The Chain Length Distribution

It is also useful to define the moments of the chain length distribution. The yth... 

A given chain will undergo i-1 propagation steps (each with probability 0) before terminating (with probability 1-0). Thus, if termination is wholly by chain transfer or disproportionation, the chain length distribution is given by cq. 17 (Figure 5.1) ... 

In conventional radical polymerization, the chain length distribution of propagating species is broad and new short chains are formed continually by initiation. As has been stated above, the population balance means that, termination, most frequently, involves the reaction of a shorter, more mobile, chain with a longer, less mobile, chain. In living radical polymerizations, the chain lengths of most propagating species are similar (i.e. i j) and increase with conversion. Ideally, in ATRP and NMP no new chains are fonned. In practice,... 

While it is, in principle, desirable to take the limiting slope of the log CLD plot, in practice the limiting slopes are very susceptible to experimental noise and baseline choice issues. Moad and Moad24 have shown that very little error is introduced by systematically taking the slope over the top 10% or the top 20% of the chain length distribution. The values for the slopes will overestimate ln(0). However, because the discrepancy is systematic, the Mayo" analysis still provides a good estimate for C ( 6% error for the example in Figure 6.1). 

The T containing sequences can be evaluated using expressions analogous to those described in Section 7.3.2.1 to provide the chain end compositions and the chain length distribution. 

Another chromatographic method is pyrolysis/gas-liquid chromatography. The sample is mixed with P205 and heated to 400°C. This technique yields the chain length distribution of the fatty acids initially used [107],... 

Fig. 5. Variation in the chain-length distribution in M. citrifolia suspensions during a typical experimental jet run [130]...

A second important characteristic is the value otj. of the elongation at which rupture occurs. The corresponding values of r/rm show that rupture generally occurred at approximately 80-90% of maximum chain extensibility (12). These quantitative results on chain dimensions are very important but may not apply directly to other networks, in which the chains could have very different configurational characteristics and in which the chain length distribution would presumably be quite different from the very unusual bimodal distribution intentionally produced in the present networks. 

Relation between the Ultimate Strength and the Chain Length Distribution... 

For the transformation of the macrocomposite model to a molecular composite model for the ultimate strength of the fibre the following assumptions are made (1) the rods in the macrocomposite are replaced by the parallel-oriented polymer chains or by larger entities like bundles of chains forming fibrils and (2) the function of the matrix in the composite, in particular the rod-matrix interface, is taken over by the intermolecular bonds between the chains or fibrils. In order to evaluate the effect of the chain length distribution on the ultimate strength the monodisperse distribution, the Flory distribution, the half-Gauss and the uniform distribution are considered. 

All computations so far have been performed with the full width of the chain length distributions, i.e. chains with aspect ratios Lld< 10 have also been included. However, the effects of stress transfer across chain ends and the stress concentrations may become important below this aspect ratio. In the theory by Yoon these effects are neglected. In particular, for the Flory distribution containing a relatively large proportion of very short chains, the effects may be considerable. Therefore, calculations are performed in which the negative effect of the very short rods is approximated by assuming that for an aspect ratio L/d<4 the contribution of these rods to the strength is set to zero. 

In the previous sections the theoretical relations describing the strength of a polymer fibre as a function of the intrinsic parameters, such as the chain modulus, the modulus for shear between adjacent chains, the orientation distribution and the chain length distribution, have been discussed. In this section the dependence of the strength on the time and the temperature will be investigated. 

The CpFe(C0)2 2-NaY adduct is active for syngas conversion. Under the standard conditions the CO conversion is quite similar to that observed for Feo(CO)-j 2-NaY (Table k). However hydrogen conversion is higher and this is reflected in the chain-length distribution which shows a better selectivity for light hydrocarbons (Figure 3). 

These adducts are more active than the iron ones in the conversion of syngas. At 250°C, a higher yield of methane is observed (Table U) and carbon dioxide is produced in smaller amounts. Inspection of Table 5 summarizing the influence of the H2/CO ratio on products selectivity also indicates a higher production of saturated hydrocarbons. This behavior is typical for cobalt catalysts in F-T synthesis (j2,25). The chain-length distribution is similar to that observed for catalysts derived... 

Two different instrumental measurements were used to test the oxidation hypothesis GC and atomic absorption (AA). GC was used to determine the chain length distribution of the EO. From the GC results, an average chain length number was calculated. The gas chromatogram is shown in Fig. 21.1. In addition, headspace GC measurement of the gaseous compounds dissolved in the liquid ethoxylate was made and compared to typical or good product. Atomic absorption spectroscopy was used to determine the levels of several trace metals known to catalyze oxidation of fatty chemicals. 

In industrial reactors, the full equilibration of the chain length distribution is prevented by incomplete mixing, as well as by the residence time distribution, thus resulting in considerable deviations from the equilibrium polydispersity index. These deviations are generally higher for continuous plants than for batch plants and increase with increasing plant capacity as demonstrated in Figure 2.2. 

A useful alternate to the Mayo plot for determining Cs (as well as other transfer constants) is the chain length distribution (CLD) method based on the molecular weight distributions obtained from size exclusion chromatography (SEC) [Christie and Gilbert, 1996, 1997 Heuts et al., 1998, 1999 Moad and Moad, 1996]. The limiting slope AHIGH of a... 

If aU propagation rate coefficients are equal and all termination rate coefficients are equal, we can obtain a simple expression for the chain length distribution, called the Schultz Flory distribution... 

Equilibrium conditions for the chain length distribution are set up in these reactions, the mean chain length n of the anion being greater the higher the temperature used in the preparation and the lower the water... 

From these considerations an important conclusion follows immediately. The structure of a network is insufficiently described if only the number of networks chains and their length distribution is specified. In addition, knowledge of the topography of the chain length distribution or the spatial distribution of the chain length distribution is required. That means that in networks of the same chain length distribution the chains may still be quite differently distributed in space (Fig. 12). 

The application of universal calibration in an investigation of the chain-length distributions in various celluloses by GPC has already been described in an earlier publication (1). In the procedure average DP values are obtained in a manner unlike that normally used in GPC. There... 

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