Zero-one system

Polymer particles contain at most one radical (zero-one system). 

The bimolecular termination reaction in free-radical polymerization is a typical example of a diffusion controlled reaction, and is chain-length-depen-dent [282-288]. When pseudobulk kinetics appUes, the MWD formed can be approximated by that resulting from bulk polymerization, and it can be solved numerically [289-291]. As in the other extreme case where no polymer particle contains more than one radical, the so-caUed zero-one system, the bimolecular termination reactions occur immediately after the entrance of second radical, so unique features of chain-length-dependence cannot be found. Assuming that the average time interval between radical entries is the same for all particles and that the weight contribution from ohgomeric chains formed... 

In a zero-one system in which a radical has just entered a polymer particle containing one polymer radical, and is terminated instantaneously, the number fraction distribution in the absence of a polymer transfer reaction is given by ... 

Assuming a simple zero-one system during Interval II, a model analysis was conducted to clarify the conditions needed to form bimodal MWD through the effects of limited space [311]. 

The instantaneous MWD of the primary chains formed during Interval II in a zero-one system (assuming combination termination) is given by the most probable distribution, whose number fraction distribution is given by ... 

Analytic solutions for Eq. (5) provide the most direct path of the prediction of PSD evolution. For batch polymerizations in Interval II, however, analytic solutions have only been achieved for the so-called zero-one system (Lichti et al., 1981). These are systems wherein negligibly few particles contain two or more free radicals because of the rapidity of the bimolecular termination reaction (e.g., in styrene emulsion polymerizations with small latex particles). In this case, Eq. (5) may be written as follows ... 

The analytic solution derived by Lichti el of. is applicable to any zero-one system in Interval IL with no restrictions on the rate coefficients except that th be independent of time (but may depend on size). Although a zero-one description is applicable to some important systems, it is not universally valid. For systons where it exceeds 4. general analytic solutions have yet to be developed, and we now examine methods for solving Eq. (5) for Systems of arbitrary n. 

PSD in Interval II hatch reactions, the results of a sample calculation are provided (see Fig. 1). In this calculation for a zero-one system, it was assumed that p = 1, fe = 0. K = I and c > p, the units being aibitrarily, but appropriately, chosen. The initial condition selected, which is typical of a s cd emulsion polymerization system, was that... 

Lin and Chiu (1979) developed a theory for the MWD in a zero-one system that correctly predicts (where M is continuous rather than discrete) the free-radical lifetime distribution Ni leading to a value of 2 for P. The formalism of Lin and Chiu involves calculating the number of chains containing m monomer units. Their final results involve sequence summations over the range K m < oo. 

Two types of systems are identified in emulsion polymerization the pseudo-bulk system and the zero-one system... 

The term zero-one designates that all latex particles contain either zero or one active free radical. The entry of a radical in a particle that already contains a free radical will instantaneously cause termination. Thus, the maximum value of the average number of radicals per particle, n, is 0.5. In a zero-one system, compartmentalization plays a crucial role in the kinetic events of emulsion polymerization processes. In fact, a radical in one particle will have no access to a radical in another particle without the intervention of a phase transfer event. Two radicals in proximity will terminate rapidly however, the rate of termination will be reduced in the process because of compartmentalization, as the radicals are isolated as separate particles. Consequently, the propagation rate is higher and the molecular weight of the polymer formed is larger than in the corresponding bulk systems. Which model is more appropriate depends primarily on the particle size. Small particles tend to satisfy the zero-one model, as termination is likely to be instantaneous. ... 

At high monomer conversion, the viscosity inside the polymer particles increases sharply and further polymerization becomes diffusion controlled. The particles are referred to as a glassy polymer and the kinetics for a zero-one system is no longer valid. To account for these changes, the propagation rate coefficient can be expressed as follows... 

S-E cases 1 and 2 correspond to what is known as zero-one systems, in which the radicals grow in isolated compartments, reaching very high molecular weights hence, this characteristic feature of emulsion polymerization is known as compartmentalization. In case 3, this characteristic is relaxed so that radicals in a given particle grow in the presence of other radicals. As more radicals coexist within the particles, the system approaches the behavior of a bulk polymerization (or pseudobulk system). 

A zero-one system is one where entry of a radical into a particle whidi already contains a growing radical causes termination at a rate much more rr id than that of overall polymerization. In such a system, n (the average number of radicals per particle) carmot exceed 0.5 (a zoo-one system is thus similar to the more traditional Smith-Ewart Cases 1 and 2 combined [2]). Termination occurs only between an entoing radical and a radical which has been growing for some time, and by the definition of the system, is not rale detomining. For this reason polymerization in a zero-one system will usually be quite different, both in properties and product, from that in an equivalent non-compartmentalized (bulk or solution) polymerization. 

The instantaneous MMD can be found from a knowledge of the rate coefficients for chain-stopping events, which are transfer (to monomer, polymer and chain-transfer agent) and termination, and for chain growth, which is propagation. In addition, since in a zero-one system entry into a particle which already contains a growing chain results (by definition) in instantaneous termination, entry is also a chain-stopping event in a zero-one system. 

In a zero-one system, the instantaneous number MMD is simply given by... 

Reliable values for the entry rate coefficient can be obtained in a number of ways [1]. The simplest is to look at the approach to steady state in a zero-one system (although this limits the range of particle size and initiator concentration that can be studied). The kinetics are then controDed only by the rate coefficients for entry and exit, since by the very definition of a zero-one system, termination within the particle is not rate-determining. If a chemical initiator is used, then in principle... 

Some results for a zero-one system are shown in Figure 5.12, and for a pseudo-bulk system in Figure 5.13. It is essential to be aware of the problem of... 

Figure S.12 Instantaneous MMD (arbitrary units) obtained by successive substration of the cumulative MMD for a styrene zero-one system at 50 °C (unswollen seed radius 44 nm, initiator 10 mol dm persulfate) recalculated from data in [57]. The range of Wp (as a percentage) for each sample is shown. The uncertainty is high at low and high M. The value of n at the start of Interval III is 0.3...

Theoretically, the description of the evolution of the CLD for an emulsion polymerization process is complex as not only the number of radicals has to be tracked per particle but also the chain lengths of these radicals. For FRP, however, the computational effort can be reduced in case a so-called zero-one system is obtained, in which the polymer particles either contain no radicals or only one radical at a given time. 

As discussed above, the MWD depends on the number of radicals per particle. Smith and Ewart [28] distinguished three limiting cases In Case 1 0.5 in Case 2 n = 0.5 and in Case 3 it 0.5. In Cases 1 and 2, the probability of having particles with more than one radical is almost negligible, and hence the system may be considered to be formed by particles with no radicals and particles with one radical (zero-one system). In Case 3, the average number of radicals is large and the kinetics is close to bulk polymerization. 

In a zero-one system, the inactive polymer chains are formed in particles containing one radical by chain transfer to monomer and by instantaneous termination upon entry of one radical. 

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