Expanded branching equation

Expanded branching equation (XB) A topological method for describing steric effects which takes into account the order of branching by using as parameters n,y, the number of j -th branching atoms bonded to the i-th atoms of the substituent. 

We may account for the effect of the order of branching by means of the expanded branching (XB) equation. 

The strongly acceleratory character of the exponential law cannot be maintained indefinitely during any real reaction. Sooner or later the consumption of reactant must result in a diminution in reaction rate. (This behaviour is analogous to the change from power law to Avrami—Erofe ev equation obedience as a consequence of overlap of compact nuclei.) To incorporate due allowance for this effect, the nucleation law may be expanded to include an initiation term (kKN0), a branching term (k N) and a termination term [ftT(a)], in which the designation is intended to emphasize that the rate of termination is a function of a, viz. 

The simplest new phenomenon induced by this mechanism is a secondary bifurcation from the first primary branch, arising from the interaction between the latter and another nearby primary branch. It leads to the loss of stability of the first primary branch or to the stabilization of one of the subsequent primary branches, as illustrated in Fig. 1. The analysis of this branching follows similar lines as in Section I. A, except that one has now two control parameters X and p., which are both expanded [as in equation (5)] about the degeneracy point (X, p.) corresponding to a double eigenvalue of the linearized operator L. Because of this double degeneracy, the first equation (7) is replaced by... 

Olvera de la Cruz and Sanchez [76] were first to report theoretical calculations concerning the phase stability of graft and miktoarm AnBn star copolymers with equal numbers of A and B branches. The static structure factor S(q) was calculated for the disordered phase (melt) by expanding the free energy, in terms of the Fourier transform of the order parameter. They applied path integral methods which are equivalent to the random phase approximation method used by Leibler. For the copolymers considered S(q) had the functional form S(q) 1 = (Q(q)/N)-2% where N is the total number of units of the copolymer chain, % the Flory interaction parameter and Q a function that depends specifically on the copolymer type. S(q) has a maximum at q which is determined by the equation dQ/dQ=0. 

If the function (x, y) has been found by solving the appropriate hydrodynamic problem, then the expression (18.11) represents the equation of the drop trajectory. The constant C is determined by the initial position of the drop at the separator entrance. As far as the diameter of the branch pipe for gas input is much smaller than the cross-sectional diameter of the separator, the gas flows into the separator as an expanding jet with the opening angle 2a 28° (Fig. 18.1). Then the distance at which the jet reaches the wall of the separator equals... 

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