Number-average functionality cross-links

The same relation is found for the end-linking of molecules of low functionality if = 3 or 4) and for the vulcanization of long molecular chains. The second-moment average number of cross-linking sites along the chain, f2, is defined as... 

The mole fraction of the monomer units that are cross-linked in the polymer is X,., and nt is Ihe number-average number of atoms in the polymer backbone between cross-links. The temperature should be expressed in absolute degrees in this equation. The constant K is predicted to be between 1.0 and 1.2 it is a function of the ratio of segmental mobilities of cross-linked to uncross-linked polymer units and the relative cohesive energy densities of cross-linked and uncross-linked polymer (88). The theoretical equation is probably fairly good, but accurate tests of it are difficult because of the uncertainty in making the correction for the copolymer effect and because of errors in determining nf. 

There are two parameters used as a measure of cross-Unk density the number of network chains, v, usually expressed as v/ V, where V is the volume of the unstrained network and the number of cross-links (p) per unit volume, p/F. The relationship between p and v is established by knowing the number of chains starting from a particular cross-linking point, (functionality). The two most important types of network are the tetrafunctional (c ) =z 4) and the trifunctional ( = 3). Another characteristic parameter of a network is the number-average molecular weight between cross-links,... 

In classic coordination chemistry the parameter n is known as the formation function, showing the number of functional groups directly bound with metal ions, whereas in the coordination chemistry of macromolecular complexes this parameter corresponds to the average number of functional groups which are excluded from the total number of ligands for binding of one metal ion. Moreover, for complexation of MX with cross-linked polymers the total concentration of macroligands is constant and does not depend on the dilution of the suspension in a solvent. 

Figure 10-23. Glass transition temperature as a function of the reciprocal number average molar mass for linear,/ = 2, and star-shaped branched poly(styrenes) with/ = 3 or/ = 10-12 arms, as well as for poly(styrenes) cross-linked with divinyl benzene with mean degrees of cross-linking per chain of/= 3, 5-7, or 10-12 (according to data from F. Rietsch, D. Daveloose and D. Froelich).

Difunctional monomers such as DVB or divinyl ether have been found to be efficient in the synthesis of star (co)polymers having a cross-linked core from which homopolymer or block copolymer arms radiate outwards. However, so far only core last method has been reported in cationic polymerization. This method is particularly suited to prepare stars with many arms. The average number of arms per molecule is a function of several experimental and structural parameters. 

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