Configurational Statistics of Highly Branched Polymer Systems

Abstract

Applications of the theory of branching processes to polymer systems can be so formulated, that all statistical parameters emerge automatically in a form which applies to the soluble part of the system, i.e., to the whole system up to the gel point, and to the sol fraction beyond. In a self-contained presentation, previous work along these lines is here extended to computing configurational statistics of systems arising most directly by condensation processes. These statistics are the mean-square radii of the molecules in such systems, averaged in various ways and useful for theories of light scattering and viscosity. Numerical calculations for random f-functional polycondensation are presented as plots of configurational parameters against conversion for f=3, 4, and 6. The general equations given extend beyond the random case to condensations with ``first-shell'' substitution effect, i.e., in which the rates of making or breaking a given bond between two repeat units depends on how many other bonds these two units carry (to further units). In terms of current theories, the intrinsic viscosity of a system at its gel point is shown to be finite. The spatial averaging of molecular size is based on random flight statistics. The well-known theorem by Kramers used in this connection is rederived on a simple topological basis, and generalized to deal with copolymerization of units of different sizes.

Two simple approaches to sol–gel transition

“…It can be shown easily [14] that for the system considered here the extinction probability can be obtained as a root of the equation…”

“…They used a laborious approach based on combinatoric considerations of the most probable composition of the system. A much more effective variant of this approach, exploiting theory of branching processes, has been developed later by Dobson and Gordon [14,15]. This approach allows also an analysis of structural details of the system behind the critical point, see, e.g.…”

“…Actually, a liquid selforganization, resulting in a sol-gel transition, is caused by the positive feedback between the strain ε and the number of cross-links N in Eq. (14). Hence, the increase of the number of cross-links should intensify the self-organization effect.…”

“…(13), (14) and (16) is known in synergetics [17] as the Lorenz system where the deformation ε, the stress σ and the number of cross-links N play roles of an order parameter, a conjugate field and a control parameter, respectively. It is very important for following considerations that the relation between micro-, meso-and macroscopic values of the relaxation times…”

Kinetic Modeling of Non‐Linear Polymerization

“…The prediction of the structure of branched polymers in the approximation of the Gaussian chain model has been carried out using the theory of branching processes [59,60] in order to compute the needed concentrations of polymer molecules or sequences. [61,62] The time dependent spatial correlations of polymer groups can be described through an elegant statistical mathematical formalism involving generating functionals of time and space coordinates.…”