Emulsion polymerization. III. Theoretical prediction of the effects of slow termination rate within latex particles

Abstract The Smith‐Ewart theory predicts that there is an interval during an isothermal homopolymerization when the conversion varies linearly with time. This prediction rests on the assumptions that, during this interval II, the particle number is constant, the monomer concentration in the particles is constant, and the termination rate within the particles is instantaneous, so that the average number of radicals per particle Q is half. In this paper this latter assumption is abandoned. If the termination rate is slow, two or more radicals can coexist in a particle. The termination rate within a particle becomes a function of the particle size because of the decreased probability that two radicals meet for termination in a given time when the volume in which these radicals are located increases. It follows that with increasing conversion the termination rate decreases. Stockmayer's calculations based on this model neglected the variation of particle volume with time, and it was assumed that a steady state of radical concentration in particles exists. In the present calculations these restrictive assumptions were not used. Stockmayer calculated only how Q should vary with conversion. In the present paper several experimentally verifiable consequences of the model are shown. The new calculations show that the interval II conversion‐time curve can be represented by the formula At 2 + Bt , where B is the Smith‐Ewart rate and is proportional to the particle number and the parameter A is independent of the particle number and depends mainly on initiation and termination rates. From A and B and propagation and termination rate constants can be calculated. With the aid of parameters A and B the conversion dependence of molecular weight and of Q can also be predicted for interval II. In the theoretical calculations the distribution of radicals among particles is established. It is shown that for a given value of Q this distribution is unique, independent of the experimental conditions leading to this Q . This distribution was derived solely from kinetic considerations and is analogous to the statistical Poisson distribution. With increasing Q , i.e., with increasing conversion, this distribution broadens. Since each particle grows proportionally to the number of radicals in it, particles must grow at greatly varying rates if there is broad distribution of radicals among them. It follows that the particle size distribution has to broaden with increasing conversion, contrary to predictions based upon the Smith‐Ewart model. At present it is not yet possible to predict quantitatively the shape of the conversion‐time curve in interval III, the interval following the disappearance of monomer droplets. The reason for this is that the functional dependence of the termination rate constant upon monomer concentration in the particles is not known. However, once the conversion‐time curve is experimentally determined, it is possible to calculate from it the interval III values of Q and of molecular weight.