Continuous distribution of chain lengths

Consider the continuous distribution of chain lengths in which the number of moles of each species is constant over the range 0 < x < a and zero outside that range. 

Consider the continuous distribution of chain lengths in Example 5. It is shown there that x /x = 2.0 for this distribution. Obtain an expression for Xg/xg for this distribution in terms of the MHS constant a in Eqs. 6.14 and 6.16. Given that 0.5 a 1, what is the possible range of x /x for this distribution ... 

Example 9.11 Which type of isothermal reactor would produce the narrowest possible distribution of chain lengths in a free-radical addition polymerization continuous stirred tank reactor (CSTR, or backmix), batch (assume perfect stirring in each of the previous), plug-flow tubular, or laminar-flow tubular ... 

In ATRP, there are reactive and dormant polymer species in equilibrium during the polymerizations, which alternate between halide-capped polymers (dormant) and growing (reactive) polymers with a free radical on the end. The choice of catalyst controls this equilibrium which in turn influences the polymerization rate and the distribution of chain lengths. The mechanism offers flexibihty to conduct reactions in bulk, solution, or emulsions/suspensions, just as fiee-radical polymerizations. Due to the capability to polymerize a large range of monomers with an inexpensive catalyst in a reactor, where purity is nearly as important as in anionic polymerizations, ATRP continues to grow in popularity. For further information, review articles written by the inventors are available [12,16]. 

Fig. 10.8. The ordering operator distribution for the three-dimensional Ising universality class (continuous line - data are courtesy of N.B. Wilding). Points are for a homopolymer of chain length r = 200 on a 50 x 50 x 50 simple cubic lattice of coordination number z = 26 [48], The nonuniversal constant A and the critical value of the ordering operator Mc were chosen so that the data have zero mean and unit variance. Reprinted by permission from [6], 2000 IOP Publishing Ltd...

Figure 6.10 (Continued) (c) average chain-length distributions of selected A-, B-, and C-type starch.79...

P — Pseudo-steady-state assumption C = Continuous variable for chain length L = Laplace transforms Z = Z-transforms E = Eigenzeit transform M = Moments of distribution N = Numerical techniques... 

Figure 5.9 is an example of an ACF showing characteristic exponential decay. In Eq. 5.7, i represents a group of particles with population q(Fi) that has the same characteristic decay constant Fj. q(Fj) and q(F) are the intensity-weighted discrete and continuous distributions of the decay rates, respectively. The parameter F is related to the translational and rotational motions of particles in a given medium. For a macromolecular chain, F is mainly determined by its chain length and conformation in the medium. For a soft particle, F relates to the softness, flexibility, dimension and shape of the particle. For a solid particle, F is characteristic of its shape and dimension. It should be noted that, except for... 

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