Average branching density

The mechanism of emulsion polymerization ensures that the polymer concentration at the polymerization locus is semidilute or concentrated, which results in a greater probability of branch chain formation [299, 300]. This effect produces unique average branching densities and unique distributions of branching densities, that are significantly different from corresponding bulk polymerization [266,301]. 

Figure 16a shows the development of average branching density in emulsion polymerization and in a corresponding bulk polymerization, both of which involve the polymer transfer reactions [301]. 

Fig. 16 Development of the average branching density a and the branching density distribution b for emulsion and bulk polymerizations. The conversion at which monomer droplets disappear in emulsion polymerization is Xc=0.5...

Combination and integration between 9 and if/ yields the average branching density p 9, y/) ... 

It follows from the steady-state variants of the balances of monomer and branching density, Eqs. (135)-(137), that the average branching density p is related to (steady-state) conversion x, according to ... 

In Figure 6, these data are plotted versus the branching density z of crosslinking molecules. Gy./G is fairly independent of network microstructure. It covers a range of 0.24 to 0.32 as a result of statistical scattering, averaging to 0.28 as in case of the networks with tetrafunctional crosslinks. 

In contrast to porosity, the pore density and the specific surface area are quantities directly related to the actual size of pores and pore walls. The pore density NP is defined as the number of pores per unit area and it usually refers to a plane normal to the pore axis. For (100) oriented substrates this plane is parallel to the electrode surface, but for other orientations or strongly branched pores, there is no preferred plane orientation and NP refers to an average of the pore density of different planes. For arrays of straight pores the pore density can be directly calculated from the array geometry. For cylindrical pores of diameter d orthogonal to the electrode surface, for example, the average pore density NP is given by ... 

Because the SCR width depends not only on doping density but also on bias, the average pore density is expected to decrease with the square root of bias. This, however, is not observed. An increase in bias often leads to the formation of breakdown-type mesopores at the macropore walls. Because these spiking pores show diameters on the order of a few tens of nanometers they are hard to identify even in an SEM. Spiking pores can be identified if they are enlarged by subsequent chemical etching, as shown in Fig. 8.10. Details of their branched morphology become visible by formation of an oxide replica and after etchback of the sub-... 

Foam (5) is a collection of gas bubbles with sizes ranging from microscopic to infinite for a continuous gas path. These bubbles are dispersed in a connected liquid phase and separated either by lamellae, thin liquid films, or by liquid slugs. The average bubble density, related to foam texture, most strongly influences gas mobility. Bubbles can be created or divided in pore necks by capillary snap-off, and they can also divide upon entering pore branchings (5). Moreover, the bubbles can coalesce due to instability of lamellae or change size because of diffusion, evaporation, or condensation (5,8). Often, only a fraction of foam flows as some gas flow is blocked by stationary lamellae (4). 

Branched macromolecules have a higher average segment density than unbranched macromolecules of the same molar mass, and have a lower coil volume. This is easily seen by comparing a star-shaped branched molecule with a linear one. The influence of the branching on the dimensions can be expressed by a g factor... 

Short-chain branching n. The presence of 2- to 4-carbon side chains along the backbone of a polymer molecule. In an average low-density polyethylene molecule, both ethyl and butyl side chains, mostly the latter, are beheved to be present, with a total of 50 in a typical molecule, in addition to one much longer branch. 

As usual in a CSTR, the independent variable is residence time in the example of Figure 9.18, do for Seg-0. Now, the average scission density 7 is a constant following by a similar argument to that for branching [Eq. (146)) gives ... 

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