Can the gel point of a cross‐linking polymer be detected by the G′ – G″ crossover?

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Abstract
Dynamic mechanical measurements allow direct determination of the instant at which a network polymer gels. In such an experiment, the evolution of G′(t,ω0) and G″ (t,ω0) is measured in small amplitude oscillatory shear as a function of cross‐linking time t. The frequency ω0 is kept constant throughout. At the beginning of the experiment, G″ is orders of magnitude larger than G′, and at completion of reaction, this order is reversed. It recently has been suggested by Tung and Dynes that the gel point (GP) might occur at the time at which G′ and G″ cross each other. However, there is much dispute whether GP occurs exactly at the crossover or just somewhere in its vicinity. This study resolves the dispute by modeling the rheological behavior at GP: There is only one class of network polymers for which GP coincides with the crossover. This class of polymers exhibits, when reaching GP, power law relaxation G(t) ∼ t−n with a specific exponent value n = 1/2. Examples are stoichiometrically balanced network polymers and networks with excess cross‐linker, however, only at temperatures much above the glass transition. Otherwise, the power law behavior would be masked by vitrification. Power law relaxation seems to be property of polymers at GP in general. However, some polymers have a different exponent value, n ≠ 1/2, in which case the crossover occurs before GP (for n < 1/2) or after GP (for n > 1/2); i.e. the crossover cannot be used for detecting GP. While there are no networks known to us with n < 1/2, recent experiments showed that network polymers that are lean on cross‐linker exhibit power law relaxation with n > 1/2. A new method is suggested for measuring GP of these imbalanced networks.