Critical branching coefficient

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. 

Network molecules can form when n chains are expected to lead to more than n chains through branching of some of them. The maximum number of chains that can emanate from the end of a single chain, such as that analysed above, is (/- 1) and so the probable number of chains emanating from the chain end is a(f- 1). Network molecules can form if this probability is not less than one, i.e. a(/-I) l. Thus the critical branching coefficient, ac, for gelation is given by... 

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. 

Such reactions have been used to explain the three limits found in some oxidation reactions, such as those of hydrogen or of carbon monoxide with oxygen, with an "explosion peninsula between the lower and the second limit. However, the phenomenon of the explosion limit itself is not a criterion for a choice between the critical reaction rate of the thermal theory and the critical chain-branching coefficient of the isothermal-chain-reaction theory (See Ref). For exothermic reactions, the temperature rise of the reacting system due to the heat evolved accelerates the reaction rate. In view of the subsequent modification of the Arrhenius factor during the development of the reaction, the evolution of the system is quite similar to that of the branched-chain reactions, even if the system obeys a simple kinetic law. It is necessary in each individual case to determine the reaction mechanism from the whole... 

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... 

Since only monomer A/ (/ > 2) is present, the branching coefficient (a), defined earlier, is simply p and the critical condition for gel formation is thus given by... 

The branching coefficient as defined by Flory is a = the probability that a certain branched unit will be joined to a second branched unit rather than to a terminal group. For example, for a trifunctional monomer if a = 1/2 the molecule is a continuous chain equivalent in theory to a gel. In this case a = 1/2 is the critical condition defining the start of the formation of an infinite tridimensional network. 

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

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. 

Since water is much more efficient than either H2, N2 or O2 as a third body in reaction (iv) (see Table 7), the simplest interpretation of the suppression of the limits for manipulation times greater than the optimum is that it is associated with water formation by the slow reaction as the limit is approached. This interpretation is supported by the fairly successful calculation [21] of critical withdrawal rates at 500 °C using rate coefficients derived from the slow reaction studies at 500 torr. At 500 °C this depression is the only effect observable, and there is never any rise in Umit above that at fast withdrawal rates. At this temperature therefore, the quadratic branching is fully developed. 

The first term in this quadratic equation is the initiation reaction rate based on the inflow concentration of the reactant. The coefficient for the term in [X]ss has something of the character of the previous net branching factor. The above equation has a single positive solution for any set of rate constants, residence time and inflow concentration a typical variation of [.ATJss with [A]o is shown in Fig. 5.3(b) and shows a rapid increase in the vicinity of some critical concentration [A]o,cr-The behaviour can be quantified if we make the approximation of ignoring the (probably small) initiation terms, setting ki =0. The steady-state condition can then be written in the form... 

The first two represent the high-temperature chemistry and the second two the branch chain of the low-temperature chemistry. Griffiths [85] has criticized the model for losing the essential feature of alkane autoignition chemistry, a switch from radical branching to non-branching reactions as the temperature increases, which is responsible for the negative temperature coefficient. The model relies on thermal feedback mechanisms only. 

A lot of work has been done to synthesize TPA chromophores with higher branches [502, 503, 507, 514, 517-520, 523]. Most of these chromophores possess both large OPA and TPA coefficients. A critical comparison of TPA... 

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. 

The increase in Kic to 6.5-7.5 is attributed to the misfit of the thermal expansion coefficients of TiC and SiC, introducing considerable radial tensile stresses at the phase boundaries and hoop compressive stresses in the matrix. These stresses enable crack deflection, crack branching, and microcracking above a critical particle size of 3 [tm. The optimum volume content of TiC ranges between 20 and 30 vol.%. 

---