Abstract
A simple macroscopic model is used to determine an optimal annealing schedule for self-assembly of binary monolayers of spherical particles. The model assumes that a single rate-controlling mechanism is responsible for the formation of spatially ordered structures and that its rate follows an Arrhenius form. The optimal schedule is derived in an analytical form using classical optimization methods. Molecular dynamics simulations of the self-assembly demonstrate that the proposed schedule outperforms other schedules commonly used for simulated annealing.
Highlights
The term self-assembly refers to a non-intrusive process that transforms a system of components from a less to a more ordered state
This work is motivated by self-assembly in model systems proposed by Grzybowski et al [5]. Those systems are horizontal monolayers of oppositely charged millimeter-size spherical particles placed inside a large square container
The electrostatic interactions alone are not sufficient for arranging randomly placed particles into ordered lattices, as the particles are trapped in meta-stable states
Summary
Where τ is the relaxation time at a given temperature. In what follows, we extend the model one step further by assuming that it holds for non-isothermal processes. For these data we obtain the Arrhenius fit with the energy barrier α = 0.043 and frequency ν = 0.012 Note that in this range of temperatures, dimensionless relaxation times estimated from (5) are in the thousands, which is well below the limitation imposed by the computational speed. To rationalize the parameters of the Arrhenius fit in terms of a microscopic mechanism, we calculated a collection of energy barriers for diffusion on the surface and in the bulk. These calculations were done with the climbing-image nudged elastic band method to determine the minimum energy pathway and saddle point between minima [10]. We suggest that the likely rate-controlling mechanism is surface diffusion
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