SYNOPSISTwo different theoretical approaches have been developed to describe network formation in a six-component, three-stage process with the stochastic theory of branching processes. Both approaches make use of probability-generating functions and cascade substitutions. In the POLYM approach, the products of the previous stage, namely, complete (branched) molecules, are used as building units in the subsequent stage. The molar masses and unreacted functionalities are identified by means of dummy variables. In the resulting formulas the sequence of the three stages is clearly visible. In the MONOM approach, the system is treated as a quasi-one-stage process; the original monomers are the building units in all three stages, the sequence of which is only apparent from the input parameters. In the MONOM approach the original monomers are used as building units, irrespective of the stage in which they react. A separate dummy variable is chosen for each possible reaction in each stage in the MONOM approach, so that the two approaches give analytically identical results, but along completely different derivations. This avoids correlation errors in the MONOM approach. Both approaches have incorporated kinetic effects by way of substitution effects in two of the monomers. The two approaches are prefaced by the treatment of several paradigms of increasing complexity to show the unity and strength of the theory and to warn against typical pitfalls.

This investigation concerns the important class of fluids whose rheological properties are described by a quasilinear viscoelastic constitutive equation of the Boltzmann superposition type. The first Cox–Merz relation is closely approximated by such a fluid if its nonlinearity in shear can be described by the strain measure $ S_{12} (\gamma) = \int_0^\gamma {J_0} (\upsilon)dv $, irrespective of the distribution of its relaxation times and, hence, its linear viscoelastic properties. Here γ equals the shear strain and J0 the zeroth‐order Bessel function. The second Cox–Merz relation is met by materials with a different nonlinearity, namely S12(γ) = Si(γ), where Si is the sine integral. Experimental data on melts of a polystyrene and a low‐density polyethylene sample were utilized to demonstrate that both Cox–Merz relations cannot hold simultaneously.

For a series of homogeneous ethylene‐propylene copolymers with mole fraction ethylene (x1) ranging between 0.40 and 0.85, the inversion (I) independent propagation probabilities (Pijs) and reactivity ratios ( rijs) and the methylene, ethylene, and propylene sequence length distributions have been determined from 13C nuclear magnetic resonance (NMR) data, using a first‐order Markovian terpolymer [ethylene (1), inverted propylene (2), and normal propylene (3)] model. For each sample, the limiting values of I (Imin, Imax) are given. Calculations of the common parameters for 19 samples show that the polymerization direction characterized by the r set and I is statistically more probable than the opposite direction, which is characterized by the r′ set and I′. Further, I = Imin appears to be close to the most probable value of I. The resulting r set is r12 = 119, r13 = 19.7 and, for I = Imin, r21 = 0, r31 = 0.034, r32 = 2.98. In the limited range 0.60 < x1 < 0.85, there appears to be no preference for either polymerization direction, so that the solutions characterized by the r set and I = Imin and by the r′‐set and I = I′max, respectively, are about equally probable. If the copolymer reactivity ratios re and rp are defined in the usual manner, it follows that for the limited range re = 17. In the model used here, rp is found to be dependent on the ethylene‐propylene ratio in the reactor. Nevertheless, it can be stated that rp ≅ 0.029 and re rp ≅ 0.50.

No abstract

A multi-purpose computer program has been developed for describing copolymerization and network formation in step reactions. It operates at two hierarchical levels: a generation level and an application level. At the generation level, the class of problems is defined and prepared, and this information is used to generate automatically a problem-specific program based on generalized concepts of the branching theory. At the application level, the problem-specific program can be used for model calculations with all possible variations in recipes and kinetic parameters within the boundaries of the specific class of processes concerned. INTRODUCTIONCopolymerization and network formation in step reactions are best described by the theories of branching processes that have been developed since the early 1940s. However, processes of practical importance are characterized by an almost unlimited diversity of types of reactive monomers or polymers,