Molecular Weight Distribution of Living Polymers

In the sense of this definition, the associated species discussed above can also be called living, even if they do not participate in a propagation reaction, but they are in equilibrium vdth the active non-associated species on a much shorter time scale than the overall reaction. For simplicity, in the following discussion the association is not considered explicitly, but if it is known quantitatively it can be considered by making the rate coefficients a function of [P-Li]. 

For fast initiation (k kp), all initiator is transferred to growing chains immediately, so monomer conversion and the kinetic chain length are given by Eqs. (21)-(23). 

The resulting frequency distribution h(i) of chain lengths i can be derived by solving the system of differential equations sequentially starting with [Pi]t=o = [f]o and the definition of the kinetic chain length given above [Eq. (24)]. 

The average number- and weight-average degrees of polymerization and the poly-dispersity for this Poisson distribution are given by Eqs. (26)-(28). 

however, k kp does not hold, meaning if the initiation rate coefficient is in the order of kp or even less, there is no immediate conversion of initiator to growing chains and the simplification of Eq. (23) becomes invahd and must be replaced by Eq. (19). The solution for this general case is given in Refs. 58 and 59. From Eqs. (15) and (16), the variation of monomer conversion as a function of initiator conversion is given by Eq. (29) with r = kp/ki 1. 

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Y. Rouault, A. Milchev. Monte Carlo study of molecular weight distribution of living polymers. Phys Rev E 55 2020, 1997. 

Eq. (17) predicts that, when Pn is 100, the polydispersity is equal to 1.01, so that the polymer is virtually monodisperse. However, such ideal monodisperse polymers have scarcely been synthesized. The lowest values of polydispersity (Mw/Mn = 1.05-1.10) have been attained in homogeneous anionic polymerization 43). Gold 41) calculated the polydispersity of a polymer in the living polymerization with a slow initiation reaction and showed that the value of Pw/Pn increases slightly to a maximum (1.33) with an increase in polymerization time, followed by a decrease toward 1.00. Other factors affecting the molecular weight distribution of living polymer have been discussed in several papers 5S 60). 

Lyssy, T. Mechanism of formation and molecular weight distribution of living polymers. Helv. Chim. Acta. 42, 2245 (1959). 

Szwarc, M., and M. Litt Molecular weight distribution of "living" polymers. Part II. Effect of impurities. J. Phys. Chem. 62, 568 (1958). 

Brown, W. B., and M. Szwarc Molecular weight distribution of living polymers. Trans. Faraday Soc. 54, 416 (1958). 

The molecular weight distribution can be calculated by solving the mass balance equations for monomer(s), initiator (catalytic sites), and polymeric species with different chain lengths. When quasisteady state assumption is applied to live polymers or propagating active centers, the molecular weight distribution of live polymers is often represented by the Schultz-Flory most probable distribution. However, the calculation of the chain length distribution of dead polymers is in general quite complicated. For some special cases such as... 

Fig. 23A-C. Isobutylene polymerization at — 25 °C. GPC traces of PIBs obtained A as a function of time B initiator ED ratio at the same time C at similar conversion. Numbers at peaks indicate molecular weight distributions of living polymers. H20 C and L/C -ED represent polymers produced by initiation by water, nonliving ionic species, and living complexed species, respectively...

Tappe, R., and F. Bandermann, Molecular Weight Distribution of Living Polymers in Semibatch Reactors, Mfliramo/. Chem., 160, 117, 1988. 

The equilibrium between monomer and living polymer is dynamic and therefore the molecular weight distribution of the polymer will change with time until the equilibrium distribution is reached. This is a peculiar process in which the amount of polymer present in the system, as well as its number average molecular weight is constant. This means also that, the number of polymeric... 

The intrinsic viscosities of the polymers prepared in tetrahydrofuran increased throughout the experiment. This system thus exhibits some of the aspects of living polymerization—that is, catalyst activity over an extended period, and increasing viscosity average molecular weights with added amounts of monomer. The rather broad molecular-weight distributions of these polymers, however, differentiates this system from that of the classical case in which polymerization proceeds in the complete absence of a termination process. 

Terminations in tetrahydrofuran polymerizations can depend upon the choice of the counterion, particularly if the reaction is conducted at room temperature. In many reactions the chain continues to grow without any considerable chain termination or transfer. This produced the term living polytetrahydrofuran. Thus, in polymerizations of tetrahydrofuran withPFe or SbFe counterions, the molecular weights of the products can be calculated directly from the ratios of the initiators to the monomers. The molecular weight distributions of the polymers from such polymerization reactions with PFe and SbF6 , however, start out as narrow, but then broaden. This is believed ... 

A variety of aspects concerned with living polymers, e.g. effect of impurities, the molecular weight distribution in living polymer systems, monomer-living polymer equilibria, etc., were treated during the years that followed. The problems of termination were reviewed in an article in Adv. Polymer Sci. 2, 275 (1960), and a paper delivered during the Wiesbaden Symposium in 1959 and published in Makromol. Chem. 35, 132 (1960)describes the state of art achieved in this field at that time. 

The kinetics of free radical polymerization and the molecular weight distribution of the polymer were already discussed in Section 1.6.2 of Chapter 1. To improve the chemical and mechanical properties of the polymer great efforts were undertaken a number of years ago to achieve narrow distributions. This is possible with anionic or cationic — so-called living — polymerization, in which chains can not terminate or transfer and grow at a rather uniform rate, thus yielding a polymer with a polydispersity close to one. This type of polymerization requires very special operating conditions and high purity of the feed, however. 

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