Direct renormalization method

Des Cloizeaux (1981) has offered another approach using the direct renormalization method on the basis of the continuous chain model in continuous f-dimensional Edwards-type space. 

Owing to the infinite number of macromoleciilar conformations, the analytical expressions of theory have different-type divergences at the first stage to be subject to renormalization. In order to study the asymptotic properties of chains with a high molecular weight, three successive renormalization procedures must be performed on the whole. 

Unperturbed chains arc Brownian chains, i.e. the continuous limit of chains with independent segments. The partition function (the functional integral) of a Brownian chain diverges due to the infinite number of degrees of freedom. To cancel this divergence, the first renormalization procedure is required. 

Introduction of interactions gives rise to ultraviolet divergences at short distances, which also need renormalization. 

Introduction of interactions to very long chains leads to infrared divergences requiring the third renormalization procedure of the analytical expressions. 

---

However, the principles and the techniques of renormalization theory are not directly related to the existence of fields. They apply whenever one deals with a critical system, i.e. whenever one has to describe large-scale phenomena which depend only globally on the chemical microstructure. Thus, because an ensemble of long polymers in a solution constitutes a critical system, renormalization principles and renormalization techniques must be directly applicable to their study. Actually, this idea appeared quite naturally. It led to the decimation method which has been described previously and which lacks efficiency. However, the same idea can be applied in a much better way. This direct renormalization method (des Cloizeaux 1980)37,38 consists in adapting to polymers methods which had been successful in field theory.39 In other words, the aim is to bypass the Laplace de Gennes transformation (see Chapter 11). This method applies to semi-dilute solutions as well as to dilute solutions. 

Entropy can also be calculated by starting from the continuous model and, in this case, we expect a similar result. However, we must note that in the continuous case, the entropy of the system is really infinite and that a finite entropy can be obtained only after performing a subtractive renormalization. Actually, this question has been tackled several times, either indirectly with the help of a zero component field theory,3,4 or by using the direct renormalization method.2 We shall now describe in detail the latter approach. 

Until now, the exponents 0, have been calculated (in powers of e) only from field theory however, it is also possible to use the direct renormalization method. Thus, as an example, we shall use this method to explicitly calculate the exponents 0, to first-order in e. For this purpose we shall use formula (13.1.140) which connects the 0, to the [Pg.580]

In the following, we consider only monodisperse systems and we study tricritical systems by using the direct renormalization method. It is also possible to proceed indirectly by introducing a tricritical field theory and the correspondence which exists between field theory and polymer theory. This approach... 

Here, we use the direct renormalization method which relies on a slightly different conceptual approach. Nevertheless the points of view are equivalent and the results obtained through field theory can be directly recovered, (as we shall now show) by using more recent results obtained (1985) by Duplantier.15... 

Thus, to calculate the swelling of an isolated chain and the osmotic pressure of a set of chains in the vicinity of the Flory point, a determination of the partition functions 3 (E, — H S) and 3 N x S) by dimensional renormalization is sufficient. The calculations will be performed to first orders in x and y for a space dimension d = 3 — e (0 < s < 1), and we note that the purely repulsive terms have been already calculated in Chapter 10. The partition functions will be represented by series in terms of x and y. Finally, in order to study the behaviour of long polymers, we shall treat these series by using the direct renormalization method. 

Des Cloizeaux and Duplantier (1985), using the direct renormalization method (Du-plantier, 1986a), have calculated the form factor of scattering... 

Since then, the theory of the tricritical phenomena has been developed by the direct renormalization methods (des Cloizcaux, 1981) (see section 5.4) in (l)uplantier, 1986bd), by the dimensional regularization in momentum. sp lce (see section 5.2) in (Kholodenko and Freed, 1984 Cherayil ct al., 1985 Douglas and Freed, 1985). 

Thus, the direct renormalization version produces the same results as does the cut-off version. All the divergent and finite components, which explicitly reduce in the cut-off version, simply do not emerge in the direct renormalization method. These operations arc performed automatically during the analytical passing from d<2to2[Pg.722]

A version of the direct renormalization method in the polymer theory was proposed by des Cloizeaux. 

We now present briefly more explicit calculations of the mutual virial coefficients obtained with the use of des Cloizeaux direct renormalization method for blends of linear flexible polymers in a common good solvent, a common 0-solvent and a selective solvent and for blends of rodlike polymers and flexible polymers in a 0-solvent (marginal behavior). These calculations enable one to find (universal) prefactors relating the mutual virial coefficient to the chain volume (in Eq. 7) in the asymptotic limit. Moreover they give the corrections to the scaling behavior which explicitly depend on the interactions between unlike monomers and are actually responsible for the phase separation of flexible polymer blends in a good solvent. 

The direct renormalization method also allows the determination of the mutual virial coefficient in a common 0-solvent (g = g = 0) and a selective solvent (g = 0, g = g ). In both cases, for symmetric polymers, when the radius ratio is equal to unity we find a hard sphere interaction characterized by a dimensionless virial coefficient... 

The interaction between flexible polymer chains of different chemical nature A and B is measured by their second virial coefficient GaB- We have calculated Gab using Descloizeaux direct renormalization method in a solvent which is either a good or a 0 solvent for the chains when the A-B interaction is repulsive. 

---