Optimised polyolefin branch quantification by 13 C NMR spectroscopy

Quantitative branch determination in polyolefins by solid- and melt-state 13C NMR has been investigated. Both methods were optimised toward sensitivity per unit time. While solid-state NMR was shown to give quick albeit only qualitative results, melt-state NMR allowed highly time efficient accurate branch quantification. Comparison of spectra obtained using spectrometers operating at 300, 500 and 700 MHz 1H Larmor frequency, with 4 and 7~mm MAS probeheads, showed that the best sensitivity was achieved at 500 MHz using a 7 mm 13C-1H optimised high temperature probehead. For materials available in large quantities, static melt-state NMR, using large diameter detection coils and high coil filling at 300 MHz, was shown to produce comparable results to melt-state MAS measurements in less time. While the use of J-coupling mediated polarisation transfer techniques was shown to be possible, direct polarisation via single-pulse excitation proved to be more suitable for branch quantification in the melt-state. Artificial line broadening, introduced by FID truncation, was able to be reduced by the use of π pulse-train heteronuclear dipolar decoupling. This decoupling method, when combined with an extended duty-cycle, allowed for significant improvement in resolution. Standard setup, processing and analysis techniques were developed to minimise systematic errors contributing to the measured branch contents. The final optimised melt-state MAS NMR method was shown to allow time efficient quantification of comonomer content and distribution in both polyethylene- and polypropylene-co-α-olefins. The sensitivity of the technique was demonstrated by quantifying branch concentrations of 8 branches per 100,000 CH2 for an industrial ‘linear’ polyethylene in only 13 hours. Even lower degrees of 3–8 long-chain branches per 100,000 carbons were able to be estimated in just 24 hours for a series of γ-irradiated polypropylene homopolymers.

Cut-first branch-and-price-second for the capacitated arc-routing problem

This paper presents the first full-fledged branch-and-price (bap) algorithm for the capacitated arc-routing problem (CARP). Prior exact solution techniques either rely on cutting planes or the transformation of the CARP into a node-routing problem. The drawbacks are either models with inherent symmetry, dense underlying networks, or a formulation where edge flows in a potential solution do not allow the reconstruction of unique CARP tours. The proposed algorithm circumvents all these drawbacks by taking the beneficial ingredients from existing CARP methods and combining them in a new way. The first step is the solution of the one-index formulation of the CARP in order to produce strong cuts and an excellent lower bound. It is known that this bound is typically stronger than relaxations of a pure set-partitioning CARP model.rnSuch a set-partitioning master program results from a Dantzig-Wolfe decomposition. In the second phase, the master program is initialized with the strong cuts, CARP tours are iteratively generated by a pricing procedure, and branching is required to produce integer solutions. This is a cut-first bap-second algorithm and its main function is, in fact, the splitting of edge flows into unique CARP tours.