Polymer gels formed from A-B type of coupling reaction were investigated vigorously as a "model network" because the mesh size of the network was defined by the size and shape of the building blocks. However, the formation of these networks has an inherent difficulty in homogeneous mixing of mutually reactive building blocks, leading to the formation of inhomogeneous network structures. Recently, we have designed and fabricated the Tetra-PEG gel by combining two well-defined symmetrical tetra-arm polymers of the same size. Although this gel was formed by simply mixing two polymer solutions, the structure was extremely homogeneous. In this study, we investigated the gelation kinetics of Tetra-PEG gel in detail. The reaction kinetics corresponded well with the theoretical prediction, suggesting the homogeneous reacting system. This homogeneous reacting system may contribute to the homogeneous network structure of Tetra-PEG gel.

The reaction rate for the polycondensation in the Tetra-PEG gel system, i.e., A−B type coupling reaction between mutually reactive two four-arm polymers, has been studied by ATR-IR spectroscopy. It was found that (1) the polycondensation kinetics of Tetra-PEG gel can be simply treated as a chemical reaction between mutually reactive end-groups in solution, (2) the reaction undergoes as a simple second-order reaction from beginning to end regardless of gelation threshold, and (3) the gelation mechanism was predicted from the thermodynamic enthalpy and entropy at the transition state estimated by temperature dependence of rate constants. The reson of smooth second-order kinetics is suspected to be that the meanfield approximation can be applied to the reactivity of terminal groups on Tetra-PEGs; i.e., the reactivity of terminal groups on Tetra-PEGs is not affected by the steric hindrance, substitution effet, and gelation threshold. ■ INTRODUCTIONGelation or cross-linking is one of the fundamental but key reactions in the production of polymeric materials. A variety of commercial products, namely rubbers, gels, paints, and adhesives, are made by way of gelation threshold, i.e., the onset of formation of infinitely large clusters. Hence, the understanding of the kinetics of network formation is of particular importance not only from basic science but also from industrial points of view.One of the simplest ways describing the cross-linking kinetics is to treat as a series of branching reactions. 1,2 Let us consider the AB-type polycondensation. When the mean-field approximation is applied to the reactivity of terminal functional group, the reaction rate equation is described as a simple diffusioncontrolled reaction rate equation.Here, C A (t), C B (t), and k are the concentrations of species A and B and reaction rate constant. It should be noted that k is constant from initiation to completion regardless of the gelation threshold because the mean-field approximation is applied to the reactivity of terminal functional group. The validity of this equation was tested and partly confirmed. For example, Yokoyama and Dusek et al. investigated the time variation of the number of cross-links on the polyurethane network formation. However, the reaction kinetics was not discussed in detail. 3,4 Rozenberg, Kambe, and Dusek et al. reported that the reaction rate constant fell down at the later stage of reaction between epoxides and amines. 5−8 These studies were on the polymer melt systems. As for the network formation of conventional chemical gels, Bates and Howard calculated the reaction rate constants for gelation reaction under assumption of third-order kinetics. 9 This reaction obeyed a third-order kinetics only at the early stage of gelation. In addition, Stepto et al. reported polycondensation reaction of low-molecular-weight precrusors around the gel point. 10,11 However, the reaction exponent or reaction rate constant was not predicted. As mentioned above, there are only a few experimental data fully supporting the eq 1....

We investigated the relationship between the elastic modulus, G and the reaction probability, p for polymer networks. First, we pointed out that the elastic modulus is expressed by G = {(fp∕2 - 1) + O((p - 1)(2))} Nk(B)T∕V (percolated network law), which does not depend on the local topology of the network structure or the existence of the loops. Here, N is the number of lattice point, V is the system volume, f is the functionality of the cross-link, k(B) is the Boltzmann constant, and T is the absolute temperature. We also conducted simulations for polymer networks with triangular and diamond lattices, and mechanical testing experiments on tetra-poly(ethylene glycol) (PEG) gel with systematically tuning the reaction probability. Here, the tetra-PEG gel was confirmed to be a potential candidate for ideal polymer networks consisting of unimodal strands free from defects and entanglements. From the results of simulations and experiments, it was revealed, for the first time, that the elastic modulus obeys this law in the wide range of p (p(c) ≪ p ≤ 1), where p(c) is the reaction probability at gelation threshold.