Modeling Aggregation Processes of Lennard-Jones particles Via Stochastic Networks

Abstract

We model an isothermal aggregation process of particles/atoms interacting according to the Lennard-Jones pair potential by mapping the energy landscapes of each cluster size N onto stochastic networks, computing transition probabilities from the network for an N -particle cluster to the one for N + 1, and connecting these networks into a single joint network. The attachment rate is a control parameter. The resulting network representing the aggregation of up to 14 particles contains 6427 vertices. It is not only time-irreversible but also reducible. To analyze its transient dynamics, we introduce the sequence of the expected initial and preattachment distributions and compute them for a wide range of attachment rates and three values of temperature. As a result, we find the configurations most likely to be observed in the process of aggregation for each cluster size. We examine the attachment process and conduct a structural analysis of the sets of local energy minima for every cluster size. We show that both processes taking place in the network, attachment and relaxation, lead to the dominance of icosahedral packing in small (up to 14 atom) clusters. 1 arXiv:1612.09599v2 [cond-mat.stat-mech] 11 Apr 2017 102, 103, and 104, the energy-minimizing configurations are non-icosahedral [37]. Some of them are highly symmetric. For example, the global minimum for N = 38, a truncated octahedron with FCC atomic packing, has the point group O h of order 48, i.e., there are 48 orthogonal transformations mapping the cluster onto itself. The global minimum for N = 75, a Marks decahedron, has point group D 5h of order 20. Remarkably, the mass spectra graphs in [15,20] do not have prominent peaks at N = 38 and N = 75. On the other hand, the mass spectra in [15,16,17,20,21] consistently exhibit peaks corresponding to the clusters of the so-called magic numbers of atoms N admitting complete icosahedra. These numbers are: N = 13, 55, 147, 309, 561, etc. The point group order of an icosahedron is 120. Evidently, atoms tend to self-assemble into highly symmetric complete icosahedra in experimental settings, while they seem to miss highly symmetric low-energy configurations based on other kinds of packing, at least for small numbers of atoms. Choosing a model and an approachIntrigued by these facts, we undertook an attempt to understand the self-assembly of free Lennard-Jones particles (atoms) into clusters on the quantitative level by means of combined analytical and computational methods. Most previous theoretical studies of Lennard-Jones clusters dealt with those of fixed numbers of atoms, i.e., atoms were allowed neither to fly away nor to join the cluster. These works can be divided into two groups, full phase-space-based (e.g. [29,34]) and network-based. The latter approach was pioneered by Wales and collaborators [31,37,11,40,41]. Their powerful computational tools for mapping energy landscapes onto networks are based on the basin-hopping method [37] and discrete path sampling [38]. Numerous networks representing energy landsca...

From sticky-hard-sphere to Lennard-Jones-type clusters

“…The N ≥ 12 SHS data gives α SHS ≈ 2.21. Figure 3 also shows the (6,12)-LJ results obtained using basin-hopping; these yield α LJ ≈ 1.10, which is close to the α = 0.8 value estimated by Wallace [59] or to the recently given value of 1.04 by Forman and Cameron [45]. Note that the rapid increase of |M SHS |/|M LJ | with N is explained by the much larger values of α for the SHS compared to the LJ clusters.…”

“…While the maximum contact number increases (sub)linearly with N, the number of non-isomorphic cluster structures |M(N)| and transition states is assumed to increase exponentially [11,44,45] (here we denote M(N) as the set of all non-isomorphic cluster structures of size N, and |M(N)| as the number of structures in M(N)). Stillinger showed that under certain conditions lim N→∞ |M(N)| ∝ exp(αN) [44].…”

Simulated annealing with adaptive cooling rates

“…This is not surprising, as it is due to the rapid increase in the number of local minima with increasing dimensionality of the energy landscape. (Table III lists the number of local minima 24,25 for the smaller clusters. This value rises steeply enough that the energy landscape has not been fully explored for even moderately large clusters.)…”

Complex Networks and Interacting Particle Systems

“…On the other hand, within fixed scales (in particular, at the molecular scale), network analyses of structure tend to either take data-analytic approaches [18,19] or impose external "control parameters" [8] built on the notion of adaptive links [20]. In this context, a change in temperature which allows certain molecular structures to emerge would be modeled by setting an external parameter which regulates how the links are created and destroyed.…”