# Bayesian Optimization of the Experimental Parameters of Material Synthesis: Application to the Magnesioreduction of Rare-Earth-Free (Mn1-xFex)5Si3 Magnetocalorics

Sylvain Le Tonquesse, Laura Agnarelli, Swathi Sakthivel

► To cite this version:

Sylvain Le Tonquesse, Laura Agnarelli, Swathi Sakthivel. Bayesian Optimization of the Experimental Parameters of Material Synthesis: Application to the Magnesioreduction of Rare-Earth-Free (Mn1-xFex)5Si3 Magnetocalorics. Chemistry of Materials, 2025, 37 (15), pp.5740-5752. 10.1021/acs.chemmater.5c00879. hal-05232344

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Bayesian Optimization of the experimental
parameters of complex material synthesis:
Application to the magnesioreduction of
rare-earth free (Mn<sub>1-x</sub>Fe<sub>x</sub>)<sub>5</sub>Si<sub>3</sub> magnetocalorics

Sylvain Le Tonquesse,\* Laura Agnarelli, and Swathi Sakthivel

Laboratoire CRISMAT, CNRS UMR 6508, ENSICAEN, Normandie Universite, Caen
14050, France

E-mail: sylvain.letonquesse@cnrs.fr

Abstract

Optimizing complex synthesis processes remains a significant challenge in materials
science, particularly when high phase purity must be achieved in complex multi-variable
systems with restricted experimental throughput. Here, we demonstrate the application
of a fully experimental Bayesian Optimization (BO) framework to efficiently optimize
the synthesis conditions of (Mn₁₋ₓFeₓ)₅Si₃, a rare-earth-free magnetocaloric material.
Using magnesioreduction as the synthesis route, seven experimental parameters were
iteratively refined via active learning to maximize phase purity. Phase-pure samples,
including the challenging peritectoid-forming Fe₅Si₃, were obtained within five active
learning iterations. The approach generalizes to unseen compositions and yields chemically
meaningful insights via SHAP analysis of the trained Gaussian Process model, identifying
maximum temperature and SiO₂ excess as key drivers of phase purity. X-ray diffraction
and scanning electron microscopy confirmed high crystallinity, uniform submicron grain
size, and accurate compositional control in the optimized samples. Magnetic measurements
revealed a near-room-temperature magnetocaloric effect in ferromagnetic Mn<sub>1.25</sub>Fe<sub>3.75</sub>Si<sub>3</sub>,
with a peak −Δ*S<sub>mag</sub>* of 1.06 J kg<sup>-1</sup> K<sup>-1</sup> at 275 K under a 2 T magnetic field change.
This study establishes BO as a powerful and data-efficient framework for accelerating
the synthesis of functional materials from small experimental datasets.

Keywords

Bayesian Optimization, Magnesio reduction Synthesis, Magnetocalorics, Intermetallics

**Introduction**

Optimizing synthesis parameters is a central challenge in materials science. Achieving
high purity, appropriate density, and controlled chemical composition and microstructure
is essential for the reproducible study and tuning of material properties. Even with well-
established synthesis techniques, multiple trials are typically required to obtain materials of
sufficient quality, making the process resource- and time-intensive. The challenge intensifies
when developing unconventional or sophisticated synthesis routes, for instance to access
new metastable compounds<sup>1,2</sup>, intricate microstructures<sup>3</sup>, complex sample shapes<sup>4</sup>, large
single crystals<sup>5</sup> or crystallographically textured materials<sup>6</sup>. Such efforts often require the
simultaneous optimization of many parameters, greatly increasing the experimental workload.

Traditionally, synthesis has relied on empirical, trial-and-error approaches guided by
classical chemistry and prior experience. While often effective for low-dimensional problems,
this approach becomes impractical in systems with many variables, especially when non-
linear or non-intuitive interactions exist among synthesis parameters. In this context, machine
learning (ML) has emerged as a powerful tool for handling such complex and multidimensional
systems<sup>7</sup>. Standard ML models are trained to map input parameters (descriptors) to a target
output. The goal is to predict this output accurately for new, unseen inputs, thus enabling
navigation of the parameter space beyond human intuition. However, most ML applications
in materials science to date have focused on property prediction based on large open-access
databases⁸,⁹, not on synthesis. A first key limitation to the application of ML to material
synthesis is the need for large datasets to train predictive models reliably (ideally hundreds
of entries), which is rarely available for novel or non-standard synthesis methods. Indeed,
sophisticated models, such as neural network, require large datasets to avoid overfitting,
a situation where the model memorizes training data rather than learning generalizable
patterns. A second key limitation is reliability of the dataset. Literature-derived data can
be incomplete, inconsistently reported, or poorly reproducible, which introduces bias and
uncertainty that severely limit predictive performance. As a result, despite its promise, ML-
driven synthesis optimization remains constrained by the fundamental issues of the "data
quantity" and "data reliability" limitations.

An effective strategy to overcome "data reliability" limitations is to train models on in-house experimental datasets. This has the advantage of providing full control over data quality and reducing systematic errors caused by variations in experimental setups between different laboratories. However, the major drawback is that these datasets are typically very small, tens of entries at maximum, thus not revolving the "data quantity" issue. A promising strategy to address both limitations simultaneously is Bayesian Optimization (BO), which differs from conventional ML approaches in two key ways. First, unlike traditional ML approaches that aim to model the entire parameter space with high accuracy, BO focuses solely on identifying the most promising combination of synthesis parameters to optimize the targeted property. Second, BO operates within an active learning framework, whereby new experimental results are incorporated iteratively to refine the model and guide future experiments toward the most informative regions of the parameter space. This makes BO particularly well-suited for experimental synthesis optimization, as it allows for relatively
small datasets while continuously adapting to new experimental observations. The general
workflow of a BO-guided synthesis strategy is illustrated in Figure 1a. A synthesis protocol
is first established, and key experimental parameters along with their optimization ranges
are identified. An initial dataset is generated, where each entry consists of a set of parameter
values and its corresponding experimental outcome to maximize. A Gaussian Process (GP)
surrogate model is then trained on this dataset, conditioning a prior function to the observed
data. A key strength of GP is that they not only provide a best-fit function that describes
the experimental data but also quantify prediction uncertainty using Bayesian inference.
This uncertainty assessment is used to compute the acquisition function, which guides the
selection of the next experiment by balancing exploration, where uncertain regions of the
parameter space are investigated, and exploitation, where known high-performing conditions
are refined. The proposed synthesis condition, corresponding to the highest acquisition
score, is then tested experimentally. If the performance metric is achieved, the optimization
stops. Otherwise, the newly obtained data is added to the dataset, and the process is
repeated iteratively. This iterative active learning cycle enables efficient exploration of high-
dimensional synthesis spaces while minimizing the number of experimental trials and avoiding
overfitting thanks to the experiments feedback.

While BO has been widely adopted in computational optimization, its application to
experimental materials synthesis remains relatively underexplored. Nevertheless, recent
studies have demonstrated its potential across various material systems. For instance,
Bourgès et al. integrated BO into a machine learning pipeline to optimize sintering parameters
for enhancing the thermoelectric figure of merit ZT of sulfide materials¹⁰. In their study, a
GP model was trained primarily on synthetic data generated by a random forest regressor,
which itself had been trained on only eleven experimental data points. After several BO
iterations, the optimized conditions yielded a reported 60 % ZT improvement. A similar
approach was employed by Lambard et al. to optimize six parameters of a hot extrusion
<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Schematic workflow of the BO approach and (b) the magnesioreduction process used in this study.</figcaption></figure>

process for enhancing coercivity and remanence in Nd-Fe-B magnets¹¹. Additionally, Ishii *et al.* used BO to optimize solely the reaction temperature of BaFe₂(As,P)₂ superconductor. Their study, based on only thirteen in-house experiments, resulted in an optimized purity of 91 %¹². Other reports have used BO to optimize different sets of parameters, such as the composition of high-entropy alloys to enhance electrocatalytic properties¹³, the in-solution growth of Au nanorods¹⁴ and the sputtering condition of nitride thin films¹⁵. Despite these promising efforts, BO's full potential for optimizing experimental solid-state synthesis processes in a truly high-dimensional space remains largely unexplored. Most prior studies have relied on computationally derived datasets or focused on the optimization of a single or few experimental variables. In contrast, the present study demonstrates a fully experimental implementation of BO for the simultaneous optimization of seven synthesis parameters. We apply BO to the magnesioreduction synthesis of (Mn₁-xFe_x)₅Si₃ a rare-earth-free magnetocaloric material, leveraging its ability to systematically refine synthesis parameters and achieve phase-pure materials with minimal experimental trials. A further
distinguishing feature of the present approach is the use of different acquisition strategies at different stages of the optimization: an initial focus on exploration is followed by a final phase of exploitation. This adaptive control over the exploration-exploitation balance allows for more efficient convergence while maintaining robustness in the early stages of learning.

The magnetocaloric effect (MCE) refers to the reversible temperature change experienced by a material upon the application or removal of an external magnetic field, resulting from a change in magnetic entropy ($-\Delta S_{mag}$) associated with magnetic ordering. This phenomenon forms the basis of solid-state magnetic refrigeration, which offers an energy-efficient and environmentally friendly alternative to conventional gas-compression cooling systems¹⁶. Materials exhibiting strong MCE near room temperature are of particular interest for practical applications and have thus become the focus of extensive research aimed at developing economically viable alternatives to existing cooling technologies. State-of-the-art MCE materials, which exhibit the highest $-\Delta S_{mag}$, and are often based on rare-earth elements, particularly Gd. For example, elemental Gd exhibits a $-\Delta S_{mag}$ of 5 J kg⁻¹ K⁻¹ at 294 K under a $\Delta B$ = 2 T magnetic field change, while $Gd_5Si_2Ge_2$ reaches 14 J kg⁻¹ K⁻¹ at 272 K under the same field conditions¹⁷. Despite their excellent performance, rare-earth-based materials face significant limitations, particularly in terms of mechanical durability under repeated magnetization-demagnetization cycles, as well as the high cost and supply concerns associated with Gd. These pose a major barrier to large-scale industrial adoption, highlighting the urgent need for alternative MCE materials that combine high efficiency with economic and sustainable viability¹⁸.

The solid solution $(Mn_{1-x}Fe_x)_5Si_3$ ($0 \le x \le 1$) has been widely investigated for magnetocaloric applications, as its magnetic transition temperature can be tuned via the Fe/Mn ratio. An optimal composition near $MnFe_4Si_3$ exhibits a $-\Delta S_{mag}$ of approximately 2.3 J kg⁻¹ K⁻¹ at 294 K for a field change of 0-5 T¹⁹. While its performance remains lower than rare-earth
compounds, this system has garnered attention due to its composition of earth-abundant, low cost elements and its good resistance to oxidation, all of which are attractive properties from an industrial standpoint. However, synthesizing this material via conventional fusion-based methods presents significant challenges. The high vapor pressure of molten Mn complicates precise stoichiometric control, while the Fe-rich end of the phase diagram is particularly problematic due to a narrow stability window of about 250 K²⁰, complicating phase purity control. To overcome these challenges, this study reports the use of a magnesioreduction process to synthesize (Mn₁-xFeₓ)₅Si₃, following the reaction:

$$x \text{MnO} + (2.5 - 0.5x)\text{Fe}_2\text{O}_3 + 3\text{SiO}_2 + (10.5 - 0.5x)\text{Mg} \rightarrow (\text{Mn}_{1-x}\text{Fe}_x)_5\text{Si}_3 + (10.5 - 0.5x)\text{MgO}$$

Magnesioreduction synthesis involves the co-reduction of metal oxides using Mg as a reductant to form intermetallics (Figure 1b). This type of reaction is used in the industry, e.g. the Kroll's process for Ti and Zr production, and has recently been adapted for complex intermetallics²¹⁻²⁵. Compared to conventional melting methods, magnesioreduction offers several advantages: (i) it directly yields highly crystalline submicronic powders, (ii) its exothermicity promotes fast reactions and reduce or eliminate the need for additional annealing, (iii) it simplify the synthesis of phases containing volatile elements like Mn, and (iv) it uses stable and cost-effective oxide precursors. Despite these advantages, achieving phase-pure (Mn₁-xFeₓ)₅Si₃ via magnesioreduction requires tight control over multiple synthesis parameters. In this study, we employ BO to systematically refine seven experimental parameters and achieve single phase purity. We demonstrate that BO not only accelerates the discovery of optimal synthesis conditions but also enables a deeper understanding of the synthesis process. Beyond this specific case, our work establishes BO as a generalizable framework for the data-efficient optimization of complex solid-state synthesis processes, advancing the broader goal of accelerating materials discovery.
# Experimental part

## Magnesioreduction Synthesis

The oxide precursors used were Fe₂O₃ (99.995 %, Strem Chemicals), MnO (99 %, Alfa Aesar, +200 mesh powder), and SiO₂ (99.9 %, Alfa Aesar, average particle size 500 nm). The precursors were weighed to match the target stoichiometry of (Mn₁-xFe_x)₅Si₃ (parameters 1). An additional excess of SiO₂ (parameter 2) was added to compensate for potential silicon loss during the high-temperature reaction due to reaction with Nb crucible. Each synthesis batch was prepared from a total of 3 g of precursor powders. The precursor mixture was transferred to a 65 mL hardened steel grinding jar containing two hardened steel balls (Ø = 12.6 mm, total weight = 7 g). Mixing was performed in air using a SPEX 8000D ball-miller for a duration optimized as parameter 3. Ball-milling served to homogenize the precursor mixture and enhance interfacial contact, thereby improving the kinetics of the magnesioreduction reaction. Approximately 600 mg of the ball-milled powder was cold-pressed into a Ø = 6 mm pellet under a uniaxial pressure corresponding to parameter 4. The pellet was transferred to a Nb crucible together with Mg turnings (99.98 %, Alfa Aesar, ~1 cm length) in a glovebox maintained under an inert Ar atmosphere (H₂O and O₂ < 1 ppm). The Mg turnings were placed in direct physical contact with the oxide precursor pellet to ensure efficient heat transfer from the highly exothermic MgO formation during the reaction. An excess of 5 mol. % Mg was systematically used to ensure complete reduction of the oxides. The crucible was sealed under Ar atmosphere by arc-melting and subsequently enclosed in an evacuated silica tube to prevent its oxidation during thermal treatment. The sealed silica tube was placed in a resistive furnace and heated to the target temperature (parameter 6) at a ramp rate defined by parameter 5. The sample was held at the maximum temperature for a duration defined by parameter 7. The tube was quenched in cold water to avoid peritectoid decomposition of Fe-rich phases. The pellet, now embedded in MgO byproducts, was recovered by opening the Nb crucible. For samples confirmed to be phase-
pure, the MgO side-product was removed by leaching the crushed pellet in 0.03 N phosphoric
acid (Alfa Aesar) for 1 hour. The powder was isolated from the slurry by centrifugation,
rinsed three times with distilled water, once with ethanol, and dried at 373 K for 1 hour. The
final product usually consisted of approximately 400 mg of $(Mn_{1-x}Fe_x)_5Si_3$ powder which is
suitable for magnetic characterization.

Structural and Microstructural Characterization

Powder X-ray diffraction (XRD) patterns were collected at room temperature using a Bruker
D8 Advance diffractometer equipped with a Cu radiation source, a Ge monochromator to
isolate the Cu<sub>Kα1</sub> radiation (λ = 1.56059 Å), and a LinxEye energy-dispersive detector
to minimize Fe and Mn fluorescence. Patterns were acquired over a 2θ range of 15-120°
with a step size of 0.015°. Instrumental broadening was calibrated using a LaB<sub>6</sub> NIST
standard. Rietveld refinements were performed using the MAUD software suite, employing
the Thompson-Cox-Hastings pseudo-Voigt function for peak shape modeling. Phase quantification
of (Mn<sub>1-x</sub>Fe<sub>x</sub>)<sub>5</sub>Si<sub>3</sub> was carried out after thermal treatment via Rietveld refinement, deliberately
excluding the MgO phase. This exclusion prevented variability in the byproduct's quantity
inside the pellet. For samples determined to be phase-pure after synthesis, XRD measurements
were repeated after MgO removal to confirm its complete dissolution and to extract refined
structural parameters. Microstructural and chemical composition analysis were conducted
using a JEOL JSM-7200F field-emission scanning electron microscope (SEM) equipped
with an energy-dispersive X-ray spectroscopy (EDS) detector. Secondary electron images
were collected at an accelerating voltage of 2 kV to assess powder morphology, while EDS
quantification were performed at 15 kV.

Magnetic Measurements

Magnetic measurements were performed using a Quantum Design SQUID magnetometer.
Temperature-dependent magnetization was recorded under both zero-field-cooled (ZFC) and
field-cooled (FC) conditions. To evaluate the MCE, magnetic isotherms were measured across a narrow temperature window centered around the magnetic ordering temperature of each sample. The magnetization was recorded under increasing and decreasing magnetic fields from 0 to 5 T to capture the first and second quadrants of the hysteresis loop. These data were subsequently used to estimate the isothermal magnetic entropy change ($-\Delta S_{mag}$) via the numerical evaluation of the Maxwell relation as described in reference$^{26}$.

## Bayesian Optimization

The BO methodology was implemented in Python using the Gpytorch$^{27}$ and Botorch$^{28}$ libraries for GP regression model and acquisition functions, respectively. Part of the initial experimental conditions was selected using Latin Hypercube Sampling (LHS) via the pyDOE library to ensure a space-filling sampling of the parameter space. Each synthesis condition was represented as a 7-dimensional input vector $x$, where each component $x_D$ ($D \in \{1, ..., 7\}$) corresponds to one of the seven synthesis parameters to be optimized. The complete training dataset is encoded as a $n \times 7$ matrix $X$, where $n$ is the number of experiments, and each row is a specific synthesis condition. The associated target values, which correspond to the resulting phase purities determined by Rietveld refinement of the samples after thermal treatment but before MgO removal (Figures SI 1-5), are collected in a $n \times 1$ column vector $y$. To ensure equal weighting of all variables, each parameter was normalized to the $[0,1]$ interval using min-max scaling, based on the bounds listed in Table 1. The unknown objective function $f(x)$, which maps synthesis conditions to phase purity, is modeled as a GP:

$$f(\mathbf{x}) \sim GP(\mu_0(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')) \quad (1)$$

with prior mean function $\mu_0(\mathbf{x})$ and covariance kernel $k(\mathbf{x}, \mathbf{x}')$. An Automatic Relevance Determination Radial Basis Function (ARD-RBF) kernel was employed:
$$k(\mathbf{x}, \mathbf{x}') = \sigma^2 \exp \left( -\sum_{D=1}^{7} \frac{(x_D - x'_D)^2}{2l_D^2} \right) \quad (2)$$

where $\sigma^2$ is the output variance and $l_D$ denotes the length scale associated with the $D^{th}$ input dimension. Unlike the standard RBF kernel, the ARD-RBF kernel allows each input parameter to have an independent length scale, enabling the model to better assess the relative importance of different synthesis parameters. A constant prior mean function $\mu_0(\mathbf{x}) = c$ was adopted, corresponding to an unbiased and purely data-driven modeling approach. The model hyperparameters ($l_D$, $\sigma^2$, and $c$) were optimized by maximizing the marginal log-likelihood using the Adam optimizer with a learning rate of 0.01 over 2000 iterations.

At each iteration, the prior mean function and covariance matrix are updated based on the available dataset $\mathbf{X}$ using Bayesian inference to construct the posterior distribution. The updated covariance matrix $K(n \times n)$ is calculated as:

$$K_{ij} = k(\mathbf{x}_i, \mathbf{x}_j) + \sigma_n^2 \delta_{ij} \quad i, j \in \{1, \dots, n\} \qquad (3)$$

where a fixed noise level $\sigma_n^2 = 1 \times 10^{-4}$ was chosen to reflect the small uncertainty associated with phase purity measurements obtained from Rietveld refinement. For a new synthesis condition $\mathbf{x}^*$, the posterior mean $\mu(\mathbf{x}^*)$ and variance $\sigma^2(\mathbf{x}^*)$ are computed as:

$$\mu(\mathbf{x}^*) = k(\mathbf{x}^*, \mathbf{X}) K^{-1} \mathbf{y} \qquad (4)$$

$$\sigma^2(\mathbf{x}^*) = k(\mathbf{x}^*, \mathbf{X}) - k(\mathbf{x}^*, \mathbf{X}) K^{-1} k(\mathbf{X}, \mathbf{x}^*) \qquad (5)$$

where $k(\mathbf{x}^*, \mathbf{X})$ is the vector of covariances between the new point and each training point. These posterior estimates $\mu(\mathbf{x}^*)$ and $\sigma^2(\mathbf{x}^*)$ are used by the acquisition functions to determine
the next experimental condition to evaluate. In Figure 1a, these correspond graphically to the red mean line and the shaded blue confidence interval, respectively.

To balance exploration and exploitation during optimization, two acquisition functions were employed: (i) Expected Improvement (EI) and (ii) Probability of Improvement (PI). EI was used during early iterations (1-2) to explore under-sampled regions, while PI was used in later iterations (3-5) to refine high-purity regions. The PI acquisition function is defined as:

$$PI(\mathbf{x}^*) = \Phi(Z) \quad \text{with} \quad Z = \frac{\mu(\mathbf{x}^*) - f_{\text{best}}}{\sigma(\mathbf{x}^*)} \qquad (6)$$

where $\Phi(Z)$ is the cumulative distribution function of the standard normal distribution, $Z$ is the normalized improvement factor, and $f_{\text{best}}$ is the highest observed phase purity so far. The EI acquisition function is given by:

$$EI(\mathbf{x}^*) = (\mu(\mathbf{x}^*) - f_{\text{best}})\Phi(Z) + \sigma(\mathbf{x}^*)\phi(Z) \qquad (7)$$

where $\phi(Z)$ the standard normal probability density function given by:

$$\phi(Z) = \frac{1}{\sqrt{2\pi}} \exp \left( -\frac{Z^2}{2} \right) \qquad (8)$$

To identify the next synthesis condition, the acquisition function was maximized using a multi-restart gradient-based optimization procedure (10 restarts from 1000 initial samples) to avoid local maxima convergence. The resulting parameter vector, initially in normalized form, was converted back to physical units via inverse min-max scaling to generate interpretable experimental settings. These conditions were then evaluated experimentally, and the corresponding purity measurements were incorporated into the dataset, updating the GP model in an iterative active learning cycle. To ensure reproducibility, fixed random seeds were used for both the LHS sampling and GP hyperparameter optimization. All computations were
performed on a computer equipped with an Intel Core i5-12500H CPU and 16 GiB of RAM. The full code and experimental dataset used in this study are available at the GitHub repository accompanying this article: https://github.com/SLTCrismat/BO_Magnesioreduction_MnFeSi_Magnetocaloric.

# Results and discussion

## Magnesioreduction synthesis of $(Mn_{1-x}Fe_x)_5Si_3$

Magnesioreduction has proven to be an effective process for synthesizing intermetallic compounds directly as well-crystallized submicrometric powders. The thermodynamic driving force of these reaction is the formation of MgO following the reaction:

$$2Mg + O_2 \rightleftharpoons 2MgO \ (\Delta G_f^\circ = -1136 \text{ kJ mol}^{-1})$$

The standard Gibbs free energy of formation for MgO is significantly lower than that of most transition metal and p-block oxides. After reduction, the metals in their elemental form react via solid-state processes to eventually form $(Mn_{1-x}Fe_x)_5Si_3$. The heat released by the exothermic MgO formation contributes to increased local temperatures, thereby enhancing atomic diffusion and promoting phase formation. Despite being thermodynamically favorable, the reaction requires external heating to initiate. Depending on the synthesis temperature, markedly different reaction mechanisms can occur. For instance, in the magnesioreduction synthesis of $CoSb_3$ from $Sb_2O_4$ and $Co_3O_4$ at 810 K (below the melting point of Mg), the reaction proceeds slowly over several days via a stepwise sequence: (i) initial reduction to lower oxides, (ii) formation of intermediate ternary oxide compounds, and (iii) eventually the intermetallic formation$^{22}$. This behavior resembles an oxygen-getter process, where Mg gradually consumes the oxygen partial pressure in the sealed crucible to drive reduction to completion. Magnesioreduction synthesis performed at higher temperatures, such as the synthesis of $MnSi_{1.73}$ or $\beta$-FeSi$_2$ at 1173 K, proceeds much more rapidly, often within a few
hours²³. In that study, the reaction was found to occur in a narrow temperature range, closely associated with the melting of Mg, and was characterized as exhibiting SHS-like kinetic features. While SHS behavior is not explicitly demonstrated in the present system, this comparison helps illustrate the complex kinetic regimes that can arise in magnesioreduction reactions depending on the temperature conditions. From a purely thermodynamic point of view, both routes should yield complete oxide reduction and intermetallic formation. However, in practice, kinetic limitations such as sluggish diffusion due to stable intermediate phases formation or poor precursors mixing can hinder complete conversion and affect phase purity. Additionally, the MgO byproduct tends to coat particle surfaces during reaction, forming a barrier that further slow down diffusion. These kinetic complexities highlight that magnesioreduction reactions, though thermodynamically straightforward, demand precise control over synthesis parameters to ensure complete conversion during reasonable time. In the present system, the additional difficulty associated with forming Fe-rich ($Mn_{1-x}Fe_x)_5Si_3$ phases, due to their narrow thermodynamic stability window, further complicates optimization. As a result, conventional trial-and-error strategies become quickly impractical in such a high-dimensional parameter space and BO was employed to efficiently explore synthesis conditions and maximize phase purity.

$Mn_5Si_3$ forms congruently at 1540 K, making it the most thermally stable phase in the Mn–Si binary phase diagram. In contrast, the Fe–Si system exhibits significantly more complex phase behavior, as illustrated in Figure 2a. $Fe_5Si_3$ forms peritectoically ($2Fe_2Si + FeSi \rightleftharpoons Fe_5Si_3$) at 1364 K and decomposes ($Fe_5Si_3 \rightleftharpoons Fe_3Si + 2FeSi$) below 1098 K. The narrow stability window of $Fe_5Si_3$, combined with sluggish solid-state diffusion between the neighboring FeSi and Fe₃Si phases, makes its synthesis particularly challenging. While FeSi is reported to be diamagnetic²⁹, Fe₃Si is ferromagnetic with $T_C < 820$ K³⁰ and might therefore impact the magnetocaloric properties measurement of the sample even in small quantities, making high phase purity essential. Another experimental challenge arises from the reaction
<figure><img src="image_3.png" /><figcaption>Figure 2: (a) Partial Fe–Si phase diagram reproduced from Han *et al.*²⁰ highlighting the narrow stability range of Fe₅Si₃. (b) Rietveld-refined XRD pattern of an initial unsuccessful attempt to synthesize Fe₅Si₃ by magnesioreduction (sample 8). The experimental data (red dots), calculated pattern (black line), and difference (blue line) are shown. The ticks represent the Bragg positions of Fe₅Si₃, MgO, Fe₃Si, FeSi and α-Fe from top to bottom.</figcaption></figure>

of Si with the Nb tube at high temperatures (T > 1200 K), forming highly stable compounds such as α-Nb₅Si₃. The extent of Si depletion depends on both temperature and dwell time, often resulting in off-stoichiometry and favor the formation of Si-deficient secondary phases. Fig. 2b shows the XRD pattern from an early synthesis attempt of Fe₅Si₃, where only 23 wt. % of the target phase was obtained (excluding MgO). The dominant impurity phase was the kinetically stable FeSi. In comparison, Mn₅Si₃ was synthesized with significantly higher purity of 86 wt. % (sample 7, Figure SI 2) under the same conditions, underscoring the greater difficulty of forming Fe-rich compositions in this system. It is the lack of substantial improvement after eight experimental trials that motivates the use of BO to systematically refine the synthesis protocol. The choice of the seven synthesis parameters to optimize (Table 1) reflects the known influence of precursor stoichiometry, the ball-milling conditions and thermal treatment on the magnesioreduction synthesis outcome. The optimization ranges were as well defined based on prior knowledge of magnesioreduction reactions and the outcomes of the initial synthesis attempts. The aim was to provide the BO framework with a sufficiently broad yet experimentally tractable parameter space. For instance, the
upper bound of the maximum temperature (parameter 6) was deliberately set slightly above the thermodynamic stability window of Fe₅Si₃ to account for kinetic barriers and allow the algorithm to explore out-of-equilibrium routes. The higher bounds for precursors ball-milling time, heating rate and dwell duration were set not to exceed five days. The objective was to maximize the phase fraction of (Mn₁⁻ₓFeₓ)₅Si₃ in order to yield phase-pure samples suitable for magnetic characterization.

<table><caption>Table 1: Overview of the seven synthesis parameters optimized by BO, along with their respective ranges and the targeted maximized output</caption><thead><tr><th>No.</th><th>Synthesis parameter</th><th>Optimization range</th></tr></thead><tbody><tr><td>1</td><td>Fe stoichiometry</td><td>0 - 1</td></tr><tr><td>2</td><td>Excess of SiO<sub>2</sub></td><td>0 - 30 mol. %</td></tr><tr><td>3</td><td>Ball-milling duration</td><td>0 - 12 h</td></tr><tr><td>4</td><td>Pelletizing pressure</td><td>10 - 300 MPa</td></tr><tr><td>5</td><td>Heating rate</td><td>40 - 300 K h<sup>-1</sup></td></tr><tr><td>6</td><td>Maximal temperature</td><td>1073 - 1473 K</td></tr><tr><td>7</td><td>Dwell duration</td><td>0 - 60 h</td></tr></tbody></table>

<table><thead><tr><th>No.</th><th>Maximized target value</th><th>Range</th></tr></thead><tbody><tr><td>1</td><td>(Mn<sub>1-x</sub>Fe<sub>x</sub>)<sub>5</sub>Si<sub>3</sub> phase purity</td><td>0 - 100 wt. %</td></tr></tbody></table>

# Bayesian Optimization

## Synthesis Conditions optimization

The initial dataset consisted of eight exploratory synthesis attempts, supplemented by seven additional samples whose conditions were selected using the LHS method. This strategy was employed to achieve a more representative and space-filling coverage of the seven-dimensional parameter space, thereby avoiding a training dataset overly concentrated in a local region of the parameters space. Such bias could lead the GP model to underperform when exploring unexplored areas. By ensuring broader initial coverage of the parameter space, the BO
model was better equipped to identify quickly optimal synthesis conditions. The resulting 15 samples constituted the initial dataset, with their synthesis conditions and measured purities summarized in Table SI 1. Notably, phase-pure samples were already achieved for two Mn-rich compositions: Mn₃.₅Fe₁.₅Si₃ (sample 12) and Mn₄.₅Fe₀.₅Si₃ (sample 15). The average phase purity across the initial dataset was 54.9 wt. %, which was used to define the constant prior mean function μ(x*) of the GP model. In each BO iteration, three new synthesis conditions were proposed, each corresponding to one of three targeted compositions: Mn₅Si₃, Mn₂Fe₃Si₃, and Fe₅Si₃. These compositions were deliberately chosen to span both end members and a central composition within the solid solution range. As previously discussed, the synthesis of Fe-rich and Mn-rich compounds is governed by distinct thermodynamic and kinetic behavior necessitating most likely different optimal synthesis conditions. For each composition, the acquisition function was maximized on a “slice” of the GP model where the stoichiometry parameter (parameter 1) was fixed, and only the remaining six synthesis parameters were optimized. This strategy resembles a batch-mode BO approach with three parallel proposals per iteration. In addition to accelerating the optimization by performing syntheses in parallel, this strategy ensured that the BO model acquired feedback across the entire stoichiometry range. Without this constraint, the optimizer would likely have favored Mn-rich compositions, which already exhibited higher purities in the initial dataset due to their more accessible synthesis. Indeed, the highest purities observed in the initial dataset (iteration 0) were 82 wt. % for Mn₅Si₃, 73 wt. % for Mn₂Fe₃Si₃, and only 13 wt. % for Fe₅Si₃. Figure 3a illustrates the evolution of phase purity over the optimization iterations for each of the three fixed compositions.

In the first two optimization iterations, the EI acquisition function was employed to guide the selection of new synthesis conditions. This choice was intended to favor exploration of under-sampled regions of the parameter space that appeared promising based on the initial dataset but had not yet been tested. As shown in Figure 3a, this exploratory phase was
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3: (a) Evolution of the phase purity for Mn₅Si₃, Mn₂Fe₃Si₃, and Fe₅Si₃ compositions across the BO iterations. (b) Euclidean distance |x<sub>best</sub> − x<sup>*</sup>| between the newly proposed synthesis parameters (x<sup>*</sup>) and the current best-known conditions (x<sub>best</sub>). The vertical dotted line differentiate the optimization phase using EI and PoI acquisition functions.</figcaption></figure>

effective since after only two iterations, high phase purity was achieved for all three targeted
compositions: 100 wt. % for Mn₅Si₃ (sample 19, iteration 2), 98 wt. % for Mn₂Fe₃Si₃ (sample
20, iteration 2) and 98 wt. % for Fe₅Si₃ (sample 18, iteration 1). These results highlight the
efficiency of BO in leveraging a small, representative initial dataset to rapidly identify near-
optimal synthesis conditions. However, the exploratory nature of the EI acquisition function
also led to the selection of conditions yielding poor outcomes. For instance, Mn₅Si₃ (sample
16, iteration 1) and Fe₅Si₃ (sample 21, iteration 2) resulted in purities of only 22 wt. % and
32 wt. %, respectively. These values are much lower than the current best-known condition
giving purity of 86 wt. % (sample 7, initial dataset) and 98 wt. % (sample 18, iteration
1), respectively. This behavior is typical of acquisition functions that strongly prioritize
exploration as they may propose experiments far from previously successful conditions,
thereby increasing the risk of failure. This trend is illustrated in Figure 3b, which shows
the Euclidean distance |x<sub>best</sub> − x<sup>*</sup>| between the new proposed condition x<sup>*</sup> and the current
best-known condition x<sub>best</sub>. This distance metric, ranging between 0 and √6 ≈ 2.44 (the
diagonal length of a 6D unit hypercube), serves as a proxy for the degree of exploration. EI resulted in large distances values, close to an average of 1, corresponding to highly exploratory steps. Optimization for $Mn_5Si_3$ was halted after iteration 3, as satisfactory conditions had already been achieved. To improve convergence efficiency for the Fe-rich compositions, the acquisition function was switched to PoI starting from iteration 3. The later favors local optimization over global exploration by selecting conditions close to $\mathbf{x}_{\text{best}}$. This shift resulted in the identification of phase-pure samples for Fe-rich compositions in iteration 5 (samples 27 and 28). As expected, Figure 3b shows a clear reduction in $|\mathbf{x}_{\text{best}} - \mathbf{x}^*|$ after the transition to PoI, indicating a more localized search strategy. As a result, high-purity samples were obtained within only a few iterations for each target composition demonstrating the data efficiency of BO.

It is important to emphasize that the synthesis conditions identified in this study are not unique. Due to the complexity and high dimensionality of the parameter space, multiple combinations of experimental parameters may lead to phase-pure ($Mn_{1-x}Fe_x)_5Si_3$. The goal of the BO approach is not to exhaustively map this space, but to efficiently identify high-performing conditions for a defined target metric. In the present study, the optimization strategy focused exclusively on maximizing phase purity. However, other practical constraints, such as total reaction time or thermal budget, were not considered in the current single-objective framework. Future work will extend this approach to a multi-objective BO strategy, where phase purity is optimized simultaneously with secondary objectives such as synthesis time. This will allow identification of synthesis conditions that balance performance with process efficiency. Regarding generalization, it is important to note that the GP model within BO is not intended to offer global predictive accuracy across the entire parameter space. Rather, it is designed to be most accurate in regions near the best-performing conditions, where the model has received the most experimental feedback. To evaluate the local generalization ability of the model, a previously unseen composition, $Mn_{1.25}Fe_{3.74}Si_3$,
was selected. This composition was chosen based on prior reports of a ferromagnetic transition
near room-temperature³¹, making it of special interest for magnetocaloric applications. The
synthesis conditions proposed by the trained BO model yielded a phase purity of 99 wt.
% (sample 28, generalization validation), confirming that the model retains relatively good
predictive performance for conditions close to those already explored.

Chemical interpretation of the trained GP model

One of the advantages of the purely data-driven optimization approach employed in this
study is its ability to propose synthesis conditions that a chemist might not intuitively
consider. A notable example is the thermal treatment strategy identified by the BO model,
which consistently converged toward a heating profile characterized by a slow temperature
ramp followed by immediate quenching upon reaching the target maximum temperature.
While such a protocol may initially seem counterintuitive, it can be rationalized from a
chemical standpoint. This rapid quenching likely limits the dwell time at elevated temperatures,
thereby minimizing undesirable side reactions like the formation of α-Nb₅Si₃ from Si with the
crucible material. Meanwhile, the slow heating phase may enable the progressive formation
of the desired phase during ramp-up, prior to reaching critical decomposition temperatures.

This insight suggests that kinetic factors, such as diffusivity, possible intermediate formation
and crucible reactivity, are as critical as thermodynamic stability in governing successful
phase formation in this system. A second noteworthy outcome is the optimal maximum
synthesis temperature of 1355 K identified for Fe₅Si₃ (sample 27, iteration 5), which is
close to the reported peritectoid decomposition temperature of 1364 K. This value was
identified solely from experimental data and without incorporating any prior thermodynamic
knowledge into the model. The model autonomously inferred this thermal limit through
successive iterations. After the purity dropped to 45 wt. % (sample 25, iteration 4) after
a reaction at 1398 K, largely above the decomposition temperature, the next proposed
condition shifted toward lower temperatures, ultimately converging near the known decomposition
point. This is a good illustration of the ability of BO to uncover thermodynamic thresholds purely through adaptive learning from experimental outcomes.

<figure><img src="image_6.png" /><figcaption>Figure 4: (a) SHAP summary plot illustrating the impact of synthesis parameters on the BO's prediction of phase purity. (b) SHAP dependence plot showing the influence of the excess of SiO₂ (parameter 2) on model predictions.</figcaption></figure>

While the principal advantage of data-driven optimization is its ability to reduce experimental workload, a greater scientific utility would lie in revealing interpretable patterns that shed light on the underlying chemistry of the system of interest. GP models are non-parametric and do not yield explicit coefficients tied to individual synthesis variables, making them inherently less interpretable than, for example, linear regression or decision-tree-based models. However, interpretability tools such as SHAP (SHapley Additive exPlanations) can be applied to GP models to assess the relative importance and directional impact of each synthesis parameter on the model's predictions. SHAP works by evaluating how the model's posterior mean varies across subsets of input features, allowing for both global interpretability (which parameters are most influential overall) and local interpretability (how specific values affect individual predictions). As the GP model is most reliable near well-sampled regions, SHAP-based interpretations in sparsely explored areas should therefore be treated with caution due to increased uncertainty. Figure 4a presents the SHAP summary plot for the seven optimized parameters. Each point corresponds to a sample in the dataset, with pink
and blue color indicating high and low parameters values, respectively. Fe stoichiometry (parameter 1) and maximum synthesis temperature (parameter 6) exhibit the broadest SHAP distributions, indicating their dominant influence on predicted purity. This aligns with Fe-rich compositions being more challenging to synthesis single phased due to their narrow thermal stability window, and temperature plays a critical role in avoiding their decomposition. Dwell duration (parameter 7) and heating rate (parameter 5) show moderate influence. In both cases, higher parameter values are frequently associated with negative SHAP contributions, suggesting that excessive dwell time or overly aggressive heating hinder phase formation as discussed above. In contrast, pelletizing pressure (parameter 4) and ball-milling duration (parameter 3) have relatively minor influence, indicating a limited role under the explored conditions. The excess of SiO₂ (parameter 2) emerges as one of the most influential parameters. As shown in Figure 4b, its SHAP value follows a concave trend, revealing an optimal value near 15 mol. %. This reflects the need to compensate for silicon loss to the crucible during high-temperature reactions. At low excess, formation of Fe₃Si secondary phases is commonly observed (sample 1 with 0 mol. % excess, Figure SI 1), while at high SiO₂ excess tends to lead to Si-rich phase like FeSi (sample 11 with 30 mol. % excess, Figure SI 2). The absence of correlation between the optimal SiO₂ excess and the Mn-to-Fe ratio is chemically expected, as the required excess is primarily governed by crucible interactions rather than by the nominal stoichiometry of the target phase.

To further investigate whether the influence of synthesis parameters varies across the composition range, we performed separate SHAP analyses on Fe-rich (Fe > 75 mol. %) and Mn-rich (Fe < 25 mol. %) subsets (Figure SI 6). While limited by small sample sizes, the results show a stronger impact of maximum temperature and SiO₂ excess in the Fe-rich regime, consistent with the narrow stability window of Fe₅Si₃. In contrast, Mn-rich compositions exhibit milder parameter sensitivity overall, though the relative importance ranking of descriptors remains broadly similar across both regimes. These SHAP-derived
insights not only validate experimentally observed trends but also offer a quantitative basis for refining future optimization strategies. In particular, they can support more informed descriptor selection in subsequent experiments on this system.

## Materials characterizations

### Structural and micro-structural analyses

<figure><img src="image_7.png" /><figcaption>Figure 5: Rietveld-refined XRD patterns of (a) Fe<sub>5</sub>Si<sub>3</sub> (sample 27), (b) Mn<sub>1.25</sub>Fe<sub>3.75</sub>Si<sub>3</sub> (sample 28) (c) Mn<sub>2</sub>Fe<sub>3</sub>Si<sub>3</sub> (sample 26) and (d) Mn<sub>5</sub>Si<sub>3</sub> (sample 19). The experimental data are plotted in red symbols, the calculated pattern with a black line, and the difference with a blue line. The black and green vertical ticks indicate the theoretical Bragg positions of the Mn<sub>5</sub>Si<sub>3</sub>-type phase and the Fe<sub>3</sub>Si impurity phase, respectively.</figcaption></figure>

The purest samples obtained after BO were selected for detailed structural characterization.

The Rietveld-refined powder XRD patterns of Fe<sub>5</sub>Si<sub>3</sub> (sample 27, iteration 5), Mn<sub>1.25</sub>Fe<sub>3.75</sub>Si<sub>3</sub>
(sample 28), Mn₂Fe₃Si₃ (sample 26, iteration 5) and Mn₅Si₃ (sample 19, iteration 2), after MgO removal, are shown in Figure 5. All samples exhibit sharp diffraction peaks, indicating a high degree of crystallinity. All the observable diffraction peaks can be indexed by the Mn₅Si₃ structure-type (P6₃/mcm, space group no. 193), with the exception of Mn₁.₂₅Fe₃.₇₅Si₃ showing minor Fe₃Si impurity (1 wt. %). The crystal structure of (Mn₁-xFeₓ)₅Si₃, shown in Figure 6a, includes two non-equivalent transition metal (TM) sites. TM atoms occupying the 4d Wickoff site are coordinated by distorted Si octahedra that share faces along the *c* axis, forming infinite chains ∞[TMSi₃]. These chains are connected via edge-sharing to analogues neighboring chains, generating a three-dimensional framework with channels running parallel to the *c* axis. The TM atoms on the 6g Wyckoff site also form infinite face-sharing chains of infinite chains of face-sharing octahedra along the *c* axis that are located inside the channel of the 3D framework. Neutron diffraction experiments have shown that Fe preferentially occupies the 4d site, while Mn favors the 6g site in (Mn₁-xFeₓ)₅Si₃³². Due to the similar X-ray scattering factors of Fe and Mn, site occupancies could not be refined independently in this study. Instead, atomic positions were modeled based on the nominal compositions, with Fe and Mn statistically distributed across both TM sites. The structural parameters obtained from Rietveld refinement of the XRD pattern are shown in Table SI 2 and Figure SI 7. The refined atomic positions and isotropic displacement parameters were consistent across the series. The refined lattice parameters are in good agreement with those reported for arc-melted samples of comparable composition³³. Notably, the individual lattice constants *a* and *c* deviate from Vegard's law and yield a non-monotonic *c/*(*a* ratio, which reaches a minimum around *x* ≈ 0.5. However, the unit cell volumes, shown in Figure 5b, follow Vegard's law closely, with the BO samples (marked as black stars) aligning well with the linear trend established by V. Johnson *et al*.³³. Similar unit cell volumes are found for comparable chemical composition, 187.40 Å³ (+0.4 %), 190.35 Å³ (+0.4 %), 192.44 Å³ (+0.4 %) and 199.11 Å³ (-0.1 %) for *x* = 1, 0.75, 0.6 and 0, respectively. Larger positive deviation from the Vegard's law are observed for sample with lower purity, which can be attributed to
heterogeneous repartition of the Mn and Fe between the main phase and the impurities.

<figure><img src="image_8.png" /><figcaption>Figure 6: (a) Crystal structure of Mn₅Si₃ shown along the *c*-axis, (b) Refined unit cell volume of the 28 samples used for BO according to the nominal compositions. The indicated purity exclude the side-product MgO.</figcaption></figure>

Secondary electron images of the BO-optimized samples are shown in Figure 7, alongside their corresponding chemical compositions obtained by EDS. All powders exhibit relatively uniform particle size distributions. The average particle size decreases from approximately 1 µm for Mn₅Si₃ to 500 nm for Fe-rich samples. The larger grain size observed for Mn₅Si₃ is likely related to its distinct synthesis conditions. Notably, the ball-milling duration for sample 19 was only 46 minutes, compared to 720 minutes for the Fe-rich samples. This reduced mechanical activation may have favored larger grain growth during the thermal treatment. Additionally, the broader thermodynamic stability window of Mn₅Si₃, as inferred from the phase diagram, may contribute to enhanced coarsening kinetics during thermal treatment. Such microstructures, characterized by submicronic grain size with narrow distribution, are commonly observed in transition metal silicides synthesized via magnesioreduction.

Similar morphologies have been reported for MnSi₁.73²³ or β-FeSi₂²⁴ prepared under comparable conditions. EDS analysis performed over large sample areas confirms chemical compositions closely matching the nominal compositions. No residual MgO signals were detected in the EDS spectra, indicating that the acid leaching step effectively removed the MgO byproduct.
<figure><img src="image_9.png" /><figcaption>Figure 7: SEM secondary electron images showing the morphology of the powder of (a) Fe<sub>5</sub>Si<sub>3</sub> (sample 27), (b) Mn<sub>1.25</sub>Fe<sub>3.75</sub>Si<sub>3</sub> (sample 28) (c) Mn<sub>2</sub>Fe<sub>3</sub>Si<sub>3</sub> (sample 26) and (d) Mn<sub>5</sub>Si<sub>3</sub> (sample 19) after MgO removal. The EDS chemical compositions is indicated on the top right-side for each samples.</figcaption></figure>

Combined with the refined unit cell's volume parameters from XRD, these results demonstrate that the magnesioreduction process enables the synthesis of high-purity and compositionally well-controlled (Mn<sub>1-x</sub>Fe<sub>x</sub>)<sub>5</sub>Si<sub>3</sub> phases that are of satisfying quality for accurate magnetocaloric characterization.

## Magnetic and magnetocaloric properties

The thermal evolution of magnetization for the BO-optimized samples, including both ZFC and FC measurements, is shown in Figure 8a. The (Mn<sub>1-x</sub>Fe<sub>x</sub>)<sub>5</sub>Si<sub>3</sub> exhibits a monotonic increase in magnetic ordering temperature with increasing Fe content, accompanied by a progressive change in the nature of magnetic ordering. For Mn<sub>5</sub>Si<sub>3</sub> and Mn<sub>2</sub>Fe<sub>3</sub>Si<sub>3</sub>, two distinct magnetic transitions are observed. A paramagnetic to collinear antiferromagnetic transition at T<sub>N1</sub> = 94 K and 214 K, respectively, followed by a second transition to a non-collinear antiferromagnetic state at T<sub>N2</sub> = 69 K and 102 K. According to Ait Haddouch
et al.³², these transitions in Mn₂Fe₃Si₃ are associated with structural symmetry reductions from hexagonal to orthorhombic at T<sub>N1</sub>, and further to monoclinic at T<sub>N2</sub>. The transition temperatures obtained here for Mn₅Si₃ are consistent with literature values, while those for Mn₂Fe₃Si₃ remain less well established, with prior reports showing inconsistent results and limited insight into the underlying mechanisms¹⁹,³¹. In addition to the main magnetic transitions, a weak anomaly labeled T* is detected near 30 K in both Mn-rich compounds, and near 333 K for Mn₂Fe₃Si₃. Further investigation is needed to clarify the origin of these anomalies, which could arise from residual impurity phases that are below the detection limit of XRD. With increasing Fe concentration, the system transitions to ferromagnetic ordering. Mn₁.₂₅Fe₄.₂₅Si₃ and Fe₅Si₃ exhibit second-order ferromagnetic transitions at 271 K and 335 K, respectively. These results are in good agreement with earlier work by Candini et al.³¹, who reported ferromagnetism from x ≈ 0.7 in this series.

<figure><img src="image_10.png" /><figcaption>Figure 8: (a) Magnetization versus temperature measurement for the BO optimized (Mn<sub>1-x</sub>Fe<sub>x</sub>)<sub>5</sub>Si<sub>3</sub> sample (x = 0, 0.6, 0.75 and 1). Orange spheres and blue diamonds represent ZFC and FC curves, respectively. (b) Magnetic entropy changes (−ΔS<sub>mag</sub>) measurement between 0 and 2 T.</figcaption></figure>

The magnetic isotherms of Fe₅Si₃ and Mn₁.₂₅Fe₄.₂₅Si₃ (Figure SI 8) confirm their ferromagnetic nature, with sharp low-field magnetization rises and saturation tendencies at higher fields.
The antiferromagnetic Mn₅Si₃ compound manifests quasi-linear isotherms and around T<sub>N1</sub>, the Arrott plot (Figure SI 9) portrays s-shaped curves indicating a first-order phase transition. Mn₂Fe₃Si₃ shows more complex behavior. Although S-shaped Arrott plots are also present near T<sub>N1</sub>, the field-dependent magnetization curves above 100 K begin to deviate from linearity and display partial saturation. This behavior is indicative of a metamagnetic transition and supports the presence of field-induced spin reorientation in this intermediate composition. To quantify the MCE in these compounds, we have estimated the isothermal entropy change (−ΔS<sub>mag</sub>) from the isothermal magnetization measurements using the expression:

$$ \Delta S_{mag} = \int_{0}^{B} \left( \frac{\partial M}{\partial T} \right)_{B} dB \qquad (9) $$

The calculated −ΔS<sub>mag</sub>(T) curves are shown in Figure 8b and summarized in Table SI 3 for the (Mn<sub>1-x</sub>Fe<sub>x</sub>)₅Si₃ series. Notably, the entropy change curves obtained from magnetization and demagnetization branches exhibit minimal hysteresis, demonstrating excellent reversibility across the series. Among all compositions, Fe₅Si₃ exhibits the highest entropy change, with −ΔS<sub>mag</sub> values of 1.30 and 2.70 J kg<sup>-1</sup> K<sup>-1</sup> for field changes of 2 T and 5 T, respectively. This result contrasts with previous reports, where MnFe₄Si₃ was identified as the peak performer with −ΔS<sub>mag</sub> = 2.0 J kg<sup>-1</sup> K<sup>-1</sup> at 305 K under ΔB = 2 T<sup>19,34</sup>. The slightly lower value measured here for Mn<sub>1.25</sub>Fe<sub>3.75</sub>Si₃ may be attributed to the presence of a minor Fe₃Si impurity phase (T<sub>C</sub> ≈ 820 K)<sup>35</sup>, as detected by Rietveld refinement. If the magnetic moments of the impurity and main phase are not aligned, the overall entropy change could be suppressed due to spin misalignment effects. Moreover, differences in synthesis route and grain size must also be considered: compared to arc-melted bulk samples, the powders synthesized here by magnesioreduction exhibit submicron grain sizes (≈ 500 nm), which are known to reduce MCE due to surface effects, strain, and magnetic domain refinement<sup>36,37</sup>. Significant variations in MCE between synthesis methods have also been reported, with differences exceeding 10 J kg<sup>-1</sup> K<sup>-1</sup> in other systems such as ErAl₂<sup>38</sup>. Interestingly, Mn₅Si₃ exhibits a pronounced inverse MCE near T<sub>N1</sub>, reaching −ΔS<sub>mag</sub> = 2.86 J kg<sup>-1</sup> K<sup>-1</sup> at 5 T. This peak
likely arises from the first-order magnetic and structural transition associated with symmetry breaking, as discussed earlier.

Overall, the $(Mn_{1-x}Fe_x)_5Si_3$ series demonstrates moderate magnetocaloric performance compared to leading Mn-based systems such as Mn-Fe-P-Si, Ni-Mn-In, and $Mn_5Ge_3$$^{39-41}$. Nevertheless, these materials offer attractive advantages, including the use of earth-abundant, low-cost elements and chemically tunable transition temperatures, which are crucial attributes for scalable and sustainable magnetic refrigeration technologies. Importantly, there remains significant room for further improvement. In particular, targeted compositional modifications, such as the insertion of light elements (e.g., H, C, B) into the structural channels, could enhance magnetic entropy changes via magnetoelastic coupling$^{42}$. Additionally, exploiting the strong crystallographic anisotropy of this system in textured polycrystalline samples may open pathways to harnessing the rotating MCE$^{43}$. The BO-optimized magnesioreduction synthesis developed in this work provides a robust and flexible platform for pursuing both of these optimization strategies in future studies.

## Conclusion

This study demonstrates the successful application of a fully experimental BO strategy to the high-dimensional problem of synthesis parameter optimization. Using the magnesioreduction synthesis of the magnetocaloric $(Mn_{1-x}Fe_x)_5Si_3$ system as a case study, we show that BO trained exclusively on experimentally acquired data can efficiently identify synthesis conditions yielding phase-pure materials within a limited number of active learning iterations. Seven experimental parameters were optimized simultaneously, illustrating the power of BO in navigating synthesis spaces that are otherwise prohibitively large for empirical optimization approaches. Notably, phase-pure samples of the challenging peritectoid-forming $Fe_5Si_3$ was obtained within only five BO iterations while the more accessible congruently melting $Mn_5Si_3$
composition required just two iterations. A batch optimization strategy, using parallel
optimization of three fixed composition, was used to accelerate the optimization and ensure
that the BO model received feedback across the whole solid solution. When applied to
an unseen intermediate composition, Mn<sub>1.25</sub>Fe<sub>3.75</sub>Si<sub>3</sub>, the trained model predicted synthesis
parameters that yielded 99 wt. % phase purity without retraining, validating its transferability.
Model interpretability, accessed via SHAP analysis, revealed chemically meaningful insights,
identifying the Fe stoichiometry, the maximum reaction temperature and the SiO<sub>2</sub> precursor
excess as the dominant factors controlling phase formation. This underscores BO's utility
not only as an optimization tool but also as a data-driven framework for revealing chemical
understanding of complex syntheses. Future work will extend the BO framework to multi-
objective optimization, incorporating factors such as total synthesis time, thermal budget, or
cost, to identify synthesis conditions that balance phase purity with industrial applicability.

Magnetic measurements revealed a systematic evolution from antiferromagnetic to ferromagnetic ordering with increasing Fe content, consistent with literature. Importantly, near room-temperature ferromagnetism was achieved in Mn<sub>1.25</sub>Fe<sub>3.75</sub>Si<sub>3</sub>, with magnetocaloric properties reaching -Δ*S<sub>mag</sub>* of 1.06 J kg<sup>-1</sup> K<sup>-1</sup> at 275 K for a field change of 2T. While the current MCE performance remains modest compared to rare-earth-based benchmarks, the demonstrated synthesis control, compositional tunability, and use of earth-abundant elements position (Mn<sub>1-x</sub>Fe<sub>x</sub>)<sub>5</sub>Si<sub>3</sub> as a promising rare-earth-free platform for further development. In particular, future improvements may arise from compositional modifications, such as the insertion of light elements into the structural channels, or from exploiting the strong anisotropy of this system in textured bulk samples. The BO-guided synthesis strategy developed here offers a robust foundation for such future optimization efforts.
Acknowledgements

The authors acknowledge the financial support of the « French Agence Nationale de la Recherche LabEx EMC3 » (ANR-10-LABX-0009) through the project *MagMatMa* and the Normandy Region « Normandie Recherche - Labels d'excellence » through the project *MagRota* (Grant n°23E06217).

Supporting Information Available

* Table summarizing the experimental conditions used to synthesize the 29 samples of this study

* Rietveld refined crystallographic parameters for the BO optimized (Mn$_{1-x}$Fe$_x$)$_5$Si$_3$ ($x$ = 0, 0.6, 0.75 and 1)

* Rietveld refined XRD pattern of all the samples used for BO optimization

* Evolution of the Rietveld refined lattice parameters with the Fe stoichiometry in BO optimized samples (Mn$_{1-x}$Fe$_x$)$_5$Si$_3$ ($x$ = 0, 0.6, 0.75 and 1)

* Isothermal magnetization measurements for the BO optimized samples (Mn$_{1-x}$Fe$_x$)$_5$Si$_3$ ($x$ = 0, 0.6, 0.75 and 1)

* Arrott plots for the BO optimized samples (Mn$_{1-x}$Fe$_x$)$_5$Si$_3$ ($x$ = 0, 0.6, 0.75 and 1)

* Summary of the temperature and maximum values of the magnetic entropy change for the BO optimized samples (Mn$_{1-x}$Fe$_x$)$_5$Si$_3$ ($x$ = 0, 0.6, 0.75 and 1) at 2 and 5 T.
References

(1) Yilmaz, T.; Yildiz, O. G.; Peighambardoust, N. S.; Baitinger, M.; Aydemir, U. One-Step Synthesis of Metastable Mo₂AlB₂ from MoAlB Using Gaseous HCl. *Inorg. Chem.* **2025**, *64*, 1139–1145.

(2) Cui, X.; Liu, Y.; Chen, Y. Ultrafast Micro/Nano-Manufacturing of Metastable Materials for Energy. *Natl Sci Rev* **2024**, *11*, nwa033.

(3) Kleinhanns, T.; Milillo, F.; Calcabrini, M.; Fiedler, C.; Horta, S.; Balazs, D.; Strumolo, M. J.; Hasler, R.; Llorca, J.; Tkadletz, M.; Brutchey, R. L.; Ibáñez, M. A Route to High Thermoelectric Performance: Solution-Based Control of Microstructure and Composition in Ag₂Se. *Adv. Energy Mater.* **2024**, *14*, 2400408.

(4) Thimont, Y.; Coderch, A. M.; Chevallier, G.; Duployer, B.; Galodé, A.; Gascoin, F.; Berthebaud, D.; Presmanes, L. Elaboration of *p*-type Ge doped MnSi$_{y}$ ($1.73 < y < 1.77$) Thermoelectric Legs with Complex Shapes by Binder Jetting Additive Manufacturing Technique. *Acta Mater.* **2025**, *282*, 120466.

(5) Becker, P.; Niewa, R. Progress in Ammonothermal Crystal Growth of Indium Nitride. *J. Cryst. Growth* **2022**, *581*, 126480.

(6) Le Tonquesse, S.; Zhang, W.; Srinivasan, B.; Fontaine, B.; Hiroto, T.; Mori, T.; Halet, J.-F.; Berthebaud, D.; Suzuki, T. S. Improvement of Thermoelectric Properties via Texturation Using a Magnetic Slip Casting Process–The Illustrative Case of CrSi₂. *Chem. Mater.* **2022**, *34*, 1143–1156.

(7) Manna, S.; Paul, P.; Manna, S. S.; Das, S.; Pathak, B. Utilizing Machine Learning to Advance Battery Materials Design: Challenges and Prospects. *Chemistry of Materials* **2025**, *37*, 1759–1787.
(8) Pöhls, J.-H.; Lo, C.-W. T.; MacIver, M.; Tseng, Y.-C.; Mozharivskyj, Y. Driving Thermoelectric Optimization in AgSbTe₂ via Design of Experiments and Machine Learning. *Chem. Mater.* **2025**, *37*, 2281–2289.

(9) Inada, Y.; Fujioka, M.; Morito, H.; Morito, H.; Sugahara, T.; Yamane, H.; Katsura, Y. Elemental Reactivity Maps for Materials Discovery. *Chem. Mater.* **2025**, *37*, 2097–2105.

(10) Bourgès, C.; Lambard, G.; Sato, N.; Tachibana, M.; Ishii, S.; Mori, T. Process Optimization on Kesterite-based Ceramics for Enhancing their Thermoelectric Performances Assisted by Active Machine Learning Approach: A Tool for Metal-sulfide Ceramics Development. *Acta Mater.* **2024**, *281*, 120342.

(11) Lambard, G.; Sasaki, T.; Sodeyama, K.; Ohkubo, T.; Hono, K. Optimization of Direct Extrusion Process for Nd-Fe-B Magnets using Active Learning Assisted by Machine Learning and Bayesian Optimization. *Scripta Mater.* **2022**, *209*, 114341.

(12) Ishii, A.; Kikuchi, S.; Yamanaka, A.; Yamamoto, A. Application of Bayesian Optimization to the Synthesis Process of BaFe2(As,P)2 Polycrystalline Bulk Superconducting Materials. *J. Alloys Compd.* **2023**, *966*, 171613.

(13) Mints, V. A.; Pedersen, J. K.; Bagger, A.; Quinson, J.; Anker, A. S.; Jensen, K. M. O.; Rossmeisl, J.; Arenz, M. Exploring the Composition Space of High-Entropy Alloy Nanoparticles for the Electrocatalytic H₂ /CO Oxidation with Bayesian Optimization. *ACS Catal.* **2022**, *12*, 11263–11271.

(14) Rao, A.; Grzelczak, M. Revisiting El-Sayed Synthesis: Bayesian Optimization for Revealing New Insights during the Growth of Gold Nanorods. *Chem. Mater.* **2024**, *36*, 2577–2587.

(15) Fébba, D. M.; Talley, K. R.; Johnson, K.; Schaefer, S.; Bauers, S. R.; Mangum, J. S.; Smaha, R. W.; Zakutayev, A. Autonomous Sputter Synthesis of Thin Film Nitrides
with Composition Controlled by Bayesian Optimization of Optical Plasma Emission.
*APL Materials* **2023**, *11*, 071119.

(16) Kitanovski, A. Energy Applications of Magnetocaloric Materials. *Adv. Energy Mater.* **2020**, *10*, 1903741.

(17) Pecharsky, V. K.; Gschneidner, K. A., Jr. Giant Magnetocaloric Effect in $Gd_5(Si_2Ge_2)$. *Phys. Rev. Lett.* **1997**, *78*, 4494–4497.

(18) Zarkevich, N. A.; Zverev, V. I. Viable Materials with a Giant Magnetocaloric Effect. *Crystals* **2020**, *10*, 815.

(19) Songlin, D.; Tegus, O.; Brück, E.; Klaasse, J.; de Boer, F.; Buschow, K. Magnetic Phase Transition and Magnetocaloric Effect in $Mn_{5-x}Fe_xSi_3$. *J. Alloys Compd.* **2002**, *334*, 249–252.

(20) Han, K.; Saito, M.; Xia, J.; Ohnuma, I.; Kainuma, R. Experimental Determination of Phase Diagram Involving Silicides in the Fe-Si Binary System. *J. Alloys Compd.* **2022**, *919*, 165810.

(21) Le Tonquesse, S.; Alleno, E.; Demange, V.; Dorcet, V.; Joanny, L.; Prestipino, C.; Rouleau, O.; Pasturel, M. Innovative Synthesis of Mesostructured $CoSb_3$-based Skutterudites by Magnesioreduction. *J. Alloys Compd.* **2019**, *796*, 176–184.

(22) Le Tonquesse, S.; Alleno, E.; Demange, V.; Prestipino, C.; Rouleau, O.; Pasturel, M. Reaction Mechanism and Thermoelectric Properties of $In_{0.22}Co_4Sb_{12}$ Prepared by Magnesiothermy. *Mater. Today Chem.* **2020**, *16*, 100223.

(23) Le Tonquesse, S.; Verastegui, Z.; Huynh, H.; Dorcet, V.; Guo, Q.; Demange, V.; Prestipino, C.; Berthebaud, D.; Mori, T.; Pasturel, M. Magnesioreduction Synthesis of Co-Doped $\beta$-FeSi$_2$: Mechanism, Microstructure, and Improved Thermoelectric Properties. *ACS Appl. Energy Mater.* **2019**, *2*, 8525–8534.
(24) Le Tonquesse, S.; Joanny, L.; Guo, Q.; Elkaim, E.; Demange, V.; Berthebaud, D.; Mori, T.; Pasturel, M.; Prestipino, C. Influence of Stoichiometry and Aging at Operating Temperature on Thermoelectric Higher Manganese Silicides. *Chem. Mater.* **2020**, *32*, 10601–10609.

(25) Le Tonquesse, S.; Dorcet, V.; Joanny, L.; Demange, V.; Prestipino, C.; Guo, Q.; Berthebaud, D.; Mori, T.; Pasturel, M. Mesostructure - Thermoelectric Properties Relationships in $V_xMn_{1-x}Si_{1.74}$ ($x$ = 0, 0.04) Higher Manganese Silicides Prepared by Magnesiothermy. *J Alloys Compd.* **2019**, 152577.

(26) Neves Bez, H.; Yibole, H.; Pathak, A.; Mudryk, Y.; Pecharsky, V. Best Practices in Evaluation of the Magnetocaloric Effect from Bulk Magnetization Measurements. *J. Magn. Magn. Mater.* **2018**, *458*, 301–309.

(27) Gardner, J. R.; Pleiss, G.; Bindel, D.; Weinberger, K. Q.; Wilson, A. G. GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration. 2021; https://arxiv.org/abs/1809.11165.

(28) Balandat, M.; Karrer, B.; Jiang, D. R.; Daulton, S.; Letham, B.; Wilson, A. G.; Bakshy, E. BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization. Advances in Neural Information Processing Systems 33. 2020.

(29) Mihalik, M.; Timko, M.; Samuely, P.; Tomašovičova-Hudákova, N.; Szabó, P.; Menovsky, A. Magnetic Properties and Gap Formation in FeSi. *J. Magn. Magn. Mater.* **1996**, *157-158*, 637–638.

(30) Ruiz, D.; Ros-Yanez, T.; Ortega, L.; Sastre, L.; Vandenbossche, L.; Legendre, B.; Dupre, L.; Vandenberghe, R.; Houbaert, Y. Effect of atomic order on the electrical and magnetic properties of Fe$_{100-x}$Si$_x$ ($6 < x < 14$) alloys. *IEEE Trans. Magn.* **2005**, *41*, 3286–3288.
(31) Candini, A.; Moze, O.; Kockelmann, W.; Cadogan, J.; Brück, E.; Tegus, O. Revised Magnetic Phase Diagram for Fe<sub>x</sub>Mn<sub>5-x</sub>Si<sub>3</sub> Intermetallics. *J. Appl. Phys.* **2004**, *95*, 6819–6821.

(32) Ait Haddouch, M.; Abboushi, N.; Sharma, N.; Eich, A.; Grzechnik, A.; Li, C.; Tolkiehn, M.; Alsamamra, H.; Voigt, J.; Friese, K. Site Dependence of the Magnetocaloric Effect in Mn<sub>x</sub>Fe<sub>5-x</sub>Si<sub>3</sub>. *J. Appl. Crystallogr.* **2022**, *55*, 1164–1172.

(33) Johnson, V.; Weiner, J. F.; Frederick, C. G.; Rogers, D. B. Magnetic and Mössbauer Effect Studies of Mn<sub>5</sub>Si<sub>3</sub>:Fe<sub>5</sub>Si<sub>3</sub> Solid Solutions. *J. Solid State Chem.* **1972**, *4*, 311–323.

(34) Hering, P.; Friese, K.; Voigt, J.; Persson, J.; Aliouane, N.; Grzechnik, A.; Senyshyn, A.; Brückel, T. Structure, Magnetism, and the Magnetocaloric Effect of MnFe<sub>4</sub>Si<sub>3</sub> Single Crystals and Powder Samples. *Chem. Mater.* **2015**, *27*, 7128–7136.

(35) Shinjo, T.; Nakamura, Y.; Shikazono, N. Magnetic Study of Fe<sub>3</sub>Si and Fe<sub>5</sub>Si<sub>3</sub> by Mössbauer Effect. *J. Physical Soc. Japan* **1963**, *18*, 797–801.

(36) Quintero, M.; Passanante, S.; Irurzun, I.; Goijman, D.; Polla, G. Grain size modification in the magnetocaloric and non-magnetocaloric transitions in La<sub>0.5</sub>Ca<sub>0.5</sub>MnO<sub>3</sub> probed by direct and indirect methods. *Appl. Phys. Lett.* **2014**, *105*, 152411.

(37) Ruan, M.; Yang, C.; Wang, L.; Jin, P.; Guo, Z.; Wei, X.; Wu, W. Size-dependent magnetocaloric effect in GdVO<sub>4</sub> nanoparticles. *J. Alloy Compd.* **2022**, *894*, 162351.

(38) Prusty, M. M.; Chelvane, J. A.; Morozkin, A. V.; Gururaj, K.; Pradeep, K. G.; Paulose, P. L.; Nirmala, R. Magnetocaloric effect in melt-spun rare earth intermetallic compound ErAl<sub>2</sub>. *AIP Advances* **2022**, *12*, 035145.

(39) Kang, K.; Lee, A.-Y.; Cha, H.; Jang, J.; Ahn, C.-W.; Choi, J.-J.; Hahn, B.-D.; Yoon, W.-H.; Choi, J.-H.; Kim, J.-W. Magnetoelastic Coupling Induced Magnetocaloric
Property Variations in Ni-doped (Mn, Fe)₂(P, Si) alloys. *J. Alloys Compd.* **2022**, *927*,
167008.

(40) Sang, Y.; Bao, L.; Zhang, H.; Ding, Y.; Gao, J.; Zhao, D.; Zhang, H.; Bai, Y.; Li, Z. The Structural, Magnetic and Magnetocaloric Properties of Mn₅₋ₓCrₓGe₃ Alloys. *J. Alloys Compd.* **2025**, *1019*, 179298.

(41) Nehan, P. Z. Z.; Baharum, A.; Samsudin, A. R.; Ghazali, M. I.; Zulkefly, K.; Salleh, M. M. The Magnetocaloric Effect Properties for Potential Applications of Magnetic Refrigerator Technology: A Review. *Phys. Chem. Chem. Phys.* **2024**, *25*, 1–10.

(42) Corbett, J. D.; Garcia, E.; Guloy, A. M.; Hurng, W.-M.; Kwon, Y.-U.; Leon-Escamilla, E. A. Widespread Interstitial Chemistry of Mn 5 Si 3 -Type and Related Phases. Hidden Impurities and Opportunities. *Chem. Mater.* **1998**, *10*, 2824–2836.

(43) Yibole, H.; Hanggai, W.; Ou, Z.; Hamane, R.; Hardy, V.; Guillou, F. Magnetic Properties, Anisotropy Parameters and Magnetocaloric Effect of Flux Grown MnFe₄Si₃ Single Crystal. *J. Magn. Magn. Mater.* **2020**, *504*, 166597.
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