Average number of free radicals per particle

Figure 1. Polydispersity index of the polymer produced in Interval II of an emulsion polymerization terminated solely by combination as a function of the average number of free radicals per particle...

M0 = molecular weight of monomer, n = average number of free radicals per particle. 

Figures 7 and 8 show typical particle size distributions for vinyl acetate emulsions produced in a single CSTR. A large number of particles,are quite small with 80 to 90% being less than 500 A in diameter. The large particles, though fewer in number, account for most of the polymer mass as shown by the cumulative volume distributions. Data are also presented on Figures 7 and 8 for the number of particles based on diameter measurements (N ), the average number of free radicals per particle, and the steady state conversion.

These iV values relate directly to the rate of polymerization through the average number of free radicals per particle, n = iiV (with the normalization condition = 1). The Smith-Ewart equations describing the time evolution of are... 

A detailed critique of the validity of Eq. (27) is given in Section III. In brief, two criteria are required for this validity, both of which are well satisfied in ordinary emulsion polymerization systems. These are (i) that the time required for formation of a single polymer chain be much less than that over which significant changes occur in the rate coefficients governing the MWD and (ii) that the average number of free radicals per particle of size steady-state value. 

We will consider the MWD in two simple cases. The first is when chain transfer is sufficiently rapid to ensure that all other chain-stopping events can be ignored. In such a situation, whereas the compartmentalized nature of the reaction may affect the rate of initiation of new chains, it will not affect the lifetime distributions of the chains once they are formed. The MWD may then be found from the bulk formulas, provided only that the average number of free radicals per particle, is known. Such an approach has been used by Friis et al. (1974) to calculate the MWD evolved in a vinyl acetate emukion polymerization. These authors included in addition the mechanisms of terminal bond polymerization and of transfer to polymer (both of which cause broadening). The formulas required for the in corporation of these mechanisms could be taken from bulk theory. 

The zero-one-two model accurately describes systems wherein n (the average number of free radicals per particle) does not exceed 0.7 it is thus applicable to small-particle styrene (Hawtett et al., 1980), vinyl acetate lUgelstad and Hansen, 1976), and vinyl chloride (Phis and Hamielec, 1975) emulsion polymerizations. In tbe Interval II steady state, the analytic solutions to the various functions involved in the MWD ohiained by Lichti et a . 1980) are as follows in these expressions, the dependence on suppressed since a steady state is assumed... 

The basic tenet of the Smith-Ewart Case 2 particle growth model is that each particle will contain an active free radical one half of the time. Thus, the average number of free radicals per particle is 0.5. The rate of polymerization is given by... 

It is generally accepted that the number of latex particles per unit volume of water, the average number of free radicals per particle (n = 0.5), and the concentration of monomer in the particles are constant for emulsion polymerization systems that follow the ideal Smith-Ewart Case 2 kinetics. As a result, a constant reaction rate period can be observed during emulsion polymerization. Monomer molecules must be transferred from the gigantic monomer droplets to the growing submicron latex particles to supply the reaction. A dynamic balance between the rate of consumption of monomer in the latex particles and the rate of diffusion of monomer molecules from the monomer droplets to the particles may thus be established, and this results... 

Beyond Interval II, the second maximal polymerization rate can be attributed to the gel effect. The bimolecular termination reaction becomes diffusion-controlled in the latex particles and the average number of free radicals per particle increases signihcantly in the latter stage of polymerization, thereby leading to an acceleration of the free radical polymerization. The rate of polymerization then decreases continuously toward the end of polymerization due to the depletion of monomer and/or the diffusion-controlled propagation reaction in the reaction loci. 

The Smith and Ewart-Stockmayer-O Toole treatments [48-50] (see Chapter 4) that are widely used to calculate the average number of free radicals per particle (n) are based on the assumption that the various components of the monomer-swollen latex particles (e.g., monomer, polymer, free radicals, chain transfer agent, etc.) are uniformly distributed within the particle volume. A latex particle in emulsion homopolymerization of styrene involves uniform distribution of monomer and polymer within the particle volume except perhaps for a very thin layer near the particle surface. In the case of free radicals, this uniform distribution would only hold in a stochastic sense. However, as illustrated in Eq. (8.1), free radicals are not distributed uniformly in the latex particles when water-soluble initiators are used to initiate the free radical polymerization. The assumption of uniform distribution of free radicals in the latex particles would be valid only if the particles are very small or chain transfer reactions are the dominate mechanism for producing free radicals. If such a nonuniform free radical distribution hypothesis is accepted, the very basis of the Smith and Ewart-Stockmayer-O Toole methods might be questioned. Despite this potential problem, the Stockmayer-O Toole solutions for the average number of free radicals per particle have been used for kinetic studies of many emulsion polymerization systems. The theories seem to work reasonably well and have been tested extensively with monomers such as styrene. 

Figure 8.2. Profiles of the calculated average number of free radicals per particle (n, or n ) as a function of monomer conversion for the experiment at SCC with the weight fractions of polystyrene seed particles (154nm in diameter), the second-stage monomer (methyl methacrylate), and the weight fraction of initiator (potassium persulfate) equal to 0.06, 0.09, and 9.62 X 10 , respectively.

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