Molar mass Poisson

Reaction mechanisms and molar mass distributions The molar mass distribution of a synthetic polymer strongly depends on the polymerization mechanism, and sole knowledge of some average molar mass may be of little help if the distribution function, or at least its second moment, is not known. To illustrate this, we will discuss two prominent distribution functions, as examples the Poisson distribution and the Schulz-Flory distribution, and refer the reader to the literature [7] for a more detailed discussion. 

Already in the study of linear chain molecules it has become evident that the shape of the molar mass distribution and its width provide a valuable guide to the mechanism of chain formation. Best known are the most probable (or Schulz-Flory) distribution and the narrow Poisson distribution. The former is often... 

The molar mass distribution of branched materials differ most significantly from those known for Hnear chains. To make this evident the well known types of (i) Schulz-Flory, or most probable distribution, (ii) Poisson, and (iii) Schulz-Zimm distributions are reproduced. Let x denote the degree of polymerization of an x-mer. Then we have as follows. 

Fig. 19. Weight fraction molar mass distributions w(x) of the Schulz-Zimm type for various numbers of coupled chains in a double logarithmic plot. Note fory=l the Schulz-Zimm distribution becomes the most probable distribution in the limit of/ l the Poisson distribution is eventually obtained. In all cases the weight average degree of polymerization was 100. The narrowing of the distribution with the number of coupled chains is particularly well seen in the double logarithmic presentation...

Many addition polymerization reactions with very low concentrations of impurities have propagation rates much faster than initiation rates and have essentially no termination. Such reactions produce narrow molar mass distributions that can be approximated by the Poisson distribution. Comparison of the polydispersity index of anionically polymerized butadiene with Eq. (1.69) is shown in Fig. 1.20. 

In the past decade, several new approaches for controlled NCA polymerization based on the classical primary amine initiation have been reported. In 2003, Dimitrov and Schlaad reported the controlled ( ammonium mediated ) polymerization of ZLL-NCA at elevated temperature using primary amine hydrochloride salts as initiators (Figure 4.3). The initiator reactivity most likely is due to the formation of a small amount of free amine by reversible dissociation of HCl. This equilibrium is strongly shifted toward the dormant amine hydrochloride species. Consequently, as soon as a free amine reacts with an NCA, the resulting adduct is immediately protonated and prevented from further reaction. The presence of protons in the system suppresses formation of unwanted NCA anions ( activated monomers ). The obtained polypeptide blocks exhibit a very narrow, close to a Poisson, molar-mass distribution. 

The Poisson distribution written in the bottom boxed equation of Fig. 3.31 approaches the binomial molar mass distribution for the case that p is small and the average kinetic chain length v = No/N is large, a condition approached for most hving polymers. It is an equation that can be calculated much more easily than the binomial distribution. Analyzing the Poisson distribution, one finds that polymerizations without termination lead to rather narrow molar mass distributions as discussed in Sect. 3.3 with Figs. 3.39 1. 

Apart from the use of uniform (or almost uniform) standards, other methods for determining the BB function have been developed. For example, by assuming a uniform and Gaussian BB function with a linear molar mass calibration, it is possible to use the mass and molar mass chromatograms for simultaneously estimating the standard deviation of the BB function and the calibration coefficients.Alternatively, if the shape of the MMD is known (e.g., it is a Poisson distribution on a linear molar mass axis), then the BB function can be estimated from the difference between the (mass or molar mass) chromatogram and its theoretical prediction in the absence of BB. Finally, the BB function can be theoretically predicted from a representative fractionation model. " Unfortunately, however, this approach is so far unfeasible due to the difficulty in determining the associated physicochemical parameters. 

Maximum extension ratio of a single chain of mass Me Maximum extension ratio of a chain strand of molar mass Mg in an entanglement network Poisson ratio Entanglement density Polar coordinate angle Polymer density Stress... 

Important differential mass-distribution functions (probability density function of mass-distribution) are the most probable distribution (Schulz-Flory), the Schulz-Zimm distribution, the Poisson distribution, Tung distribution, and logarithmic normal distribution (Wesslau distribution) [08IUP2]. Methods for the determination of distribution functions of molar mass are listed in Table 4.1.4. 

Fig. 1.3. Molar mass distributions of the Schulz-Zimm type for /3 = 2 left) and of the Poisson type right). Both correspond to the same number average degree of polymerization, A n = 10 ...

Fig. 2.8 Weight-fraction Poisson distribution of chain lengths for various values of x in a polymerization without termination. For comparison, the broken line represents the most probable distribution of molar mass when x = 11 (after Flory).

The samples obtained under such conditions exhibit a Poisson-type distribution of their molar masses. This type of distributions is obtained when one distributes in a random way m objects in n boxes, with m>>>n. 

In the case of controlled free radical polymerizations (see Section 7.5.8), the initiation period is short compared to that of propagation and all the chains created grow simultaneously at the same rate. The polymer formed under such conditions exhibits a narrow dispersity of molar masses corresponding to a Poisson distribution (see Section 8.4). 

It is a consequence of Equation (2.10) that the Mw/Mn values for an AB-block copolymer should be smaller than the values normally observed for A and B homopolymers with molar mass comparable to the blocks provided the block copolymerisation reaction proceeds in a similar manner to the homopolymerisation. The vast majority of the Mw/M data presented in the literature is based on SEC measurements. In fact SEC is problematic for the characterization of very narrow MMDs. For homopolymers the axial dispersion phenomenon is the main problem, whereas for block copolymers it is also questionable to what extent true noninteracting conditions are accessible. A development has started towards the use of alternative techniques to SEC for the characterization of diblock copolymers. Apart from the popular MALDI-TOF mass spectroscopy various newer chromatographic techniques have been used. A series of PS samples prepared under as identical conditions as possible (/CHX/sBuLi/45 C/ZCHsOH/) were analysed by SEC and TGIC and the measured Mw/M values compared with the Poisson distribution predictions. 

The living polymerization of strained three- and four-membered monomers typically provides polymers with a narrow molar mass distribution, best described by the Poisson function [91], for which the dispersity indexes (D = DP jDP = M /M ) assume values in the range -1.25 > D > 1, depending on the polymer chain length (Equahon 1.24). As discussed earlier, the polymerization of these monomers is essentially irreversible. 

---