(476g) Identification of Stable Bimetallic Nanoclusters Via a Mathematical Optimization Framework

Metallic nanoparticles (NPs) have distinct properties from bulk metals and show potential for use in a wide range of applications, including catalysis, electronics, and optics [1]. While at the nanoscale there are many possible structures that can be imagined, it is not obvious which is the most stable for a given particle size and composition. This work proposes a mathematical optimization-based design framework to address this combinatorial problem.

The cohesive energy of an NP accounts for the strength of its inter-metallic bonds [2]. Therefore, we can deduce the structure of the most stable nanocluster of a given mass by maximizing its cohesive energy. Our work is based on a bond-centric cohesive energy model for bimetallic nanoclusters proposed by Zihao et al [3]. This model relates a particle’s cohesive energy with the constituent atoms’ coordination numbers. In this work, we formalize the search for the most cohesive structure via a mathematical optimization model by introducing binary variables to indicate the presence or absence of certain type of atoms in design space. This leads to a Mixed-Integer Linear Programming (MILP) optimization model that can be solved to identify optimal structures using well-established numerical optimization techniques.

In order to improve the tractability of this design framework and be in position to address practically relevant sized NPs in reasonable computational time, we also propose a two-step “structure-then-order” heuristic approach. First, we identify stable nanoparticle geometries using a monometallic nanocluster geometry optimization model that we have developed previously [4], and which results in a pool of nanocluster geometries that may be good candidates for bimetallic NPs. The pool can be further extended by local search based on a metaheuristic search framework such as simulated annealing and genetic algorithms. Then, we use a second MILP model for searching over the space of alloy compositions on a particular geometry. This second model possesses significantly reduced combinatorial complexity compared to the fully-flexible bimetallic structure optimization counterpart, and can thus be efficiently solved to guaranteed optimality for any particle size of interest. By iteratively solving many such alloy composition optimization models, we can obtain good solutions to the fully-flexible counterpart.

We have used this optimization framework to study a wide range of transition bimetallic systems. In many cases, our results reveal segregation patterns such as well-known core-shell structures, which are supported in the literature by both experimental and computational results. However, in some cases, unintuitive trends can be observed for certain bimetallic systems. These results will help us efficiently explore the bimetallic design space and gain a deeper understanding of relationships between structure, order and stability.

[1] Ferrando, Riccardo, Julius Jellinek, and Roy L. Johnston. "Nanoalloys: from theory to applications of alloy clusters and nanoparticles." Chemical reviews 108, no. 3 (2008): 845-910.

[2] Kittel, Charles and Paul McEuen. Introduction to solid state physics. Vol. 8. New York: Wiley, 1976.

[3] Zihao Yan, Michael G. Taylor, Ashley Mascareno, and Giannis Mpourmpakis. "Size-, Shape-, and Composition-Dependent Model for Metal Nanoparticle Stability Prediction." Nano letters 18, no. 4 (2018): 2696-2704.

[4] Natalie M. Isenberg, Michael G. Taylor, Zihao Yan, Christopher L. Hanselman, Giannis Mpourmpakis, Chrysanthos. E. Gounaris, “Identification of Optimally Stable Nanocluster Geometries via Mathematical Optimization and Density-Functional Theory”, Under review