Number of branches

The number of branches in HDPE resins is low, at most 5 to 10 branches per 1000 carbon atoms in the chain. Even ethylene homopolymers produced with some transition-metal based catalysts are slightly branched they contain 0.5—3 branches per 1000 carbon atoms. Most of these branches are short, methyl, ethyl, and -butyl (6—8), and their presence is often related to traces of a-olefins in ethylene. The branching degree is one of the important stmctural features of HDPE. Along with molecular weight, it influences most physical and mechanical properties of HDPE resins. 

Investigation has shown that chain transfer to polymer occurs predominantly on the acetate methyl group in preference to the chain backbone one estimate of the magnitude of the predominance is 40-fold (92,93). The number of branches per molecule of poly(vinyl acetate) polymerised at 60°C is ca 3, at 80% conversion. It rises rapidly thereafter and is ca 15 at 95% conversion and 1-2 x lO" number-average degrees of polymerisation. 

In order to improve the physical properties of HDPE and LDPE, copolymers of ethylene and small amounts of other monomers such as higher olefins, ethyl acrylate, maleic anhydride, vinyl acetate, or acryUc acid are added to the polyethylene. Eor example, linear low density polyethylene (LLDPE), although linear, has a significant number of branches introduced by using comonomers such as 1-butene or 1-octene. The linearity provides strength, whereas branching provides toughness. 

Niobium-Tantalum Niobium and tantalum form solid-solution alloys which are resistant to many corrosive media and possess all the valuable properties of the pure metals. This could have great practical value since in a number of branches of technology it might permit the replacement of pure tantalum by a cheaper alloy of niobium and tantalum. Miller" and Argent" reported data on the resistance of the niobium-tantalum system, but the tests were only carried out under mild conditions and the data have only limited significance. However, Gulyaev and Georgieva and Kieffer, Bach and Slempkowski carried out tests at elevated temperatures and their work indicated that the corrosion rates of the alloys are substantially that of tantalum provided the niobium content does not exceed 50%. 

It will be seen that where there are a number of branches from a main duct, there will be an excess of available pressure in these... 

A common way of visualizing such spaces is through a kind of hierarchical tree (see figure 7.3), whose leaves are identified with states and the distances between leaves - measured by counting the number of branches that must be traversed in... 

Similar arguments as in Sec. 39 lead to the following result Two centric trees are congruent if and only if the same number of branches originates from their centers and the configurations of these branches are equivalent with respect to the group associated with the corona of the center. 

Examining the behavior of equation (4.3) beyond the circle of convergence lx >= p / we find that in any region of the open unit disk the continuation of r(x) is algebroid, that is, it has only finitely many branches and finitely many algebraic branch points. The number of branches might increase to infinity if the region increases to the open disk. 

The Kd(X) values thus obtained (Table 3) were analyzed by the multivariant technique using such parameters as n, a0, Bt, Ibmch, and Ihb, where Bj is a STERIMOL parameter showing the minimum width of substituents from an axis connecting the a-atom of the substituents and the rest of molecule, and Ibrnch, an indicator variable representing the number of branches in a substituent. 

Upon formulating these relationships, phenols with branched alkyl substituents were not included in the data of a-cyclodextrin systems, though they were included in (3-cyclodextrin systems. In all the above equations, the n term was statistically significant at the 99.5 % level of confidence, indicating that the hydrophobic interaction plays a decisive role in the complexation of cyclodextrin with phenols. The Ibrnch term was statistically significant at the 99.5% level of confidence for (3-cyclo-dextrin complexes with m- and p-substituted phenols. The stability of the complexes increases with an increasing number of branches in substituents. This was ascribed to the attractive van der Waals interaction due to the close fitness of the branched substituents to the (3-cyclodextrin cavity. The steric effect of substituents was also observed for a-cyclodextrin complexes with p-substituted phenols (Eq. 22). In this case, the B parameter was used in place of Ibmch, since no phenol with a branched... 

The number of branches of star molecules made this way cannot be precisely predicted. Accurate molecular weight measurements are necessary to determine the average... 

Graft copolymers are those in which a homopolymer backbone has a number of branches, which are themselves homopolymers of another monomer, grafted on ... 

SDIBS polymers have been characterized by a variety of methods. The average number of branches BR in our earlier publications, B here to conform with other publications) in the SDIBS core was determined by selective link destmction (see Figure 7.7) using the following equation ... 

Our latest efforts have been to concentrate on investigating the architecture of the DIB core. As discussed before, the average number of branches per molecule (B) determined by selective link destmction and equilibrium swelling showed good agreement with the kinetic B (Equation 7.2). However, branching analysis by SEC proved to be a challenge. 

From the topology of branched systems with trifunctional branchpoints, for any given molecule the number of branched units is equal to the number of terminal unit minus one. Thus, Eq. 1 can be further simplified to... 

X (number of branched units) (total number of units) - 1... 

The neurons from which NTs are released number more than 7 billion in the human brain. Each (Fig. 1.2) consists of a cell body, the soma or perikaryon, with one major cytoplasmic process termed the axon, which projects variable distances to other neurons, e.g. from a cortical pyramidal cell to adjacent cortical neurons, or to striatal neurons or to spinal cord motoneurons. Thus by giving off a number of branches from its axon one neuron can influence a number of others. All neurons, except primary sensory neurons with cell bodies in the spinal dorsal root ganglia, have a number of other, generally shorter, projections running much shorter distances among neighbouring neurons like the branches of a tree. These processes are the dendrites. Their... 

The degree of branching by transfer with polymer obviously will increase with the conversion since the relative incidence of branching must depend on the ratio of polymer to monomer in the system. To examine the matter from the point of view of reaction rates, let 6 represent the fraction of monomer molecules which have polymerized out of a total of iVo in the system, and let v represent the total number of branches. (At variance with the definition used elsewhere. No is the total number of units polymerized and unpolymerized as well.) The rates of generation of branches and of polymerization can then be written... 

Another parameter coinranly used for characterizing long chain branching is the number of branch points (or frequency) per 1000 repeat units. The number average number of branch points can be calculated across the chromatogram from... 

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