Chain length cumulative

Assume that this is the controlling resistance so that U=h. A kinetic model is needed for Rp and for the instantaneous values of and Iw The computer program in Appendix 13 includes values for physical properties and an expression for the polymerization kinetics. Cumulative values for the chain lengths are calculated as a function of position down the tube using... 

The data cumulatively lead to the following general conclusions. Within homologous series of n-alkaneamines, the pattern of affinity for cucurbituril is one of gradual enhancement as chain length is increased until a maximum is... 

The values of fa were calculated by Equation 4 for all ranges of the cumulative fraction. The results of the calculation enabled us to compare the characteristics of GPC performances at two different times. In this treatment any difference in the GPC performance—e.g., resolution—was expressed in terms of the correction factor, ft, to the chain length, Ai. 

The only, but essential, use we make here of Eq. (11.19) is to extract the properties of the functions W u) etc. Note that by construction these functions are independent of (L, M), of momenta, and of chain lengths. Writing down Eq. (11.19) for three different choices of (L,M) we therefore have a system of three equations linear in W, 1/V, t/, which allows us to express these functions in terms of renormalized cumulants, Three important results immediately follow. 

An isothermal batch reactor produces a fairly narrow MWD except possibly at high conversions. Conditions will vary from the start to the end of a batch reaction, and the MWD will broaden. The dynamic chain length of a free-radical polymer will because of changes in monomer concentration and initiation rate, although usually not by much when measured by the cumulative In and Iw of the finished polymer. When the catalyst is heterogeneous, an isothermal batch reactor will produce... 

Cumulative values for the chain lengths are calculated as a function of position down the tube using differential forms for Equations 13.14 and 13.15 ... 

Instantaneous chain length is statistical, but the cumulative value can be considerably broader because of changes in relative rates of propagation and termination. 

Free-radical photopolymerizations (see Chap. 10) of multifunctional acrylic monomers result in cross-linked polymeric networks. The kinetic picture of such polymerizations varies from ordinary linear polymerization because the diffusion of free radicals and functional groups becomes severely restricted. This causes growing polymer chains to rapidly cyclize and cross-link into clusters (microgels). The clusters become linked up into networks. Many free radicals become trapped, but terminations take place by combinations and by chain transferring. The cumulative chain length in such polymerizations can be calculated from the following equation [125] ... 

When the initiation rate constant is increased, the polymerization rate also rises because of the higher population or active radicals. The final conversion, however, drops. This is due to an increase in active radicals and results in lower kinetic chain length. The cumulative chain length, v, is the number of propagation steps that are carried out be a free radical. It can be calculated from the following equation... 

Principle of the random selection of a reaction based on cumulative reaction probabilities (Eq. 10.52) for macromolecular reactions in an additional step the involved chain lengths are selected here the random number t2 is 0.7 reaction 4 is selected. 

Wcxun(r) weight-fraction distribution of cumulative chains Wp(r) chain length distribution of primary chains u>s sol fraction X monomer conversion Xa functional group A conversion Xb functional group B conversion Vf free-voltune fraction a stoichiometric imbalance ratio... 

Some authorities prefer to represent molecular weight distributions in cumulative (integral) form rather than the differential form used here. Cumulative distributions are the number or weight fraction of material in the sample with chain length < x (or molecular weight < M), so that the maximum value of the y-axis is 1. 

FIGURE 9.5 Instantaneous and cumulative number-average chain lengths versus conversion for an isothermal, free-radical, batch polymerization (data of Example 9.10). 

RELATIONS BETWEEN INSTANTANEOUS AND CUMULATIVE AVERAGE CHAIN LENGTHS... 

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