Molecular weight Poisson

This relation was verified experimentally7 49 and it was shown that the degree of polymerization in a system containing "living polymers is independent of concentrations of initiator or monomer and of temperature. Furthermore, if all the growing centers were formed in a time much shorter than the time of polymerization, a Poisson molecular weight distribution would be obtained. Indeed, by using this technique samples of polystyrene were obtained for which MjMn = 1.04. 

Termination is negligible although not entirely eliminated. A "wrong monomer addition gives, e. g., urea derivatives which cease to grow. The lack of termination leads to the relation DPn of the polymer = Monomer/Initiator, and for a not too slow initiation the resulting product has a Poisson molecular weight distribution. 

Living polymers resulting from an instantaneously initiated but non-terminated polymerization, have a nearly Poisson molecular weight distribution, provided that Mo > Me. The polymerization seems to cease as the concentration of the residual monomer attains its equilibrium value - no further conversion of the monomer into polymer could be detected at that stage of the reaction. Nevertheless, the system is not yet in its ultimate equilibrium state. 

A Poisson molecular weight distribution occurs only if the following requirements are fulfilled 1) the rate of initiation is much faster than the rate of polymerization 2) propagation occurs by addition of the monomer to the polymer chain end and 3) there is no termination, chain transfer or any other secondary reaction. 

Electron-transfer initiation, developed in the course of our studies, is extremely fast and therefore all the growing centers are formed simultaneously before they had a chance to grow by more than a few monomeric units. In a living polymer system termination or chain transfer is avoided and under such conditions the resulting macromolecules have the Poisson molecular weight distribution . 

Figure 2.5 shows plots of Mw/Mn vs. conversion for polymerization systems possessing a Poisson molecular weight distribution (eqn (2.30)) and three distributions predicted by eqn (2.31) (P = kd/kp[l]f) with kpjk = 5, 2, and 0.5 L moU For the system with slow exchange kpjk = 5 L moU ), the final distribution at p=1.0 is 1.05, significantly broader than Poisson. For A p/ (j = 0.5LmoU the distribution of the reversible-deactivation polymerization is very nearly Poisson at high conversions but deviates from Poisson at low conversions. For p/ d<0-5L mol , eqn (2.31) predicts a distribution that is more narrow than Poisson, especially at low conversions. 

Figure 2.5 M fM conversion for polymerization systems possessing a Poisson molecular weight distribution (eqn (2.30)) and three distributions predicted by eqn (2.31) (p = kJkp I]o) with kjkp = 5, 2, and 0.5 mol L . For all systems, [I]o = 0.01 molL h...

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

That the Poisson distribution results in a narrower distribution of molecular weights than is obtained with termination is shown by Fig. 6.11. Here N /N is plotted as a function of n for F= 50, for living polymers as given by Eq. (6.109). and for conventional free-radical polymerization as given by Eq. (6.77). This same point is made by considering the ratio M /M for the case of living polymers. This ratio may be shown to equal... 

The Debye temperature, can be calculated from the elastic properties of the solid. Required are the molecular weight, molar volume, compressibility, and Poisson s ratio.11 More commonly, do is obtained from a fit of experimental heat capacity results to the Debye equation as shown above. Representative values for 9o are as follows ... 

The catalytic single-step Alfen process has a good space-time yield, and the process engineering is simple. The molecular weight distribution of the olefins of the single-step process is broader (Schulz-Flory type of distribution) than in the two-step Alfen process (Poisson-type distribution) (Fig. 2). As a byproduct 2-alkyl-branched a-olefins also are formed, as shown in Table 6. About... 

The molecular weight distribution of the obtained homologous fatty alcohols corresponds with a relatively narrow Poisson distribution and equals the molecular weight distribution of the olefins in the two-step Alfen process [38a]. 

Furthermore, the reaction scheme implies that the molecular weight distribution is Poisson-like — i.e. very narrow — as it had been shown earlier on theoretical basis by Flory 8), Gold 9), and Szwarc l0>. Even though two (or more) types of active species add monomer at very different rates, the polydispersity remains narrow, provided solvation/desolvation and ionic dissociation/association processes are fast U). 

The distinguishing feature of such a mechanism occurs in the fact that the growth of all polymer molecules proceeds simultaneously under conditions affording equal opportunities for all. (This will hold provided the addition of monomer to the initiator is not much slower than succeeding additions.) These circumstances are unique in providing conditions necessary for the formation of a remarkably narrow molecular weight distribution—much narrower than may be obtained by polymer fractionation, for example. Specifically, they are the conditions which lead to a Poisson distribution of the number and mole fraction, i.e. ... 

Thermal initiation and ordinary bimolecular termination also occur during polymerization in addition to initiation by the dissociation of the adduct or the active polymer chain-end dissociation and reversible temination (formation of the dormant species). Therefore, the degree of the control of the molecular weight and the molecular weight distribution is determined by the ratio of the polymer chains produced under control and uncontrol. If the contribution of the thermal initiation and bimolecular termination is very small, the molecular weight distribution is close to the Poisson distribution, i.e., Mw/Mn=1 + 1/Pn, where Pn is the degree of polymerization. It was shown that when the number of... 

Ethene oligomerisation. In view of the above limitations there is a demand for a process that selectively makes linear 1-alkenes. Three processes are available, two based on aluminium alkyl compounds or catalysts and one on nickel catalysts. The aluminium processes use aluminium in a stoichiometric fashion and they produce a narrow molecular weight distribution (a Poisson distribution, vide infra). 

Initially the polymer molecular weight distribution obeys a Poisson distribution, typical of a chain growth reaction without chain transfer. Since the reactions are reversible, at a later stage, also the equilibration between the polymers becomes important and a broad distribution of molecular weights is obtained. As can be seen from Figure 16.5 the presence of linear alkenes causes chain termination (chain transfer) and thus low molecular weights are produced if the cycloalkenes are not sufficiently pure. 

The effects of exchange depends on the relative acidities of the alcohol (or other protonic substance) and the polymeric alcohol. The exchange reaction occurs throughout the course of the polymerization if the acidities of the two alcohols are approximately the same. The polymerization rate is unaffected while the molecular weight decreases (Eq. 7-15), but the molecular weight distribution (MWD) is Poisson. 

Here, v is Poisson s ratio which is equal to 0.5 for elastic materials such as hydrogels. Rubber elasticity theory describes the shear modulus in terms of structural parameters such as the molecular weight between crosslinks. In the rubber elasticity theory, the crosslink junctions are considered fixed in space [19]. Also, the network is considered ideal in that it contained no structural defects. Known as the affine network theory, it describes the shear modulus as... 

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