Critical branching

Peniche-Covas, C. A. L., Dev, S. B., Gordon, M., Judd, M. Kajiwara, K. (1974). The critically branched state in covalent synthetic systems and the reversible gelation of gelatin. In Gels and Gelling Processes. Faraday Discussions of the Chemical Society, No. 57, pp. 165-80. 

In t-DVB/S copolymerization, Antonietti and Rosenauer isolated microgels slightly below the macrogelation point [221]. Using small angle neutron scattering measurements they demonstrated that these microgels exhibit fractal behavior, i.e. they are self-similar like the critically branched structures formed close to the sol-gel transition. 

Starburst (de Gennes) Dense Packing — A Critical Branched... 

Starburst/cascade dendrimer dense-packing critical branching state, nanoscopic steric effects. 

Clearly, this proposed congestion at the critical branched state M, should exhibit (1) sterically-inhibited reaction rates, (2) sterically-induced stoichiometry, and, quite possibly, (3) a critical phase change due to surface cooperativity (association). These phenomena have been observed experimentally for the Starburst PAMAM dendrimers and may prove to be diagnostic probes for this critical branched state in other dendrimer families. 

AdoMet is a critical branch point metabolite. In its decarboxylated form, it serves as aminopropyl group donor in polyamine synthesis as AdoMet it is the methyl group donor for most transmethylation reactions (Fig. 7.1). In eukaryotic cells AdoMet decarboxylase is activated by putrescine and is the rate-limiting step of polyamine biosynthesis. Since production of dc-AdoMet is irreversible (13) regulation of AdoMet decarboxylase by putrescine determines the overall rate of polyamine synthesis. 

Linear polyethylene (LPE or HOPE) has been blended with LLDPE with varying branch distributions. The critical branch content for phase separation was lower for mPE, where the branches were more evenly distributed, than for Ziegler-Natta LLDPE, where the branches are heterogeneously distributed. Where cocrystallization occurred, the unit cell of the crystals showed marked expansion (4). Crystallization conditions were shown to be important for the cocrystallization of blends of linear and branched polyethylenes, low isothermal temperatures promote cocrystallization as did quench cooling (5). 

A similar branching effect on the miscibility was observed in the HDPE/ LLDPE-O blends, which were prepared from HDPE (49,400 and 3.60 PDI) and LLDPE-O polymers (69,200-104,000 M and 1.8-3.40 PDI) with 2-87 hexyl branches per 1000 backbone carbons (42). It was observed that the critical branch number in the LLDPE-O component capable of causing immiscibility in the HDPE/ LLDPE-O blend was 50 branches per 1000 backbone carbons, as determined using inverse gas chromatography, rather than the SANS technique. 

When extensive branching occurs the system can gel. This occurs when —c . Under these conditions the critical branching probability, p, is related to the extent of reaction of the phenolic groups at the gel point, o-c, by... 

In this paper we report experimental results on the kinetics of the polyaddition reaction in the melt and in solution (diglyme) using two different catalysts, namely tetrabutyl ammonium hydroxide (TBAH) and benzyl triethyl ammonium chloride (BTAC) at different temperatures. By simultaneous measurements of a, and in two cases we have also evaluated the critical branching parameters. 

The method of generating functions is in concurrence with combinatorial methods and recursive techniques. Combinatorial methods have been used most early in particular by Flory and Stockmayer to derive critical branching [15],... 

When the mixture critical line is discontinuous, it is divided into two branches by a three-phase VLLE line, as shown in Figure 9.22. The high temperature end of the VLLE line is a UCEP that connects one branch of the critical line to the critical point of the more volatile component. This branch of the critical line is usually short. Based on the behavior of the critical branch emanating from the critical point of the less volatile component, we divide these mixtures into two classes. 

Class D. From the critical point of the less volatile component, the critical branch in these mixtures traces a path to high pressures without terminating at a fluid-phase... 

The gel formation can be linked to the functionality, f, and a branching coefficient, a. The branching coefficient a gives the probability that a specific functional group (of functionality > 2) is connected to another branch point. One can deduce from statistical reasoning that a gel appears at a critical branching coefficient, a. For f=3, a gel appears when is 0.5, i.e., there is a 50% chance that each branch is connected to another branch point. The network structure depends on the concentration of branch points and the degree of polymerization. 

From metabolic fluxes illustrated in Fig. 16.2, it can be seen that SAM accumulation can be maximized not only by stimulation of synthesis, but also by inhibition of SAM degradation. Located at the critical branch point of SAM metabolism. 

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