The structure and properties of molecular trees and networks

Abstract

The gel point is a well-understood critical point in polymer science (Flory, Stockmayer). A summary is given of the resolution of those aspects which may seem paradoxical at first sight. The relevant equations for the basic paradigm of f-functional polycondensation are very simple. Special attention is paid to the critically branched state of materials not far from the gel point. The quasiinvariance principle is explained according to which all solution properties appear ultimately to level off as the critical conversion is approached from below. The 'Malthusian' packing paradox is resolved by a proper treatment of the ring-chain competition situation, which also disposes of the spurious divergence of the rate of cyclization predicted by a more naive theory.Network theories not based on extinction probability (Charlesby) are not worth considering. The proper definition of an elastically active network chain (EANC) was based on this concept by Scanlan and by Case in 1960, and it greatly simplifies the graph-like-state theories of network structure (usually called 'network topology'). It allows classical rubber elasticity theory to be applied near the gel point. The point is characterised by a fifth-order Ehrenfest transition due to the contribution of long-range correlations to the configurational free energy. Though directly relevant data on reversible gelation are not available, data on isothermal crosslinking of very diverse systems support this analysis. Except possibly for some highly crosslinked systems, the parameter M (mean chain length between 'crosslinks') is an undesirable ingredient of elasticity or swelling theories. Chain-end corrections are quite undesirable (and usually done incorrectly). Everything is correctly and more simply formulated in terms of Scanlan-Case EANCs.The proper understanding of the gel point as a critical point allows the construction of reduced plots, illustrated with temperature superposition of experimental modulus-conversion plots (by M. Judd) for aqueous gelatin jellies. These fit reasonably to the basic model involving triple-helix junction zones, essentially without adjusting arbitrary parameters. Such reducedvariable treatments for critically branched materials eliminate the otherwise inescapable difficulties of characterization, purity, electrolyte content, etc., from gelatin research.

Universal Cyclic Topology in Polymer Networks

“…Theoretical efforts have addressed the delay of the gelpoint due to loop formation [25][26][27][28]; however, the intrinsic dependence of the cyclic topology on the condition under which the polymer network is formed as well as the inherent relations between different orders of loop structures are unknown [29,30]. Experimental techniques such as rheology/spectroscopy and multiple-quantum NMR can provide semi-quantitative information related to the loop structure [31][32][33].…”

Markovian approach to nonlinear polymer formation: Free-radical polymerization with chain transfer to polymer

“…(28). These values weight-average chain length development for the are plotted as a function of 1/N, as shown in Figpresent reaction system is given by 4,29 : ure 3.…”

Cross-Linking and Structure of Polymer Networks

“…Refs. [16][17][18], the tree-like structure of the molecules is formally retained, but some of the reacted functionalities are classified as engaged in ring formation while the others are active in branching and structure growth.…”