Chain end correction

Note that as M the absolute number of chain ends per unit volume decreases, as does the chain-end correction. 

Saa = -esB = e was chosen, and an effective coordination number Zcir = 2.5 occurred ) Thus one finds that the coefficient in the relation T N ,r is somewhat smaller than expected from Eq. (3), and in addition there is a "chain end"-correction. However, an integral equation theoryyielded Tc cc Vn, and th simulations showed that this result clearly is incorrect. More recent improved versions of integral equation theories now yield Tc N as well but still the theoretical understanding of the prefactor in this relation is limited. 

This equation is of the same form as eqn (3.33) but contains the term (.417) which is presumed to be near unity and N instead of N in order to take into account a chain end correction such as that given in eqn (3.38). 

The chain end correction factor obtained, 2-3(Mc,chemlM ) is slightly... 

A crossover is also found for self-diffusion coefficients in the melt state, from T> (X M for short chains (after a chain-end correction) to P a M , where d = 2 or possibly more for long chains [43 5]. 

S. Prager (University of Minnesota) You said the chain-end correction concept would not work in your case and that you calculated in another way. Could you enlarge on that ... 

If this equation is valid, M should be independent of the molecular weight, M, and the chain end correction factor, [1 - 2M / A/)], should be equivalent to the inverse of the shift factor, b. It is... 

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. 

This is an expression for the overall enthalpy and entropy divided by the volume of the complete lamella and is strictly correct. However, because the total free energy difference is calculated, the effects of the unfolded chain ends lying at the surface are implicitly included in AH[Tm(0, p)] and AS[Tra(0, p)] and it is therefore misleading to consider these as bulk per volume quantities. The proportion due to the contribution from the surfaces will be greatest for thinner crystals, that is for lower molecular weight. 

The effect of chain ends, which are elastically ineffective, has been neglegted in this calculation. But as chain ends are only formed from crosslinking molecules, whose fraction in the network is rather small, this should cause a negligible correction. 

C NMR of isotactic polypropene shows the main error is pairs of racemic dyads instead of isolated racemic dyads (Table 8-3) [Heatley et al., 1969 Resconi et al., 2000 Wolfsgruber et al., 1975]. A stereoerror in the addition of a monomer molecule is immediately corrected when stereocontrol is by the chiral active site. If stereocontrol was due to the propagating chain end, an error would continue to propagate in an isotactic manner to yield a polymer, referred to as an isotactic stereoblock, containing long isotactic all-R and all-5 stereoblocks on each side of the error. 

The functionality of the star was calculated by comparing the integrated peak area of core protons [aromatic (6=6.8 ppm), -CH2- (6=4.0 ppm)] to the chain end protons (-CH2- (6=1.95 ppm), and -CH3 (6=1.65 ppm)). This procedure gave -8.1 arms per core after correcting for the presence of -10% linear contaminant. Thus NMR spectroscopy provides direct evidence for the formation of... 

In connection with the subject of the relation between association state and kinetic order, it is germane to mention observations of Roovers and Bywater (45). They measured the dissociation constant for the tetramer , dimer case for polyiso-prenyllithium in benzene. The technique involved a study of the electronic spectra at 272 and 320 nm. If the process they measured can be directly related to the association-dissociation equilibrium, their results can be used to calculate the dissociation constant for the correct dimer monomer system. This value is ca. 2xl0-5 at 30.5°C. If this value is accepted, then the situation is encountered where the degree of dissociation of the polyisoprenyllithium chain ends varies from about 0.10 to... 

Two years later Thompson and Hinshelwood (40) after studying the kinetics of the oxidation of ethylene in silica vessels at temperatures between 400° and 500° and finding that the rate is affected by the total pressure approximately in a reaction of the third order, the effects depending very much more on the partial pressure of the ethvlenes than on that of oxygen, suggested as a via media that while there is no doubt that Bone s interpretation of the course of the oxidation as a process of successive hydroxylations is essentially correct... the first stage in the reaction is the formation of an unstable peroxide if this reacts with more oxygen the chain ends but if it reacts with ethylene, unstable hydroxylated molecules are formed which continues the chain. It should be noted, however, that they adduced no experimental proof of the actual initial formation of the assumed peroxide. 

The factor x corrects for the fact that each step in the reaction does not lead to a chain end with i units of monomer Mi, but only some do. We can safely assume that the sequential distribution of the chain lengths of the unreactive and reactive sequences is the same. Under all ordinary conditions the concentration of radicals assumes a value such that the... 

A single step of the polymerization is analogous to a diastereoselective synthesis. Thus, to achieve a certain level of chemical stereocontrol, chirality of the catalytically active species is necessary. In metallocene catalysis, chirality may be associated with the transition metal, the ligand, or the growing polymer chain (e.g., the terminal monomer unit). Therefore, two basic mechanisms of stereocontrol are possible (145,146) (i) catalytic site control (also referred to as enantiomorphic site control), which is associated with the chirality at the transition metal or the ligand and (ii) chain-end control, which is caused by the chirality of the last inserted monomer unit. These two mechanisms cause the formation of microstructures that may be described by different statistics in catalytic site control, errors are corrected by the (nature (chirality) of the catalytic site (Bernoullian statistics), but chain-end controlled propagation is not capable of correcting the subsequently inserted monomers after a monomer has been incorrectly inserted (Markovian statistics). 

Several strategies were employed in order to reduce this effect. Myers et al. described the use of a pulsed ion optic to limit the access of ions to the extraction region [28]. By application of a voltage pulse to one of the quadrupole optics, the ion beam was swept across the slit at the end of the primary optic chain. By correct choice of delay parameters, the extraction region was filled with ions only immediately prior to a repeller event. Mahoney et al. described a different approach that relied on differences in the energies of signal and continuum background ion populations [42]. 

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