VODE is a new initial value ODE solver for stiff and nonstiff systems. It uses variable-coefficient Adams-Moulton and BDF methods in Nordsieck form, as taken from the older solvers EPISODE and EPISODEB, treating the Jacobian as full or banded. Unlike the older codes, VODE hss a highly flexible user interface that is nearly identical to that of the ODEPACK solver LSODE. In the process, several algorithmic improvements have been made in VODE, aside from the new user interface. First, a change in stepsize and/or order that is decided upon at the end of one successful step is not implemented until the start of the next step, so that interpolations performed between steps use the more correct data. Secondly, a new algorithm for setting the initial step size has been included, which iterates briefly to estimate the required second derivative vector. Efficiency is often greatly enhanced by an added algorithm for saving and reusing the Jacobian matrix J, as itoccurs in the Newton matrix, under certain conditions. As an option, this Jacobian-saving feature can be suppressed if the required extra storage is prohibitive. Finally, the modified Newton iteration is relaxed by a scalar factor in the stiff case, as a partial correction for the fact that the scalar coefficient in the Newton matrix may be out of date. Independently, we have studied the fixed-leading-coefficient form of the BDF methods, and have developed a version of VODE that incorporates it. This version does show better performance on some problems, but further tuning and testing is needed to make a final evaluation of it. Like its predecessors, VODE demonstrates that multistep methods with fully variable stepsizes and coefficients can outperform fixed-stepinterpolatory methods on problems with widely different active time scales. In one comparison test, on a 1-D diurnal kinetics-transport problem with a banded internal Jacobian, the run time for VODE was 36% lower than that of LSODE without the J-saving algorithm, and 49% lower with it. The fixed-leading-coefficient version ran slightly faster, by another 12% without J-saving and 5% with it.

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In heterogenous olefin polymerization with Ziegler catalysts, the influence of monomer mass transport in the growing granule on polymer properties has been extensively modeled, but it has not been possible to clearly establish the importance of diffusion experimentally since the multisited nature of most Ziegler catalysts can produce similar effects. In this study, ethylene-propylene copolymers were made with single-sited metallocene catalysts by slurry polymerization in liquid monomers. These copolymers had a relatively narrow molecular weight distribution with a composition distribution (CD) broader than expected for a single-site catalyst. Data analysis indicates that mass-transfer limitations in the polymer particles are the most likely explanation for the observed results. For amorphous copolymers, a diffusion IntroductionThe vast majority of the billions of pounds of polyolefins produced each year by Ziegler-Natta catalysts are made in multiphase reactors where the polymer exists as granules suspended in a fluid containing the monomers. For the most part, Ziegler polyolefins have a broad molecular weight distribution (MWD) with a ratio of weight to number average molecular weight (Q) of greater than 2 (4 to 10 is typical). Early investigations recognized that likely explanations of broad MWD included multiple active catalyst sites in the catalyst and nonsteady-state effects at the active site due to mass transport limitations (Zucchini and Cecchin, 1983). Initial attempts (Bul and Higgins, 1970) to model the diffusionheaction process at an active site embedded in a matrix of polymer indicated that diffusion alone could account for the observed MWDs. However, a large body of evidence now exists (see Zucchini and Checcin, 1983;Cozewith, 1987;Yechevskaya et al., 1987;and Haejaski et al., 1988) showing that multisited Ziegler catalysts are the rule rather than the exception. Thus, diffusion may indeed have an influence on polymerization rates and polymer properties, but if so, it would be very difficult to distinguish from the complicated polymerization behavior resulting from the multiple active sites. Consequently, there is no unequivocal data published in the literature that demonstrates whether polymer phase mass transport effects are important in Ziegler polymerizations.Recently, a new class of olefin polymerization coordination catalysts was discovered (Sinn et al., 1980) based on metallocene compounds of zirconium, titanium, or hafnium combined with an activator such as methyl alumoxane. These catalyst systems are capable of making the same range of olefin homo-and copolymers as Ziegler-Natta catalysts, however, they tend to be single-sited and to produce polymers with a most probable ( M J M , = 2) molecular weight distribution. In multiphase polymerizations with single-sited metallocene catalysts, deviations from the polymerization rate behavior or polymer properties expected for kinetically controlled reactions might well be strong evidence for diffusional effects on the polymerization.We have studie...

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