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.
Highlights
For the past two decades, nanotechnology has been a major focus in science and technology
In the targeted drug delivery section, we present a review of applied mathematical modeling for targeted drug delivery
This review has focused on the use of applied mathematical modeling to determine the mechanical energy behavior of nanostructures
Summary
For the past two decades, nanotechnology has been a major focus in science and technology. Analytical expressions are presented for the interaction energies of the basic molecular building blocks, namely, points, lines, planes, rings, spheres, and cylinders, all deduced utilizing the 6-12 Lennard-Jones potential function together with the continuous approximation.
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