Structural optimization of Pt–Pd alloy nanoparticles using an improved discrete particle swarm optimization algorithm

Abstract

Pt–Pd alloy nanoparticles, as potential catalyst candidates for new-energy resources such as fuel cells and lithium ion batteries owing to their excellent reactivity and selectivity, have aroused growing attention in the past years. Since structure determines physical and chemical properties of nanoparticles, the development of a reliable method for searching the stable structures of Pt–Pd alloy nanoparticles has become of increasing importance to exploring the origination of their properties. In this article, we have employed the particle swarm optimization algorithm to investigate the stable structures of alloy nanoparticles with fixed shape and atomic proportion. An improved discrete particle swarm optimization algorithm has been proposed and the corresponding scheme has been presented. Subsequently, the swap operator and swap sequence have been applied to reduce the probability of premature convergence to the local optima. Furthermore, the parameters of the exchange probability and the ‘particle’ size have also been considered in this article. Finally, tetrahexahedral Pt–Pd alloy nanoparticles has been used to test the effectiveness of the proposed method. The calculated results verify that the improved particle swarm optimization algorithm has superior convergence and stability compared with the traditional one.

Introduction

Clusters or nanoparticles (NPs), which are aggregates of atoms or molecules ranging from a few to millions, constitute a new type of material different from the individual atoms or molecules or bulk counterpart  [1], [2]. Metal NPs have aroused intense interests due to their potential applications in biomedicine, electronics, optical and catalysis fields  [3]. Among these NPs, Pt NP is considered as one of the best catalysts because of its excellent reactivity and stability  [4]. However, the extremely high price of Pt makes it expensive to use Pt metal. Generally, there are two types of NPs to reduce the demand for Pt: One is Pt NPs with open surface structures, and the other is Pt-based alloy NPs  [5]. A representative of the former is tetrahexahedral (THH) Pt NPs with high-index facets which have been synthesized by an electrochemical route and exhibited greatly enhanced catalytic activity compared with the existing commercial Pt/C catalysts  [6], [7]. For the latter, Pt–Pd alloy NPs have been extensively investigated due to their acceptable price and enhanced catalytic activity and selectivity  [8]. To further improve the chemical activity of Pt–Pd alloy NPs, a feasible strategy is to prepare high-index-faceted Pt–Pd alloy NPs. A typical example is that the synthesized THH Pt–Pd alloy NPs, which mainly enclosed by {10  3  0} high-index facets, exhibit a catalytic activity at least three times higher than the THH Pd NPs, and six times higher than commercial Pd black catalysts for the electrooxidation of formic acid due to the synergy effects of high-index facets and electronic structures of the alloy  [9].

Since the activity and selectivity of NP catalysts are closely associated with their structures, investigation of the stable structures is crucial for understanding their catalytic performances. The nature of searching the stable structures of NPs is a typical optimization problem. Usually, the potential energy is used as a standard in finding the most stable structure of NP corresponding to the lowest-energy structure at the ground state. However, the number of local minimum rises exponentially with the increasing particle sizes in the global optimization scheme, and thus the searching procedures are computationally expensive. For example, the number of local minimum is 1505 for the Lennard-Jones clusters with 13 atoms, but the number of local minimum is at least $10^{12}$ for a larger cluster with 55 atoms  [10]. Essentially, the global optimization for searching the stable structures of NPs is a nonpolynomial-complete problem  [11]. To date, some methods have been proposed to solve this problem. They can be classified into four categories: Genetic Algorithm (GA)  [12], [13], [14], [15], [16], [17], [18], Particle Swarm Optimization (PSO) algorithm  [19], [20], [21], [22], Monte Carlo method  [4], [23], [24], [25], [26], [27] and the remaining methods such as conjugate gradients method  [28], [29], [30], [31]. However, developing a new method becomes an urgent need for the global optimization of NPs due to the following two aspects. On one hand, most of available studies focus on stable structures of small clusters while large alloy NPs have rarely been studied. On the other hand, the methods used in the structural optimization of these clusters or NPs concentrate mainly on the genetic algorithm and Monte Carlo methods. The genetic algorithm is costly and very time-consuming for the large NPs, and the convergence of the Monte Carlo methods is too slow  [2], [10]. The pioneering works of Wang et al.  [19], [20], [21] and Lv et al.  [22] have shown that the PSO is a highly efficient method for the prediction of global stable structure. However, their works mainly addressed the studies of small clusters, and it may be too time-consuming to realize the structure optimization if their computational approaches are used directly for large NPs. Therefore, development of a feasible and effective method is of great importance for the theoretical study of large NPs.

As we know, the particle swarm optimization (PSO) algorithms have been extensively used to solve the combinatorial optimization problems such as traveling salesman problem (TSP)  [32], [33], [34], [35], and the corresponding results have definitely verified their effectiveness. Actually, searching the stable structures of NPs is also a typical combinatorial optimization problem. Inspired by the applications of the PSO algorithms in the TSP, in this article we have proposed an improved PSO algorithm to carry out the structural optimization of Pt–Pd alloy NPs. The THH Pt–Pd alloy NPs, bound with high-index facets, have been chosen to be investigated because of their significance in catalysis fields. The design scheme will be systematically introduced in the following section. Furthermore, the parameters setting rules will be verified in the fourth section, and the analyses of convergence and stability will be thoroughly investigated by comparison with the traditional PSO algorithm. To the best of our knowledge, this is the first report on the PSO algorithm being used in large alloy NPs. This article is organized as follows. In Section  2, we firstly describe how to obtain the stable structures of Pt–Pd alloy NPs, and then their optimization models are presented. Section  3 introduces the coding scheme, swap operator and swap sequence to adapt to the discretization of structural optimization. The calculated results and the related discussion are presented in Section  4, and the main conclusions are summarized in Section  5. To avoid the misapprehension or confusion, we introduce the ‘particle’ to denote the particle used in the PSO method and the particle without single quotation marks to denote the nanoparticle or cluster.