Continuous distribution of molecular weights

Inhomogeneities in a real network may occur either because of a continuous distribution of molecular weight between crosslinks or due to the regions of different average molecular weights (as may be the case in randomly crosslinked networks). 

Donnelly s Method for MWD s. If one assumes a continuous distribution of molecular weights, then one can write Equation 26 as... 

The proportionality factor G may depend on many characteristics of the solute polymer,but since D is the only polymer-dependent variable in Eq. (2.5), it is reasonable here to treat G as a function of D only for a homologues series of polymers. Thus, for a polydisperse polymer sample with a continuous distribution of molecular weight M, Eq. (2.5) may be generalized as... 

If a mixture contains more than two fractions the treatment can easily be generalized. For example, a trimodal mixture gives four linear isotherm sections ). The Increasing part of fig. 5.29 is replaced by two sections of different slope in the first the lowest molecular weight is displaced from the surface, in the second the middle fraction. For a continuous distribution of molecular weights a rounded isotherm is obtained the adsorption fractionation varies smoothly with the polymer dose. 

The model described by Eqs. 8.13 can be generalized for a continuous distribution of molecular weights to give Eq. 8.15. 

Multiblock OBCs from chain shuttling polymerization have very different architectures. The overall chains and blocks within chains have distributions of molecular weights, with MJMn approaching 2.0. The statistical shuttling process produces chains with a distribution in the number of blocks per chain. The block junctions are precise since each block is grown on a different catalyst, and the compositions are homogeneous since the OBCs are produced at steady-state in a continuous reactor. 

H(x) is a continuous distribution of the logarithms of relaxation times and is called the "relaxation spectnun" by rheologists, whilst the "true" distribution of relaxation times is xH(x). We have reported on Figure 1 the normalized distribution of relaxation times for 4 polystyrene samples with polydispersity indices ranging from 1.05 to 4.2 [2]. It is clear that the distribution of relaxation times broadens with the distribution of molecular weights these features will be anal3 d in terms of molecular models in sections 3 to 6. 

Approximate reaction networks have become customary for modeling reactions in which the species are too numerous for a full accounting or chemical analysis. Lumped components or continuous distributions commonly take the place of single components in process models for refinery streams (Wei and Kuo 1969 Weekman 1969 Krambeck 1984 Astarita 1989 Chou and Ho 1989 Froment and Bischoff 1990). Polymerization processes are described in terms of moments of the distributions of molecular weight or other properties (Zeman and Amundson 1965 Ray 1972, 1983 Ray and Laurence 1977). Lumped components, or even hypothetical ones, are also prevalent in models of catalyst deactivation (Szepe and Levenspiel 1968 Butt 1984 Pacheco and Petersen 1984 Schipper et al. 1984 Froment and Bischoff 1990). 

In a classical free radical chain polymerisation, the slowest step is usually the initiation, for instance in the case of thermal decomposition of a peroxide. In the reaction medium, new radicals are continuously generated, initiating new chains. Growth and termination of chains are very fast, and the active centres are rapidly inactivated, as the termination rate is proportional to the square of radical concentration (Rf = fef[M ] ). Such a reaction is not controlled, resulting in a large distribution of molecular weight of polymers synthesised by classical free radical chain polymerisation. 

Floiy molecutai weight distribution simply because there is a distribution of residence rimes in a continuously fed reactor. This feature is often beneficial, because copolymers with broader distributions of molecular weights are typically easier to process. 

Unlike polymers, characterized by continuous functions of molecular weight distribution, oligomers are characterized by discrete functions. This means that while for polymers the numeric molecular weight distribution (M)AM represents the numeric fraction of molecules with molecular weight from M to M+AM, for oligomers, it represents the numeric fraction of molecules with molecular weight Mj, this fraction corresponds to a certain degree of... 

Such a polymer derives it name from the monomer from which it is usually manufactured. An idealized sample of polymer would consist of chains all having identical molecular weight. Such systems are called monodisperse polymers. In practice, however, all polymers are made up of molecules with molecular weights that vary over a range of values (i.e., have a distribution of molecular weights) and are said to be polydisperse. Whether monodisperse or polydisperse, the chemical formula of the polymer remains the same. For example, if the polymer is polystyrene, it would continue to be represented by... 

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