Chain-end model

As in Ziegler-Natta polymerization, steric control of the propagation step may involve either the interaction of the monomer with a chiral metal centre (enantiomorphic sites model), or the interaction of the monomer with the chiral centres in the repeating unit(s) adjacent to the metal centre (chain-end model). (The relationship to Ziegler-Natta polymerization will be considered further in Section VIII.C.3.)... 

As indicated by Tulleken/ there is a nomenclature problem with the term Bernoullian to indicate the chain-end model. As it is the last inserted monomer unit that controls the insertion of the following olefin, it would be better to use the term symmetric Markovian model. 

The use of statistical models to interpret (and to rationalize) NMR tacticity and sequence data is well established (97,98). In this volume the enantiomorphic-site model has been used by Segre et al. in their studies of polypropylene at high fields (55). A two-site model has been en loyed by Shimozawa et al. to observe the effects of internal donors m propylene polymerization (56). Other models for polyolefins have been reported in the literature, e.g., the multi-site model (99), the dual catalytic-site/chain-end model (100), the perturbed model (101), the consecutive two-site model (102), the four-component model (103), and the chain end model (104). 

The simplest GGS model for such polymer networks is the so-called fixed-chain-ends model a single chain bearing dendritic wedges (CBDW), whose ends are fixed in space (immobile) [78]. From a physical point of view, it cor-... 

Fig. 32 2-D polymer network built from chains bearing dendritic wedges. Note that in the simplified fixed-chain-ends model the cross-linking points (depicted here as open beads) are assumed to be fixed in space, see text for details...

Fig. 33 Reduced storage modulus [G ( )] versus the reduced frequency to plotted on double logarithmic scdes. Shown are results for 2-D networks bearing trifunctional dendritic wedges of fifth (dashed line) and seventh (solid line) generations. Also displayed are the results obtained from a simplified fixed-chain-ends model for the same dendritic wedges, namely g = 5 (line with stars) and g = 7 (line with circles). The chains between the cross-links consist of 51 beads each...

FIGURE 3. Percentage trans isomer in active chain end models for butadienyllithium and isoprenyllithium. DIPIP is 1,2-dipiperidinoethane and TMEDA is tetramethylethylene-diamine. 

The agreement between the calculated positions of the resonances for the different sequences is excellent. The relative fractions calculated from the chain-end model with P(m) = 0.79g also shows good agreement with the experimental values. It is clear that the high field measurements are revealing new and valuable information about the stereospecific polymerization of polypropylene. 

According to the chain-end model with f (m) = 1 (Data taken from Ref. [20, table I ].)... 

Three record types are used to group other records TER indicates the end of a chain, while MODEL/ENDMDL surround groups of ATOM, HETATM. TER and similar records. 

The next step in the development of a model is to postulate a perfect network. By definition, a perfect network has no free chain ends. An actual network will contain dangling ends, but it is easier to begin with the perfect case and subsequently correct it to a more realistic picture. We define v as the number of subchains contained in this perfect network, a subchain being the portion of chain between the crosslink points. The molecular weight and degree of polymerization of the chain between crosslinks are defined to be Mj, and n, respectively. Note that these same symbols were used in the last chapter with different definitions. 

The interdiffusion of polymer chains occurs by two basic processes. When the joint is first made chain loops between entanglements cross the interface but this motion is restricted by the entanglements and independent of molecular weight. Whole chains also start to cross the interface by reptation, but this is a rather slower process and requires that the diffusion of the chain across the interface is led by a chain end. The initial rate of this process is thus strongly influenced by the distribution of the chain ends close to the interface. Although these diffusion processes are fairly well understood, it is clear from the discussion above on immiscible polymers that the relationships between the failure stress of the interface and the interface structure are less understood. The most common assumptions used have been that the interface can bear a stress that is either proportional to the length of chain that has reptated across the interface or proportional to some measure of the density of cross interface entanglements or loops. Each of these criteria can be used with the micro-mechanical models but it is unclear which, if either, assumption is correct. 

Figs. 7 and 8 respectively show the ripples obtained by SIMS and neutron reflection during welding of HDH/DHD interfaces. Its interesting to note that all dynamics models predict ripples since in general, the chain ends move faster than the chain centers. However, their shape, amplitude and time dependence provide a... 

Studies on model compounds also suggest that unsaturated chain-end groups should not have an important influence on the thermal stability of PVC [21]. In conclu-... 

If chains are long such that the initiation and termination reactions have a negligible effect on the average sequence distribution, then according to the terminal model, PAA, the probability that a chain ending in monomer unit MA adds another unit MA, is given by eq. 22 8... 

In the classical diffusion control model it is assumed that propagation occurs according to the terminal model (Scheme 7.1). The rate of the termination step is limited only by the rates of diffusion of the polymer chains. This rate may be dependent on the overall polymer chain composition. However, it does not depend solely on the chain end.166,16... 

In eq. 68, is defined as in the chemical control model but this expression is cast in terms of the monomer feed composition rather than the radical chain end population. 

More complex models for diffusion-controlled termination in copolymerization have appeared.1 tM7j Russo and Munari171 still assumed a terminal model for propagation but introduced a penultimate model to describe termination. There are ten termination reactions to consider (Scheme 7.1 1). The model was based on the hypothesis that the type of penultimate unit defined the segmental motion of the chain ends and their rate of diffusion. 

Chains with uttdesired functionality from termination by combination or disproportionation cannot be totally avoided. Tn attempts to prepare a monofunctional polymer, any termination by combination will give rise to a difunctional impurity. Similarly, when a difunctional polymer is required, termination by disproportionation will yield a monofunctional impurity. The amount of termination by radical-radical reactions can be minimized by using the lowest practical rate of initiation (and of polymerization). Computer modeling has been used as a means of predicting the sources of chain ends during polymerization and examining their dependence on reaction conditions (Section 7.5.612 0 J The main limitations on accuracy are the precision of rate constants which characterize the polymerization. 

Studies with model compounds show that secondary benzoate esters eliminate benzoic acid to form unsaturated chain ends as shown in Scheme 8.2.15... 

Table 2 shows a list of collagen model peptides which have teen prepared. Many efforts have been made to prevent racemization. The polycondensation reaction seemed to be more sensitive to racemization than the coupling steps preparing the monomeric tripeptide. Therefore, the sequence of the monomer was selected with Gly or Pro at the C-terminal chain end, because racemization is mostly favored at the carboxy-activated amino acid, and these amino acids cannot racemize. 

The Alexander model is based on two assumptions that enable simple expressions for these two terms (1) The concentration profile of the layer is step-like. That is, the monomer volume fraction within the layer, (p Na3/d2L, is constant, independent of position (2) The chains are uniformly stretched. That is, all chain ends are positioned on a single plane at a distance L from the surface. [In this paper, we use the symbol to mean approximately equal to or equal to within a numerical factor of order one we use to mean proportional to .] The first assumption simplifies the calculation of Fin, while the second yields a simple expression for Fel. 

The Alexander model and its descendants impose strong restrictions on the allowed chain configurations within the tethered assembly. The equilibrium state thus found is subject to constraints and may not attain the true minimum free energy of the constraint-free system. In particular, the Alexander model constrains the segment density to be uniform and all the chain ends to be at the same distance from the grafting surface. Related treatments of curved systems retain only the second... 

To study processes which affect the end-to-end vector r, it is sometimes informative to consider only the two beads localized at each chain end and connected by a single spring. This model, known as the elastic dumbbell, was originally proposed by Kuhn over half a century ago [40] and constitutes the simplest model of chain dynamics in flow. 

In the very first model of molecular orientation in flow, Kuhn and Kuhn [2] depicted the polymer as having one chain end anchored in space (Fig. 39). In... 

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