Functionality of a branch point

The functionality of a monomer, oligomer or polymer molecule (or their mixtures) or of a branch point may be included in the scheme of Rule 7 by adding the symbol / and the figure corresponding to the functionality in parentheses after the name of the monomer, oligomer or polymer molecule or branch point. 

Expansion is considered for finite, regular polyethylene stars perturbed by the excluded volume effect. An RIS model is used for the chain statistics. The number of bonds in each branch ranges up to 10 240, and the functionality of the branch point ranges up to 20. The form of the calculation employed here provides a lower bound for the expansion. If the number, n, of bonds in the polymers is heid constant, expansion is found to decrease with increasing branch point functionality. Two factors dictate the manner in which finite stars approach the limiting behavior expected for very large stars, These two factors are the chain length dependence at small n of the characteristic ratio and of fa -a3) / n1/2. 

Infinite network formation becomes possible when the expected number of chains, which will succeed a certain number of chains through branching them, exceeds this number, i.e. if the product ab(f — 1) exceeds unity (/ is the functionality of the branch points). The critical value of the brandling coeffident is thus... 

A second method, using also anionic polymerization techniques was developed to achieve a better knowledge of the functionality of the branch points. In this method the bifunctional living polymer species is prepared under the same conditions as above. It is then reacted with a stoichiometric amount of a plurifunctional electrophilic reagent14,1S. The chief difficulty is to find a reagent the electrophilic functions of which are isoreactive, to ensure that the reaction will go to completion. 

Theoretically, an infinite network consisting of chains of all lengths, from i = 0 to i = oo, can be present, so a, the probability that a given functional group that is part of a branch point leads via a chain of bifunctional units to another branch unit, is simply the sum over all these values, as shown in Equations 5-24. 

Notes f is the functionality of the branch point (i.e., the number of chains attached to branch point) a is the number of unreacted functional groups X is the fraction of polymer in the backbone chain n is the number of branch points per chain n is the average number of branch points per chain is the weight-averaged number of branchpoints per chain n is 3.14159... gs andg4 denote trifunctional and tetrafunctional branch points, respectively subscripts n, w, or z represent the number, weight, and z-averages, respectively. 

Fig. 1.7. A reticulated structure (network) here the functionality of the branching points is equal to four.

Na and Nb are the numbers of A and B groups at the beginning of the polycondensation. The branch molecules consist of A groups. / is the functionality of the branch-point-forming molecules (i.e., not the mean functionality of all the molecules). The total number of monomeric units present is then... 

A nonlinear polymer in which the molecules consist of linear main chain to which there are randomly attached secondary chain branches, viz., low density polyethylene. A fraction of repeat units in a polymer that statistically contain one branch is defined as the branching density X = ab/n, where a is the branching coefficient (dependent on functionality of the branch point), b is the number of branch points, and n is the number of repeat units. 

A branched molecule has branches long (polymeric branch, >10 C-atoms) or short (oligomeric branch, 2...6 C-atoms). It can be distinguished between long-chain branched (LCB) and short-chain branched polymers (SCB). The point on a chain at which a branch is attached is termed as branch point. The functionality / of the branch point is determined by the number of branches which are connected in this point (/-functional branch, e.g. tri-functional, tetra-functional). Usually, short-chain branches affect the properties of the solid polymer (in particular crystallinity) while the long-chain branches have a great... 

If one follows an analytic function around a contour to the initial point and it does not return to the same value, then the function is multi-valued. This is associated with the presence of a branch point within the contour. A branch point is a type of singularity, distinct from a pole or essential singularity. It is not isolated since, as we shall see below, its effects are not localized at one point. The function (z-aY is, for non-integral values of y, a multi-valued function which is of considerable interest in the present context. We will, therefore, outline its properties. In the standard polar representation, it becomes... 

The case of dendrimers illustrates the efforts made to assemble supramolecular structures of nanoscopic scale that are free of any dispersity. The term dendrimers was coined by Tomalia [1] to designate three-dimensional layered arrangements of chemical bonds that arise from the introduction of a branching point at each monomer unit. More precisely, a dendrimer consists of four main components a central core, arms of identical length, linking branching points that are therefore symmetrically distributed in the dendrimer, and end-standing reactive functions. 

When monomers of the type AlA or even greater functionality are involved, the effect of their incorporation into the growing polymer chain is to introduce a branch point into the polymer. 

First of all it is necessary to determine the branching coefficient a, w hich is defined as the probability that a given functional group of a branch unit leads via a chain of bifunctional units to another branch unit. In a polymer of the type shown in Fig. 61, a is the probability that an A group selected at random from one of the trifunctional units is connected to a chain the far end of which connects to another trifunctional unit. As will be shown later, both the location of the gel point and the course of the subsequent conversion of sol to gel are directly related to a. 

Branched and crossiinked condensation polymers are produced when the reaction mixture includes tri-functional monomers as well as bi-functional ones. The incorporation of a single tri-functional monomer into a chain generates a branch point. As we increase the fraction of tri-functional monomers, branching becomes more prevalent and the resulting molecules more complex. When sufficient tri-functional monomers are present we create a three-dimensional crossiinked network. Figure 1.12 shows the general outline of the effects of tri-functional monomers on condensation polymers. 

Figure 19 (a) Peak melting temperature as a function of the branch content in ethylene-octene copolymers (labelled -O, and symbol —B (symbol, ) and -P (symbol, A) are for ethylene-butene and ethylene-propylene copolymers, respectively) and obtained from homogeneous metallocene catalysts show a linear profile, (b) Ziegler-Natta ethylene-octene copolymers do not show a linear relationship between peak melting point and branch content [125]. Reproduced from Kim and Phillips [125]. Reprinted with permission of John Wiley Sons, Inc. 

Different types of LCB are distinguished. Star polymers are the simplest branched polymers because they have only one branch point. Regular star polymers have a branch point with a constant number (functionality,/) of arms and every arm has the same molecular weight. They are therefore monodisperse polymers. Star polymers may also have arms with a most probable distribution [5], Star polymers can also be polydisperse due to a variable functionality. Palm tree [6] or umbrella polymers [7] that contain a single arm with different molecular weight (MW) than the other arms are classified under the asymmetric star [8] polymers, see Figure 3.2. 

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