Mathematical modeling of interaction energies between nanoscale objects: A review of nanotechnology applications

Abstract

In many nanotechnology areas, there is often a lack of well-formed conceptual ideas and sophisticated mathematical modeling in the analysis of fundamental issues involved in atomic and molecular interactions of nanostructures. Mathematical modeling can generate important insights into complex processes and reveal optimal parameters or situations that might be difficult or even impossible to discern through either extensive computation or experimentation. We review the use of applied mathematical modeling in order to determine the atomic and molecular interaction energies between nanoscale objects. In particular, we examine the use of the 6-12 Lennard-Jones potential and the continuous approximation, which assumes that discrete atomic interactions can be replaced by average surface or volume atomic densities distributed on or throughout a volume. The considerable benefit of using the Lennard-Jones potential and the continuous approximation is that the interaction energies can often be evaluated analytically, which means that extensive numerical landscapes can be determined virtually instantaneously. Formulae are presented for idealized molecular building blocks, and then, various applications of the formulae are considered, including gigahertz oscillators, hydrogen storage in metal-organic frameworks, water purification, and targeted drug delivery. The modeling approach reviewed here can be applied to a variety of interacting atomic structures and leads to analytical formulae suitable for numerical evaluation.

Computational Analysis of Strain-Induced Effects on the Dynamic Properties of C60 in Fullerite

“…We can see the effects of the angular velocity direction because we used the hybrid discrete-continuous mathematical model. The calculation results show that the models of smoothed molecules [40,41,47,48] applied to all fullerenes, including the central fullerene, will give a qualitatively incorrect result when the axis of rotation does not coincide with the direction of deformation (see Figure 7). Because the rotation of fullerenes is not present in these models (ω = 0).…”

“…We can see the effects of the angular velocity direction because we used the hybrid discrete-continuous mathematical model. The calculation results show that the models of smoothed molecules [40,41,47,48] applied The effects of rolling and force reduction are shown schematically in Figure 7. Figure 7a describes the rectilinear motion that occurs when the angular velocity ω and the force F lie on the same straight line passing through the center of mass of the C 60 molecule or when ω = 0.…”

“…At the initial moment of time, the distance between the centers of the two nearest molecules is around 1.002 nm [38,39]. The interaction of the central molecule with smoothed C 60 is described using interatomic interaction potentials based on the Lennard-Jones potential [40,41]. Thus, in this article, the interaction of any carbon atom of the central fullerene with any smoothed fullerene depends only on their mutual position but does not depend on the position of any other particles, as in many-particle potentials [42][43][44].…”

“…The interaction potential between a carbon atom of the central C 60 and a smoothed spherical fullerene has the following form [33,34,40,41]…”

Swarm Intelligence Steers a Global Minima Search of Clusters Bound on Carbon Nanostructures

“…The continuum approximation (CA), wherein highly symmetric chemical structures are approximated to geometries of uniform surface atom density, is a smart approach that can save a lot of computational time for optimization problems involving carbon nanostructures. 20 For example, by approximating carbon nanostructures like carbon rings, fullerenes, CNTs, and graphene by their corresponding geometries of uniform density (see Figure 1), CA has been successful in explaining nano-oscillatory mechanisms in carbon-based nano electro-mechanical systems (NEMS) devices, targeted drug delivery applications of CNTs, and the selective filtration of ions using carbon nanocones within a reasonable computational time. 20,21 Inspired by the efficiency of the PSO algorithm in tracking down the global minima on complex PESs and the advantages offered by the CA, we hypothesized that an amalgamation of PSO with CA should be a very useful strategy for probing the binding of atomic and molecular clusters to carbon nanostructures.…”