Existence of the continuous chain limit

The end-to-end distribution for a chain in a solution of arbitrary concentre tion takes the form 

Outside that region the continuous chain model is no valid representation of a polymer chain. (To avoid confusion we should recall that (T -0)/0 l does not imply z 1.) 

Universality and two-parameter scaling in the general case of finite excluded volume, Be comes about by the much more sophisticated mechanism of renormalization. As will be discussed in later chapters (see Chap. 11, in particular) both the discrete chain model and the continuous chain model can be mapped on the same renormalized theory. The renormalized results superficially look similar to expressions like Eq. (7.13), but the definition of the scaling variables iie, z is more com plica led. Indeed, it is in the definition of R ) and z in terms of the parameters of the original unrenormalized theory, that the difference in microstructure of the continuous or discrete chain models is absorbed. 

We still have to prove that the limit t — 0 exists, a task which is not completely trivial, indeed we wall find that the limit naively exists only for quantities like A 1 (Eq. (4.28)) or P(p, n) (Eq. (4.5)), defined as ratios of partition functions. For the partition function itself or for the Greensfunc-tions (Eq. (4.24)) the continuous chain limit exists only after a 

Continuous Chain Model and Naive Two Parameter Theory 

To illustrate the problem we consider the expansion of the Greensfmiction (Eq, (4.20)), which to first order is given by Eq. (4.6)  

The first order diagram is reprodiicc d in Fig. 7.1a. Taking the t t ntinuons chain limit we from Eq. (4.11) find 

---

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. 

Here Q is dimensionless. Assuming the existence of the continuous chain limit we find... 

Such a formulation indeed exists and is known as renormalized perturbation theory The construction uses a modified form of the continuous chain limit. As explained in Chap. 7, naively the continuous chain limit — 0 is taken with R2 = i2n,z = 0en f2 fixed. In results like (4.16 ii) ... 

The organization of this chapter is as follows. In Sect. 7.1 wo carefully define the continuous chain limit and we introduce the appropriate modification of the Feynman rules. We. then establish the two parameter scheme by dimensional analysis. Section 7.2 is devoted to the question whether the continuous chain limit exists. The analysis is presented on the diagrammatic level. It exploits the field theoretic representation, which also is derived on the level of diagrams. All the analysis is based on the cluster expansion. Extension to the loop expansion is not difficult, but will not be considered, since it is not needed in the sequel. 

The Edwards Hamiltonian is an appealing but most formal object. To mention a simple fact, shrinking to zero the segment size of the discrete chain model as done in the continuous chain limit, we in general get a continuous but not differentiable space curve. Strictly speaking the first part, of Vj, does not exist. Further serious mathematical problems are connected to the (5-function interaction. Hie question in which sense Ve is a mathematically well defined object beyond its formal perturbation expansion is ari interesting problem of mathematical physics. 

Concerning the first two points the answer is that in some sense they compensate each other Working with the continuous chain model the purpose of renormalization is not to eliminate the microstructure dependence, which effectively is suppressed by the limit — 0, but to make the theory finite for d = 4. It turns out that all divergencies occurring for d —> 4 in the (additively renormalized) continuous chain model can be absorbed into renormalization factors The renormalized theory exists for d < 4 It furthermore... 

The point S = 1 denotes the mid-point of the transition, there being an equal number of repeating units in each of the two conformations. The transition is relatively broad for the larger value of a. However, it becomes sharper as a decreases. Only in the limit of a = 0 does the transition actually become discontinuous. However, since a must exceed zero for any real chain the helix-coil transition is a continuous process. In this respect it differs from a true first-order phase transition. This conclusion is in accord with the general axiom enunciated by Landau and Lifshitz (78) that a one-dimensional transition must be continuous. The two phases must mix with one another to some extent. This characteristic of a one-dimensional system causes the transition to be diffuse and permits the co-existence of the two phases over a finite temperature range. Strictly interpreted, therefore, helix-coil transitions do not qualify as true phase transitions. 

---