Nonlinear polymer

Modem synthesis methods have allowed preparation of polymers with a great variety of branching architectures, such as random, comb, uniform and nonuniform star, bmsh, and dendritic [66, 67]. It is understandable that the conformations and hence several properties of these stmctures can vary remarkably among linear polymers of similar compositions and molecular weights. The compression of the branched molecule compared to linear molecules of same molecular weight can be quantified in terms of the factors defined in Eqs. (21) and (22), where the subscripts b and I represent the branched and the linear polymers. 

The factors g and g can often be related, depending on the branching architecture. For example, for star polymers with p equal arms [68, 69], the relationship is given by Eq. (23). 

Variation of polymer melt viscosity with molecular weight, (from C. W. Macosko, Rheology Principles, Measurements and Applications, Copyright 1994 VCH. Used by permission from the publisher, John Wiley Sons, Inc.). 

Such characteristics of several other branched architectures have been examined [70]. The molecular compression behavior by branching allows high loading of the polymeric additives without the excessive viscosity enhancement associated with polymers. As we shall discuss in Section 13.3.6, the rheological effects of 

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Figure C2.1.2. Polymers witli linear and nonlinear chain architectures. The nonlinear polymers can have branched chains. Short chains of oligomers can be grafted to tire main chain. The chains may fonn a. stor-like stmcture. The chains can be cross-linked and fonn a network.

Recommendations on additional aspects of macromolecular nomenclature such as that of regular double-strand (ladder and spiro) and irregular single-strand organic polymers continue to be pubHshed in I ure and Applied Chemistty (100,101). Recommendations on naming nonlinear polymers and polymer assembHes (networks, blends, complexes, etc) are expected to be issued in the near future. 

Dehydration or Chemical Stabilization. The removal of surface silanol (Si—OH) bonds from the pore network results in a chemically stable ultraporous soHd (step F, Fig. 1). Porous gel—siHca made in this manner by method 3 is optically transparent, having both interconnected porosity and sufficient strength to be used as unique optical components when impregnated with optically active polymers, such as fiuors, wavelength shifters, dyes, or nonlinear polymers (3,23). 

Macosko, CW Miller, DR, A New Derivation of Average Molecular Weights of Nonlinear Polymers, Macromolecules 9, 199, 1976. 

Nonlinear polymers are obtained from monomers at least some of which possess a functionality exceeding two. In other words, nonlinear polymers may be defined as those containing units some of which are yolyfunctional, this term being reserved for functionalities exceeding two. Thus, the condensation of glycerol, a trifunctional reactant, with phthalic acid, a bifunctional reactant, yields a nonlinear polymer comprising structures such as... 

If the condensation proceeds far enough, a network structure is easily formed. Similarly, copolymerization of a little divinyl adipate with vinyl acetate yields a nonlinear polymer in which chains of bifunctional vinyl acetate units are bridged, or cross-linked, by the tetrofunctional divinyl adipate units, as indicated on the following page, where the divinyl adipate unit is enclosed between vertical dashed lines ... 

An example of a nonlinear polymer derived by cross-linking an initially linear polymer is afforded by vulcanized natural rubber. In the usual vulcanization procedure involving the use of sulfur and accelerators, various types of cross-linkages may be introduced between occasional units (about one in a hundred) of the polyisoprene chains. Some of these bonds are indicated to be of the following type ... 

Structural units of functionality exceeding four may occur in nonlinear polymers. The terminology set forth above is easily extended to include them. It should be noted further that various polyfunctional units having differing functionalities may occur in the same structure, in the same way that bifunctional units coexist with the polyfunctional units. 

Even among nonlinear polymers, many of the materials of interest are composed of a preponderance of bifunctional units with only a minority of polyfunctional units. This applies to vulcanized rubber where no more than about 1 or 2 percent of the isoprene units are cross-linked. It also applies to the amylopectin fraction of starch which consists of chains composed of an average of about 20 glucose units, these chains being connected to one another by trifunctional units to yield an irregular branched array to wool consisting of poly-... 

As previously indicated, both condensation and addition polymers may be prepared from monomers of functionality exceeding two, with resulting formation of nonlinear polymers. Hence the distinction between linear and nonlinear polymers subdivides both the condensation and the addition polymers, and four types of polymers are at once differentiable linear condensation, nonlinear condensation, linear addition, and nonlinear addition. The distinction between linear and nonlinear polymers is clearly warranted not only by the marked differences in their structural patterns but also by the sharp divergence of their properties. 

Molecular Weight Distributions in Nonlinear Polymers and the Theory of Gelation... 

The critical conditions for the formation of infinite networks will be discussed at the outset of the present chapter. Molecular weight distributions for various nonlinear polymers will then be derived. Experimental data bearing on the validity of the theory will be cited also. 

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