Selected themes and examples from applied mathematics, nanoscale modelling, and mathematical physics.
My research focuses on the modelling of nanoscale structures, devices, and materials using classical applied mathematical techniques. This page highlights several themes that have shaped that work.
The emphasis is on mathematical modelling, geometric structure, continuum methods, and analytical techniques that help explain or predict behaviour in nanoscale systems.
Nanoscaled oscillating systems are of interest because of their potential use in ultra-high-frequency devices. Mathematical models of fullerene and nanotube-based oscillators can help identify stable configurations, predict frequencies, and understand the influence of geometry on device performance.
Many nanoscale structures exhibit cylindrical, spherical, or spheroidal forms. Because energetic constraints often favour symmetric conformations, geometrical reasoning can simplify otherwise complex molecular problems and lead to results that are both mathematically tractable and physically informative.
By assuming appropriate symmetry at the outset, it is often possible to derive useful predictions for idealised systems and novel structures in limiting cases.
Electrorheological fluids provide an example of how mathematical modelling can illuminate the behaviour of particulate systems under externally applied fields. In particular, modelling the Winslow effect helps describe how particle interactions change under electric loading and how that affects the macroscopic response of the fluid.
Such questions connect continuum modelling, particle interaction forces, and the design of materials with unusually large field-induced yield stresses.
Even at very small length scales, classical continuum mechanics can remain useful. One example is the modelling of intermolecular interaction forces between carbon nanotubes, where continuum approximations can replace computationally expensive pairwise calculations by surface-integral formulations.
In favourable cases, these integrals can be reduced to analytical expressions involving hypergeometric functions and related special functions.
The joining of carbon nanostructures such as nanotubes, graphene sheets, and fullerenes is central to the design of many novel nanoscale devices. These joins are fundamentally geometric, and they can be studied by minimising deviations in bond lengths and bond angles from ideal configurations.
Variational ideas are also useful here: one can seek join configurations that minimise total curvature squared subject to other structural constraints.
