# Anomalous SAXS and APT investigation of core-shell precipitation in Al-Mg-Sc-Zr alloys

I Amedeo, N Kirby, P Mota Santiago, Frédéric De Geuser, M Perez, Sophie

Cazottes, T Dorin

► To cite this version:

I Amedeo, N Kirby, P Mota Santiago, Frédéric De Geuser, M Perez, et al.. Anomalous SAXS and APT investigation of core-shell precipitation in Al-Mg-Sc-Zr alloys. Acta Materialia, 2025, 301, pp.121593. 10.1016/j.actamat.2025.121593. hal-05380077

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# Anomalous SAXS and APT investigation of core–shell precipitation in Al–Mg–Sc–Zr alloys

I. Amedeo<sup>a,b</sup>, N. Kirby<sup>c</sup>, P. Mota Santiago<sup>c</sup>, F. De Geuser<sup>d,e</sup>, M. Perez<sup>b,e</sup>, S. Cazottes<sup>b</sup>, T. Dorin<sup>a,f</sup>,*

## ARTICLE INFO

### Keywords:

Core–shell precipitation

Aluminium alloys

Al<sub>3</sub>(Sc,Zr)

Anomalous SAXS

Atom probe tomography

Precipitation kinetics

Strengthening mechanisms

## ABSTRACT

Understanding the compositional evolution of core–shell L1<sub>2</sub> Al<sub>3</sub>(Sc,Zr) precipitates in aluminium alloys is critical for tailoring their mechanical performance. In this study, we combine atom probe tomography (APT) and anomalous small-angle X-ray scattering (ASAXS) to characterise, for the first time, the in situ chemical evolution of these precipitates during a two-step ageing treatment. APT confirms the sequential formation of a Sc-rich core followed by a Zr-enriched shell, while ASAXS, conducted near the Zr K-edge, enables time-resolved tracking of Zr incorporation during precipitation. Two alloys with and without Mg were compared to investigate the effects of Mg on the precipitation kinetics, morphology of precipitates and strengthening behaviour.

## 1. Introduction

In the context of physical metallurgy of aluminium alloys, the addition of scandium for precipitation hardening is promising as this element displays a significant strengthening effect on a per-atom basis [1] thanks to the formation of nanoscale, coherent L1<sub>2</sub> Al<sub>3</sub>Sc precipitates [2]. It is also known that in the presence of both scandium and zirconium, mixed chemistry Al<sub>3</sub>(Sc, Zr) precipitates displaying a Sc-rich core surrounded by a Zr-rich shell can form [3–5]. This chemical heterogeneity has been explained by the large difference in diffusion coefficients of Sc and Zr in Al [6,7] and the high energy cost for vacancy creation in the L1<sub>2</sub> structure [8], leading to a low diffusion speed of solute atoms inside the precipitates. Due to the slow diffusion speed of Zr, these precipitates display a significantly slower coarsening rate than single chemistry Al<sub>3</sub>Sc precipitates which is of significant interest for the development of high-temperature aluminium alloys [9]. The addition of Sc and Zr to Al–Mg alloys has been shown to lead to a number of benefits and most notably to a significant reduction in recrystallization [10], in addition to the strengthening provided by the precipitates. Recent work has also shown the impact of Al<sub>3</sub>(Sc, Zr) on reducing plastic heterogeneities in Al–Mg alloys [11,12].

The formation kinetics of nano-scale L1<sub>2</sub> precipitates has been extensively studied using a range of characterisation techniques. Key parameters typically investigated include the evolution of precipitate

size, volume fraction, and composition. When multiple L1<sub>2</sub>-forming elements with the general stoichiometry Al<sub>3</sub>X are present, complex multi-layered “onion-like” precipitate structures—often described as core–shell—can develop. These have been most effectively characterized using atom probe tomography (APT), which enables nanoscale resolution of chemical gradients. Notably, the core–shell nature of Al<sub>3</sub>(Sc, Zr) precipitates was first revealed in Fuller’s PhD thesis using APT [5], and later confirmed by kinetic Monte Carlo simulations [8]. Since then, a wide body of APT research has reported similar multi-layered Al<sub>3</sub>X precipitates, highlighting their potential to tailor mechanical properties. High-resolution transmission electron microscopy (HR-TEM), particularly with high-angle annular dark field (HAADF) imaging and atomic-resolution energy-dispersive X-ray spectroscopy (EDX), has also been used to confirm core–shell morphologies [3,13].

While APT and TEM provide detailed snapshots of precipitate chemistry at specific ageing states, it does not capture the continuous evolution of composition during thermal treatments. In contrast, small-angle X-ray scattering (SAXS) allows monitoring of precipitate evolution in situ, but requires assumptions about scattering contrast to extract volume fraction and thus does not directly resolve chemical changes. For instance, previous work assumed a distinct core and shell regions model with fixed compositions to characterize core–shell precipitates using SAXS [14].

<footnote><sup>a</sup> Institute for Frontier Materials, Deakin University, Waurn Ponds, 3216 VIC, Australia</footnote>

<footnote><sup>b</sup> MATEIS, INSA Lyon, Villeurbanne, 69100, France</footnote>

<footnote><sup>c</sup> Australian Synchrotron, ANSTO, Clayton, 3168, Australia</footnote>

<footnote><sup>d</sup> University of Grenoble Alpes, CNRS, Grenoble INP, SIMaP, Grenoble, F-38000, France</footnote>

<footnote>* Corresponding author.
E-mail address: thomas.dorin@deakin.edu.au (T. Dorin).</footnote>
One way to overcome this limitation is through the use of anoma-
lous SAXS (ASAXS), which exploits the variation in scattering contrast
of a specific element when the X-ray energy approaches its absorption
edge. This method has proven effective in probing the composition of
scattering objects during phase transformations. ASAXS is compatible
with in situ experiments, provided that high energy resolution and
low counting times are achieved, conditions typically only met at
synchrotron sources. For example, ASAXS was used to resolve the
progressive enrichment of (Mg₂Zn) precipitates in Cu during heat treat-
ments in an Al-Zn-Mg-Cu alloy, a process controlled by the much slower
diffusivity of Cu compared to Zn and Mg [15]. This approach directly
informs the present work, where Zr plays a role analogous to Cu, and
ASAXS is exploited to continuously track its delayed incorporation into
Al₃(Sc,Zr) precipitates during multi-step ageing. Applying ASAXS to
study the precipitation kinetics of Al₃(Sc,Zr) offers a unique opportu-
nity to fully understand the development of the core–shell structure.
This insight is critical for optimising ageing treatments and improving
the mechanical performance of aluminium alloys containing scandium
and zirconium.

Additionally, the effect of Mg on the precipitation kinetics is not
fully known, especially its possible impact on the core–shell mor-
phology. Some studies have been focusing on the effect of Mg after
long ageing times. The presence of Mg has been shown to slightly
decrease the solubility of Sc in Al and to slightly reduce the coarsening
kinetics of Al₃Sc. Mg has also been shown to modify the morphology
of precipitates during coarsening, from faceted cuboids to precipitates
remaining spherical [16]. Additionally, segregation of Mg at the inter-
face of the precipitates has been observed after long ageing at 300 °C
and was explained by preferential electronic interactions of the Mg-Sc
pair [17]. This result also explained the modification of the morphology
of the precipitates during coarsening. An early study suggested a faster
decomposition rate of a supersaturated Sc solid solution in presence of
Mg content but lacks observations of the precipitates [18]. Concerning
the effect of Mg on the formation of Al₃Zr precipitates, studies have
shown a similar reduction of the solubility of Zr in Al due to the pres-
ence of Mg but the impact on the precipitation kinetics was not fully
assessed [19]. Finally, the strengthening mechanisms of an Al–Mg–Sc–
Zr alloy were studied and attributed to grain boundary strengthening
and the formation of Al₃(Sc,Zr) secondary particles [20,21] but these
precipitates were not characterized explicitly, nor was the effect of Mg
on strengthening investigated.

Magnesium was selected for this study because it is a principal alloying element in commercially important 5xxx and 6xxx series alloys, and has already been extensively combined with Sc and Zr in alloys such as Scmalloy. While earlier work has suggested that Mg can influence both solubility and morphology of Al₃(Sc,Zr) precipitates, these effects have not been examined in a systematic, time-resolved manner. Clarifying Mg's role is therefore essential to bridge the gap between model binary alloys and technologically relevant compositions. Moreover, the strong interaction recently identified between Sc and Si during nucleation [22] makes Mg particularly relevant, owing to its well-known affinity with Si, which may indirectly affect the precipitation pathways of Al₃(Sc,Zr).

The present study addresses two key knowledge gaps identified above. First, the evolution of core–shell precipitate composition in Al-(Mg)-Sc-Zr alloys will be continuously monitored during a two-step ageing using in situ anomalous SAXS, and the results directly compared with APT data. This approach enables quantitative tracking of Zr incorporation into Al₃(Sc,Zr) precipitates. Second, the potential role of Mg in influencing core–shell formation and/or strengthening mechanisms is explored by comparing two alloys with different Mg contents. Together, these insights aim to clarify the interplay between alloy chemistry and precipitation pathways in Al–Mg–Sc–Zr systems, with implications for the design of high-performance aluminium alloys.

## 2. Experimental methodology

Two aluminium alloys were prepared with nominal compositions of Al-2Mg-0.1Sc-0.1Zr (wt.%) and Al-0.1Sc-0.1Zr (wt.%), hereafter referred to as Al2Mg and Al0Mg, respectively. These compositions were selected to approach commercially relevant Mg levels while ensuring sufficient solute content for observable precipitation during ageing, and to allow full dissolution of solutes upon solution treatment.

The alloys were synthesised by induction melting of commercial-
purity Al together with Al-2Sc, Al-5Zr and Al-25Mg master alloys.
Melting was performed in an alumina crucible under an argon atmo-
sphere at 750 °C, with a holding duration of approximately 30 min to
ensure chemical homogeneity. The melt was then poured at 720 °C into
a copper mould with a cylindrical billet geometry of 30 mm diameter
and approximately 100 mm height. Based on the mould geometry and
mass, the solidification cooling rate is estimated to be on the order of
10-30 K/s, which is typical of laboratory-scale casting and comparable
to direct-chill casting rates used for aluminium billets. The ingots were
subsequently homogenised at 615 °C for 24 h to reduce solidification
microsegregation. Previous work has shown that homogenisation at
similar temperatures is sufficient to remove Zr concentration gradients
in Al alloys [23]. These low Sc/Zr levels have been reported to be fully
supersaturated in the as-cast condition and free of Sc- or Zr-bearing
primary particles [4]. To confirm this, SEM was performed on the
two alloys after the solution treatment (Fig. 1). EDS analysis on the
only observed particles reveal that they are Fe-containing intermetallics
most likely caused by contamination during casting or Fe impurities in
the commercial master alloys.

Bulk composition of the alloys after solution treatment was as-
sessed by inductively coupled plasma (ICP) spectroscopy, with surface
preparation up to 4000-grit SiC paper. Ten successive measurements
were performed per sample, with the first two discarded to avoid
surface contamination effects. Averaged results from the remaining
eight scans confirmed the compositions were close to the nominal
values, with minor Si impurity levels attributed to the use of general-
purpose casting equipment. Notably, the Zr content was slightly higher
in the Mg-containing alloy (Table 1).

To promote the sequential precipitation of Sc and Zr, a two-step
ageing treatment was applied: 2 h at 300 °C followed by 2 h at 500 °C.
The lower-temperature step enables Sc-rich Al₃Sc precipitates to form,
as Zr diffusion is limited at 300 °C. Upon heating to 500 °C, Zr diffusion
becomes active, allowing for the formation of a Zr-rich shell around the
pre-existing Al₃Sc core.

Vickers microhardness testing was used to track precipitation kinet-
ics during ageing. A 1 kgf load and 10 s dwell time were used for all
indentations.

Subsequent sections describe the atom probe tomography (APT) and
small-angle X-ray scattering (SAXS) characterisation techniques used to
monitor precipitate evolution in composition, size, and volume fraction.

2.1. *Electron microscopy*

Scanning electron microscopy (SEM) samples were cut using a SiC-
blade cutoff table and hot-mounted in conductive prior to grinding with
successively fine SiC papers of 240, 600 and 1200 grit. The samples
were then polished using polishing diamond suspension with particle
size of 6, 3 and 1 µm, followed by polishing with a colloidal silica
suspension. A JEOL 7800F field emission SEM was used for secondary
electron and energy dispersive X-ray spectroscopy (EDS) analysis.

Folios of 10 mm × 10 mm × 0.4 mm were cut with a SiC-blade cutoff
table and thinned to a thickness of approximately 100 µm using
SiC grinding disks of 600, 1200 and 4000 grit. 3 mm diameter disks
were punched in the foil and perforated via electropolishing using a
TenuPol-5 electropolishing station. The electrolyte used consists in 30%
HNO₃ in 70% methanol maintained at approximately -35 °C using
liquid nitrogen. A voltage of 15 V was used with a light sensor to
detect perforation. A JEOL 2100 LaB₆ TEM was used for conventional
bright-field imaging and selected area diffraction.
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** SEM secondary electron imaging of (a) Al0Mg and (b) Al2Mg alloys after solution treatment at 615 °C. No large Sc and Zr containing particles are visible, EDS analysis in the Al2Mg alloy in the region shown in red is presented in (c) and shows the presence of Fe-containing intermetallics. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<caption>**Table 1**</caption>

Composition of the two studied alloys from ICP spectroscopy, average of 8 successive measurements. A confidence interval of 95% corresponding to plus/minus two standard deviations is given.

<table><thead><tr><th></th><th>Al</th><th>Sc</th><th>Zr</th><th>Si</th><th>Fe</th><th>Mg</th></tr></thead><tbody><tr><td>Al2Mg</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(w%)</td><td>balance</td><td>0.124 ± 0.001</td><td>0.097 ± 0.001</td><td>0.05 ± 0.02</td><td>0.09 ± 0.006</td><td>1.83 ± 0.01</td></tr><tr><td>(at%)</td><td>balance</td><td>0.07 ± 0.0006</td><td>0.03 ± 0.0003</td><td>0.05 ± 0.002</td><td>0.04 ± 0.004</td><td>2.03 ± 0.01</td></tr><tr><td>Al0Mg</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(w%)</td><td>balance</td><td>0.123 ± 0.001</td><td>0.037 ± 0.001</td><td>0.083 ± 0.005</td><td>0.08 ± 0.002</td><td>–</td></tr><tr><td>(at%)</td><td>balance</td><td>0.07 ± 0.0006</td><td>0.01 ± 0.0003</td><td>0.08 ± 0.005</td><td>0.04 ± 0.002</td><td>–</td></tr></tbody></table>

## 2.2. Atom probe tomography

10 mm×0.4 mm×0.4 mm blanks for APT were cut and then electropolished using an electrolyte consisting in perchloric acid in glacial acetic acid following a two-step polishing protocol [24]. APT was conducted on a Cameca LEAP 5000XR with a pulse rate of 200 kHz, pulse fraction of 20%, temperature of 25 K and average detection rate of 1%. After reconstruction of the evaporated volumes, the distribution of isolated atoms in the matrix (DIAM) method is used to calculate the matrix composition in solute elements [25]. This method relies on the nearest-neighbour distribution to calculate the probability for solute atoms to be isolated (i.e. that no similar solute atoms are present at a given distance around the centre atom). Using this method gives a statistical and direct way to measure matrix composition and the fraction of solute atoms located in precipitates.

A simple method to assess the chemical composition and size distribution of the precipitates is to start by defining surfaces of equal

solute contents (isosurfaces) which help to visually identify precipitates. Then, proximity composition histograms (proxigrams) giving the average composition around all isosurfaces as a function of the distance to the isosurface are a useful tool to quantify composition and local composition variations inside and around precipitates [26]. These plots also give an estimate of the mean size of precipitates.

The main error on bulk composition measurements in APT comes from identifying background hits as evaporated ions when ranging a peak on the mass spectrum and overlapping peaks of two different species which can partially be accounted for by using peak decomposition tools. While a general accuracy of 0.01 at.% is sometimes presented for APT [27,28] and can be safely assumed for most elements in this alloy, Zr detection in APT is more complicated than other elements due to the fact that Zr is present as 5 isotopes at 2 different states of charges meaning that each peak consists in a small number of hits. This is accentuated by the fact that Zr is a relatively heavy element compared to other elements in this alloy.
## 2.3. Small-angle X-ray scattering

10 mm × 10 mm × 0.4 mm foils were cut from the ingots for SAXS. These samples were tested on the SAXS/WAXS beamline of the Australian Synchrotron on a Linkam THS600 heating stage, the two steps ageing treatment presented above was performed with a heating rate of 100 °C/min. In situ SAXS acquisition was performed at two energies close to the K absorption edge of Zr: 17.758 keV (E<sub>off</sub>) and 17.994 keV (E<sub>on</sub>) to observe the formation of the Zr-rich shell via anomalous scattering effects close to the K-absorption edge of Zr (K<sub>Zr</sub> = 17.9976 keV [29]). After acquiring 5 scans at one energy with a acquisition time of 10 s each, the energy of the beam was switched in approximately 10 s, 5 scans were obtained at this second energy, etc. This ensures that only a limited delay of approximately 1 min exists between scans obtained at two different energies which is negligible compared to the total duration of the experiment. Thus, the scans at different energies can be compared assuming they capture the same state of the sample. Additional scans were performed at the beginning of the experiment, after the first step of ageing at 300 °C and at the end of the experiment at two other energies: 17.885 keV and 17.985 keV. They were performed in order to verify that anomalous effects are consistent at more energies. The energy resolution of the beamline is advertised at 2 × 10<sup>-4</sup> thus leading to an energy dispersion of approximately ±3.5eV at the used energies. A 2D-CCD detector was used (size 169 × 179 mm; 981 × 1043 pixels) and the chosen sample-detector distance was 6.8 m, leading to an observed q-range of 0.003–0.35 Å<sup>-1</sup>. Signal intensity was normalised to 100 000 counts on the beamstop and absolute calibration was performed using a glassy carbon standard, taking into account the physical thickness of the sample via transmission, these procedures were performed at each beam energy.

The effect of anomalous scattering can be seen in the expression of the scattering contrast:

$$ \Delta\rho^* = \frac{1}{\Omega} \sum_{i \in \{\text{Sc}, \text{Zr}\}} \Delta f_i(E)(C_p^i - C^i) \quad (1) $$

Due to the low expected amount of Mg and Si in the precipitates along with low contrast due to the close atomic number with Al, the contribution of Mg and Si to the contrast of precipitates is neglected.

A common expression of the contrast in SAXS involves the matrix composition which is expected to evolve during precipitation. This evolution is sometimes low enough to be neglected in calculations, especially for highly diluted alloys [30]. Alternatively, this evolution could be estimated using for example DIAM calculations on APT samples at different stages of ageing. Here, an alternative expression of the SAXS contrast is used instead, that uses the nominal composition of the alloy instead of the matrix composition [31]. C<sub>p</sub><sup>i</sup> is the composition of i in the precipitates and C<sub>i</sub> is the global composition of i in the alloy. Both of these values can be estimated using APT. The atomic volume of precipitates is assumed to be close to that of the matrix [31]. Δf<sub>i</sub> is the difference in scattering factor between element i and the matrix, in the case of a very diluted alloy. In the general case, an element's scattering behaviour can be described by a complex scattering factor:

$$ f_i(E) = (Z_i + f_i'(E)) + i f_i''(E) \quad (2) $$

where Z<sub>i</sub> is the atomic number of element i and f<sub>i</sub>' and f<sub>i</sub>'' are corrections to the scattering factor that vary with the energy of the incident X-ray beam [15]. Far from the absorption edge of an element, these corrective factors can be considered null and the scattering factor is close to the atomic number of the element. As the energy of the beam gets closer to the absorption edge of the element, both corrective factors start varying significantly. As long as the energy of the incident beam is below the absorption edge of the considered element, variations of the complex part of the scattering factor can be neglected. When the incident beam's energy is lower than the absorption edge of the considered element, the real part of the scattering factor correction typically decreases as the energy of the beam gets closer to the absorption edge.

Thus, assuming that the scattering factor of Zr alone is varying in the energy range selected for this experiment (cf Fig. 2), any decrease in scattered intensity when changing the energy of the beam can be associated with the presence of Zr in the precipitates.

The obtained I(q) signal can be described using three contributions [33,34]: First, the contribution of large scattering objects (grain boundaries, the surfaces of the samples, etc.) outside of the observed q-range. The asymptotic behaviour of this contribution can be described following Porod's approximation in the form of K<sub>p</sub>/q<sup>n</sup> with n ranging from 3.5 to 4. The second contribution can be approximated by a simple constant and is related to Laue scattering of randomly distributed solute atoms. The remaining signal is considered to be caused by the precipitates. The total theoretical form of the signal can then be expressed:

$$ I_{\text{total}}(q) = I_{\text{precipitates}}(q) + \frac{K_p}{q^n} + I_{\text{Laue}} \quad (3) $$

Quantitative results on the population of precipitates are obtained by fitting of the experimental data. The experimental signal is fitted using the theoretical signal of a background and a distribution of precipitates. When fitting, the experimental error is used as a weight on the residuals to ensure that each point contributes to the extent of its accuracy and the entire experimental curve is fitted even when intensity ranges over several orders of magnitude [34]. A Schulz distribution of hard spheres is used in this study, leading to the following fitting function to represent the intensity scattered by precipitates:

$$ I_{\text{precipitates}}(q, \bar{\eta}^2, \bar{R}, Z) = \bar{\eta}^2 N_v 8\pi^2 \left( \frac{\bar{R}}{Z+1} \right)^6 G_1(q, \bar{R}, Z) \quad (4) $$

where $\bar{\eta}^2$ is the mean square fluctuation, $N_v = f_v/\bar{V}$ is the number density of precipitates, $\bar{R}$ is the mean radius of precipitates and, Z is a width parameter of the precipitates distribution and $G_1$ is the integrated expression of the Schulz size distribution [35]. This distribution has been chosen as it can be integrated numerically, the integrated expression used in this study can be found at: [36]. The width parameter Z is related to the standard deviation of the distribution:

$$ Z = \left( \frac{\bar{R}}{\sigma} \right)^2 - 1 \quad (5) $$

The standard deviation can be used to calculate the volume mean radius of the distribution (Eq. (6)). The volume mean radius is a good size indicator for asymmetrical distributions as large precipitates can represent a significant volume even though their frequency is low in the distribution. This is particularly interesting in SAXS as experimental SAXS signal is proportional to the square of the volume of precipitates, making it especially sensitive to large particles in the detriment of smaller ones.

$$ \bar{R}^{\text{volume}} = \bar{R} \left( 1 + 3 \left( \frac{\sigma}{\bar{R}} \right)^2 \right) \quad (6) $$

The mean square fluctuation is used a descriptor of intensity and can be linked with the volume fraction and scattering contrast of precipitates:

$$ \bar{\eta}^2 = \Delta\rho^{*2} \frac{f_v}{1 - f_v} \quad (7) $$

In the general case, SAXS is used to measure the size distribution and volume fraction of precipitates with known composition and thus contrast. In the current study, a slightly different approach is taken in which the volume fraction is fixed in the fitting function to allow determination of the precipitates composition through exploitation of the anomalous scattering effect. Then, the volume fraction is calculated from the mean square fluctuation using Eq. (7).

The ratio of the mean square fluctuation fitted at each energy can be used to get an estimate of the Zr content in the precipitates. We can assume that the precipitates follow a Al<sub>3</sub>Zr<sub>χ</sub>Sc<sub>(1-χ)</sub> composition, or
<figure><img src="image_4.png" /><figcaption>Fig. 2. Theoretical evolution of Δf for Mg, Sc and Zr close to the K absorption edge of Zr. Red dashed lines indicate the beam energies used in this study. Data from [32], calculation routine by [29]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

χ = C<sub>p</sub><sup>Zr</sup> / (C<sub>p</sub><sup>Sc</sup>+C<sub>p</sub><sup>Zr</sup>). Thus, we can write: C<sub>p</sub><sup>Zr</sup> = χ0.25 and C<sub>p</sub><sup>Sc</sup> = (1 - χ)0.25 and using Eq. (1):

$$
\begin{align}
\frac{\bar{\eta}_{E_{on}}}{\bar{\eta}_{E_{off}}} &= \frac{\Delta\rho^*_{E_{on}}}{\Delta\rho^*_{E_{off}}} \nonumber \\
&= \frac{\Delta f_{Sc}((1-\chi)0.25 - C_{Sc}) + \Delta f_{Zr}^{E_{on}}(\chi0.25 - C_{Zr})}{\Delta f_{Sc}((1-\chi)0.25 - C_{Sc}) + \Delta f_{Zr}^{E_{off}}(\chi0.25 - C_{Zr})} \tag{8}
\end{align}
$$

Δf<sub>Sc</sub> can be approximated by Z<sub>Sc</sub> - Z<sub>Al</sub> = 8 and the theoretical variation of scattering factor of Zr with energy can be calculated (cf Fig. 2) leading to Δf<sup>Eon</sup><sub>Zr</sub> = 23.24 and Δf<sup>Eoff</sup><sub>Zr</sub> = 18.99. Finally, solving Eq. (8), we can express χ as a function of the ratio of mean square fluctuations at the two beam energies:

$$
\begin{equation}
\begin{split}
\chi(r) ={}& \frac{\Delta f_{Sc}(C_{Sc} + r0.25 - rC_{Sc} - 0.25)}{\Delta f_{Sc}(r-1) + \Delta f_{Zr}^{E_{on}} - r\Delta f_{Zr}^{E_{off}}} \\
& + \frac{\Delta f_{Zr}^{E_{on}} C_{Zr} - \Delta f_{Zr}^{E_{off}} rC_{Zr}}{\Delta f_{Sc}(r-1) + \Delta f_{Zr}^{E_{on}} - r\Delta f_{Zr}^{E_{off}}}
\end{split}
\tag{9}
\end{equation}
$$

This value of χ is a global composition of precipitates taken as homogeneous spheres, i.e intensity changes from anomalous SAXS do not specify if the precipitates display a core–shell structure but only that the global composition of precipitates is changing. The possible

core–shell morphology of precipitates can however be assessed from APT.

It should be emphasised that fixing f<sub>v</sub> during fitting does not imply a constant precipitate volume fraction during ageing. This parameter was fixed only to extract η², which captures the anomalous signal. The actual volume fraction was then recalculated at each time step from Eq. (7) using the measured η² and the contrast Δρ*. Similarly, composition effects were decoupled from anomalous effects by evaluating the ratio of η² at the two incident energies (Eq. (9)), which directly yields the Zr/Sc ratio in precipitates independent of f<sub>v</sub>.

3. Experimental results

3.1. Hardness and atom probe tomography

A two-step precipitation treatment was applied following solution treatment, consisting of 2 h at 300 °C followed by 2 h at 500 °C. Vickers hardness was measured throughout the treatment to monitor precipitation kinetics (Fig. 3).

During the first step of 2 h at 300 °C, hardness increased from (48±1) HV1 after solution treatment to a peak of (72±3) HV1 in Al2Mg, and from (21 ± 0.6) HV1 to (50 ± 1.3) HV1 in Al0Mg. These results are consistent with previous observations on binary Al-Sc alloys aged under
<figure><img src="image_5.png" /><figcaption>Fig. 3. Precipitation kinetics monitoring via Vickers hardness during two step ageing of Al - 2Mg - 0.1Sc - 0.1Zr and Al - 0.1Sc - 0.1Zr with APT reconstructions after 2 h at 300 °C followed by 2 h at 500 °C. APT reconstructions are only showing Al, Sc and Zr atoms, subvolumes of the total reconstructions.</figcaption></figure>

similar conditions [37]. It can be noted that the hardening kinetics is slower in the alloy containing Mg.

The hardness is observed to decrease during the second heat treat-
ment step for both alloys. The hardness drops by about 10 HV1 and
5 HV1 for the alloy with 0 and 2 wt% Mg respectively. This shows
a significant improvement in thermal stability in the alloy containing
Mg. We note that the Al2Mg alloy also contains a slightly higher bulk
Zr content (0.03 at.% vs. 0.01 at.% in Al0Mg, Table 1), which could
contribute to this difference since Zr is a slow diffusing element that
enhances precipitate stability. However, this difference in Zr content is
relatively small, and additional effects of Mg, such as its interactions
with Si at 300 °C and its impact on Zr incorporation at 500 °C, are
also likely to contribute to the observed behaviour.

APT reconstructions were obtained at the end of each step in the
heat treatment. After 2 h at 300 °C, Sc-rich precipitates were clearly
visible in both alloys, while Zr remained in solid solution. Following the
second step at 500 °C, precipitates grew in size and developed a visible
Zr-rich shell around the Sc-rich core, consistent with a core–shell mor-
phology. Full details of the APT data analysis and matrix composition
analysis are provided below. In these dilute alloys, Zr quantification
in APT is sensitive to peak decomposition and low counting statistics,
which can bias DIAM matrix values slightly high. Note that ASAXS
contrast (Eq. (1)) uses the bulk ICP compositions, so ASAXS results are
not affected by this APT bias.

Isosurfaces of 2 at.% Sc were used to identify precipitates, and
composition profiles (proxigrams) were computed relative to these
surfaces. Proxigrams for the Al0Mg and Al2Mg alloys aged 2 h at 300 °C
and aged at 2 h at 300 °C + 2 h at 500 °C are presented in Fig. 4. The
proxigram for a sample aged 2 h at 300 °C + 15 min at 500 °C for the
Al2Mg alloy is also presented. In all 5 samples, precipitates showed a
composition close to the 25 at.% Sc expected for the Al₃Sc phase. The
presence of Si was detected after 2 h at 300 °C with respectively 2 and
5 at.%Si in the sample with 2 and 0 wt.%Mg. The sample containing
Mg shows clearly less Si in the precipitate core which suggest a smaller

involvement of Si in the formation kinetics in the presence of Mg. The
Si content decreases close to 0 after exposure at 500 °C for both alloys.
This is in line with prior studies and the now well-established effect
of Si impurities on the formation of Al₃Sc during the early stages of
precipitation [38].

After 2 h at 300 °C, no significant Zr enrichment was observed in-
side or around the precipitates, and Mg levels appeared constant. After
the second step at 500 °C, precipitates increased in size, and a Zr-rich
shell was clearly visible around the Sc-rich core. Interestingly, precip-
itate growth was most rapid during the initial ramp to 500 °C, with
limited additional coarsening observed during the remainder of the 2-
hour hold. Both Si and Mg content within the precipitates appeared to
decrease during this step.

The trends were largely similar between the Al0Mg and Al2Mg al-
loys. However, the amount of Zr detected in and around the precipitates
was lower in Al0Mg compared to Al2Mg. Additionally, the precipitates
reached slightly larger sizes at the end of the second step in Al0Mg.

APT also enabled quantification of matrix composition at various
stages of ageing using the DIAM method. Results are summarised in
Table 2. Matrix compositions measured after solution treatment agreed
well with bulk ICP data (Table 1), and confirmed the slightly higher Zr
content in Al2Mg. During ageing, Sc content in the matrix decreased
during the first step, with both Sc and Zr further depleted during the
second step, in line with ongoing precipitation.

These APT results confirm that the two-step ageing treatment effec-
tively promotes sequential precipitation: Sc-rich Al₃Sc forms during the
first step, followed by Zr enrichment at the precipitate-matrix interface
during the second step, resulting in a clear core–shell morphology.
While both alloys exhibit similar precipitation behaviour, subtle dif-
ferences in precipitate size and Zr partitioning suggest that Mg content
may influence precipitation kinetics and solute partitioning. However,
APT provides only discrete snapshots of this evolution. To gain a con-
tinuous view of precipitate growth and compositional changes during
ageing, in situ anomalous SAXS was employed, as described in the
following section.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Proxigrams computed from APT reconstructed volumes after 2 h at 300 °C (left), 2 h at 300 °C followed by 15 mins at 500 °C (centre) and 2 h at 300 °C followed by 2 h at 500 °C (right) in Al2Mg (top) and Al0Mg (bottom), reference isosurfaces are 2 at.% Sc.</figcaption></figure>

<table><caption>**Table 2**

Matrix composition (at%) calculated through DIAM at various stages of the two-step precipitation treatment in Al0Mg and Al2Mg.</caption><thead><tr><th></th><th>Sc</th><th>Zr</th><th>Si</th><th>Mg</th></tr></thead><tbody><tr><td>Al0Mg</td><td></td><td></td><td></td><td></td></tr><tr><td>Solution treated</td><td>0.09</td><td>0.04</td><td>0.03</td><td>—</td></tr><tr><td>2 h at 300 °C</td><td>0.025</td><td>0.038</td><td>0.020</td><td>—</td></tr><tr><td>2 h at 300 °C + 2 h at 500 °C</td><td>0.016</td><td>0.026</td><td>0.026</td><td>—</td></tr><tr><td>Al2Mg</td><td></td><td></td><td></td><td></td></tr><tr><td>Solution treated</td><td>0.07</td><td>0.05</td><td>0.05</td><td>2.23</td></tr><tr><td>2 h at 300 °C</td><td>0.028</td><td>0.053</td><td>0.037</td><td>2.23</td></tr><tr><td>2 h at 300 °C + 15 mins 500 °C</td><td>0.028</td><td>0.045</td><td>0.042</td><td>2.23</td></tr><tr><td>2 h at 300 °C + 2 h at 500 °C</td><td>0.032</td><td>0.041</td><td>0.05</td><td>2.24</td></tr></tbody></table>

## 3.2. Anomalous small-angle X-ray scattering results

In situ SAXS experiments were conducted during a two-step ageing treatment on the two alloys under study. During the In situ ageing, SAXS acquisitions were conducted at two different alternating incident beam energies (17.758 keV and 17.994 keV) close to the absorption edge of Zr. The applied heat treatment consisted in 2 h at 300 °C followed by 2 h at 500 °C. Traditional SAXS allows for continuous monitoring of the size distribution of precipitates and anomalous scattering effects allow for continuous monitoring of the composition of precipitates which then allows to estimate the evolution of precipitate volume fraction.

Curve fitting was performed on the scans obtained at 17.758 keV and 17.994 keV during ageing at 300 °C followed by 500 °C. Fitting results concerning the size distribution of precipitates are presented in Fig. 5. The evolution of the volume mean radius of precipitates which takes into account the width parameter of the Schulz distribution is shown in Fig. 5 (a). The evolution of the mean square fluctuation,

which is proportional to the precipitate volume fraction, is presented in Fig. 5 (b). A steady increase in mean radius of the distribution can be observed during each step of ageing with a rapid increase during temperature changes. It should be noted that the fitted radii shows a good match between the two energies during the entire run. This is encouraging as anomalous scattering effects are indeed only impacting the relative intensity of the scattering objects but should not modify their apparent scattering length. Both alloys display a similar precipitates size during the first step of ageing but a larger volume mean radius is observed during the second step of ageing for the Al0Mg alloy.

The fitted mean square fluctuation shows a similar evolution to the mean radius during the stages of ageing at 300 °C and 500 °C with a rapid increase during temperature changes and a steady increase during the stages themselves. This increase in mean square fluctuation could be caused by an increase in the volume fraction of precipitates. During the first step of ageing, the fitted mean square fluctuation is relatively similar between the two alloys but during the second step of ageing, a larger mean square fluctuation is observed for Al2Mg.

It should be emphasised that $\eta^2$ is proportional to $f_v \times \Delta\rho^*$ and thus reflects both the volume fraction of precipitates and their evolving scattering contrast. Consequently, $\eta^2$ values in Fig. 4b cannot be directly compared between alloys or ageing steps without correcting for composition. The corrected $f_v$ values, obtained via Eq. (7) using anomalous contrast analysis, are presented separately in Fig. 6.

The anomalous scattering effects can be used to estimate the evolution of precipitate composition by computing the evolution of the ratio of mean squared fluctuations at the two energies (Eq. (9)). Accordingly, the evolution of $\chi$ in $Al_3(Zr_xSc_{(1-x)})$ in precipitates is calculated and plotted in Fig. 6.
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Volume mean radius of precipitates (a) and mean square fluctuation (b) obtained from fitting of experimental SAXS scans during multi-step ageing of two alloys with different Mg content. Red line indicates the evolution of the temperature. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

During the first step of ageing, the mean square fluctuation is the same at the two beam energies for both alloys and we can thus estimate the value of $\chi$ to be 0. During the second step of ageing, the fitted mean square fluctuation is higher for the scans with beam energy far from the absorption edge of Zr which we can attribute to the enrichment of the precipitates in Zr. The value of $\chi$ increases up to approximately 0.2 for Al2Mg, for the alloy without Mg, there is more spread in the calculated value of $\chi$ but the value seems lower, around 0.05 to 0.1. These results show a clear impact of Mg on the kinetics of Zr enrichment of the Al₃(Sc,Zr) precipitates, with a marked acceleration in the enrichment when Mg is present.

The ratio of Zr to Sc inside the APT isosurfaces used previously to compute proxigrams is also shown in Fig. 6 and shows generally good agreement with anomalous SAXS results. Similarly, the APT results showed an accelerated enrichment in the alloy with Mg with the maximum ratio reached after only 15 min at 500 °C. The $\chi$ values obtained by ASAXS (≈ 0.2 for Al2Mg and ≈ 0.05–0.1 for Al0Mg) are close to the nominal Zr/(Sc+Zr) ratios of the two alloys (≈ 0.3 and ≈ 0.125, respectively), once the solid solubility of Zr at 500 °C is considered. This suggests that the final Zr enrichment is influenced by the bulk alloy composition. However, the time-resolved APT data demonstrate that Zr incorporation occurs more rapidly in the Mg-containing alloy, indicating that Mg also influences the kinetics of shell formation rather than solely the bulk composition. APT-derived global precipitate compositions (integrated within the 2 at.% Sc isosurfaces) yield $\chi$ values that agree with ASAXS within uncertainties (Al2Mg ≈ 0.2; Al0Mg ≈ 0.05–0.1), confirming the consistency of the two techniques (see Fig. 5).

Now that the evolution of the composition of precipitates has been assessed by exploiting the anomalous scattering effects, it can be used to calculate, at each timestep, the SAXS contrast of the precipitates $\Delta\rho^*$ using Eq. (1). The evolution of the volume fraction during the experiment is then calculated using Eq. (7) and plotted in Fig. 7.

For each alloy, a volume fraction is calculated at each beam energy using the mean square fluctuation fitted at this energy and the SAXS contrast calculated using the value of $\chi$ calculated using the anomalous scattering effect and the global composition of the alloy obtained from ICP (cf Table 1). Then, the average of the volume fraction at the two beam energy is calculated and presented in Fig. 7.

During the first step of the treatment, where the composition of the precipitates is assumed to be constant and following Al₃Sc, an increase in the volume fraction can be observed from 0% to approximately 0.35% for Al0Mg and 0.2% for Al2Mg. The alloy free of Mg hence displays an accelerated kinetics of formation. This higher kinetics might

be in line with the likely higher availability of Si solute in the alloy free of Mg as it is known that Si has a significant impact on Al₃Sc especially at early forming stages [38]. Indeed, Si is likely tied to Mg in the Al2Mg alloy forming MgSi rich phases or clusters and hence not as available to bind with Sc at initial cluster formation.

When moving to the second step of ageing, the precipitation of Zr occurs and should theoretically prevent the dissolution of the formed Al₃Sc. The volume fraction of precipitates reaches a similar value for the two alloys around 0.3% to 0.35%. This value is in line with equilibrium volume fractions predicted by Thermocalc for these alloy compositions. For the Al0Mg alloy, the volume fraction seems to experience a small decrease. This can likely be explained by the lower Zr content in this alloy which is not efficient enough to prevent full dissolution of the Al₃Sc driven by the increased Sc solubility at this higher temperature.

These in situ SAXS results confirm the formation and growth of Al₃Sc precipitates during the first ageing step, followed by progressive Zr enrichment during the second step—consistent with a core–shell transformation. The use of anomalous scattering enabled a quantitative assessment of precipitate composition. The Zr enrichment was revealed to be extremely rapid after switching to the higher ageing temperature as imaged by the Sc-to-Zr ratio stabilising to its final value within minutes of exposure at 500 °C, with or without Mg. Importantly, the evolution of precipitate volume fraction also suggests that Mg contributes to greater stability at elevated temperatures, likely by enhancing Zr incorporation. These findings complement the APT observations and provide a continuous picture of the precipitation process, setting the stage for further discussion of the underlying mechanisms.

# 4. Discussion

## 4.1. *Evolution of the composition of core-shell precipitates*

Reconstructed APT volumes have shown that for both alloys with different Mg content, a single step ageing treatment at 300 °C led to the formation of Al₃Sc precipitates as Zr diffusion was not activated. A two-step ageing treatment at 300 °C and 500 °C allowed the formation of core–shell precipitates in both alloys. SAXS monitoring of precipitation with quantitative exploitation of anomalous scattering during the two-step ageing treatment allowed for a measurement of the ratio of Zr to Sc in the precipitates from a value of 0 (no Zr) at the end of the first step to approximately 0.2 in Al2Mg and approximately 0.05 in Al0Mg. It should be reminded that visible anomalous effects can only be interpreted as a variation of the global composition of precipitates. The
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Evolution of $\chi$ in Al$_3$(Zr$_\chi$Sc$_{(1-\chi)}$) in precipitates obtained from ASAXS during multi-step ageing of two alloys with different Mg content. The value of $\chi$ inside 2 at.% Sc APT isosurfaces is also shown.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Evolution of the volume fraction of precipitates calculated from the fitted mean square fluctuation and evolution of the precipitates composition. The average of the volume fraction calculated from the two beam energies is presented for each alloy.</figcaption></figure>

assumption that this variation is associated with a core–shell morphol-
ogy is only possible based on the APT observations. This constitutes the
first quantitative in situ measurement of the evolution of the Zr-to-Sc
ratio during ageing in such alloys.

In contrast, prior work using SAXS required strong assumptions regarding precipitate structure—for example, a fixed 15% shell thickness during ageing and fixed compositions of Zr-to-Sc ratio of 0 in the core and 0.8 in the shell [14]. These assumptions led to a calculated constant global Zr fraction of $\chi$ = 0.3 throughout the ageing process. In the present study, anomalous SAXS directly confirms the progressive evolution of the Zr-to-Sc ratio without needing such assumptions.

In a separate study by Lefebvre et al. a large, core–shell Al₃(Sc, Zr) precipitate of approximately 20 nm in diameter was observed in APT after 15 h ageing at 475 °C [39]. This precipitate consisted in a 5 nm, Sc-rich core surrounded by a 8 nm thick shell containing Zr. The global composition of the precipitate was estimated to be 74 ± 0.5 Al - 21.5 ± 1 Sc - 4.5 ± 1 Zr (at%), thus matching the Al₃(Sc, Zr) stoichiometry and a Zr-to-Sc ratio of 0.17. This value is close to those obtained in the present study via anomalous SAXS, albeit at a slightly higher temperature and shorter ageing duration.

These findings highlight the value of in situ techniques for revealing
the dynamics of core–shell evolution. A precise understanding of this
transformation pathway is essential for designing heat treatments that
<figure><img src="image_10.png" /><figcaption>Fig. 8. (a) Al-Sc and (b) Al-Zr phase diagrams with and without the presence of 2 w% Mg drawn from thermodynamics databases (ThermoCalc 2023b TCAL8) [40]. Note the effect of Mg on the solubility of Sc and Zr at 500 °C.</figcaption></figure>

deliberately control core and shell formation, ultimately enabling the tailoring of precipitate structures to achieve targeted thermal stability, coarsening resistance, and mechanical response.

## 4.2. Effect of Mg on Al₃(Sc, Zr) formation kinetics

Our study demonstrated the impact of Mg on slowing down the formation kinetics of Al₃Sc, which is consistent with previous work [16]. This was first revealed in the hardness plot (Fig. 3), with a delayed hardening in the presence of Mg and confirmed during the in situ SAXS heat treatment which revealed a slower volume fraction evolution in the Al2Mg alloy during ageing at 300 °C (Fig. 7). This impact of Mg on precipitation kinetics is not straightforward. Indeed, based on thermodynamics, the presence of 2 wt% Mg decreases the maximum solubility at 500 °C from 0.078 wt% to 0.057 wt% for Sc and from 0.063 wt% to 0.036 wt% for Zr (see Fig. 8). The lower Sc and Zr solubility in the presence of Mg should equate with a higher driving force for precipitation and hence accelerated kinetics in the presence of Mg which is contrary to what is observed in the present study. As a result, we suggest that the presence of Mg indirectly impacts the formation kinetics of Al₃Sc through its interaction with Si, as Mg and Si are known to strongly bind and form clusters that sequester vacancies [41]. TEM was conducted in the Al2Mg alloy after 2 h at 300 °C (Fig. 9), and the microstructure was found to contain numerous rod-shaped precipitates aligned along the (100) directions, consistent with Mg₂Si, thereby confirming the hypothesis that Mg traps the available Si during the low-temperature ageing step.

Previous studies have shown that Si is crucial to the formation of Al₃Sc, especially at early stages through its strong tendency to form clusters with Sc due to attractive binding energies at first and second nearest neighbour distances [38,42]. This Si-Sc clustering is known to be the main nucleation driver for Al₃Sc in Si- and Sc-containing Al alloys, and recent work has shown that such clusters can form even at room temperature [43]. This effect is particularly prominent at low Si levels, where small differences in Si content have a disproportionate impact on precipitation kinetics [22]. The decreased Si contribution in the presence of Mg was also evident from APT, which showed Si-enriched precipitates in the Al0Mg alloy compared to Al2Mg after 2 h at 300 °C (cf Fig. 4).

Upon increasing the temperature to 500 °C, the alloy free from Mg experienced a slight decline in volume fraction, while the Al2Mg alloy exhibited a significant increase. This is likely due to Mg reducing the

solid solubility of Zr, thereby making more Zr available for shell formation. This could also explain the higher Zr content inside precipitates according to ASAXS analysis (cf Fig. 6). This effect is compounded by the fact that the bulk Zr content was higher in Al2Mg than in Al0Mg. In terms of size, Mg had only a limited effect: precipitates in the presence of Mg were slightly smaller at 500 °C, consistent with previous observations [16]. Furthermore, the Mg₂Si particles formed during ageing at 300 °C (Fig. 9) would be expected to dissolve at 500 °C, releasing Si back into solution and thereby influencing the precipitation kinetics.

Additionally, Mg atoms have been shown to segregate at the coherent interface of Al₃Sc [44,45] and core-shell Al₃(Sc, Zr) precipitates [11,46], forming a mixed Al₃Zr and Mg shell around the Sc-rich core. The driving force for this segregation has been attributed to electronic interactions between Sc and Mg based on ab initio calculations [45]. This effect was also observed in the present study (see Fig. 4), although to a limited extent. It is also likely that Mg modifies the interfacial energy between Al₃Zr and either the matrix or Al₃Sc, potentially increasing Zr shell formation.

Our results highlight the challenge of optimising Al₃(Sc, Zr) core-shell precipitation kinetics in Mg-containing alloys. On one hand, the reduced solubility of Sc and Zr due to Mg could be seen as beneficial, as it increases the maximum amount of these costly elements available for precipitation. However, in real alloys that always contain some Si—either as an impurity (< 0.5 wt%) or as an alloying addition (> 0.5 wt%)—this effect becomes more complex. Small amounts of Si are actually highly beneficial for Al₃(Sc, Zr) formation, with optimal strengthening observed at around 0.1 wt% Si [22]. The presence of Mg reduces the effectiveness of Si by forming Mg-Si clusters or Mg₂Si-type phases, particularly at intermediate temperatures, as evidenced in TEM. This impact diminishes at 500 °C, where these phases dissolve. Nevertheless, the slightly negative impact of Mg is reflected in the lower hardening response of Al2Mg compared to Al0Mg, despite higher nominal levels of Sc and Zr in Al2Mg.

As such, the accelerated kinetics in Al0Mg comes closer to the maximum equilibrium fraction predicted by Thermo-Calc of approximately 0.5 wt%, while Al2Mg remains far from this equilibrium. At 500 °C, both alloys reach values close to the equilibrium volume fractions predicted by Thermo-Calc: 0.37 wt% in Al2Mg and 0.35 wt% in Al0Mg. This supports the conclusion that Mg has a limited effect on thermodynamics but a significant effect on precipitation kinetics—especially when contrasted with the strong kinetic influence of Si.
<figure><img src="image_11.png" /><figcaption>Fig. 9. TEM conventional bright field imaging in Al2Mg after 2 h at 300 °C showing the presence of needles consistent with Mg₂Si precipitates. The needles are aligned in the < 100> directions as highlighted with yellow arrows. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

These insights are essential for designing optimal Al–Mg-Sc-Zr alloys. Achieving the right balance of Si and Mg content is critical: enough Si to promote precipitation, but not so much Mg as to hinder it. This challenge may be best addressed through high-throughput studies exploring the full Mg–Si compositional space to identify compositions that maximise the formation and benefits of Al₃(Sc, Zr) precipitates in Al–Mg alloys.

### 4.3. Strengthening model

The strength of metallic alloys can be broken down in a set of strengthening mechanisms that are defined by the ability of different features to slow down dislocation movement. Typically, the main strengthening mechanisms in aluminium alloys are solid solution strengthening, grain boundary strengthening and precipitation strengthening. As no work hardening has been performed after solution treatment in the current study, the grain size can be expected to be relatively large and constant during ageing and is combined with the intrinsic yield strength of pure Al which is also constant. These values are fitted on the experimental measurement in the solutionized state after subtracting the contribution of solution strengthening. The uncertainty on this value is thus considered to be the sum of the uncertainty on the experimental measurement and on the solution strengthening. An estimate of the yield strength of the material is made based on the hardness measurements using the relationship: $\sigma_y \approx 3H$ [47]. It should be noted that this relationship is only a rough estimate commonly used for steel but that has been verified for Al–Mg–Si alloys only. In our present study, the impact of solid solution can be estimated using the solid solution contents obtained from APT and the precipitation strengthening aspects can be obtained from the size and volume fraction obtained from SAXS.

#### 4.3.1. Solid solution strengthening

The contribution of Mg, Sc and Zr solute atoms to solution strengthening can be modelled using the solute concentration by:

$$ \sigma_{ss} = \sum_{i \in \{Mg, Sc, Zr, Si\}} k_i c_i^{3/2} \quad (10) $$

where $k_i$ is a constant defined for each solute element, the value of which is presented in Table 3.

The evolution of the matrix composition during ageing is presented in Table 2 and is used to populate the model. A relative error of 10% on the matrix composition obtained from the DIAM method is used when considering the contribution of solute strengthening [25].

<table><caption>**Table 3**
Values of $k_i$ in Eq. (10) for Mg, Sc, Zr and Si.</caption><thead><tr><th></th><th>Mg [48]</th><th>Sc [49]</th><th>Zr [50]</th><th>Si [48]</th></tr></thead><tbody><tr><td>$k_i$ (MPa w%$^{2/3}$)</td><td>29</td><td>82.5</td><td>60</td><td>66.3</td></tr></tbody></table>

#### 4.3.2. Precipitation strengthening

The ability of precipitates to strengthen a metallic alloy depends on the ability of dislocations to pass through the precipitates. This happens either by precipitate shearing—where an anti-phase boundary is created which “cuts” the precipitate—or by precipitate bypassing, where a dislocation loop is left around the precipitate. The transition between these mechanisms is usually dictated by precipitate size, with small precipitates more likely to be sheared and large precipitates more likely to be bypassed.

Considering the different contributions to precipitate strengthening presented in a previous work by Seidman et al. on Al₃(Sc, Zr) precipitates [37] and based on the size of precipitates measured from SAXS, the main strengthening mechanisms are attributed to order strengthening for sheared precipitates during the first step and Orowan bypassing during the second step of the treatment.

In principle, shearing may also involve coherency and modulus mismatch strengthening, which together can dominate at very small precipitate sizes. However, given that our precipitates after 2 h at 300 °C were already several nanometres in radius—well above the size where coherency and modulus strengthening operate effectively— these contributions are negligible in the present alloys, and order strengthening is the relevant shearing mechanism. Accordingly, order strengthening can be represented by:

$$ \Delta\sigma_1 = 0.81 M \frac{\gamma_{apb}}{2b} \left(\frac{3\pi f_v}{8}\right)^{1/2} \quad (11) $$

where $M = 3.06$ is the Taylor factor [51], $b = 0.286$ nm is the magnitude of the Burgers vector, and $f_v$ is the volume fraction of precipitates. $\gamma_{apb}$ is the energy of the anti-phase boundary. As a first approximation, a value of 370 mJ/m² can be used. This value is an average of previously reported values for Al₃Sc [37,52–56]. Concerning Al₃Zr, the actual anti-phase boundary energy is not fully known but is expected to be close to that of Al₃Sc [57]. It is thus relevant to use the same value throughout multi-step ageing, especially as we have seen that the Zr content at the interface still seems lower than the Sc content during the second step of ageing (cf. Fig. 4).

Orowan bypassing strengthening is represented by:

$$ \Delta\sigma_{or} = M \frac{0.4}{\pi} \frac{Gb}{\sqrt{1-\nu}} \frac{\ln\left(\frac{2R}{b}\right)}{\lambda} \quad (12) $$
<figure><img src="image_12.png" /><figcaption>Fig. 10. Evolution of the experimental yield strength increment based on hardness measurements during two-step precipitation treatment of the two studied alloys compared with contribution of precipitates to hardness based on anomalous SAXS results.</figcaption></figure>

where $v = 0.34$ is the Poisson's ratio of the matrix [58] and $\lambda$ is the edge-to-edge inter-precipitate distance, calculated by:

$$ \lambda = R \left[ \left( \frac{3\pi}{4f_v} \right)^{1/2} - 1.64 \right] \quad (13) $$

Using the values of volume fraction and mean radius obtained from SAXS fitting (Fig. 5), the main strengthening mechanism can be identified as order strengthening during all of the first step and Orowan bypassing during the second step. The dominant interaction mechanism is the one demanding the least energy and hence the least strength increase. The various critical resolved shear strengths for order strengthening and bypassing can be calculated using the size and volume fractions obtained from SAXS. The results are plotted in Fig. 10 as a function of ageing time and compared with the experimental increment of yield strength based on hardness measurements. It can clearly be seen that order strengthening dominates initially, with a clear transition from precipitate shearing in the first step to bypassing in the second step.

### 4.3.3. Balance of strengthening contributions

Based on the results from the previous sections, we now have enough information to estimate the strength contributions in three conditions: solution-treated samples, after 2 h at 300 °C, and after 2 h at 300 °C followed by 2 h at 500 °C. We have APT measurements for these conditions to estimate the solid solution content. Furthermore, we have now established that order strengthening dominates after 2 h at 300 °C and bypassing dominates after the subsequent step at 500 °C.

The yield strength provided by the different solute species and precipitates is compared with experimental data in Fig. 11. There is satisfying agreement between the experimental and calculated yield strength using measured properties of the precipitates and solute content.

These results support the idea that precipitates offer significant strengthening through the applied two-step precipitation treatment. Additionally, the improved stability of precipitates due to the presence of Zr in the shell helps maintain strengthening at high temperatures, as the volume fraction of precipitates is preserved. This is promising for the development of aluminium alloys with additions of Sc and Zr.

In this assessment, we only considered two ageing states to evaluate the balance between different strengthening mechanisms. After the first step at 300 °C, the dominant mechanism is order strengthening. As the precipitates grow during the temperature increase, their behaviour shifts from being sheared to being bypassed, changing the strengthening mechanism to Orowan bypassing.

It is clear that precipitation strengthening from Al₃(Sc, Zr) is significantly higher in the alloy free from Mg. After the first ageing step, a maximum precipitation strengthening of approximately 82 MPa and 57 MPa is reached in the Al0Mg and Al2Mg alloys, respectively. Following the second ageing step, these values decrease to approximately 53 MPa and 51 MPa in Al0Mg and Al2Mg, respectively. This reduction in precipitation strengthening occurs despite the significantly higher Sc and Zr contents in the Mg-containing alloy. This further confirms the challenge of maintaining full strengthening potential from Al₃(Sc, Zr) in Al-Mg alloys.

## Conclusions

This study investigated the influence of Mg on the precipitation behaviour and associated strengthening mechanisms in Al-Mg-Sc-Zr alloys, using a combination of APT, SAXS, and anomalous SAXS (ASAXS). The key findings are as follows:

* Mg additions slow down the formation of Al₃Sc precipitates during ageing at 300 °C, despite thermodynamic predictions which predict an increase in volume fraction. This counterintuitive behaviour is attributed to the consumption of Si by Mg, which inhibits Si-Sc clustering — a known precursor for Al₃Sc nucleation. The presence of Mg₂Si particles after the ageing step at 300 °C was confirmed with TEM. Atom probe tomography (APT) also reveals a lower Si content in precipitates formed in the Mg-containing alloy, supporting the proposed mechanism of reduced nucleation efficiency due to Mg-Si interactions.

* The use of anomalous SAXS (ASAXS) enabled the first in-situ, time-resolved tracking of precipitate composition evolution, specifically the Zr/Sc ratio, during complex multistep ageing
<figure><img src="image_13.png" /><figcaption>Fig. 11. Experimental estimation of the yield strength after each step of ageing compared with theoretical strengthening from pure Al and grain size, solute strengthening and precipitate strengthening.</figcaption></figure>

treatments. It was supported by APT to visualise that this com-
position evolution is indeed linked with formation of a core–
shell morphology. This revealed that the formation of core–shell
Al₃(Sc,Zr) precipitates occurs progressively, with the Al₃Sc form-
ing during the first step at 300 °C with Zr incorporation occurring
when going to the higher temperature at 500 °C where Zr dif-
fusion is enabled. The Zr-to-Sc ratio stabilised around ≈ 0.05 in
AlOMg and ≈ 0.2 in Al2Mg, values broadly consistent with the
bulk Zr levels of the two alloys. While these final values primarily
reflect effects from the bulk composition, the accelerated enrich-
ment in Al2Mg demonstrates that Mg also influences the kinetics
of Zr incorporation.

* Precipitation strengthening reached approximately 80 MPa in the Mg-free alloy after the first step, compared to 60 MPa for the Mg-containing alloy. After the second step, these values decreased to approximately 50 MPa, for both alloys. This highlights the lower strengthening achieved through Al₃(Sc,Zr) precipitation in the presence of Mg.

* A detailed analysis of the operative strengthening mechanisms showed a transition from coherency and modulus mismatch strengthening to order strengthening and Orowan bypassing.

* The integration of ASAXS, and APT provides a comprehensive framework for studying precipitate structure, composition, and mechanical impact, and offers a valuable pathway for optimising heat treatment strategies in advanced Al-Sc-Zr-Mg alloys.

CRediT authorship contribution statement

I. **Amedeo:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis. **N. Kirby:** Resources, Methodology, Investigation, Formal analysis, Conceptualization. **P. Mota Santiago:** Resources, Methodology, Investigation, Formal analysis. **F. De Geuser:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **M. Perez:** Writing – review & editing, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **S. Cazottes:** Writing – review & editing, Supervision. **T. Dorin:** Writing – review & editing, Visualization, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The SAXS experiments were undertaken on the SAXS/WAXS
beamline at the Australian Synchrotron, part of ANSTO (Beam-
time AS241/SAXS/20942). The authors are grateful for the help
of Alexandru Techeres on the SAXS experiment and on SAXS data
calibration/normalisation. Deakin University's Advanced Characteriza-
tion Facility is acknowledged for use of the Cameca LEAP 5000XR
APT, JEOL 7800F field emission SEM and JEOL 2100 LaB6 TEM
instruments. Thomas Dorin acknowledges Australian Research Council
grant DE190100614.

Data availability

The raw data required to reproduce the above findings will be
shared upon request.

References

[1] L. Toporova, D. Eskin, M. Kharakterova, T. Dobatkina, Advanced Aluminum Alloys Containing Scandium, Gordon & Breach, 1998.

[2] L. Jiang, B. Rouxel, T. Langan, T. Dorin, Fundamentals of Aluminium Metallurgy, Woodhead Publishing, 2018, pp. 439–494.

[3] A. Tolley, V. Radmilovic, U. Dahmen, Segregation in Al3(Sc,Zr) precipitates in Al-Sc-Zr alloys, Scr. Mater. 52 (7) (2005-04) 621–625.

[4] B. Forbord, W. Lefebvre, F. Danoix, H. Hallem, K. Marthinsen, Three dimensional atom probe investigation on the formation of Al3(Sc,Zr)-dispersoids in aluminium alloys, Scr. Mater. 51 (4) (2004) 333-337.

[5] C. Fuller, D. Seidman, Temporal evolution of the nanostructure of Al(Sc,Zr) alloys: Part II-coarsening of Al(ScZr) precipitates, Acta Mater. 53 (20) (2005-12) 5415-5428.

[6] S. Fujikawa, Diffusion of scandium in aluminum, Research Report of Laboratory of Nuclear Sciences, Tohoku University, 30, 1997.

[7] H. Bakker, Diffusion in Solid Metals and Alloys, Springer, 1990.

[8] E. Clouet, L. Laé, T. Épicier, W. Lefebvre, M. Nastar, A. Deschamps, Complex precipitation pathways in multicomponent alloys, Nat. Mater. 5 (6) (2006) 482–488.

[9] L. Jiang, B. Rouxel, T. Langan, T. Dorin, Coupled segregation mechanisms of Sc, Zr and Mn at θ' interfaces enhances the strength and thermal stability of Al-Cu alloys, Acta Mater. 206 (2021) 116634.
[10] V. Zakharov, Effect of scandium on the structure and properties of aluminum alloys, Metal science and heat treatment 45 (7) (2003) 246-253.

[11] F. Bakare, L. Schieren, B. Rouxel, L. Jiang, T. Langan, A. Kupke, M. Weiss, T. Dorin, The impact of L12 dispersoids and strain rate on the Portevin-Le-Chatelier effect and mechanical properties of Al-Mg alloys, Mater. Sci. Eng.: A 811 (2021) 141040.

[12] F. Bakare, L. Jiang, T. Langan, M. Weiss, T. Dorin, Impact of dispersoids' distribution on portevin-le-chatelier effect and surface quality in Al-Mg-Sc-Zr alloys, Mater. Sci. Eng.: A 875 (2023) 145108.

[13] T. Dorin, S. Babaniaris, L. Jiang, A. Cassel, C. Race, A. Eggeman, D. Kelly, S. Haigh, J. Robson, Stability and stoichiometry of L12 Al3 (Sc, Zr) dispersoids in Al-(Si)-Sc-Zr alloys, Acta Mater. 216 (2021) 117117.

[14] A. Deschamps, L. Lae, P. Guyot, In situ small-angle scattering study of the precipitation kinetics in an Al-Zr-Sc alloy, Acta Mater. 55 (8) (2007) 2775-2783.

[15] T. Marlaud, A. Deschamps, F. Bley, W. Lefebvre, B. Baroux, Influence of alloy composition and heat treatment on precipitate composition in Al-Zn-Mg-Cu alloys, Acta Mater. 58 (1) (2010) 248-260.

[16] E.A. Marquis, D.N. Seidman, Coarsening kinetics of nanoscale Al3Sc precipitates in an Al-Mg-Sc alloy, Acta Mater. 53 (15) (2005-09) 4259-4268.

[17] E. Marquis, D.N. Seidman, M. Asta, C. Woodward, V. Ozoliņš, Mg segregation at Al/Al3Sc heterophase interfaces on an atomic scale: Experiments and computations, Phys. Rev. Lett. 91 (3) (2003) 036101.

[18] V. Zakharov, Stability of the solid solution of scandium in aluminum, Met. Sci. Heat Treat. 39 (2) (1997) 61-66.

[19] F. Beledi, R. Casati, C. Andrianopoli, F. Cuccaro, M. Vedani, Investigation and characterization of an Al-Mg-Zr-Sc alloy with reduced Sc content for laser powder bed fusion, J. Alloys Compd. 924 (2022) 166519.

[20] K. Kendig, D. Miracle, Strengthening mechanisms of an Al-mg-Sc-Zr alloy, Acta Mater. 50 (16) (2002) 4165-4175.

[21] J. Shen, B. Chen, J. Wan, J. Shen, J. Li, Effect of annealing on microstructure and mechanical properties of an Al-Mg-Sc-Zr alloy, Mater. Sci. Eng.: A 838 (2022) 142821.

[22] A. Techeres, M. Perez, A. Deschamps, S. Seibt, N. Kirby, V. Dorin, High-throughput investigation of Si and Fe effects on Al-Sc-Zr precipitation kinetics, Mater. Sci. Eng. A 943 (2025) 148851.

[23] J. Zhihong, G. Hu, B. Forbord, J.K. Solberg, Effect of homogenization and alloying elements on recrystallization resistance of Al-Zr-Mn alloys, Mater. Sci. Eng.: A 444 (2007) 284-290.

[24] M. Miller, G. Smith, Atom probe microanalysis: Principles and applications to materials problems, Technical Report, Oak Ridge National Lab., 1987.

[25] F.D. Geuser, W. Lefebvre, Determination of matrix composition based on solute-solute nearest-neighbor distances in atom probe tomography, Microsc. Res. Tech. 74 (3) (2011-03) 257-263.

[26] O.C. Hellman, J.A. Vandenbroucke, J. Rüsing, D. Isheim, D.N. Seidman, Analysis of three-dimensional atom-probe data by the proximity histogram, Microsc. Microanal. 6 (5) (2000) 437-444.

[27] M. Thuvander, On the accuracy of compositional quantification for atom probe tomography, Microsc. Microanal. 22 (S3) (2016) 642-643.

[28] K. Thompson, J. Bunton, T.F. Kelly, D.J. Larson, Characterization of ultralow-energy implants and towards the analysis of three-dimensional dopant distributions using three-dimensional atom-probe tomography, J. Vac. Sci. Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. 24 (1) (2006) 421-427.

[29] S. Brennan, P.L. Cowan, A suite of programs for calculating x-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths, Rev. Sci. Instrum. 63 (1) (1992) 850-853.

[30] A. Deschamps, F. De Geuser, Z. Horita, S. Lee, G. Renou, Precipitation kinetics in a severely plastically deformed 7075 aluminium alloy, Acta Mater. 66 (2014) 105-117.

[31] F. de Geuser, Trajectoires de précipitation dans les alliages multi-constitués, 2020.

[32] D.T. Cromer, D. Liberman, Relativistic calculation of anomalous scattering factors for X rays, J. Chem. Phys. 53 (5) (1970) 1891-1898.

[33] B.J. Heuser, Small-angle neutron scattering study of dislocations in deformed single-crystal copper, J. Appl. Crystallogr. 27 (6) (1994) 1020-1029.

[34] F. De Geuser, M. Styles, C. Hutchinson, A. Deschamps, High-throughput in-situ characterization and modeling of precipitation kinetics in compositionally graded alloys, Acta Mater. 101 (2015) 1-9.

[35] G. Schulz, Molecular weight determination in macromolecular compounds. VIII. distribution function of multimolecular compounds and their determination by fractionation, Z. Phys. Chem. 47 (1949) 155-193.

[36] M. Kotlarchyk, S.-H. Chen, Analysis of small angle neutron scattering spectra from polydisperse interacting colloids, J. Chem. Phys. 79 (5) (1983) 2461-2469.

[37] D. Seidman, E. Marquis, D. Dunand, Precipitation strengthening at ambient and elevated temperatures of heat-treatable Al(Sc) alloys, Acta Mater. 50 (16) (2002) 4021-4035.

[38] C. Booth-Morrison, Z. Mao, M. Diaz, D. Dunand, C. Wolverton, D.N.V. Seidman, Role of silicon in accelerating the nucleation of Al3 (Sc, Zr) precipitates in dilute Al-Sc-Zr alloys, Acta Mater. 60 (12) (2012) 4740-4752.

[39] W. Lefebvre, F. Danoix, H. Hallem, B. Forbord, A. Bostel, K. Marthinsen, Precipitation kinetic of Al3 (Sc, Zr) dispersoids in aluminium, J. Alloys Compd. 470 (1-2) (2009) 107-110.

[40] J.-O. Andersson, T. Helander, L. Höglund, P. Shi, B. Sundman, Thermo-calc & DICTRA, computational tools for materials science, Calphad 26 (2) (2002) 273-312.

[41] J. Peng, S. Bahl, A. Shyam, J. Haynes, D.V. Shin, Solute-vacancy clustering in aluminum, Acta Mater. 196 (2020) 747-758.

[42] T. Dorin, M. Ramajayam, S. Babaniaris, T. Langan, Micro-segregation and precipitates in as-solidified Al-Sc-Zr-(Mg)-(Si)-(Cu) alloys, Mater. Charact. 154 (2019) 353-362.

[43] L. Jiang, K. Wood, A. Sokolova, T. Dorin, Solute clustering behaviour during natural ageing in Al-Cu alloys containing Sc and Zr, Scr. Mater. 265 (2025) 116764.

[44] E.A. Marquis, D.N. Seidman, Nanostructural evolution of Al3Sc precipitates in an AlScMg alloy by three-dimensional atom probe microscopy, Surf. Interface Anal. 36 (56) (2004-05) 559-563.

[45] E.A. Marquis, D.N. Seidman, M. Asta, C. Woodward, Composition evolution of nanoscale Al3Sc precipitates in an Al-Mg-Sc alloy: Experiments and computations, Acta Mater. 54 (1) (2006) 119-130.

[46] L. Liu, X.-Y. Cui, J.-T. Jiang, B. Zhang, K. Nomoto, L. Zhen, S.P. Ringer, Segregation of the major alloying elements to Al3 (Sc, Zr) precipitates in an Al-Zn-Mg-Cu-Sc-Zr alloy, Mater. Charact. 157 (2019) 109898.

[47] D. Tabor, The Hardness of Metals, Oxford University Press, 2000.

[48] O. Myhr, Ø. Grong, S. Andersen, Modelling of the age hardening behaviour of Al-Mg-Si alloys, Acta Mater. 49 (1) (2001) 65-75.

[49] C. Yang, P. Zhang, D. Shao, R. Wang, L. Cao, J. Zhang, G. Liu, B. Chen, J. Sun, The influence of Sc solute partitioning on the microalloying effect and mechanical properties of Al-Cu alloys with minor Sc addition, Acta Mater. 119 (2016) 68-79.

[50] S. Babaniaris, M. Ramajayam, L. Jiang, T. Langan, T. Dorin, Tailored precipitation route for the effective utilisation of Sc and Zr in an Al-Mg-Si alloy, Materialia 10 (2020) 100656.

[51] M. Meyers, Mechanical Metallurgy: Principles and Applications, vol. 761, Prentice-Hall, Inc, Englewood Cliffs, N. J. 07632, U. S. A, 1984, 1984.

[52] K. Kendig, D. Miracle, Strengthening mechanisms of an Al-Mg-Sc-Zr alloy, Acta Mater. 50 (16) (2002) 4165-4175.

[53] N. Kumar, R. Mishra, Additivity of strengthening mechanisms in ultrafine grained Al-Mg-Sc alloy, Mater. Sci. Eng.: A 580 (2013) 175-183.

[54] J. Røyset, N. Ryum, Scandium in aluminium alloys, Int. Mater. Rev. 50 (1) (2005) 19-44.

[55] E. George, J. Horton, W. Porter, J. Schneibel, Brittle cleavage of L12 trialuminides, J. Mater. Res. 5 (8) (1990) 1639-1648.

[56] M. Phillips, B. Clemens, W. Nix, A model for dislocation behavior during deformation of Al/Al3Sc (fcc/L12) metallic multilayers, Acta Mater. 51 (11) (2003) 3157-3170.

[57] W. Lefebvre, N. Masquelier, J. Houard, R. Patte, H. Zapolsky, Tracking the path of dislocations across ordered Al3Zr nano-precipitates in three dimensions, Scr. Mater. 70 (2014) 43-46.

[58] H. Frost, M. Ashby, Deformation-Mechanism Maps: the Plasticity and Creep of Metals and Ceramics, Pergamon Press, Oxford, 1982.