Des Cloizeaux relation

The values for gf, pf> <1 (the latter two using the numerical procedure discussed in Appendix C) shown in Fig. 5 are reported in Table 4. Within the presently available accuracy, it seems that the relation between ds and i, Eq. (21), holds. The verification of Eq. (22) is somewhat more delicate, and therefore it is crucial to determine the values of gf, Us, and (in particular) 7 as precisely as possible (this being the reason for the two methods of analysis to determine 7s discussed above). By doing this, the des Cloizeaux relation, Eq. (22), has been found to hold. Contrarily to the case of gf, a theoretical estimation for 5 is still lacking. For d-dimensional regular lattices, it is well known that 52 is given by the McKennzie-Moore relation [18], Eq. (7). Unfortunately, a naive straightforward conversion of Eq. (7) to the Sierpinski lattice does not lead to consistent results. 

The discrepancy between the numerical results and the above des Cloizeaux relation has been resolved by a careful consideration of the scale-invariant effects of disorder associated to critical percolation clusters [74]. In the following, we present a brief derivation of the so-called generalized des Cloizeaux relation which is able to explain the numerical results satisfactorily. 

This generalized des Cloizeaux relation is in very good agreement with numerically ob-tmned values [74], The second term in Eq. (43) has its origin in the (self-similar) disordered nature of the backbone of critical percolation clusters and is expected to be absent on deterministic fractals such as the Sierpinski lattice. EE results support this conclusion as shown in Section 4 for both triangular and square Sierpinski lattices. 

On the incipient percolation cluster, being a paradigm example of random fractals, SAWs do not obey the standard des Cloizeaux relation, in the form of Eq. (6). Rather, numerical results are consistent with the generalized relation, Eq. (43) [74]. The latter is based on two main features, one is the underlying multifractal nature of SAWs on such random fractals, reflected by the presence of the first-moment 71, and the second the effects of the structural disorder on the probability that the end-to-end SAW distance... 

Finally, for completeness in Appendix A 7.1 we consider the formal relation of the continuous chain model to a field theoretic Hamiltonian, used to describe critical phenomena in ferrornagnets. It was this relation discovered by de Genries [dG72] and extended by Des Cloizeaux [Clo75, which initiated the application of the renormalization group to polymer solutions and led to the embedding into the larger realm of critical phenomena. 

The picture considered in the previous section is idealised one the macromolecule does not exist in isolation but in a certain environment, for example, in a solution, which is dilute or concentrated in relation to the macromolecules (Des Cloizeaux and Jannink 1990). The important characteristic for the case is the number of macromolecules per unit of volume n which can be written down through the weight concentration of polymer in the system c and the molecular weight (or length) of the macromolecule M as... 

The simple projection relation between the right model eigenfunctions of Hg and their true counterparts is an appealing aspect of Bloch s formalism. However, the non-Hermiticity of the resulting effective Hamiltonian represents a strong drawback, as discussed in Section VII. This has led many, beginning with des Cloizeaux [7], to derive Hermitian effective Hamiltonians, des Cloizeaux s method transforms the lag)ol not the... 

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. 

In fact, as shown by Weill and des Cloizeaux,13 measurements related to dynamics should be avoided. Indeed, we pointed out in Chapter 13, Section 1.4.3, that the lengths which take part in the determination of dynamical physical quantities, include not only the mean square end-to-end distance or the radius of gyration of polymers, but also the hydrodynamic radius. Now, it is easy to show that the hydrodynamic radius reaches its asymptotic behaviour... 

Non-uniformity of swelling was predicted, by Yamakawa and later by des Cloizeaux. This effect appears in the expansion of the mean square distance, as a function of the interaction b (see Chapter 10, Section 7.2). The inhomogeneity effect is related to a fundamental property of swelling. Let us then give an experimental evaluation of this phenomenon. 

Some attempts have been made to obtain approximate expressions which describe a, and Of over an extended range of 2. Weill and des Cloizeaux [106] derived the relation... 

Some authors (for example, Weill and des Cloizeaux [106]) aigue that because of the long-range nature of hydrodynamic interactions between paired beads the theoretically predicted asymptotic slope 0.8 of the relation between log [ ] and log M will be approached so slowly that it may not be observable experimentally. This implies that actually observed HMS relations with v < 0.8 are mere segments of a non-linear dependence of logf ] on logM. 

When the actual experimental temperature used is equal to 6, xi = 1/2, at which point all excess contributions to the solution thermodynamics disappear and the solution exhibits ideal behaviour since the second virial coefficient has a value of zero. At this point the excluded volume effects that cause an expansion of the polymer molecule are exactly balanced by the unfavourable polymer-solvent interactions and the molecule adopts imperturbed, random walk dimensions. The influence on polymer dimensions and the highly detailed theories of polymer configuration in relation to the excluded volume parameter are beyond the scope of this book but are extensively covered by Yamakawa (1971) and to some extent by des Cloizeaux and Jannink (1990). 

The second virial coefficient of the macromolecular coil B T) depends not only on temperature but on the nature of the solvent. If one can find a solvent such that B T) =0 at a given temperature, then the solvent is called the 6-solvent. In such solvents, roughly speaking, the dimensions of the macromolecular coil are equal to those of an ideal macromolecular coil, that is the coil without particle interactions, so that relations of Sects. 2.1 - 2.3 can be applied in this case. However, it is a simplified description of the phenomenon. The fuller review of the theory of equilibrium properties of polymer solutions can be found in the monographs by des Cloizeaux and Jannink [29] and by Gross-berg and Khokhlov [27]. 

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. 

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