Thermodynamics of ultrarefractory condensates: Implications for the high-temperature limit of the inner dust rim of the early solar protoplanetary disk

Ultrarefractory (UR) condensates found in refractory inclusions in chondrites contain records of the early nebular thermochemistry that prevailed in the high-temperature region close to the protosun. Recent reports of the UR phases such as allendeite (Sc₄Zr₃O₁₂), tazheranite ((Zr,Ti,Ca)O_2−x), and kangite ((Sc,Ti,Al,Zr,Mg,Ca,□)₂O₃) imply, respectively, nebular condensation temperatures and origins higher than and inward of those previously deduced from calcium-aluminum-rich inclusions. However, knowledge gaps on their thermochemistry have precluded a quantitative understanding of temperatures and chemical pathways that led to their origins in the early solar protoplanetary disk. Here we use density functional theory to determine the thermochemistry of these materials for the first time. We find that allendeite is a stable phase under equilibrium conditions with its condensation temperature (1643 K at 10⁻⁴ bar) in the same range as that of nominal hibonite (CaAl₁₂O₁₉, 1637 K at 10⁻⁴ bar). Among the UR oxides, tetragonal ZrO₂ exhibits the highest condensation temperature (1739 K at 10⁻⁴ bar) and potentially reveals the high-temperature limits at which solid dust could have survived in the inner region of the disk. In comparison, we find that pure cubic ZrO₂ does not form from a cooling gas of solar composition undergoing equilibrium condensation. Similarly, we find that the stoichiometric endmember of kangite, Sc₂O₃ does not condense under equilibrium conditions, and moreover, the role of Ti and Zr as solutes is crucial to modeling its stability and origins.