Computational Modeling of the Structural Stability of Metal Nanoclusters Based on the Molecular Dynamics Method
Abstract
The paper examines the results of molecular dynamics modeling of metallic clusters of copper (Cu), silver (Ag) and cobalt (Co). The focus was on how the geometric properties and energetic stability of nanoclusters vary with size. Numerical calculations were performed using the Lammps software suite. This software package is widely used for atomistic modeling tasks and has proven itself in the study of systems with a large number of particles. The interatomic interactions were described using EAM and MEAM potentials, and the simulations were performed in a high-performance computing environment with MPI/OpenMP support. The work was conducted in two sequential stages. In the first stage, the clusters were relaxed at a temperature of 0K to obtain configurations corresponding to the minimum energy state. The systems were then gradually heated to 300K, which made it possible to trace changes in their stability and assess possible structural rearrangements during thermal evolution. The computational results showed that as the number of atoms increases, the overall geometry of the clusters approaches a spherical shape, and the system's energetic stability is enhanced due to the increase in the volume of the inner atoms. We present a systematic MD simulation study of structural evolution and energetic stability in Cu, Ag, and Co nanoclusters comprising 13 to 55 atoms. By identifying magic-number clusters and comparing compositional behavior from 0K to 300K, we reveal distinct size-dependent stability trends across the three metals. These findings offer quantitative insight into nanocluster formation mechanisms relevant to catalysis and nanomaterial engineering.
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M.M. Komilov, R. Aliev, A.A. Mirzaalimov, S.R. Aliev, M.K. Abduvohidov, N.A. Mirzaalimov, J. Ziyoitdinov, et al., “Enhancing and Optimizing Optical Properties of Bifacial Solar Cells by Incorporating Metal Nanoparticles,” East European Journal of Physics, (4), 291–297 (2025). https://doi.org/10.26565/2312-4334-2025-4-27
G.M. Poletaev, Yu.A. Gafner, S.L. Gafner, Yu.V. Bebikhov, and A.A. Semenov, “Molecular Dynamics Study of the Devitrification of Amorphous Copper Nanoparticles in Vacuum and in a Silver Shell,” Metals, 13(10), 1664 (2023). https://doi.org/10.3390/met13101664
L. Wang, Y. Zhang, X. Bian, and Y. Chen, “Melting of Cu nanoclusters by molecular dynamics simulation,” Physics Letters A, 310(2–3), 197–202 (2003). https://doi.org/10.1016/S0375-9601(03)00263-9
T. Kuruganti, P.K. Joshi, and M. Goswami, “Simulation of two nanoparticle melting to understand the conductivity drop of 3D-printed silver nanowires,” Materials and Design, 236, 112502 (2023). https://doi.org/10.1016/j.matdes.2023.112502
F. Sánchez-Pérez, F. Sánchez-Pérez, O. Borrell-Grueiro, A. Casasnovas-Melián, D.J. Ramos-Ramos, A. Guerrero-Martínez, L. Bañares, A. Prada, et al., “Formation of hollow silver nanoparticles under irradiation with ultrashort laser pulses,” Nanophotonics. 13(7), 1149-1157 (2024). https://doi.org/10.1515/nanoph-2023-0881
H. Guo, L. Zhang, Q. Zhu, C. Wang, G. Chen, and P. Zhang, “Molecular Dynamics Simulation of the Coalescence and Melting Process of Cu and Ag Nanoparticles,’ Advances in Condensed Matter Physics, 2021, Article 9945723 (2021). https://doi.org/10.1155/2021/9945723
L. Wang, Y. Zhang, X. Bian, and Y. Chen, “Melting of Cu nanoclusters by molecular dynamics simulation,” Physics Letters A, 310(2–3), 197–202 (2003). https://doi.org/10.1016/S0375-9601(03)00263-9
C. Canan, and M. Celtek, “Size dependence on melting of copper nanoparticles via molecular dynamics simulation,” in: Proceedings of the 25th International Scientific Conference UNITECH, (Gabrovo, Bulgaria, 2025).
S.-H. Oh, J.-S. Kim, C.S. Park, and B.-J. Lee, “Second nearest-neighbor modified embedded-atom method interatomic potentials for the Mo–M (M = Al, Co, Cr, Fe, Ni, Ti) binary alloy systems,” Computational Materials Science, 194, 110473 (2021). https://doi.org/10.1016/j.commatsci.2021.110473
A. Mahata, T. Mukhopadhyay, and M.A. Zaeem, “Modified embedded-atom method interatomic potentials for Al–Cu, Al–Fe and Al–Ni binary alloys: From room temperature to melting point,” Computational Materials Science, 201, 110902 (2022). https://doi.org/10.1016/j.commatsci.2021.110902
M. Muralles, J.T. Oh, and Z. Chen, “Modified embedded atom method interatomic potentials for the Fe–Al, Fe–Cu, Fe–Nb, Fe–W, and Co–Nb binary alloys,” Computational Materials Science, 230, 112488 (2023). https://doi.org/10.1016/j.commatsci.2023.112488
M.S. Nitol, M.J. Echeverria, K. Dang, M.I. Baskes, and S.J. Fensin, “New modified embedded-atom method interatomic potential to understand deformation behavior in VNbTaTiZr refractory high entropy alloy,” Computational Materials Science, 237, 112886 (2024). https://doi.org/10.1016/j.commatsci.2024.112886
A.O. Moghaddam, R. Fereidonnejad, M. Moaddeli, D. Mikhailov, A. Vasenko, and E. Trofimov, “Second nearest-neighbor modified embedded-atom method interatomic potentials for the Zr–X (X = Co, Fe, Ni) binary alloys,” Computational Materials Science, 247, 113534 (2025). https://doi.org/10.1016/j.commatsci.2024.113534
S. Garg, N. Kaur, N. Goel, M. Molayem, V.G. Grigoryan, and M. Springborg, “Properties of Naked Silver Clusters with Up to 100 Atoms as Found with Embedded-Atom and Density-Functional Calculations,” Molecules, 28(7), 3266 (2023). https://doi.org/10.3390/molecules28073266
S. Garg, N. Kaur, N. Goel, M. Molayem, V.G. Grigoryan, and M. Springborg, “Properties of Naked Silver Clusters with Up to 100 Atoms as Found with Embedded-Atom and Density-Functional Calculations,” Molecules, 28(7), 266 (2023). https://doi.org/10.3390/molecules28073266
B.J. Alder, and T.E. Wainwright, “Phase Transition for a Hard Sphere System,” The Journal of Chemical Physics, 27(5), 1208 1209 (1957).
A. Rahman, “Correlations in the Motion of Atoms in Liquid Argon,” Physical Review, 136(2A), A405–411 (1964). https://doi.org/10.1103/PhysRev.136.A405
L. Verlet, “Computer “Experiments” on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules,” Physical Review, 159(1), 98–103 (1967). https://doi.org/10.1103/PhysRev.159.98
R.S. Averback, and T. Diaz de la Rubia, H. Hsieh, and R. Benedek, “Interactions of energetic particles and clusters with solids,” Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 59 60, 709–717 (1990). https://doi.org/10.1016/0168-583X(91)95688-A
Z. Insepov, and B. Kabdiev, “Film Deposition with Cluster Beams: An Alternate Path to Epitaxial, Crystalline Films,” Proc. of Intern. Conf. on Phys. and Chem. of Finite Systems: from Clusters to Crystals, NATO ASI Series B, 2, 429 (1992).
M. Moseler, J. Nordiek, O. Rattunde, and H. Haberland, “Energetic impact of Cu-clusters on Cu-surfaces,” Radiation Effects and Defects in Solids, 142(1-4), 27-38 (1997). https://doi.org/10.1080/10420159708211594
N. Boumerdassi, Molecular dynamics simulations of multiple Ag nanoclusters deposition on a substrate: Master’s Thesis. The University of Texas at Austin, May 2014.
M. Turdimatov, F. Mukhtarov, N. Ibrokhimov, Sh. Umarov, J. Mirzayev, and R. Rakhmatov, “Mathematical approximator based on basic spline approximation,” E3S Web of Conferences. 508, 04010 (2024). https://doi.org/10.1051/e3sconf/202450804010
O.V. Bachurina, Nonlinear Spatially Localized Vibrational Modes in Metals, Ph.D. thesis (Ufa State Petroleum Technological University, Ufa, 2019).
M.S. Daw, and M.I. Baskes, “Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals,” Physical Review B, 29(12), 6443–6453 (1984). https://doi.org/10.1103/PhysRevB.29.6443
S.M. Foiles, M.I. Baskes, and M.S. Daw, “Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys,” Physical Review B, 33(12), 7983–7991 (1986). https://doi.org/10.1103/PhysRevB.33.7983
M.I. Baskes, “Application of the embedded-atom method to covalent materials: A semiempirical potential for silicon,” Physical Review Letters, 59(23), 2666–2669 (1987). https://doi.org/10.1103/PhysRevLett.59.2666
H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, A. DiNola, and J.R. Haak, “Molecular dynamics with coupling to an external bath,” The Journal of Chemical Physics, 81(8), 3684-3690 (1984). https://doi.org/10.1063/1.448118
H.C. Andersen, “Molecular dynamics at constant pressure and/or temperature,” The Journal of Chemical Physics, 72(4), 2384 2393 (1980). https://doi.org/10.1063/1.439486
S. Nosé, “A unified formulation of the constant temperature molecular dynamics methods,” The Journal of Chemical Physics, 81(1), 511-519 (1984). https://doi.org/10.1063/1.447334
Samsonov, V. M., Vasiliev, S. A., and Samsonov, M. V. (2017). A study of the energy characteristics of silver nanoclusters using molecular dynamics. Journal of Structural Chemistry, 58(7), 1415–1420. https://doi.org/10.26902/JSC20170714
LAMMPS Documentation, “LAMMPS Molecular Dynamics Simulator – User Manual,” [Online]. Available: https://docs.lammps.org/Manual.html [Accessed: 10-Jul-2025].
Jmol Documentation, “Jmol: an open-source Java viewer for chemical structures in 3D,” [Online]. Available: https://chemapps.stolaf.edu/jmol/docs/ [Accessed: 10-Jul-2025].
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