Jie Fu

Radiative–conductive systems are intrinsically nonlinear due to the quartic temperature dependence of thermal radiation. Under fixed total heating power, convexity arguments imply that nonuniform temperature distributions radiate more efficiently and therefore exhibit a lower mean temperature than their isothermal counterparts. However, this conclusion remains qualitative, and an explicit quantitative relation between temperature nonuniformity and mean temperature reduction has been lacking. Here we derive a variance-based analytical expression linking the area-averaged temperature to the corresponding isothermal equilibrium temperature in a nonuniformly heated radiative–conductive system. Within a second-order asymptotic expansion about the ambient temperature, we show that the leading reduction in the area-averaged temperature is proportional to the temperature variance. This result transforms the convexity-based inequality into a quantitative statistical relation within the perturbative regime and provides a physically transparent framework for describing nonlinear radiative averaging in thermally heterogeneous systems.

Determining the concentration of light elements in the Earth’s outer core is crucial for understanding the generation of the geomagnetic field, as well as the Earth’s internal dynamics and thermal evolution. However, as a potential dominant light element in the outer core, the precise composition of silicon (Si) is still a topic of intense debate. Due to the limited experimental data, significant controversies exist between theoretical models and experimental predictions regarding the Si content in the outer core. In this work, we have calculated pressure (P)–volume (V)–temperature (T) data of Fe-X wt.% Si, where X = 0, 2.4, 4.9, 7.5, and 10.3 at ~136–330 GPa and 4000–7000 K by first-principles molecular dynamics (FP-MD) simulations. We employed pressure correction to address the discrepancy between theoretical and experimental measurements. Based on the corrected data, we established an equation of state (EoS) for Fe-Si alloys. We calculated thermodynamic properties, including density (ρ), thermal expansivity, Grüneisen parameter, isothermal and adiabatic bulk moduli, and sound velocity (V_P). To constrain the silicon content in the outer core, the ρ and V_P of Fe-X wt.% Si computed along the outer core geotherm were compared with the Preliminary Reference Earth Model (PREM). Assuming Si is the only light element in the outer core and is constrained by PREM data, the maximum Si content at T_ICB = 5400 K is 2.4 ± 1.7 wt.%. Considering the uncertainty in T_ICB, the maximum Si content in the outer core ranges from 1.8 to 3.0 ± 1.7 wt.%. The fact that a homogeneous binary alloy cannot match all seismic observations provides evidence that the content of Si in the outer core may vary with depth. Besides, the work states that pressure correction helps bridge the gap between theoretical and experimental estimates of Si concentration. The pressure-corrected equation of state provides a robust benchmark for constraining multi-component core models.

Magnesium (Mg) partitioning behavior between solid and liquid iron (Fe) shows that Mg slightly favors liquid Fe under conditions at the Earth’s core. This means that the Earth’s outer core may contain more Mg than previously thought. However, geophysical properties such as density (ρ) and sound velocity (V_P) of liquid Fe-Mg alloys under outer-core conditions have yet to be explored. Considering that the liquid outer core includes approximately 10 wt.% light elements, here we established an equation of state (EoS) of liquid Fe-Mg alloys with Mg less than 10 wt.% to study the thermodynamic properties under outer-core conditions using first-principles molecular dynamics. The results show that adding Mg to liquid Fe will clearly reduce the ρ, isothermal bulk modulus (K_T), and adiabatic bulk modulus (K_S). Meanwhile, it will increase the V_P. In order to access the Mg content in the outer core, the ρ and V_P of liquid Fe-Mg alloys along the geotherm are compared with the preliminary reference Earth model. Assuming Mg is the only light element, the maximum content of Mg required is approximately 2.9–6.1 wt.% due to the temperature uncertainty at the inner core boundary (ICB). Further, considering the geochemical constraints (the partition coefficient between liquid and solid Fe, molten Fe-alloy, and silicate melts), the content of Mg is further constrained to below 0.5 wt.%.

Carbon (C) is regarded as a possible light element in the Earth’s outer core; however, its abundance in this region remains a subject of controversy. Here, we applied first-principles molecular dynamics (FP-MD) simulations to calculate the density (ρ), thermodynamic properties, and sound velocity (VP) of liquid iron-carbon (Fe-C) alloys (Fe-X wt% C, X = 0, 2.1, 4.7, 7.6, and 11.2) under the outer core conditions (~136-330 GPa, 4000-6000 K) to explore the composition of the outer core. Compared with seismic observations, roughly 5.1-6.6 wt% C alloyed with Fe can explain the outer core’s ρ deficit, while 2.9-3.7 wt% C aligns with the seismological VP. Given this discrepancy, we extended our investigation to the iron-carbon-oxygen (Fe-C-O) ternary alloy to provide a more in-depth understanding of the Earth’s outer core composition. The results reveal that certain Fe-C-O models (i.e. Fe-7.3 wt% O, Fe-6.5 wt% O-0.3 wt% C, and Fe-5.3 wt% O-1.1 wt% C) can satisfy seismic observations under some possible geothermal profile of the outer core. The ‘optimal’ models which can match the seismic data consistently exhibit O-rich characteristics consistently, with the maximum C content is 1.5 wt% and the minimum O content is 4.7 wt%. In conclusion, the findings in this work offer some possible C and O reserves in the Earth’s outer core, which benefits to the composition prediction of the outer core.

Slag refining is a crucial step for Si purification by the metallurgical processes. In this study, the impurity removal at Si/SiO₂–CaO–MgO slag interface was investigated experimentally and theoretically. The diffusion couple experiments show that the slag underwent the processes of sintering, solid particle dissolution, and composition homogenization. When the melted slag contacted the Si, the interfacial reaction occurred, and the impurity oxides formed and transferred into the slag phase. The thermodynamic analyses confirmed the existence of the oxidation reaction between impurities and slag. First-principles molecular dynamics simulations of the Si–X–O system (X = Al, B, Ca) clarified the binding behaviors of Al, B, Ca with Si. The calculated binding energy shows that the affinity ability of these impurities with Si are B > Al > Ca at the molecular level, and B impurity is the most difficult one to be removed from Si during the slag refining process.

Identifying the stable structures of gold (Au) clusters is a huge challenge in cluster science. In this work, we have searched the ground-state structures of neutral Au _n (n = 16–25) clusters using the potential of an artificial neural network (ANN) trained with density functional theory (DFT) data. Compared with the DFT data, the root mean square error of binding energy predicted by the ANN potential is about 8.66 meV/atom. Applying the ANN potential to search the ground-state structures by comprehensive genetic algorithm, we have found several new candidates of Au₁₈, Au₂₂, and Au₂₃, which have not been previously reported. Au₁₈ has a hollow cage structure, whereas Au₂₂ and Au₂₃ are flat cage structures. From the electronic analysis, we elucidate the stability mechanism of the newly found structures that are associated with the electronic shell closure of superatomic orbitals. Additonally, we also clarified how to clean a database to train an efficient ANN potential in detail. Overall, this work proves that applying machine learning to the description of atomic interactions can accelerate the search of ground-state structures of clusters and help to find new candidates for stable cluster structures.

The search of ground-state structures (GSSs) of gold (Au) clusters is a formidable challenge due to the complexity of potential energy surface (PES). In this work, we have built a high-dimensional artificial neural network (ANN) potential to describe the PES of Au₂₀ clusters. The ANN potential is trained through learning the GSS search process of Au₂₀ by the combination of density functional theory (DFT) method and genetic algorithm. The root mean square errors of energy and force are 7.72 meV atom⁻¹ and 217.02 meV Å⁻¹, respectively. As a result, it can find the lowest-energy structure (LES) of Au₂₀ clusters that is consistent with previous results. Furthermore, the scalability test shows that it can predict the energy of smaller size Au_16–19 clusters with errors less than 22.85 meV atom⁻¹, and for larger size Au_21–25 clusters, the errors are below 36.94 meV atom⁻¹. Extra attention should be paid to its accuracy for Au_21–25 clusters. Applying the ANN to search the GSSs of Au_16–25, we discover two new structures of Au₁₆ and Au₂₁ that are not reported before and several candidate LESs of Au_16–18. In summary, this work proves that an ANN potential trained for specific size clusters could reproduce the GSS search process by DFT and be applied in the GSS search of smaller size clusters nearby. Therefore, we claim that building ANN potential based on DFT data is one of the most promising ways to effectively accelerate the GSS pre-screening of clusters.

The crystal structure of iron, the major component of the Earth's inner core (IC), is unknown under the IC high pressure (P) (3.3–3.6 Mbar) and temperature (T) (5000–7000 K). Experimental and theoretical data on the phase diagram of iron at these extreme PT conditions are contradictory. Applying quasi-ab initio and ab initio molecular dynamics we computed free energies of the body-centered cubic (bcc), hexagonal close-packed (hcp), and liquid phases. The ionic free energies, computed for the embedded-atom model, were corrected for electronic entropy. Such correction brings the melting temperatures of the hcp iron in very good agreement with previous ab initio data. This validates the calculation of the bcc phase, where fully ab initio treatment is not technically possible due to large sizes required for convergence. The resulting phase diagram shows stabilization of the bcc phase prior to melting in the pressure range of the IC. The melting temperature of the bcc phase is equal to 7190 K at the pressure 360 GPa.