Poisson chain-length distributions

Figure 2. Poisson chain-length distributions, Equation 1, for v= 20 (left PDI = 1.05) and v= 50 (right PDI = 1.02).

Although the extent of BB could be determined with a variety of methods, a newly developed approach based on the use of samples with Poisson distribution was employed. Poisson (number chain length) distributions are characterized by one quantity, namely the location of the peak maximum, Lmax, which is easily accessible from experiment. Furthermore, it was shown by Chang and coworker that anionically prepared polystyrenes approach Poisson distributions. 

Number chain length distribution, n(L), calculated for homogeneous pulsed laser polymerization and Lo = 100. An ideal Poisson distribution centred at the same Lmax value Is included and the positions of the points of inflection are presented by circles and triangles. 

The chain length distribution in the original Ziegler Aujbaureaktion is adjusted by the ratio of ethylene to aluminum in the growth reaction in which the provided ethylene competes for the available Al-centers. After elimination in the second step this results in a Poisson-type distribution (Figure 6.16.1) that has the highest probability at the chain length that corresponds to an even distribution of all ethylene to the coordination sites at the aluminum. 

Assumption 1 treat radical chain-length distribution as if it is fully monodisperse during a TR-SP-PLP experiment and thus ignore Poisson distribution, chain transfer processes and background initiation. Assumption 2 radical chain length can be described by equation 3.3. 

This is a Hory distribution or most probable distribution and much broader (D>1) than the Poisson distribution (D=l) resulting from batch polymerization. The increase in molecular weight polydispersity is due to the fact that the chains are living, and so are directly impacted by the RTD. Recall that the polydispersity of the RTD for a CSTR is 2. For long residence time, approaches 1 and D becomes 2. Thus, the number chain length distribution (NCLD) takes on the breadth characteristics of the RTD, due to the living nature of the polymer chains. For comparison, one should recall that the lifetime of a free radical is 1-10 s. This is insignificant in comparison with the RTD and so that RTD has little to no effect on D in free radical polymerization. 

In the Figure 4 the values of the disoersion obtained from a computer experiment are compared with Poisson and linear dispersions. It can be seen from that figure that the true distribution is much wider than the Poisson distribution and that the width is increased with an increase in chain length. In the initial stage, the dispersion follows the linear approximation. For short chains at high degrees of cross-linkinc the distribution becomes narrower due to the accumulation of chains with many cross-linkages /close to the maximum value/7 and the dispersion tends to that of a Poisson. 

The diversity of the chemical structures possible in these compounds is enormous, thus making it difficult to arrive at precise structure-activity relationships. In many instances commercial preparations are mixtures of surfactants with the mean length or weight of any side group or chain being distributed around a Poisson distribution curve. Thus, there is tremendous variation possible within individual surfactants and mixtures of surfactants, often making it difficult to interpret results. 

This paradigm gives rise to a Poisson distribution of chain lengths (7) ... 

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