Graphene is a carbon molecule with the structure of a honeycomb lattice. We show how this structure can arise in two dimensions as the minimizer of an interaction energy with two-body and three-body terms. In the engineering literature, the Brenner potential is commonly used to describe the interactions between carbon atoms. We consider a potential of Stillinger-Weber type that incorporates certain characteristics of the Brenner potential: the preferred bond angles are 180 degrees and all interactions have finite range. We show that the thermodynamic limit of the ground state energy per particle is the same as that of a honeycomb lattice. We also prove that, subject to periodic boundary conditions, the minimizers are translated versions of the honeycomb lattice.

Control of the uniformity of vertically aligned carbon nanotube structures (CNT "forests"), in terms of both geometry and nanoscale morphology (density, diameter, and alignment), is crucial for applications. Many studies report complex and sometimes unexplained spatial variations of the height of macroscopic CNT forests, as well as variations among micropillars grown from lithographically patterned catalyst arrays. We present a model for chemically coupled CNT growth, which describes the origins of synergetic growth effects among CNT micropillars in proximity. Via this model, we propose that growth of CNTs is locally enhanced by active species that are catalytically produced at the substrate-bound nanoparticles. The local concentration of these active species modulates the growth rate of CNTs, in a spatially dependent manner driven by diffusion and local generation/consumption at the catalyst sites. Through experiments and numerical simulations, we study how the uniformity of CNT micropillars can be influenced by their size and spacing within arrays and predict the widely observed abrupt transition between tangled and vertical CNT growth by assigning a threshold concentration of active species. This mathematical framework enables predictive modeling of spatially dependent CNT growth, as well as design of catalyst patterns to achieve engineered uniformity.

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Abstract. Metastable configurations in condensed matter typically fluctuate about local energy minima at the femtosecond time scale before transitioning between local minima after nanoseconds or microseconds. This vast scale separation limits the applicability of classical molecular dynamics methods and has spurned the development of a host of approximate algorithms. One recently proposed method is diffusive molecular dynamics which aims at integrating a system of ordinary differential equations describing the likelihood of occupancy by one of two species, in the case of a binary alloy, while quasistatically evolving the locations of the atoms. While diffusive molecular dynamics has shown to be efficient and provide agreement with observations, it is fundamentally a model, with unclear connections to classical molecular dynamics.In this work, we formulate a spin-diffusion stochastic process and show how it can be connected to diffusive molecular dynamics. The spin-diffusion model couples a classical overdamped Langevin equation to a kinetic Monte Carlo model for exchange amongst the species of a binary alloy. Under suitable assumptions and approximations, spin-diffusion can be shown to lead to diffusive molecular dynamics type models. The key assumptions and approximations include a well defined time scale separation, a choice of spin-exchange rates, a low temperature approximation, and a mean field type approximation. We derive several models from different assumptions and show their relationship to diffusive molecular dynamics. Differences and similarities amongst the models are explored in a simple test problem. Version: September 26, 2017Key words. Diffusive molecular dynamics, metastability, kinetic Monte Carlo, quasistationary distributions, mean field approximation 1. Introduction. Metals, alloys, and other condensed matter typically exhibit behavior on at least two vastly different time scales making direct numerical simulation by classical molecular dynamics (MD) impractical. The vibrational time scale of the atoms, measured in femtoseconds (10 −15 s), sets a first time scale, and constrains the time step of MD. Physically relevant phenomena occur on time scales of microseconds (10 −6 s) or longer, setting a second time scale. Milliseconds of the laboratory time can be achieved with direct MD using special purpose hardware, but typical MD simulations only reach nanoseconds [39].The source of this scale separation is the metastability present in the physical system. By metastability, we mean the tendency of the system to persist in a particular arrangement, or conformation, of atoms for a relatively long time before rapidly transitioning to some other persistent conformation. In many condensed matter problems, these distinct metastable states correspond to local minimizers of a potential energy landscape and transitions occur through saddle points. The system fluctuates