For sub-2-nm Aln−x_{n-x}Nix_x nanoclusters, all existing stability statements rest on zero-temperature binding or potential energies of artificially constructed or incompletely sampled geometries. Blanco, Guevara, Lozada and Castro (2021) pose the problem of establishing the relative stability of these clusters as a function of Gibbs free energy: systematically explore the isomer and homotop space of each size and composition (first of all for the 10-20-atom sizes and compositions 0≤x≤n0 \le x \le n), replacing the source's artificial structural constructions, and compute G(T)G(T) including vibrational and configurational entropy over an explicitly stated temperature range, determining whether the zero-temperature ordering survives. The source separately notes that ligand or solvent environments complicate electronic-structure calculations for such clusters; environment effects on the Al-Ni hierarchy are outside the scope of this problem.

Background

Bimetallic ("nanoalloy") clusters - aggregates of a few to a few hundred atoms of two metals - can adopt structures and compositions with no bulk analogue, and their electronic structure, magnetism, and catalytic activity depend sensitively on which geometry is actually stable. For a binary cluster of fixed size and composition, the candidate structures comprise topological isomers (different atomic connectivities) and homotops, the distinct ways of distributing the two elements over the same geometric skeleton; their number grows combinatorially with size. Identifying the lowest-energy structure therefore requires systematic global-optimization methods - genetic algorithms, basin hopping (a Monte Carlo technique that hops between local minima of the potential-energy surface), particle-swarm optimization, or, recently, deep reinforcement learning - typically combined with density functional theory (DFT), the standard ab initio framework in which the interacting-electron problem is rewritten in terms of the electron density; the local density approximation (LDA) is its simplest exchange-correlation treatment. In nearly all such studies, "stability" means the zero-temperature potential or binding energy per atom. The Gibbs free energy G(T)=E−TSG(T) = E - TS additionally contains entropic contributions - vibrational entropy and configurational entropy over homotops and near-degenerate isomers - which at finite temperature can reorder a hierarchy derived from energies alone. Real clusters are also often produced in solution or coated by protective ligand molecules, which further changes their stable structures.

The source study (Blanco, Guevara, Lozada and Castro, J. Phys.: Conf. Ser. 1938, 012002 (2021)) examined Aln−x_{n-x}Nix_x nanoclusters with 10-20 atoms and 0≤x≤n0 \le x \le n using LDA-DFT (DMol3), computing densities of states, HOMO-LUMO gaps (the gap between the highest occupied and lowest unoccupied molecular orbitals), and average binding energies for each structure. Because exhaustive isomer optimization was judged computationally out of reach, the geometries were artificially designed motifs (linear, planar, Y-shaped, pyramidal, tetrahedral, octahedral, hexagonal-ring, and prism structures). The authors found overlapping Al sp and Ni 3d orbital contributions, semiconducting gaps, the largest average binding energy for Al10_{10} (a D2hD_{2h} double hexagonal ring) and the smallest for Ni20_{20} (C2vC_{2v}). They close by stating that it "is interesting to establish the stability of the nanocluster structures as a function of the Gibbs free energy", and they acknowledge two scope limitations of their study: the isomer space was not systematically searched, and ligand/solvent environments - which they note complicate electronic-structure calculations for clusters in dissolution or protected by ligand layers - were not modeled.

The surrounding literature frames what is and is not known. Genetic-algorithm searches with Gupta-type many-body potentials have produced zero-temperature structure maps for Ni-Al nanoalloy clusters (Bailey et al., 2003), and particle-swarm optimization has been applied to the same system (Zhou et al., 2008); 13-atom Ni-Al structures and dynamics were analyzed early on (Krissinel and Jellinek, 1997). DFT ground states are known for limited composition slices (Deshpande et al., 2010; Zhang et al., 2013; Acioli, 2024, who also predicted - still unmeasured - photoelectron spectra for Aln_nNi−^- anions), and basin-hopping sampling combined with spin-polarized DFT has been carried out for much larger, 79- and 201-atom fcc-derived Ni-Al clusters (Mokkath and Henzie, 2017). Finite-temperature behavior has been probed by molecular dynamics for 1-10 nm NiAl particles - melting-point depression and Ni/Al demixing (Delogu, 2007), and freezing into ordered B2 (CsCl-type) AlNi (Settem, 2019). What no study provides, for any Al-Ni cluster size, is a Gibbs-free-energy-based stability ranking over systematically sampled isomers, and environment or ligand effects on Al-Ni clusters are entirely unexplored (adjacent results exist for ligand-protected pure-Al clusters and for adsorbate effects on nanoalloy structure prediction).

Problem Statement

For Aln−x_{n-x}Nix_x nanoclusters - the 10-20-atom sizes and compositions 0≤x≤n0 \le x \le n studied by Blanco et al. - establish the relative stability of the cluster structures as a function of Gibbs free energy rather than zero-temperature binding energy alone. This requires: (1) systematic exploration of the isomer space (topological isomers and homotops) for each size and composition, replacing the artificial structural constructions of the source study; and (2) computation of the Gibbs free energy G(T)G(T) of the competing low-energy structures over an explicitly stated, physically motivated temperature range, with the vibrational and configurational entropy contributions (over homotops and near-degenerate isomers) treated explicitly, and determination of the resulting stability hierarchy - in particular, whether the zero-temperature binding-energy ordering found in the source study (largest average binding energy for Al10_{10}, smallest for Ni20_{20}, with binding energy decreasing as nickel content increases) survives at finite temperature. The overarching objective is a stability assessment that predicts which Al-Ni nanocluster structures are actually realizable at finite temperature, replacing the zero-temperature, bare-cluster, artificially-constructed picture. Ligand or solvent environments - which the source notes complicate electronic-structure calculations for such clusters, and which remain unexplored for Al-Ni - may further shift the hierarchy; assessing that shift is not a required element of this objective, though it is a natural extension.

References

  1. J. Blanco, U. Guevara, R. Lozada, O. Castro, 'Electronic and structural properties in nanocluster Aln−x_{n-x}Nix_x', Journal of Physics: Conference Series 1938, 012002 (2021). DOI: 10.1088/1742-6596/1938/1/012002, https://doi.org/10.1088/1742-6596/1938/1/012002
  2. M. S. Bailey, N. T. Wilson, C. Roberts, R. L. Johnston, 'Structures, stabilities and ordering in Ni-Al nanoalloy clusters', The European Physical Journal D 25, 41-55 (2003). DOI: 10.1140/epjd/e2003-00218-2, https://doi.org/10.1140/epjd/e2003-00218-2
  3. E. B. Krissinel, J. Jellinek, '13-atom Ni-Al alloy clusters: Structures and dynamics', International Journal of Quantum Chemistry 62, 185-197 (1997). DOI: 10.1002/(SICI)1097-461X(1997)62:2<185::AID-QUA7>3.0.CO;2-S, https://doi.org/10.1002/(SICI)1097-461X(1997)62:2%3C185::AID-QUA7%3E3.0.CO;2-S
  4. J. C. Zhou, W. J. Li, J. B. Zhu, 'Theoretical study of Ni-Al nanoalloy clusters using particle swarm optimisation algorithm', Materials Science and Technology 24, 870-874 (2008). DOI: 10.1179/174328408X307373, https://doi.org/10.1179/174328408X307373
  5. M. D. Deshpande, R. Pandey, M. A. Blanco, A. Khalkar, 'Magnetic properties of Ni13−n_{13-n}Aln_n clusters with nn = 0-13', Journal of Nanoparticle Research 12, 1129-1136 (2010). DOI: 10.1007/s11051-009-9654-6, https://doi.org/10.1007/s11051-009-9654-6
  6. X. Zhang, B.-X. Li, Z.-w. Ma, J.-j. Gu, 'First-Principles Study on Stability and Magnetism of Nim_mAln_n (mm = 1-3, nn = 1-9) Clusters', The Scientific World Journal 2013, 468327 (2013). DOI: 10.1155/2013/468327, https://doi.org/10.1155/2013/468327
  7. P. H. Acioli, 'Theoretical prediction of low-energy photoelectron spectra of Aln_nNi−^- clusters (nn = 1-13)', Journal of Molecular Modeling 30, 155 (2024). DOI: 10.1007/s00894-024-05944-x, https://doi.org/10.1007/s00894-024-05944-x
  8. J. H. Mokkath, J. Henzie, 'Chemical ordering patterns and magnetism of NiAl nanoclusters', Materials Research Express 4, 015010 (2017). DOI: 10.1088/2053-1591/aa575f, https://doi.org/10.1088/2053-1591/aa575f
  9. F. Delogu, 'Demixing phenomena in NiAl nanometre-sized particles', Nanotechnology 18, 065708 (2007). DOI: 10.1088/0957-4484/18/6/065708, https://doi.org/10.1088/0957-4484/18/6/065708
  10. M. Settem, 'On the structural analysis of ordered B2 AlNi nanoparticles obtained using freezing simulations', Intermetallics 106, 115-123 (2019). DOI: 10.1016/j.intermet.2018.12.018, https://doi.org/10.1016/j.intermet.2018.12.018
  11. S. Alnemrat, J. P. Hooper, 'Oxidation of ligand-protected aluminum clusters: An ab initio molecular dynamics study', The Journal of Chemical Physics 140, 104311 (2014). DOI: 10.1063/1.4867467, https://doi.org/10.1063/1.4867467
  12. C. Li, D. Raciti, T. Pu, L. Cao, C. He, C. Wang, T. Mueller, 'Improved Prediction of Nanoalloy Structures by the Explicit Inclusion of Adsorbates in Cluster Expansions', The Journal of Physical Chemistry C 122, 18040-18047 (2018). DOI: 10.1021/acs.jpcc.8b03868, https://doi.org/10.1021/acs.jpcc.8b03868
  13. M. Usman, M. U. Farooq, F. Chen, 'Deepcluster: An advanced design for the prediction of the bimetallic Ni-Al nanoclusters via deep reinforcement learning', Intermetallics 194, 109291 (2026). DOI: 10.1016/j.intermet.2026.109291, https://doi.org/10.1016/j.intermet.2026.109291