Average number of radicals per particle

Eigure 6.5 illustrates the mechanisms controlling n. In most emulsion polymerization systems, radicals are produced in the aqueous phase from thermal or redox 

Equation (8) is a simplification because it assumes that all radicals are able to enter the polymer particles, but the radicals directly produced from inorganic initiators are too hydrophilic to be able to enter a hydrophobic phase. On the other hand, the radicals generated from radical desorption (see below) are hydrophobic and able to enter the polymer particles regardless of their length. 

Once inside the polymer particles, the radicals undergo the classical mecha- 

One consequence of the compartmentalization of radicals in the particles is that the overall concentration of radicals in the system is much greater than in solution and bulk polymerization, and hence the polymerization rate is higher. 

Chain-transfer reactions to monomers and chain-transfer agents lead to the formation of small and mobile radicals that can exit the polymer particle. Radical desorption leads to a decrease in the average number of radicals per particle. Equation (10), where is the rate coefficient for radical exit [Eq. (11)] [25], gives the rate of radical exit from a population of particles with an average number of radicals per particle n. In Eq. (11), X is an overall mass-transfer rate coefficient, y/rj the ratio between the rate of generation of small radicals by chain transfer and the rate of consumption of these radicals (mostly by propagation), m the partition coefficient of the small radicals between the polymer particles and the aqueous phase, [M] the concentration of monomer in the aqueous phase, km, the termination rate constant in the aqueous phase, and [R] the concentration of radicals in the aqueous phase. 

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Emulsion Polymerization. Emulsion polymerization takes place in a soap micelle where a small amount of monomer dissolves in the micelle. The initiator is water-soluble. Polymerization takes place when the radical enters the monomer-swollen micelle (91,92). Additional monomer is supphed by diffusion through the water phase. Termination takes place in the growing micelle by the usual radical-radical interactions. A theory for tme emulsion polymerization postulates that the rate is proportional to the number of particles [N. N depends on the 0.6 power of the soap concentration [S] and the 0.4 power of initiator concentration [i] the average number of radicals per particle is 0.5 (93). 

However, the kinetics of PVC emulsion does not foUow the above theory. The rate shows the same increasing behavior with conversion as mass polymerization (94,95). [N depends on [3], but the relationship varies with the emulsifier type (96,97). However, the rate is nearly independent of [N (95). The average number of radicals per particle is low, 0.0005 to 0.1 (95). The high solubiUty of vinyl chloride in water, 0.6 wt %, accounts for a strong deviation from tme emulsion behavior. Also, PVC s insolubiUty in its own monomer accounts for such behavior as a rate dependence on conversion. 

Radical Concentration in Particles. The radical concentration in the particles is also needed to calculate the reaction rates. The average number of radicals per particle was calculated by the O Toole (16) equation which accounts for radical entry, desorption, and termination. 

General. In this section, a mathematical dynamic model will be developed for emulsion homopolymerization processes. The model derivation will be general enough to easily apply to several Case I monomer systems (e.g. vinyl acetate, vinyl chloride), i.e. to emulsion systems characterized by significant radical desorption rates, and therefore an average number of radicals per particle much less than 1/2, and to a variety of different modes of reactor operation. 

Case 1 h < 0.5. The average number of radicals per particle can drop below 0.5 if radical desorption from particles and termination in the aqueous phase are not negligible. The decrease in n is larger for small particle sizes and low initiation rates. 

The time evolution of the average number of radicals per particle, n, is given by ... 

Bartholme t al. (22) found for styrene with persulfate initiation and a sodium alkyl benzene sulfonate emulsifier that there was a discrepancy between their measured value of E (21.7 kJ mol- ) and that calculated (on the assumption that Att = 0) from the expression EN = /5(E - E ). However their value of E (which was derived from measuremlnts of the rate of seeded emulsion polymerization experiments in which N was the same at all temperatures) now seems to be too high probably because the average number of radicals per particle, n < 0.5 at the lower temperatures taking E =32.5 kJ mol-1 as the best estimate, AHg can be calculated rom AHg = /2(E (exp) - E (calc))... 

As is clear from Eq. l,the rate of particle growth (R /Nr) is proportional to the monomer concentration, [M]p and the average number of radicals per particle, n, respectively. Thus, n is one of the basic parameters that characterize the kinetic behavior of particle growth in an emulsion polymerization system. Early researchers devoted their efforts to deriving a quantitative description of n by solving Eq. 3 for n defined by Eq. 2 [4,119,120]. 

SE Interval II begins at the cessation of nucleation, or in hght of the nucleation theory just reviewed, when the particle number becomes relatively constant. Most theories developed for this interval assume a constant particle number and use the quasi-steady-state approximation (QSSA) for average number of radicals per particle. The kinetics and mechanisms of Interval II have been some of the most studied aspects of macroemulsion polymerization. SE Interval II ends when the monomer droplets disappear and the monomer concentration in the particles begins to decrease. 

In the present study we have established that, with radiation initiation at high particle numbers, the Smith-Ewart recursion formula can be simplified even further. By assuming rapid termination in the latex particles where the average number of radicals per particle, rf < 1/2, a rate expression is derived wherein onlv particles containing 0 or 1 radical are significant. 

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