Branched polymers, determination

Another definition, taking into account polymerization conversion, has been more recently proposed.192 Perfect dendrimers present only terminal- and dendritic-type units and therefore have DB = 1, while linear polymers have DB = 0. Linear units do not contribute to branching and can be considered as structural defects present in hyperbranched polymers but not in dendrimers. For most hyperbranched polymers, nuclear magnetic resonance (NMR) spectroscopy determinations lead to DB values close to 0.5, that is, close to the theoretical value for randomly branched polymers. Slow monomer addition193 194 or polycondensations with nonequal reactivity of functional groups195 have been reported to yield polymers with higher DBs (0.6-0.66 range). 

Relationships between dilute solution viscosity and MW have been determined for many hyperbranched systems and the Mark-Houwink constant typically varies between 0.5 and 0.2, depending on the DB. In contrast, the exponent is typically in the region of 0.6-0.8 for linear homopolymers in a good solvent with a random coil conformation. The contraction factors [84], g=< g >branched/ <-Rg >iinear. =[ l]branched/[ l]iinear. are another Way of cxprcssing the compact structure of branched polymers. Experimentally, g is computed from the intrinsic viscosity ratio at constant MW. The contraction factor can be expressed as the averaged value over the MWD or as a continuous fraction of MW. 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

Molecular Structure. Most starches consist of a mixture of two polysaccharide types amylose, an essentially linear polymer, and amylopectin, a highly branched polymer. The relative amounts of these starch fractions in a particular starch are a major factor in determining the properties of that starch. 

NMR spectroscopy can be utilized to obtain branch content information for commercial polymers in a direct quantitative manner. Polyethylene and PVC are the two commercial polymers that have been studied extensively for branch content determination by this technique [71-75]. Obtaining branch content information from such polymers has been dealt with, by using high-resolution... 

Because of their insolubility, the restricted access of chemical reagents and the influence of the neighborhood on the mobility of chain segments and functional groups of crosslinked polymers, the determination of residual reactive or functional groups in crosslinked polymers is much more difficult than in linear or branched polymers. This is especially true for densely crosslinked polymers prepared from tetrafunctional monomers, such as DVB. 

Rouse mechanism within the tube and the disengagement of the polymer from the tube. For a branched polymer the arm is tethered at one end so this restricts motion. In order for disengagement to occur the arm has to retract itself down the tube. This is the dominant timescale and determines the viscosity. We can think of this as akin to an activation energy process giving rise to an exponential dependence in the viscous process. As yet only qualitative agreement has been achieved. 

In summary, the approach outlined here is a straightforward method for determining representative values of viscosity ratios [ n ] MA /[ h ] LB I certainly g values significantly less than 1.0 are expected for such highly branched polymers (33). However, the anomalous dependence of g (v) on M[v1a suggests that 1) the core/shell hydrodynamic configuration and/or chromatographic artifacts invalidate universal calibration, and/or 2) the LB elution behavior does not conform to that of polystyrene in the assumed, constant manner. Further work is necessary to elucidate these points. 

These equations are general and apply equally for multifunctional reactions such as that of Af with B, or that of Ay with A—A and B B. Depending on which of these reactant combinations is involved, the value of a will be appropriately determined by the parameters r,f, p, and p. For convenience the size distributions in the reaction of equivalent amounts of trifunctional reactants alone, that is, where a p, will be considered. A comparison of Eqs. 2-89 and 2-166 shows that the weight distribution of branched polymers is broader than that of linear polymers at equivalent extents of reaction. Furthermore, the distribution for the branched polymers becomes increasingly broader as the functionality of the multifunctional reactant increases. The distributions also broaden with increasing values of a. This is seen in Fig. 2-17, which shows the weight fraction of x-mers as a function of a for the polymerization involving only trifunctional reactants. 

The transfer constant Cp for chain transfer to polymer is not easily obtained [Yamamoto and Sugimoto, 1979]. Cp cannot be simply determined by introducing the term Cp[P]/[M] into Eq. 3-108. Transfer to polymer does not necessarily lead to a decrease in the overall degree of polymerization. Each act of transfer produces a branched polymer molecule of... 

The architecture of hypeibranched polymers and dendrimers is connected with difficulties in determining molar mass. Many of the common characterization techniques—e.g. size exclusion chromatography (SEC)—used for polymers are relative methods where polymer standards of known molar mass and dispersity are needed for calibration. Highly branched polymers exhibit a different relationship between molar mass and hydrodynamic radius than their linear counterparts. 

In several cases the melt viscosity of a series of lightly-branched polymers has been determined as a function of MW, and compared with that of linear polymers, and it has been found or may be deduced from the published data that there is a cross-over molecular weight, below which the branched polymer is less viscous, but above which it more viscous, than the linear polymer of equal MW. This behaviour is observed with some comb-shaped polystyrenes (35) and poly(vinyl acetate)s (59, 89), star polybutadienes (57, 58, 123), and randomly-branched polyethylenes (56,61). Jackson has found (141) that if the ratio ZJZC of the number of chain atoms at the cross-over point, Zx, to the number at the kink in the log 0 — logM curve, Zc, [as given in Ref. (52)], is plotted against nb, the number of branches, a reasonable straight line is obtained, as in Fig. 5.1. 

It is now generally accepted that the GPC retention volume is a function of the product M tf, independent of the nature or structure of the polymer 46, 47) though Pannell 45) found that it failed to correlate the elution behaviour of his highly branched polystyrenes, it may be accepted that M rf will be determinable from GPC retention volumes for moderately branched polymers. To estimate branching, it is necessary to separate this product so that M and [rf are both known and the relation between them can then be used, subject to the uncertainties mentioned in Subsection 9.2.2, for this purpose. It is usual to measure rf rather than M in order to make the separation, as it is easier. The combination of GPC and intrinsic viscosity measurements is now the most usual method for studying long branching. 

Likewise, the samples can be arranged in the order of decreasing coil size and increasing branching, as determined by g COrr and g", again using the catalyst systems to identify the samples. The most linear polymers are the reference butyllithium samples followed by the nickel-based polymer, butyllithium, alfin, cobalt based, titanium based, and emulsion. The correction to the branching factor for polydispersity makes the nickel-based and alfin polybutadienes less branched with respect to the other polymers examined. 

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