Zero Branched polymers

In both this theory and that presented by Casassa, the differences between linear and branched polymers are ultimately referrable to the fact that the branched polymers have not only a lower mean-square radius for a given MW, but also a higher mean segment density, and a considerably higher central segment density, for a given mean-square radius the importance of this was pointed out by Stockmayer and Fixman in 1953 (2). Otherwise, the two treatments give different reasons for the lower 0Al of branched polymers, for the expression (6.9), used by Casassa, implies that a and b of Eq. (6.12) are zero. 

The viscoelastic properties of solutions of linear and 4-armed star polybutadienes were studied by Osaki and co-workers (117), who compared the storage and loss shear moduli extrapolated to zero concentration with theoretical values. The results for the branched polymer could be accounted for well in terms of the Zimm and Kilb theory (34) a lower value of the hydrodynamic interaction parameter was indicated for the branched polymer than for the linear one, which may be associated with the higher density of polymer segments in the former. 

Graessley s theory, though satisfactory for linear polymers, has not yet been shown to apply to branched polymers. Fujimoto and co-workers (65) attempted to apply it to comb-shaped polystyrenes, but obtained only poor agreement with experiment. They attributed this to the failure of the assumption that the state of entanglement is the same in branched polymers as in linear ones. It is not surprising that this theory fails, for (in common with earlier theories) it predicts that the zero shear-rate viscosity of all branched polymers will be lower than that of linear ones, contrary to experiment. 

The polyethenes prepared with catalyst 2 (Fig. 3a) have greatly elevated elastic modulus G values due to LCB compared to the linear polymers shown in Fig. 3b. LCB also shifts the crossover point to lower frequencies and modulus values. The measured complex viscosities of branched polymers (see also Table 2) are more than an order of magnitude higher than calculated zero shear viscosities of polymers having the same molecular weight but a linear structure. The linear polymers have, in turn, t] (0.02 radvs)... 

Note that this term is proportional to (f — 1) and goes to zero for star-branched polymers (f = 1). 

The zero shear viscosities of these randomly branched polystyrenes were measured and compared with those of linear polystyrenes and it was found that t]0 for all of the branched polymers were far lower than that of linear homologues of the same overall molecular weight. In addition, a scaling of fJo was observed for the first two generations of each branched series of... 

The zero-shear viscoelastic properties of concentrated polymer solutions or polymer melts are typically defined by two parameters the zero-shear viscosity (f]o) and the zero-shear recovery compliance (/ ). The former is a measure of the dissipation of energy, while the latter is a measure of energy storage. For model polymers, the infiuence of branching is best established for the zero-shear viscosity. When the branch length is short or the concentration of polymer is low (i.e., for solution rheology), it is found that the zero-shear viscosity of the branched polymer is lower than that of the linear. This has been attributed to the smaller mean-square radius of the branched chains and has led to the following relation... 

Takahashi, Y. Suzuki, F. Miyachi, M. Noda, L Nagasawa, M. Zero-shear viscosity of branched polymer solutions. Polymer J. 1986, 1968 (18), 1-89. 

Branched polymers with short branches have lower zero shear rate viscosity than linear polymers" of the same molecular weight. Branched polymer with long branches has a higher viscosity than linear polymers. 

The shear viscosity was found to be a decreasing function of shear rate (Fig. 3) but to have an asymptotic viscosity, rjo, at lower shear rates. Fox and Flory [F5] found the zero shear viscosity of rjo of linear polymer chains to depend on the 3.4 power of molecular weight. This was subsequently confirmed by various researchers [B39, F6, G20, P13, P14]. It was at first found that the zero shear viscosity for branched polymers was lower than that for linear polymers of the same molecular weight [S2] however, subsequently Kraus and Gruver [K20] showed that above a molecular weight, characteristic of the molecular topology, rjo for carefully prepared cruciform and Y-shaped polymers actually increases more rapidly perhaps with the sixth power. 

Both linear and side-branched polyethylenes are of technological interest of the 10 million tonnes of polyethylene sold in Europe and in the USA in 1980, 68 per cent was branched and 32 per cent linear polyethylene. As the number of side branches per 100 carbon atoms is increased from zero there are pronounced changes in physical properties (see Chapter 4), but the linear and side-branched polymers are identical in one respect, which is that they may be reversibly heated to melt and then cooled to crystallize time and time again. On melting they how as does a liquid and are thus thermoplastic (or thermal-flow) polymers. In this they are distinguished from cross-linked polyethylene which when heated will not flow. 

Star polyisopienes 6 greater for methyl i-propylketone than for melhyl-isobulylketone whidt conelates with their different < /1 values. (1/3 —X2) is essentially identical in the two solvente. In dioxane the 6 depression is zero appar tly because the (1 /3 -X2) term is complete n igible. It seems, therefore, that dilute solution properties of star-branched polymers can be at least approximated in terms of three parameters p 1,6 and X2- The smoothed density model is known to predict the variation of a and A2 rather poorly for linear polymers so that quantitative agreement for branched polymers is unlikely. 

Similar results are obtained for linear, four and six-branched polyisoprenes at a concentration of 0.145 g/ml. In this case, however, at hi r concentrations more serious deviations occur from theory. The higher molecular weight samples can have zero shear viscosities higher than linear polymers of the same molecular wei t Such behavior was first noted in a study of melt viscosity of regular star-branched polybutadienesViscosities of the order of one hundred times that of a linear equivalent could be observed, but the effect decreased rapidly on dilution with solvents i.e. the viscosities of branched polymers were more sensitive to concentration than those of linear polymers. Star-branched polyisoprenes show viscosity enhance-... 

Equation (49) formulated for blends of linear macromolecules also provides the facility to model blends of linear polymers (index L) and branched polymers (index B) synthesized from the same monomer [28]. If the end-group effects and dissimilarities of the bi- and trifunctional monomers can be neglected, the parameter a becomes zero. This means that the integral interaction parameter is determined by the parameter i lb. i c > the conformational relaxation, in combination with the intramolecular interaction parameters of the blend components. Because of the low values of and the first terms in (47) and (48) can be neglected with respect to the second terms (for molar masses of the polymers that are not too low) so that one obtains the following expression ... 

It is obvious that the conformational relaxation must be proportional to the degree of branching and approach zero upon the transition of the branched polymer to a linear polymer. For the sake of consistency and simplicity, we define the degree of branching, jS, again in terms of intrinsic viscosities (cf. (27)) as ... 

Gelation is a critical phenomenon of connectivity and as such we will use the percolation theory to describe it. As percolation theory was described in detail, here we recall the behaviors of measurable quantities which experimental results are given hereafter. Below the gelation threshold the system is composed of finite size polymers branched in the 3-dimensions of space. We shall call those polymers "polymers clusters" in order to distinguish them from other branched polymers as stars or combed polymers. Below the gelation threshold, the system is viscous at zero frequency. At the gelation threshold, there appears a giant polymer clus-... 

In terms of viscosity, branched polymers have lower zero shear viscosity than linear polymers of the same total molecular weight (Af > Me) when the branches are short (A/j, < Me). When Mb > Me, the branched chain viscosities overtake those of the equally massive linear chains. 

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