From dendrimers to fractal polymers and beyond

Abstract

The advent of dendritic chemistry has facilitated materials research by allowing precise control of functional component placement in macromolecular architecture. The iterative synthetic protocols used for dendrimer construction were developed based on the desire to craft highly branched, high molecular weight, molecules with exact mass and tailored functionality. Arborols, inspired by trees and precursors of the utilitarian macromolecules known as dendrimers today, were the first examples to employ predesigned, 1 → 3 C-branched, building blocks; physical characteristics of the arborols, including their globular shapes, excellent solubilities, and demonstrated aggregation, combined to reveal the inherent supramolecular potential (e.g., the unimolecular micelle) of these unique species. The architecture that is a characteristic of dendritic materials also exhibits fractal qualities based on self-similar, repetitive, branched frameworks. Thus, the fractal design and supramolecular aspects of these constructs are suggestive of a larger field of fractal materials that incorporates repeating geometries and are derived by complementary building block recognition and assembly. Use of terpyridine-M2+-terpyridine (where, M = Ru, Zn, Fe, etc) connectivity in concert with mathematical algorithms, such as forms the basis for the Seirpinski gasket, has allowed the beginning exploration of fractal materials construction. The propensity of the fractal molecules to self-assemble into higher order architectures adds another dimension to this new arena of materials and composite construction.

Controlled imprisonment of wave packet and flat bands in a fractal geometry

“…Moreover, it is also reported that fractals may produce Hofstadter butterfly [52][53][54] and they are also suitable prototype example for showing fractional quantum Hall states [55], energy level splitting described via fractal dimensions [56] and non-trivial quantum conductivity [57,58]. Also, fractals are noticed in self assembled polymers [59]. Although, fractal geometry does not have only theoretical point of interest, rather it has also been exploited experimentally to analyze the magneto-resistance [60], a distinct crossover of superconducting phase boundary [61].…”

Controlled imprisonment of wave packet and flat bands in a fractal geometry

“…Moreover, it is also reported that fractals may produce Hofstadter butterfly [52]- [54] and they are also suitable prototype example for showing fractional quantum Hall states [55], energy level splitting described via fractal dimensions [56] and non-trivial quantum conductivity [57,58]. Also, fractals are noticed in self assembled polymers [59]. Although, fractal geometry does not have only theoretical point of interest, rather it has also been exploited experimentally to analyze the magnetoresistance [60], a distinct crossover of superconducting phase boundary [61].…”