Weighting, Poisson

Elamrani et al. 1996] Elamrani, S., Berry, M.B., Phillips Jr., G.N., McCammon, J.A. Study of Global Motions in Proteins by Weighted Masses Molecular Dynamics Adenylate Kinase as a Test Case. Proteins 25 (1996) 79-88 [Elcock et al. 1997] Elcock, A.H., Potter, M.J., McCammon, J.A. Application of Poisson-Boltzmann Solvation Forces to Macromolecular Simulations. In Computer Simulation of Biomoleeular Systems, Vol. 3, A.J. Wilkinson et al. eds., ESCOM Science Publishers B.V., Leiden... 

In Ibis equation the P points R,- correspond to the grid used to solve the Poisson equation for l, i and Wi are weighting factors. 

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... 

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. 

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. ... 

F(x), here, is (Es + AEs)/ Er + A )), as we just noted. In the previous case, the weighting function was the Normal distribution. Our current interest is the Poisson distribution, and this is the distribution we need to use for the weighting factor. The interest in our current development is to find out what happens when the noise is Poisson-distributed, rather than Normally distributed, since that is the distribution that applies to data whose noise is shot-noise-limited. Using P to represent the Poisson distribution, equation 49-59 now becomes... 

As we did in the analysis of Poisson-distributed noise, we compute the expected value of T as the weighted sum of the transmittance described by equation 49-5 (reference [10]) ... 

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. 

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...

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