Branching coefficient critical value

Our interest from the outset has been in the possibility of crosslinking which accompanies inclusion of multifunctional monomers in a polymerizing system. Note that this does not occur when the groups enclosed in boxes in Table 5.6 react however, any reaction beyond this for the terminal A groups will result in a cascade of branches being formed. Therefore a critical (subscript c) value for the branching coefficient occurs at... 

As an example of the quantitative testing of Eq. (5.47), consider the polymerization of diethylene glycol (BB) with adipic acid (AA) in the presence of 1,2,3-propane tricarboxylic acid (A3). The critical value of the branching coefficient is 0.50 for this system by Eq. (5.46). For an experiment in which r = 0.800 and p = 0.375, p = 0.953 by Eq. (5.47). The critical extent of reaction, determined by titration, in the polymerizing mixture at the point where bubbles fail to rise through it was found experimentally to be 0.9907. Calculating back from Eq. (5.45), the experimental value of p, is consistent with the value =0.578. 

The point in the reaction at which gelation occurs has been deduced by Flory. 4 The gel point is developed in terms of the branching coefficient, a, which is the probability that a given functional group on the multifunctional monomers leads, via a chain that can contain any number of bifunctional units, to another multifunctional monomer. The critical value of the branching coefficient, denoted by ac, at which gelation occurs is... 

Because the boiling temperature of 1,4-BD is much higher than of the two reaction products and the reaction is irreversible, the bifurcation behavior is only affected by the mass transfer coefficient ratio Kwater/KTHF, if kbd is not extremely high or low. There exists a critical value of Kwater/KTHF = 2.1, above which the stable node branch approaches the THF-vertex. 

O critical value of the branching coefficient Q eff effective thermal diffusivity [m s ]... 

Many important questions and conjectures remain unresolved. It is not known whether these solutions are the only embedded //-surfaces for the five dual pairs of skeletal graphs studied, for example. An important issue is whether or not there exists a bound on the mean curvature attainable in such families for all of the branches studied here, and for the family of unduloids with a fixed repeat distance (Anderson 1986), the dimensionless mean curvature H = HX is always less than n, where X is the sphere diameter in the sphere-pack limit. It is possible that there exists an upper bound on H lower than n that depends on the coordination number, or the Euler characteristic. For the P, D, I, WP, F, and RD branches, the islands over which K > 0 coalesce wih neighboring R regions at a critical mean curvature that is the same (to within an error in H of about 0.15) as the value H corresponding to the local minimum in surface area. We have given what we suspect to be the analytical value for the area of the F-RD minimal surfaces, and for the first nonzero coefficient in both the area and volume expansions about // = 0 in the P family. 

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