Real Chains

The introduction of vectors of constant displacement length to represent the individual elements, which actually vary in length, is rendered more plausible by inquiry into the effect of incorporating this artifice in the treatment of the freely jointed chain. In this case V = m H. Upon substitution of this expression together with n nlm in Eq. (17), the previous expression for / , Eq. (6), is recovered. Hence the calculated distribution is unaff ected by an arbitrary subdivision of the chain in this manner. We conclude that the value chosen for m in the reduction of the real chain to an equivalent freely jointed chain likewise is inconsequential (within the limits on m stated above). 

The foregoing discussion of equivalent chains requires merely that its root-mean-square end-to-end distance shall equal that of the real chain. In order to define completely the equivalent chain, its contour lengths may also be required to coincide with that of the real chain. 

Thus the fully extended length of the equivalent chain will be set equal to that of the real chain with all valence angle B and ) restrictions removed. Then n and V (and therefore m) for the equivalent chain are completely defined by... 

Inasmuch as any real chain may be replaced by an equivalent freely jointed chain, provided the chain is sufficiently long, it is obvious that Eq. (14) is general. 

There are indications which fall short of definite proof, that a certain amount of chain rearrangement takes place, thus reducing the real chain expansion to values less than expected from elementary models. 

FIGURE 9.12 Ideal and real chain growth. Growth probability as a function of carbon number. 

Figure 2.8 shows part of a polyethylene glycol chain and illustrates the type of problem encountered with real chains. First, there are fixed bond angles (in this case the tetrahedral angle), and second, there is the physical volume occupied by each atom so that the walk cannot for example cross itself. Both of these constraints mean that the random... 

In a real chain segment-segment correlations extend beyond nearest neighbour distances. The standard model to treat the local statistics of a chain, which includes the local stiffness, would be the rotational isomeric state (RIS) [211] formalism. For a mode description as required for an evaluation of the chain motion it is more appropriate to consider the so-called all-rotational state (ARS) model [212], which describes the chain statistics in terms of orthogonal Rouse modes. It can be shown that both approaches are formally equivalent and only differ in the choice of the orthonormal basis for the representation of statistical weights. In the ARS approach the characteristic ratio of the RIS-model becomes mode dependent. 

Thus, to reflect correctly the state of the system at a certain temperature, one needs a simulation technique that generates an ensemble of chain conformations whose statistical properties reflect those of the real chain at thermodynamic equilibrium. The chromatin fiber models that are discussed in the following use one or the other variation of such techniques. 

Indeed, reaction (20) can be considered as a real chain transfer with the monomer. In fact, by making a comparison between the reactions (20) and (6) and (7), it can be observed that, after these reactions, the catalytic complexes [CatJCHaCHaCHs and [CatJCHaCHa are respectively formed and both these complexes can add monomeric units in the polymerization process. The reactions (6) and (7) are equivalent to a real chain transfer because the over-all polymerization rate appears to be independent of the triethylaluminum concentration. Considering the chemical analogies of the catalytic complexes, resulting from the reactions (20) and (7), it may also be assumed that the transfer reaction whose rate is of first order with regard to the monomer may be considered a real chain transfer (from a kinetic point of view). 

Monte-Carlo simulations are applied to estimate the characteristic ratios and p parameters from the RIS models for PE, POM, polybutadiene, and polyisoprene. Here the p parameter is defined as the ratio of the radius of gyration to the hydrodynamic radius. The p parameters of these real chains in the unperturbed state show only a slight dependence on the microconformation in the limit of large molecular weights and are found close to 1.504, which is the value for an idealized Gaussian chain. The estimated p parameters of the real chains appear to be correlated to the chain stiffness and increase with the characteristic ratios. 

It is possible to include real chain microstructure into consideration for the case when Gy, 1 both in the swollen and the collapsed state. Then, it becomes possible to use virial expansion for Fin, [23-25] ... 

In preceding sections it has tacitly been assumed that the behaviour of a real chain molecule can be described with the aid of a chain consisting of Z randomly jointed links each of length A. In fact, this model is so well-known that only some additional remarks will be made. The original equations of Kuhn (62) and of Kuhn and GrOn (64) read ... 

For the present purpose it should first be stated that the introduction of the subchain model does not change the character of the picture given for the elastic dumb-bell. A much more complicated situation exists, when real chain molecules are considered. It seems, however, that the statistical character of these chains, when they possess a Gaussian distribution of end-points, will suffice for an explanation of the validity of the stress-optical law. 

N is the number of skeletal bonds in one chain and k is the Boltzmann constant while b is the mean skeletal bond. The mean square end-to-end distance, Nb2, is also referred to as the square of the Gaussian correlation length between the chain ends, o(N) it reflects the effect of linkage of chemical units. The previous relationship between the force f and the extension r is extended to any real chain submitted to a small elongation provided the correlation length, o(N), includes the stiffness property of the polymer o(N)2= A.KNb2 XK is referred to as a persistence length. The related reduction of entropy is expressed as ... 

Only one other corporation in the world, Barclays Bank, contains more members of the British monarchy s most elite order among its directors. That fact alone establishes Canadian Pacific s vassalhood before the feudal rights of the British monarchy. In terms of Canada s real chain of command, it is an honor for Charles R. Bronfman to sit on Canadian Pacific s board. 

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