Two-parameter theory

At the level of the regular octahedron or tetrahedron discussed in the previous subsection the AOM appears to be at a disadvantage relative to CFT or ordinary LFT as it is a two-parameter theory, whereas there is fundamentally only one piece of data experimentally accessible — the -orbital splitting into t2(g) and sets. At best the AOM could be made a one-parameter theory by supposing that the ratio en/ea can be treated as a constant, not experimentally accessible. Then... 

T want to finish this introduction by a short sketch of the history of the subject. The physics of dilute polymer solutions by now has been ail active field for about TO years. Much of the early work is connected to the name of Flory and summarized in his classic books [Flo53, Fit>69], Up to about. 1970 much theoretical or experimental work concentrated on the behavior in the dilute limit, where via virial expansions the problem can be reduced to considering only a few interacting chains. The development led to the so-called two parameter theories , which essentially expand quantities like Rg or A2 in powers of z, In 1971 these developments were most carefully reviewed in a book by Yamakawa [Yam.71]. 

All this analysis will be based on the mathematically most simple spring and bead model. Standard two parameter theory is based on the continuous chain model, which can be derived from our model by a not quite trivial limiting process. The derivation stresses that, standard two parameter theory is expected to hold only close to the (9-temperature, In this context we also exhibit the relation of polymer theory to a special quantum held theory. 

Continuous Chain Model and Naive Two Parameter Theory... 

Even for d < 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d < 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

We must note also a second important restriction of the continuous chain model. As we will see. by construction it deals with infinitely long chains n — oo. infinitesimally close to the -point , 5C — 0. Thus naive two parameter theory is valid only very close to the -temperature. In later chapters we will see how further renormalization leads to a theory of excluded volume effects valid for all /%, > 0. 

The bust line of Eq. (7.9) follows since dimensionless quantities can depend only on dimensionless combinations of their variables. As a result of this simple dimensional analysis in the continuous chain model we have found the two parameter theory instead of the three independent parameters ,3A.nt physical observables involve only the two combinations Rq, z. Perturbation theory now proceeds in powers of 2. Thus the continuous chain limit gives a precise meaning to the simple argument presented in Chap. 6. 

In the equations for hr or / we have collected the /o-dependence into the parameters sn or S . The connection to the more familiar formulation of the two-parameter theory is established by introducing... 

This is consistent with our previous discussion (see Chap, 7), where we stressed that the continuous chain model, underlying naive two-parameter theory, is a valid representation of a general polymer model only close to the -point. 

Many other high values of y have been reported in the literature, but invariably the points for the lower molecular weights fall into the upper left area of the plot of A2M/[jj] against (a2 — 1). Some examples are seen in Fig. 2. As with the data of Fig. 1, we are obliged either to reject these trends in the data as spurious or else to abandon faith in the adequacy of any two-parameter theory. We prefer the former alternative, not only intuitively but also because the supremely careful recent measurements of Schulz and Kirste, shown in Fig. 30, give strong support to the two-parameter idea. 

Two conflicting theoretical views concerning the flexibility of polymer chains and the role of the volume effect and the draining effect on fry] are discussed in the literature polymer chains of typical flexibility such as vinyl polymer chains, and a large value of Ip] can be interpreted in terms of the excluded volume effect (view point A) polymer chains are semi- or inflexible and their large unperturbed chain dimension is mainly responsible for a large [ry] (view point B). The former has its foundation on the two parameter theory 110. Untill 1977 these inconsistencies constituted one of the most outstanding problems yet unsolved in the science of polymer solutions. 

The Two-Parameter Theory cannot explain the dilute solution properties of CD, because in CD solutions solvatation and the draining effect can never be neglected. 

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