Dynamical study of metallic clusters using the statistical method of time series clustering

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Abstract

We perform common neighbor analysis on the long-time series data generated by isothermal Brownian-type molecular dynamics simulations to study the thermal and dynamical properties of metallic clusters. In our common neighbor analysis, we introduce the common neighbor label (CNL) which is a group of atoms of a smaller size (than the cluster) designated by four numeric digits. The CNL thus describes topologically smaller size atomic configurations and is associated an abundance value which is the number of “degenerate” four digits all of which characterize the same CNL. When the cluster is in its lowest energy state, it has a fixed number of CNLs and hence abundances. At nonzero temperatures, the cluster undergoes different kinds of atomic activities such as vibrations, migrational relocation, permutational and topological isomer transitions, etc. depending on its lowest energy structure. As a result, the abundances of CNLs at zero temperature will change and new CNLs with their respective new abundances are created. To understand the temperature dependence of the CNL dynamics, and hence shed light on the cluster dynamics itself, we employ a novel method of statistical time series analysis. In this method, we perform statistical clustering at two time scales. First, we examine, at given temperature, the signs of abundance changes at a short-time scale, and assign CNLs to two short-time clusters. Quasi-periodic features can be seen in the time evolution of these short-time clusters, based on which we choose a long-time scale to compute the long-time correlations between CNL pairs. We then exploit the separation of correlation levels seen in these long-time correlations to extract strongly-correlated collections of CNLs, which we will identify as effective variables for the long-time cluster dynamics. It is found that certain effective variables show subtleties in their temperature dependences and these thermal traits bear a delicate relation to prepeaks and main peaks seen in clusters Ag14, Cu14 and Cu13Au1. We therefore infer from the temperature changes of effective variables and locate the temperatures at which these prepeaks and principal peaks appear, and they are evaluated by comparing with those deduced from the specific heat data.

Introduction

At very low temperatures, atoms in a metallic cluster execute solid-like vibration. The description of this oscillatory motion in the presence of an external probe such as temperature is quite different from that in the bulk. Whereas in a bulk system the thermal response of atoms are treated individually the same owing to the translational symmetry, atoms in a cluster under the same condition are, however, realized by their whereabouts locations since their geometrical sites have much bearing on how they respond and hence the cluster properties. Consider, for example, a 14-atom pure cluster and a 38-atom bimetallic cluster Ag32Cu6 at their respective lowest energy states. The former [1], [2] can be differentiated generally into three broad kinds of atoms, namely, floating (adatom), surface and central atoms and the latter [3], on the other hand, can be categorized coarse grainedly into surface and central atoms. Such a classification of atoms by the positions they reside is instructive at higher temperatures and provides a practical means in quantitative analysis of the microscopic dynamics of clusters. In two recent communications [2], [4], we have applied this strategy of partitioning atoms in a cluster to investigate the velocity autocorrelation function; we deduced from this dynamic quantity and its Fourier-transformed function, the power spectrum, the average temperature at which a cluster melts. We demonstrated that these dynamic variables can be employed to infer the melting temperature Tm of a cluster and the predicted value is reasonably close to that estimated from the principal peak position of the specific heat CV (widely accepted to be the melting temperature Tm). In the course of interpreting the dynamic results, an attempt was made to explain the anomalous increase of the relative root–mean–square bond length fluctuation constant [2], [4] δ=1n(n1)i=1njinrij2(t)trij(t)t2rij(t)t whose aberrant thermal response occurs not at Tm but at a much lower temperature. In these latter works [2], [4], attention was drawn to tracking down simultaneously the instantaneous relative bond length rij(t) in δ for ij-atoms. A careful and thorough analysis of rij(t) at each temperature reveal indeed the complex dynamics of individual atoms. It is now understood, for instance, that the anomalous increase of δ at lower temperatures for Cu14 or Ag14 [2], [4] is due mainly to the migrational relocations of the floating atom. As the temperature of the cluster climbs up further, the rij(t) indicates an increase in the frequency of permutational isomer transitions among the floating and surface atoms, and such exchange activities continue until the central atom participates finding its way out via permutating with a surface or floating atom. The cluster should by then approach Tm. Although the analysis using rij is dynamically fruitful, it is nevertheless a complicated procedure since a synchronous follow-up of different rijs (atomic pairs of floating-surface, surface-surface, central-surface, etc.) must be effected for a conclusive description to be reached. Moreover, the rij variable does not give much insight into the factors that govern the atomic dynamics at different temperatures.

In this work, we turn to a more practical means of studying the dynamical behavior of clusters. Rather than examining rij simultaneously for different pairs of ij-atoms, we scrutinize instead a group of atoms which comprises a root pair and its common neighbors. This kind of the common neighbor analysis (CNA) finds its application in a wide range of problems in bulk systems. We illustrate in this paper that the CNA can equally well be applied to study the dynamical motion of atoms in a cluster. Differing from the usual simulation method where the time development of the position coordinates and velocities of atoms in a cluster are used to explore the cluster dynamics from calculating such quantities as the root–mean–square displacement, velocity autocorrelation function, etc., we introduce here a new scheme — based on CNA and time series clustering — to understand the time evolution.

In a typical molecular dynamics (MD) simulation running over tens of millions of time steps, the sheer volume of data contained in the large number of displacement time series makes analyzing and understanding the underlying physics a challenging task. Simple visualization of these simulation data in the form of a movie brings forth too much subjective bias. An objective way to reduce data complexity is to effect time series clustering, a class of methods that has found applications in diverse areas of business, science and technology. In time series clustering, the objective is to group together time series data sets with similar dynamical features so that a much smaller collection of dynamically distinct clusters is obtained. Broadly, the time series clustering methods can be classified into model-based methods [5], [6], [7], [8], [9], [10], [11] and correlations-based methods [12], [13], [14], [15], [16], [17], [18]. Both these methods have been reviewed by Liao [19] and we refer the interested readers to this article for more details. Since the physics behind nanocluster melting is not yet fully understood, we cannot apply existing model-based methods to our problem, as this would introduce modeling biases. Neither can we simply adapt existing correlations-based methods, as our goal is to develop a mechanistic picture of the melting process. Instead, we choose to develop a new correlations-based time series clustering method that is a hybridization of the methods developed by Rummel et al. [14], [18] and Tumminello et al. [13], [15]. In our method, we draw attention to novel ways of analyzing the correlation matrix and demonstrate, by illustration of several small clusters using CNA, how the effective variables (to be defined below) are obtained and applied to describe the dynamics of our physical system. We should emphasize that the time series clustering method was developed for larger complex systems in mind, and has in fact been successfully applied to finance [20], [21], neuroscience [22], [23], [24], [25], [26], and meteorology [27], [28], [29]. We have, for instance, very recently applied the method to financial markets (500–3000 degrees of freedom) [30] and global positioning system networks (100 degrees of freedom) [31]. In these two systems, the dynamical time scales are not well separated. Employing the method of partial hierarchical clustering [30], we, however, managed to extract the effective variables as well as their dynamics. In contrast, the dynamical time scales are well separated in the metallic clusters chosen here, up to very high temperatures. This important qualitative difference between the financial/geophysical systems and metallic clusters confers an added advantage to the time series clustering method, because the separation of dynamical time scales translates into a separation of correlation levels in each of the long-time windows. There is thus no cause for alarm in the event that the dynamical time scales are not well separated. Armed with this time series clustering scheme, we proceed to interpret the microscopic time series data which we will obtain from an isothermal Brownian-type MD simulation on several metallic clusters.

The present paper is organized as follows. In Section 2, we give a brief summary of the CNA following the definition of Honeycutt and Andersen [32] and introduce the method of time series clustering on multiple time scales in detail. We devote a significant portion of the method to explain its advantage. In the discussions below, we refer extensively to various statistical observations made from the time series of the common neighbor labels which are sole elements in the CNA. Our intention is to explain clearly to readers how quasi-regular synchronies in the short-time dynamics can be converted into reliable long-time correlations, based on which we then identify the effective variables. We shall not describe the isothermal Brownian-type MD simulations that were used to generate the microscopic time series data. We refer the interested readers, however, to our previous works for technical details [1], [2], [4]. Throughout this section, for concreteness, we use Cu13 as an illustrative example to inaugurate the statistical time series clustering methodology. In Section 3, we give a quantitative discussion of the results derived from the time series analysis first for Ag14, and then for Cu14 and Cu13Au1; we make a comparison between Cu14 and Cu13Au1, delving into the possible interpretations of the dynamics. With regard to the melting scenario, we infer Tm from the temperature-dependences of the time series data and compare its value with corresponding one deduced from CV. Since the CV for Ag14, Cu14 and Cu13Au1 have been reported previously [1], [2], we will therefore simply cite the results or just summarize them without further description. Finally, we give a conclusion in Section 4.

Section snippets

Common neighbor analysis

In our CNA, we examine the neighborhoods around all pairs of atoms in the cluster, and note these at each time step in the form of four-digit common neighbor labels (CNL), {c1,c2,c3,c4}. In this analysis, we say that there is a bonded root-pair atoms if the distance between them is rb1.2r0,r0 being the nearest neighbor separation. Therefore, for a given root-pair atoms, we set c1=1 if they are bonded, or c1=2 if they are not. Next, we set c2 equal to the number of atoms bonded simultaneously

Results and discussions

We are now in a position to apply the time series analysis to study the dynamical properties of metallic clusters. We first present results for Ag14, and then for Cu14 and Cu13Au1, in a manner that allows comparison between pure and bimetallic clusters.

Conclusions

We have generated long-time series data sets for metallic clusters Cu13, Ag14, Cu14, and Cu13Au1 using an isothermal Brownian-type MD simulation. From the time evolution of atomic configurations recorded at different temperatures, the CNLs and their associated abundances were calculated. We performed statistical clustering on the change of CNL-abundance time series at a short-time scale, and then another statistical clustering based on the short-time correlation between CNL-abundance changes at

Acknowledgement

This work is supported by the National Science Council, Taiwan (NSC96-2112-M-008-018-MY3).

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