Observing and Modeling the Sequential Pairwise Reactions that Drive Solid-State Ceramic Synthesis

Solid-state synthesis from powder precursors is the primary processing route to advanced multicomponent ceramic materials. Designing ceramic synthesis routes is usually a laborious, trial-and-error process, as heterogeneous mixtures of powder precursors often evolve through a complicated series of reaction intermediates. Here, we show that phase evolution from multiple precursors can be modeled as a sequence of pairwise interfacial reactions, with thermodynamic driving forces that can be efficiently calculated using ab initio methods. Using the synthesis of the classic high-temperature superconductor YBa$_2$Cu$_3$O$_{6+x}$ (YBCO) as a representative system, we rationalize how replacing the common BaCO$_3$ precursor with BaO$_2$ redirects phase evolution through a kinetically-facile pathway. Our model is validated from in situ X-ray diffraction and in situ microscopy observations, which show rapid YBCO formation from BaO$_2$ in only 30 minutes. By combining thermodynamic modeling with in situ characterization, we introduce a new computable framework to interpret and ultimately design synthesis pathways to complex ceramic materials.


. Introduction
Solid-state ceramic synthesis involves heating a mixture of precursor powders at high temperatures (typically > °C) and has been used to realize countless functional materials. [ -] Recent in situ characterization studies have revealed that solid-state reactions often evolve through a variety of nonequilibrium intermediates prior to formation of the equilibrium phase. [ -] These complicated phase evolution sequences are currently difficult to understand, resulting in laborious trialand-error efforts to optimize ceramic synthesis recipes. Theory and computation could help guide synthesis planning, but computation has mostly been used to evaluate thermodynamic stability or overall reaction energies. [ -] While these quantities are valuable, they do not provide mechanistic insights into which nonequilibrium intermediates will appear during Solid-state synthesis from powder precursors is the primary processing route to advanced multicomponent ceramic materials. Designing reaction conditions and precursors for ceramic synthesis can be a laborious, trial-and-error process, as heterogeneous mixtures of precursors often evolve through a complicated series of reaction intermediates. Here, ab initio thermodynamics is used to model which pair of precursors has the most reactive interface, enabling the understanding and anticipation of which non-equilibrium intermediates form in the early stages of a solidstate reaction. In situ X-ray diffraction and in situ electron microscopy are then used to observe how these initial intermediates influence phase evolution in the synthesis of the classic high-temperature superconductor YBa Cu O +x (YBCO). The model developed herein rationalizes how the replacement of the traditional BaCO precursor with BaO redirects phase evolution through a low-temperature eutectic melt, facilitating the formation of YBCO in min instead of + h. Precursor selection plays an important role in tuning the thermodynamics of interfacial reactions and emerges as an important design parameter in planning kinetically favorable synthesis pathways to complex ceramic materials. phase evolution. The ability to rationalize and anticipate which intermediate phases form would enable solid-state chemists to design crystallization pathways that target (or avoid) specific intermediates, accelerating the design of time-and energy-efficient ceramic synthesis recipes for new materials.
The complexity of phase evolution in solid-state synthesis arises from the various pathways by which an initially heterogeneous mixture of precursor particles can transform to a homogeneous target phase. At the microscopic level, solid-state reactions initiate in the interfacial regions between precursors as the system is heated. Because interfacial reactions can only occur between two solid phases at a time, we hypothesize that by determining which pair of precursors exhibits the most reactive interface, we can anticipate which interfacial reaction initiates the overall solid-state reaction, as illustrated schematically in Figure a. Once two precursors react to form a new phase, this nonequilibrium intermediate will then react through its interface with other precursors and intermediate phases. By decomposing the overall phase evolution into a sequence of pairwise reactions, we can calculate the thermodynamics and analyze the kinetics of each reaction step separately, providing a simplified theoretical picture to conceptualize and navigate ceramic synthesis. [ , -] We demonstrate how this concept of sequential pairwise reactions enables us to model phase evolution in the ceramic synthesis of the classic high-temperature superconductor, YBa Cu O +x (YBCO). [ -] Following the discovery that YBCO remains superconducting above the boiling point of liquid N (> K), YBCO has been synthesized many thousands of times in laboratories around the world. The typical synthesis recipe for YBCO calls for three precursors-a . / / molar ratio of Y O /BaCO /CuO powders-which are ground in a mortar, then compacted, pelletized, and baked in air at °C for > h. Even after h, the synthesis reaction is often incomplete, so the pellets must be reground, repelletized, and rebaked until phase-pure YBCO is obtained. [ ] It has been reported that replacing BaCO with BaO can shorten YBCO synthesis times to h and eliminate the need for regrinding. [ , ] This dramatic difference in synthesis times offers an ideal case study to explore how precursor selection governs phase evolution in solid-state synthesis. [ ] In Figure b, we show temperature-dependent Gibbs reaction energies, ∆G rxn , for the formation of YBCO with either BaO or BaCO as the barium source. BaO is less stable than BaCO , [ ] so although both reactions are thermodynamically favorable (∆G rxn < ) above ≈ °C, the thermodynamic driving force (magnitude of ∆G rxn ) is much larger with BaO .
Naively, one might anticipate that this larger driving force explains why YBCO synthesis with a BaO precursor proceeds faster. Here, we show that the mechanism actually proceeds in multiple stages. First, the BaO precursor initiates an early BaO |CuO reaction to form a crucial Ba Cu O intermediate. This intermediate then directs phase evolution through a lowtemperature eutectic melt, which provides fast liquid diffusion to facilitate rapid YBCO formation in only min. BaO is a relatively uncommon YBCO precursor, appearing in only out of synthesis recipes for YBCO (and related phases) as text-mined from the literature, [ ] whereas BaCO is the most Figure . Schematic of sequential pairwise interfacial reactions and overall reaction energetics for YBCO synthesis. a) Schematic of the pairwise reaction concept, illustrating that phase evolution from powder precursors must initiate at the shared interface between two precursor grains. b) The temperature-dependent Gibbs reaction energies, ∆G rxn , for the formation of YBCO from precursor mixtures utilizing BaCO (dashed line) or BaO (solid line) as the Ba source.
common Ba precursor, at out of recipes (all extracted synthesis recipes are shown in Table S , Supporting Information). By better understanding how uncommon precursors promote kinetically favorable sequential pairwise reactions, [ ] we can build toward new design principles for precursor selection and rational synthesis planning.
Here, we use in situ synchrotron X-ray diffraction (XRD) to characterize the temperature-time-transformation process of YBCO formation, as well as in situ microscopy (scanning electron microscope (SEM), dark-field scanning transmission electron microscopy (DF-STEM)) to directly observe the spatiotemporal microstructural evolution from the three initial precursors. By comparing these experimentally observed phase evolution pathways against density functional theory (DFT)-calculated thermodynamics [ ] aided by a machine-learned model for temperature-dependent Gibbs free energies, [ ] we both model and observe the role of interfacial reactions in dictating phase evolution in solid-state synthesis. In particular, we show how thermodynamic calculations can predict the relative reactivity of pairwise interfaces, and can also anticipate the first intermediate that forms at the most reactive interface. Once this first intermediate forms, kinetically controlled processes play a more significant role in the subsequent phase evolution, which is directly observed using in situ characterization. Our work here provides a theoretical foundation to model phase evolution from multiple precursors and demonstrates the importance of precursor selection in governing the dynamics of phase evolution during the solid-state synthesis of complex ceramics.

. Results and Discussion
In Figure , we show in situ synchrotron X-ray diffraction patterns for phase evolution in YBCO synthesis in air with either BaCO ( When BaCO is replaced with BaO , the reaction thermodynamics change dramatically as the BaO |CuO interface has large driving force (∆G rxn < − kJ mol − ) to form ternary Ba-Cu-oxides above °C (Figure d). This is consistent with in situ XRD observations of barium copper oxides emerging at ≈ °C and the consumption of BaO by ≈ °C (Figure b). Synthesis of YBCO using a BaCO precursor usually requires > h with intermittent regrindings, [ ] so it is not surprising that YBCO did not form in our min in situ experiment (Figure a). At temperatures > °C, traces of a Y Cu O phase are observed, even though the BaCO |CuO interface has the larger thermodynamic driving force to react (Figure c). BaCO decomposition is reported to have a substantial activation barrier of kJ mol − , [ ] and the thermodynamic driving forces for all Y O -BaCO -CuO interfacial reactions have ∆G rxn less negative than − kJ mol − up to °C, which is evidently too small to overcome this kinetic barrier. These poor reaction kinetics, coupled with a small thermodynamic driving force, underlie the slow synthesis of YBCO when starting from a BaCO precursor.
The fast formation of YBCO when starting from BaO originates from the large thermodynamic driving force at the BaO |CuO interface, which is ≈ kJ mol − larger than at the BaCO |CuO interface at °C. We previously demonstrated in the synthesis of Na x MO (M = Co, Mn) that the first phase to form in an interfacial reaction is the compound with the largest compositionally-unconstrained reaction energy from the precursors. [ ] Here, our results in the YBCO system provide further evidence for this theory. We calculate that Ba Cu O has the largest reaction energy to form at the BaO |CuO interface, and indeed this is the first observed ternary phase, which is accompanied by evolution of O gas. Between and °C, Ba Cu O decomposes to form BaCuO and CuO (Figure b). The preferential reactivity of the BaO |CuO interface-instead of the Y O |BaO or Y O |CuO interfaces-provides another example that the first phase to form in an interfacial reaction is the phase with the largest thermodynamic driving force, and further suggests that when multiple competing interfaces exist, the interface with the most exergonic compositionally-unconstrained reaction energy will initiate the solid-state reaction.
Our approach here assumes that thermodynamics plays the dominant role in selecting which pairwise interface is most reactive, but kinetic considerations are also important. In previous studies of diffusion couples between metal-silicon and metalmetal interfaces, both reaction energies and interdiffusion rates governed initial phase formation. [ -] Transport kinetics are in fact intimately coupled with thermodynamic considerations, as thermodynamic driving forces appear in Fick's first law as the chemical potential gradient. When different pairwise interfaces exhibit large differences in driving forces, as they do here in the Y O -BaO -CuO system, thermodynamic considerations are likely to dominate the relative kinetics of interdiffusion. However, when the thermodynamic terms are comparable between different interfaces, a more explicit treatment of diffusion kinetics cannot be avoided. Because transport arises from a combination of bulk, dislocation, and surface diffusion mechanisms, it is today challenging to compute the relative interdiffusion kinetics between different interfaces. However, in the limit where reactions are thermodynamically controlled, our model offers a tractable way to anticipate which pairwise interface will be most reactive in a given precursor mixture, and which phase is most likely to form at those interfaces-information which is invaluable for synthesis planning.
Whereas in situ XRD measurements track the temperaturetime-transformation evolution of the system, in situ SEM/DF-STEM provides direct spatiotemporal observation of the microstructural evolution during the solid-state reaction. We next monitored the synthesis of YBCO from Y O -BaO -CuO on a hot stage using in situ electron microscopy (SEM/DF-STEM: Hitachi HF ). Although the in situ microscopy used here cannot identify crystal structure, the reaction conditions (temperature, heating rate, precursors) are similar to those characterized by in situ XRD (Figure b). One difference is that the in situ microscopy heating was conducted in vacuum as opposed to air, but we show in Figure S (Supporting Information), that this does not significantly affect the thermodynamic driving forces in the initial pairwise reactions. For this reason, we anticipate that the temperature-time-transformation progression between the two methods (XRD and electron microscopy) are comparable. We also characterize the elemental distribution in the sample using energy-dispersive X-ray spectroscopy (EDX) before and after the in situ microscopy experiment (our EDX instrument can only operate at room temperature).
In Figure a, we show DF-STEM snapshots of the particles during heating along with EDX before and after heating. A video of this reaction is also provided as Supporting Information Video S .
At room temperature, EDX shows that the three precursor powders are in intimate contact. Importantly, it is clear from EDX that all three potential pairwise interfaces (Y O |BaO , Y O |CuO, and BaO |CuO) exist in the sample. As the stage is heated to °C, the initial BaO and CuO precursors react at the BaO |CuO interface, which according to the in situ XRD experiments, results in Ba Cu O . Meanwhile, the Y O particle remains inert, as does its interface with BaO . From to °C, we observe the ejection of small bubble-like particles, which corresponds to the reaction: Ba Cu O → BaCuO + CuO + . O . In a separate in situ heating experiment, we confirm with SEM and EDX measurements that this initial reaction occurs strictly in the Ba-Cu-O subsystem ( Figure For the formation of BaCuO , we calculate a reaction energy of − kJ mol − ( BaO + CuO → BaCuO + O ), meaning that ≈ / rd of total reaction driving force is consumed before Y O becomes involved in the reaction. Only ≈ kJ mol − remain to drive the reaction to form YBCO. This is more or less comparable to the overall reaction energy from Y O , BaCO and CuO (Figure b), indicating this thermodynamic driving force does not account for the quick formation of YBCO when BaO is used. Thus, we anticipate kinetic selection to play the primary role in the formation of the next phase. Indeed, this kinetic mechanism is provided by the melting of BaCuO and CuO at ≈ °C. This liquid Ba-Cu-O melt is then rapidly consumed into the Y O particle to form YBCO. In the EDX taken after the experiment, the morphology of the Y region remains similar to the beginning of the experiment, but now Ba and Cu signals are found in the final particle.
In Figure b, we overlay the observed phase evolution sequence onto the pseudobinary BaO -CuO isopleth [ ] of the overall Y O -BaO -CuO phase diagram to reveal how the BaO precursor enables rapid YBCO synthesis. The first reaction occurs before °C and proceeds at the BaO |CuO interface to form Ba Cu O . This is consistent with our calculations in Figure  To verify the role of BaCuO and the BaCuO |CuO eutectic in enabling rapid YBCO synthesis, we performed an additional in situ synthesis starting from Y O , BaCuO , and CuO, which similarly leads to rapid YBCO formation above ≈ °C (Figure S , Supporting Information). A deviation between the total mass of crystalline phases and the thermogravimetric measurement of the total sample mass that precedes rapid YBCO formation again confirms that a liquid phase mediates YBCO formation.
If one consults the Y O -CuO or Y O -BaO phase diagrams, [ ] the lowest liquidus temperatures in these systems are ≥ °C, which is above the temperature at which YBCO decomposes ( °C). [ ] BaO therefore plays a crucial role in directing the phase evolution through the pseudobinary BaO -CuO subsystem-where a low-temperature liquid eutectic acts as a self-flux, providing the fast diffusion kinetics needed to form YBCO in min. This is in contrast to when BaCO is used as the Ba source, where the slow decomposition reaction kinetics at the BaCO |CuO interface forces the overall reaction to proceed through the Y O -CuO subsystem, and a high liquidus temperature of °C obstructs any liquid-mediated transport kinetics for YBCO formation. [ ] Although the overall reaction energies shown in Figure b suggest that the larger thermodynamic driving force is why a reaction with the BaO precursor proceeds more quickly than with BaCO , we emphasize here that the magnitude of the overall reaction energy is not the origin of the fast synthesis time. Instead, it is the initial selection of the BaO -CuO subsystem, where there is a low-temperature eutectic below the decomposition temperature of YBCO, that enables rapid YBCO synthesis by forming a self-flux. This finding highlights the need to consider computations beyond the phase stabilities of the target or overall reaction energies in order to obtain mechanistic insights into the reaction pathways by which phases can evolve during synthesis. Upon cooling the sample down from °C at a rate of °C min − , in situ XRD shows in Figure a structural transition from tetragonal to orthorhombic YBCO at °C, indicating the uptake of ambient O into YBa Cu O to form YBa Cu O +x , consistent with reports from the literature. [ , ] The synthesized product exhibits a strong diamagnetic signal below K (Figure c), indicating the successful synthesis of superconducting YBCO. From a thermodynamic perspective, it is well-characterized that YBa Cu O +x is metastable at low temperature with respect to decomposition [ ] by the reaction Cu O 100 kJ mol at 27 C 2 3 6.5 2 2 3 2 3 6 rxn 1 G ( ) However, this solid-state decomposition is kinetically limited during cooling. On the other hand, oxygen diffusion is highly mobile in the YBCO framework, [ , ] indicating that this final topotactic uptake of O gas at the YBCO|O interface is a kinetically mediated nonequilibrium reaction.
In Figure , we summarize how phase evolution during YBCO synthesis can be understood as a sequence of pair-wise reactions that result from an interplay between thermodynamics and kinetics. The initial mixture of three precursors-Y O , BaO , and CuO-produces three possible reactive interfaces. We calculated in Figure

Figure .
Topotactic O uptake and phase transition during slow cooling. a) In situ synchrotron XRD pattern for cooling of Y O + BaO + CuO precursor from to °C at °C min − . "tet" refers to the tetragonal structure and "ort" to the orthorhombic structure. b) Changes in lattice parameters during cooling. c) Magnetic susceptibility of synthesized YBCO exhibiting superconductivity above liquidus nitrogen temperature. d) The tetragonal and orthorhombic crystal structures for YBCO, where blue spheres are Y, green are Ba, orange are Cu, and red are O.

Figure .
Phase-evolution pathway for the formation of YBCO dictated by sequential pairwise reactions. The YBCO synthesis pathway is shown here along two qualitative axes-the thermodynamic driving force to form new phases along the vertical axis and the diffusion rate of reactive species along the horizontal axis. Within this framework, we understand reaction events occurring in either a thermodynamic regime, where driving forces or diffusion rates are sufficiently high that equilibrium products are observed, or a kinetic regime, where ion transport is sufficiently slow or driving forces sufficiently small such that the system becomes unreactive or nonequilibrium products are formed.
interface possesses the largest thermodynamic driving force to react, and predicted Ba Cu O to be the first reaction intermediate, which was confirmed by in situ XRD (Figure b) and in situ electron microscopy (Figure a; and Figure S , Supporting Information). The formation of Ba Cu O below °C consumes ≈ / rd of the overall reaction driving force, meaning the ensuing reactions necessarily occur with smaller driving forces. Using in situ DF-STEM, we observed that after the peritectoid decomposition of Ba Cu O into BaCuO + CuO, there is no further phase evolution in the system until the formation of a eutectic melt at the BaCuO |CuO interface. This liquid melt serves as a self-flux, providing fast Ba and Cu transport into the thus-far immobile Y O , forming YBa Cu O (Figure ). Finally, fast topotactic oxygen uptake at the YBa Cu O |O interface upon cooling yields the superconducting YBa Cu O +x phase ( Figure ), which persists kinetically as a metastable phase to room temperature, instead of decomposing to the equilibrium Y O + Ba Cu O phases.

. Conclusion
Our investigation here provides a general conceptual framework to approach the solid-state synthesis of complex multicomponent ceramics. A ceramic synthesis reaction that begins from N precursors will exhibit N C pairwise reaction interfaces. We showed here that in the early stages of synthesis, when thermodynamic driving forces are large, the first reaction will occur between the two precursors with the largest compositionally-unconstrained reaction driving force. This initial reaction interface determines which pseudobinary subsystem the ensuing phase evolution proceeds from, and we showed that this initial interface can be anticipated from ab initio calculations. For YBCO, starting with a BaO precursor leads to a large driving force to form Ba Cu O at the BaO |CuO interface; whereas starting from the traditional BaCO precursor results in slow BaCO decomposition kinetics, forcing the reaction through the Y O -CuO subsystem, where slow diffusion kinetics means manual regrinding is necessary to reintroduce interfaces between unfinished reaction intermediates.
In general, the replacement of oxide/carbonate precursors with peroxides may be an effective way to redirect the synthesis of multicomponent materials through different subsystems. In Figure S (Supporting Information), we show that the energy required to disproportionate alkali(ne) peroxides is generally less than their corresponding oxides/carbonates. By thoughtfully choosing the starting precursors [ , ] to control which pairwise interface is the most reactive, one can deliberately direct phase evolution through whichever pseudobinary subsystem exhibits the best kinetic pathway to the target material. Today, it remains difficult to anticipate which kinetic mechanisms are available in a given subsystem, especially when thermodynamic driving forces are similar between different interfaces. In the near term, in situ characterization remains the most productive approach for rationally designing solid-state synthesis recipes. In the future, a theoretical framework that embeds nucleation, diffusion, and crystal growth kinetics within a thermodynamic description of sequential pairwise reactions will pave the way toward a complete computational platform for predictive solidstate ceramic synthesis.

. Experimental Section
In Situ Synchrotron Powder X-ray Diffraction: Y O (> . %, Kojundo Kagaku), BaCO (> . %, Kojyundo Kagaku), BaO (> %, Jyunsei Kagaku), CuO (> %, Wako Chemical) were weighed in a molar ratio of Y/Ba/Cu = / / and loaded into a zirconia pot with zirconia balls with a diameter of mm. The starting materials were milled by planetary ball milling for h over rpm. The mixed powder was loaded into a quartz capillary with a diameter of . mm.
The change in crystalline phases was examined using synchrotron powder X-ray diffraction at the BL B beamline of SPring-(proposal nos.
A , B , and A ). The quartz capillary with powder mixture was settled in a furnace in air atmosphere. Heating started after setting the furnace to °C at the heating rate of °C min − till °C. The samples were kept min at °C and then cooled at °C min − to °C. The diffraction data of θ range from . ° to . ° with a step of . ° were collected using a high-resolution D semiconductor detector (MYTHEN). [ ] The wavelength of the radiation beam was determined using a CeO standard. Rietveld refinement was performed by RIETAN-FP, [ ] and the crystal structure was visualized using VESTA software. [ ] In Situ Transmission Electron Microscopy (TEM) Measurement: In an Ar-filled glove box, BaO powder (> %, Jyunsei Kagaku) was milled by planetary ball milling for h over rpm. The powder was sieved to remove particles larger than µm. In ambient atmosphere, Y O (> . %, Kojundo Kagaku), CuO nanopowder (> %, Alderich), and above BaO powder were weighed in a molar ratio of Y/Ba/Cu = / / and loaded again into a zirconia pot with mm zirconia balls. The powder was mixed by planetary ball milling for h over rpm. The sample was dispersed in dehydrated ethanol and ultrasonicated. This suspension was dropped onto a silicon nitride TEM grid.
Morphological and compositional changes were observed by TEM (HF-Hitachi High-Tech Corporation). The accelerating voltage was kV and pressure was ≈ × − Pa. The sample was initially heated at °C, then heated to °C at °C min − . The apparatus allowed three images to be recorded simultaneously: SEM, bright-field scanning transmission electron microscopy (BF-STEM), and DF-STEM. Before and after heating the sample, the compositional distribution was examined by EDX mapping at room temperature.
Magnetization Measurement: The magnetization was measured using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design MPMS-) with an applied field of Oe in order to check the Meissner effect of the synthesized sample.
Computational Details: Standard Gibbs formation energies, ∆G°f(T), for gaseous species were obtained from NIST. [ ] To account for the synthesis atmosphere (air), Gibbs formation energies of a given gaseous species, ∆G°f ,i (T), were obtained as where R is the gas constant and p i approximates the activity coefficient of gaseous species, i. The only gaseous species evolved or consumed in reactions discussed in this work are O and CO , where O 2 p was taken to be . atm and CO 2 p = .
atm. For solid-state compounds, formation enthalpies (at K) were obtained with DFT, utilizing the SCAN meta-GGA density functional. [ ] Each structure was obtained from the Materials Project database [ ] and optimized using the Vienna Ab initio Simulation Package (VASP) [ ] and the projector augmented wave method, [ ] a plane-wave energy cutoff of eV, and k-points per reciprocal atom. Standard Gibbs formation energies, ∆G°f(T), for each solid-state compound were then obtained by combining the DFT-calculated formation enthalpies, the machine-learned descriptor, [ ] and