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Two-dimensional (2D) electronic materials are of significant technological interest due to their exceptional properties and broad applicability in engineering. The transition from nanoscale physics, which dictates their stable configurations, to macroscopic engineering applications requires the use of multiscale methods to systematically capture their electronic properties at larger scales. A key challenge in coarse-graining is the rapid and near-periodic variation of the charge density, which exhibits significant spatial oscillations at the atomic scale. Therefore, the polarization density field—the first moment of the charge density over the periodic unit cell—is used as a multiscale mediator that enables efficient coarse-graining by exploiting the almost-periodic nature of the variation. Unlike the highly oscillatory charge density, the polarization varies over lengthscales that are much larger than the atomic, making it suitable for continuum modeling. In this paper, we investigate the electrostatic potential arising from the charge distribution of arbitrarily-deformed 2D materials. Specifically, we consider a sequence of problems wherein the underlying lattice spacing vanishes and thus obtain the continuum limit. We consider three distinct limits: where the thickness is much smaller than, comparable to, and much larger than the in-plane lattice spacing. These limiting procedures provide the homogenized potential expressed in terms of the boundary charge and dipole distribution, subject to the appropriate boundary conditions that are also obtained through the limit process. Furthermore, we demonstrate that despite the intrinsic non-uniqueness in the definition of polarization, accounting for the boundary charges ensures that the total electrostatic potential, the associated electric field, and the corresponding energy of the homogenized system are uniquely determined. 

We report that a dielectric polymer chain, constrained at both ends, sharply collapses when exposed to a high electric field. The chain collapse is driven by nonlocal dipolar interactions and anisotropic polarization of monomers, a characteristic of real polymers that prior theories were unable to incorporate. Once collapsed, a large number of chain monomers accumulate at the center location between the chain ends, locally increasing the electric field and polarization by orders of magnitude. The chain collapse is sensitive to the orientation of the applied electric field and chain stretch. Our findings not only offer new ways for rapid actuation and sensing but also provide a pathway to discover the critical physics behind instabilities and electrical breakdown in dielectric polymers. 

We consider electrostatic interactions in two classes of nanostructures embedded in a three dimensional space: (1) helical nanotubes, and (2) thin films with uniform bending (i.e., constant mean curvature). Starting from the atomic scale with a discrete distribution of dipoles, we obtain the continuum limit of the electrostatic energy; the continuum energy depends on the geometric parameters that define the nanostructure, such as the pitch and twist of the helical nanotubes and the curvature of the thin film. We find that the limiting energy is local in nature. This can be rationalized by noticing that the decay of the dipole kernel is sufficiently fast when the lattice sums run over one and two dimensions, and is also consistent with prior work on dimension reduction of continuum micromagnetic bodies to the thin film limit. However, an interesting contrast between the discrete-to-continuum approach and the continuum dimension reduction approaches is that the limit energy in the latter depends only on the normal component of the dipole field, whereas in the discrete-to-continuum approach, both tangential and normal components of the dipole field contribute to the limit energy. 

In this paper, a symmetry-adapted method is applied to examine the influence of deformation and defects on the electronic structure and band structure in carbon nanotubes. First, the symmetry-adapted approach is used to develop the analog of Bloch waves. Building on this, the technique of perfectly matched layers is applied to develop a method to truncate the computational domain of electronic structure calculations without spurious size effects. This provides an efficient and accurate numerical approach to compute the electronic structure and electromechanics of defects in nanotubes. The computational method is applied to study the effect of twist, stretch, and bending, with and without various types of defects, on the band structure of nanotubes. Specifically, the effect of stretch and twist on band structure in defect-free conducting and semiconducting nanotubes is examined, and the interaction with vacancy defects is elucidated. Next, the effect of localized bending or kinking on the electronic structure is studied. Finally, the paper examines the effect of 5–8–5 Stone–Wales defects. In all of these settings, the perfectly matched layer method enables the calculation of localized non-propagating defect modes with energies in the bandgap of the defect-free nanotube.