Chains distribution of lengths

For preparative purposes batch fractionation is often employed. Although fractional crystallization may be included in a list of batch fractionation methods, we shall consider only those methods based on the phase separation of polymer solutions fractional precipitation and coacervate extraction. The general principles for these methods were presented in the last section. In this section we shall develop these ideas more fully with the objective of obtaining a more narrow distribution of molecular weights from a polydisperse system. Note that the final product of fractionation still contains a distribution of chain lengths however, the ratio M /M is smaller than for the unfractionated sample. 

Using both condensation-cured and addition-cured model systems, it has been shown that the modulus depends on the molecular weight of the polymer and that the modulus at mpture increases with increased junction functionahty (259). However, if a bimodal distribution of chain lengths is employed, an anomalously high modulus at high extensions is observed. Finite extensibihty of the short chains has been proposed as the origin of this upturn in the stress—strain curve. 

If the initiation reaction is much faster than the propagation reaction, then all chains start to grow at the same time. Because there is no inherent termination step, the statistical distribution of chain lengths is very narrow. The average molecular weight is calculated from the mole ratio of monomer-to-initiator sites. Chain termination is usually accompHshed by adding proton donors, eg, water or alcohols, or electrophiles such as carbon dioxide. 

FIG. 3 Probability distributions of chain lengths at four temperatures (given as parameter) [28]. In the inset, oscillations of the MWD due to the formation of rings are shown for T = 0.2. 

Example 13.5 Determine the instantaneous distributions of chain lengths by number and weight before and after termination by combination. Apply the quasi-steady and equal reactivity assumptions to a batch polymerization with free-radical kinetics and chemical initiation. 

For long linear chains the second condition is supported by the Stockmayer bivariate distribution (8,9) which shows the bivariate distribution of chain length and composition is the product of both distributions, and the compositional distribution is given by the normal distribution whose variance is inversely proportional to chain length. 

Theoretically, if each molecule in a polymer sample were to be linked to two of its neighbors, a single highly branched molecule would form that would encompass the whole sample. In practice, due to the statistical distribution of chain lengths and the random incorporation of crosslinks, the situation is far more complex. 

When surfactant molecules contain more than one distribution, for example, a distribution of chain lengths in the hydrophobic and hydrophilic portions, two-dimensional liquid chromatography (2DLC) is a very powerful method for complete analysis. One can get the full quantitation of the distribution of molecular size by using the 2DLC technique. For example, take a surfactant molecule like alcohol ethoxylates (AE s) having a general structure of... 

Parallel to the development of the new theoretical approaches considerable experimental work was done on model networks especially synthesized, to show the effects of pendent chains, loops, distribution of chain length, functionality of crosslinks, etc. on properties (5-21). In some instances, the properties turned out... 

A Gaussian distribution of chain lengths about the mean of 34 monomer units (from the ellipsometric results on the growth). 

For the same values of ea g, r0 and u0 as for the monodisperse distribution, Figs. 45 and 46 show the results for the Flory distribution of chain lengths. The curves start at degrees of polymerisation determined by zn=[(2a)-1+l]. A comparison of Fig. 45 with Fig. 42 shows that, for a diameter equal to the chain di-... 

Fig. 45 Ultimate strength of PpPTA fibres versus the degree of polymerisation applying the Flory distribution of chain lengths for various values of the diameter 2 r calculated with Eq. 93...

VC=C n the yellow solution. In both cases the experimental data are quite well represented considering that there is only one parameter in the fit. The model also adequately describes the Raman excitation profile. The good agreement between the model and the experimental results confirms the basic premise of the model that the individual polymer chains in solution contain a distribution of chain lengths which determine the absorption characteristics of the solutions. 

The length and the distribution of chain lengths are functions of the temperature, pressure, residence time, catalyst characteristics, and the proportion of ethylene present in the reaction, A measure of this is the mole ratio of ethylene, which measures the weight of ethylene compared to the weight of triethyl aluminum in scales related to their atomic weights. As an example, Table 15-2 shows how the distribution of chain lengths can vary, using different mole ratios of ethylene to triethyl aluminum. 

The hydrocarbon base is petroleum derived and does, in fact, contain a distribution of chain lengths with the predominant species being Ci2- In addition, there can be a greater or lesser degree of chain branching. The... 

Figure 12 illustrates some catalysts that form ethylene oligomers with a Schulz-Froly distribution of chain lengths. Complexes F12-1-F12-7 afford almost exclusively a-olefms, " while complexes F12-8 and F12-9 yield a mixture with inner olefins. 

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