Topological polymers

Topological polymers, or polymer topology, could refer to a single polymeric chain with topological information or a polymer network with special junctions or connections. When the topology of a polymeric chain or network is investigated, the exact chemical composition is usually neglected, but the way of junctions and connections is more considered. Various topological structures, on one hand, could potentially change the interactions (van der Waals interaction, hydrogen bonding, etc.) between each of the polymer chain. On the other hand, topology also determines the hierarchical structures within a polymer network, from a microscopic level (<1 nm) to a macroscopic level (10-100 nm), which eventually affords polymeric materials with completely different physical properties,[2] such as mechanical property,[5] glass transition temperature,[6] gelation concentration.[7]

In early 1950s, Paul J. Flory was the pioneer who developed theories to explain topology within a polymer network, and the structure-property relationships between the topology and the mechanical property, like elasticity, was initially established afterwards.[3] Later in 1980s, Bertrand Duplantier developed theories to describe any polymer network topologies using statistical mechanics, which could help to derive topology-dependent critical exponents in a polymer network.[8] In early 2000s, Yasuyuki Tezuka and coworkers were the first ones that systematically described a single molecular chain with topological information.[9]

Adapted from Y. Tezuka and coworker's description of a topological polymer chain with more generalized rules,[9] the topology notation rules are to be introduced first, followed by three classical classifications, including linear, branched and cyclic polymer topologies, and they are classified in a table reorganized and redrawn from Y. Tezuka and coworker[9] (Copyright, 2001 by American Chemical Society).

A general polymer chain could be generalized into an undirected graph with nodes (vertices or points) and edges (lines or links) based on graph theory. In a graph theory topology, two sets of nodes are present, termini and junctions. The quantity ‘degree’ represents the number of edges linked to each node, if the degree of a certain node is larger than 3 (including 3), the node is a junction, while the degree of a node is 1, the node is a terminus. There are no nodes with a degree of 2 since they could be generalized into their adjacent nodes.

As for a certain polymer, as long as the topology is fixed, a specific topology notation could be generated using the following rules:

A general polymer chain notation could be expressed as:

${\displaystyle P_{m}(x,y)[s_{1}(s_{11},s_{12},..),s_{2}(s_{21},s_{22},...),...]}$

1. Capitalized letter ${\displaystyle P}$ represents the main topology within a polymer, ${\displaystyle A}$ represents linear or branched topology, and Roman numerals are used to represent the number of rings in the polymer chain, ${\displaystyle I}$ represents one cycle, ${\displaystyle II}$ represents two cycles, ${\displaystyle III}$, ${\displaystyle IV}$, ${\displaystyle V}$, etc. represents three, four, five cycles and so on.
2. ${\displaystyle m}$ represents the number of nodes in the graph theory topology, ${\displaystyle x}$ represents the number of termini and ${\displaystyle y}$ represents the number of junctions, and ${\displaystyle m=x+y}$is always true.
3. If ${\displaystyle P_{m}(x,y)}$ could represent an exclusive topology, there is no need to add more information to specify the notation. However, if multiple possibilities are present, extra information is needed.

i. For branched topology, a main chain is first selected, and the degree of each junction nodes along the chain should be noted as ${\displaystyle s_{i}}$connected by a hyphen. If there is a side chain on any of the main chain node, ${\displaystyle s_{ij}}$ should be noted with a bracket following the main chain notation.

ii. For monocyclic topology, the outward branch should be firstly identified with the number of branches at each of the junctions as ${\displaystyle s_{i}}$connected by a hyphen. Then the topology of each branch should be identified using the rule in i as ${\displaystyle s_{ij}}$using a bracket following the ${\displaystyle s_{i}}$notations.

iii. For multicyclic topology, superscript letter (${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ and so on) is used to describe internal connections within an existing ring.

Unlike single chain polymeric species, polymer network topology is more complicated as a result of the amorphous feature so that a simple notation is usually not feasible. To analyze the topology of a network, the crosslinkers, including the branched crosslinker and cyclic crosslinker, are considered.

The topology of a polymer chain or a polymer network is crucial in determining the macroscopic properties of a polymeric material, especially mechanical properties like elasticity and physical properties involving phase transitions. To date, several polymers with topological interest have been developed, which have been used for many applications, such as mechanical elastomer,[16] energy,[17] and so on. Below are some of the representative topological polymers or polymer networks.