Scale-free power-law distribution of emulsion-polymerized branched polymers: Power exponent of the molecular weight distribution

The molecular weight distribution (MWD), formed in emulsion polymerization that involves the polymer transfer reaction during Interval II, may approach the power-law distribution as polymerization proceeds. The power exponent, alpha, of the weight fraction distribution W(M) = M(-alpha) conforms to the relationship, alpha = 1/P(b), where P(b) is the probability that the chain end is connected to a backbone chain. The MWD of emulsion-polymerized polyethylene reported in literature agrees reasonably well with the relationship, W(M) = M(-alpha) with alpha = 1/P(b). This simple relationship could be used to estimate the Pb value from the MWD data, possibly leading to determining the polymer transfer constant under well-designed experimental conditions. Because alpha > 1, the number-average MW always approaches a finite value, but the weight- and higher order-averages of MWD may continue to increase as the particle grows without limit depending on the magnitude of P(b). The power-law distributions are self-similar, possessing the nature of fractals and lacking a characteristic scale. The i-th moment of the MWD for the present reaction system continues to increase without limit during Interval II for P(b) >= 1/i.