Modeling the early stages of self-assembly in nanophase materials. II. Role of symmetry and dimensionality

Abstract

We study the early stages of self-assembly of elementary building blocks of nanophase materials, considering explicitly their structure and the symmetry and the dimensionality of the reaction space. Previous work [Kozak et al., J. Chem. Phys. 134, 154701 (2007)] focused on characterizing self-assembly on small square-planar templates. Here we consider larger lattices of square-planar symmetry having N = 255 sites, and both hexagonal and triangular lattices of N = 256 sites. Furthermore, to assess the consequences of a depletion zone above a basal layer (λ = 1), we study self-assembly on an augmented diffusion space defined by λ = 2 and λ = 5 stacked layers having the same characteristics as the basal plane. The effective decrease in the efficiency of self-assembly of individual nanophase units when the diffusion space is expanded, by increasing the template size and/or by enlarging the depletion zone, is then quantified. The results obtained reinforce our earlier conclusion that the most significant factor influencing the kinetics of formation of a final self-assembled unit is the number of reaction pathways from one or more precursor states. We draw attention to the relevance of these results to zeolite synthesis and reactions within pillared clays.

Stochastic Signatures of Phase Space Decomposition

“…In this contribution we have studied ergodic flows of a random walker undergoing unbiased displacements in a positional phase space represented by a host lattice, and characterized quantitatively the consequences of different metric decompositions of a given, finite parent lattice. As a model for studying the early stages of self-assembly of nanoparticles, the results here complement those reported in [3,4] where the phase (reaction) space was defined by a host, periodic lattice. Specifically, by breaking the translational symmetry of the reaction space, we find that the reaction efficiency in a finite system is strongly dependent not only on whether the system is compartmentalized, but also on whether the overall reaction space of the microreactor is further decoupled into separable reactors.…”

“…To provide a physical motivation for the present study, understanding the factors influencing self-assembly in nanophase materials is a major experimental and theoretical challenge [1,2]. Modeling even the early stages of selfassembly already introduces fundamental questions, for example, whether the process is reducible to a sequence of elementary steps occurring under equilibrium conditions [3,4]. The role of different system geometries in influencing the efficiency of reactions in a finite, compartmentalized system [5] is an ongoing problem, one aspect of which is explored quantitatively here.…”

First-passage properties of mortal random walks: Ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs

“…We close by emphasizing that the mechanism of reduction of dimensionality on which we have been focusing here relies on a mortality constraint; it is therefore fundamentally different from other mechanisms discussed in the literature, such as the one originally hypothesized by Adam and Delbruck [57] and used as a source of inspiration in subsequent works (see, e.g., Refs. [58][59][60][61]). In the situation considered by Adam and Delbruck, the unconditional reaction rate of a diffusion-limited target search process is greatly enhanced, whereas in our case an increase in the conditional reaction rate is observed.…”