BRANCHING IN POLYETHYLENE
The principal purpose of this paper is to present the current status, as viewed by the authors, of our knowledge of branching in polyethylene and the effects of branching on its properties. Another purpose is to present some speculative material, and some challenging observations, in an attempt to stimulate, in some measure, future work on the structure and chemistry of polyethylene. In endeavoring to achieve these aims, the authors have drawn heavily upon their own work and liberally from the literature, particularly the published results of R. B. Richards and his collaborators. Polyethylene resins are made by polymerizing ethylene at elevated temperatures and high pressures.’, The polymer molecules formed under these conditions are branched hydrocarbon chains3* 4 . 6 * and their average molecular size varies from approximately 50 carbon atoms per molecule, for the greasy products, to more than 1000 carbon atoms per molecule for rigid resins. The degree of branching in polyethylene has been shown to play an important role in determining the properties of the resin.’ In fact, it is now generally recognized that degree of branching and molecular weight are the most important molecular variables in polyethylene resins. In this paper, degree of branching is defined as the number of branches per 100 carbon atoms, and the number of branches per molecule is defined as the number of -CH3 groups per molecule more than two. Although some chains and branches are terminated by vinyl groups (-CH=CHJ, the concentration of these is very small compared to the concentration of methyl groups. Side chain methylene groups (RR’C=CHZ) may be regarded as short branches but, as such, represent not more than 15 per cent, and generally a much smaller proportion, of the total number of branches. As a result, this configuration is not considered significant for the purposes of this paper.
CROSSLINKING IN MONOVINYL MONOMERS (I). BY CHAIN TRANSFER WITH THE POLYMER CHAIN
It is known that certain monovinyl monomers, notably methyl acrylate and vinyl acetate, form crosslinked insoluble network polymers on bulk polymerization in the absence of chain transfer agents. For methyl acrylate, macrogelation may occur at conversions (a) below 5 per cent,’. 2, and microgelation, resulting in a heterogeneous polymerization to an insoluble popcorn type of polymer, has been observed in thermal polymerizations at conversions appreciably less than 1 per cent?. In the polymerization of vinyl acetate gelation has been observed at very high conversiona6 Gelation is not observed in the bulk homopolymerization of other common monomers such as styrene or methyl methacrylate. It has been demonstrated recently that branching occurs in vinyl acetate as the result of chain transfer reactions with the monomer and with the polymer chains6 It has long been suspected2 that, in methyl acrylate, chain transfer with the “active” alpha hydrogen atom on the carbon adjacent to the carbonyl carbon atom will occur and thereby lead to branching. Recent studies of the chain transfer constants with methyl acrylate of methyl 2-ethyl butyrate and of methyl isobutyrate, bear this out? The very high molecular weights obtained even at low conversions in the polymerization of methyl acrylate,’* * on the other hand, indicate that branching via chain transfer with the monomer is unimportant in this case. Since crosslinking results if two branch radicals combine, and since a particular molecule may contain more than one branch, it will be recognized that, under certain conditions, a mechanism involving branching and termination by combination can lead to gelation. It is the purpose of this paper to consider the possibility of gelation in the particular case wherein branching occurs solely by chain transfer with the polymer chain. Predictions concerning the.ge1 point (if any) are made in terms of the number of active transfer sites per chain (ns), and the relative probability that the growth of a radical will be interrupted by combination with another radical rather than by chain transfer. The simple case of the graft copolymerization of a monomer with a polymer containing the same number of active transfer sites on each molecule is treated rigorously. An approximate treatment concerning gelation in the homopolymerization of monovinyl monomers yields results which are at least qualitatively in accord with the observations mentioned heretofore. This preliminary treatment is sufficiently precise to reveal why the tendency to gelation decreases in the order: methyl acrylate > vinyl acetate >> styrene > methyl methacrylate. Methods are suggested for treating more exactly the conditions leading to macrogelation, or microgelation and inhomogeneous polymerization. These will be developed in future publications.
The parallel waveform relaxation stochastic Runge–Kutta method for stochastic differential equations
For large-scale non-autonomous Stratonovich stochastic differential equations, we study a very general parallel waveform relaxation process which is on the basis of stochastic Runge–Kutta (SRK) method of mean-square order 1.0 in this literature. The convergence of the whole parallel numerical iterative scheme can be guaranteed and the scheme provides better properties in terms of decreasing the load of the computation and operating speed. At the same time, the related limit method is also introduced as the continuous approximation derived from the iterative scheme. In the approximation interval, it is worth noting that the mean-square order of the parallel numerical iterative scheme can be kept consistent with the previous SRK method at any arbitrary time point, not just at discrete points. Some numerical simulations are presented to elaborate the computing efficiency of the parallel numerical iterative scheme.