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Opposing this trend to a slight extent is the small decrease in the concentration of monomer in the particles at higher temperatures Thus, the value of [M] for styrene decreases $15% in going from 30 to 90 C [Smith and Ewart, 1948] The overall activation energy for emulsion polymerization is, thus, a combination of the activation energies for propagation, radical production, and [M] For the few systems studied, the overall activation energies for emulsion polymerization are approximately the same as or less than those for the corresponding homogeneous polymerization [Stavrova et al, 1965] Carrying out an emulsion polymerization requires an awareness of the krafft point of an ionic surfactant and the cloud point of a nonionic surfactant Micelles are formed only at temperatures above the Krafft point of an ionic surfactant For a nonionic surfactant, micelles are formed only at temperatures below the cloud point.



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Emulsion polymerization is carried out below the cloud temperature of a nonionic surfactant and above the Krafft temperature of an ionic surfactant..

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4-3f Molecular Weight and Particle Size Distributions Theoretical considerations indicate that compartmentalization of radicals in polymer particles does not change the polydispersity index PDI X w =X n in emulsion polymerization from its value of 2 in homogeneous polymerization when termination takes place by transfer to monomer, chain-transfer agent, or other substance [Butte et al., 2002a,b; Giannetti et al., 1988; Gilbert, 1995; Lichti et al., 1980, 1982; Mendizabal et al., 2000]. However, emulsion polymerization results in molecular weight broadening when termination involves bimolecular reaction between radicals. While short propagating chains are likely to couple or disproportionate with longer chains in homogeneous polymerization (PDI 1:5 and 2 for coupling and disproportionation, respectively) (Sec. 3-11), any two chains that undergo bimolecular termination in emulsion polymerization are not random. The broadening of PDI in emulsion polymerization is greater for disproportionation than for coupling. For case 2 behavior, coupling of the propagating chain in a polymer particle with the low-molecular-weight entering radical does not greatly affect PDI. Such coupling is equivalent to termination by chain transfer and PDI has a value of 2 compared to 1.5 for homogeneous polymerization. When termination is by disproportionation, PDI has a value of 4 at " 0:5 compared to 2 for n homogeneous polymerization [Butte et al., 2002a,b]. Low-molecular-weight radicals entering the polymer particles disproportionate with propagating radicals and increase the number of low-molecular-weight molecules; X n is decreased while X w is essentially unchanged and X w =X n increases. When " > 0:5 (case 3), the tendency toward molecular weight broadn ening decreases as the size of the radicals undergoing coupling or disproportionation

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Using (9.1 0), (9.11), and (9.16) in (9.9), we find the expected 8 Hamiltonian for the modes of the universe creation and annihilation operators (see Problem 9.2), (9.17) Before tackling our main goal, it is useful to examine .c(k) a bit more closely. As < 1, the function (9.18)

become more nearly the same size PDI tends toward the values in homogeneous polymerization (15 and 2 for coupling and disproportionation, respectively) as " increases n from 05 to 2 The preceding discussion relates primarily to polymerization in stage II of an emulsion polymerization Since this constitutes the major portion of an emulsion polymerization and the reaction conditions (N, Ri , kp , [M]) are relatively constant during stage II, the molecular weight distribution is considerable narrower than in homogeneous polymerization [Cooper, 1974; Lin and Chiu, 1979] However, there is molecular weight broadening with conversion as reaction proceeds through stage III where various reaction parameters are no longer constant Also, the molecular weights produced during stage I are not the same as in stage II The PDI for a batch polymerization taken to complete conversion can be as high as 5 7 [Butte et al.

is bounded in the interval 0 ~ A < 00 and lim~-+oo f(A) Poisson's sum formula, [599] which states that

, 2002a,b], which is still lower than for a typical homogeneous polymerization In addition to the molecular-weight distribution, there is a particle size distribution in emulsion polymerization [Chen and Wu, 1988; Gardon, 1977; Lichti et al, 1982] The particle size distribution (PSD) is expressed, analogously to the molecular weight distribution, as the ratio of the weight-average particle size to number-average particle size (Different particle sizes are calculated depending on whether one uses the particle radius, diameter, or volume as the measure of particle size) The particle size distribution is a consequence of the distribution of times at which different polymer particles are nucleated The polydispersity is maximum during interval I and narrows considerably during the subsequent period.

Then we find that (see Problem 9.3)

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