Branching molar mass distribution

Various authors reported on such phase equilibria in supercritical solutions of linear PE or LDPE in ethylene (see [90-94]). It is unfortunate, however, that in many cases essential information on the molecular characteristics of the samples used is missing (degree of branching, molar mass distribution) which renders quantitative treatment of the measured phase diagrams impossible. 

Keywords. Solution properties. Regularly branched structures. Randomly and hyperbranched polymers. Shrinking factors. Fractal dimensions. Osmotic modulus of semi-di-lute solutions. Molar mass distributions, SEC/MALLS/VISC chromatography... 

The chemical constraint reduces the number of possible reactions considerably, and consequently it leads to a much narrower molar mass distribution. Furthermore, the extent of reaction a of the A-group can cover all values from zero to unity, but the extent of reaction P of the equally reactive 5-groups cannot become larger than P=a/(f-l). One important consequence of this strict constraint is that gelation can never occur [1,13]. A much higher branching density than by random polycondensation can be achieved. For this reason one nowadays speaks of hyperbranching. 

These observations require a detailed explanation. After several unsuccessful attempts a satisfying answer was finally found. A first step was made by the ingenious derivation of the molar mass distributions of randomly branched or randomly cross-linked materials [14]. The equation, that was later rederived by Elory [13], will be given in the next section. Here it suffices to point out that the width of the distribution, or the polydispersity index MJM , increases asymptotically with the weight average degree of polymerization... 

The molar mass distribution of branched materials differ most significantly from those known for Hnear chains. To make this evident the well known types of (i) Schulz-Flory, or most probable distribution, (ii) Poisson, and (iii) Schulz-Zimm distributions are reproduced. Let x denote the degree of polymerization of an x-mer. Then we have as follows. 

The Schulz-Zimm distribution would be found for/end-to-end coupled linear chains which obey the most probable distribution, as well as for/of such chains which are coupled onto a star center. This behavior demonstrates once more the quasi-linear behavior of star branched macromolecules. In fact, to be sure of branching, other structural quantities have to be measured in addition to the molar mass distribution. 

The architecture dependence is also demonstrated in Fig. 33 by the factors of several star macromolecules, flexible cychc chains. Randomly and hyper-branched materials show a more complex behavior because of the large width in the molar mass distribution. Table 5 gives the actual values. The plot of Fig. 33 shows nicely how for a large number of arms the factor for hard spheres is approached. 

The dissolved polymer molecules are separated on the basis of their size relative to the pores of a packing material contained in a column. The chromatograms can be converted to molar mass distributions, average molar masses, Mn, M, and M, long-chain branching and its distribution. 

Fig. 1. Weight fraction Wn of backbone polymer with n branches for most probable molar mass distribution of the mother polymer parameter is the average number of branches in one mother molecule, Nt. The points in the curves denote Wn at n = 0,1,... 8...

Fig. 5. Weight fraction Wfen of graft copolymer with n branches for the most probable molar mass distribution. Parameter is the average number of brandies in one graft copolymer molecule, Ng. The points in the curves denote Wgn at n = 1, 2,..., 15...

The species -O-N-R in the scheme is a stable organic radical called a nitroxide, one type of radical that does not react with itself but which reacts with carbon radicals forming weak C-O-N bonds. This approach continues to develop and allows the synthesis of polymers with very narrow molar mass distributions, block copolymers, and polymers with different architectures such as highly branched materials (discussed shortly). This is but one of a limited number of techniques that show promise for producing improved thermoplastics, elastomers, and adhesives for packaging and automotive applications (Anon. 2002). 

Very early reports on these systems described them as polycondensates, consisting of broad molar-mass distributions with randomly branched topologies. The methods of synthesis included Friedel-Crafts coupling of benzyl alcohols [108] and the polymerization of 2,5,6-tribromophenol involving aryl ether formation [109], In addition, hyperbranched natural carbohydrate polymers, such as amylopectin, dextrin, and glycogen have been extensively studied [73-75]. 

The structural complexity of synthetic polymers can be described using the concept of molecular heterogeneity (see Fig. 1) meaning the different aspects of molar mass distribution (MMD), distribution in chemical composition (CCD), functionality type distribution (FTD) and molecular architecture distribution (MAD). They can be superimposed one on another, i.e. bifunctional molecules can be linear or branched, linear molecules can be mono- or bifunctional, copolymers can be block or graft copolymers, etc. In order to characterize complex polymers it is necessary to know the molar mass distribution within each type of heterogeneity. 

Size exclusion chromatography is the premier polymer characterization method for determining molar mass distributions. In SEC, the separation mechanism is based on molecular hydrodynamic volume. For homopolymers, condensation polymers and strictly alternating copolymers, there is a correspondence between elution volume and molar mass. Thus, chemically similar polymer standards of known molar mass can be used for calibration. However, for SEC of random and block copolymers and branched polymers, no simple correspondence exists between elution volume and molar mass because of the possible compositional heterogeneity of these materials. As a result, molar mass calibration with polymer standards can introduce a considerable amount of error. To address this problem, selective detection techniques have to be combined with SEC separation. 

Figure 9 Compared molar mass distributions of regular and star-branched bromo-butyls. (From Ref. 65.)...

Random branching and gelation 6.4.5 Molar mass distribution... 

The structure of the branched polymers produced by any random branching process is the same. Any individual hyperbranched polymer structure made from reacting ABy i monomers can also be made by reacting Ay monomers. The difference between these branching processes is the molar mass distribution—the relative amounts of each structure produced. 

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). 

The number of monomers in the characteristic branched polymer iV and its size are symmetric around the gel point e = 0). The molar mass distribution is in fact similar for the same s above and below the gel point (within the framework of mean-field scaling). 

The universality class of the randomly branched polymers can be determined by constructing a universal molar mass distribution plot, like the ones shown in Fig. 6.26. First, the number density distribution function n(p, N) is determined from the concentration and weight-average molar... 

Randomly branched polymers are of enormous importance for certain polymer processing operations, such as blow moulding and film blowing. The molar mass distribution of all randomly branched polymers is described by percolation models. For commercial randbmly branched polymers, the critical percolation model applies only very close to the gel point. The branching chemistry used in commercial randomly branched polymers is usually stopped far short of the critical region. While critical percolation does not apply to these polymers, the mean-field percolation model does a superb job of describing the molar mass distribution of randomly branched commercial polymers. 

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... 

Heterogeneity is a generic quality of polysaccharides All characteristics occur as distributions and may be handled as distributions of molar fractions, referring to the number of distributed components, or distributions of mass fractions, referring to mass contributions of distributed components. In particular, for broad distributions, the difference between mass and molar distribution becomes significant and sometimes crucial. Molar mass distribution is a central piece in this puzzle of correlating molecular characteristics with polysaccharide performance. Additionally, optional branching characteristics, substitution patterns, and responses of aqueous polysaccharide systems to different kinds of applied stress need to be determined. 

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