Abstract
This review article highlights recent developments in symmetry, combinatorics, topology, entropy, chirality, spectroscopy and thermochemistry pertinent to 2D and 1D nanomaterials such as circumscribed-cyclopolyarenes and their heterocyclic analogs, carbon and heteronanotubes and heteronano wires, as well as tessellations of cyclopolyarenes, for example, kekulenes, septulenes and octulenes. We establish that the generalization of Sheehan’s modification of Pólya’s theorem to all irreducible representations of point groups yields robust generating functions for the enumeration of chiral, achiral, position isomers, NMR, multiple quantum NMR and ESR hyperfine patterns. We also show distance, degree and graph entropy based topological measures combined with techniques for distance degree vector sequences, edge and vertex partitions of nanomaterials yield robust and powerful techniques for thermochemistry, bond energies and spectroscopic computations of these species. We have demonstrated the existence of isentropic tessellations of kekulenes which were further studied using combinatorial, topological and spectral techniques. The combinatorial generating functions obtained not only enumerate the chiral and achiral isomers but also aid in the machine construction of various spectroscopic and ESR hyperfine patterns of the nanomaterials that were considered in this review. Combinatorial and topological tools can become an integral part of robust machine learning techniques for rapid computation of the combinatorial library of isomers and their properties of nanomaterials. Future applications to metal organic frameworks and fullerene polymers are pointed out.
Highlights
Two-dimensional nanomaterials, especially those arising from different forms of graphenes, for example, circumscribed-cyclopolyarenes, tessellations of kekulenes, septulenes and octulenes, nanobelts, and various forms of macrocycles that have pores have received significant attention over the years [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]
Related one-dimensional nanomaterials such as carbon nanotubes and heteronanotubes and nanowires composed of B/N, Ga/N, C/N, Ga/As have been the topic of a number of studies because these materials exhibit very interesting electronic, geometrical, topological, chiral, polarizability, and optical properties [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59]
Cycloarenes are derived from a synthesis of angular and linear annulations of benzene rings that result macro-cyclic systems with cavities that are especially suitable for sequestration of both anions and cations
Summary
Two-dimensional nanomaterials, especially those arising from different forms of graphenes, for example, circumscribed-cyclopolyarenes, tessellations of kekulenes, septulenes and octulenes, nanobelts, and various forms of macrocycles that have pores have received significant attention over the years [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. Their structures with cavities offer intriguing possibilities for the environmental management, organic chemistry, rational drug design and delivery, as they possess optimal electronic and geometrical features for both trapping of toxic ions to transportation of heavy metal ions, halide anions, drugs and so forth [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] Carbon nanotubes and their heteroanalogs such as GaN, GaAs nanowires and related decorations with metals such as gold have opened up a new vista and a new state of matter that has culminated into significant research activities over the years [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59]. This review brings together combinatorics, group theory and topological techniques with applications to heteronanotubes, circumscribedcyclopolyarenes, tessellations of graphene fragments with cavities such as tessellations of kekulenes, and so forth
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