# Assessing Thermodynamic Selectivity of Solid-State Reactions for the
Predictive Synthesis of Inorganic Materials

Matthew J. McDermott  Materials Sciences Division, Lawrence Berkeley National
Laboratory, Berkeley, CA  Brennan C. McBride  Department of Chemistry,
Colorado State University, Fort Collins, CO  Corlyn Regier  Department of
Chemistry, Colorado State University, Fort Collins, CO  Gia Thinh Tran
Department of Chemistry, Colorado State University, Fort Collins, CO  Yu Chen
Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
CA  Adam A. Corrao  Department of Chemistry, Stony Brook University, Stony
Brook, NY  Max C. Gallant  Materials Sciences Division, Lawrence Berkeley
National Laboratory, Berkeley, CA  Gabrielle E. Kamm  Department of Chemistry,
Stony Brook University, Stony Brook, NY  Christopher J. Bartel  Department of
Chemical Engineering and Materials Science, University of Minnesota,
Minneapolis, MN  Karena W. Chapman  Department of Chemistry, Stony Brook
University, Stony Brook, NY  Peter G. Khalifah  Department of Chemistry, Stony
Brook University, Stony Brook, NY  Gerbrand Ceder  Materials Sciences
Division, Lawrence Berkeley National Laboratory, Berkeley, CA  James R.
Neilson  Department of Chemistry, Colorado State University, Fort Collins, CO
Kristin A. Persson  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, CA<EMAIL_ADDRESS>
###### Abstract

Synthesis is a major challenge in the discovery of new inorganic materials.
There is currently limited theoretical rationale for planning optimal solid-
state synthesis procedures that selectively yield desired targets with minimal
impurities. Using an interface reaction model, we propose two selectivity
metrics – primary and secondary competition – to measure the favorability of
target and impurity phase formation in solid-state reactions. We first apply
these metrics to assess the selectivity of 3,520 solid-state synthesis recipes
extracted from the scientific literature, comparing the thermodynamic
optimality of various approaches to popular targets (e.g., LiMn2O4, LiFePO4)
under our framework. Secondly, we implement these metrics in a data-driven
synthesis planning workflow and test its use in suggesting synthetic routes to
barium titanate (BaTiO3). From an 18-element chemical reaction network created
with first-principles materials thermodynamic data from the Materials Project,
we identify 82,985 synthesis reactions for BaTiO3 and select nine to test
experimentally, characterizing their reaction pathways using ex post facto
synchrotron powder X-ray diffraction over a wide range of temperatures.
Analysis of experimental results indicates that our proposed selectivity
metrics correlate with observed target and impurity formation. We identify two
favorably predicted reactions with unconventional BaS/BaCl2 and Na2TiO3
precursors that yield BaTiO3 more rapidly and with fewer impurities than
conventional solid-state approaches, highlighting the importance of precursor
selection and consideration of complex chemistries with additional elements
beyond those in the target material. Altogether, our framework serves as a
foundation for the predictive synthesis of inorganic materials, facilitating
the optimization of existing synthesis approaches and the design of syntheses
to novel materials, including those that cannot be easily made from
conventional “off-the-shelf” precursors.

Department of Materials Science and Engineering, University of California,
Berkeley, CA These authors contributed equally to this work. These authors
contributed equally to this work. Department of Materials Science and
Engineering, University of California, Berkeley, CA Department of Materials
Science and Engineering, University of California, Berkeley, CA Chemistry
Division, Brookhaven National Laboratory, Upton, NY Department of Materials
Science and Engineering, University of California, Berkeley, CA Department of
Materials Science and Engineering, University of California, Berkeley, CA

##  1 Introduction

The predictive synthesis of inorganic materials remains a grand challenge in
modern chemistry and materials science.1 Unlike organic synthesis, which is
often described via discrete reaction steps or mechanisms, inorganic materials
synthesis reactions cannot be deconstructed into elementary steps,2, 3
hindering the analogous development of retrosynthetic analysis techniques 4
and computer-aided synthesis planning.5 This lack of successful mechanistic
models has made the synthesis of predicted new materials a critical bottleneck
in high-throughput computational materials design efforts,6 with many proposed
materials having yet to be successfully synthesized. 7, 8, 9

While there are numerous inorganic synthesis methods (e.g., hydrothermal,
mechanochemical, sol-gel, intercalation, etc.),10 we limit the scope of this
work to bulk solid-state synthesis via powder reactions. This choice has been
motivated by the straightforward and scalable nature of working with bulk
powders, which makes solid-state synthesis suitable for applications ranging
from one-off laboratory synthesis to industrial mass manufacturing. In powder
reactions, product formation proceeds via nucleation and growth at interfacial
contact areas in the powder mixture (Figure 1a).11 The equilibrium phases of
the reacting system can be predicted by constructing a convex hull in free
energy and composition space, where the composition axis is a mixing ratio
between the two precursor compositions (Figure 1b).12 Here, we calculate the
convex hull exclusively using normalized compositions and energies (i.e., on a
per-atom basis). This construction, which we refer to herein as the “interface
reaction hull”, is a subsection of the compositional phase diagram for binary
systems and a “quasibinary” two-dimensional slice of the full phase diagram
for chemical systems with three or more elements. The exact product species,
and the sequence in which they appear, cannot be predicted with thermodynamics
alone; to do so requires intimate knowledge of the kinetic rates of all
physically feasible reactions. However, a commonly adopted theoretical
simplification assumes that the reaction product(s) with the most negative
pairwise reaction energy will be the first to nucleate and grow as a powder
mixture is heated.13, 11 This hypothesis is based on two principles: 1) the
random packing of solid crystallites results in very few locations where three
or more particles are simultaneously in contact, and 2) the activation energy
barrier to nucleation scales inversely with free energy as $\Delta
G^{\dagger}\propto\frac{\gamma^{3}}{\Delta G_{rxn}^{2}}$. The surface energy,
$\gamma$, is particularly important when comparing the feasibility of
reactions with similar energies.14 Despite this, many solid-state reactions
are likely not nucleation-limited but rather transport-limited due to the
relative sluggishness of solid-state diffusion and the often large driving
forces of these reactions (10-100 kJ/mol).15

Figure 1:  Modeling chemical reactions at heterogeneous solid interfaces in a
binary/quasibinary chemical system. (a) Cartoon model of a powder reaction
between the hypothetical precursors: $\alpha$ (orange) and $\beta$ (blue). The
nucleation of a new phase, $\gamma$ (green), occurs at the $\alpha|\beta$
interface according to the reaction $\alpha$ \+ $\beta$ -¿ $\gamma$. (b) A
possible reaction pathway for the powder system in which a secondary reaction
of the equilibrium phase ($\gamma$) yields an impurity phase, $\delta$ (red).
The interface reaction hull (top) shows available interfacial reactions and
their corresponding Gibbs free energies, $G$, and mixing ratios, $x$. The one-
dimensional spatial model (bottom) shows reaction steps beginning from an
equal mixture of $\alpha$ and $\beta$. The impurity phase, $\delta$, may be
kinetically retained in a local equilibrium state; however, with infinite
time, the system should approach the global equilibrium state composed
entirely of $\gamma$.

The complex interplay between thermodynamics and kinetics makes solid-state
synthesis prone to the unpredictable and undesirable formation of impurity
phases.13 A classic example of a nonselective synthesis is that of the
prototypical multiferroic bismuth ferrite, BiFeO3, via the standard reaction
from binary oxides: Bi2O3 + Fe2O3 -¿ 2 BiFeO3. This reaction typically yields
impurity phases Bi2Fe4O9 and Bi25FeO39, which are challenging to isolate and
remove.16, 17 Unfortunately, the presence of an impurity phase is difficult to
predict a priori, and is typically attributed to “kinetic” factors or changes
in phase equilibria related to precursor purity, morphology, volatility, or
processing conditions. To optimize the performance of solid-state reactions
and maximize conversion to the desired target, the experimentalist frequently
relies on intuition and heuristic rules to choose the 1) precursor
compositions (typically off-the-shelf binary phases such as carbonates,
oxides, etc.), 2) grinding/milling protocol, 3) synthesis annealing
temperature, 4) synthesis atmosphere (e.g., vacuum, flowing O2), 5) synthesis
time, and 6) cooling protocol. Heuristics include well-known rules such as
Tamman’s Rule for estimating reaction onset temperature (i.e., two-thirds the
melting temperature of the precursor with the lowest melting point),18 as well
as “chemical intuition” or human-biased experimental protocols (e.g.,
selecting synthesis times based on common increments, such as four hours,
eight hours, etc.).19 Unfortunately, these heuristics may be insufficient to
achieve successful synthesis on the first attempt(s), necessitating follow-up
experiments that can be time-intensive and costly. In the worst cases, human-
biased heuristics lead to lower success rates than randomly-generated
experimental protocols.20

The a priori calculation of reaction selectivity in solid-state synthesis
permits the ranking of synthesis approaches based on their thermodynamic
likelihood of success, thereby circumventing the current time-consuming trial-
and-error (Edisonian) approach. The assessment of reaction selectivity is
particularly relevant in the proposal of optimal synthesis precursors;21 in
several cases, improved navigation of the phase diagram was shown to lead to a
more practical synthesis 22, 23, 11, 24, 25. However, no solid-state reaction
selectivity metric has been formally established. In recent work,14 Aykol et
al. demonstrated a computational workflow for ranking solid-state synthesis
reactions by two performance metrics: 1) a catalytic nucleation barrier factor
incorporating structural similarity and epitaxial matching, and 2) the number
of known competing phases. These metrics perform well in rationalizing
successful syntheses in the literature but lack generality; for example, the
nucleation metric is derived assuming all reactions are nucleation-limited,
which, as discussed, is not true for many solid-state reactions. Additionally,
while a metric based on the total number of competing phases is significant as
it hints at a measure of reaction selectivity, such a scheme does not account
for the relative stability of these competing phases. A count-based
selectivity metric is also biased by how many phases are known to exist at the
present time and the extent to which various structural configurations (e.g.,
disordered or defective phases) have been enumerated within the data.

In this work, we address the longstanding issue of assessing the selectivity
of solid-state reactions by deriving two complementary thermodynamic metrics
measuring the degree of phase competition from the interface reaction model.
We incorporate these competition metrics into a computational synthesis
planning workflow for identifying and ranking synthesis reactions, which
builds upon the high-throughput reaction enumeration tools we previously
developed for constructing solid-state chemical reaction networks26 from large
materials databases such as the Materials Project.27 Our selectivity metrics,
computational workflow, literature analysis, and experimental findings yield a
framework for generating more optimal and efficient solid-state synthesis
routes, providing a foundation for the predictive synthesis of inorganic
materials. The suggestion of nonstandard precursors, particularly those
involving additional elements beyond those in the target composition, expands
the synthetic capabilities of the solid-state approach.

##  2 Results and Discussion

###  2.1 Derivation of Selectivity Metrics

####  2.1.1 The Interface Reaction Hull

To construct the interface reaction hull (Figure 1b), one begins with
thermodynamic data for the reacting system (i.e., a set of relevant phases and
their compositions and energies). In this work, we acquire formation
enthalpies, $\Delta H_{f}$, from the Materials Project database 27 and extend
them to Gibbs free energies of formation, $\Delta G_{f}(T)$, through the use
of a prior machine learning model 28 and supplemental experimental
thermochemistry data 29 (see Methods). For systems with two elements, the
interface reaction hull is equivalent to the binary compositional phase
diagram, where each vertex represents a single phase. However, for systems
with three or more elements, the non-precursor vertices include both single
phases and mixtures of phases. These mixtures are stoichiometric combinations
of phases representing the products of balanced reactions of the precursors.
The balanced reactions can be determined via 1) computing slices of the full
compositional phase diagram along the tie-line connecting the precursors 12 or
2) combinatorial reaction enumeration.26 More specifically, the maximum number
of products for a particular vertex is one less than the number of elements in
the system. The interface reaction hull is thus generalized such that all non-
precursor vertices correspond to reactions with coordinates given by the
atomic mixing ratio of precursors, $x$, and the Gibbs free energy of the
reaction, $\Delta G_{\text{rxn}}$. This model can be further generalized to
environmental conditions other than fixed temperature and pressure by
constructing the hull with the appropriate thermodynamic potential. For
example, in open systems, one would use the grand potential energy, $\Phi$.
Note that in these systems, the hull vertices may include additional open
elemental reactants/products (e.g., O2) that do not factor into the
determination of $x$.

As a model for solid-state reactions, the interface reaction hull construction
also rationalizes the formation of impurity phases. To demonstrate this, we
revisit the binary system in Figure 1. In this system, the target phase
($\gamma$) is predicted to form first because it is the phase with the highest
driving force of formation (most negative $\Delta G_{\textrm{rxn}}$),
irrespective of the average composition of the total system 12. When the
$\gamma$ phase forms, however, it also introduces two additional interfaces:
$\alpha|\gamma$ and $\gamma|\beta$. These secondary interfaces can be modeled
via the construction of new interface reaction hulls or, more simply, by
splitting the original hull into two subsections (i.e., to the left and the
right of the target). Figure 1b suggests that the $\gamma|\beta$ interface
should produce an impurity phase via the exergonic reaction,
$\gamma+\beta\longrightarrow\delta$. The full conversion of reactants to the
target phase is impeded while $\delta$ persists. In a “one-dimensional” solid-
state reaction (Figure 1b, bottom), local thermodynamic equilibrium may be
achieved when the system reaches a state in which all stable product phases on
the interface reaction hull have formed and the growth of the product layer(s)
slows down until it ceases entirely. This situation has been observed in
previous experimental studies on diffusion couples.30, 31 This observed
mixture of products may be kinetically “stable” (i.e., with locally stable
interfaces), but it is not the global equilibrium state of the system. Rather,
the global equilibrium state is the combination of phases that minimizes the
free energy given the composition of the entire powder mixture; in Figure 1b,
this corresponds to entirely $\gamma$. In powder reactions, access to the
global equilibrium state is often provided by re-grinding steps in which new
interfaces are exposed and mixed to facilitate complete conversion to the
equilibrium products. However, in reacting systems with significant phase
competition and slow transport, high temperatures and long heating times may
be necessary but impractical; pure target synthesis may not be achievable if
the desired products are unstable at high temperatures. These situations can
be avoided entirely by proposing alternative precursors that are more
selective (i.e., those with interface reaction hulls containing few to no
competing phases).

####  2.1.2 Measuring Phase Competition

To predict the thermodynamic selectivity of a solid-state reaction, we propose
two complementary metrics for assessing phase competition using the interface
reaction hull: primary ($C_{1}$) and secondary ($C_{2}$) competition. Although
both metrics measure the relative energetic favorability of competing
reactions, they model different mechanisms for impurity formation. The primary
competition measures the favorability of competing reactions of the original
precursors, while the secondary competition measures the favorability of
subsequent competing reactions between the precursors and target phase(s). The
origin of the two competition mechanisms is illustrated in Figure 2.

Figure 2:  Origin of primary and secondary competition in solid-state
reactions. In this simple interface reaction hull for a binary (two-element)
system, two interface reactions can form a competing phase (magenta square).
The primary reaction (yellow arrow) occurs at the interface between the two
precursors $\alpha$ and $\beta$, whereas the secondary reaction (green arrow)
occurs between the target phase (pink star) and the remaining $\beta$
precursor, leading to a smaller driving force (arrow length). The coordinates
of the competing phase, which must lie within the illustrated bounds (gray
triangle) if the target phase is to be stable, determine the relative values
of the primary and secondary reaction energies.

Primary competition, $C_{1}$, is measured via calculation of the relative
thermodynamic advantage of the most exergonic competing reaction from the
original precursors, as assessed through an energy difference:

$C_{1}=\Delta G_{\textrm{rxn}}-\min_{i}(\Delta G_{c_{i}}).$ (1)

Here, $\Delta G_{\textrm{rxn}}$ is the energy of the target synthesis
reaction, and $\Delta G_{c_{i}}$ are the energies of possible competing
reactions from the precursors. Lower $C_{1}$ values are favorable and result
in more selective target formation. When $C_{1}$ is positive, the target
reaction is less energetically favorable than the competing reaction with the
greatest driving force (most negative energy), suggesting that a competing
phase is likely to form. On the other hand, when $C_{1}$ is negative, the
target reaction is predicted to have the greatest driving force of any
reaction on the hull. By considering only the single most competitive
reaction, this functional form avoids the aforementioned bias related to using
the total number of competing reactions. When no exergonic competing reactions
are predicted for an interface, the competing reaction energy term is assigned
a value of zero, representing the scenario in which the precursors do not
react (e.g., $\alpha$ -¿ $\alpha$). This results in the limiting condition:
$C_{1}\geq\Delta G_{\textrm{rxn}}$.

Secondary competition, $C_{2}$, assesses the favorability of impurity phase
formation via secondary reactions between the target and precursor(s). This
metric is important and distinct from primary competition because it measures
the relative stability of the products of the target synthesis reaction with
respect to decomposition into the competing phase(s). Their relative stability
can be measured by computing the “inverse distance to the hull” (Figure 2),
which for systems with one competing phase, is equivalent to the secondary
reaction energy, $\Delta G_{d}$, at the precursor-target interface.

Figure 3:  Secondary reaction sequences in an interface reaction hull. (a) The
hull is divided into two subsections to the left (L) and right (R) of the
target, representing the two additional precursor-target interfaces. (b-c)
Secondary reaction sequences on each side of the target, with gray boxes
highlighting the final reaction sequences. The recursive, binary nature
results in the number of unique sequences, $N(n)$, following the Catalan
numbers $u_{n}$.

In an interface reaction hull with several competing phases, a sequence of
multiple secondary reactions may occur (Figure 3). When the target phase is
formed, it introduces two new precursor-target interfaces that divide the hull
into subsections to the left and right of the target. A secondary reaction may
occur in either subsection, exposing another two interfaces. If, at either
interface, there is a remaining driving force to form additional competing
phases, then this process may continue in a recursive fashion until all
possible secondary reactions have occurred. There are multiple ways to draw a
feasible secondary reaction sequence (Figures 3b,c). Consider a particular
secondary reaction sequence indexed $j$. This sequence has a total energy
given by the sum of the energies of its $n$ steps,

$\Delta G_{2,j}=\sum_{k=1}^{n}\Delta G_{d_{j,k}}=\Delta G_{d_{j,1}}+\Delta
G_{d_{j,2}}+...+\Delta G_{d_{j,n}},$ (2)

where the number of reaction steps ($n$) in the sequence also equals the
number of non-reactant (interior) vertices in the hull subsection.

If every secondary reaction step is required to be the one with the minimum
energy (i.e., largest driving force), then only one unique reaction sequence
exists in the left and right hull subsections. However, one must consider the
situation where the minimum-energy principle does not hold due to kinetic
limitations; this applies in particular to hulls where all secondary reactions
have similar magnitude driving forces or a particular phase is kinetically
limited from forming, perhaps due to an overall small driving force.
Therefore, we must consider the alternative secondary reaction sequences shown
in Figures 3b,c. These alternative sequences are not necessarily less
favorable; while each alternative sequence may feature a first reaction step
with a smaller driving force (i.e., small $\Delta G_{d_{j,1}}$), the latter
steps may have larger magnitude energies, resulting in comparable total energy
($\Delta G_{2,j}$) for the particular sequence.

To encompass all combinatorial possibilities in our estimation of secondary
competition, we choose to compute the mean total energy of all feasible
secondary reaction sequences:

$\overline{\Delta G_{2}}=\frac{1}{N}\sum_{j=1}^{N}\Delta G_{2,j}.$ (3)

Determining the total number of unique secondary reaction sequences, $N$, is
mathematically equivalent to calculating the total number of full binary trees
with $n$ interior nodes, which yields the Catalan number sequence,
$u_{n}=1,1,2,5,14,42,132,429,...(n=0,1,2,...)$.32 Using this connection to the
Catalan numbers, we developed a non-recursive algorithm for calculating
$\overline{\Delta G_{2}}$ that is significantly faster than the equivalent
recursive solution (see Methods).

Finally, we formulate the $C_{2}$ metric such that it accounts for all
possible reaction sequences in either hull subsection to the left (L) and
right (R) of the target phase:

$C_{2}=-\left(\overline{\Delta G_{2,L}}+\overline{\Delta G_{2,R}}\right).$ (4)

The negative factor is included so that a lower $C_{2}$ value corresponds to a
more favorable selectivity. Because our definition of a secondary reaction
assumes that $\Delta G_{d}\leq 0$, the secondary competition metric obeys the
limiting behavior: $C_{2}\geq 0$.

We note that both $C_{1}$ and $C_{2}$ implicitly assume that the target phase
is thermodynamically stable (“on the hull”) under the conditions for which the
equilibrium phase diagram is derived. However, the competition metrics are
still calculable for a metastable phase by manually decreasing its energy
until it becomes stable.

###  2.2 Application to Experimental Literature

Using the competition metrics $C_{1}$ and $C_{2}$, we now assess the
selectivities of solid-state reactions previously reported in the experimental
literature and use these to rank synthesis recipes by their predicted
thermodynamic optimality. Reaction energies and competition metrics were
calculated for 3,520 unique experiments reported in the text-mined solid-state
reaction literature dataset by Kononova et al. 33. Each unique experiment
corresponds to a particular balanced reaction, maximum synthesis temperature,
and atmospheric environment (e.g., air, flowing O2, etc.). We modeled all
reactions containing up to two solid precursors and an optional gaseous
reactant (see Methods). Reactions that were reported with no particular
atmospheric environment are denoted as “closed” and modeled with Gibbs free
energies ($\Delta G_{\textrm{rxn}}$), while those with a defined environment
are denoted as “open” and modeled using grand potential energies
($\Delta\Phi_{\text{rxn}}$). The results of these calculations are shown in
Figures 4a,b.

Figure 4:  Thermodynamic analysis of synthesis recipes in the experimental
solid-state literature. Synthesis maps of 3,520 literature reactions from the
Kononova et al. dataset, plotted on a shared axis of reaction energy, $\Delta
E_{\textrm{rxn}}$, and independent axes of (a) primary competition, $C_{1}$,
and (b) secondary competition, $C_{2}$. These selectivity metrics are
constrained by their lower bounds: $C_{1}=\Delta E_{\textrm{rxn}}$ (diagonal
parity line) and $C_{2}=0$. (c) Median transformed cost ($\Gamma_{t}$)
rankings of synthesis recipes for each of the 40 most popular targets in the
dataset. The shading marks the targets with average recipes below (green) or
above (pink) the median $\Gamma_{t}$ of all experiments in the dataset
(-0.089). Selected recipes are discussed in Tables 1 and 2. The full data set
is provided in the Supporting Information.

The synthesis “maps” (Figures 4a,b) allow one to identify favorable reactions
by comparing the relative weights of the three reaction metrics: reaction
energy ($\Delta E_{\textrm{rxn}}$), primary competition ($C_{1}$), and
secondary competition ($C_{2}$). Thermodynamically optimal reactions are ones
that minimize all three metrics, resulting in placement in the lower left
region of each plot. According to our calculations, many reactions reported in
the literature are predicted to be energetically favorable and selective.
Approximately 17.3% of reactions have no exergonic competing reactions on the
interface reaction hull (i.e., they are on the bounding line $C_{1}=\Delta
E_{\textrm{rxn}}$), and 23.0% of reactions have negligible secondary
competition ($C_{2}\leq 0.001$ eV/atom). Assuming the literature reactions are
experimentally feasible, one would expect all calculated reaction energies to
be negative. Our thermodynamic modeling captures this within reasonable error:
82.8% of reactions have a negative reaction energy, and 97.0% of reactions
have $\Delta E_{\textrm{rxn}}\leq 0.1$ eV/atom. Of the reactions with positive
energies, most (87.1%) contain one or more common gases: O2 (85.1%), CO2
(48.8%), or H2O (8.4%). A major source of error in our reaction energy
calculations is likely the disagreement between the assumed and actual gas
partial pressures; it is often challenging to model the actual environmental
conditions of synthesis, as it requires that they be both 1) accurately
reported and 2) correctly extracted from the text. Another known source of
error is systematic challenges in estimating GGA-calculated formation energies
of carbonate compounds.14 However, this has been addressed and partially
mitigated with a fitted energy correction (see Methods).

To quantitatively assess thermodynamic optimality, we follow a similar
approach to prior works26, 24 and define a cost function, $\Gamma$, that
combines the driving force and reaction selectivity evaluated at a particular
set of conditions (i.e., temperature and atmosphere). In this work, we opt to
use a simple linear weighted summation of the reaction’s energy and
competition scores,

$\Gamma=x_{0}\Delta E_{\textrm{rxn}}+x_{1}C_{1}+x_{2}C_{2},$ (5)

where $\Delta E_{\text{rxn}}$ is the reaction energy (either $\Delta
G_{\textrm{rxn}}$ or $\Delta\Phi_{\textrm{rxn}}$), and $x_{0}$, $x_{1}$, and
$x_{2}$ are user-defined weights for each parameter. Due to the different
scaling of each parameter, we find that a non-equal weighting of $x_{0}=0.10$,
$x_{1}=0.45$, and $x_{2}=0.45$ produces reasonably diverse results that do not
significantly favor one parameter over another. We note that this selection is
arbitrary and subject to further optimization.

Unfortunately, closed and open reactions cannot be rigorously compared due to
their different energy scales (Gibbs free energies vs. grand potential
energies). This is further evidenced by differences in the reaction metric
distributions (Figures 4a,b). Since no ideal solution exists for ranking and
comparing reactions under different environmental conditions, we transform the
cost function to account for the energy scale difference. To do this, we apply
a power transformation to the cost distributions for the closed and open
reactions, resulting in monotonically transformed costs, $\Gamma_{t}$, whose
distributions are closer to standard normal distributions (Figure
LABEL:supp_fig:power_transform). This new variable facilitates a fairer
comparison between closed and open reactions, allowing for a more realistic
ranking of synthesis recipes.

Figure 4c shows the median values of $\Gamma_{t}$ calculated from the
synthesis recipes of the 40 most popular targets in the literature dataset
(i.e., those with the most reactions extracted from the text). Given the
extent of coverage of our thermodynamic data, we limit our analysis to only
the targets for which we have at least five recipes successfully calculated
(see Methods). In the following sections, we select several targets with costs
below and above the median $\Gamma_{t}$ value, analyzing the factors leading
to their optimal and suboptimal synthesis recipes, respectively.

####  2.2.1 Optimal Literature Recipes

Of the 40 most popular targets in the literature dataset (Figure 4c), twenty
have synthesis recipes with an average cost value ($\Gamma_{t}$) below the
dataset’s median (-0.089), indicating generally favorable thermodynamic
optimality. A selection of these targets and their highest/lowest-ranked
recipes are provided in Table 1, along with other selected reactions of
interest. DOIs and raw (untransformed) costs for each reaction are provided in
the Supporting Information.

For many targets in Table 1, the conventional reaction involving off-the-shelf
binary precursors is predicted to be thermodynamically optimal. For example,
in the synthesis of the spinel ZnGa2O4, the reaction between the binary
oxides, ZnO + Ga2O3 -¿ ZnGa2O4, is already perfectly selective ($C_{1}=\Delta
E_{\textrm{rxn}}$ and $C_{2}=0$) over all temperatures in the dataset.
Furthermore, these precursors appear in all 17 calculated literature recipes
for this target. The favorability of the conventional route seems to apply to
several other targets presented here, including BiVO4, CoFe2O4, Y3Al5O12, and
BiFeO3. Since the precursors are already favorable for these phases, we can
instead analyze which conditions are most favorable for target phase
formation. For CoFe2O4, higher temperatures in an open oxygen environment
appear to be the most optimal. On the other hand, lower temperatures in an
open oxygen environment appear to favor the production of BiVO4 and BiFeO3.
Finally, for Y3Al5O12, there appears to be a tradeoff between selectivity
(which is optimal at lower temperatures) and driving force (which is optimal
at high temperatures); it appears that intermediate temperatures (600 ℃) in a
closed environment result in the most suitable compromise. We note, however,
that there are many reasons to use specific processing conditions outside of
the pursuit of target phase purity (e.g., improved density, annihilation of
defects, optimization of crystallite sizes, etc.). These alternative reasons
may explain some of the variability of conditions reported in the literature.

Interestingly, several targets in Table 1 feature synthesis recipes that
appear extremely unfavorable. These often involve elemental precursors, such
as Bi + 0.5 V2O5 + 0.75 O2 -¿ BiVO4, or Bi + 0.5 Fe2O3 + 0.75 O2 -¿ BiFeO3.
Referencing the original articles from which these reactions were sourced 34,
35 suggests that these precursors were not actually used, and the reaction’s
inclusion in the dataset is likely the result of an error in the literature
extraction process. This explains some other impractical and highly suboptimal
routes in the dataset, such as reactions involving alkali metal precursors
(e.g., SiO2 + 4 Li + O2 -¿ Li4SiO4).

For other targets such as LiCoO2, Li4SiO4, and LiNiO2, there appears to be
great variability in precursor selection in the literature. For LiCoO2, the
optimal synthesis recipe from our calculations is the reaction of LiOH and
Co3O4 in a flowing oxygen environment at low to intermediate temperatures
(i.e., 700 ℃). The use of Co3O4 appears to significantly outperform recipes
using CoCO3. Additionally, using LiOH may offer a thermodynamic advantage over
Li2CO3, although with little effect on the reaction selectivity. A similar
conclusion applies to the synthesis of Li4SiO4, although the use of lithium
carbonate appears to be more favorable at high temperatures in an open-oxygen
environment. LiOH (particularly the monohydrate) also appears to offer some
advantage in the synthesis of LiNiO2, especially in oxygen at low temperatures
(480-600 ℃).

Table 1: Thermodynamic analysis of optimal experimental synthesis recipes for selected popular targets in the literature. The ranking of reactions is determined by the transformed cost, $\Gamma_{t}$. The highest- and lowest-ranked reactions are shown along with selected reactions of interest. Raw costs and DOIs are available in the Supporting Information. |  |  Reaction | Temp. | Energy | Open |  C1 |  C2 | $\Gamma_{t}$
---|---|---|---|---|---|---|---|---
Target | Rank |  | (ºC) | (eV/at) |  | (eV/at) | (eV/at) | (a.u.)
ZnGa2O4 | 1 |  ZnO + Ga2O3 $\longrightarrow$ ZnGa2O4 | 1400 | -0.218 | O | -0.218 | 0.000 | -1.622
12 |  ZnO + Ga2O3 $\longrightarrow$ ZnGa2O4 | 1200 | -0.088 |  | -0.088 | 0.000 | -0.969
17 |  ZnO + Ga2O3 $\longrightarrow$ ZnGa2O4 | 500 | -0.067 |  | -0.067 | 0.000 | -0.818
BiVO4 | 1 |  0.5 V2O5 \+ 0.5 Bi2O3 $\longrightarrow$ BiVO4 | 600 | -0.529 | O | -0.210 | 0.171 | -1.156
5 |  0.5 V2O5 \+ 0.5 Bi2O3 $\longrightarrow$ BiVO4 | 500 | -0.169 |  | -0.061 | 0.034 | -0.713
12 |  Bi + 0.5 V2O5 \+ 0.75 O2 $\longrightarrow$ BiVO4 | 600 | -1.860 | O | 0.801 | 4.489 | 2.412
CoFe2O4 | 1 |  CoO + Fe2O3 $\longrightarrow$ CoFe2O4 | 1250 | -0.088 | O | -0.062 | 0.000 | -0.877
3 |  0.3333 Co3O4 \+ Fe2O3 $\longrightarrow$ CoFe2O4 \+ 0.1667 O2 | 950 | -0.058 | O | -0.018 | 0.011 | -0.662
11 |  2 Fe + Co + 2 O2 $\longrightarrow$ CoFe2O4 | 1400 | -1.738 | O | 0.223 | 2.957 | 2.111
14 |  CoCO3 \+ Fe2O3 $\longrightarrow$ CoFe2O4 \+ CO2 | 1400 | -0.215 |  | 0.156 | 0.370 | 2.781
LiCoO2 | 1 |  LiOH + 0.3333 Co3O4 \+ 0.08333 O2 $\longrightarrow$ LiCoO2 \+ 0.5 H2O | 700 | -0.120 | O | -0.012 | -0.000 | -0.727
4 |  0.5 Li2CO3 \+ 0.3333 Co3O4 \+ 0.08333 O2 $\longrightarrow$ LiCoO2 \+ 0.5 CO2 | 950 | -0.061 | O | -0.014 | -0.000 | -0.692
23 |  0.5 Li2CO3 \+ CoCO3 \+ 0.25 O2 $\longrightarrow$ LiCoO2 \+ 1.5 CO2 | 300 | -0.121 | O | 0.154 | 0.357 | 0.532
30 |  Co + LiOH$\cdot$H2O + 0.75 O2 $\longrightarrow$ LiCoO2 \+ 1.5 H2O | 900 | -0.382 | O | 1.001 | 1.454 | 1.992
Li4SiO4 | 1 |  SiO2 \+ 2 Li2CO3 $\longrightarrow$ Li4SiO4 \+ 2 CO2 | 1445 | -0.040 | O | -0.017 | -0.000 | -0.683
7 |  SiO2 \+ 4 LiOH $\longrightarrow$ Li4SiO4 \+ 2 H2O | 700 | -0.049 |  | -0.004 | -0.000 | -0.412
10 |  SiO2 \+ 2 Li2CO3 $\longrightarrow$ Li4SiO4 \+ 2 CO2 | 1150 | -0.010 |  | -0.005 | -0.000 | -0.366
18 |  SiO2 \+ 4 Li + O2 $\longrightarrow$ Li4SiO4 | 800 | -2.133 | O | 0.163 | 2.830 | 2.038
LiNiO2 | 1 |  NiO + LiOH$\cdot$H2O + 0.25 O2 $\longrightarrow$ LiNiO2 \+ 1.5 H2O | 480 | 0.008 | O | 0.010 | 0.002 | -0.559
4 |  NiO + 0.5 Li2O + 0.25 O2 $\longrightarrow$ LiNiO2 | 800 | 0.032 | O | 0.058 | 0.021 | -0.344
6 |  Ni(OH)2 \+ LiOH + 0.25 O2 $\longrightarrow$ LiNiO2 \+ 1.5 H2O | 770 | 0.074 | O | 0.074 | -0.000 | -0.331
24 |  NiO + 0.5 Li2O2 $\longrightarrow$ LiNiO2 | 800 | -0.035 |  | 0.067 | 0.147 | 0.973
27 |  Ni(OH)2 \+ Li + 0.5 O2 $\longrightarrow$ LiNiO2 \+ H2O | 700 | -0.536 | O | 1.770 | 3.044 | 2.396
Y3Al5O12 | 1 |  1.5 Y2O3 \+ 2.5 Al2O3 $\longrightarrow$ Y3Al5O12 | 600 | -0.071 |  | -0.034 | 0.000 | -0.623
15 |  1.5 Y2O3 \+ 2.5 Al2O3 $\longrightarrow$ Y3Al5O12 | 1727 | -0.240 | O | -0.051 | 0.221 | -0.260
36 |  5 Al(OH)3 \+ 1.5 Y2O3 $\longrightarrow$ Y3Al5O12 \+ 7.5 H2O | 1300 | -0.120 |  | -0.007 | 0.118 | 0.202
BiFeO3 | 1 |  0.5 Bi2O3 \+ 0.5 Fe2O3 $\longrightarrow$ BiFeO3 | 600 | -0.073 | O | 0.034 | 0.053 | -0.386
4 |  0.5 Bi2O3 \+ 0.5 Fe2O3 $\longrightarrow$ BiFeO3 | 100 | -0.011 |  | 0.000 | 0.004 | -0.312
38 |  Bi + 0.5 Fe2O3 \+ 0.75 O2 $\longrightarrow$ BiFeO3 | 900 | -1.285 | O | 1.087 | 3.558 | 2.364

####  2.2.2 Suboptimal Literature Recipes

Reactions can still be successful despite high $C_{1}$ and $C_{2}$ values.
However, our thermodynamic assessment suggests that these reactions are
suboptimal and likely require some combination of 1) long heating times to
promote thermodynamic equilibrium, 2) follow-up regrinding steps, or 3) fine-
tuning of temperature and reaction atmosphere. In Table 2, we highlight
several popular target materials that are associated with higher-than-average
costs for their synthesis recipes.

Table 2: Thermodynamic analysis of suboptimal experimental synthesis recipes for selected popular targets in the literature. The ranking of reactions is determined by the transformed cost, $\Gamma_{t}$. The highest- and lowest-ranked reactions are shown along with selected reactions of interest. Raw costs and DOIs are available in the Supporting Information. |  |  Reaction | Temp. | Energy | Open |  C1 |  C2 | $\Gamma_{t}$
---|---|---|---|---|---|---|---|---
Target | Rank |  | (ºC) | (eV/at) |  | (eV/at) | (eV/at) | (a.u.)
PbTiO3 | 1 |  PbCO3 \+ TiO2 $\longrightarrow$ PbTiO3 \+ CO2 | 1200 | -0.402 | O | 0.118 | 0.750 | 0.960
2 |  PbO + TiO2 $\longrightarrow$ PbTiO3 | 1100 | -0.360 | O | 0.186 | 0.694 | 0.987
13 |  PbO + TiO2 $\longrightarrow$ PbTiO3 | 800 | -0.163 |  | 0.136 | 0.288 | 2.172
Na2Ti3O7 | 1 |  2 NaOH + 3 TiO2 $\longrightarrow$ Na2Ti3O7 \+ H2O | 750 | -0.058 | O | 0.079 | 0.046 | -0.273
3 |  Na2CO3 \+ 3 TiO2 $\longrightarrow$ Na2Ti3O7 \+ CO2 | 1100 | 0.009 |  | 0.108 | 0.104 | 1.025
8 |  Na2CO3 \+ 3 TiO2 $\longrightarrow$ Na2Ti3O7 \+ CO2 | 80 | 0.053 |  | 0.202 | 0.200 | 2.344
NaTaO3 | 1 |  0.5 Na2CO3 \+ 0.5 Ta2O5 $\longrightarrow$ NaTaO3 \+ 0.5 CO2 | 1150 | -0.088 |  | 0.035 | 0.085 | 0.300
4 |  0.5 Na2CO3 \+ 0.5 Ta2O5 $\longrightarrow$ NaTaO3 \+ 0.5 CO2 | 1200 | -0.233 | O | 0.285 | 0.665 | 1.098
9 |  0.5 Na2CO3 \+ 0.5 Ta2O5 $\longrightarrow$ NaTaO3 \+ 0.5 CO2 | 900 | -0.194 | O | 0.441 | 0.866 | 1.435
Ca3(PO4)2 | 1 |  CaCO3 \+ Ca2P2O7 $\longrightarrow$ Ca3(PO4)2 \+ CO2 | 800 | -0.068 |  | -0.013 | -0.000 | -0.494
3 |  3 CaCO3 \+ 2 NH4H2PO4 $\longrightarrow$ Ca3(PO4)2 \+ 3 CO2 \+ 3 H2O + 2 H3N | 1150 | -0.199 | H | 0.078 | 0.269 | 0.187
4 |  3 CaCO3 \+ 2 (NH4)2HPO4 $\longrightarrow$ Ca3(PO4)2 \+ 3 CO2 \+ 3 H2O + 4 H3N | 1200 | -0.156 | N | 0.105 | 0.374 | 0.463
10 |  3 CaO + P2O5 $\longrightarrow$ Ca3(PO4)2 | 550 | -0.641 |  | -0.007 | 0.463 | 1.673
12 |  3 CaCO3 \+ 2 (NH4)2HPO4 $\longrightarrow$ Ca3(PO4)2 \+ 3 CO2 \+ 3 H2O + 4 H3N | 1200 | -0.143 |  | 0.094 | 0.336 | 2.232
Li2MnO3 | 1 |  Li2CO3 \+ MnO2 $\longrightarrow$ Li2MnO3 \+ CO2 | 975 | -0.096 | O | -0.015 | 0.015 | -0.670
4 |  2 LiOH + MnO2 $\longrightarrow$ Li2MnO3 \+ H2O | 650 | -0.092 |  | -0.024 | 0.010 | -0.533
5 |  MnO2 \+ 2 LiOH$\cdot$H2O $\longrightarrow$ Li2MnO3 \+ 3 H2O | 70 | -0.059 |  | 0.001 | 0.016 | -0.295
14 |  MnCO3 \+ Li2CO3 \+ 0.5 O2 $\longrightarrow$ Li2MnO3 \+ 2 CO2 | 500 | -0.196 | O | 0.262 | 0.545 | 0.942
15 |  MnCO3 \+ 2 LiOH$\cdot$H2O + 0.5 O2 $\longrightarrow$ Li2MnO3 \+ CO2 \+ 3 H2O | 450 | -0.187 | O | 0.253 | 0.554 | 0.943
25 |  2 Li + MnO2 \+ 0.5 O2 $\longrightarrow$ Li2MnO3 | 700 | -1.971 | O | 0.400 | 2.399 | 1.989
LiMn2O4 | 1 |  0.5 Li2CO3 \+ 2 MnO2 $\longrightarrow$ LiMn2O4 \+ 0.5 CO2 \+ 0.25 O2 | 700 | -0.032 | O | 0.041 | 0.051 | -0.347
2 |  LiOH + 2 MnO2 $\longrightarrow$ LiMn2O4 \+ 0.5 H2O + 0.25 O2 | 1000 | -0.052 | O | 0.031 | 0.067 | -0.343
12 |  0.5 Li2CO3 \+ Mn2O3 \+ 0.25 O2 $\longrightarrow$ LiMn2O4 \+ 0.5 CO2 | 950 | -0.073 | O | 0.030 | 0.110 | -0.240
47 |  2 MnCO3 \+ 0.5 Li2CO3 \+ 0.75 O2 $\longrightarrow$ LiMn2O4 \+ 2.5 CO2 | 400 | -0.319 | O | 0.148 | 0.488 | 0.664
63 |  Mn2O3 \+ 0.5 Li2O2 $\longrightarrow$ LiMn2O4 | 800 | -0.138 |  | 0.121 | 0.283 | 2.072
LiFePO4 | 1 |  0.3333 Li3PO4 \+ 0.3333 Fe3(PO4)2 $\longrightarrow$ LiFePO4 | 600 | -0.020 |  | -0.008 | 0.002 | -0.386
2 |  LiPO3 \+ 0.5 Fe2O3 $\longrightarrow$ LiFePO4 \+ 0.25 O2 | 1200 | -0.151 | O | 0.108 | 0.120 | -0.060
3 |  0.3333 Li3PO4 \+ 0.3333 Fe3(PO4)2$\cdot$8H2O $\longrightarrow$ LiFePO4 \+ 2.667 H2O | 800 | -0.065 |  | 0.000 | 0.068 | 0.012
4 |  LiOH$\cdot$H2O + Fe(PO4)$\cdot$2H2O $\longrightarrow$ LiFePO4 \+ 3.5 H2O + 0.25 O2 | 650 | -0.111 | O | 0.062 | 0.247 | 0.149
5 |  0.5 Li2CO3 \+ FePO4 $\longrightarrow$ LiFePO4 \+ 0.5 CO2 \+ 0.25 O2 | 700 | 0.006 | O | 0.189 | 0.122 | 0.207
BaTiO3 | 1 |  BaCO3 \+ TiO2 $\longrightarrow$ BaTiO3 \+ CO2 | 1050 | -0.056 | O | 0.075 | 0.065 | -0.230
16 |  BaCO3 \+ TiO2 $\longrightarrow$ BaTiO3 \+ CO2 | 1100 | -0.028 |  | 0.025 | 0.026 | -0.045
37 |  BaO + TiO2 $\longrightarrow$ BaTiO3 | 800 | -0.239 |  | 0.061 | 0.168 | 0.778
40 |  BaO + TiO2 $\longrightarrow$ BaTiO3 | 1300 | -0.225 |  | 0.080 | 0.167 | 0.916

Our findings suggest that the targets in Table 2 require a more judicious
synthesis due to inherently greater competition in their phase space. Many
targets are ranked poorly because the conventional reaction is predicted to be
suboptimal. This appears to be true for lead titanate (PbTiO3), which on
average, has the most suboptimal recipe of any target investigated (Figure
4c). The high $C_{1}$ and $C_{2}$ values for PbTiO3 synthesis seem to be
almost entirely related to the instability of the PbO precursor, both with
respect to decomposition and to the formation of a solid-solution phase. In
our calculations, the latter manifests as the predicted stabilization of a
theoretical Pb15TiO17 phase. Our modeling of PbO instability supports
experimental observations suggesting that the volatility of PbO at higher
temperatures results in a challenging PbTiO3 synthesis.36 The use of PbCO3 as
an alternative appears to perform similarly, albeit with a slightly more
favorable driving force and lower $C_{1}$.

For Ca3(PO4)2, Li2MnO3, and LiFePO4, the highest-ranked synthesis recipes
appear to be nearly thermodynamically optimal already (i.e., low $C_{1}$ and
$C_{2}$). The optimal recipe for Ca3(PO4)2 is the reaction of stoichiometric
amounts of CaCO3 and Ca2P2O7 at moderate temperatures (800 ℃) in a closed
environment. For Li2MnO3, the reaction of Li2CO3 and MnO2 in open-oxygen
environments at moderately high temperatures (900-975 ℃) is favorable, and the
use of LiOH in a closed environment at lower temperatures (650 ℃) appears to
be similarly advantageous.37 Finally, for the synthesis of LiFePO4, the use of
Li3PO4 and Fe3(PO4)2 in a closed environment at lower temperatures (600 ℃)38
appears to be highly selective and greatly outperforms the other recipes
analyzed in the literature dataset.

For the remaining targets Na2Ti3O7, NaTaO3, LiMn2O4, and BaTiO3, even the
highest-ranked literature synthesis recipes are theoretically suboptimal,
which suggests that these materials are suitable candidates for future
synthesis optimization efforts. Recipes for Na2Ti3O7 generally feature poor
driving forces and low selectivities; however, using NaOH as an alternative
precursor mitigates this some. All NaTaO3 recipes in our dataset used the same
precursors, but those with open-oxygen environments resulted in substantially
higher phase competition. LiMn2O4 synthesis appears to follow similar trends
as those discussed previously for LiMn2O3; the lithium carbonate route in open
air appears to be the most optimal of the recipes explored, despite a somewhat
low driving force and moderate competition.39 The use of a MnCO3 precursor is
not recommended due to greatly decreased selectivity. Finally, for BaTiO3, the
conventional BaCO3 route at high temperatures (1050 ℃) and open oxygen
conditions appears to be the most favorable; however, with this route, the
driving force is still somewhat low and features a moderate amount of phase
competition. The BaTiO3 system will be examined extensively in the following
sections as an experimental case study for assessing reaction selectivity.

The literature reaction data reveal an inherent tradeoff between driving force
and selectivity. Reactions with only elemental precursors (e.g., Li metal, O2
gas) typically have significantly greater driving forces but also much higher
competition scores (i.e., lower selectivity), with median values of $\Delta
G_{\textrm{rxn}}=-0.272$, $C_{1}=0.016$, and $C_{2}=0.112$ eV/atom for the
closed reactions. For reactions with at least one binary (two-element)
precursor, these values shift to $\Delta G_{\textrm{rxn}}=-0.037$,
$C_{1}=0.006$, and $C_{2}=0.020$ eV/atom. In other words, sourcing an element
from a precursor with pre-formed bonds yields an interface reaction hull
representing a more selective slice of the phase diagram, but at the expense
of sacrificing some of the available reaction energy. Fortunately, this
tradeoff is not universal and can be circumvented by considering more complex
precursor chemistries containing additional elements other than those in the
target composition. For example, in oxide synthesis, the use of hydroxides,
carbonates, and salts (as in metathesis reactions) often permits the formation
of thermodynamically favorable byproducts that increase the reaction’s driving
force and selectivity. The use of additional elements beyond those found in
the target composition has been dubbed “hyperdimensional chemistry”40 due to
its connection to phase diagram geometry; adding a new compositional axis
greatly increases the number of ways the phase space can be sliced, allowing
one to thermodynamically “shortcut” otherwise unavoidable competing impurity
phases. However, not all elemental precursors are unselective, and the
necessity of these alternative routes depends on the degree of phase
competition in the chemical system of interest. It is typically favorable to
synthesize a binary target comprised of elements commonly existing in only a
single oxidation state; for example, Cd + Te -¿ CdTe has both perfect
selectivity ($C_{1}=-0.058$, $C_{2}=0.000$ eV/atom) and a high driving force
($\Delta G_{\textrm{rxn}}=-0.576$ eV/atom) at 800 ℃.

###  2.3 Case Study: Synthesis of Barium Titanate (BaTiO3)

To investigate whether our reaction selectivity metrics accurately describe
impurity formation, we designed and performed experiments testing the
influence of precursor selection on the reaction pathway observed during
solid-state synthesis. To suggest these experiments, we developed a
computational synthesis planning workflow that integrates our proposed
selectivity metrics with previous methods for computing solid-state reaction
networks.26 The workflow has five user inputs: 1) target composition, 2)
additional elements, 3) temperature, 4) thermodynamic stability cutoff (i.e.,
maximum $E_{\textrm{hull}}$), and 5) chemical potential of an open element, if
any. The workflow returns a ranked list of possible synthesis reactions based
on the cost function implemented in Equation 5.

We selected the ferroelectric barium titanate (BaTiO3) as a case study system
for testing our predictions. In addition to being a technologically important
and well-studied material in the literature, BaTiO3 is an ideal target for
investigating the thermodynamic selectivity of solid-state reactions due to
the high number of competing phases in the Ba–Ti–O chemical system (Figure
LABEL:supp_fig:batio3_irh). Conventional routes from binary oxides are well-
known to proceed through intermediates 41, and at least 12 unique ternary
compositions have been experimentally observed, including the compositional
neighbors BaTi2O5 and Ba2TiO4, which frequently appear during synthesis (the
latter in particular). Other commonly observed minor impurities include
Ba4Ti13O30, BaTi4O9, and Ba6Ti17O40. Additional compositions in this space,
such as Ba2Ti9O20, BaTi5O11, Ba4Ti12O27, and Ba2Ti6O13, have been previously
synthesized but are less commonly observed as intermediates or impurity
phases.41 Another motivation for selecting BaTiO3 is the kinetic accessibility
of its chemical system; even with such a high number of competing Ba–Ti–O
phases, the solid-state synthesis of barium titanate is generally considered
facile. The ease with which Ba–Ti–O phases can be synthesized suggests that
kinetic factors do not significantly divert reaction outcomes from those that
are thermodynamically favorable, providing greater justification for our
attempts at assessing the likelihood of intermediate/impurity formation
through a purely thermodynamic lens.

Using our synthesis planning workflow, we computed and ranked 82,985 binary
(two-precursor) synthesis reactions producing BaTiO3, selected from a massive
18-element reaction network composed of all 2,536,160 enumerated binary
reactions among 2,417 phases with energies $\leq 0.050$ meV/atom above the
hull. To capture alternative chemistries (e.g., metathesis reactions, gas-
forming reactions, ion exchange reactions, etc.), we selected a chemical
system consisting of the target elements (Ba, Ti, O), alkali metals (Li, Na,
K) and alkaline earth metals (Mg, Ca, Sr), halogens (F, Cl, Br), chalcogens
(S), pnictogens (N, P), and other common elements (B, C, H). The $\Delta
G_{\textrm{rxn}}$, $C_{1}$, and $C_{2}$ values for all calculated synthesis
reactions are shown in the synthesis maps illustrated in Figure 5. We
calculated these values at an intermediate temperature of $T=600$ ℃, which is
near the median of our experimentally accessible temperature range (see
Methods). We also considered the modeling of open-O2 reactions; similar
results are shown in Figure LABEL:supp_fig:batio3_open for 62,133 open
reactions in a more constrained subsystem.

Figure 5:  Synthesis maps of 82,985 calculated reactions producing BaTiO3.
Target reactions and their competition scores are extracted from an 18-element
network of 2,536,160 reactions modeled at a temperature of $T=600$ ℃.
Reactions are plotted on a shared axis of reaction energy, $\Delta
G_{\text{rxn}}$, and on independent axes of (a) primary competition, $C_{1}$,
and (b) secondary competition, $C_{2}$. The sharp boundaries are lower-bound
limits of $C_{1}=\Delta G_{\textrm{rxn}}$ and $C_{2}=0$. Blue diamonds
indicate selected reactions experimentally tested in this work. Orange squares
represent reactions on the three-dimensional Pareto front of $\Delta
G_{\text{rxn}}$, $C_{1}$, and $C_{2}$.

The determination of an optimal synthesis reaction can be formulated as an
optimization problem of simultaneous minimization of the three reaction
metrics: $\Delta G_{\text{rxn}}$, $C_{1}$, and $C_{2}$. A common approach for
multi-objective optimization in synthesis planning is the identification of
the Pareto front.14 Here, we calculate a three-dimensional Pareto front for
the BaTiO3 synthesis reactions (Table 3). The Pareto front reactions for the
open-O2 system are provided in the Supporting Information.

Table 3: Pareto front reactions to BaTiO3 and their associated Gibbs free energies, $\Delta G_{\text{rxn}}$ ($T=600$ ℃), primary competition scores, $C_{1}$, secondary competition scores, $C_{2}$, and costs, $\Gamma$. All units are in eV/atom. |  Reaction | $\Delta G_{\text{rxn}}$ |  C1 |  C2 | $\Gamma$ | Theoretical
---|---|---|---|---|---|---
Rank |  |  |  |  |  |
1 |  2 TiP + 4.5 BaCO3 $\longrightarrow$ BaTiO3 \+ Ba3(PO4)2 \+ 0.5 BaTi2O5 \+ 4.5 C | -0.465 | -0.465 | -0.000 | -0.256 |
2 |  3 Ba(NO3)2 \+ 3.333 LiTi2N3 $\longrightarrow$ BaTiO3 \+ 2 BaTi2O5 \+ 1.667 Li2TiO3 \+ 8 N2 | -0.868 | -0.302 | -0.000 | -0.223 |  LiTi2N3
4 |  MgTiN2 \+ 0.6 Ba(NO3)2 $\longrightarrow$ MgO + 0.2 BaTiO3 \+ 0.4 BaTi2O5 \+ 1.6 N2 | -0.901 | -0.223 | 0.008 | -0.187 |
6 |  8 O2 \+ Ba4TiP4 $\longrightarrow$ BaTiO3 \+ Ba3P4O13 | -2.330 | 0.062 | 0.136 | -0.144 |
7 |  4 BaO2 \+ 2 Ti $\longrightarrow$ Ba3TiO5 \+ BaTiO3 | -1.489 | -0.021 | 0.036 | -0.142 |
18 |  0.25 Mg4TiN4 \+ 0.3 Ba(NO3)2 $\longrightarrow$ MgO + 0.2 BaTiO3 \+ 0.05 Ba2TiO4 \+ 0.8 N2 | -0.983 | -0.057 | 0.078 | -0.089 |  Mg4TiN4
28 |  TiO + BaO2 $\longrightarrow$ BaTiO3 | -1.044 | -0.034 | 0.099 | -0.075 |
33 |  0.75 Ti7P4 \+ 0.3333 Ba5P3O12F $\longrightarrow$ BaTiO3 \+ 0.25 Ba2TiO4 \+ 4 TiP + 0.1667 BaF2 | -0.129 | -0.129 | -0.000 | -0.071 |  Ba5P3O12F
54 |  MgTi(SO4)3 \+ 4 BaMg2 $\longrightarrow$ BaTiO3 \+ 9 MgO + 3 BaS | -1.515 | -0.005 | 0.225 | -0.052 |  MgTi(SO4)3
63 |  4 Ti(NO3)4 \+ 5 Ba4P2O $\longrightarrow$ Ba2TiO4 \+ 3 BaTiO3 \+ 5 Ba3(PO4)2 \+ 8 N2 | -1.570 | -0.018 | 0.261 | -0.047 |
85 |  2.25 TiC + 0.5 Ba3(PO4)2 $\longrightarrow$ BaTiO3 \+ 0.25 Ba2TiO4 \+ TiP + 2.25 C | -0.067 | -0.067 | -0.000 | -0.037 |
90 |  1.25 Ba5(TiN3)2 \+ 1.125 Ti(NO3)4 $\longrightarrow$ BaTiO3 \+ 2.625 Ba2TiO4 \+ 6 N2 | -1.287 | -0.029 | 0.236 | -0.035 |
108 |  0.75 Ti7P4 \+ 0.5 Ba3(PO4)2 $\longrightarrow$ BaTiO3 \+ 0.25 Ba2TiO4 \+ 4 TiP | -0.150 | -0.038 | -0.000 | -0.032 |
199 |  1.45 Ti(ClO4)4 \+ 0.8 Ba6Mg23 $\longrightarrow$ BaTiO3 \+ 0.45 Ba2TiO4 \+ 18.4 MgO + 2.9 BaCl2 | -2.540 | -0.008 | 0.527 | -0.020 |
259 |  3 BaO2 \+ Ti2N2O $\longrightarrow$ BaTiO3 \+ Ba2TiO4 \+ N2 | -0.913 | -0.099 | 0.268 | -0.015 |  Ti2N2O
498 |  TiNCl + 1.5 BaO2 $\longrightarrow$ BaTiO3 \+ 0.5 BaCl2 \+ 0.5 N2 | -0.967 | -0.150 | 0.350 | -0.007 |
547 |  Li2TiO3 \+ 0.3333 Ba3(PO4)2 $\longrightarrow$ BaTiO3 \+ 0.6667 Li3PO4 | -0.011 | -0.011 | -0.000 | -0.006 |
552 |  1.5 BaO2 \+ TiNF $\longrightarrow$ BaTiO3 \+ 0.5 BaF2 \+ 0.5 N2 | -0.960 | -0.153 | 0.354 | -0.006 |  TiNF
651 |  TiBrN + 1.5 BaO2 $\longrightarrow$ BaTiO3 \+ 0.5 BaBr2 \+ 0.5 N2 | -0.963 | -0.152 | 0.357 | -0.004 |
1850 |  Ba3(PO4)2 \+ Ca4Ti3O10 $\longrightarrow$ Ca4P2O9 \+ 3 BaTiO3 | 0.010 | 0.010 | -0.000 | 0.005 |
2825 |  0.05882 Ba2Mg17 \+ 0.1118 Ti(NO3)4 $\longrightarrow$ MgO + 0.1059 BaTiO3 \+ 0.005882 Ba2TiO4 \+ 0.2235 N2 | -1.896 | -0.004 | 0.447 | 0.010 |
6598 |  3 Ti(SO4)2 \+ 8 BaMg2 $\longrightarrow$ BaTiO3 \+ 16 MgO + 6 BaS + BaTi2O5 | -1.628 | -0.006 | 0.413 | 0.020 |
7657 |  2 BaO2 \+ 0.25 NaTi5(NCl)5 $\longrightarrow$ BaTiO3 \+ 0.25 Ba2TiO4 \+ 0.25 NaCl + 0.5 BaCl2 \+ 0.625 N2 | -0.932 | -0.079 | 0.336 | 0.023 |  NaTi5(NCl)5
8896 |  0.5 Ba5(TiN3)2 \+ 3 KNOF2 $\longrightarrow$ BaTiO3 \+ 3 KF + 1.5 BaF2 \+ 3 N2 | -1.170 | -0.040 | 0.356 | 0.025 |
13191 |  TiO2 \+ 0.5 Ba2Ca(BO2)6 $\longrightarrow$ BaTiO3 \+ 0.5 Ca(B3O5)2 | 0.060 | 0.060 | -0.000 | 0.033 |
15339 |  0.5 Ba5(TiN3)2 \+ 3 NOF $\longrightarrow$ BaTiO3 \+ 1.5 BaF2 \+ 3 N2 | -1.667 | -0.058 | 0.509 | 0.036 |
39153 |  0.5 Ba5(TiN3)2 \+ 1.5 SO2 $\longrightarrow$ BaTiO3 \+ 1.5 BaS + 1.5 N2 | -1.102 | -0.042 | 0.450 | 0.074 |
45245 |  0.5 Ti2S + 1.5 BaO2 $\longrightarrow$ BaTiO3 \+ 0.5 BaS | -1.366 | -0.068 | 0.563 | 0.086 |
57844 |  0.8 Ba5(TiN3)2 \+ 0.45 Ti(ClO4)4 $\longrightarrow$ BaTiO3 \+ 1.05 Ba2TiO4 \+ 0.9 BaCl2 \+ 2.4 N2 | -1.729 | -0.077 | 0.728 | 0.120 |
64696 |  5 MgTiH4 \+ 3 Ba(NO3)2 $\longrightarrow$ BaTiO3 \+ 5 MgO + 10 H2 \+ 2 BaTi2O5 \+ 3 N2 | -1.142 | -0.078 | 0.659 | 0.147 |
65291 |  0.8 Ba5(TiN3)2 \+ 1.8 Br2O3 $\longrightarrow$ BaTiO3 \+ 0.6 Ba2TiO4 \+ 1.8 BaBr2 \+ 2.4 N2 | -1.810 | -0.055 | 0.790 | 0.150 |
65313 |  0.5 Ba5(TiN3)2 \+ 1.5 BrO2F $\longrightarrow$ BaTiO3 \+ 1.5 BaBrF + 1.5 N2 | -1.878 | -0.053 | 0.804 | 0.150 |
66253 |  3.5 Ba5(TiN3)2 \+ 9 ClO3 $\longrightarrow$ BaTiO3 \+ 6 Ba2TiO4 \+ 4.5 BaCl2 \+ 10.5 N2 | -1.841 | -0.021 | 0.775 | 0.155 |

Many Pareto-optimal reactions feature unconventional reactants and byproducts.
BaO2 and Ba5(TiN3)2 are the most commonly appearing precursors (eight times
each), followed by Ba(NO3)2 and Ba3(PO4)2 (four times each). Nearly all
reactions (31 of 33) involve precursors containing additional elements other
than Ba, Ti, and O. In particular, nitrogen is used in over half of the
reactions (19 of 33), each featuring N2 gas formation. While unconventional,
the high prevalence of nitride precursors in the Pareto front is not
theoretically unreasonable; nitrides generally have less negative formation
energies than oxides, making oxide formation with N2 gas evolution both
energetically and entropically favorable.

However, many of the reactions appearing on the Pareto front are impractical
from an experimental standpoint. For example, the aforementioned nitride
precursors are likely challenging to synthesize and handle. Other Pareto front
reactions involve theoretical phases (e.g., LiTi2N3), uncommon or toxic
precursors (e.g., Ba5(TiN3)2, SO2), difficult-to-remove byproducts (e.g.,
Ba2TiO4), or refractory precursors (e.g., TiC). Some of these suggested
reactions can be removed easily with user-applied filters that account for
specific experimental restrictions. For example, one can supply a list of
available precursor compositions (“off-the-shelf” phases) or composition types
to be avoided (e.g., sulfides, acids, etc.). Other reactions require a more
careful manual inspection to validate experimental practicality, particularly
with regard to refractory precursors. This challenge can be mitigated by the
future inclusion of data or models relevant to solid-state reaction kinetics
(e.g., defect formation energies) and the prediction of physical properties
(e.g., melting points, hardness).42 The unorthodox nature of the unprocessed
reactions is an advantage of our approach, however, and permits synthesis
recommendations that expand beyond traditional chemical intuition.

We selected nine reactions from Figure 5 to test experimentally. The
calculated thermodynamic metrics for these reactions are provided in Table 4.
We intentionally selected reactions spanning various precursor chemistries,
free energies, and competition scores. To prioritize the study of impurity-
forming reactions and avoid the aforementioned practicality challenges, we did
not explicitly include any reactions on the Pareto front. The conventional
synthesis route, BaCO3 + TiO2 -¿ BaTiO3 + CO2, was chosen as a baseline
reference (Expt. 1). A positive energy is calculated for this reaction
($\Delta G_{\textrm{rxn}}$=+0.042 eV/atom at $T=600$ ℃); however, this is
likely due to residual uncorrected error in the calculated energy of BaCO3,
which is not included in the NIST-JANAF dataset. We did not test the analogous
reaction with BaO precursor due to its hygroscopic nature, which makes it
difficult to handle. However, we did test the alternative reaction from barium
peroxide, BaO2 (Expt. 2). We included two reactions that form BaTiO3 directly
from at least one other ternary phase (Expts. 3, 4). A reaction with a Ti
metal precursor was selected due to its extremely high $C_{2}$ score (Expt.
5). Two metathesis reactions (Expts. 6, 7) were selected due to their
predicted high performance, including the unconventional use of a sulfide
precursor (BaS). The final two reactions (Expts. 8, 9) were selected for being
endergonic ($\Delta G_{\text{rxn}}>0$) to validate the accuracy of our free
energy predictions. We note that several experiments feature precursors that
are not easily purchasable from a chemical supplier (e.g., Ba2TiO4, Na2TiO3);
these phases were synthesized following recipes reported in the literature
(see Methods).

Table 4: Selected experimental BaTiO3 synthesis reactions and their associated Gibbs free energies, $\Delta G_{\text{rxn}}$ ($T=600$ ℃), primary competition scores, $C_{1}$, secondary competition scores, $C_{2}$, and costs, $\Gamma$. | Reaction | $\Delta G_{\text{rxn}}$ |  C1 |  C2 | $\Gamma$
---|---|---|---|---|---
Expt. |  | (eV/at) | (eV/at) | (eV/at) | (eV/at)
1 |  BaCO3 \+ TiO2 $\longrightarrow$ BaTiO3 \+ CO2 | 0.042 | 0.043 | 0.000 | 0.024
2 |  BaO2 \+ TiO2 $\longrightarrow$ BaTiO3 \+ 0.5 O2 | -0.180 | 0.021 | 0.119 | 0.045
3 |  Ba2TiO4 \+ TiO2 $\longrightarrow$ 2 BaTiO3 | -0.036 | 0.030 | 0.043 | 0.029
4 |  Ba2TiO4 \+ BaTi2O5 $\longrightarrow$ 3 BaTiO3 | -0.001 | -0.001 | 0.000 | -0.001
5 |  Ba(OH)${}_{2}\cdot$H2O + 3.666 Ti $\longrightarrow$ BaTiO3 \+ 1.333 Ti2H3 | -0.530 | 0.070 | 0.534 | 0.219
6 |  BaCl2 \+ Na2TiO3 $\longrightarrow$ BaTiO3 \+ 2 NaCl | -0.075 | -0.007 | 0.040 | 0.007
7 |  BaS + Na2TiO3 $\longrightarrow$ BaTiO3 \+ Na2S | -0.048 | -0.001 | 0.041 | 0.013
8 |  2 BaS + 3 TiO2 $\longrightarrow$ 2 BaTiO3 \+ TiS2 | 0.090 | 0.090 | 0.000 | 0.050
9 |  BaSO4 \+ 2 TiO2 $\longrightarrow$ BaTiO3 \+ TiOSO4 | 0.178 | 0.178 | 0.000 | 0.098

The nine synthesis experiments were completed using a gradient furnace (see
Methods),43 allowing the observation of reaction products over a wide range of
temperatures ($\sim$200-1000 ℃) with ex post facto SPXRD. The experimental
results are summarized in Figure 6. Mole fractions of each phase were
determined via Rietveld refinement of SPXRD patterns acquired at various
positions (temperatures) along the length of the sample after heating and
subsequent cooling to ambient temperature. The ex post facto phase fraction
plots can be interpreted similarly to those constructed with in situ data, as
each effectively illustrates the reaction pathway during heating. More
rigorously, however, each point in Figure 6 is pseudo-independent and best
interpreted as the result of an isothermal (ex situ) reaction at its
associated temperature. Shorter reaction times were selected to ensure capture
of the onset of short-lived intermediate phases critical to assessing reaction
pathway selectivities (Table LABEL:tab:exp_times). Selected Rietveld analysis
results for each experiment are provided in the Supporting Information
(Figures LABEL:supp_fig:refinement1-LABEL:supp_fig:refinement9).

Figure 6: Reaction pathways for selected BaTiO3 synthesis experiments. (a-i)
Mole fractions of observed phases for Experiments 1-9, as determined through
Rietveld refinements of ex post facto SPXRD data. Phase types are
distinguished by shape: precursors (circles), targets (diamonds), byproducts
(triangles), and impurities (exes). Background shading denotes the total mole
fractions of precursor (gray), impurity (pink), and target/byproduct (green).
The median total reaction time was $\sim$67 min; exact times for each
experiment are provided in Table LABEL:tab:exp_times.

The observed reaction pathways demonstrate significant variation in target and
impurity formation. Visualizing the interface reaction hulls of the selected
experiments helps to rationalize their predicted and observed performance
(Figure LABEL:supp_fig:all_irh). The most complex of these pathways is that of
Expt. 5, which features the formation of BaTiO3 at an intermediate temperature
range (400-500 ℃) before impurities Ba2TiO4, TiH2, and TiO2 begin to dominate.
Indeed, Expt. 5 exhibits both the largest driving force (-0.530 eV/atom) and
$C_{2}$ value (0.534 eV/atom) of any reaction, supporting the observation of a
complex reaction pathway containing many impurity phases.

The conventional synthesis reaction between BaCO3 and TiO2 (Expt. 1) was
largely incomplete after 60 min at 1000 ℃ and exhibited significant formation
of BaTi2O5. The interface reaction hull for this system (Figure
LABEL:supp_fig:all_irha) suggests that the formation of BaTi2O5 is the most
favorable reaction outcome, supporting our observations. In Expt. 2, the
reaction of TiO2 with BaO2 appears to significantly decrease the reaction
onset temperature but also features substantial impurity formation. In three
of the experiments (Expts. 4, 6, and 7), the BaTiO3 synthesis reaction is the
most favorable reaction on the hull, resulting in $C_{1}<0$. Notably, the use
of ternary precursor(s) in Expts. 3 and 4 (Ba2TiO4, BaTi2O5) results in low
(or zero) $C_{2}$, but also very little driving force. As a result, we observe
near-perfect selectivity (i.e., very few visible impurities) at the expense of
slowing down the reactions substantially. When the reaction energy is above
zero, impurities associated with exergonic competing reactions may still form.
For example, the hull for Expt. 8 indicates a significant degree of
competition, including several reactions with only slightly positive energies
($\Delta G_{\textrm{rxn}}<0.01$ eV/atom). Experiments 8 and 9 are indeed
largely unreacted as predicted but feature minor impurities (BaSO4 and/or
BaO).

The metathesis reactions (Expts. 6, 7) show the overall greatest performance,
yielding primarily BaTiO3 and the predicted byproducts at moderately low
temperatures (600-700 ℃). While metathesis reactions producing alkali halides
are well-known for their optimal performance 44, it is a notable and
surprising result that the sulfide-based reaction (BaS + Na2TiO3 -¿ BaTiO3 +
Na2S) achieves such pure and direct synthesis of BaTiO3, as predicted. Its
success further highlights the importance of considering more complex
chemistries involving additional elements besides those in the target phase
(i.e., hyperdimensional chemistries) 40. Some impurity formation, however, is
evident in both metathesis reactions, particularly at lower temperatures.
BaTi2O5 forms in both experiments and small amounts of Ba2TiO4 form in Expt.
7. However, this observation is supported by the calculated $C_{1}$ and
$C_{2}$ scores, which indicate that neither reaction should be perfectly
selective. Unexpectedly, the dominant impurities in Expt. 6 are carbonate
compounds: Ba3(CO3)Cl4 and BaCO3. We presume this results from the reaction of
the original precursors with CO2 in the air. For completeness, we have
accounted for these in our experimental analysis even though they were not
explicitly considered in the selectivity calculations. However, we did exclude
from consideration any Si-containing impurities such as Ba2TiSi2O8, which
formed in small amounts due to reaction with the quartz capillaries; these Si
impurities were minor ($<$0.02 mole fraction) and did not significantly affect
our analysis.

To quantitatively assess the performance of our predictions in determining the
outcomes of experimental reactions, we propose three reaction outcome metrics
summarizing the behavior of a reaction pathway: the minimum precursor
remaining ($P$), the maximum target/byproduct formed ($T$), and the maximum
impurity formed ($I$). Each metric is a mole fraction value taken from any
data point (i.e., temperature) within the reaction pathway. This permits the
capture of key features of the pathway independent of the kinetics of that
reaction and is necessary given the range of chemistries explored. Our outcome
metrics are visualized on the Precursor-Target-Impurity (PTI) plots in Figure
7a. The first quantity, $P$ (minimum precursor remaining), gives insight into
the reactivity and kinetics of the reaction: high values indicate reactions
that did not complete at any temperature within the reaction timeframe. The
second quantity, $T$ (maximum target/byproduct formed), provides a measure of
the success of the reaction in producing BaTiO3 and predicted byproduct(s).
Finally, the third quantity, $I$ (maximum impurity formed), measures the
selectivity of the reaction pathway, indicating the maximum fraction of
intermediate/impurity phases synthesized at any temperature. Note that the
temperature-independent PTI metrics are not required to sum to one for a
particular experiment. This is intentional and advantageous because it permits
the capture of poor selectivity in even nominally well-performing reactions;
i.e., both $T$ and $I$ can be high ($\sim$1) within the same experiment.

Figure 7: Summary of experimental results and correlations with calculated
reaction metrics. (a) Precursor-Target-Impurity (PTI) plots, where the height
of each bar indicates the relevant mole fraction captured at the most
representative position (i.e., temperature) along the length of the sample
after heating (and cooling). The different color bars correspond to the
minimum amount of precursor remaining ($P$, gray), the maximum amount of
target/byproducts formed ($T$, green), and the maximum amount of impurity
formed ($I$, pink). (b) Positive correlation between $P$ and the Gibbs free
energy of the reaction, $\Delta G_{\textrm{rxn}}$. (c) Negative correlation
between $T$ and $C_{1}$. (d) Positive correlation between $I$ and $C_{2}$. The
$T$ and $I$ mole fractions have been normalized by the maximum amount of
precursor consumed in the experiment, $1-P$, for enhanced visualization of the
trend. The small impurity amounts detected in Expts. 8 and 9 are treated as
yielding $I=0$ to more accurately reflect the lack of observed reaction.

Figures 7b-d show selected correlations between reaction outcomes ($P$, $T$,
and $I$) and the calculated reaction metrics ($\Delta G_{\textrm{rxn}}$,
$C_{1}$, and $C_{2}$). The full set of all (3x3) pairwise correlation plots is
available in Figure LABEL:supp_fig:metric_performance. We observe that the
calculated reaction energy ($\Delta G_{\textrm{rxn}}$) correlates most
strongly with the minimum amount of precursor remaining ($P$) at the
conclusion of the experiment (Figure 7b). With infinite reaction times, we
would theoretically expect this distribution to resemble a step function:
$P=0$ for reactions with $\Delta G_{\textrm{rxn}}<0$, and $P=1$ for those with
$\Delta G_{\textrm{rxn}}>0$. In our work, the distribution is less defined,
given the shorter reaction times and different chemistries explored. The
coordinates of Expts. 1 and 2 appear to deviate the most from a step-like
distribution. Difficulty in modeling the energetics of carbonate reactions was
previously discussed and likely explains the deviation of Expt. 1. In
contrast, the deviation of Expt. 2 is likely kinetic in nature, as the
reaction appears to stall (Figure 6b); this may suggest that the maximum
temperature (750-800 ℃) is too low to achieve sufficient reaction completion.

The primary ($C_{1}$) and secondary ($C_{2}$) competition metrics display
negative and positive correlations with the maximum amounts of target ($T$)
and impurity ($I$) phases formed, respectively (Figures 7c,d). We note that
these trends are not strictly obeyed in a monotonic fashion. Still, the
correlations are significant and can be theoretically rationalized by their
derivation from the interface reaction hull. It is reasonable that $C_{1}$
correlates most strongly with $T$, as this competition metric effectively
measures the relative favorability of the target reaction over competing
reactions. Similarly, it is sensible that $C_{2}$ should correlate most
strongly with $I$ given its derivation as a measure of the relative stability
of impurity phases with respect to the target phase. To be precise, the
experimental correlation between $C_{2}$ and $I$ suggests that one should
consider not only the sum of the inverse hull distance energies for all
competing phases but the total energy of the entire secondary reaction
sequence containing them (Equation 2). By definition, this quantity includes,
and thus will always be larger than or equal to, the sum of all competing
inverse hull distance values for a particular interface reaction hull.
Regarding the functional form of $C_{2}$, the question arises as to whether a
more simple summation of the maximum energy secondary reactions in the left
and right hull subsections is a sufficient measure for secondary competition.
While the maximum energy secondary reactions tend to account for much of the
value of $C_{2}$, on average, the full $C_{2}$ metric is 0.077 eV/atom greater
than considering the maximum energy secondary reactions alone (Figure
LABEL:supp_fig:correlations_secondary), suggesting that $C_{2}$ is a more
conservative metric. For completeness, we also tested an alternative
formulation of the secondary competition metric using the enclosed “area” to
the hull. While this metric correlates with $C_{2}$, its calculation is more
numerically unstable, and its units are less interpretable. Hence, we
generally recommend the approach of modeling full secondary reaction
sequences, which is straightforward to implement using our secondary
competition algorithm (see Methods).

In a previous study,24 we suggested a solid-state reaction selectivity metric
based on the difference in elemental chemical potentials between precursors
and targets, measured by a distance along the chemical potential diagram
(i.e., the “total chemical potential distance”). This metric was used to
rationalize the unique selectivity of the Na-based precursor in synthesizing
pyrochlore Y2Mn2O7 from YOCl and $A$MnO2 ($A$=Li, Na, K). While
straightforward to compute using just the chemical potential diagram, the
distance metric operates in the space of chemical potentials rather than
reaction energies, rendering it less intuitive and more difficult to precisely
discern the specific competing reactions. From the results here, we generally
recommend that $C_{2}$ be used instead of total chemical potential distance
where possible. Both selectivity metrics capture similar characteristics of
the competing phase space: each effectively involves the summation of
competing phase stabilities through inverse hull distances or the
corresponding chemical potential stability ranges. More precisely, both
quantities are correlated (Figure LABEL:supp_fig:correlations) because
chemical potentials are mathematical derivatives of the convex hull in energy-
composition space. However, the total chemical potential distance is biased,
particularly by competing phases with defective elemental-like compositions
(e.g., Mg149Cl). Due to the increased weight of the entropic ($-TS$) term in
the definition of Gibbs free energy, chemical potential diagrams featuring
these compositions as competing phases may yield very high (unfavorable) total
chemical potential distance values for synthesis reactions.

We acknowledge that while $C_{1}$ and $C_{2}$ are meant to capture different,
independent mechanisms by which competing phases form, these metrics are at
least partially correlated due to the geometric constraints of the convex
hull. In particular, one situation is geometrically limited from occurring:
high $C_{2}$ and low $C_{1}$ (Figure LABEL:supp_fig:competition_extremes).
Stated explicitly: if a competing phase lies significantly below the tie-line
formed by the target and a precursor (i.e., high $C_{2}$), then both the
target and that competing phase necessarily have similar reaction energies to
form from the precursors, leading to high $C_{1}$. In general, however, this
restriction does not make the metrics redundant; while there is some
correlation between the two, the correlation is not particularly strong
(Figure LABEL:supp_fig:correlations). Therefore, we generally recommend the
tandem use of both selectivity metrics.

The major limitation of our current synthesis planning workflow is the
assumption that optimal synthesis reactions can be predicted with the
thermodynamic energy landscape alone. While this is not the case for all
chemistries, we show here that at least for chemical systems that exhibit
practical solid-state reaction kinetics, the energy landscape alone can
provide much of the rationalization for the observation of impurity phases. To
say that impurity and secondary phases are inherently “kinetic” products is a
misnomer. Rather, these phases may be the thermodynamic minima of smaller
“local” interface systems, distinct from the thermodynamic products of the
entire reaction mixture (the global thermodynamic solution). Furthermore,
these impurities are often not easily convertible to final products without
long-range mass transport or intervention (e.g., via re-grinding or subsequent
heating). This explains why impurities are often pervasive and challenging to
remove in chemical systems with lower driving forces and/or slower kinetics
(e.g., BiFeO3).

Currently, our synthesis planning workflow focuses exclusively on optimizing
product purity; however, there are many issues one must consider when
designing a synthesis recipe for a target compound: material cost, safety
concerns, stability in air, handling challenges, availability of precursors,
etc. Many routes suggested by our workflow involve the formation of byproducts
that are not easily removable from the product mixture (e.g., the formation of
BaTiO3 with byproduct Ba2TiO4). To this point, one should be thoughtful in
designing criteria by which to filter recommended synthetic routes. For
example, one can prioritize the formation of only gaseous byproducts (e.g., O2
or CO2) or those easily removable by a solvent (e.g., NaCl). The cost function
used to rank reactions can be modified to include other reaction metrics of
interest, such as the estimated economic cost of the precursor materials.
While not explicitly demonstrated here, the synthesis planning workflow can
also be extended to use for multi-step syntheses, allowing one to
retrosynthetically sequence reactions to a target material beginning with
purchasable, “off-the-shelf” precursors.

##  3 Conclusions

Using the interface reaction model for powder reactions, we proposed two
thermodynamic selectivity metrics for solid-state reactions: primary ($C_{1}$)
and secondary ($C_{2}$) competition. To systematically and critically examine
the effectiveness of our metrics, we analyzed existing successful synthesis
routes available in the literature and, leveraging a massive set of 82,985
synthesis reactions extracted from an 18-element reaction network constructed
from Materials Project data, designed and executed nine BaTiO3 synthesis
experiments with a range of selectivity values as compared to conventional
precursors (BaCO3 and TiO2). Analysis of reaction pathways in the nine
experiments via ex post facto synchrotron powder X-ray diffraction reveals
that $C_{1}$ and $C_{2}$ correlate with the maximum amounts of target and
impurity formed, respectively.

The main advantage of our approach compared to recent, existing approaches14,
25 is the ability to simultaneously consider a wide range of chemistries,
including those with unconventional additional elements. These so-called
hyperdimensional chemistries 40 allow one to bypass commonly encountered
intermediates in target systems with many competing phases. This was
demonstrated particularly for the BaTiO3 system studied in this work and is
relevant for many other materials in the literature that are conventionally
synthesized with theoretically suboptimal precursors (e.g., Na2Ti3O7, NaTaO3,
LiMn2O4, etc.).

We anticipate that the selectivity metrics presented here and our
computational synthesis planning workflow will significantly reduce the
synthesis bottleneck, providing more rapid development of synthesis approaches
for new, predicted materials. Our workflow provides a theoretical rationale
for using certain precursors and synthesis conditions over other options,
which promises to optimize existing synthesis procedures for current
technologically important materials.

We envision our approach to be particularly useful in aiding high-throughput
automated laboratory exploration efforts.45 Predictions can be used to design
and downselect the synthesis reactions tested, reducing the cost and current
trial-and-error approach to inorganic materials synthesis. The future
inclusion of models for the kinetic behavior of reactions, such as estimates
of the reactivity of precursors based on solid-state diffusivities, will
further enhance predictions.

##  4 Methods

###  4.1 Thermodynamic Data

Gibbs free energies of formation, $\Delta G_{\text{f}}(T)$, were acquired or
approximated in a similar approach to those of previous works.26, 24 We
acquired experimental $\Delta G_{\text{f}}(T)$ values from the NIST-JANAF
thermochemical tables 29 where available. Experimental values were limited to
compounds with low melting points (i.e., $T_{m}\leq 1500$ ℃), as these systems
demonstrate more complex phase change behavior over the temperature range
studied here. For predominantly solid compounds (i.e., those with melting
points above this threshold), as well as for all other phases not available in
the NIST-JANAF thermochemical tables, we estimated $\Delta G_{\text{f}}(T)$
using the machine-learned Gibbs free energy descriptor identified by Bartel et
al.28 This descriptor was applied using formation enthalpies $\Delta
H_{\text{f}}(T=298$ K) acquired from the Materials Project (MP) database,27
version 2022.10.28.

Due to the well-known and systematic formation energy error of carbonate
compounds calculated with GGA exchange-correlation functionals,14, 19 we
applied an energy correction of 0.830 eV per CO3^2- anion to all carbonate
compounds acquired from MP. This value was determined by fitting the mean
error between computed and experimental $\Delta G_{\text{f}}(T=300$ K) values
for 15 metal carbonate compounds (Figure LABEL:supp_fig:carbonates).

###  4.2 Synthesis Planning Workflow

The synthesis reaction calculation and ranking procedure was implemented as a
Python-based workflow in the existing reaction-network package.26 The code is
available on GitHub at https://github.com/materialsproject/reaction-network.
The workflow was constructed and launched on computing resources using the
jobflow46 and fireworks47 workflow packages.

The synthesis planning workflow consists of three sequential steps. First,
phases and their formation energies for the chemical system of interest are
acquired as previously described. The total number of phases can be optionally
reduced by setting a threshold for the maximum energy above hull ($\Delta
G_{\text{hull}}$). In this work, we used a moderately large threshold of
$\Delta G_{\text{hull}}\leq 50$ meV/atom, evaluated at ambient temperature
($T=300$ K). Second, reaction enumeration is performed for the acquired phases
using the combinatorial and free energy minimization approaches described in
our previous work on solid-state reaction networks.26 Note that the
combinatorial approach allows one to identify reaction product combinations
above the hull (i.e., “metastable” products), which makes the analysis more
robust to numerical error in the thermodynamic data. For systems with an open
element (e.g., O2 gas), this reaction enumeration step is performed again
using grand potential energies, where the open element has been assigned a
user-defined value for the chemical potential (often the standard state,
$\mu=\mu^{0}$ eV). Finally, $C_{1}$ and $C_{2}$ scores are calculated for all
target synthesis reactions (i.e., those that form the desired target
composition). To do this, the relevant competing reactions are extracted from
the full set of enumerated reactions. We define a competing reaction as one
whose precursors are a subset of the target reaction’s precursors. These
competing reactions are then used to compute the interface reaction hull, from
which $C_{1}$ and $C_{2}$ are calculated via Equations 1 and 4. For open
systems, this selectivity calculation procedure is performed again, including
any additional enumerated open reactions and ensuring that all reactions are
calculated with grand potential energies at the corresponding chemical
potential.

###  4.3 Secondary Competition Algorithm

The secondary competition score, $C_{2}$, is defined as the negative sum of
the mean secondary reaction sequence energies to the left and right of the
target on the interface reaction hull (Equation 4). One approach for acquiring
these quantities involves using a recursive algorithm to identify all possible
sequences and their energies. However, this strategy is too slow for the high-
throughput calculation of $C_{2}$ in systems with many competing reactions.

Instead, we have identified a non-recursive algorithm that takes advantage of
the connection between this problem and the recursive construction of binary
trees via the use of the Catalan number sequence. Our algorithm reformulates
the sum of all secondary reaction sequence energies as a sum of individual
secondary reaction energies weighted by their multiplicities, i.e., the total
number of appearances of a particular reaction within the set of all possible
secondary reaction sequences. The energy of any reaction indexed $k$ can be
calculated geometrically as the altitude, $h_{k}$, of the triangle formed by
its product vertex and two reactant vertices on the interface reaction hull.
We find that the altitude multiplicity, $m_{h_{k}}$, is determined to be the
product of three Catalan numbers, $u_{n}$, such that

$m_{h_{k}}=u_{n_{l}}\cdot u_{n_{r}}\cdot u_{(n-n_{l}-n_{r}-1)},$ (6)

where $n_{l}$ and $n_{r}$ refer to the number of interior vertices (i.e.,
within the triangle) to the left and right of the vertex of interest,
respectively, and $n$ is the total number of interior vertices for the entire
hull subsection. For example, for secondary reactions between nearest
neighbors, $n_{l}=0$ and $n_{r}=0$, resulting in an altitude multiplicity of
$m_{h}=u_{n-1}$.

The mean secondary reaction sequence energy for the hull subsection can then
be calculated as

$\overline{\Delta G_{2}}=\frac{1}{N}\sum_{k}^{V}{m_{h_{k}}\cdot h_{k}},$ (7)

where the sum occurs over all of the $V$ unique reaction energies (altitudes),
which is the number of unique triangles that can be constructed for the hull
subsection, including the two exterior vertices: $V=\binom{n+2}{3}$. The total
number of unique secondary reaction sequences equals the corresponding Catalan
number, $N=u_{n}$. Finally, once this process has been performed for both the
left and right hull subsections, the secondary competition ($C_{2}$) can be
calculated via Equation 4.

###  4.4 Literature Reactions

Solid-state literature reactions studied in this work were acquired from the
text-mined dataset of 31,782 inorganic materials synthesis recipes originally
extracted from the literature by Kononova et al.33 and available at
https://github.com/CederGroupHub/text-mined-synthesis_public (version
2020-07-13). The original dataset was filtered down to 8,530 reactions that
contain: 1) precursors composed of $\leq 2$ solids and $\leq 1$ elemental
gases (i.e., O2, H2, and N2), 2) no elements with an atomic number greater
than 94 or for which the Gibbs free energy descriptor does not apply (e.g.,
Ne, Ar, Pm, Ra), 3) ten or fewer total elements due to limitations in the
convex hull algorithm, and 4) no ions. Finally, these reactions were required
to be stoichiometrically balanceable after adjusting compositions for hydrates
and fractional formulas. For reactions containing variable compositions with
one open variable (e.g., Nd_1-$x$Sr_$x$CoO_3), we attempted to substitute all
extracted values of $x$ and retained the reactions that could be successfully
balanced.

Competition metrics and free energies were assessed for each of the remaining
reactions. For the enumerated competing reactions, metastable phases were
considered up to a maximum threshold of $\Delta G_{\text{hull}}=50$ meV/atom,
evaluated at ambient temperature ($T=300$ K). Interface reaction hulls were
constructed at the maximum temperature reported during synthesis,
$T_{\textrm{syn}}$. If this was not provided, a temperature of 800 ℃ was
assumed. Formation energies, $\Delta G_{\text{f}}(T_{\text{syn}})$, were
assigned based on the ground-state energy for a given composition; i.e., we
selected the lowest available formation energy of all polymorphs with the
composition of interest. For increased accuracy, we did not include a reaction
if any of its entries were missing from our thermodynamic data. For reactions
with an open gas (i.e., O2, H2, N2), we assigned a chemical potential of
$\mu_{\text{gas}}=0$ eV (i.e., standard state at $T_{\text{syn}}$) for that
element. For reactions completed in air, we assumed an O2 partial pressure of
0.21 atm and thus assigned a chemical potential of
$\mu_{\text{O}}=\frac{1}{2}k_{b}T_{\text{syn}}\ln(0.21)$ eV. Finally, we
removed duplicates with the same reaction equation and
temperature/environment, as well as identity reactions (e.g., $A\rightarrow
A$), These filtering steps yielded a total of 3,520 unique literature
reactions.

###  4.5 Precursor Materials

Precursors for all experiments were purchased from chemical providers or
produced via known solid-state synthesis approaches, as necessary. Precursors
acquired from chemical providers include: barium carbonate (BaCO3, J.T.Baker
99.9%), titanium(IV) oxide (anatase TiO2, Acros Organics 99.9%), barium
sulfate (BaSO4, J.T.Baker 99.9%), barium hydroxide hydrate (Ba(OH)2*8H2O,
Mathsen Colman & Bell 98%), barium chloride hydrate (BaCl2*2H2O, Fisher
Scientific 99.9%), activated carbon (C, J.T.Baker 99.9%), and titanium metal
(Ti, annealed foil, Alfa Aesar 99.7%).

Ba2TiO4 was synthesized based on an existing approach 48 using stoichiometric
amounts of BaCO3 and anatase TiO2. The chemicals were mixed, ground using a
mortar and pestle, placed in an alumina boat inside of a mullite process tube
with self-sealing endcaps, and then heated at 950 ℃ for 16 hrs under Ar flow
with a heating rate of 10 ℃/min. The powder was then reground and reheated at
1100 ℃ for another 16 hrs at a heating rate of 10 ℃/min. Handling operations
were completed in an Ar glovebox due to the hygroscopic nature of Ba2TiO4.

BaTi2O5 was synthesized with stoichiometric amounts of BaCO3 and anatase TiO2.
The chemicals were mixed, ground using a mortar and pestle, and heated in an
alumina boat at 900 ℃ for 5 hrs as a pre-treatment step. The powder was then
reground and reheated at 1220-1225 ℃ for 24 hrs with heating and cooling steps
of 3 hrs.49

Barium sulfide (BaS) was synthesized following an existing approach 50 using
BaSO4 and activated carbon. The chemicals were mixed, ground using a mortar
and pestle, pressed into a 0.5” diameter pellet with two tons of force, and
heated in an alumina boat at 1100 ℃ for 7-10 min in air, with a heating rate
of 10 ℃/min and natural cooling in the furnace.

Sodium metatitanate (Na2TiO3) was synthesized based on an existing approach 51
using stoichiometric amounts of sodium hydroxide (NaOH, Fisher Scientific
99.9%) and anatase TiO2, with a slight excess of NaOH. The chemicals were
mixed, ground using a mortar and pestle, and heated in an alumina boat at 500
℃ for 2 hrs with a heating rate of 10 ℃/min.

###  4.6 Ex Post Facto SPXRD Reactions

Synchrotron powder X-ray diffraction (SPXRD) data were collected in
transmission (i.e., Debye-Scherrer) geometry on beamline 28-ID-2 (XPD) at the
National Synchrotron Light Source-II (NSLS-II). Data were collected on a 2D
area detector (PerkinElmer XRD 1621, 2048x2048 pixel array, 200x200 $\mu$m
pixel size) at a sample-to-detector distance of 1407.1 mm using an incident
X-ray energy of 68.12 keV ($\lambda$ = 0.182 Å) with a 0.60 x 0.20 mm beam
size. A total acquisition time of 1 s was used, summing five subframes
collected for 0.2 s each.

Samples were packed into 1.1 mm OD/0.9 mm ID quartz capillaries. The capillary
ends were filled with a 3 mm plug of powder silicon (Si, Strem 99.0%) followed
by a cap of recycled silicon dioxide (SiO2). To account for possible gas
production, the capillaries for Expts. 1, 2, 5, 7, and 8 were left unsealed,
and a moderate vacuum ($P_{\textrm{gage}}=-20$ in Hg) was pulled on the
samples during heating. All other sample capillaries (Expts. 3, 4, 6, and 9)
were flame-sealed under argon.

Experiments were carried out in the gradient furnace described in Ref. 43,
which heats samples to different temperatures across a range of spatial
positions on the capillary (Figure LABEL:supp_fig:gradient_furnace). The
furnace was operated with a Eurotherm 2408 temperature controller and a TDK
Lambda 900W (30V/30A) power supply. Furnace heating elements were wound from
resistive wire (Kanthal A-1, #24 awg). A K-type thermocouple (stainless steel,
0.01” OD) placed at an intermediate position along the sample was used as
input for PID control of the furnace. We performed experiments in three
temperature ranges with setpoints of $T_{H}=550$ ℃ (Expts. 1, 3, 4, 8),
$T_{L1}=450$ ℃ (Expts. 2, 5, 6, 9), and $T_{L2}=400$ ℃ (Expt. 7). This choice
was motivated by differences in reactivity among the samples. Position-
dependent temperatures were determined using a fit of measured in situ lattice
expansion from NaCl/Si and Al2O3/MgO standards (Figure
LABEL:supp_fig:gradient_calibration). Using the root-mean-square error of the
curve fit, the estimated uncertainty for each temperature point is 11.4 ℃
($T_{H}$), 7.9 ℃ ($T_{L1}$), and 10.9 ℃ ($T_{L2}$). The experiments spanned a
total temperature range of 189-1064 ℃.

The median total time of each experiment was $\sim$67 min. Heating, holding,
and cooling times varied among experiments due to differences in sample heat
capacities, reactivities, and the sizes of investigated temperature windows;
specifically, unreactive samples (Expts. 8, 9) and samples with smaller
studied temperature windows (Expts. 3, 4) were held at elevated temperatures
for shorter times. These differences are accounted for in our analysis via the
use of relative metrics (i.e., mole fraction) and normalization by reaction
progress. The exact heating, holding, and cooling times for each sample are
shown in Table LABEL:tab:exp_times.

###  4.7 Quantitative Phase Analysis of Powder Diffraction Data

Quantitative analysis of synchrotron powder X-ray powder diffraction data was
carried out with the Rietveld method using either the TOPAS v6 (Expts. 1-3, 5,
7-9) or GSAS-II (Expts. 4, 6) software packages.52, 53 Atomic displacement
parameters were fixed to $B=1$ Å2, and peak broadening was primarily modeled
via crystal size broadening using a Lorentzian function. Site occupancies were
fixed at one.

##  5 Supporting Information

  * •

Description of cost function power transformation; interface reaction hull for
BaO$|$TiO2; open-O2 reaction data for BaTiO3; selected Rietveld refinements
from all experiments; interface reaction hull plots for selected experimental
reactions; selectivity metric correlations; details of carbonate correction;
description of gradient furnace setup and temperature calibration; table of
experimental reaction times and temperature setpoints (PDF)

  * •

Table of 3,520 reactions extracted from the literature with their calculated
performance metrics and DOIs (XLSX)

  * •

Table of 82,985 predicted/ranked closed BaTiO3 synthesis reactions (XLSX)

  * •

Table of 62,133 predicted/ranked open-O2 BaTiO3 synthesis reactions (XLSX)

  * •

Table of 133 Pareto optimal open-O2 BaTiO3 synthesis reactions (XLSX)

##  6 Acknowledgements

This work was supported as part of GENESIS: A Next Generation Synthesis
Center, an Energy Frontier Research Center funded by the U.S. Department of
Energy, Office of Science, Basic Energy Sciences under Award Number DE-
SC0019212. This research used resources of the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility operated under Contract No. DE-AC02-05CH11231.
Additional computational resources were provided by the Swift high-performance
computing (HPC) system at the National Renewable Energy Laboratory (NREL).
This research used Beamline 28-ID-2 (XPD) of the National Synchrotron Light
Source II, a U.S. Department of Energy (DOE) Office of Science User Facility
operated for the DOE Office of Science by Brookhaven National Laboratory under
Contract No. DE-SC0012704. The authors thank the Analytical Resources Core,
Center for Materials and Molecular Analysis at Colorado State University for
instrument access and training (RRID: SCR_021758). Partial support for G.E.K.
was provided by an NRT QuADS Fellowship (Quantitative Analysis of Dynamic
Structures) under NSF DGE-1922639.

The authors would like to thank Ren Borgia, Dr. Sanjit K. Ghose, Dr. Hui
Zhong, and John Trunk for their experimental assistance. M.J.M. would also
like to thank Dr. Rishi Kumar, Dr. Andrew Rosen, and Evan Spotte-Smith for
helpful discussion and feedback.

##  7 Synopsis

Selectivity analysis of massive inorganic chemical reaction networks
identifies alternative synthesis recipes for solid-state materials (e.g.,
BaTiO3) that outperform conventional approaches.

## References

  * Kovnir 2021 Kovnir, K. Predictive Synthesis. _Chemistry of Materials_ 2021, _33_ , 4835–4841, DOI: 10.1021/acs.chemmater.1c01484
  * DiSalvo 1990 DiSalvo, F. J. Solid-State Chemistry: A Rediscovered Chemical Frontier. _Science (New York, N.Y.)_ 1990, _247_ , 649–655, DOI: 10.1126/science.247.4943.649
  * Kohlmann 2019 Kohlmann, H. Looking into the Black Box of Solid‐State Synthesis. _European Journal of Inorganic Chemistry_ 2019, _2019_ , 4174–4180, DOI: 10.1002/ejic.201900733
  * Corey and Cheng 1989 Corey, E.; Cheng, X.-M. _The Logic of Chemical Synthesis_ ; John Wiley & Sons, Inc., 1989
  * Coley et al. 2018 Coley, C. W.; Green, W. H.; Jensen, K. F. Machine Learning in Computer-Aided Synthesis Planning. _Accounts of Chemical Research_ 2018, _51_ , 1281–1289, DOI: 10.1021/acs.accounts.8b00087
  * Kim et al. 2017 Kim, E.; Huang, K.; Saunders, A.; McCallum, A.; Ceder, G.; Olivetti, E. Materials Synthesis Insights from Scientific Literature via Text Extraction and Machine Learning. _Chemistry of Materials_ 2017, _29_ , 9436–9444, DOI: 10.1021/acs.chemmater.7b03500
  * Narayan et al. 2016 Narayan, A.; Bhutani, A.; Rubeck, S.; Eckstein, J. N.; Shoemaker, D. P.; Wagner, L. K. Computational and Experimental Investigation for New Transition Metal Selenides and Sulfides: The Importance of Experimental Verification for Stability. _Physical Review B_ 2016, _94_ , 045105, DOI: 10.1103/PhysRevB.94.045105
  * Rong et al. 2017 Rong, Z.; Xiao, P.; Liu, M.; Huang, W.; Hannah, D. C.; Scullin, W.; Persson, K. A.; Ceder, G. Fast Mg2+ Diffusion in Mo3(PO4)3O for Mg Batteries. _Chemical Communications_ 2017, _53_ , 7998–8001, DOI: 10.1039/c7cc02903a
  * Gardner et al. 2019 Gardner, J.; Thakre, A.; Kumar, A.; Scott, J. F. Tin Titanate—the Hunt for a New Ferroelectric Perovskite. _Reports on Progress in Physics_ 2019, _82_ , 092501, DOI: 10.1088/1361-6633/ab37d4
  * West 2014 West, A. R. _Solid State Chemistry and Its Applications_ , 2nd ed.; Wiley: Chichester, West Sussex, UK, 2014
  * Miura et al. 2021 Miura, A.; Bartel, C. J.; Goto, Y.; Mizuguchi, Y.; Moriyoshi, C.; Kuroiwa, Y.; Wang, Y.; Yaguchi, T.; Shirai, M.; Nagao, M.; Rosero‐Navarro, N. C.; Tadanaga, K.; Ceder, G.; Sun, W. Observing and Modeling the Sequential Pairwise Reactions That Drive Solid‐State Ceramic Synthesis. _Advanced Materials_ 2021, _33_ , 2100312, DOI: 10.1002/adma.202100312
  * Richards et al. 2015 Richards, W. D.; Miara, L. J.; Wang, Y.; Kim, J. C.; Ceder, G. Interface Stability in Solid-State Batteries. _Chemistry of Materials_ 2015, _28_ , 266–273, DOI: 10.1021/acs.chemmater.5b04082
  * Bianchini et al. 2020 Bianchini, M.; Wang, J.; Clément, R. J.; Ouyang, B.; Xiao, P.; Kitchaev, D.; Shi, T.; Zhang, Y.; Wang, Y.; Kim, H.; Zhang, M.; Bai, J.; Wang, F.; Sun, W.; Ceder, G. The Interplay between Thermodynamics and Kinetics in the Solid-State Synthesis of Layered Oxides. _Nature Materials_ 2020, _19_ , 1088–1095, DOI: 10.1038/s41563-020-0688-6
  * Aykol et al. 2021 Aykol, M.; Montoya, J. H.; Hummelshøj, J. Rational Solid-State Synthesis Routes for Inorganic Materials. _Journal of the American Chemical Society_ 2021, _143_ , 9244–9259, DOI: 10.1021/jacs.1c04888
  * d’Heurle 1988 d’Heurle, F. M. Nucleation of a New Phase from the Interaction of Two Adjacent Phases: Some Silicides. _Journal of Materials Research_ 1988, _3_ , 167–195, DOI: 10.1557/jmr.1988.0167
  * Kumar et al. 2000 Kumar, M. M.; Palkar, V. R.; Srinivas, K.; Suryanarayana, S. V. Ferroelectricity in a Pure BiFeO3 Ceramic. _Applied Physics Letters_ 2000, _76_ , 2764–2766, DOI: 10.1063/1.126468
  * Bernardo et al. 2011 Bernardo, M.; Jardiel, T.; Peiteado, M.; Caballero, A.; Villegas, M. Reaction Pathways in the Solid State Synthesis of Multiferroic BiFeO3. _Journal of the European Ceramic Society_ 2011, _31_ , 3047–3053, DOI: 10.1016/j.jeurceramsoc.2011.03.018
  * Merkle and Maier 2005 Merkle, R.; Maier, J. On the Tammann–Rule. _Zeitschrift für anorganische und allgemeine Chemie_ 2005, _631_ , 1163–1166, DOI: 10.1002/zaac.200400540
  * Huo et al. 2022 Huo, H.; Bartel, C. J.; He, T.; Trewartha, A.; Dunn, A.; Ouyang, B.; Jain, A.; Ceder, G. Machine-Learning Rationalization and Prediction of Solid-State Synthesis Conditions. _Chemistry of Materials_ 2022, _34_ , 7323–7336, DOI: 10.1021/acs.chemmater.2c01293
  * Jia et al. 2019 Jia, X.; Lynch, A.; Huang, Y.; Danielson, M.; Lang’at, I.; Milder, A.; Ruby, A. E.; Wang, H.; Friedler, S. A.; Norquist, A. J.; Schrier, J. Anthropogenic Biases in Chemical Reaction Data Hinder Exploratory Inorganic Synthesis. _Nature_ 2019, _573_ , 251–255, DOI: 10.1038/s41586-019-1540-5
  * He et al. 2023 He, T.; Huo, H.; Bartel, C. J.; Wang, Z.; Cruse, K.; Ceder, G. Precursor Recommendation for Inorganic Synthesis by Machine Learning Materials Similarity from Scientific Literature. _Science Advances_ 2023, _9_ , eadg8180, DOI: 10.1126/sciadv.adg8180
  * Jiang et al. 2017 Jiang, Z.; Arun Ramanathan,; Shoemaker, D. P. In Situ Identification of Kinetic Factors That Expedite Inorganic Crystal Formation and Discovery. _Journal of Materials Chemistry C_ 2017, _5_ , 5709–5717, DOI: 10.1039/C6TC04931A
  * Miura et al. 2020 Miura, A.; Ito, H.; Bartel, C. J.; Sun, W.; Rosero-Navarro, N. C.; Tadanaga, K.; Nakata, H.; Maeda, K.; Ceder, G. Selective Metathesis Synthesis of MgCr2S4 by Control of Thermodynamic Driving Forces. _Materials Horizons_ 2020, _7_ , 1310–1316, DOI: 10.1039/c9mh01999e
  * Todd et al. 2021 Todd, P. K.; McDermott, M. J.; Rom, C. L.; Corrao, A. A.; Denney, J. J.; Dwaraknath, S. S.; Khalifah, P. G.; Persson, K. A.; Neilson, J. R. Selectivity in Yttrium Manganese Oxide Synthesis via Local Chemical Potentials in Hyperdimensional Phase Space. _Journal of the American Chemical Society_ 2021, _143_ , 15185–15194, DOI: 10.1021/jacs.1c06229
  * Chen et al. 2023 Chen, J.; Cross, S. R.; Miara, L. J.; Cho, J.-J.; Wang, Y.; Sun, W. Navigating Phase Diagram Complexity to Guide Robotic Inorganic Materials Synthesis. 2023; http://arxiv.org/abs/2304.00743
  * McDermott et al. 2021 McDermott, M. J.; Dwaraknath, S. S.; Persson, K. A. A Graph-Based Network for Predicting Chemical Reaction Pathways in Solid-State Materials Synthesis. _Nature Communications_ 2021, _12_ , 3097, DOI: 10.1038/s41467-021-23339-x
  * Jain et al. 2013 Jain, A.; Ong, S. P.; Hautier, G.; Chen, W.; Richards, W. D.; Dacek, S.; Cholia, S.; Gunter, D.; Skinner, D.; Ceder, G.; Persson, K. A. Commentary: The Materials Project: A Materials Genome Approach to Accelerating Materials Innovation. _APL Materials_ 2013, _1_ , 011002, DOI: 10.1063/1.4812323
  * Bartel et al. 2018 Bartel, C. J.; Millican, S. L.; Deml, A. M.; Rumptz, J. R.; Tumas, W.; Weimer, A. W.; Lany, S.; Stevanović, V.; Musgrave, C. B.; Holder, A. M. Physical Descriptor for the Gibbs Energy of Inorganic Crystalline Solids and Temperature-Dependent Materials Chemistry. _Nature Communications_ 2018, _9_ , 4168, DOI: 10.1038/s41467-018-06682-4
  * Chase 1998 Chase, J. M. W. _NIST-JANAF Thermochemical Tables_ , 4th ed.; American Institute of Physics for the National Institute of Standards and Technology: Washington D.C., 1998
  * Schmalzried 1981 Schmalzried, H. _Solid State Reactions_ , 2nd ed.; Verlag Chemie, Weinheim: Germany, 1981
  * van Loo 1990 van Loo, F. J. J. Multiphase Diffusion in Binary and Ternary Solid-State Systems. _Progress in Solid State Chemistry_ 1990, _20_ , 47–99, DOI: 10.1016/0079-6786(90)90007-3
  * Stanley and Fomin 1999 Stanley, R. P.; Fomin, S. Enumerative Combinatorics. 1999, 159–285, DOI: 10.1017/cbo9780511609589.005
  * Kononova et al. 2019 Kononova, O.; Huo, H.; He, T.; Rong, Z.; Botari, T.; Sun, W.; Tshitoyan, V.; Ceder, G. Text-Mined Dataset of Inorganic Materials Synthesis Recipes. _Scientific Data_ 2019, _6_ , 203, DOI: 10.1038/s41597-019-0224-1
  * Cooper et al. 2016 Cooper, J. K.; Scott, S. B.; Ling, Y.; Yang, J.; Hao, S.; Li, Y.; Toma, F. M.; Stutzmann, M.; Lakshmi, K. V.; Sharp, I. D. Role of Hydrogen in Defining the N-Type Character of BiVO4 Photoanodes. _Chemistry of Materials_ 2016, _28_ , 5761–5771, DOI: 10.1021/acs.chemmater.6b01994
  * Selbach et al. 2009 Selbach, S. M.; Tybell, T.; Einarsrud, M.-A.; Grande, T. High-Temperature Semiconducting Cubic Phase of BiFe0.7Mn0.3O3+δ. _Physical Review B_ 2009, _79_ , 214113, DOI: 10.1103/PhysRevB.79.214113
  * Forrester et al. 2004 Forrester, J. S.; Zobec, J. S.; Phelan, D.; Kisi, E. H. Synthesis of PbTiO3 Ceramics Using Mechanical Alloying and Solid State Sintering. _Journal of Solid State Chemistry_ 2004, _177_ , 3553–3559, DOI: 10.1016/j.jssc.2004.06.005
  * Paik et al. 2002 Paik, Y.; Grey, C. P.; Johnson, C. S.; Kim, J.-S.; Thackeray, M. M. Lithium and Deuterium NMR Studies of Acid-Leached Layered Lithium Manganese Oxides. _Chemistry of Materials_ 2002, _14_ , 5109–5115, DOI: 10.1021/cm0206385
  * Herstedt et al. 2003 Herstedt, M.; Stjerndahl, M.; Nytén, A.; Gustafsson, T.; Rensmo, H.; Siegbahn, H.; Ravet, N.; Armand, M.; Thomas, J. O.; Edström, K. Surface Chemistry of Carbon-Treated LiFePO4 Particles for Li-Ion Battery Cathodes Studied by PES. _Electrochemical and Solid-State Letters_ 2003, _6_ , A202, DOI: 10.1149/1.1594413
  * Kitamura et al. 2009 Kitamura, N.; Iwatsuki, H.; Idemoto, Y. Improvement of Cathode Performance of LiMn2O4 as a Cathode Active Material for Li Ion Battery by Step-by-Step Supersonic-Wave Treatments. _Journal of Power Sources_ 2009, _189_ , 114–120, DOI: 10.1016/j.jpowsour.2008.10.046
  * Neilson et al. 2023 Neilson, J. R.; McDermott, M. J.; Persson, K. A. Modernist Materials Synthesis: Finding Thermodynamic Shortcuts with Hyperdimensional Chemistry. _Journal of Materials Research_ 2023, _38_ , 2885–2893, DOI: 10.1557/s43578-023-01037-2
  * Bierach 1988 Bierach, J. W. The Formation of Barium Titanate Ceramics by Solid State Reaction. Ph.D. thesis, Lawrence Berkeley National Laboratory, 1988
  * Hong et al. 2022 Hong, Q.-J.; Ushakov, S. V.; van de Walle, A.; Navrotsky, A. Melting Temperature Prediction Using a Graph Neural Network Model: From Ancient Minerals to New Materials. _Proceedings of the National Academy of Sciences_ 2022, _119_ , e2209630119, DOI: 10.1073/pnas.2209630119
  * O’Nolan et al. 2020 O’Nolan, D.; Huang, G.; Kamm, G. E.; Grenier, A.; Liu, C.-H.; Todd, P. K.; Wustrow, A.; Tran, G. T.; Montiel, D.; Neilson, J. R.; Billinge, S. J. L.; Chupas, P. J.; Thornton, K. S.; Chapman, K. W. A Thermal-Gradient Approach to Variable-Temperature Measurements Resolved in Space. _Journal of Applied Crystallography_ 2020, _53_ , 662–670, DOI: 10.1107/s160057672000415x
  * Martinolich and Neilson 2017 Martinolich, A. J.; Neilson, J. R. Toward Reaction-by-Design: Achieving Kinetic Control of Solid State Chemistry with Metathesis. _Chemistry of Materials_ 2017, _29_ , 479–489, DOI: 10.1021/acs.chemmater.6b04861
  * Szymanski et al. 2021 Szymanski, N. J.; Zeng, Y.; Huo, H.; Bartel, C. J.; Kim, H.; Ceder, G. Toward Autonomous Design and Synthesis of Novel Inorganic Materials. _Materials Horizons_ 2021, _8_ , 2169–2198, DOI: 10.1039/d1mh00495f
  * Ganose et al. 2023 Ganose, A.; Jain, A.; Rignanese, G.-M.; Waroquiers, D.; Petretto, G., et al. Jobflow. GitHub, 2023; https://github.com/materialsproject/jobflow
  * Jain et al. 2015 Jain, A.; Ong, S. P.; Chen, W.; Medasani, B.; Qu, X.; Kocher, M.; Brafman, M.; Petretto, G.; Rignanese, G.; Hautier, G.; Gunter, D.; Persson, K. A. FireWorks: A Dynamic Workflow System Designed for High‐throughput Applications. _Concurrency and Computation: Practice and Experience_ 2015, _27_ , 5037–5059, DOI: 10.1002/cpe.3505
  * McSloy et al. 2018 McSloy, A. J.; Trussov, I.; Jarvis, A.; Cooke, D. J.; Slater, P. R.; Panchmatia, P. M. Mechanism of Carbon Dioxide and Water Incorporation in Ba2TiO4: A Joint Computational and Experimental Study. _The Journal of Physical Chemistry C_ 2018, _122_ , 1061–1069, DOI: 10.1021/acs.jpcc.7b10330
  * Zhu and West 2010 Zhu, N.; West, A. R. Formation and Stability of Ferroelectric BaTi2O5. _Journal of the American Ceramic Society_ 2010, _93_ , 295–300, DOI: 10.1111/j.1551-2916.2009.03393.x
  * Murthy and Reddy 2012 Murthy, J. S. N.; Reddy, P. V. V. Solid-State Reaction Between Barium Sulfate and Carbon. _Chemical Engineering Communications_ 2012, _199_ , 966–990, DOI: 10.1080/00986445.2011.631204
  * Meng et al. 2016 Meng, F.; Liu, Y.; Xue, T.; Su, Q.; Wang, W.; Qi, T. Structures, Formation Mechanisms, and Ion-Exchange Properties of $\alpha$-, $\beta$-, and $\gamma$-Na2TiO3. _RSC Advances_ 2016, _6_ , 112625–112633, DOI: 10.1039/C6RA16984H
  * Coelho 2018 Coelho, A. A. TOPAS and TOPAS-Academic: An Optimization Program Integrating Computer Algebra and Crystallographic Objects Written in C++. _Journal of Applied Crystallography_ 2018, _51_ , 210–218, DOI: 10.1107/S1600576718000183
  * Toby and Dreele 2013 Toby, B.; Dreele, R. V. GSAS-II: The Genesis of a Modern Open-Source All Purpose Crystallography Software Package. _Journal of Applied Crystallography_ 2013, _46_ , 544–549, DOI: 10.1107/s0021889813003531