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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. 

The rapid proliferation of electronic cigarettes (ECs) has raised significant concerns about their potential health effects on both users and bystanders. This study systematically investigates the impact of EC aerosol exposure on human alveolar epithelial cells (A549), considering variations in device parameters, nicotine concentration, and exposure type. Using a gravity-based air–liquid interface exposure system, we assessed cytotoxicity and epithelial barrier integrity by measuring cell viability and transepithelial electrical resistance (TEER). Our results indicate that EC aerosol exposure significantly reduces cell viability and disrupts monolayer integrity in a dose- and device-dependent manner. Notably, VUSE (pod-type) exposure led to a 16% decrease in viability and a 41% reduction in TEER, while VOOPOO (mod-type) exposure caused a 25% viability loss and a 61% reduction in TEER. Power settings played a critical role: at 60 W, cell viability dropped by 48% at 12 mg/mL nicotine concentration compared to 29% at 0 mg/mL. Moreover, under the same number of puffs (30 puffs), firsthand exposure resulted in a 73% viability decrease, whereas secondhand exposure showed a 47% reduction, indicating substantial bystander risks associated with EC usage. These findings underscore the importance of device specifications and exposure conditions in determining EC aerosol toxicity. The observed epithelial barrier disruption suggests increased vulnerability to respiratory diseases. Given the comparable toxicity of firsthand and secondhand aerosols, regulatory measures should extend beyond direct users to include bystander protection. This study highlights the urgent need for comprehensive toxicity assessments to inform public health policies on EC use.