We consider the problem of two-dimensional freeform lens design in order to model functions defined on the unit disc. We find the exact mapping between the lens surface and the function to be modeled. We also determine a consistency relation for the model function using a geometrical argument. Because this mapping is highly nonlinear, we make an approximation where the partial derivatives of the lens surface are assumed to be small. We provide examples of lens surfaces for some model functions in this approximation. We also consider complex analytic model functions and see that if they correspond to a lens surface, the set of such complex analytical functions is quite restricted. We utilize the Schwarz lemma, when considering complex analytical model functions from unit disc to itself.

This review consolidates density functional theory insights into hydrogen embrittlement accumulated over the past decade. Fundamental hydrogen-metal interactions spanning adsorption, dissolution, and diffusion are analysed across BCC, FCC, and HCP structures, revealing order-of-magnitude differences in transport kinetics governed by interstitial site geometry rather than crystal class alone. The competing HEDE, HELP, and HESIV mechanisms are critically evaluated, with computational evidence increasingly supporting synergistic operation whose balance depends on local microstructure and stress state. The paradoxical role of hydrogen traps is reconciled: efficacy depends on spatial distribution relative to failure sites, not merely binding strength. Electronic structure descriptors, particularly the d-band centre, provide a transferable screening framework for hydrogen-resistant compositions. Material-specific design principles encompassing alloying strategies, precipitate engineering, and multilayer coatings are extracted. Methodological advances including machine learning potentials and thermodynamic integration are assessed, and five future research priorities are identified.

The surging demand for high-performance and sustainable energy storage has spurred the development of advanced anode materials for lithium- and sodium-ion batteries (LIBs and SIBs). Among emerging candidates, two-dimensional (2D) graphene-based heterostructures stand out due to their exceptional electrical conductivity, large surface area, mechanical robustness, and highly tunable interfacial properties. These features enable them to overcome intrinsic limitations of conventional anodes, including low capacity, sluggish ion transport, and severe volume changes during cycling. This review provides a comprehensive and critical overview of recent advances in graphene-based heterostructure anodes, integrating insights from first-principles calculations and experimental studies. We examine diverse systems, including carbide/graphene, sulfide/graphene, nitride/graphene, oxide/graphene, MXene/graphene, and monoelemental 2D material/graphene composites, focusing on structural stability, electronic properties, ion adsorption and diffusion, theoretical capacities, and open-circuit voltages. Particular emphasis is placed on the heterointerface, which modulates charge redistribution, enhances electron transport, reduces ion migration barriers, suppresses volume expansion, and facilitates fast, reversible Li⁺/Na⁺ storage. Recent progress in synthesis strategies and emerging machine learning-guided design approaches is also highlighted, offering new directions for accelerating the discovery of high-performance heterostructures. Finally, current challenges, including scalability, interface engineering, and experimental validation, are discussed, alongside perspectives for the rational design of next-generation anodes. This review underscores the transformative potential of graphene-based heterostructures as high-capacity, durable, and efficient anodes, advancing the frontier of sustainable energy storage.

Despite the promise of sodium-ion batteries (SIBs) for large-scale energy storage, the development of high-performance anode materials remains a critical challenge. Here, we report that the two-dimensional β-phase carbon selenide (β-CSe) monolayer exhibits remarkable Na-ion storage properties identified through comprehensive first-principles calculations. The buckled honeycomb structure demonstrates exceptional stability with positive phonon frequencies and preserved C-Se bonds during molecular dynamics at both room temperature (300 K) and elevated temperature (400 K). Na adsorption occurs preferentially at hollow sites with strong binding energies (−2.95 eV on C-side) and substantial charge transfer (0.82|e|), thermodynamically favoring uniform Na distribution which may help suppress dendrite formation. Strikingly, the material exhibits ultrafacile Na diffusion with maximum energy barriers of only 0.019-0.021 eV, among the lowest reported for SIB anodes, suggesting exceptional rate performance. Basin-hopping Monte Carlo simulations reveal a theoretical capacity of 589 mAh/g with an average insertion potential of 1.11 V, while the material advantageously transitions from semiconductor to metallic behavior upon Na insertion. The anisotropic Poisson's ratio (as low as 0.05) further minimizes volume changes during cycling. These findings establish β-CSe as a promising candidate for high-performance SIB anodes and provide valuable insights for designing advanced battery materials.

Hydrogen is a clean and renewable energy carrier, but its reversible storage near ambient conditions remains a major challenge. Here, density functional theory (DFT) combined with ab initio molecular dynamics (AIMD) is employed to assess the newly predicted 2D orthorhombic diboron dinitride (o-B ₂N ₂) monolayer, in pristine and Li-functionalized forms, as a hydrogen storage medium. On the pristine surface, H ₂ physisorbs with binding energies of −0.158 to −0.174 eV. Li atoms anchor strongly at the hexagonal hollow sites ((Formula presented.) from −0.979 to −1.321 eV, strongest at the B-rich H1 site), donate 0.65–0.84 (Formula presented.) to the substrate, and render the semiconducting monolayer metallic. A positive cluster formation energy ((Formula presented.) eV per Li pair) and a 5 ps AIMD simulation at 400 K confirm that the Li adatoms remain dispersed, without clustering. Each (Formula presented.) center polarizes and binds up to five H ₂ molecules, with average adsorption energies of −0.207 to −0.336 eV/H ₂, within the optimal window for room-temperature reversible storage. The 4Li@o-B ₂N ₂(20H ₂) system attains a theoretical gravimetric capacity of 15.12 wt% and a practical capacity of 10.99 wt% under realistic operating conditions (charging at 30 atm/25 °C; release at 3 atm/100 °C). These results establish Li-functionalized o-B ₂N ₂ as a promising hydrogen storage material that merits experimental exploration.

This study investigates the electronic and superconducting properties of Ni₃AC (A: Mg, Zn, and Cd) antiperovskites through first-principles computational methods. Importantly, Ni₃MgC has been identified as a superconductor with a transition temperature (T_c) of 8.644 K, while Ni₃ZnC and Ni₃CdC exhibit T_c values of 2.172 K and 3.861 K, respectively, in remarkable agreement with experimental. The electron–phonon interaction strength in these materials suggests medium-coupling superconductivity. This study provides significant insights into the mechanisms driving superconductivity in metal-carbide antiperovskites, identifying opportunities for their use in advanced technologies.

Photocatalytic water splitting represents a promising approach for sustainable hydrogen production, with two-dimensional Janus materials offering unique advantages through intrinsic electric fields that enhance charge separation. We present a comprehensive first-principles investigation of Janus AlXY₂ (X = Ga, In; Y = S, Se, Te) monolayers using density functional theory and ab initio molecular dynamics simulations. All six systems exhibit excellent structural, thermal, and mechanical stability with HSE06 bandgaps of 2.029–2.969 eV suitable for UV-light absorption. The asymmetric structure generates strong intrinsic electric fields of 5.391–6.437 V perpendicular to the monolayer plane, significantly enhancing photogenerated charge carrier separation. While pristine monolayers show poor hydrogen evolution reaction (HER) activity with Gibbs free energies of 1.937–2.371 eV, strategic introduction of metal vacancies dramatically improves performance, reducing ΔG_H values to −0.371 to ＋0.607 eV and approaching optimal catalytic conditions. These findings demonstrate the potential of defect-engineered 2D Janus AlXY₂ materials for efficient photocatalytic hydrogen production.