Planar Frameworks Research Articles

Rational tensegrities through the lens of toric geometry

A classical tensegrity model consists of an embedded graph in a vector space with rigid bars representing edges, and an assignment of a stress to every edge such that at every vertex of the graph the stresses sum up to zero. The tensegrity frameworks have been recently extended from the two dimensional graph case to the multidimensional setting. We study the multidimensional tensegrities using tools from toric geometry. We introduce a link between self-stresses and Chow rings on toric varieties. More precisely, for a given rational tensegrity framework F, we construct a glued toric surface XF. We show that the abelian group of tensegrities on F is isomorphic to a subgroup of the Chow group A1(XF;Q). In the case of planar frameworks, we show how to explicitly carry out the computation of tensegrities via classical tools in toric geometry.

Catalytic Enantioselective Synthesis of Inherently Chiral Molecules: Recent Advances

AbstractInherent chirality represents a distinct category of molecular chirality that does not fall within the traditional classification of four chiral elements: central, axial, planar, and helical chirality. While extensive research has been conducted on the catalytic enantioselective construction of these conventional chiralities, the corresponding synthesis of inherently chiral molecules has remained largely unexplored. This minireview provides a comprehensive summary of recent advancements in this field, focusing on the catalytic asymmetric synthesis of inherently chiral calixarenes, saddle‐shaped tetraphenylenes and their heterocycle derivatives, mechanically planar chiral rotaxanes and chiral multilayer 3D frameworks, as well as our perspective in this field.

Synthesis and Properties of Rubicene-Based Aromatic π-Conjugated Compounds as Five-Membered Ring Embedded Planar Nanographenes.

Polycyclic aromatic hydrocarbons consisting of two or three rubicene substructures were designed as π-conjugated compounds embedding five-membered rings. The target compounds with t-butyl groups were synthesized by the Scholl reaction of precursors consisting of 9,10-diphenylanthracene units, even though a partially precyclized precursor was required for the synthesis of the trimer. These compounds were isolated as stable and dark blue solids. Single-crystal X-ray analysis and DFT calculations revealed the planar aromatic framework of these compounds. In the electronic spectra, the absorption and emission bands were considerably red-shifted compared with those of the reference rubicene compound. In particular, the emission band of the trimer extended to the near-IR region while retaining the emissive property. The narrowed HOMO-LUMO gap with the extension of the π-conjugation was confirmed by cyclic voltammetry and DFT calculations.

Rh(III)-catalysed site-selective alkylation of β-carbolines/isoquinolines and tandem C[sbnd]H/C[sbnd]N functionalization to construct indolizine-indole frameworks

The rhodium-catalysed ortho-selective carbenes insertion strategy to activate the non-acidic sp2 CH bond of β-carboline and isoquinoline scaffolds was developed. This transformation endows facile fabrication of CC bond with high atom economy, pleasant yields and wide functional group tolerance exploiting directing properties of pyridinic nitrogen. In this protocol, diazo compounds of diethyl malonate, dimethyl dimedone and oxindole are tested as carbene source with the release of environmentally benign N2 gas as the by-product. Moreover, to perceive mechanistic insights, ESI-MS studies were conducted, and the key intermediates associated with this transformation were identified. In addition, the metal-free construction of planar polycyclic indolizine-indole frameworks has been accomplished from the synthesized products. Furthermore, the neoplastic and fluorescence properties of CN annulated compounds were also determined.

Physical Correlation of Topological Indices of Transition Metal Tetra-Cyano Benzene Structure Using Curve Fitting

Molecular graph theory is crucial in model development and design of any molecular topologies. The mathematical polynomials of chemical graphs are called topological indices and they are highly valuable for determining their physicochemical characteristics. In this manuscript, we have developed the physical relationship between heat of formation (and entropy) and topological indices using curve fitting method of innovative planar metal-organic frameworks - Surprisingly, the - frameworks are basically metals in every respect inside 1-turn heading and ultimately exhibit ferromagnetic coupling to appealing nanostructures, implying ideal candidates and a novel promise of unusual electromagnetic applications.

CALCULATION OF REINFORCED CONCRETE FRAMES WITH REGARD TO CRACKING

Summary. It is known that in the elements of reinforced concrete frames during the formation of cracks, the stiffness characteristics change, which in turn leads to a significant redistribution of forces between the individual elements. Software complexes usually used in the engineering of reinforced concrete constructions take into account the change in bending stiffnesses of rod finite elements as a result of cracking. In this case, in order to obtain sufficient accuracy, as a rule, one element of the frame (column, transom) is split into several finite elements, which significantly increases the number of unknowns in the system of equations. In addition, almost all software complexes do not take changes in the torsional stiffness of rod elements as a result of cracking into consideration.
 Calculation of plane and spatial frames by classical displacement method and finite element method does not practically differ in terms of the number of unknowns in the system of equations. In a frame consisting of individual rods (columns, transoms), the forces at the different ends of each rod can differ significantly. Therefore, if the stiffness changes as a result of cracking (in an iterative process), the stiffness of the entire rod has to be changed, and this also leads to significant errors. To calculate accurately, we have to divide each rod into a certain number of finite elements, which greatly increases the number of unknowns.
 To eliminate this disadvantage, the paper proposes a method for calculating spatial and planar frameworks using the classical method of displacements. But in this case the values of bending moments and reactions on the supports are determined taking into consideration the variable stiffness along the length of each element. Thus, without increasing the number of unknowns, it is possible to take into consideration the stiffness variable along the length of the rod. Moreover, dividing the length of the element into any number of sections does not increase the number of unknowns, while refining the calculation results with regard to cracking. Changing the stiffnesses can be written as a fairly simple subprogram in the general calculation program. On the example of the simplest spatial frame, the importance of dividing the rod into several sections with different stiffnesses is shown. In this case, the number of unknown systems of equations does not increase.
 Keywords. Reinforced concrete frame; displacement method; finite element; stiffness; cracking.

States of self-stress in symmetric frameworks and applications

We use the symmetry-extended Maxwell rule established by Fowler and Guest to detect states of self-stress in symmetric planar frameworks. The dimension of the space of self-stresses that are detectable in this way may be expressed in terms of the number of joints and bars that are unshifted by various symmetry operations of the framework. Therefore, this method provides an efficient tool to construct symmetric frameworks with many ‘fully-symmetric’ states of self-stress, or with ‘anti-symmetric’ states of self-stress. Maximising the number of independent self-stresses of a planar framework, as well as understanding their symmetry properties, has important practical applications, for example in the design and construction of gridshells. We show the usefulness of our method by applying it to some practical examples.

Control of Physicochemical Properties for Thiophene-Fused Naphthodiphospholes By Precise Fusion of Heterole Rings

Fused polycyclic aromatics have been vigorously pursued over the years because of their attracting structural, electronic, and photophysical features associated with the rigid and planar π-conjugated frameworks. For polycyclic aromatics with heterole-fused structures, the orientation of fused heterole rings as well as the geometry of their fused structures has a large impact on the physicochemical properties. In this regard, we expected that introducing two phosphorus atoms into polycyclic aromatics would provide an assortment of isomers originated from not only the position and orientation of the phosphole rings, but also the orientation of substituents on the phosphorus atoms (trans/cis configurations). Herein, we designed and synthesized a series of six isomers of thiophene-fused naphthodiphospholes based on our envision. Systematic investigations into the physicochemical properties revealed that the positions of heterole rings are of utmost importance to tune their electronic properties, and the orientations of heterole rings and substituents on the phosphorus atoms play critical roles in the molecular arrangements in the solid states.

Hybrid functional analysis of planar metal-organic frameworks

The electronic energy band structure and optical properties of the planar metal-organic frameworks, namely M-HAB and M3[HITP] (M = Ni, Cu; HAB = hexaaminobenzene; HITP = hexaiminotriphenylene) were investigated in this work using two hybrid exchange-correlation functionals. The materials studied here contain nickel and copper, 3d electrons of which should be considered as strongly correlated subsystem. These semi-core electrons are treated here by means of the hybrid HSE06 and PBE0 functionals. We have found that Ni-HAB and Cu-HAB are metals, whereas Ni3[HITP]2 and Cu3[HITP]2 are degenerate p-type semiconductors.

Understanding the origin of serrated stacking motifs in planar two-dimensional covalent organic frameworks.

Covalent organic frameworks (COFs) have attracted significant attention due to their chemical versatility combined with a significant number of potential applications. Of particular interest are two-dimensional COFs, where the organic building units are linked by covalent bonds within a plane. Most properties of these COFs are determined by the relative arrangement of neighboring layers. These are typically found to be laterally displaced, which, for example, reduces the electronic coupling between the layers. In the present contribution we use dispersion-corrected density-functional theory to elucidate the origin of that displacement, showing that the common notion that the displacement is a consequence of electrostatic repulsions of polar building blocks can be misleading. For the representative case of COF-1 we find that electrostatic and van der Waals interactions would, actually, favor a cofacial arrangement of the layers and that Pauli repulsion is the crucial factor causing the serrated AA-stacking. A more in-depth analysis of the electrostatic contribution reveals that the "classical" Coulomb repulsion between the boroxine building blocks of COF-1 suggested by chemical intuition does exist, but is overcompensated by attractive effects due to charge-penetration in the phenylene units. The situation becomes more involved, when additionally allowing the interlayer distance to relax for each displacement, as then the different distance-dependences of the various types of interactions come into play. The overall behavior calculated for COF-1 is recovered for several additional COFs with differently sized π-systems and topologies, implying that the presented results are of more general relevance.

Synthesis, Structures, and Properties of BN-Dinaphthothiophenes: Influence of B and N Placement on Photophysical Properties and Aromaticity.

Substitution of the C═C functionality with the isosteric and isoelectronic B-N moiety has emerged as a powerful way to expand the family of polycyclic aromatic hydrocarbons. In this paper, two types of BN-dinaphthothiophene (BN-DNT) derivatives with different B and N substitution patterns were synthesized in short steps from commercially available materials. X-ray crystallographic analysis revealed that BN-DNT 1 and 2 had rigid and planar frameworks. Their photophysical properties and the aromaticity of the BN rings of the BN-DNTs were slightly dependent on the B and N substitution patterns. However, their response toward fluoride anions was greatly dependent on the B and N substitution patterns.

A Wavy Two-Dimensional Covalent Organic Framework from Core-Twisted Polycyclic Aromatic Hydrocarbons.

A high degree of crystallinity is an essential aspect in two-dimensional covalent organic frameworks, as many properties depend strongly on the structural arrangement of the different layers and their constituents. We introduce herein a new design strategy based on core-twisted polycyclic aromatic hydrocarbon as rigid nodes that give rise to a two-dimensional covalent organic framework with a wavy honeycomb (chairlike) lattice. The concave-convex self-complementarity of the wavy two-dimensional lattice guides the stacking of framework layers into a highly stable and ordered covalent organic framework that allows a full 3D analysis by transmission electron microscopy revealing its chairlike honeycomb facets and aligned mesoporous channels. Remarkably, the waviness of the framework does not disrupt the interlayer π-π stacking that shows charge transporting properties similar to those of planar covalent organic frameworks. The implementation of core-twisted aromatics as building blocks for covalent organic frameworks brings new possibilities in the design of highly ordered organic materials.

Stable Radical Cation-Containing Covalent Organic Frameworks Exhibiting Remarkable Structure-Enhanced Photothermal Conversion.

The production of a radical cation-containing covalent organic framework (COF) has been accomplished by sequential in situ reactions, quaternization, and one-electron reduction of the 2,2'-bipyridine-based COFs. The acid-catalyzed COF formation enables the cis configuration of 2,2'-bipyridyl moieties in the structure, of which the stability arises from the eclipsed stacking of the two-dimensional layered structure. The postfunctionalization generates cyclic alkylated diquats as the sole products from the controlled quaternization. The reduction of diquat cations on the COF skeletons results in a large number of radical cations, which delocalize and uniaxially stack on top of one another by virtue of interlayered π-electronic couplings. The absorption of the near-infrared (NIR) region exhibited by the cationic radical COF is remarkably high owing to the intercharge transfer across the π-coupling interlayers. Also, the long-range array of extended and planar frameworks in such a COF leads to the extra stability of the radical cations against external stresses. The structure-enhanced performance of the COF material is witnessed with photothermal conversion efficiencies of as high as 63.8 and 55.2% when exposed to 808 and 1064 nm lasers, respectively. Further PEG modification on such a COF allows photoacoustic imaging and photothermal therapy in vivo under NIR light illumination to be manifested.

Multifaceted Geometric Assessment towards Simplified Urban Surfaces Built by 3D Reconstruction

Large-scale three-dimensional (3D) reconstruction from multi-view images is used to generate 3D mesh surfaces, which are usually built for urban areas and are widely applied in many research hotspots, such as smart cities. Their simplification is a significant step for 3D roaming, pattern recognition, and other research fields. The simplification quality has been assessed in several studies. On the one hand, almost all studies on surface simplification have measured simplification errors using the surface comparison tool Metro, which does not preserve sufficient detail. On the other hand, the reconstruction precision of urban surfaces varies as a result of homogeneity or heterogeneity. Therefore, it is difficult to assess simplification quality without surface classification. These difficulties are addressed in this study by first classifying urban surfaces into planar surfaces, detailed surfaces, and urban frameworks according to the simplification errors of different types of surfaces and then measuring these errors after sampling. A series of assessment indexes are also provided to contribute to the advancement of simplification algorithms.

First-principle study of metal-organic frameworks of the 4d and 5d transition metal series with phthalocyanine and tetracyanobenzene

Abstract In the present work, the structural, electronic and magnetic properties of planar frameworks built of Tetracyanobenze (TCNB) and Phthalocyanine (Pc) with transition metals (TM) of the 4d and 5d series (TM = Mo, Ru, Rh, Pd, Ag, Cd, Re, Ir, Pt, and Hg) have been investigated by density functional theory (DFT) including Hubbard-like Coulomb correction terms. In that series, ferromagnetism has been found for Ru-Pc, Re-Pc, Mo-TCNB and Re-TCNB which all are also at least half-metallic with the exception of Mo-TCNB being insulating. The ferromagnetic state is very stable for Mo-TCNB, Re-TCNB, and especially Ru-Pc with an energy difference between ferromagnetic and antiferromagnetic spin arrangements surpassing 100 meV per magnetic ion for Ru-Pc. The prediction of ferromagnetic and half-metallic metal-organic frameworks provides new insight into Pc/TCNB-based nanostructures with potential applications in spintronics.

Nonplanar dust-acoustic waves and chaotic motions in Thomas Fermi dusty plasmas

The properties of linear and nonlinear nonplanar dust acoustic (DA) solitary waves and chaotic behavior are investigated in an unmagnetized Thomas Fermi dusty plasma, whose components are degenerate electrons, ions, and negatively charged inertial cold dust grains. A linear dispersion relation is obtained and solved numerically. It has been observed that linear excitation characteristics are influenced by radial distance r, geometric term ν, and ion-to-electron Fermi temperature ratio σi. We have also noted that the addition of a geometrical term in dispersion relation gives damping along the radial axis. A modified Korteweg-de Vries (KdV) equation is derived by employing the reductive perturbation technique, and its numerical solutions are obtained. The modified KdV equation is discussed for cold dust grains in planar and nonplanar frameworks. Upon the introduction of external periodic perturbation, the perturbed modified KdV equation is studied in planar geometry via some qualitative and quantitative approaches. The perturbed KdV equation can give rise to the periodic, quasiperiodic, and chaotic motions for DA waves. The strength of the external perturbation and dust concentration h play the major role of the switching parameter in the transition of dynamic motion. The developed chaos can be weakened with the variation of dust concentration h. It has been observed that the dust concentration affects the dynamics of DA waves in planar geometry which is an important observation in this study.

Synthesis, Characterization, and Properties of Bis-BN Ullazines.

A series of bis-BN ullazine derivatives, including the parent species, were synthesized in a small number of steps from commercially available materials. X-ray crystallographic analysis revealed that bis-BN ullazines have rigid and planar frameworks. Most of the bis-BN ullazines are stable toward air and moisture. In addition, the absorption and emission bands of these ullazines are blue-shifted, compared to those of their carbonaceous ullazine analogs.

CF3 -Inspired Synthesis of Air-Tolerant 9-Phosphaanthracenes that Feature Fluorescence and Crystalline Polymorphs.

9-Phosphaanthracene (dibenzo[b,e]phosphorin, acridophosphine) has attracted interest as one of the heavier acenes. Herein, we demonstrate an efficient synthetic process that provides air-tolerant 1,8-bis(trifluoromethyl)-9-phosphaanthracenes. The sterically encumbered and electron-withdrawing trifluoromethyl (CF3 ) groups are quite advantageous not only to stabilize the intrinsically unstable heavier unsaturated phosphorus atom but also to facilitate construction of the phosphinine skeleton based on a putative increase in aromaticity. The isolated 9-phosphaanthracenes allowed characterization of their fluorescence functionality and planar heteroanthracene frameworks. The crystal structures of 9-phosphaanthracenes are remarkably dependent on the aryl substituents at the 10 position; anthryl-substituted 9-phosphaanthracene showed unique polymorphs that induced different-colored crystals.

The Number of Realizations of a Laman Graph

Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursive formula for the number of complex solutions of such systems.

Computing the number of realizations of a Laman graph

Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.