Molecular weight distribution in random crosslinking of polymer chains

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AbstractThe molecular weight distribution (MWD) of crosslinked polymer molecules formed during polymeric network formation is the sum of the fractional MWDs containing 0, 1, 2, 3, … crosslinkages. The MWD for polymer molecules containing 𝓀 crosslinkages is investigated for the random crosslinking of polymer chains whose initial MWD is given by the Schulz‐Zimm distribution. For a very narrow initial MWD, each fractional MWD with 𝓀 = 0, 1, 2, … is independent and a multimodal distribution is obtained for the whole distribution. When the initial MWD is uniform, the average crosslinking density within the polymer fraction whose degree of polymerization is r, ρr is simply given by ρr = ρgel,c – 2/r irrespective of the extent of crosslinking reaction where ρgel,c is the crosslinking density within gel fraction at the gel point. On the other hand, the MWDs with 𝓀 crosslinkages overlap each other with different 𝓀 values significantly for the broader initial distributions, and ρr increases with the progress of crosslinking reactions. The value of ρr increases with increasing r but levels off asymptotically at large r. The average crosslinking density of polymer molecules containing 𝓀 crosslinkages ρ𝓀 is an increasing function of k but soon reaches a plateau; sooner for the broader initial MWDs. For 𝓀 ≥ 1, ρ𝓀 is always larger than the average crosslinking density of the whole reaction system ρ in the pregelation period, i.e., in terms of the crosslinking density, the difference between polymer molecules with and without crosslinkage is most significant. In general, the average crosslinking density ρ, which is convenient to use in describing the nature of the whole reaction system, cannot be considered as a characteristic degree of crosslinking for polymer molecules containing at least one crosslinkage. Consideration of the bivariate distribution of r and k reveals important aspects of the polymeric network formation that have been obscured in the conventional theories in which the averages including linear polymers are solely considered. © 1995 John Wiley & Sons, Inc.