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where D, L, and T are the mole fractions of dendritic, linear, and terminal units. For high molecular weight polymers, D, L, and T have limiting values of 0.25, 0.25, and 0.50, respectively, and DB is 0.50. (For linear polymers, DB is zero. DB is 1 for dendrimers Sec. 2-16b.) Hyperbranched polymers are similar to linear polymers in that they show molecular weight polydispersity. There is also polydispersity due to constitutional isomerism, resulting from the different placements of monomer units. For example, LXII has many isomers, such as those arising from placement of the lower leftmost AB2 unit at any of the other positions around the perimeter of the molecule. The synthesis of hyperbranched polymers has inherent dif culties which limit molecular weight or shape. One is the occurrence of intramolecular cyclization, which decreases the functional groups available for polymerization. Another dif culty is the steric crowding that occurs within and/or at the periphery of hyperbranched molecules with increasing conversion and molecular size. (This steric crowding effect, discussed extensively for dendrimers, is called de Gennes dense packing [de Gennes and Hervet, 1983].) Although steric crowding may not limit polymer molecular weight (as it does for dendrimers), it will affect molecular shape. Functional groups in some regions become inaccessible for reaction, but growth proceeds in other regions where functional groups are accessible. Steric crowding in a region is relieved by an increase of linear (one unreacted

B group) units and fewer dendritic (no unreacted B groups) units. The polydispersity in molecular weight and isomerism, the intramolecular cyclization, and the molecular shape variations due to steric crowding contribute to the random nature of hyperbranched polymers. Hyperbranched polymers can also be synthesized from the following systems [Jikei and Kakimotoa, 2001]:

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The AB2 AB system is equivalent to AB2 except that AB2 units are separated from each other by AB units. The AB2 B3 system modi es the AB2 system by using B3 as a central core from which polymerization radiates and offers greater control of molecular shape. The A2 B3 system is one of the standard systems used to produce crosslinked polymers (Sec. 2-10). It is useful for synthesizing hyperbranched polymers only when crosslinking is minimized by limiting conversion and/or diluting the reactants with solvent. Hyperbranched polymers can also be synthesized by chain polymerization, ring-opening polymerization, and combinations of ring-opening and step polymerization [Kim, 1998; Voit, 2000] (Secs. 3-6e, 3-15b-4, 3-15b-5, 3-15c, 5-4c).

After World War II, West German yields rose with all other yields Following a statistical gap, they emerged in 1950 at around 6% and fluctuated between 550% and 650% during the 1950 s A wide variety of nominal rates, redemption terms, payment and tax provisions makes detailed yield quotations of little significance in this decade Changes in tax status and the laws governing or controlling the market prevent comparison even from one year to the next There is no doubt, however, that West German yields in the 1950 s were very high in comparison with yields in other Western nations In 1950, West German prime long yields were 355 basis points higher than similar yields in the United States By 1960 the differential had declined to 175 basis points By 1970 the differential had vanished, and in the late 1970 s and 1980 s it became negative.

Dendrimers are dendritic (highly branched) polymers with a more controlled structure than the hyperbranched polymers [Bosman et al, 1999; Fischer and Vogtle, 1999; Grayson and Frechet, 2001; Hecht, 2003; Newkome et al, 1999; Simanek and Gonzalez, 2002; Tomalia, 2001; Tomalia and Frechet, 2002; Vogtle et al, 2000] Other terms have also been used to describe various dendritic polymers arborol, cascade, cauli ower, starburst The synthesis of dendrimers differs greatly from the one-pot proces for hyperbranched polymers Dendrimers are synthesized by more complicated processes involving sequential reactions There are two different methods: divergent and convergent The divergent method starts with a core molecule and builds outward by attaching to the core successive layers (or generations) of monomer units one at a time A large variety of reactions (organometallic as well as organic) have been used in synthesizing dendrimers The divergent method is illustrated in Fig.

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2-22 for the synthesis of polyamidoamine (PAMAM) dendrimers [Tomalia et al, 1990] A repetitive sequence of two reactions are used the Michael addition of an amine to an a,b-unsaturated ester followed by nucleophilic substitution of ester by amine Ammonia is the starting core molecule The rst step involves reaction of ammonia with excess methyl acrylate (MA) to form LXIII followed by reaction with excess ethylenediamine (EDA) to yield LXIV LXV is a schematic representation of the dendrimer formed after four more repetitive sequences of MA and EDA Dendrimers are characterized by the number of generations, the number of layers or shells of repeat units, analogous to the layering of an onion The number of generations is the number of repetitive sequences of the Michael addition and nucleophilic substitution reactions LXIV and LXV correspond to generations 1 and 5, respectively LXIV has three arms or three NH2 groups.

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