AB2 system

Hyperbranched polymers are generally composed of branched (dendritic), Hn-ear, and terminal units. In contrast to AB2 systems, there are two different types of linear units in SCVP one resembles a repeat unit of a polycondensate (----A -b----) and one a monomer unit of a vinyl polymer (--a(B )---). 

Subsequently an AB2-system is assumed for reasons of simplification, even if higher branches are known to the literature. 

The AB2 + AB system is equivalent to AB2 except that AB2 units are separated from each other by AB units. The AB2 + B3 system modifies 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. 

AB2 Systems. Since layered structures are important variants in AB2 systems (e.g. the Cdl2 and CdCb families of structures), it was deemed necessary to always use a Coulomb potential without a damping term. The method chosen to perform the full Coulomb sum was the one suggested by de Leeuw66. Again, the resulting structure candidates in general did not depend on the exact values of the effective potential. The most common structure types found are summarized in table 3. We note that these include many commonly found structures like anatas,... 

Figure 7 Sketch of monolayers of metal particles of different size, (a) represents an AB2 system, (b) shows an AB superlattice...

A similar approach has been applied to many other ABj-type molecules in which an atom with valence p orbitals, or even a group of atoms, is bonded to a central atom A. For example, HjSiOSiH, may be interpreted as an 8-valence electron AB2 system with A = O and B = SiH, (Tossed, 1984b). The -SiHj group interacts with O primarily by cr bonding and contributes one valence electron to the central atom. Thus, we would qualitatively expect the Si-O-Si angle to be bent, as observed. The SiHj O orbital that interacts with the O p orbital is the lo2 MO, which would be purely Si 3p in character for a planar SiH, group, and would acquire some Si 3s and H Lv character in a pyramidal geometry. 

Fig. 1. Molecular orbital diagram for an AB2 system (a) ligands more electronegative than central atom (b) less electronegative.

Fig. 4. Effect of including s orbital interaction on the molecular orbital diagrams for AB2 systems. Dashed lines show where aj orbitals have been shifted, (a) Central atom more electronegative and (b) central atom less electronegative than the ligands.

A further parameter, introduced by Erey et al., allows a comparison to be made of the density of branching of different AB polymers. This average number of branches (ANBs) deviating from the linear direction per non-terminal monomer unit, and was 0.333 for AB2 systems, 0.421 for AB3 systems, and converges to 0.582 for growing x, again for random polymerization and full conversion [57]. 

The maximum possible number of growth directions in AB2 systems is given by... 

In case of permutation symmetry of like atoms as in AB2 systems one can modify the DVR in the 7-coordinate that leads to a modified, symmetrised L-matrix ... 

For AB2 systems, since eqn [1] fails for polymers with high DPn and low DB, leading to DB > 0 for linear, low-molecular-weight polymers. Muller and Yan at the same time developed a systematic equation to calculate the DB for the SCVP. ... 

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