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    Continuous transition from palladium clusters to nanocrystals simulated by machine learning potential

    Ziyi Liang1, Luneng Zhao1, Hongsheng Liu1,*, Junfeng Gao1,2,3,†, and Feng Ding3,‡

    • 1Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, School of Physics, Dalian 116024, China
    • 2Interdisciplinary Institute of Smart Molecular Engineering, Dalian University of Technology, Dalian 116024, China
    • 3Suzhou Laboratory, Suzhou 215123, People's Republic of China

    • *Contact author: liuhongsheng@dlut.edu.cn
    • †Contact author: gaojf@dlut.edu.cn
    • ‡Contact author: dingf@szlab.ac.cn

    Phys. Rev. Materials 10, 046001 – Published 8 April, 2026

    DOI: https://doi.org/10.1103/y1gy-9sfq

    Abstract

    The physical and chemical properties of clusters undergo significant changes with changes in size. Therefore, the evolution of cluster structure with size has always been an important issue in nanoscience. However, due to the complexity of cluster potential energy surfaces and limitations in computational simulation methods, the evolution of cluster structures has long been a challenge. Here, we developed a high-accuracy machine learning interatomic potential (MLIP) for palladium systems, including clusters, surfaces, and crystal. This MLIP enables a cross-scale (Pdn, n=12−21856) study of palladium systems from clusters to nanocrystal. For Pdn (n=12−60) clusters, simulated annealing revealed 32 new lower-energy structures. Small Pdn clusters (n < 85) show core-shell structures and repeated transitions from truncated to complete octahedra near FCC magic sizes (e.g., Pd38 to Pd44, Pd79 to Pd84). When the size exceeds 200, the Wulff construction becomes increasingly stable. There is competition among multiple structural modes (icosahedron, octahedron, and Wulff construction) within the range of 200–460. Beyond 460 atoms, Wulff constructions dominate and the system transfers from cluster to nanocrystal. Our work reveals continuous transition from Pd clusters to nanocrystal and the critical size, and paves the way for studies on other clusters.

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    This article appears in the following collection:

    Machine Learning for Materials Discovery and Understanding

    The Editors of Physical Review Materials are pleased to present the Collection on Machine Learning for Materials Discovery and Understanding, highlighting cutting-edge advances in machine learning method development and applications for materials discovery and fundamental understanding of the structure-property-function relationship. The Collection is being guest-edited by Deyu Lu of Brookhaven National Laboratory (USA) and Jinlan Wang of Southeast University (China). Every article published in this collection underwent a rigorous peer review process, adhering to the same high standards applied to all papers. The Physical Review Materials editorial team managed the peer review and made all editorial decisions.

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