Autonomous and dynamic precursor selection for solid-state materials synthesis

Solid-state synthesis plays an important role in the development of new materials and technologies. While in situ characterization and ab-initio computations have advanced our understanding of materials synthesis, experiments targeting new compounds often still require many different precursors and conditions to be tested. Here we introduce an algorithm (ARROWS3) designed to automate the selection of optimal precursors for solid-state materials synthesis. This algorithm actively learns from experimental outcomes to determine which precursors lead to unfavorable reactions that form highly stable intermediates, preventing the target material’s formation. Based on this information, ARROWS3 proposes new experiments using precursors it predicts to avoid such intermediates, thereby retaining a larger thermodynamic driving force to form the target. We validate this approach on three experimental datasets, containing results from over 200 synthesis procedures. In comparison to black-box optimization, ARROWS3 identifies effective precursor sets for each target while requiring substantially fewer experimental iterations. These findings highlight the importance of domain knowledge in optimization algorithms for materials synthesis, which are critical for the development of fully autonomous research platforms.


Introduction
Conventional high temperature synthesis based on solid-state reactions has long been used for the preparation of inorganic materials 1 .This method involves the mixing and subsequent heating of solid powders to facilitate reactions between them.Despite its apparent simplicity, the outcomes of solid-state synthesis experiments are often difficult to predict 2,3 .While density functional theory (DFT) calculations can be used to assess thermodynamic stability 4 , even materials that are stable can sometimes be difficult to synthesize owing to the formation of inert byproducts that compete with the target and reduce its yield [5][6][7][8] .Further complicating matters is the prevalence of metastable materials 9 used in countless technologies including photovoltaics 10 and structural alloys 11 .
Metastable materials are typically prepared using low-temperature synthesis routes, where kinetic control can be used to avoid the formation of equilibrium phases 12 , though recent work has shown that metastable phases can also appear as intermediates during high temperature experiments [13][14][15] .
To optimize the purity of a desired product, whether it be stable or only metastable, requires careful selection of precursors and reaction conditions.This selection process traditionally relies on domain expertise, reference to previously reported procedures for similar targets (if any exist) 16,17 , and the use of heuristics such as Tamman's rule 18 .However, there is no clear roadmap to optimize the solid-state synthesis of novel inorganic materials, which can lead to many experimental iterations with no guarantee of success.
A new opportunity exists to accelerate inorganic materials development by leveraging computer-aided optimization to plan solid-state synthesis experiments, learn from their outcomes, and make improved decisions regarding the selection of precursors and conditions that enable the formation of desired phases.Such an approach has found success in organic chemistry, where reactions can often be described by the breaking and formation of individual bonds 19,20 .This enables the use of retrosynthetic methods, which start from the target and "work backward" through stepwise reactions until a set of available starting materials is reached 21 .As many different reaction paths can lead to a given target, computer-aided optimization techniques based on Monte Carlo tree search and reinforcement learning have been successfully used to rapidly screen for promising synthesis routes [22][23][24] .In contrast, inorganic materials synthesis has yet to benefit from the widespread use of algorithms that can optimize experimental procedures.Their development is hindered by the difficulty of modeling solid-state reactions, where the corresponding phase transformations involve concerted displacements and interactions among many species over extended distances 2 .Some progress has been made in simplifying the analysis of solid-state reaction pathways by decomposing them into step-by-step transformations that take place between two phases at a time, hereafter referred to as pairwise reactions 6,15 .However, it remains difficult to predict the temperature at which a given pairwise reaction will occur, as well as what phase(s) will form as a result of that reaction.
To determine which reaction outcomes are most plausible for a given set of precursors and conditions, much of the existing work on computer-aided planning for solid-state synthesis has relied on the analysis of thermochemical data based on Density Functional Theory (DFT) calculations 25,26 .For example, McDermott et al. introduced a graph-based approach that ranks various reaction pathways by a cost function designed to account for changes in the Gibbs free energy of reaction along each path 27 .A related approach developed by Aykol et al. parameterizes reactions by two axes -one that approximates the nucleation barrier of the targeted phase and another that accounts for its competition with possible byproducts -from which optimal reactants can be identified along the Pareto front 28 .Alternatively, machine learning models can be trained on synthesis data from the literature and applied to suggest effective precursors and conditions for a given target by considering its similarity with previously reported materials 16,17 .While these methods have been successfully applied in some cases, their use remains limited as they only provide a fixed a priori ranking of synthesis routes for a given material, which is not readily updated should the initial experiments fail.Such an approach differs from that of human experts, who continuously learn from failed experiments and make informed decisions regarding which parameters should be tried next.To ensure maximal utility, computational techniques for solidstate synthesis planning and optimization should do the same.
In this work, we introduce an algorithm for Autonomous Reaction Route Optimization with Solid-State Synthesis (ARROWS 3 ), which is designed to guide the selection of precursors for the targeted synthesis of inorganic materials.Given a desired structure and composition, ARROWS 3 uses existing thermochemical data in the Materials Project to form an initial ranking of precursor sets based on their DFT-calculated reaction energies 29,30 .Highly ranked precursors are suggested for experimental validation throughout a range of temperatures, which are iteratively probed and analyzed using machine learning algorithms to identify the intermediates that form along each precursor set's reaction pathway.When such experiments fail to produce the desired phase, ARROWS 3 learns from their outcomes and updates its ranking to avoid pairwise reactions that consume much of the available free energy and therefore inhibit formation of the targeted phase.
To benchmark the performance of ARROWS 3 , we conducted 188 synthesis experiments targeting YBa2Cu3O6.5,forming a comprehensive reaction dataset that critically includes both positive and negative results.Our approach is shown to identify all the effective synthesis routes from this dataset while requiring fewer experimental iterations than Bayesian optimization or genetic algorithms.We further demonstrate that ARROWS 3 can be applied in-line with experiments to guide the selection of precursors for two metastable targets, Na2Te3Mo3O16 and LiTiOPO4, each of which were successfully prepared with high purity.

Design of ARROWS 3
The logical flow of ARROWS 3 is summarized in Fig. 1 and detailed in the Methods.Given a target specified by the user, in addition to the precursors and temperatures that may be used for its synthesis, ARROWS 3 forms a list of precursor sets that can be stoichiometrically balanced to yield the target's composition.In the absence of previous experimental data, these precursor sets are initially ranked by their calculated thermodynamic driving force (∆G) to form the target (Fig. 1a).
While many factors influence the rates at which solid-state reactions proceed 31 , those with the largest (most negative) ∆G tend to occur most rapidly 15,16,32 .However, such reactions may also be slowed by the formation of intermediates that consume much of the initial driving force 7 .To address this, ARROWS 3 proposes that each precursor set be tested throughout a range of temperatures, thereby providing "snapshots" of the corresponding reaction pathway (Fig. 1b).The intermediates formed at each step in the reaction pathway are identified using X-ray diffraction (XRD) with machine-learned analysis 33 .ARROWS 3 then determines which pairwise reactions led to the formation of each observed intermediate phase (Fig. 1c), and it leverages this information to predict the intermediates that will form in precursor sets that have not yet been tested (Fig. 1d).
In subsequent experiments, ARROWS 3 prioritizes sets of precursors that are expected to maintain a large driving force at the target-forming step (∆G′), i.e., even after intermediates have formed (Fig. 1e).This process is repeated until the target is successfully obtained with sufficiently high yield, as specified by the user, or until all the available precursor sets have been exhausted.(e) The precursor ranking is updated based on the newly calculated ∆G′.All chemical formulae shown are placeholders for arbitrary compounds, and in general there is no restriction on the compositions where ARROWS 3 is applicable.
To validate the effectiveness of ARROWS 3 , new experimental synthesis data is needed.
Existing results from the literature tend to be heavily biased toward positive results, which precludes the development of models that can learn from failed experiments 34 .We therefore built a solid-state synthesis dataset for YBa2Cu3O6.5 (YBCO) by testing 47 different combinations of commonly available precursors in the Y-Ba-Cu-O chemical space, which were mixed and heated at four synthesis temperatures ranging from 600-900 °C.Importantly, this dataset includes both positive and negative outcomes, i.e., reactions that do and do not yield sufficiently pure YBCO.
As such, it can be used as a benchmark on which to test ARROWS 3 and compare its efficacy with alternative optimization algorithms.Two additional chemical spaces are also considered, where we use ARROWS 3 to actively guide the experiments.The first set of experiments targeted Na2Te3Mo3O16 (NTMO), which is metastable with respect to decomposition into Na2Mo2O7, MoTe2O7, and TeO2 according to DFT calculations 35 .The second set of experiments targeted a triclinic polymorph of LiTiOPO4 (t-LTOPO), which has a tendency to undergo a phase transition into a lower-energy orthorhombic structure (o-LTOPO) with the same composition 36 .The features of each space tested are summarized in Table 1.Further details regarding the corresponding experiments are provided in the Methods.

YBCO
Before discussing the optimization of YBCO synthesis using ARROWS 3 , we first summarize the outcomes from all 188 experiments to give context regarding the difficulty of obtaining high-purity YBCO while using a short hold time of 4 h.Indeed, only 10 of these experiments led to the formation of pure YBCO without any prominent impurity phases (i.e., no impurity peaks within the detection limit of XRD-AutoAnalyzer).Another 83 experiments gave partial yield of YBCO, in addition to several unwanted byproducts.Fig. 2a shows the distribution of YBCO yield (wt. %) at each synthesis temperature sampled in this work.Generally, the use of higher temperature leads to increased yield of YBCO, likely due to enhanced reaction kinetics.Precursor selection also has a marked effect on the target's yield.Fig. 2b shows the success rate of each precursor, which we define as the percentage of sets where that compound was included and resulted in the formation of YBCO without any detectable impurities.This plot suggests that the less commonly used binary precursors tend to outperform their standard counterparts.For example, BaO and BaO2 have moderately high success rates of 46% and 22%, respectively, whereas sets with BaCO3 always produce impure samples (i.e., 0% success rate).Precursor sets including Y2Cu2O5 and Ba2Cu3O6 also have comparably high success rates of 33% and 31%, respectively.We will later show that these ternary phases enable the direct formation of YBCO while circumventing inert byproducts such as Y2BaCuO5.
Fig. 2c displays a pie chart containing the four most common impurity phases that coexist with YBCO, or prevent its formation entirely, at 900 °C.Each slice in the pie chart represents the fraction of experiments where the specified impurity phase appears.Most of the impure samples (28/44) contain BaCuO2 or Y2BaCuO5, which are known to be relatively inert during the synthesis of YBCO, requiring intermittent grinding to improve the sample's purity 37,38 .BaCO3 is another common impurity phase, detected in 13/44 samples, which is likely slow to react owing to its high decomposition temperature in air (1000 °C) 39,40 .While CuO is also frequently detected, it only ever appears with at least one other impurity that is Cu deficient.When such phases do not form, CuO can contribute to the formation of YBCO, as evidenced by its success rate of 20%.where each slice represents the relative number of occurrences for each compound at 900 °C.The small gray sliver includes two less commonly observed impurities, YBaCu3O7 and YBa4Cu3O9.
To determine whether ARROWS 3 can effectively distinguish between successful and failed synthesis routes, we assessed how many iterations are required to identify all 10 optimal experiments that result in the formation of YBCO without any detectable impurities.For comparison, we also applied Bayesian optimization (BO) and a genetic algorithm (GA) to the same task by using a one-hot representation of each precursor.These algorithms are known to perform well on numerical inputs such as temperature 41,42 ; however, their effectiveness with respect to categorical inputs is less well proven.To specifically probe the latter case, we constrained BO and GA to optimize the selection of precursors while sampling all temperatures for each precursor set.
Both black-box algorithms have stochastic elements and were therefore applied to the YBCO dataset 100 times, each with a random starting seed, and their results were averaged.Because ARROWS 3 is deterministic, only a single run was performed for its validation on the YBCO dataset.Further details on the BO and GA techniques used in this work are given in Supplementary Note 1.
Fig. 3a shows the number of optimal synthesis routes (those yielding pure YBCO) discovered with respect to the number of experiments that were queried by each algorithm tested here.ARROWS 3 successfully identified all 10 optimal synthesis routes after sampling 87 experiments (46% of the design space).For the same task, BO and GA required on average 164 and 167 experiments, respectively (87% and 89% of the design space).We suspect the ineffectiveness of the black-box algorithms tested in this work is related to their use of one-hot representations for the precursors, which treat each compound independently and contain no physical information regarding their composition or structure.In contrast, ARROWS 3 encodes compositional and thermodynamic information in its optimization through its ranking by ∆G.It also learns from failed experiments to avoid pairwise reactions that form inert byproducts such as BaCuO2 and Y2BaCuO5, instead prioritizing sets of precursors expected to retain a strong driving force (∆G′) to form YBCO.
Fig. 3b displays the number of pairwise reactions learned by ARROWS 3 with respect to the number of experiments that were queried.This plot includes pairs of phases that react within the temperature range considered (≤ 900 °C), denoted reactive pairs, as well as the phases that do not react within that range, denoted inert pairs.From 87 experiments, ARROWS 3 gained information regarding 34 pairwise interactions, including 24 reactive and 10 inert pairs.We find that the identification of new successful synthesis routes is often preceded by the discovery of new pairwise reactions.For example, ARROWS 3 learned from experiments 30-34 that BaO reacts with CuO to form BaCuO2 at 800 °C, which subsequently reacts with Y2O3 at 900 °C to form Y2BaCuO5.Because these pairwise reactions consume much of the driving force that remains to form YBCO, the algorithm decides to prioritize sets of precursors that do not contain such reactive pairs (BaO|CuO or BaCuO2|Y2O3).This decision leads to the successful discovery of three new synthesis routes that produce YBCO without any detectable impurities, as shown by the steep rise of the green curve in Fig. 3a between experiments 38-43.While previous work has shown that BaCuO2 can effectively contribute to YBCO formation when it melts in combination with CuO 6 , there was no evidence of melting in our samples owing to the use of low synthesis temperatures (≤ 900 °C) that ensured all products could be easily extracted.
In addition to learning which pairwise reactions should be avoided, ARROWS  with respect to the number of experiments queried.
To showcase the pairwise reactions learned by ARROWS 3 , we present in Fig. 4 a heatmap where each square represents a pair of phases.If any information was learned regarding the reactivity of that pair, the square is colored a light shade of blue according to the temperature at which a reaction proceeds.If a pair was instead found to be inert at all temperatures ≤ 900 °C, a dark shade of blue is used.We also denote reactions that produce YBCO (yellow star) or its competing phases, BaCuO2 (orange circle) and Y2BaCuO5 (red cross).This heatmap reveals that YBCO forms at 900 °C when Ba2Cu3O6 reacts with Y2O3 or Y2(CO3)3.It is separately observed that Ba2Cu3O6 reacts with Y2Cu2O5 when both are present at 800 °C, resulting in a mixture of YBCO and CuO.The direct formation of YBCO from Ba2Cu3O6 and Y2Cu2O5 provides an explanation as to why both phases have high success rates when used as precursors (Fig. 2b).In contrast, the 0% success rates associated with BaCO3 and BaCuO2 can be traced to the limited reactivity of each phase with many of the precursors tested here.This is illustrated in Fig. 3 by the dark blue shading that signifies inert reactions pairs in the rows corresponding to BaCO3 and BaCuO2.Even when BaCO3 does react, it often produces BaCuO2 or Y2BaCuO5, which are both common impurity phases that preclude the formation of YBCO.The presence of Y2BaCuO5 is particularly detrimental to the synthesis of YBCO as it does not react with any precursor in the allotted hold time of 4 h, which ARROWS 3 learns over the course of the 87 experiments we performed.For a more detailed visualization of the information gleaned from each stage in the experimental process, we plot in Supplementary Fig. 1 an evolution of the heatmap displaying which pairwise reactions were learned after 30, 60, and 90 experiments.This improvement is largely attributed to reduced Na2Mo2O7 formation when more stable Na precursors such as Na2CO3 or Na2TeO3 are used.ARROWS 3 further discovered from the outcome of the 16 th experiment that it is even more effective to use precursors (Na2O, MoO3, and TeO2) that avoid Na2Mo2O7 entirely by instead forming Na2MoO4.This was the only precursor set for which Na2Mo2O7 was not detected at any temperature, and as a result, it successfully produced a sample containing 62% NTMO by weight.It did so by forming Na2MoO4, which retains a favorable (negative) driving force to react with the remaining precursors and form the target: Given that the updated reaction energy is relatively small, we suspect that longer hold times could be used to improve the purity of the synthesis product.To confirm this, we prepared a new sample containing the same precursors (Na2O, MoO3, and TeO2) and held them at the optimized synthesis temperature 400 °C for a longer hold time of 8 h.The XRD pattern of the resulting product is shown in Fig. 5b, revealing that the use of a longer hold time led to substantially improved purity.The sample contained 94% NTMO by weight, in addition to a 6% TeO2 impurity.
For comparison, we carried out an identical synthesis procedure using a precursor mixture where MoO3 was replaced with MoO2, for which the resulting product did not contain any detectable amount of NTMO (Supplementary Fig. 2).This contrasting result highlights the importance of precursor selection and its effect on the reaction pathways that proceed during synthesis.By replacing a single precursor and thus altering which intermediate phase forms first (Na2Mo2O7 or Na2MoO4), the target yield can vary from 0% to >90%.

LTOPO
As a final demonstration, ARROWS 3 was used to direct a series of experiments targeting the triclinic polymorph of LiTiOPO4 (t-LTOPO) based on a search space consisting of 30 different precursor sets and two synthesis temperatures (400, 500, 600, 700 °C).To achieve this target, the algorithm must learn to avoid the formation of a lower-energy polymorph that exists at the same composition but adopts an orthorhombic structure (o-LTOPO) 36 .In the top panel of Fig. 6a, we plot the weight fraction obtained for each polymorph with respect to the number of precursor sets that were sampled by ARROWS 3 during its optimization of the synthesis process.These weight fractions are taken from experimental outcomes at 700 °C, which is the only temperature where either polymorph of LTOPO formed.The solid dots in Fig. 6a represent experimentally observed weight fractions, whereas the hollow dots represent predictions made based on the intermediates formed at 400 °C.A total of eight precursor sets were tested before identifying an optimal synthesis route for t-LTOPO, though many of these sets produced identical intermediates and therefore did not require sampling of temperatures > 400 °C.
A key distinguishing feature between the precursor sets tested by ARROWS As outlined in recent work 44 , preferential nucleation of o-LTOPO tends to occur when preceded by reactions with small changes in the Gibbs free energy.This is confirmed by the synthesis outcome of precursor set 1 annealed at 700 °C, which produces a sample containing 35% o-LTOPO and only 17% t-LTOPO, in addition to leftover LiTi2(PO4)3 and TiO2 impurities.
To avoid the reactions that form LiTi2(PO4)3 and thereby retain larger ∆G′ to form the target, ARROWS 3 suggests precursors where such reactions have not yet been observed.As shown by the data to the right of the dividing line in Fig. 6a, which separates experiments selected using ∆G from those selected using ∆G′, this decision successfully reduced LiTi2(PO4)3 formation and led to increased yield of t-LTOPO.The plateau in the amount of each phase formed with precursor sets 3-7 is associated with the use of less reactive Li sources -including Li2CO3, Li2TiO3, and Li4Ti5O12 -which tend to persist until higher temperature and reduce the amount of LiTi2(PO4)3 that forms as an intermediate.While this led to increased yield of the target, o-LTOPO still accompanied its formation at 700 °C.In contrast, the eighth precursor set proposed by ARROWS 3 (Li2O, TiO2, and P2O5) resulted in 54% target yield and no detectable amount of o-LTOPO.
Notably, this was also the only precursor set that did not form any LiTi2(PO4)3 at 400 °C.It instead formed a set of intermediates that maintained a stronger driving force to form the target as shown by the chemical reaction below: Li3PO4 + 2 TiO2 + 3 TiP2O7 → 3 LiTiOPO4 (∆G′ = -24 meV/atom) Because ARROWS 3 identified a synthesis route that gave 54% yield for t-LTOPO, exceeding our pre-defined objective of 50%, the optimization process was complete.Nevertheless, to verify that the target could made with higher purity using these optimized precursors, we separately performed a synthesis procedure where Li2O, TiO2, and P2O5 were ball milled prior to heating the mixture at 700 °C for 4 h.The XRD pattern of the resulting product is shown in Fig. 6b, revealing the formation of t-LTOPO without any detectable impurity phases.For comparison, the same procedure was also applied to a precursor mixture of LiOH, TiO2, and P2O5.As shown in Supplementary Fig. 3, the corresponding synthesis product contained LiTi2(PO4)3 and o-LTOPO impurities, which limited the yield of t-LTOPO to 46% when using these non-optimized precursors.
Although t-LTOPO was successfully optimized, we advise careful application of ARROWS 3 for synthesizing metastable polymorphs.Our algorithm worked effectively with LTOPO, as its desired (metastable) polymorph is favored at large reaction energies, primarily due to its stable surface energy at small particle size 44 .This makes it well-suited for ARROWS 3 , which learns to prioritize synthesis pathways with large reaction energy at the target-forming step.
However, if the stable polymorph instead had low surface energy, its formation would be enhanced by the recommended precursor sets.Therefore, our general recommendation is to use ARROWS 3 for the following cases: 1) targets that are inherently stable; 2) targets that are metastable with respect to phase separation; and 3) targets that are metastable with respect to polymorphic transition but have lower surface energies than the ground states.

Discussion
Precursor selection often has a marked effect on the outcomes of solid-state synthesis experiments, dictating whether they form desired products or unwanted impurities 6,7 .The importance of choosing optimal precursors is demonstrated by our syntheses targeting YBCO, for which only 10 precursor sets (out of 47 total) are successful in forming YBCO without any detectable impurity phases.Similarly, both NTMO and LTOPO were found to require the use of specific precursor sets that circumvent the formation of competing phases that otherwise limit the yield of the experiments to determine where a given reaction pathway "goes wrong."It does so by rationalizing each set of experimental outcomes using pairwise reaction analysis, which assumes that a mixture of solid precursors reacts two phases at a time.This assumption is justified by several previous studies 6,15,45 , where in situ XRD was used to verify that solid-state reactions often proceed in pairs owing to the limited diffusion lengths of species in the solid medium.In the current work, systematic pairwise reaction analysis was used to identify which precursors in each set reacted to consume much of the available free energy, thereby reducing the driving force (∆G′) that remains to form the target.Once this information is known, ARROWS 3 prioritizes experiments based on precursor sets that are expected to avoid such unfavorable pairwise reactions.Our tests on the YBCO dataset showed this to be an effective approach for the rapid identification of optimal synthesis routes, as ARROWS 3 identified all 10 of the best experimental procedures while sampling less than half of the search space.Similarly, it identified successful procedures for the synthesis of two metastable phases, NTMO and LTOPO, while requiring only 35% and 14% of their search spaces to be sampled, respectively.
In comparison to ARROWS 3 , black-box optimization techniques including genetic algorithms and Bayesian optimization perform relatively poorly on the YBCO dataset (Fig. 3a).
We suspect their ineffectiveness is caused by using one-hot encodings to represent each precursor set, which fails to capture the similarities and differences between various chemical compounds.
Recent work on organic synthesis has shown that black-box optimization techniques can perform well in the selection of molecular precursors when they are represented using physical descriptors such as SMILES strings 46 ; however, no such universal representation exists for crystalline materials.Further complicating matters is the fact that precursor sets used in solid-state synthesis often have varied lengths -i.e., some sets contain more precursors than others -which make them difficult to represent using a fixed-length input vector for optimization.
ARROWS 3 systematically explores the search space associated with solid-state synthesis by actively learning from failed experiments.To overcome the limitations outlined in the previous paragraph, ARROWS 3 relies on a single metric (the remaining reaction energy) that can be updated is it reconstructs the path a given synthesis procedure takes.Previous work has demonstrated that reaction energies (∆G) often dictate the selectivity of competing phases in solid-state synthesis 6,15 , and reactions with larger ∆G tend to occur more rapidly 16,32 .Initially, when no intermediates are known, the available reaction energy corresponds to the free energy difference between the target and precursors, thus motivating our choice to first prioritize experiments based on precursor sets with the largest reaction energies.Once intermediates become known, ARROWS 3 re-ranks precursor sets based on their updated reaction energies (∆G′) remaining to form the target.Using this approach, the algorithm can discard reaction pathways that become trapped in metastable states close in energy to the target.Notably, a unique feature of ARROWS 3 is that it becomes more efficient at identifying optimal experiments as it builds the size of its pairwise reaction database.
This was demonstrated by the correlation between the frequency at which optimal synthesis routes were discovered on the YBCO dataset and the number of pairwise reactions that were collected (Fig. 3b).
Further improving the utility of the pairwise reactions learned by ARROWS 3 is their transferability across materials in related chemical spaces.For example, our analysis of the YBCO experiments revealed 34 unique pairwise reactions involving common precursors for Y, Ba, and Cu.Should any of these compounds be used for the synthesis of a new material, ARROWS 3 would operate more effectively by predicting their reaction outcomes a priori.As the decision making performed by ARROWS 3 requires no manual intervention after being given its initial input (e.g., what to synthesize), it is well suited to act as the "brain" behind autonomous platforms that are currently being developed 25 .With years of continuous and autonomous experimentation, such platforms could lead to the development of a standardized pairwise reaction database that covers much of the periodic table, enabling accurate predictions regarding optimal synthesis routes for new materials without requiring additional experiments.Researchers across the field of solid-state chemistry could also contribute to this database and refer to it for their own synthesis design.

Conclusion
The development of ARROWS 3 provides a proof of concept for the use of computer-aided optimization algorithms in the synthesis of inorganic materials.Using thermochemical data from DFT calculations, this algorithm prioritizes synthesis experiments that involve precursor sets with large reaction energies.It then identifies which intermediate phases form during these experiments and determines the pairwise reactions that led to their formation.By subsequently predicting the reaction pathways in precursor sets that have not yet been tested, ARROWS 3 prioritizes sets that are expected to avoid unfavorable pairwise reactions (i.e., those that form stable intermediates) and therefore retain a strong driving force to form the target.We validated the effectiveness of this algorithm on three synthesis datasets, which demonstrated that ARROWS 3 can successfully identify optimal synthesis routes while requiring few experimental iterations.The optimization trajectory additionally provides interpretable explanations of these successes by detailing which pairwise reactions led to the formation of the target phase in each synthesis route.
We anticipate that ARROWS 3 will not only facilitate a more systematic approach to the planning of synthesis experiments performed by human researchers, but also enable the development of fully autonomous platforms for materials development 25 .An additional benefit of this algorithm in conjunction with automated synthesis platforms is that multiple successful synthesis routes can be learned for a given target.Such information on alternate experimental procedures will be valuable when more practical considerations become important, such as the optimization of morphology, synthesis cost, or the ability to industrially scale up the synthesis of a novel compound.
While we have shown that ARROWS 3 performs well on three benchmarks, there may still be room for improvement.To aid in the development of new algorithms for decision making in solid-state synthesis, all data reported in this work is made publicly available.In particular, the YBCO dataset is the first of its kind to include all possible combinations of various precursors for the solid-state synthesis of a bulk inorganic materials.Critically, this includes both positive (successful) and negative (failed) synthesis outcomes, and as such, can be used to train and validate algorithms that require both types of data.
precursor sets ( !"#! ), this information defines the search space containing  "$% points over which optimization is performed for a given target:

Initial ranking by ∆G
The thermodynamic driving force behind a chemical reaction is set by the change in the Gibbs free energy (∆G) between its products and reactants.Under constant temperature and pressure, reactions can occur spontaneously only if they reduce the Gibbs free energy (∆G < 0) of the system.
ARROWS 3 initially ranks all the available precursor sets in order of their reaction energies (∆G) to form the target.Those with the largest (most negative) ∆G are prioritized, whereas those with ∆G ≥ 0 are excluded from consideration.For each set, ∆G of the solid compounds is determined using DFT-calculated 0 K formation energies from the Materials Project 29 , along with temperaturedependent free energies approximated using the machine-learned descriptor developed by Bartel et al. 30 For gaseous compounds, ∆G is obtained from the experimental NIST database 47 .All reaction energies are normalized per atom of the product phase(s) formed to ensure a consistent comparison between different precursor sets.
The initial ranking by ∆G is intended to prioritize sets of precursors that are expected to react under short timescales; however, such precursors are not necessarily the most effective at forming the target.In addition to having a strong thermodynamic driving force to form the target, precursor sets with large ∆G often have similarly large driving forces to form unwanted impurity phases 7 .We have therefore designed ARROWS 3 to learn from the outcomes of failed experiments by determining which reactions led to the formation of such impurity phases.Details on this process are given in the next two sections.

Temperature selection for intermediate identification
To pinpoint the origin of any impurity phases that caused a synthesis procedure to fail, it is necessary to identify the intermediate phases that formed while heating.Previous work has demonstrated that precursors used in solid-state synthesis typically do not transform directly to the final products, but instead proceed through a series of pairwise reactions that form transient intermediate phases and incrementally reduce the free energy of the sample 6,15 .Characterizing these intermediates would traditionally require the use of in situ X-ray diffraction (XRD); however, we propose that similar information can be obtained by testing a range of synthesis temperatures for a given precursor set.Assuming that a fixed hold time is used at each temperature, the XRD patterns gathered from the resulting samples provide discrete snapshots of the reaction pathway, from which intermediate phases can be identified in a high-throughput and automated fashion using recently developed machine learning algorithms .
By inspecting the temperature-dependent synthesis outcomes for a given precursor set, ARROWS 3 determines which pairwise reactions occurred while heating.To this end, we assume that any phases detected at a specific temperature () may act as reactants that lead to the formation of new phases at the next highest temperature ( + ∆).Accordingly, when XRD measurements reveal a new phase that is not present in the associated precursor set nor identified as an intermediate phase at lower temperature, ARROWS 3 is tasked with identifying the precise combination of phases responsible for its formation.If a new phase is detected at  -./ , the algorithm evaluates which two-phase combination(s) from the precursor set have the appropriate compositions (i.e., can be stoichiometrically balanced) to produce that phase.In cases where there exists only one such possible combination, it is recorded as an observed pairwise reaction with an onset temperature less than  -./ .A similar procedure is followed when new phases are detected at  >  -./ , except that ARROWS 3 considers the intermediate phases detected at the next lowest temperature ( − ∆) as possible reactants.
Oftentimes, different sets of precursors can react to form identical sets of intermediates at low temperature, which subsequently result in the same products upon further heating 48 .To avoid testing all temperatures for such redundant synthesis routes, ARROWS 3 suggests that experiments first be performed at  -./ for each precursor set.It then checks whether the observed products and their associated weight fractions differ from those obtained using other precursors sets that were previously tested at  -./ .Differences as large as 10% are allowed between two sets of products while still considering them to be identical as there is often limited precision in the refinements performed using XRD patterns from multi-phase mixtures.If the observed products for a precursor set are indeed unique, the next highest temperature ( + ∆) is proposed for that set.This process is repeated until the target is successfully obtained with sufficiently high yield, as specified by the user, or until  -0$ is reached for the specified precursor set.

Updated ranking by ∆G′
ARROWS 3 learns from previously identified pairwise reactions to make informed decisions regarding optimal synthesis routes.It does so by predicting which intermediates will form upon heating precursor sets that have not yet been tested.An example of this process is given below for an arbitrary target ( 1 ): Precursor set not yet tested:  + 2 +  (∆G ./.#.02 ) Previously identified pairwise reaction:  + 2 →  1 (∆G ./#"3-) Reaction using anticipated intermediates:  1 +  (∆G 4 = ∆G ./.#.02 − ∆G ./#"3-) In this example, the anticipated intermediate phases were determined based on previous synthesis outcomes that involved a reaction between  and .The updated reaction energy (∆G′) to form the target ( 1 ) is then calculated based on the intermediates ( 1 + ) that result from this pairwise reaction.Similar analysis is applied to all precursor sets that have not yet been tested and their reaction energies are updated accordingly.In cases where no intermediates can be predicted, the reaction energy remains unchanged (∆G 4 = ∆G).Following these changes, precursor sets are ranked to prioritize reactions with the most negative ∆G 4 , i.e., those with the largest thermodynamic driving force at the presumed target-forming step.ARROWS 3 uses the updated ranking to continually suggest new precursor sets until an experiment is found that gives sufficiently high yield of the target phase (as specified by the user) or until all precursor sets have been tested.

YBCO synthesis
The synthesis of YBCO is most commonly performed using Y2O3, CuO, and BaCO3 37 .This combination of precursors requires > 12 h of annealing at 950 °C, in addition to intermittent regrinding, to eliminate the unwanted impurity phases that often appear.In contrast, recent work has shown that by replacing BaCO3 with BaO2, YBCO can be obtained with high purity while using a shorter anneal time of 30 min 6,38 .These findings highlight the importance of precursor selection and its effect on the yield of YBCO under limited hold time, making it a suitable test case for ARROWS 3 .To this end, we considered 11 common precursors from the Y-Ba-Cu-O space: Y2O3, Y2(CO3)3, BaO, BaCO3, BaO2, CuO, CuCO3, Cu2O, BaCuO2, Ba2Cu3O6, and Y2Cu2O5.All binary phases (including the carbonates) were purchased from Sigma Aldrich, whereas the ternary phases were synthesized following the procedures detailed in Supplementary work has proposed that the t-LTOPO nucleates first owing to its more stable surface energy, which dictates the relative nucleation rate of each polymorph when ∆G is large 44 .Therefore, although ARROWS 3 encodes no structural information and is not designed for the synthesis of metastable polymorphs in general, we believe it is well-suited for t-LTOPO (and similarly stabilized metastable polymorphs) since it aims to identify reaction pathways that maintain large ∆G.

Figure 1 :
Figure 1: A schematic illustrating how ARROWS 3 guides precursor selection.(a) Precursor sets are initially ranked by their driving force (∆G) to form the target.(b) Experiments are performed at iteratively higher temperatures to identify reaction intermediates.(c) Pairwise reactions are gleaned from the experimental data.(d) Using the identified pairwise reactions, intermediates are predicted for other precursor sets and their remaining driving forces (∆G′) are updated accordingly.

Fig. 2 :
Fig. 2: A summary of outcomes from the experiments targeting YBCO.(a) Distributions of YBCO yield (wt. %) at different synthesis temperatures represented using violin and box plots, where each box extends from the lower to upper quartiles.(b) The success rate of each precursor, defined as the percentage of sets where that compound is included and forms YBCO without any impurities.(c) Common impurity phases that prevent YBCO formation are shown by a pie chart,

Fig. 3 :
Fig. 3: Optimization results from the experimental YBCO synthesis dataset.(a) Number of optimal synthesis routes identified as a function of the experimental iterations required by ARROWS 3 , Bayesian Optimization (BO), and a Genetic Algorithm (GA).The dashed line represents the total number of optimal synthesis routes in the dataset.(b) Pairwise reactions discovered by ARROWS 3

Figure 4 :
Figure 4: Pairwise reactions in the Y-Ba-Cu-O chemical space, illustrated by a heatmap where the color of each square represents the temperature (°C) at which a reaction is observed.Inert pairs correspond to phases that do not react within the temperature range considered.Pairs without any data are left blank (white squares).Yellow stars denote pairs that react to produce YBCO.Orange circles and red crosses denote pairs that form impurities, Y2BaCuO5 and BaCuO2, respectively.

Figure 5 :
Figure 5: Optimization results from ARROWS 3 on the synthesis of NTMO.(a) The top panel shows the weight fraction of the target obtained from each precursor set that was tested at 500 °C.The bottom panel displays the weight fraction of a competing phase, Na2Mo2O7, obtained at 400 °C.Solid (hollow) dots represent experimental (predicted) values.(b) XRD pattern measured from the synthesis product of the optimized precursor set, Na2O + TeO2 + MoO3 after an 8 h hold at 400 °C.For comparison, a reference pattern is shown for NTMO (ICSD #171758).

Figure 6 :
Figure 6: Optimization results from ARROWS 3 on the synthesis of t-LTOPO.(a) The top panel shows the weight fractions obtained for the target and its competing polymorph based on each precursor set that was tested at 700 °C.The bottom panel displays the weight fraction of a common impurity phase, LiTi2(PO4)3, obtained at 400 °C.Solid (hollow) dots represent experimental (predicted) values.(b) XRD pattern measured from the synthesis product of the optimized precursor set, Li2O + TiO2 + P2O5, which was ball milled and subsequently heated to 700 °C for 4 h.For comparison, a reference pattern for t-LTOPO (ICSD #39761) is also shown.
metastable targets.Changing just one precursor can lead to a completely different synthesis outcome, as shown by the 94% wt.increase observed in the yield of NTMO when MoO2 is replaced by MoO3.Understanding the origin of such large changes requires a detailed inspection of their associated reaction pathways.While this would typically be accomplished by using in situ characterization techniques, we have shown that information regarding the intermediate phases formed during solid-state synthesis can be gathered by probing different annealing temperatures with fixed hold times.For example, the low-temperature (400 °C) synthesis experiments targeting LTOPO reveal whether LiTi2(PO4)3 forms as an intermediate, which subsequently controls the yield of the metastable polymorph at higher temperature (700 °C).ARROWS 3 effectively uses intermediate phase information gleaned from low-temperature

Table 1 :
Information regarding three search spaces on which ARROWS 3 was tested.!"#! and  "$%! represent the number of precursor sets and experiments, respectively.

Table 1 :
Ten commercially available phases were purchased from Sigma Aldrich and used as precursors: Li2O, Li2CO3, LiOH, TiO2, P2O5, NH4H2PO4, (NH4)2HPO4, Li3PO4, Li2TiO3, and Li4Ti5O12.A total of 30 precursor sets, listed in Supplementary Table3, were considered.Four synthesis temperatures (400, 500, 600, and 700 °C) were sampled for each set at fixed a hold time of 4 h.Synthesis experiments were performed under the guidance of ARROWS 3 until t-LTOPO was obtained with a weight fraction exceeding 50%.No black-box optimization techniques were applied.All precursor sets tested for the synthesis of YBCO.The stoichiometry of each set is determined by the composition of YBCO, in addition to gaseous byproducts that include O2 and CO2.Of the 47 precursor sets tested, only 10 resulted in the formation of YBCO without any detectable impurity phases.These 10 sets are highlighted green below.

Table 3 :
All precursor sets considered for the synthesis of LTOPO.The stoichiometry of each set is determined by the composition of LTOPO, in addition to gaseous byproducts that include O2, CO2, NH3, and H2O.The optimal set identified by ARROWS 3 is highlighted in green.