Construction and Properties of Sierpiński Triangular Fractals on Surfaces

Abstract

Fractal structures are of fundamental importance in science, engineering, mathematics, and aesthetics. Construction of molecular fractals on surfaces can help to understand the formation mechanism of fractals and a series of achievements have been acquired in the preparation of molecular fractals. This review focuses on Sierpiński triangles (STs), representatives of various prototypical fractals, on surfaces. They are inves-tigated by Monte Carlo simulations and ultra-high vacuum scanning tunneling microscopy. STs are bonded through halogen bonds, hydrogen bonds, metal-organic coordination bonds and covalent bonds. The coexistence of and competition between fractals and crystals are realized for a hydrogenbonded system. Electronic properties of two types of STs are summarized.

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

“…However, recent advances in materials science and nanotechnology allows fabrication of deterministic fractal materials with predefined structures, i.e., structures where an intrinsic pattern repeats itself exactly under scaling, such as: Sierpinski triangular [52][53][54], supramolecular [25,55], octahedral [56] or Cantor fractals [57]. In addition, theoretical developments in SAS from deterministic fractals have shown that the corresponding intensity curves are characterized by a much more complex behavior, as compared to scattering from random fractals.…”

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

“…26,27 In principle, the combination of three-fold and 120º V-shaped nodes is the basic requirement for creating Sierpiński triangle selfassembly. [28][29][30] Molecular based Sierpiński triangles have been experimentally achieved on single-crystalline surfaces 31 via supramolecular structures such as halogen bonding, 29 hydrogen bonding, 32 and metal−organic coordination [33][34][35][36][37] or in the form of covalently-bonded polymers. 30,38,39 To grow a COF from a low-symmetry monomer, we selected 1,3benzene diboronic acid (1,3-BDBA) with a reduced D1 symmetry as the monomer (Fig.…”

Structural Properties of Molecular Sierpiński Triangle Fractals

“…where G 1 (q) is given by Equation (19). Then, the total intensity, or the fractal form factor, is obtained by using the expression of the triangle form factor given by Equation ( 14) in Equation (8).…”

“…Fabrication of such structures is a complex process involving an interplay between halogen, hydrogen, metal-organic coordination and covalent bonds, depending on the specific type of atoms used to build the fractal, as well as on the substrate [7]. Up to now, only ST fractals can be prepared on surfaces in a controlled way [8]. As such, ST fractals are constructed on both Ag(111) and symmetry-mismatched fourfold Ag(100) surfaces through chemical reactions between H3PH molecules and Fe atoms in vacuum [9].…”