QUANTITATIVE ASPECTS 4-2a Rate of Polymerization in .NET

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4-2 QUANTITATIVE ASPECTS 4-2a Rate of Polymerization
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An expression for the rate of polymerization is obtained by considering rst the rate in a single polymer particle in which propagation is occurring (i.e., a particle containing a radical) and then the number of such particles. At the start of polymerization in a typical system where the concentration of micelles is 1021 L 1 and the initiation rate for the formation of radicals is 1016 L 1 s 1 , a radical diffuses into a micelle every 105 s at the start of interval I. As the system progresses through interval I, this time period decreases sharply, since the concentration of micelles is decreasing. A radical enters each polymer particle on an average of every 10 s during intervals II and III, where N is typically 1017 L 1 . Once inside the micelle or polymer particle, a radical propagates in the usual manner at a rate rp dependent on the propagation rate constant kp and the monomer concentration [M] in the particle.
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The monomer concentration is usually quite high since in many cases the equilibrium swelling of the particle by monomer is of the order 50 85% by volume. Values of [M] as high a 5 M are common. [M] varies only weakly with the size of the polymer particles. Consider now what occurs on the entry of a radical into a particle that already has a radical. For most reaction systems, the radical concentration in a polymer particle is 10 6 M or higher. This is a higher radical concentration than in the homogeneous polymerization systems and the radical lifetime here is only a few thousandths of a second. The entry of
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a second radical into the polymer particle results in immediate bimolecular termination. Thus the polymer particle will have either one or zero radicals. The presence of two radicals in one particle is synonymous with zero radicals, since termination occurs so quickly. The particle is then dormant until another (the third) radical arrives. The particle is again activated and propagation proceeds until the next radical. The cycle of alternate growth and inactivity of the polymer particle continues until the monomer conversion is essentially complete. Compartmentalization (or segregation) of a propagating radical in a polymer particle isolates it from other propagating radicals and allows growth to higher molecular weight than might be achieved in solution polymerization, provided that the rate of entry of another radical from the aqueous phase is suf ciently slow. The rate of polymerization Rp at any instant is given by the product of the concentration of active particles [P] and the rate of propagation in a particle.
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[P] is conveniently expressed by
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where N 0 is the concentration of micelles plus particles, " is the average number of radicals n per micelle plus particle, and NA is the Avogadro number. The use of NA in Eq. 4-3 and in subsequent equations expresses [P] in mol L 1 and Rp in mol L 1 s 1 . Combination of Eqs. 4-2 and 4-3 yields the polymerization rate as
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" " n N 0 n is zero at the start of interval I, since " 0. N 0 decreases, n increases, and the product " N 0 n increases with time during interval I. At the start of interval II, N 0 has reached its steadystate value N. " may or may not reach an absolutely constant value. Behavior D in interval II n usually involves a steady-state " value, while behavior E usually involves a slow increase in " n n with conversion. " will remain approximately constant or increase in interval III although a n decrease will occur if the initiation rate decreases sharply on exhaustion of the initiator concentration. Most texts show Eq. 4-5 for the polymerization rate instead of the more general Eq. 4-4. Equation 4-5 applies to intervals II and III where only polymer particles exist (no micelles). It is during intervals II and III that the overwhelming percent of monomer conversion to polymer takes place. In the remainder of Sec. 4-2, the discussions will be concerned only with these intervals.
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The value of " during intervals II and III is of critical importance in determining Rp n and has been the subject of much theoretical and experimental work. Three cases can be distinguished cases 1, 2, and 3. The major differences between the three cases are the occurence of radical diffusion out of the polymer particles (desorption), the particle size, modes of termination, and the rates of initiation and termination relative to each other and to the other reaction parameters. The quantitative interplay of these factors leading to case 1, 2, or 3 behavior has been discussed [Gao and Penlidis, 2002; Gilbert, 1995; Nomura, 1982;
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