Contraction factor

Since DN(UV-- -XY)Z = N(DUV- - -XY)Z, the theorem is proved for n + 1 factors. This lemma can be generalized by multiplying both sides of Eq. (10-196) by an arbitrary number of contracted factors, and using Eq. (10-195) to bring these factors within the N products. Wick s theorem now states that a T product can be decomposed into a unique sum of normal products as follows ... 

Relationships between dilute solution viscosity and MW have been determined for many hyperbranched systems and the Mark-Houwink constant typically varies between 0.5 and 0.2, depending on the DB. In contrast, the exponent is typically in the region of 0.6-0.8 for linear homopolymers in a good solvent with a random coil conformation. The contraction factors [84], g=< g >branched/ <-Rg >iinear. =[ l]branched/[ l]iinear. are another Way of cxprcssing the compact structure of branched polymers. Experimentally, g is computed from the intrinsic viscosity ratio at constant MW. The contraction factor can be expressed as the averaged value over the MWD or as a continuous fraction of MW. 

Highly branched PMMA synthesized by SCVCP of MMA with the inimer 12 via GTP was characterized by GPC using universal calibration and MALS [28]. The corresponding Mark-Houwink plots, log[ ]] versus logM, and contraction factor,g =[ ]]b nched/[fl]iinear> a function of logM are presented in Fig. 3 for different comonomer ratio y=[MMA]o/[12]o. For M>10 the viscosity of the... 

Fig. 3. Mark-Houwink plot (a) and contraction factors (b),g =[ j]b,a ched/[ lliineaD as a function of the molecular weight for the copolymerization of the methacrylate-type inimer 12 with MMA under different comonomer ratios, y=[MMA]o/[12](,=1.2 (+), 5.2 (0), 9.8 (V), 26 (A), 46.8 (0),86.5 ( ),respectively.TheintrinsicviscositiesofPMMA(—) are given for comparison. (Reproduced with permission from [28]. Copyright 2001 American Chemical Society.)...

Fig. 6. Mark-Houwink plots ( ) and contraction factors (A) of branched Pt-BuA (a=0.47) obtained by SCVP of the macroinimer 8. ( ) linear Pt-BuA (a=0.80). (Reproduced with permission from [46]. Copyright 2000 WILEY-VCH.)...

Fig. 7. a Mark-Houwink plot of highly branched PMMA obtained by SCVCP of MMA with the inimer 12. (-)RI signal ( ) intrinsic viscosity of feed, ( ) intrinsic viscosity of linear PMMA (O) contraction factor, g. b Separation of feed polymer into fractions by preparative SEC. (-) RI signal of fractions (-) accumulated RI signals (.) RI signal of feed polymer. (Repro-... 

In evaluating the geometrical factor g of the strips, a contraction factor of 0.33% from room temperature has been introduced [28]. 

The various quantities can be compared with that of the linear analogue at constant molar mass. This leads to so called contraction factors, which are significant quantities for a quantitative estimation of the number of branching points per macromolecule. 

For the contraction factors g of the mean square radii of gyration for fractions of randomly branched materials Zimm and Stockmayer [49] obtained... 

Fig. 24. Contraction factors for star-branched macromolecules as a function of the number of arms. The full line represents strictly regular stars, the dashed line one that for stars with polydisperse arms (MyM =2), the dotted line according to Daoud and Cotton [29]. The symbols represent the data from the literature. The deviation at large / represents the stretching of the arms due to overcrowding...

Such behavior has not been observed. Kurata et al. [ 129] developed an empirically based suggestion after comparing the results available in those days for star molecules. He also assumes power law correlation between the two contraction factors as given by... 

Table 3. Experimentally determined contraction factors g = Rgi,IRgu and various star molecules in 0-and good solvents ...

Fig. 25. The viscosity contraction factor g as a function of the geometric contraction factor g for star-branched macromolecules ( 3-128). No distinction was made between the chemical nature of the various arms and between the thermodynamic quality of the solvents used. See Table 3...

This section on the contraction factor may be concluded with an example of a comb macromolecule [136]. Due to the route of preparing this comb, unattached side chains also occurred in the system. Figure 27 shows the result of the molar mass dependence of [rj] which was obtained from a SEC/LALLS/VISC fractionation. One observes at low molar masses a straight line with an exponent of 11 =0 JO that coincides with the exponents of linear side chains. It follows a... 

If the optimal solution obtained in steps 4 (or 5) is an internal point to the original hypercube then select a contraction factor and reduce the size of hypercube. This new hypercube is centered on the optimal solution of previous step. 

Fig. 4 Contraction factors, g=R2g,star/R2g,iinear> as a function of the number of arms, /, for three different PAA stars (closed symbols) and non-hydrolyzed PtBuA precursor stars (open symbols) [110]...

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