Fracture in hexagonal honeycomb lattices undergoing large deformation

We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem.

Polymeric honeycomb structures are used in impact mitigation applications due to their high specific strain energy dissipation capacity at extended strains. The load bearing and energy absorption capacities of honeycomb lattices can be concurrently improved by tuning their Poisson's ratio. For instance, the inward-facing cavities in reentrant honeycomb geometries result in an auxetic behavior, effectively increasing the impact energy absorption performance. This study investigates the interrelationship between cell wall thickness, the Poisson effect, and load-bearing capacities of hexagonal and reentrant honeycomb lattices fabricated from thermoplastic polyurethane (TPU) by fused filament fabrication (FFF) additive manufacturing technique. The wall thickness of the samples is varied to investigate its effect on the mechanical performance of the honeycomb structures. The fabricated samples are characterized under quasi-static compression. The results indicate that the load-bearing responses of conventional and reentrant honeycomb structures improve with increasing wall thicknesses, while the energy absorption efficiency is inversely related to the cell wall thickness. Hexagonal honeycomb lattices exhibit superior stiffness and strength compared with their reentrant counterparts, whereas reentrant lattices are lighter and show a more compliant nature due to their more open geometry and inward-facing cavities. We observe that the reentrant lattices tend to fail due to shear deformation while honeycomb lattices fail by the bending and stretching of the cell walls. The findings suggest that the honeycomb lattice structure can be optimized by tuning the wall thickness based on the desired mechanical properties informed by a particular application.

Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice

Particles adsorbed at a fluid-fluid interface induce capillary deformations that determine their orientations and generate mutual capillary interactions which drive them to assemble into 2D ordered structures. We numerically calculate, by energy minimization, the capillary deformations induced by adsorbed cubes for various Young's contact angles. First, we show that capillarity is crucial not only for quantitative, but also for qualitative predictions of equilibrium configurations of a single cube. For a Young's contact angle close to 90°, we show that a single-adsorbed cube generates a hexapolar interface deformation with three rises and three depressions. Thanks to the threefold symmetry of this hexapole, strongly directional capillary interactions drive the cubes to self-assemble into hexagonal or graphenelike honeycomb lattices. By a simple free-energy model, we predict a density-temperature phase diagram in which both the honeycomb and hexagonal lattice phases are present as stable states.

Using the Van der Hoff--Benson method, general relations are derived for electric potentials and dipole-dipole interaction tensors of two-dimensional lattice systems which consist of charges or dipole moments changing periodically from site to site. Numerical values of the potentials and dipole-dipole interaction tensors as well as their asymptotics near the symmetric points of the first Brillouin zone are presented for square, triangular, and hexagonal (honeycomb) lattices. In particular, an asymptotic responsible for the formation of an incommensurate phase in triangular antiferromagnets is given. Dispersion laws for collective excitations of these lattices are calculated and compared to those derived in the nearest-neighbor approximation. The symmetry properties and lattice-sublattice relations are determined, which enables us to minimize the number of independently calculable lattice sums. For a two-dimensional Bravais lattice constituted by similar charges, at a constant unit-cell area, a dependence of the Coulomb energy on geometrical parameters of a lattice is derived. It is shown that with parameters of a triangular lattice, this dependence reaches the minimum which specifies all coefficients of long-wavelength quadratic asymptotics of the dipole-dipole interaction tensor. The low-temperature states are considered for a triangular lattice of dipoles with their orientations degenerate in the lattice plane. The temperature of the orientational phase transition is estimated by various methods and its significant characteristics are discussed. \textcopyright{} 1996 The American Physical Society.

Edge diffraction on triangular and hexagonal lattices: Existence, uniqueness, and finite section

We discuss the phase diagram of the quantum dimer model on the hexagonal (honeycomb) lattice. In addition to the columnar and staggered valence-bond solids which have been discussed in previous work, we establish the existence of a plaquette valence-bond solid. The transition between the plaquette and columnar phases at $v/t=\ensuremath{-}0.2\ifmmode\pm\else\textpm\fi{}0.05$ is argued to be first order. We note that this model should describe valence-bond-dominated phases of frustrated Heisenberg models on the hexagonal lattice and discuss its relation to recent exact diagonalization work by Fouet et al. on the ${J}_{1}\ensuremath{-}{J}_{2}$ model on the same lattice. Our results also shed light on the properties of the transverse field Ising antiferromagnet on the triangular lattice and the classical Ising antiferromagnet on the stacked triangular lattice, which are related to dimer models by duality.

Energy-based fracture mechanics of brittle lattice materials

We propose the use of a projection method based on a honeycomb lattice to enhance the geometrical understanding of general fullerenes. An arbitrary fullerene consisting of pentagonal and hexagonal arrangements of carbon atoms is completely specified by the distribution of twelve pentagonal defects on an otherwise hexagonal honeycomb lattice. Utilizing this projection method, we demonstrate the geometric specification of icosahedral hyperfullerenes and general chiral fullerene tubules suitably capped on each end.

The development of advanced spintronic devices requires ultrathin two-dimensional (2D) ferromagnetic (FM) materials with high Curie temperature (T C) and large out-of plane magnetic anisotropy energy (MAE). However, the number of high-T C 2D ferromagnets synthesized through top-down experimental methods is very limited. Here, we propose a bottom-up approach for constructing 2D ferromagnets with high T C by assembling magnetic superatoms. The MnSr9 superatom was first selected as building blocks to construct a series of 2D materials with square, triangular and hexagonal honeycomb lattices. First-principles studies show that all the MnSr9 self-assembled films are thermodynamically stable and exhibit ferromagnetism, unfortunately, they lack the necessary magnetic anisotropy. By substituting one Sr atom with a heavy 5d transition metal (5d-TM) atom, all these 5d-TM@MnSr8 clusters show enhanced stability and symmetry, and their self-assembled hexagonal honeycomb crystals exhibit significant magnetic anisotropy and enhanced ferromagnetism from 5d-TM atoms. Taking the PtMnSr8 superatom as an example, we have demonstrated these characteristics in detail, and the T C and out-of-plane MAE of its honeycomb structure reach up to 253 K and 3.47 meV per unit cell under biaxial tensile strain. Moreover, the PtMnSr8 honeycomb structure on hexagonal boron nitride monolayer substrate exhibit further enhanced ferromagnetism (T C ≈ 327 K) and distinctive antioxidant properties. This study highlights that assembling magnetic superatoms on suitable substrates is an effective way for constructing high-performance 2D FM materials.

Clustered speckle, optical lattices, and their optical vortex array are subjects of interest in optical wave manipulation. In this study, disordered optical lattices and vortex arrays with different unit structures were found in the clustered speckles generated by a circularly-distributed multi-pinhole scattering screen when it was illuminated by coherent light. These structures included hexagonal lattices, kagome lattices, and honeycomb lattices. Moreover, optical lattices with asymmetric units generated by modulation of phases with non-integer multiples of 2π were discussed. Theoretical analysis and numerical calculations demonstrated that optical lattices in clustered speckles in the observation plane were generated by the phase modulations of the random scattering screen. The lattice type depended on the number of 2π multiples of the summed phase difference between the pinholes. Additionally, the conditions for the formation of periodical optical lattices and their vortex arrays were given. Different optical lattices and their vortex arrays appearing simultaneously in the clustered speckle were difficult to generate using the common multi-beam interference system. This phenomenon is of great significance in the study of the orbital angular momentum of photons and other fields.

We report on a systematic study of two dimensional, periodic, frustrated Ising models with a quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagome lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet as well as a two quasi one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson (LGW) approach from large fields as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagome). We also discuss the effect of a finite temperature, as well as the interplay between longitudinal and transverse fields--in the kagome problem the latter gives rise to a non-trivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence bond phases that have been invoked in discussions of high-temperature superconductivity.

Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices

Recent advances in nanoporous graphene membrane for gas separation and water purification

A designer of metallic energy absorption structures using additively manufactured cellular materials must address the question of which of a multitude of cell shapes to select from, the majority of which are classified as either honeycomb, beam-lattice, or Triply Periodic Minimal Surface (TPMS) structures. Furthermore, there is more than one criterion that needs to be assessed to make this selection. In this work, six cellular structures (hexagonal honeycomb, auxetic and Voronoi lattice, and diamond, gyroid, and Schwarz-P TPMS) spanning all three types were studied under quasistatic compression and compared to each other in the context of the energy absorption metrics of most relevance to a designer. These shapes were also separately studied with tubes enclosing them. All of the structures were fabricated out of AlSi10Mg with the laser powder bed fusion (PBF-LB. or LPBF) process. Experimental results were assessed in the context of four criteria: the relationship between the specific energy absorption (SEA) and maximum transmitted stress, the undulation of the stress plateau, the densification efficiency, and the design tunability of the shapes tested—the latter two are proposed here for the first time. Failure mechanisms were studied in depth to relate them to the observed mechanical response. The results reveal that auxetic and Voronoi lattice structures have low SEA relative to maximum transmitted stresses, and low densification efficiencies, but are highly tunable. TPMS structures on the other hand, in particular the diamond and gyroid shapes, had the best overall performance, with the honeycomb structures between the two groups. Enclosing cellular structures in tubes increased peak stress while also increasing plateau stress undulations.