Poisson polymers

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. 

The integration of Eq. (6.106) is central to the kinetic proof that living polymers follow Poisson statistics. The solution of this differential equatior is illustrated in the following example. 

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 behavior of the strain softened material resembles the behavior of rubberlike polymers. For instance, the Poisson s ratio of an ideally plastic material is also close to 0.5 [94, 95], Proper understanding of crack propagation involves the microscopic level. Apparently, the load is transmitted by the molecular strands [97] from one crosslink to the next crosslink, exactly, as it is in rubberlike materials. However, two things are different in strain softened polymers as compared to rubberlike materials ... 

Sjoerdsma S.D., Bleijenberg A.C.A.M., and Heikens D., The Poisson ratio of polymer blend, effects of adhesion and correlation with the Kemer packed grain model. Polymer, 22, 619, 1981. 

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. ... 

When retention ordering can be established, the theoretical peak capacity could be effectively utilized in a multidimensional separation system in a far more efficient manner. However, one is reminded that with the exception of synthetic polymers and a few other special cases of small molecules, real samples have almost random retention time distributions. It is rare when the free energy, enthalpy, and entropy of interaction are determined in LC for molecules utilized in retention mechanism studies. However, the retention energetics have been determined in GC studies by Davis et al. (2000) who found that many complex samples will exhibit Poisson distributions of retention times due to a Poisson distribution in enthalpy and a compensating distribution in entropy. 

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. 

Anisotropic materials have different properties in different directions (1-7). 1-Aamples include fibers, wood, oriented amorphous polymers, injection-molded specimens, fiber-filled composites, single crystals, and crystalline polymers in which the crystalline phase is not randomly oriented. Thus anisotropic materials are really much more common than isotropic ones. But if the anisotropy is small, it is often neglected with possible serious consequences. Anisoiropic materials have far more than two independent clastic moduli— generally, a minimum of five or six. The exact number of independent moduli depends on the symmetry in the system (1-7). Anisotropic materials will also have different contractions in different directions and hence a set of Poisson s ratios rather than one. 

A second type of anisotropic system is the biaxially oriented or planar random anisotropic system. This type of material is illustrated schematically in Figure 2A. Four of the five independent elastic moduli are illustrated in Figure 2B in addition there are two Poisson s ratios. Typical biaxially oriented materials are films that have been stretched in two directions by either blowing or tentering operations, rolled materials, and fiber-filled composites in which the fibers are randomly oriented in a plane. The mechanical properties of anisotropic materials arc discussed in detail in following chapters on composite materials and in sections on molecularly oriented polymers. 

What is the Young s modulus of the polymer in Problem 1 if Poisson s ratio is 0.35 ... 

Another effect o(f orientation shows up as changes in Poisson s ratio, which can be determined as a function of time by combining the results of tension and torsion creep tests. Poisson s ratio of rigid unoriented polymers remains nearly constant or slowly increases with time. Orientation can drastically change Poisson s ratio (254). Such anisotropic materials actually have more than one Poisson s ratio. The Poisson s ratio as determined when a load is applied parallel to the orientation direction is expected to... 

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... 

Nonsolvent bath, polymer precipitation by immersion in, 15 808-811 Nonspecific elution, in affinity chromatography, 6 398, 399 Nonstationary Poisson process, in reliability modeling, 26 989 Non-steady-state conduction, 9 105 Nonsteroidal antiinflamatory agents/drugs (NSAIDs) 21 231 for Alzheimer s disease, 2 820 for cancer prevention, 2 826 Nonsulfide collectors, 16 649 Nonsulfide flotation, 16 649-650 Nonsulfide mineral flotation collectors used in, 16 648-649t modifiers used in, 16 650, 651t Nonsulfide ores, 16 598, 624... 

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. 

Here the first term is the usual diffusive current, with Dc being the usual cooperative diffusion constant of the polymer molecule. The second term is a convective current due to the presence of induced electric field arising from all charged species in the system, p is the electrophoretic mobility of the polymer molecule derived in the preceding section. From the Poisson equation, we obtain... 

Gas compression in closed-cell polymer foams was analysed, and the effect on the uniaxial compression stress-strain curve predicted. Results were compared with experimental data for a foams with a range of cell sizes, and the heat transfer conditions inferred from the best fit with the simulations. The lateral expansion of the foam must be considered in the simulation, so in subsidiary experiments Poisson s ratio was measured at high compressive strains. 13 refs. 

When there is no volume change, as when an elastomer is stretched, Poisson s ratio is 0.5. This value decreases as the Tg of the polymer increases and approaches 0.3 for rigid solids such as PVC and ebonite. For simplicity, the polymers dealt with here will be considered to be isotropic viscoelastic solids with a Poisson s ratio of 0.5, and only deformations in tension and shear will be considered. Thus, a shear modulus (G) will usually be used in place of Young s modulus of elasticity E Equation 14.2) where E is about 2.6G at temperatures below Tg. 

Carswell and Nason assigned five classifications to polymers (Figure 14.8). It must be remembered that the ultimate strength of each of these is the total area under the curve before breaking. The soft and weak class, such as PIB, is characterized by a low modulus of elasticity, low yield (stress) point, and moderate time-dependent elongation. The Poisson ratio, i.e., ratio of contraction to elongation, for soft and weak polymers is about 0.5, which is similar to that found for many liquids. 

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