Construction and Properties of Sierpiński Triangular Fractals on Surfaces

Emergence of fractal geometries in the evolution of a metabolic enzyme

Fractals are patterns that are self-similar across multiple length-scales1. Macroscopic fractals are common in nature2-4; however, so far, molecular assembly into fractals is restricted to synthetic systems5-12. Here we report the discovery of a natural protein, citrate synthase from the cyanobacterium Synechococcus elongatus, which self-assembles into Sierpiński triangles. Using cryo-electron microscopy, we reveal how the fractal assembles from a hexameric building block. Although different stimuli modulate the formation of fractal complexes and these complexes can regulate the enzymatic activity of citrate synthase in vitro, the fractal may not serve a physiological function in vivo. We use ancestral sequence reconstruction to retrace how the citrate synthase fractal evolved from non-fractal precursors, and the results suggest it may have emerged as a harmless evolutionary accident. Our findings expand the space of possible protein complexes and demonstrate that intricate and regulatable assemblies can evolve in a single substitution.

Second-order superatoms: Au52-PAP featuring a three-dimensional cluster-of-clusters core

The recent characterization of Au52-PAP cluster can be viewed as a three-dimensional arrangement featuring four Au13 motifs. As a result, a new set of superatomic orbitals are built up from the superatomic shell of each constituent unit, denoted by 1S'21P'62S'21D'102P'61F'6 and, thus, referred to as a second-order superatomic shell structure. This favors the rationalization of larger species toward the formation of cluster-assembled materials of different sizes.

Multiquanta flux jumps in superconducting fractal

We study the magnetic field response of millimeter scale fractal Sierpinski gaskets (SG) assembled of superconducting equilateral triangular patches. Directly imaged quantitative induction maps reveal hierarchical periodic filling of enclosed void areas with multiquanta magnetic flux, which jumps inside the voids in repeating bundles of individual flux quanta Φ0. The number Ns of entering flux quanta in different triangular voids of the SG is proportional to the linear size s of the void, while the field periodicity of flux jumps varies as 1/s. We explain this behavior by modeling the triangular voids in the SG with effective superconducting rings and by calculating their response following the London analysis of persistent currents, Js, induced by the applied field Ha and by the entering flux. With changing Ha, Js reaches a critical value in the vertex joints that connect the triangular superconducting patches and allows the giant flux jumps into the SG voids through phase slips or multiple Abrikosov vortex transfer across the vertices. The unique flux behavior in superconducting SG patterns, may be used to design tunable low-loss resonators with multi-line high-frequency spectrum for microwave technologies.

Structural Quantification of the Surface-Confined Metal-Organic Precursors Simulated with the Lattice Monte Carlo Method

The diversity of surface-confined metal-organic precursor structures, which recently have been observed experimentally, poses a question of how the individual properties of a molecular building block determine those of the resulting superstructure. To answer this question, we use the Monte Carlo simulation technique to model the self-assembly of metal-organic precursors that precede the covalent polymerization of halogenated PAH isomers. For this purpose, a few representative examples of low-dimensional constructs were studied, and their basic structural features were quantified using such descriptors as the orientational order parameter, radial distribution function, and one- and two-dimensional structure factors. The obtained results demonstrated that the morphology of the precursor (and thus the subsequent polymer) could be effectively tuned by a suitable choice of molecular parameters, including size, shape, and intramolecular distribution of halogen substituents. Moreover, our theoretical investigations showed the effect of the main structural features of the precursors on the related indirect characteristics of these constructs. The results reported herein can be helpful in the custom designing and characterization of low-dimensional polymers with adjustable properties.

Simple and Tailorable Synthesis of Silver Nanoplates in Gram Quantities

Due to plasmonic and catalytic properties, silver nanoplates are of significant interest; therefore, their simple preparation in gram quantities is required. Preferably, the method is seedless, consists of few reagents, enables preparation of silver nanoplates with desired optical properties in high concentration, is scalable, and allows their long-term storage. The developed method is based on silver nitrate, sodium borohydride, polyvinylpyrrolidone, and H₂O₂ as the main reagents, while antifoam A204 is implemented to achieve better product quality on a larger scale. The effect of each component was evaluated and optimized. Solution volumes from 3 to 450 mL and concentrations of silver nanoplates from 0.88 to 4.8 g/L were tested. Their size was tailored from 25 nm to 8 μm simply by H₂O₂ addition, covering the entire visible plasmon spectra and beyond. They can be dried and spontaneously dispersed after at least one month of storage in the dark without any change in plasmonic properties. Their potential use in modern art was demonstrated by drying silver colloids on different surfaces in the presence of reagents or purified, resulting in a variety of colors but, more importantly, patterns of varying complexity, from simple multi-coffee-rings structures to dendritic forms and complex multilevel Sierpiński triangle fractals.

Steering Metal-Organic Network Structures through Conformations and Configurations on Surfaces

Molecular adsorption conformations and arrangement configurations on surfaces are important structural aspects of surface stereochemistry, but their roles in steering the structures of metal-organic networks (MONs) remain vague and unexplored. In this study, we constructed MONs by the coordination self-assembly of isocyanides on Cu(111) and Ag(111) surfaces and demonstrated that the MON structures can be steered by surface stereochemistry, including the adsorption conformations of the isocyanide molecules and the arrangement configurations of the coordination nodes and subunits. The coordination self-assembly of 1,4-phenylene diisocyanobenzene afforded a honeycomb MON consisting of 3-fold (isocyano)₃-Cu motifs on a Cu(111) surface. In contrast, geometrically different chevron-shaped 1,3-phenylene diisocyanobenzene (m-DICB) failed to generate a MON, which is ascribable to its standing conformation on the Cu(111) surface. However, m-DICB was adsorbed in a flat conformation on a Ag(111) surface, which has a larger lattice constant than a Cu(111) surface, and smoothly underwent coordination self-assembly to form a MON consisting of (isocyano)₃-Ag motifs. Interestingly, only C₃-Ag nodes with heterotactic configurations could grow into larger subunits; those subunits with heterotactic configurations further grew into Sierpiński triangle fractals (up to fourth order), while subunits with homotactic configurations afforded a triangular MON.

Mathematical Modeling of Layered Nanocomposite of Fractal Structure

A model of a layered hierarchically constructed composite is presented, the structure of which demonstrates the properties of similarity at different scales. For the proposed model of the composite, fractal analysis was carried out, including an assessment of the permissible range of scales, calculation of fractal capacity, Hausdorff and Minkovsky dimensions, calculation of the Hurst exponent. The maximum and minimum sizes at which fractal properties are observed are investigated, and a quantitative assessment of the complexity of the proposed model is carried out. A software package is developed that allows calculating the fractal characteristics of hierarchically constructed composite media. A qualitative analysis of the calculated fractal characteristics is carried out.

Sierpiński Structure and Electronic Topology in Bi Thin Films on InSb(111)B Surfaces

Deposition of Bi on InSb(111)B reveals a striking Sierpiński-triangle (ST)-like structure in Bi thin films. Such a fractal geometric topology is further shown to turn off the intrinsic electronic topology in a thin film. Relaxation of a huge misfit strain of about 30% to 40% between Bi adlayer and substrate is revealed to drive the ST-like island formation. A Frenkel-Kontrova model is developed to illustrate the enhanced strain relief in the ST islands offsetting the additional step energy cost. Besides a sufficiently large tensile strain, forming ST-like structures also requires larger adlayer-substrate and intra-adlayer elastic stiffnesses, and weaker intra-adlayer interatomic interactions.

Boronic ester Sierpiński triangle fractals: from precursor design to on-surface synthesis and self-assembling superstructures

Herein, we designed and synthesized a precursor with a three-fold node and successfully constructed covalent Sierpiński triangle (ST) fractals with boronic ester linkages both at the liquid/solid interface at room temperature and by thermal annealing in a water atmosphere under ambient conditions. Remarkably, large-scale ordered superstructures of covalent STs are constructed by thermal annealing, which paves the way for property investigation of STs.

Structural Properties of Molecular Sierpiński Triangle Fractals

The structure of fractals at nano and micro scales is decisive for their physical properties. Generally, statistically self-similar (random) fractals occur in natural systems, and exactly self-similar (deterministic) fractals are artificially created. However, the existing fabrication methods of deterministic fractals are seldom defect-free. Here, are investigated the effects of deviations from an ideal deterministic structure, including small random displacements and different shapes and sizes of the basic units composing the fractal, on the structural properties of a common molecular fractal—the Sierpiński triangle (ST). To this aim, analytic expressions of small-angle scattering (SAS) intensities are derived, and it is shown that each type of deviation has its own unique imprint on the scattering curve. This allows the extraction of specific structural parameters, and thus the design and fabrication of artificial structures with pre-defined properties and functions. Moreover, the influence on the SAS intensity of various configurations induced in ST, can readily be extended to other 2D or 3D structures, allowing for exploration of structure-property relationships in various well-defined fractal geometries.

Programming and simulating chemical reaction networks on a surface

Models of well-mixed chemical reaction networks (CRNs) have provided a solid foundation for the study of programmable molecular systems, but the importance of spatial organization in such systems has increasingly been recognized. In this paper, we explore an alternative chemical computing model introduced by Qian & Winfree in 2014, the surface CRN, which uses molecules attached to a surface such that each molecule only interacts with its immediate neighbours. Expanding on the constructions in that work, we first demonstrate that surface CRNs can emulate asynchronous and synchronous deterministic cellular automata and implement continuously active Boolean logic circuits. We introduce three new techniques for enforcing synchronization within local regions, each with a different trade-off in spatial and chemical complexity. We also demonstrate that surface CRNs can manufacture complex spatial patterns from simple initial conditions and implement interesting swarm robotic behaviours using simple local rules. Throughout all example constructions of surface CRNs, we highlight the trade-off between the ability to precisely place molecules and the ability to precisely control molecular interactions. Finally, we provide a Python simulator for surface CRNs with an easy-to-use web interface, so that readers may follow along with our examples or create their own surface CRN designs.

Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures

Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific theoretical models which exploit the fractal symmetry have been developed to extract additional information from SAS data. Although this problem can be solved exactly for many particular fractal structures, due to the intrinsic limitations of the SAS method, the inverse scattering problem, i.e., determination of the fractal structure from the intensity curve, is ill-posed. However, fractals can be divided into various classes, not necessarily disjointed, with the most common being random, deterministic, mass, surface, pore, fat and multifractals. Each class has its own imprint on the scattering intensity, and although one cannot uniquely identify the structure of a fractal based solely on SAS data, one can differentiate between various classes to which they belong. This has important practical applications in correlating their structural properties with physical ones. The article reviews SAS from several fractal models with an emphasis on describing which information can be extracted from each class, and how this can be performed experimentally. To illustrate this procedure and to validate the theoretical models, numerical simulations based on Monte Carlo methods are performed.

Covalent organic frameworks from a monomer with reduced symmetry: polymorphism and Sierpiński triangles

We report on the synthesis of a covalent organic framework based on the low-symmetry 1,3-benzenediboronic acid precursor. Two distinct polymorphs are obtained, a honeycomb network and Sierpiński triangles, as elucidated by scanning tunneling microscopy. Control over polymorph formation was achieved by varying the precursor concentration for on-surface synthesis.