
Similarly, a linear relationship between the measured
HV and calculated sp (valid for both the original and
revised radii) was established
HV ¼ 0:11sp þ 85
ð7Þ
The Vicker’s hardness calculated from this formula was
compared in Fig. 5b with that directly measured. There is a
relatively large spread in the data, but the correlation factor improved from 0.66 for the original radii to 0.72 for
the revised ones. This supports the hypothesis that the radii
of the precipitates are effectively extended by the coherency
strain field.
Nevertheless, the results demonstrate that the hardness
cannot be explained solely by the amount, fraction or size
of the precipitates. The scatter in hardness for similar particle populations indicates that other particle-related features influence the mechanical properties. Some data sets
had a relatively large spread in cross-section areas, which
was not contained in the model calculations leading to
Eq. (8)[29]. Also, those calculations were based on circular
cross sections; this is not representative of b00 precipitates,
which exhibit anisotropic strain fields (Fig. 3a and b).
Finally, the model calculations in Ref. [29] assumed infinitely long rods, while the measured average length of b00
precipitates was between 37 and 73 nm in the present case.
Edge effects and interaction between precipitates as well as
the surrounding strain fields should be included in model
calculations to improve their predictive power.
elastic strain field in the Al matrix and inside the
precipitates.
The HAADF-STEM investigations of the b00 structure
indicated that Mg4Al3Si4 is the most likely composition
of b00 . This was in apparent conflict with DFT calculated
stabilities of different precipitate compositions, indicating
that Mg5Al2Si4 was more stable than Mg5Si6, followed by
Mg4Al3Si4. However, anisotropic HAADF intensities suggested a higher Al content of Mg sites along the {1 3 0}
interface, consistent with elastic contractions within the
precipitate predicted by DFT calculations. This means that
strain leads to Al being more stable in the interface region
than in the interior of the precipitate. Given that the
growth direction must be inside-outwards, kinetic restrictions lead to a metastable composition close to Mg4Al3Si4
surviving throughout most of the precipitate.
The strain field in Al around the precipitates was also
calculated by DFT. It exhibited an exponential decline with
the distance from the precipitate. This gives a new way of
estimating the effective size of a precipitate seen by moving
dislocations; the active radius of a b00 precipitate should be
given a value up to 1 nm larger than the radius measured
by high-resolution microscopy.
The strength of Al–Mg–Si alloys that only contained b00
precipitates was quantified by Vicker’s hardness and yield
strength measurements. The number density and size of
the precipitates were measured with TEM, and new empirical relationships between precipitate morphology and
alloy strength were established. Increasing the effective
radius of the hardening precipitates by 1 nm led to
improved correlation between the calculated and measured
strength parameters, which means that the strain field
should be included in future improvements of precipitate
strengthening models.
Acknowledgements
Computation time from the NOTUR consortium and
discussions with Lars Nils Bakken are gratefully acknowledged. W.L. acknowledges the Agence Nationale pour la
Recherche for financial support through the Programme
Jeune Chercheur – Jeune Chercheuse TIPSTEM.
Appendix A. Methodology
5. Summary and conclusions
This paper has presented a combined experimental and
modelling study of b00 -precipitates in Al–Mg–Si alloys
using TEM and HAADF-STEM investigations, DFT simulations and hardness measurements. Atomic resolution
and Z-contrast data from the microscopy measurements
made it possible to estimate the site-resolved chemical composition of b00 from the intensity of single atomic columns.
Periodic atomistic models of the precipitate were constructed as semi-infinite precipitate needles embedded in
an Al matrix. This made it possible to compare the stability
of different precipitate compositions and to quantify the
HAADF-STEM observations were performed on a
JEOL ARM 200F microscope equipped with a Schottky
field emitter and operating at 200 kV. This instrument
was also equipped with a spherical aberration Cs-probe
corrector. The following parameters were used: probe
diameter 0.1 nm, objective aperture semi-angle 22.5 mrad,
detector half-collection angle between 40 and 150 mrad.
For the quantitative analysis, Gaussian smoothing over 4
pixels and Wiener deblurring over 3 3 pixels were performed using the MatLab image processing toolbox. The
deconvolution of the electron probe was performed for a
calculated electron probe, following the expression
P.H. Ninive et al. / Acta Materialia 69 (2014) 126–134
available in Ref. [30], using the set of acquisition parameters and a Cs = 0.1 mm to account for residual probe
imperfections. For the extraction of column intensities,
an expectation model of the experimental image was built
and optimised following the procedure described in
Ref. [11].
The DFT calculations were carried out using VASP
(Vienna Ab initio Simulation Package) [31–33] with the
projector augmented-wave method [34], within the PBE
generalised gradient approximation [35]. The plane wave
cut-off energy was 250 eV. The accuracy of the results
was most sensitive to the density of k-points, and much
care was used to obtain well-converged energies. A C-centred Monkhorst Pack k-point grid of 1 1 8 for the precipitate models and 4 4 4 for the 256-atom reference
calculations were found sufficient for a numerical precision
of 0.05 eV for the reported energies. The atomic positions
and cell parameters were simultaneously relaxed, followed
by an additional self-consistent calculation to determine
the total electronic energy with high precision. The quasiNewton method was used for the relaxation, with remaining forces <0.01 eV Å1. The precipitate models contained
576 atoms and were based on a 12 12 1 aluminium
supercell. Al atoms were replaced by a whole number of
precipitate formula units according to the interfacial relations given by Eq. (1). Convergence of the distance between
the precipitate needles dn was assessed by varying dn
between 0.8 and 4 nm, and it was found that dn > 2 nm
was needed to avoid overlap of strain fields. The precipitate
models with NFU = 16 had dn = 1.7 nm, and a strain field
of 3% remained at the supercell boundary. The other models had dn between 2.4 and 4 nm, and the strain fields vanished completely at the supercell boundaries. It was thus
appropriate to relax the unit cell parameters according to
calculated pressures. The typical experimental spacing
between b00 needles is 10 nm. The coherence in the [0 0 1]
direction was good enough to avoid spurious results from
the small unit cell size in that direction.
The precipitate population and hardness of three Al–
Mg–Si alloys were studied. The total amount of silicon
and magnesium was 1.6 at.%, and the Si/Mg ratio was
0.71, 1.34 and 2.58, respectively. Three different artificial
ageing procedures were performed at 175, 200 and
260 °C, following Ref. [36]. The microstructure of the
alloys after artificial ageing was analysed in bright field,
with a CM30 TEM operated at 150 kV. The instrument
was equipped with a Gatan parallel electron energy loss
spectrometer that enabled determination of the specimen
thickness in the central area of the thin foils used for
TEM imaging. TEM pictures at different magnifications
were recorded both with a CCD camera and on film for
measuring particle number densities, average needle
lengths, average cross-section areas, and widths of the precipitate free zone at grain boundaries. The precipitate volume fraction for each of the analysed samples is the
product of the precipitate number density, average needle
133
length and average cross-section area. A detailed description of the method can be found elsewhere [37,38].
For each alloy and temper, the hardness was measured
by Vickers hardness with a 3 kg load and five indentations
per sample. For each alloy, five round tensile specimens
with a diameter of 6 mm were prepared parallel to the
extrusion direction. All the specimens were taken from
the middle section of the extrusion press and in centre of
the rod. A tensile testing machine was operated with a constant strain rate of 1.3 103 s1, and a long travel contact
extensometer with a fixed original gauge length of 25 mm
was used for strain measurements.
The strengthening stress originating from the precipitates sp was calculated by a model based on rod-like precipitates [29]
Gb 1=2
2:632rr
sp ¼ 0:055
fr þ 0:93f r þ 2:43fr3=2 ln
ð8Þ
rr
r0
where G is the shear modulus of the aluminium matrix
(28 GPa [37]), b is the Burgers vector in aluminium
(0.286 nm [39]), r0 is the inner cut-off radius of the dislocation line tension, rr is the average cross-sectional radius of
the rod precipitates (measured by TEM), and fr is the volume fraction of the rod precipitates (calculated from the
number density measured by TEM). The equation differs
from that presented in Ref. [29] because of an error in that
publication (A.W. Zhu, private communication).
Appendix B. Supplementary material
Supplementary data associated with this article can be
found, in the online version, at http://dx.doi.org/10.1016/
j.actamat.2014.01.052.
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