Randomly distributed branching point

SYSTEMS WITH RANDOMLY DISTRIBUTED BRANCHING POINTS... 

For heterogeneous polymers with larger, uniform numbers of randomly distributed branch points per molecule with a random distribution of branch lengths, they derived Eqs. 221 and 2.22 for tri- and tetra-functional branch points, respectively ... 

Eor m randomly distributed branch points in each molecule of a sample that has been fractionated by molecular weight, i.e., that is monodisperse, Zimm and Stockmayer [8] showed that for larger values of m, average values of g can be approximated by Eq. 2.25 for a functionality of 3 and by Eq. 2.26 for a functionality of 4. 

Fig. 18. Reciprocal particle-scattering factors of star-molecules with polydisperse rays, where f denotes the number of rays per molecule. The same functions are obtained also for the ABC-type polycondensates, where nb denotes the number of branching points per molecule. The case f = 1 or nb = 0 is identical to linear chains obeying the most probable lengths distribution. It also represents the scattering behaviour of randomly branched f-functional polycondensates 1...

Universal scaling curves for the molar mass distribution of randomly branched polyesters with many different samples in each class. The filled symbols have Nq = 2 monomers between branch points and correspond to critical percolation in three dimensions. The open symbols have Nq — 900 monomers between branch points and obey the mean-field percolation model. Data of C. P. Lusignan ei al., Phys. Rev. E 52, 6271 (1995) 60, 5657(1999). 

As an illustration of the Rouse model, consider the polydisperse mixture of polymers produced by random branching with short chains between branch points. The molar mass distribution and size of the branched polymers in this critical percolation limit were discussed in Section 6.5. Close to the gel point, some very large branched polymers (with M> 10 ) are formed and the intuitive expectation is that such large branched polymers would be entangled. However, recall that hyperscaling requires... 

If the primary chains possess a molar mass distribution, then the probability of finding a tetrafunctional branch point in a primary molecule has to be calculated. Every primary molecule—even the very smallest—must have at least one tetrafunctional branch point for network formation. Since the branch points are assumed to be randomly distributed, large primary molecules will consequently have more than one branch point. The expectation of finding one branch point per primary molecule thus depends on both the average size of the primary molecules and the number of monomeric units that carry branch points. If this number is too small, then below a given size of the primary molecules it is impossible for all the primary molecules to be cross-linked to a network. The mass fraction (wm)i, not the mole fraction (Xm)h must therefore be used for the fraction of cross-link-carrying monomeric units. 

Since branch points are randomly distributed in amylopectin, and since starch consists of amylopectin and amylose, starch is likewise not a single substance, but a mixture of compounds. The amylose content of various starches is generally about 15-25%, but can approach 34% (lily bulbs), or even 67% (steadfast pea). Amylose can be separated from amylopectin by precipitation from an aqueous solution with butanol or by dissolution in liquid ammonia. 

Fig. 1. Graft Copolymers (1) random graft copolymer (identical branches randomly distributed along the backbone) (2) regular graft copolymer (identical branches equally spaced along the backbone) (3) simple graft copolymer (3-miktoarm star copolymer) and (4) graft copolymer with two trifunctional branch points. Exact graft copolymers.

Some of the principles as well as problems involved in the melting of random copolymers are found in olefin type copolymers. The melting temperatures of a large number of random type ethylene copolymers, as determined by differential scanning calorimetry, are plotted as a function of the mole percent branch points in Fig. 5.11. The samples represented in this figure are either molecular weight and compositional fractions or those with a narrow composition distribution with a most probable molecular weight distribution.(74) These samples were crystallized and heated rapidly. In this set of data there are two different copolymers that contain ethyl... 

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